IEEE MTT-V055-I06B (2007-06B) [55, 06B ed.]

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JUNE 2007

VOLUME 55

NUMBER 6

IETMAB

(ISSN 0018–9480)

PART II OF TWO PARTS

SPECIAL ISSUE ON 2006 EUROPEAN MICROWAVE WEEK

Manchester, U.K., was the location for European Microwave Week, 10–15 September 2006

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 $14.00, plus an annual subscription fee of $20.00 per year for electronic media only or $40.00 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 J. S. KENNEY, President L. BOGLIONI D. HARVEY S. M. EL-GHAZALY J. HAUSNER M. HARRIS K. ITOH

J. MODELSKI, President Elect L. KATEHI T. LEE B. KIM J. LIN N. KOLIAS

K. G. GARD, Secretary A. MORTAZAWI B. PERLMAN V. J. NAIR A. ROSEN

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T. S. SAAD P. STAECKER

W. SHIROMA R. SNYDER

N. KOLIAS, Treasurer K. VARIAN K. WU R. WEIGEL R. YORK

Distinguished Lecturers K. TOMIYASU L. YOUNG

G. BOECK W. HOEFER T. ITOH

B. KIM J. LASKAR V. LUBECKE

J. C. RAUTIO D. ROOT D. RYTTING

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K. VARIAN (2006) K. C. GUPTA (2005) R. J. TREW (2004)

MTT-S Chapter Chairs Albuquerque: S. BIGELOW Atlanta: D. LEATHERWOOD Austria: R. WEIGEL Baltimore: A. D. BROWN Beijing: Z. FENG Beijing, Nanjing: W. X. ZHANG Belarus: A. GUSINSKY Benelux: D. V.-JANVIER Brasilia: A. KLAUTAU, JR. Buenaventura: C. SEABURY Buffalo: E. M. BALSER Bulgaria: K. ASPARUHOVA Cedar Rapids/Central Iowa: D. JOHNSON Central New England: K. ALAVI Central & South Italy: S. MACI Central No. Carolina: T. IVANOV Chicago: Z. LUBIN Cleveland: G. PONCHAK Columbus: F. TEIXEIRA Connecticut: C. BLAIR/R. ZEITLER Croatia: Z. SIPUS Czech/Slovakia: P. HAZDRA Dallas: R. EYE Dayton: A. TERZOUOLI, JR. Denver: M. JANEZIC Eastern No. Carolina: D. PALMER Egypt: I. A. SALEM Finland: T. KARTTAAVI Florida West Coast: K. O’CONNOR

Foothill: C. ANTONIAK France: P. EUDELINE Germany: W. HEINRICH Greece: I. XANTHAKIS Hong Kong: W. Y. TAM Houston: J. T. WILLIAMS Houston, College Station: C. MADSEN Hungary: T. BERCELI Huntsville: H. G. SCHANTZ India/Calcutta: P. K. SAHA India Council: K. S. CHARI Israel: S. AUSTER Japan Council: Y. TAKAYAMA Kitchener-Waterloo: R. R. MANSOUR Lithuania: V. URBANAVICIUS Long Island/New York: J. COLOTTI Los Angeles, Coastal: A. SHARMA Los Angeles, Metro: J. WEILER Malaysia: Z. AWANG Melbourne: J. KRALOVEC Milwaukee: S. G. JOSHI Mohawk Valley: P. RATAZZI Monterrey Mexico: R. M. RODRIGUEZ-DAGNINO Montreal: K. WU New Hampshire: T. PERKINS New Jersey Coast: D. REYNOLDS New South Wales: G. TOWN New Zealand: J. MAZIERSKA North Italy: G. GHIONE

North Jersey: K. DIXIT North Queensland: A. TSAKISSIRIS Northern Nevada: B. S. RAWAT Norway: S. E. WHEATLEY Orange County: H. J. DE LOS SANTOS Oregon: T. RUTTAN Orlando: P. WAHID Ottawa: Q. YE Philadelphia: J. NACHAMKIN Phoenix: C. WEITZEL Poland: M. P. MROZOWSKI Portugal: C. A. CARDOSO FERNANDES Princeton/Central Jersey: W. CURTICE/A. KATZ Queensland: A. ROBINSON Rio de Janeiro: J. R. BERGMANN Rochester: S. M. CICCARELLLI/J. VENKATARAMAN Romania: I. SIMA Russia, Nizhny-Novgorod: Y. BELOV Russia, Saint Petersburg: M. SITNIKOVA Russia, Moscow: V. KALOSHIN Russia, Saratov-Penza: N. RYSKIN Saint Louis: D. MACKE San Diego: J. TWOMEY Santa Clara Valley/San Francisco: J. J. SOWERS Seattle: K. POULSON Seoul Council: H.-Y. LEE Siberia, Novosibirsk: V. SHUBALOV Siberia, Tomsk: O. STUKACH Singapore: O. B. LEONG

Editors-In-Chief DYLAN WILLIAMS NIST Boulder, CO 80305 USA Phone: +1 303 497 3138 Fax: +1 303 497 3970 email: [email protected] AMIR MORTAZAWI Univ. of Michigan Ann Arbor, MI 48109-2122 USA Phone: +1 734 936 2597 Fax: +1 734 647 2106 email: [email protected]

South Africa: P. W. VAN DER WALT South Australia: H. HANSEN South Brazil: L. C. KRETLY Southeastern Michigan: L. M. ANNEBERG Southern Alberta: S. BOUMAIZA Spain: L. FE HARO Springfield: P. R. SIQUEIRA Sweden: A. RYDBERG Switzerland: J. HESSELBARTH Syracuse: E. ARVAS Taipei: C.-S. LU Thailand: M. KRAIRIKSH Toronto: G. V. ELEFTHERIADES Tucson: N. BURGESS/S. MORALES Turkey: O. A. CIVI Twin Cities: M. J. GAWRONSKI UK/RI: A. REZAZADEH Ukraine, Central Kiev: Y. POPLAVKO Ukraine, East: A. A. KIRILENKO Ukraine, Rep. of Georgia: R. ZARIDZE Ukraine, Vinnitsya: V. DUBOVOY Ukraine, West: I. ISAYEV Venezuela: M. PETRIZZELLI Victoria: A. MITCHELL Virginia Mountain: D. MILLER Washington DC/Northern Virginia: J. QIU Winnipeg: V. OKHMATOVSKI Yugoslavia: B. MILOVANOVIC

Associate Editors

DANIEL DE ZUTTER ZOYA POPOVIC YOSHIO NIKAWA Universiteit Gent Kokushikan Univ. Univ. of Colorado, Boulder Belgium Japan USA email: [email protected] email: [email protected] email: [email protected] KENJI ITOH JOSÉ PEDRO SANJAY RAMAN Mitsubishi Electronics Univ. of Aveiro Virginia Polytech. Inst. and State Univ. Japan Portugal USA email: [email protected] email: jcp.mtted.av.it.pt email: [email protected] JENSHAN LIN Univ. of Florida USA email: [email protected] M. GOLIO, Editor-in-Chief, IEEE Microwave Magazine G. E. PONCHAK, Editor-in-Chief, IEEE Microwave and Wireless Component Letters

RICHARD SNYDER RS Microwave Company USA email: [email protected] RUEY-BEEI WU National Taiwan Univ. Taiwan, R.O.C. email: [email protected] ALEXANDER YAKOVLEV Univ. of Mississippi USA email: [email protected] T. LEE, Web Master

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Digital Object Identifier 10.1109/TMTT.2007.901542

JUNE 2007

VOLUME 55

NUMBER 6

IETMAB

(ISSN 0018-9480)

PART II OF TWO PARTS

SPECIAL ISSUE ON 2006 EUROPEAN MICROWAVE WEEK

Guest Editorial .... ......... ........ ......... ......... ........ ......... ......... ........ ....... C. M. Snowden and R. D. Pollard

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PAPERS

Intermodulation Distortion of Third-Order Nonlinear Systems With Memory Under Multisine Excitations .... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ..... J. P. Martins, N. Borges Carvalho, and J. C. Pedro -Band Miniaturized Quasi-Planar High- Resonators ........ ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ..... K. J. Vanhille, D. L. Fontaine, C. Nichols, Z. Popovic´, and D. S. Filipovic´ Demonstration of Negative Refraction in a Cutoff Parallel-Plate Waveguide Loaded With 2-D Square Lattice of Dielectric Resonators ..... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... T. Ueda, A. Lai, and T. Itoh UWB Array-Based Sensor for Near-Field Imaging ...... ......... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... A. G. Yarovoy, T. G. Savelyev, P. J. Aubry, P. E. Lys, and L. P. Ligthart Composite Right/Left-Handed Metamaterial Transmission Lines Based on Complementary Split-Rings Resonators and Their Applications to Very Wideband and Compact Filter Design ..... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ M. Gil, J. Bonache, J. García-García, J. Martel, and F. Martín Analysis of Electromagnetic Response of 3-D Dielectric Fractals of Menger Sponge Type ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... . E. Semouchkina, Y. Miyamoto, S. Kirihara, G. Semouchkin, and M. Lanagan Sub-Microsecond RF MEMS Switched Capacitors ...... ......... ......... ........ ......... ......... ........ ......... ......... .. .. .. B. Lacroix, A. Pothier, A. Crunteanu, C. Cibert, F. Dumas-Bouchiat, C. Champeaux, A. Catherinot, and P. Blondy Alternating-Direction Implicit Formulation of the Finite-Element Time-Domain Method ....... ........ ......... ......... .. .. ........ ......... ......... ........ ......... .... M. Movahhedi, A. Abdipour, A. Nentchev, M. Dehghan, and S. Selberherr Analysis of Wideband Dielectric Resonator Antenna Arrays for Waveguide-Based Spatial Power Combining . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ........ Y. Zhang, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson

1264 1272 1280 1288 1296 1305 1314 1322 1332

(Contents Continued on Page 1262)

(Contents Continued from Page 1261) Concurrent Dual-Band Class-E Power Amplifier Using Composite Right/Left-Handed Transmission Lines ... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... S. H. Ji, C. S. Cho, J. W. Lee, and J. Kim Electrically Controllable Artificial Transmission Line Transformer for Matching Purposes ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ....... C. Damm, J. Freese, M. Schüßler, and R. Jakoby An 11-Mb/s 2.1-mW Synchronous Superregenerative Receiver at 2.4 GHz ..... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ....... F. X. Moncunill-Geniz, P. Palà-Schönwälder, C. Dehollain, N. Joehl, and M. Declercq A Wideband CMOS Variable Gain Amplifier With an Exponential Gain Control ..... H. D. Lee, K. A. Lee, and S. Hong High Precision Radar Distance Measurements in Overmoded Circular Waveguides ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ..... N. Pohl, M. Gerding, B. Will, T. Musch, J. Hausner, and B. Schiek On the Robustness of Digital Predistortion Function Synthesis and Average Power Tracking for Highly Nonlinear Power Amplifiers ...... ......... ........ ......... ......... ........ ......... ........ O. Hammi, S. Boumaiza, and F. M. Ghannouchi Study and Design Optimization of Multiharmonic Transmission-Line Load Networks for Class-E and Class-F -Band MMIC Power Amplifiers ...... ......... ......... ........ ......... ......... . R. Negra, F. M. Ghannouchi, and W. Bächtold Integrated Receiver Based on a High-Order Subharmonic Self-Oscillating Mixer ... . S. A. Winkler, K. Wu, and A. Stelzer Global Modeling Analysis of HEMTs by the Spectral Balance Technique ...... ......... ....... G. Leuzzi and V. Stornelli Information for Authors .. ........ ......... ......... ........ ......... .......... ........ ......... ......... ........ ......... ......... .

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Guest Editorial

T

HE 36th European Microwave Conference (EuMC) was held at the International Conference Centre, Manchester, U.K., 10–15 September 2006, as part of the 2006 European Microwave Week (EuMW2006), which also comprised the European Radar Conference (EuRAD), the European Microwave Integrated Circuits Conference (EuMIC), and the European Conference on Wireless Technology (ECWT). The 2006 event, organized on behalf of the European Microwave Association, saw a record number of paper submissions with 1171 manuscripts, from which 794 papers were accepted for the 123 oral and poster sessions. Over 3500 delegates attended EuMW and 274 companies exhibited in the GMEX Exhibition Hall. The 16 workshops and short courses proved very popular with nearly 600 attendees, a highlight being the trip to the Jodrell Bank Radio Telescope. Places were in high demand for the short course on “Fundamentals of Microwave Power Amplifier Design” and the workshops on “Microwave Sensors and Imaging Systems” and “RF MEMS and RF Microsystem Applications and Development in Europe.” The social side of the week was, as ever, a great success; the conference dinners were all sold out and the partner programs proved very popular. European Microwave Week has become a truly international event with papers submitted from 52 countries. While the U.K. as host nation submitted the largest number (13%), and Europe as a whole accounted for a total of 54%, Asia–Pacific submitted 26% and North America submitted 10% of the papers. The EuMC attracted 737 submissions, EuRAD 118, EuMIC 192, and ECWT 124 papers. The most popular topic area, with 178 submissions, embraced filters, multiplexers, and passive components, while antennas, phased arrays, and wave propagation also attracted very large numbers of submissions. Power amplifiers, linearizers, and linearization techniques were hot topics attracting 74 papers. Active circuits, RF modules, and wireless systems continued to prove to be a major area of activity. Microwave and millimeter-wave monolithic integrated circuits (ICs) for industrial applications, interconnects, packaging, and multichip modules attracted 56 submissions. Research in radar antenna systems transmit/receive technology, and radar technology and systems proved equally popular. During the conference week, all the sessions were well attended with the most popular sessions in the area of metamaterials, with two rooms filled to capacity, and with multiplexers and couplers continuing to be of great interest. Other very popular topics were RF microelectromechanical systems (MEMS), GaAs and GaN power amplifier technologies, and radar applications. Plenary presentations by Prof. Tatsuo Itoh on metamaterial technologies and applications and Prof. Sir Martin Sweeting on low-cost satellite technologies stimulated a great deal of interest with the large auditorium well filled.

The prizes for the best papers during the week went to T. Nagatsuma, H. Ito, and K. Iwatsuki for their work on “Generation of Low-Phase Noise and Frequency-Tunable Millimeter/Terahertz-Waves Using Optical Heterodyning Techniques with Uni-Traveling Carrier Photodiodes” (EuMC Microwave Prize), with two EuMC Young Engineer Prizes going to C. M. Siegel and C. Lee. The ECWT Conference Prize went to John F. Gerrits for his paper on “A Wideband FM Demodulator Circuit for a Low-Complexity FM-UWB Receiver” and the Young Engineer Prize went to Andre Kruth. The EuMIC Conference Prize was awarded to M. V. Heijningen for a paper on “De-Band GaN High Power sign and Analysis of a 34 dBm Amplifier MMIC,” and the EuMIC Young Engineer Prize going to G. Posada Quijano. The EuRAD Conference Prize went to P. Ries, F. D. Lapierre, and J. G. Verly for their paper on “RANSAC-Based Flight Parameter Estimation for Registration-Based Range-Dependence Compensation in Airborne Bistatic STAP Radar with Conformal Antenna Arrays” and the EuRAD Young Engineer Prize was awarded to A. Meta. The overwhelming view of delegates was that EuMW2006 had attracted technical papers of a very high standard. This TRANSACTIONS’ Special Issue on 2006 European Microwave Week has drawn on a selection of papers that were submitted in expanded form and will give the reader a flavor of the breadth and quality of the technical sessions held during EuMW2006. The organizers and Technical Program Committees (TPCs) are very grateful to Richard Ranson and George Heiter for their huge contribution and to Jeff Pond for the many hours of support offered to help us with the electronic Technical Paper Management System. We are also very grateful to the members of the TPCs for their time and dedication in producing a very high-quality program, which led to a very successful conference week. Finally, we are in debt to the reviewers for this TRANSACTIONS’ Special Issue who help maintain the high quality of this publication. We pass on our very best wishes to the organizers of the 2007 EuMW event to be held in Munich, Germany, which we are sure delegates will find to be an outstanding event.

Digital Object Identifier 10.1109/TMTT.2007.896798

0018-9480/$25.00 © 2007 IEEE

CHRISTOPHER M. SNOWDEN, Guest Editor Chair, 2006 European Microwave Week Vice-Chancellor’s Department University of Surrey Guildford, GU2 7XH U.K. ROGER D. POLLARD, Guest Editor Co-Chair, 2006 European Microwave Week School of Electronic and Electrical Engineering The University of Leeds Leeds, LS2 9JT U.K.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

Intermodulation Distortion of Third-Order Nonlinear Systems With Memory Under Multisine Excitations João Paulo Martins, Student Member, IEEE, Nuno Borges Carvalho, Senior Member, IEEE, and José Carlos Pedro, Fellow, IEEE

Abstract—This paper presents a methodology to compute the distortion output of a class of third-order nonlinear dynamic systems from only standard two-tone test results. Closed-form expressions are presented to compute the distortion output and metrics as adjacent channel power ratio and co-channel power ratio/noise tones. The impact power ratio for an arbitrary multisine with of memory effects in a multisine excitation is also addressed, improving the design of RF components by a careful understanding of memory effects mechanism in real modulated signals. An experimental validation is presented to prove the proposed theory. Index Terms—Measurements, memory effects, nonlinear systems, waveform analysis.

I. INTRODUCTION

M

EMORY effects have a strong impact on the design of RF system components for the new wireless scenarios. This type of phenomenon can deeply impact any form of linearization mechanism, or even obviate its implementation in wideband systems, due to the difficulty in controlling the correspondent wide baseband characteristics. For these reasons, it is imperative that a deep study is undertaken on the memory effect’s mechanisms noticed in the telecommunication systems when they are driven by real modulated excitations. Memory effects can be divided into short and long term, with “short” and “long” referring to the time constants involved in the impulse response tail of the nonlinear dynamic system. The long-term memory time constants impact the signal’s envelope, while the short time constants affect the RF carrier. Since in a communication system the information is carried by the envelope, the understanding of the long-term memory effect mechanisms is a fundamental topic for predicting the system’s performance degradation. Most of the research on memory effects of nonlinear dynamic systems was based on swept frequency two-tone tests [1]–[7], characterizing these dynamic responses through the variation of amplitude and phase of the observed intermodulation distortion (IMD) products.

Manuscript received September 29, 2006; revised March 14, 2007. This work was supported in part by the European Union under the Network of Excellence TARGET Contract IS-1-507893-NoE and under Project ColteMepai POSC/EEA-ESE/55739/2004. The work of J. P. Martins was supported by the Portuguese Science Foundation, Fundação para a Ciência e Tecnologia under Ph.D. Grant 22056/2006. The authors are with the Instituto de Telecomunicações, Universidade de Aveiro, 3810-193 Aveiro, Portugal (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.896794

However, little was done to extrapolate this knowledge to more complex excitation forms such as multisine signals [8]. In fact, the relation between two-tone and multisine signal tests presented in [9] was restricted to a memoryless third-order nonlinearity. It was then extended in [10] for systems of fifth order, but still static. In [11], we proved that, for a simplified model of an RF system describing the third-order behavior including memory effects, it is possible to relate two-tone IMD measurements with nonlinear distortion evaluation under a multisine excitation. This was then demonstrated by simple simulations with a five-tone multisine signal. Nevertheless, the presented results only stated the idea that it is possible to infer a multisine characterization of a class of nonlinear third-order dynamic systems from a set of two-tone tests, no quantification was given for these multisine responses. Moreover, the results presented in [11] addressed, exclusively, spectral regrowth distortion, while co-channel distortion was left uncovered. In this paper, we expand the results presented in [9], deriving closed-form expressions for the calculation of spectral regrowth and co-channel distortion of a class of third-order nonlinear dynamic systems under multisine signal excitation from a set of two-tone test observations. Using this formulation, it is possible to study and explain the memory effect mechanisms of these types of systems, driving the RF design engineer to an optimized design of baseband filtering. Following this introduction, this paper presents a brief overview of what was presented in [11] in order to put its subject in context. In Section III, mathematical support for the relation between two-tone and multisine IMD components is given both for spectral regrowth and co-channel distortion. In Section IV, we present some results based on the developed formulas, and identify the relation between the amplifier baseband filtering characteristics and its nonlinear memory effects. Finally, in Section V, we demonstrate the proposed theory through experimental results. This paper concludes in Section VI by summarizing the main achievements of this study. II. RELATIONSHIP BETWEEN TWO-TONE IMD AND MULTISINE NONLINEAR DISTORTION IN A THIRD-ORDER DYNAMIC NONLINEARITY In [11], we discuss the relationship between the two-tone excitation IMD and multisine nonlinear distortion in a third-order nonlinearity presenting memory. Here, we summarize the most

0018-9480/$25.00 © 2007 IEEE

MARTINS et al.: IMD OF THIRD-ORDER NONLINEAR SYSTEMS WITH MEMORY UNDER MULTISINE EXCITATIONS

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Fig. 1. Third-order dynamic nonlinearity model. Fig. 2. Five-tone multisine and correspondent IMD distortion. TABLE I FIVE-TONE THIRD-ORDER MIXING PRODUCTS

Thus, for instance, we can see that the spectral regrowth tone in the output signal depends on identified as (2) and

(3) important results from [11] to contextualize the following sections. In order to model this phenomenon, we assumed that the nonlinear in-band response of a third-order nonlinear system presenting memory to a narrowband signal can be decomposed as the sum of a cubic polynomial direct path response with an up-converted baseband component [13]. With such a model in mind, the baseband component is demodulated from the RF signal in a second-order nonlinearity and then pressed with memory in a low-pass filter that mimics the baseband response of the nonlinear system (Fig. 1) [2], [12], [13]. According to this model, the in-band IMD transfer function for a two-tone signal is given by [2]

If we manage to characterize each of those terms individually, we could get all the long-term memory effects that we need for a multisine excitation. In order to clearly identify each of those components, a twotone test is performed and the result is computed according to (1). Since the most important terms are the ones that vary with tone spacing, we start by first identifying the constant part of the expression. That is done from the asymptotic behavior of at very low-frequency separations, i.e., in the limit tends to zero Hz. Thus, the two-tone output diswhen tortion becomes (4) where

is1

(1) where is the third-order nonlinear transfer function arising directly from the third-order static conversion, and are the second-order and nonlinear transfer functions responsible for the baseband and second harmonic signal components that will then be remixed to fall onto the system’s first zone output. If we now consider an uncorrelated multisine excitation, i.e., when the tones do not share the same phase reference, the output distortion from a nonlinear dynamic system will be the vector addition of the several components whose amplitude and phase depend on the tone spacing. Table I presents these components obtained for a five-tone signal (see also Fig. 2). If the multisine excitation could be considered narrowband, i.e., if the system’s bandwidth is greater than the signal’s bandwould be approximately constant width, then and equal to . The mixing product arising from , is at the second harmonic, could also be considwhere ered constant ( ) since the relative bandwidth change with the tone spacing is very small.

(5) is the remaining term that varies with tone and spacing representing the memory contribution. This way, by changing the tone spacing, the different components can be obtained. If the tone spacing is made sufficiently small, can also be extracted by continuity in zero separation frequency if a smooth frequency response is assumed. However, in the multisine case are accounted since the terms in vector additions (2) and (3), we must have them characterized in both amplitude and phase. Thus, for each frequency component, we need to solve the following equation: (6) The computation of the system nonlinear transfer functions is achieved by using higher order statistics (HOS), considering 1This formula is presented in a compact way contrary to what is in (5) [11]. represents the constant part of the In the current form, the complex value second term of (5) [11]

K

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Fig. 3. Co-channel and adjacent-channel spectral components.2

a two tone as the input test signal [14]. This way, we can obtain the transfer function both in amplitude and phase. These equations should be calculated for each tone spacing, different tone spacing’s involved, and thus at least having a linear system of equations should be solved, corresponding tone separations plus the constant . to This system of equations is built by measuring the HOS for each two-tone signal at each different tone spacing. In a test with an arbitrary number of tones, the system to be solved can be represented in matrix form as

(7)

where is the third-order statistic for an tone is the second-order transfer function for an spacing, tone spacing, and is the number of tone spacings considered. In Section III, we show how to use this matrix for the calculation of an overall nonlinear distortion response to an -tone multisine excitation.

will land on a specific position of the spectral regrowth. Nevertheless, this study is somehow simplified considering an equally . spaced multisine since, in that case, any The main objective will thus be to obtain, in an automatic way, every mixing product for an arbitrary number of tones for a specific spectral regrowth mixing position. This tool will allow the computation of the multisine nonlinear distortion very efficiently by exclusively calculating the number of baseband components to be analyzed. In order to understand how to apply the proposed formulation, consider an equally spaced -tone input signal (Fig. 3) described by , . The response of a third-order nonlinear system to this input excitation is obtained combining all three tones of the input signal with , , and representing the input spectral positions. The output will be given by (8) where (9) as was previously seen and as in [11]. for Despite the large number of combinations that fall in a multisine excitation, a precise account of those combinations is needed before proceeding with the overall power calculation of the mixing product. In [7], the number of different combinations that fall on each position was calculated for a memoryless nonlinearity. The different arrangements were then considered for the ones falling within the input signal components and, in that case, there were some contributions that were correlated with the input, and some others uncorrelated. Beyond these, there were some others that fall at the adjacent channel, where two types of mixing products and were also considered: the ones presenting equal . the ones where However, in that case, due to memoryless nonlinearity, no is affected by a importance was given to the fact that each different . This coefficient is dependent on a binomial function that depends on two baseband values ( and ), responsible for the nonlinear system’s memory can be effects. In fact, we are considering that expressed as

III. IMD COMPUTATION- GENERALIZATION TO AN ARBITRARY NUMBER OF TONES In Sections I and II, we discussed the inherent relation between multisine and two-tone responses in a nonlinear thirdorder system presenting memory. Now we will discuss how to quantify this relation deriving analytical results amenable to predict the multisine characterization outcome. In order to do so, we need to account for every nonlinear mixing product that falls on each output spectral regrowth com. ponent (Fig. 3) This task is somehow complex since it will demand an extensive account of every combination of three frequencies that

Nb

k

2The , , and symbols denote the number of tones of the input multisine, half the number of tones, and the index of a specific component, respectively.

(10) where

Thus, within this model, accounting for the system’s nonlinear dynamic effects means to calculate each different and values that falls on each position for an arbitrary number of tones . In order to obtain closed formulas to calculate these different arrangements, we will divide the nonlinear mixing products in three different classes, i.e.: 1) the ones falling onto the adjacent-

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corresponds to the multinomial coefficient [6] that where and 3/4 if , and is the specvalues 3/8 if tral regrowth tone accordingly to Fig. 3. The amplitude of each multisine component is represented by and is a constant. B. Co-Channel Signal-Correlated Mixing Products As was explained earlier, the co-channel mixing products are divided into signal-correlated and uncorrelated components. The correlated ones are the products that obey the constraint when two involved frequencies are equal. at all times, and spans from 1 to . Thus, Now the output power at each co-channel mixing product becomes (12)

Fig. 4. Different matrices of spectral regrowth mixing products that fall on the frequency position ! for a five-tone excitation.

channel; 2) the co-channel products that are correlated to the signal; and 3) the co-channel uncorrelated products. The measurement procedure starts with a set of swept tone spacings, and frequency two-tone tests covering recording the amplitude and phase values of each IMD component [14], [15]. Using (7), the function of the tone spacing is then computed. With these functions, it is then possible to calculate each distortion component that falls at a specific frequency position according to (9). Finally, the various components are added together, according to the characteristics of each mixing product. The power of the correlated distortion products must be accounted for as the average power of the addition of phasor quantities, while the power of the uncorrelated ones can be computed by the direct addition of the powers of the individual components. A. Adjacent-Channel Mixing Products For a giving adjacent spectral position , we must account for all different values of and . In that respect, and in order to simplify the calculation, we have developed a matrix scheme, (in the ilshown in Fig. 4, where for each spectral regrowth, lustrated case for a five-tone multisine excitation), the binomial combinations were calculated— and pairs. The combinations marked with an “ ” correspond to different and values, while the ones marked with an “o” correspond to equal and . This separation is important since the amplitude of mixing products with equal and must be multiplied by the multinomial coefficient 3/8, while the ones with different and must be scaled by 3/4. By expanding these matrices for an increasing number of tones, from 2 to , and then adding up the corresponding mixing products, we were able to derive a formula that automatically generates the output power of the component at and for each pair of and values. The obtained formula is

(11)

where is the frequency position of the desired mixing product according to Fig. 3. In this case, each contribution is accounted for in a vector addition since the distortion products are all correlated to each other. C. Co-Channel Signal-Uncorrelated Mixing Products Since we are assuming that there is no phase correlation between any of the input tones, the generation mechanism of signal-uncorrelated co-channel mixing products is similar to the one responsible for the ones falling at the adjacent channel. Indeed, as any input frequency combithat is different from one of nation has a phase is different from both any other combination (as long as and ), the mixing products that fall at a certain will all be uncorrelated. Therefore, the final result of the power at each position is, on average, equivalent to the addition of each individual component’ power. This implies that, while in (12) the was given as the square of the voltage sum, output power it must now be given as the sum of the squares of voltage

if

if

vertice

if

vertice

if

(13)

where vertice

(14)

is a function that rounds the elements of to the and nearest integers towards minus infinity. Thus, if the overall output nonlinear distortion is to be calculated from a two-tone measurement, the procedure is the one presented below. First, a two-tone measurement is made and the is calculated for all values of corresponding spanning from 0 to . This will then be stored into a

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Fig. 5. Tone spacing contributions for the adjacent-channel power.

Fig. 6. Tone spacing contributions for the signal-uncorrelated co-channel power.

matrix. The next step involves the calculation of for all binomials at each position, and, using and , obfrom the previous taining the value of the function matrices. The solution of the nonlinear distortion to a multisine excitation then becomes straightforward. IV. IMPACT OF MEMORY IN THE IMD OF MULTISINE SIGNALS Here, we aim to analyze the impact of memory effects on the output of a nonlinear system presenting memory when driven by a random multisine signal. For performing this task, the formulation presented above will be very important since by accounting for the values of and , we can infer information of what are the most important characteristics of the baseband that impact the overall nonlinear distortion. Beginning with the adjacent-channel scenario, we calculate the number of times that each binomial function appears for each . This result is visible in Fig. 5. An interesting observation from this analysis is that the main contribution to the adjacent-channel power is at the middle of and the figure, which means that the binomial , is the most important term to be considered in order to minimize the adjacent channel power ratio (ACPR) figure-of-merit [6]. The same analysis was done for the co-channel distortion components, as is presented in Fig. 6. In this case, the most important components, i.e., having stronger impact in the co-channel distortion, are the baseband components of lower frequency. Considering now the correlated co-channel case (Fig. 7), we can conclude that the most important components are again the baseband components of lower frequency. Another interesting aspect is that all the correlated distortion components have at least one dependence with zero tone spacing, corresponding to the dc value of the baseband filter. In order to validate these hypotheses a computer-aided design (CAD)/computer-aided engineering (CAE) simulation was run by using different baseband filters. The operating bandwidth of 1.7 MHz was split into three bands with equal bandwidth, as shown in Fig. 8.

Fig. 7. Tone spacing contributions for the signal-correlated co-channel power.

Fig. 8. Baseband filters for memory effect generation.

The three filters considered are: 1) a low-pass filter for the baseband components; 2) a filter reinforcing the middle of the baseband; and finally 3) a filter accounting for the remaining part of the baseband.

MARTINS et al.: IMD OF THIRD-ORDER NONLINEAR SYSTEMS WITH MEMORY UNDER MULTISINE EXCITATIONS

Fig. 9. System output and predicted output for a baseband low-pass filter.

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Fig. 11. System output and predicted output for the higher band bandpass filter.

Fig. 10. System output and predicted output for the middle passband filter. Fig. 12. System output and predicted output for a memoryless system.

The input signal used in the simulation is an uncorrelated multisine of 100 tones. To simulate a system with memory, a direct implementation of the model of Fig. 1 was done. Fig. 9 presents the obtained results for the multisine output and the output obtained by our formulation. The ACPR and co-channel power ratio (CCPR) [6], were computed and the corresponding values are 41.2 and 24.7 dB. A good agreement is visible at the spectral regrowth. Fig. 10 presents the obtained results for a passband baseband filter and with all the other configurations remaining unchanged. The obtained ACPR and CCPR results are 38.2 and 32.6 dB, respectively. Finally, a last test was run with the higher frequency passband filter. In this case, the ACPR and CCPR results are 40.9 and 34.9 dB (Fig. 11). Based on previous observations, we can conclude that the low-pass and higher band bandpass filters impose a similar ACPR. On the contrary, the passband filter has a higher contribution to the adjacent distortion degrading the ACPR values. This validates our previous hypothesis based on the formulation developed. The same analysis was done for the co-channel distortion. In this case, the most important components are the ones with lower tone spacing, as can be seen in the lower CCPR values for the low-pass filter, and a decrease for the other two subsequent filters.

Fig. 12 is a reference test. It presents the value of the output spectrum for a memoryless nonlinearity, which, compared to the results presented in [8], guarantees that the formulation now obtained is consistent with previous results. In order to deeply test our technique, we have also simulated a system where the third-order direct path is near zero, and the second harmonic filter presents a flat, but complex, response. In this case, some spectral regrowth asymmetry is clearly visdB and ible from Fig. 13, and a different dB is obtained. In all the previous results, we have showed superimposed both the multisine output arising for an intensive CAD/CAE simulation and the results arising from our proposed two-tone approach; a remarkable match is visible. In summary, our analysis points to several conclusions, which are: 1) the middle frequency components of the baseband are determinant to the behavior of the adjacent-channel distortion and, thus, to the ACPR and 2) for the in-channel distortion, the most important baseband contributions are the ones arising from components located around dc. V. EXPERIMENTAL RESULTS An experimental test was also run in order to validate the theory presented above. The setup comprises a low-power

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Fig. 13. System output and predicted output for a system presenting notorious asymmetry.

Fig. 15. Measured output and predicted response for the amplifier with memory.

Fig. 16. Measured output and predicted response for the memoryless amplifier. Fig. 14. Input impedance of the drain biasing network.

amplifier based on the ATF 55143 pseudomorphic HEMT (pHEMT) device biased in classes A and B. A drain bias network was designed to present a varying baseband response in order to press memory effects in the output response. A memoryless network was also designed to prove the validity of the presented algorithm in the standard memoryless situation. The test was run at a central frequency of 900 MHz with a tone spacing of 20 kHz. The input signal is composed by 20 tones leading to a 400-kHz bandwidth signal obtained from an arbitrary waveform generator (AWG). A record of 1000 waveform segments with random phase was used as a random multisine signal to predict the output of the system to a narrowband Gaussian noise input. The measurement procedure comprised the synchronous acquisition of both the input and output signals by a high-speed sampler and the post-processing of the data in order to get system output [14]. of the biasing network that In Fig. 14, we present the was applied to the drain of the device. As can be seen, the impedance of the memoryless bias is almost constant and close to a short circuit. The bias presenting memory has varying impedance with a resonance in the middle of the band. A first test was done in the class A operation with the bias network that presents a varying baseband characteristic.

Fig. 17. Measured output and predicted response for the amplifier presenting memory and asymmetry.

Fig. 15 presents the computed output distortion obtained by our method and the measured results. A good agreement can be noticed between the computed values and measured output. The increasing error observed in the low-power tones was attributed to the fifth-order distortion that is always present in a real system, but was not accounted for in our third-order analysis. Fig. 16 presents the obtained results for the memoryless case. A good agreement can be noticed, even in this ideal case, indicating the applicability of the method.

MARTINS et al.: IMD OF THIRD-ORDER NONLINEAR SYSTEMS WITH MEMORY UNDER MULTISINE EXCITATIONS

To test the practical ability to predict asymmetry (strong evidence of long-term memory effects), the amplifier was biased in class B. The output is presented in Fig. 17. As can be seen, the predicted response and measured output are in perfect agreement. The memory effects can also be noticed in the shape of the correlated and uncorrelated distortion components falling in the co-channel band. VI. CONCLUSIONS This paper has presented an analytical methodology to compute the output distortion of a certain class of dynamic thirdorder systems to a multisine excitation using only two-tone tests. A qualitative explanation of the impact of the baseband filter in the co-channel and adjacent channel has also given, permitting a better bias network design for power amplifiers. The presented methodology has been tested against simulated and laboratory experiments, demonstrating its validity even in the case of adjacent-channel asymmetric responses. The computed results enable the computation of multitone figures-of-merit as ACPR and CCPR from only a small set of two-tone measurements, which are standard in every RF laboratory. This study is a step forward toward the understanding of the memory generation mechanisms and in the extrapolation of the usual standard RF test results to the prediction of the dynamic system’s output to a multisine signal excitation. REFERENCES [1] W. Bosch and G. Gatti, “Measurement and simulation of memory effects in predistortion linearizers,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1885–1890, Dec. 1989. [2] K. J. Vuolevi and T. Rahkonen, Distortion in RF Power Amplifiers. Norwood, MA: Artech House, 2003. [3] K. Remley, D. Williams, D. Schreurs, and J. Wood, “Simplifying and interpreting two-tone measurements,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2576–2584, Nov. 2004. [4] H. Ku and J. Kenney, “Behavioral modeling of nonlinear RF power amplifiers considering memory effects,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2495–2504, Dec. 2003. [5] N. B. de Carvalho and J. C. Pedro, “A comprehensive explanation of distortion sideband asymmetries,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2090–2101, Sep. 2002. [6] J. Pedro and N. Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits. Norwood, MA: Artech House, 2003. [7] A. Soury, E. Ngoya, J. M. Nebus, and T. Reveyrand, “Measurement based modeling of power amplifiers for reliable design of modern communication systems,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 795–798. [8] D. Schreurs, M. Myslinski, and K. A. Remley, “RF behavioural modelling from multisine measurements: Influence of excitation type,” in 33th Eur. Microw. Conf., Munich, Germany, Sep. 2003, pp. 1011–1014. [9] J. Pedro and N. Carvalho, “On the use of multitone techniques for assessing RF components’ intermodulation distortion,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2393–2402, Dec. 1999. [10] N. Boulejfen, A. Harguem, and F. M. Ghannouchi, “New closed-form expression for the prediction of multitone intermodulation distortion in fifth-order nonlinear RF circuits/systems,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 121–132, Jan. 2004. [11] J. P. Martins, N. B. Carvalho, and J. C. Pedro, “Multi-sine response of third order nonlinear systems with memory based on two-tone measurements,” in 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 263–268. [12] J. C. Pedro, N. B. Carvalho, and P. M. Lavrador, “Modeling nonlinear behavior of bandpass memoryless and dynamic systems,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 2133–2136. [13] A. Walker, M. Steer, K. Gard, and K. Gharaibeh, “Multi-slice behavioral model of RF systems and devices,” in Radio Wireless Conf., Atlanta, GA, Sep. 2004, pp. 71–74.

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[14] J. C. Pedro and J. P. Martins, “Amplitude and phase characterization of nonlinear mixing products,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 3237–3245, Aug. 2006. [15] J. P. Martins, P. M. Cabral, N. B. Carvalho, and J. C. Pedro, “A metric for the quantification of memory effects in power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4432–4439, Dec. 2006.

João Paulo Martins (S’06) was born in Sever do Vouga, Portugal, on May 13, 1973. He received the B.Sc. and M.Sc. degrees from the Universidade de Aveiro, Aveiro, Portugal, in 2001 and 2004, respectively, and is currently working toward the Ph.D. degree at in memory effects in nonlinear systems at the Universidade de Aveiro. From 2001 to 2005, he was a Researcher with the Instituto de Telecomunicações, Universidade de Aveiro. Since 2006, he has been with Chipidea Microelectrónica, Lisbon, Portugal. His main research interests are wireless systems and nonlinear microwave circuit design.

Nuno Borges Carvalho (S’92–M’00–SM’05), was born in Luanda, Angola, in 1972. He received the Diploma and Doctoral degrees in electronics and telecommunications engineering from the Universidade de Aveiro, Aveiro, Portugal, in 1995 and 2000, respectively. From 1997 to 2000, he was an Assistant Lecturer with the Universidade de Aveiro, in 2000 a Professor, and is currently an Associate Professor. He is also a Senior Research Scientist with the Instituto de Telecomunicações, Universidade de Aveiro. He was a Scientist Researcher with the Instituto de Telecomunicações, during which time he was engaged in different projects on nonlinear CAD and circuits and RF system integration. He coauthored Intermodulation in Microwave and Wireless Circuits (Artech House, 2003). He has been a reviewer for several magazines. His main research interests include CAD for nonlinear circuits, design of highly linear RF-microwave power amplifiers (PAs), and measurement of nonlinear circuits/systems. Dr. Borges Carvalho is a member of the Portuguese Engineering Association. He is a reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and an IEEE MTT-11 Technical Committee member. He was the recipient of the 1995 Universidade de Aveiro and the Portuguese Engineering Association Prize for the best 1995 student at the Universidade de Aveiro, the 1998 Student Paper Competition (third place) presented at the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), and the 2000 Institution of Electrical Engineers (IEE), U.K., Measurement Prize.

José Carlos Pedro (S’90–M’95–SM’99–F’07) was born in Espinho, Portugal, in 1962. He received the Diploma and Doctoral degrees in electronics and telecommunications engineering from the Universidade de Aveiro, Aveiro, Portugal, in 1985 and 1993, respectively. From 1985 to 1993, he was an Assistant Lecturer with the Universidade de Aveiro, and a Professor since 1993. He is currently a Senior Research Scientist with the Instituto de Telecomunicações, Universidade de Aveiro, as well as a Full Professor. He coauthored Intermodulation Distortion in Microwave and Wireless Circuits (Artech House, 2003) and has authored or coauthored several papers appearing in international journals and symposia. His main scientific interests include active device modeling and the analysis and design of various nonlinear microwave and opto-electronics circuits, in particular, the design of highly linear multicarrier PAs and mixers. Dr. Pedro is an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and is a reviewer for the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). He was the recipient of the 1993 Marconi Young Scientist Award and the 2000 Institution of Electrical Engineers (IEE) Measurement Prize.

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Ka-Band Miniaturized Quasi-Planar High-Q Resonators

Kenneth J. Vanhille, Student Member, IEEE, Daniel L. Fontaine, Christopher Nichols, Member, IEEE, Zoya Popovic´, Fellow, IEEE, and Dejan S. Filipovic´, Member, IEEE

Abstract—Air-filled copper -band resonators with heights between 250–700 m are demonstrated with measured unloaded factors between 440–829. The low-profile quasi-planar resonators are fabricated using a sequential metal deposition process. Miniaturization of up to 70% with respect to a TE101 quasi-planar resonator is accomplished by reducing the footprint of the resonator while maintaining the height. Finite-element simulations predict the measured resonant frequency within less than 1%. Due to fabrication imperfections, un is measured to be approximately 15% less than predicted. The resonators are compatible with air-filled rectangular microcoaxial feed lines fabricated in the same metal-deposition process. Index Terms—Cavity resonator, coaxial transmission line, photolithography, quality ( ) factor.

I. INTRODUCTION

T

HE miniaturization of microwave resonators, including reentrant cavity designs, has been of interest since the development of microwave engineering [1]. Work with filters using evanescent waveguide for miniaturization and suppression of spurious modes with the ability to do wideband filters is presented in [2]. More recently, interest in microfabricated microwave and millimeter-wave resonators using various techniques has been seen. The loading of -band surface micromachined cavity resonators with barium titanate and alumina is examined in [3]; ’s in the hundreds are demonstrated with fabrication tolerances being the most important limiting factor on performance. Silicon micromachining of Ku-band resonators with drain holes used to release the sacrificial material are fabricated as outlined in [4]; the electrical performance is decreased by a small amount of leftover silicon in the resonator after the processing. Silicon micromachined and metal-plated layered polymer resonators with miniaturization factors up to -bands are presented in [5]. More recently, 70% at - and tuning from 5.4 to 10.9 GHz of an evanescent-mode resonator was demonstrated [6]. The resonators presented in [5] and [6] use rib-shaped interdigital capacitor loading (among other Manuscript received September 29, 2006; revised February 3, 2007. K. J. Vanhille, Z. Popovic´, and D. S. Filipovic´ are with the Department of Electrical and Computer Engineering, University of Colorado at Boulder, Boulder, CO 80309-0425 USA (e-mail: [email protected]; [email protected]; [email protected]). D. L. Fontaine is with the Advanced Systems and Technology Division, BAE Systems, Nashua, NH 03060 USA (e-mail: [email protected]). C. Nichols is with Rohm and Haas Electronic Materials LLC, Blacksburg, VA 24060 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.2007.895232

techniques), and are an excellent starting point for the designs - and -band rectangular coaxial transpresented here. mission line filters with nickel walls using the electrochemical fabrication (EFAB) process have been shown in [7] and [8], respectively. In this study, the resonators are not characterized alone; however, the largest theoretical ’s that can be obtained from nickel coaxial resonators with the same dimensions as in [7] and [8] are less than 80. The same technology with gold plating has recently demonstrated factors near 250 at 60 GHz [9]. Laser etching of copper has been used to fabricate rect-band filters [10]. 2.5-D resonators using angular coaxial U-shaped half-wavelength resonators in a cavity operating band have been fabricated using a low-temperature in the co-fired ceramic (LTCC) implementation [11]. This method allows compact filter design by including multiple resonators in the same cavity. Presented here are loaded-cavity resonators fabricated using a planarizing sequential-layering process that enables simultaneous fabrication of integrated rectangular coaxial transmission lines. Microfabrication of air-filled rectangular coaxial transmission lines and other components allow high isolation, dense circuit integration, multilayer topologies, low-loss performance, and dominant TEM propagation into the 400-GHz range [12]. Rectangular coaxial transmission lines, as well as other air-loaded surface-micromachined passive structures, such as high-quality factor inductors, have been demonstrated in the millimeter-wave frequency range [13]. Transmission lines, multiport couplers, and bandpass filters have been fabricated using nickel air-filled coax lines [14]. Quasi-planar microfabricated copper branch-line couplers and transmission-line resonators have been recently presented in -band. [15] with improved electrical properties up to Results for several quasi-planar resonators fabricated using the process presented here are given in [16]. Initial results for a miniaturized quasi-planar resonator are introduced in [17]. This paper gives a comprehensive analysis, design, and measurement study for miniaturized resonators at different frequencies and with different loadings. Sensitivity studies of the release holes, new resonator topologies, and behavioral circuit models are presented. Fig. 1(a) shows an example of a miniaturized cavity resonator with an arbitrary loading mechanism. Two types of predominately capacitive loading are presented in this paper. Fig. 1(b) shows what we refer to as the “puck-loading” used at 26 GHz and Fig. 1(c) shows the 36-GHz “rib-loading” structure, described in detail in Section II. It is worth clarifying the meaning of “quasi-planar” in the for a context of this study. Fig. 2 shows simulated values of

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-BAND MINIATURIZED QUASI-PLANAR HIGH-

RESONATORS

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Fig. 3. Basic fabrication steps for miniaturized quasi-planar resonators. The first eight steps (S1–S8) are identical for the five- and ten-layer structures.

Fig. 1. (a) Sketch of a loaded quasi-planar resonator showing the loading, relevant dimensions, and resonator features. (b) Sketch of the metal puck loading in the 26-GHz resonator. The four support posts are also shown. (c) Half-view sketch of the ribs in the 36-GHz loaded cavity resonator. The ribs are cut along the xy -plane to simplify their presentation. The relevant dimensions of the resonators are given in Table I.

allows resonators and microcoaxial feeds to be fabricated in the same process, while simultaneously providing factors higher than those of coaxial transmission line resonators of the same height. This paper is organized as follows. • Section II details the design of the resonators. Brief descriptions of the fabrication process, and the resonator feeding and probing structures are also given. • Section III provides a portion of the analysis necessary to understand the various design parameter effects on the performance of the resonators. • Section IV describes a method for deriving a behavioral circuit model of the loaded cavity resonators. • Section V shows the measurement results for the designed resonators. • Section VI discusses a few final points and indicates potential directions for future research. II. FABRICATION-DRIVEN ANALYSIS AND DESIGN

Fig. 2. Percentage of Q of a resonator with respect to the maximum Q of metallic resonator as a function of normalized cavity height h =w for a TE several metals. The measured Q values for 26- and 36-GHz fabricated cavity resonators are given.

square cavity with a footprint on the side, and with different metal conductivity values. The height of the cavity ( ) is varied from a small value to . The measured unloaded factors for two unloaded quasi-planar cavity resonators are indicated in Fig. 2 for comparison. The normalized heights of the and , justifying the terminology two cavities are “quasi planar.” “Low profile” describes these resonators with regard to their height equally well. The small electrical height

The PolyStrata process developed by Rohm and Haas Electronic Materials LCC, Blacksburg, VA, consists of sequentially depositing layers of metal and photoresist with high aspect ratios (2 : 1) [18]. The resonators are fabricated on a low-resistivity silicon substrate using a sequence of standard photolithographic steps, as depicted in Fig. 3. Each copper layer is chemically polished before the next layer is deposited. The final step involves removing the photoresist, leaving microfabricated airfilled copper structures. Devices with up to ten layers are demonstrated in this paper. The fabrication and resulting mechanical parameters dictate the electrical design. The major fabrication-related factors and their influence on component design are as follows. 1) Total resonator height is limited by the number of layers and thickness of each layer. In our case, with ten layers, a maximal cavity height of 700 m is possible, which will factor as presented in Fig. 2. The electrical limit the height of the tallest resonator presented in this paper is one twelfth of a free-space (TEM) wavelength.

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Fig. 5. Normalized Q for different sized release holes on a cavity resonator. The surface area of the release holes is kept constant with respect to the top surface area of the cavity. This is shown for the 250-m-tall cavities at 26 GHz and 700-m-tall cavities at 36 GHz. Fig. 4. (a) Puck-loaded cavity resonator taken from [17]. The ports are indi200 m. (b) Two cavity cated by P1 and P2. The release hole size is 200 resonators fabricated for operation at 36 GHz. The resonator at left (R1) is full sized, and the resonator at right (R2) uses miniaturization. The release hole size is 400 400 m. Probing of the structure is done on the ends of the resonator at ports indicated by P1 and P2. The loading region in each resonator is indicated by the dashed lines.

2

2

2) Photoresist release holes on the top wall and on the sidewalls limit the factor. However, a relatively large number of holes are required to completely evacuate the lossy dielectric from the cavity. The size and number of release holes were carefully examined within fabrication parameters and are described in Section II-A. 3) The design of the loading to reduce the size of the resonator is limited to what can be fabricated with this technology. The main limitation was the size of the loading structure, which needed to be mechanically fixed to the top wall while providing enough loading. We could not use previously published designs, as given the mechanical constraints, they would not provide a sufficiently large loading reactance. This constraint resulted in a 3-D capacitance, which is unique to this paper and differs from the fins presented in [5]. 4) The aspect ratio of a thin cavity is such that the top wall can sag or bulge. The mechanical support posts, which increase the footprint of the 26-GHz loaded cavity, are eliminated in the 36-GHz design because of the miniaturization with the 3-D loading. 5) The inductive coupling between the input -coaxial cable and cavity is constrained by the height of the center conductor and the required impedance match; the analysis is detailed in Section II-C. Results from two sets of wafers are given in order to show the fabrication constraints on the resonator factor, i.e., 1) a wafer with coaxial components fabricated in a five-layer process with design frequencies near 26 GHz and 2) a wafer with 36-GHz components made using a ten-layer process. Resonators from the two wafers are shown in Fig. 4.

A. Design of Release Holes In order to remove lossy photoresist after the top metal layer is deposited, release holes in the top layer and on the sidewalls are necessary. Electrically, these holes perturb the current flow and, therefore, affect the resonant frequency, as well as the factor. While no holes or small holes will give the highest factor, many larger holes are needed for high fabrication yield, thus an optimization study is required. The release holes on the sides of these resonators do not have a large effect on the electrical performance because of the cavity heights used so only the effects of the top holes will be examined. Fig. 5 shows the results of a 3-D finite-element method (FEM) study, using the Ansoft High Frequency Structure Simulator (HFSS) version 10, of a cavity resonator versus the number of rows of holes of ). The on the cavity resonator top surface (total holes ratio of the surface area of the release holes to the total top-wall surface area of the resonator is kept constant. For what is fabricated, this ratio is 8.74% for the five-layer 26-GHz designs (250- m cavity height) and 21.80% for the ten-layer 36-GHz designs (700- m cavity height). These numbers are dictated 200 m square release holes at 26 GHz by the use of 200 and 400 400 m square release holes at 36 GHz . Fig. 5 shows curves obtained numerically for both ratios at both design frequencies. The resonant frequency of the resonators changes slightly as the number of holes increases. To for a fixed frequency, it is necescompare these changes in sary to normalize the values. We begin with the formula for due to the conductor losses of a rectangular metallic cavity resonator, as can be found in [19] (1) where is the wavenumber, is the wave impedance of the is the surface resistance of the cavity-filling medium, and cavity walls. The length, width, and height of the cavity are ,

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TABLE I DESIGN VALUES FOR THE PARAMETERS AS NOTED IN FIG. 1

, and , respectively. We assume that and are equal and that the cavity is air filled as follows: (2) To compare two resonators with the same size, but different configurations of release holes, it is necessary to normalize the calculated unloaded quality factors based on the frequency of resonance. If the cavity dimensions and materials do not change, we find the ratio of quality factors to be related to a power of the ratio of the resonant frequencies (3) The data shown in Fig. 5 are simulated for two frequencies and two top plate surface area to release hole surface area ratios using the normalization in (3). The cavity is 250- m tall for the 26-GHz resonator and 700- m tall for the 36-GHz resonator. We see that the resonator performance is minimally affected by the release hole size when the cavity is 250- m tall; however, the unloaded of the cavity is large enough in the 700- m-tall case in that an effect on the is seen for large hole sizes. B. Mechanical Support The values of the dimensions labeled in Fig. 1 are given in Table I. The puck-loaded cavity contains four mechanical support posts that connect the top and bottom cavity walls. The added mechanical strength comes with a penalty in size, as the posts behave as inductances in parallel, decreasing the overall inductance of the cavity and increasing the necessary cavity footprint for a given frequency [17]. The metal rib loaded cavity contains two ribs connected to the bottom cavity wall and a single rib hangs from the top cavity wall. A cross configuration is used with the ribs to increase the strength of the connection of the ribs to the top and bottom cavity walls. The ribs have the effect of increasing the capacitance of the cavity, operating much like an interdigitated capacitor. The rib widths that are not explicitly labeled in Fig. 1(c) are 100 m in size. C. Coupling The resonators are measured in a two-port topology with probing structures in the regions marked P1 and P2 in Fig. 4. 150- m-pitch ground–signal–ground (GSG) probes are placed in contact with a short section of vertical rectangular coax, as shown in the figure, where the relatively complex geometry

Fig. 6. (a) Top-view photograph of the probing area for the 36-GHz resonator. (b) Cross-sectional view of the port area (courtesy of the Mayo Foundation). (c) 3-D model of the resonator feed.

serves for impedance matching by reactively tuning the discontinuity parasitics. The center conductor of the rectangular coax in the probing region bends downwards to connect to the bottom wall of the cavity, resulting in inductive coupling, as shown in Fig. 1(a). A top view of one of the probe structures of the 36-GHz resonator is shown in Fig. 6(a). A wafer is diced down the center line of the rib-loaded resonators to reveal the cross-sectional view shown in Fig. 6(b), giving the sidewall profiles. An indication of the detailed analysis involved in producing high-quality transitions to the on-wafer environment is shown in Fig. 6(c), which gives the computer-aided design (CAD) model of the resonator probe port. III. RESONATOR MINIATURIZATION Comprehensive numerical studies are conducted to examine the effects of several of the design parameters that affect the resonator performance. The more efficient eigenmode analysis is applied to examine parameter effects and trends, while the driven analysis provides frequency-dependent -parameters. The loading effect of the capacitive puck and the more complex 3-D metallic ribs is addressed here. Fig. 7 shows the results of a numerical study of the size of the capacitive loading puck of the 26-GHz resonator for puck heights of 75 m and 175 m, corresponding to the two possible values using the five-layer process. The position and size of the support posts and the overall dimensions of the resonator are kept constant. The fabrication process allows three possible discrete heights (0, 75, and 175 m), as there are three layers that make up the actual cavity (two more layers exist, one for the top wall and one for the bottom wall, making a total of five layers). The tallest height is chosen to maximize the factor, whose theoretical value is less than 2% lower than the highest ideal value for a copper resonator without release holes.

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Fig. 7. Results of a parametric analysis of a simplified model of the 26-GHz resonator. The release holes are not included for this analysis. The FEM eigenmode analysis examines how different diameters for the loading puck affect the Q and f of the resonator. Two puck heights (75 and 175 m) corresponding to the two possible heights using the five-layer fabrication process are examined. Values corresponding to the fabricated resonator are indicated on the graph.

Fig. 9. Equivalent circuit is generated for the resonator with feed combination for the miniaturized resonator using a loading puck and support posts. (a) Circuit topology used to create the model. (b) Measured input impedance of the resonator/feed combination compared to the input impedance of the circuit model derived from the measured data.

IV. CIRCUIT-MODEL DEVELOPMENT

Fig. 8. Results of a parametric analysis of the 36-GHz resonator. The analysis examines different spacings of the loading ribs S , as given in the inset. Eigenmode analysis using the FEM is employed, and values corresponding to the fabricated resonator are indicated on the graph.

Fig. 8 presents the results of a parametric study examining the effect of the rib spacing on and of the resonator. The values corresponding to the dimensions used for the fabrication are highlighted on the graph. As the rib separation is increased, the capacitance between the ribs diminishes; however, the capacitive coupling from the ribs to the opposite wall of the cavity remains strong. The scaling of the two -axes is the same relative to the first point of each data set, emphasizing the frequency changes more rapidly than over the range of separation values considered. Returning to Fig. 4, the puck-loaded resonator, shown at the top of the figure, has a footprint that is 15% smaller than a normal cavity resonator and 50% smaller than a resonator using the four support posts [17]. A photograph of an unloaded ten-layer 36-GHz resonator is shown on the left in Fig. 4(b). A miniaturized rib-loaded resonator operating at the same frequency, but with a 71% reduction in surface area, is shown on the right-hand side of this figure.

Though full-wave FEM simulations are accurate and flexible for detailed studies, they are computationally expensive. For example, on a dual-processor Intel Xeon 3.4 machine, it takes four minutes per frequency point with a 70-k tetrahedron mesh. Therefore, behavioral circuit models are developed for the two loaded resonators and compared with the measured results to validate this approach. A circuit model, as shown in Fig. 9(a), is developed for the two-port resonator. The steps for extracting the model are similar to those outlined in [20]. Once the capacitance of the gap between the puck and top wall is approximated using quasistatic analysis, the other circuit parameters can be determined. The other parameters of interest are the series resistance of the , the turns ratio , which is approximately the coufeed pling coefficient, the total inductance of the support posts , the characteristic impedance of the lines connecting to the resonator , and the nominal capacitance, resistance, and inductance of the cavity , , and , respectively. Starting values for these parameters are found using full-wave analysis, and then the parameters are fit to correspond to the measured results. Good agreement can be seen between the measured response and the derived equivalent circuit [see Fig. 9(b)]. The calculated , , values for the circuit parameters are turns, pF, pF, pH, pH, and k . A similar exercise using the simulated -parameter data from the full-wave response yields a circuit model whose input impedance is indistinguishable from that calculated in the simulation. Fig. 10(a) shows the model and Fig. 10(b) shows the input impedance predicted by the model, as compared with the sim-

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TABLE II SUMMARY OF THE SIMULATION AND MEASUREMENT RESULTS IS GIVEN FOR THE TWO RESONATORS

Fig. 10. Equivalent circuit is generated for the resonator with feed combination for the miniaturized resonator using three ribs for miniaturization. (a) Circuit topology used to create the model. (b) Simulated input impedance of the resonator/feed combination and the input impedance of the circuit model derived from the simulated data.

ulated input impedance for the rib-loaded resonator. A comparison with the simulated results demonstrates that the circuit model predicts the measured response and agrees with the full-wave simulations. The circuit model has one inductor less than the model in Fig. 9(a), as there are no support posts in this , is caldesign. The loading capacitance from the ribs, i.e., culated using Ansoft’s Maxwell 3D, as the approximation of the capacitance is more involved than what was done for the loading puck. This method calculates the electrostatic capacitance, but this is a good approximation since the dimensions of at the resonant frequency. the loading ribs are less than is 0.068 pF. The nominal capacitance of the resonator is pF, the nominal inductance is nH, k , the turns ratio is the nominal resistance , and the series resistance of the feed is . An examination of the derived circuit models of the two miniaturized resonators presented in Figs. 9(a) and 10(a) reveals that quite similar behavioral models describe the two resonators with seemingly different loading topologies. This approach can be generalized to other possible quasi-planar loading topologies. V. MEASUREMENT RESULTS The resonators are measured using 150- m-pitch probes from Cascade Microtech on a Cascade Summit 9000 probe station connected to an HP-8510C network analyzer. An external short-open-load-thru (SOLT) calibration on an alumina substrate is performed. A comparison of the measured and simulated results for the two miniaturized resonators is given for the in Table II. The simulated and measured values of puck-loaded resonator are 508 and 442, respectively. Compared

Fig. 11. S -parameter transmission data for the 26-GHz resonator taken from [17]. A comparison of measured, circuit model, and simulated results is shown. The simulated resonator is modeled using HFSS. The offset in frequency between the measured and simulated results is less than 0.7%.

Fig. 12. S -parameter transmission data for the 36-GHz resonator. A comparison of measured, circuit model, and simulated results is shown. The simulated resonator is modeled using HFSS. The offset in frequency between the measured and simulated results is less than 0.4%.

to a cavity resonator with four support posts, but no capacitive loading, the 50% reduction in footprint results in virtually no for a cavity resreduction in quality factor. The simulated onator using this technology is 541; therefore, a 15% footprint reduction with four supporting posts is realized by sacrificing . The values of for the rib-loaded resonator 6% of are 995 and 829 for the simulated and measured results. This of 1308 for a full-sized cavity compares to a simulated resonator with the same cavity height. A 71% reduction in . footprint is thus achievable for a 25% reduction in The measurement results of the puck-loaded 26-GHz resfrom the full-wave simulation onator are shown in Fig. 11.

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and the equivalent-circuit model of Fig. 9(a) are also plotted for comparison. The resonant frequency of the measured resonator differs by less than 0.7% from the predicted value. The for the rib-loaded 36-GHz resonator measurement results of are given in Fig. 12. Again, a comparison is made between the full-wave simulation and the derived equivalent-circuit model; the deviation of the measured from predicted resonant frequency is less than 0.4%. VI. CONCLUSION Results from two miniaturized resonators have been presented and their performance and size has been compared to that of full-sized cavity resonators operating at the same frequencies. The puck-loaded resonator achieves a 15% miniaturization in the footprint of the resonator, while including mechanical supporting posts (this gives a 50% reduction in footprint compared to a cavity with just the support posts). The rib-loaded resonator has a 71% miniaturization in the cavity footprint compared to a stanresonator. The electrical effects of release holes and dard their size and placement are studied. The electrical phenomena resulting in the miniaturization of the resonators are explored and the quality factors and cavity footprints are compared to standard cavity resonators. Behavioral equivalent circuits are derived and their usefulness validated by measurements. From here, one could develop filters, using the circuit models as the building blocks for rapid synthesis. Such resonators can also be integrated with rectangular coaxial lines and active elements for circuits that require high- factors such as low phase-noise oscillators and frequency diplexers. ACKNOWLEDGMENT The authors would like to thank G. Potvin, BAE systems, Nashua, NH, D. Sherrer and the Rohm and Haas Microfabrication Team, Blacksburg, VA, W. Wilkins and her measurement team, Mayo Clinic, Rochester, MN, Dr. J. Evans, Defense Advanced Research Projects Agency (DARPA), Arlington, VA, and E. Adler, Army Research Laboratory (ARL), Adelphi, MD, for their support. The authors would also like to thank M. Lukic´ , and Dr. S. Rondineau, both with the University of Colorado at Boulder, for helpful technical input. REFERENCES [1] G. L. Ragan, Ed., Microwave Transmission Circuits, ser. MIT Radiat. Lab. New York: McGraw-Hill, 1948, vol. 9. [2] R. V. Snyder, “New application of evanescent mode waveguide to filter design,” IEEE Trans. Microw. Theory Tech., vol. MTT-25, no. 12, pp. 1013–1021, Dec. 1977. [3] C. A. Tavernier, R. M. Henderson, and J. Papapolymerou, “A reducedsize silicon micromachined high-Q resonator at 5.7 GHz,” IEEE Trans. Microw. Theory Tech, vol. 50, no. 10, pp. 2305–2314, Oct. 2002. [4] K. Strohm, F. Schmuckle, O. Yaglioglu, J.-F. Luy, and W. Heinrich, “3D silicon micromachined RF resonators,” in IEEE MTT-S Int. Microw. Symp. Dig, Philadelphia, PA, Jun. 2003, pp. 1801–1804. [5] A. Margomenos, B. Liu, S. Hajela, L. Katehi, and W. Chappell, “Precision fabrication techniques and analysis on high-Q evanescent-mode resonators and filters of different geometries,” IEEE Trans. Microw. Theory Tech, vol. 52, no. 11, pp. 2557–2566, Nov. 2004. [6] S. Hajela, X. Gong, and W. J. Chappell, “Widely tunable high-Q evanescent-mode resonators using flexible polymer substrates,” in IEEE MTT-S Int. Microw. Symp. Dig, Long Beach, CA, Jun. 2005, pp. 2139–2142.

[7] R. Chen, E. Brown, and C. Bang, “A compact low-loss Ka-band filter using 3-D micromachined integrated coax,” in Proc. IEEE Int. MEMS Conf., Maastricht, The Netherlands, Jan. 2004, pp. 801–804. [8] J. Reid and R. Webster, “A compact integrated V -band bandpass filter,” in Proc. IEEE AP-S Int. Symp., Monterey, CA, Jul. 2004, pp. 990–993. [9] E. D. Marsh, J. Reid, and V. S. Vasilyev, “Gold-plated micromachined millimeter-wave resonators based on rectangular coaxial transmission lines,” IEEE Trans. Microw. Theory Tech, vol. 55, no. 1, pp. 78–84, Jan. 2007. [10] I. Llamas-Garro, M. Lancaster, and P. Hall, “Air-filled square coaxial transmission line and its use in microwave filters,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 152, pp. 155–159, Jun. 2005. [11] L. Rigaudeau, P. Ferrand, D. Baillargeat, S. Bila, S. Verdeyme, M. Lahti, and T. Jaakola, “LTCC 3-D resonators applied to the design of very compact filters for Q-band applications,” IEEE Trans. Microw. Theory Tech, vol. 54, no. 6, pp. 2620–2627, Jun. 2006. [12] M. Lukic´ , S. Rondineau, Z. Popovic´ , and D. Filipovic´ , “Modeling of realistic rectangular  -coaxial lines,” IEEE Trans. Microw. Theory Tech, vol. 54, no. 5, pp. 2068–2076, May 2006. [13] I. Jeong, S.-H. Shin, J.-H. Go, J.-S. Lee, and C.-M. Nam, “High-performance air-gap transmission lines and inductors for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech, vol. 50, no. 12, pp. 2850–2855, Dec. 2002. [14] J. Reid, E. D. Marsh, and R. T. Webster, “Micromachined rectangular coaxial transmission lines,” IEEE Trans. Microw. Theory Tech, vol. 54, no. 8, pp. 3433–3442, Aug. 2006. [15] D. S. Filipovic´ , Z. Popovic´ , K. Vanhille, M. Lukic´ , S. Rondineau, M. Buck, G. Potvin, D. Fontaine, C. Nichols, D. Sherrer, S. Zhou, W. Houck, D. Fleming, E. Daniel, W. Wilkins, V. Sokolov, and J. Evans, “Modeling, design, fabrication, and performance of rectangular -coaxial lines and components,” in IEEE MTT-S Int. Microw. Symp. Dig, San Francisco, CA, Jun. 2006, pp. 1393–1396. [16] K. J. Vanhille, D. L. Fontaine, C. Nichols, D. S. Filipovic´ , and Z. Popovic´ , “Quasi-planar high-Q millimeter-wave resonators,” IEEE Trans. Microw. Theory Tech, vol. 54, no. 6, pp. 2439–2446, Jun. 2006. [17] ——, “A capacitively-loaded quasi-planar Ka-band resonator,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 495–497. [18] D. Sherrer and J. Fisher, “Coaxial waveguide microstructures and the method of formation thereof,” U.S. Patent 7 012 489, Mar. 14, 2006. [19] D. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998, pp. 300–350. [20] F. Gardiol, Introduction to Microwaves. Norwood, MA: Artech House, 1983, pp. 136–142.

Kenneth J. Vanhille (S’00) received the B.S. degree in electrical engineering from Utah State University, Logan, in 2002, the M.S.E.E. degree from the University of Colorado at Boulder, in 2005, and is currently working toward the Ph.D. degree at the University of Colorado at Boulder. From 2000 to 2003, he was with the Space Dynamics Laboratory, Logan, UT, where he designed space science instrumentation for sounding rocket campaigns. In 2002, he was a member of the National Aeronautics and Space Administration (NASA) Academy, Goddard Space Flight Center. His current interests include millimeter-wave components and antenna design.

Daniel L. Fontaine was born in Holyoke, MA, on February 17, 1966. He received the B.S. and M.S. degrees in electrical engineering from the University of Massachusetts at Amherst, in 1988 and 1991, respectively. From 1988 to 1996, he was a Senior Design Engineer with the Raytheon Company, Tewksbury, MA. Since 1996, he has been a Principal Design Engineer with the Advanced Systems and Technology Division, BAE Systems, Nashua, NH. His professional design experience and interests include microwave and millimeter-wave patch antennas and arrays, quasi-optical feed networks, transmit/receive (T/R) modules, and frequency-selective surfaces.

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Christopher Nichols (M’03) received the B.S. degree in physics from Arkansas State University, State University, in 1990, and the M.S. degree in physics and Ph.D. degree in applied science from The College of William and Mary, Williamsburg, VA, in 1992 and 1996, respectively. His doctoral dissertation involved the engineering of a novel hyperthermal neutral stream etch process tool for charge-free wafer stripping. Prior to graduation, he was with IBM, Yorktown Heights, NY, where he was involved with ionized physical vapor deposition. He is currently a Senior Engineer and Microfabrication Program Manager with Rohm and Haas Electronic Materials LLC, Blacksburg, VA.

Zoya Popovic´ (S’86–M’90–SM’99–F’02) received the Dipl.Ing. degree from the University of Belgrade, Belgrade, Serbia, in 1985, and the Ph.D. degree from the California Institute of Technology, in 1990. She is currently the Hudson Moore Jr. Chaired Professor of Electrical and Computer Engineering with the University of Colorado at Boulder. Her research interests include high-efficiency and low-noise microwave circuits, quasi-optical millimeter-wave techniques, smart and multibeam antenna arrays, intelligent RF front ends, RF optics, and wireless powering for batteryless sensors.

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Dr. Popovic´ was the recipient of the 1993 and 2006 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Microwave Prize for best journal papers. She was also the recipient of the 1996 URSI Issac Koga Gold Medal, a 2000 Humboldt Research Award for Senior U.S. Scientists, and a 1993 National Science Foundation (NSF) Presidential Faculty Fellow Award.

Dejan S. Filipovic´ (S’97–M’02) received the Dipl. Eng. degree in electrical engineering from the University of Nis, Nis, Serbia, in 1994, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1999 and 2002, respectively. From 1994 to 1997, he was a Research Assistant with the University of Nis. From 1997 to 2002, he was a Graduate Student with the University of Michigan at Ann Arbor. He is currently an Assistant Professor with the University of Colorado at Boulder. His research interests are in the development of millimeter-wave components and systems, multiphysics modeling, antenna theory and design, as well as in computational and applied electromagnetics.

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Demonstration of Negative Refraction in a Cutoff Parallel-Plate Waveguide Loaded With 2-D Square Lattice of Dielectric Resonators Tetsuya Ueda, Member, IEEE, Anthony Lai, Student Member, IEEE, and Tatsuo Itoh, Fellow, IEEE

Abstract—A 2-D negative-refractive index metamaterial is proposed and investigated, which is composed of a parallel-plate waveguide loaded with a square lattice of disc-type dielectric resonators. Collective and macroscopic behavior of the dielectric resonator lattice under the fundamental TE resonance gives negative effective permeability, whereas the parallel-plate waveguide below the cutoff for TE modes provides negative effective permittivity. Thus, the double-negative condition for the propagated waves with the appropriate polarizations can establish the left-handedness. Equivalent-circuit models are shown to give a good insight into the physical mechanism of the guided waves. Numerical simulation of the 2-D dispersion diagram verifies the existence of the left-handed (LH) guided modes along with the isotropic characteristics. A triangular prism of the proposed LH structure that is sandwiched by the right-handed parallel-plate waveguides is designed and fabricated to directly observe the negative refractive index of the LH waveguide. The numerical and experimental results validate the negative refraction for the proposed structure. Index Terms—Dielectric resonators, lenses, parallel-plate waveguides, periodic structures.

I. INTRODUCTION EGATIVE-REFRACTIVE-INDEX metamaterials or left-handed (LH) metamaterials [1], [2] have been intensively investigated for applications to planar super lenses, near-field imaging, antenna, and so forth that covers microwave, millimeter-wave, terahertz, as well as optical regions. The LH metamaterials have negative effective permittivity and permeability simultaneously, resulting in the negative refractive index and support of backward wave propagation. Most conventional LH metamaterials are classified mainly into two types of structures: one is the combination of high- polarization-selective magnetic and electric resonators, such as a

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Manuscript received October 17, 2006; revised March 14, 2007. This work was supported in part by the Lockheed Martin Corporation and in part by the U.S. Department of Defense under Grant N00014-04-10762, monitored by the U.S. Office of Naval Research. T. Ueda was with the Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 90095-1594 USA. He is now with the Department of Electronics, Kyoto Institute of Technology, Kyoto 606-8585, Japan (e-mail: [email protected]). A. Lai and T. Itoh are with the Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 90095-1594 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.2007.897753

combination of split-ring resonators and thin wires [2], and the other is a transmission line loaded with series capacitances and shunt inductances [3]–[6]. It is noted that they are both made of metals. However, metals are no longer a good conductor and can cause some serious insertion loss when they are used in resonant sections at higher frequencies such as the terahertz region and beyond. Therefore, it is desirable to propose and design new configurations of LH materials in these frequency regions. To date, LH metamaterials have been proposed without metals by using dielectric resonators. One is the two dielectric-resonator scheme, which is a combination of TE and TM resonances of dielectric resonators, the operation of which corresponds to split-ring resonators and thin wires, respectively [7]–[9]. However, the operation requires high quality ( ) factors for the dielectric resonators, i.e., a high ratio of dielectric constants between dielectric resonators and host medium, in order to avoid perturbation of the resonances due to their coupling. From a fabrication tolerance point of view, it is difficult to adjust both the TE and TM resonant frequencies of dielectric resonators because of their narrow operational bands. The other proposal is a one dielectric-resonator scheme [10]. It is constructed based on the mutual coupling between dielectric resonators. Negative-refractive-index metamaterials based on photonic crystals may be in the scheme [11], although both the basic concepts are quite different in that the operational bands in photonic crystals are considered as derivatives from the Bragg scattering due to periodicity of the lattice. To maintain the coupling between dielectric resonators in the one dielectricresonator scheme, the contrast of the dielectric constants is set smaller than that for the two dielectric-resonator scheme. The one dielectric-resonator scheme can have a wide operational band due to the coupling between dielectric resonators. However, the operation is sensitive to the arrangements of lattice structures because the coupling strength directly depends on the distances between the dielectric resonators. On the other hand, it is well known that the effective dielectric constant of a rectangular metallic waveguide becomes negative at frequencies below the cutoff. A rectangular waveguide below the cutoff loaded with split-ring resonators has been proposed and demonstrated as a 1-D LH transmission waveguide since the split-ring resonators provide negative effective permeability [12]. Recently, we have proposed another scheme, i.e., a one dielectric-resonator scheme in the cutoff background, using a 2-D dielectric resonator lattice structure under the TE resonance

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that is inserted in a cutoff parallel-plate waveguide, and demonstrated numerically and experimentally negative refraction for this 2-D structure [13]. However, the detail discussion and comparison between numerical and measurement results were not done in it. In this paper, we further investigate the propagation characteristics and negative refraction of the 2-D dielectric-resonatorbased LH parallel-plate waveguide, and provide additional numerical and experimental results, along with their comparison. In addition, equivalent-circuit models for the corresponding 1-D structure are introduced in order to give a good insight into the LH operation. The use of the proposed configuration has some advantages compared to other 1-D or 2-D LH structures. First of all, the fabrication tolerance is larger than that of the two dielectric-resonator scheme since the cutoff parallel-plate waveguide for TE modes provides negative effective permittivity over a wide frequency range. Second, the LH operation is not sensitive to the coupling strength between dielectric resonators. The coupling can make the operational band broad. In addition, we can design the negative effective permittivity and permeability separately, which is attractive for designing and quite different from the above-mentioned one dielectric-resonator scheme. Finally, it should be emphasized that the proposed dielectric-resonator-based parallel-plate structure is expected to reduce the conducting loss compared to the split-ring-resonator-based cutoff rectangular waveguides, mainly from two reasons, which are: 1) it does not require complicated metallic structures to realize negative permeability and 2) the use of the parallel-plate TE mode instead of the corresponding rectangular waveguide mode removes metallic walls along the primary magnetic field component from the waveguide. II. GEOMETRY AND BASIC CONCEPT [13] The geometry of the problem is shown in Fig. 1. It is composed of a parallel-plate waveguide and a 2-D square lattice of disc-type dielectric resonators. The latter is placed in the center plane between parallel plates. The parallel-plate waveguide is composed of two conducting plates with the distance of and a . The disc-type host medium with the dielectric constant of , the height dielectric resonators has a dielectric constant of of , and the diameter of . The period of the lattice is , and the axis of the dielectric resonators is set perpendicular to the parallel plates. The electric field vector of the incident waves is designated to be in the plane of the parallel plates so that the TE mode is dominant. The effective permittivity for the parmode for the case without dielectric resonators allel-plate is given by (1) with

where is an integer, represents the speed of light in vacuum, and is the angular frequency of the wave. Therefore, the TE

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Fig. 1. 2-D lattice structure for one dielectric-resonator scheme in the cutoff waveguide (after [13]). (a) Perspective view. (b) Side view.

modes in the parallel-plate waveguide have negative effective permittivity below the cutoff frequencies of On the other hand, the fields of a fundamental TE resonant mode for each dielectric resonator behave like a magnetic dipole, and the collective and macroscopic behavior for the dielectric resonator lattice structure can give negative effective permeability on the upper side of the resonant frequency to the propagated wave when the magnetic field of the incident wave is parallel to the magnetic dipoles of the dielectric resonators under the TE resonance [14]. In the following discussion, the height of the dielectric resonator disc is set to be small commode, which pared to the radius of the disc so that the behaves like a magnetic dipole directing along the axis of the dielectric resonator disc, i.e., normal to the parallel plates, is the fundamental resonant mode [15]. It should be emphasized that the effective permeability does not depend significantly on the lattice type, but on the density of the dielectric resonators. Thus, the proposed structure can operate as a 2-D LH metamaterial by keeping both the effective permittivity and permeability negative simultaneously. III. EQUIVALENT-CIRCUIT MODELS Here, we will introduce an equivalent-circuit model for the 1-D proposed structure in order to provide a good insight into the physical mechanism of the LH operation, comparing it with the conventional circuit model. The equivalent circuit for a unit cell of the 1-D structure is shown in Fig. 2(a). It is composed of two sections; a transmission line for the fundamental TE mode in the parallel-plate waveguide and an LC resonator section for resonance of dielectric resonators. The LC resonator is magnetically coupled to a series branch of the transmission line

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Fig. 3. Dispersion diagram for the 2-D structure (after [13]).

Fig. 2. Equivalent-circuit models for 1-D LH transmission line. (a) Circuit based on the physical phenomena. (b) Circuit based on the mutual coupling often used in conventional evanescent mode filter designing. (c) Schematic frequency dependence of effective series inductance and shunt capacitance. Solid and dotted lines show those of the circuits shown in (a) and (b), respectively.

permeability is negative at frequencies near and above the res. onant frequency of dielectric resonators It is noted that the other equivalent-circuit model for conventional evanescent mode filters using dielectric resonators is described in terms of the mutual magnetic coupling between LC resonators, as shown in Fig. 2(b) [15]. The circuit in Fig. 2(b) can also be reduced to the LH transmission line, and the estimated effective inductance and capacitance and are given in the following: (4)

with the mutual inductance . The circuit model for the coupling is the same as that for the coupling between split-ring resonators and conventional transmission lines [16]. The transmission line for the parallel-plate waveguide can be described and the shunt resonant secin terms of a series inductance and inductance . The resonant tion with a capacitance frequency of the shunt circuit section corresponds to the cutoff frequency for the fundamental TE mode in parallel-plate waveguide , whereas the resonant frequency of the LC resonator . The escorresponds to that of dielectric resonators with timated effective inductance in the series branch , and capacitance in shunt branch , for the total transmission line in Fig. 2(a), which correspond to the effective permeability and permittivity respectively, are given in the following: (2) with

(3) From (2), the effective permittivity of the transmission line becomes negative below the cutoff within , while the

with (5) and the capacitance Although the effective inductance can take the frequency dependences similar to (2) at frequencies below the cutoff, as shown in Fig. 2(c), this conventional circuit model cannot give any zeros for the effective shunt capacitance or singular points for the series inductance. Thus, the circuit model in Fig. 2(a) can describe transmission characteristics both below and above the cutoff regions, whereas the circuit model in Fig. 2(b) can apply only below the cutoff region. IV. DISPERSION DIAGRAMS Here, we will show the dispersion diagram of the proposed structure under the periodic boundary condition in order to numerically validate the existence of LH modes. Between the two parallel metallic plates, a 2-D periodic array of dielectric resonators are placed within a background material . The various parameters with a dielectric constant of used in the numerical calculation are as follows: the dielectric constant, height, and diameter of the disc-type dielectric res, mm, and mm, reonators are mm and spectively. The distance of the parallel plates is mm. the length of the unit cell is The dispersion curve for the TE mode in the proposed structure along the high symmetry Brillouin zone points [3] are shown in Fig. 3. The fundamental quasi-TEM mode and the TM guided modes (polarization normal to the parallel plate) are not plotted in Fig. 3 in order to clearly display

UEDA et al.: DEMONSTRATION OF NEGATIVE REFRACTION IN CUTOFF PARALLEL-PLATE WAVEGUIDE

Fig. 4. Equi-frequency map for the 2-D structure.

the LH TE mode. The obtained TE mode is a fundamental TE mode and no other modes coexist in the same band or below the frequency region, except for the fundamental quasi-TEM mode. Fig. 3 clearly shows that an LH TE mode is supported (i.e., antiparallel group and phase velocities) from 10.4 to 11.15 GHz. mode in the parIt is noted that the cutoff frequency of the allel-plate waveguide without dielectric resonators is estimated to be 20.2 GHz from (1), and that the resonant frequency of the mode for a single dielectric resonator is numerically estimated to be 10.0 GHz for the isolated case without parallel plates where the dielectric resonator is surrounded by the host medium. As shown in Fig. 3, LH characteristics are obtained for and branches near and above the resonant freboth the quency of the dielectric resonator mode, but below the mode in the parallel-plate wavecut off frequency of the guide. Fig. 3 also shows that from 10.75 to 11.15 GHz , the and curves coincide, which indicates that the structure is isotropic within this frequency range. The isotropic characteristic is also verified by the equi-frequency map, as shown in Fig. 4. The quantities and in Fig. 4 represent the and components of the wavenumber vectors, respectively. The results of Figs. 3 and 4 numerically confirm the existence of the LH TE mode for all the propagation directions in the proposed 2-D structure. V. NUMERICAL VERIFICATION OF NEGATIVE REFRACTION Based on the dispersion diagram obtained in Section IV, the numerical demonstration of negative refraction of the specific 2-D LH structure will be given. First of all, we need to design the right-handed (RH) waveguide structures to set up the interfaces between LH and RH regions where the negative refraction is found. As the specific RH mode in the following discussion, we consider a parallel-plate guided mode in the waveguide filled with a high dielectric constant of 10.2. In this case, cutoff frequency of the funmode in the parallel-plate waveguide is 9.4 GHz damental mm and, therefore, it is the guided for the distance of mode above the frequency. The effective refractive indices for the RH and LH structures are estimated from the dispersion diagrams and are shown in Fig. 5. As seen from this figure, it is found that the effective refractive index for RH structure is approximately 1.5 at 10.8 GHz

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Fig. 5. Effective refractive indices for the 2-D LH and RH structures.

and the corresponding value is around 1.0 for the LH structure for both and . Refraction of the propagated beam at an interface between LH and RH regions is governed by Snell’s law (6) where and denote refractive indices in the RH and and denote LH regions, respectively. The quantities the incident (transmitted) angle of the beam at the interface in the LH region, and the transmitted (incident) angle at the same interface in the RH region, respectively. Thus, when the incident , , and the transmission angle is positive with is permitted, the transmitted angle is negative, as seen from (6). The top view of the specific configuration used in the fullwave simulation and the simulated field profiles are shown in Fig. 6(a). The structure to be simulated is composed of three separate regions, i.e., Regions 1–3. In what follows, we consider a triangular prism of the proposed 2-D LH metamaterials, which can clearly show negative refraction when it has a negative index for the propagated waves. The LH prism is placed in Region 2. The configuration parameters of the LH waveguide are the same as those given in the above discussion, except for its finite size and shape. There are 15 dielectric resonator discs inserted in Region 2. The LH prism is placed between two RH regions, i.e., Regions 1 and 3. The dimensions of the structure are as follows; the total longitudinal length of the structure between a top edge line of Region 1 and a bottom edge line of Region 3, as well as the width are both 130 mm, the length of Region 1 is 50 mm, the maximum longitudinal length and width of Region 2 are both 30 mm corresponding to five cells, and the minimum longitudinal length of Region 3 is 50 mm. The 5 mm 5 mm cross-sectional rectangular waveguide that is filled with the same dielectric material as Regions 1 and 3 is connected to mode Region 1. The fundamental rectangular waveguide with a polarization in the plane of the parallel plates is launched as a quasi-point source, and the cylindrical wave is propagated in Region 1, as shown in Fig. 6(a). In the numerical simulation, commercial full-wave field simulation software Ansoft’s High Frequency Structure Simulator (HFSS) version 10 is used. Fig. 6(a) illustrates the field patterns for the normal component of the magnetic field on the center plane between the parallel GHz. In the numerical simulation, plates operating at absorbing boundary conditions have been assumed on all the sidewalls, except for the rectangular waveguide section at the

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Fig. 7. Fabricated structures for measurement. (a) Top view. (b) Fabricated LH prism to be inserted in Region 2.

Fig. 6. Top view of the geometry and the calculated magnetic field profiles. (a) 2-D RH-LH-RH structure (after [13]). (b) 2-D RH-RH-RH structure.

input port. The propagated wave has a relatively flat phase contour at the first interface between the RH region 1 and LH region 2 I-1, and the incidence is almost normal to the boundary I-1. After the transmission into Region 2 with a small reflection at I-1, the wave meets the second interface between Region 2 and Region 3 I-2. The interface I-2 is set oblique by 45 with regard in Fig. 6(a) denotes to the first interface I-1. The dotted line the direction normal to the interface I-2. From the numerical results, it is found that the transmitted wave into Region 3 propagates at a negative refractive angle of approximately 30 with regard to the normal to the interface . For comparison, the simulated I-2, and meets the edge line field profile is shown in Fig. 6(b) for the case where the RH prism is inserted in Region 2. In this case, transmitted beam is found to be propagated and go straight down to the center of the bottom edge of Region 3 without the serious influence of the finite size of the prism.

Fig. 8. Measured and simulated magnetic field profiles along the edge lines of Region 3 when the RH prism is inserted in Region 2.

The above numerical result of negative refraction can be estimated by using the refractive indices in Fig. 5 and (6) In this and in (6) correspond to the incase, the quantities cident angle of the beam at I-2 in LH region 2, and the transmitted angle at the same interface in RH region 3, respectively. is 45 for Fig. 6. The transmitted angle Of course, can be estimated to be approximately 28 with , at GHz from Fig. 5, which is in a good agreement with the numerical simulation. In addition, as the operational frequency increases, the magnitude of the refractive

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Fig. 9. Measured and simulated magnetic field profiles along the edge lines of Region 3 when the LH prism is inserted in Region 2. (a) f = 10:7 GHz. (b) f =

10:8 GHz. (c) f = 10:9 GHz. (d) f = 11:0 GHz.

index for the proposed LH prism decreases dramatically compared to the flat value for the RH structures, according to Fig. 5. As a result, the transmitted negative angle in Region 3 should should approach zero and the beam spot on the edge line move to the point , as seen from (6). The detail discussion and comparison between numerical simulation and prediction from Snell’s law will be given in Section VI. Thus, numerical simulation results verify the left-handedness for the proposed structure. VI. DEMONSTRATION OF NEGATIVE REFRACTION AND DISCUSSION Based on the numerical simulation results, we experimentally demonstrate negative refraction for the same 2-D RH-LH-RH structure, as shown in Fig. 6(a). The configuration parameters of the fabricated structure are the same as those in the numerical simulation. The photographs of the fabricated 2-D RH-LH-RH structure, as well as the LH prism, are shown in Fig. 7. An RT/Duroid 5880 substrate with a dielectric constant of 2.2 was used as a host medium between the parallel plates in the LH region, and an RT/Duroid 6010 substrate with a dielectric constant of 10.2 was chosen as a dielectric medium between parallel plates in the RH region including the rectangular waveguide at rectanthe input port. The 5 mm 5 mm cross-sectional gular waveguide mode with the polarization in the parallel plates was excited with an -band rectangular waveguide (WR-90) at the input port. A vector network analyzer (HP 8510C) was used as the input signal generator. Plastic screws were used in the vicinity of corners of each region to support the structures and minimize the influence on the beam transmission where the parallel plates had holes with a diameter of 3 mm.

The 1-D field profiles of the magnetic field component normal to the parallel plates were measured at positions outside 5 mm away along two edge lines of Region 3; one is the edge line where the field concentration is expected when Region 2 is RH, where the field is concenand the other is the edge line trated when Region 2 is LH. A loop antenna with the diameter of 3.35 mm was used as a magnetic probe for the normal component of the magnetic field instead of electric probes because the primary axis of the magnetic field directs normal to the parallel plates, while the electric field vectors are in the various directions in the plates depending on the measured positions. While measuring one of the edge lines, the sidewall was open to the air, but all the other sidewalls were covered with an absorber to avoid undesired wave reflections and scatterings. The magnetic probe was moved by a 1-mm step with an – scanner Newport Corporation motorized linear stage. To obtain the magnitude and phase of the measured fields, the probe was connected to the network analyzer that was used as the input source. In Fig. 8, the measured and simulated magnetic field profiles for the case where the RH prism is inserted in Region 2 are shown. The dimension of the RH prism is the same as that of the LH prism. The negative and positive values of the position in the horizontal axis in Fig. 8 represent the distance of the meaand , respectively. For the sured positions from along RH prism case, the transmitted beam goes straight down to the , as expected from Fig. 6(b). From Fig. 8, bottom edge line the measured profile of the main beam is in a good agreement with the numerical simulation results. Therefore, the influence of air gaps between three regions in the fabricated structure on the main beam direction is found to be small. The high sidelobe

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Fig. 10. Position of the beam spot versus the operational frequency. (a) Beam spot on the edge line QB under the assumption of absorbing boundary condition on QB . (b) Beam spot at positions 5 mm away from QB in the air region.

on is due to the undesired reflection and scattering from the . sidewall at the edge line In Fig. 9, measured and simulated field profiles as a function of the operational frequency are shown for the case where the fabricated LH prism is inserted in Region 2. As for the simulated fields in Fig. 9, it is noted that one of the edge lines or , along which the fields are calculated, is assumed to be open to the air region and that absorbing boundary conditions are assumed on all the other sidewalls in order to coincide with the measurement setup. Experimental results generally have reasonable agreement with simulations. From the measured profiles in Fig. 9, all the peak values of the field profiles were found on the edge . However, as seen from the measured field profiles at line GHz GHz, and GHz , discrepancies of the profiles between the measurement and simulation become significant around at point as the frequency is increased, i.e., as the beam spot moves to . This discrepancy may be caused by the nylon screw at , which is used to hold the structure. Further, the calculated and measured beam positions are compared and shown in Fig. 10. In Fig. 10(a), the calculated beam are shown under the assumppositions on the edge line for compartion of an absorbing boundary condition on ison. Fig. 10(b) shows the positions of the beam spot 5 mm outin the air region, as found in side away along the edge line Fig. 9. In Fig. 10, solid lines show the beam position estimated from Snell’s law in terms of the refractive indices in Fig. 5, and square and round dots show the positions of the simulated

and measured beam spot, respectively. As the beam positions, we have basically taken the peak value of the magnitude in the . simulated or measured field profiles along the edge line When the main beams were split into two, the center position between the two peaks has been taken as the beam position. As seen from Fig. 10(a), the full-wave simulation results are in a good agreement with the prediction from Snell’s law under the . This reassumption of absorbing boundary conditions on sult provides the validation of the numerical simulation results. However, there are significant discrepancies between them for the case in Fig. 10(b). Since the refractive index of air is smaller than that of RH region 3, the ray theory-based beam spot significantly bends toward in the air region, especially for oblique . As a result, the slope of the curve incidence at the interface estimated from Snell’s law in Fig. 10(b) becomes steeper compared to Fig. 10(a). On the other hand, full-wave simulation results in Fig. 10(a) and (b) show that the condition of the open does not provide significant beam bending toboundary on ward in the air region. On the contrary, it makes the curve of the simulated beam positions flatter. Therefore, the discrepancies between the numerical simulation and the prediction from Snell’s law in Fig. 10(b) may originate from the fact that, in the full-wave simulation, we extract the near fields in the vicinity rather than the transmitted waves from the of the interface boundary. Finally, by comparing the simulated and measured beam spots in Fig. 10(b), the measured beam spots are confirmed to be around the numerically predicted positions. Thus, negative refraction of the propagated waves through the proposed LH prism and the beam scanning with the operational frequency were verified experimentally and numerically. VII. CONCLUSION In this study, we have proposed and investigated the 2-D negative-refractive-index metamaterial, which is composed of a parallel-plate waveguide below the cutoff for the fundamental TE mode and a square lattice of dielectric resonators inserted in the waveguide. An equivalent-circuit model for the 1-D structure has been introduced to explain the physical mechanism of the LH transmission line. Dispersion diagrams have been obtained to show the existence of the LH TE mode and the isotropic characteristics. In addition, negative refraction was numerically and experimentally demonstrated for the 2-D LH triangular prism sandwiched by RH parallel-plate waveguides. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of " and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan. 1968. [2] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, pp. 4184–4187, May 2000. [3] A. Lai, C. Caloz, and T. Itoh, “Composite right/left-handed transmission line metamaterials,” IEEE Micro, pp. 34–50, Sep. 2004. [4] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley, 2006. [5] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002.

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[6] G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials: Fundamental Principles and Applications. Piscataway, NJ: IEEE Press, 2005. [7] C. L. Holloway, E. Kuester, J. Baker-Javis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2596–2601, Oct. 2003. [8] O. G. Vendik and M. S. Gashinova, “Artificial double negative (DNG) media composed by two different dielectric sphere lattices embedded in a dielectric matrix,” in Proc. 34th Eur. Microw. Conf., Oct. 2004, pp. 1209–1212. [9] L. Jylhä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys., vol. 99, pp. 043102-1–043102-7, Feb. 2006. [10] E. A. Semouchkina, G. B. Semouchkin, M. Lanagan, and C. A. Randall, “FDTD study of resonance processes in metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1477–1486, Apr. 2005. [11] M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B, Condens. Matter, vol. 62, no. 16, pp. 10696–10722, Oct. 2000. [12] R. Marques, J. Martel, F. Mesa, and F. Medina, “Left handed media simulation and transmission of EM waves in sub-wavelength SRRloaded metallic waveguides,” Phys. Rev. Lett., vol. 89, pp. 1839011–183901-4, Oct. 2002. [13] T. Ueda, A. Lai, and T. Itoh, “Negative refraction in a cut-off parallelplate waveguide loaded with a two dimensional lattice of dielectric resonators,” in Proc. 36th Eur. Microw. Conf., Sep. 2006, pp. 435–438. [14] L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Elect. Eng., vol. 94, pt. III, pp. 65–68, 1947. [15] S. B. Cohn, “Microwave bandpass filters containing high- dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 4, pp. 218–227, Apr. 1968. [16] F. Martin, F. Falcone, Bonache, R. Marques, and M. Sorolla, “Miniaturized coplanar waveguide stop band filters based on multiple tuned split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 12, pp. 511–513, Dec. 2003.

Q

Tetsuya Ueda (M’97) received the B.E., M.E., and Ph.D. degrees in communication engineering from Osaka University, Osaka, Japan, in 1992, 1994, and 1997, respectively. Since 1997, he has been a Research Associate with the Department of Electronics, Kyoto Institute of Technology, Kyoto, Japan. From 2005 to 2006, he was a Visiting Scholar with the Department of Electrical Engineering, University of California at Los Angeles (UCLA). His current research interests include metamaterials and their applications.

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Anthony Lai (S’98) received the B.E. degree in electrical engineering from The Cooper Union for the Advancement of Science and Art, New York, NY, in 2001, the M.S. degree in electrical engineering from the University of California at Los Angeles (UCLA), in 2005, and is currently working toward the Ph.D. degree in electrical engineering at UCLA. His research interests include planar antennas and metamaterial applications.

Tatsuo Itoh (S’69–M’69–SM’74–F’82) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1969. From September 1966 to April 1976, he was with the Electrical Engineering Department, University of Illinois at Urbana-Champagin. From April 1976 to August 1977, he was a Senior Research Engineer with the Radio Physics Laboratory, SRI International, Menlo Park, CA. From August 1977 to June 1978, he was an Associate Professor with the University of Kentucky, Lexington. In July 1978, he joined the faculty of The University of Texas at Austin, where he became a Professor of electrical engineering in 1981 and Director of the Electrical Engineering Research Laboratory in 1984. During Summer 1979, he was a Guest Researcher with AEG-Telefunken, Ulm, Germany. In September 1983, he was selected to hold the Hayden Head Centennial Professorship of Engineering with The University of Texas at Austin. In September 1984, he was appointed Associate Chairman for Research and Planning of the Electrical and Computer Engineering Department, The University of Texas at Austin. In January 1991, he joined the University of California at Los Angeles (UCLA), as Professor of Electrical Engineering and Holder of the TRW Endowed Chair in Microwave and Millimeter Wave Electronics. He was an Honorary Visiting Professor with the Nanjing Institute of Technology, Nanjing, China, and with the Japan Defense Academy. In April 1994, he became an Adjunct Research Officer with the Communications Research Laboratory, Ministry of Post and Telecommunication, Tokyo, Japan. He currently holds a visiting professorship with The University of Leeds, Leeds, U.K. He has authored or coauthored 350 journal publications and 650 refereed conference presentations. He has authored 40 books/book chapters in the area of microwaves, millimeter-waves, antennas, and numerical electromagnetics. He generated 66 Ph.D. students. Dr. Itoh is a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI. He served as the editor-in-chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (1983–1985). He serves on the Administrative Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was vice president of the IEEE MTT-S in 1989 and president in 1990. He was the editor-in-chief of the IEEE MICROWAVE AND GUIDED WAVE LETTERS (1991–1994). He was elected an honorary life member of the IEEE MTT-S in 1994. He was the chairman of USNC/URSI Commission D (1988–1990), and chairman of Commission D of the International URSI (1993–1996). He was chair of the Long Range Planning Committee, URSI. He serves on advisory boards and committees for numerous organizations. He was elected a member of the National Academy of Engineering in 2003. He has been the recipient of numerous awards including the Shida Award presented by the Japanese Ministry of Post and Telecommunications (1998), the Japan Microwave Prize (1998), the IEEE Third Millennium Medal (2000), and the IEEE MTT-S Distinguished Educator Award (2000).

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UWB Array-Based Sensor for Near-Field Imaging Alexander G. Yarovoy, Senior Member, IEEE, Timofey G. Savelyev, Pascal J. Aubry, Pidio Ekoue Lys, Member, IEEE, and Leo P. Ligthart, Fellow, IEEE

Manuscript received October 4, 2006; revised March 16, 2007. This work was supported by the Dutch Ministry of Defense under Contract DMO/HDM/DR&D N01/26-10. The authors are with the Faculty of Electrical Engineering, Mathematics and Computer Science, International Research Centre for Telecommunications–Transmission and Radar, Delft University of Technology, 2628CD Delft, The Netherlands (e-mail: [email protected]; [email protected]; [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.2007.896791

array-based GPR has been realized,2 where a simple antenna beam focusing has been realized using programmable time delays in four transmit and four receive antennas. Remarkably, all the above-mentioned vehicle-based systems, except for those of [4], were designed for detection of antitank mines, which typically have a cylindrical shape and a diameter larger than 25 cm. Detection of antipersonnel (AP) mines, which might be just a few centimeters in diameter, is a much more difficult task and, thus, requires much more complicated hardware and software. Numerous field trials of different GPR sensors have proven that, while for most soils, the GPR sensor can achieve a desirable detectability level, the decrease of the false alarm rate remains the most important task for GPR developers. This decrease can be achieved by improving the signal-to-clutter ratio and via classification of detected targets with differentiation between landmines and mine-like objects. The former depends on the footprint size of the transmit and receive antennas and is determined mainly by the hardware. The latter is done mainly in signal processing, and one of the possible approaches here is target classification based on features extracted from 3-D images of the subsurface [5]. Other approaches for target classification can be found, for example, in [6]–[8]. To verify the image-based classification approach, the International Research Centre for Telecommunications–Transmission and Radar (IRCTR), Delft University of Technology, Delft, The Netherlands, has developed a number of GPR systems dedicated to landmine detection [9]–[11]. These systems have demonstrated a high potential of developed GPR technology for classification of surface-laid and buried targets. However, all the developed systems required 2-D mechanical scanning for data acquisition. This paper describes a qualitatively new design of an arraybased radar. In this design, a single radar sensor simultaneously uses all antennas within the array in order to image the subsurface. The antenna system consists of a single transmit antenna and a linear array of receive ones. The imaging is done by electronic scanning along the array with the receive-array footprint when the near-field focusing is done using true-time digital delays. This paper is organized as follows. Design considerations for the system are presented in Section II. The development of a novel antenna array is described in Section III, while the development of electronics is described in Section IV. The imaging algorithm is presented in Section V, and the first measurement results are presented in Section VI. Main outputs of this study are then summarized in Section VII.

1[Online]. Available: http://www.humanitarian-demining.org/demining/detection/niitek.asp?version=view

2GeoCenters EFGPR. [Online]. Available: http://www.geo-centers.com/ products/EFGPR-Overview-1-03.pdf

Abstract—In this paper, the development of an ultra-wideband (UWB) array-based time-domain radar sensor for near-field imaging is described. The radar sensor is designed to be used within a vehicle-mounted multisensor system for humanitarian demining. The main novelty of the radar lies in the system design with a single transmitter and multichannel receiver. Design of the UWB antenna array is also novel. The radar produces 3-D images of subsurface by 1-D mechanical scanning. The imaging capability of the radar is realized via electronic steering of the receive antenna footprint in a cross-scan direction and synthetic aperture processing in an along-scan direction. Imaging via footprint steering allows for a drastic increase in the scanning speed. Index Terms—Ground penetrating radar (GPR), landmine detection, near-field imaging, ultra-wideband (UWB) antenna array.

I. INTRODUCTION

I

T HAS been demonstrated that ground penetrating radar (GPR) is a useful sensor for the multisensor system intended for landmine detection, especially for humanitarian demining [1]. In order to avoid 2-D mechanical scanning over the surface and to speed up the ground survey, a number of array-based GPRs have been developed (see, e.g., [2]–[4] and online1). In these systems, the array is formed by a number of parallel radars in which the number of transmit antennas is equal to the number of receive ones. The mechanical scanning in the plane of the array is replaced by sequential operation of the constituting radars. Despite a substantial increase of the survey speed in comparison with 2-D mechanical scanning, this approach still limits the scanning speed up to a few kilometers per hour. Furthermore, the sequential operation of the radars keeps the signal-to-clutter ratio for the whole system at the same level as for a single system because the total footprint of the whole array remains a sum of the footprints of separate antennas, and the low value of the signal-to-clutter ratio typically results in the high false alarm rate. A more advanced approach to the

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Fig. 2. Developed antenna array. The total aperture of the array is 84 cm.

Fig. 1. Block scheme of the array radar.

II. DESIGN CONSIDERATIONS We are aiming to a vehicle-based multisensor system for landmine detection. For the GPR sensor, it means that the radar front-end should include an antenna array consisting of a number of transmit and receive elements. In a traditional approach, the number of transmit antennas is equal to the number of receive antennas. Keeping in mind that for AP mine detection the scattered by subsurface field should be measured every 5–7 cm in both horizontal directions, the antenna array requires a large number of transmit–receive antenna pairs. This results in a bulky antenna system (see, e.g., [4]) and complicated hardware. To avoid these problems, we suggest a novel approach to a system design [12], [13]. Firstly, the antenna array consists of a single transmit antenna and an array of receive antennas. The swath of such an array is limited by the footprint of the transmit antenna and the antenna array configuration. The scanning along the array is then done not by sequential measurements via different Tx/Rx antenna pairs, but via focusing of the receive array footprint in the near field and scanning with this footprint along the array. Such an approach drastically reduces the data acquisition time, reduces the number of transmit antennas, simplifies the antenna system, and considerably simplifies the electronics. Secondly, in order to improve the power budget of the radar at low frequencies (which are important for sufficiently deep penetration into the ground), we select a special waveform of the pulse fired by a generator. This waveform has a large spectral content at low frequencies to compensate for the decrease of the transmit antenna gain. Based on the above-mentioned design considerations, we developed a system that comprises a pulse generator, an antenna array, a seven-channel signal conditioner, and an eight-channel sampling converter. The generator also has the second (calibration) output, which is used as a reference for compensation of time jitter and time drift. A block scheme of the array radar is shown in Fig. 1.

III. ANTENNA ARRAY A. General Approach The main tasks for the antenna array are to measure target responses in the ultra-wideband (UWB) frequency range and to provide near-field imaging of the subsurface by means of electronic steering of the received field along the array. Since in the near field the antenna’s radiation patterns are not yet formed, the near-field focusing can be done only in terms of near-field footprints, which describe spatial distribution of the electromagnetic (EM) field around the antenna in some particular plane. An ideal antenna element for such focusing should have a single radiation center, which results in a quasi-circular footprint. Following the Huygens principle, by combining a number of such antenna elements in an array and controlling the radiation time, it is possible to focus the EM field in a certain spatial area. If the radiated signals are short transient pulses, then the focusing can be achieved not only in space, but also in time [14]. Practical implementation of such an approach is, however, difficult due to the need to generate a number of identical waveforms and to synchronize them in time. Thus, we have decided to implement the same focusing idea, but only for EM field reception. For transmission, we have decided to use a single UWB transmit antenna that should illuminate the whole area under investigation with an ultra-short pulse. In order to remove influence of the transmit antenna on the performance of the receive array and to decrease direct coupling between the transmit and receive antennas (which, due to the finite dynamic range of a receiver, limits the receiver sensitivity), we separated the transmit antenna and the receive array in the same way as we have done in our previously designed radars [9], [11]. As a result, the receive array is placed in the -plane of the transmit antenna with 27-cm separation between the transmit antenna aperture and the receive loops (Fig. 2). High elevation of the transmit antenna gives us a number of advantages, among them the following. 1) The waveform of the radiated field on the surface and in the ground remains the same within the antenna footprint at 10-dB level.

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Fig. 3. Experimentally determined impulse response of the transmit antenna.

Fig. 4. Footprint (peak-to-peak value of the received voltage, shown in decibels).

2) The waveform of the radiated field does not depend on the type of the ground (i.e., its dielectric permittivity) as the transmit antenna is fully decoupled from the ground. 3) The size of the antenna footprint at the air-ground interface can be changed by varying the antenna elevation. B. Transmit Antenna For the transmit antenna, we chose the dielectric wedge antenna [15] for which operational bandwidth, impulse response, and footprint shape are best suited to the application in mind. To make the footprint at low frequencies fully symmetrical, a UWB balun [16] has been designed and added to the antenna. When the antenna is excited with an ultra-short step pulse (with a rise time of approximately 25 ps), it produces a response with a duration of approximately 150 ps (on the half-amplitude level) (Fig. 3). This response is determined mainly by the properties of the antenna itself and is very close to the impulse response of the antenna. As it can be seen in Fig. 4, the footprint (at a 10-dB level) of the transmit antenna has a diameter of approximately 80 cm. The observed spatial distribution of the radiated signal is valid not only for the peak-to-peak magnitude of the time-domain signal, but also for different spectral components of the signal. This figure gives an indication for the maximal size of the received array. C. Receive Array Design The goals of the receive array design procedure were minimal mutual coupling between the elements (while keeping receive antenna separation smaller than a half-wavelength at the highest operational frequency), maximal sensitivity of a single antenna element, and minimal end effects. We have decided to use the shielded loop antenna [17] as a basic element of the array. At the frequencies above its first resonance, the loop has a relatively flat sensitivity over a large frequency band [18]. Furthermore, the loop antenna has an aperture of the same order as a linear dipole, but unlike the dipole, the loop possesses a very small ringing in

Fig. 5. Sensitivity (in decibels with respect to 1 V) of different loops within the array. Loop diameter is 3 cm. The loop sensitivity is defined as a voltage in a perfectly matched load of the loop produced due to an incident electric field with strength of 1 V/m.

the time domain. Finally, the loop antenna has a single radiation center, of which the position is constant with frequency. As the loops are transparent for the incident wave, they can be placed beneath the transmit antenna in its -plane. While being aligned in their -plane, the loops exhibit strong mutual coupling. The coupling results in waveform distortion and correlation of signals, received by different loops in the array, and at the end, it reduces focusing possibilities of an imaging system. In order to analyze the impact of the coupling in detail, an array of 13 loop antennas has been simulated. It has been found that the coupling results in an increase of the loop sensitivity (with respect to an isolated loop) up to 10 dB in some relatively narrow frequency band (Fig. 5). The allocation of this frequency band depends on the loop separation and loop diameter. With an increase of the loop separation or loop diameter, the sensitivity peak due to the coupling shifts

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Finally, the number of loops in the array has been selected based on the size of the transmit antenna footprint. The latter determines how fast the amplitude of the illumination field decreases from the center towards the edges of the array. The receive antennas measure the field scattered by a target, and the more measurements are done, the better should detection and classification of the target be. However, if the target is situated far from the transmit antenna, both the incident and scattered fields will be too low to provide a sufficient signal-to-noise ratio (SNR) in the receiver. Based on this analysis and on the footprint of the transmit antenna, we conclude that the swath of the miniarray can be as large as 84 cm. This size has determined the total number of loops in the array to be 13. IV. RADAR ELECTRONICS Fig. 6. Measured ground reflection by the central loop in the array for different separation between the loops. Coupling in the array results in long ringing.

towards lower frequencies. Furthermore, the coupling slightly (up to 300 MHz) shifts the dip in the sensitivity towards low frequencies. Finally, the discrepancy between sensitivities of different loops does not vary considerably: the outmost side loops have 4–5 dB smaller magnitude of the peak due to the coupling and smaller dip shift. In the time domain, the loop response in free space to a short pulse, radiated by the transmit antenna, qualitatively repeats the waveform of the incident pulse [18]. The mutual coupling of the loops within the array results in additional ringing. This ringing looks like damped oscillations with a frequency determined by the spike in the loop sensitivity due to the coupling (Fig. 6). The magnitude and duration of the ringing for the central loop is, in general, larger that for the side loops. EM simulations of the array have been used to optimize the array. The optimizing parameters were the loop radius and separation between the loops. The loops’ radius determines loop sensitivity and bandwidth: the smaller the loop radius, the larger its bandwidth and the smaller is its sensitivity. It should be noticed that technologically it is very difficult to manufacture loops with a diameter smaller than 3 cm from a semirigid cable. The loop’s diameter and separation between the loops also determine the frequency at which the loop’s sensitivity has a spike, and in this way, they determine magnitude and duration of the ringing. If the upper frequency of the pulse is lower than the frequency at which the loop’s sensitivity has a spike, the resonant coupling mechanism between the loops is not excited and the ringing is low. If the resonant coupling is excited, then the smaller the distance between the loops is, the shorter the duration of the ringing due to the coupling between the antennas, and the larger the magnitude of the ringing. Trading off the operational bandwidth, the loop sensitivity, the duration of the ringing, and its magnitude, we selected 5 cm as an optimal separation between the antenna elements. However, due to mechanical constrains related to the positioning of the feeding cables within the array, it was possible to achieve the minimal separation of 7 cm.

A. Pulse Generator The choice of the operational bandwidth is one of the most critical issues in GPR design. On one hand, at the frequencies below 1 GHz, attenuation losses in the ground are small [19] and considerable penetration depth can be achieved. On the other hand, landmine detection requires down-range resolution of the order of several centimeters in the ground, which can be achieved using operational bandwidth of more than 3 GHz. Based on these considerations, we have selected an operational bandwidth from 500 MHz to 3 GHz as the minimum desired. This bandwidth can be covered using different waveforms, which have different energy distribution over this band. To improve the radar power budget at low frequencies, which are not efficiently radiated by the transmit antenna due to the relatively small size of its physical aperture, we wanted from the generator output the highest spectral density at low frequencies. The second demand for the waveform selection was duration of the received pulse. In order to find the optimal waveform, we compared three pulse generators with a step pulse, double exponential pulse, and monopulse output [see Fig. 7(a)] in order to determine the optimal output waveform for the developed antenna array. The waveforms of the probing pulses (pulses radiated by the transmit antennas) and their spectra are shown in Fig. 7(b) and (c), respectively. The waveforms of the radiated pulses have been calculated from the signals measured by a 3-cm loop using a deconvolution procedure [20]. The double-exponential pulse provides the largest spectral content at low frequencies (while normalized to the common absolute maximum), the largest bandwidth, the largest amplitude of the radiated pulse, moderate late-time ringing, and the heaviest early-time ringing. In order to achieve the same peak amplitude as by the double exponential pulse, the magnitude of the step pulse should be increased 5 , becoming 2.5 larger than the double-exponential one. The step pulse provides the maximal spectral content at low frequencies (while normalized to the relative maximum), the second largest bandwidth, the smallest amplitude of the radiated pulse, and the heaviest late-time ringing. In order to achieve the same peak amplitude as by the double exponential pulse, the magnitude of the step pulse should be increased 5 becoming 2 larger than the double-exponential one.

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lowest early- and late-time ringing. In order to achieve the same peak amplitude as by the double exponential pulse, the magnitude of the monopulse should be increased 1.625 , becoming 0.8 larger than the double-exponential one. Trying to maximize the radar power budget at frequencies below 1 GHz, we tuned the rise time and pulse duration of the double-exponential generator and ended with a duration of approximately 41 ps (on half-amplitude level) and magnitude of approximately 40 V. As a result, the probing pulse has a bandwidth from 0.5 to 3.3 GHz at a 3-dB level of the bandwidth and moderate ringing. This generator also has the second output (for calibration), which is used as a reference for compensation of time jitter and time drift. B. Receiver (a)

(b)

(c) Fig. 7. Waveforms of generator’s: (a) outputs, (b) waveforms, and (c) spectra of the transmitted field for different generators.

The 50-ps monopulse provides the smallest spectral content at low frequencies (for any normalization), the smallest bandwidth, the moderate amplitude of the radiated pulse, and the

The receiver chain consists of a switch, a seven-channel signal conditioner, and an eight-channel sampling converter (built by GeoZondas Ltd., Vilnius, Lithuania). The receiver chain has an analog bandwidth from 200 MHz up to 6 GHz and a linear dynamic range of 53 dB (without averaging). The switch commutates 12 receive antennas to six channels of the signal conditioner. The central loop is connected directly (by passing the switch) to the fourth channel of the signal conditioner. The other six channels of the signal conditioner are connected either to the six most internal loops (forming together with a central loop an equidistant array of seen loops) or to the three most left and three most right loops. An important part of the receiver chain is the signal conditioner. It consists of seven low-noise amplifiers (LNAs) with a gain of 15 dB and a soft clipping. The signal conditioner improves the SNR and makes it possible to use the whole dynamic range of the ADC. The equivalent noise floor (which includes the discretization noise of ADC) of the receiver is less than 0.75-mV rms without averaging. The spectrum of the noise almost corresponds to white noise; thus, it can be efficiently suppressed by averaging. The signal conditioner decreases the noise floor and improves the SNR by almost 20 dB. The sampling converter operates with a sampling rate of 525 kHz per channel. This high sampling rate results for fast data acquisition and allows for fast scanning with an array. The maximal scanning speed might be as high as 148 km/h, which is too high for such application as humanitarian demining. However, this substantial reserve in the scanning speed allows for implementing averaging over a number of received pulses and, thus, for reduction of the noise floor of the radar (which is determined by the discretization noise of the ADC) proportionally to the square root of the averaging number. The observation time window can be varied from 32 ps to 20 ns with a number of acquisition points available from 16 to 4096. Large flexibility in selection of the observation time window and the sampling time allows us to adjust the system to various soils and data acquisition scenario. A very important feature of the sampling converter is its high measurement accuracy. The maximal error in the amplitude scale and the time scale linearity of the sampling converter is approximately 1%.

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Fig. 8. Power budget of the radar. The diamond corresponds to the reflection from a sand surface, while the rectangle and star correspond to scattering from a metal cylinder, which is flash laid beneath the central loop (rectangle) and beneath the right-most loop of the array (star).

C. Overall Hardware Performance The power budget of the radar is described in Fig. 8. The solid line corresponds to the largest signal (peak-to-peak value), which can be linearly processed in the analog part of the front-end and accurately measured. The dotted line corresponds to the direct coupling between the transmit and receive antennas. This is the signal with the largest magnitude. In order to efficiently use the linear dynamic range of the receiver, this signal is softly clipped by the signal conditioner. Dashed lines 1 and 2 correspond to the weakest signal, which can be distinguished from the discretization noise in the receive mode without averaging. Dashed line 1 corresponds to the maximal scanning speed of 148 km/h, while dashed line 2 corresponds to scanning speed of 5 km/h. It can be seen that the ground reflection (in this case, dry sand) lies very close to the level of the maximal linearly processed signal, thus allowing accurate measurements of surface-laid targets. The magnitude of a target return depends on a target position with respect to the transmit antenna, but as can be seen from this figure, these variations are of the order of a few decibels. The signal processing gain is not included in this power budget. However, the imaging algorithm by simultaneous processing of all 13 received signals gives an additional improvement up to 22 dB. V. IMAGING ALGORITHM The imaging procedure combines a synthetic aperture radar (SAR) algorithm with the digital steering of the receive array footprint. The synthetic aperture is being constructed in the direction of mechanical scanning for each array channel separately with a so-called diffraction stacking algorithm [21] (1) where is a travel time for the grid-point , expresses travel time for the same depth grid points and . In this processing, the coordinate system starts in the transmit antenna’s phase center: -axis expresses depth, -axis corresponds to the mechanical scan direction, and and -axis represents the cross-scan direction. The travel times are computed as a two-way time delay between the transmit antenna, a grid point, and every receive antenna (Fig. 9). Every

Fig. 9. Imaging geometry for the array GPR.

after which we array channel is being focused onto line perform summation of the focused signals of all channels for this line. In terms of phased antenna arrays, summation of the channels represents digital synthesis of the array pattern with its max. For the near-field case, it is equivimum directed at alent to the digital steering of the array footprint. Instead of linearly progressive phase shift, our footprint steering is based on the travel times, which do not have a linear relationship between the array channels. A combination of the SAR in the scan direction and footprint steering in the array’s plane delivers a 3-D image of the target. Prior to imaging, we perform data pre-processing that includes low-pass filtering of the raw data to suppress uncorrelated noise, alignment of the direct coupling in every B-scan to compensate for time drift, and background subtraction to remove direct coupling and ground surface reflection. A very important issue in array imaging is calibration of the system. For that we use a small metal sphere (2-cm diameter) placed under the central loop. A key SAR parameter, a so-called “zero time,” is adjusted for every channel to image the sphere in the correct position. VI. MEASUREMENT RESULTS The performance of the system has been tested at the TNO DS-S premises, The Hague, The Netherlands [22]. The receive array has been elevated approximately 20 cm above the ground, which is a dry sand. The measured dielectric permittivity of the sand is 3.03. The short duration of the transmitted signal [see Fig. 7(b)] results in high down-range resolution of the radar. The waveform of the target response keeps its shape while measured by different loops in the array. This feature is very important for application of the focusing algorithms to the measured data. Fig. 10 illustrates a resolution capability of the array. Two metal discs of 5-cm diameter placed on the ground surface and separated by 7 cm between the edges are completely resolved. The minimum intensity between two images is more than 20 dB

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Fig. 10. Image of two metal discs with 5-cm diameter separated by 7 cm. Image intensity is in a linear scale.

The approach has been verified experimentally by manufacturing a “proof-of-principle” demonstrator based on the timedomain technology. The application in mind is humanitarian demining. The radar possesses a bandwidth of approximately 3.56 GHz and the operational band starts at 240 MHz. This results in a combination of fine down-range resolution with sufficient penetration into the ground for detection of buried targets. Using the near-field footprint formation in the cross-scan direction and synthetic aperture focusing in the scan direction, the radar provides 3-D focused images of the near-field area. The maximal scanning speed of the system is 148 km/h, which is almost two orders of magnitude higher than by existing systems. The cross-range resolution of the images is approximately 5 cm, which is sufficient for an application such as humanitarian demining. The swath of these images is approximately 84 cm. A combination of the above-mentioned figures-of-merits allows us to discuss a technological breakthrough in the field. The first experimental results have shown that the system successfully images AP mines. In the next stage of the research program, dedicated software for image processing, localization, and classification of targets will be embedded into the system. The developed approach can be successfully used in other applications such as medical imaging, area surveillance, road and runway inspection, and others. ACKNOWLEDGMENT

Fig. 11. Image of two PMN2 AP landmines buried at a depth of 5 cm in sand. Image intensity is given in decibels.

below the intensity in the centers of the disks. Furthermore, it is still possible to resolve the discs for 5-cm separation at a 6-dB level. Fig. 11 demonstrates a capability of the array to image buried landmines. Two PMN-2 landmines were buried in dry sand at a depth of 5 cm. One mine is in the middle of the imaging strip, the other is at the right-most position. Both targets can be clearly seen. Despite that the landmines are identical, their images look different due to the difference in energy received from a target under the transmit antenna and from a target that is close to the array’s edge. VII. CONCLUSIONS A novel approach for an array-based imaging radar has been proposed. The approach comprises the system design with a single transmitter and multichannel receiver, design of a receive antenna array, and selection of a special waveform fired by the pulse generator. The proposed approach aims at realization of fast 3-D imaging of subsurface by 1-D mechanical scanning over it. Drastic improvement in the scanning speed over the existing systems should be gained by electronic steering of the receive antenna footprint in a cross-scan direction. At the same time, the suggested approach considerably simplifies the radar sensor in general and its antenna system compared to the known array GPR for detection of AP mines, e.g., [4].

The authors would like to thank Dr. B. Levitas, GeoZondas Ltd., Vilnius, Lithuania, for fruitful cooperation and help during the development of the radar electronics and J. H. Zijderveld, P. Hakkaart, and M. van der Wel, all with the Delft University of Technology, Delft, The Netherlands, for their technical assistance during antenna measurements, system assembling and testing. REFERENCES [1] J. McDonald et al., “Alternatives for landmine detection,” Rand Corporation, Arlington, VA, 2003. [2] J. McFee et al., “A multisensor, vehicle-mounted, teleoperated VMine detector with data fusion,” in Proc. SPIE Detection and Remediation Technol. for Mines and Minelike Targets III, 1998, vol. 3392, pp. 1082–1093. [3] R. B. Cosgrove, P. Milanfar, and J. Kositsky, “Trained detection of buried mines in SAR images via the deflection-optimal criterion,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 11, pp. 2569–2575, Nov. 2004. [4] R. J. Chignell and M. Hatef, “LOTUS—A real time integrated sensor suite for anti-personnel mine detection, incorporating the MINEREC GPR,” in Proc. 10th Int. Ground Penetrating Radar Conf., Delft, The Netherlands, Jun. 21–24, 2004, pp. 665–668. [5] E. E. Ligthart, A. G. Yarovoy, F. Roth, and L. P. Ligthart, “Landmine detection in high resolution 3-D GPR images,” in Proc. MIKON 2004, 2004, vol. 2, pp. 423–426. [6] Q. Zhu and L. M. Collins, “Application of feature extraction methods for landmine detection using Wichmann/Niitek ground penetrating radar,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 1, pp. 81–85, Jan. 2005. [7] T. Wang, O. Sjahputera, J. M. Keller, and P. D. Gader, “Feature analysis for forward-looking landmine detection using GPR,” in Proc. SPIE Detection and Remediation Technol. for Mines and Minelike Targets X, 2005, vol. 5794, pp. 1233–1244. [8] F. Roth, P. van Genderen, and M. Verhaegen, “Radar scattering models for the identification of buried low-metal content landmines,” in Proc. 10th Int. Ground Penetrating Radar Conf., Delft, The Netherlands, Jun. 21–24, 2004, pp. 689–692.

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[9] A. G. Yarovoy et al., “Ground penetrating impulse radar for detection of small and shallow-buried objects,” in Proc. IGARSS’99, Hamburg, Germany, Jun. 1999, vol. 5, pp. 2468–2470. [10] P. van Genderen et al., “A multi-frequency radar for detecting landmines: Design aspects and electrical performance,” in Proc. 31st Eur. Microw. Conf., Oct. 2001, vol. 2, pp. 249–252. [11] A. G. Yarovoy, L. P. Ligthart, A. Schukin, and I. Kaploun, “Polarimetric video impulse radar for landmine detection,” Subsurface Sens. Technol. and Applicat., vol. 3, no. 4, pp. 271–293, Oct. 2002. [12] A. Yarovoy, P. Aubry, P. Lys, and L. Ligthart, “UWB array-based radar for landmine detection,” in Proc. 3rd Eur. Radar Con., Manchester, U.K., Sep. 13–15, 2006, pp. 186–189. [13] A. Yarovoy, P. Aubry, P. Lys, and L. Ligthart, “Array-based GPR for landmine detection,” in Proc. 11th Int. Ground Penetrating Radar Conf., Columbus, OH, Jun. 19–22, 2006, pp. 1–6. [14] R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag., vol. 40, no. 8, pp. 888–905, Aug. 1992. [15] A. G. Yarovoy, A. D. Schukin, I. V. Kaploun, and L. P. Ligthart, “The dielectric wedge antenna,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1460–1472, Oct. 2002. [16] K. D. Palmer, J. H. Zijderveld, and A. G. Yarovoy, “Differential antenna feeding system for short range UWB radar,” in Eur. Radar Conf., Amsterdam, The Netherlands, 2004, pp. 21–24. [17] J. J. Goedbloed, Electromagnetische compabiliteit; analyse en onderdrukking van storproblemen. Norwell, MA: Kluwer, 1990. [18] A. G. Yarovoy, R. V. de Jongh, and L. P. Ligthart, “Ultra-wideband sensor for electromagnetic field measurements in time domain,” Electron. Lett., vol. 36, no. 20, pp. 1679–1680, Sep. 2000. [19] R. J. Chignell, H. Dabis, N. Frost, and S. Wilson, “The radar requirements for detecting anti-personnel mines,” in 8th Int. SPIE Ground Penetrating Radar Conf., May 2000, vol. 4084, pp. 861–866. [20] T. G. Savelyev, L. van Kempen, and H. Sahli, “Deconvolution techniques,” in Ground Penetrating Radar, ser. Radar, Sonar, Navigation, and Avion. 15, D. J. Daniels, Ed., 2nd ed. London, U.K.: IEE Press, 2004, pp. 298–310. [21] J. Groenenboom and A. G. Yarovoy, “Data processing and imaging in GPR system dedicated for landmine detection,” Subsurface Sens. Technol. Applicat., vol. 3, no. 4, pp. 387–402, Oct. 2002. [22] V. Kovalenko, A. Yarovoy, L. P. Ligthart, P. Hakkaart, and J. Rhebergen, “Joint IRCTR/TNO-DS&S measurement campaign for AP-mine detection with VIR GPR,” in Proc. 3rd Int. Adv. Ground Penetrating Radar Workshop, Delft, The Netherlands, May 2005, pp. 31–36.

Alexander G. Yarovoy (M’96–SM’04) received the Diploma degree in radiophysics and electronics (with honors) and the Cand. Phys., Math. Sci., and Dr. Phys. degrees in radiophysics from the Kharkov State University, Kharkov, Ukraine, in 1984, 1987, and 1994, respectively. In 1987, he joined the Department of Radiophysics, Kharkov State University, as a Researcher, and became a Professor in 1997. From September 1994 to 1996, he was a Visiting Researcher with the Technical University of Ilmenau, Ilmenau, Germany. Since 1999, he has been with the International Research Centre for Telecommunications–Transmission and Radar (IRCTR), Delft University of Technology, Delft, The Netherlands, where he coordinates all UWB-related projects. His main research interests is UWB technology and its applications (in particular, UWB radars) and applied electromagnetics (in particular, UWB antennas). Prof. Yarovoy served as the co-chairman and the Technical Program Committee chair for the 10th International Conference on Ground Penetrating Radar (GPR2004), Delft, The Netherlands, and as the secretary of the 1st European Radar Conference (EuRAD’04), Amsterdam, The Netherlands. He was the recipient of a 1996 International Union of Radio Science (URSI) Young Scientists Award and the 2001 European Microwave Week Radar Award for the paper that best advances the state-of-the-art in radar technology.

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Timofey G. Savelyev was born in Frunze, U.S.S.R., in 1974. He received the Dipl.-Eng. degree (cum laude) in radio electronics from the Baltic State Technical University, St. Petersburg, Russia, in 1997, and the Ph.D. degree in electrical engineering from the State University of Aerospace Instrumentation, St. Petersburg, Russia, in 2000. His doctoral research concerned adaptive array radar systems. He has since been involved in the field of GPR landmine detection as a Visiting Scientist with Vrije Universiteit Brussel, Brussels, Belgium, in 2002, and as a research associate with the Centre for Northeast Asian Studies, Tohoku University, Sendai, Japan, from 2003 to 2005. He is currently a Post-Doctorate Researcher with the International Research Centre for Telecommunications–Transmission and Radar (IRCTR), Delft University of Technology, Delft, The Netherlands. His current research interests include GPR landmine detection, UWB array radars, UWB signal processing and analysis, and radar imaging.

Pascal J. Aubry was born in Fontenay-aux-Roses, France, on the March 8, 1969. He received the D.E.S.S. degree in electronics and automatics from the Université Pierre et Marie Curie (Paris 6), Paris, France, in 1993. Following his military service in the Air Force in 1994, he was a Young Graduate Trainee with the European Space Research and Technology Centre (ESTEC), in 1996, where he was involved with antenna measurements. Since 1997, he has been with the International Research Centre for Telecommunications–Transmission and Radar (IRCTR), Delft University of Technology (TUD), Delft, The Netherlands. His interests include antenna measurement techniques in the frequency and time domains and GPR system testing.

Pidio Ekoue Lys (M’05) was born in Kara, Togo, in 1978. He received the B.S. degree in applied physics and M.S. degree in electronics for telecommunications from Paris VI University, Paris, France in 2002 and 2004, respectively. In 2004, he was a Visiting Scholar with the Telecom-Paris School. He developed a UWB antenna for pulsed applications. He then joined Completude-Objectif Maths, Paris, France, as Educational Support in Mathematics and Physical Sciences for high school students from 2003 to 2005. In 2005, he joined the International Research Centre for Telecommunications–Transmission and Radar (IRCTR), Delft University of Technology, Delft, The Netherlands, where he is currently an Engineer. His research interests include antenna system design and signal processing for polarimetric radar.

Leo P. Ligthart (M’94–SM’95–F’02) was born in Rotterdam, The Netherlands, on September 15, 1946. He received the Engineer’s degree (cum laude) and Doctor of Technology degree from the Delft University of Technology, Delft, The Netherlands, in 1969 and 1985, respectively, the Doctorate degree (honoris causa) from the Moscow State Technical University of Civil Aviation, Moscow, Russia, in 1999, and the Doctorate degree (honoris causa) from Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia, in 2001. Since 1992, he has held the Chair of Microwave Transmission, Radar and Remote Sensing with the Department of Information Technology and Systems, Delft University of Technology. In 1994, he became Director of the International Research Center for Telecommunications and Radar, Delft University of Technology. His principal areas of specialization include antennas and propagation and radar and remote sensing. He has also been active in satellite, mobile, and radio communications.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

Composite Right/Left-Handed Metamaterial Transmission Lines Based on Complementary Split-Rings Resonators and Their Applications to Very Wideband and Compact Filter Design Marta Gil, Student Member, IEEE, Jordi Bonache, Member, IEEE, Joan García-García, Member, IEEE, Jesús Martel, Member, IEEE, and Ferran Martín, Member, IEEE

Abstract—In this paper, we discuss in detail the transmission characteristics of composite right/left-handed transmission lines based on complementary split-rings resonators. Specifically, the necessary conditions to obtain a continuous transition between the left- and right-handed bands (balanced case) are pointed out. It is found that very wide bands can be obtained by balancing the line. The application of this technique to the design of very wideband and compact filters is illustrated by means of two examples. One of them is based on the hybrid approach, where a microstrip line is loaded with complementary split-rings resonators, series gaps, and grounded stubs; the other one is a bandpass filter, also based on a balanced line, but in this case, by using only complementary split-rings resonators and series gaps (purely resonant-type approach). As will be seen, very small dimensions and good performance are obtained. The proposed filters are useful for ultra-wideband systems. Index Terms—Complementary split-rings resonators, metamaterials, microwave filters, transmission lines.

I. INTRODUCTION OMPLEMENTARY split-rings resonators were introduced by Falcone et al. in 2004 as new resonant particles for the synthesis of metamaterials with negative effective permittivity [1]. It was first demonstrated that by etching these elements in the ground plane of a microstrip line, the structure was able to inhibit signal propagation in the vicinity of their resonance frequency. Later, the first left-handed line based on complementary split-rings resonators was implemented by

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Manuscript received September 29, 2006; revised February 20, 2007. This work was supported by Spain–Ministerio de Educación y Ciencia under Project Contract TEC2004-04249-C02-01, by the Seiko Epson Corporation, by the European Union under the Network of Excellence METAMORPHOSE, and by the Catalan Government under the Centre d’Investigació en Metamaterials per a la Innovació en Tecnologia Electrònica i de Comunicacions. The work of M. Gil was supported by the Ministerio de Educación y Ciencia under Formación de Profesorado Universitario Grant AP2005-4523. M. Gil, J. Bonache, J. García-García, and F. Martín are with the Grup d’Enginyeria de Microones i Milimètriques Aplicat/Centre d’Investigació en Metamaterials per a la Innovació en Tecnologia Electrònica i de Comunicacions, Departament d’Enginyeria Electrònica, Escola Tècnica Superior d’Enginyeria, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). J. Martel is with the Grupo de Microondas, Departamento de Física Aplicada 2, Escuela Técnica Superior de Arquitectura. Universidad de Sevilla, 41012 Seville, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.897755

etching series capacitive gaps in the conductor strip, above the positions occupied by the complementary split-rings resonators [2]. The series gaps were then responsible for the negative effective permeability of the structure. Thus, by combining these elements (gaps and complementary split-rings resonators), a narrow band with simultaneously negative permittivity and permeability appeared in the vicinity of the resonance frequency of the resonators and, hence, a left-handed behavior in that band. In the design of such structures, the main attention was directed to achieve a left-handed behavior in the desired band. However, these structures also exhibit a forward wave (right-handed) behavior at higher frequencies due to the parasitic elements of the host line. Similar behavior was previously demonstrated by Sanada et al. [3] by simply loading a host transmission line with series gaps and shunt inductors. This nonresonant structure was called a composite right/left-handed line to clearly point out its composite (left- and right-handed) nature. The main goal of this paper is to study the left- and right-handed transmission in complementary split-rings resonators loaded metamaterial transmission lines, and to obtain practical implications for the design of wideband and ultra-wideband (UWB) pass filters based on them. Two types of metamaterial transmission lines will be considered, which are: 1) those lines where complementary split-rings resonators are simply combined with series gaps (purely a resonant-type approach [2]) and 2) lines including complementary split-rings resonators, series gaps, and grounded stubs in the unit cell (hybrid approach [4]). As will be shown in Section II, the purely resonant metamaterial transmission line can be modeled by means of a lumped element circuit model, which is very similar to that circuit that describes the composite right/left-handed transmission line implemented by series capacitors and shunt inductors [3]. With regard to the hybrid approach, it provides a further degree of flexibility due to the presence of additional elements (i.e., grounded stubs acting as shunt connected inductors). It will be shown that by balancing the lines, a continuous transition between the left- and right-handed band results, and wide or ultrawide passbands appear. For the hybrid approach, it will be shown that these wide bands can be allocated either below or above the typical transmission zero, intrinsic to the presence of complementary split-rings resonators. In section III, two prototype device examples are given. One is a three-stage purely resonant balanced structure, which ex-

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Fig. 1. Topologies of the: (a) resonant-type and (b) hybrid left-handed cells and equivalent circuits (c) and (d). Ground plane metal is depicted in gray, whereas the upper metal level is depicted in black. In the topology of (b), two series gaps are included and shunt stubs are grounded through metallic vias.

hibits a wide transmission band. This structure is very useful as a high-pass structure (though transmission is limited at high frequencies). The second one is a bandpass filter, which was presented in [5], and is based on the hybrid approach. Details on the design procedures are given here for the first time. As will be seen, to describe the behavior of such a structure, it is necessary to include in the lumped element circuit model the necessary elements to account for the second resonance of complementary split-rings resonators. It is the first time that the first and second resonance of a complementary split-rings resonator are used in a circuit. The main conclusions of this study are highlighted in Section IV. It is clear that composite right/left-handed transmission lines based on complementary split-rings resonators are of actual interest for the synthesis of microwave filters with wide bands or UWBs. The small dimensions of the unit cells make this approach very attractive for the fabrication of low-cost and miniature modules where such filters are necessary. II. TOPOLOGY, CIRCUIT MODEL, AND ANALYSIS OF COMPOSITE RIGHT/LEFT-HANDED COMPLEMENTARY SPLIT-RINGS RESONATOR-BASED TRANSMISSION LINES The layouts of the purely resonant and hybrid left-handed cells, as well as their corresponding lumped-element equivalent-circuit models are depicted in Fig. 1. The purely resonant unit cell consists of a microstrip line with a series gap etched in the strip

and a complementary split-rings resonator printed in the ground plane. The structure is described by the circuit model shown in Fig. 1(c), where the complementary split-rings resonators are modeled by means of the resonant tank formed by the and the inductance , whereas the gaps are capacitance described by means of the capacitance . is the inductance of the line, whereas models the electric coupling between the line and complementary split-rings resonators. The topology corresponding to the hybrid line [see Fig. 1(b)] is analogous to that of Fig. 1(a), but with the addition of two shunt stubs. These elements are described through a shunt connected inductance , as can be appreciated in the model of Fig. 1(d). These circuit models are indeed simplified versions of the more general circuit model reported in [6]. In the model reported in [6], inter-resonator’s coupling and the effects of the line capacitance corresponding to that portion of the host line outside the influence of the rings are also considered. However, as was discussed in [6], coupling between neighboring complementary split-rings resonators can be neglected if circular geometries are used (as is the case in this study), and the effects of the line capacitance can also be neglected, unless the complementary split-rings resonators are substantially separated. According to these comments, the circuit models depicted in Fig. 1 are justified (these models and the layouts have been previously reported [4], but they are reproduced here for completeness of this paper). An inspection of these circuit models reveals that there are regions where the series reactance and shunt susceptance are

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both negative (left-handed regions), zones where they are both positive (right-handed bands), and intervals where they have opposite signs (forbidden bands). These topologies and the corresponding circuit models have been previously used by the authors [2], [4], [7], but not for the synthesis of bandpass filters with wide bands based on the composite behavior of the lines and balanced designs. Let us now analyze in detail the transmission characteristics of these composite right/left-handed transmission lines on the basis of their equivalent-circuit models. This analysis can be carried out through the Bloch theory. The phase shift per cell and the characteristic impedance , which are the key parameters for microwave circuit design, are given by [8] (1) (2) and are the series and shunt impedances, respecwhere tively, for the circuits of Fig. 1. Propagation is allowed in that are real numbers. The cells can regions where both and be designed to be either unbalanced or balanced [9]. In the first case, the series impedance and shunt admittance are null at different frequencies. Conversely, the series and shunt resonance frequencies are identical for balanced designs. Thus, for unbalanced structures, a frequency gap appears between the following frequencies: (3) (4) where and are the series and shunt resonance frequencies, respectively, and the structure is left/right-handed below/above that gap. For the balanced case, these frequencies are identical, , and the transition between the leftnamely, and right-handed band is continuous (i.e., it takes place at the transition frequency ). For the purely resonant complementary split-rings resonatorloaded line, the general expression providing the phase shift per cell and the characteristic impedance are given by

(5)

(6) and these expressions are simplified to (7) (8)

. Thus, for the for the balanced case, where balanced design, signal transmission changes from left- to righthanded at , whereas the lower limit of the left-handed region and the upper limit of the right-handed band are obtained by forcing (8) to be zero (the calculation is tedious and, hence, it is not reproduced here). As occurs in balanced lines implemented through nonreso(although the nant elements, the group velocity is finite at phase constant is null at that frequency). The characteristic impedance is null at the extremes of the propagation band and it varies smoothly in the vicinity of the transition frequency. However, for the cells described by the circuit of Fig. 1(c), is maximized above . This does not represent any limitation of these complementary split-rings resonator based lines, as compared to those implemented through nonresonant elements. Diagrams corresponding to the dispersion relation and Bloch impedance for the balanced case in purely resonant composite right/left-handed transmission lines are thus similar to those of composite right/left-handed lines based on nonresonant elements [9]. However, a transmission zero located near the lower edge of the transmission band and given by (9) is present in purely resonant metamaterial transmission lines (this transmission zero is at the origin for nonresonant composite right/left-handed lines). Let us now consider the hybrid cell, described by the circuit model of Fig 1(d). In this case, the analysis is more complicated . Specifically, there due to the presence of the inductance are three relevant frequencies relative to the shunt impedance: namely, two frequencies that null the corresponding admittance, and a transmission zero frequency in between [also given by (9)]. Let us now consider the balanced case, which is of particular interest to achieve broadband structures. Since there are two shunt resonances, there are actually two alternatives to achieve the balanced condition: that where the series resonance is identical to the lower resonance frequency of the shunt impedance, and that case where the higher resocoincides with the series nance of the shunt impedance resonance. Rather than obtaining the analytical expressions for the dispersion relation and characteristic impedance, it is more useful to illustrate the two balance solutions mentioned by means of a representation of the characteristic impedance and dispersion diagram (see Fig. 2). For both alternatives, the characteristic impedance and phase constant in the vicinity of the transition frequency exhibit similar behavior to that of the purely resonant-type approach. However, there is an additional , it appears allowed band. In the former case above the transition frequency, and this band exhibits forward , this additional wave propagation. Conversely, for allowed band is left-handed and it is located below the transition frequency. Nevertheless, these bands are typically very narrow and they are not useful in practical applications. The transmission zeros are more valuable since they can be used to suppress undesired harmonics that may appear as consequence of parasitic resonances [7], or to enhance frequency selectivity at the lower edge of the transmission band.

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Fig. 2. Representation of the series, shunt, and characteristic impedance for the hybrid composite right/left-handed line model for the two situations described in the text. (a) Balanced case with the series resonance identical to the lower resonance frequency of the shunt impedance; electrical parameters are: L = 20 nH, C = 1:28 pF, C = 2 pF, C = 10 pF, L = 0:6 nH, and L = 12 nH. (b) Balanced case with the series resonance identical to the higher resonance frequency of the shunt impedance electrical parameters are: L = 20 nH, C = 0:28 pF, C = 2 pF, C = 10 pF, L = 0:6 nH, and L = 12 nH. (c) and (d) depict the dispersion diagrams corresponding to the cases considered in (a) and (b), respectively. In (a) and (b), only the real part of Z is depicted. For Z and Z , only the absolute value has been represented (these impedances are purely reactive).

In Section III, these structures are applied to the design of wideband microwave filters. Actually, the presented models [see Figs. 1(c) and (d)] are able to explain device behavior up to a limited frequency. At the upper limit of the composite right/lefthanded transmission band, either the lumped-element approximation fails, or it is necessary to include additional elements to the models in order to account for the second resonance frequency of the complementary split-rings resonators. This latter aspect will be discussed in Section III since it is fundamental to explain the characteristics of one of the presented filters. III. DESIGN EXAMPLES To illustrate the potentiality of balanced composite right/lefthanded lines loaded with complementary split-rings resonators to the design of wideband filters, two prototype device examples are provided. The first one is a three-stage bandpass filter based on the purely resonant-type approach. The layout of the filter is depicted in Fig. 3. The structure is roughly balanced (contrary to previous bandpass filters based on similar topologies [4]), as revealed by the dispersion diagram depicted in Fig. 4, which has been inferred from the simulated (through Agilent Momentum) and measured -parameters of a single cell. This figure points out a wide band for signal transmission. To achieve the quasi-balance condition, optimization has been required; namely, the interdigital capacitors are tuned until the series resonance (given by and ) coincides with the shunt resonance tank) to a good approximation. Under perfect bal( is located in the ance, the input port is matched and, hence,

Fig. 3. Layout of the filter formed by three balanced purely resonant cells. The metallic parts are depicted in black in the top layer, and in gray in the bottom layer. The rings are etched on the bottom layer. Dimensions are: total length l = 55 mm, linewidth W = 0:8 mm, external radius of the outer rings r = 7:3 mm, ring width c = 0:4 mm and ring separation d = 0:2 mm; the interdigital capacitors are formed by 28 fingers separated 0.16 mm.

center of the Smith chart. In practice, perfect balance (or perfect matching) is not actually achieved and we adopt that solureaches the closest position to the center of the tion where (possible reasons for this imperfect balance Smith chart at will be discussed later in reference to the next example). The design of the filter to satisfy any given specifications can be done with the help of the electrical circuit model of the unit cell and the parameter-extraction method published in [10]. Nevertheless, this has not been the case and we simply have designed a balanced line to demonstrate the possibilities of the approach. We have simulated the frequency response of two-, three-, and four-stage structures, and we have fabricated the device with three complementary split-rings resonators (the Rogers RO3010 substrate has been used with thickness mm and dielectric constant ) in order to obtain measured data for one case. These results are depicted in Fig. 5. The structure exhibits high-frequency selectivity at the

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Fig. 4. Dispersion diagram for the structure of Fig. 3.

Fig. 6. (a) Simulated reflection and transmission coefficient of a single balanced cell corresponding to the filter shown in Fig. 10. (b) Phase of S . Backward wave propagation corresponds to positive values of phase in (b). Hence, the left-handed band is extended from 3.3 up to 5.5 GHz.

Fig. 5. Simulated: (a) insertion and (b) return losses of the three filters composed of two-, three-, and four-stages identical to those of Fig. 3. The measured frequency response of the fabricated prototype (Fig. 3) is also depicted.

lower band edge and controllable rejection below the cutoff frequency. The lower limit of the passband can be accurately controlled since the electrical model perfectly describes the frequency response up to regions well beyond the transition frequency. However, in general, the model does not properly fit the measured or simulated (full-wave electromagnetic) device responses in the vicinity of the upper band edge, as has been previously discussed relative to unbalanced structures [11] (the limitation is related to the fact that the lumped-element model is valid only in a limited frequency interval). Thus, though the device is by nature a bandpass structure, it must be actually considered as a high-pass filter, useful to eliminate interfering signals present below the lower limit of the band (conventional microwave high-pass filters do also exhibit rejection at high frequencies). The second example is a bandpass filter, which was presented in [5], and subjected to the following specifications: active area below 1 cm , bandwidth covering the 4–6-GHz range or wider,

at least 80-dB rejection at 2 GHz, in-band ripple lower than 1 dB, and group-delay variation smaller than 1 ns. This device has been designed by means of the hybrid approach [see Fig. 1(b)], which has been demonstrated to offer small size solutions in moderate or narrow bandpass filters [4], [7]. In this case, the with thickness substrate is the Rogers RO3010 mm to achieve the required rejection at 2 GHz (as is explained in [5]). For the design of the unit cell, we have chosen the hybrid complementary split-rings resonator loaded line with the series resonance frequency as close as possible to the higher . This gives a resonance of the shunt impedance transmission zero below the passband of interest, which is useful to obtain high rejection, as required, in the vicinity of 2 GHz. Although below that transmission zero there is a left-handed passband present (see Fig. 2), it is very narrow for the designed structures and its effects are irrelevant (as will be seen). To achieve the required specifications, the purpose is to design a balanced unit cell exhibiting a flat response covering the required bandwidth. From simulation, the required number of stages to achieve 80-dB rejection level at 2 GHz can be determined. With the help of the equivalent-circuit model of Fig. 1(d), we have designed a balanced unit cell satisfying the previous requirement. The simulated frequency response of this unit cell is depicted in Fig. 6. From the phase response, it is clearly seen that backward wave transmission is switched to forward wave transmission within the band. Above the passband of interest, which extends approximately from 3 to 10 GHz, there is an additional transmission zero (between 12 GHz–13 Hz), which is not explained by means of the equivalent-circuit model [see Fig. 1(d)] for the situation that we

GIL et al.: COMPOSITE RIGHT/LEFT-HANDED METAMATERIAL TRANSMISSION LINES BASED ON COMPLEMENTARY SPLIT-RINGS RESONATORS

Fig. 7. Equivalent-circuit model of the unit cell of the considered filter (Fig. 10) that includes the effects of the second resonance frequency of the complementary split-rings resonators. For coherence, we have added a sub-index that indicates the resonance order that each shunt branch models.

Fig. 8. Electrical simulation of the circuit of Fig. 7. Parameters are: L = 2:6 nH, C = 0:2 pF, L = 5:4 nH, C = 305 pF, C = 0:65 pF, = 0:55 nH, C = 0:21 pF, C = 0:32 pF, and L = 0:27 nH. L

are considering . In addition, a transmission peak is also present at 13.7 GHz. This behavior can be explained by considering the effects of the second resonance frequency of the complementary split-rings resonators. Namely, these particles exhibit several resonance frequencies [12], [13]. The first one, the quasi-static resonance, is the resonance of interest for most of the applications of complementary split-rings-resonator-based circuits. However, at higher frequencies, there are additional resonance frequencies (dynamic), which may play a role under certain circumstances. This is the case in the current design. Namely, if we add to the circuit of Fig. 1(d), an additional parallel branch to account for the second resonance of the complementary split-rings resonator, the transmission zero, as well as the transmission peak above it, are perfectly explained. The equivalent circuit that includes the effects of the second resonance is depicted in Fig. 7. The parameters of the new parallel branch have been adjusted to obtain the behavior observed in Fig. 6. From the electrical simulation of this circuit model (obtained by means of Agilent ADS), we obtain a frequency response (see Fig. 8), which is qualitatively very similar to that depicted in Fig. 6. It is interesting to mention , is that the coupling capacitance of this new branch, i.e., very small. This is consistent with the fact that electric coupling is very weak at the second resonance of complementary split-rings resonators [12], [13]. We would like to clarify that optimization of the filter unit cell has been done directly from

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layout due to the difficulty of handling with such a large number of parameters. Rather than obtaining an accurate description of the frequency response of the filter unit cell through the circuit model, our intention has been to justify the presence of the second resonance of the complementary split-rings resonators, which plays a fundamental role in order to obtain a sharp cutoff above the passband of interest, and this is clear from the qualitative comparison of Figs. 6 and 8. The position of the two transmission zeros can be controlled with the dimensions of the complementary split-rings resonators and gaps. From the circuit model of Fig. 7, both transmission zeros are given by (9) by by the corresponding reactive elements replacing , , and , and for the lower transmission zero and , ( , , and for the upper transmission zero). Obviously, the small frequency value of the first transmission zero can be only explained by a large coupling capacitance , which has been estimated to be in the vicinity of 300 pF (see caption of Fig. 7). This capacitance is much larger than the coupling capacitance corresponding to the second resonance, as expected since the electric excitation of the second resonance is very weak. Howis not intuitive and we have verified ever, the large value of it from the analysis of the gap pair alone (i.e., by excluding the complementary split-rings resonator and removing the vias). From the simulated -parameters of this structure, we have inferred the capacitance values of the well-known -model of the gap, and through -T transformation, the capacitances of the T-model have been inferred. These capacitances are in and , qualitative agreement with the capacitor values of given in the caption of Fig. 8 (perfect agreement is not expected due to the effects of the complementary split-rings resonators and vias). Hence, the element values of the circuit model given in the caption of Fig. 8 are reasonable, including the large value of the capacitance . The return loss of the structure (Fig. 6) exhibits two reflection zeros. To explain the origin of such reflection zeros, the characteristic impedance of the structure (inferred from the simulated -parameters according to standard formulas [8]) has been obtained (Fig. 9). According to this illustration, the reflection zero located at 7.5 GHz is due to impedance matching (the characteristic impedance is 50 at that frequency). Two additional reflection zeros would be expected: one of them at that frequency where the impedance again takes the value of 50 ; the other one at the transition frequency (phase matching). Since these frequencies do almost coincide, the two reflection zeros merge. Actually this second reflection zero is obscured by the fact that the device is not exactly balanced, as Fig. 9 reveals. Nevertheless, the effect of this imperfect balance is not appreciable in the frequency response of the structure, which is flat in the region of interest. We would like to clarify that the imperfect balance is not related to an improper design. The balance condition is achieved when the series resonance coincides with the . This resonance roughly coshunt resonance identified as incides with the first (quasi-static) resonance of the complemen). Howtary split-rings resonator (given by the tank ever, this quasi-static resonance can also be excited magnetically, as was discussed in [12], through the component of the magnetic field contained in the plane of the particle. This magnetic coupling is weak, but it may suffice to prevent “perfect”

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Fig. 9. Characteristic impedance inferred from electromagnetic simulation of the unit cell of the filter depicted in Fig. 10.

Fig. 10. Layout of the fabricated filter, formed by cascading four balanced hybrid cells. The metallic parts are depicted in black in the top layer, and in gray in the bottom layer. The rings are etched on the bottom layer. The dashed rectangle has an area of 1 cm . Dimensions are: linewidth W = 0:126 mm, external radius of the outer rings r = 1:68 mm, rings width c = 0:32 mm and rings separation d = 0:19 mm; inductor width is 0.10 mm and the distance between the metals forming the gap is 0.4 mm. From [5].

balance since its effect is to open the series branch just at the transition frequency [14]. This may explain the behavior of the characteristic impedance in the vicinity of 5.5 GHz. The fact that perfect balance has not been achieved in spite of the fine tuning of the series elements support this argument. Nevertheless, this aspect requires further study, which is out of the scope of this paper. The fabricated filter is depicted in Fig. 10. Four stages have been enough to achieve the required rejection at 2 GHz. Fig. 11 illustrates the simulated (through Agilent Momentum) and measured (by means of the Agilent 8720ET Vector network analyzer) frequency response of the filter. Finally, Fig. 12 depicts the simulated and measured group delay. Reasonable agreement between simulation and experiment has been achieved. The size cm , as well as the measured results indicate that specifications are satisfied, with the exception of ripple, which is slightly higher than 1 dB. This excess of ripple is attributed to fabrication related tolerances. Nevertheless, the combination of size and performance for this periodic filter is a relevant aspect to highlight. A very wide measured bandwidth (3.5–10 GHz) has been achieved with high selectivity at both band edges, and the first spurious located at 17 GHz (measurement). This has

Fig. 11. Simulated (dashed line) and measured (solid line) frequency response of the fabricated filter depicted in Fig. 10. Measured in-band losses are better than 3 dB in the transmission band. From [5].

Fig. 12. Simulated (dashed line) and measured (solid line) group delay of the filter of Fig. 10.

been achieved by means of a planar implementation with an area clearly below 1 cm . Group-delay variation is also very small in the allowed band ( 1 ns). A similar filter has been reported in [15] by loading a composite right/left-handed line with capacitively coupled resonators. The equivalent circuit of the unit cell is similar to the circuit of Fig. 7. The main difference is the presence of the inductive element , which is absent in the structure of [15] and the two coupling capacitances, and , responsible for the presence of two transmission zeros (only one coupling capacitance is present in the structure reported in [15]). The performance of the filters is similar, though dimensions are clearly smaller in our design. The effect of the complementary split-rings resonator in the filter of Fig. 10 can be easily evaluated by means of full wave electromagnetic simulations (not shown in this paper). If the complimentary split-rings resonators are removed from the structure, the frequency response dramatically changes (both the passband and transmission zeroes disappear). This fact demonstrates that the shunt susceptance of the unit cells is strongly affected by the presence of the complimentary split-rings resonators. In other words, though the series gaps and shunt stubs do produce a composite right/left-handed behavior by themselves, complimentary split-rings resonators are clearly needed to achieve the required performance. In the opinion of the authors, these UWB metamaterial filters based on balanced cells, including both the purely resonant and hybrid models, are of interest in practical applications since

GIL et al.: COMPOSITE RIGHT/LEFT-HANDED METAMATERIAL TRANSMISSION LINES BASED ON COMPLEMENTARY SPLIT-RINGS RESONATORS

they seem to be competitive in terms of dimensions and performance. Other UWB pass filters based on other approaches, recently proposed, can be found in [16]–[19]. IV. CONCLUSION In conclusion, it has been demonstrated that resonant type metamaterial transmission lines based on complementary split-rings resonators exhibit a composite right/left-handed behavior that is useful for the synthesis of compact size and high-performance planar filters in terms of bandwidth. Two models have been analyzed, which are 1) the purely resonant approach, where complementary split-rings resonators are simply combined with series gaps and 2) the hybrid model, where grounded stubs are added to the previous model. The key point to achieve wideband or UWB is the design of balanced cells, where the transition between the left- and the right-handed band is continuous. Two prototype device examples have been reported in order to illustrate the design procedure and the achievable results by means of the two mentioned models. It has been found that rejection and cutoff at the lower edge of the band are perfectly controllable for the purely resonant metamaterial filters, whereas at high frequencies, the cutoff is not easily controllable due to the limitations of the electric model to properly describe device behavior at high frequencies. Nevertheless, a fractional bandwidth higher than 100% has been measured in the fabricated prototype. These filters are useful to eliminate interfering signals present below the frequency region of interest. For the metamaterial filter based on the hybrid approach, we have roughly obtained the required specifications and size. In this case, device behavior has been explained also including the frequency response above the passband of interest. To this end, it has been necessary to include in the electric model of the unit cell the effects of the second resonance frequency of the complementary split-rings resonators. This is the first time that the second resonance of complementary split-rings resonators is used in the design of a filter (a symmetric frequency response for the final filter has been obtained). To our knowledge, the combination of dimensions and performance for the filter based on the hybrid cell is unique. These results demonstrate the potentiality of metamaterial based filters in applications requiring wide band and UWBs.

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[5] J. Bonache, J. Martel, I. Gil, M. Gil, J. García-García, F. Martín, I. Cairó, and M. Ikeda, “Super compact (< 1 cm ) bandpass filters with wide bandwidth and high selectivity at C -band,” in Proc. Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 599–602. [6] I. Gil, J. Bonache, M. Gil, J. García-García, F. Martín, and R. Marqués, “Accurate circuit analysis of resonant type left handed transmission lines with inter-resonator’s coupling,” J. Appl. Phys., vol. 100, Oct. 2006, Paper 074908-1-10. [7] J. Bonache, I. Gil, J. García-García, and F. Martín, “Novel microstrip bandpass filters based on complementary split rings resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 265–271, Jan. 2006. [8] D. M. Pozar, Microwave Engineering. Reading, MA: Addison-Wesley, 1990. [9] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. New York: Wiley, 2005. [10] J. Bonache, M. Gil, I. Gil, J. Garcia-García, and F. Martín, “On the electrical characteristics of complementary metamaterial resonators,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 10, pp. 543–545, Oct. 2006. [11] I. Gil, J. Bonache, M. Gil, J. García-García, and F. Martín, “Left handed and right handed transmission properties of microstrip lines loaded with complementary split rings resonators,” Microw. Opt. Technol. Lett., vol. 48, no. 12, pp. 2508–2511, Dec. 2006. [12] J. D. Baena, J. Bonache, F. Martín, R. Marqués, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García, I. Gil, and M. Sorolla, “Equivalent circuit models for split ring resonators and complementary split rings resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1451–1461, Apr. 2005. [13] J. García-García, F. Martín, J. D. Baena, and R. Marqués, “On the resonances and polarizabilities of split rings resonators,” J. Appl. Phys., vol. 98, pp. 033103-1–033103-9, Sep. 2005. [14] F. Martín, F. Falcone, J. Bonache, R. Marqués, and M. Sorolla, “A new split ring resonator based left handed coplanar waveguide,” Appl. Phys. Lett., vol. 83, pp. 4652–4654, Dec. 2003. [15] H. V. Nguyen and C. Caloz, “Broadband highly selective bandpass filter based on a tapered coupler resonator (TCR) CRLH structure,” in Proc. Eur. Microw. Assoc., Mar. 2006, vol. 2, pp. 44–51. [16] L. Zhu, S. Sun, and W. Menzel, “Ultra wide band (UWB) bandpass filters using multiple mode resonator,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 796–708, Nov. 2005. [17] H. N. Shaman and J.-S. Hong, “A compact ultra-wideband (UWB) bandpass filter with transmission zero,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 603–605. [18] D. Packiaraj, M. Ramesh, and A. T. Kalghatgi, “Broad band filter for UWB communications,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 606–608. [19] J. García-García, J. Bonache, and F. Martín, “Application of electromagnetic bandgaps (EBGs) to the design of ultra wide bandpass filters (UWBPFs) with good out-of-band performance,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4136–4140, Dec. 2006.

REFERENCES [1] F. Falcone, T. Lopetegi, J. D. Baena, R. Marqués, F. Martín, and M. Sorolla, “Effective negative- " stop-band microstrip lines based on complementary split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 280–282, Jun. 2004. [2] F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, R. Marqués, F. Martín, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett., vol. 93, Nov. 2004, 197401. [3] A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left handed transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 2, pp. 68–70, Feb. 2004. [4] J. Bonache, M. Gil, I. Gil, J. García-García, and F. Martín, “Limitations and solutions of resonant-type metamaterial transmission lines for filter applications: The hybrid approach,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 2006, pp. 939–942.

Marta Gil (S’07) was born in Valdepeñas (Ciudad Real), Spain, in 1981. She received the Physics degree from the Universidad de Granada, Granada, Spain, in 2005, and is currently working toward the Ph.D. degree in subjects related to metamaterials and microwave circuits at the Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain. She studied for one year at the Friedrich Schiller Universität Jena, Jena, Germany. She is currently with the Universitat Autònoma de Barcelona under the framework of METAMORPHOSE. Ms. Gil was the recipient of a Formación de Profesorado Universitario Research Fellowship (Reference AP2005-4523) presented by the Spanish Government (MEC).

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Jordi Bonache (S’05–M’06) was born in Cardona (Barcelona), Spain, in 1976. He received the Physics and Electronics Engineering degrees and Ph.D. degree in electronics engineering from the Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain, in 1999, 2001, and 2007, respectively. In 2000, he joined the High Energy Physics Institute of Barcelona (IFAE), where he was involved in the design and implementation of the control and monitoring system of the MAGIC telescope. In 2001, he joined the Department d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, where he is currently an Assistant Professor. His research interests include active and passive microwave devices and metamaterials.

Joan García-García (M’05) was born in Barcelona, Spain, in 1971. He received the Physics degree and Ph.D. degree in electrical engineering from the Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain, in 1994 and 2001, respectively. He then became a Post-Doctoral Research Fellow with the Institute of Microwaves and Photonics, The University of Leeds, Leeds, U.K., under the INTERACT European Project. In 2002, he was a Post-Doctoral Research Fellow with the Universitat Autònoma de Barcelona, under the Ramon y Cajal Project of the Spanish Government. In November 2003, he become an Associate Professor of electronics with the Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona.

Jesús Martel (M’07) was born in Seville, Spain, in 1966. He received the Licenciado and Doctor degrees in physics from the University of Seville, Seville, Spain, in 1989 and 1996, respectively. Since 1992, he has been with the Department of Applied Physics II, University of Seville, where, in 2000, he became an Associate Professor. His current research interest is focused on the numerical analysis of planar transmission lines, modeling of planar microstrip discontinuities, design of passive microwave circuits, microwave measurements, and artificial media.

Ferran Martín (M’05) was born in Barakaldo (Vizcaya), Spain, in 1965. He received the B.S. degree in physics and Ph.D. degree from the Universitat Autònoma de Barcelona (UAB), Bellaterra (Barcelona), Spain, in 1988 and 1992, respectively. From 1994 to 2006, he was an Associate Professor of electronics with the Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, and since January 2007, he has been a Full Professor of electronics. He is the Head of the Microwave and Millimeter Wave Engineering Group, UAB. He is a partner of the Network of Excellence (NoE), European Union METAMORPHOSE. Since January 2006, he has been the Head of Centre d’Investigació en Metamaterials per a la Innovació en Tecnologia Electrònica i de Comunicacions (CIMITEC), a Research Center on metamaterials funded by the Catalan Government, which has been created for technology transfer on the basis of metamaterial concepts. He has authored or coauthored over 200 technical conference, letter and journal papers. He is currently coauthoring the monograph on metamaterials Metamaterials with Negative Parameters: Theory, Design and Microwave Applications. He has been Guest Editor for two Special Issues on metamaterials in two international journals. He is member of the Editorial Board of the IET Proceedings on Microwaves Antennas and Propagation. He has filed several patents on metamaterials and has headed several development contracts. In recent years, he has been involved in different research activities including modeling and simulation of electron devices for high-frequency applications, millimeter-wave and terahertz generation systems, and the application of electromagnetic bandgaps to microwave and millimeter-wave circuits. He is currently also very active in the field of metamaterials and their application to the miniaturization and optimization of microwave circuits and antennas. Dr. Martín has organized several international events related to metamaterials, including two workshops of the 2005 and 2007 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). He is a member of the Technical Program Committee of the International Congress on Advanced Electromagnetic Materials in Microwave and Optics (Metamaterials). He and the members of CIMITEC were the recipients of the 2006 Duran Farell Prize for technological research.

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Analysis of Electromagnetic Response of 3-D Dielectric Fractals of Menger Sponge Type Elena Semouchkina, Member, IEEE, Yoshinari Miyamoto, Soshu Kirihara, George Semouchkin, and Michael Lanagan, Member, IEEE

Abstract—Experimental studies and finite-difference time-domain simulations of electromagnetic (EM) response of the second-stage Menger sponge dielectric structures have been performed with different types of excitation in order to gain deeper insight into the phenomenon of EM wave localization in these fractals. Analysis of simulated amplitude distributions of electric field oscillations in the Menger sponges has revealed bandgap-like effects caused by resonances in the front part of the structures, as well as formation of the full-wave resonance mode in the central cavity at the localization frequency. It is demonstrated that penetration of the waves inside the structure at the localization frequency leads to equalizing of the EM response from different parts of the 3-D fractal, however, no high- eigenmode is formed in the second-stage Menger sponge. Simulations of the modified fractal structures have been used to show the potential of formation of a bandgap with defect-related localized photon states by 3-D fractals. Index Terms—Ceramics, electromagnetic (EM) fields, finite-difference time-domain (FDTD) method, fractals, resonance.

I. INTRODUCTION

L

OCALIZATION of electromagnetic (EM) waves in periodic and nonperiodic structures has been a subject of theoretical and practical interest over the past two decades. At first, Anderson localization of the same type as electron localization in random lattices was the main point of interest [1], [2]. This type of localization is related to multiple scattering and appears when every field oscillation of the wave is accompanied by a scattering event. Most reliable evidence of Anderson localization was obtained by measurements of transmission fluctuations for alumina spheres that were randomly positioned in a waveguide [3]. In the latter study, EM wave localization was related to a drop in photon density of states in a narrow frequency window just above the first Mie resonance. Light localization in photon states within the pseudogaps, where density of states for transmitted waves should be vanishing, was predicted earlier by John [4], who investigated disordered superlattices, such as 3-D photonic bandgap crystals.

Manuscript received September 30, 2006; revised March 1, 2007. This work was supported by the National Science Foundation under Award DMI-0339535. E. Semouchkina, G. Semouchkin, and M. Lanagan are with the Materials Research Institute, The Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]; [email protected]; [email protected]). Y. Miyamoto and S. Kirihara are with the Joining and Welding Research Institute, Osaka University, Osaka 567-0047, Japan (e-mail: miyamoto@jwri. osaka-u.ac.jp; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.897816

McGurn et al. [5] have studied transmission properties of a disordered photonic crystal built of rods with a square cross section and have observed strong confinement of EM energy in a disordered structure, while perfect periodic structure did not support light localization. Sigalis et al. [6] have shown that localized states form tail-like distributions near the edges of the crystal band gaps, and can overlap the gaps. Yablonovitch et al. [7] have investigated single donor- and acceptor-like defects in a 3-D photonic crystal designed from intersecting drilled holes. They considered the passbands for extended waves located above and below the bandgap as ones similar to the conduction and valence bands in semiconductors. Later, this concept was shared and developed by other researchers [8]. Smith et al. [9] have demonstrated an appearance of highly localized photon states in a 2-D photonic bandgap crystal with a single defect of periodicity created by removal of one rod. Villeneuve et al. [10] have shown that the properties of the defect modes can be controlled by changing the nature and size of the defects and that they originate from the resonances in the vicinity of the defects, the ones that resemble whispering gallery modes in a microdisk laser. The factor for these resonances increases with the size of the crystals to 10 000 and up since the only energy loss in the structure occurs by tunneling through the edges of the crystal. Yariv et al. [11] have proposed to create an optical waveguide based on evanescent-field coupling between the defect cavities in 2-D photonic crystals. Bayindir et al. [12] have presented the results of experiments confirming propagation of photons at frequencies within the bandgap of a wood-pile photonic crystal by a mechanism of hopping between the defect modes that provided for nearly 100% transmission. Recently, Qi et al. [13] have presented a 3-D photonic crystal with the designed point defects and have shown that they can create up to three bands for localized photons within the crystal bandgap. The defects were identified by their signature in the reflection and transmission spectra of the material slab. Earlier, EM wave guiding through linear defects in 2-D crystals was predicted and experimentally demonstrated [14]. These findings have caused an increased interest to defects and microcavities in photonic crystals and led to the development of lossless waveguides and other useful optical devices [15], [16]. Essentially less attention was paid to EM wave localization in quasi-periodic photonic structures—fractals, although it was demonstrated [17], [18] that planar fractals possess with a series of self-similar resonances leading to multiple gaps and passbands for EM waves over an ultrawide frequency range. Especially important is that fractals can efficiently reflect (and conduct) EM waves with wavelengths much larger than fractal di-

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mensions. It gives them a noticeable advantage over photonic crystals, whose dimensions should be several times larger than the wavelength in order to generate a bandgap. Strong localization of dipole radiation in fractals resulting in very high local fields was predicted earlier by Shalaev [19]. Recently, Shalaev has discussed surface-enhanced Raman scattering from metal films of fractal nature and have noticed that local field intensities are several orders larger than ones of the incident wave that confirms light localization [20]. Takeda et al. have reported experimental observation of EM localization in 3-D Fractals of Menger sponge type [21]. They have revealed clear attenuation of both reflection and transmission ( and ) at some specific frequency and have detected electric field intensity profile in the center cavity of the Menger sponge. Later, the same team has presented extended data about the new phenomenon [22], and has shown that specific frequencies of the localized modes in a Menger sponge depend on the fractal stage and on dielectric permittivity of the material. However, investigation of transmission and reflection characteristics for the bulk wall built from Menger sponges did not reveal any difference from a dielectric sheet of the same dimensions, therefore, no bandgaps have been detected for the structure. Previously, we have presented the first results of the FDTD modeling [23] confirming a part of experiments described in [21]. In this study, new simulations and experimental studies have been performed for Menger sponges excited in different ways in order to gain deeper understanding of the processes responsible for EM wave localization in 3-D fractals. Section II describes prototyping and experimental procedures and compares measurement results for different excitation types. Section III presents the simulation model and the simulated scattering parameter spectra of the second-stage Menger sponge. Section IV is devoted to the analysis of amplitude and phase distributions of electric field oscillations that were obtained for various cross sections of the structure in simulations of real and imaginary experiments. In Section V, we demonstrate reflection and transmission characteristics, and the amplitude distributions of electric field oscillations for a photonic crystal designed by transformation of a Menger sponge in order to elucidate the problem of gap formation in photonic fractals. Conclusions presented in Section VI include a short discussion on the nature of the observed phenomena. II. PROTOTYPES AND EXPERIMENTAL RESULTS The Menger sponge is a 3-D version of the Cantor bar fractal. The Cantor bar fractals are formed by extracting the central one of three equal segments of the bar and by repeating this process with remaining segments to form higher stages [see Fig. 1(a)]. The Menger sponge structure is obtained from a monolithic dielectric cube that is divided in 27 identical cubic pieces. The extraction of six such pieces from the centers of all faces of the large dielectric cube and of one piece from the center of the cube volume produces the first-stage Menger sponge. To get the second stage, every one of the remaining 20 cubic pieces is divided in 27 smaller pieces and the extracting procedure is repeated. Similar operations should be repeated in order to get the third stage, etc. The fractal complexity can be described , by a parameter , which is defined by the relation

Fig. 1. (a) Cantor bar fractals of various stages. (b) Prototype of the secondstage Menger sponge (from [23]).

where is the number of the self-similar units newly created . In the Menger when the size of the initial unit decreases to sponge, used in this paper, and so that . Menger sponge prototypes were fabricated by using a stereolithographic machine (SCS-300P, D-MEC Ltd., Tokyo, Japan). The titania–silica ceramic particles with average size of 10 m were dispersed into the photosensitive liquid epoxy resin up to 10 vol.%. The beam of an ultraviolet laser of 350-nm wavelength and spot diameter of 100 m was scanned along the liquid surface at 90 mm/s. The layers of 150- m thickness were solidified through photo polymerization and stacked layer by layer to create a 3-D object. The dimensional accuracy of the object compared to the model was within 0.15% [24]. The dielectric constant of the bulk solidified epoxy with ceramic particles was measured by using dielectric measurement kit (HP85070B, Agilent Technologies, Tokyo, Japan) and was equal to 8.8. Edge length of the dielectric cube was 81 mm, while the central air cavity had 27 27 mm cross sections. The cross sections of the smaller channels were of 9 9 mm. In the first experimental setup, the microwave response of the prototype was measured in free space by using a network analyzer (E8364B, Agilent Technologies) and two pyramidal horn antennas [see Fig. 2(a)]. The sample was placed between the horns at a distance of 40 mm from their openings. This distance was chosen to correspond to 1.5 wavelengths at the localization frequency. Horn antennas were attached to rectangular waveguide parts with dimensions of 80-mm length, 23-mm width, and 10-mm height. The horns were 170-mm long and their openings had the cross section of 130 80 mm. The dimensions of the horns allowed for illumination of the entire front surface of the Menger sponge by the incident plane wave and for wave propagation along the side surfaces of the structure. The measured frequency range exceeded -band and was 6–20 GHz. The second setup was developed to couple EM energy directly into the central cubic cavity of the Menger sponge through the aperture of 27 27 mm in the aluminum plate placed between the horn transmitter and the sample at the distance of 70 mm from the Menger sponge surface [see Fig. 2(b)]. Such a setup was also supposed to allow for separation of the response of the central region of the sponge from the responses of other parts, which could have different reflection and transmission characteristics and could contribute to the measurement results since the Menger sponge dimensions were almost three times larger than the expected wavelength at the localization frequency (28 mm at 10.5 GHz). Additional advantage of the

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Fig. 2. Schematics of transmission experiments with: (a) two horn antennas and (b) one horn and two monopole antennas.

second setup was an opportunity to avoid reflections from the receiver horn as the latter was replaced by a monopole antenna. A similar antenna was used to measure the 90 scattering. The experimental scattering parameter data obtained in the first setup are presented in Fig. 3(a). The well-pronounced minima of both reflection and transmission coefficients at the same frequency are well seen in this figure. Deeper and sharper minima of similar type far exceeding other irregularities of the spectra have been observed for the Menger sponges of third and fourth stages [22]. As follows from Fig. 3(a), the specific frequency that could be related to localization of EM waves in the second-stage sponge is approximately 10.5 GHz, which agrees with the estimates based on an empirical equation used to predict the frequency of localized mode in dielectric Menger sponges [22]. Fig. 3(b) and (c) presents the data obtained in the second experimental setup. As seen in Fig. 3(b), the transmission spectrum looks similar to the spectrum observed in the first setup and demonstrates deep minimum at the same frequency of approximately 10.5 GHz. The spectrum of 90 scattering instead has a peak at this frequency. Peaks in 90 scattering spectra can usually be related to increased reflection from a sample caused, for example, by its large scattering cross section at high- resonance. However, large scattering cross section would contribute so that the above-described minima would not be obto served. It is worth noticing that the peak of 90 scattering at the localization frequency is only approximately 2.5 times higher are than other irregularities in the spectra, while the dips in of several orders larger. This observation allows for assuming that the peak in 90 scattering spectrum is not related to scattering and is instead caused by radiation from those big channels of the sponge, which are directed at a 90 angle with respect to the propagation vector of the incident wave.

Fig. 3. (a) Transmission (black) and reflection (grey) spectra for the secondstage Menger sponge obtained in the first setup (from [23]). (b) Transmission and (c) 90 scattering spectra obtained for the same structure in the second setup by using the monopole antennas.

III. SIMULATIONS OF -PARAMETER SPECTRA A. Simulation Model The finite-difference time-domain (FDTD) model of the second-stage Menger sponge was built from approximately 150 cubic and rectangular blocks with the dielectric permittivity of 8.8. The transmitting horn was introduced in the model with exactly the same dimensions as those in experiments, while the receiving horn was reduced in dimensions to be screened by the sample from the direct transmitter radiation. The pyramidal shapes of the horns were reproduced by fine staircasing so that horn models were capable of operating in the same radiation band as one of the experimental horns. The transmitting horn was excited by a current in the central conductor of a coax used to feed the input waveguide of the transmitter. The schematic of the simulation model is depicted in Fig. 4. Perfectly matched layer (PML) boundary conditions have been applied at all boundaries of the computational domain. The cell size used in this model was usually of 1.5 1.5 1.5 mm, which exceeded mm at a frequency of 10.5 GHz for a dielectric with permittivity of 8.8. Using a finer mesh with a cell size of 0.9 mm in the model of the setup with two

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Fig. 4. Schematics of the model (from [23]) used for simulation transmission and reflection spectra.

horns led to a number of cells exceeding 10 , which made simulations problematic. However, we have used the cell size of 0.9 0.9 0.9 mm in simulations of another setup with a smaller computational domain described in Section IV, and have found that fine meshing provided for the results are quite similar to the ones obtained at the cell size of 1.5 mm for both the amplitude distributions of electric field oscillations and the frequency of the deep drop in transmission. It justified the employment of meshing with the cells dimension of 1.5 mm. At such meshing, the fractal elements of the Menger sponge included 216 cubic cells. Field patterns for the cross sections of such elements were built by using 36 cells that provided acceptable quality of the image. The -parameters were calculated in the frequency range of 7.5–15.0 GHz, while the field distributions were sampled in a range of 9.5–11.75 GHz. B. Simulation Results The results of -parameters spectra simulations for the second-stage Menger sponge are presented in Fig. 5 in a frequency range of 10.0–12.1 GHz, which included a localization frequency of 10.5 GHz observed in experiments. Similar to the experimental data, the average value of the simulated reflection coefficient increases, while the average value of transmission coefficient decreases with frequency increase. No specific features that could be related to bandgap formation have been observed. The average level of reflection in simulations was slightly less than in experiments. Besides, the spectrum contained a set of deeper drops not observed in experiments. Part of these features could be related to resonances inside the transmitter, between the transmitter and the front surface of the sample, and between the receiver horn and the back surface of the sample or the transmitter. However, from comparand spectra obtained for the setup with the ison of the Menger sponge and the spectra obtained for the setup without the structure (Fig. 5), it follows that the spectra obtained for the setup with the sponge contain features related to the structure itself. For lossless EM wave transmission through the samples, a dip is expected to be accompanied by an increase of and in vise versa, however, as seen in Fig. 5, the combined dips in reflections and transmission are observed at some frequencies. For

Fig. 5. Simulated S (top) and S (bottom) for the model depicted in Fig. 4 (grey curves) and for the same setup without the fractal structure (black curves with markers). Dashed–dotted lines indicate correlation between the reflection and transmission dips.

example, it takes place at a frequency of 10.55 GHz, i.e., almost at the “localization frequency” found in experiments. However, essentially deeper drops can be seen at a frequency of 10.25 GHz drops down to below 40 dB, which is consistent with when the 50-dB dip observed experimentally. Fig. 5 indicates that these dips could be related to the Menger sponge rather than to resonances in the setup without the sponge. Slightly different than in experiments, the value of the frequency, supposedly providing for the EM wave localization in simulations, could be caused by using a staircasing approximation in the model for representation of pyramidal horns. We found that decreasing the staircasing step in this approximation caused noticeable, though relatively small, changes in the scattering parameters’ spectra. Even at the small step size, the employed approximation could lead to formation of a wavefront at the aperture of the horn antenna, which was slightly different in the model from the one in the experimental setup. As mentioned earlier in [23] and demonstrated in more detail below, the fractal structures under study do not provide for an EM response typical for a resonance mode in a single resonator at frequencies around 10 GHz. Instead, interplay between formation and destruction of local resonances takes place, which are sensitive to the specifics of the incident wavefront. It follows that we could expect slightly different distributions of resonance field oscillations in the model and in the experiments at the same frequency. It could be the reason for some differences in local and integrated reflection and transmission coefficients and could cause the observed shift of the frequency of EM wave localization. Another reason for the slight frequency shift could be a difference of the values of dielectric permittivity of the Menger sponge material in the model and in experiments. The value of the dielectric permittivity was measured with the accuracy not

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Fig. 7. Schematics of the model and electric field pattern in the median (vertical) cross section pictured at one of the moments to show the wavefront after the wave exits the aperture. Fig. 6. Simulation model without the horn antennas. A frame behind the structure depicts the electric wall source.

exceeding 0.1. In addition, an imaginary part of the permittivity on the level of 0.1 was detected experimentally. Changing the permittivity value in simulations by 0.1 really led to shifting in the positions of the deep drops in -parameter spectra by approximately 0.1 GHz. Addition of conductivity to the dielectric on the level of 0.01 S/m (that corresponded to the worst measured loss), however, was found to produce a negligible effect on the frequency location of the dips. IV. AMPLITUDE DISTRIBUTIONS OF FIELD OSCILLATIONS A. Model for Field Pattern Simulations In order to separate EM response of the Menger sponge from responses of other parts of the setup, we have simulated a modified excitation model without the horn antennas. In this model, a plane-wave source of “electric wall” type was placed at the same distance from the sample as the distance between the sample and the transmitting horn aperture in the second experimental setup shown in Fig. 2(b). An “electric wall” source is often used in FDTD simulations to excite a plane wave. In our case, this source could be described as a rectangular area in vertical plane with uniform distribution of planar electric field having only component. The dimensions of the rectangular area of an the source were equal to the dimensions of the aperture of the transmitting horn antenna used in the previous model and in the experiments so that the excited wave was expected to be similar to the experimental one. An aperture mask with planar dimensions equal to the dimensions of the aluminum mask used in the second experimental setup was introduced between the source and sample (Fig. 6). Two monopole antennas were also employed in this model similar to how it was done in the experimental setup. To control relative values of transmission and 90 scattering, we simulated the voltages between the inner and outer conductors of the antennas. Excluding the horn antennas from the new model provided for space saving of the computational domain and gave us an opportunity to use smaller sized cells of 0.9 0.9 0.9 mm. The simulations of the model shown in Fig. 6 have revealed a deep drop in transmission at approximately 10.35 GHz and a

peak of the signal for 90 scattering at close frequency that was quite similar to experimental observations presented in Fig. 3. However, the results of time-domain investigation of the wave propagation in the new setup did not confirm the expectations that the aperture mask could be used to direct the wave packet into the center of the Menger sponge through the big channel of the fractal structure. As seen in Fig. 7, which is a snapshot at one of the time moments of the wave propagating in the model, the wavefront has a spherical shape after exiting the aperture mask so that it illuminates the whole front surface of the structure. Spherical shape of the wavefront could complicate the character of the sponge response; therefore, no aperture mask was used in the following simulations of the fields excited by the electric wall source. B. Field Amplitude Patterns in the Sponge Cross Sections Fig. 8 presents the amplitude and phase distributions of oscillations in median and cross sections of the sponge at different frequencies. The areas of highest amplitudes on the amplitude distributions are shown via black spots, while white areas indicate the areas of lowest fields. White and black areas on the phase distributions correspond to the phase values 3.14), respectively, while large areas of gray have the ( phase value close to zero. It means that gray areas have the 180 phase difference with respect to black or white areas. The patterns sampled at 10.3–10.375 GHz were chosen for presentation in Fig. 8 since the lowest transmitted and/or reflected field magnitudes have been observed at these frequencies. The minimum field values were found at 10.325 GHz, however, at this frequency, 90 scattering was also low. An increased scattering appeared instead at 10.375–10.4 GHz. As seen in Fig. 8, two almost equal field antinodes are observed in the central cavity at this frequency. At higher frequencies, the intensity of fields in antinodes dropped down. Oscillations in these two antinodes were found to be opposite in phase as they should be at the full wavelength cavity resonance. The described field distribution corresponded to the two-peaked field intensity profiles revealed in the central cavity of the Menger sponge in experiments at 10.5 GHz [21]. Since the setup used in simulations of the field distributions did not include horn antennas, we could not relate the ob-

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Fig. 9. Experimental results for the field intensities in the crossings of the small channels of the Menger sponge in the vicinity of the localization frequency.

XY

Fig. 8. Amplitude distributions of electric field oscillations in median (first and third rows) and (second and fifth rows) cross sections of the Menger sponge (sample locations are shown by dashed lines). The fourth row illustrates phase distributions in the -plane for two frequencies. Oscillations in white and black areas, from one side, and in gray areas, from the other side, have opposite phases.

XZ

XY

served difference of 0.1 GHz between the frequencies of cavity resonance in simulations and experiments to the difference in incident wavefronts and some other factors mentioned in Section III-B. This difference could be related to the slight inaccuracy of determination of the dielectric permittivity at the measurements. Insertion of the monopole antenna in the cavity could also perturb the field distribution and cause some discrepancies. Another fact that we would like to accentuate is that a noticeable increase of the signal in the antenna controlling the 90 scattering has been observed at the frequencies, which provided for formation of cavity resonance. It could be considered in favor of the assumption made at the end of Section II that the 90 radiation from the Menger sponge is not related to scattering, and, instead, is defined by radiation through the side-directed big air channels. Such radiation could be expected when high intensity fields are formed in the center cavity, and we have really observed the side beams in field distributions at frequencies of 10.375–10.4 GHz. Analysis of the areas with high amplitude of field oscillations in Fig. 8 has shown that, although field amplitudes in the central cavity become relatively high at 10.375–10.4 GHz, they are still essentially lower than the ones in some dielectric islands

and near the front crossings of the small channels of the Menger sponge. It is worth mentioning that strong fields in the front part of the Menger sponge were also observed in field simulations of the model with vis-á-vis located horn antennas [23]. Moreover, field patterns at 10.375 GHz in Fig. 8 have features similar to ones observed at frequency of 10.25 GHz in simulations of the model with two horn antennas [23]. It follows that the model with two horn antennas provided for occurrence of the same phenomena at 10.25 GHz as the model with an “electric wall” source at 10.375 10.4 GHz, i.e., 0.15 GHz closer to experimental data. It led us to conclude that, at least 0.15 GHz of the shift in localization frequency in simulations made for the setup with two vis-á-vis horn antennas could be related to imperfections of the source models, and should not cause any troubles. A detailed analysis of the amplitude distributions for the component in the vicinity of the front crossings of the small channels has shown that strong fields in these areas are related to overlapping of the tails of resonance modes formed in the nearest dielectric islands. The dimensions of these islands are quite close to the wavelength in dielectric of the fractal structure at 10.35–10.375 GHz, which is approximately 9.75 mm. Classification of resonances in the dielectric islands demands using a finer mesh in the simulations. As well seen in Fig. 8, field patterns at a frequency of 10.3 GHz demonstrate high field amplitudes only in the front part of the sponge. A similar situation is observed in the frequency range between 10.2–10.3 GHz and at higher frequencies of approximately 10.5 GHz. Besides strong fields in the vicinity of front crossings of the small channels, one can see antinodes in the front part of the big channel, while antinodes in the central cavity disappear. It is worth mentioning that the front part of the big channel between the surface of the Menger sponge and the entrance to the central cavity has dimensions equal to the dimensions of the central cavity. This means that the same type of full wavelength resonance as the one observed in the central cavity could be formed in the front part of the big channel. Fig. 9 demonstrates the difference between the electric field intensities in the front and back crossings of the small channels of the fractal structure measured experimentally by using monopole antennas inserted in the small channels in the vicinity of the localization frequency. A 15-dB difference between field

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Fig. 11. Transmission characteristic (S ) of the modified structure (without big channels in the side centers, as shown in the inset) in comparison with the characteristic of an “empty” setup (marked by dots). Arrows on the frequency axis mark the location of the gap in transmission.

Fig. 10. Upper row: field patterns in the median XY -plane for the sample with the cutoff 1/3 of the volume (left) and for the whole Menger sponge (right) at frequencies corresponding to maximal average fields near the front crossings of the small channels (dashed lines show the geometry of the samples), and lower figure: frequency dependences of the field amplitudes in the front crossings (in relative units) for the samples with the cut off 2/3 of the volume (1), 1/3 of the volume (2), and for the whole sponge (3).

amplitudes in the front and back parts of the sponge was detected. The simulated field patterns demonstrate that at frequencies below 10.3 GHz and above 10.425 GHz, the front part of the Menger sponge blocks the EM waves from propagation into the inner parts of the sample. This conclusion has been confirmed by comparison of EM response of the second-stage Menger sponge with responses of the partial fractal structures obtained by cutting off either 1/3 or 2/3 back parts of the sponge. As seen in Fig. 10, field patterns obtained for the partial fractal structures and for the Menger sponge look very similar. Frequency dependencies of the average field amplitudes in the areas of the front crossings of small channels also look similar for these structures and have peaks at the above-mentioned frequencies below 10.325 GHz and above 10.425 GHz. It follows from the presented data that the front parts of the Menger sponge create a type of stopband for wave propagation through the structure at these frequencies. On the contrary, the central channel becomes open for wave penetration into the center of the Menger sponge in the narrow frequency band of 10.325–10.425 GHz and, as a sequence, an equalizing of the EM response in different parts of the structure occurs. Although such equalizing still does not allow for formation of an ideally symmetric eigenmode predicted by Sakoda [25] for the Menger sponge structures, there is a possibility that such modes could be formed in the Menger sponges of higher

stages possessed with a higher level of self-symmetry. Our future study will be extended to simulations of the third-stage Menger sponges. V. BANDGAP FORMATION IN A PHOTONIC CRYSTAL DESIGNED BY TRANSFORMATION OF A 3-D FRACTAL The presented data lead us to conclude that EM wave localization in a quasi-periodic dielectric structure such as the second-stage Menger sponge possibly has a resonance nature. However, the opportunity for stopband formation and light localization in 3-D fractals by the same mechanism as in dielectric photonic crystals still needs to be evaluated. We have used the structural similarity of the second-stage Menger sponge and the photonic crystal to transform the fractal structure into a photonic-bandgap crystal with the central cavity becoming a defect of periodicity inside the structure. These changes could be described as an insertion of cubic species similar to any of 20 regular cubic species used to compose the second-stage Menger sponge (27 27 27 mm in size) into the openings of the big channels of the structure at the center of each side. This procedure made outer parts of the sponge periodic and, at the same time, conserved the center cubic cavity unchanged. To make the “periodicity” more perfect, we have also increased the dimensions of the structure by 9 mm from every side. As a result, the front surface of the modified structure looked as shown in the inset of Fig. 11. The transformed structure is too small to be capable of demonstrating the properties of a photonic crystal. However, the simulated transmission characteristic of this structure by using the setup with vis-à-vis location of two horn antennas have shown interesting results. Although the spectrum (Fig. 11) did not demonstrate a perfect bandgap similar to the one expected for photonic crystals, a clear drop in the average level of transmission in the band between 11.3–12.2 GHz is seen. It can be considered as a signature of the bandgap formation. The observed band cannot be related to any specifics

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of the setup model since, in the “empty” setup, a relatively high transmission at the same frequencies takes place (compare the graphs obtained in the setup models with and without the fractal structure in Fig. 11) As seen from Fig. 11, transmission through the fractal structure demonstrates a sharp increase, almost reaching the values characteristic for the “empty” setup near the low frequency edge of the stopband. It possibly points out an existence of a passband in the transformed structure. In addition, we have found that formation of the full-wave resonance in the central cubic cavity of the modified structure occurs at a frequency of 11.525–11.55 GHz, i.e., closer to the low frequency edge of the band with decreased transmission. Such position of the defect mode has been observed for the EM wave localization caused by defects in photonic crystals. In a future study, we will examine the specifics of the energy band structure of fractals and will conduct simulations and measurements of fractal and modified structures combined of multiple units to reproduce conditions typical for photonic crystals. VI. CONCLUSION Experimental study of the 3-D dielectric fractals of the second-stage Menger sponge type performed by using two different excitation setups with and without a horn receiver have confirmed deep attenuation of both reflection and transmission characteristics of the Menger sponge and an increase in 90 scattering at a previously found localization frequency. A similar response was obtained from the FDTD simulations of scattering parameters spectra of the Menger sponge, although some effects related to external parts of the setup were found to be capable of affecting localization phenomena. FDTD study of the amplitude distributions of electric field oscillations has shown that at frequencies beyond the narrow frequency range, in which the EM wave localization takes place, the front part of the Menger sponge blocks the waves from propagation inside the structure. Experiments with monopole antennas inserted into small cavities of the Menger sponge have confirmed field enhancement in the front part of the sample in the vicinity of the localization frequency. Penetration of the EM energy inside the fractal structure in the narrow frequency band, when it is not blocked by the resonances in the front part of the sponge, leads to equalizing and symmetrization of the EM response from different parts of the structure. However, the amplitude distribution of the field at this frequency still does not correspond to any high- eigenmode predicted for the Menger sponge fractals. In agreement with the earlier experimental observations of the field profiles in the central cavity of the Menger sponge at the localization frequency, the simulations have demonstrated a formation of the full wavelength resonance in the cavity. However, it was found that the magnitude of electric field oscillations in the central cavity is much lower than in some dielectric islands of the Menger sponge and near the front crossing of small channels. It is possible that the stage number of the investigated sponge was not high enough to support resonance eigenmode at nonsymmetric radiation. While the obtained results lead us to conclude that the EM wave localization is possibly related to the high- resonance in the Menger sponge, the simulations performed for modified fractal structures have demonstrated some resemblance between the EM

wave localization in fractals and the phenomenon of light localization on local resonance modes near the defects of periodicity in photonic-bandgap crystals. A high- factor of the latter localization is provided due to the bandgap for transmission through photonic crystals, and in fractals, we have also observed phenomena leading to stopband formation near the localization frequency. Future work should be aimed at the investigation of energy band structures of the fractals and at the study of localization phenomena in the Menger sponges having a higher stage number. The results of this study confirm that simulation of amplitude and phase distributions of field oscillations proposed in [26] is a very valuable tool for investigating the nature of resonance processes in photonic materials and structures.

REFERENCES [1] S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett., vol. 53, pp. 2169–2172, 1984. [2] P. W. Anderson, “The question of classical localization: A theory of white paint?,” Philos. Mag., vol. B52, pp. 505–509, 1985. [3] A. A. Chabanov and A. Z. Genack, “Photon localization in resonant media,” Phys. Rev. Lett., vol. 87, no. 15, pp. 153901-1–153901-4, Oct. 2001. [4] S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett., vol. 58, no. 23, pp. 2486–2489, Jun. 1987. [5] A. R. McGurn, P. Sheng, and A. A. Maradudin, “Strong localization of light in two-dimensional disordered dielectric media,” Opt. Commun., vol. 91, pp. 175–179, 1992. [6] M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B, Condens. Matter, vol. 53, no. 13, pp. 8340–8348, Apr. 1996. [7] E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett., vol. 67, no. 24, pp. 3380–3383, Dec. 1991. [8] A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 60, no. 5, pp. 6118–6127, Nov. 1999. [9] D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Amer. B, Opt. Image Sci., vol. 10, no. 2, pp. 314–321, Feb. 1993. [10] P. Villeneuve, S. Fan, and J. D. Joannopoulis, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B, Condens. Matter, vol. 54, no. 11, pp. 7837–7842, Sep. 1996. [11] A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: A proposal and analysis,” Opt. Lett., vol. 24, no. 11, pp. 711–713, Jun. 1999. [12] M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: A waveguiding mechanism through localized coupled cavities in three dimensional photonic crystals,” Phys. Rev. B, Condens. Matter, vol. 61, no. 18, pp. R11855–R11858, May 2000. [13] M. Qi, E. Lidorikis, P. T. Rakich, S. Johnson, J. Joannopoulos, E. P. Ippen, and H. I. Smith, “A three-dimensional optical photonic crystal with designed point defects,” Nature, vol. 429, pp. 538–542, Jun. 2004. [14] S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in photonic crystal,” Science, vol. 282, pp. 274–276, Oct. 1998. [15] S. Kawashima, M. Okano, M. Imada, and S. Noda, “Design of compound-defect waveguides in three-dimensional photonic crystals,” Opt. Express, vol. 14, no. 13, pp. 6303–6307, Jun. 2006. [16] N. C. Panoiu, M. Bahl, and R. M. Osgood, Jr., “All-optical tunability of nonlinear photonic crystal channel drop filter,” Opt. Express, vol. 12, no. 8, pp. 1605–1610, Mar. 2004.

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[17] W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett., vol. 89, no. 22, pp. 223901/1–223901/4,, Nov. 2002. [18] L. Zhou, C. T. Chan, and P. Sheng, “Theoretical studies on the transmission and reflection properties of metallic planar structures,” J. Phys. D, Appl. Phys., vol. 37, pp. 368–373, 2004. [19] V. Shalaev, R. Botet, and A. Butenko, “Localization of collective dipole excitation on fractals,” Phys. Rev. B, Condens. Matter, vol. 48, no. 9, pp. 6662–6664, Sep. 1993. [20] V. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-Dielectric Films. Berlin, Germany: Springer-Verlag, 2000. [21] M. W. Takeda, S. Kirihara, Y. Miyamoto, K. Sakoda, and K. Honda, “Localization of electromagnetic waves in three-dimensional fractal cavities,” Phys. Rev. Lett., vol. 92, no. 9, pp. 093902/1–093902/4, Mar. 2004. [22] A. Mori, S. Kirihara, Y. Miyamoto, M. W. Takeda, K. Honda, and K. Sakoda, “Integration of ceramic/epoxy photonic fractals with localization of electromagnetic waves,” in Proc. 29th Int. Adv. Ceram. and Composites Conf., Jan. 2005, pp. 361–366, 2005. [23] E. Semouchkina, Y. Miyamoto, S. Kirihara, G. Semouchkin, and M. Lanagan, “Simulation and experimental study of electromagnetic wave localization in 3-D dielectric fractal structures,” in Proc. 36th Eur. Microw. Conf., Sep. 2006, pp. 776–779, 2006. [24] S. Kirihara, Y. Miyamoto, K. Takenaga, M. W. Takeda, and K. Kajiyama, “Fabrication of electromagnetic crystals with a complete diamond structure by stereolithography,” Solid State Commun., vol. 121, pp. 435–439, 2002. [25] K. Sakoda, “Electromagnetic eigenmodes of a three-dimensional photonic fractal,” Phys. Rev. B, Condens. Matter, vol. 72, pp. 184201/ 1–184201/7, 2005. [26] E. Semouchkina, W. Cao, R. Mittra, and W. Yu, “Analysis of resonance processes in microstrip ring resonators by the FDTD method,” Microw. Opt. Technol. Lett., vol. 28, no. 5, pp. 312–321, Mar. 2001. Elena Semouchkina (M’04) received the M.S. degree in electrical engineering and Ph.D. degree in physics and mathematics from Tomsk State University, Tomsk, Russia, in 1978 and 1986, respectively, and the Ph.D. degree in materials from The Pennsylvania State University, University Park, in 2001. She was a Scientist with Russian academic centers such as the Siberian Physics–Technical Institute, St. Petersburg State Technical University, and Ioffe Physics–Technical Institute, where she was involved with the investigation of metal–oxide–semiconductor devices and the development of infrared photodetectors. Since 1997, she has been with the Materials Research Institute, The Pennsylvania State University, initially as a Graduate Research Assistant, then as a Post-Doctoral Scholar, later as a Research Associate, and, since 2006, as a Senior Research Associate and Associate Professor. She has authored or coauthored approximately 70 publications in scientific journals. Her current research interests are focused on computational analysis of EM processes in microwave materials, metamaterials, and devices. Dr. Semouchkina was a recipient of the Xerox 2001 Research Award of The Pennsylvania State University for the best Ph.D. thesis and the National Science Foundation 2004 Advance Fellows Award.

Yoshinari Miyamoto received the M.S. and Ph.D. degrees in materials science and engineering from Osaka University, Osaka, Japan, in 1969 and 1976, respectively. Since 1997, he has been a Professor with the Joining and Welding Research Institute, Osaka University. He has authored or coauthored approximately 470 publications in scientific journals, monographs, and books. His current research interests are the development of 3-D photonic fractals and crystals by computer-aided design (CAD)/CAD stereolithography, freeform fabrication of metals by 3-D microwelding, and combustion synthesis of nitride ceramics. Dr. Miyamoto was a recipient of the 1989 Fulrath Award, 1992 Academic Award from the Ceramic Society of Japan, and the 2003 ECD Bridge Building Award. He was elected a Fellow of The American Ceramic Society in 2004.

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Soshu Kirihara received the M.S. and Ph.D. degrees in materials science and engineering form Ibaraki University, Ibaraki, Japan, in 1996 and 1999, respectively. Since 1999, he has been with the Joining and Research Institute, Osaka University, Osaka, Japan, initially as a Research Fellow of the Japan Society for the Promotion of Science, later as a Research Associate of Osaka University, and, since 2006, as an Associate Professor. He has authored or coauthored approximately 50 publications in scientific journals. His current research interests include EM wave properties of dielectric ceramics with 3-D structures fabricated by using computer-aided design and manufacturing (CAD/CAM) process. Dr. Kirihara was a recipient of the 2003 Japan Society of Powder and Powder Metallurgy (JSPM) Award for Innovative Research.

George Semouchkin received the M.S. degree in electrical engineering, Ph.D. degree in materials, and Doctor of Science degree in physics and mathematics from the Leningrad Polytechnic Institute (now St.-Petersburg State Technical University), St. Petersburg, Russia, in 1962, 1970, and 1990, respectively. Prior to joining The Pennsylvania State University, University Park, in 1999, he was with the St. Petersburg State Technical University, St. Petersburg, Russia, as a Professor, a Leading Scientist, a Head of the Laboratory, and earlier as a Senior Scientist, where he studied ionic crystals, ceramic materials, inorganic dielectrics, and developed microelectronic devices. He is currently a Visiting Professor of materials with the Materials Research Institute, The Pennsylvania State University. He has authored over 150 technical publications. His current research interests include designing low-temperature co-fired ceramic (LTCC)-based microwave devices and all-dielectric metamaterials.

Michael Lanagan (M’99) received the B.S. degree in ceramic engineering from the University of Illinois at Urbana-Champaign, in 1982, and the Ph.D. in ceramic science and engineering from The Pennsylvania State University, University Park, in 1987. He is currently a Professor of materials science and engineering and Associate Director of the Materials Research Institute with The Pennsylvania State University. Prior to joining The Pennsylvania State University, he was a Staff Scientist for 12 years with the Argonne National Laboratory, where he studied materials for superconductors, molten carbonate fuel cells, and high-energy density capacitors. He has authored over 150 technical publications. He holds eight patents. His current research interests include the development of new dielectric materials for high energy density capacitors and microwave metamaterials. Dr. Lanagan is a member of The American Ceramic Society and The International Microelectronics and Packaging Societies. He was an invited participant to the National Academy of Engineering’s “Frontiers of Engineering,” which recognizes promising young scientists in all areas of research.

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Sub-Microsecond RF MEMS Switched Capacitors Benjamin Lacroix, Arnaud Pothier, Aurelian Crunteanu, Christophe Cibert, Frédéric Dumas-Bouchiat, Corinne Champeaux, Alain Catherinot, and Pierre Blondy, Member, IEEE

Abstract—This paper presents fast switching RF microelectromechanical systems (MEMS) capacitors with measured switching times between 150–400 ns. By introducing bent sides on a planar microbeam, it is shown experimentally that the resonance frequency of aluminum bridges is increased by a factor of 25 compared to standard RF MEMS components. In addition, this original shape can be implemented easily in post-processing of complementary metal–oxide–semiconductor circuits. Several designs are presented with measured mechanical resonance frequencies between 1–3 MHz and measured switching times under 400 ns. Using this original approach, sub-microsecond RF MEMS switched capacitors have been designed and fabricated on quartz substrate. Their resulting RF performance is presented with a measured capacitance ratio of 2.3. Reliability tests have also been performed and have demonstrated no significant mechanical behavior variation over 14 billion cycles and a moderate sensitivity to temperature variation. Index Terms—Capacitors, high-speed integrated circuits, microelectromechanical systems (MEMS).

The tensile stress inside the structure allows to increase the mechanical resonance frequency since the effective spring constant is higher and the mass remains constant. Using a different approach, Mercier et al. showed that miniature membrane-based RF MEMS capacitors have many advantages like very high spring constant, resulting in a very fast switching speed, large pull-up pressure, and little sensitivity to temperature variation and charging in the dielectric layer [8]. The authors have previously improved this concept working on a 3-D geometry approach for the movable beam. It has been demonstrated that adding simple bent sides on conventional beam edges strongly increases the mechanical structure stiffness [9]. Since the effective mass is kept low, the structure is able to actuate in sub-microsecond switching times. This paper aims to expand the concept of fast switching to MEMS capacitor designs for high-speed reconfiguration applications. II. MECHANICAL CONCEPT FOR FAST SWITCHING

I. INTRODUCTION

W

HENEVER using RF microelectromechanical systems (MEMS) capacitors or switches, many microwave circuits and subsystems such as phase shifters [1], control circuits, and tunable components [2]–[5] have demonstrated better performance than semiconductors in terms of losses and power consumption. Other recent applications like smart antennas or voltage-controlled oscillators (VCOs) require low loss and low power tuning elements integrated with silicon RF electronics. However, when fast modulation is required for radar, reconfigurable antennas, or other tunable components [6], microwave systems mainly use semiconductors since RF MEMS components are not currently able to reach switching times less than 1 ms. Even if the switching speed is one of the main limitations compared to semiconductors, little work has been done to improve the speed of RF MEMS capacitors and switches. Mercier et al. [7] have demonstrated sub-microsecond switching times using dielectric membrane switches with built-in tensile stress.

Manuscript received October 24, 2006; revised February 16, 2007. B. Lacroix, A. Pothier, A. Crunteanu, and P. Blondy are with the XLIM–Centre National de la Recherche Scientifique, Unité Mixte de Recherche 6172, University of Limoges, 87060 Limoges, France (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). C. Cibert, F. Dumas-Bouchiat, C. Champeaux, and A. Catherinot are with the Sciences des Procédés Céramiques et de Traitements de Surface–Centre National de la Recherche Scientifique, Unité Mixte de Recherche 6638, University of Limoges, 87060 Limoges, France (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.897760

Most metallic membrane-based MEMS switches in the literature can operate in the microsecond range, mainly limited by their low mechanical resonance frequency , which is directly linked to the beam effective mass and the spring constant , shown as follows in (1), if a simple spring mass model is considered: (1) Indeed for electrostatic actuated devices, it has been demonstrated [10] that, in first approximation, the switching time for inertia-limited systems (as in the case of vacuum conditions) mainly depends on the applied actuation voltage ’s and on the component mechanical characteristics [pull-in voltage , mechanical resonance frequency , as shown in (2)]. Consequently, devices with higher mechanical resonance frequencies result in faster switching components (2) To obtain high resonance frequencies, a first approach is to use a specific 3-D bridge, which is actually derived from macroscopic U-beam profiles well known in conventional mechanical constructions for better stiffness than classical beams. Fig. 1 presents a scanning electron microscope (SEM) photograph of such a shape, here referred to as a boat-like beam. Indeed, as shown in Fig. 2, the main idea consists of implementing simple bent metallic sides on a conventional MEMS planar microbeam to strongly stiffen the movable membrane since the angle presented by the bridge membrane and its bent sides and their corresponding width have a major influence on the structure mechanical behavior.

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Fig. 1. Proposed boat-like bridge structure geometry.

Fig. 3. (a) Computed mechanical resonance frequency as a function of a for our proposed 40 10 0.35 m beam for different bent side widths and (b) computed pull-in voltages as a function of t for different bending angles.

2

Fig. 2. Geometry of a conventional bridge and a boat-like bridge where l; w; t; and h represents the planar beam length, width, thickness, and the bent side additional width, respectively.

Due to our technological limitations, the length and the width of the beam have been limited, respectively, to 40 and 10 m to have beams as miniature as possible. These beams are suspended only 0.3 m above their pull-down electrode to keep reasonable actuation voltages. The bent side geometry and the membrane thickness have been optimized using 3-D finite-element method ANSYS mechanical simulations1 to reach high mechanical resonance frequency values combined with moderate pull-in voltages. As presented in Fig. 3(a), both - and -parameters induce a strong mechanical resonance frequency improvement until the global structure mass increase becomes too important for higher bent side width values. On the other end, as the beam becomes stiffer, the needed pull-in voltage also quickly increases. Using a liftoff process, we can obtain 3- m-wide stiffening sides bent at , with 40 10 0.35 m bridges that can be easily fabricated using a conventional lithography process. Hence, the expected structure mechanical resonance frequency is 3260 kHz for a pull-in voltage in the 60-V range, whereas it is only 1210 kHz for a conventional beam. As a result, it corresponds to a spring constant 12 times higher (250 N/m) than a planar beam (20 N/m) with a device mass increase of 60%. The comparison between computed mechanical resonance frequencies of conventional miniature bridges and boat-like bridges is also shown in Table I for longer bridges until 80 m. 1ANSYS

Inc., Canonsburg, PA. [Online]. Available: www.ansys.com

2

TABLE I COMPARISON OF MECHANICAL RESONANCE FREQUENCIES BETWEEN CONVENTIONAL MINIATURE AND BOAT-LIKE ALUMINUM BRIDGES

TABLE II COMPARISON OF MECHANICAL RESONANCE FREQUENCY BETWEEN STANDARD, CONVENTIONAL MINIATURE, AND BOAT-LIKE BRIDGES

It is shown than the resonance frequency is approximately 2.8 higher for boat-like shape geometry. Table II presents simulated mechanical results also computed with ANSYS. For instance, in the case of a 40- m–long, 10- m –wide, and 0.35- m-thick aluminum bridge, ANSYS predicts a mechanical resonance frequency around 20–60 times higher for a boat-like geometry than standard RF MEMS bridges.

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Fig. 4. Experimental setup for mechanical resonance frequency.

These results show the interest in working on mechanical structure geometry to increase the mechanical resonance frequency, and consequently, to improve the switching speed. Such structures are potentially able to reach switching times around a hundred of nanoseconds or so, as we will see in Section III. III. MECHANICAL CHARACTERIZATION AND RELIABILITY TESTS To validate the previous computed results, simple mechanical test structures have been fabricated and implemented as a series tunable capacitor on a coplanar waveguide (CPW) line using the fabrication process described in [9]. A. Mechanical Resonance Measurements To measure boat-like structure mechanical resonance frequencies, a specific microwave test bench, presented in Fig. 4 (from [9]) has been used [11]. The measurement principle consists of detecting an amplitude modulation induced by the movable bridge motion on a continuous wave (CW) RF signal. A 10-GHz RF carrier signal is applied to the component, which is actuated using a 40-V magnitude periodic sinusoidal actuation signal. This low-frequency (LF) signal mechanically excites the movable bridge and makes it vibrate inducing a periodic capacitance variation between the component input and output. This capacitance modulation creates a coming out RF carrier signal amplitude modulation that can be easily detected using a spectrum analyzer. Under vacuum, at the mechanical resonance frequency, the membrane displacement becomes maximum inducing a maximum intermodulation level detected. Fig. 5 (from [9]) shows typical modulated signal spectrums observed at test structure output for 60- and 40- m-long bridges. We can observe the carrier signal frequency (normalized here to 0 Hz) and two intermodulation peaks on each side. The first one corresponds to the applied actuation , which is detected because of bias T signal frequency signal leakages, whereas the second one is generated by the bridge vibration, which occurs at a frequency corresponding to

Fig. 5. Detected modulated RF signal at bridge mechanical resonance frequency.

(linked to the electrostatic actuation [11]). The bridge resonance frequency is experimentally found when this peak magnitude reaches its maximum value corresponding to the maximum displacement of the membrane. The measurement technique resolution is estimated to be below 1 kHz. Bridges of different lengths, corresponding to Table I, have been fabricated and tested. The measured results are summarized in Table III. 80- and 60- m-long bridges both combine good performance in mechanical resonance frequency and actuation voltage: measured pull-down voltages are 40 and 60 V, respectively, for 0.3- m air gap MEMS structures. For these bridges, we have obtained very good agreement with mechanical simulations. Even if 40- m-long bridges require higher actuation voltages (around 120 V) because of their higher stiffness, they achieve very high mechanical resonance frequencies better than 2 MHz. Measured pull-in voltages are higher than the computed ones in Section II due to the multilayer effects not taken into account in simulations. Moreover, the measured mechanical resonance frequencies for the 50- and 40- m-long bridges are less important than the simulated ones, attributed to

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TABLE III MEASURED MECHANICAL CHARACTERISTICS FOR 10-m-WIDE AND 0.35-m-THICK ALUMINUM MADE BOAT-LIKE STRUCTURES

Fig. 7. Measured: (a) switching time and (b) release time for a 40-m-long bridge structure.

Fig. 6. Experimental setup for switching speed measurement.

a lack of accuracy in our simulations to model the real anchor pads influence on shorter beams. B. Switching Speed Measurements For switching speed measurements, another specific test bench has been used and is shown in Fig. 6 (from [9]). The measurement principle consists in detecting an amplitude modulation on a CW RF signal due to the capacitive impedance change induced by the aluminum bridge motion between its up- and down-state positions. However, measurements are performed in the time domain by using a diode-based RF detector connected to a standard oscilloscope. A selective bandpass filter has been introduced just before the RF detector to eliminate all parasitic signals. Consequently, the recorded signal at the detector output only relies on the RF signal amplitude modulation at the applied CW frequency, i.e., in our case, 10 GHz, increasing the detection accuracy. MEMS structures have been actuated using a 100-Hz monopolar square signal, which can be amplified until 200 V. One can notice that no bias T has been used in this setup, mainly due to the additional rise time they can induce on the

applied actuation voltage that could be problematic for sub-microsecond switching times measurement. Nevertheless, our LF amplifier induces a rise time on applied actuation voltage in the microsecond range, as can be seen in Fig. 7. As a result, the bridge motion is affected and the measured signal at RF detector output is quite difficult to be accurately measured since it is a sub- s switching time. However, for 40- m-long bridge structures (Fig. 7), we can reasonably estimate that these switching times are at least in the 400-ns range. Further development is in progress to improve the measurement technique. of these structures has been Likewise, the release time measured and is shown in Fig. 7(b). The measured release time of 40- m-long bridges is around 500 ns, demonstrating the high pull-up pressure induced by the high boat-like bridge stiffness. One can notice that high pull-up pressure will improve the reliability of such devices, limiting stiction phenomena. C. Reliability Tests The different structures have also been subjected to extensive cycling tests to evaluate their mechanical reliability. They have been periodically cycled using a 60-kHz bipolar square signal between their up- and down-state positions, while their mechanical resonance frequency has been measured every 100 million cycles. The tested beams have demonstrated no significant changes in mechanical resonance frequency ( 1 kHz) until 14 billion cycles when tests have been stopped without seeing any notable degradation. Structures have been also subjected to thermal cycling under vacuum from 20 C to 140 C. The mechanical resonance frequency has also been recorded versus the temperature, as shown

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Fig. 8. Measured mechanical resonance frequency variation as a function of temperature.

in Fig. 8 for 50- m-long boat-like beams. It can be seen that the induced mechanical resonance frequency variation is only 5% (1375–1305 kHz) in the 20 C to 140 range. IV. SUB-MICOSECOND RF MEMS SWITCHED CAPACITORS A. RF Design

Fig. 9. Top- and cross-view sections of our proposed RF capacitor with its modeled parasitic capacitance.

Due to such small dimensions of the microbeams, the RF design of the capacitors based on boat-like bridges has been especially studied using Agilent Momentum2 full-wave electromagnetic simulator. The switched capacitors presented here are implemented in a series configuration on a low permittivity quartz substrate to limit losses through the substrate. As the beams are suspended only 0.3 m above the capacitive contact electrode, the maximum capacitance contrast between up- and down-state positions cannot be more than 4 with our current fabrication process. Consequently, the actuation electrode has to be especially designed in order to achieve a parasitic capacitance as low as possible, but ensuring a moderate actuation voltage. Indeed, this actuation electrode induces a capacitive coupling between the design input and output, as shown in Fig. 9, strongly increasing the global parasitic capacitance. This parasitic capacitance has been lowered as allows our fabrication process, but is still in the range of the capacitor up-state capacitance (ten femtofarads) and, therefore, constitutes the main capacitor capacitance ratio limitation in this design. B. Fabrication Process The used fabrication process is described in Fig. 10. It starts with the capacitor biasing network fabrication using a resis) to pattern the pull-down electrode, comtive material (4 k bined to the evaporated gold biasing line. This network is then fully passivated using a 400-nm-thick alumina layer deposited by pulsed laser deposition. The following step consists in a 150-nm-thick gold metallization deposition and patterning to form the capacitor CPW RF lines and RF contact electrodes (only evaporated gold made). These RF contact electrodes are then locally covered with a 100-nm-thick alumina layer also deposited by pulsed laser deposition to ensure a capacitive contact 2Agilent

Technol., Palo Alto, CA. [Online]. Available: www.agilent.com

Fig. 10. Fabrication process of proposed RF MEMS capacitors.

in the capacitor down-state position. A 0.3- m polymethylglutarimide (PMGI) photoresist layer is deposited and patterned as a sacrificial layer for the bridge membrane. The next step consists of a 3- m standard positive photoresist layer deposition, which is exposed using a specific process in order to realize a liftoff resist mold with 20 angle edges. A 0.35- m-thick aluminum layer is then deposited by thermal evaporation and lifted off to form the boat-like shape. Finally, bridge structures are fully released removing the sacrificial layer and are dried in a CO (carbon dioxide) critical point dryer. Fig. 11 shows a closer

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Fig. 11. Fabricated RF capacitor implemented over a CPW line on quartz substrate.

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Fig. 13. Comparison of measured reflection coefficient for no actuated 60-m-long bridge capacitor and corresponding test structure fabricated without bridge.

Fig. 14. Lumped model of the capacitor implemented on CPW line. Fig. 12. Measured RF performance of 60-m-long bridge capacitor and fitted down- and up-state capacitances.

view of fabricated boat-like structures following the presented process. C. RF Measurements The fabricated switched capacitors RF performance has been measured on-wafer with an HP 8722ES network analyzer using -paa single open-load-thru calibration technique. Resulting rameters are presented in Fig. 12 for the up- and down-state positions actuated using a 2-Hz bipolar signal. Resulting capacitances in both states have been extracted by fitting techniques. Hence, the up-state capacitance is only 21 fF and the down-state capacitance is 49 fF, which results in a capacitance ratio of 2.3, considering the parasitic capacitance effect. The parasitic capacitance has been also evaluated using test structures fabricated without a bridge. The extracted value of this capacitance is around 11 fF and is in the same range of the bridge up-state capacitance, as discussed earlier. The capacitor has been also evaluated using reflection coefficient measurements. Indeed, as one can see in Fig. 13, the measured characteristics progressively leave the Smith chart edge

when the frequency increases. The same behavior has also been observed on test structures fabricated without a bridge, permitting to conclude that the main source of losses does not come from the MEMS structure, but mainly from the capacitor CPW line resistive losses and biasing network. Fig. 13 shows that test structures without bridge and capacitor structures present very similar losses levels and only a small capacitance change, as we have seen before. As the measured capacitance is very low, the electrical figure of merit ( ) of this device is difficult to be accurately measured and extracted, especially for low frequencies. Consequently, a lumped model, presented in Fig. 14, has been considered to evaluate both test and capacitor structures and to evaluate the MEMS intrinsic value by fitting the measurement performances. In this figure, the MEMS intrinsic capacitor is modeled by a associated in series with a bridge resiscapacitance . Other losses are considered using a series tance close to 110 for the CPW line. This important resistance value is mainly due to the small metallization thickness used for the CPW line in our case. Whereas the biasing influence , is taken into account considering parasitic capacitances , and and a resistance representing the resistive actuation electrode (several kiloohms). As shown in Fig. 15,

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Fig. 15. Measured and extracted intrinsic capacitor.

Q for capacitor structure, test structure, and

[7] D. Mercier, P.-L. Charvet, P. Berruyer, C. Zanchi, L. Lapierre, O. Vendier, J.-L. Cazaux, and P. Blondy, “A DC to 100 GHz high performance ohmic shunt switch,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, vol. 3, pp. 1931–1934. [8] D. Mercier, K. Van Caekenberghe, and G. M. Rebeiz, “Miniature RF MEMS switched capacitors,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, 4 pp. [9] B. Lacroix, A. Pothier, A. Crunteanu, C. Cibert, F. Dumas-Bouchiat, C. Champeaux, A. Catherinot, and P. Blondy, “CMOS compatible fast switching RF MEMS varactors,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 1072–1075. [10] G. M. Rebeiz, RF MEMS Theory, Design, and Technology. New York: Wiley, 2003. [11] D. Mercier, A. Pothier, and P. Blondy, “Monitoring mechanical characteristics of MEMS switches with a microwave test bench,” presented at the 4th Micro and Nano Technol. for Space Round Table, Noorwijk, The Netherlands, Jun. 2003. Benjamin Lacroix received the Master degree in electrical engineering from the University of Limoges, Limoges, France, in 2005, and is currently working toward the Ph.D. degree at the XLIM Research Institute, University of Limoges. His current research interests mainly include design and fabrication of RF MEMS components and systems as tunable devices (phase shifters, reconfigurable filters).

despite the small presented by the measured component, the can be reasonably estimated to be higher intrinsic MEMS than 100 at 10 GHz and 50 at 20 GHz because losses induced only by the aluminum bridge are low. Improvement on the fabrication process of these switched capacitor components (i.e., thicker metallic CPW line layer use) is in progress to confirm this assumption. V. CONCLUSION We have shown an original geometrical optimization concept for RF MEMS components in order to improve the switching speed of RF MEMS components. The proposed mechanical design have demonstrated very high mechanical resonance frequencies, between 1–3 MHz, and fast measured switching times under 400 ns. Reliability tests have been performed showing a stable mechanical behavior over 14 billion cycles and moderate temperature sensitivity. Based on a very simple liftoff technique, this devices fabrication can be easily post-processed, for example, on CMOS circuits. This concept has been extended to RF MEMS capacitors fabrication. The presented components have demonstrated a capacitance ratio of 2.3. Tunable devices as phase shifters or matching networks can be designed with such a capacitance ratio allowing fast reconfiguration. Arrays of our switched capacitors (to obtain higher capacitance values) are also suitable for other reconfigurable systems as VCOs or resonators requiring fast modulation. REFERENCES [1] G. M. Rebeiz, G.-L. Tan, and J. S. Hayden, “RF MEMS phase shifters: Design and applications,” IEEE Micro, vol. 3, no. 2, pp. 72–81, Jun. 2002. [2] E. R. Brown, “RF-MEMS switches for reconfigurable integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1868–1880, Nov. 1998. [3] J. Danson, C. Plett, and N. Tait, “Design and characterization of a MEMS capacitive switch for improved RF amplifier circuits,” in Proc. IEEE Custom Integrated Circuits, Sep. 2005, pp. 251–254. [4] D. Mercier and P. Blondy, “Millimeter-wave tune-all bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1175–1181, Apr. 2004. [5] A. Pothier et al., “Low-loss 2-bit tunable bandpass filters using MEMS DC contact switches,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 354–360, Jan. 2005. [6] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005.

Arnaud Pothier received the Ph.D. degree in electrical engineering from the University of Limoges, Limoges, France, in 2003. He is currently a Full-Time Researcher with the Centre National de la Recherche Scientifique (CNRS), XLIM, University of Limoges. His current research activity is focused on the tenability functions development and implementation for analogical communication modules using RF MEMS components especially for reconfigurable/programmable filters and phase shifters. Aurelian Crunteanu received the Phys. Eng. degree in optics and optical technologies, Master’s degree, and Ph.D. degree in physics from the University of Bucharest, Bucharest, Romania, in 1995, 1996, and 2000, respectively, and the Ph.D. degree in material sciences from the Claude Bernard University, Lyon 1, France, in 2001. His activities were oriented to laser-based deposition of thin films and microstructures from gas and liquid phase. From 2001 to 2003, as a Post-Doctoral Fellow with the Institute of Imaging and Applied Optics, Swiss Federal Institute of Technology, Lausanne, Switzerland, his research was oriented on fabrication and characterization of microstructures and nanostructures in laser-host materials, optical investigation of planar and channel waveguides, and laser-assisted thin-film deposition. Since 2003, he has been a Researcher with the Centre National de la Recherche Scientifique (CNRS), XLIM Research Institute, University of Limoges, Limoges, France. His current research activities are focused on the development of new materials for microelectronics and optics, RF-MEMS reliability, and optical switching using MEMS technology. Christophe Cibert received the Master degree in ceramic materials and surface treatments from the University of Limoges, Limoges, France, in 2003, and is currently working toward the Ph.D. degree at the University of Limoges. He is currently with the Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS) Laboratory, University of Limoges. His main research concerns the characterization of dielectric thin films deposited by pulsed laser deposition and plasma enhanced chemical vapor deposition for MEMS applications.

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Frédécric Dumas-Bouchiat received the Master degree in ceramic materials and surface treatment from the Université de Limoges, Limoges, France, in 2002, and the Ph.D. degree in material sciences from the Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS) Laboratory, Unité Mixte de Recherche (UMR) 6638, Centre National de la Recherche Scientifique (CNRS), Université de Limoges, in 2005. He is currently a Post-Doctoral Researcher involved with the oxide materials field with the Laboratoire d’Electrodynamique des Materiaux Avances (LEMA), Unité Mixte de Recherche (UMR) 6157, Centre National de la Recherche Scientifique (CNRS)–Commissariat à l’Énergie Atomique (CEA), Université de Tours, Tours, France. His main research concerns laser ablation, pulsed laser deposition of thin films, physical characterizations, and development of new materials for microelectronics and optics (metallic nanoparticles, dielectrics, and dielectric matrices doped with nanoparticles).

Corinne Champeaux received the Ph.D. degree in electrical engineering from the University of Limoges, Limoges, France, in 1992. Since 1992, she has been an Assistant Professor with the Faculty of Science, University of Limoges. She currently conducts research with the Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS) Laboratory, Unité Mixte de Recherche (UMR) 6638, Centre National de la Recherche Scientifique (CNRS), University of Limoges. Her main research interests are laser–matter interactions and pulsed-laser thin-films deposition techniques. She is involved with the development and fabrication of MEMS components through the elaboration of new materials and fabrication processes.

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Alain Catherinot received the Doctorate degree in physical sciences from the University of Orléans, Orléans, France, in 1980. He is currently a Professor with the Process and Material Sciences Department, University of Limoges, Limoges, France, where he conducts research on pulsed laser deposition (PLD) techniques with the Sciences des Procédés Céramiques et de Traitements de Surface (SPCTS) Laboratory, Unité Mixte de Recherche (UMR), Centre National de la Recherche Scientifique (CNRS), University of Limoges. His research interests include plasma and laser materials interactions, PLD of thin films and nanostructured materials, and their characterizations. He is also involved in MEMS fabrication using innovative deposition methods and materials.

Pierre Blondy (M’99) received the Ph.D. and Habilitation degrees from the University of Limoges, Limoges, France, in 1998 and 2003, respectively. From 1998 to 2006, he was with the Centre National de la Recherche Scientifique (CNRS), as a Research Engineer with the XLIM Laboratory, where he began research on RF MEMS technology and applications to microwave circuits. He is currently a Professor with the University of Limoges, leading a research group on RF MEMS. In 1997, he was a Visiting Scholar with The University of Michigan at Ann Arbor, and a Visiting Researcher with the University of California at San Diego, La Jolla, in 2006. He has authored or coauthored over 150 papers in referred journals and conferences and 19 invited presentations since 1988. He is a Reviewer for various journals in the area of MEMS and microwaves. Dr. Blondy was an Associate Editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS in 2006. He is a member of the IEEE International Microwave Conference Technical Program Committee since 2003.

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Alternating-Direction Implicit Formulation of the Finite-Element Time-Domain Method Masoud Movahhedi, Student Member, IEEE, Abdolali Abdipour, Senior Member, IEEE, Alexandre Nentchev, Mehdi Dehghan, and Siegfried Selberherr, Fellow, IEEE

Abstract—In this paper, two implicit finite-element time-domain (FETD) solutions of the Maxwell equations are presented. The first time-dependent formulation employs a time-integration method based on the alternating-direction implicit (ADI) method. The ADI method is directly applied to the time-dependent Maxwell curl equations in order to obtain an unconditionally stable FETD approach, unlike the conventional FETD method, which is conditionally stable. A numerical formulation for the 3-D ADI-FETD method is presented. For stability analysis of the proposed method, the amplification matrix is derived. Investigation of the proposed method formulation shows that it does not generally lead to a tri-diagonal system of equations. Therefore, the Crank–Nicolson FETD method is introduced as another alternative in order to obtain an unconditionally stable method. Numerical results are presented to demonstrate the effectiveness of the proposed methods and are compared to those obtained using the conventional FETD method. Index Terms—Alternating-direction implicit (ADI) technique, Crank–Nicolson (CN) method, finite-element time-domain (FETD) method, instability, Maxwell’s equations, unconditional stability.

I. INTRODUCTION VER THE past few years, considerable attention has been devoted to time-domain numerical methods to solve Maxwell’s equations for the analysis of transient problems. Due to their potential to generate wideband data and model nonlinear materials, numerical simulation schemes for simulating electromagnetic transients have grown increasingly popular in recent years. Several methods can be used to calculate the time-domain solution of electromagnetic problems. The well-known one is the finite-difference time-domain (FDTD) algorithm, introduced by Yee in 1966 [1]. The FDTD method discretizes the time-dependent Maxwell curl equations using central differences in time and space and a leap-frog explicit scheme for time integration. Its principal advantage is ease of implementation. However, this method suffers from the well-known staircase problem, and its removal requires much more effort in the sacrifice of computational resources. The

O

Manuscript received September 30, 2006; revised January 5, 2007 and March 3, 2007. This work was supported in part by the Iran Telecommunication Research Center. M. Movahhedi and A. Abdipour are with the Department of Electrical Engineering, AmirKabir University of Technology, Tehran, Iran (e-mail: [email protected]; [email protected]). A. Nentchev and S. Selberherr are with the Institut für Mikroelektronik, Technische Universität Wien, A-1040 Vienna, Austria. M. Dehghan is with the Department of Applied Mathematics, AmirKabir University of Technology, Tehran, Iran. Digital Object Identifier 10.1109/TMTT.2007.897777

finite-element time-domain (FETD) method combines the advantages of a time-domain technique and the versatility of its spatial discretization procedure [2]. In contrast, the FETD method can easily handle both complex geometry and inhomogeneous media, which cannot be achieved by the FDTD scheme. Over the past few years, a variety of FETD methods have been proposed [2]–[18]. These schemes fall into two categories. One directly discretizes Maxwell’s equations, which typically results in an explicit finite-difference-like leap-frog scheme. These approaches are conditionally stable [4]–[8]. The other discretizes the second-order vector wave equation, also known as the curl–curl equation, obtained by eliminating one of the field variables from Maxwell’s equations [9]–[18]. These solvers can be formulated to be unconditionally stable [9]–[13] or conditionally stable [14]–[18]. In an unconditionally stable scheme, the time step is not constrained by a stability criterion. However, it is limited by the required numerical accuracy in implementing the time derivatives of the electromagnetic fields. Therefore, if the minimum cell size in the computational domain is required to be much smaller than the wavelength, these schemes can be more efficient in terms of computer resources such as CPU time. In some simulations using the FETD method, it is preferred that the first-order Maxwell equations are directly considered and solved. For instance, implementation of the complex frequency shifted perfectly matched layer in open-region electromagnetic problems, which has better performance than the conventional perfectly matched layer, is easier and more efficient when directly applied to Maxwell’s curl equations [19]. However, the unconditionally stable methods for the FETD solution of the second-order vector wave equation are usually used. In this paper, we introduce two unconditionally stable vector FETD methods based on the alternating-direction implicit (ADI) and Crank–Nicolson (CN) schemes to directly solving first-order Maxwell’s equations. The ADI technique was first introduced to solve Maxwell’s curl equations using the finite-difference method. This algorithm is called the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method [20], [21]. We previously applied the ADI-FETD method for solving the 2-D TE wave [22]. In this paper, we extend this approach to the 3-D wave and introduce the 3-D ADI-FETD method. Moreover, another alternative for time discretization to obtain an unconditionally stable method for the FETD solution of the Maxwell equations based on the CN scheme is presented. It will be shown that the ADI method can be considered as a perturbation of the implicit CN formulation.

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ADI and CN schemes involve solving a linear system at each time step. When the ADI scheme is applied to a standard FDTD approach, the matrices are well structured and it renders a linear system that is only semi-implicit and easy to solve. That is, it leads to a 1-D solution of a tri-diagonal system of equations that must be factorized at each time step. However, this is neither the case for the ADI scheme, nor for the CN scheme with a finite-element approach. They lead to a fully implicit system. Several lumping techniques have been proposed in order to obtain explicit schemes without solving a linear system at each time step [14], [23]–[25]. Moreover, a recently developed approach avoids lumping altogether by constructing a set of orthogonal vector basis functions that yield a diagonal mass matrix [26], [27]. A most recent explicit FETD method, which is fundamentally different from traditional explicit FETD formulations for solving Maxwell’s equations, has been introduced [28]. This new explicit FETD is derived from a recently developed FETD decomposition algorithm [29] by extending domain decomposition to the element level. With the element-level decomposition, no global system matrix has to be assembled and solved as required in the implicit FETD, and each element is related to its neighboring elements in an explicit manner. Here, we explain the details of numerical formulations of the ADI- and CN-FETD solutions of the 3-D Maxwell equations. Moreover, some numerical results are provided to validate the proposed methods. This paper is organized in the following manner. In Section II, the essential principle of the ADI scheme for time discretization of time-dependent partial differential equations is presented. Section III describes the formulations of the proposed 3-D ADI-FETD method. Section IV presents and investigates the stability condition of the conventional and proposed schemes. The CN-FETD method as a more accurate unconditionally stable method is presented in Section V. In Section VI, the numerical results are shown. Finally, conclusions are presented in Section VII. II. ADI PRINCIPLE The ADI technique is well reported in the study of parabolic equations with finite elements [30]–[32]. In this paper, we use this technique to solve Maxwell’s curl equations and the contribution is relevant to wave propagation (hyperbolic equations). The ADI technique takes its name from breaking up a single implicit time step into two half time steps. In the first half time step, an implicit evaluation is applied to one dimension and an explicit evaluation is applied to the other, assuming two dimensions in the problem statement. For the second half time step, the implicit and explicit evaluations are alternated, or switched, between the two dimensions. The dimensions to alternate between are typically spatial; however, temporal variables can also be used [33]. For explanation of the ADI method as a technique for the development of an implicit integration scheme, the time-dependent curl vector equations of Maxwell’s equations are considered

(1)

These equations can be cast into six scalar partial differential equations in Cartesian coordinates. We consider the following scalar equation from the above given system: (2) By applying the ADI principle, which is widely used in solving parabolic equations [34], the computation of (2) for the FETD solution marching from the th time step to the th time step is broken up into two computational subadvancements: the advancement from the th time step to the th time step and the advancement from the th time step to the th time step. More specifically, the two substeps are as follows. th time step, 1) For the first half time step, i.e., at the the first partial derivative on the right-hand side of (2), i.e., , is replaced with its unknown pivotal values at the th time step; while the second partial derivatives , is replaced with its on the right-hand side, i.e., known values at the previous th time step. In other words, (3) th time 2) For the second half time step, i.e., at the , step, the second term on the right-hand side, i.e., is replaced with its unknown pivotal values at the th , is replaced time step; while the first term, i.e., th time step. with its known values at the previous In other words, (4) The above two substeps represent the alternations in the FETD recursive computation directions in the sequence of the terms, i.e., the first and second terms. They result in the implicit formulations, as the right-hand side’s of the equations contain the field values unknown and to be updated. The technique is then termed “the alternating direction implicit” technique. Attention should also be paid to the fact that no time-step difference (or lagging) between electric and magnetic field components is present in the formulations. Applying the same procedure to all of the other five scalar differential equations of Maxwell’s equations, one obtains the complete set of the implicit formula. III. FORMULATIONS OF THE 3-D ADI-FETD SCHEME The ADI-FETD solution of Maxwell’s equations for analyzing full 3-D electromagnetic problems is described here. The Maxwell curl equations governing the solution of a 3-D problem in a lossless medium have been given by (1). In these equais the electric field and tions, is the magnetic flux density. According to the ADI procedure for the time discretization, the following equations are obtained.

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th time step

is the electric field circulation along the th edge, where is the flux of the magnetic flux density through the th face, is the Whitney one-form vector basis function associated to the th edge, and is the Whitney two-form vector basis function associated to the th face [35] such that

(8) The Lebesgue spaces are defined by where

(9)

(5) •

th time step

The space is the space of all functions on domain that are square integrable, which is often referred to as the space of functions with finite energy [36]. For vector functions, , the corresponding space is denoted . For Whitney one-forms, the basis functions are well known , where and are by now. For example, for the edge nodes of the edge, it is (10) is the Lagrange interpolation polynomial at vertex where [35]. Similarly, the vector basis functions for Whitney two, where , , forms associated with a particular facet and are nodes of the face, can be written as (11) The Galerkin method is applied to the Maxwell curl equations (5) and (6) using the field approximations (7). Testing the first and three scalar equations of (5) and (6) with basis function yield the following the second three with basis function equations in the matrix form. th time step •

(6) Now we consider the finite-element solution of the above -volume is equations. The examined 3-D domain in the assumed to be discretized by a finite-element mesh composed of tetrahedral elements, edges, and faces. In each point of the element, , the electric field , and the magnetic flux density are approximated by edge and facet elements, respectively, as

(12) •

th time step

(7) (13)

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where the matrix entries are given by

(16) In this step, as in the previous one, the values of the magnetic flux density are first computed and then the values of the electric field are computed. IV. STABILITY ANALYSIS

(14) For tetrahedral elements, it can be easily seen that so and . Equations (12) and (13) can be further simplified for efficient computation. By substituting the expressions for and presented by the second equation of (12) and (13) into their first equations and transferring the local equations to a global system, one obtains the following. th time step •

In the linear theory of grid-based methods for the numerical solution of ordinary and partial differential equations, the success of numerical schemes is summed up in the well-known Lax equivalence theorem [37]: “consistency and stability imply convergence.” For general FETD methods, the consistency is selfevident and assured by their formulations [2]. Subsequently, the question of stability is of paramount importance. Here, we present and investigate the stability condition of both conventional and proposed schemes for time discretization of the finite-element method. One of the factors that affect the performance of a computational method is numerical dispersion. The proposed finite-element method, like other grid-based methods for solving Maxwell’s equations, such as the FDTD method [38] and ADI-FDTD method [39], [40], exhibits numerical dispersion and numerical anisotropy due to the finite grid and finite time sampling. The numerical dispersion relation for the time-domain vector finite-element method has been derived on a 3-D hexahedral grid [36]. Investigation of the numerical dispersion of the ADI finite-element method will be considered as future work. A. Conventional FETD Scheme In the conventional method for time discretization of the Maxwell equations, time is discretized such that the electric degrees of freedom will be known at whole time steps and the magnetic degrees of freedom will be known at the half time steps. This is often refereed to as leap-frog method. Using this method for time discretization and following the analysis of [41] gives the equations

(17) (15) By solving the above equations, we first obtain the values of the magnetic flux density at the th time step. Thereafter, the values of the electric field can be directly . For the second calculated using the values of half time step, we have the following. th time step •

and . The where and are symmetric positive definite. The source matrices term can be neglected for the stability analysis. These equations can be expressed in matrix form as (18) where

(19)

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The matrix in (18) is called the amplification matrix of the method. Stability of the above equation requires , where is the spectral radius of . A tedious, but straightforward calculation shows that the eigenvalues of are given by

and have been defined as (25), shown The matrices at the bottom of this page. and It is easy to show that . Thus, . Combination of the above two equations reads

(20) where

and is an eigenvalue of the matrix . Equivalently, and satisfy the generalized eigenvalue problem (21) is symmetric positive definite and the matrix is symmetric positive semidefinite. Thus, and the eigenvalues of the amplification matrix will have unit magnitude if and only if [36] The matrix

(22) A similar bound on the time step for stability of the nonorthogonal grid finite-difference time-domain (NFDTD) schemes and the generalized Yee (GY) methods was derived in [41]. In these methods, structure of the amplification matrix is similar to the for the conventional FETD scheme. structure of the matrix B. Proposed ADI-FETD Scheme For investigation of the stability condition of the proposed FETD method, we first derive the amplification matrix of the method. In general, (15) and (16) can be summarized as the following matrix form: (23) (for advancement from the th to

th time step) (24)

(for advancement from

th to

th time step).

(26) or simply (27) By checking the magnitude of the eigenvalues of , one can determine whether the proposed scheme is unconditionally are equal stable; if the magnitudes of all the eigenvalues of to or less than unity, the proposed scheme is unconditionally stable; otherwise it is potentially unstable [30]. Direct finding of eigenvalues of the amplification matrix appears to be very difficult. Therefore, an indirect approach can be used with which the ranges of the eigenvalues can be determined. For instance, the Schur–Cohn–Fujiwa criterion can , with be applied, where the characteristic polynomial of its roots being the eigenvalues, is examined [42]. This investigation and analytical proof for unconditional stability can be considered as an open problem and will remain a topic for future research. It is important to note that numerical results obtained from many simulations show that the scheme is stable even for large time steps. Thus, from the implementation aspect, the method can be considered as an unconditionally stable scheme. V. ALTERNATIVE DESCRIPTION OF THE ADI METHOD One of the principal advantages of the ADI-FDTD method is that it renders a system that is only semi-implicit. That is, it leads to a 1-D solution of a tri-diagonal system of equations that must be factorized at each time step. Since only the 1-D problem is being solved, the additional computational cost is significantly reduced. As a result, there is a tendency to sacrifice accuracy of the ADI time-integration to maintain a semi-implicit solution procedure for the FDTD method [43]–[47]. Generally, in the presented method (the ADI-FETD method), this process is lost

(25)

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since, for an arbitrary tessellation, the field coefficients cannot be decoupled into Cartesian projections. Consequently, a fully implicit procedure is resulted. However, this is not the case with the special case of an orthogonal Cartesian mesh (orthogonal hexahedral element) for the finite-element method. The use of mass lumping techniques in this situation leads to a system with well-structured matrices [36] and, hence, the resulting linear systems can be readily solved by using methods such as ADI-FDTD. In fact, the use of mass lumping techniques for solving the proposed time-domain finite-element method with the orthogonal hexahedral element results in a semi-implicit procedure. Moreover, the proposed method can be more efficient than the other possible unconditionally stable schemes in some applications such as hybrid methods. Hybrid methods, which combine the desirable features of two or more different techniques, are being developed to analyze complex electromagnetic problems that cannot be otherwise resolved conveniently and/or accurately by using the methods individually. One of the most efficient hybrid methods is the finite-element finite-difference time-domain (FE-FDTD) hybrid method. In this method, the FDTD is used to treat large regions with less complexities, while using the FETD method is used to handle complex boundaries and structures [48]–[50]. An alternative to this hybrid method is the combination of unconditionally stable FDTD and FETD methods. The ADI-FDTD method is always used as an unconditionally stable method for the finite-difference method. By using the finite-element method whose time discretization is derived from the same scheme as in the FDTD method (i.e., the proposed method in this paper), the overall performance of the unconditionally stable hybrid method is improve. Therefore, the proposed FETD method can be efficiently used in the unconditionally stable FE-FDTD hybrid method. Another application where the ADI scheme can be efficiently used is the hybrid finite-element boundary-integral (FE-BI) method. This hybrid method is a powerful numerical technique for solving open-region electromagnetic problems [51], [52]. This method uses an artificial boundary to divide the infinite solution domain into interior and exterior regions in which fields are represented using finite elements and boundary integrals, respectively. The resulting hybrid method permits accurate and efficient analysis of complex electromagnetic phenomena, especially those involving inhomogeneous media. In [53] and [54], it has been shown that the ADI method is an efficient method for solving problems with nonlocal boundary conditions (integral boundary conditions). Therefore, the proposed method, which is based on the ADI method, can also be efficiently implemented for the FE-BI method. Recent analysis shows that although the ADI scheme for time discretization has second-order accuracy in time, the method exhibits a splitting error associated with the square of the timestep size [45]. Having an asymptotic second-order accuracy, the splitting error becomes dominant in regions with larger spatial derivatives. This can be detrimental for modeling problems where strong near-field coupling occurs and/or structures contain field singularities such as in tips and corners. It can be shown that the ADI method applied to the Maxwell equations is a perturbation of the implicit CN formulation [43]. The CN

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scheme is a well-known implicit algorithm in numerical computation. The CN algorithm advances time by a full time-step size with one marching procedure. 3-D Maxwell’s equations can be written as (28) where tial derivative operators defined in [47]. Applying the CN scheme at time step

and

are par, we get

(29) The above equation can be factorized as

(30) or

One-step approximation (31) Error The “One-step approximation” can be written in two stages, which become equivalent to the two stages of the ADI scheme [47]. Since the ADI method solves “One-step approximation” instead of (31) (CN method), it introduces a splitting error of the form (32) to the solution. The effect of this splitting error depends on three factor); 2) the spatial factors, i.e.: 1) the time-step size ( derivatives of the field ( factor); and 3) the temporal varifactor). When field variaation of the field tion and/or the time step size is large, the splitting error becomes more pronounced. To obtain the CN-FETD method formulation, the Maxwell curl equations is projected into its weak form using the Galerkin method and spatially discretizing using finite elements. Based on the CN formulation for time discretization, the following equations are derived:

(33)

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Fig. 1. Time variations of current density, I used as an excitation for the 3-D rectangular cavity.

where (33) can be further simplified as

and

. Equation

(34) As can be seen from (34), the CN formulation is very similar to the ADI scheme for the FETD solution of Maxwell’s equations. From Section IV, in the ADI formulation [see (15) and (16)], we and . have

Fig. 2. Time-domain electric fields at the center of the cavity recorded with the conventional FETD method and the proposed ADI-FETD method. (a) Conven: t . tional FETD solution that becomes unstable with t (b) Proposed 3-D ADI-FETD solution with t . t

1 = 1 05 2 1 1 = 15 2 1

VI. NUMERICAL RESULTS Here, to validate the accuracy and stability of the proposed 3-D ADI- and CN-FETD methods, a simple example of the excitation of a lossless 3-D rectangular cavity with perfectly conducting walls is studied. The dimensions of the cavity are 1.0 mm 0.5 mm 1.5 mm. For comparative purposes, this cavity is simulated and its resonant frequencies are obtained with both the conventional and proposed FETD methods. In the conventional FETD method, a leap-frog scheme is applied for time discretization. The computation was carried out using a mesh with tetrahedral elements. This mesh contains 27550 tetrahedra and 51710 edges, which is used for the ADI- and CN-FETD methods and also for the conventional FETD method. On the cavity surfaces, which are assumed to be perfectly conducting walls since the tangential components of the electric field is zero, we impose a homogeneous Dirichlet boundary condition, i.e., we set to zero all the electric field circulations associated with edges belonging to these surfaces.

For each half time step, the updating of the electric and magnetic fields requires solving two matrix equations. The system . Since this mawill be written generically as trix is not time dependent, it can be factorized once before the step-by-step procedure to obtain an explicit scheme. For matrix factorization, we use the direct solver in the SGI’s scientific computing software library (SCSL), which turns out to be a highly efficient direct solver. Note that the factorization is performed only once and that only forward and backward substitutions are needed in each time step. Noting this on (15) and (16) shows that for forming the general matrix system to update the magnetic field, the matrix must be computed. We also use the SCSL routines, which calculate the infactorization, to compute the matrix inverse, diverse after rectly. The resonant frequencies can be obtained by launching a time signal and applying the Fourier transform on the time response.

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TABLE I PROPOSED ADI-FETD, CN-FETD, AND CONVENTIONAL FETD SIMULATION RESULTS WITH DIFFERENT TIME STEPS

An excitation sinusoidally modulated Gaussian is used as current density in this simulation given by (35) where and current density.

,

, , . Fig. 1 shows the time variations of this

A. Numerical Verification of the Stability First, we investigate the stability of the proposed ADI- and CN-FETD methods. Simulations were run for the homogeneous 3-D rectangular cavity with both the conventional and proposed methods having a time step that exceeds the limit determined by the stability condition for the conventional FETD method, i.e., s. Fig. 2 shows the in our case, electric field recorded at the center of the cavity. was used with the conventional FETD method, was while a 15 times larger time step used with the ADI-FETD scheme. As can be seen, the conventional FETD quickly becomes unstable [see Fig. 2(a)], while the ADI FETD remains with a stable solution [see Fig. 2(b)]. We also extended the simulation time to a much longer period with the proposed scheme. No instability was observed. For another scheme, the CN-FETD method, the same large time step was used and the stable solution was shown. B. Numerical Accuracy Versus Time Step Since the proposed ADI- and CN-FETD methods are shown to be stable for very large time steps, the selection of the time step is no longer restricted by stability, but by modeling accuracy. As result, it is interesting and meaningful to investigate how the time step will affect accuracy. For comparative purposes, both the conventional FETD method and proposed methods were used to simulate the cavity s was chosen again. The time step and fixed with the conventional FETD method, while different were used with the proposed ADI- and values of time step CN-FETD methods to check for the accuracy. Table I presents the simulation results for the dominant mode, which is in the cavity. The dominant mode is determined according to the frequency components of the excitation and its position. As can be seen, the relative errors of the proposed unconditionally stable methods increase with the time step, while this increment is more for the ADI method in contrast with the CN scheme. These errors are completely due to the modeling accuracy of the numerical algorithm such as the numerical dispersion. The

Fig. 3. Relative errors of the conventional FETD method, proposed ADI-FETD, and CN-FETD methods as a function of relative time step t= t . The dashed line represents the unstable point of the conventional FETD scheme.

1 1

tradeoff to the increased errors is, however, the reduction in the number of iterations and CPU time. By increasing the time step, the conventional FETD solutions become unstable, while the proposed FETD method continues to produce stable results. Fig. 3 illustrates a plot of the errors for another excited versus the discrete time step mode of the cavity computed using the conventional FETD method and the proposed ADI- and CN-FETD methods. For clarity, a relative is used. As can be seen, at low time-step , the errors of both the conventional FETD method and the proposed FETD methods are almost the same. , the conventional However, after FETD solutions diverge, while the proposed FETD methods continue to produce stable results with increasing errors that may or may not be acceptable depending on the applications and users’ specifications. VII. CONCLUSION This paper has introduced the 3-D ADI-FETD method for solving first-order Maxwell’s equations. Using an ADI scheme, which is applied directly to the Maxwell curl equations for time discretization, leads to an implicit FETD method. For stability analysis, the amplification matrix of the proposed method was derived, and it was shown, with numerical simulations, that an unconditionally stable scheme is achievable. Moreover, numerical simulation shows that this method is very efficient and the

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results agree very well with those of the conventional FETD method, which uses a leap-frog scheme for time discretization. Investigation of the ADI-FETD formulation shows that this method, unlike the ADI-FDTD method, does not generally lead to a tri-diagonal system of equations. Hence, we used a CN time integration and introduced unconditionally stable CN-FETD method. When field variation and/or the time step size is large, the CN-FETD method becomes more accurate than the ADI-FETD method. Since in some electromagnetic simulations it is preferred that the first-order Maxwell equations are directly solved, the proposed unconditionally stable methods can be very efficient and useful. They can decrease the simulation time because the time step is no longer restricted by the numerical stability, but by the modeling accuracy of the FETD algorithm. ACKNOWLEDGMENT The authors would like to thank the three reviewers of this paper for their constructive comments and helpful suggestions. REFERENCES [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. 8, pp. 302–307, Aug. 1966. [2] J. F. Lee, R. Lee, and A. C. Cangellaris, “Time-domain finite element methods,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 430–442, Mar. 1997. [3] K. Mahadevan, R. Mittra, and P. M. Vaidya, “Use of Whitney’s edge and face elements for efficient finite element time domain solution of Maxwell’s equations,” J. Electromagn. Waves Applicat., vol. 8, no. 9/10, pp. 1173–1191, 1994. [4] K. Choi, S. J. Salon, K. A. Connor, L. F. Libelo, and S. Y. Hahn, “Time domain finite analysis of high power microwave aperture antennas,” IEEE Trans. Magn., vol. 31, no. 5, pp. 1622–1625, May 1995. [5] M. Feliziani and F. Maradei, “An explicit-implicit solution scheme to analyze fast transients by finite elements,” IEEE Trans. Magn., vol. 33, no. 3, pp. 1452–1455, Mar. 1997. [6] A. C. Cangellaris, C. C. Lin, and K. K. Mei, “Point-matched time-domain finite element methods for electromagnetic radiation and scattering,” IEEE Trans. Antennas Propag., vol. AP-35, no. 10, pp. 1160–1173, Oct. 1987. [7] J. T. Elson, H. Sangani, and C. H. Chan, “An explicit time-domain method using three-dimensional Whitney elements,” Microw. Opt. Technol. Lett., vol. 7, pp. 607–610, Sep. 1994. [8] M. F. Wong, O. Picon, and V. F. Hanna, “A finite-element method based on Whitney forms to solve Maxwell equations in the time-domain,” IEEE Trans. Magn., vol. 31, no. 5, pp. 1618–1621, May 1995. [9] G. Mur, “The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media,” IEEE Trans. Magn., vol. 28, no. 3, pp. 1130–1133, Mar. 1992. [10] J. F. Lee and Z. Sacks, “Whitney elements time domain (WETD) methods,” IEEE Trans. Magn., vol. 31, no. 5, pp. 1325–1329, May 1995. [11] S. D. Gedney and U. Navsariwala, “An unconditionally stable finite-element time-domain solution of the vector wave equation,” IEEE Microw. Guided Wave Lett., vol. 5, no. 5, pp. 332–334, May 1995. [12] W. P. Capers, Jr., L. Pichon, and A. Razek, “A 3-D finite element method for the modeling of bounded and unbounded electromagnetic problems in the time domain,” Int. J. Numer. Modeling, vol. 13, pp. 527–540, 2000. [13] R. Fernades and G. Fairweather, “An alternating direction Galerkin method for a class of second order hyperbolic equations in two space dimensions,” SIAM J. Numer. Anal., vol. 28, no. 5, pp. 1265–1281, 1991. [14] D. R. Lynch and K. D. Paulsen, “Time-domain integration of the Maxwell equations on finite elements,” IEEE Trans. Antennas Propag., vol. 38, no. 12, pp. 1933–1942, Dec. 1990. [15] J. F. Lee, “WETD—A finite-element time-domain approach for solving Maxwell’s equations,” IEEE Microw. Guided Wave Lett., vol. 4, no. 1, pp. 11–13, Jan. 1994.

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[40] F. Zheng and Z. Chen, “A study of the numerical dispersion relation for the 2-D ADI-FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 9, pp. 405–407, Sep. 2003. [41] S. D. Gedney and J. A. Roden, “Numerical stability of nonorthogonal FDTD method,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 231–239, Feb. 2000. [42] E. I. Jury, Inners and Stability of Dynamic Systems. New York: Wiley, 1974. [43] S. G. Garcia, T.-W. Lee, and S. C. Hagness, “On the accuracy of the ADI-FDTD method,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 31–34, Dec. 2002. [44] G. Sun and C. W. Trueman, “Approximate Crank–Nicolson schemes for the 2-D finite-difference time-domain method for TE waves,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2963–2972, Nov. 2004. [45] G. Sun and C. W. Trueman, “Unconditionally-stable FDTD method based on Crank–Nicolson scheme for solving three-dimensional Maxwell equations,” Electron. Lett., vol. 40, no. 10, pp. 300–301, May 2004. [46] S. Wang, F. L. Teixeira, and J. Chen, “An iterative ADI-FDTD with reduced splitting error,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 92–94, Feb. 2005. [47] J. Lee and B. Fornbergb, “Some unconditionally stable time stepping methods for the 3-D Maxwell’s equations,” J. Comput. Appl. Math., vol. 166, pp. 479–523, 2004. [48] D. Koh, H. Lee, and T. Itoh, “A hybrid full-wave analysis of via-hole grounds using finite-difference and finite-element time-domain methods,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2217–2223, Dec. 1997. [49] T. Rylander and A. Bondeson, “Application of stable FEM-FDTD hybrid to scattering problems,” IEEE Trans. Antennas Propag., vol. 50, no. 2, pp. 141–144, Feb. 2002. [50] N. V. Venkatarayalu, G. Y. Beng, and L.-W. Li, “On the numerical errors in the 2-D FE/FDTD algorithm for different hybridization schemes,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 4, pp. 168–170, Apr. 2004. [51] D. Jiao, M. Lu, E. Michielssen, and J. M. Jin, “A fast time-domain finite-element-boundary integral method for electromagnetic analysis,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1453–1461, Oct. 2001. [52] D. Jiao, A. A. Ergin, B. Shanker, E. Michielssen, and J. M. Jin, “A fast higher-order time-domain finite element boundary integral method for 3-D electromagnetic scattering analysis,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1192–1202, Sep. 2002. [53] M. Dehghan, “Alternating direction implicit methods for two-dimensional diffusion with a non-local boundary condition,” Int. J. Comput. Math., vol. 27, no. 3, pp. 349–366, 1999. [54] M. Dehghan, “A new ADI technique for two-dimensional parabolic equation with an integral condition,” Comput. Math. Appl., vol. 43, pp. 1477–1488, 2002. Masoud Movahhedi (S’06) was born in Yazd, Iran, in 1976. He received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1998, the M.Sc. degree in electrical engineering from AmirKabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2000, and is currently working toward the Ph.D. degree in electrical engineering at AmirKabir University of Technology. From December 2005 to September 2006, he was a Visiting Student with the Institute for Microelectronics, Technische Universität Wien, Vienna, Austria. His research interests are computer-aided design of microwave integrated circuits, computational electromagnetic, semiconductor high-frequency RF modeling, and interconnect simulations. Mr. Movahhedi is a reviewer for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He was the recipient of the GAAS-05 Fellowship presented by the GAAS Association to young graduate researchers for his paper presented at GAAS2005. He was also the recipient of the Electrical Engineering Department Outstanding Student Award in 2005 and 2006.

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Abdolali Abdipour (M’97–SM’06) was born in Alashtar, Iran, in 1966. He received the B.Sc. degree in electrical engineering from Tehran University, Tehran, Iran, in 1989, the M.Sc. degree in electronics from Limoges University, Limoges, France, in 1992, and the Ph.D. degree in electronic engineering from Paris XI University, Paris, France, in 1996. He is currently a Professor with the Electrical Engineering Department, AmirKabir University of Technology (Tehran Polytechnic), Tehran, Iran. He is also currently Head of the Electrical Engineering Department, AmirKabir University of Technology and Director of the Radio Communication Center of Excellence. He has authored or coauthored over 115 papers in refereed journals and local and international conferences. He authored Noise in Electronic Communication: Modeling, Analysis and measurement (AmirKabir Univ. Press, 2005, in Persian) and Transmission Lines (Nahre Danesh Press, 2006, in Persian). His research areas include wireless communication systems (RF technology and transceivers), RF/microwave/millimeter-wave circuit and system design, electromagnetic (EM) modeling of active devices and circuits, high-frequency electronics (signal and noise), nonlinear modeling, and analysis of microwave devices and circuits.

Alexandre Nentchev was born in Sofia, Bulgaria, in 1971. He received the Diplomingenieur degree in electrical engineering from the Technische Universität Wien, Vienna, Austria, in 2004, and is currently working on his Ph.D. degree at the Institut für Mikroelektronik, Technische Universität Wien, Vienna, Austria. His scientific interests include 3-D interconnect simulation of multilevel wired very large scale integration (VLSI) circuits and software technology.

Mehdi Dehghan was born in Eghlid-Fars, Iran, in 1957. He is currently an Associate Professor with the Department of Applied Mathematics, AmirKabir University of Technology, Tehran, Iran. His research interests include numerical solutions of partial differential (and integral) equations, numerical integration, numerical linear algebra, and difference equations. He has authored or coauthored approximately 100 papers in refereed journals and has presented several papers at local and international conferences.

Siegfried Selberherr (M’79–SM’84–F’93) was born in Klosterneuburg, Austria, in 1955. He received the Diplomingenieur degree in electrical engineering and Doctoral degree in technical sciences from the Technische Universität Wien, Vienna, Austria, in 1978 and 1981, respectively. Since 1984, he has held the Venia Docendi on computer-aided design. Since 1988, he has been the Chair Professor of the Institut für Mikroelektronik, Technische Universität Wien. From 1998 to 2005, he was Dean of the Fakultät für Elektrotechnik und Informationstechnik, Technische Universität Wien. His current research interests are modeling and simulation of problems for microelectronics engineering.

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Analysis of Wideband Dielectric Resonator Antenna Arrays for Waveguide-Based Spatial Power Combining Yizhe Zhang, Student Member, IEEE, Ahmed A. Kishk, Fellow, IEEE, Alexander B. Yakovlev, Senior Member, IEEE, and Allen W. Glisson, Fellow, IEEE

Abstract—Wideband probe-fed dielectric resonator antenna (DRA) arrays in an oversized dielectric loaded waveguide with hard horn excitation and radiating in free space are investigated for their use in waveguide-based spatial power-combining systems. A single embedded DRA element excited by a coaxial probe is analyzed first inside a hollow rectangular waveguide and a TEM waveguide. 1-D embedded DRA arrays are then studied inside the - and -plane sectoral hard horns. Finally, an entire spatial power-combining system with a 2-D embedded DRA array radiating in free space is analyzed. The analysis of the entire system is based on the finite-difference time-domain method with region-by-region discretization and subgridding schemes. The system’s prototype of 2-D 3 3 embedded DRA array inside a pyramidal hard horn was fabricated and simulation results were compared with measurement results, showing a good agreement. Numerical results for radiation patterns are demonstrated for the examples of 2-D embedded dielectric resonator arrays excited by a pyramidal hard horn and radiating in free space. Index Terms—Dielectric loaded waveguide, dielectric resonator antenna (DRA) arrays, finite-difference time-domain (FDTD) method, pyramidal hard horn, sectoral hard horn, spatial power combining.

I. INTRODUCTION

N RECENT years, spatial power combining has been proposed in order to combine in free space the power produced by many solid-state devices at microwave and millimeter-wave frequencies [1]–[11]. This represents an alternative to conventional power combining, which becomes less useful at these frequencies due to high conduction and material losses. In general, spatial power combiners can be grouped into open- and closed-boundary (waveguide based) [8]–[11] amplifier arrays. Open-boundary (also called free space) spatial power combiners were designed with the use of optical elements (lenses) and Gaussian beam excitation, where receive and transmit antennas were placed in the far-field region. In waveguide-based spatial

I

Manuscript received October 24, 2006; revised January 16, 2007. This work was supported in part by the National Science Foundation under Grant ECS0220218. The authors are with the Department of Electrical Engineering Center for Applied Electromagnetic Systems Research, The University of Mississippi, University, MS 38677-1848 USA (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.2007.896777

power combiners, the amplifier array was placed inside an oversized hard-walled waveguide in close proximity to receive and transmit hard horn antennas. Due to their compact structure, the waveguide-based spatial power combiners can be analyzed more accurately and efficiently. In spatial power-combining systems, the antenna array elements must be appropriately chosen and accurately modeled in order to obtain wide frequency bandwidth and high powercombining efficiency. Dielectric resonator antennas (DRAs) became potentially useful as antenna elements a few decades ago [12]–[14] due to their attractive characteristics, including low profile, light weight, low cost, and inherently wide bandwidth. Compared with microstrip antennas, which suffer from conduction loss and surface waves in antenna array applications, DRAs have high radiation efficiency and high power-handling capability. DRA arrays also require a smaller horn aperture compared with microstrip patch antenna arrays, which can decrease the size of the whole system or further increase the number of the elements inside a horn of the size used with microstrip patches. All these features make DRAs advantageous as antenna elements for array applications. Recently, extensive research has been done to achieve DRA bandwidth enhancement by using different dielectric constant materials and special shaping such as embedded DRAs [15], stacked DRAs [16], and disc-ring DRAs [17], [18]. Composed of two different dielectric materials, the embedded DRA element can be optimized for wideband operation over the band of interest. The spatial power-combining system is formed by an array of amplifying unit cells with each cell receiving, amplifying, and then transmitting a signal or radiating a signal into free space. The incident signal from the transmit horn couples to an array of embedded DRA elements (with geometry shown in Fig. 1). After amplification, the signal is coupled back to the receiving antenna elements in the same manner, re-radiated through the DRA array, and then collected by the receive horn or radiated into free space directly. In order to achieve high efficiency from the power-combining system, the incident power distribution must be uniform with the same magnitude and phase across the plane of the antenna array. To obtain a uniform excitation of amplifier arrays, hard electromagnetic walls have been recently realized by dielectric loading along the narrow sides of the waveguide to create a TEM waveguide and horns in order to obtain a uniform field distribution in the waveguide cross section and the horn aperture [19]–[21]. Therefore, all the array elements can be illuminated with approximately the same power, resulting in high power-combining

0018-9480/$25.00 © 2007 IEEE

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Fig. 2. Implementation of the subgridding scheme in the FDTD technique. Fig. 1. Geometry of coaxial probe-fed embedded DRA array inside a dielectric loaded horn.

efficiency. It should be noted that this paper concerns the analysis of a passive system, and the inclusion of active elements is out of the scope of this paper. In this paper, a coaxial probe-fed single embedded DRA element is studied first in a hollow rectangular waveguide and waveguide loaded with hard walls. 1-D embedded DRA arrays are then studied inside the - and -plane sectoral hard horns. Finally, an entire spatial power-combining system with a 2-D embedded DRA array, as shown in Fig. 1, is analyzed. The system consists of a coaxial probe-fed embedded DRA array, dielectric loaded oversized waveguide, and pyramidal hard horn. A uniform field distribution is achieved across the horn aperture at the design frequency by using hard walls in the oversized waveguide and pyramidal horn. The finite-difference time-domain (FDTD) method is utilized as the numerical method to analyze this electrically large and complicated spatial power-combining system. The region-by-region method [22] and subgridding method [23] are combined with the FDTD algorithm to significantly reduce memory requirements and simulation time. The scattering parameters are computed using the proposed FDTD method and show a good agreement with the measured results and results obtained from the Ansoft’s High Frequency Structure Simulator (HFSS) commercial software [24]. II. THEORY Spatial power combiners are electromagnetically large and complicated structures, which include different dielectric materials. Therefore, modeling of spatial power-combining systems is a very challenging task. Analysis of these passive arrays creates a basis for the investigation of active spatial power-combining/dividing systems. A general geometry of a waveguide-based spatial power combiner is shown in Fig. 1, where a coaxial probe-fed embedded DRA array inside an oversized hard-walled rectangular waveguide is excited by a pyramidal hard horn. A numerical analysis of this problem is performed by the FDTD method [25], [26]. For the excitation of the coaxial and waveguide ports, a sinusoidally modulated Gaussian pulse is applied, and the perfectly matched layer (PML) absorbing boundary [27] is implemented to terminate

both the waveguide and coaxial line ports. Since the memory requirements and simulation time to model the whole geometry are tremendous, a region-by-region calculation method and subgridding method are used, instead of modeling the whole structure with the same size of fine FDTD cells. In this method, the entire structure is decomposed into four electromagnetically coupled regions, which include the -band rectangular waveguide, hard pyramidal horn, oversized hard-walled waveguide with the embedded DRA array, and all the coaxial lines. These four regions are updated separately and data communications between each region are performed for every FDTD time step. Memory utilization was reduced by approximately half with the use of our region-by-region FDTD method in comparison to a traditional FDTD method. Due to the requirement that the mesh size in the conven(at the highest tional FDTD needs to be approximately frequency), the embedded high dielectric materials are included in a small region of the whole geometry. The application of the subgridding scheme (Fig. 2), which meshes finely inside the high-permittivity region, has the advantages of large savings of computation time and storage and at the same time has very little accuracy loss. In the proposed geometry, a local grid (LG) with finer mesh is applied to the embedded DRA array elements and the coaxial lines, and a main grid (MG) is applied to the other regions, where the mesh size ratio between the MG and LG is 1/3. In order to update the electric and magnetic fields in the LG region, knowledge of the boundary conditions for the refined volume is needed. The known field values in the MG are used to estimate the unknown field values in the LG on the main-grid–local-grid (MG–LG) boundary depicted in Fig. 3 at every time step by fields interpolation. The LG values can then be coupled back to the MG at the collocated positions. The method is used to allow for finer tuning of some geometry parameters (e.g., the feeding probe length). At the same time, this scheme can increase the simulation accuracy without increasing the numerical cost significantly. By the use of a subgridding scheme, similar accuracy compared with a fine grid in the whole computation region can be achieved with approximately 1/3 of simulation time cost. The scattering parameters are calculated in terms of port voltages normalized by characteristic impedances of the waveguide and coaxial ports, respectively. The detailed description can be found in [28].

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Fig. 3. Electric fields on the MG–LG boundary with a refinement factor of 3.

Fig. 5. S -parameters of the waveguide-based coaxial probe-fed embedded DRA element.

Fig. 4. Waveguide-based coaxial probe-fed embedded DRA. A waveguide is loaded with hard walls made by dielectric slabs with thin printed dipoles.

III. SINGLE EMBEDDED DRA ELEMENT IN RECTANGULAR WAVEGUIDE A full-wave analysis of a single coaxial probe-fed embedded DRA element inside an -band rectangular waveguide was performed by the proposed FDTD method. The initial dimensions of the embedded DRA are chosen so that the resonant frequenmode for the two individual resonators are cencies of the tered at approximately 10 GHz. In reality, the two resonators form a coupled system so that the size of one resonator also influences the resonant frequency of the other one. The geom, etry of the embedded DRA is shown in Fig. 4 with , mm, mm, mm, mm, mm. The embedded DRA configuration is creand ated by embedding the material with high dielectric permitinto the material with low dielectric permittivity tivity (as shown in Fig. 4). The embedded DRA was placed inside a standard -band waveguide with cross-sectional dimensions 22.86 mm 10.16 mm and excited by an off-centered coaxial probe by which the correct mode can be excited and energy can be coupled between the waveguide and coaxial line. The coaxial mm, radius mm, and probe has a length of mm away from the center of the dielectric is located resonator. The -parameters for the proposed design are shown

in Fig. 5, where a matching bandwidth of 23% (based on a 10-dB reflection coefficient level) is achieved. To simplify the measurement process, the embedded dielectric resonator was fabricated by a perforation technique [29], [30]. The DRA element is built using Rogers RT/Duroid 3210 LM. Both simulated and measured results have wide frequency bandwidth showing a good agreement. To design an efficient spatial power-combining system, it is important to achieve a uniform power division among the antenna elements in the embedded DRA array. Hard walls are used in the proposed spatial power-combining system in order to create a uniform excitation of all antenna elements in the array, and the study of a single element placed at different positions in the waveguide cross section is important to predict the uniformity of coupling for the array case. Due to the interaction between the embedded DRA and the hard walls, the embedded DRA characteristics are significantly affected by the presence of hard walls. In order to have a sufficient distance to offset the single embedded DRA from the center of an -band waveguide, very thin hard walls are required instead of two quarter-wavelength dielectric-loaded walls. Since two quarterwavelength dielectric-loaded walls with a dielectric constant of 2.2 occupy approximately half of the waveguide cross section, there is no space for the embedded DRA to be offset from the center. For the one-element design, hard electromagnetic walls in the hollow rectangular waveguide were created by loading its narrow walls with a frequency-selective surface (FSS) composed of dipoles printed on a grounded dielectric slab [31], as shown in Fig. 4. The substrate has dielectric constant and thickness mm. The dipoles printed on a thin dielectric substrate have the following dimensions: length mm, width mm, and periodicity mm. For the resonant frequency of 10 GHz, where the surface acts as a high-impedance surface, the structure is designed to be compatible with TEM propagation and a uniform field distribution is achieved in the inner waveguide region (between dielectric slabs). The embedded DRA is placed in two different positions: in the middle of waveguide and offset 4 mm from the waveguide

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2

Fig. 6. Reflection coefficients of the waveguide-based coaxial probe-fed embedded DRA element.

Fig. 8. Reflection coefficient at the waveguide port for the 1 3 DRA array with coaxial probe excitation operating inside a dielectric loaded -plane sectoral hard horn.

H

Fig. 7. DRA array with coaxial probe excitation inside a dielectric loaded -plane sectoral hard horn as a 1 : 3 power combiner/divider.

H

center line. Fig. 6 demonstrates that the resonance frequency and scattering parameters of these two cases are not changed significantly, which verifies that the uniform field distribution is achieved across the waveguide aperture. The thin hard walls were built using Rogers RT/Duroid 6002 LM dielectric material. Wide bandwidth and good agreement are achieved for both simulated and measured results. IV. EMBEDDED 1-D DRA ARRAY WITH SECTORAL HARD HORN EXCITATION Here we concentrate on the analysis of 1-D DRA arrays excited by sectoral - and -plane hard horns before we proceed with the analysis of 2-D DRA arrays with pyramidal hard horn excitation. The effects of array parameters, such as inter-element antenna spacing and the distance of the horn walls to the array edges, are investigated in order to obtain high power-combining/dividing efficiency. A 1 3 embedded DRA array excited by an -plane sectoral hard horn is analyzed with the geometry shown in Fig. 7. At the resonant frequency, a uniform field distribution is obtained by loading the sidewalls of the -plane sectoral horn with thin hard walls composed of printed dipole FSS elements. The inter-eland the distance of ement spacing the array to the dielectric hard walls with aperture dimensions 62 mm 10.16 mm are chosen in order to achieve a minimum reflection and to reduce mutual coupling among the elements. Both the simulation results from HFSS and measurement results

Fig. 9. Transmission coefficients from the waveguide port to the DRA array elements operating inside a dielectric loaded -plane sectoral hard horn.

H

are shown in Fig. 8 for the reflection coefficient at the input of the horn, where 17% matching bandwidth (based on 10-dB return loss level) is achieved and good agreement between simulation and measurement results is obtained. The measured transmission coefficients from the input of the horn to the three DRA elements of the array are shown in Fig. 9, where the coupling from the -plane sectoral hard horn to each of the three DRA elements is approximately 5 dB, which indicates that an approximately equal power is divided among the array elements and a uniform field distribution is achieved across the -plane sectoral hard horn aperture in the presence of the DRA array. Operating in an -plane sectoral hard horn, a 4 1 embedded DRA array is analyzed with the geometry shown in Fig. 10. Inside the -plane sectoral hard horn, a thin dielectric suband thickness mm is applied to strate with create the hard walls in order to achieve a uniform field distribution around the resonant frequency. The vertical inter-elebetween the DRA elements in the array ment spacing with aperture dimensions 22.86 mm 70 mm is chosen to obtain a minimum reflection and to reduce mutual coupling among

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Fig. 10. DRA array with coaxial probe excitation inside a dielectric loaded E -plane sectoral hard horn as a 1 : 4 power combiner/divider. Fig. 13. Top view of 2-D DRA array with rectangular lattice.

Fig. 14. Top view of 2-D DRA array with triangular lattice.

2

Fig. 11. Reflection coefficient at the waveguide port for the 4 1 DRA array with coaxial probe excitation operating inside a dielectric loaded E -plane sectoral hard horn.

Fig. 15. Coaxial probe-fed 3 midal hard horn.

2 3 embedded DRA array operating in a pyra-

elements is approximately 6 dB, which indicates that an approximately equal power is divided among the array elements within the matching range. Fig. 12. Transmission coefficients from the waveguide port to the four DRA array elements inside the E -plane sectoral hard horn.

the DRA elements. Figs. 11 and 12 show the reflection coefficient from the waveguide port and transmission coefficients from the waveguide to the four DRA elements. It can be seen that a good agreement between our codes and HFSS is obtained and a matching bandwidth of 13.2% is achieved. The coupling from the -plane sectoral hard horn to each of the four DRA

V. EMBEDDED 2-D DRA ARRAY WITH PYRAMIDAL HARD HORN EXCITATION Based on the analysis of a single embedded DRA element operating in waveguide and 1-D embedded DRA array in - and -plane sectoral hard horns, 2-D arrays of coaxial probe-fed embedded DRA elements excited by a pyramidal hard horn are investigated. Two specific array arrangements are used in antenna arrays for receiving and radiating a signal. The simplest and most obvious is the rectangular lattice, as illustrated in Fig. 13. In this configuration, the antenna elements

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Fig. 16. Experimental and simulation results of reflection coefficient at the waveguide port of the embedded DRA array with rectangular lattice fed by a pyramidal hard horn with geometry shown in Fig. 15.

Fig. 18. Experimental results of transmission coefficients from the waveguide port to the embedded DRA array elements (5–7) with rectangular lattice fed by a pyramidal hard horn.

Fig. 17. Experimental results of transmission coefficients from the waveguide port to the embedded DRA array elements (2–4) with rectangular lattice fed by a pyramidal hard horn.

Fig. 19. Experimental results of transmission coefficients from the waveguide port to the embedded DRA array elements (8–10) with rectangular lattice fed by a pyramidal hard horn.

are simply placed in a row/column configuration, where each and each row column is separated by the horizontal spacing by the vertical spacing . The other common choice is the triangular lattice, as shown in Fig. 14. This configuration is characterized by the placement of the antennas along a diagonal axis, where the spacing between DRA elements is defined by the diagonal distance between elements . In order to model the DRA array in the same environment as the single DRA element and decrease the mutual coupling between the DRAs in the vertical direction, two horizontal metal plates are inset into the oversized waveguide, as shown in Fig. 1. Since the plates do not affect the vertical electric field, both the uniform field distribution at the horn aperture and the reflection coefficient at the waveguide port are not changed, and at the same time, the horn aperture size and length of the horn can be reduced. Fig. 15 illustrates the prototype of the 2-D embedded DRA array with the rectangular lattice operating in the pyramidal hard horn. The DRA dimensions and the probe parameters are the same as in the single-element design, the oversized

and , and waveguide aperture dimensions are denoted as the distances between the elements of the DRA array and the waveguide walls are denoted as and . The array is centered in the oversized waveguide cross section (with the dimensions mm and mm) with a uniform horiand a uniform vertical spacing zontal spacing of , where is the wavelength in free space at of 10 GHz. An -band hard horn with length of 15 cm is created by loading dielectric materials along its two -plane walls, which are parallel to the -field. A uniform field distribution at the aperture of the hard horn is obtained when the dielectric mm and dielectric constant are thickness used in the design of hard walls. Our results, the results from HFSS, and experimental results for the reflection coefficient at the input of the horn and the transmission coefficients from the input of the horn to the nine elements of the array are shown in Figs. 16–19. It can be seen in these figures that the coupling from the horn to each of the nine DRA elements is approximately 10 dB, which indicates that every element receives 1/9

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Fig. 20. Architecture of waveguide-based spatial power divider/radiator with embedded DRA array.

Fig. 23. Radiation pattern of the waveguide-based spatial power divider/radiator for 2-D DRA array with triangular lattice at 10 GHz.

Fig. 21. Reflection coefficients at the waveguide port for the three cases: 2-D array with rectangular lattice, 2-D array with triangular lattice, and 2-D array with rectangular lattice, where the receiving and radiating arrays are connected by coaxial lines.

Fig. 24. Architecture of waveguide-based spatial power divider/radiator with receiving and radiating embedded DRA array connected by coaxial lines.

Fig. 25. Radiation pattern at 10 GHz for the geometry in Fig. 24. Fig. 22. Radiation pattern of the waveguide-based spatial power divider/radiator for 2-D DRA array with rectangular lattice at 10 GHz.

power and the uniform field distribution is achieved across the oversized waveguide aperture in the presence of the array. In order to radiate the power into free space, the second set of the embedded DRA array is placed on the back of the horn aperture at the same corresponding positions as the receiving elements, as depicted in Fig. 20. For the rectangular and triangular

lattices, the reflection coefficients from the waveguide port are demonstrated in Fig. 21. Figs. 22 and 23 show the radiation patterns for these two cases at 10 GHz, respectively, where it can be seen that the rectangular lattice results in smaller values of cross polarization compared to those obtained with the use of triangular lattice. Since the vertical distance between the array , nine coaxial lines are added to elements is smaller than connect two sets of embedded DRA arrays, as shown in Fig. 24. vertical and By this way, the radiating antenna array has

ZHANG et al.: ANALYSIS OF WIDEBAND DRA ARRAYS FOR WAVEGUIDE-BASED SPATIAL POWER COMBINING

horizontal distances between elements. Fig. 25 depicts the radiation pattern for this geometry. Due to the coaxial line probe-feed offset from the center line, some higher order mode are excited causing the radiation patterns in Figs. 22, 23, and 25 to show some asymmetry in the -plane. VI. CONCLUSION In this paper, we have presented the FDTD analysis of a single probe-fed embedded DRA element inside a rectangular waveguide, 1-D embedded DRA arrays fed by the H- and -plane sectoral hard horns, and 2-D embedded DRA arrays fed by a pyramidal hard horn and radiating in free space. The embedded DRA element was optimized for wideband performance and, consequently, 23% bandwidth for a single DRA element and 15% bandwidth for the 2-D DRA array were achieved over the frequency band of interest. The analysis method was based on the FDTD technique combined with a region-by-region method and subgridding scheme to reduce memory and simulation time. A uniform field distribution at the horn aperture was achieved by using hard walls along a pyramidal horn, and two metal plates inset into the oversized waveguide were used to decrease the -plane mutual coupling between the array elements. The analysis provided the necessary information for the design parameters such as embedded DRA dimensions, feeding probe, inter-element spacing, and the distance of the array to the oversized waveguide hard walls in order to increase the power coupling efficiency from the hard horn to the array elements. REFERENCES [1] M. Kim, J. J. Rosenberg, R. P. Smith, R. M. Weikle, J. B. Hacker, M. P. D. Lisio, and D. B. Rutledge, “A grid amplifier,” IEEE Microw. Guide Wave Lett., vol. 1, no. 11, pp. 322–324, Nov. 1991. [2] C. Y. Chi and G. M. Rebeiz, “A quasi-optical amplifier,” IEEE Microw. Guide Wave Lett., vol. 3, no. 6, pp. 164–166, Jun. 1993. [3] M. Kim, E. A. Sovero, J. B. Hacker, M. P. D. Lisio, J. C. Chiao, S. J. Li, D. R. Gagnon, J. J. Rosenberg, and D. B. Rutledge, “A 100-element HBT grid amplifier,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 10, pp. 1762–1771, Oct. 1993. [4] T. Ivanov, A. Balasubramaniyan, and A. Mortazawi, “One- and twostage spatial amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 9, pp. 2183–2143, Sep. 1995. [5] H. S. Tsai and R. A. York, “Quasi-optical amplifier array using direct integration of MMICs and 50 ohm multi-slot antennas,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1995, pp. 593–596. [6] S. Ortiz, T. Ivanov, and A. Mortazawi, “CPW fed microstrip patch quasi-optical amplifier array,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 2, pp. 276–280, Feb. 2000. [7] B. Deckman, D. S. Deakin, Jr., E. Sovero, and D. B. Rutledge, “A 5-watt, 37 GHz monolithic grid amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, pp. 805–808. [8] A. B. Yakovlev, S. Ortiz, M. Ozkar, A. Mortazawi, and M. B. Steer, “A waveguide-based aperture-coupled patch amplifier array—Full-wave system analysis and experimental validation,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2692–2699, Dec. 2000. [9] S. Ortiz, J. Hubert, L. Mirth, E. Schlecht, and A. Mortazawi, “A highpower -band quasi-optical amplifier array,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 2, pp. 487–494, Feb. 2002. [10] M. Ozkar, “Electromagnetic modeling for the optimized design of spatial power amplifiers with hard horn feeds,” Ph.D. dissertation, Dept. Elect. Eng., North Carolina State Univ., Raleigh, NC, 2001. [11] M. Ozkar and A. Mortazawi, “Electromagnetic modeling and optimization of spatial power combiners/dividers with hard horns,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 144–150, Jan. 2005. [12] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. AP-31, no. 5, pp. 406–412, May 1983.

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[13] M. W. McAllister, S. A. Long, and G. L. Conway, “Rectangular dielectric resonator antenna,” Electron. Lett., vol. 19, pp. 219–220, Mar. 1983. [14] A. A. Kishk, H. A. Auda, and B. Ahn, “Radiation characteristics of cylindrical dielectric resonator antenna with new applications,” IEEE AP-S Newslett. , vol. 31, pp. 7–16, Feb. 1989. [15] A. A. Kishk, “Experimental study of broadband embedded dielectric resonator antennas excited by a narrow slot,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 79–81, 2005. [16] A. A. Kishk, X. Zhang, A. W. Glisson, and D. Kajfez, “Numerical analysis of stacked dielectric resonator antennas excited by a coaxial probe for wideband applications,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1996–2006, Aug. 2003. [17] S. H. Ong, A. A. Kishk, and A. W. Glisson, “Rod-ring dielectric resonator antenna,” Int. J. RF Microw. Comput.-Aided Eng., vol. 14, no. 5, pp. 441–446, Sep. 2004. [18] S. H. Ong, A. A. Kishk, and A. W. Glisson, “Wideband disc-ring dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 35, no. 6, pp. 425–428, Dec. 2002. [19] P. S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,” Electron. Lett., vol. 24, pp. 168–170, 1988. [20] M. A. Ali, S. C. Ortiz, T. Ivanov, and A. Mortazawi, “Analysis and measurement of hard-horn feeds for the excitation of quasi-optical amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 4, pp. 479–487, Apr. 1999. [21] M. N. M. Kehn and P. S. Kildal, “Miniaturized rectangular hard waveguides for use in multifrequency phased arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 100–109, Jan. 2005. [22] Y. Zhang, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “FDTD analysis of a probe-fed dielectric resonator antenna array with hard horn for spatial power combiner,” in Proc. Asia–Pacific Microw. Conf., 2005, vol. 3, no. 12, pp. 1669–1672. [23] L. Kulas and M. Mrozowski, “Low-reflection subgridding,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1587–1592, May 2005. [24] High Frequency Structure Simulator (HFSS) Based on the Finite Element Method. ver. 9.2.1, Ansoft Corporation, Pittsburgh, PA, 2004. [25] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [26] K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics. Boca Raton, FL: CRC, 1993. [27] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, pp. 185–200, 1994. [28] Y. Zhang, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Analysis and design of wideband dielectric resonator antenna arrays for waveguide-based spatial power combining,” in 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006. [29] J. B. Muldavin and G. M. Rebeiz, “Millimeter-wave tapered-slot antennas on synthesized low permittivity substrates,” IEEE Antennas Wireless Propag. Lett., vol. 47, no. 8, pp. 1276–1280, Aug. 1999. [30] Y. Zhang and A. A. Kishk, “Study of embedded dielectric resonator antennas using perforated dielectric materials for wideband applications,” in Proc. IEEE Int. AP-S Symp., Albuquerque, NM, Jul. 2006, pp. 1321–1324. [31] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 70–81, Jan., 2005.

Yizhe Zhang (S’05) received the B.Sc. and M.Sc. degrees in electrical engineering from Southeast University, Nanjing, China, in 2000 and 2003, respectively, and is currently working toward the Ph.D. degree in electrical engineering at The University of Mississippi, University. From 2000 to 2003, she was a Research Assistant with the Department of Electrical Engineering, Southeast University. Her research interests include DRAs, microstrip antennas, numerical methods in electromagnetics, and modeling of high-frequency amplifier arrays for spatial power combining.

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Ahmed A. Kishk (S’84–M’86–SM’97–F’98) received the B.S. degree in electronic and communication engineering from Cairo University, Cairo, Egypt, in 1977, the B.S. degree in applied mathematics from Ain-Shams University, Cairo, Egypt, in 1980, and the M.Eng. and Ph.D. degrees in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 1983 and 1986, respectively. From 1977 to 1981, he was a Research Assistant and an Instructor with the Faculty of Engineering, Cairo University. In 1981, he joined the Department of Electrical Engineering, University of Manitoba. From 1981 to 1985, he was a Research Assistant with the Department of Electrical Engineering, University of Manitoba. From December 1985 to August 1986, he was a Research Associate Fellow with the Department of Electrical Engineering, University of Manitoba. In 1986, he joined the Department of Electrical Engineering, The University of Mississippi, University, as an Assistant Professor. From 1994 to 1995, he was on sabbatical leave with Chalmers University of Technology for an academic year. Since 1995, he has been a Professor with The University of Mississippi. He was Co-Editor of the “Special Issue on Advances in the Application of the Method of Moments to Electromagnetic Scattering Problems” of the ACES Journal. He was also an Editor of the ACES Journal in 1997. From 1998 to 2001, he was an Editor-in-Chief of the ACES Journal. His research interest includes the areas of design of millimeter frequency antennas, feeds for parabolic reflectors, DRAs, microstrip antennas, soft and hard surfaces, phased-array antennas, and computer-aided design for antennas. He has authored or coauthored over 160 refereed journal papers and 15 book chapters. He coauthored Microwave Horns and Feeds (IEE Press, 1994; IEEE Press, 1994). He also coauthored Chapter 2 of the Handbook of Microstrip Antennas (Peregrinus, 1989). Dr. Kishk is a Fellow of the IEEE Antennas and Propagation Society (IEEE AP-S) and the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) since 1998. He is a member of Sigma Xi, the U.S. National Committee of International Union of Radio Science (URSI) Commission B, the Applied Computational Electromagnetics Society, the Electromagnetic Academy, and Phi Kappa Phi. He was the chair of the Physics and Engineering Division, Mississippi Academy of Science (2001–2002). He was a guest editor of the January 2005 “Special Issue on Artificial Magnetic Conductors, Soft/Hard Surfaces, and Other Complex Surfaces” of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. From 1990 to 1993, he was an associate editor for the IEEE Antennas and Propagation Magazine. He is currently an editor for the IEEE Antennas and Propagation Magazine. He was the recipient of the 1995 and 2005 Outstanding Paper Awards for papers published in the Applied Computational Electromagnetic Society Journal. He was the recipient of the 1997 Outstanding Engineering Educator Award presented by the Memphis Section of the IEEE. He was the recipient of the Outstanding Engineering Faculty Member of the 1998, and 2001 and 2005 Faculty Research Award. He was the recipient of the Award of Distinguished Technical Communication for an entry of the IEEE Antennas and Propagation Magazine, 2001. He was the recipient of The Valued Contribution Award for Outstanding Invited Presentation, “EM Modeling of Surfaces with STOP or GO Characteristics—Artificial Magnetic Conductors and Soft and Hard Surfaces” presented by the Applied Computational Electromagnetic Society. He was also the recipient of the IEEE MTT-S Microwave Prize, 2004.

Alexander B. Yakovlev (S’94–M’97–SM’01) was born in the Ukraine, in 1964. He received the Ph.D. degree in radiophysics from the Institute of Radiophysics and Electronics, National Academy of Sciences, Kharkov, Ukraine, in 1992, and the Ph.D. degree in electrical engineering from the University of Wisconsin at Milwaukee, in 1997. From 1992 to 1994, he was an Assistant Professor with the Department of Radiophysics, Dniepropetrovsk State University. From 1994 to 1997, while working toward his doctoral degree,

he was a Research and Teaching Assistant with the Department of Electrical Engineering and Computer Science, University of Wisconsin at Milwaukee. From 1997 to 1998, he was a Research and Development Engineer with the Compact Software Division, Ansoft Corporation, Paterson, NJ, and with the Ansoft Corporation, Pittsburgh, PA. From 1998 to 2000, he was a Post-Doctoral Research Associate with the Electrical and Computer Engineering Department, North Carolina State University, Raleigh. In Summer 2000, he joined the Department of Electrical Engineering, The University of Mississippi, University, as an Assistant Professor, and became an Associate Professor in 2004. He coauthored Operator Theory for Electromagnetics: An Introduction (Springer, 2001). He is an Associate Editor-in-Chief for the ACES Journal. His research interests include mathematical methods in applied electromagnetics, analysis of artificial magnetic conductor surfaces and guided-wave structures, modeling of high-frequency interconnection structures and amplifier arrays for spatial and quasi-optical power combining, microstrip and waveguide discontinuities, integrated-circuit elements and devices, theory of leaky waves, transient fields in layered media, and catastrophe and bifurcation theories. Dr. Yakovlev is a senior member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the IEEE Antennas and Propagation Society (IEEE AP-S). He is a member of URSI Commission B. In September 2005, he became an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the recipient of the Young Scientist Award presented at the 1992 URSI International Symposium on Electromagnetic Theory, Sydney, Australia, and the Young Scientist Award presented at the 1996 International Symposium on Antennas and Propagation, Chiba, Japan. He was also the recipient of the 2003 Junior Faculty Research Award of the School of Engineering, The University of Mississippi.

Allen W. Glisson (S’71–M’78–SM’88–F’02) received the B.S., M.S., and Ph.D. degrees in electrical engineering from The University of Mississippi, University, in 1973, 1975, and 1978, respectively. In 1978, he joined the faculty of The University of Mississippi, where he is currently a Professor and Chair of the Department of Electrical Engineering. He has served as Associate Editor for Radio Science and as Co-Editor-in-Chief for the Applied Computational Electromagnetics Society Journal. His current research interests include the development and application of numerical techniques for treating electromagnetic radiation and scattering problems, and modeling of dielectric resonators and DRAs. Dr. Glisson is a member of Commission B of the International Union of Radio Science and a member of the Applied Computational Electromagnetics Society. Since 1984, he has served as the associate editor for Book Reviews and Abstracts for the IEEE Antennas and Propagation Society Magazine. He has also served on the Board of Directors of the Applied Computational Electromagnetics Society (ACES). He is currently treasurer of ACES. He is a member of the IEEE Antennas and Propagation Society (AP-S) IEEE Press Liaison Committee. He was a member of the IEEE AP-S Administrative Committee (AdCom), secretary of Commission B of the U.S. National Committee of URSI. He was editor-in-chief for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He was selected as the Outstanding Engineering Faculty Member in 1986, 1996, and 2004. He was the recipient of a 1989 Ralph R. Teetor Educational Award. He was the recipient of the 2002 Faculty Service Award of the School of Engineering, The University of Mississippi. He was a recipient of a Best Paper Award presented by the SUMMA Foundation, a two-time recipient of a citation for excellence in refereeing from the American Geophysical Union, and a recipient of the 2004 Microwave Prize presented by the IEEE Microwave Theory and Techniques Society (IEEE MTT-S).

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Concurrent Dual-Band Class-E Power Amplifier Using Composite Right/Left-Handed Transmission Lines Seung Hun Ji, Choon Sik Cho, Member, IEEE, Jae W. Lee, Member, IEEE, and Jaeheung Kim, Member, IEEE

Abstract—A concurrent dual-band class-E power amplifier using composite right/left-handed transmission lines (CRLH TLs) is proposed. Dual-mode operation is achieved by using the frequency offset and nonlinear phase slope of CRLH TLs for the matching network of power amplifiers. The frequency ratio of two operating frequencies is not necessarily an integer. Two operating frequencies are chosen as 836 MHz and 1.95 GHz for simulation. Three methods for designing a CRLH TL power amplifier are proposed. The measured results based on one method show that output powers of 22.4 and 22.2 dBm were obtained at 800 MHz and 1.70 GHz, respectively. In terms of maximum power-added efficiency, we obtained 42.5% and 42.6% at 800 MHz and 1.70 GHz, respectively. Index Terms—Class-E power amplifier, composite right/lefthanded transmission line (CRLH TL), dual band.

I. INTRODUCTION

R

ECENTLY, RF equipment is required to operate seamlessly using different wireless communications standards and spectra that are in use around the world. Various efforts have been made to realize multiband operation. Adaptable RF circuits whose performance can be changed without loss of performance according to the wireless environment will be necessary to achieve this concept [1]. Power amplifiers are a key component in mobile terminals and must have high operation efficiency in order to maximize the battery life, and reduce the size and cost. In several power amplifiers, the switched-mode class-E tuned power amplifiers with a shunt capacitor have found widespread application due to their design simplicity and high-efficiency operation. The drain efficiency of the class-E power amplifier theoretically reaches 100% [2]. Concurrent dual-band operation is beneficial to reduce the number of circuit components in modern wireless communication systems requiring two frequency bands. However, dual-band power amplifiers are difficult to design because Manuscript received October 3, 2006; revised January 12, 2007. This work was supported by Korea Aerospace University under a 2006 Faculty Research Grant and by the Korea Science and Engineering Foundation under the ERC Program through the Intelligent Radio Engineering Center Project, Information and Communications University. S. H. Ji, C. S. Cho, and J. W. Lee are with the School of Electronics, Telecommunication and Computer Engineering, Korea Aviation University, Goyang 412–791, Korea (e-mail: [email protected]; [email protected]; jwlee1 @hau.ac.kr). J. Kim is with the Information and Communications University, Daejeon 200701, Korea (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.2007.895236

matching networks of power amplifier are usually made to operate at one specific frequency. The need for good matching networks arises in order to deliver maximum power to a load [3]. Since matching networks are fixed to one operating frequency, concurrent dual-band power amplifiers need a novel matching network. Dual-band power amplifiers using the low-pass Chebyshev-form impedance transformer were presented in [4] and [5]. In this paper, the simple features of metamaterial [6] based on transmission lines are used to implement a matching network for class-E power amplifier for concurrent dual-mode operation. As described in [7], composite right/left-handed transmission lines (CRLH TLs) possess interesting phase characteristics such as antiparallel phase and nonlinear phase slope. Thus far, this novel transmission media have been used in the implementation of passive devices such as couplers, resonators, and antennas [7]–[9]. The use of CRLH TLs allows for the manipulation of phase slope and phase offset at zero frequency [8]. This attribute can be used to specify the phase delay of a CRLH TL at different frequencies to create the necessary impedance for proper matching network. Using this method, a CRLH TL network can be used to design a dual-band class-E power amplifier [10]. Another key point in RF equipment is its size. The size of the power amplifier is an important feature in evaluating its performance. RH TL parts of the proposed CRLH TL in [10] are composed of microstrip lines. Proposed dual-band power amplifier in this study, using only the negative phase response of the CRLH TL, has long electrical length of RH TLs. Elongated RH TLs cause increased size and power loss of the power amplifier. In this paper, the relationship between the left-handed (LH) and right-handed (RH) parts is analyzed. The electrical length of RH TLs can be shortened using the positive phase response of the CRLH TL. The design of the dual-band class-E power amplifier using the CRLH TL was originally introduced in [10]; here, a more shortened CRLH TL and a lumped- element CRLH TL have been considered. II. CLASS-E POWER AMPLIFIER In the class-E power amplifier, the transistor operates as an on–off switch and the shapes of the current and voltage waveforms provide a condition that minimizes the power dissipation and maximizes the power amplifier efficiency. The circuit topology for the class-E power amplifier [2] is shown in Fig. 1. It consists of a transistor acting as a switch, a shunt capacitor across the switch, and the matching network using microstrip lines. When the switching frequency of the transistor

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Fig. 1. Class-E power amplifier circuit topology.

Fig. 2. Lumped elements model for the CRLH TL when

TABLE I IMPEDANCES TOWARD SOURCE AND LOAD

is at the fundamental frequency, the impedance of the output network including the load is found to be [11]

N = 1.

lead). On the other hand, the RH TL has negative phase response (phase lag). Therefore, the CRLH TL can substitute for the matching network shown in Fig. 1. Fig. 2 shows the lumped is element model for the CRLH TL when one unit cell used [8]. and are inductive and capacitive elements of the RH , are the inductive and capacitive elements of the TL and LH TL. When the series resonance and shunt resonance are equal, (2) (3) (4)

(1) assuming the impedances of all other harmonics are open. In practice, an open-circuit termination at the second harmonic is sufficient to give class-E operation [11]. can also be opIn the mean time, the output impedance timized using the load–pull technique [12]. This study obtains and using the source–pull and load–pull the optimized techniques, as shown in Table I. At first, two different class-E power amplifiers are designed individually at two different operating frequencies ( , ). At this time, dc-bias voltages such and drain bias voltage are set at as gate bias voltage the same values in two different class-E power amplifiers. The transistor must operate as an on–off switch and the typical duty for producing maximum output power cycle is 50% [13]. takes different values at two frequencies [2]. The input matching circuit is realized by source–pull conjugate matching and output matching circuit by load–pull conjugate matching using (1) as a starting point. The input matching circuit and output matching circuit can be composed of two microstrip lines, respectively. Each matching section has a phase ) at two frequencies. CRLH TL matching netresponse ( , and , can be designed. In this paper, three works, using methods for designing a dual-band power amplifier are proposed. The first design method uses the negative phase response of the CRLH TL [10]. The second design method uses the positive phase response of the CRLH TL. Meanwhile, the third design method uses an LC (inductor and capacitor) lumped network CRLH TL. III. DESIGN OF DUAL-BAND POWER AMPLIFIER USING CRLH TL A. First Method The CRLH TL, which is the combination of an LH TL and an RH TL, is proposed in [14]. The equivalent lumped element model of the LH TL exhibits positive phase response (phase

the structure is said to be balanced [14]. characteristic impedances defined as

and

are the

(5) (6) (7) , , and are usually fixed as 50 . The phase response can approximately be expressed in the balanced condition (8) (9) (10) Unlike the ideal case, the CRLH TL has innate LH and RH cutoff frequencies as [15] (11) (12) Since the phase response of the CRLH TL is set to and at , the phase response of the CRLH TL at can be written as

at and (13) (14)

In (13) and (14), the negative phase response of the CRLH TL was used, as shown in Fig. 3, where is a positive number and

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Fig. 4. Proposed concurrent dual-band class-E power amplifier using CRLH TLs. Fig. 3. Phase response of CRLH TL.

can be chosen arbitrarily for minimizing and (14) can be written as

. From (5)–(10), (13)

(15) (16) (17) (18) For the given obtain [8],

and

, solving for

and

in (12) and (13) to Fig. 5. Phase response of CRLH TL using the second method.

(19)

(20)

If , and are calculated with a large . For the next is calculated from (11). If , the design step, is completed. Otherwise, the design is performed again with a , and are used to determine and larger [8]. , , from (5) and (15), and the physical length of the RH TL from (6) and (8). Finally, CRLH TLs are substituted for the matching network instead of microstrip lines to realize a dual-band operation. Fig. 4 shows the proposed concurrent dual-band class-E power amplifier using CRLH TLs. and of the designed class-E power amFor suitable plifier proposed here, the value of needs to be greater than 2. When , the length of RH TLs is almost . B. Second Method The problem with the first design method is the long physical length of the RH TL. The positive phase response of the CRLH TL can be used in order to decrease the size of the power

amplifier in the second design method. Instead of (13), the phase response of the CRLH TL can be set as (21) (22) has a positive value because and is a where positive number. Fig. 5 shows the phase response of the CRLH TL using the second method. The next steps are the same as the first method. Fig. 6 shows the difference between the phase responses of the first and second methods. In Fig. 6, the phase slope of the CRLH TL simulated using the first method is steeper than that using the second method and is larger in because the difference between the first method. Such difference increases the phase slope of the CRLH TL. As the phase slope of the CRLH TL increases, the phase slope of the RH TL also increases. The electrical length of the RH TL increases in proportion to the phase slope of the RH TL. Therefore, electrical length of the RH TL, using the first method, is longer than that using the second method. However, since the electrical length of the RH TL is not proportional to

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Fig. 8. Performances of proposed single-band class-E power amplifiers at the two operating frequencies. Fig. 6. Phase responses of CRLH TL using the first and second methods.

Fig. 7. Unit cell of LC CRLH TL.

, the number of unit cell increases. As the phase slope of the CRLH TL increases, the phase slope of the LH TL increases. increases, decreases in (9) and (11). As the value of As becomes smaller, the number of unit cells in the CRLH TL is lowered. C. Third Method The LC CRLH TL was proposed in [14]. Fig. 7 shows the LC CRLH TL composed of lumped elements only. The third design method of a dual-mode power amplifier uses the LC CRLH TL. The third method can make the size of the CRLH TL as small as possible. The third method is similar to the second method in and can be set as in (21) and (22). Rest of the that steps are the same as the first method and cutoff frequency conto obtain the same results from the dition is added as and . Therefore, the dual-band first method and value of power amplifier can also be designed using the LC CRLH TL. IV. SIMULATION AND MEASUREMENT The proposed concurrent dual-band class-E power amplifiers were simulated using Agilent ADS at cellular (836 MHz) and 3G (1.95 GHz) frequencies. The transistor model used is Mitsubishi MGF2415. At first, two different single-band class-E power amplifiers at the two frequencies are designed individis chosen as 3 pF. Practically, ually as shown. The value of

Fig. 9. Voltage and current waveforms of dual-band class-E power amplifier using the first method at 836 MHz.

the method depicted in [12] was used for matching in two different single-band class-E power amplifiers at the two frequencies. The output impedance was optimized using the load–pull technique to produce class-E operation, as shown in Table I. Thereafter, each microstrip lines in the matching network are substituted for the CRLH TL. Fig. 8 shows the simulated output power and power-added efficiency (PAE) of a single-band class-E power amplifier designed individually at 836 MHz and 1.95 GHz. In this case, maximum output powers of 24.9 and 24.8 dBm, and PAEs of 50.07% and 50.04% at 836 MHz and 1.95 GHz were obtained, respectively. Using this configuration, dual-band class-E power and is obtained using amplifiers are designed. Each source–pull and load–pull at 836 MHz and 1.95 GHz, as shown in Table I. Suitable CRLH TLs are designed for the input and output matching networks. The CRLH TLs are optimized to proand phase response closer to those for singleduce a band matching networks. Fig. 9 shows the voltage and current waveform of the simulated power amplifier of a dual-band class-E power amplifier using the first method at 836 MHz. As the voltage and current hardly overlap each other, this is sufficiently operating as class-E operation. In Fig. 9, the transistor operate as an on-off

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Fig. 10. Simulation layout of proposed dual-band class-E power amplifier using the first method.

Fig. 11. Fabricated layout of proposed dual-band class-E power amplifier using the first method.

switch and the duty cycle is approximately 50%, enabling the simulated power amplifier to operate well for class-E operation. Fig. 10 shows the simulation layout of the proposed dual-band class-E power amplifier using the first method at the two operating frequencies. In the simulation layout, a different microstrip radial stub size for biasing is used to operate at 836 MHz and 1.95 GHz. The device is biased with a drain voltage of 2.8 V and a gate voltage of 1.9 V. Fig. 11 shows the fabricated layout of the proposed dual-band class-E power amplifier using the first method. Its size is 165 mm 150 mm. Fig. 12 shows the output power and PAE of simulated and measured dual-band class-E power amplifier at the two operating frequencies using the first method. In this case, maximum PAE of 45.3% and 44.7% were obtained at 830 MHz and 1.80 GHz for measured results, respectively. The output powers of 20.6 and 19.5 dBm were obtained at 830 MHz and 1.80 GHz. Since the operating frequencies are chosen for the maximum output power, a gap between simulation and measurement is observed for some input powers. This error can be mitigated if a post-tuning for the matching section is carried out. Fig. 13 shows the fabricated layout of proposed class-E power amplifier using the second method at the two operating frequencies. Its size is 90 mm 110 mm. Fig. 14 shows measured results using the second method. Maximum output powers of 22.4 and 22.2 dBm were obtained and maximum PAEs of 42.5% and 42.6% at 800 MHz and 1.70 GHz, respectively. With input power less than 12 dBm, performance obtained is not satisfactory due to the occurrence of frequency shift. The reason it op-

Fig. 12. Output powers and PAEs of dual-band class-E power amplifier using the first method at the two operating frequencies. (a) 836 MHz for simulation, 830 MHz for measurement. (b) 1.95 GHz for simulation, 1.80 GHz for measurement.

Fig. 13. Fabricated layout of proposed dual-band class-E power amplifier using the second method.

erates in an unexpected manner is also caused from the shifted operating frequency, which was chosen for maximum output

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Fig. 16. Simulation result of power amplifier using the third method. TABLE II SUMMARY OF DESIGNED MATCHING SECTIONS

TABLE III PERFORMANCE COMPARISON

Fig. 14. Output power and PAE of dual-band class-E power amplifier using the second method at the two operating frequencies. (a) 836 MHz for simulation, 800 MHz for measurement. (b) 1.95 GHz for simulation, 1.70 GHz for measurement.

Fig. 15. Dual-band class-E power amplifiers using the first method versus the second method.

power. A post-tuning for the matching section can also ameliorate this error.

Using the second method, the physical size of the dual-band power amplifier was reduced as shown in Fig. 15. Dual-band power amplifier using the third method is also designed using simulation only (see Fig. 16). Maximum output power was obtained almost at the same values as those of each single-band class-E power amplifier. Table II shows the summary of matching sections, which were designed using the electrical and physical lengths for all three cases (in Fig. 1). Table III shows performance comparison with other studies. At high frequencies, the proposed dual-band class-E power amplifiers employing CRLH TLs shows comparable performance.

JI et al.: CONCURRENT DUAL-BAND CLASS-E POWER AMPLIFIER USING CRLH TLs

V. CONCLUSION A concurrent dual-band class-E power amplifier using CRLH TLs was proposed. Dual-band operation was achieved by the frequency offset and phase slope of the CRLH TL for matching networks. The frequency ratio of two operating frequencies is not necessarily an integer. We can control the phase response of the CRLH TL as needed at two operating frequencies. Two operating frequencies are originally chosen, i.e., 836 MHz and 1.95 GHz, in this study. In the proposed dual-band class-E power amplifier using the first method, the output powers of 20.6 and 19.5 dBm were obtained at 830 MHz and 1.80 GHz. In case of maximum PAE, we obtained 45.3% and 44.7% at two operating frequencies. The measured results of the proposed dual-band class-E power amplifier using the second method showed that output power of 22.4 and 22.2 dBm was obtained at 800 MHz and 1.70 GHz, respectively. In case of maximum PAE, we obtained 42.5% and 42.6% at two operating frequencies. The PAE of proposed dual-band class-E power amplifiers using the first and second methods reaches almost 90% performance of a normal class-E power amplifier at two individual operating frequencies. Therefore, CRLH TLs can be applied to other circuits requiring multiband operation. REFERENCES [1] A. Fukuda, H. Okazaki, T. Hirota, Y. Yamao, Y. Qin, S. Gao, A. Sambell, and E. Korolkiewicz, “Novel 900 MHz/1.9 GHz dual-mode power amplifier employing MEMS switches for optimum matching,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, pp. 121–123, Mar. 2004. [2] N. O. Sokal and A. D. Sokal, “Class E—A new class of high-efficiency tuned single-ended switching power amplifiers,” IEEE J. Solid-State Circuits, vol. SC- 10, no. 6, pp. 168–176, Jun. 1975. [3] G. Guillermo, Microwave Transistor Amplifiers. Upper Saddle River, NJ: Prentice-Hall, 1997. [4] F. Bohn, S. Kee, and A. Hajimiri, “Demonstration of a switchless class E=F dual-band power amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 3, pp. 1631–1634. [5] K. Uchida, Y. Takayama, T. Fujita, and K. Maenaka, “Dual-band GaAs FET power amplifier with two-frequency matching circuits,” in Asia–Pacific Microw. Conf., Dec. 2005, vol. 1, pp. 4–7. [6] V. Veselago, “The electrodynamics of substances with simultaneously negative values of  and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [7] C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 980–992, Mar. 2004. [8] I. Lin, M. Devincentis, C. Caloz, and T. Itoh, “Arbitrary dual-band components using composite right/left handed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1142–1149, Apr. 2004. [9] I. H. Lin, C. Caloz, and T. Itoh, “A branch line coupler with two arbitrary operating frequencies using left-handed transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, vol. 1, pp. 325–328. [10] S. H. Ji, C. S. Cho, J. W. Lee, and J. Kim, “836 MHz/1.95 GHz dualband class-E power amplifier using composite right/left-handed transmission lines,” in IEEE Eur. Microw. Conf., 2002, vol. 36, pp. 356–359. [11] T. B. Mader and Z. Popovic´ , “The transmission-line high-efficiency class-E amplifier,” IEEE Microw. Guided Wave Lett., vol. 5, no. 9, pp. 29–292, Sep. 1995. [12] S. Pajic, N. Wang, and Z. Popovic´ , “Comparison of X -band MESFET and HBT class-E power amplifiers for EER transmitters,” in IEEE AP-S Int. Symp. Dig., Jun. 2005, pp. 2031–2034. [13] T. Suetsugu and M. K. Kazimierczuk, “Comparison of class-E amplifier with nonlinear and linear shunt capacitance,” IEEE Trans. Circuits Syst. I, Fundamen. Theory App., vol. 50, no. 8, pp. 1089–1097, Aug. 2003.

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[14] C. Caloz and T. Itoh, Electromagnetic Metamaterials. New York: Wiley, 1999. [15] ——, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH transmission line,” in IEEE AP-S Int. Symp. Dig., 2002, vol. 2, pp. 412–415.

Seung Hun Ji was born in Seoul, Korea, in 1981. He received the B.S. degree in electronics, telecommunication, and computer engineering from Hankuk Aviation University, Goyang, Korea, in 2006, and is currently working toward the M.S. degree at Hankuk Aviation University. His research interests include microwave power amplifiers, monolithic microwave integrated circuits (MMICs)/RF integrated circuits (RFICs), and digitally controlled microwave power amplifiers.

Choon Sik Cho (S’98–M’99) received the B.S. degree in control and instrumentation engineering from Seoul National University, Seoul, Korea in 1987, the M.S. degree in electrical and computer engineering from the University of South Carolina, Columbia, in 1995, and the Ph.D. degree in electrical and computer engineering from the University of Colorado at Boulder, in 1998. From 1987 to 1992, he was with LG Electronics, where he was involved with communication systems. From 1999 to 2003, he was with Pantec & Curitel, where he was principally involved with the development of mobile phones. In 2004, he joined the School of Electronics, Telecommunication and Computer Engineering, Hankuk Aviation University, Goyang, Korea. His research interests include the design of RFICs/MMICs, especially for power amplifiers, oscillators, low-noise amplifiers (LNAs), antennas and passive circuit design, and the computational analysis of electromagnetics.

Jae W. Lee (S’92–M’98) received the B.S. degree in electronic engineering from Hanyang University, Seoul, Korea, in 1992, and the M.S. and Ph.D. degrees in electrical engineering (with an emphasis in electromagnetics) from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1994 and 1998, respectively. From 1998 to 2004, he was a Senior Member with the Advanced Radio Technology Department, Radio and Broadcasting Research Laboratory, Electronics and Telecommunications Research Institute (ETRI), Taejon, Korea. He then joined the School of Electronics, Telecommunication and Computer Engineering, Hankuk Aviation University, where he is currently an Assistant Professor. His research interests include high power amplifier design, computational electromagnetics, electromagnetic interference (EMI)/electromagnetic compatibility (EMC) analysis on printed circuit boards (PCBs), and component design in microwave and millimeter waves.

Jaeheung Kim (S’98–M’02) received the B.S. degree in electronic engineering from Yonsei University, Seoul, Korea, in 1989, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Colorado at Boulder, in 1998 and 2002, respectively. From 1992 to 1995, he was with the DACOM Corporation, where he was involved with wireless communication systems. From 2002 to 2006, he was with the Department of Electrical and Electronic Engineering, Kangwon National University, Chuncheon, Korea. In 2006, he joined the Information and Communications University, Daejeon, Korea. His research interests include beam-forming arrays and high-efficiency active circuits.

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Electrically Controllable Artificial Transmission Line Transformer for Matching Purposes Christian Damm, Student Member, IEEE, Jens Freese, Martin Schüßler, and Rolf Jakoby, Member, IEEE

Abstract—In this paper, we present the application of a tunable artificial line as an impedance matching element. It is based on the transmission line approach of metamaterials and features separate control over line impedance and propagation constant. We show the useful exploitation of the special properties of a fully tunable composite right/left-handed line used as a transmission line transformer, which explicitly includes the use of the bandgap region. Corresponding transmission line theory illustrating the benefit of an artificial line is given. Measurements of a built device operating at 5 GHz and based on semiconductor varactor diodes are presented and compared to simulated results. A procedure to obtain the matchable impedance region from the measured scattering parameters is outlined and applied to measured and simulated data. Index Terms—Artificial line, composite right/left-handed (CRLH) line, matching network, metamaterials.

3.1-dB insertion loss was obtained by tuning the electric and . magnetic properties symmetrically, meaning that This paper will focus on a different operation mode found in [9], but not yet studied. This operation mode was called a combined phase shifter and attenuator, which is the result of a tunable line with independent control over electrical length and line impedance. Therefore, the application as line transformer for a matching circuit is proposed and studied. It is very interesting because it takes advantage of all wave propagation regions of the CRLH line including the bandgap region. Hence, it differs greatly from the application as phase shifter because the tuning states are now 2-D and explicitly include the band gap, which essentially augments the operative range compared to a classic tunable line, as will be shown below. II. MOTIVATION

I. INTRODUCTION

T

HE transmission line approach to realize artificial media called left-handed (LH) media was first presented at almost the same time by Iyer and Eleftheriades [1], Caloz and Itoh [2] and Oliner [3]. The terminology “left-handed” is not related to chirality, but to the LH coordinate system formed by and the wave vector . electromagnetic field vectors and This approach was further developed to the concept of composite right/left-handed (CRLH) metamaterials, which incorporates the inevitable parasitic right-handed (RH) components in all practical LH medium realizations [4]. Many interesting applications like backfire to endfire scannable leaky wave antennas [5], branch line [6] and directional couplers [7], and compact antennas [8] have been found based on this theory. In [9], we presented the first realization of a fully controllable artificial line with the ability to tune propagation constant and and line impedance separately by two controlling voltages . This was obtained by artificially synthesizing magnetic and electric properties according to the transmission line approach enabling their separate control. The aim was to have a phaseshifting device that is not detuned in the line impedance while changing the phase. A differential phase shift of up to 234 with Manuscript received October 16, 2006; revised February 19, 2007. This work was supported by the German Research Foundation DFG within the Research Training Group 1037 “Tunable Integrated Components in Microwave Technology and Optics.” C. Damm, M. Schüßler, and R. Jakoby are with the Institut für Hochfrequenztechnik, Technische Universität Darmstadt, 64283 Darmstadt, Germany (e-mail: [email protected]). J. Freese is with Tesat Spacecom GmbH & Co. KG, 71522 Backnang, Germany. 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.2007.896771

A transmission line with propagation constant and line terminated with an arbitrary complex load impedance will transform the load impedance into the input impedance according to the well-known formula (e.g., [10]) (1) Case A: For a conventional lossless transmission line, the propagation constant is purely imaginary and the is purely real. This line can line impedance to a given source impedance be used to match certain loads . For real load and source impedances, this would be the classic quarter-wavelength transformer. Complex loads can also be matched, as can be seen from Fig. 1(a), which shows all impedances in the Smith chart that can be transformed into a given by varying the electrical length from 0 to 180 and the line impedance from 0 to . Impedances outside of those circular areas cannot be matched with this arrangement. To match impedances of the outer regions, a second transmission line would be needed to transform them into the two circular areas in a first step and then match them with the first line in a second step. Case B: Let us consider now having a transmission line is purely real together with a purely imaginary where . It is very important to note that line impedance the line is still lossless even though the attenuation constant is not zero. This is due to the fact that the line impedance is imaginary at the same time. The line can be regarded as a lossless capacitive or inductive voltage divider, depending on . There is no phase progression due to . The the sign of nonunitary ratio of output voltage to input voltage described by is not due to loss of energy. The sum of power reflected

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Fig. 3. CRLH unit cell of finite length dz . The discrete values follow from the per-/times-unit length values of an infinitesimal line element by multiplication with dz for the RH components and division by dz for LH components, respectively. Fig. 1. Shaded areas show the impedances that can be transformed into the reference impedance using a line transformer. (a) Case A for a line with purely j and real line impedance Z R . imaginary propagation constant (b) Case B for real  with purely imaginary Z jX .

=

=

=

=

which represent the resonance frequencies of the series and parallel resonator of the CRLH unit cell. In the LH and RH bands, and , whereas inside we have Case A with and , respecthe bandgap, we have Case B tively. These properties can be seen from the equations for the propagation constant [12] and the line impedance

(3)

(4)

Fig. 2. Example for the trajectories (solid black lines) of eight impedances (dots at line ends) to reach the matched condition. Characteristic properties of corresponding line transformers are given in shaded boxes. Dashed circles mark the borders between the two Cases A and B shown in Fig. 1.

at the input and transmitted to the output is still equal to the incident power. According to (1), such a line would be able to match the impedances shown in Fig. 1(b) to a real valued by varying from 0 to and from to . This is exactly the complementary area of the first described Case A. Combining both Cases A and B therefore enables matching of arbitrary impedances to a real source impedance. Fig. 2 visualizes the trajectories for different load impedances transforming them into the source impedance. The line properties needed for that are given as well. Since a conventional transmission line covers only Case A due to their line impedance being always real, we consider now an artificial line because it gives more degrees of freedom and can have the required properties of both Cases A and B, as will be shown in the following. The corresponding unit cell of the so-called CRLH line [11] with finite length is shown in Fig. 3. In general, this line shows a bandgap in between the two frequencies [12] (2)

for a unit cell of length , considering the cases when the bracketed expressions change sign by exceeding their self-resonance (2). Fig. 4 shows exemplary plots frequencies for an arbitrary unit cell configuration visualizing these properties. The matching scenarios of Cases A and B are demonstrated now for two of the normalized impedances shown in Fig. 2 and the CRLH line of Fig. 4. For Case A, we consider the , marked by a solid line frame load impedance in Fig. 2, which needs an electrical line length of rad to be matched to 50 . The line and line impedance GHz in the RH shows the appropriate line impedance for regime. To obtain the correct electrical length, the line has to be mm, giving tailored to a physical length of rad/m rad. For Case B, we consider , marked by the dashed line frame, needing a line with and . The line impedance is obtained inside the bandgap for GHz. From /m follows a 10.7 mm. Due to the dispersive character physical length of of the CRLH line, the corresponding properties are frequency dependent and, therefore, have a limited bandwidth. III. TUNABLE ARTIFICIAL LINE To match different impedances with one circuit at a fixed frequency, it has to be tunable. An artificial line can also only be tailored to discrete values of the electrical length because of the discrete unit cells. Therefore, we replace the fixed elements and of Fig. 3 by variable elements and , . The propagatunability denoted by multiplicative factors tion constant and line impedance now become functions of the . tuning state

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Fig. 4. Visualization of the described properties for an ideal CRLH line ac: nH, C : pF, L nH, cording to Fig. 3 with L C : pF, and dz : mm. The upper two plots show the propagation R jX . constant  j and the lower two the line impedance Z 2.8 GHz to f : GHz, the line impedance In the bandgap region from f gets imaginary and the propagation constant is real.

= 0 58 = +

= 1 33 = 6 33 

= 0 34

58

= 10 = +

For phase-shifting applications reported in [9], we used symspanning a 1-D tuning space, metric tuning states having which enables constant line impedance together with a variable phase shift, as can be seen from rewriting (4) as

(5)

where the factors in brackets in the numerator and denominator cancel each other for the balanced condition [12] (6) We obtained a figure-of-merit (FoM) of 60 /dB\ –\ 75 /dB at 6.5 GHz with 30% bandwidth for the application as a matched phase shifter. The FoM is defined as differential phase shift of and divided by the maximal occurring two tuning states attenuation when going from the first to second state

(7)

Fig. 5. (a) Equivalent-circuit representation of a CRLH artificial line with separate control of electric and magnetic properties. The dotted line marks one unit cell of the periodic structure with varactor diodes as tunable elements and additional biasing elements to apply the dc feed voltages. The boxes labelled “T-Line” symbolize the transformer line transforming the capacitors into the needed shunt inductors. RH contributions are omitted for clarity, but considered in all simulations. (b) Photograph of the built artificial line with overall dimensions of 18 mm 20 mm. From [9].

2

The set of tuning states has to be linearly ordered to have a defined path from the first to second state, which is fulfilled for the 1-D tuning. For the 2-D tuning treated later on, (7) is to , neither applicable due to infinitely many paths from nor needed because we do not consider the application as phase shifter, but as a matching circuit. As tunable elements, we used varactor diodes from Infineon, Munich, Germany, for the capacitances, and since there are no easily tunable inductances available, we used the same varactors together with a transformer line to realize tunable inductances for the shunt branch. Fig. 5(a) shows the equivalentcircuit model, neglecting RH contributions and parasitics for clarity. The special biasing scheme that has been used enables separate biasing of series and shunt varactors. This is essential to the application we propose, and distinguishes this paper from prior publications, which either tune only the series varactors, e.g., [12]–[14], or use a biasing scheme supporting only application of the same voltage to series and parallel loading ele-

DAMM et al.: ELECTRICALLY CONTROLLABLE ARTIFICIAL TRANSMISSION LINE TRANSFORMER FOR MATCHING PURPOSES

Fig. 6. (a) Photograph of the microstrip test fixture used to contact the circuit. Calibration was done using a pseudo-zero-length TRL calibration with the built standards shown in (b) to remove the influence of connector transitions and microstrip feeding lines.

ments [15], [16], prohibiting the application we present here. Fig. 5(b) shows the actual device with overall dimensions of 18 mm 20 mm we designed on a Rogers RO3003 substrate. Now we proceed to investigate the general case of varying both tuning voltages independently, leading to a 2-D tuning space spanned by both parameters. Therefore, this case is called 2-D tuning. This tuning strategy gives a line with the propagation constant imaginary or real together with a real or imaginary line impedance, respectively,

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Fig. 7. Measured transmission phase versus transmission attenuation for 5 GHz and with 50- load and source impedance. Solid curves represent tuning states with constant voltage U and dashed curves states with constant voltage U . Numbers at the ends of the curves are the corresponding voltages. Data from [9].

Fig. 8. Circuit schematic of generator with source voltage U , source impedance Z , and load impedance Z . The two-port box symbolizes the matching network described by the chain parameter matrix .

A

(8) We essentially need the unbalanced bandgap region, previously excluded for the phase shifter, to cover the matching Case B shown in Fig. 1(b). The operation frequency for this mode is 5 GHz with a bandwidth chosen to be 10%. There is no strict criterion for the bandwidth because increasing the operation region will decrease the matchable impedance region, and vice versa. Measurements are done in a microstrip environment using an Anritsu vector network analyzer and a Rosenberger microstrip test fixture to contact the device. A standard thru-reflect-line (TRL) calibration with custom standards was used to remove the influence of feeding lines enabling exact phase evaluation without the need of further deembedding. Test fixture and calibration standards are shown in Fig. 6. The measured tuning range for the circuit is shown for 5 GHz in Fig. 7. It gives in relation to the the phase of the forward transmission . Solid lines belong magnitude of the forward transmission to constant values of the first tuning voltage and dashed lines to constant . The numbers at the ends of the lines represent the corresponding voltage. Each intersection between both line , the area formed by all types gives one tuning state intersections equals the obtainable tuning range. Moving along a horizontal path in this area corresponds to a constant phase with variable line impedance, along a vertical path, vice versa. Ideally the line covered area should be a rectangle with edges parallel to the axes so that every phase value could be combined

Fig. 9. Shaded area in the Smith chart represents the impedances that can be perfectly matched to 50 using the build device. Every black dot corresponds to a matchable impedance calculated from a measured tuning state of the line. Solid lines describe path of constant voltage U and dashed lines describe path of constant U . The reference impedance is 50 .

with all values of the line impedance, but due to parasitic contributions like packaging, soldering, and frequency-dependent behavior this cannot be reached. To evaluate the matchable impedance region, a rigorous procedure such as described in [17] is not possible due to nonideal element behavior and numerous parasitics. Instead we use the following way: the scattering parameters of the device are measured for all tuning states, and then converted into chain parameters using the well-known formulas (e.g., [18]). The definition

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=

Fig. 10. Visualization of the real power ratio transmission coefficient t P =P , describing the power dissipation in the matching network in dependence of the load impedance. (a) Measured data. (b) Simulated values. The scale is linear and unit free.

Fig. 11. Obtained gain concerning the delivered power using the matching network. The gain is defined as the ratio of delivered real power with a matching network to the directly delivered real power without a matching network. Upper plot: (a) based on measured data. Lower plot: (b) for simulation results.

of currents and voltages at the ports are according to Fig. 8. The and current at the chain parameter matrix links voltage output terminal to the input terminal as follows:

for given input variables. Since matching to the real valued is assumed at the input port 1, it holds source impedance (12)

(9) Inserting (10) and (12) into (11) yields for the load impedance that can be matched to To calculate the load impedance the inverse chain matrix of the artificial line is used

,

(10) since the load impedance

is now explicitly given by

(11)

(13) Evaluating (13) for all measured tuning states gives all impedances that can be perfectly matched to the source impedance. In is always assumed to the following, the source impedance be 50 without loss of generality. Fig. 9 shows the impedances obtained in that way, which and can be matched to 50 . Solid lines represent constant dashed lines represent constant , respectively. The dots are the actual measured points. The impedance region is quite large, which corresponds to a large tuning range of line impedance and electrical length. For all calculated impedances, the input

DAMM et al.: ELECTRICALLY CONTROLLABLE ARTIFICIAL TRANSMISSION LINE TRANSFORMER FOR MATCHING PURPOSES

reflection is below or equal to 70 dB. Allowing a less redB will further increase strictive input matching of the matchable area. It is interesting to note that all circular lines corresponding to one constant voltage run through the center of the Smith chart. This is due to the fact that the circuit was originally designed to have an impedance of 50 for the 1-D tuning. Calculating the real power transmission coefficient gives a measure for the power dissipated in the matching network

(14)

The results are shown in Fig. 10 in dependence of the corresponding impedances. For an ideal lossless circuit, all impedances will lead to a value of one, but due to inherent insertion losses of approximately 2.5–4 dB, this is not obtained. The efficiency decreases towards the border of the Smith chart. This is because the idle power traveling back and forth on the line is getting larger the farther away the load impedance is located from the matching point in the center of the chart. The lower picture/plot of Fig. 10 shows results obtained by simulation of the circuit with AWR’s Microwave Office. Since the periodic structure is sensitive to small parameter changes, all used surface mount device (SMD) elements have been characterized individually on the same substrate as the circuit itself, again using a TRL calibration for well-defined reference planes. The varactors were characterized throughout the whole tuning voltage range. The obtained data was then included into the simulator by scattering parameter files. The simulated results agree very well with the measured data, showing only a small rotation and compression of the matchable impedance area compared to measurements. The values of the power transmission is also slightly higher in the simulation. This is due to the lack of radiation losses in the simulation, as well as additional losses in the real circuit due to the manual soldering. Since the matching network is lossy, a reduction of input reflection does not necessarily mean an increased power delivered to the load. Therefore, to evaluate the gain obtained by using the lossy matching network, the ratio of the power delivered to the and the power delivered to the load with matching network is calculated load when directly connected to the source

(15) and plotted in Fig. 11, again for measured and simulated values. In an area around the center, the gain is smaller than 1 and, therefore, a larger real power is delivered to the load without matching network. This is due to the network losses because a gain larger than one means that the mismatch losses of the impedance without matching network must be higher than the inherent losses in the network. Therefore, the losses of the network must be reduced to increase the gain, but there is an advantage even for an area with gain smaller than 1 not expressed in the plot because the load is well matched to the source using the network and there is no reflected power that travels back into the generator.

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IV. CONCLUSION AND OUTLOOK The successful realization of a fully controllable artificial line using semiconductor varactors and application as a novel matching network has been shown in this paper. A broad matchable impedance region was obtained at 5 GHz for 10% bandwidth, although the circuit was primarily optimized to for the case of symmetric maintain an impedance of 50 . Therefore, a redesign for the more general aptuning plication of arbitrary tuning can further increase the bandwidth as well as the matchable impedance region. A transfer of this proof of concept to MMIC technologies is very promising because the shortening of the unit cell will lead to increased bandwidth and the reduction of parasitic effects due to packaging and soldering will give better performance in terms of insertion loss and easier design because these parasitics dramatically degrade the ideal CRLH properties. The possible direct integration into existing MMIC applications also seems very interesting and will be pursued in further study. The power-handling capability of the circuit has not yet been studied, but is expected to be low due to the nonlinearity of the semiconductors. A replacement of the varactor diodes by barium–strontium–titanate varactors, which are known for their high power-handling capabilities, e.g., [19], is straightforward and would alleviate this problem. Liquid crystals (LCs) were also recently found to be quite linear in high power regions [20] and are, therefore, an excellent option to realize the proposed structure with LC varactors giving augmented power-handling capabilities, as well as higher usable frequency regions. REFERENCES [1] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index media supporting 2-D waves,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, 2002, pp. 1067–1070. [2] C. Caloz and T. Itoh, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” in IEEE AP-S/URSI Int. Symp. Dig., San Antonio, TX, Jun. 2002, pp. 412–415. [3] A. A. Oliner, “A periodic-structure negative-refractive-index medium without resonant elements,” in IEEE AP-S/URSI Int. Symp. Dig., San Antonio, TX, Jun. 2002, pp. 41–44. [4] G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [5] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” Electron. Lett., vol. 38, no. 23, pp. 1414–1416, Nov. 2002. [6] R. Islam and G. V. Eleftheriades, “Phase-agile branch-line couplers using metamaterial line,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 7, pp. 340–342, Jul. 2004. [7] C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 980–992, Mar. 2004. [8] M. Schüßler, J. Freese, and R. Jakoby, “Design of compact planar antennas using LH-transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 209–212. [9] C. Damm, M. Schüßler, J. Freese, and R. Jakoby, “Artificial line phase shifter with separately tunable phase and line impedance,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 423–426. [10] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 3.6, p. 95. [11] A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Micro, vol. 5, no. 3, pp. 34–50, Sep. 2004. [12] M. Antoniades and G. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 103–106, 2003.

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[13] A. Megej and V. F. Fusco, “Low-loss analog phase shifter using varactor diodes,” Microw. Opt. Technol. Lett., vol. 19, no. 6, pp. 384–386, Dec. 1998. [14] H. Kim, A. Kozyrev, A. Karbassi, and D. van der Weide, “Linear tunable phase shifter using a left-handed transmission line,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 5, pp. 366–368, May 2005. [15] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 161–173, Jan. 2005. [16] D. Kuylenstierna, A. Vorobiev, P. Linner, and S. Gevorgian, “Composite right/left handed transmission line phase shifter using ferroelectric varactors,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 167–169, Apr. 2006. [17] M. Thompson and J. Fidler, “Determination of the impedance matching domain of impedance matching networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 10, pp. 2098–2106, Oct. 2004. [18] G. D. Vendelin, A. Pavio, and U. Rhode, Microwave Circuit Design. New York: Wiley, 1990, ch. 1, pp. 16–17. [19] A. Kozyrev, A. Ivanov, T. Samoilova, O. Soldatenkov, and K. Astafiev, “Nonlinear response and power handling capability of ferroelectric TiO film capacitors and tunable microwave devices,” J. Ba Sr Appl. Phys., vol. 88, no. 9, pp. 5334–5342, Nov. 2000. [20] F. Goelden, S. Mueller, P. Scheele, M. Wittek, and R. Jakoby, “IP3 measurements of liquid crystals at microwave frequencies,” in Proc. 36th Eur. Microw. Conf., Sep. 2006, pp. 971–974.

Christian Damm (S’03) was born in Marburg, Germany, in 1977. He received the Dipl.-Ing. degree from the Technische Universität Darmstadt, Darmstadt, Germany in 2004, and is currently working toward the Dr.-Ing. degree at the Institut für Hochfrequenztechnik, Technische Universität Darmstadt. He is currently a Research Assistant with the Institut für Hochfrequenztechnik, Technische Universität Darmstadt. His research interests include tunable microwave components and antennas based on metamaterials and liquid crystals.

Jens Freese was born in Wittmund, Germany, in 1973. He received the Dipl.-Ing. degree from the University of Siegen, Siegen, Germany, in 1999, and is currently working toward the Dr.-Ing. degree at the Institut für Hochfrequenztechnik, Technische Universität Darmstadt, Darmstadt, Germany. In January 2007, he joined Tesat Spacecom GmbH & Co. KG, Backnang, Germany, where he is involved in the development of microwave circuits for satellite applications. His research interests included planar and conformal antennas, as well as active array antennas and adaptive beamforming.

Martin Schüßler was born in Hösbach, Germany, in 1967. He received the Dipl.-Ing. and Ph.D. degrees from the Technische Univerität Darmstadt, Darmstadt, Germany, in 1992 and 1998, respectively. He is currently a Researcher with the Institut für Hochfrequenztechnik, Technische Universität Darmstadt, where he is involved with small antennas and metamaterials.

Rolf jakoby (M’97) was born in Kinheim, Germany, in 1958. He received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the University of Siegen, Siegen, Germany, in 1985 and 1990, respectively. In January 1991, he joined the Research Center, Deutsche Telekom, Darmstadt, Germany. Since April 1997, he holds a Chair in Wireless Communications with the Technische Universität Darmstadt, Darmstadt, Germany. His research deals with channel modeling and network planning for broadband wireless systems, smart/compact antennas and microwave sensors, tunable passive microwave components based on agile materials, and metamaterial-based applications in the microwave region. He is Editor-in-Chief of Frequenz. Dr. Jakoby was the recipient of a 1992 award presented by the Chamber of Commerce and Industry (CCI) Siegen and a 1997 Society for Information Technology (ITG) Prize.

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An 11-Mb/s 2.1-mW Synchronous Superregenerative Receiver at 2.4 GHz F. Xavier Moncunill-Geniz, Pere Palà-Schönwälder, Member, IEEE, Catherine Dehollain, Member, IEEE, Norbert Joehl, and Michel Declercq, Fellow, IEEE

Abstract—This paper presents a low-voltage low-power high-speed superregenerative receiver operating in the 2.4-GHz industrial–scientific–medical band. The receiver uses an architecture in which, thanks to the presence of a phase-locked loop, the quench oscillator is operated synchronously with the received data at a quench frequency equal to the data rate. This mode of operation has several benefits. Firstly, the traditional problem of poor selectivity in this type of receiver is to a large extent overcome. Secondly, considerably higher data rates can be achieved than with classical receivers. Thirdly, the bit envelope can be matched to the superregenerative oscillator, which improves sensitivity. The receiver includes an RF front end optimized to support high quench frequencies at low supply voltages, responding to today’s increasing demand for high speed and low power consumption. The prototype implemented is very simple and achieves a data rate of 11 Mb/s with a current consumption of 1.75 mA at a supply voltage of 1.2 V—an excellent tradeoff between cost, performance, and power consumption. Index Terms—Low power, radio receiver, RF oscillator, superregenerative receiver.

I. INTRODUCTION

S

UPERREGENERATIVE receivers are well suited to short-range wireless communications due to their exceptional simplicity, reduced cost, and low power consumption. Typical applications for this type of receiver are remote control systems (such as garage door openers, robots, model ships, airplanes, etc.), short distance telemetry, and wireless security [1]–[7]. The interest of industry in the superregenerative receiver is made evident by recent patent applications [8]–[10], more than eighty years after it was presented by Armstrong in 1922 [11]. Hence, superregenerative architectures are present today in commercial products, in which they typically operate at low data rates.1 Research on superregeneration has been carried out in unlicensed RF bands up to millimeter-wave frequencies [12], [13]. However, only very recent papers report operation Manuscript received October 16, 2006. This work was supported by the Spanish Dirección General de Investigación under Grant TIC2003-02755 and Grant TEC2006-12687/TCM, and by the École Polytechnique Fédérale de Lausanne. F. X. Moncunill-Geniz and P. Palà-Schönwälder are with the Department of Signal Theory and Communications, Technical University of Catalonia, 08034 Barcelona, Spain (e-mail: [email protected]). C. Dehollain, N. Joehl, and M. Declercq are with the Electronics Laboratory, Swiss Federal Institute of Technology, CH-1015 Lausanne, Switzerland (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.896796 1Telecontrolli Wireless Products—RF Receivers, Telecontrolli Srl, Casoria, Italy. [Online]. Available: http://www.telecontrolli.com

in the 2.4-GHz industrial–scientific–medical (ISM) band [3], [5]–[7], [14], [15], which, compared to lower frequency bands, allows larger signal bandwidth to be used and is available worldwide. It is well known that classical superregenerative receivers suffer from poor frequency selectivity when they are applied to narrowband communications and, consequently, are more vulnerable to noise and interference than other systems [2], [16]. This behavior is caused by the characteristic pulsating operation of the receiver, in which the superregenerative oscillator (SRO) controlled by the quench oscillator samples the envelope of the input signal asynchronously at a rate (quench frequency) that is considerably higher than the modulation bandwidth. During the sampling process, the SRO is sensitive to the input signal for a relatively small fraction of the quench period. Hence, the sensitivity periods are much shorter than the data period and, consequently, the RF bandwidth, which is inversely proportional to the duration of the sensitivity periods, becomes much larger than the modulation bandwidth. The selectivity of a superregenerative receiver can be improved by decreasing the quench frequency. In practice, this limits its use to low data-rate applications. Although the selectivity can also be improved by using special quench wave shapes, the RF bandwidth will continue to be considerably larger than the modulation bandwidth [2]. The use of stable and high- frequency references such as surface acoustic wave (SAW) or bulk acoustic wave (BAW) devices can also decrease the reception bandwidth, although this unavoidably reduces the quench frequency and, therefore, the data rate [4], [16]. Significant improvements can be obtained by using smart -enhancement techniques, as reported recently in [5] and [6]. Responding to today’s increasing demand for high-speed and low-power consumption, in this paper we present a superregenerative receiver prototype designed to support high quench frequencies. We also make use of a synchronous mode of operation, which has been successfully applied in spread-spectrum communications [14], [15] and yields significant advantages over conventional receivers. Thanks to the presence of a phase-locked loop (PLL), the SRO is quenched synchronously with the received data so that the quench frequency equals the data rate. The usefulness of this mode of operation is twofold: on the one hand, for a given data rate, the quench frequency can be reduced in comparison with a conventional receiver to obtain a more selective receiver with an RF bandwidth comparable to that of the input signal and, on the other, for a fixed quench frequency (e.g., limited by resonator ), the synchronous operation allows higher data rates to be achieved. Receiver sensitivity

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the logarithmic mode, the amplitude of the oscillations is allowed to reach its limiting equilibrium value, which is determined by the nonlinearity of the active devices. In this mode of operation, the amplitude of the RF pulses remains constant, but the incremental area under the envelope is proportional to the [3]. In both modes, the modlogarithm of the amplitude of ulating input signal can be retrieved by low-pass filtering the envelope of the RF pulses, as depicted in Fig. 1(a). The low-noise amplifier (LNA) improves signal reception and minimizes SRO reradiation through the antenna. According to the analysis presented in [3], the SRO can be modeled as a frequency-selective network fed back through a variable-gain amplifier. Typically, the selective network has two dominant poles that provide a bandpass response centered on a certain frequency , which is characterized by the transfer function (1) is the maximum ampliwhere is the damping factor and fication. The feedback amplifier controlled by the quench oscilthat varies periodically with time. lator provides a gain The closed-loop operation of the system can be characterized from the instantaneous damping factor, which is defined as (2)

Fig. 1. (a) Block diagram of a conventional superregenerative receiver. (b) SRO RF input signal, closed-loop damping function, and output voltage in the linear mode of operation.

can also be improved when specially shaped symbols are transmitted. In [17], the design of a superregenerative RF front end conceived to support high data rates is presented. The front end operates at 3 V of voltage supply and reaches 10 Mb/s. In this paper, we describe the entire receiver architecture and present an improved design in which the supply voltage has been decreased to 1.2 V and the transfer speed increased to 11 Mb/s. The design achieves very low consumed energy per bit (0.19 nJ/bit), thus offering excellent performance at low cost.

In the periods in which is negative [see Fig. 1(b)], the system becomes unstable and the amplitude of the RF oscillachanges to positive, the system stabilizes tion rises. When and the oscillation is damped. The behavior of the receiver can be characterized from its response to a single RF pulse applied within the limits defined by and in Fig. 1(b), which can be expressed as (3) where is a positive value representing the peak amplitude and is a normalized shaping function that equals zero outside . The voltage generated at the output of the the interval (i.e., SRO is another RF pulse that starts increasing at crosses zero with a negative slope) and achieves the when maximum amplitude at (i.e., at the zero crossing of with a positive slope). The expression of the output pulse within the interval (0, ) is [3]

II. BASIC THEORY OF SUPERREGENERATION The block diagram of a conventional narrowband superregenerative receiver is shown in Fig. 1(a). The core of the receiver is the SRO. It is an RF oscillator that is controlled by a low-frequency quench generator or quench oscillator, which causes the RF oscillations to rise and die out repeatedly. The signal generated in the SRO is composed of a series of RF pulses separated by the quench period , in which the periodic buildup of the , as depicted in oscillations is controlled by the input signal Fig. 1(b). In the linear mode of operation, the oscillations are damped before they reach their limiting equilibrium amplitude, and their peak amplitude is proportional to that of the injected signal. In

(4) where is an amplification factor, is a normalized bandis pass frequency-response function centered on , and , and are the normalized output pulse envelope. , mainly determined by the frequency and shape of the quench signal, and their expressions can be found in [3]. In particular, can be calculated through (5) where

is the superregenerative gain, which is associated with

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Fig. 2. Time and frequency diagrams representing the SRO input and output signals with OOK modulation. (a) Taking several samples per bit (classical receiver). (b) Taking one sample per bit (implemented receiver).

an exponential buildup of the oscillation (6) and is the sensitivity curve, a normalized function that takes [3] the maximum of unity at (7) Fig. 3. Block diagram of the synchronous superregenerative receiver.

at is finite, can be approxiWhen the slope of mated by a Gaussian function, as expressed in (7), whose stanis considerably smaller than the quench pedard deviation riod. Simulation carried out in [18] predicts for normal conditions of operation. Consequently, (5) shows that the receiver is especially sensitive to the input signal , which is called in a certain environment at the instant the sensitivity period. This point confirms the behavior of the SRO as a sampling device. The RF bandwidth of the receiver is proven to be inversely proportional to , in accordance with [3], [18] dB

(8)

Fig. 2(a) illustrates operation in the linear mode of a conventional receiver with a narrowband on–off keying (OOK) modulated input signal. The SRO is asynchronously quenched several to satisfy the Nyquist criterion times during each bit period so that the information can be retrieved by low-pass filtering the baseband bit samples. As shown in Fig. 2(a), since the sensitivity period is much shorter than the bit period, the resulting RF bandwidth is much greater than the modulation bandwidth. ; therefore, . Roughly, dB This is the main reason why this receiver exhibits poor selectivity. III. SYNCHRONOUS OPERATION OF THE RECEIVER Fig. 2(b) illustrates the mode of operation employed in the current receiver for detecting OOK-modulated signals.

The SRO is quenched synchronously with the received signal so that a single sample of each bit pulse is taken. The synchronous operation of the SRO has already been exploited in spread-spectrum communications. However, unlike the architectures described in [14], in which the input signal is sampled at a rate of one sample per chip with subsequent integration of the chip samples to retrieve the bit values, in the current approach, the bits are sampled directly. This means that the quench frequency equals the bit frequency. Since the duration of the sampled bit is closer to that of the sensitivity period, the bandwidth of the modulated signal and that of the receiver become similar. Moreover, the synchronous operation allows the use of special bit envelopes that concentrate the signal energy in the sensitivity periods. In particular, (5) indicates that the receiver operates as a matched filter when the signal . envelope matches the sensitivity curve, i.e., when In this way, the receiver can make more efficient use of the incoming signal power. The block diagram of the receiver is shown in Fig. 3. It incorporates a PLL that controls the quench voltage-controlled oscillator (VCO) to ensure bit sampling is properly carried out. The error signal is obtained from the output of the envelope detector, which is passed through a baseband amplifier (BBA) and a loop filter. Fig. 4 shows the normalized envelope of the received bit pulse and the sensitivity curve of the SRO under normal operation. Once acquisition has been achieved, the loop performs tracking by centering the sensitive periods of the SRO on the

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Fig. 4. Normalized envelope of the input bit pulse p (t), shifted sensitivity curve of the SRO s(t +  ), and normalized envelope of the SRO output oscillation.

ascending flanks of the bit pulses received. Thus, the phase deviation of the quench VCO with regard to the received pulse results in an amplitude variation at the loop-filter output that tends to correct the error. Operation on a single flank allows a simpler architecture to be implemented. IV. USE OF THE SRO AS A PHASE DETECTOR The classical theory of PLLs is applicable to the current design. The main novelty in this scheme is in the way in which the phase is detected by the SRO. Proper PLL performance requires that the sensitive periods of the SRO be centered on the rising flanks of the bit envelopes, which means that a certain tracking error must exist for a given input-signal amplitude when the input bit frequency is locked (Fig. 4). From (4) and (5), the incremental voltage of a tuned receiver at the envelope detector s with respect output when the sensitivity curve is delayed to the equilibrium point in Fig. 4, as well as for a generic peak amplitude of the input bit pulse, is defined by (9) where the function is subsequently averaged by the loop filter and provides the discrimination characteristic (10) is the cross-correlation of (normalized bit enve(sensitivity curve of the SRO). lope) and As has been mentioned, a return-to-zero bit envelope matched is optimum to the sensitivity curve of the SRO for the SRO. Assuming this condition and also the Gaussian ex(7), the expression of the discrimination charpression of acteristic becomes (11) Fig. 5 shows the normalized discrimination curves for sev, where the origin corresponds eral values of with to a stable equilibrium point of operation. Generally, as is confirmed in practice, the best performance from the point of view of tracking occurs when is approximately 1.7 , as it provides a linear behavior with a larger dynamic range around the origin. However, a smaller value may be preferred in order to increase the output signal-to-noise ratio, especially at low signal levels

Fig. 5. Normalized discrimination characteristic for several values of  and V = V .

from the point of view of data detection). There(ideally, fore, a compromise must be established in practice. V. DESCRIPTION OF THE IMPLEMENTED PROTOTYPE A schematic diagram of the implemented receiver is shown in Fig. 6. It is a discrete-element prototype based on the diagram in Fig. 3 and built on a printed circuit board (PCB) with a 0.8-mm FR-4 substrate. The RF part (LNA, SRO, and envelope detector) is based on the design presented in [17], which was developed and electromagnetically simulated with Agilent Technologies’ Advanced Design System (ADS). The BFP405 transistor is used in the RF stages to take advantage of its high gain at low biasing currents and low parasitic capacitances. The LNA is a cascode configuration providing high reverse isolation of approximately 38 dB. It includes an input matching network to achieve a minimum noise figure below 3 dB. The SRO is built on a Colpitts oscillator operating at 2.45 GHz, in which the inductance is provided by a small microstrip line. The quench is applied through the base by means of an RF choke to avoid oscillator loading effects. The LNA and the envelope detector are connected to the SRO transistor base to take advantage of higher signal amplitude. The impedance of the choke and of the inter-stage capacitors is carefully chosen to avoid unwanted filtering of the quench signal at high quench frequencies. The resonance frequency of the loaded SRO is adjustable around 2.45 GHz, and the quiescent loaded , when the quench is disabled, equals 38. This value is the result of a compromise between achieving high quench frequencies and low current consumption. The envelope detector is built on a common-collector configuration in which the emitter capacitor is charged by the transistor and discharged through the emitter resistor. It includes a filter in the biasing network to remove the quench components transferred from the SRO. Limited by the cascode, the minimum supply voltage of this architecture is approximately 0.95 V. Further details on the RF part are described in [17]. The BBA makes use of a common-emitter inverting amplifier to increase the amplitude provided by the envelope detector from approximately 50 mVpp to 0.5 Vpp on an output load of

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Fig. 6. Complete schematic of the implemented receiver (11-Mb/s data rate).

10 pF. A clamper is included at the output to restore the upper base line of the signal. The output pulses can be used directly to perform the data detection and, unlike classical receivers, no low-pass filter is required to integrate the bit samples and remove the quench components. A common-base Colpitts oscillator acts as the quench VCO, in which a resistive feedback network and the coupling capacitor of the varicap act as a first-order loop filter. The quench signal generated is sinusoidal and has a relatively small amplitude, which ensures that a more selective receiver is obtained. The quench signal is taken from the emitter because of the low output impedance in this node. The signal inversion caused by the BBA amplifier means that the PLL leads the SRO to operate on the descending flank of the received pulses instead of the ascending one (Fig. 4). A modified version of the scheme in Fig. 6 adapted to operate at a lower rate of 1 Mb/s was also evaluated. VI. EXPERIMENTAL RESULTS The performance of the architecture presented was evaluated at 1- and 11-Mb/s data rates. Fig. 7 shows a sequence of the modulating signal and the corresponding voltages measured at the receiver side at 11 Mb/s. Fig. 8 shows the frequency selectivity curves at 1 and 11 Mb/s and Table I summarizes the receiver’s main features. The signal modulation was, in all cases, an OOK modulation with a matched Gaussian bit envelope to

Fig. 7. Signals at the transmitter and receiver sides at 11-Mb/s data rate.

improve sensitivity and facilitate PLL operation. The selectivity curves were obtained from the rejection of a continuous wave (CW) signal with regard to the reception center frequency. The 3-dB bandwidth is 5 and 38 MHz, respectively. Note that the ratio between the receiver RF bandwidth and the data rate is much smaller than in a classical receiver. For instance, a classical receiver operating at 1 Mb/s and taking ten samples per bit would require a quench frequency of 10 MHz, thus exhibiting a bandwidth close to 38 MHz. With the synchronous operation,

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TABLE II DISTRIBUTION OF THE CURRENT CONSUMPTION IN THE RECEIVER IN MICROAMPERES

TABLE III COMPARISON OF SUPERREGENERATIVE RECEIVER PERFORMANCE Fig. 8. Selectivity curves at 1- and 11-MHz quench frequency.

TABLE I SUMMARY OF RECEIVER PERFORMANCE

the RF bandwidth is only 5 MHz. At 11 Mb/s, the ratio between the RF bandwidth and the data rate, which is equal to 3.5, is even more favorable. In comparison with a Bluetooth receiver, the current prototype at 1 Mb/s turns out to be less selective [19]. However, the use of a matched modulation consisting of a bit envelope with relatively small duty cycle will generally introduce additional rejection to both adjacent channel and cochannel interference. A characteristic of the implemented receiver is that the demodulated output becomes independent from the input signal level within a certain range. The reason for this is that the loop advances or delays the phase of the quench VCO to regulate the amount of input signal power that falls into the sensitivity pe-

riods of the SRO. This ensures that, in the steady state of operation, the control signal of the VCO exhibits a fixed level to maintain the loop in lock. This self-regulation mechanism acts as an automatic level control. The supported input dynamic range depends in practice on the characteristics and adjustment of the loop filter. Table II shows the distribution of consumption in the receiver and Table III compares the implemented receiver with other reported superregenerative architectures. The data rate achieved by the current prototype exceeds that of previous designs by more than one order of magnitude, and the energy per received bit is substantially lower. Its sensitivity is also remarkable in comparison with the data rate. The net sensitivity improvement due to the use of a matched bit envelope taking into account the tracking error of the PLL is in the order of 7 dB. Finally, we mention that the implemented SRO can be operated at higher data rates (nearly 18 Mb/s). However, sensitivity decreases progressively due to hangover limitations [3]. VII. CONCLUSION In this paper, a high-performance superregenerative receiver has been presented that allows the limits of superregeneration in the ISM band of 2.4 GHz to be assessed. The receiver uses a moderate resonator to support high quench frequencies and operates in a synchronous mode, the latter yielding a significant number of advantages as follows. 1) Increased data rate: since this rate equals the quench frequency and not a fraction of it. In particular, 11 Mb/s is

MONCUNILL-GENIZ et al.: 11-Mb/s 2.1-mW SYNCHRONOUS SUPERREGENERATIVE RECEIVER AT 2.4 GHz

a landmark value for this type of receiver that clearly surpasses previously reported values [1]–[7]. 2) Improved selectivity: the RF bandwidth of the receiver is much closer to the bandwidth of the received signal, overcoming one of the traditional drawbacks of the receiver. 3) Improved sensitivity: the use of a matched bit envelope allows the bit energy to be concentrated in the sensitivity periods of the receiver. The improvement typically ranges from 5 to 10 dB. 4) Simplicity: since each received bit can be detected in a single quench period, the synchronous receiver does not require the relatively high-order low-pass filter that is commonly included in conventional receivers to remove quench components. 5) Reduced power consumption: for a given data rate, the receiver can be quenched at a lower frequency, which reduces current consumption in the SRO. 6) Data clock: the quench signal can be used as a bit-synchronous clock reference in subsequent circuits, which renders additional clock recovery circuits unnecessary. In summary, the current design demonstrates that superregenerative receivers can be successfully operated in the 2.4-GHz ISM band with an outstanding tradeoff between cost, performance, and power consumption. The synchronous operation is also applicable to high- SROs operating at low data rates.

REFERENCES [1] A. Vouilloz, M. Declercq, and C. Dehollain, “A low power CMOS super-regenerative receiver at 1 GHz,” IEEE J. Solid-State Circuits, vol. 36, no. 3, pp. 440–451, Mar. 2001. [2] N. Joehl, C. Dehollain, P. Favre, P. Deval, and M. Declercq, “A low power 1 GHz super-regenerative transceiver with time-shared PLL control,” IEEE J. Solid-State Circuits, vol. 36, no. 7, pp. 1025–1031, Jul. 2001. [3] F. X. Moncunill-Geniz, P. Palà-Schönwälder, and O. Mas-Casals, “A generic approach to the theory of superregenerative reception,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 1, pp. 54–70, Jan. 2005. [4] B. Otis, Y. H. Chee, and Y. Rabaey, “A 400 W-RX, 1.6 mW-TX super-regenerative transceiver for wireless sensor networks,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., San Francisco, CA, Feb. 2005, vol. 1, pp. 396–397, 606. [5] J. Y. Chen, M. P. Flynn, and J. P. Hayes, “A 3.6 mW 2.4-GHz multichannel super-regenerative receiver in 130 nm CMOS,” Proc. IEEE Custom Integrated Circuits Conf., pp. 361–364, Sep. 2005. [6] J. Y. Chen, M. P. Flynn, and J. P. Hayes, “A fully integrated autocalibrated superregenerative receiver,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2006, pp. 1490–1499. [7] I. McGregor, G. Whyte, K. Elgaid, E. Wasige, and I. Thayne, “A 400 W Tx=380 W Rx 2.4 GHz super-regenerative GaAs transceiver,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 1523–1525. [8] V. Leibman, “Superregenerative low-power receiver,” W.O. Patent 03 009 482, Jan. 30, 2003. [9] K. L. Addy, “Thermostat having a temperature stabilized super-regenerative RF receiver,” U.S. Patent 6 810 307, Oct. 26, 2004. [10] G. M. Vavik, “Transponder, including transponder system,” U.S. Patent 2005/0 270 222, Dec. 8, 2005. [11] E. H. Armstrong, “Some recent developments of regenerative circuits,” Proc. IRE, vol. 10, no. 8, pp. 244–260, Aug. 1922. [12] W. R. Day, “Superregenerative backward wave amplifiers for millimeter waves,” Proc. IEEE, vol. 52, no. 6, pp. 711–712, Jun. 1964. [13] N. B. Buchanan, V. F. Fusco, and J. A. C. Stewart, “A Ka band MMIC super regenerative detector,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, vol. 3, pp. 1585–1588.

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[14] F. X. Moncunill-Geniz, P. Palà-Schönwälder, and F. del Águila-Lopez, “New superregenerative architectures for direct-sequence spread-spectrum communications,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 52, no. 7, pp. 415–419, Jul. 2005. [15] F. X. Moncunill-Geniz, P. Palà-Schönwälder, C. Dehollain, N. Joehl, and M. Declercq, “A 2.4-GHz DSSS superregenerative receiver with a simple delay-locked loop,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 8, pp. 499–501, Aug. 2005. [16] D. L. Ash, “A low cost superregenerative SAW stabilized receiver,” IEEE Trans. Consumer Electron., vol. CE-33, no. 3, pp. 395–403, Aug. 1987. [17] F. X. Moncunill-Geniz, C. Dehollain, N. Joehl, M. Declercq, and P. Palà-Schönwälder, “A 2.4-GHz low-power superregenerative RF front-end for high data rate applications,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 1537–1540. [18] F. X. Moncunill Geniz, “New super-regenerative architectures for direct-sequence spread-spectrum communications,” Ph.D. dissertation, Dept. Signal Theory Commun., Univ. Politécnica Catalunya, Barcelona, Spain, 2002. [19] H. A. Alzaher and M. K. Alghamdi, “A CMOS bandpass filter for low-IF Bluetooth receivers,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 8, pp. 1636–1647, Aug. 2006.

F. Xavier Moncunill-Geniz received the Telecommunications Engineering and Ph.D. degrees from the Technical University of Catalonia, Barcelona, Spain, in 1992 and 2002, respectively. He is currently a Collaborating Professor with the Department of Signal Theory and Communications, School of Telecommunications Engineering of Barcelona (ETSETB), Technical University of Catalonia, where he is involved in the field of circuit theory and analog electronics. His main research area is RF circuit design with a particular emphasis on the theory and implementation of new superregenerative receiver architectures.

Pere Palà-Schönwälder (S’89–A’95–M’05) received the Telecommunications Engineering and Ph.D. degrees from the Technical University of Catalonia, Barcelona, Spain, in 1989 and 1994, respectively. He is currently an Associate Professor with the Department of Signal Theory and Communications, School of Engineering of Manresa (EPSEM), Technical University of Catalonia. He has managed several Spanish research projects. His research interests include computer-aided circuit design, nonlinear circuits, and RF communication electronics.

Catherine Dehollain (M’93) received the M.Sc. degree in electrical engineering and Ph.D. degree from the Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, in 1982 and 1995, respectively. Since 1995, she has been with the Electronics Laboratory (LEG), EPFL, where she is responsible for RF activities. Since 1998, she has been a Lecturer with the EPFL in the area of RF circuits, electric filters and computer-aided design (CAD) tools. Since 2006, she has been Maître d’Enseignement et de Recherche (MER) with the EPFL. She has authored or coauthored four scientific books and 50 scientific publications. Her research interests include low-power analog circuits and electric filters.

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Norbert Joehl was born in Geneva, Switzerland, in 1959. He received the M.S. and Ph.D. degrees in electrical engineering from the Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, in 1985 and 1992, respectively. Since 1985, he has been with the Electronics Laboratory, EPFL, where he has been involved in the field of low-power and high-performance RF analog CMOS and BiCMOS integrated circuit design and is currently responsible for research activities. He was a Lecturer of electronic systems with the École d’Ingénieurs de l’Etat de Vaud. He has authored or coauthored approximately 20 scientific publications.

Michel Declercq (S’70–M’72–SM’97–F’00) received the Electrical Engineering degree and Ph.D. degree from the Catholic University of Louvain, Louvain, Belgium, in 1967 and 1971, respectively. In 1985, he joined the Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, where he is currently a Professor, Dean of the School of Engineering, and Director of the Electronics Laboratory. He is an Expert of the European Commission for scientific research programs in information technologies. He has authored or coauthored over 220 scientific publications and three books. He holds several patents. His research activities are related to mixed analog-digital integrated-circuit design and design methodologies.

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A Wideband CMOS Variable Gain Amplifier With an Exponential Gain Control Hui Dong Lee, Student Member, IEEE, Kyung Ai Lee, and Songcheol Hong, Member, IEEE

Abstract—A CMOS wideband cascaded variable gain amplifier (VGA) with a temperature-independent exponential gain control characteristic is presented in this paper. The exponential gain control function is realized using parasitic bipolar transistors and a control signal converter. The bandwidth is extended using an inductive peaking technique for high-frequency operations. The gain of the VGA varies from 38.8 to 55.3 dB in relation to the control voltage that varies from 0 to 1.8 V. The bandwidth of the proposed VGA is approximately 900 MHz with a gain control range of 94.1 dB. The proposed VGA includes a dc offset cancellation loop to avoid amplification of the dc offset. The VGA is powered by 1.8 V with 11.4 mA. The VGA chip including bondpads occupies an area of 850 m 490 m. Index Terms—CMOS analog integrated circuit, gain control, linear-in-decibel gain characteristic, temperature compensation, variable gain amplifier (VGA), wideband systems.

I. INTRODUCTION HE variable gain amplifier (VGA) is an indispensable building block to maximize the dynamic range of modern wireless communication systems [1]–[3], as well as medical equipment, hearing aids, disk drives, and so on [4]–[7]. A VGA is typically employed in a feedback loop to realize automatic gain control (AGC). The VGA of an AGC loop is used to control the transmission signal power or to adjust the received signal amplitude. There are two possible approaches to build the VGA. One is to build a discrete gain step VGA with a digital control signal [8]–[10], and the other is to build a continuous VGA controlled by an analog gain control signal [1]–[7]. In general, digitally controlled VGAs use binary weighted arrays of resistors or capacitors for gain variations [11] and analog VGAs adopt a variable transconductance or a variable resistance to control the gain. For a code division multiple access (CDMA) system requiring a power control range larger than 80 dB, the VGA with continuously variable gains is preferred because it avoids signal phase discontinuity that is expected to cause problems [3], [12]

T

Manuscript received October 16, 2006; revised January 3, 2007. This work was supported by the Korea Science and Engineering Foundation under the Engineering Research Center Program through the Intelligent Radio Engineering Center, Information and Communications University, Korea. The authors are with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305701, Korea (e-mail: [email protected]; [email protected]; schong@ee. kaist.ac.kr). 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.2007.896787

and it reduces the large number of control bits required with digitally controlled VGAs. Until now, VGA circuits based on various technologies such as bipolar, BiCMOS, and CMOS have been introduced [1]–[12]. Recently however, CMOS VGAs are preferred due to the low cost and easy integration with other CMOS analog/digital parts. An important requirement for a CMOS VGA is a decibellinear gain control characteristic, where the gain of the VGA changes exponentially with the control signal. The exponential gain control is required to achieve a wide dynamic range and to maintain the AGC loop settling time independent of the input signal level [12], [13]. However, it is difficult to realize this exponential function due to its inherent square or linear characteristics in CMOS technology. Although a transistor operating in a subthreshold region has an exponential characteristic, it is generally not preferred due to other unfavorable effects such as noise and bandwidth [14]. Another possible method is to use parasitic bipolar transistors to generate the desired exponential function. The linear-in-decibel gain control signal is generand ated using the relationship between a collector current . This is strongly dependent on base-to-emitter voltage the temperature and processes. Therefore, various compensation techniques that guard against the temperature and the process variations are required to have an accurate signal power control mechanism. Obviously, temperature compensation is needed to control the output level in a linear-in-decibel manner over a wide dynamic range [15]. It is also preferable to minimize the resolution of the digital-to-analog converter, which drives the gain control terminal [12]. The method used to generate the temperature-independent exponential function requires complex circuits However, this method achieves a linear-in-decibel controlled range of more than 30 dB per stage, which is difficult to achieve using a pseudoexponential function in a short-channel CMOS. Another important aspect of a wideband VGA is a large bandwidth. There are many systems for high-speed data communications such as ultra-wideband (UWB) systems, wireless local area networks (LANs), and Bluetooth [16], [17]. These systems provide a high data rate with relatively low power consumption in short-range wireless communications. For high-speed data communication, the bandwidth of a VGA must be very wide. Therefore, a wideband VGA is a key component. In this paper, we introduce a wideband CMOS VGA with an exponential function generation. We discuss the VGA implementations in the view of the bandwidth and the obtainable gain range and, for that purpose, we compare some recently reported VGAs in Section II. In Section III, the method required to generate the exponential function is discussed and the simulation results are shown to prove the validity of the proposed

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Fig. 1. Four types of VGA topologies.

approach. The proposed VGA is presented in Section IV, which also includes a detailed description of the circuit topology. The measured performance and conclusion follow in Sections V and VI, respectively. II. COMPARISON OF VGA CIRCUITS There have been some challenges in the design of the VGAs to meet gain-control accuracy, stability, and linearity requirements. Here, the design techniques from previously presented CMOS VGAs focusing on both VGA cells with various gain control schemes and linear-in-decibel gain variation characteristics are discussed. Most CMOS VGAs use a pseudoexponential function [4]–[6] for decibel-linear gain expressed as control characteristics. As shown in Fig. 1(a), the core of the VGA consists of a differential amplifier and diode connected loads. The differential gain of the VGA in Fig. 1(a) is equal to , where is the transconductance of the is the output impedance. Since input differential pair and the output is a diode connected load, is proportional to . Since and are functions of the bias current, the gain variation is obtained by controlling the and diode loads . CMOSbias currents of the input pair based VGAs that adopt this function offer less than 15 dB of gain range with a linearity error of less than 0.5 dB [4]–[6]. Owing to the limited gain range in recent research, multiple stage VGAs have been used to satisfy the required dynamic gain range of

wireless systems. To improve the gain control range, the control currents ( and ) are generated by (1) based on the Taylor series approximation function [18]. The core VGA circuit is the same, i.e., (1) where and are fixed and is the independent control variable. The VGA exhibits a gain control range of more than 35 dB with a gain error of less than 1 dB through the optimization of the values of and ; however, its operating frequency varies with the control currents. In Fig. 1(b), the quasi-exponential function of a successive attenuation approximation was realized using a simple of the rigid with a combination of the constant resistance resistor and the variable resistance of the combined triode attenuation can be exof MOSFETs [19], [20]. The . Accordingly, we can approximate pressed as to . It is well known that is proportional to the gate source voltage in the triode attenuator in region. Thus, we can control the gain of the the exponential function. In this scheme, as the control voltage increases, the quasi-exponential curve expands more and more by successive approximations. In addition, the more MOSFETs that are turned on, the wider the piecewise exponential curve becomes. This results from the overlap conductance of the MOSFETs previously turned on. However, using only a single stage

LEE et al.: WIDEBAND CMOS VGA WITH EXPONENTIAL GAIN CONTROL

of the attenuation is not satisfactory to obtain the exponential function in the required dynamic range. Fig. 1(c) shows a signal-summing VGA, which has advantageous low noise and low distortion characteristics [21], [22]. The signal-summing VGA can operate at a high frequency because the gain control stages are common base transistors. However, an unusable gain control range of approximately 20 dB remains around the maximum gain in this type of linear-in-decibel VGA. This unusable range damages the noise performance and reduces the operable gain range. In a bipolar linear-in-decibel VGA, a temperature stabilizing technique with an additional temperature-dependent current has been proposed [23]. However, this stabilizing technique itself is sensitive to the device parameters of temperature, and the reported results have shown that the variable gain characteristic is still sensitive to temperature. A MOSFET equivalent circuit, which can use the entire MOSFET’s operation region from the square law region to the exponential law region, has also been reported [24]. However, its performance in high-frequency response, noise, gain, and gain error was lower than that of the bipolar linear-in-decibel VGA when compared [23]. To achieve both a high bandwidth over several hundred megahertz and a wide dynamic range, the CMOS VGA cell based on a differential cascode structure with a resistor load [25], as shown in Fig. 1(d), has been reported. By controlling the operating point of the input transistors (transistors M1 and M2) of the cascode amplifier in the VGA cell from saturation to the linear region, it is possible to control the transconductances of the MOSFETs and the linear range of input transistors with a low distortion without sacrificing bandwidth. It is also possible to maintain a good high-frequency characteristic by applying a . This has achieved a high bandconstant tail current source width characteristic using a cascode structure. However, it is difficult for a structure with a resistor load to operate with a comparable performance at a low voltage because there is no headroom for a larger output voltage swing. As reported in [2], the CMOS VGA adopts a differential cascode structure and a differential structure using the pseudoexponential function in Fig. 1(a). For a low-voltage and high-frequency operation, the CMOS VGA uses an active load instead of a resistive load. The current sharing bias scheme with a constant current also improves the dynamic range and linearity. However, the bandwidth of 350 MHz is insufficient for CDMA UWB operations. From the viewpoint of a high-frequency and low-power opattenuation eration, the cascode VGA and the VGA with are superior among the four types of VGAs presented here. For a wide dynamic range, the cascode VGA and the modified pseudoexponential function VGA are preferred. We adopted a cascode VGA with active loads because it is expected to provide the high-frequency operation and wide gain control range. III. PROPOSED EXPONENTIAL FUNCTION GENERATOR Fig. 2 depicts the block diagram of the exponential function generator with compensation circuits. It consists of a voltage-tocurrent converter (VIC), a linear current multiplier, and an exponential function generator. An external control voltage is converted to a control current by the VIC. The current is then linearly multiplied and compensated by the linear

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Fig. 2. Block diagram of the proposed exponential current generator.

Fig. 3. Exponential function generator using parasitic NPN transistors (Q1 and Q2).

multiplier. Finally, the compensated linear current is transusing the exponential genformed to an exponential current erator. A. Exponential Function Generator Fig. 3 shows the exponential function generator and the cross section of a parasitic NPN transistor, which was used as Q1 and Q2. Today, most of the CMOS foundries provide the deep n-well technology. The deep n-well CMOS gives a chance to apply a different substrate bias to nMOS residing in another p-well, as well as excellent isolation against the substrate coupling noise between digital logic circuits and analog circuits [26]. We could obtain a high-performance NPN transistor whose current gain is almost 20, better than that of a conventional parasitic PNP transistor, 2 3. Thus, we used this parasitic NPN transistor. The operation of exponential function generator is as follows. generates a voltage drop of The input control current between the bases of parasitic NPN transistors Q1 and Q2. Due to the exponential nature of Q1 and Q2, and are and the collector currents , respectively, where indicates the reverse denotes the base saturation current of Q1 and Q2, and emitter voltage of Q1. As the current mirroring the constant

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Fig. 4. Linear current multiplier.

current is also constant, the current has an exponential . The MOSFET transistors characteristic as a function of M2 and M4 perform the role of the current source mirroring the obtained same current . The output exponential current through the current mirroring is then given by (2) between Accordingly, the output voltage difference and has an exponential function, which is given by

(3) B. Linear Current Multiplier The linear current multiplier is realized using a current divider consisting of a differential pair, as shown in Fig. 4. Equations (2) and (3) include the thermal voltage . This implies that the should be current varies with the temperature. The current proportional to absolute temperature (PTAT) current to compenindicates a PTAT current from sate for the gain variation. and the the PTAT bias circuit [27]. The control current are independent of the temperature and constant current supply voltage. These features will be explained in Section III-C regarding the VIC block. The gate-to-gate voltage of the differential pair (M3 and M4) is copied from the differential pair M1 tracks and M2. Thus, the current dividing ratio for , where the input current is a control signal and is the size ratio of the multiplier is of the mirroring CMOS devices. Thus, the output current given by (4) is fed to the exponential current genThis output current erator. It is independent of the supply voltage variation and is linearly dependent on the absolute temperature. C. VIC Fig. 5 shows the schematic of the VIC. The constant voltages and can be generated from the bandgap reference circuit and its buffers; the external control voltage

Fig. 5. VIC: VVC and VC denote the voltage-to-voltage converter and voltage copier, respectively.

can be varied from 0 to 1.8 V. In the figure, VVC indicates the voltage-to-voltage converter and VC denotes the voltage copier is converted to the control curand its current generation. by the VVC and VC. For symmetry, the compared rent is obtained. These currents are subconstant current tracted; then the supply voltage and temperature-independent is achieved. Additionally, ancurrent is generated from the same circuit, other constant current as shown in Fig. 4. Here, is fixed at 1.24 V, and is fixed at 0.83 V. Finally, the current of is obtained in place of . The normalized current portion is insensitive to temperature, process, and voltage variations. Its relation sets the maximum and minimum values of the control voltage. From the above currents and Fig. 2, we can rewrite the current in (4). If , then V V

(5) The procedure of the control signal conversion is summarized as follows. In Fig. 2, the external control voltage is a constant voltage independent of the temperature and supply voltage; thus, the VIC block needs an absolute voltage circuit to compare the absolute control voltage. The constant current and constant voltages and are obtained using the bandgap reference voltage circuit and an operational transconand ductance amplifier (OTA) circuit. The currents are constant currents at a fixed external voltage independent of supply-voltage and temperature, as in Fig. 2, while the current is linearly dependent on the temperature. The current is linearly dependent on the temperature and independent of the supply voltage. The temperature and supply-voltage compensain (5). tion is achieved by making a proper in (3) is expressed as Finally, the exponential voltage (6) This voltage is a function of the resistance and constant voltage, and the size ratio of mirroring the CMOS devices regardless

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a linearity error of less than 1.2 dB. Fig. 6(b) shows the simulated exponential current characteristics versus an external gain control voltage at 30 C, 25 C, and 80 C. As with the differing resistances, the linear control ranges of 20, 40, and 60 dB were obtained with a linearity error of less than 1.2 dB. Fig. 6 indicates that the proposed exponential function generator operates successfully with the temperature and supply-voltage compensation circuits and accurately operates in a linear-in-decibel fashion. The simulated results confirm that the current generator can obtain a wide control range ( 40 dB) with a small linearity error and the temperature and supply voltage compensation techniques for gain variation are valid. In considering the within the voltage scaling and simulated data, the voltage required voltage control range is expressed by (7) where , , and is the coefficient of the voltage scaling and normalized process. In summary, a new CMOS exponential function generator with compensation circuits has been developed. The exponential function is based on the relation between the emitter–base voltage and the collector current of a vertical NPN transistor. The inherent temperature dependence and supply-voltage variation are greatly reduced using compensation circuits. The exponential function generator is applied to the following CMOS VGA circuit. IV. DESIGN OF THE WIDEBAND CMOS VGA

Fig. 6. Simulated V–I characteristics: (a) with VDD variations and (b) with temperature variations.

As mentioned above, a VGA maintains a constant power level in the output of an AGC loop so that the subsequent circuitry has the better performance. The loop gain of the AGC remains constant across the entire operating range. This condition ensures identical loop transient responses regardless of the input signal level [13]. The VGA is a critical component of the AGC loop. Thus, it is very important to design the VGA considering specifications such as bandwidth, linearity, and so on. We will describe a wideband CMOS VGA that adopts the proposed exponential function generator. A. Architecture of the Proposed VGA Circuit

of the temperature and supply voltage. The exponential current also has the same function. D. Simulation Results for the Exponential Function Generator The feasibility of the proposed method was verified through simulations using a 0.18- m CMOS process. For the CMOS triple-well process, we adopted vertical NPN bipolar transistors, as in Fig. 3, and a bandgap reference circuit. Fig. 6(a) shows the simulated exponential current characteristics of the proposed method versus an external gain control voltage at different supply voltages (VDD) of 1.7, 1.8, and in (6) was used, a greater control 1.9 V. When the larger k , the exponential currange was obtained. When rent changed through two decades. This indicated that the conand trol range was more than 40 dB. In other cases ( k ), linear control ranges of 20 and 60 dB were obtained with

The architecture of the proposed CMOS VGA for wideband systems is shown in Fig. 7. The VGA circuit features a control voltage generator with the proposed exponential function generation and a main VGA circuit consisting of a dc offset canceller, three-stage VGA cells, and a fixed gain amplifier [28]. The difis amplified by the VGA cells and then amferential signal plified again by the fixed gain amplifiers to meet the targeted signal level at the output. The control voltage generator conto the required internal verts the external control voltage control signal . In order to remove the dc offset voltage, a dc offset canceller is introduced between the output of the VGA and the output of the first VGA cell. B. VGA Circuit Fig. 8 shows the schematic of the proposed VGA circuit. For operation at high frequencies, a differential cascode topology is

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Fig. 9. Small-signal equivalent circuit to calculate the output impedance.

the triode and saturation regions are given by (8) and (9), respectively [28], as follows: (8) (9) Fig. 7. Architecture of the proposed VGA circuit.

Fig. 8. Schematic of the proposed VGA circuit.

used to achieve a high gain and reduce the Miller effect. Conventionally, a differential structure has several advantages: it suppresses even harmonics, rejects common mode noises, and doubles the signal swing for a given supply voltage. Additionally, an active inductive load is exploited for wideband and low-voltage applications and can greatly improve the bandwidth [2], [28]. The VGA in Fig. 8 is constructed from a differential cascode structure (M1 and M6, M2 and M7) with a constant tail current . The input transistors (M1 and M6) are designed to operate in a triode region or in a saturation region based on the internal . When the input signal is small and is control voltage in a high state, the two input transistors remain in the saturation region and a high gain is obtained. The distortion is relatively low due to a small input signal. In contrast, when the input signal is in a low state, the input transistors remain in is large and the triode region and a low gain with low distortion is obtained. A variable gain can be achieved by controlling the gate voltage . In other words, by changing the of transistors M2 and M7, , the transconductance ( or ) of the cascode voltage or ) in topology can be varied. The transconductance (

W/L, is the drain–source voltage of M1, where is the drain current of M1. and , ), A load network consists of active loads (M3, M8, bleeding current sources (M4, M9), and dc level sensing transistors (M5, M10) [28]. The dc level sensing transistors monitor , ); then a common the dc level of the output signals ( ) compares this mode feedback (CMFB) network ( dc level with a reference voltage and sets the current by automatically adjusting the bias currents dc level to of the bleeding current sources (M4, M9). With this bleeding technique, the current through the active load transistors (M3, M8) can be reduced. The output impedance is determined dominantly by the active loads because the impedances of the bleeding current sources and dc level sensing transistors are relatively large compared to those of the active loads. Fig. 9 shows a small-signal equivalent circuit that estimates the output impedance in Fig. 8. It is assumed that M3 and M8 are identical. From the small-signal equivalent circuit, the output impedance at the operating frequency is given by

(10) is the transconductance, is the output resistance, where is the gate–drain capacitance, and is the gate–source capacitance of M3 or M8. is expressed by in the low operating freFrom (10), quency region. In the high-frequency region near the 3-dB frequency of the gain, the output impedance can be approximated as (10). Gain boosting frequency is the zero point . It shows that this active load effectively acts like an inductive load in the high-frequency region, of the active as shown in Fig. 10. The effective inductance load equivalently forms a parallel resonance circuit associated at the output node. Thus, by adwith parasitic capacitance justing the size of capacitor , it is possible to extend 3-dB frequency to a somewhat higher region by boosting the gain at

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to the negative feedback loop. Therefore, the VGA gain can be expressed by (13) Equation (13) shows that the gain of the VGA cell has an exponential function for the external control voltage . D. DC Offset Canceller

Fig. 10. Frequency response of the VGA using a gain boosting technique with various C values (10 50 fF).



high frequency. By simulation, we decided the optimal capacitor value as 20 fF to increase frequency range. Compared to the inductive active load in [28], this active load is simple and do not need additional current path. The capacitance value of is also much smaller (20 versus 200 fF). of the circuit given Finally, we can obtain the overall gain by (11) The gain also has a peaking point around the zero frequency and the bandwidth of the circuit can be improved. C. Control Voltage Generator We described the method for the temperature-independent exponential function in Section II: we utilize this method to generate the internal control voltage. The control voltage generator to an internal converts an external control voltage signal . This is consistent with the theory control voltage signal illustrated in [25]. The exponential voltage from Section II is applied to the inand is then amputs of the scaled VGA cell that has gain plified by the VGA cell, as shown in Fig. 7. A constant voltage is generated by an external current source and then subtracted from the output of the VGA cell. Therefore, if the gain of , the output of the VGA cell can be expressed a VGA cell is as

(12) The voltage of the VGA cell is amplified by an operational amof the amplifier plifier (OP-AMP); then the output voltage is fed back into the control node of the VGA cell. In a steady of the amplifier approaches 0 due state, the input voltage

In the proposed VGA, the cascaded gain cells may have a dc voltage problem due to a device mismatch and oxide gradients. The amplified offset voltage at the output of the entire VGA is cancelled out by an offset canceller with a negative feedback to the gain stage. The dc offset canceller shown in Fig. 7 is inserted between the output of the VGA and the output of the first VGA cell. The dc offset canceller is composed of an OTA, a low-pass filter (LPF), and a voltage-to-current (V–I) converter, which is included regardless of whether it is needed to cope with the device mismatch due to process deviations or not. In general, a large capacitance is used to keep the cutoff frequency of the LPF low. In the design of the LPF, effort has been made to maintain the capacitor value as low as possible so that the capacitor area is small. The operation of the dc offset canceller can be explained as follows. The differential input signals, which contain the dc offset between the positive and negative components, is amplified by the OTA. The LPF extracts the dc levels of the output differential signals; the dc offset is then cancelled at the output of the first VGA cell via a negative feedback loop. E. Noise Analysis The input referred noise of the VGA is dominated by the noise of the first variable gain circuit shown in Fig. 7, and the noise of the control stage and common mode feedback circuit is typically negligible. At relatively low frequencies, cascode devices contribute negligible noise [29]. The noise components of the variable gain circuit in Fig. 8 are illustrated in Fig. 11(a). In Fig. 11(a), the thermal noise of the transistors is modeled by the current sources connected between the drain and source terminals with a spectral density of . To calculate , we first obtain the total output the thermal component of noise with the input shorted together. Since the noise sources in the circuit are uncorrelated, superposition of the noise power quantities is possible. Furthermore, the source terminals of transistors M1 and M6 cannot be considered virtual ground, making it difficult to use the half-circuit concept. Thus, the effect of noise sources in each branch of the differential amplifier must is obtained by be derived individually. The contribution of reducing the circuit in Fig. 11(b). As shown in Fig. 11(c), we can into two (correlated) current sources and calcudecompose late their effect at the output. Thus,

(14) Note that the two voltages are added directly because and are, therefore, correlated. If they both arise from , the total

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Fig. 11. Calculation of the input-referred noise of the proposed VGA.

input thermal noise in Fig. 11(a) is obtained taking into account the noise of M3, M4, M8, and M9 as follows: (15) Considering the effect of flick noise, the total input referred noise voltage per unit bandwidth of the variable gain circuit, as shown in Fig. 11(a), can be given as

Fig. 12. Microphotograph of the fabricated VGA (core chip size: 0.59 mm 0.33 mm, total size: 0.85 mm 0.49 mm).

2

2

(16) where and are process-dependent constants of the corV F [29]. responding MOS transistors in the order of 10 is maximized in As shown in (16), at the maximum gain, order to minimize the total input-referred noise voltage given reduces; thus, the total in (16). As the gain decreases, input-referred noise voltage as given in (16) increases. Consequently, the noise of the proposed VGA is a decreasing function of the gain. Initially, we designed our VGA circuit in consideration of parasitic effects such as imperfect ground (substrate losses) and parasitic capacitors along the main signal path. We confirmed that initial simulation results were very similar to post simulation results with parasitic capacitors and resistors automatically generated by the Cadence Assura RCX tool. V. SIMULATION AND MEASUREMENT RESULTS The proposed VGA integrated chip (IC) was fabricated using 0.18- m CMOS technology. Fig. 12 shows the chip photograph of the implemented VGA IC. The active area occupies 0.85 mm 0.49 mm including bondpads. The VGA is tested with a VDD supply of 1.8 V, and the dc current drawn with no applied input

Fig. 13. (a) Internal control voltage generation and (b) gain characters with internal control voltage from [28].

signal is 11.4 mA. The VGA IC was measured with a vector network analyzer (Agilent 8753ES), a signal source generator (Agilent E4433B), and a spectrum analyzer (HP 8564E). The input and output of the VGA were connected with a signal source and spectrum analyzer, respectively, through commercial balun (balanced to unbalance or vice versa) devices to convert the single-ended signal into the differential signals. Fig. 13(a) shows the relationship between the external control voltage and the generated internal control voltage through simulation using Cadence Spectre. This simulation shows the slope is linear from approximately 0.72 to 1.52 V according to

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Fig. 16. Accurate linear-in-decibel gain curve at 25 C, gain error, and gain deviation from the gain at 25 C; a test frequency of 100 MHz was applied.

Fig. 14. Frequency response of the VGA under different control voltages; at all gain levels, the 3-dB bandwidth is larger than 900 MHz.

0

Fig. 17. Input P1dB and NF characteristics at a test frequency of 100 MHz.

TABLE I PERFORMANCE SUMMARY OF THE PROPOSED VGA

Fig. 15. Measured and stimulated gain characteristics with temperature variations at a test frequency of 100 MHz.

the external control voltage . The internal control voltage is applied to each VGA gain cell. Fig. 13(b) shows the meacharacteristics at the frequency of 100 MHz as sured gain . The VGA has linear-in-decibel gain characa function of teristics when the input transistors (M1 and M2 in Fig. 8) are set V [28]. From Fig. 13, it in the triode region is possible to obtain a linear-in-decibel gain feature in the range of 0 1.8 V, which will be described in the following two paragraphs. Fig. 14 shows the frequency response of the proposed VGA at different control voltages. The measured bandwidth of the VGA is up to 900 MHz with a gain control range of 94.1 dB 38.8 55.3 dB . The large bandwidth was achieved using gain boosting around the 3-dB frequency (around 600 700 MHz). At the minimum gain of 38.8 dB, the 3-dB bandwidth reached its minimum value of 750 MHz because the input transistors deviated from the operating point in the triode

Simulation data

region and the bias current was reduced in relation to the change of the operating point. Moreover, we can see that the measured gain shows an average gain with 1.5-dB gain flatness from 10 to 900 MHz at various control voltages. These results ensure that the constant current bias scheme can successfully prevent bandwidth variation according to the control bias, except in the minimum gain state. Fig. 15 shows the measured and simulated forward gain characteristics as a function of the control voltages at 30 C,

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TABLE II PERFORMANCE COMPARISON OF THE PROPOSED AND PREVIOUSLY REPORTED VGAs

Simulation results Signal-summing VGA with control signal converter (CSC) Master–slave control circuits with MOSFETs in a subthreshold region

25 C, and 80 C when the test signal frequency is 100 MHz. The measured gain characteristics are in good agreement with the simulated ones. The difference between the gain curves is a result of the errors of current mirroring and reference voltages. The simulation results show that the gain deviations from the gain characteristics at room temperature are within 3 dB over a 96-dB gain range at various control voltages. In the case of measurement, gain errors of 3 dB occurred within the 94.1-dB gain control range. These results confirm the validity of the proposed ideas for obtaining linear-in-decibel variable gain characteristics with temperature compensation. However, the exact linear range is limited to 79.4 dB, as shown in Fig. 16, because the nonlinear range of the internal control voltage, V in Fig. 13, is utilized in order to obtain a higher gain. Additionally, the imperfect effects in the process of the control signal conversion, such as inaccurate current mirroring and device modeling, particularly, the lateral NPN transistors, degraded the gain characteristics. In the gain range of 79.4 dB, the linearity errors from the normalized linear curve and the gain deviations from the gain at 25 C were 1.0 and 1.8 dB, respectively. The measured input P1dB and measured/simulated noise figures (NFs) of the proposed VGA are presented as a function of the control voltage at a test frequency of 100 MHz in Fig. 17. The input P1dB varies from 10.8 to 59.1 dBm. It is maximized at the minimum gain state and, as the gain increases, it decreases so that decreasing the allowable input signal swing results in the reduction of P1dB, as shown in Fig. 17. The minimum NF is 6.8 dB at a maximum gain of 55.3 dB and the NF increases as the gain decreases. Through simulations, the total spot output noise is V Hz at 100 MHz and the main noise sources are the input transistors (M1 and M6) and load transistors (M3, M4, M8, and M9) described in (16). The measured NF data were obtained in

the gain range of 5 40 dB due to the limitation of the NF equipment. The measurement results are summarized in Table I and a comparison with previous studies is given in Table II. The proposed VGA achieves the best performance in terms of high-frequency performance with a high gain, wide decibel-linear gain controllability with temperature compensation, and chip area. VI. CONCLUSION A wideband CMOS VGA with a temperature compensation circuit has been presented. The proposed VGA utilizes an exponential function generator with PVT compensation circuits. In addition, wideband operation with a high gain was achieved using a cascode configuration and a gain boosting technique. The current bleeding technique in the load network was also used to improve the linearity. The VGA has a 94.1 dB ( 38.8 55.3 dB) gain control range and, particularly, 79.4-dB linear-indecibel gain characteristics. Furthermore, the VGA operates up to a 3-dB frequency of 900 MHz. It consumes a total current of 11.4 mA under a single power of 1.8 V and occupies an area of 0.85 0.49 mm . Temperature-compensated performance was confirmed from 25 C to 80 C. ACKNOWLEDGMENT The authors would like to thank to Dr. C.-H. Kim, Teltron Inc., Daejeon, Korea, for his useful discussions. REFERENCES [1] H. D. Lee, C.-H. Kim, and S. Hong, “An SiGe BiCMOS transmitter module for IMT2000 applications,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 8, pp. 371–373, Aug. 2004. [2] J. K. Kwon, K. D. Kim, W. C. Song, and G. H. Cho, “Wideband high dynamic range CMOS variable gain amplifier for low voltage and low power wireless applications,” Electron. Lett., vol. 39, no. 10, pp. 759–760, Mar. 2003.

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[3] T. Yamaji, N. Kanou, and T. Itakura, “A temperature-stable CMOS variable-gain amplifier with 80-dB linearly controlled gain range,” IEEE J. Solid-State Circuits, vol. 37, no. 5, pp. 553–558, May 2002. [4] R. Harjani, “A low-power CMOS VGA for 50-Mb/s disk drive read channels,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 42, no. 6, pp. 370–376, Jun. 1995. [5] W. M. Christopher, “A variable gain CMOS amplifier with exponential gain control,” in VLSI Circuits Tech. Dig. Symp., Jun. 2000, pp. 146–149. [6] P. Huang, L. Y. Chiou, and C. K. Wang, “A 3.3-V CMOS wideband exponential control variable-gain-amplifier,” in Proc. IEEE Int. Circuits Syst. Symp., May 1998, pp. I-285–I-288. [7] M. M. Green and S. Joshi, “A 1.5 V CMOS VGA based on pseudodifferential structures,” in Proc. IEEE Int. Circuits Syst. Symp., May 2000, pp. IV-461–IV-464. [8] S. Otaka, H. Tanimoto, S. Watanabe, and T. Maeda, “A 1.9-GHz Si-bipolar variable attenuator for PHS transmitter,” IEEE J. Solid-State Circuits, vol. 32, no. 9, pp. 1424–1429, Sep. 1997. [9] P. Orsatti, F. Piazza, and Q. Huang, “A 71-MHz CMOS IF-baseband strip for GSM,” IEEE J. Solid-State Circuits, vol. 35, no. 1, pp. 104–108, Jan. 2000. [10] H. O. Elwan and M. Ismail, “Digitally programmable decibel-linear CMOS VGA for low-power mixed-signal applications,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 47, no. 5, pp. 388–398, May 2000. [11] J. Hauptmann, F. Dielacher, R. Teiner, C. C. Enz, and F. Krummenacher, “A low-noise amplifier with automatic gain control and anticlipping control in CMOS technology,” IEEE J. Solid-State Circuits, vol. 27, no. 7, pp. 974–981, Jul. 1992. [12] F. Carrara and G. Palmisano, “High-dynamic-range VGA with temperature compensation and linear-in-dB gain control,” IEEE J. Solid-State Circuits, vol. 40, no. 10, pp. 2019–2024, Oct. 2005. [13] J. M. Khoury, “On the design of constant settling time AGC circuit,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 45, no. 3, pp. 283–294, Mar. 1998. [14] Y. Zheng, J. Yan, and Y. P. Xu, “A CMOS dB-linear VGA with predistortion compensation for wireless communication applications,” in Proc. IEEE Int. Circuits Syst. Symp., May 2004, pp. I-23–I-26. [15] B. Gilbert, “The multi-tanh principle: A tutorial overview,” IEEE J. Solid-State Circuits, vol. 33, no. 1, pp. 2–17, Jan. 1998. [16] R. Harjani, J. Harvey, and R. Sainati, “Analog/RF physical layer issues for UWB systems,” in Proc. Int. VLSI Design Conf., Jan. 2004, pp. 941–948. [17] R. Yao, Z. Chen, and Z. Guo, “An efficient multipath channel model for UWB home networking,” in Proc. IEEE Radio Wireless Conf., Sep. 2004, pp. 511–516. [18] Q.-H. Duong, L. Quan, and S.-G. Lee, “An all CMOS 84-dB linear low-power variable gain amplifier,” in VLSI Circuits Symp. Tech. Dig., Jun. 2005, pp. 114–117. [19] Y.-S. Youn, C.-S. Kim, N.-S. Kim, and H.-K. Yu, “A 1 GHz-band low distortion up-converter with a linear in dB control VGA for digital TV tuner,” in IEEE Radio Freq. Integrated Circuits Symp. Dig., May 2001, pp. 257–260. [20] Y.-S. Youn, J.-H. Choi, M.-H. Cho, S.-H. Han, and M.-Y. Park, “A CMOS IF transceiver with 90 dB linear control VGA for IMT-2000 application,” in VLSI Circuits Tech. Symp. Dig., Jun. 2003, pp. 131–134. [21] W. M. C. Sansen and R. G. Meyer, “Distortion in bipolar transistor variable-gain amplifiers,” IEEE J. Solid-State Circuits, vol. SC-8, no. 8, pp. 275–282, Aug. 1973. [22] W. M. C. Sansen and R. G. Meyer, “An integrated wideband variablegain amplifier with maximum dynamic range,” IEEE J. Solid-State Circuits, vol. SC-9, no. 8, pp. 159–166, Aug. 1974. [23] S. Otaka, G. Takemura, and H. Tanimoto, “A low-power low-noise accurate linear-in-dB variable-gain amplifier with 500-MHz bandwidth,” IEEE J. Solid-State Circuits, vol. 35, no. 12, pp. 1942–1948, Dec. 2000. [24] O. Watanabe, S. Otaka, M. Ashida, and T. Itakura, “A 380-MHz CMOS linear-in-dB signal-summing variable gain amplifier with gain compensation techniques for CDMA systems,” in VLSI Circuits Tech. Symp. Dig., Jun. 2002, pp. 136–139.

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[25] W. C. Song, C. J. Oh, G. H. Cho, and H. B. Jung, “High frequency/ high dynamic range CMOS VGA,” Electron. Lett., vol. 36, no. 13, pp. 1096–1098, Jun. 2000. [26] I. Nam and K. Lee, “High-performance RF mixer and operational amplifier BiCMOS circuits using parasitic vertical bipolar transistor in CMOS technology,” IEEE J. Solid-State Circuits, vol. 40, no. 2, pp. 392–402, Feb. 2005. [27] W. Claes, W. Sansen, and R. Puers, “A 40- A/channel compensated 18-channel strain gauge measurement system for stress monitoring in dental implants,” IEEE J. Solid-State Circuits, vol. 37, no. 3, pp. 293–301, Mar. 2002. [28] H. D. Lee, K. A. Lee, and S. Hong, “Wideband VGAs using a CMOS transconductor in triode region,” in Proc. 36th Eur. Microw. Conf., Sep. 2006, pp. 1449–1452. [29] B. Razavi, Design of Analog CMOS Integrated Circuits. New York: McGraw-Hill, 2001, pp. 233–239, 377–381.

Hui Dong Lee (S’02) received the B.S. and M.S. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree at KAIST. His research interests include analog, RF, and microwave integrated circuit design for wireless communications in CMOS and BiCMOS technologies. His focus is on the analysis and design of various VGAs for multistandard applications.

Kyung Ai Lee received the B.S. degree in electrical engineering from Kyungpook National University, Dae-gu, Korea, in 2002, the M.S degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2004, and is currently working toward the Ph.D. degree at KAIST. Her research interests include analog, RF, and microwave integrated circuit design for wireless communications in CMOS and HBT and HEMT technologies. Her focus is on the analysis and design of a power amplifier for military and commercial applications.

Songcheol Hong (S’87–M’88) received the B.S. and M.S. degrees in electronics from Seoul National University, Seoul, Korea, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1989. In May 1989, he joined the faculty of the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea. In 1997, he held short visiting professorships with Stanford University, Palo Alto, CA, and Samsung Microwave Semiconductor, Gyeonggi-do, Korea. His research interests are microwave integrated circuits and systems including power amplifiers for mobile communications, miniaturized radars, millimeter-wave frequency synthesizers, and novel semiconductor devices.

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High Precision Radar Distance Measurements in Overmoded Circular Waveguides Nils Pohl, Michael Gerding, Bianca Will, Thomas Musch, Josef Hausner, Member, IEEE, and Burkhard Schiek, Member, IEEE

Abstract—Distance measurements in overmoded waveguides are an important application for industrial radar systems. The accuracy of the measurements is deteriorated by the appearance of higher order modes in the metal tube, although the frequency-modulated continuous-wave method is used with a large bandwidth. This paper describes the problems caused by dispersion and multimode propagation and presents a solution in the form of mode-matched antennas for feeding the overmoded waveguide. It is shown that different modes, e.g., the 11 and 01 modes, are equally well suited for precision distance measurements, as is demonstrated both by simulations and measurements. Index Terms—Frequency modulated continuous wave (FMCW), higher order mode suppression, low-loss mode, overmoded circular waveguide, oversized waveguide, radar distance measurement, waveguide transition.

Fig. 1. Test setup of the FMCW free-space radar system with a metallic reflector in free space for the measurement of the distance to the reflector.

ramps and a high dynamic range of the measurement system, the distance measurement error may be below 1 mm for a free-space system.

I. INTRODUCTION

T

HE state-of-the-art offers more and more accurate measurements in different areas of engineering. The measurement precision performed by industrial radar systems has steadily increased. Many industrial systems include metallic tubes, which are permanently built in. These metallic tubes provide the only access to the system and the possibility of radar distance measurements. The metallic tubes have a fixed diameter. In the frequency range of 24–28 GHz, which is a typical frequency range for radar distance measurements, the tubes behave as overmoded waveguides. The properties of the reflecting object can differ considerably. On the one hand, the objects may be of a metallic or solid dielectric structure, on the other hand, the reflecting object may be a dielectric liquid. Furthermore, the metallic tube can show a poor conductivity, e.g., caused by depositions inside the metal tube or rust and oxidation of the metal tube. In this case, the use mode may be advantageous. of the The precision and robustness of a free-space radar measurement system is not easily achieved by distance measurements in an overmoded circular hollow metallic waveguide. Due to large bandwidths and the use of the frequency-modulated continuous-wave (FMCW) principle with highly linear frequency

Manuscript received October 16, 2006; revised March 22, 2007. N. Pohl and J. Hausner are with the Institut für Integrierte Systeme, RuhrUniversität Bochum, 44801 Bochum, Germany (e-mail: [email protected]). M. Gerding, B. Will, T. Musch, and B. Schiek are with the Arbeitsgruppe Hochfrequenzmesstechnik, Ruhr-Universität Bochum, 44801 Bochum, Germany. 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.2007.896784

II. FMCW FREE-SPACE RADAR SYSTEM For the purpose of a comparison to guided waves in a circular waveguide, measurements in free space have been performed with a system as shown in Fig. 1. The measurement bandwidth is 4 GHz, the frequency range is 24–28 GHz. The aerial is a circular horn antenna with a 3-dB beamwidth of approximately 10 . The reflecting object is a metallic plate. The reflector can be moved by means of a stepper motor into different positions. The stepper motor also delivers accurate position data that can be used as a reference. Here, the FMCW radar was realized by a network analyzer. The network analyzer measures the complex transfer function of the measurement section. In order to simulate an FMCW system, only the real part of the measured transfer function is evaluated. A Fourier transformation of this transfer function yields a quasi-impulse response of the free-space section. The time delay of the reflected pulse, which is directly proportional to the distance, is determined by means of a pulse center algorithm. The measured distance is normalized to a reference measurement in order to eliminate offset and slope errors. Fig. 2 shows distance errors of a free-space measurement. The maximum distance error of this single target environment is below 1 mm. These measurement results show that an FMCW radar with the cited bandwidth can deliver very accurate results. The impulse response (Fig. 3) of the system confirms that the disturbances in the system are quite small. III. DISTANCE MEASUREMENTS IN OVERMODED WAVEGUIDES Compared to [1], an improved measurement setup is used. The considered setup also uses the above discussed realization of an FMCW radar on the basis of a network analyzer. However, in this setup (Fig. 4), the electromagnetic (EM) waves are

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Fig. 2. Measured distance errors of the test setup free-space FMCW radar system versus the distance to the reflector.

Fig. 5. Distance errors of the FMCW radar system measured in an circular waveguide versus distance to the reflector.

Fig. 3. Typical plot of a free-space FMCW impulse response of a metallic reflector.

Fig. 6. Plot of the impulse response in an overmoded circular waveguide measured with an FMCW radar system.

Fig. 4. System setup for the distance measurements in overmoded waveguides with an FMCW radar system.

guided by a metallic tube to the metallic reflector. The metallic tube guides the EM wave like a circular waveguide with a diameter of the tube of 80 mm. Therefore, the circular waveguide no longer has the properties of a monomode waveguide. The antenna is the same circular horn with a diameter of 72 mm. The aim of the measurement is to determine the distance to the metallic reflector acting as a sliding short. The position of the

stepper motor is used for a reference measurement. The measurement error as a function of the distance (Fig. 5) shows that this circular waveguide structure leads to significantly higher maximum measurement errors of 4 mm, despite the fact that the used FMCW system has a high fundamental precision. The impulse response (Fig. 6) of this arrangement shows that the determination of the precise delay time of the pulse can only be accomplished under the influence of some major disturbances. IV. CAUSES OF MEASUREMENT ERRORS AND APPROACHES OF SOLUTIONS The differences in the used reflectors can be excluded as a source of errors because the metallic reflector acts similar to a short in both cases. One cause for the significantly higher measurement errors may be the dispersion of the waveguide because the group and

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Fig. 8. Cross section of a monomoded to overmoded waveguide transition realized by a linear horn antenna.

Fig. 7. Simulated distance errors of two simple mathematical antenna models; an ideal monomode antenna denoted by the dashed–dotted line and a simple multimode antenna model denoted by the solid line.

phase velocity in a waveguide differ from free space and furthermore depend on frequency. As is well known, the wavelength in the waveguide is depending on the frequency and is infinity at the cutoff frequency of the corresponding mode in the waveguide. By the use of suitable algorithms it is, however, possible to include the known dispersion into the evaluation model so that the measurement results will not be influenced by dispersion. Further causes for measurement errors can be given by the transition of the antenna to the metal tube. Between the antenna and tube there is a gap of 4 mm, which may lead to multiple reflections of the transmitted pulses. Furthermore, in a metallic waveguide, a large number of modes, which have different cutoff frequencies, but are able to propagate, may be excited. Since the cutoff frequency determines the speed of propagation, the different impulse responses superimpose with slightly different delay times and phases and each delay time has a different frequency dependence. Therefore, the resulting impulse response may look rather deteriorated. Fig. 7 shows the simulated error contribution of two different antenna models. The first one is represented by the dashed line, which is the result of a perfect monomode antenna for the mode of a circular waveguide. The propagation of the wave is only influenced by the dispersion of the cutoff frequency, which is compensated by the algorithm. The error is noticeably below 1 mm. This proves that the selected signal processing algorithm is robust against dispersion. The solid curve shows the deviation of the distance with an antenna, which excites two unand , with relative amplitudes of 8 wanted modes, i.e., and 10 dB, respectively, leading to much higher distance deviations. Further system simulations have shown that the only a dominant error contribution is caused by the excitation of higher order waveguide modes with relevant amplitudes. The excitation of higher order modes leads to a mode dispersion of the impulse response.

Fig. 9. Simulated S -parameters of the linear horn antenna as a waveguide transition.

V. MODE-MATCHED ANTENNA IMPROVEMENT In order to minimize the measurement error, the major goal is to employ a waveguide transition with a better suppression of the unwanted higher order modes. Fig. 8 shows the cross section of the linear horn antenna, which was used for the previous measurements. Fig. 9 depicts the -parameters of the EM simulation of this antenna. The dashed curves show the conversion into higher order modes. The simulations were performed with the transient-solver of the 3-D EM software Microwave Studio of CST. The simulation shows that the excitation of higher order modes is considerable with a maximum of approximately 9 dB. One way to improve the suppression of higher modes is to use a longer horn antenna, but in many applications, the length of the antenna is limited. Another possibility is to use a parabolic horn antenna for the waveguide transition. Fig. 10 shows the geometry of the simulated parabolic horn antenna. As can be seen in Fig. 11, the suppression of the higher order modes is much better, although the antenna is shorter. In general, the measurement errors can be reduced by minimizing the mode conversion at the aperture of the antenna. Such

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Fig. 12. Cross section of a monomoded to overmoded waveguide transition realized by an improved mode-matched lens antenna. Fig. 10. Cross section of a monomoded to overmoded waveguide transition realized by a parabolic horn antenna.

Fig. 13. Simulated S -parameters of the improved mode-matched lens antenna as a waveguide transition. Fig. 11. Simulated S -parameters of the parabolic horn antenna as a waveguide transition.

a mode matched antenna has a plane phase front as a main criteria. The concept of a plane phase front is based on geometrical ray optics. All rays must have the same delay time from the feed of the metallic tube in order to excite a plane phase front. This can be achieved by using a dielectric lens. Here the lens consists of Teflon due to its good mechanical and electrical properties. In the first iteration, the dielectric lens was constructed according to geometrical optics and it was then numerically optimized by a 3-D EM simulation tool. In addition, the whole antenna is filled with Teflon. Thus, the combination of a linear horn antenna and a dielectric lens leads to an improved mode-matched lens antenna, as shown in Fig. 12. EM simulations of the mode-matched antenna to a waveguide transition show a suppression of higher order modes of approximately 20 dB in the given frequency range, as shown in Fig. 13, which yields an improvement of 5–10 dB in comparison to the parabolic horn antenna. VI. SIMULATIONS AND MEASUREMENTS WITH DIFFERENT ANTENNAS With EM simulations of the antennas as an -port device with one port for every mode, it is possible to simulate the distance error of the full measurement system. Therefore, the waveguide is analytically described as an ideal loss free waveguide with the

Fig. 14. Block diagram of the simulation model for distance measurements based on the simulated S -parameters of the antenna.

related cutoff frequency for every mode. The basic idea of this simulation is outlined in Fig. 14. The simulation of the linear horn antenna leads to the distance errors plotted in Fig. 15. These simulated distance errors are in the same range as the measurement error in Fig. 5 and the errors of the simple model in Fig. 7. Fig. 16 shows that the improved mode suppression of the parabolic horn antenna leads to lower measurement errors. The error is approximately reduced by a factor of 3 relative to the linear horn antenna. Fig. 17 shows the resulting simulated distance errors achieved with the mode-matched lens antenna. In this case, the system simulation leads to a measurement error of less than 0.8 mm. Thus, a further improvement of the accuracy by approximately a factor of 2 can be observed and, thus, it may be concluded that

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Fig. 15. Simulated distance errors of an electromagnetically simulated linear horn antenna.

Fig. 16. Simulated distance errors of an electromagnetically simulated parabolic horn antenna.

Fig. 17. Simulated distance errors of an electromagnetically simulated improved mode-matched lens antenna.

Fig. 18. Measured distance errors of a circular waveguide versus distance to the reflector with a parabolic horn antenna.

the results have the same accuracy as the measurements in free space. Fig. 18 shows the measured distance errors of a parabolic horn antenna. Compared to Fig. 5, an improvement of the accuracy by a factor of 3 can be observed. Fig. 19. General field distribution of the

VII. SIMULATIONS WITH THE

H

mode in a circular waveguide.

MODE

The metallic tube allows the propagation of a number of modes, namely, 276, in the given frequency range of mode. The field 24–28 GHz. One of these modes is the distribution of this mode, shown in Fig. 19, is advantageous for the setup because there are no currents in the direction of propagation. Thus, gaps in the metallic tube and especially a gap between the antenna and tube only cause small disturbances. mode has a low attenuation, therefore, Furthermore, the propagation losses are low even for a poor conductivity of the metallic wall.

The mode is not the fundamental mode of a circular waveguide and an arbitrary excitation may generate a number of modes. Thus, a mode converter is needed, which converts of a monomode circular waveguide the fundamental mode mode of a circular with a diameter of 8.1 mm into the waveguide with a larger diameter (in this case, 18 mm). For the realization of the mode converter, a combination of different waveguides is used. The arrangement of these waveguides was chosen in such a way that all modes with the exception mode, which are able to propagate in the circular of the

POHL et al.: HIGH PRECISION RADAR DISTANCE MEASUREMENTS IN OVERMODED CIRCULAR WAVEGUIDES

Fig. 21. Simulated distance errors using the guide.

Fig. 20. Schematic diagram and realized step structure of the mode converter to in a circular waveguide.

H

H

waveguide, are not excited due to the symmetry properties mode of of the mode converter. In a first step, the the monomode circular waveguide is converted into the mode of a monomode rectangular waveguide with a cross 10 mm. The rectangular waveguide is section of 5 mm divided into two monomode waveguides in a next step. Each of these rectangular waveguides is then rotated by 90 , which provides a phase shift of 180 between both waveguides. In a further step, each waveguide is divided into two rectangular waveguides once again. Each of these four waveguides is then rotated by 45 . In a last step, these four monomode rectangular waveguides are combined in one circular waveguide with a mode diameter of 18 mm. In this waveguide, only the exists. All other propagatable modes are not excited due to the symmetry properties of the four rectangular waveguides, which feed this circular waveguide. In Fig. 20, a schematic diagram of the different steps is shown. Thus, the mode converter can be described by six elements according to the above-described steps, which are also shown in Fig. 20.

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H

mode in an overmoded wave-

The mode converter shall have a match versus frequency better than 20 dB and additionally the construction of the mode converter should be simple, compact, and broadband. The different parts of the mode converter can basically be realized in two different ways. On the one hand, the different parts can be realized by a very smooth tapered structure. For this type of realization, the length of the taper is approximately inversely proportional to the match. Thus, a good match involves a long structure. On the other hand, the different parts can be realized by a stepped structure, in which the different elements have a number of steps with a width in the vicinity of a quarter-wavelength. The exact width of the single steps, which depends amongst others on the guided wavelength, was found by numerical optimization calculated with CST’s EM simulation tool Microwave Studio. The different step widths were limited to three different widths, i.e., 2.6, 4.4, and 5.9 mm, to simplify the construction. The reflections of the single steps cancel each other. Thus, a good broadband match can be realized with a few steps and, consequently, a short length. Indeed, the mode converter has a match better than 20 dB versus the frequency range of 24–28 GHz. This stepped structure is relatively simple to fabricate because the cross section of each step is constant and can be cut out of a metal plate with the appropriate width. Afterwards, the different metal plates are fused together. The final structure, shown in Fig. 20, has a square cross section with an edge length of 3 cm and a height of 12.2 cm. By comparison, a mode converter realized by a smooth tapered structure with a comparable cross section and match has a length of approximately 41 cm and is more difficult to fabricate. mode converter combined A simulation of the with the improved mode-matched lens antenna also shows a mode, generated by the high measurement accuracy. The mode converter, which feeds the antenna, is more robust against disturbances of the waveguide. The results of the simulation, shown in Fig. 21, obtain a distance measurement error of approximately 0.6 mm. This result is comparable with the results simulated with the lens antenna fed with the mode.

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This result was expected due to the fact that there is no prinand modes concerning the cipal difference between the distance measurements because the simulation were performed with loss free waveguide models. Thus, for practical systems, setup it is expected that measured distance errors with the setup, will show an improved accuracy, as compared to the mode is a low-loss mode. because the VIII. CONCLUSION Distance measurements in circular overmoded waveguides by means of a standard free-space radar setup lead to suboptimal results. Rather significant modifications of the antenna used as a waveguide transition are necessary in order to adapt the antenna to this special arrangement. Minimizing the multimode propagation in the overmoded circular waveguide for a low mode dispersion is an important goal in order to obtain a high precision measurement system. A parabolic horn instead of a linear horn leads to a higher accuracy. A still improved measurement precision is obtained with a mode-matched lens antenna. For lossy mode may have advantages. waveguides, the If instead of the pulse center algorithm the phase-slope algorithm [3] is used, the measurement results, as well as the simulated results improve approximately by the factor of 0.6. REFERENCES [1] T. Musch, N. Pohl, M. Gerding, B. Will, J. Hausner, and B. Schiek, “Radar distance measurements in over-sized circular waveguides,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., 2006, pp. 1036–1039. [2] M. I. Skolnik, Introduction to Radar Systems, 3rd ed. New York: McGraw-Hill, 2000. [3] T. Musch, “A high precision 24 GHz FMCW-RADAR using a phaseslope signal processing algorithm,” in Proc. 32nd Eur. Microw. Conf., Milan, Italy, 2002, pp. 945–948. [4] M. Gerding, T. Musch, and B. Schiek, “A novel approach for a high precision multi target level measurement system based on time-domain-reflectometry,” in Proc. 35th Eur. Microw. Conf., Paris, France, 2005, pp. 737–740. [5] T. Musch, M. Küppers, and B. Schiek, “A multiple target high precision laser range measurement system based on the FMCW concept,” in Proc. 33rd Eur. Microw. Conf., Munich, Germany, 2003, pp. 991–994. [6] T. Rosenberg and M. Schneider, “High-performance transitions for overmoded operation of elliptical waveguides,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 10, pp. 1749–1755, Oct. 2000. [7] E. L. Holzman, “A simple circular-to-rectangular waveguide transition,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 1, pp. 25–26, Jan. 2005. [8] F. Sporleder and H.-G. Unger, Waveguide Tapers Transitions and Couplers. London, U.K.: Peregrinus, 1979. [9] G. G. Gentili, “Properties of TE–TM mode-matching techniques,” IEEE Trans. Antennas Propag., vol. 39, no. 9, pp. 1669–1673, Sep. 1991. [10] M. Gerdine and H. Lenzing, “Reduction of delay distortion in a horn-reflector antenna system employing overmoded-waveguide feeder,” IEEE Trans. Commun., vol. 18, no. 2, pp. 21–26, Feb. 1970. [11] B. Plaum, D. Wagner, W. Kasparek, and M. Thumm, “Optimization of waveguide bends and bent mode converters using a genetic algorithm,” in 25th Int. Infrared Millim. Waves Conf. Dig., Beijing, China, 2000, pp. 219–220. [12] J. P. Quine, “E - and H -plane bends for high-power oversized rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 1, pp. 54–63, Jan. 1965. [13] S. L. Choon, L. Shung-Wu, and C. Shun-Lien, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 7, pp. 773–785, Jul. 1986.

[14] J. L. Doane, “Low-loss twists in oversized rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 6, pp. 1033–1042, Jun. 1988.

Nils Pohl was born in Aachen, Germany, in 1980. He received the Dipl.-Ing degree in electrical engineering from Ruhr-Universität Bochum, Bochum, Germany, in 2005. Since 2006, he has been a Research Assistant with the Institut für Integrierte Systeme, Ruhr-Universität Bochum. His current fields of research are concerned with frequency synthesis and radar systems in integrated circuits.

Michael Gerding was born in Herne, Germany, in 1975. He received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from Ruhr-Universität Bochum, Bochum, Germany, in 2000 and 2005, respectively. Since 2000, he has been a Research Assistant with the Arbeitsgruppe Hochfrequenzmesstechnik, RuhrUniversität Bochum. His current fields of research are concerned with frequency synthesis, time-domain reflectometry (TDR), and industrial applications of microwaves.

Bianca Will was born in Marburg, Germany, in 1980. She received the Dipl.-Ing degree in electrical engineering from Ruhr-Universität Bochum, Bochum, Germany, in 2006. Since 2006, she has been a Research Assistant with the Arbeitsgruppe Hochfrequenzmesstechnik, Ruhr-Universität Bochum. Her current fields of research are concerned with multiport measurements, calibration methods, waveguide transitions, and the measurement of dielectric profiles.

Thomas Musch was born in Mülheim, Germany, in 1968. He received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from Ruhr-Universität Bochum, Bochum, Germany, in 1994 and 1999, respectively. Since 1994, he has been a Research Assistant with the Arbeitsgruppe Hochfrequenzmesstechnik, RuhrUniversität Bochum. His current fields of research are concerned with frequency synthesis, fractional divider techniques, radar systems for microwave range finding, and industrial applications of microwaves.

Josef Hausner (M’88) was born in 1961. He received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering (in the field of microwave technology) from the Technical University Munich, Munich, Germany, in 1986 and 1991, respectively. He began his career in industry with Siemens AG, where he was involved with high-speed access systems on digital subscriber lines (HDSLs) and with Infineon Technologies, where his interest focused on system-on-chip designs for wireless communications. In 2004, he became a Full Professor with the Institut für Integrierte Systeme, Ruhr-Universität Bochum, Bochum, Germany. His current research interests are integrated systems and circuits for multistandard wireless communications. Prof. Hausner is a member of the associations Verband der Electrotechnik (VDE) and Informationstechnische Gesellschaft (ITG).

POHL et al.: HIGH PRECISION RADAR DISTANCE MEASUREMENTS IN OVERMODED CIRCULAR WAVEGUIDES

Burkhard Schiek (M’85) was born in Elbing, Germany, in 1938. He received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the Technische Universität Braunschweig, Braunschweig, Germany, in 1964 and 1966, respectively. From 1964 to 1969, he was an Assistant with the Institut für Hochfrequenztechnik, Technische Universität Braunschweig, where he was involved with frequency multipliers. From 1969 to 1978, he was with the Microwave Application Group, Philips Forschungslaboratorium Hamburg GmbH, Ham-

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burg, Germany, where he was involved with solid-state oscillators, oscillator noise, microwave integration, and microwave systems. Since 1978, he has been a Professor with the Department of Electrical Engineering, Ruhr-Universität Bochum, Bochum, Germany, where he is involved with high-frequency measurement techniques and industrial applications of microwaves.

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On the Robustness of Digital Predistortion Function Synthesis and Average Power Tracking for Highly Nonlinear Power Amplifiers Oualid Hammi, Student Member, IEEE, Slim Boumaiza, Member, IEEE, and Fadhel M. Ghannouchi, Fellow, IEEE

Abstract—In this paper, a comprehensive study of the robustness of the digital predistortion function synthesis is presented. This study covers two aspects: the processing of the power amplifier (PA) input and output measured data intended for the extraction of the corresponding predistortion function, and the optimal setting of the predistorter’s small-signal gain to adaptively track the average power variation between the input and output of the predistorter. First, the accuracy of the polynomial curve fitting and the lookup table’s scheme in mimicking the measured AM/AM and AM/PM characteristics of the PA is investigated. To address the high dispersion of coefficients of the polynomial function, which limits the order that can be implemented and the fitting capabilities, a pre-processing technique is proposed. Second, the fitted PA curves are used to investigate the effects of the small-signal gain of the predistortion function on the linearization performance. An automated average power tracking technique is introduced in order to maintain a unit average gain of the predistorter. The measured spectra at the output of the amplifier show an additional 12-dBc improvement in the output spectrum regrowth. Index Terms—Digital predistorter (DPD), lookup table (LUT), nonlinearity, polynomial model, power amplifier (PA).

I. INTRODUCTION

I

N MODERN wireless communication systems, advanced modulations and access techniques are employed to increase the spectrum efficiency of the overcrowded and limited RF spectrum. These access techniques, which include wideband code division multiple access (WCDMA) and orthogonal frequency division multiplexing (OFDM), result in envelope varying signals that set stringent requirements on the linearity performance of the transmitter and, especially, the power amplifier (PA). In order to meet these linearity requirements, linearization techniques are needed to lower the backoff level at the output of the PA and thus improve the achievable power efficiency [1]–[6]. Predistortion is among the more intuitive linearization techniques, and its implementation can be either analog [1]–[4] or digital [5], [6], [9]–[12]. However, digital implementation is being widely preferred due to the accuracy that can be achieved

Manuscript received October 16, 2006; revised January 24, 2007. This work was supported by the Informatics Circle of Research Excellence, by the Natural Sciences and Engineering Research Council of Canada, and by Canada Research Chairs. The authors are with the iRadio Laboratory, Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada T2N 1N4 (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.2007.895237

in synthesizing the predistortion function when digital signal processors are used. Digital predistortion is, therefore, a cost-effective approach in the linearization of RF PAs, particularly for base-station applications. This technique consists of adding a complementary nonlinear function upstream of the PA so that the cascade of both nonlinear functions behaves as a linear system. In addition to its inherent conceptual simplicity, digital predistortion offers a moderate implementation complexity, while achieving good performance by taking advantage of the accuracy of digital signal processing. Furthermore, the digital implementation of the predistortion function gives it the ability to synthesize highly nonlinear shapes. Accordingly, it becomes possible to use highly nonlinear PAs, such as Doherty amplifiers and amplifiers biased in deep class AB. Indeed, these PAs have increased power efficiency compared to that of the widely deployed mildly nonlinear class AB PAs. The use of digital predistorters (DPDs) along with highly nonlinear PAs raises new challenges that limit system performance. First, for memoryless predistorters, two approaches can be used to process the PAs measured AM/AM and AM/PM characteristics and extract the fitted curves: polynomial functions and lookup tables (LUTs) [6]–[9]. This is a critical step in the design of DPDs since the predistortion function will be the complementary nonlinear function of the fitted AM/AM and AM/PM curves of the PA. The polynomial functions showed good performance when used with weakly nonlinear PAs [9]. However, a comparative study between the robustness of the polynomial functions and the LUTs, when applied to linearize highly nonlinear PAs, revealed that the polynomial functions have limited fitting and numerical accuracies [10]. Second, the predistorter’s gain at the operating average power is a major consideration that needs to be taken into account when DPDs are used with highly nonlinear PAs. This issue is not critical in designing DPDs for weakly nonlinear PAs since the corresponding predistorter’s average output power is fairly close to its average input power. Conversely, highly nonlinear PAs exhibit significant gain variation at low input power levels, which translate into a steeper gain variation of the corresponding DPD. Consequently, the DPD’s average output power differs from its average input power that was used in the PA characterization step. Accordingly, the PA will operate at an average input power level that is different from the one used in the characterization process. This leads to a mismatch between the PA’s nonlinear characteristics and the predistorter’s nonlinearities and, ultimately, to a degradation in the achievable performance.

0018-9480/$25.00 © 2007 IEEE

HAMMI et al.: ROBUSTNESS OF DIGITAL PREDISTORTION FUNCTION SYNTHESIS AND AVERAGE POWER TRACKING FOR HIGHLY NONLINEAR PAs

Predistorters, based on multi-LUTs, which take into account the variation of the PA’s average input power, were proposed in [6], but, this approach does not compensate for the average input power variation due to the predistortion function shape. Thus, for highly nonlinear PAs, it is necessary to adjust the predistorter architecture to automatically monitor the predistortion function and ensure a unit gain at the operating average power level. This will guarantee a perfect match between the predistortion function and the PA nonlinearity and, thus, achieve better linearity performance. In this paper, the robustness of the digital predistortion function synthesis process applied to linearize highly nonlinear RF PAs is investigated. The extraction of the PA’s nonlinear characteristics from the measured data and the predistorter’s ability to track the transmitted average power are considered. Section II presents the comparative study of the robustness of the polynomial and LUT approaches. In Section III, a data pre-processing technique is introduced, and its outcome on the polynomial approach robustness is evaluated. The augmented predistorter architecture, which includes an automated average power control, is presented in Section IV, and its effectiveness is assessed experimentally. The conclusions are presented in Section V.

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Fig. 1. Measured AM/AM and AM/PM characteristics of PA1.

II. COMPARATIVE STUDY OF THE ROBUSTNESS OF THE POLYNOMIAL AND LUT TECHNIQUES The quality of the linearity improvement achieved by DPDs depends on the accurate extraction of the PA’s nonlinear characteristics. Two approaches have been proposed in the literature to extract the PA’s memoryless nonlinearities from the raw measured AM/AM and AM/PM data. In the first method, the measured data is fitted using two polynomial functions, whereas in the second technique, the measured data is smoothed using an averaging technique, and the resulting curves are stored into two LUTs. To study the robustness of these two approaches when applied to highly nonlinear PAs, two PAs biased in deep class AB close to the pinch off were used. Both amplifiers were built using LDMOS transistors. The first PA (PA1) was a 100-W peak power amplifier that operated around 2140 MHz. The second amplifier (PA2) was a 10-W peak power amplifier that operated around 1950 MHz. The two amplifiers were characterized under a two-carrier WCDMA signal excitation. Each carrier had a chip rate of 3.84 Mc/s, and the separation between the two carriers was equal to 5 MHz. The resulting signal had a total bandwidth of 10 MHz and a peak-to-average power ratio of 10.4 dB. The experimental setup used to measure the instantaneous waveforms at the input and output of each PA is described in [10]. The measured AM/AM and AM/PM characteristics of PA1 and PA2 are presented in Figs. 1 and 2, respectively. These figures show that both amplifiers exhibited highly nonlinear characteristics that had different shapes. The polynomial fitting of the nonlinear characteristics presented in Figs. 1 and 2 calls for the use of high-order polynomial functions to achieve a satisfactory fitting accuracy. However, as the polynomial order increases, the dispersion of the polynomial coefficients, identified using the least square error (LSE)

Fig. 2. Measured AM/AM and AM/PM characteristics of PA2.

algorithm, increases. This dispersion is defined as the magnitude of the ratio between the maximum and minimum values of the polynomial coefficients. Thus, increasing the polynomial order to improve the fitting quality makes the identification of the coefficients more computationally demanding and results in a numerical inaccuracy once the polynomial order gets above a given value. In order to avoid such inaccuracy, the dispersion of the polynomial coefficients should be kept to less than the resolution of the digital signal processor. In this study, a 32-bit digital signal processor is used to implement the identification processes required by both approaches. The highest dispersion value that could be handled by this dig2 for a 32-bit data ital signal processor would be 2.15 10 format.

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Fig. 4. Measured and fitted AM/AM and AM/PM characteristics of PA1. Fig. 3. Coefficients’ dispersion versus the polynomial order.

The identification of the polynomial functions that best fit the measured characteristics of both amplifiers was performed using the LSE algorithm available in a commercial digital signal processing software (MATLAB, The MathWorks Inc., Natick, MA). For each polynomial order, the dispersion of the coefficients was calculated. The results are plotted in Fig. 3, which presents the coefficients’ dispersion of the polynomial fitting versus the polynomial order. It demonstrates that the implementation of polynomial functions with an order higher than 7 cannot be supported within the available resolution. In the LUT-based approach, the measured data is first smoothed using the dynamic weighted moving average technique presented in [11]. The resulting data is saved into a LUT that models the PA’s nonlinear behavior. The averaged version is designated and expressed by of the measured data (1) . where is the weight factor The polynomial functions that best fit the measured data, while meeting the resolution constraint, are presented in Figs. 4 and 5 for PA1 and PA2, respectively. These figures also illustrate the results obtained using the LUT approach. It is shown that the LUT approach leads to an accurate fitting of the measured data for all the considered characteristics. Conversely, the polynomial approach was unable to precisely fit the measured data over the entire power range. Also, since the fitting accuracy of PA2 was better than that of PA1, Figs. 4 and 5 demonstrate that the inaccuracy of the polynomial fitting depends on the nonlinear shape to fit. Such inaccuracy in extracting the PA behavior leads to a mismatch between the actual PA nonlinearity and the synthesized predistortion function. This will ultimately lead to poor linearity performance, as reported by the authors in [10]. Consequently, the polynomial fitting can only be used

Fig. 5. Measured and fitted AM/AM and AM/PM characteristics of PA2.

over a limited power range, which will reduce the achievable power efficiency of the linearized PA. III. DATA PRE-PROCESSING FOR ENHANCING POLYNOMIAL FITTING ROBUSTNESS The polynomial coefficients’ dispersion, as previously explained, limits the order of the polynomial functions that can be used to fit the measured data. However, it is necessary to increase the polynomial order to enhance the fitting accuracy. The use of floating point processors would overcome the limitation on the acceptable dispersion value at the expense of higher cost. However, it is worth mentioning that as the coefficients’ dispersion increases, the Vandermonde matrix conditioning worsens. Bad conditioning of the Vandermonde matrix leads to

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a more computationally consuming matrix inversion process. Accordingly, the use of a floating point processor is not a valuable solution for practical implementations. Here, a data pre-processing technique is proposed in order to reduce the coefficients’ dispersion. The considered polynomial function is given by

(2)

where is the estimated value of using the polynomial is the polynomial order, and are fitting, the polynomial coefficients. The dispersion of the coefficients can be reduced provided , which modifies that there is a pre-processing of the signal its distribution. The signal pre-processing technique performed in this study consists of centering and scaling the input data , as described by the following equation: stream (3)

Fig. 6. Approximation error versus the order of the polynomial function.

where and are the mean value and standard deviation of the signal , respectively. The proposed pre-processing procedure changes the statis. The resulting signal is centered at zero tics of mean and is scaled to unit standard deviation. By centering , the spreading of the values of and scaling the signal is reduced, and the accuracy of the subsequent numeric computations is improved. Consequently, if we consider a given polynomial order, the use of the data pre-processing technique will result in the same fitting accuracy, but will decrease the dispersion of the polynomial coefficients. The implementation of higher order polynomials is then feasible within the available resolution constraint stated in Section II. This increase in the polynomial order will improve the fitting accuracy, as reported in Fig. 6. This figure presents the approximation error on both the AM/AM and AM/PM functions versus the polynomial function order for PA1. The approximation error plotted in this figure is given by

Error dB

mean

(4) Fig. 7. Coefficients’ dispersion versus the polynomial order after data preprocessing.

and are the estimated nonlinear (AM/AM or, where equivalently, AM/PM) characteristics of the PA using the LUT approach and the polynomial fitting approach, respectively. The refers to the polynomial order. index in To evaluate the usefulness of the data pre-processing technique, the polynomial coefficients’ dispersion was calculated for the measured AM/AM and AM/PM curves of both amplifiers. The results are summarized in Fig. 7, which shows the polynomial coefficients’ dispersion versus the polynomial order. This figure illustrates that, with the data pre-processing technique, high-order polynomial functions can be implemented with sufficient calculation accuracy. Such polynomials

have better fitting capabilities for PA1 and PA2, as presented in Figs. 8 and 9, respectively. The orders of the polynomial functions presented in these figures are 14 and 16 for the AM/AM and AM/PM characteristics, respectively. The results of the LUT-based approach are also shown in these same figures for reference. According to the results reported in Figs. 8 and 9, one can conclude that the use of the data pre-processing technique, along with the polynomial fitting, leads to the same fitting accuracy as the LUT approach, by enabling the use of high-order polynomial functions. However, such orders require high computational complexity in the identification process

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Fig. 8. Measured and fitted AM/AM and AM/PM characteristics of PA1.

Fig. 10. AM/AM characteristics of the DPD versus its small-signal gain.

Fig. 9. Measured and fitted AM/AM and AM/PM characteristics of PA2.

compared to the LUT approach. Moreover, a monitoring algorithm that implements an accuracy evaluation metric, such as the normalized mean squared error between the original and the predicted data, is required to choose the appropriate polynomial order. Otherwise, a sufficiently high polynomial order can be specified for the fitting function, but this will result in an unjustified increase in the computational complexity. IV. AUTOMATED PREDISTORTION FUNCTION SYNTHESIS A comparative study was carried out above for the LUT and polynomial function approaches that can be considered to process the measured PA data. This demonstrated that the polynomial functions can achieve similar fitting accuracy as the

LUT-based approach; however, the polynomial fitting involves higher computation complexity, which is attributed to the LSE algorithm. Here, we will consider the LUT approach when applied to the linearization of the PA, PA2, using a memoryless DPD. The effectiveness of the nonlinearity cancellation in digital predistortion-based linearizers depends on two key considerations. First, the PA nonlinear behavior should be accurately characterized. This has been ensured in this study by using the LUT approach combined with the realistic characterization technique using the input and output waveforms proposed in [12]. Second, the optimal predistortion function needs to be synthesized based on the PA’s nonlinear characteristics without disturbing the behavior of the PA. Yet the PA’s behavior depends on the signal statistics and its average power. Consequently, it is crucial for the predistorter not to change the transmitted average power at which the PA characterization was performed. Thus, an adequate choice needs to be done when deriving the predistortion function. Indeed, for the considered PA nonlinearity, several predistortion functions can be synthesized depending on the PA’s small-signal gain. Fig. 10 shows various predistortion functions having different small-signal gains that are directly related to the chosen small-signal gain of the PA. These predistorters lead to different linearity performances, as proven by the measured spectra at the output of the linearized PA that are presented in Fig. 11. The spectra shown in Fig. 11 were obtained for a constant power level at the linearized PA’s input. This power level was chosen so that the predistorter with the lowest maximum input power level (this is the predistorter that corresponds to Gain dB in Fig. 10) does not saturate. The input backoff of this predistorter was equal to the input signal’s peak-to-average power ratio.

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Fig. 12. DPD architecture with built-in automated average power tracking.

, which has a small-signal gain tion equation:

Fig. 11. Measured spectra at the output of the PA.

Fig. 11 demonstrates that the predistorter derived with a 0-dB small-signal gain has limited performance, and that the appropriate choice of the predistorter’s small-signal gain improves the linearity by more than 12 dBc. Herein, the optimal predistorter refers to the predistorter presented in Fig. 10 that has 5 dB of small-signal gain. This predistorter is optimal in the sense that its choice was based on the criterion of unit gain at the operating average power. Indeed, the nonoptimal predistorter, with 0 dB of small-signal gain, shifts the average power of the transmitted signal by 3.77 dB. Such a change in the average power stimulates a PA behavior that is different from the behavior that was determined in the characterization process. This causes a discrepancy between the synthesized predistortion function and the actual PA nonlinear characteristics. This discrepancy is at the origin of the limited improvement in the linearity performance. To ensure the highest linearity improvement, it is necessary to choose from among the different predistorters that have different small-signal gains, the one that has a unit gain at the operating average power. Accordingly, the optimal predistortion function depends on the PA’s nonlinearity shape, the signal statistics and the operating average power. Still, the optimal predistorter does not necessarily correspond to the one that has an instantaneous unit gain at the operating average power. An automated DPD function synthesis process that blindly identifies the optimal predistorter is proposed in Fig. 12. The proposed architecture aims to dynamically adjust the actual predistortion function, depending on the input average power. This consists of designing the conventional predistortion function, which has a nonoptimal small-signal gain, and then mapping it into the optimal predistortion function. , which As shown in Fig. 10, a predistortion function has a small-signal gain is related to the predistortion func-

by the following

(5) where and . In the proposed architecture, the automated predistortion function synthesis is done without any prior knowledge of the value of the required optimal predistorter small-signal gain . For this purpose, (5) is approximated by

(6) . where In this latter equation, the AM/PM function does not include . Indeed, this the small-signal phase-shift compensation results in a constant phase shift that is independent of the instantaneous input power level and, thus, does not affect the linearity performances. In (6), the value of the small-signal gain shift is approximated by the average power variation through for both the AM/AM and AM/PM the predistorter functions. Even though the average power variation through the predistorter is a nonlinear function of the small-signal gain shift, this variable can be used to approach the optimal predistortion function. Since the PA behavior is not very sensitive to an average power variation that is less than 0.5 dB, a decision component is added in the proposed architecture to determine whether or not the predistortion function should be changed. The input and output average powers are calculated using the weighted moving average technique in which the average power is estimated as follows: (7) where is the weight factor . The proposed predistorter architecture was evaluated under the two-carrier WCDMA signal presented in Section I. The

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TABLE I AVERAGE POWER VARIATION THROUGH THE PREDISTORTER

average input power was set to 7 dBm. Two predistortion functions, which had small-signal gains of 0 and 2 dB, respectively, were considered. The proposed predistorter architecture was used to adjust the predistortion function iteratively, by controlling the predistorter’s small-signal gain to minimize the average power variation through the predistorter. The results are presented in Table I. The th iteration (it. ) reported in Table I corresponds to the adjustment of the predistortion ) by the function obtained in the preceding iteration (it. , which is also obtained from the preceding value of ), according to (6). iteration (it. The linearity performances obtained at the output of the amplifier for the second case, where the initial small signal of the DPD was set to 2 dB, are presented in Fig. 11. The iteration 1 corresponds to the cases where the predistorter’s small-signal gain is set to 4 dB. This corroborates the usefulness of the decision block since the adjustment of the predistortion function was not required when the average power variation through the predistorter was less than 0.5 dB. The validity of the approximation in (6) was also assessed. Finally, the measurement results demonstrate the effectiveness of the proposed architecture and its ability to adjust the predistortion function within one or two iterations. V. CONCLUSION In this paper, critical issues related to the linearization of highly nonlinear PAs using the digital predistortion technique are presented. First, a comparative study between the robustness of the polynomial and LUT approaches in processing the PA’s measured data for the predistortion function synthesis was conducted. This demonstrated the ability of the LUT approach to accurately fit the measured data. Conversely, the polynomial approach has limited fitting capabilities due to the polynomial coefficients’ dispersion. This limitation was overcome by introducing a signal pre-processing technique that centers and scales the measured data. The fitted PA curves were then used to synthesize various predistortion functions that have different small-signal gains. An automated average power tracking solution, which ensures a unit average gain through the predistorter, was proposed to adaptively select the optimal predistortion function. This kept the average power of the predistorted signal at the PA input close enough to that used in the PA characterization process to match the predistortion function with the actual behavior of the nonlinear PA. The effectiveness of the proposed architecture was assessed experimentally. The results showed that the linearity performances were significantly enhanced, and that the system converges in less than three iterations.

ACKNOWLEDGMENT The authors would like to thank C. Simon, University of Calgary, Calgary, AB, Canada, for providing technical support during the measurements. The authors also want to acknowledge Agilent Technologies, Palo Alto, CA, for the donation of their Advanced Design System (ADS) software. REFERENCES [1] F. H. Raab, P. Asbeck, S. Cripps, P. B. Kenington, Z. B. Popovic´ , N. Pothecary, J. F. Sevic, and N. O. Sokal, “Power amplifiers and transmitters for RF and microwave,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 814–826, Mar. 2002. [2] J. Yi, Y. Yang, M. Park, W. Kang, and B. Kim, “Analog predistortion linearizer for high-power RF amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2709–2713, Dec. 2000. [3] S. Y. Lee, Y. S. Lee, and Y. H. Jeong, “Fully-automated adaptive analog predistortion power amplifier in WCDMA applications,” in 35th IEEE Eur. Microw. Conf., Oct. 2005, vol. 2, 4 pp. [4] S. Y. Lee, Y. S. Lee, S. H. Hong, H. S. Choi, and Y. H. Jeong, “Independently controllable 3rd and 5th order analog predistortion linearizer for RF power amplifier in GSM,” in IEEE Asia–Pacific Adv. Syst. Integrated Circuits Conf., Aug. 2004, pp. 146–149. [5] Y. Nagata, “Linear amplification technique for digital mobile communications,” in IEEE Veh. Technol. Conf., May 1989, vol. 1, pp. 159–164. [6] W. J. Jung, W. R. Kim, and K. B. Lee, “Digital predistorter using multiple lookup tables,” Electron. Lett., vol. 39, no. 19, pp. 1386–1388, Sep. 2003. [7] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology, and Techniques. New York: Kluwer, 2000. [8] E. G. Jeckeln, S. Huei-Yuan, E. Martony, and M. Eron, “Method for modeling amplitude and bandwidth dependent distortion in nonlinear RF devices,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 3, pp. 1733–1736. [9] L. Ding and G. T. Zhou, “Effects of even-order nonlinear terms on power amplifier modeling and predistortion linearization,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 156–162, Jan. 2004. [10] O. Hammi, S. Boumaiza, and F. M. Ghannouchi, “On the robustness of the predistortion function synthesis for highly nonlinear RF power amplifiers linearization,” in 36th IEEE Eur. Microw. Conf., Sep. 2006, pp. 145–148. [11] T. Liu, S. Boumaiza, and F. M. Ghannouchi, “Deembedding static nonlinearities and accurately identifying and modeling memory effects in wideband RF transmitters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3578–3587, Nov. 2005. [12] S. Boumaiza and F. M. Ghannouchi, “Realistic power-amplifiers characterization with application to baseband digital predistortion for 3G base stations,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 3016–3021, Dec. 2002.

Oualid Hammi (S’03) received the B.Eng. degree in electrical engineering from the École Nationale d’Ingénieurs de Tunis, Tunis, Tunisia, in 2001, the M.Sc. degree from École Polytechnique de Montréal, Montréal, QC, Canada, in 2004, and is currently working toward the Ph.D. degree at the University of Calgary, Calgary, AB, Canada. His current research interests are in the area of microwave and millimeter-wave engineering in general. His particular research activities are related to the design of intelligent and highly efficient linear transmitters for wireless communications and the development of digital signal processing (DSP) techniques for PA linearization purposes.

HAMMI et al.: ROBUSTNESS OF DIGITAL PREDISTORTION FUNCTION SYNTHESIS AND AVERAGE POWER TRACKING FOR HIGHLY NONLINEAR PAs

Slim Boumaiza (S’00–M’04) received the B.Eng. degree in electrical engineering from the École Nationale d’Ingénieurs de Tunis, Tunis, Tunisia, in 1997, and the M.Sc. and Ph.D. degrees from the École Polytechnique de Montréal, Montréal, QC, Canada, in 1999 and 2004, respectively. In May 2005, he joined the Electrical and Computer Engineering Department, The University of Calgary, Calgary, AB, Canada, as an Assistant Professor and faculty member with the iRadio Laboratory. His research interests are in the areas of design of RF/microwave and millimeter-wave components and systems for wireless communications. His current interests include RF/digital signal processing (DSP) mixed design of intelligent transmitters, design, characterization, modeling and linearization of high-power amplifiers, reconfigurable and multiband RF transceivers, and adaptive DSP.

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Fadhel M. Ghannouchi (S’84–M’88–SM’93–F’07) received the Ph.D. degree in electrical engineering from the University of Montréal, Montréal, QC, Canada, in 1987. He is currently an iCORE Professor, a Canada Research Chair, and the Director of the iRadio Laboratory, Department of Electrical and Computer Engineering, The University of Calgary, Calgary, AB, Canada. He has held invited positions with several academic and research institutions in Europe, North America, and Japan. His has authored or coauthored over 300 publications. He holds seven patents. His research interests are in the areas of microwave instrumentation, modeling of microwave devices and communications systems, design and linearization of RF amplifiers, and SDR radio systems.

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Study and Design Optimization of Multiharmonic Transmission-Line Load Networks for Class-E and Class-F -Band MMIC Power Amplifiers

K

Renato Negra, Member, IEEE, Fadhel M. Ghannouchi, Fellow, IEEE, and Werner Bächtold, Fellow, IEEE

Abstract—A design-oriented analysis of microwave transmission-line class-E and class-F amplifiers is presented in this paper. Multiharmonic transmission-line load networks are analyzed and compared in terms of harmonic suppression and their effects on output power and efficiency. Based on this study, a design of highly efficient monolithic-microwave integrated-circuit amplifiers has been carried out. To allow circuit optimization and to simplify the design process, analytic expressions were derived for the most practical multiharmonic transmission-line networks. Fabricated amplifiers achieve state-of-the-art efficiency of 56.2% and 59.0% for class-E and class-F operation at -band for power levels of 19.1 and 20.0 dBm, respectively. Moreover, without the need for supplementary filtering sections, harmonic suppression for operation well into compression is better than 25 and 30 dBc for the transmission-line class-F and class-E amplifiers, respectively. Index Terms—Class E, class F, harmonic suppression, high efficiency, MODFETs, monolithic-microwave integrated-circuit (MMIC) power amplifiers (PAs).

I. INTRODUCTION IGHLY efficient circuits require specific harmonic content to be present in the voltage and current waveforms at the output terminal of the transistor. To minimize the simultaneous presence of high currents and high voltages in the active device, the product of current and voltage should ideally be zero at all frequencies, except at the fundamental frequency. This precludes power dissipation in the device, resulting in highly efficient operation. Depending on the application and its frequency range, the load transformation network may be realized using lumped or distributed components, or a deliberate compound of both. In the frequency range up to -band, harmonic tuning networks can still be implemented monolithically using on- and off-chip lumped components [1], [2]. Besides the performance of the components at the design frequency, their behavior must also be scruntinized at the relevant harmonic frequencies. Transmission lines are often preferred over lumped elements at microwave and millimeter-wave frequencies [3]–[6] because high-performance

H

Manuscript received October 16, 2006; revised March 6, 2007. R. Negra and F. M. Ghannouchi are with the iRadio Laboratory, Electrical and Computer Engineering Department, University of Calgary, Calgary, AB, Canada T2N 1N4 (e-mail: [email protected]; [email protected]). W. Bächtold, retired, was with the Laboratory for Electromagnetic Fields and Microwave Electronics, Eidgenössische Technische Hochschule, Zürich, 8092 Zürich, Switzerland. Digital Object Identifier 10.1109/TMTT.2007.896769

Fig. 1. Analyzed transmission-line load circuits suitable for class-E multiharmonic impedance termination.

and high-quality lumped components are difficult to fabricate. Distributed elements typically provide better broadband performance and higher factors, and are easier to fabricate with high precision at microwave and millimeter-wave frequencies than lumped elements. Various transmission-line load networks have been proposed to approximate ideal class-E [7], [8] and class-F [9], [10] operation. Since the number of circuit elements in the high power side of the final stage should be kept at a minimum, a rigorous study is carried out to assess the effects, in terms of power-output capability, efficiency, and harmonic suppression of the number of harmonics considered by different transmission-line load-coupling networks. It is also of practical interest to identify the impact of circuit tuning on harmonic suppression, as it is necessary to assure compliance with out-of-band emission regulations without the need of additional filtering sections at the output of the amplifier. In [1], the impact of circuit tuning on harmonic suppression has been analyzed for transmission-line class-F networks. This paper extends this study by also including different transmission-line class-E load-coupling topologies, and by providing closed-form expressions for the most practically relevant class-E and class-F output matching topologies. Thus far, reported performances of transmission-line class-E and class-F power amplifiers (PAs) were mainly achieved by dimensioning the circuit based on standard lumped-element topologies, which are then transformed to equivalent distributed networks. Without the availability of analytic expressions, tuning of multiharmonic circuits is very time consuming, if not impossible, since highly efficient operation requires controlled changes of fundamental and harmonic impedances at the same time. Measurements on manufactured MMIC amplifiers [3], [4] were performed, in order to validate the theoretical findings. Using the same standard GaAs pseudomorphic HEMT (pHEMT) process for both class-E and class-F monolithic-microwave integrated-circuit (MMIC) PA chips provides a direct

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Fig. 2. Analyzed transmission-line load circuits suitable for class-F multiharmonic impedance termination.

performance comparison of state-of-the-art highly efficient -band amplifiers.

and high impedances at odd harmonics [1] at

II. HIGH-EFFICIENCY CLASSES OF OPERATION

at at

A. Class-E Operation Class E employs only a single transistor, and the passive load circuit is designed to overcome the device parasitic capacitance problem of class D, by means of providing zero voltage switching (ZVS). Zero voltage across the device during the switching instants eliminates the capacitive loss associated with the periodic charging and discharging of the parasitic output capacitance. Moreover, class-E operation also has the distinct advantage of being able to accommodate the nonzero switching times of real devices. The output matching is designed to displace in time voltage and current switching instants so that a finite transition time between the two discrete states can be accommodated. This not only allows for the achievement of unity efficiency in theory, but also obtains very high efficiency with practical implementations at microwaves. The drawback of delaying the rise of the device voltage is a reduction of output and power utilization factor (PUF). power capability The load conditions for nominal class-E operation are [11] at at where the optimal class-E load resistance is given by design frequency

(1)

at the fundamental

(2) The output shunt capacitance of the load network is composed, in whole or in part, of the parasitic device output capac. itance in (1) ensures that both the voltage across The impedance the device output terminals and its derivative are zero at the instant when the switch is closing. B. Class-F Operation The principle of class F is to operate the transistor as a saturated controlled current source. By multiharmonic tuning, current and voltage waveforms at the drain are shaped to minimize power dissipation in the device. In order to approximate a square-wave voltage and a half sine-wave current waveform, the class-F network provides low impedances at even harmonics

(3)

denotes the maximum device drain curReferring to (3), denotes the applied supply voltage, and denotes rent, the knee voltage. In this way, the product of voltage—containing only odd harmonic components—and current—composed of only even harmonics—is zero at all frequencies, except at the fundamental. Therefore, no power is theoretically dissipated at higher harmonics, while the overlap of high current and high voltage in the device is minimized throughout the entire RF cycle. III. HARMONIC TERMINATION EFFECTS Theoretical values for incremental inclusive control of harmonics have been calculated in [12] for different classes of operation. Those theoretical results show that the influence of harand decreases with the increasing monic termination on order of the considered harmonic component. To assess the effects of harmonic loading on practical circuits, a GaAs pHEMT model was loaded with different output matching circuits suitable for class-E and class-F tuning and was designed using ideal lines. The pHEMT model provided by the foundry and used for the analyses and designs is an analytic charge-conservative large-signal model suitable for linear and switched-mode circuit design [13]. The model has been proven to be very accurate for linear and nonlinear designs beyond 90 GHz [14]. At the input, the transistor is matched for maximum power transfer using realistic coplanar waveguide (CPW) models [15]. The load-coupling transmission-line networks for class-E and class-F approximation used in this study are shown in Figs. 1 and 2, respectively. A. Class E Each of the networks shown in Fig. 1 was engineered to provide the optimum class-E load impedance at the design frequency GHz. The total switching capacitance in this design is composed of the device inand a metal–insulator–metal (MIM) caternal capacitance pacitor, which was added to boost the power-output capability of the circuit. The respective impedances, presented by through at multiples of , are given in Table I along , dc-to-RF efficiency , with simulated output power power-added efficiency (PAE), and higher harmonic suppression at the load.

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TABLE I IMPACT OF DIFFERENT IDEAL TRANSMISSION-LINE LOAD MATCHING NETWORKS ON THE PERFORMANCE OF A REALISTIC GaAs PHEMT DEVICE MODEL IN CLASS-E OPERATION

TABLE II IMPACT OF VARIOUS TRANSMISSION-LINE LOAD MATCHING NETWORKS ON THE PERFORMANCE OF A CLASS-F AMPLIFIER USING A REAL pHEMT DEVICE MODEL

The analysis shows that including proper termination has the biggest impact on all analyzed parameters. As expected, power-output capability drops from the class-AB value, attained - , to values closer to ideal class-E operation, as through more harmonic impedances are controlled. On the other hand, a - . If is taken substantial increase in is observed for into account, efficiency still improves, but to a much smaller has been designed to provide the same abextent. and . In this way, solute reactive impedance for both harmonic suppression at the load at these frequencies is excellent, as demonstrated in Table I. Tuning the circuit for optimum second or third harmonic termination results in slightly lower . Since provides optimum efficiencies, but higher impedance, harmonic output at this frequency is very low. However, filtering of higher harmonic components is worse than - . Adding has an impact only on third harwith monic attenuation, while leaving the output at other frequencies basically unchanged. From the results in Table I it appears that the topology provides the best compromise between performance improvements and circuit complexity. The addition of more stubs, or even sections, has only minor beneficial effects, brings a valuable amelioration compared to whereas - , not only in terms of harmonic filtering, but also in regard to electrical performance.

, PAE, fundamental impedance transformation reduces compared to a broadband resistive loading. Being a and half-wavelength, or multiple of it, at all even-order harmonics, the resulting impedance at the output terminal of the transistor . Instead of ideally proat these frequencies is equal to at is even higher viding a low impedance to ground, than at , by a factor equal to the load transformation ratio . This explains why both and efficiency are than for any other analyzed load network. lower for The values listed in Table II confirm that second harmonic tuning also has the biggest impact in practice. The extra stub , used for third-harmonic voltage peaking in - , only gives slight improvement in efficiency, while for the analyzed stays the same as for - . class-F amplifier, is limited by the applied supply voltage after introducing the even harmonic stub in Fig. 2(c). Besides its valuable stub, contribution to harmonic filtering, the addition of a as shown in Fig. 2(d), also slightly enhances efficiency. Proper fifth harmonic termination leaves the amplifier performance suppression. basically unchanged, except for and , in theory, increase monotonically for Whereas an increasing number of considered harmonics [12], in simulation based on a realistic high-frequency device model and ideal transmission lines, improvements could be observed only for proper harmonic tuning up to the third harmonic. Consideration of additional frequency components does not further enhance circuit performance. Adding extra stubs only improves higher harmonic suppression. In spite of the utilization of ideal transmission lines, both the rate of change and the absolute values of output power and efficiency improvements are considerably smaller than the theoretical values in [12]. , , and are made Since the electrical lengths of , , and , respectively, no a quarter-wavelength at harmonic component is observed in the load voltage at these frequencies. This, however, changes if the characteristic imand electrical lengths of the transmission lines pedances are optimized to account for device output parasitics. Unlike class E, where the high reactive impedances at higher harmonics have to be provided at the connection point of the total , the reference plane for optimum output shunt capacitance fundamental and harmonic impedance termination in class F . Hence, is to the left of the device internal capacitance when determining the actual lengths of the transmission lines

B. Class F Each of the matching circuits in Fig. 2 provides the loadline at . The actual optimum load for the used tran. Such a low loadline is inherent with poor sistor is efficiency, due to the large impedance transformation ratio and the high loss of on-chip transmission lines. In order to achieve has been increased accordingly. a competitive efficiency, Harmonic impedance termination of the networks differs considerably, as shown in Table II. As for the class-E analysis , , and PAE, as well as above, Table II lists simulated harmonic suppression for the class-F amplifier as a function of the number of harmonics considered by the load-matching to - . networks serves only as reference Load-coupling network since fundamental load transformation is always required in and practical applications. Comparison of shows that using a simple quarter-wave transmission line for

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TABLE III SIMULATED PERFORMANCE OF THE MICROWAVE CLASS-F AMPLIFIER TUNED FOR MAXIMUM 

Fig. 4. Equivalent circuit diagrams: (a) at the switching signal frequency, (b) at the second, and (c) third harmonic frequency for the transmission-line class-E load coupling network.

Fig. 3. Circuit diagram of the 22–25-GHz coplanar class-E PA.

forming the class-F load-coupling network, the device output parasitics have to be taken into consideration. to Four of the five load-coupling networks, i.e., - , were tuned to maximize of the class-F amplifier. During the optimization process, the tuning range for all characteristic impedances and electrical lengths was restricted to realizable values for on-chip CPWs. The impedances at the first five harmonics provided by the tuned load networks are tabulated in Table III, alongside with , PAE, and . the corresponding simulated IV. TRANSMISSION LINE LOAD NETWORK DESIGN A. Class E Fig. 3 shows the complete schematic of the class-E PA [3] based on the single section output network - . This network was selected for the experimental prototyping of a 24-GHz amplifier since it provides a good compromise between circuit complexity and performance. Due to its relative simple structure, this network is also well suited for monolithic integration in the upper microwave spectrum. and The electrical parameters of the capacitive stubs can be engineered to simultaneously provide the reactance required for fundamental load transformation, as well as a low and . By input impedance at selected harmonics, i.e., at making the electrical length of the stubs exactly one quarterwavelength at a particular harmonic, this harmonic is short circuited to ground and, thus, suppressed at the load. The wave impedance of the stubs can then be chosen to provide the desired capacitive reactance for load transformation at the fundamental are deterfrequency. The transmission-line parameters of mined by the requirement to provide the appropriate class-E

load angle of 49.05 at the fundamental and to transform the low input impedance of the stubs toward higher reactive impedances at the selected harmonics. The equivalent circuits for the output matching network at the frequencies of interest are shown in Fig. 4. For the signal frequency , the network behaves like a second-order low-pass filter, as depicted in Fig. 4(a). This structure is used to provide at the design the transformation of the 50- load down to and , the output network can be reprefrequency. At sented as shown in Fig. 4(b) and (c). The second and third harand , remonic content is short circuited to ground by transforms spectively, and is thus suppressed at the load. these shorts toward higher reactive impedances at the indicated frequencies. Analytic expressions for the design and optimization of the selected load-coupling circuit are very useful in calculating the starting values for a new design. Moreover, they are an essential tool for tuning multiharmonic circuits. Specific load conditions have to be maintained at a given frequency, while the impedances presented to other frequencies may be deliberately adjusted. Without analytic expression, class-E circuit optimization is only possible by going through a lumped-element equivalent circuit [7] or by adopting an empirical approach. Both of theses methods are time consuming and fail to guarantee certain , operating conditions, e.g., keeping a load angle of throughout the optimization process. The analytic design equations for the single-section transmission-line output network in Fig. 3 are derived using standard transmission-line and network theory. Referring to Fig. 3, admittance , resulting from the parallel , , and the load is connection of the two open-stubs written as (4) The capacitive part , expressed in terms of the transmission, and is line parameters , , (5) where is the propagation constant. The class-E impedance in (1), which has to be present at the connection point of the shunt , is transformed toward the load by . At the capacitance , the transformed admittance can be written as output of (6)

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By choosing a moderate to high characteristic impedance for , the line length can be calculated by the parameter equating the real parts of (4) and (6)

(7) and solving for the electrical length of

Fig. 5. Circuit diagram of the 24-GHz coplanar MMIC class-F PA.

(8) The second set of equations, obtained by equating the imag, is used to calculate in inary parts (5). To fulfill the harmonic termination conditions, the electrical length and of the open stubs are made exactly one and , respectively. By selecting an quarter-wavelength at appropriate value for one of the characteristic impedance, e.g., , the other one, in this example, can be derived from (5) as follows: (9)

Fig. 6. Equivalent-circuit diagrams: (a) at the switching signal frequency and (b) at the second and (c) third harmonic frequency for the class-F transmissionline output network.

loaded in Fig. 5 was chosen to be one quarter-wavelength at , thus providing a low impedance at even harmonics and a suitable means to inject the drain bias. At higher frequencies, the parasitics at the output of the device have to be taken into , account when calculating line lengths. Transmission line for example, has to be adjusted in length by [4] (10)

and can be chosen deThe two design parameters liberately, or adjusted with the aid of a linear circuit simulator, to obtain high reactive impedances at both selected harmonics, while allowing for the selection of practical values for all characteristic impedances. The derived analytic formulas ensure class-E switching conditions throughout the tuning process. B. Class F The selected output network for the MMIC -band class-F amplifier is shown in the schematic in Fig. 5. As for the class-E topology, three transmission lines also provide the best compromise for class-F approximation at high frequencies. The electrical lengths of these transmission lines are determined by the harmonic termination requirements of the class-F operation. The characteristic impedances of the three lines can be engineered to obtain the desired load transformation at the fundamental frequency. The equivalent circuits for the output matching network at the harmonic frequencies of interest are shown in Fig. 6. The , which is one quarter-wave at , provides a low open stub impedance at its input terminal at this frequency. Transmission transforms the low impedance, ideally a short circuit, line into an appropriate high reactive impedance at the terminals of the transistor-internal current source. Therefore, its line length and then tuned to accommodate the output is set to parasitics of the device. The electrical length of the capacitively

to compensate for the parasitic drain inductance , and the . Capacitor in Fig. 5 drain-to-source output capacitance must be made large enough to ensure a low impedance to ground and . at both is a tradeoff The choice for the characteristic impedance between obtaining a low dc resistance and a broad operational bandwidth. A low dc resistance demands a large conductor cross section, while bandwidth is maximized by selecting a high char. To determine the characteristic imacteristic impedance for and , the same principle as for the class-E pedances transmission-line load network is used. The parallel connection with the open stub results in of the load (11) For this network, the capacitive susceptance so that by

is made up only

(12) The fundamental class-F loadline given in (3) is transformed toward load by . The admittance at the connection point between and may be written as (13)

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whereas can be calculated by equating the real parts of (11) and (13), leading to (14) and

is obtained by equating the imaginary parts (15)

If the output parasitics and are neglected, and can be selected. However, these device parasitics have to be taken into account at miand need to be corrected crowaves, and the values for accordingly.

Fig. 7. Chip microphotograph of the monolithic integrated low-voltage class-E PA. The chip dimensions are 1 m 1.5 m.

2

V. EXPERIMENTAL VERIFICATION Based on the third harmonic voltage peaking networks and - , MMIC class-E and class-F PAs for operation at 24 GHz have been designed and fabricated using the Fraunhofer IAF P42 GaAs pHEMT process. The active device is a 0.12- m-wide T-gate GaAs pHEMT with a transit of 115 GHz and an off-state drain-to-source frequency of more than 5 V. breakdown voltage Air bridges are used at all junctions and discontinuities to effectively suppress any conversion from the CPW to the unwanted coplanar slotline (CSL) mode. Due to the presence of a metallic plane on the backside of the 625- m-thick GaAs substrate formed by the heat sink, it was required that all unused chip area was covered with metal and connected to ground. In this way, propagation of a parallel-plate mode in this grounded CPW geometry is prevented.

TABLE IV SUMMARY OF SIMULATED AND MEASURED RESULTS FOR THE COMPACT CPW CLASS-E PA

A. Class E Fig. 3 shows the schematic of the implemented -band class-E PA using CPWs. The output network design is a compromise between low insertion loss and the requirements for class-E operation. The outcome of the study on multiharmonic matching networks in Section III shows that no substantial efficiency improvement can be attained by including more than three harmonics. This is due to the high attenuation associated with on-chip transmission lines. Including RF signal and dc pads, as well as a 10-pF on-chip bypassing MIM capacitor, the chip area occupied by the circuit is 1 m 1.5 m. The performance of the fabricated MMIC class-E PA shown in Fig. 7 is summarized in Table IV. The detailed large-signal performance of the -band class-E amplifier can be found in [3]. Input and output matching networks have also been fabricated separately in order to be able to verify the harmonic termination. Fig. 8 shows that the load-coupling network provides a high reactive impedance at the second and third harmonics, as well as the required load impedance for class-E operation at the fundamental frequency. Agreement between simulation and measurements of the output matching network is also shown. The harmonic output measured at 22, 23, and 24 GHz is plotted in Fig. 9. Due to modeling inaccuracy and fabrication

Fig. 8. Comparison of measured (solid line) and simulated (dashed line) smallsignal input reflection coefficient of the separately fabricated class-E load-coupling network. Square markers are placed at 24 GHz, circles at 48 GHz, and triangles at 72 GHz.

tolerances, the peak performance of the circuit is slightly shifted toward lower frequencies. The effects of this can be seen in and is lowest for an input Fig. 9. Harmonic output at signal frequency of 23 GHz, although attenuation is best at 22 GHz. However, both second and third harmonic components are suppressed by more than 30 and 35 dBc, respectively, at the three measurement frequencies and for all measured drive conditions, up to around 4-dB gain compression. B. Class F Fig. 5 shows the schematic of the 24-GHz transmission-line class-F PA. The same RF device as for the class-E amplifier in

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Fig. 9. Measured harmonic output of the CPW class-E PA at 22 GHz (dashed–dotted line), 23 GHz (dotted line), and 24 GHz (solid line). Square markers denote the output at f , circles at 2f , and triangles at 3f .

Fig. 11. Comparison of measured (solid line) and simulated (dashed line) small-signal input reflection coefficient of the separately fabricated class-F load-coupling network. Square markers are placed at 24 GHz, circles at 48 GHz, and triangles at 72 GHz.

Fig. 10. Chip microphotograph of the monolithic integrated low-voltage class-F PA. The chip dimensions are 1.5 m 2 m.

2

TABLE V SUMMARY OF SIMULATED AND MEASURED RESULTS FOR THE COMPACT HIGHLY EFFICIENT CPW CLASS-F AMPLIFIER

Section V-A, i.e., a 480- m-wide GaAs pHEMT, is also used in this design. A chip microphotograph of fabricated coplanar MMIC class-F PA is shown in Fig. 10. The circuit occupies an area of 2 m 1.5 m on the chip. The performance of the fabricated MMIC class-F PA is summarized in Table V. A detailed design description of the circuit has been reported in [4]. The agreement between the designed and fabricated two-stub output matching network is shown in Fig. 11. The implemented load matching network provides high reactance for the third , and the calculated loadline harmonic, low impedance at impedance at the fundamental frequency, as required for class-F operation. The harmonic output measured at 22, 23, and 24 GHz is plotted in Fig. 12. The effect of harmonic termination can be

Fig. 12. Measured harmonic output of the CPW class-F amplifier at 22 GHz (dashed–dotted line), 23 GHz (dotted line), and 24 GHz (solid line). Square markers denote the output at f , circles at 2f , and triangles at 3f .

seen as a function of frequency. Since all line lengths were optimized for operation at 24 GHz, harmonic suppression is more pronounced at this frequency than at 22 and 23 GHz. Due to the of the third harmonic stub high characteristic impedance , its effect on harmonic suppression is more wideband than for the low-impedance stub. VI. CONCLUSION A comprehensive study on the effect of multiharmonic termination on the performance of -band switching-mode and harmonic-controlled PAs has been presented in this paper. The effects of harmonic tuning on output power, efficiency, and harmonic filtering of transmission-line load-coupling circuits suitable for class-E and F approximation were investigated. Both simulations and measurements showed that it is sufficient to consider only a small number of harmonics when designing highly efficient amplifiers in the upper microwave frequency range. Unlike ideal load networks, harmonic suppression has to be considered in practice for transmission-line class-F networks in order to avoid additional filtering sections. Class-E designs

NEGRA et al.: STUDY AND DESIGN OPTIMIZATION OF MULTIHARMONIC TRANSMISSION-LINE LOAD NETWORKS

are not as critical in this regard, as the lengths of the stubs can be kept very close to the ideal values. Analytic design equations have been derived for the load-coupling networks that have been found to provide the best compromise between circuit performance and complexity. The closedform expressions for the two-stub class-E and class-F transmission-line networks facilitate the design procedure and enable optimization of component parameters, while ensuring proper high-efficiency operation. A switching-mode class-E PA and a harmonic-controlled amplifier have been designed at 24 GHz using a GaAs pHEMT technology and CPWs. Using the derived formulas, careful tuning of the circuits in terms of efficiency, as well as harmonic filtering, resulted in state-of-the-art efficiency for low-voltage amplifiers at -band. Without additional filtering, measured levels of the most relevant higher harmonics, i.e., and at the load are better than 25 and 30 dBc for the fabricated MMIC class-F and class-E PAs, respectively. REFERENCES [1] R. Negra and W. Bachtold, “Lumped-element load-network design for class-E power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2684–2690, Jun. 2006. [2] R. Tayrani, “A broadband monolithic -band class-E power amplifier,” in Proc. IEEE RFIC Symp., Seattle, WA, Jun. 2002, pp. 53–56. [3] R. Negra and W. Bachtold, “Switched-mode high-efficiency -band MMIC power amplifier in GaAs pHEMT technology,” in Proc. IEEE Int. Electron Devices for Microw. Optoelectron. Applicat. Symp., Krüger Nat. Park, South Africa, Nov. 2004, pp. 15–18. [4] R. Negra and W. Bachtold, “ -band coplanar class-F MMIC power amplifier in a GaAs pHEMT technology,” in Proc. Int. Microw. Opt. Technol. Symp., Fukuoka, Japan, Aug. 2005, pp. 771–774. [5] S. Pajic, W. Narisi, P. M. Watson, T. K. Quach, and Z. Popovic´ , “ -band two-stage high-efficiency switched-mode power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2899–2907, Sep. 2005. [6] P. Colantonio, F. Giannini, G. Leuzzi, and E. Limiti, “Very high efficiency microwave amplifier: The harmonic manipulation approach,” in Proc. 13th Int. Microw., Radar, Wireless Commun. Conf., Wroclaw, Poland, May 2000, pp. 33–46. [7] A. Wilkinson and J. Everard, “Transmission-line load-network topology for class-E power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1202–1210, Jun. 2001. [8] T. Mader, E. Bryerton, M. Markovic, M. Forman, and Z. Popovic´ , “Switched-mode high-efficiency microwave power amplifiers in a freespace power-combiner array,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 10, pp. 1391–1398, Oct. 1998. [9] M. Reece, C. White, J. Penn, B. Davis, M. Bayne, N. Richardson, W. Thompson, and L. Walker, “A -band class F MMIC amplifier design utilizing adaptable knowledge-based neural network modeling techniques,” in in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 615–618. [10] A. V. Grebennikov, “Circuit design technique for high efficiency class F amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, pp. 771–774. [11] N. Sokal and A. Sokal, “Class E—A new class of high-efficiency tuned single-ended switching power amplifiers,” IEEE J. Solid-State Circuits, vol. 10, no. 3, pp. 168–176, Mar. 1975. [12] F. Raab, “Class-E, class-C, and class-F power amplifiers based upon a finite number of harmonics,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 8, pp. 1462–1468, Aug. 2001.

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[13] R. Osorio, M. Berroth, W. Marsetz, L. Verweyen, M. Demmler, H. Massler, M. Neumann, and M. Schlechtweg, “Analytical charge conservative large signal model for MODFETs validated up to millimeterwave range,” in in IEEE MTT-S Int. Microw. Symp. Dig., 1998, pp. 595–598. [14] S. Kudszus, W. H. Haydl, M. Neumann, and M. Schlechtweg, “94/47-GHz regenerative frequency divider MMIC with low conversion loss,” IEEE J. Solid-State Circuits, vol. 35, no. 9, pp. 1312–1317, Sep. 2000. [15] W. Haydl, A. Tessmann, K. Zuefle, H. Massler, T. Krems, L. Verweyen, and J. Schneider, “Models for coplanar lines and elements over the frequency range 0–120 GHz,” in Proc. 35th Eur. Microw. Conf., Prague, Czech Republic, 1996, pp. 996–1000. Renato Negra (S’06–M’07) received the M.Sc. degree in telematics from the Graz University of Technology, Graz, Austria, in 1999, and the Ph.D. degree in electrical engineering from the Eidgenössische Technische Hochschule (ETH) Zürich, Zürich, Switzerland, in 2006. From 1998 to 2000, he was with Alcatel Space Norway AS (formerly AME Space AS), Horten, Norway, where he was involved in the design and characterization of space-qualified RF equipment. In April 2000, he joined the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich. Since January 2006, he has been a Post-Doctoral Fellow with the iRadio Laboratory, University of Calgary, Calgary, AB, Canada. His research interests are linearization techniques, highly efficient PAs, and advanced wireless transmitter architectures.

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Fadhel M. Ghannouchi (S’84–M’88–SM’93–F’06) received the Ph.D. degree in electrical engineering from the University of Montréal, Montréal, QC, Canada, in 1987. He is currently an iCORE Professor, a Canada Research Chair, and the Director of the iRadio Laboratory, Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. He has held invited positions at several academic and research institutions in Europe, North America, and Japan. He has authored or coauthored over 300 publications. He holds seven patents. His research interests are in the areas of microwave instrumentation, modeling of microwave devices and communications systems, design and linearization of RF amplifiers, and SDR radio systems.

Werner Bächtold (M’71–SM’99–F’00) received the Diploma and Ph.D. degrees in electrical engineering from the Eidgenössische Technische Hochschule (ETH) Zürich, Zürich, Switzerland, in 1964 and 1968, respectively. From 1969 to 1987, he was with the IBM Zürich Research Laboratory, where he was involved with device and circuit design and analysis with GaAs MESFETs, design of logic and memory circuits with Josephson junctions, and semiconductor lasers for digital communication. He has had several assignments with the IBM T. J. Watson Research Center, Yorktown Heights, NY. From December 1987 to March 2005, he was a Professor of electrical engineering with ETH Zürich. He headed the Microwave Electronics Group, Laboratory for Electromagnetic Fields and Microwave Electronics, and was engaged in the design and characterization of MMICs, design and technology of InP HEMT devices and circuits, and microwave photonics.

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Integrated Receiver Based on a High-Order Subharmonic Self-Oscillating Mixer Simone A. Winkler, Student Member, IEEE, Ke Wu, Fellow, IEEE, and Andreas Stelzer, Member, IEEE

Abstract—In this paper, a novel integrated receiver front-end at 5.8 GHz using a high-order subharmonic balanced self-oscillating mixer (SOM) with a high conversion gain of 11.1 dB is presented, to the authors’ knowledge, for the first time. SOMs are excellent choices for a low-cost receiver design as both the oscillating source and the mixer are combined in one single device. Moreover, the presented structure is of high interest for high-end millimeter-wave systems, as subharmonic operation reduces local oscillator (LO) frequency requirements. The balanced concept of the shown circuit offers inherent RF-LO isolation, lower AM noise, and easy suppression of harmonics. In this paper, the design and performance results of the complete receiver structure are presented both with simulation and measured results and a circuit design model is also developed. Index Terms—Balanced structure, harmonics, millimeter wave, oscillator, receiver, self-oscillating mixer (SOM).

I. INTRODUCTION IGH-END millimeter-wave systems such as receiver front-ends demand for low-cost and high-performance designs. Self-oscillating mixers (SOMs) are excellent choices for such a design as both the oscillating source and the mixer are combined in one single device. In addition to the reduction in component number, they offer numerous advantages such as lower power consumption and the possibility of integration into monolithic microwave integrated circuits (MMICs). SOMs have been implemented using both diodes and transistors. However, the capability of obtaining conversion gain and the development of transistors well into the millimeter-wave region has made the latter the mainly used device. The first FET SOM was designed in [1]. A common problem referring to SOM solutions has been the lack of RF-to-local oscillator (LO) isolation emerging by the combination of the two circuit functions into one device. This problem has been addressed in numerous publications, and a common solution to bypass this issue is given by the use of a

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Manuscript received September 30, 2006; revised January 15, 2007. This work was supported in part by the Austrian Academy of Science and the Natural Sciences and Engineering Research Council of Canada under a Discovery Grant. S. A. Winkler is with the Poly-Grames Research Center, École Polytechnique, Montréal, QC, Canada H3T 1J4, and also with the Institute for Communications and Information Engineering, University of Linz, 4040 Linz, Austria (e-mail: [email protected]). K. Wu is with the Poly-Grames Research Center, École Polytechnique, Montréal, QC, Canada H3T 1J4 (e-mail: [email protected]). A. Stelzer is with the Institute for Communications and Information Engineering, University of Linz, 4040 Linz, Austria (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.2007.896775

dual-gate FET, using one port for the oscillation feedback network and injecting the RF input signal at the second port [2]–[4]. Another possibility to eliminate this problem is the use of a balanced circuit structure [5]–[8], which offers inherent RF-LO isolation, as well as a number of other important advantages such as lower AM noise, suppression of unwanted harmonics, etc. Furthermore, the input does not have to be matched both at LO and RF frequencies, which simplifies the input matching requirements. Efforts have been made to integrate SOMs with antennas to form receivers [2], [4], [9]. This allows for building a simple receiver circuit without the use of an input RF balun, as it would be required in conventional balanced mixers. Thus, the presented solution results in a planar design, which is highly desired in many commercial low-cost applications. An extension to the standard SOM technique is subharmonic operation [3]–[6], [10]: instead of mixing the RF signal with the fundamental LO frequency, a harmonic of the LO frequency is used. In this way, transistor gain only has to occur until the fundamental LO frequency, which drastically lowers the transistor requirements [10]. maximum frequency In this paper, the design of a complete receiver front-end using a third-harmonic SOM with high conversion gain is presented, to the authors’ knowledge, for the first time. It is realized by using a balanced structure, which simplifies subharmonic mixing, and furthermore offers the above-stated well-known performance advantages of balanced structures. Receivers as proposed in this study can be used for a large number of applications such as radar systems, radiometry, imaging applications, communication systems, transceiver front-ends, etc. The elimination of the input balun compared to conventional designs saves space and allows for a monolithic-microwave integrated-circuit (MMIC) implementation of the proposed circuit. Furthermore, the presented solution is very suitable for millimeter-wave applications [3], [6], [10] because the requirements for of the transistors are highly reduced. Performance results of this study are presented both in simulation and measurement and a design model is developed that is used to optimize circuit performance. II. SOM DESIGN Here, we introduce the concept of the SOM by separating the design procedure into: 1) the mixer design (see Section II-A); 2) the oscillator configuration (see Section II-B); and 3) the combination of the two designs into an SOM (see Section II-C). The circuit concept of the SOM used in this receiver design is shown in Fig. 1. A single balanced mixer is built up

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WINKLER et al.: INTEGRATED RECEIVER BASED ON HIGH-ORDER SUBHARMONIC SOM

Fig. 1. Balanced SOM concept: the gates of a single balanced gate mixer are connected by a resonating line to form a balanced oscillator. Applying RF power at the center of the line allows for self–oscillating mixing.

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gates, while the LO is out of phase. The input filter serves both as input matching and as a means to improve RF-LO isolation and to enhance oscillation. It is to be noted that the input only has to be matched at RF frequency, which eases the wideband matching requirements at both RF and LO frequencies that are a common problem in conventional mixer designs. An LO short circuit at the drains is required to retain the drain voltage from dropping into the transistor’s linear region. Furthermore, unwanted mixing products, RF leakage, and other spurious responses should be eliminated from the output. In the presented design, these two requirements are fulfilled by broadband stepped impedance output filters. Short-circuit stubs at the sources prevent instability at the gate at RF frequency. B. Oscillator Configuration

Fig. 2. Circuit concept of the subharmonic balanced SOM: for odd-order harmonic mixing, the two outputs are combined by an IF balun, whereas for even–order mixing, the outputs are combined in phase.

by two transconductance gate mixers with their outputs combined either in-phase or out-of-phase for enhancing the desired harmonic output. The two gates of the transistors are then connected by a transmission line, which acts as a resonating element in order to create a balanced oscillator. Injecting the RF signal at the center of this line, we can obtain the desired self-oscillating mixing operation. The circuit concept of the SOM used in this receiver design is shown in Fig. 2. The active devices used in this study are NE34018 FETs. A. Mixer Design Balanced mixers were developed to take advantage of enhancing and suppressing even- and odd-order harmonic products, respectively. The presented work is based on an active transconductance gate mixer design. By combining two such mixers, a single balanced mixer can be created [11]. The main design requirements for a single balanced mixer are the proper phase relations between RF and LO signals at the two gates, as shown in Fig. 1. Subsequently, superposing the IF outputs either in or out of phase suppresses either even- or odd-order harmonic products, respectively, which makes it easy to implement subharmonic operation. Moreover, in this way, conversion gain is increased due to the power combination of the two branches. The RF power is injected via an input filter at the center of the transmission line. This causes the RF to be in phase at the

The oscillating part of the SOM follows a straightforward negative resistance oscillator design [12] and capacitive feedback at the sources introduces negative resistance at the gates. To make the design balanced, the balanced oscillator theory in [13] based on the extended resonance technique [14] is used: a balanced oscillator can be built by creating two equal oscillator branches with negative resistance at the transistors’ gates and letting them resonate with each other through a simple transmission line [6]. In this way, a virtual ground point at LO frequency is created at the center of the line and the oscillation at the transistors’ gates occurs out of phase. It is to be noted that the drain short circuit at LO frequency required for the gate mixer configuration already determines the oscillator load. C. Combination into SOM Finally these two concepts can be combined into a SOM circuit. The proper phase relations of RF and LO signals at the gates are fulfilled by creating oscillation out of phase at the gates, as described above, and by injecting the RF input signal at the center of the transmission line. A balun (an in-phase combination) then rejects the in-phase even-order (out-of-phase odd-order) mixing products. The RF input has to be matched without influencing the LO virtual ground at the injection point, which can be solved by a simple filter structure. In this way, inherent RF-LO isolation at the input is at first guaranteed through the virtual ground property that strongly suppresses the LO signal, and furthermore, by the additional filtering through the matching network. An important advantage of this balanced SOM structure over conventional balanced mixers is the elimination of the input balun/hybrid from the design since the LO signal is already generated out of phase at the gates of the two transistors. III. RECEIVER DESIGN In a further design step, the described SOM circuit was combined with a patch antenna to form a third-harmonic SOM receiver circuit. Due to the inherent RF-LO isolation of the presented SOM structure, we do not need to take particular care of termination impedances at LO frequency for the antenna and a simple design can be used, which presents a good advantage of the novel design.

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Fig. 3. Fabricated antenna used for the third harmonic SOM receiver circuit.

Fig. 6. Output spectrum of the third–harmonic SOM at an RF input of 5.8 GHz and 30 dBm. The oscillation built up to 1.863 GHz, which yields an IF frequency of 160 MHz.

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Fig. 4.

jS j of the antenna used for building the complete receiver.

Simulations have been performed using Agilent ADS 2005A and harmonic balance simulation. A special approach to analyze autonomous circuits has been adapted [15], [16]. A. SOM

Fig. 5. SOM circuit prototype (fabricated in microstrip technology, substrate RT5880 with " = 2:2).

A patch antenna with rectangular shape, shown in Fig. 3, was simulated and measured separately in order to verify its functionality. The input matching is shown in Fig. 4. Simulation and measurement agree very well, and the slight deviation observed in this figure is due to the lack of losses in the simulation. The presented antenna shows a gain of 7.25 dB in the simulation and a 10-dB bandwidth of 4.3% in the measurement.

A separate SOM circuit has been fabricated prior to the design of the complete receiver circuit repeating and optimizing the design in [5]. In this way, performance could be investigated in detail. Compared to the circuit in [5], an optimization of the circuit design parameters and the input matching network has led to an increase of conversion gain to 11.1 dB at an RF input power of 30 dBm. Fig. 6 shows the measured IF output spectrum of the optimized SOM. The input RF signal at 5.8 GHz is mixed with the oscillation signal that builds up at 1.863 GHz and produces an IF signal of 160 MHz. The power consumption of the balanced SOM is 32 mW (16 mW per transistor) at a drain bias voltage of 2 V. This value agrees both in measurement and simulation. Fig. 7 shows the measured conversion gain of the original and the optimized circuit versus simulation results. The main contribution to the simulation of conversion gain is given by the transconductance waveform description of the transistor model used in the simulation [11]. Its inaccurate representation explains the difference between simulated and measured conversion gain. In addition, simulation shows instability behavior and convergence failure at input powers larger than 20 dBm. B. Receiver

IV. EXPERIMENTAL RESULTS The presented circuit topology has been fabricated in microstrip technology on a Rogers substrate RT5880 with a rel. The final prototype is ative dielectric permittivity of shown in Fig. 5. In order to optimize the required LO short circuit at the drain, additional radial stubs have been inserted.

The receiver has been tested with a transmitting patch antenna connected to a signal generator with an RF power of 5 dBm. The distance between the antennas was 1 m. Fig. 8 shows the IF output spectrum for the measurement of the complete receiver circuit. It is to be noted that the spectrum in Fig. 8 compares very well to the standalone SOM result in

WINKLER et al.: INTEGRATED RECEIVER BASED ON HIGH-ORDER SUBHARMONIC SOM

Fig. 7. Conversion gain versus RF input power for simulation and two fabricated circuits.

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Fig. 9. Fourier analysis of the transconductance waveform: the maximum of the third harmonic component lies at around 0.7 V, which complies with our gain performance results and the experimental results (gain maximum is at a different location due to matching).

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of the transistor, the choice of the source feedback, the output terminations at the drain (also realized by the output filter), and the applied input power. For illustration, these parameters are indicated in Fig. 2. A. Harmonic Output

Fig. 8. Output spectrum of the complete receiver circuit. Performance is very similar to the standalone SOM.

Fig. 6, which proves that the SOM performance is not affected by the added antenna. With the above-stated values, the Friis formula gives a theoretical power of 28.2 dBm at the receiver input. Referring this value to the IF power level at the receiver output, a theoretical conversion gain of 7.2 dB is achieved. This value is lower compared to the gain of the standalone SOM due to an antenna gain lower than the simulated value used in the Friis formula. V. RECEIVER PERFORMANCE MODEL Here, a detailed performance investigation of the developed subharmonic SOM receiver has been done. In a SOM, the main performance goals are given by its conversion gain, its IF frequency, noise figure, LO leakage, and its intermodulation distortion (IMD). In the above-described structure, the primary design parameters to achieve these goals are the gate and drain bias voltages

In harmonic mixers, the gate voltage can be chosen in order to maximize the desired harmonic output: a Fourier analysis shows that maximizing the desired harmonic component of the transconductance waveform by the proper choice of the gate voltage enhances the desired harmonic mixing order [17]. Therefore, in the proposed design, the third harmonic component must be maximized. In Fig. 9, a comparison of the third harmonic transconductance waveform component with the simulated and measured conversion gain is shown. It is clear that there is a direct relationship between these two parameters. The maximum of conversion gain lies at 0.6 V due to a conjugate matching at this bias point, which has been chosen in accordance with the oscillator design. B. Conversion Gain Conversion gain is mainly influenced by maximizing the desired harmonic component of the transconductance waveform, as shown in Fig. 9. This is at first ensured by the proper choice of the gate voltage, but it is also very important to provide a drain short circuit at LO frequency and its harmonics in order to maintain the transistor from dropping into the linear region and, therefore, maximizing transconductance variation. Moreover, conversion gain is improved by increased drain bias and by the power combination due to the balancing of the design, as shown in Fig. 10. C. Oscillation Frequency Tuning Oscillation frequency is determined by the source feedback capacitance, choice of gate bias voltage, and the output termination of the transistors.

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Fig. 10. Comparison between the measured single–ended and balanced IF output power. Many undesired spurious responses, harmonics, and the LO leakage can be suppressed, whereas the conversion gain is increased.

Fig. 12. Noise figure compared to conversion gain over the IF bandwidth. A minimum noise figure of 6.9 dB is measured.

Fig. 11. Measured oscillation frequency versus drain bias voltage: oscillation frequency can be tuned linearly in a relative bandwidth of 2.3%.

Fig. 13. Pulling of the oscillation frequency by the RF input signal and the relative frequency shift.

Fig. 11 shows the tunability of the oscillation frequency versus drain bias voltage providing a relative tuning bandwidth of 2.3%. It is to be noted that, in spite of the operation as a mixing and oscillating device at the same time, we can obtain a linear tuning behavior. Tuning is also possible by varying the source capacitance.

Fig. 12 shows the measured noise figure of the circuit over the IF bandwidth. The measurement has been performed with an Agilent noise figure analyzer that allows for simultaneous gain measurement, which is shown in Fig. 12 as well. A minimum noise figure of 6.9 dB for an IF frequency of 140 MHz is measured. This value is very low and practical for almost any application. It is to be noted that the measured conversion gain is slightly lower compared to the above-shown results due to the insertion loss of an additional output filter that is required for the band selection in the noise measurement.

D. Noise Figure In an active FET mixer, the main contribution to noise is given by the high-field diffusion noise at the drain [18]. Therefore, unlike as in an amplifier design, mismatch does not significantly improve noise behavior since thermal noise at the input and output is not high and the device shows lower gain due to a smaller value of transconductance variation. Noise figure optimization in our design is, therefore, given by short–circuit termination of gate and drains at LO frequency and its harmonics. Moreover, the gate should be short circuited at IF frequency and the drain at RF frequency to prevent amplifier–mode gain. At the same time, both ports are conjugately matched to maximize conversion gain. Furthermore, inherent improvement of the noise figure is given by a balanced design, as presented in this paper.

E. Frequency Pulling Contrary to conventional mixer circuits, the presented receiver involves an autonomous oscillation circuit, which makes the LO frequency a variable parameter whose changes can affect circuit performance. The RF input power especially needs to be considered, as it pulls the oscillation frequency and, therefore, can affect the dynamic range of the designed receiver circuit. The measurement in Fig. 13 shows a constant 30 dBm) and a very LO frequency for low input powers ( low pulling effect of maximum 2.8 MHz or 0.15% relative bandwidth for an RF power of 10 dBm, which proves very

WINKLER et al.: INTEGRATED RECEIVER BASED ON HIGH-ORDER SUBHARMONIC SOM

Fig. 14. Conversion gain and LO leakage versus drain capacity: Up to around 10 pF, the LO leakage can be effectively reduced without affecting conversion gain.

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more effective suppression of any higher order unwanted mixing products. Fig. 14 shows the influence of the drain capacitor: at low values, it reduces LO leakage, but leaves the conversion gain unaffected. For optimum values, it can even enhance the gain due to the more effective LO short circuit at the drain. In Fig. 15, the LO leakage versus RF power is shown, and simulation and measurement results agree well. The deviation can be explained by the difference in conversion gain of simulation and measurement. LO leakage for this third harmonic SOM is also drastically decreased by balancing the design, as can be seen from Fig. 10. Combining all these aspects, we can achieve an LO reduction by the drain capacitance of 31 dB and a suppression of 18 dB by balancing the design, which yields a total suppression of 39 dB. VI. CONCLUSION In this paper, a complete receiver at 5.8 GHz using a thirdharmonic SOM is presented. It is based on a balanced structure allowing for easy implementation of the subharmonic operation and offering inherent RF-LO isolation. A very high conversion gain of 11.1 dB and a noise figure of 6.9 dB were achieved. The SOM is integrated into a complete receiver by adding a patch antenna. Moreover, a detailed performance study for providing a good design model has been developed and presented both in simulation and measurement. ACKNOWLEDGMENT

Fig. 15. LO leakage versus RF input power for simulation and fabricated circuit.

The authors wish to acknowledge J. Gauthier and R. Brassard, both with the Poly-Grames Research Center, École Polytechnique, Montréal, QC, Canada, for the fabrication of the circuit. REFERENCES

well the functionality of the presented circuit for practical applications. F. IMD IMD has become increasingly important in microwave mixer design in terms of dynamic range performance. Unlike harmonic and second-order distortion products, third-order IMD products can be very close to the IF signal and cannot be easily eliminated by filtering. Usually in receivers the mixer constitutes the greatest contributor to intermodulation. In subharmonic mixers, the various performance advantages are outweighed by a generally lower third-order IMD. For the designed third-order self–oscillating mixer, we measured an output–referred third-order intercept point (TOI) of 4.5 dB. G. LO Leakage LO leakage is mainly reduced by the output filtering and the LO short-circuit terminations, as described above. However, to further improve LO leakage, a capacity between the two outputs is introduced, which forms a low-pass filter that suppresses the LO signal, but leaves the IF output and, therefore, the conversion gain unaffected. In addition, this technique allows for a

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[11] S. A. Maas, Microwave Mixers, 2nd ed. Norwood, MA: Artech House, 1993. [12] R. W. Rhea, Oscillator Design & Computer Simulation, 2nd ed. New York: McGraw-Hill, 1997. [13] K. S. Ang, M. J. Underhill, and I. D. Robertson, “Balanced monolithic oscillators at - and -band,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 2, pp. 187–193, Feb. 2000. [14] A. Mortazawi and B. C. De Loach, Jr., “Multiple element oscillators utilizing a new power combining technique,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 12, pp. 2397–2402, Dec. 1992. [15] S. ver Hoeye, L. Zurdo, and A. Suarez, “New nonlinear design tools for self-oscillating mixers,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 8, pp. 337–339, Aug. 2001. [16] J. Morales, A. Suarez, E. Artal, and R. Quere, “Global stability analysis of self-oscillating mixers,” in Proc. 25th Eur. Microw. Conf., Bologna, Italy, Sep. 1995, pp. 1216–1219. [17] K. Schmidt von Behren, D. Pienkowski, T. Mueller, M. Tempel, and G. Boeck, “77 GHz harmonic mixer with flip-chip Si-Schottky diode,” in 14th Int. Microw., Radar, Wireless Commun. Conf., 2002, vol. 3, pp. 743–746. [18] S. A. Maas, Noise in Linear and Nonlinear Circuits, 1st ed. Norwood, MA: Artech House, 2005.

K

Ka

Simone A. Winkler (S’05) was born in Mittersill, Austria, on May 4, 1982. She received the Dipl.–Ing. (M.Sc.) degree in mechatronics (with distinction) from the Johannes Kepler Universität Linz, Linz, Austria, in 2005, and is currently working toward the Ph.D. degree in electronics and microwave engineering at the École Polytechnique Montréal, QC, Canada. He doctoral project concerns a self-oscillating mixing technique for millimeter-wave radiometer applications. From July to November 2004, she spent a research term with the University of Waikato, Hamilton, New Zealand. Her main research interests are radar system engineering, active receiver circuits, millimeter-wave radars, and radar signal processing. Ms. Winkler is a student member of the European Microwave Association (EuMA) and the Austrian Electrotechnical Association (ÖVE). She was selected to the first rank throughout Québec for the FQRNT 2007–2008 Merit Scholarship Program. Moreover, she was the recipient of the 2006 Best Paper Award presented at the IEEE Canadian Conference. on Electrical and Computer Engineering (CCECE) and the 2006 Hedy-Lamarr Award for her ongoing doctoral research. She was also the recipient of the 2005 Erwin–Wenzl–Preis, the 2005 GIT–Förderpreis, and the 2005 tech2b Award for her diploma thesis. She was also selected for a long-term doctoral fellowship from the Austrian Academy of Science.

Ke Wu (M’87–SM’92–F’01) is Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering with the École Polytechnique (University of Montréal), Montréal, QC, Canada. He also holds a Cheung Kong endowed chair professorship (visiting) with Southeast University, and an honorary professorship with the Nanjing University of Science and Technology, Nanjing, China, and the City University of Hong Kong, Hong Kong. He has been the Director of the Poly-Grames Research Center. He has authored or coauthored over 530 referred papers and several books/book chapters. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers and sensors. He is also interested in the modeling and design of microwave photonic circuits and systems. Dr. Wu is a Fellow of the Canadian Academy of Engineering (CAE) and the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He is a member of Electromagnetics Academy, Sigma Xi, and the URSI. He has held key positions in and has served on various panels and international committees including the chair of Technical Program Committees, International Steering Committees, and international conferences/symposia. He has served on the Editorial/Review Boards of many technical journals, transactions, and letters including being an editor and guest editor. He is currently the chair of the joint IEEE Chapters of Microwave Theory and Techniques Society (MTT-S)/Antennas and Propagation Society (AP-S)/Lasers and Electro-Optics Society (LEOS), Montréal, QC, Canada. He is an elected IEEE MTT-S Administrative Committee (AdCom) member for 2006–2009 and serves as the chair of the IEEE MTT-S Transnational Committee. He was the recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award.

Andreas Stelzer (M’00) was born in Haslach an der Mühl, Austria, in 1968. He received the Diploma Engineer degree in electrical engineering from the Technical University of Vienna, Vienna, Austria, in 1994, and the Dr.techn. degree (Ph.D.) in mechatronics (with honors sub auspiciis praesidentis rei publicae) from Johannes Kepler University Linz, Linz, Austria, in 2000. Since 2000, he has been with the Institute for Communications and Information Engineering, University of Linz, Linz, Austria, and became an Associate Professor in 2003. Since 2003, he has been a key Researcher with the Linz Center of Competence in Mechatronics, where he is responsible for numerous industrial projects. Since 2007, he is Head of the Christian Doppler Laboratory for Integrated Radar Sensors, University of Linz. He has authored or coauthored over 150 journal and conference papers. His research focuses on microwave sensor systems for industrial applications, RF and microwave subsystems, ultra-wideband technology, surface acoustic wave (SAW) sensor systems and applications, as well as digital signal processing for sensor signal evaluation. Dr. Stelzer is a member of the Austrian Electrotechnical Association (ÖVE). He has been a reviewer for international journals and conferences. He has served as an associate editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He was the recipient of several awards including the Electrical and Electronic Engineering for Communication (EEEfCOM) Innovation Award and the European Microwave Association (EuMA) Radar Prize at the European Radar Conference.

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Global Modeling Analysis of HEMTs by the Spectral Balance Technique Giorgio Leuzzi and Vincenzo Stornelli

Abstract—A global physical/electromagnetic high electronmobility transistor (HEMT) simulation approach, entirely in the frequency domain, is here described for microwave computeraided design applications. The frequency-domain spectral balance technique for the solution of steady-state nonlinear differential equations is applied to the moments of Boltzmann’s transport equation for the analysis of the intrinsic active part of the device, yielding a very simple formulation. A numerical electromagnetic solver in the frequency domain is used for the analysis of the extrinsic passive embedding and access structure. The two analyzes are coupled, and give a self-consistent global description of the device. The frequency-domain formulation allows easy inclusion of frequency-dependent parameters of the semiconductor, and a natural extension to multitone analysis, without the need for cumbersome time-frequency transformations. The technique is applied to a quasi-2-D hydrodynamic modeling of the active device for simplicity, but is suitable for more comprehensive approaches as well. DC and small-signal microwave results up to 40 GHz are obtained for a 0.3- m gate-length AlGaAs–InGaAs–GaAs pseudomorphic HEMT transistor, and compared to experimental data. Index Terms—Electron device modeling, frequency-domain analysis, global modeling, monolithic microwave integrated circuit, physic-based analysis.

I. INTRODUCTION

H

IGH-FREQUENCY devices and circuits are sensitive to electromagnetic coupling, process variation, and propagation effects [1]. Therefore, a substantial effort has lately been devoted to the development of numerical techniques that take into account the physical and electromagnetic effects within an active device or circuit. This approach is often referred to as “global modeling” [2]–[6]. The active and passive parts of the devices are described by physical a priori models, and then coupled for consistency. This approach allows the prediction of the performances of the device prior to fabrication, and a preliminary optimization of its structure. At circuit level, this approach allows prediction of coupling between different devices or elements, and a better chance of first-pass success with high performances. However, a considerable amount of memory and CPU time is required, and this extensive load on the computational side is a considerable drawback. Currently, global simulation algorithms have not yet reached the stage of practical applicability to circuit design; nonetheless, several commercial comManuscript received October 17, 2006; revised January 12, 2007. The authors are with the Department of Electrical Engineering, University of L’Aquila, L’Aquila 67100, Italy (e-mail: [email protected]; stornelli@ing. univaq.it). Digital Object Identifier 10.1109/TMTT.2007.895233

puter-aided design (CAD) developers are beginning to integrate the physical and electromagnetic approaches, even though with many limitations. A wise choice of the basic algorithms can considerably ease the task for future applications. Simulation in the time domain is a popular technique for both charge transport analysis in the semiconductor and electromagnetic field analysis in the passive structures. However, it has some drawbacks: first of all, the analysis of circuits or devices with different time constants requires a long analysis time before the steady state is reached. This problem becomes more and more critical when a two-tone or multitone excitation is applied. Secondly, the dispersive behavior of some parameters in high-frequency structures (e.g., the dielectric permittivity of the semiconductor) is not efficiently taken into account in the time domain. Another popular approach makes use of the harmonic balance (HB) technique [7], where the passive embedding circuit is analyzed in the frequency domain, then coupled to the time-domain analysis of the active nonlinear device by means of time-domain transformations. Assuming a steady-state behavior with limited bandwidth, the equations can then be solved either in the sampled time or discrete frequency domain. The HB method has several advantages that have determined its success: first of all, the steady-state solution is directly obtained with a single analysis, whatever the time constants present in the device/circuit. Secondly, the linear part of the structure is reduced to an equivalent network with a frequency-domain representation (e.g., an -parameter matrix), thus minimizing the number of nodes where the balancing equations are written. Moreover, the evaluation in the frequency domain is very practical for many microwave components, both lumped and distributed. There are, however, also drawbacks: first of all, memory requirements increase substantially. Secondly, it is difficult to detect instabilities of the circuit at spurious frequencies. Also, taking into account the frequency dispersive behavior of the nonlinear device requires a convolution, as a consequence of the time-domain analysis of the nonlinear sub-network. Moreover, analysis with multitone excitations is not very practical. In order to overcome at least some of these problems, a full frequency-domain approach can be adopted also for the nonlinear part of the device or circuit. To this goal, the so-called spectral balance (SB) technique has been developed in the past and applied to nonlinear microwave circuit analysis [9]. The approach is similar to that of the HB; however, a basic requirement for the SB technique is that the nonlinear relations among the unknown quantities must be in the form of power series or rational functions. If this is the case, the time-domain products between quantities in the nonlinear device become frequency-

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iterative method. Given the assumption of periodic regime, the variables at the connecting nodes are expanded in Fourier series as follows: (1) Fig. 1. Partitioning of the nonlinear system.

domain convolutions among quantities with discrete spectra, which can be directly evaluated in the frequency domain [10]. In this way, the need for time-frequency transformations is removed, and a complete frequency-domain analysis is achieved. The requirement of a polynomial or rational-function form of the nonlinearities is a real limitation for circuit analysis applications with empirical models relating voltages and currents in an active device. For good accuracy, empirical models usually adopt transcendental functions as the exponential or hyperbolic tangent. Contrariwise, transport equations in the semiconductor only include rational functions of the physical quantities (charge density, particle momentum and energy, electrical potential, etc.), and therefore, are natural candidates for the SB scheme [12]. In this paper, the SB technique is applied to the quasi-2-D (Q2-D) hydrodynamic physical analysis of a high electron-mobility transistor (HEMT), and coupled to a frequency-domain electromagnetic (EM) field analysis of the passive embedding structure. Within the active device, the electron transport in the channel is assumed to take place essentially in the direction from source to drain, while the channel charge control is due to the gate–channel voltage. The electron transport is modeled by the first three moments of the Boltzmann’s transport equation (BTE), i.e., the charge, momentum, and energy conservation equations, in one dimension [15]–[20]. Poisson’s equation in the propagation direction is also coupled to the transport equations. The charge in the channel is computed by the a priori self-consistent solution of Schrödinger’s and Poisson’s equations in the vertical direction for all gate–channel voltages; the results are then fitted by a polynomial or by a rational function of the gate–channel voltage, and coupled to the transport equations. This arrangement accounts for the device layer structure and material composition, and allows application of this model to several types of field-effect transistors (FETs). A code for a PC has been developed for the physical analysis, and then coupled to a commercial frequency-domain EM simulator for the passive part. Experimental data at dc and smallsignal microwave frequencies for a 0.3- m pseudomorphic high electron-mobility transistor (pHEMT) manufactured by Selex have been compared to the simulations, confirming the capabilities of the proposed approach. II. SB TECHNIQUE The general frame of the SB technique is similar to that of the HB, both based on the assumption of periodic steady-state regime. The device or circuit is first divided into linear and nonlinear parts, and connected at a given number of nodes, as in Fig. 1. The two parts are analyzed separately, and then the quantities at the connecting ports are matched by means of an

where the variable can be any physical quantity in the semiconductor, as the charge density, velocity or energy, or the potential at a given point in space. The fundamental angular freis expressed as in (2), and the coefficients are quency complex phasors that, due to the assumption of real signals in time, are Hermitian, as reported in (3) as follows: (2) (3) The coefficients of the Fourier series expansion are expressed as (4) If we assume that the first few harmonics are sufficient to describe the behavior of the electrical and physical quantities or, in other words, if we assume that the signal has a limited bandwidth (or a limited number of harmonics), (5) we, therefore, can truncate the Fourier series expansion after terms, obtaining a finite number of equations (and harmonics). In this case, only a limited number of samples in time is required to evaluate the coefficients of the Fourier series, according to Nykvist’s sampling theorem. We can then rewrite the variable in (1) as (6) and express the coefficients as (7) with (8) and finally, replacing (8) in (7), we obtain (9) The key point in the SB method is that the Fourier series expansion of the product of two variables has an explicit Fourier series expansion representation

(10)

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equations for the layers between the gate and channel. The passive embedding structure (pads, metallizations, substrate, etc.) is analyzed by a means of a numerical frequency-domain commercial EM solver, and connected to the gate, source, and drain electrodes. The 1-D hydrodynamic electron transport in the channel is then analyzed by the SB technique, including the charge-control model and the effect of the embedding structure. A. Channel Charge Control

Fig. 2. pHEMT cross-sectional geometry.

where the tion of the

coefficients are computed as a discrete convoluand coefficients in the frequency domain (11)

If the nonlinear functions of the variables in the system equations include only multiplications and divisions, i.e., only polynomials or rational functions, then the equations can be expressed explicitly as polynomial or rational functions of the unknown coefficients of the Fourier series expansion of the variables. For the sake of illustration, let us refer to Fig. 1, where there are nodes connecting the linear and nonlinear subsystems. We can truncate the Fourier series expansion of each variable at each terms. We must then equate the quanconnecting node after tities at the nodes (e.g., equate the currents at the connecting nodes, as for the Kirchhof Current Law); this implies equating the phasors at each harmonic frequency separately. Therefore, complex equations, one for each we obtain a set of harmonic frequency, plus real equations for the dc

The HEMT charge-control model [13], [14] relates the channel electron density to the physical parameters used in the device simulation and to the channel-to-surface potential. Several values of the channel-to-surface potentials are simulated, and the resulting charge densities are stored in a lookup table (LUT). The charge-control model implemented here, which includes DX centers and trapping effects, is obtained by iteratively and self-consistently solving the Schrödinger’s and Poisson’s equations (15) and (16) as follows:

(13)

(14) is the effective mass, and are the energy where level and the wave function of the th subband, respectively ( th eigenvalue and eigenfunction of Schrödinger’s equation), is the electrostatic potential through the semiconductor, is the position-dependent dielectric constant, and is the ionized donor density. The electron concentration in Poisson’s equation is computed from the wave function (15)

(12) Equation (12) is the nonlinear complex equation system in the unknown voltage phasors that yields the solution to our nonlinear problem. The system is solved numerically by an iterative method, where the values of the voltage phasors are first estimated, and then iteratively corrected until the error is negligible. In this approach, the inversion of the Jacobian matrix is a computationally heavy step of the algorithm, especially when the size of the matrix is large. Several numerical procedures are used for the reduction of the computational burden of this step [8]. In our case, we have used an LU factorization algorithm. III. PHYSICAL/EM MODEL DESCRIPTION In this study, the SB technique is applied to a Q2-D model of the transport within the semiconductor, coupled to a numerical EM analysis of the passive embedding structure. A pHEMT device has been used as application, but the technique can be easily extended to other devices and modeling approaches. Fig. 2 shows the vertical cross section of the pHEMT considered for this study. The channel charge-control model in the vertical dimension is found by solving Schrödinger’s and Poisson’s

is a 3-D density of states. In general, this is comwhere puted by integration of the Fermi–Dirac distribution function, but this cannot be done in closed form in this case. Since a numerical integration would be too computationally intensive, approximations are commonly used [11]. We have used Boltzmann statistics, making use of the Unger approximation for mild degeneracy and of the Sommerfeld approximation for deep degeneracy. Dopant ionization is also taken into account. The conduction band edge at the gate electrode is determined by the sum of the Schottky barrier height and the gate voltage. The wave functions at both boundaries are fixed to zero since the high barrier at the heterointerface strictly bounds electrons within the channel. The unknowns are the electrostatic potential and the electron concentration/wave functions. The equations are solved as follows: in the th step, the solution of Schrödinger’s equation (13) , using the gives the energy level and the wave function from the th step (or an initial electrostatic potential guess for the first step). Poisson’s equation (14) is then solved, , which together with (15), giving the electrostatic potential

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will be used in the th step to obtain the new values of and . The step is repeated until convergence is reached. The system is solved by a classical finite-difference time-domain method, and the results are stored in a LUT, and then fitted by a Lagrange polynomial. The need for polynomial interpolation stems from the requirements of the SB technique, but is not difficult or critical. B. Electromagnetic Analysis The EM field distribution in the embedding structure is found by solving the Maxwell’s equations in the frequency domain by a commercial CAD program. The program is the AWR EM-Sight that uses the Galerkin method of moments in the spectral domain. The embedding passive structure is thus taken into account in the analysis, removing the need for equivalent lumped parasitic extraction. Results in the form of admittance parameters at the interconnecting nodes on the electrodes are then coupled to the physicbased analysis in the channel. In our structure, the fingers of the HEMT are divided into several elementary slices. In each slice the transport phenomena in the channel are connected to the embedding structure via the admittance parameters at the connecting nodes on the electrodes. The admittance parameters of the embedding structure connect the different slices among them, taking into account the propagation effects along the gate fingers, the fringing fields at the electrodes, etc. In HEMT devices the thickness of the active layer is usually much smaller (less than 0.1 m) than the total thickness of the device (on the order of tens to hundreds of micrometers). This assumption allows to consider the active region as an infinitely thin layer for the EM field solver.

(18) where is the electron density, is the electron velocity, is the longitudinal electric field, is the can be electron energy, and the relaxation terms as well as obtained by steady-state Monte Carlo analysis. These equations are then coupled to Poisson’s equation in the -direction [see (19)] between source and drain (19) is the effective channel charge density and is provided by the channel charge-control model, as explained above. All physical quantities in the channel are discretized in the space domain with a sufficiently fine step (in the order of 10 nm); at each space point, the quantities are expressed as Fourier series in the time variable, truncated to a suitable maximum harmonic component, as illustrated in Section II. The external applied potential (the driving signal) appears as a boundary condition to Poisson’s equation between source and drain and to the vertical charge-control equation between gate and channel, together with the admittance parameters, as a Norton equivalent. The applied potential is also expressed in the frequency domain, i.e., by the amplitude of the phasors at the harmonic frequencies. where

IV. ALGORITHM DESCRIPTION

C. Active Layer Transport Model The basic assumption in our model is that the current flows from source to drain, parallel to the heterointerface in the 2-D electron gas of the pHEMT. Carrier dynamics are assumed to be adequately described by the BTE [15]–[22], or rather, by its first three moments over the momentum space; the assumption is also made that the exchange of momentum and energy in the electron gas due to collision terms can be approximated by a relaxation time model [23], [24]. The model is, therefore, that of a single equivalent electron gas, with “hydrodynamic” particle, momentum, and energy conservation [see (16)–(18)]. • Continuity equation (16) •

• Energy conservation

-momentum conservation

(17)

The outline of the implemented simulation process is as follows. -direction Schrödinger–Poisson 1) The self-consistent system is solved for different bias conditions, and the channel charge density is expressed by a polynomial approximating function of the actual potential between gate and source ( ). 2) The EM analysis of the extrinsic part of the device and the embedding structure is evaluated, and the results stored in the form of a -matrix connecting the spatial points along the active channel and at the source and drain terminals. 3) The charge-control law is coupled to the active channel transport model in the -direction by replacing the interpolated charge dependence on gate–channel voltage into (19). The EM analysis is also taken into account in (19) for each discretized spatial point and at the drain and source boundary points. 4) The resulting nonlinear system of equations is solved iteratively. The external dc and microwave characteristics are obtained by first computing voltages and currents at the electrodes, then computing the corresponding quantities at the external contacts by means of the admittance matrix of the embedding structure, and finally transforming them into -parameters for ease of presentation.

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TABLE I pHEMT LAYER STRUCTURE

Fig. 4. Total charge density calculated for a gate bias voltage V

Fig. 3. Band energy diagram calculated for a gate bias voltage V

= 00:1 V.

= 00:1 V.

As said above, each quantity in (16)–(19) is expanded in Fourier series. Space derivatives are discretized by means of finite central differences, while the time derivatives are expressed as the derivative of the Fourier series expansions. The frequency-dependent behavior of the dielectric constant has been approximated by means of a polynomial, and therefore, easily taken into account in the algorithm. For the sake of brevity, only (16) is shown in its frequency domain form

Fig. 5. Envelope of the first two wave functions (1: dashed line, 2: solid line) calculated for a gate bias voltage V = 0:1 V.

0

(20) By expanding the product terms and balancing each harmonic separately, we obtain the nonlinear equation system (21). The real equations complete system of is solved by means of a standard Newton–Raphson iterative algorithm. It is important to remark that a suitable arrangement of the variables ensures a convenient structure of the Jacobian matrix, yielding reduced computation time and easier convergence

Fig. 6. Drain current for V = 3 V and different gate bias voltages. Solid line: measured. Dashed line: simulated.

V. RESULTS

(21)

The proposed technique has been used to analyze a multifinger pHEMT device manufactured by Selex S.I., Rome, Italy. The layer structure is reported in Table I. The device has two gate fingers with a gate length of 0.3 m and a gatewidth of 25 m each.

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Fig. 7. Electron energy along the device channel as a function of space and time in dc conditions. (left) Time-domain. (right) SB analysis.

Fig. 8. Electron velocity as a function of space and time with applied RF drain signal and gate and drain dc bias. (left) Time-domain. (right) SB analysis.

A. DC Results By solving the Schrödinger–Poisson system, the conduction band profile (Fig. 3), charge density (Fig. 4), and the envelope wave function behavior (Fig. 5) have been evaluated for the device for gate–channel voltage values from 1 to 0.1 V. This calculation also allows the subband energy levels and their fractional occupation to be evaluated. As a verification, the dc device characteristics have been computed. Fig. 6 shows the simulated transcharacteristic compared to the measured one for the considered pHEMT: the agreement is good. In Fig. 7, the electron energy distribution along the channel is shown as a function of space and time. A classical time-domain result, implemented with a standard finite-difference time-domain (FDTD) method and showing the response to a voltage step, is shown on the left. The SB analysis is shown for comparison on the right: the steady-state dc solution is directly obtained without the transient necessary for the time-domain analysis. B. RF Results In Fig. 8, the electron velocity is shown for an applied RF signal over gate and drain dc bias, with a time-domain analysis (left) and the SB algorithm (right). It is apparent that the SB approach yields directly the steady-state solution without the transient. From the time-domain analysis, the initial velocity overshoot is clearly shown before the energy builds up to the steady-state value.

Small-signal RF parameters have been evaluated by driving the gate or drain port with a sinusoidal source (with 50impedance) superimposed on the dc bias and terminating the other port with a 50- resistance plus the dc bias, as in (21) (22) The admittance parameters at angular frequency computed from two analyses

are then

(23)

(24) Small-signal scattering parameters are then obtained by applying standard - to -parameter transformations. Results over the 4–39-GHz frequency range are shown in Fig. 9 for the considered pHEMT device, where they are compared to the measured ones at different bias points, with good agreement. The scattering parameters are obtained from on-wafer measurements with Microprobe on-wafer probes, an HP8510 vector network analyzer, and short-open-load-thru (SOLT) on-wafer

LEUZZI AND STORNELLI: GLOBAL MODELING ANALYSIS OF HEMTs BY SB TECHNIQUE

S

Fig. 9. Comparison between simulated and measured -parameters for the considered pHEMT for range is 4–39 GHz in 7-GHz steps.

Fig. 10. Simulated and Measured

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V

= 3 V and three different gate bias voltage. The frequency

at the three different bias point.

calibration. In Fig. 10, comparisons between simulated and behavior are illustrated for all the accounted measured bias point. The results illustrate the ability of the model to predict the dependence of small-signal microwave parameters on bias voltages.

VI. CONCLUSIONS In this study, the feasibility of the application of the SB technique to the physical/electromagnetic analysis of pHEMTs has been demonstrated. This approach allows the analysis in the frequency domain of the intrinsic active part of the device cou-

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pled to the passive embedding environment including substrate, pads, metallizations, and access structures. The frequency-domain approach proves especially valuable at frequencies and for device geometries for which the propagation effects and EM field interactions in the embedding passive structure become important. These effects become more severe when approaching the millimeter-wave frequency region. The frequency dispersive behavior of the device parameters can also be easily accounted for. The frequency-domain approach also allows a relatively straightforward extension to the analysis with multitone excitations. The method allows the simulation of the dc and microwave characteristic of the pHEMT starting from the knowledge of its physical and geometric parameters, allowing the designer to predict the behavior of the device and potentially optimizing its performance with limited computer-time consumption. In order to validate this technique, comparisons between experimental and simulated dc and small-signal -parameters results up to 40 GHz have been presented for a pHEMT device manufactured by Selex, confirming the feasibility of the approach. REFERENCES [1] M. Alsunaidi, S. Sohel Imtiaz, and S. El-Ghazaly, “Electromagnetic wave effects on microwave transistors using a full-wave time domain model,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 6, pp. 799–807, Jun. 1996. [2] R. Grondin, S. El-Ghazaly, and S. Goodnick, “A review of global modelling of charge transport in semiconductors and full-wave electromagnetics,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 1244–1253, Jun. 1999. [3] A. Witzig, C. Schuster, P. Regli, and W. Fichtner, “Global modeling of microwave applications by combining the FDTD method and a general semiconductor device and circuit simulator,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 919–928, Jun. 1999. [4] A. Cidronali, G. Leuzzi, G. Manes, and F. Giannini, “Physical/electromagnetic pHEMT modeling,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 830–838, Mar. 2003. [5] A. Cidronali, G. Collodi, G. Leuzzi, and G. Manes, “Numerical analysis of a 0.2 m AlGaAsnGaAs HEMT including electromagnetic effects,” in Proc. IEEE Int. Compound Semiconduct. Symp., San Diego, CA, Sep. 7–11, 1997, pp. 635–638. [6] I. Witzig, C. Schuster, P. Regli, and W. Fichtner, “Global modeling of microwave applications by combining the FDTD method and a general semiconductor device and circuit simulator,” IEEE Trans. Microw. Theory Tech., vol. . 47, no. 6, pp. 919–928, Jun. 1999. [7] R. W. Dutton et al., “Device simulation for RF applications,” in Proc. Int. Electron Devices Meeting, Dec. 1997, pp. 301–304. [8] B. Troyanovsky, “Frequency domain algorithms for simulating large signal distortion in semiconductor devices,” Ph.D. dissertation, Dept. Elect. Eng., Stanford Univ., Stanford, CA, 1997 [Online]. Available: www-tcad.stanford.edu/tcad/pubs/theses/boris_thesis.pdf [9] C. Chang, M. Steer, and G. Rhyne, “Frequency-domain spectral balance using the arithmetic operator method,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 11, pp. 1681–1688, Nov. 1989. [10] G. W. Rhyne, M. B. Steer, and B. D. Bates, “Frequency-domain nonlinear circuit analysis using generalized power series,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 2, pp. 379–387, Feb. 1988. [11] I.-H. Tan, G. L. Snider, and E. L. Hu, “A self-consistent solution of Schrödinger–Poisson equations using a nonuniform mesh,” J. Appl. Phys., vol. 68, pp. 4071–4071, Oct. 1990. [12] G. Leuzzi and V. Stornelli, “Frequency-domain physics-based analysis of semiconductor devices by a spectral-balance approach,” in Proc. Eur. Microw. Week, Manchester , U.K., Sep. 2006, pp. 410–413. [13] P. Lugli, A. Neviani, and M. Saraniti, “Physical models for heterostructure FET simulation,” J. Appl. Phys., vol. 1.1, no. 4, pp. 447–456, Aug. 1990. [14] T. Wang and C. H. Hsieh, “Numerical analysis of nonequilibrium electron transport in AlGaAs/InGaAs/GaAs pseudomorphic MODFET’s,” IEEE Trans. Electron Devices, vol. 37, no. 9, pp. 1930–1938, Sep. 1990.

[15] F. Giannini, G. Leuzzi, M. Kopanski, and G. Salmer, “Large-signal analysis of Q2-D physical model of MESFETs,” Electron. Lett., vol. 29, no. 21, pp. 1891–1893, Oct. 1993. [16] P. A. Sandborn, J. R. East, and G. I. Haddad, “Quasi two-dimensional modeling of GaAs MESFET’s,” IEEE Trans. Electron Devices, vol. ED-34, no. 5, pp. 981–985, May 1987. [17] R. Drury and C. M. Snowden, “A quasi-two-dimensional HEMT model for microwave CAD applications,” IEEE Trans. Electron Devices, vol. 42, no. 6, pp. 1026–1032, Jun. 1995. [18] R. Singh and C. M. Snowden, “A quasi-two-dimensional HEMT model for DC and microwave simulation,” IEEE Trans. Electron Devices, vol. 45, no. 6, pp. 1165–1169, Jun. 1998. [19] T. Shawki, G. Salmer, and O. El-Sayed, “MODFET 2-D hydrodynamic energy modelling: Optimization of subquarter-micron-gate structures,” IEEE Trans. Electron Devices, vol. 37, no. 1, pp. 21–30, Jan. 1990. [20] R. Singh and C. M. Snowden, “Small-signal characterization of microwave and millimeter-wave HEMT’s based on a physical model,” IEEE Trans. Microw. Theory Tech, vol. 44, no. 1, pp. 114–121, Jan. 1996. [21] Y. Hussein, A. El-Ghazaly, and S. Goodnick, “Numerical technique for modeling and optimization of high-frequency multifinger transistors,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2334–2346, Dec. 2003. [22] A. Scholze, A. Schenk, and W. Fichtner, “Single-electron device simulation,” IEEE Trans. Electron Devices., vol. 47, no. 10, pp. 1811–1818, Oct. 2000. [23] B. Camez, A. Cappy, A. Kaszynski, E. Constant, and G. Salmer, “Modeling of a submicrometer gate field-effect transistors including effects of nonstationary electron dynamics,” J. Appl. Phys., vol. 51, pp. 784–790, Jan. 1980. [24] P. Sandborn, A. Rao, and P. Blakey, “An assessment of approximate nonstationary charge transport models used for GaAs device modeling,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 7, pp. 817–829, Jul. 1989.

Giorgio Leuzzi received the Electronic Engineering degree (cum laude) from the University of Roma “La Sapienza,” Rome, Italy, in 1982. His thesis concerned transmission lines for microwave integrated circuits. Following military service, he became a Teaching and Research Assistant with the University of Rome “Tor Vergata,” Rome, Italy, where, since 1991, he has taught microwave electronics. In 1990, he spent several months with the Centre Hyperfréquences et Semiconducteurs, University of Lille, Lille, France. In 1998, he became an Associate Professor and, in 2001, a Full Professor of electronic devices with the University of L’Aquila, Monteluco di Roio, L’Aquila, Italy. He regularly teaches international summer courses of the ITSS and has regular collaborations with corporations such as Alenia Marconi Systems and Agilent Technologies. His research activities involve the simulation of semiconductor devices for microwaves and millimeter waves, linear and nonlinear characterization of microwave and millimeter-wave active devices, and development of methods and algorithms for microwave nonlinear circuit design. He participated in several research projects, both Italian (CNR MADESS, MURST Microelettronica, and ASI) and international (ESPRIT Projects COSMIC, MANPOWER, FUSE, EDGE, and ESA).

Vincenzo Stornelli was born in Avezzano (AQ), Italy. He received the Electronic Engineering degree (cum laude) from the University of L’Aquila, L’Aquila, Italy, in 2004. In October 2004, he joined the Department of Electronic Engineering, University of L’Aquila, where he is currently involved with problems concerning physics-based simulation, computer-aided design (CAD) modeling characterization and design analysis of active microwave components, circuits, and subsystems projects, and design of integrated circuits for RF and sensor applications. He regularly teaches courses in the European computer patent. He has had professional collaborations with national corporations such as Thales Italia. His research interests include several topics in computational electromagnetic, including microwave antennas analysis for outdoor ultra-wideband (UWB) applications.

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Digital Object Identifier 10.1109/TMTT.2007.901545

Digital Object Identifier 10.1109/TMTT.2007.901546

Digital Object Identifier 10.1109/TMTT.2007.901547

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S. Kang P. Kangaslahtii V. S. Kaper B. Karasik N. Karmakar A. Karwowski T. Kashiwa L. Katehi H. Kato K. Katoh A. Katz R. Kaul R. Kaunisto T. Kawai K. Kawakami A. Kawalec T. Kawanishi S. Kawasaki H. Kayano M. Kazimierczuk R. Keam S. Kee L. C. Kempel P. Kenington A. Kerr A. Khalil A. Khanifar A. Khanna F. Kharabi R. Khazaka J. Kiang J. F. Kiang Y. W. Kiang B. Kim C. S. Kim D. I. Kim H. Kim H. T. Kim I. Kim J. H. Kim J. P. Kim M. Kim W. Kim S. Kimura N. Kinayman A. Kirilenko V. Kisel M. Kishihara A. Kishk T. Kitamura K. I. Kitayama T. Kitazawa T. Kitoh M. Kivikoski G. Kiziltas D. M. Klymyshyn R. Knochel L. Knockaert Y. Kogami T. Kolding B. Kolundzija J. Komiak G. Kompa A. Konczykowska H. Kondoh Y. Konishi B. Kopp K. Kornegay T. Kosmanis P. Kosmas Y. Kotsuka A. Kozyrev N. Kriplani K. Krishnamurthy V. Krishnamurthy C. Krowne V. Krozer J. Krupka W. Kruppa D. Kryger R. S. Kshetrimayum H. Ku H. Kubo A. Kucar A. Kucharski W. B. Kuhn T. Kuki A. Kumar M. Kumar C. Kuo J. T. Kuo H. Kurebayashi K. Kuroda D. Kuylenstierna M. Kuzuhara Y. Kwon G. Kyriacou P. Lampariello M. Lancaster L. Langley U. Langmann Z. Lao G. Lapin L. Larson J. Laskar M. Latrach C. L. Lau A. Lauer J. P. Laurent D. Lautru P. Lavrador G. Lazzi B. H. Lee C. H. Lee D. Y. Lee J. Lee J. F. Lee J. H. Lee J. W. Lee R. Lee S. Lee S. G. Lee S. T. Lee S. Y. Lee T. Lee T. C. Lee D. M. Leenaerts Z. Lei G. Leizerovich Y. C. Leong R. Leoni P. Leuchtmann G. Leuzzi A. Leven B. Levitas R. Levy G. I. Lewis H. J. Li L. W. Li X. Li Y. Li H. X. Lian C. K. Liao M. Liberti E. Lier L. Ligthart S. T. Lim E. Limiti C. Lin F. Lin H. H. Lin

J. Lin K. Y. Lin T. H. Lin W. Lin Y. S. Lin E. Lind L. Lind L. F. Lind D. Linkhart P. Linnér D. Linton A. Lipparini D. Lippens V. Litvinov A. S. Liu C. Liu J. Liu J. C. Liu Q. H. Liu S. I. Liu T. Liu T. P. Liu O. Llopis D. Lo J. LoVetri N. Lopez Z. Lou M. Lourdiane G. Lovat D. Lovelace H. C. Lu K. Lu L. H. Lu S. S. Lu Y. Lu V. Lubecke S. Lucyszyn R. Luebbers N. Luhmann A. Lukanen M. Lukic A. D. Lustrac J. F. Luy C. Lyons G. Lyons G. C. M H. Ma J. G. Ma Z. Ma P. Maagt S. Maas G. Macchiarella P. Macchiarella J. Machac M. Madihian A. Madjar V. Madrangeas A. Maestrini G. Magerl S. L. Mageur A. A. Mahmoud S. Mahmoud F. Maiwald A. H. Majedi M. Makimoto S. Makino J. Malherbe G. Manara R. Manas G. Manes T. Maniwa R. Mansour D. Manstretta J. Mao S. G. Mao A. Margomenos R. Marques G. Marrocco J. Martel J. Martens J. Marti G. Martin E. Martinez K. Maruhashi J. E. Marzo H. Masallaei N. Masatoshi D. Masotti G. D. Massa B. Matinpour T. Matsui A. Matsushima S. Matsuzawa H. Matt G. Matthaei L. Maurer J. Mayock J. Mazierska S. Mazumder G. Mazzarella K. McCarthy G. McDonald R. McMillan D. McNamara D. McQuiddy F. Medina C. Melanie A. Á. Melcon F. Mena C. C. Meng H. K. Meng W. Menzel F. Mesa A. C. Metaxas R. Metaxas P. Meyer E. Michielssen A. Mickelson D. Miller P. Miller B. W. Min R. Minasian J. D. Mingo J. Mink B. Minnis F. Miranda D. Mirshekar C. Mishra S. Mitilineos R. Mittra K. Miyaguchi M. Miyakawa H. Miyamoto R. Miyamoto M. Miyashita M. Miyazaki K. Mizuno S. Mizushina J. Modelski W. V. Moer S. Mohammadi H. Moheb J. Mondal M. Mongiardo P. Monteiro C. Monzon A. D. Morcillo J. Morente T. Morf D. R. Morgan M. Morgan

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