IEEE MTT-V053-I03A (2005-03) [53, 3A ed.]

Table of contents :
010 - 01406269......Page 1
020 - [email protected] 3
030 - [email protected] 4
I. I NTRODUCTION......Page 7
A. Mode Spectrum of NRD-Guide and $H$ -Guide......Page 8
B. Spectral Dyadic Green's Function and Volume-Integral Equation......Page 9
C. Current Discretization and System Matrix Structure......Page 10
D. Analytical Evaluation in Space-Domain Integrations and ${ m}=......Page 12
A. NRD-Guide Open Ends......Page 13
Fig. 5. Frequency characteristics of the phase of ${ S} _{11}$ f......Page 14
C. NRD-Guide Resonators......Page 15
Fig.€11. Resonant frequency characteristics of an inhomogeneous......Page 16
IV. C ONCLUSION......Page 17
C. D. Nallo, F. Frezza, and A. Galli, Full-wave modal analysis o......Page 19
I. I NTRODUCTION......Page 21
Fig.€1. (a) Schematic of the popular source inductor feedback am......Page 22
B. Design Procedure of the Source-Inductor-Feedback LNA......Page 24
Fig.€4. Calculated $K_c$ versus $ R/P$ characteristics of modern......Page 25
Fig.€6. Flowchart of the design procedures for finding optimized......Page 26
Fig.€9. Detailed process flows of silicon substrate thinning.......Page 27
Fig.€11. (a) Measured and simulated quality factors versus frequ......Page 28
Fig. 13. Measured transducer gain ( ${ S}_{21}$ ) versus frequen......Page 29
D ERIVATION OF THE NF FOR AN FET W ITH G ATE AND S OURCE I NDUCT......Page 30
D. Melendy, P. Francis, C. Pichler, K. Hwang, G. Srinivasan, and......Page 31
Fig.€1. Generic schematic of optimum single-ended dual-fed distr......Page 33
II. E FFECT OF FET I NPUT AND O UTPUT C APACITANCE ON THE SE DFD......Page 34
IV. A MPLIFIER D ESIGN......Page 35
Fig. 5. Embedded simulated ${s}$ -parameters of the 50- $\Omega$......Page 36
Fig.€9. Measured load power versus frequency.......Page 37
M. R. Moazzam and C. S. Aitchison, High gain microwave amplifier......Page 38
C. Cammacho-Penalosa and C. S. Aitchison, Modeling frequency dep......Page 39
I. I NTRODUCTION......Page 40
II. W EIGHTED T WO -D IMENSIONAL (2-D) FDTD F ORMULATION......Page 41
Fig.€2. Numerical dispersion of the AOM optimized at 10 CPW and......Page 42
Fig.€4. Numerical dispersion of the DOM optimized at 10 CPW and......Page 43
A. Optimal Parameter......Page 44
D. Relative RMS Error Comparison......Page 45
F. Three-Dimensional (3-D) Case......Page 46
G. CM Techniques......Page 47
Fig.€10. Numerical dispersion of the HAM-FW method, HAM-S method......Page 48
G. Sun and C. W. Trueman, Quantification of the truncation error......Page 49
II. C ROSS -C OUPLED $E$ -P LANE F ILTERS......Page 51
Fig. 3. Coupling coefficient values versus: (a) iris width $({l}......Page 52
TABLE I D ATA FOR $Ka$ -B AND C ROSS -C OUPLED F OLDED $E$ -P LA......Page 53
C. Four-Resonator Cross-Coupled Filter With Asymmetric Response......Page 54
Fig.€12. Calculated (dashed line) and measured (solid line) resp......Page 55
Fig.€15. (a) Response of a direct-coupled $E$ -plane filter with......Page 56
IV. M ILLIMETER -W AVE D IPLEXER......Page 57
A. E. Williams, A four-cavity elliptic waveguide filter, IEEE Tr......Page 58
Y. Rong, H. Yao, K. A. Zaki, and T. Dolan, Millimeter-wave $Ka$......Page 59
Fig.€1. Schematic diagram of the classical Doherty amplifier.......Page 60
A. Principle of Operation......Page 61
B. Efficiency Calculation......Page 62
Fig.€5. Fundamental current as a function of conduction angle $(......Page 63
Fig.€9. Comparison of PAE measurement results of the three-stage......Page 64
Fig. 13. Bias voltage adjustment of peak amplifier #2 with incre......Page 65
Y. Yang, J. Yi, Y. Y. Woo, and B. Kim, Optimum design for linear......Page 66
S. C. Cripps, RF Power Amplifiers for Wireless Communications .......Page 67
Fig.€2. Dual-mode ring resonator topology using shunt-capacitanc......Page 69
B. Odd-Mode Analysis......Page 70
III. F ILTER A NALYSIS......Page 71
IV. D ISCUSSIONS ON V ARIOUS T YPES OF D UAL -M ODE R INGS......Page 72
TABLE II D ESIGN P ARAMETERS FOR R INGS A AND B......Page 73
Fig.€12. (a) Frequency response of varactor-tuned filter at vari......Page 74
M. Makimoto and M. Sagawa, Varactor tuned bandpass filters using......Page 75
A. Radial Electric-Current Loop in Free Space......Page 76
III. E LECTRIC -F IELD C OMPONENTS D UE TO A D ISK OF R ADIAL C......Page 77
IV. N UMERICAL S OLUTION OF A T OP -L OADED C OAXIAL T RANSITION......Page 78
Fig.€4. Input impedance as a function of frequency for a coaxial......Page 79
VI. C ONCLUSIONS......Page 80
M. Sierra-Pérez, M. Vera, A. G. Pino, and M. Sierra-Castañer, An......Page 81
II. L INEARIZATION P RINCIPLE......Page 82
A. Amplifier Modeling......Page 83
IV. G ENERAL C RITERIA FOR D IMENSIONING THE L INEARIZER......Page 84
V. E VALUATION OF D ISTORTION F ROM THE LD AND B UFFER......Page 85
VI. E XPERIMENTAL V ALIDATION......Page 86
F. H. Raab, P. Asbeck, S. Cripps, P. B. Kenington, Z. B. Popovic......Page 87
C. G. Rey, Adaptive polar work-function predistortion, IEEE Tran......Page 88
II. C-C OLPITTS O SCILLATOR T OPOLOGY AND O NE -P ORT A NALYSIS......Page 89
Fig. 3. $\vert G_{r} (\omega_{n})/G_{m0}\vert$ over $\omega_{n}$......Page 90
III. P HASE N OISE IN THE P ROPOSED C-C OLPITTS O SCILLATOR......Page 91
Fig. 9. Measurement results. (a) $L_{\rm tank} = 10$ nH, $C_{g}......Page 93
TABLE III M EASUREMENT R ESULTS OF THE P ROPOSED C-C OLPITTS VCO......Page 94
P. Kinget, A fully integrated 2.7 V 0.35 $\mu$ m CMOS VCO for 5......Page 95
II. D ESIGN S TRATEGY......Page 96
Fig.€1. Single-mode equivalent circuits for the principal polari......Page 97
III. R ESULTS......Page 98
Fig.€7. Phase difference between the transmission coefficients f......Page 99
IV. C ONCLUSION......Page 100
R. Tascone, R. Orta, M. Baralis, A. Olivieri, and O. Peverini, C......Page 101
A. CMRC Filter......Page 103
B. T-Shaped Patches......Page 104
B. ACMRC BPF......Page 105
C. CPW-Fed BPF......Page 106
Q. Xue, K. M. Shum, and C. H. Chan, Low conversion loss fourth-s......Page 107
1) Low-Frequency Noise Sources Near DC:......Page 109
2) Noise Sources and Noisy Output Signal [ 4 ]: Any physical cir......Page 110
IV. N UMERICAL S IMULATION OF A B IPOLAR T RANSISTOR O SCILLATOR......Page 111
V. C ONCLUSION......Page 112
Phase-Noise Calculation......Page 113
J. C. Nallatamby, M. Prigent, M. Camiade, and J. Obregon, Phase......Page 114
A. Identification of PCB Discontinuities in the Wavelet Domain......Page 115
Fig.€1. Discretization of the PCB in the wavelet domain.......Page 116
B. Adopted Full-Wave Model......Page 117
D. Inclusion of the Two-Port Equivalent Into a Quasi-TEM TL Mode......Page 118
Fig.€3. Right-angle-bend geometry in a microstrip line.......Page 119
A. Validation of the Identification Process for a Right-Angle-Be......Page 120
B. Inclusion of the Discontinuity Model Into a Quasi-TEM Model......Page 121
C. Validation of the Methodology Through Comparison With Measure......Page 122
Fig.€12. Time-domain waveforms of the: (a) transmitted power wav......Page 123
E. Choice of the Basis......Page 124
P. Kok and D. De Zutter, Capacitance of a circular symmetric mod......Page 125
J. R. Phillips, L. Daniel, and L. M. Silveira, Guaranteed passiv......Page 126
II. D IRECT P ARAMETER E XTRACTION......Page 127
A. Emitter, Base, and Collector Resistances......Page 128
C. Base Emitter and Base Collector Capacitances......Page 129
Fig. 6. Measured (triangles) and calculated (dashed line) ${ C}_......Page 130
D. Transport Factor......Page 131
III. H IGH -F REQUENCY N OISE M ODELING......Page 132
Fig.€13. Measured and simulated: (a) minimum noise figure, (b) m......Page 133
Fig.€15. Simplified layout of the transistor and equivalent circ......Page 134
Fig.€17. Noise figure versus temperature plot for a constant bas......Page 135
J. M. O'Callaghan and J. P. Mondal, A vector approach for noise......Page 136
Fig.€2. Block diagram of the dual-loop OEO.......Page 137
Fig.€4. Experimental data for the oscillator output taken from a......Page 138
B. Phase-Noise Measurement......Page 139
D. B. Sullivan, D. W. Allan, D. A. Howe, and F. L. Walls, Eds.,......Page 140
W. F. Walls, Cross-correlation phase noise measurements, in Proc......Page 141
II. M EASUREMENT S ETUP......Page 142
IV. R ESULTS: A MPLITUDE AND P HASE I MBALANCE......Page 143
Fig.€7. Mode conversion dependent on phase error.......Page 144
C. R. Curry, How to calibrate through balun transformers to accu......Page 145
Fig.€1. Planar transmission-line network with deembedding arms a......Page 146
A. Application of the Lorentz Reciprocity Theorem to the Deembed......Page 147
B. Procedure for Extracting Scattering Parameters From the Ortho......Page 148
Fig.€4. Positions of two probe arms with respect to the deembedd......Page 149
III. D ISCUSSION OF P ERFORMANCE......Page 150
Fig. 6. End conductance of a $w/h = 1.79$ and $\epsilon _{r} = 2......Page 151
Fig.€9. Attenuation constant of a lossless microstrip with $\eps......Page 152
J. Martel, R. R. Boix, and M. Horno, Static analysis of microstr......Page 153
E. Drake, R. R. Boix, M. Horno, and T. K. Sarkar, Comparison amo......Page 154
II. A NALYSIS OF C OUPLED S PIRAL R ESONATORS......Page 155
Fig.€3. (a) Coupling coefficient $k$ as a function of coupling g......Page 156
TABLE€I N ORMALIZED C OUPLING M ATRIX FOR THE F OUR -P OLE F IL......Page 157
IV. E XPERIMENTAL P ERFORMANCE......Page 158
EM User's Manual, Version 8.0, Sonnet Software Inc., North Syrac......Page 159
II. LNB T ASKS......Page 160
IV. D OWN -C ONVERTER......Page 161
B. Proposed VCO......Page 162
A. Conventional Divide-by-Two Prescaler......Page 163
Fig.€12. Cross section of the technology adopted.......Page 164
VIII. E XPERIMENTAL R ESULTS......Page 165
TABLE€II IC P ERFORMANCE S UMMARY......Page 166
R. G. Meyer and A. K. Wong, Blocking and desensitization in RF a......Page 167
I. I NTRODUCTION......Page 168
III. B ROYDEN -B ASED D IRECT SM......Page 169
V. C OMPARISON B ETWEEN B ROYDEN, NISM, AND LISM......Page 170
Fig.€3. Results after applying Broyden-based direct SM optimizat......Page 171
Fig.€6. Fine model minimax objective function values at each ite......Page 172
VII. CMOS D RIVERS FOR A L ONG M ICROSTRIP L INE......Page 173
Fig.€14. Fine model minimax objective function values at each LI......Page 174
B. Razavi, Design of Analog CMOS Integrated Circuits . New York:......Page 175
II. T IME -D OMAIN M ODELING W ITH L UMPED -E LEMENT D EVICES OF......Page 177
III. N EGATIVE -F EEDBACK -B ASED F OURIER -T RANSFORM M ETHOD......Page 178
IV. H ILBERT -T RANSFORM -B ASED F OURIER -T RANSFORM M ETHOD......Page 179
A. Triangular Pulse......Page 180
B. Rectangular Pulse......Page 181
Fig.€10. Time-domain waveforms and their spectra extracted with......Page 182
VI. E RROR AND C ONVERGENCE T EST......Page 183
K. Murakami, S. Hontsu, and J. Ishii, Transient analysis of a cl......Page 184
II. T HEORY......Page 185
A. LNA/180 $^{\circ}$ -bit Design......Page 186
Fig.€6. Simulated insertion and return losses at different analo......Page 187
IV. P HYSICAL D ESIGN......Page 188
Fig. 12. Measured output return loss $(S_{22})$ from 11 to 12 GH......Page 189
VI. C ONCLUSION......Page 190
J.-S. Rieh, D. Greenberg, M. Khater, K. T. Schonenberg, S.-J. Je......Page 191
II. S INGULAR E LEMENTS......Page 192
B. Singular Vector Bases......Page 193
Fig.€2. Effects of material property and wedge angle on the sing......Page 194
A. Rectangular Slit Waveguide......Page 195
Fig.€4. Electric field of the first mode along the slit in Fig.€......Page 196
D. Microstrip Transmission Line......Page 197
A. Definition of the Interpolation Operators......Page 198
b) Proof of (13): Equation (10) implies $$ \int_{\Delta}\nabla \......Page 199
L. Vardapetyan, $hp$ -adaptive finite element method for electro......Page 200
I. I NTRODUCTION......Page 201
II. 2-D NUFFT A LGORITHM......Page 202
III. I NCORPORATING THE 2-D NUFFT I NTO THE SDA......Page 203
Fig.€4. Measured and calculated $S$ -parameters of the hairpin r......Page 204
Fig.€7. Measured and calculated $S$ -parameters of the interdigi......Page 205
W. Menzel, L. Zhu, K. Wu, and F. Bögelsack, On the design of nov......Page 206
Website......Page 208
300 - 01406298......Page 209
310 - 01406299......Page 210
320 - 01406300......Page 211
330 - 01406301......Page 212
340 - 01406302......Page 213

Citation preview

MARCH 2005

VOLUME 53

NUMBER 3

IETMAB

(ISSN 0018-9480)

PART I OF TWO PARTS

Editorial: How to Get Your Manuscript Published in this TRANSACTIONS in Six Months or Less . . . . . . . . D. F. Williams

797

PAPERS

An Order-Reduced Volume-Integral Equation Approach for Analysis of NRD-Guide and -Guide Millimeter-Wave Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D. Li, P. Yang, and K. Wu A 2.17-dB NF 5-GHz-Band Monolithic CMOS LNA With 10-mW DC Power Consumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-W. Chiu, S.-S. Lu, and Y.-S. Lin Compact Dual-Fed Distributed Power Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .K. W. Eccleston Optimized Finite-Difference Time-Domain Methods Based on the Stencil . . . . . . . . . .G. Sun and C. W. Trueman Novel -Plane Filters and Diplexers With Elliptic Response for Millimeter-Wave Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Ofli, R. Vahldieck, and S. Amari Analysis and Design of a High-Efficiency Multistage Doherty Power Amplifier for Wireless Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .N. Srirattana, A. Raghavan, D. Heo, P. E. Allen, and J. Laskar An Analysis of Miniaturized Dual-Mode Bandpass Filter Structure Using Shunt-Capacitance Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.-F. Lei and H. Wang Evaluation of the Input Impedance of a Top-Loaded Monopole in a Parallel-Plate Waveguide by the MoM/Green’s Function Method . . . . . . . . . . . . . . . . . A. Valero-Nogueira, J. I. Herranz-Herruzo, E. Antonino-Daviu, and M. Cabedo-Fabres Study of an Active Predistorter Suitable for MMIC Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Iommi, G. Macchiarella, A. Meazza, and M. Pagani A Complementary Colpitts Oscillator in CMOS Technology. . . . . . . . . . . . . . . . . . . . . . . . . . .C.-Y. Cha and S.-G. Lee A Novel Design Tool for Waveguide Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Virone, R. Tascone, M. Baralis, O. A. Peverini, A. Olivieri, and R. Orta A Compact Bandpass Filter With Two Tuning Transmission Zeros Using a CMRC Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. M. Shum, T. T. Mo, Q. Xue, and C. H. Chan On the Role of the Additive and Converted Noise in the Generation of Phase Noise in Nonlinear Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.-C. Nallatamby, M. Prigent, and J. Obregon Two-Port Equivalent of PCB Discontinuities in the Wavelet Domain . . ..R. Araneo, S. Barmada, S. Celozzi, and M. Raugi

799 813 825 832 843 852 861 868 874 881 888 895 901 907

(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Small-Signal and High-Frequency Noise Modeling of SiGe HBTs. . . . .U. Basaran, N. Wieser, G. Feiler, and M. Berroth Injection-Locked Dual Opto-Electronic Oscillator With Ultra-Low Phase Noise and Ultra-Low Spurious Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Zhou and G. Blasche Pure-Mode Network Analyzer Concept for On-Wafer Measurements of Differential Circuits at Millimeter-Wave Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Zwick and U. R. Pfeiffer An Orthogonality-Based Deembedding Technique for Microstrip Networks . . . . . . . . . . M. P. Spowart and E. F. Kuester Superconducting Spiral Filters With Quasi-Elliptic Characteristic for Radio Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Zhang, F. Huang, and M. J. Lancaster A Monolithic 12-GHz Heterodyne Receiver for DVB-S Applications in Silicon Bipolar Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Girlando, S. A. Smerzi, T. Copani, and G. Palmisano A Linear Inverse Space-Mapping (LISM) Algorithm to Design Linear and Nonlinear RF and Microwave Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. E. Rayas-Sánchez, F. Lara-Rojo, and E. Martínez-Guerrero Iterative Methods for Extracting Causal Time-Domain Parameters . . . . . . . . . . . . . . . . . . . . . . . . . S. Luo and Z. Chen A 12-GHz SiGe Phase Shifter With Integrated LNA . . . . . . . . . . . . . . . . . . . . . . . . . . T. M. Hancock and G. M. Rebeiz Two-Dimensional Curl-Conforming Singular Elements for FEM Solutions of Dielectric Waveguiding Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D.-K. Sun, L. Vardapetyan, and Z. Cendes Application of Two-Dimensional Nonuniform Fast Fourier Transform (2-D NUFFT) Technique to Analysis of Shielded Microstrip Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.-Y. Su and J.-T. Kuo

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Information for Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ANNOUNCEMENTS

IEEE MTT-S Undergraduate/Pre-Graduate Scholarships. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Issue on Microwave Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2005 IEEE Compound Semiconductor IC Symposium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 and will receive this TRANSACTIONS upon payment of the annual Society membership fee of $14.00 plus an annual subscription fee of $24.00. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only.

ADMINISTRATIVE COMMITTEE K. C. GUPTA, President M. P. DE LISO S. M. EL-GHAZALY M. HARRIS T. ITOH

K. VARIAN, Vice President D. HARVEY J. HAUSNER L. KATEHI

A. MORTAZAWI, Secretary T. LEE D. LOVELACE J. MODELSKI

S. KAWASAKI J. S. KENNEY N. KOLIAS

Honorary Life Members A. A. OLINER K. TOMIYASU T. S. SAAD L. YOUNG

L. E. DAVIS W. GWAREK W. HEINRICH W. HOEFER

M. HARRIS, Treasurer

V. J. NAIR B. PERLMAN D. RUTLEDGE Distinguished Lecturers T. ITOH B. KIM J. LASKAR J. C. RAUTIO

K. VARIAN R. WIEGEL S. WETENKAMP

W. SHIROMA R. SNYDER R. SORRENTINO D. RYTTING M. SHUR P. SIEGEL R. J. TREW

Past Presidents R. J. TREW (2004) F. SCHINDLER (2003) J. T. BARR IV (2002)

MTT-S Chapter Chairs Albuquerque: G. WOOD Atlanta: J. PAPAPOLYMEROU Austria: R. WEIGEL Baltimore: B. MCCARTHY Beijing: Y.-R. ZHONG Beijing, Nanjing: W.-X. ZHANG Belarus: A. GUSINSKY Benelux: D. V.-JANVIER Buenaventura: L. HAYS Buffalo: M. R. GILLETTE Bulgaria: F. FILIPOV Cedar Rapids/Central Iowa: D. JOHNSON Central New England/Boston: F. SULLIVAN Central & South Italy: R. TIBERIO Central No. Carolina: T. IVANOV Chicago: R. KOLLMAN Cleveland: G. PONCHAK Columbus: J.-F. LEE Croatia: J. BARTOLIC Czech/Slovakia: P. HAZDRA Dallas: P. WINSON Dayton: A. TERZOULI, JR. Denver: K. BOIS East Ukraine: A. KIRILENKO Eastern No. Carolina: D. PALMER Egypt: I. A. SALEM Finland: T. KARTTAAVI Florida West Coast: S. O’BRIEN

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IEEE Periodicals Transactions/Journals Department Staff Director: FRAN ZAPPULLA Editorial Director: DAWN MELLEY Production Director: ROBERT SMREK Managing Editor: MONA MITTRA Senior Editor: CHRISTINA M. REZES IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (ISSN 0018-9480) is published monthly by the Institute of Electrical and Electronics Engineers, Inc. Responsibility for the contents rests upon the authors and not upon the IEEE, the Society/Council, or its members. IEEE Corporate Office: 3 Park Avenue, 17th Floor, New York, NY 10016-5997. IEEE Operations Center: 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. NJ Telephone: +1 732 981 0060. Price/Publication Information: Individual copies: IEEE Members $20.00 (first copy only), nonmember $69.00 per copy. (Note: Postage and handling charge not included.) Member and nonmember subscription prices available upon request. Available in microfiche and microfilm. Copyright and Reprint Permissions: Abstracting is permitted with credit to the source. Libraries are permitted to photocopy for private use of patrons, provided the per-copy fee indicated in the code at the bottom of the first page is paid through the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. For all other copying, reprint, or republication permission, write to Copyrights and Permissions Department, IEEE Publications Administration, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. Copyright © 2005 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved. Periodicals Postage Paid at New York, NY and at additional mailing offices. Postmaster: Send address changes to IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. GST Registration No. 125634188. Printed in U.S.A.

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Editorial: How to Get Your Manuscript Published in this TRANSACTIONS in Six Months or Less

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T IS both more prestigious and more difficult to publish in this TRANSACTIONS than in nearly any other journal or conference digest. The competition is fierce and the standards are high. Here are a few common-sense tips on how to prepare your paper to give it the greatest chance of acceptance. Of course, there is no substitute for a brilliant solution to an important engineering problem. But, there is more to a good TRANSACTIONS paper than meets the eye. Read on. You will be surprised at how many very simple things you can do to ease your paper through the review process and get that coveted acceptance notification delivered straight to your in-box. For me, the most important stage of writing a paper starts before I have put down a single word on paper. This is the planning stage of the paper. Some authors begin by writing up an outline. I prefer to gather my graphs together and look them over. Others look over similar papers that have been previously published in this TRANSACTIONS. Whichever way you approach writing a paper, ask yourself what story you have to tell, and whether that story is complete. The planning stage is essential because it helps you gain important perspective and set up a logical organizational framework for your paper. It also keeps you from starting in on a paper before you have gathered sufficient measurements or completed the analysis. Once you have gathered your ideas, download the Word or LaTeX IEEE template for this TRANSACTIONS’ submissions from http://www.mtt.org/publications/For_Authors/for_authors.htm. Read the template before you begin. It is chock full of sound advice on grammar and style, and contains many useful tips. Starting your paper in the template will also result in a more professional-looking submission. This favorably impresses the reviewers. I cannot overemphasize the importance of grammar and organization of your ideas. Reviewers are very busy people and they were chosen because they have made important contributions to the field. Thus, your number 1 goal is to get the reviewer to understand and appreciate your technical contribution as quickly as possible. The last person you want commenting on your technical work is a grumpy reviewer who has just spent an hour marking up your paper with a red pen or, worse yet, struggling to understand the point you are trying to make. Remember that nearly all reviewers put many hours and sometimes days into reviewing a paper, and you want to make their job as easy as possible. Good grammar and exposition are difficult to come by, and you probably did not go into electrical engineering because

Digital Object Identifier 10.1109/TMTT.2004.842512

prose flows from your pen. It helps to revise and then re-revise over a period of two months or more. It is often surprising how many weaknesses you can find this month in last month’s brilliant tour-de-force. Whether you are a native English speaker or not, take your paper to an expert for grammatical proofreading and correction: a native-speaking English literature, history, or philosophy professor or graduate student. Go over the paper together with your grammatical advisers. Be inquisitive, and try to understand not just what they suggest changing, but why. If you see this TRANSACTIONS’ reviewer as your most important adversary in the publishing process, you need to learn editorial jujitsu. Start by lining up your own set of technical reviewers well before you submit. Just as with your grammatical advisers, arrange a meeting with each of your technical reviewers. Try to use this process as a way of getting them to talk about the paper. You will find what they say to be far more useful than what they wrote. Sometimes you will find that you simply did not think of writing down some key points. Above all, keep your cool. What your reviewers and grammatical advisers tell you will be hard to hear. However, if you do this right, your editorial jujitsu will have put the reviewers to work for you. You see, reviewers often provide exactly what you lack the most, which is perspective—perspective on all sorts of things, from the most subtle technical issues to the most obvious (in retrospect) organizational problems. If you want your paper to be accepted in the first review cycle, you need to get this TRANSACTIONS’ reviewer to focus on your brilliant technical contributions. You do not want your reviewer sorting through your previous publications trying to decide what is new and what is old. Keep in mind that reviewers and readers alike prize both originality and completeness, and you will do far better pleasing your reviewers and readers with new ideas and with complete and original papers than trying to explain why your current paper differs significantly enough from your last to deserve publication. No discussion of this TRANSACTIONS would be complete if it did not touch on conference Special Issues and the relationship of “expanded” to conference papers. Much has changed in recent years, with conference papers generally becoming archival and available electronically. Writing a good paper for a conference Special Issue is far more difficult than writing any other TRANSACTIONS paper. The root of the problem is that some topics are well suited to a conference paper, and others to a TRANSACTIONS paper, but only a very few to both. However, maybe you have your heart set on one of these Special Issue papers. “What to do?” I hear you ask. First, start early.

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Contrary to popular belief, writing two good complementary papers really does take twice as long as writing one paper. Be sure to choose a subject that is actually large enough to justify two separate submissions, and that contains a subtopic suitable for a conference paper. The conference paper should be a short vignette, completely and thoroughly treating an important aspect of the entire study, but no more. I am not talking about simply putting highlights in the conference paper or “dumbing it down.” These papers have very little long-term value. It should be short, complete, and standalone, and readers should still be able to read the paper in future years and learn something that never appeared anywhere else. When you write up the main body of the paper for a Special Issue, try summarizing the conference paper in a few paragraphs or a short section, rather than repeating the conference paper in its entirety. The expanded TRANSACTIONS paper is not just the conference paper with more words and equations. It should build on the conference paper, but in a way that both publications are worth reading and so that a reader learns different things from each. This strategy will help you write two truly distinct and complementary crowd pleasers, is sure to wow the reviewers, and will give you a great additional opportunity to advertise your work. Finally, there is a possibility that a reviewer will still object to your paper. In this case, there are only two possible courses of action that will steer you clear of the endless review-cycle vortex. You can determine that the reviewer was right to begin with, and fix your paper, or you can figure out why the reviewer misunderstood you, and fix your paper. Trying to convince a TRANSACTIONS reviewer, who is typically an expert in the field, that he or she never should have objected to your work in the first place is guaranteed to send you straight into the maw of the vortex! Using these simple tips will not cover up technical blunders or ensure acceptance in this TRANSACTIONS. But it will most assuredly put you in a far better position to get your work and insights out to your most important audience, your peers in the microwave community that this TRANSACTIONS serves. Finally, I need to talk about properly referencing papers. Not only do incomplete and inaccurate references create an impression of carelessness, but errors and the time required to correct them are the major reason for delays in copy editing (the time between the manuscript being sent to the IEEE and its being published). Some rules and examples for IEEE references are as follows. Always use month and year of publication in references and abbreviate months. Use initials for the first names of authors in your list of references, and include all authors’ names. If the periodical is an IEEE publication, the issue number and month of publication is necessary. Any IEEE TRANSACTIONS that was published prior to 1988 (with the exception of the

PROCEEDINGS OF THE IEEE) must carry the TRANSACTIONS’ acronym, e.g., vol. MTT-25). Note that the correct reference for this TRANSACTIONS is IEEE Trans. Microw. Theory Tech. The correct reference for the 2004 IEEE International Microwave Symposium (IMS) is “in 2004 IEEE MTT-S Int. Microwave Symp. Dig.” Do not use acronyms for conferences: spell out the full name of the conference (e.g., use Int. Electron Devices Meeting instead of IEDM). If references carry online information, the author should include this (i.e., http information, etc.) at the end of the reference. Finally, keep your eyes open for the automated reference checker being developed by the IEEE! • Periodicals: Author(s) Initial(s), Surname(s), “Title of paper,” Title of Periodical, vol #, issue #, pp. xx-xx, Abbrev. Month, Year. • Books: Author(s) Initial(s), Surname(s), “Title of chapter in book (if applicable),” Title of Book, xth ed. City of Publisher, State/Country: Abbrev. name of Publisher, Year, Chapter X (if applicable), Section X (if applicable), pp. xx-xx. • Reports: Author(s) Initial(s), Surname(s), “Title of report,” Name of Company, City of Company, State/Country of Company, Report number, Year. • Handbook (generally a “book” published by a company, as opposed to a publisher): Title of Manual/Handbook, x edition, Abbrev. Name of Company, City of Company, State/Country of Company, Year, pp. xx-xx. • Published Conference Proceedings: Author(s) Initial(s), Surname(s), “Title of paper,” Unabbreviated Name of Conference, City of Conference, State/Country, Abbrev. Month Year, pp. xx-xx (published conference proceedings MUST include page numbers). • Unpublished Papers Presented at Conference: Author(s) Initial(s), Surname(s), “Title of paper,” presented at the Name of Conference, City of Conference, State/Country, Year. • Patents: Author(s) Initial(s), Surname(s), “Title of patent,” U.S. Patent # xxxxx, Abbrev. Month Day, Year. • Theses (Masters) and Dissertations (Ph.D.): Author(s) Initial(s), Surname(s), “Title of thesis/dissertation,” Abbrev. Department, University, City of Univ., State/Country, Year. • Unpublished References: Author(s) Initial(s), Surname(s), private communication, Abbrev. Month, Year. Or as applicable: Author(s) Initial(s), Surname(s), “Title of paper,” unpublished. • Standards: Title of Standard, Standard number, Date/Year. Admittedly, the tips and tricks in this paper were learned the hard way by the author. They do not necessarily represent the editorial policy of this TRANSACTIONS. DYLAN F. WILLIAMS, Associate Editor, TMTT National Institute of Standards and Technology Boulder, CO 80305 USA

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An Order-Reduced Volume-Integral Equation Approach for Analysis of NRD-Guide and -Guide Millimeter-Wave Circuits

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Duochuan Li, Ping Yang, and Ke Wu, Fellow, IEEE

Abstract—An order-reduced volume-integral equation approach is proposed for modeling and analyzing of nonradiative dielectric (NRD)-guide and -guide millimeter-wave circuits that involve arbitrarily shaped planar geometry and inhomogeneous dielectric. A half-sinusoidal vertical variation of fields is used so that the discretization of current for the volume-integral equation is made in the parallel plane. Combined basis functions of propagating and local modes are used in the Garlerkin’s method of moments on the basis of spectrum analysis of the NRD-guide and -guide. A vertical integration in the space domain is carried = 1 out analytically and a first-order Green’s function with is developed. The solution for the volume-integral equation in modeling NRD-guide and -guide circuits is then reduced to a two-dimensional planar problem. This technique can be applied for calculating the characteristics of various waveguide components and multiport circuits such as resonant frequencies and -parameters. The framework of this technique is demonstrated through its application to an NRD-guide open-end. In addition, an -guide open-end, three types of resonators, and three-pole gap-coupled NRD-guide filters are modeled and analyzed. The results are in good agreement with the measurements and results obtained by other methods. Index Terms—Green’s function, method of moments, millimeter wave, nonradiative dielectric (NRD)-guide and -guide circuits, order-reduced volume integral equation (ORVIE), vertical integration.

I. INTRODUCTION

M

ILLIMETER-WAVE techniques become increasingly important because of emerging wireless communication and noncommunication applications. Over this frequency range, the commonly used planar techniques, namely, microwave integrated circuit (MIC), miniaturized hybrid microwave integrated circuit (MHMIC), and monolithic microwave integrated circuit (MMIC), generally suffer from problems of prohibitive conductor loss and critical dimensional tolerance. Therefore, nonplanar technologies, which include metallic and dielectric waveguides, should be considered. Among those nonplanar schemes, the nonradiative dielectric (NRD) waveguide is promising for the making of passive components because it can effectively suppress radiation loss along circuit bends and discontinuities. Since its inception in 1981 [1], this technology has been used in the design and fabrication of a large class of Manuscript received February 9, 2004; revised May 24, 2004. The authors are with the Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, QC, Canada H3V 1A2 (e-mail: liduo@ grmes.polymtl.ca; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842511

integrated circuits and antennas that have demonstrated superior electrical performance at millimeter-wave frequencies [2]–[4]. Several numerical methods were used to analyze discontinuities and components in NRD circuits. The mode-coupling theory was applied in [5] to analyze the loss characteristics of the NRD-guide bends. A rigorous expression for coupling and modes (also referred to coefficient between as , or , in some publications) was derived and then used in the two-mode coupling equations to be solved for the bending loss analysis. This theory was reexamined in [6] to improve the design technique of NRD-guide bends. An efficient design technique of an NRD-guide filter based on a variational principle has been developed in [7], and both gap-coupled-type and alternating-width-type filters were successfully designed. However, those methods are geometrically specific and hard to extend for general modeling problems of NRD-guide discontinuity and components. The mode-matching technique has also been used to solve a large class of NRD-guide components and discontinuities, which include open ends, junctions, steps, gaps, T-junctions, and diplexer [8]–[10]. Nevertheless, it will fail once the planar section (the top view of dielectric circuit) of the guiding structure is of arbitrary shape and/or has inhomogeneous dielectric permittivity. In this paper, an order-reduced volume-integral equation (ORVIE) method is developed, which is able to solve not only NRD circuits of an arbitrarily shaped planar section and inhomogeneous dielectric permittivity in the planar section, but also -guide circuits in which radiation loss must be considered since the spacing between the two parallel plates is larger than a half-wavelength in free space [11]. An electric-field integral-equation (EFIE) method was used to obtain eigenmodes of dielectric guide in [12] and [13]. The EFIE in its standard, also referred to as the domain integral equation, treats dielectric strip domains as local perturbation of the configuration, replacing them with equivalent polarization currents. An electric dyadic Green’s function is then used for the integral representation of fields in the layer in which the strips are embedded. Due to its rigorous full-wave formulation, the EFIE technique is capable of handling both open and closed structures and describing physical effects such as wave leakage. However, when this method extends to three dimensions, referred to as the volume integral-equation method, only a few simple problems with finite dielectric regions can be solved because of large memory and CPU time requirements [14], [15]. This method has not been used in the modeling of NRD-guide

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Fig. 1. Cross section of NRD-guide and

The mode spectrum in this structure can be divided into a discrete number of surface waves, which is classified into LSE and LSM modes, and a continuous set of radiation modes. As far as the eigenvalue problem is concerned, the presence of the parallel metal plates implies a discretization of the vertical wavenumber , with null or integer ), while the horizontal ( wavenumbers can be derived from transcendental equations related to the dielectric slab guide. The transcendental equations for the eigenvalues along are expressed as

H -guide.

and -guide circuits. In this study, this method is applied to the modeling of NRD-guide and -guide circuits by reducing the original three-dimensional (3-D) problems to two-dimensional (2-D) planar problems. In the NRD-guide and -guide structures, a half-sinusoidal vertical ( ) variation of fields remains unchanged all over the circuit with the - or -mode excitation. Therefore, the current discretization in the volume integral equation can be implemented in the – -plane just as in planar circuits. With such basis functions, the vertical integration in the space dofirst-order main can be obtained analytically and an Green’s function can be constructed. Thus, the element integrations are carried out in two dimensions in the spectral domain. The solution of NRD-guide and -guide circuits are reduced to 2-D planar problems. This largely reduces calculation effort and makes it possible to simulate NRD circuits of any shaped planar section and any varied dielectric permittivity in the planar section. In particular, it can accurately calculate radiation loss along bends and discontinuities of the -guide by extracting poles in the Green’s function. In what follows, a complete description of the mode spectrum for the NRD-guide and -guide is presented first, followed by the formulation of the volume integral-equation and spectral dyadic Green’s function. Afterwards, the ORVIE technique is outlined. Subsequently, techniques for the integration are briefly described. In Section III, the solution procedure of an NRD-guide open-end is presented in detail to demonstrate features of the ORVIE scheme. The -guide open-ends, resonators, notched square, and rectangular sections with inhomogeneous dielectric variation, and three-pole gap-coupled NRD filter are then modeled and discussed. II. FORMULATIONS A. Mode Spectrum of NRD-Guide and

-Guide

To understand the proposed ORVIE method, a detailed description of the mode spectrum of the NRD-guide and -guide is given in the following. The NRD-guide and -guide have the same structure as in Fig. 1, which can be viewed as a rectangular dielectric rod (width , height , and relative permittivity ) sandwiched between two parallel metal plates. The desired mode in this structure. The prinoperating mode is the cipal advantage of the -guide is known for its low transmission loss, which is achieved by keeping the plate spacing more than , but it suffers radiation loss at bends and discontinuities. On the other hand, the advantage of the NRD guide is its ability to suppress such radiation loss by keeping the plate spacing less . than

symmetric modes antisymmetric modes (1) for LSE modes and for LSM modes. The where , is the -directed wavenumber propagation factor is inside the dielectric, and is the decay constant in the air region. , , and have an analytical relationship as follows: (2) (3) Solving the transcendental equations gives rise to a system of eigenvalues with odd integers ( ) in symmetrical ) in antisymmetric modes and even integers ( modes. The propagation constants are then obtained by

Surface-wave modes occur at , where satisfies . Radiation modes appear while , where satisfies . In this case, the waves can propagate away at an angle from the dielectric strip on both sides. The space of the mode spectrum can be divided into surfacewave modes, continuous radiation modes, and prohibited regions, as depicted in Fig. 2 by simply defining two boundary and in the dialines gram where is a decay constant, is a propagation constant, and

LI et al.: ORVIE APPROACH FOR ANALYSIS OF NRD-GUIDE AND

-GUIDE MILLIMETER-WAVE CIRCUITS

Fig. 3.

Top view of the dielectric circuits of NRD-guide and

801

H -guide.

as an NRD-guide. The radiation mode is partly evanescent region where the and partly propagating in the structure works as an -guide. Only the propagating radiation mode can cause radiation loss and this is why an NRD-guide can suppress radiation loss at bends and discontinuities with . The case of is similar to the operating mode where the boundaries and shift to a higher that of frequency range and the radiation mode cuts off at , as shown in Fig. 2(c). B. Spectral Dyadic Green’s Function and Volume-Integral Equation Formulation

H m

m

Fig. 2. Mode spectrum of NRD-guide and -guide. (a) = 0 (spectral condition). (b) = 1 (spectral condition). (c) = 2 (spectral condition).

m

A full spectrum for was generated (with mm mm and , as shown in Fig. 2. , as described in Fig. 2(a), the LSE In the case of modes reduce to modes and no modes exist. mode, and The fundamental guided TE mode, i.e., the the propagating radiation mode exist over all frequency range. , , and The higher order TE modes, namely, modes, merge to a continuous radiation mode at a low-frequency range where . The boundaries and shift , as shown in to a high-frequency region in the case of Fig. 2(b). The continuous radiation mode becomes completely region where the structure works evanescent in the

Fig. 3 shows the top-view (planar section) of a multiport NRD-guide or -guide circuit that is sandwiched between two parallel infinitely extended metal plates. The core circuit relates to the external circuits through physical ports attached to an equal number of intrinsic standard NRD-guide or -guide feed lines. These feed lines are assumed to be semi-infinitely long with the same height , different width , and constant dielectric permittivity . The core circuit may be in the form of an NRD-guide or -guide as the dielectric permittivity varies . The whole circuit is surrounded by a in the – -plane medium with a constant dielectric permittivity . Suppose that the total field is , and according to the volume equivalent theorem, the resulting field will remain the same if we replace the inhomogeneous dielectric body with the surrounding medium, but assume there is a specific electric current density in its volume with the following condition (a time dependency is considered): of (4) The electric field generated by the polarized volume current is (5)

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where is the space occupied by the dielectric strips. The in the spectral domain is dyadic Green’s function formulated as

(6) The explicit form of for the two parallel metal plates can be obtained by letting the reflection coefficient at the two plates [16], and can be expressed as (7), shown at the bottom of this page, where

and

In this Green’s function, the poles at which the denominator is equal to zero represent the parallel-plate waveguide modes, which form the radiation loss mechanism in discontinuities -guide circuits. This phenomenon is in NRD-guide and similar to the surface wave in microstrip circuits [17], [18]. , , where These poles occur at ( ). There is always at least continuous radiation mode, one pole that represents the as shown in Fig. 2(a). Another pole, representing the propagating radiation mode, as shown in Fig. 2(b), appears in the Green’s function at . With the increasing of frequency ( ) or increasing of the distance between the two ) will appear in this Green’s plates ( ), more poles ( function.

C. Current Discretization and System Matrix Structure The volume integral equation (4) can be solved with Garlerkin’s method of moments. Generally, the volume polarized current discretization should be made in three dimensions and this would make the matrix too large and make this method inefficient because of large memory and CPU time requirement. However, the NRD-guide and -guide circuits in Fig. 3 can be viewed as sections of dielectric waveguides that are short circuited by two infinite parallel metal plates placed perpendicularly to the longitudinal axis ( ) at a certain distance apart. In this is fixed by case, the propagating constant along , namely, , with null or integer ). the presence of the plates ( eigenmode excitation is considIn this paper, only the modes, including surface-wave modes ered. Thus, only and a continuous radiation wave mode, appears in this circuit modes. due to the orthogonal properties between different The eigenmode excitation with other values is similar to the case. With the eigenmode excitation, a half-siin nusoidal vertical ( ) variation of fields ( and components and in the component) remains unchanged over the entire circuits. The discretization can be made only in the – -plane just as planar circuits. In this case, the matrix is greatly reduced and becomes solvable for complicated circuits. In Fig. 3, the volume current of polarization can be represented by combined local and propagating modes basis functions. Local mode means that the field is restricted in a finite region of the circuit while the propagating mode means that the field will propagate out of the circuit or propagate into the circuit from outside through the feed lines. The local mode can be either entire mode or sub-sectional mode and the propagating mode, of course, is the entire mode. The field in the core circuit can be represented by the local modes that either can be entire modes or sub-sectional modes with rectangular or triangular meshes to fit different planar sections. The field along the feed lines consists of a discrete number of surface-wave modes and a continuous radiation mode, as shown in Fig. 2(b). The propagating surface-wave modes can be represented by propagating mode basis functions and the exponent terms in propagating modes are truncated after several cycles. The evanescent surface wave modes, if they appear, can be represented by local modes, and the entire-domain basis functions are more efficient for them since they decay much slower than the evanescent radiation mode. The radiation mode is totally evanescent in the frequency region of the NRD-guide, and becomes partly evanescent, and partly propagating in the operating frequency of the -guide. The evanescent radiation mode can be represented by local modes for which the sub-sectional basis functions are more efficient. The propagating part of the radiation mode that

(7)

LI et al.: ORVIE APPROACH FOR ANALYSIS OF NRD-GUIDE AND

-GUIDE MILLIMETER-WAVE CIRCUITS

will cause radiation loss in bends and discontinuities is reprepole in the Green’s function (7). sented by the With the introduced basis functions, the polarized volume current in the circuits and feed-lines can be written as

803

These entries can be obtained by a spectral-domain integration with the vertical part still in the space domain as follows:

(10)

(11) (8)

where ( ) are planar parts of the propagating modes in the th feed line including LSE or LSM modes, are planar which have exact analytical expressions. are planar parts of the inparts of the local modes. mode cidental propagating mode that can be either the mode. We call , , and or as planar basis functions. are the unknown transmission coefficients of propagating modes at port l. is the number of is the coefficients of the the unknown planar local modes. unknown planar local modes. , Only the local modes , and are used as testing functions in this Galerkin’s method of moments. The discretized integral-equation system is solved and a linear system of equations can be obtained as

Here, we merge the incidental terms with the transmission terms in (11) and use the subscript to represent and . and are Fourier transforms of the and . The functions planar basis functions and are of and . The power flow at ports is formulated by (12) Since the surface waves have exact analytic expressions, the power of the propagating modes at port l can be obtained analytically as

(13)

for the

mode and

(9) are impedance sub-matrices of self and mutual cou) and plings between the two local modes of ( components. and are impedance sub-matrices of mutual couplings between the local modes and propaand gating modes where . and are the coupling between the local mode and ’s component of the propagating mode. One of the crucial points of the method of moments lies in effective and accurate evaluation of the system matrix entries.

(14)

mode. is the propagating constant of the mode, is the transverse wavenumber inside the dielectric, and is the transverse wavenumber outare coefficients in the analytical side the dielectric. and and modes. expressions of the for the

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Suppose that the circuit is excited by an or mode at port l, then the -parameters can be obtained by

Once again, results of the vertical integration of the ( , ) function in (7) can be obtained component dealing with the by (15)

where

can be obtained in (9).

D. Analytical Evaluation in Space-Domain Integrations and Green’s Function The space-domain integrals within (10) and (11) in connection with the variables and can be evaluated analytically. The vertical integration of ( , ), ( , ), ( , ) and ( , ) components that are concerned with the function in (7) is

(19)

and results of the vertical integration concerned with source in in (7) can be obtained by

(20) (16)

The results of the vertical integration of ( , ) and ( , ) components related to the function in (7) can be obtained by

The component in the square brackets in (7) can then be simplified by combining the source term (20) with the principle term (19) and using the separation condition as

(21) The vertical integration in connection with the second term in (10) and (11) can be obtained by (17)

Similarly, results of the vertical integration of ( , ) and ( , ) components related to the function in (7) can be obtained by

(22) With the help of (16)–(19), a new Green’s function can be constructed as

(18) (23)

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805

where . With this Green’s function, the integration in (10) and (11) can be written as

Fig. 4. NRD-guide and

(24)

H -guide open ends.

can also be applied in the rectangular coordinate. The symmetric property and redundancy reduction techniques can also be used to reduce calculation effort in this method. Some of these properties will be shown in the following examples. III. NUMERICAL RESULTS AND DISCUSSION

(25) The Green’s function in (23) is simpler than (7) without triancomponent in the square bracket gular functions in it. The pole apis greatly simplified as a constant. Only the . pears in this Green’s function in the condition of Other poles, especially the ever-existing pole, are eliminated through the vertical integrations in (20)–(23). Since this case for NRD-guide Green’s function only governs the Green’s function and -guide circuits, we call it the and refer to (7) as the full-scale Green’s function. The integrations in (24) and (25) are 2-D in the spectral domain with respect and , as in planar circuits. With the 2-D discretization to technique, the solution of the volume integral equation for NRDguide and -guide circuits has been reduced to a completely 2-D planar problem. We call this method the ORVIE method. E. Some Technique for Numerical Integrations The integrations in (24) and (25) are similar to those used in planar circuits, but much simpler because of the simple Green’s functions. Many kinds of planar basis functions, such as piecewise sinusoidal functions, pulse basis functions with a rectangular or triangular mesh scheme, or the entire mode can be used for the local modes to fit different cross sections of circuits just as used in planar circuits. When poles appear, the numerical integration must be carand ried out in the polar coordinate with because the poles appear at . However, when there are no poles in the integration, they can be carried out either in the polar coordinate or in the rectangular coordinate. In the case of planar basis functions in an NRD-guide circuit, integrands and can be rearranged as and , and oscillate with a sinusoidal behavior. Thus, the integrations in the rectangular coordinate is more efficient than its polar counterpart. The integrations in the polar coordinate with poles were discussed in [17] and [19]. A general asymptotic subtraction technique was used in [19]. In this technique, the asymptotic behavior of integrands was searched prior to performing the numerical computations. Subtracting these representations leads to a fast decaying integrand, and the integral thus becomes well suited for a numerical integration; the complete analytical solutions can be determined for the asymptotic part. This technique

Several examples are discussed here to demonstrate properties and efficiency of the ORVIE method. The main emphasis focuses on NRD-guide circuits because of their important applications at millimeter-wave frequencies. The following solution procedure of the NRD-guide open end is presented to showcase the features of the ORVIE method. A. NRD-Guide Open Ends The most difficult part of the ORVIE approach lies in dealing with feed lines in which both propagating- and local-mode basis functions are used. Here, we will discuss the calculation procedure of an NRD-guide open-end whose field distribution is similar to that along the feed lines. The NRD-guide open end itself is also frequently encountered in the design of passive components and active devices. The effects of the open end are studied first in [8] with a mode-matching method. The planar section of an NRD-guide open end with width mm, thickness mm, and is mode and a continshown in Fig. 4. Only a reflected uous evanescent radiation mode are produced in this structure mode is used as the incident mode with the when an above parameters. Fields of the radiation mode can be represented by sub-sectional basis functions over a certain region of calculation. The length of region of calculation , as shown in Fig. 4, depends on the average decay constant of the radiation mode. By defining a weighting average decay constant , generally

is

enough. We divide the calculation region into by cells along the of equal area. The length of each cell is -direction, and along the -direction. The center , is denoted by ( , ) with coordinate of the ( , ) cell, and . Planar basis functions for the radiation mode can thus be represented by a set of pulse basis functions as follows: (26) where elsewhere (27) elsewhere and

.

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The planar propagating mode basis functions are (28) where

(29) The upside sign in (28) and (29) represents the reflected wave and the downside sign represents the incident wave. Fourier transand are given in the Appendix . The exforms of ponent terms in (28) are truncated after several cycles. It is found that choosing a length of the exponent as an integral number of wavelength speeds the convergence of integrals. The length of depends on the length of the calculation region the exponent of the radiation mode. Typically, the solution is insensitive to the wave exponent length for greater than three or four times . We choose the same set of sub-sectional basis functions as the testing functions. The even and odd properties of the integrand can also be used to reduce the integration range to the first quadrant in the – coordinate, the integration of the element in (24) and (25) can be written as

Fig. 5. Frequency characteristics of the phase of open-end structure with different mesh schemes.

S

for the NRD-guide

, is equal to . To make The asymptotic term valid for all , should be much greater than . Since the integrand with large contributes little to the total integrais enough. Subtracting the asymptotic terms tion, and , which are given in the Appendix , in the first integral of (32) leads to a fast decaying integrand. The integral thus becomes well suited for numerical integration. For the second integral, called the asymptotic part, complete analytical solutions can be determined as (33) (34)

(30)

(31) The integrands , , and terms are given in the Appendix. The integrations in (30) and (31) can be carried out in the rectangular coordinate ( , ) since the integrands oscillate in coordinates and , respectively, as shown in the Appendix . The integrations are defined through infinite spectral integrals and these can be evaluated numerically by restricting the unto the finite region bounded integration interval and , where and are sufficiently large numbers. The integrands exhibit a rapidly oscillating behavior, and amand . For the plitude of the integrands decreases as integer number , it is found that , and the convergence of the integrals is thus ensured. The weakest convergence characterizes the second terms of and . At this point, it would be beneficial to search for asymptotic behavior of the integrands prior to performing the numerical computation. The can be written as integration term relating to

(32)

have a similar solution for Integration terms concerning that is changed to . In the case of a uniform mesh on a rectangular structure with basis functions, there are shifting terms and , where and in (46), (47), (52), and (53). Thus, only different system matrix must be calculated. matrix elements of the In this case, only several seconds of CPU time is needed for one frequency point. The NRD open-end is simulated with a frequency sweep between 26–30 GHz. To estimate the error introduced by the discretization of the structure, the simulation was repeated on the basis of four different meshes with increasing cell density , 6 3, 9 5, 12 7. The calculated phases of at the reference plane are plotted in Fig. 5 as a function of frequency, while the magnitudes of remains almost unit constant in all the cases. The convergence is very satisfactory at low frequencies. However, more meshes should be used at high frequency since more fields due to the radiation mode are produced and decay slowly in the high frequencies’ region. In Fig. 6, our theoretical results are verified by comparing the amwith calculated and measured results plitude and phase of in [8], showing a very good agreement. Measurement errors observed over the frequency range of interest were explained in [8], i.e., the 26.5-GHz cutoff effect of the mode launcher from the rectangular waveguide, to a large extent, causes the phase deviation and a difference in results. Around the upper part of the frequency range, the errors may come from the fact that is

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Fig. 8.

Fig. 6. Calculated and measured results of S -parameters for the NRD-guide open-end discontinuity. (a) Amplitude of S . (b) Phase of S .

Calculated amplitude of

807

S

of

H -guide open end.

appears in the integrations, which must be carried out in the as a funcpolar coordinate. Fig. 8 shows the magnitude of tion of frequency. It can be seen that the magnitude decreases very rapidly just after the radiation loss appears, then slowly as the frequency continues to increase. This phenomenon can continuous radiation mode be well explained from the configuration in Fig. 2(b). The level of radiation loss depends on the continuous spectrum width of the propagating radiation . The wider the spectrum is, the higher the mode radiation loss becomes. The slope rate of is infinite at the start , and this point of the propagating radiation mode indicates that the increasing rate of the spectrum width is infinite. Therefore, the decreasing rate of radiation loss is infinite at GHz, where , and the slope the critical point curve is also infinite at this critical point. As frerate of the becomes smaller and the quency increases, the slope rate of increasing of the continuous spectrum width becomes slower. Therefore, the decreasing of the radiation loss becomes slower. C. NRD-Guide Resonators The ORVIE approach can easily be used to solve the NRD-guide resonators. A homogeneous linear system of equations can be obtained as follows for the core circuit in Fig. 3:

(35)

Fig. 7. Field distribution of the end.

m = 1 radiation mode in the NRD-guide open

very close to the spacing limitation governed by the nonradiative condition of the guide, which is a half-wavelength in free space. Fig. 7 shows the field distribution of the radiation mode in the – -plane inside the dielectric medium. It can be seen that the field varies sinusoidally in the -direction and exponentially in the -direction. B.

-Guide Open-End

As the frequency increases, the structure in Fig. 4 will become an -guide. In this case, a pole at

Resonance frequencies can be obtained by letting the determinant of the coefficient matrix be zero as follows: (36) Four rectangular planar section resonators with different dimensions are calculated for comparison with measured results to validate the proposed method. Table I summarizes the measured and simulated resonant frequencies of the four different resonators. Theoretical results obtained with a numerical approach in [20] are also presented in Table I for comparison, and this approach effectively combines the method of lines with the mode-matching method. It is found that those results coincide well with each other for both LSE and LSM modes.

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COMPARISON

OF

TABLE I THEORETICAL AND EXPERIMENTAL RESULTS FOR THE RESONANT FREQUENCY (IN GIGAHERTZ) OF DIFFERENT NRD RESONATORS (" = 2:58)

Fig. 10. Variation of the quasi-dual resonant frequencies f and f for a square-section NRD-guide resonator [illustrated in Fig. 9(b)] as the corner’s cut d varies with parameters " = 2:53, a = 12:3 mm, and b = 10 mm.

Fig. 9. NRD-guide resonators. (a) Rectangular-section resonator. (b) Notched square-section resonator.

A notched square-section resonator is then modeled, as shown in Fig. 9(b), which was first calculated precisely with a boundary-element method (BEM) in [21]. A symmetric cut is considered along a corner of the square section (height , un). In this perturbed sides , and notched sides so that case, a pair of resonance is related to perfect electric and perfect magnetic symmetry walls placed along the notched-square minor diagonal [dashed line in Fig. 9(b)]. Fig. 10 shows a and as a variation of the quasi-dual resonant frequencies function of in the notched square-section NRD resonator for the lowest pair of resonance. It is shown that results from BEM and ORVIE methods agree very well. The maximum difference between the two methods is less than 0.01 GHz and it mainly comes from the finite discretization in these two methods. The BEM can solve resonators of arbitrary shape, but with a constant dielectric-permittivity profile. The method in [20] can model an inhomogeneous dielectric-resonator problem, but the permittivity can vary in one direction only. Our proposed ORVIE method has no such limitation, and it can handle a resonator with any shape of planar section and with dielectric permittivity varying in both the - and -direction. Fig. 11 depicts resonant characteristics of an inhomogeneous resonator

Fig. 11. Resonant frequency characteristics of an inhomogeneous dielectric resonator against the exponential decaying factor  for both LSM and LSE fundamental modes (a = 2:25 mm, b = 0:6 mm, c = 1:25 mm).

with a profile function of the dielectric permittivity in the following: (37) (in this in which is an exponentially decaying factor and example, ) the maximum permittivity of the dielectric resonator. It indicates that both the LSE and LSM modes have a quasi-linear increase with an exponential factor. This has also verified the analysis in [20] in which only the LSM mode has

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on the analysis of the mode spectrum in the NRD-guide and -guide. Vertical integration in the space domain can be carGreen’s function can be ried out analytically, and an constructed. The solution of the volume integral equation for NRD-guide and -guide circuits has then been reduced to a 2-D planar modeling problem. This method can be applied to the calculation of both the resonant frequency and scattering parameter for various dielectric circuits. The framework of the proposed technique was demonstrated through its application to an NRD-guide open-end structure. A number of examples include -guide open-end, resonators, and an air gap-coupled NRD-guide filter have been successfully modeled, and theoretical results have also been compared with other available simulations and measurements. APPENDIX The Fourier transforms of

and

are

Fig. 12. (a) Top view of a three-pole gap-coupled NRD-guide filter (a = 3:5 mm b = 2:7 mm d = d = d = 2:72 mm. L = l = 1:60 mm, l = l = 3:5 mm " = 2:04). (b) Calculated and measured transmission loss and return loss of the air gap-coupled filter. Theory (——). Measured (, – –*– –), Theory in [7] (–.–.–.).

(38) this behavior, while the LSE mode is not sensitive to a high decaying factor of the inhomogeneous dielectric. This phenomenon was explained in [20] because the dielectric permittivity only varies in the -direction, but remains constant in the -direction in that case.

(39) (40)

D. NRD-Guide Filter The last example is an air gap-coupled type three-pole 0.1-dB Chebyshev ripple bandpass filter with a 2% bandwidth at a center frequency of 49.5 GHz, which was designed in [7] based on a variational technique and fabricated with a Teflon dielectric. The ORVIE, of course, can be used to design this kind of NRD-guide filter. In this study, it was used to model the whole filter and to find its frequency characteristics. Fig. 12(a) shows the configuration of the gap-coupled NRD-guide filter with gemm, mm, ometrical dimensions mm, and mm. Simulated filter response is shown in Fig. 12(b), together with measured and simulated results [7] for comparison. Agreement between modeling and measurements of insertion loss is quite satisfactory. The excess insertion loss was measured and found to be 0.3 dB, i.e., 0.06 dB in our method, and it is more close to the designed value of 0.1 dB. The return loss in our method is a little shift to the low-frequency region.

(41)

(42) The integrands

and

are

IV. CONCLUSION In this paper, an ORVIE approach has been proposed and presented for modeling and analysis of NRD-guide and -guide circuits having any kind of section shape and inhomogeneous dielectric media. By introducing a half-sinusoidal vertical variation of field, the discretization can be made only in the parallel plane. Combined propagating mode and local mode basis functions are used in the Garlerkin’s method of moments based

(43) and (44)–(53), shown on the following page.

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(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

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has the following expressions:

(54)

(55)

The

asymptotic are

equations

of

and

811

[12] J. S. Bagby, D. P. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 10, pp. 906–915, Oct. 1985. [13] J. F. Kiang, S. M. Ali, and J. A. Kong, “Integral equation solution to the guidance and leakage properties of coupled dielectric strip waveguides,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 2, pp. 193–203, Feb. 1990. [14] S. L. Lin and G. W. Hanson, “An efficient full-wave method for analysis of dielectric resonators possessing separable geometries immersed in inhomogeneous environments,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 1, pp. 84–92, Jan. 2000. [15] S. Y. Ke and Y. T. Cheng, “Integration equation analysis on resonant frequencies and quality factors of rectangular dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 571–574, Mar. 2001. [16] K. A. Michalski and J. R. Mosig, “Multilayed media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 508–519, Mar. 1997. [17] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-30, no. 11, pp. 1191–1196, Nov. 1982. [18] R. W. Jackson and D. Pozar, “Full-wave analysis of microstrip open-end and gap discontinuities,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 10, pp. 1036–1042, Oct. 1985. [19] T. Vaupel and V. Hansen, “Electrodynamic analysis of combined microstrip and coplanar/slotline structures with 3-D components based on a surface/volume integral-equation approach,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1788–1800, Sep. 1999. [20] K. Wu, “A combined efficient approach for analysis of nonradiative dielectric (NRD) waveguide components,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 4, pp. 672–677, Apr. 1994. [21] C. D. Nallo, F. Frezza, and A. Galli, “Full-wave modal analysis of arbitrary-shaped dielectric waveguides through an efficient boundary-element-method formulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 2982–2990, Dec. 1995.

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REFERENCES [1] T. Yoneyama and S. Nishida, “Nonradiative dielectric waveguide for millimeter-wave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 11, pp. 1188–1192, Nov. 1981. , “Nonradiative dielectric waveguide,” in Infrared and Millimeter [2] Wave. New York: Academic, 1984, vol. 11, ch. 2, pp. 61–98. , “Recent development in NRD-guide technology,” Ann. [3] Telecommun., vol. 47, no. 11–12, pp. 508–514, 1992. [4] F. Furoki and T. Yoneyama, “Nonradiative dielectric waveguide circuit components using beam-lead diodes,” Electron. Commun. Jpn., pt. 2, vol. 73, no. 9, pp. 35–40, 1990. [5] T. Yoneyama, H. Tamaki, and S. Nishida, “Analysis and measurements of nonradiative dielectric waveguide bends,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 8, pp. 876–882, Aug. 1986. [6] H. Sawada, T. Yoneyama, and F. Kuroki, “Size reduction of NRD-guide bend,” in Proc. Asia–Pacific Microwave Conf., Seoul, Korea, Nov. 4–7, 2003, pp. 1450–1453. [7] T. Yoneyama, F. Kuroki, and S. Nishida, “Design of nonradiative dielectric waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 12, pp. 1659–1662, Dec. 1984. [8] F. Boone and K. Wu, “Mode conversion and design consideration of integrated nonradiative dielectric (NRD) components and discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 4, pp. 482–492, Apr. 2000. , “Full-wave model analysis of NRD guide T-junction,” IEEE Mi[9] crow. Guided Wave Lett., vol. 10, no. 6, pp. 228–230, Jun. 2000. , “Nonradiative dielectric (NRD) waveguide diplexer for mil[10] limeter-wave applications,” in IEEE MTT-S Int. Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1471–1474. [11] F. J. Tischer, “A waveguide structure with low losses,” Arch. Elektr. Uebertrag., vol. 7, pp. 592–596, Dec. 1953.

Duochuan Li was born in Huainan, Anhui Province, China. He received the B.Sc. degree in physics from Peking University, Beijing, China, in 1990, the M.Sc. and Ph.D. degrees in controlled nuclear fusion and plasma physics from the Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, Anhui, China, in 1993 and 1998 respectively, and is currently working toward the Ph.D. degree in electrical engineering at the École Polytechnique de Montréal, Montréal, QC, Canada. His research interests include computational electromagnetics, NRD waveguides, 3-D hybrid planar/nonplanar integration techniques, and substrate integrated waveguides.

Ping Yang was born in Hunan Province, China. He received the B.Eng. and M.Eng degrees in radio engineering from the Nanjing Institute of Technology (now Southeast University), Nanjing, Jangsu, China, in 1986 and 1989, respectively, and is currently working toward the Ph.D. degree in electronic engineering at the École Polytechnique de Montréal, Montréal, QC, Canada. From 1989 to 2000, he was with the Department of Information and Control Engineering, Shanghai Jiao Tong University, Shanghai, China, where he was formerly an Assistant Professor and then an Associate Professor. His current research interests involve computational electromagnetic and modeling of multilayered integrated circuits.

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Ke Wu (M’87–SM’92–F’01) was born in Liyang, Jiangsu Province, China. He received the B.Sc. degree (with distinction) in radio engineering from the Nanjing Institute of Technology (now Southeast University), Nanjing, China, in 1982, and the D.E.A. and Ph.D. degrees in optics, optoelectronics, and microwave engineering (with distinction) from the Institut National Polytechnique de Grenoble (INPG), Grenoble, France, in 1984 and 1987, respectively. He conducted research with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, prior to joining the École Polytechnique de Montréal (Engineering School affiliated with the University of Montreal), Montréal, QC, Canada, as an Assistant Professor. He is currently a Professor of Electrical Engineering and Canada Research Chair in Radio-Frequency and Millimeter-Wave Engineering. He has been a Visiting or Guest Professor with the Telecom-Paris, Paris, France, INPG, the City University of Hong Kong, Hong Kong, the Swiss Federal Institute of Technology (ETH-Zurich), Zurich, Switzerland, the National University of Singapore, Singapore, the University of Ulm, Ulm, Germany, and the Technical University Munich, Munich, Germany, as well as many short-term visiting professorships with other universities. He also holds an honorary visiting professorship and a Cheung Kong endowed chair professorship (visiting) with Southeast University, Nanjing, China, and an honorary professorship with the Nanjing University of Science and Technology, Nanjing, China. He has been the Director of the Poly-Grames Research Center, as well as the Founding Director of the Canadian Facility for Advanced Millimeter-Wave Engineering (FAME). He has authored or coauthored over 390 referred papers and also several books/book chapters. His current research interests involve hybrid/monolithic planar and nonplanar integration techniques, active and passive circuits, antenna arrays, advanced field-theory-based computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers. He is also interested in the modeling and design of microwave photonic circuits and systems. He serves on the Editorial Board of Microwave and Optical Technology Letters, Wiley’s Encyclopedia of RF and Microwave Engineering, and Microwave Journal. He is also an Associate Editor of the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE). Dr. Wu is a member of the Electromagnetics Academy, the Sigma Xi Honorary Society, and the URSI. He is a Fellow of the Canadian Academy of Engineering (CAE). He has held numerous positions in and has served on various international committees, including the vice-chairperson of the Technical Program Committee (TPC) for the 1997 Asia–Pacific Microwave Conference, the general cochair of the 1999 and 2000 SPIE International Symposium on Terahertz and Gigahertz Electronics and Photonics, the general chair of the 8th International Microwave and Optical Technology (ISMOT’2001), the TPC chair of the 2003 IEEE Radio and Wireless Conference (RAWCON’2003), and the general co-chair of the 2004 IEEE Radio and Wireless Conference (RAWCON’2004). He has served on the Editorial or Review Boards of various technical journals, including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He served on the 1996 IEEE Admission and Advancement Committee and the Steering Committee for the 1997 joint IEEE Antennas and Propagation Society (AP-S)/URSI International Symposium. He has also served as a TPC member for the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). He was elected into the Board of Directors of the Canadian Institute for Telecommunication Research (CITR). He served on the Technical Advisory Board of Lumenon Lightwave Technology Inc. He is currently the chair of the joint chapters of the IEEE MTT-S/AP-S/LEOS in Montreal, QC, Canada, and the vice-chair of the IEEE MTT-S Transnational Committee. He was the recipient of a URSI Young Scientist Award, the Oliver Lodge Premium Award of the Institute of Electrical Engineer (IEE), U.K., the Asia–Pacific Microwave Prize, the University Research Award “Prix Poly 1873 pour l’Excellence en Recherche” presented by the École Polytechnique de Montréal on the occasion of its 125th anniversary, and the Urgel-Archambault Prize (the highest honor) in the field of physical sciences, mathematics, and engineering from the French–Canadian Association for the Advancement of Science (ACFAS). In 2002, he was the first recipient of the IEEE MTT-S Outstanding Young Engineer Award.

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A 2.17-dB NF 5-GHz-Band Monolithic CMOS LNA With 10-mW DC Power Consumption Hung-Wei Chiu, Shey-Shi Lu, Senior Member, IEEE, and Yo-Sheng Lin, Member, IEEE

Abstract—Design principles of CMOS low-noise amplifiers (LNAs) for simultaneous input impedance and noise matching by are introduced. It is found tailoring device size for opt that opt close to 50 can be obtained by using small devices (110 m) and small currents (5 mA). Based on the proposed approach, CMOS LNAs with on-chip input and output matching 20 m) and normal (750 m) substrates networks on thin ( are implemented. It is found that the noise figure (NF) (3.0 dB) of the CMOS LNA at 5.2 GHz with 10-mW power consumption on the normal (750 m) substrate can be reduced to 2.17 dB after m. The reduction of NF the substrate is thinned down to is attributed to the suppression of substrate loss of the on-chip inductors. The input return loss ( 11 ) is smaller than 22 dB across the entire band of interest (5.15–5.35 GHz). An input 1-dB compression point ( 1 dB ) of 8.3 dBm and an input third-order intercept point of 0.8 dBm were also obtained for the LNA on the thin substrate.



= 50

20

Index Terms—Low-noise amplifier (LNA), MOSFET amplifier, noise figure (NF), thin substrate.

I. INTRODUCTION IGH data-rate (up to 50 Mb/s) wireless local area networks (LANs), which exploit the 300-MHz bandwidth in the 5-GHz frequency band (5.15–5.35/ 5.725–5.825 GHz) released by the Federal Communications Commission (FCC) for the unlicensed national information infrastructure (UNII) have become increasingly popular and important for mobile computing devices such as notebook computers. The allocated frequencies overlap the European standard for the highperformance radio local area network (HIPERLAN), which also operates in the 5-GHz band (5.15–5.35/5.47–5.725 GHz). Recently, many -band low-noise amplifiers (LNAs) have been implemented in various technologies for these LAN systems with excellent noise performance [1]–[14]. However, some of the LNAs with extremely low noise figures (NFs) were achieved at the expense of very high dc power consumption, and others suffered from high input/output return losses 10 dB), insufficient dynamic range ( dBm), (

H

Manuscript received March 16, 2004; revised May 11, 2004. This work was supported under Grant NSC92-2212E002-091, Grant 91EC17A05-S10017, and Grant NSC93-2752-E002-002-PAE and by the United Microelectronics Company (UMC) under the UMC University Program. H.-W. Chiu was with the Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. He is now with the Mixed-Mode and RF Library Division, Taiwan Semiconductor Manufacturing Company, Taiwan, R.O.C. S.-S. Lu is with the Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. (e-mail: [email protected]). Y.-S. Lin is with the Department of Electrical Engineering, National Chi-Nan University, Puli, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842510

or low linearity [input third-order intercept point (input IP3)]. Ultra-low NF is generally not necessary in short-range wireless applications, while low power dissipation to extend the battery life is strongly demanded for portable wireless data applications [5]. Table I is a summary of recent reports on -band LNAs mW). From Table I, it is with low dc power consumption ( clear that CMOS integrated circuits, which are receiving much attention due to their potential for low cost and the prospect of system-level integration [15], must equal to or surpass the mW) and low NF ( 3.0 dB) of low power consumption ( the bipolar and GaAs circuits in order to compete with them. Besides, as previously mentioned, good input/output match, sufficient linearity, and high dynamic range are required as well. In commercial and consumer applications, the cost must also be kept low. One solution to this is the use of a foundry process with proven yield and reliability to fabricate the circuits with on-chip input/output matching networks. Therefore, there is a need to implement an LNA, which can simultaneously balance all of the following constraints: 1) low NF ( 3.0 dB); 2) low dc power consumption ( 15 mW); 10 dB); 3) low input/output return losses ( ( 20 dBm); 4) sufficient input 5) high input IP3 ( 10 dBm); 6) low cost. In this paper, we describe 5-GHz-band CMOS LNAs on both thin ( 20 m) and normal (750 m) substrates, which meet all the above requirements. The purpose of silicon substrate thinning of the LNA is to study the effect of silicon substrate thickness on the NF performance. The emphasis of this study is to reduce the power consumption of the CMOS LNA while still retaining acceptable noise performance, good input/ output match, sufficient linearity, and a high dynamic range. A single-stage cascode amplifier topology with inductive degeneration at the source was used. In the course of developing such an LNA, a formula of the NF for a field-effect transistor (FET) with source and gate inductors was derived to facilitate circuit design. Based on the derived formula and the careful selection of the device size and bias, the gate inductor and source inductor, a CMOS on-chip matched LNA with a normal (750 m) substrate for 802.11a and HIPERLAN2 receivers exhibiting NFs of 2.7, 3.0, and 3.3 dB, input return losses of 27, 30, and 29 dB, output of 8.36, 8.3, return losses of 15, 15, and 16 dB, and 8.33 dBm, input IP3 of 0.3, 0.3, and 0.4 dBm with power consumptions of 12, 10 and 3.6 mW, respectively, were demonstrated experimentally (see Table I). In addition, it was found that the NF (2.17 dB) of the LNA with power consumption of 10 mW on a thin substrate (20 m) was better than that (3 dB)

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TABLE I SUMMARY OF THE STATE-OF-THE-ART C -BAND LNAs

of the LNA on a normal substrate (750 m) [16]. The reduction of the NF of an LNA with a thin substrate is mainly due to reduction of the substrate loss of the inductors in the LNA. II. PRINCIPLES OF CIRCUIT DESIGN The schematic of the popular source–inductor–feedback amplifier with the gate inductor for input impedance matching is shown in Fig. 1(a). Fig. 1(b) shows an equivalent circuit of is used to Fig. 1(a) for noise calculation. The source inductor achieve simultaneous input and noise matching [15]–[18] and to provide the desired input resistance (50 ) [19]. To thoroughly understand the effect of source inductance on the NF of an FET quantitatively, a formula of the NF for an FET with source and gate inductors was derived and is detailed as follows. A. NF of an FET With Source and Gate Inductors By extending the noise theory for an FET without a source inductor published by Pucel et al. [20] and referring to Fig. 1(b), for an FET with source and gate we have derived the NF inductors as follows:

Fig. 1. (a) Schematic of the popular source–inductor–feedback amplifier with an input matching gate inductor. (b) Equivalent circuit of Fig. 1(a) for noise calculation.

(1) where is the signal source resistance (usually 50 ), and represent gate and source parasitic resistances of the FET,

respectively, and stand for the series resistance of the and , respectively, inductors , is the source impedance of the is the transconductance of the FET, is signal source,

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the gate-to-source capacitance of the FET, is the Boltzmann is the constant, is absolute temperature, is frequency, induced gate noise current, and is the drain noise current. denotes “the real part of.” and can be written as (2) (3) where and are the coefficients of gate and drain noise, reand if the symbols in spectively, and are equal to [21] and [22] are used, where is the coefficient of gate noise, is the coefficient of drain noise, and is the ratio of device transconductance to zero-bias drain conductance. In fact, the expression for can be put in the following useful form by inserting (2) and (3) into (1): (4) where

,

, and

are defined as follows:

(10)

is the correlation coefficient of

and

defined by (11)

Theoretically, the value of is 0.395 for a long channel device [21]. Since and are difficult to extract from the measured results, we shall consider these functions as our fundamental noise coefficients, which can be estimated from the , , , and ), as will measured noise parameters ( in (4) is usube described shortly. The expression of NF ally written in terms of minimum NF , and the optimal or source admittance source impedance as follows:

(5) (12) (6)

(13)

(7)

(14)

Comparing (5)–(7) with (35)–(37) derived in the Appendix, it , , and are the functions of , , and can be seen that as follows:

where , , , and are given by (15)–(19), shown at the bottom of this page. Now, several observations are described in order here. For the time being, let us suppose that . From (18), we know that the introduction of , to a , which is consisfirst-order approximation, does not affect tent with the experimental results [23]. According to (15) and is independent of (17), it is also interesting to note that , while is reduced when is introduced; i.e., in the Smith chart should follow a constant resistance ( ) ) when increases, as circle with decreasing reactance ( shown in Fig. 2(a), which, to a first-order approximation, agrees

(8)

(9)

(15)

(16)

(17) (18) (19)

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tion condition applied to the drain is immaterial). The locus with inof the complex conjugate of the input impedance is also shown in Fig. 2(a). The two loci intercept creasing at some point, which stands for the simultaneous impedance and if a suitable source inand noise matching if is chosen. Note that, if , it is impossible ductor to achieve impedance matching and noise matching simultaneously. Therefore, it is important for a circuit designer to check of the CMOS process that will be used is close whether the to 1 or not before the source inductor feedback technique can be used for simultaneous input and noise matching. We will discuss later. For the time being, if we assume , the value of and can be used to then a matching network consisting of transform the signal source impedance (usually 50 ) into this intercept point, as shown in Fig. 2(b). For the ease of matching, if one may even eliminate the need of is chosen to be . According to (15), it is doable to by a careful selection of transistor size ( ) and modify is then selected so that transconductance ( ) or bias. ( ) is also 50 and and are coincident at a point on the constant 50- circle in the Smith chart. Nevertheless, from (17), it is, in general, not possible to to zero by merely changing transistor parameters or reduce by the use of . Hence, a gate inductor must be added to to zero in order to achieve the minimum NF conreduce ). The loci of and cordition ( responding to this situation are shown in Fig. 2(c). Also note ( ) is reduced, is reduced as well acthat when cording to (19). However, one must be cautious that should not be too large or may become negative, and both will increase accordingly. the NF and B. Design Procedure of the Source-Inductor-Feedback LNA

Fig. 2. Loci of Z and Z in the Smith chart with increasing L and L . (a) To a first-order approximation, Z follows a constant resistance circle with decreasing reactance (X ) when L and L are increased and the loci of Z and Z intercept at some point. (b) The loci of Z and Z in the Smith chart and Z in the Smith chart with L with L , C , and L . (c) The loci of Z and L when R = 50 is required.

with the published experimental data [17]. On the other hand, of the LNA is given by the input impedance

(20) if gate-to-drain capacitance ( ) and output resistance ( ) of the FET are neglected (and, correspondingly, the termina-

Suppose that an LNA in an RF receiver is preceded by a high-frequency bandpass filter whose output resistance is 50 , . As previously mentioned, has to then has to be zero in order to achieve the minimum be 50 and if we want to avoid additional input matching compoNF nents other than . According to (15), can be set to 50 by choosing suitable and if and are known. Simcan be reduced to zero by choosing suitable ilarly, if is given. , , and can be estimated from the measured , , and of an FET without source and gate ) based on the inductors (i.e., can be determined by following proposed steps. First of all, can be determined by (6). Third, can be (19). Second, determined by (16). Finally, can be determined by (17). , , and were estimated from a test device before the design , , and of the LNAs studied in this paper. It is found that are weak functions of frequency [see Fig. 3(a)]. In addition, is 1 within an experimental uncertainty. Therefore, at least for the current CMOS process, impedance matching and noise matching can be achieved simultaneously. Besides, as shown in Fig. 3(b)–(d), the agreements between the predicted and experi, , and are quite satisfactory for mental values of the design of LNAs. As mentioned before, in an average modern CMOS process, is always close to 1 or not is crucial. From (9), is whether

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Fig. 3. (a) Extracted K , K , and K versus frequency characteristics of a MOSFET without source and gate inductors (i.e., L = L = R = R = 0). (b) Comparison between predicted and experimental R . (c) Comparison between predicted and experimental X . (d) Comparison between predicted and experimental F .

a function of the ratio , and . Since in the short-channel regime is not currently known, we have assumed to be equal to its long channel value (i.e., 0.395), as was done in [22]. The versus based on (9) are plotted in Fig. 4. values of is beyond 0.042, From this figure, it is clear that when is a monotonically decreasing function of . Since the ratio can be interpreted physically as the relative contribution of gate noise and drain noise to the total noise, we may say deviates from 1 substantially if the gate noise plays an that important role in a device. Some specific values of ( ) may be given to get a feeling of the values of . For a long-channel FET, and (twice ) [21]. For a short-channel device, the assumptions made in [22] are adopted, i.e., continues to be approximately twice as large as and, hence, although and are taken as 1–2 and 2–4, respectively, the ratio is still a constant 2. Now is and were used for longonly a function of . If and short-channel devices [22], respectively, then the calculated can be summarized as ratio (long channel) (short channel). (21) From (21) and Fig. 4, the corresponding is 0.833 and 0.911 for long- and short-channel devices, respectively. Therefore, for should be close to 1. In an average modern CMOS process, , , and are functions of and from addition, (8)–(10). If is assumed to be equal to its long channel value

Fig. 4. Calculated K technologies.

versus R=P characteristics of modern CMOS

(i.e., 0.395), then (the indication of the gate length) is the only parameter that most strongly affect the values of the three -parameters, i.e., process parameters highly related to gate length ( ) will most strongly affect the values of the three -parameters. The complete schematic of our LNA is shown in Fig. 5. A single-stage topology is chosen to minimize the power dissipa) and tion and to improve input 1-dB compression point ( input IP3 [15]. A cascode configuration is used to improve stability and to reduce the Miller effect. The source inductor is used for simultaneous input and noise matching, while inductors and and capacitor are used for output matching. Fig. 6 presents a flowchart of the design procedures for finding the size and bias of the transistor. First, the current [or gate overdrive voltage ( - )] and current density

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Fig. 5. Complete schematic of the LNA.

gain cutoff frequency of the transistor are set to the values of the test device. As corresponding to the minimum and corresponding to the minimum shown in Fig. 7, (1.0 dB) of the test device are 49 A m and 26 GHz, meets the specification, respectively. Second, check if which is set to 10 dB. If yes, go to the next step. If no, return to in order to make meet the the first step, i.e., sacrifice of the LNA is given by specification. It can be shown that (22) where is the resonant frequency of is the resistance the input/output matching networks and is not seen at point D2 at the resonant frequency. Since known a priori, a conservative and worse case value of 50 is chosen for the initial estimate of . Third, for noise matching, is set to 50 and, from (16), the required can be deis decided by and . Fifth, cided. Fourth, the required is decided by and . Finally, if the the driving current power consumption and linearity meet the specification, then the finding of the size and bias of the transistor is completed. Otherwise, go back to step 1 and use lower current density. After deciding the size and bias of the transistor, the other components of the LNA can be determined as follows. The input of the LNA is known to be [19] and, resistance is determined when is set to 50 . is calculated thus, to zero. , , and capacitor by setting (17), i.e., are determined by the standard matching technique. The calcum and lated result shows that the best condition is mA for . The measured result shows that is achieved when m and mA, which not only agree with the calculated results, but also expericlose to 50 can be obtained by mentally demonstrate that using small device and small current and, therefore, the power constraint method [19], [22] is not always necessary. III. CIRCUIT IMPLEMENTATION AND MEASUREMENT RESULTS The LNA under study was fabricated with a standard 0.25- m mixed-signal/RF CMOS technology on a p-type

Fig. 6. Flowchart of the design procedures for finding optimized transistor’s size and bias.

20 cm provided substrate with substrate resistivity by the commercial foundry United Microelectronics Corporation (UMC), Hsinchu, Taiwan, R.O.C. The main features of the backend processes are as follows. There are five metal layers. Metal-4 (M4) and Metal-5 (M5) were used as the underpass metal layer and the top metal layer of the inductors, respectively. The top metal thickness and underpass metal thickness were 2 and 0.6 m, respectively. The oxide thickness between top metal and underpass metal, and the oxide thickness between underpass metal and silicon substrate were 1 and 6.2 m, respectively. No patterned ground shield was implemented below the inductors. Normal substrate thickness was approximately 750 m. Low- dielectric material was used

CHIU et al.: 2.17-dB NF 5-GHz BAND MONOLITHIC CMOS LNA WITH 10-mW DC POWER CONSUMPTION

Fig. 7. F 410 m.

and f versus current density J of a test device with gatewidth

Fig. 9.

Fig. 8.

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Die photograph of the LNA.

as the inter-metal-dielectric (IMD) layers for high-performance mixed-signal/RF-CMOS applications. Fig. 8 presents a die photograph of the LNA. This circuit, which occupies an area of 500 795 m , is operated with V. The substrate of the LNA was thinned down to approximately 20 m for the purpose of studying the effect of silicon substrate thickness on the NF performance of the LNA. Fig. 9 shows the detailed process flow to thin down the substrate of the LNA. First, stick the front side (the side with the LNA) of the chip (5 mm 5 mm) to the glass substrate with wax. Second, polish the back of the chip mechanically to the target thickness, i.e., 20 m, by holding the glass substrate facing down onto a rotating pad with a diamond sand paper. Third, apply photosensitive epoxy to the back of the chips and stick another glass substrate to it, followed by a 3-s UV exposure for epoxy curing and activation. Viscosity is enhanced as the exposure time increases. Fourth, soften the wax by heating so that the glass substrate on the front side of the chip can be removed. Finally, clean the chip by acetone. The thickness of the overall die plus photosensitive epoxy is approximately 40 m. Since a glass substrate has

Detailed process flows of silicon substrate thinning.

been attached to the backside of the chip, it is easy to handle the chip. For the current small 5 mm 5 mm chip, it is found that the yield is 100% in the laboratory. It is not clear to us that if this process will be a low-cost and high-yield process in the industry. However, as technology advances, we believe that the yield of this process should also be high even for large diameter wafers used in the industry. The noise and scattering parameters were measured on wafer using an automated NP5 measurement system from ATN Microwave Inc., Palo Alto, CA. The measured characteristics of NFs versus frequency for the LNA on a normal substrate with different power consumptions are shown in Fig. 10(a). Minimum NFs of 2.7, 2.9. and 3.3 dB were obtained at the frequency 5.2 GHz with power consumptions of 12, 7.6, and 3.6 mW, respectively. The frequency (5.2 GHz) where minimum NF hap. Fig. 10(b) compares the measured charpens is denoted by acteristics of NFs versus frequency for the LNA on both normal and thin substrates with power consumption of 10 mW. Minimum NFs of 2.17 and 3.0 dB are obtained around the frequency GHz for the LNAs on the thin and normal substrates, respectively. Our results unequivocally demonstrate that low NFs and low power consumptions can be achieved simultaneously with on-chip input and output matching networks in CMOS technology at 5-GHz band. In fact, the difference between minimum and 50- NF is only 0.2 dB at 5.2 GHz, indicating an approximation to optimum noise matching due to the to the desired value of 50 ( ). From our design of

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Fig. 10. (a) Measured characteristics of NFs versus frequency for the LNA on a normal substrate (750 m) with different power consumptions. (b) Measured characteristics of NFs versus frequency for the LNA on both normal (750 m) and thin (20 m) substrates with 10-mW power consumption.

experimental data, we conclude that setting does (device size) or large current, not necessarily require large at least for the CMOS process we used. Fig. 11(a) shows the measured and simulated quality factors versus frequency characteristics of a 5.5-turn testing inductor fabricated at the same time with the LNAs on both normal (750 m) and thin substrates (50 and 20 m). The track width and gap between tracks of this inductor were 8 and 2 m, respectively. The inner dimension inside the inner coil was 60 m 60 m. As can be seen, the measured maximum ) were 6.9, 8.5, and 11.3, respectively, quality factors ( for silicon substrate thicknesses of 750, 50, and 20 m. The ) were 13.5, 14.7, measured self-resonance frequencies ( and 16.7, respectively, for silicon substrate thicknesses of 750, 50, and 20 m, i.e., 63.8% (from 6.9 to 11.3) and 23.7% (from and , re13.5 to 16.7) performance improvements of spectively, are achieved if the silicon substrate is thinned down from 750 to 20 m. This means the silicon substrate thinning is effective in improving both the quality factor and resonant frequency of the inductors due to the reduction of silicon substrate loss. It is interesting to note that silicon substrate thinning can largely improve both the and of an inductor, while metal or polysilicon pattern-ground-shield (PGS) can only , but deteriorate of an inductor [26]. Taking improve the inductor with the best performance (i.e., with polysilicon PGS on a silicon substrate with resistivity cm) in is only 33% (from [26], for example, the improvement in

Fig. 11. (a) Measured and simulated quality factors versus frequency characteristics. (b) Measured minimum NFs and the inverse of maximum available power gains versus frequency characteristics of a 5.5-turn testing inductor fabricated at the same time with the LNAs on both normal (750 m) and thin (50 and 20 m) substrates.

5.08 to 6.76), but the deterioration in is up to 47.1% (from 6.8 to 3.6). The measured of a 5.5-turn testing gate inductor obtained was compared to the calculated from the measured -parameters at various silicon substrate thicknesses (750, 50, and 20 m), as shown in Fig. 11(b). conformed well to the As can be seen, the measured calculated [24]. In addition, the measured at 5.4 GHz was 1.07, 0.83, and 0.52 dB, respectively, for silicon substrate thicknesses of 750, 50, and 20 m. This means the silicon substrate thinning is effective in improving of the inductors due to the reduction of silicon substrate loss. Besides, the measured -parameters of a testing MOSFET with gate length of 0.25 m fabricated at the same time with the LNAs on both normal (750 m) and thin substrates (50 and 20 m) are nearly the same, i.e., silicon substrate thinning shows no effects on the performance of MOSFETs. Based on the above results, the reason why silicon substrate thinning can improve the NF of an LNA can be explained as follows. Since the thickness of the thin substrate ( 20 m) is much smaller than the diameter of the inductor (on the order of 100 m) [26], [27] and much larger than the gate length of the MOSFETs (0.25 m), the reduction of the NF of an LNA with a thin substrate should be mainly due to the reduction of substrate loss of the inductors in the LNA. The measurement of the NFs on the 5.5-turn testing inductor shows a 0.55-dB reduction of NF

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Fig. 12. Measured characteristics of input return loss (S ) versus frequency for the LNA on both normal (750 m and thin (20 m) substrates with 10-mW power consumption.

(from 1.07 to 0.52 dB) after substrate thinning to 20 m [see Fig. 11(b)]. This result demonstrates the 0.83-dB reduction of the NF of the LNA (from 3.0 to 2.17 dB) is reasonable. versus Fig. 12 shows the measured characteristics of frequency for the LNAs on both normal and thin substrates with power consumption of 10 mW. Minimum return losses of 5.1 GHz (thin substrate) and 30 dB at of 45 dB at of 5.3 GHz (normal substrate) were achieved as a result of . The in-band (5.15 5.35 GHz) return the right choice of losses were below 18 dB and 22 dB for normal and thin substrates, respectively, indicating a very good match even over is very close to . This is important the band. Note that for simultaneous input matching and noise matching. In fact, it is given by can be shown from (4) that the frequency

(23) If

and

is not far from unity, (23) reduces to

(24) Take, for example, the experimental values of the LNA on a normal substrate at power consumption of 10 mW, i.e., GHz and GHz, as shown in Figs. 10(b) and 12, respectively. A value of was determined, which is approximates 1 explains why close to 1. The fact that and are so close to each other. Thus, can be interpreted physically as a measure of the separation between and . In other words, if a CMOS process does not result in close to 1, it is almost hopeless for circuit designers to dea sign a circuit with simultaneous input impedance matching and noise matching over the narrow band of interest. The measured transducer gain ( ) for the LNA on a normal substrate with different power consumptions is shown in Fig. 13(a). The bandpass nature of this amplifier is evident from the plot. Under the bias condition of 12 mW, the gain has

Fig. 13. Measured transducer gain (S ) versus frequency of the LNA: (a) on a normal substrate (750 m) with different power consumptions and (b) on both normal (750 m) and thin (20 m) substrates with 10-mW power consumption.

a peak value of 11.5 dB. For a direction conversion receiver, the load resistance is not necessarily 50 and a higher gain can be expected. For lower power consumptions, the characteristics are similar, but with lower gains of 10.4 dB at 7.6 mW and 8 dB at 3.6 mW. One figure-of-merit is the ratio of gain to dc power consumption [2]. The values of gain to dc power consumption attained by this CMOS LNA were 0.95, 1.4, and 2.2 dB/mW at three different power consumptions of 12, 7.6 and 3.6 mW, and available gain for the LNA respectively. The measured on both normal and thin substrates with power consumption of 10 mW are shown in Fig. 13(b). The available gain has peak values of 10.9 and 11.2 dB for normal and thin substrates, respectively. The values of gain to dc power consumption attained are 1.12 (thin substrate) and 1.09 (normal substrate) dB/mW. As can be seen clearly in Table I, the obtained ratios of gain to dc power consumption are comparable to the other state-of-the-art -band LNAs shown in Table I. Microwave power performances were measured by a load–pull ATN system with automatic tuners. The measured and input IP3 data for the LNA on a thin (20 m) substrate under power consumption of 10 mW are shown in Fig. 14. As can be seen, an input of 8.3 dBm and an and input IP3 of 0.3 dBm were obtained. Other measured input IP3 data are summarized in Table I. A summary of the measured amplifier characteristics at different bias conditions is also included in Table I.

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, senoise generators from transistor gate resistance noise , induced gate noise current ries resistance of gate inductor , transistor source resistance noise , series resistance of source inductor , drain noise current , and signal source resistance noise . The noise generators representing the extrinsic thermal sources are given by their mean square values , , and yields the following equations: Fig. 14. Measured input P and input IP3 for the LNA on a thin substrate (20 m) with 10-mW power consumption.

,

, . A careful analysis

(26)

IV. CONCLUSIONS to a desired value by the selecA method of designing tion of transistor device size and transconductance or bias is pro, NFs of 2.17 and 3.0 dB, and posed. By setting input return losses of 45 and 30 dB at the 5-GHz band from a monolithic CMOS LNA with 10-mW dissipation on both thin ( 20 m) and normal (750 m) substrates are demonstrated with a standard 0.25- m CMOS process provided by a commercial foundry. The fact that is close to 1 in the CMOS process we used can achieve low NF and low input return loss simultaneously in the narrow band of interest. In addition, substrate thinning is effective in reducing the NF. The result also unequivocally demonstrates that low NFs and low power consumption can be achieved simultaneously with on-chip input and output matching networks in CMOS technology at the 5-GHz band. From both a performance and a cost perspective, these experimental results show that CMOS is very competitive with silicon bipolar and GaAs technologies.

(27)

(28)

(29)

(30)

(31) APPENDIX DERIVATION OF THE NF FOR AN FET WITH GATE AND SOURCE INDUCTORS A noise analysis of the circuit shown in Fig. 1(b) quantifies the effects of the gate and source inductors. An equivalent circuit for noise calculation is depicted in Fig. 1(b). By definition the NF, can be expressed in the form

(25) where stands for the total noise current in the short-circuited drain–source path originated from the noise current com, , , , , , and produced by the ponents

(32) Inserting (26)–(32) into (25) yields

(33) Substituting into (33), we get (34), shown at the bottom of this page. By

(34)

CHIU et al.: 2.17-dB NF 5-GHz BAND MONOLITHIC CMOS LNA WITH 10-mW DC POWER CONSUMPTION

analogy with the results derived by Pucel et al., (34) can be , , and are defined as put in the form of (4) if follows:

(35)

(36)

(37)

and, thus, the NF of an FET with gate and source inductors are derived. ACKNOWLEDGMENT The authors gratefully acknowledge Dr. G. W. Huang, W. Wang,, Nano-Device Laboratory (NDL), Hsinchu, Taiwan, R.O.C., for high-frequency measurements and J. J. Yu, UMC, Hsinchu, Taiwan, R.O.C., forhelpful discussions. REFERENCES [1] K. W. Kobayashi, A. K. Oki, L. T. Tran, and D. C. Streit, “Ultra low dc power GaAs HBT S - and C -band low noise amplifiers for portable wireless applications,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 3055–3061, Dec. 1995. [2] H. Morkner, M. Frank, and D. Millicker, “A high performance 1.5 dB low noise GaAs PHEMT MMIC amplifier for low cost 1.5-8 GHz commercial applications,” in IEEE Microwave Millimeter-Wave Monolithic Circuits Symp. Dig., 1993, pp. 13–16. [3] Y. Tsukahara, S. Chaki, Y. Sakaki, K. Nakahara, N. Andoh, H. Matsubayasi, N. Tanino, and O. Ishihara, “A C -band 4 stage low noise miniaturized amplifier using lumped elements,” in IEEE MTT-S Int. Microwave Symp. Dig., 1995, pp. 1125–1128. [4] U. Lott, “Low dc power monolithic low noise amplifier for wireless applications at 5 GHz,” in IEEE Microwave Millimeter-Wave Monolithic Circuits Symp. Dig., 1998, pp. 81–84. [5] S. Yoo, D. Heo, J. Laskar, and S. S. Taylor, “A C -band low power high dynamic range GaAs MESFET low noise amplifier,” in IEEE Microwave Millimeter-Wave Monolithic Circuits Symp. Dig., 1999, pp. 81–84. [6] J. J. Kucera and U. Lott, “A 1.8 dB noise figure low dc power MMIC LNA for C -band,” in IEEE GaAs IC Symp. Dig., 1998, pp. 221–224. [7] M. Soyuer, J.-O. Plouchart, H. Ainspan, and J. Burghartz, “A 5.8 GHz 1-V low noise amplifier in SiGe bipolar technology,” in IEEE Radio Frequency Integrated Circuits Symp. Dig., 1997, pp. 19–22. [8] H. Samavati, H. R. Rategh, and T. Lee, “A 5-GHz CMOS wireless LAN receiver front end,” IEEE J. Solid-State Circuits, vol. 35, no. 5, pp. 765–772, May 2000. [9] E. Westerwick, “5-GHz band CMOS low noise amplifier with a 2.5 dB noise figure,” in IEEE Int. VLSI Technology, Systems, Application Symp. Dig., 2001, pp. 224–227. [10] P. Leroux and M. Steyaert, “High-performance 5.2-GHz LNA with on-chip inductor to provide ESD protection,” Electron. Lett., vol. 37, pp. 467–469, Mar. 2001. [11] H. Hashemi and A. Hajimiri, “Concurrent multiband low-noise amplifiers-theory, design, and application,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 288–301, Jan. 2002.

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[12] D. J. Cassan and J. R. Long, “A 1-V transformer-feedback low-noise amplifier for 5-GHz wireless LAN in 0.18-m CMOS,” IEEE J. SolidState Circuits, vol. 38, no. 3, pp. 427–435, Mar. 2003. [13] C.-Y. Cha and S.-G. Lee, “A 5.2-GHz LNA in 0.35-mm CMOS utilizing inter-stage series resonance and optimizing the substrate resistance,” IEEE J. Solid-State Circuits, vol. 38, no. 4, pp. 669–672, Apr. 2003. [14] T. P. Liu and E. Westerwick, “5-GHz CMOS radio transceiver front-end chipset,” IEEE J. Solid-State Circuits, vol. 35, no. 12, pp. 1927–1933, Dec. 2000. [15] B. A. Floyd, J. Mehta, C. Gamero, and K. O. Kenneth, “A 900-MHz, 0.8-m CMOS low noise amplifier with 1.2-dB noise figure,” in Proc. IEEE CICC, 1999, pp. 661–664. [16] H. W. Chiu and S. S. Lu, “A 2.17 dB NF, 5 GHz band monolithic CMOS LNA with 10 mW DC power consumption,” in IEEE VLSI Symp. Dig., 2002, pp. 226–229. [17] M. Aikawa, T. Oohira, T. Tokumitsu, T. Hiroda, and M. Muraguchi, Monolithic Microwave Integrated Circuits (in Japanese). Tokyo, Japan: EIC, 1997, p. 90. [18] Y. Konishi and K. Honjo, Microwave Semiconductor Circuits (in Japanese). Tokyo, Japan: Nikan Industrial News Publisher, 1993, p. 114. [19] D. K. Shaeffer and T. H. Lee, “A 1.5 V, 1.5 GHz CMOS low noise amplifier,” IEEE J. Solid-State Circuits, vol. 32, no. 5, pp. 745–759, May 1997. [20] R. A. Pucel, H. A. Haus, and H. Statz, “Signal and noise properties of Gallium arsenide field effect transistors,” in Advances in Electronics and Electron Physics, L. Morton, Ed. New York: Academic, 1975, vol. 38, pp. 195–265. [21] A. V. D. Ziel, Noise in Solid State Devices and Circuits. New York: Wiley, 1986, p. 90. [22] T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits. Cambridge, U.K.: Cambridge Univ. Press, 1998, pp. 272–303. [23] A. Anastassiou and M. J. O. Strutt, “Effect of source lead inductance on the noise figure of a GaAs FET,” Proc. IEEE, vol. 62, no. 3, pp. 406–408, Mar. 1974. [24] B. Razavi, RF Microelectronics. Upper Saddle River, NJ: PrenticeHall, 1998, p. 47. [25] H. Fukui, “Design of microwave GaAs MESFET’s for broadband low noise amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 27, no. 7, pp. 643–650, Jul. 1979. [26] C. P. Yue and S. S. Wong, “On chip spiral inductors with patterned ground shields for Si-based RF IC’s,” IEEE J. Solid-State Circuits, vol. 33, no. 5, pp. 743–752, May 1998. [27] D. Melendy, P. Francis, C. Pichler, K. Hwang, G. Srinivasan, and A. Weisshaar, “Wide-band compact modeling of spiral inductors in RF-IC’s,” in IEEE MTT-S Int. Microwave Symp. Dig., 2002, pp. 717–720.

Hung-Wei Chiu was born in Taipei, Taiwan, R.O.C., in 1976. He received the B.S. degree from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 1998, and the M.S. and Ph.D. degrees from National Taiwan University, Taipei, Taiwan, R.O.C., in 2000 and 2003, respectively, all in electrical engineering. In 2004, he joined the Taiwan Semiconductor Manufacturing Company, Taiwan, R.O.C., as a Designer with the Mixed-Mode and RF Library Division. Since then, he has been involved in the area of automation of mixed-mode and RF circuit design.

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Shey-Shi Lu (S’89–M’91–SM’99) was born in Taipei, Taiwan, R.O.C., on October 12, 1962. He received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1985, the M.S. degree from Cornell University, Ithaca, NY, in 1988, and the Ph.D. degree from the University of Minnesota at Minneapolis–St. Paul, in 1991, all in electrical engineering. His M.S. thesis concerned the planar doped barrier hot electron transistor. His doctoral dissertation concerned the uniaxial stress effect on the AlGaAs/GaAs quantum well/barrier structures. In August 1991, he joined the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. His current research interests are in the areas of radio-frequency integrated circuits (RFICs)/monolithic microwave integrated circuits (MMICs), and micromachined RF components.

Yo-Sheng Lin (M’02) was born in Puli, Taiwan, R.O.C., on October 10, 1969. He received the Ph.D. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1997. His Ph.D. degree concerned the fabrication and study of GaInP/ InGaAs/GaAs doped-channel FETs and their applications on monolithic microwave integrated circuits (MMICs). In 1997, he joined the Taiwan Semiconductor Manufacturing Company (TSMC), as a Principle Engineer for 0.35/0.32 dynamic random access memory (DRAM) and 0.25 embedded DRAM technology developments with the Integration Department of Fab-IV. Since 2000, he has been responsible for 0.18/0.15/0.13-m CMOS low-power device technology development with the Department of Device Technology and Modeling, Research and Development. In 2001, he became a Technical Manager. In August 2001, he joined the Department of Electrical Engineering, National Chi-Nan University, Puli, Taiwan, R.O.C., where he is currently an Associate Professor. His current research interests are in the areas of characterization and modeling of RF active and passive devices, and radio-frequency integrated circuits (RFICs)/monolithic microwave integrated circuits (MMICs).

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Compact Dual-Fed Distributed Power Amplifier Kimberley W. Eccleston, Member, IEEE

Abstract—The dual-fed distributed amplifier (DFDA) allows efficient combining of field-effect transistors (FETs) at the device level without using -way power combiners. However, the FETs must be spaced 180 , resulting in physically large circuits. In this paper, meandered artificial transmission lines (TLs) comprised of microstrip lines periodically loaded with short open-circuit stubs can be used in place of TLs to reduce the size. The approach incorporates FET input and output capacitances with the artificial TLs, thereby eliminating their detrimental effects on bandwidth and performance. Both simulation and experimental results of a class-A three-FET single-ended DFDA designed to operate at 1.8 GHz demonstrate the feasibility of this approach and the validity of the design method. The size reduction is approximately one-third compared to realization using microstrip lines only, and the maximum efficiency is greater than 35% over a bandwidth of 15%. Index Terms—Distributed amplifiers, microstrip circuits, microwave field-effect transistor (FET) amplifiers, microwave power FET amplifiers.

Fig. 1. Generic schematic of optimum single-ended dual-fed distributed power amplifier.

I. INTRODUCTION

T

O OBTAIN high microwave powers using solid-state devices, it is necessary to combine the output power of several transistors. Parallel power combining is a popular method and is performed: 1) at the system level, involving separate design of amplifier modules and -way combiners [1] or 2) at the device level, involving combiners that integrate both combining and impedance matching [2]. However, inter-connecting transmission lines (TLs) between amplifier modules and combiners add to the size and have nonzero insertion loss in the case of the former, while the necessity of using wide microstrip lines limits the performance of the latter [3]. Distributed amplification [4] is a method where power combining is performed directly at the device level without the need for -way combiners. The conventional distributed amplifier (CDA) offers ultra-wide bandwidth, but has low efficiency and some of the transistors are under utilized—particularly at microwave frequencies [5]. The dual-fed distributed amplifier (DFDA) [6] comprises a pair of TLs that are periodically coupled by transistors, a hybrid to drive both ends of the gate (input) line, and another hybrid to combine waves appearing at the ends of the drain (output) line. There is no idle gate and drain line terminations in the DFDA, which are the bane of gain and efficiency of the CDA. Early investigations of the DFDA have demonstrated its gain [6] and efficiency [7], [8] advantages over the CDA.

Manuscript received February 29, 2004; revised May 13, 2004. This work was supported in part by the National University of Singapore under Academic Research Fund Grant R-263-000-249-112. The author was with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260. He is now with the Department of Electrical and Computer Engineering, University of Canterbury, Christchurch, New Zealand (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842508

The gain of the DFDA is maximized when 180 hybrids are used [9]. For the case of an even number of field-effect transistors (FETs) and use of 180 hybrids, 180 electrical spacing between the FETs results in all FETs contributing equal output power [10] and operating into identical load trajectories [11]. These identical load trajectories can be made optimum, thereby ensuring efficient operation and full utilization of all FETs [11]. Under this condition, the combining efficiency is 100% [10]. However, despite the above advantages, the DFDA has severe input and output match. The single-ended dual-fed distributed amplifier (SE DFDA) is a half-circuit equivalent of the DFDA [12]. Two SE DFDAs can be combined using 90 hybrids to form a balanced amplifier with its inherently good port match [12], [13]. Similar to the DFDA, the SE DFDA operates efficiently when ideal FETs are electrically spaced 180 [13]. The number of FETs in this optimum SE DFDA is, in principle, arbitrary. The SE DFDA has been demonstrated in practice to be a viable and efficient method of combining FET output power without using -way combiners for both class-A operating conditions [14] and class-B operating conditions [15]. The optimum SE DFDA, however, requires 180 electrical spacing between the FETs [13], and means that the circuit is physically large at low microwave frequencies. Meandering of the TLs can be used to achieve compactness [16]. The use of artificial transmission lines (ATLs) in place of TLs can be used to reduce the length [17]–[19]. Photonic-bandgap structures that can be implemented in a microstrip line allow reduction of length [20], but explicit design formulas are as yet unavailable. Another approach is to replace a long length of TL with a shorter length of TL with shunt capacitances at each end [21], [22]. In this paper, we demonstrate that meandered ATLs result in significant reduction of circuit size. The proposed

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TABLE I EFFECT OF FET INPUT AND OUTPUT CAPACITANCE ON THREE-FET SE DFDA PERFORMANCE

method incorporates FET input and output capacitances into the ATLs, thereby eliminating the detrimental effects of these capacitances upon bandwidth and performance. II. EFFECT OF FET INPUT AND OUTPUT CAPACITANCE ON THE SE DFDA Fig. 1 shows a schematic of an -FET SE DFDA. It is important to note that there are no idle resistive gate and drain line terminations, as is the case in the CDA. The short-circuit terminations of both the gain and drain lines emulate the dual feeding of the gate and drain lines [15]. Other pure reactive terminations can be used instead of short circuits [23], [24], though shortcircuit terminations, as shown in Fig. 1, allow convenient application dc-bias voltages to the FETs [13]. In the case of the circuit of Fig. 1, the generator Thevenin impedance is equal to the gate line characteristic impedance , and the load impedance is equal to the drain line character. The electrical length of each TL section is istic impedance indicated in Fig. 1. Namely, the electrical spacing between FET gates and FET drains is , and the short-circuit terminations of the gate and drain lines are located from the th FET. We assume that the FETs are identical. If we now assume that FET input and output capacitances are zero (at least at the center frequency), then with equal to 180 : 1) the FET drain current magnitudes are equal; 2) the FET drain voltage magnitudes are equal; and 3) for each FET, the drain voltage is antiphase with respect to the drain current [11], [13]. In other words, the FET load trajectories are identical when is equal to 180 [11], [13]. It has been shown [10] that the dual-feeding mechanism partially compensates for line loss, resulting in similar load trajectories in the presence of line loss. For efficient operation and full utilization of the FET, we want the drain current to swing between zero and its maximum al), and the drain voltage to swing between lowable value ( ) and the maximum allowable voltage the knee voltage ( ). Hence, for class-A operation of the SE DFDA, the ( FET load trajectories are optimum when [13] (1) The above principles and design method assume that the FET input and output capacitances are zero. In practice, the input and output capacitances of microwave FETs are nonzero, but their effects are minimal at low microwave frequencies, especially if the characteristic impedance of the gate and drain lines is sufficiently low [14]. The gate characteristic impedance can be

made arbitrarily low, but the drain line characteristic impedance is constrained by (1). Fortunately, the output capacitance is less than the input capacitance. Alternatively, the input and output of each FET can be parallel resonated by shunt inductances to eliminate the effects of FET input and output capacitances at the center frequency [11], [13]. However, the inclusion of parallel resonant circuits into the SE DFDA limits the bandwidth. Simulations of a three-FET SE DFDA were performed to investigate the effect of FET input and output capacitance on the circuit of Fig. 1, and to investigate the feasibility of reducing the size of the SE DFDA using capacitive loading [21], [22] at the FET ports. To eliminate other artifacts such as nonlinearity, parasitic inductances, etc., the FETs were represented by a simple linear model comprising a transconductance and input and were both equal to 50 . and output capacitances. The following four cases were considered: Case 1) Zero FET capacitances. Case 2) 0.5-pF FET input and output capacitance and equal to 180 . Case 3) 0.5-pF FET input and output capacitance and adjusted to compensate for these capacitances. Case 4) Shunt capacitances at FET ports adjusted so that a one-third circuit size reduction is achieved. The results of these simulations are summarized in Table I. In all cases, the load trajectories are essentially resistive about the small-signal center frequency ( ). The right-most column of Table I gives a qualitative comparison of the FET load trajectories. Identical load trajectories mean that all FETs are equally utilized and, hence, all FETs can be made to operate optimally and be fully utilized. When the load trajectories differ significantly, at most, only one FET can operate optimally. As expected, Case 1—being the ideal case—allows for optimum operation and has the largest small-signal bandwidth. Cases 2 and 3 represent the cases of capacitance of a typical medium power microwave FET. The effect of the capacitance is a noticeable reduction in small-signal bandwidth. Case 4 demonstrates that considerable circuit size reduction can be achieved, but is at the expense of bandwidth and the load trajectories are unsatisfactory. The conclusion of this investigation is that the effects of the FET input and output capacitances need to be eliminated for maximum bandwidth and optimum performance. Further, the use of shunt capacitive loading at the FET ports is unfeasible for reducing the size of the SE DFDA. This is quite different from the CDA, where FET input and output capacitances are normally absorbed into the gate and drain line, since the electrical separation between FETs is small [25].

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Fig. 2. Application of ATLs in the SE DFDA. Only one FET section is shown. In this case, is 7 for both input and output ATLs.

m

III. SE DFDA USING ATLs Meandering of the TLs is a commonly used method to reduce the area occupied by the circuit (e.g., [16]), but meandering alone has limited scope since: 1) the gate line has to be wide to mitigate the effect of FET input capacitance [14] and 2) a minimum separation between parallel sections of a meandered line is necessary to minimize the effects of spurious coupling [26]. ATLs are periodic structures that can be employed in place of TLs, resulting in size reduction. Earlier work on the application of ATLs in branch-line and rat-race couplers have demonstrated realization methods in microstrip line that makes optimum use of circuit area [19]. The proposal is to replace the gate and drain line TLs with ATLs. Fig. 2 shows the method to apply ATLs to one FET (180 ) section of the SE DFDA. The spacing between shunt capacitances (including FET input and output capacitances) is identical. The shunt capacitances on the middle ATL sections of Fig. 2 are provided by the FET input and output capacitances. The number of ATL unit cells for a 180 section and, hence, is 7 for both input and output ATLs in is Fig. 2. The characteristic impedances and phase velocities of the input and output TL elements are and and and , respectively. If the input and output capacitances of the FET are and , respectively, then and should be equal to and , respectively, to ensure that the periodic structures behave as electrically smooth TLs (i.e., ATLs) [27]. The effective charand acteristic impedances of the input and output ATLs are , respectively, and are smaller than and , respectively [19], [28]. The effective phase velocities of the ATLs are also smaller than and [19], [28]. It is possible to have greater than and greater than , in which case, capacitances of – and – must be placed in shunt with the gate and drain terminals of the FET, respectively. It can be shown using the theory for periodic structures [27] , , and are constrained so that the that the values of , is obtained as follows: desired value of (2)

Fig. 3. Physical realization of the ATL. Only the output line is shown and, in this case, is 9. Dashed box indicates portion of ATL simulated.

m

and

ATL sections have an electrical length of 180 as follows: (3)

where is the center angular frequency of the amplifier. Similar constraints apply for the drain ATL. and phase veThe microstrip characteristic impedance locity are dependent on its width , as well as substrate height and permittivity, and conductor thickness [26]. Equations resulting in in (2) and (3) can be solved to eliminate terms of , , and . Hence, a relation between and can be established, which can be used to obtain a suitable value (e.g., 50 ). Finally, (2) of that gives the desired value of can be solved for . In previous research, we have realized compact ATLs entirely from microstrip-line elements to implement compact hybrids [19]. Fig. 3 shows the proposal for application in the SE DFDA and involves meandering for both compactness and layout compatibility with the FETs, i.e., the “Y”-shaped three-way junctions minimizes interaction between the FET package and ATL. In the case of Fig. 3, is 9. The shunt capacitances are realized with stubs. When realizing the ATLs, it is important that dimensions , (space between stubs), and , shown in Fig. 3, are at least three times the substrate height so that coupling between elements is minimized [26]. The dimensions and and stub lengths need to be less than one-tenth the guide wavelength [19], [27]. IV. AMPLIFIER DESIGN We now consider the design of a three-FET SE DFDA that operates at 1.8 GHz, uses the Fujitsu FLK012WF packaged GaAs FET [29], and has a substrate with 31-mil height and dielectric constant of 2.22. As is the case in all power-amplifier design, it is necessary to have a large-signal model of the FETs. In this study, ADS1 1Advanced Design System 2002 (ADS), Agilent Technol., Palo Alto, CA, 2002. [Online]. Available: http://www.agilent.com

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was used to simulate the amplifier and, hence, the standard ADS GaAs FET large-signal model was used with the following options invoked: 1) Statz drain current model [30]; and ; 2) junction capacitance model for both 3) diode junctions to model breakdown and gate forward conduction. The model also included linear parasitic elements and a constant series RC network in parallel with the drain current source to account for drain current dispersion [31]. The parameters for the model were obtained from information provided in the data sheets [29] and the model was fitted to the data sheet dc characteristic and -parameters. The optimized model gave good representation of the dc characteristics and the -parameters up to 10 GHz (though prediction became less accurate above 6 GHz). From the FET model (and data sheet [29]), design parameV, V, and ters were extracted, i.e., mA, and the pinchoff voltage is 2 V. Application equal to 50 , and the gate and drain bias of (1) results in voltages should be 1 and 5.5 V, respectively, for class-A operation [14]. Under this condition, the FET model reveals that is 0.47 pF and is 0.34 pF. The theoretical load power when the FETs are driven to their maximum undistorted class-A extent is 23.1 dBm, and the corresponding dc efficiency (ratio of load power to dc power drawn from the power supply) is 41%. It is easy to show that the dc efficiency of a single FET operated under exactly the same conditions would have a dc efficiency of 41%. and are similar, the approach taken in this As both study was to design identical input and output ATLs, and an FET loading capacitance of 0.41 pF (being the average of 0.47 and 0.34 pF) was assumed. That is both ATLs were designed to have an effective characteristic impedance of 50 . With 50- ATLs, coupling to a 50- generator and load is simplified. The width of the stubs was set to 2 mm, and a length of 5.02 mm gives an input admittance equal to that of a 0.41-pF capacitance at 1.8 GHz. In the layout, the actual stub length was and no additional shortening of the stub was shortened by required to compensate for end-effect fringing. With equal to 7, equal to 1.3 mm results in equal to 50 , and equal to 6.1 mm. An electromagnetic (EM) simulation was performed of a 180 section of the planar ATL structure using Sonnet.2 The outcome of the simulation was three-port -parameters representing the geometry enclosed in the dashed box of Fig. 3 over a range of frequencies. To ensure that the simulation boundaries used by Sonnet do not affect the results, the sidewalls were placed at least 3 mm away from the structure. Sonnet can be setup to remove the effects of microstrip lines used to connect the structure to the simulation ports on the sidewalls. To confirm that the simulated geometry and the FET input (or output capacitance) behaves as a 50- 180 TL, the simulated results were embedded with a lumped capacitance of 0.41 pF to represent the FET input or output capacitance, as 2Sonnet Lite 8.51, Sonnet Software Inc., Liverpool, NY, 2002. [Online]. Available: http://www.sonnetusa.com

Fig. 4. Embedding circuit to confirm response of ATL geometry.

Fig. 5. Embedded simulated s-parameters of the 50- /180 ATL.

shown in Fig. 4. The result of this embedding is two-port -parameters. For the initial design, it was found that the effective characteristic impedance was satisfactory, but the electrical and stub length length was only 162 at 1.8 GHz. Thus, were both lengthened by 11%, and, hence, the final optimized values of and stub length are equal to 6.8 and 5.6 mm, respectively. The structure was resimulated and the embedded results are shown in Fig. 5. Fig. 5 shows that the ATL return loss is greater than 30 dB below 2.2 GHz. As a comparison, if microstrip TLs had been used, their length would have been around 63 mm to achieve an electrical length of 180 at 1.8 GHz. On the other hand, the 180 ATL is 48 mm in length and has been meandered. The three-port data for the ATL section was exported to ADS for harmonic-balance and small-signal -parameter simulations. The three-port data represented the ATLs up to a frequency of 18 GHz in steps of 0.2 GHz, which is sufficient for harmonic-balance simulation with up to eight harmonics. Chip capacitors of 10-pF capacitors were used for both dc blocking and to short-circuit terminate the gate and drain ATLs at RF frequencies. The chip capacitors have a series inductance of 300 pH, and this effect was included in the simulations. The gate and drain power supplies were fed to the end of the ATLs. A 50- chip resistor was included in series with the gate bias power supply to ensure stability of the circuit at low frequencies. For convergence of the harmonic-balance simulations, eight harmonics was found to be sufficient. The simulations confirmed that the amplifier would work in accordance with theoretical expectations, and is stable. The amplifier was fabricated, and Fig. 6 shows a photograph of the amplifier. The size of the amplifier is about the same as a two-FET SE DFDA [14] with the same operating frequency, but uses 180 microstrip TLs between each FET. Thus, the size of an amplifier section had been reduced by about one-third.

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Fig. 6. Prototype three-FET SE DFDA. Fig. 8. Measured and simulated load power and PAE versus input generator available power at 1.8 GHz.

Fig. 7. Measured and simulated small-signal S -parameters of the amplifier. Heavy lines are measurements, thin dashed lines are simulations.

V. EXPERIMENT Due to manufacturing variations, the actual FETs will have characteristics that differ from published data [29]. DC measurements of the amplifier were first performed and established that the optimum gate bias voltage for class-A operation to be 1.1 V when the drain bias voltage is 5.5 V. Small-signal -parameter measurements were made under class-A operating conditions using a vector network analyzer. , , Fig. 7 shows the measured and simulated values of . The measured small-signal gain ( ) has a peak of and 17 dB and is consistent with the peak simulated small-signal gain. Aside from a shift in center frequency, the measurements are consistent with the simulations. The shift in center frequency is due to uncertainties such as the interaction between the body of the FET packages and the ATL (that was not considered in the simulations) [15], circuit etching tolerance, tolerance of FET location, and FET model uncertainty. The bandwidth is, however, 350 MHz and is close to the bandwidth obtained for the ideal case (Case 1) investigated in Section II. The mismatch at both the input and output ports (Fig. 7) is normal for SE DFDAs, which are inherently mismatched. In practice, one would use two identical SE DFDAs in a balanced amplifier whose ports would be matched [12], [14]. Scalar large-signal measurements of the amplifier were obtained using a computer-controlled system comprising a spectrum analyzer, signal generator, and multiplexed multimeter. The spectrum analyzer was calibrated against a power meter. The software managed the calibration and accounted for cable insertion loss. The measurement system permitted mea-

Fig. 9. Measured load power versus frequency.

surement of the input generator available power, load power, and the power drawn from the power supply. The facilities we had did not permit accurate measurement of the amplifier input power. Instead, the input power was taken as the input generator available power and this is justified since, in practice, the SE DFDA would be used in a balanced amplifier that has matched ports [12], [14]. Due to the mismatched port of the SE DFDA, the actual input power will be considerably smaller than the generator available power and, therefore, the measured power-added efficiency (PAE) will be smaller than the true PAE; but is a true reflection of the PAE that would be encountered in practice when used in a balanced amplifier [12], [14], [15]. Fig. 8 shows the measured and harmonic-balance simulation results of the load power and PAE at 1.8 GHz. Fig. 8 shows good correspondence between simulation and measurements, as well as consistency with theoretical predictions. Figs. 9 and 10 show, respectively, the measured load power and PAE as a function of frequency for various input generator power levels. The measurements show that the optimum large-signal performance is centered at 1.7 GHz. It is clear that the amplifier saturates at around 23 dBm over a bandwidth of 200 MHz. The maximum efficiency is greater than 35% over a bandwidth of 250 MHz (or 15%). The efficiency of the experimental three-FET SE DFDA is as good as a conventional single-FET power amplifier that uses the same FET.

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in the large-signal measurements while he was with the National University of Singapore as a participant of a student exchange program. REFERENCES

Fig. 10. Measured PAE versus frequency.

VI. CONCLUSION In this paper, we have demonstrated a distributed combined power amplifier with an efficiency similar to a conventional single-FET power amplifier employing the same FET. The amplification principle is based upon the DFDA approach to efficiently combine FET output power. Meandered ATLs assembled from microstrip-line elements used in place of TLs results in compactness, and also incorporates FET input and output capacitances, thereby eliminating their detrimental effect on bandwidth and performance. The experimental results of a class-A three-FET SE DFDA designed to operate at 1.8 GHz demonstrated the feasibility of the proposed method and the validity of the design method. The level of size reduction is one-third per amplifier section, and maximum efficiency was greater than 35% over a bandwidth of 15%. The experimental amplifier was realized using a soft substrate, but the principles of the method could be applied to other realization technologies. The proposed method allows the realization of an SE DFDA on a monolithic microwave integrated circuit (MMIC) where 180 microstrip lines would otherwise be unfeasible. In this case, the shunt capacitances could be realized with metal–insulator–metal (MIM) capacitors available in standard MMIC processes [26]. This would be necessary for high-power FETs due to their significant input and output capacitances [28]. The application of ATLs in active MMICs has been demonstrated by others [17]. The proposed method assumed that FET parasitic gate and drain inductances have a negligible effect. This assumption is valid at 1.8 GHz for the FETs used in the prototype (operating at 1.8 GHz). This assumption becomes increasingly invalid with increasing frequency. The author is currently investigating methods to design the SE DFDA that consider FET parasitic inductances, as well as input and output capacitances, and would enable the SE DFDA to be applied at higher frequencies. ACKNOWLEDGMENT The author wishes to express gratitude to S. C. Lee, C. H. Sing, and L. H. Chan, all of the National University of Singapore, Singapore, for their assistance in fabricating the amplifier. The author would also like to thank I. Gunasinghe, University of Adelaide, Adelaide, Australia, for his assistance

[1] M. E. Bialkowski, “Power combiners and dividers,” in Wiley Encyclopaedia of Electrical and Electronic Engineering, J. G. Webster, Ed. New York: Wiley, 1999, vol. 16, pp. 585–602. [2] Y. Taniguchi, Y. Hasegawa, Y. Aoki, and J. Fukaya, “A C -band 25 watt linear power FET,” in IEEE MTT-S Int. Microwave Symp. Dig., 1990, pp. 981–984. [3] Y. Ikeda, H. Yukawa, H. Uchida, Y. Tarui, H. Koide, H. Utsumi, Y. Itoh, and K. Tsubouchi, “A Ku-band power amplifier using impedance matching circuit coupled to Wilkinson power divider,” in Asia–Pacific Microwave Conf., 2003, pp. 1801–1804. [4] T. T. Y. Wong, Fundamentals of Distributed Amplification. Boston, MA: Artech House, 1993. [5] J. L. B. Walker, “Some observations on the design and performance of distributed amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 1, pp. 164–168, Jan. 1992. [6] C. S. Aitchison, N. Bukhari, C. Law, and N. Nazoa-Ruiz, “The dual-fed distributed amplifier,” in IEEE MTT-S Int. Microwave Symp. Dig., 1988, pp. 911–914. [7] C. S. Aitchison, M. N. Bukhari, and O. S. A. Tang, “The enhanced power performance of the dual-fed distributed amplifier,” in Eur. Microwave Conf., 1989, pp. 439–444. [8] S. D’Agustino and C. Paoloni, “Design of high-performance power-distributed amplifier using Lange couplers,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2525–2530, Dec. 1994. [9] I. A. Botterill and C. S. Aitchison, “Effect of hybrids on gain performance of dual-fed distributed amplifiers,” Electron. Lett., vol. 30, no. 13, pp. 1067–1068, Jun. 1994. [10] K. W. Eccleston, “Output power performance of dual-fed and single-fed distributed amplifiers,” Microwave Opt. Technol. Lett., vol. 27, no. 4, pp. 281–284, Nov. 2000. , “Design considerations for the dual-fed distributed power ampli[11] fier,” in Asia–Pacific Microwave Conf., 2000, pp. 205–208. [12] M. R. Moazzam and C. S. Aitchison, “A high gain dual-fed single stage distributed amplifier,” in IEEE MTT-S Int. Microwave Symp. Dig., 1994, pp. 1409–1412. [13] K. W. Eccleston, “Design and performance of a balanced single-ended dual-fed distributed power amplifier,” in Asia–Pacific Microwave Conf., 2001, pp. 1187–1190. [14] K. W. Eccleston and O. Kyaw, “FET power combining with the use of the balanced single-ended dual-fed distributed amplifier approach,” Microwave Opt. Technol. Lett., vol. 35, no. 1, pp. 42–45, Oct. 2002. , “Analysis and design of class-B dual-fed distributed power ampli[15] fiers,” Proc. Inst. Elect. Eng., pt. H, vol. 151, no. 2, pp. 104–108, Apr. 2004. [16] T. Hasegawa, S. Banba, and H. Ogawa, “A branchline hybrid using valley microstrip lines,” IEEE Microw. Guided Wave Lett., vol. 2, pp. 76–78, Feb. 1992. [17] P. Kangaslahti, P. Alinikula, and V. Porra, “Miniaturized artificial-transmission-line monolithic millimeter-wave frequency doubler,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 4, pp. 510–518, Apr. 2000. [18] A. Görür, C. Karpuz, and M. Alkan, “Characteristics of periodically loaded CPW structures,” IEEE Microw. Guided Wave Lett., vol. 8, pp. 278–280, Aug. 1998. [19] K. W. Eccleston and S. H. M. Ong, “Compact planar microstrip line branch-line and rat-race couplers,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 10, pp. 2119–2125, Oct. 2003. [20] Q. Xue, K. M. Shum, and C. H. Chan, “Novel 1-D microstrip PBG cells,” IEEE Microw. Guided Lett., vol. 10, no. 10, pp. 403–405, Oct. 2000. [21] T. Hirota, A. Minakawa, and M. Muraguchi, “Reduced-size branch-line and rat-race hybrids for uniplanar MMIC’s,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 3, pp. 270–275, Mar. 1990. [22] I. Toyoda, T. Hirota, T. Hiraoka, and T. Tokumitsu, “Multilayer MMIC brach-line coupler and broad-side coupler,” in Microwave and Millimeter-Wave Monolithic Circuits Symp. Dig., 1992, pp. 79–82. [23] M. R. Moazzam and C. S. Aitchison, “High gain microwave amplifier tunable over a decade bandwidth,” Proc. Inst. Elect. Eng., pt. H, vol. 142, no. 6, pp. 489–491, Dec. 1995.

ECCLESTON: COMPACT DUAL-FED DISTRIBUTED POWER AMPLIFIER

[24] K. W. Eccleston, “Multiband power amplifier for multiband wireless applications,” in 3rd Int. Microwave and Millimeter Wave Technology Conf., Beijing, China, Aug. 17–19, 2002, pp. 1142–1145. [25] Y. Ayasli, R. L. Mozzi, J. L. Vorhaus, L. D. Reynolds, and R. A. Pucel, “A monolithic GaAs 1 – 13-GHz traveling-wave amplifier,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 7, pp. 976–981, Jul. 1982. [26] I. Bahl and P. Bhartia, Microwave Solid-State Circuit Design. New York: Wiley, 1988. [27] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1992, ch. 8. [28] K. W. Eccleston and O. Kyaw, “Design of a dual-fed distributed power amplifier using artificial transmission lines,” in Progress in Electromagnetics Research Symp., Singapore, Jan. 7–10, 2003, p. 218. [29] Fujitsu Microwave Semiconductor Databook, Fujitsu Compound Semiconduct. Inc., San Jose, CA, 1997. [Online]. http://www.fujitsu.com. [30] H. Statz, P. Newman, W. Smith, R. A. Pucel, and H. A. Haus, “GaAs FET device and circuit simulation in SPICE,” IEEE Trans. Electron Devices, vol. MTT-34, no. 2, pp. 160–169, Feb. 1987. [31] C. Cammacho-Penalosa and C. S. Aitchison, “Modeling frequency dependence of output impedance of a microwave MESFET at low frequencies,” Electron. Lett., vol. 21, no. 13, pp. 528–529, Jun. 1985.

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Kimberley W. Eccleston (S’85–M’90) was born in Melbourne, Australia, in 1964. He received the B.E. degree (Electronic Engineering) (with distinction) from the Royal Melbourne Institute of Technology (RMIT), Melbourne, Australia, in 1986, and the Ph.D. degree from the University of Queensland, Brisbane, Australia, in 1991. From 1990 to 1992, he was a Research Scientist with the Defence Science and Technology Organization, Adelaide, Australia. From 1992 to 2004, he was with the Department of Electrical and Computer Engineering, National University of Singapore (NUS). Since 2005, he has been with the Department of Electrical and Computer Engineering, University of Canterbury, Christchurch, New Zealand, where he is currently a Senior Lecturer. He has taught students at various levels, and has developed specialized courses in microwave active circuit design at both the undergraduate and graduate levels. He has authored over 40 technical papers. He holds one patent. His research is currently focused on active and passive microwave circuit design. He possesses experience in microwave measurements and microwave device modeling. Dr. Eccleston has been a key member of organizing committees of several IEEE conferences hosted in Singapore.

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Optimized Finite-Difference Time-Domain Methods Based on the (2; 4) Stencil Guilin Sun, Student Member, IEEE, and Christopher W. Trueman, Senior Member, IEEE

Abstract—The higher order (2 4) scheme optimized in terms of Taylor series in the finite-difference time-domain method is often used to reduce numerical dispersion and anisotropy. This paper investigates optimization of the numerical dispersion behavior for a square Yee mesh based on the (2 4) computational stencil. It is shown that, for one designated frequency, numerical dispersion can be eliminated for some directions of travel, such as the coordinate axes or the diagonals, or numerical anisotropy can be eliminated entirely, resulting in a constant “residual” numerical dispersion. Using a coefficient-modification technique, the residual numerical dispersion can then be completely eliminated at that frequency, or for a wide-band signal, the numerical dispersion error and the averaged-accumulated phase error can be minimized. The stability of the method is analyzed, the numerical dispersion relation is given and validated using numerical experiments, and the relative rms errors are compared to the standard (2 4) scheme for the proposed methods. The optimized methods are second-order accurate in space and have higher accuracy than the standard (2 4) scheme. It has been found that the dispersion error of the (2 4) scheme is like that of a second-order accurate method, though it behaves like a fourth-order accurate method in terms of anisotropy. Index Terms—Computational electromagnetics, finite-difference time-domain (FDTD) method, higher order method, numerical anisotropy, numerical dispersion.

I. INTRODUCTION

T

HE finite-difference time-domain (FDTD) method originally proposed by Yee [1] has been widely used to solve Maxwell’s equations in numerous electromagnetic applications [2]. However, the numerical dispersion and anisotropy inherent in Yee’s FDTD method prohibits its application to electrically large objects because of the phase-error accumulation. In addition, the anisotropic velocity makes compensation for the phase error in the near-to-far transformation complex [3]. To overcome this limitation, numerous methods have been proposed. Cole [4], [5] described a high-accuracy algorithm with a nonstandard finite-difference method and adjacent nodes. Forgy and Chew [6] presented a nearly isotropic method on an overlapped lattice. Nehrbass et al. [7] demonstrated a reduced dispersion method without increasing the order. Juntunen and Tsiboukis [8] presented a method using artificial anisotropy. Rewienski and Mrozowski [9] introduced an iterative algorithm in cylindrical coordinates. Wang et al. Manuscript received March 8, 2004; revised May 25, 2004. The authors are with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada H4B 1R6 (e-mail: Trueman@ ece.concordia.ca; [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842507

[10] suggested a parameter-optimized method for the alternating-direction-implicit finite-difference time-domain (ADI FDTD) method. Taflove and Hagness [2] analyzed several scheme methods, including the standard higher order [11]. Wang and Teixeira [12]–[15] proposed several methods such as angle-optimized and dispersion-relation-preserving schemes using complicated filters. Zygiridis and Tsiboukis [16] presented a method based on Taylor-series analysis of the numerical dispersion relation. Xie et al. [17] developed a scheme with numerical dissipation. Shao et al. [18] proposed a generalized higher order method. Zingg [19] compared some high-accuracy methods. Shlager and Schneider [20] compared the performance of several low-dispersion methods. Other methods may be found in the references of the above-mentioned papers. For a square mesh, Yee’s FDTD method can have no numerical dispersion along the diagonals and at the Courant limit time step size [2]. The standard higher order method at small Courant numbers can have no numerical dispersion along the diagonals or along the axes. The fact that a higher order method can reduce the numerical error in the FDTD implies that the larger computational stencils used in higher order methods can cancel some of the errors in Yee’s second-order formulation. With the optimization presented in this paper, better results than the scheme are obtained. The proposed methods are more flexible: by choosing special optimal parameters at certain mesh densities and time-step sizes, the numerical dispersion can be eliminated along a pre-assigned direction of travel such as along the axes or along the diagonals of a square mesh, or the anisotropy can be eliminated altogether at one frequency, with “residual” dispersion error, which can then be removed completely with a simple coefficient-modification (CM) technique, or, for a broadband signal, the numerical dispersion error and averaged-accumulated phase error can be minimized. Another feature of the proposed method is that it can be incorporated into the current algorithm directly including the ADI FDTD method [21] without additional computational cost. Since the proposed methods are second-order accurate in space, the material interface and boundary conditions can be easily treated using existing second-order methods. In contrast, the scheme must use higher order methods to treat the material interfaces in order to maintain the overall accuracy. Notice that [11]–[16] provide methods to reduce the numerical dispersion based on the same stencil with approximate dispersion relation analysis and using coefficient modifi-

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cation. This paper treats the weight parameter and the CM parameter separately. Methods presented in this paper are given that eliminate or minimize dispersion, or eliminate anisotropy, using the exact numerical dispersion relation. This paper is organized as follows. Section II gives the formulation, amplification factor, and numerical dispersion relation for all the proposed methods in terms of a weight parameter. Sections III–V describe how to obtain the optimal values of the weight parameter in various senses. Section VI discusses the stability, time-step size limit, accuracy, and the CM technique to reduce the residual numerical dispersion. Section VII proposes methods to minimize the dispersion error over a broad bandwidth. (2) II. WEIGHTED TWO-DIMENSIONAL (2-D) FDTD FORMULATION wave in a linear isotropic nondisFor simplicity, a 2-D persive medium is assumed. Based on the Taylor-series analysis [2], [11], the first-order spatial derivatives in the scheme are approximated by the use of the conventional Yee’s elements plus “one-cell-away” elements. Both elements use a second-order finite-difference formula to eliminate third-order and higher odd-order terms. The fourth-order accuracy is obtained by the cancellation of the two second-order error terms from the Yee’s elements and the “one-cell-away” elements. In this paper, we do not pursue fourth-order accuracy based on the Taylor series. Instead, the methods are based on the optimization of the numerical dispersion error. To achieve this, is introduced to optimize the relative a weight parameter contributions of Yee’s elements and the “one-cell-away” elements. The new update equations are given by (1)–(3) as follows:

(3) where , , is the time step size, and are the permittivity and permeability of the material, reand are the spatial meshing sizes, and spectively, are the indices of the computational cells, and is the index of time step. The weight parameter is to be determined. If , (1)–(3) becomes the conventional Yee’s method [1], , the above formulation is exactly the same as [2]. If the scheme [2], [11]. In Sections III-V, the weight parameter will be chosen to optimize the dispersion behavior in various ways. With the Fourier analysis method [22], the amplification factor for this formulation can be obtained as

(4)

where (1)

,

, ,

, and

.

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The phase constant of the numerical wave is . The angle is the direction of travel with respect to the -axis, and and . The amplification factor can be expressed in terms of the and the angular frequency as . time-step size Thus, with some manipulation, the numerical dispersion equation can be obtained as

Fig. 1. Optimal value of weight parameter as a function of the mesh density for various methods.

(5)

Since the numerical dispersion depends on the weight parameter, the dispersion behavior can be optimized in various ways. III. AXES-OPTIMIZED METHOD (AOM) The AOM has no numerical dispersion along the axes for any time-step size within the stability limit at a designated frequency. To eliminate the numerical dispersion error along the ), use (5) to examine the nuaxes for a square mesh ( merical dispersion along the -axis to obtain Fig. 2. Numerical dispersion of the AOM optimized at 10 CPW and the (2; 4) method.

(6)

where is the numerical phase constant along the axes. Solving (6) obtains the “axes-optimized” value of the weight as parameter

(7)

To have a solution of (7), set equal to the theoretical phase at the designated frequency. Since the mesh constant size can be expressed in terms of mesh density as , is a function of mesh density (7) shows that the optimal value , the signal frequency, and time-step size. Fig. 1 shows the optimal values of the weight parameter as a function of mesh density for four Courant numbers , , , , where is the Courant limit in the 2-D and case. As the Courant number increases, i.e., as the time-step size at a fixed mesh density increases, the optimal value of the weight parameter decreases. The optimal value also decreases monotonically as the mesh density increases. Fig. 2 graphs the numerical dispersion (quantified as the relwhere is the numerical velocity) versus the ative velocity direction of travel , at a mesh density of 10 cells per wave, , , length (CPW) for Courant number . For comparison, the numerical dispersion of the and scheme is also shown. For the AOM, Fig. 2 shows that, indeed, there is no numerical dispersion along the axes. Both

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Fig. 3. PNDE for the AOM along 0 and 10 , the DOM along 45 and 55 , optimized at 10 CPW, and the minimum dispersion error of the (2; 4) scheme along the axes.

the AOM and methods have velocity larger than the physfor the scheme. Both ical speed, except at methods have larger velocity along the diagonals than along the axes, similar to Yee’s method. Fig. 2 shows the results from numerical experiments as small circles, with good agreement with the theory. All the numerical experiments in this paper are performed in a 2000 2000 . The excitation cell space with mesh density source is located at the center. As time advances, the field values are recorded every five degrees on two quarter-circles: an inner circle located at 50 wavelengths away; and an outer circle located 60 wavelengths away from the source. The numerical velocity is then calculated from the time delay for propagation between the two circles. To compare accuracy, the numerical dispersion error can be defined as . Fig. 3 shows the percentage numerical disfrom mesh persion error (PNDE) along the axis and for densities from 8 to 45 CPW for the AOM at the time step sizes and . The AOM method is optimized of at 10 CPW, where the axial dispersion error is zero. This figure also shows the minimum dispersion error along the axis of the method. It can be seen that the numerical dispersion error for the AOM in the sector of 10 is much smaller than that of the scheme. For example, at , the maximum error for the AOM at approximately 14 CPW is 0.016% along , whereas the minimum disthe axis, and 0.0218% at method is persion error at the same mesh density for the much larger at 0.8624%. Note that the AOM is intended for use in problems where most waves travel within a sector around the axis, such as the laser cavity and other examples indicated in [14]. For problems where the prominent direction of wave is along the diagonal, the method in Section IV can be used. IV. DIAGONALLY OPTIMIZED METHOD (DOM) A DOM has no numerical dispersion error along for a square mesh. To derive a formula for calculating the optimal

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Fig. 4. Numerical dispersion of the DOM optimized at 10 CPW and the (2; 4) method.

weight parameter, use (5) to examine the numerical dispersion along 45 to obtain

(8)

Set , and solve (8) for the optimal value of the to obtain weight parameter

(9)

The optimal value as a function of the mesh density is graphed in Fig. 1 and has similar behavior to that of the AOM method. Fig. 4 graphs the numerical dispersion at four different Courant numbers for the DOM. It can be seen that, as expected, there is no numerical dispersion along the diagonal. Different from the scheme, shown in Fig. 4, all the velocities AOM and the in the DOM method are below or equal to the physical speed. The small circles are the results of numerical experiments and agree well with the theory. Fig. 3 graphs the PNDE for the DOM and at the time step sizes of along the diagonal and optimized at 10 CPW. It can be seen that the PNDE is much smaller than that of , and smaller , the maxthan that of the AOM. For example, at imum dispersion error for the DOM at approximately 14 CPW , is 0.00336% along the diagonal, and 0.00937% at

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which is approximately 25.7 and 9.2 times smaller than that of scheme, respectively, the minimum dispersion error of the at the same mesh density. Note that the AOM and DOM are special cases which eliminate numerical dispersion in a designated direction of travel. The formulation of (1)–(3) allows the elimination of the nu. The merical dispersion at any specific angle of interest weight parameter can be found from

Fig. 5. Numerical dispersion for the IOM optimized at 10 CPW.

(10) (11a) (11b) The solution is straightforward, but messy and will not be given here. V. ISOTROPIC OPTIMIZATION METHOD It has been shown that the weight parameter can be chosen to make the numerical phase constant exactly equal to the theoretical value in one direction of travel. The value of the weight parameter depends on the mesh density and time step size, as shown in Fig. 1. If the AOM and DOM are combined together, the numerical velocity can be made independent of the direction of travel. After eliminating the time-step size term, (6) and (8) can be solved to obtain the “isotropic” optimal value of the weight parameter, given by (12), shown at the bottom of this page. This optimal parameter is only a function of mesh density, and is independent of both the time step size and the signal frequency. The numerical dispersion using is graphed in Fig. 5 with solid lines for a mesh density of 10, at four different Courant numbers. The “curves” are, in fact, horizontal lines, demonstrating that the numerical velocity is indeed independent of the direction of travel. There is no anisotropy, as expected, but there is numerical dispersion. This method is termed

the isotropically optimized method (IOM). The IOM obtains a circular numerical wavefront since the numerical velocity is uniform in all directions. The residual numerical dispersion in the IOM can be corrected with the technique that will be briefly discussed in Section VI-G. This isotropic method makes the phaseerror compensation easier for the near-to-far transformation. VI. DISCUSSION A. Optimal Parameter The above analysis shows that, by suitable choice of the weight parameter in (1)–(3), numerical dispersion and anisotropy can be made to exhibit different properties: zero velocity error along a designated direction of travel, such as the coordinate axes (AOM) or along the diagonals (DOM) or zero anisotropy (IOM) for a square mesh. The update (1)–(3) for the AOM, DOM, and IOM are the same, except for the value of the weight parameter . Fig. 6 shows that the numerical velocity along the diagonals (solid curves) and along the axes (dashed curves) increases monotonically with the value of the weight parameter, for mesh density of 10 CPW at four different Courant numbers. Where each curve intersects with the horizontal line for relative velocity 1.000, there is no numerical dispersion. Where two of the curves for the same time-step size intersect one another, there is no anisotropy for that value of the weight parameter. These “special” optimal values correspond to the values given in (7), (9), and (12). Other optimal values of the weight parameter can be found in different ways, which will be shown in Section VII.

(12)

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Fig. 6. Relative numerical velocity along the axes and along the diagonals as a function of the weight parameter.

Fig. 7. Anisotropy at different optimized wavelength with the IOM and the (2; 4) method.

B. Stability and the Time-Step Size Limit

the anisotropy is less than a pre-set maximum. The mesh density in Fig. 7 can be transformed into wavelength by

Numerical stability relies on the magnitude of the amplification factor in (4) [22]. Equation (4) implicitly assumes that the following inequality holds: (13) Following the analysis for the Yee’s method [2], set the sine term equal to unity. The time step size is then bounded by (14a) where (14b)

is the same Courant limit as the Yee’s method. For where scheme, and the time-step size limit is 6/7 the of the Courant limit, which is the expected result. Since the optimal value of weight parameter of the IOM at mesh density 10 is 1.133733, its corresponding time-step size limit is 0.84867 times of the Courant limit, which is a little smaller than 6/7. is used in this This is why the Courant number paper. In the limit of zero mesh size, the optimal value of is 9/8, thus, the time-step size limit is the same as the standard scheme.

s

= 0:848s for

(15) Hence, the bandwidth can be estimated. For optimization at cells per wavelength, the anisotropy increases sharply as increases above , reaches the maximum , and then decreases. If the of 1.4 10 at upper bound on the anisotropy is, e.g., 1.4 10 , then the . The shortest mesh density must be larger than wavelength for which the anisotropy is less than 1.4 10 is . For comparison, Yee’s FDTD requires and scheme requires for the same anisotropy. Note that the anisotropy for Yee’s is nearly independent of the time-step size, hence, it is about the same at . For optimization of the IOM the Courant limit and , the anisotropy is very large at coarse mesh at sizes. For anisotropy less than 1.4 10 , we must choose ; hence, the shortest wavelength is . , the anisotropy is less than Note also that, for 9.8 10 , much smaller than the chosen bound, whereas the scheme requires for anisotropy 1.4 10 . increases, the residual numerical dispersion deNote that as has the largest discreases. Optimization at , the dispersion is about persion error and, at scheme. In all the cases, the the same as the standard dispersion error decreases monotonically with increasing mesh density.

C. Anisotropy for Other Frequencies The IOM is designed for zero anisotropy at one frequency of operation. Fig. 7 shows the anisotropy as a function of mesh density for three different optimizations: optimized for zero , , and , respectively, anisotropy at with fixed Courant number , which results in the largest numerical dispersion. Suppose at the frequency of and the mesh density . optimization, the wavelength is At higher and lower frequencies, there will be some numerical anisotropy, and it is useful to know the bandwidth over which

D. Relative RMS Error Comparison For the three methods presented in this paper, their numerical dispersion errors are quite different because the numerical velocity can be either larger or smaller than the theoretical velocity. Thus, it is difficult to compare them. The rms truncation (or discretization) error of (1)–(3) based on the Fourier method [23] is a useful way to compare methods, and provides more information than the numerical dispersion error. Fig. 8 gives the maximum relative rms truncation errors for the AOM, DOM,

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Fig. 8. Largest rms errors for the AOM, DOM, IOM, (2; 4), and Yee methods. TABLE I LARGEST RELATIVE rms ERRORS FOR DIFFERENT METHODS IN PERCENTAGE

accuracy increases are calculated to be 4 for Yee, 2.83 and 3.91 scheme, respectively, and 4 for the optimized for the methods. These results are quite close to those in Table II. The larger increase in Table II for the optimized methods may imply that higher order spatial terms have more effects in canceling errors at small Courant numbers. Further computations show scheme, the anisotropy error obeys the that, for the fourth-order accuracy, but the numerical dispersion error is second order. For example, from Fig. 4, the anisotropy error scheme, decreases 15.5 (approximately 2 ) times for the and 4.13 (approximately 2 ) times for Yee’s FDTD method, AOM, and DOM when the mesh density increases from 10 to . On the other 20 CPW at the Courant number hand, the numerical dispersion error decreases 3.92 times for scheme, 4.13 times for the AOM and DOM, and 4.21 the for the IOM from Figs. 2, 4, and 5. Thus, the scheme functions as a second-order method in terms of dispersion error. Note that the numerical dispersion error is defined as the difference between unity and the relative numerical velocity. F. Three-Dimensional (3-D) Case

TABLE II ACCURACY INCREASE FROM MESH DENSITY 10–20 CPW FROM TABLE I

IOM, Yee’s FDTD method, and methods at Courant numand . The maximum error occurs along the bers diagonals for the AOM, and occurs along the axes for DOM. Their smallest dispersion errors along their designated direction were given previously. As the time-step size increases, Yee’s FDTD method has a smaller error and the other methods have larger errors. Table I gives some typical values for the largest percentage rms error, and will be used to demonstrate the properties of different methods in the following. E. Accuracy By Taylor-series analysis, the optimized methods in this paper are second-order accurate. Due to the partial cancellation of the higher order terms, the truncation error is compensated to a certain extent. By extracting information from Table I, the accuracy increase at the mesh density of 20 CPW compared to that at 10 CPW is listed in Table II. scheme has the lowest increase It seems strange that the in accuracy in Table II. This is because the truncation error is a function of both the time-step size and the spatial increment. Analysis based on the Taylor series with the fundamental plane-wave solution of Maxwell’s equations shows that the for Yee’s FDTD accuracy is on the order of method, and for the scheme, where is from the fourth-order term of the Taylor-series expansion, and for the optimized methods. Thus, for the same Courant number, from mesh density 10–20 and at , according to the above analysis, the at

The optimized methods are readily extended to 3-D. The update equations for the 3-D case are similar to (1)–(3), but will not be given here for brevity. The amplification factor is given by (16) as

(16) which has unity magnitude within the time-step size limit given term. The numerical in (14a). Equation (14b) needs a similar dispersion relation is

(17)

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Based on (17), the optimal values of the weight parameter for the 3-D AOM and DOM cases are similar to the 2-D case. For the IOM, the optimal value is given by (18), shown at the bottom , the optimal weight parameter of this page. If is 1.1327, thus, it has a time-step size limit of 0.84965 times the 3-D Courant limit. In contrast, the filter scheme [13] has a limit . 0.765 times the Courant limit with

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For example, to eliminate the axial numerical dispersion, the parameters are chosen to be

G. CM Techniques Many techniques mentioned in Section I modify the coeffiand in the update equations in order to reduce cients of the numerical dispersion, thus, they are refereed as CM techand , and niques. In general, can be replaced by can be replaced by and , respectively, where the parameters , , , and can be modified differently. , , and ; [10] [8] uses , , and ; [4] and [5] uses use and in conjunction with adjacent cells. Reference [16] uses in addition to modifying the spatial difference. References [6], [7], [12]–[14] also provide methods to modify these parameters. The parameter values are obtained according to different criteria such as low numerical dispersion. These methods all change the “speed”: in ; the standard FDTD method, the physical speed is in the CM techniques, the new speed is for a , , , , square mesh ( ). Note that, from (14b), it can be seen that such modifications change the time-step size limit, which has been pointed out in times the original [2] and [20]. The new stability limit is limit before modification. For the IOM, from Fig. 5, the numerical velocity is larger than the speed of light. Therefore, the coefficient modification will increase the time-step size limit. If the problem contains only one material, these techniques are the same. However, if the problem uses different materials, then unless all the materials are modified by the same factor, the reflection coefficient at a material boundary changes and there is artificial reflection. Thus, this paper uses coefficient modification for all the update equations. In addition to the methods mentioned above, we have derived several formulas to optimize the values of these “ ”-parameters in different senses. A simple, straightforward way to find the values of -parameters is to reduce or eliminate the numerical dispersion at one mesh density along one direction of travel.

(19a)

(19b) Note that is the theoretical phase constant at a designated frequency to have zero numerical dispersion, and is not necessarily to be the same as , where zero anisotropy is desired. When they are the same, the combined IOM and the simple CM technique can have zero anisotropy and zero numerical dispersion at one designated frequency. Such a CM technique does not change the anisotropy. Note that different methods to put the -parameters into the update equations are equivalent in terms of numerical dispersion, but have slightly different rms errors. In practical applications, it is the accumulated phase error, not the numerical dispersion error, that affects the result. Thus, evaluation of the accumulated phase error is important. It is defined as (20) in degrees, where is the distance of wave travel, which is in this paper, is the wavelength that changes with the signal frequency. Anisotropy makes the accumulated phase error in (20) dependent on the direction of travel. Define the averagedaccumulated phase error for all the directions of travel as (21) where ( ) is the numerical velocity along . Fig. 9 shows the averaged-accumulated phase error for the scheme and IOM at time-step size , and for the IOM with

(18)

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Fig. 9. Accumulated phase errors of the (2; 4) and IOM methods, the IOM method with coefficient modification (IOM-CM), and the HAM FW method with coefficient modification, optimized at 11.5 CPW.

coefficient modification (IOM CM) using (19), optimized at 10 and 11.5 CPW, respectively. The phase error for at its step size limit is larger, which is not given. It can be seen that both the have large phase errors at coarse mesh density IOM and (high frequency), and the error monotonically decreases as the mesh density becomes large. With coefficient modification for the IOM, the phase error is zero at the optimized frequency, but increases to a maximum at a somewhat lower frequency, and then decreases at much lower frequencies. This is a phenomenon common to all the CM techniques. VII. HIGH-ACCURACY METHODS WITH LARGER TIME-STEP SIZES In previous sections, several methods have been proposed based on the optimization of the numerical dispersion in different senses, and the formulas for the optimal values of the weight parameter are given. Other “optimal” values for the scheme is weight parameter can also be found. The obtained from the Taylor series with a fixed value of the weight parameter for whatever the mesh density is based on the order of accuracy analysis. Next, we will find a similar constant value of the weight parameter that is “optimal” in terms of numerical dispersion error. , AOM, DOM, and IOM methods have larger disThe persion error as the time-step size increases, which is contrary to the Yee’s method. In addition, their numerical phase velocity is generally larger than the physical speed. Since these methods are based on the weighted contributions of Yee’s elements and the “one-cell-away” elements, the larger numerical velocity indicates that the methods are “over-weighted.” When the weight parameter becomes smaller, the numerical velocity becomes smaller, and approaches to the physical speed, as can be seen from Figs. 1–5. Expanding the sine terms in (12) with the Taylor’ series up to the third order, a constant value of the weight parameter is . Using (14a), the time-step size limit is obtained as scheme. 39/41 (0.9512195), which is larger than 6/7 of the The averaged-accumulated phase errors at the time step sizes

Fig. 10. Numerical dispersion of the HAM-FW method, HAM-S method, and the (2; 4) scheme at different time step sizes.

and are shown in Fig. 9 using a CM method. It can be seen that the phase error of this method is and IOM methods at coarse much smaller than that of the mesh sizes. This method is termed the “high-accuracy method with fixed-weight parameter” (HAM FW). The numerical dispersion is shown in Fig. 10, where the results from the numerical experiments are in small circles, with very good agreement to the theory. The numerical dispersion error is smaller than that and Yee’s methods. For example, for a mesh denof the sity of 10 CPW, at their corresponding time-step size limits, the largest dispersion error is 0.81%, 0.59%, and 0.43% for the Yee, , and HAM FW methods, respectively, without coefficient modification. As the time-step size decreases, the error increases for the HAM FW method, which is similar to the Yee’s FDTD method. The optimal value of the weight parameter can also be found by a search algorithm once the optimization criterion to is set. For a range of weight parameter from with a increment of 0.0001, and for Courant numto with a increment of 0.0001, a bers from where simple search algorithm was used to find values of is less the error than 10 . The search finds pairs, which meet the error criterion. However, the time-step size should not be larger than the upper bound given from (14a) . This eliminates many pairs, as leaving useful pairs with in the range from 1.0546 to 1.0743, which correspond to a time-step size limit larger than 0.9 times the Courant limit of Yee’s FDTD method. For the , , which pair is greater than , thus, the method is stable. For the pair , , thus, this method is also stable. With the increase of the weight parameter’s value, the limit of the time-step size decreases. Note that, at the time-step size limit, the dispersion error is not the least with the values of the weight parameter found by search. This method is termed the high-accuracy method by searching (HAM-S). and , the dispersion from Using theory and numerical experiment is graphed in Fig. 10 for

SUN AND TRUEMAN: OPTIMIZED FDTD METHODS BASED ON THE

STENCIL

HAM-S. It can be seen that the absolute numerical dispersion error is 0.247% along the axis, and 0.246% along the diagonal. The averaged-accumulated phase error is 0.00886 at 10 CPW, with a minimum of 0.0001 at 10.7 CPW, and a maximum of 0.0091 at 13.8 CPW, too small to be shown in Fig. 9. and was also verified The case of with numerical experiments. The maximum dispersion error is approximately 0.184%. Since the two cases have very close dispersion, the result is not shown in Fig. 10. However, caution must be taken when using the searching method. The time-step size limit for the IOM can be found from (14a) directly since the value of the optimal weight parameter (12) is independent of the time-step size. However, in the searching method, the value of the optimal weight parameter depends on the time-step size. One must make sure that the time-step size used in practice is smaller than the bound given by (14a) according to the value of the weight parameter. The HAM method, particularly the HAM-S method, is a broad-band more efficient method, with a time-step size larger method. The IOM method is relatively than that of the narrow band, but can have no numerical error at one designated or obfrequency. Note that using (10) at , and eliminates error along the corresponding tains angles, which is close to the search method. The search method is more general. VIII. CONCLUSION Starting with the computational stencil, a weighted FDTD method was formulated, which can be optimized by choosing a proper value of weight parameter based on the exact numerical dispersion relation, which includes all higher order terms. Methods to find the specific values of the weight parameter were given to eliminate or minimize numerical errors in various senses. The CM technique was used to reduce the numerical dispersion error further. The methods presented in this paper can have better accuracy than the standard scheme even though they are second-order accurate in space. In particular, the HAM-S has much higher accuracy than the scheme. Numerical experiments were used in this paper to validate the numerical dispersion predicted by theory in 2-D and good agreement was found. The optimized FDTD methods are simple, and easy to be incorporated into existing scheme. No additional comFDTD codes employing the putational effort is involved because the value of the weight parameter is computed before the FDTD algorithm is run. scheme is It is shown that, in terms of anisotropy, the fourth-order accurate; however, in terms of dispersion error, it is only second-order accurate. This paper has shown that, with the same computational stencil, the numerical dispersion, anisotropy, and the averaged accumulated phase errors are quite different with different values of weight parameter, coefficient modification, and timestep size. The optimization can be in different senses. A method can be designed having suitable optimization for a specific practical problem. For a nonsquare mesh, one can first partially use the CM technique to lift up the velocity along the axis with the coarse cell size, and then use the methods in this paper.

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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. 5, pp. 302–307, May 1966. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics—The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000. [3] T. Martin, “An improved near-to-far-zone transformation for the finitedifference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 9, pp. 1263–1271, Sep. 1998. [4] J. B. Cole, “A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 9, pp. 2053–2058, Sep. 1995. [5] , “High-accuracy Yee algorithm based on nonstandard finite differences: New development and validations,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1185–1191, Sep. 2002. [6] E. A. Forgy and W. C. Chew, “A time-domain method with isotropic dispersion and increased stability on an overlapped lattice,” IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 983–996, Jul. 2002. [7] J. W. Nehrbass, J. O. Jevtic, and R. Lee, “Reducing the phase error for finite-difference methods without increasing the order,” IEEE Trans. Antennas Propag., vol. 46, no. 8, pp. 1194–1201, Aug. 1998. [8] J. S. Juntunen and T. D. Tsiboukis, “Reduction of numerical dispersion in FDTD method through artificial anisotropy,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 4, pp. 582–588, Apr. 2000. [9] M. Rewienski and M. Mrozowski, “An iterative algorithm for reducing dispersion error on Yee’s mesh in cylindrical coordinates,” IEEE Microw. Guided Wave Lett., vol. 10, no. 9, pp. 353–355, Sep. 2000. [10] M. Wang, Z. Wang, and J. Chen, “A parameter optimized ADI-FDTD method,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 9, pp. 118–121, Sep. 2003. [11] K. Lan, Y. Liu, and W. Lin, “A higher order (2; 4) scheme for reducing dispersion in FDTD algorithm,” IEEE Trans. Electromagn. Compat., vol. 41, no. 2, pp. 160–165, May 1999. [12] S. Wang and F. L. Teixeira, “A three-dimensional angle-optimized finitedifference time-domain algorithm,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 811–817, Mar. 2003. [13] , “Dispersion-relation-preserving FDTD algorithms for large-scale three-dimensional problems,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 8, pp. 1818–1828, Aug. 2003. [14] , “A finite-difference time-domain algorithm optimized for arbitrary propagation angles,” IEEE Trans. Antennas Propag., vol. 51, no. 9, Sep. 2003. [15] , “Grid-dispersion error reduction for broadband FDTD electromagnetic simulations,” IEEE Trans. Magn., vol. 40, no. 2, pp. 1440–1443, Mar. 2004. [16] T. T. Zygiridis and T. D. Tsiboukis, “Low dispersion algorithms based on the higher order (2; 4) FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1321–1327, Apr. 2004. [17] Z. Xie, C.-H. Chan, and B. Zhang, “An explicit fourth-order staged finite-difference time-domain method for Maxwell’s equations,” J. Comput. Appl. Math., vol. 147, pp. 75–98, 2002. [18] Z. Shao, Z. Shen, Q. He, and G. Wei, “A generalized higher order finitedifference time-domain method and its application in guided-wave problems,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 856–861, Mar. 2003. [19] D. W. Zingg, “Comparison of high-accuracy finite-difference methods for linear wave propagation,” SIAM J. Sci. Comput., vol. 22, no. 2, pp. 476–502, Jul. 2000. [20] K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 642–653, Mar. 2003. [21] J. Chen, Z. Wang, and Y. Chen, “higher order alternative direction implicit FDTD method,” Electron. Lett., vol. 38, no. 22, pp. 1321–1322, Oct. 2002. [22] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. Pacific Grove, CA: Brooks/Cole, 1989. [23] G. Sun and C. W. Trueman, “Quantification of the truncation errors in finite-difference time-domain methods,” in Can. Electrical and Computer Engineering and Humane Technology Conf., Montreal, QC, Canada, May 4–7, 2003.

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Guilin Sun (S’02) was born in Henan, China, in 1962. He received the B.Sc. degree from the Xi’An Institute of Technology, Xi’An, China, in 1982, the M.Sc. degree from the Beijing Institute of Technology, Beijing, China, in 1988, respectively, both in optical engineering, and is currently working toward the Ph.D. degree in electrical and computer engineering at Concordia University, Montreal, QC, Canada. From 1988 to 1994 he was an Assistant Professor with the Xi’an Institute of Technology. From 1994 to 2000, he was an Associate Professor with the Beijing Institute of Machinery. From 1998 to 1999, he was a Research Associate with the University of Southern California, Los Angeles, where he was involved with the characterization of electrooptical (E/O) polymers used in photonic devices. He has authored or coauthored approximately 70 journal and conference papers in optical engineering and several papers on FDTD methods. He appears in Marquis Who’s Who in the World, 16th edition, 1999. His current research interest is in computational electromagnetics, particularly in new methods of the FDTD method in microwave and optical frequencies. Mr. Sun was the recipient of several awards and honorable titles.

Christopher W. Trueman (S’75–M’75–SM’96) received the Ph.D. degree from McGill University, Montreal, QC, Canada, in 1979. His doctoral dissertation concerned wire-grid modeling aircraft and their high-frequency (HF) antennas. He is currently a Professor with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada. His research concerns computational electromagnetics uses moment methods, the FDTD method, and geometrical optics and diffraction. He has been involved with electromagnetic compatibility (EMC) problems with standard broadcast antennas and high-voltage power lines, the radiation patterns of aircraft and ship antennas, EMC problems among the many antennas carried by aircraft, and on the calculation of the radar cross section (RCS) of aircraft and ships. He has studied the near and far fields of cellular telephones operating near the head and hand. He has recently been concerned with indoor propagation of RF signals and electromagnetic interference (EMI) with medical equipment in hospital environments.

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Novel E -Plane Filters and Diplexers With Elliptic Response for Millimeter-Wave Applications Erdem Ofli, Member, IEEE, Rüdiger Vahldieck, Fellow, IEEE, and Smain Amari, Member, IEEE

Abstract—A new class of -plane filters based on cross-coupled resonators or over-moded cavities is introduced. The filters exhibit pseduoelliptic transfer functions with steep attenuation slopes. The metal inserts and separating wall between waveguide sections can be fabricated using electro-deposition techniques with the accuracy required for millimeter-wave applications. The filters are mass-producible, are much shorter than traditional -plane filters, and have less insertion loss. Based on this new type of -plane filter, highly compact diplexer structures are designed. A comparison between prototype filters and design prediction shows excellent agreement. Index Terms—Diplexers, elliptic filters, guide filters.

-plane filters, waveFig. 1. Cross-coupled folded E -plane metal-insert filter structure with source– load coupling (four resonators).

I. INTRODUCTION

F

OR OVER 20 years, -plane filters have been known as a cost-effective solution for low-to-moderate filter requirements. Their ease of manufacturing and the availability of accurate design software have made direct-coupled -plane filters the structure of choice for many applications in the frequency range from 10 to 150 GHz (e.g., [1]–[4]). The electrical performance of -plane filters is mainly determined by the pattern of the metal inserts. Keeping those thin enough, say, between 30–100 m, the pattern can be fabricated with high precision using photolithographic or electro-deposition techniques. A commonly known disadvantage of direct-coupled -plane filters is that, for a given number of resonators, the attenuation slopes are relatively moderate. Steeper attenuation slopes are feasible only by adding more resonators, thus, also adding insertion loss and increasing the filter length. To avoid these disadvantages, but maintain the advantage of ease of fabrication of -plane filters, an alternative approach was suggested in [5]. By placing inductively coupled stopband stubs, the attenuation slope could be improved. The drawback of this solution is, however, that the waveguide split-block housing to clamp the metal insert became larger in size. This paper introduces yet another technique to improve the slope selectivity of -plane filters without increasing the number of resonators. This solution is based on the well-known technique of cross-coupling of resonators [6], [7], which, except in [8], has never been applied to -plane filters. Cross-coupling Manuscript received March 9, 2004; revised November 1, 2004. E. Ofli and R. Vahldieck are with the Laboratory for Electromagnetic Fields and Microwave Electronics, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland (e-mail: [email protected]). S. Amari is with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada K7K 7B4 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842506

between resonators generates transmission zeros at finite frequencies resulting in pseduoelliptic filter functions. As shown in the following, this technique allows very compact filter and diplexer designs based on -plane technology. II. CROSS-COUPLED

-PLANE FILTERS

The basic -plane structure that allows the realization of pseduoelliptic filter functions is shown in Fig. 1. The sketch shows a fourth-order -plane filter, which is folded between the second and third resonators. In this case, the source and load are coupled through a slot in the common wall separating both filter halves. The height of the slot is commensurate with the wavemodes. Cross-couguide height and, thus, exciting only plings between the resonators and the source and load lead to transmission zeros number of resonators at finite frequencies. The computed filter response of the source–load coupled -plane filter is shown in Fig. 2 [8]. For comparison, the same figure illustrates the response from a traditional -plane filter. Five finite transmission zeros are obtained with this configuration. Four of those are due to the source–load coupling and cross-coupling between resonators. The additional zero can be explained by the frequency dependence of the coupling coefficients, especially those implementing the cross-coupling between the source and load and resonators 1 and 4, respectively. In fact, it is possible to generate finite transmission zeros even in direct-coupled resonator filters if the coupling coefficients are strongly dispersive [9]. The simulation of the cross-cou-mode approach using the pled structure is based on a mode-matching technique (MMT). For the successful implementation of transmission zeros with cross-coupled resonators, shown in Fig. 1, it is important that both negative and positive coupling coefficients can be realized

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Fig. 2. Response of the cross-coupled folded E -plane metal-insert filter with source–load coupling (solid line) and straight E -plane single metal-insert filter (dashed line) with four resonators.

utilizing only magnetic coupling. If the source (besides its coupling to the load, Fig. 1) is coupled only to resonator 1 and the load only to resonator 4, then the coupling between resonators 1 and 4 must be negative in order to generate two transmission zeros on the imaginary axis of the complex -plane [10]. More precisely, the product of the three direct couplings and cross-coupling must be negative. This is not possible in an -plane structure if only the fundamental resonance is used in all resonators. In this case, the cross-coupling would be positive thereby placing the transmission zeros onto the real axis in the complex -plane. To achieve negative coupling between resonance is excited in resonator 3 source and load, a only. By placing the coupling slot between resonators 2 and 3 in the second half of resonator 3, the coupling between resonators 1 and 4 becomes negative. This shifting technique is well described in [11]. To design cross-coupled -plane filters, the first step is to synthesize a low-pass prototype, which fits the specifications and determine the appropriate coupling matrix. A synthesis technique of canonical filters with source–load coupling was developed by Bell [12]. A more recent technique suggests that filters with source–load coupling of orders higher than two are practically unrealizable given the large numerical differences between the entries of their coupling matrices [13]. The discussion in [13] overlooks, however, that a more useful solution of the coupling matrix exists and that the coupling coefficients can indeed be easily implemented for filters of any order. This was shown in [14]. From the synthesis of a pseduoelliptic filter, the target function is known. What follows is an electromagnetic (EM)-based optimization approach to find the optimum geometrical structure to match the target response. Before optimizing the structure, initial starting dimensions for the coupling and resonator sections of a direct-coupled filter are determined following the procedure given in [2]. Subsequently cross-coupling is introduced for which the size and location of the opening in the separating wall coupling resonators 1 and 4 are approximately determined. To do so, the coupled

Fig. 3. Coupling coefficient values versus: (a) iris width (l ) and (b) iris thickness of the coupled resonator structure.

resonator structure (resonators 1 and 4) is analyzed as sketched and are found in Fig. 3(a) and their resonant frequencies using the MMT. The coupling coefficient between both resonators ( ) is then calculated as [15] (1) Fig. 3(a) and (b) shows the coupling coefficient versus the width and thickness of the coupling iris, respectively. Enlarging the iris width or reducing the iris thickness increases the coupling value. Note that the height of the coupling slot is the same as the waveguide height in order not to complicate the computation. The opening size of the coupling slot and the lengths of resonators 1 and 4 are obtained by equating the value obtained from (1) to the corresponding entry in the coupling matrix. A similar procedure is applied to find the opening size between the second and third resonators (Fig. 1), except that now nonidentical cavities are considered. To illustrate the results of this initial design,

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Fig. 4. Computed response of the filter before optimization (dashed line), after optimization (solid line) and ideal response of the prototype filter (dotted line).

the response of a four-resonator filter with cross-coupling between the first and fourth resonators is plotted in Fig. 4. The dotted line in the same figure shows the ideal response of the prototype filter. Starting with these initial dimensions, the structure is optimized using a hybrid optimization technique based on a surrogate model [16]. For the example in Fig. 4, the sensitivities of the filter parameters [frequency shift of each single resonator with , input and output respect to the center frequency couplings m and coupling coefficients between res(direct and cross couplings)] with respect to the onators geometrical parameters (position and size of resonators, septa, and coupling slots in the separating wall) are computed. The resonant frequency shifts and the variations of coupling coefficients versus geometrical changes are shown in Fig. 5(a) and (b), respectively. The change in one dimension mainly changes the frequency and coupling parameter associated with the closest resonator. However, depending on the loading, slight changes also occur in neighboring resonators and respective coupling coefficients. The solid line in Fig. 4 shows the optimum response of the filter after 600 optimization steps (in the model space) and only 13 EM simulations. The optimized filter dimensions are shown in Table I. In the following, all filters are designed on the basis of the above procedure.

Fig. 5. Sensitivities of: (a) normalized frequency shifts and (b) normalized l, 2: l l, coupling coefficients with respect to geometrical changes (1: l l; l m). 3: l

+1

+ 1 . .. ; 1 = 10

+1

TABLE I

DATA

FOR

Ka-BAND CROSS-COUPLED FOLDED E -PLANE

METAL-INSERT FILTER (IN MILLIMETERS)

A. Advantages and Disadvantages of Filters With Source–Load Coupling Filters with source/load-multiresonator coupling show significantly improved slope selectivity compared to standard all-pole -plane filters with the same number of resonators. Fig. 6 shows the performances of three four-resonator filters of the same center frequency and bandwidth: a direct-coupled filter (solid line) and a cross-coupled filter with (dotted line) and without (dashed line) source–load coupling. A comparison of the asymmetric response of the cross-coupled filter with source–load coupling that of a cross-coupled filter without source–load coupling demonstrates further improvement in the stopband.

The main advantage of these filters is that they allow better control of the stopbands since they can implement up to finite transmission zeros (in a filter of order ) instead of only when the source and load are coupled to only one resonator each. The additional two transmission zeros can be used to either

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Fig. 6. Response of three four-resonator filters: direct-coupled filter (solid line) and cross-coupled filter with (dotted line) and without (dashed line) source–load coupling.

Fig. 7. Response of the cross-coupled folded E -plane metal-insert filter (solid line) and straight E -plane single metal-insert filter (dashed line) with four resonators.

improve the stopband or the group delay. The positions of the transmission zeros are very sensitive to small changes in the dimensions of the slot between source and load, which makes these filters sensitive to manufacturing tolerances. B. Four-Resonator Cross-Coupled Filter Fig. 7 shows a four-resonator filter structure with a crosscoupling between resonators 1 and 4. The in-band return loss is 20 dB with a center frequency of 30.25 GHz and a bandwidth of 500 MHz. The two finite transmission zeros are located at 29.50 and 31.00 GHz. The coupling matrix of this filter is given as

Fig. 8.

(2) To implement the negative coupling coefficient between resonators 1 and 4, the -mode resonance is used in resonator resonance. 3. All other resonators are based on the The response of the filter is compared with that of an ordinary -plane filter with the same number of resonators (Fig. 7). The dramatic improvement due to the two finite transmission zeros (solid line) in the cutoff slope is evident. Several response simulations of the filter were obtained applying random variations to the geometrical dimensions of the structure, as shown in Fig. 8. The analysis indicates that the waveguide width and insert dimensions must stay within 10 m of their nominal values to achieve repeatable results over a large number of samples, which is compatible with the tolerance requirements for direct-coupled filters. This accuracy for the metal inserts can easily be achieved, also in mass production, using an electro-deposition/plating technique.

6

Tolerance analysis ( 10 m) of the filter of Fig. 7.

C. Four-Resonator Cross-Coupled Filter With Asymmetric Response An asymmetric filter response with a steeper attenuation slope either on the left- or right-hand side from the passband can be quite useful for a number of applications. To achieve such a response with a cross-coupled -plane filter requires the coupling of resonators 1 and 4 via evanescent modes and resonators 1 and 3 via the fundamental mode. The corresponding coupling matrix is the following:

(3) This is demonstrated in Fig. 9. This figure shows a comparison between the MMT and, for verification, a finite-ele-

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Fig. 10. Structure of a cross-coupled folded waveguide bends.

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E -plane metal-insert filter with

Fig. 11. (a) Cross-coupled folded E -plane metal-insert filter with waveguide bends. (b) Metal inserts.

Fig. 9. (a) Structure of a cross-coupled folded E -plane metal-insert filter with four resonators (cross-coupling between first and third resonators). (b) Response of the filter by MMT (solid line) and HFSS (dashed line).

ment analysis (HFSS). Both simulations are in good agreement. The transmission zero on the left-hand side of the passband can be independently shifted closer to the passband by placing the septum (between resonators 3 and 4) closer to the coupling slot (between resonators 1 and 3). The position of the shorting wall (resonator 3) strongly affects the position of the transmission zero in the higher stopband. To shift both zeros at the same time, one has to change the dimension of the coupling slot between resonators 1 and 3. D. Four-Resonator Cross-Coupled Filter With Waveguide Bends The input and output waveguides of the original cross-coupled -plane filter (Fig. 7) are on the same side and are separated by only a 100- m-thick separating wall. With this configuration, it is not possible to connect standard waveguide adaptors to the input and output of the filter. One solution is to add 90 -plane bends in front and behind the filter, as shown in Fig. 10. For optimum filter design, the dimensions of the bends must be included in the optimization. The filter of Fig. 10 was fabricated and measured. A photograph of the filter and the metal inserts are shown in Fig. 11(a) and (b). The measured response is shown in Fig. 12, showing excellent agreement with the computed response. It is shown that not only the improvement in the attenuation slope is still maintained, but in addition, the interaction of the bends with the filter results in a better spurious response of the overall structure. E. Triplet The structure shown in the inset of Fig. 13 also allows the realization of filter functions with finite transmission zeros. This

Fig. 12. Calculated (dashed line) and measured (solid line) response of the filter (Fig. 11) and calculated response of the cross-coupled folded E -plane filter without waveguide bends (dashed–dotted line).

structure contains three resonators that are coupled to the load and the source, but not to each other. Similar structures with two iris-coupled resonators have been introduced and discussed in detail in [17]. The widths of the three waveguide parts are not equal. The dimensions are used to place the transmission zeros according to the application. Such a filter block can be used with other filter blocks to design higher order filters, and for applications where two transmission zeros are required in the immediate vicinity of the passband. The initial design of the filter is found by following the procedure outlined in [17]. The performance of the filter is illustrated in Fig. 13. The dashed line shows the initial response. The presence of three separate paths for the signal provides two transmission zeros at both sides of the passband. In order to achieve destructive interference becavity and two cavities are tween the paths, one employed.

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Fig. 13. Computed response of the triplet (inset) before optimization (dashed line) and after optimization (solid line).

Fig. 14. Direct-coupled E -plane metal-insert filter structure with over-moded cavity (four resonators).

III.

-PLANE FILTERS WITH OVER-MODED CAVITIES

In some applications, only one transmission zero either on the left- or right-hand side of the passband is required. This is possible by using dispersive stubs, as suggested in [18] for iriscoupled filters, and later also shown for -plane filters [19]. The design of -plane filters with over-moded cavities is also initially based on standard -plane filter design to meet given specifications roughly [2]. Here, one assumes all resonators coupled via metal inserts. From this procedure, the coupling coefficient between resonator 2 and the over-moded cavity is approximately known. The same coupling coefficient must now be realized by the slot width in the separating wall coupling resonator 2 into 3 (Fig. 14). To obtain a transmission zero either on the left- or right-hand side from the passband the length of the third resonator is increased until the transmission zero occurs. A transmission zero occurs to the right-hand side of the passband when the coupling slot is placed in the first half of , the over-moded cavity (counting from the shorting wall of Fig. 14), and on the left-hand side of the passband when the slot is positioned in the second half of the over-moded cavity. Two design examples for four-resonator filters are given in Fig. 15 with transmission zeros placed on either side of the passband. The in-band return loss in both cases is 23 dB (bandwidth 500 MHz) for center frequencies of 38.8 and 39.5 GHz, respectively. Both filters are used in Section IV in a diplexer application with transmission zeros placed within the guard band.

Fig. 15. (a) Response of a direct-coupled E -plane filter with over-moded cavity and transmission zero on the right-hand side of the passband computed with the MMT (solid line) and verified by HFSS (dashed line). (b) The same as (a), but with transmission zero on the left-hand side of the passband.

The thickness of the metal inserts and the separating wall is chosen to be 100 m (suitable for metal etching or electro-deposition/plating techniques). The filter of Fig. 15(a) was fabricated and measured to verify the design. The photograph of the filter is shown in Fig. 16(a) and (b); the metal inserts are of 100- m thickness. A comparison between measured and simulated response is given in Fig. 17. The principle of coupling-resonator sections by over-moded cavities is applied to a direct-coupled ridged waveguide filter, as shown in Fig. 18. The dashed lines in the filter sketch illustrate

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(a) Direct-coupled E -plane filter with over-moded cavity. (b) Metal

Fig. 17. Calculated (dashed line) and measured (solid line) response of the filter (Fig. 16). Fig. 19. Top view of the new diplexer structures: (a) with cross-coupled folded E -plane metal-insert filters and (b) with direct-coupled E -plane metal-insert filters with over-moded cavities.

Fig. 18. Response of the direct-coupled ridged E -plane metal-insert filter structure with an over-moded cavity calculated with HFSS.

the ridged waveguide sections of the structure. Direct coupling between the second and third resonators is achieved by introducing a full-height opening in the separating wall. An initial design of a direct-coupled ridged -plane filter with an over-moded cavity is obtained by adjusting the dimensions of the original optimized direct-coupled -plane filter shown in Fig. 15(a). The lengths of each coupling septum in the ridged filter are adjusted to obtain the same two-port scattering

parameters as the corresponding septum in the original filter structure without ridged sections at the center frequency of the filter. The lengths of the ridged resonator sections are then calculated using the propagation constants of the fundamental mode in the ridge sections, i.e., making the electrical lengths of resonators in both filters equal. The final dimensions are then found by optimization. A -band direct-coupled ridged -plane filter with an over-moded cavity is shown in Fig. 18. The thickness of the metal inserts and the separating wall is, again, 100 m. The improvement in the stopband due to the presence of a finite transmission zero at 39.92 GHz is obvious. The resulting filter is more compact than the full-height directcoupled -plane filter with an over-moded cavity and has a wider spurious-free range. IV. MILLIMETER-WAVE DIPLEXER Both filter structures shown in Figs. 10 and 14 are well suited for compact and high-performance millimeter-wave diplexers [20]. This is shown in Fig. 19(a) where two cross-coupled folded -plane filters (Fig. 10) are combined with a tapered broadband -plane power divider. Fig. 19(b) illustrates a diplexer composed of an -plane T-junction with a metal slab and two -plane filters with over-moded cavities (Fig. 14).

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Fig. 20. Response of the diplexer of Fig. 19(a) computed with the MMT (solid line) and verified by HFSS (dashed line).

The initial design and optimization of the channel filters, the -plane transformer [21], and the -plane T-junction [22] is done separately. In a final optimization run, the entire structure is fine tuned to account for the interaction effects at the various interfaces. For the structure in Fig. 19(a), the cross-coupled folded -plane filters from Fig. 10 are utilized. In particular, the following dimensions are subject to optimization: taper diand lengths mensions like the individual waveguide widths , the distance between the bifurcation [ , Fig. 19(a)] and the taper, the distance between the filters and the start of the bifurcation [ , Fig. 19(a)], and the first septum of the individual filters [ , Fig. 19(a)]. Other dimensions of the filters remain unchanged throughout the optimization. The overall structure is analyzed and optimized by using the MMT. The performance of the entire diplexer is illustrated in Fig. 20. The dashed line in the same figure shows the response obtained by an HFSS analysis to verify the MMT results. The agreement between both simulations is very good. It should be noted that due to long CPU run times, the optimization could not be done directly with HFSS, which requires over 90 000 tetrahedras to discretize the entire structure. Following the same procedure as in the previous diplexer design, another compact diplexer is designed using two direct-coupled -plane filters with only one transmission zero each [see Fig. 19(b)]. The channel filters are designed and pre-optimized such that the attenuation poles provided by each filter are placed in the guard band to improve the isolation between the two chanand , as well as the dimensions of the nels. The lengths -plane T-junction [ and , Fig. 19(b)] were optimized to compensate for the mutual loading effect between the filters and to ensure a good match between the input and output waveguides. The lengths of the first septa of the individual filters [ and , Fig. 19(b)] are also included in the optimization. Fig. 21 shows the performance of the entire diplexer structure after optimization. The return loss is better than 15 dB in the passbands of the channels and can be improved further by including the other dimensions of the filters as independent variables in the optimization.

Fig. 21.

Response of the diplexer of Fig. 19(b) computed with the MMT.

V. CONCLUSION Novel -plane metal-insert filters and diplexers based on cross-coupled resonators or over-moded cavities have been introduced. These structures are suitable for application in broadband millimeter-wave links as low-cost, high-performance, and compact components. Utilizing source–load coupling, over-moded cavities or combinations thereof leads to a highly flexible class of filter structures suitable for applications in the frequency range from 10 to 100 GHz. Experimental filters have been fabricated and tested. Measured results are in good agreement with computed results. REFERENCES [1] R. Vahldieck, J. Bornemann, F. Arndt, and D. Grauerholz, “Optimized waveguide E -plane metal insert filters for millimeter wave applications,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 1, pp. 65–69, Jan. 1983. [2] Y. C. Shih, “Design of waveguide E -plane filters with all-metal inserts,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 7, pp. 695–704, Jul. 1984. [3] L. Q. Bui, D. Ball, and T. Itoh, “Broad-band millimeter-wave E -plane bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 12, pp. 1655–1658, Dec. 1984. [4] R. Vahldieck and W. J. Hoefer, “Finline and metal insert filters with improved passband separation and increased stopband attenuation,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 12, pp. 1333–1339, Dec. 1985. [5] J. Bornemann, “Selectivity-improved E -plane filter for millimeter-wave applications,” Electron. Lett., vol. 27, no. 21, pp. 1891–1893, 1991. [6] A. E. Atia and A. E. Williams, “Nonminimum-phase optimum-amplitude bandpass waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 4, pp. 425–431, Apr. 1974. [7] S. Amari and J. Bornemann, “CIET-analysis and design of folded asymmetric H -plane waveguide filters with source–load coupling,” in Proc. 30th Eur. Microwave Conf., Paris, France, Oct. 2000, pp. 270–273. [8] E. Ofli, R. Vahldieck, and S. Amari, “Analysis and design of mass-producible cross-coupled, folded E -plane filters,” in IEEE MTT-S Int. Microwave Symp. Dig., Phoenix, AZ, May 2001, pp. 1775–1778. [9] S. Amari and J. Bornemann, “Using frequency-dependent coupling to generate finite attenuation poles in direct-coupled resonator bandpass filters,” IEEE Microw. Guided Wave Lett., vol. 9, no. 10, pp. 404–406, Oct. 1999. [10] A. E. Williams, “A four-cavity elliptic waveguide filter,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 12, pp. 1109–1114, Dec. 1970.

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[11] U. Rosenberg, “New planar waveguide cavity elliptic function filters,” in Proc. 25th Eur. Microwave Conf., Bologna, Italy, Sep. 1995, pp. 524–527. [12] H. C. Bell, Jr., “Canonical lowpass prototype network for symmetric coupled-resonator bandpass filters,” Electron. Lett., vol. 10, no. 13, pp. 265–266, 1974. [13] J. R. Montejo-Garai, “Synthesis of N -even order symmetric filters with N transmission zeros by means of source–load cross-coupling,” Electron. Lett., vol. 36, no. 3, pp. 232–233, 2000. [14] S. Amari, “Direct synthesis of folded symmetric resonator filters with source–load coupling,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 6, pp. 264–266, Jun. 2001. [15] K. R. Sturley, Radio Receiver Design. London, U.K.: Chapman & Hall, 1945, ch. 7. [16] P. Harscher, E. Ofli, R. Vahldieck, and S. Amari, “EM-simulator based parameter extraction and optimization technique for microwave and millimeter wave filters,” in IEEE MTT-S Int. Microwave Symp. Dig., Seattle, WA, Jun. 2002, pp. 1113–1116. [17] S. Amari and U. Rosenberg, “The doublet: A new building block for modular design of elliptic filters,” in Proc. 32th Eur. Microwave Conf., Milan, Italy, Sep. 2002, pp. 405–407. [18] W. Menzel, F. Alessandri, A. Plattner, and J. Bornemann, “Planar integrated waveguide diplexer for low-loss millimeter wave applications,” in Proc. 27th Eur. Microwave Conf., Jerusalem, Israel, Sep. 1997, pp. 676–680. [19] E. Ofli, R. Vahldieck, and S. Amari, “Compact E -plane and ridge waveguide filters/diplexers with pseudo-elliptic response,” in IEEE MTT-S Int. Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 949–952. , “A novel compact millimeter-wave diplexer,” in IEEE MTT-S Int. [20] Microwave Symp. Dig., Seattle, WA, Jun. 2002, pp. 377–380. [21] J. Bornemann, “Design of millimeter-wave diplexers with optimized H -plane transformer sections,” Can. J. Elect. Comput. Eng., vol. 15, no. 3, pp. 5–8, 1990. [22] Y. Rong, H. Yao, K. A. Zaki, and T. Dolan, “Millimeter-wave Ka-band H -plane diplexers and multiplexers,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2325–2330, Dec. 1999.

Erdem Ofli (M’02) received the B.Sc. and M.Sc. degrees in electrical engineering from Bilkent University, Ankara, Turkey, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from Swiss Federal Institute of Technology (ETH) Zürich, Zürich, Switzerland, in 2004. His research interests include microwave and millimeter-wave components and systems design and numerical techniques in electromagnetics.

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Rüdiger Vahldieck (M’85–SM’86–F’99) received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the University of Bremen, Bremen, Germany, in 1980 and 1983, respectively. From 1984 to 1986, he was a Post-Doctoral Fellow with the University of Ottawa, Ottawa, ON, Canada. In 1986, he joined the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, where he became a Full Professor in 1991. During the fall of 1992 and the spring of 1993, he was a Visiting Scientist with the Ferdinand-Braun-Institute für Hochfrequenztechnik, Berlin, Germany. In 1997, he accepted an appointment as a Professor of EM-field theory with the Swiss Federal Institute of Technology (ETH) Zürich, Zürich, Switzerland, and became Head of the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH) in 2003. His research interests include computational electromagnetics in the general area of electromagnetic compatibility (EMC) and, in particular, for computer-aided design of microwave, millimeter-wave, and opto-electronic integrated circuits. Since 1981, he has authored or coauthored over 230 technical papers in books, journals, and conferences, mainly in the field of microwave computer-aided design. Prof. Vahldieck is the past president of the IEEE 2000 International Zürich Seminar on Broadband Communications (IZS’2000). Since 2003, he has been president and general chairman of the International Zürich Symposium on Electromagnetic Compatibility. He is a member of the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. From 2000 to 2003, he was an associate editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, and in January 2004, he became the editor-in-chief. Since 1992, he has served on the Technical Program Committee (TPC) of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), the IEEE MTT-S Technical Committee on Microwave Field Theory, and in 1999, on the TPC of the European Microwave Conference. From 1998 to 2003, he was the chapter chairman of the IEEE Swiss Joint Chapter on Microwave Theory and Techniques, Antennas and Propagation, and EMC. Since 2005, he has been president of the Research Foundation for Mobile Communications. He was the recipient of the J. K. Mitra Award of the Institution of Electronics and Telecommunication Engineers (IETE) (in 1996) for the best research paper in 1995 and was corecipient of the Outstanding Publication Award of the Institution of Electronic and Radio Engineers in 1983.

Smain Amari (M’98) received the DES degree in physics and electronics from Constantine University, Constantine, Algeria, in 1985, and the Masters degree in electrical engineering and Ph.D. degree in physics from Washington University, St. Louis, MO, in 1989 and 1994, respectively. From 1994 to 2000, he was with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada. From 1997 to 1999, he was a Visiting Scientist with the Swiss Federal Institute of Technology, Zürich, Switzerland, and a Visiting Professor in Summer 2001. Since November 2000, he has been with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada, where he is currently an Associate Professor. He is interested in numerical analysis, numerical techniques in electromagnetics, applied physics, applied mathematics, wireless and optical communications, computer-aided design (CAD) of microwave components, and application of quantum field theory in quantum many-particle systems.

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Analysis and Design of a High-Efficiency Multistage Doherty Power Amplifier for Wireless Communications Nuttapong Srirattana, Student Member, IEEE, Arvind Raghavan, Member, IEEE, Deukhyoun Heo, Member, IEEE, Phillip E. Allen, Fellow, IEEE, and Joy Laskar, Senior Member, IEEE

Abstract—A comprehensive analysis of a multistage Doherty amplifier, which can be used to achieve higher efficiency at a lower output power level compared to the classical Doherty amplifier, is presented. Generalized design equations that explain the operation of a three-stage Doherty amplifier, which can be easily extended -stage Doherty amplifier, are derived. In addition, the to an optimum device periphery, which minimizes AM–AM distortion for perfect Doherty amplifier operation, is analyzed. For the first time, a multistage Doherty power amplifier that meets wide-band code-division multiple-access (WCDMA) requirements is demonstrated. The designed power amplifier exhibits a power-added efficiency (PAE) of 42% at 6-dB output power backoff and 27% at 12-dB output power backoff. These PAEs are more than 2 and 7 better, respectively, than that of a single-stage linear power amplifier at the same output power backoff levels. The power amplifier is capable of delivering up to 33 dBm of output power, and has a maximum adjacent channel power leakage ratio of 35 and 47 dBc at 5- and 10-MHz offset, respectively. To the best of the authors’ knowledge, these represent the best reported results of a Doherty amplifier for WCDMA application in the 1.95-GHz band to date. Index Terms—Amplifiers, code division multiple access, MESFETs, microwave Doherty amplifiers, microwave power amplifiers, power amplifiers.

I. INTRODUCTION

P

OWER-AMPLIFIER efficiency enhancement techniques have become critical in the design of modern communication systems in which one of the main concerns is power consumption. It is known that the power amplifier is one of the most power-consuming components of the system. Several modulation techniques employed in wireless communication today require linear amplification with a high peak-to-average ratio. Unfortunately, linear power amplifiers generally have poor efficiency, especially when not operated at the maximum output power condition. Significant effort has been expended in the development of techniques to improve the efficiency of linear RF power amplifiers, especially in the low-power (or “backed-off”) region, where the power amplifier normally operates. Several methods, such as envelope elimination and Manuscript received March 10, 2004; revised July 3, 2004. N. Srirattana, P. E. Allen, and J. Laskar are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]). A. Raghavan is with Quellan Inc., Atlanta, GA 30318 USA. D. Heo is with the Electrical and Computer Science Department, Washington State University, Pullman, WA 99164 USA. Digital Object Identifier 10.1109/TMTT.2004.842505

Fig. 1.

Schematic diagram of the classical Doherty amplifier.

restoration (EER) and bias adaptation have been extensively explored [1]–[3]. However, these methods require the use of external control circuits and signal processing, resulting in an increased level of design complexity. The Doherty amplifier is a technique for improving the efficiency of backed-off linear amplifiers. In a Doherty amplifier, the output powers of two amplifiers operating at a proper phase alignment and bias level are combined using a quarter-wave transmission line without the use of any additional components [4]. The “self-managing” characteristic of the Doherty amplifier has made its implementation attractive for various applications [5]–[21]. In the classical Doherty amplifier configuration, shown in Fig. 1, the saturation power of the carrier amplifier is one-fourth of the maximum system output power. This results in an efficiency peak at 6-dB output power backoff from the normal peak efficiency power level. In the past few years, researchers have focused on the design of asymmetrical Doherty amplifiers, where the saturation of the carrier amplifier is at a lower level compared to the classical design [12], [13], [18], [19]. Theoretically, asymmetrical Doherty amplifier designs exhibit a significant drop in efficiency in the region between the efficiency peaking points, with the extent of efficiency reduction being proportional to the backoff level, as shown in Fig. 2. Nevertheless, it is possible to use more than two amplifiers to maintain the efficiency without significant dropping throughout the backoff region and extend the backoff level far beyond the classical design. This is the so-called multistage Doherty amplifier [20], [21]. The concept of the multistage Doherty amplifier is similar to the conventional Doherty amplifier, yet no detailed analysis, particularly for the design of the generalized multistage Doherty amplifier, has been presented. This paper presents a set of novel and comprehensive design equations for the multistage Doherty amplifier and, for the first time, an implementation of a three-stage Doherty power amplifier (DPA) system to meet

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Fig. 2. Drain efficiency of the multistage DPA using class-B amplifiers ( = 2; = 4; = 4; = 4 for asymmetrical DPA [13]).

Fig. 4. Equivalent circuit of the three-stage Doherty amplifier at: (a) lowpower operation, (b) medium-power operation, and (c) high-power operation.

(2) Fig. 3.

Schematic diagram of the multistage Doherty amplifier.

the wide-band code-division multiple-access (WCDMA) uplink standard specifications, with significant efficiency enhancement at backed-off conditions. It will be shown that, by following the given analysis, practical performance closely approaching the promise of theory can be achieved. The practical issue of the correlation of AM–AM distortion with the choice of the device periphery has been explored and described mathematically, and proven through the design prototype. II. ANALYSIS OF MULTISTAGE DOHERTY AMPLIFIER The multistage Doherty amplifier (Fig. 3) uses more than one peak amplifier, with quarter-wave transmission lines to combine their output power. An active load–pulling effect is created as different amplifiers turn on at different power levels. In the design of a multistage Doherty amplifier, it is important to find the characteristic impedance of the quarter-wave transmission lines required for Doherty operation. These impedance values depend on the level of backoff and can be calculated using the following set of equations: (1)

where (for odd ) or (for is the total number of amplifier stages, and is even ), the backoff level (positive value in decibels) from the maximum output power of the system at which the efficiency will peak. is set by the carrier amThe maximum level of backoff plifier. The number of efficiency peaking points is directly proportional to the number of amplifier stages used in the design. A. Principle of Operation To simplify the analysis, the equivalent circuit of the threestage Doherty amplifier with an ideal current source to represent each amplifier, as shown in Fig. 4, is used in the derivation. The design equations for an -stage Doherty amplifier can easily be generalized from this analysis. From (1) and (2), the characteristic impedance of each output quarter-wave transmission line is found to be (3) (4) where (5) (6) The phase of the peak amplifier output currents must lag that of the carrier amplifier output current

and by 90

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and 180 , respectively, for proper Doherty amplifier operation. This is achieved by inserting additional transmission lines at the input of each peak amplifier. By doing so, the signal from each amplifier will be added constructively at the output load. The operation of a three-stage Doherty amplifier can be separated into three regions, which are: 1) low-power operation, where only the carrier amplifier is turned on; 2) medium-power operation, where carrier amplifier and peak amplifier #1 are both turned on; and 3) high-power operation, where all the power amplifiers are turned on. At low-power operation [see Fig. 4(a)], where all the peak amplifiers are in the off state and appear as open circuits, the carrier amplifier will see an impedance given by (7) where (8) , the maximum Assuming the system supply voltage is , when the carrier amplifier is in output power at the load saturation, is (9) For the case of the three-stage Doherty amplifier, with peak efficiency at 6- and 12-dB backoff ( and ), the values of and are 2 and 4, respectively. At this point, the maximum power from the carrier amplifier will be 1/16 of the maximum possible system power. Since the other amplifiers are still turned off and the carrier amplifier is in saturation, the entire output power will be delivered only from this amplifier and, therefore, the overall efficiency is equal to the maximum efficiency of the carrier amplifier. In the medium-power operation region [see Fig. 4(b)], the impedances presented at the output of each amplifier can be analyzed using power conservation (see the Appendix) and are given by (10)

In the high-power region [see Fig. 4(c)], peak amplifier #2 is in operation and produces a load–pulling effect on the other amplifiers. An analysis of the terminating impedance to the output of each amplifier can be done similarly to the medium power case, resulting in the following design equations: (12) (13) (14) where , which has a value between (when peak amplifier #1 saturates) and 1 (when all amplifiers satuand rate). For the three-stage Doherty amplifier with decreases from to when peak amplifier #2 has reached saturation, while remains unchanged. is equal to , which enables half the total At saturation, system output power to be delivered from peak amplifier #2. The rest of the output power is delivered from the carrier amplifier and peak amplifier #1 in the ratio of 1 : 3. should be determined by It is suggested that the value of from considering the amount of power delivered to the load since this peak amplifier will the peak amplifier stage # contribute more power compared to other stages. Using (14), will be terminated with the optimum peak amplifier # is calculated by matching condition when the value of # where amplifier #

(15)

is the optimum termination of the peak

# .

B. Efficiency Calculation The efficiency of the three-stage Doherty amplifier can be calculated using power conservation analysis. For low-power operation [see Fig. 4(a)], using (7) and the relationship of the and voltage across the load carrier amplifier dc current given by [20] (16) the ideal drain efficiency using a class-B amplifier for the carrier amplifier can be formulated as

(11) , which has a value between (when where only the carrier amplifier saturates) and 1 (when both carrier and peak amplifier #1 saturate), assuming every amplifier uses . For the case of the threethe same drain bias voltage of stage Doherty amplifier, with and will to at the saturation of peak amplifier #1 reduce from because of the load–pulling effect resulting from the turning on of peak amplifier #1. Also, peak amplifier #1 is presented with a of . Therefore, the total output transformed impedance power, which is delivered equally from both amplifiers, will be 1/4 of the maximum output power (i.e., 6-dB backoff) and will result in an efficiency peak (equal to the maximum efficiency) since both amplifiers are in saturation.

(17) This implies a peak efficiency at or of the maximum output power of the system ( % for a class-B amplifier). For medium-power operation [see Fig. 4(b)], dc current consumption of carrier amplifier and peak amplifier #1 can be calculated as (18) (19)

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Therefore, for class-B carrier and peak amplifiers, the drain efficiency can be formulated as

(20) From (20), there exists another peak in efficiency at or of the maximum output power of the system. Similarly, for high-power operation [see Fig. 4(c)], the carrier and peak amplifier dc currents are derived as (21) (22) (23) Finally, the drain efficiency is given by

Fig. 5. Fundamental current as a function of conduction angle (A relative device periphery).

=

gain reduction in a class-C amplifier with minimum alteration of the input power division. To illustrate this, the three-stage Doherty amplifier described in Section II is used in the following analysis. To maintain the gain level until the maximum power operation, the periphery of the peak amplifiers needs to be increased to compensate for fundamental current reduction (assuming a fixed bias configuration is used). The fundamental component is given by the following expression [22]: of RF current (25)

(24) , which Equation (24) shows a peak in efficiency at is at the maximum system power. For the general case of the -stage Doherty amplifier, the drain efficiency can be calculated similarly to the above analysis. The simulated drain efficiency of the multistage Doherty amplifier using class-B amplifiers is illustrated in Fig. 2.

is the maximum current swing for class-A operation where and is the conduction angle. Since is proportional to the device periphery, using (21), (22), and (23), and assuming that, in a general case, peak amplifier transistors are approaching class-B bias at the peak system operation, the relative device periphery can be calculated to be Device periphery ratio carrier amplifier: peak amplifier #1: peak amplifier #2 (26)

III. PRACTICAL CONSIDERATIONS IN MULTISTAGE DOHERTY AMPLIFIER DESIGN In the Doherty amplifier, peak amplifiers are biased below the threshold so that they will not be turned on before the input power has reached a predetermined level. This means the operation of peak amplifiers has been forced into class C, which is known to typically have lower gain compared to the class-A/AB operation of the carrier amplifier. Therefore, to achieve the required output power in each operation region, two factors can be controlled: the input power to each amplifier and the size of each transistor, which is related to current gain. It is possible to provide greater input power to the peak amplifiers, which are biased in class C, to achieve the required output power without creating AM–AM distortion. Nevertheless, the unequal division of input power will reduce the amount of output power from other transistors, thus reducing the overall gain. To avoid this, the current gain of each transistor must be adjusted to compensate for the

In practice, the periphery of the transistor for peak amplifiers may need to be larger than the above calculated values, as they may still be in class-C bias at maximum system operation, as shown in Fig. 5. This can be determined experimentally depending on the extent of backoff efficiency improvement and bias point design. Nevertheless, the periphery of the carrier amplifier must be adjusted to provide sufficient output power at the system’s maximum operating level and used as a reference for the peak amplifiers. The periphery of the carrier amplifier device can be calculated by (27) where

is the device periphery of the carrier amplifier and is the device periphery of a class-A biased transistor . that can deliver the system’s maximum power to the load

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Schematic diagram of the three-stage WCDMA DPA. Fig. 8. Measured output power, gain, and PAE of the three-stage WCDMA DPA using a device periphery ratio of 1 : 2 : 4 at 1.95 GHz.

Fig. 7. Board layout of the three-stage WCDMA DPA prototype using a device periphery ratio of 1 : 2 : 4. Each GaAs FET device has the same package size of 3.8 mm 4.2 mm.

2

IV. MEASUREMENT RESULTS and To verify the analysis, a three-stage DPA with was designed using GaAs field-effect transistor (FET) devices and microstrip-based power-combining elements on an FR-4 printed circuit board. The schematic diagram is shown in Fig. 6, with the actual board layout shown in Fig. 7. The three-stage DPA is targeted for a class-1 WCDMA uplink standard with 33-dBc ACLR1 and 43-dBc ACLR2 linearity requirements with 33-dBm maximum output power. The input signal is divided equally by a microstrip Wilkinson power divider network before feeding to the input matching network of each amplifier. Delay lines of 90 and 180 are inserted at the input of peak amplifiers #1 and #2, respectively. The design is tested with a single-tone signal at 1.95 GHz, the results of which are shown in Fig. 8. The choice of device periphery is calculated from (26) with the predetermined backoff level improvement, which results in a ratio of 1 : 3 : 4. However, because of the limitation of device availability, a device size ratio of 1 : 2 : 4 was chosen with the being 2400 m/0.6 m. The drain carrier amplifier device bias voltage for all GaAs FET devices in this design is 10 V. The gate bias voltage of the carrier amplifier is set to 1.64 V, which is above the pinchoff voltage of 2 V for class-AB biasing. The

Fig. 9. Comparison of PAE measurement results of the three-stage WCDMA DPAs using different device periphery ratios at the frequency of 1.95 GHz (0-dB backoff corresponds to output power of 33 dBm).

gate-bias voltages of peak amplifiers #1 and #2 are adjusted so that peak amplifiers #1 and #2 are turned on at the backed-off levels of 12 and 6 dB, respectively. The design achieved a linear power gain of 12.2 dB with 1-dB output compression at 33 dBm. The 33-dBm output power level is defined as the 0-dB backoff level in this study. At that point, the power-added efficiency (PAE) was measured to be 48.5%. The PAE was also measured to be 42% at 6-dB backoff, and 27% at 12-dB backoff, which represents a PAE improvement of 2.5 and 7.5 , respectively, compared to a single-stage class-AB design. The PAE at 12-dB backoff, shown in Fig. 9, is lower than the ideal simulation mainly because of the soft turn-on characteristic of the peak amplifier devices, which contributes to more dc power consumption at the low power level. Nevertheless, these results show an impressive PAE improvement in the and backoff region. In Fig. 9, a three-stage DPA with using the device periphery ratio of 1 : 1 : 1 was designed and measured for giving a comparison. It is seen from Fig. 9 that a lower device periphery ratio (1 : 1 : 1) results in poorer PAE improvement at the same backoff level.

SRIRATTANA et al.: HIGH-EFFICIENCY MULTISTAGE DOHERTY POWER AMPLIFIER FOR WIRELESS COMMUNICATIONS

Fig. 10. Measured ACLRs of the three-stage WCDMA DPAs using different device periphery ratios at 1.95 GHz.

Fig. 11. Measured output power of the three-stage WCDMA DPA with a device periphery ratio of 1 : 2 : 4 versus frequency at the maximum output power level (33 dBm at 1.95 GHz).

The three-stage DPA is tested with a real-time WCDMA signal using a chip rate of 3.84 Mc/s generated from the Agilent E4438C vector signal generator. The output signal is measured with a raised root cosine (RRC) filter with of 0.22 and a bandwidth equal to the chip rate. The adjacent channel power leakage ratio (ACLR) measurement results in Fig. 10 show that the design meets the WCDMA ACLR requirements of 33 dBc (ACLR1) and 43 dBc (ACLR2) at 5- and 10-MHz offset, respectively, up to a power output of 34 dBm. ACLR1 and ACLR2 at 33-dBm output power are measured to be 35 dBc and 47 dBc, which provide a few decibels of margin over the linearity requirement. Moreover, it can be noticed that the ACLR levels of the three-stage DPA with the device periphery ratio of 1 : 2 : 4 are higher than that of the 1 : 1 : 1 design since each amplifier stage is operated closer to its saturation, resulting in better overall PAE. The 3-dB bandwidth is measured to be 160 MHz, dominated by the quarter-wave transformer characteristic, with only 0.5-dB output power variation from center frequency in the WCDMA uplink frequency range (1.92–1.98 GHz), as shown in Fig. 11. The ACLR is still within the specification in this frequency range (Fig. 12).

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Fig. 12. Measured ACLR of the three-stage WCDMA DPA with a device periphery of 1 : 2 : 4 versus frequency at the maximum output power level (33 dBm at 1.95 GHz).

Fig. 13. Bias voltage adjustment of peak amplifier #2 with increasing input drive level to alleviate AM–AM distortion.

Thus, the device size ratio of 1 : 2 : 4 obtained from ideal calculations results in a performance sufficient to meet the stringent WCDMA requirements. However, in reality, the peak amplifier devices may not have reached class-B operation, as assumed in the ideal-case calculations. It is possible to further enhance the performance of the DPA by moving the bias point of the peak amplifier from class C in the backoff region toward class B as much as possible by adjusting the gate voltage with increasing input drive. In this design, peak amplifier #2, which was biased in very deep class C, is now biased manually with a dynamic bias profile, as illustrated in Fig. 13. The measurement results of output power characteristics and linearity of the dynamic-biased three-stage DPA are shown in Figs. 14 and 15. It is seen that the results have improved in all respects. The ACLR levels at the output power of 33 dBm have reduced from 35 to 40 dBc and 47 to 55 dBc for ACLR1 and ACLR2, respectively. This is expected from the improved AM–AM characteristics of the dynamically biased amplifier compared with the fixed bias amplifier, as observed from the output power curves in Fig. 14. It can be summarized

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APPENDIX [in Fig. 4(b)] is equal The output power delivered to the load to the combination of the output power delivered from carrier amplifier and peak amplifier #1, which can be expressed as

(28) or (29)

where and

. is the parallel combination of , which is equal to . Since is equal to , the expression for can be written as

Fig. 14. Measured output power, gain, and PAE of the three-stage WCDMA DPA with dynamic biasing applied to peak amplifier #2 (at 1.95 GHz).

(30)

Using (29) and (30),

and

can be derived as

(31) (32)

REFERENCES [1] G. Hanington, P.-F. Chen, P. M. Asbeck, and L. E. Larson, “High-efficiency power amplifier using dynamic power-supply voltage for CDMA applications,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1471–1476, Aug. 1999. [2] F. H. Raab, B. E. Sigmon, R. G. Myers, and R. M. Jackson, “ -band transmitter using Kahn EER technique,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2220–2225, Dec. 1998. [3] 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. [4] W. H. Doherty, “A new high efficiency power amplifier for modulated waves,” in Proc. IRE, vol. 24, Sep. 1936, pp. 1163–1182. [5] R. J. McMorrow, D. M. Upton, and P. R. Maloney, “The microwave Doherty amplifier,” in IEEE MTT-S Int. Microwave Symp. Dig., 1994, pp. 1653–1656. [6] D. M. Upton, “A new circuit topology to realize high efficiency, high linearity, and high power microwave amplifiers,” in Proc. Radio Wireless Conf., 1998, pp. 317–320. -band Doherty amplifier [7] C. F. Campbell, “A fully integrated MMIC,” IEEE Microw. Guided Wave Lett., vol. 9, no. 3, pp. 114–116, Mar. 1999. [8] C. P. McCarroll, G. D. Alley, S. Yates, and R. Matreci, “A 20 GHz Doherty power amplifier MMIC with high efficiency and low distortion designed for broad band digital communication systems,” in IEEE MTT-S Int. Microwave Symp. Dig., 2000, pp. 537–540. [9] K. W. Kobayashi, A. K. Oki, A. Guitierrez-Aitken, P. Chin, L. Yang, E. Kaneshiro, P. C. Grossman, K. Sato, T. R. Block, H. C. Yen, and D. C. Streit, “An 18–21 GHz InP DHBT linear microwave Doherty amplifier,” in IEEE Radio Frequency Integrated Circuits Symp. Dig., 2000, pp. 179–182. [10] Y. Yang, J. Yi, Y. Y. Woo, and B. Kim, “Optimum design for linearity and efficiency of microwave Doherty amplifier using a new load matching technique,” Microwave J., vol. 44, no. 12, pp. 20–36, Dec. 2001.

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Fig. 15. Measured ACLR of the three-stage WCDMA DPA with dynamic biasing applied to peak amplifier #2 (at 1.95 GHz).

that with a correct choice of device periphery and appropriate biasing, the performance of the multistage design can be as good as a perfect class A/AB single-stage amplifier at the maximum operating power, with improved PAE in the low power region. V. SUMMARY This paper has presented the design and analysis of a threestage DPA for WCDMA application, which shows a significant improvement in PAE in the low power region, compared to a single-stage design, while satisfying all the WCDMA requirements. The design equations derived in this paper can easily be generalized to the design of an -stage Doherty amplifier that mitigates efficiency degradation up to even higher output power backoff levels. The analysis of power device selection enables the achievement of near-perfect Doherty amplifier operation. For the design of communication systems where efficiency enhancement is needed, the multistage Doherty amplifier can be an interesting alternative to existing reported techniques.

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SRIRATTANA et al.: HIGH-EFFICIENCY MULTISTAGE DOHERTY POWER AMPLIFIER FOR WIRELESS COMMUNICATIONS

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20] [21]

[22]

, “Experimental investigation on efficiency and linearity of microwave Doherty amplifier,” in IEEE MTT-S Int. Microwave Symp. Dig., 2001, pp. 1367–1370. S. Bousnina and F. M. Ghannouchi, “Analysis and experimental study of an L-band new topology Doherty amplifier,” in IEEE MTT-S Int. Microwave Symp. Dig., 2001, pp. 935–938. M. Iwamoto, A. Williams, P.-F. Chen, A. Metzger, L. Larson, and P. Asbeck, “An extended Doherty amplifier with high efficiency over a wide power range,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2472–2479, Dec. 2001. J. Lees, M. Goss, J. Benedikt, and P. J. Tasker, “Single-tone optimization of an adaptive-bias Doherty structure,” in IEEE MTT-S Int. Microwave Symp. Dig., 2003, pp. 2213–2216. Y. Zhao, M. Iwamoto, L. E. Larson, and P. M. Asbeck, “Doherty amplifier with DSP control to improve performance in CDMA operation,” in IEEE MTT-S Int. Microwave Symp. Dig., 2003, pp. 687–690. J. Cha, Y. Yang, B. Shin, and B. Kim, “An adaptive bias controlled power amplifier with a load-modulated combining scheme for high efficiency and linearity,” in IEEE MTT-S Int. Microwave Symp. Dig., 2003, pp. 81–84. S. Bae, J. Kim, I. Nam, and Y. Kwon, “Bias-switching quasi-Dohertytype amplifier for CDMA handset applications,” in IEEE Radio Frequency Integrated Circuits Symp. Dig., 2003, pp. 137–140. Y. Yang, J. Cha, B. Shin, and B. Kim, “A fully matched N -way Doherty amplifier with optimized linearity,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 986–993, Mar. 2003. , “A microwave Doherty amplifier employing envelope tracking technique for high efficiency and linearity,” IEEE Microw. Wireless Comp. Lett., vol. 13, no. 9, pp. 370–372, Sep. 2003. F. H. Raab, “Efficiency of Doherty RF power-amplifier systems,” IEEE Trans. Broadcast., vol. BC-33, no. 3, pp. 77–83, Sep. 1987. N. Srirattana, A. Raghavan, D. Heo, P. E. Allen, and J. Laskar, “Analysis and design of a high-efficiency multistage Doherty power amplifier for WCDMA,” in Proc. IEEE Eur. Microwave Conf., Munich, Germany, Oct. 2003, pp. 1337–1340. S. C. Cripps, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House, 1999.

Nuttapong Srirattana (S’00) was born in Bangkok, Thailand. He received the B.Eng. degree (with first class honors) in telecommunication engineering from the King Mongkut’s Institute of Technology, Ladkrabang, Bangkok, Thailand, in 1999, the M.S. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 2002, and is currently working toward the Ph.D. degree in the area of silicon-based transistor modeling and its applications to RF power amplifiers and novel architectures for high-efficiency linear RF power amplifiers at the Georgia Institute of Technology. From 2001 to 2002, he was a Part-Time Design Engineer with the National Semiconductor Corporation, Atlanta, GA, where he was involved in the development of SiGe HBT power amplifier for global system for mobile communications (GSM) handsets. Since 2001, he has been a Research Assistant with the Analog Integrated Circuit and Signal Processing Laboratory and the Microwave Application Group, Georgia Institute of Technology. His research interests include microwave/RF device characterization and modeling, linearization techniques, and high-efficiency linear RF power amplifier design.

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Arvind Raghavan (S’00–M’04) received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Madras, India, in 1999, and the M.S. and Ph.D. degrees from the Georgia Institute of Technology, Atlanta, in 2002 and 2003, respectively. He is currently with Quellan Inc., Atlanta, GA. He has held part-time and research and development positions with National Semiconductor, IBM, and RF Micro Devices. His research interests include RF integrated circuit (RFIC) design, microwave device characterization and modeling, and integrated circuit (IC) design for high-speed digital communication applications.

Deukhyoun Heo (S’97–M’01) received the B.S.E.E. degree in electrical engineering from Kyoungpuk National University, Daegu, Korea, in 1989, the M.S.E.E. degree in electrical engineering from the Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1997, and the Ph.D. degree in electrical and computer engineering from the Georgia Institute of Technology, Atlanta, in 2000. For six years, he was a Senior Design Engineer with LG Information and Communications Ltd. In 2000, he joined the National Semiconductor Corporation, where he was a Senior Design Engineer involved with the development of SiGe RFICs for cellular applications. After two years of industry experience with the National Semiconductor Corporation, he joined the Yamacraw Research Center, Georgia Institute of Technology to perform research that focused on integration of high-frequency electronics with integration of mixed technologies for next-generation wireless. Since Fall 2003, he has been an Assistant Professor with the Electrical and Computer Science Department, Washington State University, Pullman. His research interests include RF/microwave/opto devices, circuits, and their applications. He has primarily been concerned with RF/microwave/opto transceiver design based on CMOS, BiCMOS, SiGe, and GaAs technologies for wireless data communications, the development of active device nonlinear model for power amplifiers, and the multilayer module development for RF/microwave/opto system-on-package solution. Dr. Heo was the recipient of the 2000 Best Student Paper Award presented at the IEEE Microwave Theory and Techniques Society (MTT-S) International Microwave Symposium (IMS).

Phillip E. Allen (M’62–SM’82–F’92) received the Ph.D. degree in electrical engineering from the University of Kansas, Lawrence, in 1970. He is currently a Professor with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, where he holds the Schlumberger Chair. He has been with the Lawrence Livermore Laboratory, Pacific Missile Range, and Texas Instruments Incorporated, and has consulted with numerous companies. He has taught at the University of Nevada at Reno, University of Kansas, University of California at Santa Barbara, and Texas A&M University. His technical interest include analog integrated circuit and systems design with a focus on implementing lowvoltage and high-frequency circuits and systems in standard CMOS technology. He has authored or coauthored over 60 refereed publications in the area of analog circuits. He coauthored Introduction to the Theory and Design of Active Filters (New York: McGraw-Hill, 1980), Switched Capacitor Filters (New York: Van Nostrand, 1984), CMOS Analog Circuit Design (New York: Holt, Rienhart and Winston, 1987), and VLSI Design Techniques for Analog and Digital Circuits (New York: McGraw-Hill, 1990). His current research concerns frequency synthesizers, RF, IF, and baseband filters compatible with standard IC technology.

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Joy Laskar (S’84–M’85–SM’02) received the B.S. degree (highest honors) in computer engineering with math/physics minors from Clemson University, Clemson, SC, in 1985, and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign, in 1989 and 1991 respectively. Prior to joining the Georgia Institute of Technology, Atlanta, in 1995, he has held faculty positions with the University of Illinois at Urbana-Champaign and the University of Hawaii. At the Georgia Institute of Technology, he holds the Joseph M. Pettit Professorship of Electronics and is currently the Chair for the Electronic Design and Applications Technical Interest Group, the Director of Georgia’s Electronic Design Center, and the System Research Leader for the National Science Foundation (NSF) Packaging Research Center. With the Georgia Institute of Technology, he heads a research group with a focus on integration of high-frequency electronics with optoelectronics and integration of mixed technologies for next-generation wireless and opto-electronic systems. In July 2001, he became the Joseph M. Pettit Professor of Electronics with the School of Electrical and Computer Engineering, Georgia Institute of Technology. He has authored or coauthored over 210 papers. He has 10 patents pending. His research has focused on high-frequency IC design and their integration. His research has produced numerous patents and transfer of technology to industry. Most recently, his research has resulted in the formation of two companies. In 1998, he cofounded the advanced wireless local area network (WLAN) IC company RF Solutions, which is now part of Anadigics. In 2001, he cofounded the next-generation interconnect company Quellan Inc., which develops collaborative signal-processing solutions for enterprise applications. Dr. Laskar has presented numerous invited talks. For the 2004–2006 term, he has been appointed an IEEE Distinguished Microwave Lecturer for his Recent Advances in High Performance Communication Modules and Circuits seminar. He was a recipient of the 1995 Army Research Office’s Young Investigator Award, 1996 recipient of the National Science Foundation (NSF) CAREER Award, 1997 NSF Packaging Research Center Faculty of the Year, 1998 NSF Packaging Research Center Educator of the Year, 1999 corecipient of the IEEE Rappaport Award (Best IEEE Electron Devices Society journal paper), the faculty advisor for the 2000 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) Best Student Paper Award, 2001 Georgia Institute of Technology Faculty Graduate Student Mentor of the Year, a 2002 IBM Faculty Award, 2003 Clemson University College of Engineering Outstanding Young Alumni Award, and 2003 Outstanding Young Engineer of the IEEE MTT-S.

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861

An Analysis of Miniaturized Dual-Mode Bandpass Filter Structure Using Shunt-Capacitance Perturbation Ming-Fong Lei, Student Member, IEEE, and Huei Wang, Senior Member, IEEE

Abstract—A dual-mode bandpass ring filter using two pairs of shunt capacitances has been analyzed and designed. The size of the ring resonator can be reduced significantly by increasing the shunt capacitances. Since the resonator size is not an integer multiple of operating frequency, harmonic suppression is easily achieved. Analysis shows that the two pairs of shunt capacitances independently control the even- and odd-mode resonant frequency of the ring resonator. Moreover, the shunt capacitance also allows easier biasing, making tunable filters using varactors realizable. Various configurations of dual-mode bandpass filters have been reported, and the advantages and disadvantages of each structure are discussed. Several filters of various capacitances have been designed and tested at 1.8 GHz, and show significant size reduction of over 67%. A varactor-tuned filter has also been designed, and demonstrated a measured tunable center frequency of 20%. This study is a solution to the miniaturization of ring filters, and allows performance tuning, making ring filters more attractive in monolithic microwave integrated circuits and system-on-chip applications.

Fig. 1. Basic structure of a dual-mode ring resonator.

Index Terms—Bandpass, dual mode, filter, miniature, ring.

I. INTRODUCTION

M

ICROSTRIP ring resonators have many applications in microwave and millimeter-wave systems, such as filters, couplers, antennas, and frequency-selective surfaces. The theory of ring resonators on these various applications is well documented [1]. Ring resonators used as bandpass filters have received much attention due to its simple design and narrow bandpass response. For each resonant frequency of a ring resonator, two orthogonal modes exist [2], [3]; ring filters that exploit both modes for operation are often referred to as “dualmode ring resonators/filters.” General conditions to form a dualmode filter are described in [4], namely, 90 between input and output ports, a discontinuity inside the resonator, and symmetry. A basic structure of a dual-mode filter is shown in Fig. 1: the two ports are fed 90 relative to each other, and perturbations can be added on the -plane ( for symmetry) or the -plane ( for antisymmetry). Previous studies have focused on perturbations on the -plane [4]–[7]; it is possible to control both the even- and odd-mode frequencies by a stepped-impedance perturbation on the -plane [6], but a small impedance ratio may be difficult to realize, and is more sensitive to fabrication variations. A dual-mode filter with series capacitive perturbation was proposed [8]; the new topology uses series capacitances on both Manuscript received March 15, 2004. This work was supported in part by the National Science Council under Grant NSC 92-2219-E-002-016,Grant NSC 92-2219-E-002-024, Grant NSC 02-2219-E-002-033, and Grant NSC 93-2752-E-002-PAE. The authors are with the Graduate Institute of Communication Engineering and Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842504

Fig. 2. Dual-mode ring resonator topology using shunt-capacitance perturbations.

the - and -plane to control the even- and odd-mode resonant frequency individually. However, there is a subtle disadvantage in this structure: in order to satisfy the resonant conditions, the ring circumference must be greater than one wavelength. A novel perturbation technique has been recently demonstrated [9] using square patches and notches to achieve shunt capacitances and series inductances. The author found that both Chebyshev and quasi-elliptic responses could be practically realized by a single dual-mode resonator. The nature of the coupling between the modes is studied by using a full-wave electromagnetic (EM) simulator, and various cases have been analyzed. However, the analysis is mostly descriptive, and is difficult to implement in actual circuit design since a full-wave EM simulator is required. In this paper, we analyzed the characteristics of a dual-mode filter using shunt capacitances, as shown in Fig. 2. Using even–odd-mode analysis [10], we have shown that the evenand odd-mode resonance of the ring resonator is governed independently by and , respectively. The analysis also reveals that the ring is less than a full wavelength, and can be reduced significantly by increasing and , the upper limit of size reduction depends on the resonant frequency of the capacitors. Since the resonator must be connected to the main circuit via some sort of capacitance, the effect of a tapping

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A. Even-Mode Analysis The even-mode equivalent circuit of the ring resonator is shown in Fig. 3, where a magnetic wall is applied along the -plane, which is an open circuit, and divides the capacitance into one-half. The value is one-eighth of the ring circumference. The normalized even-mode input admittances and can be derived, and are as shown in (2a) , and (2b) at the bottom of this page. The parameters , and . is the characteristic impedance of the transmission line, and is the phase constant. Applying the resonance condition of (1), the even-mode resonance happens when

Fig. 3. Even-mode equivalent circuit of the ring resonator in Fig. 2.

(2c) B. Odd-Mode Analysis The analysis for the odd-mode case is similar to the even and mode, the normalized odd-mode input admittances are (3a) (3b)

Fig. 4. Odd-mode equivalent circuit of the ring resonator in Fig. 2.

Using (1), the odd-mode resonance appears when capacitor to the resonator is also discussed. A brief discussion of the advantages and disadvantages of each type of dual-mode filter is presented. Several filters of varying capacitances are designed, and show a 67% size reduction compared to the conventional one-wavelength ring filters. A varactor-tuned filter has also been designed, and is tunable both in center frequency and bandwidth. The significant size reduction and the possibility of tunable frequency response of this structure make it attractive for monolithic microwave integrated circuits (MMICs) and system-on-chip (SOC) applications. II. ANALYSIS OF RESONATOR The structure in Fig. 2 can be analyzed using even–odd-mode analysis [10]. Dividing the circuit along -plane, the even- and odd-mode equivalent circuit are shown in Figs. 3 and 4, respectively. Since the equivalent circuit is in a shunt configuration, the ring is resonant when its input admittance is zero, which is expressed as

(1) Subscripts 1 and 2 represent the input admittance of the two separate arms.

(3c) Several important observations can be made from the above analysis. The first observation is that the odd- and even-mode and resonant frequency can be controlled independently by , respectively. The second one is that the condition for evenand odd-mode resonant frequency is in the same mathematical form, as shown in (2c) and (3c), which we will refer to as the characteristic equation of the resonator. The first two observations are the same as [8]. The third observation, which is less obvious, is that the ring resonator can be significantly miniaturized. Observing (2c) and (3c), since is a positive number, must , therefore, the total circumference of the ring be less than is less than one wavelength. The resonant condition for a series capacitance ring resonator [7] is (4) must be greater than , therefore, the total In this case, circumference of the ring for series capacitance perturbation is greater than one wavelength. Equations (2c), (3c), and (4) can be viewed graphically. If we set the resonant frequency as a constant, the right-hand side of (2c) and (3c) can be plotted as a function of the circumference of the ring, and the susceptance

(2a) (2b)

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Fig. 5. (a) Equation (2c) plotted as two lines as functions of ring circumference. (b) Equation (4) plotted as two lines as functions of ring circumference. The x-axis is the circumference of the ring, and the y -axis is the normalized susceptance. The frequency f is set as a constant, and  is the wavelength at f . The intersection of the solid line and the dotted line represents a solution of resonance: the horizontal axis is the circumference of the ring, and the vertical axis is the normalized susceptance of the perturbation capacitor.

is a constant line, varying in value by the choice of the capacitance, as shown in Fig. 5(a). Equation (4) can be similarly plotted, as shown in Fig. 5(b). The intersection of the two lines is the solution for a given capacitance (horizontal axis) and ring circumference (vertical axis). Since is positive, there is a forand . In both bidden region for negative values of cases, an increase in the perturbation capacitance will reduce the ring size, but due to the nature of the cotangent function and tangent function, the circumference of the shunt-capacitance ring has an upper limit of one wavelength, and the series capacitance ring has a lower limit of one wavelength. Therefore, it is possible to shrink the ring size significantly by increasing the shunt-capacitance values. In the limiting case when the ring size is one wavelength, reduces to zero for (2c), and increases to infinity for (4), which is the case of an ideal ring resonator. The characteristic (2c) and(3c) can be expressed as

Fig. 6. Relationship between perturbation capacitance and resonant frequency for various ring sizes; the ring sizes are relative to a 2-GHz one-wavelength ring with intrinsic impedance of 90 .

the curves are steeper for small capacitances than large capacitances; this means it is easier to realize narrow bandwidth designs using large capacitances as opposed to small capacitances. The center frequency of the resonator can be approximated from the average of the even- and odd-mode frequencies

(6) The coupling coefficient [2], [11] can be computed from the even- and odd-mode frequencies as follows:

(5) where is the even- or odd-mode resonant frequency, is the average radius of the ring, is the capacitance of , , or is the characteristic impedance common capacitor , and of the ring. Equation (5) can be solved numerically for a given capacitance, ring size, and characteristic impedance of the ring. A set of solutions has been derived for rings of various circumferences; the relationship between the perturbation capacitance and the resonant frequency is plotted in Fig. 6. The ring is modeled as ideal transmission lines with intrinsic impedance of 90 ; the various lines represent different ring circumferences, relative to a 2-GHz one-wavelength ring. For a set ring size and the characterand resonant frequency, the capacitance istic impedance of the line is inversely proportional, therefore, the perturbation capacitance can be decreased at the expense of or vice versa. increasing Many details regarding this structure can be found by expressing (5) graphically into Fig. 6. First of all, we have already seen from (2) and (3), for a determined resonant frequency, the reduction of the ring size can be achieved by increasing the perturbation capacitance, and is shown explicitly in Fig. 6. Second, the resonant frequency converges to the natural frequency of the ring as the capacitance is reduced to zero. Thirdly, the slopes of

(7)

In actual implementation, and should be relatively large, but they are not large enough for a good Taylor approximation or asymptotic approximation. Therefore, the relationship between resonant frequency and perturbation capacitance should be numerically derived as shown in Fig. 6 for various ring sizes, and the values read from the chart to find the coupling coefficient . III. FILTER ANALYSIS When the resonator is weakly coupled, the resonator characteristics are basically the same as the analysis in Section II since the coupling capacitance affects the resonator characteristics very little. In filter applications, a minimum insertion loss is desired at passband, which means that the resonator is connected to the signal path via relatively large capacitors. Fig. 7 shows

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has an effect of shifting the resonant frequency by capacitor . Furthermore, the relationa factor of has been numerically found, and can be ship between and approximated as a straight line. Take for example, the rings of relationship can be expressed as Fig. 6, where the pF

Fig. 7.

Schematic of the ring resonator filter.

the schematic of the resonator used as a filter, which is similar to Fig. 2 with the tapping capacitance explicitly drawn. The depends on maximizing of the entire filter at choice of passband. The relatively large capacitance will affect the resonant frequencies of the two modes, possibly affecting the coupling coefficient and, thus, the effects of the tapping capacitors must be taken into account for the filter analysis. The shift can be found by the of the resonant frequencies caused by following analysis. into the even- and odd-mode analysis The inclusion of is to add a series capacitance to Figs. 3 and 4. The resonant condition for a series connection is when the input impedance is zero as follows:

(8) Due to the complexity of (8), no simple analytical relation (or ), and can be easily derived. The addibetween , tional will only alter the even- and odd-mode frequencies by a small amount; therefore, the even- and odd-mode admittances can be approximated by a first-order Taylor expansion at their and can resonant frequencies. The admittances be expressed as first-order Taylor expansions near the resonant and as frequencies

(9) Applying (9) and (8), the shift in even- and odd-mode frequencies due to the tapping capacitor can be expressed as (10)

We can solve for numerically from (2) and (3) using mathematical software or from circuit simulation software. From and are equal for the simulation, for the same ring, same resonant frequency; therefore, the addition of a coupling

Hz

(11)

The shift of resonant frequencies can be calculated by applying (11) into (10), which shows that the effect of the tapping capacitor reduces the resonant frequencies at most 12.8%; since the tapping capacitor affects the even- and odd-mode frequency by the same ratio, the coupling coefficient of the resonator remains the same. The altered frequencies represented and will be the actual characteristic frequencies of as the filter structure. The design procedure for a dual-mode filter is summarized as follows: 1) decision of center frequency and bandwidth determines and ; 2) decision of tapping capacitance ; calculate and from and using (10) and (11); and 3) for a determined ring size and line impedance, find that corresponds to and using Fig. 6. IV. DISCUSSIONS ON VARIOUS TYPES OF DUAL-MODE RINGS Dual-mode ring resonators can be characterized into two types: one using series perturbations on the - and -planes [8] and the other using shunt perturbations on the - and -planes. The characteristic equation for series perturbation is (4), and the characteristic equation for shunt perturbation is (2c). Even though the analysis for (2c) and (4) were performed using capacitances, the analysis can be applied for inductances. The characteristic equations for the dual problem are (12) (13) Equation (12) is for shunt inductance, and (13) is for series inductance. Like the capacitive case, the two modes can be controlled individually by the - and -plane inductances. In the case of inductive perturbations, a series perturbation will reduce the length of the resonator to less than a wavelength, and a shunt perturbation will increase the length of the resonator to more than a wavelength. In reality, both of these two structures using inductances as perturbation elements have disadvantages and limitations. For the shunt inductance case, the resonator itself is dc shorted, both to the ground and to the two ports, thus limiting its usefulness in filter applications. For the series inductance case, a smaller will have a smaller circuit size, but this will require larger inductances, which are usually not realizable in millimeter-wave frequencies. The series capacitance has the advantage of inherent dc blocking between the two ports, but has a larger resonator size. On the other hand, the shuntcapacitance perturbation has a smaller resonator size, and can be miniaturized significantly by increasing the capacitance. Furthermore, the shunt configuration allows easier bias, making varactor-tuned filters realizable for this type of resonator. The

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TABLE I SUMMARY OF VARIOUS TYPES OF DUAL-MODE RING RESONATOR

insertion loss is mostly due to the transmission-line loss, and since the ring circumferences of various configurations are approximately the same size, the insertion losses are similar, with slightly lower insertion losses for the series inductive and shunt capacitive configurations, and a slightly higher insertion loss for the series capacitive configuration. The advantages and disadvantages of each structure are summarized in Table I. Comparing with other types of planar circuit filters using a coupled-line filter as an example, a coupled-line filter of similar performance to a dual-mode ring filter will require numerous sections, and will also need additional stubs to introduce attenuation poles in the response. This translates to a very large board size in the order of several wavelengths. This large size also makes coupled-line filters very sensitive to the loss tangent of the board; a slight degradation of loss tangent will degrade insertion-loss response significantly. On the other hand, dual-mode ring filters have a very small size with a narrow bandpass response, and are less sensitive to the substrate loss.

Fig. 8. Ring filters. Clockwise from top left: ring A, ring B, and varactor-tuned filter.

TABLE II DESIGN PARAMETERS FOR RINGS A AND B

V. EXPERIMENT RESULTS Several filters were designed at 2 GHz using the equations derived in Section IV in order to validate the usefulness and advantages of our structure. The substrate we used was FR4, with a thickness of 1.6 mm, relative dielectric constant between 4.2–4.4, and loss tangent of 0.022. The capacitances were realized either using standard 0603 surface-mount ceramic capacitors (5% tolerance) or inter-digit capacitors. The linewidths of the rings are 1 mm, corresponding to an intrinsic impedance of 90 . Since the capacitance of the surface-mount capacitors have discrete values, it is difficult to set the design goals first and then choose the required capacitance values. Therefore, for the experiments, we chose the capacitance values first, and calculate the filter performance afterwards. Note that this is by no means a proper design procedure, it was implemented in this manner due to the discrete capacitance values and to verify our theories. Two ring filters of different sizes were designed: one with a ring radius of 7.8 mm, and one with a radius of 11 mm. To simplify further explanations, the 7.8-mm ring will be referred to as ring A, and the 11-mm ring will be referred to as ring B. Fig. 8 shows a photograph of rings A and B. The circumference of the two rings corresponds to a full wavelength at 3.5 and 3 GHz, respectively. We used surface-mount capacitors to realize the

perturbation capacitances for ring A and interdigital capacitors for ring B. The capacitance values for rings A and B are summarized in Table II. Simulation was performed using AWR’s Microwave Office software with the effects of parasitic and junctions considered. The use of an EM simulator was kept to a minimum in that is was only used for the characterization of interdigital capacitors and via-holes. The measured -parameters of rings A and B are shown in Figs. 9 and 10, respectively. The dotted lines are simulation results, and the solid lines are measurements. It can be observed from Figs. 9 and 10 that both circuits show good results: ring A has a best insertion loss of 1.8 dB at 1.87 GHz and a 3-dB bandwidth of 210 MHz (11%); ring B has a best insertion loss of 2.1 dB at 1.9 GHz and a 3-dB bandwidth of 240 MHz (12.6%). Since the ring is not a full wavelength, harmonic bandpass response is not observed. Compared with the original design parameters of Table II, both results of

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Fig. 11. (a) Perturbation capacitance shunted on the transmission line. (b) Tunable perturbation capacitance using a reverse-biased diode. The dotted region is equivalent to C of (a).

Fig. 9. Measured S -parameters of ring A plotted against simulation results. Simulations are dotted line, measurements are solid lines.

Fig. 10. Measured S -parameters of ring B plotted against simulation results. Simulations are dotted line, measurements are solid lines.

rings A and B have shifted approximately 150 MHz to the lower end of the spectrum. This is due to two reasons: for ring A, the frequency shift is due to the inductance of the via-holes and the short transmission line connections between; for ring B, it is because of the variation of , and the slight difference in ring size during fabrication. The simulation results in Figs. 9 and 10 have taken these effects into account, and show good agreement against measurements. We have also observed in simulation that, for the transmission should be greater than , which is just the zeros to appear, opposite of [7] with series capacitances. Compared with conventional ring filters that require a full-wavelength ring, ring A shows a 67% size reduction, and ring B shows a 55% size reduction. Electrically tunable filters usually use varactors for variable capacitances are usually implemented in combline filters [12], [13]. Varactor-tuned filters using ring resonators have also been reported [14]. To demonstrate the tunable filter concept of this structure, a filter using varactors was also designed. The circuit of the tunable perturbation capacitance is shown in Fig. 11(b), which uses a reverse-biased Schottky barrier diode in series with another capacitor for tunable capacitance. This complex

Fig. 12. (a) Frequency response of varactor-tuned filter at various center frequencies. The dotted lines are S s and the solid lines are S s. (b) Tunable bandwidth of varactor-tuned filter.

circuit was chosen for two reasons: the first reason is because the capacitance of the diode was too large for our design. The second reason is that the varactor diodes have a large series resistance, resulting in a poor insertion loss and low- performance, which will limit the usefulness of our filters. The circuit in Fig. 11(b) decreases the overall perturbation capacitance to our design values, and reduces the effects of the diode resistance on insertion loss and bandwidth. A large resistor was shunted to ground to provide a dc short for the diodes. The values chosen for Fig. 11(b) are : 0.47 pF and : 1.8 pF or 2.2 pF depending or . The tuning range for is 1.3–1.9 pF and is on

LEI AND WANG: ANALYSIS OF MINIATURIZED DUAL-MODE BANDPASS FILTER STRUCTURE

1.4–2.2 pF. The relationship between the bias voltage and resonant frequency can be derived via simulation software and the data in Fig. 6. From this relationship, tunable filters with variable center frequency and constant bandwidth or constant center frequency and variable bandwidths can be designed. The measured results are shown in Fig. 12. In Fig. 12(a), the center frequency can be tuned continuously from 1.63 to 1.88 GHz (only four points are shown for clarity); in Fig. 12(b), the 3-dB bandwidth is tunable from 190 to 280 MHz. VI. CONCLUSION The dual-mode ring resonator using shunt capacitances as perturbations has been described and analyzed. The self-resonant frequency, as well as the even- and odd-mode frequencies, can be described by a single characteristic equation. The evenand odd-mode frequencies of the resonator can be individually controlled. Furthermore, the characteristic equation shows that this structure can reduce the ring size significantly. The shunt structure of the resonator also allows biasing, making voltage tunable filters possible using varactors. A brief discussion on various configurations of dual-mode ring resonators shows that the shunt-capacitance structure has numerous advantages compared to others. Filters have been designed with this new structure, with a significant size reduction of 67%. A varactor-tuned filter has also been designed, with a tuning range of 20%. The measured results agree with the simulation well, proving significant size reduction, and the concept of tunable filter is possible using this structure.

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[7] Kwok-Keung and M. Cheng, “Design of dual-mode ring resonators with transmission zeros,” Electron. Lett., vol. 33, no. 16, pp. 1392–1393, Jul. 1997. [8] B. T. Tan, S. T. Chew, M. S. Leong, and B. L. Ooi, “A dual-mode bandpass filter with enhanced capacitive perturbation,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 8, pp. 1906–1910, Aug. 2003. [9] A. Gorur, “Description of coupling between degenerate modes of a dualmode microstrip loop resonator using a novel perturbation arrangement and its dual-mode bandpass filter applications,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 671–677, Feb. 2004. [10] J. Reed and G. H. Wheeler, “A method of analysis of symmetrical fourport networks,” IRE Trans. Microw. Theory Tech., vol. MTT-4, no. 10, pp. 246–252, Oct. 1956. [11] I. Bhal and P. Bhartia, Microwave Solid State Circuit Design. New York: Wiley, 2003, ch. 6. [12] J. Uher and W. J. R. Hoefer, “Tunable microwave and millimeter-wave bandpass filters,” IEEE Trans. Microw. Theory and Tech., vol. 39, no. 4, pp. 643–653, Apr. 1991. [13] I. C. Hunter and J. D. Rhodes, “Electronically tunable microwave bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 30, no. 9, pp. 1354–1360, Sep. 1982. [14] M. Makimoto and M. Sagawa, “Varactor tuned bandpass filters using microstrip-line ring resonators,” in IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2, 1986, pp. 411–414.

Ming-Fong Lei (S’00) was born in Taipei, Taiwan, R.O.C., on December 20, 1980. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2002, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University. His research interests include the design and analysis of microwave and millimeter-wave circuits, microwave device modeling, and filters.

ACKNOWLEDGMENT The authors will like to acknowledge the helpful suggestions of Dr. C.-H. Wang and Prof. Y.-S. Lin, both of National Taiwan University, Taipei, Taiwan, R.O.C. REFERENCES [1] K. Chang, Microwave Ring Circuits and Antenna. New York: Wiley, 1996. [2] M. Makimoto and S. Yamashita, Microwave Resonators and Filters for Wireless Communication. Berlin, Germany: Springer-Verlag, 2000. [3] I. Wolff, “Microstrip bandpass filters using degenerate modes of a microstrip ring resonator,” Electron. Lett., vol. 8, no. 12, pp. 302–303, Jun. 1972. [4] H. Yabuki, M. Sagawa, M. Matsuo, and M. Makimoto, “Stripline dualmode ring resonators and their application to microwave devices,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 5, pp. 723–729, May 1996. [5] A. C. Kundu and A. Awai, “Control of attenuation pole frequency of a dual-mode microstrip ring resonator bandpass filter,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1113–1117, Jun. 2001. [6] M. Matsuo, H. Yabuki, and M. Makimoto, “Dual-mode steppedimpedance ring resonator for bandpass filter application,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 7, pp. 1235–1240, Jul. 2001.

Huei Wang (S’83–M’87–SM’95) was born in Tainan, Taiwan, R.O.C., on March 9, 1958. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1980, and the M.S. and Ph.D. degrees in electrical engineering from Michigan State University, East Lansing, in 1984 and 1987, respectively. During his graduate study, he was engaged in research on theoretical and numerical analysis of EM radiation and scattering problems. He was also involved in the development of microwave remote detecting/sensing systems. In 1987, he joined the Electronic Systems and Technology Division, TRW Inc. He was a Member of the Technical Staff and Staff Engineer responsible for monolithic-microwave integrated-circuit (MMIC) modeling of computer-aided design (CAD) tools, MMIC testing evaluation, and design. He then became the Senior Section Manager of the Millimeter Wave Sensor Product Section, RF Product Center, TRW Inc. In 1993, he visited the Institute of Electronics, National Chiao-Tung University, Hsin-Chu, Taiwan, R.O.C., and taught MMIC-related topics. In 1994, he returned to TRW Inc. In February 1998, he joined the faculty of the Department of Electrical Engineering, National Taiwan University, as a Professor. Dr. Wang is a member of Phi Kappa Phi and Tau Beta Pi.

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Evaluation of the Input Impedance of a Top-Loaded Monopole in a Parallel-Plate Waveguide by the MoM/Green’s Function Method Alejandro Valero-Nogueira, Member, IEEE, Jose I. Herranz-Herruzo, Member, IEEE, Eva Antonino-Daviu, Member, IEEE, and Marta Cabedo-Fabres, Member, IEEE

Abstract—An accurate modeling of a top-hat monopole transition in a parallel-plate waveguide is performed using the method of moments/Green’s function method. The selection of the appropriate source–field relationship to override divergence series is discussed in detail. Numerical results are given for the input impedance of a top-loaded coaxial transition. A mode-matching solution is used as a reference to validate the results. Index Terms—Green’s functions, method of moments (MoM), numerical analysis, waveguide arrays, waveguide transitions.

I. INTRODUCTION

C

OAXIAL probes are often used as transitions to rectangular or parallel-plate waveguides. Such transitions have been studied in great detail by numerous authors [1]–[7]. When the geometry is separable, there are basically two ways of facing the problem. The first way requires dividing the problem in a number of canonical regions, expanding the fields in terms of the solutions of the Helmholtz equation in these regions, and enforcing the tangential-field continuity at the boundaries between them. This technique is commonly known as mode matching and some good examples of application to the problem of our concern are discussed in [1]–[4]. The second approach entails the use of the specialized Green’s functions of the sources present within the guide [7]. Conventionally, mode-matching solutions are adopted over specialized Green’s functions. A number of reasons can be alleged for this preference. Firstly because, for separable geometries, field solutions may be more straightforward to formulate. Secondly, this type of methods can cope with a fairly large variety of geometries of practical interest, such as coaxial sleeves [4], top-hat loading [1], multilayer insulation [5], etc., and thirdly, due to the recognized accuracy and computational efficiency of these methods. However, there are problems of a nonseparable nature in which the probe is close to other probes [8] or to some radiating elements such as the slots of a radial line slot array (RLSA) antenna [9], which may be strongly coupled to the probe. In such cases, tasks such as optimization of the transition dimensions should take all these couplings into consideration. Accordingly, Manuscript received March 19, 2004; revised June 4, 2004. This work was supported in part by the Comision Interministerial de Ciencia y Tecnología under Project TIC2001-2364-C03-02. The authors are with the Departamento de Comunicaciones, Universidad Politécnica de Valencia, 46022 Valencia, Spain. Digital Object Identifier 10.1109/TMTT.2004.842501

a current-based method involving specialized Green’s functions and the method of moments (MoM) should be more effective than mode matching without losing its accuracy. In [7], a monopole transition to a parallel-plate waveguide was studied using specialized Green’s functions. Even though this is the most common type of transition in practice, in some cases, a simple monopole fails to achieve the required impedance matching and it may be convenient to load the probe with a disk or hat on top of it. One of such cases is exemplified by those RLSA antennas designed to operate at the lower microwave bands (2–6 GHz). At these bands, usual guides are too bulky to find application in low-profile or even mobile antennas and the making of thinner guides is imperative. Hence, for those instances, the Green’s function of an electric loop of radial current is required. Moreover, the currents at the attachment region of the top-hat monopole should be managed carefully. In Section II, the construction of the required Green’s functions is treated. It then follows a discussion on the appropriate source–field relationship to avoid a divergent series solution of the impedance matrix entries. This paper continues with the documentation of the details involved in the numerical implementation. Finally, Section V is devoted to the analysis of the top-hat monopole transition, validation of numerical results, measurements, and its use in applications where employing the MoM/Green’s function method is the right choice. II. CONSTRUCTION OF GREEN’S FUNCTIONS In a top-hat monopole transition, axial as well as radial currents are involved. In addition, the excitation field originated in a coaxial aperture has to be properly handled. Green’s functions derivation for a unit circular cylindrical axial electric current and a loop magnetic current in a parallel-plate waveguide is described in detail in [7] and it will not be repeated here. Furthermore, a general Green’s function for a ring of radial electric current was also derived earlier to analyze annular discontinuities in a radial waveguide [10]. Yet, in our study, the Green’s function for the radial currents will be constructed from a different perspective, following the approach described in [7] for the sake of clarity when we combine axial and radial currents. A. Radial Electric-Current Loop in Free Space Before facing a parallel-plate environment, let us construct the Green’s function for a radial electric-current loop source in

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VALERO-NOGUEIRA et al.: EVALUATION OF INPUT IMPEDANCE OF TOP-LOADED MONOPOLE IN PARALLEL-PLATE WAVEGUIDE

integer. Hence, the function by Fig. 1.

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in (5) must be substituted

Disk of radially directed current in free space.

(6) free space first. The solution might be useful in itself to model wire to plate junctions. Besides, the solution for a parallel-plate waveguide can be obtained easily from it using image theory. The geometry under consideration is shown in Fig. 1. and are the disk’s inner and outer radii, respectively. A cylindrical coordinate system is selected so that the -axis is the disk’s symmetry axis. The electric-current density is given by

Using the transformation (7) the integral in (5) becomes a summation and the Green’s function for the radial electric-current loop in the parallel plate can be written as

(1) The current density is radially directed, therefore, the electromagnetic fields can be derived from a single component of the satisfying vector potential

(8) where

(2) The solution of (2) can be written in the form (3)

(9) After a few manipulations of , the exponentials are trans. formed to more convenient sine functions and can be definitively written as Hence,

As stated above, we assume a uniform radial current, therefore, . The Green’s function of a unit radial current loop is the solution to the differential equation

(10) where

(4) (11) The solution to (4) is very well documented. Notice that this differential equation is formally identical to the one associated to a ring of azimuthal magnetic current whose solution can be found among others in [11] as follows:

(5) being ,

, and for

, and where for ,

B. Radial Electric-Current Loop in an Infinite Parallel-Plate Waveguide Once the Green’s function for the loop source in free space is known, the solution for the loop in a parallel-plate waveguide is fairly straightforward to derive using the image theory. Consider and , respectively. The the plates are located in between the plates. unit radial current loop is located in As a result of applying the image theory, the original problem is substituted by an infinite set of parallel loops in free space along direction are the -axis. Loops with current pointing in the , while those loops whose currents are located in directed to are placed in , with being an

III. ELECTRIC-FIELD COMPONENTS DUE TO A DISK OF RADIAL CURRENTS IN A PARALLEL-PLATE WAVEGUIDE Once the Green’s function for a unit radial electric-current loop is known, the next step is obtaining the electric-field components for the problem of our concern. We are dealing with a disk of radial currents on top of a monopole, therefore, we are interested in determining the axial and radial electric-field and , recomponents produced by the disk, namely, spectively. There exist, however, different ways of establishing a source–field relationship [13]. The correct selection of such a relationship will greatly condition the computational efficiency of the final numerical implementation. In particular, it has been observed that the formulation (12) compoproduces a highly divergent series solution for the nent due to the double derivative in . Therefore, it would be advantageous to use a formulation where the derivatives were in the currents. One such operator is the mixed potential formulation requiring vector potential and scalar potential . As is well known, all derivatives are moved to the basis and testing

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is the axial segment length. The basis functions for where decay the radial current density should include a

Fig. 2. Top-loaded monopole transition to an infinite parallel-plate waveguide.

functions when the impedance operator is formulated. However, in our case, this formulation leads to a very involved and inefficient code comprising many integrals that have to be solved numerically. Keep in mind that the Green’s function obtained in Section II will have to be combined with the Green’s function for an axial current in [7] to model the top-hat probe. component relies on The approach adopted to obtain the a third source–field formulation. It can be shown [13] that the source–field relationship can also be written as (13) As can be seen, differentiation is solely on the current’s part this time. The numerical implementation of this formulation is particularly simple as far as the component is concerned, and only some care must be taken in selecting the basis functions to expand the unknown current density and manipulating Dirac delta functions and their derivatives.

(16) is the radial segment length. Finally, an extra basis where function must be added to enforce current continuity at the monopole-to-disk junction. The attachment mode defined in [12] is used to this purpose as follows:

(17) To compute the impedance matrix entries , we have to calculate the electric field on the monopole and on the disk surface using two different Green’s functions. , the component of the electric field due to the monopole current on the monopole itself, and , the component of the electric field produced by the monopole current on the disk, are obtained using the Green’s functions derived in [7] and the source–field relationship in (12). Application of Galerkin method circumvents the computation of , the component of the electric field on the monopole due to the radial current on the disk, because the reciprocity theorem assures that (18)

IV. NUMERICAL SOLUTION OF A TOP-LOADED COAXIAL TRANSITION IN AN INFINITE PARALLEL-PLATE WAVEGUIDE Expressions obtained above are now applied to solve for the electric currents on the coaxial-to-waveguide monopole transition loaded with a disk on top of it. The geometry is shown in Fig. 2. The transition consists of a monopole of radius and height . The disk on top has an outer radius and is considered infinitely thin. This means no loss of generality, as thick disks can be contemplated as a combination of axial and radial currents. The whole structure is coaxially fed by an annular aperture of outer radius and the parallel-plate waveguide height is . The unknown electric currents are obtained formulating the corresponding electric-field integral equation and solving it by the MoM in its Galerkin specialization (14)

, the radial component of the electric field due Finally, to the disk current on the disk itself, will be computed using the Green’s function in (10) and the source–field formulation in (13). Most electric-field components above are obtained without calculation deserves commenmajor difficulties. Only the tary, as the derivatives involved in the formulation must be interpreted as generalized functions [13]. Substituting (16) in (13) and differentiating the current basis functions leads to

(19)

Subsectional basis/testing functions are used for both the axial and radial currents. For the axial current, the expressions are

Similarly, introducing the radial portion of the attachment mode (17) in (13) gives

(15)

(20)

VALERO-NOGUEIRA et al.: EVALUATION OF INPUT IMPEDANCE OF TOP-LOADED MONOPOLE IN PARALLEL-PLATE WAVEGUIDE

Fig. 3. Input impedance as a function of frequency for a coaxially driven top-hat monopole in a parallel-plate waveguide of height h = 8 cm.

Note that, this time, the Green’s function has to be convolved with the derivative of a Dirac delta function. This is solved performing an integration by parts. It can be shown [14] that

(21) Now the differentiation is on the Green’s function and the electric field can be readily obtained. To complete the numerical specification of the problem, the excitation vector (22) is defined using the expressions developed in [7] for the electric field produced by a coaxially fed annular aperture in a parallelplate waveguide. V. RESULTS Several cases have been run to validate the code developed using the expressions derived above. For all cases, a mode-matching code [1] has been used as a reference. In all cases, the number of waveguide modes needed to reach a conor less, depending on the waveguide vergent result was electrical height. The first case considered has been taken from [6]. There, a top-hat monopole is used as a transition to a parallel-plate wavecm, cm, guide having Fig. 2 dimensions of cm, cm, and cm. A very similar problem was also faced in [3], the only difference being that the waveguide height was set variable with frequency, . To analyze this structure, the commercial code IE3D1 has also been run for comparison. Fig. 3 shows the results for the three methods mentioned. Good agreement is observed between our model and Bialkowski’s solution. Furthermore, the antiresonance frequency for both solutions is close to what a 1MoM-Based electromagnetic simulator, IE3D, Zeland Software, Fremont, CA. [Online]. Available: http://www.zeland.com

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Fig. 4. Input impedance as a function of frequency for a coaxially driven top-hat monopole in a TEM parallel-plate waveguide of height h = 6 mm.

first-order approximation would suggest. Note that the transiis approximately a half-wavelength long tion’s total length for GHz. This result also matches that of [3]. As for the IE3D solution, it is clearly shifted from the other two and from the expected solution, therefore, we have disregarded it as a reference for the subsequent cases. A second example is shown in Fig. 4. It considers a TEM waveguide filled with dielectric material of permittivity . Dimensions now are mm, mm, mm, mm, and mm. This time the frequency range and dimensions has been selected to produce a lower antiresonance and compare the impedance values in more detail than in the previous case. As can be seen, five unknowns (two of them in the axial part, two more in the disk and the attachment mode) are enough for most part of the frequency range. However, convergence is not reached until nine unknowns are used. However, even so, our results differ slightly from Bialkowski’s as we approach the antiresonance. Yet another result compares the evolution of the input impedance as the disk radius increases. The other dimensions are those of the previous example. The working frequency is GHz. Very good agreement is attained once again, as can be observed in Fig. 5. To check experimentally the validity of the model developed, we have built a radial waveguide fed by a top-hat monopole at its center. The guiding structure was built up as a sandwich made by two grounded dielectric slabs. One was 6-mm thick, while the other was 1 mm. Hence, the resultant radial guide was 7 mm in height. The dielectric material used was polypropylene with a . This value was obtained permittivity of experimentally. The probe consists of a 6-mm-long monopole topped with a disk, 3 cm in diameter. The disk was made using copper foil, which was adhered to the thicker grounded slab and then soldered to the monopole probe. The radial guide was obviously finite (49.8 cm in diameter). Hence, the air-dielectric discontinuity at the guide end had to be included in the electromagnetic model. To that purpose, we obtained the equivalent admittance of the aperture using Harrington’s basic formulation for apertures in ground planes [15] and then adapted it to our

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Fig. 5. Input impedance as a function of disk radius for a coaxially driven top-hat monopole in a TEM parallel-plate waveguide of height h= = 0:38. Fig. 7.

Fig. 6. S -parameter for a finite radial line 49.8 cm in diameter. Other parameters are: h = 0:7 cm, l = 0:6, b = 1:5 cm, a = 0:65 mm, a = 2 mm, and " = 2:25 j 8 10 .

0 1

cylindrical one. The -parameter was then measured using an Agilent 8510C Network Analyzer, obtaining the results shown in Fig. 6. Despite the difficulty in reproducing the ideal situation with the setup depicted above, good correspondence between measured and simulated data can be observed. Finally, to illustrate the effectiveness of top-hat probes for practical applications, we consider the standing-wave linearly polarized radial line slot array antenna (SW LPRLSA) of Fig. 7. In this antenna, slots are arranged in concentric rings spaced . Their orientation is selected so that they all contribute to the same polarization. The field distribution structure is a radial guide short circuited at its outer end. A full description of this antenna is beyond the scope of this paper. The details involved in the full-wave analysis of such antennas can be found in [9] or [16] among others. This antenna was designed to operate at 5.3 GHz for Hiperlan/2 uses. The most challenging goal in the design of such antenna at that frequency band is making the guide as thin as possible. For thin guides, monopole probes are unable to match the coaxial line to the radial guide

SW LPRLSA antenna.

Fig. 8. SW LPRLSA return-loss performance using two types of feed probes.

unless the slots are cut so electrically short that the antenna bandwidth is unacceptably narrow. On the other hand, a top-hat monopole has proven to provide excellent impedance matching even with resonant slots. Fig. 8 shows the return-loss performance of both the monopole probe and the top-hat monopole after running an optimization process on them. The guide height cm . Needless to say, the optimization rouwas tine demanded the longest feasible monopole probe cm and, even so, a good is far from being reached. cm, However, the optimized top-hat probe ( cm) shows excellent operation. The other antenna parammm, mm, , slot length, eters are cm, slot width, mm, azimuthal spacing becm, antenna radius, cm, and tween slots, number of slots . VI. CONCLUSIONS A top-loaded monopole transition has been numerically modeled using a MoM/Green’s function method. The appropriate

VALERO-NOGUEIRA et al.: EVALUATION OF INPUT IMPEDANCE OF TOP-LOADED MONOPOLE IN PARALLEL-PLATE WAVEGUIDE

source–field relationship for the radial currents has been discussed. Results have shown a very good match to those obtained with mode matching. Consequently, structures involving top-loaded probes in the presence of other probes or slots in a parallel-plate waveguide can be modeled accurately, as was illustrated. ACKNOWLEDGMENT The authors would like to thank Prof. M. Bialkowski, The University of Queensland, Brisbane, Australia, who kindly provided his mode-matching codes PROBE.FOR and CPROBE. FOR to compare and validate our results, and to Prof. J. E. Page de la Vega, of the Universidad Politécnica de Madrid, Madrid, Spain, who measured the polypropylene permittivity for us. REFERENCES [1] M. E. Bialkowski, “Analysis of a coaxial-to-waveguide adaptor including a discended probe and a tuning post,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 2, pp. 344–349, Feb. 1995. [2] Z. N. Chen, K. Hirasawa, and K. Wu, “A broad-band sleeve monopole integrated into parallel-plate waveguide,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1160–1163, Jul. 2000. [3] M. A. Morgan and F. K. Schwering, “Eigenmode analysis of dielectric loaded top-hat monopole antennas,” IEEE Trans. Antennas Propag., vol. 42, no. 1, pp. 54–61, Jan. 1994. [4] Z. Shen and R. H. MacPhie, “Rigorous evaluation of the input impedance of a sleeve monopole by modal-expansion method,” IEEE Trans. Antennas Propag., vol. 44, no. 12, pp. 1584–1591, Dec. 1996. , “Input admittance of a multilayer insulated monopole antenna,” [5] IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1679–1686, Nov. 1998. [6] M. E. Bialkowski, “On the link between top-hat monopole antennas, disk-resonator diode mounts, and coaxial-to-waveguide transitions,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 1011–1013, Jun. 2000. [7] B. Tomasic and A. Hessel, “Electric and magnetic current sources in the parallel plate waveguide,” IEEE Trans. Antennas Propag., vol. AP-35, no. 11, pp. 1307–1310, Nov. 1987. [8] , “Linear array of coaxially fed monopole elements in a parallel plate waveguide—Part I: Theory,” IEEE Trans. Antennas Propag., vol. 36, no. 4, pp. 449–462, Apr. 1988. [9] J. I. Herranz-Herruzo, A. Valero-Nogueira, and M. Ferrando-Bataller, “Optimization technique for linearly polarized radial-line slot-array antennas using the multiple sweep method of moments,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1015–1023, Apr. 2004. [10] B. Azarbar and L. Shafai, “Field solution for radial waveguides with annular discontinuities,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 11, pp. 883–890, Nov. 1979. [11] D. Dudley, Mathematical Foundations of Electromagnetic Field Theory. Piscataway, NJ: IEEE Press, 1998, pp. 171–171. [12] D. M. Pozar and E. H. Newman, “Electromagnetic modeling of composite wire and surface geometries,” IEEE Trans. Antennas Propag., vol. AP-28, no. 1, pp. 121–125, Jan. 1980. [13] A. F. Peterson, Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press, 1999, pp. 8–9. [14] B. Friedman, Principles and Techniques of Applied Mathematics. New York: Dover, 1990, pp. 140–140. [15] R. F. Harrington, Time–Harmonic Electromagnetic Fields. Piscataway, NJ: IEEE Press, 2001, pp. 180–180. [16] M. Sierra-Pérez, M. Vera, A. G. Pino, and M. Sierra-Castañer, “Analysis of slot antennas on a radial transmission line,” Int. J. Microwave Millimeter-Wave Computer-Aided Eng., vol. 6, no. 2, pp. 115–127, 1996.

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Alejandro Valero-Nogueira (S’92–M’97) was born in Madrid, Spain, on July 19, 1965. He received the M.S. degree in electrical engineering from the Universidad Politecnica de Madrid, Madrid, Spain, in 1991, and the Ph.D. degree in electrical engineering from the Universidad Politécnica de Valencia, Valencia, Spain, in 1997. In 1992 he joined the Departamento de Comunicaciones, Universidad Politecnica de Valencia, where he is currently an Associate Professor. During 1999, he was on leave with the ElectroScience Laboratory, The Ohio State University, where he was involved in fast solution methods in electromagnetics and conformal antenna arrays. His current research interests include computational electromagnetics, Green’s functions, waveguide slot arrays, and automated antenna design procedures.

Jose I. Herranz-Herruzo (M’04) was born in Valencia, Spain, in 1978. He received the M.S. degree in electrical engineering from the Universidad Politecnica de Valencia, Valencia, Spain, in 2002, and is currently working toward the Ph.D. degree at the Universidad Politecnica de Valencia. Since 2002, he has been with the Electromagnetic Radiation Group, Departamento de Comunicaciones, Universidad Politecnica de Valencia. His main research interests include slot array antenna design and optimization and efficient computational methods for printed structures.

Eva Antonino-Daviu (M’00) was born in Valencia, Spain, on July 10, 1978. Following development of her M.S. thesis at the University of Stuttgart, Stuttgart, Germany, she received the M.S. degree in electrical engineering from the Universidad Politecnica de Valencia, Valencia, Spain, in 2002, and is currently working toward the Ph.D. degree at the Universidad Politecnica de Valencia. In 2002, she joined the Electromagnetic Radiation Group, Departamento de Comunicaciones, Universidad Politecnica de Valencia. Her doctoral research interests include planar and microstrip antenna design and optimization and computational methods for printed structures.

Marta Cabedo-Fabres (M’02)was born in Valencia, Spain, in 1976. She received the M.S. degree in electrical engineering from the Universidad Politecnica de Valencia, Valencia, Spain, in 2000, and is currently working toward the Ph.D. degree at the Universidad Politecnica de Valencia. Since 2000, she has been with the Electromagnetic Radiation Group, Departamento de Comunicaciones, Universidad Politecnica de Valencia. Since January 2004, she has been a Lecturer with the Escuela Superior Politecnica de Gandia, Gandia, Spain. Her technical interests include numerical methods for solving electromagnetic problems and optimization techniques for wide-band antenna design.

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Study of an Active Predistorter Suitable for MMIC Implementation Roberto Iommi, Giuseppe Macchiarella, Member, IEEE, Andrea Meazza, and Maurizio Pagani

Abstract—A linearizer architecture, based on predistortion and well suited for monolithic-microwave integrated-circuit (MMIC) implementation is presented in this paper. A mathematical model for the linearizer is described, which allows an optimum choice of the various design parameters (taking into account several constraints due to the specific implementation environment). The power amplifier to be linearized is described by nonlinear functions experimentally derivable from large signal -parameters; the mathematical model is based on the complex power series expansion of these functions and on the definition of a suitable scaling rule illustrated in this paper. The overall linearizer model has been implemented in a system simulator (working with complex signals), also allowing the investigation of second-order distortion effects (not taken into account during the dimensioning of the structure). A preliminary prototype of the linearizer operating in the 12–16-GHz band has been implemented as a MMIC device (GaAs 0.25- m technology); the measured performances have confirmed the promising potential of the novel architecture introduced. Index Terms—Intermodulation distortion, linearization, predistortion, RF power amplifier (PA).

I. INTRODUCTION

S

EVERAL approaches have been proposed to achieve the more and more stringent requirements on linearity of RF and microwave power amplifiers (PAs) imposed by the diffusion of broad-band multilevel modulation schemes [1]–[4]. However, when the practical implementation of these devices is required in a monolithic environment, only a few architectures satisfy the several constraints imposed by this type of realization [5]. In this paper, we begin from a general two-loop architecture to derive an active predistortion linearizer particularly suited for monolithic-microwave integrated-circuit (MMIC) implementation. Fig. 1 shows the block diagram of a general two-loop architecture, where the first loop extracts the distortion generated by the nonlinear driver (NLD) block; a second loop adds this distortion, with the correct amplitude and phase (determined by the and ) to the main signal, canceling complex coefficients the nonlinear distortion generated by the PA. In order to obtain the PA linearization, the distortion generated by the NLD must be similar to that produced by the PA; a solution (well suited for MMIC implementation) is to realize the NLD as a scaled-down replica of the PA and to drive it with the same backoff. For the

Manuscript received March 19, 2004; revised September 14, 2004. R. Iommi is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, 32-20133 Milan, Italy. G. Macchiarella is with the Dipartimento di Elettronica e Informazione and the Dipartimento di Elettronica, Politecnico di Milano, 32-20133 Milan, Italy (e-mail: [email protected]). A. Meazza and M. Pagani are with Ericsson Lab Italy, 20090 Vimodrone, Italy. Digital Object Identifier 10.1109/TMTT.2004.842499

Fig. 1.

Block diagram of a two-loop predistorter.

delay block, an active implementation [5] is compatible with the MMIC environment; in particular, the same NLD block can be employed, however, imposing a much larger backoff. In this paper, the linearization principle of the proposed solution is first described in detail (Section II). In Section III, a mathematical analysis of the novel architecture (based on the complex signal notation) is presented, and the results are employed in Section IV for deriving suitable criteria for the linearizer dimensioning. A system simulator is then employed to assess the second-order effects (not included in the analytical model) on the linearizer performances; the numerical results obtained are illustrated in Section V. Finally, in Section VI, the measured results obtained from a MMIC prototype of the proposed architecture are presented; the prototype has been fabricated using GaAs technology and operates at 15.3 GHz. II. LINEARIZATION PRINCIPLE The scheme of the proposed linearizer is presented in Fig. 2 [6]. The block is a power divider with arbitrary transfer factors and ; the blocks represent couplers with coupling factors (where denotes ). Note that the parameare either real or pure imaginary, depending on the ters kind of coupler implemented; in absence of losses, it holds the . The blocks and represent the condition cascade of magnitude and phase controllers, which multiply the input signals by a complex constant. Finally, the two blocks, i.e., linear driver (LD) and NLD, are identical amplifiers, which are suitable scaled-down version of the PA to be linearized (the adopted scaling rule will be discussed in Section III). The purpose of the first loop is to extract the distortion produced by the NLD, assuming that the LD operates in the linear region; this can be obtained by assigning suitable values to the and deriving the value of for which the linear factors vanishes as follows: part of

0018-9480/$20.00 © 2005 IEEE

(1)

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Fig. 2. Scheme of the active predistorter.

Now it can be observed that, if the NLD operates with the same extracted from the backoff of the PA, the distortion signal first loop is a scaled-down replica of the distortion generated by the PA. In the second loop, the distortion signal generated in loop 1 is added, with opposite phase, to the signal passed through is determined by imposing that the the LD; the coefficient relative amplitudes of linear and distortion signals at the output of loop 2 are the same at the NLD output. This is obtained for the following value of : (2) It can be observed that the signal at the output of loop 2 could be employed, in principle, to drive the PA; actually, several constraints have to be imposed on the absolute signal levels in the two loops and then on the coupling factors values, which make it very difficult to fulfill the power-level requirement at the input of the PA (especially for a MMIC implementation). For these reasons, a linear buffer amplifier has been introduced for obtaining the required power level at the PA input (this buffer must operate with a sufficiently large backoff in order to produce a negligible distortion). Let us now consider the signal at the PA input; it can be expressed as follows: (3) where is proportional to the input signal and deteris the dismines the required power level at the PA output; tortion produced by the PA (at the output power level) scaled by the PA gain. is sufficiently small, there will be no disObserve that if tortion at the PA output [i.e., the distortion produced by the PA ]. is canceled by the part of the input signal equal to One can wonder the reason of using, in loop 1, an LD (which is an active element) instead of a delay line (as in the error loop of a feedforward linearizer; these loops, in fact, are very similar and perform a similar operation). The specific MMIC environment for which the proposed linearizer has been developed justifies this choice: a monolithic delay line would, in fact, be very difficult to be realized with sufficient accuracy. Moreover, LD

and NLD being identical devices, a sort of implicit adaptation is obtained here (not possible with a delay line), which maintains the balance of loop 1 for small variations of environmental parameters (such as temperature, power level, etc.). III. DERIVATION OF A MATHEMATICAL MODEL Here, a mathematical model for the active predistorter is introduced; this model will allow investigating the criteria for assigning suitable values to the parameters in the two loops. In particular, these criteria must concern: 1) the divider and couplers parameters, which must also satisfy the balancing conditions expressed by (1) and (2) and 2) the scaling rule, defining LD and NLD size with respect to the PA size. Note that, in the following analysis, the LD and buffer will be assumed perfectly linear; however, once a suitable set for the system parameters have been found, the power capability of these amplifiers must be determined by imposing that the be negligible distortion they introduce in the output signal (these evaluations can be performed with a commercial system simulator). The model adopts the complex envelope notation, assuming a narrow-band modulated signal as exciting input (for simplicity, is actually a real signal). A. Amplifier Modeling The PA and NLD are characterized by their large-signal complex gain , defined as the ratio of the output and input signal envelopes; it can be represented as the sum of a constant term (linear part), plus a nonlinear term depending on the input power (square magnitude of the input signal envelope) [7]. It then has (4) . The function can be expanded in a power with series (with complex coefficients ) as follows: (5) The coefficients can be found by matching the AM–AM and AM–PM characteristic of a given amplifier [7].

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B. Scaling Rule

After some algebraic manipulation, the expression of the first can be evaluated as follows: three coefficients

Using the above model for the NLD and PA, it has

(6) Let us now introduce a specific scaling rule for the two amplifiers, which defines a relationship between their large-signal gains. This rule is expressed as follows. • Equal small-signal gains: • Same distortion when the NLD is excited with an input power scaled down by a factor with respect to the PA input:

. where It can now be observed that the exact linearization implies for However, imposing this condition , a simple expression for the only to the first term can be derived as follows: buffer gain

(7)

(13)

This scaling rule has a simple representation when AM–AM and AM–PM conversions are considered; in fact, the scaling factor can be considered as the ratio between two output power levels where AM–AM or AM–PM have the same value. In particular, may represent the difference (in decibels) between of the two amplifiers.

determines the PA It is interesting to note that this value of input power equal to times the power at the input of the NLD; for the scaling rule adopted, this condition assure that the distortion produced by the NLD is exactly equal to that produced by the PA. The distortion generated at the PA output by the input is then exactly cancelled; what remains (represented signal terms) is the distortion due to the part by the higher order of the input signal constituting the correction produced by the linearizer. The proposed scheme cannot generally correct this distortion, which then represents a bottom reference when evaluating the improvements obtained in practical implementation of the proposed architecture.

C. Computation of the Linearizer Response Let us consider now the signal at the output of the linear buffer ); from the scheme in Fig. 2, applying (assumed with gain the balancing conditions (1) and (2) and imposing the same to the LD and the NLD, the following exsmall-signal gain pression is obtained:

(8) in a power Applying the scaling rule (7) and expanding series, as shown before, (8) can be rewritten as follows: (9) where

(12)

IV. GENERAL CRITERIA FOR DIMENSIONING THE LINEARIZER A first-order dimensioning of the linearizer can be performed using (1), (2), and (13); however, suitable values must be a and to the priori assigned to the coefficients scaling factor (it is assumed that the PA model is given and, and are known). Performing this thus, the parameters assignment, it must be considered that some constraints exist on the coupling coefficients values. First of all, the coefficients are implemented through passive devices so their magnitude must be . The linearity assumption on LD operation also requires that its input signal must be suffiat the NLD input. Another ciently smaller than the signal of the linearizer, aspect to be considered is the overall gain which is generally required to be about unity; the value of can be easily derived from the scheme in Fig. 2 and from (13) (14)

Now the PA output input signal

can be expressed as a function of its

(10) Substituting (9) in (10), a general expression of the output as function of the input of the predistorter can be obtained as follows:

(11)

, the Equation (14) shows that, imposing unit value to coefficient must be given by . The first choice to be faced concerns the scaling factor ; selecting a large value for this parameter allows a higher efficiency for the linearizer, but also requires a large value for the buffer gain; moreover, if the scaling factor is too large, the scaling rule considered in this paper could not be followed with sufdepends ficient accuracy by the physical devices. Note that and ; for the choice of the couplings pathrough (13) on rameters, it is convenient to refer to the normalized parameter .

IOMMI et al.: STUDY OF ACTIVE PREDISTORTER SUITABLE FOR MMIC IMPLEMENTATION

Fig. 3. Locus of ( p 10. ( = ) =

;

) or given values of

j



j

. Assigned parameter:

Fig. 5. (

=

877

G

) =

G G = k) ; " = 0 :5 .

= ( 10

p

p

as a function of

. Assigned parameters:

TABLE I DESIGN PARAMETERS VALUES

Fig. 4. 0:5.

Locus of (

;

) or given values of

j



j

. Assigned parameter: " =

It has been decided to also assign a priori a value to the coupling coefficient ; this parameter should not be too small because it is inversely proportional to ; however, for values of too close to 1, and could become greater than 1 (which is not acceptable). The compromise choice has been ( 6 dB). For the choice of the parameters and , it should be represents the ratio of the input powers observed that at the NLD and LD; it is assumed that a value of 10 dB for this ratio determines negligible nonlinear distortion at the LD output. Moreover, the imposition of unit gain for the linearizer value, as shown by (14); then determines the and . To now allow a choice for the remaining parameters ( and ), some graphs have been realized; the first two report values determining constant values of the locus of (Fig. 3) and (Fig. 4). The following graph (Fig. 5) reports the normalized buffer (in decibels) as a function of . gain

Examination of the above graphs allows us to select several values for the pair; however, it should be observed that the practical implementation of the blocks requires to have and sufficiently smaller than 1. Another aspect to be consid, which is convenient to ered concerns the normalized gain be as small as possible (especially for MMIC implementation). are reported toIn Table I, some compromise values for and . gether with the corresponding values of Note that it should be possible if the PA gain is large enough, to select a value for the scaling factor determining unit value (which means to remove this element from the linearizer for architecture). V. EVALUATION OF DISTORTION FROM THE LD AND BUFFER In the previous analysis of the linearizer, the blocks representing the LD and buffer have been assumed perfectly linear. However, in practical implementation, these amplifiers actually contribute to the overall residual distortion at the output of the linearized PA. Moreover, it is necessary to have a criterion for ). It assigning the power capability to the buffer (i.e., the is assumed that the LD has the same nonlinear characteristic of the NLD (i.e., the two amplifiers are identical). In order to evaluate the performances of the proposed architecture when considering the above distortion contribution, a commercial system simulator has been used (Visual System Simulator (VSS) from AWR, El Segundo, CA). The PA to be linearized is based on a pseudomorphic high electron-mobility transistor (pHEMT) device (developed by Ericsson, Vimodrone, Italy, and fabricated using GaAs technology of TriQuint,

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Fig. 6. Measured PA large-signal gain and AM–PM. best fit (4): F : : i; F :

1 049 0 0318 . :

i; F

:

i

F

coefficients for the

= 01 48 0 5 87 = 0 082 + 0 27 = 0 186 + 0 506 = 00 251 + 0 039 :

:

i; F

:

:

:

i; F

i; F

= 0 379 0 = 0 048 0 :

:

Fig. 7. Comparison of the linearizer performances (including the nonlinearity contribution from the buffer and LD).

Hillsboro, OR) operating in the 12–16-GHz band; in Fig. 6, the AM–AM and AM–PM characteristics of the PA at 13.9 GHz are shown, together with the power-series complex coefficients obtained for the best match. The LD and NLD have the same large-signal gain obtained from that of the PA by applying the scaling rule (7) with the assigned factor. Finally, the large-signal gain of the buffer of the PA, suitably employs the same nonlinear term is, however, assigned scaled down; the small-signal gain arbitrarily (the scaling factor is then determined for a required and for the specified gain). A linearizer has then been dimensioned according to the first for the considered PA is row of Table I; considering that 15.64 dB (Fig. 6) and assigning a scaling factor of 3 dB for the NLD, the following values are obtained for the linearizer parameters: Fig. 8.

dB Note that the phases of and obtained from (1) and (2) can be 0 or 90 or 180 (depending on the couplers employed). A quadrature phase-shift keying (QPSK) modulated signal with a symbol rate of 3.84 MHz and root-raised-cosine shaping has been used for exciting the linearized amplifier; the results of the simulations have shown that a compromise value for of the buffer is approximately 6–7 dB lower than that of PA (this produces a worsening of the adjacent power adjacent channel power ratio (ACPR) level of 7–8 dB with respect to the ideal buffer case). The LD contribution to the overall distortion was found to be absolutely negligible (this means that the assumpratio is adequate). tion on the In Fig. 7, the performances of the linearizer are illustrated. The bottom curve (linearized PA—ideal) represents the result with both LD and buffer perfectly linear; the middle curve (linearized PA) refers to the nonideal LD and buffer (the latter has 6.8 dB lower than that of PA); the top curve is the nona linearized PA response. It can be observed the relevant increase

ACPR of the linearized PA as function of the output power.

of linearity produced by the novel scheme: the ACPR of the linearized PA is approximately 18 dB greater than that of the PA; the worsening in the ACPR produced by the buffer with respect to ideal case is approximately 7 dB (as previously said). The very low out-of-band distortion level in the ideal case (bottom curve) is produced (as previously explained) by the error signal injected in the PA (not corrigible by the linearizer). An interesting peculiarity of the linearizer architecture introduced here is its intrinsic capability to operate with good performances over a large range of the output power (typically, predistortion linearizers tend to fail at low levels of input signal, where the overall distortion may become even larger than that of the PA alone [4]). To illustrate this peculiarity, Fig. 8 shows the computed ACPR for the linearized PA and for the PA alone, as function of the output mean power of the QPSK signal: the performances increase produced by the linearizer is maintained well below the output power design level (approximately 23.5 dBm). VI. EXPERIMENTAL VALIDATION In order to validate the novel linearizer architecture, a preliminary MMIC prototype has been designed and fabricated in

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Fig. 9. Chip layout. Fig. 11.

Measured PSD of the fabricated prototype at P

= 19 dBm.

depending on the output power level; this confirm the capability of the novel architecture to operate with a variable output power level, as pointed out by the simulations. From the reported results, it can be also observed that the linearizer performance deteriorates for distant out-of-band frequency components. This is probably due to the fifth-order distortion produced by the active phase shifter in the first loop, determining the distortion “background noise” characterizing the linearizer. In conclusion, the experimental results have confirmed the validity of the linearizer architecture introduced and studied here. Fig. 10. Measured PSD of the fabricated prototype at P = 14 dBm. The spectrum with and without linearizer is reported; as a reference, the spectrum of the input signal (translated by the PA gain) is also shown.

GaAs 0.25- m technology (TriQuint foundry). The device operates at approximately 15 GHz and the PA to be linearized is similar to the one considered in Section V. In Fig. 9, the chip layout is shown (actual dimensions: 5.2 3.2 mm); it includes the LD and NLD, while the buffer is implemented out of the chip. To simplify this preliminary realization, the controllers imand employ voltage-controlled phase shifters plementing and fixed attenuators; all the other coupling elements are implemented as a Lange coupler with fixed attenuators of suitable value (when required). The design of the linearizer has been performed following the guidelines discussed above; the final circuit has been optimized by employing Agilent ADS software (with TriQuint library models). To test the realized prototype, a 128 quadrature amplitude modulation (QAM) signal has been employed (with 155.52-Mbit/s modulation speed); the output spectrum [power spectral density (PSD)] has been then measured for different levels of the input power. Two examples of the measurements performed are reported in Figs. 10 and 11, which show the PSD at an output mean power of 14 and 19 dBm, respectively; note that the tuning of the phase shifters was not optimized for each power level (the linearizer alignment has been performed at 14 dBm). The measured results show a reduction of the out-of-band distortion with the linearizer in the 6–12-dB range,

VII. CONCLUSION This paper has introduced a linearizer architecture, based on predistortion, which is particularly suited for monolithic integration (i.e., MMICs). A mathematical model for the linearizer has been described, which has allowed to derive dimensioning criteria (taking into account various constraints posed by the practical implementation of the scheme). Using a system simulator operating with complex signals, the linearity requirements of the output buffer has been assessed; simulations have also confirmed some assumptions made during the model development (i.e., LD distortion contributions). Finally, a prototype MMIC device implementing the studied linearizer has been fabricated in GaAs technology; the measured results have validated the novel architecture introduced (in particular, it has been verified its capability to operate on a relatively large dynamic range while maintaining good linearity improvements). ACKNOWLEDGMENT The authors wish to thank G. Procopio and F. Palomba, both of Ericsson Lab Italy, Vimodrone, Italy, for the precious help in the mounting and experimental characterization of the MMIC prototype. REFERENCES [1] P. B. Kenington, High-Linearity RF Amplifier Design. Norwood, MA: Artech House, 2000. [2] 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.

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[3] J. Cha, J. Yi, J. Kim, and B. Kim, “Optimum design of a predistortion RF power amplifier for multicarrier WCDMA applications,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 655–663, Feb. 2004. [4] 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. [5] P. Bianco, S. G. Donati, G. Ghione, M. Pirola, C. U. Naldi, C. Florian, G. Vannini, F. Filicori, and L. Manfredi, “Optimum design of a new predistortion scheme for high linearity -band MMIC power amplifiers,” Microwave Eng., pp. 51–55, Jan. 2002. [6] R. Iommi, G. Macchiarella, A. Meazza, and M. Pagani, “A new active predistortion linearizer suitable for MMIC power amplifier,” in GaAs Symp., Munich, Germany, Oct. 2003, pp. 121–124. [7] C. G. Rey, “Adaptive polar work-function predistortion,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 722–726, Jun. 1999.

K

Andrea Meazza received the Electronic Engineering degree from the Politecnico di Milano, Milan, Italy, in 1990. From 1990 to 1998, he was an Analog and Digital Circuits Designer with Alcatel Italia, Vimercate, Italy. His research has mainly focused on RF circuits design for telecommunication equipments. Since 1998, he has been with Ericsson Lab Italy, Vimodrone, Italy, where he has been engaged in the development of high-linearity MMIC PAs in GaAs technology for microwave and millimeter-wave radio systems. He is currently involved in MMIC implementations of linearization techniques for PAs devoted to highly linear applications. He has authored or coauthored six technical papers.

Roberto Iommi was born in Novara, Italy, in 1978. He received the Laurea degree in telecommunication engineering from the Politecnico di Milan, Milan, Italy, in 2003. His thesis concerned monolithic PA linearization. He is currently concluding a postgraduate fellowship on MMIC techniques with the Politecnico di Milan.

Giuseppe Macchiarella (M’88) was born in Milan, Italy, in 1952. He received the Laurea degree in electronic engineering from the Politecnico di Milano, Milan, Italy, in 1975. From 1977 to 1987, he was a Researcher with the National Research Council of Italy, where he was involved in studies on microwave propagation. In 1987, he became an Associate Professor of microwave engineering with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy. He is also the Scientific Coordinator of the PoliEri Laboratory, a MMIC research laboratory, which is jointly supported by the Politecnico di Milano and Ericsson Lab Italy. His current research concerns the field of microwave circuits with special emphasis on microwave filters synthesis and power-amplifier linearization. He has authored or coauthored over 80 papers and conference presentations.

Maurizio Pagani was born in Borgomanero, Italy, in 1963. He received the Laurea degree in electronic engineering from the Politecnico di Torino, Turin, Italy, in 1988. In 1988, he joined Telettra S.p.A., where he was Researcher involved in the characterization and modeling of microwave GaAs devices and circuits. From 1992 to 1998, he was responsible for the microwave integrated-circuit design of Alcatel Italia. In 1998, he joined the Microwave Product Design Centre, Ericsson, Milan, Italy where he is currently Manager of the MMIC design unit of Ericsson Lab Italy, Vimodrone, Italy. Since 2002, he has been Contract Professor of microwave integrated circuits with the University of Florence, Florence, Italy. His main research interests are integrated-circuit design solutions for high-volume applications, linear PAs, and linearization techniques. In these research areas, he has been involved in several cooperation activities with the main Italian universities.

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A Complementary Colpitts Oscillator in CMOS Technology Choong-Yul Cha, Member, IEEE, and Sang-Gug Lee, Member, IEEE

Abstract—A new complementary Colpitts (C-Colpitts) oscillator topology is introduced and the oscillation mechanism as a oneport model is analyzed. Based on the one-port analysis and the existing phase-noise model, the phase-noise equation of the proposed C-Colpitts oscillator is derived as the function of the oscillation frequency, factor of tank circuit, and bias current. The phase-noise equation provides the design guideline to optimize the phase noise of the proposed Colpitts oscillator, of which the property is proven with simulation and measurement results. The proposed Colpitts voltage-controlled oscillators are fabricated using 0.35- m CMOS technology for 2-, 5-, 6-, and 10-GHz bands. Measurement shows that the phase noise is 118.1 dBc at 1-MHz offset from 6-GHz oscillation while dissipating 4.6 mA of current from a 2.0-V supply. Index Terms—CMOS, Colpitts, complementary, phase noise, optimization, voltage-controlled oscillator (VCO).

I. INTRODUCTION

W

ITH advancements in submicrometer CMOS technology, CMOS technology has become widely used for low-cost and highly integrated RF integrated circuits (RFICs). Recently, single-chip transceivers that integrate both digital and RF circuits using CMOS technology have been introduced. Among the building blocks in single-chip RFICs, design and implementation of fully integrated low-noise CMOS voltage-controlled oscillators (VCOs) is known as a challenging block because of the inborn limitations of silicon CMOS process technology. Most of the previously reported publications about CMOS VCOs describe the negativedifferential topology. In these publications, in order to optimize the phase-noise performance, researchers stress the importance of layout issues such as active and passive device design, and the floor plan of layout to reduce the side effects of the parasitics in CMOS technology [1]–[3]. In negative-based submicrometer CMOS differential VCOs, the complementary structure shows a better performance than the NMOS-only structure, as a result of the reduced hot carrier effect, better up/down swing symmetry, and higher transconductance of the constituting transistors [3]. Thus, by using low parasitic simple and high transconductance oscillator topology, there exists more potential in the design of a low-noise oscillator in high frequency or with low Manuscript received March 21, 2004; revised May 31, 2004. This work was supported in part by the Institute of Information Technology Assessment, which is funded by the Korean Ministry of Information and Communication. C.-Y. Cha is with the Department of Engineering, Information and Communications University, Daejeon 305-714, Korea (e-mail: netcar@ icu.ac.kr). S.-G. Lee was with Harris Semiconductor, Melbourne, FL 32955 USA. He is now with the Department of Engineering, Information and Communications University, Daejeon 305-714, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842498

Fig. 1. (a) Proposed C-Colpitts oscillator core. (b) Equivalent circuit of C-Colpitts core including parasitics.

power. Traditionally, the Colpitts oscillator, which has a simple oscillator core, has been the most favored topology for low phase noise [4]. However, since the conventional Colpitt oscillator needs additional circuits for bias and buffer interfaces, its oscillation performances may be degraded by the parasitics in high frequency. In this paper, a complementary Colpitts (C-Colpitts) oscillator topology [5] is introduced that is effectively composed of two components, a complementary NMOS and PMOS transistor pair and an inductor, and requires no additional circuits for bias and buffer interfaces. Since the proposed C-Colpitts oscillator is simple, has a complementary structure, and provides high transconductance, better oscillation performance can be achieved. In Section II, the operational principle of the proposed C-Colpitts oscillator is analyzed as a one-port oscillator model. In Section III, the phase-noise equation of the proposed C-Colpitts oscillator is derived. The phase-noise property is confirmed with simulation and provides the design guideline to optimize the phase-noise performance. In Section IV, the phase-noise property of C-Colpitts VCOs is proven with experimental results. Next, the performances of C-Colpitts VCOs with 0.35- m CMOS technology are compared with the findings from previous research. Conclusions are finally presented in Section V. II. C-COLPITTS OSCILLATOR TOPOLOGY AND ONE-PORT ANALYSIS Fig. 1(a) and (b) shows the core of the proposed C-Colpitts oscillator and the small-signal equivalent circuit with parasitic represents the overall transconcomponents. In Fig. 1(b), represents the overall gate–source ductance and represent the drain to substrate capacitance, and parasitics of and in Fig. 1(a), respectively. In Fig. 1(b), represents the series resistance of inductor . In Fig. 1(a) and , inductor , gate–source and (b), the transistors , and substrate parasitc constitutes a Colpitts capacitors

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oscillator [6]. As described earlier, since the simple and complementary structure of the proposed C-Colpitts oscillator decreases the parasitic components and increases the negative conductance, the potential of high-performance oscillation will increase. For better understanding and design optimization, the oneport analysis for the small-signal equivalent circuit, shown in Fig. 1(b), is described in the following. From Fig. 1(b), the can be given by equivalent conductance

Fig. 2. Equivalent one-port oscillator circuit.

(1) where and represent the small-signal drain current and node voltage of the NMOS and PMOS transistors. From (1), the can be re-expressed as a comequivalent conductance bination of the equivalent real conductance and inductance as follows:

(2) Fig. 3.

(3) where

and . In (2) and (3), and represent the quality ( ) factor, series resonance frequency of the series – – circuit, and normalized frequency over , respectively. From (2) and (3), it is obvious that the real value of the becomes negative when equivalent conductance . This means that the proposed oscillator can oscillate only the frequencies above . The negative conductance generation behavior can be explained as follows: , due to the second-order phase tranat frequencies above – – circuit, the phase transition sition of the series at the gate node of the transistor with respect to the drain node becomes larger than 90 , and this leads to the drain current inversion. As shown in (2) and (3), the imaginary part of can be represented by an equivalent inductor at all frequencies, and by the similar mechanism, the inductance increases rapidly . From (2) and (3), and can be for re-expressed as follows: (4)

(5) Using (4) and (5), the equivalent one-port oscillator circuit can be configured as shown in Fig. 2. As shown in (4) and (5),

j

G (! )=G

j

over ! .

when and only has an equivalent value, which means that the phase difference between the gate and drain node in Fig. 1(a) is exactly 90 . At frequencies , the phase between the gate slightly higher than and drain node rapidly approaches 180 , leading to negative real conductance. In this frequency region, the voltage drop across the gate–source capacitor is amplified by the amount of the quality – – circuit, which means a factor of the series over . As frequency increases sharp increase in further, the phase between the gate and drain node of the series circuit approaches almost 180 , and the imaginary term of is negligibly small, leading to a huge amount of active inductance. In this frequency region, the -factor multiplication in the series resonance circuit can no longer be applied any more and the voltage division mechanism dominates in the – – circuit. The negative conductance will series decrease with an increase in frequency as a result of a decrease of voltage drop in the gate–source capacitor. The normalized conductance and active inductance and , shown in (4) and (5), are plotted in Figs. 3 and 4. over for the different Fig. 3 shows the variation of . As can be seen in Fig. 3, the proposed oscillator value of topology provides a large negative conductance over a wide frequency band of operation and shows a peak near the frequency . The large absolute value of in Fig. 3 indiof cates that the proposed C-Colpitts topology is suitable for very low-power oscillation, particularly when the high- factor inductor is combined. over for Fig. 4 shows the variation of , where and are constant. the different value of (1–30 mS) and Considering the typical values for (1–5 GHz), it can be shown that the values for ranges

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From (6), it is known that the oscillation frequency of the . To proposed C-Colpitts oscillator is always higher than find the equivalent parallel tank resistance at a given oscillation frequency, we must keep an eye on the fact that the series – – can be changed to an equivalent series cir). cuit (note: At the oscillation frequency , the impedance of the series – – circuit is given as

Fig. 4.

L (!

)

2G

!

over ! .

(7)

(8) is the equivalent inductance of the equivalent seIn (8), circuit. From (8), the equivalent factor, i.e., , of ries circuit at the oscillation frequency is given as the series

(9) Fig. 5. (a) Equivalent one-port oscillator with C . (b) Simplified equivalent one-port oscillator circuit with an equivalent parallel tank.

from a few to a few hundred nanohenry, depending on the fre. Therefore, at operation frequenquency of operation and can play a major role in determining the cies near frequency of resonance, but less of a role as the operation fre. quency moves away from In the proposed C-Colpitts oscillator, the oscillation frequency can be controlled by adding high- loading capacitance at the drain node of and , as shown in Fig. 1(a). The loading capacitance will not affect the frequency behavior , and the factor of the – – of and – branch. For the equivalent circuit including , which is shown in Fig. 5(a), the oscillation frequency can be determined by the combined resonance frequency of and . In Fig. 5(a), for simplificaand ) and tion, by ignoring the substrate parasitics ( , the oscillation frequency is given as

Using (9), for given oscillation frequency is derived as equivalent parallel tank resistance

, the

(10) From (8) and (10), the equivalent one-port oscillator circuit with the parallel tank circuit is given as shown in Fig. 5(b), where the substrate parasitics and active inductance has been omitted. III. PHASE NOISE IN THE PROPOSED C-COLPITTS OSCILLATOR According to Leeson’s [7] and Hajimiri and Lee’s [3] phasenoise model, there are two components that contribute to phase noise, i.e., phase perturbation and amplitude fluctuation. In the low-offset frequency, the phase noise is dominated by the phase perturbation term. According to Hajimiri and Lee’s phase-noise region, the phase noise is given as [3] model, in the (11)

(6)

In (11), is the equivalent resistance of the parallel tank is the maximum charge swing across the equivacircuit. of the parallel tank circuit. If the lent tank capacitance maximum voltage swing of the tank circuit is represented as

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, then . and is the rms value of the impulse sensitivity function and the offset frequency for phase-noise measurement, respectively. From (11), it is obvious of the that the phase noise can be improved by maximizing parallel tank capacitance. oscillator, it is known that In the conventional negativea constant negative conductance is provided over a wide frequency range for the given bias current. Interestingly, as can be seen in Fig. 3, the negative conductance of the proposed C-Colpitts oscillator shows significant changes over the operation frequency. Since the bigger negative conductance requires a more bias current, the equivalent dc-bias current of the C-Colpitts oscillator can be derived for the given oscillation frequency. Assuming the long-channel MOSFET, the equivalent dc-bias curin a steady oscillation condition of C-Colpitts can be rent easily derived using (4) as follows:

From (14) and (15), the maximum charge swing the drain loading capacitance is given as

across

(16) Finally, using (10), (11), and (16), the phase noise of the proregion is derived as posed C-Colpitts oscillator in the

where (17) (12)

(13) and is the mobility, where oxide capacitance, channel width, and channel length of the is the bias NMOS and PMOS transistors, respectively, and current. can Using (13) and (10), the maximum voltage swing be derived as

(14) From (6), the drain loading capacitance as sented as the function of

can be repre-

(15)

In (17), the channel noise, noise, hot carrier effect, and the nonlinear characteristics of the CMOS device are not included. From (17), it can be seen that . To verify the above argument, a simulation is carried out for the C-Colpitts oscillator, as shown in Fig. 1(a) using 0.35- m CMOS technology with 2.0 V and 2.93 mA of supply voltage and bias current. The width of the NMOS and PMOS transistor is 150 and 300 m, respectively. The overall capacitance from the gate source is 1.645 pF, which includes the intrinsic gate–source capacitance of NMOS and PMOS transistor and an ideal 1 pF at gate-to-ground. of series resistance. By The inductor is 5 nH with 3 , the oscillation changing the drain loading capacitance and of 14.66 frequency is controlled. Simulation shows and 1.755 GHz, respectively. Using the circuit parameters and , the phase-noise performance from phase-noise equation (17) and circuit simulation at 100-kHz offset frequency . As is plotted in Figs. 6 and 7, as a function of was previously discussed, since the noise of the tank resistance is only considered, the phase-noise performance from (17) differs from the circuit simulation result. The channel noise, noise, hot carrier effect, and the nonlinear characteristics of the CMOS device are not considered in (17). leads to lower phase noise, As shown in Fig. 6, the higher approaches . As and the phase noise is improved as approaches , the negative conductance and loading capacitance increase at the same time, which leads to an increase in the maximum charge swing across the oscillator tank and, therefore, the phase noise improves. From Fig. 6, the best phase (see Fig. 3). When noise is achieved near the peak of approaches near , the phase noise decreases abruptly as

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Fig. 8. (a) Test circuit of C-Colpitts oscillator. (b) Core chip micrograph (top) and VCO module (bottom).

Fig. 6. Phase noise at 100-kHz offset frequency from (17).

Fig. 7. Simulated phase noise at 100-kHz offset frequency.

the negative conductance quickly decreases (see Fig. 3). However, both the phase noise characteristics shown in Figs. 6 and 7 are well matched in behavior over the oscillation frequency and achieve the minimum phase noise at the almost same frequency approaching , the region. As discussed in Section II, with phase difference between the gate and drain node in Fig. 1(a) approaches 90 , which means that the cyclo-stationary phasenoise contribution of the MOSFET can be suppressed [4]. In the proposed C-Colpitts oscillator, the loading capacitance can oscillator since a greater be larger than that of the negativenegative conductance is provided with the same power consumption. In other words, the phase noise that can be achieved oscillator. is better than that of the negativeFig. 7 plots the phase-noise performance at the offset frequency of 100 kHz, the ratio between the drain loading and gate–source capacitance, and the maximum charge swing as a . In Fig. 7, the phase-noise behavior shows function of nearly the same behavior as what is predicted in Fig. 6. The difference in the slope of Fig. 7 compared to Fig. 6 may have resulted from the complex combination of the supply voltage pushing to the output swing, and the nonlinearity of active devices for the large-signal swing. IV. VCO DESIGN AND EXPERIMENTAL RESULTS To evaluate the phase-noise characteristics of the C-Coplitts oscillator described in Section III, the C-Colpitts oscillator

Fig. 9. Measurement results. (a) L

5:6 nH, C = 2 pF.

= 10 nH, C = 4 pF. (b) L

=

schematic shown in Fig. 8(a) is composed as a test module with a C-Colpitts core chip and external chip components, as (NMOS), (PMOS), shown in Fig. 8(b). In Fig. 8(a), , and constitute the C-Colpitts oscillator, and (NMOS) and (PMOS) constitute the inverter buffer. is an accumulation-mode varactor, and and are the bypass and coupling capacitor, respectively. The C-Colpitts core is integrated using 0.35- m CMOS technology. The performance of the VCO module is measured with 2.5 V of by changing the value of supply voltage and for the given , and . DC current flows from 4.3 to 6.3 mA, which has approximately 40% of variation depending on the value of . Fig. 9(a) and (b) shows measured performances for the case nH and pF and (b) nH of (a) and pF over the oscillation frequency. In Fig. 9(a), the

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TABLE I MEASUREMENT RESULTS WITH L = 10 nH AND C

= 4 pF

Fig. 10.

Complete VCO schematic of the proposed C-Colpitts oscillator.

TABLE III MEASUREMENT RESULTS OF THE PROPOSED C-COLPITTS VCOs OTHER PUBLICATIONS TABLE II = 5:6 nH AND C MEASUREMENT RESULTS WITH L

AND

= 2 pF

capacitance of ranges 0 open 22 pF and the oscillation pF) to 1290 MHz frequency covers from 688 MHz ( open . In Fig. 9(b), the capacitance of ranges from 0(open) 10 pF and the oscillation frequency covers from 1150 pF) to 1747 MHz open . ( and of the – In both cases, measuring branch is impossible. However, from the measurement results in Fig. 9(a) and (b), the frequency of can be estimated as approximately 688 and 1150 MHz. Even though the measurement results in Fig. 6(a) and (b) are , both measured results are well not as normalized as matched with the phase-noise trend of the previous plots in Figs. 6 and 7. The best phase-noise performance is achieved near , 688 and 1150 MHz, respecthe frequency of the estimated tively. Since the oscillation frequency changes over the value of , the power-frequency normalized figure-of-merit (FOM) is provided for an objective performance comparison. As shown in Fig. 9, the FOM also follows the same trend with the phase-noise property, as shown in Fig. 6. In Tables I and II, the measurement results are summarized. Fig. 10 shows another complete schematic of the proposed C-Colpitts oscillator applied for a VCO, which includes a represents directly coupled inverter as a buffer. In Fig. 10, represents the ac an accumulation-mode MOS varactor, represents the bypass capacitor to an blocking resistor, and ac ground. In Fig. 10, the varactor is connected on the gate side of the oscillator core in order to obtain a wider tuning range.

and are added for impedance matching and dc blocking. Several VCOs with different frequencies of oscillation, i.e., 2-, 5-, 6-, and 10-GHz bands, have been fabricated based on 0.35- m CMOS technology. With 2- and 10-GHz-band VCOs, the inductors are implemented as an external printed circuit board (PCB) spiral and on-chip bond wire, respectively, while on-chip spiral inductors are used for the 5- and 6-GHz-band VCOs. The performances of the VCOs are evaluated for various power dissipations by changing the supply voltage. Table III summarizes the measurement results in comparison with other reported 5-GHz-band VCOs. The fabricated VCOs were not optimized for minimum phase noise following analysis, as described in Section III. However, the fully integrated 5- and 6-GHz-band VCOs, shown in Table III, present better performances than that of the corresponding previous study that is implemented using 0.25- m CMOS technology, as they are designed closer to the optimum point. With the 2-GHz design, the loading capacitance is significantly smaller (only around 0.5 pF) than the optimum point, which leads to higher phase noise. Fig. 11 shows the measurement result of the phase noise at 6 GHz. In Fig. 9, the phase noise at 1-MHz offset is 118.1 dBc while dissipating 4.6 mA of current from a 2.0-V supply. Fig. 12 shows the micrograph of the fabricated VCOs: 2-, 5-, 6-, and 10-GHz bands, respectively.

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Several VCOs of 2-, 5-, 6-, and 10-GHz bands are fabricated using 0.35- m CMOS technology. Even though the fabricated VCOs were not optimized for the low phase-noise performance following the design guideline in Section III, the 5- and 6-GHzband VCO shown in Table I presents better performances than that of the corresponding previous work that is implemented using 0.25- m CMOS technology. The measured phase noise of 6.0 GHz @1-MHz offset is 118.1 dBc while dissipating 4.6 mA of current from a 2.0-V supply. REFERENCES

Fig. 11.

Measured phase noise at 6.0 GHz as a function of offset frequencies.

[1] C. R. C. De Ranter and M. S. J. Steyaert, “A 0.25 m CMOS 17 GHz VCO,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., San Francisco, CA, Feb. 2001, pp. 370–371. [2] C.-M. Hung, B. A. Floyd, N. Park, and K. K. O, “Fully integrated 5.35-GHz CMOS VCOs and prescalers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 17–22, Jan. 2001. [3] A. Hajimiri and T. H. Lee, The Design of Low Noise Oscillators. Norwell, MA: Kluwer, 1999. [4] T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits. Cambridge, U.K.: Cambridge Univ. Press, 1998. [5] C.-Y. Cha and S.-G. Lee, “A complementary Colpitts oscillator based on 0.35 m CMOS technology,” in Proc. Eur. Solid-State Circuit Conf., Sep. 2003, pp. 691–694. [6] B. Razavi, RF Microelectronics. Upper Saddle River, NJ: PrenticeHall, 1998. [7] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, Feb. 1966. [8] J. Bhattacharjee, E. Gebara, J. Laskar, D. Mukherjee, and S. Nuttinck, “A 5.8 GHz fully integrated low power low phase noise CMOS LC VCO for WLAN applications,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, Jun. 2002, pp. 585–588. [9] V. Boccuzzi, S. Levantino, and C. Samori, “A 94 dBc/Hz@100 kHz, fully-integrated, 5-GHz, CMOS VCO with 18% tuning range for Bluetooth applications,” in Proc. IEEE Custom Integrated Circuits Conf., May 2001, pp. 201–204. [10] P. Kinget, “A fully integrated 2.7 V 0.35 m CMOS VCO for 5 GHz wireless applications,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 1998, pp. 226–227.

0

Fig. 12. Micrograph of fabricated oscillator. (a) 2-, (b) 5-, (c) 6-, and (d) 10-GHz bands.

V. CONCLUSION A new C-Colpitts CMOS oscillator topology is proposed and the oscillation mechanism based on a small-signal one-port oscillator model isanalyzed. The one-portanalysis shows that the large – negative conductance originated from the series – resonance network can be provided in the proposed C-Colpitts osof the series – – cillator, which has a peak near the network. Considering the simple and complementary structure and the generation of high negative conductance in the proposed C-Colpitts oscillator, it is adequate for high-performance oscillation in high frequency or with low power. Based on the result of the one-port analysis and reported phase-noise model, the phase-noise equation of the proposed C-Colpitts oscillator is derived as a function of the oscillation factor of tank circuit , and bias current. The frequency, derived phase-noise equation shows that the best phase noise can be achieved by adjusting the oscillation frequency of the proposed Colpitts oscillator near the peak frequency of the negative conductance. Through the simulation and measurement, it is proven that the derived phase-noise property of the proposed Colpitts oscillator has good correlation.

Choong-Yul Cha (M’04) was born in Gyungnam, Korea, in 1972. He received the B.S. degree in electronic engineering from Yeungnam University, Kyungpook, Korea, in 1995, the M.S. degree in engineering from the Information and Communications University, Daejeon, Korea, in 2002, and is currently working toward the Ph.D. degree at the Information and Communications University. From 2002 to 2003, he was an Analog, RF, and Opto-Electrical Circuit Designer with GAINTECH, Daejeon, Korea. His main research interest is highspeed analog and digital circuit design for wireless and wire line (optical) communications using CMOS and BiCMOS technology. Sang-Gug Lee (M’04) was born in Gyungnam, Korea, in 1958. He received the B.S. degree in electronic engineering from Gyungbook National University, Gyungbook, Korea, in 1981, and the M.S. and Ph.D. degrees in electrical engineering from the University of Florida at Gainesville, in 1989 and 1992, respectively. In 1992, he joined Harris Semiconductor, Melbourne, FL, where he was engaged in silicon-based RF integrated-circuit (IC) designs. From 1995 to 1998, he was an Assistant Professor with the School of Computer and Electrical Engineering, Handong University, Pohang, Korea. Since 1998, he has been with the Information and Communications University, Daejeon, Korea, where he is currently an Associate Professor. His research interests include the silicon technology-based (bipolar junction transistor (BJT), BiCMOS, CMOS, and SiGe BICMOS) RFIC designs such as low-noise amplifiers (LNAs), mixers, oscillators, power amps, etc. He is also active in designing high-speed ICs for optical communication such as transimpedance amplifiers (TIAs), driver amps, limiting amps, clock data recovery (CDR), mux/demux, etc.

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A Novel Design Tool for Waveguide Polarizers Giuseppe Virone, Riccardo Tascone, Member, IEEE, Massimo Baralis, Oscar Antonio Peverini, Augusto Olivieri, and Renato Orta, Senior Member, IEEE

Abstract—This paper presents a novel method for the design of broad-band waveguide polarizers. It consists of an interactive procedure where the designer places the zeros of the reflection coefficient for one polarization, simultaneously also controlling the frequency response of the other one and vice versa. In this way, the best matching condition for both polarizations can be found. This design tool is based on an automated phase control procedure and on an algorithmic characterization of the relationship between the two frequency responses. The spurious effects of multimodal interactions, frequency dispersion, and losses are taken into account and compensated by characterizing them with suitable transfer functions. Designs of iris-loaded polarizers in square and circular waveguides are described, with operative bandwidths up to 30%. -band prototype show reMeasurements on a 10% bandwidth flection and cross-polarization levels of approximately 50 dB, with a phase error of approximately 0.4 . Index Terms—Design methods, synthesis techniques, waveguide components, waveguide polarizers.

I. INTRODUCTION

M

ANY KINDS of waveguide polarizers have been presented in the literature. As reported in [1], the most appealing configuration for broad-band low-loss applications consists of a circular or square waveguide loaded with discontinuities such as irises [1], [2], septa [3], grooves [4], or corrugations [5]. In some cases, two input/output transition sections are also included [1], [6]. In order to obtain the polarization conversion, the waveguide structure has to provide a phase difference of 90 between the transmission coefficients of two orthogonal linearly polarized degenerate modes (called principal polarizations) without coupling them. The main requirements for waveguide polarizers are compact size, low cross-polarization level, and good return loss over a broad frequency band. In order to fulfill all these specifications, design methods for waveguide polarizers have been widely investigated in the past. Generally, these methods are based on parametric analyses of the device [2] or on optimization procedures applied to a starting guess configuration [1], [3]–[5]. However, because of the large number of parameters involved in the polarizer design, and if a good starting guess is not available, these approaches can require a long computation time, sometimes without successful results, because of the presence of local minima. Manuscript received March 19, 2004; revised May 13, 2004 and June 30, 2004. This work was supported by the Italian Space Agency (ASI) under the Sky Polarization Observatory (SPOrt) Project. The authors are with the Istituto di Elettronica ed Ingegneria dell’Informazione e delle Telecomunicazioni, Italian National Research Council, 10129 Turin, Italy. Digital Object Identifier 10.1109/TMTT.2004.842491

In order to avoid the above-mentioned problems, a novel design strategy has been developed. With the help of an interactive graphic computer tool, it directly yields the geometry of the device without the necessity of any optimization algorithm. This method is useful for the design of waveguide polarizers with transverse discontinuities like irises or grooves. Longitudinal structures, such as profiled longitudinal ridges [3] and waveguides with cut corners or flats [7], are not considered. Using this interactive tool, the designer satisfies the specification on return loss by positioning the reflection zeros, and simultaneously controlling the frequency responses of both polarizations. This strategy is very similar to that used in the antenna array design, where the radiation pattern can be shaped by positioning the zeros of the array factor [8]. In particular, through an automated phase control procedure (PCP), the level of the reflection coefficient for both polarizations is dynamically adjusted in order to obtain the required 90 phase difference. In this way, the highest return loss for both principal polarizations can be easily achieved without worrying about the phase difference. In the PCP, the distributed-parameter-model synthesis technique (DPMST) reported in [9] is applied. In fact, a polarizer can be considered as a pair of waveguide filters (one for each principal polarization) where the filtering properties are not particularly stressed, but the most important feature is the matching. The DPMST allows one to synthesize an arbitrary frequency response, hence the reflection zeros can be placed at will. Through the DPMST and a proper full-wave analysis method (see for example [1], [3], [4], [10]–[13]), the geometrical parameters of the discontinuities (iris apertures or corrugation heights, etc.) and the spacings between them are evaluated. The PCP is based on a single-mode model of the polarizer. However, the design procedure takes full account of multimodal interactions, frequency dispersion of the discontinuities and losses by means of a generalization of the compensation technique presented in [9]. Two examples of iris-loaded polarizers are presented. The first is a square-waveguide device with a 30% bandwidth. The -band polarizer, designed second is a circular-waveguide in the framework of the SPOrt (Sky Polarization Observatory) project [14]. For the latter, measurements are compared to the predicted data. II. DESIGN STRATEGY The frequency responses of the polarizer for the principal polarizations are conveniently represented by the parameters and , i.e. the elements of the two transmission matrices of the device. These quantities are related

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difference and

Fig. 1. Single-mode equivalent circuits for the principal polarizations.

to the scattering parameters , and respectively, by the following expressions:

,

Notice that the zeros of coincide with the . zeros of the reflection coefficient According to a single-mode model, the polarizer is equivalent to the two uncoupled distributed-parameter circuits depicted in Fig. 1, where and are the scattering matrices of the discontinuities for the two principal polarizations and are their spacings. In the case of reciprocal lossless structures and assuming that the magnitude (but not the phase) of the transmission coefficients of the discontinuities is constant in the operative frequency band, the parameters and have the following polynomial form ([9]):

where is the number of discontinuities, and are complex coefficients, and . The angle is related to the round-trip phase shift (RTPS) of the waveguide sections defined by two consecutive discontinuities. Since the lengths of these sections are generally different, is the average of the various RTPS. One can observe that the position of the zeros defines the shape of the frequency response whereas the maximum degree coefficient is related to the level of the reflection coefficient . The parameters and have another important role in the design, because, they are monotonically related to the phase

between the transmission coefficients

In fact, for a distributed structure of the type of Fig. 1: 1) an increase in the level of the input reflection coefficient (without changing the zero location and, hence, the overall bandwidth) leads to an increase of the reflection coefficient of each discontinuity ; 2) each discontinuity introduces a which increases with phase difference ; and 3) the phase differits reflection coefficient ence introduced by the whole structure is approximately equal to the sum of these phase differences . Hence, a higher level of the input reflection coefficient of the whole structure leads to a larger phase difference . This monotonic relationship is used to obtain the 90 phase difference by controlling . dynamically the coefficient Starting from a polynomial frequency response defined by its zeros (for example Chebyshev zeros) and by the coefficient , the distributed-parameter-model synthesis technique (DPMST) reported in [9] is applied. This technique directly yields the transmission coefficients for the polarization of each discontinuity and a set of phase terms related to the electrical spacings between the discontinuities. At this point, the geometrical parameters of the discontinuities (e.g., iris apertures, corrugations heights or widths, etc.) are determined from the transmission coefficients . In particular, this task is carried out by a bisection method applied to a full-wave analysis tool (notice that only the transmission coefficients at the central frequency are required) or to a pre-calculated lookup table. Finally, the spacings are also evaluated [9]. Since the geometry of the discontinuities has been determined, the full-wave analysis tool is used to also compute the scattering for the polarization. This allows the computation matrices (at the central frequency), as well as of the phase difference of , which will be used later in the compensation stage. At this point, the sum of the partial phase differences is (initially set at 90 ), and the error compared with the goal is used to modify the initial value of . At the end of this task, called the PCP, a first rough waveguide polarizer is available and a full-wave analysis of the synthesized device is carried out. Clearly, due to the phenomena neglected in the single-mode model of Fig. 1, the phase difference between the two polarizations is not 90 over the whole band and the -polarization matching is different from . Moreover, generally, the return loss for the polarization is not satisfactory. Hence, to improve the preliminary design, a generalization of the compensation technique presented in [9] is introduced. Let us first consider the phase difference, whose average value, in the operative band, can be written as . is of the order of few degrees, the value of is Since dynamically modified in order to obtain the 90 target

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w

Fig. 3. Iris cross sections. is the geometrical parameter used in the design. (a) Rectangular iris in square waveguide used for the 30% bandwidth waveguide -band polarizer in circular waveguide. polarizer. (b) Iris of the

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Fig. 2. Scheme of the abstract system identification. (a) Whole synthesis procedure. (b) Linear system identification. (c) Modified linear system identification.

the linear system approximation could be poor in this case, and a better representation is required. However, within the PCP stage, the single-mode response was also computed. Therefore, for the polarization it is possible to use this signal as input of the system that maps single-mode to full-wave response for the polarization. This leads to the definition of the transfer function

Turning now to the problem of the return loss for the two polarizations, the results of the full-wave analysis are interpolated with two -degree polynomials

At this point, as reported in [9], it is very useful to consider , , and as periodic signals with respect to the variable (the phase of the complex variable ) and the coefficients , , and as the discrete spectra of these signals. Hence, the whole synthesis scheme, depicted in Fig. 2(a), composed by the PCP, full-wave analysis, and polynomial interpolation procedure can be considered as an abstract system with as the input and and as outputs. Generally, in the case of devices without strongly dispersive discontinuities (such as waveguide polarizers), the full-wave freis not very far from . This fact sugquency response gests the representation of the above-mentioned system in terms of a linear and -invariant one. Thanks to this assumptions, it is [see Fig. 2(b)] possible to introduce a transfer function between the full-wave frequency response and the singlemode frequency response . In this way, incorporates the effects of multimodal interactions, frequency dispersion of the discontinuities, and losses. In particular, this transfer function is easily evaluated as the ratio between the spectra of the output and input signals, and , respectively,

In order to characterize the relationship between the frequency responses of the two principal polarizations, another could be introduced [see Fig. 2(b)]. transfer function This function would relate the full-wave response to the . However, since these signals corsingle-mode response respond to two different polarizations (two different circuits), they could be quite different from each other. Consequently,

as depicted in Fig. 2(c). In this way, the relationship between the two frequency responses of the polarizer has been found. All the system blocks of Fig. 2(c) are invertible. In fact, the PCP, which contains the DPMST, can be performed starting inor . As for the other blocks, and differently from have a magnitude near the unity because the full-wave responses are not very far from the single-mode ones. Thanks to the fast execution of the numerical code, which implements the whole system of Fig. 2(c), an interactive graphic interface was built. With this computer tool, the designer drags with the mouse, directly observing the zeros of the effect of this action on both frequency responses ( and ). In particular, when the designer drags a zero of , all the zeros of move. Generally, circle some of them could go away from the arc of the corresponding to the operative bandwidth. It is a convenient choice to operate only on the zeros that are close to this arc, leaving the others at their positions, because only the former have a significant effect on the reflection coefficients. At the end of this zero-positioning procedure, the final geomand are good etry of the polarizer is available. Since approximations of the full-wave responses of the device, no optimization is required. III. RESULTS As a design example, a broad-band polarizer in a square waveguide with seven irises is presented. The iris thickness, normalized to the waveguide dimension , is equal to 0.02. The iris cross section is depicted in Fig. 3(a). For this geometry, a preliminary coarse parametric analysis was carried out by the method of moments with weighted Gegenbauer polynomials as basis functions to represent the aperture field [15]. This analysis showed that the phase difference introduced by these irises has . For a minimum (quadratic-like behavior) close to this reason, was chosen as the central frequency of the design. The transmission coefficients for the principal

VIRONE et al.: NOVEL DESIGN TOOL FOR WAVEGUIDE POLARIZERS

Fig. 4. Transmission coefficients for the principal polarizations and phase difference introduced by a single iris of the type of Fig. 3(a) at a= = 0:814. w is the iris aperture. The iris thickness is equal to 0:02a.

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Fig. 6. Reflection coefficient for the principal polarizations of the squarewaveguide polarizer with a 30% frequency bandwidth. Solid line: vertical polarization (capacitive). Dashed line: horizontal polarization (inductive). Full-wave analysis.

Fig. 7. Phase difference between the transmission coefficients for the principal polarizations of the square-waveguide polarizer with a 30% frequency bandwidth. Full-wave analysis. Fig. 5. Reflection zeros in the complex z -plane of the square-waveguide polarizer with a 30% bandwidth. (): vertical polarization (capacitive). (): horizontal polarization (inductive).

polarizations, and the phase difference introduced by a single iris, are reported in Fig. 4. This data were used as lookup tables by the polarizer design tool. With the aim of obtaining a 30% bandwidth, the design procedure started from a Chebyshev frequency response with a 50-dB return loss for the vertical polarization. After the dragging of the , reflection zeros (which remain near the arc of the circle highlighted in Fig. 5), a 48-dB return loss was achieved for both polarizations, as can be seen in Fig. 6, where the full-wave analysis curves are reported. It has to be noted that only three zeros for each polarization remained near the arc defined by the bandwidth, whereas the other ones moved away to maintain the required phase difference.

The phase difference is shown in Fig. 7. Thanks to the PCP, the average value of 90 was automatically obtained. The maximum deviation from 90 is 3.8 corresponding to a maximum cross-polarization level of 29.5 dB. This value, which essentially depends on the iris frequency dispersion, can be somewhat improved by increasing the number of , discontinuities [1]. The iris apertures are , , , and with . The spacings between the irises , , , and are with . The present tool was also used to design a high-performance circular-waveguide polarizer for -band correlation radiometer of the Sky Polarization the Observatory (SPOrt) project [14]. The iris cross section is depicted in Fig. 3(b). A similar geometry was used in [7] for polarizers in circular waveguide with cut corners. In this

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Fig. 8. Computed and measured reflection coefficient of the manufactured -band waveguide polarizer for the vertical polarization (capacitive).

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Fig. 10. Computed and measured phase difference between the transmission coefficients for the principal polarizations of the -band polarizer prototype.

Fig. 9. Computed and measured reflection coefficient of the manufactured -band waveguide polarizer for the horizontal polarization (inductive).

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Fig. 11. factured

case, a full-wave analysis technique based on the method of moments in the spectral domain has been applied [16]. With this technique, a coarse parametric analysis was carried out showing that, for a given waveguide diameter of 9.38 mm, a suitable iris thickness is 0.5 mm. In fact, for this thickness, the have a minimum at approximately the phase differences central frequency of 32 GHz. The number of discontinuities has been chosen equal to eight because of the severe specifications (reflection coefficients and cross-polarization below 48 dB). mm, After the design stage, the iris apertures were mm, mm, mm, and with . The spacings between the mm, mm, mm, irises were mm, and with . From this design, a one-block aluminum prototype has been manufactured by the spark–erosion technique. The 30-mm-long polarizer exhibits a measured return loss for both polarizations better than 50 dB in a 10% frequency band, which is in good

agreement with simulations (Figs. 8 and 9). Some slight differences between the predicted and measured results appear at levels of reflection coefficients below 40 dB. At those levels, the manufacturing uncertainties can play a significant role in the -band. It has to be noted that, using a spark–erosion process, the resulting uncertainties are of the order of 0.01 mm. The predicted and measured maximum deviation from 90 for the phase is 0.42 (Fig. 10), which leads to a cross-polardifference ization level lower than 48 dB, as reported in Fig. 11.

Ka

Computed and measured cross-polarization level for the manu-

Ka-band waveguide polarizer.

IV. CONCLUSION An interactive tool for the design of waveguide polarizers has been presented. This method directly yields the geometry of the device without making use of optimization techniques so that local minima problems are avoided. A design example of broad-band iris-loaded square-waveguide polarizer has been used to present the characteristic of

VIRONE et al.: NOVEL DESIGN TOOL FOR WAVEGUIDE POLARIZERS

the proposed design tool. The method takes full account of frequency dispersion of the discontinuities, losses, and multimodal -band prototype interactions. In fact, measurements on the are in excellent agreement with the predictions. The high electrical performances of the synthesized devices confirm that a direct design approach allows one to better exploit the potentialities of the consolidated polarizer configurations. Finally, with this method, the input/output transitions (if necessary) can be designed in conjunction with the polarizer itself. In fact, the discontinuities of these transitions can be described with their scattering parameters and can be included in the equivalent circuits of Fig. 1. In this way, more compact structures could be obtained. REFERENCES [1] G. Bertin, B. Piovano, L. Accatino, and M. Mongiardo, “Full-wave design and optimization of circular waveguide polarizers with elliptical irises,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1077–1083, Apr. 2002. [2] K. K. Chan and H. Ekstrom, “Dual band/wide band waveguide polarizer,” in Asia–Pacific Microwave Conf., Dec. 3–6, 2000, pp. 66–69. [3] J. Bornemann, S. Amari, J. Uher, and R. Vahldieck, “Analysis and design of circular ridged waveguide components,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 3, pp. 330–335, Mar. 1999. [4] N. Yoneda, M. Miyazaki, H. Matsumura, and M. Yamato, “A design of novel grooved circular waveguide polarizers,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2446–2452, Dec. 2000. [5] J. Rebollar and J. Esteban, “CAD of corrugated circular–rectangular waveguide polarizers,” in 8th Int. Antennas Propagation Conf., vol. 2, 1993, pp. 845–848. [6] J. Rebollar and J. de Frutos, “Dual-band compact square waveguide corrugated polarizer,” in IEEE AP-S Int. Symp., vol. 2, Jul. 11–16, 1999, pp. 962–965. [7] R. Levy, “The relationship between dual mode cavity cross-coupling and waveguide polarizers,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 11, pp. 2614–2620, Nov. 1995. [8] R. S. Elliot, Antenna Theory and Design. Englewood Cliffs, NJ: Prentice-Hall, 1981. [9] R. Tascone, P. Savi, D. Trinchero, and R. Orta, “Scattering matrix approach for the design of microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 3, pp. 423–430, Mar. 2000. [10] F. Arndt, R. Beyer, M. Reiter, T. Sieverding, and T. Wolf, “Automated design of waveguide components using hybrid mode-matching/numerical EM building-blocks in optimization-oriented CAD frameworks—State-of-the-art and recent advances,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 747–760, May 1997. [11] T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. New York: Wiley, 1989. [12] J. Uher, J. Bornemann, and U. Rosenberg, Waveguide Components for Antenna Feed Systems: Theory and CAD. Norwood, MA: Artech House, 1993. [13] O. A. Peverini, R. Tascone, M. Baralis, G. Virone, and D. Trinchero, “Reduced-order optimized mode-matching cad of microwave waveguide components,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 311–318, Jan. 2004. [14] S. Cortiglioni, G. Bernardi, E. Carretti, L. Casarini, S. Cecchini, C. Macculi, M. Ramponi, C. Sbarra, J. Monari, A. Orfei, M. Poloni, S. Poppi, G. Boella, S. Bonometto, L. Colombo, M. Gervasi, G. Sironi, M. Zannoni, M. Baralis, O. Peverini, R. Tascone, G. Virone, R. Fabbri, V. Natale, L. Nicastro, K.-W. Ng, E. Vinyajkin, V. Razin, M. Sazhin, I. Strukov, and B. Negri, “The Sky Polarization Observatory,” New Astronomy, no. 9, pp. 297–327, May 2004. [15] A. Morini, T. Rozzi, A. Angeloni, and M. Guglielmi, “Accurate calculation of the modes of the circular multiridge waveguide,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, Jun. 8–13, 1997, pp. 199–202. [16] R. Tascone, R. Orta, M. Baralis, A. Olivieri, and O. Peverini, “Compact stepped twisted rectangular waveguide transition for microwave feeding networks,” in Electromagnetics in Advanced Applications, Turin, Italy, Sep. 2001, pp. 623–626.

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Giuseppe Virone was born in Turin, Italy, in 1977. He received the Electronic Engineering degree (summa cum laude) from the Politecnico di Torino, Turin, Italy, in 2001. Since February 2002, he has been with the Applied Electromagnetics Section, Istituto di Elettronica e di Ingegneria Informatica e delle Telecomunicazioni (IEIIT), Italian National Research Council (CNR), Politecnico di Torino. His research interests are in the area of the analysis and design of microwave and millimeter-wave passive components, feed systems and antennas, waveguide discontinuities and transitions, numerical methods, and measurement techniques.

Riccardo Tascone (M’02) was born in Genoa, Italy, in 1955. He received the Laurea degree (summa cum laude) in electronic engineering from the Politecnico di Torino, Turin, Italy, in 1980. From 1980 to 1982, he was with the Centro Studi e Laboratori Telecomunicazioni (CSELT), Turin, Italy, where his research mainly dealt with frequency-selective surfaces, waveguide discontinuities, and microwave antennas. In 1982, he joined the Centro Studi Propagatione e Antenne (CESPA), Turin, Italy, of the Italian National Research Council (CNR), where he was initially a Researcher and, since 1991, has been a Senior Scientist (Dirigente di Ricerca). He has been Head of the Applied Electromagnetics Section, Istituto di Ricerca sull’Ingeneria delle Telecomunicazioni e dell’Informazione (IRITI), an institute of the CNR. Since September 2002, he has been with the Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni (IEIIT), a newly established institute of the CNR. He has held various teaching positions in the area of electromagnetics with the Politecnico di Torino. His current research activities are in the areas of microwave antennas, dielectric radomes, frequency-selective surfaces, radar cross section, waveguide discontinuities, microwave filters, multiplexers, optical passive devices, and radiometers for astrophysical observations.

Massimo Baralis was born in Turin, Italy, in 1970. He received the Laurea degree in electronics engineering from the Politecnico di Torino, Turin, Italy, in 2001. Since May 2001, he has been with the Electromagnetics Group, Istituto di Ricerca sull’Ingegneria delle Telecomunicazioni e dell’Informazione (IRITI), Italian National Research Council (CNR), Turin, Italy. His research is focused on the application of numerical methods and measurements techniques and concerns the analysis and design of microwave and millimeter-wave passive components, feed systems, and antennas.

Oscar Antonio Peverini was born in Lisbon, Portugal, in 1972. He received the Laurea degree (summa cum laude) in telecommunications engineering and Ph.D. degree in electronics engineering from the Politecnico di Torino, Turin, Italy, in 1997 and 2001, respectively. From August 1999 to March 2000, he was a Visiting Member with the Applied Physics/Integrated Optics Department, University of Paderborn, Paderborn, Germany. In February 2001, he joined the Istituto di Ricerca sull’Ingegneria delle Telecomunicazioni e dell’Informazione (IRITI), an institute of the Italian National Council (CNR). Since December 2001, he has been a Researcher with the Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni (IEIIT), a newly established institute of the CNR. He teaches courses on electromagnetic field theory and applied mathematics at the Politecnico di Torino. His research interests include numerical simulation and design of surface acoustic wave (SAW) waveguides and interdigital transducers (IDTs) for integrated acoustooptical devices, of microwave passive components and radiometers for astrophysical observations, and microwave measurements techniques.

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Augusto Olivieri was born in Courmayeur (AO), Italy, in 1942. He received the Diploma degree in telecommunication from the Istituto A. Avogadro di Torino, Turin, Italy, in 1963. From 1964 to 1967, he was with Poste Telecomunicazioni e Telegrafi (PTT). From 1964 to 1971, he was a Laboratory Technician with the Department of Electronics, Politecnico di Torino. In 1971, he joined the Centro Studi Propagatione e Antenne, Turin (CESPA), Italian National Research Council (CNR). He is currently with the Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni (IEIIT), Turin, Italy. His primary interests cover a range of areas of telecommunication, radio propagation, antennas, measurement of microwave components, and instrumentation for advanced astrophysical observations.

Renato Orta (M’92–SM’99) received the Laurea degree in electronics engineering from the Politecnico di Torino, Turin, Italy, in 1974. Since 1974, he has been a member of the Department of Electronics, Politecnico di Torino, initially as an Assistant Professor, then as an Associate Professor and, since 1999, as a Full Professor. In 1985, he was a Research Fellow with the European Space Research and Technology Center (ESTEC–ESA), Noordwijk, The Netherlands. In 1998, he was a Visiting Professor (CLUSTER Chair) with the Technical University of Eindhoven, Eindhoven, The Netherlands. He currently teaches courses on electromagnetic (EM) field theory and optical components. His research interests include the areas of microwave and optical components, radiation and scattering of EM and elastic waves, and numerical techniques.

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A Compact Bandpass Filter With Two Tuning Transmission Zeros Using a CMRC Resonator Kam Man Shum, Member, IEEE, Ting Ting Mo, Student Member, IEEE, Quan Xue, Senior Member, IEEE, and Chi Hou Chan, Fellow, IEEE

Abstract—This paper presents a new compact, low-insertion loss, sharp-rejection, and narrow-band microstrip bandpass filter (BPF) using a compact microstrip resonant cell (CMRC) as the resonator. Experimental results show that the filter only has 1.3and 1.5-dB insertion losses when using the symmetrical and asymmetrical CMRC resonators, respectively. The effect of varying the length of the resonator to the notch frequencies of the filter has been studied. Furthermore, due to the CMRC characteristic, the size of this filter is only 0 1 by 0 29 . Based on the same design methodology, a mircostrip BPF with a coplanar-waveguide feed is also designed with an increased bandwidth. All the proposed filters have been verified by simulation and measurement with good agreement. These filters are attractive for hybrid microwave integrated circuit and monolithic-microwave integrated-circuit implementation. Index Terms—Bandpass filter (BPF), compact microstrip resonating cell, coplanar waveguide (CPW), microstrip filter.

I. INTRODUCTION

B

ANDPASS FILTER (BPF) is one of the most important components in microwave circuits. To meet the size requirement of modern microwave communications systems, compact microwave BPFs with low insertion loss and low cost are in high demand. Recently, there has been an increasing interest in planar BPFs due to their ease of fabrication. Filters using various planar resonators such as the open loop, miniaturized hairpin, stepped-impedance, quarter-wave, and quasi-quarter-wave resonators have been proposed for either performance improvement or size reduction [1]–[7]. However, all of them have an insertion loss of over 2 dB and some are cascading structures with the disadvantage of increased filter size. To reduce the insertion loss, high-temperature superconductor (HTS) thin-film materials are used in [8], [9]. On the other hand, active devices are used in conjunction with the resonator to provide sufficient gain to compensate for high insertion loss [10], [11]. These methods, however, will certainly increase the cost and circuit complexity. Therefore, there are still ample opportunities for research on designing low-loss and compact BPFs. Moreover, due to the rapid growth of the spectrum utilization and higher receiver sensitivity, filters with a wider

Manuscript received April 1, 2004; revised June 16, 2004. This work was supported by the Hong Kong Research Grant Council under Grant CityU 1237/02E. The authors are with the Wireless Communications Research Center, Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842492

upper stopband may be required to reduce interference. However, many planar filters are designed with a half-wavelength resonator that can generate harmonics by itself. A low-pass or bandstop filter can be cascaded to eliminate this problem, albeit with an increase in circuit size and transmission loss. A compact microstrip resonant cell (CMRC) has been reported in [12]. At the resonant frequency, the cell exhibits a bandstop characteristic, making it a low-pass filter. In [13], we have demonstrated that the same CMRC resonator can be used for the BPF without providing the design methodology. By introducing a direct coupling configuration with the CMRC, the BPF design and the subsequent parameter analyses are provided systematically in this paper. By properly tuning the resonant frequency of the T-shaped patches of the CMRC, a new narrow-band BPF with two transmission zeros and a low insertion loss of only 1.5 dB is developed. The design methodology of the proposed filter will be presented step by step in this paper. The proposed CMRC filter with two poles in the passband and a single transmission zero is given in Section II. Detail analysis of the new filter is presented in Section III. By properly tuning the resonant frequencies of the upper and lower T-shaped patches, a filter having three transmission poles within the passband and two transmission zeros located, respectively, at the low and high rejection bands is introduced and implemented in Section IV. Currently, there is a growing demand for coplanar waveguide (CPW) in microwave integrated circuits (MICs) and monolithic microwave integrated circuits (MMICs). A CPW filter can provides large bandwidth and low loss (e.g., as demonstrated in [14]). To extend the application of the design methodology, an example using the CMRC resonator with the CPW feed to form a BPF is also presented in Section IV. All the presented filters are fabricated on a substrate with a relative dielectric constant of 1.5 mm. of 2.65, a loss tangent of 0.001, and a thickness These new filters are rigorously modeled by IE3D [15]. II. CMRC FILTER A. CMRC Filter Fig. 1 shows the proposed geometry of the CMRC BPF. It with the consists of the input and output coupling line CMRC bandstop filter in the middle of the structure. The width of the structure is identical to that of a 50- microstrip line on the same substrate. The first step in the design is to select an appropriate attenuation notch of the CMRC bandstop filter. By tuning the length of the CMRC and the size of the etched pattern, different bandstop frequencies can be achieved. Coupling lines connected to 50- lines are added on both sides

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Fig. 1. Geometry of the CMRC resonator coupled with input and output couple lines. Dimensions of the filters are W = 12 mm, W = 0:55 mm, W = 0:7 mm, l = 1:1 mm, l = 1:9 mm, l = 25:3 mm, l = l = 3:8 mm, g = 0:2 mm, g = 0:4 mm, and g = g = 0:8 mm.

Fig. 3. Frequency response of the SITTL. (a) Comparison with the CMRC BPF. (b) Parametric study of the SITTL. Fig. 2. filter.

Comparison between the simulated and measured results of the CMRC

of the resonator to enhance the coupling line effect. It should of the couple line is reduced to be noted that if the width , the CMRC filter (direct end-coupled that of the feed line approach) has a high insertion loss. Fig. 2 shows the simulated and measured results of the CMRC BPF. The filter has an asymmetrical frequency response and two poles in the passband. Besides, it has a minimum insertion loss of 1.3 dB and a return loss nearly of 15 dB at 1.7 GHz. The fractional 3-dB bandwidth of this filter is 5%, while the attenuation pole at 1.83 GHz has an attenuation level of 47 dB. The attenuation rate for the upper band is 10 dB/22 MHz. Furthermore, by , where is the the size of this filter is only guided wavelength on this substrate at the center frequency. III. FILTER ANALYSIS A. Stepped Impedance Transformer With Tapped Line (SITTL) As seen from Fig. 1, after removing the upper and lower T-shaped patches in the middle of the CMRC resonator, the remaining structure can be regarded as a stepped-impedance transformer with two coupling lines, which is a series LC resonant circuit. Fig. 3(a) shows the comparison of the simulated frequency responses between the SITTL and CMRC BPFs depicted in Fig. 1. It is found that the resonant frequency of the SITTL only shifts by 1.5% from the frequency of the minimum return loss of the CMRC BPF and both structures have a similar insertion loss. Therefore, the simple SITTL structure can

be used to estimate the center frequency of the filter. Fig. 3(b) against shows the variation of the midband frequency , and while keeping all other didifferent values of , , mensions constant. These parameters are changed one at a time mm, mm, from the reference values of mm, and mm at which the center frequency is defined as . According to Fig. 3(b), the center frequency is mainly controlled by and , but is slightly affected . by and B. T-Shaped Patches From Fig. 3(a), we can conclude that the inclusion of the T-patch resonator provides the transmission zero of the filter. However, the T-shaped patches cannot be analyzed separately, as they are coupled with the H-shaped stepped-impedance transform. Fig. 4(a) shows the comparison of the frequency response of the CMRC BPF of Fig. 1 and the CMRC bandstop filter with the same CMRC resonator. The location of the transmission zero of the CMRC BPF and that of the bandstop filter are very close to each other. Therefore, we can predict the transmission zero from the analysis of the CMRC bandstop filter. Fig. 4(b) against difshows the variation of the null frequency , , , and while keeping ferent values of all other dimensions constant. These parameters are changed mm, one at a time from the reference values of mm, mm, and mm at which the . According to Fig. 4(b), the null frequency is defined as null frequency is mainly controlled by and is slightly

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Fig. 6. Comparison of the frequency response of the conventional CMRC resonator and the new ACMRC resonator.

Fig. 4. Frequency response of the CMRC BPF. (a) Comparison with the CMRC bandstop filter. (b) Parametric study of the T-shaped patches. Fig. 7. Comparison of the simulated frequency response of the ACMRC stopband and ACMRC BPFs. Dimensions of the filters are W = 3:9 mm, W = 12 mm, W = 0:55 mm, W = 0:7 mm, l = 1:1 mm, l = 1:9 mm, l = 25:3 mm, l = 2:9 mm, l = 2:2 mm, g = 0:2 mm, g = 0:6 mm, and g = g = 0:6 mm.

Fig. 5. Simulated frequency response of the transmission zero in the lower band of the CMRC BPF.

affected by , , and . Following the analysis above, the transmission zero of the CMRC BPF shown in Fig. 3 can be mm and mm. The tuned by setting transmission zero will be shifted from the upper to lower stopband, as shown in Fig. 5. IV. ASYMMETRICAL COMPACT MICROSTRIP RESONANT CELL (ACMRC) FILTER A. ACMRC This CMRC BPF is useful for diplexer design, as it has a controllable transmission zero, as well as a fast rolloff. However,

for most applications, BPFs having symmetrical frequency response are preferable. The presented CMRC filter can only provide a single transmission zero. To alleviate this difficulty, an ACMRC is introduced. Results shown in Fig. 4(b) indicate that the T-shaped patch notch frequency is controlled by , , , , and . Hence, the upper and lower T-shaped patches of the CMRC are tuned with different dimensions in order to achieve two different notch frequencies, yielding the ACMRC resonator. The frequency responses of the ACMRC and CMRC bandstop filter are compared in Fig. 6. They have similar responses, except that the ACMRC has one additional null located at the stopband. B. ACMRC BPF By properly tuning the resonant frequencies of the upper and lower T-shaped patches, the ACMRC filter can provide three transmission poles within the passband and two transmission zeros located respective at the low and high rejection bands, as shown in Fig. 7. As observed from the frequency response, the filter has an insertion loss of 1.3 dB and a return loss of 25 dB at the midband frequency of 1.7 GHz. Furthermore, the two transmission zeros are at 1.66 and 1.76 GHz with a 22and 39-dB rejection, respectively. The notches of the frequency

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Fig. 9. Comparison of the simulated and measured results of the ACMRC filter.

Fig. 10.

Simulated and measured results of the ACMRC filter from 1 to 6 GHz.

Fig. 8. Variations of the filter response versus the lengths: (a) l and (b) l .

response are similar to the ACMRC resonator. As mentioned in Section III-B, the notch frequency is mainly controlled by the lengths and . Therefore, the ACMRC filter with different lengths for the upper and for the lower T-shaped patches are studied and the simulation results are shown in Fig. 8(a) and (b). As observed in Fig. 8, increasing the length (or ) results in lowering the notch frequency. Furthermore, if only or is altered, one transmission zero will be changed and the other transmission zero and the center frequency of the filter will remain unaffected. Therefore, the filter bandwidth can be controlled by properly selecting the resonant frequencies of the T-shaped patches. The ACMRC BPF has been fabricated and measured, and the results are illustrated in Fig. 9. This filter has a symmetrical frequency response and three poles in the passband and two transmission zeros are obtained. It has insertion and return losses of 1.5 and 17 dB at the midband frequency of 1.68 GHz, respectively. In addition, this symmetric response filter has a fractional 3-dB bandwidth of 4.8%. The two transmission zeros are located respectively at 1.61 GHz with an attenuation level of 30 dB and 1.75 GHz with an attenuation level of 35 dB. The attenuation rate for both the lower and upper bands is 10 dB/12 MHz. The difference in the fractional filter bandwidths between the simulation and experimental results in Fig. 9 mainly comes from the shift of the transmission zero. It is due to the fabrication tolerances and the assumptions of infinite dielectric and ground plane in the simulation.

Fig. 10 shows the comparison of the simulated and measured filter responses from 1 to 6 GHz. It can be seen that the filter has second and third harmonic rejection of 29 and 17 dB, respectively. As the new filters are designed using the CMRC resonator having sections of thin microstrip line, it is worthwhile to investigate the power-handling capability of the filter. It is observed that the filter can accept 20-W continuous wave (CW) input power, which is sufficient for most applications. C. CPW-Fed BPF A new BPF design using the microstrip structure has been demonstrated in the previous sections. It is suitable for narrowband applications. In order to expand the application of the proposed design methodology, we modify the end-coupling region of the filter and combine it with the CPW as a broadside coupling to increase the filter’s bandwidth. This method paves the way to incorporate the microstrip and CPW structure without any transition or modification of the resonator. The modified CPW-fed BPF is illustrated in Fig. 11. It can be seen that the resonator is etched on the top of the substrate. The CPW feed is connected with a rectangular patch etched on the bottom side of the substrate. The coupling regions are controlled by the upper and lower rectangular patches . The simulated and measured results are shown in Fig. 12. This filter has the same characteristics of the micrsotrip structure, i.e., it has a symmetrical frequency response and three poles in the passband and two transmission zeros. It has insertion and return losses of 1.5 and 21 dB, respectively, at the mid-

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V. CONCLUSION

Fig. 11. Structure of proposed CPW-fed BPF and the parameters are W = 8:9 mm, W = 0:3 mm, W = 0:2 mm, W = 3:6 mm, W = 9:3 mm, g = 0:2 mm, g = 0:77 mm, g = 1 mm, g = 0:3 mm, l = 1:3 mm, l = 17:7 mm, l = 3:7 mm, l = 3:4 mm, and l = 3:8 mm.

In this paper, two types of new microstrip BPFs incorporating the CMRC resonator have been presented. By using the symmetrical CMRC structure, two poles and one transmission zero are obtained. In contrast, using the asymmetrical CMRC structure, three poles and two transmission zeros, as well as a symmetrical frequency response are provided. Both filters have the salient features of compact size, low insertion loss, fast rolloff, and high harmonic rejection. The CMRC and ACMRC BPFs have the insertion losses of 1.3 and 1.5 dB, respective. In addition, the bandwidth of the narrow-band filter can be tuned by properly selecting the resonant frequency of the T-shaped patches. We have also proposed a CPW feed to enhance the bandwidth of the filter for wide-band applications. Filter design guidelines are given, which can be combined with the distributive equivalent circuit to speed up the simulation time. Good agreements between experimental and simulation results have been obtained. REFERENCES

Fig. 12. Comparison between the simulated and measured results of the proposed CPW BPF.

band frequency of 2.45 GHz. In addition, this filter has a fractional 3-dB bandwidth of 9%. The two transmission zeros are at 2.32 GHz with an attenuation level of 27 dB and 2.59 GHz with an attenuation level of 29 dB. The attenuation rate for both the lower and upper bands is 4 dB/10 MHz. D. Design Procedure Here, we summarize the design procedure of the CMRC BPF as follows. 1) Find out the desirable resonant frequency of the SITTL without the presence of the upper and lower T-patches. Generally, the outer width of the filter is tuned to three times the width of the 50- feed line. 2) Tune the transmission zero of the CMRC resonator close to the desirable frequency and combine the CMRC and SITTL to form a BPF. The same dimensions of upper and lower patches should be used at the initial design stage. 3) Tune the identical T-shaped patches together to making the transmission zero to the desired location. 4) Lastly, tune either the lower or upper T-shaped patch to make the resonator as an ACMRC and to create another transmission zero to the opposite side of the passband to form a symmetrical bandpass frequency response. Distributive lumped model similar to [16] can be develop to speed up the simulation to obtain the initial design. Fine tuning of the filter can be done by IE3D.

[1] L. Zhu, P. Wecowski, and K. Wu, “New planar dual filter using cross-slotted patch resonator for simultaneous size and loss reduction,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 5, pp. 650–654, May 1999. [2] J. S. Hong and M. J. Lancaster, “Theory and experiment of novel microstrip slow-wave open-loop resonator filter,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2358–2365, Dec. 1997. [3] S. Y. Lee and C. M. Tsai, “New cross-coupled filter design using improved hairpin resonators,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2482–2490, Dec. 2000. [4] A. Gorur, “A novel dual mode bandpass filter with wide stopband using the properties of microstrip open-loop resonator,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 10, pp. 386–388, Oct. 2002. [5] C. C. Yu and K. Chang, “Novel compact elliptic-function narrow-band bandpass filters using microstrip open-loop resonator with coupled and crossing lines,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 7, pp. 952–957, Jul. 1998. [6] J. S. Hong and M. J. Lancaster, “Bandpass characteristics of new dualmode microstrip square loop resonators,” Electron. Lett., vol. 31, no. 24, pp. 891–892, Nov. 1995. [7] R. W. Jackson and H. C. Hsu, “Multilayer elliptic filter,” Electron. Lett., vol. 37, no. 13, pp. 838–839, Jun. 2001. [8] G. Tsuzuki, M. Suzuki, and N. Sakakibara, “Superconducting filter for IMT-2000 band,” in IEEE MTT-S Int. Microwave Symp. Dig., 2000, pp. 669–672. [9] M. Reppel and H. Chaloupka, “Novel approach for narrowband superconducting filters,” in IEEE MTT-S Int. Microwave Symp. Dig., 1999, pp. 1563–1566. [10] C. Yang and T. Itoh, “A varactor-tuned active microwave band-pass filter,” in IEEE MTT-S Int. Microwave Symp. Dig., 1990, pp. 499–502. [11] A. Griol and J. Marti, “Microstrip multistage coupled ring active bandpass filters with harmonic suppression,” Electron. Lett., vol. 35, no. 7, pp. 575–577, Apr. 1999. [12] Q. Xue, K. M. Shum, and C. H. Chan, “Novel 1-D microstrip PBG cells,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 403–405, Oct. 2000. [13] K. M. Shum, Q. Xue, C. Y. Chiu, and C. H. Chan, “Compact bandpass filter using CMRC,” presented at the Asia–Pacific Microwave Conf., Seoul, Korea, 2003. [14] J. S. Hong and M. J. Lancaster, “Coplanar quarter-wavelength quasielliptic filters without bond-wire bridges,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1150–1156, Apr. 2004. [15] IE3D Manual, Version 10.1, Zeland Software Inc., Fremont, CA, 2003. [16] Q. Xue, K. M. Shum, and C. H. Chan, “Low conversion loss fourthsubharmonic mixers incorporating CMRC for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1449–1454, May 2003.

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Kam Man Shum (M’02) was born in GuangXi, China, in 1974. He received the B.Eng. and M.Phil. degrees in electronic engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 1998 and 2001, respectively, and is currently working toward the Ph.D. degree at the City University of Hong Kong. In 1998, he joined the Wireless Communications Research Center, City University of Hong Kong, where he is currently an Assistant Engineer. His research interests include microwave and millimeter-wave circuits, electromagnetic-bandgap (EBG) structures and their applications, and microstrip antennas.

Quan Xue (M’02–SM’04) received the B.S., M.S., and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1988, 1990, and 1993, respectively. In 1993, he joined the Institute of Applied Physics, University of Electronic Science and Technology of China, as a Lecturer. In 1995, he became an Associate Professor and, in 1997, he became a Professor. From October 1997 to October 1998, he was a Research Associate and then a Research Fellow with the Chinese University of Hong Kong. Since June 1999, he has been with the Wireless Communications Research Center, City University of Hong Kong, where he is currently an Associate Professor. His research interests include microwave circuits and antennas.

Ting Ting Mo (S’04) was born in Shanghai, China, in 1978. She received the B.S. degree in electronic engineering from Shanghai Jiao Tong University, Shanghai, China, in 2001, and is currently working toward the Ph.D. degree at the City University of Hong Kong. She is currently with the Wireless Communications Research Center, City University of Hong Kong. Her research interest is focused on the applications of CPWs in microwave devices and RF circuits.

Chi Hou Chan (S’86–M’86–SM’00–F’02) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1987. Since April 1996, he has been with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. He is currently Dean of the Faculty of Science and Chair Professor of Electronic Engineering. He is also an Adjunct Professor with the University of Electronic Science and Technology of China, Peking University, and Zhejiang University. His research interests include computational electromagnetics, electronic packaging, antennas design, and microwave and millimeter-wave communications systems. Prof. Chan was the recipient of the 1991 U.S. National Science Foundation (NSF) Presidential Young Investigator Award.

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On the Role of the Additive and Converted Noise in the Generation of Phase Noise in Nonlinear Oscillators Jean-Christophe Nallatamby, Michel Prigent, Member, IEEE, and Juan Obregon, Senior Member, IEEE

Abstract—In a conventional approach, the oscillator phase noise due to noise sources near carrier is defined as additive phase noise by assuming that the oscillator operates in a near linear fashion. Nevertheless, fundamentally, an oscillator circuit is inherently nonlinear. In this paper, we show that the phase noise generated by noise sources around the fundamental frequency of oscillation is due to two simultaneous and correlated phenomena of the same order of magnitude: additive phase noise and converted phase noise due to conversion from one sideband to another. An analytical calculation applied to a simple purely theoretical circuit allows evaluation of the respective influence of the two above-mentioned phenomena. Numerical simulations performed on a realistic transistor oscillator circuit then confirms the importance of the conversion phenomenon already shown by the analytical evaluation. The converted noise results to be of the order of 6 dB higher than the additive noise. The term of additive phase noise must be intended to characterize the phase noise generated in linear components located out of the nonlinear oscillation loop and, for example, in buffer amplifiers following the oscillator itself. Index Terms—Additive noise, converted noise, oscillators, phase noise.

I. INTRODUCTION

I

T IS well known that the low-frequency noise sources near dc [1] and white-noise sources around the carrier and its harmonics are the main contributors to the phase noise in an oscillator circuit. With the advent of high-frequency and high-quality bipolar transistors, the phase noise is mainly due to internal white-noise sources at a frequency as close as 10 kHz from the carrier [2]. The interactions from which these white-noise sources generate phase noise in free-running transistor oscillators have scarcely been analyzed and often misunderstood. In a conventional approach [3], [4], the phase noise of an oscillator is defined as additive phase noise by assuming that the oscillator operates in a near linear fashion: each noise source near carrier generating an output noise signal at each own frequency offset. Noise conversion between the lower and upper sidebands near carrier is then neglected.

In this paper, we show that, in oscillators, the phase noise generated by noise sources around the fundamental frequency of oscillation is due to two simultaneous and correlated phenomena of the same order of magnitude: • additive phase noise; • converted phase noise due to conversion from one sideband to another. An analytical calculation applied to a simple purely theoretical circuit allows to evaluate the respective influence of the two phenomena mentioned above. Numerical simulations performed on a realistic transistor oscillator circuit allow to conclude that the phase noise resulting from the conversion phenomenon applied to the noise sources near the carrier is generally more important than the one generated by additive noise, which demonstrates the importance of the conversion phenomenon already shown by the analytical evaluation. This paper is organized as follows. In Section II, we introduce the definition of oscillator phase noise, the noise sources and output signals involved in this phenomenon. In Section III, we define the additive phase noise spectral density alone and then the additive converted phase-noise spectral density due to noise sources near carrier. An analytical calculation on a simple one-port oscillator circuit, detailed in the Appendix, gives a first response on the respective roles of the additive and converted phenomena in the generation of the phase noise. In Section IV, a numerical calculation performed on a realistic example of transistor oscillator including: linear thermal noise sources and cyclostationary shot-noise sources, which allows to confirm the analytic previsions. Finally, conclusions are drawn in Section V. II. OSCILLATOR PHASE-NOISE DEFINITION Phase-noise simulation tools in free-running nonlinear oscillators are well established. These simulation tools rightly consider phase noise to be generated by low-level noise sources. Consequently, the noise sources can be considered as perturbing small signals [2]. A. Main Noise Sources

Manuscript received March 26, 2004; revised June 25, 2004. The authors are with the Institut de Recherche en Communications Optiques et Microondes, Institut Universitaire de Technologie Génie Electrique et Informatique Industrielle, Universite de Limoges, 19100 Brive, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842493

Noise sources may be classified over the frequency range. 1) Low-Frequency Noise Sources Near DC: • Noise sources, called colored noise sources, have a spectral density inversely proportional to . The most signifi-

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cant colored noise sources are (flicker) noise and – noise sources present in the semiconductor devices. • White-noise sources have low-frequency components near dc. These sources are due to diffusion noise, which gives rise to the well-known thermal and shot noise. 2) High-Frequency Noise Sources: In microwave oscillators, near carrier, only the white-noise sources are present. These noise sources feature a constant power spectral density over the harmonics of the fundamental oscillation frequency.

These noise sources generate AM and PM modulations. AM noise always being lower than PM noise [2], we may neglect it for sake of clarity. The output waveform of an oscillator can be then expressed as (6) with (7)

B. Phase-Noise Generation Process With the advent of high-frequency and high-quality bipolar transistors, microwave oscillators present a phase noise that mainly results from the internal white-noise sources located near carrier. 1) Noiseless Output Signal: Fundamentally, an oscillator circuit is inherently nonlinear: the constant oscillation level is due to saturation so that the amplitude reaches a steady-state value and remains practically constant afterwards. A purely noiseless oscillator generates a periodic signal with a specified fundamental frequency

where is a random amplitude, and is a random phase. is also taken as an For numerical simulation purposes, equivalent deterministic quantity by defining its power spectral density as (8) Let us introduce (7) into (6) and expand this expression by assuming, as usual, that the modulation index is small. We obtain the well-known expression

(1) The pure oscillator output signal can be expressed at the fundamental frequency by (2) are nominal amplitude, frequency, and where , , and phase, respectively. 2) Noise Sources and Noisy Output Signal [4]: Any physical circuit generates noise that can be modeled by electrical noise sources. These can later be classically represented by an infinite sum of pseudosinusoidal current components (in a 1-Hz bandwidth) with random amplitudes and phases. They take the generic form (3) For numerical simulation purposes, is taken as an equivalent deterministic quantity by defining the power spectral density of this component as

(9) with , amplitude of the lower noise sideband; , amplitude of the upper noise sideband; , phase of the lower noise sideband; , phase of the upper noise sideband. Finally, let us write

The phase-noise spectral density (8) can be written for numerical simulation purposes as

(4) We will focus primarily on noise sources located at frequencies (lower sideband) and (upper sideband), where is the offset frequency from the oscillation frequency . The oscillator phase noise will be analyzed from that offset. More generally, all the noise sources generating phase noise at an offset away from the carrier will be written as

(5) with

. is the number of harmonics components chosen for the numerical noise simulation.

Note that, in the following, equal to zero, and then

(10) is the phase of reference taken

(11) Expressions (10) and (11) are general expressions allowing to away from the carcalculate the phase noise at an offset rier. These expressions are function of the peak amplitude of and of two noise sidebands: the lower sideband the carrier and upper sideband . They can be applied whatever the phase-noise generation process.

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III. RELATIVE INFLUENCE OF ADDITIVE AND CONVERTED NOISE IN OSCILLATOR PHASE-NOISE GENERATION1 A. Additive Noise [2], [3] In a conventional approach, the additive phase noise of an oscillator is described by assuming that the oscillator operates in a near linear fashion: uncorrelated noise sources located at and generate output noise signals at their own frequency’s offset. Additive phase noise is defined as the sum of the spectral densities of phase-noise modulation components resulting from the addition of each output noise sideband with the carrier signal. Let us apply this noise formalism to a nonlinear oscillator. In this approach, two uncorrelated white-noise sources, located at and , generate output phase-modulated signals by addition with the carrier at . Thus, the conventional calculation of the additive noise assumes the following. generates a single-sideband output noise voltage • at its own frequency . generates a single-sideband output noise voltage • at its own frequency . • Near carrier,

Fig. 1. Simulated oscillator circuit V

= 5 V.

From (11), a straightforward calculation gives

(12) (14)

From the definition of the additive phase noise [3], [4], may be written as (13) where

is the peak amplitude of the output carrier signal.

B. Converted Noise In the previous calculation of the phase noise due to and , the nonlinear conversion around the carrier is neglected and, more generally, the conversion from a noise frequency to another is ignored. Nevertheless, even if all the noise sources, except the two and uncorrelated noise sources near carrier are neglected, it remains that the phase-noise generation process cannot be reduced beyond the following irreducible phenomenon. • generates two correlated signals by addition by conversion. •

generates two correlated signals by addition by conversion.

1Throughout this paper, the subscript term: “near carrier,” applied to the phase noise, refers to the phase noise at offset from oscillation frequency ! only due to the noise sources at ! + and ! near carrier, whereas S indicates the total phase noise at offset from oscillation frequency ! by taking into account all noise sources centered at k!

with k = 0; . . . ; N where N is the number of harmonics chosen for numerical nonlinear oscillator analysis.

0

6

The results obtained using (13) and (14) must be compared to evaluate the real relative influence of the addition and conversion phenomenon near the carrier in a nonlinear oscillator. This evaluation will be done numerically on a bipolar transistor oscillator circuit. Beforehand, however, we used to carry out an analytical evaluation applied to a simple purely theoretical circuit with an active one-port, as described in the Appendix. Calculations carried out are based on the methodology described in [5]. The result (see the Appendix) shows that : in this example, the additive phase noise is lower than the phase noise due to conversion. Note that the converted noise increases with the gain compression of the transistor in oscillation. As a rule: a gain compression of 3 dB is considered as a low value in oscillator circuits. Nevertheless, this gain compression leads to a highly nonlinear amplifier so the converted noise is as important as it appears in simulation. IV. NUMERICAL SIMULATION OF A BIPOLAR TRANSISTOR OSCILLATOR CIRCUIT Fig. 1 shows the analyzed circuit diagram. The transistor model has five nonlinear elements [6]. The osGHz. Our software, whose cillation frequency is fundamentals can be found in [7], allows us to introduce as well noise sources with discrete components described by pseudosinusoids centered at the desired frequency than a whole range of noise sources centred on the whole set of frequencies with , where is the number of harmonics chosen for the numerical nonlinear oscillator analysis. All these noise

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sources can be either correlated or uncorrelated and linear or cyclostationary sources. In order to find out the relative influence of noise sources in the additive/converted phase noise, be they linear or cyclostationary, several numerical simulations are carried out. The steady state of the oscillator circuit is first determined; the phase-noise simulation is then performed. In the following, we detail the different phase-noise simulations. V. However, it must be Results are given for noted that the same conclusions can be deduced over the full range of tuning voltage with practically the same differences , , and between

TABLE I SIMULATIONS RESULTS WITH LINEAR THERMAL NOISE SOURCES

TABLE II SIMULATIONS RESULTS WITH LINEAR THERMAL CYCLOSTATIONARY SHOT-NOISE SOURCES

AND

A. Simulation With Linear Thermal Noise Sources The linear white-noise sources are located across all the linear resistive elements as current noise sources of spectral density Hz

(15)

represents the conductance of the resistive element where across which the noise source is located. 1) Noise Sources Limited to Frequency Components Located Around the Oscillation Frequency: Let us introduce across each resistive element two uncorrelated pseudosinusoidal thermal noise sources. They are located at frequencies and , where is the oscillation frequency; they can be written as

B. Simulation With Linear Thermal Noise Sources and Cyclostationary Shot Noise Sources and are Cyclostationary shot-noise sources now introduced, respectively, across the base–emitter and collector–emitter junctions with the generic form (17) (18) (19) (20)

(16) The resulting phase noise is calculated at the load resistance: 1) due to additive noise, equivalent to (13) and 2) due to additive and converted noise, equivalent to (14). 2) Simulation With All the Uncorrelated Thermal Noise Sources: We then introduce all the uncorrelated thermal noise with ; we choose for sources at the simulation, and the total phase noise due to this whole set of linear thermal noise sources is also calculated. Results obtained: Table I indicates the results obtained away from the carrier, for each for three frequencies , previous phase-noise calculation, namely, , and 3) Important Remarks: Several points have to be noted as follows. • Additive noise alone underestimates the phase noise. The converted noise around the carrier cannot be neglected. and • The comparison between shows that the noise sidebands other than those located at play a role in the generation of the total phase noise. • Complementary simulations show that the white-noise (near dc) generate a phase noise sources located at 24 dB below the final result ( ). It must be noted that this later result is dependent on the circuit architecture and cannot be taken as a general result valid for all architectures

We suppose that and are uncorrelated noise sources. Simulations are performed following the same procedure as for the linear noise sources case, i.e., first by introducing in the simulator, then by introducing (17)-(20) with to (17)-(20) with the whole set of frequencies from . Table II indicates the simulations results obtained for , , and with linear thermal and cyclostationary shot-noise sources present together, for three frequencies away from the carrier. The same remarks as those obtained in the first simulation with linear noise sources alone have to be pointed out for the last three simulations: converted as well as additive noise must be taken into account for a correct evaluation of the phase noise of the oscillator. V. CONCLUSION A detailed analysis on the generation of phase noise in transistor oscillators shows that additive and converted noise phenomena occur at the first order and that none of them can be neglected. The term “additive phase noise” is insufficient to characterize the influence of noise sources around the fundamental frequency of oscillation. It must be intended to characterize the phase noise induced by noise sources in linear devices. Nonlinear phenomena occur in oscillator phase-noise generation, and a simple expression such as the one developed by Leeson,

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• The oscillation conditions (A5)

Phase-Noise Calculation Fig. 2.

Simplified representation of a one-port oscillator.

and improved since [8], can permit to calculate the tendencies provided that it is used with care by applying a rule-of-thumb, taking into account the conversion and cyclostationary noise sources. Nevertheless, an accurate calculation of phase noise in microwave oscillators should include in the simulation all the phase-noise generation processes associated to all the cyclostationary noise sources for an optimized simulation. The simulation shown in this paper can be undertaken on commercial software packages.

In order to calculate the phase noise, we assume that the perturbations due to the noise sources are very small. Two uncorrelated noise sources are considered as follows: at at

(A6)

The nonlinear element can be replaced by its linear periodically time-varying counterpart [4]. It follows a matrix equation, called the conversion matrix, representing the nonlinear element for the small perturbations around the oscillator steady state. The resulting conversion matrix is

APPENDIX Let us consider the following simple equivalent circuit of an oscillator described by a parallel configuration of an active biased one-port and a linear passive circuit made up by an induc. tance , capacitance , and conductance Around the fundamental frequency of oscillation given by

(A7) The admittance of the passive circuit can be written around as follows: (A8) where At

the circuit can be drawn as shown in Fig. 2.

,

and is the distance from the carrier. becomes

Oscillation Conditions Let us suppose that the active biased device by a Van der Pol equation

(A9) is described At

,

becomes

(A1)

(A10)

(A2)

. In the following, we will use Now taking into account the current Kirchoff law for the perturbations, we obtain from (A6) and (A10) the noise voltages at the output load

If the harmonic voltages can be neglected,

The differential conductance of the active device can be written as

(A11) (A3) Introducing (A2) in (A3) and in (A1), and taking into account Kirchoff’s current law, we obtain the following. • The differential conductance (A4) with

(A12) From (13) and (14), we can now calculate the phase noise by setting and noting that the two noise sources are uncorrelated. The additive phase noise and additive converted phase noise can be evaluated by taking into account the following conditions. • • Oscillator is nonlinear: • Near carrier:

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Additive Phase Noise is calculated by From (A11), the noise voltage in the load setting ; then the phase noise is calculated from (13). We obtain dB

(A13)

Additive and Converted Phase Noise From (A11), (A12) and from (11) or (14) of the main text

dB

[7] J. M. Paillot, J. C. Nallatamby, M. Hessane, R. Quere, M. Prigent, and J. Rousset, “A general program for the steady-state, stability and FM noise analysis of microwave oscillators,” in IEEE MTT-S Int. Microwave Symp. Dig., 1990, pp. 1287–1290. [8] J. C. Nallatamby, M. Prigent, M. Camiade, and J. Obregon, “Phase noise in oscillators—Leeson formula revisited,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1386–1394, Apr. 2003.

(A14)

It may be concluded that, in this nonlinear oscillator,

Jean-Christophe Nallatamby received the D.E.A. degree in microwave and optical communications and Ph.D. degree in electronics from the Universite de Limoges, Brive, France, in 1988 and 1992, respectively. He is currently a Lecturer with the Institut Universitaire de Technologie Génie Electrique et Informatique Industrielle, Universite de Limoges. His research interests are nonlinear noise analysis of nonlinear microwave circuits, the design of the low phase noise oscillators, and the noise characterization of microwave devices.

(A15) More detailed calculations leading to (A11) and (A12) can be found in the excellent paper [5]. It is worth noting that the generation of phase noise in an oscillator due to near-carrier noise sources does not need any nonlinear capacitance. Nevertheless the presence of such capacitance enhances the phase-noise level, its role in the conversion process may be of first order, as was clearly demonstrated in [1] REFERENCES [1] H. Siweris and B. Schiek, “Analysis of noise upconversion in microwave FET oscillators,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 3, pp. 233–242, Mar. 1985. [2] J. C. Nallatamby, R. Sommet, M. Prigent, and J. Obregon, “Semiconductor device and noise sources modeling, design methods and tools, oriented to non linear H.F. oscillator CAD,” presented at the SPIE 2nd Int. Fluctuations and Noise Symp., Maspalomas, Canary Islands, May 2004. [3] W. Robins, Phase Noise in Signal Sources. London, U.K.: Perigrinus, 1982. [4] D. Scherer, “Learn about low-noise design,” Microwaves, pp. 116–122, Apr. 1979. [5] K. F. Schüneman and K. Behm, “Nonlinear noise theory for synchronized oscillators,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, pp. 452–458, May 1979. [6] J. P. Fraysse, D. Floriot, P. Auxemery, M. Campovecchio, R. Quere, and J. Obregon, “A nonquasi-static model of GaInP/GaAs HBT for power applications,” in IEEE MTT-S Int. Microwave Symp. Dig., 1997, pp. 379–382.

Michel Prigent (M’93) received the Ph.D. degree from the Universite de Limoges, Brive, France, in 1987. He is currently a Professor with the Universite de Limoges. His field of interest are the design of microwave and millimeter-wave oscillator circuits. He is also involved in characterization and modeling of nonlinear active components (field-effect transistors (FETs), pseudomorphic high electron-mobility transistors (pHEMTs), HBTs, etc.) with a particular emphasis on low-frequency noise measurement and modeling for the use in monolithic microwave integrated circuit (MMIC) computer-aided design (CAD).

Juan Obregon (SM’91) received the E.E. degree from the Conservatoire National des Arts et Métiers (CNAM), Paris, France, in 1967, and the Ph.D. degree from the Universite de Limoges, Brive, France, in 1980. He then joined the Radar Division, Thomson-CSF, where he contributed to the development of parametric amplifiers for radar front-ends. He then joined RTC Laboratories, where he performed experimental and theoretical research on Gunn oscillators. In 1970, he joined the DMH Division, Thomson-CSF, and became a Research Team Manager. In 1981, he was appointed Professor at the Universite de Limoges. He is currently Professor Emeritus with the Universite de Limoges. His fields of interest are the modeling, analysis, and optimization of nonlinear microwave circuits, including noisy networks.

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Two-Port Equivalent of PCB Discontinuities in the Wavelet Domain Rodolfo Araneo, Member, IEEE, Sami Barmada, Member, IEEE, Salvatore Celozzi, Senior Member, IEEE, and Marco Raugi

Abstract—A novel wavelet-based technique is presented for the extraction of two-port equivalents of common printed circuit board (PCB) discontinuities. The wavelet-transformed scattering parameters of the discontinuity can be included in a wavelet equivalent of TEM wave propagation paths along the PCB in order to obtain an overall equivalent of the whole structure under analysis. The wavelet representation minimizes the CPU time and computer storage requirements while maintaining excellent accuracy, thus, it proves to be a very useful modeling tool, especially in the design stage. Index Terms—Multiresolution analysis, printed circuits, wavelet transforms.

I. INTRODUCTION

A

CTUAL configurations of printed circuit boards (PCBs) for high-speed signal transmission are extremely complex and, generally, a full-wave electromagnetic analysis is required since the use of circuit simulators may result in predictions frustrated by unacceptable uncertainties. The frequency range of interest is often in the order of 10 GHz or above and even though some parts of the PCB behave like lumped or quasi-TEM distributed parameter components, very often others, the so-called discontinuities, depart in various ways from this ideal conditions. It is the case, for instance, of vias, bends, and so forth; even the presence of variations in close proximity (e.g., slit in the ground plane) may considerably affect the signal propagation. In the past, especially in dealing with band-limited signals, simple lumped equivalent circuits have been proposed accounting for the presence of the discontinuity [1]–[5] and this approach has its own validity and is still the subject of research and improvements. SPICE-like circuit simulators are widely used in academic and industrial research centers for their versatility, especially when a time-domain analysis is needed. Actually, equivalent circuits of realistic structures are very complex, although suitable for a direct implementation in a SPICE model. Since the original simplicity and intuitiveness of equivalent circuits have been lost, different equivalents, which can be easily implemented in popular computational Manuscript received March 26, 2004; revised July 12, 2004 and September 29, 2004. This work was supported in part by the Italian Ministry of University under a Program for the Development of Research of National Interest PRIN Grant 2002093437. R. Araneo and S. Celozzi are with the Department of Electrical Engineering, University of Rome “La Sapienza,” 00184 Rome, Italy. S. Barmada and M. Raugi are with the Department of Electrical Systems and Automation, University of Pisa, 56126 Pisa, Italy. Digital Object Identifier 10.1109/TMTT.2004.842496

environments and can compete with SPICE in terms of CPU time and precision, can be proposed as effective alternatives. A well-known alternative method, on which a great deal of research has focused, e.g., [6] and [7], proposes the characterization of general discontinuities by means of the scattering matrix computed in the frequency domain. Knowing the transverse field pattern of the particular mode of interest, the scattering matrix is extracted from time-domain simulations via Fourier transform and, if necessary, inserted in a circuit simulator. The method, widely applied to waveguide discontinuities, has been proven to be versatile and accurate in many applications. However, when a time-domain analysis is necessary, it can be used only at the cost of an inverse Fourier analysis with difficulties in dealing with possible nonlinearities. In this paper, the attempt of representing the discontinuity behavior through a transfer function in the wavelet domain is presented. The resulting equivalent representation can easily include lumped equivalent circuits, thus representing a self-standing alternative of SPICE-like equivalent-circuit simulators. Wavelet expansion (WE) and MRA are becoming a popular tool in the microwave area, and recent research has demonstrated that the wavelet approach can be conveniently applied to a wide class of problems. In particular, wavelets can be used as two-dimensional bases for the description of the spatial behavior of electromagnetic quantities, as in the multiresolution time-domain method (MRTD) [8], [9] or they can be used as a matrix compression tool in a method of moments (MoM) solution [10]. In the approach presented in this paper, the wavelet basis is used in the time domain as a tool for the identification of PCB discontinuities. A discontinuity is represented by its own input–output characteristic without the approximations inherent in any equivalent-circuit representation. Another advantage is represented by the efficiency of such a wavelet approach, which retains many characteristics of the real behavior in reasonable and compact sets of data. II. COMBINED WAVELET-FULL-WAVE ANALYSIS A. Identification of PCB Discontinuities in the Wavelet Domain Recently, WE techniques have been used for the numerical solution of multiconductor transmission lines (TLs) [11], and they have been demonstrated as being powerful tools, especially in terms of simulation efficiency when fast transients are analyzed. In [11], the quasi-TEM model of a general nonuniform TL is discretized in elementary cells and conveniently solved

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Fig. 1. Discretization of the PCB in the wavelet domain.

in the wavelet domain. The result is then inverse transformed, giving the unknown quantities in the time domain. A similar procedure may be applied when arbitrary discontinuities representing areas of departure from quasi-TEM propagation conditions are encountered along the signal path. The model of the discontinuity as a two-port equivalent in the wavelet domain can be easily included in a quasi-TEM model representing the remaining parts of the microstrip line. The whole PCB can be modeled in the wavelet domain as a cascade of cells where the discontinuity is one of the cells, as depicted in Fig. 1. As shown in [11], the relation between the physical quantities at the ports of an elementary cell are represented by a (sparse) matrix of constant coefficients; on the same principle, WE is used here as a tool through which the identification of the PCB discontinuity is performed in the time domain. The final result is a low-dimensional sparse matrix of constant coefficients representing the electromagnetic behavior of the discontinuity. Let us consider the discontinuity as a two-port system, represented by its scattering parameters, whose input and output , , , and . Given quantities are, respectively, , the WE of each a wavelet basis of the above quantities is an -dimensional vector, i.e., , , and so forth, where , , , and are vectors of coefficients in terms of wavelet basis. The relationship between input and output power waves can be written in the wavelet domain as a matrix–vector product as follows: (1) For simplicity, the above matrix will be indicated as , the input vector as , the output vector as , and (1) can be rewritten in the wavelet domain as (2) where the dimension of the matrix is , while the length of vectors and is . In applications where the parameters of the system are is known, while in this case, the matrix known, matrix in the wavelet domain has to be determined. The determination of , i.e., the identification of the PCB discontinuity, is then performed by the use of the three-dimensional (3-D) full-wave results in the following three steps. Step 1) The th column of is equal to (in the wavelet domain) when . is inverse transformed in the time domain and Step 2) used as input for the full-wave model. Step 3) Output just obtained is transformed in the wavelet domain yielding vector corresponding to the th column of the matrix .

This scheme is performed for all the columns following the standard -parameters determination, i.e., determining the four sub. This procedure gives the complete representation matrices of the two-port equivalent in the wavelet domain for a given basis dimension. Finite difference time domain (FDTD) has been used for the full-wave simulations, as reported in Section II-B. is actuIt has to be noted that the inverse transform of ally the th element of the chosen wavelet basis. It could seem that an overwhelming and cumbersome computational effort is necessary to apply the above-illustrated identification procedure requires one insince each column of the wavelet operator dependent and separated FDTD run. On the contrary, it is easy to note that the number of 3-D FDTD simulations is not equal . to, but drastically less than From the multiresolution analysis (MRA) theory [12]–[16], it is known that each subspace is characterized by a basis of functions that differ from each other by a translation factor (this is true for all but the border functions). Hence, in case of a timeinvariant system, a single function of a subspace can be used to determine the output for all the functions of the subspace. This simple consideration implies a strong reduction of the number of simulations required for the characterization. There are several wavelet bases available in the literature, and in this study, we use Daubechies wavelets on the interval. The reason of this choice is that they are very effective in representing signals such as fast transients (as widely addressed in the literature) and they are divided into families according to the number of vanishing moments, i.e., the degree of the polynomials they can represent with no error. The choice between wavelets on the real line and wavelets on the interval is driven by the fact that we are solving a time-bounded problem. vanishing Daubechies wavelets on the interval with moments have been adopted for this problem. Daubechies wavelets on the interval are characterized by having, for each subspace of the MRA, NV (where NV is the number of vanishing moments) functions defined on the left border and NV on the right border of the interval . Furthermore, the dimension of the basis characterizing the coarser resolution is also related to the number of vanishing moments NV. In the case under analysis, the lower resolution is characterized by means of 32 functions. Higher resolutions mean higher CPU times (and, in this case, also a higher number of 3-D simulations in order to characterize the system) but, obviously, a higher accuracy. Based on the great number of tests performed, a good compromise between accuracy and CPU time is achieved with bases of or functions. Looking at a basis of 128 functions in detail, it can be observed that it has a structure like , i.e., is composed of three subspaces, respectively, characterized by 32 scaling functions at resolution 5, 32 wavelets at resolution 5, and 64 wavelets at resolution 6. Among those 128 functions, the only independent ones for each subspace are the 12 functions (equal to 2NV) on the border and the central function (the one that is translated in order to form the basis). Hence, referring to the basis of dimension 128, the independent functions to . In the same be considered are way, for the basis composed by 256 functions, four different

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subspaces are obtained and the independent simulations are . only A further reduction of the number of the full-wave runs can be ; achieved when the input signals are equal to zero at time as a matter of fact, in these cases, it is not necessary to evaluate the response of the system to the left border basis functions, which characterize the signals for which . Hence, the number of runs is reduced, e.g., in the case of dimension 128, to . Since the system is reciprocal and symmetric, only runs of the 3-D FDTD code are necessary to determine the four submatrices of the matrix . The following three steps summarize the identification procedure. Step 1) Create the input signals, i.e., the wavelet basis functions. simulations in order to determine Step 2) Perform the the transmitted and reflected waves. Step 3) Perform WE of the signals obtained at the previous step and cast the results in matrix . Once the matrix chain has been determined, in order to compute the output at any input signal, the following three steps must be performed. Step 1) WE of the time domain input signal. Step 2) Solution of the system, i.e., product of matrices in the wavelet domain. Step 3) Inverse WE to determine the output in the time domain. It is important to point out that, once the wavelet operator is obtained with a reasonable computational effort, an extremely efficient and powerful representation is available, which enables the analysis of the propagation of any time-domain signal by means of a simple desktop computer to invert a sparse square matrix of leading dimension (much less than one second, typically, compared with around 1 min required by SPICE and generally from 10 min up to more than 1 h for an accurate FDTD run, all performed on the same computer). This can be useful, for instance, in optimization procedures where a huge number of simulations have to be carried out. The numerical implementation of the procedure for the extraction of the matrix has been implemented in MATLAB. B. Adopted Full-Wave Model In order to account for the non-TEM nature of the traveling waves in proximity of a generic discontinuity occurring along a microstrip line, a full-wave 3-D FDTD code has been adopted due to its well-known capability of modeling complex wave interaction problems in inhomogeneous regions [17]. The FDTD method has been widely used in the past to tackle the problem of the characterization of discontinuities [18]. Initially [2] its accuracy has been not excellent since voltages and currents on the feeding TLs have been considered as superposition of TEM waves even in the proximity of the discontinuity. Hereafter, much research has focused [6] on the -parameter extraction for any possible propagating or evanescent mode, extending the definition of modal voltages and currents along with the concept of incident and reflected waves. According to the above-mentioned procedure, the FDTD method has been applied to calculate the time signals reflected

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by and transmitted through the overall structure, composed by the discontinuity and the feeding microstrip lines in a broad-band frequency range. The FDTD method considered here is aimed at isolating only the effect of the discontinuity, meanwhile minimizing the physical and numerical reflections due to either radiation boundary conditions or mismatch of the terminal impedances. Its meaningful characteristics can be summarized as follows. 1) The ground plane and signal strips have been considered perfectly conductive and no loss tangent has been assigned to the dielectric substrate (lossless dielectric), which is characterized by only the relative dielectric constant. This choice has been made in order to simulate only the effect of the lumped discontinuity on the propagation of the dominant mode since possible dielectric losses, due to the dispersion of the substrate, or conductive ones, due to the finite conductivity of the planes, can be inserted and well modeled in the TL model of the feeding lines [1], [19]. 2) The perfectly conductive ground plane of the PCB constitutes the bottom boundary of the lattice grid so as to minimize the dimensions of the computational domain. 3) A complex frequency-shifted perfectly matched layer (CFS PML) has been used to terminate the computational domain [20] and the PML layers have been placed adherent to the substrate so as to minimize reflections and fringing effects around the edges due to the finite dimensions of the ground/power plane. 4) Since the enclosed medium is inhomogeneous, an effective relative permittivity estimated via quasi-static theory [21] has been used to compute the optimal value of the maximum conductivity of the PML [20, eq. (10)]. 5) A lumped voltage generator and a resistor have been used, respectively, to excite and close the structure. The value of the internal resistance of the generator along with the resistor has been set equal to the characteristic impedance of the microstrip line, which has been computed means of accurate analytical expressions [21] under quasi-TEM assumption. A Gaussian pulse with frequency content from 100 MHz up to 15 GHz has been used to drive the source. 6) The voltage and current time signals obtained from the 3-D FDTD simulation at the voltage generator and the resistor are used to compute the time trend of the scattering power waves [22], which are then transformed in the wavelet domain according to the proposed procedure. Hence, the method is intrinsically based on the assumption that the dominant mode traveling along the line is quasi-TEM. Although this assumption is well posed due to the fact that the transversal dimensions of the printed circuits are much smaller than the wavelength in the dielectric at the frequencies of interest, extensions to superior order modes are possible at the cost of introducing modal voltages and currents [6] and more sophisticated techniques for exiting the finite-difference (FD) scheme [23]. In this way, it is possible to isolate, and hereafter to model, only the effects of the discontinuity under analysis, reducing the numerical errors, which could affect the overall accuracy of the equivalent extraction methodology, while speeding up the effi-

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ciency of the simulation. Besides, the method allows a reliable comparison between the full-wave results and the results computed with either the wavelet representation or with the classical approach of the equivalent-circuit model by which the system behavior is modeled in terms of a network composed by the connection of equivalent-circuit elements, lumped or distributed, which can be easily solved by means of circuit simulators. This latter approach is obviously prone to the accuracy achieved in the extraction of the equivalent circuit. All the equivalents obtained in the wavelet domain, either two-port or multiports, can interact only through their terminal ports, as the most of the equivalents that can be computed in the frequency and time domain with different procedures. Attempts to consider coupling between neighboring elements can be found in [7] under the assumption that they must be small compared to the wavelength and to the distance between them. Anyway, the proposed method ends with the computation of a new global equivalent, thus, it does not bring substantial changes into the theoretical background. If two different discontinuities are placed so close to each other that the coupling between them takes place in several modes, radiation effects and the excitement of higher order modes, and not only through the fundamental TEM mode, it should be better to compute one whole equivalent. In the utmost case that no single lumped discontinuity can be isolated, which is not the purpose of this paper, then the usage of a full-wave approach, such as the partial-element equivalent circuit (PEEC), which computes a more complex and larger network, may be the correct choice. C. Increasing the Accuracy of the Model In order to perform an efficient WE, the number of time samples of the function to be expanded is equal to the dimension of the basis, i.e., equal to the number of coefficients, which are always a power of two. Hence, when the vector is inverse transformed, a time signal, corresponding to the th function of the basis and represamples, is obtained. It should be underlined that sented by Daubechies wavelets are defined by a recursive algorithm (the dilation equation) so that their representation in terms of a power of two number of samples does not lead to loss of information, but it is implicit in their nature. The FDTD model requires, in general, a much higher number of time steps in order to converge to the solution; usually . The exciting signal (an element of the basis) must samples; this can be obtained by also be characterized by of samperforming the inverse WE using a number ples with . time The output signal of the FDTD run (characterized by samples in order samples) then require a down-sampling to to be able to perform the WE with a basis of dimension . This down-sampling of the output signals obviously leads to a loss of information, which is then imported in the matrix representing the input-output behavior of the discontinuity. The easiest way to overcome this problem would be to use higher resolution basis to have a number of time samples closer to , but this approach suffers from two main drawbacks: it would lead to a higher dimension of the basis and, hence, to a higher number of simulations to calculate the

; subspaces of higher resolutions columns of the matrix are characterized by wavelet functions with higher frequency content, which could require a too fine and dense grid to be accurately simulated in the FDTD scheme. For the above-mentioned reasons, a different strategy has been utilized: one or more additional subspaces of wavelet functions are added to the basis, e.g., lengthening the basis from 128 to 256 functions. This new basis is then fully utilized in the representation of the input and output signals, i.e., the WE of the input and output signals is now performed by the use of this new basis and, thus, the down-sampling of the FDTD results is characterized by a lower loss of information (the time samples is double in the new basis from number of 128 to 256). In the same way, the WE of the input signals is performed on the same higher dimension basis, leading to a better representation of the signal itself. On the other hand, the additional functions are not taken into to account in the identification process, then the new matrix be used for the 256 functions basis is derived from the FDTD runs already obtained for the 128 functions basis, and no additional FDTD runs are required. In this way, the time required to is the same as before. obtain the new wavelet matrix This technique of increasing the basis dimension must be performed when the original basis is already capable of accurately characterizing the discontinuity behavior. For example, beginning from a very low-resolution basis (for instance, composed by only 32 functions, requiring 13 FDTD runs) and implementing the above-described procedure up to the desired number of time samples, will not, of course, give satisfactory results. A quantitative analysis of this assertion will be performed in Section II-E, where guidelines for the choice of the wavelet basis are also given. D. Inclusion of the Two-Port Equivalent Into a Quasi-TEM TL Model In order to cascade-connect several two-port circuits using scattering parameters, it is necessary to use the so-called scattering transfer parameters, which can be obtained from the scattering parameters by the use of the following relations: (3) and are matrices of wavelet In our case, the quantities coefficients, but relation (3) can be used straightforwardly. The quasi-TEM TL model in the wavelet domain (for timedomain WE) has been previously obtained in [11] and it is based on the definition of the differential operator in the wavelet domain for the Daubechies wavelets on the interval. The details of its development can be also found in [11]. Here, we recall only the basic principle of its usage: once a wavelet basis of dimension is chosen, the differential operator is represented . by a sparse matrix of constant entries of dimension Differentiation of a function is then performed in the wavelet domain by a simple matrix–vector product. In fact, if , and and are, respectively, the WE of the funcand in the wavelet domain, it yields . tions This shows that the operator can be also used as a symbolic operator in the sinusoidal analysis. operator as the

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For example, the matrix for a lossless TL, characterized by a length and distributed inductance and capacitance per unit length, is in the frequency domain (4) As demonstrated in [11], the corresponding matrix in the wavelet domain is simply obtained by replacing the operator with the operator , which yields (5) . where the four blocks have dimension Hence, a cascade between a lossless TL and the discontinuity is represented in the wavelet domain in terms of scattering . transfer parameters by the product

Fig. 2.

Frequency content of scaling and wavelet functions.

E. Choice of the Basis Dimension In order to better understand the relation between the accuracy of the results and the dimension of the chosen wavelet basis, a study of the frequency content of the basis is performed. In this study, the authors follow an intuitive approach to define the guidelines for the choice of the dimension of the basis in order to obtain accurate results. As stated before, the choice of the Daubechies wavelets on the interval leads to a basis that can be divided in the following subsets: 32 scaling functions, 32 wavelets, 64 wavelets, 128 wavelets, and so forth. Considering a linear system, the frequency content of the output signal is related to the frequency content of the excitation; operating the proposed identification procedure, any input and output signal is represented as a linear combination of the wavelet basis functions; hence, the model is characterized in the frequency range of the chosen wavelet basis. For these reasons, the choice of the basis is related to its capability to properly represent only the input signals in terms of their frequency content. This trend has been verified in several simulations and some results are reported below. In Fig. 2, the normalized spectra of four bases functions are reported on the same graph. Only the central functions have been analyzed since the functions on the border are strictly related to the central ones, and their frequency content is coincident with a good approximation. III. NUMERICAL SIMULATIONS To illustrate meaningfully the advantages and possibilities of the wavelet analysis, the right-angle bend discontinuity is initially investigated because of its simplicity and the availability of various modeling techniques. A comprehensive comparison of the simulated results with measurements has been conducted to assess the accuracy of the proposed methodology. Another application regarding the more challenging problem of the characterization of the discontinuity represented by a via through a hole in a ground plane is then presented.

Fig. 3. Right-angle-bend geometry in a microstrip line.

A comparison between the results obtained by means of the proposed method, a full-wave model, and equivalent circuits is reported. Due to the impossibility of implementing all the equivalent circuits available in literature, three typical configurations have been considered. The first one represents simple circuits with intuitive configurations, while the second and third ones represent complicated circuits and have been chosen because they have shown good results as equivalents of PCB discontinuities [4], [5], [24]–[26]. The simple circuit has been used in the first application where it maintains some validity, while for the second complex test, it is necessary to adopt higher order circuits to have meaningful results. The first equivalent circuit has been derived under quasi-TEM approximation: this assumption is well posed if the frequency components carried by the signal are not too high so that the dimensions of the discontinuities are much smaller than the minimum wavelength in the dielectric, as it is proven by the appreciable amount of work present in literature focused on the extraction of the capacitance and inductance of lumped discontinuities. The higher order circuits computed with the fitting procedure described in [4] and [5] are derived

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Fig. 4. Equivalent SPICE circuit of the right-angle-bend discontinuity.

Fig. 6. Second right-angle-bend geometry with its representation in the wavelet domain.

Fig. 5. Time-domain waveforms of the: (a) transmitted power waves and (b) reflected ones in the case of a modulated Gaussian voltage source.

only assuming that the dominant mode is quasi-TEM, but all the nontransverse and radiation effects in the neighborhood of the discontinuity are simulated and accounted for. When dealing with very high-order circuits, necessary to accurately approximate the computed or measured transfer functions, the use of a model order reduction [27], [28] algorithm becomes quite an imperative step in order to reduce the discontinuity model to a minimum number of poles. Besides, complex procedures, often iterative, are required to ensure the passivity and the causality of the new reduced model. It is easy to conclude that, in general, the computational effort required by the equivalent-circuit parameter-extraction procedure is remarkably longer than that needed by the proposed wavelet approach.

Fig. 7. Time-domain waveforms of the: (a) transmitted power waves and (b) reflected ones in the case of a modulated Gaussian voltage source.

A. Validation of the Identification Process for a Right-Angle-Bend Discontinuity The geometric configuration is depicted in Fig. 3: the bend if formed up by two microstrip lines of equal characteristic

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Fig. 8. (a) Configuration and (b) picture of the PCB used for the measurements.

impedance . The structure is driven by a voltage generator with internal resistance equal to 50 and is terminated on an equal resistive load . The results of the whole structure simulated with the wavelet representation are compared with the full-wave results obtained via the FDTD code and the results computed by means of the equivalent-circuit model. The simple equivalent circuit, which is shown in Fig. 4, is considered in this test. The circuit is synthesized under quasi-TEM assumptions and quasi-static formulations have been employed to evaluate the lumped parameters and the primary distributed constants of the network. Closed-form expressions have been used to compute the characteristic impedance and the effective dielectric constant [10], which are necessary to characterize the wave propagation along the strips, modeled as lossless uniform TLs with a delay time ps. The lumped inductances nH and capacitance fF of the bend are evaluated by means of the approximate expressions reported in [9] and [11]. At first, a modulated Gaussian pulse with frequency content from 100 MHz up to 15 GHz is considered to drive the PCB structure. Fig. 5 show the comparison among the time domain reflected and transmitted power waves computed with

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Fig. 9. Time-domain waveforms of the: (a) reflected power waves and (b) transmitted ones in the case of a modulated Gaussian incident pulse.

the proposed wavelet approach, the full-wave FDTD method, and the equivalent circuit inserted into SPICE. It is possible to note the good agreement between the full-wave data and results of the proposed approach for both the reflected and transmitted waves. On the contrary, remarkable discrepancies can be observed in the SPICE results, especially as concerns the waves reflected on the feeding port. This is due to the low accuracy of the equivalent circuit, which is entirely based on an approximated quasi-static formulation. Anyway, from an engineering point-of-view, the simplicity of the equivalent circuits remains convenient if only the transmission channel has to be modeled. Adopting the higher order equivalent circuit [5], [6], obtained with a comparable computational effort, the equivalent-circuit results have similar precision of the wavelet model. The results are not reported here for the sake of conciseness. B. Inclusion of the Discontinuity Model Into a Quasi-TEM Model To highlight the advantages and smoothness of the wavelet approach, which rise from the fact that the obtained equivalent can be efficiently included into a quasi-TEM TL model, thus allowing to simulate any complex system, the previous right-angle-bend

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Fig. 10. (a) Via discontinuity configuration and proposed equivalent lumped circuits of the via under analysis. (b) Equivalent circuit #1 [12], [13]. (c) Equivalent circuit #2 [14], [15].

configuration has been simulated a second time, doubling the length of the feeding arms, as shown in Fig. 6. In the wavelet domain, the whole structure is modeled as a cascade of cells, as reported in Fig. 6. The matrix represents the discontinuity formed up by the two previous striplines of length mm, and it is the same as the previous case since the geometrical parameters have been not changed. The additional length of the microstrip lines is simulated as a lossless TL whose wavelet-domain matrix is analytically available. The resulting global matrix is, hence, TWT. In the case of SPICE, the same equivalent circuit reported in Fig. 4 is used, where the time delay along the two feeding striplines is now twice the previous one. Fig. 7 show the transmitted and reflected power waves in the case of the aforementioned modulated Gaussian pulse, computed with the wavelet approach and the lumped circuit equivalent and checked against the full-wave results. Good agreement between the full-wave data and the wavelet results, generally better than that with SPICE, especially as regards the reflected power waves, corroborates the suitability of the proposed approach and proves the possibility of modeling complex systems once the equivalent wavelet-domain matrix of every cell is known. C. Validation of the Methodology Through Comparison With Measurements Fig. 8 shows the board configuration and measurement setup. The chosen board consists of 70-mm-wide square laminate,

with a low dielectric-loss FR-4 core, suitable for high-frequency performance, with a standard thickness of 1.6 mm and an electrodeposited copper cladding of 35 m on both sides. The dielectric material shows a relative permittivity, which is quite constant around 4.5 over a wide frequency range and is very stable versus temperature. Furthermore, a very low dissipation factor of 0.0023 is shown at 10 GHz, 23 C. The feeding microstrips, 3-mm wide and 50-mm long, are designed to exhibit a 50- characteristic impedance constant in the frequency range from 1 up to 8 GHz, where only the fundamental TEM mode is traveling. The measurements have been conducted by using the two-port vector network analyzer (VNA) Anritsu model 37369C, which is designed to make accurate -parameter measurement across the 40-MHz–40-GHz range. The line-reflect-match (LRM) calibration method has been applied to remove besides the classical systematic errors such as signal leakage, impedance mismatches, and frequency response of the cables and connectors, the effects of the physical transition necessary to connect the microstrip lines to the coaxial cables. In this way, the magnitude and phase of the scattering parameters in the frequency domain have been measured in the 8 : 1 range from 1 up to 8 GHz by using 1601 points and an averaging of 1%. In a post-processing session, the time waveforms have then been computed from the measured data. Considering an incident modulated Gaussian pulse with the same frequency content, the reflected and transmitted power waves have been computed through the following three steps: Step 1) Discrete Fourier transform of the incident signal.

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Fig. 11. Time-domain waveforms of the: (a) transmitted power waves and (b) reflected ones in the case of a modulated Gaussian voltage source.

Step 2) Multiplication with the forward or reverse measured scattering parameter. Step 3) Inverse Fourier transform. The results are reported in Fig. 9, which shows the comparison of the reflected and transmitted power waves computed by simulating the whole structure with the wavelet representation, as described in Section III-B, versus the full-wave results obtained via the FDTD code and measurements. The fairly good agreement in terms of both magnitude and delay time is a good indication of the validity of the proposed procedure. D. Validation of the Identification Process for a Via-Hole Discontinuity The analyzed configuration is drawn in Fig. 10(a), and the dimensions are as follows. • mm. mm. • mm. • mm. •

Fig. 12. Time-domain waveforms of the: (a) transmitted power waves and (b) reflected ones in the case of a trapezoidal voltage source.

• mm. mm. • . • As already noted, it is now necessary to adopt higher order equivalent circuits shown in Fig. 10(b) [5], [6] and Fig. 10(c) [25], [26]. Fig. 11 show the comparison between the time-domain reand transmitted power waves computed with flected the proposed wavelet approach, the full-wave FDTD method, and the equivalent lumped circuit inserted into SPICE, when the structure is driven by the modulated Gaussian pulse. It is possible to note the excellent agreement between the full-wave data and the results of the proposed approach in terms of both absolute values and time delays due to the fact that a number of 512 wavelet coefficients derived from a basis of 128 wavelets following the procedure of Section II-C has been used. Besides, the SPICE results are in good agreement with the previous ones

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Fig. 13.

Reflected signal of an incident Gaussian pulse.

Fig. 14.

Reflected signal of an incident modulated Gaussian pulse.

since the high-order lumped equivalent circuit is able to account for all the distributed effects in the neighborhood of the discontinuity. It should be noted that the first equivalent circuit is more accurate than second one since it has been derived specifically in the frequency range from 1 to 10 GHz, i.e., it can be thought of as a wide range approximation, while the latter is founded on quasi-static approximations. This inefficiency of the lumped circuits clearly appears in all its importance when the structure is driven with the trapezoidal pulse. It is easy to verify that the frequency content ranges from dc up to around 50 GHz. Fig. 12 show the comparison between and transmitted the computed time domain reflected power waves. Once again, the accuracy of the proposed method, based on 512 wavelet coefficients derived from a basis of 128 wavelets following the procedure of Section II-C, is good for either the transmitted and reflected wave. On the contrary, SPICE equivalent circuit #1 clearly shows its limits in the late time behavior of the waveforms, which is to say the low-frequency behavior of the discontinuity up to approximately 1 GHz, which is

Fig. 15.

Reflected signal of an incident trapezoidal pulse.

Fig. 16. Frequency content of Gaussian (1), modulated Gaussian (2), and trapezoidal pulses (3) compared with the frequency content of the previous scaling and wavelet functions.

the lower limit of the lumped circuit at low frequencies. Besides, lumped circuit #2 shows a stable behavior in the late times, but its accuracy is quite coarse, especially in the modeling of the reflected signal. E. Choice of the Basis In Figs. 13–16, some simulation results for the previously introduced via-hole model are reported; the chosen excitations are: 1) a pure Gaussian pulse; 2) a modulated Gaussian pulse; and 3) a trapezoidal pulse, in ascending order of their frequency content. These simulations have been performed using bases with different dimensions, respectively, by using: 1) the first 32 functions; 2) the first 64 functions (32 scaling 32 wavelets); and 3) the first 128 functions (32 scaling 32 wavelets 64 wavelets).

ARANEO et al.: TWO-PORT EQUIVALENT OF PCB DISCONTINUITIES IN WAVELET DOMAIN

For the sake of conciseness, only the reflected signal is reported since the differences between the bases are more evident in this case than in the transmitted signal. 1) Gaussian Pulse: It can be seen that the three bases give results that are indistinguishable so even the lower dimension (resolution) basis is capable of achieving the desired accuracy. 2) Modulated Gaussian Pulse: As can be seen, the results are accurate and coincident for the basis of 64 and 128 functions, while they differ for basis of 32 functions. 3) Trapezoidal Pulse: It can be seen that the results now differ among the three bases; the lower dimension one (32 functions) is now much less accurate, and comparing these figures with that of the same via-hole above shows that the middle resolution basis is more accurate than the low resolution one, but less than the full basis. It is evident that, in order to characterize the system in a wider frequency band, a higher dimension basis is required. The frequency range of interest can be chosen by analyzing the exciting signals. In Fig. 16, the spectra of the Gaussian, modulated Gaussian, and trapezoidal pulses are added. It is now evident that the frequency content of the Gaussian is included in one of the first 32 scaling functions, hence, the response to this pulse is accurate with this low-dimensional basis, as obtained before in the simulations. Vice versa the modulated Gaussian has a higher frequency content, hence, it is necessary to also include the first 32 wavelet functions. The same remark is valid for the trapezoidal pulse, but now the third block of functions is also necessary to have a good accuracy, as also shown in the previous simulations. The frequency analysis of the wavelets basis, related to the accuracy of the proposed method, shows that the choice of the dimension of the wavelet basis must be performed by analyzing the frequency content of the input signals of the systems. IV. CONCLUSIONS An efficient procedure, based on a combined 3-D FDTD and wavelets analysis, has been proposed for the analysis of PCB interconnects with discontinuities. The procedure allows the extraction of the transfer matrix describing the discontinuity behavior in the wavelet domain starting from full-wave results. In this way, an efficient analysis can be conducted by means of the wavelet representation, when the same discontinuity has to be simulated in different configurations or under different excitations, without resorting to other computationally cumbersome 3-D simulations. The procedure has demonstrated being satisfactorily sound and reliable through different configurations whose results have been validated by means of comparisons with measured data, full-wave results, and SPICE results obtained by simulating the discontinuity with equivalent lumped circuits available in literature. REFERENCES [1] K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines. Norwood, MA: Artech House, 1996. [2] M. A. Schamberger, S. Kosanovich, and R. Mittra, “Parameter extraction and correction for transmission lines and discontinuities using the finitedifference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 6, pp. 919–925, Jun. 1996.

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[3] T. Mangold and P. Russer, “Full-wave modeling and automatic equivalent-circuit generation of millimeter-wave planar and multilayer structures,” IEEE Trans. Microw. Theory Tech, vol. 47, no. 6, pp. 851–858, Jun. 1999. [4] R. Araneo and S. Celozzi, “Extraction of equivalent lumped circuits of discontinuities using the finite-difference time-domain method,” in IEEE Int. Electromagnetic Compatibility Symp., Minneapolis, MN, Aug. 2002, pp. 119–122. , “A general procedure for the extraction of lumped equivalent cir[5] cuits from full-wave electromagnetic simulations of interconnect discontinuities,” in Proc. 15th Int. Zurich Symp., Zurich, Switzerland, Feb. 2003, pp. 419–424. [6] W. K. Gwarek and M. C. Marcysiak, “Wide-band S -parameter extraction from FD-TD simulations for propagating and evanescent modes in inhomogeneous guides,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 8, pp. 1920–1928, Aug. 2003. [7] B. L. A. Van Thielen and G. A. E. Vandenbosch, “Method for the calculation of mutual coupling between discontinuities in planar circuits,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 155–164, Jan. 2002. [8] E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “Multiresolution time-domain (MRTD) adaptive schemes using arbitrary resolutions of wavelets,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 2, pp. 501–516, Feb. 2002. [9] T. Dogaru and L. Carin, “Scattering analysis by the multiresolution timedomain method using compactly supported wavelets systems,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1752–1760, Jul. 2002. [10] E. Y. Shifman and Y. Leviatan, “Analysis of transient interaction of electromagnetic pulse with an air layer in a dielectric medium using wavelet-based implicit TDIE formulation,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 2018–2022, Aug. 2002. [11] S. Barmada and M. Raugi, “Transient numerical solution of nonuniform MTL equations with nonlinear loads by wavelet expansion in time or space domain,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 47, no. 8, pp. 1178–1190, Aug. 2000. [12] S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 7, pp. 674–693, Jul. 1989. [13] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992. [14] G. Kaiser, A Friendly Guide to Wavelets. Boston, MA: Birkhauser, 1999. [15] A. Cohen, I. Daubechies, and P. Vial, “Wavelets on the interval and fast wavelet transforms,” Appl. Comput. Harmon. Anal., vol. 1, no. 1, pp. 54–81, Dec. 1993. [16] S. G. Mallat, A Wavelet Tour of Signal Processing. New York: Academic, 1998. [17] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 2000. [18] R. Araneo, C. Wang, X. Gu, J. Drewniak, and S. Celozzi, “Efficient modeling of discontinuities and dispersive media in printed transmission lines,” IEEE Trans. Magn., vol. 38, no. 3, pp. 765–768, Mar. 2002. [19] R. Araneo, F. Maradei, and S. Celozzi, “Digital signal transmission thru differential interconnects: Full-wave vs. SPICE modeling,” in IEEE Int. Electromagnetic Compatibility Symp., Boston, MA, Aug. 2003, pp. 855–858. [20] J. A. Roden and S. Gedney, “An efficient FDTD implementation of the PML with CFS in general media,” in IEEE Antennas and Propagation Society Int. Symp., vol. 3, Jul. 2000, pp. 1362–1365. [21] E. Hammerstad and O. Jensen, “Accurate models for microstrip computer-aided design,” in IEEE MTT-S Int. Microwave Symp. Dig., May 1980, pp. 407–409. [22] K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 3, pp. 194–202, Mar. 1965. [23] S. Van den Berghe and D. De Zutter, “Efficient FDTD S-parameter calculation of microwave structures with TEM ports,” in IEEE Antennas Propagation Society Symp., vol. 2, Jul. 11–16, 1999, pp. 1078–1081. [24] T. K. Sarkar, Z. A. Marícevic´ , J. B. Zhang, and A. R. Djordjevic´ , “Evaluation of excess inductance and capacitance of microstrip junctions,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 6, pp. 1095–1097, Jun. 1994. [25] T. Wang, R. F. Harrington, and J. R. Mautz, “Quasi-static analysis of a microstrip via through a hole in a ground plane,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 6, pp. 1008–1013, Jun. 1988. [26] P. Kok and D. De Zutter, “Capacitance of a circular symmetric model of a via hole including finite ground plane thickness,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1229–1234, Jul. 1991.

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[27] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: Passive reducedorder interconnect macromodeling algorithm,” IEEE Trans. ComputerAided Design Integr. Circuits Syst., no. 8, pp. 645–654, Aug. 1998. [28] J. R. Phillips, L. Daniel, and L. M. Silveira, “Guaranteed passive balancing transformations for model order reduction,” IEEE Trans. Computer-Aided Design Integr. Circuits Syst., no. 8, pp. 1027–1041, Aug. 2003.

Rodolfo Araneo (M’03) was born in Rome, Italy, on October 29, 1975. He received the M.S. (cum laude) and Ph.D. degrees in electrical engineering from the University of Rome “La Sapienza, ” Rome, Italy, in 1999 and 2002, respectively. In 1999, he was a Visiting Student with the National Institute of Standards and Technology (NIST) Boulder, CO, where he was involved with TEM cells and shielding. During the spring semester of 2000, he was a Visiting Researcher with the Department of Electrical and Computer Engineering, University of Missouri–Rolla (UMR), where he was involved with PCBs and FDTD techniques. His research is mainly in the field of electromagnetic compatibility (EMC) and includes numerical and analytical techniques for modeling high-speed PCBs, shielding, and TL analysis. Dr. Araneo was the recipient of the 1999 Past President’s Memorial Award presented by the IEEE EMC Society.

Sami Barmada (M’01) was born in Livorno, Italy, on November 18, 1970. He received the Master and Ph.D. degrees in electrical engineering from the University of Pisa, Pisa, Italy, in 1995 and 2001, respectively. From 1995 to 1997, he was with ABB Teknologi, Oslo, Norway, where he was involved with distribution network analysis and optimization. He is currently an Assistant Professor with the Department of Electrical System and Automation, University of Pisa, where he is involved with numerical computation of electromagnetic fields, particularly on the modeling of multiconductor transmission lines (MTLs) and to the application of the WE to computational electromagnetics.

Salvatore Celozzi (M’92–SM’97) is currently a Full Professor of electrotechnics with the University of Rome “La Sapienza,” Rome, Italy. He has authored or coauthored over 90 papers in international journals or conferences, mainly in the fields of printed circuits, electromagnetic shielding, and TL theory. Prof. Celozzi was the founder of the EMC Chapter of the IEEE Central and South Italy Section in 1997. He served as the associate editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY from 1996 to 2000. He was vice-chair of the International Symposium EMC EUROPE 2002. He was the recipient of the 2002 IEEE EMC Society Award Certificate of Technical Achievement and several Best Paper Awards presented at international conferences.

Marco Raugi received the Electronic Engineering degree and Ph.D. degree in electrical engineering from the University of Pisa, Pisa, Italy, in 1985, and 1990, respectively. He is currently a Full Professor of electrical engineering with the Department of Electrical Systems and Automation, University of Pisa. His research concerns numerical methods for the analysis of electromagnetic fields in linear and nonlinear media. His main applications have been devoted to nondestructive testing, electromagnetic compatibility in TLs, and electromagnetic launchers. He has authored or coauthored approximately 100 papers in international journals and conferences. He has served as chairman for various Editorial Boards. Dr. Raugi was the general chairman of the Progress In Electromagnetic Research Symposium (PIERS), Pisa, Italy, 2004. He has served as chairman and session organizer of international conferences. He was the recipient of the 2002 IEEE Industry Application Society Melcher Prize Paper Award.

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Small-Signal and High-Frequency Noise Modeling of SiGe HBTs Umut Basaran, Student Member, IEEE, Nikolai Wieser, Gernot Feiler, and Manfred Berroth, Senior Member, IEEE

Abstract—An improved and complete method for the smallsignal and high-frequency noise modeling of SiGe HBTs in a BiCMOS process is presented. A comprehensive survey of the suggested parameter-extraction methodology for the SiGe HBT transistor model in conjunction with the analytically derived equations is given. The suggested transistor model is compatible with BiCMOS processes and takes into account the parasitic effects like the extrinsic capacitances and substrate effect. The accuracy of the proposed transistor model is verified by on-wafer -parameter measurements up to 40 GHz. An important proof of the accuracy of the proposed parameter-extraction methodology is the presented physical noise model that can accurately predict the measured noise parameters up to our measurement frequency limit of 18 GHz. The noise model accuracy at various temperatures and biases is examined. Finally, the effect of the on-wafer contact pads on the noise performance of the transistor is investigated in detail. Index Terms—BiCMOS, high-frequency noise modeling, noise parameters, parameter extraction, SiGe HBT.

I. INTRODUCTION

D

UE TO the continuously increasing transistor operating frequencies and the advances in the SiGe BiCMOS technology, bipolar transistor models have undergone a tremendous evolution in recent years. In most cases, the Gummel–Poon model is insufficient for today’s modern and fast bipolar transistors. The advanced transistor models [1], [2] can characterize the transistor operation in a large bias and frequency range at the cost of more complicated extraction methods and measurement effort due to a larger number of unknowns of the transistor equivalent circuit. Many of the straightforward parameter-extraction methods applicable to III–V HBTs may yield frustrating results for SiGe HBTs. This is generally due to the lossy behavior of the substrate requiring more accurate deembedding procedures for the contact pads with additional test structures and due to a variety of inner parasitic effects like the extrinsic capacitances and substrate effect. Hence, the analytical equations derived for the parameter extraction of the III–V HBT transistor models lose their validity for SiGe HBTs. The suggested parameter-extraction flow will be discussed in Section II. This paper is organized as follows. In Section II, the suggested parameter-extraction methods based on the transistor Manuscript received April 19, 2004; revised July 13, 2004. The authors are with the Institute of Electrical and Optical Communication Engineering, University of Stuttgart, D-70569 Stuttgart, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842490

Fig. 1. Small-signal equivalent circuit of the SiGe HBT under the forwardactive operation. The total base resistance R consists of an intrinsic and an extrinsic part (R = R + R ). The emitter is connected to the ground.

equivalent circuit in Fig. 1 are discussed in detail. In Section III, a physics-based noise model is introduced and the noise model accuracy is evaluated by a comparison of the simulated and measured noise parameters for three transistor emitter lengths. Finally, the effect of temperature variation and on-wafer contact pads on the noise performance of the transistors is discussed. II. DIRECT PARAMETER EXTRACTION The device-under-test is an SiGe HBT from AMI Semiconductor Belgium, Oudenaarde, Belgium, with an emitter length of 9.4 m in the common-emitter configuration with a grounded substrate contact. The transistor was measured using a Cascade semiautomatic on-wafer probe station, an Agilent 4155 semiconductor parameter analyzer to obtain the dc characteristics, and an HP 8510 network analyzer to obtain the -parameters up to 40 GHz. Since the bipolar transistor performance is strongly dependent on the temperature, the temperature was held constant at 298 K with a temperature control unit. This enables one to have consistent measurement results throughout the dc, -parameter, and noise parameter measurements. The measured -parameters have been deembedded with open and short test structures to take into account the parallel and series contact pad parasitics. This will be discussed in Section III.

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Fig. 2. Extraction of the bias-independent series resistances. (a) Frequency dependence of the series resistances. (b) Base current dependence of Re(Z Z ). (d) Emitter current dependence of Re(Z ). (c) Collector current dependence of Re(Z

0

A. Emitter, Base, and Collector Resistances The base–emitter diode junction of an npn transistor is forward biased under the forward-active operation in contrast to the depleted base–collector diode where the electrons are drifted from the base to the collector via the electric field. In the saturation mode of the transistor, the base–collector diode is also forward biased and the electrons injected from the emitter flow to the base node due to the absence of the electric field at the V, as the base–emitter base–collector junction. For voltage is increased, the base–emitter and base–collector diodes become gradually short circuited and the linearized small-signal and base–collector diode resistances fall base–emitter off, linearly simplifying the transistor equivalent circuit in Fig. 1 to a conventional T-model for sufficiently large base–emitter voltages [3]. Hence, the T-model consists of the bias-independent series resistances due to the polysilicon contacts n buried collector layer and also series inductances due to the metallization layers remained at the top of the transistor. The series resistances of the transistor are then simply calculated by (1) (2) (3) . The right-hand sides of (1)–(3) have where a flat frequency response, as illustrated in Fig. 2(a), verifying

0Z

).

the validity of the T-model under the saturation operation of the transistor. The linearized small-signal base–collector and base–emitter diode resistances fall off linearly with increasing collector and emitter currents, as shown in Fig. 2(c) and (d), respectively. The -axis interception for infinite large currents yields the bias-independent emitter and collector series resis, , tances. In the same manner, and yield the emitter, base, and collector inductances arising from the metallization at the top of the transistor. However, the effect of these remaining parasitic inductances were found to be small, up to 40 GHz, after deembedding the transistor with open and short test structures. Therefore, they will be neglected throughout this study. Note that is a diffusion current and has a negative sign in the saturation mode. consists of an extrinsic and intrinsic The base resistance part. Although is treated as a single resistance in the smallsignal transistor model in Fig. 1, the intrinsic base-resistance beneath the emitter is bias dependent and subject to base width, base-conductivity modulation, base push-out, and emitter current crowding [4]. Due to the interplay of these several effects, the bias dependence of the base resistance is a complicated phenomenon. All these effects cause a decrease of the intrinsic base resistance with increasing base current due to an increase in the base conductivity and the base cross-sectional area. For suffiV, the right-hand side of ciently large base currents and (2) seems to approach a constant value, which is assigned to the extrinsic base resistance as shown in Fig. 2(b).

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number of unknowns. The transistor equivalent circuit can be substantially simplified by operating the transistor under the cutoff operation where both junctions are depleted. Provided , , and can be calcuthat lated by [6] (8) (9)

Fig. 3. Simplified cross section of a vertical bipolar transistor with basic parasitic capacitances noted.

The substrate–collector depletion capacitance can be expressed as (10)

B. Base Resistance and Substrate Network In the forward-active mode of the transistor, however, the total base resistance cannot be simply extracted from (2). Especially the extrinsic base–collector and base–emitter capacitances start to influence the high-frequency characteristics of the transistor and prevent a straightforward extraction of the base resistance from the -parameters (see Fig. 3). A simplified cross section of the bipolar transistor is shown in Fig. 3. The extrinsic capacitances do not have any contribution to the real part of the . For simplicity, is calculated from the input admittance -model of the transistor [5]

where is the zero-bias capacitance, is the built-in pois the grading coefficient of the substrate–coltential, and lector depletion capacitance. At high frequencies where the prestarts to viously made assumption can be directly lose its validity, the substrate capacitance extracted by

(11) (4) where and . Note of the -model is not equal that the base–emitter resistance [6]. and are given by to

To avoid any confusions and emphasize the bias dependence of , the substrate–collector capacitance is denoted by a biasin (11). For today’s modern bipolar dependent term transistors, the value of can be comparable with that of and, therefore, the validity of (8), (9), and (11) can be further from the -parameter matrix improved by subtracting

(5) (12) and (6) is the common-base current gain, is the commonwhere is the ideality factor. From (4), at emitter current gain, and high frequencies, is calculated by (7) versus gives The -axis intercept of according to (7). Since the bias-independent emitter resistance is already known from the previous steps, it can be subtracted from the -axis intercept value yielding only . The selection of a proper equivalent circuit that takes into account all the relevant physical effects of the substrate is a prerequisite for its broad-band characterization [7]. The main difficulties arise from the insufficient information obtained from the -parameter and dc measurement data to determine the substrate network elements. The transistor equivalent circuit given in Fig. 1 describes the transistor operation under the forward-active operation where an accurate direct extraction of the substrate network elements is not straightforward due to a large

where denotes the -parameters deembedded using open and short test structures under the cutoff operation of the transistor. C. Base–Emitter and Base–Collector Capacitances In the equivalent circuit of the transistor shown in Fig. 1, both base–collector and base–emitter capacitances are divided into an extrinsic and intrinsic part. This accounts in a first approximation for the distributed nature of the base–emitter and base–collector capacitances over the base resistance and the bias-independent extrinsic capacitances resulting from the overlap between the emitter polysilicon, extrinsic base polysilicon, and n collector, as illustrated in the device cross section in Fig. 3. The transistor operation can be characterized with the cutoff equivalent V and the base voltages below the circuit in Fig. 4 for built-in potential of both junctions. The total base–collector and base–emitter junction capacitances can be calculated by (13) and (14)

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Fig. 4. Cutoff equivalent circuit for the extraction of the base–collector and base–emitter capacitances.

respectively. Under the assumption that the bias dependence of the extrinsic capacitances and is negligible, and are given by

Fig. 5. Extraction of the intrinsic and extrinsic base–emitter capacitance by means of a linear regression.

(15)

and (16)

where and are the zero-bias capacitances, and are the built-in potentials, and and are the grading coefficients of the base–collector and base–emitter depletion capacitances, respectively. To limit the number of unknowns, the and are assumed to be bias inextrinsic capacitances dependent in (15) and (16). Several methods to determine the extrinsic and intrinsic caand pacitances directly from the real parts of [8], [9] have been suggested in the recent years. Although these methods are based on a correct theoretical analysis, one major drawback of these methods is the requirement of very accurate measurement results to obtain physically correct values. This condition is difficult to realize, especially if the real part of is very low and can be noisy due to the limited accuracy of the measurement setup. In this paper, an iteration method [10] has been applied to extract the extrinsic and intrinsic base–collector and base–emitter capacitances after calculating the logarithm of both sides of (15) and (16). The resulting linear equation for the base–emitter junction can be written as

Fig. 6. Measured (triangles) and calculated (dashed line) junction parameters. C is 5.6 fF.

C

and extracted

Under the forward-active operation of the transistor, the base–emitter junction is forward biased and governed by the diffusion capacitance term. The base–collector junction is depleted to shift the minority carriers in the base by an electric field to the collector. The total transit time from the emitter to the collector [11] is given by (18) is the ideality factor and is the sum of the biaswhere independent transit times. The equations used in [11] do not take . Therefore, a correction of the measurement into account data by means of the -parameters is necessary as follows:

(17)

(19)

where , , and have been extracted by the extraction tool so that (17) yields a linear equation whose -axis intercept is equal to zero (see Fig. 5). The CV characteristics of with extracted junction parameters , , and are illustrated in Fig. 6. Also, the same optimization procedure has been applied to the base–collector junction to extract the extrinsic and intrinsic base–collector capacitances.

where denotes the corrected -parameter matrix. According to (18), the transit time decreases linearly with increasing collector current at low collector current densities. At high collector current densities, however, the Kirk effect [4] causes a widening of the intrinsic base region, which results in a drastic increase of the base and the total transit time, as shown in Fig. 7. At low current densities, the base–emitter capacitance can be assumed

BASARAN et al.: SMALL-SIGNAL AND HIGH-FREQUENCY NOISE MODELING OF SiGe HBTs

Fig. 8. Fig. 7. Transit time versus collector current. The calculated value of the base–emitter capacitance from the slope is 83 fF.

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Measured (crosses) and calculated (solid line) magnitude of .

TABLE I EXTRACTED BIAS-INDEPENDENT TRANSISTOR EQUIVALENT-CIRCUIT PARAMETERS

Fig. 9. Measured (crosses) and calculated (solid line)  .

TABLE II EXTRACTED BIAS-DEPENDENT TRANSISTOR EQUIVALENT-CIRCUIT PARAMETERS (V = 1:5 V, I = 1:1 mA)

D. Transport Factor The transport factor

is given by (20)

and has low-pass characteristics with the 3-dB cutoff frequency . is a time delay associated with the base–collector depletion region and base transit times [12]. From (20), the magnitude of in the low-frequency range yields , as depicted in Fig. 8. can be derived from the -parameters [12], [13] (21)

to be constant [11], which is justified by the linear decrease of the transit time with increasing collector current. The sum of and is obtained at low collector current densities from the slope of the versus plot in Fig. 7. The extracted extrinsic and intrinsic base–collector capacitances for a collector current of 1.1 mA and a collector voltage of 1.5 V is given in Tables I and II.

. Since with the previously determined collector resistance the transistor equivalent circuit used in [10] does not include , (21) becomes valid after applying (19). According to (20), is responsible for the fall off of , as shown in Fig. 8. can be calculated as Hence, the 3-dB cutoff frequency (22)

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Fig. 10. Measured (triangles) and simulated (solid lines) S -parameters of the SiGe HBT with an emitter length of 9.4 m up to 40 GHz. The operating point was selected for a typical low-noise operation of the transistor (V = 1:5 V, I = 1:1 mA, and I = 6 A).

Fig. 11.

High-frequency noise parameter (26 GHz) and S -parameter (40 GHz) measurement setup in an isolated chamber.

Having calculated

,

can be calculated by (23)

is the phase of , as shown in Fig. 9. where In the forward-active operation of the transistor, the base–collector junction is reverse biased. Therefore, the linearized smallis usually too high and is signal base–collector resistance difficult to determine directly from the -parameters. Having already determined the rest of the transistor model parameters can be obfrom the previous steps, an accurate value of tained by a final optimization procedure. This method yields, in most cases, satisfactory results, especially if the percentage error between the simulated and measured imaginary part of , which is governed by the RC constant formed by and , is constrained to values below 5%.

The extracted bias-independent and bias-dependent transistor equivalent circuit parameters are given in Tables I and II, respectively. The measured and simulated -parameters are shown in Fig. 10. For all the -parameters, the error is below 10%. III. HIGH-FREQUENCY NOISE MODELING Noise parameter measurements have been carried out with an in-house developed measurement system (see Fig. 11) to investigate the accuracy of the noise model. The high-frequency noise model including the thermal and shot noise sources is illustrated in Fig. 12. The model accuracy can be improved up to the transit frequency by taking into account the noise transit time [14], which can be defined as the time delay of the electrons that reach the collector after being injected from the emitter and

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Fig. 12.

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High-frequency noise model with thermal and shot noise sources.

Fig. 13. Measured and simulated: (a) minimum noise figure, (b) magnitude, (c) phase of the optimum noise reflection factor, and (d) normalized noise resistance of the transistor with an emitter length of L = 9:4 m up to 18 GHz (V = 1:5 V, I = 1:1 mA).

has, in fact, the same value of the base transport factor time delay in (20). The phase shift associated with the noise transit time, however, is negligible within our measurement frequency range of 2–18 GHz. This gives rise to the assumption (see Fig. 12), which is verified by the good agreement between the simulated and measured noise parameters up to 18 GHz, as

shown in Figs. 13 and 14. The highest deviation of the simulated minimum noise figure from the directly measured one is less than 0.2 dB, which lies within the measurement equipment inaccuracy range (0.1–0.3 dB). The power loss due to the RF coupling to the substrate and the thermal noise generated by the resistive substrate can severely alter the noise performance of

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Fig. 14. Measured and simulated: (a) minimum noise figure, (b) magnitude, (c) phase of the optimum noise reflection factor, and (d) normalized noise resistance of the transistor at 5 GHz for three different transistor emitter lengths.

Fig. 15. Simplified layout of the transistor and equivalent circuit of the contact pads used to simulate the embedded noise parameters. All resistances generate thermal noise.

the intrinsic transistor. Neglecting the contact pad parasitics may lead to a misleading information about the transistor noise characteristics even if a ground shield is used under the contact pads to isolate the signal from the lossy silicon substrate. The transistor noise model depicted in Fig. 12 has been extended by the equivalent circuit of the contact pads (see Fig. 15) using open and short test structures. This enables one to simulate the overall noise characteristics of the transistors including the contact pads. The intrinsic transistor noise parameters can be obtained by removing the equivalent circuit of the contact pads from the transistor noise model. The metallization layers of the test structures should be identical to those of the contact pads and the interconnection lines of the transistors.

represents the capacitance due to the oxide layer between the top metal level and the substrate. and represent the resistivity and permittivity of the substrate, respectively. The series emitter resistance and inductance due to the contact pad have been neglected. The results in Fig. 13 indicate that the contact pads give rise to a minimum noise figure increase of 1.2 dB at 5 GHz and 2 dB at 18 GHz. The extent of the noise-figure degradation becomes even larger at higher frequencies due to an increase of the RF coupling to the lossy substrate. Noise-parameter measurements and simulations have been performed for three transistor emitter lengths, as shown in Fig. 14. The transistors are biased at a constant collector current density of 0.1 mA m . Fig. 14 shows that the directly

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measured minimum noise figure decreases with increasing emitter length. In contrast to the embedded noise parameters, the deembedded minimum noise figure is less sensitive to the differs from changes in the emitter length. Since due to the contact pad, the directly measured does not correspond to the intrinsic optimum noise reflection coefficient and the noise matching of the transistor is, therefore, degraded. The noise figure of a transistor for an arbitrary source reflection coefficient is given by the well-known formula [15] (24) where is the reflection coefficient seen by the transistor. Equation (24) indicates that the extent of the noise-figure degradation . This becomes more critical for transisdepends on and tors with smaller emitter lengths since they feature larger and , as can be seen in Fig. 14(b) and (d). The measured noise parameters have been deembedded by using the noise correlation matrices [16] besides modeling the contact pads. The intrinsic -parameters of the transistors are calculated as [17]

Fig. 16. Noise figure versus temperature plot for a constant collector current of 1 mA. The transistor emitter length is 9.4 m. The equivalent circuit of the contact pads has been included in the transistor noise model.

(25) (26) (27) where corresponds to the intrinsic -parameters of the transistors. The intrinsic -parameters can be obtained by using the electrical transformation matrices of two-ports. The deembedded noise parameters of the transistor are calculated as (28) (29) (30) (31) (32) where is the Boltzmann constant and is the ambient temperis the -representation of the deembedded ature. transistor noise correlation matrix and can be transformed into the chain representation by using the transformation matrices of noisy two-ports [18]. The deembedding procedure with the noise correlation matrices should be handled with great care, as this method requires very accurate measurement results. Noise-figure measurement errors even in the range of a measurement equipment inaccuracy range (0.1–0.3 dB) may lead to large deviations from the simulation results after deembedding the noise parameters with open and short test structures. This is confirmed by the measurement and simulation results plotted in Fig. 13, where the directly measured noise parameters are in good agreement with the simulation results in contrast to the deembedded noise parameters with the noise correlation matrices. These measurement inaccuracies are attributed to the calibrated excess noise ratio (ENR) uncertainty of the noise source, limited dynamic range of the noise figure meter and the mismatch between input tuner and the transistor at large magnitudes of the reflection coefficient seen by the transistor. This

Fig. 17. Noise figure versus temperature plot for a constant base voltage of 0.83 V. The transistor emitter length is 9.4 m. The equivalent circuit of the contact pads has been included in the transistor noise model.

leads to errors arising from the least square fitting algorithm to solve the noise-parameter equations [19]. These sources of error cannot be eliminated, but minimized by a careful calibration of the measurement system. Fig. 16 shows the directly measured and simulated noise figure in a temperature range of 273–343 K. The simulated noise figure has been obtained from the transistor noise model including the equivalent circuit of the contact pads. The source impedance is 50 and the collector current is held constant at 1 mA. The noise figure increases with temperature as a result of increasing thermal noise arising from the series resistances of the transistor and decreasing gain. In contrast to Fig. 16, the noise figure shown in Fig. 17 decreases with increasing temperature due to the insufficient gain at low temperatures if the base supply voltage is held constant at 0.83 V. The thermal noise contribution of each resistor in the transistor noise model in, , , and in the equivalent circuit of cluding the pads is given by the mean quadratic value of the noise current. is the ambient temperature and has been varied by a temperature control unit between 273–343 K during the measurements. The transistor noise model shows a good agreement with the measurement results, as shown in Figs. 16 and 17.

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IV. CONCLUSION Small-signal and high-frequency noise-modeling methods for SiGe HBTs have been presented. The transistor equivalent circuit is compatible with BiCMOS processes and takes into account the inner parasitics like the RF coupling to the substrate and the extrinsic base–emitter and base–collector capacitances. The consideration of these parasitic effects is a prerequisite for obtaining the physically correct values of the transistor model parameters like the base resistance, intrinsic junction capacitances, and transport factor. The high-frequency noise-modeling accuracy of the proposed physics-based transistor noise model for three transistor emitter lengths has been investigated. Having employed the proposed parameter-extraction procedure, the presented high-frequency noise model proves to be accurate at various temperatures and biases provided that the change in the noise characteristics of the transistor due to the contact pad parasitics is accurately taken into account. ACKNOWLEDGMENT The authors would like to thank E. Gerhardt and W. Graudszus, both with Alcatel Stuttgart, Stuttgart, Germany, for fruitful discussions. The authors are also indebted to R. Gillon and E. Vestiel, both of AMI Semiconductor Belgium, Oudenaarde, Belgium, for constructive support of this research. REFERENCES [1] W. J. Kloosterman, J. A. M. Geelen, and D. B. M. Klaassen, “Efficient parameter extraction for the MEXTRAM model,” in Proc. IEEE Bipolar/BiCMOS Circuits Technology Meeting, 1995, pp. 70–73. [2] M. Schroeter, S. Lehmann, H. Jiang, and S. Komarow, “HICUM/ Level0—A simplified compact bipolar transistor model,” in Proc. IEEE Bipolar/BiCMOS Circuits Technology Meeting, 2002, pp. 112–115. [3] Y. Gobert, P. J. Tasker, and K. H. Bachem, “A physical yet simple smallsignal equivalent circuit for the heterojunction bipolar transistor,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 1, pp. 149–153, Jan. 1997. [4] J. S. Yuan, SiGe, GaAs and InP Heterojunction Bipolar Transistors. New York: Wiley, 1999. [5] U. Basaran and M. Berroth, “An accurate method to determine the substrate network elements and base resistance,” in Proc. IEEE Bipolar/BiCMOS Circuits Technology Meeting, 2003, pp. 93–96. [6] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits. New York: Wiley, 2001. [7] M. Pfost, H. M. Rein, and T. Holzwarth, “Modeling substrate effects in the design of high-speed Si-bipolar IC’s,” IEEE J. Solid-State Circuits, vol. 31, no. 10, pp. 1493–1501, Oct. 1996. [8] B. Ardouin, T. Zimmer, H. Mnif, and P. Fouillat, “Direct method for bipolar base–emitter and base–collector capacitance splitting using high frequency measurements,” in Proc. IEEE Bipolar/BiCMOS Circuits Technology Meeting, 2001, pp. 114–117. [9] D. Berger, N. Gambetta, D. Celi, and C. Dufaza, “Extraction of the base– collector capacitance splitting along the base resistance,” in Proc. IEEE Bipolar/BiCMOS Circuits Technology Meeting, 2000, pp. 180–183. [10] B. Li, S. Prasad, L. Yang, and S. C. Wang, “A semianalytical parameterextraction procedure for HBT equivalent circuit,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 10, pp. 1427–1435, Oct. 1998. [11] K. Aufinger, J. Boeck, T. F. Meister, and J. Popp, “Noise characteristics of transistors fabricated in an advanced silicon bipolar technology,” IEEE Trans. Electron Devices, vol. 43, no. 9, pp. 1533–1538, Sep. 1996. [12] A. P. Laser and D. L. Pulfrey, “Reconciliation of methods for estimating f for microwave heterojunction transistors,” IEEE Trans. Electron Devices, vol. 38, no. 8, pp. 1685–1692, Aug. 1991. [13] B. Li and S. Prasad, “Basic expressions and approximations in smallsignal parameter extraction for HBTs,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 5, pp. 534–539, Oct. 1999. [14] G. Niu, J. D. Cressler, S. Zhang, W. E. Ansley, C. S. Webster, and D. L. Harame, “A unified approach to RF and microwave noise parameter modeling in bipolar transistors,” IEEE Trans. Electron Devices, vol. 48, no. 11, pp. 2568–2574, Nov. 2001.

[15] H. A. Haus, W. R. Atkinson, W. H. Fonger, W. W. Mcleod, G. M. Branch, W. A. Harris, E. K. Stodola, W. B. Davenport, Jr., S. W. Harrison, and T. E. Talpey, “Representation of noise in linear two ports,” Proc. IRE, vol. 48, no. 1, pp. 69–74, Jan. 1960. [16] G. Knoblinger, “RF-noise of deep-submicron MOSFETs: Extraction and modeling,” presented at the Proc. IEEE Eur. Solid-State Device Research Conf., 2001. [17] M. C. A. M. Koolen, J. A. M. Geelen, and M. P. J. G. Versleijen, “An improved de-embedding technique for on-wafer high frequency characterization,” in Proc. IEEE Bipolar/BiCMOS Circuits Technology Meeting, 1991, pp. 188–191. [18] H. Hillbrand and P. Russer, “An efficient method for computer aided noise analysis of linear amplifier networks,” IEEE Trans. Circuits Syst., vol. CAS-23, no. 4, pp. 235–238, Apr. 1976. [19] J. M. O’Callaghan and J. P. Mondal, “A vector approach for noise parameter fitting and selection of source admittances,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 8, pp. 1376–1382, Aug. 1991.

Umut Basaran (S’04) was born in Malatya, Turkey, in 1976. He received the B.S. degree in electrical engineering from the Yildiz Technical University, Istanbul, Turkey, in 1998, the Dipl.Ing. degree in electrical engineering from the University of Kassel, Kassel, Germany, in 2001, and is currently working toward the Ph.D. degree at the University of Stuttgart, Stuttgart, Germany. He is currently with the Institute of Electrical and Optical Communication Engineering, University of Stuttgart, where he is involved in the modeling of SiGe HBTs, MOSFETs, and design of CMOS low-noise amplifiers.

Nikolai Wieser was born in Stuttgart, Germany, in 1970. He received the Diploma degree in physics (with honors) from the University of Stuttgart, Stuttgart, Germany, in 1996, and the Ph.D. degree in physics from the Walter Schottky Institute of the Technische, University of Munich, Munich, Germany, in 2000. He was the German Aerospace Research Center, Stuttgart, Germany, to perform research in the field of the III nitrides. Since 2000, he has been with the Institute of Electrical and Optical Communication Engineering, University of Stuttgart.

Gernot Feiler was born in Billed, Romania, in 1977. He is currently working toward the Dipl.-Ing. degree in electrical engineering at the University of Stuttgart, Stuttgart, Germany.

Manfred Berroth (SM’02) was born in Obersontheim, Germany, in 1956. He received the Dipl.-Ing. degree from the University of the Federal Armed Forces, Munich, Germany, in 1979. He then developed microprocessor systems and dedicated image-processing software as a consultant. In 1987, he joined the Institute for Applied Solid State Physics, Freiburg, Germany, where he was engaged in the development of circuit simulation models for GaAs field-effect transistors, as well as integrated-circuit design. Since 1996, he has been a Professor with the Institute of Electrical and Optical Communication Engineering, University of Stuttgart, Stuttgart, Germany. His research interests and activities are electronic and opto-electronic devices and circuits at high frequencies.

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Injection-Locked Dual Opto-Electronic Oscillator With Ultra-Low Phase Noise and Ultra-Low Spurious Level Weimin Zhou, Member, IEEE, and Gregory Blasche

Abstract—We report a new injection-locked dual opto-electronic oscillator (OEO) that uses a long optical fiber loop master oscillator to injection lock into a short-loop signal-mode slave oscillator, which showed substantial improvements in reducing the phase noise and spurs compared to current state-of-the-art multiloop OEOs operating at 10 GHz. Preliminary phase-noise measurement indicated approximately 140-dB reduction of the spurious level. Index Terms—Injection locked, opto-electronic oscillator (OEO), phase noise, spurious level.

I. INTRODUCTION Fig. 1. Block diagram of the OEO.

H

IGH-PERFORMANCE microwave oscillators require a high quality factor ( ) cavity in order to reduce the phase noise. However, the is limited in traditional microwave electronic devices due to size and power constraints. In 1995, an opto-electronic oscillator (OEO) was introduced by Yao and Maleki [1], [2], which used a long optical fiber as a delay line in a feedback loop completed both by optical and electronic paths, as shown in Fig. 1. The basic concept is to convert the microwave oscillations into modulated laser light that is sent into a long optical fiber. A photodetector converts the modulated light signal back into microwave signals that are amplified and filtered by a microwave filter, which is then fed into the optical modulator closing the feedback loop. Several kilometers of low-loss optical fiber in the OEO loop can generate a cavity with values more than 10 , which is several orders of magnitude higher than that from the best commercial microwave filters. In the OEO, the mode spacing is inversely proportional to the cavity . Therefore, the RF filter is not able to filter out many of the unwanted modes, especially those close to the carrier. Multiloop OEOs were recently reported [3]–[5], which suppress the spurs by adding a second loop in the cavity. As shown in Fig. 2, the modulated laser light is split into two optical fibers, a long fiber and a short one. Two photodetectors convert the light signals into separate microwave signals that are combined using a microwave power combiner. The combined signal is sent to the RF filter, amplifier, and fed back to the optical modulator. Using Manuscript received March 31, 2004; revised July 9, 2004. The work of G. Blasche was supported by the Army Research Laboratory (ARL) under ARL Cooperative Agreement DAAD19-00-2-004. W. Zhou is with the Sensors and Electron Devices Directorate, U.S. Army Research Laboratory, Adelphi, MD 20783 USA (e-mail: [email protected]). G. Blasche is with the Physics Department, Boston University, Boston, MA 02215 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842489

Fig. 2. Block diagram of the dual-loop OEO.

the Vernier caliper effect, one can use an RF phase shifter to tune one mode from the short loop close to a mode from the long loop within the filter band. This combined mode will be enhanced in the oscillator, forming a strong mode, which becomes the carrier signal. Due to the energy competing effect, all the other mismatched modes will be suppressed. A 30-dB reduction of the spurious level has been reported [5] using this scheme. However, the spurious modes are still supported by either the longor short-loop cavity, making it hard to further reduce the spurious level. In addition, this parallel dual-loop OEO sacrifices the high produced from the long fiber. The overall is “averaged” between the long loop’s high and the short loop’s low

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Block diagram of an injection-locked dual OEO.

so that the phase noise increases compared with the single-loop long-fiber OEO. As shown in [5], the phase-noise level of a double-loop OEO with 8.4- and 2.2-km fibers is only equivalent to that from a 4.4-km fiber single-loop OEO. II. EXPERIMENTAL RESULTS A. Injection-Locked Dual OEO To solve the problem of maintaining the high of the multiloop system while eliminating the spurious modes that are supported by the cavity loops, we introduce a new injectionlocked dual OEO scheme. Injection-locking schemes have been used and studied previously in nonoptical RF oscillators [6], [7], which demonstrated an improvement in phase-noise reduction for their low- slave oscillators. Here, in our OEO, we use the injection scheme differently where the slave OEO is used to filter out the multimode spurs generated by the high- master OEO and to maintain the high by the injection locking. As shown in Fig. 3, the RF output signal from a high- long-fiber single-loop master OEO is injected into a short fiber slave OEO to lock in the oscillation frequency and phase. The length of the slave OEO’s optical fiber is chosen such that only one mode is allowed within the RF-filter bandwidth in that single loop OEO, therefore, suppressing the spurious modes from the master OEO by the destructive interference in the slave OEO’s cavity. Thus, and the slave the master OEO’s long fiber builds the high OEO’s short-loop filter out the spurs. To make a proof-of-principle demonstration, we built a master OEO using greater than 6 km of Corning SMF28 optical fiber having an effective index of refraction of 1.46 at 1550 nm, which is the wavelength of the single-mode laser used to carry the signal in the optical path. In the first approximation, , where the frequency spacing of the modes is is the speed of light and is the fiber length. Therefore, in the master oscillator is approximately 34 kHz. The RF filter used in the master OEO has a center frequency at 10 GHz and a filter bandwidth of 8 MHz, allowing hundreds of modes to oscillate in the master OEO. Fig. 4(a) shows the spectrum of the master OEO measured using an Advantest-3271A microwave spectrum analyzer, which indicates a 34.8-kHz spacing between each oscillation peak. The envelope shape of the multimode

Fig. 4. Experimental data for the oscillator output taken from a RF spectrum analyzer for the: (a) master OEO alone, (b) slave OEO alone, and (c) injection-locked OEO. (Spectra (a)–(c) are taken with the same span, resolution, and reference level.)

amplitudes reflects the passband characteristic of the multisection RF filter. Fig. 4(b) shows the single peak mode spectrum of

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Fig. 5. Phase-noise measurement data of the injection-locked dual OEO, which shows the relative phase-noise intensity versus offset frequency from the 10-GHz center carrier. 60-Hz noise from the power supply is denoted by a dashed line for clarity. The doted line indicates a range of a worst uncompressed noise level.

the slave OEO (composed of a 50-m optical fiber length) free running without injection lock, which has a broader linewidth compared to the peaks of the master loop shown in Fig. 4(a). After the multimode signals of the master OEO are injected into the slave OEO, an RF phase shifter is used to bring the slave OEO’s oscillation into the locking range with one of the strong modes of the master OEO. When locked, the side modes are drastically reduced. Fine tuning of the slave loop phase makes the multimode spurs disappear from the measured RF spectrum, as shown in Fig. 4(c). We use the same settings, 200-kHz span, 10-dBm reference level, and 10-Hz resolution bandwidth, for all three measurements. The single peak signal after the injection locking becomes sharp and clean. The spurs at multiples of 34.8 kHz disappear from the output. A 4-MHz span continuation spectrum is inserted into Fig. 4(c) to show no other spurs within the RF filter bandpass. (Since the spurs are symmetric with respect to the center peak frequency, we only need to show the spectrum from the center peak to the higher frequency end of the filter.) The inserted spectrum was taken separately because different resolution bandwidth has to be used for the longer span. The noise floor after injection lock, shown in Fig. 4(c), is even lower than that from the master OEO [see Fig. 4(a)]. Notice that the noise level of the RF spectrum analyzer is much higher than that from our OEO, therefore, a more sophisticated phase-noise measurement system is required in order to measure the true phase noise of the OEO. B. Phase-Noise Measurement A preliminary phase-noise measurement has been performed using a precision phase measurement technique developed at the National Institute of Standards and Technology (NIST), Boulder, CO [8]–[10]. The phase-noise measurement equipment is commercially provided by Femtosecond System Inc., Denver, CO, which is capable of dual-channel cross-correlation measurements [11]. However, due to the unavailability of two identical RF reference sources at this time, we have performed a noise measurement using a two-source single-channel method. For this measurement, a reference source is frequency/phase locked to the OEO under test. Phase noise is detected after the

carrier signal is canceled at a mixer by tuning the reference into the opposite phase. The measured phase noise represents the highest noise of the two oscillators. We have used another double-loop OEO with effective 4 km of fiber length as the reference source. As explained in Section II-A, when the reference OEO is locked by our high- OEO under test, the phase noise from the reference OEO could be lower than that when it is free running. However, spurs from the reference OEO will remain in this case. In Fig. 5, we show the preliminary measured phase-noise data. There are a few peaks expressed by dashed lines, which are associated with the 60-Hz ac power sources used on all the voltage supplies of our OEO. We verified from the raw data that the frequencies of these peaks are exact multiples of 60 Hz. We believe that if we replace our voltage sources for the photodetectors and optical modulators with batteries, we can eliminate those peaks from the noise spectrum. The periodic noise oscillation below 60 Hz was present in a noise floor measurement taken without the OEO under test. We also know that if we have any spurs, they must be located at 34.8 and 69.6 kHz in our phase-noise spectrum. We can see some small peaks that may be associated with the spurs, but their intensity level is well below 140 dBc/Hz, which is much lower than the spur level reported from the double-loop OEO scheme in [4] and [5]. The first one or two spurs closest to the carrier should be the strongest. Since the slave OEO’s short cavity allows only single-mode oscillation, when the phase shifter is tuned to lock the oscillation to center frequency, the other spur modes will be out-of-phase, the further from the center frequency, the more the phase mismatch will be. Secondly, the RF filter profile will also reduce the magnitude of any mode away from the center frequency. This result shows that our spur reduction concept of using destructive interference of the unsupported spur modes in the short slave OEO cavity provides greater reduction of the spurious modes than the double-loop OEO configuration, which uses energy competition between the supported spur modes and selected carrier mode. The preliminary noise data also indicates a low phase-noise level below 110 dBc/Hz at a low offset frequency (10–100 Hz). This data demonstrates that the high

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from our master OEO is preserved in the slave OEO after the injection locking. However, the frequency tuning of our reference oscillator is somewhat difficult due to the poor design of the tuning mechanism. This makes phase locking difficult using a low-gain phase-locked loop. Therefore, noise compression is possible at the low offset-frequency range due to the relative high gain of the phase-locked loop. To be safe in the interpretation of the data, we have drawn a straight dotted line denoting the upper range in the noise spectrum, under which we believe the real noise level should be. The injection-locked OEO was laid out on an optical table during the measurement in an environmental controlled laboratory, therefore, thermal instability is thought to be at a minimum. III. DISCUSIONS A. Injection-Locking Conditions Different physical states of the injection-locked OEO have been observed during the inject-locking process under different conditions. The phase-noise level and spurious level may change depending on the relative RF signal power level injected into the slave OEO with respected to the slave OEO’s free-running power level. When the spur level of the injected signal from the master OEO matches that from the same spur after one cycle feedback in the slave OEO, destructive interference may work the best to cancel the spur. We have also noticed that, when the frequency of the free-running slave OEO is tuned at the exact frequency of one of the master OEO’s modes, after the injection lock, the oscillation frequency may hop to another neighboring mode. Only after additional fine tuning of the phase shifter, will we observe a certain drop of spur level and noise level. This hints that there may be a self-cleaning process occurring under certain injection-locking conditions. Additional investigation and theoretical studies are needed to confirm this. B. Comparison The major architectural difference from the previous multiloop OEO is that the resonant cavity of the long loop of the master OEO is isolated and independent from the cavity in the slave OEO so there is no feedback for the spurious modes. This will make a fundamental difference in the physics for the oscillation signal created in the injection-locked dual OEO. First, unlike the multiloop OEO, which is in a parallel configuration having an “average” , the injection-locked dual OEO is in a series configuration. It has been demonstrated [6], [7] that the phase noise of a low- microwave oscillator can be reduced by injection locking from a high- source. Therefore, we believe that, at the injection-locked condition, the high of the master OEO is preserved in the slave OEO. Secondly, since the slave OEO cavity is designed to allow only single-mode oscillation, once the phase shifter is tuned such that the slave OEO’s mode is matched to one of the master OEO’s modes for injection locking, the super-mode spurs (within the RF filter band) from the master OEO that are injected into the slave OEO cannot be supported by the slave OEO’s short-loop oscillator cavity. Therefore, these spurious modes will die out due to the destructive interference within the short loop. This has a better result compared with the multiloop OEO in which their interlinked multiloop cavity still supports the spur modes.

We can also compare our injection-locked OEO with many conventional microwave oscillators. With the high , our OEO phase noise compared favorably with the best nonoptoelectronic commercial microwave oscillators in the low offset-frequency range (up to 600 Hz, indicated by the preliminary data). Since the low offset-frequency phase noise is dominate by the oscillator’s value, in the large offset-frequency range, the OEO noise figure is slightly worse than the best commercial oscillator. Since we have not yet focused on lowering the noise floor of the electronic circuitry in this project, and the higher offset-frequency noise is attributed to the electronics, we believe that it is a solvable engineering problem to further reduce the noise figure in the higher offset-frequency range by improvement of the electronics. Besides the phase-noise comparison, the OEO technology has a major advantage over conventional microwave oscillators by offering great frequency agility over a very wide operating range. This is due to the fact that even a large change in microwave frequencies represents a very small fractional bandwidth when compared to the optical carrier frequency. IV. CONCLUSIONS In conclusion, we have developed an injection-locked dualOEO architecture, which maintains the high produced by a long fiber loop master OEO and uses a short-loop slave OEO to filter out the spurs produced by the master OEO so that the oscillator output has ultra-low phase noise and an ultra-low spurious level. This oscillator can be built using commercially available opto-electronic and microwave components at a reasonably low cost. ACKNOWLEDGMENT The authors wish to thank Dr. C. Fazi for providing his leadership and support for the Frequency Control Program at the U.S. Army Research Laboratory (ARL), Adelphia, MD, as well as his many helpful technical discussions. The authors also thank Dr. W. Walls, Femtosecond System Inc., Denver, CO, for his assistance with phase-noise measurement and training. Author G. Blasche would also like to thank Dr. B. Goldberg, Boston University Photonics Center, Boston, MA. REFERENCES [1] X. S. Yao and L. Maleki, “Converting light into spectrally pure microwave oscillation,” Opt. Lett., vol. 21, pp. 483–485, Apr. 1996. [2] , “Optoelectronic microwave oscillator,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 13, no. 8, pp. 1725–1735, Aug. 1996. [3] , “Dual microwave and optical oscillator,” Opt. Lett., vol. 22, no. 24, pp. 1867–1869, Dec. 1997. [4] , “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron., vol. 36, no. 1, pp. 79–84, Jan. 2000. [5] D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multiloop optoelectronic oscillator,” in Proc. IEEE Int. Frequency Control Symp., 2003, p. 405. [6] H.-C. Chang, X. Cao, M. J. Vaughan, U. K. Mishra, and R. A. York, “Phase noise in externally injection-locked oscillator array,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 11, pp. 2035–2041, Nov. 1997. [7] K. Kurokawa, “Injection locking of microwave solid-state oscillator,” Proc. IEEE, vol. 61, no. 10, pp. 1386–1410, Oct. 1973. [8] Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology—Random Instabilities, IEEE Standard 1139-1999, 1999. [9] D. B. Sullivan, D. W. Allan, D. A. Howe, and F. L. Walls, Eds., “Characterization of clocks and oscillators,” NIST, Boulder, CO, Tech. Note 1337, Mar. 1990.

ZHOU AND BLASCHE: INJECTION-LOCKED DUAL OEO

[10] D. A. Howe, D. W. Allan, and J. A. Barnes, “Properties of signal sources and measurement methods,” in Proc. 35th Annu. Frequency Control Symp., 1981, pp. A1–A47. [11] W. F. Walls, “Cross-correlation phase noise measurements,” in Proc. IEEE Frequency Control Symp., 1992, pp. 257–261.

Weimin Zhou (M’04) received the B.S. and M.S. degrees in physics from the Universite de Toulouse, Toulouse, France, in 1982 and 1983, respectively, and the Ph.D. degree in physics from Northeastern University, Boston, MA, in 1991. He is currently a Research Physicist and Team Leader with the U.S. Army Research Laboratory, Adelphi, MD. His team is involved with the design and fabrication of novel opto-electronic devices, opto-electronic integrated circuits, and development of RF microwave-photonic devices and systems including RF-photonic oscillators and optical-controlled phased-array antennas. Dr. Zhou was the recipient of a National Research Council Research Associateship Award (1991–1994) while with the U.S. Army Electronic Technology and Devices Laboratory, Fort Monmouth, NJ.

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Gregory Blasche received the B.A., M.A., and Ph.D. degrees in physics from Boston University, Boston, MA, in 1999, 2001, and 2004, respectively. His doctoral research involved the development of a highpower line-narrowed laser diode array for the generation of hyperpolarized noble gases. He is currently with the Physics Department, Boston University, Boston, MA.

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Pure-Mode Network Analyzer Concept for On-Wafer Measurements of Differential Circuits at Millimeter-Wave Frequencies Thomas Zwick, Member, IEEE, and Ullrich R. Pfeiffer, Member, IEEE

Abstract—A measurement concept based on a two-port vector network analyzer has been developed, which enables puremode on-wafer measurements of differential circuits in the millimeter-wave frequency range. An error model for the measurement system is derived as required for future calibration algorithms. Based on WR15 waveguide components, together with 1.85-mm coaxial probes, a setup has been built and its amplitude and phase imbalances have been characterized in the frequency range from 50 to 65 GHz. Index Terms—Differential measurements, millimeter-wave (MMW) measurements, vector network analyzer (VNA).

I. INTRODUCTION

up to 110 GHz and adds relatively small cost to the two-port VNA itself. Additionally there is no commercial four-port VNA above 50 GHz yet available. A 50–65-GHz version of the setup has been built and used to measure gain and output power of a differential amplifier in its nonlinear region [6]. Another version of the setup based on WR10 WG components has been used to supply a pure differential signal to an SiGe frequency divider to enhance its sensitivity up to 96 GHz [7]. In Section II, the setup is described and an error model for the system is derived in Section III. Measured amplitude and phase imbalances are presented in Section IV for a 50–65-GHz version of the setup. Conclusions are given in Section V.

D

IFFERENTIAL circuits are becoming increasingly common at microwave and, more recently, also at millimeter-wave (MMW) frequencies [1]. Measurement systems, which allow the determination of the differential-mode, common-mode, and any mode conversion responses are required. For passive devices or active devices operating in their linear region, individual single-ended responses can be measured with a four-port vector network analyzer (VNA) and combined later to obtain the balanced response mathematically [2]. However, many active devices do not follow such a model for their behavior, as shown in [3], once operating in or close to their nonlinear region. In [4], a pure-mode vector network analyzer (PMVNA) has been developed, which allows measuring balanced devices by exciting pure differential or common-mode signals. In [5], it has additionally been shown that the PMVNA has a higher accuracy than the four-port VNA when differential devices are measured. The setup described in [4] is based on two modified commercial test sets and operates up to a maximum frequency of 20 GHz. Thereby one port of each test set has to be paired to form a mixed mode port, which will become nonpractical at MMW frequencies where flexible cables are not available anymore or have substantial losses. Therefore, in this project, a new PMVNA concept was developed, which is based on a commercial two-port VNA and two baluns built of waveguide (WG) components. The new concept can be realized with commercially available equipment Manuscript received April 5, 2004. This work was supported in part by the National Aeronautics and Space Administration. T. Zwick was with the IBM T. J. Watson Research Center, Yorktown Heights, NY 10598 USA. He is now with Siemens VDO Automotive AG, 88138 Weissensberg, Germany (e-mail: [email protected]). U. R. Pfeiffer is with the IBM T. J. Watson Research Center, Yorktown Heights, NY 10598 USA. Digital Object Identifier 10.1109/TMTT.2004.842488

II. MEASUREMENT SETUP The measurement setup is based on a commercial two-port VNA. Each port of the VNA is connected to a special balun structure based on a WG magic tee, as shown in Fig. 1. A manual WG switch is used to switch the VNA port between the port and the differential-mode port of common-mode the magic tee. The unused port of the magic tee is terminated by the switch. The two output ports of the magic tee ( and in Fig. 1) are connected to a differential (double coplanar) probe (ground–signal–ground–signal–ground (GSGSG) configuration) via phase shifters, WG to coax adapters, and very short coaxial cables to contact the balanced device-under-test (DUT) on-wafer. The coaxial cables allow small movement of the probe against the WG construction for leveling. The phase shifters are used to compensate any length differences in the two paths. To provide the shortest possible connections (cable losses are substantial at MMW frequencies), all equipment shown in Fig. 1 is mounted on the positioner together with the probes. In case of MMW VNAs, the two ports/reflectometers are usually separated in two MMW modules with WG ports. These MMW modules will be mounted onto the positioners together with the balun. The setup can easily be realized with commercially available parts for the standard WG bands WR15 (50–75 GHz), WR12 (60–90 GHz), and WR10 (75–110 GHz) in combination with 65-GHz cables having 1.85-mm coaxial connectors or 110-GHz cables having 1.0-mm connectors. To further improve the performance, one could also think about building special dual probes with WG connectors to eliminate the need for any cables and adapters in the setup. The WG switch allows switching between common- and differential-mode signals on both DUT ports. Therefore, this setup allows the measurement of all combinations of differential-

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Fig. 1. Schematic of the balun.

and common-mode reflection and transmission -parameters, except for the mixed-mode reflection parameters. This does not impose a big restriction since the latter are not relevant in most applications. Mixed-mode reflection parameters can only be enabled in the setup shown here with an extremely high additional effort. Either two additional reflectometers have to be added with some additional modifications to built a MMW version of the concept shown in [4] or a set of WG switches has to be used to and built a four-port VNA connected to the common-mode ports of the magic tees. The latter solution differential-mode has the problem that the setups on the two positioners would have to be connected by WG components, which is undesirable since the two positioners need to be moved independently. III. ERROR MODEL OF THE BALUN By assuming that the whole balun is passive and reciprocal, an error model for the configuration can be defined. Additionally, switching terms are neglected. If switching terms are required, the model can be extended based on the method described in [8]. An error model for the left balun (the error model for the right balun can be defined the same way) can be written as

Fig. 2.

50–75-GHz balun setup on the probe station.

Fig. 3.

Calibration substrate specially fabricated for phase adjustment.

(1)

where and belong to the differential-mode port and and belong to the common-mode port ( and , respectively, in Fig. 1). The power waves at the probe ports are given by , , , and . The reflection coefficients at the differential- and common-mode input ports are denoted by and , respectively. The coupling between these two ports is . The same and are used for both paths toward the probe thru terms with the amplitude and phase imbalance accounted for in and . and denote the reflection coefficients of the two is the coupling between the two coplanar probes, whereas probes (mainly through the magic tee). Combinations of different on-wafer terminations (short, open, load, offset short, etc.) and on-wafer thru lines of different length have to be used to determine the ten error terms in (1) for both baluns. In [9], a calibration algorithm is shown for the case of just one balun where the two ports and are connected to the two ports of a VNA simultaneously without using a switch. Once solved, the four-port -parameters of the DUT can be converted to mixed-mode -parameters, as explained in [2].

IV. RESULTS: AMPLITUDE AND PHASE IMBALANCE Based on WR15 WG components (50–75 GHz) and 1.85-mm coaxial components (dc–65 GHz), a measurement setup (see Fig. 2) has been built, as described in Section II. Using a 65-GHz two-port VNA, the imperfections of the balun have been determined in the frequency range from 50 to 65 GHz. First, the two phase shifters have to be adjusted to provide the optimal phase balance at the probe tips over the frequency range of interest. Therefore, a special calibration substrate has been designed (see Fig. 3), which consists of a 50- coplanar transmission thru line and adjacent 50- terminations on either side of the line. By shifting the probe (see Fig. 3), the two signals of the differential probe can be alternatingly contacted to the thru line, while the other side is automatically terminated by the 50- load on the substrate. On the other end of the thru line, a standard single-ended coplanar probe is used. The VNA is calibrated to the connectors at the single-ended probe and coax to

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Magnitude error over frequency for both modes.

Fig. 6. Mode conversion dependent on magnitude error.

Fig. 5. Phase error over frequency for both modes.

Fig. 7.

WG adapter, which had to be added at the input to the balun to connect the coaxial VNA. This procedure allows phase adjustment while the WG switch is set to common mode. Now, after the phase adjustment, the remaining imbalance have been measured using the same for both modes and setup. Figs. 4 and 5 show the magnitude and phase error for both modes. Since the VNA could only be calibrated to the connector of the single-ended measurement probe, but not to the end of the coplanar waveguide (CPW) line, the measured magnitude and phase errors shown in Figs. 4 and 5 do not perfectly represent and in (1). Due to small mismatches between the calibrated port and the desired port at the probe tips of the differential probes, the measurement results shown here can be expected to be worst cases of and . Maximum magnitude and phase errors for both modes of approximately 1.5 dB and 15 have been observed. If less bandwidth is required (as in [6]), the phase error can be reduced to approximately 10 (see Fig. 5). The slight slope of the phase error over frequency is caused by the magic tee itself, while the ripple in both errors is caused by mismatches between all the components behind the magic tee. This was confirmed by measuring the magic tee itself. Since the isolation between the magic tee input ports ( and in Fig. 1) is high (usually better than 30 dB), one would expect to be in the same range. Here, has been estimated by measuring the isolation in the whole setup while terminating both

probes with 50- on-wafer terminations. The worst isolation observed in the frequency range from 50 to 65 GHz in both baluns was 12.5 dB. This is caused by mismatches between components (mainly the WG to coax adapters) in both signal paths together with different lengths resulting in mode conversion. The return loss (up to 15 dB for both switch positions) was found to be limited by the poor matching of the adapters as well. WG loads as used on the unused input port usually have a return loss better than 30 dB over the full WG band (here, 50–75 GHz). The worst case loss through the balun structures to the probe tips was 6.5 dB of which over 2-dB loss is caused by the used differential probe wedge (better versions with less than 1-dB loss are also available). Both the return and insertion losses could be further improved by using WG probes connected directly to the phase shifters, as already mentioned in Section II. A second balun, which was built using components from partly different sources, showed a very similar performance. , defined by the amplitude For a certain imbalance error and the phase error , the mode conversion can be cal. Thereby all reflection coefficients in (1) culated by are again neglected. The mode-conversion dependent on amplitude and phase error are given in Figs. 6 and 7, respectively. This clearly shows that, with an amplitude error of up to 1.5 dB and a phase mismatch of up to 15 , a calibration algorithm is required to accurately measure the common-mode gain or the

Mode conversion dependent on phase error.

ZWICK AND PFEIFFER: PURE-MODE NETWORK ANALYZER CONCEPT FOR ON-WAFER MEASUREMENTS OF DIFFERENTIAL CIRCUITS

mode conversion terms of a differential device. The differential gain can be measured fairly accurate by just treating the setup as a normal two-port VNA and using a standard differential calibration substrate, as shown in [6]. V. CONCLUSIONS A pure-mode network analyzer concept for on-wafer measurements of differential circuits at MMW frequencies has been presented. The setup has been described in detail and an error model, as required for calibration algorithms, has been derived. A special calibration substrate has been fabricated to allow proper phase adjustment. Based on WR15 WG components (50–75 GHz) and 1.85-mm coaxial components (dc–65 GHz) two baluns have been built and characterized by measurements. An amplitude error of up to 1.5 dB and a phase mismatch of up to 15 have been observed in the frequency range from 50 to 65 GHz. This results in mode conversion of up to 15 dB, which clearly shows that a calibration algorithm is required to accurately measure mixed-mode terms with this setup. ACKNOWLEDGMENT The authors would like to thank B. Gaucher, Dr. M. Soyuer, and Dr. M. Oprysko, all of the Communications Department, IBM T. J. Watson Research Center, Yorktown Heights, NY, for the friendly support, and Dr. L. Hayden, Cascade Microtech Inc., Beaverton, OR, for very helpful discussions.

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[6] U. R. Pfeiffer, S. K. Reynolds, and B. A. Floyd, “A 77 GHz SiGe power amplifier for potential applications in automotive radar systems,” in IEEE RFIC Symp., Fort Worth, TX, Jun. 2004, pp. 91–94. [7] A. Rylyakov and T. Zwick, “96 GHz static frequency divider in SiGe bipolar technology,” in IEEE Compound Semiconductor Integrated Circuit Symp., Monterey, CA, Oct. 2003, pp. 288–290. [8] R. B. Marks, “Formulations of the basic vector network analyzer error model including switching terms,” in IEEE Automatic RF Techniques Group Conf., Denver, CO, Jun. 1997, pp. 115–126. [9] C. R. Curry, “How to calibrate through balun transformers to accurately measure balanced systems,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 961–965, Mar. 2003.

Thomas Zwick (M’00) received the Dipl.-Ing. (M.S.E.E.) and Dr.-Ing. (Ph.D.E.E.) degrees from the Universität Karlsruhe (TH), Karlsruhe, Germany, in 1994 and 1999, respectively. From 1994 to 2001, he was Research Assistant with the Institut für Höchstfrequenztechnik und Elektronik (IHE), TH. From February 2001 to September 2004, he was with the IBM T. J. Watson Research Center, Yorktown Heights, NY. Since October 2004, he has been with Siemens VDO Automotive AG, Weissensberg, Germany. His research interests include wave propagation, stochastic channel modeling, channel measurement techniques, material measurements, microwave techniques, wireless communication system design, and MMW antenna design. Dr. Zwick has participated in the European COST231 Evolution of Land Mobile Radio (Including Personal) Communications and COST259 Wireless Flexible Personalized Communications. For the Carl Cranz Series for Scientific Education, he served as a lecturer for wave propagation. He was the recipient of the 1998 Best Paper Award presented at the International Symposium on Spread Spectrum Technology and Applications (ISSSTA).

REFERENCES [1] S. Reynolds, B. Floyd, U. Pfeiffer, and T. Zwick, “60-GHz transceiver circuits in SiGe bipolar technology,” in IEEE Int. Solid-State Circuits Conf., San Francisco, CA, Feb. 2004, pp. 442–443. [2] D. E. Bockelman and W. R. Eisenstadt, “Combined differential and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995. [3] J. Dunsmore, “New methods and nonlinear measurements for active differential devices,” in IEEE MTT-S Int. Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1655–1658. [4] D. E. Bockelman and W. R. Eisenstadt, “Pure-mode network analyzer for on-wafer measurements of mixed-mode S -parameters of differential circuits,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1071–1077, Jul. 1997. [5] D. E. Bockelman, W. R. Eisenstadt, and R. Stengel, “Accuracy estimation of mixed mode scattering parameter measurements,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 1, pp. 102–105, Jan. 1999.

Ullrich R. Pfeiffer (M’02) received the Diploma degree in physics and Ph.D. degree in physics from the University of Heidelberg, Heidelberg, Germany, in 1996 and 1999 respectively. In 1997, he was a Research Fellow with the Rutherford Appleton Laboratory, Oxfordshire, U.K., where he developed high-speed multichip modules. In 2000, his research was based on high-integrated real-time electronics for a particle physics experiment with the European Organization for Nuclear Research (CERN), Zurich, Switzerland. In 2001, he joined IBM and is currently a Research Staff Member with the IBM T. J. Watson Research Center, Yorktown Heights, NY. His research involves RF circuit design, power-amplifier design at 60 and 77 GHz, and high-frequency modeling and packaging for MMW communication systems.

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An Orthogonality-Based Deembedding Technique for Microstrip Networks Michael P. Spowart and Edward F. Kuester, Fellow, IEEE

Abstract—This paper introduces a new orthogonality-based method of extracting scattering parameters (deembedding) from numerical current distributions on microstrip networks. All deembedding methods require sufficient length in the network feed lines that only a fundamental mode arrives at the discontinuity. In our new method, the length of feed lines used to excite a network with only a fundamental mode can be shortened compared with other methods. On that basis, this new method can be used to improve other deembedding methods. Estimates of end susceptance and end conductance of the open-end discontinuity are used for performance evaluation. End conductance computations are highly sensitive to errors in computed numerical reflection coefficients, making the accurate analysis of the open-end discontinuity a particularly challenging deembedding example. Results show that the orthogonality-based method is both stable and accurate. In the case of the open-end discontinuity, significant improvement in performance compared to other deembedding techniques can be achieved. Index Terms—Deembedding, end capacitance, end conductance, end susceptance, Lorentz reciprocity, method of moments (MoM), microstrip networks, modal analysis, orthogonal decomposition, -parameters.

I. INTRODUCTION

T

HE FULL-WAVE method of moments (MoM) [1] is widely used to find the total two-dimensional (2-D) surface current distribution on a planar transmission-line circuit limited in accuracy only by numerical errors. Total current density is provided rather than the amplitudes and propagation constants of the modes at the ports. The designer is most often interested in finding a parameter matrix, e.g., scattering parameters, of the circuit, as well as propagation constant and characteristic impedance at the circuit feed ports. Therefore, the lowest order incident and scattered modes are of primary interest. An accurate method of extracting circuit parameters from samples of the total current at and near a chosen reference plane is needed. Fig. 1 depicts a passive microstrip circuit with three feed ports as an example. The circuit in Fig. 1 is excited by voltage gap sources shown at the ends of the feed lines. The metal surface of the circuit is divided into a grid. The surface current density in each cell is approximated by basis functions multiplied Manuscript received April 1, 2004; revised July 8, 2004. This work was supported by the National Center for Atmospheric Research. M. P. Spowart is with the Atmospheric Technology Division, National Center for Atmospheric Research, Boulder, CO 80301 USA (e-email: spowart @ucar.edu). E. F. Kuester is with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309 USA (e-mail: kuester@schof. colorado.edu). Digital Object Identifier 10.1109/TMTT.2004.842487

Fig. 1. Planar transmission-line network with deembedding arms attached to each port and voltage gap sources at the ends of each feed line.

by constant amplitudes (one for each side of the cell). Rooftop basis functions are used in this study. Surface current density can , which, for the planar mibe expressed generally as , where is the direction percrostrip becomes pendicular to the substrate surface. When the circuit grid contains only a single cell in the transverse direction, as is the case with the feed lines in Fig. 1, surface current is taken to be con,a stant in the transverse direction and can be expressed as scalar directed along the axis of the strip. Numerical results have shown that higher order evanescent modes, excited by junctions and other discontinuities (such as voltage gap sources) in a microstrip circuit, decay rapidly and are not measurable more than a few tenths of a wavelength (or possibly much less) away from the discontinuity [2]. Therefore, it can be assumed that the total current density more than approximately 0.3 wavelengths away from any discontinuity may be written as (1) where and are the forward and backward complex current is the complex propagation constant, amplitudes, and is a linear coordinate along the line. The process of finding , , and at each port and, in turn, the scattering matrix of the network, is called deembedding. Wave amplitudes can be defined in terms of the current amplitudes using (2) is the incoming power to the port, is the width where of a cell (and, thus, also of the strip), and the characteristic is defined as a ratio of power to current. Using impedance and to denote the incoming wave and outgoing wave at port

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in the th excitation state (voltage gap in the th port only) of an -port network yields (3) are the network scattering parameters. The objective where of this paper is to propose a new method of obtaining network scattering parameters and feed-line propagation constants from numerical surface currents on a microstrip circuit. In this new method, the property of orthogonality of the modes at the ports or in the feed lines attached to the ports is used to extract information about the fundamental modes. The method utilizes overlapping, finite length, probe, and feed port microstrip lines. II. ORTHOGONALITY-BASED METHOD OF DEEMBEDDING WITH ELECTRICALLY SHORT FEED LINES Here, a new orthogonality-based deembedding method is developed for microstrip networks with an arbitrary number of input/output ports. Attached to each network port is an electrically short extension line, as shown in Fig. 1. In order to account for transverse current in the deembedding arm or a transverse variation in longitudinal current, the feed line might be divided into any number of cells in the transverse direction. The feed line is capable of supporting a set of waveguide modes whose fields are

(4) is the propagation constant for the th where and are the transverse -demode and indicates a pendent mode fields. The convention that indicates a backward travforward traveling mode and eling mode is used. The following relationship exists between forward and backward modes of a reciprocal waveguide:

(5) components of The subscript indicates the transverse the respective vectors. Although the nature of the above assumption about the fields of the waveguide suggests that only discrete modes are possible, included in the above set are modes that represent the continuum of “radiation” modes [3]–[5]. In a microstrip, discrete modes by themselves generally do not provide a complete description of the fields. A more complete description would involve adding the following integrals to the right-hand sides of (4), respectively:

(6)

Fig. 2. Lorentz reciprocity theorem applied to a circuit enclosed in the box with volume V . Surface S is defined by the semi-infinite Y –Z -plane, which extends above the circuit ground plane.

A. Application of the Lorentz Reciprocity Theorem to the Deembedding Problem The general -port circuit shown in Fig. 1 is enclosed in a box, as shown in Fig. 2, and the Lorentz reciprocity theorem is applied to the volume enclosed by the box. The box encloses the entire network with the exception of one of its deembedding arms. This deembedding arm is approximately bisected by the surface of the box. The remaining surfaces of are either far removed from the network such that contributions to the surface integral are approximately zero (due to the radiation condition) on these surfaces or the tangential electric field on the bottom surface of the box is zero, as this surface lies on the perfectly conducting ground plane. The deembedding reference plane is . The “a” circuit, shown completely in Fig. 2 is located at the network being analyzed and the “a” state refers to the fields produced by sources in the “a” circuit. A second (or “b”) circuit is needed in order to apply the Lorentz reciprocity theorem. The “b” state designates the fields of an electrically short probe arm, a length of identical microstrip to that in the deembedding arm of the circuit being modeled that is excited on one end by a voltage gap and left open on the other end. The probe arm, or circuit “b,” occupies the same space as the “a” circuit extension arm, but is shorter in length, as shown in Fig. 3. The integral form of the Lorentz reciprocity theorem is [6], [7]

(7) The integral over on the right-hand side of (7) becomes a and exist only on the network and surface integration, as probe arm metal surfaces. Henceforth, the electric fields on the right-hand side of (7) are assumed to be the – components, on the plane of the strip conductors. The integral over on the left-hand side becomes an integral over in the – -plane due to the radiation condition and ground plane. Furthermore, only the transverse (to ) components of

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Since the probe arm has the same transverse geometry and physical properties as the deembedding arm, the orthogonality property of the modes yields (11) (12) , but the mode fields are not the same, the When and are said to be degenerate. In lossless wavegmodes uides, modes that are not already orthogonal may be made orthogonal so that (11) and (12) hold with the requirements on the and , repropagation constants replaced by spectively, yielding Fig. 3. Positions of the probe arm with respect to the deembedding arm of a one-port microstrip open-end.

(13)

the electric and magnetic fields contribute to the integral on the left-hand side of (7). The left-hand side of (7) becomes (8) where the subscript refers to the transverse components in the –plane. By (4), the transverse fields on are

(14) is a norm of the discrete mode and is the Krowhere , necker delta. In the case of a lossless waveguide, the norm as defined above, is equal to the power carried by the th mode. yields Substituting (14) into (10) with

(15) (9)

is the -location of the plane . A similar represenwhere tation holds for the fields of the probe arm with the wave ampliand replaced by and , respectively, and the tudes replaced by the position on the probe arm. Subposition stituting (9) into the left-hand side of (7) and expanding yields

(10) where the indices and refer to the modes of the network and probe arm, respectively. By conveniently restricting the “b” probe arm circuit to lie completely within the “a” circuit feed arm such that the probe arm does not overlay the gap voltage evaluates source of the circuit feed arm, the integral of to zero as the electric field tangential to the perfectly conducting probe arm is zero. Without loss of generality, we choose the . coordinate system of the deembedding arm such that

The superscript “ ” has been dropped from the circuit electric field and current density and the subscript has been added in of the circuit. reference to the surface current density B. Procedure for Extracting Scattering Parameters From the Orthogonality-Based Reciprocity Equation The integral on the right-hand side of (15) evaluates to a complex number. It will turn out not to be necessary to evaluate the . Therefore, (15) becomes a single equation with three norm -type products and a unknowns for each mode : two . propagation constant When the amplitudes of the forward and backward traveling fundamental modes , , , and are much larger than all and , the same will be true of the product of the other compared to the other products to an even larger degree. This is typically true for any waveguide. Based upon this assumption, the first term in the sum on the left-hand side of (15) is sufficiently greater than the sum of the remaining terms such that the remaining terms may be neglected. Thus, if (16) Equation (15) becomes (17)

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is a complex number found by evaluating the electric-field integral equation (EFIE) (18) Equation (18) is solved by the usual Galerkin implementation of the MoM. , , and . Equation (17) contains three unknowns At least two more independent equations are needed. One approach to obtaining additional equations is to shift the probe arm to the right- or left-hand side, and/or reverse its position relative to the arm of the original circuit in order to generate additional equations. Fig. 3 shows a probe arm in five different positions below a simple one-port open-end network. In this example, the network consists of an open-end line illustrated with triangular cells, and a deembedding arm made up of 14 rectangular cells attached to the left-hand-side port. For the purpose of clearly illustrating the probe arm in its shifted and reversed positions, the probe arm is not shown lying on top of and in the “same space” as the deembedding arm, but rather in five separate positions below the deembedding arm. The shaded cells at the ends of the probe and deembedding arms represent voltage gap sources. Applying (17) to each of the five probe arm positions yields (19) (20) (21) (22) (23) In (22) and (23), the probe arm has been reversed. Equations , (19)–(21) may be used to solve for the three unknowns , and . Since the network scattering parameters are ratios of the wave variables and , (22) and (23) are used to solve and , which can then be used with the and for products above (by taking ratios) to find the network scattering parameters. Solving (19)–(23) for the unknown products and the port propagation constant yields (24)

Fig. 4. Positions of two probe arms with respect to the deembedding arms of a two-port microstrip line. Shaded cell represents a voltage gap source.

-port network. Fig. 4 shows a two-port line with extension arms attached to the left- and right-hand-side ports. The local -coordinate bisects the extension arms and is directed toward the circuit for ease of calculations. The respective probe arm positions are again shown below the extension deembedding arms. For the two-port example, (24)–(28) are evaluated for each port and for each excitation state. For an -port network, each port is excited separately from the others. Evaluating ratios of wave variables for each set of equations associated with each of the source and probe arm locations yields (29) are known complex numbers that are functions of where – and , refers to the port number and refers to the excitation state. Rearranging (3) yields (30) equations in unknowns. For the Equation (30) is a set of two-port example in Fig. 4, the scattering parameters are

(25) (26) (27) (28) By appropriately taking ratios of (25)–(28), the wave variables associated with the probe arm (superscripted with ) are elimi. nated from the result as are the references to the norm C.

-Port Networks

The orthogonality-based deembedding method illustrated in the one-port open-end example above is easily applied to an

(31) (32) (33) (34) If two or more of the ports are connected to sufficiently closely coupled side-by-side microstrips, the orthogonality method may need to be modified by considering normal modes of the coupled microstrips instead of the modes of an isolated single microstrip. This extension is left for future work.

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III. DISCUSSION OF PERFORMANCE While geometrically simple, the open-end discontinuity presents one of the most challenging problems for a deembedding method. An ideal lossless stub with a true open-circuit loading presents a pure susceptance at the junction of the stub and the main line. However, the actual boundary condition at the open end of the stub is not an idealized open circuit, but a complex load that can be represented by a conductance resulting from radiation into space and surface waves, and a susceptance resulting from energy stored in the higher order modes corresponding to the fringing fields at the open-end. The susceptance can be represented by a hypothetical extension of the stub length by an amount usually between 0.3–0.5 substrate thicknesses beyond the physical open end of the mirepresents crostrip [8]–[23]. The hypothetical line extension effective capacitance at the open end caused by the fringing electric field. End capacitance has been computed using various methods by other investigators [24]–[28]. Often the power lost due to conductance at the open end is of interest in the analysis of the open-end line, e.g., if the of a resonator must be determined. Power loss is proportional , where is the reflection coefficient, which to is very nearly unity for most lines. As a result of subtracting two numbers that are nearly equal, accuracy may be lost even when the error in the estimate of is small. Consequently, accurate estimates of are required. Wider lines with lower dielectric-constant substrate materials operating at higher frequencies produce greater radiation losses. Often, even small radiation losses in the vicinity of other junctions and lines are of concern to microwave designers for reasons such as parasitic coupling. Open-circuit microstrip stubs are frequently used in filters, matching networks, and antennas. Therefore, many investigators have attempted to accurately compute the open-end radiation conductance using a wide variety of methods [29], [8]–[16]. The various techniques used to study the open-end discontinuity can be broadly classified as numerical methods, methods involving radiation from the open-end aperture, and methods that compute radiation from a distribution of electric and polarization currents along the line. Although numerical methods may not provide the best results in the case of the open-end discontinuity, the purely analytical methods are not generally applicable to other discontinuities. For this reason, the numerical methods are most often used in computer-based analysis tools. General numerical deembedding techniques include the three-point method [30], transmit reflect line (TRL) method [31], short-open calibration (SOC) method [24]–[26], [32], nonlinear least squares methods [33], standard Prony and least squares Prony methods [34]–[39], and generalized pencil-of-functions (GPOF) and matrix-pencil methods [40]–[43]. The three-point method uses three samples of total current (provided by MoM results) at locations , , and in Fig. 1. Three equations of the type given by (1) are used to solve for , , and . More than three samples of current can be used to solve an overdetermined set of nonlinear equations in a manner that

minimizes errors in computed estimates of , , and . Several nonlinear least squares methods exist including the Newton, Gauss–Newton, Levenberg–Marquardt, and secant methods. Not all of the nonlinear least squares methods have been applied to the deembedding problem and more work in this area could certainly be done. The least squares Prony and GPOF methods both involve a modal analysis of sampled data. In the Prony method, (frequency-domain) data are modeled by a sum of poles multiplied by residues. The residues and poles are found by solving a linear prediction filter problem for a set of coefficients, finding the poles by factoring a polynomial with the coefficients as amplitudes, and finally inverting a rectangular matrix to yield estimates of the residues. The GPOF modal analysis method is similar to the Prony method, but involves finding the solution of an eigenvalue problem rather than a linear prediction filter. The GPOF method is reported to have advantages over the Prony method in low signal-to-noise conditions and is shown in [43] to perform better than the TRL method for computing the angle of the reflection coefficient of the open-end microstrip discontinuity. The TRL method models the higher order cutoff modes excited by a microstrip circuit discontinuity by formulating an error matrix using TRLs or single and double delay lines. The inverse of the error matrix is then cascaded with a circuit param) matrices are conveniently used in eter matrix. Chain ( this application. By calibrating the source discontinuity in this way, a shorter deembedding arm may be used. In the SOC method, network discontinuities are calibrated using SOC circuits [24]. Network feed lines are excited with voltage sources and the network itself is replaced with electric and then magnetic walls such that only fundamental modes end up at the network discontinuity. Feed lines must be separately analyzed using the MoM, first with electric walls at the ports and then with magnetic walls at the ports to produce an error matrix for each port. The error matrices are then cascaded with a network matrix found by the same calibration technique. Based on the insertion of feed-line error matrices, the network discontinuity is excited with a fundamental mode only. While there are significant advantages to this method, it still requires the use of feed lines, which must be long enough when separately analyzed by the MoM to produce only a fundamental mode at the network junction. Also, each feed line must be separately analyzed twice (once each with idealized open and short loads) and then the junction itself analyzed with further MoM runs. For example, three SOC MoM runs are required for a one-port network. One of the advantages of the SOC method is that the feed lines used in the final MoM run(s) can be very short, as seen in [24, Fig. 5]. The orthogonality method needs two MoM runs for a one-port network, one for the actual circuit and one for the probe arm, which is short. The orthogonality method derives the port propagation constant ( ) as part of the deembedding process and feed-line characteristic impedance ( ) is not needed, which is advantageous because at high frequencies becomes ambiguous. On the other hand, when using the SOC method, both the feed-line propagation constants and characteristic impedances must be precalculated using a separate 2-D simulator.

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TABLE I ORTHOGONALITY VALUES OF OPEN-END CAPACITANCE FOR THE SAME SUCCESSIVELY SHORTER OPEN-END MICROSTRIP LINES IN [24] NORMALIZED BY THE WIDTH OF THE LINE ( = 9:6, w=h = 1:0)

VALUES

OF

TABLE II NORMALIZED OPEN-END CAPACITANCE FROM VARIOUS METHODS

Fig. 5. stub.

Table I provides normalized end capacitance values for an open-end line with and at GHz for various open-end line lengths. This is the same line analyzed by Zhu and Wu, except that is specified in [24] as the substrate used in the SOC calculations. However, the “static” value in [24, Fig. 5(b)] can be derived by the method of Silvester and Benedek [22] using an and line. Results of the orthogonality method in Table I are provided for comparison with SOC and “static” values given in [24, Fig. 5(b)]. Normalized capacitance for the 5.6-mm line is not shown in Table I. This is because the 5.6-mm line consists of only five cells with the grid size of used in [24], which does not permit the required shifting of the probe arm in orthogonality deembedding. The 5.6-mm line could have been analyzed with finer gridding, but would not result in data that could be directly compared with [24]. Reference [24, Fig. 5(b)] gives 51.8 pF/m as the normalized end capacitance value from the SOC method and 57.6 pF/m for the “static” method. This represents a difference of approximately 10% at GHz (without knowing whether Zhu and Wu actually used ). Table I yields an average value of 55.2 pF/m for the orthogonality method (a difference of 4.1% at 2.0 GHz when compared with the static value). A static analysis has also been performed by Martel et al. for the same , line with end capacitance of 57.2 pF/m reported in [28]. Table II provides a summary comparison of the above methods. If Zhu and Wu truly used , then the SOC rewill be lower than 51.8 pF/m by approximately sult for 2% due to using a slightly lower characteristic impedance line. As a further means of assessing the accuracy of computed open-end capacitance, the results of Oh et al. [27] for the mm, mm, and line are compared here with the orthogonality method. Oh et al. report 17.33 fF and the orthogonality method yielded 16.62 fF as the open-end capacitances for this line. While the SOC results in [24] indicate stability of the method, the accuracy of the SOC for microstrip open-end capacitance results is inferior to that of the orthogonality method when compared to static results presented here. Open-end conductance

Normalized length extension of a w=h = 1:79,  = 2:32 microstrip

Fig. 6. End conductance of a w=h = 1:79 and  = 2:32 microstrip stub.

results are not provided in [24], thus, we have turned to other sources to validate our method for this quantity. computed by the Fig. 5 compares the length extension orthogonality method, three-point deembedding method, and analytical method of James and Henderson [9]. Results of the orthogonality method and the analytical method of James and Henderson compare well at lower frequencies where the length extension approaches a constant. At higher frequencies, both numerical methods significantly underestimate the effective line extension, indicating the need for further work in this area of numerical deembedding. The experimental open-end conductance measurements of Wood et al. [10] [obtained by measuring voltage standing-wave ratio (VSWR)] are plotted in Fig. 6 along with results from numerical methods implemented in this study for an , mm, and mm microstrip stub. This is a relatively wide line with a correspondingly thick substrate that produces measurable radiation. Many of the investigators cited in this study have reported results using this line [8]–[11], [14]–[16]. This has made the task

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Fig. 8. w=h

Effects of shortening the network extension arms of an  = 12:9 and

= 0:37 line with =20 length cells.

Fig. 7. End conductance and end susceptance of an open-end  = 12:9 and w=h = 0:37 line.

of comparing methods convenient. In all of these cases, the experimental measurements of Wood et al. are plotted as a benchmark data set. Thus, it is straightforward to compare results of the orthogonality method with those cited above. The analytical method of Jackson and Pozar and the orthogonality method provide better agreement with the experimental data in Fig. 6 than the numerical methods shown. The analytical methods of Sobol [16], James and Henderson [9], and Shabunin [8] all compare well with the measured data. However, these methods are not general and are limited to the analysis of the open-end discontinuity. Fig. 7 compares normalized end conductance and susceptance values obtained by three numerical methods. The orthogonality method performs better than the Prony and three-point techniques, which typically underestimate the radiated power from the open-end. Overall, the orthogonality method provides the best results of the numerical methods compared here. The primary motive for developing the orthogonality-based method was to improve upon the accuracy that is obtained by methods that assume that the amplitudes of forward and backward traveling fundamental modes are much greater than all other modes such that the presence of all other modes can be ignored. The orthogonality method extracts the fundamental modes from samples of current that are allowed to contain higher order modes. In this method, the assumption is made

Fig. 9. w=h

Attenuation constant of a lossless microstrip with 

= 0:37.

= 12:9 and

that the magnitude of the product of the fundamental-mode amplitudes is sufficiently greater than the sum of all other higher order products that the sum of higher order products can be neglected in a Lorentz reciprocity evaluation. Fig. 8 provides an example of performance degradation when the network extension arm and probe arm are reduced in length such that (16) becomes invalid. Guide wavelength is computed using different numbers of length cells in the deembedding arm. In Fig. 8, it is seen that the error in the three-point deembedding method [30] is approximately 0.05 mm for a 14-cell extension arm and converges to the value computed by the orthogonality method at 18 cells. While the orthogonality method computes constant guide-wavelength values from 18 cells down to 12 cells, errors in guide wavelength rapidly increase as errors in the wave variable product criterion (16) increase as the length of the feed line is decreased. Another way of measuring the accuracy of a deembedding method is to evaluate the attenuation constant extracted for a lossless line. Ideally, for a lossless line. Fig. 9 compares the attenuation constant computed with the three-point and orthogonality methods and reveals that, in this case, the orthogo-

SPOWART AND KUESTER: ORTHOGONALITY-BASED DEEMBEDDING TECHNIQUE FOR MICROSTRIP NETWORKS

TABLE III COMPUTED VALUES OF GUIDE WAVELENGTH BY THE ORTHOGONALITY AND THREE-POINT DEEMBEDDING METHODS FOR SUCCESSIVELY SHORTER GRID CELLS

nality method provides improved performance as its attenuation constant curve is lower by at least 10% (and often much more) than the three-point curve. Table III shows the results of a convergence analysis of guide wavelength extracted from a microstrip line that is gridded with successively shorter cells while keeping the overall length of the line constant. Guide wavelength is computed for the and line at 10 GHz. Table III is offered as evidence of the numerical stability of the orthogonality method as grid cell size is reduced. IV. CONCLUSION In this study, a new method is developed for extracting scattering parameters from numerical values of the surface current distribution in the feed lines of -port passive microstrip circuits. The Lorentz reciprocity theorem is applied to the circuit under study in such a way as to develop an EFIE for products of wave amplitudes of the forward and reverse modes at the ports of the circuit under study and in circuit stubs used to probe the circuit feed lines. The property of orthogonality of the modes of identical feed lines is then used to extract from the EFIE the fundamental mode from the sum of all modes in a manner that produces less error than would result from simply neglecting the existence of all higher order modes. The MoM is used to solve the EFIE for wave amplitude products. Ratios of these products directly yield the scattering parameters and propagation constants of the circuit ports. Improved results are obtained with very little extra computational effort. Electrically short feed lines and probe arms are used, as they result in less computational work for a given grid size. As discussed in Section III, the orthogonality method yields superior results when feed lines are made longer than approximately . Conversely, circuit feed lines may be shortened by approximately while maintaining the same accuracy as provided by other commonly used numerical deembedding methods. All deembedding approaches require feed lines with enough length that only the fundamental mode arrives at the discontinuity. In this regard, the orthogonality method can be used to improve other deembedding methods. ACKNOWLEDGMENT The authors wish to thank the National Center for Atmospheric Research, Boulder, CO, for supporting this work and the reviewers of this paper’s manuscript for their valuable suggestions.

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REFERENCES [1] R. F. Harrington, Field Computations by Moment Methods. New York: Macmillan, 1968. [2] D. I. Wu, D. C. Chang, and B. I. Brim, “Accurate numerical modeling of microstrip junctions and discontinuities,” Int. J. Microwave MillimeterWave Computer-Aided Eng., vol. 1, no. 1, pp. 48–58, 1991. [3] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1973. [4] V. V. Shevchenko, Continuous Transitions in Open Waveguides. Boulder, CO: Golem, 1971. [5] D. Marcuse, Light Transmission Optics, 2nd ed. New York: Van Nostrand, 1982. [6] R. F. Harrington, Time–Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [7] F. Sporleder and H. G. Unger, Waveduide Tapers Transitions and Couplers. New York: IEEE Press, 1979. [8] S. N. Shabunin, “Radiation from the open-end of a microstrip line,” Izv. Vyssh. Uchebn. Zaved Radioelektron., vol. 27, no. 2, pp. 76–78, 1984. [9] J. R. James and A. Henderson, “High-frequency behavior of microstrip open-circuit terminations,” IEE J. Microwaves, Opt., Acoust., vol. 3, no. 5, pp. 205–218, 1979. [10] C. Wood, P. S. Hall, and J. R. James, “Radiation conductance of opencircuit low dielectric constant microstrip,” Electron. Lett., vol. 14, pp. 121–123, 1978. [11] G. Kompa, “Approximate calculation of radiation from open-ended wide microstrip lines,” Electron. Lett., vol. 12, no. 9, pp. 222–224, 1976. [12] L. J. Van Der Pauw, “The radiation of electromagnetic power by microstrip configurations,” IEEE Microw. Theory Tech., vol. MTT-25, no. 9, pp. 719–725, Sep. 1977. [13] E. J. Denlinger, “Radiation from microstrip resonators,” IEEE Microw. Theory Tech., vol. MTT-17, no. 4, pp. 235–236, Apr. 1969. [14] L. Lewin, “Radiation from discontinuities in strip-line,” Proc. Inst. Elect. Eng., pt. H, vol. 107, pp. 163–170, 1960. [15] R. W. Jackson and D. M. Pozar, “Full-wave analysis of microstrip open-end and gap discontinuities,” IEEE Microw. Theory Tech., vol. MTT-33, no. 10, pp. 1036–1042, Oct. 1985. [16] H. Sobol, “Radiation conductance of open-circuit microstrip,” IEEE Microw. Theory Tech., vol. MTT-19, no. 11, pp. 885–887, Nov. 1971. [17] M. Kirschning, R. H. Jansen, and N. H. L. Koster, “Accurate model for open end effect of microstrip lines,” Electron. Lett., vol. 17, no. 3, pp. 123–124, 1980. [18] R. H. Jansen and N. H. L. Koster, “Accurate results on the end effect of single and coupled microstrip lines for use in microwave circuit design,” Arch. Elektr. Ubertragung, vol. 34, no. 2, pp. 453–459, 1980. [19] S. S. Toncich and R. E. Collin, “Characterization of microstrip discontinuities by a dynamic source reversal method using potential theory,” in Proc. URSI Int. Electromagnetic Theory Symp., Sydney, Australia, Aug. 17–20, 1992. [20] R. H. Jansen, “Hybrid mode analysis of end effects of planar microwave and millimeter wave transmission lines,” Proc. Inst. Elect. Eng., pt. H, vol. 128, pp. 77–86, 1981. [21] S. S. Toncich, R. E. Collin, and K. B. Bhasin, “Full-wave characterization of microstrip open end discontinuities patterned on anisotropic substrates using potential theory,” IEEE Microw. Theory Tech., vol. 41, pp. 2067–2073, Dec. 1993. [22] P. Silvester and P. Benedek, “Equivalent capacitance of microstrip open circuits,” IEEE Microw. Theory Tech., vol. MTT-20, no. 4, pp. 511–516, Aug. 1972. [23] E. O. Hammerstad, “Equations for microstrip circuit design,” in IEEE MTT-S Int. Microwave Symp. Dig., 1980 , pp. 407–409. [24] l. Zhu and K. Wu, “Unified equivalent-circuit model of planar discontinuities suitable for field theory-based CAD and optimization of M(H)MIC’s,” IEEE Microw. Theory Tech., vol. 47, no. 9, pp. 1589–1601, Sep. 1999. [25] , “Short-open calibration technique for field theory-based parameter extraction of lumped elements of planar integrated circuits,” IEEE Microw. Theory Tech., vol. 50, no. 8, pp. 1861–1869, Aug. 2002. [26] R. S. Chen, D. X. Wang, and E. K. N. Yung, “Application of the shortopen calibration technique to vector finite element method for analysis of microwave circuits,” Int. J. Numer. Modeling, vol. 16, pp. 367–385, 2003. [27] K. S. Oh, J. E. Schutt-Aine, and R. Mittra, “Computation of excess capacitances of various strip discontinuities using closed-form Green’s functions,” IEEE Microw. Theory Tech., vol. 44, no. 5, pp. 783–788, May 1996. [28] J. Martel, R. R. Boix, and M. Horno, “Static analysis of microstrip discontinuities using the excess charge density in the spectral domain,” IEEE Microw. Theory Tech., vol. 39, no. 9, pp. 1623–1631, Sep. 1991.

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[29] T. K. Sarkar, Z. A. Maricevic, M. K. Djordjevic, and A. R. Djordjevic, “Frequency dependent characterization of radiation from an open end microstrip line,” Arch. Elektr. Ubertragung, vol. 48, no. 2, pp. 101–107, May 1994. [30] D. C. Chang and J. X. Zheng, “Electromagnetic modeling of passive circuit elements in MMIC,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 9, pp. 1741–1747, Sep. 1992. [31] J. C. Rautio, “A de-embedding algorithm for electromagnetics,” Int. J. Microwave Millimeter-Wave Computer-Aided Eng., vol. 1, no. 3, pp. 282–287, 1991. [32] L. Zhu and K. Wu, “Characterization of unbounded multiport microstrip passive circuits using an explicit network-based method of moments,” IEEE Microw. Theory Tech., vol. 45, no. 12, pp. 2114–2124, Dec. 1997. [33] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, 1983. [34] L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis. Reading, MA: Addison-Wesley, 1991. [35] S. L. Marple, Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: Prentice-Hall, 1987. [36] S. M. Kay, Modern Spectral Estimation, Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1988. [37] M. L. Van Blaricum and R. Mittra, “A technique for extracting the poles and residues of a system directly from its transient response,” IEEE Trans. Antennas Propag., vol. AP-23, no. 11, pp. 777–781, Nov. 1975. , “Problems and solutions associated with Prony’s method for pro[38] cessing transient data,” IEEE Trans. Antennas Propag., vol. AP-26, no. 1, pp. 174–182, Jan. 1978. [39] A. J. Poggio, M. L. Van Blaricum, E. K. Miller, and R. Mittra, “Evaluation of a processing technique for transient data,” IEEE Trans. Antennas Propag., vol. AP-26, no. 1, pp. 165–173, Jan. 1978. [40] Y. Hua and T. K. Sarkar, “Generalized pencil-of- function method for extracting poles of an electromagnetic system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989. [41] Y. Hua and . K. Sarkar, “Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,” IEEE Trans. Acous., Speech, Signal Process., vol. 38, no. 5, pp. 814–824, May 1990. [42] T. K. Sarkar, Z. A. Marlcevic, and M. Kahrizi, “An accurate de-embedding procedure for characterizing discontinuities,” Int. J. Microwave Millimeter-Wave Computer-Aided Eng., vol. 2, no. 3, pp. 135–143, 1992. [43] E. Drake, R. R. Boix, M. Horno, and T. K. Sarkar, “Comparison among different approaches for the full-wave MOM characterization of openended microstrip lines,” Microwave Opt. Technol. Lett., vol. 21, no. 4, pp. 246–248, 1999.

Michael P. Spowart received the B.S.M.E. degree from the University of Hawaii, Honolulu, in 1971, the M.S.M.E degree from the University of Colorado at Boulder, in 1972, the M.S.E.E. degree from the University of Washington, Seattle, in 1975, and the Ph.D. degree in electrical engineering (in the field of electromagnetics) from the University of Colorado at Boulder, in 1996. Since 1980, he has been with the Atmospheric Technology Division, National Center for Atmospheric Research, Boulder, CO, where he is currently Head of Instrumentation Engineering with the Research Aviation Facility. His professional interests include electronics, signal processing, and applied mathematics.

Edward F. Kuester (F’98) received the B.S. degree from Michigan State University, East Lansing, in 1971, and the M.S. and Ph.D. degrees from the University of Colorado at Boulder, in 1974 and 1976, respectively, all in electrical engineering. Since 1976, he has been with the Department of Electrical and Computer Engineering, University of Colorado at Boulder, where he is currently a Professor. In 1979, he was a Summer Faculty Fellow with the Jet Propulsion Laboratory, Pasadena, CA. From 1981 to 1982, he was a Visiting Professor with the Technische Hogeschool, Delft, The Netherlands. From 1992 to 1993, he was a Professeur Invité with the École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. In 2002 and 2004, he was a Visiting Scientist with the National Institute of Standards and Technology (NIST), Boulder, CO. He coauthored one book, two book chapters, and has translated two Russian books. He has authored or coauthored over 60 papers in refereed technical journals. He coholds two U.S. patents. His research interests include the modeling of electromagnetic phenomena of guiding and radiating structures, applied mathematics, and applied physics. Dr. Kuester is a member of the Society for Industrial and Applied Mathematics and Commissions B and D of the International Union of Radio Science (URSI).

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Superconducting Spiral Filters With Quasi-Elliptic Characteristic for Radio Astronomy Guoyong Zhang, Frederick Huang, and Michael J. Lancaster, Senior Member, IEEE

Abstract—To produce a filter small enough to fit a 2-in wafer at 408 MHz while maintaining high-quality performance, half-wavelength single spiral microstrip resonators are introduced. New coupling structures make both positive and negative coupling available by changing the directions of spiral winding. An eight-pole high-temperature superconducting bandpass spiral filter with 3.7% bandwidth at 408-MHz band is presented for radio astronomy applications at the Jodrell Bank Radio Observatory, Macclesfield, Cheshire, U.K. A quasi-elliptic characteristic with four transmission zeros is realized by adding three cross-couplings to the standard Chebyshev filter. The filter shown is designed and fabricated on a 32 mm 18 mm 0.508 mm MgO substrate. The untuned measured results of the filter at 30 K show a maximum passband insertion loss 0.35 dB (ripple 0.27 dB), a minimum return loss 13.2 dB, and minimum out-of-band rejection of 65 dB, which have good agreement with its electromagnetic full-wave simulation results. Index Terms—Bandpass, cross-coupling, high-temperature superconducting (HTS), spiral resonator.

I. INTRODUCTION

A

T THE ultrahigh-frequency (UHF) band, the radio astronomical observation at the Jodrell Bank Radio Observatory, Macclesfield, Cheshire, U.K., is seriously contaminated by local TV signals. Since the radio telescope receivers are already cooled to cryogenic temperatures to reduce the noise in the semiconductor electronics, high-temperature superconducting (HTS) bandpass filters, with extremely high selectivity and low passband insertion loss, offer a convenient effective way to eliminate the interference. Generally, there are two different design approaches to realize filters with high selectivity. The first one increases the number of resonators in the filter design, which has the undesired effects of increasing insertion loss, producing an asymmetrical response because of the strong unwanted cross-couplings, and increasing the filter size [1]–[4]. The second approach deliberately introduces additional cross-coupling between nonadjacent resonators to realize quasi-elliptic filter characteristics [5]–[7]. However, it is impossible to design a filter using conventional large half-wavelength microstrip resonators at UHF band because of the 2–3-in limited size of a substrate. Single spiral resonators have been known to have numerous advantages, including highly efficient utilization of limited area, high quality factor, good power handling, and are insensitive to fabrication Manuscript received March 31, 2004; revised June 28, 2004. This work was supported by the U.K. Engineering and Physical Sciences Research Council. The authors are with the Department of Electronic and Electrical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842485

procedures. Single spiral resonators have been used for noncross-coupled filters in [8]–[11]. In this paper, to achieve a filter small enough to fit a 2-in wafer at the specified 408 MHz by the Jodrell Bank Observatory, while maintaining high-quality performance, single spiral resonator filters are designed. The coupling of coupled single spiral resonators is analyzed. Different types of coupling structures make both positive coupling and negative coupling available by changing the directions of spiral winding, clockwise or anticlockwise, following a track from the outside to the center. Quasi-elliptic characteristic bandpass filters can be easily realized without changing the forms of spirals [12]. One design example of a four-pole filter with one cross-coupling that exhibit a single pair of transmission zeros at finite frequencies is demonstrated with simulation results. In addition, an eight-pole HTS spiral bandpass filter with 3.7% bandwidth at 408-MHz band fabricated on a 32 mm 18 mm 0.508 mm MgO substrate is presented. A quasi-elliptic characteristic with four transmission zeros is realized by adding three cross-couplings to the standard Chebyshev filter. Untuned experimental results have good agreement with the simulation results. II. ANALYSIS OF COUPLED SPIRAL RESONATORS Microstrip resonators can be coupled electrically, magnetically, or there can be mixed coupling with both electric and magnetic coupling. The mixed coupling coefficient can be written as [5]

(1)

where and are the vector electric fields of resonators 1 and are the vector magnetic fields, and and and 2, are the volume of the air and substrate, respectively. Note the negative sign in the equation. Coupling structures of coupled spiral resonators are shown in Fig. 1. The potential outside and inside of the spirals is shown for the symmetric mode at one of the resonant frequencies. In the time domain, charge density is simultaneously positive at both ends of the resonators, and negative at the other ends of the tails. and indicate the currents in the spiral winding. Fig. 2 shows two resonators, which are weakly coupled as in Fig. 1(a).

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Fig. 1. Coupling structures of coupled spiral resonators with: (a) and (b) the same directions of spiral winding and (c) and (d) the opposite directions. This diagram is not to scale.

Fig. 3. (a) Coupling coefficient k as a function of coupling gap s between coupled spiral resonators in correspondence with the coupling structure of Fig. 1(a) and (b). (b) Magnitude (in decibels) (solid line) and phase (DEG) of S (dotted line) when the coupling gap is 0.8 mm in Fig. 1(a). Fig. 2. Top and side views of coupled spiral resonators like Fig. 1(a). Parts of the electric and magnetic fields are indicated.

and are both predominantly in the -direction. In a cycle, currents flow away from the positive charged ends. The and are both significantly only in the magnetic fields space between the coupled resonators. Here, their -components are in the opposite directions. The -component of the magnetic field is much weaker compared to the -component in the microstrip structure and, therefore, can be ignored. Hence, in (1), the dot product of the magnetic fields and is negand is ative, while the dot product of the electric fields positive. The conclusion is drawn that the electric coupling and magnetic coupling in Figs. 1(a) and 2(b), where coupled spirals resonators are wound in the same directions, tend to add. Similarly, the electric coupling and magnetic coupling in Fig. 1(c) and (d), where coupled spirals resonators are wound in the opposite directions, tend to cancel. As it is difficult to obtain accurate explicit equations for resonance and coupling coefficient of spiral resonators, full-wave electromagnetic simulation by Sonnet EM [15] is used in our studies. The coupling coefficients between two frequency synchronous resonators are calculated by the well-known technique described in [13].

In simulations, the dielectric layer is an MgO substrate with . The thickness of the air 0.508-mm thickness and layer is 5 mm. Lossless conductor is assumed for the case of superconducting thin film. The linewidth of the resonator is 0.05 mm and gap between tracks is 0.05 mm, giving a rectanMHz with an gular spiral resonator with resonance area of only 2.75 mm 5.05 mm. To accumulate more charge at the outside end of the spiral resonator, which gives rise to stronger electric coupling as the coupling structure, as shown in Fig. 1(a), the width of the outside end is increased to 0.25 mm. Considering both the accuracy and simulation time [10], [14], 50- m cell size in the - and -axis directions is chosen in the simulation. The coupling coefficient as a function of coupling gap , between coupled spiral resonators in Fig. 1, is shown in Figs. 3(a) and 4(a). Phase is shown, as well as the magnitude of in Figs. 3(b) and 4(b). Since electric coupling and magnetic coupling are 180 out-of-phase, the sign of the electric coupling should be opposite to that of magnetic coupling, which can be defined as positive. As concluded above, the electric coupling and magnetic coupling tend to add when two coupled spiral resonators are wound in the same direction. Thus, in Fig. 3(a), the coupling coefficients, which keep negative and converge to zero as coupling gaps increase, indicates that electric and magnetic

ZHANG et al.: SUPERCONDUCTING SPIRAL FILTERS WITH QUASI-ELLIPTIC CHARACTERISTIC FOR RADIO ASTRONOMY

Fig. 4. (a) Coupling coefficient k as a function of coupling gap s between coupled spiral resonators in correspondence with the coupling structure of Fig. 1(c) and (d). (b) Magnitude (in decibels) (solid line) and phase (DEG) of S (dotted line) when the coupling gap is 0.8 mm in Fig. 1(c).

coupling enhance each other. On the other hand, the electric and magnetic coupling tend to cancel when two coupled spiral resonators are wound in the opposite direction. As indicated in Fig. 4(a), the coupling in correspondence with the structure in Fig. 1(c), for small gaps, which is negative, should be mainly electric coupling, while for large gaps, the coupling should be dominated by magnetic coupling. The coupling is positive because magnetic coupling is much stronger than electric coupling when the spiral resonators are coupled, as shown in Fig. 1(d). Note the phase difference (DEG) of different coupling structures, where electric coupling dominates [see Fig. 3(b)] or magnetic coupling dominates [see Fig. 4(b)]. III. DESIGN OF SPIRAL FILTERS By controlling of directions of spiral winding and distance of coupling gap, different signs of coupling can be realized. This is important for the quasi-elliptic characteristic filter. A. Design Example I The first design is a four-pole cross-coupled bandpass filter with the specification a center frequency of 408 MHz and a 1.2% bandwidth (3 dB). Fig. 5(a) shows the layout and topology of , , and are realthe filter. The negative couplings ized using resonators oriented as shown in Fig. 1(b), (a), and

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Fig. 5. (a) Layout and topology of one four-pole quasi-elliptic spiral filter, which is designed for 408.7-MHz center frequency and 1.2% bandwidth using 16 mm 18 mm 0.508 mm MgO substrate. (b) Simulated performance of the four-pole spiral filter.

2

2

TABLE I NORMALIZED COUPLING MATRIX FOR THE FOUR-POLE FILTER WITH ONE CROSS-COUPLING

(b), respectively. Resonators 1 and 4 are oriented as shown in [dotted line in the Fig. 1(d) to realize the positive coupling topology of the filter shown in Fig. 5(a)]. The normalized coupling matrix with one cross-coupling for this filter is shown in Table I. and are The normalized input and output admittances both 1.04. Two additional quarter-wavelength spirals shown separate from the four isolated spiral resonators in Fig. 6(a) are to provide external magnetic couplings. The filter is symmetrical and the gaps (with reference to Fig. 5(a), , , and ) are 0.15, 0.85, 0.30, and 2.10 mm. The quasi-elliptic filter could be implemented on a 16 mm 18 mm MgO substrate, whose thickness is 0.508 mm and dielectric constant is 9.66 in the simulation. The designed results of the filter after optimization using simulation are shown in Fig. 6(b). The asymmetrical frequency positions of transmission zeros mainly result from the undesired nonadjacent coupling between resonators 1 and 3, 6 and 8, 2 and 4, 5 and 8, etc.

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Fig. 6. (a) Layout of an eight-pole spiral filter with three cross-coupling on 32 mm 18 mm 0.508 mm MgO substrate. (b) Simulated performance of the filter with four transmission zeros. The solid line is S (in decibels) and the dotted line is S (in decibels).

2

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TABLE II NORMALIZED COUPLING MATRIX OF THE EIGHT-POLE FILTER WITH THREE CROSS-COUPLINGS

Fig. 7. (a) Measured performance of the eight-pole spiral filter with four transmission zeros. The solid line is S (in decibels) and the dashed line is S (dB). (b) S (in decibels) response in wide-band.

B. Design Example II One eight-pole quasi-elliptic spiral filter with a specified center frequency of 408 MHz, and 3.7% bandwidth (3 dB) is designed as indicated in Fig. 6(a). Resonators 1 and 2 are oriented relative to each other, as shown in Fig. 1(b), while resonators 3 and 4 are as shown Fig. 1(a) to realize negative couplings and . The positive couplings and are realized as shown in Fig. 1(d). Due to the symmetrical structure of the filter, the resonators and couplings can be mirrored horizontally. The coupling matrix for this filter we chose with three crosscouplings , , and is shown in Table II. For a quasi-elliptic filter response, the cross-couplings have to be negative according to the coupling matrix. The cross-coupling paths are introduced by adding transmission lines between the ends of the spirals. The strength of the cross-couplings is mainly determined by the gap between these lines and the spiral resonators, but is also influenced by the characteristics (e.g., length and width) of the transmission lines. By adjusting the physical parameters, the transmission zeros can be moved. The spiral resonator is the same as designed in Section II. Gaps of the symmetrical filter corresponding to the coupling

coefficients from to are 0.15, 0.30, 0.35, 0.65, and 0.60 mm. Fig. 6(b) shows the three cross-couplings producing four transmission zeros located at 1.7 and 7.5 MHz from the lower band edge, and 2.6 and 15.4 MHz from the upper band edge. The asymmetrical positions of transmission zeros are mainly caused by the undesired nonadjacent couplings. It is believed that this can be tuned by placing tuning screws above all cross-coupling area. The designed passband is from 398.9 to 414.8 MHz. The maximum insertion loss is 0.07 dB throughout the passband due to imperfect matching. The filter only occupies a 32 mm 18 mm area on an MgO substrate. IV. EXPERIMENTAL PERFORMANCE The eight-pole quasi-elliptic spiral filter is manufactured using HTS YBCO films of 600-nm thickness on a 0.508-mm-thick MgO substrate. Films are deposited on both sides of the substrates. The same package technique is taken as shown in [14]. Fig. 7(a) shows the measured response at 30 K. No tuning was performed in the measurement. The experimental performance of the filter shows good agreement with the simulated response. The minimum insertion loss of the device is 0.075 dB and includes the loss of K connectors, indicating an average unloaded

ZHANG et al.: SUPERCONDUCTING SPIRAL FILTERS WITH QUASI-ELLIPTIC CHARACTERISTIC FOR RADIO ASTRONOMY

resonator value of 25 000. The passband ripple is approximately 0.27 dB and the corresponding return loss is better than 13.2 dB. Four transmission zeros appears clearly, which are located 1.5 and 6.6 MHz from the lower band edge, and 2.2 and 8.7 MHz from the upper band edge. The out-of-band rejection is kept below 70 dB up to the second harmonic response at 930 MHz [see Fig. 7(b)]. However, the measured center frequency is shifted 0.8 MHz lower than the simulated one, which is only 0.2% of the center frequency. This may be caused by the variations of materials, in this case, mainly from the dielectric constant and thickness of the substrate. V. CONCLUSION Different types of coupling structures allow easy implementation of positive and negative coupling in the design of a quasi-elliptic spiral filter by changing the directions of spiral winding. Design examples have been presented with respect to different filter topologies. Considering the application at the UHF band on radio astronomy, an eight-pole spiral bandpass filter with quasi-elliptic characteristic has been designed, fabricated, and measured. The experimental results show good coincidence with simulated results.

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[12] F. Huang and X. Xiong, “Very compact spiral resonator implementation of narrow-band superconducting quasielliptic filters,” in 33rd Eur. Microwave Conf. Dig, Oct. 2003, pp. 1059–1062. [13] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters Impedance-Matching Networks and Coupling Structures. Norwood, MA: Artech House, 1980. [14] G. Zhang, M. J. Lancaster, F. Huang, M. Zhu, and B. Cao, “Accurate design of high Tc superconducting microstrip filter at UHF band for radio astronomy front end,” in IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2004, pp. 1115–1118. [15] EM User’s Manual, Version 8.0, Sonnet Software Inc., North Syracuse, NY, 2002.

Guoyong Zhang was born in Tianjin, China, on November 26, 1976. He received the B.Sc. degree in physics from Nankai University, Tianjin, China, in 1999, the M.Sc. degree in physics from Tsinghua University, Beijing, China, in 2002, and is currently working toward the Ph.D. degree (part time) at the University of Birmingham, Edgbaston, Birmingham, U.K. From 1999 to 2002, he was engaged in research on HTS filters for mobile communication base-station systems as a Research Assistant with Tsinghua University. Since July 2002, he has been a Research Fellow with the School of Electronic and Electrical Engineering, University of Birmingham. His current interests include HTS RF and microwave filters and couplers from UHF band to C -band for radio astronomy observatory applications and mobile communication applications.

ACKNOWLEDGMENT The authors would like to thank N. Roddis, Jodrell Bank Radio Observatory, Macclesfield, Cheshire, U.K., for providing the specification and also wish to thank D. Holdom for fabricating the HTS circuit, and C. Ansell for his technical support. REFERENCES [1] D. Zhang, G.-C. Liang, C. F. Shih, Z. H. Lu, and M. E. Johansson, “A 19-pole cellular bandpass filter using 75-mm-diameter high-temperature superconducting thin films,” IEEE Microw. Guided Wave Lett., vol. 5, no. 11, pp. 405–407, Nov. 1995. [2] G. Zhang, M. Zhu, and B. Cao, “Design and performance of a compact forward-coupled HTS microstrip filter for a GSM system,” IEEE Trans. Appl. Supercond., vol. 12, no. 4, pp. 1897–1901, Dec. 2002. [3] M. S. Gashinova, M. N. Goubina, G. Zhang, I. A. Kolmakov, Y. A. Kolmakov, and I. B. Vendik, “High-Tc superconducting planar filter with pseudo-Chebyshev characteristic,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 792–795, Mar. 2003. [4] G. Tsuzuki, M. Suzuki, and N. Sakakibara, “Superconducting filter for IMT-2000 band,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2519–2525, Dec. 2000. [5] J. S. Hong and M. J. Lancaster, “Couplings of microstrip square openloop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 11, pp. 2099–2109, Nov. 1996. [6] M. Reppel and J.-C. Mage, “Superconducting microstrip bandpass filter on LaAlO with high out-of-band rejection,” IEEE Microw. Guided Wave Lett., vol. 10, no. 5, pp. 180–182, May 2000. [7] G. Tsuzuki, S. Ye, and S. Berkowitz, “Ultra selective 22-pole, 10-transmission zero superconducting bandpass filter surpasses 50-pole Chebyshev rejection,” in IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2002, pp. 1963–1966. [8] E. Gao, S. Sahba, H. Xu, and Q. Y. Ma, “A superconducting RF resonator in HF range and its multi-pole filter applications,” IEEE Trans. Appl. Supercond., vol. 9, no. 2, pp. 3306–3069, Jun. 1999. [9] C. K. Ong, L. Chen, J. Lu, C. Y. Tan, and B. T. G. Tan, “High-temperature superconducting bandpass spiral filter,” IEEE Microw. Guided Wave Lett., vol. 9, no. 10, pp. 407–409, Oct. 1999. [10] F. Huang, “Ultra-compact superconducting narrow-band filters using single- and twin-spiral resonators,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 487–491, Feb. 2003. [11] E. Sakurai, Z. Ma, and Y. Kobayashi, “Coupling characteristics of microstrip spiral resonators and the design of a 4-pole bandpass filter,” IEICE, Tokyo, Japan, Tech. Rep. MW2001-67, Sep. 2001.

Frederick Huang was born in Singapore, in 1955. He received the B.A. degree in engineering science and D.Phil. degree from the University of Oxford, Oxford, U.K., in 1980 and 1984, respectively. His doctoral research concerned surface acoustic wave (SAW) devices, mainly dot-array pulse compressors. He spent two years with Racal Research Ltd., where he was involved with the processing of speech signals, including analog voice scramblers. At the end of 1985, he joined Thorn EMI, and was seconded to Oxford University, where he studied the use of Langmuir–Blodgett films in SAW devices. Since 1989, he has been a Lecturer with the University of Birmingham, Edgbaston, Birmingham, U.K., where he is currently with the Electronic, Electrical, and Computer Engineering Department, School of Engineering. He has been involved with superconducting delay-line filters including linear phase and chirp devices. His minor interests are microstrip and waveguide discontinuities. His more recent research areas include superconducting switched filters, slow-wave structures, quasi-lumped element filters, and spiral bandpass filters.

Michael J. Lancaster (M’91–SM’04) received the Physics and Ph.D. degrees from Bath University, Bath, U.K., in 1980 and 1984, respectively. His doctoral research concerned nonlinear underwater acoustics. Upon leaving Bath University, he joined the Surface Acoustic Wave (SAW) Group, Department of Engineering Science, Oxford University, as a Research Fellow. His research concerned the design of new novel SAW devices including filters and filter banks. These devices worked in the 10-MHz–1-GHz frequency range. In 1987, he became a Lecturer of electromagnetic (EM) theory and microwave engineering with the School of Electronic and Electrical Engineering, University of Birmingham, Edgbaston, Birmingham, U.K. Shortly upon joining the School of Engineering, he began the study of the science and applications of high-temperature superconductors, involved mainly with microwave frequencies. He currently heads the Emerging Device Technology Research Centre. His current personal research interests include microwave filters and antennas, as well as the high-frequency properties and applications of a number of novel and diverse materials.

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A Monolithic 12-GHz Heterodyne Receiver for DVB-S Applications in Silicon Bipolar Technology Giovanni Girlando, Santo A. Smerzi, Tino Copani, and Giuseppe Palmisano

Abstract—A monolithic heterodyne receiver for digital video broadcasting via-satellite (DVB-S) applications is presented. The integrated circuit consists of a down-converter block and a phase-locked-loop-based local-oscillator synthesizer to translate the DVB-S RF-band (10.7–12.75 GHz) to an IF ranging in the -band (0.95–2.15 GHz). The receiver exhibits a conversion gain of 38 dB, a single-sideband noise figure of 7 dB, and an output 1-dB compression point of 5 dBm. A 2.2-GHz-wide voltage-controlled oscillator (VCO) tuning range, extending from 8.6 to 10.8 GHz, is achieved adopting a transformer-based topology. The VCO phase noise is as low as 95 dBc/Hz at 100-kHz offset from a 10.6-GHz carrier. The integrated receiver draws 160 mA from a 3.3-V supply voltage. This paper demonstrates the feasibility -band DVB-S heterodyne receiver integrated in low-cost of a silicon bipolar technology. 46-GHz-

+

Index Terms—Direct broadcast satellites, low-noise amplifier, microwave bipolar integrated circuits (ICs), mixers, phase-locked loops (PLLs), phase noise, receivers.

I. INTRODUCTION

H

ISTORICALLY, appealing communication mass markets were born when an increasing demand of new services and cheap equipments were available at the same time. Over the last decade, the need of frequency allocation for new wide-band digital telecommunication services (e.g., wireless local area network (WLAN), HyperLAN) has led to a crowding in the band below 10 GHz. Moreover, low-cost integrated circuits (ICs) operating in the - and -band, e.g., for the synchronous optical network platform (SONET) or industrial scientific and medical (ISM) unlicensed band, have been recently proposed, thus demonstrating the continuous development of low-cost technology processes [1]–[3]. In this scenario, low-cost and low-complexity monolithic approaches will be more and more mandatory for telecommunication systems. This paper focuses on the design and implementation of a 12-GHz heterodyne receiver for digital video broadcasting via-satellite (DVB-S) applications, integrated in a low-cost silicon bipolar technology. DVB-S systems are gaining a renewal interest, as they can support high-data rate internet services via satellite, e.g., digital video broadcasting return channel via-satellite (DVB-RCS) or high-definition television (HDTV). The block diagram of a DVB-S receiver is shown in Fig. 1. The -band satellite signal, which is a linearly polarized wave (horizontal or vertical), is picked up by a parabolic dish and Manuscript received April 21, 2004; revised September 22, 2004. G. Girlando and S. A. Smerzi are with STMicroelectronics, Catania 95100, Italy (e-mail: [email protected]; [email protected]). T. Copani and G. Palmisano are with the Electrical, Electronic, and Systems Engineering Department, University of Catania, Catania 95100, Italy (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842484

Fig. 1. Block diagram of a DVB-S receiver.

conveyed to the feed horn. Successively, the low-noise block (LNB) down-converter amplifies and shifts these signals to an IF. The IF signal is sent to the integrated receiver decoder (IRD), through the antenna cable for tuning and digital demodulation. Demanding system requirements have traditionally forced a discrete approach for LNBs [4]. Usually, GaAs high electron-mobility transistor (HEMT) and field-effect transistor (FET) devices implement the down-converter block, while dielectric resonant oscillators (DROs) provide local oscillator (LO) frequencies. LNB fabrication costs are due to components, assembling, and the manual tuning of DRO frequencies. This paper presents a single-chip receiver for DVB-S implementing a built-in phase-locked loop (PLL) that guarantees proper LO oscillation frequencies, thus eliminating DROs. To meet the severe DVB-S LO specifications, a transformer-based voltage-controlled oscillator (VCO) was used. Moreover, the integrated receiver adopts a single-ended structure for both input and output terminals, thus drastically simplifying the application board and reducing current consumption. However, this choice required a very accurate analysis of substrate and wire-bonding parasitic effects at the -band. This paper is organized as follows. A brief overview of LNB tasks is given in Section II. The proposed LNB architecture is presented in Section III. The circuit design of the down-converter is discussed in Section IV. The proposed transformerbased VCO is described in Section V. The design of a high-speed divide-by-two prescaler is dealt with in Section VI. Highlights on layout and assembling issues are discussed in Section VII. Finally, experimental results and conclusions are reported in Sections VIII and IX, respectively. II. LNB TASKS Typically, the satellite signal power at the input of the LNB ranges from 100 to 90 dBm, while the power level at the input of the IRD goes from 65 to 25 dBm. Considering path

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TABLE I LNB SPECIFICATIONS

Fig. 2.

Block diagram of the proposed LNB.

losses due to weather conditions and the antenna cable, an LNB mounted on a common 60-cm-diameter antenna has to guarantee a noise figure lower than 0.6 dB and provide a power gain of 60 dB. Since losses of the coaxial cable at 12 GHz are unacceptable, -band the LNB translates the RF satellite signals from the (RF: 10.7–12.75 GHz) to an IF ranging in the -band (IF: 0.95–2.15 GHz). More specifically, the RF DVB-S band is conventionally divided into low band (LB: 10.7–11.7 GHz) and high band (HB: 11.7–12.75 GHz). The LB and HB are translated to 0.95–1.95 and 1.1–2.15-GHz bands, respectively. The former down-conversion is acted by a 9.75-GHz LO, while the latter by a 10.6-GHz LO. Thus, the required IF band on the antenna cable is reduced from 2.05 to 1.2 GHz, avoiding the superposition with the terrestrial TV bands at VHF and UHF. III. PROPOSED LNB ARCHITECTURE The block diagram of the proposed LNB is shown in Fig. 2 [5]. Two stages of external HEMT amplifiers are placed at the input to provide low noise-figure performance and a gain around 25 dB. The band/polarization block sets the PLL frequency to convert the LB or HB to the IF. Moreover, it also turns on HEMT1V or HEMT1H for vertical or horizontal polarization, respectively. The grayed box encloses the IC. It is based on single-ended architecture for both input and output terminals. Thus, the number of external devices is reduced and the required microstrip structures on the printed circuit board (PCB) are simplified. The IC consists of two LNAs (LNA1 and LNA2), a mixer (MIX), an output buffer (BUF), and a PLL-based LO synthesizer. In the IC, LB and HB are selected by simply programming the PLL frequency divider. Besides avoiding the use of DROs, the integrated PLL eliminates the routing of LO signals on the application board of the LNB. This point is particularly appealing in multiuser LNBs (e.g., twin or quad) where two or more downconverter chains have to share the same LO signal. To fulfill the LNB specifications summarized in Table I, the IC has to provide a conversion gain of 35 dB with an in-band gain variation of 4 dB, a single-sideband (SSB) noise figure of 8 dB and an output 1-dB compression point of 5 dBm. The LO phase noise has to be as low as 95 dBc/Hz at 100-kHz offset from the carrier. Since a single VCO is adopted to synthesize both the 9.75- and 10.6-GHz frequencies, a tuning range of approximately 9% is needed. In addition to system requirements, the VCO tuning range has to cover process variations. For this reason, a tuning range of 20% was used as a specification.

Fig. 3. Schematic of the LNA1–LNA2–MIX.

IV. DOWN-CONVERTER The schematic of the LNA1-LNA2-MIX block is shown in Fig. 3. To boost the signal level and reduce the noise degradation due to the mixer MIX, two inductive degenerated cascode low-nosie amplifiers (LNAs) were used at the input of the IC. was chosen to achieve the Inductive emitter degeneration 50- input matching according to (1) as follows: (1) and are the parasitic base resistance and effective transition frequency of the input transistor, respectively. The bias current was designed to achieve simultaneously 50- input matching and minimum noise figure [6]. A 0.15-nH emitter inductor was adopted. Since (1) is frequency independent, a wide-band matching for the real part of the input impedance was obtained. To increase the power gain of LNA1, a resonant load at 11.75 GHz between the two LNAs was implemented by using and the series capacitor a 0.92-nH integrated inductor . Nearby the resonance at the operating frequency , the inductive and capacitive susceptance parts cancel out each other models the LNA load. If is and an equivalent resistance at the operating frequency the quality factor of the inductor

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Fig. 4.

Schematic of the output buffer (BUF).

,

is given by . Thus, to enhance the power gain, has to be as high as possible. The quality factor of was as high as 10 at 12 GHz. the integrated inductor On the other hand, the 3-dB bandwidth (BW) of the inter, where the quality factor of the matching network is is given by (2) as follows: network (2) , which is the real part of the input impedance of the LNA2, was increased to 50 by means of the emitter degeneration of 0.15 nH. Thus, a reduction of and, consequently, an enhancement of BW, were obtained to meet in-band gain flatness requirements. The mixer MIX consisted of a V–I converter and a Gilbert quad. The double-balanced structure of the mixer reduces the LO to RF feed-through, avoiding the high-power LO signal’s, which is very close to the RF band, desensitization of the receiver. An integrated transformer , placed between LNA2 and mixer MIX, produced the unbalanced-to-balanced signal conversion. Moreover, the transformer and metal–insulator–metal and provided an 11.75-GHz reso(MIM) capacitors nant inter-matching network. To bias the mixer MIX, two exload ternal chokes (RFC) were used at the output. A 400resistor was used to decrease the quality factor of the mixer load, thus allowing in-band gain variation requirement to be achieved. The schematic of the output buffer (BUF) is shown in Fig. 4. The buffer implemented a fully integrated balanced-to-unbalanced converter, avoiding the need for an external balun. Resisand and bias currents were set to provide power tors gain, a BW as large as the DVB-S IF band, and the output matching at 75 . V. LO SYNTHESIZER The LO was implemented by a PLL-based frequency synthesizer. The PLL adopts a digital tri-state phase-frequency detector and an external second-order passive loop filter [7]. A dual-modulus divider set to 390/424 selects the LO for LB and HB, respectively. The block consists of a divide-by-two prescaler followed by a cascade of programmable 2/3 dual-modulus dividers [8]. The DVB-S standard poses challenging LO specifications for both tuning range and phase noise. Indeed, at high operating frequencies, fully integrated VCOs suffer from poor resonator quality factors and parasitics that affect performance. Therefore, a new design approach was developed to overcome the drawback of conventional VCOs.

Fig. 5. Schematic of a conventional bipolar differential VCO.

A. Conventional VCOs A schematic of a commonly used bipolar differential VCO is shown in Fig. 5 [9]–[11]. It is based on a varactor-tuned LC resonator and a differential pair to restore energy losses of integrated passive devices (e.g., inductors and capacitors). Although junction varactors are universally adopted as tuning devices in silicon bipolar technologies, they suffer from parabecause of the sitic capacitances at the cathode terminal junction between epitaxial layer and -doped bulk. In Fig. 5, capacitors and resistors are used to provide varactors with the required reverse voltage , while avoiding parasitic capacitance to affect oscillation frequency. Nevertheless, series capacitors do not allow exploiting the full tuning capability of diode varactors. Indeed, a high value of is required to make its impact on overall resonator capacitance negligible. However, capacitance cannot be arbitrarily high because of plate parasitics of integrated capacitors. One more drawback affecting the circuit in Fig. 5 is the bias resistors , which give a significant contribution to the VCO phase noise. Usually, high ’s are employed to avoid reduction of the LC resonator quality factor. In contrast, high ’s increase the AM–PM noise conversion because of varactor modulation by low-frequency thermal noise. B. Proposed VCO In this study, inductive coupling was exploited in the VCO design to overcome previously discussed drawbacks. The schematic of the proposed transformer-based VCO is shown in Fig. 6. All three metal layers offered by the technology process were used to implement a vertical-stacked integrated transformer. Inductive couplings provided the feedback between the LC resonator and active gain stage, avoiding the use of series capacitors and bias resistors. Therefore, both the tuning range and phase noise performance are improved compared with common integrated VCOs. In the proposed topology, the required gain of the active stage to sustain oscillation is inversely proportional to the coupling

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Fig. 6. Schematic of the proposed VCO.

factors of the transformer. Thus, the vertical-stacked approach was preferred over planar transformers because it allows higher coupling factors and to be achieved [12] and, hence, current consumption to be reduced. To avoid tuning-range wasting capability, the VCO free-running oscillation frequency must be predicted with great accuracy. Therefore, the integrated transformer and metal connections up to the active stage were carefully designed by using electromagnetic (EM) simulations. The transformer consisted of three single-turn differential inductors and the inductance of each turn was 0.7 nH. Moreover, the coupling factors were 0.87, while the quality factor of the resonator inductance resulted as high as 17 at a 5-GHz frequency. The available junction varactors exhibit an excellent tuning capability of 50%, but they suffer from a poor quality factor at 10 GHz. Therefore, the LO was implemented by using a 5-GHz VCO followed by a frequency doubler. The frequency doubler consists of a full-wave rectifier and comparator. The frequency response of the comparator . was enhanced by using a zero-peaking technique A theoretical 6-dB loss in the phase-noise performance is expected because of the frequency doubling. To avoid further impairment, the noise injection of on-state diodes was carefully minimized. In contrast, since the required tuning range is halved (e.g., 4.875–5.300 GHz), the VCO sensitivity to control voltage is halved, thus improving noise performance. VI. VCO PRESCALER In the integrated PLL, a fixed digital divide-by-two prescaler was included before a programmable dual-modulus (195/212) divider. Thus, a 25-MHz reference oscillator was used to lock the VCO frequencies for the LB and HB selection. The design of the first stage of the frequency divider is critical because it works at the highest frequency. The implementation of such high-speed digital dividers in a silicon bipolar technology is an issue of great concern. Indeed, although bipolar digital circuits can still work at frequencies approaching transistor , a high supply voltage is required for successful operation. Indeed, at higher frequencies, as the transistor current gain diminishes, multiple emitter follower (EF) stages are used to decouple the driver stage from the capacitive load due to the subsequential stages [13].

Fig. 7.

Schematic of a conventional divide-by-two prescaler.

A. Conventional Divide-by-Two Prescaler Usually, a D-type master/slave (M/S) flip-flop is employed to perform a divide-by-two prescaler thanks to its high-speed capability and straightforward design. The schematic of a common bipolar implementation is shown in Fig. 7. In the D-type M/S flip-flop, the inverted slave–latch output feeds the master–latch input. EFs are employed to drive the data and clock inputs of the latches. Each latch consists of a differential stage for the read-data operation and a cross-coupled differential stage for the hold operation. For this prescaler, high-speed operation is impaired whenever the cross-coupled stage of each latch fails to accomplish the hold-data phase. Indeed, due to bipolar junction transistor (BJT) base resistance and the collector–base capacitance , the conductance of a simple cross-coupled stage, starting from the dc value of , increases with frequency until it reaches a zero value, as is shown in Fig. 8. If minimum-size devices are employed to reduce parasitic capacitances, can be neglected, but extrinsic base and a shorter emitter width lead to a high value of . In this case, if is the base–emitter capacitance, turns to positive at a frequency given by (3) as follows: (3) The use of EFs in the feedback path of the cross-coupled stage increases the value of . However, a higher supply voltage is required and large transistors must be used in the

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Fig. 8. Conductance versus frequency for a conventional and capacitive degenerated cross-coupled pair.

Fig. 10.

LO-to-RF and RF-to-RF feed-through.

Fig. 11.

Die micrograph of the receiver.

Fig. 12.

Cross section of the technology adopted.

Fig. 9. Schematic of the proposed divide-by-two prescaler.

emitter–follower stages to avoid ringing over pulse edge and unwanted oscillations. Those large devices could result, in turn, in a high loading for the preceding stages, leading to limitations on high-speed operation. B. Proposed Divide-by-Two Prescaler The improved divide-by-two prescaler used in the PLL is shown in Fig. 9 [14]. The circuit is still based on a M/S D-type flip-flop, but now the cross-coupled differential stage employs capacitive degeneration . In this case, the conductance of the pair is negative up to the frequency given by (4) as follows:

(4) The behavior of versus frequency is shown and compared with the conductance of a simple cross-coupled pair in Fig. 8. At low frequencies, the conductance has a smaller magnitude and reaches a zero value at dc where the circuit in Fig. 7 . At higher frequencies and as dihas a value of minishes, the magnitude of the conductance is higher than the common cross-coupled pair and reaches a zero value at a higher frequency. VII. LAYOUT AND ASSEMBLING ISSUES - and In designing of monolithic circuits operating at -bands, unwanted high-frequency couplings must be carefully taken into account to avoid IC performance degradation [15], [16]. Indeed, substrate parasitic effects and bonding

wire inductive couplings can lead to desensitization [17] and instability. Thus, EM simulations of the layout of each critical block are a mandatory task in the design flow. Typically, integrated inductors and transformers are high die area devices, thus leading to high capacitive parasitics toward the substrate. Since they usually operate in resonant conditions, they introduce significant disturbances in the substrate through the parasitics. In particular, the LO-to-RF and RF-to-RF feedthrough were considered. These coupling phenomena are depicted in Fig. 10. The former is due to the injection of part of the strong LO signal, developed at the VCO transformer, which reach the input pad of LNA1 through the substrate. The latter

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Fig. 13.

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Multishielding technique for feed-through reduction.

represents the RF signal return from the LNA2 output transformer to the input of the receiver. The die micrograph of the receiver is shown in Fig. 11. The floor plan of the integrated receiver was the first fundamental step to limit feed-through effects. For this purpose, the VCO resonator and the output transformer of the LNA2 were positioned as far as possible from the input pad of the LNA1. The receiver was integrated in a 0.8- m self-aligned emitter low-cost silicon bipolar technology. The process features a BJT, three metal layers and trench isolation. A 46-GHzsimplified cross section of the technology adopted is shown in Fig. 12. Buried layer (BL) contacts connect the first metal layer (ML1) to the n BL. Thanks to the oxide trench (ISO), it is possible to break off the low-impedance path for disturbances injected into the BL. However, the parasitic capacitance allows parasitic signals between the n BL and the p-bulk flowing through the substrate. To further enhance the isolation among the critical parts of the IC, a multishielding technique was adopted, as illustrated in Fig. 13. This picture shows the RF-to-RF feed-through reduction technique, which was also applied to decrease the LO-to-RF feed-through. Each block (LNA1, LNA2, MIX, BUF, PLL, and VCO) has its own ML1 ground plane connected below the BL by means of huge numbers of BL contacts. These ground planes are insulated from each other by means of wide areas of ISO and connected to the PCB ground reference through several down-bonding wires. This approach allows collecting noise and interferences flowing in both the BL and bulk. Further wide ground planes, filled with BL contacts and connected to the PCB ground reference with down-bonding wires, were placed among every block. This allows a further amount of disturbances flowing in the substrate to be rejected. The last shield of the input pad of the receiver was represented by their own LNA1 ground plane, which is joined to the PCB ground through eight down-bonding wires. Thus, the multishielding technique permits only minimum residual couplings reaching the receiver input pad. Furthermore, the input pad of the LNA1 was as small as possible and implemented with the third metal layer (ML3), thus reducing the parasitic capacitance and avoiding the noise-figure degradation

Fig. 14.

Snapshot of bond-wire arrangement.

due to substrate losses. Measurements revealed an excellent LO-to-RF feed-through rejection of 55 dB. In addition to substrate disturbances, great care must be taken to avoid high mutual inductance between the input bond wire and the bond wires that feed the supply voltage to LNA1 and LNA2. Indeed, these couplings produce instability risks. Several dispositions for the IC bond wires were studied with three-dimensional (3-D) EM simulations. A snapshot of the best arrangement for the bond wires of the two LNAs is given in Fig. 14. An optimum placement for the input and supply voltage bond wires of the LNA1, forming an angle of 45 , was observed. Four down-bonding wires were inserted at each side of the input bond wire to produce a Faraday-cage protection around the central bond wire. EM simulations revealed that the Faraday-cage technique produced an isolation enhancement of 8 dB. VIII. EXPERIMENTAL RESULTS The measured SSB noise figure and conversion gain versus LB and HB frequencies are reported in Fig. 15. For an input signal at 12 GHz, the receiver exhibits a conversion gain of 38 dB and an SSB noise figure of 7 dB. Commercial HEMT

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Fig. 17.

Measured phase noise for a 10.6-GHz LO.

Fig. 18.

Measured VCO tuning range.

Fig. 15. Measured SSB noise figure and conversion gain versus LB and HB RF frequencies.

TABLE II IC PERFORMANCE SUMMARY

Fig. 16.

j

S

j

versus the RF band and jS

j

versus the IF band.

devices typically show a noise figure of 0.4 dB with an associated gain of 12 dB. By using two of these amplifier stages, the proposed LNB performs an overall noise figure as low as 0.6 dB. The conversion gain variation is within 7 dB over the entire RF frequency range. The IC in-band gain variation is mainly due to inter-matching networks between the LNAs and between the second LNA and mixer. Actually, the external HEMT stages can be designed to recover the high-frequency loss of the IC conversion gain, thus smoothing the overall gain variation. In Fig. 16, the input and output reflection coefficients of the reand ) are reported versus the RF and IF band, ceiver ( respectively. The DVB-S standard requires a minimum output return loss (RL) of 8 dB. Actually, measurements revealed a minimum RL of 7 dB, which can easily be improved by an external matching network. The output 1-dB compression point is greater than 5 dBm over the whole IF band. The VCO exhibits a phase noise of 95 dBc/Hz at 100-kHz offset from a 10.6-GHz carrier, as shown in Fig. 17. Measurements revealed a

VCO tuning range of 22%, as depicted in Fig. 18. It ranges from is swept from 8.6 to 10.8 GHz when the control voltage 0 to 3.3 V, which is fully within the design target. The current consumption of the integrated receiver is 160 mA from a 3.3-V supply voltage. A summary of the IC performance is given in Table II.

GIRLANDO et al.: MONOLITHIC 12-GHz HETERODYNE RECEIVER FOR DVB-S APPLICATIONS

IX. CONCLUSION A fully integrated heterodyne 12-GHz DVB-S receiver in silicon bipolar technology has been presented. The receiver has shown a conversion gain of 38 dB and an SSB noise figure of 7 dB. To achieve a phase noise as low as 95 dBc/Hz at 100-kHz offset from a 10.6-GHz carrier, an innovative VCO topology based on a three-stacked layer transformer was developed. The VCO tuning range was 2.2-GHz wide. The receiver was implemented in a 46-GHz- silicon bipolar process, thus demonstrating the feasibility of this low-cost technology in typical GaAs-field applications. ACKNOWLEDGMENT The authors thank A. Castorina, STMicroelectronics, Catania, Italy, for his support in IC measurements. REFERENCES [1] K. Watanabe et al., “A low-jitter 16 : 1 MUX and a high-sensitivity 1 : 16 DEMUX with integrated 39.8 to 43 GHz VCO for OC-768 communication systems,” in IEEE Int. Solid-State Circuits Conf. Dig., Feb. 2004, pp. 166–167. [2] H. Werker et al., “A 10 Gb/s SONET-compliant CMOS transceiver with low cross-talk and intrinsic jitter,” in IEEE Int. Solid-State Circuits Conf. Dig., Feb. 2004, pp. 172–173. [3] E. Sönmez, A. Trasser, K.-B. Schad, P. Abele, and H. Schumacher, “A single-chip 24 GHz receiver front-end using a commercially available SiGe HBT foundry process,” in IEEE Radio Frequency Integrated Circuits Symp. Dig., vol. I, Jun. 2002, pp. 159–162. [4] “Satellite earth stations and system (SES): Television receive-only (TVRO) satellite earth station operating in 11/12 GHz frequency bands,” ETSI, Sophia-Antipolis, France, ETS 300 784, 1997. [5] S. Smerzi, T. Copani, G. Girlando, A. Castorina, and G. Palmisano, “A 12-GHz heterodyne receiver for digital video broadcasting via satellite applications in silicon bipolar technology,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, Jun. 2004, pp. 25–28. [6] S. P. Voinigescu, M. C. Miliepaard, J. L. Showell, G. E. Babcock, D. Marchesan, M. Schroter, P. Schvan, and D. L. Harame, “A scalable highfrequency noise model for bipolar transistors with application to optimal transistor sizing for low-noise amplifier,” IEEE J. Solid-State Circuits, vol. 32, no. 9, pp. 1430–1439, Sep. 1997. [7] M. Soyeur and R. G. Meyer, “Frequency limitations of a conventional phase-frequency detector,” IEEE J. Solid-State Circuits, vol. 25, no. 8, pp. 1019–1022, Aug. 1990. [8] C. S. Vaucher, I. Ferencic, M. Locher, S. Sedvallson, U. Voegeli, and Z. Wang, “A family of low-power truly modular programmable dividers in standard 0.35 m CMOS technology,” IEEE J. Solid-State Circuits, vol. 35, no. 7, pp. 1039–1045, Jul. 2000. [9] M. Zannoth, B. Kolb, J. Fenk, and R. Weigel, “A fully integrated VCO at 2 GHz,” IEEE J. Solid-State Circuits, vol. 33, no. 12, pp. 1987–1991, Dec. 1998. [10] J. T. H. Lee and A. Hajimiri, “Oscillator phase noise: A tutorial,” IEEE J. Solid-State Circuits, vol. 35, no. 3, pp. 326–336, Mar. 2000. [11] Q. Huang, “Phase noise to carrier ratio in LC oscillators,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 7, pp. 965–980, Jul. 2000. [12] J. R. Long, “Monolithic transformers for silicon RF IC design,” IEEE J. Solid-State Circuits, vol. 35, no. 9, pp. 1368–1382, Sep. 2000. [13] H.-M. Rein and M. Möller, “Design considerations for very high-speed Si-bipolar IC’s operating up to 50 Gb/s,” IEEE J. Solid-State Circuits, vol. 31, pp. 1076–1090, Aug. 1996. [14] T. Copani, S. A. Smerzi, and G. Palmisano, “A novel prescaler for silicon bipolar multi-gigahertz applications,” in IEEE Radio Frequency Integrated Circuits Symp. Dig., Jun. 2004, pp. 595–598. [15] W. K. Chu et al., “A substrate noise analysis methodology for large-scale mixed-signal ICs,” in Proc. IEEE Custom Integrated Circuits Conf., 2003, pp. 369–372. [16] J. H. Wu, J. Scholvin, J. A. del Alamo, and K. A. Jenkins, “A Faraday cage isolation structure for substrate crosstalk suppression,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 10, pp. 410–412, Oct. 2001.

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[17] R. G. Meyer and A. K. Wong, “Blocking and desensitization in RF amplifiers,” IEEE J. Solid-State Circuits, vol. 30, no. 8, pp. 944–946, Aug. 1995.

Giovanni Girlando was born in Gela, Italy, in 1971. He received the Laurea degree in electronics engineering from the University of Catania, Catania, Italy, in 1997. Since 1998, he has been with STMicroelectronics, Catania, Italy, where he is with the Radio Frequency Advanced Design Group (RF-ADG), a joint research group supported by the University of Catania and STMicroelectronics. He is involved in the design and development of Ku-band receiver for digital video broadcasting applications and RF IC for wireless telecommunications. His current research interests concern high-linearity low-noise amplifiers and building blocks for RF integrated transceivers.

Santo A. Smerzi received the Laurea degree in electronic engineering from the University of Catania, Catania, Italy, in 1998. Since 1998 he has been with STMicroelectronics, Catania, Italy, where he is with the Radio Frequency Advanced Design Group (RF-ADG), a joint research group of the University of Catania and STMicroelectronics. He is involved in the design and development of receiver for satellite applications and RF ICs for wireless telecommunications. His current research interests concern fractional-N PLLs, VCOs, and building blocks for RF integrated transceivers.

Tino Copani was born in Catania, Italy, in 1972. He received the Laurea degree in electronics engineering and Ph.D. degree in electronics and automation engineering from the University of Catania, Catania, Italy, in 1998 and 2004, respectively. In 2001, he attended the Istituto Superiore per la Formazione di Eccellenza of Catania, Catania, Italy, where he completed the Advanced Course on Microelectronics and Systems. Since 2000, he has been with STMicroelectronics, Catania, Italy, where he is with the Radio Frequency Advanced Design Group (RF-ADG), a joint research group supported by the University of Catania and STMicroelectronics. He is involved in the design and development of RF ICs for satellite communications. His current research interests lie in the design of microwave VCOs, frequency synthesizers and high-speed digital circuits.

Giuseppe Palmisano received the Laurea degree in electronics engineering from the University of Pavia, Pavia, Italy, in 1982. From 1983 to 1991, he was Researcher with the Department of Electronics, University of Pavia, where he was involved in CMOS and BiCMOS analog IC design. In 1992, he was a Visiting Professor with the Universidad Autonoma Metropolitana (UAM), Mexico City, Mexico, where he taught a course on microelectronics for Ph.D. students. In 1993 and 2000, he was with the Faculty of Engineering, University of Catania, as an Associate Professor and Full Professor, respectively, where he taught microelectronics. Since 1999, he has been with STMicroelectronics, Catania, Italy, where he has led the Radio Frequency Advanced Design Group (RF-ADG), a joint research group supported by the University of Catania and STMicroelectronics. He has designed several innovative analog ICs within the framework of national and European research projects and in collaboration with electronics industries. He has coauthored over 150 papers in international journals and conference proceedings and a book on current operational amplifiers. He holds several international patents. His current research interest lies in the design of RF ICs for portable communications equipment.

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A Linear Inverse Space-Mapping (LISM) Algorithm to Design Linear and Nonlinear RF and Microwave Circuits José Ernesto Rayas-Sánchez, Senior Member, Fernando Lara-Rojo, Member, IEEE, and Esteban Martínez-Guerrero, Member, IEEE

Abstract—A linear inverse space-mapping (LISM) optimization algorithm for designing linear and nonlinear RF and microwave circuits is described in this paper. LISM is directly applicable to microwave circuits in the frequency- or time-domain transient state. The inverse space mapping (SM) used follows a piecewise linear formulation, avoiding the use of neural networks. A rigorous comparison between Broyden-based “direct” SM, neural inverse space mapping (NISM) and LISM is realized. LISM optimization outperforms the other two methods, and represents a significant simplification over NISM optimization. LISM is applied to several linear frequency-domain classical microstrip problems. The physical design of a set of CMOS inverters driving an electrically long microstrip line on FR4 illustrates LISM for nonlinear design. Index Terms—Aggressive space mapping (ASM), Broyden, computer-aided design (CAD), high-speed digital design, interpolating neural networks, inverse space mapping (SM), neural models, nonlinear transient design, optimizing expensive functions, RF and microwave design, surrogate models.

I. INTRODUCTION

S

PACE-MAPPING (SM) optimization techniques have been proposed in numerous innovative ways to efficiently design microwave circuits using very accurate, but computationally expensive models, typically full-wave electromagnetic (EM) simulators. A comprehensive review on SM for microwave modeling and design is the research by Bandler et al. [1]. All of the algorithmic SM approaches to microwave engineering design have been illustrated with linear frequency-domain design problems, although the original formulation of SM [2], as well as some other more advanced versions [3]–[6], consider a general formulation that, in principle, could also be applied for transient-domain design. Furthermore, in some of the SM algorithms, the frequency variable is intelligently manipulated to improve the parameter-extraction (PE) process, as in [7] and [8], making these particular SM techniques applicable only for frequency-domain linear problems. An interesting formulation to nonlinear EM optimization by SM has been recently developed [9], where the mapping inversion process is merged with the harmonic-balance analysis into the solution of a nonlinear system

Manuscript received April 22, 2004; revised August 2, 2004. This work was supported in part by the Consejo Nacional de Ciencia y Tecnología, Mexican Government under Grant 010581, Grant I39341-A, and Grant PFPN-03-42-8. The authors are with the Department of Electronics, Systems and Informatics, Instituto Tecnológico y de Estudios Superiores de Occidente, Tlaquepaque, Jalisco 45090, México (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842482

of equations. In this paper, we describe an integrated transientand frequency-domain SM-based design algorithm. Artificial neural networks (ANNs) have also been extensively used for efficient electromagnetics-based design and optimization of microwave circuits [10]. All of these ANN techniques for microwave design have been developed either for the frequency or transient domains. An integrated transientand frequency-domain ANN-based algorithmic (online) design approach has not yet been reported. Powerful techniques for developing EM-based neural models in the frequency domain [11]–[13] and time domain (trained from frequency-domain EM data [14]–[16] or from time-domain measured samples [17]) have been proposed. Advanced ANN techniques for EM-based modeling of passive microwave components for frequency-domain and transient analysis have been recently developed [18]. Once these neural models are trained, they can be added to linear and nonlinear circuits to efficiently incorporate EM effects during optimization. Neural inverse space-mapping (NISM) optimization was the first SM algorithm that explicitly made use of the inverse of the mapping from the fine to the coarse model parameter spaces [6], [19]. A statistical procedure to PE is employed in NISM to avoid the need for multipoint matching and frequency mappings. An ANN whose generalization performance is controlled through a network growing strategy approximates the inverse mapping at each iteration. The ANN starts from a two-layer perceptron and automatically migrates to a three-layer perceptron when the amount of nonlinearity found in the inverse mapping becomes significant. The NISM step consists of evaluating the current neural network at the optimal coarse model solution. In this paper, we describe in detail the linear inverse spacemapping (LISM) algorithm to design by optimization proposed in [20], and compare its performance with other SM-based optimization algorithms. LISM follows a piecewise linear formulation to implement the inverse of the mapping, avoiding the use of neural networks. LISM approximates the inverse of the mapping function at each iteration by linearly interpolating the last pairs of coarse and fine model design parameters, where is the number of optimization variables. This change significantly simplifies the implementation of the algorithm with respect to the NISM version. It also allows us to generalize the algorithm to be directly applicable for frequency-domain problems, for time-domain steady-state problems, and for time-domain transient-state problems, the latter being particularly relevant in high-speed digital design. The same statistical procedure

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RAYAS-SÁNCHEZ et al.: LISM ALGORITHM TO DESIGN LINEAR AND NONLINEAR RF AND MICROWAVE CIRCUITS

to PE is used in LISM as in NISM. LISM also follows an aggressive formulation in the sense of not requiring up-front fine model evaluations. LISM can be applied to design linear circuits and nonlinear circuits. A rigorous comparison between Broyden-based direct SM, NISM, and LISM is realized using a synthetic example in the frequency domain. LISM optimization is contrasted with NISM, and further illustrated by a classical problem of highspeed digital signal propagation: the physical design of a set of CMOS inverters driving an electrically long microstrip line on FR4. II. SM FOR DESIGN BY SOLVING A SYSTEM OF EQUATIONS Here, we follow the SM notation [1]. Let the vectors and represent the design parameters of the coarse and fine models, respectively . The corresponding optiand . The fine model is mizable responses are in vectors assumed to be a high-fidelity (or high accuracy) representation whose evaluations are expensive, while the coarse model is a low-fidelity representation that can be intensively evaluated with no significant cost.

set of characterizing responses available in the two models and , as described in [19]. An implicit assumption in (2) is that the PE process is unique (given a set of fine model characterizing responses, there is a unique vector of coarse model parameters whose coarse model responses match those of the fine model). If PE is nonunique, alternative formulations to (2) can be followed, as in [21]. C. Solving a System of Nonlinear Equations An SM-based optimization algorithm can be formulated to find the fine model parameters that make the fine model response sufficiently close to the optimal coarse model response. , i.e., by This is realized by iteratively solving finding an approximate root of the system of nonlinear equations (3) At the space-mapped solution, so that . Clearly, this SM formulation does not aim at finding the actual optimal fine model solution , which corresponds to the direct minimization of the original objective function using the fine model

A. Optimizing the Coarse Model SM-based algorithms start by directly optimizing the coarse model using conventional optimization methods that typically require many function evaluations (1) where is the objective function (usually minimax) is the expressed in terms of the design specifications, and optimal coarse model design. Vector contains the operating conditions, which consists of any required combination of independent variables according to the nature of the simulation, such as the operating frequencies, time samples, bias levels, excitation levels, rise time, fall time, initial conditions, temperature, etc. B. PE In the SM context, the PE process consists of a local alignment of the two models, i.e., consists of finding the coarse model design , whose corresponding responses are as close as possible to the fine model responses at the current fine model design . The PE process can be formulated as a nonlinear multidimensional vector function , where is evaluated by solving (2) PE is a key sub-problem in any SM algorithm, and many different techniques have been proposed to realize it [1]. In this paper, we take an statistical approach when solving (2), aligning and , but the complete not only the optimizable responses

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(4) For most practical problems, solving (4) by conventional optimization methods is prohibitive. In spite of the fact that solving the system of nonlinear equadoes not guarantee to find , it can still tions be used as an efficient practical design procedure to find the fine model parameters that yields a desired fine model response. Other SM algorithms have been formulated [22]–[24] that aim at finding at the expense of a larger number of fine model evaluations. III. BROYDEN-BASED “DIRECT” SM In a Broyden-based “direct” SM, the mapping equation is directly solved by using Broyden’s rank 1 updating formula [25]. This formulation corresponds to the so called aggressive space mapping (ASM) [26]. Here, the next iterate is predicted by (5) where the step

solves the linear system (6)

and matrix is an approximation of the Jacobian of with respect to at the current iterate. It is initialized by the identity matrix and updated by using

(7)

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The Broyden-based “direct” SM optimization algorithm can be summarized as follows.

where (11)

Algorithm: Broyden-based “Direct” SM

A detailed description on how to choose the complexity of the ANN is found in [19].

begin find

solving (1) ,

B. LISM

using (2)

Here, the interpolating function piecewise linear mapping defined by

repeat until stopping criterion for

solve

is implemented by (12)

using (2)

and the columns of are stored in . where The linear inverse mapping is “trained” by interpolating the last pairs of designs by solving the optimization problem

end

(13) IV. NISM AND LISM TECHNIQUES

where the th error vector is given by

In the inverse SM techniques, an approximation of the inverse of the mapping function is developed at each iteration. This is over a realized by optimizing an interpolating function set of corresponding designs, where is a vector of weights. The next iterate is predicted by simply evaluating the current inverse mapping at the optimal coarse model solution (8) contains the optimal weights for the current inverse where mapping. The inverse SM algorithms can be summarized as follows. Algorithm: Generic Inverse SM begin find

solving (1) , initialize

repeat until stopping criterion

(14) C. Common Aspects in NISM and LISM In both NISM and LISM algorithms, the vector of weights is initialized to implement a unit inverse mapping. Both algorithms also use the same optimization method (the scaled conjugate gradient available in the MATLAB Neural Network Toolbox1) for training the inverse mapping, i.e., for solving (10) and (13). Since NISM uses a two-layer perceptron during the first iterations [19], if LISM and NISM are implemented with the same PE method, then they predict the same iterates during the first iterations. Clearly, LISM represents a significant simplification over NISM, mainly because the problem of controlling the generalization performance of the ANN is avoided, which is done in NISM by controlling the complexity of the ANN (the number of hidden neurons) and the training error (the number of epochs and the magnitude of the minimum acceptable training error).

using (2) train

V. COMPARISON BETWEEN BROYDEN, NISM, AND LISM

end

A. NISM Here, the interpolating function ANN defined by

is implemented by an

(9a) (9b)

It is seen that Broyden-based “direct” SM, NISM, and LISM algorithms require a fine model evaluation per iteration. To make a fair comparison between the three algorithms, they are implemented using exactly the same PE procedure following the statistical approach described in [19]. They also use the same termination criteria: 1) when the maximum absolute error in the solution of the system of nonlinear equations is small enough; 2) when the relative change in the fine model design parameters is small enough; or 3) when a maximum number of iterations is reached as follows: (15a)

(9c) contains and the columns of and . The ANN is trained to interpolate all the accumulated pairs of designs by solving

(15b)

where

(10)

(15c) where

and

.

1MATLAB Neural Network Toolbox, ver. 4.0 (R12), The MathWorks, Natick, MA, 2000.

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Fig. 1. Synthetic example to compare the performance of Broyden-based “direct” SM optimization, NISM optimization, and LISM optimization. (a) Coarse model consisting of a canonical lumped RLC parallel resonator x = [R ( ) L (nH) C (pF)] . (b) Fine model consisting of the same resonator with some parasitic elements, with R = 0:5 ; L = 0:13 nH, x = [R ( ) L (nH) C (pF)] .

Fig. 2. Starting point for the SM optimization of the lumped RLC parallel resonator: optimal coarse model response (-); R (x ) and fine model response ( ) at the optimal coarse model solution R (x ), where x = [50 0:2370 11:8792] .



Consider the following synthetic example. Both the coarse and fine models are illustrated in Fig. 1. The coarse model consists of a canonical parallel lumped resonator [see Fig. 1(a)], nH pF . whose design parameters are The fine model consists of the same parallel lumped resonator with some parasitic elements: a parasitic series resistor assoand a parasitic series inductance ciated to the inductance associated to the capacitor [see Fig. 1(b)]. The fine model design parameters are nH pF . We nH. These parasitic values imtake pose a severe misalignment between the two models, and makes very nonlinear. the mapping function The design specifications are (assuming a reference impedance of 50 ) from 1 to 2.5 GHz and from 2.95 to 3.05 GHz. from 3.5 to 5 GHz, and Performing direct minimax optimization of the coarse model by conventional methods (we used a sequential quadratic programming (SQP) method available in the MATLAB Optimization Toolbox2), we find the optimal coarse model solution . The optimal coarse model reand the fine model response at the optimal sponse coarse solution are illustrated in Fig. 2. We first apply the Broyden-based direct SM algorithm described in Section III. The algorithm cannot solve this problem. 2MATLAB Optimization Toolbox, ver. 2.1 (R12), The MathWorks, Natick, MA, 2000.

Fig. 3. Results after applying Broyden-based “direct” SM optimization to the lumped RLC parallel resonator, for three different attempts. (a) Coarse model response (-) at x and fine model responses at the three SM solutions found ( ; ; ). (b) Fine model minimax objective function values at each iteration, for the three attempts (- -; - -; - -). Broyden-based “direct” SM optimization fails at solving this problem, becoming unstable.

2



 2

Fig. 3 shows the final responses and the fine model objective function values at each iteration for three different attempts. It is seen that the Broyden-based “direct” SM algorithm becomes calculated in (7) beunstable. This is because matrix calculated in comes ill conditioned, making the next step (6) unreliable. We then apply the NISM algorithm described in Section IV-A. Since the mapping function for this problem is very nonlinear, to illustrate the difficulty in controlling the generalization performance of the ANN at each iteration, we use 3000, 5500, and 10 000 epochs. Fig. 4 shows the final responses and the fine model objective function values at each iteration for each different number of epochs. In the three cases, NISM terminates due to the number of iterations that have reached the maximum. NISM finds an excellent fine model response when 3000 epochs are used at the space mapped-solution . From Fig. 4(b), it is confirmed that too large a number of epochs yields too small a training error, which makes the ANN “over fit” the training samples, and to deteriorate its generalization performance, predicting the following iterates with a large error.

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Fig. 4. Results after applying NISM optimization to the lumped RLC parallel resonator. (a) Coarse model response (-) at x and fine model response at the NISM solution using 3000 epochs ( ) using 5500 epochs ( ) and using 10 000 epochs ( ). (b) Fine model minimax objective function values at each NISM iteration using 3000 epochs (- -) using 5500 epochs (- -) and using 10 000 epochs (- -). This problem illustrates the difficulty in controlling the generalization performance of the ANN when the mapping between the two models is severely nonlinear.



2



2

Finally, we apply the LISM algorithm described in Section IV-B. Fig. 5 shows the final responses and the fine model objective function values at each iteration. LISM finds an excellent fine model response at the SM solution and terminates at the tenth iteration due to a sufficiently small error in the solution of the system of nonlinear equations. Since the fine model used is computationally very efficient, we can apply direct minimax optimization (using a conventional SQP optimization method). After 121 fine model evaluations starting from , the optimal fine model solution found is and the corresponding objec. The fine model tive function value is minimax objective function values at each iteration are shown in Fig. 6. Fig. 7 compares the optimal fine model response , , and the fine model the optimal coarse model response response at the space-mapped solution after applying LISM op. A comparison between and timization the fine model minimax objective function values at each itera-

Fig. 5. Results after applying LISM optimization to the lumped RLC parallel resonator. (a) Coarse model response (-) at x and fine model response ( ) at the LISM solution. (b) Fine model minimax objective function values at each LISM iteration (- -). An excellent solution is found after ten iterations.





Fig. 6. Fine model minimax objective function values at each iteration (-) of the conventional optimization method (SQP) directly applied to the fine model version of the lumped RLC parallel resonator.

tion after applying Broyden-based direct SM, NISM, and LISM optimization is shown in Fig. 8. It is seen that LISM outperforms NISM and Broyden-based “direct” SM. Table I summarizes the numerical results obtained for this problem after applying the four optimization methods described.

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Fig. 9. Block diagram of a set of CMOS inverters driving a capacitive load through an electrically long microstrip line. The segments of microstrip lines have a length L . The buffers are biased from a single dc voltage source. The effects of the dc power bus on the integrity of the propagated signal are usually neglected; they should be considered in a high-fidelity model (fine model). Fig. 7.

Comparing results after direct conventional optimization of the lumped , fine model response () at the LISM solution, and coarse model response (-) at x .

RLC parallel resonator: fine model response (- -) at x

specifications, termination criteria, coarse and fine models, and optimal coarse model solutions were used as in [19]. The coarse and fine models for the first problem were implemented in MATLAB. For the second and third problems, the coarse models were implemented in APLAC,3 while the fine models were implemented in Sonnet.4 The results found after applying LISM optimization to these three problems were the same as those found in [19], as expected, since the number of NISM iterations needed for solving these three problems were no . greater than VII. CMOS DRIVERS FOR A LONG MICROSTRIP LINE

Fig. 8. Comparing the optimal fine model minimax objective function value U (R (x )) with the fine model minimax objective function values at each iteration after applying Broyden-based direct SM, NISM, and LISM optimization to the lumped RLC parallel resonator. TABLE I RESULTS OF THE OPTIMIZATION OF THE LUMPED RLC PARALLEL RESONATOR

VI. FREQUENCY-DOMAIN LINEAR EXAMPLES We applied LISM optimization to the following design problems: a two-section impedance transformer, a bandstop microstrip filter with open stubs, and a high-temperature superconducting (HTS) microstrip bandpass filter. The design

Consider the problem of designing a set of CMOS inverters to drive an electrically long microstrip line. The most common practice to reduce the delay and other signal integrity problems when driving a long line is to place buffer stages at different locations along the line [27]. As an illustrative example, two intermediate buffer stages are inserted between the initial and final drivers on a long microstrip line, as shown in Fig. 9. is a trapezoidal pulse with a 3-V amThe input voltage plitude, 2.5-ns duration, and 100-ps rise time and fall time. The microstrip lines are on an FR4 half-epoxy half-glass substrate mil, loss tangent of 0.025, and dielecwith thickness using a width mil (50- lines). tric constant A typical 0.5- m CMOS process technology is assumed for all the inverters. The dc power line biasing each inverter is typically neglected (see Fig. 9), although it can have a significant impact on the integrity of the propagated signal. Both the coarse and fine models are implemented in APLAC. The high-fidelity model is shown in Fig. 10. It uses the Berkeley short-channel IGFET model (BSIM) (level 4) for each MOSFET, and the built-in component Mlin available in APLAC for all the microstrip line segments. The dc power line biasing each inverter is also modeled as shown in Fig. 10, assuming that it follows a completely different path than that one of the signal path. The low-fidelity model is shown in Fig. 11. It uses the level-1 (Shichman and Hodges) model for all the transistors. The main driven microstrip line (signal path) is modeled by segments of ideal lossless transmission lines using the built-in component Tlin available in APLAC. The effects of the dc power line bus are completely neglected. Due to the well-known inaccuracy of 3APLAC ver. 7.70b, APLAC Solutions Corporation, Helsinki, Finland, 2002. 4em

Suite, ver. 8.52, Sonnet Software Inc., 1020 North Syracuse, NY, 2002.

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Fig. 10. High-fidelity model for the CMOS buffers driving an electrically long microstrip line. All the MOSFETs are using the BSIM (level 4) model. The built-in component Mlin available in APLAC is used for all the microstrip line segments on FR4 half-epoxy half-glass substrate with thickness H = 10 mil, width W = 19 mil, loss tangent of 0.025, and dielectric constant " = 4:5 (50- lines). The dc power line biasing each inverter is also modeled (assuming it follows a completely different path).

Fig. 12. Starting point for LISM optimization of the CMOS inverters driving a long microstrip line: input trapezoidal voltage (v ), optimal coarse model output voltage (v ), and fine model output voltage (v ) at the optimal coarse model solution x = [11 11:5 11 10:5] (in micrometers).

Fig. 13. Final results after applying LISM optimization to the CMOS inverters driving a long microstrip line: input trapezoidal voltage (v ), optimal coarse model output voltage (v ), and fine model output voltage (v ) at the LISM solution x = [23 18 21 19:5] (m).

Fig. 11. Low-fidelity model for the CMOS buffers driving an electrically long microstrip line. All the MOSFETs are using the Shichman and Hodges (level 1) model, the microstrip lines are modeled by segments of ideal lossless transmission lines (" = 4:5; Z = 50 ), and the effects of the dc power line are neglected.

the level-1 MOSFET model for submicrometer devices [28], the channel length in the coarse model was taken as a preassigned parameter [5] and adjusted to realize an initial compensation of the coarse model (notice that the coarse model is using channel lengths of 0.8 m instead of 0.5 m). V from 0 to The design specifications are 1 ns, V from 3 to 4.5 ns, and V from 6.5 to 8.5 ns. The design parameters are the channel widths for all the MOSFETs, assuming symmetric inverters . After applying conventional optimization (using an SQP method ), the following optimal coarse (in model solution is found: micrometers). The excitation signal, as well as the coarse and fine model responses at the optimal coarse solution , are shown in Fig. 12. After six LISM iterations, a solution is found for this problem. The fine model response at the LISM solution (in micrometers) is compared with

Fig. 14. Fine model minimax objective function values at each LISM iteration during the optimization of the CMOS inverters driving a long microstrip line (--).

the optimal coarse model response in Fig. 13. The fine model minimax objective function values at each LISM iteration is shown in Fig. 14. LISM requires seven fine model simulations to solve this problem. The algorithm took 1 h 53 min 11 s to find the solution using a 1.6-GHz Pentium 4 computer with 512 MB of RAM.

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VIII. CONCLUSION We have described an LISM optimization algorithm for designing microwave circuits in the frequency- or time-domain transient state. The inverse SM follows a piecewise linear formulation, avoiding the use of neural networks. LISM is rigorously compared with Broyden-based direct SM and NISM. LISM optimization outperforms the other two methods, and represents a significant simplification over NISM optimization. LISM is applied to several linear frequency-domain classical microstrip problems, and is illustrated by a classical problem of high-speed digital signal propagation. ACKNOWLEDGMENT The authors thank M. Kaitera, APLAC Solutions Corporation, Helsinki, Finland, for making APLAC available. The authors also thank Dr. J. C. Rautio, Sonnet Software Inc., North Syracuse, NY, for making em available. REFERENCES [1] J. W. Bandler, Q. Cheng, S. A. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 337–361, Jan. 2004. [2] J. W. Bandler, R. M. Biernacki, S. H. Chen, P. A. Grobelny, and R. H. Hemmers, “Space mapping technique for electromagnetic optimization,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2536–2544, Dec. 1994. [3] M. H. Bakr, J. W. Bandler, R. M. Biernacki, S. H. Chen, and K. Madsen, “A trust region aggressive space mapping algorithm for EM optimization,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2412–2425, Dec. 1998. [4] M. H. Bakr, J. W. Bandler, N. Georgieva, and K. Madsen, “A hybrid aggressive space-mapping algorithm for EM optimization,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2440–2449, Dec. 1999. [5] J. W. Bandler, M. A. Ismail, and J. E. Rayas-Sánchez, “Expanded space mapping EM based design framework exploiting preassigned parameters,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 12, pp. 1833–1838, Dec. 2002. [6] , “Neural inverse space mapping EM-optimization,” in IEEE MTT-S Int. Microwave Symp. Dig., Phoenix, AZ, May 2001, pp. 1007–1010. [7] , “Neuromodeling of microwave circuits exploiting space mapping technology,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2417–2427, Dec. 1999. [8] M. H. Bakr, J. W. Bandler, M. A. Ismail, J. E. Rayas-Sánchez, and Q. J. Zhang, “Neural space mapping optimization for EM-based design,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2307–2315, Dec. 2000. [9] V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 362–377, Jan. 2004. [10] J. E. Rayas-Sánchez, “EM-based optimization of microwave circuits using artificial neural networks: The state of the art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 420–435, Jan. 2004. [11] P. Burrascano and M. Mongiardo, “A review of artificial neural networks applications in microwave CAD,” Int. J. RF Microwave Computer-Aided Eng., vol. 9, pp. 158–174, May 1999. [12] A. Patnaik and R. K. Mishra, “ANN techniques in microwave engineering,” IEEE Micro, vol. 1, pp. 55–60, Mar. 2000. [13] Q. J. Zhang and K. C. Gupta, Neural Networks for RF and Microwave Design. Norwood, MA: Artech House, 2000. [14] V. K. Devabhaktuni, M. C. E. Yagoub, Y. Fang, J. J. Xu, and Q. J. Zhang, “Neural networks for microwave modeling: Model development issues and nonlinear modeling techniques,” Int. J. RF Microwave ComputerAided Eng., vol. 11, pp. 4–21, Jan. 2001. [15] J. Xu, M. C. E. Yagoub, R. Ding, and Q. J. Zhang, “Neural-based dynamic modeling of nonlinear microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2769–2780, Dec. 2002.

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[16] Y. Cao, J. J. Xu, V. K. Devabhaktuni, R. T. Ding, and Q. J. Zhang, “An adjoint dynamic neural network technique for exact sensitivities in nonlinear transient modeling and high-speed interconnect,” in IEEE MTT-S Int. Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 165–168. [17] T. Liu, S. Boumaiza, and F. M. Ghannouchi, “Dynamic behavioral modeling of 3G power amplifiers using real-valued time-delay neural networks,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 1025–1033, Mar. 2004. [18] X. Ding, V. K. Devabhaktuni, B. Chattaraj, M. C. E. Yagoub, M. Deo, J. Xu, and Q. J. Zhang, “Neural-network approaches to electromagneticbased modeling of passive components and their application to highfrequency and high-speed nonlinear circuit optimization,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 436–449, Jan. 2004. [19] J. W. Bandler, M. A. Ismail, J. E. Rayas-Sánchez, and Q. J. Zhang, “Neural inverse space mapping for EM-based microwave design,” Int. J. RF Microwave Computer-Aided Eng., vol. 13, pp. 136–147, Mar. 2003. [20] J. E. Rayas-Sánchez, F. Lara-Rojo, and E. Martínez-Guerrero, “A linear inverse space mapping algorithm for microwave design in the frequency and transient domains,” in IEEE MTT-S Int. Microwave Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 1847–1850. [21] M. H. Bakr, J. W. Bandler, and N. Georgieva, “An aggressive approach to parameter extraction,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2428–2439, Dec. 1999. [22] M. H. Bakr, J. W. Bandler, K. Madsen, J. E. Rayas-Sánchez, and J. Søndergaard, “Space mapping optimization of microwave circuits exploiting surrogate models,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2297–2306, Dec. 2000. [23] J. W. Bandler, Q. Cheng, D. H. Gebre-Mariam, K. Madsen, F. Pedersen, and J. Søndergaard, “EM-based surrogate modeling and design exploiting implicit, frequency, and output space mappings,” in IEEE MTT-S Int. Microwave Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1003–1006. [24] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, D. M. Hailu, K. Madsen, A. S. Mohamed, and F. Pedersen, “Space mapping interpolating surrogates for highly optimized EM-based design of microwave devices,” in IEEE MTT-S Int. Microwave Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 1565–1568. [25] C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput., vol. 19, pp. 577–593, 1965. [26] J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 12, pp. 2874–2882, Dec. 1995. [27] R. J. Baker, H. W. Li, and D. E. Boyce, CMOS Circuit Design, Layout, and Simulation. Piscataway, NJ: IEEE Press, 1998. [28] B. Razavi, Design of Analog CMOS Integrated Circuits. New York: McGraw-Hill, 2001.

José Ernesto Rayas-Sánchez (S’88–M’89–SM’95) was born in Guadalajara, Jalisco, México, on December 27, 1961. He received the B.Sc. degree in electronics engineering from the Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO), Guadalajara, México, in 1984, the Masters degree in electrical engineering from the Instituto Tecnológico y de Estudios Superiores de Monterrey (ITESM), Monterrey, México, in 1989, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 2001. Since 1989, he has been Professor with the Department of Electronics, Systems, and Informatics, ITESO. In 1997, he spent his sabbatical with the Simulation Optimization Systems Research Laboratory, McMaster University. He returned to ITESO in 2001. His research focuses on the development of novel methods and techniques for computer-aided modeling, design and optimization of analog wireless and high-speed electronic circuits and devices exploiting SM and ANNs. Dr. Rayas-Sánchez is a member of the Mexican National System of Researchers, Level I. He is currently the IEEE Mexican Council Chair, as well as the IEEE Region 9 Treasurer. He was the recipient of a 1997–2000 Consejo Nacional de Ciencia y Tecnología (CONACYT) Scholarship presented by the Mexican Government, as well as a 2000–2001 Ontario Graduate Scholarship (OGS) presented by the Ministry of Training for Colleges and Universities in Ontario, Canada. He was the recipient of a 2001–2003 CONACYT Repatriation and Installation Grants presented by the Mexican Government. He was also the recipient of a 2004–2006 SEP-CONACYT Fundamental Scientific Research Grant presented by the Mexican Government.

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Fernando Lara-Rojo (M’98) was born in Guadalajara, México, on June 5, 1935. He received the B.Sc. degree in electronics engineering from the Instituto Politécnico Nacional (IPN), México City, México, in 1958, and the Ph.D. degree in physics from the Institute of Theoretical Physics, University of Kiel, Kiel, Germany, in 1973. From 1958 to 1968, he was an Engineer in telecommunication systems with Philips, Siemens, and the Secretaría de Comunicaciones y Transportes, Mexican Government (SCT). From 1973 to 1975, he was a Professor with Centro de Investigación y de Estudios Avanzados (CINVESTAV), México City, México, and from 1975 to 1980, with the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Tonantzintla, México. Since 1984, he has been a Professor with the Electronics, Systems and Informatics Department, Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO), Guadalajara, México. He is mainly interested in machine intelligence and automated reasoning. Over the last seven years, his research has focused on applications of fuzzy logic and neurofuzzy systems. Dr. Lara-Rojo was the recipient of a 1968-1972 Instituto Nacional de la Investigación Científica [now the Consejo Nacional de Ciencia y Tecnología (CONACYT)] Scholarship presented by the Mexican Government. He has also been the recipient of three scholarships for scientific collaboration sabbaticals at German universities by the German Academic Exchange Service (DAAD).

Esteban Martínez-Guerrero (M’03) was born in Puebla, México, on November 28, 1965. He received the B.Sc. degree in electronics engineering from the Universidad Nacional Autonoma de México (UNAM), México City, México, in 1990, the Masters degree in electrical engineering from the Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV—IPN), México City, México, in 1996, and the Ph.D. degree in integrated electronic devices from the Institut National des Sciences Appliquées de Lyon (INSA-Lyon), Lyon, France, in 2002. From 1998 to 2002, he collaborated with the Nanophysics and Semiconductor Research Group, Commissariat à l Energie Atomique (CEA)-Grenoble, Grenoble, France. Since 2002, he has been Professor with the Electronics, Systems, and Informatics Department, Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO), Guadalajara, México. His research focuses on physical modeling, simulation and design of semiconductor devices, and growth and characterization of thin-film and semiconductor nanostructures, namely, quantum dots and quantum wells for optical telecommunication systems. Dr. Martínez-Guerrero is a member of the Mexican National System of Researchers, Level I. He is a member of the IEEE Electron Devices Society. He was the recipient of 1992–1995 and 1998–2002 Consejo Nacional de Ciencia y Tecnología (CONACYT) Scholarships presented by the Mexican Government.

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Iterative Methods for Extracting Causal Time-Domain Parameters Shuiping Luo, Student Member, IEEE, and Zhizhang (David) Chen, Senior Member, IEEE

Abstract—Recent interest in time-domain modeling techniques has been largely motivated by the demands for simulating broad-band electronic systems and high-speed digital circuits. These techniques normally require strict causality of any parameters to be used with them. However, most of parameters, such as the -parameters of a transistor, are given only in the frequency domain and over a limited frequency range of interest. Direct applications of regular transformation techniques to these band-limited frequency-domain parameters, such as inverse Fourier transform, often lead to noncausal time-domain correspondents. Therefore, schemes need to be carefully developed for extraction of the time-domain parameters that are causal while retaining the original frequency-domain information within the frequency range of interest. In this paper, two iterative methods are proposed for the causal extraction, and numerical examples are given to validate the effectiveness. The errors of the methods are found to be approximately 1%. Index Terms—Causality, fast Fourier transform (FFT), frequency-domain parameters, Hilbert transform, iterative method, network parameters, time-domain parameters.

I. INTRODUCTION

R

ECENT interest in time-domain modeling methods has been mostly motivated by the demands for simulating broad-band electronic systems and high-speed digital circuits. However, when they are used with black-box types of devices and components, care needs to be taken. For instance, when the finite-difference time-domain (FDTD) method [1] is used, two approaches are normally employed: one is to incorporate the FDTD into the governing voltage–current equations of a device [2], [3] and another is to incorporate the governing voltage–current equations of a device into the FDTD marching equations [4], [5]. In the first approach, the nonlinear circuit model of a device has to be known and solved with the current terms being updated with the field FDTD. In the second approach, however, a nonlinear circuit model is not necessarily needed. Instead, the network parameters of a device, such as the - or -parameters, are used. They are then incorporated or interfaced with the FDTD equations. The implementation of the second approach is relatively simpler, but it has a stringent requirement for the network parameters of the devices: the parameters need to be in the time domain and have to be causal because the FDTD models are causal. Manuscript received May 10, 2004; revised August 17, 2004. This work was supported in part by the Natural Science and Engineering Research Council of Canada and the Atlantic Innovation Fund of Canada. The authors are with the Microwave and Wireless Laboratory, Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, Canada B3J 2X4 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842480

In a normal circumstance, network parameters are given or obtained only in the frequency domain and over a limited frequency range or band of interest. For instance, most manufacturers provide the -parameters for a field-effect transistor (FET) over only a limited frequency range under certain biasing conditions. To obtain the time-domain correspondents, transformation techniques such as the inverse Fourier transform can be applied. However, direct application of these transformation techniques to the band-limited frequency-domain parameters usually leads to noncausal parameters in time. Perry and Brazil [6] and Chen [7] studied such phenomena and proposed the uses of the Hilbert transform to tackle the problem. The method proposed by Perry and Brazil achieves the causality by maintaining the magnitudes of the original parameters, but modifying their phases for the minimum-phase systems [6]. It may present errors due to the short artificial period on the frequency response and the simple direct interpolation of the method. It may also have problems in general phase-sensitive or narrow-band cases. The technique proposed by Chen has difficulties in handling the singularity of the Hilbert transformation integrals [7]. The errors were found somewhat large because of the simple direct extrapolation. Narayana et al. also applied the property of Hilbert transform and proposed an iterative method to interpolate/extrapolate missing frequency-domain data outside the frequency range of interest [8]. In this paper, two iterative techniques are proposed for general extractions of causal time-domain parameters from their band-limited frequency counterparts by forcing the causality directly in the time domain. They are conceptually simple and easy to implement. The extracted time-domain parameters not only are causal, but also contain the same frequency-domain information as the original parameters over the given limited frequency range in both magnitude and phase. This paper is organized in the following way. First, we brief the needs for causal time-domain parameters extraction and the ways of including them in a causal time-domain simulation. The computation steps of the proposed two iterative techniques are then described. After it, three numerical examples are presented to numerically demonstrate the validity and effectiveness of the proposed techniques. Finally, discussions and conclusions are made on the limitations and future work. II. TIME-DOMAIN MODELING WITH LUMPED-ELEMENT DEVICES OF CAUSAL NETWORK PARAMETERS Numerous time-domain numerical methods, such as the widely used FDTD method [1], are causal numerical models. When used to incorporate a lump-element device, they require time-domain network parameters of the device to be causal [4].

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Fig. 1. Equivalent two-port network of an active device.

More specifically, assume that a device can be represented with a two-port network, as shown in Fig. 1. Also assume that the network parameters are known only in the frequency domain over a certain frequency band. Take the admittance network parameters ( -parameters) as our example. They can then be used to describe the relationship between the terminal voltand in the freages and currents quency domain as follows:

Fig. 2. Time-domain y (t) obtained with the direct inverse Fourier transform of the frequency-domain Y -parameters.

(1) where and are the admittance parameters expressed in the frequency domain. The corresponding relationship in the time domain is then (2) where

represents causal time-domain convolution, and are the time-domain correspondents of and , and and are the causal time-domain correspondents of the freand . quency domain In theory, and can be obtained through the inverse Fourier transform of and . However, in most cases, and are known or given over only a limited frequency range or band. Direct and simple inverse Fourier transform of them will often lead to noncausal time-domain -parameters. For instance, Fig. 2 shows of the of NE425S01 given by California Eastern Laboratories (CEL), Santa Clara, CA, over the frequency range of 0.5–18 GHz [9]. is noncausal with significant values at As can be seen, . Such noncausal values cannot be implemented negative or incorporated in a causal time-domain numerical model such as FDTD. Therefore, the simple inverse Fourier transform is not adequate due to the fact that most parameters are given or known over a limited frequency band. In other words, transformation schemes need to be carefully developed that can extract the causal time-domain parameters from the band-limited frequency-domain parameters. In the following sections, two iterative methods are described to obtain the correct causal -parameters. The first method applies the Fourier transform in combination with an error feedback technique and the second method applies the Fourier transform in conjunction with the Hilbert transform.

Fig. 3.

Flowchart of the negative-feedback-based method.

III. NEGATIVE-FEEDBACK-BASED FOURIER-TRANSFORM METHOD Suppose that is the given frequency parameter that is known over a finite frequency range of interest. An iterative procedure, as illustrated by the flowchart in Fig. 3, is proposed to obtain its causal time-domain correspondents. is an intermediate In the flowchart, and the original variable parameter. The error between over the frequency range of interest can be defined in numerous ways. In our case, they are defined in the following

LUO AND CHEN: ITERATIVE METHODS FOR EXTRACTING CAUSAL TIME-DOMAIN PARAMETERS

way to ensure that errors for both the real and imaginary parts are small enough: error

error

error

(3)

with error

(4a)

error

(4b)

where (5a) (5b) denotes the maximum value within the frequency where range. are done differently within and outside Corrections of the frequency range. The reason is that the parameters inside to the range can be compared with the reference value obtain the feedbacks, while the values of the parameters outside the frequency range can not. within the frequency range, For (6) with

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outside the frequency range as the frequency goes up. In our case, 0.9 is an empirical optimal value from our numerical simulations. In the above-described procedure (see Fig. 3), the negative is introduced to the iterative loop. Therefore, difference we call the method the negative-feedback-based Fourier transform method. The convergence of the method is proven numerically in Secbecomes tion IV and can be reasoned by the fact that when over the limsufficiently close to the original parameter ited band, exit conditions are satisfied and the iteration will terminate with the causal time-domain parameter. In the worst sceis not a causal parameter and the iterations do not nario, if converge, the computation will terminate once the number of it. In the numerical erations exceeds the prescribed number examples that are described in Section V, the prescribed number is set to be 500 with errors in (3) set to less than 1%. In principle, there are variations in the way the extraction is carried out, i.e., the iterative methods are not unique. In Section IV, we will describe another approach that is based on the Hilbert transform. It provides another alternative method in case the negative-feedback Fourier-transform method does not converge adequately fast. IV. HILBERT-TRANSFORM-BASED FOURIER-TRANSFORM METHOD Suppose that domain parameter

is the Fourier transform of a causal time, i.e., (11)

(7) where is a coefficient used to weight error and error in the feedback. It is computed with the following equation with an initial value being set to 1.0: error

error

represents the Fourier transform. The following where Hilbert transform relation should then exist [10]: (12a)

(8)

error , it means that the imaginary part has smaller If error errors than the real part. will then reduce during the iterations to decrease the weight of the imaginary part of the parameters; otherwise, it would increase. outside the frequency range, we take For

(12b) Since (13)

(9) with

with

(14)

(10) . is a frequency sampling point Here, taken outside the frequency range of interest. The first frequency is the border frequency point of the frequency range point of interest. is the total number of frequency points taken is the last frequency point and outside the frequency range. is neglishould be large enough so that the magnitude of gibly small beyond it. The factor of 0.9 ( 1.0) in (10) is chosen so that the new has a smooth transition from inside to outside the frequency range; at the same time, it also damps

(12) can be rewritten as (15a) (15b) The above relations form the basis for the proposed iteration method; therefore, the method is called the Hilbert-transformbased Fourier-transform method.

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. Again, the factor of 0.9 in (17) is where and chosen to allow smooth transitions of the new from inside to outside of the known frequency range while their magnitudes damped as frequency increases. In the flowchart, the error functions or criteria are defined the same as that used in the negative feedback technique described in Section III, i.e., (3) is again used to measure the error. Once again, a number is preset as the maximum allowable number of iterations. If the iteration number exceeds that number, the iterations will be terminated and the procedure is considered not converged within the allowable number of iterations. V. NUMERICAL VALIDATIONS To validate the two proposed methods, we first apply them to two theoretically known cases: a temporal triangular pulse and a temporal rectangular pulse with their spectra only partially known. Once the methods are proven valid, they are applied to a real FET amplifier. A. Triangular Pulse The triangular pulse considered is expressed as follows:

(18) otherwise

Fig. 4. Flowchart of the Hilbert-transform-based Fourier-transform method.

Suppose again that is the original given frequency parameter that is known over a limited frequency band of interest. The procedure for the Hilbert-transform-based Fourier-transform method is illustrated by the flowchart shown in Fig. 4. and within and outside the Again, in correcting frequency range, different scheme functions are used. For and within the frequency range, the difference and are computed using (5) as follows:

where (ns) is the duration of the pulse in time. By taking the Fourier transform, a complete spectrum of the triangular pulse is obtained as (19) The pulse has a finite duration in the time domain, but extends to infinity in the frequency domain. is given or known only over Assume now that 0 Hz–6 GHz, i.e., we now have GHz

(16a) (16b) . is a frequency sampling point inside where is the the frequency range of interest. The frequency point border frequency point of the frequency range of interest. For and outside the frequency range, (17a) (17b) where (17c) (17d)

unknown

GHz.

(20)

To extract the pulse in the time domain with the above limited information, the simplest way is to directly take the inverse with assumption of being zero Fourier transform of beyond 6 GHz. The resulted time-domain signal is not causal. To make it causal, one can simply cut off the time-domain values (i.e., set the values for to be zero). We call for this approach the direct cutoff method. The causal time-domain signal obtained with such a direct cutoff method can be converted to the frequency domain to check its spectrum. The results are shown in Fig. 5. From Fig. 5, one can see both the time-domain signal and its spectrum are quite different from the original. In particular, within the frequency range of 0–6 GHz of interest, the spectrum is very much different from the original. The relative difference is nearly 100% at 6 GHz. of real

LUO AND CHEN: ITERATIVE METHODS FOR EXTRACTING CAUSAL TIME-DOMAIN PARAMETERS

Fig. 5. Time-domain pulse and its Fourier transform obtained with the direct cutoff method for the triangular pulse.

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Fig. 7. Time-domain waveforms and their spectra extracted with the two proposed iterative methods when the known frequency range of interest is from 0 to 60 GHz for the triangular pulse.

time-domain signals extracted with the two proposed methods improve the accuracy significantly. Even though the extracted signals have some differences from the original triangular signal in the time domain, their spectra are almost the same in the range of 0–6 GHz (of interest). If the known frequency range is further extended to 60 GHz, the results are shown in Fig. 7. In comparisons of Figs. 6 and 7, one can see that the accuracy in the time domain is dramatically improved when the known frequency range of interest is increased. In particular, as shown in Fig. 7, the time-domain waveforms extracted with the proposed methods are extremely close to the original triangular signal. This is expected as over 99% of the energy of the triangular pulse is covered within 0–60 GHz. However, the accuracy in the frequency domain is not improved as much as that in the time domain. If the frequency range concerned is only from 0 to 6 GHz, the extracted pulse shown in Fig. 6 are the adequate representations of the original triangular pulse even though the two pulses look different in the time domain. B. Rectangular Pulse The second example is a rectangular pulse defined as Fig. 6. Time-domain waveforms and its spectra extracted with the proposed two iterative methods when the known frequency range is from 0 to 6 GHz for the triangular pulse.

Now the two methods proposed in this paper are applied. Fig. 6 shows the results. As can be seen, the extracted time-domain pulses are causal and much closer to the original triangular pulse. The differences in the frequency domain within the range of 0–6 GHz are less than 1%. In other words, the causal

otherwise

(21)

(ns) is the duration of the pulse. Fourier transwhere form of the rectangular pulse gives (22) Now assume that are known only from 0 to 6 GHz, application of the direct cutoff approach, as described in Sec-

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Fig. 8. Time-domain waveform and its spectra extracted with the direct cutoff approach for the rectangular pulse.

tion V-A, leads to the results shown in Fig. 8. As can be seen, although the imaginary part of the Fourier transform of the extracted pulse seems to have small errors, the real part of the Fourier transform, as well as the waveform in the time domain, is quite different from the original rectangular pulse. However, the results with the proposed methods have much smaller errors. Fig. 9 presents the results when the known frequency range of interest is from 0 to 6 GHz, and Fig. 10 presents the results when the range is from 0 to 60 GHz. As can be seen, the conclusion is similar to those for the triangular pulse extraction: when the known frequency range is only from 0 to 6 GHz, the extracted waveform, although having differences from the rectangular pulse, contains almost the same frequency components as those of the rectangular pulse within the concerned frequency range of 0 to 6 GHz. Therefore, the extracted pulses shown in Fig. 9 are the adequate representations of the original rectangular pulse in terms of the frequency range of interest. In summarizing the above results, we conclude that the proposed iterative methods are effective and useful in extracting time-domain signals from a limited frequency range of interest. The time-domain signals extracted as such are causal and can be used to represent the original pulses in light of the frequency range of interest. In Section V-C, we apply the proposed iterative methods to a linear FET amplifier.

Fig. 9. Time-domain waveforms and their spectra extracted with the two proposed iterative method methods when the known frequency range of interest is from 0 Hz–6 GHz for the rectangular pulse.

Fig. 10. Time-domain waveforms and their spectra extracted with the two proposed iterative methods when the frequency range of interest is from 0 to 60 GHz for the rectangular pulse.

C. FET Amplifier The FET amplifier used NE425S01 as its active device [9]. As indicated before, the -parameters of the active device are given by CEL only over the frequency range of 0.5–18 GHz. They are converted to the -parameters in the frequency-domain for our computation.

Now the proposed methods are applied to extract the causal time-domain -parameters of the device. Figs. 11 and 12 show the extracted time-domain parameters. The parameters extracted with the two methods are similar in general shapes in the time domain, but different in some details. For example,

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Fig. 13.

Layout of the FET amplifier circuit.

Fig. 14.

Computed S 21 of the amplifier in Fig. 13 from different methods.

Fig. 11. Time-domain Y -parameters extracted with the negative-feedbackbased method for the FET used in the amplifier.

TABLE I MAXIMUM PERCENTAGE ERRORS WITHIN THE FREQUENCY RANGE OF INTEREST

Fig. 12. Time-domain Y -parameters extracted with the Hilbert-transformbased method for the FET used in the amplifier.

extracted with the Hilbert-transform-based method has one more small ripple at the initial time than the extracted with the negative-feedback-based method. These differences can be attributed to the fact that the different values of outside the frequency range are extracted with the two methods. After the causal time-domain -parameters of the FET are obtained, they are included in an in-house time-domain simulator that is based on the modified central difference (MCD) method [11] in modeling the overall amplifier. Fig. 13 shows the layout of the amplifier. In it, the characteristic impedances of the main transmission line TL1 and open-circuited stub line TL2 are both 50 . The phase velocities on TL1 and TL2 are both 2.12535 10 (m/s). The length of TL1 is 6.4 (mm) and the length of TL2 is 3.6 (mm).

Fig. 14 is the computed overall of the whole amplifier. For reference, Agilent’s Advanced Design System (ADS) was also employed to simulate the circuit. The results are also plotted in Fig. 14. As can be seen, the results from the MCD method that employs the extracted causal time-domain -parameters are very close to the results from ADS. VI. ERROR AND CONVERGENCE TEST To assess the errors and convergence of the proposed method, we have run numerical tests on them. Table I and Fig. 15 show the results. It can be seen from Table I that the proposed methods give very good accuracy. The error with the negative feedback method is, in general, smaller than those with the Hilbert transform. Nevertheless, one should, therefore, not discard

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REFERENCES

Fig. 15.

Convergence curve for extracting the triangular pulse.

the Hilbert-transform-based method as it may present better solutions in other cases. Fig. 15 gives the convergence of the proposed methods for a triangle pulse (0–6 GHz). The Hilbert-transform-based method converges faster than the negative-feedback-based method at the initial 100 iterations, but slower thereafter.

VII. DISCUSSION AND CONCLUSIONS In this paper, two iterative approaches are proposed for extracting causal time-domain parameters from their frequencydomain counterparts known only over a limited frequency range of interest. Both methods are shown to be effective and useful with a good accuracy of approximately 1%. The time-domain parameters extracted as such are completely causal and can be included in an explicit time-marching simulator such as FDTD that has a stringent requirement for time causality. It is noted that the causal extraction processes described in this paper is a preprocessing procedure for parameter inclusions in time-domain simulators. Therefore, their computation time is not very critical for the time-domain simulations. Even so, in all the cases we computed in this paper, the solutions were obtained within 3 min with 500 iterations on a laptop Pentium-IV PC with 1.8-GHz CPU and 512-MB RAM. The programming platform used was MATLAB. Like any other iterative methods (e.g., those used in optimizations), one method may not converge fast enough, while the others may. Therefore, a preset number is required in order to terminate the iterations and switch the method. The two methods proposed in this paper provide such alternations to each other, albeit other iterative methods may also be developed. Finally, it should be pointed out that the methods proposed in this paper are limited to linear systems at this stage. Extensions to nonlinear systems will be the subject of future research.

[1] A. Taflove, Computational Electrodynamics. Norwood, MA: Artech House, 1998. [2] M. J. Piket-May, A. Taflove, and J. Baron, “FD-TD modeling of digital signal propagation in 3-D circuits with passive and active loads,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 8, pp. 1514–1523, Aug. 1994. [3] C. N. Kuo, V. A. Thomas, S. T. Chew, B. Houshmand, and T. Itoh, “Small signal analysis of active circuits using FDTD algorithm,” IEEE Microw. Guide Wave Lett., vol. 5, no. 7, pp. 216–218, Jul. 1995. [4] W. Sui, D. Christensen, and C. Durney, “Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 4, pp. 724–730, Apr. 1992. [5] J. Z. Zhang and Y. Y. Wang, “FDTD analysis of active circuit based on the S -parameters,” in Proc. Asia–Pacific Microwave Conf., vol. 3, Hong Kong, Dec. 1997, pp. 1049–1052. [6] P. Perry and T. Brazil, “Forcing causality on S -parameter data using the Hilbert transform,” IEEE Microw. Guided Wave Lett., vol. 8, no. 11, pp. 378–380, Nov. 1998. [7] Y. Chen, “Design and simulation of active integrated antennas,” M.A.Sc. thesis, Dept. Elect. Comput. Eng., Dalhousie Univ., Halifax, NS, Canada, 2003. [8] S. M. Narayana, G. Rao, R. Adve, T. K. Sarkar, V. C. Vannicola, M. C. Wicks, and S. A. Scott, “Interpolation/extrapolation of frequency domain responses using the Hilbert transform,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 10, pp. 1621–1627, Oct. 1996. [9] “Data sheet of NE425S01,” CEL, Santa Clara, CA, May 2004. [Online]. Available: http://www.cel.com/pdf/datasheets/ne425s01.pdf. [10] A. Papoulis, The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962. [11] K. Murakami, S. Hontsu, and J. Ishii, “Transient analysis of a class of mixed lumped and distributed constant circuits,” in Proc. Int. Circuits Systems Symp., Jun. 1988, pp. 2843–2846.

Shuiping Luo (S’04) received the B.Eng. degree from Chengdian University, Chengdu, China, in 1989, the M.A.Sc. degree from Shanghai Jiao Tong University, Shanghai, China, in 1992, and is currently working toward the Ph.D. degree at Dalhousie University, Halifax, NS, Canada. From March 1992 to March 1996, he was an Instructor with the Department of Electronic Engineering, Shanghai Jiao Tong University. From March 1996 to April 2000, he was a Software Engineer with the Shanghai Bell Company, Shanghai, China. From November 2000 to May 2001, he was a Software Developer with Nortel Networks, Ottawa, ON, Canada. His research interests include numerical modeling and simulation, RF circuit design and optimization, antenna designs for wireless and satellite communications, and real-time embedded system.

Zhizhang (David) Chen (S’92–M’92–SM’96) received the B.Eng. degree from Fuzhou University, Fuzhou, China, in 1982, the M.A.Sc. degree from Southeast University, Nanjing, China, in 1986, and the Ph.D. degree from the University of Ottawa, Ottawa, ON, Canada, in 1992. From January of 1993 to August of 1993, he was a Natural Science and Engineering Research Council Post-Doctoral Fellow with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada. In 1993, he joined the Department of Electrical and Computer Engineering, Dalhousie University, Halifax, NS, Canada, where he is currently a Full Professor. He has authored or coauthored over 100 journal and conference papers, as well as industrial reports in the areas of computational electromagnetics and RF/microwave electronics for wireless communications. His current research interests include RF/microwave electronics, time-domain numerical modeling and simulation, smart antenna systems, and wireless technologies and applications.

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A 12-GHz SiGe Phase Shifter With Integrated LNA Timothy M. Hancock, Member, IEEE, and Gabriel M. Rebeiz, Fellow, IEEE

Abstract—This paper presents the design and measurement of a 12-GHz active phase shifter with an integrated low-noise amplifier in SiGe. The phase is controlled with a 1-bit (180 ) digital phase shifter and a 180 continuous analog phase shifter. The 1-bit digital phase shifter is implemented by switching between a 90 low-pass network and a 90 high-pass network. The analog phase shifter is implemented using a lumped-element loaded line technique where both series and shunt elements are adjusted to continuously vary the phase while maintaining a constant port impedance. Design equations for this constant-impedance phase-shift network are derived and presented. The phase shifter achieves 3.7 0.5 dB of gain, with a noise figure of 4.4 dB at a center frequency of 11.5 GHz. The total chip size including pads is 1920 m 780 m, and the area of the active circuitry without pads is 1680 m 660 m (1.1 mm2 ).

+

Index Terms—All-pass networks, direct broadcast satellite (DBS), phased arrays, phase shifters, RF integrated circuit (RFIC), satellite communications, silicon germanium.

I. INTRODUCTION

P

HASE shifters are an invaluable component of microwave systems that have traditionally been limited to military applications or high-end commercial applications due to cost constraints. Phase shifters have mostly been implemented with III–V planar integrated-circuit (IC) technologies, but recently, the performance of silicon technology has improved so that compact low-cost phase shifters are a distinct possibility. A popular design for microwave and millimeter-wave phase shifters uses an electronic means to change the phase velocity along a section of transmission line. The concept was introduced in the 1950s at Bell Laboratories, Murray Hill, NJ, and implemented using Schottky diodes by Nagra and York [1]. This technique was further explored by Barker and Rebeiz using microelectromechanical systems (MEMS) varactors to load a transmission line with excellent results [2]. The same concept was used in [3], where the MEMS bridges were replaced with varactors fabricated using thin-film ferroelectric barium strontium titanate (BST). In commercial IC processes, phase shifters can be implemented using loaded lines where the transmission line is synthesized with lumped inductors [4]. These can give good Manuscript received April 8, 2004; revised June 27, 2004. This work was supported by the Boeing Company under a contract and by M/A-COM Research and Development under a gift. T. M. Hancock was with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109-2122 USA. He is now with the Analog Device Technology Group, Massachusetts Institute of Technology Lincoln Laboratory, Lexington, MA 02420-9108 USA (e-mail: [email protected]). G. M. Rebeiz was with the Radiation Laboratory, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109-2122 USA. He is now with the Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA 92093-0407 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842479

Fig. 1. Schematic of ideal tunable T-network.

performance, but there is tradeoff between the matching performance and the amount of phase per section that can be obtained. Therefore, a large number of inductors are required to maintain a good match while achieving the desired phase shift. The remaining types of phase shifters in commercial IC processes are digital and require the use of a solid-state switch that is typically implemented with a p-i-n diode or field-effect transistor (FET). Large phase shifts (180 ) are usually implemented with a switched high-pass/low-pass topology. For the remainder of the bits 90 , an embedded FET topology can be used [5], [6]. These designs offer good performance for their high level of integration, but rely on the use of an FET as a low-loss switching element. To further reduce the cost of phase shifters, it would be highly desirable to use silicon technology. A monolithic-microwave integrated-circuit (MMIC) implementation similar to [5] and [6] was implemented in silicon using a thick polyimide on top of the silicon chip so that a low-loss microstrip layer could be used for the inductors and transmission lines [7], [8]. This study details the implementation of a phase shifter in SiGe with an integrated low-noise amplifier (LNA) without any post-fabrication processing. A digital 180 phase shift is integrated into the LNA and is cascaded with an analog phase shifter using SiGe varactors that provides 180 of continuously variable phase shift. The analog phase shifter uses a constantimpedance tuning technique that tunes the phase while simultaneously keeping a constant port impedance. The SiGe phase shifter with integrated LNA is designed using extensive electromagnetic simulation. II. THEORY Low-pass - and T-networks can provide up to 90 of delay while remaining matched; conversely, the dual high-pass network can provide up to a 90 phase advance. However, the inductance and capacitance must be changed simultaneously to keep a constant port impedance. For the T-network in Fig. 1, the required values of the inductance and capacitance can be derived for a given phase delay and the condition that the network is matched to a characteristic impedance . The matrix for the network is

0018-9480/$20.00 © 2005 IEEE

(1)

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Fig. 2. Schematic of proposed tunable T-network.

where and

(2)

are the normalized values for and . Neglecting any resistive parasitics, to remain matched, , therefore, using (1), and . An expression for can be determined as follows: (3)

Fig. 3. Simulated: (a) return loss and (b) insertion phase of the proposed constant-impedance phase shift network over a 2 : 1 change in capacitance.

The insertion phase of the network is Solving (8) and (9) for

and

results in

(4) (10) Substituting (3) into (4), (11)

(5) Using (3) and (5), the value of inductance and capacitance can be calculated as and

(6)

From (6), for a capacitance tuning range of 2 : 1, one can achieve a phase shift of 60 . This is only possible if the inductance can be simultaneously tuned over a 3.7 : 1 tuning ratio. In an IC process, tunable inductors are not available; however, over a small bandwidth (10%–20%), a variable inductance can be approximated by a fixed inductor in series with a variable capacitance because the series capacitance subtracts reactance from the inductor. This leads to the proposed circuit of Fig. 2. has the same capacitance ratio as can Assuming that be designed to be times larger than to facilitate easy control of tuning. and can The reactance of the series combination of be written as (7) The equivalent series inductance of this network is determined by dividing (7) by . If the equivalent inductance is forced to equal the ideal inductance at the maximum and minimum values, the following two equations can be written: (8) (9)

For a 50- port impedance, 12.2-GHz center frequency and a capacitance ratio of 2 : 1, the phase-shift network can be easily and of 260 and designed using (10) and (11). For are found to be 130 fF, given by the technology, and 1.36 and 1.13 nH, respectively. Simulation shows that a capacitance ratio of 2 : 1 results in a phase shift of 60 and a return loss of better than 15 dB (Fig. 3). Notice that the network is perand , as fectly matched when the capacitance equals per (8) and (9). In between these values, the equivalent inductance does not equal the ideal inductance, but is close enough that the matching is better than 15 dB. III. CIRCUIT DESIGN A. LNA/180 -bit Design The topology for the 1-bit digital phase shifter with an integrated LNA is shown in Fig. 4(a). The first stage is simultaneously matched for noise and return loss by sizing the device for inductive emitter degenproperly and through the use of eration [9]. Since a 5-V supply voltage was required, a resistive load of 1 k was used instead of an RF choke. This saves area with only a small impact on the noise performance. The has four emitter fingers (5 0.5 m ) and input transistor is biased at 3.6 mA for low noise. Using a cascode in the first stage was an option and would result in more gain, but with more noise. Between the first and second stage of the LNA, a dc blocking capacitor and interstage matching inductor is used for maximum power transfer. and is the same The second stage in the LNA consists of size as . It is also biased for low-noise performance at 3.6 mA, but does not use inductive degeneration. It uses a cascode for

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Q

Fig. 4. (a) Digital 180 phase bit with integrated LNA. The input impedance of is not matched to the characteristic impedance of the phase shift networks. , which results in a gain mismatch between the two phase states. (b) When the delay networks are not 90 , there is a difference in the impedance presented to

6

increased gain and to implement a single-pole–double-throw and (SPDT) switch in the signal path. The bias of are controlled such that only one transistor pair is turned on at a time and the signal is directed either through the high-pass 90 or low-pass 90 network to provide a net and are biased at 1.9 mA). The com180 phase shift ( plete LNA/180 bit consumes 10.2 mA. The simulated gain is 14.5 dB with a noise figure of 3.4 dB at 12.2 GHz. The 1-dB input compression point and input third-order intercept point (IIP3) is 25 and 15 dBm, respectively. The high- and low-pass networks are designed to have a characteristic impedance of approximately 100 . Using a highimpedance network improves the power transfer between and by providing some interstage matching. The values of the low-pass T-network can be chosen by using (6). For the high-pass -network, the derivation is similar, resulting in the following dual expressions:

and

Q

Fig. 5. Passive varactor-tuned phase-shifter network. The shunt capacitors are parasitics, which cause the performance to deviate from the ideal.

(12)

where is 90 in (6) for the low-pass network and is 90 in (12) for the high-pass network. The shunt capacitor in the low-pass T-network was implemented as a parallel-plate structure in the oxide between metal 1 and metal 2 [metal–insulator–metal (MIM)]. The series in the high-pass -network was implemented capacitor as two series capacitors to maintain symmetry in the network. These two series capacitors were done using the MIM capacitor layer in the process that uses silicon nitride as a dielectric. In both networks, the inductors are placed next to each other and simulated as a three-port element to take into account the mutual coupling. The high-pass/low-pass networks, including , are full-wave simulated and optithe blocking capacitors mized to provide a net phase shift of 180 with a small amplitude and . imbalance when integrated between the transistors There is a subtlety that can be overlooked when designing the switched high-pass/low-pass networks. At first glance, this topology could be used to create 90 , 45 , and 22.5 bits, but these are not implemented as easily. The input impedance of and the output impedance of are not matched to the

Fig. 6. Simulated insertion and return losses at different analog control voltages for the constant-impedance phase shifter.

characteristic impedance of the high- and low-pass networks. When the input impedance of does not equal , it will be transformed along a constant impedance circle and be rotated . by In the case of the 180 bit, where 90 networks are used, the input impedance of both signal paths are rotated 180 and both end up at the same point on the Smith chart. This is not the case if the other phase-shifter bits are implemented in this manner [see Fig. 4(b)], and results in a gain imbalance because the gain of the two signal paths are proportional to the impedances preand . The gain imbalance can sented to the outputs of and the input of be corrected by matching the output of to before inserting the high- and low-pass networks, but this requires the use of four additional matching networks, leading to an increased area and narrower bandwidth. For this reason, the remainder of the phase shifter is implemented with an analog constant-impedance phase-shift network.

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Fig. 7.

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Microphotograph and schematic of 12-GHz active phase shifter with integrated LNA.

B. Constant-Impedance Design The first step in designing the analog constant-impedance phase shifter is to determine the sizing of the shunt and series varactors. Equation (6) is used to determine the size of the shunt are then calculated using (10) varactor. The value of and and (11). In reality, the cathodes of the series varactor diodes have a shunt substrate diode associated with them resulting in a parasitic junction capacitance. In addition, there are also shunt parasitics at the two terminals of the inductors (Fig. 5). The effect of these parasitics can be compensated by adjusting the through simulation. The paravalues of value of and sitics reduce the tuning range to approximately 45 , requiring the complete analog phase shifter to use four sections to achieve the required 180 phase shift. The varactors have a simulated in the range of 30–45 at 12 GHz and the inductors have a simulated of approximately 15–18 at 12 GHz, depending on how close they are to the coplanar waveguide (CPW) ground plane. This results in approximately 1.7 dB of insertion loss per 45 section. The full-wave simulations of the four cascaded sections are shown in Fig. 6. The insertion loss is 6.8 dB, matched around the design frequency, and the phase delay is linear with frequency. The constant-impedance network is easily scalable to higher frequencies, but care must be taken to minimize and correctly model the substrate parasitics associated with the varactors and inductors. The power-handling capability of the network was also simulated and the phase begins to deviate from the nominal value due to self-biasing of the varactors at high input powers. When the diodes are reverse biased at 2 V, the phase shifts from its nominal value by 2 for an input power of 20 dBm. Therefore, the overall linearity of the phase shifter is limited by the input LNA.

IV. PHYSICAL DESIGN The circuit was implemented via Atmel’s SiGe2-RF process. The technology offers two HBTs with a unity current gain of 50 and 80 GHz and a collector–emitter cutoff frequency of 4 and 2.5 V, respectively. Both breakdown voltage of transistors have a maximum oscillation frequency device was used in the design of 90 GHz.1 The 80-GHz the phase shifter. There are no through wafer vias or backside metallization so the phase shifter is laid out in a CPW fashion. A complete ADS2 design kit for the active devices is provided by the Atmel Corporation, Heilbronn, Germany. The phase shifter was first designed in ADS with lumped components to determine the topology and bias points of the devices. The physical layout is then simulated in pieces using Sonnet3 for planar electromagnetic simulations to take into account the coupling between adjacent inductors and the interaction with the ground plane. The multiport -parameters are used in ADS in conjunction with the models of the active devices to simulate the phase-shifter performance and iterate as necessary. Once completed, the final layout is done in Cadence4 using Virtuoso. Fig. 7 shows a microphotograph of the phase shifter with an integrated LNA. The dimensions of the total chip, including pads, is 1920 m 780 m; however, without pads, the area of the active circuitry is 1680 m 660 m, an area of 1.1 mm . 1Atmel SiGe HBT Foundry, Heilbronn, Germany. [Online]. Available: http://www.atmel.com/products/SiGeBipolar/, 2004. 2ADS 2003A, Agilent Technology Inc., Palo Alto, CA, 1983–2003. 3Sonnet 9.52, Sonnet Software Inc., North Syracuse, NY, 1986–2003. 4Cadence 4.4.6, Cadence Design Systems Inc., San Jose, CA, 1992–2000.

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Fig. 10. Measured phase difference flatness; the phase versus frequency curves are normalized to their value at 11.5 GHz.

Fig. 8. Measured phase at 11.5 GHz as a function of control voltage. The insertion loss is high and is not used at 1–2 V.

Fig. 11. Measured gain and return loss of the phase shifter with integrated LNA in both digital states of the 180 bit and for analog control voltages from 5 to 2.5 V (0.5-V steps). Fig. 9. Measured absolute phase for analog control voltages from 5 to 2.5 V (0.5-V steps).

V. MEASURED RESULTS The phase shifter was measured on-chip using RF probes and a line-reflect-reflect-match (LRRM) calibration to the probe tips to measure the -parameters. There is one digital control and three analog phase controls. For most of the testing, the three analog controls were shorted together to form a single analog control. The digital control introduces a 180 phase delay, while the analog control voltage provides continuous phase control over a 180 range. Fig. 8 shows that a phase shift of 0 –180 is achieved with the digital control high and the analog voltage swept from 5 to 2.4 V. When the digital control is low and the analog voltage swept from 5 to 2.4 V, the insertion phase is 180 –360 . The constant-impedance network was designed based on the CV curve of the varactor model, which indicated that a reverse bias from 5 to 1 V would be needed. However, a reverse bias of only 5–2.4 V is found to achieve the required phase shift, and degrades quickly below 2 V. This is due to a combination of increased capacitance and a decrease in varactor , and indicates that the scalable foundry varactor model underestimates the tuning ratio and overestimates the . The absolute phase performance is shown in Fig. 9 and, as expected, increases linearly with frequency. Fig. 10 shows the same curves normalized to their relative phase values at 11.5 GHz. This shows that the relative phase shift is constant and flat to within 10 over more than 1 GHz of bandwidth. The measured gain and return loss are presented in Fig. 11. The overall gain of the phase shifter is simulated to be 7.2 dB at

Fig. 12. Measured output return loss (S ) from 11 to 12 GHz, showing that a constant impedance is maintained.

12.2 GHz and measured to be 3.7 0.5 dB at 11.5 GHz. From 11–12 GHz, the gain varies by less then 1.2 dB. Both the input and output return losses are better than 10 dB over the band of interest. The was designed to be better than 15 dB, and the difference is most likely due to the inaccuracies in the varactor model. However, the concept of using series varactors to implement a constant-impedance phase shift network is working properly. Fig. 12 shows the output return loss for different phase states from 11 to 12 GHz and the impedances are concentrated in a very small area. The measured noise figure is 4.4 dB at 11.5 GHz, which agrees well with the simulated value of 3.7 dB at 12.2 GHz. The difference is most likely due to increased loss in the analog phase shifter that follows the 180 digital bit. This would also account for the 3.5-dB reduction in overall gain. In future designs, the noise figure could be improved by increasing the of the inductors used in the LNA stage ( and ) by using the parasitic inductance of a package (bond wires, etc.). This design was not intended for packaging and, therefore, this design

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TABLE I COMPARISON BETWEEN SIMULATED AND MEASURED PERFORMANCE OF THE PHASE SHIFTER WITH INTEGRATED LNA

TABLE II ANALOG VOLTAGES TO PRODUCE DIGITAL-LIKE BEHAVIOR BY CONTROLLING THE PHASE-SHIFTER CELLS INDEPENDENTLY. ALL CELLS RESULT IN 0 RELATIVE PHASE WHEN BIASED AT 5 V

approach was not an option. Beyond improving the of the passive devices in the LNA stage, the device would need to scale to improve the noise performance. This technology uses a 0.5- m emitter width that can obtain a 1.5-dB noise figure with 14 dB of associated gain at 10 GHz. However in a state-of-the-art technology where the emitter width could be as small as 0.12 m, the minimum noise figure is 0.45 dB with an associated gain of 14 dB at 10 GHz [10]. There was a shift in center frequency from 12.2 to 11.5 GHz; however, a more accurate varactor model would eliminate these errors in a second iteration. The measured input 1-dB compression point and IIP3 is 27.3 and 17.3 dBm, respectively. The linearity requirement is dependent on the application, but the input power could be as high as 20 dBm. The linearity of this design is dominated by the performance of the LNA and could with an RF choke and using a be improved by replacing lower supply voltage to avoid transistor breakdown. As designed, the circuit has a bandwidth of 1 GHz around 11.5 GHz. This is limited by the allowable amplitude and phase ripple over changes in phase state and is determined by the phase per stage of the constant impedance network. Currently, four stages are used with 45 per stage. The bandwidth could be increased by making the network more distributed and using eight stages with 22.5 per stage at the expense of more area. A summary of the simulated and measured results are shown in Table I. Although this phase shifter has an analog control function, individual control of the phase-shift cells can be done to synthesize a 4-bit digital phase shifter. Referring to Fig. 6, the analog control lines for two of the phase shift cells are tied together to implement a 90 bit. The other two cells can correspond to the 45 and 22.5 bits. The required bias voltages are presented in Table II.

Fig. 13. Measured relative phase: (a) of the 16 states when the phase shifter is tested in a digital fashion and (b) normalized to the corresponding 4-bit digital values (0 , 22.5 , 45 , etc.).

The voltages of Table II were applied manually to simulate a 4-bit phase shifter. The measured relative phase shift is shown in Fig. 13(a). Fig. 13(b) shows the phase variation when the phases are normalized to their ideal digital values. This reveals that there is an 18 phase error on the 180 bit, and may be due to the fabrication tolerance of the MIM capacitance layer used for . This could easily be corrected in a second iteration. VI. CONCLUSION This paper has presented the design and measurement of a 12-GHz phase shifter with an integrated LNA in SiGe. A digital 180 phase shift is integrated into the LNA using a switched high-pass/low-pass network. The remaining phase shift is achieved with an analog phase shifter using varactors that implement a constant-impedance tuning technique that tunes the phase while simultaneously compensating for the dB matching. The phase shifter has a measured gain of 3.7 at 11.5 GHz with a noise figure of 4.4 dB. The phase shifter has more than 360 of phase shift and the relative phase performance is flat with frequency within 10 over more than 1 GHz of bandwidth. The analog control lines for individual phase shifter cells were also adjusted independently to demonstrate that a 4-bit digital phase shifter could also be implemented with good performance.

HANCOCK AND REBEIZ: 12-GHz SiGe PHASE SHIFTER WITH INTEGRATED LNA

ACKNOWLEDGMENT The authors would like to thank Dr. J. P. Lanteri, M/A-COM, Lowell, MA, for supporting this program through generous access to the ATMEL mask sets, and Dr. I. Gresham, M/A-COM, and Dr. Ming Chen, Boeing Company, Seattle, WA, for technical discussions. REFERENCES [1] A. S. Nagra and R. A. York, “Distributed analog phase shifters with low insertion loss,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1705–1711, Sep. 1999. [2] N. S. Barker and G. M. Rebeiz, “Optimization of distributed MEMS transmission-line phase shifters U -band and W -band designs,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 1957–1966, Nov. 2000. [3] B. Acikel, T. R. Taylor, P. J. Hansen, J. S. Speck, and R. A. York, “A new TiO thin films,” IEEE high performance phase shifter using Ba Sr Microw. Wireless Compon. Lett., vol. 12, no. 7, pp. 237–239, Jul. 2002. [4] F. Ellinger, H. Jäckel, and W. Bächtold, “Varactor-loaded transmissionline phase shifter at C -band using lumped elements,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1135–1140, Apr. 2003. [5] J. Wallace, H. Redd, and R. Furlow, “Low cost MMIC DBS chip sets for phase array application,” in IEEE MTT-S Int. Microwave Symp. Dig., 1999, pp. 677–680. [6] C. F. Campbell and S. A. Brown, “A compact 5-bit phase-shifter MMIC for K -band satellite communication systems,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2652–2656, Dec. 2000. [7] M. A. Teshiba, R. V. Leeuwen, G. Sakamoto, and T. Cisco, “A SiGe MMIC 6-bit p-i-n diode phase shifter,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 12, pp. 500–501, Dec. 2002. [8] R. Tayrani, M. A. Teshiba, G. Sakamoto, Q. Chaudhry, R. Alidio, Y. Kang, I. S. Ahmad, T. C. Cisco, and M. Hauhe, “Broad-band SiGe MMICs for phased-array radar applications,” IEEE J. Solid-State Circuits, vol. 38, no. 9, pp. 1462–1470, Sep. 2003. [9] O. Shana’a, I. Linscott, and L. Tyler, “Frequency-scalable SiGe bipolar RF front-end design,” IEEE J. Solid-State Circuits, vol. 36, no. 6, pp. 888–895, Jun. 2001. [10] J.-S. Rieh, D. Greenberg, M. Khater, K. T. Schonenberg, S.-J. Jeng, F. Pagette, T. Adam, A. Chinthakindi, J. Florkey, B. Jagannathan, J. Johnson, R. Krishnasamy, D. Sanderson, C. Schnabel, P. Smith, A. Stricker, S. Sweeney, K. Vaed, T. Yanagisawa, D. Ahlgren, K. Stein, and G. Freeman, “SiGe HBT’s for millimeter-wave applications with of 300 GHz,” in Proc. IEEE simultaneously optimized f and f RFIC Symp., 2004, pp. 395–398.

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Timothy M. Hancock (S’96–M’05) received the B.S. degree in electrical engineering from the Rose-Hulman Institute of Technology, Terre Haute, IN, and the M.S. and Ph.D. degrees in electrical engineering from The University of Michigan at Ann Arbor, where he was involved with the development of SiGe integrated microwave components. During the summers of 2000 and 2001, he was involved with a fully integrated global positioning system (GPS) solution and power control circuitry for handset power amplifier applications. In 2004, he was with M/A-COM, where he was involved with the design of SiGe components for an automotive radar solution at 24 GHz. He is currently a Technical Staff Member with the Analog Device Technology Group, Massachusetts Institute of Technology (MIT) Lincoln Laboratory, Lexington, where he is involved with the development of integrated microwave circuits and systems.

Gabriel M. Rebeiz (S’86–M’88–SM’93–F’97) received the Ph.D. degree in electrical engineering from the California Institute of Technology, Pasadena. He is a Full Professor of electrical engineering and computer science (EECS) with the University of California at San Diego, La Jolla. He authored RF MEMS: Theory, Design and Technology (New York: Wiley, 2003). His research interests include applying MEMS for the development of novel RF and microwave components and subsystems. He is also interested in SiGe RFIC design, and in the development of planar antennas and millimeter-wave front-end electronics for communication systems, automotive collision-avoidance sensors, and phased arrays. Prof. Rebeiz was the recipient of the 1991 National Science Foundation (NSF) Presidential Young Investigator Award and the 1993 International Scientific Radio Union (URSI) International Isaac Koga Gold Medal Award. He was selected by his students as the 1997–1998 Eta Kappa Nu EECS Professor of the Year. In October 1998, he was the recipient of the Amoco Foundation Teaching Award, presented annually to one faculty member of The University of Michigan at Ann Arbor for excellence in undergraduate teaching. He was the corecipient of the IEEE 2000 Microwave Prize. In 2003, he was the recipient of the Outstanding Young Engineer Award of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He is a Distinguished Lecturer for the IEEE MTT-S.

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Two-Dimensional Curl-Conforming Singular Elements for FEM Solutions of Dielectric Waveguiding Structures Din-Kow Sun, Member, IEEE, Leon Vardapetyan, and Zoltan Cendes, Fellow, IEEE

Abstract—This paper proposes a curl-conforming singular element for modeling electromagnetic fields around singular points. Similar to the Nédélec types of regular vector elements, the space of the proposed singular elements consists of gradient and rotational subspaces. The proposed singular elements have arbitrary singularity orders that are precomputed analytically according to local geometry and material properties. The singularity orders of the gradient bases depend on the electric-field behavior; the rotational bases on magnetic-field behavior. Assigning integer singularity orders transforms singular elements into regular elements. Since the gradient subspace is properly modeled, the proposed singular elements are free from contamination by spurious modes. By including the singular elements in the solution space, deterioration of convergence rates often encountered with waveguides containing singular corners is avoided. Validation of the proposed singular elements is provided both theoretically in terms of the de Rham diagram and numerically by solving canonical singular dielectric waveguiding structures. Index Terms—Curl-conforming, finite elements, Maxwell’s equations, singular element, waveguides.

I. INTRODUCTION

T

HE finite-element method based on curl-conforming vector bases provides robust solutions of dielectric waveguides of arbitrary shape. However, such structures often contain sharp metal edges or sharp dielectric corners for which regular curl-conforming vector bases do not converge as rapidly as anticipated. This slow convergence is a consequence of the nonanalytical nature of the solution near sharp corners. The field is singular at an infinitely sharp corner so that it is necessary to refine the mesh around singular points if traditional curl-conforming polynomial finite elements [1] are used. Several publications [2]–[8] have proposed using singular elements to improve convergence in the vicinity of singular points. References [2]–[4] employ scalar singular elements to analyze the quasi-TEM modes of transmission lines, and [5]–[8] propose high-order curl-conforming singular vector elements to solve the full vector wave equation. However, high order in these references means only high-order variations in the azimuthal direction; in the radial direction, where the field exhibits singularity, only the lowest term is accurately taken into account. Although the authors of [6] and [8] allow multiple singularity orders, all higher orders are assumed to differ from Manuscript received April 23, 2004; revised August 17, 2004. The authors are with the Ansoft Cooperation, Pittsburgh, PA 15219 USA. Digital Object Identifier 10.1109/TMTT.2004.842477

Fig. 1.

3 cm

2 2 cm rectangular slit waveguide showing the mesh.

the lowest order by an integer. This assumption may not catch the true singularity in the physical field. In the case of metal edges, the singularity orders are integers times a fraction; in the case of dielectric wedges, they are completely arbitrary [9], [10]. Not including higher order terms in the radial basis functions results in lost rates of convergence for the polynomial orders used in the approximation. Furthermore, the singularity orders between the electric- and magnetic-field modes may not be related. This degree of freedom must be incorporated; otherwise, the new singular element is not able to simultaneously predict both electric- and magnetic-field modes accurately. In the following, we derive curl-conforming singular vector bases with arbitrary singularity orders that capture the solution behavior in the vicinity of singular points and, therefore, yield high rates of convergence. To simultaneously solve for the electric- and magnetic-field singularity, two interdependent conditions are satisfied: the new singular bases matches the radial profile of the electric-field singularity in the vector space itself, and the range space of the curl operator acting on the electric field matches the physical singularity in the magnetic field. Validation of the proposed singular elements is provided both in terms of theory and by solving canonical singular dielectric waveguiding structures. II. SINGULAR ELEMENTS An example of a waveguide to be solved is shown in Fig. 1. The polygon surrounding the singular point will be called the singular region and the rest of the area will be called the regular region. The local fields near the singular points of a waveguide are static and can be characterized as electric field ( ) modes

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SUN et al.: 2-D CURL-CONFORMING SINGULAR ELEMENTS FOR FEM SOLUTIONS OF DIELECTRIC WAVEGUIDING STRUCTURES

and magnetic field ( ) modes. Longitudinal components of the fields around singular points are written as [3]

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’s that work with the regular elements are defined. The scalar bases in [12] are as follows:

modes modes where and ’s are positive and are sorted by magnitude and specify the singularity orders (i.e., the singularity coefficients) modes, respectively. Also, are triangular for the and polar coordinates [11] that are related to simplex coordinates through the following relations:

Here, vertex 0 is assumed to be the singular point, and both and . The singularity orders depend on the angles of the corners and the ratios of material properties across the interfaces. In [12], a numerical procedure is proposed for finite-element solutions of dielectric waveguiding structures. The longitudinal component of the electric field is expanded in terms of regular scalar bases, and the transverse components are expanded in terms of regular vector bases. To extend this procedure to the presence of singular points, we need to derive a set of singular scalar bases and a set of singular vector bases for triangles touching singular points. We give the proposed basis vectors below and defer the discussion of their construction until Section II-C. Let be the singularity index of the singular element and write the highest order of -mode singularity as . As discussed later, -mode singularities are related to the range of the curl operator; therefore, the highest order of -mode singu, is one index higher. This is similar to regular bases larity, where the range order is one order less than the highest order of bases. Further, as discussed in Section III-B, we choose to equal in numerical studies.

B. Singular Vector Bases As is the case with regular vector basis functions, singular vector basis functions consist of gradient and rotational bases. Singular gradient bases are obtained by taking the gradient of the preceding singular scalar bases. Rotational bases are solely responsible for the range space of the curl operator, which is . It follows that radial powers of the range space of related to . Rotational singular the curl operator should match those of bases are classified and built as follows. for edge 12 only. The 1) Edge-associated: curl of this basis vector provides a constant vector. 2) Face-associated:

for

, and , where the coefficients ’s are chosen such that the tangential projection of the basis vectors on edge 12 vanishes. The curl of the . face-associated vectors is The coefficients of ’s that match the regular vector bases in [12] are as follows:

A. Singular Scalar Bases Singular scalar bases are classified and built as follows. 1) Vertex-associated: . 2) Edge-associated: for . a) Edges 01 and 02:

b) Edge 12:

, where

, and is the regular st order associated with edge scalar basis of the 12. This ensures the continuity between singular and regular elements. for 3) Face-associated: , and . The total number of singular scalar bases is . Vertex-associated scalar bases are the same as those used in [5]. Edge- and face-associated bases provide higher order variations in the radial and azimuthal directions. The explicit form of cannot be written until the bases of the adjacent regular

Note that these coefficients are independent of the subscript . Also, in the construction of face-associated bases, we impose zero tangential components on three edges by removing them from the existing singular gradient bases. Therefore, no extra polynomials are introduced into the vector space. Since the three vertex-associated singular gradient bases are linearly dependent, the number of degrees of freedom of sin. The number gular gradient bases is of singular rotational bases is ; therefore, the total number of degrees of freedom of singular vector bases is , identical to that of Nédélec’s vector bases. The tangential continuity of the field is guaranteed by the property that each individual basis is tangentially continuous. Singular

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gradient bases provide field variations along edges 01 and 02. Edge-associated rotational bases on edges 01 and 02 are not needed because the edge vector on edge 12 already furnishes the necessary constant vector in the range space. This is consistent to the tree–cotree splitting in [12]. Since in [12] the vertex-associated vector bases are employed together with the edge-associated vector bases, a tree–cotree splitting procedure is performed to eliminate the redundant unknowns on the tree edges. In this case, the tree edges are chosen to be edges 01 and 02. C. Spurious Modes For the proposed elements to be free of spurious-mode contamination, they need to satisfy the discrete compactness property. In [13], Boffi links the de Rham diagram to the discrete compactness of the finite-element spaces. For a waveguiding structure uniform in the -direction, one may assume the field , where is quantities have a -dependence of the form the propagation constant. According to the de Rham diagram of two-and-one-half dimensions, the proposed elements must satisfy the following relationships in both continuous ( ) and discrete ( ) spaces:

where the spaces are defined to be

The discrete spaces are the subspaces of continuous spaces; the interpolation operators ’s are defined in the Appendix. is the solution In the following, we show that space for , and is the solution space for . The proof of the commutativity of the de Rham diagram (1) is provided in the Appendix. For a waveguide, the elec. tric potential can be written as Here, and in the Appendix, we use uppercase and lowercase letters to differentiate between field variables with and without -dependence, respectively. The first inclusion in (1) tells us that the vector space contains . Since should contain , and belongs to . With the rotational vectors ’s, we have and . By choosing the basis vectors for to be and , and for to be in , conthe second inclusion in (1) is satisfied, i.e., tains . Therefore, we need (2)

Fig. 2. Effects of material property and wedge angle on the singularity orders. The permittivity of the dielectric wedge increases from 4 (solid line) to 10 (dashed line).

Consequently, we have and , which completes the third inclusion. From the above arguments, we also observe the singularity orders in the gradient subspace of the vector bases are related to , while the singularity orders in the rotaand tional subspace come from . Thus, the relationships between the singularity orders of the bases and the physical quantities can be derived purely from a mathematical point-of-view. One can perceive these relationships from a physical point-of-view, as is done in the literature and at the beginning of this section. Note that, if we start with the de Rham diagram corresponding to two dimensions, as given in the Appendix, we cannot fully see the relationships from a mathematical point-of-view. Since the proposed singular elements are constructed to satisfy the requirements of the de Rham diagram, spurious modes are properly modeled. Their corresponding eigenvalues are zero and, most important, they do not mix with physical , singular bases turn into regular modes. When bases. This means that singular bases are more general than regular bases and can be used throughout the problem domain, although they are less efficient in the calculation of the system and deviate from integers by a small matrix. When amount, singular bases deviate from the corresponding regular bases by a small amount as well. This suggests that singular bases are continuously convergent to regular ones. By studying singular corners arising from dielectric interfaces, we reassure the singularity orders are continuously and smoothly varied away from their regular counterparts. This is demonstrated in Fig. 2 where either the permittivity or the angle of the wedge is adjusted to increase the severity of singularity. In a dielectric modes are singular, but modes are regular. waveguide, Fig. 2 shows that, as the wedge angle increases, each of the first three modes of a zero-degree wedge splits into two singular modes and recombines into a regular mode at some wedge angles. It also indicates that the splitting is more significant with larger permittivities.

SUN et al.: 2-D CURL-CONFORMING SINGULAR ELEMENTS FOR FEM SOLUTIONS OF DIELECTRIC WAVEGUIDING STRUCTURES

III. PERFORMANCE

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Consequently,

A. Computing the Singularity Orders The singularity orders can be predicted by the method described in [10], where dielectric wedges are considered. This method can be extended to more general cases in which any number of interfaces meets at one point. This is done by asand for suming a linear combination of each region and imposing continuity and derivative continuity across interfaces. The result is a transcendental equation to be solved numerically for the singularity orders. B. Choosing the Singularity Index for Singular Elements There are two ways to choose the singularity index of the basis vectors for singular elements. One way is to let the largest radial power of singular elements approximate the largest radial power of the adjacent regular elements. This usually requires that the singular elements have more terms than the regular elements and, consequently, both the total number of unknowns and the highest angular order of the singular elements will be larger than those of the regular elements. This choice fully recovers the rate of convergence of the regular elements. However, it is much more complicated to implement this choice because the number of basis vectors varies from one singular point to another. The other way is to employ the same highest angular order everywhere in the waveguide. Accordingly, the singularity index is equal to the range order of the adjacent regular elements. This choice exactly inherits the same configuration as the regular elements and, hence, the total number of unknowns is equal, as is the computational cost. Although this choice does not fully recover the optimal rate of convergence of the regular elements, it is easier to implement. We will only study this set of bases numerically. C. Calculation of Matrix Entries To secure the expected rate of convergence, the element matrix must be integrated analytically. This is done by symbolically forming the bases in triangular polar coordinates and symbolically performing the multiplication and curl/grad operations to compute the integrands and, finally, a trivial analytical integration is performed. It should be noted that only one symbolic form for all singularity orders is required. Some typical integration examples follow. We write . Thus,

It follows that

D. Limitations of the Proposed Singular Elements The newly proposed singular elements share the following two fundamental limitations with the references in [2]–[8]. 1) A triangle cannot have more than one singular point. This limitation may be removed by employing multiple sets of bases—one for each singular point. However, we have not implemented or tested this scheme. In three-dimensional domains, where point singularities coexist with edge singularities, some way of managing multiple singularities in one mesh element is necessary. 2) The size of the singular region must be on the order of one wavelength, and it must not be refined, or else accuracy may degrade. While singular bases faithfully produce accurate results, regular bases fail to do so. This is because as the singular region becomes smaller, the solution profile of its nearest neighbors is also nearly singular. Regular bases cannot model the rapid change of field in this transition region [14]. Consequently, further mesh refinement is required in the region. Shrinking the size of the singular region cuts the reach of the singular bases for reducing the total error in the singular region. On the other hand, if the size of the singular region is more than one wavelength, the assumption of the static limit is no longer valid. Further study is needed to determine the optimal size. IV. NUMERICAL RESULTS The proposed singular elements are validated with four examples including a rectangular slit waveguide, a double-ridged waveguide, a centrally loaded dielectric waveguide, and a microstrip transmission line. We compute the propagation constants for each example by employing or not employing singular elements around singular points, and by refining mesh or increasing basis orders. To compare the rates of convergence between two solutions, we plot the relative error of the propagation defined as versus matrix size. Refconstant erence values are computed by using Ansoft’s HFSS [15] for a much denser mesh. To study the rates of convergence, a series of successively refined meshes is created. The singular regions are never refined in any of our examples, or else the solution may get worse. Other than the identifiable zero-eigenvalue modes, no spurious modes are observed in any of numerical examples. Also, we use to denote both and since we choose to equal as discussed in Section III-B. A. Rectangular Slit Waveguide

where is to be integrated analytically. For the curl terms, we have

The first example is the rectangular slit waveguide shown in Fig. 1, which contains a vertical metallic edge. Both orders of - and -mode singularities are . The reference values for the first three modes are 282.48576 m , 247.83117 m , and 237.04824 m . Fig. 3(a) compares the rates of convergence for

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Fig. 3. Convergence rates for p refinement with the rectangular slit waveguide in Fig. 1 at 14 GHz. (a) Mode 1. (b) Mode 2. (c) Mode 3. The lines are produced by the regular element method; the dots by the singular element method. Along each line or group of dots, the range order p rises from 0 to 3.

the ground mode. It shows that singular elements give a more accurate solution, and their rates of convergence are much steeper. Fig. 3(b) and (c) compares the next two modes. The third mode in Fig. 3(c) displays a similar improvement as the ground mode. The second mode, however, is a nonsingular mode, and Fig. 3(b) indicates that, with the nonsingular mode, the rates of convergence are essentially identical whether the proposed singular elements are used or not. This is because, in this case, some singular bases actually have integer radial powers and, hence, indeed provide solutions for the nonsingular modes. Fig. 4 plots the electric field of the first mode along the slit for the mesh in Fig. 1. The size of the singular region is 0.2 cm, approx. As shown, regular bases struggle to catch the imately rapid change of the field profile while singular bases have done very well even with only the lowest order term. Higher order regular bases and singular bases continue to improve the solution in their own regions. This example demonstrates that if the singularity orders in the singular elements are accurately set, the singular elements truthfully provide both nonsingular and singular modes. B. Double-Ridged Waveguide The second example is the double-ridged waveguide shown in . As in the preFig. 5, which has singularity orders equal to vious example, the results in Fig. 6 show that the error is greatly

Fig. 4.

Electric field of the first mode along the slit in Fig. 1.

reduced and the rates of convergence are improved. The reference value for the ground mode, a TE mode, is 152.43963 m . A difficulty with this type of waveguide is that the width of the ridge can be so narrow that the proposed singular elements are not applicable because either some triangles will have two singular points or the sizes of singular regions are too small to give an improved solution.

SUN et al.: 2-D CURL-CONFORMING SINGULAR ELEMENTS FOR FEM SOLUTIONS OF DIELECTRIC WAVEGUIDING STRUCTURES

Fig. 5.

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Double-ridged waveguide. Fig. 8. Convergence rates for p refinement with the centrally loaded dielectric waveguide in Fig. 7 at 14 GHz. See Fig. 3 for the description of the data markers. The range order p rises from 0 to 4.

Fig. 9. Microstrip transmission line.

Fig. 6. Convergence rates for p refinement with the double-ridged waveguide in Fig. 5 at 10 GHz. See Fig. 3 for the description of the data markers. The range order p rises from 0 to 3.

Fig. 7.

Centrally loaded dielectric waveguide.

C. Centrally Loaded Dielectric Waveguide Consider next the centrally loaded dielectric waveguide shown in Fig. 7. The dimensions of the waveguide are 4 cm 2 cm. The middle of the waveguide is filled with a medium having dielectric constant 3 of dimensions 2 cm 1 cm. We examine whether or not the singular fields at the four dielectric corners degrade the convergence rates. The computed first five orders of -mode sinmodes gularities are 0.839, 1.161, 2, 2.839, and 3.161. The are nonsingular since there is no variation in permeability. The reference propagation constant for the ground mode, a quasi-TE mode, is 438.71020262 m . From Fig. 8, we do not see any significant improvement until the error is less than 10 . This result is consistent with the result in [12], where only regular elements are used in the entire domain and the convergence slows down around 10 . This is because the total error is initially dominated by the error in the regular region. The decline in the error in the singular region can only be detected after the error in the regular region is sufficiently reduced. This example shows that the traditional polynomial finite-element method may be good enough to compute the propagation constants of the problems involving dielectric objects.

Fig. 10. Convergence rates for p refinement with the microstrip transmission line in Fig. 9 at 10 GHz. See Fig. 3 for the description of the data markers. The range order p rises from 0 to 4.

D. Microstrip Transmission Line The final problem is the microstrip transmission line shown in Fig. 9, where an infinitively thin perfectly conducting strip is laid on top of a substrate with a dielectric constant of 9.6 having a ground plane on the bottom. Both the width of the strip and the height of the substrate are 25 mil. Although the problem domain is unbounded, the field is concentrated in the area between the strip and ground plane. We, therefore, truncate the problem domain with a conducting box. The orders of -mode ; the -mode sinsingularities at ends of the strip are . The reference value for the ground mode, a gularities are quasi-TEM mode, is 545.20636 m . Again, we compare the rates of convergence with and without singular elements, and the results are shown in Fig. 10. As in the previous problem, significant improvement is seen only after the error from the regular region has been greatly reduced. However, unlike the previous problem, without singular elements, the accuracy is low. Thus, for microstrip problems, it is crucial to improve the solution around metal edges by using either mesh refinement or by employing singular elements.

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V. CONCLUSION A general form of curl-conforming vector finite elements has been presented for the treatment of singular fields at sharp metal edges and sharp dielectric corners. The proposed elements are more general than the Nédélec types of elements in two respects: first, radial powers of the proposed bases are rational with absolutely no relationship among radial powers of the different bases; second, the gradient and rotational subspaces are formed from individual considerations. They are not tied together, as is the case with Nédélec elements. The gradient subspace is used to satisfy the field behavior of the primary (electric) field, while the rotational subspace is used to satisfy the dual (magnetic) field. As a result, the radial powers of two subspaces are not related either. The proposed singular scalar bases may be readily applied to the three-dimensional scalar Helmholtz equation with singular points present. However, the application to the three-dimensional vector Helmholtz equation will require further research because vector singularities in three dimensions have edge singularities as well as point singularities.

number of testing functions is automatically equal to the finite dimension of the discrete space. Also, the resulting linear functionals are linearly independent as long as the testing functions are represented by a set of linearly independent bases. Thus, the unisolvent property is assured. Three operators, when operated on a whole mesh, produce a globally conforming discretization: produces a field with continuity, with tangential continuity, and with normal continuity. In addition, we stress that no relationships between the bases of the different discrete spaces are assumed here. Relationships are brought in via the inclusion relationships of the de Rham diagram. is defined as as follows. The operator 1) For the vertex-associated bases, (3) and the corresponding basis at the vertex

is

2) For the edge-associated bases, (4)

APPENDIX Vector finite-element discrete spaces are classified into three groups, which are: 1) vertex-associated ( ); 2) edge-associated ( ); and 3) face-associated ( ). Here, vertex associated indicates that the continuous components of the basis vectors vanish on the edge opposite the vertex to which the basis vector is assospace, the continuous component is the field ciated. In the space, it is the tangential component. itself, while in the The continuous components of edge-associated bases vanish on the other two edges, and those of face-associated bases vanish on all edges. Furthermore, each group can be subdivided into two subgroups, i.e., null ( ) and supplement ( ). Both classifications are denoted by subscripts. Therefore, the spaces observe the following relationships:

3) For the face-associated bases, (5) The operator is defined as 1) For the vertex-associated bases,

as follows. (6)

, and where vector at the vertex . 2) For the edge-associated bases: • rotational:

is the basis

(7) where • gradient: (8) where

Notice that is a pure gradient space. Also, the same subscripts are used to indicate the components of the fields, e.g.,

satisfies

is defined by

and is related to by 3) For the face-associated bases,

(9)

A. Definition of the Interpolation Operators Adopting the definitions in [16], we develop a similar methodology for interpolation operators. The interpolation is constructed such that the testing functions for each operator span the entire corresponding discrete space. Therefore, the

.

(10) Due to (2), the definition of the operator by replacing by and from that of

can be obtained by .

SUN et al.: 2-D CURL-CONFORMING SINGULAR ELEMENTS FOR FEM SOLUTIONS OF DIELECTRIC WAVEGUIDING STRUCTURES

The operator triangle itself,

is defined as

such that, for the

Since

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is irrotational, (19)

(11)

and since (4) and (8), there results

, by the constructions in (20)

B. Proof of the Commutativity of the de Rham Diagram (1) Using (5) and (9) provides

Theorem 1: If and only if (12) (13) (14) then

(21)

(15) (16)

Since

, by the construction in (10), we have

(17) where (22)

Proof: It is straightforward to derive (12) from (15) and (14) from (17). We only derive (13) from (16) here as follows:

Since satisfies (18)–(22), which matches operator, it follows that the definition of the . However, from (1’), we know and, thus, it follows that . b) Proof of (13): Equation (10) implies (23) By the constructions in (6) and (7), we obtain

Due to (2), the first and third terms in the each side of the equation cancel out. This proves Theorem 1. and by transforms (13) to (14). Replacing by Therefore, Theorem 1 implies that the de Rham diagram (1) can be reduced to the following diagram where the operator only works on the transverse directions

Stokes’s theorem provides (24) Combining (23) and (24) gives (25) Due to (25) and (11), we have

Following the proof of the two-dimensional (2-D) de Rham diagram in [17], we reach a similar result for (1’) in Theorem 2. Theorem 2: The de Rham diagram (1’) is commutative. , by a) Proof of (12): Since the constructions in (3) and (6), we have (18)

From (1’), we know and, , which completes the proof. therefore, In defining the interpolation operators, we specify the properties the bases must satisfy, but not the specific forms of the

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bases. Therefore, the proof of the de Rham diagram is general and is not limited to a specific basis. To prove the commutativity of (1’), we employ the following properties that can be taken as guidelines for achieving the commutativity of the de Rham diagram. 1) Satisfy the inclusion relationships of the de Rham diagram at the discrete level, i.e.,

2) The basis set is unisolvent. 3) The basis set satisfies the continuity (conforming) requirement. It should be noted that satisfying the de Rham diagram does not guarantee high rates of convergence. To obtain a higher rate of convergence, the bases must match the regularity of the local field that comes from the physics. This is where the singularity order of the field plays an important role. Due to the , we employ a Whitney basis. However, for constant term in some other purpose, one may use other bases, such as a rational without vioWhitney basis, i.e., lating the proof of the de Rham diagram. This rational Whitney . On edge basis reduces to the regular Whitney basis when , its tangential component is identical to that of 12, where , the regular Whitney basis. The difference is that its curl is which is generally not a constant. REFERENCES

[1] J. C. Nédélec, “Mixed finite elements in R ,” Numer. Math., vol. 35, pp. 315–341, 1980. [2] J. M. Gil and J. Zapata, “Efficient singular element for finite element analysis of quasi-TEM transmission lines and waveguides with sharp metal edges,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 1, pp. 92–98, Jan. 1994. [3] B. Schiff and Z. Yosibash, “Eigenvalues for waveguides containing re-entrant corners by a finite-element method with super elements,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 2, pp. 214–220, Feb. 2000. [4] J. S. Juntunen and T. D. Tsiboukis, “On the FEM treatment of wedge singularities in waveguide problems,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 6, pp. 1030–1037, Jun. 2000. [5] J. M. Gil and J. P. Webb, “A new edge element for the modeling of field singularities in transmission lines and waveguides,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2125–2130, Dec. 1997. [6] Z. Pantic-Tanner, J. S. Savage, D. R. Tanner, and A. F. Peterson, “Twodimensional singular vector elements for finite-element analysis,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 2, pp. 178–184, Feb. 1998. [7] R. D. Graglia and G. Lombardi, “Hierarchical singular vector bases for the FEM solution of wedge problems,” in Proc. URSI Int. Electromagnetic Theory Symp., Pisa, Italy, 2004, pp. 834–836. [8] , “Singular higher order complete vector bases for finite methods,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1672–1685, Jul. 2004. [9] J. Van Bladel, “Field singularities at metal-dielectric wedges,” IEEE Trans. Antennas Propag., vol. AP-33, no. 4, pp. 450–455, Apr. 1985. [10] J. B. Andersen, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag., vol. AP-26, no. 7, pp. 598–602, Jul. 1978. [11] M. Stern and E. B. Becker, “A conforming crack tip element with quadratic variation in the singular fields,” Int. J. Numer. Methods Eng., vol. 12, no. 2, pp. 279–288, Feb. 1978. [12] D.-K. Sun and Z. J. Cendes, “Fast high-order FEM solutions of dielectric wave guiding structures,” Proc. Inst. Elect. Eng., pt. H, vol. 150, pp. 230–236, Aug. 2003. [13] D. Boffi, “A note on the de Rham complex and a discrete compactness property,” Appl. Math. Lett., vol. 14, pp. 33–38, Jan. 2001.

[14] J. M. Gil and J. Zapata, “A new scalar transition finite element for accurate analysis of waveguides with field singularities,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 8, pp. 1978–1982, Aug. 1995. [15] HFSS User’s Guide, Version 8.5, Ansoft Corporation, Pittsburgh, PA, 2002. [16] L. Demkowicz and I. Babuska, “p interpolation error estimates for edge finite elements of variable order in two dimensions,” in SIAM J. Numer. Anal., vol. 41, Aug. 2003, pp. 1195–1208. [17] L. Vardapetyan, “hp-adaptive finite element method for electromagnetics with applications to waveguiding structures,” Ph.D. dissertation, Dept. Comput. Appl. Math., The University of Texas at Austin, Austin, TX, 1999.

Din-Kow Sun (M’89) was born in Taipei, Taiwan, R.O.C., in 1956. He received the B.S. degree from the National Taiwan University, Taiwan, R.O.C., in 1978, and the Ph.D. degree from Carnegie–Mellon University, Pittsburgh, PA, in 1984, both in physics. From 1984 to 1986, he was a Research Associate with the Department of Electrical and Computer Engineering, Carnegie–Mellon University. Since 1986, he has been a Research Engineer with the Ansoft Corporation, Pittsburgh, PA. His current research projects include the construction of vector singular bases and applications of domain decomposition and nonconforming finite-element methods.

Leon Vardapetyan was born in Saint Petersburg, Russia, in 1966. He received the Ph.D. degree in computational and applied mathematics from The University of Texas at Austin, in 1999. From 2000 to 2002, he was with Bell Laboratories, Lucent Technologies, Murray Hill, NJ, where he modeled propagation and amplification of signals in optical fibers. In 2003, he joined the Ansoft Corporation, Pittsburgh, PA. His current research interests include analysis and development of numerical methods for time-domain electromagnetics.

Zoltan Cendes (S’67–M’73–F’03) received the B.S.E. degree from The University of Michigan at Ann Arbor, in 1968, and the M.S. and Ph.D. degrees in electrical engineering from McGill University, Montreal, QC, Canada, in 1970 and 1972, respectively. Following graduation, he joined the General Electric Corporation, initially with the Large Steam-Turbine Generator Department and then with the Corporate Research and Development Center. During a portion of this time, he was also an Adjunct Associate Professor with Union College, Schenectady, NY. In 1980, he became an Associate Professor of electrical engineering with McGill University. In 1982, he joined the faculty of Electrical and Computer Engineering, Carnegie–Mellon University, Pittsburgh, PA, where served as Professor until 1996 and, since then, as Adjunct Professor. He is also founder and Chairman of the Ansoft Corporation, Pittsburgh, PA, where he currently serves as their Chief Technology Officer and is responsible for directing the company’s product and technology research. He has authored or coauthored over 200 journal and conference papers. He is on the Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering. Dr. Cendes is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Technical Committee on Computer-Aided Design. He has served on the International Steering Committee of the COMPUMAG Conference and is a past chairman of the IEEE Conference on Electromagnetic Field Computation.

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Application of Two-Dimensional Nonuniform Fast Fourier Transform (2-D NUFFT) Technique to Analysis of Shielded Microstrip Circuits Ke-Ying Su and Jen-Tsai Kuo, Senior Member, IEEE

Abstract—A two-dimensional nonuniform fast Fourier transform (2-D NUFFT) technique is developed for analysis of microstrip circuits in a rectangular enclosure. The 2-D Fourier transform of a nongrid point is approximated by Fourier bases in a square neighborhood with ( + 1) by ( + 1) grid points. The square neighborhood can be reduced to an octagonal region with 2 2 + 3 + 1 grid points without sacrificing accuracy if is sufficiently large. This technique allows an arbitrary discretization scheme on conductors and shows a great flexibility for the analysis. Asymmetric rooftop functions are inevitably used to expand surface current densities on conductors. Based on the spectral-domain approach, all elements of the final method-of-moments matrix are double summations of products of a weighted Green’s function and trigonometric functions. By using the proposed technique, the double summations at all sampled points can be obtained via the 2-D NUFFT. The scattering parameters of a compact miniaturized hairpin resonator, an interdigital capacitor, and a wide-band filter are calculated. The calculated results show good agreement with measurements. Index Terms—Method of moments (MoM), nonuniform fast Fourier transform, spectral-domain approach (SDA).

I. INTRODUCTION

C

HARACTERIZATION of microstrip discontinuities is an important task in computer-aided design (CAD) of microstrip circuits. Many methods for modeling the discontinuities have been developed, such as the finite-difference time-domain (FDTD) method [1], [2], spectral-domain approach (SDA) [3], [4], finite-element method (FEM) [5], method of lines [6], [7], integral equation (IE) [8], [9], and mode-matching technique [10]. The method of moments (MoM) is a core engine for analysis of microstrip circuits [3], [4], [8], [9], [11]–[15]. Most of CPU time is consumed in evaluation of the MoM matrix elements since the Green’s functions converge slowly and a large number of basis functions are required for expanding surface current densities on conductors. In [11], the MoM matrix elements are linear combination of elements of precomputed index tables that are computed from the two-dimensional discrete fast Manuscript received April 14, 2004; revised June 20, 2004. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 91-2213-E-009-126 and Grant NSC 93-2752-E-009-002-PAE. K.-Y. Su was with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. He is now with the Design Service Division, Taiwan Semiconductor Manufacturing Company Ltd., Hsinchu 300, Taiwan, R.O.C. J.-T. Kuo is with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842475

Fourier transform (2-D FFT). In this method, however, the mesh scheme is restricted to be uniform. Obviously, uniform grids are very inefficient for analysis of a general microstrip circuit because electric currents have rapid variations along microstrip edges, thus, fine local discretization becomes a must for accurate analysis of the whole circuit. In [12], for reducing the number of unknowns, the currents are expanded by a linear combination of the current distributions at the first few resonant modes of the circuit. However, very fine discretization and a 2-D FFT of large size are still required to find solutions at the resonant modes. In [13], microstrip discontinuities on a lossy multilayered substrate are analyzed based on the electric-field integral-equation (EFIE) formulation. The conductors are uniformly discretized. At least two important points in regard to convergence of results are reported. One is that required number of summation terms in calculating the MoM matrix elements must be at least 1.25 times the total number of segments, and the other is that the mesh sizes and , respectively, in transmust be no larger than verse and longitudinal directions for accurate calculations. Nonuniform meshes are used in the mixed-potential integral equation (MPIE) [14] to overcome the large-size matrix problem caused by uniform discretization. The acceleration procedure is at a Green’s function level, and efficient MoM techniques with rectangular, but nonuniform and nonfixed, meshes can be constructed. In this paper, a two-dimensional nonuniform fast Fourier transform (2-D NUFFT) [15] incorporated with the SDA is developed for analysis of microstrip circuits. The mesh scheme for the microstrip circuit can be very flexible, although each subdivision must be rectangular. This idea is extended from the NUFFT algorithms in [16] and [17]. The concept for evaluating the FFT of one-dimensional (1-D) nonuniform data is to approximate the exponential function at each nonuniform sampled point by interpolating oversampled uniform Fourier bases with coefficients. The order of arithmetic operations is found to be with being the oversampling rate and being the FFT size. The accuracy of the approximation is increased as and are increased. The increase of , however, will increase , the size of the FFT, thus, is usually chosen to be 2. When accuracy of the data obtained by a least square error sense [17] is more than one order higher than that calculated by the method in [16]. In [18], the authors apply the NUFFT technique to analysis of multiple coupled microstrip lines. The 2-D NUFFT for two-dimensional nonuniform data in the – -plane can be established by employing two 1-D NUFFTs interpolated coefficients in the - and -directions. The of oversampled 2-D FFT bases are simply the products of two

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and denote the intewhere and , respectively. The accugers nearest to ’s are chosen to minimize error of (2) in a least racy factors square sense [17]. Substituting (2) into (1) yields (3) where

(4)

Fig. 1. 2-D NUFFT algorithm: exponential function at a nonuniform sample point (x ; y ) is approximated by Fourier bases at (q + 1) (q + 1) uniform oversampled grids (X ; Y ) in a square neighborhood or by those in an octagonal neighborhood.

2

sets of coefficients. However, it is found that the 1-D coefficients decay exponentially, as the bases are far away from the sample point [16]. It means that these coefficients have only negligible contribution and the number of the required coefficients can be significantly reduced to save a lot of operations without deteriorating accuracy. In this paper, Section II formulates the 2-D NUFFT algorithm and Section III incorporates the 2-D NUFFT in the SDA for microstrip circuit analysis. Section IV addresses the accuracy of the 2-D NUFFT and presents calculation examples including a microstrip hairpin resonator, an interdigital capacitor, and a wide-band filter. The results are also validated by measurements.

Calculation of (4) can be performed by a regular 2-D FFT of . In (3), the 2-D interpolated coefficients can size be obtained by two sets of 1-D NUFFT coefficients, i.e., the square neighborhood in Fig. 1. It is found that some of have negligible magnitudes, as they are away from , and these points have to be removed from the approximation for computation efficiency. If the coefficients associated with these insignificant grid points are directly removed from the square neighborhood, however, the accuracy of (3) can be significantly reduced. The accuracy can be recovered if the coefficients of selected grid points are derived in a least square error sense, as the 1-D case in [17]. Let the coefficients be expressed as (5) One efficient way to obtain closed forms of the reduced square regular Fourier matrix and reduced column vector is to extract them from a full regular Fourier matrix and a full with grid points. Here, and column vector can be obtained by extending the 1-D method in [17]. The proand is as follows. cedure for determining 1) Define a vector product as

II. 2-D NUFFT ALGORITHM

(6)

In Fig. 1, the cross is a nonuniform sample point and , and the circles and large black dots are uniformly oversampled , which are called the square neighborhood of grids herein. We are going to evaluate the following 2-D Fourier transform:

and let

(1) (7) and are finite complex sequences and and where are even integers. The first step of the 2-D NUFFT algorithm is to determine the interpolation coefficients for approximating the following exponential function with accuracy ’s: factors

(2)

and

(8)

SU AND KUO: APPLICATION OF 2-D NUFFT TECHNIQUE TO ANALYSIS OF SHIELDED MICROSTRIP CIRCUITS

be rows of the regular Fourier matrix for 1-D problem, . The th row where then equals . It is noted that depends on of and , but is independent of and . ; then the 2) Choose th element of can be derived as

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and are unknown constants to be determined. The where Fourier transform of the basis functions can be easily derived. For example, let the th basis function for the current in the -direction be (12) where

otherwise (13a) otherwise. (13b)

(9) where . and from and , respectively. The 3) Extract indices of square grid points are , and the indices of octagonal grid points, the large black dots in Fig. 1, called octagonal or nonsquare neighborhood herein, are . Let this index sequence be , then the th is the th element of , and the element of th element of is the th element of . 4) Evaluate in (4) by a regular 2-D FFT of size . in (3) using and the 5) Calculate interpolated coefficients . generated by the random number We test 2 pairs of generator of MATLAB software, and compare the 2-D NUFFT results with that obtained by direct summation (1). It is found , the coefficients associated with grid points in that, when the reduced neighborhood are sufficient to provide results with accuracy of the same order as that in the square neighborhood. Additional results will be presented and discussed in Section IV. III. INCORPORATING THE 2-D NUFFT INTO THE SDA

Their 2-D Fourier transforms can be derived as

(14a)

(14b) It is important to note that the transforms (14) are trigonometric functions weighted by powers of or . It can be validated that the transforms of basis functions for currents in the -direction can be expressed in a similar way. The Galerkin’s procedure is used to set up the final MoM th element can be expressed as matrix, of which the (15) After some algebraic manipulations with simple trigonometric identities, evaluation of can be reduced to

For a microstrip circuit enclosed in a rectangular shielded box of dimensions , one of the spatial-domain Green’s functions can be written as [11] (16) (10) where , and is the Green’s function in the spectral domain [19]. Other Green’s functions of the structure can be expressed in a similar manner. In the solution procedure, asymmetric rooftop functions are used to expand current densities on conductors, and the halfrooftop functions in [20] are used for modeling those at source and load terminals. Let the current densities be expressed as (11)

where or and and or . Fig. 2 summarizes the procedure for establishing the final MoM matrix. First, partition the circuit and find the required 2-D NUFFT interpolation coefficients for four sets of sampling . Second, evaluate the double summapoints tions of products in (16) by the 2-D NUFFT. If the impressed and load currents are in the same directions, only five calls of 2-D NUFFTs will be needed. Finally, recombine the five summations to set up the final MoM matrix. When the currents on input and output feed lines are obtained, the complex amplitudes of the incident and reflected current

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TABLE I COMPARISON OF CPU TIME AND L ERROR OF ONE CALL OF THE 2-D NUFFT IN ANALYSIS OF A HAIRPIN RESONATOR

Fig. 2. Solution procedure for evaluating the MoM matrix.

Fig. 4. Measured and calculated S -parameters of the hairpin resonator.

Fig. 3. Hairpin resonator in shielded box and its mesh scheme in analysis. Structure parameters are " = 10:2; L = 0:7; L = 1:01; L = 2:74; L = 8; L = 6; w = 1; w = 1:19; g = 0:2; and g = 0:8. All dimensions are in millimeters.

waves can be extracted by using the generalized pencil-of-function (GPOF) method [21], and the scattering parameters can be obtained via standard circuit theory. IV. RESULTS Three examples are used to demonstrate the proposed technique for analyses of microstrip circuits. They include a compact miniaturized hairpin resonator [22], microstrip interdigital capacitor, and wide-band bandpass filter [23]. Consider a hairpin resonator with a mesh scheme shown in Fig. 3. The mm, and the dimenthickness of dielectric substrate is sions of shielding box are 23.6 18.15 16 mm . To reflect rapid current density variations near edges of conductors, the , sampling points are chosen according to , and can be the length or width of where the conductor. A uniform sampling, however, is used herein on feeding lines in the -direction, as indicated in the plot, which is required by the GPOF method. errors in analysis of the Table I lists the CPU seconds and hairpin resonator obtained by the 2-D NUFFTs with octagonal and . The CPU and square neighborhoods for time is measured with a MATLAB program of version 5.3 on a personal computer with a Pentium IV processor of 1.6 GHz.

The calculation of the error is based on the results obtained by direct computation of (1). Both the square and octagonal schemes have identical regular Fourier matrices and column vectors for determining the interpolated coefficients, which have closed-form expressions and are independent of frequency. It means that the interpolation coefficients can be stored in computer memory once the mesh scheme is defined. Thus, the CPU time for steps 1–3 is not included in Table I. In step 4, the CPU time for one regular 2 , i.e., and , 2-D FFT with size 2 takes 3.02 s. The CPU time in Table I accounts for the calculain (3) using and the interpolated tion of coefficients. Comparing the CPU seconds listed in Table I, one can see that the octagonal 2-D NUFFT uses only 76%, 72%, and 67% of that and , respectively. for the square neighborhood for In comparison with the errors, the octagonal and square 2-D NUFFTs have very close values for all listed values. Note errors decrease about one order in magnitude as that the is increased from 4, 6, 8, to 10. Fig. 4 compares the simulation results with measurement data of the hairpin resonator. The shielding box is included in the experiment. The peak and deep frequencies of the curve for have the least accuracy, while those for and have close values and a good agreement with the measured data. Away from these two frequencies, the three curves have 4–5 dB and 3–4 dB away from the measurement at 2.1 and 2.9 GHz, reand have very good spectively. Since the results for is used for simulation herein. consistency,

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Fig. 5. Normalized currents on the hairpin resonator at resonant frequency f = 2:473 GHz. (a) jJ (x; y )j. (b) jJ (x; y )j.

Fig. 6. Microstrip interdigital capacitor and its discretization in analysis. Structure parameters are " = 10:2; L = 8; L = 1:6; L = 0:8; L = 1:2; L = 7:9; d = 0:4; e = 0:4; g = 0:2; and s = 0:2. All dimensions are in millimeters.

Fig. 5 plots magnitudes of the current densities on the hairpin and are normalized with respect to at resonance. Both in the circuit. The currents show relatively the maximum large magnitudes at edges and corners of the resonator. It reflects flexibility and necessity of the nonuniform mesh scheme in efficient analysis of a planar microstrip circuit. Fig. 6 plots the discretization of the interdigital capacitor. The dimensions of the shielded box are 18.6 18.6 16 mm . This circuit is chosen for demonstration due to its electrically small alternative fingers and gaps, and it is tough for simulation. In our nonuniform discretization, 327 cells are generated. Fig. 7 shows the -parameters of the interdigital capacitor ob. The simulation tained by the octagonal 2-D NUFFT with results obtained by the proposed method have a good agreement with the measurements including the enclosure. The CPU time for a frequency point is 123.69 s. The structure is also simulated with the commercial software SONNET and the results are incorporated into Fig. 7. In SONNET simulation, two uniform discretizations are used. In mm and mm, and it results in the first, 1274 subdivisions. One frequency point takes 5 s in total. The mm, and it results second discretization uses in 3865 subdivisions. The Fourier transform, matrix filling, and matrix solver take 12, 55, and 63 s, respectively. One frequency point takes 130 s. Both cases have very close results. It can be

Fig. 7. Measured and calculated S -parameters of the interdigital capacitor.

seen from Fig. 7 that the three sets of plots are in a reasonable agreement. Comparing with the CPU seconds used in SONNET, we suggest that source codes for the FFT and matrix solver be implemented in a low-level computer language for saving the computation time. It is noted that our codes are developed under the MATLAB environment. The third example is a microstrip wide-band filter in a shielded box of sizes 30.352 30.352 16 mm . The dimensions of circuit layout are adopted from [23]. Fig. 8 shows the circuit division in our analysis. The reason why the circuit is chosen for demonstration is that the structure consists of two pairs of coupled microstrips with a very narrow linewidth and gap size of 0.125 mm and a section of wide microstrip of 7.74-mm width. Such a structure consists of strong discontinuities at impedance junctions, which definitely need a fine discretization for simulation. Again, if a uniform mesh is used, either the narrow coupled lines will have insufficient cells for accurate analysis or the final MoM matrix may have an unacceptably large size for fine discretizations. Here, only 353 nonuniform cells are used, and 344 and 306 basis functions are and , respectively. used for Fig. 9 plots the simulation and measured -parameters for the is over filter. The distance between the circuit and sidewalls 1.45 times the width of the low-impedance microstrip, and the top cover height is over 24 times the substrate thickness. These two sizes are chosen to approximate the circuit in an open space, as in [23]. The calculated results show a reasonable agreement

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REFERENCES

Fig. 8. Shielded microstrip wide-band filter and its associated mesh. L = 8; L = 0:56; L = 0:576; L = 0:69; L = 0:3605; L = 0:125; L = 0:125; L = 0:125; L = 5:19; L = 4:88; L = 0:38; L = 2:06; L = 1:9; L = 7:75; L = 11:3; t = 0:635; c = 16; " = 10:8. All dimensions are in millimeters.

Fig. 9. Measured and calculated S -parameters of the wide-band filter.

with the measurements given in [23]. The CPU time for generating a frequency point is 122.31 s. V. CONCLUSION A 2-D NUFFT technique incorporated with the SDA has been proposed for efficient analysis of microstrip circuits in a rectangular enclosure. In this method, the mesh scheme has good flexibility since conductors can be discretized into fine cells near the edges and relatively large cells in regions with smooth current densities. The 2-D NUFFT algorithm can be implemented with grid points in a square or octagonal, i.e., reduced neighborhood. The convergence of the 2-D NUFFT algorithm has been discussed. As compared with a square neighborhood, the approximation with octagonal grid points leads to a smaller MoM matrix and preserves accuracy of results. In analysis of three microstrip circuits, each entry in the final MoM impedance matrix has five types of double summations, and all entries of each type can be obtained via one call of the 2-D NUFFT. The scattering parameters of the hairpin resonator, an interdigital capacitor, and a wide-band filter are calculated and validated by measurements. The 2-D NUFFT algorithm has been proven as a useful advance in the efficiency of MoM calculation. It may have widespread applications in science and engineering.

[1] X. Zhang and K. Mei, “Time-domain finite difference approach to a calculation of the frequency-dependent characteristics of microstrip discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1775–1787, Dec. 1988. [2] D. Bica and B. Beker, “Analysis of microstrip discontinuities using the spatial network method with absorbing boundary conditions,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 7, pp. 1157–1161, Jul. 1996. [3] R. H. Jansen, “The spectral-domain approach for microwave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 10, pp. 1043–1056, Oct. 1985. [4] , “A 3D field theoretical simulation tool for the CAD of millimeter wave MMICs,” Alta Freq., vol. LVII, pp. 203–216, 1988. [5] R. W. Jackson, “Full wave, finite element analysis of irregular microstrip discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 1, pp. 81–89, Jan. 1989. [6] Z. Q. Chen and B. X. Gao, “Deterministic approach to full-wave analysis of discontinuities in MIC’s using the method of lines,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 3, pp. 606–611, Mar. 1989. [7] S. B. Worm, “Full-wave analysis of discontinuities in planar waveguides by the method of lines using a source approach,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 10, pp. 1510–1514, Oct. 1990. [8] E. Drake, R. R. Boix, M. Horno, and T. K. Sarkar, “Effect of substrate dielectric anisotropy on the frequency behavior of microstrip circuits,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 8, pp. 1394–1403, Aug. 2000. [9] G. V. Eleftheriades, J. R. Mosig, and M. Guglielmi, “An efficient mixed potential integral equation technique for the analysis of shielded MMIC’s,” in Proc. 25th Eur. Microwave Conf., Sep. 1995, pp. 825–829. [10] C. N. Capsalis, N. K. Uzunoglu, C. P. Chronopoulos, and Y. D. Sigourou, “A rigorous analysis of a shielded microstrip asymmetric step discontinuity,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 3, pp. 520–523, Mar. 1993. [11] A. Hill and V. K. Tripathi, “An efficient algorithm for the three dimensional analysis of passive microstrip components and discontinuities for microwave and millimeter-wave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 1, pp. 83–91, Jan. 1991. [12] C. J. Railton and S. A. Meade, “Fast rigorous analysis of shielded planar filters,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 5, pp. 978–985, May 1992. [13] E. S. Tony and S. K. Chaudhuri, “Analysis of shielded lossy multilayered-substrate microstrip discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 4, pp. 701–711, Apr. 2001. [14] G. V. Eleftheriades, J. R. Mosig, and M. Guglielmi, “A fast integral equation technique for shielded planar circuits defined on nonuniform meshes,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2293–2296, Dec. 1996. [15] K. Y. Su and J. T. Kuo, “A two-dimensional nonuniform fast Fourier transform (2-D NUFFT) method and its applications to the characterization of microwave circuits,” in Asia–Pacific Microwave Conf., Seoul, Korea, Nov. 4–7, 2003, pp. 801–804. [16] A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput., vol. 14, pp. 1368–1393, Nov. 1993. [17] Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microw. Guided Wave Lett., vol. 8, no. 1, pp. 18–20, Jan. 1998. [18] K. Y. Su and J. T. Kuo, “An efficient analysis of shielded single and multiple coupled microstrip lines with the nonuniform fast Fourier transform (NUFFT) technique,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 90–96, Jan. 2004. [19] T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. New York: Wiley, 1989, ch. 5. [20] R. C. Hsieh and J. T. Kuo, “Fast full-wave analysis of planar microstrip circuit elements in stratified media,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 9, pp. 1291–1297, Sep. 1998. [21] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989. [22] M. Sagawa, K. Takahashi, and M. Makimoto, “Miniaturized hairpin resonator filters and their application to receiver front-end MIC’s,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1991–1996, Dec. 1989. [23] W. Menzel, L. Zhu, K. Wu, and F. Bögelsack, “On the design of novel compact broad-band planar filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 364–369, Feb. 2003.

SU AND KUO: APPLICATION OF 2-D NUFFT TECHNIQUE TO ANALYSIS OF SHIELDED MICROSTRIP CIRCUITS

Ke-Ying Su was born in Tainan, Taiwan, R.O.C., on March 16, 1974. He received the B.S. degree in applied mathematics from the National Sun Yet-Sen University (NSYSU), Taiwan, R.O.C., in 1996, the M.S. degree in mathematics from the National Central University (NCU), Taiwan, R.O.C., in 1998, and the Ph.D. degree in communication engineering from the National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C. He is currently with the Design Service Division, Taiwan Semiconductor Manufacturing Company (TSMC) Ltd., Hsinchu, Taiwan, R.O.C. His research interests include the analysis of microwave circuits and numerical techniques in electromagnetics.

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Jen-Tsai Kuo (S’88–M’92–SM’04) received the Ph.D. degree from the Institute of Electronics, National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1992. Since 1984, he has been with the Department of Communication Engineering, NCTU, as a Lecturer in both the Microwave and Communication Electronics Laboratories. He is currently a Professor with the Department of Communication Engineering, and serves as the Chairman of the Degree Program of Electrical Engineering and Computer Science (EECS), NCTU. During the 1995 academic year, he was a Visiting Scholar with the University of California at Los Angeles. His research interests include the analysis and design of microwave circuits, high-speed interconnects and packages, field-theoretical studies of guided waves, and numerical techniques in electromagnetics.

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W. Jang R. Jansen J. Jargon B. Jarry P. Jarry A. Jelenski W. Jemison S.-K. Jeng M. Jensen E. Jerby G. Jerinic T. Jerse P. Jia D. Jiao J.-M. Jin J. Johansson R. Johnk W. Joines K. Jokela S. Jones U. Jordan L. Josefsson K. Joshin J. Joubert R. Kagiwada T. Kaho M. Kahrs D. Kajfez S. Kalenitchenko B. Kalinikos H. Kamitsuna R. Kamuoa M. Kanda S.-H. Kang P. Kangaslahtii B. Kapilevich K. Karkkainen M. Kärkkäinen A. Karpov R. Karumudi A. Kashif T. Kashiwa L. Katehi A. Katz R. Kaul S. Kawakami S. Kawasaki M. Kazimierczuk R. Keam S. Kee S. Kenney A. Kerr O. Kesler L. Kettunen M.-A. Khan J. Kiang O. Kilic H. Kim I. Kim J.-P. Kim W. Kim C. King R. King A. Kirilenko V. Kisel A. Kishk T. Kitamura T. Kitazawa M.-J. Kitlinski K. Kiziloglu R. Knerr R. Knöchel L. Knockaert K. Kobayashi Y. Kobayashi G. Kobidze P. Koert T. Kolding N. Kolias B. Kolner B. Kolundzija J. Komiak A. Komiyama G. Kompa B. Kopp B. Kormanyos K. Kornegay M. Koshiba T. Kosmanis J. Kot A. Kraszewski T. Krems J. Kretzschmar K. Krishnamurthy C. Krowne V. Krozer J. Krupka W. Kruppa H. Kubo C. Kudsia S. Kudszus E. Kuester Y. Kuga W. Kuhn T. Kuki M. Kumar J. Kuno J.-T. Kuo P.-W. Kuo H. Kurebayashi T. Kuri F. KurokI L. Kushner N. Kuster M. Kuzuhara Y.-W. Kwon I. Lager R. Lai J. Lamb P. Lampariello M. Lanagan M. Lancaster U. Langmann G. Lapin T. Larsen J. Larson L. Larson J. Laskar M. Laso A. Lauer J.-J. Laurin G. Lazzi F. Le Pennec J.-F. Lee J.-J. Lee J.-S. Lee K. Lee S.-G. Lee T. Lee K. Leong T.-E. Leong Y.-C. Leong R. Leoni M. Lerouge K.-W. Leung Y. Leviatan R. Levy L.-W. Li Y.-M. Li

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