[Journal] IEEE Transactions on Microwave Theory and Techniques. Vol. 63. No 12 [2]

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DECEMBER 2015

VOLUME 63

NUMBER 12

IETMAB

PART II OF TWO PARTS SPECIAL ISSUE ON 2015 INTERNATIONAL MICROWAVE SYMPOSIUM

2015 Symposium Issue

Phoenix, AZ, USA, site of the 2015 IEEE MTT-S International Microwave Symposium

(ISSN 0018–9480)

IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $17.00, plus an annual subscription fee of $25.00 per year for electronic media only or $46.00 per year for electronic and print media. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE T. LEE, President A. ABUNJAILEH S. BARBIN

R. HENDERSON, Secretary

K. WU, President Elect

T. BRAZIL M. GOUKER

R. GUPTA W. HONG

J. LASKAR G. LYONS

A. JACOB S. KOUL

M. MADIHIAN S. PACHECO

Honorary Life Members T. ITOH R. SPARKS

G. PONCHAK S. RAMAN

M. GOUKER, Treasurer J. RAUTIO S. REISING

M. SALAZAR-PALMA A. SANADA

D. SCHREURS J. WEILER

Distinguished Lecturers

P. STAECKER K. TOMIYASU

R. CAMERON R. H. CAVERLY G. CHATTOPADHYAY

T.-W. HUANG M. JARRAHI J. J. KOMIAK

E. MCCUNE A. MORTAZAWI T. OHIRA

D. WILLIAMS

Past Presidents J. PAWLAN J. C. PEDRO A. STELZER

J. YAO H. ZIRATH T. ZWICK

R. WEIGEL (2014) M. GUPTA (2013) N. KOLIAS (2012)

MTT-S Chapter Chairs Albuquerque: E. FARR Argentina: A. M. HENZE Atlanta: K. NAISHADHAM Austria: A. SPRINGER Baltimore: I. AHMAD Bangalore/India: K. VINOY Beijing: Z. FENG Belarus: S. MALYSHEV Benelux: G. VANDENBOSCH Boston: C. GALBRAITH Bombay/India: M. V. PITKE Brasilia: J. BEZERRA/ M. VINICIUS ALVES NUNES Buenaventura: C. SEABURY Buffalo: M. R. GILLETTE Bulgaria: K. ASPARUHOVA Canada, Atlantic: Z. CHEN Cedar Rapids/Central Iowa: C. G. XIE Central & South Italy: L. TARRICONE Central No. Carolina: Z. XIE Central Texas: J. PRUITT Centro-Norte Brasil: M. V. ALVES NUNES Chengdu: Z. NEI Chicago: D. ERRICOLO Cleveland: M. SCARDELLETTI Columbus: A. O’BRIEN Connecticut: C. BLAIR Croatia: D. BONEFACIC Czech/Slovakia: J. VOVES Dallas: R. SANTHAKUMAR Dayton: A. TERZUOLI Delhi/India: A. BASU

Denver: M. JANEZIC Eastern No. Carolina: T. NICHOLS Egypt: E. HASHEESH Finland: V. VIIKARI Florida West Coast: J. WANG Foothills: M. CHERUBIN France: D. BAJON Germany: G. BOECK Greece: R. MAKRI Gujarat/India: S. CHAKRABARTY Harbin: Q. WU Hawaii: K. MIYASHIRO Hong Kong: H. WONG Houston: S. A. LONG Houston, College Station: G. H. HUFF Hungary: L. NAGY Huntsville: H. SCHANTZ Hyderabad/India: S. R. NOOKALA India: D. BHATNAGER India/Kolkata: S. SANKARALINGAM Indonesia: E. T. RAHARDJO Israel: S. AUSTER Japan: N. SUEMATSU Kansai: T. ISHIZAKI Kingston: S. PODILCHAK Kitchener-Waterloo: R. R. MANSOUR Lebanon: E. NASSAR Lithuania: B. LEVITAS Long Island/New York: S. PADMANABHAN Los Angeles, Coastal: V. RADISIC Los Angeles, Metro/San Fernando: T. CISCO

Macau: C. C. PONG Madras/India: S. SALIVAHANAN Malaysia: M. K. M. SALLEH Malaysia, Penang: B. L. LIM Melbourne: R. BOTSFORD Mexican Council: R. M. RODRIGUEZ-DAGNINO Milwaukee: S. G. JOSHI Monterrey/Mexico: R. M. RODRIGUEZ-DAGNINO Morocco: M. ESSAAIDI Montreal: K. WU Morocco: M. ESSAAIDI Nagoya: J. BAE Nanjing: W. HONG Nanjing, Hangzhou: L. SUN New Hampshire: E. H. SCHENK New Jersey Coast: J. SINSKY New South Wales: Y. RANGA New Zealand: A. WILLIAMSON North Italy: G. OLIVERI North Jersey: A. K. PODDAR Northern Australia: J. MAZIERSKA Northern Canada: M. DANESHMAN Northern Nevada: B. S. RAWAT Norway: M. UBOSTAD Orange County: H. J. DE LOS SANTOS Oregon: K. MAYS Orlando: K. KARNATI Ottawa: Q. ZENG Philadelphia: A. S. DARYOUSH Phoenix: S. ROCKWELL

DOMINIQUE SCHREURS KU Leuven B-3001 Leuven, Belgium

Editorial Assistants MARCIA HENSLEY USA ENAS KANDIL Belgium

Sweden: A. RYDBERG Switzerland: M. MATTES Syracuse: D. MCPHERSON Taegu: Y.-H. JEONG Tainan: H.-H. CHEN Taipei: C. MENG Thailand: C. PHONGCHAROENPANICH Toronto: G. V. ELEFTHERIADES Tucson: H. XIN Tunisia: A. GHARSALLAH Turkey: B. SAKA Twin Cities: C. FULLER UK/RI: A. REZAZADEH Ukraine, East: N. K. SAKHNENKO Ukraine, Kiev: Y. PROKOPENKO Ukraine, Rep. of Georgia: K. TAVZARASHVILI Ukraine, Vinnitsya: V. M. DUBOVOY Ukraine, West: I. IVASENKO United Arab Emirates: N. K. MALLAT Uttar Pradesh/India: M. J. AKHTAR Vancouver: S. MCCLAIN Venezuela: J. B. PENA Victoria: K. GHORBANI Virginia Mountain: T. A. WINSLOW Washington DC/Northern Virginia: T. IVANOV Western Saudi Arabia: A. SHAMIM Winnipeg: P. MOJABI Xian: X. SHI

Associate Editors

Editors-In-Chief JENSHAN LIN Univ. of Florida Gainesville, FL32611-6130 USA

Pikes Peak: K. HU Poland: W. J. KRZYSZTOFIK Portugal: J. CALDINHAS VAZ Princeton/Central Jersey: W. CURTICE Queensland: K. BIALKOWSKI Rio de Janeiro: J. R. BERGMANN Rochester: M. SIDLEY Romania: T. PETRESCU Russia, Moscow: V. A. KALOSHIN Russia, Nizhny-Novgorad: G. L. PAKHOMOV Russia, Novosibirsk: A. YAROSLAVTSEV Russia, Saratov/Penza: M. D. PROKHOROV Russia, Saint Petersburg: S. P. ZUBKO Russia, Siberia: V. V. SUHOTIN Russia, Tomsk: D. ZYKOV San Diego: J. TWOMEY Santa Clara Valley/San Francisco: N. SHAMS Seattle: S. EBADI Seoul: C. SEO Serbia and Montenegro: B. MILOVANOVIĆ Shanghai: J. MAO Singapore: Z. YANG South Africa: A. LYSKO South Australia: T. KAUFMANN South Brazil: J. R. BERGMANN Southeastern Michigan: T. OZDEMIR Southern Alberta: E. FEAR Spain: J. I. ALONSO Springfield: P. R. SIQUEIRA Sri Lanka: A. U. A. W. GUNAWARDENA St. Louis: D. BARBOUR

NUNO BORGES CARVALHO Universidade de Aveiro Aveiro, Portugal

J.-C. CHIAO Univ. of Texas at Arlington Arlington, TX USA

JIASHENG HONG Heriot-Watt Univ. Edinburgh, UK

LUCA PERREGRINI Univ. of Pavia Pavia, Italy

OLGA BORIC-LUBECKE Univ. of Hawaii at Manoa Manoa, HIUSA

GILLES DAMBRINE Univ. of Lille Lille, France

T.-W. HUANG Nat. Taiwan Univ. Taipei, Taiwan

CARLOS SAAVEDRA Queen’s Univ. Kingston, ON, Canada

SHENG-FUH R. CHANG Nat. Chung Cheng Univ. Chiayi County, Taiwan

ROBERTO GOMEZ-GARCIA Univ. Alcala Madrid, Spain

JON MARTENS Anritsu Morgan Hill, CA USA

MARTIN VOSSIEK Friedrich-Alexander Univ. Erlangen-Nuremburg Erlangen, Germany

FRANCISCO MESA Universidad de Sevilla Seville, Spain

X. CHEN Nat. Univ. Singapore Singapore

A. RIDDLE, Editor-in-Chief, IEEE Microwave Magazine J. PAPAPOLYMEROU, Editor-in-Chief, IEEE Microwave and Wireless Component Letters HOWARD E. MICHEL, President BARRY L. SHOOP, President-Elect PARVIZ FAMOURI, Secretary JERRY L. HUDGINS, Treasurer ROBERTO DE MARCA, Past President

P. H. SIEGEL, Editor-in-Chief, IEEE Trans. Terahertz Science and Technology R. MIYAMOTO, Web Master

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Digital Object Identifier 10.1109/TMTT.2015.2503944

DECEMBER 2015

VOLUME 63

NUMBER 12

IETMAB

(ISSN 0018-9480)

PART II OF TWO PARTS SPECIAL ISSUE ON 2015 INTERNATIONAL MICROWAVE SYMPOSIUM

2015 Symposium Issue

Guest Editorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. T. Rodenbeck

4199

MICROWAVE SYMPOSIUM PAPERS

EM Theory and Analysis Techniques Accurate and Stable Matrix-Free Time-Domain Method in 3-D Unstructured Meshes for General Electromagnetic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Yan and D. Jiao Alternative Method for Making Explicit FDTD Unconditionally Stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Md. Gaffar and D. Jiao Accurate Parametric Electrical Model for Slow-Wave CPW and Application to Circuits Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Bautista, A.-F. Franc, and P. Ferrari Devices and Modeling High-Speed Antenna-Coupled Terahertz Thermocouple Detectors and Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. A. Russer, C. Jirauschek, G. P. Szakmany, M. Schmidt, A. O. Orlov, G. H. Bernstein, W. Porod, P. Lugli, and P. Russer Reliable Microwave Modeling by Means of Variable-Fidelity Response Features . . . . . . . . . . . . . . . . . . . S. Koziel and J. W. Bandler Consistent DC and RF MOSFET Modeling Using an -Parameter Measurement-Based Parameter Extraction Method in the Linear Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Zárate-Rincón, R. Torres-Torres, and R. S. Murphy-Arteaga Bayesian Optimization for Broadband High-Efficiency Power Amplifier Designs . . . . . P. Chen, B. M. Merrick, and T. J. Brazil Theory and Implementation of RF-Input Outphasing Power Amplification . . . . . . . . . . . . . . . . . . . . T. W. Barton and D. J. Perreault Hysteresis and Oscillation in High-Efficiency Power Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . J. de Cos, A. Suárez, and J. A. Garc´ıa Digitally Assisted Analog/RF Predistorter With a Small-Signal-Assisted Parameter Identification Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Huang, J. Xia, A. Islam, E. Ng, P. M. Levine, and S. Boumaiza Digital Compensation for Transmitter Leakage in Non-Contiguous Carrier Aggregation Applications With FPGA Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Yu, W. Cao, Y. Guo, and A. Zhu

4201 4215 4225

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(Contents Continued on Page 4198)

(Contents Continued from Page 4197) Passive Circuits Coupling-Matrix-Based Design of High- Bandpass Filters Using Acoustic-Wave Lumped-Element Resonator (AWLR) Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Psychogiou, R. Gómez-García, and D. Peroulis Ultra-Miniature SIW Cavity Resonators and Filters . . . . . . . A. Pourghorban Saghati, A. Pourghorban Saghati, and K. Entesari Design and Multiphysics Analysis of Direct and Cross-Coupled SIW Combline Filters Using Electric and Magnetic Couplings . . . . . . . . . . . . . . . S. Sirci, M. A. Sánchez-Soriano, J. D. Mart´ınez, V. E. Boria, F. Gentili, W. Bösch, and R. Sorrentino Exact Design of a New Class of Generalized Chebyshev Low-Pass Filters Using Coupled Line/Stub Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Musonda and I. C. Hunter Propagating Waveguide Filters Using Dielectric Resonators . . . . . . . . . . . . . . . . . . . . . . . . . C. Tomassoni, S. Bastioli, and R. V. Snyder Self-Biased Y-Junction Circulators Using Lanthanum- and Cobalt-Substituted Strontium Hexaferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Laur, G. Vérissimo, P. Quéffélec, L. A. Farhat, H. Alaaeddine, E. Laroche, G. Martin, R. Lebourgeois, and J. P. Ganne A 2.45 GHz Phased Array Antenna Unit Cell Fabricated Using 3-D Multi-Layer Direct Digital Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. P. Ketterl, Y. Vega, N. C. Arnal, J. W. I. Stratton, E. A. Rojas-Nastrucci, M. F. Córdoba-Erazo, M. M. Abdin, C. W. Perkowski, P. I. Deffenbaugh, K. H. Church, and T. M. Weller Hybrid and Monolithic RF Integrated Circuits A Reconfigurable K-/Ka-Band Power Amplifier With High PAE in 0.18- m SiGe BiCMOS for Multi-Band Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Ma, T. B. Kumar, and K. S. Yeo A Broadband GaN pHEMT Power Amplifier Using Non-Foster Matching . . . . . . . . . . . . . . . S. Lee, H. Park, K. Choi, and Y. Kwon Generalized Stability Criteria for Power Amplifiers Under Mismatch Effects . . . . . . . . . . . . . A. Suárez, F. Ramírez, and S. Sancho Highly Efficient and Multipaction-Free P-Band GaN High-Power Amplifiers for Space Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Ayllon and P. G. Arpesi A Highly Efficient LTE Pulse-Modulated Polar Transmitter Using Wideband Power Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-S. Yang, C.-W. Chang, and J.-H. Chen A Transformer-Based Poly-Phase Network for Ultra-Broadband Quadrature Signal Generation . . . . . . . J. S. Park and H. Wang Optimized Design of Frequency Dividers Based on Varactor-Inductor Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Pontón and A. Suárez GaN Microwave DC–DC Converters . . . . . . . . . . . . . . . . I. Ramos, M. N. Ruiz Lavín, J. A. García, D. Maksimovi´c, and Z. Popovi´c Common-Base/Common-Gate Millimeter-Wave Power Detectors . . . . . . . . . . . A. Serhan, E. Lauga-Larroze, and J.-M. Fournier Instrumentation and Measurement Techniques Modelling and Measurements of the Microwave Dielectric Properties of Microspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A. Abduljabar, X. Yang, D. A. Barrow, and A. Porch Hybrid Nonlinear Modeling Using Adaptive Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Barmuta, G. Avolio, F. Ferranti, A. Lewandowski, L. Knockaert, and D. M. M.-P. Schreurs RF Systems and Applications A 2.45-GHz Energy-Autonomous Wireless Power Relay Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Del Prete, A. Costanzo, A. Georgiadis, A. Collado, D. Masotti, and Z. Popovi´c 3D-Printed Origami Packaging With Inkjet-Printed Antennas for RF Harvesting Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Kimionis, M. Isakov, B. S. Koh, A. Georgiadis, and M. M. Tentzeris Ambient RF Energy Harvesting From a Two-Way Talk Radio for Flexible Wearable Wireless Sensor Devices Utilizing Inkjet Printing Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Bito, J. G. Hester, and M. M. Tentzeris Breaking the Efficiency Barrier for Ambient Microwave Power Harvesting With Heterojunction Backward Tunnel Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. H. P. Lorenz, S. Hemour, W. Li, Y. Xie, J. Gauthier, P. Fay, and K. Wu Cooperative Integration of Harvesting RF Sections for Passive RFID Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Andia Vera, D. Allane, A. Georgiadis, A. Collado, Y. Duroc, and S. Tedjini A Distributed Positioning System Based on a Predictive Fingerprinting Method Enabling Sub-Metric Precision in IEEE 802.11 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Maddio, M. Passafiume, A. Cidronali, and G. Manes Real-World Implementation Challenges of a Novel Dual-Polarized Compact Printable Chipless RFID Tag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Islam and N. C. Karmakar Gesture Sensing Using Retransmitted Wireless Communication Signals Based on Doppler Radar Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.-K. Wang, M.-C. Tang, Y.-C. Chiu, and T.-S. Horng 2015 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Available online at http://ieeexplore.ieee.org

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 63, NO. 12, DECEMBER 2015

Guest Editorial

T

HIS Special Issue of this TRANSACTIONS is comprised of expanded papers from the 2015 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS 2015), held May 17–22, 2015, in Phoenix, AZ, USA, under the leadership of Steering Committee General Chair Vijay Nair. 8626 conference participants enjoyed outstanding weather and good fellowship in this uniquely scenic destination. Numerous activities included 75 technical sessions, panels, and workshops; an exhibition featuring 620 companies, including 67 first-time exhibitors; the vibrant Student Paper Contest and Student Design competitions; and a new RF Bootcamp course. As always, the highlight of the conference was the Technical Program, representing the latest advances in the state-ofthe-art in microwave theory and techniques. As illustrated in Table I, IMS 2015 was truly international in scope. A total of 818 manuscripts from 43 countries were submitted in December 2014 to the IMS 2015 Technical Program Committee, chaired by Chuck Weitzel. Of those manuscripts, 453 papers (a 55% acceptance rate) from 36 countries were accepted for presentation in podium and interactive forum settings. These conference papers are available for download on IEEE Xplore. All authors were invited to significantly expand their IMS 2015 papers for consideration in this TRANSACTIONS’ Special Issue. Of 102 papers submitted for consideration, 37 have been accepted for publication. This year, as in previous years, the editorial process for this TRANSACTIONS’ Special Issue has been entirely handled by the same Editors-in-Chief and Associate Editors who are responsible for the regular issues of this TRANSACTIONS. This policy ensures that all the papers presented in this Special Issue have been evaluated, not only using the same process as regular-issue papers, but also by the same Editors and Editorial Review Board as for regular-issue papers. Please join us in thanking the Editors, the reviewers, and most importantly, the authors for preparing this issue for publication according to a highly constrained publication timeline. Their efforts ensure that the IEEE MTT-S IMS and this TRANSACTIONS continue to represent the latest advances of significance to our profession.

TABLE I IMS 2015 ACCEPTED PAPERS BY COUNTRY

CHRISTOPHER T. RODENBECK IMS 2015 Special Issue U.S. Naval Research Laboratory Washington, DC 20375 USA

Digital Object Identifier 10.1109/TMTT.2015.2496698

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Christopher T. Rodenbeck (S’97–M’04–SM’09) received the B.S. (summa cum laude), M.S., and Ph.D. degrees in electrical engineering from Texas A&M University, College Station, TX, USA, in 1999, 2001, and 2004, respectively. In December 2014, he joined the U.S. Naval Research Laboratory, Washington, DC, USA, where he is currently jump starting a new department focused on millimeter-wave airborne radar. From 2004 to 2014, he was with Sandia National Laboratories, Albuquerque, NM, USA, where he led a multidisciplinary advanced/exploratory technology development effort for microwave and sensor applications. During the summers of 1998–2000, he was with TriQuint Semiconductor, Dallas, TX, USA, as an Intern with the Monolithic Microwave Integrated Circuit (MMIC) Design Group. He has authored or coauthored more than 83 papers and government reports in the microwave and millimeter-wave fields. He holds several patents. His publication topics concern the design of GaAs and silicon-on-insulator RF integrated circuits (RFICs), low-temperature co-fired ceramic (LTCC) multichip modules, radar receiver optimization, microwave power combining, ultra-wideband radar, the design of antennas at frequencies from UHF through millimeter wave, radiation effects, and semiconductor device modeling. Since 2011, he has been an associate editor for the Encyclopedia of Electrical and Electronics Engineering (Wiley), responsible for the microwave theory and techniques subject area. Dr. Rodenbeck was the recipient of fellowships from NASA, the State of Texas “to advance the state of the art in telecommunications,” Texas A&M, and TxTEC in support of his graduate studies. He has also been supported by Raytheon, TriQuint Semiconductor, the Office of the Secretary of Defense, NASA Jet Propulsion Laboratory, NASA Glenn Research Center, and the U.S. Army Space Command under research grants. He was the Principal Investigator for a research and devlopment program that received the 2012 NNSA Award of Excellence. He was the recipient of the 2015 IEEE Microwave Thoery and Techniques Society (IEEE MTT-S) Outstanding Young Engineer Award, a 2013 Sandia Innovator Award, and a 2011 internal citation for “Excellence in Radar Technology Leadership.”

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Accurate and Stable Matrix-Free Time-Domain Method in 3-D Unstructured Meshes for General Electromagnetic Analysis Jin Yan, Student Member, IEEE, and Dan Jiao, Senior Member, IEEE

Abstract—We develop a new time-domain method that is naturally matrix free, i.e., requiring no matrix solution, regardless of whether the discretization is a structured grid or an unstructured mesh. Its matrix-free property, manifested by a naturally diagonal mass matrix, is independent of the element shape used for discretization and its implementation is straightforward. No dual mesh, interpolation, projection, and mass lumping are required. Furthermore, we show that such a capability can be achieved with conventional vector basis functions without any need for modifying them. Moreover, a time-marching scheme is developed to ensure the stability for simulating an unsymmetrical numerical system whose eigenvalues can be complex-valued and even negative, while preserving the matrix-free merit of the proposed method. Extensive numerical experiments have been carried out on a variety of unstructured triangular, tetrahedral, triangular prism element, and mixed-element meshes. Correlations with analytical solutions and the results obtained from the time-domain finite-element method, at all points in the computational domain and across all time instants, have validated the accuracy, matrix-free property, stability, and generality of the proposed method. Index Terms—Electromagnetic analysis, finite-difference time domain (FDTD) methods, matrix-free methods, time-domain finite-element methods, time-domain methods, unstructured mesh.

I. INTRODUCTION

M

ANY engineering challenges demand an efficient computational solution of large-scale problems. If a computational method can be made matrix free, i.e., free of matrix solutions, then it has a potential of solving very large scale problems. Among existing computational electromagnetic methods, the explicit finite-difference time-domain (FDTD) method [1], [2] is free of matrix solutions. However, it requires a structured orthogonal grid for space discretization. To overcome this limitation, many nonorthogonal FDTD methods have been developed such as the curvilinear FDTD [3]–[5], contour and conformal FDTD [6]–[8], discrete surface integral (DSI) methods

Manuscript received June 15, 2015; revised September 05, 2015; accepted October 19, 2015. This work was supported in part by the NSF under Grant 1065318, and by DARPA under Grant HR0011-14-1-0057. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. The authors are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495257

[9], generalized Yee-algorithms [10]–[15], and the Finite Integration Technique with affine theories [16]. Needless to say, they have significantly advanced the capability of the original FDTD method in handling unstructured meshes. In existing nonorthogonal FDTD methods, a dual mesh is generally required. The dual mesh needs to satisfy a certain relationship with the primary mesh. Such a dual mesh may not exist in an unstructured mesh. For cases where the dual mesh exists, the accuracy of many nonorthogonal FDTD methods can still be limited. This is because in these methods, the field unknowns are placed along the edges of either the primary mesh or the dual mesh, and are assumed to be constant along the edges. Restricted by such a representation of the fields, one can only obtain the dual field accurately (second-order accurate) at the center point of the loop of the primary field, and along the direction normal to the loop area. Elsewhere and/or along other directions, the accuracy of the dual field cannot be ensured. However, the points and directions, where the dual fields can be accurately obtained, are not coincident with the points and directions of the dual fields located on the dual mesh, in an unstructured mesh. Actually, the only mesh that can align the two is an orthogonal grid, which is used by the traditional FDTD method. As a result, the desired dual fields have to be obtained by interpolations and projections, the accuracy of which is difficult to control in an arbitrary unstructured mesh. It is observed that many interpolation and projection schemes lack a theoretical error bound. The same is true to the primary fields obtained from the dual fields. In addition to accuracy, stability is another concern since the curl operation on is, in general, not reciprocal to that on in existing methods developed for irregular meshes. It can be proved that such a nonreciprocal operation can result in complex-valued or negative eigenvalues in the underlying numerical system. They make a traditional explicit time-marching absolutely unstable. This fact was also made clear in [15]. As a consequence, it remains a research problem how to ensure both accuracy and stability of an FDTD-like method in an unstructured mesh. The finite-element method in time domain (TDFEM) [17] has no difficulty in handling arbitrarily shaped irregular meshes, but it requires the solution of a mass matrix, thus not being matrix-free in nature. The mass-lumping has been used to diagonalize the mass matrix in TDFEM, and also finite integration technique [16]. But it requires well-shaped elements to be accurate [18]. In addition to mass lumping, orthogonal vector basis functions have been developed to render the mass matrix diagonal [19], [20]. These bases are element-shape dependent. They

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also rely on an approximate integration to make the mass matrix diagonal. In recent years, Discontinuous Galerkin time-domain methods [21], [22] have been developed, which only involve the solution of local matrices of small size. However, this is achieved by not enforcing the tangential continuity of the fields across the element interface at the same time instant. Certainly, an accurate result would still have to satisfy the continuity conditions of the fields. Not satisfying them has implications in either accuracy or efficiency. For example, it is observed that a Discontinuous Galerkin time-domain method typically requires a time step much smaller than that of a traditional explicit time-domain method for accurate transient analysis. Recently, a new time-domain method is developed in [23], which requires no matrix solution regardless of whether the discretization is a structured grid or an unstructured mesh. Since the curl operation on and that of are enforced to be reciprocal of each other in [23], although the stability is guaranteed for an arbitrary unstructured mesh, the accuracy remains to be a strong function of mesh quality. In this paper, we develop an accurate and stable matrix-free time-domain method that is independent of the element shape used for discretization. The tangential continuity of the fields is satisfied across the element interface at each time instant. No dual mesh, interpolation, projection, and mass-lumping are needed. The accuracy and stability are both guaranteed for an arbitrary unstructured mesh. This method is also made very easy to implement. In addition, in a structured grid and with zerothorder vector bases, the proposed method reduces exactly to the FDTD. The basic idea of this paper was outlined in [24], where 2-D formulations are provided, and modified higher-order bases are developed to achieve a matrix-free method. In this paper, we present 3-D formulations of [24] for general electromagnetic analysis. We also show the proposed matrix-free method can be formulated without modifying the traditional vector basis functions. In addition, a comprehensive analysis is conducted on the accuracy and stability of the proposed method. Numerical results on various highly unstructured triangular, tetrahedral, triangular prism meshes as well as meshes with mixed-elements are presented to demonstrate the accuracy, matrix-free property, and generality of the proposed method.

II. PROPOSED FRAMEWORK FOR CREATING A MATRIX-FREE TIME-DOMAIN METHOD In this section, we present a general framework for creating a matrix-free time-domain method independent of the shape of the elements used for discretization. We separate the presentation of the framework from that of the detailed formulations (to be given in next section) because the formulation corresponding to the proposed framework is not unique. Under the proposed framework, we can develop different formulations to achieve a matrix-free time-domain method. Consider a general electromagnetic problem involving arbitrarily shaped geometries and materials. For such a problem, an unstructured mesh with arbitrarily shaped elements is more accurate and efficient for use, as compared to an orthogonal grid. The elements do not have to be of the same type. They can be a mix of different types of elements such as tetrahedral, triangular

prism, and brick elements. Starting from the differential form of Faraday's law and Ampere's law (1) (2) we pursue a discretization of the two equations in time domain, which can yield a numerical system free of matrix solutions independent of the element shape used for discretization. A. Discretization of Faraday's Law To discretize Faraday's law, we propose to expand the electric field in each element by a set of vector bases as the following: (3) where is the unknown coefficient of the th vector basis , and is the number of vector bases in each element. The degrees of freedom of the vector bases are defined not only on the faces of the element but also inside the element. Such a choice of vector bases permits accurate generation of the other field unknown at any point along an arbitrary direction, without a need for interpolation and projection. Substituting the expansion of into (1), computing at points , and then taking the dot product at each point respectively, of the resultant with unit vector we obtain

(4) which can be compactly written into the following linear system of equations: (5) where is a diagonal matrix of the permeability, is a global vector of length whose th entry is (6) and

is a sparse matrix, the nonzero entries of which are (7)

where denotes the global index of the -point, and is the global index of the 's vector basis function. Let be the is of total number of vector bases used to expand . The size . During the procedure of constructing , the tangential continuity of is enforced since the tangential electric fields at the element interface are uniquely defined in global vector , and shared in common by all elements. B. Discretization of Ampere's Law To discretize Ampere's law, we apply it at points, and then take the dot product of the resultant with unit vector at each point, where and are

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associated with the degrees of freedom of the vector bases used in (3). We obtain (8) where

3

where is the time step, and the time instants for and , denoted by superscripts, are staggered by half. Neither (11) nor (12) involves a matrix solution. Equations (5) and (10) can also be solved in a second-order based way. Taking another time derivative of (10) and substituting (5), we obtain

(9) which is at point along the direction. The at point in (8) is generated by using the fields [obtained from is located at an element (5)] encircling . For example, if interface, the fields used to generate it are sampled across the elements sharing . A detailed formulation with guaranteed accuracy will be given in next section. As a result, we obtain the following discretization of Ampere's law (10) where is a sparse matrix of size , and notes the discretized eration, the th entry of is , and the are the diagonal matrices of permittivity, and ductivity respectively.

deopand con-

C. Connecting Ampere's Law to Faraday's Law In order to connect (10) to (5), we need to find the relationship between and . In [24], by making a minor modification . In this of the traditional vector bases, we make work, we show the traditional vector bases can also be kept as they are without any need for modification. In this case, we can find an analytical relationship between and as , with an extremely simple block diagonal matrix whose diagonal blocks are either of size 1 1 or 2 2. The detailed formulation of will be given in next section. In addition, when generating (5), apparently, we have an infinite number of choices of the points and the directions for computing the discrete . However, to connect (5) to (10), we need to keep in mind that the -points and directions we choose should facilitate accurate generation of the desired in (5) so that we can march on in time step by step—from to via (5), and then from back to through (10). D. Time Marching A leap-frog-based time discretization of (5) and (10) clearly yields a time-marching scheme free of matrix solutions as follows:

(11)

(12)

(13) where (14) It is evident that the above numerical system is also free of matrix solutions with a central-difference based discretization in time. This is because the matrix in front of the second-order time derivative, which is known as mass matrix, and the matrix before the first-order time derivative are both naturally diagonal. Since the matrices are made naturally diagonal in the proposed method, no approximation-based mass-lumping is needed. It is also worth mentioning that the leap-frog-based time discretization shown in (11) and (12) is the same as the central-difference-based explicit discretization of the second-order system (13). This can be readily seen by writing the counterpart of (12) for evaluating , i.e., replacing by in (12), subtracting the resultant from (12), and then substituting (11) to replace the term. Since (11) and (12) are the same as the explicit discretization of (13), we can directly solve (13), which also has only half a number of unknowns. If unknowns are needed, they can readily be recovered from through (11). E. Remark In the framework described above, we expand into certain vector basis functions in each element, while sampling the unknowns at discrete points to generate desired unknowns. One can also switch the roles of the electric and magnetic fields: expand the into vector basis functions in each element, while sampling the unknowns. Which way to use depends on the convenience for solving a given problem. III. PROPOSED FORMULATIONS In this section, we present detailed formulations to realize the aforementioned matrix-free framework with guaranteed accuracy and stability. Since 2-D formulations have been presented in [24], 3-D formulations will be the focus of this section. A. Accurate Construction of

and

's Degrees of Freedom

A common choice of the vector basis functions for expanding the fields is the zeroth-order curl-conforming bases (edge elements) [25]. These bases have constant tangential components along the edges where they are defined. The field representation in the traditional FDTD is, in fact, a zeroth-order vector basis representation in an orthogonal cell. However, the zeroth-order vector bases have a constant curl in every element. Using such bases to represent , the resultant is a constant in each element, and the is only second-order accurate at the center point of each element. From such discrete -fields, we cannot

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Fig. 1. (a) Locations of point . (b) Locations of

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points required for the accurate evaluation of points with zeroth-order vector bases.

at

reversely obtain the unknowns associated with the zerothorder vector bases accurately in an arbitrarily shaped element. To help understand the aforementioned point more clearly, take a 2-D problem discretized into arbitrarily shaped triangular elements as an example. Consider an arbitrary th edge. With the zeroth-order vector bases to expand , the shown in (9) has the unit vector tangential to the th edge, and the center point of the th edge, as illustrated in Fig. 1. To obtain such an accurately from the discrete (now only since the problem is 2-D), the two -points should be located on the line that is , as perpendicular to the th edge and centered at the point shown in Fig. 1(a). In this way, the edge is perpendicular to the -loop (in the plane defined by -direction and the line normal to the edge), and resides at the center of the loop. As a result, an accurate can be obtained from a space derivative of the two unknowns. However, using the zeroth-order edge elements, the curl of is constant in every element, thus we cannot generate at the desired points accurately. From another perspective, we can view the obtained at the center point of every element to be accurate. However, in an arbitrary unstructured mesh, the line segment connecting the center points of the two elements sharing an edge may not be perpendicular to the edge, and the two center points may not have the same distance to the edge either, as illustrated in Fig. 1(b). To overcome the aforementioned problem, we propose to use higher-order curl-conforming vector bases to expand in each element. With an order higher than zero, the curl of and hence is at least a linear function of , , and in each element. With this, the can be obtained at an arbitrary point along an arbitrary direction accurately from (5). We hence can use this freedom to choose points and directions in such a way that they can reversely generate unknowns accurately from (10). First-order bases are sufficient for use. Certainly, one can employ bases whose order is even higher. This is one of the reasons why the detailed formulations corresponding to the proposed framework are not unique. In this work, first-order bases are used, since they satisfy the need of the proposed matrix-free method and they minimize computational overhead as compared to other bases. For completeness of this paper, in Appendix, we list all the twenty first-order bases in a tetrahedral element [26] together with their degrees of freedom defined in terms of locations and projection directions . B. Relationship Between

discrete electric fields at points along directions as defined in (9). If , then . Hence, (10) and (5) are directly connected to each other. Among higher-order vector basis functions [26], the vector bases associated with edges satisfy naturally. However, the bases defined on the faces and those inside the element, in general, do not. This problem can be solved by modifying the original higher-order vector bases to make , as done in [24]. We can also keep the original higher-order vector bases as they are, but find the relationship between and as follows. Substituting (3) into (9), we have (15) from which we obtain (16) where

matrix obviously has the following entries: (17)

is of size but an extremely simple matrix—It is a The block diagonal matrix with each diagonal block of size either 1 or 2. To be specific, for the vector basis function whose degree of freedom is associated with edges, the and elsewhere in the th row ; for the vector basis function whose degree of freedom is not associated with edges, it is either defined on faces or inside the element. Such a basis function comes in as a pair, for which there are two nonzero elements on the th row of , and two nonzero elements on the th row of , forming a 2 2 diagonal block in as the following (18) The off-diagonal terms in the above do not vanish because for face or internal degrees of freedom, the basis function pair associated with each point are not perpendicular to each other in terms of the vector basis's direction. Overall, the can be written as

(19)

where each diagonal block is equal to either 1 or a 2 2 matrix shown in (18), which can be readily inverted to obtain , denoted by . Obviously, is also a block diagonal matrix whose diagonal blocks are of size either 1 or 2. As a result, we find a closed-form relationship between from as (20) Equation (5) hence can be rewritten as

and

The vector contains the unknown coefficients of vector contains the basis functions as shown in (3), while vector

(21) Thus, (10) and (5) are connected to each other.

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Fig. 2.

points and directions for generating

C. Accurate Construction of and Directions

.

and Choice of

's Points

To construct (10) accurately, we propose to use an -loop uniquely defined for each 's degree of freedom to obtain the desired in (5). This loop centers each 's degree of freedom, and is also positioned perpendicular to the 's degree of freedom. This -loop can be chosen in its simplest manner: a 1-D line segment in 2-D settings, and a 2-D rectangular loop centering and normal to the 's degree of freedom in 3-D problems, as shown in Fig. 2. Regardless of the shape of the element, such a rectangular loop can always be defined for each unknown. Along this loop, we select the middle points of the four sides as -points and the four unit vectors tangential to each side as -directions to generate . As a result, each unknown is associated with four -points and directions. These -points are all located inside the elements that share the unknown, instead of being selected on the faces of the elements. In this way, each point is located only in one element, and hence the -field at the point can be readily found from (5). The set of -points and -directions defined for each makes the whole set of -points denoted by , and the whole set of -directions denoted by . With the aforementioned choice of -points and directions, in (8) can be accurately discretized with secondthe order accuracy as the following (22) where is the distance between and , while is the distance between and as illustrated in Fig. 2. With (22), we obtain (23) where denotes the global index of the -point associated with is simply two times the distance between the the , and -point ( ) and the -point . Each row of has only four nonzero elements. Obviously, there is no need to construct a dual mesh for as the -points and -directions we select are individually defined for each unknown, which do not make a mesh. In addition, regardless of the choice of -points and directions, there is no difficulty in generating corresponding from (5) accurately, due to the use of higher-order basis functions. D. Imposing Boundary Conditions The proposed method, in its first-order form (11)–(12), conforms to that of the FDTD numerical system; in its secondorder form (13), conforms to the second-order wave equation

5

based TDFEM. Hence, the boundary conditions in the proposed method can be implemented in the same way as those in the TDFEM and FDTD. Below we provide more details. For closed-region problems, the perfect electric conductor (PEC), the perfect magnetic conductor (PMC), or other nonzero prescribed tangential or tangential are commonly used at the boundary. To impose prescribed tangential at boundary points, in (5), we simply set the entries at the points to be the prescribed value, and keep the size of the same as before to produce all discrete from the discrete . In (10), since the entries at the points are known, the updating of (10) only needs to be performed for the rest entries. As a result, we can remove the rows from corresponding to the boundary fields, the same as before. while keeping the column dimension of The above treatment, from the perspective of the second-order system shown in (13), is the same as keeping just rows of , providing the full-length (with the boundary entries specified) for the multiplied by , but taking only the rows of all the other terms involved in (13). To impose a PMC boundary condition, the total unknown number is without any reduction. Equation (5) is formulated as it is since the -points having the PMC boundary condition can be placed outside the computational domain. As for (10), there is no need to make any change either since the tangential is set to be zero outside the computational domain. The end result is the same as a TDFEM numerical system subject to the second-kind boundary condition. For open-region problems, the framework of (5) and (10) in the proposed method is conformal to that of the FDTD. As a result, the various absorbing boundary conditions that have been implemented in FDTD such as the commonly used PML (perfectly matched layer) can be implemented in the same way in the proposed matrix-free method. IV. TIME MARCHING FREE OF MATRIX-SOLUTION WITH GUARANTEED STABILITY A leap-frog-based time marching shown in (11)–(12) as well as a central-difference based time discretization of (13) is absolutely matrix-free, i.e., free of a matrix solution. However, both are absolutely unstable since the curl-curl operator here is an unsymmetrical matrix. This is not only true for the proposed method but also true for any method whose curl operation on one field unknown is not the reciprocal of the curl operation on the other field unknown. To prove, we can perform a stability analysis of (11)–(12) and (13) [27], [28]. The -transform of the central-difference based time marching of (13), or (11)–(12) after eliminating , results in the following equation: (24) where is the eigenvalue of . The two roots of (24) can be readily found as (25) If is Hermitian positive semidefinite like that resulting from TDFEM or FDTD in an orthogonal grid, all its eigenvalues are nonnegative real. Thus, we can always find a time step to make

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in (25) bounded by 1, and hence the explicit simulation of (13) as well as (11)–(12) is stable. Such a time step satisfies , where is the maximum eigenvalue of , which is also 's spectral radius. However, if is not symmetrical, which is the case in the proposed method and many existing nonorthogonal FDTD methods, its eigenvalues either are real (can be negative) or come in complex-conjugate pairs. For complex-valued eigenvalues as well as negative ones, the , and neither two roots and shown in (25) satisfy of them has modulus equal to 1. As a result, the modulus of one of them must be greater than 1, and hence the explicit time-domain simulation of (13) and (12) must be unstable. However, if we choose to make symmetric, the accuracy cannot be guaranteed in a general unstructured mesh. This dilemma is solved as follows without sacrificing the matrix-free merit of the proposed method. Basically, we can start with the following backward-difference based discretization of (13) [17]:

Since is not diagonal, (29) requires a matrix solution. To avoid that, we can solve this problem as follows. Let the diagonal part of be , which means (31) Front multiplying both sides of (29) by

, we obtain (32)

where

is the right hand side of (29), and (33)

Although (29) permits the use of any large time step, when we choose the time step based on that of a conventional explicit method, the time step satisfies (34) and therefore (35)

(26) associated with is chosen at the th time where the step instead of the th step. Performing a stability analysis of (26) for lossless cases, we find the two roots of as (27) As a result, the can still be bounded by 1 even for an infinitely large time step. However, this does not mean the backward difference is unconditionally stable since now the can be complex-valued or even negative. To make the magnitude of (27) bounded by 1, we find that the time step needs to satisfy the following condition (28)

where denotes the imaginary part of . It is obvious to see that the scheme is stable for large time step, but not stable for small time step. Such a requirement happens to align with preferred choices of time step, since a large time step is desired for an efficient simulation. Rearranging the terms in (26), we obtain

(29) where (30)

This time step is also the time step required by accuracy when space step is determined by accuracy. Since in (31) is diagonal, the norm of its inverse can be analytically evaluated as (36) We therefore obtain from (35) and (36) (37) As a result, the inverse of as a series expansion

can be explicitly represented (38)

which can be truncated after the first few terms without sacrificing accuracy due to (37). Thus, the system matrix has an explicit inverse, and hence no matrix solution is required in the proposed method. The final update equation becomes (39) is a diagonal matrix which is 's inverse. The number where of terms is guaranteed to be small (less than 10) since (37) holds true, and the central-difference-based time step (34) is usually not chosen right at the boundary, , but smaller for better sampling accuracy. Notice that the spectral radius of , as revealed in (37), is essentially the square of the ratio of the actual time step used to the largest time step permitted by the stability of a conventional explicit scheme . It is a constant irrespective of the mesh quality. Therefore, the convergence of (38) is guaranteed and the convergence rate does not depend on the mesh quality. Notice that using (38) does not change the stability analysis since it is used to obtain the inverse of system matrix, which does not change the backward difference based time marching scheme.

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The computational cost of (39) is sparse matrix-vector multiplications since each term can be computed from the previous term. For example, if we first compute , then the second term in (39) can be obtained from . Let the resultant be . The third term relating to is nothing but . Therefore, the cost for computing each term in (39) is the cost of multiplying by the vector obtained at the previous step, thus efficient. V. RELATIONSHIP WITH FDTD In a regular orthogonal grid and with the zeroth-order vector bases, the proposed method reduces exactly to the FDTD. This is very different from the mixed formulation like [29] where mass lumping has to be used to prove equivalency. To explain, for a 2-D rectangular grid and a 3-D brick-element based discretization, with a zeroth-order edge vector basis used in each rectangular or brick element, the operation of in the proposed method is the same as how the curl of is discretized in the FDTD; and the operation of with is the same as how the curl of is discretized in the FDTD. naturally satisfies in an orthogFurthermore, since onal grid, the resulting numerical system is symmetric and positive semidefinite. Hence the original leap-frog explicit time marching is stable without any need for special treatment. That is also why in a traditional FDTD with an orthogonal grid, an explicit time marching is never observed to be absolutely unstable because the system matrix is symmetric. To see the above point more clearly, take the 2-D rectangular grid as an example. The is simply a union of at the center point of each edge, with being either or along each edge; and the is nothing but the vector containing at the center point of each rectangular patch. Each row of has four nonzero elements as each element has four bases. Multiplying the th row of by is nothing but (40) where , , , are the global indexes of the four edge basis functions in the rectangular element where the point is located, and and are the two side lengths of the rectangular element. It is evident that (40) is the same as that performed in the FDTD to produce the at the center of each -loop. With , the operation of is to do (41) where is simply the length of the side that is perpendicular to edge in a rectangular element. Obviously, the above is the same as that used in the FDTD to calculate fields, which is an accurate discretization of of second-order accuracy at the center point of an edge for along the edge. In addition, even in an orthogonal grid, the implementation of the proposed method is more convenient, since no dual grid is needed. After is formed for the grid, is known as without any construction cost. For unstructured meshes, the FDTD method would fail, whereas the proposed method is accurate and stable regardless of how irregular and unstructured the mesh is.

Fig. 3. Simulation of wave propagation and reflection in a 2-D triangular mesh. (a) Mesh. (b) Illustration of incident wave and truncation boundary conditions.

VI. NUMERICAL RESULTS In this section, we simulate a variety of 2- and 3-D unstructured meshes to demonstrate the validity and generality of the proposed matrix-free method in analyzing arbitrarily shaped structures and materials discretized into unstructured meshes. The accuracy of the proposed method is validated by comparing with both analytical solutions and the TDFEM method that is capable of handling unstructured meshes but not matrix-free. A. 2-D Triangular Mesh The first example is a wave propagation and reflection problem in an 2-D triangular mesh shown in Fig. 3(a). Some mesh elements are very skewed due to fine features in a narrow gap whose size is less than a few . The dielectric in the red shaded region and 1 elseconstant is where. The incident is specified as , where , , s, and denotes the speed of light. The top, bottom and right boundaries are terminated by PEC, while the left boundary is truncated by the sum of the incident and reflected fields as illustrated in Fig. 3(b). Since the left boundary is not close to the dielectric discontinuity, the reflected field at the left boundary can be analytically approximated as , is the -coordinate at the left boundary, and is the where width of the computational domain. In the proposed method, the number of expansion terms used is 9 in (38). For comparison, we simulate the same example by TDFEM since it is capable of handling unstructured meshes. The time step used in both methods is . In Fig. 4(a), the electric fields at two

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Fig. 5. Illustration of the tetrahedron mesh of a 3-D structure.

Fig. 4. Simulation of a 2-D triangular mesh. (a) Electric fields at two points. (b) Entire solution error versus time.

points

and randomly selected are plotted in comparison with TDFEM results. The directions of the two fields are respectively , and . Excellent agreement can be observed with TDFEM results. Such an agreement is also observed at all points for all time. As shown in Fig. 4(b), the entire solution error as compared with the TDFEM solution is less than 2% at all time instants. A few peak errors are due to the comparison with close-to-zero fields. The entire solution error is defined by (42)

where denotes the entire unknown vector of length solved from the proposed method, and denotes the reference solution, which is TDFEM result in this example. B. Wave Propagation in a 3-D Box Discretized into Tetrahedral Mesh A 3-D box discretized into tetrahedral elements is simulated in free space. The mesh details are shown in Fig. 5. The discretization results in 544 edges and 350 elements. To investigate the accuracy of the proposed method in such a mesh, we consider that the most convincing comparison is a comparison with analytical solution. We hence study a free-space wave propagation problem whose analytical solution is known. To simulate such an open-region problem, we impose an analytical boundary condition, i.e., the known value of tangential , on the outermost boundary of the problem; we then numerically simulate

Fig. 6. Simulation of a 3-D box discretized into tetrahedral elements. (a) Simulated two electric fields in comparison with analytical results. (b) Entire solution error for all unknowns versus time.

the fields inside the computational domain and correlate results with the analytical solution. The structure is illuminated by a plane wave having , where , , and . The time step used in the proposed method is , which is the same as what a traditional central-difference based TDFEM has to use for stability. The number of expansion terms is 9 in (38). In Fig. 6(a), we plot the first and 1832th entry randomly selected from the unknown vector, which represent , with , and 1832 respectively. From Fig. 6(a), it can be seen clearly that the electric fields solved from the proposed method have an excellent agreement with analytical results. To further verify the accuracy of the proposed method in the entire computational domain, we assess the entire solution error (42) as a function of time, where the reference solution is analytical result . In Fig. 6(b), we plot across the

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Fig. 8. Illustration of the tetrahedron mesh of a sphere structure.

Fig. 7. (a) Entire solution error versus time of all unknowns obtained from -rows of equations. (b) Entire solution error versus time of all obtained -rows of equations. from

whole time window in which the fields are not zero. It is evident that less than 4% error is observed at each time instant, demonstrating the accuracy of the proposed method. The center peak in Fig. 6(b) is due to a comparison with close to zero fields. In addition to the accuracy of the entire method, we have also examined the accuracy of the individual , and separately, since each is important to ensure the accuracy of the whole scheme. First, to solely assess the accuracy of , we perform the time marching of (5) only without (10) by providing an analytical to (5) at each time step. The resultant is at each time step. As can then compared to analytical be seen from Fig. 7(a) where the following -error

Fig. 9. Simulation of a 3-D sphere discretized into tetrahedral elements. (a) Two electric fields in comparison with analytical results. (b) Entire solution error for all unknowns versus time.

(43) unknowns is is plotted with respect to time, the error of all less than 3% across the whole time window, verifying the accuracy of . Similarly, in order to examine the accuracy of , we perform the time marching of (10) only without (5) by providing an analytical to (10) at each time step. The relative error of all unknowns shown in (42) as compared to analytical solutions is plotted with time in Fig. 7(b). Again, very good accuracy is observed across the whole time window, verifying the accuracy of . C. Wave Propagation in a Sphere Discretized into Tetrahedral Mesh The third example is a sphere discretized into tetrahedral elements in free space, whose 3-D mesh is shown in Fig. 8.

The discretization results in 3183 edges and 1987 tetrahedrons. Again, we set up a free-space wave propagation problem in the given mesh to validate the accuracy of the proposed method against analytical results. The incident has the same form in as that of the first example, but with accordance with the new structure's dimension. The outermost boundary of the mesh is truncated by analytical fields. The , which is the same as that time step used is used in a traditional TDFEM method. The number of expansion terms is 9 in (38). The two degrees of freedom of the electric field, whose indices in vector are 1 and 9762, respectively, are plotted in Fig. 9(a) in comparison with analytical data. Excellent agreement can be observed. In Fig. 9(b), we plot the entire solution error shown in (42) versus time. Less than 3% error is observed in the entire time window. It is evident that the proposed method is not just accurate at certain points, but

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Fig. 10. Top view of the triangular prism mesh of an coaxial cylinder structure.

Fig. 11. Simulation of a 3-D coaxial cylinder discretized into triangular prism elements. (a) Two electric fields in comparison with analytical results. (b) Entire solution error for all unknowns versus time.

accurate at all points in the computational domain for all time instants simulated. D. Coaxial Cylinder Discretized Into Triangular Prism Mesh The fourth example has an irregular triangular prism mesh, the top view of which is shown in Fig. 10. The structure has two layers of triangular prism elements (into the paper) with each layer being 0.05 m thick. The discretization results in 3092 edges and 1038 triangular prisms. Both the innermost and outermost boundaries are terminated by exact absorbing boundary condition, which is the analytical tangential on the boundary. The incident has the same form as that in the first example, but with . The used is and the number of expansion terms is 9. Two observation points, whose indices in vector are 1 and 11 272 respectively, are chosen

Fig. 12. Simulation of a mesh having different types of elements. (a) Illustration of the mesh. (b) Two electric fields in comparison with analytical results. (c) Entire solution error for all unknowns versus time.

to plot the electric fields in Fig. 11(a). Excellent agreement with analytical solutions can be observed. In Fig. 11(b), we plot the entire solution error shown in (42) versus time in comparison with the reference results which are analytical solutions. Again, excellent accuracy (less than 0.7% error) is observed at all points in the computational domain for all time instants simulated. E. Mesh With Mixed Elements We have examined the capability of the proposed method in handling meshes made of different types of elements. This mesh is illustrated in Fig. 12(a), which consists of 1312 triangular elements in the center and 84 rectangular elements surrounding it. In each triangular element, there are eight first-order vector bases; and in each rectangular element, there are 12 first-order vector bases. The interface between a rectangular and a triangular element is an edge, where the degrees of freedom from both elements are shared in common to ensure the tangential

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Fig. 13. Illustration of materials and geometry of a package inductor.

continuity of the fields. A wave propagation problem is simulated in this mixed-element mesh. The incident field is a plane wave having , where , and . The time step used is . In Fig. 12(b), the electric fields at two randomly selected points are plotted in comparison with analytical data. Excellent agreement can be observed. In Fig. 12(c), the entire solution error is plotted as a function of time. Again, excellent accuracy is observed, which verifies the capability of the proposed method in handling meshes having mixed types of elements. Such a capability also facilities a convenient implementation of various absorbing boundary conditions such as the perfectly matched layer. F. S-Parameter Extraction of a Lossy Package Inductor In this example, we simulate a package inductor made of lossy conductors of conductivity 5.8e+7 S/m, and embedded in a dielectric material of relative permittivity 3.4. Its geometry and material parameters are illustrated in Fig. 13. The inductor is discretized into five layers of triangular prism elements, the thickness of each of which is 6.5, 30, 6.5, 8.5, and 30 from bottom to top, respectively. The top view of the mesh is shown in Fig. 14(a). The boundary conditions are PEC on the top and at the bottom, and PMC on the other four sides. A current source is launched respectively at the two ports of the inductor. It has a Gaussian derivative pulse of , with , and . The number of expansion terms is 10 used in this simulation. The voltages obtained at both ports with port 1 (upper port) excited and port 2 open are plotted in Fig. 14(b) in comparison with the TDFEM results. Excellent agreement can be observed. The -parameters are also extracted and compared with those generated from the TDFEM. Very good agreement can be seen from Fig. 14(c) and (d) across the entire frequency band. G. CPU Time and Memory Comparison Among existing time-domain methods for handling unstructured meshes, the TDFEM only requires a single mesh like the proposed method. The TDFEM also has guaranteed stability and accuracy, and it ensures the tangential continuity of the fields across material interfaces. We hence choose the TDFEM to benchmark the performance of the proposed method. The example considered is a large-scale example having millions of unknowns, since small examples are not challenging to solve, which is true to almost every time-domain method. The computational domain is a circular cylinder of radius 1 m

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and height 5 m, which is discretized into 25 layers of triangular prism elements. The thickness of each layer is 0.02 m. The incident field is a plane wave having , where , and . The time step used is , which is the same in the TDFEM and the proposed method. The number of expansion terms used in the proposed method is nine in (38). The zeroth-order vector bases are employed in the TDFEM, whereas the first-order bases are used in the proposed method. This comparison is, in fact, disadvantageous to the proposed method since the sparse pattern resulting from a higher-order-bases based discretization is much more complicated and the system matrix has many more nonzeros, as compared to the zeroth-order-based discretization. However, if the proposed method is able to show advantages even for such a disadvantageous comparison, then its efficiency gain over the same-order TDFEM would become even more obvious. The triangular prism discretization results in 3 718 990 unknowns in the zeroth-order TDFEM. We find that the TDFEM simulation cannot be performed on our desktop PC that has 16-GB memory due to the TDFEM's large memory requirement. This is because although the explicit TDFEM only requires solving a mass matrix, which is sparse and simple, its and factors are generally dense. Although the mass matrix is time independent, and hence we only need to factorize it once. The TDFEM still has to be equipped with sufficient memory to store and factors to carry out the following backward and forward substitutions for the matrix solution at each time step. Certainly, iterative solvers can be used to reduce memory usage, however, they are not cost-effective in time-domain analysis since many right hand sides need to be simulated, and the number of right hand sides is equal to the number of time steps. We hence find a computer that has 128-GB memory so that the TDFEM simulation can be successfully performed on this example. On this computer, it takes the TDFEM 2109.44 s and more than 72-GB memory to finish the LU factorization of the mass matrix. The CPU time cost at each time marching step is 9.31 s, which is one backward and forward substitution time. For a fair comparison, a similar number of unknowns is generated in the proposed method. The resulting system matrix size is 3 741 700. In contrast to the 2109.44 s cost by TDFEM for factorization, the proposed method has no factorization cost since it is free of matrix solution. In contrast to the 72-GB memory required by the TDFEM, the proposed method only takes 6.2-GB memory to store the sparse matrices, as it does not need to store and since the mass matrix is diagonal. The CPU run time of the proposed method at each time step is 3.76 s, which is spent on a few matrix-vector multiplications. From the aforementioned comparison, the computational efficiency of the proposed method can be clearly seen. Recently, advanced research has also been developed to reduce the computational complexity of a direct matrix solution [30]. However, not solving a matrix always has its computational advantages as compared to solving a matrix. We have also compared the accuracy between the two methods using the analytical data as the reference, since the example is set up to have an analytical solution. The entire solution error of the proposed method measured by (42) is shown to across the entire time window. The entire be less than

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Fig. 14. Simulation of a 3-D package inductor with dielectrics and lossy conductors. (a) Top view of the triangular prism element mesh. (b) Time-domain voltages at the two ports. (c) Magnitude of -parameters. (d) Phase of -parameters.

. The solution error of the TDFEM is shown to be less than accuracy of the proposed method is satisfactory. Meanwhile, the slightly better accuracy of the Galerkin-based TDFEM could be attributed to the fact that it satisfies the Maxwell's equations in an integration sense across each element, whereas the proposed method let the Maxwell's equations be satisfied only at discrete and points. Furthermore, in the TDFEM, both Faraday's law and Ampere's law are satisfied in the same element, whereas in the proposed method, the second law is satisfied across the elements over the loops orthogonal to the first field unknowns. In addition, the time discretization scheme may also contribute to the difference in accuracy. VII. CONCLUSION In this paper, a new matrix-free time-domain method with a naturally diagonal mass matrix is developed for solving Maxwell's equations in 3-D unstructured meshes, whose accuracy and stability are theoretically guaranteed. Its property of being free of matrix solution is independent of element shape, thus suitable for analyzing arbitrarily shaped structures and materials discretized into unstructured meshes. The method is neither FDTD nor TDFEM, but it possesses the advantage of the FDTD in being naturally matrix free, and the merit of the TDFEM in handling arbitrarily unstructured meshes. No dual mesh, mass-lumping, interpolation, and projection are required. In addition, the framework of the proposed method permits the use of any higher-order vector basis function, thus allowing for any desired higher order of accuracy in both electric and magnetic fields. Moreover, the formulations presented in this paper do not require any modification on the traditional vector bases. Extensive numerical experiments on unstructured triangular, tetrahedral, triangular prism meshes, and mixed elements have validated the accuracy, matrix-free property, stability, and

generality of the proposed method. Comparisons have also been made with the TDFEM in unstructured meshes in CPU time, memory consumption, and accuracy, which demonstrate the merits of the proposed method. APPENDIX FIRST-ORDER CURL-CONFORMING VECTOR BASIS FUNCTIONS In a tetrahedral element, among the 20 first-order vector bases [26], there are 12 edge vector basis functions, which are defined as

(44) where

are volume coordinates, and denote the normalized zeroth-order edge bases as

follows:

(45) in which is the length of the th edge. The degrees of freedom of the 12 edge vector bases shown in (44) are located respec-

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tively at the following points in each element, with their corresponding projection directions defined as: (48)

REFERENCES

(46) where denotes the vector pointing from node to node . There are also two vector basis functions whose degrees of freedom are located at the center point of each face. In total, there are eight such bases, which are

(47) and corresponding unit The locations vectors associated with the above eight face vector bases are

[1] K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp. 302–307, May 1966. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA, USA: Artech House, 2000. [3] R. Holland, “Finite-difference solution of maxwell's equations in generalized nonorthogonal coordinates,” IEEE Trans. Nucl. Sci., vol. NS-30, no. 6, pp. 4589–4591, Dec. 1983. [4] M. Fusco, “FDTD algorithm in curvilinear coordinates [EM scattering],” IEEE Trans. Antennas Propag., vol. 38, no. 1, pp. 76–89, Jan. 1990. [5] J.-F. Lee, R. Palandech, and R. Mittra, “Modeling three-dimensional discontinuities in waveguides using nonorthogonal FDTD algorithm,” IEEE Trans. Microw. Theory Techn., vol. 40, no. 2, pp. 346–352, Feb. 1992. [6] T. G. Jurgens and A. Taflove, “Three-dimensional contour FDTD modeling of scattering from single and multiple bodies,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1703–1708, Dec. 1993. [7] S. Dey and R. Mittra, “A locally conformal finite difference time domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Microw. Guided Wave Lett., vol. 7, no. 9, pp. 273–275, 1997. [8] Y. Hao and C. J. Railton, “Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes,” IEEE Trans. Microw. Theory Techn., vol. 46, no. 1, pp. 82–88, Jan. 1998. [9] M. Madsen, “Divergence preserving discrete surface integral methods for maxwells equations using nonorthogonal grids,” J. Comput. Phys., vol. 119, pp. 34–45, 1995. [10] C. Chan, J. Elson, and H. Sangani, “An explicit finite-difference timedomain method using whitney elements,” in Proc. IEEE Int. Symp. Antennas Propag. (AP-S), 1994, vol. 3, pp. 1768–1771. [11] S. Gedney, F. Lansing, and D. Rascoe, “A full-wave analysis of passive monolithic integrated circuit devices using a generalized yee-algorithm,” IEEE Trans. Microw. Theory Techn., vol. 44, no. 8, pp. 1393–1400, Aug. 1996. [12] A. Bossavit and L. Kettunen, “Yee-like schemes on a tetrahedral mesh, with diagonal lumping,” Int. J. Numer. Modelling-Electron. Networks Devices Fields, vol. 12, no. 1, pp. 129–142, 1999. [13] C. F. Lee, B. J. McCartin, R. T. Shin, and J. A. Kong, “A triangle grid finite-difference time-domain method for electromagnetic scattering problems,” J. Electromagn. Waves Appl., vol. 8, no. 4, pp. 1429–1438, Aug. 1994. [14] M. Hano and T. Itoh, “Three-dimensional time-domain method for solving maxwells equations based on circumcenters of elements,” IEEE Trans. Magn., vol. 32, no. 3, pp. 946–949, May 1996. [15] S. Gedney and J. Roden, “Numerical stability of nonorthogonal fdtd methods,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 231–239, Feb. 2000. [16] M. Cinalli and A. Schiavoni, “A stable and consistent generalization of the fdtd technique to nonorthogonal unstructured grids,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1503–1512, May 2006. [17] D. Jiao and J. Jin, “Finite element analysis in time domain,” in The Finite Element Method in Electromagnetics. Hoboken, NJ, USA: Wiley, 2002, pp. 529–584. [18] M. Feliziani and F. Maradei, “Hybrid finite-element solutions as time dependent maxwells curl equations,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1330–1335, May 1995. [19] D. A. White, “Orthogonal vector basis functions for time domain finite element solution of the vector wave equation,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1458–1461, May 1999. [20] D. Jiao and J. Jin, “Three-dimensional orthogonal vector basis functions for time-domain finite element solution of vector wave equations,” IEEE Trans. Antennas Propag., vol. 51, no. 1, pp. 59–66, Jan. 2003. [21] S. D. Gedney et al., “The discontinuous galerkin finite element time domain method (DGFETD),” in Proc. IEEE Int. Symp. Antennas Propag., 2008, p. 4. [22] S. D. Gedney, J. C. Young, T. C. Kramer, and J. A. Roden, “A discontinuous galerkin finite element time-domain method modeling of dispersive media,” IEEE Trans. Antennas Propag., vol. 60, no. 4, pp. 1969–1977, Apr. 2012.

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[23] J. Yan and D. Jiao, “A matrix-free time-domain method independent of element shape for electromagnetic analysis,” in Proc. IEEE Int. Symp. Antennas Propag. (AP-S), 2014, pp. 2258–2259. [24] J. Yan and D. Jiao, “Accurate matrix-free time-domain method in unstructured meshes,” in Proc. IEEE Int. Microw. Symp. (IMS), 2015, pp. 1–4. [25] J. Jin, The Finite Element Method in Electromagnetics. Hoboken, NJ, USA: Wiley, 2014. [26] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 329–342, Mar. 1997. [27] D. Jiao and J. M. Jin, “A general approach for the stability analysis of time-domain finite element method,” IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1624–1632, Nov. 2002. [28] Q. He, H. Gan, and D. Jiao, “Explicit time-domain finite-element method stabilized for an arbitrarily large time step,” IEEE Trans. Antennas Propag., vol. 60, no. 11, pp. 5240–5250, Nov. 2012. [29] M.-F. Wong, O. Picon, and V. F. Hanna, “A finite element method based on Whitney forms to solve maxwells equations in the time domain,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1618–1621, May 1995. [30] B. Zhou and D. Jiao, “Direct finite-element solver of linear complexity for large-scale 3-d electromagnetic analysis and circuit extraction,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3066–3080, Oct. 2015.

Jin Yan received the B.S. degree in electronic engineering and information science from the University of Science and Technology of China, Hefei, China, in 2012. She is currently working toward the Ph.D. degree in electrical engineering at Purdue University, West Lafayette, IN, USA. She currently works in the On-Chip Electromagnetics Group at Purdue University. Her research is focused on computational electromagnetics, high-performance VLSI CAD, and fast and high-capacity numerical methods. Ms. Yan was the recipient of an Honorable Mention Award of the IEEE International Symposium on Antennas and Propagation in 2015.

Dan Jiao (S'00–M'02–SM'06) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, Urbana, IL, USA, in 2001. She then joined the Technology Computer-Aided Design (CAD) Division, Intel Corporation, until September 2005, where she was a Senior CAD Engineer, Staff Engineer, and Senior Staff Engineer. In September 2005, she joined Purdue University, West Lafayette, IN, USA, as an Assistant Professor with the School of Electrical and Computer Engineering. She is currently a Professor with Purdue University. She has authored 3 book chapters and over 230 papers in refereed journals and international conferences. Her current research interests include computational electromagnetics; high-frequency digital, analog, mixed-signal, and RF integrated circuit (IC) design and analysis; high-performance VLSI CAD; modeling of microscale and nanoscale circuits; applied electromagnetics; fast and high-capacity numerical methods; fast time-domain analysis; scattering and antenna analysis; RF, microwave, and millimeter-wave circuits; wireless communication; and bio-electromagnetics. Dr. Jiao has served as the reviewer for many IEEE journals and conferences. She is an Associate Editor of the IEEE TRANS. ON COMPONENTS, Packaging, and Manufacturing Technology. She received the 2013 S. A. Schelkunoff Prize Paper Award of the IEEE Antennas and Propagation Society, which recognizes the Best Paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION during the previous year. She was among the 21 women faculty selected across the country as the 2014–2015 Fellow of ELATE (Executive Leadership in Academic Technology and Engineering) at Drexel, a national leadership program for women in the academic STEM fields. She has been named a University Faculty Scholar by Purdue University since 2013. She was among the 85 engineers selected throughout the nation for the National Academy of Engineerings 2011 US Frontiers of Engineering Symposium. She was the recipient of the 2010 Ruth and Joel Spira Outstanding Teaching Award, the 2008 National Science Foundation (NSF) CAREER Award, the 2006 Jack and Cathie Kozik Faculty Start up Award (which recognizes an outstanding new faculty member of the School of Electrical and Computer Engineering, Purdue University), a 2006 Office of Naval Research (ONR) Award under the Young Investigator Program, the 2004 Best Paper Award presented at the Intel Corporations annual corporate-wide technology conference (Design and Test Technology Conference) for her work on generic broadband model of high-speed circuits, the 2003 Intel Corporations Logic Technology Development (LTD) Divisional Achievement Award, the Intel Corporations Technology CAD Divisional Achievement Award, the 2002 Intel Corporations Components Research the Intel Hero Award (Intel-wide she was the tenth recipient), the Intel Corporations LTD Team Quality Award, and the 2000 Raj Mittra Outstanding Research Award presented by the University of Illinois at Urbana-Champaign.

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Alternative Method for Making Explicit FDTD Unconditionally Stable Md. Gaffar and Dan Jiao, Senior Member, IEEE

Abstract—An alternative method is developed to make an explicit FDTD unconditionally stable. In this method, given any time step, we find the modes that cannot be stably simulated by the given time step, and deduct these modes directly from the system matrix (discretized curl-curl operator) before the explicit time marching. By doing so, the original FDTD numerical system is adapted based on the desired time step to rule out the root cause of instability. The resultant explicit FDTD marching is absolutely stable for the given time step no matter how large it is, and irrespective of space step. The accuracy is also guaranteed for time step chosen based on accuracy. Numerical experiments have validated the accuracy, efficiency, and unconditional stability of the proposed new method for making an explicit FDTD unconditionally stable. Index Terms—Explicit methods, finite-difference time-domain method (FDTD), stability, unconditionally stable methods.

I. INTRODUCTION

F

INITE-DIFFERENCE TIME-DOMAIN (FDTD) method [1], [2] is one of the most popular time domain methods for electromagnetic analysis. This is largely attributed to its simplicity and optimal computational complexity at each time step gained by not solving a matrix. However, the time step of a traditional FDTD is restricted by space step for stability, as dictated by the well-known Courant-Friedrichs-Lewy (CFL) condition. When the space step can be chosen based on accuracy for sampling the working wavelength, the time step dictated by the CFL stability condition agrees well with the time step required by accuracy. Hence, the dependence of time step on space step does not become a concern. However, when the problem being simulated involves fine features relative to working wavelengths such as an on-chip nanometer integrated circuit working at microwave frequencies, or a multiscaled system spanning a wide range of geometrical scales, the time step determined by space step for a stable FDTD simulation can become many orders of magnitude smaller than the time step required by accuracy. Due to such a small time step, a tremendous number of time steps must be simulated to reach the time corresponding to the working frequency, which is computationally prohibitive. From

Manuscript received June 19, 2015; revised September 10, 2015, October 21, 2015; accepted October 23, 2015. Date of publication November 11, 2015; date of current version December 02, 2015. This work was supported in part by a grant from NSF under award No. 1065318, and a grant from DARPA under award HR0011-14-1-0057. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, 17–22 May 2015. The authors are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2015.2496255

the accuracy point of view, such a choice of time step is not necessary, and hence the time step’s dependence on space step is a numerical problem that must be overcome. Implicit unconditionally stable FDTD methods [3]–[13] have been developed to overcome the dependence of time step on space step. In these methods, the time integration technique is changed to a different way such that the resulting time marching scheme has an error amplification factor bounded by 1, thus ensuring stability. However, the implicit methods require a matrix solution, the efficiency of which is not desired when a large problem size is encountered. In addition, it is observed that the accuracy of the implicit methods can degrade greatly with the increase of time step. Late-time instability has also been observed among existing implicit unconditionally stable FDTD methods. Recently, advanced research [14]–[18] has been pursued to address the time step problem in the framework of the original explicit time-domain methods. In [15], [17], [18], the root cause of instability is identified for explicit time-domain methods, based on which an explicit and unconditionally stable time-domain finite-element method (TDFEM) is successfully developed in [15], [17] and the same capability is demonstrated for FDTD in [18]. The root-cause analysis shown in [15], [17], [18] is different from a conventional stability analysis [2], [19]. In a conventional stability analysis, the time step required for a stable time-domain simulation is derived and used to guide the choice of time step. From such a stability analysis, apparently, except for choosing the time step based on the stability criterion, there is no other way forward to make an explicit method stable. On the contrary, the root-cause analysis given in [17], [18] reveals that when an explicit time-domain method becomes unstable, not every eigenmode present in the field solution becomes unstable. Only a subset of eigenmodes is unstable, while the rest of the eigenmodes are still stable. This subset of eigenmodes is the root cause of instability, which are termed unstable modes. These modes have eigenvalues (characterizing the rate of field variation in space) greater than that can be accurately captured by the given time step, thus causing instability. When the time step is chosen based on accuracy, the unstable modes are not required by accuracy. Hence, they can be removed without affecting the accuracy. Based on the root-cause analysis, in [18], an explicit FDTD that is unconditionally stable is developed. It has also been extended to analyze general lossy problems in [21], [22]. In this method, the field solution is expanded into stable eigenmodes, and the numerical system is also projected onto the space of stable eigenmodes. The resulting explicit time-marching is absolutely stable for the given time step no matter how large it

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is. Comparisons with state-of-the-art implicit unconditionally stable FDTD methods have also shown clear advantages of the new explicit method in accuracy, dispersion error, and stability in addition to computational efficiency [22]. This is mainly because in an implicit unconditionally stable method, the root cause of instability that makes an explicit method unstable is not removed from the numerical system. When a large time step beyond CFL condition is used, the root cause, which is unstable modes, cannot be accurately simulated by the given time step. Although they are suppressed to be stable, they can still negatively affect the overall accuracy and stability of the implicit method. To preserve the advantage of the explicit FDTD in avoiding solving a matrix equation, a preprocessing algorithm is developed in [18] to extract the stable eigenmodes from the field solutions obtained from the traditional explicit FDTD. The time window simulated in the preprocessing step is much smaller than that of the entire time window to be simulated. However, since the time step required by a traditional FDTD method is used in the preprocessing step, the speedup of the overall scheme can become limited by the preprocessing step. In this work, we develop a new explicit and unconditionally stable FDTD method. This new method eliminates the traditional FDTD-based preprocessing in [18]. Meanwhile, it permits the use of a large time step upfront in the explicit time marching by deducting the unstable modes directly from the FDTD system matrix. The unstable modes have the largest eigenvalues of the system matrix, and hence they can be efficiently found in complexity, with the number of unstable modes. Using the proposed method, one only needs to perform a very minor modification on the traditional FDTD to make it unconditionally stable. Hence, the proposed method is convenient for use. The basic idea of this work has been presented in our IMS conference paper [20]. In this paper, we expand [20] to address aspects that have not been addressed before, including algorithm details, complexity and accuracy analysis, open-region problems, how to efficiently find unstable modes, and comparisons with the previous explicit and unconditionally stable FDTD method [18]. Extensive numerical experiments and comparisons with existing methods have demonstrated the unconditional stability, accuracy, and efficiency of the proposed alternative explicit and unconditionally stable method. II. PRELIMINARIES Before presenting the proposed work, it is necessary to review the root cause of instability [17], [18]. Using a matrix notation, we can rewrite the FDTD updating equations into the following compact form: (1) (2) where denotes the vector of electric field unknowns placed along the edges of the primary grid, denotes the vector of magnetic field unknowns along the edges of the dual grid, is the vector of current sources whose entry is with being current density, is time step, superscripts such as , and denote the time instant, and are

sparse matrices representing the discretized , and operators, respectively. As can be seen from (1) and (2), the computations involved in the FDTD are sparse matrix-vector multiplications. Equations (1) and (2) solve both and . We can also eliminate one field unknown to see the root cause of instability more easily. Rewriting (2) for , we find (3) Subtracting (3) from (2), and using (1) to replace the term of in the resultant, we arrive at (4) where

, which is actually at the -th time instant. Equation (4) is nothing but a central-difference based discretization of the following second-order wave equation (5) where (6) The solution to (5) at any time is a time-dependent superposition of the eigenmodes of . Performing a -transform of (4), it can be found the eigenmodes, whose eigenvalues satisfy the following condition, can always be stably simulated by the given time step (7) The root cause of instability is thus the eigenmodes whose eigenvalues are greater than , which are termed unstable modes. In a traditional explicit time-domain method, the underlying numerical system and thereby the eigenmodes governing the field solution are not changed, but the time step is adjusted based on the CFL condition so that a time-domain simulation can be made stable. The CFL condition essentially requires the time step to be chosen based on the largest eigenvalue of , so that (7) is satisfied for all eigenmodes present in the numerical system. In an explicit and unconditionally stable method like [17], [18], the desired time step is not changed, but the numerical system is changed so that only those eigenmodes that can be stably simulated by the given time step are kept, while the unstable modes are discarded. In this way, the dependence of the time step on space step is removed, and an explicit method can also be made unconditionally stable. III. PROPOSED METHOD From the root-cause analysis reviewed in the previous section, it is evident that once a space discretization is done, whether there exist unstable modes or not is known for a given time step, regardless of time marching. Therefore, the source of instability is inherent in the system matrix resulting from the space discretization, rather than in the field solution. To completely remove such a source, the system matrix has to be

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changed. In this section, we present proposed method, explain how it works, and analyze its complexity and accuracy. In addition, we also describe how to handle open-region problems in the proposed method.

Since , where is the diagonal matrix of eigenvalues, using (13), we obtain

A. Method

which can further be written as

(14)

Let denote the matrix formed by all the unstable modes, with each column being an eigenvector of whose eigenvalue is greater than . How to efficiently find will soon be given in next section. Right now, assume has been generated. In the proposed method, we use to directly change the original system matrix to a new system matrix

is the eigenvector matrix of stable eigenmodes, where and are diagonal matrices containing stable and unstable eigenvalues, respectively. Multiplying both sides of (15) by , since ’s eigenvectors are orthogonal, we obtain

(8)

(16)

For an that is Hermitian and positive semi-definite, the above is equal to

Thus, substituting (16) into (8) and using (15), (8) is nothing but

(9) i.e., multiplying by from the right. We then perform an explicit FDTD simulation on the new system matrix . If a second-order based system shown in (4) is employed, we simply modify it to (10) and march on in time step by step. If the original FDTD-based first-order system given in (1) and (2) is used, we update them to

(11) which is the same as (10). This can be readily verified by eliminating unknowns from (11). One can also eliminate unknowns to obtain an equation for , which is the -based counterpart of (10). In this case, becomes , and in (11), the term does not exist in the first row, but appears in front of in the second row. After obtaining the solution of from (10) or (11), we need to add one more important step to make the solution correct, which is (12) Obviously, the aforementioned method only requires a very minor modification in the traditional FDTD, and hence the method is convenient for use. Now, we shall explain how the proposed method works. B. How It Works? The new system matrix consists of the stable eigenmodes only, and hence the source of instability is completely removed. To prove, we first utilize the property of the eigenvectors of . Since is Hermitian positive semi-definite, its eigenvectors are orthogonal. Hence, the following property holds true: (13)

(15)

(17) and hence the space of stable modes only. Since is symmetric as can be seen from (17), (8) is the same as (9). However, for non-symmetric such as the one resulting from a lossy analysis [22], (8) is different from (9), and (8) is the correct one to use since it is still made of modes only, while (9) is not. In addition, in this case, the should be orthogonalized to satisfy before being removed from , and the in (8) is replaced by . After updating the system matrix from to that is free of the source of instability, we can perform the explicit FDTD time-marching on with absolute stability. However, after implementing (10) and (11), we found the result is indeed stable but not accurate. Interestingly, if is found by first obtaining all eigenvectors of , and then selecting from them, the accuracy is good. However, if is found by computing the unstable eigenvectors of only, the results do not match the reference data. Certainly, the first approach that finds all eigenvectors is not practical for use when problem size is large, and the second approach is the one that can truly make the proposed method useful in practice. To figure out the problem, we compare the modes found by the second approach with those found by computing all eigenvectors of . They show good agreement with each other. Therefore, the accuracy of is not a problem. This has led to the finding that the updated matrix has additional zero eigenvalues, the eigenvectors corresponding to these additional zero eigenvalues are not the eigenvectors of the original matrix , and they make the result wrong. To explain, the as shown in (17) has rank , where is the number of stable eigenmodes of . On the other hand, is a matrix of size . Hence, the additional eigenvalues are zero, whose eigenvectors make an additional nullspace. As a result, when computing (10) or (11), the field solution is not only the superposition of the modes, but also the additional nullspace modes as the following: (18) are corresponding coefficient vectors. The true where and solution should satisfy . Hence, the result of (18) is wrong. When is found by computing all the eigenvectors of

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and from them choosing , (17) matches that in (15) very well. Hence, the contribution of the additional nullspace in (17) is almost zero. However, this is not the case when is found from computing only the eigenvectors of that violate (7). To solve this problem, after obtaining the solution of from (10) or (11), we need to add (12) to make the solution correct. This treatment removes the second term in (18). This is because must be in the space of since it is not in . By deducting from , all of the -components and thereby -components are removed, making ’s solution correct. It is also worth mentioning that if nullspace, whose eigenvectors are also termed DC modes, does not make an important contribution in the field solution such as in the case of an electrically large antenna, the step of (12) is not needed. To use an infinitely large time step without making the FDTD unstable, we simply remove all the eigenmodes of whose eigenvalues are nonzero. To use other time step sizes, we remove eigenmodes adaptively based on the given time step. As a result, the proposed method flexibly permits the use of any time step independent of space step, thus being explicit and unconditionally stable.

hence , thus beyond the maximum frequency required to be captured by accuracy. The above accuracy analysis is for source-free problems. The same holds true for problems with sources, as shown by the analysis given in Section IV.B, and specifically (40) of [18]. D. Treatment of Open-Region Boundary Conditions In open-region problems, the computational domain can be truncated by various Absorbing Boundary Conditions (ABC) such as Perfectly Matched Layers (PML). Since the field solution inside PML is fictitious, and there is no fine feature inside the PML region either, we do not perform any special treatment in the PML region, but to conduct the FDTD simulation as it is. In the solution domain, we update the system matrix by deducting the unstable modes from it. Basically, we divide the unknown into two groups, one inside the solution domain denoted by , and the other elsewhere such as boundary, PML or other ABCs, denoted by . The same is done for unknown . Subsequently, the sparse matrices and are cast into the following form:

C. Complexity and Accuracy Analysis 1) Complexity Analysis: As compared to the original FDTD, the only additional computation involved in the proposed method is the computation of at each time step, as shown in (10) and (11). The can be efficiently evaluated by two matrix-vector multiplications: first, computing , the cost of which is ; second, multiplying the resultant by , the cost of which is also . If one computes first, the resultant matrix is a dense matrix of size . Multiplying such a dense matrix by would cost operations, which is expensive when is large. Therefore, the approach of doing two matrix-vector multiplications should be used to obtain . 2) Accuracy Analysis: When the time step is chosen based on accuracy, the unstable modes are not required by accuracy, and hence they can be deducted from the system matrix without affecting accuracy. To explain, in the proposed method, we expand the space dependence of the field solution using the eigenmodes of as follows: (19) is the time-dependent coefficient of the -th eigenwhere mode . In a source-free problem, the is analytically known as [18] (20) arbitrary coefficients. Hence, the square root of with the eigenvalue is also the frequency of the field’s time variation, i.e., . This is, in fact, dispersion relation. In free space, the is analytically known as , where is free-space wave number. In inhomogeneous problems, the is not analytically known but can be numerically found. When the time step is chosen based on accuracy such as . The unstable modes have , and

(21) With the above, rewriting (11) separately for , we obtain

and

(22) IV. FINDING UNSTABLE MODES For any given time step , the unstable modes are the eigenmodes of whose eigenvalues are greater than . Hence, the unstable modes have the largest eigenvalues of . Since is sparse, the computing task becomes how to find the largest eigenpairs of a sparse matrix. The Arnoldi method is particularly suited for this computing task [23]. In steps, it can find a complete set of largest eigenvalues and eigenvectors. When the matrix is Hermitian, the Arnoldi process reduces to Lanczos method. A -step Arnoldi method on matrix is to carry out the following computation: (23) where is a unitary matrix of size is a small upper Hessenberg matrix of dimension is the -th column of identity matrix , and is a column vector of length yielding an matrix. When the norm of and hence the norm of goes to zero, the eigenvalues of are the eigenvalues of the original matrix , and multiplied by the eigenvectors of are the eigenvectors of . The detailed algorithm for realizing (23) can be found from Algorithm 3.7 of [23]. The overall computational cost is simply sparse matrix-vector multiplications, each of which is multiplied by an intermediate vector

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one has to use a shift-invert technique to transform the eigenvalues of interest (now smallest eigenvalues) to the largest eigenvalues of a new matrix. This new matrix can be written as , where is a shift value chosen to be small so that the smallest eigenvalues can become the largest ones of the new matrix. It is evident that finding the eigenvalues of is computationally much more expensive as compared to finding the eigenvalues of , since a matrix solution is involved. Furthermore, in steps, we cannot guarantee finding a complete set of smallest eigenvalues since is empirical. Moreover, has a nullspace whose eigenvalues are zero. The size of nullspace grows with . In other words, when matrix size increases, the number of eigenvectors whose eigenvalues are zero also increases. This further increases the computational cost. In contrast, the preprocessing algorithm developed in [18] is an efficient and reliable algorithm for finding a complete set of stable eigenmodes. The problem of the increasing size of the nullspace is also well handled in this preprocessing algorithm. This is because all the nullspace eigenvectors share the same eigenvalue (zero) in common. Given a right hand side (source) vector, the contributions from the nullspace eigenvectors are grouped together and become a single vector. Hence, the algorithm in [18] does not suffer from the issue of increasing nullspace size. V. COMPARISON WITH PREVIOUS METHOD First, we prove the proposed new method is mathematically equivalent to the previous method [18]. In previous method [18], the fields are expanded in the space of stable modes, and the numerical system is projected onto the space of stable modes. Consider the solution of (4), the is expanded as , and the time-dependent unknown coefficient vector is solved from the following equation: generated during the -step process, and the orthogonalization of the resulting vectors. The complexity of the sparse matrix-vector multiplications is , while the complexity of orthogonalization is , and hence the overall complexity is . This is much more efficient than a brute-force eigenvalue solution. A straightforward -step Arnoldi process cannot ensure the largest eigenpairs to be found in steps. Spurious eigenvalues may also be produced. We hence employ the implicitly restarted Arnoldi method [23] to systematically drive the residual of (23) to be zero. For completeness of this paper, we give the algorithm of implicitly restarted Arnoldi method as shown in Algorithm 1, which is modified to suit the need of this work. In this algorithm, from Step 4 to 12 is to shift unwanted eigenvalues so that the next initial vector is rich in the wanted eigenvectors. The computational cost from Step 4 to 12 is negligible as compared to Step 3, since these steps are performed on small matrices of size . The computational complexity of Step 3 is , where is proportional to . The cost of Step 13 is again negligible since it is performed on a small matrix of size . Overall, the complexity of Algorithm 1 is for finding largest eigenpairs of . One may wonder why we do not use the same procedure to find the stable eigenmodes. The stable eigenmodes turn out to have the smallest eigenvalues of . To find them efficiently,

(24) In the proposed method, we solve (10). Substituting (17) into it, we obtain (25) , since in the new method we do not Here, let explicitly expand the field solution in the space. Vector hence consists of the coefficients corresponding to both the , and the modes. Multiplying (25) by , with (12), we obtain , and satisfying the same equation as (24). Hence, the proposed new method is mathematically equivalent to the previous method. Therefore, its accuracy and stability are both ensured. However, the two methods are computationally different. In the previous method [18], a traditional FDTD-based preprocessing is developed to find the space of stable modes. In the proposed method, no such preprocessing is required, and hence the method is not subject to the constraint of the traditional FDTD's time step. In the previous method, the numerical system is projected onto the space of stable modes; in the proposed method, the unstable modes are directly deducted from the numerical system to eradicate the root cause of instability. In the

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previous method, the explicit marching is performed on a reduced order system since the number of stable modes is smaller than the original system size; in the proposed method, there is no reduction in system order. In the previous method, if the number of stable modes is large, the modes can be broken into bands and found band by band independent of each other; in the proposed method, if the number of unstable modes is many, the computational cost for finding them remains to be and, in general, cannot be made smaller. Overall, when the number of unstable modes is not large, the proposed method is efficient for use. This is typically true in many problems solved by the FDTD: the fine features only occupy a small portion of the entire space discretization. The proposed method is also much more convenient for implementation. In addition, by removing just one unstable mode whose eigenvalue is the largest, one already can use a time step larger than the CFL time step using the proposed method; by removing the highest two unstable eigenmodes, one can use an even larger time step; and so on. Hence, with negligible computational cost, the proposed method allows for the use of a time step beyond the stability criterion. In contrast, in the method of [18], the time step in the preprocessing procedure is restricted by the time step required by stability. The computational overhead is more for one to use a time step beyond the CFL condition. Certainly, the two methods can be combined to accentuate the advantages of both methods. For example, when the number of unstable modes is many, from the proposed method, we can still remove a certain number of unstable modes within feasible run time, based on which the time step can be immediately enlarged although it has not been enlarged to the time step allowed by accuracy yet. Using the resulting updated system matrix, and hence a much increased time step, the preprocessing step in [18] can be accelerated greatly to identify the stable modes. The proposed method hence does not need to finish the simulation of the entire time window, but a small window simulated in the preprocessing step. The previous method can then be used to carry out explicit marching efficiently: the system has a much reduced order and is diagonal, in addition, the computation of the term in the proposed method is also avoided. VI. NUMERICAL RESULTS A. Demonstration of Unconditional Stability First, we demonstrate the unconditional stability of the proposed method using an example that has an analytical solution. It is a 3-D parallel plate structure whose dimension is 900 m, 6 m, and 1 m along -, -, and -direction, respectively. The space step is 0.2 m, 0.85714 m, and 90 m, respectively along -, -, and -direction. This structure is excited at the near end by a current source that has a very low frequency pulse of with s, and s. The time step required by sampling accuracy thereby is at the level of s, while that dictated by the CFL condition for stability is s. Hence, there is a more than 160 orders of magnitude difference in the two time steps. With a time step of s, the proposed method stably simulated the structure with excellent accuracy. As can be seen from Fig. 1(a), the voltages generated from the proposed method and

Fig. 1. Demonstration of unconditional stability. (a) Voltage waveforms. (b) Entire solution error as a function of time.

the analytical solution are on top of each other. Notice that the structure behaves as a capacitor at very low frequencies, and hence the near- and far-end voltages are identical to each other. The s appears to be already an extremely large time step. In fact, the proposed method allows for a time step of infinity. The includes all the eigenmodes whose eigenvalues are nonzero, leaving zero eigenvalues only, and hence permitting an infinitely large time step. The number of modes is 561. In addition to examining the solution accuracy at selected points, we have also assessed the entire solution error by measuring , where consists of all electric field unknowns in the computational domain solved from the proposed method, whereas is obtained from the analytical solution. The entire solution error is shown in Fig. 1(b) as a function of time, verifying the accuracy of the proposed method at all points in the computational domain at each time instant. Notice that the error is plotted as it is instead of a percentage error. It takes the proposed method 2.99 s to finish the entire simulation including the time for finding unstable modes. To finish the same simulation, the FDTD would have to take more than s (the expanding universe time). This example appears to be dramatic, however, it is necessary to examine whether a

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Fig. 3. Simulation of a mm-level cavity. (a) Electric fields obtained from the proposed method at two points using different tme steps. (b) Comparison with the ADI and HIE methods for the electric field simulated.

B. Parallel Plate Excited by a Current Source at Higher Frequencies

Fig. 2. Simulation of a 3-D parallel plate structure. (a) Voltage waveforms. (b) Entire solution error as a function of time. (c) The ratio of the unstable modes component to the entire field solution.

method is truly unconditionally stable. This example also shows clearly that the dependence of time step on space step is a numerical problem, instead of a fundamental physical law one has to obey.

Next example is the same structure but with a fast Gaussian derivative pulse having a maximum input frequency 34 GHz. Since the space discretization remains the same, the time step required by a stable FDTD simulation remains to be s, while the time step required by sampling accuracy is s. The proposed method is able to generate accurate and stable results using the time step of s. As shown in Fig. 2(a), the voltage waveforms simulated by the proposed method are in excellent agreement with those from the conventional explicit FDTD. The accuracy is further demonstrated by the entire solution error plotted in Fig. 2(b) as a function of time. The CPU time cost by the proposed method is approximately 6.013 s with 3.251 s for time marching, and the rest for finding the unstable modes. Compared with 6875.4 s required by the conventional explicit FDTD, the speedup of the proposed method is 1145.9. In this example, we have also examined the weights of the unstable modes in the field solution. Let the weights be denoted by . It can be computed at each time step from , where is the reference FDTD solution. As can be seen

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Fig. 5. Entire solution error for simulating a 3-D on-chip bus structure.

s solely determined by accuracy, whereas the time step used by the conventional FDTD is s. The number of modes removed is 288. In Fig. 3(a), we plot the electric fields at two points, mm, and mm, respectively, in comparison with the FDTD solutions. Excellent agreement can be observed. The total simulation time of the proposed method is 5.74 s including the time for finding , and that for performing (11)–(12), in contrast to the 27.37 s cost by FDTD. The average entire solution error is found to be less than 3%. We have also simulated this example using the ADI and the HIE method [24] using the time step of s. In Fig. 3(b), we compare the voltages at point simulated by the three methods, which further verifies the accuracy of the proposed method. The proposed method flexibly adapts the eigensystem based on the required time step. For example, if the required time step is s instead of s, the eigenvalues are accordingly removed from the largest down to . The results are equally accurate as can be seen from Fig. 3(a). The CPU time for this case is 19.03 s. D. Open-Region Radiation

Fig. 4. Simulation of an open-region problem. (a) Structure. (b) Electric field at the observation point. (c) Entire solution error.

clearly from Fig. 2(c), the weights of the discarded unstable modes are small as compared to the entire field solution. C. Millimeter-Level Cavity Previous structures have very fine features relative to working wavelengths. Next, we consider a millimeter cavity whose space discretization is comparable to that required by the input spectrum. The overall dimension is 19.4 mm 12.4 mm 0.14118 mm. The space step along -, -, and -direction is respectively 1.8 mm, 1.8 mm, and 0.03529 mm. A current element of length 0.0334 mm is located in middle of the cavity along -direction. The proposed method uses a time step of

Next, we simulate an open-region problem with a dipole antenna radiating in presence of multiple dielectric cylinders, as illustrated in Fig. 4(a). The solution domain is 15.7 mm by 10.3 mm, surrounded by a 10-layer PML region. The maximum space step size is m. The smallest space step is m. There are three cylinders situated on the left side of the solution domain and a current source along -direction located on the upper right corner. The pulse of the current source is a Gaussian derivative with a maximum input frequency of Hz. The cylinders have relative permittivity . The time step required by stability is s, whereas the time step used by the proposed method is s. The proposed method takes s to finish the entire simulation, while the FDTD costs s. The electric field at the observation point shown in Fig. 4(a) is plotted and compared with that of traditional explicit FDTD in Fig. 4(b). The entire solution error is shown in Fig. 4(c). Excellent accuracy is observed.

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able to use the time step required by accuracy, s, to obtain accurate results. In contrast, the conventional FDTD must use a time step of s to ensure stability. The total CPU time of the proposed method is 7.12 s, whereas that of the conventional FDTD is 7 426 s. The speedup of the new method over the FDTD is approximately 1 043. The number of removed eigenmodes is 536 in this example. The entire solution error is plotted in Fig. 5 as a function of time, revealing good accuracy of the proposed method. The speedup of the method in [18] over the traditional FDTD is 47. Hence, the proposed method is more efficient in simulating this example. F. Electromagnetic Interference (EMI) Example In the last example, we simulate an EMI example as illustrated in Fig. 6(a), and compare the performance of the proposed method with the method of [24]. The structure is a cube of side length 11 cm truncated by perfect electric (PEC) boundary conditions all around. In the center, there is a PEC sheet with five slots. The thid slot has a width of 0.25 mm, and others are of width 1 cm. Then we set the total at the center point of the lower-half domain to be with s, and s. The third slot is discretized along into 5 uniform cells. The cell size is 1 cm along -, and -direction, respectively, and 0.02 cm along in the areas other than the third slot. We compare the results obtained from the conventional FDTD, the proposed method, and the HIE in [24], for a time step of 0.166 ps which is the time step of the conventional FDTD, and the time step of 1.66 ps, respectively. The obtained for the two choices of the time step at the center point of the upper domain are plotted in Fig. 6(b), and (c), respectively. The HIE is shown to be unstable for the time step of 1.66 ps, whereas the proposed method still generates stable and accurate results.

VII. CONCLUSION

Fig. 6. Simulation of an EMI problem. (a) Illustration of the structure. (b) Electric field at the observation point using a time step of 0.166 ps. (c) Electric field at the observation point using a time step of 1.66 ps.

E. On-Chip 3-D Bus Next, an on-chip 3D bus structure embedded in an inhomogeneous stack of dielectrics is simulated. The proposed method is

In this paper, an alternative method is developed to achieve unconditional stability in an explicit FDTD simulation. It retains the strength of FDTD in avoiding matrix solutions, while eliminating its shortcoming in time step. The unstable modes are directly deducted from the original FDTD numerical system to eradicate the root cause of instability. Since the unstable modes have the largest eigenvalues and the FDTD system matrix is sparse, the unstable modes can be efficiently and reliably found in complexity, where is the number of unstable modes. The proposed method only requires a very minor modification on the traditional FDTD to make it unconditonally stable. Its implementation is hence convenient. Numerical experiments and comparisons with existing explicit FDTD methods have demonstrated the superior performance of the proposed method in stability, accuracy, and efficiency. The essential idea of the proposed method can also be applied to other time domain methods. The proposed method complements the capability offered by the original explicit and unconditionally stable FDTD [18]. Recently, this work has also been extended for analyzing general lossy problems in [25].

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REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. 14, no. 5, pp. 302–307, May 1966. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA, USA: Artech House, 2000. [3] T. Namiki, “A new FDTD algorithm based on alternating-direction implicit method,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 10, pp. 2003–2007, Oct. 1999. [4] F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the courant stability conditions,” IEEE Microwave Guided Wave Lett., vol. 9, no. 11, pp. 441–443, Nov. 1999. [5] G. Sun and C. W. Trueman, “Unconditionally stable Crank-Nicolson scheme for solving two-dimensional Maxwell's equations,” Electron. Lett., vol. 39, no. 7, pp. 595–597, Apr. 2003. [6] J. Lee and B. Fornberg, “A splitting step approach for the 3-D Maxwell's equations,” J. Comput. Appl. Math, vol. 158, no. 2, pp. 485–505, 2003. [7] G. Zhao and Q. H. Liu, “The unconditionally stable pseudospectral time-domain (PSTD) method,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 11, pp. 475–477, Nov. 2003. [8] J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one dimentional scheme,” Electron. Lett., vol. 41, no. 19, pp. 1046–1047, Sep. 2005. [9] Y. S. Chung, T. K. Sarkar, B. H. Jung, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 1, pp. 56–64, Jan. 2011. [10] Z. Chen, Y. T. Duan, Y. R. Zhang, and Y. Yi, “A new efficient algorithm for the unconditionally stable 2-D WLP-FDTD method,” IEEE Trans. Antennas Propag., vol. 61, no. 7, pp. 3712–3720, Jul. 2013. [11] Z.-Y. Huang, L.-H. Shi, B. Chen, and Y. H. Zhou, “A new unconditionally stable scheme for FDTD method using associated hermite orthogonal functions,” IEEE Trans. Antennas and Propag., vol. 62, no. 9, pp. 4804–4808, Sep. 2014. [12] E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propagat., vol. 56, no. 1, pp. 170–177, Jan. 2008. [13] M. Gaffar and D. Jiao, “A simple implicit and unconditionally stable FDTD method by changing only one time instant,” in Proc. IEEE Int. Symp. Antennas Propagat., Jul. 2014, pp. 1–2. [14] A. Ecer, N. Gopalaswamy, H. U. Akay, and Y. P. Chien, “Digital filtering techniques for parallel computation of explicit schemes,” Int. J. Computat. Fluid Dynamics, vol. 13, no. 3, pp. 211–222, 2000. [15] Q. He and D. Jiao, “An explicit time-domain finite-element method that is unconditionally stable,” in Proc, 2011 IEEE Int. Symp. Antennas Propag., Jul. 2011, pp. 4–. [16] C. Chang and D. S. Costas, “A spatially filtered finite-difference time-domain scheme with controllable stability beyond the CFL limit,” IEEE Trans. Microw. Theory and Tech., vol. 61, no. 3, pp. 351–359, Mar. 2013. [17] Q. He, H. Gan, and D. Jiao, “Explicit time-domain finite-element method stabilized for an arbitrarily large time step,” IEEE Trans. Antennas Propag., vol. 60, no. 11, pp. 5240–5250, Nov. 2012. [18] Md. Gaffar and D. Jiao, “An explicit and unconditionally stable FDTD method for electromagnetic analysis,” IEEE Trans. Microw. Theory Tech., vol. 62, no. 11, pp. 2538–2550, Nov. 2014. [19] D. Jiao and J. M. Jin, “A general approach for the stability analysis of time-domain finite element method,” IEEE Trans. Antennas Propagat., vol. 50, no. 11, pp. 1624–1632, Nov. 2002. [20] Md. Gaffar and D. Jiao, “An alternative method for making an explicit FDTD unconditionally stable,” in Proc. IEEE Int. Microwave Symp. (IMS), May 2015, pp. 1–4. [21] Md. Gaffar and D. Jiao, “An explicit and unconditionally stable FDTD method for the analysis of general 3-D lossy problems,” in Proc. IEEE Int. Microwave Symp. (IMS), Jun. 2014, pp. 1–4. [22] M. Gaffar and D. Jiao, “An explicit and unconditionally stable FDTD method for the analysis of general 3-D lossy problems,” IEEE Trans. Antennas Propag., vol. 63, no. 9, pp. 4003–4015, Sep. 2015. [23] D. C. Sorensen, “Implicit application of polynomial filters in a k-step arnoldi method,” SIAM J. Matrix Analysis Appl., vol. 13, no. 1, pp. 357–385, 1992.

[24] J. Chen and J. Wang, “A three-dimensional semi-implicit FDTD scheme for calculation of shielding effectiveness of enclosure with thin slots,” IEEE Trans. Electromagn. Compat., vol. 49, no. 2, pp. 354–360, 2007. [25] M. Gaffar and D. Jiao, “A new explicit and unconditionally stable FDTD method for analyzing general lossy problems,” in Proc. IEEE Int. Symp. Antennas Propag., Jul. 2015, pp. 1–2. Md. Gaffar received the B.Sc. degree in electrical engineering from the Bangladesh University of Engineering and Technology (BUET), Dhaka, Bangladesh in October, 2009. Since 2011, he has been pursuing the Ph.D. degree in school of electrical and computer engineering at Purdue University, West Lafayette, IN, USA. His research interests include computational electromagnetics and semiconductor physics. Mr. Gaffar has received academic awards in recognition of his research achievements, including the Best Poster Award (among all groups) and Best Project Award in communication and Electromagnetic in EEE Undergraduate Project Workshop (EUProW) 2009. At Purdue, his research has been recognized by the IEEE International Microwave Symposium Best Student Paper Finalist Award in 2013 and 2015, and the 2014 IEEE International Symposium on Antennas and Propagation Honorable Mention Paper Award.

Dan Jiao (S’00–M’02–SM’06) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, IL, USA, in 2001. She then worked at the Technology Computer-Aided Design (CAD) Division, Intel Corporation, until September 2005, as a Senior CAD Engineer, Staff Engineer, and Senior Staff Engineer. In September 2005, she joined Purdue University, West Lafayette, IN, USA, as an Assistant Professor with the School of Electrical and Computer Engineering, where she is now a Professor. She has authored three book chapters and over 230 papers in refereed journals and international conferences. Her current research interests include computational electromagnetics, high-frequency digital, analog, mixed-signal, and RF integrated circuit (IC) design and analysis, high-performance VLSI CAD, modeling of microscale and nanoscale circuits, applied electromagnetics, fast and high-capacity numerical methods, fast time-domain analysis, scattering and antenna analysis, RF, microwave, and millimeter-wave circuits, wireless communication, and bio-electromagnetics. Dr. Jiao has served as the reviewer for many IEEE journals and conferences. She is an Associate Editor of the IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY. She received the 2013 S. A. Schelkunoff Prize Paper Award of the IEEE Antennas and Propagation Society, which recognizes the Best Paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION during the previous year. She was among the 21 women faculty selected across the country as the 2014–2015 Fellow of ELATE (Executive Leadership in Academic Technology and Engineering) at Drexel, a national leadership program for women in the academic STEM fields. She has been named a University Faculty Scholar by Purdue University since 2013. She was among the 85 engineers selected throughout the nation for the National Academy of Engineerings 2011 US Frontiers of Engineering Symposium. She was the recipient of the 2010 Ruth and Joel Spira Outstanding Teaching Award, the 2008 National Science Foundation (NSF) CAREER Award, the 2006 Jack and Cathie Kozik Faculty Start up Award (which recognizes an outstanding new faculty member of the School of Electrical and Computer Engineering, Purdue University), a 2006 Office of Naval Research (ONR) Award under the Young Investigator Program, the 2004 Best Paper Award presented at the Intel Corporations annual corporate-wide technology conference (Design and Test Technology Conference) for her work on generic broadband model of high-speed circuits, the 2003 Intel Corporations Logic Technology Development (LTD) Divisional Achievement Award, the Intel Corporations Technology CAD Divisional Achievement Award, the 2002 Intel Corporations Components Research the Intel Hero Award (Intel-wide she was the tenth recipient), the Intel Corporations LTD Team Quality Award, and the 2000 Raj Mittra Outstanding Research Award presented by the University of Illinois at Urbana-Champaign.

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Accurate Parametric Electrical Model for Slow-Wave CPW and Application to Circuits Design Alfredo Bautista, Anne-Laure Franc, and Philippe Ferrari, Senior Member, IEEE

Abstract—In this paper, a predictive electrical model of the slow-wave coplanar waveguide structure (S-CPW) is presented. The model was developed under the assumption of Quasi-TEM propagation mode. This assumption allows treating separately the electric field from the magnetic field. Therefore, inductive and capacitive effects are processed apart. Within this context, analytical formulas, parameterized by S-CPW geometric dimensions, are given for each electric parameter in the model, including resistances that account for losses. The model was validated with electromagnetic simulations and measurement results on several integrated technologies. An excellent agreement was achieved over a wide frequency band from DC up to 110 GHz, with a maximum error of 10%. Consequently, the model provides a fast and powerful tool for designing circuits based on S-CPW. The developed model enables a better insight of how geometries influence the overall S-CPW performance. The model was applied to the design of a quarter-wave length transmission lines and tunable phase shifter. The transmission lines were optimized in terms of performance, minimum length or surface. The tunable phase shifter was designed by embedding varactors in the S-CPW floating shield. These designs highlight the efficiency of the model for complex optimization or complex circuits design, respectively. Index Terms—Electromagnetic modeling, millimeter-wave integrated circuits, slow-wave coplanar waveguide (S-CPW).

I. INTRODUCTION

I

N RECOGNITION of ongoing demands for higher data rate communication systems, special attention has been focused on millimeter-wave (mm-wave) frequencies. Communication systems working at this frequency range include a wide variety of passive circuits such as matching networks, baluns, phase-shifter, couplers, power dividers among others. Unfortunately, it is well known that losses in passive devices rise as frequency increases, degrading the circuits' quality factor, and thus affecting the entire system performance. In consequence, designing high performance passive circuits is a fundamental task. Previous works reported in [1]–[4] suggest that slow-wave coplanar waveguide (S-CPW) structure is the best basic cell for Manuscript received July 02, 2015; revised September 22, 2015, October 16, 2015; accepted October 18, 2015.Date of publication November 17, 2015; date of current version December 02, 2015. This paper is an extended version from the IEEE MTT-S International Microwave Symposium, 17-22 May 2015, Phoenix, USA. A. Bautista and P. Ferrari are with the Université de Grenoble Alpes, IMEP-LAHC, F-38010, Grenoble, France and also with CNRS, IMEP-LAHC, F-38000, Grenoble, France (e-mail: [email protected]; [email protected]). A.-L. Franc is with the Université de Toulouse, INPT, UPS; LAPLACE, ENSEEIHT, BP7122, F-31071, Toulouse, France, and also with CNRS, LAPLACE, F-31071 Toulouse, France (e-mail: anne-laure.franc@laplace. univ-tlse.fr). Digital Object Identifier 10.1109/TMTT.2015.2495242

elite passive devices at mm-wave. S-CPW gathers both high quality factor and compactness, providing a solution in the development of high performance building blocks. Most of current efforts for improving circuits' performance are based on S-CPW. For example, recent works at mm-wave use S-CPW for designing power splitters [2], mixers [5], band-pass filters [6], Lange couplers [7] or high directivity coupled-line couplers [8] and switched transmission lines [9]. They are also implemented in active circuitry like power amplifiers [10]–[12]. In order to find the most suitable S-CPW, all of these designs were carried out by deep optimization steps using full-wave 3D-EM tools. Designers are forced to recursively face the same procedure for each different CMOS/BiCMOS technology owing to the dissimilarities of their Back-End-Of-Line (BEOL). That leads to long development times, without any degree of certainty of getting the best design. From that it is highly important to provide designers with an accurate electrical model based on S-CPW geometries parameters, allowing proper optimizations based on very fast circuit simulations. The S-CPW concept was first introduced in [13], and since then many papers have been focused on developing an appropriate model [14]–[18]. Despite all these efforts, most of the proposed models are not based upon physics but on fitting equations with several correction parameters. For instance [14] presents a simplistic model were the inductance depends on many fitting coefficients. A more complex model can be found in [15]. However, the RLCG topology was employed inappropriately since it is not S-CPW compliant, as shown in [18]. Indeed, most of the electric field is stored between the CPW strips and the floating strips. So, no electric field flows through the bulk silicon, leading to negligible substrate losses. In the counterpart, losses in the floating shield are not taken into account in the conventional RLCG model, whereas they are important in advanced CMOS technologies since the thickness of the metal layers used for the S-CPW shielding ribbons is only hundreds of nanometers. A physical compatible model was presented in [16]. This complex topology is the result of a non-optimal structure. The model proposed is not predictive since it depends on many fitting parameters. A better tentative of modeling shielded transmission line was presented in [17]. The model is based on a time domain approach. Even if this model considers losses, they are not based on the physical interpretation. Furthermore, losses in the shielding strips are not considered. On the other hand, a physics based electrical model with a RLRC topology was presented in [18], with a deep understanding of the physics. Although it gives a detailed model, closed forms formulas for electrical parameter estimation were not carried out. In [19], the au-

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Fig. 2. RLRC electrical equivalent model for S-CPW.

Fig. 1. A 3D schematic view of the S-CPW topology in CMOS or Bi-CMOS technologies.

thors presented a brief explanation of the parametric model. The modeling of the inductance and capacitance were briefly described, but the resistances' modeling was not addressed. In this work, an accurate and detailed formulation for each element of the RLRC model proposed in [18] is developed. It is based on the assumption of Quasi-TEM propagation mode. The developed equations are wide band and accurate in all current CMOS/BiCMOS technologies without the need of fitting factors. They are mostly based on physical models already presented in the literature, refined for particular S-CPW needs. In Section II the RLRC topology is briefly described. The model formulation is then developed in Section III. Results are presented in Section IV and compared with measurements and 3D-EM simulations for several CMOS/BiCMOS technologies. Section V presents two practical applications of the model for design optimization purposes. Finally, discussion and conclusions are given in Section VI. II. RLRC MODEL A S-CPW is composed of a conventional CPW transmission line loaded by a patterned shield, which is electrically floating (Fig. 1). In CMOS/BiCMOS technologies, the periodic shield is implemented below the CPW strips. Therefore, the electric field is trapped in between the CPW strips and the shield preventing conductive losses in the bulk silicon. In contrast, the magnetic field is not perturbed due to its conservative properties. The RLRC model shown in Fig. 2 was proposed in [18], except the strip-to-strip capacitance that was added here in order to consider cases with small gaps too. It consists of an inductance due to the current propagation in the CPW, a capacitance related to the capacitive effect between the due to CPW strips and the shield, a coupling capacitance the electric coupling between the signal and the ground strips, created by the current flowing in the floating shield, and resistances. The latter reflect the conductive losses in the CPW , and the strips , the conductive losses in the shield . In contrast eddy current losses in the patterned shield with the RLCG model, this model does not include conductance since the electric field does not penetrate the bulk substrate, and losses in the insulating layers between the metals of the BEOL are negligible compared to conductive losses. So far, to extract the component values, two simulations were still needed [18]: an electric and a magnetic simulation. The

Fig. 3. Photography of a S-CPW fabricated in 28-nm CMOS STMicroelectronics technology.

electric simulation allows and calculation, whereas the magnetic simulation gives , and . Hereinafter, will not be considered because we do not have accurate model in the literature. Eddy currents are negligible on a well-designed S-CPW up to more than 100 GHz. However, they should be taken into account at higher frequencies. From these components, the parameters of the transmission line can be calculated, e.g., propagation constant and characteristic impedance. All the required equations are detailed in [18]. In the following, the study focuses on developing accurate parametric formulations for all the model components (except in order to avoid electromagnetic simulations and, thus, allowing fast design optimization, as for microstrip lines or conventional CPW. III. QUASI-TEM ANALYSIS The work reported in [19] presented a brief explanation of the parametric model and gave the method for the calculus of the inductance and the capacitance , respectively. In this section, a much more detailed derivation of each model component is exposed. In particular the calculus of the resistances modeling losses is addressed. The development of the scalable predictive model is based on the assumption of Quasi-TEM propagation mode. This assumption allows inductive and capacitive elements to be calculated independently. In similar fashion, conductive losses are splitted in the CPW strips and in the patterned shield. This approach allows a better understanding of how the S-CPW's geometries influence each electrical component in the model. A. Inductance Inductance is created by the magnetic field generated by current flowing in the central strip and ground. Currents flowing in the floating shield do not play any role in the inductance as they are flowing perpendicularly to the propagation direction. Therefore, adding a floating shield to the CPW does not affect the inductance. Based on this fact, classical CPW's inductance formulation can be used without lack of accuracy.

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Fig. 5. Cross-section showing the electric field lines considered for the capacitance calculus. Fig. 4. Inductance of two S-CPWs fabricated in 28-nm CMOS STMicroelectronics technology. Model presented in dashed line. Measurements presented in continuous line.

Hence, inductance segmentation approach proposed in [20] is used. This method divides the spectrum in four domains. When the frequency increases, the first domain is limited by the frequency at which currents approximate to DC distribution. The second limit is associated to the frequency at which the internal current distribution in the center becomes uniform. The last limit is fixed by the skin-effect. In Fig. 4, this method is compared to the measurements of two S-CPW transmission lines implemented in the 28-nm CMOS STMicroelectronics technology. The inductance was extracted from the measured parameters by using the well-known relation . It can be noticed that the prediction is excellent in the whole band of interest (DC to 110 GHz). Resonances appearing in the measurement results are due to the long electrical length of the S-CPW, because of the slow-wave effect. They correspond to electrical lengths which are multiples of a half wavelength. However, except resonances, this result validates the calculus of the S-CPW inductance by using the formulation used for CPW. B. Capacitance The estimation of the S-CPW capacitance is an issue. So far, few attempts of capacitance evaluation have been reported. For example, the work reported in [14] exhibits a weak understanding of the S-CPW that leads to a misinterpretation of the electric field behavior. Effectively, the fitted equations describe a null shielding effect. In the work presented in [17], authors have clearly neglected the capacitance estimation. Consequently, the characteristic impedance is misreckoning when implementing the patterned shield in different BEOL levels. To the best author's knowledge, the S-CPW capacitance estimation was presented for the first time in [19]. However, just a brief of the methodology was exposed. The current work presents an extended and deeper explanation. In addition, the designed rule for preventing electric field leakage is explained in detail. The capacitance estimation is based on the work presented in [21]. As shown in Fig. 5, specific field lines topology is assumed, depending on the considered region. In [21], the capacitance is calculated by solving integral equations, whereas in our

Fig. 6. Capacitance convergence for m.

m,

m and

case it is calculated with a sum for more flexibility. The upper point charge capacitance of [21] is not considered here since it is not based on correct physical interpretation. Moreover, the estimation of the lower point charge capacitance is different. Then, the capacitance is separated in four regions (Fig. 5), named bottom plate, point charge, fringe, and upper plate capacitances. Notice that signal-to-ground and leakage electric fields through the floating shield are neglected in this first step. For each region, the electric field is discretized in electric field lines created by point charges. Accordingly, the capacitance at each region is given, in the general case, by the sum of these electric field lines (1) (1) is the dielectric constant in vacuum, is the relawhere tive dielectric constant, is the width of the conductor, is its length, and is the height or distance between conductors. This generalization simplifies the calculation for each region and the problem becomes just a matter of: (i) defining the path described by each electric field line, and (ii) choose the number of field lines that must be considered for proper convergence. Fig. 6 shows the capacitance error in function of the number of electric field lines . Here, the capacitance is calculated for different values of . Then, when is large enough, the computed capacitance remains constant. At this point the convergence is achieved. From this convergence study, as a rule of thumb is set to 100.

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Fig. 7. Fringe region. Path description of the electric field distribution for the fringe capacitance calculus.

Fig. 9. Upper plate region. Path description of the electric field distribution for the upper plate capacitance calculus.

The electric field going from the top of the strip towards the floating shield creates the region called upper plate (Fig. 9). This region contributes less in the total capacitance, however it is not negligible. The electric field lines describe a circular path, with a radius going from to (6) (6)

Fig. 8. Angle point charge region. Path description of the electric field distribution for the fringe capacitance calculus.

The bottom plate region has a constant height, thus (1) becomes the well-known plate capacitance (2), when considering the whole bottom plate width

The radius can be expressed as (7) Therefore, the upper plate capacitance is given by (see Appendix C)

(2) (8) For fringe and upper plate regions, the formulation is based on the assumption that the electric field lines describe a circular path as illustrated in Fig. 7. In the fringe region, the height ( depends on the radius, so is computed as follows: (3) substituting (3) instead of in (1) and as the width, the fringe capacitance becomes (see Appendix A) (4) A particular case is the angle point charge capacitance (Fig. 8), named . This capacitance is created by the electric field lines at the bottom edges of the strips. The electric field lines describe a parabolic path and they are delimited by the height ( . By considering this trajectory and substituting in (1), this capacitance can be approximated by (5). Note that is independent of (see Appendix B) (5)

Finally, the total signal strip capacitance is computed by the sum of the contribution of all regions (9) Fig. 10 shows the comparison between the herein proposed model and the model proposed in [21], when only the capacitance linked to the signal strip is considered. In this case the capacitance is the same as the microstrip one. Then, the electromagnetic simulation was performed by considering a conventional microstrip line. When the main contribution of the signal strip relies in the plate capacitance, i.e., small heights or big widths, the influence of the upper plate is lower, therefore the error in both methods are less than 5%. However, when the influence of the upper plate capacitance starts to be important, the method proposed in [21] overestimates the capacitance and the error increases up to 20%, whereas the error is still lower than 5% with the proposed formulation. Since the floating shield ends at the external edge of the ground, the external electric field behaves slightly different from the one previously discussed (Fig. 5). Therefore, for computing the capacitance created between the ground and the floating shield, the four regions are separated into internal and

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TABLE I CAPACITANCE ZONES

STRIP-TO-STRIP

Fig. 10. Signal strip capacitance

estimation versus dioxide height.

Fig. 12. Electric field lines in zone B.

Fig. 11. Ground capacitance estimation.

external. The internal capacitances are the same than the signal capacitances, the same equations can be safely used. External electric field lines describe circular paths and they are segmented in the same different regions. Then, the aforementioned methodology applies for defining the path that each electric field line describes. It has been assumed that all electric field lines are centered in O (Fig. 5). Therefore, finding the of each external region is a trivial task. Fig. 11 shows the importance of computing the external capacitance contribution. When the lower plate and the floating shield are close, the electric field is mainly stored between them: the main contribution is given by the plate capacitance and the external electric field contributes less in the total capacitance. As a consequence, the external electric field can be neglected for smaller . In the other hand, as the distance increases, the electric field is distributed proportionally among all regions: plate capacitance decreases and the total capacitance starts to depend upon all regions. Therefore, the external relative contribution increases, leading to a misestimating of the total capacitance if the external electric field is not considered. The accuracy of the model is also depicted in Fig. 11. While considering the contribution of the external electric field, the error remains lower than 5% for all heights. This confirms the hypotheses of the electric field behavior. So far, strip-to-strip capacitance and leakage through the shield have not been taken into account. In practical cases, the strip-to-strip capacitance is negligible for bigger

than , but this is no longer the case when . In order to calculate , four zones are considered, in function of , as indicated in Table I. In each zone it is assumed that the electric field lines tend to take the shortest path to the ground. Zone A is delimited for bigger than two times , leading to a neglegible . In other words, when this conditions holds true, all the electric field lines of the upper plate go directly towards the floating shield, and there is no direct coupling between signal strip and ground strip. As soon as the gap is smaller than two times (Zone B), the electric field lines are splitted into two different directions: a part of the electric field lines of the top plate goes directly to the ground strip and the other part goes towards the floatting shield (Fig. 12). Therefore, the width of the upper plate ( in (8) has to be substituted by the effective width : (10) The electric field lines that go from the signal strip to the . This capacitance ground strip, lead to the capacitance can be approximated by using the fringe capacitance (4), when replacing by , and by . Then the strip-to-strip capacitance is equal to (11) with (12)

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Fig. 14. Electric field lines in zone D.

Fig. 13. Electric field lines in zone C.

(13) conFig. 13 depicts the behaviour in zone C. Here, the sidered in Fig. 12 is zero since all the electric field lines of the upper plate region are going from the top plate of the signal strip directly to the top plate of the ground strip. Therefore, in is equal to . In addition, the thickness considered to calculate is modified, since a portion of the electric field lines goes straight from the signal strip to the ground strip. This portion creates a capacitance named . is the sum of and , where

Fig. 15.

coupling capacitance estimation.

(14) (15) where and

is the width and are equal to

is the height.

(22) Then (23)

(16) where (17) For estimating stitued by (18) and

and is calculated as (19)

(24)

is sub-

(18) (19) The zone D (Fig. 14) considers that (4) and (8), as defined in Figs. 12 and 13, are zero, since all the electric field lines in these regions go directly to the ground. Accordingly, the width in is equal to . In this zone, part of the electric field lines that leads to the capacitance goes direclty to the ground. Therefore, and become (20) (21)

Fig. 15 shows the comparison of a simulation carried out with HFSS, with the proposed model. The height was kept constant while the gap varies from 5 m to 28 m, which correspond to usual values, and for two different cases. It can be seen that the coupling capacitance increases with the reduction of the gap. The model accurately predicts the capacitance along all variations of the gap with an error lower than 5%. So far, the capacitance created between the bottom plate and the floating shield has been approximated as a single element . This approximation is almost true if the floating shield is well designed so that no electric field lines leak to the substrate. However, this is not always the case, and the model can be improved. To consider this leakage, the electric field is separated in different regions, as shown in Fig. 16. For SS smaller than , the electric field will be completely confined between the strip and the floating shield. If SS in-

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Fig. 18. Photography of the S-CPW fabricated in the BiCMOS 130-nm STMicroelectronics technology. Fig. 16.

capacitance advanced estimation.

C. Losses There are three main contributions of losses as previously explained: conductive losses in the CPW strips, conductive losses in the floating shield and losses created by eddy currents flowing in the floating shield. The latter are not considered since no physical model is really consistent. However, they can become non negligible when the working frequency is greater than about 100 GHz, depending on the S-CPW geometry. Conductive losses in the CPW are modeled by the method proposed in [20]. As already stated, this method holds as the magnetic field in the S-CPW and conventional CPW behaves almost identically. Conductive losses in the floating shield are simply modeled as (26) where is the shield period distance between the ground strips

, and D is the .

D. Floating Shield: Inductance The current flowing in the shield ribbons introduces an inductive effect . This inductance has to be taken into account when wide gaps and/or high frequencies are considered [23]. This inductance is approximated by Fig. 17. (a) Capacitance estimation in function of . tenuation constant in function of

, (b) Normalized at-

(27)

creases beyond this limit, the electric field leaks to the substrate. This can be seen by the drop in the total capacitance as SS increases (Fig. 17(a)). The leaked field penetrates the lossy substrate causing an increase in the losses. This behaviour is illustrated in Fig. 17(b), where the normalized attenuation constant was extracted from HFSS simulations carried out for different ratios. The normalized attenuation constant was calculated as follows: the attenuation constant in function of the ratio is divided by the attenuation constant when there is no electric field leakage to the substrate (25). An important rule can be derived: a ratio greater than 0.5 is needed in order to ensure that leakage represents less than 5% of the total attenuation constant value

(25)

where must be extracted for each technology. As explained in [23], the value of has been fixed by fitting several measurement results and taking the average value. IV. RESULTS Fig. 18 shows the fabricated S-CPWs in the BiCMOS 130-nm technology from STMicroelectronics. In Fig. 19 the comparison between the model and the measurements results are presented. The model was also validated on the 28-nm (Table II), and 55-nm from STMicroelectronics. The mm-wave measurement setup is presented in Fig. 20. In the results presented in Fig. 19, three different topologies were implemented in the BiCMOS 130-nm technology, with high, low and medium (50 ) characteristic impedance, respectively. The S-CPWs were implemented using different metal layers. The names of the S-CPWs are given as follows:

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Fig. 20. Photography of the measurement set-up.

TABLE II MEASUREMENTS VERSUS MODEL RESULTS AT 60 GHZ FROM VARIOUS TECHNOLOGIES

V. REAL USER CASES Fig. 19. Comparison between the model and the measurement results. (a) Characteristic impedance, (b) effective dielectric constant, and (c) attenuation constant (measurement results in black, model in gray).

“CPW M6M5 Shield M3” means that the CPW strips were fabricated by the stack of metal 5 and metal 6 layers, whereas the floating strips were fabricated in Metal 3 layer. Table II presents the comparison between the model and the measurements at 60 GHz, for S-CPWs having different characteristic impedances implemented in the 28 nm from STMicroelectronics, 0.35 m AMS, and 0.13 m and 0.25 m technology from IHP, respectively. The purpose of these two comparisons is to show that the model accurately predicts the S-CPW's performance when using different geometries implemented in different BEOL and different metal layers from various technologies. In all cases, the prediction error remains lower than 10% (for all the parameters, i.e., the characteristic impedance, effective dielectric constant, and attenuation constant).

This section provides the reader with practical cases to exemplify how the model can be exploited. Two cases are shown: the first addresses the optimization of a 50 S-CPW with electrical length of 90 while the second allows the user to be directly involved with the RLRC model. A. 50

S-CPW Optimization

There are plenty of circuits that require transmission lines. For example power dividers, branch line couplers, etc. Consequently, this section presents the case where a quarter wavelength 50- S-CPW is needed. Due to the topology (geometries and metal layers), any S-CPW will have a lot of different ways to be implemented. Therefore, designers have to firstly set out the requirements, in terms of performance and/or surface. These requirements will define the geometries that fit the best the S-CPW. For example Table III presents three different solutions for a 50- S-CPW in the BiCMOS 55-nm technology. In the first case, the transmission line is selected to reach the highest

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50

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TABLE III S-CPW PHYSICAL AND ELECTRICAL (60 GHZ) PARAMETERS FOR THREE DIFFERENT DESIGN REQUIREMENTS

quality factor achievable in the given technology. In the second case, the transmission line length has been minimized , and the last one is the less bulky (Surface min). The quality factor at 60 GHz, physical length and surface area are given in Table III, highlighting the different targeted goal. A maximum Q factor of 50 can be reached for an overall surface of 0.058 mm , whereas a minimum surface equal to 0.023 leads to a Q-factor limited to 20. The importance of showing these three different cases is not to present a set of rules but to show the flexibility of the model. The model allowed obtaining these results in a matter of seconds. When comparing to the developing time of any 3D EM, that presents a clear advantage. In addition, the user can set up any desired goal for any desired S-CPW.

Fig. 21. Phase shifter electrical model.

B. Tunable Phase-Shifter The synthesis tool allows the user to manipulate directly the RLRC model in a CAD tool (like Cadence for example). To illustrate this point, a S-CPW based tunable phase shifter is considered (see Fig. 21(a)). Here, the floating shield is broken between the signal and the ground, to allow connecting varactors in between. As a consequence, the total S-CPW's capacitance is variable and is equal to (28) is the varactor's capacitance value and is the parwhere asitic capacitance coming from the coupling between the two extremities of the broken floating shield. The equivalent model is shown in Fig. 21(b). Notice that the other elements in the electrical model remain unaltered. The variation in the total capacitance modifies the phase constant of the transmission line. Thus, the absolute S-CPW's phase can be changed. Normally, for designing this type of tunable phase shifter, a 3D EM simulation tool is needed, together with a deep understanding of the S-CPW's physical behavior, and round-trips between 3D-EM tools and CAD tool (Cadence for example) are necessary. This procedure drives to long developing times. The model proposed in this paper allows the designer to make a quick optimization by scripting it directly in any CAD tool. In this example, a 30 tunable phase shifter at 60 GHz working frequency was designed, while targeting the maximum allowed Figure of Merit (FoM) defined by the ratio of , and the maximum insertion the phase shift in degrees (29). First, the varactors where chosen to loss in dB

Fig. 22. Photograph of the phase shifter, with DC control pads on each side of the phase shifter.

have the maximum tuning range (TR) given by the technology ( in the 55-nm technology from STMicroelectronics) with a quality factor of 8. Once the varactor has been chosen, the next step is to find the geometries and stack that gives the desired phase shift with the maximum FoM. By using the model and replacing the equivalent capacitance of the S-CPW by (20), it is possible to make a quick parametric analysis for finding the desired results. (29) The phase shifter was then fabricated in the 55-nm technology from STMicroelectronics (Fig. 22). The performance of the phase shifter is shown in Fig. 23, where measurement and simulation results (obtained from the model proposed in this paper) are compared. Fig. 23(a) shows the maximum phase shift versus frequency. It is possible to see that the model accurately predicts the behavior across the whole frequency band. Fig. 23(b) shows the . The FoM is shown in phase shift versus the tuning voltage Fig. 23(c). Measurement and simulation results show very good agreement. VI. CONCLUSION In this work, a full parametric electrical model, which is derived from physical analysis, is proposed for S-CPW transmis-

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APPENDIX A For computing the fringing capacitance, it has been considered that the electric field lines describe a quarter of circle. Then (3) can be rewritten as (A-1) where

is (A-2)

Substituting (A-1), (A-2) in (1), and the width being equal to the thickness of the strip ( the fringe capacitance is (A-3)

APPENDIX B For deriving the angle point charge capacitance, it is considered that the electric field lines describe a parabolic path and they are delimited by the height . Then, by taking the perimeter of a parabola we obtain the path that each electrical field line describes

(A-4) Substituting (A-4) in (1) and considering the width as angle point charge capacitance becomes

, the

(A-5) and (A-5) can be approximated as (A-6) Fig. 23. Phase shifter measurements versus model results: (a) Maximum phase shift,(b) phase shift as a function of the control voltage at 60 GHz, and (c) FoM.

sion lines. This tool was validated in several CMOS technologies: 0.35 m AMS, 0.25 m IHP, 0.13 m STMicroelectronics, 0.13 m IHP, 55-nm STMicroelectronics, and 28-nm STMicroelectronics. It successfully estimates the behavior of the S-CPW, without the need of any fitting or correction parameter. Even more, it allows fast optimization without the need of 3D-EM simulators. The tool was implemented in MATLAB and can be provided to any user by request. This will help designers to deal with the several criteria of S-CPW, i.e., characteristic impedance, electrical performance (Q factor), physical length (effective dielectric constant), and total width.

APPENDIX C The same approach as is used to derive the upper capacitance. Here, it is considered that the electric field describes a circular path (A-7) By substituting (A-7) in (1) and the width with plate capacitance is

the upper

(A-8)

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ACKNOWLEDGMENT The work presented here has been performed in the RF2THZ SiSoC project of the EUREKA program CATRENE, in which the French partners are funded by the DGCIS. REFERENCES [1] J. J. Lee and C. S. Park, “A slow-wave microstrip line with a high-Q and a high dielectric constant for millimeter-wave CMOS application,” IEEE Microw. Wireless Componen. Lett., vol. 20, no. 7, pp. 381–383, Jul. 2010. [2] A.-L. Franc, E. Pistono, N. Corrao, D. Gloria, and P. Ferrari, “Compact high-Q, low-loss transmission lines and power splitters in RF CMOS technology,” in Proc. Int. Microwave Symp., Baltimore, MD, Jun. 7–9, 2011, pp. 1–4. [3] K. Kim and C. Nguyen, “An ultra-wideband low-loss millimeter-wave slow-wave wilkinson power divider on 0.18 SiGe bicmos process,” IEEE Microw. Wireless Componen. Lett., vol. 25, no. 5, pp. 331–333, May 2015. [4] T. Cheung and J. R. Long, “Shielded passive devices for silicon-based monolithic microwave and millimeter-wave integrated circuits,” IEEE J. Solid-State Circuits, vol. 41, no. 5, pp. 1183–1200, May 2006. [5] I. C. H. Lai, Y. Kambayashi, and M. Fujishima, “60-GHz CMOS down-conversion mixer with slow-wave matching transmission lines,” in Proc. IEEE Asian Solid-State Circuits Conf., Nov. 2006, pp. 195–198. [6] A.-L. Franc et al., “High-performance shielded coplanar waveguides for the design of CMOS 60-GHz bandpass filters,” IEEE Trans. Electron Devices, vol. 59, no. 5, pp. 1219–1226, May 2012. [7] M. Kärkkäinen, D. Sandström, M. Varonen, and K. A. I. Halonen, “Transmission line and lange coupler implementations in CMOS,” in Proc. 5th European Microwave Integrated Circuits Conf., Paris, France, Sep.–Oct. 26–1, 2010, pp. 357–360. [8] J. Lugo, A. Bautista, F. Podevin, and P. Ferrari, “High-directivity compact slow-wave CoPlanar waveguide couplers for millimeter-wave applications,” in Proc. EuMC, Oct. 2014, pp. 1072–1075. [9] T. LaRocca, S.-W. Tam, D. Huang, Q. Gu, E. Socher, W. Hant, and F. Chang, “Millimeter-wave CMOS digital controlled artificial dielectric differential mode transmission lines for reconfigurable ICs,” in Proc. IEEE Int. Microw. Symp., Atlanta, USA, Jun. 15–20, 2008, pp. 181–184. [10] X. Tang, E. Pistono, P. Ferrari, and J.-M. Fournier, “Enhanced performance of 60-GHz power amplifier by using slow-wave transmission lines in 40 nm CMOS technology,” Int. J. Microw. Wireless Technol., vol. 4, pp. 93–100, Feb. 2012. [11] M. Varonen, M. Kärkkäinen, M. Kantanen, and K. A. I. Halonen, “Millimeter-wave integrated circuits in 65-nm CMOS,” IEEE J. Solid-State Circuits, vol. 43, no. 9, pp. 1991–2001, Sep.. 2009. [12] T. La Rocca, J. Y.-C. Liu, and M.-C. F. Chang, “60 GHz CMOS amplifiers using transformer-coupling and artificial dielectric differential transmission lines for compact design,” IEEE J. Solid-State Circuits, vol. 44, no. 5, pp. 1425–1435, May 2009. [13] S. Seki and H. Hasegawa, “Cross-tie slow-wave coplanar waveguide on semi-insulating Ga-As substrates,” Electron. Lett., vol. 17, no. 25, pp. 940–941, Dec. 1981. [14] T. Masuda, N. Shiramizu, T. Nakamura, and K. Washio, “Characterization and modeling of microstrip transmission lines with slow-wave effect,” in Proc. Topical Meeting on Silicon Monolithic Integrated Circuits in RF Syst., Orlando, FL, Jan. 23–25, 2008, pp. 155–158. [15] A. Sayag, D. Ritter, and D. Goren, “Compact modeling and comparative analysis of silicon-chip slow-wave transmission lines with slotted bottom metal ground planes,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 4, pp. 840–847, Apr. 2009. [16] I. C. H. Lai, Y. Kambayashi, and M. Fujishima, “Characterization of high Q transmission line structure for advanced CMOS processes,” IEICE Trans. Electron., vol. E89-C, no. 12, pp. 1872–1879, Dec. 2006. [17] L. F. Tiemeijer, R. M. T. Pijper, R. J. Havens, and O. Hubert, “Low-loss patterned ground shield interconnect transmission lines in advanced IC process,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 3, pp. 561–570, Mar. 2009.

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[18] A.-L. Franc, E. Pistono, G. Meunier, D. Gloria, and P. Ferrari, “A lossy circuit model based on physical interpretation for integrated shielded slow-wave CMOS coplanar waveguide structures,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 2, pp. 754–763, Feb. 2013. [19] A. Bautista, A. Franc, and P. Ferrari, “An accurate parametric electrical model for slow-wave CPW,” in International Microwave Symposium 2015, Phoenix, USA, May 17–22, 2015, pp. 1–4. [20] W. Heinrich, “Quasi-TEM description of MMIC coplanar lines including conductor-loss effects,” IEEE Microw. Theory Techn., vol. 41, no. 1, pp. 45–42, Jan. 1993. [21] W. Zhao, X. Li, S. Gu, S. H. Kang, M. Nowak, and Y. Cao, “Field-based capacitance modeling for sub-65 nm on-chip interconnect,” IEEE Trans. Electron Devices, vol. 56, no. 9, pp. 1862–1872, Sep. 2009. [22] A.-L. Franc, E. Pistono, and P. Ferrari, “Design guidelines for high performance slow-wave transmission lines with optimized floating shield dimensions,” in Proc. Eur. Microw. Conf., Paris, France, Sep. 28–30, 2010, pp. 1190–1193. [23] A.-L. Franc, E. Pistono, and P. Ferrari, “Dispersive model for the phase velocity of slow-wave CMOS coplanar waveguides,” in Proc. 45th EuMC, Paris, France, Sep. 6–11, 2015.

Alfredo Bautista was born in Victoria, Mexico, in 1980. He received the M.Sc. degree from the Instituto Tecnológico y de Estudios Superiores de Monterrey, Monterrey, Nuevo León, México, in 2005 and the Ph.D. degree from the Joseph Fourier University, Grenoble, France in 2009. From 2010 to 2012, he held a postdoctoral position in the CEA-LETI, Grenoble, France, and from 2012 to 2015, in the INP, Grenoble, France. His current technical interests focus in modelling passive devices for mm-wave, design of phase-shifter, VCOs, and analog integrated circuits.

Anne-Laure Franc was born in France in 1985. She received the engineer and M.Sc. degrees from the Cergy-Pontoise University, Paris, France, in 2008 and the Ph.D. degree from Grenoble University, Grenoble, France, in 2011. In 2012, she had a postdoctoral position with Darmstadt University, Darmstadt, Germany and she held a temporary lecturer and research assistant position in the Grenoble-Alpes University, Grenoble, France in 2013. Since September 2013, she has been an Associate Professor with the University of Toulouse, Toulouse, France. Her research interests focus on tunable microwave components.

Philippe Ferrari (SM’03) received the Ph. D. degree from INP Grenoble, Grenoble, France, in 1992. He joined the University of Savoy, France, as an Assistant Professor, and was involved in the development of RF characterization techniques and nonlinear transmission lines. Since 2004, he is a Professor at the Grenoble-Alpes Universitym Grenoble, France. His main research interests concern tunable devices, and new circuits based on slow-wave transmission lines. He is author or co-author of more than 180 international papers, and co-holder of five patents. He is a member of the editorial board of the International Journal on RF and Microwave Computer-Aided Engineering, and an Associate Editor of the International Journal of Microwave and Wireless Technologies.

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High-Speed Antenna-Coupled Terahertz Thermocouple Detectors and Mixers Johannes A. Russer, Member, IEEE, Christian Jirauschek, Member, IEEE, Gergo P. Szakmany, Mark Schmidt, Alexei O. Orlov, Gary H. Bernstein, Fellow, IEEE, Wolfgang Porod, Fellow, IEEE, Paolo Lugli, Fellow, IEEE, and Peter Russer, Life Fellow, IEEE

Abstract—An antenna-coupled nanothermocouple (ACNTC) is an integrated structure consisting of a dipole nanoantenna and a nanothermocouple (NTC). ACNTCs are excellently suited as polarization-sensitive detectors and mixers for the long-wavelength far-infrared range around 30 THz. Radiation collected by the integrated nanoantenna and fed into the hot junction creates a temperature difference between the hot and cold junctions of the thermocouple, which results in open-circuit voltage due to the Seebeck effect. Due to the geometry-dependence of the Seebeck coefficient in nanowires, we realize single-metal ACNTCs. The fundamentals of single-metal NTCs are discussed. The thermal dynamics of NTCs is investigated showing that NTCs could exhibit mixer and detector intermediate frequency and low frequency cutoffs beyond 100 GHz. We provide experimental evidence of ACNTCs. Index Terms—Heat conduction, Seebeck effect, terahertz (THz) detector, thermocouple (TC).

I. INTRODUCTION

T

HERMOCOUPLEs (TCs) are excellently suited as detectors for the long-wavelength far-infrared range around 30 THz. The detector operation of TCs is based on Joule heating of the TCs by the incident radiation and the Seebeck effect, i.e., the thermoelectric effect that directly converts a temperature difference to an electric voltage [1], [2]. The TC, like any thermodynamic device, is subject to the restrictions of Carnot's theorem stating that the efficiency of any heat engine operating beis tween two heat reservoirs at absolute temperatures [3]. In limited to a maximum value of order to enhance the efficiency of the TC, the radiation to be detected should be collimated within a small area of a nanothermocouple (NTC) so that the electron temperature at the NTC junction gains a strong increase. In this work, we show that NTCs are extremely fast detectors with response times in the picosecond area. This property will Manuscript received July 01, 2015; revised September 22, 2015 and October 22, 2015; accepted October 23, 2015. Date of publication November 12, 2015; date of current version December 02, 2015. C. Jirauschek thanks the Deutsche Forschungsgemeinschaft for funding under Project DFG JI 115/4-1. G. P. Szakmany gratefully acknowledges financial support from the Notre Dame Joseph F. Trustey Fund for Postdoctoral Scholars. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium Phoenix, AZ, USA May 17–22, 2015. J. A. Russer, C. Jirauschek, M. Schmidt, P. Lugli, and P. Russer are with the Institute for Nanoelectronics, Technische Universität München, 80333, Munich, Germany. G. P. Szakmany, A. O. Orlov, G. H. Bernstein, and W. Porod are with the Center for Nano Science and Technology, University of Notre Dame, Notre Dame, IN 46556 USA.. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2496379

open new areas of applications in the terahertz (THz) frequency bands, so far accessible only in the microwave and millimeterwave bands so that the THz bands will gain additional interest from system engineers so far mainly concerned with applications up into the millimeterwave region only. Fast NTC detectors may be employed in imaging, communications and sensing applications. Currently, TCs are commonly used in applications for which fast response times are not critical, and cut-off frequencies of some kHz and below are sufficient, and they receive little consideration as elements in RF or THz engineering. However, fast NTCs have also a rather interesting potential for applications in microwave engineering. High efficiency is desirable in communications and sensing applications since it governs the sensitivity of the device. Antenna-coupled nanothermocouples (ACNTCs) are integrated structures consisting of a dipole nanoantenna and a NTC connected to the antenna feed [4]. The nanoantenna collimates the IR energy incident within the effective aperture of the antenna and feeds it into the thermocouple. The dipole nanoantenna yields polarization-sensitive detection. This induces a temperature difference between the hot junction at the nanoantenna feed and the remote cold junction. Due to the Seebeck effect, an open-circuit voltage arises across the antenna feed port. The hot junction of the NTC is located at the center of the dipole-antenna where the antenna current exhibits its maximum, resulting in optimum device response. In [5], we have shown that NTCs can also be formed by two nanowires of the same metal but different cross-sectional areas. Compared with conventional bi-metallic TCs, fabrication of such shape-engineered mono-metallic nanowire TCs is much easier, and mass-production of mono-metallic TCs could be accomplished by simple manufacturing technologies. Since the NTC junction exhibits transverse dimensions of a few tens of nanometers and the TC junction is metallic without depletion layer, there would be a frequency limitation due to a junction capacitance at the RF side in case of an ideal broadband nanoantenna shape. Due to the non-perfect conductivity of the antenna and the finite antenna cross-sectional area at the feed point there will be some cutoff frequency, however somewhere in the far-infrared region. Since the Seebeck effect is conveyed by Joule heating, the TC is essentially a square-law detector. Due to the small thermal volume of the TC hot junction, the detector and mixer time-constant is on the order of picoseconds. Therefore, we can expect extremely high bandwidths in detector and mixer applications. In [6], we discussed the possible application of ACNTCs as detectors in far-infrared sensor and communication systems. The dynamics of ACNTCs has been investigated in [7]–[9] and it was shown there that due to its low thermal capacity, an ACNTC is an extremely fast square-law detector with cutoff frequencies up to several hundred GHz.

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RUSSER et al.: HIGH-SPEED ANTENNA-COUPLED TERAHERTZ THERMOCOUPLE DETECTORS AND MIXERS

In this extended paper, we also provide experimental results confirming response times of ACNTC in the RF regime. In our investigations, we expand the theoretical model for the Seebeck coefficient of single-metal thermocouples and on the modeling framework for ACNTC in THz detector and mixer applications. In Section II we describe the properties of single-metal nanothermocouples (SMNTCs) and discuss the theoretical fundamentals of SMNTCs on the basis of the Seebeck theory and the Fuchs-Sondheimer model. The fabrication of ACNTC is described in Section III. In Section IV, the thermal diffusion in the thermocouple is modeled and compact lumped-element equivalent-circuit models are established describing the electric and thermal behavior of ACNTCs. The pulse-response for the NTC is computed. In Section V, the detector and mixer properties of ACNTCs are investigated on the basis of the lumped element equivalent circuit models of ACNTCs. Detection measurements laser operating at 10.6 are presented with a modulated in Section VI. II. THEORETICAL MODEL FOR THE SEEBECK COEFFICIENT OF A SINGLE-METAL THERMOCOUPLE TCs composed by two conductors and with Seebeck and , containing two junctions show an opencoefficients circuit Seebeck voltage (1) if there is a temperature difference between the two junctions. Usually thermocouples are made from two different metals exhibiting different Seebeck coefficients. Demodara et al. [10] have shown that in thin films the reduction of the electron mean free path by additional scattering yields a thickness dependence of the Seebeck coefficients. The calculation of this geometry-dependent Seebeck coefficient is essential in order to determine the efficiency of single metal thermocouples, as used for detecting radiation in the terahertz regime from a nanoantenna [4]. As compared to thin films, thin wires that are used for the NTCs only have two additional surface scattering planes; thus, the size effect can be expected to be a scaled problem. Calculations for the conductivity of a thin wire were done by Dingle [11] and MacDonald and Sarginson [12] based on the Fuchs-Sondheimer model. In the following, we first use this model to investigate the Seebeck coefficient of thin films rather than wires due to more available experimental data for films. Afterwards we provide a formula for the Seebeck coefficient of wires based on the conductivity model of Dingle, and use it to confirm our measured results. Elaborate microscopic models for the theoretical calculation of the Seebeck coefficient have been developed [13]–[18]. However, here our focus is on more-compact, semi-empirical models that provide insight into the geometric dependence and serve as a guidance to experimentalists. For a monocrystalline thin metal film with thickness the Seebeck coefficient can be calculated in the framework of the Fuchs-Sondheimer model based on the first order solution of the linearized Boltzmann equation to [19]

(2)

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Here, is the (absolute) ambient temperature, the charge of the Fermi energy, the Boltzmann constant, an electron, with the electron mean free path of the bulk material

and

(3)

(4) The surface scattering is modeled by the specularity parameter [20], which is independent of the direction of motion of the electrons and the energy (a direction-dependent model was developed by Stoffer [21]). It is assumed that a fraction of electrons is reflected specularly at the boundary and still contributes of the electrons is to the current, whereas the fraction reflected diffusely and does not contribute to the current anyand , given by more [20]. The parameters (5a) (5b) describe the derivative of the electron mean free path of the bulk material and the area of constant energy at the Fermi energy. These parameters cannot be extracted from a compact model and have to be measured in addition to the transport parameters and ; thus, the measured values are highly dependent on the structure of the fabricated material and the production process [22]. Several investigations to determine those paand as shown rameters yielded contradicting values of below. It can be concluded that correct annealing is a significant factor in generating monocrystalline structures with a negligible amount of impurities [23]–[29]. All of these authors found significant effects in the change of the transport parameters when the film structure differed from a monocrystalline structure. The available data for the transport properties have therefore to be seen in a critical manner. An asymptotic approximation for relatively large films is obtained from (2) as [19] (6) It should be noted that the previous expressions refer to unsupported metal films, hence substrate effects are not considered. A discussion of such effects can be found in [30]. For practical purposes it is more convenient to express the Seebeck coefficient in terms of the temperature coefficient of resistance (t.c.r.) [31], yielding (7) where the subscripts and 0 refer to the film and the bulk material, respectively. Because the t.c.r. can be easily extracted from experiment, the determination of and is not necessary to obtain a value for . In addition, to eliminate the unknown it is advantageous to calculate the difference between the thin-film and the bulk-material Seebeck coefficients given by (8)

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FOR

AND ABSOLUTE

TABLE I FOUND BY DIFFERENT SOURCES OBTAINED BY THERMOPOWER MEASUREMENTS

with the Seebeck coefficient of the bulk material (9) This makes it possible to determine by interpreting as . can be found by two meaa linear function in and or from the slope of , given by surements of . The parameter can then be determined using (9). If the structure of the metal is not monocrystalline, the grain boundary scattering has to be taken into account. This is the case when the electron mean free path is equal to or smaller [19]. Models for grain than the average grain diameter boundary scattering were developed by Mayadas-Shatzkes [32] and Pichard, Tellier, and Tosser [33]. The Fuchs-Sondheimer model is derived from the Boltzmann equation, which is a semiclassical diffusive transport equation based on the free electron model. In this free electron model and [23]. Here, the Fermi surface is assumed to be spherical, and the isotropic relaxation time is constant over the Fermi surface. This is not the case for real metals such as silver, copper, or gold for which the Fermi surfaces are distorted at eight of the fourteen Brillouin zone planes. As expected, experimental results differ from the values predicted by the free electron model because of this deviation. Table I summarizes and for different metals. selected measured values for Copper, silver, and gold are chosen because these metals have a similar shape of the Fermi surface. and As mentioned above, the contradicting values of can be ascribed to differences in the production process of the of all the presented metals can film. The negative sign of be explained by the decrease in area of the distorted Fermi surfaces with increasing energy because of the contact zones of the Brillouin boundary [29]. By comparing the experimental data for silver in Fig. 1 and copper in Fig. 2 from Narasimha Rao, Mohan and Reddy with the exact Fuchs-Sondheimer model (2) and the thick film approximation (6), a qualitatively good match between theory and data can be observed. However, the weak slope of the curve of the exact calculation based on (2) produces a large error for a thickness below the electron mean free path. This implies that the Fuchs-Sondheimer model is an inadequate description of the very thin film behaviour. It is surprising that the thick film approximation curve shows a better agreement than the exact curve since it is . The opposite sign of for copper and only valid for silver is due to the different energy dependence of the electron mean free path.

Fig. 1. Difference of the Seebeck coefficient between the bulk material and a nm [29], , thin film with thickness for silver, [36], ; other parameters and measurement data from [27].

Fig. 2. Difference of the Seebeck coefficient for copper, nm, , [36], ; other parameters and measurement data from [25].

It should be pointed out that the Fuchs-Sondheimer model only provides a good match with experiment for extremely is a perfectly rough surface) according rough surfaces ( to [22]. But a more critical problem is the availability of a complete set of measurement data including all transport parameters for different metals providing consistent results. In summary the presented data fits relatively well the measurements of Narasimha Rao, Mohan, and Reddy for a monocrystalline film with a rough surface. For very thin films, a more accurate model is needed that also takes quantum effects into account to describe the size effect more precisely. In the case of wires, an approximate formula for thick wires (width and height greater than the electron mean free path) of arbitrary cross-section was developed by Dingle [11] for the , given by wire conductivity (10) is the perimeter and is the wire cross-section Here, area. By inserting above formula for the wire conductivity into the general expression for the Seebeck coefficient

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Fig. 3. Calculated Seebeck coefficient of the single-metal NTC as a function of width of the first wire segment for various widths of the second wire segment. is set to 17.22. The height is 45 nm,

and taking into account the energy dependence of the electron mean free path, we obtain (11) with the geometry factor (12) As expected, (11) can be simplified to (6) for films with thick. For single-metal NTCs with different wire cross nesses sections, one has to subtract the Seebeck coefficients of the two and , yielding single wires with different geometry factors (13) This formula is evaluated based on two available measurement results for Pd/Pd single-metal NTCs with a height of 45 nm and widths of 80 nm/470 nm [8] and 50 nm/200 nm (Section VI of this paper), respectively. The measured Seebeck coefficients for [8] and 0.86 , these two NTCs are respectively. , In the following, we assume the parameter values [37], , and . The material parameters and are taken from [38], but vary with the production process conditions as discussed above. From (13), and 11.66 by using the data of the first we obtain and second thermocouple, respectively. This uncertainty in the may again be partially due to the production exact value of process conditions, as also discussed above in the context of Table I. In addition, substrate effects and the fact that the wire segments are connected at a 90 angle at the hot junction, as discussed in Section III, affect the measurement results, but are not explicitly considered in the theoretical model. Thus, fitting (13) to the measurement results yields an effective value for , which also accounts for these effects. Fig. 3 shows the geometry dependence of the Seebeck coefficient, where we here take the . averaged value From Figs. 1, 2, and 3 it can be concluded that the efficiency of the NTC is maximized when a thin segment, with a thickness on the order of the electron mean free path, is connected to bulk material or to dimensions which are much greater than the electron mean free path. A large geometrical difference of the two segments leads to a large relative Seebeck coefficient and thus,

Fig. 4. Scanning electron micrograph of (a) an ACTC and (b) a thermopile constructed from several ACTCs connected in series.

according to (1), to a higher Seebeck voltage. Quantum size effects beyond the electron mean free path length could improve the Seebeck coefficient even more and should be further investigated. III. FABRICATION OF THE ANTENNA-COUPLED SINGLE-METAL NANOTHERMOCOUPLE Now we present the fabrication method of single-metal ACNTCs using the same metal for both the antenna and the NTC. Fig. 4(a) shows that the hot junction of the NTC is located at the center of the antenna, where the radiation-induced antenna current is at maximum and hence the heating is largest. The hot junction of the NTC is formed from a junction of narrow and wide wire segments from the same metal. The single-metal NTC operation is based on the structure size dependence of the absolute Seebeck coefficient in nanowires [5]. When the mean free path of electrons is comparable to the physical dimensions of a conductor, the absolute Seebeck coefficient is decreased from its bulk value due to the increased electron scattering. We exploit this property for single-metal NTCs by constructing them from the same metal nanowires, but with two different cross sections. As a result, the relative Seebeck coefficient between the two wire segments is nonzero, because the reduction of the absolute Seebeck coefficient is more pronounced in the narrow wire segment. Similar to our previous work [5], [4], [39], the antenna-coupled single-metal nanothermocouples were patterned by electron beam lithography, and metalized by 45-nm-thick electron beam evaporated Pd. The 50 nm and 200 nm wide wire segments are joined at 90 deg at one end, and connected to the bonding pads at their other ends, as shown in Fig. 4. The dipole antenna, long, is connected to the NTC at its center. The which is 2.4

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Fig. 5. Perspective view on the disk with heated patch.

measured electrical resistance of these devices is 2.1 . Fabrication complexity of such devices is greatly reduced compared to bi-metallic ACNTCs [4], since only one lithography and one deposition step is required. We constructed antenna-coupled thermopiles by connecting several ACNTCs in series to increase the output voltage of the detector. Fig. 4(b) shows such a linear thermopile array, where each individual ACNTC possesses its own read-out circuit, allowing thermopile-length-dependent measurements. IV. THERMAL CONDUCTION

Fig. 6. Time dependence of (normalized) for variable circular patch radius , for an initial penetration depth nm at nm for nm nm nm, nm computed from (14) and from the . Foster topology equivalent circuit (Fig. 8(b)) for

Fig. 7. Equivalent circuit of the thermocouple.

For the purpose of investigating the thermal heat conduction, we approximate the TC structure by a circular thermocouple patch. This patch models the hot junction of the ACTC. The , and its thickpatch is sitting on a substrate, its diameter is as shown in Fig. 5. Excited by an impulsive heating ness is , the temperature time dependence at by the patch at time is obtained for this structure by solving the diffusion equation [8] as

(14) The thermal diffusivity obtained for silicon as substrate material , considering a thermal conductivity is , density , and specific . We have varied the circular patch heat radii and have assumed an initial penetration depth nm at nm, nm. The time dependence of obtained is plotted in Fig. 6 and shows that is independent of for small times below 1 ps. In this case, the time evolution is predominantly determined by diffusion in -direction. As time advances, and spatial extension of the diffusion spread of gets on the order of , we find a strong dependence of the time on the junction radius . evolution of In order to model the dynamics of the TC, we have implemented in a previous work [6] an equivalent circuit that acelement, for the temperature decay counted, using a single with a single time constant. A single time constant, however, is not sufficient for modeling the thermal subsystem accurately, since there are several regions with variable heat capacity and

variable heat conductivity that govern the process. Hence, introducing several time constants will improve accuracy [40]. Using Foster or Cauer equivalent circuit topologies, we can synthesize lumped element equivalent circuit models with the desired accuracy [41], [42]. The TC is modeled by an equivalent circuit consisting of three parts, as shown in Fig. 7: the left-hand side models the (radio frequency) part, the center part represents electric the thermal equivalent circuit, and on the right-hand side, the (low frequency) part is equivalent circuit for the electric given. The antenna's open-circuit voltage, caused by the incielectrical field, is denoted as , and its radiation dent , resistance, its inductance, and capacitance are denoted as , and , respectively. Joule heating occurs in the TC juncpower , where tion due to the dissipation of the is the junction resistance and is the current in the junction resistance. The junction capacitance is denoted as . The dissipated power controls the current in the thermal equivalent circuit, which governs together with the thermal impedance the temperature enhancement . This temperature enhancement induces a Seebeck voltage in the circuit modeling of the electric low-frequency part. The inner resistance for the LF . Having obtained the solution of the diffucircuit model is sion equation, equivalent circuits can be used to model the . For this purpose we have to consider thermal impedance the frequency poles of this solution. In the complex frequency plane, the poles are located on the negative real axis. Two topologies for lumped element equivalent circuits are depicted in Fig. 8: The Cauer type topology (a) yields a ladder network with capacitors and resistors as the parallel and the series elements, respectively. The Foster type topology (b) yields -circuits connected in series. parallel

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For the case of negligible junction capacitance obtain

we (21)

In the tuned case we obtain

with harmonic excitation (22)

The square-law rectification is accomplished in the thermal part of the equivalent circuit. The heat generated in the theris related to mocouple junction with the electric resistance the flowing electric current via Fig. 8. Circuit modeling the thermal transfer function of the thermocouple: a) Cauer type RC equivalent circuit, b) Foster type RC equivalent circuit.

(23) In the complex frequency domain this equation becomes

For the Cauer topology equivalent circuit, given in Fig. 8(a), is of the form the thermal impedance (15)

where is the complex frequency. of the Foster topology equivalent circuit The impedance (Fig. 8(b)) is given by (16) with the time constant inverse Fourier transform of

. The pulse response is the , given by (17)

We used the analytic solution (14) to obtain the thermal time response of the TC with parameters stated above, and we compared these results to the matched equivalent Foster topology model. The equivalent circuit model exhibits three elements , , , , with , . This comparison is given in Fig. 6. V. THZ DETECTORS AND MIXERS Consider the equivalent circuit of the thermocouple depicted in Fig. 7. This circuit represents a quadratic detector. The right-hand and left-hand electric equivalent circuits are linear, whereas the thermal circuit in the center exhibits quadratic transfer characteristics. The left-hand electric equivalent circuit can be described in frequency domain by (18) (19) This yields

(20)

(24) and are the electric junction current and the where heat, respectively in the frequency domain and the operator denotes the convolution operation. The time derivation is considas ered as a thermal current and the temperature increase a thermal voltage. Its Fourier transform we denote with . is given by The temperature increase due to the heat (25) is given by (15) or (16), where the thermal impedance respectively. The right-hand electric equivalent circuit in Fig. 7 is the linear circuit describing the intermediate frequency or low-freaccounts quency output. The controlled voltage source for the thermal voltage excited via the Seebeck effect. The is given by output voltage (26) Inserting (25), (24), and (22) yields

The open-circuit output voltage given by

for

(27) is (28)

Fig. 9 shows the frequency response for nm computed from the a circular patch radius Foster topology equivalent circuit (Fig. 8(b)) using (16). The elements with equivalent circuit model exhibits three , , , , , . For the chosen parameters the computed 3 dB cutoff frequency is 136.6 GHz in the mixer mode and 206.8 GHz in the detector mode. the thermoDue to the broad-band characteristics at the couple is perfectly suited for mixer applications for frequencies input signal voltage with up to 30 THz. Consider an amplitude modulated with the signal the carrier frequency , (29)

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The autoconvolution of

is given by

Fig. 9. Frequency response for a circular patch radius nm computed from the Foster topology equivalent circuit (Fig. 8(b)) using (16); 3 dB cutoff frequency marker for mixer and detector mode .

The Fourier transform of

is symbolically denoted by (30)

and defined via (31) From (29), (30), and (31) we obtain (32) We superimpose a harmonic local oscillator signal . Its Fourier transform is

(39) Assuming the magnitude of the frequency difference to be small compared to the frequencies and we have in the first line of (39) the low-frequency and dc components of and the local oscillator the direct rectification of the signal signal . The second and third lines exhibit the intermediate and the lines frequency part with frequencies around 4 to 7 contain spectral parts around twice the local oscillator . These frequencies around are frequency, i.e., around . We need only to consider the low-frefiltered out by quency part playing a role in direct detector applications (40)

(33)

and the intermediate frequency part playing a role in mixer applications

(34)

(41)

The total input signal of the thermocouple is

This yields in the frequency domain

Inserting (40) into (28) yields the open-circuit output voltage for (35)

and The Fourier transform of the product of two signals is given by the convolution of the respective spectra, i.e., (36) where the convolution

We note

is defined by

(42) The output frequency characteristics of the detector is deterand represented in Fig. 9 for a NTC diameter mined by cutoff frequency equal to the of 25 nm, showing a detector thermal cutoff frequency of 260.8 GHz. In general the expresdenotes the spectrum of the square-law recsion tified AM signal envelope. For the case of mixer operation we insert (41) into (28) and obtain the open-circuit output voltage

(37) (43)

(38)

signal with the frequency is shifted by the local osThe to the intermediate frequency . cillator frequency The intermediate frequency band again is limited by the thermal cutoff frequency of 136.6 GHz. In the mixer case, the envelope

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Fig. 10. Open-circuit voltage as a function of thermopile length.

spectrum distortion.

is frequency shifted but exhibits no nonlinear

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Fig. 11. Scanning electron micrograph of the NTCs for frequency-dependent . measurements. The hot and cold (not shown) junctions are separated by 400 A thermopile was constructed from the NTCs by connecting them in series at their terminals.

VI. MEASUREMENTS Infrared measurements of the devices were performed by a laser operating at 28.3 THz. The laser linearly polarized beam was square-wave modulated by a mechanical chopper at 1 kHz. In order to maximize the device response, the polarization angle of the laser beam was set parallel to the antenna axis by a half-wave plate. The open-circuit voltage response was measured by a custom made pre-amplifier and with standard lock-in techniques. We measured the open-circuit voltage response of the thermopile with various lengths, as shown in Fig. 10. The response followed the thermopile addition rule, i.e., is a linear function of the number of ACNTCs in the thermopile. This confirms our assumption that the radiation-induced antenna currents heat the hot junction of the NTC, and shows that antennacoupled single-metal thermocouples and thermopiles function , as IR detectors. The relative Seebeck coefficient is 0.86 as measured by the characterization platform introduced in [39] for a single-metal NTC constructed from 50 and 200 nm wide Pd wire segments. Our IR experimental setup does not allow direct measurement of the frequency-dependent response of ACNTCs. Therefore, the frequency-dependent response of the NTCs was determined by pulse heating of the hot junctions using an acousto-optic modulator and a laser diode. To characterize the frequency-dependent response of NTCs to pulsed heating, we built the structure shown in Fig. 11. The geometry of the hot junctions is nominally identical in both structures (ACNTC and NTC) to avoid the impact of the thermal conductivity and geometry of the antenna on the response time of the NTC. The cold junctions for the frequency-dependent away from the hot junctions measurements are located 400 to ensure that the laser beam does not illuminate the cold junctions and so they remain at ambient temperature. We also constructed a thermopile by connecting four NTCs in series at their terminals. Now we discuss the various components of our experimental setup as shown in Fig. 12. Our heat source is an AlGalnP laser diode (HL6545MG), operating at 660 nm in a constant emission mode. The laser beam is collimated and focused to form a

Fig. 12. Measurement set-up.

200- -diameter spot at the devices, which was determined by the widely-used knife edge measurement technique. In order to perform frequency-dependent measurements, the beam is square-wave modulated with an acousto-optic modulator (AOM) between 30 kHz and 6.5 MHz. The signal generated by the NTCs in response to the laser-induced temperature oscillations is detected by an SR844 high frequency lock-in amplifier, which was synchronized to the AOM driver. In order to avoid the intrinsic bandwidth limitation of the measurement due to the low-pass filter by the source resistance and the input capacitance, the input impedance of the lock-in was set to 50 . Fig. 13 shows the open-circuit voltage response of a NTC and a four-NTC-long thermopile as a function of modulation frequency from 30 kHz to 6.5 MHz. From the figure we can determine that the 3 dB level of the measured signal is about 3.9 MHz. However, there is no physical reason to believe that the attenuation of the measured NTC signal is due to the thermal time constant of the NTC; rather it is caused by: 1) parasitic low-pass filtering, and 2) frequency-dependent intensity of the modulated laser beam. First, the resistance of the lead lines of the NTC (8 ) along with the cable capacitances form a low-pass filter, which limits the bandwidth of the measurements. This effect is evident form the different 3 dB levels of the single NTC (3.9 MHz) and the thermopile (3.5 MHz), which has about four times larger resistance and cable capacitance. Second, the intensity of the modulated laser beam is frequency dependent, as shown in Fig. 13 by the output beam

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and low-frequency bandwidths in detector applications, both on the order of 100 GHz will be feasible. This will open the door for novel future applications in the THz bands. Future detector and mixer experiments with ACNTCs excited by two-frequency far-infrared signals will push forward the experimental validation of the broadband detector and mixer properties of ACNTCs. REFERENCES

Fig. 13. Frequency-dependent response of the NTCs and a four-NTC-long thermopile. The reference measurement shows frequency-dependent intensity of the laser beam caused by the AOM.

intensity of the AOM measured by a photo diode (PD). Therefore, the incident power on the NTC as a function of frequency is not uniform, which results in an attenuation of the measured signal. Fig. 13 also shows that, by comparing the NTC response to the PD response, the fluctuation in the measured thermal response is not the property of the NTC, but is rather a frequency-dependent intensity fluctuation of the laser beam introduced to the system by the AOM. VII. CONCLUSION AND OUTLOOK In this work we describe the design, fabrication, and theoretical and experimental investigation of single-metal ACNTCs. ACNTCs are excellently suited as polarization-sensitive detectors and mixers for the long-wavelength far-infrared range around 30 THz. The theoretical fundamentals of the geometry-dependent Seebeck effect facilitating single-metal ACNTCs have been discussed. In thin films, the influence of surface electron scattering on the mean free path of the electrons yields a geometry dependence of the Seebeck effect, and makes SMTCs possible. We have given the experimental evidence for single-metal ACNTCs. The theoretical analysis of the thermal dynamics of NTCs has shown that detector low-frequency bandwidths and mixer intermediate-frequency beyond 100 GHz could be expected. This makes TCs presently the fastest THz detectors, suitable for broadband detector and heterodyne receiver applications. The combination with state of the art coherent terahertz sources, such as quantum cascade laser structures for room temperature terahertz frequency conversion [43]–[45], will pave the way for many innovative applications. We experimentally demonstrated that ACNTCs are able to detect and rectify long-wave infrared radiation at 10.6 wavelength. We also demonstrated that such NTCs are able to follow thermal oscillations in the MHz range; however, the bandwidth limitations of our experimental setup does not allow the determination of the cut-off frequency of these devices. Although our experimental investigations proved that NTCs exhibit detection bandwidths at least in the MHz range and consequently will be applicable for many sensing and communications applications, our theoretical investigations have shown that intermediate frequency bandwidths in mixer applications

[1] N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys. London: Oxford Univ. Press, 1936. [2] F. L. Bakker, J. Flipse, and B. van Wees, “Nanoscale temperature sensing using the Seebeck effect,” J. Appl. Phys., vol. 111, no. 8, pp. 084306–084306-4, 2012. [3] D. C. Agrawal and V. J. Menon, “The thermoelectric generator as an endoreversible Carnot engine,” J. Phys. D: Appl. Phys., vol. 30, no. 3, p. 357, 1997. [4] G. P. Szakmany, P. M. Krenz, A. O. Orlov, G. H. Bernstein, and W. Porod, “Antenna-coupled nanowire thermocouples for infrared detection,” IEEE Trans. Nanotechn., vol. 12, no. 2, pp. 163–167, Dec. 2013. [5] G. P. Szakmany, A. O. Orlov, G. H. Bernstein, and W. Porod, “Singlemetal nanoscale thermocouples,” IEEE Trans. Nanotechn., vol. 13, no. 6, pp. 1234–1239, 2014. [6] G. P. Szakmany, A. O. Orlov, G. H. Bernstein, W. Porod, M. Bareiss, P. Lugli, J. A. Russer, C. Jirauschek, P. Russer, M. T. Ivrlač, and J. A. Nossek, “Nano-antenna arrays for the infrared regime,” in 18th Int. ITG Workshop on Smart Antennas (WSA), Erlangen, March 2014. [7] J. A. Russer and P. Russer, “Dynamics of long-wave infrared range thermocouple detectors,” in Proc. XXXIst URSI General Assembly and Scientific Symp., Beijing, China, Aug. 17–23, 2014, pp. 1–4. [8] J. A. Russer, C. Jirauschek, G. P. Szakmany, A. O. Orlov, G. H. Bernstein, W. Porod, P. Lugli, and P. Russer, “A nanostructured long-wave infrared range thermocouple detector,” IEEE Trans. Terahertz Sci. Technol., vol. 5, no. 3, pp. 335–343, May 2015. [9] J. A. Russer, C. Jirauschek, G. P. Szakmany, A. O. Orlov, G. H. Bernstein, W. Porod, P. Lugli, and P. Russer, “Antenna-coupled terahertz thermocouples,” in Proc. IEEE MTT-S IMS, May 2015, pp. 1–4. [10] V. Damodara and N. Soundararajan, “Size and temperature effects on the Seebeck coefficient of thin bismuth films,” Phys. Rev. B, vol. 35, no. 12, pp. 5990–5996, Apr. 1987. [11] R. Dingle, “The electrical conductivity of thin wires,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 201, no. 1067, pp. 545–560, 1950. [12] D. MacDonald and K. Sarginson, “Size effect variation of the electrical conductivity of metals,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 203, no. 1073, pp. 223–240, 1950. [13] A. T. Ramu, L. E. Cassels, N. H. Hackman, H. Lu, J. M. Zide, and J. E. Bowers, “Rigorous calculation of the Seebeck coefficient and mobility of thermoelectric materials,” J. Appl. Phys., vol. 107, no. 8, p. 083707, 2010. [14] R. Kim, S. Datta, and M. S. Lundstrom, “Influence of dimensionality on thermoelectric device performance,” J. Appl. Phys., vol. 105, no. 3, p. 034506, 2009. [15] B. Wang, J. Zhou, R. Yang, and B. Li, “Ballistic thermoelectric transport in structured nanowires,” New J. Phys., vol. 16, no. 6, p. 065018, 2014. [16] L. Hicks and M. Dresselhaus, “Thermoelectric figure of merit of a onedimensional conductor,” Phys. Rev. B, vol. 47, no. 24, p. 16631, 1993. [17] D. Broido and T. Reinecke, “Theory of thermoelectric power factor in quantum well and quantum wire superlattices,” Phys. Rev. B, vol. 64, no. 4, p. 045324, 2001. [18] M. Cattani, M. Salvadori, A. Vaz, F. Teixeira, and I. Brown, “Thermoelectric power in very thin film thermocouples: Quantum size effects,” J. Appl. Phys., vol. 100, no. 11, p. 4905, 2006. [19] C. R. Tellier and A. J. Tosser, Size Effects in Thin Films. Amsterdam: Elsevier, 1982. [20] E. Sondheimer, “The mean free path of electrons in metals,” Adv. Phys., vol. 1, no. 1, pp. 1–42, Jan. 1952. [21] S. B. Soffer, “Statistical model for the size effect in electrical conduction,” J. Appl. Phys., vol. 38, no. 4, pp. 1710–1715, 1967. [22] J. Sambles, “The resistivity of thin metal films some critical remarks,” Thin Solid Films, vol. 106, no. 4, pp. 321–331, 1983. [23] W. F. Leonard and H.-Y. Yu, “Thermoelectric power of thin copper films,” J. Appl. Phys., vol. 44, no. 12, pp. 5320–5323, 1973.

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[24] R. Suri, A. Thakoor, and K. Chopra, “Electron transport properties of thin copper films. I.,” J. Appl. Phys., vol. 46, no. 6, pp. 2574–2582, 1975. [25] V. V. R. Narasimha Rao, S. Mohan, and P. J. Reddy, “Electrical resistivity, TCR and thermoelectric power of annealed thin copper films,” J. Phys. D: Appl. Phys., vol. 9, no. 1, p. 89, 1976. [26] H.-Y. Yu and W. F. Leonard, “Thermoelectric power of thin silver films,” J. Appl. Phys., vol. 44, no. 12, pp. 5324–5327, 1973. [27] V. V. R. Narasimha Rao, S. Mohan, and P. J. Reddy, “The size effect in the thermoelectric power of silver films,” Thin Solid Films, vol. 42, no. 3, pp. 283–289, 1977. [28] S. F. Lin and W. F. Leonard, “Thermoelectric power of thin gold films,” J. Appl. Phys., vol. 42, no. 9, pp. 3634–3639, 1971. [29] M. Hubin and J. Gouault, “Resistivity and thermoelectric power beand of gold and silver thin films formed tween and studied in ultrahigh vacuum,” Thin Solid Films, vol. 24, no. 2, pp. 311–331, 1974. [30] C. Pichard, A. Tosser, and C. Tellier, “Thermoelectric power of supported thin polycrystalline films,” J. Mater. Sci., vol. 17, no. 1, pp. 10–16, 1982. [31] W. Leonard and S. Lin, “Thermoelectric power of thin metal films,” J. Appl. Phys., vol. 41, no. 4, pp. 1868–1868, 1970. [32] A. Mayadas and M. Shatzkes, “Electrical-resistivity model for polycrystalline films: The case of arbitrary reflection at external surfaces,” Phys. Rev. B, vol. 1, no. 4, p. 1382, 1970. [33] C. Pichard, C. Tellier, and A. Tosser, “A three-dimensional model for grain boundary resistivity in metal films,” Thin Solid Films, vol. 62, no. 2, pp. 189–194, 1979. [34] R. Angus and I. Dalgliesh, “Thermopower and resistivity of thin metal films,” Phys. Lett. A, vol. 31, no. 5, pp. 280–281, 1970. [35] M. Angadi and S. Shivaprasad, “Thermoelectric power measurements in thin palladium films,” J. Mater. Sci. Lett., vol. 1, no. 2, pp. 65–66, 1982. [36] H. Ibach and H. Lüth, Solid-State Physics: An Introduction to Principles of Material Science. Berlin: Springer, 2003. [37] F. Mueller, A. Freeman, J. Dimmock, and A. Furdyna, “Electronic structure of palladium,” Phys. Rev. B, vol. 1, no. 12, p. 4617, 1970. [38] S. Shivaprasad, L. Udachan, and M. Angadi, “Electrical resistivity of thin palladium films,” Phys. Lett. A, vol. 78, no. 2, pp. 187–188, 1980. [39] G. P. Szakmany, P. M. Krenz, L. C. Schneider, A. O. Orlov, G. H. Bernstein, and W. Porod, “Nanowire thermocouple characterization platform,” IEEE Trans. Nanotechn., vol. 12, no. 3, pp. 309–313, May 2013. [40] V. Székely, “A new evaluation method of thermal transient measurement results,” Microelectron. J. , vol. 28, pp. 277–292, 1997. [41] V. Székely and T. Van Bien, “Fine structure of heat flow path in semiconductor devices: A measurement and identification method,” SolidState Electron., vol. 31, no. 9, pp. 1363–1368, Sep. 1988. [42] Y. C. Gerstenmaier, W. Kiffe, and G. Wachutka, “Combination of thermal subsystems modeled by rapid circuit transformation,” in Proc. THERMINIC, 2007, pp. 115–120. [43] K. Vijayraghavan, Y. Jiang, M. Jang, A. Jiang, K. Choutagunta, A. Vizbaras, F. Demmerle, G. Boehm, M. C. Amann, and M. A. Belkin, “Broadly tunable terahertz generation in mid-infrared quantum cascade lasers,” Nat. Commun., vol. 4, p. 2021, 2013. [44] C. Jirauschek, A. Matyas, P. Lugli, and M.-C. Amann, “Monte carlo study of terahertz difference frequency generation in quantum cascade lasers,” Opt. Express, vol. 21, no. 5, pp. 6180–6185, 2013. [45] Q. Lu, N. Bandyopadhyay, S. Slivken, Y. Bai, and M. Razeghi, “Continuous operation of a monolithic semiconductor terahertz source at room temperature,” Appl. Phys. Lett., vol. 104, no. 22, pp. 221105–221105-5, 2014.

Johannes A. Russer (M’09) received the Dipl.-Ing. (M.S.E.E.) degree in electrical engineering and information technology from the Universität Karlsruhe, Karlsruhe, Germany, in 2003. In 2004, he joined the University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA, as a Research Assistant, where he received the Ph.D. E.E. degree in 2010. From 2007 to 2010, he was with Qualcomm Inc. as an intern. Since 2010, he has been a Postdoctoral Research Fellow at the Institute of Nanoelectronics of the Technische Universität München (TUM), Munich, Germany.

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Dr. Russer received the Best Student Paper Award at the IEEE International Microwave Symposium in 2008. In 2015, he received the best paper award from the ITG (German Society for Information Technology). He is a member of VDE, and of the Eta Kappa Nu honor society.

Christian Jirauschek (S’03–M’04) received the Dipl.-Ing. and Ph.D. degrees in electrical engineering from the Universität Karlsruhe (TH), Karlsruhe, Germany, in 2000 and 2004, respectively. From 2002 to 2005, he worked at the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA. He then joined the Institute of Nanoelectronics at Technische Universität München (TUM), Munich, Germany, where, since 2007 he headed an independent junior research group within the Emmy Noether program of the Deutsche Forschungsgemeinschaft (DFG). In 2014, he received a Heisenberg professorship grant from the DFG. His research interests include modeling in the areas of photonics and nanoelectronics.

Gergo P. Szakmany received the Diploma in electrical and computer engineering from the Pazmany Peter Catholic University, Budapest, Hungary in 2007, and the M.S. and Ph.D. degrees in electrical engineering from the University of Notre Dame, Notre Dame, IN in 2011 and 2013, respectively. He is continuing his research at the University of Notre Dame on antenna-coupled infrared detectors as a Postdoctoral Research Associate. His research interests focuses on submicron device fabrication and characterization.

Mark Schmidt received the B.Sc. degree from the Technische Universität München (TUM), Munich, Germany, in 2014, where he specialized on high frequency engineering. He is currently pursuing a M. Sc. degree at TUM. He is working at the Institute of Nanoelectronics of the Technische Universität München (TUM), Munich, Germany, on the modeling of infrared-range thermocouple detectors. His research interests include theory, design, and applications in the areas of photonics, radio-frequency engineering, and energy harvesting.

Alexei O. Orlov received the M.S. and the Ph. D. degrees in physics from the Moscow State University, Moscow, Russia, in 1983 and 1990, respectively. He is currently is a Research Professor at the University of Notre Dame, Notre Dame, IN, USA. From 1983 to 1993, he worked at the Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Moscow, Russia. During this time, he conducted research on mesoscopic and quantum ballistic effects in electron transport of GaAs field-effect transistors. He was a visiting fellow at the University of Exeter, Exeter, UK in 1993, and joined the Department of Electrical Engineering at the University of Notre Dame in 1994. His research interests include experimental studies of mesoscopic, single-electron, and molecular electronic devices and sensors, nanomagnetics, and quantum-dot cellular automata. He has authored or co-authored more than 150 journal publications.

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Gary H. Bernstein (S’78–M’79–SM’95–F’06) received the B.S.E.E. degree from the University of Connecticut, Storrs, CT, USA, (with honors), in 1979, and the M.S.E.E. degree from Purdue University, West Lafayette, IN, USA, in 1981. He received the Ph.D. in electrical engineering from Arizona State University, Tempe, AZ, USA, in 1987, after which he spent a year there as a postdoctoral fellow. During the summers of 1979 and 1980, he was a Graduate Assistant at Los Alamos National Laboratory Los Alamos, NM, USA, and in the summer of 1983, interned at the Motorola Semiconductor Research and Development Laboratory, Phoenix, AZ, USA. He joined the Department of Electrical Engineering at the University of Notre Dame, Notre Dame, IN, USA, in 1988 as an Assistant Professor, and was the Founding Director of the Notre Dame Nanoelectronics Facility (NDNF) from 1989 to 1998. He has authored or co-authored more than 200 publications in the areas of electron beam lithography, nanomagnetics, quantum electronics, high-speed integrated circuits, electromigration, MEMS, and electronics packaging. He was promoted to the rank of Professor in 1998, and served as the Associate Chairman of his department from 1999 to 2006. Dr. Bernstein received an NSF White House Presidential Faculty Fellowship in 1992, and was named a Frank M. Freimann Professor of Electrical Engineering in 2010.

Wolfgang Porod (M’86–SM’90–F’01) received the M.S. and Ph.D. degrees from the University of Graz, Graz, Austria, in 1979 and 1981, respectively. After appointments as a Postdoctoral Fellow at Colorado State University, Fort Collins, CO, USA, and as a Senior Research Analyst at Arizona State University, Tempe, AZ, USA, he joined the University of Notre Dame (UND) in 1986 as an Associate Professor. He now also serves as the Director of Notre Dame's Center for Nano Science and Technology. He is currently a Frank M. Freimann Professor of Electrical Engineering at the UND. His research interests are in the area of nanoelectronics, with an emphasis on new circuit concepts for novel devices. He has authored some 300 publications and presentations. He has served as the Vice President for Publications for the IEEE Nanotechnology Council (2002–2003), and he was appointed an Associate Editor for the IEEE TRANSACTIONS ON NANOTECHNOLOGY (2001–2005). He has been active on several committees, in organizing special sessions and tutorials, and as a speaker in IEEE Distinguished Lecturer Programs.

Paolo Lugli (M’01–SM’07–F’11) received the Degree in physics from the University of Modena, Modena, Italy, in 1979. He received the Master of Science degree in 1982, and the Ph.D. degree in 1985, both in electrical engineering, from the Colorado State University, Fort Collins, CO, USA. In 1985, he joined the Physics Department of the University of Modena as a Research Associate. From 1988 to 1993, he was Associate Professor of Solid State Physics with the engineering faculty of the 2nd University of Rome “Tor Vergata”. In 1993, he was appointed as Full Professor of Optoelectronics at the same university. In 2002, he joined the Technical University of Munich, Munich, Germany, where he was appointed Head of the newly created Institute for Nanoelectronics. His current research interests include nanoimprint lithography, the modeling, fabrication, and characterization of organic devices for electronics and optoelectronics applications, the design of circuits and architectures for nanostructures and nanodevices, the numerical simulation of microwave semiconductor devices, and the theoretical study of transport processes in nanostructures. He is author of more than 350 scientific papers and co-author of the books “The Monte Carlo Modelling for Semiconductor Device Simulations” (Springer: 1989) and “High Speed Optical Communications” (Kluver Academic: 1999). Dr. Lugli served as General Chairman of the IEEE International Conference on Nanotechnology held in Munich in 2004. He is a mmber of the German National Academy of Science and Engineering (ACATECH).

Peter Russer (F’94–LF’13) received the Dipl.-Ing. (M.S.E.E.) degree in 1967 and the Dr. Techn. (Ph. D.E.E.) degrees in 1967 and 1971, respectively, both from the Vienna University of Technology, Vienna, Austria. In 1971, he joined the Research Institute of AEG-Telefunken in Ulm, Germany. From 1981 to 2008, he was Professor and Head of the Institute for High Frequency Engineering at the Technische Universität München (TUM), Munich, Germany. From October 1992 to March 1995, he was Director of the Ferdinand-Braun-Institut für Höchstfrequenztechnik, Berlin. From 1997 to 1999, he was Dean of the Department of Electrical Engineering and Information Technology of the TUM. His current research interests include electromagnetic fields, numerical electromagnetics, metamaterials, integrated microwave and millimeter-wave circuits, statistical noise analysis, electromagnetic compatibility, and quantum nanoelectronics. He has published more than 800 scientific papers and five books including “Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering” (Boston: 2006). Dr. Russer was elected Member of ACATECH in 2006. In 2006, he received the Distinguished Educator Award and in 2012, the Microwave Pioneer Award, both of the IEEE MTT Society, and in 2009, the Distinguished Service Award from the European Microwave Association. In 2007, he received an honorary Doctor degree from the Moscow University of Aerospace Technologies. In 2010 he was awarded the Golden Ring of Distinction of the German Association for Electrical, Electronic and Information Technologies (VDE).

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Reliable Microwave Modeling by Means of Variable-Fidelity Response Features Slawomir Koziel, Senior Member, IEEE, and John W. Bandler, Life Fellow, IEEE

Abstract—In this work, methodologies for low-cost and reliable microwave modeling are presented using variable-fidelity response features. The two key components of our approach are: 1) a realization of the modeling process at the level of suitably selected feature points of the responses (e.g., -parameters versus frequency) of the structure at hand and 2) the exploitation of variable-fidelity EM simulation data, also for the response feature representation. Due to the less nonlinear dependence between the coordinates of the feature points on the geometrical parameters of the structure of interest, the amount of training data can be greatly reduced. Additional cost reduction is obtained by means of generating the majority of the training data at a coarse-discretization EM simulation level and exploiting the correlations between the EM models of various fidelities. We propose two ways of combining the low- and high-fidelity data sets: 1) an external approach, through space mapping (simpler to implement) and 2) an internal approach, using co-kriging (more flexible and potentially offering better accuracy). The operation and performance of our modeling techniques are demonstrated by three microstrip filter examples and a compact rat-race coupler. A comprehensive verification and comparisons with several benchmark techniques, as well as application examples (filter optimization) are also provided. Index Terms—Co-kriging, computer-aided design, feature-based modeling, kriging, microwave component modeling, space mapping (SM), surrogates modeling.

I. INTRODUCTION

A

CCURATE evaluation of the electrical performance of microwave structures can be obtained through high-fidelity full-wave electromagnetic (EM) analysis. Unfortunately, this comes at considerable computational cost, particularly for complex devices/circuits and when interactions (EM couplings) Manuscript received June 19, 2015; revised September 03, 2015; accepted October 20, 2015. Date of publication November 11, 2015; date of current version December 02, 2015. This work was supported in part by the Icelandic Centre for Research (RANNIS) under Grant 130450051 and Grant 141272051, and in part by the Natural Sciences and Engineering Research Council of Canada under Grant RGPIN7239-11 and Grant STPGP447367-13, and in part by Bandler Corporation. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. S. Koziel is with the School of Science and Engineering, Reykjavík University, 101 Reykjavík, Iceland, and also with the Faculty of Electronics, Telecommunications, and Informatics, Gdańsk University of Technology, 80-233 Gdańsk, Poland (e-mail: [email protected]). J. W. Bandler is with the Simulation Optimization Systems Research Laboratory and Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4K1, Canada, and also with the Bandler Corporation, Dundas, ON L9H 5E7, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495266

with the environment are included. Consequently, the utilization of EM simulations in a design process, e.g., for parametric optimization, uncertainty quantification, or tolerance-aware design (design centering), may be prohibitive, as multiple evaluations of the structure at hand are involved. Executing such tasks in a reasonable timeframe requires fast and accurate replacement models (surrogates). Fast surrogate models can be constructed using two classes of techniques: 1) response surface approximation (RSA) [1] and 2) physics-based surrogate modeling [2], [3]. The first relies on approximating sampled high-fidelity EM simulation data. The most popular methods include neural networks [4], [5], kriging interpolation [6], support vector regression [7], [8], radial-basis function interpolation [9], the Cauchy method [10], as well as Gaussian process regression (GPR) [11], [12]. The advantages of the RSA surrogates include their versatility (as data-driven models they are easily transferable between various problem domains) and speed (once established, the RSA model is computationally cheap to evaluate). On the other hand, approximation models are not suitable for handling multi-dimensional parameter spaces. If the number of parameters exceeds just a few, the amount of training data necessary to ensure sufficient accuracy of the model (typically, below 5% of the relative RMS error [13]) grows very quickly so that the effort for model construction may not be practically justified (unless the model is to be reused under various design scenarios) or even feasible. The second class of techniques for constructing fast surrogates—physics-based modeling—relies on appropriate correction of an underlying low-fidelity model such as an equivalent circuit (popular method: space mapping (SM) [14], [15]). Physics-based surrogates require less training data and-due to the problem-specific knowledge embedded in the low-fidelity model-offer better generalization. However, they are less generic, more complex to implement, and their applicability is typically limited to cases when fast low-fidelity models are available; their accuracy depends on the reliability of the low-fidelity model; and it might not be straightforward to accommodate additional training data (if available) [16]. To some extent, these issues can be alleviated by combining SM with an approximation-based correction layer (e.g., [17], [18]). Reduction of the number of training points for approximation-based surrogates can be achieved by realizing the modeling process in an alternative representation of the system responses, where the dependence of the alternative responses on the designable parameters is less nonlinear. This approach has been explored, e.g., in the shape-preserving response prediction (SPRP) technique [19] or in [20] for inverse modeling of filters. A recent modeling technique [21] utilizes the concept of feature points

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similar to those in SPRP but with considerably simpler implementation (achieved by abandoning the use of so-called reference designs [21]). Feature-based modeling has been demonstrated to ensure good accuracy using a fraction of the training points required by conventional methods [21]. As indicated in [21], further reduction of the setup cost of the surrogate model can be achieved by using variable-fidelity EM simulation within the feature-based modeling framework. In [22], densely sampled low-fidelity EM model data was supplemented by sparsely sampled high-fidelity EM data with the two data sets blended (at the level of response features) into a single surrogate using co-kriging [23]. In this work, we provide extensive numerical validation of this approach. In addition, in Section II, we propose an alternative (external) way of combining low- and high-fidelity feature-based surrogates using SM. The latter is simpler and easier to implement than [22] while ensuring similar accuracy. Our test cases provided in Section III include three microstrip filter examples and a compact rat-race coupler. Application of the variable-fidelity feature-based models for design optimization is also demonstrated. II. VARIABLE-FIDELITY FEATURE-BASED MODELING In this section, we formulate the surrogate modeling problem, recall the concept of feature-based surrogates, and redefine the concept within a variable-fidelity setting. A. Surrogate Modeling , the response We denote by vector of the microwave device of interest. In particular, may represent at chosen frequencies, to , i.e., . is assumed to be evaluated using high-fidelity EM analysis. Consequently, it is computationally expensive. The task is to build a fast surrogate (replacement) model that represents in . Given sufficient accuracy of , it can be used in place of for solving design tasks that require multiple, high-cost evaluations of the latter. Let be the training set so that the responses of the high-fidelity model at , are known. Conventional response surface modeling attempts to directly model , . In many cases, the surrogate is created as an ensemble of RSA models constructed for individual frequencies, i.e., obtained by approximating the data sets , for . Sometimes [23], frequency is treated as an additional designable parameter, so that the RSA surrogate is constructed by approximating the data pairs . B. Variable-Fidelity Response Features Given the high nonlinearity of typical responses of microwave devices with respect to their designable parameters, particularly for filters, the direct modeling of the high-fidelity model responses is a challenging task that requires large data sets using (cf.

Fig. 1. Family of responses for a microstrip bandpass filter evaluated , : high-fidelity model along a selected line segment (—) and low-fidelity model . Selected feature points and groups and for . of corresponding points marked (o) for

Fig. 2. Selected feature point plots between designs and : (a) frequency and (b) levels. They correspond to two feature points: center frequency of the filter (– – –) and 10 dB level on the left-hand side of the passband (—); thick and thin lines are used for high- and low-fidelity model feature points, respectively.

Section II-A) and which is virtually impossible in high-dimensional design spaces. The key concept behind the modeling techniques considered here that allow reduction of the number of training points in the modeling process are certain response features [21]. The feature points (cf. Fig. 1) may include points corresponding to specific response levels (e.g., 10 dB, 3 dB), as well as those allocated in between fixed-level points (e.g., uniformly spaced in frequency). As indicated in Fig. 2 the dependence of the feature points on the design parameters is much less nonlinear than those of the original responses (here, -parameters), and thus easier to model. Feature-based modeling was originally introduced in [21]. It relies on extracting the feature points from the sampled EM-simulated responses at the training locations, constructing the RSA models of the individual feature points, and synthesizing the surrogate response at a design of interest from these RSA models. In this work, we utilize training data acquired from variable-fidelity simulations: from sparsely-sampled points and from densely-sampled data obtained from coarse-discretization EM simulations (low-fidelity model ), . Although and are misaligned (cf. Fig. 1), they are also well correlated so that the initial surrogate model obtained from the data can be enhanced by using a few points to construct the accurate, final surrogate model. We use the notation , , and to denote the th feature point of , and to denote the th feature ; and denote the frequency and point of magnitude (level) components of (similarly for ).

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with (4) and are predicted feature point locations corresponding to , the design being considered; , , is a discrete set of frequencies at which the response of the structure is being evaluated (cf. Section II-A). According to the internal approach, both and are implemented through co-kriging as in (2). denotes a function that interpolates the level vector and frequency vector into the response at a given frequency . Because and (i.e., frequencies and levels of the feature points) are less nonlinear than the original responses , a substantially smaller number of training points is necessary to ensure faithful modeling. Also, excellent correlation between and (cf. Fig. 2) allows for further reduction of the surrogate model setup cost because a very limited number of samples is sufficient to elevate the -based kriging model to high-fidelity accuracy through co-kriging. where

Fig. 3. Co-kriging concept: model (—), model (– – –), model sam, model samples . Kriging interpolation of model samples ples is not an adequate representation of the model (limited data set). of blended and data provides better acCo-kriging interpolation curacy at a lower computational cost.

C. Variable-Fidelity Feature-Based Modeling: The Internal Approach Multi-fidelity feature-based modeling is a two-step process. First, we construct approximation surrogates and , , corresponding to the feature points. In the internal approach presented in this section, the construction of and is based on both high-fidelity and low-fidelity training points and their corresponding feature points , and , . Blending these two types of data sets is realized through co-kriging [20] as described in the next paragraph. Let be the set of responses associated with the training set (i.e., low-fidelity feature points through ). The kriging interpolant is given as [13]

D. Variable-Fidelity Feature-Based Modeling: The External Approach The internal approach presented in Section II-C combines the low- and high-fidelity training data at the level of the feature points. In the external approach outlined below, the high-fidelity data is included through a SM correction of the initial featurebased surrogate model obtained as in (1), (2), however, using the low-fidelity training data only. The SM correction is realized at the level of the original responses as follows [15]: (5)

(1) where and are Vandermonde matrices of the test point and the base set , respectively; is determined by generalized least squares (GLS), is an vector of correlations between the point and the base set , where the entries are , and is a correlation matrix, whose entries are given by ). We use the exponential correlation function . The regression function is constant, and . Co-kriging is a type of kriging where the and model data are combined to enhance the prediction accuracy (cf. Fig. 3). Co-kriging is a two-step process: first a kriging model of the coarse data is constructed and, on the residuals of the fine data , a second kriging model is applied, where ; can be approximated as . The co-kriging interpolant is defined as [25] (2) , and can be found in [25]. Definitions of , , The multi-fidelity feature-based surrogate (the internal approach) is defined as (3)

such that , with , , being components of . Here, the model parameters (diagonal matrix), (square matrix), (column vector), and are obtained from the standard parameter extraction procedure needed by SM, namely, where

is a frequency scaled model

(6) If the correlation between the low- and high-fidelity models is sufficient (which is normally the case when both models are based on EM analysis), the correction given by (5) and (6) should significantly improve the accuracy of the surrogate. It should also be noted that the computational cost of surrogate model identification [i.e., solving the parameter extraction process (6)] can be neglected compared to the high-fidelity data acquisition because it is realized at the level of a fast feature-based model . E. Internal Versus External Approach Apart from considerable conceptual differences between the internal and external approaches, i.e., blending in the high-fidelity model data at the level of the response features rather than

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at the level of the original responses, the external approach is simpler to implement. On the other hand, the internal approach can be more flexible because the inclusion of a larger amount of high-fidelity training data should lead to an improvement of model accuracy. This may not be the case for the external approach (beyond a certain number of points) due to the fixed number of degrees of freedom of the surrogate (5) (the number of model parameters depends on the problem's dimensionality but not on the cardinality of the data set). III. VERIFICATION EXAMPLES In this section, we provide a comprehensive benchmarking of the multi-fidelity feature-based modeling techniques of Section II and demonstrate applications of the feature-based surrogate models in design optimization. A. Test Cases and Experimental Setup In order to illustrate the operation and performance of the proposed modeling techniques we consider three microstrip filter examples and a compact coupler. The first example (Filter 1) is the stacked slotted resonators bandpass filter [26] shown in Fig. 4(a). The high-fidelity filter model is simulated in Sonnet em using a grid of 0.05 mm 0.05 mm. The low-fidelity model is also simulated in Sonnet on a 2 mm 2 mm grid. The substrate parameters are thickness 0.635 mm, and permittivity . The designable parameters are . The region of interest is defined as the interval with and . The second structure (Filter 2) is the fourth-order ring resonator bandpass filter [27] shown in Fig. 4(b). The high-fidelity filter model is simulated in FEKO using 952 triangular meshes. The low-fidelity FEKO model utilizes 174 meshes. The substrate parameters are thickness 1.52 mm, and permittivity . The designable parameters are . The region of interest is defined as the interval with and . The last filter structure considered here (Filter 3) is the microstrip bandpass filter with open stub inverter [28]shown in Fig. 4(c). The high-fidelity filter model is simulated in FEKO using 432 triangular meshes. The low-fidelity FEKO model utilizes 112 meshes. The substrate parameters are thickness 0.508 mm, and permittivity . The designable parameters are . The region of interest is defined as the interval with and . The final test structure is a folded rat-race coupler (RRC) [29] shown in Fig. 4(d). The structure is implemented on RF-35 substrate , 0.762 mm, ). The designable parameters are given by: , with , fixed (all dimensions in millimeters). The high- and low-fidelity models of the structure are both implemented in CST Microwave Studio ( mesh cells, simulation time 15 min for and 8000 mesh cells, simulation

Fig. 4. Filter structures used for feature-based modeling verification: (a) stacked slotted resonators filter [26]; (b) fourth-order ring resonator bandpass filter [27]; (c) bandpass filter with open stub inverter [28]; (d) rat-race coupler [29].

time 20 s for terval

). The region of interest is defined as the inwith and . Model accuracy is verified using the relative error measure expressed in percent and averaged over 100 random test designs. The multi-fidelity feature-based models (both the internal and external versions) are compared with the following modeling methods: • regular (single-fidelity) feature-based modeling [21]; • generalized shape-preserving response prediction (GSPRP) [19]; • direct kriging interpolation of the high-fidelity data [6]; • response surface modeling using radial-basis functions [9]. The kriging model utilizes a Gaussian correlation function [6], whereas the radial-basis function model uses Gaussian basis functions [6]. The length-scale parameter of the latter is optimized using cross-validation [6]. For all the above modeling

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TABLE I MODELING RESULTS FOR FILTER 1

TABLE II MODELING RESULTS FOR FILTER 2

Fig. 5. Model responses at the selected (random) test designs: high-fidelity and model (—) and multi-fidelity feature-based model (set up with points) (o): (a) Filter 1; (b) Filter 2; and (c) Filter 3; and (d) RRC.

methods, five different cases are considered with the number of training points varying between and . B. Numerical Results and Comparisons With Benchmark Methods The results are gathered in Tables I–IV. Fig. 5 shows the highfidelity and feature-based model responses at selected test points for Filters 1 to 3 and for the RRC. The following observations can be made.

• Both the internal and external multi-fidelity feature-based models ensure excellent accuracy even with a very small number of high-fidelity training samples (specifically, 20 and 50). • The internal feature-based modeling approach is generally better than the external approach, however, the latter is still considerably better than a single-level feature-based surrogate for a small number of training samples and comparable or better overall. • Asymptotically (i.e., for the number of high-fidelity training points increasing to 400), both multi-fidelity feature-based modeling methods are comparable or better than the single-fidelity feature-based models and ones based on GSPRP. • All modeling approaches exploiting the concept of response features are considerably more accurate than

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TABLE III MODELING RESULTS FOR FILTER 3

TABLE IV MODELING RESULTS FOR THE RRC

Fig. 6. Optimization results: (a) Filter 1; (b) Filter 2; (c) Filter 3; (d) RRC. For (a)–(c), responses shown as (– – –) and (—) for the multi-fidelity feature-based surrogate model responses (the internal approach) at the initial design and at the optimized design; corresponding verification by high-fidelity model responses shown as (o). For (d), thin lines show surrogate responses at the initial design, thick lines show the surrogate responses as the optimized design; corresponding verification by high-fidelity model responses shown as (o); 280-MHz bandwidth of the optimized coupler marked with horizontal line.

conventional modeling techniques working directly with the original system responses, here, -parameters versus frequency. The above conjectures are consistent throughout all the test cases considered in this paper. It should also be emphasized that the benchmarking is quite comprehensive, both with respect to the competitive methods (four techniques) and with respect to the training set size (from 20 to 400 samples). Perhaps the most important point is that—according to the authors' knowledge—multi-fidelity feature-based modeling (both the internal and external approaches) are the only methods that

result in excellent (and practically usable) accuracy for an extremely small number of training points (20 and 50 samples). At the same time, one should bear in mind the limitations of the method, namely, the necessity of maintaining consistency of the feature points across the entire training set. For certain structures, such as the ones utilized in this work, as well as other structures with well-defined response “shapes” (e.g., coupling structures, narrow-band antennas, certain integrated photonic components such as microring resonators) it is easy to achieve. For other structures, such as high-order filters, feature-based modeling may be the method of choice for local modeling for, e.g., statistical design purposes [30].

KOZIEL AND BANDLER: RELIABLE MICROWAVE MODELING BY MEANS OF VARIABLE-FIDELITY RESPONSE FEATURES

C. Application Examples: Design Optimization As an additional verification, the multi-fidelity feature-based surrogate models have been utilized for parametric optimization of the Filters 1 through 3 as well as the RRC. The following design specifications are considered. • Filter 1: 1 dB for 2.35 GHz 2.45 GHz, 20 dB for 2.2 GHz and 2.6 GHz. • Filter 2: 1 dB for 1.75 GHz 2.25 GHz, 20 dB for 1.5 GHz and 2.5 GHz. • Filter 3: 1 dB for 1.95 GHz 2.05 GHz, 20 dB for 1.8 GHz and 2.2 GHz. • RRC: Obtain an equal power split, i.e., at the operating frequency of 1 GHz; simultaneously maximize the 20 dB bandwidth (symmetrically) around for and . In all cases, the multi-fidelity feature-based surrogate constructed using the internal approach and 50 high-fidelity training samples (200 low-fidelity samples) has been used in the process. The initial and final responses obtained by optimizing the feature-based surrogate model (the final design is verified by the high-fidelity model) are shown in Fig. 6. The design specifications for the filter structures are marked using thick horizontal lines. Because of the very good accuracy of the surrogates, no further design tuning was found necessary. In the case of RRC, the bandwidth of the optimized coupler is 280 MHz with the power split error 0.2 dB at 1 GHz. IV. CONCLUSION Variable-fidelity feature-based techniques for low-cost surrogate modeling of microwave structures have been proposed. Reduction of the computational cost associated with setting up surrogate models has been achieved by combining two basic components: 1) the exploitation of certain feature points, which allows us to move the modeling process to an alternative representation of the system response, where the functional landscape is much less nonlinear than for the original responses (in particular, the frequency-dependent -parameters) and 2) the utilization of variable-fidelity EM simulations, where an initial surrogate created with densely sampled coarse-discretization EM simulation data is enhanced by sparsely sampled high-fidelity EM data. Two approaches to blending the variable-fidelity EM data into the final surrogate have been proposed, i.e., an internal one (based on co-kriging at the level of the feature points), and an external one (based on SM). As demonstrated by three microstrip filters and a rat race coupler example and comparisons with several benchmark techniques, both of our multi-fidelity feature-based approaches outperform not only conventional approximation modeling methods but also feature-based approaches that exploit single-fidelity EM simulations. Significant improvement of the predictive power of the surrogate is especially observed for small high-fidelity training sets. This opens new opportunities for construction of quasi-global surrogates for applications such as parametric design optimization (also demonstrated in this work). According to our knowledge, no surrogate modeling technique reported in the literature so far exhibits comparable

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performance. At the same time, one needs to bear in mind the limitations of the method, namely, the necessity of maintaining consistency of the feature point sets across the surrogate model domain. Consequently, our method is less versatile than general-purpose approximation techniques. On the other hand, with careful definition of the response features, as well as for numerous cases where the system response is well-defined in terms of its shape (microwave couplers, low-order filters, narrow-band antennas, phased array antennas, various classes of integrated photonic devices, wireless power transfer systems, etc.) but also higher-order filters in terms of local modeling for statistical/robust design application and uncertainty quantification, multi-fidelity feature-based modeling may be the method of choice for rapid construction of fast, accurate and reusable surrogates. REFERENCES [1] T. W. Simpson, J. Peplinski, P. N. Koch, and J. K. Allen, “Metamodels for computer-based engineering design: Survey and recommendations,” Eng. With Comput., vol. 17, no. 2, pp. 129–150, Jul. 2001. [2] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 1, pp. 337–361, Jan. 2004. [3] S. Koziel and J. W. Bandler, “Recent advances in space-mapping-based modeling of microwave devices,” Int. J. Numer. Modelling, vol. 23, no. 6, pp. 425–446, Nov. 2010. [4] H. Kabir, Y. Wang, M. Yu, and Q. J. Zhang, “Neural network inverse modeling and applications to microwave filter design,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 4, pp. 867–879, Apr. 2008. [5] Y. Cao, X. Chen, and G. Wang, “Dynamic behavioral modeling of nonlinear microwave devices using real-time recurrent neural network,” IEEE Trans. Electron Devices, vol. 56, no. 5, pp. 1020–1026, May 2009. [6] N. V. Queipo, R. T. Haftka, W. Shyy, T. Goel, R. Vaidynathan, and P. K. Tucker, “Surrogate-based analysis and optimization,” Progr. Aerosp. Sci., vol. 41, no. 1, pp. 1–28, Jan. 2005. [7] L. Xia, J. Meng, R. Xu, B. Yan, and Y. Guo, “Modeling of 3-D vertical interconnect using support vector machine regression,” IEEE Microw. Wirel. Comp. Lett., vol. 16, no. 12, pp. 639–641, Dec. 2006. [8] A. J. Smola and B. Schölkopf, “A tutorial on support vector regression,” Statist. Comput., vol. 14, no. 3, pp. 199–222, Aug. 2004. [9] M. D. Buhmann and M. J. Ablowitz, Radial Basis Functions: Theory and Implementations. Cambridge, U.K.: Cambridge University, 2003. [10] A. G. Lamperéz, P. K. Sarkar, and M. S. Palma, “Generation of accurate rational models of lossy systems using the Cauchy method,” IEEE Microw. Wirel. Comp. Lett., vol. 14, no. 10, pp. 490–493, Oct. 2014. [11] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. Cambridge, MA, USA: MIT Press, 2006. [12] J. P. Jacobs and J. P. De Villiers, “Gaussian-process-regression-based design of ultrawide-band and dual-band CPW-fed slot antennas,” J. Electromagn. Waves Appl., vol. 24, pp. 1763–1772, 2010. [13] I. Couckuyt, F. Declercq, T. Dhaene, H. Rogier, and L. Knockaert, “Surrogate-based infill optimization applied to electromagnetic problems,” Int. J. RF Microw. Comput.-Aided Eng., vol. 20, no. 5, pp. 492–501, Sep. 2010. [14] J. W. Bandler, N. Georgieva, M. A. Ismail, J. E. Rayas-Sánchez, and Q. J. Zhang, “A generalized space mapping tableau approach to device modeling,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 1, pp. 67–79, Jan. 2001. [15] J. W. Bandler, Q. S. Cheng, and S. Koziel, “Simplified space mapping approach to enhancement of microwave device models,” Int. J. RF Microw. Comput.-Aided Eng., vol. 16, no. 5, pp. 518–535, Sep. 2006. [16] S. Koziel, J. W. Bandler, and K. Madsen, “Theoretical justification of space-mapping-based modeling utilizing a data base and on-demand parameter extraction,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 12, pp. 4316–4322, Dec. 2006. [17] S. Koziel and J. W. Bandler, “A space-mapping approach to microwave device modeling exploiting fuzzy systems,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 12, pp. 2539–2547, Dec. 2007.

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[18] S. Koziel and J. W. Bandler, “Modeling of microwave devices with space mapping and radial basis functions,” Int. J. Numer. Modell., vol. 21, no. 3, pp. 187–203, May 2008. [19] S. Koziel and L. Leifsson, “Generalized shape-preserving response prediction for accurate modeling of microwave structures,” IET Microw., Ant. Prop., vol. 6, no. 12, pp. 1332–1339, Sep. 2012. [20] E. Menargues, S. Cogollos, V. E. Boria, B. Gimeno, and M. Guglielmi, “An efficient computer-aided design procedure for interpolating filter dimensions using least squares methods,” in Eur. Microw. Integr. Circuits Conf., 2012, pp. 250–253. [21] S. Koziel, J. W. Bandler, and Q. S. Cheng, “Low-cost feature-based modeling of microwave structures,” presented at the IEEE MTT-S Int. Microw. Symp., Tampa, FL, USA, 2014. [22] S. Koziel and J. W. Bandler, “Accurate modeling of microwave structures using variable-fidelity response features,” presented at the IEEE MTT-S Int. Microw. Symp., Phoenix, AZ, USA, 2015. [23] M. C. Kennedy and A. O'Hagan, “Predicting the output from complex computer code when fast approximations are available,” Biometrika, vol. 87, pp. 1–13, 2000. [24] J. P. Jacobs and S. Koziel, “Two-stage framework for efficient Gaussian process modeling of antenna input characteristics,” IEEE Trans. Antennas Propag., vol. 62, no. 2, pp. 706–713, Feb. 2014. [25] S. Koziel, S. Ogurtsov, I. Couckuyt, and T. Dhaene, “Variable-fidelity electromagnetic simulations and co-kriging for accurate modeling of antennas,” IEEE Trans. Antennas Propag., vol. 61, no. 3, pp. 1301–1308, Mar. 2013. [26] C. L. Huang, Y. B. Chen, and C. F. Tasi, “New compact microstrip stacked slotted resonators bandpass filter with transmission zeros using high-permittivity ceramics substrate,” Microw. Opt. Tech. Lett., vol. 50, no. 5, pp. 1377–1379, May 2008. [27] M. K. M. Salleh, G. Pringent, O. Pigaglio, and R. Crampagne, “Quarter-wavelength side-coupled ring resonator for bandpass filters,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 1, pp. 156–162, Jan. 2008. [28] J. R. Lee, J. H. Cho, and S. W. Yun, “New compact bandpass filter resonators with open stub inverter,” IEEE Miusing microstrip crow. Guided Wave Lett., vol. 10, no. 12, pp. 526–527, Dec. 2000. [29] S. Koziel, A. Bekasiewicz, P. Kurgan, and J. W. Bandler, “Expedited multi-objective design optimization of miniaturized microwave structures using physics-based surrogates,” presented at the IEEE MTT-S Int. Microw. Symp., Phoenix, AZ, USA, 2015.

[30] S. Koziel and J. W. Bandler, “Rapid yield estimation and optimization of microwave structures exploiting feature-based statistical analysis,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 1, pp. 107–114, Jan. 2015. Slawomir Koziel (M'03–SM'07) received the M.Sc. and Ph.D. degrees in electronic engineering, the M.Sc. degree in theoretical physics, and the M.S. degree in mathematics from Gdańsk University of Technology, Gdańsk, Poland, in 1995, 2000, 2000, and 2002, respectively, and the Ph.D. degree in mathematics from the University of Gdańsk, Gdańsk, Poland, in 2003. He is currently a Professor with the School of Science and Engineering, Reykjavík University, Reykjavík, Iceland. He is also a Visiting Professor with Gdańsk University of Technology. His research interests include CAD and modeling of microwave circuits, simulation-driven design, surrogate-based optimization, space mapping, circuit theory, analog signal processing, evolutionary computation, and numerical analysis.

John W. Bandler (S'66–M'66–SM'74–F'78–LF'06) studied at Imperial College, London, U.K., and received the B.Sc. (Eng.), Ph.D., and D.Sc.(Eng.) degrees from the University of London, London, U.K., in 1963, 1967, and 1976, respectively. He joined McMaster University, Hamilton, ON, Canada, in 1969, where he is now a Professor Emeritus. He founded Optimization Systems Associates Inc. in 1983 and sold it to Hewlett-Packard in 1997. He is President of Bandler Corporation, Dundas, ON, Canada. Dr. Bandler is a Fellow of several societies, including the Royal Society of Canada and the Canadian Academy of Engineering. In 2004, he was a recipient of the IEEE MTT-S Microwave Application Award. He was a recipient of the IEEE Canada McNaughton Gold Medal in 2012, the Queen Elizabeth II Diamond Jubilee Medal in 2012, the IEEE MTT-S Microwave Career Award in 2013, and the McMaster University's Faculty of Engineering Research Achievement Award in 2014.

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Consistent DC and RF MOSFET Modeling Using an -Parameter Measurement-Based Parameter Extraction Method in the Linear Region Fabián Zárate-Rincón, Student Member, IEEE, Reydezel Torres-Torres, Senior Member, IEEE, and Roberto S. Murphy-Arteaga, Senior Member, IEEE

Abstract—This paper demonstrates the feasibility of using data extracted from experimental -parameters to implement models to accurately represent MOSFET behavior under DC and RF regimes with consistency. For this purpose, a method to extract the MOSFET's drain-to-source conductance, the subthreshold swing, the source/drain resistance, the effective gate length, and the threshold voltage is proposed. The method is based on the determination of the inverse of the channel resistance that is directly related to the previously mentioned parameters. Hence, two-port -parameter measurements of common-source configured RF-MOSFETs with different gate lengths are performed, varying the gate-to-source voltage from the subthreshold region to strong inversion. In order to verify the validity and consistency of the method, excellent correlation of DC and RF models with experimental data is achieved. Index Terms—MOSFET parameters, RF-MOSFET, ters, small-signal model.

-parame-

I. INTRODUCTION

I

N ORDER TO determine MOSFET parameters such as the drain-to-source conductance , the subthreshold swing , the source and drain resistances, the effective gate length , and the threshold voltage , DC measurements are conventionally performed with the device biased in the linear region to guarantee a uniformly inverted channel [1], [2]. Nevertheless, since and present a strong bias dependence due to the LDD (lightly-doped drain) regions [3], experimentally obtaining MOSFET's fundamental parameters from solely DC measurements by using regressions involving experimental data at multiple gate-to-source voltages is very difficult. Moreover, since the channel resistance is inversely proportional to , and become comparable to Manuscript received June 25, 2015; revised September 15, 2015; accepted October 17, 2015. Date of publication November 13, 2015; date of current version December 02, 2015. This work was supported in part by CONACyT, Mexico, under Grant 83774-Y and IMEC, Leuven-Belgium, supplied the test structures. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. The authors are with the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Department of Electronics, Tonantzintla, Puebla, 72840, Mexico (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495363

in strong inversion, particularly for nanometer devices. Alternatively, regressions of the experimentally determined total drain-to-source resistance as a function of the gate mask-length can be performed for parameter extraction. However, the variation of the MOSFET parasitic resistances with impedes the accurate determination of , and using data measured from several devices [3]. Furthermore, DC methods commonly underestimate due to the difficulty of completely removing the effect of the parasitic series resistances, implying that methods relying on curve extrapolations involving obtained in this regime will be inaccurate. Due to this fact, it is common to find discrepancies in the values obtained for a particular parameter when using different DC-measurement based approaches. Therefore, the intrinsic properties of a MOSFET have to be obtained by considering its bias-dependent and geometry-dependent characteristics, which will allow obtaining physically based parameters for the implementation of scalable models. This can be achieved by means of RF-measurements [4]. In a previous work on multi-fingered transistors with different gate lengths [5], we presented the correlation of DC extracted parameters to -parameter based methods using the inverse of . In that paper, we describe the noticeable variation of the bias-dependent source and drain resistances for devices with different layouts, which points out the importance of using measurements performed on a single device to correctly determine the corresponding parameters. In fact, using this concept, in [6] the series parasitic resistances obtained from DC and RF data are consistent. However, the dependence of these resistances with voltage is not considered. Using a straightforward small-signal equivalent circuit element extraction method for n-channel RF-MOSFETs with different gate lengths, this work shows how to reproduce DC measurements from RF data in the linear region. This can be correctly achieved from high frequency measurements by performing data regressions involving the experimentally dependent , and at different bias conditions. Even when the methodology is developed and verified here with experiments performed on planar bulk MOSFETs, it can be applied to other types of transistors presenting a similar small-signal representation, such as finFETs [7]. In fact, with the appropriate assumptions, the proposal can be adapted to analyze the impact of less studied effects impacting the performance of diverse advanced devices.

0018-9480 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Fig. 1. Conceptual depiction of the small-signal equivalent circuit for a . MOSFET at strong inversion and Fig. 2. Micrograph of the device under test for

nm.

II. TEST STRUCTURES AND RF MEASUREMENTS To develop and validate the modeling and parameter extraction methodology proposed in this paper, a group of multi-finger microwave MOSFETs in a common source configuration, with nm, 90 nm, and 120 nm, were fabricated gate lengths in a conventional CMOS process. For these devices, the finger and the number of fingers were fixed at width and 64, respectively. The devices under test (DUTs) present a shallow trench isolation scheme and a ground shield with the purpose of increasing the bulk resistance to avoid undesirable coupling through the substrate. The pads built for probing purposes are made of aluminum and are placed at the top metal layer. The performance of the transistors is improved by using a polysilicon/SiON gate and a guard ring. With the purpose of performing two-port -parameter and DC measurements on the DUTs, vector network analyzer (VNA) and semiconductor device analyzer (SDA) setups were used, respectively. This also requires the use of two pitch ground-signal-ground (GSG) probes for the RF expericurves. ments, and DC probe needles for obtaining Before measuring the -parameters, the VNA setup was calibrated using an off-wafer line-reflect-match (LRM) algorithm to remove the undesirable effects of the cables and probes as well . In addition, as for establishing the reference impedance to a two-step de-embedding procedure using the measurements of on-wafer “open” and “short” dummy structures was carried out to subtract the effect of the pad parasitics from the measurements. Bear in mind, however, that the effect of the extrinsic MOSFET's parasitics (e.g., the multifingered gate electrode resistance) is not removed when applying this de-embedding procedure, which is described, as well as the dummy structures in [8]. These measurements were performed up to 60 GHz under from the subthreshold redifferent bias conditions, varying , gion to strong inversion. This allowed the extraction of and . The considered small-signal equivalent circuit is presented in Fig. 1, where is the gate resistance, is the is the gate-to-source capacitance, is bulk resistance, and are the junction cathe gate-to-drain capacitance, pacitances, and is the drain-to-source capacitance. A microphotograph of one of the DUTs and the experimental setup are presented in Figs. 2 and 3, respectively.

Fig. 3. Experimental setup used to perform RF measurements under different bias conditions, illustrating the vector network analyzer (VNA), the power supply and the probe station.

III. DETERMINATION OF THE CHANNEL RESISTANCE AND THE SERIES PARASITIC RESISTANCES The procedure relies on the extraction of the small-signal from the -parameters under different bias-conditions. This parameter is related to the drain-to-source current and to . There are several works focused on the extraction of it [9]–[13]. In this work, the device is biased from sub-threshold to strong inversion while maintaining . This ensures that the transistor is operating in the linear region, in which the channel can be considered as uniformly inverted. Under this regime, and can be appropriately obtained at different . Bear in mind, however, that the substrate losses in CMOS technologies become significant at high-frequencies, which requires the extraction of the substrate parasitics previous to finding and . Thus, after obtaining these parasitics represented by means of , and , the corresponding effects can be removed from the experimental data. The most suitable bias condition to perform the extraction of the substrate elements is set to and (i.e., the zero-bias cold-FET condition). In this regime, there is no inversion channel under the gate oxide, and thus is not defined and can be neglected in the small-signal equivalent circuit, which allows easily determining and by means of [14]. Consecutively, is approximately equal to . Once obtaining these values, their undesirable effect can be subtracted from the experimental -parameters, collected in the

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continue with the procedure, is transformed into -parameters . Hence, from for transistors operating within the strong inversion regime at , the following relations can be established [15] (3) (4)

Fig. 4. Real part of and , and with the corresponding linear regresnm at and sion (continuous line) of MOSFETs with .

In addition, the imaginary part of written as

-parameters

can be (5) (6)

where

(7)

(8)

Fig. 5. Simulated (continuous line) and measured (symbols) -parameters up and to 60 GHz of MOSFETs with two different gate lengths at .

matrix, by using the following equation that expresses the corrected -parameter matrix (1) is associated with the parallel between where the impedance and that occurs since the source terminal is tied to the bulk terminal, and the superscript is used to define - or -parameters at after removing the substrate parasitic network. Since the inversion channel acts as a conductive shield, becomes much less than the other capacitances and then, it can be neglected. On the other hand, can be expressed as (2) The additional consideration of including and in comparison with our previous work in [5] allows one to achieve better results when frequency becomes higher. Furthermore, it is important to remark that the only bias dependence to be taken into account in the substrate elements due to its noticeable impact on the device's output characteristics is related to the bulkto-source voltage . In fact, as a result of the weak dependence of the rest of the substrate network components on the gate and drain biasing voltages, the removal of , and extracted at the zero-bias FET condition is performed on measurements under different at . In addition, to

In this equation, and are the -parameters at zero-bias. Furthermore, can be obtained using (5) through (8). Then, and can be extracted by means of (3) and (4). Fig. 4 shows the real part of the -parameters and the obtained values for at and . Notice that the plotted -parameters are in the order of ohms, and include the contribution of the channel resistance as well as that of the series resistances. Moreover, can be determined from the linear regression of as a function of at lower frequencies. Hence, when the value for the series resistances is determined, it is possible to assess the corresponding impact on the device's input and output impedances by carrying out a comparison involving these curves. In order to verify the extracted equivalent circuit elements, a comparison between curves obtained from simulations using the model in Fig. 1 and measured -parameters is performed up to 60 GHz for different and , which is presented in Fig. 5 through Fig. 8. This shows the validity of reproducing small-signal data at microwave frequencies using the extracted parameters. It is important to remark the fact that although model-experiment correlation at the higher measured frequencies is acceptable, some data dispersion is observed in the experimental data beyond 50 GHz since the open-short de-embedding procedure exhibits limited accuracy at these frequencies [8]. The procedure previously mentioned can be also used in the case of similar technologies such as finFets since the equivalent circuit does not significantly vary. However, the parasitic capacitances associated with the fin should be taken into account in and as follows [16] (9) (10)

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Fig. 6. Magnitude and phase of simulated (continuous line) and measured up to 60 GHz of MOSFETs with two different gate lengths at (symbols) and .

Fig. 7. Simulated (continuous line) and measured (symbols) -parameters up nm at different and . to 60 GHz of MOSFETs with

where and are intrinsic capacitances, is the external capacitance and is the internal capacitance. IV. EXTRACTION OF MOSFET PARAMETERS As mentioned before, the bias dependence of and complicates the MOSFET's parameter extraction from DC data in short-channel transistors because these parasitic resistances are of the same order of magnitude as when the inversion channel is formed. This happens due to the use of LDD regions, which is also an issue in bulk finFETs [7]. So, due to the corresponding voltage drop across and , which also depends on , when there is a current flowing through the channel, it is not straightforward to accurately model . For this reason, based on the obtained value by using RF measurements, the MOSFET's parameters can be found in an alternative and efficient fashion. An important advantage of the RF procedure in the linear region is that the resistances can be determined at , which avoids the voltage drop across . First, let us define a current that is flowing through the channel when a given intrinsic voltage is applied to it. For this assumption, and were previously removed from the transistor. In the case of the subthreshold swing , the following expression can be used:

Fig. 8. Magnitude and phase of simulated (continuous line) and measured up to 60 GHz of MOSFETs with nm at different (symbols) and .

(11) Fig. 9. Determination of the subthreshold swing.

which can be rewritten as (12) This can be done because is fixed at a given value and thus it does not alter the result. Furthermore, notice that , which allows defining as

of the previously mentioned curves in the subthreshold region. After applying this methodology, the obtained values are 114 mV/dec for nm and 77 mV/dec for nm. On the other hand, can be normalized by in order to find a mathematical relation between this one and as follows:

(13) (14) as a function Fig. 9 shows the curves of the inverse of of in a semi-logarithmic plot for MOSFETs with two different gate lengths. Based on (14), corresponds to the slope

This expression avoids using the total resistance extracted from DC data (i.e., that includes not only the

ZÁRATE-RINCÓN et al.: CONSISTENT DC AND RF MOSFET MODELING USING AN EXTRACTION METHOD

Fig. 10. Linear extrapolation of nm. imum slope point with

Fig. 11. Extraction of

and

against

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at its maxFig. 12. Linear regressions to obtain obtained from RF measurements at

at different .

voltages using

through the SDL method.

characteristics of the channel but also those of the pads, source and drain terminals. Therefore, using (14), which is corrected from the drain and source series parasitics, can be obtained by applying the well-known linear extrapolation method. As shown in Fig. 10, is obtained in this case. For comparison purposes, was also determined using the second derivative of the logarithm (SDL) method [2], obtaining practically the same value for as shown in Fig. 11. This is due to the fact that the SDL method also removes the effect of the MOSFET's series parasitics from the experimental data. Bear in mind, however, that the later approach is highly sensitive to the measurement noise since a second order derivative of experimental data greatly magnifies this noise. Furthermore, one expects that the versus data would present a linear trend to find related to the point in which but this does not happen in practice due to the dependence of and on . This problem can only be solved by subtracting these parasitic resistances from for each value of . In the linear region, can be expressed as (15) as a function of are preThe linear regressions of sented in Fig. 12, showing an excellent agreement with the extracted values. In this way, can be found from the extrapolation to , where . This procedure can be followed for all values of voltage. Once is known, is easily determined from (15). Fig. 13 shows the resulting data for devices with different gate mask length. The considerable variation between the curves in this figure points out the importance of a methodology not relying on regressions varying assuming negligible changes in the parameters for devices with

Fig. 13. versus . presenting different

curves showing the noticeable difference for devices

different geometries. On the other hand, notice that for the considered devices approaches Lg only at when part of the LDD regions under the gate become inverted. Thus, even though different gate lengths are considered in Fig. 13, the trend of the corresponding curves is approximately the same since the doping profile is the same for all the devices. In order to compare the previous results, is also obtained by using the RF capacitance method presented in [4]. For this purpose, the gate-to-channel capacitance related to must be determined as follows: (16) where (17) In Fig. 14, the linear regression of as a function of for different is shown. In a similar way to , can be found from the extrapolation to . Moreover, the comparison between the proposed and RF capacitance methods is presented in Fig. 15 which shows that a good agreement is achieved for values greater than 0.4 V when the inversion channel is completely formed. It is important to mention that the capacitance method is not accurate at lower voltages in which depends not only on intrinsic characteristics but also on extrinsic components. V. VALIDATION OF THE PROPOSED METHODOLOGY In order to prove the validity of the extracted parameters, it is necessary to use an expression that allows reproducing the DC

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Fig. 16. Correlation of (19) with experimental data for different Fig. 14. Linear regressions to obtain obtained from RF measurements at

at different .

.

voltages using

Fig. 17. Inverse of extracted (symbols) and simulated (continuous lines) as a function of for different . Fig. 15. Comparison of RF capacitance method.

versus

obtained from proposed method and

measurements from the obtained from RF data. To accomplish this, a semi-empirical equation can be used [2]:

(18) is the main branch of the Lambert W function, and where are fitting parameters, is the mobility degradation coefficient and is equal to the ideality factor times the thermal voltage . Notice that the previous equation can be used from the subthreshold region to strong inversion. Since is related to , can be found from (18), which is given by

(19) . However, it is more convenient to express with as a function of with the aim of simplifying the representation of since the Lambert W function cannot be expressed in terms of elementary functions. Accordingly, can be determined from

Fig. 18. Derivative of the inverse of extracted (symbols) and simulated (conagainst for different . tinuous lines)

to experimentally obtained can be achieved, as is illustrated in Fig. 16 for different . Due to the non-linear form of (20), the Levenberg-Marquardt algorithm was used for obtaining the corresponding parameters through least squares optimization. The resulting values for these parameters are listed in Table I. Moreover, in order to gain insight from this model implementation, it is more illustrative to plot the inverse of and its derivative obtained from the explicit expression (19). The corresponding curves are respectively plotted in Figs. 17 and 18. On the other hand, it is known that the parasitic resistances can be modeled as follows [17], [18]: (21)

(20)

(22) Based on this expression, it is possible to accurately represent for a wide range of values. In this the extracted fashion, a good agreement between (20) and data corresponding

where nents,

and and

are the bias-independent compoare the bias-dependent components,

ZÁRATE-RINCÓN et al.: CONSISTENT DC AND RF MOSFET MODELING USING AN EXTRACTION METHOD

TABLE I PARAMETERS USED FOR IMPLEMENTING THE (20).

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MODEL GIVEN BY

Fig. 21. Measured (symbols) and simulated (continuous lines) for different at . of

Fig. 19. Linear regressions (continuous lines) of and as a . These regressions are used to analytically obtain function of the parameters in (20) and (21).

as a function

However, it is well known that is higher than for since there is no inversion channel, and thus, . The comparison between measured and simulated can be seen in Fig. 20. After determining from (24), related to can be accurately obtained by using RF data, fixing at a given value that guarantees the operation of the device in the linear region. The corresponding curves of measured and simulated are presented in Fig. 21, achieving a good agreement with experimental data. In this way, it is demonstrated that DC and RF measurements can be correlated through semi-empirical equations, taking into account the bias-dependent component of parasitic resistances. This allows characterizing the transistor with LDD regions, maintaining the coherence between RF and DC data. VI. CONCLUSION

Fig. 20. Measured (symbols) and simulated (continuous lines) for different . tion of

and for

as a func-

and

are fitting parameters. The linear regressions are presented in Fig. 19, from which and are determined for nm. The difference between and is due to the variation of the fabrication process when LDD regions are made and the design of the multi-fingered transistors in which drain and source have different areas. Finally, can be expressed as and

(23)

REFERENCES

or

(24) where is valid for

Model implementations for representing MOSFETs DC and high-frequency small-signal behavior were carried out in the linear operation region directly using -parameter measurements. Excellent model-experiment correlations were achieved at several bias conditions and for devices presenting different geometry. It was shown that MOSFET's parameters such as the drain-to-source conductance, the subthreshold swing, the effective gate length and the threshold voltage can be obtained using RF measured data allowing one to implement DC models in a consistent way. In order to verify this fact, the paper presented in the IMS 2015 was enriched here by showing the feasibility of representing current–voltage curves measured with a semiconductor-device analyzer using parameters obtained by processing experimental -parameters. In addition, thorough comparisons were also performed to point out the advantages of directly implementing DC models using experimental RF data.

. The previous equation , since it is not defined at .

[1] K. Terada, “Reconsideration of effective MOSFET channel length extracted from channel resistance,” in Proc. Int. Microelectron. Test Structures Conf., Udine, Italy, Mar. 2014, pp. 3–7. [2] A. Ortiz, F. J. García, J. Muci, A. Terán, J. J. Liou, and C. S. Ho, “Revisiting MOSFET threshold voltage extraction methods,” Microelectron. Rel., vol. 53, no. 1, pp. 90–104, Jan. 2013. [3] J. Kim, J. Lee, I. Song, Y. Yun, J. D. Lee, B. G. Park, and H. Shin, “Accurate extraction of effective channel length and source/drain series resistance in ultrashort-channel MOSFETs by iteration method,” IEEE Trans. Electron Devices, vol. 55, no. 10, pp. 2779–2784, Oct. 2008.

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[4] S. Lee, “A new RF capacitance method to extract the effective channel length of MOSFET's using -parameters,” in IEDM Tech. Dig., Hong Kong, 2000, pp. 56–59. [5] F. Zárate-Rincón, R. S. Murphy-Arteaga, and R. Torres-Torres, “Correction of DC extracted parameters for microwave MOSFETs based on -parameter measurements,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, May 2015, pp. 1–4. [6] J. A. Reynoso-Hernandez, F. E. Rangel-Patiño, and J. Perdomo, “Full RF characterization for extracting the small-signal equivalent circuit in microwave FET's,” IEEE Trans. Microw. Theory Techn., vol. 44, no. 12, pp. 2625–2633, Dec. 1996. [7] P. Magnone, V. Subramanian, B. Parvais, A. Mercha, C. Pace, M. Dehan, S. Decoutere, G. Groeseneken, F. Crupi, and S. Pierro, “Gate voltage and geometry dependence of the series resistance and of the carrier mobility in FinFET devices,” Microelectron. Eng., vol. 85, no. 8, pp. 1728–1731, Aug. 2008. [8] R. Torres, R. S. Murphy, and J. A. Reynoso, “Analytical model and parameter extraction to account for the pad parasitics in RF-CMOS,” IEEE Trans. Electron Devices, vol. 52, no. 7, pp. 1335–1342, Jul. 2005. [9] J. P. Raskin, G. Dambrine, and R. Gillon, “Direct extraction of the equivalent circuit parameters for the small signal SOI MOSFET's,” IEEE Microw. Guided Wave Lett., vol. 7, no. 12, pp. 408–410, Dec. 1997. [10] A. Pascht, M. Grözing, D. Wiegner, and M. Berroth, “Small-signal and temperature noise model for MOSFETs,” IEEE Trans. Microw. Theory Techn., vol. 50, no. 8, pp. 1927–1934, Aug. 2002. [11] G. Crupi, D. M. M.-P. Schreurs, B. Parvais, A. Caddemi, A. Mercha, and S. Decoutere, “Scalable and multibias high frequency modeling of multi fin FETs,” Solid-State Electron., vol. 50, no. 10/11, pp. 1780–1786, Nov./Dec. 2006. [12] J. Wood, D. Lamey, D. C. M. Guyonnet, D. Bridges, N. Monsauret, and P. H. Aaen, “An extrinsic component parameter extraction method for high power RF LDMOS transistors,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, Jun. 2008, pp. 607–610. [13] F. Zarate, G. A. Alvarez, R. Torres, R. S. Murphy, and S. Decoutere, “Characterization of RF-MOSFETs in common-source configuration at different source-to-bulk voltages from -parameters,” IEEE Trans. Electron Devices, vol. 60, no. 8, pp. 2450–2456, Aug. 2013. [14] R. Torres and R. S. Murphy, “Straightforward determination of smallsignal model parameters for bulk RF-MOSFETs,” in Proc. IEEE Int. Caracas Conf. on Devices, Circuits and Syst., Dominican Republic, Nov. 2004, pp. 14–18. [15] D. Schreurs, Y. Baeyens, B. Nawelaers, W. De Raedt, M. Van Hove, and M. Van Rossum, “ -parameter measurement based quasistatic large-signal cold HEMT model for resistive mixer design,” Int. J. Microw. Millimetre-Wave Comp.-Aided Eng., vol. 6, no. 4, pp. 250–258, Jul. 1996. [16] J. Alvarado, J. C. Tinoco, V. Kilchytska, D. Flandre, J. Raskin, A. Cerdeira, and E. Contreras, “Compact small-signal model for RF FinFETs,” in Proc. Int. Caribbean Conf. on Devices, Circuts and Syst., Playa del Carmen, Mar. 2012, pp. 1–4. [17] E. Torres, R. Torres, G. Valdovinos, and E. Gutiérrez, “A method to determine the gate bias-dependent and gate bias-independent components of MOSFET series resistance from -parameters,” IEEE Trans. Electron Devices, vol. 53, no. 3, pp. 571–573, Mar. 2006.

[18] K. Y. Lim and X. Zhou, “A physically based semi-empirical series resistance model for deep-submicron MOSFET I-V modeling,” IEEE Trans. Electron Devices, vol. 47, no. 6, pp. 1300–1302, Jun. 2000.

Fabián Zárate Rincón (S’14) received the B.S. degree in electronic engineering from the University of Quindio, Armenia, Colombia, in 2006, and the M.S. degree in electronics from the National Institute for Astrophysics, Optics and Electronics (INAOE), Puebla, Mexico, in 2012. He is currently pursuing the Ph.D. degree in electronics at INAOE. From 2006 to 2010, he was a Research Assistant with the University of Quindio. His research interest includes the study of semiconductor devices operating at microwave frequencies.

Reydezel Torres-Torres (S'01–M'06–SM'15) received the Ph.D. degree from from the National Institute for Astrophysics, Optics and Electronics (INAOE), Puebla, Mexico. He is a senior researcher in the Electronics Department of INAOE in Mexico. He has authored more than 70 journal and conference papers and directed six Ph.D. and 15 M.S. theses, all in experimental high-frequency characterization and modeling of materials, interconnects, and devices for microwave applications and has worked for Intel Laboratories in Mexico and IMEC in Belgium.

Roberto S. Murphy-Arteaga (M'92–SM'02) received the B.Sc. degree in physics from St. John's University, Collegeville, MN, USA, and the M.Sc. and Ph.D. degrees from the National Institute for Research on Astrophysics, Optics and Electronics (INAOE), Tonantzintla, Puebla, México. He has taught graduate courses at the INAOE since 1988, totaling over 100 taught undergrad and graduate courses. He has given over 80 talks at scientific conferences and directed seven Ph.D. theses, 13 M.Sc. and 2 B.Sc. theses. He has published more than 120 articles in scientific journals, conference proceedings and newspapers, and is the author of a text book on electromagnetic theory. He is currently a Senior Researcher with the Microelectronics Laboratory, INAOE, and the Director of Research of the INAOE. His research interests are the physics, modeling and characterization of the MOS Transistor and passive components for high frequency applications, especially for CMOS wireless circuits, and antenna design. He is the President of ISTEC, a member of the Mexican Academy of Sciences, and a member of the Mexican National System of Researchers (SNI).

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Bayesian Optimization for Broadband High-Efficiency Power Amplifier Designs Peng Chen, Student Member, IEEE, Brian M. Merrick, Member, IEEE, and Thomas J. Brazil, Fellow, IEEE

Abstract—This paper proposes a novel, optimization-oriented strategy for the design of broadband, high-efficiency power amplifiers (PAs) using Bayesian optimization (BO). The optimization algorithm optimizes the drain waveforms by maximizing the fundamental output power while minimizing the harmonic and dissipated components. The optimization process is automated using simulation software. Circuit-based BO and electromagnetic-based (EM-based) BO are applied to design 10 and 30 W PAs. The 10 W PA designed using circuit-based BO achieves a drain efficiency higher than 60% with output power greater than 39.8 dBm from 1.5 to 2.5 GHz, while the 30 W PA designed using EM-based BO offers a drain efficiency higher than 57% with output power greater than 43.8 dBm across the band. Upon comparison of results, it is revealed that the proposed strategy outperforms a commercial electronic design automation (EDA) software's built-in optimizer, thus demonstrating that the EM-based BO is well-suited to the challenge of high power designs. Index Terms—Bayesian optimization, broadband, drain waveforms, Gaussian process, high efficiency, power amplifier (PA).

I. INTRODUCTION

M

ODERN wireless communication systems, which have high transmission rates and spectral efficiency, are creating a growing demand for broadband high-efficiency power amplifiers (PAs). A series of high-efficiency PA modes, such as Class-F and Class-J, have been developed over the past few decades [1]–[5]. These new PA modes have received widespread interest since they allow designers to analyze PA performance from the point of view of waveform engineering [6], [7]. The non-overlapping of the drain voltage and drain current waveforms has been recognized as a key criterion in achieving high efficiency in PA designs. Based on this concept, continuous modes have been studied in order to give greater flexibility to the drain waveforms for wideband designs [8]–[12]. In practical PA designs, the desired waveforms are often achieved through tuning the harmonic impedances presented by the matching networks. In order to realize a broadband design, the matching networks require multiple

Manuscript received July 01, 2015; revised September 10, 2015; accepted October 17, 2015. Date of publication November 11, 2015; date of current version December 02, 2015. This work was supported by a research grant from Science Foundation Ireland (SFI) under its Investigators Programme 2012, Grant 12/IA/1267. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. The authors are with the School of Electrical and Electronic Engineering, University College Dublin, Dublin 4, Ireland (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495360

sections of transmission lines, generally greater than 10 in total. This tuning work places a great demand on design experience, especially when the efficiency needs to remain at a high level across the band. The broadband matching problem can be regarded as a high-dimensional optimization problem. In view of this, it is helpful to apply an optimization-oriented technique to tackle this problem. Although commercial EDA software packages, such as ADS, AWR, etc., are quite useful in microwave circuit optimization, their built-in optimizers may fail to find a satisfactory solution when the dimension of variables is very high or the optimization goals are stringent. To overcome this limitation, it is necessary to develop modifiable, intelligent algorithms tailored to specific optimization tasks. To date, a number of optimization algorithms, such as space mapping [13], [14], artificial neural networks [15]–[17], and shape-preserving response [18], have been successfully exploited to the applications of filters, couplers, antennas, and transistor models. In the field of intelligent algorithms, Bayesian optimization (BO) is an example of a supervised learning algorithm, used for the global optimization of an expensive objective function in a high-dimensional space [19], [20]. It is particularly useful when the objective function is non-convex and multi-modal, especially when closed-form expressions for the objective function or its derivatives do not exist. Since the objective function is an unknown “black-box,” BO typically assumes that the function is sampled from Gaussian processes (GP) [21]. Due to its good properties of flexibility and tractability, BO is able to capture the characteristics of the unknown function by building a GP model based on the training data. After that, BO exploits an acquisition function to determine the next point to evaluate [22]. By repeating these steps, BO has the ability to achieve the global optimum value for the unknown objective function. In [23], we applied Gaussian processes regression (GPR) to optimize the harmonic impedances for a continuous Class-F power amplifier design. Compared to [23], this paper expands GPR into the framework of BO, and applies BO to optimize the drain waveforms of an active circuit rather than of a passive circuit. In addition, the dimension of variables is increased up to 28, which is double that of [23]. Moreover, circuit simulation and electromagnetic (EM) simulation-based optimization are both implemented to design 10 and 30 W broadband high-efficiency PAs. In this paper, Section II first discusses the high-efficiency PA theory which serves as the objective function for the PA design optimization. Section III is devoted to describing the framework of Bayesian optimization, which is composed of Bayesian inference, modeling, prediction, and acquisition

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100% efficiency requires (3), (4) to satisfy the following condition [2]:

(5)

III. BAYESIAN OPTIMIZATION

Fig. 1. Power conversion in a PA system.

A. Bayesian Inference functions. Section IV describes the proposed automatic optimization-oriented strategy for PA design. Section V validates the proposed method through the presentation of 10 and 30 W broadband high-efficiency PA designs. In this section, the algorithm convergence and measured results are shown to demonstrate that the proposed method is well-suited to broadband high-efficiency PA designs. Comparison designs using the ADS simulated annealing optimizer are also provided. Finally, in Section VI, conclusions are given. II. HIGH-EFFICIENCY PA THEORY

In this paper, the problem of interest is to find the best design parameters for the matching networks such that the PA achieves high efficiency. The optimization problem can be viewed as a probability event , where denotes the design parameters and denotes the observed PA efficiency. Suppose that the “black-box” function is defined as (6) where is a weighting vector and is a feature space mapping. Given a set of training design parameters, the accumulated observations are . Let's define as the prior probability which represents the belief about prior to the observation of , and as the likelihood function which expresses how probable high efficiency is given the function . According to Bayes' theorem, the posterior probability for the function after observing takes the form

The high efficiency PA modes, for instance the Class, achieve high efficiency due to the non-overlapping voltage and current waveforms present at the internal drain. It is of no doubt that waveform engineering has become a critical guiding principle in high-efficiency PA designs. In order to fully understand how the drain voltage and drain current significantly affect the efficiency, it is necessary to analyze the power conversion in a PA system. In Fig. 1, the PA receives power both from the input signal and the DC supply. Since the input power is assumed to be totally dissipated in the input network and the transistor, it does not contribute directly to the output power. In (1), the DC power supplied to the device exits the device in three parts: in the form of heat dissipated in the active devices, , as a radio frequency (RF) signal at the fundamental and RF signals at the harmonics . As a consequence, the drain efficiency is calculated by (2) and and are defined in (3), (4)

indicates the probability to evaluate the where weighting vector after the observations. Although the integrated term (8) cannot be expressed in a closed-form, it is a normalization constant. Thus, the posterior probability is proportional to the likelihood of given multiplied by the prior probability of

(1)

(9)

(2)

In practice, the maximum a posteriori (MAP) metric is a common approximation used to estimate (9), and is written in the following form [19]:

(7) (8)

(3) (10) (4) B. Modeling and Prediction and represent the drain voltage and drain current where at the th harmonic with intersection angle . It was pointed out by Colantonio that a PA with zero dissipated power, i.e., non-overlapping drain waveforms, may only have an efficiency of approximately 80%, depending on the waveforms. The ideal

1) “Black-Box” Modeling: As discussed before, a “black-box” function which represents the PA system needs to be found for the optimization design. The weighting vector is determined by fitting (6) to the training data. When training the model, an error term is introduced to measure the misfit

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Fig. 2. Exponential family distribution.

Fig. 3. GP model with training data.

between the function and the training data. One simple error function, which is widely applied [19], is given by

Substituting (12) into (13) and (14), the negative log likelihood function for (13) and (14) are derived as

(11) is a penalty term used to control the over-fitting where phenomenon, called regularization, and governs the balance between the sum-of-squares error term and the regularization term. As can be seen from (10), the probability distribution plays a critical role in Bayesian optimization. There are many distributions which can be used to form building blocks for complex models, such as the beta distribution, the Dirichlet distribution, and the exponential family distribution [19]. Fig. 2 describes the variation in correlation with the distance between two points for the exponential family distribution. The correlation decreases when two points move far away, while it increases when they are close. In the case of , it is called a Gaussian distribution, which has a smoothness correlation over different points. The Gaussian distribution is written in the following form:

(15)

(16) Similarly, taking the negative logarithm of (10) and combining with (15) and (16), the maximum of the posterior probability is equal to the minimum of (17)

Due to these good properties, a Gaussian distribution is wellsuited for the distribution of continuous variables in a high-dimensional space [20]. Assuming that the observed high efficiency, , has a Gaussian distribution with a mean equal to , given the design parameters, , then the likelihood function of (10) is expressed as

Comparing (11) with (17), we can see that maximizing the posterior probability is equivalent to minimizing the error function with the regularization parameter . As a consequence, an accurate model for the PA system can be constructed based on the training data. This regression method is known as Gaussian processes regression (GPR), fitting the training data with a mean and a standard deviation , as depicted in Fig. 3. 2) Prediction from GP Model: A Gaussian process is a probability distribution over the function such that the values of evaluated at different points jointly have a Gaussian distribution. Suppose that can be separated into two joint Gaussian distributed subsets, and

(13)

(18)

(12)

where is the inverse variance of the Gaussian distribution. The prior probability is commonly selected as a Gaussian distribution with a mean equal to 0 and an inverse variance of , and can be written as (14)

(19) (20) where (19) and (20) represent the corresponding partitions of the mean vector and the covariance matrix , respectively.

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In Bayesian optimization, there are four commonly-used acquisition functions: probability of improvement (PI), expected improvement (EI), upper confidence bound, and entropy search [24]–[27]. In this paper, the first two acquisition functions are applied in order to search for the optimal values for the design parameters. 1) Probability of Improvement: PI evaluates at the point where the improvement will most likely occur [24]. However, PI focuses on local optimization such that the points have high uncertainty. Given the current maximum point , PI is calculated by

TABLE I SUMMARY OF BAYESIAN OPTIMIZATION

The conditional distribution lowing mean and covariance:

is expressed with the fol(30) (21) (22)

When applying the GP model for new predictions, it is necessary to add a noise term which serves as the unmatched error. Here, the function (6) is modified to the following form: (23) where is specified by a mean function and a kernel function , such that , and the noise also has a Gaussian distribution . According to the GP properties, the values from the observation will have a multivariate Gaussian distribution , with the kernel matrix given by .. .

..

.

.. .

(24)

where the kernel function denotes the covariance matrix between and . The new value , which is predicted from the GP model at the new point has a joint Gaussian distribution with (25)

where represents the current best value, and are calculated from (27), (28), and is the normal cumulative distribution function. 2) Expected Improvement (EI): EI has been shown to be an efficient criterion for finding the global optimum in many “black-box” functions [25]. It allows us to make a tradeoff between the local optimization and the global search. This acquisition can be computed analytically as if if (31) (32) where denotes the probability density function. In (31), the first term chooses the points where the mean is high while the second term targets the points where the variance is large. Considering constraint conditions for the objective function, PI can be used to calculate the probability of being greater than the constraint limit [24]. In this situation, one model is built for the objective function; the other is for the constraint function. Since the two models are independent, the new point in (29) is obtained by maximizing (33)

(26) Referring to (21) and (22), the predicted data the GP model is specified as a Gaussian distribution

from (27) (28)

C. Acquisition Functions for Searching

Table I summarizes the algorithm steps of Bayesian optimization. It is worth noting that two major choices significantly affect the global search of Bayesian optimization. The first is selecting an appropriate GP distribution and a kernel function for the modeling; the second is choosing an acquisition function to guide the search. IV. OPTIMIZATION-ORIENTED STRATEGY

After building the GP model, a non-trivial task is how to search for the next point of interest from the constructed GP model. Acquisition functions are exploited to determine where to next evaluate the GP model. Since our goal is to find a set of design parameters such that high efficiency is obtained, the objective function is defined as (29)

The automatic optimization strategy for the PA designs is accomplished by combining ADS with MATLAB or R. As is shown in Fig. 4, ADS is used to run the PA simulation so that training data is obtained. MATLAB or R is responsible for the Bayesian optimization since there are many related codes written in these two programming languages, which are available for public use. MATLAB or R starts the Bayesian optimization when the training data is exported from ADS.

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Fig. 4. Automatic optimization strategy.

Fig. 5. Matching network topology.

Fig. 6. Optimization process for the PA designs.

Similarly, ADS activates the simulation as soon as it receives new design parameters. Finally, the optimization strategy outputs the best design parameters to the users after the iterations finish. In this paper, the design parameters are the widths and the lengths of the matching network transmission lines. The simplified real frequency technique (SRFT) is exploited to determine the matching network topology and the initial guess for the design parameters [28], [29]. The S-parameters of the transistor are used as the inputs to the SRFT in order to generate this initial guess. As shown in Fig. 5, there are six steppedimpedance transmission lines and a bias line for both the input and output matching networks, resulting in 28 design parameters in total. The output targets in the optimization are the fundamental output power, and the harmonic and dissipated components described in Section II. It is important to note that it is the voltage and current waveforms at the current-generator plane that we are interested in optimizing, rather than the waveforms

Fig. 7. Fabricated 10 W PA designs. (a) EM-based BO with 50 training data points. (b) Circuit-based BO with 50 training data points. (c) Circuit-based BO with 200 training data points.

seen at the package plane. The objective function is defined as the root mean square (RMS) of the fundamental output power across the band (34) where denotes the number of the frequency points. The constraint function is the sum of the RMS of the harmonic and dissipated power (35) Harmonics greater than the fifth are neglected in this paper due to their minor impact upon efficiency improvement. As per (2)

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Fig. 8. Convergence for the 10 W PA designs.

Fig. 9. Simulated intrinsic drain waveforms at 2 GHz for the 10 W PA design using EM-based BO.

and (5), this paper optimizes the drain waveforms by maximizing (34) while minimizing (35), guaranteeing that high efficiency is achieved in a wide band. Since the 28-dimensional space is very high, it requires a vast number of training data to capture the characteristics of the PA system, leading to quite expensive computation during the optimization. In order to balance the model accuracy and the computational time, this paper uses sub-models to search for design parameters instead of using the whole model. A sub-model is constructed around the current point, representing a portion of the whole model. After finding a better point, a new sub-model is constructed and is then applied to predict the next point. In this way, the sub-models are able to find the optimum point for the whole model. However, it should be pointed out that there is a risk of the optimization converging on a local minimum, rather than the desired global minimum. Fig. 6 further describes the optimization process more specifically for the PA designs. First, we set the iteration number and apply the SRFT to obtain the initial guess for the design parameters. The sampling points are created within a 10% range of the current point using the Latin hypercube sampling technique [30]. By sweeping the sampling design parameters, ADS runs the simulation and exports the RMS of those in (34) and (35). With the available training data, the optimization algorithm builds the GP sub-models and predicts new design param-

Fig. 10. Compared measurements for 10 W PA designs. (a) Drain efficiency. (b) Output power. (c) Gain.

eters by maximizing the acquisition function in (33) using the DIRECT algorithm [31]. For the sake of good convergence, the optimization algorithm rejects predicted points which worsen performance and keeps the current best one. In order to avoid sub-optimal local minima in the optimization, the random seed is changed such that the sampling points and the sub-models are renewed at each iteration. The optimization algorithm repeats the above steps and finally outputs the optimal design parameters when the iterations are terminated. In this way, we apply Bayesian optimization to optimize a PA design automatically.

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Fig. 11. Measurement setup.

V. PRACTICAL PA OPTIMIZATION A. Broadband 10 W PA Design In order to demonstrate the validity of this design approach, we optimized the designs of broadband high-efficiency 10 W PA using circuit- and EM-based Bayesian optimization, respectively. Cree CGH40010F Gallium Nitride (GaN) high-electron mobility transistors (HEMTs) were used for the designs. In the circuit-based BO, the sub-models were built with the training data from circuit simulation. Similarly, in the EM-based BO, EM simulation supplied the training data for building the submodels, which are more accurate than those of the circuit simulation. Considering that the EM simulation requires expensive computational time, we used 50 training data points to build the GP sub-models in the EM-based BO. In order to test the sensitivity of the optimization to the number of training data, 50 and 200 training data points were used in the circuit-based BO. For comparison, the ADS simulated annealing optimizer with 1000 iterations was employed to optimize the PAs while keeping the same variable settings and objective functions. The transistor model is a critical part of the “black-box” modeling, affecting the deviation between the simulated results and measurements to a great extent. Clearly, if the model is not of sufficient accuracy, then the optimization process will not result in good performance upon measurement. Cree's dynamic load-line model has been shown in a large number of papers to be of good accuracy [10], [11], and, hence, was used for the designs. Additionally, the model in question gives access to the intrinsic current and voltage waveforms, as desired. The PA designs were implemented on Taconic RF35 substrate with 3.5 and a thickness of 1.52 mm. The iteration number was set to 20 and the initial design parameters of the matching networks of Fig. 5 were obtained using the SRFT, , where [5.75, 1.63, 15.30, 2.21, 27.91, 26.15, 2.09, 11.34, 1.57, 11.95, 1.83, 5.78, 1, 1] mm, and [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 20, 20] mm. The bounded range for the width in the matching networks was 0.7 mm 30 mm, corresponding to the characteristic impedance within 9 105 on the RF35 board. Following the optimization process shown in Fig. 6, the 10 W PAs were optimized from 1.5 to 2.5 GHz, with a step of 50 MHz. The optimization process ran on a computer featuring an Intel Core i7-3770 CPU @ 3.40 GHz with 16.0 GB RAM. After 20 iterations, the EM-based BO outputted the optimized parameters, =[5.07, 1.51, 15.38, 2.79, 22.69, 26.12, 2.16,

Fig. 12. Fabricated 30 W PA designs. (a) EM-based BO with 50 training data points. (b) Circuit-based BO with 50 training data points. (c) Circuit-based BO with 200 training data points.

11.19, 1.60, 10.63, 1.43, 5.96, 1.28, 1.11] mm, and [5.20, 5.29, 5.47, 4.42, 5.76, 6.24, 3.54, 7.90, 4.31, 4.03, 5.66, 4.94, 18.54, 15.46] mm. In the circuit-based BO with 50 training data points, the optimized design parameters were [4.77, 1.57, 18.24, 2.54, 28.58, 27.92, 2.19, 13.08, 1.46, 13.86, 1.42, 6.62, 1.41, 1.23] mm, and [5.26, 5.55, 5.68, 3.41, 7.67, 6.33, 4.25, 7.89, 4.98, 4.02, 5.43, 4.67, 18.89, 19.22] mm. As for the circuit-based BO with 200 training data points, we got the optimized parameters, [5.01, 1.60, 20.14, 3.02, 26.05, 28.62, 2.10, 12.62, 1.57, 12.81, 1.64, 4.97, 1.32, 1.03] mm, and [4.82, 4.55, 5.10, 2.96, 7.05, 7.16, 4.07, 9.80, 5.88, 4.46, 5.09, 5.37, 22, 21.04] mm. Fig. 7 shows the fabricated PAs which were optimized using the three approaches described above. The PAs were biased at a drain voltage of 28 V with a quiescent drain current of 70 mA. The convergence of the proposed optimization algorithm is shown in terms of a normalized error in Fig. 8. The

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Fig. 13. Convergence for the 30 W PA designs.

normalized errors drop considerably over the 20 iterations, indicating that the algorithm converges very quickly. It was observed that the normalized error reduced when the training data number was increased from 50 to 200. Therefore, increasing the training data number is helpful in obtaining better design parameters from accurate models. The simulated drain waveforms of the PA using EM-based BO, depicted in Fig. 9, show only a small amount of overlap, suggesting high performance. Fig. 10 shows the measurement results of the optimized PAs, together with comparison designs created using the SRFT and the ADS simulated annealing optimizer. These results were obtained with the measurement setup shown in Fig. 11, when an available input power of 28 dBm was provided across the band. The optimized PAs all showed large improvements compared with the initial design using the SRFT. The optimized PA using EM-based BO delivered drain efficiency above 60.0% from 1.5 to 2.5 GHz, with output power greater than 39.9 dBm and gain larger than 11.8 dB. The PA using circuit-based BO with 200 training data points showed better performance than that using 50 training data, corresponding to the lower normalized error of Fig. 8. It achieved greater than 60.8% drain efficiency over the entire band, with output power greater than 39.8 dBm and gain larger than 11.4 dBm. Although the ADS simulated annealing optimizer achieved a drain efficiency of greater than 61.4% over the band, the output power decreased to 39.1 dBm, with gain dropping down to 11.0 dB. It should be noted that the EM-based design provided superior output power and gain when compared with the other designs, particularly at the upper end of the frequency scale.

Fig. 14. Compared measurements for 30 W PA designs. (a) Drain efficiency. (b) Output power. (c) Gain.

B. Broadband 30 W PA Design We also applied the above approaches to optimize the designs of 30 W PA operating from 1.5 to 2.5 GHz. Cree CGHV40030F GaN HEMTs were used for the designs. Again, the Cree's dynamic load-line model was utilized in the ADS simulation. The SRFT provided the initial design parameters, [7.01, 1.44, 20.79, 3.07, 26.86, 30, 1.66, 20.29, 4.01, 17.61, 3.08, 3.66, 0.8, 0.8] mm, and [7, 7, 7, 7, 7, 7, 6.5, 6.5, 6.5, 6.5, 6.5, 6.5, 15, 20] mm. As shown in Fig. 12, the first PA using EM-based BO had the dimensions of [5.51, 1.84, 11.71, 1.50, 19.14,

24.64, 1.67, 17.57, 3.50, 18.40, 2.63, 6.51, 1.03, 1.17] mm, and [3.37, 3.92, 5.65, 3.20, 11.79, 5.20, 3.34, 8.59, 8.92, 5.43, 8.28, 8.15, 20.21, 20.49] mm. The second PA using circuit-based BO with 50 training data points had the optimized parameters, [8.33, 1.59, 17.11, 2.61, 27.13, 28.98, 1.55, 19.85, 3.62, 18.30, 2.85, 5.90, 1.04, 1.22] mm, and [6.82, 7.41, 5.77, 6.60, 8.09, 6.96, 4.14, 7.60, 7.8, 5.77, 7.86, 7.12, 18.68, 15.59] mm. As for the last design using circuit-based BO with 200 training data points, the optimized parameters were,

CHEN et al.: BAYESIAN OPTIMIZATION FOR BROADBAND HIGH-EFFICIENCY POWER AMPLIFIER DESIGNS

Fig. 15. Simulated intrinsic drain waveforms at 2 GHz for the 30 W PA design using EM-based BO.

TABLE II COMPARISONS OF THE 30 W PA DESIGNS

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as minimize the harmonic and dissipated components, resulting in similar non-overlapping drain waveforms. The optimization algorithm builds the GP sub-models for the PA and utilizes an acquisition function to predict design parameters with which high efficiency is achieved. The 10 and 30 W PA designs were optimized using circuit-based BO and EM circuit-based BO. Measured results show that the 10 W PA using circuit-based BO with 200 training data points obtained better than 60% drain efficiency and greater than 39.8 dBm output power from 1.5 to 2.5 GHz. In the 30 W PA designs, EM-based BO performed much better than circuit-based BO, due to the higher power designs being more sensitive to simulation accuracy. The 30 W PA using EM-based BO operated at a drain efficiency of greater than 57%, with output power greater than 43.8 dBm across the band. Comparison results show that the proposed strategy greatly surpasses the ADS simulated annealing optimizer in terms of the gain and output power achieved across the band, with comparable efficiency shown with both methods. Moreover, it should be stressed that the performances of the optimized PAs also rely on the accuracy of the transistor models applied in the ADS simulation. ACKNOWLEDGMENT

The measurements are compared from 1.5 to 2.5 GHz. The cost time is the approximate average time for each optimization iteration.

[7.54, 1.58, 17.76, 3.70, 27.06, 28.52, 1.91, 19.93, 3.56, 16.64, 2.91, 5.75, 0.91, 1.26] mm, and [7.22, 6.04, 6.0, 6.56, 9.23, 6.17, 4.58, 7.86, 7.60, 5.39, 6.35, 5.45, 17.33, 17.33] mm. The algorithm convergence of the 30 W PA designs are shown in Fig. 13. The 30 W PAs were biased at a drain voltage of 50 V with a quiescent drain current of 95 mA, and were measured with 32 dBm available input power. As shown in Fig. 14, the optimized PA using EM-based BO obtained drain efficiency greater than 57.0% across the band, with output power greater than 43.8 dBm and gain larger than 11.6 dBm. As high power PAs have a smaller range of ideal harmonic impedances, circuit-based optimization designs show worse performances. The PA using the circuit-based BO with 200 training data gave a minimum drain efficiency of 51.5%, which was still better than that of using the ADS simulated annealing optimizer. The EM-based PA again showed significantly greater output power and gain than the other designs. The non-overlapping drain waveforms for the 30 W PA design using EM-based BO are shown in Fig. 15. Table II gives the comparison of the 30 W PAs using the above different methods. From the compared results, we can see that EM-based BO is more preferable than circuit-based BO in high power designs although it requires expensive computational time. VI. CONCLUSION A Bayesian optimization-based method is presented in this paper to design broadband high-frequency power amplifiers from 1.5 to 2.5 GHz. This method is implemented automatically by combining ADS with MATLAB or R programming. Its aim is to maximize the fundamental output power, as well

The authors would like to acknowledge the support of Cree, Inc. REFERENCES [1] S. C. Cripps, RF Power Amplifier for Wireless Communications, 2nd ed. Boston, MA, USA: Artech House, 2006. [2] P. Colantonio, F. Giannini, and E. Limiti, High Efficiency RF and Microwave Solid State Power Amplifiers. New York, NY, USA: Wiley, 2009. [3] P. Wright, J. Lees, J. Benedikt, P. J. Tasker, and S. C. Cripps, “A methodology for realizing high efficiency Class-J in a linear and broadband PA,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 12, pp. 3196–3204, Dec. 2009. [4] S. C. Cripps, P. J. Tasker, A. L. Clarke, J. Lees, and J. Benedikt, “On the continuity of high efficiency modes in linear RF power amplifiers,” IEEE Microw. Wirel. Compon. Lett., vol. 19, no. 10, pp. 665–667, Oct. 2009. [5] M. Eron, B. Kim, F. Raab, R. Caverly, and J. Staudinger, “The head of the class,” IEEE Microw. Mag., vol. 12, no. 7, pp. S16–S33, Dec. 2011. [6] P. J. Tasker and J. Benedikt, “Waveform inspired models and the harmonic balance emulator,” IEEE Microw. Mag., vol. 12, no. 2, pp. 38–54, Apr. 2011. [7] Y. Y. Woo, Y. Yang, and B. Kim, “Analysis and experiments for high efficiency Class-F and inverse Class-F power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 5, pp. 1969–1974, May 2006. [8] V. Carrubba, A. L. Clarke, M. Akmal, J. Lees, J. Benedikt, P. J. Tasker, and S. C. Cripps, “On the extension of the continuous class-F mode power amplifier,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 5, pp. 1294–1303, May 2011. [9] V. Carrubba, A. L. Clarke, M. Akmal, J. Lees, J. Benedikt, S. C. Cripps, and P. J. Tasker, “The continuous inverse class-F mode power amplifier with resistive second-harmonic impedance,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1928–1936, Jun. 2012. [10] N. Tuffy, L. Guan, A. Zhu, and T. J. Brazil, “A simplified broadband design methodology for linearized high-efficiency continuous Class-F power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1952–1963, Jun. 2012. [11] K. Chen and D. Peroulis, “Design of broadband highly efficient harmonic-tuned power amplifier using in-band continuous Classmode transferring,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 4107–4116, Dec. 2012. [12] S. Preis, D. Gruner, and G. Boeck, “Investigation of class-B/J continuous modes in broadband GaN power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., 2012, pp. 1–3.

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[13] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 1, pp. 337–361, Jan. 2004. [14] S. Koziel, J. W. Bandler, and K. Madsen, “A space mapping framework for engineering optimization: Theory and implementation,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 10, pp. 3721–3730, Oct. 2006. [15] Q. J. Zhang, K. C. Gupta, and V. K. Devabhaktuni, “Artificial neural networks for RF and microwave design—From theory to practice,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 4, pp. 1339–1350, Apr. 2003. [16] J. E. Rayas-Sánchez, “EM-based optimization of microwave circuits using artificial neural networks: The state-of-the-art,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 1, pp. 420–435, Jan. 2004. [17] L. Zhang, K. Bo, Q.-J. Zhang, and J. Wood, “Statistical space mapping approach for large-signal nonlinear device modeling,” in 36th Eur. Microw. Conf. Dig., Manchester, U.K., Sep. 2006, pp. 676–679. [18] S. Koziel, “Shape-preserving response prediction for microwave design optimization,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 11, pp. 2829–2837, Nov. 2010. [19] C. M. Bishop, Pattern Recognition and Machine Learning. New York, NY, USA: Springer, Aug. 2006. [20] E. Brochu, V. M. Cora, and N. de Freitas, “A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning,” Univ. British Columbia, Vancouver, BC, Canada, Tech. Rep., 2010. [21] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. Cambridge, MA, USA: MIT Press, 2006. [22] J. Snoek, H. Larochelle, and R. P. Adams, “Practical Bayesian optimization of machine learning algorithms,” presented at the Neural Inform. Process. Syst., Lake Tahoe, CA, USA, 2012. [23] P. Chen and T. J. Brazil, “Gaussian processes regression for optimizing harmonic impedance trajectories in continuous Class-F power amplifier design,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2015, pp. 1–3. [24] A. I. J. Forrester and A. J. Keane, “Recent advances in surrogate-based optimization,” Progr. Aerosp. Sci., vol. 45, pp. 50–79, 2009. [25] D. R. Jones, M. Schonlau, and W. J. Welch, “Efficient global optimization of expensive black box functions,” J. Global Optimization, vol. 13, pp. 455–492, 1998. [26] N. Srinivas, A. Krause, S. Kakade, and M. Seeger, “Gaussian process optimization in the bandit setting: No regret and experimental design,” presented at the Int. Conf. Machine Learning, Haifa, Israel, 2010. [27] P. Hennig and C. J. Schuler, “Entropy search for information-efficient global optimization,” J. Mach. Learning Res., vol. 13, pp. 1809–1837, 2012. [28] B. S. Yarman, Design of Ultra Wideband Power Transfer Networks. New York, NY, USA: Wiley, 2010. [29] B. S. Yarman and H. J. Carlin, “A simplified “real frequency” technique applied to broadband multistage microwave amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 30, no. 12, pp. 2216–2222, Dec. 1982.

[30] M. D. McKay, R. J. Beckman, and W. J. Conover, “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,” Technometrics, vol. 21, no. 2, pp. 239–245, 1979. [31] D. E. Finkel, “Direct optimization algorithm user guide,” North Carolina State Univ., Raleigh, NC, USA, CRSC Tech. Rep. CRSC-TR03-11, Mar. 2003.

Peng Chen (S'15) received the B.E. degree in communication engineering and the M.E. degree in electronic engineering from Harbin Institute of Technology, Harbin, China, in 2010 and 2012, respectively. He is currently pursuing the Ph.D. degree in electronic engineering from the RF & Microwave Research Group, University College Dublin, Dublin, Ireland. His research interests include nonlinear device modeling and optimization algorithms applied to power amplifier design.

Brian M. Merrick (S'12–M'15) received the B.E. and Ph.D. degrees from University College Dublin (UCD), Dublin, Ireland, in 2010 and 2015, respectively. He is currently a postdoctoral researcher with the RF and Microwave Research Group, UCD. His research interests include device characterization and nonlinear modeling, and the design of high-efficiency, broadband power amplifiers.

Thomas J. Brazil (F'03) received the Ph.D. degree from the National University of Ireland, Dublin, Ireland, in 1977. He is head of the RF and Microwave Research Group, UCD. He holds the Chair of Electronic Engineering at the School of Electrical and Electronic Engineering, University College Dublin (UCD). His research interests include the fields of nonlinear modelling and characterization techniques at the device, circuit, and system level within high frequency electronics. Dr. Brazil was elected a Member of the Royal Irish Academy (RIA) in 2004. He is a serving member of IEEE MTT-S AdCom.

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Theory and Implementation of RF-Input Outphasing Power Amplification Taylor W. Barton, Member, IEEE, and David J. Perreault, Fellow, IEEE

Abstract—Conventional outphasing power amplifier systems require both a radio frequency (RF) carrier input and a separate baseband input to synthesize a modulated RF output. This work presents an RF-input/RF-output outphasing power amplifier that directly amplifies a modulated RF input, eliminating the need for multiple costly IQ modulators and baseband signal component separation as in previous outphasing systems. An RF signal decomposition network directly synthesizes the phase- and amplitude-modulated signals used to drive the branch power amplifiers (PAs). With this approach, a modulated RF signal including zero-crossings can be applied to the single RF input port of the outphasing RF amplifier system. The proposed technique is demonstrated at 2.14 GHz in a four-way lossless outphasing amplifier with transmission-line power combiner. The RF decomposition network is implemented using a transmission-line resistance compression network with nonlinear loads designed to provide the necessary amplitude and phase decomposition. The resulting proof-of-concept outphasing power amplifier has a peak CW output power of 93 W, peak drain efficiency of 70%, and performance on par with a previously-demonstrated outphasing and power combining system requiring four IQ modulators and a digital signal component separator. Index Terms—Base stations, Chireix, LINC, load modulation, outphasing, power amplifier (PA), signal component separator (SCS), transmission-line resistance compression network (TLRCN).

I. INTRODUCTION

O

UTPHASING architectures use phase-shift control of multiple saturated or switched-mode branch power amplifiers (PAs) to create a modulated radio frequency (RF) output. Interaction through a lossless non-isolating power combiner produces load modulation of the branch amplifiers, which in turn modulates the system output power. When realized with efficient saturated or switched-mode branch amplifiers, the outphasing approach has the potential to provide high operating efficiency over a wide range of output power levels, making it Manuscript received June 30, 2015; revised September 22, 2015; accepted October 23, 2015. Date of publication November 10, 2015; date of current version December 02, 2015. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. T. W. Barton is with the Department of Electrical Engineering, The University of Texas at Dallas, Richardson, TX 75080 USA. (e-mail: taylor.barton@ utdallas.edu). D. J. Perreault is with the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495358

ideal for high peak-to-average power ratio (PAPR) signals such as those found in modern communications systems. Examples of this approach include Chireix power combining systems [1]–[5], and the multi-way lossless outphasing system [6]–[11] which improves upon the achievable operating efficiency of the Chireix system by providing nearly ideal resistive loading conditions to the branch PAs. Since its introduction by Chireix in 1935, a major limitation in the outphasing approach has been the need for signal component separation of the desired RF signal into multiple phase- and amplitude-modulated signals driving the branch PAs. For example, in an early commercial outphasing amplifier, Ampliphase, two sets of dynamic phase-modulating amplifiers were used to modulate the carrier signal by the (baseband) audio signal and drive the branch PAs [12]. This property has made outphasing systems less attractive compared to the Doherty approach with its ability to operate directly on a modulated RF input signal [13], [14]. The signal component separator (SCS) and the associated need for multiple baseband-to-RF upconverting paths incurs excessive complexity, cost, and power consumption to the outphasing system compared to power amplifiers that can operate directly on a modulated RF input. The digital signal decomposition requirement can also complicate any digital correction scheme. SCS approaches performing the required computation have been proposed in various domains, including analog baseband [15] and analog IF operating in feedback [16]–[18] or open-loop [19] topologies. The most common approach in modern outphasing systems, however, is to use some form of digital signal processing based on lookup tables or other means of computing the nonlinear relationship between the input signal and the phase-modulated branch PA drives [20]–[25]. As illustrated in the block-diagram comparison in Fig. 1, performing the signal component separator (SCS) in the RF domain instead of the digital domain allows for decoupled design of the digital and RF elements, reduced system complexity, and for the resulting outphasing system to be treated as a “black box” drop-in replacement for another PA. This work presents an RF-input/RF-output outphasing PA, shown in block diagram form in Fig. 2. An RF-domain signal decomposition network directly synthesizes the multiple branch PA drive signals from a modulated RF input signal, performing both the phase and amplitude modulation required for outphasing systems to control the output power over a wide range [3], [9]. This signal decomposition network (or RF-domain signal component separator) is based on use of a resistance compression network (RCN) terminated

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Fig. 2. Conceptual schematic of the four-way RF-input outphasing system, with example waveforms sketched assuming the PA is operating in the outphasing regime. The phasor relationship among the four branch voltages – is a result of the decomposition network structure. The non-isolating combining and decomposition networks are abstracted for simplicity.

Fig. 1. Comparison of signal component separation techniques for conventional outphasing systems and the RF-domain SCS proposed in this work. Baseband signals are indicated with black lines while RF paths are shown in red. The RF-input outphasing PA has reduced cost, complexity, and power consumption and can operate directly on a modulated RF input. (a) Digital SCS. (b) RF-domain SCS (this work).

with nonlinear elements. The resulting system is a true RF amplifier in the sense that its input is a modulated RF signal, and its output is an amplified version of that signal. This paper expands on the brief conference paper [26] with a complete analysis of the theory and development of this approach, and presents additional experimental data including modulated output spectrum and input impedance characterization of the 2.14-GHz prototype system. In Section II we derive the theoretical behavior of the RF decomposition network, and describe how a nonlinear termination network is used to implement mixed-mode phase- and amplitude-modulation of practical outphasing systems. Design of the nonlinear termination network and RF decomposition networks is discussed in Section III. Finally, the experimental system showing proof-of-concept operation at a 2.14 GHz carrier frequency with peak output power of 95 W is described in Section IV. II. SYSTEM THEORY Fig. 3 shows a simplified schematic of the proposed RF-input/RF-output system, in which a passive RF decomposition network performs signal separation of a modulated input to produce the four modulated signals required to drive the four branch PAs. The system consists of the decomposition network (based on a four-way transmission-line resistance compression network (TLRCN) terminated with nonlinear components), driver and branch PAs, and a lossless multi-way power combiner. A. Conceptual Overview In the outphasing operating mode, the function of the RF signal decomposition network is to convert amplitude modulation at the system input into relative phase modulation among the inputs to the four branch PAs. As can be seen from the struc-

ture in Fig. 3, the signal decomposition is related to the combining network through symmetry. Conceptually, the decomposition network's function can be thought of as being opposite of that of the combining network. That is, in the power combining network, phase-modulated signals interact to produce amplitude modulation at the output, whereas in the signal decomposition network, input amplitude modulation is converted to four phase-modulated drive signals. The four outphased drive signals are chosen such that the branch PAs “see” loading conditions that vary (nearly resistively) over some pre-determined range. Extending the conceptual symmetry argument, then, varying the decomposition network's loads over that same range of resistance values should (at least approximately) generate the desired outphasing relationship among the four port phases. The nonlinear loads ( in Fig. 3) are designed to vary as a function of input amplitude so that amplitude modulation at the input is converted “automatically” to phase modulation among the four PA drive signals. This inverse resistance compression network (IRCN) outphasing control strategy forms the basis of the RF signal decomposition in the outphasing regime. The implementation described in this work and used for the experimental prototype is based on an all-transmission-line approach as described for the power combining system in [10], and for resistance compression networks (RCNs) in [27], but we note that versions are also possible using discrete components, microstrip techniques, or a combination of those, related to the lumped-element [7], [8], microstrip with shunt reactive element [9], and all-transmission-line [10] variations of the four-way outphasing power combiner. Likewise, the approaches described in this work may be applied two-way (Chireix) outphasing, including its relatively wideband variations [4], [5]. The four-way outphasing architecture is selected as the basis for this work due to both its nearly-resistive loading conditions presented to the branch PAs (which lends itself to an intuitive understanding of the RF-input approach by arguments of symmetry) and because the impact of eliminating the multiple upconvering paths including mixers and filters is even greater in the four-way system compared to the conventional Chireix approach. B. Outphasing Operation An IRCN outphasing control law was originally introduced as one basis for selecting the input phases and for a

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Fig. 3. Simplified schematic of the new RF-input/RF-output outphasing system showing the transmission-line-based implementation used in the experimental – , given also by (1)–(4). prototype in this work. The vector diagram at upper left describes the relative phase relationship among the four branch signals

baseband-input, lumped-element four-way outphasing system [6]. This control law is based on the approximate relationship between the phase angles of a multi-way RCN and a corresponding multi-way power combining network with appropriate controls: the power combiner can be (approximately) derived from an RCN by applying the principle of time-reversal duality [6]. In essence, the direction of power flow in the power combining network is reversed by changing the sign of each reactance, resistance, and electrical length, replacing sources (approximated as negative resistors) with resistors (and vice versa). This transformation is illustrated for the lumped-element combiner of [6] in Fig. 4. An analogous inverse-RCN approach can be seen in the relationship between the transmission-line combining network [10] and transmission-line resistance compression network (TLRCN) [27]. As described in detail in [27], a TLRCN may be constructed as a binary tree of transmission-line sections, with the two branch lengths at the th branch point a deviation from a base length (typically or ). The ends of the final branches are typically terminated in identical loads, and a quarter-wave transmission line may also be employed before the initial branch point to provide impedance matching into the TLRCN. Examining the RF decomposition network of Fig. 3, we can see that it is based on a passive network (derived from a TLRCN) terminated with four varying (but equal) resistances . In the following analysis, we assume that the TLRCN terminating impedance is , neglecting any effects of the input impedance . In practice, as will be shown in Section III below, the terminating impedance can be designed to include effects of . The four output port voltages – can be found in terms of the input , terminating resistances , and network parameters through analysis of the decomposition network (1)–(2). These relationships are derived in the Appendix. Note that in (2), represents the impedance into the transmission-line pairs closest to the

Fig. 4. Dual relationship between RCN and combiner (shown here in lumped-element form) means that wide range output power control of the combiner system is possible through an IRCN control scheme.

nonlinear loads (see Fig. 3) and is a function of . From [27], will be purely resistive when is resistive

(1)

(2) The phases of the four port voltages are related as indicated in (1) and the vector diagram in Fig. 3, and can be shown to be related to the load resistance as in (3)–(4) where , , , and , with and

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Fig. 5. Plot of the port relationship from (1) when is varied, assuming is constant, and the decomposition network is designed with parameters , , , , and . Note that in practice, varies as a function of .

representing differences in base line lengths as illustrated in Fig. 3 (3) (4) The magnitudes and phases of the four voltages at the output of the RF decomposition network, – , are plotted as a function of load resistance in Fig. 5. Note that in this plot, is held constant. In practice, as will be described below, the value of will vary as a function of , and the magnitude relationship will therefore be modified from that shown in Fig. 5. Throughout this work we will assume that the PA output amplitudes are equal to each other. The further assumption that they are constant (in the outphasing operating mode) is enforced by both a limiter-based implementation of the nonlinear terminations and the saturating characteristics of the drivers and branch PAs, and will be examined in more detail in the next section. After amplification, the four branch PA outputs ( – ) (Fig. 3) are assumed to have the same outphasing relationship given by the vector diagram in Fig. 3 and with and from (3)–(4). The magnitudes – are furthermore assumed to be equal with value . In this case, the output power of the combining system is described by (5) [10] (5) is a system propNote that, although the expression for erty (and therefore unchanged from [10]), the selection of outphasing angles and is different in this work from that of the Optimal Susceptance (OS) control law in [10]. From (5) it can be seen that the output power of the amplifier system can be modulated by controlling either and (through control of ), the branch PA drive amplitude , or a combination of these methods.

Fig. 6. Plot of the theoretical loading impedances on the ports based on the IRCN (solid) and OS (dashed) control laws for the transmission-line system in this work, assuming a four-way transmission-line power combining as in [10] , , , , with values . and

The load impedances seen by the four branch PAs is found following the methodology in [10] but with the outphasing angles given by (3)–(4). In this analysis, the magnitudes of the four PA outputs – are assumed to be equal and to have a constant value ; that is, we ignore any amplitude variation in the PA drive signals due to variation in the amplitudes of – (the load impedances seen by the branch PAs do not depend on the value of , only on the relative phases of the four PA outputs). The resulting effective branch PA loading impedances (each including the effect of load modulation from the action of the other PAs) are shown in Fig. 6. Note that because the RCN and combiner networks are not exact duals, the actual combiner output power does not precisely track the (scaled) input power. However, this and other nonlinearities in the implemented system can be addressed through pre-distortion of the input signal [6], [11], [28]. C. Mixed-Mode Operation The above system analysis assumes that the branch PAs are operated in saturated or switched mode, with a constant-envelope output voltage and output power control achieved only through modulation of the effective load impedance seen by each PA. In principle, the output power of an outphasing system can be modulated in this way through phase-only control of the multiple signals driving the branch PAs. Practical implementations, however, use both phase and amplitude modulation of the signals driving the branch PAs for two reasons [3], [9]. First, as the system output power is reduced (by increasing the load resistance seen by each branch PA), the input power required to drive each PA into saturation is reduced as well. In this case, drive amplitude modulation can be used to improve efficiency of the overall RF lineup (and power-added efficiency [PAE]) while maintaining outphasing operation [9]. Second, and more importantly for many communications applications, amplitude modulation of the branch PA drive signals can be used to extend the system's output power range to include zero crossings. These two effects are summarized in Fig. 7.

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Fig. 7. Benefits of mixed-mode operation; (a) measured PAE of the inverse class F PAs used in this work showing PAE over load modulation; PAE is maintained by reducing drive amplitude as the loading impedance increases (reproduced from [9]), (b) theoretical efficiency of four-way outphasing (assuming branch PAs are implemented in class B) with (solid) and without (dashed) back-off control at low power levels.

Note that in Fig. 7(a), the individual final stage PAs are operated in compression (and driven into saturation) for most of the outphasing range, leading to flat power gain versus output voltage. However, at the highest output powers/lowest PA loading impedance (owing to load modulation), the drive amplifiers no longer drive the final stage hard enough to saturate it, so the PA comes out of compression and overall power gain increases slightly. Below the shown range, the drive amplitude to the PAs is reduced, and so efficiency is expected to decrease as shown in Fig. 7(b) due to this mixed-mode operation. In the pure outphasing mode (dashed lines in Fig. 7(b)), by contrast, the efficiency drops off in a near-vertical way as a result of operating outside the nominal range of the combiner. The combining network and associated control law is designed for operation over a fixed range (e.g., 10 dB as in Fig. 6), and outside this range the loading impedances of the branch PAs rapidly become highly reactive. Both Chireix and four-way outphasing systems are designed for a particular dynamic range over which the effective load impedance of the branch PAs is well-controlled, but present highly reactive loads in the limit as the output power goes to zero (resulting in problematic loading conditions for the branch PAs). In practical outphasing systems requiring accurate zerocrossings, therefore, the output power range can be extended by holding the outphasing angles constant at the phases corresponding to the lower extreme of the desired outphasing range, and backing off on the drive amplitudes to operate the branch PAs in a class-B or other non-saturated mode [3], [9]. This approach has the additional advantage that it does not rely on exact cancellation of the outputs of the multiple branch PAs to produce zero output power. In this work, we realize mixed-mode phase with amplitude modulation through the design of the termination networks at the RCN output ports. In the outphasing mode of operation, the effective loading resistance to the RCN varies over the desired range (i.e., the range of loading impedances that the branch PAs are designed to operate well over). This load resistance variation is a function of input drive amplitude, so that amplitude modulation of the RF input is decomposed into phase modulation for the four branch PA signals. Below the outphasing range, the

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Fig. 8. Nonlinear load element used to terminate the RCN: (a) equivalent schematic, and (b) effective input resistance at the fundamental (idealized diode and ON-resistance , assumed). with turn-on voltage

phases should be held constant at the value corresponding to the low end of the outphasing range, or in other words the effective load resistance seen by the individual PAs becomes fixed. Now amplitude modulation at the input produces amplitude modulation of the branch PA drives. Note that the port impedance derivation in the previous section assumes that the drive amplitudes – are equal, and this operating condition is also maintained in mixed-mode operation. D. Nonlinear Termination The amplitude to phase conversion of the decomposition network is produced by realizing its terminations as nonlinear networks whose effective resistance is a function of input power. That is, the variable resistances of Fig. 3 are implemented using nonlinear passive networks having an effective one-port impedance that varies as a function of the applied voltage. At the upper range of input power, the nonlinear loads behave as variable resistors, generating outphasing control angles corresponding to the IRCN control law. Below a threshold level the terminating resistance is fixed and the four signals are amplitude-modulated with the input signal, with the input signal split evenly among the four branches. The nonlinear load network used in this work is shown in Fig. 8. The implemented impedance variation of the load network is not optimized, but it has the general required characteristics to demonstrate the RF-input outphasing concept, namely that (in the high-power range) the resistance posed by the load network decreases as the power driving it increases. When the applied (sinusoidal) current driving the load network is sufficiently large, the voltage waveform across the load network will be a clipped version of the input current. As the input power is increased, the output voltage will remain at the clipped amplitude, but the fundamental component of the current will increase with input power. As a result, neglecting the effect of and any parasitic resistance, the effective input impedance of this network will be an inverse function of input power. When the parallel resistance is included, then, the input resistance will be limited to a value at low drive (drive levels insufficient to turn on the diodes). The resistor corresponds to an impedance-transformed version of the 50- impedance into the driver amplifiers ( , Fig. 3). Fig. 9 shows the simulated amplitude and phase relationships among the four branches when the nonlinear network in Fig. 8 is used to terminate the decomposition network. The

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Fig. 9. Simulated amplitude and phase of voltages at the output of the decomposition network, as a function of system input voltage. The mixed-mode outphasing and amplitude control can be seen: between a threshold voltage the outphasing angles are fixed and the amplitude of the branch voltages is proportional to the input voltage. Note that only the fundamental component of the amplitude is shown.

Fig. 10. Effective loading impedances seen by the four branch PAs in the RF-input/RF-output system ( , Fig. 3) as a function of system input voltage, when the outphasing angles resulting from the decomposition network are applied to the power combiner (simulated). The diode threshold voltage is . simulated as

mixed-mode outphasing and amplitude control can be clearly seen; below a threshold voltage the outphasing angles are fixed, and the amplitude of the branch voltages is proportional to the input voltage amplitude. Above that threshold, the four voltages – follow the IRCN control law. Similarly, the simulated port voltage amplitude (fundamental component only) shows the limiting effects of the diode termination. Note that these voltage signals approach square waves as the drive input level increases, i.e., there is significant additional harmonic content. The driver and RF stage amplifiers also have limiting characteristics, further enforcing constant-envelope behavior at the output of the branch PAs (e.g., the input of the power combining network). The effective load impedances of the four branch PAs, shown in Fig. 10, is simulated using the idealized diode-based model. III. IMPLEMENTATION This section describes the design and implementation of the RF decomposition network used for experimental validation of the approach, including the implementation of the TLRCN, and the nonlinear loading network. The experimental system is designed to operate at 2.14 GHz. A. Microstrip TLRCN The signal decomposition network is implemented as an alltransmission-line RCN [27], although alternative implementations including could be designed as in the related lumped-element [8] or microstrip [9] power combining networks. The layout (Fig. 11) is made up of curved microstrip segments; right angles and “T” junctions are avoided in order to most closely match the theoretical behavior of the network. The layout of this and other system components are not optimized for size, although more compact versions are possible by using e.g., serpentine layout. This network is implemented on a 1.52-mmthick Rogers RO4350 substrate.

Fig. 11. Layout of RF signal decomposition board implemented on a 1.52mm-thick Rogers RO4350 substrate. The board dimensions are 14.5 cm 14.2 is synthesized from the (50- ) cm. As indicated, the parallel resistance impedance into the driver stage using a quarter-wavelength impedance transformation.

The TLRCN component of the decomposition network is designed assuming that the nonlinear loads vary over an approximately 10–85 range, corresponding to the range that the combining network is designed to present to the branch power amplifiers. The layout therefore uses the same parameters as the power combining network [10], with characteristic impedances , and delta-electrical lengths and . (The original paper describing the transmission-line combiner [10] indicated values and in error; the lengths and were used in both the transmission-line combiner and TLRCN.) A secondary benefit of the TLRCN structure is that it provides a narrow-range resistive input impedance even when its load im-

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pedances ( , Fig. 3) vary. As in the corresponding combiner design, an impedance-transforming quarter-wave transmission line is included at the input port, with . Combined with the resistance compression behavior of the TLRCN, this impedance transformation establishes a nominally 50- input impedance to the power amplifier system. The simulated reflection coefficient at the input port has over the entire range of operation. B. Nonlinear Load Element The anti-parallel diode pair is implemented using Avago HSMS-286C detector diodes, which are available as a single packaged pair. Matching to reduce the effects of parasitic reactance is incorporated into the quarter-wavelength transmission line before the diode. For other choices of diodes or frequency, a more complicated impedance match may be necessary. The shunt resistance is designed to be 85 , and is generated by transforming the 50- input impedance of the following (driver) stage with a quarter-wavelength impedance transformer.

Fig. 12. Comparison of measured relative phases at the outputs of the decomposition network (grey), the full PA paths (black; branch PAs 50- terminated), and the ideal phase relationship calculated based on commanded power using the optimal susceptance control law [10] (dashed, top axis). Net phase shift has been removed from the plot for clarity. The mixed-mode behavior can be seen below around 7 dB input commanded power, below which the measured outphasing angles are held constant.

IV. EXPERIMENTAL SYSTEM A. Decomposition Network The decomposition network is characterized by measuring the phase relationship at its four output ports (ports A, B, C, D in Fig. 11) when the input power is varied. For this experiment, the decomposition network is first terminated (at reference plane D, Fig. 3) with 50- loads representing the input impedances to the PA drivers. The relative phases at the four output ports of the decomposition network ( – ) are shown in Fig. 12 (grey curves). This figure shows the relative phases among the four branches only; net phase shift has been removed from the plot for clarity. Next, the full RF paths are characterized up to reference plane E; for this measurement, the branch PAs are terminated in 50 and the relative phase of signals – at their outputs is characterized with system input power (Fig. 12, black curves). Note that the actual phase relationship into the combiner may vary if the PAs have load-modulation to phase-modulation nonlinearity. Static phase offsets are trimmed out using phase shift tuners (Aeroflex Weinchsel 980–4) in the four paths. The transition between outphasing control and drive modulation can also be seen near 7 dB normalized input power. The measured phase characteristics are compared to those calculated from the OS control law (which in principle yields more optimal performance than the IRCN control law), shown in Fig. 12 as dashed curves (and referring to the top axis). A close match can be observed between the phase relationships among the measured and theoretical control angles. At the same time, it can be seen by comparing the top and bottom axes that there is not a one-to-one correlation between the two scales, and that the input power range of the experimental system is approximately twice (in dB) that of the calculated power command range. This means that the output power is not a linear function of linear power (as will be shown in the system measurements below); this nonlinearity is related to the implementation of the nonlinear load network used in this work.

Fig. 13. Measured transmission coefficient (normalized) of each PA path (terminated by 50 ) as a function of system input power. The variation in AM/AM performance between the four paths was not corrected for in the system characterization.

The amplitude characteristics of all four ports are likewise measured over swept input power. This measurement, made into 50- loads only, reveals amplitude mismatch among the four branches, particularly at low drive levels (see Fig. 13). This imbalance was not corrected for in the system characterization. The resistance compression property of the decomposition network can be seen in Fig. 14, which shows the measured input impedance to the RF decomposition network ( in Fig. 3) over the swept power characterization of Fig. 12. The input reflection coefficient remains better than over the entire range of applied power. Furthermore, the impedance is nearly constant, with an improved 50-ohm match clearly possible if desired. B. RF-Input/RF-Output Outphasing System 1) Overview: A photograph of the complete system including (left to right) the decomposition network, phase shift tuners, two predriver stages, the inverse class F branch PAs, and the power combining network [10], is shown in Fig. 15. The input CW or W-CDMA signal is generated using a Rohde

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Fig. 14. Input reflection coefficient for the signal decomposition network as the input power is swept over the 20-dB range as in Fig. 12. This measurement in Fig. 3. The nearly constant corresponds to the reference plane indicated by ) is a result of the resistance compression input impedance (showing network used as the basis for the signal decomposition network.

& Schwarz SMJ100A vector signal generator, and is amplified by a pre-amplifier to reach the required input level before the signal decomposition network. The nature of the nonlinear loads in the decomposition network constrains the possible range of input powers to the system for a given design. Here, the decomposition network is designed for input powers up to 23 dBm, while a 50 dBm system output power is desired. After the RF-domain decomposition, phase-shift tuners are employed to trim out static phase offsets (these are largely due to mismatched delays in the driver amplifiers). The two driver amplifiers, based on demonstration boards for the Hittite HMC455 and Freescale MW7IC2020N parts respectively, have not been optimized for the system, and so are excluded from the efficiency characterization. In fact, as can be seen in the photograph, excess gain in the driver chain is adjusted for using fixed attenuators. This arrangement is due to the available driver stages and is clearly undesirable for a practical system. Note that the driver PAs have been replaced compared to the related work in [26], although efficiency performance is similar. The branch PAs are based on the CGH40025 device from CREE and the design in [14], and are the same PAs as used in the baseband-input work demonstrating the all-transmission-line multi-way outphasing power combiner [10]. The power combining network is likewise the same as used in [10], so that the performence of that baseband-input system can be directly compared to this work. Drain efficiency of the final-stage power amplifiers (which have approximately 10 dB gain [9]) is measured using an Agilent N6705A power supply and Rohde & Schwarz NRT power meter. A block diagram of the measurement setup is shown in Fig. 16.

Fig. 15. System photograph showing: (a) the testbench including instrumentation; (b) details of the RF decomposition network, drivers, and RF power stage.

Fig. 16. Block diagram of the measurement setup, with reference planes indicated based on the definitions in Fig. 3.

2) CW Measurements: A CW measurement of drain efficiency of the final RF power stage versus output power is shown in Fig. 17 (black curve). The power amplifier has a peak output power of 49.7 dBm and peak drain efficiency of 70%. Also shown (grey curve) is a system characterization in which the branch PA drive signals (corresponding to reference plane in Fig. 3) are generated using four separate IQ modulators with

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Fig. 18. CW characterization of the power amplifier. The inflection point at the transition between outphasing and amplitude modulation for output power control can be seen around an output power of approximately 44 dBm. Fig. 17. Measured outphasing power amplifier performance (black), and measured performance of a baseband-input system comprising the same branch PAs and power combining network [10]. The new RF-input PA has nearly identical performance but operates directly on a modulated RF signal with a simple passive network replacing the complex baseband signal processing setup of [10]. This variation occurs due to differences in the active devices, passive devices, interconnects and pcb manufacturing among the PA and driver chains of the four paths.

the Optimal Susceptance control law that selects outphasing angles such that the reactive component of the branch PA load impedance is minimized (reproduced from [10]). The same combiner and RF power stage are used for both measurements. Note that the inclusion of phase shift tuners compared to the initial prototype in [26] allows for static phase adjustment of the four paths and a slightly improved efficiency characteristic. The excellent match in the efficiency performance of these systems demonstrates the effectiveness of the RF signal decomposition network. The reduced peak power of the new system is likely due to a combination of the higher load susceptance associated with the IRCN law and the phase mismatch observed in Fig. 12. Compared to the system using four IQ modulators, this proof-of-concept RF-input/RF-output implementation has significant advantages in system complexity without degradation in peak efficiency. The RF-input/RF-output outphasing amplifier is also characterized in terms of CW input/output characteristics as shown in Fig. 18. The two regions of output power control, amplitude modulation and outphasing control, are apparent from this measurement. The clear “knee” between the two regimes, and the compressive behavior in the outphasing control region, indicate that the nonlinear termination for this design is not optimal. This and other nonlinearities can be addressed through pre-distortion of the input signal or further refinement of the nonlinear load characteristic used in the decomposition network. 3) Modulated Measurements: A preliminary characterization of the outphasing PA was performed using a W-CDMA input signal. The measured output spectrum, with no linearization applied, is shown in Fig. 19. For this measurement, the average output power was 24 W, and average drain efficiency of the final PA stage was . The measured output PAPR was 6.18 dB. A comparison to other works of related technology and power level is given in Table I. The evident nonlinearity of the RF-input outphasing PA may be attributed to several factors including the nonlinear

Fig. 19. Output spectrum of the RF-input outphasing PA when no linearization is applied, measured over a 40 MHz span for a 3.84 MHz W-CDMA signal.

TABLE I COMPARISON TO OTHER WORKS: W-CDMA PERFORMANCE.

overall AM-AM characteristic shown in Fig. 18, which is most likely caused by the non-ideal nonlinear termination element used in the implemented system. Improvements in this nonlinear characteristic, along with a complete characterization of the load-modulation-to-amplitude-modulation characteristics of the branch PAs, could improve the observed “knee” characteristic in Fig. 18. The linearity of outphasing systems has additionally been shown to be sensitive to amplitude, phase, and delay mismatch among the branches, and limited bandwidth of the combining network [31]–[33]. We note that a common linearization approach for outphasing systems (e.g., [32], [33]) treats the signal component separator, branch

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PAs, and power combining network as a “black box” model, and applies digital predistortion (DPD) based on the overall input-output characteristic of the system. Although linearization is outside the scope of this work, which focuses on the initial proof-of-concept demonstration of RF-input outphasing, we therefore expect that this architecture is compatible with conventional linearization techniques. Fig. 20. Three-port network forming the building blocks of both the transmission-line combining network and the TLRCN-based decomposition network.

V. CONCLUSION The RF signal decomposition network presented in this work exploits the relationship between resistance compression networks and lossless outphasing power combiners in order to create an RF-input/RF-output outphasing power amplifier. This approach eliminates the digital signal component separator and multiple IQ modulators required for prior outphasing system implementations. Advantages of the approach include reduced system cost and baseband signal processing complexity, and the ability to work with many existing calibration and digital pre-distortion schemes. The proof-of-concept prototype in this work is implemented using transmission-line techniques, and demonstrates the feasibility of this approach through CW measurements at 2.14 GHz. The system achieves a peak output power of 93 W and a peak drain efficiency of 70%, performance that is on par with the previously-demonstrated outphasing system [10] requiring four IQ modulators. The excellent match between these two systems demonstrates the effectiveness of the RF signal decomposition approach. This approach can be extended to a range of frequencies and implementation types including lumped element implementations as in [8], or microstrip versions [9]. Future development of this technique will focus on design of the nonlinear termination in the signal decomposition network to generate a more overall-linear characteristic.

ports. The voltage magnitudes are equal and given by (9), while the two port phases are given by (10)–(11) (9) (10) (11) When is as defined graphically in Fig. 20, then, this outphasing angle can be written as (12) To form the four-way decomposition network, the stage in Fig. 20 is “stacked” in a coporate combining structure. The behavior of the second stage is identical to the first, and it is loaded with a variable resistance ( in Fig. 3) that is a function of . This impedance is the input impedance to the first stage ( in Fig. 20) and can be calculated following the analysis in [27] to be

APPENDIX

(13)

This appendix gives the derivation of (1)–(4) relating the port voltages of the decomposition network when the terminating impedance varies. The three-port network shown in Fig. 20 can be thought of as the fundamental building block of both the transmission-linebased power combining network and the TLRCN. The port relationship of this network can be shown through transmission-line analysis to be: (6) where . When ports 1 and 2 are terminated resistively, , the port voltages can be written as

The relative angle resulting at the plane indicated by in Fig. 3 is obtained by substituting the expression for into (12), producing (3)) (14) as As shown in Fig. 3, we denote the segments closer to having differential electrical length , and the segment closer to the input as having differential electrical length . The port voltage magnitudes after the second stage (i.e., at the reference plane indicated by in Fig. 3) can be found by substituting the expression for into (9) (15)

(7) (8) In this application, the parameters of interest are the magnitude of port voltages and , and the relative phase between the

Substituting (15) into (9) yields the port voltage amplitude given in (1). REFERENCES [1] H. Chireix, “High power outphasing modulation,” Proc. IRE, vol. 23, no. 11, pp. 1370–1392, Nov. 1935.

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[2] D. Calvillo-Cortes, M. van der Heijden, M. Acar, M. de Langen, R. Wesson, F. van Rijs, and L. de Vreede, “A package-integrated Chireix outphasing RF switch-mode high-power amplifier,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 10, pp. 3721–3732, Oct. 2013. [3] J. Qureshi, M. Pelk, M. Marchetti, W. Neo, J. Gajadharsing, M. van der Heijden, and L. de Vreede, “A 90-W peak power GaN outphasing amplifier with optimum input signal conditioning,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 8, pp. 1925–1935, Aug. 2009. [4] M. van der Heijden, M. Acar, J. Vromans, and D. Calvillo-Cortes, “A 19 W high-efficiency wide-band CMOS-GaN class-E Chireix RF outphasing power amplifier,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), Jun. 2011, pp. 1–4. [5] M. Pampin-Gonzalez, M. Ozen, C. Sanchez-Perez, J. Chani-Cahuana, and C. Fager, “Outphasing combiner synthesis from transistor load pull data,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), May 2015, pp. 1–4. [6] D. Perreault, “A new power combining and outphasing modulation system for high-efficiency power amplification,” IEEE Trans. Circuits Syst. I: Reg. Papers, vol. 58, no. 8, pp. 1713–1726, Feb. 2011. [7] A. Jurkov, L. Roslaniec, and D. Perreault, “Lossless multi-way power combining and outphasing for high-frequency resonant inverters,” in Proc. Int. Power Electron. Motion Control Conf., Jun. 2012, vol. 2, pp. 910–917. [8] T. Barton, J. Dawson, and D. Perreault, “Experimental validation of a four-way outphasing combiner for MICROWAVE power amplification,” IEEE Microw. Wireless Compon. Lett., vol. 23, no. 1, pp. 28–30, Jan. 2013. [9] T. Barton and D. Perreault, “Four-way microstrip-based power combining for MICROWAVE outphasing power amplifiers,” IEEE Trans. Circuits Syst. I: Reg. Papers, vol. 61, no. 10, pp. 2987–2998, Oct. 2014. [10] T. W. Barton, A. S. Jurkov, and D. J. Perreault, “Transmission-linebased multi-way lossless power combining and outphasing system,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), Jun. 2014, pp. 1–4. [11] A. Jurkov, L. Roslaniec, and D. Perreault, “Lossless multi-way power combining and outphasing for high-frequency resonant inverters,” IEEE Trans. Power Electron., vol. 29, no. 4, pp. 1894–1908, Apr. 2014. [12] A. Miller and J. Novik, “Principles of operation of the Ampliphase transmitter,” Broadcast News, no. 104, Jun. 1959. [13] W. Doherty, “A new high efficiency power amplifier for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [14] A. Grebennikov, “A high-efficiency 100-W four-stage Doherty GaN HEMT power amplifier module for WCDMA systems,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), Jun. 2011, pp. 1–4. [15] L. Panseri, L. Romano, S. Levantino, C. Samori, and A. Lacaita, “Lowpower signal component separator for a 64-QAM 802.11 LINC transmitter,” IEEE J. Solid-State Circuits, vol. 43, no. 5, pp. 1274–1286, May 2008. [16] A. Rustako and Y. Yeh, “A wide-band phase-feedback inverse-sine phase modulator with application toward a LINC amplifier,” IEEE Trans. Communications, vol. 24, no. 10, pp. 1139–1143, Oct. 1976. [17] D. Cox and R. Leck, “Component signal separation and recombination for linear amplification with nonlinear components,” IEEE Trans. Commun., vol. 23, no. 11, pp. 1281–1287, Nov. 1975. [18] B. Shi and L. Sundstrom, “A 200-MHz IF BiCMOS signal component separator for linear LINC transmitters,” IEEE J. Solid-State Circuits, vol. 35, no. 7, pp. 987–993, Jul. 2000. [19] B. Shi and L. Sundstrom, “A translinear-based chip for linear LINC transmitters,” in Proc. Symp. VLSI Circuits, Jun. 2000, pp. 58–61. [20] L. Sundstrom, “The effect of quantization in a digital signal component separator for LINC transmitters,” IEEE Trans. Vehicular Tech., vol. 45, no. 2, pp. 346–352, May 1996. [21] S. Hetzel, A. Bateman, and J. McGeehan, “A LINC transmitter,” in Proc.. IEEE Vehicular Tech. Conf., May 1991, pp. 133–137. [22] W. Gerhard and R. Knoechel, “LINC digital component separator for single and multicarrier W-CDMA signals,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 1, pp. 274–282, Jan. 2005. [23] Y. Li, Z. Li, O. Uyar, Y. Avniel, A. Megretski, and V. Stojanovic, “High-throughput signal component separator for asymmetric multilevel outphasing power amplifiers,” IEEE J. Solid-State Circuits, vol. 48, no. 2, pp. 369–380, Feb. 2013.

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[24] T.-W. Chen, P.-Y. Tsai, D. De Moitie, J.-Y. Yu, and C.-Y. Lee, “A low power all-digital signal component separator for uneven multi-level LINC systems,” in Proc. European Solid-State Circuits Conf., Sep. 2011, pp. 403–406. [25] M. Heidari, M. Lee, and A. Abidi, “All-digital outphasing modulator for a software-defined transmitter,” IEEE J. Solid-State Circuits, vol. 44, no. 4, pp. 1260–1271, Apr. 2009. [26] T. Barton and D. Perreault, “An RF-input outphasing power amplifier with RF signal decomposition network,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), May 2015, pp. 1–4. [27] T. Barton, J. Gordonson, and D. Perreault, “Transmission line resistance compression networks and applications to wireless power transfer,” IEEE J. Emerging Sel. Topics Power Electron., vol. 3, no. 1, pp. 252–260, Mar. 2015. [28] A. Jurkov and D. Perreault, “Design and control of lossless multi-way power combining and outphasing systems,” in Proc. Midwest Symp. Circuits Syst., Aug. 2011, pp. 1–4. [29] D. Calvillo-Cortes, M. van der Heijden, and L. de Vreede, “A 70 W package-integrated class-E Chireix outphasing RF power amplifier,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), Jun. 2013, pp. 1–3. [30] J. Kim, J. Moon, Y. Y. Woo, S. Hong, I. Kim, J. Kim, and B. Kim, “Analysis of a fully matched saturated Doherty amplifier with excellent efficiency,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 2, pp. 328–338, 2008. [31] T. Hwang, K. Azadet, R. Wilson, and J. Lin, “Linearization and imbalance correction techniques for broadband outphasing power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 7, pp. 2185–2198, Jul. 2015. [32] P. Landin, J. Fritzin, W. V. Moer, M. Isaksson, and A. Alvandpour, “Modeling and digital predistortion of class-D outphasing RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1907–1915, Jun. 2012. [33] A. Aref, T. Hone, and R. Negra, “A study of the impact of delay mismatch on linearity of outphasing transmitters,” IEEE Trans. Circuits Syst. I: Reg. Papers, vol. 62, no. 1, pp. 254–262, Jan. 2015. Taylor W. Barton (S'07–M'12) received the Sc.B., M.Eng., E.E., and Sc.D degrees from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, in 2012. In 2014, she joined The University of Texas at Dallas (UT Dallas), Dallas, TX, where she is currently an Assistant Professor. Prior to joining UT Dallas, she was a Postdoctoral Associate in the MIT Microsystems Technology Laboratories. Her research interests include high-efficiency RF, power, and analog circuit design, and classical control theory.

David J. Perreault (S'91–M'97–SM'06–F'13) received the B.S. degree from Boston University, Boston, MA, USA, and the S.M. and Ph.D. degrees from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA. In 1997, he joined the MIT Laboratory for Electromagnetic and Electronic Systems as a Postdoctoral Associate, and became a Research Scientist in the laboratory in 1999. In 2001, he joined the MIT Department of Electrical Engineering and Computer Science, where he is presently Professor and Associate Department Head. His research interests include design, manufacturing, and control techniques for power electronic systems and components, and in their use in a wide range of applications. He also consults in industry, and is co-founder of Eta Devices, a startup company focusing on high-efficiency RF power amplifiers. Dr. Perreault received the Richard M. Bass Outstanding Young Power Electronics Engineer Award, the R. David Middlebrook Achievement Award, the ONR Young Investigator Award, and the SAE Ralph R. Teetor Educational Award, and is co-author of seven IEEE prize papers.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 63, NO. 12, DECEMBER 2015

Hysteresis and Oscillation in High-Efficiency Power Amplifiers Jesús de Cos, Student Member, IEEE, Almudena Suárez, Fellow, IEEE, and José A. Garc´ıa, Member, IEEE

Abstract—Hysteresis in power amplifiers (PAs) is investigated in detail with the aid of an efficient analysis method, compatible with commercial harmonic balance. Suppressing the input source and using, instead, an outer-tier auxiliary generator, together with the Norton equivalent of the input network, analysis difficulties associated with turning points are avoided. The turning-point locus in the plane defined by any two relevant analysis parameters is obtained in a straightforward manner using a geometrical condition. The hysteresis phenomenon is demonstrated to be due to a nonlinear resonance of the device input capacitance under near optimum matching conditions. When increasing the drain bias voltage, some points of the locus degenerate into a large-signal oscillation that cannot be detected with a stability analysis of the dc solution. In driven conditions, the oscillation will be extinguished either through synchronization or inverse Hopf bifurcations in the upper section of the multivalued curves. For an efficient stability analysis, the outer-tier method will be applied in combination with pole-zero identification and Hopf-bifurcation detection. Departing from the detected oscillation, a slight variation of the input network will be carried out so as to obtain a high-efficiency oscillator able to start up from the noise level. All the tests have been carried out in a Class-E GaN PA with measured 86.8% power-added efficiency and 12.4-W output power at 0.9 GHz. Index Terms—Bifurcation, class-E, harmonic balance (HB), GaN, hysteresis, power amplifier (PA), stability, UHF.

I. INTRODUCTION

C

LASS-E power amplifiers (PAs) have been receiving increased attention due to their potential for simultaneously providing linear and efficient amplification when employed in bias- or load-modulation architectures [1]. However, it is not uncommon to observe instability phenomena in these amplifiers, some of which have been reported in [2] and [3]. Indeed, hysteresis can be found under near-optimum input-matching conditions, which gives rise to sudden transitions or jumps [2], [4], [5] between different sections of the power-transfer curve Manuscript received July 03, 2015; revised September 27, 2015; accepted October 13, 2015. Date of publication October 29, 2015; date of current version December 02, 2015. This work was supported by the Spanish Ministry of Economy and Competitiveness (MINECO) under Project TEC2014-60283-C3-1-R and Project TEC2014-58341-C4-1-R, with FEDER co-funding, the Parliament of Cantabria (12.JP02.64069) and by the Predoctoral Fellowship for Researchers in Training of the University of Cantabria and the Regional Ministry of Education of the Government of Cantabria. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. The authors are with the Communications Engineering Department, Escuela Técnica Superior de Ingenieros Industriales y de Telecomunicación (ETSIIT), University of Cantabria, 39005 Santander, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2492968

when either increasing or decreasing the input power. From a geometrical viewpoint, the hysteresis is due to the presence of turning points or infinite-slope points [5], [6] in the solution curve, which, in designs based on commercial harmonic balance (HB), have been detected [2], [4], [7] with the aid of an auxiliary generator (AG). However, there is little insight into the mechanism for the appearance of these turning points, often observed when approaching the intended operation conditions [2], [3]. In the GaN HEMT-based Class-E PA studied here, the nonlinear input capacitance resonates with the inductive input matching network, as will be demonstrated analytically. For an accurate prediction/suppression of the phenomenon, the outer-tier method presented in [8] will be adapted to the case of the PA with hysteresis, which will allow tracing the multivalued solution curves in an efficient manner, with no need for parameter switching [7], [9]–[11]. This method will also enable a direct calculation of the turning-point locus in terms of any practical analysis parameter, such as the gate bias voltage, the input matching capacitor, or the input power, by simply imposing a geometrical condition [8]. The observation of the hysteresis phenomenon studied in detail in [3] is often empirically associated with the onset of oscillations under a relatively small variation of the circuit parameters or element values. This paper expands [3] by studying the relationship between these two phenomena, apparently quite different, which will be done through a detailed analysis of the impact of the drain bias voltage on the turning-point locus. As will be shown, the turning-point locus, which at zero drain bias voltage is solely due to the input nonlinear capacitance, spreads over lower input power values. From certain , some discrete points of the locus reach zero input power and degenerate into free-running oscillations that coexist with a stable dc solution. As will be demonstrated in this work, under most input matching conditions, this oscillation cannot be detected with any standard stability analysis, as it coexists with a stable dc regime even for gate bias voltages above the conduction threshold. When injecting the input power, it will give rise to a stable self-oscillating mixer regime, coexisting with the stable periodic solution at the frequency of the input source. The oscillation will be extinguished either through synchronization [5], [12], for input frequencies near the free-running oscillation frequency, or through inverse Hopf-type bifurcations [5], [6], [12]. This paper expands [3] with a detailed analysis of the oscillatory solution and the mechanisms for the oscillation extinction. Synchronization of an oscillation with an injection source occurs at particular types of turning points, at which a local-global

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DE COS et al.: HYSTERESIS AND OSCILLATION IN HIGH-EFFICIENCY PAs

bifurcation takes place [5], [12], [13], also known as a limit cycle on saddle node in the Poincaré map. Therefore, these bifurcations will belong to the turning-point locus that is efficiently detected with the outer-tier method. Another contribution relative to [3] is the derivation of a methodology for the Hopf bifurcations, which will be carried out combining this outer-tier method with a zero-amplitude oscillation condition, imposed with the aid of a small-signal AG [2], [5], [11]. Since the Hopf bifurcations may take place in any section of the multivalued curves, the combination of this limit-oscillation condition with the outer-tier method is very advantageous since it avoids the need for a large-signal nonperturbing AG to sustain solutions in sections of the multivalued curve to which commercial HB does not converge by default. Another extension with respect to [3] will be the combination of the outer-tier methodology with pole-zero identification to obtain the whole evolution of the stability properties of the multivalued solution curve in a single sweep with no need to perform any parameter switching. The ease of application of the bifurcation detection methodologies will allow an in-depth investigation of the impact of the most relevant parameters, such as the gate bias voltage, the input matching capacitor, and the input power, on the global stability of the PA. This will provide insight into the various instability mechanisms observed in the PA and the relationships between them. A final contribution expanding [3] is the derivation of an oscillator design methodology based on a controlled selection of the element values that should turn the Class-E PA into a highly efficient oscillator [14], able to start up from the noise level. RF power oscillators may be of interest for the implementation of high-power density and fast response resonant dc/dc converters [15], wireless power transmission links [16], etc. This paper is organized as follows. Section II presents the study of the hysteresis using a high-efficiency Class-E PA demonstrator. The stability of the resulting multivalued curves is analyzed in Section III with the aid of the outer-tier method. In Section IV, the detected free-running oscillation is related to the hysteresis phenomenon. Finally, a high efficient RF power oscillator is presented. II. STUDY OF HYSTERESIS A. Class-E PA Demonstrator A PA at 900 MHz was designed in [3] with the aim of obtaining a very high value of power-added efficiency (PAE). Among many choices, the high efficiency of Class-E operation was exploited, minimizing the switching loss associated with the transistor output capacitance. A CGH35030F GaN on SiC HEMT from Cree Inc. was selected as the switching element due to the very low value of its on resistance times output capacitance product and high breakdown voltage (over 120 V). Initially, the transistor was experimentally characterized in an Arlon 25N substrate ( mm, m) for a typical drain bias voltage V, after verifying the peak value of the voltage waveform, [17], would stay below the process breakdown figure. The gate bias voltage was set slightly below pinch-off: V, while the output capacitance and off-state resistance were estimated

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Fig. 1. (a) Schematic of the Class-E PA demonstrator [3]. Values in the table are for Colcraft Air Core inductors and ATC 100B capacitors. (b) Simulated reflection coefficient of the output network and load–pull contours. (c) Photograph of the measurement setup.

from the measured parameter [3]. The measured values of on-state resistance, output capacitance, and off-state resistance at 900 MHz are 0.6 , 3.5 pF, and 5.1 k , respectively. An ideal dc voltage source, a capacitor, and a resistance were added to the already accurate and reliable Cree’s proprietary large-signal transistor model to finely adjust, respectively, the values of the gate threshold voltage, output capacitance, and off-state resistance in simulations to the measured device parameters. As a first approximation, the output network was designed based on the optimum or nominal conditions for a 50% switching duty cycle and maximum output power found by Raab [17],

(1) Unfortunately, the required inductance value in the classic Class-E output network may be too large to be obtained with commercially available coils, as their self-resonant frequency could be below the most significant higher order harmonics to be properly terminated [18]. One not uncommon solution for lumped-element Class-E implementations at UHF band [19] is to take advantage of a high- coil with a self-resonant frequency in between the second- and third-order harmonics [20] so that a high enough impedance (either inductive or capacitive) is presented. Here, a slightly different strategy was adopted. A smaller inductor, in the schematic of Fig. 1(a), was selected, with an associated smaller equivalent series resistance at dc, trying to minimize its contribution to the circuit conduction loss [21]. In addition, advantage was taken from its higher self-resonant frequency at about the fifth-order harmonic for properly terminating most of them. The optimum reactance value in (1) was adjusted with a section of microstrip transmission line, while the capacitor to ground allowed synthesizing the desired . The reflection coefficient of the resulting output network, , is represented in the Smith chart in Fig. 1(b), together with the simulated load–pull contours for drain efficiency (solid line) and output power (dashed line).

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TABLE I RF CLASS-E PAs IN THE LITERATURE

Fig. 2. PAE of the Class-E PA versus input power. Solid lines are the results of a default simulation in commercial HB. The curve is completed (dashed line) with the method described in Section II-C. Symbols are measurements.

The contours were obtained at the fundamental frequency with ideal terminations (open circuit) to other harmonics. A typical single low-pass section was used to match the input of the PA to increase the gain and thus the PAE. The experimental value of the input matching capacitor was 10 pF, a bit lower than the one used in simulations. A resistance was introduced in the gate biasing path to improve stability at lower frequencies. Although not included in the schematic for simplicity, a capacitor of 56 pF and a bank of high-valued capacitors (1, 10, and 100 nF and 1 and 10 F) were added in the gate and drain dc lines. The employed measurement setup is shown in Fig. 1(c). As can be seen, two low-pass filters with cutoff frequency at 1 GHz are included just before the power sensor to exclude any possible contribution of harmonics different than the fundamental to the output power. Table I includes the measurement results and a comparison with other RF Class-E PAs in the literature. In Fig. 2, the PAE of the amplifier is represented versus the input power . When increasing , a jump is observed for 10 dBm. When decreasing the input power there is no longer a jump at 10 dBm, but at a lower value of about 7 dBm. This behavior is evidence of hysteresis. Looking at the simulated curve, two turning points or infinite slope points are found, responsible for the undesired phenomenon. The multivalued curve has been traced with the method described in Section II-C. Note that default HB is unable to pass through the infinite slope points. Any other solution curve (output power, power gain, dc consumption, etc.) represented versus the input power will fold at the same values of as the curve in Fig. 2.

carried out short circuiting the transistor drain and source terminals, and modeling the transistor input with only its nonlinear gate-to-channel capacitance, as depicted in Fig. 3(a). The input matching network considered is the one in the original design in Fig. 1(a), but neglecting the impact of the parasitics in the lumped elements, the transmission line, and the RF choke. A describing function of the corresponding nonlinear charge will be used, assuming a sinusoidal input waveform . This allows formulating the circuit at the fundamental frequency,

(2) has been defined. Calculating where a complex function the derivatives of the real and imaginary part of (2) with respect to and , the Jacobian matrix is shown in (3) at the bottom of this page, where . The singularity condition is

(4) The expression of the input power (in dBm) is obtained squaring and adding the real and imaginary parts of (2),

B. Study of the Parametric Hysteresis The hysteresis phenomenon observed in the solution curve of Fig. 2 is caused by a nonlinear resonance of the device input capacitance (due to the gate-to-source and the input-reflected gate-to-drain Miller capacitances [29]) with the inductive impedance of the input network. An analytical study will be

(5) Results obtained with the analytical formulation (5) and with HB are compared in Fig. 3(b). As shown in the figure, the simplified analytical study of the input network is able to predict

(3)

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Fig. 4. Circuit used for obtaining the outer-tier admittance function

Fig. 3. (a) Circuit used to study the input network of the PA. (b) Comparison between the analytical result and HB simulation. The turning points have been marked.

.

Fig. 5. PAE of the Class-E PA versus input power for some values of the gate bias voltage. Solid lines are simulation results obtained with the outer-tier method. The dashed line superimposed is the turning-point locus. Symbols are measurements.

the existence of two turning points. Each of these two turning points fulfills the condition (4). Discrepancies come from contributions of the extrinsic parameters inside the transistor model, not considered when calculating the nonlinear charge in the frequency domain. C. Analysis Method The multivalued solution curves will be obtained by adapting the outer-tier method, developed in [8] for injection-locked oscillators, to the case of PAs. The input generator is suppressed using, instead, an AG [5], [11] at the gate terminal, which operates at the input frequency (Fig. 4). This enables the calculation of the outer-tier admittance function , where is the current through the AG at the fundamental frequency and is the AG amplitude. The admittance function will be combined with the Norton equivalent of the input network at the gate terminal, which can be calculated from its scattering matrix. The combination of both functions provides an outer-tier equation, which relates the gate voltage amplitude at the fundamental frequency (agreeing with ) and the input generator current , (6) where

To obtain a power-transfer curve, is swept, calculating the function at each sweep step. The input generator current enabling each voltage is determined with the outer tier equation given by (6). Note that the outer-tier (6) is combined with the full HB system, acting as an inner tier, so all the circuit variables are available at each sweep step. Therefore, relevant magnitudes such as the output power or

Fig. 6. Turning-point locus in the plane defined by the gate bias voltage and the input power. Square symbols are measurements (for the particular value V, it was also measured when decreasing ).

efficiency can be calculated in a straightforward manner. The outer-tier method provides the multivalued solution curves with no need of parameter switching unlike previous works [5], [30]. The method has been applied to complete the solution curve in Fig. 2 and to trace the whole one in Fig. 3 in simulation. It has also been used to analyze the impact of the gate bias voltage on the PAE curves (Fig. 5). It must be emphasized that here a complete description of the input network is considered, including full models of the passive components, the transmission line, and the RF choke. The evolution of the hysteresis region versus a relevant parameter , affecting the nonlinear resonance, can be efficiently investigated by tracing the turning-point locus in the plane defined by the particular parameter and the input power, following the method in [8]. One of these analysis parameters

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Fig. 7. Zero-level contours of the total conductance and susceptance obtained when introducing an AG at the gate terminal for V dBm. The intersections determine the three solutions in the correand sponding solution curve of Fig. 2. Note the multivalued nature of the contours.

is , already considered in the analysis of Fig. 5, and the other is the input matching capacitance . To obtain the turning-point locus one should consider the surface described by (6) on the plane defined by and . The turning-point locus is the set of points of satisfying zero partial derivative with respect to , i.e., the zero-level contour, (7) as a parameter The locus resulting from (7) when using has been superimposed on the solution curves of Fig. 5 via the dashed line. As can be seen, the locus passes through all the turning points of the solution curves. Fig. 6 shows the same turning-point locus represented in the plane defined by and . The hysteresis region decreases with and vanishes to zero near the pinch-off voltage . Measurement points are superimposed. To get some insight into the reasons for the observation of the phenomena in the lower range, one must take into account that in these subthreshold conditions the transistor gain increases with the excitation amplitude. As a result, the conductance function, when looking into the circuit nonlinear section from the AG terminals (Fig. 4), initially increases with the excitation amplitude and then decreases as expected in any physical device. Since the frequency of each coexisting solution agrees with the frequency of the input source, the phase shift between the driving source and the AG excitation voltage is a relevant variable. Fig. 7 shows the contour plots of total conductance and total susceptance equal to zero in the plane defined by the AG phase and amplitude, when the input power is set to 8 dBm. The total conductance/susceptance functions include contributions from both the linear and nonlinear sections of the PA at the observation node. There is a steady-state solution for each intersection of the two contour plots so three steady-state solutions , , and coexist for this value, in agreement with the results of Fig. 2. The high dependence of the total admittance function on the excitation amplitude gives rise to two zero susceptance contours and a bending of the zero conductance contour. This favors the occurrence of three intersections,

Fig. 8. Evolution of the turning-point locus versus the drain bias voltage, repand . This diagram illustrates the conresented in the plane defined by nection between the hysteresis and a free-running oscillation. The values in the upper axis correspond to the quality factor of the input network for each value in the label.

corresponding to the three solutions that coexist for the particular value. Next, the impact of the matching capacitance will be analyzed. In [3], the turning-point locus was represented in the plane defined by and for different values of . This allowed to suppress the hysteresis. The final measurements results for an experimental capacitor value of 8.2 pF are 85.4% of PAE, 16.1 dB of power gain, and 12.3 W of output power. Here, the turning-point locus is represented in the plane defined by and for different values of the drain bias voltage . For V, one obtains the small-size locus in Fig. 8, existing for capacitor values between 31.3 and 52.3 pF. This locus characterizes the hysteresis phenomenon that is solely due to the nonlinear input capacitance, as studied in Section II-B. For each constant value, the locus provides the input power values at which the hysteresis jumps are produced. When increasing , there is also an influence of the nonlinear transfer characteristic [29], and the locus expands over larger intervals of and . From a certain value, the locus decays to zero value and this will give rise to an oscillation phenomenon. For a detailed analysis, the typical drain bias voltage V will be considered, as this is the one selected for the high-efficiency design. The analysis versus will be extended to the quality factor of the input network . For estimating , the transistor has been represented by a resistance in series with a linear capacitance, whose values were obtained from the device small-signal input impedance at V below pinchoff when short circuiting the drain and source terminals. Despite neglecting the nonlinear nature of the device input capacitance, this approximation may be appropriate enough as long as the signal excursion does not take the gate-to-source junction into conduction, as described in [31]. In these conditions, the general definition of the quality factor is applied, (8)

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where and are, respectively, the resistance and reactance of the circuit and . The frequency of resonance is found for each value of , computing through a finite difference for calculating the derivative. An additional axis is included in Fig. 8 with the values of corresponding to each value in the label. Fig. 9(a) presents the turning-point locus for V. In the results presented so far and in [3], a model of the input matching capacitor, including the parasitics, is used. For those analysis using as a parameter, a lower value is selected, for which no hysteresis is found, and then an ideal capacitor is connected in parallel. In this way, the value of the capacitance can be varied continuously. As can be seen in Fig. 9(a), there are two capacitor values for which the turning-point locus reaches the axis (zero input power value). At these two particular points, and , the locus should degenerate into two free-running oscillations. An interesting fact is that the dc solution of the PA is stable for all the capacitor values considered in Fig. 9(a), as the device is biased below pinch-off. In measurements, the dc solution was found stable for all the capacitor values tested, even when biasing the transistor above pinch-off. In simulations, the dc solution only becomes unstable in a very reduced region when biasing the transistor above pinch-off for large capacitor values. Therefore, the oscillations in Fig. 9(a) cannot be detected with a small-signal stability analysis of the PA. The relationship between the hysteresis and the oscillations obtained in Fig. 9(a) will be investigated with a stability analysis methodology adapted to the case of multivalued solutions. III. STABILITY ANALYSIS THROUGH MULTIVALUED SOLUTIONS

THE

The application of a complementary stability analysis through multivalued solution curves is demanding since HB will converge to the default solution, usually corresponding to the one with the smallest output power, or will not converge at all. To cope with this problem, the outer-tier method will be combined with pole-zero identification [32], [33] and bifurcation detection [2], [5], [11], [30], [34], as explained in the following. A. Stability Analysis In the analysis of Fig. 4, the whole solution curve is traced suppressing the input source and using an AG to calculate the outer-tier admittance function considered in (6). The curve is then obtained by sweeping the AG amplitude in HB. The input power is calculated from using the Norton equivalent. Thus, turning points only result from the composition of any of the circuit state variables with the input power. In this way, the equivalent system (6) provides the same solutions that would be obtained with a suitably initialized HB system. Therefore, the stability can be analyzed with the AG connected to the circuit, instead of the driving source (Fig. 4). Note that unlike other previous methods the AG does not fulfill a nonperturbation condition. In the presence of this AG, a small-signal current source at the incommensurate frequency is introduced at a sensitive circuit node (the gate terminal). This source is used to linearize the circuit about the large-signal periodic regime at

Fig. 9. Synchronization with free-running oscillations. (a) Turning-point (solid line) and Hopf-bifurcation (dashed–dotted line) loci: input power versus the in free-running input matching capacitor value. (b) Output power versus W). (c) Autonomous frequency versus in free-runoperation ( values for which the autonomous frequency ning operation. Note that the is 0.9 GHz agree with the degenerate points of the turning-point locus ( W). Stable (unstable) sections in solid (dashed) line. Square symbols are mea). surements (attention was only paid to values close to

each HB sweep step with the conversion-matrix approach [35], [36]. The following calculation is performed: (9)

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Fig. 10. Comparison between the transfer function obtained in the original circuit with the one obtained using the circuit of Fig. 4.

small input power, this oscillation will give rise to a quasi-periodic regime. Therefore, when injecting input power, one may expect the occurrence of Hopf bifurcations, leading to a transition between the periodic and quasi-periodic regime or vice versa. In previous works [2], [5], [30], Hopf bifurcations are detected by introducing an AG at the oscillation frequency and solving the steady-state oscillation condition for oscillation amplitude tending to zero. Again, a problem arises when dealing with multivalued solution curves as the Hopf bifurcation may occur in the curve sections to which commercial HB does not converge by default. Here, the outer-tier method will be used to avoid the need of an extra AG to initialize/sustain these coexisting solutions. Let a Hopf bifurcation leading to the generation/extinction of an oscillation at the frequency be considered. The oscillation amplitude tends to zero at the bifurcation point so this bifurcation can be detected linearizing the circuit about the largesignal periodic solution at each HB sweep step with the conversion-matrix approach [35], [36]. A small-signal current source at is connected to a sensitive node of the circuit in Fig. 4. At each Hopf bifurcation, the following system of combined steady-state plus bifurcation equations must then be fulfilled: (10)

Fig. 11. Comparison between the Hopf-bifurcation locus obtained with the previous method and with the new method.

where is the gate node voltage and is the current of the small-signal source. The stability analysis is performed applying pole-zero identification [32], [33] to the function (9) obtained for each value. For the stability analysis to work properly, a difference with respect to [3] and [8] must be remarked: the filter should only stop the frequency of the input generator (0.9 GHz), allowing for a proper impedance termination at other harmonic mixing terms. For validation, the stability analysis has been applied to the solution point in Figs. 2 and 7. Note that it is a point to which the HB method converges by default, as gathered from the solid line simulation in Fig. 2. The transfer functions obtained with the original circuit and performing the topology change (suppression of the input source plus introduction of the outer-tier AG) are compared in Fig. 10. The results in the two different analysis conditions overlap. B. Hopf-Bifurcation Detection The degenerate points and in the turning-point locus of Fig. 9(a) evidence the existence of a free-running oscillation for some capacitor values. In the presence of a relatively

where is the sensitive node voltage and is the current of the small-signal source. System (10) is solved through optimization of and . Note that the standard AG-based procedure [11] would also require the optimization of the amplitude and phase of the extra AG that is used to achieve convergence to the upper section(s) of the solution curve, and therefore, it is a more demanding optimization process. Parameter switching in this extra AG would be needed for the calculation of the whole Hopf-bifurcation locus versus two relevant parameters, such as and . The Hopf-bifurcation locus obtained with (10) has been superimposed in the plane defined by and in Fig. 9(a). As can be seen, the Hopf locus is composed by three different sections. Section 1 corresponds to Hopf bifurcations in the lower sections of the multivalued curves. Section 2 corresponds to Hopf bifurcation in the upper sections of the multivalued curves and Section 3 to Hopf bifurcations obtained for capacitor values at which the solution curve is no longer multivalued. The accuracy of the Hopf-bifurcation locus calculated with (10) has been validated through comparison with the one obtained with the previous method [11], which, by default, can only detect Hopf bifurcations in the nonmultivalued sections of the curves. As shown in Fig. 11, the results of both methods are totally overlapped. In Section IV, the described approach will be applied for a detailed investigation of the relationship between the hysteresis and the oscillatory phenomena detected in Fig. 9(a). IV. RELATIONSHIP BETWEEN HYSTERESIS SELF-OSCILLATION

AND

The turning-point locus in Fig. 9(a) indicates the presence of free-running oscillations that cannot be detected with an ordi-

DE COS et al.: HYSTERESIS AND OSCILLATION IN HIGH-EFFICIENCY PAs

nary stability analysis of the dc solution. This is because the periodic solution at the input drive frequency is stable when the transistor is biased below pinch-off. The interval for which these oscillations exist will be analyzed using one of the two degenerate turning points at W as an initial value for a free-running oscillator analysis. Using an AG [5], the free-running oscillation curve has been traced versus at constant V, in Fig. 9(b). As can be seen, the solution curve exhibits a turning point at pF. For , there is no oscillatory solution. For , there are two coexisting steady-state oscillations for each value. The infinite slope point at implies that a real pole passes through zero at this particular capacitor value [5]. Therefore, the two sections of the oscillation curve must exhibit different stability properties. As has been verified with pole-zero identification [32], applied through the oscillation curve, the upper section of this curve is stable and the lower section is unstable. At each of the two points of the turning-point locus in Fig. 9(a) obtained for W, the PA solution degenerates into a free-running oscillation, having approximately the same frequency as the input generator (0.9 GHz). In fact, there are two capacitor values for which the free-running frequency agrees with this precise value, as gathered from Fig. 9(c). Each of them is responsible for one of the two degenerate points in the turning-point locus of Fig. 9(a). With the values considered in the measurements and under a full variation of , the dc solution was always stable so dc solutions were physically obtained in the whole bias voltage range going from 8 to 2.3 V. Despite this fact, a free-running oscillation coexists with each dc solution, which in measurements could only be observed by injecting input power up to a certain level and then decreasing this power to zero. The described situation prevents the detection of this oscillation when performing a stability analysis of the dc solution. As an example, Fig. 12(a) presents the closed oscillation curve obtained for pF versus . The oscillation amplitude does not decay to zero for any value so there are no Hopf bifurcations in dc regime. This is why this oscillation does not start up from the noise level. Arguably, the oscillation in Fig. 9(b) should arise at a Hopf bifurcation from the dc regime, obtained versus the gate bias voltage for some combinations of the capacitor and other circuit element values. Fig. 12(b) presents the variation of the dominant poles of the dc solution versus the gate bias voltage. The poles approach the imaginary axis, but do not cross this axis, which prevents the detection of the coexisting high amplitude oscillation of the PA. As already stated, to obtain this oscillation in measurements it was necessary to inject the circuit with enough input power from the driving source and then reduce to zero for a particular value of . Once oscillating, can be varied to obtain the measurement points. The measurement points obtained in this way are superimposed in Fig. 12(a). The reason why the input power is able to start the oscillation will be understood after a thorough stability analysis of the periodic solution curves obtained versus the input power. The impact of the free-running oscillation in Fig. 12(a) on the stability properties of the PA power-transfer curves will be analyzed considering variations in the capacitor , which di-

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pF in freeFig. 12. (a) Output power versus gate bias voltage for running operation. Stable (unstable) sections are in solid (dashed) line. Square symbols are measurements. (b) Stability analysis of the dc solution versus the gate bias voltage.

rectly affects the input matching and oscillation conditions, as gathered from Fig. 9. This capacitor will be varied from 12 to 23 pF. Fig. 13 presents the turning-point and Hopf-bifurcation loci, calculated with (7) and (10), respectively, in the plane defined by the input power and the capacitance . The values (starting at pF) that give rise to a free-running oscillation, as detected from the solution curve in Fig. 9(b), are indicated with a thick vertical line at W. For these capacitor values when increasing from zero, the oscillation associated will give rise to a self-oscillating mixing regime coexisting with the periodic stable solution. This quasi-periodic regime has never been observed experimentally when increasing the input power from zero. Once the circuit is operating in the upper section of a given periodic solution curve, it is observed when reducing the input power. There is then a transition from a periodic regime at to a self-oscillating mixer solution. This solution is mathematically extinguished at certain input power so it only exists in the lower input power range. Depending on the input frequency value, it can be extinguished in two different manners: through synchronization or through an inverse Hopf bifurcation. Synchronization will occur for capacitor values such that the free-running oscillation frequency is close enough to the fre-

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Fig. 13. Turning-point (solid line) and Hopf-bifurcation (dashdot line) loci repand . The capacitor values giving rise resented in the plane defined by to a free-running oscillation have been marked with a thick solid line. The values analyzed in Fig. 14 are marked.

quency of the input generator . Due to the similarity of the two frequencies, the input source will have a significant influence over the self-oscillation. Therefore, the synchronization will take place for rather low input power values. Inverse Hopf bifurcations will be obtained for capacitor values such that the free-running oscillation frequency is farther away from . Due to the larger frequency difference, a higher input power will be required for extinction of the oscillation. As described in the following, the parameter region for the occurrence of each of the two phenomena is easily determined from inspection of Fig. 13. As already stated, a stable free-running oscillation will exist for all the capacitor values in the thick solid line in Fig. 13. In the neighborhood of the degenerate point , the transition to periodic regime will take place when crossing the turning-point locus as the input power is increased. In fact, turning points do not always indicate jump phenomena, but may also correspond to synchronization (limit cycle on a saddle node in the Poincaré map), as described in [6] and [12]. As a general rule, turning points near the free-running oscillation, having no Hopf bifurcations in the neighborhood, will correspond to synchronization [5]. In the diagram of Fig. 13, this will be the case for values between 15.8 and 20.8 pF. Oscillation extinction through an inverse Hopf bifurcation takes place for a higher difference between and the original free-running value. In the diagram of Fig. 13, this occurs when Section 2 of the Hopf-bifurcation locus is crossed when increasing the input power. This is the case for capacitor values pF. It is important to emphasize that the free-running oscillation in Fig. 13 cannot be detected with a small-signal stability analysis. Section 1 of the Hopf locus corresponds to Hopf bifurcations in the lower section of the multivalued curves. Their implications on the circuit solution will be better understood when superimposing the Hopf-bifurcation locus on the power-transfer curves obtained for different values. Fig. 14(a) and (b) presents the power-transfer curves obtained for pF and pF, respectively. The turning-point locus obtained with (7) passes through all the infinite-slope points of the solution curves (marked in the figure with a ), as can be verified through simple inspection. On the

Fig. 14. Periodic solution curves of the PA: output power versus input power . The stable sections are highlighted. The turningfor different values of point (dashed line) and Hopf-bifurcation (dashdot line) loci, together with the pF. Square symbols are bifurcations points, have been included. (a) pF. (c) and pF. measurements. (b)

other hand, the Hopf locus obtained with (10) passes through all the Hopf bifurcaton points (marked with ), as will be validated later with pole-zero identification. The case of pF will be initially considered. The lower section of the power-transfer curve, up to the point , is stable. This is because, when injecting the input power, it emerges from a stable dc regime. At the Hopf bifurcation, a

DE COS et al.: HYSTERESIS AND OSCILLATION IN HIGH-EFFICIENCY PAs

quasi-periodic regime is generated, which should be extinguished from certain input power, due to the natural reduction of the negative resistance with the input amplitude. When the oscillation is extinguished, a jump takes place to the upper section of the periodic curve, with stable behavior. Now, reducing the input power, the circuit remains in the stable periodic solution up to the Hopf bifurcation , where an oscillation is generated. Note that when increasing the input power from zero, the self-oscillation [due to the existence of free-running solutions in Fig. 9(b)] is extinguished at this same bifurcation. However, because of the coexistence of the stable periodic solution with this oscillatory regime, it will be rare to observe this oscillation when increasing the input power from zero. The upper section of the periodic curve is unstable between the turning point and the Hopf bifurcation so the potential jump point is never reached physically. In the case of pF, when increasing from zero, the periodic solution curve will be stable up to the Hopf-bifurcation point . In a manner similar to the previous case, a transition to quasi-periodic regime will occur at this point and then to the upper section of the periodic curve, with stable behavior. When decreasing the input power, the upper section of the curve keeps stable up to the turning point , which is, in fact, a synchronization point. The reason for the different behavior is that the input frequency 0.9 GHz is quite close to the free-running oscillation frequency obtained for pF so the transition to quasi-periodic regime is through a loss of synchronization. For pF, there is no longer a free-running oscillation, as gathered from Fig. 9(b), so, as can be expected, there is no Hopf bifurcation in the lower section of the periodic solution curve in Fig. 14(c), in agreement with Fig. 13. The two turning points will give rise to jumps between different sections of the multivalued periodic curve (hysteresis). For pF, a regular curve showing gain expansion (with neither oscillation nor jumps) is obtained. As gathered from the bifurcation loci in Fig. 13, there can be Hopf bifurcations in the lower section of the curves (Section 1 of the Hopf-bifurcation locus) even when there are no free-running oscillations. The quasi-periodic regime generated at these points should exhibit a turning point when increasing the input power (as those reported in [5] and [11]) and be extinguished in a saddle-connection bifurcation in the Poincaré map [12]. This is a global bifurcation [12] that requires the presence of a saddle point such as those in the intermediate section of the multivalued solution curves. The saddle-connection bifurcation is also associated with co-dimension two bifurcations, at which the turning-point and the Hopf-bifurcation loci merge, such as the ones indicated with in Figs. 9(a) and 13. The accuracy of the Hopf-bifurcation detection has been validated with pole-zero identification. Fig. 15 shows the evolution of the critical poles of the solution curve corresponding to pF. The pole locus indicates that the lower section is stable up to dBm, where the poles cross to the right-hand side of the complex plane (RHP), giving rise to a Hopf bifurcation from periodic regime. The poles merge and split into two real poles, and for dBm, one of the real poles crosses to the left-hand side of the complex plane (LHP) at a turning point

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Fig. 15. Pole locus of the solution curve in Fig. 14(a) obtained for pF.

in the solution curve. This real pole crosses again to the RHP at dBm, giving rise to the second turning point in the solution curve. When further increasing the input power, the two real poles merge on the RHP and split into two pairs of complex-conjugate poles (associated with a same pair of Floquet multipliers [5], [6], [37]). At dBm, the poles cross to the LHP giving rise to an inverse Hopf bifurcation from which the periodic solution curve becomes stable. The pole analysis is in total agreement with the results of the bifurcation analysis in Figs. 13 and 14(a). The PA measurements for an input matching capacitor of 12 pF have been superimposed in Fig. 14(a). The capacitor value is significantly lower in measurements that the one used in simulations. As mention in Section II-A, the capacitor value that best matched the input of the PA in measurements was already lower than the one used in simulations. The shift in the capacitor value is attributed to modeling inaccuracies. One must also take into account that the value used in the simulations was a combination of a lower capacitor value and an ideal capacitor connected in parallel, to be able to vary the capacitance continuously, as explained in Section III-A. Even under this unavoidable accuracy limitations, the qualitative behavior and power levels are reproduced satisfactory. The measured spectra at different input power values are shown in Fig. 16. At low input power, the spectrum is periodic, as shown in Fig. 16(a). When continuously increasing the input power, a Hopf bifurcation is obtained at dBm, which gives rise to the quasi-periodic spectrum in Fig. 16(b). When further increasing the input power, the self-oscillation vanishes due to a turning point in the quasi-periodic solution curve so a jump takes place at dBm to the upper section of the periodic solution curve. Fig. 16(c) shows the spectrum obtained for dBm. Now, when reducing the input power, the PA keeps behaving in the periodic regime up to the input power dBm, at which the circuit undergoes a Hopf bifurcation. Below dBm, the circuit operates in a quasi-periodic regime [see Fig. 16(d)] that is never observed when increasing the input power from zero. The results are in total agreement with the two Hopf-bifurcation points that were detected with (10) and displayed in Figs. 13 and 14(a).

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Fig. 16. Sequence of output spectra measured for pF. (a) When dBm). increasing the input power from very low value it is periodic ( dBm). (c) Jump to the periodic (b) Jump to a quasi-periodic solution ( dBm). (d) Now, when decreasing the input power, a quasisolution ( periodic solution, resulting from the mixing with the free-running oscillation, is dBm). Synchronization with this last one is obtained observed ( reducing the input frequency. (e) Adler spectrum near synchronization ( GHz).

With the discrete capacitor values available during the experimental tests, it was not possible to observe a synchronization phenomenon with the input frequency GHz. This is because, for those capacitor values, the frequency of the free-running oscillation that coexists with the dc solution is too different from GHz. To validate the existence of the phenomenon, a small shift was applied to the input frequency, using, instead, the value GHz, together with the capacitor pF. With these parameter values, synchronization could be measured. The typical near-synchronization spectrum, with a triangular shape [38], is shown in Fig. 16(e). As previously shown, the input power is able to start an oscillation in the PA, which persists when reducing the power to zero, even when the dc solution is stable in the whole range considered. In Section IV-A, the above study will allow turning the PA into a highly efficient free-running power oscillator. A. High-Efficiency RF Class-E Power Oscillator In simulation, a shift of the pole locus in Fig. 12(b) to the right is observed when suppressing the 50- load of the input source. This gives rise to an interval of values for which the dc solution becomes unstable, allowing the oscillation to

Fig. 17. (a) Output power and efficiency of the RF Class-E power oscillator, represented versus the gate bias voltage. Square symbols are measurements. V. (b) Phase noise spectral density measured at

start up from the noise level. Fig. 17(a) presents the free-running oscillation curve obtained for the original capacitor value pF in the design of Fig. 1(a). Note that the oscillation persists in a large gate bias voltage interval. The Hopf bifurcation from the dc regime is subcritical [5], [12], [39] so the oscillation amplitude grows for decreasing values of . In measurements, the value of for which the oscillation starts up ( 2.8 V) agrees with the Hopf-bifurcation point obtained in simulation. The evolution observed versus is in total agreement with the predicted results. In comparison with other configurations, requiring an accurate synthesis of a feedback network [39], the oscillator topology is greatly simplified. Indeed, the feedback path is provided here by the device gate-to-drain capacitance . In Fig. 17(a), the efficiency variation has also been represented versus the gate bias voltage. A peak value as high as 86.4% was measured for V, staying above 80% for V. The gate bias voltage provides a simple way to control the oscillation frequency. For the voltage interval considered in Fig. 17(a), it varies between 0.826 and 0.98 GHz. Finally, the phase noise spectral density was captured for several oscillation frequencies and no noticeable difference was appreciated. In Fig. 17(b), it is represented at V. Phase noise values of 114.8 and 141.4 dBc/Hz were estimated for frequency offsets of 100 kHz and 1 MHz, respectively,

DE COS et al.: HYSTERESIS AND OSCILLATION IN HIGH-EFFICIENCY PAs

in the ranges reported in the literature for GaN HEMT based oscillators [40]. V. CONCLUSION An in-depth investigation of hysteresis in a Class-E PA has been presented, demonstrating that it is due to a nonlinear resonance of the transistor input capacitance with the inductive input matching network. The different set of circuit parameters and operating conditions that give rise to turning points in the solution curves are efficiently detected with an outer-tier method, under a geometrical condition for infinite slope. Under an increase of the drain bias voltage, the locus evolves so as to give rise to a free-running oscillation that for the most usual circuit element values cannot be detected with any standard stability analysis, even under an exhaustive variation of the bias voltages. Under input power injection, this oscillation will give rise to an undesired self-oscillating mixer regime, extinguished either through synchronization or inverse Hopf bifurcations. The Hopf bifurcations in the multivalued curves can be efficiently detected combining the outer-tier method with a limit-oscillation condition imposed with the aid of a small-signal AG. The great flexibility in the bifurcation analysis has enabled a thorough investigation of the circuit stability properties under extensive variations of bias voltages, input power, and circuit element values. In this way, it has been possible to modify the original PA design, close to the state-of-the-art for the UHF band so as to make a practical use of the oscillation originally associated with the degenerated turning points. It has been possible to obtain a high-efficiency Class-E oscillator with a slight variation of the input network, providing 154 MHz of frequency coverage with an efficiency figure above 80%. ACKNOWLEDGMENT The authors would like to thank M. N. Ruiz, University of Cantabria, for her assistance with measurements, J. Ma Salmón (retired) for kindly crafting the aluminum carriers for the PA, S. Pana, University of Cantabria, for her help with the fabrication of the prototype, and R. Baker, Cree Inc., for the support provided. REFERENCES [1] F. H. Raab et al., “Power amplifiers and transmitters for RF and microwave,” IEEE Trans. Microw. Theory Techn., vol. 50, no. 3, pp. 814–826, Mar. 2002. [2] S. Jeon, A. Suárez, and D. B. Rutledge, “Analysis and elimination of hysteresis and noisy precursors in power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 3, pp. 1096–1106, Mar. 2006. [3] J. de Cos, A. Suárez, and J. A. Garc´ıa, “Parametric hysteresis in power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., 2015, pp. 1–4. [4] N.-Ch. Kuo et al., “DC/RF hysteresis in microwave pHEMT amplifier induced by gate current—Diagnosis and elimination,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 11, pp. 2919–2930, Nov. 2011. [5] A. Suárez, Analysis and Design of Autonomous Microwave Circuits. Hoboken, NJ, USA: Wiley, 2009. [6] T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems. New York, NY, USA: Springer-Verlag, 1989. [7] E. Palazuelos, A. Suárez, J. Portilla, and F. J. Barahona, “Hysteresis prediction in autonomous microwave circuits using commercial software: Application to a Ku-band MMIC VCO,” IEEE J. Solid-State Circuits, vol. 33, no. 8, pp. 1239–1243, Aug. 1998.

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[8] J. de Cos and A. Suárez, “Efficient simulation of solution curves and bifurcation loci in injection-locked oscillators,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 1, pp. 181–197, Jan. 2015. [9] D. Hente and R. H. Jansen, “Frequency-domain continuation method for the analysis and stability investigation of nonlinear microwave circuits,” Proc. Inst. Elect. Eng., vol. 133, no. 5, pt. H, pp. 351–362, Oct. 1986. [10] L. O. Chua and A. Ushida, “A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits,” Int. J. Circuit Theory Appl., vol. 4, no. 3, pp. 215–239, Jul. 1976. [11] A. Suárez, J. Morales, and R. Quéré, “Synchronization analysis of autonomous microwave circuits using new global-stability analysis tools,” IEEE Trans. Microw. Theory Techn., vol. 46, no. 5, pp. 494–504, May 1998. [12] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamic Systems, and Bifurcations of Vector Fields. New York, NY, USA: Springer-Verlag, 1983. [13] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York, NY, USA: Springer-Verlag, 1990. [14] J. Ebert and M. Kazimierczuk, “Class E high-efficiency tuned power oscillator,” IEEE J. Solid-State Circuits, vol. SSC-16, no. 2, pp. 62–66, Apr. 1981. [15] H. Hase, H. Sekiya, J. Lu, and T. Yahagi, “Resonant DC/DC converter with class E oscillator,” in IEEE Int. Circuits Syst. Symp., 2005, pp. 720–723. [16] A. N. Laskovski and M. R. Yuce, “Class-E oscillators as wireless power transmitters for biomedical implants,” in Int. Appl. Sci. Biomed. Commun. Tech. Symp., 2010, pp. 1–5. [17] F. H. Raab, “Idealized operation of the class E tuned power amplifier,” IEEE Trans. Circuits Syst., vol. CS-24, no. 12, pp. 725–735, Dec. 1977. [18] F. H. Raab, “Class-E, class-C, and class-F power amplifiers based upon a finite number of harmonics,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 8, pp. 1462–1468, Aug. 2001. [19] J. A. Garc´ıa, R. Marante, and M. N. Ruiz, “GaN HEMT class E2 resonant topologies for UHF DC/DC power conversion,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 4220–4229, Dec. 2012. [20] R. Negra and W. Bächtold, “Lumped-element load-network design for class-E power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 6, pp. 2684–2690, Jun. 2006. [21] N. O. Sokal and A. Mediano, “Redefining the optimum RF class-E switch-voltage waveform, to correct a long-used incorrect waveform,” in IEEE MTT-S Int. Microw. Symp. Dig., 2013, pp. 1–3. [22] N. D. Lopez, J. Hoversten, M. Poulton, and Z. Popović, “A 65-W highefficiency UHF GaN power amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., 2008, pp. 65–68. [23] J. Cumana, A. Grebennikov, G. Sun, N. Kumar, and R. H. Jansen, “An extended topology of parallel-circuit class-E power amplifier to account for larger output capacitances,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3174–3183, Dec. 2011. [24] A. Al Tanany, A. Sayed, and G. Boeck, “Broadband GaN switch mode class E power amplifier for UHF applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 2009, pp. 761–764. [25] T. B. Mader, E. W. Bryerton, M. Markovic, M. Forman, and Z. Popović, “Switched-mode high-efficiency microwave power amplifiers in a free-space power-combiner array,” IEEE Trans. Microw. Theory Techn., vol. 46, no. 10, pp. 1391–1398, Oct. 1998. [26] Y. Qin, S. Gao, P. Butterworth, E. Korolkiewicz, and A. Sambell, “Improved design technique of a broadband class-E power amplifier at 2 GHz,” in Eur. Microw. Conf., 2005, pp. 4–6. [27] H. G. Bae, R. Negra, S. Boumaiza, and F. M. Ghannouchi, “High-efficiency GaN class-E power amplifier with compact harmonic-suppression network,” in Eur. Microw. Conf., 2007, pp. 9–12. [28] M. P. van der Heijden, M. Acar, and J. S. Vromans, “A compact 12-watt high-efficiency 2.1–2.7 GHz class-E GaN HEMT power amplifier for base stations,” in IEEE MTT-S Int. Microw. Symp. Dig., 2009, pp. 657–660. [29] L. C. Nunes, P. M. Cabral, and J. C. Pedro, “AM/AM and AM/PM distortion generation mechanisms in Si LDMOS and GaN HEMT based RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 799–809, Apr. 2014. [30] A. Suárez and R. Quéré, Stability Analysis of Nonlinear Microwave Circuits. Boston, MA, USA: Artech House, 2003. [31] S. Maas, The RF and Microwave Circuit Design Cookbook. Norwood, MA, USA: Artech House, 1998. [32] J. Jugo, J. Portilla, A. Anakabe, A. Suárez, and J. M. Collantes, “Closed-loop stability analysis of microwave amplifiers,” Electron. Lett., vol. 37, no. 4, pp. 226–228, Mar. 2001.

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[33] N. Ayllon, J. M. Collantes, A. Anakabe, I. Lizarraga, G. SoubercazePun, and S. Forestier, “Systematic approach to the stabilization of multitransistor circuits,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 8, pp. 2073–2082, Aug. 2011. [34] S. Jeon, A. Suárez, and D. B. Rutledge, “Global stability analysis and stabilization of a class-E/F amplifier with a distributed active transformer,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 12, pp. 3712–3722, Dec. 2005. [35] J. M. Paillot, J. C. Nallatamby, M. Hessane, R. Quéré, M. Prigent, and J. Rousset, “A general program for steady state, stability, and FM noise analysis of microwave oscillators,” in IEEE MTT-S Int. Microw. Symp. Dig., 1990, pp. 1287–1290. [36] V. Rizzoli, F. Mastri, and D. Masotti, “General noise analysis of nonlinear microwave circuits by the piecewise harmonic-balance technique,” IEEE Trans. Microw. Theory Techn., vol. 42, no. 5, pp. 807–819, May 1994. [37] J. M. Collantes, I. Lizarraga, A. Anakabe, and J. Jugo, “Stability verification of microwave circuits through Floquet multiplier analysis,” in Proc. IEEE Asia–Pacific Circuits Syst., 2004, pp. 997–1000. [38] R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE, vol. 61, no. 10, pp. 1380–1385, Oct. 1973. [39] S. Jeon, A. Suárez, and D. B. Rutledge, “Nonlinear design technique for high-power switching-mode oscillators,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 10, pp. 3630–3640, Oct. 2006. [40] M. Horberg, L. Szhau, T. N. T. Do, and D. Kuylenstierna, “Phase noise analysis of a tuned-input/tuned-output oscillator based on a GaN HEMT device,” in Eur. Microw. Conf., 2014, pp. 1118–1121.

Jesús de Cos (S’15) was born in Santander, Spain. He received the Telecommunications Engineering degree and M.Sc. degree from the University of Cantabria, Santander, Spain, in 2010 and 2011, respectively, and is currently working toward the Ph.D. degree at the University of Cantabria. His research interests include stability analysis, bifurcation theory, and circuit simulation techniques applied to microwave circuits. Mr. de Cos was a finalist in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium Student Paper Competition in 2015.

Almudena Suárez (M’96–SM’01–F’12) was born in Santander, Spain. She received the Electronic Physics and Ph.D. degrees from the University of Cantabria, Santander, Spain, in 1987 and 1992, respectively, and the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1993. She is currently a Full Professor with the Communications Engineering Department, University of Cantabria. She coauthored Stability Analysis of Nonlinear Microwave Circuits (Artech House, 2003). She authored Analysis and Design of Autonomous Microwave Circuits (IEEE, 2009). Dr. Suárez is a member of the Technical Committees of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) and the European Microwave Conference. She was an IEEE Distinguished Microwave Lecturer (2006–2008). She has been a member of the Board of Directors of the European Microwave Association (EuMA) since 2012. She is the editor-in-chief of the International Journal of Microwave and Wireless Technologies.

José A. Garc´ıa (S’98–A’00–M’02) was born in Havana, Cuba. He received the Telecommunications Engineering degree from the Instituto Superior Politécnico “José A. Echeverría” (ISPJAE), Havana, Cuba, in 1988, and the Ph.D. degree from the University of Cantabria, Santander, Spain, in 2000. From 1988 to 1991, he was a Radio System Engineer with a high-frequency (HF) communication center, where he designed antennas and HF circuits. From 1991 to 1995, he was an Instructor Professor with the Telecommunication Engineering Department, ISPJAE. From 1999 to 2000, he was with Thaumat Global Technology Systems, as a Radio Design Engineer involved with base-station arrays. From 2000 to 2001, he was a Microwave Design Engineer/Project Manager with TTI Norte, during which time he was in charge of the research line on SDR while involved with active antennas. From 2002 to 2005, he was a Senior Research Scientist with the University of Cantabria, where he is currently an Associate Professor. During 2011, he was a Visiting Researcher with the Microwave and RF Research Group, University of Colorado at Boulder. His main research interests include nonlinear characterization and modeling of active devices, as well as the design of power RF/microwave amplifiers, wireless powering rectifiers, and RF dc/dc power converters.

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Digitally Assisted Analog/RF Predistorter With a Small-Signal-Assisted Parameter Identification Algorithm Hai Huang, Student Member, IEEE, Jingjing Xia, Anik Islam, Eric Ng, Student Member, IEEE, Peter M. Levine, Member, IEEE, and Slim Boumaiza, Senior Member, IEEE

Abstract—This paper proposes a digitally assisted analog/radio frequency predistorter (ARFPD) and a linear small-signal-assisted parameter identification algorithm suitable for the linearization of power amplifiers driven with wideband and carrier aggregated communication signals. It starts by describing the newly proposed finite-impulse-response assisted envelope memory polynomial (FIR-EMP) model which allows for reduction of hardware implementation complexity while maintaining good linearization capacity and low power overhead. Furthermore, a linear two-step small-signal-assisted parameter identification algorithm is devised to estimate the parameters of the two main blocks of the FIR-EMP model. Measurement results obtained by using the FIR-EMP predistorter demonstrate its excellent linearization capacity when used to compensate for distortion exhibited by gallium nitride Doherty power amplifiers driven by digitally modulated signals with a bandwidth up to 80 MHz. This confirms the potential of ARFPD as a very promising candidate for the linearization of small cell base stations power amplifiers while simultaneously reducing the power overhead compared to the popular digital predistortion technique. Index Terms—Analog/radio frequency predistortion (ARFPD), digital predistortion (DPD), envelope memory polynomial (EMP), linearization, small-signal assisted parameter identification (SSAPI).

I. INTRODUCTION

T

HE NEED TO provide higher data rates and increased capacity has driven modern wireless communication systems to shift towards highly dense networks composed of large numbers of small-cells using advanced modulation schemes. For carrier-aggregated signals with modulation bandwidths of up to 100 MHz, and high peak-to-average power ratios (PAPR), high efficiency power amplifiers (PAs) for radio frequency (RF) wireless front-ends often require advanced efficiency enhancement techniques such as envelope tracking [1] and Doherty [2]. Such techniques unavoidably introduce significant distortions,

Manuscript received July 02, 2015; revised September 27, 2015, October 21, 2015; accepted October 22, 2015. Date of publication November 09, 2015; date of current version December 02, 2015. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, 17–22 May 2015. The authors are with the Emerging Radio Systems Group (EmRG), Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2015.2495362

which require sophisticated linearization systems to compensate for both the static nonlinearity and memory effects in order to meet the linearity requirements. Given the significantly lower transmitted power of small-cell base stations (a few watts or less down from tens of watts in macro-cell base stations), the power overhead of the linearization system can no longer be neglected. Digital predistortion (DPD), as shown in Fig. 1(a) is a popular choice to compensate for the distortion exhibited in PAs. It consists of pre-adjusting the communication signal magnitude and phase in baseband by applying a nonlinear function which mimics the inverse nonlinear behaviour of the PA [3]–[5]. To compensate for in-band distortions and inter-modulation products produced by the dynamic nonlinear behaviour of the PA, the output signal of the DPD has a wider bandwidth (typically five times) than that of the input baseband signal. Hence, the DPD engine needs to operate at much higher clock rates compared to the original digital baseband signal, and the digitalto-RF converter needs much faster digital-to-analog converters (DACs) and wider instantaneous bandwidth. The power overhead of DPD is only a minor concern for high power macro-cell base stations. However, as the power overhead increases with the signal bandwidth and does not scale down with the PA output level, it becomes a significant factor for small-cell base stations with much lower transmitted power, compromising the practicality of DPD. Some attempts have been made to bring more of the signal processing in the predistorter to the analog domain [6]–[8]. The digital/RF architecture uses a digital processor to synthesize the predistortion function and applies it to the RF input signal through a vector multiplier. However, the predistortion engine is still implemented in the digital domain and its performance is limited due to the static polynomial nature of the predistortion function, which becomes insufficient in the presence of significant PA memory effects. Exploiting the inherent low power property of analog circuits, several works [9]–[14] have investigated fully analog predistortion solutions. In [9]–[12], static polynomials with nonlinearity orders of up to five are synthesized using analog circuits and applied directly to the intermediate frequency (IF) or RF signals. Although being truly lowpowered solutions, their practical adoption is hindered by the limited linearization capacity in regards to static nonlinearity and the lack of a viable solution to address the memory effects. In [13], [14], the concept of analog/RF predistortion (ARFPD), as shown in Fig. 1(b), is investigated. The pre-

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the performance of the proposed model and identification algorithm, followed by the conclusion in Section V. II. LINEAR FILTER ASSISTED ENVELOPE MEMORY POLYNOMIAL MODEL OVERVIEW In an ARFPD system, the generation of predistorted signals usually involves two steps: a) synthesis of a nonlinear function (e.g., memory polynomial (MP)) that is complementary to the distortion of the PA and b) application of the synthesized predistortion function output to the RF input signal to generate the required predistorted outputs. One popular approach to address the memory effects is to use the MP model whose conventional baseband representation is expressed as (1)

Fig. 1. Block diagram of (a) a conventional DPD system and (b) the ARFPD system.

distorted signal generated by the analog engine, which uses complex analog signal processing circuits capable of handling memory effects, is applied to the RF signal through vector multipliers. Compared to the conventional DPD as shown in Fig. 1(a), the ARFPD scheme eliminates the power hungry DPD engine and reduce the bandwidth requirement of the DAC and IQ modulator, at the cost of additional analog hardware such as the analog predistortion engine and the RF vector multipliers. An ARFPD system capable of linearizing PAs with up to 20 MHz modulated signals while consuming as little as 0.2 W of power has been reported [14]. The linearization capacity of the aforementioned works is still limited, however, due to the difficulty in designing a predistortion model of comparable linearization capacity to DPD using analog hardware. In [15], a study of the solutions presented in [13], [14] revealed their inability to handle the linear memory effects of PAs. Accordingly, a new finite-impulse-response filter assisted envelope memory polynomial (FIR-EMP) predistorter is proposed. Preliminary linearization results where the predistortion signal is computed in software confirms the validity of the FIR-EMP model. In this work, a two-step, small-signal-assisted parameter identification (SSAPI) algorithm is proposed to extract the coefficients of the FIR-EMP structure for which the conventional least square errors (LSE) technique cannot be used. A new ARFPD test bench including the key component RF vector multiplier is devised to realistically assess the performance of the FIR-EMP model and the SSAPI algorithm in the presence of actual analog hardware. This paper is organized as follows. Section II briefly reintroduces the FIR-EMP model proposed in [15]. Section III presents the proposed linear parameter identification algorithm of the cascaded FIR filter and EMP model. Section IV presents the ARFPD test bench and the experimental results used to verify

where and are baseband samples before and after the predistortion, are the model coefficients, is the nonlinearity order, and is the memory depth. Reformulating the MP of (1) in the continues time domain for the ARFPD scheme, can be expressed as

(2) (3) where is a nonlinear function corresponding to a memory term of order , and denotes the time delay. The block diagram corresponding to (2) is illustrated in Fig. 2(a). The envelope signals are generated by the envelope detector followed by analog delay elements (D), and the delayed RF inputs are generated using RF delay elements (RF-D). The many RF-D elements and vector multipliers required for this architecture would consume a large amount of physical space and increase the hardware complexity. To simplify the hardware implementation, attempts have been made to remove the dependency of the memory paths on receiving phase information from the preceding input signals by using the envelopes of the signals only. The EMP in [16] is one such approach. The baseband complex output is given by (4) where , and are defined in the same way as (1). A reformulation of (4) for ARFPD results in (5) The performance of the EMP predistorter is assessed using a wideband gallium nitride (GaN) Doherty PA. Measurement

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Fig. 3. AM/AM and AM/PM characteristics, measured and modelled using the EMP model of a GaN Doherty PA driven with a 20 MHz WCDMA signal at 2 GHz [15].

sate for the memory effects, both the amplitude and phase information of preceding signals are required [18]. To address the shortcomings of EMP, linear filter is added before the EMP engine [15]. As shown in the block diagram in Fig. 2(b), the linear filter is implemented as a finite impulse response (FIR) filter in digital baseband. The resultant signal from the linear filter is then predistorted using EMP to compensate for the static and dynamic nonlinearities. The advantages of the proposed FIR-EMP model in the ARFPD setup over the conventional MP model can be seen clearly by comparing Figs. 2(a) and (b). The FIR filter in the digital baseband can be clocked at a much lower speed than a full DPD, thus requiring lower power overhead, as it must only cover the bandwidth of the baseband signal rather than the typical five times factor imposed by a DPD. EMP is also much simpler to realize in analog hardware than the MP based predistorter as it does not require multiple RF delay elements and vector multipliers. Fig. 2. Block diagram of (a) the MP function (b) the proposed FIR filter assisted EMP when applied to the ARFPD.

III. FIR-EMP PARAMETER IDENTIFICATION ALGORITHM A. Challenges in Identifying the Parameters

results indicate that the performance of the EMP model is adequate when the signal bandwidth is limited to 20 MHz, but it significantly degrades as the modulation bandwidth extends beyond that. Such shortcomings of EMP model are attributed to its limited ability to model the linear memory distortion. To illustrate the point, the measured AM/AM and AM/PM plots of the GaN Doherty PA, and those predicted by EMP, are shown in Fig. 3. It is clear that the EMP model results have good agreement with the measured data at high input powers where the nonlinearity of the PA is more pronounced, but fails to predict the distortion at low powers where the linear memory distortion dominates (also where there is wider signal bandwidth). This limitation of EMP is a result of the simplification used to derive the EMP from the Volterra series. The simplification assumes flat behaviour by the fundamental and its harmonics [17], and that the passband of the PA is much larger than the RF signal bandwidth [18]. Wider modulation bandwidth signals challenge this narrowband assumption. To accurately model and compen-

In the proposed FIR-EMP scheme outlined in the previous section, the FIR filter block is implemented in the digital domain and the EMP block is implemented in the RF domain. The cascade of a linear filter and a highly nonlinear block makes identifying the respective coefficients a major challenge. The commonly used LSE algorithm, popular for identifying the coefficients of a single-block predistortion model (e.g., MP or low pass equivalent Volterra), cannot be used to identify the respective coefficients of such a two-block nonlinear system due to the lack of linear relationships between the coefficients and the output signals. One potential solution is to use a nonlinear optimization algorithm such as the quasi-Newton method but this can be computationally intensive and is not suitable for realtime applications. B. Proposed Identification Algorithm In this section, a small-signal-assisted parameter identification (SSAPI) algorithm is proposed to efficiently identify the

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Fig. 4. Training scheme for identifying the coefficients of the FIR-EMP predistorter.

coefficients of the FIR filter and the EMP block, respectively without resorting to nonlinear optimization. Fig. 4 depicts the training scheme for the FIR-EMP predistorter. During the predistorter training (inverse modelling), and are the input and output envelope of the predistorter respectively, and the following expression can be obtained Fig. 5. Steps of the proposed SSAPI algorithm for the FIR-EMP predistorter (a) FIR parameter identification using the forward model of the PA (b) EMP parameter identification using the intermediate output of the FIR filter.

(6) is the intermediate training data at the output of where the FIR filter block, is the order of the FIR filter, and are the coefficients of the FIR filter. Expanding (6) leads to the following expression

(7) generally According to (7), estimating the values of and requires an advanced nonlinear optimization algorithm, which is possible but impractical in terms of the required time and computation resources. In the proposed SSAPI algorithm, the basic principle is to use a small-signal to probe the linear memory effects of the nonlinear device, which avoids the static nonlinearity and the nonlinear memory effects associated with the device, and leaves the linear memory effects the dominant source of distortion. To illustrate this, assume that the magnitude of the output is small enough such that all of the higher order terms (e.g., ) can be approximated as zero. Equation (7) can be simplified to (8) can be absorbed by the FIR coefficients and the linear gain . With (8), coefficients of the FIR filter block are estimated using the LSE algorithm. In order to obtain the small-signal training data, and , one obvious approach is to operate the PA at a significant back-off region, where the linear memory effects dominate

Fig. 6. Modelling accuracy versus the number of iterations for the quasi-Newton nonlinear optimization and the proposed SSAPI algorithm.

and the nonlinearity of the PA can generally be neglected. However, this obvious approach has three limitations. First, the linear memory effects of the PA might change due to a shift in the PA's region of operation. Second, the small-signal stimulus requires a high dynamic range in the transmitter observation receiver and the recorded samples could be sensitive to measurement noise. Lastly, forcing the PA to operate in the small-signal region necessitates off-line training and is not suitable for field applications. In consideration of the aforementioned limitations, this obvious approach is deemed unsuitable for obtaining the smallsignal training data. To address such limitations, it is proposed that the small-signal training data (e.g., and ) can be deduced using the forward model of the PA (e.g., Volterra). As the small-signal training data are obtained using the forward model in the software without forcing the PA to operate at a significant back-off region, the aforementioned limitations are less

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Fig. 7. Block diagram of the ARFPD engine test bench and the PAs under test.

problematic. Fig. 5 summarizes the steps of the proposed SSAPI algorithm and the detailed algorithm is summarized below: 1) The forward model of the PA is estimated using and . 2) The small-signal training data are estimated by using a small-signal stimulus as the input to the forward model. 3) The coefficients of the FIR block, are estimated using the LSE algorithm based on and . 4) The intermediate training data are estimated according to (6) 5) The coefficients of the EMP model are estimated using the LSE algorithm based on and . C. Evaluation of the Proposed SSAPI Algorithm In order to assess the validity of the proposed parameter identification algorithm, coefficients of the FIR-EMP predistorter are estimated using 1) the quasi-Newton algorithm from the MATLAB nonlinear optimization toolbox and 2) the proposed SSAPI algorithm. The training data is the same test data used in Fig. 3. Out of the 50000 available points, 10000 points are used to train the proposed FIR-EMP model and the other 40000 points are used for model validation purpose. Fig. 6 compares the modelling normalized mean square error (NMSE) versus the number of iterations corresponding to the two algorithms. Note that each of the two algorithms takes approximately the same computation time per iteration. According to Fig. 6, the proposed SSAPI algorithm achieved an NMSE of dB and only requires one iteration. In contrast, the nonlinear optimization requires iterations to reach the same level of modelling accuracy. It is worth mentioning that the described SSAPI algorithm is specifically proposed for the FIR-EMP model and is fundamentally different from the well-known parameter identification algorithm used by the Hammerstein or Wiener models [19], [20]. The two-box Hammerstein or Wiener model that consists of a static nonlinearity and an FIR filter is typically trained by

Fig. 8. Photograph of the ARFPD measurement setup.

first identifying the static nonlinearity, either through a continuous wave (CW) test [19] or a moving average method [20], before the FIR block can be identified. However, in the case of the proposed FIR-EMP model, the linear and nonlinear memory effects modelled by the FIR and EMP blocks respectively, are coupled and cannot be separated for conventional CW testing or the moving average method. On the other hand, the proposed SSAPI algorithm begins by first identifying a PA forward model, which is then used to generate new sets of input-output signals at the small-signal, so that the FIR filter coefficients can be identified. The coefficients of the EMP block are then identified using typical LSE algorithms. IV. VALIDATION AND MEASUREMENT RESULTS To realistically assess the performance of the proposed FIR-EMP model in an ARFPD system, a digitally assisted ARFPD test bench is built. Its corresponding block diagram is shown in Fig. 7 and the photograph of the testbench is shown in Fig. 8. The test bench consists of five main parts: the digital signal processing (DSP) unit, IQ modulator, RF vector multiplier, PA under test and the transmitter observation receiver.

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Fig. 9. (a) AM/AM and (b) AM/PM characteristics for PA1 without predistortion (red) and with the proposed FIR-EMP in ARFPD test bench (blue) of a 30 MHz WCDMA signal.

The DSP unit synthesizes the predistortion function (i.e., FIR-EMP) and generates the corresponding signals in the digital baseband. Two Keysight N8241A arbitrary waveform generators (AWGs) are synchronized using the same sampling clock (625 MHz). AWG-1 sends the baseband analog in-phase and quadrature signals and to the transmitter (ADL5375) for up-conversion. AWG-2 generates baseband signals and and sends them to the RF vector multiplier (ADL5390) in order to apply the predistortion function to the incoming RF signal. The output of the PA under test is captured using a Keysight N9030A signal analyzer with a maximal observation bandwidth of 160 MHz. All the equipment is synchronized using a 10 MHz reference clock. Compared to a complete ARFPD system, the current digitally assisted ARFPD test bench implements the major RF building blocks (i.e., IQ modulator and RF vector multiplier) but uses an emulated predistortion engine. Although implementation of the predistortion engine in hardware is feasible, this function would require a dedicated integrated circuit design. As a proofof-concept evaluation, the described ARFPD test bench is used to evaluate the proposed FIR-EMP model at the system level. The proposed FIR-EMP model and the proposed SSAPI algorithm are evaluated in the ARFPD test bench described previously to linearize different PAs driven by various wideband and intra-band carrier-aggregated signals. Two PAs under test are used, consisting of: 1) A GaN push-pull PA with 85-W peak envelope power operating at a carrier frequency of 900 MHz (PA1) 2) A GaN Doherty PA with 20-W peak envelope power operating at a carrier frequency of 1.9 GHz (PA2) [21].

Fig. 10. (a) AM/AM and (b) AM/PM characteristics for PA2 without predistortion (red) and with the proposed FIR-EMP in ARFPD test bench (blue) of a 30 MHz WCDMA signal.

The nonlinearity order is found out to be 7 and the memory depth is found out to be 5 based on the PAs characteristics through iterative parameter sweeping. For comparison purposes, a DPD test bench is also built. This DPD test bench is identical to Fig. 7 but without the RF vector multiplier and AWG-2. Three predistortion test cases are designed: 1) using the MP model in the DPD setup 2) using the dynamic deviation reduction based Volterra series (DDR-Volterra) model [4] in the DPD setup 3) using the proposed FIR-EMP model in the ARFPD setup. The validity of the proposed FIR-EMP model is first assessed in the ARFPD test-bench to linearize PA1 and PA2, driven by two different wideband modulated signals: 1) A 20 MHz LTE signal with a PAPR of 8.9 dB 2) A 30 MHz 4-carrier 110011 WCDMA signal with a PAPR of 8.6 dB. Tables I summarize the linearization results for PA1 and PA2, respectively, corresponding to the different test cases. The PA distortions (AM/AM and AM/PM) corresponding to PA1 and PA2 driven under the 30 MHz signal are plotted in Figs. 9 and 10. The output spectra of PA1 and PA2 under the three test cases are plotted in Figs. 11 and 12. For PA1, the proposed FIR-EMP model in the ARFPD setup achieved an adjacent channel leakage ratio (ACLR) improvement of 20 dBc when linearizing a 20 MHz LTE signal. Compared with the MP model based DPD setup, up to 2.7 dB ACLR improvement is noticed.

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TABLE I SUMMARY OF THE LINEARIZATION PERFORMANCE OF PA1 AND PA2 DRIVEN BY WIDEBAND MODULATED SIGNALS

Fig. 11. Output spectra of PA1 driven with 30 MHz bandwidth signal.

Fig. 12. Output spectra of PA2 driven with 30 MHz bandwidth signal.

For PA2, the advantages of the proposed model are evident according to Table I. The ACLR measured using the proposed FIR-EMP are 4.6 dB better than the MP model based DPD for

Fig. 13. Output spectra of PA2 driven by 80 MHz signal.

the 30 MHz WCDMA signal. Linearization results from the DDR-Volterra model are provided as the baseline for comparison, and the FIR-EMP model in the ARFPD test bench achieves comparable linearization capacity. Fig. 12 shows the excellent linearization capacity of the proposed FIR-EMP model by comparing the spectra recorded for PA2. The proposed FIR-EMP test case has significantly lower in-band and out-of-band distortion compared to the spectra obtained using the MP-DPD. It achieves comparable linearization results to the sophisticated DDR-Volterra model based DPD setup. To further evaluate the performance of the proposed FIR-EMP model under newer 4G communication signals, a 40 MHz mixed standard carrier aggregated signal, consisting of a 15 MHz 3-carrier WCDMA signal and a 15 MHz LTE signal with a combined PAPR of 8.4 dB is synthesized. For PA1, the proposed FIR-EMP in the ARFPD test-bench achieves an ACLR of dBc, compared to an ACLR of dBc found using the MP in DPD. For PA2, the proposed FIR-EMP test case achieve an ACLR of dBc, compared to the MP test case which records an ACLR of dBc. Compared with the sophisticated DDR-Volterra model, the proposed FIR-EMP model achieves a similar level of linearization (i.e., about dBc ACLR).

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To further assess the performance of the proposed FIR-EMP under even wider bandwidths, an 80 MHz mixed-standard intraband carrier aggregated signal, consisting of a 20 MHz 4-carrier WCDMA signal and a 20 MHz LTE signal with a combined PAPR of 9.5 dB. As summarized in Table I, when compared with the MP test case, the proposed FIR-EMP model achieves a 4 dB improvement in the measured ACLR. Significant in-band distortion is observed in the MP-DPD case, which leads to a high error vector magnitude (EVM) of 3.5% while the proposed FIR-EMP model has a measured EVM of 1.4%. The output spectra are presented in Fig. 13. V. CONCLUSION In this paper, an ARFPD system using the FIR-EMP model along with a linear SSAPI algorithm has been presented. An ARFPD test bench, which incorporates major RF components, has been built to assess the validity of the proposed FIR-EMP scheme and the SSAPI algorithm. It has been demonstrated that the SSAPI algorithm can extract coefficients with excellent modelling accuracy for the cascaded blocks of the FIR-EMP in a single iteration. Measurement results have shown that the proposed FIR-EMP model using the SSAPI algorithm can successfully linearize multiple PAs driven with various wideband and carrier-aggregated signals of up to 80 MHz instantaneous bandwidth. Linearization performance comparable to a DDR-Volterra based DPD scheme indicates the viability of the proposed FIR-EMP model for implementing ARFPD modules capable of mitigating the distortions exhibited by PAs driven by communication signals with up to 80 MHz modulation bandwidth. This confirms the potential of ARFPD as a very promising candidate for the linearization of small-cell base station PAs, which would reduce the power overhead compared to using the popular digital predistortion technique. REFERENCES [1] S. Jin, K. Moon, B. Park, J. Kim, Y. Cho, H. Jin, D. Kim, M. Kwon, and B. Kim, “CMOS saturated power amplifier with dynamic auxiliary circuits for optimized envelope tracking,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 12, pp. 3425–3435, Dec. 2014. [2] A. Mohamed, S. Boumaiza, and R. Mansour, “Doherty power amplifier with enhanced efficiency at extended operating average power levels,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 12, pp. 4179–4187, Dec. 2013. [3] J. Kim and K. Konstantinou, “Digital predistortion of wideband signals based on power amplifier model with memory,” Electron. Lett., vol. 37, no. 23, pp. 1417–1418, Nov. 2001. [4] A. Zhu, J. Pedro, and T. Brazil, “Dynamic deviation reduction-based Volterra behavioral modeling of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 12, pp. 4323–4332, Dec. 2006. [5] B. Fehri and S. Boumaiza, “Baseband equivalent Volterra series for behavioral modeling and digital predistortion of power amplifiers driven with wideband carrier aggregated signals,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 11, pp. 2594–2603, Nov. 2014. [6] S. Boumaiza, J. Li, M. Jaidane-Saidane, and F. Ghannouchi, “Adaptive digital/RF predistortion using a nonuniform LUT indexing function with built-in dependence on the amplifier nonlinearity,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 12, pp. 2670–2677, Dec. 2004. [7] W. Kim, K. Cho, S. Stapleton, and J. Kim, “Baseband derived RF digital predistortion,” Electron. Lett., vol. 42, no. 8, pp. 468–470, Apr. 2006.

[8] W. Woo, M. Miller, and J. Kenney, “A hybrid digital/RF envelope predistortion linearization system for power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 1, pp. 229–237, Jan. 2005. [9] T. Rahkonen, O. Kursu, M. Riikola, J. Aikio, and T. Tuikkanen, “Performance of an integrated 2.1 GHz analog predistorter,” in Proc. International Workshop on Integrated Nonlinear Microwave and Millimeter-Wave Circuits, Jan. 2006, pp. 34–37. [10] E. Westesson and L. Sundstrom, “Low-power complex polynomial predistorter circuit in CMOS for RF power amplifier linearization,” in Proc. ESSCIRC, Sep. 2001, pp. 486–489. [11] N. Mizusawa, S. Tsuda, T. Itagaki, and K. Takagi, “A polynomial-predistortion transmitter for WCDMA,” in Proc. IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2007, pp. 350–608. [12] A. Kidwai and B. Jalali, “Power amplifier predistortion linearization using a CMOS polynomial generator,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., Jun. 2007, pp. 255–258. [13] R. N. Braithwaite, “Memory correction for a WCDMA amplifier using digital-controlled adaptive analog predistortion,” in Proc. IEEE Radio and Wireless Symp., 2010, pp. 144–147. [14] F. Roger, “A 200 mW 100 MHz-to-4 GHz 11th-order complex analog memory polynomial predistorter for wireless infrastructure RF amplifiers,” in Proc. IEEE Int. Solid-State Circuits Conf. Tech. Dig., 2013, pp. 94–95. [15] H. Huang, A. Islam, J. Xia, P. Levine, and S. Boumaiza, “Linear filter assisted envelope memory polynomial for analog/radio frequency predistortion of power amplifiers,” in Proc. IEEE MTT-S IMS, May 2015, pp. 1–3. [16] O. Hammi, F. Ghannouchi, and B. Vassilakis, “A compact envelope-memory polynomial for RF transmitters modeling with application to baseband and RF-digital predistortion,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 5, pp. 359–36, May 2008. [17] C. C. Cadenas, J. R. Tosina, M. J. M. Ayora, and J. M. Cruzado, “A new approach to pruning Volterra models for power amplifiers,” IEEE Trans. Signal Process., vol. 58, no. 4, pp. 2113–2120, 2010. [18] E. G. Lima, T. R. Cunha, and J. C. Pedro, “PM-AM/PM-PM distortions in wireless transmitter behavioral modeling,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., 2011, pp. 1–4. [19] J. Pedro and S. Maas, “A comparative overview of microwave and wireless power-amplifier behavioral modeling approaches,” IEEE Trans. Microwave Theory Techn., vol. 53, no. 4, pp. 1150–1163, Apr. 2005. [20] T. Liu, S. Boumaiza, and F. Ghannouchi, “Augmented Hammerstein predistorter for linearization of broad-band wireless transmitters,” IEEE Trans. Microwave Theory Techn., vol. 54, no. 4, pp. 1340–1349, Jun. 2006. [21] H. S. A. Jundi and S. Boumaiza, “An 85-W multi-octave push-pull GaN HEMT power amplifier for high efficiency communication applications at microwave frequencies,” IEEE Trans. Microw. Theory Techn., vol. PP, no. 99, pp. 1–10, Sep. 2015. Hai Huang (S'15) received the B.A.Sc. degree in electrical and computer engineering from the University of Waterloo, Waterloo, Ontario, Canada, in 2013 and is currently working towards the Ph.D. degree at the University of Waterloo. His research interests include Analog/digital signal processing, analog and RF integrated circuits and linearization of RF power amplifiers.

Jingjing Xia received the B.Eng. and Ph.D. degree in electrical and electronics engineering from the Nanyang Technological University, Singapore, in 2008 and 2013, respectively and the M.A.Sc. degree in electrical and computer engineering from the University of Waterloo, Waterloo, ON, Canada in 2013. He is currently a research engineer in the Emerging Radio Research Group Lab., University of Waterloo. His current research interests include microwave and millimeter-wave transmitters as well as analog and digital predistortion techniques.

HUANG et al.: DIGITALLY ASSISTED ANALOG/RF PREDISTORTER WITH A PARAMETER IDENTIFICATION ALGORITHM

Anik Islam received the B.A.Sc. degree from the University of Waterloo, Waterloo, ON, Canada, in 2013 and is currently working towards an M.A.Sc. degree at the University of Waterloo. In 2014, he joined the Emerging Radio Systems Group at the University of Waterloo as a Research Associate. His research focuses on the construction and identification of nonlinear dynamic models for the predistortion of radio frequency power amplifiers.

Eric Ng (S’14) is currently pursuing the B.Sc. and M.Sc. in electrical engineering at the University of Waterloo, Waterloo, Canada. In 2014 he joined the Emerging Radio Systems group as an undergraduate student research assistant, working on hybrid predistortion techniques in RF power amplifiers. His research interests include digitally assisted analog RF front end design and ultra-low power transceiver architecture.

Peter M. Levine received the B.Eng. degree in computer engineering and the M.Eng. degree in electrical engineering from McGill University, Montreal, QC, Canada in 2001 and 2004, respectively, and the Ph.D. in electrical engineering from Columbia University, NY, USA in 2009. From 2009 to 2010, he worked as a Research Engineer in integrated circuit and sensor design for Ion Torrent, Guilford, CT, USA. In 2011, he joined the Department of Electrical and Computer Engineering at the University of Waterloo, Waterloo, ON, Canada as an Assistant Professor. His research interests include CMOS-integrated bio-

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chemical assays, integrated microsystems for clinical and environmental monitoring, and the design of precision analog/mixed-signal integrated circuits for sensor interfaces. He is an inventor or co-inventor on six issued U.S. patents. Prof. Levine has served as a technical program committee member for several conferences, including the IEEE/ACM International Symposium on Low Power Electronics and Design (ISLPED) and the IEEE Biomedical Circuits and Systems Conference (BioCAS). He was a recipient of the Intel Foundation Ph.D. Fellowship in 2005.

Slim Boumaiza (S'00–M'04–SM'07) received the B.Eng. degree in electrical engineering from the Ecole Nationale Ingnieurs de Tunis, Tunis, Tunisia, in 1997, and the M.S. and Ph.D. degrees from the Ecole Polytechnique de Montreal, Montreal, QC, Canada, in 1999 and 2004, respectively. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada, where he leads the Emerging Radio Systems Group, which conducts multidisciplinary research activities in the general area of RF/microwave and millimeter component and system design for wireless communications. His current research interests include RF/digital signal processing mixed design of intelligent RF transmitters; design, characterization, modeling and linearization of high-efficiency RF power amplifiers; and reconfigurable and software-defined transceivers.

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Digital Compensation for Transmitter Leakage in Non-Contiguous Carrier Aggregation Applications With FPGA Implementation Chao Yu, Member, IEEE, Wenhui Cao, Student Member, IEEE, Yan Guo, Student Member, IEEE, and Anding Zhu, Senior Member, IEEE

Abstract—In this paper, a generalized dual-basis envelope-dependent sideband (GDES) distortion model structure is proposed to compensate the distortion induced by transmitter leakage in concurrent multi-band transceivers with non-contiguous carrier aggregation. This model has a generalized structure that is constructed via first generating a nonlinear basis function that maps the inputs to the target frequency band where the distortion is to be cancelled, and then multiplying with a second basis function that generates envelope-dependent nonlinearities. By combining these two bases, the model keeps in a relatively compact form that can be flexibly implemented in digital circuits such as field programmable gate array (FPGA). Experimental results demonstrated that excellent suppression performance can be achieved with very low implementation complexity by employing the proposed model. Index Terms—Behavioral model, carrier aggregation, multi-band, power amplifiers, transmitter leakage suppression.

I. INTRODUCTION ON-CONTIGUOUS carrier aggregation (CA) technique [1] has been proposed to effectively combine multiple frequency bands to conduct high-speed data transmission in wireless communications. To support CA operation, high-efficiency concurrent multi-band transmitters are often deployed [2]. Due to nonlinear characteristics of RF power amplifiers (PAs), distortion is normally added into the transmit signal after amplification [3]. In multi-band operation, the distortion is usually located not only near the transmission bands, but also at the intermodulation frequencies. These intermodulation frequency bands, e.g., the third-order intermodulation (IM3) bands, sometimes can overlap with the receiver bands in the frequency-division duplex (FDD) mode, as illustrated in Fig. 1. Ideally, the duplexers shall have enough attenuation to avoid the distortion generated by the transmitter that falls into the receiver band. In practice, however, it is not easy to design such duplexers to meet

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Manuscript received June 29, 2015; revised September 02, 2015; accepted October 18, 2015. Date of publication November 11, 2015; date of current version December 02, 2015. This work was supported in part by the Science Foundation Ireland under the Principal Investigator Award Grant Number 12/IA/ 1267. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, 17–22 May 2015. C. Yu is with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). W. Cao, Y. Guo, and A. Zhu are with the School of Electrical and Electronic Engineering, University College Dublin, Dublin 4, Ireland (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2015.2495144

Fig. 1. Transmitter leakage in 3-carrier carrier aggregation scenario.

the requirement. Some intermodulation products can therefore leak to the receiver band and introduce serious spurious emission to the receiver, causing significant quality degradation of the received signal. Various compensation schemes have been proposed either in transmitter (Tx) or in receiver (Rx) to resolve the problem. Because of low cost and great accuracy, digital predistortion (DPD) [4]–[10] in the transmitter has been widely employed to remove the sideband distortion. In [4], C. Yu et al. proposed a full-bandwidth DPD method by treating the multiband signal including the sideband signal as a single signal to effectively remove the unwanted sideband distortion. In [5], [6], P. Roblin and J. Kim et al. proposed a frequency selective DPD method to successfully cancel the sideband separately by employing a large signal network analyzer (LSNA) to extract the device under test (DUT) information. In [7], S. A. Bassam et al. proposed a filtering-based sideband distortion modeling technique to inject the anti-phase sideband distortion for distortion suppression. To reduce complexity, M. Abdelaziz et al. in [8], [9] proposed simplified methods by only picking the modeling terms falling into the specified distortion bands. In [10], Z. Fu et al. proposed a sideband compensation scheme based on evaluating and minimizing the power spectral density (PSD) of PA output signal around the pre-specified frequency. The DPD-based sideband compensation methods work well in general but they require extra bandwidths to transmit the sideband information in multi-band transmitters, which is often not desirable in many applications.

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YU et al.: DIGITAL COMPENSATION FOR TRANSMITTER LEAKAGE

Instead of removing the distortion in Tx, some other compensation schemes [11]–[20] are realized in Rx. The main idea is to build a distortion model to generate the replica of the distortion that falls into the receiver band and subtract it from the received signal to obtain the original signal, as shown in Fig. 1. In [11], [12], A. Frotzscher et al. analyzed the impact on system performance in zero-IF receiver impaired by transmitter leakage. In [13], M. Kahrizi et al. proposed a digital method to suppress the second-order intermodulation (IM2) of Tx leakage in WCDMA direct-conversion receivers. M. Omer et al. in [14] created the replica of the sideband distortion by assuming that the frequency response of duplex filter is known, while A. Kiayani et al. in [17] proposed a method to estimate the transmitter leakage channel including both duplexer and PA. The same authors in [18] extended this method to deal with concurrent dual-band signal. In [19] and [20], H.-T. Dabag et al. proposed an all-digital cancellation technique to mitigate the receiver desensitization in uplink CA in cellular handsets. In [21], we proposed a novel dual-basis envelope-dependent sideband distortion model to characterize the transmitter leakage in the receiver band in concurrent dual-band transceivers. Experimental results showed that only a very small number of model coefficients with narrowband digital signal processing are required to achieve satisfactory cancellation performance. Due to limited space, in [21], only the basic concept and the verification of suppression of distortion in concurrent dual-band transceivers were given. In this paper, we provide a detailed analysis of transmitter leakage and give comprehensive derivations for the model development in more complex scenarios, such as 3-Carrier (3-C) CA applications. Based on the analysis, a generalized dual-basis envelope-dependent sideband (GDES) distortion model structure is then proposed to provide a uniform architecture to suppress various distortions that appeared in such systems. By employing the proposed model structure, different distortion components can be accurately characterized and compensated with the same digital circuit module. The distortion overlapping issue can also be easily resolved. Compared to the existing methods, the proposed model is in a compact format and can be easily extended to different scenarios without increasing much complexity. A generalized FPGA architecture with detailed hardware implementation is also given. The rest of the paper is organized as follows: In Section II, the transmitter leakage analysis focusing on the 3-C CA application is provided. A generalized model structure is then proposed in Section III with FPGA implementations given in Section IV. The experimental results for the different scenarios are provided in Section V, followed by a conclusion in Section VI.

II. TRANSMITTER LEAKAGE ANALYSIS With increasing demands for high data rates, carrier aggregation techniques will be widely employed in wireless cellular communications and the number of aggregated carriers will inevitably keep increasing in the future. As discussed earlier, the distortion generated in transmitters can leak to receiver bands

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and cause quality degradation of the received signals. This situation becomes worse when CA is employed. In this section, we take a 3-carrier CA scenario as an example to illustrate how the transmitter leakage is generated. A. 3-Carrier Carrier Aggregation Considering the frequency plan for LTE FDD mode [22] and the transmitter architecture employing CA techniques [23], three cases of frequency allocations might be assigned for three-carrier carrier aggregation. As shown in Fig. 2(a), in the first case, three bands are located at MHz (LTE band 5), MHz (LTE band 1), and MHz (LTE band 22). Since these three bands are spanned in a very large frequency range, multiple RF chains and power amplifiers may be employed. In this case, only the distortions near main carriers are our concern. The intermodulation products crossing the multiple bands may not cause severe problems. However, in the second case, shown in Fig. 2(b), if three bands are located at MHz (LTE band 3), MHz (LTE band 1), and MHz (LTE band 7), the intermodulation products will spread over to nearby receiver bands which can cause problems. For example, the upper 3rd-order intermodulation product generated from band 1 and 3 is located around 2510 MHz , which overlaps with the uplink band for LTE band 7 allocation (2500 MHz–2570 MHz). In the third case, shown in Fig. 2(c), three bands are located at MHz (LTE band 3), MHz (LTE band 2), and MHz (LTE band 1). In this case, all three bands are located over a frequency range of 400 MHz and thus it is possible to only employ one wideband PA to transmit this multi-band signal. Similar to the second case, the intermodulation products also affect receiver bands and in this case the situation becomes much more complex since not only the intermodulation products generated from two frequency bands, but also the ones generated from three frequency bands can affect receivers. For instance, one of the IM3 generated from these three bands are 1950 MHz , that overlaps with the uplink band for LTE band 1 allocation (1920 MHz–1980 MHz). B. Derivation for Tx Leakage Let's assume the aggregated input signal sented as

can be repre-

(1) where are the baseband representations of the signals located at the carrier frequencies . If the input signal passes through a nonlinear system, the output will contain many distortion products that can spread over multiple frequency bands. Here, to simplify the derivation, a memoryless polynomial model is taken for example, that is (2)

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From (3), we can see that the center frequency for each distortion item can be calculated from the main carrier frequencies, that is

(4) For example, the center frequency of the distortion at IM3 bands can be obtained from

(5) By using (5), the distortion at IM3 at the frequency can be expressed as

(6)

Fig. 2. 3-Carrier carrier aggregation allocation.

where and is the input and output, respectively and is the nonlinear order. Substituted (1) into (2), all the distortion can be obtained

Removing the carrier frequency, the baseband information can be represented as

(7) For better illustration, the distortion terms are listed below

.. .

(3)

(8)

From (8), we can see that the distortion is generated from combinations of the signals located at different bands. If we treat these signals at each band as independent inputs, constructing the distortion model is straightforward: simply generate each term by combining the baseband signals at different bands, as shown in Fig. 3. The disadvantage of this approach is that

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Fig. 3. Model structure based on picking terms for 3-C CA application.

the model complexity will increase quickly with the number of bands and the situation becomes worse when higher order nonlinearity and memory terms are included into the model.

can be divided into two sections as listed in Table I. Looking closely, we can find that the two parts of the modeling terms have distinct functionalities: the common part is related to the band of the distortion to be cancelled while the changing part depends on the envelope of the inputs only. We will explain this phenomenon in detail in next two sections. Nevertheless, based on this finding, in this work, we propose to decompose the model into two basis functions: the first basis function is to locate the frequency components in the target bands, denoted as Basis 1; and the second basis is to create an accurate mapping from the input to the output by using envelope dependent nonlinear terms, denoted as Basis 2. The model structure can be described as

(12)

III. PROPOSED MODEL To overcome the disadvantage of the existing models, a novel model structure is proposed in this section.

where

A. Model Basis Decomposition

As mentioned earlier, if the input signal passes through a nonlinear PA, the output will contain many distortion products that can spread over multiple frequency bands. To cancel transmitter leakage, we only concern the distortion that falls at the receiver frequency band, for example, the distortion at band in 3-C CA Case in Fig. 2(b). Because this frequency band is different from where the original input signals are located, to generate this distortion, we must “inter-modulate” inputs between two bands and thus generate the new frequency band. The basic description for the dual-band scenario can be found in [9]. For instance, as illustrated in Fig. 4(a), to generate the distortion at band, we can multiply two inputs from one band with the conjugate of the input from the other band

For better illustration, two special cases for IM3 in (5) are chosen. One is for the case in Fig. 2(b) where the IM3 is generated from LTE band 1 and 3 and the other is for the case in Fig. 2(c) where the IM3 is generated from all three bands, that is (9) Then, (7) can be transformed to (10), shown at the bottom of the page. Looking at (10), although the term combinations change with the order of nonlinearities, there are some “common” unchanged terms in each equation. To illustrate this, we can re-write (10) as (11), shown at the bottom of the page, where we can see that and appear in all the modeling terms, for each case, respectively. By separating the common parts from the changing parts in each term, the model

and

represent Basis 1 and 2, respectively.

B. Model Basis 1

(13) where is the conjugate operation. This term is corresponding to the carrier frequency change, i.e., . As shown in Fig. 3, (13) only generates the 3rd-order distortion.

(10)

(11)

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TABLE I CONVENTIONAL MODEL DECOMPOSITION

TABLE II PROPOSED MODEL DECOMPOSITION

For higher order distortion, we need to add in more terms, e.g., for the 5th order. To ensure the model output stay in the target band, these extra added terms should not “move” the band. In other words, they should only affect the frequency components within the target band, but not crossing the bands. For instance, is only used to weight in . In the frequency domain, is corresponding to the frequency , which indicates that multiplying this term does not change the carrier frequency. Therefore, from the frequency selection point of view, (13) is the key element that “selects” the target bands in the model construction. Follow the same logic, the band selection element for IM3 band in the 3-C CA case in Fig. 2(c) is (14) as illustrated in Fig. 4(b). In summary, Basis 1 of the above cases can be described as

(15) Based on the same idea, the distortion located at other frequency bands can also be constructed by simply changing the combination of the signal terms. C. Model Basis 2 To model high order nonlinearities, (15) can be multiplied with different high order terms as shown in Table I, which can be describe as Basis 2, that is

(16)

This polynomial extension is straightforward, but this operation will lead to considerable increase of the model complexity when strong nonlinear distortion is involved as discussed earlier. As mentioned in [24], [25] for the low-pass equivalent model, once the relationship between the input and output meets the requirement of odd parity and the mapping is located at the specified frequency band, it is not necessary to build the high-order

nonlinearities using conventional polynomials. Instead of using each individual envelope, in this work, we propose to construct the second basis function using the average envelope of the signal, that is

(17)

This structure keeps the even-parity, which satisfies the oddparity rule of the low-pass equivalent model construction, when multiplied with the Basis 1 function that satisfies the odd-parity. At the first glance, we may think the square root operations can be very complex in hardware implementation compared to the conventional polynomials. Surprisingly, with the assistance of coordinate rotation digital computer (CORDIC) technique [26] in FPGA, the complexity can be significantly reduced and becomes lower than that for the polynomials. We will discuss this in detail in Section IV D. Model Basis Re-Combination To simplify the model expression, we move the power operation out of the basis in (17) and re-define the terms inside the power operation as Basis 2. Thus the new model can be expressed as (18) is the basis deciding which band to be compenwhere sated, and is the basis for generating the high-order nonlinearities. represents the nonlinear order. In the dual-band and tri-band cases, specific examples are given in Table II. The derivation above is only for the memoryless nonlinear systems. To further characterize a wider range of nonlinear systems, memory effects need to be taken into account. To do so, the model can be constructed as

(19) where and is Basis 1 and Basis 2, respectively. again represents the nonlinear order and and represent the memory length for each basis, respectively.

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Fig. 6. Proposed model structure.

and then multiplied with the Basis 2 function respectively and finally combined together to form the full model. In summary, there are three basis functions used in the model for this case

(22)

Fig. 4. Basis 1 generation. (a) 3-C CA Case in Fig. 2(b). (b) 3-C CA Case in Fig. 2(c).

To generalize this procedure, we reformat the model as

(23)

Fig. 5. Even-spaced case for 3-C CA application.

E. The Generalized Model The model proposed above can be easily extended to general cases without structure changes. For instance, in the 3-C CA case, if the three carrier frequencies are evenly allocated, multiple intermodulation products may fall into the same frequency band, as shown in Fig. 5, where the carrier frequency MHz (LTE band 3), MHz (LTE band 2), MHz (LTE band 1) are evenly spaced, which leads that the distortion located at 2245 MHz can be generated from two different IM3 products. One is generated from two carriers, 1985 MHz and 2115 MHz, and the other from all three carriers, that is

(20) Because both distortion bands are located at the same frequency, the total distortion component should consist of two parts, which requires two different modeling terms. As discussed earlier, the frequency components can be easily selected with Basis 1 functions in the proposed model. In this case, we simply construct two Basis 1 functions to model the two IM3 products, i.e.,

(21)

is the basis for the th distortion band or where term to be compensated, and is the basis for modeling high-order nonlinearities. represents the nonlinear order and represent the memory length for each basis, respectively. We call this model the generalized dual-basis envelope-dependent sideband (GDES) distortion model. The model structure is illustrated in Fig. 6, where a frequency analysis block is added to select distortion components and bands before constructing the model. Compared to the existing solutions, this new model structure provides many advantages. Firstly, the signal processing bandwidth is only related to the baseband signals at each band, leading to the narrow bandwidth requirement. Secondly, because only one average envelope is involved, the number of model coefficients is significantly reduced and thus low-complexity implementation can be realized. Thirdly, the target band can be arbitrarily changed by replacing the terms in Basis 1 without significantly changing the model structure, which brings great flexibilities for future extension. IV. FPGA IMPLEMENTATIONS To evaluate the practical application of the proposed cancellation structure, the proposed model is implemented in FPGA and compared with the existing models in terms of resource consumption. A generalized FPGA implementation architecture for TX leakage suppression in 3-C CA application is also proposed in this section.

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Fig. 8. FPGA implementation for mode selection.

Fig. 7. PGA implementation architecture for (a) the existing model structure and (b) the proposed model structure.

A. FPGA Resource Consumption Comparison Two types of structures are employed to make a fair comparison. The first model is an existing model based on termspicking approach in the dual-band case as

(24) The second model is a typical example of the proposed model for the same dual-band scenario as

(25) The objective is to compare the resources consumption when the similar performance is achieved. Based on (24) and (25), two FPGA implementation architectures can be built as shown in Fig. 7. In Fig. 7(a), the common part can be implemented by employing two complex multipliers. To implement the changing part, four square operations and two adders are required to calculate and . Multiplexing technique can be employed to reuse the hardware resource and thus reduce resource consumption. Different orders of nonlinear terms are then fed into multiplication and combination module to construct all the possible combinations for the two inputs, e.g., . The different outputs will then be multiplied with the common part, and fed into memory structure (equivalent to the FIR structure). Finally, all these terms can be added together to construct the full distortion model. Since there are many possible combinations, a large number of multipliers are usually involved in this implementation.

In Fig. 7(b), the proposed structure mainly consists of three parts: Basis 1 generation, Basis 2 generation and the combination of these two bases including different orders and memory. Firstly, in Basis 1 generation, is equivalent to the common part in Fig. 7(a). Secondly, Basis 2 is generated in a different way from the conventional method. At the first glance, one may think more complex computation will be involved in the envelope calculation, since there is square root operation. However, by using CORDIC [26], this step becomes very simple with only shift and addition operations involved, which significantly reduces the implementation complexity. The details for the square root implementation are given in Appendix. Two CORDIC modules are employed to generate and , which take the I signal as one input and Q signal as the other input to calculate the . To reduce the resource consumption, multiplexing technique can also be employed, which is also discussed in Appendix. After this operation, we can continue to employ a CORDIC module to realize the implementation of . Then, the implemented function can be used to generate high order terms, combined with the Basis 1, and delayed and multiplied with different coefficients. Finally, all these terms can be added together to construct the full distortion model. In practice, the memory structure can be further simplified based on the practical requirement. For example, if memory terms are few, all the different memory structure can be added first and then delayed together, which will reduce the number of multipliers required. B. The Generalized Architecture for 3-C CA Application One big advantage of the proposed structure is that the envelope term is in a generalized format which can be easily extended to various multi-band cases. For instance, to extend from dual-band to tri-band, only one more CORDIC needs to be employed in the implementation. The detail of this implementation is given in Appendix. Furthermore, because the model is in a generalized structure, all the distortions located at different IM3 bands can be generated by using the same hardware block. It also allows multiplexing to be employed to save resources in FPGA. For instance, as illustrated in Fig. 8, the input may consist of three original inputs . To generate , we can simply select from branch 1 and select from branch 2 and

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Fig. 9. General FPGA implementation architecture for tri-band intermodulation product cancellation.

3. To generate and should be selected from different branches, respectively. The resource consumption for this module will be discussed in next section with a practical example. Based on the discussion above, the general FPGA implementation architecture for tri-band intermodulation product cancellation is shown in Fig. 9. In selection block, the input “mode” is used to select the cancellation band. For example, in the 3-C CA application, there are 9 options for the IM3 bands selection. With this operation, the distortion located at different bands can be cancelled by controlling the single variable of the mode. V. EXPERIMENTAL RESULTS To effectively validate the proposed method, a test bench was setup as shown in Fig. 10. In the transmit chain, the baseband signals with different carrier aggregation allocations are generated in PC by software MATLAB, then up-converted to RF frequency, and fed into a high power LDMOS PA operated at 2.14 GHz with average output power of 37.5 dBm. Due to the limitation of the platform, we conduct the test without a real duplexer but using a digital filter instead. In other words, the transmitter distortion at the receiver band is not attenuated by a duplexer before down-conversion. In our test, the full transmitter signal is fed into the receiver, then down-converted, sampled and finally demodulated back to the baseband. The sideband distortion is obtained by applying a digital filter on the received signal. The distortion suppression model was implemented in FPGA and can run in real-time, but the model extraction was conducted in MATLAB by using the standard least squares (LS) algorithm. Furthermore, in these tests, the receiver chain was considered being linear and had a fixed gain. Due to the bandwidth limitation of the platform, we only used 5 MHz signals at each band to conduct the “proof-of-concept” tests. A. 3-C CA Case 1: Fig. 2(b) In this test, the baseband signal combines two 5 MHz signals located at MHz and MHz and with peak-to-average power ratio (PAPR) of 7.8 dB. The sideband distortion is located at

Fig. 10. Test bench setup.

MHz. The model configuration in (25) is set as . Fig. 11 shows the measured power spectrum density with and without the transmitter leakage suppression. From Fig. 11, it can be clearly seen that 25 dB suppression can be achieved by employing the proposed model, which confirms the model accuracy. The signal processing bandwidth required is only 46 MHz that is corresponding up to 9th order nonlinearities with the two 5 MHz baseband signals, regardless of the frequency spacing. It is also worth mentioning that only 8 coefficients are required in this proposed model, which leads to a very low-complexity in practical implementations. The model was implemented in real hardware FPGA board. The measurement results from FPGA implementation are compared with the one from the simulation in Fig. 12, where we can see that the hardware performance is almost as good as that simulated in MATLAB. For the conventional model, to obtain the similar performance, 12 coefficients are required, that is, the model configuration in (24) is set as . The performance is also illustrated in Fig. 11. The FPGA resource utilization for this case is listed in Table III to compare the resource consumption. The implementation of Basis 1, , in both models are the same, which occupies 2068 slice LUTs and 2036 slice registers.

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TABLE III FPGA RESOURCE UTILIZATION COMPARISONS FOR THE CASE IN FIG. 2(B)

Fig. 11. Measured performance comparison for distortion suppression at IM3 band in 3-C CA case 1: Fig. 2(b).

Fig. 12. Measured performance for proposed method in distortion suppression at IM3 band in 3-C CA case 1: Fig. 2(b).

The differences are in the implementations of the step 2, 3 and 4, whose resource consumptions are listed in Table III in details Basis 2 of in the proposed algorithm is accomplished by using CORDIC, which saves 57% LUTs and 38% registers in contrast with that in step 2 in the conventional model. As mentioned earlier, the conventional algorithm requires 4 more coefficients than the proposed one based on the similar calibration performance, resulting in great amount hardware occupation in step 3 and 4 to implement coefficients multiplication. Since the memory structures of two models are identical, the resource usages for both approaches are the same in step 5. In summary, compared to those in the conventional model, the numbers of slice LUT and slice register used in the proposed model decrease by 3426 and 3309, respectively. As discussed in Section IV, multiplexing technique can be employed to further reduce the FPGA resource consumption. The simplified cases are illustrated in Table IV. In step 2, the generation block of and by CORDIC can be multiplexed, which is the same case as to obtain and by adders and multipliers in the conventional model In the final step 5, the summation with different memory consists of current terms and delayed terms can share one structure. Moreover, the resource consumption of step 4 in Table IV is dramat-

ically reduced in the low-cost implementation compared with that in Table III. This is because the specific mode of the multiplications between coefficients and input terms in step 4 is employed. When the multiplier model is set as constant coefficient model, the consumption will be calculated depended on fixed coefficient, which is normally less than common (parallel) multiplier mode. Therefore, both implementations for step 4 in Table IV employ fixed coefficient strategy to further save hardware dissipation. The difference of resource utilization between the proposed and the conventional method in the low-cost multiplexing implementation is smaller than the previous structure without simplification in Table III, but the proposed model still shows advantages over the conventional model. Furthermore, comparisons with other models published in the literature are also listed in Table V, in terms of suppression performance and hardware resource usage. From the results, we can see that our model can achieve great suppression performance with relatively low hardware resources. B. 3-C CA Case 2: Fig. 2(c) In this test, the baseband signal combines three 5 MHz signals located at MHz, MHz, MHz to form a 3-C CA signal. Although the scenario is changed compared to Part A, the model configuration is still set as with 8 coefficients. Fig. 13 shows the measured power spectrum density with and without the transmitter leakage suppression. In Fig. 13, two typical examples of IM3 distortion bands in 3-C CA are expected to be compensated: (1) the IM3 distortion generated from all three carriers, e.g., the target frequency is located at , as shown in Fig. 13(a); (2) the IM3 distortion generated from any two carriers, e.g., the target frequency is located at , as

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TABLE IV FPGA RESOURCE UTILIZATION MULTIPLEX FOR THE CASE IN FIG. 2(B)

TABLE V COMPARISONS OF THE IMD3 CANCELLATION IN THE DUAL-BAND SCENARIO

Fig. 13. Measured performance for distortion suppression at IM3 band in 3-C (2) target frequency CA case 2: Fig. 2(c). (a) target frequency .

shown in Fig. 13(b). From Fig. 13, it can be clearly seen that 2 dB suppression can be achieved for both cases by employing the proposed model. Also the measurement results from FPGA implementations perform as good as the ones from MATLAB. The FPGA resource utilizations for both cases are listed in Table VI. Compared to the resource utilization in Table IV, the consumptions in both cases listed in Table VI only increase slightly Also it can be easily seen that there is slight difference in FPGA resource utilization for both cases. The reason is that due to the different values of the coefficients in these two cases, the FPGA implementation will lead to slight different hardware occupations. Based on the results, it can be seen that the proposed methods will save more FPGA resources when more carriers involved. It is also worth mentioning that the resource consumptions for the selection module in Fig. 8 have also been investigated in this section. The FPGA resource utilization comparison is listed in Table VII Compared to the case without Mux implementation in Table VI, the one with Mux implementation only increases 96 slice LUTs which is insignificant. However, this module will largely enhance the flexibility to form a uniform structure to cancel any sideband distortion located in IM3 bands.

TABLE VI FPGA RESOURCE UTILIZATION FOR THE CASE IN FIG. 2(C)

TABLE VII FPGA RESOURCE UTILIZATION COMPARISON FOR THE SELECTION MODULE

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TABLE VIII FPGA RESOURCE UTILIZATION COMPARISON FOR SQUARE ROOT OPERATION

C. 3-C CA Case 3: Fig. 5 In this test, the baseband signal combines three 5 MHz signals located evenly at MHz, MHz, MHz. One of the sideband distortions is located at 2200 MHz, which can be generated from and , that is and also from all three carriers, that is, . Both distortion components will be overlapped with each other. The model configuration in (22)–(23) is set as , in which two memory parameters are simplified to one. Compared to the cases in Part A and B, the number of the coefficients will be doubled, that is, 16 coefficients, since there are two different basis 1 functions in the model. Fig. 14 shows the measured power spectrum density with and without the transmitter leakage suppression. From Fig. 14, it can be clearly seen that again 20 dB suppression can be also achieved by employing the proposed model. VI. CONCLUSION In this paper, a generalized dual-basis envelope-dependent sideband distortion model, which is further developed from the basic concept in [21], was proposed to model and suppress transmit leakage for non-contiguous CA applications. The proposed model structure provides great flexibility for dealing with different intermodulation products in a uniform structure, which has been validated by FPGA implementation. Experimental results demonstrated excellent model performance with very low model complexity, which provides a promising application in future carrier aggregation applications.

APPENDIX FPGA IMPLEMENTATION OF SQUARE ROOT OPERATION CORDIC is a technique that calculates the trigonometric functions of sine, cosine, magnitude and phase to a desired precision via iteratively rotating the phase of the complex number by multiplying it with a succession of constant values. In this Appendix, FPGA implementation for the square root operation of complex numbers employing CORDIC is provided. To find the magnitude of a complex number, , we can simply rotate it to have a phase of zero and then the magnitude of this complex number is just the real part since the imaginary part is zero. To do this in digital circuits using CORDIC, we first need to make sure its phase is less than degrees. This can be achieved by rotating the complex number by 90 degrees first if its phase is greater than 90 degrees: at the first step, we need to determine if the complex number has a positive or negative phase by looking at the sign of the value. If the phase is positive, rotate it by degrees otherwise by degrees. To rotate by degrees, swap and , and

Fig. 14. Measured performance for distortion suppression at IM3 band in 3-C CA case 3: Fig. 5.

change the sign of , i.e., ; to rotate by degrees, swap and , and change the sign of , i.e., . The phase of is now less than degrees, and we then further rotate the phase iteratively using CORDIC. Since the phase of a complex number is , the phase of “ ” is and likewise, the phase of “ ” is . To add phases, we can multiply by “ ” while to subtract phases, we can use “ ”. In the following iterations, we rotate the phase of the complex number using numbers of the form of “ ”, where is decreasing with powers of two after each iteration, starting with and thereafter , etc., until the phase goes to zero. The operations can be expressed as (A.1) is the real and imaginary part of the complex where and number, respectively, and represents th rotation. can have the value of or 1, which is used to determine the direction of the rotation depends on the sign of . represents the gain of each rotation, that is (A.2) To simplify the operation, the gain can be compensated together by using a scaling factor in the end of iterations, that is (A.3) Since the multiplies are powers of two, CORDIC can be implemented in binary arithmetic logic using just shifts and adds without using actual multipliers [26]. For instance, at each iteration, the real part is obtained via which is only involving shifting to right and adding with . A. Implementation of Since (A.4)

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One CORDIC module can be directly employed and reused, that is (A.5)

However, in this operation, the scaling factor for is , while the one for is . Therefore, both operations require multipliers. In order to reduce the number of multipliers, a new method is proposed below, that is

Then employing the CORDIC to calculate again, we can obtain (A.10) (A.6) To reduce complexity, the scaling factor of CORDIC and compensated later, that is

Finally, we can obtain

can be moved out

(A.7) where we can see that only two CORDICs are involved to conduct the square root operation. Because CORDIC module only uses adders and shifters, the FPGA resource consumed in the proposed approach is much less than that in the conventional polynomial implementation. Let's compare the resource consumption of with that of . At first glance, one may think the implementation of should be more complex than that of , since there is one extra square root operation. However, after careful investigation, the actual resource consumptions are totally different, as shown in Table VIII. The implementation of will require four complex multipliers and three adders. Due to the multipliers, the resource consumption will be costly, which will require 1172 slice LUTs and 1211 slice registers in FPGA. Even if the multiplexing technology is employed, e.g., and may share the same resources, the total resource consumption is still very high. On contrary, the implementation of will only require three CORDIC module, which only employs 495 slice LUTs and 726 slice registers, leading to 40% saving of the resource consumption. Also, if multiplexing is employed, and can share the same CORDIC module and thus only two CORDIC modules will be required, which will further reduce the resource consumption. In summary, because the multipliers will consume more resources than CORDIC, the total FPGA resources consumed for the implementation of is actually large than the one of . B. Implementation of Based on the implementation (A.7) of more input is added into CORDIC module. Firstly

, one (A.8)

then

(A.9)

(A.11) Although one more CORDIC is employed, it can be also multiplexed, which significantly reduces the total implementation cost. REFERENCES [1] M. Iwamura, K. Etemad, M.-H. Fong, R. Nory, and R. Love, “Carrier aggregation framework in 3GPP LTE-advanced [WiMAX/LTE update],” IEEE Commun. Mag., vol. 48, no. 8, pp. 60–67, Aug. 2010. [2] W. Chen, S. A. Bassam, X. Li, Y. Liu, K. Rawat, M. Helaoui, and F. M. Ghannouchi, “Design and linearization of concurrent dual-band Doherty power amplifier with frequency dependent power ranges,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 10, pp. 2537–2546, Oct. 2011. [3] P. B. Kennington, High Linearity RF Amplifier Design. Norwood, MA, USA: Artech House, 2000. [4] C. Yu, M. Allegue-Martinez, Y. Guo, and A. Zhu, “Output-controllable partial inverse digital predistortion for RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 62, no. 11, pp. 2499–2510, Nov. 2014. [5] P. Roblin, S. K. Myoung, D. Chaillot, Y. G. Kim, A. Fathimulla, J. Strahler, and S. Bibyk, “Frequency selective predistortion linearization of RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 65–76, Jan. 2008. [6] J. Kim, P. Roblin, D. Chaillot, and Z. Xie, “A generalized architecture for the frequency-selective digital predistortion linearization technique,” IEEE Trans. Microw. Theory Tech., vol. 61, no. 1, pp. 596–605, Jan. 2013. [7] S. A. Bassam, M. Helaoui, and F. M. Ghannouchi, “Channel-selective multi-cell digital predistorter for multi-carrier transmitters,” IEEE Trans. Commun., vol. 60, no. 8, pp. 2344–2352, Aug. 2012. [8] M. Abdelaziz, L. Anttila, J. R. Cavallaro, S. S. Bhattacharyya, A. Mohammadi, F. Ghannouchi, M. Juntti, and M. Valkama, “Low-complexity digital predistortion for reducing power amplifier spurious emissions in spectrally-agile flexible radio,” in Proc. 9th Int. Conf. Cognitive Radio Oriented Wireless Networks Commun. (CROWNCOM), Jun. 2–4, 2014, pp. 323–328. [9] M. Abdelaziz, L. Anttila, A. Mohammadi, F. Ghannouchi, and M. Valkama, “Reduced-complexity power amplifier linearization for carrier aggregation mobile transceivers,” in Proc. IEEE ICASSP, Florence, Italy, May 2014, pp. 3908–3912. [10] Z. Fu, L. Anttila, M. Abdelaziz, M. Valkama, and A. M. Wyglinski, “Frequency-selective digital predistortion for unwanted emission reduction,” IEEE Trans. Commun., vol. 63, no. 1, pp. 254–267, Jan. 2015.

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[11] A. Frotzscher and G. Fettweis, “Baseband analysis of Tx Leakage in WCDMA zero-IF receivers,” in Proc. IEEE Int. Symp. Control, Commun. Sig. Proc. (ISCCSP'08), 2008, pp. 129–134. [12] A. Frotzscher, M. Krondorf, and G. Fettweis, “On the performance of OFDM in zero-IF receivers impaired by Tx Leakage,” in Proc. IEEE Int. Conf. Commun. (ICC'09), Jun. 2009, pp. 1–5. [13] M. Kahrizi, J. Komaili, J. Vasa, and D. Agahi, “Adaptive filtering using LMS for digital TX IM2 cancellation in WCDMA receiver,” in Proc. IEEE Radio Wireless Symp., Orlando, FL, USA, Jan. 22–24, 2008, pp. 519–522. [14] M. Omer, R. Rimini, P. Heidmann, and J. S. Kenney, “A compensation scheme to allow full duplex operation in the presence of highly nonlinear microwave components for 4 G systems,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Baltimore, MD, USA, Jun. 5–10, 2011, pp. 1–4. [15] M. Omer, R. Rimini, P. Heidmann, and J. S. Kenney, “All digital compensation scheme for spur induced transmit self-jamming in multi-receiver RF frond-ends,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Montreal, QC, Canada, Jun. 17–22, 2012, pp. 1–3. [16] A. Kiayani, L. Anttila, and M. Valkama, “Modeling and dynamic cancellation of TX-RX leakage in FDD transceivers,” in Proc. IEEE Int. Midwest Symp. Circuits Syst. (MWSCAS), Aug. 2013, pp. 1089–1094. [17] A. Kiayani, L. Anttila, and M. Valkama, “Digital suppression of power amplifier spurious emissions at receiver band in FDD transceivers,” IEEE Signal Process. Lett., vol. 21, no. 1, pp. 69–73, Jan. 2014. [18] A. Kiayani, M. Abdelaziz, L. Anttila, V. Lehtinen, and M. Valkama, “DSP-based suppression of spurious emissions at RX band in carrier aggregation FDD transceivers,” in Proc. 22nd Eur. Sig. Proc. Conf. (EUSIPCO), Sep. 2014, pp. 591–595. [19] H.-T. Dabag, H. Gheidi, P. Gudem, and P. M. Asbeck, “All-digital cancellation technique to mitigate self-jamming in uplink carrier aggregation in cellular handsets,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, USA, Jun. 2013, pp. 1–3. [20] H.-T. Dabag, H. Gheidi, S. Farsi, P. Gudem, and P. M. Asbeck, “Alldigital cancellation technique to mitigate receiver desensitization in uplink carrier aggregation in cellular handsets,” IEEE Trans. Microw. Theory Tech., vol. 61, no. 12, pp. 4754–4765, Jan. 2013. [21] C. Yu and A. Zhu, “Modeling and suppression of transmitter leakage in concurrent dual-band transceivers with carrier aggregation,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, USA, May 2015. [22] “3rd Generation Partnership Project; LTE; Evolved Universal Terrestrial Radio Access (E-UTRA); Base Station (BS) Radio Transmission and Reception (Release 10),” Tech. Spec. 3GPP TS 36.104 V10.2.0 (2011-05), 3GPP, Sophia Antipolis Cedex, France, May 2011. [23] S. Bassam, W. Chen, M. Helaoui, and F. Ghannouchi, “Transmitter architecture for CA: carrier aggregation in LTE- Advanced systems,” IEEE Microw. Mag., vol. 14, pp. 78–86, Aug. 2013. [24] E. G. Lima, T. R. Cunha, H. M. Teixeira, M. Pirola, and J. C. Pedro, “Base-band derived Volterra series for power amplifier modeling,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2009, pp. 1361–1364. [25] E. G. Lima, T. R. Cunha, and J. C. Pedro, “A physically meaningful neural network behavioral model for wireless transmitters exhibiting PM-AM/PM-PM distortions,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 12, pp. 3512–3521, Dec. 2011. [26] J. E. Volder, “The CORDIC trigonometric computing technique,” IRE Trans. Electron. Compute., vol. EC-8, no. 3, pp. 330–334, Sep. 1959.

Chao Yu (S'09–M'15) received the B.E. degree in information engineering and M.E. degree in electromagnetic fields and microwave technology from Southeast University (SEU), Nanjing, China, in 2007 and 2010, respectively, and the Ph.D. degree in electronic engineering from University College Dublin (UCD), Dublin, Ireland, in 2014. He is currently an Associate Professor with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, SEU. His research interests include RF power amplifiers modeling and linearization, high-speed ADC digital correction. He is also interested in antenna design, FPGA hardware implementation and RF wireless system design.

Wenhui Cao (S'15) received the B.E. degree in automation from Beijing University of Chemical Technology, Beijing, China, in 2013 and is currently working toward the Ph.D. degree in University College Dublin (UCD), Dublin, Ireland. She is currently with RF and Microwave Research Group, UCD. Her research interests include nonlinear behavioral modeling of RF power amplifiers, digital post-correction of high speed ADCs, and high performance field-programmable gate-array (FPGA) implementation methodologies.

Yan Guo (S'13) received the B.E. degree in information science and engineering from East China Jiaotong University, Nanchang, Jiangxi Province, China, in 2007, the M.E. degree in communication and information systems from Southeast University, Nanjing, China, in 2011, and is currently working toward the Ph.D. degree with University College Dublin (UCD), Dublin, Ireland. He is currently with the RF and Microwave Research Group, UCD. His research interests include spectrum sensing for cognitive radio, digital predistortion for RF power amplifiers, and field-programmable gate-array (FPGA) hardware implementations.

Anding Zhu (S'00–M'04–SM'12) received the B.E. degree in telecommunication engineering from North China Electric Power University, Baoding, China, in 1997, the M.E. degree in computer applications from the Beijing University of Posts and Telecommunications, Beijing, China, in 2000, and the Ph.D. degree in electronic engineering from University College Dublin (UCD), Dublin, Ireland, in 2004. He is currently a Senior Lecturer with the School of Electrical, Electronic and Communications Engineering, UCD. His research interests include high frequency nonlinear system modeling and device characterization techniques with a particular emphasis on Volterra-series-based behavioral modeling and linearization for RF power amplifiers (PAs). He is also interested in wireless and RF system design, digital signal processing, and nonlinear system identification algorithms.

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Coupling-Matrix-Based Design of High- Bandpass Filters Using Acoustic-Wave Lumped-Element Resonator (AWLR) Modules Dimitra Psychogiou, Member, IEEE, Roberto Gómez-García, Senior Member, IEEE, and Dimitrios Peroulis, Senior Member, IEEE

Abstract—This paper presents an original and simple coupling-matrix-based synthesis methodology for the design of a new class of bandpass filters (BPFs) that employ hybrid acoustic-wave-lumped-element resonator (AWLR) modules with improved out-of-band isolation (IS). The proposed BPFs feature quasi-elliptic-type frequency response—shaped by poles and transmission zeros (TZs) for an th-order transfer function, compact physical size, and high effective quality factors of the order of 1000. Despite the use of acoustic wave (AW) resonators, passbands exhibiting fractional bandwidths (FBWs) that are no longer limited by the electromechanical coupling coefficient of the constituent AW resonators are obtained. A coupling-matrix-based model of a multi-mode AW resonator is also reported. It facilitates the incorporation of high- and low-frequency spurious modes that are present in a realistic filter response so that they can be anticipated at the synthesis and simulation levels. For proof-of-concept validation purposes, two BPF prototypes at 418 MHz made up of commercially-available surface acoustic wave (SAW) resonators and surface mounted devices (SMD) were built and measured. They perform first(one pole and two TZs) and second-order (two poles and four TZs) transfer functions with measured passband insertion losses (IL) between 2.4–5.4 dB, between 1600–1900, 3-dB absolute ), bandwidths ranging from 0.52 to 1 MHz (i.e., 1.6–3.2 times and minimum IS levels between 25–46 dB. Index Terms—Acoustic wave (AW) filter, bandpass filter (BPF), enhancement, quasi-elliptic-type filter, RF/microwave filter, surface acoustic wave (SAW) filter, SAW resonator.

I. INTRODUCTION

E

MERGING wireless communication systems are more and more evolving towards multi-frequency and multi-standard communication services which in turn result in

Manuscript received June 29, 2015; revised September 28, 2015; accepted October 11, 2015. Date of publication November 18, 2015; date of current version December 02, 2015. This work was supported in part by the National Science Foundation under Award 1247893 and in part by the Spanish Ministry of Economy and Competitiveness under Project TEC2014-54289-R. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. D. Psychogiou and D. Peroulis are with the School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]; [email protected]). R. Gómez-García is with the Department of Signal Theory and Communications, University of Alcalá, Madrid 28871, Spain (e-mail: roberto.gomez. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2494597

a highly-congested frequency radio spectrum [1], [2]. A typical example of this trend is the UHF band (300–3000 MHz), which is currently exploited by numerous military and commercial applications sharing closely-spaced or even co-located frequency bands. In order to efficiently utilize this region of the electromagnetic (EM) spectrum without sacrificing receiver sensitivity, bandpass filters (BPFs) with increased selectivity, large out-of-band rejection, narrow fractional bandwidth (FBW), and low insertion loss (IL) need to be employed in the RF front-end module in order to properly preselect the desired band of frequencies [3], [4]. When size is of critical importance in these systems, RF BPFs using EM wave resonators (e.g., microstrip-line [5], lumped-element [6], [7], and waveguide [8] resonators) cannot adequately meet the aforementioned performance requirements due to the inversely proportional relationship between resonator size and unloaded quality factor . On the other hand, filtering topologies made up of acoustic wave (AW) resonators [e.g., surface acoustic wave (SAW) and bulk acoustic wave (BAW) type] exhibit high values (1000–10 000) in a highly miniaturized volume [9]–[13]. However, their FBW is inherently limited by the electromechanical coupling coefficient 0.05%–0.1%) of their structural materials. It is typically between 0.4–0.8 for conventional lattice or ladder AW resonator arrangements [11]. Furthermore, as an additional shortcoming to be mentioned, they require AW resonators of alternative impedances and resonant frequencies. This makes their response sensitive to manufacturing and tolerance variations. A number of bandwidth-enlargement techniques have been reported in the open technical literature to date. In [14]–[19], the development of piezoelectric materials with high is discussed. However, the frequency response of these BPFs is limited by increased IL (i.e., reduced ), temperature sensitivity, and unwanted spurious modes [17], [19]. In an alternative approach, all-AW BPFs with closely-spaced passbands are effectively combined to increase the BPF bandwidth [20]. Their frequency response is nevertheless constrained by fabrication tolerances and material properties variations, which result in excessive ripples within the passband leading to perceptible amplitude-distortion levels for in-band processed signals. A hybrid-technology BPF design concept in which AW resonators are effectively combined with lumped-element (LE) components is reported in [21] for bandwidth-broadening purposes. It allows the realization of large-FBW (up to 5 ) transfer func-

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tions whose passband preserves the high- (e.g., of 4000) characteristics of the constituent AW resonators. However, their RF response is limited by poor out-of-band isolation (IS) when large FBW transfer functions are actualized. In this paper, a novel acoustic-wave-lumped-element (AWLR) resonator module that comprises one AW resonator (two-terminal one-port type) and three LE resonators is reported. It permits the realization of high- BPFs with quasi-elliptic-type frequency response, , and IS levels that can be arbitrarily designed—without being constrained by the realizable FBW state—as opposed to the BPF architecture in [21]. A further relevant contribution of this work is the development of an original and simple coupling-matrix (CM)-based synthesis methodology for this class of BPFs. It facilitates the design of AW-resonator-based BPFs through conventional coupled-resonator filter design techniques and improves the flexibility of existing AW-filter synthesis methodologies. Preliminary results on this approach were first demonstrated in [22]. It is the intention of this work to further investigate this concept and present an in-depth theoretical analysis of its merits by expanding its applicability to higher-order BPF transfer functions. Furthermore, the CM-based model of a multi-mode AW resonator that facilitates the incorporation of spurious modes (high- and low-frequency ones) already at the theoretical design level is reported and validated through various synthesis examples. Lastly, new first- and second-order BPF prototypes are presented as proof-of-concept demonstrators. The content of this work is organized in the following manner. In Section II, the engineered BPF design concept that consists of AWLR modules with improved IS is introduced. The theoretical foundations of the CM-based synthesis technique are expounded through various BPF design examples. Moreover, the multi-mode CM-based model of two-terminal one-port AW resonators is also presented. In Section III, the design and experimental validation of BPF prototypes that feature first- and second-order transfer functions are reported as proof-of-concept demonstrators. Lastly, a summary of the major contributions and the most relevant conclusions of this work is provided in Section IV. II. THEORETICAL FOUNDATIONS The devised BPF concept is based on a new class of AWLR modules with improved IS that are synthesized through a hybrid combination of one high- resonant node , three lowresonant nodes ( and ), and one non-resonant node (N) as detailed in the coupling matrix diagram (CMD) in Fig. 1(a). Each AWLR module features a quasi-elliptic-type frequency response that is shaped by one pole in-between two transmission zeros (TZs) [see Fig. 1(a)]. The location of the pole is primarily defined by the resonant node whereas the TZs stem from the parallel combination of a passband resonance—created by node —and a stopband resonance—produced by the combination of nodes N and . Note that, in this configuration, the resonant nodes are only used to define the IS of the AWLR module as it will be further explained in this section. In the practical BPF developments of this work, two-terminal one-port type AW resonators (e.g., SAW or BAW type) are considered as highresonant elements whereas the rest of the low- resonant nodes

Fig. 1. (a) th-order AWLR BPF that comprises AWLR modules with improved IS. The characteristic response (S-parameters) of the single AWLR module with improved IS is also shown. (b) Circuit schematic of a realistic th-order BPF model that consists of two-terminal one-port type AW res, and impedance inverters (K-inv). onators, LE resonators

and coupling sections are realized by means of conventional LE components. This is illustrated in Fig. 1(b) for an example circuit schematic of an th-order BPF with quasi-elliptic-type response. The theoretical foundations of the AWLR-based BPF concept are described in this section starting from the detailed analysis of the AWLR module with improved IS. Its application to the synthesis of high-order bandpass filtering transfer functions is subsequently addressed. A. Acoustic Wave Resonator Modeling The circuit details of the two-terminal one-port type AW resonator [23] that is typically represented by the Butterworth Van-Dyke (BVD) model [see Fig. 2(a)], along with its corresponding frequency response—for an AW resonator example with H, fF, , and pF—are depicted in Fig. 2. Its transfer function consists of one series resonant frequency and one anti-series resonant frequency that can be specified using (1) [23, p. 864]. Note that these frequencies can be calculated from the geometrical and material properties of the AW resonator for a given resonator geometry as discussed in [11]. Such a frequency response can be synthesized from a CMD in which the parallel combination of one resonant node and one non-resonant node N is considered as depicted in Fig. 2(b). Its corresponding self-coupling coefficients and

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BANDPASS FILTERS

Fig. 2. AW resonator. (a) Butterworth Van-Dyke (BVD) circuit model [23]. (b) Equivalent CMD for the representation of one resonant and one antiresonant , (i.e., ), frequency with . (c) Synthesized S-parameters using the BVD model in and H, (a) for a two-terminal one-port type AW resonator with fF, , pF, , and the CMD in (b).

can be specified through (2) [24], [25] whereas the inter-node coupling coefficients and — for the example in Fig. 2—are obtained by fitting the synthesized CMD response to the one obtained by RF-measurements or EM-simulations assuming a FBW equal to . Note that and in (2) refer, respectively, to the FBW and of the series resonant branch [ , , in Fig. 2(a)] of the AW resonator that can be determined from (3), in which denotes the reference impedance of the two-port network. Fig. 2(c) illustrates a comparison between the simulated response that stems from the BVD circuit and the synthesized transfer function using the CMD in Fig. 2(b). As can be seen, they are in fairly-close agreement and, thus, validate the devised CMD for the two-terminal one-port type AW resonator

(1)

(2)

(3)

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Fig. 3. (a) CMD representation of a multi-mode AW resonator. The parallel cascaded resonant nodes represent the fundamental series resonant frequency ) and the spurious resonant frequencies (nodes ), respectively. (b) (node , , Example matrix of an AW resonator with five resonant nodes (nodes , , ) and one non-resonating node (node N). (c) RF-measured response for a commercially-available SAW resonator from Abracon Corp. (ASR418S2) , and synthesized response from a CMD with , , , , , (i.e., ), , , , and .

As an additional original contribution of this work, the CMD of the AW resonator can be further extended to include the undesired signal contributions of low- and high-frequency spurious modes. These modes are present in the measured frequency response and typically appear in a close proximity to the fundamental resonances ( , ), hence limiting the filter's performance. A generic multi-mode CMD is depicted in Fig. 3(a). In this scenario, the higher-order modes are taken into consideration by adding extra resonant nodes— - , where is the number of the existing spurious modes—in-parallel to the primary resonant node as for the case of representing multimode resonators in multi-band filter designs [24, Ch. 9], [26]. The self-coupling coefficients and the inter-node coupling coefficients can be calculated through (4), in which , , and are, respectively, the FBW, , and resonant frequency of each spurious resonance that can be derived from the RF-measured or EM-simulated response. In order to verify the aforementioned model, a comparison between the CM-synthesized and the RF-measured response of a commercially-available two-terminal one-port type AW resonator from Abracon Corp. (part ASR418S2) is depicted in Fig. 3. Note that the synthesized response was obtained from the CM in Fig. 3(b), in which one fundamental and four spurious modes are considered. As can be seen, a very close agreement is obtained

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among the two traces, hence successfully verifying the conceived multi-mode CMD for two-terminal one-port type AW resonators. Its applicability to the synthesis of bandpass transfer functions that take into consideration the multi-mode nature of AW resonators will be discussed in the next sections

(4) B. Acoustic-Wave-Lumped-Element Resonator Module Using as a reference the CMD in Fig. 2(b) and by readily adding a resonant node below the non-resonant node , is effectively decoupled from as shown in the examples in Fig. 4. It results in a high- passband that is located in-between two TZs with FBW (0.032% for the AW resonator of Figs. 2 and 3) that can be calculated through (3). In a practical RF design scenario, it can be implemented by adding an inductance in parallel to the parasitic capacitance of the AW resonator as shown in Fig. 1(b). It is apparent that for a given AW resonator topology the coupling coefficients , , , and are fixed. As such, the location (normalized frequency) of the out-of-band TZs can be only adjusted by modifying and as described in (5) that respectively correspond to altering the resonant frequency (through the variation of ) or its relevant , values—for the same resonant frequency as the high- resonant node—in relation to the series resonant branch , . A set of power transmission response examples that are synthesized using the aforementioned CMD for alternative levels of and are plotted in Fig. 4(b) and (c), respectively, to illustrate the aforementioned TZ control capability. The power transmission response dependence (IS and passband IL) on the finite of is also shown at the same figure. It can be observed that the IS at the location of the TZs only depends on ( that corresponds to ), which is practically defined by the of since the element of the AW resonator is considered lossless. As such, the resulting of the grouped resonator ( , ) is much higher than that of a conventional LE resonator in which both the inductive and capacitive resonator components feature a finite . Noteworthy is that even for a of the order of 100, the IS levels at the location of the TZs are over 45 dB. In addition, as can be seen in the inset of Fig. 4(a), the passband IL remains unchanged. This demonstrates the suitability of the AWLR module for the realization of high- BPFs with quasi-elliptic-type frequency response

Fig. 4. (a) CMD of the AWLR and detail of its power transmission response : 70, 70, and 31.4i (i.e., ). (b) Power for various levels of : 70, 70, and 31.4i (i.e., transmission response for various levels of ). The rest of the coupling coefficients are: , , (i.e., ), and . (c) Power : 55, 85, and 105. The rest of the transmission response for various levels of , , coupling coefficients are: (i.e., ), , and .

augmented to theoretically any value despite the finite of the AW resonator which typically limits the FBW in all-AW BPFs to 0.4–0.8 . As a trade-off to be mentioned, the referred FBW increase comes at the expense of reduced out-of-band rejection as typically observed in all-AW filters

(5) The FBW of the AWLR module can be further increased by inserting impedance inverters at its input/output ports, as illustrated in the CMD of Fig. 5(a). Its corresponding coupling coefficients can be specified though (6), in which is the designed FBW. Fig. 5(b) illustrates the synthesized response of the CMD in Fig. 5(a) for various sets of . As can be seen, the FBW of the obtained BPF response can be arbitrarily

(6)

C. Acoustic-Wave-Lumped-Element Resonator Module With Improved Out-Of-Band Isolation The out-of-band rejection levels of the AWLR with improved FBW can be arbitrarily increased by readily integrating

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Fig. 5. (a) CMD of the AWLR with impedance inverters at its input/output and accesses. (b) Power transmission response for various levels of , , (i.e., ), and .

two low- resonant nodes as illustrated in the CMD in Fig. 6(a). Note that both nodes need to resonate at the center frequency of the passband while featuring different inductance and capacitance values than those of the remaining CMD nodes as shown in the practical implementation circuit in Fig. 1(b). In this manner, the inter-node coupling coefficients that are adjacent to (i.e., , , and ) are scaled by a factor that represents the inductance scaling among the resonators as well as the nodes transformation in relevance to the CMD in Fig. 5(a). For a given inductance , it can be calculated using (7) which in turn results in a quasi-elliptic-type frequency response with significantly improved out-of-band IS as demonstrated in Fig. 6(b). A comparison between the synthesized response of the AWLR with improved IS and that of Fig. 5(a) is illustrated in Fig. 6(b) for various levels of . As can be seen, the decrease of produces an increase of the out-of-band rejection. The finite effect of the resonant nodes (with equal to ) is also shown through a comparison between the synthesized transfer function of the CMD in Fig. 6(a) for infinite and of 100. It can be observed that it comes at the expense of increased IL. However, the corresponding of the resulting passband is of the order of 2300 despite the use of low- resonant nodes with around 100. This corroborates the potential of this design approach for the actualization of high- BPF transfer functions despite the inclusion of the low- nodes for IS enhancement (7)

Fig. 6. (a) CMD and equivalent CM of the AWLR module with improved IS. 12.8 (b) Synthesized power transmission response for various levels of ( and 31.4i (i.e., infinite and 100). The values and 25.7) and , , , of the rest of the coupling coefficients are: . (c) Synthesized power transmission response of the AWLR and module with improved IS using the multi-mode model of the AW resonator for 1, 3, 6, and 9), , , , various levels of FBW ( , , and . Note that in all examples the is considered lossless. resonant node

D. Filter Synthesis with Incorporated Spurious Modes In the analysis detailed in Sections II-B and II-C, the utilized AW resonator is represented through the simplified CMD in Fig. 2(a) (i.e., only the fundamental modes are included) in

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order to expound the most relevant filter synthesis principles in a comprehensive manner. However, taking into consideration that the presence of spurious modes can severely affect the filter's performance, their prediction at the theoretical synthesis level becomes of critical importance. Various examples of synthesized transfer functions that stem from an expanded version of the CMD in Fig. 6(a), in which the spurious modes of the AW resonator are incorporated as parallel-cascaded resonant nodes - [see Fig. 3(a)], are demonstrated in Fig. 6(c). As can be seen, the impact of the spurious resonant modes becomes more prominent with the increase of FBW. In particular, when considering large FBW states, the spurious modes that are located in a close proximity to the fundamental modes might appear as in-band notches. Thus, although the FBW of the AWLR module with improved IS is not theoretically limited by the , it is however restricted in practice by the separation and magnitudes of the lower and upper closest-to-passband spurious modes. In addition, it should be noticed that broader FBWs can be realized by either suppressing the undesired spurious modes or by increasing their separation from the fundamental modes as reported by the authors in [27], [28]. However, such an investigation falls out of the scope of this work. E. Extension to Higher-Order Transfer Functions The AWLR concept with improved IS can be readily extended to the realization of higher-order BPF transfer functions by cascading in-series multiple AWLR modules ( for an th-order BPF response) as shown in the th-order CMD in Fig. 1(a) and its corresponding circuit schematic in Fig. 1(b). The devised CMD facilitates the synthesis of BPF transfer functions with and quasi-elliptic-type frequency response that is shaped by poles and 2 TZs. The obtained TZs can be located around the passband in a symmetric or in an asymmetric fashion. Note that each pair of TZs is controlled by the coupling coefficients and (frequency characteristics of the resonant node ) of each AWLR module as described in (5). In order to better illustrate the aforementioned functionalities, the synthesis of second-order transfer functions with symmetrically located TZs is considered. They stem from the CMD and its equivalent CM representation in Fig. 7(a). Note that, in this case, the coupling sections with normalized coefficients and tailor the BPF's FBW and power matching levels, respectively. Furthermore, the BPF's IS is controlled by as in the first-order transfer function in Fig. 6. An example of a synthesized second-order transfer function that realizes a 3-dB absolute bandwidth of 0.8 MHz (lossless case) and features symmetrically located TZs is depicted in Fig. 7(b) for various levels of . Note that, in this implementation, lossless resonant nodes are considered. As can be seen, the cascade of two AWLR modules with improved IS gives rise to spurious passbands that are symmetrically located around the BPF's center frequency. However, their intensity can be significantly mitigated by introducing a finite in the resonant nodes , which is typically the case in a practical BPF development in which LE resonators are considered. It is apparent that there is a tradeoff between passband IL and the magnitude of the spurious modes as observed in Fig. 7(b). However, in most realistic implementations, the use of resonant

Fig. 7. Second-order BPF using two series-cascaded AWLR modules with improved IS. (a) CMD and CM. (b) Frequency response of a second-order transfer function that stems from the CMD in (a) with 3-dB absolute bandwidth of 0.8 (lossless is conMHz and alternative levels of the of the resonator , sidered in these examples) using the following coupling coefficients: , , , (i.e., ), , and . The frequency response of a second-order BPF with similar bandwidth and of 1200 is also shown for comparison purposes.

nodes with offers a relatively good compromise in terms of the above parameters. Such an example is shown in Fig. 7(b), in which a second-order BPF response with of the order of 1200 and minimum IS of 55 dB is obtained by using nodes with of only 75, making the proposed BPF synthesis approach suitable for LE development. Alternative examples of synthesized transfer functions are plotted in Fig. 8. Note that, in all the cases, resonant nodes with finite ( and ) are considered. It can be observed in Fig. 8(a) that for the same levels of IS (e.g., 50 dB at 405 MHz) narrower bandwidth states feature lower levels of IL—which correspond to larger levels of . Furthermore, it should be further noticed that for the same bandwidth state the increase of IS (smaller ) comes at the expense of some IL deterioration as also shown in Fig. 8(a). Synthesized transfer

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Fig. 9. (a) Schematic circuit of the AWLR module with improved IS. (b) Manufactured prototype for the realization of a 3-dB absolute bandwidth of 1 MHz and minimum IS 25 dB. The utilized LE components are as follows: Johanson 251R14S110GV4T, Coilcraft 1206CS100, Coilcraft 165-05-A06, and SAW=Abracon ASR418S2.

Fig. 8. Second-order BPF using two series-cascaded AWLR modules with improved IS. (a) Frequency response for alternative levels of 3-dB absolute bandwidth (e.g., 0.46 and 0.8 MHz) and IS (e.g., 50 and 70 dB at 405 MHz) for symmetrically located (around the passband) double pairs of TZs . (b) Frequency response of a quasi-elliptic BPF with 0.8-MHz 3-dB absolute bandwidth (labelled as “BW in this figure”) and four symmet. Unless rically located (around the passband) TZs , , explicitly stated in the legend of each figure, , (i.e., ), (i.e., ), (i.e., ), , and . Note that for the synthesis of these transfer functions, resonators with realistic values of have been considered. The indicated 3-dB absolute bandwidth of the BPF refers to the bandwidth before the RF losses are added.

functions for different positions of symmetrically located TZs around the desired passband (located at 418 MHz) are also shown in Fig. 8(b). They have been obtained through AWLR modules that either feature symmetric [e.g., grey trace in Fig. 8(b)] or asymmetric [e.g., red and black trace in Fig. 8(b)] TZs [see Fig. 4(b)]. This demonstrates the controllability of TZs for improved IS realization. III. RF MEASUREMENTS AND DISCUSSION First- and second-order BPF prototypes have been designed, manufactured, and characterized in order to evaluate the practical usefulness of the proposed CM-based synthesis formalism for mixed-technology AWLR modules and its applicability to the realization of high- BPFs with quasi-elliptic-type frequency response. All filters prototypes have been implemented on a Rogers RO 4003C dielectric substrate with the following characteristics: dielectric permittivity , dielectric thickness 1.524 mm, dielectric loss tangent , and 35- m-thick Cu-cladding. They have

been designed for a center frequency of 418 MHz and a 50reference impedance. Note that, for the proposed implementation approach, external matching networks which would lead to increased IL and physical size are not required as opposed to conventional all-AW filter architectures. Commercially-available two-terminal one-port type SAW resonators (ASR418S21) from Abracon Corporation were employed as resonant nodes. Furthermore, for the realization of the low- resonant nodes and and their adjacent impedance inverters (coupling coefficients and ), LE components from Johanson Tech. and Coilcraft Inc. were utilized as listed in the captions of Figs. 9 and 11. The RF design of the BPF prototypes starts by defining the filter CM by using the design methodology described in Section II. Afterwards, the coupled-resonator approach in [24, pp. 215–216] is employed in order to specify the realistic impedance inverter values (90 -length-at- transmission lines) which are in turn implemented by their first-order -type LE lowpass equivalent circuit. Lastly, the final values of the LEs that need to be utilized in the actual BPF prototype are obtained through full-wave simulation analysis of the realistic filter geometry (landing pads for the LE components and RF excitation interface) using the EM solver of the Advanced Design Systems (ADS) software from Keysight Technologies. In the next subsections, the RF performance metrics of two—firstand second-order—BPF prototypes are discussed in terms of RF measurements and simulations. A. First-Order BPF Using AWLR Modules With Improved IS An example of an experimental BPF prototype that features a first-order quasi-elliptic-type frequency response with 3-dB absolute bandwidth of 1 MHz and minimum IS of 25 dB is illustrated in Fig. 9. Its LE components, along with its equivalent

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Fig. 10. (a) RF-measured, EM-simulated (multi-mode resonator model), and CM-synthesized (multi-mode resonator model) power transmission response of the first-order prototype in Fig. 9. (b) RF-measured, EM-simulated (multi-mode resonator model), and CM-synthesized (multi-mode resonator model) input power reflection response of the first-order prototype in Fig. 9.

schematic circuit, are also shown at the same figure. The RF performance of this filter assembly was experimentally evaluated in terms of S-parameters with an Agilent E8361A network analyzer. Its measured power transmission and reflection responses are shown in Fig. 10. Based on these curves, the measured RF performance parameters can be summarized as follows: 3-dB absolute bandwidth of 0.95 MHz, 2.4 dB (SMA connector loss and RF excitation interface are included), (SMA connector loss and RF excitation interface are included), , IS > 25 dB, and return loss (RL) 21.6 dB. A comparison between the RF-measured, EM-simulated, and the CM-synthesized response (multi-mode SAW resonator model) is also depicted in Fig. 10. As can be seen, all responses are in close agreement, thus successfully validating the suggested filter design concept. As discussed in Section II, the presence of spurious modes is attributed to the multi-mode nature of the SAW resonator as typically observed in all-AW resonator filters—as, for example, in [10]–[22]. B. Second-Order BPF Using AWLR Modules With Improved IS The realization of higher-order BPF transfer functions was experimentally validated through a second-order filter assembly. Its corresponding manufactured prototype, equivalent circuit model, and its actual commercially-available LE components are presented in Fig. 11. The RF-measured response of

Fig. 11. (a) Schematic circuit of a second-order BPF using the AWLR module with improved IS. (b) Manufactured prototype of the second-order BPF for the actualization of a 3-dB absolute bandwidth of 0.5 MHz and 50 dB. The utilized LE components are as follows: minimum IS Johanson 251R14S2R0BV4T, Coilcraft 0805HQ-39N, JoCoilcraft 0805CS-6N9, Johanson hanson 251R14S1R2BV4T, Coilcraft 0806SQ-6N9, 251R14S0R3BV4T//251R14S180JV4T, Johanson 251R14S0R5BV4T//251R14S160GV4T, Coilcraft 165-05-A06, and SAW=Abracon ASR418S2.

the second-order BPF prototype is plotted in Fig. 12. Its main characteristics can be summarized as follows: 3-dB absolute bandwidth equal to 0.52 MHz, 5.4 dB (SMA connector loss and RF excitation interface are included), (SMA connector loss and RF excitation interface are included), , IS > 46 dB, and 33 dB. As can be seen, it agrees well with the one obtained through EM simulations and CM synthesis, hence confirming the applicability of the conceived concept to the realization of higher-order transfer functions featuring high levels of and FBW that is no longer limited by the of the constituent AW resonators. IV. CONCLUSION This work has presented a new and straightforward CM-based RF-design approach that allows the realization of AW-resonator-based BPFs with the following features: 1) FBW that is no longer limited by the of the utilized AW resonators; 2) controllable levels of IS; 3) AW resonators of identical frequency response and impedance characteristics; and 4) of the order of 1000 for a compact physical size. The CMD of a multi-mode AW resonator has been extracted to allow for the incorporation of spurious modes already in the filter analytical synthesis and simulation stages. A new class of AWLR modules that enable the actualization of quasi-elliptic-type BPFs with controllable levels of IS has also been proposed. Its experimental usefulness and the underlying CMD synthesis approach have been verified through the design of two—first- and second-order—BPF prototypes at 418 MHz

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Fig. 12. (a) RF-measured, EM-simulated (multi-mode resonator model), and CM-synthesized power transmission response of the second-order prototype in Fig. 11. (c) RF-measured, EM-simulated (multi-mode resonator model), and CM-synthesized input power reflection response of the second-order prototype in Fig. 11.

that employ commercially-available SAW resonators and SMD components. For these circuits, transfer functions with passband IL between 2.4–5.4 dB were measured. They correspond to between 1650–1900, which are approximately 20–50 larger than in conventional LE filters for comparable physical size. The FBW enhancement of the aforementioned BPF prototypes was measured between 1.6–3.2 , which is 2–16 times larger than that of state-of-the-art all-AW BPFs which feature FBWs between 0.2–0.8 —as, for example, the ones in [11], [12], [15], [19], [29] 0.17-0.29 . As an added advantage to be emphasized, the conceived BPF design approach facilitates the realization of stopbands with continuous increase in the IS levels when considering frequencies away from the passband. Such a feature is not feasible in conventional all-AW-resonator BPFs (e.g., ladder type) [15], [31] as well as in the AWLR filter in [21], which typically suffer from decreased levels of IS. These RF performance merits highlight the potential of the engineered mixed-technology filter principle for the actualization of miniaturized highBPFs with quasi-elliptic-type filtering profile. REFERENCES [1] M. Rais-Zadeh, J. T. Fox, D. D. Wentzloff, and Y. B. Gianchandani, “Reconfigurable radios: A possible solution to reduce entry costs in wireless phones,” Proc. IEEE, vol. 103, no. 3, pp. 438–451, Mar. 2015.

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[2] R. Gómez-García, F. M. Ghannouchi, N. B. Carvalho, and H. C. Luong, “Advanced circuits and systems for CR/SDR applications,” IEEE J. Select. Emerging Topics Circuits Syst., vol. 3, no. 4, pp. 485–488, Dec. 2013. [3] W. J. Chappell, E. J. Naglich, C. Maxey, and A. C. Guyette, “Putting the radio in “Software-defined radio”: Hardware developments for adaptable RF systems,” Proc. IEEE, vol. 102, no. 3, pp. 307–320, Mar. 2014. [4] J. Shellhammer, A. K. Sadek, and W. Zhang, “Technical challenges for cognitive radio in the TV white space spectrum,” Inform. Theory Appl. Workshop, pp. 323–333, Feb. 8–13, 2009. [5] Z. Wang, J. R. Kelly, P. S. Hall, A. L. Borja, and P. Gardner, “Reconfigurable parallel coupled band notch resonator with wide tuning range,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 11, pp. 6316–6326, Nov. 2014. [6] K. Zeng, D. Psychogiou, and D. Peroulis, “A VHF tunable lumpedelement filter with mixed electric-magnetic couplings,” presented at the IEEE Wirel. Microw. Tech. Conf., Cocoa Beach, FL, USA, Apr. 13–15, 2015. [7] Y. C. Ou and G. M. Rebeiz, “Lumped-element tunable bandstop filters for cognitive radio applications,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 10, pp. 2461–2468, Oct. 2011. [8] D. Psychogiou and D. Peroulis, “Tunable VHF miniaturized helical filters,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 2, pp. 282–289, Feb. 2014. [9] C. Campbell, Surface Acoustic Wave Devices for Mobile and Wireless Communications, 1st ed. Orlando, FL, USA: Academic Press, 1998. [10] R. Aigner, “SAW and BAW technologies for RF filter applications: A review of the relative strengths and weaknesses,” in Proc. IEEE Ultrason. Symp., Beijing, China, Nov. 2–5, 2008, pp. 582–589. [11] S. Gong and G. Piazza, “Design and analysis of Lithium-Niobate-based high electromechanical coupling RF-MEMS resonators for wideband filtering,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 403–414, Jan. 2013. [12] S. Gong and G. Piazza, “An 880 MHz ladder filter formed by arrays of laterally vibrating thin film Lithium Niobate resonators,” in IEEE 27th Int. MEMS Conf., San Francisco, CA, USA, Jan. 26–30, 2014, pp. 1241–1244. [13] M. Pijolat, S. Loubriat, D. Mercier, A. Reinhardt, E. Defaÿ, C. Deguet, M. Aïd, S. Queste, and S. Ballandras, “LiNbO3 film bulk acoustic resonator,” in Proc. IEEE Int. Freq. Control Symp., Newport Beach, CA, USA, Jun. 1–4, 2010, pp. 661–664. [14] G. Endoh, O. Kawachi, and M. Ueda, “A study of leaky SAW on piezoelectric substrate with high coupling factor,” in Proc. IEEE Ultrason. Symp., Caesars Tahoe, NV, USA, Oct. 17–20, 1999, pp. 309–312. [15] T. Omori, Y. Tanaka, K. Hashimoto, and M. Yamaguchi, “Synthesis of frequency response for wideband SAW ladder type filters,” in Proc. IEEE Ultrason. Symp., New York, NY, USA, Oct. 28–31, 2007, pp. 2574–2577. [16] S. Gong and G. Piazza, “Multi-frequency wideband RF filters using high electromechanical coupling laterally vibrating lithium niobate MEMS resonators,” in Proc. IEEE 26th Int. MEMS Conf., Taipei, Taiwan, Jan. 20–24, 2013, pp. 785–788. [17] S. Gong and G. Piazza, “An 880 MHz ladder filter formed by arrays of laterally vibrating thin film Lithium Niobate resonators,” in Proc. IEEE 27th Int. MEMS Conf., San Francisco, CA, USA, Jan. 26–30, 2014, pp. 1241–1244. [18] M. Pijolat, S. Loubriat, D. Mercier, A. Reinhardt, E. Defaÿ, C. Deguet, M. Aïd, S. Queste, and S. Ballandras, “LiNbO3 film bulk acoustic resonator,” in Proc. IEEE Int. Freq. Control Symp., Newport Beach, CA, USA, Jun. 1–4, 2010, pp. 661–664. [19] H. Nakanishi, H. Nakamura, and R. Goto, “High-electromechanical-coupling-coefficient surface acoustic wave resonator on Ta2O5/Al/LiNbO3 structure,” Jpn. J. Appl. Phys., vol. 49, no. 7S, p. 07HD21, Jul. 2010. [20] J. Meltaus, V. P. Plessky, A. Gortchakov, S. Harma, and M. M. Salomaa, “SAW filter based on parallel-connected CRFs with offset frequencies,” in Proc. IEEE Ultrason. Symp., Honolulu, HI, USA, Oct. 5–8, 2003, pp. 2073–2076. [21] D. Psychogiou, R. Gómez-García, R. Loeches-Sánchez, and D. Peroulis, “Hybrid acoustic-wave-lumped-element resonators (AWLRs) for highbandpass filters with quasi-elliptic frequency response,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 7, pp. 2233–2244, Jun. 2015. [22] D. Psychogiou, R. Gómez-García, and D. Peroulis, “Highbandpass filters using hybrid acoustic-wave-lumped-element resonators (AWLRs) for UHF applications,” presented at the IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, USA, May 17–22, 2015.

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[23] J. D. Larson, III, R. C. Bradley, S. Wartenberg, and R. C. Ruby, “Modified Butterworth-Van Dyke circuit for FBAR resonators and automated measurement system,” in Proc. IEEE Ultrason. Symp., San Juan, Puerto Rico, Oct. 22–25, 2000, pp. 863–868. [24] J.-S. Hong, Microstrip Filters for RF/Microwave Applications, 2nd ed. Hoboken, NJ, USA: Wiley, 2011. [25] D. Swanson and G. Macchiarella, “Microwave filter design by synthesis and optimization,” IEEE Microw. Mag., vol. 8, no. 2, pp. 55–69, Apr. 2007. [26] J.-S. Hong, H. Shaman, and Y.-H. Chun, “Dual-mode microstrip openloop resonators and filters,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 8, pp. 1764–1770, Aug. 2007. [27] H. Nakamura, H. Nakanishi, R. Goto, and K.-Y. Hashimoto, “Suppression of transverse-mode spurious responses for saw resonators on SiO2/Al/LiNbO3 structure by selective removal of SiO2,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 58, no. 10, pp. 2188–2193, Oct. 2011. [28] R. H. Olsson, J. Nguyen, T. Pluym, and V. M. Hietala, “A method for attenuating the spurious responses of aluminum nitride micromechanical filters,” J. Microelectromech. Syst., vol. 23, no. 5, pp. 1198–1207, Oct. 2014. [29] H. Wang, H. Zhong, Y. Shi, and K.-Y. Hashimoto, “Design of narrow bandwidth elliptic-type SAW/BAW filters,” Electron. Lett., vol. 48, no. 10, pp. 539–540, May 2012. [30] D. Morgan, Surface Acoustic Wave Filters: With Applications to Electronic Communications and Signal Processing, 2nd ed. San Diego, CA, USA: Academic Press, 2010. [31] O. Ikata, T. Miyashita, T. Matsuda, T. Nishihara, and Y. Satoh, “Development of low-loss band-pass filters using SAW resonators for portable telephones,” in Proc. IEEE Ultrason. Symp., Tucson, AZ, USA, Oct. 20–23, 1992, pp. 111–115. Dimitra Psychogiou (S'10–M'14) received the Dipl.-Eng. degree in electrical and computer engineering from the University of Patras, Patras, Greece, in 2008, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology (ETH) Zürich, Switzerland, in 2013. Since 2013, she has been with Purdue University, where she is currently a Senior Research Scientist at the Department of Electrical and Computer Engineering, West Lafayette, IN, USA. In 2008, she joined the Wireless Communication Research Group (WiCR), University of Loughborough, Loughborough, U.K., as a Research Assistant. From 2009 to 2013, she was a Teaching and Research Assistant with the Laboratory of Electromagnetic Fields and Microwave Electronics (IFH), ETH Zürich. Her main research interests include RF design and characterization of reconfigurable microwave and millimeter-wave passive components, tunable filter synthesis, and frequency agile antennas.

Roberto Gómez-García (S'02–M'06–SM'11) was born in Madrid, Spain, in 1977. He received the Telecommunication Engineer and Ph.D. degrees from the Polytechnic University of Madrid, Madrid, Spain, in 2001 and 2006, respectively. Since April 2006, he has been an Associate Professor with the Department of Signal Theory and Communications, University of Alcalá, Alcalá de Henares, Madrid, Spain. He has been for several research stays in the C2S2 Department of the XLIM Research Institute (formerly IRCOM), University of

Limoges, France, Telecommunications Institute of the University of Aveiro, Portugal, the U. S. Naval Research Laboratory (NRL), Microwave Technology Branch, Washington, DC, USA, and Purdue University, West Lafayette, IN, USA. His current research interests include the design of fixed/tunable high-frequency filters and multiplexers in planar, hybrid and MMIC technologies, multi-function circuits and systems, and software-defined radio and radar architectures for telecommunications, remote sensing, and biomedical applications. Dr. Gómez-García is an Associate Editor for the IEEE TRANSACTIONS OF MICROWAVE THEORY AND TECHNIQUES, IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, and IET Microwaves, Antennas and Propagation. He is a Guest Editor of the IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS 2013 Special Issue on “Advanced Circuits and Systems for CR/SDR Applications”, the IET Microwaves, Antennas and Propagation 2013 Special Issue on “Advanced Tuneable/Reconfigurable and Multi-Function RF/Microwave Filtering Devices,” and the IEEE Microwave Magazine 2014 Special Issue on “Recent Trends on RF/Microwave Tunable Filter Design”. He is a reviewer for several IEEE, IET, EuMA, and Wiley Journals. He serves as a member of the Technical Review Board for several IEEE and EuMA conferences. He is also a member of the “IEEE MTT-S Filters and Passive Components” (MTT-8), “IEEE MTT-S Biological Effect and Medical Applications of RF and Microwave” (MTT-10), “IEEE MTT-S Wireless Communications” (MTT-20), and “IEEE CAS-S Analog Signal Processing” (ASP) Technical Committees.

Dimitrios Peroulis (S'99–M'04–SM’15) received the Ph.D. degree in electrical engineering from the University of Michigan at Ann Arbor, Ann Arbor, MI, USA, in 2003. Since August 2003, he has been with Purdue University, West Lafayette, IN, USA, where he is currently Professor of the Department of Electrical Engineering and the Deputy Director of the Birck Nanotechnology Center. His current research projects are focused on the areas of reconfigurable electronics, RF MEMS, and sensors in harsh environment applications. He has been a key contributor on developing very high quality ( > 1000) RF MEMS tunable filters in mobile form factors. Furthermore, he has been investigating failure modes of RF MEMS and MEMS sensors through the DARPA M/NEMS S&T Fundamentals Program, Phases I and II) and the Center for the Prediction of Reliability, Integrity, and Survivability of Microsystems (PRISM) funded by the National Nuclear Security Administration. Dr. Peroulis was a recipient of the National Science Foundation CAREER Award in 2008, the Outstanding Young Engineer Award of the IEEE Microwave Theory and Techniques Society (MTT-S) in 2014, the Outstanding Paper Award from the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society (Ferroelectrics section) in 2012. He has co-authored over 220 journal and conference papers. His students have received numerous Student Paper Awards and other student research-based scholarships. He is a Purdue University Faculty Scholar and has also received ten teaching awards including the 2010 HKN C. Holmes MacDonald Outstanding Teaching Award and the 2010 Charles B. Murphy Award, which is Purdue University's highest undergraduate teaching honor.

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Ultra-Miniature SIW Cavity Resonators and Filters Ali Pourghorban Saghati, Student Member, IEEE, Alireza Pourghorban Saghati, Student Member, IEEE, and Kamran Entesari, Member, IEEE

Abstract—This paper presents ultra-miniature substrate integrated waveguide (SIW) cavity resonators, and filters by employing ramp-shaped slots as interdigital capacitors (IDCs) to force the structure to operate in the first negative-order resonance mode. Additionally, a metal patch in the middle metal layer along with disconnected vias is used to increase the equivalent series capacitance of the resonator and accordingly the miniaturization factor. By applying this method to half-mode SIW (HMSIW) and quarter-mode SIW (QMSIW) resonators, 90%, and 95% of miniaturization is achieved, respectively, compared to conventional full-mode SIW resonators. Afterwards, a two-pole bandpass filter is proposed based on the presented QMSIW resonator. Also, by using a combination of HMSIW and QMSIW ultra-miniature resonators, two trisection filters with a controllable transmission zero on either side of the passband are presented. To the best of authors' knowledge, this is the first disclosure of simultaneous use of HMSIW and QMSIW resonators to achieve asymmetric filter response with an ultra-compact size. Index Terms—Bandpass filter, cavity filter, composite right/left handed (CRLH), miniaturization, substrate integrated waveguide (SIW).

I. INTRODUCTION

L

OW loss bandpass filters, constructed using compact high- resonators are one of the essential blocks of modern wireless communication systems. Ease of fabrication, high-power handling, high linearity, and integration compatibility of substrate integrated waveguide (SIW) cavity resonators make them a good candidate for high-performance microwave filters [1]. Despite their advantages, their use in compact microwave devices is hindered due to their large size. As a result, miniaturization techniques need to be employed to reduce the size of SIW filters, while maintaining the high performance characteristics [2]. Manuscript received June 26, 2015; revised September 14, 2015; accepted October 11, 2015. Date of publication November 11, 2015; date of current version December 02, 2015. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. A. Pourghorban Saghati is with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]). A. Pourghorban Saghati was with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA. He is now with Ossio Inc., Bellevue, WA 98004 USA (email: [email protected]). K. Entesari is with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2494023

The existing work in the area of miniature SIW filters includes loading the SIW resonator using defected ground planes (DGS) and ring gaps [3], [4], using complementary split-ring resonators to change the characteristic cutoff frequency of the SIW structure [5], forcing the SIW resonator to operate at lower-order modes (e.g., -1st mode) using metamaterial-inspired components [6], and cutting the SIW resonators on their fictitious magnetic walls to achieve half-mode SIW (HMSIW) or quarter-mode SIW (QMSIW) resonators [7]–[9]. Among these, the largest miniaturization factor belongs to the QMSIW with roughly 75% of size reduction. The miniaturization factor for the other methods such as the CSRR-loaded resonators, and the negative order resonance-mode ones is limited by the geometrical dimensions of the employed loading structures and the area available on the top/bottom walls of the SIW cavity. This paper proposes ultra-miniature HMSIW and QMSIW resonators in which the first negative order resonance is excited to further decrease the size of the resonators. The mentioned issue of size limitations for loading structures becomes more severe for the case of HMSIW/QMSIW resonators. Accordingly, based on the method first proposed in [10], ramp-shaped slots as inter-digital capacitors (IDC) on the top metal layer of the cavity structure are employed to efficiently use the available resonator area. Also, by employing an additional middle metal layer, a loading patch is employed to increase the capacitance value of the loading IDCs. Finally, disconnected via posts are inserted in the locations of maximum E-field distribution to increase the miniaturization beyond the limits defined by the size of the IDCs. Using these elements, 90% miniature HMSIW, and 95% miniature QMSIW resonators are achieved. The miniature QMSIW resonator was first introduced by the authors in [2], and was used toward the design of a two-pole filter operating at a center frequency of 690 MHz with an area of and a fractional bandwidth of 5.9%. Here, a more in-depth study on the miniaturization method is conducted. Also, the half-mode SIW version of the resonator is provided. Moreover, a combination of an HMSIW and two QMSIW resonators is uniquely employed to design two different trisection filters with controllable transmission zeros on either side of the passband. In addition to the miniaturization due to area reduction (HMSIW-QMSIW combination instead of three SIW resonators), frequency shift due to the loading IDC structure and disconnected vias results in 70% of size reduction in comparison with a normal full-mode SIW resonator. As a result, the entire area of the trisection filter consumes roughly 30% of the area for only one full-mode SIW cavity resonator at the operating frequency. The two trisection filters have midband frequencies of 912 and 754 MHz with total areas of , and , respectively. In comparison with the existing

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Fig. 1. (a) Res. I: Normal cylindrical SIW cavity resonator. (b) Res. II: Ring gap-loaded SIW cavity resonator (37% miniaturization). (c) Res. III: Ramp-shaped IDC loaded SIW cavity resonator (57% miniaturization). (d) Res. IV: IDC-loaded SIW cavity resonator employing a rectangular patch in an additional middle metal layer (66% miniaturization). (e) 3D cross-section view of Res. IV (and the simplified LC resonator model). (f) Res. V: IDC-loaded SIW cavity resonator loaded with floating patch and disconnected vias (73% miniaturization). (g) A-A' cross-section view of Res. V.

SIW-based trisection filters in literature [11]–[14], the proposed trisection filters are the first ones which are ultra-compact, while having controllable transmission zeros, to the best of authors' knowledge. II. RESONATOR DESIGN A. Full-Mode SIW Resonator Fig. 1 demonstrates the procedure used to achieve the miniaturized SIW cavity resonator. Res. I, shown in Fig. 1(a), is a conventional cylindrical SIW cavity resonator which its dimensions are determined based on the fundamental TM010 mode [15]. Based on a Rogers RT/Duroid 6010 ( , )1 dielectric layer with a thicknesses of 3.135 mm, is calculated to be 17.4 mm. As a result a fundamental TM010 mode exists around 2 GHz. For a fixed resonator size, inserting proper elements into the structure will reduce the resonance frequency. The miniaturization factor for a particular miniaturized resonator operating at a lowered frequency of is computed using Miniaturization factor 1Rogers

Corp.. Brooklyn, CT, USA.

(1)

where is the area of a conventional cylindrical SIW resonator, which fundamentally operates at and is the area of the proposed resonators. This miniaturization factor contribution for each element is mentioned in the caption of Fig. 1. 1) Miniaturization Elements: a) Capacitive Ring Gap: First, the SIW cavity resonator is loaded with a shunt via at the center [see Fig. 1(b)]. Afterwards, a ring slot is etched on the top wall to disconnect this wall from the bottom wall. The ring gap can be modeled as small series capacitance which loads the resonator and results in a lower resonance frequency. The capacitance value, as a function of physical dimensions, can be approximated based on the approach provided in [4]. The frequency down-shift due to this capacitive loading is mainly limited by the size of the annular gap. To better understand this limitation, the structure is simulated using commercial high frequency structure simulator (HFSS),2for various , and values, while is fixed to 0.75 mm, and the results are shown in Fig. 2(a). As can be seen, increasing the width and length of the ring gap results in higher miniaturization factors. However, these two dimensions are limited by the size of the cavity which is assumed to be fixed for all the resonators. Fine tuning of the resonance frequency is also feasible by adjusting the spacing value of the ring-gap . Considering 2Ansys

HFSS ver. 15, Ansys Inc.. Canonsburg, PA, USA, 2013.

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Fig. 2. Miniaturization factor of Res. II for different ring slot dimensions.

4 mm, 8 mm, and 0.75 mm, 37% miniaturization is achieved. b) Ramp-Shaped Interdigital Capacitor (IDC): Fig. 1(c) shows Res. III in which the ring slot on the top metal layer is replaced by ramp-shaped slot [10]. In this figure, is the total width of the ramp-shaped slot etched on the top metal wall of the cavity, while is the angle between the adjacent slots. Similar to Res. II, stands for the spacing value of the slot-gap, and its value remained 0.75 mm for Res. III, IV, and V. The cavity-wall vias and center via can be modeled as shunt inductors. Therefore, utilizing IDC, as an equivalent series capacitance, forces the SIW resonator to operate at resonance [6]. This is basically different from Res. II, where the fundamental resonance frequency is shifted down due to the capacitive loading effect of the annular gap, and as a result, the operating mode is quasi-TEM [4]. In order to observe the 1st, 0th, and resonances, Res. III is simulated as a two-port resonator, and the resonance peaks on are plotted and shown in Fig. 3(a). In this figure, the frequency is normalized to the fundamental TM010 resonance frequency. For this resonator, the zeroth and first negative order resonances happen at 0.8 and , respectively. In this case, increasing the series capacitive loading results in further shifting down the resonance frequency rather than the fundamental mode. Similar to traditional IDC structures, for a constant total area of the IDC, higher equivalent series capacitance values are achievable by increasing the number of fingers [16]. For the ramp-shaped configuration, assuming the total size of the IDC is constant, increasing the number of fingers is feasible by decreasing the angle between the adjacent slots . The effect of the ramp-shaped slot physical dimensions on the miniaturization factor is studied by simulating the resonator for various , and values, while the number of fingers are changed accordingly. The results are shown in Fig. 3(b). According to the available area on the cavity top-metal wall (for 17.4 mm), by decreasing from 50 to 30 , the maximum possible number of fingers is increased from 4 to 8, and consequently, higher miniaturization factors are achieved. However, further decreasing to lower than 30 means drastic reduction in the fingers' metal width . As a result, higher capacitance values cannot be achieved for values less than 30 , and the miniaturization factor drops for . Moreover, this trade-off between , and results in another bottleneck. Reduction in the metal width of each finger will reduce the quality factor

Fig. 3. (a) Resonance peaks on of Res. III, IV, and V shown in of Res. III for various number of Fig. 1. (b) Miniaturization factor and fingers for the IDC (solid lines are miniaturization factor and dashed lines ). are

of the IDC, and the resonator [16]. This is also shown in Fig. 3(b), where the unloaded quality factor of Res. III is plotted for each case. Therefore, the optimum values of , and are employed for this design. On the other hand, has small effect on the resonance frequency, and can be used for fine tuning. The spacing value has minor effect on the miniaturization factor of Res. III, while very large values of can cause the of the resonator to drop. This is mainly due to increased leakage loss from the top interdigital slot. By increasing from 0.75 to 1.75 mm, the resonance frequency only reduces by 3.1%, however, the value drops from 253 to 221. On the other hand, for 0.75 mm, the leakage loss form the top IDC slot is roughly constant, and neither the miniaturization factor, nor value change considerably. Therefore, considering 0.75 mm, and 15.5 mm, the miniaturization factor is roughly 57% for Res. III. c) Floating Metal Patch: To further increase the capacitive loading and thus the frequency shift, a floating metal patch is inserted in an additional middle metal layer beneath the rampshaped slot [Res. IV shown in Fig. 1(d)]. As can be seen in Fig. 3(a), by employing this metal patch, the resonance frequency is shifted down to a lower value compared to Res. III. To better demonstrate the capacitive loading effect of this floating patch, a 3D cross-section view of Res. IV is shown in Fig. 1(e). In this figure, the shunt capacitance to the ground (between the patch and the cavity bottom wall) and the two fringing capacitances (between the ramp-shaped slot edges and the patch) are named as , and , respectively, and stands for the equivalent series capacitance introduced by the ramp-shaped interdigital slot. For simplicity, the metal planes are assumed perfect electric conductors, and parasitics are excluded. As can be

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SIMULATED

Fig. 4. (a) Miniaturization factor of Res. IV for various widths and heights of the patch. (b) The effect of the patch openings and disconnected vias on the miniaturization factor for different heights of the patch.

seen, the patch increases the effective series capacitance as follows: (2) While should be minimized, needs to be maximized to increase the overall series capacitance and achieve more miniaturization. The effect of the size and location of the patch on shifting down the resonance frequency is studied by simulating the structure for various widths and heights for the patch, and the results are shown in Fig. 4(a). While can be used for fine adjustment, needs to be maximized to reduce , increase two values, and to achieve the highest miniaturization factor. However, this dimension is limited by the height of the standard substrate thickness used for the floating patch. Therefore, considering 7.5 mm for highest miniaturization, is chosen to be 2.5 mm, while is 0.635 mm, based on the availability of standard Rogers RT/Duroid 6010 substrate thicknesses and fabrication constraints. By using this structure, a miniaturization factor of 66% is achieved. d) Disconnected Vias: Disconnected vias are the last loading elements [see Fig. 1(f)] to increase the miniaturization factor. These capacitive disconnected via posts will further load the cavity and push the resonance frequency to even lower values [see Fig. 3(a)]. The loading effect of disconnected vias and their design considerations are well-studied in [17]. They need to be placed where the E-field distribution is maximum inside the cavity to have the highest loading effect. Fig. 1(f) and (g) show the top and 2D cross-section views of the final resonator (Res. V), respectively, in which the patch

TABLE I FACTOR OF THE RESONATORS SHOWN IN FIG. 1

and the disconnected vias are both inserted. The loading via posts are disconnected from the bottom wall of the cavity by using disk-slot openings. These openings are relatively small compared to wavelength at operating frequencies, and the leakage is negligible [17]. Also, capacitive via posts need to be disconnected from the floating metal patch to avoid shortening it to the top metal wall. This can be done by etching either circular disk- or triangular-shaped slots on the floating patch and around the vias. The opening slots on the floating patch should have minimum effect on the equivalent fringing capacitances between the patch and the ramp-shaped slot edges . In order to better study the effect of the opening slots on the miniaturization factor, the simple patch in Res. IV is replaced with a patch with triangular-shaped slots, and a patch with disk-shaped slots, respectively. Then, the two modified resonators are simulated for different heights of the patch and the results are compared to the original Res. IV in Fig. 4(b). As can be seen, etching the slots on the patch will slightly reduce the miniaturization factor of Res. IV, however, there is no significant difference between utilizing patch with diskor triangular-shaped slots. In this design, triangular slots are etched on the metal patch to disconnect it from the loading vias, and avoid shortening it to the top metal wall [see Fig. 1(f)–(g)]. Afterwards, disconnected vias are inserted, and the final resonator is also simulated for different heights of the patch. Inserting disconnected vias results in higher miniaturization factors compared to Res. IV for any height of the patch with triangular slots. By employing all the miniaturization elements, 73% of miniaturization is achieved. Ramp-shaped IDC has the most effect on miniaturization as it forces the resonator to operate at resonance mode. 2) Loss: To study the loss mechanism of the presented resonators, the unloaded quality factor is extracted from two-port s-parameters simulation, based on the approach in [18]. The top, middle, and bottom metal layers are all considered as copper (1 oz), and as mentioned before, both substrate layers are Rogers RT/Duroid 6010 ( , ) with different thicknesses of 2.5 mm and 0.635 mm. For Res. III to V, the two dielectric layers are bonded together using Rogers RO4450B pre-preg material ( , , 0.09 mm). Each single resonator is weakly coupled at the input/output ports, and simulated. The resulting unloaded quality factors are shown in Table I. The loss mechanism of a regular SIW cavity resonator (Res. I) has been investigated in literature [19]. Using a low-loss dielectric layer, the leakage loss from the apertures between the side-wall vias is negligible if the vias are placed close enough to each other [3]. Therefore, for such resonators, conductor loss

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Fig. 5. Magnitude of electric field distribution inside the resonators shown in Fig. 1 at the corresponding resonance frequency.

and dielectric dissipation have the main contribution to degradation. The -factor of a conventional rectangular SIW cavity resonator is reported 500 at the fundamental resonance frequency [3], while the and the of the employed dielectric material are 3, and 0.001, respectively. However, using high-permittivity dielectric materials such as Rogers 6010, with higher loss tangent of 0.0023, -factor drops to 312 for Res. I in this design. Moreover, other loss factors need to be considered carefully as different miniaturization elements are added to the resonator. As can be seen in Table I, Res. II has a lower value compared to Res. I. This is mainly because of leakage loss from the annular ring gap on the top metal wall. The leakage loss can be minimized by increasing the equivalent capacitance introduced by the ring-gap. This means that by increasing the miniaturization factor, the leakage loss decreases. To better discern this relation, the magnitude of E-field distribution is plotted for all resonators in Fig. 5. The center via post and the capacitive elements around it force the maximum E-field distribution to be around the capacitive elements. As this capacitive loading effect increases by inserting IDC, metal patch, and the disconnected vias, the operating frequency of the resonator is pushed down to lower values, and hence, the wavelength becomes larger. This means that the size of the resonator relative to its operating wavelength becomes smaller as miniaturization increases. As a result, the leakage loss decreases for resonators with higher miniaturization factors, and Res. III, IV, and V have larger values compared to Res. II. Based on this argument, Res. V is expected to have less radiation loss and better compared to Res. III, or IV. However, this resonator has another source of leakage which is the disk-slot openings on the bottom wall of the cavity. As a result, the is degraded compared to Res. III and IV. B. Half-Mode SIW Resonator The miniaturization method introduced in the previous subresonance of a regular section was based on exciting the SIW cavity, and shift that resonance down to the lowest possible value by loading the cavity. In order to design a more compact resonator, the proposed method is applied to a HMSIW cavity resonator (see Fig. 6). First, the regular SIW resonator (Res. I) is bisected on its fictitious magnetic wall. For a large diameter to height ratio of the cavity, the leakage from the open wall would be negligible, and the resonator will operate at roughly a same resonance frequency [7]. Then, all the above miniaturization elements are inserted to achieve a HMSIW resonator

Fig. 6. (a) Original HMSIW resonator. (b) Modified HMSIW resonator. (c), (d) Magnitude of magnetic field distribution inside the original and modified HMSIW resonators, respectively.

which operates at resonance. While, the frequency of operation remains approximately the same compared to Res. V [see Fig. 1(e)], the size is reduced by half. 1) Effect of the Center Via Position: Fig. 6(a) and (b) show the top view of two possible configurations for the miniaturized HMSIW cavity resonator. The difference between these two resonators is the place of the center via. As Res. V [see Fig. 1(f)] is bisected to achieve the HMSIW resonator, the center via can be either kept at the same position, or replaced by two other shunt vias at the two ends of the ramp-shaped slot [see Fig. 6(b)]. While the original HMSIW resonator is useful to achieve positive coupling, the modified HMSIW resonator enables negative coupling. This is very useful in filter design process, where cross-coupling between the resonators is needed. The magnitude of H-field distribution is plotted for the two resonators in Fig. 6(c) and (d). For the original HMSIW resonator, the magnetic field is maximum at the center and around the shunt via, while the E-field is maximum around the capacitive elements [see Fig. 6(c)]. As a result, by changing the shunt via position, the maximum of the magnetic field happens at the two ends of the ramp-shaped slot instead of the center of the cavity [see Fig. 6(d)]. By controlling the position of the

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FINAL DIMENSIONS (mm)

Fig. 7. (a) Ultra miniature QMSIW resonator. (b) Magnitude of magnetic field distribution inside the original and modified QMSIW resonators.

magnetic field concentration, it is possible to couple the resonators either positively or negatively (see Section III). However, changing the center via position slightly lowers miniaturization factor for the modified HMSIW resonator. Considering 11.75 mm, the miniaturization factor for the modified HMSIW resonator is 90%, while for the original HMSIW resonator, it is 92%. 2) Loss: The loss mechanism of the two HMSIW resonators is investigated by calculating the value based on the same approach described in the previous subsection. Using same substrate layers as before, values of 216 and 221 are achieved for the original and modified HMSIW resonators, respectively. The degradation is mainly contributed to the leakage loss from the open wall of the cavity, even though, the conductor loss would be less for HMSIW resonators compared to their SIW counterparts. C. Quarter-Mode SIW Resonator QMSIW resonator can be obtained by bisecting the HMSIW resonator on its fictitious magnetic wall [8]. The overall size of the QMSIW resonator is roughly 25% of its SIW counterpart [9]. Again, inserting all the above miniaturization elements inside the QMSIW resonator forces the structure to resonate at resonance mode. Therefore, an ultra miniature resonator can be achieved which has approximately the same resonance frequency as Res. V [see Fig. 1(e)], while its size is reduced by 75% [2]. Fig. 7(a) shows the top, middle, and bottom metal layers of the final ultra miniature QMSIW resonator. The shunt center via used in the SIW resonators is now placed on the corner of the resonator, and the ramp-shaped inter-digital slot, and the floating patch are precisely quarter of their SIW version counterparts. 1) Effect of the Corner Via Position: Similar to the HMSIW resonator presented in the previous subsection, it is possible

TABLE II ULTRA-MINIATURE QMSIW FILTER

OF THE

to control the magnetic field concentration inside the cavity to some degree by modifying the location of the shunt corner via. The magnitude of H-field distribution is shown in Fig. 7(b) for two QMSIW resonators. As can be seen, the magnetic field is concentrated around the corner via. By placing the shunt via on the other end of the ramp-shaped slot, the concentration of the H-field is modified. Again, altering the shunt via position results in slightly lower miniaturization factors. The dimensions of the final resonator are tabulated in Table II. Based on these dimensions, and using same dielectric layers, the resonance happens at 731 MHz. Compared to a full mode SIW resonator with a fundamental mode at 731 MHz, roughly 95% miniaturization is achieved for the original QMSIW resonator in Fig. 7(a) [2]. However, considering 11.75 mm, the miniaturization factor for the modified QMSIW resonator is 93%. 2) Loss: The loss mechanism of this resonator is studied based on the same approach used for the SIW and HMSIW resonators. The values are 186 and 189 at 731 MHz for the original and modified QMSIW resonators, respectively. Although the conductor loss would be less than the SIW and HMSIW versions of the resonator, because of two open walls of the cavity the values are degraded. Ultra compact size, relatively high , and capability of cross coupling of this resonator make it a suitable candidate for ultra miniature filter design. III. FILTER DESIGN AND IMPLEMENTATION The ultra miniature HMSIW, and QMSIW resonators, introduced in the previous section, are used to design and fabricate a two-pole bandpass filter, and two trisection filters with transmission zeros on either side of the passband. First, the design values of the low-pass prototype response is determined for each filter based on the target specifications. Afterwards, the required coupling coefficient matrices and external quality factors are calculated for each filter based on the design values achieved in the first step. Since the method is based on the coupled-resonators filter design, the next step is to establish the relationship between the coupling coefficient matrix and the physical dimensions of the coupled resonators. The coupling factor for synchronously tuned coupled resonators can be approximated using [20] (3) , and are the two split resonance frequencies where extracted from full-wave simulation of the coupled resonators with weakly-coupled input/output ports. The positive or negative sign of the coupling factor determines whether the resonators are positively or negatively coupled, respectively.

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Fig. 8. Top and bottom views of the miniaturized two-pole QMSIW filter. The 4 mm, 4.4 mm, 8.9 mm, parameter values are: 10.9 mm, and 18 mm.

The value is extracted from full-wave simulation of the singly loaded resonator using [20] (4)

Fig. 9. (a) Coupling factor of the two-pole filter as a function of the spacing values as a function of the L-shaped slot length. between the resonators. (b)

where is the simulated resonance frequency and the is the difference of the frequencies at which a phase shift of occurs in the phase response of the resonator. A. Two-Pole Bandpass Filter Fig. 8 shows the top, and bottom views of the two-pole bandpass filter using the proposed QMSIW resonator implemented with two Rogers RT/Duroid 6010LM substrates that are bonded together with Rogers RO4450B pre-preg material as described before [2]. The two-pole coupled-resonator filter is designed to operate at center frequency of 700 MHz. For an in-band return loss better than 20 dB with pass-band ripple of 0.01 dB and 3-dB bandwidth (FBW) of 6%, using the designed values of the low-pass Chebyshev prototype response, the required coupling factor is , and the external quality factor is . While the spacing between the two resonators ( , and ) is used to obtain the proper coupling, the L-shaped slot ( and ) is used at the input/output to adjust the required external quality factor. To better show the relationship between the physical dimensions and the required theoretical design values, the coupling factor as a function of the spacing between the resonators, and the value as a function of the L-shaped slot length is plotted in Fig. 9(a) and (b), respectively. The horizontal distance between the two resonators, , can be used for coarse adjustment of the coupling factor. However, the vertical distance, , can be utilized for fine tuning. As shown in Fig. 7(b), the H-field distribution inside the original QMSIW cavities are maximized around the shunt corner via. As decreases, and increases, the two shunt corner vias will become closer to each other, and as a result, higher coupling factors are achievable ( 4 mm, and 4.4 mm for ). For the external quality factor, the total length of the L-shaped slot needs to be adjusted accurately. In this design, value becomes close to 7.4 when is 9 mm and is 11 mm.

Fig. 10. Top (left) and bottom (right) views of the two-pole filter prototype.

Fig. 10 shows the top and bottom views of the fabricated prototype. First, the top and bottom metal walls are etched on the upper metalization side of the top substrate, and lower metalization side of the bottom substrate, respectively. Afterwards, the bottom-side metalization of the top substrate is completely removed, while the top-side metalization of the bottom substrate is etched to create the floating patch. The two substrate layers are then bonded to each other using Rogers pre-preg material, and finally, the plated via holes are drilled through both substrates [2]. The overall size of the presented QMSIW two-pole filter, excluding the microstrip feed lines, is , where is the wavelength in free space at the frequency of operation. Fig. 11(a) shows the measured and simulated narrow-band s-parameters responses of the proposed filter [2]. Also, the ideal synthesized response of the standard two-pole Chebyshev prototype is plotted for and [20]. As can be seen, good agreement between the theory, simulation, and measurement results is achieved. The return loss is better than 30 dB, while the measured in-band insertion loss is 2.1 dB. The measured 3-dB fractional bandwidth is 5.9%, which is almost the same as the designed value. The measured and simulated wide-band s-parameters response of the filter are shown in Fig. 11(b). Since the resonance mode of the QMSIW

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Fig. 12. Two possible single-unit trisection topologies.

TABLE III TRISECTION FILTERS SPECIFICATIONS AND REQUIRED DESIGN VALUES

Fig. 11. (a) Ideal, simulated, and measured narrow-band responses of the twopole filter. (b) Measured and simulated wide-band filter responses.

resonators is used to design the two-pole filter, there is a transmission zero occurred around the 0th-order resonance mode. Therefore, the first spurious harmonic does not appear up to the 1st resonance mode of the QMSIW resonators. This provides an out-of-band rejection better than 30 dB up to 2.1 GHz, which is almost three times the center frequency of the operating band. To the best of authors' knowledge, this is the most compact two-pole SIW-based bandpass filter, which also has improved upper stopband rejection. B. Trisection Filters In some applications high selectivity is required on solely one side of the passband [20]. Hence, for asymmetric bandpass filter response with minimum possible insertion loss, trisection filters can be used, which are capable of having transmission zero on either side of the passband that high selectivity is required [21]. The transfer function of a trisection filter can be expressed as [20] (5) (6) where is the ripple constant, is the frequency variable of the lowpass prototype filter, is the th transmission zero, and is the degree of the filter. It is noteworthy that since there is only one finite transmission zero in one-unit trisection filter, the other two will be placed at infinity in the domain. 1) Filter Synthesis: Fig. 12 shows two general topologies for one-unit trisection filter. Both are three-pole arrangements with cross-coupling between the first and the third resonators. Positive (dashed line) and negative (solid line) cross couplings are used in Trisection I and II topologies, respectively. As a result,

Trisection I topology has its transmission zero on the lower side of the passband, while for Trisection II, the transmission zero appears on the upper side of the passband. Although, the response of the trisection filter is basically asymmetrical, it is possible to keep the physical configuration of the filter symmetrical [20]. Therefore, for each topology in this design, the two coupling factors and and, also, the two external quality factors and are equal, respectively. Trisection I and II filters are designed to operate at midband frequencies of 920 and 760 MHz with out-of-band rejection 20 dB for frequencies 890 and 790 MHz, respectively. For both filters, the 3-dB fractional bandwidth is considered 4.4%, while the in-band return loss is better than 20 dB. Based on these specifications, the element values of the low pass prototype filters are first calculated based on the approach discussed in [20]. Afterwards, the required resonance frequency of each resonator ( , , ), the external quality factors, and the coupling matrix elements are extracted for both filters. The specifications and the required theoretical design values of both filters are summarized in Table III. For the Trisection I case is higher and is lower than the filter's midband frequency . For the Trisection II filter, however, is lower than its , while is higher. The general coupling matrix for both Trisection filters and are calculated based on the required specifications summarized in Table III

The two matrixes have close values, while the signs are reverse. This will guarantee mirrored frequency responses for the

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Fig. 13. Synthesized responses of the two trisection filters.

two filters compared to each other [21]. Using and the synthesized responses of two ideal trisection filters are shown in Fig. 13. The lumped-element equivalent circuits of the trisection filters are comprehensively discussed in [20] and the precise value of each lumped element can be extracted using , and . Based on the presented ultra-compact QMSIW and HMSIW resonators, two trisection filters are designed and implemented. 2) Filter Design: Figs. 14 and 15 show the top and bottom views of the two designed trisection filters. The two QMSIW resonators are the first and the third resonators of the topologies shown in Fig. 12, while the HMSIW resonator acts as the second resonator. For simplicity, the required resonance frequencies , , and are adjusted by finely tuning the radius of the cavities (R1, and R2), while miniaturization parameters are kept the same as tabulated in Table II. The two structures are excited using microstrip feed lines. However, to make enough room for SMA connectors, the distance between the input/output ports is increased to value by using additional microstrip lines. Similar to the two-pole filter and based on Fig. 9(b), the length of the L-shaped slots ( , ) at the input/output ports are adjusted to achieve the required value. As was shown in Figs. 6 and 7, it is possible to control the location of the maximum H-field distribution inside the cavities by changing the shunt corner via position. As a result, either positive or negative couplings can be achieved based on adjusting the location of the shunt via. When the two shunt vias of the two QMSIW resonators are close enough to each other, the concentration of the maximum of H-field distribution of the two resonators are in close proximity, and as a result, negative coupling can be achieved between the two resonators. However, by placing the shunt vias far from each other ( 11.75 mm), the concentration of maximum H-field distribution of the two cavities are far enough to make positive coupling between the two QMSIW resonators possible. In Trisection I configuration (see Fig. 14), the two QMSIW resonators are coupled to the HMSIW resonator negatively from the open-wall of the cavities where the shunt vias are placed. However, they are coupled to each other positively from their other open-wall where there are no shunt vias. On the other hand, for Trisection II configuration shown in Fig. 15, the two QMSIW resonators are coupled to each other negatively as their shunt vias are placed in the corner, and in close proximity to each other. However, the shunt vias of the HMSIW resonator are removed from its open edge, and instead, two of the previously disconnected vias are now connected to act as center shunt vias (see Fig. 15). This way the

Fig. 14. Top and bottom views of the designed Trisection I filter with a con19 mm, trollable transmission zero on lower side of the passband ( 17.4 mm, 4.35 mm, 3 mm, 7 mm, 9.5 mm, 1.36 mm, and 25.07 mm).

Fig. 15. Top and bottom views of the designed Trisection II filter with a con17.4 mm, trollable transmission zero on higher side of the passband ( 20 mm, 3.3 mm, 3.6 mm, 9 mm, 7.5 mm, 1.36 mm, and 25.07 mm).

undesired negative coupling between the HMSIW and QMSIW resonators is minimized, and the two QMSIW resonators are coupled positively to the HMSIW resonator from its open wall. The distance between the resonators ( and ) is used to adjust the required coupling values. In order to better show the cross-coupling behavior, each two resonators are weakly coupled at the input/output ports and simulated separately.

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Fig. 16. Inter-resonator coupling factors of the two trisection filters as a function of the spacing between the resonators.

Fig. 16 shows the relationship between the physical dimensions ( , and ), and the coupling matrix elements ( , , and ) for both trisection filters. For Trisection I topology, by increasing , lower coupling coefficient values are achievable between the QMSIW and the HMSIW resonators. However, increasing results in higher coupling coefficient values between the two QMSIW resonators in this filter. This is an evidence that the two coupling coefficients are out of phase [22]. Inversely, for Trisection II topology, increasing results in lower coupling coefficients, while for larger values, higher coupling factors are achievable. The final dimensions of the proposed filters are summarized in the caption of Figs. 14 and 15. 3) Fabrication and Experimental Results: Fig. 17(a) and (b) show the top and bottom views of the two fabricated trisection filters, respectively. The fabrication process is same as the two pole filter. The overall size of Trisection I prototype [see Fig. 17(a)], excluding the microstrip feed lines, is 39.4 mm 42.3 mm, which is equivalent to . Similarly, the size of Trisection II prototype is 41 mm 41.5 mm, which is equal to . The fabricated prototypes are measured using a two-port network analyzer (Agilent N5230A) after short-open-load-thru (SLOT) calibration. Fig. 18(a) and (b) show the simulated and measured narrow-band s-parameter responses of the two trisection filters, which are in reasonably good agreement with the ideal responses of the two filters shown in Fig. 13. The measured in-band return loss is better than 17 dB for both filters, while the insertion loss is 2.45 dB, and 2.1 dB for Trisection I and II filters, respectively. The midband frequencies are 912 MHz for Trisection I and 754 MHz for Trisection II, while the FBW is 4.2% for both prototypes, which is very close to the designed value.

Fig. 17. Top (left) and bottom (right) views of fabricated (a) Trisection I and (b) Trisection II prototypes.

Fig. 18. Measured and simulated narrow-band filter responses of: (a) Trisection I prototype and (b) Trisection II prototype.

POURGHORBAN SAGHATI et al.: ULTRA-MINIATURE SIW CAVITY RESONATORS AND FILTERS

TABLE IV COMPARISON OF RELATED WORKS

IN

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slight discrepancy in insertion loss between the measurement and simulation results corresponds to the effect of practical errors such as variations in via diameter, or soldering and SMA connector's loss, which were not considered in simulations. The quality factor error between the measured and simulated results of the proposed filters is less than 6%. Table IV compares the measured performance characteristics of the proposed two-pole and trisection filters with the previously presented works in literature. In order to perform a fair size comparison, filter sizes are computed relative to the air-wavelength at the center operating frequency for each filter. As can be seen, the proposed method achieves the highest miniaturization factor, while it meets similar or better performance specifications. V. CONCLUSION

Fig. 19. Measured and simulated wide-band filter responses of: (a) Trisection I prototype and (b) Trisection II prototype.

Fig. 19(a) and (b) show the simulated and measured wideband response of the two filters, respectively. Similar to the two pole filter, and as the resonance frequency is excited to design an ultra compact filter, a transmission zero occurs around the 0th resonance mode for both filters. Hence, a rejection level of 19 dB is achieved in the frequency range of 0.965–2 GHz and 0.8–1.5 GHz for Trisection I, and II filters, respectively. IV. DISCUSSION As can be seen in Figs. 11(a) and 18, the measured and simulated results of the proposed prototypes are relatively in good agreement. The frequency-shift error is less than 1.5%, which probably comes mainly from the dielectric constant tolerances. For the Rogers 6010 dielectric material, the reported tolerance of value is 2.45% over 10.2 from the datasheet.3 Moreover, 3RT/duroid

6010LM data sheet. [Online]. Available: www.rogerscorp.com

Ultra-miniature two-pole and trisection filters based on QMSIW and HMSIW resonators are introduced. The proposed trisection filters have controllable transmission zeros on either side of the passband. The maximum in-band loss for the two-pole and trisection filters is 2.1 and 2.45 dB, respectively. First negative order resonance is excited by using metamaterial inspired ramp-shaped interdigital capacitors on the top metal wall of the cavity. The equivalent series capacitance is increased by the aid of a floating metal patch, and disconnected via posts. By applying this method to half- and quarter-mode SIW resonators, roughly 90%–95% of miniaturization is achieved, compared to conventional SIW cavity resonators. To the best of authors' knowledge the proposed filters are the most compact SIW-based bandpass filters today. Also the combinational use of QMSIW-HMSIW to achieve an ultra-compact trisection response is proposed for the first time. REFERENCES [1] K. Entesari, A. Pourghorban Saghati, V. Sekar, and M. Armendariz, “Tunable SIW structures: Antennas, VCOs, filters,” IEEE Microw. Mag., vol. 16, no. 5, pp. 34–54, Jun. 2015. [2] A. Pourghorban Saghati, A. Pourghorban Saghati, and K. Entesari, “An ultra-miniature quarter-mode SIW bandpass filter operating at first negative order resonance,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2015, pp. 1–3.

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[3] Y. L. Zhang, W. Hong, K. Wu, J. X. Chen, and H. J. Tang, “Novel substrate integrated waveguide cavity filter with defected ground structure,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 4, pp. 1280–1287, Apr. 2005. [4] J. Martinez, S. Sirci, M. Taroncher, and V. Boria, “Compact CPW-fed combline filter in substrate integrated waveguide technology,” IEEE Microw. Wirel. Compon. Lett., vol. 22, no. 1, pp. 7–9, Jan. 2012. [5] Y. D. Dong, T. Yang, and T. Itoh, “Substrate integrated waveguide loaded by complementary split-ring resonators and its applications to miniaturized waveguide filters,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 9, pp. 2211–2223, Sep. 2009. [6] Y. Dong and T. Itoh, “Substrate integrated waveguide negative order resonances and their applications,” IET Microw. Antennas Propag., vol. 4, no. 8, pp. 1081–1091, Aug. 2010. [7] V. Sekar and K. Entesari, “A novel compact dual-band half-mode substrate integrated waveguide bandpass filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2011, pp. 1–4. [8] Z. Zhang, N. Yang, and K. Wu, “5-GHz bandpass filter demonstration using quarter-mode substrate integrated waveguide cavity for wireless systems,” in Proc. IEEE Radio Wirel. Symp., Jan. 2009, pp. 95–98. [9] C. Jin and Z. Shen, “Compact triple-mode filter based on quarter-mode substrate integrated waveguide,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 1, pp. 37–45, Jan. 2014. [10] A. Pourghorban Saghati, M. Mirsalehi, and M. Neshati, “A HMSIW circularly polarized leaky-wave antenna with backward, broadside, forward radiation,” IEEE Antennas Wirel. Propag. Lett., vol. 13, pp. 451–454, Mar. 2014. [11] L. Szydlowski, A. Jedrzejewski, and M. Mrozowski, “A trisection filter design with negative slope of frequency-dependent crosscoupling implemented in substrate integrated waveguide (siw),” IEEE Microw. Wirel. Compon. Lett., vol. 23, no. 9, pp. 456–458, Sep. 2013. [12] J. Martinez, S. Sirci, and V. Boria, “Compact SIW filter with asymmetric frequency response for C-band wireless applications,” in Proc. Int. Wirel. Symp., Apr. 2013, pp. 1–4. [13] P.-J. Zhang and M.-Q. Li, “Cascaded trisection substrate-integrated waveguide filter with high selectivity,” Electron. Lett., vol. 50, no. 23, pp. 1717–1719, 2014. [14] S. Sirci, J. Martinez, J. Vague, and V. Boria, “Substrate integrated waveguide diplexer based on circular triplet combline filters,” IEEE Microw. Wirel. Compon. Lett., vol. 25, no. 7, pp. 430–432, July 2015. [15] E. Arnieri and G. Amendola, “Analysis of substrate integrated waveguide structures based on the parallel-plate waveguide green's function,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 7, pp. 1615–1623, Jul. 2008. [16] J. Hobdell, “Optimization of interdigital capacitors,” IEEE Trans. Microw. Theory Techn., vol. 27, no. 9, pp. 788–791, Sep. 1979. [17] A. Pourghorban Saghati and K. Entesari, “A reconfigurable SIW cavity-backed slot antenna with one octave tuning range,” IEEE Trans. Antennas Propag., vol. 61, no. 8, pp. 3937–3945, Aug. 2013. [18] D. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ, USA: Wiley Global Education, 2011. [19] H. J. Tang, W. Hong, J.-X. Chen, G. Q. Luo, and K. Wu, “Development of millimeter-wave planar diplexers based on complementary characters of dual-mode substrate integrated waveguide filters with circular and elliptic cavities,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 4, pp. 776–782, Apr. 2007. [20] J.-S. G. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. Hoboken, NJ, USA: Wiley, 2004. [21] C.-C. Yang and C.-Y. Chang, “Microstrip cascade trisection filter,” IEEE Microw. Guided Wave Lett., vol. 9, no. 7, pp. 271–273, Jul. 1999. [22] X.-P. Chen and K. Wu, “Substrate integrated waveguide cross-coupled filter with negative coupling structure,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 1, pp. 142–149, Jan. 2008.

Ali Pourghorban Saghati (S'11) received the M.Sc. degree (honors) in electrical engineering from Ferdowsi University, Mashhad, Iran, in 2014. He is currently pursuing the Ph.D. degree in electrical and computer engineering at Texas A&M University, College Station, TX, USA. His research interests include miniaturized RF/microwave antennas and filters, reconfigurable multiband and broadband antennas, microwave interferometric chemical sensors for lab-on-chip applications. Mr. Pourghorban Saghati was a recipient of the Texas A&M University ECE Departmental Graduate Student Scholarship in Fall 2014.

Alireza Pourghorban Saghati (S'08) received the M.Sc. degree (honors) in electrical engineering from Urmia University, Urmia, Iran, in 2010 and the Ph.D. degree from Texas A&M University, College Station, TX, USA, in 2015. He is currently an RF/Antenna Engineer with Ossia Inc., Bellevue, WA, USA. His research interests include RF communication systems, RF energy harvesting, wireless power transfer, and wearable wireless devices. Dr. Pourghorban Saghati was a recipient of the Texas A&M University ECE Departmental Graduate Student Scholarship in Fall 2011. He was also a recipient of the Student Paper Award (honorable mention) at the Student Paper Competition presented at the 2013 IEEE AP-S International Symposium, Orlando, FL, USA.

Kamran Entesari (S'03–M'06) received the B.S. degree in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 1995, the M.S. degree in electrical engineering from Tehran Polytechnic University, Tehran, Iran, in 1999, and the Ph.D. degree from The University of Michigan, Ann Arbor, MI, USA, in 2005. In 2006, he joined the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA, where he is currently an Associate Professor. His research interests include microwave chemical/biochemical sensing for lab-on-chip applications, RF/microwave/millimeter-wave integrated circuits and systems, reconfigurable RF/microwave antennas and filters, and RF micro-electromechanical systems (MEMS). Prof. Entesari currently serves as an Associate Editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS (MWCL), and is on the Technical Program Committee (TPC) of the IEEE RFIC Symposium. He was a recipient of the 2011 National Science Foundation (NSF) CAREER Award. He was a co-recipient of the 2009 Semiconductor Research Corporation (SRC) Design Contest Second Project Award for his work on dual-band millimeter-wave receivers on silicon and the Best Student Paper Awards of the IEEE RFIC Symposium in 2014 (second place), IEEE Microwave Theory and Techniques Society (IEEE MTT-S) in 2011 (third place), and IEEE AP-S in 2013 (honorable mention).

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Design and Multiphysics Analysis of Direct and Cross-Coupled SIW Combline Filters Using Electric and Magnetic Couplings Stefano Sirci, Student Member, IEEE, Miguel Ángel Sánchez-Soriano, Member, IEEE, Jorge D. Mart´ınez, Member, IEEE, Vicente E. Boria, Senior Member, IEEE, Fabrizio Gentili, Associate Member, IEEE, Wolfgang Bösch, Fellow, IEEE, and Roberto Sorrentino, Fellow, IEEE

Abstract—In this paper, combline substrate integrated waveguide (SIW) filters using electric and magnetic couplings are thoroughly studied. Thus, a negative coupling scheme consisting on an open-ended coplanar probe is proposed and analyzed in detail. Several in-line 3-pole filters at C-band are designed, manufactured and measured showing how the presented approach can be used for implementing direct couplings while enabling an important size reduction and improved spurious-free band compared to conventional magnetic irises. A fully-packaged quasi-elliptic 4-pole filter is also designed at 5.75 GHz showing how the negative coupling structure can be used for introducing transmission zeros by means of cross-couplings between non-adjacent resonators. Finally, average and peak power handling capabilities of these filters have been also analyzed from a multiphysics point of view. Measured results validate the theoretical predictions confirming that combline SIW filters can handle significant levels of continuous and peak power, providing at the same time easy integration, compact size and advanced filtering responses. Index Terms—Cross-coupled filters, electric and magnetic mixed couplings, multiphysics analysis, power handling capabilities, quasi-elliptic filter, substrate integrated waveguide.

I. INTRODUCTION

S

UBSTRATE INTEGRATED WAVEGUIDE (SIW) has already demonstrated to be a successful approach for implementing microwave and mm-wave filters with high Q-factor, easy integration with planar circuits, and mass production manufacturing processes in PCB and LTCC technology [1]. HowManuscript received July 03, 2015; revised October 02, 2015; accepted October 11, 2015. Date of publication November 12, 2015; date of current version December 02, 2015. This work was supported in part by MINECO (Spanish Government) under projects TEC2013-47037-C5-1-R and TEC2013-48036-C3-3-R. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. S. Sirci, M. A. Sánchez-Soriano, and V. E. Boria are with the iTEAM, Universitat Politècnica de València, Camino de Vera s/n, E-46022 Valencia, Spain (e-mail: [email protected]; [email protected]; [email protected]. es). J. D. Mart´ınez is with the I3M, Universitat Politècnica de València, E-46022 Valencia, Spain (e-mail: [email protected]). F. Gentili and W. Bösch are with the IHF, Graz University of Technology, 8010 Graz, Austria (e-mail: [email protected], [email protected]). R. Sorrentino is with the University of Perugia, 06100 Perugia, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495287

ever, other potential features that, combined with the former advantages, could be of huge interest in a wide range of wireless and mobile applications are a lively subject of research, like compactness, advanced filtering responses, and recently power handling capabilities. The realization of compact SIW filters has been approached from different perspectives. Several authors have proposed more compact alternatives to conventional SIW cavities by bisecting the resonator at quasi-perfect magnetic walls. Among the most relevant techniques that can be identified in the literature, there are folded [2], half-mode [3] and quarter-mode [4] SIW bandpass filters. Other approaches have focused on loading the SIW resonator with complementary split-ring resonators [5], dielectric rods [6] and more recently combline SIW filters, which were proposed as a translation of the well-known 3D coaxial resonator concept to a substrate integrated scheme [7]. Even with some trade-offs in terms of Q-factor or manufacturing complexity, most of the former approaches can obtain significant size reduction, but keeping fabrication and integration easiness (i.e., preferably single-layer batch-fabrication processes with solid bottom ground planes) that are usually of major importance from a practical point of view. At the same time, filtering functions including transmission zeros (TZs) at finite or imaginary frequencies are of great interest in many applications, enabling to achieve higher selectivity with a reduced footprint. In this sense, the use of crosscouplings between resonators is a well-known and extended technique for the introduction of TZs, based on the generation of multiple paths between the filter input and output, and therefore allowing for signal cancellation [8], [9]. Even if positive and negative couplings are generally required, magnetic coupling using irises between adjacent resonators has been the preeminent scheme, and the use of all inductive couplings has been already demonstrated for implementing linear-phase [10] and trisection [11] band-pass SIW filters. On the other hand, electric coupling mechanisms in SIW structures have been investigated by several authors. In [12] and [13], a negative coupling structure between SIW cavity resonators is proposed using a balanced microstrip line with a pair of plated via holes. However, in both structures, slots need to be etched at the top and bottom layers of the substrate, limiting the integration and packaging possibilities of the devices. In [14], a controllable mixed coupling is created using an embedded short-ended stripline com-

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bined with a wall iris at the expense of a multi-layer fabrication process that increases the complexity of the structure. On the other hand, grounded coplanar lines [15], microstrip lines [16] and oversized cavities [17] have been proposed as single-layer solutions, although some of the former structures are limited in terms of flexibility (i.e., coupling level or resonator positioning) or they require a larger area. Following this approach, the authors have recently proposed a solution for obtaining electric coupling in combline SIW resonators using a single-layer open-ended coplanar probe [18]. Lastly, even if higher power capacity is usually accepted as an advantage of SIW structures, power handling capabilities of SIW filters has not been broadly studied yet from a multiphysics perspective considering electrical, thermal and mechanical effects at the same time. In this work, the electric coupling structure proposed in [18] is studied in detail both for implementing direct and cross-couplings in combline SIW filters. As it is shown in the paper, this electric coupling scheme can be used to allow a further reduction of the filter size and increased bandwidth compared to conventional magnetic irises. Moreover, a fully packaged quasi-elliptic filter is designed, manufactured and measured. Finally, average and peak power handling capabilities of the device are theoretically studied and experimentally validated. The obtained results show that combline SIW filters can be a self-packaged, compact solution capable of incorporating both positive and negative couplings while keeping low-cost and batch-fabrication processes. The paper is structured as follows. In Section II, the negative coupling structure is presented and investigated. Section III provides the design, manufacturing and measurement of in-line three-pole combline SIW filters using the proposed negative coupling scheme. A comparison between magnetic and electric inter-resonator coupling is carried out, showing how the latter can be used to obtain more compact implementations while keeping the filter bandwidth. In Section IV, a fully packaged quasi-elliptic filter using a negative cross-coupling between non-adjacent resonators is presented, while the power handling capabilities of the filter are studied in Section V using a multiphysics approach to obtain the average and peak power capacities of the device. Finally, Section V presents the main conclusions. II. NEGATIVE COUPLING STRUCTURE A. Coaxial SIW Resonator The building block resonator to be considered for the design of the in-line and quasi-elliptic bandpass filters is a combline resonator implemented in SIW technology [7]. The inductive section is obtained by a metallic post, implemented by a metallized via-hole, connected to ground at one of its ends. At the other side, a metallic disk having a radius much higher than the post diameter is connected. Between this disk (which can have a square or a circular shape) and the top ground plane a small air gap is inserted, so generating a high capacitance towards ground (termed loading capacitance ), which represents the capacitive section of the coaxial resonator. Fig. 1 shows the layout of a coaxial SIW resonator, including its main design parameters.

Fig. 1. (a) 3D view and (b) top view of the coaxial SIW resonator including its main design parameters.

Such resonator can be modelled as a TEM-mode combline resonator embedded into the dielectric substrate. The TEM-mode resonant frequency is given by the condition , where the susceptance of the coaxial SIW resonator can be expressed as (1) where is the propagation constant of the TEM-mode and the thickness of the substrate. Then, the characteristic impedance of the coaxial resonator, and the ratio between the outer cavity side and the via diameter are obtained from [7] (2) (3) where is the resonator electrical length at the design frequency , that in our case is . The synthesis procedure starts by choosing the resonator slope parameter at the center angular frequency of the filter. The susceptance slope parameter for resonators having zero susceptance at is defined by [19] (4) The slope parameter has to be conveniently chosen as a trade-off between quality factor , compactness and physical feasibility of synthesized values of and . In order to increase component compactness, it is very important to set the inner via hole diameter to the minimum diameter allowed

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Fig. 2. Topology of the proposed electric coupling for coaxial SIW resonators.

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for Fig. 3. Coupling coefficient variation versus iris width mm, mm and mm. As the iris width increases, the total coupling decreases until it changes from electric to magnetic coupling and then increases again. In this figure, mm.

for the fabrication technology, as well as to keep as small as possible. This allows us to reduce radiation from apertures, thus helping to preserve a reasonable value for the resonator -factor. B. Negative Inter-Resonator Coupling The proposed solution to realize an electrical coupling between coaxial SIW resonators is based on a capacitive probe implemented by a high impedance coplanar line (CPW). The two ends of such a line penetrates into the head of the resonator capacitive patches, i.e., in the region with the highest intensity of electric field. A gap between the capacitive circular patch of the resonators and the probe is ensured. Part of the field is then collected by the probe and transferred to the adjacent resonator, so realizing an electric coupling. In absence of the probe described above, the main source of coupling would be due to the magnetic field inside the SIW cavities. For the considered structure, the sign of the magnetic coupling is opposite to the electrical one. Therefore in order to boost the effect of the probe and get higher values of coupling, the magnetic coupling must be minimized. Such condition can be achieved with a wall of via holes placed across the resonators, which strongly confine the magnetic field coupling. The probe can still be realized if the width of the CPW line is sufficiently smaller than the gap between the central via-holes realizing the wall, as shown in Fig. 2. Fig. 3 shows the total coupling vs. (obtained from the contribution of both electric and magnetic couplings), where is a generic -th resonator. Such trace was derived with a constant value for the insertion (i.e., mm) of the probe inside the head of the resonator. For increasing values of , the wall becomes more similar to a coupling iris with reference to classical waveguide structures. The curve shows that as the iris opens up, the total coupling starts to decrease until a certain point after which it starts to rise again. This effect can be explained by considering that when is minimum, the iris is completely closed and we are in the condition of minimum magnetic coupling between the two resonators. Therefore, the major contribution to the coupling is given by the probe (i.e., electric coupling). When starts to increase, the magnetic coupling between the two resonators is not negligible anymore and it starts to counteract the effect of the elec-

Fig. 4. Coupling coefficient variation versus CPW probe dimensions: conductor width , spacing and insertion on capacitive disks. Red line is the mm, mm and mm). The simulation baseline (i.e., mm. post-wall iris width is

trical coupling, until they reach the same magnitude and the total coupling collapses to zero . From this point on, if is increased the magnetic coupling starts to prevail, and the effect of the electric probe is not visible anymore. This analysis demonstrates that, in order to obtain electrical coupling between the two resonators, it is not sufficient to implement the probe described above, but it is also necessary to minimize the magnetic coupling by means of a post-wall iris. The insertion of the probe inside the head of the resonators is the main parameter to control the magnitude of the coupling . The behaviour of vs. for the structure of Fig. 2 is shown in Fig. 4, which considers two resonators implemented on a substrate with and thickness mm. As expected, becomes stronger as the insertion within the head of the resonator is increased. Clearly, also depends on and . The behaviour of vs. and is also reported in Fig. 4. Nevertheless, the contribution of these parameters has a lower impact on the coupling magnitude. Thus, we can settle these values, and controlling the coupling by means of . A main advantage of the proposed electric coupling is to allow increasing direct inter-resonator coupling, especially for strongly loaded coaxial SIW resonators. As a way of comparison, the coupling strength capabilities of the magnetic and electric coupling schemes (Figs. 5 and 6) are compared. First, in

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Fig. 5. (a) Coupling coefficient variation versus the relation between and the SIW cavity side . (b) Inter-resonator the post-wall iris width coupling system based on post-wall iris, showing design parameters.

Fig. 5, the coupling provided by a post-wall iris is depicted for two different values of slope parameter. As it is shown, by increasing the slope parameter of such structures, the magnetic coupling is strongly reduced between coaxial SIW resonators. That variation of , from 0.029 to 0.034, corresponds to diminish the SIW cavity side from 15 mm to 11.6 mm while of the capacitive disks increases from 0.825 pF to 0.95 pF. Both coaxial SIW resonators resonate in all cases at the same frequency, that is 5.5 GHz. Now, by applying the proposed electric coupling for the same pair of coaxial resonators, it is possible to see how higher coupling values are easily implementable in more compact structures. Fig. 6 depicts the inter-resonator coupling values that can be obtained using the proposed electric CPW probe, and below the scheme of that solution is shown with its main design parameters. It is worth mentioning that the insertion of the coupling of the capacitive patches, thus probes modifies the value of their sizes have to be slightly modified to meet specifications in terms of frequency. The proposed coupling scheme presents different advantages: single layer implementation, accurate control of the coupling level and it is well suited for tunable cross coupling using tuning elements, which can be easily mounted on the top metal layer.

III. THREE-POLE IN-LINE SIW FILTER WITH ELECTRIC COUPLING As it has been demonstrated in the previous Section II, the proposed electric coupling is an efficient approach for ensuring higher couplings between coaxial SIW resonators, and this is

Fig. 6. (a) Coupling coefficient variation versus electric coupling probe mm and mm. (b) Interinsertion . Other parameters are: resonator coupling system created at the top metal layer of the SIW resonators, showing design parameters.

especially true when the resonator compactness has to be increased. The CPW probe used to obtain an electric coupling between coaxial SIW resonators can generate high values of coupling if the insertion of the probe inside the capacitive disks is opportunely chosen, independently of the SIW cavity size. Indeed, if the building block resonator is designed with a higher slope parameter , the SIW cavity dimensions can be strongly reduced to compensate for the needed higher capacitive contribution of while maintaining the same resonant frequency, as deduced from (1)–(4). This circuit area reduction also leads to shift up the first spurious mode of the SIW cavity, widening the stopband bandwidth, so that, an improvement in terms of size and filter response can be really achieved. In order to validate the aforementioned statements three third-order coaxial SIW filters having center frequency 5.5 GHz, passband bandwidth of 250 MHZ (that corresponds to 4.6%) and return losses better than 15 dB have been designed. The main difference between those filters shown in Fig. 7 is the coupling configuration used to couple resonators: filter in Fig. 7(a) presents magnetic couplings based on post-wall iris, meanwhile filters in (b) and (c) are based on the proposed electric coupling. In particular, the filter shown in Fig. 7(c) is designed to present a higher resonator slope parameter (which leads to a small size cavity) in order to demonstrate the improved design characteristics added by using the electric coupling. It should be noted that this size reduction is incompatible with the use of the magnetic coupling scheme, since a smaller cavity leads to lower coupling values limiting the maximum achievable passband bandwidth. In this context, the electric coupling provides a higher design flexibility in order to fulfil requirements in terms of bandwidth, size and rejection

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Fig. 9. Comparison of three simulated responses with same in-band characteristics but different stop-band behaviour.

Fig. 7. Structure of the (a) magnetic-coupled, (b) electric-coupled and (c) compact electric-coupled coaxial SIW filters and main design dimensions.

Fig. 8. Multipath coupling diagram for the three-pole in-line coaxial SIW filter based on (a) magnetic (i.e., filter of Fig. 7(a)) and (b) electric coupling system (i.e., filters of Figs. 7(b)–(c)).

band. Fig. 8 shows the coupling schemes of such three-pole coaxial SIW filters with magnetic and electric couplings. The chosen dielectric substrate in these designs is now the 1.524 mm-thick Rogers R4003 with permittivity . So, the resonator electrical length corresponds to 18.9 , i.e., 0.05 , where is the guided wavelength at 5.5 GHz. In the first two filters, the coaxial SIW resonators have been designed to present S that gives a loading capacitance of fF while . By using both values and choosing a diameter of mm for the inner via hole, the SIW resonator cavity size is mm (i.e., and ) whilst the square patch side is 4.2 mm with an air gap of 0.24 mm between the patch and the ground plane. The final structure of those coaxial SIW filters in Figs. 7(a) and (b) have a footprint of 15 45 mm , i.e., and . When the coaxial SIW resonator has S, and become now 950 fF and 89.4 , respectively. The resonator cavity size consequently diminishes down to mm , i.e., and ,

having a square disk side of 4.8 mm and an air gap of 0.24 mm. Finally, the compact electric coupling based filter shows a footprint of 11.6 34.8 mm , i.e., and . The value of inter-resonator coupling for all filters are while the external coupling coefficient is , as it is shown in Fig. 8. Fig. 9 shows the simulation results of the three aforementioned filters. As it can be seen, the filter frequency responses above the passband show a higher order spurious band, which is due to the excitation of the mode of the SIW cavity. The best stopband performance is obtained by the filter of Fig. 7(c), as expected, with a wide stopband of up to 8.5 GHz with more than 25 dB of rejection. In particular, this filter presents the same level of insertion losses ( dB at 5.5 GHz for all filters) with a 40% of total size reduction compared to the other two coaxial SIW filters of Figs. 7(a)–(b) and a 70% of area reduction with respect to a standard -mode SIW filter centered at 5.5 GHz. It should be noted that this passband bandwidth along with such a compact circuit area could not be implemented by using the magnetic coupling based on post-wall iris. The reason is that, as shown in Fig. 5, the maximum coupling values between two coaxial SIW resonators with slope parameters S coupled by a magnetic coupling scheme results less than 0.03, which is below the values needed for the third-order filter considered (see Fig. 8). Table I gives the relation between unloaded quality factor and miniaturization degree of a coaxial SIW resonator implemented in 1.524 mm-thick Rogers R4003, when its slope parameter is increased. Those values have been compared to the area and of a standard SIW cavity resonator centered at 5.5 GHz. As Table I shows, the coaxial SIW topology allows high miniaturization of SIW structures with moderate degradation of . A photograph of the filter prototypes of Fig. 7(b) and (c) is shown in Fig. 10, while the simulated and measured responses of these filters are shown in Figs. 11 and 12, respectively. Measurements for both designs have validated the proposed concept, showing good insertion losses (i.e., 1.57 dB and 1.97 dB at , respectively), and return losses better than 11 dB for both filters. The measured response of filter prototype of Fig. 7(b) undergoes a frequency shift of 1.2% towards lower frequency, from

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TABLE I VERSUS MINIATURIZATION FOR SIW CAVITY RESONATORS IMPLEMENTED IN ROGERS R4003 RESONATING AT 5.5 GHZ (UNIT: MM)

Fig. 13. Multipath coupling diagram for the cross-coupled filter.

response, which is not possible for an iris-based magnetic coupling. IV. FOUR-POLE CROSS-COUPLED SIW FILTER A. Filter Design

Fig. 10. Photography of the proposed three-pole in-line coaxial SIW filters S, and (b) S. based on electric coupling, where (a)

Fig. 11. Simulated (dashed line) and measured (solid line) wideband responses S (Fig. 7(b)). of the proposed three-pole in-line SIW filter with

Fig. 12. Simulated (dashed line) and measured (solid line) wideband responses of the proposed three-pole in-line coaxial SIW filter with S (Fig. 7(c)).

5.5 GHz to 5.43 GHz (see Fig. 11). For the compact three-pole filter of Fig. 7(c), the measured response has been shifted up to 5.9 GHz, as it can be observed in Fig. 12. These frequency deviations have been caused by an increase of the square patch air gap during fabrication process. The simulation results have taken into account such a variation in the air gap. Among others, the advantage of using such coupling scheme is that tuning devices can be opportunely used to create a reconfigurable filter

In order to further demonstrate the proposed coupling solution, its application on a 4-pole narrow-band filter with a quasielliptic frequency response is now considered. The coupling scheme of the filter is given in Fig. 13. The coupling between resonators 1 and 4, termed , has opposite sign compared to all the inter-resonator couplings. This configuration is particularly interesting because it improves the selectivity generating TZs located both above and below the filter passband. The filter center frequency is chosen to be 5.75 GHz with an equi-ripple fractional bandwidth FBW of 2% (114 MHZ), which results in a simulated 1-dB bandwidth of 94 MHZ % . The TZs are set at 5.63 and 5.87 GHz, respectively, corresponding to the center frequencies of adjacent filters in a multiplexer application with contiguous channels. In this design, the slope parameter of the combline SIW resonator is chosen to be S, while the substrate presents and thickness mm. As it was previously mentioned, the resonator electrical length corresponding to substrate thickness is 46.5 , i.e., , where is the guided wavelength at 5.75 GHz. This gives a loading capacitance of fF while the characteristic coaxial impedance is . By taking both previous values and choosing a diameter of mm for the metallic via hole implementing the coaxial topology, the resonator structure can be optimized with EM full-wave simulations, by adjusting the size of the loading capacitive disk and the gap that separates it from the top metal layer. The resonators size is mm (i.e., ) whilst the capacitive disk radius is 0.915 mm with an air gap of 0.15 mm between the disk and the ground plane. The final structure of the filter presents a very compact footprint of 21.2 21.2 mm , which corresponds to 63% of area reduction compared to a quasi-elliptic four-pole filter based on -mode SIW resonators and implemented in the same dielectric substrate. The input-output coupling is realized by means of coplanar waveguide-to-SIW transition with 90 bend slots, which are etched from the metal on top of the first (last) cavity, see Fig. 14. The value of the can be easily controlled by modifying the dimensions of the probe. The coupling coefficients are and . Fig. 14 shows the geometric configuration of the 4-pole crosscoupled filter, including the proposed electric cross-coupling and some main filter dimensions. The coupling between the SIW

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TABLE II LAYOUT DIMENSIONS OF THE DESIGNED FILTER (UNIT: MM)

Fig. 14. Layout of the cross-coupled SIW filter assembled on a carrier substrate. Open-ended CPW line provides the electric coupling.

resonators 1 and 4 and the input/output ports are realized by means of CPW current probes that are etched on the top metal layer. The decreases with increasing size of ground plane openings that are created at the end of the CPW line. The filter dimensions are given in Table II. From simulations, the minimum insertion loss is 2.2 dB and the two TZs are located at 5.63 and 5.87 GHz providing 35 dB and 27 dB of minimum rejection, respectively.

Fig. 15. Side views of the vertical CPW-CPW transition that allows SMD assembly of the packaged filter on a carrier substrate.

B. Vertical Transition In Fig. 14, below the SIW filter substrate, it can be seen a host board with two isolated coplanar lines which were implemented on a 1.524 mm-thick Rogers R4003. In fact, the considered four-pole filter has been designed for allowing surface mounting device (SMD) assembly considering a PCB fabrication process. In order to enable SMD packaging of the filter, connection must be provided to the filter input and output. This kind of self-packaged solution for coaxial SIW filters, which provides access to the input/output ports through castellated plated via holes, was originally proposed in [20] showing promising results. This solution offers potential advantages in terms of design flexibility, low loss and enabling the SMD integration of the device. The proposed system is illustrated in Fig. 15, where a scheme of the vertical transition structure is shown. The bottom layer will interface between the carrier board and the packaged filter through a vertical pseudo-CPW structure. Since the CPW structure has both signal and ground on the same plane, the connection of the device to the host board is simple. In fact, the device can be easily placed and soldered onto the carrier substrate. The diameter and spacing of the transition half-a-hole vias are the most important design parameters for controlling the EM performance [20]. C. Experimental Results The designed filter has been fabricated on a 3.175 mm-thick Rogers TMM4 substrate using a standard single-side PCB process. The diameter and center-to-center pitch of the external via holes are 0.6 mm and 1.1 mm, respectively. A photograph of the filter prototype is shown in Fig. 16 while Fig. 17 shows the frequency responses of the cross-coupled SIW filter. It is worth mentioning that simulated and measured -parameters include the vertical transition for connecting input/output ports. The measured 1-dB bandwidth is 102.5 MHZ that corresponds to 1.8%, while the band-

Fig. 16. Photography of the filter prototype assembled on its substrate carrier.

Fig. 17. Simulated (dashed line) and measured (solid line) wideband responses of the proposed SIW filter.

width at dB corresponds to 200 MHZ. The S-parameters of the fabricated filter prototype were measured with an Agilent E8364B PNA series network analyzer, and a TRL calibration was performed using a home-made CPW cal-kit. The frequency shift of 70 MHZ % between simulations and results is due to the fabrication tolerances which, however, can be corrected in the filter design process. The minimum

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COMPARISON

OF

TABLE III CROSS-COUPLED FILTERS

IN

SIW TECH.

IL has increased up to 3.6 dB due to fabrication issues of the vertical transitions, resulting in an estimated Q-factor of 225, being 320 the simulated resonator . The measured return losses are better than 25 dB. The upper stop-band is in very good agreement with simulations up to 12 GHz, being the attenuation always better than 30 dB, as shown in Fig. 17. Table III shows the comparison between this work and other cross-coupled SIW filters shown in the references, where it can be observed the high compactness degree obtained with the proposed solution.

Fig. 18. 3D Electric Field distribution for a SIW Combline cavity.

From the Newton's Law of cooling, the generated heat is equal to the total heat delivered to the environment by means of convection and/or thermal radiation. Thus, this energy balance can be written as

V. MULTIPHYSICS STUDY FOR THE POWER HANDLING CAPABILITY EVALUATION OF SIW COMBLINE FILTERS In this section the power handling capability (PHC) of the SIW combline resonator is studied in detail. Two different kind of studies are addressed: a) Average Power Handling Capability (APHC), where it is assumed that a CW signal is applied to the circuit and the electro-thermo-mechanical coupling is analyzed in order to determine the maximum temperature and thermal stress of the circuit as a function of the input signal power. And b) Peak Power Handling Capability (PPHC), in such a case pulsed signals are applied to the circuit and the corona discharges are examined as a function of the input signal power and pressure. As an example of analysis, the PHC study is focused on the quasi-elliptic filter presented in Section IV. A. Average Power Handling Capability For moderated CW signal powers (1–5 W) high temperatures can be achieved in planar circuits due to self-heating, which limits the APHC [22]–[24]. Planar circuits present three loss mechanisms: conductive, dielectric and radiation losses. They are linearly proportional to the input power, but only the two former loss mechanisms produce heat in the circuit, so that they can be defined as the internal heat sources of the structure. In particular, dielectric loss is treated as a volumetric heat source whereas conductive loss as a surface heat source. In pure planar technologies as microstrip, the gradient of temperature due to self-heating is computed by determining the conductor and dielectric losses, then, the heat flow distribution in the microstrip cross section is derived to finally obtain the temperature rise that defines the maximum working input power [22], [23], [25]. In SIW technology, due to its large aspect ratio , and the fact that conductive losses are equal both in top and bottom planes, the whole heat source can be assumed to be homogeneous in a differential region , consequently providing a homogeneous temperature in this volumetric region. This assumption is also favoured by the use of via holes conforming the SIW, which propagate the heat between the top and bottom layers.

(5) where (6) being the loss factor of the structure and the input signal power. is any external heat source (such as solar radiation), is the temperature in the surroundings, and are the surface layers exposed to convection and radiation boundary conditions, respectively, and and are the convection and radiation coefficients. The values and depend on the external conditions of each layer, i.e., if there is a heat sink attached, natural or forced convection, emissivity, etc. From (5), the maximum temperature can be computed for a given and if the environment conditions ( and ) are known. An expected conclusion from this equation is that, if is kept, the bigger the surface , the lower the is, and therefore, the higher the APHC. For the SIW combline resonators, the electric-magnetic field distribution which defines the heat source pattern is different to that presented by standard SIW rectangular cavities, as seen in Fig. 18. However, the large aspect ratio as well as the use of the outer via holes and the inner conductor, make the temperature in the SIW combline cavity nearly homogeneous, and therefore, (5) can be used as a first order approximation. For the APHC analysis of the filter, firstly, the power loss per resonator is computed from the equivalent lumped element circuit whose behaviour is modeled by the coupling matrix shown in Fig. 13. GHz and % are used for such a computation. As seen from Fig. 19, the total highest power loss is found at the inband corners, at 5.68 and 5.82 GHz, with . In particular, resonator #2 presents the highest level of losses, so that it is expected that this resonator limits the APHC. Equation (5) can be used in order to calculate the average temperature in the whole circuit. For W at 5.82 GHz, assuming natural convection with

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TABLE IV THERMO-MECHANICAL PROPERTIES OF TMM 4

Fig. 19. Power loss in each resonator per watt from the equivalent lumped el. ement circuit model.

Fig. 21. Maximum equivalent Von-Mises Stress as a function of GHz and assuming natural convection on the circuit with

Fig. 20. Thermal profile of the filter presented in Section IV. 5.82 GHz with natural convection around the circuit with C. and

W/m

C and

W/m

W at C

C, with a whole circuit area of mm and neglecting thermal radiation, the average temperature in the circuit is found to be 176 C. ANSYS Multiphysics is used in order to accurately obtain the thermal profile for such an example. Fig. 20 shows its thermal profile where the hottest spot is found in resonator #2, as predicted, with a value of 205 C. The simulated average temperature in the circuit is very close to the computed value according to (5). The APHC is limited by that which creates such a thermal gradient leading to exceed the glass transition temperature of the substrate or, that which generates a thermal stress able to destroy (or deform) the circuit. The minimum of those values limits the APHC. For the electro-thermo-mechanical coupling needed to evaluate the thermal stress, ANSYS Multiphysics is used again. Table IV summarizes the thermo-mechanical parameters required for the study. It should be remarked that the thermal conductivity in the substrate both in SIW and SIW coaxial cavities is not a critical parameter for the reduction of the gradient of temperature as it is in microstrip technology. The yield strength point gives the maximum stress that a material can afford before permanent deformation. It defines the transition between the linear mechanical behaviour (elastic behaviour) of the material and the non-linear one. Fig. 21 shows the maximum equivalent stress (Von-Mises) as a function of at the frequency where losses are the highest

. At 5.82 C.

(5.82 GHz) and for two different load cases (bottom of the circuit is fixed or unfixed). Obviously, when the bottom is fixed the stress produced is higher and the elastic limit is reached for lower power levels. The maximum stress happens in the metal, both in layers and the via holes, whereas the stress provoked in the substrate is much lower. From these figures, it can be concluded that W in order to keep the circuit in safety ranges of applied power. It should be noted that for this circuit, due to the high value of , APHC is limited by thermal stress. With respect to the deformation produced in the circuit while the CW signal is applied, in the case of the bottom is fixed, deformation is insignificant (maximum deformation m for W). In the case when the bottom is unfixed, although the maximum deformation is still low, it can produce a small frequency shift in the transfer function for high (maximum deformation m for W). B. Peak Power Handling Capability The SIW combline resonator presents an important density of electric field through the air, between the loading capacitive disk and the top layer ground. Thus, this resonator can be susceptible to present corona discharges for high power applied signals. The situation is different for standard SIW technology, since the electric field is mainly distributed through the dielectric and the critical areas for air ionization are the slot edges of the coplanar line feeding the SIW cavities [1]. In order to evaluate any possible corona discharge, the maximum voltage (and consequently, the expected maximum electric field) of the SIW combline filter should be found as a function of the input signal power. From the equivalent lumped element circuit, the voltage and current in each resonator of the filter can be computed as (7) (8)

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threshold (peak value) can be determined by following the rule [31] (13)

Fig. 22. Scheme of the SIW combline resonator where the current and voltage distributions are shown at resonance frequency.

where is the generator voltage, the normalized external quality factor, is the reference impedance and is the normalized impedance/admittance matrix defining the filter network [19]. Once the equivalent voltages/currents are known, the stored energy can be found as (9) where is the available power at the input port and BW is the absolute bandwidth of the filter in rad/s. At resonance, the stored energy by every resonator of the equivalent lumped element network should be the same as that associated to the distributed resonators [26], [27]. Fig. 22 shows the scheme of a combline resonator where the voltage and current standing waves are also plotted at resonance. From the standing waves the stored energy by the distributed resonator can be calculated as [28], [29] by integrating the standing waves (in form of sinusoidal functions) along the resonator. Thus, by making the peak voltage in the SIW combline resonator can be found at the capacitive patch as (10) where (11) It should be noted that for the case is small, the SIW combline resonator behaves as a lumped element resonator, and therefore, the values found with (10) are the same as those obtained with (7). The air ionization is a phenomenon linked to the electric field strength rather than to the voltage [27], [30]. So, the electric field strength must be estimated in the capacitive patch of the resonators forming the filter. Although the electric field distribution along the air gap is not evident, as a fast approximation the maximum electric field in each resonator can be computed as (12) where is the annular gap of the resonator . For high pressure regime (pressures mbar), the air ionization breakdown

where is the pressure in torr and is the operation frequency in GHz. At this point, from (12) and (13) the PPHC can be analytically computed in filters based on combline resonators for high pressure regime. For low pressure regime, the continuity equation describing the electron density evolution must be solved. This arduous task must be done numerically, in this work the software tool AURORASAT™ SPARK3D® is employed. This tool uses the real electromagnetic field distribution of the device under test in order to solve the continuity equation, and provides the power breakdown threshold of the device from some input parameters such as pressure, kind of gas (air or nitrogen) or temperature. As a validation example, the PPHC is computed for the quasi-elliptic filter by using the equations derived in the previous lines and by using SPARK3D. The input parameters are GHz (where the voltage is maximum, which happens at the capacitive patch of resonator #2; this frequency coincides with the frequency of maximum losses) and mbar. From the derived equations, PPHC is analytically calculated as 7.3 W whereas from SPARK3D is 11.2 W. If the maximum electric field strength of the structure is simply taken from an electromagnetic tool (such as HFSS or CST; note that a normalization could be needed depending on the software used), and by applying the rule (13), the obtained PPHC is 0.9 W, which is a value considerably lower than those previously obtained. From these results, it can be concluded that taking the maximum electric field from the EM simulation results, which is indeed a strategy commonly used for waveguides or coaxial resonators, gives a very conservative limit for PPHC in SIW combline resonators. This is due to the fact that the maximum electric field values are very concentrated around the capacitive patch edges, in a region involving just some microns, which is not enough to alter the electron density. Additionally, some field singularities could appear in the resolution of the field around such corner edges. Another conclusion is that the approximation made in (12) gives a reasonable value of electric field strength in order to determine the PPHC. Fig. 23 shows the Paschen curves of the filter for each resonator, obtained from SPARK3D at the frequency where losses and voltage magnification are maximum. Resonator #2 is limiting the PPHC, as expected. The critical pressure is around 5–10 mbar, where PPHC is just 0.6 W. So, it is presumed that in a low pressure regime corona limits PHC rather than the thermo-mechanical effects. An interesting effect has been also detected in this filtering structure: due to the sharp shape of the first (and fourth) resonator corners (see Fig. 24), it is found that there is a high concentration of electric field density around those corners, providing higher values of electric field strength than those of resonator #2. Thus, resonator #1 may limit the PPHC of the filter even though the voltage in this resonator is lower than that in resonator #2 at the frequency of analysis. However, after fabrication, those corners are rounded, as seen in Fig. 24(b), leading to a reduction of the electric field density around them. Fig. 24(c) shows the simulated Paschen Curves for

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Fig. 24. Zoom in of the resonator #1. (a) Layout. (b) Fabricated. (c) Simulated Paschen Curves comparison between the sharp and rounded corner cases.

Fig. 23. Simulated Paschen Curves for the resonators involved in the filter under test. (a) Critical pressure region. (b) The whole pressure range. In this GHz and C. simulation,

the first resonator when the corners are sharped and rounded. As deduced from this figure, rounding the corners can considerably increase PPHC in this filter topology. In Fig. 23 the Paschen Curve of resonator #1 corresponds to the rounded corner case. Obviously, for both cases the filter response remains constant. C. APHC and PPHC Measurements In order to validate the calculated and simulated results, high power measurements have been performed at the European High-Power RF Space Laboratory (Valencia, Spain). Fig. 25 shows a schematic diagram of the employed test-bed. Several methods have been used for the corona discharge detection: third harmonic detection, nulling of the forward/reverse power at the operation frequency, electron probe and by visual inspection by recording the circuit with a video camera. The applied signal to the filter has been a pulsed signal with a carrier frequency of 5.75 GHz (frequency where the voltage magnification and losses are maximum in the measured filter), a pulse width of 20 s and a duty cycle of 1%. These pulsed signal characteristics avoid any self-heating effect in the device, whereas the pulse width is wide enough to assume that the pulse breakdown threshold converges to the CW one. The PPHC has been evaluated for different pressures in order to obtain the Paschen Curves of the device. In Fig. 23 the measured data can be also observed. For all scenarios corona breakdown has firstly appeared in resonator #2, as expected. There is a reasonable good agreement (especially at

lower pressures) with the simulated results what validates the theoretical prediction and the study done. The maximum difference between the measured and simulated breakdown field levels has been of 45%, which has been found for a pressure of 600 mbar. Fig. 26 shows the capture from the video camera at the moment of a corona discharge has occurred. As seen, the spark is uniformly originated along the annular slot. With respect to the APHC measurement, a CW signal at the same frequency has been applied to the circuit at ambient pressure ( mbar) and temperature. The temperature has been measured by means of thermocouples attached to the resonators in regions where their impact is low. The circuit was suspended so that natural convection can be assumed on all faces. The thermal steady-state was reached after around 5 minutes the signal is switched on. The measured temperatures for the different CW applied signal power samples have been close to the analytically calculated and simulated values, with differences lower than 20%. Up to 5 W there were no evidences of any rupture point in the circuit, however W should be avoided since temperatures higher than 150 C were measured. Furthermore, for W a frequency shift started being noticed in the measured response as well as a considerable increase of losses. After switching on/off the applied signal, the circuit could recover its original electromagnetic performance, although probably with a slight permanent material deformation. The CW applied power was increased up to obtain the complete failure of the device. The circuit was finally destroyed by a continuous corona discharge which occurred for W. This value is lower than the one measured at ambient pressure for the pulsed signal case and shown in Fig. 23. This can be due to the following reasons: the air temperature was much higher because of self-heating, which reduces the corona breakdown threshold for high pressure regime and, because corona breakdown thresholds are normally lower for the CW case than for

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Fig. 25. Scheme of the testbed configuration used for corona breakdown detection.

the power handling measurements. They also acknowledge Val Space Consortium for its contribution—Laboratories funded by the European Regional Development Fund—A way of making Europe. We would like also to thank the “Lab-STICC Computing Platform” at the University of Brest (France) for their support in order to perform the multiphysics simulations. REFERENCES Fig. 26. Capture of a corona discharge at the second resonator of the filter under test.

the pulsed signal case, unless a very wide pulsed signal width is used. VI. CONCLUSION In this paper the design of SIW combline resonator filters with advanced performances has been investigated. For this aim, different coupling configurations involving magnetic and electric coupling mechanisms have been proposed and studied in detail. It has been demonstrated that the proposed electric coupling configuration provides a high flexibility in filter design, which allows us to design very compact filters by keeping a reasonable level of losses, as well as to increase the maximum achievable passband bandwidth in SIW coaxial technology. Additionally, it has been proposed such a filter where the magnetic and electric coupling schemes has been arranged in order to design a very compact quasi-elliptic narrowband bandpass filter, presenting a very selective response with a wide rejection band. The power handling capability (both average and peak) has been also studied for such filtering structures from a multiphysics point of view. As it has been shown, these filters can afford moderate levels of power in spite of their small circuit area and narrow band. All concepts have been validated by the simulation and measurement of some fabricated proof-of-concept filtering devices. High power measurements have been also carried on, which have verified the proposed multiphysics study. ACKNOWLEDGMENT The authors would like to thank M. Reglero and M. Taroncher for their valuable help and discussions with the realization of

[1] X.-P. Chen and K. Wu, “Substrate integrated waveguide filters: Practical aspects and design considerations,” IEEE Microw. Mag., vol. 15, no. 7, pp. 75–83, Nov. 2014. [2] N. Grigoropoulos, B. Sanz-Izquiredo, and P. Young, “Substrate integrated folded waveguides (sifw) and filters,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 829–831, Dec. 2005. [3] Y. Wang, W. Hong, Y. Dong, B. Lui, H. Tang, J. Chen, X. Yin, and K. Wu, “Half mode substrate integrated waveguide (hmsiw) bandpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 4, pp. 265–267, Apr. 2007. [4] C. Jin and Z. Shen, “Compact triple-mode filter based on quarter-mode substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 62, no. 1, pp. 37–45, Jan. 2014. [5] Y. D. Dong, T. Yang, and T. Itoh, “Substrate integrated waveguide loaded by complementary split-ring resonators and its applications to miniaturized waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 9, pp. 2211–2223, Sep. 2009. [6] L.-S. Wu, L. Zhou, X.-L. Zhou, and W.-Y. Yin, “Bandpass filter using substrate integrated waveguide cavity loaded with dielectric rod,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 8, pp. 491–493, Aug. 2009. [7] J. D. Mart´ınez, S. Sirci, M. Taroncher, and V. E. Boria, “Compact CPW-Fed combline filter in substrate integrated waveguide technology,” IEEE Microw. Wireless Compon. Lett., vol. 22, no. 1, pp. 7–9, Jan. 2012. [8] R. Levy, “Filters with single transmission zeros at real or imaginary frequencies,” IEEE Trans. Microw. Theory Tech., vol. MTT-24, no. 4, pp. 172–181, Apr. 1976. [9] J. Thomas, “Cross-coupling in coaxial cavity filters-A tutorial overview,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1368–1376, Apr. 2003. [10] X. Chen, W. Hong, T. Cui, J. Chen, and K. Wu, “Substrate integrated waveguide (SIW) linear phase filter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 787–789, Nov. 2005. [11] J. D. Mart´ınez, S. Sirci, and V. E. Boria, “Compact substrate integrated waveguide filter with asymmetric frequency response for c-band wireless applications,” in Proc. IEEE MTT-S Int. Wireless Symp. (IWS), Beijing, China, Apr. 2013, pp. 1–4. [12] X. P. Chen and K. Wu, “Substrate integrated waveguide cross-coupled filter with negative coupling structure,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 142–149, Jan. 2008.

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[13] G. H. Lee, C. S. Yoo, J. G. Yook, and J. C. Kim, “Siw quasi-elliptic filter based on LTCC for 60-GHz application,” in Proc. 4th Eur. Microw. Integ. Circuits Conf., Sep. 2009, pp. 204–207. [14] K. Gong, W. Hong, Y. Zhang, P. Chen, and C. J. You, “Substrate integrated waveguide quasi-elliptic filters with controllable electric and magnetic mixed coupling,” IEEE Trans. Microw. Theory Tech., vol. 60, no. 10, pp. 3071–3078, Oct. 2012. [15] B. Potelon, J. Favennec, C. Quendo, E. Rius, C. Person, and J. Bohorquez, “Design of a substrate integrated waveguide (SIW) filter using a novel topology of coupling,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 9, pp. 596–598, Sep. 2008. [16] W. Shen, W. Y. Yin, X. W. Sun, and L. S. Wu, “Substrate-integrated waveguide bandpass filters with planar resonators for system-on-package,” IEEE Trans. Compon. Packag. Manufact. Technol., vol. 3, no. 2, pp. 253–261, Feb. 2013. [17] F. Zhu, W. Hong, J. X. Chen, and K. Wu, “Cross-coupled substrate integrated waveguide filters with improved stopband performance,” IEEE Microw. Wireless Compon. Lett., vol. 22, no. 12, pp. 633–635, Dec. 2012. [18] S. Sirci, F. Gentili, J. D. Mart´ınez, V. Boria, and R. Sorrentino, “Quasi-elliptic filer based on SIW combline resonators using a coplanar line cross-coupling,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, USA, May 2015, pp. 1–4. [19] J. S. Hong, Microstrip Filter for RF/Microwave Applications, 2nd ed. Hoboken, NJ, USA: Wiley, Feb. 2011. [20] S. Sirci, J. Mart´ınez, R. Stefanini, P. Blondy, and V. Boria, “Compact SMD packaged tunable filter based on substrate integrated coaxial resonators,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Tampa Bay, FL, USA, Jun. 2014, pp. 1–4. [21] J. Lee, E. J. Naglich, H. Sigmarsson, D. Peroulis, and W. J. Chappell, “Tunable inter-resonator coupling structure with positive and negative values and its application to the field-programmable filter array (FPFA),” IEEE Trans. Microw. Theory Tech., vol. 59, no. 12, pp. 3389–3400, Dec. 2011. [22] K. C. Gupta, R. Garg, I. J. Bahl, and P. Bhartia, Microstrip Lines and Slotlines, 2nd ed. Boston, MA, USA: Artech House, 1996. [23] M. Sanchez-Soriano, Y. Quere, V. Le Saux, C. Quendo, and S. Cadiou, “Average power handling capability of microstrip passive circuits considering metal housing and environment conditions,” IEEE Trans. Compon., Packag., Manufact. Technol., vol. 4, no. 10, pp. 1624–1633, Oct. 2014. [24] M. Sanchez-Soriano, M. Edwards, Y. Quere, D. Andersson, S. Cadiou, and C. Quendo, “Mutiphysics study of rf/microwave planar devices: Effect of the input signal power,” in Proc. 15th EuroSime, Apr. 2014, pp. 1–7. [25] I. Bahl and K. Gupta, “Average power-handling capability of microstrip lines,” IEE J. Microw. Opt. Acoust., vol. 3, no. 1, pp. 1–4, 1979, UK. [26] C. Ernst and V. Postoyalko, “Prediction of peak internal fields in directcoupled-cavity filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 64–73, Jan. 2003. [27] M. Yu, “Power-handling capability for rf filters,” IEEE Microw. Mag., vol. 8, no. 5, pp. 88–97, Oct. 2007. [28] R. E. Collin, Foundations for Microwave Engineering. Hoboken, NJ, USA: Wiley, 2007. [29] M. Sánchez-Soriano, E. Bronchalo, and G. Torregrosa-Penalva, “Parallel-coupled line filter design from an energetic coupling approach,” IET Microw., Antennas, Propag., vol. 5, no. 5, pp. 568–575, Apr. 2011. [30] R. J. Cameron, C. M. Kudsia, and R. R. Mansour, Microwave Filters for Communication Systems. Hoboken, NJ, USA: Wiley-Interscience, 2007. [31] W. Woo and J. DeGroot, “Microwave absorption and plasma heating due to microwave breakdown in the atmosphere,” Phys. Fluids, vol. 27, no. 2, pp. 475–487, 1984. Stefano Sirci (S'14) received the B.S. and M.S. degrees (with distinction) in electronic engineering from the University of Perugia, Perugia, Italy, in 2006 and 2009, respectively. In 2009, he received the M.S. degree in efficient modal computation in arbitrarily shaped waveguides by BI-RME Method with the Polytechnic University of Valencia, Valencia, Spain, where he is currently working toward the Ph.D. degree. From 2010 to 2015, he has been with the Microwave Application Group (GAM) at the Institute of Telecommunications and Multimedia Applications (iTEAM) at the Polytechnic

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University of Valencia. His working is focused on emerging technologies for reconfigurable microwave components with emphasis on designing, fabrication and measurement of tunable microwave SIW filters, in PCB and LTCC technologies.

Miguel Ángel Sánchez-Soriano (S'09–M'13) was born in Yecla (Murcia), Spain, in 1984. He received the Telecommunications Engineer degree (with a Special Award) and the Ph.D. degree in electrical engineering from the Miguel Hernandez University (UMH), Spain, in 2007 and 2012, respectively. In 2007 he joined the Radiofrequency Systems Group, UMH, as a research assistant. He was a Visiting Researcher with the Microwaves Group headed by Prof. Jia-Sheng Hong at Heriot–Watt University, Edinburgh, U.K., in 2010. In January 2013 he joined the LabSTICC group, Université de Bretagne Occidentale, Brest, France, as a Postdoctoral Researcher, where he worked for 2 years. Since January 2015, he is a “Juan de la Cierva” research fellow at the “Grupo de Aplicaciones de Microondas” (GAM), Polytechnic University of Valencia, Spain. His research interests cover the analysis and design of microwave planar devices, especially filters and their reconfigurability, and the multiphysics study of high frequency devices. Dr. Sánchez-Soriano was the recipient of the runner-up HISPASAT award to the Best Spanish Doctoral Thesis in New Applications for Satellite Communications, awarded by the Spanish Telecommunication Engineers Association (COIT/AEIT), and the Extraordinary Ph.D. award from the Miguel Hernndez University. He serves as a reviewer for various journals and conferences, including the IEEE TRANSACTIONS ON MICROWAVES, THEORY AND TECHNIQUES, the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS and the IET Microwaves, Antennas and Propagation.

Jorge D. Mart´ınez (M'09) was born in Murcia, Spain. He received the degree in telecommunication engineering and the Ph.D. degree from Polytechnic University of Valencia (UPV), Valencia, Spain, in 2002 and 2008, respectively. He is currently Associate Professor at the School of Telecommunication Engineering of the Polytechnic University of Valencia (UPV) since 2012. He joined the Department of Electronics Engineering of the UPV in 2002 as a Research Fellow, and became Assistant Professor in 2009. During 2007 he was a Research Visitor at XLIM, CNRS and University of Limoges, France, where he worked on the design and fabrication of RF MEMS components under the advice of Prof. Pierre Blondy. He is a Researcher of the I3M R&D institute at UPV, where he actively collaborates with the Microwave Applications Group (GAM). At I3M premises, he is now the Technical Responsible of the Laboratory for High Frequency Circuits Fabrication (LCAF) of the UPV, focused on Low Temperature Co-fired Ceramics (LTCC) and other related multi-layered technologies. His current research interests are focused on emerging technologies for reconfigurable microwave components with emphasis on tuneable filters and RF MEMS, and the design and fabrication of advanced microwave filters in planar and substrate integrated waveguide technologies, as well as the application of multi-layer fabrication technologies to RF/microwave and millimetre-wave applications.

Vicente E. Boria (S'91–A'99–SM'02) was born in Valencia, Spain. . He received the Ingeniero de Telecomunicacin degree (first-class honors) and the Doctor Ingeniero de Telecomunicacin degree from the Polytechnic University of Valencia, Valencia, Spain, in 1993 and 1997, respectively. In 1993 he joined the Departamento de Comunicaciones, Polytechnic University of Valencia, where he has been Full Professor since 2003. In 1995 and 1996, he was holding a Spanish Trainee position with the European Space Research and Technology Centre, European Space Agency (ESTEC-ESA), Noordwijk, The Netherlands, where he was involved in the area of EM analysis and design

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of passive waveguide devices. He has authored or co-authored 10 chapters in technical textbooks, 125 papers in refereed international technical journals, and over 175 papers in international conference proceedings. His current research interests are focused on the analysis and automated design of passive components, left-handed and periodic structures, as well as on the simulation and measurement of power effects in passive waveguide systems. Dr. Boria has been a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the IEEE Antennas and Propagation Society (IEEE AP-S) since 1992. He is member of the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, Proceeding of the IET (Microwaves, Antennas and Propagation), IET Electronics Letters and Radio Science. Since 2013, he serves as Associate Editor of IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He is also a member of the Technical Committees of the IEEE-MTT International Microwave Symposium and of the European Microwave Conference.

Fabrizio Gentili (GSM'12–A'14) was born in Foligno, Italy. He received the Master Laurea degree (with distinction) in electronic engineering and the Ph.D. degree from the University of Perugia, Perugia, Italy, in 2010 and 2014, respectively. His doctoral dissertation concerned innovative solutions for RF and microwave filters. In 2009, he carried out his Master thesis on photonic nanoantennas with the University of Bristol, Bristol, U.K. In March 2010, he joined RF Microtech, where his research concerned the design of antennas and passive microwave components. Since April 2014, he has been a Post-Doctoral Researcher with the Institute of Microwave and Photonic Engineering, Graz University of Technology, Graz, Austria.

Wolfgang Bösch (F’13) received the engineering degrees from the Technical University of Vienna, Vienna, Austria, and Technical University of Graz, Austria. He received the M.B.A. (with distinction) at Bradford University School of Management, U.K., in 2004. He joined the Graz University of Technology, Austria, in March 2010 to establish a new Institute for Microwave and Photonic Engineering. Previously he was the CTO of the Advanced Digital Institute in the U.K., a not-for-profit organization to promote research activities. He has also been the Director of Business and Technology Integration of RFMD UK. For almost 10 years he has been with Filtronic plc as CTO of Filtronic Integrated Products and Director of the Global Technology Group. Prior to joining Filtronic, he held positions in the European Space Agency (ESA) working on amplifier linearization techniques, MPR-Teltech in Canada working on MMIC technology projects and the Corporate R&D group of M/A-COM in Boston, MA, USA, where he worked on advanced topologies for high efficiency power amplifiers. For four years he

was with Daimler-Chrysler Aerospace in Germany, working on T/R Modules for airborne radar. He has published more than 80 papers and holds 4 patents. Dr. Bösch is a Fellow of the IET. He was a Non-Executive Director of Diamond Microwave Devices (DMD) and the Advanced Digital Institute (ADI). Currently he is a Non-Executive Director of VIPER-RF (U.K.).

Roberto Sorrentino (LF'90) is a Professor at University of Perugia, Perugia, Italy, where he was the Chairman of the Electronic Department, Director of the Computer Center (1990–1995), and Dean of the Faculty of Engineering (1995–2001). His research activities have been concerned with various technical subjects, but mainly with numerical methods and CAD techniques for passive microwave structures and the analysis and design of microwave and millimetre-wave circuits including filters and antennas. In recent years he has been involved in the modelling and design of Radio Frequency Microelectromechanical Systems (RF-MEMS) and their applications on tuneable and reconfigurable circuits and antennas. He is the author or co-author of more than 150 technical papers in international journals and 200 refereed conference papers. He has edited a book Numerical Methods for Passive Microwave Structures (IEEE Press, 1989) and co-authored four books: Advanced Modal Analysis (with M. Guglielmi and G. Conciauro) (Wiley, 2000); RF and Microwave Engineering (in Italian) (with G. Bianchi) (McGrawHill, 2006); Electronic Filter Simulation and Design (with G. Bianchi) (McGrawHill, 2007); RF and Microwave Engineering, (with G. Bianchi) (J. Wiley, 2010). Dr. Sorrentino 1990 he became a Fellow of the IEEE for contribution to the modelling of planar and quasi-planar microwave and millimetre-wave circuits. He has received several international awards and recognitions: in 1993 the IEEE MTT-S Meritorious Service Award, in 2000 the IEEE Third Millennium Medal, in 2004 the Distinguished Educator Award from IEEE MTT-S, in 2010 the Distinguished Service Award from the European Microwave Association, in 2012, together with S. Bastioli and C. Tomassoni, the Microwave prize for the paper “A New Class of Waveguide Dual-Mode Filters Using TM and Nonresonating Modes.” In 2015 he was awarded the Microwave Career Award from the IEEE MTT-Society. He has been active within the IEEE MTT Society. From 1984 through 1987 he was the Chairman of the IEEE Section of Central and South Italy and was the founder of the local MTT/AP Chapter that he chaired from 1984 to 1987. From Jan. 1995 through April 1998 he was the Editor-in-Chief of the IEEE MICROWAVE AND GUIDED WAVE LETTERS. From 1998 to 2005 he has served on the Administrative Committee of the IEEE Microwave Theory and Techniques Society. He was elected again in MTT AdCom for the term 2011–2013. He is also a member of Technical Committees MTT-15 on Field Theory and MTT-1 on Computer-Aided Design, which he chaired in 2003–04. He served the International Union of Radio Science (URSI) as Vice Chair (1993–1996) then Chair (1996–1999) of the Commission D (Electronics and Photonics). Since 2007 he is the President of the Italian Commission of URSI. In 2002 he was among the founders and first President of the Italian Electromagnetic Society (SIEm) that he chaired until 2008. From 1998 to 2005 he was a member of the High Technical Council of the Italian Ministry of Communications. In 1998 he was one of the founders of the European Microwave Association (EuMA) and was its President till 2009. In 2007, he founded RF Microtech, a spin-off company of the University if Perugia dealing with RF MEMS, microwave systems and antennas.

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Exact Design of a New Class of Generalized Chebyshev Low-Pass Filters Using Coupled Line/Stub Sections Evaristo Musonda, Graduate Student Member, IEEE, and Ian C. Hunter, Fellow, IEEE Abstract—A method for the design of a new class of distributed low-pass filters enables exact realization of the series short-circuited transmission lines, which are normally approximated via unit elements in other filter realizations. The filters are based upon basic sections using a pair of coupled lines, which are terminated at one end in open-circuited stubs. The approach enables realization of transmission zeros at the quarter-wave frequency, hence giving improved stopband performance. A complete design theory starting from a distributed generalized Chebyshev low-pass prototype filter is presented. A design example demonstrates excellent performance in good agreement with theory. Index Terms—Distributed low-pass filters, generalized Chebyshev, meander line, selectivity, TEM.

I. INTRODUCTION

L

OW-PASS filters are often needed in microwave systems to “clean up” spurious responses in the stopband of coaxial and dielectric resonator filters. The most important driving factors are compact size, sharp roll-off, and wide stopband. Some of the recent works have addressed some of these problems [1], [2]. Although it is relatively easy to obtain theoretical circuit models, the challenge in practical low-pass filters lies in achieving good approximation using real transmission-line components. There exist many realizations for low-pass filters. One popular type is the stepped impedance low-pass filter consisting of interconnections of commensurate lengths of transmission lines of alternating low and high impedance [3]. This type of filter has low selectivity for a given network order because the transmission zeros are all at infinity on the real axis in the complex plane. In order to increase selectivity, transmission zeros may be placed at finite frequencies using distributed generalized Chebyshev low-pass prototype filters [4]. The problem is that there is no direct realization of the series short-circuited stubs associated with this prototype filter. In the existing physical realization [5], the series short-circuited stubs are approximated by short lengths of the high-impedance transmission line (forcing those transmission zeros at a quarter-wave frequency to move Manuscript received June 24, 2015; revised September 15, 2015; accepted October 11, 2015. Date of publication October 29, 2015; date of current version December 02, 2015. This work was supported in part by the Beit Trust, in part by The University of Leeds, in part by RS Microwave Inc., in part by Radio Design Limited, and in part by The Royal Academy of Engineering. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. The authors are with the Institute of Microwave and Photonics, School of Electronic and Electrical Engineering, University of Leeds, LS2 9JT Leeds, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2492969

Fig. 1. Proposed layout of meander-like low-pass filter: (a) composed of a section of high-impedance parallel coupled lines short circuited by a low-impedance open-circuited stub at alternate ends and (b) graphical line equivalent circuit.

to infinity on the real axis), while the shunt series foster is realized exactly as an open-circuited stub of double unit length. The approximation involved results in relatively poor stopband rejection. As an expansion to the work described in [6], this paper presents two solutions in which the series short-circuited stubs are exactly realized within the equivalent circuit of the filter. In Section II, the synthesis techniques for the two physical realizations have been developed. In the previous paper, only the equivalent circuit for the second low-pass filter physical realization was known and the element values were obtained via optimization in a circuit simulator. In this revised and expanded paper, the synthesis is developed and presented together with the required canonical low-pass circuit forms and corresponding transmission zeros that the transfer functions may realize. The procedure for different low-pass filter degrees is included with the required circuit transformations, which was a significant piece of work. Design examples are included to illustrate the synthesis technique. II. DESIGN THEORY In Fig. 1, the general physical layout is given for the proposed method. The structure consists of a middle section of high-impedance coupled lines terminated at every alternate end in a low-impedance open-circuit stub forming a “meander-like” structure, as in Fig. 1. All the transmission lines are of commensurate length. Grounded decoupling walls

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Fig. 2. Derived equivalent circuit transformations (*may be a hanging node).

must be utilized to eliminate coupling between the open-circuited stubs. In this work it is shown how a general Chebyshev transfer function may be used to implement two alternative realizations

arising from Fig. 1 via a series of derived circuit transformations and one of the earlier transformation derived by Sato in [7, Table I]. The synthesis of distributed low-pass filter networks is based on work done in [8]. Fig. 2 shows the derived equiv-

MUSONDA AND HUNTER: EXACT DESIGN OF A NEW CLASS OF GENERALIZED CHEBYSHEV LOW-PASS FILTERS

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TABLE I 7 -DEGREE LOW-PASS FILTER SYNTHESIZED ADMITTANCE VALUES

Fig. 4. Graphical representation of the equivalent circuit of Fig. 3 after transformation of the 3 -degree basic sections.

Fig. 3. Generalized Chebyshev distributed low-pass prototype.

alent-circuit transformations and the required admittance relationships. The next two sections describe how these transformations were used to derive the equivalent circuit for the two possible physical realizations from their canonical low-pass filter derivatives.

Fig. 5. Basic section containing a pair of coupled line and a stub and its equivalent circuits.

A. Physical Realization I The first physical realization realizes the equivalent circuit for the general Chebyshev distributed network given in Fig. 3. By using the synthesis technique given in [8], the network of Fig. 3 may be synthesized directly in the distributed domain from an ( odd) degree Chebyshev transfer function with pairs of symmetrically located transmission zeros and a single transmission zero at a quarter-wave frequency . simply refers to an -deIn this work, gree low-pass filter with number of transmission zeros at some general frequency in the complex plane, number of transmission zeros at quarter-wave frequency (i.e., ), and number of real axis half transmission zero pairs at infinity (i.e., ). Where exists, these transmission zeros may either be symmetrically pure imaginary frequency pairs (i.e., ) or in general paraconjugated pairs on the complex plane (i.e., ) is always even. Therefore, in general, the dissuch that tributed network of Fig. 3 is of the form . Using circuit transformation I on each of the 3 -degree sections, Fig. 3 may be transformed into Fig. 4. It is then clear from Fig. 4 that each of the 3 -degree sections is just the equivalent circuit of a pair of two parallel coupled lines with one end terminated in an open circuited stub, as depicted in Fig. 5. The overall network after the transformation is illustrated in Fig. 6. This is equivalent to Fig. 1(a), but with every second coupling between the parallel coupled lines section removed [i.e., couplings , , removed in Fig. 1(b)], such that the structure is composed of cascaded 3 -degree basic sections of Fig. 5, as shown in Fig. 6.

Fig. 6. Physical layout for generalized Chebyshev distributed low-pass prototype filter of physical realization I. TABLE II 7 -DEGREE LOW-PASS FILTER IMPEDANCE VALUES TRANSFORMATION II (A)–(C) IN 50- SYSTEM

AFTER

1) Design Example: A 7 -degree low-pass filter was designed using the techniques described above with cutoff frequency at 1 GHz, 20-dB minimum passband return loss with pairs of finite transmission zeros at and (2.18, 1.72, and 2.18 GHz), and a single quarterand electrical length at wave transmission zero

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Fig. 8. Physical layout of the striplines for the general meander-like low-pass filter with allowed coupling between parallel lines of the adjacent basic sections.

Fig. 7. Circuit and HFSS simulation response for example I.

the cutoff frequency, . The element values are shown in Table I corresponding to the circuit of Fig. 3 where symmetry is assumed for the element values. Using circuit transformation I, the circuit was transformed to the final form of Fig. 4 with the element values shown in Table II. All the element values are clearly realizable. Fig. 7 shows the circuit simulation of the design example. In this realization I, is approaching practical limits for realizable element values. Reducing the electrical length at the cutoff frequency further causes the impedance values of the high- and low-impedance lines to become unrealizably high and low, respectively. However, in reality the practical stopband bandwidth is perturbed due to high-order modes and spurious couplings between the basic sections, as shown by the HFSS simulation in Fig. 7 with small resonance peaks around . The decoupling walls do not give exact circuit realization for realization I. In the second physical realization, however, some of the couplings are allowed between basic sections. B. Physical Realization II—“Meander-Like” Low-Pass Filter A more general realization is achieved by using the layout of Fig. 1. The only difference with the previous physical realization I is that, in the general case, physical realization II, all the couplings between high-impedance coupled lines of Fig. 1 are allowed. The structure is built from the basic 3 -degree section of Fig. 5 by adding a parallel line to the parallel coupled lines section and an open-circuited stub at one end to form an interconnect each time to increase the network degree by 2. The stripline layout for physical realization II is given in Fig. 8 and its derived equivalent circuit is shown in Fig. 9 with the unit element impedance values named sequentially from input to output. This realization is optimal since an -degree filter requires commensurate length transmission lines. At the quarter-wave frequency, all the series short-circuited stubs become open circuited while all the open-circuited stubs become short circuited so that the alternate ends of the parallel coupled lines are shorted to ground. Thus, the meander-like low-pass filter of Fig. 8 has at least one transmission zero at the quarter-wave frequency. The other transmission zero pairs may exist at infinity on the real axis or as symmetrical pure imag-

Fig. 9. Graphical representation of the equivalent circuit of Fig. 8 for a meander-like low-pass filter.

inary frequency pair or, in general, as paraconjugated pairs on the complex plane due to multipath in the structure. It is now shown how the meander-like low-pass filter network of Figs. 8 and 9 may be synthesized from suitable low-pass filter networks and then using appropriate circuit transformation to transform the canonical low-pass filter network forms to a meander-like low-pass filter. The canonical low-pass filter network forms were obtained by the synthesis method in [8] and then applying cascaded synthesis. The 3 , 5 , 7 , and 9 -degree meander-like low-pass filter are examined next, as depicted in Fig. 10. The 3 -degree filter is simply a trivial case corresponding to circuit transformations I, as shown in Fig. 10 (I). In this case, a 3-0-3 or 3-2-1 low-pass filter may be realized. There are two possible cases for the 5 -degree filter, namely, a 5-0-4 and 5-2-2 low-pass filter. For the first case of Fig. 10 (II), a 5-0-4 low-pass filter has a single real axis half transmission zero pair at infinity and four transmission zeros are at a quarter-wave frequency . Beginning with the canonical network form, step 1 is to split the transmission lines into two equal parts between ports 2 and 4. In step 2, transformation III is carried out on each of the branches 3,2,4 and 2,4,5 respectively. Finally in step 3, two separate transformations II are carried out on each of the three port subnetworks of 1,3,5 and 3,5,6 to derive the final equivalent circuit, as illustrated in Fig. 10 (II). The second meander-like low-pass filters, the 5-2-2 low-pass filter in Fig. 10 (III and IV), have a single pair of symmetrical finite frequency transmission zeros , a single real axis half transmission zero pair at infinity , and two transmission zeros at the quarter-wave frequency . For an asymmetrical 5-2-2 case of Fig. 10 (III), step 1 utilizes Sato’s transformation [7, Table I] to eliminate branch 1,2 and in turn creates branches 1,3 and 1,5. In Step 2, Sato’s transformation is used again to eliminate branch 2,3 and creates branch

MUSONDA AND HUNTER: EXACT DESIGN OF A NEW CLASS OF GENERALIZED CHEBYSHEV LOW-PASS FILTERS

Fig. 10. Derived network transformation for

and

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meander-like low-pass filters.

3,5. In step 3, transformation III is applied on branch 1,5,6 and two sequential transformations III are applied on branch 1,3,5. The final equivalent circuit is obtained by application of transformation II on a three-port subnetwork of 4,6,7, as illustrated in Fig. 10 (III). For the symmetrical 5-2-2 low-pass filter of Fig. 10 (IV), step 1 splits the inductor between node 5 and 7 such that the admittances of the outermost inductors in branches 1,2 and 7,8 are identical. In step 2, transformation IV is carried out on the two-port network between nodes 2 and 7, and finally, two separate transformations II are carried out on three-port subnetworks 1,9,10 and 9,8,10, respectively, to obtain the final symmetrical form, as illustrated in Fig. 10 (IV). Note that a negative sign on in transformation IV should be taken for realizable impedance values. For a 7 -degree filter, one possible realization derived is a 7-2-5 low-pass filter of Fig. 10 (V) with a single pair of symmetrical finite frequency transmission zero pair and all the remaining five transmission zeros at the quarter-wave frequency , which is now illustrated. From the canonical low-pass filter form, transformation I is applied on the two-port network between nodes 2 and 7 in step 1. This is followed by transformation V again on the same two port network between nodes 2 and 7 in step 2. In step 3, two separate transformation III are then applied on branches 3,2,6 and 8,7,9 and finally, in step 4, three separate transformation II are applied on three port subnetworks of 1,3,5, 3,5,8, and 5,8,9, respectively, to obtained the final form of the meander-like low-pass filter of Fig. 10 (V). For a 9 -degree filter, two derivative low-pass filter networks were examined whose core subnetworks were derived from a 5 -degree network discussed above. The first one is a 9-0-8 filter with a single real axis half transmission zero pair at infinity and all the remaining eight transmission zeros at a quarter-wavelength frequency

. Beginning with the canonical low-pass filter in Fig. 10 (VI) in step 1, a two-port network between nodes 2 and 8 is replaced by a derived circuit for a 5-0-4 circuit as explained above [see Fig. 10 (II)]. In step 2, two separate transformations III are then applied to branches 3,2,7 and 5,8,9. This is followed by two separate transformations II, which are applied on three-port subnetworks of 1,3,5 and 7,9,10, respectively, to give the final equivalent circuit, as illustrated in Fig. 10 (VI). The second 9 -degree low-pass filter is a 9-2-6 low-pass filter with a single symmetrical pair of finite frequency transmission zeros , a single real axis half transmission zero pair at infinity , and all remaining six transmission zeros at a quarter-wave frequency . Beginning with the canonical low-pass filter form in Fig. 10 (VII) in step 1, a two-port network between nodes 2 and 9 is replaced by a derived circuit for a 5-2-2 circuit, as explained above [see Fig. 10 (III or IV)]. In step 2, two separate transformations III are then applied to branches 3,2,6 and 8,9,10, respectively. This is followed by two separate transformations II, which are applied on three-port subnetworks of 1,3,8 and 6,10,11, respectively, to give the final equivalent circuit shown in Fig. 10 (VII). Note that a positive sign on in transformation IV should be taken for realizable impedance values. Multiple solutions do exist depending on the transformations used. Impedance levels are within positive realizable values as long as the electrical length at the cutoff is chosen to be around as the two examples will show. Moving further away form , either direction tends to lead to extreme element’s impedance values, some of which may assume negative values arising from the formulas used in some of the cases of Fig. 2. As its canonical circuit form, the meander-like low-pass filter is relatively unaffected by small changes in the impedances values because it requires an optimal number of elements . Thus, small mismatch in the impedance values only slightly degrades the passband return loss.

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Fig. 10. (Continued.) Derived network transformation for

and

1) Design Example I: A 7 -degree (7-2-5) meander-like low-pass was designed with a symmetrical pair of finite frequency transmission zero at (1.625 GHz) and five transmission zeros at a quarter-wave frequency , 20-dB minimum passband return loss and electrical length of at a cutoff frequency of 1 GHz. The synthesized element values for the canonical low-pass filter is shown in Table III [assuming symmetry with impedance values assigned

meander-like low-pass filters.

sequentially from left to right of Fig. 10 (V)]. Using the technique as explained in Section II-B and Fig. 10 (V) by a sequence of circuit transformations, the meander-like element values were then obtained as shown in Table IV. The circuit simulation shown in Fig. 11 validates the synthesis process. 2) Design Example II: An experimental 9 -degree meander-like low-pass filter was designed with cutoff frequency at 1 GHz, 20-dB minimum return loss, and . The stop-

MUSONDA AND HUNTER: EXACT DESIGN OF A NEW CLASS OF GENERALIZED CHEBYSHEV LOW-PASS FILTERS

Fig. 10. (Continued.) Derived network transformation for

and

band insertion loss was defined to be above 70 dB between 1.3 and 2.7 GHz and this was achieved by placing a symmetrical transmission zero pair at (1.294 GHz), a single real axis half transmission zero pair at infinity and six transmission zeros at quarter-wave frequency

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yielding a 9-2-6 low-pass filter of Fig. 10 (VII). The synthesized element values for the canonical 9 -degree low-pass filter are shown in Table V. These values were transformed to the meander-like circuit with the element values shown in Table VI.

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Fig. 10. (Continued.) Derived network transformation for

and

TABLE III 7 -DEGREE CANONICAL LOW-PASS FILTER IMPEDANCE VALUES

The low-pass filter is then realized using rectangular bars or striplines. The technique by Getsinger [9], [10] was used to determine the initial physical dimensions. The final optimized dimensions are given in Table VII. The nomenclature used in Table VII corresponds to Figs. 13 and 14. Fig. 12 shows good correspondence between the measured and theoretical simulation using HFSS.

meander-like low-pass filters.

TABLE IV 7 -DEGREE MEANDER-LIKE LOW-PASS FILTER IMPEDANCE VALUES

The overall length of the low-pass filter realization is three times the electrical length at the cutoff frequency. High-order modes do exist in the structure that potentially could worsen the stopband response, especially with relatively larger groundplane spacing. The effect is to shorten the effective stopband frequency window as the design example shows in Fig. 12. The choice of the ground-plane spacing affects the spurious resonances within the filter structure, which in this case appeared

MUSONDA AND HUNTER: EXACT DESIGN OF A NEW CLASS OF GENERALIZED CHEBYSHEV LOW-PASS FILTERS

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Fig. 11. Circuit simulation of a 7-2-5 meander-like low-pass filter in example II B (1). TABLE V SYNTHESIZED 9 -DEGREE CANONICAL LOW-PASS FILTER IMPEDANCE VALUES

Fig. 13. Diagram showing the layout of the fabricated 9 -degree meander-like low-pass filter. Dimension shown are as given in Table VII. TABLE VI SYNTHESIZED 9 -DEGREE MEANDER-LIKE LOW-PASS FILTER IMPEDANCE VALUES

TABLE VIII IMPROVED 9 - DEGREE LOW-PASS FILTER OPTIMIZED DIMENSIONS (IN MILLIMETERS)

TABLE VII 9 - DEGREE LOW-PASS FILTER OPTIMIZED DIMENSIONS (IN MILLIMETERS)

Fig. 14. Physical hardware of the fabricated 9 -degree “meander-like” lowpass filter (top cover removed).

Fig. 12. Comparison of simulated response of the synthesized, HFSS, and measurement of meander-like low-pass filter.

above 2.6 GHz. The insertion loss is fairly low across the passband with a peak at 0.3476 dB at the cutoff frequency in the

measured response, as depicted in Fig. 15. The slight discrepancy in the insertion loss between the simulated and measurement results in Fig. 15 is due to a slight mismatched response, as evident from the return-loss plot in Figs. 12 and 16. Improvement in the stopband response may be achieved by reducing the ground-plane spacing from mm to mm at the expense of slightly increased insertion loss (Fig. 15). Fig. 16 shows HFSS simulations for different ground-plane spacing versus the ideal circuit response. Clearly

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Fig. 15. Comparison of insertion losses between HFSS simulations for mm and mm and measured response with mm.

Fig. 16. Comparison of optimized equivalent-circuit simulation and HFSS simulations of meander-like low-pass filter with ground-plane spacing of 15 and 25 mm.

the stopband performance matches very well with the prediction for mm. Table VIII shows the corresponding optimized physical dimensions. Notice that much smaller ground-plane spacing is limited by realizability of the physical dimensions as the dimensions of the low-pass filter are proportional to the ground-plane spacing. C. Comparison A 9 -degree meander-like low-pass filter was compared to other low-pass filter realizations. To achieve the same selectivity, a 9 -degree generalized Chebyshev low-pass filter would be required while a 15 -degree stepped-impedance low-pass filter would be required as depicted in Fig. 17 with electrical length at the cutoff frequency of 1 GHz. Thus, for the same selectivity, the proposed structure requires a much fewer number of filter elements than the stepped-impedance low-pass filter. Although the generalized Chebyshev low-pass filter may be designed with the same degree as the meander-like low-pass filter, its stopband performance is much poorer in its physical realization, as shown in Fig. 17, because the series short-circuited stubs are approximated by high-impedance transmission lines [3]. Furthermore, both the generalized Chebyshev and stepped-impedance low-pass filters’ effective stopband response is much worse in practice because it is difficult to realize ideal commensurate transmission-line elements and often discontinuities, high-order modes, and mode

Fig. 17. Circuit simulation comparison of 9 -degree meander-like low-pass filter with a 9 -degree generalized Chebyshev low-pass filter and 15 -degree stepped-impedance low-pass filter.

Fig. 18. Circuit insertion-loss simulation comparison of 9 -degree meanderlike low-pass filter with a 9 -degree generalized Chebyshev low-pass filter and 15 -degree stepped impedance low-pass filter.

conversion occurs within the filter structure [11]. These reduce the effective stopband width of practical low-pass filters to as much as half of the predicted width! Even though effective stopband width may be widened by using a lower electrical length at cutoff frequency, it is often limited by element realization as the variations in element values tend to be extreme. By utilizing relatively smaller ground-plane spacing, as described in Section II-B, the proposed low-pass structure offers superior stopband performance. Fig. 18 shows the circuit-level insertion-loss analysis of the three low-pass filters being compared above with the same ground-plane spacing of 25 mm assuming copper conductors in air. It is quite obvious that the stepped-impedance low-pass filter fairs worse because of the highest number of unit elements required to achieve the selectivity. The generalized Chebyshev low-pass filter passband insertion compares well with the proposed meander-like low-pass filter with the losses increasing towards the cutoff frequency. The proposed structure has an optimal number of unit elements equal to the degree of the network regardless of the number of finite frequency transmission zeros. The generalized Chebyshev low-pass filter, on the other hand, requires 12 unit elements to achieve the selectivity requirements. Thus, the proposed meander-like low-pass filter is much more compact with low insertion loss than the other two low-pass filters. The meander-like low-pass filter has a

MUSONDA AND HUNTER: EXACT DESIGN OF A NEW CLASS OF GENERALIZED CHEBYSHEV LOW-PASS FILTERS

high achievable roll-off rate of 246.7 dB/GHz with an achievable relative stopband bandwidth of 0.883 [12] and could be advantageous where a much deeper out-of-band rejection is required. III. CONCLUSION An exact design technique for realizing generalized Chebyshev distributed low-pass filters using a coupled line/stub without approximating the series short-circuited stubs has been demonstrated. The physical realization I has a simple equivalent circuit; however, it requires isolation walls to eliminate coupling between basic sections. Since a single basic section may realize a pair of finite frequency transmission zero, a maximum of pairs of symmetrically located transmission zeros is achievable. A more general meander-like structure, physical realization II, with an optimal number of elements and simple physical layout of transmission lines, has also been presented. Its physical realization does not require decoupling walls between the parallel coupled line section, and hence, is easier to construct. However, only certain forms of transfer functions are realizable as described above. Synthesis of a few realizations up to 9 -degree together with the required transmission zeros locations of their canonical forms have been illustrated. A low-pass filter design example utilizing the later physical realization was fabricated and measurement results showed good agreement with theory. Comparison with other low-pass filter realizations reviewed that the proposed low-pass filter has a much higher roll-off rate and deeper effective stopband. REFERENCES [1] H. M. Jaradat and W. M. Fathelbab, “Selective lowpass filters realizing finite-frequency transmission zeros,” in IEEE Radio Wireless Symp., 2009, pp. 252–255. [2] C. J. Chen, C. H. Sung, and Y. D. Su, “A multi-stub lowpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 8, pp. 532–534, Aug. 2015. [3] I. Hunter, Theory and Design of Microwave Filters. Stevenage, U.K.: IET, 2001. [4] J. D. Rhodes and S. Alseyab, “The generalized Chebyshev low-pass prototype filter,” Int. J. Circuit Theory Appl., vol. 8, pp. 113–125, 1980. [5] S. A. Alseyab, “A novel class of generalized Chebyshev low-pass prototype for suspended substrate stripline filters,” IEEE Trans. Microw. Theory Techn., vol. MTT-30, no. 9, pp. 1341–1347, Sep. 1982. [6] E. Musonda and I. Hunter, “Design of generalised Chebyshev lowpass filters using coupled line/stub sections,” in IEEE MTT-S Int. Microw. Symp. Dig., 2015, pp. 1–4.

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[7] R. Sato, “A design method for meander-line networks using equivalent circuit transformations,” IEEE Trans. Microw. Theory Techn., vol. MTT-19, no. 5, pp. 431–442, May 1971. [8] E. Musonda and I. Hunter, “Synthesis of general Chebyshev characteristic function for dual (single) bandpass filters,” in IEEE MTT-S Int. Microw. Symp. Dig., 2015, pp. 1–4. [9] W. J. Getsinger, “Coupled rectangular bars between parallel plates,” IRE Trans. Microw. Theory Techn., vol. MTT-10, no. 1, pp. 65–72, Jan. 1962. [10] M. A. R. Gunston, Microwave Transmission-Line Impedance Data. London, U.K.: Van Nostrand, 1972. [11] R. J. Cameron, C. M. Kudsia, and R. R. Mansour, Microwave Filters for Communication Systems: Fundamentals, Design, and Applications. Hoboken, NJ, USA: Wiley, 2007. [12] J. Wang, L. J. Xu, S. Zhao, Y. X. Guo, and W. Wu, “Compact quasielliptic microstrip lowpass filter with wide stopband,” Electron. Lett., vol. 46, pp. 1384–1385, 2010.

Evaristo Musonda (GSM’13) received the B.Eng degree (with distinction) from The University of Zambia, Lusaka, Zambia, in 2007, the M.Sc. degree in communication engineering (with distinction) from The University of Leeds, Leeds, U.K., in 2012, and is currently working toward his Ph.D. degree at The University of Leeds. In early 2008, he joined Necor Zambia Limited, an ICT company, prior to joining the country’s largest mobile telecommunication services provider, Airtel Zambia, in June 2008, where he was involved in core network planning, optimization, and support roles for three years. He is currently involved in research for new microwave filters synthesis techniques for digital wireless communication systems at The University of Leeds. His research interests include microwave filters and network synthesis.

Ian Hunter (M’82–SM’94–F’07) received the B.Sc. degree (first-class honors) and Ph.D. degree from Leeds University, Leeds, U.K., in 1978 and 1981, respectively. Early in his career, he was with Aercom, Sunnyvale, CA, USA, KW Engineering, San Diego, CA, USA, and Filtronic, Shipley, U.K., where he was involved with the development of broadband microwave filters for electronic warfare (EW) applications. From 1995 to 2001, he was with Filtronic Comtek, where he was involved with advanced filters for cellular radio. He currently holds the Royal Academy of Engineering/Radio Design Ltd. Research Chair in Microwave Signal Processing with the School of Electronic and Electrical Engineering, The University of Leeds, Leeds, U.K., where he currently leads a team involved with the research of new microwave filters for mobile communications systems. He has authored Theory and Design of Microwave Filters (IEE Press, 2001). Prof. Hunter is a Fellow of the IET and the U.K. Royal Academy of Engineering. He was general chair of 2011 European Microwave Week, Manchester, U.K. He is the chair of the 2016 European Microwave Conference, London, U.K.

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Propagating Waveguide Filters Using Dielectric Resonators Cristiano Tomassoni, Member, IEEE, Simone Bastioli, Member, IEEE, and Richard V. Snyder, Life Fellow, IEEE

Abstract—In this paper a new class of propagating waveguide in-line pseudoelliptic filters exploiting dielectric resonators is presented. The basic structure is a singlet, which is implemented by a mode dielectric resonator placed into a rectangular waveguide above cutoff. Couplings are controlled by a proper positioning of the puck. The fundamental propagating mode of the waveguide is exploited to both excite and bypass the resonator so as to obtain bypass coupling capability, allowing the generation of transmission zeros. Higher order filters are obtained by cascading singlets through quarter-wave or half-wave waveguide sections. The quarter-wave section behaves as an admittance inverter, while the half-wave section behaves as a resonator, the latter resulting in filters which combine dielectric and cavity resonators. Thanks to this combination, unique performances such as wide bandwidth and sharp transition bands can be obtained. To validate the proposed method a third-order filter with 2.3% fractional bandwidth (FBW) and a fifth-order filter with 8.15% FBW have been designed and manufactured, thus demonstrating the approach feasibility. Index Terms—Bandpass filters, dielectric resonators, elliptic filters, rectangular waveguide, transmission zeros (TZs).

I. INTRODUCTION

T

HE dramatic increasing of communication market demands for compact light filters with even more higher performances. High permittivity dielectric resonators are widely employed because of their compactness and superior performance in terms of Q-factor and temperature stability [1]. Their application ranges from very narrowband dielectric-loaded cavity filters for satellites and cellular base-stations, to highly stable oscillators [2], [3]. Most common dielectric-loaded cavity filters exploit the fact that the field is mainly confined within the high-permittivity dielectric puck to reduce ohmic losses. For this reasons pucks are generally suspended in a metallic enclosure. They can be axially located along the enclosure [4]–[7], or mounted in a planar configuration [8], [9]. Filter poles are obtained by exploiting the fundamental resonant mode or higher order mode in combination with its degenerate mode to obtain dual-mode filters [6], [7], the Manuscript received July 01, 2015; revised October 01, 2015; accepted October 11, 2015. Date of publication November 13, 2015; date of current version December 02, 2015. This paper is an expanded version from the IEEE International Microwave Symposium, Phoenix AZ, May, 17–22, 2015. C. Tomassoni is with the Electronic and Information Engineering Department (DIEI), University of Perugia, 06125 Perugia, Italy (e-mail: tomassoni@diei. unipg.it). S. Bastioli and R. V. Snyder are with RS Microwave, Butler, NJ 07405 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495284

latter allowing for pseudoelliptic responses while maintaining an in-line configuration. Pseudoelliptic filters exploiting mode have been lately obtained in an in-line configuration [10] by using pucks with orthogonal orientation to realize cross-coupling among nonadjacent resonators by exploiting different polarizations of the waveguide evanescent modes. The in-line topology is in fact convenient for mechanical and size considerations and in the last years several filters exploiting nonresonating modes have been proposed in order to obtain pseudoelliptic responses without using folded cross-coupled architectures. As an example, the nonresonating mode technique has been used in [11] to cascade transverse magnetic (TM) dualmode cavies, in [12] where the TM dual-mode cavities have been loaded by dielectric pucks in order to reduce the size, as well as in [13] by using dual-post resonators in propagating waveguide. A new class of in-line filters with dielectric pucks properly located in propagating rectangular waveguide has been proposed in [14]. The basic structure is a singlet, which is implemented by a mode dielectric resonator placed into a rectangular waveguide above cutoff. The fundamental propagating mode of the waveguide is exploited to both excite and bypass the resonator so as to obtain bypass coupling capability, allowing the generation of transmission zeros. Specifically, singlets having the same coupling amplitude in input and output (with no control of the source to load coupling) have been proposed in [14], where higher order filters have been obtained by cascading multiple singlets through nonresonating quarter-wave waveguide sections. This paper significantly extends the above idea by introducing new singlet configurations with a noticeably increased capability in term of coupling control. By properly positioning the dielectric pucks with respect to the waveguide, an independent control of input and output couplings is now possible. A more advanced technique to obtain a precise control of source to load coupling by using a capacitive post is also shown. Furthermore, a new filter configuration exploiting such singlets is proposed where the singlets are cascaded through half-wave waveguide sections. The half-wave waveguide sections behave as additional resonators, thus increasing the filter order. The strong bypass coupling of the singlet and the weak coupling between cavities and dielectric pucks allow for the design of wideband filter having transmission zeros very close to the pass-band edges, thus resulting in extremely sharp transition bands. This strong coupling allows wideband filters, while the weak coupling between cavities and dielectric puck together with the strong bypass allows transmission zeros very close to the filter band resulting in extremely sharps transition bands.

0018-9480 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Fig. 1. Dielectric puck in rectangular waveguide. (a) Three-dimensional view and (b) top view and coupling pattern.

This configuration overcomes the typical problem of limited bandwidth capability of dielectric filters while maintaining the high performance in terms of temperature stability of dielectric based structures. Detailed design procedure for the exact evaluation of coupling coefficients are introduced for both structures using quarter-wave and half-wave waveguide sections. Finally, a new experimental result of a fifth-order filter is presented to show the feasibility of filters which combine dielectric and cavity resonators.

Fig. 2. Singlet composed by a dielectric puck in rectangular waveguide. Comparison between full-wave response (continuous dark lines) and coupling matrix response (dotted red lines). In the figure inset, the topology and the relevant coupling matrix as well as 3-D sketch of the singlet are shown. The modal E-field and modes (yellow arrows) and modal H-field (green arrows) for are drawn. The waveguide is a WR90, while the dielectric puck has a radius mm, a height mm, and a relative permittivity .

II. SINGLET The singlet structure is here used as basic building block for the design of higher order filters. According to Fig. 1, the singlet consists of a dielectric puck suspended within a propagating waveguide. The coupling paths are illustrated in Fig. 1(b): the propagating waveguide mode excites the dielectric resonant mode, resulting in sequential couplings and . Part of the power carried by the fundamental mode bypasses the resonator and creates the input-to-output coupling . In the following the mechanisms of the couplings and the way to control them are explained in detail for the basic singlet structure (Section II-A), and for a more complex singlet structure exploiting a capacitive post as coupling section (Section II-B). A. Basic Configuration Let us first consider the case where the dielectric puck resonator is centered with respect to the waveguide. With reference to Fig. 2, for symmetry reasons related to and modal field distributions, the resonant mode is not excited and the most of the power carried by the bypasses the resonator and creates a strong source-to-load coupling. The simulated S parameters confirm that no poles are present in the response and about 80% of the power bypasses the puck ( ). In the inset of Fig. 2 the coupling matrix representation of the singlet is also shown and its response (dotted lines) is compared to the full-wave response (continuous lines). The extraction of the coupling matrix from the response is obtained by optimizing the coupling matrix response. To excite the resonant mode it is necessary to break the symmetry of the structure. This can be done in two ways: by rotating the puck as shown in Fig. 3, or by shifting the puck as shown in Fig. 4. In both cases, first-order pseudoelliptic responses with one pole and one transmission zero are obtained. Observe that the transmission zero is located below the pole when the puck is rotated, while it is located above the pole if the puck is offset. This is due to the fact that in the case of a rotation , while in the case of a shift as can be easily understood considering the field distribution shown in

Fig. 3. As in Fig. 2 but with the dielectric puck rotated around the -axis of an . angle

Fig. 4. As in Fig. 2 but the puck is no longer centered to the waveguide, and mm. the puck offset is

the figure insets. The control of and amplitude is carried out by using the rotation angle (or the offset): the larger the angle (or the shift) the stronger the coupling. Note that in both shift and rotation cases, because of the high source to load coupling, the response actually is closer to a

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Fig. 6. Comparison between full-wave and coupling matrix response of (a) and (b) puck shifted with mm and centered puck rotated of . rotated of

Fig. 5. As in Fig. 2 but the puck is shifted and rotated. The puck offset is mm (as that of Fig. 4), while the puck rotation is (as that of Fig. 3).

stop-band filter. For this reason, a fractional bandwidth (FBW) reported in figure insets is referred to a FBW of the stop-band. As can be noted, the input-to-output coupling is stronger in the case of rotation than in the case of shift ( instead of ). This is due to the fact that the field of the is higher in the center of the waveguide and is more influenced by a centered puck. In some applications the fact that the amplitude of and are the same is a limitation. To overcome this limitation a shifted and rotated puck is considered. For the estimation of the coupling matrix when shift and rotation are combined, we consider a sort of superposition property. The scattering superposition is applied to the denormalized coupling coefficients (Appendix I)

(1) Here the superscript indicates that the coupling coefficient is referred to the singlet with rotated puck (no shift), the superscript is referred to the singlet with shifted puck (no rotation), while the superscript , is referred to the singlet where the puck is both shifted and rotated. Note that the superposition property has not been applied to the direct input-to-output coupling. This is due to the fact that this coupling is mainly dependent on the puck position with respect to the center of the waveguide. Formulas have been tested in Fig. 5 where the response of a dielectric rotated 25 (as in Fig. 3) and shifted 3.8 mm (as in Fig. 4) is shown. In this case the coupling matrix has been evaluated by applying the superposition property (1) to the scattering matrices of Figs. 3 and 4. Although the superposition property is just an approximation, Fig. 5 shows that the approximation is quite accurate. Obviously, by inverting (1) it is possible to design a singlet having prescribed and

(2)

Let us suppose to design a singlet with and . According to (2) this leads to and . The singlet of Fig. 4 having mm, realizes the desired , while the singlet of Fig. 6(a) having realizes the desired . This leads to a structure with a dielectric puck shifted of mm and rotated of . The expected coupling matrix for this structure has been calculated applying the scattering superposition property (1) to those in Figs. 4 and 6(a). In Fig. 6(b) the response of the calculated coupling matrix is plotted along with the full-wave response, showing an excellent agreement. The main limitation of the present structure is that there is no control on source-to-load coupling. To overcome this limitation it is possible to use additional coupling sections. As an example irises at the input and output of the singlet can be used to control the source-to-load coupling. This leads to a very simple structure where couplings are still controlled by shifting and rotating the dielectric. Unfortunately, irises decrease also the coupling of the dielectric to source and load, and this structure can be used just in the case where no strong couplings are needed (as in the case of the manufactured filter in Section IV-B). B. Singlet With Posts Another possible coupling section can be realized by a capacitive post. As shown in Fig. 7, we consider a singlet realized by a dielectric puck and a capacitive post in propagating waveguide. The post is placed at a distance and a shift with respect to the dielectric resonator. Let us consider the case where the dielectric is centered ( ) to the waveguide. This configuration guarantees a complete control of couplings. In fact, in contrast with the singlet without post, in this case the coupling between dielectric resonator and input and output is different ( ) due to the presence of the post, and there is no necessity of combining shift and rotation of the dielectric puck. To illustrate the features of this structure let us consider a nonrotated puck ( ) and a post centered with respect to the waveguide ( ). According to Fig. 7, the symmetry of the electric field of the centered post does not allow the excitation of the resonant mode. In Fig. 8, the response of such a structure as a function of the post height is shown. As expected, there are no resonances, and the power flowing from input to output is due to the fundamental mode that bypasses the resonator: the higher the post the lower the . Of course, when the post height tends to zero, the value of tends to those of Fig. 2.

TOMASSONI et al.: PROPAGATING WAVEGUIDE FILTERS USING DIELECTRIC RESONATORS

Fig. 7. Singlet composed by a dielectric puck and a metallic capacitive post. (a) Front view, (b) top view, and (c) 3-D view. In (c) the -field for waveguide mode and puck resonant mode as well as the E-field in the transverse section of the waveguide in correspondence of the post is shown.

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Fig. 9. Singlet composed by a puck and a post. Responses of the singlet for different values of (continuous lines) compared to the coupling matrix response calculated for different FBW (dotted lines). In the figure inset the structure and the electric field distribution are shown.

Fig. 8. Singlet with centered post and nonrotated dielectric puck. Transmission coefficient as a function of the post height.

Once the distance is fixed, all the coupling parameters can be independently controlled: the post height mainly controls , the rotation angle mainly controls , while the post offset mainly controls . Observe that is chosen so that the post is close enough to influence the dielectric field. To show the design procedure let us implement a singlet having , and different from zero. To obtain we fix the post height to 7.5 mm. Different curves have been obtained in Fig. 9 by varying from 0 to 60 , resulting in different values of the coupling between load and resonator. For each the position of the post has been selected so that . This value has been obtained by exploiting the field perturbation due to the presence of the post, as shown in the figure inset in the case of . In fact, because of the radial nature of the field around the post, the field in the left and right side of the post contribute to the excitation of the resonant mode with different sign. In this case, the position of the post is chosen so that the sum of all contributions is zero, thus leading to . By increasing , a negative coupling is obtained, while decreasing the coupling becomes positive. This is clearly shown in Fig. 10, where mm produces a positive coupling [Fig. 10(c)] while mm produces a negative coupling [Fig. 10(d)], being in this specific case mm the position for which the coupling is zero [Fig. 10(b)]. As can be noted from Fig. 9, the difference between fullwave responses and equivalent matrix responses increases ap-

Fig. 10. (a) Singlet and relevant dimensions. Response for the singlet with the mm (b), mm (c), and mm (d). post in position

proaching 9.5 GHz. This is due to a spurious resonance of the post appearing above 9.5 GHz. This spurious resonance also influences responses of Fig. 8. The higher the post height, the lower the spurious frequency. Another possible singlet configuration is the one in Fig. 11(a), where the offset (instead of the rotation) is exploited to control : the higher the the higher the is instead still controlled by the post position , as in the case of the centered puck. In Fig. 12 some responses of the above singlet structure have been obtained. Different stop-bands have been obtained by varying the offset and readjusting the post offset in order to obtain . Finally, a generalized singlet configuration is shown in Fig. 11(b) where the dielectric resonator is both rotated and shifted. In that case both offset and rotation contribute to . is instead still controlled by the post position . In Fig. 13 the response of a singlet having both and equal zero is shown. Note that the rotation is chosen so that its contribution to eliminates the contribution of the shift , thus leading to . Starting from this point, by increasing ,we obtain , while decreasing we obtain .

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Fig. 14. Three singlets cascaded through admittance inverters and . In the figure, the path connecting source to resonator 3 (res, 3) avoiding all other resonators is shown (red dotted line). From this path it is possible to calculate of Fig. 16. the admittance related to

Fig. 11. (a) Noncentered puck configuration and (b) noncentered rotated puck configuration.

Fig. 15. Singlets cascaded through a quarter wave length.

Fig. 12. Noncentered puck resonator. Responses obtained for different values . of puck shift . All responses have

Fig. 16. Canonical equivalent circuit of three pole pseudoelliptic filter. Formulas allow the evaluation of admittance inverters starting from those of Fig. 14.

Fig. 13. Response of the structure illustrated in the figure inset with puck radius 3.25 mm and pack height 3 mm. The relative dielectric constant is 30.

III. SINGLET CASCADED THROUGH QUARTER-WAVE WAVEGUIDE SECTIONS A. Design Procedure Higher order filters can be obtained by cascading singlets through quarter wave transmission line lengths, as shown in Fig. 15. An important property of such a structure is that the transmission zero of each singlet is preserved exactly in the same position after the cascade. In other words, each singlet controls its own zero, allowing for a modular design and a precise positioning of each transmission zero. A quarter wave length behaves like as a unitary admittance inverter. This means that the cascade of three singlets

can be represented with the equivalent circuit of Fig. 14 with and equal 1. The coupling matrix associated to this equivalent circuit is a 9 9 matrix. In fact in this circuit, 7 nodes are present: three resonant nodes and four nonresonating nodes, as will be illustrated later on in the example of Fig. 17. Nonresonating nodes can be removed, leading to a 5 5 coupling matrix and the relevant equivalent circuit of Fig. 16. The procedure to find the -inverter connecting resonant node to resonant node is very easy and involves the path connecting node to node that avoids all other resonant nodes. The same procedure applies for the evaluation of coupling involving source and load. As an example the path related to is illustrated in Fig. 14 (red dotted line). The formula consists in the product of all encountered -inverter values pertaining to the singlets divided by the product of all -inverters values (with changed sign) used to connect the singlet. Referring to the example of Fig. 14, the inverter pertaining to the singlet are

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Fig. 18. Singlet exploited in the manufactured filters. The waveguide is a WR137. Puck dielectric constant is 35.62. Puck diameter 8.89 mm. Thickness of low-permittivity dielectric holding the puck is 2 mm.

Fig. 17. Coupling matrix obtained cascading three singlets. The highlighted submatrices are the coupling matrices of the relevant singlets. Nodes 2, 3, 5, and 6 are nonresonating nodes. Nonresonating nodes corre. spond to sources and loads of cascaded singlets.

,

and , those related to the cascade are , thus resulting in

and

(3) This formula can be easily found by considering that the cascade of three admittance inverters , and is still an admittance inverter of value , as demonstrated in the Appendix II As a filter synthesis example, let us consider the coupling matrix (4) to design a 3pole-3zero filter with 2% FBW and central frequency 5.78 GHz. To synthesize the coupling matrix (4), the response of the equivalent circuit of Fig. 14 has been first optimized. Then the associated coupling matrix of Fig. 17 has been evaluated from the circuit parameters by exploiting formulas in Appendix I. Finally formulas in Fig. 16 has been applied to remove the nonresonating nodes from the coupling matrix of Fig. 17, obtaining matrix (4)

(4) To realize the filter we first design each singlet. As is evident from Fig. 17 the design of only two singlets is required as first and third singlets are equal. Singlets of Fig. 17 have been implemented by using the structure of Fig. 18. In the singlet design, the low-permittivity dielectric holding the high permittivity dielectric puck and the tuning metal disk are also taken into account. The tuning metal disk is held by low-permittivity stick and is used to tune the resonant frequency by varying its distance to the puck. Note that, according to the coupling matrix, singlets have slightly different resonant frequency. In this case the tuning disk is also used to obtain the different resonant frequencies in singlets having identical pucks. The singlets of Fig. 17 have been designed and their responses are shown in Fig. 19 along with the relevant coupling matrix responses. Note that the reference plane of the input and output

Fig. 19. Implementation of (left) singlets 1 and 3 (left) and (right) singlet 2 of Fig. 17. Comparison between coupling matrix responses (dotted lines) and full-wave responses (continuous lines).

ports have been chosen in order to match the phase of the coupling matrix response. As can be seen, because of the dispersive behavior of the waveguide, the matching of the phase is not as good as the amplitude, and the port distance has been chosen in order to minimize much as possible the differences in the filter band. With reference to Fig. 17, a unitary coupling ( ) connects singlets. This unitary coupling corresponds to a unitary -inverter that can be implemented by a waveguide length of 90 . At the frequency of 5.78 GHz, this corresponds to a waveguide length of 19.41 mm. Considering the reference planes position, this leads to a distance of 30.25 mm between the puck centers, as reported in Fig. 20 where the response of the whole filter is also shown. Note that the full-wave simulation of the filter has been obtained just cascading the three singlets and no further optimization has been required, demonstrating the high modularity of the method. As mentioned before, another advantage is that the transmission zeros of the filter are the same of that of the cascaded singlets, thus allowing a very precise control of the transmission

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Fig. 20. Third-order filter with three transmission zeros in the upper stop-band obtained cascading the structures of Fig. 19. Comparison between coupling matrix response (dotted lines) and full-wave response (continuous lines). The transmission zero at higher frequency is a double-zero.

Fig. 22. Equivalent circuit of doublets cascaded through half-wave lengths. (a) Third-order filter. (b) Fifth-order filter.

Fig. 21. Third-order filter with a FBW of 2.3%. Comparison between measurements (dotted lines) and full-wave analysis (continuous lines).

zero position. In the filter of Fig. 20 the first and last singlet are identical and each of them produces a transmission zero exactly at 9.06 GHz, resulting in a double zero. For this reason the filtering response shows two transmission zeros only, but one of them is a double zero.

resonators, thus increasing the filter order. The half-wave section can be substituted by a lumped resonator so as to obtaining the more standard equivalent circuit of Fig. 22(b). Impedance inverter values change according to the equations in Fig. 22. Note that the presence of regular half-wave resonators (without source-to-load couplings) allows transmission zeros at infinity. Let us consider a first example where the coupling matrix (5) has been implemented by using two identical singlets with capacitive posts, exploiting the capability of such a singlet to produce different input and output coupling values. Coupling matrix (5) has been extracted from the equivalent circuit of Fig. 22(b), derived from that of Fig. 22(a), and the desired filter response has been obtained by optimization.

B. Experimental Result The manufactured filter design starts form that of Fig. 20, but the distance between puck centers has been increased of 1.65 mm (about 8 ) increasing the bandwidth from 2% to 2.3%. The mismatching in the filter band has been compensated by increasing the rotation of puck 2 from 47 to 56 . leaving the puck 1 in the same position. The modified filter has then been manufactured and in Fig. 21 measurements have been compared to full-wave simulations, showing an excellent agreement. The manufactured filter has high tuning capability: resonant frequencies are tuned by the metal disk while couplings are tuned by rotating pucks. IV. SINGLET CASCADED THROUGH HALF-WAVE WAVEGUIDE SECTIONS A. Design Procedure In contrast with the previous configuration where quarter wave waveguide sections are used, a different technique consists of cascading singlets by means of half-wave sections. With reference to Fig. 22(a), half-wave sections behave as additional

(5) As in the previous example a modular design approach has been followed by first designing individual singlets. In Fig. 23 the comparison between full-wave and equivalent circuit response of the singlet is shown. Design parameters for the equivalent circuit have been calculated by using equations in Appendix I and the formulas of Fig. 22 for . This value of is due to the fact that, for the physical realizability of the structure, the two singlets have been cascaded by adding a wave length section. In fact, as already explained in the previous example, the position of the reference planes are taken so as to obtain the best phase matching. In this case, because of reference plane positions, the cascade of singlets through a half-wave length would lead to unfeasible overlapped pucks. In the resulting filter, shown in Fig. 24, the distance between puck centers along the longitudinal direction is 53.25 mm. After the cascade a minor optimization is needed. According to Fig. 22, the final optimized parameters are: mm, post height 9.25 mm, puck rotation 37 , while a moderate tuning of

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Fig. 23. Comparison between coupling matrix response (dotted lines) and fullwave simulation (continuous lines) for the singlet with metal post sketched in mm, post height is 9.7 mm, post radius is the figure inset. Puck shift mm and mm. 1.5 mm, and, according to symbols in Fig. 7, Fig. 25. Response of fifth-order filter. Nominal response (continuous lines) compared to the response obtained by detuning the first and last cavity of 15 MHz and to the response obtained by detuning the central cavity of 15 MHz (dotted lines).

Fig. 24. Third-order filter with a double transmission zero. Comparison between coupling matrix response (dotted lines) and full-wave response (continuous lines). In the figure the nonoptimized response (continuous lines) obtained just cascading the singlet of Fig. 23 is also shown.

the frequency has been carried out by varying the position of the tuning metal disks. The values of parameters and are instead those of Fig. 23 as they were not changed during the optimization.

Fig. 26. Filter of order 5 with (a) double transmission zero and (b) relevant equivalent circuit.

B. Experimental Result A fifth-order filter implementing the coupling matrix (6) with 8.15% FBW centered at 6.26 GHz has been designed and manufactured. Coupling matrix (6) has been extracted from the equivalent circuit of Fig. 26(b) and the filtering response has been obtained by optimization.

Fig. 27. Manufactured fifth-order filter.

(6) According to Fig. 26, the filter has been implemented by using three cavities and two identical singlets realized with resonant pucks. A double transmission zeros appears in the filtering function because of the two identical singlets. In this case irises have been used instead of posts to control the source to load coupling in singlets. This configuration overcomes the typical problem of limited bandwidth capability of dielectric filters while maintaining the high performance in terms of temperature stability of dielectric based structures. This is illustrated in

Fig. 25 where the lower cut-off of the filter is shown in case of resonators detuned of 15 MHz. As is evident, in the case of the detuned cavities the position of the lower cut-off does not change and the return loss remains above 15.5 dB. The filter has been designed starting from the design of the singlet by exploiting formulas in Appendix I and in Fig. 26(b). The filter is in WR137, its length is 108 mm, and its weight is 300 grams. In any case, this is only a prototype and mass and envelope have not been optimized here.

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V. CONCLUSIONS

Fig. 28. Comparison between simulation (dotted lines) and measurement (continuous lines) for the fifth-order manufactured filter.

In this paper a new class of in-line pseudoelliptic filters using dielectric pucks in propagating waveguides has been presented. The singlet configuration has been extensively analyzed and described by means of several examples. The precise and independent positioning of transmission zeros, the modularity property, as well as the wide pass-band capability, of these structures have all been demonstrated. Detailed design procedure for higher order filter employing quarter-wave or half-wave sections between dielectric resonators have been proposed. In particular the configuration with quarter wave sections allows for simple modular design of narrow band filters where each singlet individually controls its transmission zero. On the other hands, half-wave sections introduce additional cavity resonators which allow for design of wide band filters having transmission zero very close to the (temperature stable) pass-band edges. Both design techniques have been successfully demonstrated by the experimental results of two prototype filters. APPENDIX I Admittance inverter are related to the normalized coupling matrix elements in the following way:

Fig. 29. Measurements of the fifth-order manufactured filter at three different C, and C. temperatures: room temperature,

where FBW represents the Fractional Bandwidth. In the case of the coupling between two synchronous resonators:

where and resonators.

are the resonant frequencies of the coupled APPENDIX II

Fig. 30. (Top) Two nodes connected by three j-inverters. The three j-inverters are equivalent to the j-inverter in the figure bottom. The equivalence is obtained by the equation reported in the figure.

In Fig. 27 the photograph of the manufactured filter is shown, while the comparison between full-wave simulations and measurement is shown in Fig. 28. Here the measurement has been extended to 8 GHz to show the out-of-band behavior. As can be seen, spurious resonances appear at 7.3 GHz. In any case they are still far enough from the passband to be easily removed by a simple low-pass filter. The filter has a band of 510 MHz and a very sharp transition band at the lower cutoff: an attenuation of 10 dB is obtained within 6–7 MHz. Finally, in order to show the temperature stability, the filter has been measured in a temperature range of 125 C and the relevant responses are shown in Fig. 29. As can be seen, in all cases the return loss is higher than 15 dB.

To demonstrate the formula of Fig. 30 let us consider the transmission matrix of a -inverter of value (7) By exploiting the property of the transmission matrices, the transmission matrix of the cascade of the three -inverters of Fig. 30 can be written as

The resulting transmission matrix value .

is that of a -inverter of

REFERENCES [1] C. Wang and K. A. Zaki, “Dielectric resonators and filters,” IEEE Microw. Mag., vol. 8, no. 5, pp. 115–127, Nov. 2007.

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[2] S. J. Fiedziuszko and S. Holmes, “Dielectric resonators raise your high- ,” IEEE Microw. Mag., vol. 2, no. 3, pp. 50–60, Sep. 2001. [3] R. R. Mansour, “Filter technologies for wireless base stations,” IEEE Microw. Mag., vol. 5, no. 1, pp. 68–74, Mar. 2004. [4] S. B. Cohn, “Microwave bandpass filters containing high- dielectric resonators,” IEEE Trans. Microw. Theory Techn., vol. MTT-16, no. 4, pp. 218–227, Apr. 1968. [5] W. H. Harrison, “A miniature high- bandpass filter employing dielectric resonators,” IEEE Trans. Microw. Theory Techn., vol. MTT-16, no. 4, pp. 210–218, Apr. 1968. [6] S. J. Fiedziuszko, “Dual-mode dielectric loaded cavity filters,” IEEE Trans. Microw. Theory Techn., vol. MTT-30, no. 9, pp. 1311–1316, Sep. 1982. [7] K. A. Zaki, C. Chen, and A. E. Atia, “Canonical and longitudinal dual mode dielectric resonator filters without iris,” IEEE Trans. Microw. Theory Techn., vol. MTT-35, no. 12, pp. 1130–1135, Dec. 1987. [8] J.-F. Liang and W. D. Blair, “High- TE01 mode DR filters for PCS wireless base stations,” IEEE Trans. Microw. Theory Techn., vol. 46, no. 12, pp. 2493–2500, Dec. 1998. [9] H. Rafi, R. Levy, and K. Zaki, “Synthesis and design of cascaded trisection (CT) dielectric resonator filters,” in Proc. 27th Eur. Microw. Conf., 1997, vol. 2, pp. 784–791. modedielec[10] S. Bastioli and R. S. Snyder, “Inline pseudoelliptic tric resonator filters using multiple evanescent modes to selectively bypass orthogonal resonators,” IEEE Trans. Microw.Theory Techn., vol. 60, no. 12, pp. 3988–4001, Dec. 2012. [11] C. Tomassoni, S. Bastioli, and R. Sorrentino, “Generalized TM dualmode cavityfilters,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3338–3346, Dec. 2011. [12] L. Pelliccia, F. Cacciamani, C. Tomassoni, and R. Sorrentino, “Ultracompact high-performance filters based on TM dual-mode dielectricloaded cavities,” Int. J. Microw. Wireless Techn., pp. 151–159, Apr. 2014. [13] C. Tomassoni and R. Sorrentino, “A new class of pseudoelliptic waveguide filters using dual-post resonators,” IEEE Trans. Microw.Theory Techn., vol. 61, no. 6, pp. 2332–2339, Jun. 2013. [14] C. Tomassoni, S. Bastioli, and R. S. Snyder, “Pseudo-elliptic in-line filters with dielectric resonators in propagating waveguide,” in Proc. IMS2015, Int. Microwave Symp., Phoenix, AZ, USA, May 17–22, 2015. Cristiano Tomassoni (M’15) was born in Spoleto, Italy. He received the Laurea degree and Ph.D. degree in electronics engineering from the University of Perugia, Perugia, Italy, in 1996 and 1999, respectively. His dissertation concerned the mode-matching analysis of discontinuities involving elliptical waveguides. In 1999, he was a Visiting Scientist with the Lehrstuhl für Hochfrequenztechnik, Technical University of Munich, Munich, Germany, where he was involved with the modeling of waveguide structures and devices by using the generalized scattering matrix (GSM) technique. From 2000–2007, he was a Postdoctoral Research Associate with the University of Perugia. In 2001, he was a Guest Professor with the Fakultät für Elektrotechnik und Informationstechnik, Otto-von-Guericke University, Magdeburg, Germany. During that time, he was involved with the modeling of horn antennas having nonseparable cross sections by using hybrid methods combining three different techniques: the finite-element method, mode-matching technique, and generalized multipole technique. He was also involved in the modeling of low-temperature co-fired ceramics by using the method of moments. He studied new analytical methods to implement boundary conditions in the transmission-line matrix method, and he modeled aperture antennas covered by dielectric radome by using spherical waves. Since 2007, he has been an Assistant Professor with the University of Perugia. His main area of research concerns the modeling and design of waveguide devices and antennas. His research interests also include the development of reduced-size cavity filters, reconfigurable filters, and printed reconfigurable antenna arrays. Dr. Tomassoni was the recipient of the 2012 Microwave Prize presented by the IEEE Microwave Theory and Technique Society.

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Simone Bastioli (S'10–M'11) received the Master's and Ph.D. degrees in electronic engineering from the University of Perugia, Perugia, Italy, in 2006 and 2010, respectively. In 2005, he was an Intern with Ericsson AB, Mölndal, Sweden, where he was involved with waveguide filters and transitions for RF applications. In 2006, he joined the University of Perugia under a scholarship funded by the Italian Space Agency (ASI). In 2009, he was with RF Microtech Srl, Perugia, Italy, where he was responsible for the design of advanced microwave filters for private and European Space Agency (ESA)-funded projects. In 2010, he joined the RS Microwave Company Inc., Butler, NJ, USA, where he is currently Acting Chief Engineer involved with reduced-size multimode cavity filters, advanced high-power evanescent-mode filters, as well as dielectric resonator and lumped-element filters. His research activities have resulted in more than 20 publications in international journals and conferences. He has four patent applications pending. Dr. Bastioli is a member of the MTT-8 Filters and Passive Components Technical Committee. He was the recipient of the 2012 IEEE Microwave Prize. He was the recipient of the Best Student Paper Award (First Place) of the IEEE Microwave Theory and Techniques Society (MTT-S) International Microwave Symposium (IMS), Atlanta, GA, USA, in 2008, and the Young Engineers Prize of the European Microwave Conference, Amsterdam, The Netherlands, also in 2008. He was the recipient of the Hal Sobol Travel Grant presented at the IEEE MTT-S IMS, Boston, MA, USA, in 2009.

Richard V. Snyder (LF'05) received the B.S. degree from Loyola-Marymount University, Los Angeles, CA, USA, the M.S. degree from the University of Southern California (USC), Los Angeles, and the Ph.D. degree from the Polytechnic Institute, New York University, New York, NY, USA. He is President of RS Microwave, Butler, NJ, USA, which was founded in 1981. He teaches and advises at the New Jersey Institute of Technology. He is a Visiting Professor with The University of Leeds, Leeds, U.K. He has authored or coauthored 117 papers and three book chapters. He holds 21 patents. His interests include electromagnetic (EM) simulation, network synthesis, dielectric and suspended resonators, high-power notch and bandpass filters, and active filters. Dr. Snyder served the IEEE North Jersey Section as Chairman and 14-year Chair of the IEEE Microwave Theory and Techniques–Antennas and Propagation (MTT–AP) Chapter. He chaired the IEEE North Jersey Electron Device Society and Circuits and Systems chapters for ten years. He served as General Chairman for IMS2003, Philadelphia, PA, USA. He will be Emeritus Chair of IMS2018, Philadelphia, PA, USA. He was elected to the Administrative Committee in 2004, where he served as Chair of the TCC and liaison to the European Microwave Association. He served as an IEEE MTT Society Distinguished Lecturer (2007–2010), as well as continuing as a member of the Speakers Bureau. He was an Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (T-MTT), during which time he responsible for most of the submitted filter papers. He is a member of the American Physical Society, the American Association for the Advancement of Science, and the New York Academy of Science. He was the IEEE MTT Society President in 2011. He has been a Reviewer for the IEEE T-MTT, the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, the Progress in Electromagtnetic Research Symposium, and IET Microwave. He served seven years as chair of MTT-8 and continues in MTT-8/TPC work. He is the organizer of the annual IWS Conference in China and continues as a member of the IWS EXCOM. He was a two-time recipient of the Region 1 Award. He was the recipient of the IEEE Millennium Medal in 2000.

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Self-Biased Y-Junction Circulators Using Lanthanum- and Cobalt-Substituted Strontium Hexaferrites V. Laur, G. Vérissimo, P. Quéffélec, Senior Member, IEEE, L. A. Farhat, H. Alaaeddine, E. Laroche, G. Martin, R. Lebourgeois, and J. P. Ganne

Abstract—In this paper, we propose the application of polycrystalline lanthanum- and cobalt-substituted strontium hexaferrites in the realization of self-biased circulators. These materials present a high anisotropy field, dependent on the substitution rate, which makes it possible to reach operating frequencies in the millimeter-wave range. A first demonstrator was successfully designed and realized using a 20% rate of substitution Sr La Fe Co O . This circulator showed insertion losses of 1.79 dB and an isolation level of 28.1 dB at 41.4 GHz without magnets. Performances can be significantly improved by applying a low magnetic field H Oe . According to the literature, increasing the substitution rate makes it possible to increase the anisotropy field, and thus, the internal field. Consequently, a 30% substituted strontium hexaferrite was tested. It appears that the anisotropy field was not higher in this case. However, magnetic losses are much lower and enabled us to halve insertion losses of the self-biased circulator (0.87 dB at 41 GHz). Index Terms—Ceramics, circulators, ferrites, millimeter wave measurements, rectangular waveguide, Y-junction.

I. INTRODUCTION

C

IRCULATORS are still very important microwave components in modern telecommunication systems. These three-port nonreciprocal devices are genereally used to allow full-duplex communications (transmission and reception at the same time) with a single antenna. Isolators (one of the port connected to a matched load) are also used to protect the transmission equipment, especially amplifiers, from parasitic radiations or impedance mismatch. However, in spite of much progress over recent decades, these devices remain quite costly, heavy, and bulky, which limits their integration in telecommunication systems, especially in future millimeter-wave satellite front-ends at Q-band. Manuscript received June 30, 2015; revised August 17, 2015 and October 21, 2015; accepted October 22, 2015. Date of publication November 13, 2015; date of current version December 02, 2015. This work was supported by the French DGCIS in the framework of MM-WIN (Advanced millimeter-wave interconnects) European Euripides project. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. V. Laur, G. Vérissimo, and P. Quéffélec are with Lab-STICC, University of Brest, Brest, 29290 France (e-mail: [email protected]). L. A. Farhat, H. Alaaeddine, E. Laroche, and G. Martin are with Chelton Telecom & Microwave, Villebon-sur-Yvette, 91140, France. R. Lebourgeois and J. P. Ganne are with Thales Research & Technology, Palaiseau, 91120 France. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495218

Consequently, filter-diplexers or switches are sometimes preferred for duplexing applications as they provide a better means of integration. Thus, new technologies need to be developed to decrease the size of circulators and be competitive with passive (planar diplexer) and active (MMIC switches) technologies. The bulkiness of circulators can be reduced in two main ways: using planar technologies (microstrip for example) and/or removing permanent magnets through the use of preoriented materials. The use of hard ferrites, whose uniaxial particles have been oriented during pressing, is the most classical solution to avoid using magnets [1]–[6]. Demonstrators in the Ku [6] and Ka [3], [4] bands have been realized on this principle. However, the published performances of these circulators are often corrected [2], [5] and, thus, do not correspond to the performances “as-measured” due to the difficulties encountered in the modeling of such devices and in the control of magnetic ceramics properties. Ferromagnetic nanowires [7] or soft ferrite nanowires [8], embedded in porous dielectric substrates, were also studied to design self-biased circulators. Nevertheless, these technologies have not reached maturity and need to be improved in order to be integrated in telecommunication systems. In a previous paper, presented during the 2015 IEEE International Microwave Symposium [9], we proposed to use lanthanum and cobalt doped strontium hexaferrites to realize millimeter-wave circulators. It is of particular interest that we obtained insertion losses of 1.79 dB and an isolation level of 28.1 dB at 41.4 GHz without magnets by using Sr La Fe Co O hexaferrites, here called La Co -SrM. We also observed that a low external magnetic field allowed performances to be significantly improved. As a consequence, we concluded that a material with a higher anisotropy field should make it possible to get lower insertion losses. According to the literature, increasing the substitution rate makes it possible to increase the anisotropy field. Consequently, a 30% substituted strontium hexaferrite was tested. In this paper, in Section III-A., we will first make a recap of the results obtained using La Co -SrM hexaferrites. Then, in Section III-B. we will show that using a higher substitution rate Sr La Fe Co O enabled us significantly improve the performances of the mm-wave self-biased circulator. These results will be discussed and compared with those of the literature. II. CHARACTERIZATION AND SIMULATION METHODS In this study, we used polycrystalline preoriented hexaferrites to design and realize self-biased circulators at millimeter-

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wave frequencies. Strontium hexagonal ferrites with a magnetoplumbite structure exhibit a very high anisotropy field of about 19 kOe and a high remanence to saturation ratio, making it possible to realize mm-wave self-biased circulators. This material is the ferrite the most used for such applications in the literature [1]–[6]. It has allowed the successful realization of selfbiased circulators up to 30 GHz. Some experimental demonstrations around 40 GHz have also been carried out. However, performances at this frequency in self-biased working mode are slightly degraded because of the proximity to the natural (without an applied magnetic field) gyromagnetic resonance frequency (FMR). In order to reach a working frequency of 40 GHz, two solutions can be considered: applying an external magnetic field in order to shift FMR at higher frequencies or using materials with higher anisotropy fields. Lanthanum and cobalt substitutions make it possible to increase the anisotropy field of strontium hexaferrites. Sr La Fe Co O which will be called La Co -SrM was therefore used in this work. The preparation route is based on a conventional ceramic process. First, powder with the targeted composition is prepared starting from SrCO , La O and Co O raw materials and by using solid-state reaction. Then, the hexaferrite powder is magnetically oriented and pressed before sintering. The magnetostatic properties of (La,Co)-SrM hexaferrites were studied through Superconductive Quantum Interference Device (SQUID) measurements (Quantum Design MPMS XL). These measurements enabled us to investigate the magnetization saturation level of the samples and the squareness of the hysteresis loops. Ansys HFSS software was used to model the circulators. This electromagnetic (EM) software integrates Polder's model [10] which makes it possible to predict the permeability spectra of magnetized ferrites from their saturation magnetization , . However, this internal field and magnetic losses model is well-suited to the case of saturated ferrites and therefore needs to be modified in the case of ferrites in their remanent state, where the magnetization level to be considered is (1) is the remanent magnetization along z-axis and is the relative remanent level that will be measured using SQUID measurement. Contrary to saturated soft ferrites, the internal field in preoriented hexaferrites in their remanent state is not mainly governed by an applied magnetic field but rather by anisotropy and demagnetizing fields. In our case, the internal field is calculated by the following equation:

where

(2) is the internal magnetic field along z-axis, the where applied magnetic field, the anisotropy field and the demagnetizing coefficient calculated by using Aharoni equations [11]. A previous version of a 40-GHz Y-junction rectangular waveguide circulator integrating spinel ferrites was used to test La Co -SrM materials. EM simulations were performed to optimize the dimensions of the La Co -SrM pucks for a self-biased working mode at 40 GHz (Fig. 1).

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Fig. 1. Simulation model of the self-biased circulator in rectangular waveguide technology integrating La Co -SrM hexaferrite pucks.

Fig. 2. Measurement setup used for the microwave characterization of the circulators.

Then, hexaferrite pucks were machined by keeping their c-axis (direction of the magnetization) perpendicular to the plane of the pucks. These flat cylinders were glued to the center of the Y-junction circulator. Circulators were measured with a Vector Network Analyzer (Rhode&Shwarz ZVA67) in the 40–60 GHz for the first circulator and between 35 and 50 GHz for the second one (Fig. 2). Thru-Reflect-Line (TRL) calibration procedure was performed in order to shift the reference plane after the coaxial-to-waveguide transitions. Self-biased circulators were measured in isolator mode (a load was connected to one of the ports). An external magnetic dc field, applied using an electromagnet, was also used to observe the behavior of the circulator as a function of the applied biasing field. The static magnetic field was measured with a gaussmeter during the measurement. III. CHARACTERIZATION AND SIMULATION RESULTS OF MATERIALS AND DEVICES A. Results Using La Co

-SrM

La Co -SrM preoriented ceramics were synthesized by the process described in part II. Pure strontium hexaferrites were synthesized using the same process in order to compare their static magnetic properties. Fig. 3 presents M(H) hysteresis loops of a 20% substituted SrM Sr La Fe Co O and of a nonsubstituted one. We observed a slight increase of the coercive field of doped SrM

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Fig. 3. M(H) hysteresis loops measured with a SQUID magnetometer on par-SrM materials (along the allelepiped samples made of SrM and La Co mm mm mm c-axis). Dimensions of the samples: SrM and La Co -SrM mm mm mm .

TABLE I DIELECTRIC AND MAGNETIC PROPERTIES OF La Co

Fig. 4. Internal field of a La Co -SrM cylinder as a function of its aspect ratio without an applied external magnetic field.

-SrM HEXAFERRITES

hexaferrites compared with the nonsubstituted ones H Oe and H La Co -SrM Oe). Moreover, the La Co -SrM cycle presents a higher squareness: the remanent magnetization reaches 90% of the saturation one RRL while this value is only 85% of the saturation magnetization for a nonsubstituted SrM material. Even without magnets, magnetic dipoles remain well aligned along the c-axis (direction perpendicular to the plane of the sample in our case) proving the strong potential of La Co -SrM ceramics for the design of self-biased circulators. The properties of La Co -SrM ceramics, slightly refined compared to [9], are given in Table I. Saturation magnetization and squareness values were extracted from SQUID measurements. The other values were at first approximated by using published results [8], [12], [13], and then refined by using retro-simulations of measurements. The dielectric constant value was also measured by coaxial line measurements in the 1–18 GHz frequency band that gave a slightly lower value of permittivity than the one given in [8]. This material presents a higher anisotropy field than pure SrM but its magnetic losses near resonance remain quite high. Equation (2) enabled us to calculate the internal field of a La Co -SrM cylinder as a function of its aspect ratio (height divided by radius). It appears that the internal field is much lower than the anisotropy field because of the high magnetization level of this material (Fig. 4). A previous constructed rectangular waveguide structure, optimized for spinel ferrites at 50 GHz, was modified to test the La Co -SrM materials. The evolution of the internal field was taken into account for the simulations. The optimized dimensions of La Co -SrM disks are a radius of 0.87 mm and a height of 0.2 mm. These dimensions lead to a demagnetizing factor of . By using (1) and (2), we calculated the and self-magnetization of the disks along z axis their internal field Oe. Polder's model thus allows us to obtain an approximate value of the permeability spectra (Fig. 5). The ferrite disk shows a

Fig. 5. Calculated real and imaginary parts of the permeability spectra of La Co -SrM disks ( mm and mm) without an applied field.

Fig. 6. Simulated S-parameters of the self-biased circulator integrating La Co -SrM hexaferrite pucks.

self-FMR (without applied field) at 49.1 GHz. Even if the broadening of magnetic losses due to dispersion in magnetic moments inside the sample is not taken into account here, this high value of FMR appears to be sufficient to design a self-biased circulator at around 40 GHz. Fig. 6 presents the simulated S-parameters of the self-biased circulator integrating La Co -SrM disks. A maximum isolation level Iso dB appears at 41.3 GHz while insertion losses are minimum at 42.2 GHz IL dB . However, return losses remains moderate in the operating band (RL dB at 42.2 GHz). The simulated performances appear to be sufficient to experimentally demonstrate the potential of La Co -SrM materials, and thus it was decided to realize this structure as a first proof of concept. The circulator is shown in Fig. 7. This Y-junction circulator was realized by using WR-19 rectangular waveguides.

LAUR et al.: SELF-BIASED Y-JUNCTION CIRCULATORS USING LANTHANUM- AND COBALT-SUBSTITUTED STRONTIUM HEXAFERRITES

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Fig. 9. Measured (solid lines) and simulated (dashed lines) S-parameters of the -SrM hexaferrites with an applied self-biased circulator integrating La Co biasing field of 2100 Oe. Fig. 7. Photograph of the circulator in rectangular waveguide technology (in-SrM pucks integrated into the circulator). sert: internal view of La Co

isolation and return losses were 16.5 dB and 24.6 dB, respectively. One should note that isolation remained above 15 dB from 42.58 to 56.06 GHz (more than 13 GHz) resulting in a relative bandwidth of 27.3%. However, insertion losses reached 3 dB from 47.62 GHz and thus limit the working frequency band of the device to 11.2% (42.58–47.62 GHz). Comparisons between simulation and measurement were in a good agreement. One should note that the parameters of the ferrite were kept the same for this simulation (cf. Table I) and that only the internal field was modified. The overestimation of the relative bandwidth seems again to be due to the inhomogeneity of the internal field not taken into account in the simulation. B. Results Using La Co

Fig. 8. Measured (solid lines) and simulated (dashed lines) S-parameters of -SrM hexaferrites without an the self-biased circulator integrating La Co applied biasing field.

Impedance matching is achieved through the change of sections of the WR-19 waveguide near the Y-junction. La Co -SrM disks were glued at the center of the circulator. Measured performances of the circulator in the 40–60 GHz frequency band without an applied dc field are shown in Fig. 8. Insertion losses are the lowest dB at 41.4 GHz. At this frequency, isolation is 28.1 dB. However, at the same time, return losses remain low RL dB and limit the performances of the device. The circulator demonstrates a 15 dB . A very good bandwidth of 3 GHz RBW dB agreement was demonstrated between the simulated and measured S-parameters (Fig. 8). However, the simulated fractional bandwidth appears to be slightly higher than the measured one. This phenomenon could be due to the inhomogeneity of the internal field (spatial dispersion of the magnetic moments in the polycrystalline hexaferrite disks), which is not taken into account in the simulation. This device was placed into an electromagnet to study the effect of an external biasing field (applied perpendicularly to the plane of the ferrite pucks) on its performances. Measured S-parameters when a 2100-Oe magnetic field was applied to the circulator are presented in Fig. 9. In this case, minimal insertion losses of 1.23 dB were observed at 43.2 GHz. At the same time,

-SrM

We demonstrated above that a low static magnetic field allows us to significantly improve the performances of a La Co -SrM-based circulator. Now, Grössinger et al. demonstrated in [12], [13] that anisotropy field increases as lanthanum and cobalt rate increases in Sr La Fe Co hexaferrites at room temperature. As can be seen in (2), increasing the anisotropy field could make it possible to increase the internal field without applying an additional external magnetic field. As a consequence, we undertook to replace the La Co -SrM disks by hexaferrites with a higher rate of substitution. Thus, Sr La Fe Co hexaferrites, here called La Co -SrM, were selected. Ferrite disks with the same dimensions as those made from La Co -SrM, were machined and glued in the circulators. The circulator was then measured between 35 and 50 GHz, first without an applied field, and then with increasing biasing field using an electromagnet. Fig. 10 presents the measured S-parameters of the La Co -SrM-based circulator without an applied biasing field. No increase in frequency was observed, as expected for a material with a higher anisotropy field. However, a significant decrease of insertion losses was demonstrated. Thus, minimum insertion losses of 0.87 dB at 41 GHz were measured. At this frequency, an isolation level of 16.5 dB together with return losses of 12.6 dB were demonstrated. This circulator also presents a 15 dB bandwidth of 1.3 GHz RBW . dB Retro-simulations using Ansys HFSS enabled us to extract the properties of La Co -SrM hexaferrites. A good agreement between measurements and simulations was achieved

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TABLE III MEASURED PERFORMANCES OF La Co -SrM-BASED CIRCULATORS WITH AND WITHOUT AN APPLIED BIASING FIELD

Fig. 10. Measured (solid lines) and simulated (dashed lines) S-parameters of -SrM hexaferrites without an the self-biased circulator integrating La Co applied biasing field.

TABLE II DIELECTRIC AND MAGNETIC PROPERTIES OF La Co

Relative bandwidth calculated for Isolation level less than

15 dB.

-SrM HEXAFERRITES

Fig. 12. State of the art. “As-measured” insertion losses of hexaferrite-based circulators as a function of frequency.

As observed earlier, the measured bandwidth is significantly lower than that predicted by simulations, probably because of the broadening of FMR losses linked to a slight spatial inhomogeneity of the internal field. Fig. 11. Measured (solid lines) and simulated (dashed lines) S-parameters of -SrM hexaferrites with an apthe self-biased circulator integrating La Co plied field of 1600 Oe.

with the parameters listed in Table II (Fig. 10). It appears that the decrease of insertion losses is mainly related to the strong decrease of magnetic losses for this composition ( about three times lower than that of La Co -SrM). We also observed that the anisotropy field is quite similar to that of La Co -SrM, which is consistent with the similar working frequencies in both cases but not in agreement with previous published studies concerning this family of ferrites [12], [13]. Additional material characterizations will have to be performed to explain this difference of behavior compared to the literature. This circulator was placed in an electromagnet in order to investigate its behavior with an increasing biasing field. The best performances were achieved for an applied 1600-Oe magnetic field. In these conditions, insertion losses of 0.21 dB were measured at 42.9 GHz. At the same time, isolation and return losses were 41.3 and 25.6, respectively. Isolation level remained lower than 15 dB between 41.5 and 45.1 GHz leading to a relative bandwidth of 9.7%. One should note that, in contrast to La Co -SrM, insertion losses were kept quite low in this bandwidth, with a maximum value of 0.78 dB. Simulated S-parameters were once again in good agreement with the measured performances (Fig. 11).

C. Discussion The measured performances of La Co -SrM-based circulators are listed in Table III. A self-biased operating mode was demonstrated by using La Co -SrM hexaferrites. When La Co -SrM were used, insertion losses without applied biasing field were significantly decreased due to lower magnetic losses . In both cases, a low external magnetic field made it possible to generally improve performances except for the isolation level of La Co -SrM-based circulator which was decreased. We are now working on the optimization of the Y-junction in order to improve the self-biased performances of the circulator. Fig. 12 allows us to compare our results with those found in the literature. Only “as-measured” performances are presented [1]–[4], [6], [14]–[17], i.e., external tuning or post-treatments (feed lines or removal of connector losses) are not considered. This figure presents measured insertion losses of hexaferritebased circulators as a function of frequency. The circulator designed by Wang et al. [6] is distinguished from other results by its very low frequency of operation which is, to our knowledge, the lowest one achieved for a self-biased circulator. Circulators made of preoriented hexagonal ferrite composites have also been realized [16], [17], but these technologies still need to progress and mature in order to provide lower insertion losses. The other results show operating frequencies between 24 and 40 GHz. In this context, our performances are remarkable owing

LAUR et al.: SELF-BIASED Y-JUNCTION CIRCULATORS USING LANTHANUM- AND COBALT-SUBSTITUTED STRONTIUM HEXAFERRITES

to the low level of insertion losses and the high frequency of operation. Such high working frequencies seem to be unreachable using pure SrM materials below FMR.

IV. CONCLUSION This work concerns the modeling and the characterization of a self-biased circulator around 40 GHz. The use of lanthanum-cobalt substituted strontium hexaferrites made it possible to increase the operating frequency of the circulator. Static properties were measured to confirm the increase of squareness (remanent magnetization) and anisotropy field compared to a nonsubstituted SrM. A circulator was realized in rectangular waveguide technology and characterized using Sr La Fe Co O preoriented ferrites. Without magnets, insertion losses of 1.79 dB and isolation of 28.1 dB were measured at 41.4 GHz. The application of a low magnetic field noticeably improves the performances of the device. The use of lanthanum-cobalt doped SrM with a higher substitution rate Sr La Fe Co O decreased insertion losses signifidB at 41 GHz). cantly without an applied field (IL We are convinced that we can improve the performances of these self-biased circulators. A new Y-junction specifically designed for La Co -SrM materials will soon be developed. In parallel, planar circulators will be realized and characterized in order to demonstrate the high potential of these materials for the design of integrated self-biased circulators.

REFERENCES [1] M. A. Tsankov and L. G. Milenova, “Design of self-biased hexaferrite waveguide circulators,” J. Appl. Phys., vol. 73, no. 10, pp. 7018–7020, May 1993. [2] S. A. Oliver, P. Shi, W. Hu, H. How, S. W. McKnight, N. E. McGruer, P. M. Zavracky, and C. Vittoria, “Integrated self-biased hexaferrite microstrip circulators for millimeter-wavelength applications,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 2, pp. 385–387, Feb. 2001. [3] X. Zuo, H. How, S. Somu, and C. Vittoria, “Self-biased circulator/ isolator at millimeter wavelengths using magnetically oriented polycristalline strontium M-type hexaferrite,” IEEE Trans. Magn., vol. 39, no. 5, pp. 3160–3162, Sep. 2003. [4] N. Zeina, H. How, C. Vittoria, and R. West, “Self-biasing circulators operating at Ka-band utilizing M-type hexagonal ferrites,” IEEE Trans. Magn., vol. 28, no. 5, pp. 3219–3221, Sep. 1992. [5] B. K. O'Neil and J. L. Young, “Experimental investigation of a selfbiased microstrip circulator,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 7, pp. 1669–1674, Jul. 2009. [6] J. Wang, A. Yang, Y. Chen, Z. Chen, A. Geiler, S. M. Gillette, V. G. Harris, and C. Vittoria, “Self-biased Y-junction circulator at Ku band,” IEEE Microw. Wireless Components Lett., vol. 21, no. 6, pp. 292–294, Jun. 2011. [7] M. Darques, J. Medina, L. Piraux, L. Cagnon, and I. Huynen, “Microwave circulator based on ferromagnetic nanowires in an alumina template,” Nanotechnology, vol. 21, pp. 145208.1–145208.4, 2010. [8] J. Wang, A. Geiler, P. Mistry, D. R. Kaeli, V. G. Harris, and C. Vittoria, “Design and simulation of self-biased circulators in the ultra high frequency band,” J. Magn. Magn. Mater., vol. 324, pp. 991–994, 2012.

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[9] V. Laur, G. Vérissimo, P. Quéffélec, L. A. Farhat, H. Alaaeddine, J. C. Reihs, E. Laroche, G. Martin, R. Lebourgeois, and J. P. Ganne, “Modeling and characterization of self-biased circulators in the mm-wave range,” in Proc. IEEE Int. Microwave Symp., 2015. [10] D. Polder, “On the theory of ferromagnetic resonance,” Phil. Mag., vol. 40, pp. 99–115, Jan. 1949. [11] A. Aharoni, “Demagnetizing factors for rectangular ferromagnetic prims,” J. Appl. Phys., vol. 83, no. 6, pp. 3432–3434, 1998. [12] R. Grössinger, J. C. Tellez Blanco, F. Kools, A. Morel, M. Rossignol, and P. Tenaud, “Anisotropy and coercivity of M-type Ba and Sr-ferrites containing La and Co,” in Proc. Int. Conf. Ferrites, 2000, pp. 428–430. [13] R. Grösssinger, C. Tellez Blanco, M. Küpferling, M. Müller, and G. Wiesinger, “Magnetic properties of a new family of rare-earth substituted ferrites,” Phys. B: Condensed Matter, vol. 327, pp. 202–207, Apr. 2003. [14] P. Shi, H. How, X. Zuo, S. A. Oliver, N. E. McGruer, and C. Vittoria, “Application of single-crystal scandium substituted barium hexaferrite for monolithic millimter-wavelength circulators,” IEEE Trans. Mag., vol. 37, no. 6, pp. 3941–3946, Nov. 2001. [15] J. A. Weiss, N. G. Watson, and G. F. Dionne, “New uniaxial-ferrite millimeter-wave junction circulators,” in Proc. IEEE Int. Microwave Symp., 1989, pp. 145–148. [16] C. Blengeri, T. Casad, A. Abburi, D. N. McIlroy, W. J. Yeh, and J. L. Young, “Fabrication of bulk, self-bias barium ferrites for microwave circulator applications,” J. Mater. Sci. Eng., vol. 5, pp. 314–318, 2011. [17] T. Boyajian, D. Vincent, S. Neveu, M. Leberre, and J. Rousseau, “Coplanar circulator made from composite magnetic material,” in Proc. IEEE Int. Microwave Symp., 2011. V. Laur, photograph and biography not available at the time of publication.

G. Vérissimo, photograph and biography not available at the time of publication.

P. Quéffélec, (M’99–SM’07) photograph and biography not available at the time of publication.

L. A. Farhat, photograph and biography not available at the time of publication.

H. Alaaeddine, photograph and biography not available at the time of publication..

E. Laroche, photograph and biography not available at the time of publication.

G. Martin, photograph and biography not available at the time of publication.

R. Lebourgeois, photograph and biography not available at the time of publication.

J. P. Ganne, biography and photograph not available at the time of publication.

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A 2.45 GHz Phased Array Antenna Unit Cell Fabricated Using 3-D Multi-Layer Direct Digital Manufacturing Thomas P. Ketterl, Member, IEEE, Yaniel Vega, Nicholas C. Arnal, John W. I. Stratton, Student Member, IEEE, Eduardo A. Rojas-Nastrucci, Student Member, IEEE, María F. Córdoba-Erazo, Student Member, IEEE, Mohamed M. Abdin, Student Member, IEEE, Casey W. Perkowski, Paul I. Deffenbaugh, Kenneth H. Church, Member, IEEE, and Thomas M. Weller, Senior Member, IEEE

Abstract—This paper reports on the design, fabrication and characterization of a 3-D printed RF front end for a 2.45 GHz phased array unit cell. The printed unit cell, which includes a circularly-polarized dipole antenna, a miniaturized capacitive-loaded open-loop resonator filter and a 4-bit phase shifter, is fabricated using a direct digital manufacturing (DDM) approach that integrates fused deposition of thermoplastic substrates with micro-dispensing for deposition of conductive traces. The individual components are combined in a passive phased array antenna unit cell comprised of seven stacked substrate layers with seven conductor layers. The measured return loss of the unit cell is dB across the 2.45 GHz ISM band and the measured gain is dBi including all components. Experimental and simulation-based characterization is performed to investigate electrical properties of as-printed materials, in particular the inhomogeneity of printed thick-film conductors and substrate surface roughness. The results demonstrate the strong potential for fully-printed RF front ends for light weight, low cost, conformal and readily customized applications. Index Terms—Additive manufacturing, dipole antenna, direct digital manufacturing, open loop resonator filter, switched-line phase shifter, 3-D printing.

I. INTRODUCTION

A

DDITIVE MANUFACTURING (AM) is a technology that is maturing from a cost-effective rapid prototyping solution to one suitable for low volume production in a diverse array of fields that includes RF and microwave design. As reported in [1], the AM market has grown to over 3000 organizations with an estimated $9.46 billion for printed and thin film electronics. In the microwave field, frequency selective surManuscript received July 02, 2015; revised September 22, 2015, October 21, 2015, October 23, 2015; accepted October 23, 2015. Date of publication November 11, 2015; date of current version December 02, 2015. This work was supported in part by US Air Force Research Laboratoryunder contract #FA865014-C-2421. The material was assigned a clearance of CLEARED on 18 Feb 2015 (Case Number: 88ABW-2015-0603). The work was also supported by the National Science Foundationunder grant #ECCS-1232183. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, 17–22 May 2015. T. P. Ketterl, Y. Vega, N. C. Arnal, J. W. I. Stratton, E. A. Rojas-Nastrucci, M. F. Cordoba-Erazo, M. M. Abdin, and T. M. Weller are with the Department of Electrical Engineering, University of South Florida, Tampa, FL 33620 USA (e-mail: [email protected]). C. W. Perkowski, P. I. Deffenbaugh, and K. H. Church are with Sciperio Inc., Orlando, FL 32826 USA. Digital Object Identifier 10.1109/TMTT.2015.2496180

faces, electrically-small antennas and RF MEMS devices are among the demonstrated applications [2]–[10]. While there are numerous AM processes, including a variety of all metal and all plastic methods, in this work the focus is on a multi-material approach that integrates fused deposition of thermoplastics with micro-dispensing of conductive pastes. This integrated direct digital manufacturing (DDM) process enables the realization of multi-layer, high frequency structural electronic designs in a single build. Conceptually, the approach merges the flexible 3-D design capability afforded by low temperature co-fired ceramic (LTCC) technology with the large area format achievable with multi-layer printed circuit boards (PCBs). The feature sizes currently achievable with DDM are equivalent to those of PCBs, with m minimum layer thickness and 50–100 m line widths on smooth surfaces, though the arbitrary volumetric design capability is specific to DDM. With the integration of chip and packaged semiconductor devices, printed electronic systems are realizable. This paper reports on the demonstration of components of a 2.45 GHz RF front-end that are fabricated using the DDM approach. These components comprise a unit cell of a phased array antenna (PAA) (Fig. 1) which is the first known demonstration of an AM-produced multi-layer microwave circuit. One of the presented components is a 2.45 GHz circular polarized (CP) antenna consisting of a pair of crossed-dipole elements that fits on a 6 6 cm 3-D-printed substrate. Another CP crosseddipole antenna on a 10 10 cm substrate is introduced that includes a high impedance surface (HIS) to isolate the antenna from the back ground plane, and consists of four metal layers, five substrate layers and thru conductive vias. A second component is a square open-loop resonator 2.45 GHz band-pass filter that is miniaturized using discrete capacitive loading. Finally, a multi-bit phase shifter using packaged MMIC switches is demonstrated. The printed designs use acrylonitrile butadiene styrene (ABS) thermoplastic as the substrate and DuPont CB028 thick-film Ag paste for the conductors. Preliminary results of this work are presented in [6] and are expanded in this paper. The 3-D fabrication process described in Section II now includes a detailed review of selected dielectric and conductive materials used in additive manufacturing. The section also includes DC and RF analysis of

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KETTERL et al.: 2.45 GHZ PHASED ARRAY ANTENNA UNIT CELL

Fig. 1. Phased array antenna (PAA) unit cell fabricated with direct digital manufacturing (DDM). The front-end consists of a phase shifter, band-pass filter, balun and antenna (on bottom of stack). The unit cell dimensions are 6 cm in length and width, with a height of 5.2 mm.

the conductive material used in this investigation in order to provide an understanding of the high frequency performance of the individual components described in following sections. Section III expands on the design and testing of the printed CP antennas introduced in [6] by providing return loss and radiation pattern measurements for antennas that include lumped element phase shifters in the feed network and a new HIS layer design that provides more bandwidth and is more compatible with DDM processing. A new filter topology that is miniaturized using a DDM-printed coupling capacitor is described in Section IV, along with an analysis of surface roughness effects on filter performance. Section V adds the design and analysis of a printed multi-bit shifter to the switched-line phase shifter design introduced in [6] and a detailed analysis of the phase shifter performance has been added. A major contribution to the material in [6] is presented in Section VI, which describes the first results for the PAA unit cell. II. 3-D FABRICATION Numerous additive manufacturing methods are applicable for RF/microwave applications. Fused deposition modeling (FDM), stereolithography and different jetting techniques are used to form polymer structures that can act as substrates and packaging. For (thick or thin) films such as conductive interconnects, micro-dispensing, ink-jet, aerosol jet and selective laser structuring are among the available approaches. In this work a DDM printing system that integrates micro-dispensing and fused deposition is used. Details of the DDM fabrication approach and various materials that are compatible with DDM are described in this section. A. Multi-Material Micro-Dispensing/Fused Deposition Fabrication Approach An nScrypt 3-Dx-300 printing system is the primary fabrication tool used in this work. This tool combines a micro-dispensing head with picoliter control for depositing pastes

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Fig. 2. ABS-CB028 printing process: (1) 0.2 mm thick ABS on a 90 C bed, (2) CB028 antenna elements, (3) 3 mm ABS antenna substrate, (4) CB028 ground plane with vias, (5) 0.5 mm ABS microstrip substrate, and (6) CB028 microstrip feedline. The photograph at the bottom shows the CB028 line edge quality; each tick mark is 400 um [6].

with viscosities ranging from 1.0 to 1.0e6 cP, with a fused deposition system that uses tip orifices down to 12.5 m. Both deposition systems are on a common gantry with 2 m position accuracy. The printing process used for the circularly-polarized dipole antenna (Section III) is illustrated in Fig. 2. It consists of 3 FDM steps to deposit ABS (UltiMachine) for a base layer, a substrate layer for the antenna elements, and a substrate layer for the feedline. There are also 3 micro-dispensing steps to deposit CB028 for the antenna elements, an intervening ground plane layer and the microstrip feedline. The printing bed temperature is held at 90 C which allows the CB028 to be cured in place. No additional steps to smooth the ABS surfaces are required to obtain excellent line edge quality (see bottom of Fig. 2). With the described fabrication method, the main limiting factor to edge definition is the tendency of the CB028 silver paste to flow laterally in a somewhat non-uniform fashion as the surface topography of the ABS varies. CB028 is a particle flake loaded thixotropic material, and its time dependent shear thinning feature implies that the material will move from a thick state to a thin state and then back again to a thick state. If a surface is perfectly smooth, a thixotropic material will still be micro shaped along the edges due to surface tension. The range of viscosity for CB028 is from 15 000 to 30 000 cP, and because this is not extremely high the effects of surface tension will have an impact on edge definition. For the ABS surface roughness typical of this work, it is found that line width variations can be controlled to within m, independent of nominal line width. Accordingly, minimum line dimensions on the order of 100 m are possible to achieve with an optimized process, without noticeable variation in the characteristic impedance of the lines. To improve yield, a minimum line width of m has been followed in this work. On smoother substrates such

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TABLE I RF MATERIAL PROPERTIES OF FDM-COMPATIBLE MATERIALS

TABLE II TENSILE STRENGTH OF FDM-COMPATIBLE MATERIALS

TABLE III HEAT DEFLECTION TEMPERATURE (HDT) OF FDM COMPATIBLE MATERIALS

TABLE IV BULK DC CONDUCTIVITY OF CB028 SILVER PASTE

C. DC Properties of Conductive Material The material used for conductive lines in this work is DuPont CB028. This is a thick-film paste that consists of silver flakes in a polymer matrix. When cured, the solvent in the paste evaporates and the polymer shrinks, drawing the flakes into contact. Typical film thickness in this work is m. One basic parameter which controls the conductivity of CB028 is the temperature at which it is cured. Table IV shows four key data points on the DC conductivity of CB028 measured using a four point probe and van der Pauw [18] equations on a 10 mm square patch. As shown in the table the nominal DC conductivity for the CB028 films used in this work is 2.62e6 S/m. D. RF Properties of Conductive Material

as Kapton, line dimensions down to 25 achieved.

m are consistently

B. Properties of Dielectric Materials Material selection is critical when using 3-D printing to manufacture functional devices. Several material property areas are examined herein, including the RF dielectric properties, strength and heat deflection temperature (HDT). The RF material properties of ABS, polycarbonate (PC), and ULTEM™ (polyetherimide (PEI), Stratasys ULTEM 9085) are similar to one another [13] but the loss tangent of polylactic acid (PLA) is higher [12]. A summary is provided in Table I. The tensile strength of ABS, PC, ULTEM™ and PLA is compared using the ASTM D638 standard (Table II). Measured data of maximum tensile strength is obtained using a digital pull tester of the actual materials used in the phased array antenna unit cell (see Section VI). PC and ULTEM tensile strength from Stratasys is stated as bulk, i.e., not printed, which is slightly stronger than as-printed. The heat deflection temperature (HDT) of a given material (Table III) must fulfill end use and manufacturing requirements. Softening occurs as a material passes its HDT and internal stresses introduced by the non-uniform cooling of the layers, when relieved by softening can cause warping. For instance, the HDT of ABS is 88 C @ 1.8 MPa so for the purposes of this paper, 90 C is the maximum process temperature used in order to achieve good results. PLA is not a good choice because of its low HDT. ABS, with an HDT temperature of 88 C, is selected for this work due to its printability, moderate temperature range, and low dielectric loss.

FDM printing generates relatively rough surfaces and the curing process of the micro-dispensed conductive paste results in an inhomogeneous particle distribution on the micron scale. Due to these features of the printed materials, the effective RF conductivity of the silver paste can be substantially different from the DC values shown in Table IV and can also be spatially dependent. Since conventional methods used to measure the electrical conductivity of printed traces provide an averaged value of the electrical resistivity, a measurement technique that combines the ability to spatially resolve variations in conductivity and surface relief is desirable. These variations can affect the performance of the printed structures particularly for high frequency applications. Near-Field Microwave Microscopy (NFMM) has proven to be an important tool to image electromagnetic properties of homogenous and uniform bulk insulators [19], conductors [20], [21] and semiconductors [22] with subwavelength resolution. Recently, NFMM has also been used to image the electrical resistance of resistors fabricated using micro-dispensing [23]. Advantages of using NFMM to characterize printed samples are that variations in conductivity and topography of a sample can be tracked with spatial resolution that is about the size of the sensing tip. Additionally, NFMM is a non-contact, non-destructive technique and provides the electromagnetic properties in the microwave frequency regime rather than DC values. NFMM systems commonly utilize a resonant technique for sensing changes in material properties. The basic components in this type of instrument consist of a source and a resonator terminated in a sensing element which can be an aperture or a conductive tip whose size is smaller than the wavelength at the operating frequency. The material under test (MUT) is placed in close proximity to the probe tip at a distance smaller than the tip

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Fig. 4. Cross-section SEM micrograph of (a) CB028 interconnect and (b) bim [6]. nary matrix after image processing. The silver flake size is

Fig. 3. Conductivity (a) and topography (b) images of CB028 over an area of 100 m 100 m obtained using the NFMM.

size in order for the resonator to be sensitive to the MUT properties. Changes in the resonant properties of the probe can be correlated with the material properties by calibrating the NFMM using samples with known material properties [24], [25]. Herein, a dielectric resonator-based NFMM operating at 5.73 GHz is used for imaging simultaneously the electrical conductivity and topography of CB028 on glass. Design, experimental setup and a lumped element circuit model of the NFMM used in this work are described in [23], [26], [27]. The NFMM is calibrated using samples with known electrical conductivity in order to correlate the measured Q with . The scanning is performed at a tip-sample distance of 3 m in steps of 2 m. Fig. 3(a) and (b) show the electrical conductivity and topography images obtained, respectively. The distribution in Fig. 3(a) indicates that the conductivity varies between 0.6e6 S/m and 2e6 S/m. Higher conductivity regions are observed over thinner areas in the topography. The conductivity variation observed in the NFMM data can be understood using image processing to analyze cross-sectional SEM micrographs of CB028 films. A typical micrograph for a 25.19 m thick sample is shown in Fig. 4(a). An intensity histogram for the sample can be used to isolate the silver flakes from the polymer matrix and convert the micrograph to a binary matrix (Fig. 4(b)), and from there the silver flake density distribution can be determined. Fig. 5 shows the measured silver ink particle density as a function of the sample height for films of several thicknesses.

Fig. 5. Silver particle density versus distance from bottom of film for CB028 films of varying thickness.

The results confirm that the conductive layer is not homogenous. Furthermore, this data shows particle settling, where the density decreases as a function of distance from the bottom of the film. Also, thicker samples (40 m–50 m) show lower density for the top layer, when compared with thinner samples (20 m–40 m). The results of the cross-section analysis and NFMM imaging confirm the inhomogeneous nature of the micro-dispensed CB028 films, and the reduction in silver particle density to % near the upper 5–8 um can explain the differences between the measured DC conductivity (2.6e6 S/m) and the range of conductivities observed in the NFMM images (0.6e6 to 2.0e6 S/m). This inhomogeneity will extend to the edges of printed lines where high RF current densities exist, thus contributing to an effective RF conductivity that must be accounted for in circuit design and numerical electromagnetic simulation. This effect, combined with surface roughness effects, is discussed in more detail in the following sections. It is noted that although the NFMM characterization data applies to GHz, the extracted values of conductivity are expected to remain stable at lower frequencies as long as the skin depth is much less than the conductor thickness. This conclusion is supported by results for several demonstration circuits tested in the 2–6 GHz frequency range.

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III. CIRCULARLY-POLARIZED DIPOLE ANTENNA In this section 3-D printed antennas are described to demonstrate the multilayer DDM process. Tight control of the antenna and substrate dimensions is required in order to achieve the correct resonant frequency and operation bandwidth. The manufacturing process also includes the fabrication of via sections that connect the antennas through one or more layers to the bottom feed layer, as well as buried vias that connect the bottom layer to the internal ground plane. The antennas presented in this study are circular polarized (CP) designs consisting of a pair of crossed-dipole antenna elements. To provide circular polarization, two arms of one dipole are fed by a 0–180 network and each is connected through a 90 phase shift section to a neighboring arm of the second dipole to accomplish the quadrature phase between the two elements. The antennas are required to be low profile with a thickness of less than 4 mm, and include a ground plane to separate the bowtie elements on top of the substrate and the signal feed layer fabricated on the bottom surface. The feed layer will also include the individual front-end components for the unit cell described in Section VI. Since placing an antenna in close proximity to a ground plane (much less than ) generally degrades the overall performance [28], a high impedance surface (HIS) layer will be used in one design to mitigate the ground effects and achieve high directivity in the broadside direction [29]. The two variations of the CP antenna described herein are shown in Fig. 6 and denoted as Design 1 (without an HIS layer) and Design 2 (with an HIS layer). Design 1 is a simpler geometry that requires fewer printing steps and provides a baseline for the radiation efficiency determination, and particularly the impact of the HIS layer used in the more complex Design 2. Both designs use lumped element pi network phase shifters to implement the 90 phase shift sections between antenna arms. A surface mount balun provides the 0–180 phase for the differential feed. Each bowtie element is designed to have a 50 input impedance, including the 90 lumped phase shifter. The resonant frequency, besides being dependent on the bowtie length and via height, is very sensitive to the antenna via diameter and a value of 0.8 mm is needed to achieve the required frequency of 2.45 GHz. Simulations and tuning of the antenna models were performed in Ansys HFSS. Substrate-scalable Modelithics models of Johanson L-07C series inductors and R07S series capacitors with a 0402 body style were used in simulations to achieve the proper phase shift at the design frequency. Fig. 7 shows a CAD model of the bowtie elements and the feed layer, along with location of the components that are added after the printing process. Design 1 has a ground plane that only covers the area directly underneath the feed components and is designed to have minimal effect on the antenna radiation pattern while still providing a ground to the feed network components. Fig. 8 shows the CAD model of the simulated antenna and a stack up of the printed layers. The phase shifter element values are pF and nH for the capacitor and inductor, respectively. The data are given in Fig. 9, showing close correlation between measured and simulated

Fig. 6. Two types of 3-D printed crossed-bowtie antennas: Design 1 without HIS (top); and Design 2 with HIS (bottom) [6].

Fig. 7. CAD model of the crossed-bowtie antenna element (top) and the stack-up of the printed ABS and CB028 layers (bottom) for the design without the HIS (Design 1) [6].

results; the slight frequency offset in the measured data is most likely due the fact that the antenna via diameter is not exactly 0.8 mm. The antenna includes a surface mount U.FL connector which connects to the input of the balun component. The simulation includes the balun component and also the microstrip feed line. Simulated and measured radiation patterns are also shown in Fig. 9, demonstrating an axial ratio of less than 2 dB with a measured gain of dBi at 2.45 GHz. These results compare well to the simulated axial

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Fig. 8. CAD model of the crossed-bowtie antenna element (top) and the stack-up of the printed layers for the design with HIS (bottom) [6].

ratio of 0.6 dB and gain of dBi. The gain and radiation efficiency of the antenna are not strongly dependent on the effective conductivity of the CB028 silver paste as simulation results show that the efficiency is reduced by % as the conductivity changes by an order of magnitude from 1e7 S/m to 1e6 S/m (Table V). The broadband characteristic of the antenna results in the low sensitivity to the silver film conductivity, in contrast to the behavior of Design 2 and the narrowband filter described in the following section. Based on the NFMM characterization presented earlier, the assumed CB028 conductivity in the simulations is 1e6 S/m. To compare the performance of the printed antenna to an antenna of the same design fabricated using conventional printed circuit board (PCB) materials, simulations of a CP antenna design using 0.5 oz. copper clad Rogers Duroid for the substrate and conductive material were also performed. The simulated return loss comparisons are shown in Fig. 9. Even though the PCB antenna provides about a 30% larger bandwidth, the gain and efficiency at the design frequency of 2.45 GHz, shown in Table V, are comparable to the printed antenna design. For Design 2, a high impedance surface layer is included in the design and requires an additional CB028 and ABS layer in the printing process. Also, the ground plane covers the entire substrate. A high impedance surface is generally composed of periodic and identical elements arranged in a single layer and has a bandlimited response [30]–[32]. A HIS layer consisting of square conductive patches that are capacitively coupled is used in this design, as shown in Fig. 10(top). The HIS layer was optimized using Ansys HFSS to achieve a minimum bandwidth of 100 MHz centered at 2.45 GHz. Over this bandwidth the phase of the reflection coefficient is within with a magnitude near unity, as excited using a Floquet port. A very narrow gap between the HIS patches is required to achieve a bandwidth close to 100 MHz, while keeping within the overall

Fig. 9. Comparison between measured and simulated data for antenna Design 1: return loss (top) and radiation patterns (bottom). The return loss of the antenna using copper clad graph also includes the simulated Duroid for the substrate. For the pattern plot: solid line—simulated horizontal cut; dashed line—measured horizontal cut; dash-dotted line—simulated vertical cut; dotted line—measured vertical cut.

TABLE V ANTENNA DESIGN 1 GAIN AND EFFICIENCY AT 2.45 GHZ AS A FUNCTION OF SIMULATED PRINTED SILVER CONDUCTIVITY. A COMPARISON TO A SIMULATED ANTENNA USING CONVENTIONAL PCB MATERIALS IS ALSO INCLUDED

substrate thickness constraints which are equivalent to . The required gap spacing between the HIS plates is 300 m, and the length and width of each square patch is 9.7 mm. The cross section of the antenna is shown in Fig. 10 (bottom). The phase shifter values in this case are 10 pF and 0.5 nH for the capacitor and inductor, respectively. The measured for Design 2 is given in Fig. 11 and shows an operational bandwidth shift from 2.45 GHz to GHz. Through simulation it was found that the HIS performance is highly sensitive to variations in the antenna via diameter, the HIS patch size and the gap spacing between patch elements.

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Fig. 10. CAD model of the crossed-bowtie antenna element (top) and the stack-up of the printed ABS and CB028 layers for the design with an HIS (bottom) [6].

Therefore the frequency shift is most likely due to these dimensional tolerances in the printing process. To reduce the frequency shift, tighter control of the gap width between the HIS plates and the diameter of the printed vias that connect the antenna elements to the feed layer is required. Simulation show that a mm change in the HIS gap spacing corresponds to a MHz shift in the operational frequency. A shift of 120 MHz was observed when the diameter of the printed vias was also varied mm. The differences seen in the measured and simulated out-of-band may be due to surface wave effects, which can be difficult to predict accurately in electrically-large HIS structures. Radiation pattern measurements at frequencies across the measured bandwidth show that a best case axial ratio of 2.1 dB is obtained at 2.53 GHz with a measured gain of dBi (Fig. 11, bottom). Assuming the HIS layer introduced no additional loss, the dBi gain of Design 1 would be expected to increase to approximately 0.5 dBi with the change to a more uni-directional pattern due to the HIS. This 5.4 dB gain reduction contributed by the HIS layer is equivalent to an efficiency factor of %, which is at least 3 dB worse than the % typically achieved with low profile, HIS-backed antennas that use high quality, copper-clad commercial laminates; the 3 dB degradation is similar to the results obtained for the insertion loss of DDM filters presented in the following section, when compared to PCB filters on copper-clad laminates. The gain and efficiency compare well to the efficiency predicted in simulation which are dBi and 31%, respectively. IV. MINIATURIZED OPEN-LOOP RESONATOR FILTER The performance of microstrip band-pass filters is examined in this section in order to further assess the impact of the effective RF conductivity of the CB028 film, as well as the ABS

Fig. 11. Measured and simulated data for antenna Design 2: (top) and radiation patterns (bottom). For the pattern plot: solid line - simulated horizontal cut; dashed line - measured horizontal cut; dash-dotted line - simulated vertical cut; dotted line - measured vertical cut.

substrate surface roughness on the high frequency characteristics of DDM circuits. The filter topology is comprised of square open loop resonators (SOLR) which are often used in communications systems due to their compact size [35]. SOLR filters have been miniaturized to an area of 20% that of a conventional SOLR filter by loading the resonators with surface mount capacitors [34]. Herein, four versions of a 2.45 GHz 10% bandwidth SOLR filter that is loaded with series capacitors are presented. Filters 1–3 use only discrete chip capacitors across the gaps in the resonators while Filter 4 uses a new DDM-enabled design approach to increase coupling between adjacent resonators. All filters use 18–20 m-thick CB028 conductors on the front side and for the ground plane. A. Filter Designs A photograph of Filters 1–3 is given in Fig. 12 along with an outline drawing of Filter 2; all three filters have similar dimensions, with the minimum feature size being the 240 m spacing

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Fig. 13. Top: Illustration of metal-insulator-metal overlay capacitor used to increase coupling between adjacent resonators; Bottom: Outline drawing of Filter 4 (in mm) with dashed lines indicating the MIM capacitor locations [6].

Fig. 12. Top: Band-pass Filters 1–3 printed using CB028 Ag paste; Bottom: Outline drawing of Filter 2 (in mm) [6].

between resonators. Each filter is also printed on a 32 mil-thick substrate, with ABS used for Filters 1 and 2 and Rogers 4003c microwave laminate used for Filter 3. The resonators of each filter are loaded with ATC 600S (0603 body style) 0.5 pF capacitors for miniaturization. In order to achieve greater miniaturization of the filter footprint the value of the loading capacitors placed across the resonator gaps must be increased. Numerical electromagnetic simulations reveal that the maximum value that can be used for the loading capacitors is approximately 0.7 pF, as larger values result in such small resonator dimensions that effective input/output tapping (which controls the external quality factor) cannot be obtained. A further consequence of the reduced resonator dimensions is that the coupling region between adjacent resonators is shortened, thus requiring significantly reduced resonator spacing. For the 0.7 pF design, the required spacing is 70 m as compared to the 240 m used for the designs with 0.5 pF capacitors. As noted above, on relatively rough surfaces such as the printed ABS, feature sizes at this scale cannot be consistently realized using micro-dispensing. To circumvent this problem an alternative approach to achieving the necessary coupling coefficient was developed [14]. The new approach uses 3-D-printed metal-insulator-metal overlay capacitors that bridge adjacent resonators and are formed by depositing a thin, localized layer of ABS on top of the filter and printing the top capacitor plates with silver paste (Fig. 13). Using numerical electromagnetic simulations it is found that using an ABS insulator thickness m and a resonator spacing m, the required resonator coupling coefficient is achieved with a total top plate width of m and length

m. The approximate coupling capacitance with these dimensions is 0.18 pF using the parallel plate approximation. B. Analysis of Filter Performance The measured and simulated S-parameters for the four filters are plotted in Fig. 14 and a summary of key parameters is given in Table VI. The simulations were performed using Ansys HFSS with and for the ABS substrate and and for the Rogers 4003c substrate. Keysight Advanced Design System was also used in order to include the discrete chip capacitors using Modelithics ATC600S substrate-scalable models. For Filters 1–3 it is assumed that the actual values of the capacitors are 0.45 pF due to the 10% tolerance because a frequency shift was observed in the measurements. As indicated in the table, Filter 4 achieves the greatest miniaturization factor (21% the size of a conventional filter with no capacitive loading) due to the use of 0.7 pF chip capacitors and the overlay MIM capacitors. It can also be observed in Fig. 14 that the out-of-band rejection of Filter 4 is approximately 10 dB lower than that of Filters 1–3. This difference is attributed to the MIM capacitor loading approach and the smaller footprint of the resonators, as simulations of a similar geometry that assumed 70 m resonator spacing and no overlay capacitors predict a 5 dB reduction in the out-of-band rejection. Finally, a filter identical to Filter 3 but using 0.5-oz. copper instead of the CB028 conductor demonstrates a measured insertion loss of 1.5 dB. An important difference among the filters is the surface roughness (Ra) of the ABS substrate and the corresponding impact on the pass-band insertion loss. The roughness for Filter 1 is 7.1 m versus 4.9 m Filter 2, with the improvement coming from the use of a smaller diameter FDM head and other adjustments in the printing parameters. Although there

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Fig. 15. Profilometer Scan Over a Typical FDM ABS Substrate (Left) and Cross Section of a CB028 Filter on ABS Showing Ink Valleys (Right) [6].

Fig. 14. Simulated (solid lines) and measured (markers) data for Filters 1–4 (starting clock-wise from upper left).

TABLE VI PARAMETERS FOR FILTERS 1–4. C—CHIP CAPACITOR VALUE; IL—PASS-BAND INSERTION LOSS; RA—SURFACE ROUGHNESS; BW–3-DB BANDWIDTH; AR—AREA REDUCTION COMPARED TO FILTER WITH NO CAPACITIVE LOADING; —EFFECTIVE RF CONDUCTIVITY OF CB028 FILM

Fig. 16. Switched-line phase shifters (left—PCB, right—DDM) [6].

maximum conductivity remains below the bulk DC value for CB028 (2.6e6 S/m) due to the edge effects and inhomogeneous silver flake distribution discussed in Section II.C. V. MULTI-BIT PHASE SHIFTER

are also minor geometrical differences between these filters and the 3-dB bandwidth increased, numerical electromagnetic simulation data confirm that the primary reason for improved insertion loss is an increase in the effective RF conductivity of the CB028 film due to the reduction in surface roughness. This conclusion is further supported by simulation studies on Filter 3, which has a narrower 3-dB bandwidth than Filter 2 but lower surface roughness and slightly improved insertion loss. The surface roughness in the filter simulations is represented by adjusting the conductivity of the CB028 film to match measured filter performance, and assuming the conductors to be on a perfectly flat surface. (An alternative, more computationally expensive approach of representing the surface profile of the ABS in the simulation model is discussed in Section V). A profilometer scan of a representative sample is shown in Fig. 15 (left) illustrating the filament-like characteristic of the FDM print as well as relatively deep ( m) periodic valleys between filaments. Fig. 15 (right) is an SEM micrograph of a filter cross-section which demonstrates how the CB028 paste seeps into the valleys, creating points of high current concentration and lowering the effective conductivity. Table VI lists the RF conductivity values for the silver paste that produced simulation results equal to the measured insertion loss, showing a range from 0.6e6 to 1.2e6 S/m for the different Ra values. It is important to use common profilometer settings and scan lengths in order to consistently measure surface roughness for multiple samples of these filament-like materials. The

A multi-bit 2.45 GHz phase shifter design which is integrated as part of the phased array unit cell is described in this section. The performance of the design closely matches that of an identical circuit fabricated using a high quality commercial microwave laminate. In addition, a detailed analysis of the loss is performed to study the impact of the filament-like surface of the ABS substrate. The results demonstrate the close correlation between modeled and simulated insertion loss that is achievable using full-wave simulation of the non-planar surface and accurate values for the CB028 conductivity. A. Phase Shifter Design Two versions of a single-bit 45 switched-line phase shifter are presented; one fabricated using the described DDM process and one using a conventional PCB (copper-clad Rogers 4003c) process (Fig. 16). The Skyworks AS179-92LF SPDT SC-70 is used to switch between the thru- and delay-states on both circuits. The switches and surface mount inductors and capacitors are attached using H20E, an Ag nanoparticle conductive epoxy. The measured insertion loss in the delay state of the DDM and PCB circuits at 2.45 GHz is 1.55 dB and 1.25 dB, respectively (Fig. 17). This loss includes that of the switches and surface mount components ( dB total) and the interconnect lengths, where the latter differs by 0.3 dB. As shown in Section V.B. this difference is explained by the lower conductivity of the CB028 paste compared to copper, and the roughness of the ABS surface. For the phased array unit cell, a 4-bit design combining highpass low-pass and switched-line sections is used (Fig. 18). The least significant bits (45 and 22.5 ) are implemented using the switched-line approach to take advantage of the simplicity

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Fig. 17. of thru- and delay-states of phase shifters. Solid lines depict the thru-state and dotted lines the delay sate.

Fig. 18. DDM 4-bit high pass/low pass, switched line phase shifter [6].

and manufacturability of the design. The 180 and 90 bits are implemented using a high-pass low-pass approach in order to minimize the footprint. MA-COM MASWSS0115 SPDT RF switches are used in these bits. The insertion loss of the 16 states of the 4-bit phase shifter is shown in Fig. 19(left), showing an average state insertion loss of 5.8 dB at 2.45 GHz; the total contribution from the switches is dB. The phase shift of the individual states is given in Fig. 19 (right). A 2.45 GHz power sweep on a DDM test fixture that includes a MASWSS0115 SPDT switch yields a 1 dB compression point of dBm, which is within the switch specifications and indicates that the power handling capability is not limited by the DDM fabrication approach.

Fig. 19. Measured results for the 16 states of the 4-bit phase shifter: magnitude (top) and phase (bottom). The S21 plot shows that the insertion loss from dB for all 16 phase states. The phase shift plot shows 2.4 to 2.5 GHz is about the 22.5 relative phase difference between the 16 phase states at 2.45 GHz.

B. Analysis of Phase Shifter Performance To understand the difference in the measurements of the single-bit switch-line phase shifter using the two technologies, EM simulations of CB028-on-ABS (DDM) and copper-on-RO4003C (PCB) microstrip lines are performed using Ansys HFSS. The cross section of the microstrip lines used in the simulations are shown in Fig. 20(a) and (b). The microstrip line lengths are identical to those in the phase shifters ( mm) however the switches and capacitor pads are excluded. Although both conductive materials are assumed to be flat on the top surface, the CB028 is made to conform to the eggshell (filament-like) surface on the bottom. The eggshell surface is similar to that in Fig. 15. The conductivity of CB028 is selected to be 1e6 S/m, which is in the range measured using NFMM (Fig. 5(a)). Simulated surface current density plots at 3 GHz are shown in Fig. 20(c) and (d). As expected for this frequency, the current is concentrated at the edges of the copper

Fig. 20. Cross sections of microstrip lines simulated in HFSS for (a) copper on RO4003C (PCB) and (b) CB028 on ABS (DDM). Surface current density for (c) PCB and (d) DDM.

and CB028 traces. However, for the DDM circuit high current density is also observed in the valleys. The simulated of the DDM and PCB microstrip lines is shown in Fig. 21. The predicted losses are higher for the CB028/ABS circuit. In particular, at 2.45 GHz the difference

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Fig. 21. Simulated

for the PCB and DDM microstrip lines.

Fig. 23. Measured return loss of the 2.45 phased array unit cell showing a redB for all 16 phase states across the entire 2–3 GHz turn loss of less than frequency band.

Fig. 22. Printed unit cell in an anechoic chamber during antenna pattern measurements. The unit cell with control board is shown on the right and the linearly polarized transmit antenna can be seen on the left. The center insert shows a front view of the unit cell front end with Arduino controller and wire interface board.

is 0.32 dB, in close agreement with the measurement data presented above. VI. PHASED ARRAY UNIT CELL The printing process used for the phased array unit cell is the same as used for the circularly-polarized dipole antenna (Section III) but with 14 total steps. The process consists of 7 FDM steps to deposit ABS for a base antenna layer, a dielectric separation to the HIS, a HIS-ground separation, and 4 substrate heights for the various circuits. No post-processing is required to smooth the ABS. There are also 7 micro-dispensing steps to deposit CB028 for the antenna elements, the HIS, the ground, the antenna feed, and 3 additional substrate heights. Fig. 22 shows the unit cell used for the return loss and radiation pattern measurements. The phase shifts were controlled using an Arduino microprocessor. A 6 6 cm unit cell for a 2.45 GHz phased array antenna was assembled by integrating the individual components described in the preceding sections; namely, the circular-polarized HIS-backed dipole antenna, a square open-loop resonator filter and a 4-bit phase shifter. A reduced-size Rat-Race hybrid coupler is also inserted between the filter and antenna feed points, acting as a balanced-to-unbalanced transformer (balun) to provide the required 0 –180 differential feed for the antenna. The coupler design is adopted from [35]. As illustrated Fig. 1 a different substrate thickness is used for each of the four circuit components and the required topography is realized by varying the number of ABS layers deposited in different regions of the top surface. A transition with a 45 slope is used at the boundary

Fig. 24. Measured radiation patterns of the 2.45 GHz phased array unit cell. The vertical and horizontal gain pattern plots are shown.

between different substrate thicknesses and the width of the deposited CB028 trace is varied to maintain a 50 characteristic impedance. The filter design used in the unit cell is that of Filter 2 (see Section IV). Return loss measurements of the unit cell referenced to the input of the phase shifter are shown in Fig. 23 at the 16 phase states. Again, a resonant frequency shift occurred from 2.45 to 2.53 GHz, as observed with the HIS antenna in Section III Radiation pattern measurements at 2.45 GHz in the vertical and horizontal direction to obtain the axial ratio and maximum gain in the broadside direction are given in Fig. 24. The measured 2.53 GHz gain of the unit cell varies from dBi to dBi over all phase shifter states. VII. CONCLUSION The work presented herein demonstrates the potential of direct digital manufacturing for realizing high quality, light weight and conformal microwave structural electronics. By utilizing the multi-layer and multi-material capability of the DDM process, complex 3-D printed electronic systems can be produced when combined with pick-and-place assembly

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of packaged and surface mount devices. The assemblies can have buried conductors, local variations in substrate thickness and both distributed and lumped elements that are additively manufactured. Furthermore, the minimum layer thickness and conductor feature size of the DDM process are sufficient for circuit design at least in the lower microwave frequency bands. The materials characterization and analysis indicate that thermoplastic substrates produced via FDM have surface roughness features that remain a challenge, and the impact of transmission loss of the relatively low conductivity of micro-dispensed thick-film conductors can be magnified by the roughness of the surfaces they are printed upon. Compared to typical high-performance copper-clad microwave laminates, the degradation in transmission loss is marginal in low-Q circuits. However, as expected the loss effects are more substantial in narrow-band, high-Q circuits such as the HIS-backed dipole antenna and band-pass filters. Continuing research will address improvements in surface roughness and conductivity of the printed conductive materials which are necessary to make DDM fabrication competitive with state of the art PCB technology and viable for mm-wave applications. Improvements in surface roughness may be achieved using localized and/or global in-situ post-processing using mechanical, thermal or other means. REFERENCES [1] R. Das and P. Harrop, Printed, Organic & Flexible Electronics Forecasts, Players and Opportunities 2012–2022, Business Report. [2] B. Sanz-Izquierdo and E. A. Parker, “3-D printing technique for fabrication of frequency selective structures for built environment,” Electron. Lett., vol. 49, no. 18, pp. 1117–1118, Aug. 2013. [3] I. T. Nassar and T. M. Weller, “An electrically-small, 3-D cube antenna fabricated with additive manufacturing,” in Proc. IEEE Topical Conf. Wireless Sensors Sensor Networks (WiSNet), 2013, pp. 58–60. [4] A. M. N. Al-Mobin, R. Shankar, W. Cross, J. Kellar, K. W. Whites, and D. E. Anagnostou, “Advances in direct-write printing of RF-MEMS using M3-D,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), 2014, pp. 1–4. [5] P. I. Deffenbaugh, R. C. Rumpf, and K. H. Church, “Broadband microwave frequency characterization of 3-D printed materials,” IEEE Trans. Components, Packaging Manufacturing Technol., vol. 3, no. 12, pp. 2147–2155, Dec. 2013. [6] N. Arnal, T. Ketterl, Y. Vega, J. Stratton, C. Perkowski, P. Deffenbaugh, K. Church, and T. Weller, “3-D multi-layer additive manufacturing of a 2.45 GHz RF front end,” in IEEE MTT-S Int. Microw. Symp. (IMS), 2015, pp. 1–4. [7] J. A. Paulsen, M. Renn, K. Christenson, and R. Plourde, “Printing conformal electronics on 3-D structures with aerosol jet technology,” in Proc. Future Instrum. Int. Workshop (FIIW), 2012, pp. 1–4. [8] L. Min, C. Shemelya, E. MacDonald, R. Wicker, and X. Hao, “Fabrication of microwave patch antenna using additive manufacturing technique,” in Proc. USNC-URSI Radio Sci. Meeting (Joint With AP-S Symp.), 2014, pp. 269–269. [9] L. Min, C. Shemelya, E. MacDonald, R. Wicker, and X. Hao, “3-D printed microwave patch antenna via fused deposition method and ultrasonic wire mesh embedding technique,” IEEE Antennas Wireless Propagat. Lett., vol. 14, pp. 1346–1349, 2015. [10] L. Min, X. Yu, C. Shemelya, E. MacDonald, and X. Hao, “3-D printed multilayer microstrip line structure with vertical transition toward integrated systems,” in Proc. IEEE MTT-S Int. Microw. Symp. (IMS), 2015, pp. 1–4. [11] M. Liang, “Three-dimensionally printed/additive manufactured antennas,” in Handbook of Antenna Technologies, Apr. 7, 2015, pp. 1–30. [12] T. Nakatsuka, “Polylactic acid-coated cable,” Fujikura Tech. Rev., pp. 39–45, 2011, Rev. 40. [13] D. Espalin, D. Muse, E. MacDonald, and R. Wicker, “3-D printing multifunctionality: Structures with electronics,” Int. J. Adv. Manuf. Tech., vol. 72, no. 5, pp. 963–978, Mar. 2014.

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[14] N. Arnal, T. Ketterl, T. Weller, G. Wable, T. Hue, W. Garon, and D. Gamota, “3-D digital manufacturing and characterization of antennas integrated in mobile handset covers,” in Proc. IEEE 16th Annual Wireless Microw. Technol. Conf. (WAMICON), Jun. 2015, pp. 1–5. [15] B. M. Tymrak, M. Kreiger, and J. M. Pearce, “Mechanical properties of components fabricated with open-source 3-D printers under realistic environmental conditions,” Materials & Design, vol. 58, pp. 242–246, 2014. [16] J. E. Mark, Physical Properties of Polymers Handbook, 2nd ed. New York, NY, USA: Springer Science, 2007, p. 489. [17] A. Bagsik, V. Schoeppner, and E. Klemp, “FDM part quality manufactured with ultem *9085,” in Proc. 69th Ann. Tech. Conf. Soc. Plastics Engineers (ANTEC), 2011, pp. 1294–1298. [18] L. J. Van der Pauw, “A Method of Measuring Specific Resistivity and Hall Effect of Discs of Arbitrary Shape,” Philips Research Reports, 12.1, Feb. 1958, vol. 13, pp. 1–9, no. 1. [19] M. Tabib-Azar, D. P. Su, A. Pohar, S. R. LeClair, and G. Ponchak, “0.4 um spatial resolution with 1 GHz ( cm) evanescent microwave probe,” Rev. Sci. Instrum., vol. 70, pp. 1725–1729, 1999. [20] A. Imtiaz, T. Baldwin, H. T. Nembach, T. M. Wallis, and P. Kabos, “Near-field microwave microscope measurements to characterize bulk material properties,” Appl. Phys. Lett., vol. 90, p. 243105, 2007. [21] M. Córdoba-Erazo, E. Rojas-Nastrucci, and T. Weller, “Measurement of electrical conductivity of direct digital printed conductive traces using near-field microwave microscopy,” in Proc. 42nd Int. Symp. Microelectron. (IMAPS), 2014, pp. 898–904. [22] J. C. Weber, J. B. Schlager, N. A. Sanford, A. Imtiaz, T. M. Wallis, L. M. Mansfield, K. J. Coakley, K. A. Bertness, P. Kabos, and V. M. Bright, “A near-field scanning microwave microscope for characterization of inhomogeneous photovoltaics,” Rev. Sci. Instrum., vol. 83, pp. 083702–083702-7, 2012. [23] M. F. Cordoba-Erazo and T. M. Weller, “Noncontact electrical characterization of printed resistors using microwave microscopy,” IEEE Trans. Instrum. Meas., vol. 64, no. 2, pp. 509–515, Feb. 2015. [24] V. V. Talanov, A. Scherz, and A. R. Schwartz, “A microfabricated near-field scanned microwave probe for noncontact dielectric constant metrology of low-k films,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., 2006, pp. 1618–1621. [25] V. V. Talanov, A. Scherz, R. L. Moreland, and A. R. Schwartz, “Noncontact dielectric constant metrology of low-k interconnect films using a near-field scanned microwave probe,” Appl. Phys. Lett., vol. 88, no. 19, p. 192906, May 2006. [26] M. F. Cordoba-Erazo and T. M. Weller, “Liquids characterization using a dielectric resonator-based microwave probe,” in Proc. 42nd Eur. Microw. Conf. (EuMC), 2012, pp. 655–658. [27] M. F. Cordoba-Erazo, E. A. Rojas-Nastrucci, and T. Weller, “Simultaneous RF electrical conductivity and topography mapping of smooth and rough conductive traces using microwave microscopy to identify localized variations,” in Proc. IEEE 16th Ann. Wireless Microwave Technol. Conf. (WAMICON), 2015, pp. 1–4. [28] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York, NY, USA: Wiley, 2000. [29] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Raisanen, and S. A. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1624–1632, Jun. 2008. [30] F. Costa, A. Monorchio, and G. Manara, “An equivalent-circuit modeling of high impedance surfaces employing arbitrarily shaped FSS,” in Proc. Int. Conf. Electromagn. Adv. Appl. (ICEAA), 2009, pp. 852–855. [31] S. R. Best and D. L. Hanna, “Design of a broadband dipole in close proximity to an EBG ground plane,” IEEE Antennas Propag. Mag., vol. 50, no. 6, pp. 52–64, Dec. 2008. [32] G. Bianconi, F. Costa, S. Genovesi, and A. Monorchio, “Optimal design of dipole antennas backed by a finite high-impedance screen,” Progr. Electromagn. Res. C, vol. 18, pp. 137–151, 2011. [33] H. Jia-Sheng and M. J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 11, pp. 2099–2109, Nov. 1996. [34] L. Ledezma and T. Weller, “Miniaturization of microstrip square open loop resonators using surface mount capacitors,” in Proc. IEEE 12th Ann. Wireless Microw. Technol. Conf. (WAMICON), 2011, pp. 1–5. [35] M. M. Abdin, J. Castro, W. Jing, and T. Weller, “Miniaturized 3-D printed balun using high-k composites,” in Proc. IEEE 16th Ann. Wireless Microw. Technol. Conf. (WAMICON), 2015, pp. 1–3.

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Thomas P. Ketterl (S'98–M'01) received the B.S. degree in ocean engineering from Florida Atlantic University, Boca Raton, FL, USA, in 1994 and the M.E. and Ph.D. degrees from the University of South Florida, Tampa, FL, USA, in 2000 and 2006, respectively. From 2001 to 2010, he was a Research Engineer with the Center of Ocean Technology at the University of Florida, St. Petersburg, FL, USA. Since 2011, he has been working as a Research Associate with the University of South Florida's Department of Electrical Engineering. His research interests include electronic hardware design for medical applications, 3-D printing of RF circuits and antennas, and RF MEMS.

Yaniel Vega received the B.S. degree in electrical engineering from the University of South Florida, Tampa, FL, USA, in 2012, where he is currently pursuing the M.S.E.E. degree in RF/microwave engineering. Before joining the Wami group in 2014 he worked for Space Machine and Engineering Co. His current work is focused on the Direct Printing of 3-D Structural RF Electronics.

Nicholas C. Arnal received the B.S. and M.S. degrees in electrical engineering in 2014 and 2015, respectively, from the University of South Florida, Tampa, FL, USA. During this time he worked as a graduate research assistant for Dr. Thomas Weller and performed research in the area of RF/microwave circuits produced using additive manufacturing. He now works as an RF Engineer at Lockheed Martin Space Systems in Denver, CO, USA.

John Stratton received the B.S.E.E. degree (high honors) in May 2014 from the University of South Florida, Tampa, FL, USA, where he is currently pursuing the M.S.E.E. degree, focusing on RF/microwave engineering, under the guidance of Dr. Thomas Weller. His Master's thesis is related to the direct printing of 3-D structures with embedded RF circuits.

Eduardo A. Rojas Nastrucci received the B.S. degree in electrical engineering from the Universidad de Carabobo, Valencia, Venezuela, in 2009, and the M.S. E.E. degree from the University of South Florida, Tampa, FL, USA, in 2014, where he is currently pursuing the Ph.D. degree with the WAMI group. His doctoral research is focused on additive manufactured microwave circuits and antennas. He worked as Assistant Professor from 2011–2012 at Universidad de Carabobo. Specifically, his work is oriented in developing new structures, materials, and techniques with the objective of developing 3-D printed devices with improved performance.

María F. Córdoba-Erazo (S'11) received the B.S. degree in engineering physics from Universidad del Cauca, Colombia, in 2005 and the M.S. degree in electrical engineering from University of Puerto Rico at Mayagüez in 2009. Currently, she is working toward the Ph.D. in electrical engineering at University of South Florida, Tampa, FL, USA.

Ms. Córdoba-Erazo has been recipient of the IEEE MTT-S Graduate Fellowship in 2015, the ARFTG Roger Pollard Memorial Student Fellowship in Microwave Measurement in 2015 and the DOE-UPRM Scholarship in 2008.

Mohamed Mounir Abdin received the B.Sc. degree in electrical engineering from Arab Academy for Science and Technology (AAST), Cairo, Egypt, in 2010. Prior to joining the WAMI Research Center, University of South Florida, Tampa, FL, USA, in 2013, he worked for three years as a Process Engineer at a leading MEMS Foundry called Innovative Micro Technology (IMT), Santa Barbara, CA. Currently he is exploring the possibilities of designing RF/MW Systems using additive manufacturing.

Casey W. Perkowski is currently pursuing the B.S. degree in mechanical engineering at the University of Central Florida, Tampa, FL, USA. He has been an employee at Sciperio Inc., Orlando, FL, USA, since 2013 where he has worked done research in additive manufacturing techniques for both electrical and mechanical applications.

Paul I. Deffenbaugh received the UTEP Ph.D. degreee collaborating with Dr. Weller and his team at the WAMI Research Center, University of South Florida, Tampa, FL, USA, (electromagnetics group) in 2014, making measurements of 3-D printed RF/microwave devices. He worked at nScrypt, Inc., Orlando, FL, USA, designing and building 3-D printing equipment. For two years from 2011–2013 he conducted fundamental research in the 3-D printing of RF parts at the W.M. Keck Center for 3-D Innovation in El Paso. His research interests are 3-D printing and its applications in microwave electromagnetics. He currently works at Sciperio, Inc., Orlando, FL, USA, producing RF circuits using high precision 3-D printing equipment.

Kenneth H. Church received the B.S. degree in physics and electrical engineering in 1988 and 1989, respectively, and the M.S. and Ph.D. degrees in electrical engineering in 1991 and 1994, respectively, from Oklahoma State University, OK, USA. His areas of research include laser-materials interactions, RF materials and designs, antennas and 3-D printed electronics. He currently holds professional positions at Sciperio and nScrypt, Inc., Orlando, FL, USA. He is also a Research Professor with the University of Texas at El Paso, TX, USA.

Thomas M. Weller (S'92–M95–SM'98) received the B.S., M.S., and Ph.D. degrees in electrical engineering in 1988, 1991, and 1995, respectively, from the University of Michigan, Ann Arbor, MI, USA. From 1988 to 1990 he worked at Hughes Aircraft Company, El Segundo, CA, USA. He joined the University of South Florida, Tampa, FL, USA, in 1995 where he is currently Professor and Chair in the Electrical Engineering Department. His current research interests are in the areas of RF/microwave applications of additive manufacturing, development and application of microwave materials, and integrated circuit and antenna design.

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A Reconfigurable K-/Ka-Band Power Amplifier With High PAE in 0.18- m SiGe BiCMOS for Multi-Band Applications Kaixue Ma, Senior Member, IEEE, Thangarasu Bharatha Kumar, Student Member, IEEE, and Kiat Seng Yeo, Senior Member, IEEE

Abstract—This paper presents a high power efficient broad-band programmable gain amplifier with multi-band switching. The proposed two stage common-emitter amplifier, by using the current reuse topology with a magnetically coupled transformer and a MOS varactor bank as a frequency tunable load, achieves a 55.9% peak power added efficiency (PAE), a peak saturated power of 11.1 dBm, a variable gain from 1.8 to 16 dB, and a tunable large signal 3-dB bandwidth from 24.3 to 35 GHz. The design is fabricated in a commercial 0.18- m SiGe BiCMOS technology and measured with an output 1-dB gain compression point which is better than 9.6 dBm and a maximum dc power consumption of 22.5 mW from a single 1.8 V supply. The core amplifier, excluding the measurement pads, occupies a die area of 500 m 450 m. -band, Index Terms—Current reuse, dual band, -band, power added efficiency (PAE), SiGe BiCMOS, transformer coupled load, tunable amplifier, variable gain amplifier (VGA).

I. INTRODUCTION

I

N THE RECENT decade, the rising demand in short-range high-speed wireless communication systems have nurtured the development of radio frequency (RF) integrated circuit design for the diversified -band (18–27 GHz) and -band (26.5–40 GHz) applications such as 24 GHz industrial scientific and medical (ISM) band gigabit-per-second wireless systems [1], wireless sensor network [2], point-to-point communication (18–23 GHz) [3], local multipoint distribution service (27.5–29.5 GHz) [4], and short-range (22–29 GHz) automotive radar applications for anti-collision detection [5], [6]. The power amplifier (PA) design often involves the tradeoff between the efficiency and linearity. This limitation can be mitigated by using several design techniques such as the stacked Manuscript received June 19, 2015; revised September 29, 2015; accepted October 11, 2015. Date of publication November 11, 2015; date of current version December 02, 2015. This work was supported in part by the Exploit Technologies Pte. Ltd. (ETPL), Singapore and Nanyang Technological University (NTU), Singapore. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 16–23 2015. K. Ma is with the School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: kxma@ieee. org). T. B. Kumar and K. S. Yeo are with the Singapore University of Technology and Design, 138682 Singapore (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2495129

amplifier design proposed in [7] that supports a large signal swing to increase the output power by alleviating the high frequency device's low at the cost of large supply voltage. An alternative design technique is the switched mode PA in which the circuit operation adapts dynamically based on instantaneous characteristics (amplitude, phase, frequency) of the input signal such as the class E in [8] and class in [9] that minimizes power in the amplifying transistors by avoiding overlap between the current and voltage waveforms by using the digital ON/OFF switches. However, such designs require special linearization techniques to reduce the nonlinearities introduced by the output harmonics of the switching distorted waveforms. In contrast, PA design is also based on continuous power control such as adaptive biasing technique implemented in [10] and by using load impedance modulation technique of the Doherty PA in [11]. However, these PA design techniques involve large dynamic range complex circuitry that consumes additional dc power. An implementation technique to enhance PA efficiency as well as linearity is by reducing the losses involved in the interconnects and passive components by using distributed structure equivalents in physical layout such as Wilkinson couplers [12] and the thin-film micro-strip lines (MSL) [13] as the power splitter/ combiner between multiple PA stages, the substrate-shielded coplanar waveguide (CPW) structures [14], the transmission line transformers (TLT) [3], and the substrate-shielded MSL [6], [15]. However, these distributed structures occupy a large die area. The amplifier power efficiency can be improved by using various current-reused topologies [16] such as the capacitive-coupling [17], the inter-stage LC series resonance, and the transformer-coupling. The transformer-based LC tank has been already proposed in the design of various transceiver building blocks such as the voltage controlled oscillator (VCO) [18], low noise amplifier (LNA) [19], and Class-F PA [20]. The work in [17] presents an inductor load based current-reused LNA at -band with an efficiency of up to 37%. In this paper, a SiGe BiCMOS based -/ -band digitally controlled variable gain amplifier (DVGA) with multiple frequency tunable bands is proposed, designed and verified by on-wafer measurement. This work is extended from [21] to include the physical implementation details of the loaded tank circuit, together with a detailed analysis of its quality factor enhancement, and henceforth the resulting improvement in the overall amplifier PAE of 55.9% and the linearity performance. The proposed design improves power efficiency by simultane-

0018-9480 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Fig. 1. Circuit schematic of (a) the proposed reconfigurable multi-band amplifier and (b) half circuit equivalent.

Fig. 2. Hybrid- small signal half circuit equivalent of the proposed reconfigurable multi-band amplifier.

ously reducing the overall dc power consumption using the current reuse technique and improving the amplifier linearity by using a loaded tank circuit's enhanced . The proposed design provides a variable gain control, frequency-band switching, wideband small signal gain flatness, low power consumption, improved linearity, and high PAE. This paper is organized as follows. Section II describes the detailed analysis of the proposed amplifier circuit. The detail implementation of the frequency tunable load is explained in Section III. Section IV discusses about the load tank circuit's -factor enhancement resulting in improvement of proposed amplifier's linearity and PAE performance. The design analysis is verified in Section V by on-wafer measurement with the performance comparison of this work against the state-of-the-art -band amplifiers. Finally, the paper conclusion is provided in Section VI. II. CIRCUIT ANALYSIS The circuit schematic of the proposed multi-band tunable amplifier is a fully differential two-stage ac cascaded and dc stacked common emitter amplifier as shown in Fig. 1(a). The first stage is a DVGA with a four bit digital gain control and the second stage is a frequency tunable amplifier. The block depicted as the frequency tunable load is the tank circuit that consists of an integrated 2-coil spiral transformer

and the varactor bank. The inductors and the transformer used in the proposed differential circuit are chosen with center tap configuration to significantly improve the die area utilization. The capacitors provide the ac ground for second stage differential amplifier. The circuit in Fig. 1(a) is symmetric and can be folded along the vertical axis of symmetry as shown in Fig. 1(b). By replacing the transistors with the hybrid- model of the and transistors and re-arranging the circuit components in Fig. 1(b), the small signal equivalent half-circuit of the proposed tunable amplifier is obtained as shown in Fig. 2. A. Variable Gain Amplifier Stage Analysis The output power of the proposed tunable amplifier can be adjusted by varying the gain of the first stage amplifier using the four digital bits . Based on Figs. 2 and 3, the amplifier gain of first stage is given as (1) where transconductance , early voltage , the commonemitter forward current gain , and the output resistance's degradation factor due to transistor saturation with are indicated for the transistor pair , the constant coefficients of the estimated

MA et al.: RECONFIGURABLE K-/Ka-BAND POWER AMPLIFIER WITH HIGH PAE

linear gain function to 3), the digital gain control configuration as received from the digital controller or the digital baseband ( 0 to 3), and which is the dc current corresponding to the amplifier minimum gain when all the digital control bits are reset V . The gain control based on the base bias current variation of the first stage amplifier in the stacked structure mainly affects the output resistance and does not significantly alter the transconductance of the HBT pair [17]. From the variable gain control bias circuit shown in Fig. 3, the mirrored variable base biasing currents through the nodes ( 1 and 2) has to be properly matched to avoid offset errors in the first stage differential amplifier. Unlike voltage biasing of amplifiers, the current biasing circuit provides a gain control along with the power down capability. The maximum limited input signal level at and nodes provides a small voltage fluctuation at the drain of the PMOS current mirror transistors which continues to operate in the strong saturation region by generating almost same bias currents. Hence the induced nonlinearity effect due to input signal on the input transistor's bias currents is very minimal.

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Fig. 3. Variable gain control base biasing circuit.

B. Frequency Tunable Amplifier Stage Analysis The second stage amplifier gain from Fig. 2 is based on the fixed biasing transconductance of the transistor pair [21] and frequency tunable load impedance as (2) From Fig. 4(a), the load impedance of the second stage amplifier is determined by a high -factor transformer with the primary coil which is magnetically coupled to an LC tank circuit built by using the transformer secondary coil and the varactor bank . This load impedance is connected in parallel to the output matching network and the influence of the tank circuit on output matching is described in the following section. The transformer used in the load circuit can be represented as a T-network based on [22] and also the varactor bank can be modeled as a series combination [23] of the capacitor and varactor loss as shown in Fig. 4(b). By using loop analysis of the network in Fig. 4(b), we obtain the load impedance as (3) shown at the bottom of the page. 1) Varactor Bank Q-Factor: The varactor bank, as viewed from the transformer secondary coil [shown in Fig. 4(a)], consists of the parallel connected variable capacitors with equivalent impedance . The equivalent lumped circuit model of the varactor bank impedance as determined by [23] is shown in Fig. 5. The varactor bank impedance consisting of identical varactors is given by (4)

Fig. 4. (a) Tunable load (transformer and MOS-varactor bank) with output matching network. (b) T-section model of transformer with MOS-varactor.

Fig. 5. Varactor bank with equivalent lumped circuit model.

From (4), the varactor bank

-factor is (5)

(3)

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Fig. 6. Simulation plots of output matching by tuning the tank circuit load with

Based on (5) it is deduced that by connecting any number of varactors in parallel for the tunable tank circuit, the -factor contribution from the varactor bank is almost fixed. 2) Transformer Secondary Coil Q-Factor: The transformer secondary coil's -factor is given as

varied together from 0 V to 1.8 V

.

By assuming a negligible effect of the large shunt bias resistors and on the small signal analysis, the overall multi-band tunable amplifier gain is determined by the gain product of the cascaded stages as (12)

(6) 3) Overall Tank Circuit Q-Factor: For the tunable load impedance transfer function as given in (3), we can deduce that (7) (8) where is the angular resonant frequency and is the overall tank circuit -factor. By re-arranging (8), the overall tank -factor is given as (9) The overall tank -factor [24] based on the individual tors of the inductance and the varactor bank is given by

-fac(10)

By using (5) and (6) in (10) we get (11) The overall tank -factor as determined by (9) is the same as (11). Furthermore, the -factor of second-stage amplifier response in (2) is mainly determined by transformer secondary coil and a single identical varactor of the varactor-bank. Hence, the overall -factor of the second stage amplifier load is obtained by considering the mutual magnetic coupling due to the in-phase currents of the transformer primary coil and the induced current from the secondary coil shunted with the varactor bank. This enables the design to be scalable in frequency with a voltage controlled tuning range.

From (12), we can infer that the frequency response of the overall proposed amplifier gain is a function of the four bit digital gain control configuration as well as the tunable varactor control voltage . C. Impedance Matching Analysis The amplifier input and output terminals are matched to differential impedance by using T-networks as shown in Fig. 1(a). The intermediate matching network which is also the first stage amplifier's load is L-network and its passive component values, including the inductor and de- resistors , are optimized for better amplifier gain flatness. Based on the amplifier circuit topology, the output return loss is mainly determined by the output matching network as well as the load tunable tank circuit. By using the frequency band selection bits of the load tank circuit, an adaptive output return loss is achieved as shown in Fig. 6. This allows the output return loss to be frequency band reconfigurable unlike the input return loss which is almost unaffected by the tank circuit tuning. The bandwidth for both the frequency bands are governed by the output matching network's loaded -factor as well as the tank -factor. III. FREQUENCY TUNABLE LOAD DESIGN The frequency tunable load in the second stage amplifier consists of a 2-coil center tap transformer and a MOS varactor bank connected to the secondary coil as shown in Fig. 7. The primary and secondary coils of the transformer are built on the same plane with edge coupling to avoid formation of a broadside coupled parasitic capacitance between the coils. Hence it shifts the tank circuit self-resonance to a higher frequency. The transformer coils are oriented in non-inverting mode [22] and are implemented by using the top metal layer (with thickness

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Fig. 7. Transformer layout (a) 3D view and (b) top view with interface to the varactor bank.

Fig. 8. EM simulation plot of the designed transformer's primary and secondary coil inductance along with the coupling coefficient against the standalone inductance.

) as supported of 2.81 m and sheet resistance of 10.5 m in the fabrication process to achieve a high -factor. The transformer's primary outer coil size is 85 m, the secondary coil size is 105 m, the uniform coil width is 8 m, and the coil spacing is 2 m (shown in Fig. 7). The transformer is designed and optimized using the Agilent ADS Momentum 2.5D EM simulator in RF mode to estimate the desired primary and secondary coil inductance over the entire operating frequency range as shown in Fig. 8. The transformer design does not include any kind of shielding structures. When compared to the scheme with a standalone inductive load, the transformer by using a magnetic coupling coefficient provides an enhanced -factor that increases the effective secondary coil inductance as given by (13) The transformer in the proposed design has turns ratio of 2:1 which is also evident in Fig. 7. Hence the ratio of primary coil inductance to secondary coil inductance by assuming almost equal sized coils is 4:1. This assumption is also validated by the inductance plot obtained from the EM simulation as shown in Fig. 8.

The work in [18] claims that for a same area constraint, there is no -improvement for transformer tank resonator as compared to a standalone inductor tank resonator which is not completely agreeable for the planar transformer with in-phase current orientation. Since in this work [18], the limitations such as reduction of the coil inductance by increasing its width and spacing as well as the loss contributions of the tank circuit capacitors are neglected. Moreover, due to the increased effective coil inductance by the magnetic coupling, the length of the coil can be reduced to provide the same inductance value as a standalone inductor in the required frequency range. This reduces the resistive loss associated with the transformer coil and eventually enhances the -factor due to the transformer coupled tank circuit as compared to the standalone load inductor. The varactor bank is parallel-connected MOS varactors that are operating in the accumulation mode with the gate-source tuning voltage ranging from 0.9 to 0.9 V to traverse across the to value, respectively. To avoid negative external tuning voltages at the varactor source/drain terminals , the varactor gate voltage is level-shifted to 0.9 V through the secondary transformer's center tap as shown in Fig. 7(b). This ensures a positive external voltage ranging from 0 to 1.8 V to be used as the varactor tuning voltage with the capacitance tuning characteristics as shown in Fig. 9(a). The simulation plots in Fig. 9(a) and (b) are obtained for identical varactors connected in parallel against the same varactor tuning voltage . From Fig. 9(a), the equivalent capacitance adds up as the number of identical varactors in parallel increases. An interesting observation noticed in the varactor bank -factor plot, shown in Fig. 9(a), is the equivalent -factor of the varactor bank which is unaffected by any number of parallel connected varactors . This behavior agrees with (5) as described in Section III. This is one of the merits of this proposed design and the amplifier operating frequency range can be easily reconfigured based on the number of varactors in parallel as shown by the intrinsic tank frequency plot in Fig. 9(b). As the number of varactors in the varactor-bank increases, the main design trade-off taken into consideration is the shrinking of the tunable frequency range due to the increased tank minimum capacitance value which is limited by .

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Fig. 10. Simulated single-ended open circuit (self-inductance) and varactor bank loaded tank -factor for two versions of the transformer primary coil size and the MOS varactor length as m m and m m .

Fig. 9. Simulation plots at 30-GHz frequency of the designed varactor's (a) capacitance and quality factor and (b) intrinsic tuned tank frequency.

The variation [23]of the varactor capacitance tuning voltage ( 1, 2, etc.) is defined as

by the

(14) is the minimum capacitance value of varactor, is the varactor tuning range, is the range of gate bias at which has a maximum variation for , determines the normalized bias voltage range. From (3), (7), (12) and (14) we can infer that by changing using a corresponding varactor tuning voltage ( 1, 2, etc.), the center frequency of the overall amplifier frequency response can be tuned. where,

IV.

-FACTOR ENHANCEMENT

As evident from the simulation plot in Fig. 10, the loaded tank -factor is enhanced under two conditions namely, A. Loaded Tank Q-Enhancement From

to

The loaded tank -factor curves from to in Fig. 10 are obtained by reducing the transformer size of the primary outer coil from 94 to 85 m and the secondary coil

Fig. 11. Microphotograph of proposed multi-band amplifier with tunable load m , Total area with I/O pads: m ). (Core area:

size from 114 to 105 m as well as by concurrently decreasing the MOS varactor-bank's channel length from 650 to 500 m. From Fig. 10, the curves from to are upshifted both along the frequency as well as the peak value. Both these observations can be illustrated by considering the simultaneous reduction of the tank circuit component dimensions ( and ) from to that results in a frequency upshift due to the decreased product according to (7) as well as an increased peak value by the resulting decreased resistive losses and as discussed in (9) and (13).

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Fig. 13. Large signal measurement plot of the proposed tunable amplifier at “11”). 29.5 GHz frequency, maximum gain setting, and band #2 (

ering the fact that the peak of the loaded resonant tank -factor is positioned within the pass-band of the amplifier's interested frequency range. Hence we can mitigate the complicated tank circuit analysis by neglecting the high-order effects of transformer secondary coil loaded by the capacitive varactor bank [19]. V. EXPERIMENTAL RESULTS

Fig. 12. Measured -parameter variation based on (16 steps), band (2 steps) (a) gain and (b) return loss and isolation. switch based on

B. Loaded Tank Q-Enhancement By Band-Select Input VSW From “00” to “11” Configuration From Fig. 10, we also observe that, for either of the curves or , the frequency upshifts for band-select input configuration switching from “00” to “11” as well as the peak value is enhanced. This can be analytically justified by considering the intrinsic tank frequency characteristics and the MOS varactor bank's -factor as shown in Fig. 9(b) and (a), respectively for an increase in the voltage from 0 to 1.8 V. Both these -factor enhancements simultaneously improves the proposed amplifier's PAE, linearity and the peak gain performance by reducing the tank circuit losses which are also evident from the on-wafer measurement results. The choice of the proposed transformer coil dimensions and the varactor bank aspect ratios are mainly determined by consid-

The proposed -band frequency tunable DVGA is implemented in a 0.18- m SiGe BiCMOS process from Tower Jazz Semiconductors. The microphotograph of the proposed amplifier in the fabricated wafer is shown in Fig. 11 which occupies an overall die area of 0.89 mm 0.81 mm including the on-wafer probing pads. The proposed design performance is experimentally verified by using on-wafer probing with the Agilent E8364B PNA network analyzer that supports a 4-port calibration and mixed mode scattering parameter measurement by avoiding the use of any balun. The proposed amplifier consumes a total dc current ranging from 9.9 to 12.5 mA for the maximum to minimum gain variation, respectively during its normal operation mode V and during the power down mode V dissipates a dc current of 106 A from a single 1.8 V supply voltage. The gain reduction with an increase in the bias current is due to the transition of first stage amplifying transistors from active region (maximum gain) towards saturation region (minimum gain). The measurement setup consists of wafer probe station with two RF GSSG probes for probing the amplifiers' differential input and output, along with a 7-pin dc probe that provides a 4-bit digital gain control signal along with a 2-bit frequency band selection input, and individual dc probes for supply voltage V , the base bias circuit constant current A , and the power down mode control as marked in Fig. 11. The digital pins namely , and are applied with dc voltages of either 0 V (for bit “0”) or 1.8 V (for bit “1”).

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Fig. 15. Measured stability factors of the proposed amplifier design for the maximum gain and band #1 condition.

Fig. 14. Measured linearity performance over the two bands “0000”) (a) proposed design with maximum gain ( PAE.

of the and (b)

The variable gain control using the four gain control bits of the proposed amplifier is verified by the measured -parameters shown in Fig. 12(a) and (b). Additionally, Fig. 12 indicates the frequency band switching functionality of the proposed amplifier that is achieved by providing a same voltage to pins (together depicted as ). By providing option for dual-band switching, the center frequency of the gain response can be changed. The small-signal gain's 3-dB bandwidth can support -band by selecting band#1 ( “00”) and -band using band#2 selection ( “11”). Additionally, a 0.75 dB small signal gain flatness is achieved for a frequency ranging from 22.67 to 30.2 GHz across both the frequency bands. The high transformer coupled load along with the current reuse topology of the proposed design provides a high gain and an improved linearity performance together with low dc power consumption. This results in an output 1-dB gain compression point ) of 9.6 dBm for the band#2 ( “11”), and the maximum gain condition ( “0000”) at a measured frequency of 29.5 GHz as shown in Fig. 13. The measured and PAE (with a 11.1 dBm peak saturated power and 55.9% peak PAE) of the proposed tunable amplifier over the two switchable frequency bands based on

Fig. 16. Simulated (dashed line) and measured (solid line) gain of proposed “00”). design for maximum/minimum gain conditions band#1 (

and maximum gain setting ( “0000”) are shown in Fig. 14(a) and (b), respectively. Both these plots indicate that the linearity and PAE performance are improved for band#2 ( “11”) by considering the tank circuit load enhancement as illustrated in Fig. 10. Although, the peak output power in both the configurations is comparative by only about 1.9 dB difference as well as the amplifier performance with “00” configuration is also comparable with the state-of-the-art performance to support dual band reconfigurability. Hence the configuration of “00” is necessary which also highlights the merit of this dual-band amplifier design. Based on this proposed power down method, there is a possibility to turn on the input transistors, under power down mode by a large input signal power level of at least 8 dBm. This signal level is much larger than the input point of the proposed amplifier design under normal dc bias condition. Hence by providing the same limitation on the input power

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TABLE I PERFORMANCE SUMMARY OF WIDEBAND K-/KA-BAND DRIVE POWER AMPLIFIERS

Fig. 17. Simulated load voltage and load current waveforms at 28 GHz with “00” configuration for band #1 against input power sweep from 35 to 10 dBm (step = 1 dB).

Fig. 18. Simulated load voltage and load current waveforms at 30 GHz with “11” configuration for band #2 against input power sweep from 35 to 10 dBm (step = 1 dB).

level as boundary condition we can still achieve the power down using the proposed method. The stability factors ( and ) extracted from measured S-parameters shown in Fig. 15 indicates that the proposed amplifier has unconditional stability over the operating frequency range. The difference between the simulation and measurement results as shown in Fig. 16 for the maximum and minimum gain condition can be attributed to the transistor and varactor model inaccuracy at such high frequencies. The simulation load voltage and load current waveforms at each of the band based on configuration for input power sweep indicates that the proposed dual band amplifier design operates in linear region as shown in Figs. 17 and 18 for a large

input power level of 10 dBm which is close to the input point. The performance of the proposed multi-band amplifier is consolidated in Table I and compared with the state-of-the-art -band monolithic amplifier designs. By using dc current reuse topology along with a high frequency tunable transformer coupled load and a variable gain control option, the PAE and the overall amplifier performance is improved as compared to other works in Table I. It is also evident that the amplifier's linearity performance ( and ) in other works are mainly achieved by using a larger supply voltage (increased headroom) and also high dc power consumption that supports the large signal drive capability.

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Although the proposed amplifier design has enhanced PAE performance as compared to the state-of-the-art works, the high linearity performance becomes a hard limit and a major concern in power amplifier specifications and reconfigurable devices. VI. CONCLUSION This paper presents a high power efficient broadband DVGA with reconfigurable dual-band switching capability to support -band (18–27 GHz) satellite communication, short range 24 GHz ISM (22–29 GHz) automotive radar system and -band (26.5–40 GHz) applications. In this work, the tunable loaded tank circuit -factor enhancement along with the variable gain control, and the frequency-band switching of the proposed amplifier, together with their resulting improvement on PAE and linearity performance are theoretically analyzed and experimentally verified. ACKNOWLEDGMENT The authors would like to thank the Tower Jazz Semiconductors Inc., Newport Beach, CA, USA, for providing fabrication service of the design. The authors would also like to thank W. Yang of Nanyang Technological University (NTU), Singapore for assisting in the on-wafer measurement of the proposed design. REFERENCES [1] T. Tokumitsu, “K-band and millimeter-wave MMICs for emerging commercial wireless applications,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 11, pp. 2066–2072, Nov. 2001. [2] I. Gresham et al., “Ultra-wideband radar sensors for short-range vehicular applications,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 9, pp. 2105–2122, Sep. 2004. [3] C.-W. Kuo, H.-K. Chiou, and H.-Y. Chung, “An 18 to 33 GHz fullyintegrated darlington power amplifier with guanella-type transmissionline transformers in 0.18 m CMOS technology,” IEEE Microw. Wirel. Compon. Lett., vol. 23, no. 12, pp. 668–670, Dec. 2013. [4] M. K. Siddiqui, A. K. Sharma, L. G. Callejo, and R. Lai, “A high power and high efficiency monolithic power amplifier for LMDS applications,” IEEE Trans. Microw. Theory Techn., vol. 46, no. 12, pp. 2226–2232, Dec. 1998. [5] FCC, Washington, DC, USA, “First report and order, revision of part 15 of the commission's rules regarding ultra wideband transmission systems ET Docket 98–153,” 2002. [6] J.-W. Lee and S.-M. Heo, “A 27 GHz, 14 dBm CMOS power amplifier using 0.18 m common-source MOSFETs,” IEEE Microw. Wirel. Compon. Lett., vol. 18, no. 11, pp. 755–757, Nov. 2008. [7] J.-H. Chen, S. R. Helmi, and S. Mohammadi, “A fully-integrated Ka-band stacked power amplifier in 45 nm CMOS SOI technology,” in Proc. IEEE Topical Meet. Silicon Monolithic Integr. Circuits RF Syst., Jan. 2013, pp. 75–77. [8] C. Cao, H. Xu, Y. Su, and O. K. Kenneth, “An 18-GHz, 10.9-dBm fully-integrated power amplifier with 23.5% PAE in 130-nm CMOS,” in Proc. 31st Eur. Solid-State Circuits Conf., Sep. 2005, pp. 137–140. [9] S. Y. Mortazavi and K.-J. Koh, “A class F-1/F 24-to-31 GHz power amplifier with 40.7% peak PAE, 15 dBm OP 1 dB, and 50 mW Psat in 0.13 m SiGe BiCMOS,” in Int. Solid-State Circuits Conf. Tech. Dig., San Francisco, CA, USA, Feb. 2014, pp. 254–255. [10] N. -. Kuo, J.-C. Kao, C.-C. Kuo, and H. Wang, “K-band CMOS power amplifier with adaptive bias for enhancement in back-off efficiency,” in IEEE MTT-S Int. Microw. Symp. Dig., Baltimore, MD, USA, Jun. 2011, pp. 1–4. [11] E. Kaymaksut and P. Reynaert, “Transformer-based uneven Doherty power amplifier in 90 nm CMOS for WLAN applications,” IEEE J. Solid-State Circuits, vol. 47, no. 7, pp. 1659–1671, Jul. 2012. [12] K. Kim and C. Nguyen, “A 16.5–28 GHz 0.18- m BiCMOS power amplifier with flat 19.4 1.2 dBm output power,” IEEE Microw. Wirel. Compon. Lett., vol. 24, no. 2, pp. 108–110, Feb. 2014.

[13] P.-C. Huang, J.-L. Kuo, Z.-M. Tsai, K.-Y. Lin, and H. Wang, “A 22-dBm 24-GHz power amplifier using 0.18- m CMOS technology,” in IEEE MTT-S Int. Microw. Symp. Dig., Anaheim, CA, USA, May 2010, pp. 248–251. [14] A. Komijani, A. Natarajan, and A. Hajimiri, “A 24-GHz, 14.5-dBm fully integrated power amplifier in 0.18- m CMOS,” IEEE J. SolidState Circuits, vol. 40, no. 9, pp. 1901–1908, Sep. 2005. [15] P. J. Riemer, J. S. Humble, J. F. Prairie, J. D. Coker, B. A. Randall, B. K. Gilbert, and E. S. Daniel, “Ka-band SiGe HBT power amplifier for single chip T/R module applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1071–1074. [16] V. Giammello, E. Ragonese, and G. Palmisano, “A transformer-coupling current-reuse SiGe HBT power amplifier for 77-GHz automotive radar,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1676–1683, Jun. 2012. [17] T. B. Kumar, K. Ma, and K. S. Yeo, “A Ku-band variable gain LNA with high PAE in 0.18 m SiGe BiCMOS technology,” IEEE Microw. Wirel. Compon. Lett., submitted for publication. [18] H. Krishnaswamy and H. Hashemi, “Inductor- and transformer-based integrated RF oscillators: A comparative study,” presented at the IEEE Custom Integr. Circuits Design Conf., San Jose, CA, USA, Sep. 2006. [19] Y. Xiaohua and N. M. Neihart, “Analysis and design of a reconfigurable multimode low-noise amplifier utilizing a multitap transformer,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 3, pp. 1236–1246, Mar. 2013. [20] K. K. Sessou and N. M. Neihart, “An integrated 700–1200-MHz class-F PA with tunable harmonic terminations in 0.13- m CMOS,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 4, pp. 1315–1323, Apr. 2015. [21] T. B. Kumar, K. Ma, and K. S. Yeo, “A low power programmable gain high PAE K-/Ka-band stacked amplifier in 0.18 m SiGe BiCMOS technology,” in IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, USA, May 2015, pp. 1–4. [22] J. R. Long, “Monolithic transformers for silicon RF IC design,” IEEE J. Solid-State Circuits, vol. 35, no. 9, pp. 1368–1382, Sep. 2000. [23] K.-H. Tsai and S.-I. Liu, “A 104-GHz phase-locked loop using a VCO at second pole frequency,” IEEE Trans. Very Large Scale Integr. (VLSI) Syst., vol. 20, no. 1, pp. 80–88, Jan. 2012. [24] L. Li, P. Reynaert, and M. Steyaert, “Design and analysis of a 90 nm mm-wave oscillator using inductive-division LC tank,” IEEE J. SolidState Circuits, vol. 44, no. 7, pp. 1950–1958, Jul. 2009. [25] J.-L. Kuo and H. Wang, “A 24 GHz CMOS power amplifier using reversed body bias technique to improve linearity and power added efficiency,” in IEEE MTT-S Int. Microw. Symp. Dig., Montreal, QC, Canada, Jun. 2012, pp. 1–3. [26] P. C. Huang, Z. M. Tsai, K. Y. Lin, and H. Wang, “A 17–35 GHz broadband, high efficiency pHEMT power amplifier using synthesized transformer matching technique,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 1, pp. 112–119, Jan. 2012. Kaixue Ma (M'05–SM'09) received B.E. M.E. degree from Northwestern Polytechnolgical University (NWPU), China, and the Ph.D. degree from Nanyang Technological University (NTU), Singapore. From August 1997 to December 2002, he was with China Academy of Space Technology (Xi'an), where he was Group Leader of millimeter-wave group for space-borne microwave and millimeter-wave components and subsystems of satellite payload and VSAT ground station. From September 2005 to September 2007, he was with MEDs Technologies as an R&D Manager and project leader, where he provides design services and product development. From September 2007 to March 2010, he was with ST Electronics (Satcom & Sensor Systems) as R&D Manager, Project Leader, and Technique Management Committee of ST Electronics. From 2010 to 2013, he was a Senior Research Fellow and millimeter-wave IC team leader for 60 GHz Flagship Chipset project. From 2013 to the present, he is a full Professor with the University of Electronic Science and Technology of China (UESTC). As a PI/Technique Leader, he did projects with fund more than $12 million (excluding projects done in China). His research interests include RFIC Design, satellite communication, software defined radio, microwave/millimeter-wave circuits and system using CMOS, MEMS, MMIC, and LTCC. He filed 10 patents and authored/co-authored over 140 journal and conference papers. He is review board of over 10 international Journals. He gave invited talks or keynote addresses over 20 times. Dr. Ma was a recipient of Best Paper Awards from IEEE SOCC2011, IEEK SOC Design Group Award, Excellent Paper Award from International Confer-

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ence on HSCD2010, Chip Design Competition Bronze Award from ISIC2011, and Special Mention Award of Emerging Technology, Singapore Inforcomm Technology Federation for the development of the Singapore Next Generation Wi-Fi Chipset 2012 and Named in Precious China “Thousand Young Talent Program” in 2012.

Kiat Seng Yeo received the B.Eng. (EE) and Ph.D. (EE) degrees from Nanyang Technological University (NTU), Singapore, in 1993 and 1996, respectively. He is currently an Associate Provost (Graduate Studies and International Relations) at Singapore University of Technology and Design (SUTD) and Member of Board of Advisors of the Singapore Semiconductor Industry Association. Professor Yeo is a widely known authority in low-power RF/millimeter-wave IC design and a recognized expert in CMOS technology. He has secured over $30 million of research funding from various funding agencies and the industry in the last three years. Before his new appointment at SUTD, he was Associate Chair (Research), Head of Division of Circuits and Systems, and Founding Director of VIRTUS of the School of Electrical and Electronic Engineering, NTU. He has published 6 books, 5 book chapters, over 400 international top-tier refereed journal and conference papers, and holds 35 patents. Dr. Yeo served in the editorial board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and holds/held key positions in many international conferences as Advisor, General Chair, Co-General Chair and Technical Chair. He was a recipient of the Public Administration Medal (Bronze) on National Day 2009 by the President of the Republic of Singapore and was also a recipient of the Distinguished Nanyang Alumni Award in 2009 for his outstanding contributions to the university and society.

Thangarasu Bharatha Kumar (S'12) received the B. E. (E&C) degree from Ratreeya Vidyalaya College of Engineering (RVCE), Bangalore, India, which is affiliated with the Visvesvaraya Technological University (VTU), in 2002, the M.Sc. degree from German Institute of Science and Technology (GIST), Singapore, (a joint master's degree programme by NTU, Singapore and Technische Universitaet Muenchen (TUM), Germany), in 2010, and the Ph.D. (EE) degree from NTU, Singapore, in 2015. From January 2010 to August 2015, he was with VIRTUS, IC Design Centre for Excellence, NTU, Singapore, as a Research Associate where he worked on SiGe HBT and CMOS-based reconfigurable amplifiers for microwave and millimeter wave RF integrated circuit design. He is now a researcher with Singapore University of Technology and Design (SUTD), Singapore. His research interests include RF and millimeter-wave reconfigurable integrated circuit design. He has authored/coauthored over 25 journals and conference papers. Dr. Kumar was a recipient of the DAAD scholarship during his M.Sc. study.

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A Broadband GaN pHEMT Power Amplifier Using Non-Foster Matching Sangho Lee, Student Member, IEEE, Hongjong Park, Student Member, IEEE, Kwangseok Choi, Student Member, IEEE, and Youngwoo Kwon, Senior Member, IEEE

Abstract—Non-Foster matching is applied to design a multi-octave broadband GaN power amplifier (PA) in this paper. The bandwidth limitation from high-Q interstage matching is overcome through the use of negative capacitor, which is realized with a negative impedance converter (NIC) using the cross-coupled GaN FETs. For high power operation over the entire bandwidth, the natural interstage matching is optimized for the upper subfrequency band and the lower subfrequency band is compensated for by the negative capacitance presented by non-Foster circuit (NFC). Detailed analysis is presented to understand the frequency and power limits of NIC circuits for PA applications. Two negative impedance matched PAs (NMPAs) are fabricated with 0.25- m GaN pHEMT process. The implemented PA with combining shows the output powers of 35.7–37.5 dBm with the power added efficiencies of 13–21% from 6 to 18 GHz. combining PA achieves over 5 W output power from The 7 to 17 GHz. The NFC boosts the efficiencies and power below 12 GHz to achieve broadband performance without using any lossy matching or negative feedback. To our knowledge, this is the first demonstration of NIC-based broadband amplifiers with multi-watt-level output power. Index Terms—GaN, negative capacitance, negative impedance converter, non-Foster circuit, power amplifier (PA).

I. INTRODUCTION

A

WATT-LEVEL power amplifier (PA) with multi-octave bandwidth is required for broadband applications such as electronic warfare system. GaN device is suitable for this application due to its high power density and high voltage operation, which results in relatively large load impedance. The distributed amplifier (DA) is a commonly used topology for multi-octave PA. The input and output capacitances are absorbed into the artificial transmission line to overcome the frequency limitation. However, DAs suffer from small gain and requires a relatively large die area. The reactive matched PAs (RMPAs) are also widely used as multi-octave PAs. RMPAs utilize a multiple-stage design to realize high gain. However, it has the bandwidth (BW) problem coming from Bode–Fano criterion [1], [2]. To achieve Manuscript received July 01, 2015; revised August 26, 2015; accepted October 01, 2015. Date of publication November 17, 2015; date of current version December 02, 2015. This work was supported in part by the Electronic Warfare Research Center at the Gwangju Institute of Science and Technology, by the Defense the Acquisition Program Administration, and the Agency for Defense Development. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015. The authors are with the School of Electrical Engineering and Computer Engineering, Institute of New Media and Communications, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2015.2495106

broadband characteristics, lossy matching and negative feedback method are used to lower the Q-factor. This results in low efficiency and requires large chip size and has difficulty in achieving the required gain flatness over a wide bandwidth. Reconfigurable matching concept has also been proposed for multi-stage PAs to overcome the bandwidth limitation coming from Bode–Fano criteria in the interstage matching [3], [4]. However, the overall PA efficiency may be degraded due to the switch loss. Moreover, the linearity and power handling capability of the switch may limit its use for multi-watt-level PAs. A potential alternative for wideband matching is the use of a non-Foster circuit (NFC). Typical examples of non-Foster components are the negative inductors and negative capacitors. On the Smith chart, S-parameter traces of the non-Foster components move in counterclockwise direction as the frequency increases. The first non-Foster circuit using the transistors was proposed by Linvill in 1954 [5]. The NFC was used to compensate for the parasitic effects of various circuits such as filters, varactors, and VCOs [6]. For example, the negative slope of the reactance versus frequency is used to overcome the limitation of the antenna size and Q-factor at 800 MHz [7]. The gain-bandwidth enhancement of DA has been demonstrated using the negative capacitance in [8]. However, its application was limited to small-signal operation. Although the non-Foster matching is effective in cancelling out the reactance over a broad bandwidth, there are three major challenges using non-Foster circuit, noise, stability and power handling capability. Due to the power handling issues, little work has been presented to demonstrate a broadband PA using non-Foster circuit. In this work, two-stage GaN power amplifiers have been developed using NFC for broadband operation ranging from 7 to 17 GHz. The non-Foster circuit realized with the cross-coupled GaN FETs is applied to the interstage matching to cancel out the large input capacitance of the power-stage FETs over multi-octave bandwidth. This paper is an extended version of our previous work [9], which was the first demonstration of negative impedance converter (NIC) based broadband amplifiers with watt-level output power. This paper presents more detailed small-signal analysis to understand the high frequency limitation of the NIC. In addition, the large-signal behavior of the NIC is analyzed by using the full nonlinear GaN FET models. The combined results of small- and large-signal analyses are used to find the optimum transistor size and bias points to extend the operation frequency and power range of the NIC. This paper is organized as follows. Section II presents the overall design concept of the negative impedance matched PA

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Fig. 1. Block diagram of the two-stage PA with non-Foster matching network.

(NMPA), the operation principle of the NFC, the frequency and power limits of NFC, and the detailed NFC-based PA circuit design methodology. In Section III, the measurement results of the two GaN PAs are presented including the updated results with higher output power. The power and power added efficiency (PAE) enhancement with NFC are also explained using the comparison data. II. TWO-STAGE GAN PA WITH NFC A. NFC Operation Principle and Circuit Design A simplified block diagram of the two-stage GaN PA with the proposed non-Foster circuit is shown in Fig. 1. The unit FET size of the PA is 6 125 m, which has a maximum available gain (MAG) of 12 dB at 18 GHz. The GaN FET with a size of m shows a cut-off frequency of 21.7 GHz, an of 74.6 GHz, and a transconductance of 227 mS/mm at 28 V drain bias. The load-pull measurement on a 6 125 m FET cell shows a maximum output power density of 2.8 W/mm, a PAE of 41.6%, and a power gain of 9.3 dB at 15 GHz under continuous wave conditions. Even though the transistor performance merits do not match those of the state-of-the-art GaN transistors [17], [18], [21], the concept of NIC-based PA can be proven by comparing the PA performance with and without NICs. To achieve overall gain higher than 15 dB, a two-stage design is required, in which case the bandwidth limitation often comes from the high-Q interstage matching rather than the output matching. For wide bandwidth, lowering the Q-factor of interstage matching network is essential. Since Q-factor is the ratio between the stored energy and power loss, it can be reduced by increasing the power loss and decreasing the stored energy. The example of the former is the RMPA with lossy matching and the latter is - resonance matching. But, - resonance reduces the stored energy only over a narrow bandwidth. On the other hand, non-Foster matching can reduce the stored energy over a broad bandwidth. In this work, the NFC is employed in the interstage matching to cancel out the large input capacitance of the power-stage FETs, lowering the Q-factor of interstage matching. Output matching is realized with a conventional two-section matching network. The output powers from each FET are combined through a Wilkinson power combiner to achieve W over the target frequency range of 6–18 GHz. In theory, the addition of NFCs in the output matching network can further improve the power BW. However, the power handling capability

Fig. 2. Simulated interstage impedance matching with NFC.

of NFC has to scale with the RF power, which results in the excessive dc power consumption from the NFC and degrades the PAE of the overall PA. Therefore, in this work, NFC is employed in the interstage matching circuit only. One of the key practical issues with NFC is the limited operating frequency. The bandwidth limitation comes from the self-resonance. The self-resonating frequency (SRF) is limited by the of the device. 6 125 m GaN FET used as the unit transistor in the NFC has a gate length of 0.25 m and a of 21.7 GHz. As will be shown in the next subsection, the SRF is limited to GHz, which is not high enough to cover the entire bandwidth up to 18 GHz. Therefore, the interstage matching network is optimized separately for two subfrequency regions. The natural interstage matching is optimized for the upper subfrequency band above 11 GHz, where the negative capacitance is not available. The impedance mismatch in the lower subfrequency band below 11 GHz is compensated for by the negative capacitance presented by NFC. To better understand the benefit of NFC in the interstage matching, the input impedance of the power stage is simulated together with the optimum load impedance of the driver-stage FET in Fig. 2. The optimum driver-stage impedance moves along a constant-g circle in a counterclockwise direction as the frequency increases. The input impedance of the power stage is highly capacitive due to the large input gate capacitance of 1.4 pF. It is thus very difficult to provide optimum impedance matching across the entire bandwidth. Upper frequencies above 11 GHz are matched using an inductive line and shunt stubs. The impedance mismatch in the lower frequency subband is mitigated by employing a negative shunt capacitance of pF, which basically reduces the Q-factor of the input impedance of the power stage impedance from 10.30 to 1.62 at 6 GHz and from 4.39 to 0.78 at 8 GHz. In this way, the same natural matching circuit consisting of the inductive line and the shunt inductance can be used to match the lower subband frequencies below 11 GHz as well. B. Frequency and Power Limitation of the NFC The non-Foster circuit used in the interstage matching is based on the Linvill's NIC [Fig. 3(a)]. The NIC is composed

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(a)

(b)

Fig. 4. Reduced equivalent circuit of the NIC.

Fig. 3. (a) Circuit of Linvill's NIC and (b) the simplified equivalent circuit of the NIC.

of the cross-coupled FETs with the loaded capacitor. The main operation of the NIC is to invert the polarity of to an effective capacitance, . The coefficient, , is a function of frequency. However, the NIC cannot generate the negative capacitance in the low condition. For example, high-frequency operation or large signal-operation leads to low gain conditions. In this case, NIC operates just like normal inductor. For analysis, it is useful to represent the NIC with a simplified equivalent circuit of a series resistor and either negative capacitance or inductance , depending on the frequency, as shown in Fig. 3(b). Small-signal characteristics of the NIC can be understood from a simple analysis using and . With the two identical transistors, the input impedance, , and the can be expressed by the following equations [10]:

(1) (2) is . The equivalent circuit shown in Fig. 4 is where derived from the first equation. The equivalent circuit of NIC is composed of a complicated combination of resistors, inductors, and capacitors. As a result, the input impedance is a strong function of frequency. At frequencies higher than the transistor cut-off frequency, , NIC cannot convert the loaded capacitor to negative capacitance. However, as can be seen from (2), the SRF is much lower than due to the additional capacitances and inductances. For example, SRF is only 11.6 GHz even if is 21.7 GHz for a 6 125 m GaN FET ( pF and pF). Fig. 5 shows the simulated S-parameter of the NIC using the simplified equivalent circuit of Fig. 4 and the full FET model provided by the foundry. The equivalent circuit of the intrinsic FET model contains the low-frequency dispersion effect with the thermal and trap subcircuits [11]and the equations for , and are based on the Angelov model.

Fig. 5. Simulated S-parameters of the NIC using the full device model and the reduced equivalent circuit.

The full device model simulation predicts an SRF of 13 GHz while the simplified model shows an SRF of 11.6 GHz. The difference between the two results comes from and other parasitic effects not accounted for in the simplified equivalent circuit. The detailed circuit schematic of NFC is shown in Fig. 6. The cross coupled FETs used in the NFC are 6 125 m GaN FETs. The resistor , and are used to prevent instability. helps to reduce the equivalent series resistance, , due to the impedance inversion effect of the cross-coupled pair. The simulated frequency-dependent reactance of the NFC is plotted in Fig. 7. The reactance decreases as the frequency increases up to 11 GHz, which clearly shows non-Foster characteristics. The equivalent capacitances of the NFC in this frequency range are to pF. The equivalent series resistance varies between 4.4 and 11.4 . Above 11 GHz, where the self-resonance occurs, the positive reactance slope is observed and the circuit follows Foster's theorem. In this case, the NFC is represented with a series combination of a resistance and a positive inductance as shown in Fig. 3(b). Another limitation in NFC operation may come from the power handling capability in the PAs. The NFCs cease to

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(a)

Fig. 6. Detailed schematic of the proposed NIC.

(b) Fig. 8. (a) Simulation results of the power division ratio between the NIC and power stages. (b) Simulated susceptance of the NIC versus injected RF power.

(a)

(b)

Fig. 7. Simulated (a) reactance of the NIC as a function of frequency and (b) equivalent capacitance or inductance.

show negative impedance when they are driven with large RF power. As shown in Fig. 1, NFC shares the same node as the power-stage FETs. The RF power from the driver stage is divided into three paths with a different division ratio depending on the operating frequencies. Fig. 8(a) shows the simulated

power division ratio into the NFC versus the power FETs as a function of frequency. As the frequency decreases, more power is delivered to the NFC than to the power stages, which eventually reduces the loop gain in the NFC below a threshold level required to generate the negative impedance. To show the power handling capability, we have simulated the large-signal response of the NFC with 15 V drain voltage [Fig. 8(b)]. The negative susceptance, which is required to cancel out the positive susceptance due to the large power-stage input capacitance, decreases rapidly as the power increases. If the output power is 5 W and the power gain of the main stage is 3 dB, then the expected power delivered to the NFC is 1 W at 6 GHz. It is sufficient to cease the effect of NFC. Combining the results of Fig. 8(a) and (b), it is predicted that the low frequency operation of the NFC is vulnerable to large-signal operation. So, it is expected that the effective frequency range of the NIC in our PA is limited to 6–11 GHz, corresponding to the previously mentioned “low-frequency subband.” C. Detailed NFC Design for Broadband PAs Key design parameter is the value of the load capacitance, , in the NFC [see Fig. 3(a)], since it determines in Fig. 3(b). Larger results in a larger negative capacitance, which allows the compensation of larger input transistor capacitance. This means that a smaller number of NFCs are required to compensate for the given power-stage transistors. The dc power consumption of the NFC can be reduced in this way. However,

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(a)

(a)

(b) Fig. 9. Simulated (a) power handling capability of the NFC and (b) SRF with various unit transistor size.

(b)

SRF is inversely proportional to and one needs to carefully select and considering the required bandwidth. The transistor size used in the NFC determines . The transistor size also determines the power handling capability of the NFC. Fig. 9 shows the simulated power handling capability of the NFC with the various transistor sizes. The negative susceptance represents the effect of the NFC and the power capability can be determined by the input power when the susceptance crosses zero. The power limit of the NFC improves with the transistor size. This can be understood from the load seen by the cross-coupled transistors in the NFC. Each transistor has the output load composed of the and the other transistor, which presents relatively low impedance. Due to the large bias voltage of the GaN transistors, the power limitation of the NFC arises from the current clipping rather than the voltage clipping. Larger transistors provide higher current driving capability and thus improved power handling capability. However, one cannot increase the transistor size indefinitely since it will negatively impact the high-frequency operation limit of the transistors and their cut-off frequencies. To find the optimal transistor size for SRF, we have calculated SRF as the transistor size is varied in Fig. 9(b). and the unit gate finger width are fixed at 1.5 pF and 125 m, respectively, and the number of gate fingers is increased from 2 to 8. The simulation is repeated for two drain bias voltages, 8 and 15 V. As shown in Fig. 9(b), it is clear that the optimal SRF can be achieved with 6 125 m transistors at both 8 and 15 V bias conditions. In this work, a transistor size of 6 125 m is used together a value of 1.5 pF. The resulting is 1.27, and SRF is around 11.6 GHz.

Fig. 10. Simulated negative susceptance of NFC versus the input power at various (a) gate bias and (b) drain bias.

The side effect of using larger transistors in NFC is the increased power consumption. The NFCs employed for antenna matching have employed Class-C or Class-B biasing to avoid the dc power consumption under small-signal operation [12]. However, a similar biasing may not be applicable for large signal operation as in PAs. In an attempt to find the optimum bias points, the power handling capability of the simple NFC is simulated by sweeping the gate and drain biases at 8 GHz in Fig. 10. The large-signal transistor model employed in this work predicts the PA power performance with reasonable accuracy. However, it has limited accuracy in predicting the bias dependence of the PA. The modeling accuracy can be improved by using more sophisticated GaN FET models [13], [14]. As shown in Fig. 8, the NFC used in the interstage matching should present the negative susceptance up to dBm input power to cover the frequency range down to 6 GHz. Fig. 10(a) clearly shows that Class-B biasing provides insufficient power limit. In this work, Class A–AB biasing is used. Unlike the case of the gate bias, the drain bias can be reduced to 8 V without impacting the power handling capability as shown in Fig. 10(b). This can be understood from the fact that the current clipping is the main limiting mechanism rather than voltage clipping, and that the transistor transconductance peaks around 6–10 V and degrades slightly as the drain bias is increased. At high drain bias, GaN power transistors can show degradation due to self-heating [15], [16]. In this work, a drain bias of 8 V and 15

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(a) Fig. 12. Measured small-signal gain of NIC.

combined PA with and without the

(b) Fig. 11. (a) Fabricated chip photograph and (b) block diagram of PA with non-Foster matching network.

combined

V are employed for the cross-coupled transistors used in the NFC. III. MEASUREMENT RESULTS Two PAs are designed with non-Foster matching network and fabricated using a commercial 0.25- m GaN pHEMT foundry process. Both PAs are based on a two-stage design with the NFC employed in the interstage matching circuit. The first PA combines the output powers from two power FETs, each with a transistor size of 6 125 m, using a Wilkinson combiner to achieve an output power of 36–37 dBm. A single NFC consisting of the cross-coupled FETs with the same unit transistor size is used to cancel out the input capacitance of two power FETs. NIC and interstage matching for this PA are designed for optimum power performance in the lower subfrequency band from over 6 to 10 GHz. The second PA is designed to achieve higher output power by combining the output powers of four 6 125 m GaN FETs. It consists of two parallel PA chains, each with two power FETs combined using a shared output matching network. Wilkinson coupler is employed to combine the output powers from each PA chain. The expected output power is 1–2 dB higher instead of 3 dB with power combining due to the insertion and mismatch losses of the on-chip broadband Wilkinson coupler. The interstage matching for the second PA is optimized to show more pronounced NFC effect in a slightly shifted frequency range with focus on 8–12 GHz. For testing, the PA chips are mounted on 5-mm-thick Au-plated Cu carrier with eutectic bonding to mitigate self-heating. On-wafer probing using ground-signalground (GSG) probes (the Infinity probes from Cascade Microtech) is used to measure both small-signal and large-signal

Fig. 13. Measured return losses of

combined PA with and without the NIC.

characteristics. Continuous wave signal is used to measure the power characteristics. A. PA With

Combining

Fig. 11 shows the die photograph and the block diagram of the fabricated PA chip with two-FET combining. As shown in Fig. 11(b), a single NFC cancels out the input capacitance of two power FETs. Fig. 12 shows the measured small-signal gain with and without the NIC. The drain bias voltage to the PA is 28 V while that to the NIC is set to 15 V for this measurement. To show the effect of NIC on the return losses, and are also measured in Fig. 13 with NIC turned on and off. There is very little change in the return losses when NIC is turned on. The effect of NIC on the noise performance is also measured in Fig. 14. At the frequencies where the overall gain improves with NIC, the noise figure also improves. More meaningful improvement can be observed in the power characteristics shown in Fig. 15. Since NIC-based interstage matching circuit is designed to provide the optimum load impedance to the driver stage in the low-frequency subband, the output power and PAE improves by up to dB and %, respectively, between 6 and 11 GHz when NIC is used. At 8 GHz, the output power is increased from 36 to 37.3 dBm and the PAE is improved by 4.5%. The peak PAE of 13–21% and the output power of 35.7–37.5 dBm are achieved across the

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Fig. 14. Measured noise figure of

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 63, NO. 12, DECEMBER 2015

combined PA with and without the NIC.

(a)

(b)

Fig. 15. Measured output power and PAE of the NIC.

combined PA with and without

Fig. 16. (a) Fabricated chip photograph and (b) block diagram of PA with NFC.

combined

6–18 GHz frequency bandwidth. The overall PAE degradation due to the power consumption of NIC is estimated to be %. B. Parallel Combined PA With

Combining

comThe die photograph and the block diagram of the bining PA are shown in Fig. 16. For this PA, the drain bias to the power FETs is 28 V while that to the NFC FETs is reduced to 8 V based on the analysis presented in the previous section. Fig. 17 represents the measured small-signal gain with and without NIC. Similar to the result of the combining PA, increases in the low-frequency subband from 7 to 12 GHz with the NIC. The gain improvement is more pronounced at slightly higher frequencies in this design since the interstage circuit is further optimized to achieve better matching near 8–12 GHz. The gain improvement up to 3.9 dB can be observed at 11 GHz. The measured power characteristics with and without NIC are shown in Fig. 18. Unlike the case of combining PA which showed the power improvement ( dB) only up to GHz, this circuit showed significant power improvement ( dB) up to 12 GHz. Over a slightly shifted frequency range of 7 to 17 GHz, the output power higher than 5 W is achieved. The lower frequency limit for this design is around 7 GHz, below which no power improvement is observed with the NIC. The PAE degradation due to the power consumption of NIC is estimated to be less than 0.5%. Fig. 19 compares the power sweep characteristics at 10 GHz between the two PA circuits. With NIC, the output power is increased by as much as 2.1 dBm from 36.1 to 38.2 dBm and

Fig. 17. Measured small-signal gain of NIC.

combined PA with and without the

the PAE is improved by 4.5% in combining PA while the improvement is limited to 0.66 dBm and 1.6% for combining PA. This is again attributed to the further optimized interstage network at this frequency. The output power increase in combining PA compared with PA is 1–2 dB from 7 to 17 GHz. At the band edges, the output power improvement is limited due to the increased loss of the power combiner. The measured PAE of the combining PA is also lower than the PA due to the power combiner loss. Table I compares the performance of the PA of this work with the state-of-the-art DA and RMPA using GaN pHEMTs. This work is an extended version of our previous work [9], which is the first demonstration of the NMPA. Even though the measured

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TABLE I PERFORMANCE COMPARISON TABLE OF GAN BROADBAND PAS

Fig. 18. Measured output power of

combined PA with and without the NIC.

The bandwidth limitation due to high-Q interstage matching has been mitigated through the use of a shunt negative capacitance. To guarantee the high power operation over the entire bandwidth, natural interstage matching is optimized for the upper subfrequency band and the lower subfrequency band is compensated for by the negative capacitance presented by NFC. Detailed analysis is performed to understand the frequency limitation of NIC approach, which shows that high-frequency limit comes from the self-resonance and the low-frequency limit from the power handling capability. The fabricated combining PA shows output powers higher than 5 W from 7 to 17 GHz. At frequencies, where NFC is optimized for interstage matching, the power improvement by 2.1 dBm and PAE improvement by 4.5% have been achieved. This work demonstrates that non-Foster matching can provide a new perspective in designing the broadband PAs. REFERENCES

Fig. 19. Measured power sweep characteristics of

and

PAs at 10 GHz.

power and PAE do not match the state-of-the-art results due to the limited device performance, this work shows that NIC can be an effective method to realize a broadband PA in a small die area without lossy matching or feedback. IV. CONCLUSION In this work, two-stage GaN PAs with non-Foster circuit have been developed for multi-octave broadband power applications.

[1] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” J. Franklin Inst., vol. 249, no. 1, pp. 57–83, 1950. [2] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” J. Franklin Inst., vol. 249, no. 2, pp. 139–154, 1950. [3] S. Park, J. Woo, U. Kim, and Y. Kwon, “Broadband CMOS stacked RF power amplifier using reconfigurable interstage network for wideband envelope tracking,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 4, pp. 1174–1185, Apr. 2015. [4] U. Kim, K. Kim, J. Kim, and Y. Kwon, “A multi-band reconfigurable power amplifier for UMTS handset applications,” in Proc. IEEE RF Integr. Circuits Symp. Dig., May 2010, pp. 175–178. [5] J. G. Linvill, “Transistor negative impedance converters,” in Proc. IRE, Jun. 1953, vol. 41, no. 6, pp. 725–729. [6] Q. Wu, S. Elabd, T. K. Quach, A. Mattamana, S. R. Dooley, J. McCue, dBc/Hz FOMT P. L. Orlando, G. L. Creech, and W. Khalil, “A wide tuning range Ka-band VCO using tunable negative capacitance and inductance redistribution,” in Proc. IEEE RF Integr. Circuits Symp. Dig., Jun. 2013, pp. 199–202. [7] O. O. Tade, P. Gardner, and P. S. Hall, “Antenna bandwidth broadening with a negative impedance converter,” Int. J. Microw. Wirel. Techn., vol. 5, no. 3, pp. 249–260, Jun. 2013. [8] A. Ghadiri and K. Moez, “Gain-enhanced distributed amplifier using negative capacitance,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 11, pp. 2834–2843, Nov. 2010. [9] S. Lee, J. Kim, H. Park, and Y. Kwon, “A 6–18 GHz GaN pHEMT power amplifier using non-foster matching,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2015.

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[10] Q. Wu, T. Quach, A. Mattamana, S. Elabd, S. R. Dooley, J. J. McCue, P. L. Orlando, G. L. Creech, and W. Khalil, “Design of wide tuning-range mm-wave VCOs using negative capacitance,” in Proc. IEEE Compound Semiconductor Integr. Circuit Symp., Oct. 2012, pp. 1–4. [11] O. Jardel, F. De Groote, T. Reveyrand, J.-C. Jacquet, C. Charbonniaud, J.-P. Teyssier, D. Floriot, and R. Quéré, “An electrothermal model for AlGaN/GaN power HEMTs including trapping effects to improve large-signal simulation results on high VSWR,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 12, pp. 2660–2669, Dec. 2007. [12] S. E. Sussman-Fort and R. M. Rudish, “Non-Foster impedance matching of electrically-small antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2230–2241, Aug. 2009. [13] A. Raffo, V. Vadalà, D. M. M. Schreurs, G. Crupi, G. Avolio, A. Caddemi, and G. Vannini, “Nonlinear dispersive modeling of electron devices oriented to GaN power amplifier design,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 4, pp. 710–718, Apr. 2010. [14] H. Jang, P. Roblin, and Z. Xie, “Model-based nonlinear embedding for power-amplifier design,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 9, pp. 1986–2002, Sep. 2014. [15] S. Nuttinck, E. Gebara, J. Laskar, and H. M. Harris, “Study of selfheating effects, temperature-dependent modeling, and pulsed load-pull measurements on GaN HEMTs,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 12, pp. 2413–2420, Dec. 2001. [16] L. Ardaravicius, A. Matulionis, J. Liberis, O. Kiprijanovic, M. Ramonas, L.-F. Eastman, J.-R. Shealy, and A. Vertiatchikh, “Electron drift velocity in AlGaN/GaN channel at high electric fields,” Appl. Phys. Lett., vol. 83, no. 19, pp. 4038–4040, Nov. 2003. [17] C. Campbell, T. Lee, V. Williams, M. Kao, H. Tserng, P. Saunier, and T. Balisteri, “A wideband power amplifier MMIC utilizing GaN on SiC HEMT technology,” IEEE J. Solid-State Circuits, vol. 44, no. 10, pp. 2640–2647, Oct. 2009. [18] J. J. Komiak, K. Chu, and P. C. Chao, “Decade bandwidth 2 to 20 GHz GaN HEMT power amplifier MMICs in DFP and no FP technology,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2011, pp. 1–4. [19] V. Dupuy, E. Kerherve, N. Deltimple, J.-P. Plaze, P. Dueme, B. MalletGuy, and Y. Mancuso, “A 39.7 dBm and 18.5% PAE compact X to Ku band GaN travelling wave amplifier,” in Proc. 57th IEEE Int. Midwest Symp. Circuits and Systems, Aug. 2014, pp. 611–614. [20] R. Santhakumar, B. Thibeault, H. Masataka, S. Keller, Z. Chen, U. K. Mishra, and R. A. York, “Two-stage high-gain high-power distributed amplifier using dual-gate GaN HEMTs,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 8, pp. 2059–2063, Aug. 2011. [21] U. Schmid, H. Sledzik, P. Schuh, J. Schroth, M. Oppermann, P. Bruckner, F. van Raay, R. Quay, and M. Seelmann-Eggebert, “Ultra-wideband GaN MMIC chip set and high power amplifier module for multi-function defense AESA applications,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 8, pp. 3043–3051, Aug. 2013. [22] Y. Niida, Y. Kamada, T. Ohki, S. Ozaki, K. Makiyama, N. Okamoto, M. Sato, S. Masuda, and W. Watanabe, “X-Ku wide-bandwidth GaN HEMT MMIC amplifier with small deviation of output power and PAE,” in Proc. IEEE Compound Semiconductor Integr. Circuit Symp., Oct. 2014, pp. 1–4. [23] G. Mouginot, Z. Ouarch, B. Lefebvre, S. Heckmann, J. Lhortolary, D. Baglieri, D. Floriot, M. Camiade, H. Blanck, M. Le Pipec, D. Mesnager, and P. Le Helleye, “Three stage 6–18 GHz high gain and high power amplifier based on GaN technology,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2010, pp. 1392–1395. [24] E. Kuwata, K. Yamanaka, H. Koyama, Y. Kamo, T. Kirikoshi, M. Nakayama, and Y. Hirano, “C-Ku band ultra broadband GaN MMIC amplifier with 20 W output power,” in Asia-Pacific Microw. Conf. Proc., Dec. 2011, pp. 1558–1561. Sangho Lee (S'13) was born in Suwon, Korea, in 1986. He received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea, in 2011, and is working toward the Ph.D. degree in electrical and computer engineering at Seoul National University. His research activities include millimeterwave/RF integrated circuits and system design for wireless communication and radar, especially high-power and broadband PA design.

Hongjong Park (S'13) was born in Incheon, Korea, in 1988. He received the B.S. degree in electrical and computer engineering from Seoul National University, Seoul, Korea, in 2012, and is working toward the Ph.D. degree in electrical and computer engineering at Seoul National University. His research interests include large-signal modeling of GaN HEMT and millimeter-wave GaN MMICs.

Kwangseok Choi (S'15) was born in Seoul, Korea, in 1986. He received the B.S. and M.S. degree in electrical engineering from Sogang University, Seoul, Korea, in 2008, and 2010 respectively, and is currently working toward the Ph.D. degree in electric and computer engineering at Seoul National University. From 2010 to 2013, he was with Gigalane, Suwon, Korea, as a Research Engineer. From 2013 to 2014, he was with the System Integrated Circuit Laboratory, LG Electronics, Seoul, Korea, as a Junior Research Engineer.

Youngwoo Kwon (S'90–M'94–SM'04) was born in Seoul, Korea, in 1965. He received the B.S. degree in electronics engineering from Seoul National University, Seoul, Korea, in 1988, and the M.S. and Ph.D. degrees in electrical engineering from The University of Michigan at Ann Arbor, Ann Arbor, MI, USA, in 1990 and 1994, respectively. From 1994 to 1996, he was with the Rockwell Science Center, as a Member of Technical Staff, where he was involved in the development of millimeterwave monolithic integrated circuits (ICs). In 1996, he joined the faculty of the School of Electrical Engineering, Seoul National University, where he is currently a Professor. He is a coinventor of the switchless stage-bypass power amplifier architecture “CoolPAM.” He cofounded Wavics, a power amplifier design company, which is now fully owned by Avago Technologies. In 1999, he was awarded a Creative Research Initiative Program by the Korean Ministry of Science and Technology to develop new technologies in the interdisciplinary area of millimeter-wave electronics, MEMS, and biotechnology. He has authored or coauthored over 150 technical papers in internationally renowned journals and conferences. He holds over 20 patents on RF MEMS and power amplifier technology. Dr. Kwon has been an Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He has also served as a Technical Program Committee member of various microwave and semiconductor conferences including the IEEE International Microwave Symposium, RF Integrated Circuit Symposium, and the International Electron Devices Meeting. Over the past years, he has directed a number of RF research projects funded by the Korean Government and U.S. companies. He was the recipient of a Presidential Young Investigator Award from the Korean Government in 2006.

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Generalized Stability Criteria for Power Amplifiers Under Mismatch Effects Almudena Suárez, Fellow, IEEE, Franco Ramírez, Member, IEEE, and Sergio Sancho, Member, IEEE

Abstract—Potential instability of power amplifiers (PAs) under mismatch effects is analyzed, with emphasis on the ease and generality of application of the stability criteria. The methodology is based on the evaluation of a large-signal version of the factor, considering mismatch effects in the fundamental frequency and three relevant sidebands: the baseband, the lower sideband and the upper sideband. This requires an outer-tier scattering-type conversion matrix of order 3 3 to be obtained, with the rest of sideband equations acting as an inner tier. It is taken into account that the circuit behaves nonlinearly with respect to the termination at the fundamental frequency. The consideration of three sidebands will enable the prediction of the two major forms of large-signal instability: incommensurable oscillations and frequency divisions by two. The analysis is preceded by an evaluation of the circuit own stability properties (proviso) under open and short circuit terminations at the sidebands, for all possible values of the termination at the fundamental frequency. Three different factors can be defined between any two ports of the scattering matrix. The analysis of the relationships between these factors and their continuity properties will allow the derivation of a single number able to characterize the PA potential instability for each fundamental-frequency termination. Results have been exhaustively validated with independent circuit-level simulations based on pole-zero identification and with measurements, using a variable output load and loading the PA with an antenna. Index Terms—Antenna mismatch, bifurcation, stability analysis.

I. INTRODUCTION

I

NSTABILITY of power amplifiers (PAs) under termination conditions other than 50 , usually due to antenna mismatch [1]–[4], can give rise to severe malfunctioning, as reported in many previous works [5]–[11]. Furthermore, some applications impose stable operation even under highly reflective loads [6], [8]. The stability analysis under output mismatch is involved since it must be carried out under unknown termination impedances. To be precise, the practical stability analysis of a periodic solution with harmonic components , where goes from to , is based on the introduction of Manuscript received July 02, 2015; revised September 17, 2015; accepted October 11, 2015. Date of publication November 18, 2015; date of current version December 02, 2015. This work has been supported in part by the Spanish Government under contract TEC2014-60283-C3-1-R and in part by the Parliament of Cantabria (12.JP02.64069). This paper is an expanded version from the IEEE MTT-S International Microwave Symposium, Phoenix, AZ, USA, May 17–22, 2015 The authors are with the Communications Engineering Department, University of Cantabria, 39005 Santander, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2015.2494578

a perturbation at the positive frequency [12]–[15], which will give rise, through mixing effects, to the sideband frequencies . The aim is to predict the reaction of the periodic solution to small perturbations, so the circuit will be linearized about this solution and its frequency response will be obtained by sweeping . Under mismatched conditions, the frequency-dependent load impedance will exhibit unknown values at and , where . However, some considerations can be made. In the PA, the harmonic amplitudes are generally significant at the device output terminals, but quite low at the final 50 termination of the output network [16]–[23]. For instance, in a Class-E amplifier [16], [17] a nearly sinusoidal fundamental-frequency current flows through the output series resonator, so the impedance at the fundamental frequency is the most influential one. As stated in [17], the load network may include a low-pass or a band-pass filter to suppress harmonics of the switching frequency at the final output 50 load [19], [20]. On the other hand, in a class-F amplifier [21]–[23] the output network forces the output voltage to be ideally sinusoidal and additional resonators are added to tune the harmonic components. The mismatch effects occur after the PA output network, at the reference plane indicated in Fig. 1(a), so, in general, they will have a negligible effect at harmonic frequencies , where . Taking all the above into account, the approach in [9]–[11] assumes a bandpass filtering action of the PA output network, such that the particular values of the load impedances at frequencies other than the fundamental frequency and its lower and upper sidebands, and , have a negligible impact on the stability properties. With this in mind, the analysis of mismatch effects is limited to the three frequencies , at which the termination impedances may take any value. Then, a two-tier conversion matrix analysis is carried out [10], [11]. The outer-tier system is based on a 2 2 scattering-type matrix at the two mismatched sideband frequencies and , defined at the PA termination plane. The inner-tier system accounts for the rest of sideband frequencies, whose termination impedance values should have a negligible impact on the stability properties. Thus, they can be arbitrarily terminated in 50 . The two sideband frequencies and act as two virtual ports, which has enabled the extension of the Rollet stability criteria [24], [25] to large-signal operation under output mismatch effects [10]. However, [10] assumed a particular (matched) termination condition at the fundamental frequency . The generalization to arbitrary terminations implies some analysis difficulties, since any change of leads to a different steady-state solution, which must be calculated with harmonic

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Fig. 1. Power amplifier under output mismatch effects. The transistor is an , Avago ATF-50189, and the element values are , , , , and . In the modified PA, the output inductor is . (a) Circuit schematic. (b) Sketch of the termination impedances (replacing the original 50 load) at the analysis frequencies, including harmonics, in a solid line, and sidebands, in a dashed line. (c) Photograph.

balance (HB). Practical use and interpretation of data resulting from multiple amplitude and phase sweeps in require a judicious technique. In this sense, [11] proposed the calculation of a large-signal factor, obtained from the outer-tier scattering-type matrix at and , and the use of constantcontours, traced on the Smith chart corresponding to (with respect to which the circuit behaves nonlinearly). A limitation of this method lies in the fact that only particular values of the perturbation frequency are considered, though the stability analysis of a periodic regime at , must take into account all the values between 0 and . In fact, an upper frequency higher than will be necessary to detect the subharmonic resonance leading to a frequency division by 2, which is one of the main forms of large-signal instability [26]. In view of this problem, one of the objectives here will be derivation of a new analysis method accounting for this whole interval of perturbation frequencies. In [11], a preliminary investigation including the baseband termination in the set of relevant mismatched terminations was

presented. This relied on the calculation of a 3 3 outer-tier scattering matrix at the three sideband frequencies , and . However, the three-sideband analysis in [11] was only used for a final validation of the results obtained with the 2 2 scattering matrix, due to the difficulties involved in the evaluation of the large-signal for three possible combinations of two sidebands, , and , considering, in each case, all possible values of the complex reflection coefficient at the remaining sideband, denoted as . Furthermore, the evaluation of each large-signal must be carried out for each fundamental-frequency termination and each perturbation frequency , which will lead to prohibitive computational cost, unless some useful mathematical properties are identified. This work will present a thorough methodology for the stability analysis of PAs under mismatch effects that is mathematically consistent for all the possible values of the perturbation frequency . It will be derived from an in-depth investigation of the relationships between the different factors that can be defined in a three-sideband analysis, and a detailed study of their frequency dependences. The aim will be to obtain a single real quantity defining the PA potential stability properties in the whole perturbation-frequency interval, for each termination at . Unlike the two-sideband case, the analysis at accurately deals with situations in which the dangerous frequency intervals in and are close to or comprise this frequency. Therefore, it should enable a prediction of frequency divisions by two, often encountered in unstable PAs [27], [28]. This work will also take into account the need to verify the fulfilment of a proviso, with identical meaning to Rollet's proviso [29], [30] in a small-signal analysis, ensuring the observability of mismatched-induced instabilities from the output reference plane. This will require verification of the circuit stability under both open and short circuit terminations at the three relevant sidebands for all the possible values of the fundamentalfrequency termination . The analysis strategy, based on polezero identification, will take advantage of the continuity of the circuit equations, in order to avoid an unmanageable amount of data of difficult interpretation. The methods will be illustrated by means of its application to a PA at with 80% efficiency at 22 dBm output power. The paper is organized as follows. Section II presents the calculation of the three-sideband scattering matrix. Section III describes the potential instability analysis at three sidebands. Section IV presents a validation based on the calculation of stability circles. Section V proposes a new global stability parameter that is exhaustively validated with measurements. II. CALCULATION OF THE THREE-SIDEBAND SCATTERING MATRIX The stability analysis of a periodic solution at relies on the introduction of a perturbation at a frequency , to obtain the frequency response of the circuit linearized about this solution [12]. This will give rise to the mixing frequencies , where goes from to [12]–[15], [31], [32]. The opposite frequencies , though not considered in the analysis, will also exist and their components will be complex-conjugates of those at . In the case of a stability anal-

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output network) are only relevant at , , and . Fig. 1(b) shows a sketch of the analysis frequencies and the termination impedances at the output reference plane, with the harmonic components in a solid line and the sideband frequencies in a dashed line. For each termination at the fundamental frequency ,a full HB analysis is carried out considering harmonic terms. Next, the circuit is linearized about the resulting steady-state solution with the conversion-matrix approach [31]–[33]. The circuit's linearized equations are decomposed into an outer-tier system at and an inner-tier system at the rest of the frequencies , where , terminated in . The outer-tier system is formulated at the circuit's output reference plane [Fig. 1(a)] by means of 3 3 impedance matrix , later transformed into a scattering matrix. The matrix is obtained through the simultaneous conversion-matrix analysis of three circuits, terminated in at , in open circuit at , and , and in at the rest of the frequency components, as shown in Fig. 2(b). Each circuit will contain an independent small-signal current source at one of the sidebands . Then, the parameters of a 3 3 impedance matrix are obtained from the three respective circuits, as

(1)

Fig. 2. Two-tier conversion-matrix analysis. For each termination at , the perturbed circuit is represented with a 3 3 scattering matrix, calculated at the PA output terminals, as shown in Fig. 1. (a) Three circuit replicas used for . (b) Sketch of the outer-tier the calculation of the 3 3 impedance matrix scattering matrix and load impedances at the three sideband frequencies with mismatch effects.

ysis under output mismatch effects, and taking into account the low-pass characteristic of the output network, the mismatched conditions at the output reference plane [Fig. 1(a)] can be restricted to the fundamental frequency and the three frequencies , and . These mismatched frequencies are respectively terminated in the arbitrary reflection coefficients at each perturbation frequency [Fig. 1(a)]. The load impedances at the rest of the harmonic frequencies and sideband frequencies , where , should have no impact on the stability properties and can be arbitrarily terminated in . Note that the analysis method takes into account all the harmonic and sideband frequencies without any restrictions. The sole assumption is that mismatch effects (at the reference plane, after the

The above calculation is performed with a full conversion matrix approach, taking into account all the sideband frequencies , where goes from to . Note that we will have a different impedance matrix for each termination at and each perturbation frequency . The system is linear with respect to the terminations at the sideband frequencies, but nonlinear with respect to the termination at . Thus, a harmonic-balance analysis must be performed for any variation of the termination at . The (3 3) impedance matrix in (1) can be transformed into a (3 3) matrix of scattering type , which will relate reflected and incident power waveforms at the three sidebands

(2) where the asterisk denotes complex conjugation. This scattering matrix will allow a generalization of Rollet's criteria [24], [25] to predict potential instability under mismatch effects. The PA will be potentially unstable under these effects if for any pair of terminations at the sidebands , at the sidebands or at the sidebands , it exhibits negative resistance when looking into the circuit output at , or , respectively. This verification must be performed for every passive termination at and every perturbation frequency . However, the

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analysis described will only be able to detect the circuit instability under fulfilment of the Rollet proviso [29], [30], [34], [35], which must be adapted here to the problem of three mismatched sideband frequencies. To fulfil the proviso, the circuit terminated in at must be stable on its own, or equivalently, it must not exhibit any poles on the right-hand side of the complex plane (RHP) when the three sidebands are in open and short circuit conditions. The proviso must be verified for each passive termination at , with the three sideband frequencies , and in all possible combinations of short-circuit and open-circuit terminations. Indeed, short-circuit terminations facilitate the detection of unstable series resonances, which might not be observable from the analysis reference plane, whereas open-circuit terminations facilitate the detection of unstable parallel resonances. The verification of the proviso can be carried out with pole-zero identification [8], [15], fully applicable under open/short circuit terminations, since the load remains fixed at real impedance values at all the sideband frequencies , given by zero, near infinite or 50 . In fact, any complex-impedance (with a non-zero imaginary part) must physically vary with , so pole-zero identification should not be applied under constant complex termination impedances at the sideband frequencies. The pole-zero identification will be carried out versus variations in the termination at . A double sweep in the amplitude and phase of provides disconnected circles, which impedes taking advantage of the continuity properties of the harmonic-balance system. Indeed, this set of nonlinear algebraic equations is continuous with respect to all of its variables and parameters [12], [14]. Thus, it will also exhibit a continuous dependence on the load reflection coefficient at the fundamental frequency . The analysis can be carried out following a single spiral curve, depending on a single parameter , which will define both the amplitude and phase of the reflection coefficient . For a smaller step, and higher values of , the Smith chart will be covered in a finer way. Additionally the unit circle can be considered for a detailed analysis of the effect of purely reactive impedances. The analysis of the Rollet proviso will be applied to the Class-E PA in Fig. 1(a), with specified output power 22.5 dBm and efficiency 80% at and . The original values of the output inductor and load resistance are and . The resistance is implemented through an L-C matching section, terminated in 50 (Fig. 1). The analysis has been carried out with harmonic terms. This number of harmonic terms will be considered through the whole manuscript, for both the circuit-level simulations and the two-tier conversion-matrix analysis, based on in. Fig. 3(a) shows the spiral considered in the Smith Chart. Fig. 3(b) evidences that the circuit does not fulfil the proviso. This figure shows the variation of the real part of the dominant poles versus the parameter when using short-circuit terminations at the sidebands. When these sidebands are short-circuited, the circuit is unstable even under a 50 termination at . Note, however, that it is stable when fully matched, that is, when operating under a 50 final-termination load at all the harmonic and sideband frequencies. With

Fig. 3. Application of the proviso to the original PA in Fig. 1, following the , (a) Spiral spiral curve curve considered, traced on the Smith chart. (b) Pole evolution versus the paunder short-circuit terminations at the relevant sidebands. rameter in

the spiral curve, advantage is taken of the continuity of the circuit equations for an undemanding analysis. To improve the robustness of the circuit under mismatch-induced instability, some modifications have been performed in the output network. The output inductor has been changed to and the new load resistance, also implemented with an L-C section, is . When repeating the analysis of the proviso through the spiral curve , the circuit is stable under open and short circuit terminations at the three relevant sidebands, as shown in Fig. 4. Therefore, the potential instability analysis described in the next section will be applicable. III. POTENTIAL INSTABILITY ANALYSIS THREE-SIDEBANDS

AT THE

The works [36] and [37] demonstrate an extension of Rollet's analysis to three-port linear networks, which is based on the sequential definition of three different two-port networks. In each case, the two ports are taken from those of the original threeport network, under a variable passive termination in the remaining port. In [11], this procedure is adapted to the three-band stability analysis under mismatch effects. Under a termination at any of the three sideband frequencies , the 3 3 matrix in Fig. 2 can be reduced to a 2 2 matrix [36], [37]. Three different 2 2 matrixes can be defined. The first matrix is , at the virtual ports at and , depending on . The second matrix is , at the

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virtual ports and , depending on trix is , at the virtual ports . Using (2), the matrix

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. The third maand , depending on is obtained as

(3) where the subindex stands for reduced matrix. An analogous calculation is carried out for the other two matrixes and . For each of the three 2 2 matrixes, six large-signal equivalents of the factor [38] can be defined. This will be done using the same expressions as in [38], but considering, in each case, the two virtual ports of the reduced matrix instead of the two physical ports 1 and 2. The factors and , calculated from , respectively provide the distance to the stability circle in the and planes at each value. Analogous factors and , calculated from , and and , calculated from , will also be considered. In each case, the and factors provide the same stability information. Thus, the conditional stability analysis can be carried out in terms of three factors: , and , globally denoted as factors. This three-sideband analysis is demanding since each factor depends on (agreeing in each case with the reflection coefficient inside the parentheses), together with and the perturbation frequency . Due to this complexity, the three-band analysis was used in [11] only for validation purposes, under two specific values. In the following, the properties of matrix (2) and the three factors will be studied in order to simplify the analysis methodology. A. Consideration of all Possible Passive Values of Let any of the three factors , and be considered, which for simplicity will be denoted , where 1, 2 and 3 may correspond to any of the three sidebands . The terminations at Port 1, Port 2 and Port 3 will be denoted as , and , respectively. A potentially unstable case will be assumed. By definition, corresponds to the distance from the centre of the Smith chart to the stability circle, which must be evaluated for all the passive values of . Let the set of values providing be denoted as . Then, for any , there will be a set of loads, denoted by , such that for any the input reflection coefficient when looking into Port 1, , fulfils . The set , which will include passive loads, is delimited by a stability circle in , expressed as . Now a reduction of the 3 3 scattering matrix to Port 1 and Port 3, depending on , will be considered, performing the analysis in terms of the factor . Because the first analysis port (Port 1) is the same as in the previous case, for any load connected to Port 2 and connected to Port 3 we will have . Thus, condition must be fulfilled for any pair of loads . Therefore, if there are values such that , there must be values such that .

Fig. 4. Application of the proviso to the modified PA with and , following the spiral curve in Fig. 3(a). (a) Under open circuit terminations at the relevant sidebands. (b) Under short-circuit terminations.

Next, the connection of a passive load to Port 1, where , and the load to Port 2 will be assumed. By Kirchoff's laws, the reflection coefficient when looking into Port 3 will necessarily fulfil . In an analogous manner, when connecting the passive loads to Port 1 and to Port 3, the reflection coefficient when looking into the Port 2 will fulfil . One can conclude that if any of the factors , and is smaller than one for some passive values of the reflection coefficient within the brackets, the other two will also be smaller than one for certain passive values of the reflection coefficient on which they depend. Therefore, to determine whether the amplifier is potentially unstable under mismatch effects it will be sufficient to evaluate exhaustively one of the factors for all the possible passive values of its corresponding . B. Relationship Between the Three Passivity Boundary

Factors at the

Let particular terminations at the two sidebands , given by , be assumed. Then, it will be possible to write (4) The above relationships can be combined with (2) to obtain the input reflection coefficient at the baseband, given by . This provides the following expression, depending on and the scattering parameters (5) where the parameters and , depending only on the scattering parameters, are given by

(6)

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Fig. 5. Formation of the limit set

in the

plane.

The amplifier will be unconditionally stable under mismatch effects if for any pair of values of and , fulfilling and . For notation simplicity, the circles delimiting the passivity boundaries in each of the two planes will be denoted and . To determine the images of the passive regions and , one can take into account that for each constant value of , (5) defines a bilinear transformation [17] in the other parameter . In an analogous way, for each constant value of , (5) defines a bilinear transformation in the other parameter . Thus, the image of the circle , given by , will be the boundary of the images of all the passive loads , so all the images are either inside or outside the transformed circle . The same applies for the bilinear transformation in terms of , depending on . Applying a similar reasoning, the images of the two circles and in the plane, obtained through (5), form a global boundary of the images of all the possible passive combinations of and . This can be rigorously demonstrated as follows. First, let us consider all the pairs fulfilling: (7) where the function

is given by (5). Solving for

one obtains (8)

Now the following set

will be defined: (9)

The above mapping can give rise to values fulfilling either or . In particular, application of the mapping to the passivity boundary , , denoted as , will provide a set of circles (see Fig. 5), depending on the phase (10) For each , the whole region is mapped either inside or outside the circle , which constitutes a frontier between points belonging or not to the set for that value. When performing this operation , the boundary of the region is constituted by points belonging to the circles. These points will agree with those in the whole set of stability

circles traced in the plane for , . The factor provides the distance [28] to the stability circle in the plane for each . Because the set of circles , constitutes the boundary of the points such that , it will be sufficient to evaluate the factor through the circle to determine the potential instability properties. In fact, a relevant function will be the one providing the minimum value of when evaluated through . The resulting value will be denoted as . From the analysis of the mapping in (8), when evaluating through a circle , where , the minimum will be larger than and smaller than the one obtained for any other magnitude . However, when using instead of the passivity boundary , one must be aware that the three factors will provide different potential stability predictions, as the analysis is not exhaustive and the evaluation of each of the three factors will leave out some regions of , and . For the potential stability analysis, the three parameters , and will be initially considered, where it has been taken into account that (in agreement with the properties discussed in subsections and ) the minima resulting from and should be identical, although obtained for a different phase in each case. The parameters and respectively agree with the minimum distance to the stability circle in the and planes under phase variations in . In order to obtain the minimum distance to the stability circle in the plane, the parameter , agreeing with , must also be considered. Note that factors provide the distance to the stability circle in Smith chart corresponding to the “source” termination. The analysis of the three parameters , and has been applied to the PA in Fig. 1 operating at 0.8 GHz. In an initial study, a load describing a spiral curve, , has been considered. The analysis procedure is as follows. At each and for each perturbation frequency , the phase of the reflection coefficient is swept from 0 to 360 , in a fine step, evaluating the factors , and at each step. Note that harmonic components are taken into account in the calculation of the 3 3 scattering matrix that enables the determination of these factors. This is the number considered for all the analyses presented in this work. For each , only the minimum values versus are kept, agreeing with the parameters , and , where the frequency dependence of these parameters is indicated explicitly. The results obtained for four particular values in the spiral curve are shown in Fig. 6. As expected, the minima of and are overlapped for all the values. The three parameters , and provide the same information on the potential instability of the PA, in agreement with the derivations in subsections and . Indeed, the three parameters are either larger or smaller than 1 in the same intervals of perturbation frequency . They cross unity at exactly the same frequency values [see the expanded view in Fig. 6(d)].

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Fig. 7. Frequency variation of the parameter for going from 0 to 1, in steps of 0.1. The termination considered at the fundamental fre, in Fig. 6(c). The number of harmonic compoquency is . The parameter , agreeing with provides nents is at each . the minimum

Fig. 6. Variation versus the perturbation frequency the minima of the three , and with , for five particular values factors . (a) of . The number of harmonic components considered is , (b) , (c) , (d) , and (e) . An expanded view is shown in Fig. 6(d) to show the simultaneous crossing through 1.

For illustration, Fig. 7 presents the values taken by the parameter versus the perturbation frequency , for going from 0 to 1, in steps of 0.1. The termination considered at the fundamental frequency is , in Fig. 6(c). As can be seen, the parameter , agreeing with provides the minimum value at each . Identical results are obtained when analyzing the other two factors and . C. Frequency Variation of the Parameters

,

and

For the stability analysis of a periodic regime at one should consider variations in the perturbation frequency

between 0 and a value larger than , to enable the detection of frequency divisions by 2. In Fig. 6, the parameters , and have been evaluated in the whole frequency range 0 to . The frequency is an offset with respect to dc, and . Therefore, the information obtained for the higher frequency values should be redundant. Indeed, when increasing , the parameter provides an “image” of the predictions by the parameter at lower frequencies. This is very clear in the analyses of Fig. 6(a) and Fig. 6(c). The frequency variation observed in Fig. 6 is in agreement with the stability properties of periodic solutions. In fact, the poles of periodic solutions, agreeing with the Floquet exponents, are not univocally related to the Floquet multipliers, which do define the stability properties of periodic solutions in a unique manner [29]–[31]. Equivalent poles, associated with the same pair of complex-conjugate Floquet multipliers, are symmetrically located about the spectral lines of the harmonic frequencies of the original periodic regime , that is, they are distributed as . Therefore, the two bands with observed in the analysis in Fig. 6 are linked and correspond to the same potential instability, at and . Note that the poles at and are symmetrically located about , and may tend to this value under variation of a circuit parameter [26], [41]. From Fig. 6, one can expect potential frequency divisions by 2 when the fundamental frequency is terminated at the values considered in Fig. 6(a) to Fig. 6(d). However, the region of subharmonic impedances leading to this division is expected to be small, since the values of and are close to 1. The potential frequency division by two predicted in Fig. 6 has been validated with an independent simulation. A small signal current source at is introduced into the circuit at the output reference plane [Fig. 8(a)]. At , the circuit is loaded, in each case, with one of the values considered in Fig. 6. Instead of using a modified conversion-matrix approach as in [33], [42], a harmonic balance analysis at is carried out. It is taken into account that there must be a phase relationship between the subharmonic current source and the input generator, due to the coherency of the two signals. The phase of the subharmonic source is set to zero (phase origin). Then, the phase of the input source is swept, using the

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Fig. 9. Comparison of the results obtained with the new potential-stability analysis, using the three factors , and , with those obtained using the single factor considered in [11].

the subharmonic load

Fig. 8. Validation of the capability to predict frequency divisions by 2. (a) In(b) Bounddependent simulations using a small-signal current source at aries of passive loads at the subharmonic frequency giving rise to frequency values considered in Fig. 6. (c) Spectrum of the subhardivisions for three monic solution obtained with an independent HB simulation for and .

small-signal current source to calculate the input admittance at the subharmonic component at each phase step. The opposite admittance values fulfill a limit condition for frequency division by 2, with subharmonic amplitude tending to zero, and provide the boundary in the Smith chart corresponding to the subharmonic load. The points of this boundary correspond to flip bifurcations [26], [41]. To obtain the frequency-division boundary in the Smith chart at , the reflection coefficient associated to must be calculated. The above method has been applied for the values considered in Fig. 6(b), (d) and (e), corresponding to , and . The division boundary, obtained with the technique in Fig. 8(a), has been traced in Fig. 8(b) for the three cases. It only intersects the Smith chart for and . The passive loads enabling the frequency division are inside the division boundary. This division region is small, in agreement with the quantitative predictions of Fig. 6. This correspondence is found, despite the fact that the two types of analysis are fundamentally different, since the subharmonic current source in Fig. 8(a) has a phase relationship with the input generator. The capability to obtain frequency divisions for passive loads within the boundaries obtained in Fig. 8(a) has also been validated with an independent HB simulation. For

and , one obtains the spectrum of Fig. 8(c). The results of the three-band analysis have been compared in Fig. 9 with that obtained when using the single factor considered in [11], for which the analysis is restricted to the lower and upper sidebands and . This two-band factor should agree with the one obtained with , when imposing a particular termination at baseband, such as , which was considered in [11] and also here. Therefore, it has more limited prediction capabilities. As an example, in Fig. 9, corresponding to the fundamental-frequency termination in Fig. 6(d), the two-sideband method predicts stable behavior, whereas the three-sideband one predicts potential instability. To extend the method to mismatch effects at higher harmonic terms one should consider all possible sets of harmonic impedance terminations, and obtain an M-port scattering matrix for each combination of harmonic impedances, where is the total number of mismatched sidebands. All the factors defined between any two ports of the M-port scattering matrix will provide the same information on the potential instability, under the condition that the reflection coefficients of the termination loads at all the other ports describe a unit circle. Such computational effort will not be worth in most cases due to the filtering action of the output network. The three-sideband stability analysis will be validated with measurements, in Section V, and with independent simulations through pole-zero identification, in the next section. Verification through simulation enables a high accuracy, under the certainty that the passive and active component models are identical. This validation will rely on the calculation of stability circles in the plane corresponding to the baseband termination . The predictions obtained with these circles will be compared with the results of an accurate pole-zero identification at circuit level [15], [28], [39]. IV. STABILITY CIRCLES IN THE THREE-SIDEBAND ANALYSIS In this section, the use of stability circles when considering three sideband frequency terminations (besides the fundamental-frequency termination ) is presented and applied for an independent validation of the new outer-tier methodology. Selecting particular sets of values with different stability properties would require a demanding implementation of a frequency dependent load , exhibiting the values at the corresponding frequencies. Instead, a simple passive network will be considered here, which under

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Fig. 10. Variation of the real part of the dominant pair of complex-conjugate , at constant input power poles versus the inductor , with .

TABLE I REFLECTION COEFFICIENT OF THE R-L LOAD AT THE BIFURCATION POINT

modification of an element value should give rise to different stability conditions of the PA. Next, the values exhibited by the load at the four frequencies will be calculated to check the consistency with the potential instability analysis. The load will consist of an inductor in series with a resistor , which fits commonly used antenna models. When connected to the amplifier output (replacing the nominal 50 load) and under variations of the inductor , at constant , it gives rise to different stability conditions, as predicted by the pole-zero identification method. Fig. 10 presents the variation of the real part of the dominant pair of complex-conjugate poles versus the inductor , at constant input power . As gathered from Fig. 10, the PA is stable for , and unstable for . The circuit exhibits a Hopf bifurcation at the inductor value . The first validation will be carried out at the Hopf bifurcation point obtained for . The poles crossing the imaginary axis have the critical frequency , as obtained from the pole-zero identification. At the baseband, lower sideband, fundamental and upper sideband frequencies given by , , and , the R-L load exhibits the reflection coefficients shown in Table I. Next, the outer-tier matrix in (2) will be used to obtain the stability circles in the Smith chart, when the perturbation frequency varies in the interval from to about the critical value . At the fundamental frequency , the load exhibits the reflection coefficient . For this particular , the stability circles will be obtained reducing the matrix to a 2 2 matrix at the sideband frequencies and , depending on the termination . This reduced matrix will be expressed as . At each the 2 2 matrix is calculated for the precise value exhibited by the series R-L load at .

Fig. 11. Stability conditions at (bifurcation point in Fig. 10). (a) Stability circles for traced in the plane for values corresponding to those exhibited by the R-L load in the frequency into about the critical value terval . For perturbation frequencies such that , the stability circle is traced in solid (dashed) line. The variation of the exhibited by the R-L load in the same frequency interval has also been repre, calculated sented. (b) Total admittance function with the three-sideband outer-tier analysis and with a full conversion-matrix approach in HB, using 7 harmonic terms.

The stability circles obtained from the matrix are traced in the plane , for perturbation frequencies from to [Fig. 11(a)]. For perturbation frequencies such that , the stability circle is traced in solid line. When , the circle is traced in a dashed line. This allows distinguishing between the potentially unstable and stable regions. The circle obtained for the critical perturbation frequency is traced in a bolder line. The variation of the reflection coefficient exhibited by the series R-L load through the interval to has also been represented. Around the critical frequency , the load values are located, as expected, in the potentially unstable region. Next, the bifurcation condition at will be validated. At this frequency, the R-L load exhibits the coefficient , shown in Table I. Setting the reflection coefficient at the baseband frequency to and using the matrix one obtains at the upper sideband frequency the input reflection coefficient . This value fully agrees with the inverse of the reflection coefficient exhibited by the R-L load at the upper sideband frequency, i.e.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 63, NO. 12, DECEMBER 2015

(see Table I). Thus, the bifurcation condition is fulfilled, in total consistency with the results of pole-zero identification. As an additional validation, the total admittance function , calculated with the three-sideband outer-tier analysis has been compared with the one obtained with a full conversion-matrix approach in commercial harmonic balance, using 7 harmonic terms. For the full conversion-matrix approach, a small-signal current source is introduced in parallel with the output R-L load to calculate the total admittance function, as the ratio between the source current and the node voltage. Results are shown in Fig. 11(b). As can be seen, the curves for both the real and imaginary parts are overlapped and display a bifurcation point at , where the real and imaginary parts of the total admittance are equal to zero. Next, inductor values in the unstable and stable ranges predicted by the pole-zero identification will be considered. From Fig. 10, for , the amplifier is unstable. The stability circles are calculated for the new reflection coefficient , exhibited by the modified load , at . They have been traced in the Smith chart [Fig. 12(a)] for the values exhibited by the new R-L load in the perturbation-frequency interval to , which includes the frequency of the dominant pair of complex-conjugate poles at . The stability circle at has been traced in a bolder line and the unstable region corresponds to the outside of this circle, where the exhibited by the R-L load is located. This indicates, as expected, potential instability for this baseband termination. Next, the input admittance at the upper sideband is calculated using and . This allows the evaluation of the impedance-type transfer function , analogous to the transfer function usually chosen for pole-zero identification [8], [15] [Fig. 12(b)]. This function has been compared with the one obtained through the full conversion matrix approach in harmonic balance (with 7 harmonic terms), obtaining a full overlap of both the amplitude and phase. The impedance exhibits a clear resonance at the frequency of the dominant poles, with a positive phase slope that indicates unstable behavior, in total agreement with the pole-zero analysis in Fig. 10. Next value is , in the stable region predicted by the pole analysis of Fig. 10. Most of the stability circles obtained under variations of about the frequency of the dominant poles at , lie outside the Smith chart [Fig. 13(a)]. The circle corresponding to this precise frequency value has been traced in bolder line and indicates absolute stability. The impedance transfer function [shown in Fig. 13(b)], obtained with the three-sideband outer tier analysis, is fully overlapped with the one resulting from a full conversion-matrix approach in HB, using 7 harmonic terms. It has a negative phase slope, in agreement with the stable behavior predicted by pole-zero identification. V. DEFINITION OF A GLOBAL STABILITY PARAMETER As has been shown, the potential stability properties exhibit a multi-parameter dependence, changing with the fundamental-

Fig. 12. Stability conditions for . (a) Stability circles for traced in the plane for values corresponding to those exhibited by the R-L load about the frequency of the dominant poles . For perturbation frequencies such that , the stability circle is traced in solid (dashed) line. The variation of the exhibited by the R-L load in that frequency interval has also been represented. calculated with the three(b) Impedance function sideband outer-tier analysis and with a full conversion-matrix app