# Analysis and implementation of multi-layered configurations to reduce mode coupling problems in conductor-backed coplanar waveguides for wide frequency-band microwave integrated circuit applications

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ANALYSIS AND IMPLEMENTATION OF MULTI-LAYERED CONFIGURATIONS TO REDUCE MODE COUPLING PROBLEMS IN CONDUCTOR-BACKED COPLANAR WAVEGUIDES FOR WIDE FREQUENCY-BAND MICROWAVE INTEGRATED CIRCUIT APPLICATIONS A Dissertation by MARK ALEXANDER MAGERKO Submitted to the Office o f Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 1996 Major Subject: Electrical Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9634803 UMI Microform 9634803 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ANALYSIS AND IMPLEMENTATION OF MULTI-LAYERED CONFIGURATIONS TO REDUCE MODE COUPLING PROBLEMS IN CONDUCTOR-BACKED COPLANAR WAVEGUIDES FOR WIDE FREQUENCY-BAND MICROWAVE INTEGRATED CIRCUIT APPLICATIONS A Dissertation by MARK ALEXANDER MAGERKO Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved as to style and content by: Kai Chang (Co-Chair of Committee) Steve Wright (Member) Cam1 mNguyen (Co-Chair of Committee) Jianxin Zhou (Member) A.D. Patton (Head o f Department) Mark Weichold (Member) May 1996 Major Subject: Electrical Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Analysis and Implementation of Multi-Layered Configurations to Reduce Mode Coupling Problems in Conductor-Backed Coplanar Waveguides for Wide Frequency-Band Microwave Integrated Circuit Applications. (May 1996) Mark Alexander Magerko, B.S., University of Illinois Champaign-Urbana, Illinois; M.S., University of Illinois Champaign-Urbana, Illinois Co-Chairs o f Advisory Committee: Dr. Kai Chang Dr. Cam Nguyen Packaged conductor-backed coplanar waveguides (CBCPW) for microwave integrated circuits (MICs) can support rectangular waveguide and cavity modes within the substrate regions which can couple with the dominant quasi-TEM mode and produce several undesirable results. Lateral sidewalls connect the upper and lower ground planes of the structure together at the substrate edges in a wrap-around configuration using copper tape or through the package itself. Strong mode coupling can cause the dominant mode field pattern to spread out across the entire waveguide width instead of being confined to the slot area. This phenomenon translates into a significant loss of power and the occurrence of strong resonances in the transmission measurements which limits the bandwidth of the waveguide even at lower frequencies. Multi-layered configurations with dielectrics loaded above and below the circuit conductors are employed to reduce the mode coupling without restrictions on the cross section (S+2W) or the lateral width (2A) of the waveguide which is particularly useful for MICs. The above effects associated with CBCPW are demonstrated and analyzed using waveguide theoiy, experimental data, and the spectral domain numerical method in one, two, and three dimensions. Leakage calculations for the dominant mode in the one-dimensional case are related to the mode coupling effects in two-dimensions. Summary tables and practical guidelines describing design tradeoffs for wide frequency-band operation are also presented. The inclusion o f lossy damping layers to reduce the residual resonances associated with the substrate cavity without affecting the CBCPW resonator circuit Q are described with a three-dimensional procedure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Experimental data to 40GHz is utilized to verify the bandwidths for several of the configurations, confirm the explanations o f the mode and cavity coupling effects, and demonstrate the improved responses with the incorporation of the absorbing material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my parents and my friend Donna Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to express my appreciation to Dr. Kai Chang for his guidance and encouragement throughout this work. My thanks are also extended to the members o f my committee especially my co-chair Dr. Cam Nguyen for their interest in this study. I would also like to thank the U.S. Department o f Education - Areas of National Need Program through the Department of Engineering at Texas A&M University for providing a fellowship and the Army Research Office for supporting this work. I am also appreciative to Lu Fan for his invaluable assistance in this research effort, Rogers Corporation for the donation of the Duroid™ dielectric boards used in the experimental characterization, and technical discussions with Professors N.K. Das and A. A. Oliner o f Polytechnic University, Dr. M.L. Riaziat of Varian Research Center, Professor R. W. Jackson of the University of Massachusetts. Finally, I would like to dedicate this dissertation to my friend Donna whose patience and understanding during our marriage made the goal of obtaining this degree possible and to my parents John Sr. and Mary Ann for their constant support, guidance, and sacrifice made throughout my life. I would also like to thank Nan Vaughn for her emotional encouragement and assistance during the writing of this manuscript. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii TABLE OF CONTENTS Page ABSTRACT........................................................................................................................ iii DEDICATION.................................................................................................................... v ACKNOWLEDGMENTS.................................................................................................. vi TABLE OF CONTENTS.................................................................................................. vii LIST OF TABLES.............................................................................................. xi LIST OF FIGURES........................................................................................................... xv CHAPTER I INTRODUCTION........................................................................................ A. B. C. D. E. II Comparison Between Microstrip and Coplanar Waveguides............. Overview of Conductor-Backed Coplanar Waveguides...................... Problem Definition and Research Objective........................................ Literature Review and Discussion....................................................... Dissertation Organization..................................................................... I 2 4 10 18 20 THEORETICAL ANALYSIS OF CBCPW USING THE SPECTRAL DOMAIN METHOD...................................................................................... A. B. C. D. E. F. G. General Spectral Domain Method Formulation.................................. Basis Function Selection...................................................................... Two-Dimensional Packaged Structures.............................................. Spectral Domain Method Field Formulation....................................... Characteristic Impedance Derivation................................................... Analysis of Microstrip and Finite Ground Plane CBCPW.................. Spectral Domain Method Numerical Analysis and Results................ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 24 33 36 39 43 47 52 v iii CHAPTER IE IV Page ONE-DIMENSIONAL ANALYSIS AND DESIGN OF CBCPW INCLUDING LEAKAGE EFFECTS........................................................ 63 A. Experimental Demonstration of the Leakage Effects in CBCPW B. Explanation o f Leakage Effects in Printed-Circuit Transmission Lines............................................................................... C. Spectral Domain Method Including Leakage Analysis...................... D. Numerical Results of Leakage Effects for CBCPW.......................... E. Limitations of the One-Dimensional Leakage Analysis for CBCPW................................................................................................. 64 90 TWO-DIMENSIONAL ANALYSIS OF CBCPW WITH LATERAL SIDEWALLS INCLUDING MODE COUPLING EFFECTS................. 96 ....................................................... CBCPW Mode Identification Mode Coupling Effects in CBCPW with Lateral Sidewalls.............. Additional Examples o f the Mode Coupling Effects in CBCPW Experimental Results Confirming the Two-Dimensional SDM Analysis o f CBCPW............................................................................ E. Mode Coupling Analysis o f Finite Ground Plane CBCPW................. 99 105 112 A. B. C. D. V 67 76 82 115 119 DESIGN SUMMARY INFORMATION FOR THE PROPAGATION CHARACTERISTICS OF MULTI-LAYERED CBCPW....................... A. Propagation Characteristics o f Upper Dielectric Loaded CBCPW................................................................................................. B. Design Summary Information for Multi-Layered CBCPW MICs C. Additional Configuration Considerations for Multi-Layered CBCPW MICs..................................................................................... D. Minimizing the Mode Coupling Effects in Single-Layer CBCPW MICs...................................................................................................... E. Design Procedures for Single-Layer CBCPW................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 124 130 138 146 154 ix CHAPTER VI Page THREE-DIMENSIONAL ANALYSIS OF MULTI-LAYERED CBCPW INCLUDING CAVITY EFFECTS............................................ A. Overview of the Cavity Effects in CBCPW........................................ B. SDM Analysis of Three-Dimensional CBCPW Resonator Circuits.................................................................................................. C. Verification o f the Three-Dimensional SDM Procedure................... D. Design Examples for Multi-Layered CBCPW Resonator Circuits with Damping.......................................................................... E. Experimental Demonstration of Cavity Damping for CBCPW Resonators............................................................................................ VII 158 163 173 179 183 189 CONCLUSION............................................................................................. 193 A. Original Contributions.......................................................................... B. Suggestions for Future Work............................................................... 194 196 REFERENCES................................................................................................................... 197 APPENDIX A SPECTRAL DOMAIN IMMITTANCE FORMULATION...................... B DERIVATION OF THE FIELD COEFFICIENTS FOR THE 204 SPECTRAL DOMAIN METHOD IMMITTANCE METHOD.............. 209 C DERIVATION OF THE POYNTING VECTOR INTEGRAL................ 212 D NUMERICAL METHODS WITHIN THE SPECTRAL DOMAIN METHOD..................................................................................................... A. B. C. D. E. Bessel Function Approximation........................................................... Numerical Integration........................................................................... Determinant of a Matrix...................................................................... Root Searching Algorithm................................................................... Basis Function Expansion Coefficients Solution................................. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 216 216 219 221 223 APPENDIX E Page PROPAGATION CHARACTERISTICS OF IDEAL PARALLEL PLATE WAVEGUIDE MODES.............................................................. 228 F DERIVATION OF RESIDUE CALCULUS THEOREM......................... 231 G IEEE PERMISSION TO REPRINT COPYRIGHTED MATERIAL 234 VITA................................................................................................................................ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 xi LIST OF TABLES TABLE I Page Comparison o f the effective dielectric constant for an infinite-width CBCPW with Ref. [46] for £rl= l, Er4= 13> h^lOmm, h4=0.75mm, h5=10mm, h2=h3=h6=0, S=2mm, W=l.5mm............................. 55 Comparison o f the effective dielectric constant with different basis function for the example of Table I with S=0.6mm, W=0.45mm, f=20GHz............................................................................................................ 55 SDM integration convergence o f the propagation constant for the example of Table II with tan 54=0.001 (loss tangent)..................................... 55 Comparison of the effective dielectric constant for packaged CBCPW with Ref. [21] for srl= l, er4=9.6, er5=l, h,=3mm, h4=lmm, hs=3mm, h2=h3=h6=0, S=2mm, W=lmm, A=7.5mm. presented work refers to an infinite-width structure with the same parameters for comparison purposes.......................................................................................... 58 Comparison o f the effective dielectric constant for the example of Table IV with Ref. [21] at f =30GHz for the various modes.......................... 58 Comparison of the characteristic impedances for packaged CBCPW with Ref. [43] for erl= l, sr4=I3, er5=l, 1^=5mm, h4= 1mm, h5=5mm, h2=h3=h6=0, S=lmm, W=0.4mm, A=7.5mm................................................... 60 Convergence results for the effective dielectric constant and impedance in terms of the number of spectral terms for packaged CBCPW with srl=l, er4=10.2, erS=2.2, 1^=5mm, h4=0.635mm, h5=0.635mm, h2=h3=h6=0, S=0.254mm, W=0.89mm, A=12.5mm, f =20GHz.......................................... 61 VIII Basis function expansion coefficients for example in Table VII for M=3 and N=2............................................................................................................ 61 II III IV V VT VII Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xii TABLE IX X XI XII XIII XIV XV XVI Page Comparison of the effective dielectric constant for FGP packaged CBCPW with Ref. [47] for erl=I, sr4=10, er5= l, h,=10mm, h4=lmm, h2=h3=h5=h6=0, S=lmm, W=lmm, WG= lmm, A=12.5mm.......... 62 Identification of PPMs for multi-layered structure of Fig. E.l with sr4 =10.8, er5 =2.33 and h4=2.54mm, h5=0.71mm at f=20GFIz........................ 78 SDM integration convergence o f the leaky propagation constant for CASE G in Fig. 19 for single-layer CBCPW at 20GHz................................... 81 Comparison of the normalized leakage rate for conductor-backed slotline with Ref. [31] for sr=2.25, h=8mm (dielectric thickness), f=10GHz as a function of the slot width d............................................................................... 85 Comparison of critical frequencies up to 36GHz for CBCPW for measured data, SDM analysis, predicted with maximum dimensional uncertainties, and predicted with maximum dimensional uncertainties and air gap at h5.................................................................................................. 93 Measured and predicted resonant cavity frequencies for CBCPW CASEL from Fig. 30 and for T E 0I/b modes................................................. 95 Lower dielectric loaded multi-layered CBCPW MIC design summary. erl= l, sr4=10.2, er5=2.2 and h,=5mm, h2=h3=h6=0 and 2A=25.4mm and S+2W<2.032mm, Smax= 10W, Wmin=0.127mm and refer to Fig. 13 for dimensional parameters and CASE 20: S=2W=0.635mm CASE 21: S=W=0.635mm CASE 22: S=10W=1.27mm CASE 23: S=0.254mm, W=0.889mm......... 132 Upper dielectric loaded multi-layered CBCPW MIC design summary. srl= l, sr3=10.2, sr4=2.2 and h,=5mm, h2=h5=h6=0 and 2A=25.4mm and S+2W<2.032mm, Smax=10W, Wmin =0.127mm and refer to Fig. 13 for dimensional parameters and CASE 24: S=2W=0.635mm CASE 25: S=W=0.635mm CASE 26: S=10W=1.27mm CASE 27: S=0.254mm, W=0.889mm......... 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x iii TABLE Page XVII Cutoff frequencies for the TM„, rectangular waveguide mode as a function o f the lateral width or various CBCPW configurations. Fig. 13 with h6=0 is referenced for these lower loaded multi-layered examples. *arbitrary height requires h4<2A.................................................... 139 XVIII Specification of the dielectric constant for the lower loaded multi-layered CBCPW structures using the closed form approximation procedure at. f=20GHz. Fig. 13 is referenced with erl =1, h,=5mm, h2=h3=h6=0 and 2A=25.4mm.......................................................................... 140 XIX XX XXI Dielectric uncertainty effects on the upper usable frequency of multi-layered CBCPW for S=W=0.635mm and 2A=25.4mm. CASES 40-42 are the lower loaded configuration of example 2C) from Table XV and CASES 43-45 are the upper loaded waveguide example 2C) from Table XVI............................................. ................................................... 143 Measured and calculated resonant cavity frequencies for multi-layered CBCPW of CASEQ from Fig. 42 fortheTM0<1^ modes...................... 159 Resonant frequency verification of the TM0 u mode in a packaged microstrip structure with microwave absorber damping. Reference Fig. 61 for dimensions with sr4=10.5(l-y'0.0023), sr5 =1, =21(1-/0.02), ^ = 1 . 1( 1 - 71 .4 ), and h4=1.27mm, h5=10.16mm, h6=1.27mm and 2A=15mm and 2B=24mm................................................................................. 172 XXII Q verification for various modes in a packaged microstrip structure with doped silicon damping. Reference Fig. 61 for dimensions with ' 15 1 s r4=12.7, sr5= l, = 12 and «e=4.0 x 10 cm' and h4=0.1mm, h5=0.4mm, h6=0.25mm and 2A= 12mm and2B=20mm.................................. 173 XXIII Resonant Q verification o f the dominant mode in a microstrip structure with Si damping. Reference Fig. 58 and Fig. 64(b) for dimensions / with 2A=12mm, 16 2B=20mm and WG=0 with = 12 and , n= 3.0 x 10 cm ', sr2= l, sr4=12.7 and h,=h4=0.1mm, h3=h5=h6=0 and S=0.2mm and 2L=0.775mm...................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 xiv TABLE XXIV Resonant Q verification of the TM0 22 mode in a CBCPW packaged structure with loss tangent damping at h5. Reference Fig. 58 and Fig. 64(a) for dimensions with srl= l, s r4=Er5 = 12.8, 5 4 =0.002 and h4=h5=0.125mm, h2=h3=h6=0 and S=0.025mm, W=2S, 2L= 1.0mm and 2A=2mm, 2B=3mm....................................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Page 183 XV LIST OF FIGURES FIGURE 1 2 3 4 5 6 Page Active device grounding in coplanar waveguides and microstrip, (a) Top view, (b) Side view.......................................................................................... 3 Circuit photograph of GaAs conductor-backed coplanar waveguide (CBCPW) MMIC distributed amplifier............................................................ 5 Basic CBCPW configurations, (a) Infinite-width. (b) Packaged with shorting sidewalls, (c) Packaged with finite ground planes. S, W, WG, and 2A are the widths of the center strip conductor, slots, ground plane conductors, and the lateral sidewall separation respectively. The conductor-backed plane exists at x=-h and the cover plate conductor at x=5mm.............................................................................................................. 7 Fundamental modes of CBCPW along with the source topologies including the cross-sectional vector electric field plots for the two-layer configuration, (a) Dominant or odd mode, (b) Slotline or even mode. (c) Microstrip-like or parallel plate mode....................................................... 8 Multi-layered CBCPW configurations, (a) Longitudinal view o f experimental CBCPW with shorting sidewalls, (b) Cross-sectional view of CBCPW used in the numerical analysis..................................................... 9 Approximate representations o f packaged CBCPW. (a) Ideal rectangular waveguide, (b) Ideal rectangular cavity......................................................... 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xvi FIGURE 7 8 9 10 Page Experimental responses demonstrating waveguide and cavity modes in CBCPW with shorting sidewalls and 2A=2B=38mm, S=2W=0.635mm, srl=Er2=er3=er6= l, h,=oo (open structure), h,=h3=h6=0. Refer to Fig. 5(b) for the dimension parameters. Case A 50Q microstrip line sr4=10.8, h4=0.635mm and sr5= l, h5=0, CaseB same as Case A for CBCPW, Case C same as Case B except WG=lmm, Case D same as Case B except h4=l.27mm and 8^=2.33, h5=0.381mm, Case E same as Case D except h4=0.635mm and h5=0.711mm Case F same as Case E except broadband absorber placed in the connector housing blocks. Cases A,B,C,D are lOdB/div while Cases E,F are 5dB/div and all are referenced to OdB 13 Test fixture illustrations incorporating CBCPWtransmission lines. (a) Shorting bar application to ground input and output ports of CBCPW. Connector housing blocks............................................................ 14 Alternative coplanar waveguide configurations, (a) Channelized coplanar waveguide, (b) Via hole structure.................................................. 16 Examples of structures easily simulated using the spectral domain method, (a) Cross-section of a multi-layered planar structure with multi-metallization layers, (b) Cross-sections o f CBCPWs that are open, laterally open, and shielded or packaged from left to right, respectively 23 11 Multi-layered infinite-width CBCPW cross-section....................................... 25 12 Shapes of the electric field basis functions for CBCPW with S=0.635mm, W=0.3175mm. (a) Eym basis functions, (b) E.n basis functions........................................................................................................... 35 13 Multi-layered finite-width CBCPW cross-section......................................... 38 14 Additional structures analyzed using the SDM with air above the conductors at x=0. (a) Microstrip waveguide, (b) Finite ground plane CBCPW............................................................................................................ 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xvii FIGURE 15 16 17 18 19 Page J2 idealized current components for the waveguides under analysis. (a) Dominant mode for microstrip with S=0.635mm. (b) Coplanar waveguide mode for the FGP CBCPW with S=0.635mm, W=0.3175mm, WG=0.635mm. (c) Coplanar-microstrip mode for the FGP CBCPW................................................................................................. 50 Flowchart for computing the propagation constant of the infinite-width CBCPW......................................................................................................... 54 Flowchart for computing the propagation constant of the packaged CBCPW......................................................................................................... 57 Cross-sectional vector magnetic fields plot for packaged CBCPW. erI= 1, sr4=10.2, sr5=2.2, h1=3.73mm, h4=0.635mm, h5=0.635mm, h,=h3=h6=0, S=2W=0.635mm, A=12.5mm, f =5GHz................................................... 59 Experimental data of the leakage effects in CBCPW MIC with open sidewalls and 2B=38mm, S=2W=0.508mm, WG=18.492mm, h,=oo (open structure), h2=h3=h5=h6=0. Refer to Fig. 11 for the dimension parameters. CASE G 50Q line Er4= l 0.8, h4=0.635mm, CASE H same as CASE G except h4=0.71mm and sr5=2.33 for 95Q line, and CASE 1 is the numerical representation for the dielectric loss of CASE G. All cases are referenced to 0 dB.. 66 Conductor-backed slotline leaking power to a parallel plate mode for (a) the top view and (b) the side view. The angle o f leakage 0 into the the parallel plate mode of wave number kc is also shown, k is the transverse wave number in the ^-direction of the excited parallel plate mode and k2 is the wave number o f the dominant mode guided along the r-direction...................................................................................... 69 Three-dimensional illustration of leakage in a semi-cone region from a pulse propagating on a coplanar stripline..................................................... 71 Leakage effects in conductor-backed slotline demonstrated by probing the Ex field distribution transversely cross the j-axis. The leakage peak angle is 31.8° which translates to a maximum field component at ^ = ±12.6cm. Experimental data approximately confirms the predicted leakage angle from Ref. [26].......................................................................... 75 e rl=E r2= s r3=Er5= 8 r6= l , 20 21 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xviii FIGURE 23 24 25 26 27 28 Page Contours of spectral domain integrations Cs in complex k plane. The respective poles positions P's are indicated. P0 and P'0 are for no loss case while Pj and P'j correspond to P 0 and P'0 fo ra lossy waveguide. P2 and P'2 correspond to P3 and P'3 for a highly lossy structure . All of these poles refer to a nonleaky waveguide and utilize the contour C, along the real axis. P3 and P'3 and P4 and P'4 refer to leaky waveguide case and follow the integration path C3 coinciding with the real axis and deformed round the leaky poles in a residue calculus sense............................................................................................................... 79 Normalized propagation constants for the CBCPW dominant mode (P) and the TEM PPM (&.). One-dimensional SDM analysis for experimental data of Fig. 19 with CASE 2 and CASE 3 corresponding to CASE G and CASE H, respectively...................................................... 83 Normalized propagation constants for multi-layered CBCPW dominant mode(p) and the TMq PPM (kc ) with erl=l, sr4 =10.8 and h,=5mm, h2=h3=h6=0 and S=2W=0.635mm. CASE 4 with srS=6 and h4=0.635mm, h5=5mm. CASE 5 with srS=2.33and h4=0.635mm, h5=0.71mm. CASE 6 same as CASE 5 except h4=1.27mm............. 84 Normalized leakage rate comparisons with Ref. [32] and [33] with srl =1 and h,=5mm, h2=h3=h5=h6=0. CASE 7 with er4 =13 and h4=0.2mm and S=W=0.1mm. CASE 8 with sr4 = 10 and h4=0.4mm and S=2W=2.0mm................................................................................................ 86 Normalized leakage rates (aIk0) for CBCPWs. CASE 2 and CASE 3 are referenced from Fig. 24. CASE 6 is from Fig. 25. CASE 9 same as CASE 2 except h4=2.54mm. CASE 2 is scaled by a factor o f three 88 Leakage design curves for various CBCPW examples with erl= l and h,=5mm. (a) Proper dielectric thickness at 20GHz for upper and lower loading structures with S=2W=0.635mm. CASE 10 with sr3 =10.8, er4 =2.33 and h4=0.71mm, h2=h5=h6=0 and references the upper dielectric thickness scale. CASE 11 is the same as CASE 6 of Fig. 25 with h5 as the variable and references the lower scale (b) Proper relative dielectric constant at 30GHz for lower loading waveguide. CASE 12 is the same as CASE 6 except with er5 as the the variable. CASE 13 same as CASE 7 except h5=0.635mm. Log scales are used for the leakage rate.......................................................... 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xix FIGURE 29 30 31 32 33 34 35 Page Experimental data of the leakage effects in multi-layered CBCPW MICs with open sidewalls and 2B=38mm, S=2W=0.635mm, WG=18.49mm, Brl=sr2=Er3=srt=l, er4=10.8, sr5=2.33 and h,=oo (open structure), h2=h3=h6=0. CASE I is measured data for CASE 5 h4=0.635mm, h5=0.71mm. CASE J is the experimental results for CASE 6 h4=1.27mm, h5=0.71mm. CASE K same as CA SEJ except h5=0.381mm. Refer to Fig. 11 for the dimension parameters. CASE K is referenced 5dB down from CASE J S21 and CASE J S,, corresponds to the top scale............................................... 91 Experimental data demonstrating the resonant cavity modes within an overmoded single-layer CBCPW. CASEL with srl=sr2=er3=er5=sr6=l, sr4=10.8 and h,= oo (open structure), h2=h3=h5=h6=0, h4=0.635mm and S=W=0.508mm, A=19mm, B= 12.7mm. Refer to Fig. 13 and Fig. 5(a) for dimensional parameters............................................................. 94 Idealized dispersion curves for single-layer CBCPW with lateral sidewalls for CASE 14 and erl= l, sr4=10.8 and h2=h3=h5=h6=0, h,=5mm, h4=0.635mm and S=2W=0.508mm and A=19mm. Refer to Fig. 13 for dimensions. The CBCPW mode response is from CASE 2 of Fig. 24 and superimposed onto the graph. Ideal RW are the TM0* rectangular waveguide modes............................................................................ Example for the variation o f the SDM determinant with fictitious propagation constants at 7.5GHz for CASE 15 with erl= l, sr4=10.8, er5=2.33 and hj=5mm, h4= 1.27mm, h5=0.381mm, h2=h3=h6=0 and S=2W=0.635mm and A=19mm....................................................................... Cross-sectional vector electric field plots for CASE 15 at 15GHzfor (a) CBCPW mode and (b) T M ^ mode..................................................... 98 100 102 Cross-sectional vector electric field plots of CASE 15 at 15GHz for TMqj mode, (a) Ideal rectangular waveguide analysis, (b) SDM analysis............................................................................................................... 103 Electric field plot of CBCPW and first three waveguides modes of CASE 15 at 15GHz and x=-0.01mm................................................................ 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. XX FIGURE 36 37 38 39 40 41 Page Dispersion curves of CASE 15 from Fig. 32 for CBCPW and waveguide modes demonstrating the mode coupling effects. SDM RW and Ideal RW refers to rectangular waveguide modes from the two different analysis. Mode A is the top continuous dispersion curve and Mode B is the second continuous curve........................................................................ Field plots at various frequencies for CBCPW mode o f CASE 15 demonstrating the field spreading effects due to mode coupling with the waveguide modes............................................................................................. 107 109 Additional mode coupling effects in two-dimensional CBCPW CASE 15. (a) Characterisitc impedance plots for the modes indicated in Fig 36. Modes A and B refer to the lower frequency range (8-10.1 GHz), (b) Calculated coupling coefficient of the waveguide modes to the CBCPW mode............................................................................................... 111 Additional examples of field spreading mode coupling effects in CBCPW. (a) Single-layer structure CASE 14 o f Fig. 31. (b) Multi-layered structure CASE 16 same as CASE 15 except h4=2.54mm, hs=0.71mm............................................................................... 113 Multi-layered CBCPW example demonstrating a bound dominant mode to 40GHz. CASE 5 here is same as CASE 15 except h4=0.635mm, h5=0.71mm. (a) Dispersion curves, (b) Plot for Ex field component..................................................................................................... 114 Experimental data o f the mode coupling effects in CBCPW MICs with lateral sidewalls with erl= l and h,= oo (open structure), h2=h3=h6=0 and 2B=38mm. Refer to Fig. 13 and Fig. 5(a) for the dimension parameters. For CASE L sr4=10.8 and h4=0.635mm, h5=0 and S=W=0.508mm and 2A=5mm. CASE M same as CASE 15 with er4=10.8, sr5=2.33 and h4= 1.27mm, h5=0.381mm and S=2W=0.635mm and 2A=38mm. CASE N (CASE 16) same as CASE M except h4=2.54mm, h5=0.71mm. CASE O (CASE 14) same as CASE L except 2A=38mm. All cases are 50f2 through lines and referenced to OdB................................................................................................................. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xxi Page FIGURE 42 43 44 45 46 Experimental data demonstrating the cavity mode effects in CBCPW MICs with erl= l, sr4=10.8, er5=2.33 and h,=oo (open structure), h2=h3=h6=0 and S=2W=0.635mm. Refer to Fig. 13 and Fig. 5(a) for the dimension parameters. CASE P with h4= l ,27mm, h5=0.381mm and 2A=38mm, 2B=25.4mm. For CASE Q with h4=0.635mm, h5=0.711mm and 2A=20mm, 2B=38mm. CASE R same as CASE Q except 2B=25.4mm. CASE S same as CASE Q except with broadband absorber in the connector housing blocks. All cases are 50Q through lines and referenced to OdB......................................................................... 118 Dispersion curves showing mode coupling effects in FGP CBCPW as a function of the ground plane width (WG) at 30GHz with srl= l, sr4=10.2 and h[=5mm, h4=0.635mm, h2=h3=h6=0 and S=2W=0.635mm and 2A=25.4mm. (a) Single-layer structure with h5=0. (b) Multi-layered waveguide with srS=2.33 and h5=0.635mm. CPM refers to the coplanar-microstrip modes........................................................................... 121 Dispersion curve plot of the normalized propagation constants for an upper dielectric loaded multi-layered CBCPW. CASE 19 with srl= l, sr3=10.2, sr4=2.2 and h,=5mm, h3=0.635mm, h4= 1.27mm, h2=h5=h6=0 and S=2W=0.635mm and 2A=25.4mm. Refer to Fig. 13 for parameter dimensions. BSW is the boxed surface wave modes and the IDEAL case corresponds to the grounded dielectric slab layer h3. LOWER RW’ refers to the maximum propagation constant of single-layer rectangular waveguide h4............................. 126 Cross-sectional electric field plot for the dominant CBCPW mode of the upper loaded example CASE 19 at 40GHz. The field components are normalized to the maximum values............................................................. 127 (a) Experimental data for CBCPW MICs with lower dielectric value er4=2.33. erl= l and h,= oo (open structure), h4=1.42mm, h2=h5=h6=0 and S=W=0.508mm and 2B=38mm. Refer to Fig. 13 and Fig. 5(a) for the dimensions. For CASE T h3=0 and 2A=38mm (single-layer). CASE U same as CASE T except 2A=5mm. CASE V upper loaded structure same as CASE T except er3=10.8 and h3=0.635mm and S=2W=0.635mm and with connector problems. CASEW same as CASE V except with a cutout o f the substrate round the inputs and output ports and illustrated in (b). All cases are referenced to OdB 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xxii Page FIGURE 47 48 49 50 51 52 53 Cross-sectional vector electric field plot of the TM0 , surface wave mode for the upper loaded multi-layered CBCPW of CASE 19 at 40GHz.......................................................................................................... 129 Bounded mode behavior for lower loaded multi-layered dominant CBCPW mode, (a) Normalized dispersion curves for CASES 20 and 21 of Table XV and the first waveguide mode calculated by the SDM and the ideal rectangular waveguide analysis, (b) Cross-sectional vector electric field plot (Ex and Ey) o f the dominant CBCPW mode for CASE 20 at 40GHz............................................................................... 135 Characteristic impedance verification with [57] for an air-suspended multi-layered CBCPW for power-voltage, voltage-current, and power-current definitions. Refer to Fig. 13 for dimensions with srl=l, er4 =13, sr5= l and h1=hs=5mm, h4= 1mm, h2=h3=h6=0 and S=lmm, W=0.4mm and 2A=25.4mm........................................................................ 136 Characteristic impedances of the multi-layered CBCPW MICs for the three definitions, (a) Lower loaded example 2B) o f Table XV. (b) Upper loaded example 3B) of Table XVI............................................ 137 Dispersion effect analysis for the determination o f (S+2W) of the dominant CBCPW mode for the lower loaded example 2B) o f Table XV. Refer to Fig. 13 with srl=l, sr4=10.2, sr5 =2.2 and hj=5mm, h4=0.635mm, h5=0.635mm, h2=h3=h6=0 and 2A=25.4mm. CASE 36 is S=2W=0.635mm, CASE 37 is S=2W=0.381mm, CASE 38 is S=2W=0.1905mm, and CASE 39 is S=2W=0.0953mm............................ 142 Effects of an air gap between the substrates for the multi-layered CBCPW MICs with srl=l and hj=5mm and S=2W=0.635mm and 2A=25.4mm at f=20GHz. Fig. 13 is referenced for the dimensions. CASE 46 is the lower loaded example with er4= l0.2, sr5= l, 8^=2.2 and h4=0.635mm, h5 is the air gap, h6=0.635mm, h2=h3=0. CASE 47 is the upper loaded example with er2=10.2, sr3= l, er4=2.2 and h2=0.635mm, h3 is the air gap, h4=1.27mm, h5=h6=0................................ 144 Leakage rate versus relative dielectric constant for single-layer CBCPW. Fig. 3(a) is referenced here with S=2W=0.3175mm and h= 1.27mm and f=40GHz............................................................................. 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission xxiii Page FIGURE 54 55 56 57 58 59 Normalized leakage rate curves for single-layer CBCPW. (a) Plots versus frequency with er=10.2 and h=0.635mm with CASE 49 S=2W=0.3175mm, CASE 50 S=2W=0.15875mm, and CASE 51 S=2W=0.079mm. (b) Curves versus cross-sectional circuit area with sr=10.2 and f=40GHz with CASE 52 h=0.3175mm, CASE 53 h=0.635mm, and CASE 54 h=l.27mm. Fig. 3(a) is referenced for the parameters...................................................................................................... 149 Normalized electric field plots for dominant CBCPW mode demonstrating the mode coupling effects for a single-layer structure at x=-0.01mm. Refer to Fig. 3(a) for dimensions with er =10.2 and 2A=25.4mm and f approximately 40GHz. (a) Results as substrate height (h) is decreased with (S+2W)=h/4 for each case. CASE 51 h=0.635mm, CASE 55 h=1.27mm, CASE 56 h=2.54mm, and CASE 57 h=5.08mm. (b) Plots as function o f the cross section with h= 1.27mm. CASE 58 S=2W=0.635mm, CASE 59 S=2W=0.3175mm, CASE 55 S=2W=0.1588mm, and CASE 60 S=2W=0.0794mm............... 152 Normalized dispersion curves for single-layer CBCPW. With er =10.2 and h=1.27mm and 2A=25.4mm and refer to Fig. 3(b) for parameters, (a) CASE 58 with S=2W=0.635mm. (b) CASE 60 with S=2W=0.0794mm. The crossed points are the CBCPW mode in each case................................................................................................................. 153 Normalized electric field plots for the dominant CBCPW mode demonstrating the mode coupling effects as a function of frequency for a single-layer structure at x=-0.01mm. CASE 50 with s r=10.2 and h=0.635mm and S=2W=0.1588mm and 2A=25.4mm. Refer to Fig. 3(b) for dimensions................................................................................................ 155 Cross-section o f the multi-layered CBCPW with damping material for the upper and lower resonant cavities. The complex permittivities and complex permeabilities for the lossy layers are shown......................... 161 Application of the damping material with a thickness t on the perimeter of the symmetric lower dielectric loaded CBCPW to reduce the cavity Qs . (a) Side view, (b) Top view............................................................. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xxiv FIGURE 60 61 62 Page Equivalent circuit representation for resonance mechanism in a packaged CBCPW circuit with cavity effects. Three cavity modes exists within the frequency range o f interest for this example and are modeled using -RjLjCj lumped elements and mutual coupling with turns ratio «;................... 165 Ideal rectangular cavity representation of the multi-layered packaged CBCPW. PEC shorting facewalls exist at z=0 and z=2B............................... 168 Attenuation (dielectric) losses for the dominant CBCPW mode examining the effects of the silicon lossy layer. Refer to Fig. 58 for dimensions with srI= l, sr4=10.2, sr5=2.2 and h,=5mm, h4=h5=0.635mm, h,=h,=0 and S=2W=0.635mm and 2A=25.4mm. CASE 61 h,=0 and / CASE 6 2 e rt = 12 andh6=0.25mm and«e=8.35 x 10 63 15 , c m '...................... 171 Demonstration o f the damping effects within a rectangular cavity for a MIC example using doped silicon. Refer to Fig. 61 for parameters with er4=10.2, s r5=2.2, = 12 and h4=h5=0.635mm and 2A=2B=25.4mm. Dielectric loss data for h4 and h5 substrates is listed in 16 the text. CASE 63 h6=0, CASE 64 h6=0.1mm and rt =5.2 x 10 cm"3, and CASE 65 h6=0.25mm and «e=8.35 x 10 64 65 66 cm'3........................ 174 Top view representations at x=0 of the center-fed half-wavelength resonators considered in the analysis, (a) CBCPW and (b) microstrip 175 Shapes of the electric field basis functions for a CBCPW resonator as a function of z with 2L=12.7mm for the right hand slot at _y=(S/2+W/2). (a) basis functions, (b) E.n basis functions......................................... 180 Convergence demonstration o f the resonant frequency from the SDM with additional basis functions in the z-direction for M=N o f a resonator circuit, (a) Microstrip example CASE 66 with dimensions from Fig. 58 and Fig. 64(b) o f erl= l, sr4=9.4 and h ^ m m , h4=0.6mm, h2=h3=h5=h6=0 and S=0.575mm,WG=0, 2L=4.5mm and 2A=2B=10mm. (b) CBCPW example CASE 67 with dimensions from Fig. 58 and Fig. 64(a) of erl= l, er4=10.2, sr5=2.2 and h,=5mm, h4=h5=0.635mm, h2=h3=h6=0 and S=2W=0.635mm, 2L=6.2mm and 2A=2B=25.4mm..... 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FIGURE 67 Plot of the E space domain field from the SDM for the dominant CBCPW mode of the resonator circuit o f CASE 67 with 2L=6.2mm at ,y=(S+W)/2 for the right hand slot............................................................. 68 CBCPW resonator and TM,, 3, cavity mode Qs using doped Si layer (s r = 12). a) Lower loaded multi-layered structure CASE 68 same as CASE 67 as a function of the Si layer at h6 with doping 16 3 n = 1.0 x 10 cm . b) Upper loaded CBCPW CASE 69 as a function of the dielectric thickness h4 with fixed Si layer at hs=0.2mm with 16 3 doping n = 2.1 x 10 cm and erl= l, er3=10.2, er4=2.2, h,=5mm, h3=0.635mm, h2=h6=0 and S =2W=0.635mm, 2L=4mm and 2A=2B=25.4mm. Log scales are implemented.......................................... 69 CBCPW resonator and TM q3 , cavity mode Qs for a lower loaded multi-layered structure using microwave absorber as a function o f the dielectric thickness h5. Refer to Fig. 58 for the dimensions o f CASE 70 with erl= l, sr4=10.2, erS=2.2, 8^= 21 ( 1-70 .02 ), ^ = 1. 1( 1-71 .4 ) and hj=5mm, h4=h6=0.635mm, h2=h3=0 and S=2W=0.635mm, 2L=6.2mm and 2A=2B=25.4mm.................................................................................... 70 Dimension parameters for multi-layered CBCPW straight gap-coupled resonator with damping material, (a) Top view with gap width (G) and resonator length (L). (b) Cross-sectional view with absorber placed at layer h6........................................................................................................... 71 Experimental data response with absorber material to dampen cavity Qs in straight gap-coupled CBCPW resonator. Reference Fig. 70 for dimensions with G=0.17mm and L= 12.7mm. CASE X with srl= l, er4=10.8, erS=2.33 and h,= oo (open structure), h2=h3=0, h4=0.635mm, h5= 1.09mm and S=2W=0.635mm and 2A=20mm and 2B=38mm. Absorber layer at h6 and electrical data is not known................................ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Page FIGURE 72 Experimental data o f the transmission responses for the microstrip and the multi-layered CBCPW MIC 50f2 through lines. A comparison with Fig. 7 indicates CBCPW MIC is now a viable alternative to microstrip. Refer to Fig. 5 for dimensions with erl=l and h,= oo (open structure) and 2A=2B=38mm. CASE A is microstrip line with sr4=10.8 and h4=0.635mm and S=0.635mm. CASE B is CBCPW with er4=10.8, Er5=2.33 and h4=0.254mm, h5=0.71mm, h2=h3=h6=0 and S=2W=0.635mm with absorber material in the connector blocks. Both cases are referenced to OdB........................................................................ 195 Coordinates and components of the basic TM and TE modes for the SDM immittance method............................................................................ 205 Equivalent transmission line models of the multi-layered CBCPW for the TM and TE modes................................................................................ 206 E. 1 Parallel plate waveguide representation with two dielectric substrates 229 F. 1 Illustrative example of the residue calculus theorem with a number of poles a,b,c interior to a simple closed curve C.......................................... 232 A. 1 A. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 CHAPTER I INTRODUCTION The classical uses of microwaves in terrestrial radio and radar have expanded during the past decade into many new applications due to technological advances including satellite and wireless communications, superconducting circuits, vehicular navigational systems, and phased and active antenna arrays. The use of high-speed pulse circuits for wideband communications and ultra-fast computers has pushed microwaves into the area of digital technology. In the early days o f microwaves, circuits consisted of coaxial and waveguide systems that were produced separately and then connected together by screws to form the system. These circuits were large, heavy, and not suited for high volume systems. The rapid growth of modem microwaves can be attributed to advances in microwave semiconductors and in hybrid microwave integrated circuits (MICs) and monolithic MICs (MMICs). An integrated circuit combines all of the active devices and circuit components in a planar fashion and results in smaller and cheaper systems and are easily reproducible for large volume applications. In MICs the transmission line components are first etched on a circuit board and then the remaining elements are surface mounted and wire bonded to the substrate. MMICs process the passive components (transmission lines, capacitors, resistors, inductors) and the semiconductor devices (transistors and diodes) together on the same semiconductor substrate. Transmission lines are one of the most essential elements o f microwave circuits. Analysis, modeling, and design of transmission lines are important for any component and subsystem development. The desirable characteristics of any planar transmission line are low dispersion (change in the effective dielectric constant over frequency), large characteristic impedance range, and low dielectric and conductor losses. The circuits should provide sufficient circuit surface area, easily couple to antenna systems, easily integrate active and passive components, and This dissertation follows the style and format o f the IEEE Transactions on Microwave Theory and Techniques. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 avoid vertical discontinuities within the structure. As more efforts have been spent on MMICs, interest in coplanar waveguides has been renewed. The primary interest in this waveguide is the elimination of via holes that are required to ground nodes within the circuit and this is especially important at higher frequencies of operation. A. Comparison Between Microstrip and Coplanar Waveguides For many years MICs and MMICs have predominantly used microstrip as a transmission line medium. Microstrip is a simple waveguide consisting o f a conducting strip placed on a dielectric substrate and backed by a conducting ground plane. This transmission line is well understood, flexible in terms o f circuit elements, and can operate to very high frequencies. However, the main disadvantages with microstrip are that it requires via holes to ground active devices and other nodes as depicted in Fig. 1 and at higher frequencies the dielectric thickness must be reduced to cutoff the characteristic grounded surface wave modes. The via holes also require thinner substrates to increase the yield and behave inductively at high frequencies which can create problems at the source electrode of a mounted MESFET (metal-semiconductor field-effect transistor). Coplanar waveguide (CPW) has been suggested as an alternative to microstrip [1-2] but was not widely used initially due to the mistaken assumption that it possesses a higher conduction loss than microstrip [3], CPW is a surface oriented planar transmission line suspended in air with two ground planes running parallel to the center strip and all of the conductors are in the same plane of the dielectric. This waveguide can be viewed as a coupled-slot configuration with the electromagnetic energy o f the dominant mode guided by these slots. The dominant mode is guided by the transmission line and should be TEM or quasi-TEM (transverse electromagnetic with a small field component in the direction of propagation) since the dominant source mode o f the excitation coaxial measurement system is TEM. It is also desirable to only have a single propagating mode within the frequency-band of interest. With the availability o f the ground planes, via holes are not necessary in coplanar waveguides and mounting o f active devices and other shunt elements is easily achieved as in Fig. 1. The elimination o f the via holes in GaAs (gallium arsenide) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 c o p l a n a r waveguide m ic ro strip via hole ground planes 23 (a) hole jvia . m s s / ground plane (b) Fig. 1. Active device grounding in coplanar waveguides and microstrip, (a) Top view, (b) Side view. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 MMICs can result in dramatic savings in cost as well as improved process yields on the order of 40% [4], The presence of a ground plane between the strip conductors allows a compact circuit layout and reduces the crosstalk of adjacent lines. Also, the dielectric thickness is not required to be thin for high frequency operation and the elimination of the via holes allows for thicker and structurally stronger substrates. The characteristic impedance of coplanar waveguides is determined from the ratio o f the strip to slot widths so structures with different cross sections can have the same impedance. As the number of conductors increases, the propagation characteristics become more complex. One major disadvantage of this waveguide is the existence of an unbalanced or slotline mode that is excited at discontinuities and asymmetries within the circuit. The effects of this mode can be reduced through the use of air bridges which extend the cutoff frequency o f this mode by equalizing the potentials on the top ground conductors. However, these air bridges also represent a vertical discontinuity and are utilized extensively within a circuit at the vicinity of the discontinuities to eliminate this mode as demonstrated in Fig. 2 for a GaAs MMIC amplifier [5], The darkened connections are the air bridges and maintain all of the grounded regions. CPWs are not very well characterized in terms of circuit elements and discontinuities. The effectiveness of coplanar waveguides has been demonstrated in the work of a 5-100GHz GaAs MMIC amplifier [6] and the Cascade Microtech™ wafer probe system [7], An additional conductor plane on the other side o f the dielectric is utilized to support the structure and for heat sinking purposes and forms the conductor-backed coplanar waveguide (CBCPW) This conducting plane introduces several problems with regard to mode coupling effects. B. Overview of Conductor-Backed Coplanar Waveguides The basic two-dimensional, symmetrical (with respect to the y-direction) CBCPW configurations with an infinite-length in the z-direction are shown in Fig. 3 and includes an infinite-width, packaged (shielded or boxed) with shorting sidewalls (connecting the conductor-backed plane, the ground planes, and the cover plate plane), and packaged with finite ground planes. If none o f the conductors are electrically connected (total of N=4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 2. Circuit photograph o f GaAs CBCPW MMIC distributed amplifier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 conductors excluding the cover plate conductor), then three normal modes of propagation exist (N -l) within such a system [ 8 ] and are depicted in Fig. 4. A mode is a solution to the system of equations describing the waveguide, satisfies Maxwell's equations and the boundary conditions, and corresponds to a field configuration supported by the transmission line. Associated with each mode is a cutoff frequency. If the frequency of operation is above the cutoff frequency, the mode propagates and carries energy away from circuit discontinuities. If the frequency is below the cutoff, the mode is referred to as evanescent and energy is stored at the discontinuity and is attenuated away from the discontinuity. The dominant or odd mode is a balanced, zero-cutoff, and quasi-TEM one and corresponds to a magnetic wall placed at y=0. Most o f the electric fields for this mode are concentrated within the slot areas. If the ground conductor planes at x=0 are not at the same potential, the even or slotline mode is present. This mode is an unbalanced, nonzero-cutoff, and non-TEM one and corresponds to an electric wall placed at y= 0. An example o f how this mode can be excited at a right angle bend discontinuity is also shown in Fig. 4(b). The path difference between the electromagnetic waveforms exists on the inner and outer ground planes and translates into a phase difference between the planes. Air bridges are required to extend the cutoff frequency for this mode. If the ground conductor planes are not connected to the conductor-backed plane at x=-h, the microstrip-like or parallel plate mode can propagate. This mode is a balanced, zero-cutoff mode with most of the fields concentrated within the dielectric. The cross-sectional vector electric field plots (Erx and £„) for these modes are presented in Fig. 4 and are normalized y to the largest field component for each case. The even mode can be eliminated by maintaining symmetry and an example is a through transmission line. The microstrip-like mode can be excluded by connecting the ground plane conductors at x=0 and the conductor backing plane at r=-h 4-h 5-h6 along the line length of the transmission line to achieve a uniform ground connection as in Fig. 5(a). This configuration also represents the boundary effects o f the CBCPW within a package and demonstrates the actual physical waveguide. WG extends to the substrate edge unless noted. This wrap-around connection is achieved using copper tape, conductive epoxy, or through the package. This structure is modeled with shorting lateral sidewalls at _y=±A as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 X air E E m w s w y €r (a) air E E in y jC 2A (b) X E E m air w s w wG f - f - -,n=a------------ ■------2A— -] (C) Fig. 3. Basic CBCPW configurations, (a) Infinite-width. (b) Packaged with shorting sidewalls, (c) Packaged with finite ground planes. S, W, WG, and 2 A are the widths of the center strip conductor, slots, ground plane conductors, and the lateral sidewall separation respectively. The conductorbacked plane exists at x=-h and the cover plate conductor at x=5mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 magnetic wall (a) electric wall (b) v z t K2 y y 1t i f I i —— — 1 ' I l 1 i XI t ' i 1 ' ‘ I I » » 1 I ' ar / i i t i t i j i ii vi vi i» i* »i ii vi \\ vl l k 1 k k k V\ ' magnetic wall (c) Fig. 4 Fundamental modes of CBCPW along with the source topologies including the cross-sectional vector electric field plots for the two-layer configuration, (a) Dominant or odd mode, (b) Slotline or even mode, (c) Microstrip-like or parallel plate mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 conductor X £n ■ ^ »,•-A' r'•,-A' r■-A +•,•A'^T2 ■'>V^•/•.#.^•-•A' *r■•> • • ■A ,*•r■ m m m m b Fig. 5. Multi-layered CBCPW configurations, (a) Longitudinal view o f experimental CBCPW with shorting sidewalls, (b) Cross-sectional view of CBCPW used in the numerical analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 in Fig. 5(b) and resembles a rectangular waveguide with two aperture slots to guide the electromagnetic energy of the dominant mode. The CBCPW mode or dominant mode will be referenced interchangeably throughout this dissertation. The structures considered here have cross sections (S+2W) that are much smaller than the lateral width (2A). Waveguide modes exist within the dielectric or substrate regions at layers 4,5,6 and/or 1,2,3 in Fig. 5(b) that can couple to the dominant mode and produce undesirable results. Air filled rectangular waveguides are usually designed with the ratio of the lateral width to the height (2A/h)=2 for a single mode o f operation [9] as shown in Fig. 6 (a) and requires ten separate rectangular waveguide bands (each operating in a single mode) to cover the 0.75-40GHz range. CBCPW MICs can have a ratio between 15<(2A/h)<60 for sufficient circuit surface area which translates to a multi-mode operation. The finite-length CBCPW of Fig. 5(a), within a package formed from the connector housing blocks, produces a rectangular cavity and introduces a resonant structure as demonstrated in Fig. 6 (b). A cavity can be considered as a volume enclosed by a conducting surface and within electromagnetic fields can be excited and stored. How well the energy is stored within the cavity corresponds to the Q (ratio of energy stored to the energy dissipated). With a high Q (quality factor associated with each resonance) cavity, a significant current can flow into the resonator and produce significant resonances which can interfere with the CBCPW operation. The structures considered here have a large electrical size so that many resonant frequencies exist within the band. An example for a typical multi-layered cavity structure with dielectrics s r=10.2 (relative dielectric constant) and er=2.2 with a thickness of 0.635mm each with dimensions 1.27 (h) x 20 (2A) x 38 (2B) mm possesses 23 cavity modes to 40GHz. C. Problem Definition and Research Objective The objective o f this dissertation is to explain, predict, and reduce the effects o f the waveguide and cavity modes on the operation of the CBCPW MICs for wide frequencyband applications. The CBCPW was intended to be utilized with a millimeter-wave active antenna array system to eliminate the need for via holes within the circuit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The 11 ♦ -C * 2A (a) ♦ XL 2A (b) Fig. 6 . Approximate representations of packaged CBCPW. (a) Ideal rectangular waveguide, (b) Ideal rectangular cavity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 consequence of these modes was demonstrated via experimental measurements on approximate 50ft through lines to 40GHz with an in-house developed test fixture using Omni-Spectra Inc. OS-50 connectors as shown in Fig. 7. The measurements were taken on a HP8510B network analyzer with 401 data points and calibrated at the coaxial cable ends. The network analyzer measures the S-parameters or scattering coefficients (magnitude and phase) for two-port networks. S 21 represents the forward transmission coefficient or the ratio of the transmitted wave (on output port # 2 ) to the incident wave (on input port # 1) with the output port terminated in the characteristic impedance o f the system (50ft) [10]. The magnitude of S21 for a lossless and matched transmission line is 1 or OdB and is e_az with losses where a is the attenuation constant. The conductor ground planes and the conductor-backed plane along the length o f the CBCPW were connected with electrical tape from 3M Co. as in Fig. 5(a). The ground contacts at the input and output ports were obtained using a shorting bar within the fixture and good contacts are critical for proper mode excitation. The connector housing blocks complete the cavity with a uniform ground connection as depicted in Fig. 8 . This structure is equivalent to that presented in Fig. 5(a) with the wrap-around connection (shorting sidewalls) at z=±B. The dielectrics are Duroid™ 6010 (er=10.8) and 5870 (sr=2.33) from Rogers Corp. The experimental results are described in Fig. 7 with CASE A representing a microstrip line for fixture verification, CASE B shows a typical response for a single-layer CBCPW using 6010 substrate, and CASE C depicts a three coupled-strip (finite ground planes) transmission line. As demonstrated the single-layer CBCPW MICs do not possess any appreciable bandwidth. These results were totally unanticipated and were not previously described in the literature and the presence o f the strong resonances were particularly disturbing. An entire and important class of MIC transmission lines was unusable even at lower frequencies. All of the line parameters were varied to eliminate these undesired effects but these changes still did not produce favorable results. Such attempts included a structure with a small cross-section (S+2W=0.75mm) compared to the dielectric thickness (2.54mm) for the 6010 substrate. The structures considered in this dissertation have a maximum lateral (transverse in ^-direction) or longitudinal (z-direction) dimension of 38mm and correspond to realistic MICs. A multi-layered configuration (lower dielectric Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CASE A MAGNITUDE S21 ( d B ) CASE S CASE C V \ CASE 0 > '~0 00 ■O CASE E tO CASE F 0 4 8 12 16 FREQ Fig. 7. w O 20 24 28 32 36 40 (G H z ) Experimental responses demonstrating waveguide and cavity modes in CBCPW with shorting sidewalls and 2A=2B=38mm, S=2W=0.635mm, s ^ s ^ s ^ e ^ l , h,=oo (open structure), h;,=h3=h6=0. Refer to Fig. 5(b) for the dimension parameters. Case A. 50C2 microstrip line s r4=10.8, h4=0.635mm and sr5= l, h5=0, Case B same as Case A for CBCPW, Case C same as Case B except WG=lmm, Case D same as Case B except h4= 1.27mm and srS=2.33, h5=0.381mm, Case E same as Case D except h4=0.635mm and h5=0.711mm Case F same as Case E except broadband absorber placed in the connector housing blocks. Cases AB,C,D are lOdB/div while Cases E,F are 5dB/div and all are referenced to OdB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 ^ shorting bar conductor shorting bar «o‘ 2A (b) Fig. 8 . Test fixture illustrations incorporating CBCPW transmission lines, (a) Shorting bar application to ground input and output ports of CBCPW. (b) Connector housing blocks. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 loaded) improved the experimental results as shown in Fig. 7 CASH D and produced a 26GHz bandwidth, CASE E extended the upper usable frequency of operation above 40GHz but demonstrated the effects o f the cavity resonances and modes especially above 20GHz, and CASE F is the same as CASE E with broadband absorbing material from Arlon Inc. (LS series dc-40GHz) within the connector housing blocks to dampen the cavity resonances. CBCPW must be utilized in one of four configurations for successful operation. The lateral sidewall dimension can be reduced to cutoff the rectangular waveguide modes and prevent mode coupling as in Fig. 9(a) for the channelized coplanar waveguide [11], but the available circuit surface area and frequency range is limited. The cutoff frequency of the TE0 j mode (first waveguide mode and the subscript integers pertain to the number of standing wave maxima in the field solutions describing the field variations in the x and ^-directions, respectively) for a single-layer CBCPW with er=10.2 and lateral width (2A) of 5mm is 9.46GHz. Also, this configuration requires a grooved fixture to support the dielectric structurally. An alternative CBCPW utilizes via holes along the transmission line length to eliminate the microstrip-like mode and almost forms an electric wall to cutoff the waveguide modes similar to channelized CPW is described in Fig. 9(b) [17], This structure eliminates the primary advantage of coplanar waveguides over microstrip, increases the modeling complexity, and enhances coupling effects between the vias. The cross-section dimension (S+2W) can be made much smaller than the substrate thickness to reduce the mode coupling effects and can be easily achieved for MMICs. Keeping (S+2W) less than one-fourth the dielectric thickness for GaAs and one-twentieth the wavelength is a design estimate o f the leakage rate for the CBCPW mode to achieve a useful circuit [12], Leakage is the loss of power (or attenuation) from the CBCPW mode coupled to the parallel plate modes within the substrate region between the ground plane conductors and the conductor-backed plane. width and length. The power is leaked away if the waveguide is infinite in For MICs, these line/slot requirements may not be attainable with respect to circuit etching capabilities or possible for certain applications. Multi-layered structures can be utilized for these "general purpose" systems and modify the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 CENTER STRIP GROUND PLANES DUROID METAL CHANNEL (a) Fig. 9. Alternative coplanar waveguide configurations, (a) Channelized coplanar waveguide, (b) Via hole structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 propagation characteristics of the waveguide modes and prevent mode coupling effects in CBCPWs [13-16], Strong mode coupling with the waveguide modes can cause the dominant mode field pattern to spread out across the entire waveguide width instead of being confined to the slot area. This phenomenon translates into a loss o f power and the occurrence of strong resonances in the transmission measurements of the waveguide. Both lower and upper dielectric loading structures are proposed here as realistic alternative configurations as described in Fig. 5(b) with the various layers representing dielectrics, air gaps, bonding film, and damping material. The multi-layered coplanar waveguide is still referred to as CBCPW since the thickness o f the dielectrics is not so large that the electromagnetic effects of the conductor-backed plane are still present. The lateral and longitudinal sidewalls are extended for available surface area and multiple propagating modes exists within the circuits. From the experimental data if strong mode coupling occurs within the CBCPW, absorber or damping material will not remedy this situation. The disadvantages of the multi-layered waveguides are a more complex structure, possible air gaps between the substrates, and difficulty in configuring the upper dielectric loaded CBCPW within a test fixture. The physical phenomenon behind the above mode coupling problems can be explained and modeled via the spectral domain method (SDM) in one, two, and three dimensions. SDM is a very efficient full electromagnetic wave numerical method [18-19] (especially suited for multi-layered planar transmission lines) that can predict complex propagation characteristics, leakage rates, impedance variations, mode coupling, field patterns, and resonator Qs for the various modes. This modeling capability has been utilized to predict the upper usable frequency (onset frequency of strong mode coupling between the dominant and first waveguide mode) up to 40GHz, explain in detail the mode coupling mechanism, select multi-layered structures to extend the upper usable frequency or bandwidth, and suggest configurations to reduce the substrate cavity Qs without affecting the CBCPW circuit. Experimental data is used to verify the upper usable frequencies, confirm the explanation o f the mode coupling mechanism, and demonstrate the necessity of some type of damping material to reduce the residual resonances associated with the cavity stmcture even when the dominant mode is bound to the slots. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 To summarize the questions that this dissertation will answer involving the transmission and propagation characteristics of CBCPW MICs are: 1) explain what mechanisms are causing the effects in the experimental data; 2) explain how and why these mechanisms are interfering with the CBCPW operation; 3) propose realistic alternative CBCPW configurations to minimize the above effects (multi-layered configurations); 4) develop a numerical model to predict the bandwidth of operation; 5) verify the model with experimental data to 40GHz; 6) present design summary information for multi-layered CBCPW configurations; 7) demonstrate how CBCPW MMICs can work with the mode coupling effects present. D. Literature Review and Discussion The coplanar waveguide was invented by Wen [1] in 1969 as an alternative transmission line for MIC applications, recognized the need for a shorting bar to connect all of the ground regions, and analyzed the infinite-thickness structure with a quasi-static (low frequency) method. Pucel [2] suggested the use o f coplanar waveguides for GaAs MMICs but made no mention of the possible mode coupling problems. The SDM was utilized by several authors to solve for the propagation characteristics o f air suspended coplanar waveguides. [20], The infinite-width structure was analyzed by Knorr and Kuchler Fujiki et al. [21] modeled a packaged CPW with the first few higher order waveguide modes and produced dispersion curves but these modes were not identified and the coupling effects were not discussed. Davies and Mirshekar-Syahkal [22] developed a procedure to analyze multi-layered CPW in the SDM but only considered air suspended structures with no waveguide modes propagating. The air suspended configurations are not practical because it is difficult to dangle a dielectric or semiconductor in the air. The analysis of infinite-width CBCPW was demonstrated by Shih and Itoh [23] but the leakage effects associated with the parallel plate modes were neglected. The coupling effects of the waveguide modes due to the lateral sidewalls for a GaAs dielectric were investigated with a mode-matching technique by Leuzzi et al. [24] and demonstrated the field spreading Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 of the dominant CBCPW mode. However, the authors misidentified the modes in the propagation curves. The realization that losses and dispersion in CPW could be less than those of microstrip was described by Jackson [3]. Jackson [25] also first discussed the possibility of the leakage of power from the dominant CBCPW mode and suggested a multiple dielectric configuration (lower dielectric loading) to prevent this mechanism in the infinite-width case. The analysis of leakage in conductor-backed slot lines and potential problems associated with CBCPWs were described by Shigesawa et al. [26], Mode conversion in CBCPW discontinuities with finite ground planes from the dominant mode into the microstrip-like mode was explained by Jackson [27], Channelized coplanar waveguide MICs were proposed by Simmons et al. [11]. Propagation characteristics for several CPW configurations and an approximate closed form relation for the leakage rate of the dominant mode for a single-layer CBCPW were discussed by Riaziat et al. [12], The most impressive applications of CPW circuits were demonstrated in a 5-100GHz MMIC amplifier by Majidi-Ahy et al. [6 ] and the Cascade Microtech™ wafer probe system [7], MMICs have a distinct advantage over MICs in the capability to make the cross-section much smaller than the dielectric thickness which reduces the leakage coupling effects [12], [28], Cascade Microtech™ has incorporated absorbing material in the probe heads [29] to reduce spurious mode effects and studied surface wave coupling problems in CPWs [30], Das and Pozar [31] incorporated the leakage analysis of the dominant mode into the SDM for several waveguide configurations. Leakage calculations for CBCPWs were published by McKinzie and Alexopoulous [32] and Chou et al. [33], Magerko et al. [13] presented experimental data for CBCPW MICs demonstrating the effects of moding and resonance problems, incorporated a multi-layered (lower dielectric loading) to extend the upper usable frequency, and approximately predicted frequency using a one-dimensional SDM analysis. The mode coupling effects this of packaged multi-layered CBCPWs were described in detail by Magerko et al. [14] using a two-dimensional SDM including field spreading of the dominant mode and impedance variations of the modes. The mode coupling mechanism was discussed in general terms for packaged transmission lines by Carin et al. [34], Jackson [35] has established a circuit Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 model to approximate inter-circuit coupling effects due to substrate resonances in CBCPWs. The resonance effects associated with finite ground planes in a test fixture have been predicted by Tien et al. [36], packaged CBCPW MICs Configuration considerations for multi-layered including design summary tables, upper dielectric loading structures, and the analysis of lossy layers to reduce the substrate Qs without affecting the CBCPW resonator appears by Magerko et al. [15], [28], Leakage calculations for multi-layered CBCPWs have been published by Liu and Itoh [16]. Attempts by Yu et al. [17] have included via holes along the length of the CBCPW to connect the ground plane conductors and the conductor-backed plane. However, the main advantage of coplanar waveguides over microstrip has been eliminated. A commercial CPW test fixture to 20GHz has been offered by Wiltron Inc. [37] but no published or listed information exists. E. Dissertation Organization The content o f the dissertation is organized in the following chapters. In Chapter II a description o f the SDM for multi-layered CBCPWs in one and two dimensions will be presented. The SDM is the numerical method utilized in this work. The concept of the leakage o f power from the dominant mode into the parallel plate modes is introduced in Chapter III and the one-dimensional SDM analysis is applied to multi-layered CBCPWs. described. Dispersion and leakage curves for several structures will be The one-dimensional analysis (infinite width and length) can approximately predict the bandwidth of operation from experimental data in many cases but is not accurate in others. Also, this procedure provides no explanation o f the measurement effects above the upper usable frequency. The effects o f the lateral sidewalls on the CBCPW operation are included in Chapter IV as a two-dimensional (infinite-length) SDM procedure is presented. Realistically, all waveguides will be finite in the transverse and lateral directions. Rectangular waveguide modes are propagating within the substrate regions and can cause mode coupling problems with the CBCPW mode. These effects include field spreading of the dominant mode and changes in the characteristic impedances. The two-dimensional analysis can predict the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 bandwidth more accurately and explain the occurrence of the loss of power and the strong resonances in the transmission measurements. Design characteristics of various multi-layered configurations for upper and lower dielectric loading are discussed in Chapter V including the upper usable frequencies, characteristic impedances, and effective dielectric constants. Summary tables demonstrating design trade-offs, air gap sensitivity, dielectric uncertainty analysis, cross-section dimensions to limit dispersion effects, multi-layered microstrip interfacing, closed form procedure to determine the lower dielectric constant in a multi-layered arrangement, and parameter selection for 50Q lines are all presented and discussed. The mode coupling effects in packaged CBCPWs can be empirically related to the leakage rate of an infinite-width waveguide and this is also detailed in Chapter V. If leakage is not present in a transmission line, mode coupling will not occur. Furthermore, a small leakage rate corresponds to small mode coupling effects. A closed form leakage rate or constant from [ 12] is verified and dimension recommendations to reduce the mode coupling effects in terms of this constant are detailed. A design procedure incorporating these concepts is described. This material will also demonstrate how CBCPW MMICs can operate with the mode coupling effects present. The effects of the residual resonances still present in cases when the dominant mode is bound to the slots must be addressed (see Fig. 7 CASE E). Some of these resonances are on the order of 2-4dB and are unacceptable for a transmission line. As mentioned previously, the CBCPW within a test fixture or package forms a rectangular cavity as a substructure. The effects of lossy damping layers (doped silicon or microwave absorber) on the Qs of the CBCPW resonator circuit and the cavity modes are simulated using the three-dimensional SDM and are developed and discussed in Chapter VI. To limit the effects of the damping material on the CBCPW resonator while still reducing the Qs of the cavity modes, the lower dielectric layer thickness can be increased or the lossy material applied on the perimeter o f the structure. Chapter VII concludes the dissertation by summarizing the work, presents the original contributions produced in the area of multi-layered CBCPW MICs for wide frequency-band applications, and makes suggestions for future research work in this area. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 CHAPTER H THEORETICAL ANALYSIS OF CBCPW USING THE SPECTRAL DOMAIN METHOD The spectral domain method (SDM) is widely used for the analysis o f planar integrated circuits (MICs and MMICs) and is a very efficient full-wave electromagnetic numerical method that can simulate the propagation and transmission characteristics of these structures. The parameters predicted by the SDM in this work for one, two, and three dimensions include the complex propagation characteristics, leakage rates, impedance variations, mode coupling effects, field patterns, and the resonator and cavity Qs. This method was originally proposed by Itoh and Mittra in 1973 [38], The SDM can be applied to planar transmission lines including microstrip, slot lines, finlines, coupled strips, and coplanar waveguides. The analysis can also simulate well-shaped discontinuities such as transmission line steps, disk, triangular, or ring resonators, and planar periodic structures. From a mathematical point o f view, the SDM is simply an integral transformation method. Under special circumstances the differential equations governing the behavior o f a system can be transformed into a new domain, the spectral or Fourier domain, where the transformed equations can be easily solved as algebraic equations. The alternative method to solve for the propagation characteristics o f the waveguide is the integral equation technique and is performed in the space domain [39], The problem with this procedure is the Green's functions (functions relating the currents on the strips to the fields in the slots for the CBCPW) are not available in closed form for the inhomogeneous structure which translates into very slowly convergent integrals and prohibitively long (computation time) numerical solutions. One of the most important aspects o f the SDM is the ability to handle multi-layered structures and waveguides with various metalization levels as pictured in Fig. 10(a). Examples of structures suited for the SDM analysis are depicted in Fig. 10(b). The main reason the SDM is numerically efficient is that it requires a significant amount o f analytical preprocessing of the unknown currents on the strips or the fields in the slots. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 Metallization Substrates and Su p erstates (a) • £r Er - £r (b) Fig. 10. Examples of structures easily simulated using the spectral domain method, (a) Cross-section of a multi-layered planar structure with multi-metallization layers, (b) Cross-sections o f CBCPWs that are open, laterally open, and shielded or packaged from left to right, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 This requirement imposes some restrictions on the applicability of the method. The procedure has difficulty with finite conductor thickness especially when the thickness is on the order of the skin depth (the distance a wave, as part of the field localized in a surface layer, is attenuated to Me or 36.8% of the initial value) and cannot handle structures with dielectrics that are perpendicular in the lateral direction. A mode-matching method [40] would be utilized in the latter case. The conductors in this work are assumed to be perfect electric conductors (PECs) with infinitesimal thickness and infinite conductivity. Also, the dielectrics are assumed to be homogeneous, isotropic (electric permittivity e and magnetic permeability |i are scalar constants and do not vary with position) materials. A. General Spectral Domain Method Formulation The general SDM formulation will be developed following [18-19] for the CBCPW of Fig. 11 showing the cross-section of a symmetric, infinite-width (y-direction) and infinite-length (z-direction) structure. The CBCPW consists o f three dielectric regions below the circuit conductors at x=0 for lower dielectric loaded configurations of heights (thickness) h6 , h5, and h4 with relative dielectric constants , s r5 , and s r4 respectively, and three dielectric regions above the circuit conductors at Jt=0 for upper dielectric loaded configurations o f heights h3 , h2 , and h, with dielectric constants er3 , s r2 , and srl respectively. These layers represent dielectrics, air gaps, bonding film, and absorbing or damping material. CBCPW also includes a center metallic strip of width S, slots of width W, and ground plane conductors (WG extends to infinity). The relative permeability for all of the dielectrics is |a.r= l unless otherwise noted. A conductor-backed plane exists at x=-h6-h5-h 4 and a cover plate conductor is placed at x=h3+h2+h, to simplify the analysis by eliminating the radiation effects at discontinuities. Due to the inhomogenity of the medium, the coplanar modes can not be pure TE (transverse electric) to z or TM (transverse magnetic) to z. Longitudinal components (£, and H ,) for the fields exist. These hybrid modes can be expressed as a superposition of TE to x (TEX) and TM to x (TMX) fields since the discontinuities in the dielectric layer are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 / / V / ✓ / / ' / ' / Fig. 11. Multi-layered infinite-width CBCPW cross-section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 in the x-direction. The fields in the space domain set are expressed from [9] as E —— y \dx r 2 ¥ (2 . 1a) \d x d¥ dz i a 2¥ e y dxdy e l S 'P y dx dz £ , „ = -3- -2;— + k + k d¥ Hy = dz h -r H ,= - dy e 2 h , i d ¥ ~z d xdy d¥ dy I z dx dz (2.1b) (2 . 1c) for each region i=l,2,...,5,6 where ¥ and ¥ are the electric and magnetic scalar 2 2 potential functions respectively, k = co s r;e 0 prfi0 , y =j(aeris 0 , z=y'cop.rp 0 and j= V =T. co is the radian frequency and co=27tf where f is the frequency of operation (FREQ) and e0=8.854 x 10' 15 F/mm and |i 0=47t x 10‘10 H/mm. The time convention is implied and the z dependence e*1* is assumed where y is the complex propagation constant in the z-direction and y=3 -jo.. P is the phase constant and a is the attenuation constant due to losses in the transmission line. These loss mechanisms can include material or dissipative (dielectric), modal (leakage), or radiation (open structures at discontinuities). However, the waveguides analyzed here have upper and lower conductor planes so radiation in the 6 h x-direction is not present. ¥ and ¥ are functions of x and y and are solutions of the Helmholtz or wave equation ( 2 .2 ) where £, represents ¥ or ¥ . In the SDM the potential functions and field components are transformed into the Fourier or spectral domain. The Fourier transform is taken parallel to the substrates (see Fig. 11). The transform of a function <)>with a continuous spectrum (infinite-width in the ^-direction and z-direction) can be expressed as 00 %{x,ky) = ^ J <|>(x,y) eikyy dy (2.3) -0 0 or in short notation as $ = F(<f>) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.4) 27 where ~ represents the transformed function and k is the continuous spectral variable in they-direction. The inverse Fourier transform is written as oo <t>(x,y) = j $< x,ky)e~jkyy dky . (2.5) —oo Sufficient conditions on <{> and <J) to exist are 00 J l<t> —00 dy < oo (2 .6 ) and (j) (x,y) satisfies the Dirchlet conditions over (-oo <y < go) which are: (i) <t>(*>.V) and ^ {<{>(x,>>)} are piecewise continuous; (ii) (j) (x,^) has a finite number o f maxima and minima. Equations (2.1) and (2.2) are transformed into the spectral domain using the following relations f ( | 5 ) = ~jkrl (2.7a) * ■ < !? « > = -* :? • (2-7b) and The solution to the transformed homogeneous differential equation of (2.2) is expressed as £, = i?coshK X + rsinhK X ~e h where % represents 'F or *F and k where k 2 2 2 = ky + k . - k (2.8) 2 (2.9) is the propagation constant in the x-direction and K=jkx and R, T are constants. k_ isthe continuous spectralvariable in the z-direction. For aninfinite line with variation efp and from (2.3), y=k,. The scalar potentials can be written for each region o f Fig. 11 using ( 2 .8) by incorporating the boundary conditions at the upper and lower conductor planes (tangential electric fields vanish) as; Region 6 : ~,e e Tg = A c o sh K 6 (x + h 6 + h 5 + h 4) ~h <P6 - A h sinhic6(x + h 6 + h 5 + h 4) (2.10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 Region 5: Q Q Q 5 = B sinhK5(x + h 5 + h4) + C coshK5(x + h 5 + h4) ~ h ^ (2.11) h 'Rj = B coshic5(x + h5 + h4) + C sinhK5(x + h 5 + h 4) Region 4: C g Q i¥ ,4 - D sinhic4(x + h4) + E coshic4(x + h 4) ^ h ¥ 4 = h (2.12) h D c o sh ic 4 (x + h4) + E sinhic4(x + h 4) Region 3: ~ e e h h e 'r 3 = F sinhKjX + G c o sh ic 3x ^ (2.13) h ¥ 3 = F coshKjX + G sinhK3x Region 2: g g ft 'f ' 2 = H sinhK 2 (x - h3) + / co sh K 2(x - h 3 ) h t 2 -H h (2.14) h coshK2(x - h 3) + / sinhK2(x - h3) Region 1: ~ e e ^ h h T j = J c o s h K ,( h 3 + h 2 + h , - x ) sinhK 1(h 3 + h 2 + h , - x) (2.15) where Ae , Ah, ... , f , J 1 are the unknown coefficients. The electric and magnetic fields in the spectral domain can be obtained for each region by substituting (2.10)-(2.15) into (2 . 1) and utilizing the relationships in (2.7). Twenty unknowns exist and require twenty boundary conditions to solve for the above coefficients. These conditions result from the continuity o f the tangential electric and magnetic fields between the dielectric interfaces and the presence of an electric surface current at the PEC (x=0) as follows: At x = -h5-h 4 E y6 E yS for all y (2.16a) E=6=E:5 for all.y (2.16b) HX>=Hys for all y (2.16c) 5 for all y (2.16d) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 At x = -h4 for all y (2.17a) Ezr E z4 for all y (2.17b) Hy5=Hy4 h z5=h :4 for all y (2.17c) for all y (2.17d) Ey*=Ey3 for all-V (2.18a) E,a =E.3 for all y (2.18b) A tx = 0 ~ H y4 =JZ =0 H:3-H :4 = -Jy =0 H yl \S/2\ > M aIld Ivl > IS/2 + W| otherwise |S/21 > Lv| and \y\ > |S/2 + W| otherwise (2.18c) (2.18d) At x = h3 for all y (2.19a) e z3=e :2 for all y (2.19b) £ II for all y (2.19c) e :3=h :2 for all y (2.19d) Ey2 Eyl for all y ( 2.20 a) Ezr E zl for all y (2.20 b) II for all y ( 2.20 c) h z2=h :1 for all y ( 2 .20 d) a? Ey3 Ey2 At x = h3+h2 a? where Jy(y) and J.(y) are the unknown current distributions on the strips at x=0 and the boundary conditions are specified for the entire range of y. The boundary conditions in the spectral domain are obtained as the Fourier transforms of the conditions in the space domain. The unknown currents are related to the unknown electric fields across the slots (|S/2| < [y| < |S/2 + W|) (or treating these electric fields as equivalent magnetic currents) through the admittance Green's functions which rigorously incorporates the effects of the layered medium as [J\ = [Y\[E\. (2.21) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 In terms of system theory, the Green's function is basically the transfer function. It is very desirable that the Green's function be formulated in closed form. In the general solution approach for finding the Green's functions, the field formulation is derived by solving 4(N-1) unknown coefficients from 4(N-1) coupled equations for an N dielectric layer structure. As the number of layers increases, this process becomes substantially more complicated. An alternative procedure called the immittance approach [18] enables an easy solution for multi-layer structures by decoupling the TEXand TMX components. In the immittance approach, the formulation o f (2 .21 ) is possible without knowledge of the field coefficients. The basic concept of the immittance method is a special mapping or decomposition relating the currents and fields in the Fourier domain. Appendix A describes this procedure in detail and the dyadic spectral domain admittance Green's function are written as ( 2 .22 ) (2.23) (2.24) where Ye and Y h correspond to the TM and TE modes respectively, and are determined from (A.5), (A.7), and (A. 14-A.19). Dyadics are used for representing Green's functions relating an arbitrarily oriented source to the fields and currents that it creates. Equation (2.21) can be written in the spectral domain using (2.22-2.24) in matrix form as ' J y ' J z . 'Yyy V 'V For instance, the Green's function Yyz relates the ^-directed current on the strips at (x=0, z) to the source slot electric field in the 2-direction at (jc=0, z). The unknowns of the above system are the complex propagation constant y and E y,E z,J y, J .. A large number of basis functions would be required for the current expansions (J y and J . ) due to the wide conductor area for the CBCPW (see Fig. 11). Hence, it is more numerically efficient to apply the procedure across the unknown electric fields over the slots as expressed in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 (2.25). The current expansions can be eliminated by applying the Galerkin method which is a moment method procedure. In this method the testing functions are the same as the basis functions. The first step is to expand the unknown electric fields Ey and E, across the slots at x=0 in terms of known basis or trial functions Em and E;n in the space domain as Ey = £ c mE ym(y) m=\ E . = 2 d nE :n(y) n= 1 where cm and dn are the unknown expansion coefficients. (2.26) The basis functions in the spectral domain are obtained from (2.26) and utilizing (2.3) oo oo E y= I cmE ym(ky) m=\ E._= s d nE :n(ky) . n=\ (2.27) The basis functions must bemembers of a complete set and the Fourier transforms must exist,preferable inclosed form. Any set o f functions can be usedbut those that represent the physical electric field distribution will reduce the number of functions, enhance the accuracy of the solution, and reduce the matrix size and computation time. Applying the boundary condition that the tangential electric fields at a PEC must vanish, each basis function is chosen so that it is nonzero only in the slot areas ( |S/21 < [y| < |S/2 + W|). Due to computer limitation in handling infinite size matrices, the number o f basis functions must be truncated to a finite value and the accuracy of the solution will depend on the number o f functions. After substituting (2.27) into (2.25) yields ~ ~ M ~ N ^ y = ^yy ^ c mEym + Yyz ^ d nE_n m=l n= 1 (2.28a) ~ ~ M J-. = yv X c . E m=\ (2.28b) + ~ N 2 </„£,. n= 1 Let the inner product required for the application o f the Galerkin technique be defined as < X ,Y > = J XY*dky (2.29) —oo where (*) specifies the complex conjugate. The Galerkin technique can now be applied using (2.29) with the inner product o f (2.28a) with E yp and similarly (2.28b) with E :g and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 yields oo _ _ M E*„Y„ S < J y 'K > = f —00 m -\ „ ^ C„ £ + „ £ ^ N 2 ^ d , E ;n dky p = 1,2, ...,M n= 1 (2.30a) 00 = J -0 0 ~ M £ zg -0^zv 2“ zy m -\ ~ ~ ~ N ~ £?o r _ S , c/„n e :n dfc n= 1 + ym m q = \,2,...,N. (2.30b) The left hand sides of (2.30) are zero using Parseval's theorem because the product of the currents and the slot electric fields at x=0 is zero since the currents exist only on the strips and the electricfields exist only in the slots. Parseval's theorem is written as oo oo J J(y)E*(y)dy = J J (ky)E *(ky)dky = 0 . -o o (2.31) 71 - o o A homogeneous system of equations is now formed and is described in compact form as c d M(y)] (2.32) = [0] w h ere [/1( y)] is a m atrix o f order p+q w h o s e elem ents are listed as 30 Jf E * Y WE —oo yp y> ym dk y (2.33a) 00 (2.33b) —00 oo 21 22 f E J —00 00 * Y . VE dk y zq - y ym J K * (2.33c) (2.33d) -0 0 Considering (2.32), nontrivial solutions can only occur when the matrix is singular as det [/4(y>] = 0 (2.34) where det is the determinant of the matrix and the complex propagation constant or eigenvalue y can be computed at each frequency co. Associated with each eigenvalue or mode are the eigenfunctions Q and fa . An infinite number of modes exist or are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 supported by the waveguide and depending on the structural parameters and operating frequency, one or more modes will be propagating and attenuating (P positive, real and a positive, real) in the z-direction. Most of the modes will be evanescent or non-propagating (P negative, imaginary and a positive, real). B. Basis Function Selection The basis functions describe the behavior o f the electric fields across the slots for the CBCPW. Any type of basis function can be used as long as the function satisfies the boundary conditions of the PECs (nonzero only on the slots) at x=0. The numerical accuracy and efficiency of the SDM depends on the basis function selection. In an electromagnetic problem involving strips with sharp edges and modeled by infinitely thin PECs, some of the field components may possess an unbounded behavior in the vicinity of these edges [19], In this case, the transverse electric and magnetic field components are unbounded at the strip edge and approach infinity (singular fields) with |r| 2 variation where r =(y-S/2-W/2) for the right hand slot of Fig. 11. The longitudinal field I components are bounded near the edge and vary as |r l 2 . The above conclusions are general and hold at the edge of an infinitely thin PEC placed between two dielectrics and the dielectric constants do not enter the given asymptotic expressions. Spurious solutions (non-physical) may be eliminated by choosing basis functions which are twice continuously differentiable [19], The set of basis functions should be complete to enable approximation o f the exact solution to any degree desired by simply increasing the number o f terms o f the expansion. In this manner, the numerical solutions o f the SDM can be checked for convergence. In recent years, most investigators have employed a combination of sinusoidal [18] or Chebyshev [32] functions with the edge condition or correction term to model the unbounded fields at the conductor edges for the basis functions. As described earlier in Fig. 4(a), the dominant mode for the CBCPW has an nonsymmetrical characteristic in the transverse direction. This translates to odd functions for E (dominant field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 component) while the longitudinal component E. can be described with even basis functions. Sinusoidal basis functions are used in this dissertation and are expressed as cos [(m—1)7t(y-/ft)/W] cos [ ( aw- 1)7t(y+l>)/W] J\-[2(y-b)/W ]2 sin [(wi-l)7t(y-l>)/W] J l-[2(y+b)/W)2 sin [ ( aw- 1)7t(y+6)/W] J l-[2(y-6)/W ]2 cos [»7i(y-6)/W] + m= 1,3,... (2.35) m = 2,4,... Jl-[2(y+b)/W\: cos [»7r(y+Z>)AV] aa J l-[2(y-Z»)AV]2 J l-[2(y+b)/W\2 sin [mc(y-fr)/W] _ sin [AA7t(y+6)/W] E zn(y) = ^ l-[2(y-£)/W ]2 = 1,3,... (2.36) /i = 2,4,... J \-[2(y+b)W]2 where the first terms are valid for ^>0 (right hand slot) and the second terms represent ><0 (left hand slot) and b=S/2+W/2 and all of the terms are scaled by a factor which is utilized to simplify the Fourier transforms and the voltage calculation and presented later in section E. The selected functions satisfy the criteria earlier described for basis functions. The shapes of the first three electric field basis functions are depicted in Fig. 12. The Fourier transform for the functions o f (2.35) and (2.36) can be obtained by applying (2.3) and are expressed as js m (k y b) Eymtty) = jcos(k b) j 0 m = 1,3,... ^ | W ^ + ( w - l ) 7 t l '~] _ f f |W ^ - ( /» - l ) 7 l | Jo aw = 2 ,4 ,... ( 2 .3 7 ) cos(kyb) Jo E M = -sin(AyA) Jo + «7tp + rm + J0 -J n I W A y - W 7t| 2 |W A ,. - W 7t| n= 1,3, n - 2 ,4 ,... (2.38) where J 0 is the zeroth-order Bessel function o f the first kind. The integrals are evaluated using a substitution method, product-fiinction relations [41], and recognizing that Eym is an even function across the slots for a w = 1 ,3 ,... and is odd for a w = 2 , 4,... and similarly E.n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 m-1 m=2 m=J 1.0r 0.5 4-> o c D co co o _Q 0.0 -0.5 — 1. 0 * - 1 .0 - 0 .8 - 0 .6 - 0 .4 - 0 .2 0.0 0.2 0.4 0.6 0.8 1.0 (a) n -1 »= 2 I.Or 71=3 CO C O 0.5 -t-> O c D 0.0 CO *to o -Q -0 .5 N - 1. 0 * —1.0 - 0 .8 -0 .6 -0 .4 —0.2 0.0 0.2 0.4 0.6 0.8 y (mm) 1.0 (b) Fig. 12. Shapes of the electric field basis functions for CBCPW with S=0.635mm, W=0.3175mm. (a) E>mbasis functions, (b) E,n basis functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is an even function for w=l,3,... and is odd otherwise. An important relationship aiding the evaluations of (2.37) and (2.38) with a as a constant is (2.39) The only problem with the selected basis functions is the Fourier transform contains the special Bessel function which is a disadvantage from a computational point of view. J 0(0)= 1 and JQ(y) looks qualitatively like sine or cosine waves whose amplitude decays as y - 1/2 . The number of basis functions for an accurate solution is largely dependent on the aspect ratios. For small W/S and W/D (where D=h6+h5+h4), only a single basis function is needed since both the slot coupling and the conductor-backing plane (at r=-h6-h5-h4) effects are small. As the aspect ratios increase so does the number o f basis functions required to accurately simulate the CBCPW and the structure becomes more dispersive. The Chebyshev basis functions for the aperture fields will be used later to verify the results of the sinusoidal basis functions and are included here as (H P ) E„ = N ( 2(y-b)-) [ E d . U„_, I W n= 1 (2.40) (2(y+b)') ( 2 (y+b)'] r w -J f-\T v T J (2.41) where TmA and UnA are the Chebyshev polynomials o f the first and second kinds, respectively. The Fourier transform of these basis functions may be written in terms of integer order Bessel functions [42], C. Two-Dimensional Packaged Structures In order to connect the ground plane conductors at x=0 and the conductor-backed plane at x=-h6-h5-h4 throughout the length of the CBCPW to achieve a uniform ground Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 contact (see Fig. 11), PEC lateral shorting sidewalls are used to model this wrap around structure. The waveguide is assumed to be placed within a packaged assembly and the sidewalls are extended in regions 1, 2, and 3 (see Fig. 13) to connect the cover plate conductor at x=h3+h2+h1. The lateral distance (A) is finite in width and the infinite-length (r-direction) waveguide is analyzed with a two-dimensional SDM. For structures with sidewalls, the Fourier transform o f (2.3) is replaced by a discrete finite Fourier transform where <J>(y) is defined over the interval [-A.A] and satisfies the Dirichlet conditions o f (2.6) with A J l<K*,.y)l dy < 00 -A (2.42) with $(*,kyi) = i ^ A j § ( x ,y ) J k>'y dy -A (2.43) and the inverse Fourier transform is defined as Z $(x,/cyt) e Jky y . = i (2.44) = -o o The solutions for finite regions (waveguides) are characterized by discrete spectra of eigenvalues. Accordingly, the integrals with respect to k are all replaced by summations in terms of discrete values of kyj from /'=-oo to oo for (2.30) and (2.33). In the above expansions, k^ is the discrete spectral parameter and is determined by examining the field behavior along the ^-direction. Since the field components Ex and Hx are proportional to and T* respectively as from (2.1a), examination o f these two field components is sufficient for determining kyi. As mentioned earlier, the dominant or odd mode (E ) for the CBCPW corresponds to a magnetic wall placed at .y=0 and electric PECwalls exist at ,y = ±A. Therefore, is equal to sin becomes cos f o r /'= - q o , ...,-5 ,-3 ,-1 ,1 ,3 , 5, ...,q o and T* ( /7tv') j . The discrete spectral parameter for the j-direction can be rewritten as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Fig. 13. Multi-layered finite-width CBCPW cross-section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 k >" = (/ + 2> A for / = - o o , ...,-2 ,-1 ,0 ,1 ,2 ,..., oo. (2 45) Recall from (2.1b) that Ey is proportional to J^T ^and VJ Parseval's theorem of (2.31) is also modified in the packaged case as 4 i J J(y)E*(y) dy = ± -A ,=0° ~ ~ I J(kyt)E*(kyi) = 0 . i=~™ Allof t.heelements of the matrix [A] of (2.33) are even functions (2.46) of k r Yyy, are even and Yy,, Y ^ are odd functions of kyj (see A. 8-A. 10) while Eym and E :nare odd and even functions, respectively. An even function in the space domain (y) becomes an even function in the spectral domain and similarly for odd functions. Therefore, the summations in terms of k are represented as / = oo X / —oo A = 2 Z / = —oo A. (2.47) /'= 0 An infinite number of summations terms is not possible with a computer and (2.47) is truncated and checked for convergence. Likewise, equations (2.33) are similarly modified. D. Spectral Domain Method Field Formulation The spectral domain immittance method can be efficiently extended to determine the electric and magnetic field components of the CBCPW. As stated earlier, this part of the derivation is not necessary to solve for the propagation constant of the waveguide but is required for mode identification, demonstration of the mode coupling effects on the dominant mode, boundary condition verifications, and calculation of the mode characteristic impedances. Once the propagation constant y is found from (2.34), the electric field basis function expansion coefficients (cm and dn ) are derived using (2.32). The spectral domain versions of Ey and E z at x=0 can be determined via (2.27) and then J y and J . are calculated from (2.28). The next step is rewriting the boundary conditions of (2-16)-(2.20) in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 spectral domain as At x = -h5-h4 Ey 6 = E y5 £ -6 = £ .j H y6=H y 5 (2.48a) (2.48b) (2.48c) ^ , 6 = #.-s (2.48d) £ 3,5 = ^ £ -5 = £ .4 H y, = H y4 H :5= H :4 (2.49a) (2.49b) (2.49c) (2.49d) E y4 = Ey,j £ -4 = £ -3 (2.50a) (2.50b) (2.50c) (2.50d) At x = -h4 A tx = 0 #.-3 - H :4 = ~Jy At x = h3 E y 3 = E y2 £ .3 = £ .2 ^ 3 = ^ .2 H . 3 = H :2 (2.51a) (2.51b) (2.51c) (2.5 Id) Ey2 = E yl E : 2= E . 1 H y2= H yX H ^ = H Zi (2.52a) (2.52b) (2.52c) (2.52d) At x = h3+h2 where and J , are the Fourier transforms o f the unknown current components on the strips atx=0 and are given in(2.25). Using the coordinate transform of the immittance method of (A. 2), J y and J z arelinear combinations of J u and J v andare expressed from (A.3) as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 _ k ZJ : y jk y ^ k y j z J k j + k y jy + “k + (2.53) k \ z J y (2.54) k \ In the immittance approach, the fields are decomposed into the TM, and TE, components. The electric and magnetic fields are now presented in the spectral domain via (2.1) and using (2.7) with (2.10)-(2.15) as TE, TMX Ex = y In ^ ■•n i n •-> I II tl Hx = 0 Ex = 0 (2.55a) Ey = jT V (2.55b) ~h E. = -jR V (2.55c) 1 —h Hx = I t z (2.55d) I II litT &y = H : = jR % e H. = ■R dT * z dx ^ h ■T <3T T dx (2.55e) (2.55f) ve -yh for each region i=l,2,...,5,6 and t and T are the transformed electric and magnetic scalar potential functions given in (2.10)-(2.15) and ky + R 1 T — kz ~ k y2 + kl (2.56) Equations (2.53) and (2.54) are written in terms o f the field components utilizing the boundary conditions of (2.50c) and (2.50d) as 7 “ J + NK> < \ 7 } 4e x4 - - y 3E « ] i . . 'dH x4 dx j k 2y + k 2 s h x3' dx (2.57) (2 58) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Substituting the field components of (2.55) into the boundary conditions of (2.48)-(2.52) with (2.57) and (2.58), results in two sets o f ten independent equations with ten unknowns, respectively. This method leads to less complexity in the derivation of the coefficients as compared to the general brute-force approach regardless o f the number o f dielectric layers. The final derived coefficients ( A e , A h , ... , f , f ) of the electric and magnetic fields are described in Appendix B. The field components for Ey and Hy are expressed respectively as TM_ -jRKysA sinhic6(x + h 6 + h 5 + h4) E y* = K = - j R K yS^B coshK5(x + h5 + h4) + C sinhK5(x + h s + (2.59a) h 4) J (2.59b) Ey4 = - j R K y^ D coshic4(x + h 4) + E sinhx4(x + h4)J (2.59c) E * = - j R K y3^F coshx3x + G sinhK3x J (2.59d) = -jR K yJ^H coshic2(x - h 3) + / sinhK2(x - h3)J e Z fl = jR \cylJ sin h K 1( h 3 + h 2 + h , - x) (2.59e) (2.59f) TE_ 's, n <— E y6 = j T A sinhK6(x + h6 + h 5 + h4) h h B coshK5(x + h 5 + h 4) + C sinhKs (x + h 5 + h 4) (2.60a) (2.60b) h h Ey* = j T D coshK4(x + h 4) + E sinhx4(x + h 4) (2.60c) h h Ey3 = j T F coshK3x + G sinhK3x (2.60d) n h h H coshK2(x - h 3) + / sinhK2(x - h 3) ~ h E yl = j T J sinhK,(h3 + h 2 + h , - x) (2.60e) (2.60f) and ™x H y6 = - j T A coshK6(x + h6 + h 5 + h 4) (2.61a) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 /V B C H v5 = - j T B sinhic5(x + h 5 + h4) + C cosh»c5(x + h s + h 4) (2.61b) H y4 = (2.61c) sinh>c4(x + h 4) + E coshlc4(x + h4)J H yi = - j T ^ F sinhic3x + G coshic3x J (2.6 Id) H y2 = - j T ^ H sinhic2(x - h 3) + / coshK2(x - h3)J (2.6 le) H yX = - j T J coshK1(h3 + h 2 + h, - x) (2.61 f) = - j R k :6A co sh K 6 (x + h 6 + h 5 + h 4 ) h h H yS = - j R k-3 B sin h K 5(x + h 5 + h 4) + C c o sh K s (.r + h 5 + h4) (2.62a) TE. h Hy* = ~ J R K r4 (2.62b) h D sinhK 4 (x + h 4) + E c o s h K 4 (x + h 4) h h H y3 = - j R k .3 F sinhK-,x + G coshK,x h (2.62c) (2.62d) h H y2 = - j R Kja H sinhK2(x - h3) + I coshK; (x - h3) (2.62e) H yi = j R K :lJ coshic^hj + h 2 + h t - x). (2.62f) The total field is the sum of the TM and TE components and for E v becomes „ ~ TM _ TE E ■= E ■ + E ■ y> y> y> (2.63) for i = l,2,...,5,6. Finally, the space-domain (x,>/) representation o f the fields can be obtained for plotting purposes using the inverse Fourier transforms of (2.5) or (2.44) depending on whether the lateral waveguide dimension is infinite or finite, respectively. E. Characteristic Impedance Derivation In microwave circuit design, knowledge o f the characteristic impedance of the transmission line is critical especially in the development of matching networks or antenna systems. A wide range of impedances is also desirable for various networks and an easily achievable 50Q impedance is required to interface with network analysis measurement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 equipment. The SDM can be extended to calculate the frequency dependent characteristic impedance of the CBCPW. For pure TEM structures, the voltage and current in the transmission line model have the same meaning as the voltage and current in a circuit representation. The characteristic impedance (Z) is unique. The three possible definitions for the impedance give the same result and are defined as the power-voltage, power-current, and the voltage-current respectively [43] as (2.64a) (2.64b) (2.64c) where the magnitudes o f V and / are the voltage difference between the conductors and the total longitudinal current on the center signal conductor (S) respectively, and P is the time-average complex Poynting power flow in the z-direction. For non-TEM (hybrid-mode) structures such as CBCPW, the characteristic impedance calculation is more difficult and is not unique. This is due to the fact that, unlike a TEM transmission line, for a non-TEM transmission line ambiguous voltage and current waves exist. The voltage cannot be uniquely defined independent of path for a non-TEM line. Continuity of power across the transition from a TEM to a non-TEM line is always valid. For a general transmission line the definition o f Z may depend on how the transmission line is fed [44], For the dominant CBCPW mode, the exciting source is two voltage sources oppositely directed across the slots as in Fig. 4(a) and hence definition (2.64a) would seem to approximate the characteristic impedance most closely. Also at lower frequencies, the three definitions of (2.64) should correspond to the quasi-static (zero-frequency) value. The current I in the characteristic impedance calculation is defined as [25] S/2 1 = 2 \ J : (0,y)dy 0 (2.65) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 where •/_ (O^y) is derived from the inverse Fourier transform o f (2.28b) and using either (2.5) or (2.44) depending on whether the lateral dimension is infinite or finite (packaged waveguide). Careful consideration in the integral evaluation must be applied near the edge o f the conductor as the current experiences an unbounded or singular characteristic. The voltage V in (2.64a) and (2.64c) corresponds to the negative of the path integral of the electric field across the slots for the CBCPW and is stated for the right hand slot as V= and E S/2+W J E y(0,y)dy S/2 (2.66) is taken from the basis function expansion o f (2.35) for m= 1,3,5,... is an odd function across the slots for m=2,4,6........... After only, since Ey applying the substitution method, the integral of (2.66) becomes M (m- 1)n v = - £ c mj 0 m=\ (2.67) with m=l,3,5,... . The propagated power P can be found as the integration over the cross-section of the CBCPW of the Poynting's vector projected onto the longitudinal z-direction and for the infinite-width lateral case is given by P = Re where z h3+h2+hi OO j ( (E xH *).z^* -(h 6+h5+h4) -°° (2.68) is the unit vector in the z-direction, E x H* is a time-averaged quantity cross-product, the asterisk (*) indicates complex conjugate, and Re ( ) represents the real part of. By applying Parseval's theorem o f (2.31) to (2.68), the integral is expressed as J (ExH*) • zdydx = CS oo h3+h2+hi J j K (_E x H*J • z dx dky (2.69a) -(h 6+h5+h4) 00 = t 'Z j -oo [-^ h 6 + ^h5 + ^h4 + ^*h3 + ^h2 + ^h l] ^y (2.69b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 where CS is the waveguide cross-section and _ (hs+h4) P* = J -(h 6+h5+h4) P - lu = J P 70a) . . (2.70b) -(h 5+h4) 0 P h4 = J[Ex4H*y4 -h 4 h3 , Ey4H*4)d x (2.70c) . P * = J (.2 ,3 * ;, - Ey3H*3)itc 0 h 3 + i> 2 = J . (2.70d) v (2.70e) h3 h 3 + h 2 + h ^*hl “ / ' ( X i# ? , - £ , ! # : ■ ) * h3+h2 (2 70f) where the field components are stated in (2.55a), (2.55d), (2.59), (2.60), (2.61), and (2.62) and the final results for (2.70) are presented in Appendix C. v e -v h ¥ and ¥ From section C where are even and odd respectively from (2.10), (2.55d), and (2.55e), it is clear to see that H x and H y are odd and even Sanctions, respectively. For packaged CBCPW, Parseval's theorem o f (2.46) is applied to (2.68) and (2.69a) becomes . , oo h3+h2+h, j( E x H *) • idycbc = — 2 £ J (jE x s i=0 -(h 6+h5+h4) _ H*J • z dxdkyi (2.71) and utilizing the (2.45) definition for the spectral variable with kyi the calculations of (2.70). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 F. Analysis of Microstrip and Finite Ground Plane CBCPW The results o f the analysis applied to microstrip are used to validate the procedure for calculating the resonant frequencies o f the three-dimensional SDM for CBCPW design examples in Chapter VI. The finite ground plane (FGP) CBCPW [27] is an alternative structure in which the ground conductor widths do not extend to the edge o f the substrate (WG of Fig. 13 and Fig. 3) and can be viewed as two sets of coupled microstrip lines. This type of waveguide is utilized in MMICs where high line density is required to maximize circuit functions and minimize costs. These two structures are depicted in Fig. 14. Also, note that the ground planes at x=0 (WG) for the (FGP) CBCPW are not connected electrically to the conductor-backed plane at x=-h6-h5-h4. The analysis o f the above waveguides can be performed using the SDM with modifications in the process outlined earlier. First of all, these transmission lines are fundamentally different from the conventional CBCPW of Fig. 13. Now the strip configuration (total strip area) is significantly less than for the conventional CBCPW and therefore it would be more efficient to expand the basis functions as currents on the strips as compared to the electric fields in the slots. This is reflected by modifying (2.25) at x=0 by taking the inverse of the admittance matrix as A ‘V E. A -1 h Yzz ' "V (2.72) where the dyadic spectral domain Green's function become an impedance matrix as (2.73) i -y ys l i i ' I A ^ i t& s11 Y zz Zyy 1 A V -1 A and the elements can be derived from Appendix A. The current basis functions for microstrip are zero only outside the strip area ([y|>S/2) as shown in Fig. 15(a) and can be expanded in the spectral domain as M „ J z ~ 2 c mJzm(ky) m= 1 N Jy~ 2 d nJ ,.n(k ) . (2.74) n= 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 14. Additional structures analyzed using the SDM with air above the conductors at x=0. (a) Microstrip waveguide, (b) Finite ground plane CBCPW. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 Substituting (2.74) into (2.72) yields (2.75a) m= 1 M n= 1 „ N (2.75b) and the Galerkin procedure is again applied to produce M „ „ N (2.76a) ~ M „ _ N (2.76b) The left hand sides o f (2.76) are zero using Parseval's theorem o f (2.31) and a homogeneous system o f equations is now formed to solve for the complex propagation constant y. The dielectric losses due to the presence o f dissipation in the dielectric media can be taken into account by the introduction of a complex relative constant (2.77) where tan 8- is the loss tangent of the dissipative dielectric region and (2.77) would be substituted into (A. 12), (A. 13), and the k wave number calculation. For the FGP CBCPW, three strips are involved and the current expressions are written as Jy(y) = JyA O) + Jy,2(y) + J y3 0 ) (2.78a) d : (y) = ^ ,iO ) + J , 2(y) + J =,i(y) (2.78b) where J , (y), Jy 2 (y), Jy 3 (y) and J. , (y), J:2 (y), J, 3(y) are the components o f the currents on the first, second, and third strips respectively (from left to right), and are zero outside the strip regions. The above equations are now in appropriate form and each of the unknown elements in these equations may be expressed in terms o f a set of suitable Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 l.OL 0.5 -0.5 - 1. 0, y (mm) (a) 1.0 r 0.5 -0.5 1.0 - --2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 y (mm) (b) 0.5 M S -0.5 - 1.0 y (mm) <c) Fig. 15. J „ idealized current components for the waveguides under analysis, (a) Dominant mode for microstrip with S=0.635mm. (b) Coplanar waveguide mode for the FGP CBCPW with S=0.635mm, W=0.3175mm, WG=0.635mm. (c) Coplanar-microstrip mode for the FGP CBCPW. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 basis functions. In expanded form these equations become J :(y) R S T = £ a rg r(y) + £ b s hs(y) + i C, i,(y) r= 1 5=1 t= 1 (2.79a) J y(y) U V = £ d J J y ) + £ e v*v(y) u= 1 v=l (2.79b) W +£ f j w(y) W=1 where the basis functions hs(y) and kv(y) are both equal zero for.y outside the center strip ([y| >S/2) and the functions g r(y), it(y), j u{y), IJy) are zero outside the ground strips (| S/2 + W| < \y\< |S/2 + W + W g |. Due to the symmetry consideration o f these structures, a magnetic wall can be placed at y =0 which corresponds to even modes for the dominant current component J. . For the packaged structure in Fig. 14(b), (2.45) will be invoked. Again, the basis functions for the currents on the strips include the singular behavior o f the magnetic field components normal to the stripline edge. For microstrip, the following basis set is utilized [18] cos [2(/w-l)7ry/S] ) = iw=l,2,... (2.80) « = 1, 2 ,. (2.81) ,/i-(2y/S)2 sin [2«7ry/S] Jyn(y) = /M W and note that these expressions only exist over the strip and are zero elsewhere (y > | S/21) 2 and are scaled by a factor of —r-. Also, the boundary condition at the magnetic wall 7lu (tangential magnetic field H. is zero at >=0) is incorporated into Jy . The idealized current representation in Fig. 15(a) corresponds to m=\. The Fourier transforms of the above basis set are Jzm&y) = Ji JyM y) = 7 J. Sk v + Sk v ( m - + nit l) 7 t Sky ~1 Skv ) '- H (im - l ) 7 t - tm (2.82) (2.83) where J0 is the zeroth-order Bessel function of the first kind. For the FGP CBCPW, the basis functions are derived from [45] with (2.80) and (2.81) utilized for hs(y) and kv(y) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 respectively in (2.79) while g (y) = Ml/Wo] ^ l-[2 0 + M )A V (j]J l,<y) = (2.84) (2 85) y i-[2 (y -W )A V G]2 = s in ^ t^ /W o L ^ l - [ 2 ( y + W )A V G] 2 / w(y) = fn t^ Q ^ y W o ], (2 g7) l-[2(y - M>)/WG]2 2 where AZ»=S/2+W+Wn/2. All of the relations above are scaled by a factor o f . Since ° ttWG the ground conductor planes at x=0 (WG) are not electrically connected to the conductor-backed plane at jc=-h6-h5-h4 , an additional even mode exists which also has a zero-cutoff frequency. This mode is referred to as the coplanar-microstrip (CPM) or parallel plate one as in Fig. 15(c) and the field characteristic has already been plotted in Fig. 4(c). The fields o f this mode are concentrated between the conductors at x=0 and x=-h6-h5-h4. The coplanar waveguide mode for this structure as shown in Fig. 15(b), has been depicted in Fig. 4(a) and again the fields exist primarily across the slots. The idealized current representations in Fig. 15(b) and Fig. 15(c) are for R,S,T=l in (2.79). G. Spectral Domain Method Numerical Analysis and Results In order to solve the complex propagation constant y and the expansion coefficients of the electric field basis functions o f (2.27) to plot the field patterns for the CBCPW, numerical routines must be invoked. These routines include the polynomial approximation for the Bessel functions, evaluation o f the integrals in (2.33), calculation of the determinant of the characteristic matrix [A] in (2.34), the implementation of a root searching algorithm, and the determination o f the expansion coefficients using a least mean squares method. An explanation of these numerical routines is presented in Appendix D. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 A detailed flowchart representation o f the SDM applied to the infinite-width CBCPW of Fig. 11 is presented in Fig. 16. The parameter (IN) from the inputs box refers to the number of basis functions used to perform the simulation. The program consists of three main routines: ASYMP, GAMMA, and SOLVE. T h e ® mark continuation point in the figure will be described in Chapter III. The program was written in Fortran with a Microsoft™ 5.0 compiler using double precision complex variables and simulated on a PC Pentium 90MHz system. Numerical results are presented in Table I and compared with [46] which implemented a full-wave analysis of multi-layered coplanar lines based on a hybrid/SDM approach. A common parameter utilized by microwave circuit designers based on the propagation characteristics of a transmission line is the effective dielectric constant defined as R2 R2 s <# T n = 7T k\ = -co2|a0s 0• (288) The hybrid mode of propagation along the CBCPW leads to a dispersive medium. As the frequency increases, the fields become more concentrated in the region beneath the strips. Since the fields are forced into the dielectric substrate to an increasing extent as the frequency rises, the effective dielectric constant increases with frequency. As Table I indicates, good agreement exists between these two numerical methods. Unless noted, all examples in this dissertation will refer to the propagation characteristics o f the dominant odd mode of the CBCPW from Fig. 4(a) and utilize a total of 5 basis functions (A/=3 and N=2) for (2.27). In the table, presented work refers to the results o f this dissertation. Table II shows the convergence of the effective dielectric constant and compares the sinusoidal basis functions of (2.37) and (2.38) with the Chebyshev functions of (2.40) and (2.41). A value of 7.08 was predicted by [46] for this structure. The number of basis functions is M - N where N=M-1 for the number of E. basis functions. The selection of the edge-corrected sinusoidal basis functions (used in this dissertation ) is verified. An example of the integration convergence' for the complex propagation constant y for the waveguide of Table II with dielectric loss is provided in Table III. INT PTS and ITR # refer to the number of integration points within each subinterval and the iteration value Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 Q tarQ inputs ' £ r l » ®t2 > P r3 > >®i*5 - h , . ^ , h3.A , h5 h* ‘ ° U ~2, ?4’ 59 6 dielectnc loss tang - S, W, IN (# basis mts) CALL ASYMP - calculates the asymptotic SDM integral o fjp .9 ) with upper limit for lower mtegral region CALL INITIAL - 3 initial guesses of y for root searching routine Muller's Method (D.25) CALL GAMMA -CALLGAULEG SDM integration procedure - CALL DARA SDM dyadic Green's functions o f (2.22-2.24) - CALL DARB calculates Fourier Transforms of basis fills o f (2.37-2.38) - CALL DARC computes [A] m atrix of (2.33) - CALL LUDCMP finds determinant o f (2.34) - update y using (D.24) CALL SOLVE - write v - solve for basis fht coeffs of(2.32) using has y converged? increase # integration pts SVDCMP. S \ Q I e n d j^ Fig. 16. Flowchart for computing the propagation constant o f the infinite-width CBCPW. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 within the root searching algorithm (Muller's method) respectively. The units for (3 and a are rad/mm and nepers/mm, respectively. TABLE I Comparison of the effective dielectric constant for an infinite-width CBCPW with Ref. [46] forerI= l, sr4= 13, erS= l, h^lO m m , h4=0.75mm, h5= 10mm, h2=h3=h6=0, S=2mm, W= 1.5mm. f(GH z) 5 Seff 10 15 20 Ref. [52] 4.2 4.6 5.1 5.7 presented work 4.29 4.67 5.16 5.73 TABLE II Comparison of the effective dielectric constant with different basis function for the example of Table I with S=0.6mm, W=0.45mm, f =20GHz . Seff 1 3 5 7 9 sinusoidal 6.921 6.941 6.944 6.945 6.948 Chebyshev 6.921 6.941 6.947 6.948 6.951 # basis functions TABLE III SDM integration convergence of the propagation constant for the example o f Table II with tan 54=0.001 (loss tangent). INT PTS ITR# y 5 4 (1.1030, -5.6078D-4) 5 5 (1.1035, -5.5250D-4) 5 6 (1.1035, -5.5252D-4) 15 4 (1.1038, -5.5279D-4) 15 5 (1.1038, -5.5279D-4) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 Convergence for this structure was very fast and the simulation time was 6 seconds to execute the solution. The numerical procedure for the packaged (two-dimensional) CBCPW o f Fig. 13 follows the flowchart o f Fig. 16 with four modifications. The SDM integrals are replaced with the summations o f (2.46). The integration loopback about the routine GAMMA is eliminated. The calculation of the Fourier transforms o f the basis functions (routine DARB) can be performed outside the GAMMA routine iteration since the spectral variable values are known and fixed and the Fourier transforms are not a function of y. The presence of the lateral sidewalls can now support waveguide modes within the CBCPW. If the lateral dimension is electrically large and the operating frequency is sufficiently high, a number of waveguide modes can propagate within the structure. To account and identify all of these modes, a bisection routine (BISECT) is used to bracket these solutions and replaces the routine INITIAL in Fig. 16. This routine proceeds by dividing up the difference between upper (emax) and lower ( s ^ ) dielectric constant values by an arbitrary integer (MP or mesh points) and generating a number of fictitious dielectric constants and determinants o f the characteristic matrix o f (2.34). The imaginary part of the determinant value is inspected for the phase constant o f the complex propagation constant. Dielectric losses contribute to the real part o f the determinant. The determinant vanishes at propagating mode solutions of the CBCPW and the determinant sign is examined to bracket and identify these roots. The identification and solution o f the fundamental modes in Fig. 4 by the bisection approach is straightforward. The determination o f the waveguide modes supported by the packaged CBCPW is more involved and requires substantially more mesh points to solve. Muller's method is utilized to accelerate and locate the actual complex roots and the third initial guess value for this method is the bisected value of the two endpoints. The flowchart for computing the propagation constants for a packaged CBCPW is included in Fig. 17. The parameter (A) from the inputs box refers to the dimension o f the lateral width of the package. The convergence criterion for the asymptotic summations and the propagation constant is the same as for the infinite-width case. The routine ASYMP is applied in an identical manner, determining the asymptotic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 C starQ inputs ? rl > ? r2 I 1? r 3 > ®r4> > £ rfi h , , ^ hj h4 h5 h, °U§2> ° jj 84, 6,, 06 dielectnc loss tang S .W .IN .A ^ .e ^ .M P l CALL ASYMP - calculates the asymptotic SDM summations CALLDARB - calculates Fourier Transfroms of basis &ts (2.37-2.38) CALL BISECT - bracket/identify possible mode solns. for y CALL SOLVE - writev - solve for basis frit J C END CALL GAMMA - CALL GAULEG SDM summation procedure -CALLDARA SDM dyadic Green's functions o f (2.22-2.24) - CALL DARC computes \A] matrix o f (2.33) - CALL LUDCMP finds determinant of (2.34) - update y using (D.24) all y's found? Fig. 17. Flowchart for computing the propagation constant of the packaged CBCPW. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 contributions of the matrix equations, calculating the upper summation limit, and predicting the range for the lower summation following (D.9). The two-dimensional SDM is verified in Table IV and Table V using the results of [21] for air suspended coplanar waveguides. Table IV analyzes the dominant CBCPW mode and Table V presents the first two harmonic modes (waveguides modes). Agreement between the two different procedures (moment method and SDM) is demonstrated. The simulation time for the two-dimensional example of Table III (A=10mm) was 4 seconds. TABLE IV Comparison of the effective dielectric constant for packaged CBCPW with Ref. [21] for erl=l, er4=9.6, er5=l, h,=3mm, h4=lmm, hs=3mm, h2=h3=h6=0, S=2mm, W=lmm, A=7.5mm. presented work* refers to an infinite-width structure with the same parameters for comparison purposes. S eff f (GHz) 1 10 20 30 40 3 .6 4 .3 5 5 .9 6 .7 presented work 3 .7 1 4 .2 9 5 .0 2 5 .8 2 6 .6 2 presented work* 3 .7 5 4 .3 1 5 .0 3 5 .8 3 6 .6 4 Ref. [2 1 ] TABLE V Comparison of the effective dielectric constant for the example of Table IV with Ref. [21] at f =30GHz for the various modes. e eff modes fundamental 1st harmonic 2nd harmonic Ref. [21] 5.82 4.37 3.03 presented work 5.819 4.366 3.025 An example o f the cross-sectional vector magnetic field plots (Hx and Hv) is depicted in Fig. 18 using (2.44), (2.45), (2.55d), (2.55e), (2.61), (2.62) and Appendix B. The plot Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 0.9 0. 6 , 0.3 0.0 £ -0 .3 0.6 - - 0 .9 - 1.2 1-5 - 1 . 2 - 0 . 9 - 0 . 6 - 0 . 3 0.0 0.3 0.6 y 0.9 1.2 1.5 (m m ) Fig. 18. Cross-sectional vector magnetic fields plot for packaged CBCPW. srl= l, er4=10.2, sr5=2.2, h,=3.73mm, h4=0.635mm, h5=0.635mm, h2=h3=h6=0, S=2W=0.635mm, A=12.5mm, f =5GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 is the real part of the magnetic field components and verifies the boundary condition of the magnetic wall at y=0 for the CBCPW dominant mode (tangential components of the magnetic field along a magnetic wall are zero). Hx (xj^O) vanishes. The field arrow lengths are scaled to the largest field component. The convergence criterion for the field plots is the change in the real and imaginary parts less than 5% for increments in the spectral variable kyi of 50 with a minimum o f 250 spectral terms. Evaluation o f the field components localized around the source for the CBCPW (electric fields across the slots) require more spectral terms for convergence than for removed points. The characteristic impedance calculations using (2.64)-(2.71) and Appendix C are presented in Table VI for comparison purposes with [43] which analyzed the CBCPW with a variational conformal mapping technique. Again, the power-voltage, voltage-current, and power-current definitions are included. The worst case difference between the two techniques is 1.3Q. TABLE VI Comparison of the characteristic impedances for packaged CBCPW with Ref. [43] fo rsrl= l, sr4=13, sr5=l, h[=5mm, h4=lmm, h5=5mm, h2=h3=h6=0, S=lmm, W=0.4mm, A=7.5mm. Z (fi) 2 10 20 Zpv Ref. [43] 46.5 45.5 42.9 Zpv presented work 45.5 44.2 42 ZPI Ref. [43] 45.9 45.9 43.7 45 45.2 43 Zv, Ref. [43] 45.2 46.3 44.6 Zy, presented work 44.9 46 43.7 ffeq (GHz) ZPI presented work The procedure involved to evaluate the current in the characteristic impedance calculations is complicated. The singularity in the current at the edge of the center conductor at ,y=S/2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 as in Fig. 15(b) is determined by dividing the space-domain integral into two regions. The second integral being confined directly at the conductor edge. The variation in the cos fit y) summations for the SDM field plot calculations (H ) ranges as G(x,y,kyi, y ) -= — yjKy, where G(x,y,kyj ,y) is the corresponding product including the Green's functions while the summations for the propagation constant o f (2.33) varies as G(x,y,k ,y) j - . Kyt This translates to a significant number of additional spectral terms especially near the x=0 plane. A convergence check with respect to the number of spectral terms (/') in (2.45) is demonstrated in Table VII for the effective dielectric constant and the characteristic impedance (power-voltage definition). Each o f these parameters converge quickly. TABLE VII Convergence results for the effective dielectric constant and impedance in terms of the number o f spectral terms for packaged CBCPW with erl=l, er4=10.2, sr5=2.2, h,=5mm, h4=0.635mm, h5=0.635mm, h2=h3=h6=0, S=0.254mm, W=0.89mm, A=12.5nun, f =20GHz. # spectral terms ZpV S e ff ( ^ ) 25 50 75 100 125 150 175 200 296.1 99.8 98.2 97.6 96.7 96.2 96.1 96 5.3 5.306 5.31 5.311 5.31 5.309 5.309 5.31 The expansion coefficients for the basis functions from (2.26) for the example of Table VII are presented in Table VIII for M=3 and N=2 with Cj=1.0. Results o f the SDM for the finite ground plane (FGP) structure of section F is confirmed with [47] in Table IX using a total o f 3 current basis functions on each strip for Jz and Jy . The coplanar-microstrip mode is designated CPM (from [27] and CBCPW is the coplanar waveguide mode. Agreement between these two procedures (SDM and full-wave space domain) is excellent. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 TABLE VIII Basis function expansion coefficients for example in Table VII for M=3 and N=2. coefficients C! (1.0+/0) C2 (0.414+/1.0D-10) (-0.165+/7.92D-11) (9.5D-11+/3.76D-2) (9.5D-ll-y3.22D-3) d2 TABLE IX Comparison o f the effective dielectric constant for FGP packaged CBCPW with Ref. [47] for sri=l, er4= 10, er5= l, h,=10mm, h4=lmm, h2=h3=h5=h6=0, S=lmm, W=lmm, WG=lmm, A= 12.5mm. £eff 2 4 6 8 10 CPM Ref. [47] 7.75 7.94 8.125 8.31 8.5 CPM presented work 7.74 7.97 8.17 8.33 8.49 CBCPW Ref. [47] 5.83 5.88 5.92 6.02 6.11 CBCPW presented work 5.84 5.87 5.93 6.01 6.11 f(GHz) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 CHAPTER HI ONE-DIMENSIONAL ANALYSIS AND DESIGN OF CBCPW INCLUDING LEAKAGE EFFECTS Despite the many attractive features o f CBCPW, design problems exist in the transmission line. In this chapter, the one-dimensional infinite-width CBCPW is analyzed with the ground planes not connected to the conductor-backed plane as in Fig. 11. The problem with a single-layer CBCPW (h2=h3=h5=h6=0) is the leakage of power from the dominant mode to a zero-cutoff, TEM parallel plate mode in the homogeneous substrate region. This phenomenon produces nonconventional losses and can generate undesirable coupling effects and an ineffective circuit. This leakage (attenuation) effect unconditionally occurs for the dominant mode of CBCPW at all frequencies for single-layer, infinite-width, and isotropic substrates. Leakage can occur for microstrip when the substrates are isotropic at higher frequencies and may exist for anisotropic structures [48], Similar leakage loss from the dominant mode can occur at higher frequencies for conventional CPW to conductor-backed dielectric slab modes [49], The SDM can be extended to calculate the leakage loss for the CBCPW. Three configurations exist to reduce the leakage rate or to eliminate it altogether up to a frequency (critical frequency) in CBCPW. The first method requires reducing the circuit cross-sectional area (S+2W) relative to the dielectric thickness or increasing the substrate thickness itself [12], However, in MICs the available line dimensions can be restricted and increasing the dielectric thickness may excite additional parallel plate modes that can enhance the leakage problems. An alternative method is using substrates with a low relative dielectric constant (sr =2.33) but this will increase the overall size o f the circuit as the effective wavelength is increased. A third configuration is the use of a multi-layered Part of the data reported in this chapter is © 1992 EEEE. Reprinted, with permission, from IEEE Microwave and Guided Wave Letters, vol. 2, no. 6, pp. 257-259, 1992 (see Reference [13] and Appendix G). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 CBCPW [13] to modify and control the propagation characteristics of the dominant mode or the parallel plate modes to extend the critical frequency and bandwidth of operation for the intended application. Both lower and upper dielectric loaded configurations are analyzed and presented in this dissertation. The upper loaded structure includes the placement of the substrate on top o f the ground plane (h3). The lower loaded configuration inserts the dielectric between the existing layer and the conductor-backed plane (h5) forming a inhomogeneous medium between the parallel plates. For this waveguide the first order parallel plate mode is non-TEM and has a cutoff frequency. This method places little restriction on the circuit cross-sectional area but may on the dielectric thickness, depending on the frequency o f operation. The SDM is employed to predict the critical frequency (fcrit). In this chapter, experimental and theoretical data are presented demonstrating the leakage effects and indicating that the loss mechanism is not due to dielectric loss. A physical description of the leakage behavior within CBCPW and other printed-circuit transmission lines is presented along with the mathematical extensions to the SDM for the calculation of the leakage rate. Leakage curves for various waveguides are presented and verified with other published results. Design examples of multi-layered CBCPW along with measurement data to correlate the model are introduced. The shortcomings o f the one-dimensional model for the CBCPW are discussed with regard to the accuracy o f fcrit and explanation and prediction of the resonance effects in the experimental data. A. Experimental Demonstration of the Leakage Effects in CBCPW As a precursor to the development o f CBCPW for the proposed active antenna array system operating at 3 5GHz, an in-house test fixture was developed to characterize the transmission line and circuit elements. The infinite-width CBCPW was approximately modeled with the physical structure depicted in Fig. 8(a) without the wrap-around conductor connection (shorting sidewalls) along the length at _y=±A. The perpendicular interface to the substrates at _y=±A is air. The ground planes are connected to the conductor-backed plane only at the input and output ports and the open lateral sidewall Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 dimension is extended sufficiently away from the coplanar slots (A=19mm which is over five wavelengths at 20GHz for a dielectric susbstrate with sr=10.8) as not to interfere with the dominant propagating mode which should be confined to the slots area. Figure 19 presents the experimental results for single-layer CBCPW (refer to Fig. 11) for both high (CASE G) and low (CASE H) dielectric constant substrates. CASE G corresponds to a 50Q through line and CASE H represents a 95Q through line (notice the standing wave transmission response). The high impedance line of CASE H was necessary so that the cross-sectional circuit area matched with the coaxial connector dimensions. In this dissertation alphabetically marked cases refer to experimental data and numerically marked cases depict modeled simulation results. The low frequency, single-layer CBCPW impedances were obtained using a quasi-TEM, closed-form relationship from [50]. Problems for these transmission lines exist above 3 and 10GHz for CASE G and CASE H, respectively. These results were totally unexpected and were initially assumed to be connector or grounding problems. However, additional data confirmed this phenomenon as repeatable and had not been previously reported. All o f the line parameters were varied to minimize these experimental effects but these changes still did not produce a waveguide with any appreciable bandwidth. Such attempts included a structure with a small circuit cross-sectional area (S+2W=0.75mm) compared to the dielectric thickness for the 6010 Duroid™ (h4=2.54mm) as proposed in [12]. The photolithography limitation in the circuit etching equipment in our lab was 0.25mm for a slot or line width. The transmission line (CASE G) exhibited high loss and included sharp resonance effects. The resonance effects were not as pronounced for the low dielectric case. The data was affected by changes in the transmission line length (z=2B) and width (y=2A) as well as the placement of absorber material at the substrate sides at ,y=±A. By reducing the length or width, the basic CBCPW response was similar for CASE G but the resonance effects were spaced further apart. The effects of the absorber material would smooth out the loss response and reduce the sharpness of the resonances. This was interesting as the CBCPW response was a function of the boundary conditions enclosing the structure. CASE 1 in Fig. 19 is the SDM dielectric loss calculation o f CASE G indicating that some other loss mechanism was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 C AS E CD "O C A S E 1 G 00 CASE H o < 0 4 8 12 16 20 24 28 32 36 FR E Q ( G H z ) Fig. 19. Experimental data of the leakage effects in CBCPW MIC with open sidewalls and h,=x 2B=38mm, S=2W=0.508mm, W0 = 18.492mm, e.|=s.,=sr3=ef5=ero.:=l, (open structure), h,=h3=h5=h5=0. Refer to Fig. 11 for the dimension parameters. CASE G 500. line sr4=ri0.8, h4=0.635mm, CASE H same as CASE G except h=0.71mm and sr4=2.33 for 95f2 line, and CASE 1 is the numerical representation for the dielectric loss o f CASE G. All cases are referenced to 0 dB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 contributing to these measurement effects. The loss tangent (tan 54) for the 6010 substrate at 1, 5, 10, 15, 20, 25, 30, 35, and 40GHz is 1.4, 1.7, 2.1, 2.6, 3.2, 3.8, 4.5, 5.2, and 5.9 x 10‘3, respectively. The dielectric loss calculation for CBCPW will be verified with an example later in Chapter VI. The phenomenon existed within the substrate region and could be initially explained in the one-dimensional case as leakage from the dominant CBCPW to parallel plate modes [26], B. Explanation of Leakage Effects in Printed-Circuit Transmission Lines On a uniform length printed-circuit transmission line, it is assumed that the dominant mode (source excited mode) is purely bound to the strip area for microstrip and to the slots for coplanar waveguides. For lossless structures, the propagation constant o f the dominant mode should be real. Under certain conditions, microstrip, slotlines, coplanar stripline, CPW, and CBCPW will all leak power above some critical frequency. These leakage effects typically occur at higher frequencies and are usually assumed important only for millimeter-wave integrated circuits. However, this is not a proper assumption for CBCPW MICs. A leaky dominant mode associated with a printed-circuit transmission line has electromagnetic energy which is not entirely confined to the strip or slot regions and this mode is no longer bound in this region (flow of power in the substrate away from the strip/slot region). The performance of a leaky transmission line is significantly different than that described by ideal analysis as the line would be strongly dependent on the surroundings even far from the guiding slots (in a two-dimensional configuration). Leaked energy propagating in an integrated circuit results in undesirable and possibly catastrophic cross-talk (unintentional coupling between transmission lines within an integrated circuit). The leakage loss or rate of the dominant mode (power leaked per unit length along the guiding structure) is usually far greater than the conductor and dielectric loss. Depending on the particular geometry, power leakage can occur in the form o f a surface wave (microstrip, coplanar stripline, CPW) on conductor or nonconductor-backed dielectric slabs with one or two open boundaries, a parallel plate wave (slotline and CBCPW) on a structure enclosed by PECs, or a space wave (for configurations without a top cover Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 plate). These waves are also referred to as characteristic source free modes which are supported by the waveguide in the absence o f an excitation source and appear as poles in the Green's functions. A surface wave is a wave bound or trapped to the surface of the waveguide and decays exponentially in the direction normal to and away from the guiding structure. The leakage of power from the strip/slot region occurs in the form of the lowest surface wave or parallel plate wave from the power of the dominant mode for the waveguide. At higher frequencies, leakage to higher order surface or parallel plate waves occurs which can enhance the leakage rate. The power will radiate away from the strip/slot region in the lateral direction at an angle to the longitudinal axis and the leakage rate is dependent on the frequency and dimensional parameters [51], The propagation constant becomes complex (non-spectral) for lossless waveguides if the structure is unbounded laterally or open (infinite extent in the ^-direction). The fields of an infinite transmission line with leakage propagate and exponentially grow in the transverse (lateral) direction [31], This exponential field growth phenomenon does not occur for physically real transmission lines which are finite-length. However, once the leakage mechanism is properly understood, the undesirable effects may be eliminated or minimized and for certain applications the leakage can be used to advantage in leaky wave antennas. For the various printed-circuit transmission lines mentioned, the leakage behavior and the critical frequency are different but the fundamental physical theory is common to all the cases. For the waveguides considered in this section, the top cover plate is not present and does not affect the analysis in a qualitative sense. As an example, from Fig. 20 the top view of a slot on an air-dielectric configuration with conductor-backing is shown where this slot can represent the slot of a slotline or one of the pair o f coupled slots for a CBCPW. Assume the structure is lossless (no material or conductor losss). From the figure the transverse wave number o f the excited characteristic source-free parallel plate mode (PPM) in the ^-direction is related to the other wave numbers by 2 2 2 kyN = k CJ - k z (3.1) where kc t is the propagation wave number (real quantity) of the relevant PPMs (TEM, TM, or TE and f = l , 2 , ...,a> depending on the operating frequency) that can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 y p (a) y air (b) Fig. 20. Conductor-backed slotline leaking power to a parallel plate mode for (a) the top view and (b) the side view. The angle of leakage 0 into the parallel plate mode of wave number kc is also shown. kyp is the transverse wave number in the ^-direction o f the excited parallel plate mode and k, is the wave number of the dominant mode guided along the z-direction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 supported by the structure and k, =y is the propagation constant of the dominant mode in the z-direction. The poles of the admittance Green's functions for CBCPW from (2.22-2.25) occur at kypt . A pole is a singular point o f a function fiz ) at a value of z where f[z) fails to be analytic. The propagation of the PPM in the y-direction is . For lossless structures, k=$ (the phase constant). If p >k c t , kypt is imaginary from (3.1) and the dominant mode guided by the transmission line in the z-direction is purely bound and the field decays transversely in the j-direction away from the slot and k_ is real. However, if P < kct then kypt is real from (3.1) and as Fig. 20 indicates, power leaks from the dominant mode at an angle 0 in the form of a PPM within the dielectric region bounded by the conductors. Under leakage conditions, the leaky mode propagates slower than the dominant mode with respect to the phase velocity.. The leakage occurs within the dielectric and exists as a continuous function of frequency above fcrit (critical frequency). The leakage (radiation) peak is at an angle 0 given by where this approximation is not strictly true for complex k_ but provides a good qualitative description for small loss [52], The three-dimensional leakage phenomenon is illustrated in Fig. 21 for a leaky coplanar stripline with a pulse propagating along the z-direction and producing a semi-cone of leakage within a thick substrate. The power associated with the pulse is no longer confined to the strip area but spread out with the peak leakage angle given by (3.2). Again, the consequence of the leakage mechanism is crosstalk and interference within the microwave circuit from this propagating energy and the possible significant loss o f power from the dominant mode pulse. The condition for leakage of power to occur in transmission lines can then be written as P < k eJ. (3.3) Each mode of a waveguide has a frequency in which the leakage criterion o f (3.3) can occur and this point is referred to as the critical frequency. At higher frequencies, leakage to higher order PPMs can also be present. The attenuation constant o f the dominant mode due to leakage is cll and results in a complex propagation constant o f k=$-j<xL . Under Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. propagation semi—cone of leakage Fig. 21. Three-dimensional illustration of leakage in a semi-cone region from a pulse propagating on a coplanar stripline. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 this circumstance, (3.1) is modified as k ypj= J K j - K = JKj - p 2 + a l + 2J a L P = ± C ± j D (3.4) where C and D are constants and recall the double value o f the square root and kyp t may lie in the first or third quadrant of the complex ky spectral plane. Under these conditions, the PPM propagates and exponentially grows in the outward direction and this can be explained from Fig. 20 for a lossless structure. Points a and b receive the characteristic parallel plate wave originating from the points a' and b' along the slot following (3.2). When leakage is present, the electric field of the dominant mode on the slotline has an exponential decay along the z-direction which results in a larger field value at point b' than at a'. Hence, the field magnitude at point b tends to be larger than at point a [31], For such an exponentially growing field component, the Fourier transform in the ^-direction as in (2.3) does not exist for real values of the spectral argument k (nonspectral solution) and the implementation within the SDM will be discussed in the next section. With regard to the definition of impedance on a leaky transmission line, for a finite center strip current for the CBCPW, the total power flow across the transverse cross section is infinite. This is due to the growing field in the transverse direction. Using the power-current definition of (2.64b), the impedance is infinite. It should be noted that a leaky infinite width and length transmission line does not fundamentally violate the radiation condition which assumes a finite radiated power in the far field due to a finite input source o f power. This is not the case for an ideal infinite-length leaky line since the source power itself with an variation e ~az along the line is infinite at z = -o o . The leakage criterion for a general printed-circuit transmission line with loss is presented here. The dominant mode propagation constant is k_ = (3-y'a and that for the PPM (or substrate mode) becomes kc t = u -/t and (3.4) is rewritten as kypj= J v 2 - X2 - p 2 + a.2 + 2 j(a |3 - ux) = ± E ± j F . (3.5) Again, for leakage to exist and for the PPM to propagate and exponentially grow in the outward direction, the following conditions must all be met: (1) the pole corresponds to a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 guided PPM with propagation constant kc ( , (2) Re (K.i) > Re (k. ) (3.6a) and (3) kypt pole may only lie in the first or third quadrant of the complex ky plane R e(A :^*Im (*p , ) > 0 . (3.6b) The leakage effects will now be explained for various transmission lines including coplanar stripline, microstrip, CPW, conductor-backed slotline, and CBCPW for single-layer waveguides [12] and [51], For the coplanar stripline (CPS) structure of Fig. 21, the region under the strips fills up with the electromagnetic energy of the dominant mode and under leakage conditions, the power is contained within this region instead of being leaked away. Outside the strip region, the structure consists o f a dielectric slab which supports surface waves. The lowest order surface wave is the TE0 and has a zero-cutoff frequency. Leakage will occur at fcrit and this frequency can be extended higher by reducing the dielectric thickness. The dominant CPS mode has relatively little dispersion but the surface waves are strongly dispersive. At the critical frequency the dispersion curves for the CPS mode and the relevant surface wave mode cross each other as the phase velocities along the r-direction between these modes match (phase match). For the conductor-backed CPS, the leakage mechanism is the same as above except the lowest order zero-cutoff frequency surface wave is the TMqon the conductor-backed dielectric slab surrounding structure. When the substrate is isotropic, the dominant mode on a microstrip line may or may not leak depending on the waveguide parameters and the frequency. The leakage to the dominant TMq grounded surface wave usually occurs at higher frequencies and can be extended by reducing the dielectric thickness. Again, examination of the dispersion curves for the structure will aid in the determination. If the substrate is anisotropic, leakage to TE0 surface wave can occur and this is due to the fact that this mode is affected primarily by e x (the principal crystal axis coincides with the x-axis). For substrates with e j_> e ||, leakage into the dielectric layer outside the strip region exits since the propagation curve kc of the TE0 mode will exceed p of the dominant mode on the microstrip line at a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 sufficiently high frequency. CPW can be constructed from Fig. 3(a) without the conductor-backing at x=-h and suspended in air. The supporting structure for this waveguide is essentially the same as the conductor-backed CPS with the conductor-backing reversed and the guiding strips of the CPS removed. Again, leakage from the dominant CPW mode to the TMq surface wave mode will occur at and above fcrit. For the conductor-backed slotline with isotropic substrates (pictured in Fig. 20), the region away from the slot on both sides (y-direction) resembles dielectric-filled parallel plate waveguides. The lowest order mode is a zero-cutoff TEM (E:=H,=0). A homogeneously filled transmission line with two conductors at different potentials will support a TEM mode) which has a normalized phase constant (kc , /k0 ) equal to J e J . The normalized phase constant of the dominant slotline mode is between 1 < PlkQ < ^ s 7 since the fields are partly in air and partly in the dielectric medium. Consequently, conductor-backed slotline possesses an unconditional leakage effect independent of frequency, dielectric thickness, dielectric constant, and slot width. Fig. 22 depicts a set o f measurements published in [26] on a conductor-backed slotline. The slot was excited by a metal loop that curved over the slot and was fed by a coax line at one end and short-circuit at the other. Absorber material was placed around the edges to approximately simulate an infinite-width structure. The field was probed across the far end parallel to the >>-axis to determine the leaky wave field distribution. The leakage peak angle was calculated to be 31° from (3.2) and the experimental data approximately confirms the relationship. If the waveguide was not leaky, then the magnitude of the probed transverse field distribution should decrease away from the slot. However, the leakage loss can be reduced by increasing the slot width or increasing the dielectric thickness [31]. The CBCPW leakage mechanism is analogous to that of the conductor backed slotline. CBCPW possesses an unconditional leakage effect independent o f frequency, dielectric thickness, dielectric constant, and strip/slot width. The leakage loss can be reduced by increasing the dielectric thickness or reducing the cross-sectional circuit area relative to the dielectric thickness [12], Leakage from the CBCPW odd mode is much less Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 x (d»5mm q( IQGH z ) conductor short xixial cable 9' = 31. 0' proOe y Fig. 22. Leakage effects in conductor-backed slotline demonstrated by probing the Ex field distribution transversely cross the .y-axis. The leakage peak angle is 31.8° which translates to a maximum field component at y = ±12.6cm. Experimental data approximately confirms the predicted leakage angle from Ref. [26], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 than from the CBCPW even mode (if excited due to nonsymmetry within the circuit) because the odd mode electric fields are oriented in opposite directions and produce a partial cancellation (see Fig. 4). The propagation characteristics o f the PPMs for conductor-backed slotline or CBCPW can be accurately approximated from waveguide theory and are presented in Appendix E. Unless the CBCPW substrate thickness and circuit cross-sectional area are much smaller than the wavelength, leakage and the corresponding cross-talk cannot be analyzed using conventional quasi-TEM analysis but rather a full-wave analysis such as the Spectral Domain Method with special considerations for the non-spectral nature o f leaky modes is required. C. Spectral Domain Method Including Leakage Analysis The derivation of the SDM was detailed in Chapter II and the homogeneous set o f equations to solve for the CBCPW propagation constant were presented in (2.31) and integrated in the spectral domain as ± f j ( k y) E * ( k y)dky = 0 (3.7) J ( k y) = E(ky) Y ( k y, k :) (3.8) where is from (2.28) and E (ky) is the Fourier transform of the electric field expansion functions across the slots, J (ky) is the Fourier transform o f the currents on the strips, Y (ky , k. ) is the spectral Green's functions and the Fourier transform relating the currents at (x=0, z) to the source slot electric fields at (x=0, z), ky is the continuous spectral variable, and C is the contour o f integration in the complex spectral plane of k . For nonleaky transmission lines (fields bounded in the transverse directions), the contour of integration in (3.7) can be simply along the real k axis from - qo +jO to qo +jO which is the normal procedure. However, for leaky wave solutions the fields grow exponentially in the transverse direction from the slot area and the integration contour must be appropriately deformed in the complex k plane to account for such nonspectral modes since the Fourier transform o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 (2.3) is not defined [31], The deformation o f the integration path was not recognized in the first SDM analysis of CBCPW in [53] and no experimental measurement data was performed. By not selecting the proper integration path, the phase constant can be determined somewhat accurately but the leakage loss (coupling effects) is totally neglected. The deformation of the integration path can be identified by considering the poles associated with the Green's functions of (3.8) and (2.22)-(2.25) and can be stated in general terms as (3.9) The poles of the Green's function are related to the transcendental equations for the TEr and TMx modes supported by the source-free layered dielectric PPM structure o f Fig. 11 for CBCPW and can be written respectively as (3.10) (3.11) with m = 0,1,2,... , oo and can be solved from (A.5), (A.7), and (A. 14-A. 19) in Appendix A. For structures not bounded on the top or bottom by conducting plates, the above transcendental equations would represent surface waves. The solutions o f (3.10) and (3.11) can be verified using (E.6) and (E.8) in Appendix E. Equations (3.10) and (3.11) will have an infinite number o f roots at kc t corresponding to an infinite number o f poles in (3.9) at kypt where / is the index for the TM and TE modes. For propagating PPMs in the z-direction, Ke(kc.i)> 0 and for nonpropagating modes Im(kc l) < 0. For a single layer CBCPW, (3.10) with rh=0 solves for the TEM and the next order modes are the TM, and TE, which are degenerate modes (modes with the same cutoff frequency). For multi-layered waveguides, the first TM mode is the TMq and the first TE mode is the TE0. Table X includes an example of the propagating PPMs with effective dielectric constants greater than 1.0 for a multi-layered structure at 20GHz. The spectral integrand in (3.8) is the product of the Green's function and the Fourier transform of the electric field expansion functions. The transforms of the electric field basis functions are analytic over Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 TABLE X Identification o f PPMs for multi-layered structure of Fig. E. 1 with sr4 =10.8, sr5 =2.33 and h4=2.54mm, h5=0.71mm at f=20GHz. e eff q 9.061 TE0 5.027 TM, 2.261 TM the entire complex spectral plane as indicated from (2.37) and (2.38) and the integration contours are determined only by the pole locations of the Green's functions. The poles in the Green's functions are simple (first order) poles [60], For nonleaky transmission lines (assume lossless waveguide), P > Re(£c, ) for the PPMs for all t and from (3.4) all o f the poles {k ,) exist exclusively on the imaginary axis in the complex ky plane. Again, the Fourier transform of (2.3) exists and the contour of integration can be along the real k axis. This situation is pictured in Fig. 23 for symmetric poles P0 and P0' (recall the double value o f the square root for kyp ) with C, as the path for the spectral domain integrations for various pole cases. Note in Fig. 23 that no branch cuts exist in the complex ky plane which is always true for structures bounded on the top and bottom by conducting plates [54] as for the CBCPW o f Fig. 11. Branch cuts are due to multi-valued square root expressions for outward propagation for infinite mediums and satisfying the radiation condition. Poles P1 and P,' in Fig. 23 correspond to a lossy transmission line with Re(Arr) > Re(Arc () and hence a nonleaky condition again exists and the integration contour C, is again chosen. These poles are displaced off the imaginary axis to the second and fourth quadrants proportionally to the attenuation rate. Poles P2 and P2' were originally in the first and third quadrants o f the complex ky plane but for a sufficiently lossy waveguide (conductor or material losses), these poles have moved to the second and fourth quadrants according to (3.5) and do not contribute to leakage. Now assume leakage occurs with the PPMs corresponding to poles P3 and P3' and P4 and P4' in Fig. 23 and satisfies the leakage criteria for a lossless structure. Any integration contour can be chosen containing the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 + lm • p; Fig. 23. Contours of spectral domain integrations C s in complex kv plane. The respective poles positions P's are indicated. P0 and P'0 are for no loss case while P, and P', correspond to P0 and P'0 for a lossy waveguide. P2 and P'; correspond to P3 and P'3 for a highly lossy structure. All of these poles refer to a nonleaky waveguide and utilize the contour C 1along the real axis. P3 and P'3 and P4 and P'4 refer to leaky waveguide case and follow the integration path C3 coinciding with the real axis and deformed around the leaky poles in a residue calculus sense. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 poles at Pj and P3' and P4 and P4' above and below these poles as in C , . By further deforming C2 , a simple contour C3 that covers the entire real axis and encloses poles at P3 and P3' and P4 and P4' in a residue calculus sense is selected. A mathematical description of the residue calculus theory is presented in Appendix F. By using residue calculus, (3.7) is expressed in terms of a sum of residues evaluated along the C3 integration contour as ^ { J ( k y ) £ * (ky ) dky + j t C3 E(kypq) Y( k ypqt k:) EM(kypq) <7=1 0 ~j X E{ - kypq) Y { - k ypq, k: ) E \ - k ypq) = 0 (3.12) q= 1 where O is the total number of leaky PPMs, and in the evaluation of the integrals k v € iR [54], The contour of integration satisfies the existence and validity o f the transforms of the three components of (3.12). The second and third terms of (3.12) denote the residue contribution at ky- k ypq for the poles located in the first and third quadrants in the complex plane, respectively. Equation (3.12) is represented as an infinite sum of decaying fields (transversely) for the PPMs plus the exponentially increasing fields for the PPMs. The PPMs corresponding to exponential growth are the modes to which leakage occurs. The SDM algorithm presented in Chapter II and Fig. 16 needs to be amended to include the calculations of the leakage effects in CBCPW. The continuation mark ® in this figure corresponds to the determination of the propagation constants for the PPMs for the structure solved from (3.10) and (3.11). These solutions are identified for the TM and TE modes as zero crossings for the real part of the functions for effective dielectric constant values from 0.1 to er max and then the program utilizes Muller's method to find the complex propagation constants kc t . A check is also made that the identified pole is not a removable singularity by determining whether a zero in the function occurs at kypl . An additional verification is included such that procedure does not converge to a previous solution. The above computation is included in the routine PPMID. The conditions from (3.4)-(3.6) are implemented to determine whether the PPMs are leaky. If the poles are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 leaky modes, then the routine GAULEG is modified to include the residue calculus terms from (3.12). For an assumed lossless structure, the presence o f the leaky modes and (3.12) cause the spectral domain matrix elements o f (2.32) to become complex which translates to a complex propagation constant y. As described in Chapter II, the spectral domain integrand functions in (2.32) and (3.7) are even functions o f ky and the integration interval exists from 0 < k y <ao. ensure convergence. Near the poles, the integration subintervals are increased to The above enhancement of the SDM integration algorithm is included in routine LEAKYM. The loss tangent for one o f the dielectrics layers should be input with a nominal value for the leaky mode analysis to perturb the solution in the Muller's method to converge to a complex propagation constant. An example o f the integration convergence for the complex propagation constant y for the leaky waveguide of Fig. 19 CASE G is listed in Table XI and required 10 seconds to simulate. TABLE XI SDM integration convergence o f the leaky propagation constant for CASE G in Fig. 19 for single-layer CBCPW at 20GHz. integration points iteration # Y 5 4 (1.0552, -2.6628D-8) 5 7 (1.0487,-7.6852D-2) 5 10 (1.0459,-1.081 ID-2) 5 13 (1.0951, -4.9788D-2) 5 16 (1.0910, -6.4428D-2) 5 19 (1.0907,-6.4317D-2) 15 4 (1.0829, -6.0784D-2) 15 7 (1.0838, -5.9980D-2) 25 4 (1.0840, -5.9962D-2) 25 5 (1.0840, -5.9955D-2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 D. Numerical Results of Leakage Effects for CBCPW The SDM incorporating the leakage effects from the previous section is utilized to analyze various CBCPWs, present leakage curves for the structures, verify the mathematical algorithm, and predict some o f the experimental data results. Unless noted, all o f the numerical examples assume no conductor or dielectric loss and the waveguide parameters chosen are typical for a CBCPW MIC. The first analysis is the dispersion curves for the normalized propagation constants o f the CBCPW dominant mode and the PPMs. This will determine the frequency range o f the leakage effects or fcrjt. Again, recall that for single-layer CBCPW, leakage occurs unconditionally regardless of frequency. This is depicted in Fig. 24 for two different dielectric substrates. This figure demonstrates that k jk 0 > |3/&0 across the entire frequency-band and the leakage from the dominant CBCPW mode to the zero-cutoff TEM PPM (lowest order mode) is a continuous function o f frequency. Other PPMs are propagating within the waveguide for this frequency range but do not contribute to the leakage effects and are not shown. The unconditional leakage effects partially explain the experimental results o f Fig. 19 especially for CASE G in which this transmission line is unusable across the entire frequency-band. A multi-layered CBCPW is proposed here to extend the critical frequency and produce a waveguide with an appreciable bandwidth. Examples of multi-layered (lower dielectric loading) CBCPW are presented in Fig. 25. The placement o f a lower dielectric constant substrate (sr5) between the high dielectric constant substrate (sr4=10.8) and the conductor-backing effectively modifies the propagation characteristics of the waveguide. The lowest order PPM is now the TM,, PPM which has a cutoff frequency. The propagation constant of the PPMs is reduced below that of the CBCPW dominant mode up to some frequency. This occurs since the field pattern for these modes is concentrated between the circuit conductors and the conductor-backing as in Fig. 4(c) and the field pattern for the bound CBCPW mode exists primarily within the slot region and is not significantly affected with this additional dielectric layer as depicted in Fig. 4(a). The multi-layered CBCPW does not place any restriction on the cross-sectional circuit area but is a slightly more complex structure to manufacture. CASE 4 o f Fig. 25 demonstrates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 3.50r —T— .Q3.25 ■ 3.00 ■ o < •4> 2.75 2.50 2 .2 5 2.00 ■ -4? \ CO. 1.75- O V -9 -9 - CASE 2 CBCPW CASE 3 CBCPW CASE 2 TEM CASE 3 TEM 20 24 1.50-„V1.251.00- 8 12 16 28 32 36 40 FREQ (G H z ) Fig. 24. Normalized propagation constants for the CBCPW dominant mode (P) and the TEM PPM (kc ). One-dimensional SDM analysis for experimental data of Fig. 19 with CASE 2 and CASE 3 corresponding to CASE G and CASE H, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 o CASE 4 CASE 5 CASE 6 ~o— CASE 4 CASE 5 -X CASE 6 V X CBCPW CBCPW CBCPW TM0 TM0 TM0 o o o ■se 0 4 8 12 16 20 24 28 32 36 FREQ (G H z ) Fig. 25. Normalized propagation constants for multi-layered CBCPW dominant mode (0) and the TMq PPM (kc ) with srl=l, er4=10.8 and h ^ m m , h2=h3=h6=0 and S=2W=0.635mm. CASE 4 with er5=6 arid h4=0.635mm, h5=5mm. CASE 5 with sr5=2.33 and h4=0.635mm, h5=0.71mm. CASE 6 same as CASE 5 except h4=l.27mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 that a lower dielectric constant of sr5=6 is too high even with a thick substrate (h5=5mm) and leakage occurs throughout the frequency-band. A nonleaky CBCPW (50Q transmission line) occurs for CASE 5 for the entire band of interest. A critical frequency of 21 GHz exist for CASE 6 which is the same as for CASE 5 but with a larger substrate thickness h4. For the frequency region above fcrit, the leakage o f power continues from CBCPW dominant mode into TM,, PPM. These numerical results will be compared with experimental data in section E. The leakage analysis is verified in Table XII and Fig. 26. In this table for the slotline example, the SDM analysis from [31] is compared and the results are acceptable (see Fig. 22) for this extreme case. TABLE XII Comparison o f the normalized leakage rate for conductor-backed slotline with Ref. [31] for sr=2.25, h=8mm (dielectric thickness), f=10GHz as a function of the slot width d. a/kn d (mm) 3.75 7.5 15 22.5 Ref. [31] 0.11 0.123 0.1 0.0114 presented work 0.108 0.155 0.074 0.0112 Fig. 26 compares the calculated leakage rates with references [32] (GaAs MMIC example) and [33]. Good agreement is demonstrated for both cases (except for the peak leakage point for CASE 8) and validates the leakage results presented here. CASE 8 is an extreme example and exhibits an interesting behavior as the large cross-sectional circuit area (S+2W) compared to the dielectric thickness, behaves more like microstrip than CPW. The fields are concentrated primarily below the center conductor and do not strongly couple with the PPMs. The leakage analysis for CBCPW for this dissertation was developed prior to [32] but not published as it did not fully explain the experimental Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 0.090 0 .0 8 0 0 .0 7 0 • - - CASE v CASE CASE o CASE 7 Ref. [323 7 p re se n te d work 8 Ref. 1333 8 p re se n te d work 0 .0 6 0 o 0 .0 5 0 5 0 .0 4 0 0 .0 3 0 0.020 0.010 0.000 20 30 40 FREQ (G H z) 50 60 Fig. 26. Normalized leakage rate comparisons with Ref. [32] and [33] with srl =1 and h,=5mm, h2=h3=h5=h6=0. CASE 7 with er4 = 13 and h4=0.2mm and S=W=0.1mm. CASE 8 with er4 = 10 and h4=0.4mm and S=2W=2.0mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 results of Fig. 7. Leakage curves for CBCPW MIC examples are shown in Fig. 27. CASE 2 corresponds to CASE G of Fig. 19 and the leakage has a second-order frequency dependency. CASE G from the experimental data depicts an increase in loss and in the resonance peaks with frequency but the calculated leakage significantly overestimates this loss. CASE 6 corresponds to the results in Fig. 25 with a fcrit of 21 GHz and the leakage increases quickly beyond that point. CASE 3 corresponds to CASE H of Fig. 19 and the leakage increases linearly with frequency and is much lower than for the higher dielectric constant of CASE 2. CASE 9 represents a thick dielectric example of CASE 2 and the leakage is reduced and reaches an inflection point at 30GHz, then increases substantially. The reason for this phenomenon is the leakage to additional PPMs (TM, and TE,) above 30GHz. The leakage rate can be directly related to the field coupling or overlap of the CBCPW mode and the PPMs. Consider only single-layer structures with an air dielectric above the circuit conductors. The leakage rate increases with frequency because the CBCPW mode field pattern is concentrated more and more into the higher dielectric substrate. This is evidenced by the increasing propagation phase constant on a dispersion curve and the mode interacts strongly with the PPMs. For a lower dielectric constant substrate, the leakage rate is smaller than for the equivalent higher dielectric since a greater portion o f the CBCPW mode field pattern exists in the air above the circuit conductors and does not couple with the PPMs. For CBCPW with a large cross-sectional circuit area, the leakage rate increases as the field pattern is spread out over a larger area and more efficiently couples to the PPMs. Another analogy for the leakage rate for this example is that of an aperture antenna which for a smaller aperture (S+2W), reduces the radiated energy available to couple to the PPMs. Design examples for multi-layered CBCPW using the leakage analysis are demonstrated in Fig. 28 for various leakage turn-on parameters. Fig. 28(a) is the design of the proper dielectric thickness to prevent leakage at 20GHz. CASE 10 is for an upper dielectric loading structure with a dielectric constant o f sr3=10.8 and requires a thickness Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 0 .1 8 0.16 0.14 CASE CASE CASE CASE 2 6 3 9 0.33*ct/k, 0.12 0.10 's* 0.08 0.06 0.04 0.02 0.00 FREQ (G H z ) Fig. 27. Normalized leakage rates (o<Jk0) for CBCPWs. CASE 2 and CASE 3 are referenced from Fig. 24. CASE 6 is from Fig. 25. CASE 9 same as CASE 2 except h4=2.54mm. CASE 2 is scaled by a factor o f three. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 l O o •a* CASE 10 h j CASE 11 h5 * I 0Y— <a 1 O a OQO OJOOS 0.010 0.015 0.020 0.025 0.030 0.0 3 5 Eo 0.1 0.2 0 .3 0.4 0.5 0.6 0.7 DIELECTRIC THICKNESS (m m ) (a) CASE 12 CASE 13 (b) Fig. 28. Leakage design curves for various CBCPW examples with srl=l and h ^ m m . (a) Proper dielectric thickness at 20GHz for upper and lower loading structures with S=2W=0.635mm. CASE 10 with er3=10.8, er4=2.33 and h4=0.71mm, h2=h5=h6=0 and references the upper dielectric thickness scale. CASE 11 is the same as CASE 6 of Fig. 25 with h5 as the variable and references the lower scale, (b) Proper relative dielectric constant at 30GHz for lower loading waveguide. CASE 12 is the same as CASE 6 except with er5 as the variable. CASE 13 same as CASE 7 except h5=0.635mm. Log scales are used for the leakage rate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 of h3 >0.03175mm which is a very thin substrate. CASE 11 is for a lower loaded waveguide with a dielectric constant of er5=2.33 and stipulates a minimum thickness o f h5 >0.63mm. Recall from Fig. 25 that fcrjt was 21GHz with h5=0.71mm. Fig. 28(b) demonstrates the design o f the proper relative dielectric constant value to prevent leakage at 30GHz for a lower loaded CBCPW. CASE 12 presents er5 <3.6 as the necessary condition to prevent leakage and CASE 13, for a carrier substrate o f a GaAs dielectric from CASE 7, requires er5 <6.3. Note for all o f the examples o f Fig. 28, the leakage response has a sharp rise at the leakage turn-on parameter. E. Limitations of the One-Dimensional Leakage Analysis for CBCPW Fig. 29 lists experimental data curves with open sidewalls and A=19mm to simulate an infinite-width waveguide for multi-layered lower dielectric loaded CBCPW to compare with the leakage analysis for sr4=10.8, erS=2.33 and S=2W=0.635mm (approximate 50Q through transmission lines). CASE I corresponds to CASE 5 in Fig. 25 with h4=0.635mm, h5=0.71mm and no leakage is present (bounded dominant mode response) and is confirmed from the analysis. CASE J is CASE 6 with h4= 1.27mm, h5=0.71mm and demonstrates a leakage or unbounded dominant mode characteristic around 25GHz. Note the reflection coefficient (Sn ) for this example behaves unusually at this corresponding frequency point of the S21 response. The data includes the effects o f the connectors as the network analyzer was calibrated at the cable ends. The numerical analysis for this structure predicted fcrit of 21 GHz with a sharp increase in the leakage (attenuation) rate beyond that point. CASE K is the same as CASE J except with h5=0.381mm and shows a leakage frequency around 20GHz and the SDM calculates an unconditional leakage effect with frequency. These experimental data results and numerical confirmations were initially published as part of this dissertation from [13], As shown above, the one-dimensional SDM leakage analysis could only partially explain the measurement results. The fcrit could be predicted fairly accurately in some cases and not in others. A qualitative physical description above fcrit was represented only Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 CASE I S2i MAGNITUDE (d B ) ■ CASE J Sn CASE J S 21 10 . CASE K S 21 4 8 12 16 20 24 28 32 36 FREQ (G H z ) Fig. 29. Experimental data of the leakage effects in multi-layered CBCPW MICs with open sidewalls and 2B=38mm, srl=er2=er3=ert=l, sr4=10.8, h2=h3=h6=0. sr5 =2.33 S=2W=0.635mm, and h,=oo WG= 18.49 mm, (open CASE I is measured data for CASE 5 structure), h,=0.635mm, h5=0.71mm. CASE J is the experimental results for CASE 6 h4= 1.27mm, h,=0.71mm. CASE K same as CASE J except h5=0.381mm. Refer to Fig. 11 for the dimension parameters. CASE BC is referenced 5dB down from CASE J S21 and CASE J Su corresponds to the top scale. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 by the leakage rate and could not describe the resonance peaks in the measured data. A high leakage rate translated to a fairly strong unbounded transmission response and likewise for a small rate. Also, the measured fcrit selection is somewhat arbitrary. Other CBCPW examples not shown include h4= 1.27mm, h5= 1.42mm with measured fcrjt of 28GHz and predicted fcrit of 23.5GHz, h4=2.54mm, h5=0.71mm with an experimental fcrjt of 14GHz and simulated unconditional leakage with frequency, and h4=2.54mm, h5=0.381mm with measured fcrit o f 12GHz and calculated unconditional leakage with frequency. One possible reason for the inconsistencies for the critical frequencies could be attributed to the uncertainty of the dielectric thickness (±0.102mm, ±0.051mm for 2.54mm and 1.27mm boards, respectively using 6010 er=10.8 Duroid™ substrate and ±0.0254mm for 5870) and the relative dielectric constants (±0.25 for 6010 and ±0.02 for 5870) and the possible existence o f a small air gap (0.0254mm) between the substrates (sr4=10.8, er5=l, sr6=2.33). Including these maximum dimensional uncertainties with respect to the maximum critical frequency for CASE K, produce sr4=10.55, sr5=2.35, h4=1.22mm, and h5=0.41mm. The results of the uncertainty analysis for fcrit are summarized in Table XIII. The unmarked responses in the table are nonsimulated cases. The one-dimensional analysis utilized in this section assumes an infmite-width structure which under certain conditions can constitute a source of power loss from the CBCPW dominant mode to the PPMs in the absence of lateral confinement. However, this is an unrealistic physical structure. The substrate edges aty=±A are terminated in some manner. If the side termination does not allow leakage to occur (loss of energy from the waveguide), for instance a conducting sidewall, the outwardly traveling PPM will be reflected and a transverse standing wave will be created. This standing wave can place an admittance termination to the slot region and influence the value of the dominant mode CBCPW propagation constant [26], Assume the CBCPW of Fig. 13 with PEC sidewalls that are not shorting the ground planes and the conductor-backing and a single-layer structure. The propagating leaky TEM wave becomes totally reflected from the sidewalls and the waveguide is no longer leaky but instead the CBCPW dominant mode is purely real. If the sidewalls are shorting then the TEM does not exist and the lateral confinement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 TABLE XIII Comparison of critical frequencies up to 36GHz for CBCPW for measured data, SDM analysis, predicted with maximum dimensional uncertainties, and predicted with maximum dimensional uncertainties and air gap at h5. measured fcrit (GHz) predicted CASE G 3 unconditional unconditional CASE H 10 unconditional unconditional CASE I >36 >36 CASE J 24 21 — — 23.6 CASE K 20 unconditional 12 16.5 h4= 1.27mm, h6= 1.42mm 28 23.5 — 25.9 h4=2.54mm, h6=0.71mm 14 unconditional — 7.4 h4=2.54mm, h6=0.381mm 12 unconditional — 4 max uncertainty max uncertainty with air gap — — — of this waveguide gives rise to mode coupling effects associated with rectangular waveguide modes. This is the reason why the predicted critical frequencies for the examples in Table XIII are not always accurate. The leakage analysis of infmite-width CBCPW provides insight and can correlate the degree of mode coupling effects in a two-dimensional waveguide and this will be described in Chapter V. For instance, a larger leakage rate implies a stronger coupling effect with the characteristic waveguide modes. An example of the sidewall termination effects is presented in Fig. 30 for a single-layer CBCPW with shorting sidewalls along the waveguide length with the wrap around copper tape to provide a uniform ground connection within the circuit as in Fig. 5(a). This three-dimensional waveguide produces a cavity resonator substructure which explains the presence of the strong resonance peaks for the measured data as the resonant cavity modes are above cutoff. Again, the one-dimensional leakage analysis can not predict these Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 MAGNITUDE S21 ( d B ) -5 -1 0 CASE -1 5 -20 -2 5 -3 0 FREQ ( G H z ) Fig. 30. Experimental data demonstrating the resonant cavity modes within an overmoded single-layer CBCPW. CASE L with srl=sr2=sr3=er5=ert=l, er4 = 10.8 and hj=oo (open structure), h2=h3=h5=h6=0, h4=0.635mm and S=W=0.508mm, A=19mm, B=12.7mm. Refer to Fig. 13 and Fig. 5(a) for dimensional parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 cavity modes. The dominant TE. resonant cavity mode frequencies can be approximated for a single-layer rectangular cavity o f height a, width b, and length c from [9] as fe w l® + ^b) + (= ) (3 13) with rh=0,1,2,... , ^=1,3,5,... ,p = 1,2,3,... . The placement o f a magnetic wall excitation at _y=0 places a restriction on the value of the mode number ft. The identified resonant mode frequencies to 25GHz from Fig. 30 and the probable equivalent resonant cavity modes (TE01^,) o f (3.13) are listed in Table XIV. TABLE XIV Measured and predicted resonant cavity frequencies for CBCPW CASE L from Fig. 30 and for TE0>1^ modes. predicted T E o .,i 2.2 2.16 TE01.2 3.8 3.79 T E o .1.3 5.5 5.52 T E o .1 .4 7.3 7.29 T E q .1.5 9 9.07 10.9 10.85 TF 0 .1 .7 12.9 12.63 14.6 14.43 T E 0 . i .9 16.4 16.2 T E 0 . , . 10 18.1 18.02 ^E 0.i.n 19.8 19.8 T£ 0.1.12 TE0.1.13 21.5 21.61 23.2 23.39 ©3 measured o JL 00 mode on fr(GHz) The identification of the resonant cavity modes for CBCPW with sidewalls was originally observed as part o f this work [14], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 CHAPTER IV TWO-DIMENSIONAL ANALYSIS OF CBCPW W ITH LATERAL SIDEWALLS INCLUDING MODE COUPLING EFFECTS The one-dimensional infmite-width analysis of the previous chapter was not adequate to explain and predict all o f the experimental data effects within CBCPW MICs. This discrepancy was due to the finite dimensions of the physical transmission lines. The waveguides analyzed in this chapter are again illustrated in Fig. 5 and Fig. 13 with the ground planes connected to the conductor-backing along the length (r-direction) to achieve a uniform ground connection within the test fixture. The waveguide now has terminating lateral shorting sidewalls in the transverse y-direction and could also model a CBCPW placed within a package. This structure (with a small cross-sectional circuit area compared with a large lateral width) resembles a rectangular waveguide with two aperture slots to guide the electromagnetic energy. Characteristic rectangular waveguide modes operating above the cutoff frequency are excited at circuit discontinuities (coaxial connector feed) and will propagate and couple with the dominant CBCPW mode (multi-mode propagation exists). These mode interactions are different from the leaky wave situation because the coupling effects occur only at discrete frequencies instead of existing as a continuous function of frequency. Also, a longitudinal phase match condition between the CBCPW and waveguide modes does not occur. The presence o f these modes can cause the CBCPW mode field distribution to spread out across the entire waveguide width instead of being confined to the slot regions and modify the propagation characteristics. This phenomenon is similar to the leaky wave case except the propagation constant of the CBCPW dominant mode is real and the exponentially growing wave in the transverse direction is not present. The existence of higher-order waveguide modes for the Part of the data reported in this chapter is © 1993 IEEE. Reprinted, with permission, from 1993 IEEE Microwave Theory and Techniques International Microwave Symposium Digest, pp. 947-950 (see Reference [14] and Appendix G). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 two-dimensional air-suspended CPW was originally presented in dispersion curves from [21]. The field spreading within a single-layer packaged CBCPW was published in [24], however the authors did not precisely identify the modes and did not present any experimental data demonstrating these effects. According to coupled-mode theory [55], power can be exchanged between the modes along the propagation path and is dependent on the transmission line length. Energy outside the cross-sectional circuit area will not be detected by the output coaxial connector and will resonate in the finite-length cavity structure. The connector housing blocks complete the cavity at z= 0 and z=2B of Fig. 5(a). The more this energy component spreads out, the stronger the resonance effects present. To completely avoid the coupling problems, the CBCPW dimensions can be selected to cutoff the waveguide modes. This is most easily achieved by reducing the lateral sidewall separation [11], However, to operate at higher frequencies, the narrow wall separation (y=±A) drastically reduces the available MIC surface area. For practical circuits the lateral sidewalls walls must be extended allowing rectangular waveguide modes to propagate. Conventional air-filled rectangular waveguides are designed for a single mode o f operation over a designated frequency bandwidth. An example o f the possible problems due to propagating rectangular waveguide modes in CBCPW with PEC lateral sidewalls is depicted as CASE 14 in Fig. 31 for CASE G (single-layer sr=10.8) o f Fig. 19 with A=19mm. This figure presents the first 17 ideal waveguide modes (no TEM exists since the top and bottom conductors are electrically connected) following the analysis of Appendix E and Fig. E. 1 with lateral sidewalls at y=±A. The propagation constants for the nonzero-cutoff TMXmodes for a single-layer substrate are written as (4.1) where M - 0 , 1 , 2 , . . . , qo and ft = 1 , 3 , 5 , . . . , the dominant CBCPW mode). ao with a magnetic wall excitation at _y=0 (for The phase constant for the TE modes are described by (4.1) with th = 1 ,2 ,..., qo and h - 1,3,5, ...,oo and are degenerate modes with the TM modes. The TMr and TEr modes are alternative mode set representations for the waveguide and for the single-layer substrate, the TM ^ h modes are the T E .0 h [9], The wave number in they-direction becomes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 3 .6 3 .2 2.8 o •se o o -se 1.2 0.8 Ideal RW CBCPW 0 .4 0.0 FREQ (G H z) Fig. 31. Idealized dispersion curves for single-layer CBCPW with lateral sidewalls for CASE 14 and erl= l, er4=10.8 and h;=h3=h5=h6=0, h ^ m m , h4=0.635mm and S=2W=0.508mm and A=19mm. Refer to Fig. 13 for dimensions. The CBCPW mode response is from CASE 2 of Fig. 24 and superimposed onto the graph. Ideal RW are the TM0h rectangular waveguide modes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and this term is added to the left hand side of (E.7) and (E.9) in Appendix E to determine the propagation constants for the ideal multi-layered characteristic rectangular waveguide modes. The CBCPW mode response in Fig. 31 is obtained from the Re(£.) for the infinite-width case of Chapter II. As demonstrated, many waveguide modes are propagating which can couple to the dominant mode and produce the unexpected measurement results of CASE B in Fig. 7. For instance at 20GHz, 10 waveguide modes propagate with a slower phase velocity than the CBCPW mode. This chapter utilizes the two-dimensional SDM to predict the coupling effects for CBCPW with lateral sidewalls. The SDM is a full electromagnetic wave numerical method which can directly simulate these effects. The analysis includes dispersion curves to properly identify the modes, mode impedances, coupling coefficients, field plots, and the field spreading due to the mode coupling effects o f the dominant CBCPW mode. A qualitative description of mode coupling, improved explanation of the CBCPW experimental data, and the coupling effects of finite ground plane (FGP) CBCPW are also presented. The culmination of the above analysis was an original contribution and published as part of this dissertation from [14], A. CBCPW Mode Identification The SDM for the analysis of the packaged two-dimensional (infinite-length) CBCPW follows from sections C-E of Chapter II. The spectral variable kyj of (2.45) becomes a discrete parameter and the integral analysis for the infinite-width structures is replaced by spectral summations. The bisection method is invoked to properly isolate and identify the CBCPW and waveguide modes. This routine proceeds by dividing up the propagation constant values and generating a number o f fictitious propagation constants and determinants o f the characteristic matrix [A] o f (2.34). The determinant vanishes at mode solutions o f the waveguide. An example o f this procedure is depicted in Fig. 32 for CASE 15 for a lower loaded multi-layered CBCPW and will be utilized in sections B and C of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 50 CASE 15 40 (det U I ) 30 20 CBCPW spurious -1 0 £ -20 -3 0 -4 0 .390 2.398 2.406 2.414 2.422 2.430 Re ( k z /k o ) Fig. 32. Example for the variation of the SDM determinant with fictitious propagation constants at 7.5GHz for CASE 15 with srl= l, er4=10.8, er5=2.33 and h ^ m m , h4= 1.27mm, h5=0.381mm, h2=h3=h6=0 and S=2W=0.635mm and A=19mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 this chapter. The second root in the figure is a spurious solution associated with the pole of the Green's function while the proceeding root is the proper TM q , mode. This characteristic response in the determinant is repeated for the other waveguide modes. The spurious modes can usually be identified by having extremely large basis function coefficients. The determination o f the rectangular waveguide modes using the bisection method is not a straightforward procedure. The propagation constants calculated by the ideal waveguide analysis of Appendix E are a necessary addition to the SDM to locate the modes. The dominant waveguide mode for the multi-layered structure is the TMX since 2A>h6+h5+h4. The same SDM basis functions for the dominant CBCPW mode are employed for the waveguide modes since the same boundary conditions apply and the fields should fringe across the slots due to the potential difference. Field plots generated by the SDM are necessary to identify the various waveguide modes within the CBCPW and the following examples correspond to CASE 15 of Fig. 32. The vector electric field plots (Ex and Ey) within the cross-sectional circuit area are shown in Fig. 33 for the CBCPW mode and the TMX0 5 waveguide mode. The field components for all the plots are normalized to the maximum values. The full waveguide field plot for the TMq j mode is depicted in Fig. 34. The SDM field plot of Fig. 34(b) is verified by the inclusion of the fields determined from the ideal two-layered rectangular waveguide analysis between the ground planes and the conductor-backing using Appendix E, (2.1), and [9], The TM waveguide modes possess a predominantly vertical electric field orientation in the x-direction. The couple of field points at the bottom o f the figure do not exist outside the waveguide but are just scaled in order to observe the other points. The field plots for the Ex component o f the CBCPW and the first three waveguide modes are pictured in Fig. 35. The second subscript fi in T M m o d e s refers to the number of half-sinusoids (standing wave maxima) in the transverse direction. The field response of the waveguide modes is perturbed in the vicinity o f the slot regions and satisfy the boundary condition that the tangential electric field does not exist on a PEC at y=±A. With air above the circuit conductors, the waveguide mode cutoff frequencies are increased proportionally to the cross-sectional circuit area (S+2W). Also, note in Fig. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 1.6 1.2 0.8 0.4 X-N. E o.o E * Y N r* * "'’ * — v y t v — *- s / i \ \ 0.8 - - 1.2 - 1.6 — 2.0 n n n u V V * V I* n - 2 .0 - 1 .6 - 1 .2 - 0 .8 - 0 .4 0.0 0.4 0.8 1.2 1.6 2.0 l/( m m ) (a) 1.6 1.2 0.8 0.4 £ \ \ r / o.o E 0.8 - - 1.2 - 1.6 r\ \ \ \ s v < ' - ' / ' / M M T TM \ \ \ ' i >/ / / / rr t t T1 T 1 1 \ 1 t i / / ; t r i t fr f rtr i t t t 1t t i ti t ti rr r? tt ii iT t w -0.4 • I T T T I T ' T T I I T T -2.0. T T - 2 .0 - 1 .6 - 1 .2 - 0 .8 - 0 .4 0.0 0.4 0.8 y t t i i t I 1.2 1.6 2.0 (m m ) (b) Fig. 33. Cross-sectional vector electric field plots for CASE 15 at 15GHz for (a) CBCPW mode and (b) TMg 5 mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 0.0 - 0.2 -0 .4 -0.6 £ - 0.8 -1.0- -1.6 -1.8 y (m m ) (a) 1.& 0.8/"~N 0.4 E E -0 .4 -0.8 -1.2 - 1.6 1 -2.0 (b) Fig. 34. Cross-sectional vector electric field plots of CASE 15 at 15GHz for TM „, mode, (a) Ideal rectangular waveguide analysis, (b) SDM analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 CBCPW 1.0 TM0.1 TMqj 0 .8 •0— TMo^ 0 .6 0.4 Q 2 0.2 CL 0 .0 < - 0 .2 H ^ —0.4 -0 .6 - 0 .8 - 1.0 V (m m ) Fig. 35. Electric field plot of CBCPW and first three waveguides modes of CASE 15 at 15GHz and x=-0.01mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 that the CBCPW mode is confined to the slot regions at 15GHz as expected. All o f the waveguides modes discussed are propagating. The bound CBCPW mode is quasi-TEM while the waveguide modes are either TM or TE modes. B. Mode Coupling Effects in CBCPW with Lateral Sidewalls In two-dimensional printed-circuit transmission lines, classical coupled-mode interaction between the dispersion curves o f the transmission line dominant mode and those of the supporting substructure is possible. In CBCPW with lateral sidewalls these modes are rectangular waveguides modified by the slot regions. Using classical coupled-mode theory [55], the unperturbed modes associated with the two independent structures are first identified. The two guiding structures are then combined and the new perturbed modes o f the composite waveguide are described and represented in terms of the unperturbed modes of the original independent guides. This technique was presented in Fig. 31 for CBCPW. Regions o f classical mode coupling in the perturbed modes occur at and near frequencies at which the dispersion curves o f the unperturbed modes cross. The composite structure is the finite-width CBCPW placed within a package and the lateral sidewalls connect the ground planes to the conductor-backing. The unperturbed modes of the package are those associated with the rectangular waveguide in the absence of the slots, and the dominant CBCPW mode of the infinite-width structure. The perturbed modes are the propagating rectangular waveguide modes in the presence of the slots, and the dominant CBCPW mode in the presence of the package. The rectangular waveguide supports an infinite number of modes that are propagating or nonpropagating (depending on the structural parameters and frequency). The phase constants of the rectangular waveguide modes (PRW) are related to the parallel plate waveguide (Ppp) as (4.3) with ih = 0,1,2,... ,qo and fi = 1 ,3 ,5 ,..., oo . The unperturbed transmission line can be leaky and could a complex propagation constant. However, the attenuation constant associated with the leaky dominant mode is usually several orders of magnitude smaller Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 than the phase constant (see Table XI in section C of Chapter III). Under these conditions, classical mode coupling occurs at frequencies where the unperturbed propagation constant of the CBCPW dominant mode equals that o f the unperturbed package modes. The perturbed dominant CBCPW mode has a real propagation constant for a lossless structure [56], The dispersion curves for the normalized propagation constants for the multi-layered CASE 15 are calculated by the SDM for the two-dimensional CBCPW and shown in Fig. 36 and the waveguide modes (TMo,, TMq 3, ...., TM,, 17, TMq 19) are represented from left to right as the frequency is increased. Above 32.5GHz, the TM l h and TE0 h modes also propagate but are not included as this would clutter the graph. The SDM RW and the Ideal RW refer to the waveguide mode propagation constants calculated from the spectral domain method and from the Appendix E analysis, respectively. As indicated in the figure, frequencies at which the perturbed modes approach, do not cross as for the leaky wave one-dimensional case, but rather bend away from each other in a classical coupled-mode behavior. In this frequency region of coupling, the mode field distributions are dissimilar to those for the same modes on either side of this transition region and cannot be clearly identified with either mode as energy is exchanged between the modes. As indicated in Fig. 36, strong mode coupling between the dominant CBCPW mode and the waveguide modes happens at frequencies where the dispersion curves approach and have the smallest separation and occurs approximately at 9.2GHz with TMq, mode, 17GHz with TM03, 21 GHz with TM05, 26GHz with TM0J, and 31 GHz with TM09, respectively. The coupling regions are not a continuous function of frequency but are localized. For instance, the strong mode coupling region between the dominant CBCPW mode and the TMqj mode is from 7.5-12GHz. An indication o f the degree o f mode coupling between the dominant and waveguide modes can be approximated from the dispersion curves. The greater the separation between the two dispersion curves in the mode coupling region, the stronger the coupling between the dominant and waveguide mode. From Fig. 36, the mode coupling effects between the CBCPW mode and the TMq7 mode is much more enhanced than the coupling between the dominant mode and the TMq , mode. Mode A in the figure represents the top continuous dispersion curve response of the CBCPW Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 3.0 x CBCPW mode SOM RW o I d e a l RW 2.6 o 2. 4 2.0 -B 1.0 FREQ (G H z ) Fig. 36. Dispersion curves o f CASE 15 from Fig. 32 for CBCPW and waveguide modes demonstrating the mode coupling effects. SDM RW and Ideal RW refers to rectangular waveguide modes from the two different analysis. Mode A is the top continuous dispersion curve and Mode B is the second continuous curve. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 108 mode before the mode coupling region and of the TMq^ mode after the coupling. Mode B represents the second continuous curve from the cutoff frequency of the TM q , mode before the coupling at 9.2GHz and o f the CBCPW mode after the coupling. The propagation constant of the dominant CBCPW mode does not seem to be affected by the mode coupling mechanism outside the transition region across the frequency-band. The fields associated within the coupling region become a combination of the dominant CBCPW mode and the rectangular waveguide mode. The fields are distributed or spread across the entire waveguide width instead of being confined to the slot regions even though the propagation constant of the dominant CBCPW mode is real for a lossless structure (recall for the leaky wave case the propagation constant was complex). The residual coupling effects on the dominant CBCPW mode for frequencies outside the transition regions where the modes can still be identified is also important. The cumulative effect o f the waveguide mode coupling causes the dominant mode fields to spread out across the waveguide and not bound to the slot regions. The coupling efficiency between the modes determines how the dominant mode fields spread and resembles a rectangular waveguide mode distribution (non-TEM). This effect is demonstrated for CASE 15 in Fig. 37 at x=-0.01mm for the CBCPW dominant mode at frequencies before and after the strong mode coupling regions. The field values are normalized to the maximum values. Figure 35 showed at 15GHz after coupling with the TMq, mode, the CBCPW mode is still bound to the slot regions. After coupling with the TM0 3 mode, the dominant mode energy is starting to spread across the waveguide width as indicated at 20GHz. This situation is enhanced after coupling to the TMq5 mode at 23 GHz and to the TM q7 mode at 29GHz. Energy outside the CBCPW slot regions will not detected by the coaxial connector and will resonate in the finite-length three-dimensional cavity structure. The analysis would predict the beginning o f the mode coupling effects approximately after 20GHz, higher insertion loss at 23 GHz with some resonant peaks, and probably an unusable transmission line at 29GHz. These coupling effects also depend on the transmission line length. Measurement results verifying the analysis will be presented in section D later in this chapter. Outside the transition regions, the mode coupling effects Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 1.0 0.8 CASE 15 0.6 0.4 3 ti 0 -2 -I 0 .0 CL - 0.6 - 0.8 — 20GHz 23GHz - - 29GHz - 1 .0 V (m m ) Fig. 37. Field plots at various frequencies for CBCPW mode of CASE 15 demonstrating the field spreading effects due to mode coupling with the waveguide modes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 on the rectangular waveguides modes due to the CBCPW mode were not observed in the SDM. The presentation of the mode coupling effects to the various waveguide modes with respect to the field spreading was an original contribution of this dissertation o f [14]. Additional demonstrations of the mode coupling effects are illustrated for CASE 15 in Fig. 38. Characteristic impedances for Modes A and B and the CBCPW mode are shown from Fig. 36 using the power-voltage definition of (2.64a). This impedance definition is a proper choice since the rectangular waveguide modes energy is distributed across the entire waveguide width. The impedance o f the TMXrectangular waveguide modes should be small since the fields have a large amplitude and are extended across the entire waveguide width which translates to a much higher power flow. As indicated in the transition region, the energy is exchanged between the modes and becomes a combination o f the modes and the impedance of the two modes become equal at some frequency. In other words, the fields o f the CBCPW mode and the TMq j mode in Fig. 38(a) become nearly identical at 9.2GHz. The impedance of the CBCPW mode should decrease with frequency and the field distribution resembles a waveguide mode from these coupling effects. The CBCPW dominant mode impedance varies from 50Q to 20Q over the frequency-band primarily due to the mode coupling. This impedance demonstration of the mode coupling effects was first published in [14], Coupling coefficients between the waveguide and the CBCPW modes are also pictured in Fig. 38. The response demonstrates a peak value near the strong mode coupling regions and a smaller coefficient outside these frequency points. This behavior follows from the dispersion curve plots of Fig. 36. The coupling coefficient between modes a and b describing the waveguide mode field overlaps is calculated using the following power flow relationship from [33] as Co (4.4) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill x CBCPW moda Mode A Mode B O 40 I- 0 8.0 5 8.3 10 8.6 15 8.9 20 9.2 25 9.5 FREQ (GHz) 30 9.8 35 10.1 (a) -g 0.060 o E ^ 0.050 CL O g 0.040 O *4-* ^ 0.030 Ll. LlI O 0.020 O 2 —I 0.010 CL u o 0.000 o FREQ (GHz) 25 35 (b) Fig. 38. Additional mode coupling effects in two-dimensional CBCPW CASE 15. (a) Characteristic impedance plots for the modes indicated in Fig. 36. Modes A and B refer to the lower frequency range (8-10.1GHz). (b) Calculated coupling coefficient o f the waveguide modes to the CBCPW mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 where t represents the transverse field components, CS is the entire waveguide cross-section, and C aa = C bb = 1. The integrals in the denominator of (4.4) can be determined from the spectral domain analysis of (2.70) and Appendix C and the integral in the numerator is evaluated in the space domain. C. Additional Examples o f the Mode Coupling Effects in CBCPW To confirm the SDM analysis o f the mode coupling effects in CBCPW with lateral sidewalls, three additional examples will be presented in this section and compared with experimental data in section D. Figure 39(a) includes the coupling effects for a single-layer (er=10.8) CBCPW of CASE 14 in Fig. 31 and jc=-0.01mm. At 5GHz, the dominant mode has experienced mode coupling with the TM,,, mode. Additional field spreading occurs at 8GHz after coupling to the TMq 3 mode. The CBCPW mode has coupled with the TMq 5 mode and significant field spreading across the waveguide occurs at 12GHz. At 25GHz, the dominant mode has coupled with modes TM q pT M ^, ... ,TM09,TM0 n and is no longer a quasi-TEM. A multi-layered example of CASE 16 which is similar to CASE 15 except with thicker substrates (h4=2.54mm, hs=0.71mm) is presented in Fig. 39(b). The mode coupling problems become pronounced after coupling with the TMq 5 mode as indicated at 15.5GHz. The multi-layered example of CASE 5 (same as CASE 15 except h4=0.635mm, hs=0.71mm) from Chapter III (Fig. 25) is analyzed with lateral sidewalls in Fig. 40(a). The normalized dispersion curves demonstrate that no strong mode coupling regions exists with the waveguide modes to 40GHz. As a result, the dominant mode should be bound to the slots at 40GHz which is confirmed in part (b) o f the figure. This outcome was stated in Chapter III for this example as this structure was not leaky to 40GHz. In other words, mode coupling effects will not occur in a perturbed transmission line if the unperturbed transmission line dominant mode is bound and not leaky. Also, the phase constant of the perturbed waveguide modes will always be less than the phase constant o f the unperturbed parallel plate waveguide modes from which the waveguide is composed as indicated in (4.3). Therefore, the leaky critical frequency o f an infinite-width structure will always be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 1.0t 0.8 CASE 14 — 5GHz BGHz - - 12GHz r— 25GHz 0.6 lu 0.4 p 0.2 Q a! o.o 0.2 < - H C o -0.4 - 0.6 - 0.8 -1.01 (a) 1.0 0.8 CASE 16 12.5GHz 15.5GHz 19GHz 0.6 UJ 0.4 O 0.2 CL 0.0 0.2 < - H c*q-o.4 - 0.6 - 0.8 -1.G -2 0 -16 -12 - 8 -4 0 4 y (m m ) 8 12 16 20 (b) Fig. 39. Additional examples of field spreading mode coupling effects in CBCPW. (a) Single-layer structure CASE 14 of Fig. 31. (b) Multi-layered structure CASE 16 same as CASE 15 except h4=2.54mm, h5=0.71mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 2.8 2.6 a 2.4 2.2 2.0 c* 1.8 1.2 1.0. FREQ (GHz) (a) 1.0( 0.8 40GHz 0.6 UJ 0.4 O ^ 0.2 a o.o < - 0.2 C ^-0.4 - 0.6 - 0.8 -1.01 (b) Fig. 40. Multi-layered CBCPW example demonstrating a bound dominant mode to 40GHz. CASE 5 here is same as CASE 15 except h4=0.635mm, h5=0.71mm. (a) Dispersion curves, (b) Plot for Ex field component. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 less than the first mode coupling frequency for the two-dimensional structure. D. Experimental Results Confirming the Two-Dimensional SDM Analysis o f CBCPW Measurement data of CBCPW through lines is presented to validate the two-dimensional SDM analysis of the previous sections and is shown in Fig. 41. The S21 transmission coefficient data was taken with an HP8510 network analyzer using 401 points. CASE L in the figure is the single-layer channelized CBCPW with 2A=5mm and the cutoff frequency of the first waveguide mode is 10.6GHz which is confirmed in the figure. CASE M corresponds to CASE 15 and the transmission line characteristics initially change above 12GHz with the inclusion of a small standing wave pattern which continues to about 21 GHz. A significant insertion loss and the presence of strong resonant peaks appear after 27GHz. This response is closely predicted by the SDM in Fig. 37 for this case. The improvement of the two-dimensional CBCPW model over the one-dimensional leaky wave analysis for this example is readily apparent from Table XIII o f Chapter III which stated 0<fcrit<12GHz without an air gap layer between the dielectrics. CASE N in Fig.^41 is the measurement results for CASE 16 and the data presents problems beginning at 14GHz and an unbounded behavior occurring above 17GHz. This result is in good agreement with the numerical plots of Fig. 39(b). The single-layer example o f CASE 14 is depicted by CASE O and illustrates an initial trouble point at 5GHz with strong resonances beginning at 9GHz. The data is truncated for responses 20dB down. The experimental data for CBCPW MICs in which mode coupling effects are present, exhibits a low-pass filter characteristic with a soft comer frequency. The coupling o f the CBCPW mode to the TMq5 mode for the multi-layered lower dielectric loaded examples studied in this dissertation marks the frequency point where the dominant mode has significant energy spreading, and increases with coupling to the higher-order rectangular waveguide modes. This conclusion is verified with the experimental data. For the single-layer CBCPW, coupling with the TMq3 mode provides an approximate upper usable frequency limit o f operation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 CASE L S2 i (d B ) CASE M > '•a CASE N MAGNITUDE QQ TO O CASE 0 0 4 8 12 16 20 24 28 32 36 40 FREQ ( G H z ) Fig. 41. Experimental data o f the mode coupling effects in CBCPW MICs with lateral sidewalls with srl= l and h,=oo (open structure) , h2=h3=h6=0 and 2B=38mm. Refer to Fig. 13 and Fig. 5(a) for the dimension parameters. For CASE L er4=10.8 and h4=0.635mm, h5=0 and S=W=0.508mm and 2A=5mm. CASE M same as CASE 15 with er4=10.8, srS=2.33 and h4=1.27mm, h5=0.381mm and S=2W=0.635mm and 2A=38mm. CASE N (CASE 16) same as CASE M except h4=2.54mm, h5=0.71mm. CASE O (CASE 14) same as CASE L except 2A=38mm. All cases are 50Q through lines and referenced to OdB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 The effects of the transmission line length in the overmoded CBCPW are pictured in Fig. 42. CASE P is the same as CASE M of Fig. 41 with a shorter length and the response is similar except for the lower insertion loss at higher frequencies. CASE 5 o f Fig. 25 is represented by CASE Q and as indicated the dominant mode is bound to the slots up to 40GHz as described in Fig. 40. Resonances in the data on the order of 2-4dB exist in the upper ffequency-band. Energy is coupled to the waveguide modes at the coaxial feed or by the CBCPW mode over the waveguide length. By reducing the cavity (transmission line) length, the resonances should be shifted upward in frequency and the cavity Qs modified. This statement is verified for the lower half o f the band for CASE R but the upper band resonances have effectively disappeared. The resonance at 23GHz (beginning of the upper band resonances) o f CASE Q could correspond to either TM0112, TM0 3 u , or TMgjg modes for an ideal multi-layered cavity with a length (2B) o f 38mm. The resonant cavity frequencies are determined using the transcendental expressions from Appendix E and (4.2) and are written for the TM r modes as k k c xl - ta n ^ .h s = - P xi r5 0 r4 tanAri4h 4 0 (4.5) 4 with ^ +(S) **4 + where TM =(®™J s*eo^o (f£) + Of) cocAthj} is the unknown and (46a) (4.6b) fi= 1,3,5,... ,oo and p = 1,2,3,... ,oo. For 2B=25.4mm the resonant frequencies for these three modes are 32.9GHz, 31.9GHz, and 28.7GHz, respectively which are within the measurement band but are not present in CASE R. Broadband absorber is placed within the coaxial connector blocks o f CASE S and reduces the resonances by dampening the cavity for CASE Q and produces a 40GHz transmission line which is the widest bandwidth CBCPW MIC reported to date [14]. In summary, the two-dimensional SDM can determine the transmission line bandwidth and explain the mechanism producing the line loss and the strong resonances. However, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 CASE P CO X> CN 00 CASE Q > -O • Q ID QD ■O O m CASE R < CASE S 0 4 8 12 16 20 24 28 32 36 40 FREQ (GHz) Fig. 42. Experimental data demonstrating the cavity mode effects in CBCPW MICs with erl=l, er4=10.8, srS=2.33 and ht= oo (open structure), h2=h3=h6=0 and S=2W=0.635mm. Refer to Fig. 13 and Fig. 5(a) for the dimension parameters. CASE P with h4=1.27mm, h5=0.381mm and 2A=38mm, 2B=25.4mm. For CASE Q with h4=0.635mm, hs=0.711mm and 2A=20mm, 2B=38mm. CASE R same as CASE Q except 2B=25.4mm. CASES same as CASE Q except with broadband absorber in the connector housing blocks. All cases are 50Q through lines and referenced to OdB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 the Q of the resonant peaks and the resonant frequencies in the experimental data cannot be explained by the two-dimensional analysis and a finite-length (three-dimensional) structure is required to fully predict the measurement results. The SDM will be extended to three-dimensions for CBCPW and will be presented in Chapter VI. E. Mode Coupling Analysis of Finite Ground Plane CBCPW The effects o f mode coupling on the CBCPW mode with finite ground planes (FGP) will be analyzed in this section. Referring to Fig. 14(b) of section F in Chapter II, two fundamental zero-cutoff modes (CBCPW and coplanar-microstrip) are supported by the structure and are shown in Fig. 4(a) and (c). connected to the conductor-backing. The ground planes are not electrically Measurement data for a FGP CBCPW was illustrated for CASE C in Fig. 7. At lower frequencies, the parallel plate region between the conductors at x=0 and x=-h6-h5-h4 can be viewed as bounded in the transverse direction by magnetic sidewalls at y=±(S/2+W+W0). These boundaries form a rectangular waveguide with magnetic sidewalls and waveguide modes can propagate and couple with the CBCPW dominant mode and produce similar mode coupling effects as described in previous sections of this chapter. The cutoff frequencies of these waveguide modes can be adjusted by the lateral sidewall separation (width of the ground planes WG). The wavenumber in they-direction becomes 2 - where A= S/2+W+WG and ft = 0 ,2 ,4 ,..., ( a ) oo . This expression would be added to the left hand sides of (E.7) and (E.9) in Appendix E to determine the propagation constants o f the multi-layered parallel plate waveguide o f Fig. E. 1 with magnetic sidewalls. This structure could be analyzed using the SDM for CBCPW with the same basis functions o f Chapter II section B and the above spectral parameter in (4.7) for k . At higher frequencies, this magnetic sidewall model is no longer valid and the coupled strip SDM analysis of section F o f Chapter II must be employed by expanding the unknown current distributions on each Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 strip. The FGP CBCPW has another interesting phenomenon associated with microstrip-like resonances associated with the two conductor ground planes behaving like two-dimensional patch antennas [36]. At these resonant frequencies, most of the electromagnetic energy is carried by the microstrip-like modes and the dominant CBCPW mode will not effectively propagate along the slots. These frequencies must be avoided by adjusting the patch antenna dimensions (three-dimensional problem) so the resonances occur out of the ffequency-band. An example of the mode coupling effects in FGP CBCPW is illustrated in Fig. 43(a) for a single-layer structure CASE 17 at 30GHz. A total of two and five basis functions on the center strip and the ground planes, respectively were employed for the Jy and Jz currents. Strong mode coupling exists between the CBCPW and CPM2 mode (identified as TMq2 mode) at a ground plane width (WG) o f 2.75mm. To prevent this coupling effect from occurring, the ground plane width should be kept less than 2mm or a multi-layered lower loading structure could be implemented as in Fig. 43(b) for CASE 18. As indicated from the figure, no mode coupling exists between the CBCPW and the CPM2 regardless of the ground plane width. Note that the CBCPW mode now has a higher propagation constant than the CPM0 mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.501 if i i i i 0.5 1.0 1.5 2.0 2.5 3.0 i_____■ 3.5 *.0 4.5 5.0 Wg (m m ) (a) 2.50 2.40 CBCPW 2.30 2.20 s* 2.10 •** 2.00 CPMo 1.90 1.70 1.60 CASE 18 (b) Fig. 43. Dispersion curves showing mode coupling effects in FGP CBCPW as a function of the ground plane width (WG) at 30GHz with srl= l, er4=10.2 and h ^ m m , h4=0.635mm, h2=h3=h6=0 and S=2W=0.635mm and 2A=25.4mm. (a) Single-layer structure with h5=0. (b) Multi-layered waveguide with srS=2.33 and h5=0.635mm. CPM refers to the coplanar-microstrip modes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 CHAPTER V DESIGN SUMMARY INFORMATION FOR THE PROPAGATION CHARACTERISTICS OF MULTI-LAYERED CBCPW The previous two chapters demonstrated how multi-layered CBCPW MICs could be analyzed and employed to produce a transmission line such that the dominant mode would propagate bound to the slots (quasi-TEM) with a bandwidth depending on the dimensional parameters. Additional design criterion is necessary to determine whether the multi-layered structure (both upper and lower dielectric loaded) is a viable and practical waveguide. This information for the propagation characteristics of the dominant CBCPW mode includes the upper usable frequency or bandwidth, characteristic impedance range and variation, parameter values for transmission lines with 5012 impedances, and effective dielectric constant range and variation for different configurations. Designers of multi-layered CBCPW MICs must also understand the effects o f the possible air gap between the dielectrics, the dielectric constant uncertainty value on the propagation constant, dimensions for the cross-sectional circuit area to minimize dispersion, closed form procedures to estimate the dielectric constant value and thickness for the multi-layered loaded substrate, parameter values of multi-layered microstrip that may be necessary to interface within a circuit, and other practical physical suggestions for usage. An additional matter of interest is whether CBCPW can operate properly with the mode coupling effects present. The experimental and numerical data already presented for CBCPW MICs indicated that this was not possible beyond the critical or upper usable frequency. The dominant mode resembled a rectangular waveguide mode with a field pattern spread out across the entire waveguide width. The multi-layered waveguide was Part of the data reported in this chapter is © 1994 DEEE and submitted to IEEE for publication. Reprinted, with permission, from 1994 IEEE Microwave Theory and Techniques International Microwave Symposium Digest, pp. 1697-1700 (see References [15], [28] and Appendix G). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 utilized without placing any restrictions on the cross-sectional circuit area (S+2W) or the lateral width. In other words, (S+2W) could be the same size or larger than the substrate thickness. The question o f how CBCPW MMICs can function in the presence o f the coupling effects must be investigated. Recall that a single-layer CBCPW unconditionally leaks energy to the TEM parallel plate mode which translates to mode coupling effects existing at lower frequencies for electrically wide two-dimensional waveguides. For CBCPW MMICs the line/slot widths can be made much smaller than the substrate thickness. This parameter guideline for MMICs reduces the leakage constant for the one-dimensional structure [12], [32] and reduces the coupling to the cavity modes in finite-length waveguides [35], Keeping (S+2W) less than one-fourth the dielectric thickness and one-twentieth the dielectric wavelength (A.g /20) for GaAs is an estimate for the leakage rate of the CBCPW mode for useful results [12], although this statement was not quantitatively demonstrated. A circuit designer may not have a two-dimensional SDM numerical program with field plotting capabilities to analyze the coupling effects (as presented in Chapter IV). The designer requires a procedure and a closed form expression in terms of the cross-section, substrate thickness, dielectric constant, and operating frequency for reduced mode coupling and a usable transmission line for the single-layer CBCPW. This process will also relate the mode coupling in CBCPW with lateral sidewalls (realistic two-dimensional structure) to the leakage rate of an infinite-width structure (nonphysical waveguide) and hence unite the analysis from Chapters III and IV. It has already been shown that if leakage is not present in a transmission line, mode coupling will not occur and a small leakage rate corresponds to a small mode coupling effect. This chapter will demonstrate the propagation characteristics of multi-layered CBCPW with upper loading, summary tables illustrating operational trade-offs, and additional design information for CBCPW MICs. This complete design summary is an original contribution from [15] and [28] as part o f this dissertation. The basic theory relating leakage in infinite-width to coupling effects in packaged CBCPW will be described along with appropriate leakage curves and mode coupling plots. Finally, a design procedure for CBCPW single-layer structures will be detailed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 A. Propagation Characteristics of Upper Dielectric Loaded CBCPW The upper loaded multi-layered CBCPW is another configuration alternative to extend the mode coupling effects to higher frequencies and hence the bandwidth o f the transmission lines. This CBCPW configuration was an original contribution as part of this dissertation and published in [15], Referring to Fig. 13, the substrate layer (h3) is placed on top of the circuit conductors and ground planes. This structure may cause problems in circuits with elements possessing a vertical thickness component but is well suited to transmission line elements such as matching and feeding networks, antennas, and filters. The basic idea behind this structure is the placement o f a higher dielectric constant substrate above the CBCPW with a lower dielectric constant value and hence raising the propagation constant of the dominant CBCPW mode above that of the single-layer waveguide modes. Recall that the conventional single-layer CBCPW has an unconditional leakage condition in the one-dimensional case and mode coupling effects occurring at lower frequencies within the two-dimensional transmission line. One additional matter of concern is the presence of surface waves on the grounded dielectric slab (h3) above the circuit conductors at x=0 (characteristic modes associated with microstrip) and the mode coupling effects that can occur to the dominant CBCPW mode due to these modes. These surface wave modes will limit the frequency range of operation and in the two-dimensional structure with lateral sidewalls, these modes are referred to as boxed surface wave modes [56], As the thickness of the upper layer increases (h3), the mode coupling effects with the surface wave modes exist at lower frequencies. Referring to Fig. 13, the surface wave in the air region (h,) travels in the r-direction and is attenuated in the x-direction (kxl is imaginary). The rectangular waveguide (RW) modes of the lower single-layer homogeneous structure (h4) will not couple with the dominant CBCPW mode since P cbcpw > K. rw as l°nS as h3 *s sufficiently thick. The upper loaded multi-layered configurations used in this chapter for numerical cases consist of er3=10.2 and sr4=2.2. A dispersion curve plot with normalized phase constants for the dominant, waveguide, and surface wave modes for an upper dielectric loaded CBCPW with lateral sidewalls is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 pictured in Fig. 44 (CASE 19) and demonstrates no mode coupling effects to 40GHz. The BSW IDEAL (boxed surface wave) response corresponds to a grounded dielectric slab (h3) without slots. The LOWER RW refers to the maximum propagation constant of the modes within the lower single-layer rectangular waveguide (ee(r=2.2). Note that the propagation response o f the BSW TM,,, is different than for the rectangular waveguide modes of Fig. 31 and Fig. 36. A vector cross-sectional electric field plot for CASE 19 at 40GHz is illustrated in Fig. 45 and shows that the dominant mode is bound to the slot areas as predicted from the dispersion curves. Measurement data for CBCPW high impedance through lines with sr4 =2.33 is depicted in Fig. 46(a). CASE T represents a single-layer structure and problems in the response occur at approximately 10GHz. CASE U is the same as CASE T except the lateral dimension (2A) is reduced to 5mm and the cutoff frequency of the first waveguide mode is 22.7GHz and the data shows some variations at about 33GHz. An upper loaded configuration is applied in CASE V and can be compared to CASE T. Note at the lower frequencies an approximate 50Q. transmission line impedance is present and then a large insertion loss occurs which is due to connector problems associated with placing the upper layer substrate on top of the coaxial feeds. This was already mentioned as one of the physical shortcomings for this multi-layered waveguide for circuit elements with height. A modification was made for CASE W by cutting away a semi-cone in the upper layer substrate (h3) at the port connectors as not to interfere with the coaxial pin feeds and is diagrammed in Fig. 46(b). Therefore, CASE W is not exactly modeled in Fig. 44 which predicts no mode coupling to 40GHz. The presence o f the air region above the connectors requires a three-dimensional analysis for this example. The impedance of the transmission line is still high but demonstrates a much improved response over CASE T. Fig. 47 depicts the cross-sectional vector electric field plot (Ex and Ey) at a z=constant plane for the TM„ , surface wave of CASE 19 at 40GHz. As indicated most of the energy associated with the mode is bound to the dielectric and the air interface with a predominantly vertical field component and the field is attenuated in the x-direction. This response is compared to the TM0 , rectangular waveguide mode of Fig. 34(b) for a lower loaded CBCPW in which the field exists between the conductor planes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 2.75 CASE 19 2.50 2.25 V 2.00 o 1.75 — cbcpw TMo.1 BSW SDM x TM0.1 BSW IDEAL • - - TMq.i LOWER RW (MAX) 1.50 1.251.00 FREQ (G H z ) Fig. 44. Dispersion curve plot of the normalized propagation constants for an upper dielectric loaded multi-layered CBCPW. CASE 19 w itherl=l, er3=lO,2, er4=2.2 and h^Smm, h3=0.635mm, S=2W=0.635mm and 2A=25.4mm. h4=1.27mm, h,=h5=h6=0 and Refer to Fig. 13 for parameter dimensions. BSW is the boxed surface wave modes and the [DEAL case corresponds to the grounded dielectric slab layer h3. LOWER RW refers to the maximum propagation constant of single-layer rectangular waveguide h4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 1.5 1.2 0.9 0.6 0.3 0.0 H -0 .3 0.6 - -0 .9 - 1.2 -1 .5 y (m m ) Fig. 45. Cross-sectional electric field plot for the dominant CBCPW mode of the upper loaded example CASE 19 at 40GHz. The field components are normalized to the maximum values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 CASE T CASE U CD T3 CASE V Ld O 3 Z o < 2 CASE W 0 4 8 12 16 20 24 FREQ (GHz) 28 32 36 40 (a) (b) Fig. 46. (a) Experimental data for CBCPW MICs with lower dielectric value er4=2.33. erl= l and h^oo (open structure), h4=1.42mm, h;=h5=h6=0 and S=W=0.508mm and 2B=38mm. Refer to Fig. 13 and Fig. 5(a) for the dimensions. For CASE T h3=0 and 2A=38mm (single-layer). CASE U same as CASE T except 2A=5mm. CASE V upper loaded structure same as CASE T except er3=10.8 and h3=0.635mm and S=2W=0.635mm and with connector problems. CASE W same as CASE V except with a cutout of the substrate around the inputs and output ports and illustrated in (b). All cases are referenced to OdB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 4 .5 4.0 3.5 3.0 E E 2.5 2.0 H 1.5 1. 0 - 0.5 0.0 -1 4 -1 0 -6 y (m m ) Fig. 47. Cross-sectional vector electric field plot of the TMg [ surface wave mode for the upper loaded multi-layered CBCPW of CASE 19 at 40GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 within the substrates. B. Design Summary Information for Multi-Layered CBCPW MICs Two basic multi-layered configurations for CBCPW MICs have been proposed in this dissertation. These transmission lines were analyzed using the SDM and verified numerically and experimentally to extend the frequency of operation and in some cases up to 40GHz. The lower dielectric loaded case is the preferred structure for circuits with active devices or elements with a sufficient thickness and the upper dielectric loaded waveguide is recommended for passive circuits involving feed or matching networks and antennas. In order for these multi-layered configurations to be an accepted transmission line class, the dispersive behavior, impedance range and variation, and additional design information requires investigation. Summary tables of various structures demonstrating operational trade-offs for wide frequency-band design are necessary. Examples o f these summary tables for lower and upper loaded CBCPW MICs with dielectric thickness ranging from 0.254mm to 1.27mm are illustrated in Tables XV and XVI, respectively. All of the numerical results were derived from the SDM o f Chapter II. The configurations are limited to two layers with relative dielectric constants 10.2 and 2.2 which approximately correspond to widely available substrates in Duroid™ (6010 and 5870). Four possible cross-sections (S+2W) are considered in this analysis. For Tables XV and XVI, CASES 20 and 24 correspond to 5 0 0 lines, CASES 21 and 25 have equal strip and slot widths, CASES 22 and 26 are low characteristic impedance lines, and CASES 23 and 27 produce high impedances. The minimum strip and slot widths were chosen at Smin=0.254mm and Wmin=0.127mm, respectively. The maximum cross-sectional value (S+2W) of 2.032mm was selected based on impedance variations versus frequency. The examples in these tables correspond to the maximum cross-sectional area that could be implemented in a multi-layered CBCPW MIC to 40GHz without significant dispersion effects. The listed frequency (f) is the upper usable frequency to 40GHz in which strong mode coupling occurs between the dominant CBCPW mode and the first rectangular Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 waveguide or surface wave mode (TMq ,) for all the presented cases. It was already shown in Chapter IV that strong mode coupling occurs with the TM0 5 mode for the lower loaded structure so the table responses are conservative estimates of the bandwidth. The listed frequency (f) is the point where the CBCPW mode impedance, using the power-voltage definition Zpv of (2.64a), decreases by 10% and indicates the onset of strong mode coupling with this mode. This mode coupling effect was shown in Fig. 38(a). A circuit designer requires a large effective dielectric constant to reduce the circuit size and small impedance and eefr variations over frequency. increases, the upper usable frequency decreases. Referencing Table XV, as h4 This behavior occurs because the waveguide modes are affected to a larger degree to changes in the vertical properties of the structure than the dominant CBCPW mode and hence the cutoff frequencies o f the waveguide modes are reduced. This statement is supported by Fig. 33(a) and Fig. 34. No substantial change in performance exists by increasing the lower substrate thickness (h5) since for a bounded CBCPW mode, most o f the electromagnetic energy is located in the slot regions between the air above the conductors and h4 below. One exception to this statement is for thin upper substrates such as example 1A). For 40GHz operation with a lower loaded waveguide, examples 2B) and 3B) are recommended (due the higher effective dielectric constant values) with 2B) utilizing a smaller overall thickness. Examples 2C) and 3C) could be used to 20GHz. Results from Table XVI are similar to those in Table XV. As h3 increases, the upper usable frequency (f) decreases as the cutoff frequency of the first grounded slab surface wave mode is reduced and couples to the dominant CBCPW mode at lower frequencies. The propagation characteristics o f this surface wave mode are primarily influenced by the h3 dielectric while the CBCPW mode is affected by the air, h3, and h4 layers. In both tables, increasing the thickness of the er =10.2 substrate, increases the effective dielectric constant to a diminishing point. From Table XVI, an upper usable frequency o f 40GHz could be realized by examples 3B) and 2B) with 3B) providing a closer 50Q characteristic impedance. Examples 3C) and 2C) could be selected to achieve at least a 20GHz bandwidth for the CBCPW. Fig. 48 includes dispersion curves for CASES 20 and 21 of Table XV and a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 TABLE XV Lower dielectric loaded multi-layered CBCPW MIC design summary. erl=l, er4=10.2, er5=2.2 and h,=5mm, h2=h3=h6=0 and 2A=25.4mm and S+2W<2.032mm, Smax=10W, Wmin=0.127mm and refer to Fig. 13 for dimensional parameters and CASE 20: S=2W=0.635mm CASE 21: S=W=0.635mm CASE 22: S=10W=1.27mm CASE 23: S=0.254mm, W=0.889mm. FREQ (GHz) f (mm) h4 h5 sefr(min) @ 1GHz Zp v{min), Zp v (max) (Q) @ 1GHz Zpy (£2) CASE 20 @ 1GHz eefr(min) @f Zp v (min), Zpv (max) (Q) @40GHz 40.7, 90.8 1A) 0.254 0.254 40 3.87 30.3, 80.3 48.9 4.34 IB) 0.635 0.254 36.9 5.00 31.5, 82.3 49.3 5.96 1C) 1.27 0.254 4.8 5.46 32.5, 85 50.2 5.56 ID) 2.54 0.254 3.5 5.58 33.1, 86.7 50.9 5.68 2A) 0.254 0.635 40 3.61 35.5, 95.2 55.8 4.21 45.4, 117.8 2B) 0.635 0.635 40 4.78 33.7, 88.5 51.8 5.98 33.8, 88.5 2C) 1.27 0.635 20.8 5.33 33.3, 87.2 51.1 5.97 - 2D) 2.54 0.635 6.4 5.53 33, 87 51 5.68 - 3A) 0.254 1.27 40 3.58 37.7, 102.5 58.5 4.2 45.8, 123.3 3B) 0.635 1.27 40 4.75 34.7, 91.8 53 5.97 33.4, 89.9 3C) 1.27 1.27 24 5.3 33.8, 88.7 51.6 6.08 - 3D) 2.54 1.27 10.3 5.51 33.5, 88 51.4 5.77 - - - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 TABLE XVI Upper dielectric loaded multi-layered CBCPW MIC design summary. eri= l, sr3=10.2, er4=2.2 and 1^=5mm, h2=h5=h6=0 and 2A=25.4mm and S+2W<2.032mm, Smax= 10W, Wmjn =0.127mm and refer to Fig. 13 for dimensional parameters and CASE 24: S=2W=0.635mm CASE 25: S=W=0.635mm CASE 26: S=10W=1.27mm CASE 27: S=0.254mm, W=0.889mm. (mm) f I S g 2 Zpv(Q) CASE 24 @ 1GHz 1A) 0.254 0.254 40 3.21 24.1, 65.7 39.6 3.66 40.9, 92.5 IB) 0.635 0.254 40 3.92 22.4, 60.3 36.2 5.43 32, 76 1C) 1.27 0.254 24.3 4.23 21.7, 58.6 35.3 5.39 - ID) 2.54 0.254 11.8 4.35 21.5, 58.1 35 4.85 - 2A) 0.254 0.635 40 3.54 32.6, 88.5 52 4.21 45.7, 124 2B) 0.635 0.635 40 4.65 30.3, 80 46.6 6.19 35, 92.5 2C) 1.27 0.635 28.6 5.15 28.7, 75.4 45.1 6.51 - 2D) 2.54 0.635 13.7 5.34 28.3, 74.3 44.7 6.05 - 3A) 0.254 1.27 40 3.71 35.8, 98.5 56 4.33 45.7, 127.7 3B) 0.635 1.27 40 5.0 32.5, 86.2 49.9 6.29 35, 93.8 3C) 1.27 1.27 29 5.6 31.3, 82.3 48.2 6.65 - 3D) 2.54 1.27 14.3 5.83 30.8, 81 47.7 6.35 - h3 h4 FREQ (GHz) eeff(min) @ 1GHz eeff (min) @ f Zpy (min), Zpv (m ax) (Q) @40GHz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 cross-sectional vector electric field plot at 40GHz for the lower loaded example. A small variation exists between the phase constants for the CBCPW modes in Fig. 48(a). At higher frequencies, the CBCPW mode has a larger effective dielectric constant and more of the electromagnetic energy will be concentrated in the higher dielectric substrate (h4) as illustrated in Fig. 48(b) as compared to Fig. 33(a). The term dispersion relates the rate o f change of the propagation characteristics o f transmission lines with frequency and is primarily applied to the effective dielectric constant and the characteristic impedance. Smaller dispersion for a waveguide corresponds to a more straightforward circuit analysis without significant design compromises. The characteristic impedances for the multi-layered configurations must also be analyzed in detail. An impedance verification example is first presented in Fig. 49 comparing the three SDM impedance definitions from this work and (2.64a), (2.64b), and (2.64c) with the results o f [57], Recall that for non-TEM structures such as CBCPW, the characteristic impedance is not unique and the power-voltage, voltage-current, and the power-current calculations are invoked. Approximately a IQ difference occurs between the impedances in Fig. 49. The impedances of example 2B) from Table XV for CASES 20-23 and example 3B) from Table XVI for CASES 24-27 are pictured in Fig. 50 for the three impedance definitions. Similar behavior for the two multi-layered configurations is apparent. A 30-90H impedance range exists for these cases. The variation in the impedance responses is similar to those described in [57] and Fig. 49. The larger cross-sectional circuit area in CBCPW results in a more dispersive impedance behavior and this is the reason for the limitation o f (S+2W)<2.032mm for these examples. The impedance curves demonstrate that the voltage-current has the smallest dispersive effects among the definitions for CBCPW MICs. The work presented in this section has shown that the upper and lower dielectric loaded multi-layered CBCPW MICs are valid transmission lines with wide bandwidths, sufficient impedance range and effective dielectric constant values, and tolerable dispersive effects. These structures are produced with no limitation on the lateral width and only a small restriction on the cross-sectional circuit area and translates to a very flexible transmission Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 2.50 2.25 Q . 2.00 1.75 X CASE 20 CASE 21 TM0.1 CASE 20 SDM TMo.1 IDEAL RW 1.50 FREQ (GHz) (a) 0.9 0.6 0.3 0.0 E -0 .3 0.6 - -0.9 - 1.2 —1.51 - 1 .5 - 1 .2 - 0 .9 - 0 .6 - 0 .3 0.0 0.3 0.6 0.9 y (mm) 1.2 1.5 (b) Fig. 48. Bounded mode behavior for lower loaded multi-layered dominant CBCPW mode, (a) Normalized dispersion curves for CASES 20 and 21 of Table XV and the first waveguide mode calculated by the SDM and the ideal rectangular waveguide analysis, (b) Cross-sectional vector electric field plot (Ex and E ) of the dominant CBCPW mode for CASE 20 at 40GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IMPEDANCE ( 0) 136 50 49 48 ZPv Z vi ■- - Z p t o Zpy v Zvi x Zpt Ref. 157] Ref. (57] Ref. [57] p re se n te d work p re se n te d work p re s e n te d work 47 46 CHARACTERISTIC 45 44 43 42 40 20 FREQ ( G H z ) Fig. 49. Characteristic impedance verification with [57] for an air-suspended multi-layered CBCPW for power-voltage, voltage-current, and power-current definitions. Refer to Fig. 13 for dimensions with srl= l, er4=13, er5= l and h1=hs=5mm> h4=lmm, h2=h3=h6=0 and S=lmm, W=0.4mm and 2A=25.4mm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 LU o z < 90 OL'JJ CASE 23 ............. ................. 80 Q 70 LU 0. 60 2 O 50 Ltn 40 o: lu 30 HO 20 < oc < 10 X o 0 CASE 21 CASE 20 CASE 22 Z pv Zvi Z pi B 12 16 20 24 28 32 36 40 FREQ (GHz) (a) Q 80 CASE 27 CASE 25 CASE 24 a: LU 30 CASE 26 ■ ■ ■ ■ 12 16 20 ■ ■ 24 ■ ■ 28 FREQ (GHz) (b) Fig. 50. Characteristic impedances of the multi-layered CBCPW MICs for the three definitions, (a) Lower loaded example 2B) of Table XV. (b) Upper loaded example 3B) of Table XVI. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 line platform. C. Additional Configuration Considerations for Multi-Layered CBCPW MICs The multi-layered CBCPW has been shown in the previous chapters to be a useful transmission line for MICs with sufficient bandwidth, impedance range, tolerable dispersion effects, minimum structural complexity, and minimum restrictions on the lateral width and circuit cross-sectional area. This section will include additional design information including the lateral width requirements to cutoff the waveguide modes, a closed form procedure to determine the dielectric constant and thickness for a lower loaded structure, dielectric uncertainty analysis on the propagation constants for the CBCPW modes, (S+2W) values to minimize dispersion effects for the multi-layered cases, the effects of air gaps between the substrates, and parameter values necessary to interface to a multi-layered microstrip feed circuit [15], [28], These tools provide the designer the necessary information to design CBCPW microwave circuits. The channelized CBCPW configuration [17] is one method for extending the usable bandwidth by reducing the lateral width dimension and increasing the cutoff frequencies of the rectangular waveguide modes. Equations (4.1), (E.6), and (E.7) in Appendix E are utilized to determine the parameter dimensions for this method. Table XVII includes the cutoff frequency o f the dominant TMj,, mode for the multi-layered configuration in Fig. 13 and assumes that (h4+h5)<2A and h6 =0. For the lateral width (2A) at 10mm, marginal bandwidths exist for CASES 29 and 31. By operating at frequencies where the rectangular waveguide modes are cutoff (nonpropagating), a single mode o f propagation for the dominant CBCPW mode occurs and the mode coupling effects illustrated in Chapters IV and V are avoided. However, the low dielectric constant utilized in CASE 29 will increase the wavelength and the circuit area and CASE 31 with an air suspended configuration is difficult to realize as a practical waveguide solution. A closed form procedure for determining the dielectric constant and thickness for a lower loaded multi-layered CBCPW configuration is now presented using Fig. 13 with h6=0. The analysis is a low frequency approximation requiring (h4+h5) « X4 where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 TABLE XVII Cutoff frequencies for the TMg , rectangular waveguide mode as a function of the lateral width for various CBCPW configurations. Fig. 13 with h6=0 is referenced for these lower loaded multi-layered examples. *arbitrary height requires h4<2A. Cutoff freq (GHz) 5 lateral width 2A (mm) 20 CASE 28 sr4=10.2, h4=arbitrary* h5=0 9.4 4.7 2.35 CASE 29 sr4 =2.2, h4=arbitrary* h5=0 20.2 10.1 5.05 CASE 30 sr4=10.2, h4=0.635mm er5 =2.2, h5=0.635mm 15.46 7.85 3.94 CASE 31 8r4=10.2, h4=0.635mm sr5=1.0, h5=l.905mm 24.6 13.02 6.58 ^ and f is the frequency of operation. 20GHz. 10 = T ^ (mm) ( 51) With (h4+h5)=X4/5, the approximation is valid to The dispersive effects associated with the dominant CBCPW mode can be reduced by stipulating (S+2W)<h4/2 and this requirement will be demonstrated later in this section. With this assumption the phase constant for the CBCPW mode can be approximated as (p/*o)2 * (er4 + 0 /2 (5.2) with most of the electromagnetic energy existing within the air (srI=l, h,=5mm in Fig. 13) and the h4 dielectric region. The low frequency approximations are invoked in the determination of the propagation constant for the TMq , waveguide mode from [9] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where x=h5/(h4+h5). The dielectric constant and thickness o f the lower layer is determined to avoid the mode coupling effects between the dominant CBCPW mode and the first rectangular waveguide mode. The mode coupling effects are the strongest between two modes at the phase match condition and this situation is realized by equating (5.2) and (5.3). Table XVIII summarizes the lower dielectric constant values for various CBCPW examples including GaAs (er4=13) and a high temperature superconducting dielectric YBa2Cu30 7^ (YBCO) with sr4=26 [58] utilizing the above procedure. TABLE XVIII Specification of the dielectric constant for the lower loaded multi-layered CBCPW structures using the closed form approximation procedure at f=20GHz. Fig. 13 is referenced with srl =1, hj=5mm, h2=h3=h6=0 and 2A=25.4mm. £r5 CASE 32 sr4 =10.2, h4=0.47mm h5=0.47mm (S+2W)=0.235mm <3.86 CASE 33 er4 =2.2, h4=lmm h5=lmm < 1.26 CASE 34 er4 =13, h4=0.278mm h5=0.556mm <5.69 CASE 35 er4 =26, h4=0.294mm h5=0.294mm <9.12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 An investigation to determine the values of (S+2W) to minimize dispersion effects in multi-layered CBCPW MICs is presented in Fig. 51 for the lower loaded example 2B) of Table XV. From the figure, the following condition on the cross-sectional circuit area should be made to minimize the influence o f the second or lower layer to maintain the non-dispersive characteristic o f the dominant CBCPW mode to 40GHz as h 4/2 < (S+2W) < h4 . (5.4) As expected, reducing the cross-section decreases the variation in the phase constant versus frequency. The reason for a low dispersive transmission line is a simpler circuit design for wide band applications. The effects of the uncertainties of the dielectric substrate parameter dimensions on the upper usable frequency are illustrated. The uncertainties in the substrate thickness and relative dielectric constant were presented in Chapter III for calculating the sensitivity of the leakage criticalfrequency and are applied to the CBCPW with lateral sidewalls. The variations again are ±0.051mm for the 1.27mm using the 6010 board (er=10.8 Duroid™ substrate) and ±0.0254mm for 5870 (er=2.33 Duroid™ substrate). The relative dielectric constant variations are ±0.25 for 6010 and ±0.02 for 5870. Table XIX depicts the uncertainties for both upper and lower loaded configurations for examples 2C) from Tables XV and XVI for CASES 21 and 25, respectively. As indicated in the table, a 4GHz range for the upper usable frequency is present for the lower loaded structure with the substrate variations which is significant and this sensitivity analysis should be implemented for bandwidth calculations. The effects on the propagation characteristics o f the CBCPW mode due to the possible presence of an air gap between the substrates of the multi-layered configurations requires evaluation. This air gap configuration is used in the sensitivity analysis o f the structure. The results are presented in Fig. 52 for both upper and lower loaded waveguides for examples 2B) from Table XV and 3B) from Table XVI. The air gap exists between the substrates (h5) for the lower loaded case and between the top substrate and the circuit conductor ground planes (h3) for the upper loaded case. As indicated in the figure, the lower loaded configuration is not affected by the air gap while the upper loaded case is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 2 .5 CASE •• CASE - - CASE — CASE 36 37 38 39 2.4 2.3 2.2 FREQ (G H z ) Fig. 51. Dispersion effect analysis for the determination of (S+2W) of the dominant CBCPW mode for the lower loaded example 2B) of Table XV. Refer to Fig. 13 with srl= l, sr4=T0.2, sr5 =2.2 and h[=5mm, h4=0.635mm, hs=0.635mm, h2=h3=h6=0 and 2A=25.4mm. CASE 36 is S=2W=0.635mm, CASE 37 is S=2W=0.381mm, CASE 38 is S=2W=0.1905mm, and CASE 39 is S=2W=0.0953mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 TABLE XIX Dielectric uncertainty effects on the upper usable frequency o f multi-layered CBCPW for S=W=0.635mm and 2A=25.4mm. CASES 40-42 are the lower loaded configuration of example 2C) from Table XV and CASES 43-45 are the upper loaded waveguide example 2C) from Table XVI. Upper usable ffeq (GHz) CASE 40 er4 =10.45, h4=l.321mm sr5 =2.18, hs=0.610mm 19.6 CASE 41 sr4 =10.20, h4=1.27mm sr5 =2.2, h5=0.635mm 22 CASE 42 sr4 =9.95, er5 =2.22, significantly modified. 23.66 h4= 1.219mm h5=0.660mm CASE 43 er3 =10.45, h3=l.321mm sr4 =2.18, h4=0.610mm 27.9 CASE 44 sr3 =10.20, h3=l ,27mm er4 =2.2, h4=0.635mm 29.2 CASE 45 sr3 =9.95, er4 =2.22, 30.3 h3= 1.219mm h4=0.660mm This is explained from transmission line theory from the discontinuity o f the dielectric constants o f the substrates as presented to the dominant mode. In CASE 46 the air layer is a close match to the lower substrate layer (8^=2.2) while for CASE 47 the air dielectric is a large step discontinuity to the upper substrate (er3=10.2) and the field characteristics are greatly affected. As expected, increasing the air gap thickness reduces the phase constant of the dominant mode since more of the fields reside within this dielectric region and the phase constant converges to l<p/£0<1.48. In the manufacturing of multi-layered CBCPW MICs, a bonding film would be applied Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 2.50 fi/k o 2.25 2.00 — CASE 45 - - CASE 47 1.75 0.04 0.08 AIR GAP T H I C K N E S S 0.12 (m m ) 0.16 Fig. 52. Effects of an air gap between the substrates for the multi-layered CBCPW MICs with £rl= l and h[=5mm and S=2W=0.635mm and 2A=25.4mm at f=20GHz. Fig. 13 is referenced for the dimensions. CASE 46 is the lower loaded example with sr4=l0.2, er5=l, srt=2.2 and h4=0.635mm, h5 is the air gap, h6=0.635mm, h2=h3=0. CASE 47 is the upper loaded example with er,= l0.2, sr3=l, er4=2.2 and h2=0.635mm, h3 is the air gap, h4=1.27mm, h5=h6=0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 between applied between the substrates to strengthen the structure and eliminate the air gap. A bonding film (model 3001) for microwave circuits is offered by Rogers Inc. for Duroid™ boards with er=2.28, a high melting point, and low dissipation loss. This bonding material would provide a match between the substrates in the lower loaded configuration presented here with respect to the microwave properties. Rogers also provides a ceramic-filled paste material for high dielectric boards and would be ideal for the upper loaded multi-layered structure to eliminate the air gap and the effects on the propagation constant of the dominant mode. In some circumstances, the multi-layered CBCPW MIC may be interfaced to a microstrip transmission line for a feed network or other circuit applications. The microstrip line is referenced from Fig. 14(a) and the strip width is determined by the SDM to provide an impedance match to the CBCPW transmission line. For the lower loaded example of 2B) from Table XV, a width of S=2.286mm is required for a 50fi characteristic impedance line which is wider than the maximum cross-section restriction from this table of (S+2W)<2.032mm. An approximate 50Q line in the multi-layered CBCPW corresponds to a strip to slot width of 2:1. The microstrip width would be reduced approximately in half to match the center strip width of the CBCPW line and this modification increases the microstrip impedance. The effective dielectric constant for the dominant quasi-TEM microstrip mode is 3.62 and 5.66 at 1 and 40GHz, respectively. For the upper loaded waveguide of example 3B) from Table XVI, a strip width of 3mm is needed for a 50Q characteristic impedance. This strip dimension is again significantly above the maximum cross-section width limitation for the CBCPW. Also, the microstrip mode experiences a large dispersive effect with the effective dielectric constant ranging from 2.67 to 5.25 at 1 and 40GHz, respectively. In summary, an average match condition between the multi-layered microstrip and CBCPW for the lower loaded case exists with respect to VSWR (voltage standing wave ratio) and the effective dielectric constants, but a poor match is presented in the upper loaded configuration. A design procedure for multi-layered CBCPW MICs would utilize the summary results from Tables XV and XVI for both lower and upper dielectric loaded structures with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 s=10.2 and er=2.2. The effective dielectric constant values and the bandwidths are obtained for various substrates thickness. The characteristic impedances can be determined from Fig. 50 for low, middle, high, and 50Q cases. For other substrates, (5.1)-(5.3) are implemented with the restrictions (h4+h5)= \4/5 and (S+2W)<h4/2 to calculate the dielectric parameters up to 20GHz. No closed form relationships exist for the seff or the characteristic impedance in multi-layered CBCPW. D, Minimizing the Mode Coupling Effects in Single-Layer CBCPW MICs Wide bandwidth applications utilizing CBCPW have been successfully developed in [5-7] without the use of a multi-layered configuration. The single-layer waveguide would be attractive compared to the multi-layered CBCPW with respect to manufacture complexity and expense. The question is how the CBCPW circuits (MMICs) can function properly with the coupling effects present. A single-layer CBCPW unconditionally leaks energy to the TEM parallel plate mode which translates to mode coupling effects existing at lower frequencies for electrically wide two-dimensional waveguides. For CBCPW MMICs the line/slot widths can easily be made much smaller than the substrate thickness. This parameter guideline reduces the leakage constant for the one-dimensional structure [12], [32], decreases the coupling to the rectangular waveguide modes in finite-width structures [28], and reduces the coupling to the cavity modes in finite-length waveguides [35], The procedure presented here will also relate the mode coupling in CBCPW with lateral sidewalls (realistic two-dimensional structure) to the leakage rate or constant of an infinite-width waveguide (nonphysical transmission line) [28], The method calculates the leakage constant and plots the effects o f mode coupling on the dominant CBCPW mode to 40GHz. Parameters recommendations for single-layer CBCPW MICs in terms of the cross-sectional circuit area, substrate thickness, dielectric constant, and operating frequency to significantly reduce the mode coupling effects will be described with supported examples. This work will also demonstrate that the one-dimensional (nonphysical structure) analysis can be an important tool in the design of single-layer CBCPW. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 The procedure follows the one and two dimensional SDM analysis for calculating the leakage rate and mode coupling effects from Chapters III and IV, respectively. A total of nine basis functions (five for E and four for E,n) are employed in the analysis since the fields are tightly coupled to the slots for small cross sections (S+2W) and the slot fields strongly interact. The CBCPW is assumed to be symmetric, lossless, single-layer, infinite-length and follows the configurations depicted in Fig. 3(a) and 3(b) and an ideal magnetic wall is placed at y=0 plane to excite the dominant odd mode. For the infinite width waveguide, a zero cutoff TEM parallel plate mode exists. The dominant CBCPW mode leaks energy into this PPM at all frequencies for any dielectric constant and substrate thickness and is no longer bound to the slot area. The CBCPW mode also has a complex propagation constant. For CBCPW with lateral sidewalls, rectangular waveguide modes with finite cutoff frequencies are supported. Leakage does not occur but mode coupling can cause the dominant mode field to spread out across the waveguide instead of being confined to the slot region and the dominant propagation constant is real. The SDM determines the degree of mode coupling by plotting the dominant mode field pattern. The one-dimensional infinite-width analysis is first considered. The examination o f the dielectric constant to reduce the leakage rate (a /k0 ) is investigated. Fig. 53 illustrates this calculation at 40GHz with h=1.27mm for CASE 48 and shows that a lower er is desired to reduce the leakage constant. The leakage for er=10 is ten times greater than that for er=2. The rate varies approximately as er at lower dielectric constant values and becomes more a linear function with higher dielectric constants. The desire for a smaller guide wavelength to decrease the circuit size dictates a larger dielectric constant and an er=10.2 (approximately corresponding to Duroid™ 6010) will be utilized throughout this section. The leakage rate as a function of the frequency and the cross-sectional circuit area is shown in Fig. 54. The (S+2W) parameter is decreased in half with h=0.635mm in Fig. 54(a) for the three plot curves (CASES 49-51) while the substrate thickness is successively increased by a factor of two in Fig. 54(b) at f=40GHz (CASES 52-54). The larger cross section and higher frequency of operation enhance the leakage. The leakage rate varies linearly with frequency and approximately as (S+2W)2. A 2:1 strip to slot ratio (S:W) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 0 .0 5 0 0.040 CASE 48 0.030 Q 0.020 0.010 0.000 14 Fig. 53. Leakage rate versus relative dielectric constant for single-layer CBCPW. Fig. 3(a) is referenced here with S=2W=0.3175mm and h=1.27mm and f=40GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 o .i iQr i i x i CASE CASE CASE CASE i — 49 50 51 50 Eqn. (5.5) o 0.050 0.050 0.040 0.030 0.020 0.010 0.000 10 15 20 25 30 FREQUENCY (GHz) (a) 0.150 0.135 0.120 • 0.105 • Q ------------......... x CASE 52 CASE 53 CASE 54 CASE 53 Eqn. (5.5) / / 0.090 X / 0.075 iS / S s s 0.060 s / y 0.045 / 0.030 / .•••***** ............. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 S+2W (m m ) (b) Fig. 54. Normalized leakage rate curves for single-layer CBCPW. (a) Plots versus frequency with e=10.2 and h=0.635mm with CASE 49 S=2W=0.3175mm, CASE 50 S=2W=0.15875mm, and CASE 51 S=2W=0.079mm. (b) Curves versus cross-sectional circuit area with sr= 10.2 and f=40GHz with CASE 52 h=0.3175mm, CASE 53 h=0.635mm, and CASE 54 h=l.27mm. Fig. 3(a) is referenced for the parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 approximately corresponds to a 50H characteristic impedance line. From Fig. 54(b), keeping the cross-sectional circuit area small compared to the dielectric thickness and increasing the substrate height (h) reduces the leakage rate as observed in [12] and [32]. The leakage varies approximately inversely with the substrate thickness as indicated in Fig. 54(b). This inverse relationship between the leakage and the substrate height holds for CASES 53 and 54 however, CASE 52 does not exactly follow this pattern at such large attenuation rates. For h=2.54mm, leakage to the TMt and TE, modes (degenerate pair) exists above 29GHz which will increase the rate. For h=5.08mm, leakage to the TM2 and TE2 modes (degenerate pair) also occurs above 30GHz. For (S+2W) small compared to the wavelength, a closed form relation for the normalized leakage rate of the dominant CBCPW mode into the TEM PPM has been derived from [12] using the reciprocity theorem as a . *Z (S+2 W f k0 - 4h377>.j. ' ' where Z is the characteristic impedance, seff is the effective dielectric constant, and Xg is the guide wavelength. The accuracy of (5.5) is shown in Fig. 54 and verifies the relationships between the leakage rate and the cross-sectional circuit area, thickness, relative dielectric constant, and frequency as determined from the SDM calculations. The results from (5.5) overestimate the leakage on the order of 20%. As described in Fig. 54 and (5.5), decreasing (S+2W) reduces the leakage constant most dramatically among the parameters o f the CBCPW. This complete analysis of the leakage rate of CBCPW in terms of these waveguide dimensions and validation o f (5.5) with the full electromagnetic wave SDM is an original contribution of this dissertation as part of [28], The finite-width, single-layer CBCPW with lateral sidewalls which two-dimensional version of the infinite-width structure is analyzed. is the This waveguide is referenced from Fig. 3(b) with 2A=25.4mm and corresponds to a realistic MIC circuit dimension. The field plots described here are for the dominant mode at x=-0.01mm. As mentioned earlier, increasing the substrate thickness reduces the leakage rate but additional PPMs must be considered. These PPMs now become rectangular waveguide modes. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 effects of these additional modes coupling to the dominant CBCPW mode are presented in Fig. 55(a) for various substrate heights. The dominant rectangular waveguide mode is the TM 0ft with h = 1,3,5,... ,00 . As indicated, the dominant mode spreads out across the waveguide as the fields become a combination o f the dominant and waveguide modes. Each curve is normalized to the maximum field component and the plots are for the dominant mode outside the regions o f strong coupling with the waveguide modes, with respect to frequency (see Chapter IV). The values o f (S+2W) for CASES 51, 55-57 are modified so that (S+2W)=h/4 ratio is maintained for each plot to demonstrate the mode coupling effects with the higher order modes for this example (TM AA and T E ^ for m = \,2 and h= 1,3,5,...) . Recall that these higher order TM and TE modes are degenerate modes. For a constant (S+2W)/h value, the leakage and mode coupling effects should be identical for each case. With h=2.54mm, the mode coupling effects occur with the TM W mode and with h=5.08mm the coupling effects are present with the TM, h and TM 2,h modes. The following relationship is invoked to determine the cutoff frequency of the next higher order waveguide mode as a function of the substrate height as 2 2 (& ) + © (5'6) As the lateral width (2A) is narrowed, the degree o f mode coupling is also decreased as fewer waveguide modes couple with the dominant mode. For 40GHz operation, h<1.17mm is required to cutoff the T M ,, and T E , , modes. Fig. 55(b) demonstrates the cross section necessary to minimize the mode coupling for h=1.27mm. The dominant mode is plotted around 40GHz between the mode coupling regions before or after the TMq ,3 mode (the curves are plotted at 36, 37, 38, and 39GHz, respectively). A maximum cross-sectional circuit area between h/4 and h/8 is necessary to significantly reduce the mode coupling for this case. It is interesting to note that the mode coupling effects are lessened as (S+2W) is decreased in CASES 55, 58-60 in Fig. 55(b). The dispersion curves for CASES 58 and 60 are plotted in Fig. 56(a) and Fig. 56(b), respectively. Concentrating only in the regions of strong coupling about 40GHz, the closer the dispersion curves approach each other (dominant and waveguide modes), the smaller the coupling effects. Strong mode coupling occurs for a large separation between the propagation curves and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 1.0i CASE 51 CASE 55 CASE 56 CASE 57 0.8 0.6 0.4 O 0.2 Q. 0.0 2 0.2 < -H ■**-0.4 - 0.6 - 0.8 - 1.0 " y (m m ) (a) 1.0i CASE 58 CASE 59 CASE 55 CASE 60 0.8 0.6 Ui 0.4 yi r. 0.2 0 . 0.0 2 < - 0.2 - 0.6 - 0.8 -W y (m m ) (b) Fig. 55. Normalized electric field plots for dominant CBCPW mode demonstrating the mode coupling effects for a single-layer structure at x=-0.01mm. Refer to Fig. 3(a) for dimensions with er =10.2 and 2A=25.4mm and f approximately 40GHz. (a) Results as substrate height (h) is decreased with (S+2W)=h/4 for each case. CASE 51 h=0.635mm, CASE 55 h=1.27mm, CASE 56 h=2.54mm, and CASE 57 h=5.08mm. (b) Plots as function of the cross section with h= 1.27mm. CASE 58 S=2W=0.635mm, CASE 59 S=2W=0.3175mm, CASE 55 S=2W=0.1588mm, and CASE 60 S=2W=0.0794mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 2.8 2.7 TMo.13 - 2.6 o 2.5 CBCPW ■ r 03CPW •Si 2.4 o -V 2.3 - 2.2 TMo,17 ‘ 15 36 37 38 39 40 41 42 43 44 4! FREQ (GHz) (a) 1 *'1 2.8 1 1 I 1 2.7 . TMo.ii ^ o * TMo.13 2.6 TMo.fi^, 2.5 2.4 . : bcpw CBCPW 2.3 2.2 2.1 2. • 36 37 38 TM0,17 yS ‘Mo.fi 39 40 41 42 43 44 45 FREQ (GHz) (b) Fig. 56. Normalized dispersion curves for single-layer CBCPW. With sr =10.2 and h=1.27mm and 2A=25.4mm and refer to Fig. 3(b) for parameters, (a) CASE 58 with S=2W=0.635mm. (b) CASE 60 with S=2W=0.0794mm. The crossed points are the CBCPW mode in each case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 and this is verified from Fig. 55(b). Hence, the degree o f mode coupling can be qualitatively estimated from the dispersion curves. The frequency for strong coupling between the CBCPW mode and the TMq 13 mode encompasses a wide band from 3 7-43 GHz in Fig. 56(a) while for CASE 60 this coupling range is concentrated only at 36GHz. The dispersion results from Fig. 56(a) are similar to the responses and mode coupling effects described in Chapter IV. To determine the mode coupling versus frequency, CASE 50 is presented in Fig. 57. This example shows that significant mode coupling commences between 10 and 15GHz. E. Design Procedures for Single-Layer CBCPW On the basis of the examples presented in the previous section and several other simulations, an empirical estimate relating strong mode coupling in CBCPW with lateral sidewalls to a high leakage rate in the infinite-width structure can be expressed. An estimation of the leakage rate will be suggested with supported cases from the previous section to minimize the leakage and mode coupling effects. The cross-sectional circuit area for a given substrate thickness, dielectric constant, and frequency can be approximated from the closed form relationship o f (5.5). A complete design procedure will be described to determine the parameters for the single-layer CBCPW for successful operation with minimal mode coupling effects, without requiring the user to possess a one and two dimensional SDM algorithm. A normalized leakage rate for (aJkQ) of approximately less than 0.007 or a<0.155dB/A,g is recommended for minimum leakage and mode coupling effects. This estimated value was obtained by observing the mode coupling effects as a function of (S+2W)/h in Fig. 55(b) which indicated a significant reduction in the field spreading in CASES 59 and 55. These examples correspond to h/4<(S+2W)<h/8 and the average value o f (S+2W)=h/6 was selected and then mapped onto the leakage curve plot for CASE 54 of Fig. 54(b) with an expanded log scale. The mode coupling as a function o f frequency for CASE 50 was identified from Fig. 57 translated to the leakage graph o f to occur between 10 and 15GHz which was Fig. 54(a) (again using a log scale). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These 155 1.0 0.8 FREQ-10GHz FREQ-15GHz FREQ=20GHz CASE 50 E x AM PLITUDE 0.6 0 .4 0.2 0.0 0.2 - - 0 .4 - 0.6 - 0.8 - 1.0 -1 4 -1 2 -1 0 -8 -6 -4 -2 y 0 2 (m m ) 4 6 8 10 12 14 Fig. 57. Normalized electric field plots for the dominant CBCPW mode demonstrating the mode coupling effects as a function of frequency for a single-layer structure at x=-0.01mm. CASE 50 with er=10.2 and h=0.635mm and S=2W=0.1588mm and 2A=25.4mm. Refer to Fig. 3(b) for dimensions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 examples support the estimate of (S+2W)<^.g/16 to minimize the leakage and mode coupling effects in a single-layer CBCPW. This design parameter for the leakage rate is intended for dielectric constants of er =10 which could apply to alumina, Duroid™ 6010, and GaAs substrates. The parameter specifications of [12] suggested (S+2W)<h/4<X.g /20 to reduce the leakage effects for useful CBCPW operation in GaAs circuits, however no examples were given to substantiate this guideline as compared to the presented material in the last two sections o f this work. The leakage rate of (5.5) overestimates the value derived by the SDM approximately 15-20% so a normalized constant of 0.008 is suggested. For the example o f Fig. 55(b) with h=1.27mm at f=40GHz, (5.5) calculates a value of h/(S+2W)=5.21. CASE 50 with h=0.635mm and h/(S+2W)=2 would predict an upper frequency of operation with minimum coupling effects to 11.8GHz. Both o f these results are confirmed by the field spreading plots of the dominant CBCPW mode. A characteristic impedance of 50Q and Eeff =6 were assumed in the above calculations. An iterative design procedure to determine the dimensions for single-layer CBCPW with reduced mode coupling based upon the above work would be as follows: 1) Select a dielectric constant (preferably a high value around 7<sr <13); 2) Generate a set of design curves for the characteristic impedance and the effective dielectric constant using the closed form (quasi-static) relation from [50] for various (S/W) and (S+2W)/h values; 3) Determine the upper frequency of operation for the application; 4) Use the cutoff relation for the higher order waveguide modes (5.6) to find the upper limit for the substrate thickness and then choose a height; 5) Select Z; 6) Select an initial value of (S+2W) and obtain a value for eefr from step 2; 7) Utilize (5.5) with (a /k0 )=0.008 to determine (S+2W); 8) Compare (S+2W) with that estimated from step 6 and verify the characteristic impedance. If both parameters are sufficiently close, goto step 9. If not, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 determine the effective dielectric constant and Z from the design table of step 2 for the new (S+2W) value and goto step 7; 9) Select a value for W based upon the circuit etching resolution available and then determine S. The above procedure will probably require a few iterations to converge but can be calculated manually. The effective dielectric constant value obtained from the quasi-static method from [50] can be implemented to higher frequencies. This statement is valid because the cross sections utilized in these applications are smaller than the dielectric thickness and the fields will be tightly coupled to the slot region and dispersion effects should be small. Using the above procedure for h=0.635mm with sr=10.2 at f=40GHz, a (S+2W)=0.1588mm was predicted. This line/slot restriction demonstrates the necessity for the use o f multi-layered CBCPW to reduce the mode coupling effects in wide ffequency-band MIC applications since this resolution is almost five times smaller than the available dimensions in our etching facility. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 CHAPTER VI THREE-DIMENSIONAL ANALYSIS OF MULTI-LAYERED CBCPW INCLUDING CAVITY EFFECTS In an actual circuit implementation with input and output port (finite width and length), a three-dimensional structure is produced. This situation is pictured in Fig. 8(b) for the waveguide formed with the coaxial connector blocks. The rectangular waveguide formed in two-dimensions for CBCPW with lateral sidewalls becomes a rectangular cavity in three-dimensions with longitudinal PEC shorting walls at z=0 and z=2B as viewed in Fig. 8(b). For a packaged CBCPW, a resonant cavity is produced as a substructure of the waveguide. A cavity stores electric and magnetic energy in the volume and the energy is coupled to an external circuit via a probe or aperture. Resonator circuits are very useful for some applications such as antennas, oscillators, and filter networks but not in the transmission properties of CBCPW. Coupling to the cavity modes should be avoided for the CBCPW transmission line application. For electrically large configurations, resonant frequencies fall within the frequency range of interest (dc-40GHz in this dissertation). The resonances occur at discrete frequencies (the fields within the cavity exist only at these frequencies) and will limit the bandwidth of operation. If the resonances are present within the frequency range (depending on the dimensional parameters of the waveguide), two basic outcomes are possible. For a through transmission line as shown for CASE Q of Fig. 42, sharp insertion losses are present at several o f the resonant frequencies even though the dominant CBCPW mode is bound to the slots. A portion o f the power from the CBCPW circuit couples to the cavity modes instead of propagating the energy to the output port. The amount o f the insertion loss depends on the coupling between the CBCPW circuit and the cavity modes (coupling coefficient or factor) and the Q (quality factor) o f the resonant Part of the data reported in this chapter is © 1994 EEEE and submitted to IEEE for publication. Reprinted, with permission, from 1994 IEEE Microwave Theory cmd Techniques International Microwave Symposium Digest, pp. 1697-1700 (see References [15], [28] and Appendix G). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 mode. The Q describes the efficiency o f the energy storage within the cavity for a particular mode. desirable. For the typical resonator applications discussed above, a high Q is The second outcome is the coupling/interference effects between isolated circuits within a system via the substrate resonances for a CBCPW and disturbing the intended operation. This problem includes coupling between the circuits as a function of position on the substrate [35], creating locations with enhanced coupling with the cavity. The residual resonances shown in CASE Q o f Fig. 42 for the multi-layered CBCPW must be suppressed as the 2-4dB insertion losses are unacceptable for a wide-band transmission line. The resonances o f this example are identified with possible rectangular cavity modes which are calculated with an ideal analysis and described in the next section and are listed in Table XX. This procedure was performed in Table XIV with Fig. 30 for a single-layer CBCPW. A close correlation between the measured and modeled resonant frequencies is evident (the small differences between the frequencies are attributed to the uncertainties in the relative dielectric constants) and not all o f the possible resonant modes appear in the data. TABLE XX Measured and calculated resonant cavity frequencies for multi-layered CBCPW o f CASE Q from Fig. 42 for the TM0 ,^ modes. fr (GHz) measured calculated TM o.1.2 5.8 5.6 TM o.1.3 7.5 7.2 TM o.,4 9.3 9 TH ...1, 22.5 21.9 TM o.1.13 TM o.1.15 25.9 25.2 29 28.4 TM o.1.17 31.8 31.4 TM o.1.19 34.3 34.2 TM o.1.21 36.8 36.7 mode Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 energy is coupled to the cavity at the coaxial feeds and by the CBCPW mode as it propagates along the transmission line length. For a bound CBCPW mode, the coupling from the above possible sources to the cavity should be very small since the fields of the dominant mode are localized in the slot areas and the cavity mode field representations extend across the waveguide width and length. Therefore, the low loss dielectrics for this case produce a sufficiently high Q to draw over half o f the power into the cavity (for resonances greater than 3dB). CASE R of Fig. 42 showed that the resonance effects were also a function o f transmission line length and this observation would indicate that the majority of the energy coupled to the cavity modes was from the dominant CBCPW mode and not the coaxial feeds. The use o f a damping (lossy) material to reduce the residual resonances within CBCPW by decreasing the substrate cavity Qs is necessary and this configuration was demonstrated successfully in CASE S of Fig. 42. The absorber was placed on the perimeter of the waveguide (within the test fixture blocks outside the coaxial connectors in CASE S) which may not be feasible in some applications and an alternative layered (planar) waveguide is suggested. This multi-layered damped waveguide is detailed in Fig. 58 as the h6 substrate for a lower dielectric loaded CBCPW and as the h, layer for an upper loaded circuit. A lower cavity is formed below the ground planes encompassing layers h4, h5, and h6 and an upper cavity is generated above the ground planes between substrates h3, h2, and h,. The damping planar layers are represented by complex permittivities (erI, srt) and complex permeabilites (|a.rI, pr6). Fig. 59 illustrates an implementation o f the damping material on the perimeter of a circuit. With this configuration, connecting the ground planes to the conductor-backing would probably require shorting bars on the sides as well as input and output ports of the CBCPW. The results from the measured data o f the single-layer CBCPW of Fig. 30 depicted resonances (sharp insertion losses) on the order o f 10-30dB which can be explained due to the mode coupling effects with the rectangular waveguide modes. The dominant mode field pattern resembles rectangular waveguide modes and efficiently couples to the rectangular cavity modes. The coupling coefficients from the CBCPW circuit to the cavity modes are not addressed in this work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 X £ rl = 5 r \ ~ l € rl Mrl = ^ r1 ~ J ^ r l Sf2 I , 6 uL, t - V da mpi ng material / / i . £ r4 da mpi ng material £ r6 = £ ’r 6 ~ J E r€ Mr6 = V r S ~ 3 ^ r t Fig. 58. Cross-section of the multi-layered CBCPW with damping material for the upper and lower resonant cavities. The complex permittivities and complex permeabilities for the lossy layers are shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 £n = l d a m p in g m a te ria l y (a) y d a m p in g m a te r ia l m (b) Fig. 59. Application of the damping material with a thickness t on the perimeter of the symmetric lower dielectric loaded CBCPW to reduce the cavity Qs . (a) Side view, (b) Top view. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 The presence of the damping (lossy) material in CBCPW is proposed to reduce the cavity Qs without significantly affecting the propagation characteristics of the CBCPW circuits. The layered damping configuration can be easily modeled by the multi-layered SDM solving for the propagation losses in the two-dimensional case and the resonant frequencies and the Qs in the three-dimensional structure. In particular, effects of the damping material on a half-wavelength CBCPW resonator and the cavity modes are investigated. The inclusion of a CBCPW resonator in the SDM analysis is an original contribution from [15] and [28] as part of this dissertation. This chapter will include a basic overview of the idealized CBCPW cavity resonator and introduce the damping material utilized in this chapter (doped silicon and microwave absorber). The electrical properties o f the absorber material used in the experimental data o f this dissertation (see Fig. 42) were not known so the characteristics of a microwave absorber example from [59] and doped silicon o f [60] were implemented in the SDM analysis. Doped silicon is ideally suited within a MMIC application but is included in the analysis o f the MICs here as a design demonstration. The extension of the SDM to three-dimensions for the investigation o f the CBCPW resonator is developed. The three-dimensional SDM outputs are verified with a microstrip example and additional results are presented. Design examples with the damping materials are described for both upper and lower loaded multi-layered CBCPW configurations. Tradeoffs to maximize the CBCPW resonator circuit Q and minimize the cavity mode Qs are realized with the lossy layer thickness, doping, and dielectric heights. This methodology is similar to [59] and [60] which applied the above damping materials to packaged microstrip circuits and was suggested in [35] for use in CBCPW circuits. The presentation o f the Q design curves for CBCPW is also an original contribution from [15] and [28], Finally, experimental data is included for a straight gap-coupled CBCPW resonator with absorber material and illustrates the improvement in the response. A. Overview o f the Cavity Effects in CBCPW The resonance mechanism in packaged CBCPW is illustrated with the equivalent circuit Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 in Fig. 60 from [35], The packaged CBCPW is represented by the circuit and the cavity modes which are modeled by transformers and series RLC elements. For this example, three cavity modes are assumed present within the frequency band of interest. Associated with each cavity mode is an equivalent lumped RLC circuit with the resistance R accounting for the power loss (conductor, dielectric, radiation, and coupling losses) in the resonant circuit. The inductance L and capacitance C elements determine the resonant frequency for the cavity modes. The Q for each cavity mode can be described by the R, L, and C elements. The transformers with turns ratio (n) for the mutual coupling, approximate the coupling coefficient from the CBCPW circuit to each of the cavity modes. The coupling from the CBCPW circuit to the cavity is a function o f the structural parameters and the type of discontinuity. Efficient coupling to the cavity occurs when the dominant CBCPW mode is not bound to the slots and for vertical discontinuities (dominant rectangular cavity modes have primarily vertical field components). For the cases considered here, the turns ratio o f the transformers is very small (light coupling). For CBCPW, the coupling o f energy to the unintentionally formed rectangular cavity should be avoided since a loss o f energy in the transmission response of the line exists and interference effects within the circuit are possible. The RLC equivalent circuit is modeled with an input admittance. For frequencies away from resonance, the RLC circuit has a low admittance. At resonance with a high Q cavity, the RLC admittance is very high allowing a large current to flow into the resonator cavity circuit (YV=I where Y is the admittance presented at the terminals o f the transformer, V is the voltage across the terminals, and / is the current within the RLC circuit). The admittance presented at the input terminals of the RLC resonant circuit model is n (6 . 1) for each cavity mode (/'= 1,2,3) for the example considered here. The resonant frequency becomes ( 6 .2 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 CBCPW C irc u it inputt « • UAAJ 1:ni* R 1 £r, 0 output « 11 1:no* C } 1 : n 3* ^3 ^ 3 ^3 Fig. 60. Equivalent circuit representation for resonance mechanism in a packaged CBCPW circuit with cavity effects. Three cavity modes exists within the frequency range of interest for this example and are modeled using R<LiCl lumped elements and mutual coupling with turns ratio n y Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 and the input admittance of (6.1) is purely real at resonance. The Q or quality factor for each cavity mode is an important parameter specifying the frequency selectivity and performance of a resonant circuit. In general terms, the Q is the ratio of the energy stored to the energy loss in the resonant circuit. The Q discussed here is the unloaded Q of the resonant circuit and is expressed in terms o f the RLC circuit as n Q' _ 03o.i L i Ri 1 cd0 O.i CiR i (6,3) The use o f damping lossy material to reduce the cavity Qs is modeled by increasing the equivalent resistance R of the cavity circuit. In the vicinity of resonance, co = co0 + Aco and the input admittance from (6.1) can be written as Y = —p X ----------- =r /? [ 1 +y'2Q Aco/co0] where the following approximation has been used (6.4) l/(co0 + Aco) » (1 - Aco/co0)/co0 . In microwave systems, sections of transmission lines or metallic enclosures are implemented as resonators in place of the lumped parameter circuit. The input admittance is modified to account for the losses in the resonator by replacing the resonant frequency with an equivalent complex resonant frequency. The admittance of (6.4) can be described with this frequency as Y (m0/2Q )* j [co - oo0(l +.//2Q)] r6 5 , and the complex resonant frequency becomes cor = <00(1 + jl 2Q). (6.6) If the cavity is filled with a lossy dielectric material, then the permittivity becomes complex and is described as / er = s r // - j & r / = e r(l -ytanS) (6.7) and if the damping material has a lossy magnetic susceptibility characteristic then the permeability is complex by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 / n Hr = Hr - y 'H r . (6.8) The resonant frequencies and the Qs for the cavity modes in the packaged CBCPW can be approximated with an ideal analysis for the structure in Fig. 61. PEC shorting facewalls exist at z=0 (cut away to show the internal dielectrics) and z=2B in the figure and no slots are present at x=0. Instead of solving for the propagation constants for the waveguide modes for the two-dimensional structure, the resonant frequency is calculated for the three-dimensional cavity as cor = co0 + Jco" (6.9) and from (6.6), the Q can be determined as CDn Q*-T72co (6.10) The resonant frequency can be derived for the multi-layered cavity (two dielectric layers to control the propagation characteristics o f the CBCPW mode and the rectangular waveguide modes and a damping layer to reduce the cavity mode Qs) of Fig. 61. The equivalent admittance expressions for the Y4 (TM modes) and Y4 (TE modes) from (A.14)-(A.16) in Appendix A are utilized as the admittance at x=0 for layers h4, h5, and h6 and with the conductor-backing. A magnetic wall at y=0 is assumed for the excitation of the dominant CBCPW mode. The wavenumber in the ^-direction becomes ky - fm/2A with fi= 1,3,5,... ,oo and that in the z-direction becomes k. = piz!2B with p = 0 ,1 ,2 ,..., oo for TE and TM modes. The transverse resonance method is employed along with Muller's method to locate the complex zeros and hence the resonant frequencies for the cavity modes supported in Fig. 61. This approach for the cor calculation is necessary due to the complexity of the analysis for the three-layer configuration. The transverse resonance method is an application o f transmission line theory to the equivalent circuit of a transverse section of the guiding structure and is applicable to any waveguide that is uniform in the direction o f propagation. The transverse resonance method leads to the required eigenvalue equation in a direct manner. For the structure of Fig. 61 with a ^ h short circuit at x=0, the equivalent impedances are calculated for Z 4 and Z4. The transverse resonance condition requires that the sum of the impedances looking toward the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 \N ^ c o n d u c to r $ jp j W , Fig.61. Ideal rectangular cavity representation of the multi-layered packaged CBCPW. PEC shorting facewalls exist at r=0 and z=2B. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 short circuit at x=0+ and that at the input to the three-layer transmission line at x=0’ for the T M ^ and TE modes vanish. The determination of the cavity Qs using (6.10) is similar to the procedure in [61]. The dominant cavity modes analyzed here for the CBCPW MICs are the T M ^ . The mode field patterns o f the rectangular cavity are similar to the TM or TE waveguide modes in a z=constant plane. The Q varies approximately as the ratio o f the volume to the surface area of the cavity. No conductor losses are included in the cavity analysis in this chapter. Since the electrical properties of the broadband absorber material used in the experimental data are not known, a microwave absorber example from [59] was implemented. The parameters stated for this material for 9-12GHz were er=21( l-y'0.02) and |j.r= l.l(l-y l.4 ) with h=1.27mm (thickness on the order of MIC dimensions). Microwave absorber is inexpensive and flexible to various circuit conditions (used in horizontal or verticai locations and small gaps). The drawbacks with absorber are the electrical properties are not well characterized and there can be difficulty in configuring this material as a full width layer (y=2A) within a multi-layered structure especially with tape shorting sidewalls. The other lossy material utilized in this chapter is doped silicon / (Si) with s r = 12. This lossy volume is well characterized, inexpensive, and provides damping with a small thickness [60], Damping with doped silicon within a cavity would be optimally applied to MMICs but is introduced here to demonstrate some o f the design procedures within CBCPW MIC circuits. The doping level is selected so that the skin depth (5S) is equal to the substrate thickness at the center of the band of interest (damping has bandwidth characteristic) and presented as «. = where co is the radian frequency, p0 =4te <«•"> x 10'9H/cm, and a is the conductivity and is expressed as a =W „ »e where q is the electroncharge (6-12) 1.6 x 10'19C, |xn is the electron mobility for Si 1350cm2/V-sec, and ne is thedoping level in cm'3. The dielectric loss tangent term is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with e r = 12 and e0=8.854 x 10'14F/cm. As an example o f the above design procedure with f=22.5GHz and a Si thickness of 0.025cm, the skin depth is set 5s=0.025cm. A lower doping level results in less damping while higher doping starts to make the Si layer behave like a metal withan increase in the enclosure Q.The conductivity is derivedfrom (6.11) as a =1.8mho/cmand the doping level from(6.12) becomes ne =8.35 x10l5cm'3. Finally u / applying (6.13), s r =12. Silicon has no magnetic properties, therefore p. =1 and n Hr r = 0. The effects of the damping layers on the propagation characteristics of the CBCPW and waveguides modes must first be considered before the cavity analysis. In particular, the lossy materials should not substantially increase the attenuation loss or enhance the mode coupling effects (relative dielectric constants for the microwave absorber and doped silicon are very high). Fig. 62 includes the attenuation loss for the dominant CBCPW mode without (CASE 61) and with (CASE 62) the doped Si layer. The loss tangents for the upper dielectric (6010 Duroid™ substrate er=10.2) are listed as a function of frequency at the end of section A in Chapter III. The loss tangents for the 5870 board (er=2.2) at 1, 10, 20, 30, and 40GHz are 0.6, 1.1, 1.55, 1.9, and 2.25 x 10'3, respectively. The attenuation loss calculations are verified by the PCAAMT™ program [62] in Fig. 62. For CASE 62, 8S=0.025mm and a =1.8mho/cm for f=22.5GHz (maximum damping at center of band). As indicated from the figure, the attenuation loss for the dominant CBCPW mode is not significantly increased which is expected since the fields are localized about the slot areas as depicted in Fig. 48(b). The propagation characteristics o f the dominant rectangular waveguide mode TMq j must also be investigated with the incorporation o f the lossy materials. The effective dielectric constant for this mode in CASE 61 at 40GHz is 5.12. seff increases to 5.29 with the same Si thickness o f CASE 62. An interesting response is observed with the microwave absorber for CASE 61 with h6=0.635mm and extending the electrical properties of the absorber previously stated between 9 and 12GHz to 20GHz. An initial assumption would be an increase in the effective dielectric constant Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 7 x O o CASE 61 CASE 61 Ref. C62J CASE 62 6 5 4 3 Q. 2 0 0 4 8 12 16 20 24 28 32 36 40 F R EQ ( G H z ) Fig. 62. Attenuation (dielectric) losses for the dominant CBCPW mode examining the effects of the silicon lossy layer. Refer to Fig. 58 for dimensions with 6rl= l, er4=10.2, er5=2.2 and h^Smm, h4=h5=0.635mm, h2=h3=0 and S=2W=0.635mm and 2A=25.4mm. CASE 61 h6=0 and CASE 62 and h6=0.25mm and «e=8.35 x 10 cm'3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = 12 172 from seff=3.79 since e r = 21, however the analysis shows eeff=3.1. This result is due to / the presence of the magnetic properties o f the absorber with p. =1.1. The determination for the resonant frequencies and the cavity Qs o f the ideal rectangular cavity of Fig, 61 is presented and verified by example cases. The microwave absorber case is implemented in a packaged microstrip structure from [59], The configuration consists of a dielectric layer, air, and damping material within a rectangular cavity (package). The calculated resonant frequency compares well with the result from this reference and is presented in Table XXI. TABLE XXI Resonant frequency verification of the TMq , , mode in a packaged microstrip structure with microwave absorber damping. dimensions with er4=10.5(l - y'0.0023), Reference Fig. 61 for er5= l, er6=21(l - y0.02), prt=l.l(l-yl-4), and h4= 1.27mm, h5=l0.16mm, h6= 1.27mm and 2A=15mm and 2B=24mm. complex fr (GHz) mode TMo.,., Ref. [58] 11.051+/0.249 presented work 11.058+/0.259 An example for the doped silicon case is adopted from [60] for a packaged microstrip structure and presented in Table XXII. The results compare closely with those from this reference. The application of the doped Si layer for a CBCPW MIC configuration (with no slots) is depicted in Fig. 63. CASE 63 of this figure represents only the dielectric losses for 6010 and 5870 Duroid™ substrates without the doped layer present. CASE 64 corresponds to a silicon layer thickness o f 0.1mm and CASE 65 for h6=0.254mm. Note that a thicker Si layer reduces the Qs as an increase in the lossy volume exists within the cavity. A significant reduction in the cavity Qs is obtained using the Si damping over a broad frequency-band centered at about 20GHz. Resonances with different mode numbers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 TABLE XXII Q verification for various modes in a packaged microstrip structure with doped silicon damping. Reference Fig. 61 for dimensions with sr4=12.7, sr5= l, ' 15 3 = 12 and n = 4.0 x 10 cm' and h4=0.1mm, h5=0.4mm, h6=0.25mm and 2A= 12mm and 2B=20mm. mode fr (GHz) Q Ref. [59] presented work Q H u 10.75 18 17.9 TMo.1.2 14.4 14 13.3 TM o.,.3 19 10 10 TM o.,.4 24 8 7.79 TM q.,5 29.3 6 6.27 TH n* 34.8 5 5.16 but similar resonant frequencies have approximately the same Q. B. SDM Analysis of Three-Dimensional CBCPW Resonator Circuits The spectral domain method (SDM) from Chapter II will be extended to simulate three-dimensional resonator circuits. The finite-length structure is analyzed to demonstrate the effects of lossy damping layers on the unloaded Qs o f a CBCPW resonator and rectangular cavity modes, which exist when the CBCPW circuit is placed in a package. The resonator circuits for CBCPW and microstrip are illustrated in Fig. 64. The microstrip circuit is introduced to validate the numerical procedure with published results. The resonators are ideal half-wavelength center-fed circuits and are shorted on both ends for the CBCPW structure and are open on both ends for microstrip. The circuits are also symmetrically positioned with respect to the y and r-directions. The physical resonator length is 2L from Fig. 64 and the physical length of the ideal package is 2B. The multi-layered resonator circuit (see Fig. 58) is then placed within the package as in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. --------CASE 63 ..........CASE 64 - CASE 65 o 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 FREQ (G H z ) Fig. 63. Demonstration of the damping effects within a rectangular cavity for a MIC example using doped silicon. Refer to Fig. 61 for parameters with / sr4=l0.2, er5=2.2, = 12 and h4=h5=0.635mm and 2A=2B=25.4mm. Dielectric loss data for h4 and h5 substrates is listed in the text. CASE 63 16 . h6=0, CASE 64 h6=0.1mm and n= 5.2 x 10 cm , and CASE 65 h6=0.25mm and rt,=8.35 x 10 cm*3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 y -rS? >/ / / / / / / > s s / (a) < co 2L 2B (b) Fig. 64. Top view representations at x=0 of the center-fed half-wavelength resonators considered in the analysis, (a) CBCPW and (b) microstrip. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 Fig. 8(b) connecting the ground planes at (x=0, y,z), the conductor-backing at (x=-h4-h5-h6, y,z), and the cover plate at (x=h3+h2+hj, y,z) with lateral PEC shorting sidewalls at z) and longitudinal PEC shorting facewalls at (x j\ z=±B). For a transmission line is shorted at the end, standing waves develop and both the electric and magnetic fields have null points at periodic intervals along the line. The input current for this lossless, ideal transmission line (analogy applied to the center-fed CBCPW resonator) is Im (z=0) = where Z is the characteristic impedance, Vs is the cos source voltage across the input terminals, and L is the length o f the transmission line from the source at z=0 to the shorted end. If the goal is to minimize the power transfer to the resonator circuit (one-port device), then 7in(z=0) should be minimized (P=I 2IZ). This can be achieved by requiring L=njkgl4 with ^=1,3,5,... and nt is the integer harmonic number for the resonator. The full length of the resonator in Fig. 64(a) is 2L so that 2L=w/A.g/2 and the half-wavelength resonator is derived («f=l). If the goal is to maximize the power transfer to the resonator circuit, then 7in(z=0) should be maximized by requiring n ,-2, For structures with sidewalls and facewalls, the Fourier transform of (2.3) is replaced by a discrete, finite, double transform where <|> is defined over the interval [-A.A] in the ^-direction and <J> is defined over the interval [-B,B] in the z-direction and satisfies the Dirichlet conditions of (2.6) with B A (6.14) -B -A and 2 a 2B -f -f ^ X,y' -B -A ^ ky'y ^ k 'J “ ^ ^ (615) (Mx.j/.z) = X X %{x,kyi, k :l) e -jKk*yy e ^k -iZ . I = - o o i = -o o (6.16) where the inverse Fourier transform is defined as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 The solutions for waveguides with finite dimensions are characterized by discrete spectra of eigenvalues. Accordingly, the integrals in Chapter II are all replaced by double summations in terms of discrete values o f kyi (spectral variable in the ^-direction) from /=oo to qo and in terms of discrete values o f kzl (spectral variable in the z-direction) from /=- to oo for (2.30) and (2.33). The values o f kH are determined by examining the field oo behavior o f Ex and Hx along the z-direction. The dominant CBCPW mode for a resonator shorted on both ends o f Fig. 64(a) has a bounded behavior for the longitudinal components of the electric field (£_x and EyJ at z=±L for the ends of the slot (see section B of Chapter II). The E field is zero outside the slot regions at x-0. The fundamental mode (nt = \) and odd harmonics (^= 3,5,...) of the CBCPW resonator are excited by a magnetic wall at z=0. The even harmonics (nl =2,4,6,... o f the CBCPW resonator) are excited by an electric wall at z=0. The fundamental mode of the resonator is assumed in this analysis. The basis functions for the unknown electric fields in the slots are even and odd functions with respect to z for Ey and E„ respectively. PEC walls exist at z=0,±L and cos and 'P* e is equal to sin (^ jj|) for I = - o o , ..., e becomes -5 ,- 3 ,-1 ,1 ,3 , 5,..., oo. h ¥ and *P are the electric and magnetic scalar potential functions, respectively. spectral variable in the z-direction can be rewritten * - - ' = ( / + 2} B for /= -q o , ...,-2 ,-1 ,0 ,1 ,2 , and T *. ...,o o . The (617) Recall from (2.1b) that Ey is proportional to Parseval's theorem of (2.31) is also modified for the three-dimensional packaged structure with ? j i i { J(y,z)E*(y,z)cfydz = - L - L -B -A ^=0° *=Q0 ~ ~ X J(kyi ,k :i)E*(kyi ,k :i ) = 0. /=-°° /=-<» (6.18) £ All of the elements of the matrix [/I] of (2.33) are even functions of and the summations are represented by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 /—oo i—co /=oo / — GO £ £ ,4 = 2 £ /= —oo /= —oo £ /= 0 ,4. (6.19) /= 0 A homogeneous system of equations is now formed following the procedure in section A of Chapter II and is described in compact form here as c d (6 .20) = [0] and nontrivial solutions can only occur when the matrix is singular with det [i4(co r)] = 0 (6.21) and the complex resonant frequency cor is solved for each mode and the Q can be calculated from (6.9) and (6.10). For the excitation of the fundamental mode and the odd harmonics in the CBCPW resonator with a magnetic wall at 2=0, the rectangular cavity modes are the TM )hhp and T E Mhp modes and specified with p = l,3,5,...,oo. The poles of the Green's functions in (2.22)-(2.24) now correspond to the rectangular cavity modes and the resonant frequencies. The unknown electric fields E and E, across the slots at x=0 are expanded in terms of known basis functions E and E„n in the space domain as O O oo E . = £ d nE :n(y,z) n= 1 m=\ where cm and dn are the unknown expansion coefficients. The basis functions E ( 6 . 22 ) and E.n are separable in the y and z directional variables and are written for the CBCPW resonator defined only in the slots at r=0 and S/2< [y| < (S/2+W) and 0< \z\ <L by M X m=\ E y ,i( y ) i COS \ 2L ) nz (6.23) N 2 n= 1 E z,\ 00 L sin L (6.24) where M and N are the number o f expansion terms for the y and z-directed electric fields, respectively. Eyl (y) and E , x (y) are the basis functions used in the one and two-dimensional SDM analysis from (2.35) and (2.36) and only one ^-dependent term is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 utilized with respect to numerical computation considerations [63], This implementation for the y-directed functions can be justified in CBCPW MICs by the phase constant being accurately modeled with a single expansion term. The above basis functions are plotted as a function of z in Fig. 65. The expansion functions in the spectral domain are obtained from (6.22) and (6.15) with a magnetic wall excitation at z=0 (fundamental and odd harmonics for the resonator) and are written as 7r fir Ir \ - ¥ £ ,(* * •* -< > - J E Tr fir \ c mEt .,(* „ ) ~ N E .i k ^ k - i) = Z d nE ^ ,Z \ where \jk . ^ ' sin[(2m-1)tt/2] • cos(*.;L)] ( * • sin[(2«-l)7r/2] • cos(£_.L)1 ------ V - 1-------------------------------(6.26) [(2«-1)7t/2] - (>t;/L)2 j (kyi) and E z l (kyj) are the spectral basis functions from (2.36) and (2.37). The microstrip resonator in Fig. 64(b) and Fig. 14(a) (with shorting sidewalls at _y=±A) is also investigated to verify the three-dimensional SDM procedure. The basis functions for the currents on the strip are derived from [63] and defined only on the strip at .v=0 and 0 < [y| <S/2 and 0< \z\ <L as J;(y,z) = I ITl—l Jy(y,z) = Z J y , (V) Sin [ ( 2 ^ 1 ) n= 1 00 c o s [(2 ^ -i)jB Lv AL-, (6.27) y to ] (6.28) where J. j(y) and Jy j(y) are the basis functions used in the one and two-dimensional SDM analysis. These functions incorporate the singular (unbounded) behavior in the ^-direction o f the magnetic fields normal to the edges o f the strip and again only one ^-dependent term is implemented. The SDM for microstrip with the expansion of the currents on the strip follows from section F of Chapter II. C. Verification of the Three-Dimensional SDM Procedure Before design examples of multi-layered CBCPW MIC resonator circuits are presented, the three-dimensional SDM procedure must be validated. The analysis will include Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 m = 1 m-2 I.Or m =3 CO c o 0.5 o c 3 co *o35 jQ 0.0 -0 .5 - 1.0 1 -1 0 2 (m m ) (a) 71= 1 71=2 71=3 I.Or CO C 0.5 O 3 co *o55 -O 0.0 o c —1.0 2 (m m ) (b) Fig. 65. Shapes of the electric field basis functions for a CBCPW resonator as a function of z with 2L=l2.7mm for the right hand slot at j/=(S/2+W/2). (a) E basis functions, (b) E:n basis functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 microstrip cases published in the literature. A microstrip example from [60] is illustrated in Table XXIII for a center-fed resonator in a laterally and longitudinally open structure with a doped Si cover plate. The resonant frequency for this example was calculated by the SDM at 57.23GHz with 75 spectral terms for kyi and z-direction for the E and 2 basis functions in the and Ezn electric fields in the slots (total of four). A frequency of 57GHz was listed in the reference. The calculated Qs correspond to those listed for this microstrip resonator in the table. The discrepancy for the h2=0.4mm case is due to the fact that the analysis in [60] includes surface-wave radiation loss (open structure) and the simulations here are for closed cavities. At 57GHz, radiation losses are significant which would decrease the Q of the resonator. TABLE XXIII Resonant Q verification of the dominant mode in a microstrip structure with Si damping. Reference Fig. 58 and Fig. 64(b) for dimensions with 2A=12mm, ' 16 i 2B=20mm and WG=0 with e rl = 12 and«=3.0 x 10 cm , er2 = l, s r4=12.7 and hj=h4=0.1mm, h3=h5=h6=0 and S=0.2mm and 2L=0.775mm. Q Ref. [59] presented work 0.2 30 32.4 0.4 90 121 h2 (mm) The convergence properties of the half-wavelength resonators with respect to the number of basis functions will be investigated. An example from [63] is presented in Fig. 66(a) for a microstrip resonator (CASE 66) and for a CBCPW structure (CASE 67) in Fig. 66(b). Again, 75 spectral terms were implemented for each spectral variable and M=N=2 for the basis function expansions. The results agree closely with [63] and the convergence behavior is evident with the additional expansion functions. A similar response is depicted for the CBCPW resonator in Fig. 66(b). The resonant frequency (fr) can be approximated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 13.0i Ideal half-wavelength point N I o -• '— O QC ,2-6 - ^ 12.4 ■ ui h2 < 2 O GO Ul x 12.2 presented work Ref. 1631 • CASE 66 12. 0 - ^ 11.9 TOTAL z - D I R BASIS FUNCTIONS (a) 10.60 ideal half-wavelength point N 3 10.45 o LlJ QL Li_ f- 10.30 2 < 2 O C/0 Ul 10.15cc CASE 67 10.0Q. TOTAL z - D I R BASIS FUNCTIONS (b) Fig. 66 . Convergence demonstration o f the resonant frequency from the SDM with additional basis functions in the z-direction forM =N of a resonator circuit, (a) Microstrip example CASE 66 with dimensions from Fig. 58 and Fig. 64(b) of srl=l, er4=9.4 and ht=5mm, h4=0.6mm, h2=h3=h5=h6=0 and S=0.575mm, WG=0 , 2L=4.5mm and 2A=2B=10mm. (b) CBCPW example CASE 67 with dimensions from Fig. 58 and Fig. 64(a) o f s rl= l, er4=10.2, er5=2.2 and h ^ m m , h4=h5=0.635mm, h2=h3=h6=0 and S=2W=0.635mm, 2L=6.2mm and 2A=2B=25.4mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 from the effective dielectric constant for this example. For the CBCPW of CASE 67, seff=5.21 at 10GHz. The following relationship is utilized 2L=A.g/2 with 2L=6.2mm, and fr =10.6GHz. This is the ideal half-wavelength point for the zero total number of basis functions in Fig. 66. Another confirmation of the numerical method is the tracking of the resonant frequency as the resonator length is doubled for CASE 67. The fr should decrease in half for this configuration and the SDM predicts a frequency o f 5.1832GHz. The simulation o f CASE 67 with M=N and 75 spectral terms for k and k_t required 30 seconds on a Pentium 90MHz PC. Fig. 67 is a plot o f the Ey electric field distribution for the right hand slot and is determined by calculating the inverse Fourier transform to produce the space domain field. The expected response for this field component in the figure also confirms the SDM analysis for the resonators. One final verification example is derived from [35] for the TM022 mode for two layers of GaAs with the lower layer providing the damping (lossy dielectric) to the cavity. The configuration in [35] is not symmetrical in the y or z-directions so neither a magnetic or electric wall is incorporated at y =0 or z=0, respectively. The resonant frequency from the SDM of this cavity mode was calculated at 50.4GHz and f=50GHz in the reference. A comparison of the Qs for this mode is depicted in Table XXIV and the responses again agree closely. The resonant frequency for the center-fed dipole of [35] was determined to be 54GHz and the SDM here predicted a frequency o f 53.3GHz. D. Design Examples for Multi-Layered CBCPW Resonator Circuits with Damping The three-dimensional SDM applied to resonators has been developed and verified in the previous two sections. Now multi-layered CBCPW circuits with cavity damping can be designed with the goal to maximize the circuit Q for a half-wavelength resonator, minimize the cavity Qs, minimize the dielectric thickness, and minimize the lossy layer thickness. Both upper and lower dielectric loaded CBCPW configurations will be presented along with the doped Si and microwave absorber planar lossy materials. A relationship is utilized to approximate the circuit Qs from the dielectric and conductor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 0.2 0.0 Q. - 0.2 E o v > - 0 .4 o <D - — 0.6 1.0 -1 5 -1 0 -5 0 5 10 15 Z Fig. 67. Plot of the Ey space domain field from the SDM for the dominant CBCPW mode of the resonator circuit of CASE 67 with 2L=6.2mm at _y=(S-i-W)/2 for the right hand slot. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 TABLE XXIV Resonant Q verification of the TM 0 2 2 mode in a CBCPW packaged structure with loss tangent damping at h5. Reference Fig. 58 and Fig. 64(a) for dimensions with srl =1, er4=er5 =12.8, 54 =0.002 and h4=h5=0.125mm, h2=h3=h6=0 and S=0.025mm, W=2S, 2L=1.Omm and 2A=2mm, 2B=3mm. Q Ref. [35] presented work 0.01 170 167 0.1 20 19.7 loss tangent 5S losses for the dominant CBCPW propagating mode from [64] by 71 Q = X g (ctc + OLd) (6.29) where clc and a d are the attenuation coefficients (calculated by the SDM) of the conductors and dielectrics, respectively. The above equation is applied to half-wavelength resonators. Assuming the conductors are PECs, then a c=0. For the MIC examples presented, the conductor losses should not dominate the dielectric losses in an actual circuit operating from 9-17GHz. The effects of a constant doped Si layer on the CBCPW resonator (CBCPW CIRCUIT) and the cavity modes in a lower loaded configuration for a 9-12GHz application are studied for CASE 68 in Fig. 68 (a). The fundamental mode of the resonator (w ^l) is excited with a magnetic wall at z=0 and the TMg 3, mode is the only cavity mode with a resonant frequency within the band. fr of this cavity mode is 9.53GHz and the resonant frequency of the CBCPW resonator is approximately 9.6GHz, as shown in Fig. 66(b) with h6=0. CASE 68 is the same as CASE 67 except with the presence o f the silicon layer. The doping level was arbitrarily selected for this simulation. The CBCPW CIRCUIT IDEAL response in Fig. 68 is derived from (6.29) and the TMo 3 , IDEAL is obtained from Fig. 61 (no slots configuration). As the Si damping layer is increased, the Q Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 CASE 68 CBCPW X CBCPW TMq.3.1 * TMq.3.1 0.00 0.05 CIRCUIT (SDM) CIRCUIT (IDEAL] (SDM) (ideal) 0.15 0.10 0.20 0.25 LOSSY LAYER THICKNESS h6 (mm) (a) CASE 69 X A CBCPW CIRCUIT (SDM) CBCPW CIRCUIT (IDEAL) TM(w,i (SDM) TMoj.t (IDEAL) 1.00 2.00 3.00 0.00 DIELECTRIC LAYER THICKNESS h 4 ( m m ) (b) f Fig. 68 . CBCPW resonator and TM,, 3 j cavity mode Qs using doped Si layer (er = 12). a) Lower loaded multi-layered structure CASE 68 same as CASE 67 as a e 16 i function o f the Si layer at h6 with doping «,=1.0 x 10 cm . b) Upper loaded CBCPW CASE 69 as a function o f the dielectric thickness h4 with fixed Si layer at hs=0.2mm with doping n = 2.1 x 10 cm'3 and srl=l, er3=10.2, Er4= 2 .2 , h[=5mm, h3=0.635mm, h2=h6=0 and S=2W=0.635mm, 2L=4mm and 2A=2B=25,4mm. Log scales are implemented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 187 of the cavity mode is decreased at a faster rate than for the CBCPW resonator. A Si thickness of approximately 0 .15mm would be an appropriate tradeoff between the CBCPW and cavity Qs for this structure. The accuracy of (6.29) is also depicted in Fig. 68 (a). The dominant CBCPW mode is still bound to the slots with the incorporation of the lossy layers for all the cases presented. For this frequency band, the loss tangents of tan84=2.1 x 10'3 and tan6 s= l.l x 10'3 are applied to the er= 10.2 and sr=2.2 substrates, respectively. An upper loaded structure is implemented for CASE 69 in Fig. 68 (b) and the lower dielectric thickness is varied for a fixed Si layer in the 12-17GHz range. For this case 2L=4mm, h4= 1.27mm, h5=0, and the resonant frequencies for the CBCPW, TM03, , and TMq j j (from the lower cavity between h4, h5, and h6) modes are 14.46GHz, 12.59GHz, and 16.9GHz, respectively. The TM031 mode (or the TM0] 3 degenerate mode) of the upper cavity (formed between h3, h2, and h,) has a fr=17.36GHz and is outside the band of interest. Otherwise, the Si would also be applied at h, to dampen the upper cavity for this mode. The Si layer at h5 is designed to maximize the damping effects at the center o f the band between the two cavity modes. By increasing the thickness of the lower dielectric layer (h4), the resonator Q increases as shown in Fig. 68 (b). The cavity mode Qs also increase (but at a slower rate than the CBCPW resonator) as the percentage of the lossy volume in the lower cavity is reduced with a thicker substrate (h4). Recall that the electric field distributions of the cavity modes have a predominate vertical component and are affected more by the damping materials than the CBCPW mode. A design thickness of h4=l ,5mm is recommended here. The Q of the TMq 3 3 mode is very close to the TMqj j . An example of the microwave absorber is illustrated in Fig. 69 o f CASE 70 (similar to CASE 68) for a lower loaded CBCPW waveguide. The parameters stated for a 9-12GHz bandwidth are h6=0.635mm and 2L=6.2mm. The TMq3 j is the only cavity mode within the range of interest for this structure. As compared to the doped Si layer, the microwave absorber is a more effective damping material for the same volume. One configuration proposed to minimize the dielectric loss effects of the absorber on the CBCPW mode, would be to increase the thickness of the lower substrate (h5). This strategy would reduce Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 188 x CBCPW CIRCUIT (SDM) CBCPW CIRCUIT (IDEAL) O CM O CASE 70 o o O 0.00 1.00 2.00 3.00 4.00 5.00 D IE L E C T R IC LA Y E R T H IC K N E S S h 5 ( m m ) Fig. 69. CBCPW resonator and TMq 31 cavity mode Qs for a lower loaded multi-layered structure using microwave absorber as a function of the dielectric thickness h5. Refer to Fig. 58 for the dimensions of CASE 70 with srl=l, sr4=10.2, £^=2.2, ert=21(l-_/0.02), |j.rt= l.l(l-y l.4 ) and h,=5mm, h4=h6=0.635mm, h2=h3=0 and S=2W=0.635mm, 2L=6.2mm and 2A=2B=25.4mm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 the field coupling of the CBCPW mode to the lossy layer and further concentrate the fields into the h4 and h 5 dielectrics. Fig. 69 demonstrates the results for this configuration and shows the thickness necessary to minimize these effects. The response o f CASE 70 is very similar to CASE 69 of Fig. 68 (b). A 2.0mm thickness for h5 is suggested as a tradeoff between the circuit and cavity Qs and the dielectric size. E. Experimental Demonstration of Cavity Damping for CBCPW Resonators Some type of damping configuration is necessary within multi-layered wide frequency-band CBCPW MICs to reduce the resonance effects associated with the rectangular cavity modes. A measurement example incorporating absorber (damping) material within a CBCPW resonator is presented. The circuit is a series gap-coupled straight resonator and is diagrammed in Fig. 70(a). This circuit is often employed in the measurement o f the effective dielectric constant for a transmission line and the Q of a resonator. The gap width (G) of Fig. 70(a) is selected sufficiently wide to lightly couple energy to and from the resonator and not load the resonator circuit with the measurement system. The resonator length (L) is designed from the following relationship [65] L + 2 l eo = n. Xg where nt=l, 2 ,3,... is the integer order o f the resonance, (6.30) is the guide wavelength, leo is the length extensions due to the fringing fields at the open ends o f the resonator line. The implementation o f the damping/absorber material at the h6 substrate is pictured in Fig. 70(b) as a layered configuration. This is the same absorber utilized in Fig. 42. The conductor-backed plane is modified to effectively house the lossy material for this application as packaging the absorber as a full width layer proved somewhat difficult due to the shifting or unstableness of this layer. The effective dielectric constant for the CBCPW of this section (see Fig. 71 CASE X) from the SDM was 5.43 at 5GHz. The goal was for the first resonant frequency (nj= 1) to occur near 5GHz. Applying (6.30) with this information and neglecting the end length fringing term, yielded a resonator length Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 ■* L y y (a) X 'M y Fig. 70. Dimension.parameters for multi-layered CBCPW straight gap-coupled resonator with damping material, (a) Top view with gap width (G) and resonator length (L). (b) Cross-sectional view with absorber placed at layer h6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 L=12.87mm which was set to L=12.7mm for the measurement data. The experimental result for this configuration is presented in Fig. 71 and demonstrates the expected response with a smooth curve to 40GHz. The sharpness o f the resonance reduces for each order (^=1,2,3,... ) as the losses increase and the Q decreases for subsequent modes. The first resonance occurs at 4.7GHz which could be predicted when the uncertainty in the dielectric dimensions is factored into the analysis. The absorber layer traverses the entire length of the waveguide (z=0 to z=2B). When the damping material was applied only in the vicinity o f the connector feeds at h6, the presence of the cavity modes on the CBCPW resonator response was readily apparent. The absorber volume for this case was not sufficient to dampen the substrate mode Qs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 S2 1 log REF —1 . 8 0 S dB 7 . 0 dB/ MAG * u START STOP 0 .1 0 0 0 0 0 0 0 0 4 0 .1 0 0 0 0 0 0 0 0 GHz GHz Fig. 71. Experimental data response with absorber material to dampen cavity Qs in straight gap-coupled CBCPW resonator. Reference Fig. 70 for dimensions with G=0.17mm and L= 12.7mm. CASE X with erl= l, s r4 = 10.8, er5 =2.33 and hj=oo (open structure), h2=h3=0, h4=0.635mm, h5= 1.09mm and S=2W=0.635mm and 2A=20mm and 2B=38mm. Absorber layer at h6 and electrical data is not known. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 CHAPTER VII CONCLUSION This dissertation has demonstrated the configurations necessary to achieve a wide frequency of operation for CBCPW MICs. Unless the user has access to an etching facility capable of generating very fine line/slot widths or a circuit that requires very little surface area that can cutoff the waveguide modes, a multi-layered configuration (either upper or lower dielectric loading) is needed to reduce the waveguide mode coupling effects and generate a useful transmission line. The CBCPW with lateral shorting sidewalls and placed in a package or test fixture resembles a rectangular waveguide in two-dimensions and a rectangular cavity in three-dimensions. The leakage and mode coupling effects were described in detail using both one and two dimensional SDM numerical procedures. Design recommendations concerning the upper usable frequency, impedance range and variation, effective dielectric constant range and variation, inclusion of air gaps between the dielectrics, and uncertainty analysis in the dielectric constant and thickness have been presented for dielectrics with sr=10.2 and sr=2.2. The propagation characteristics of the upper loaded structure are quite sensitive to the possible air gap existing between the circuit conductors and dielectric, and the waveguide is difficult to configure within a test fixture. The use of some type of damping material is required in many cases to reduce the residual resonances associated with the cavity structure whose resonant frequencies are within the band of interest. This damped waveguide was demonstrated experimentally with broadband absorber placed within the connector housing blocks and as a dielectric layer. Simulations using a three-dimensional SDM were presented with the effects of doped silicon and microwave absorber on the Qs of the cavity modes and the CBCPW resonator circuit. The experimental data conducted in this research played a vital role in the identification and understanding of the mode coupling problems in the CBCPW MICs. Without the measurement data, this research topic would not have been investigated. The modeling procedure was extended from one, to two, and to three dimensions to more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 194 accurately predict and describe the experimental effects. Fig. 72 depicts a microstrip and a multi-layered CBCPW through lines and as compared to Fig. 7, demonstrates that these two transmission lines are now both available for wide frequency-band MIC applications. After twenty-five years since the invention by Wen [1], coplanar waveguides (in particular the conductor-backed form) have developed to the point where they will rival microstrip as a transmission line choice for MICs. Hopefully, this work has contributed in some small way to this advancement. A. Original Contributions This dissertation has provided an improved understanding of the propagation and transmission properties of multi-layered CBCPW MICs. Specific contributions from this work are listed as: 1) introduced measurement data showing the problems with single-layered structures; 2) incorporated multi-layered (lower and upper dielectric loading) to increase the bandwidth; 3) utilized a one-dimensional SDM to approximately predict this bandwidth; 4) explained the mode coupling effects in detail with a two-dimensional SDM in a packaged configuration; 5) modeled the finite-length CBCPW as a resonator and identified the cavity modes; 6) presented design summary information for multi-layered CBCPW MICs including the upper dielectric loaded structure; 7) incorporated lossy layers to dampen the cavity mode Qs without significant affect on CBCPW mode and simulated this arrangement with a three-dimensional SDM; 8) described the simulation o f a shorted CPW resonator in the SDM; 9) demonstrated how CBCPW MMICs can operate with mode coupling effects and presented specific and supported guidelines for reduced coupling effects; 10) produced the widest frequency-band CBCPW MIC to date (40GHz). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 CASE A lQ T) CN in > LlI "O Q 3 CD TD CASE Y O < 0 4 8 12 16 20 24 28 32 36 40 FREQ (G H z ) Fig. 72. Experimental data of the transmission responses for the microstrip and the multi-layered CBCPW MIC 5 0 0 through lines. A comparison with Fig. 7 indicates CBCPW MIC is now a viable alternative to microstrip. Refer to Fig. 5 for dimensions with erl=l and hj=oo (open structure) and 2A=2B=38mm. CASE A is microstrip line with er4=10.8 and h4=0.635mm and S=0.635mm. CASE B is CBCPW with er4=10.8, sr5=2.33 and h4=0.254mm, h5=0.71mm, h2=h3=h6=0 and S=2W=0.635mm with absorber material in the connector blocks. Both cases are referenced to OdB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 196 B. Suggestions for Future Work The required research work on CBCPW is still immense. The majority o f this effort is and will concentrate on discontinuity modeling [66 ] and [67], Important considerations involving the discontinuities must be given to the analysis, design, and placement o f the air bridges associated with the slotline mode and to minimize the coupling to the cavity modes of the packaged (finite-length) structure. The moment method, SDM, finite-element, and finite-difference time domain are the numerical procedures that will be utilized in this endeavor. However, closed form expressions or simplistic design relations must be presented from the above fiill-wave procedures for the convenience of the user. Additional considerations to reduce the number of the air bridges are also extremely important. In the analysis of this dissertation, it was assumed that a perfect connection existed between the upper and lower ground conductors with the lateral sidewalls but in reality floating potential regions within a circuit can occur due to inexact grounding and this issue must be addressed. Finally, the presence of more than one propagating mode on the calibration and deembedding procedures associated with the network analyzer reference planes involving CBCPW MICs must be investigated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 REFERENCES [1] C.P. Wen, "Coplanar waveguide: A surface strip transmission line suitable for nonreciprocal gyromagnetic device applications," IEEE Trans. Microwave Theory Tech., vol. MTT-17, pp. 1087-1090, December 1969. [2] R. Pucel, "Design considerations for monolithic microwave circuits", IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 513-534, April 1981. [3] R. Jackson, "Coplanar waveguide vs. microstrip for millimeter wave integrated [4] [5] circuits," in 1986IEEEM TT-SInt. Microwave Symp. Dig., pp. 699-702. M. Riaziat, private communication, 1991, Varian Research Center, Palo Alto, CA. M. Riaziat, I. Zubeck, S. Bandy, G.Zdasiuk, "Coplanar waveguides used in 2-18 GHz distributed amplifier," in 1986 IEEE MTT-S Int. Microwave Symp. Dig., pp. 337-338. [6 ] R. Majidi-Ahy, C. Nishimoto, M. Riaziat, M. Glenn, S. Silverman, S. Weng, Y. Pao, G. Zdasiuk, S. Bandy, Z. Tan, "5-100GHz InP coplanar waveguide distributed amplifier," IEEE Trans. Microwave Theory Tech., vol. MTT-38, pp. 1986-1993, December 1990. [7] E. Strid, "26GHz wafer probe MMIC development and manufacture," Microwave J., vol. 29, no. 8, pp. 71-82, 1986. [8 ] L. Carin and K. Webb, "Isolation effects in single- and dual-plane VLSI interconnects," IEEE Trans. Microwave Theory Tech., vol. MTT-38, pp. 396-404, April 1990. [9] R. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, [10] 1961, pp. 143-190. R. Collin, Foundations fo r Microwave Engineering, McGraw-Hill, New York, 1966, pp. 170-179. [11] R. Simmons, G. Ponchak, K. Martzaklis, R. Romanofsky, "Channelized coplanar waveguides: discontinuities, junctions, and propagation characteristics," in 1989 IEEE MTT-S Int. Microwave Symp. Dig., pp. 915-918. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 [12] M. Riaziat, R. Majidi-Ahy, I. Feng, "Propagation modes and dispersion characteristics of coplanar waveguides," IEEE Trans. Microwave Theory Tech., vol. MTT-38, pp. 245-251, March 1990. [13] M.A. Magerko, L. Fan, K. Chang, "Multiple dielectric structures to eliminate problems in conductor-backed coplanar waveguide MICs," IEEE Microwave Guided Wave Lett., vol. 2, no. 6 , pp. 257-259, 1992. [14] M.A. Magerko, L. Fan, K. Chang, "A discussion on the coupling effects in conductor-backed coplanar waveguide MICs with lateral sidewalls," in 1993 IEEE MTT-S Int. Microwave Symp. Dig., pp. 947-950. [15] M.A. Magerko, L. Fan, K. Chang, "Configuration considerations for multi-layered packaged conductor-backed coplanar waveguide MICs," in 1994 IEEE MTT-S Int. Microwave Symp. Dig., pp. 1697-1700. [16] Y. Liu and T. Itoh, "Leakage phenomena in multi-layered conductor-backed coplanar waveguides," IEEE Microwave Guided Wave Lett., vol. 3, no. 11, pp. 426-427, 1993. [17] M. Yu, R. Vahldieck, J. Huang, "Comparing coax launcher and wafer probe excitation for lOmil CBCPW with via holes and airbridges," in 1993 IEEE MTT-S Microwave Symp. Dig., pp. 705-708. [18] T. Uwano and T. Itoh, "Spectral Domain Approach" in T. Itoh (ed.) Numerical Techniques fo r Microwave and Millimeter-Wave Passive Structures. John Wiley & Sons, New York, 1989, pp. 324-380. [19] D. Mirshekar-Syahkal, Spectral Domain Method fo r Microwave Integrated Circuits. John Wiley & Sons, New York, 1990. [20] J. Knorr and K. Kuchler, "Analysis o f coupled slots and coplanar strips on dielectric substrates," IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 541-548, July 1975. [21] Y. Fujiki, M. Suzuki, Y. Hayashi, "Higher-order modes in coplanar-type transmission lines," Electronics and Communications in Japan, Vol. 58-B, no. 2, pp. 74-81, 1975. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 199 [22] J. Davies and D. Mirshekar-Syahkal, "Spectral domain solution of arbitrary coplanar transmission lines with multi-layer substrate," IEEE Trans. Microwave Theory Tech., vol. MTT-25, pp. 143-146, July 1977. [23] Y. Shih and T. Itoh, "Analysis of conductor-backed coplanar waveguides," Electronics Letters, vol. 18, no. 12, pp. 538-540, 1982. [24] G. Leuzzi, A. Silbermann, R. Sorrentio, "Mode propagation in laterally bounded conductor-backed coplanar waveguides," in 1983 IEEE MTT-S Int. Microwave Symp. Dig., pp. 393-395. [25] R. Jackson, "Considerations in the use o f coplanar waveguides for millimeter-wave integrated circuits," IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 1450-1456, December 1986. [26] H. Shigesawa, M. Tsuji, A. Oliner, "Conductor-backed slot line and coplanar waveguide: dangers and full-wave analyzes," in 1988 IEEE MTT-S Int. Microwave Symp. Dig., pp. 199-202. [27] R. Jackson, "Mode conversions due to discontinuities in modified grounded coplanar waveguide," in 1988 IEEE MTT-S Int. Microwave Symp. Dig., pp. 203-206. [28] M. A. Magerko, L. Fan and K. Chang, "Analysis of multi-layered structures to reduce mode coupling problems in packaged conductor-backed coplanar waveguide MICs," submitted for publication to IEEE Trans. Microwave Theory Tech. [29] K. Jones, "Suppression of spurious propagation modes in microwave wafer probes," Microwave J., vol. 32, no. 11, pp. 173-174, 1989. [30] E. Godshalk, "Generation and observation of surface waves on dielectric slabs and coplanar structures," in 1993 MTT-S Int. Microwave Symp. Dig., pp. 923-926. [31] N. Das and D. Pozar, "Full-wave spectral domain computation of material, radiation, and guided wave losses in infinite multilayered printed transmission lines," IEEE Trans. Microwave Theory Tech., vol. MTT-39, pp. 54-63, January 1991. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 [32] W. McKinzie and N. Alexopoulos, "Leakage losses for the dominant mode of CBCPW," IEEE Microwave Guided Wave Lett., vol. 2, no. 2, pp. 65-66, 1992. [33] L. Chou, R. Rojas, P. Pathak, "A WH/GSMT based full-wave analysis of the power leakage from conductor-backed coplanar waveguides," in 1992 MTT-S Int. Microwave Symp. Dig., pp. 219-222. [34] L. Carin, G. Slade, K. Webb, A. Oliner, "Packaged printed transmission lines: modal phenomena and relation to leakage," in 1993 IEEE MTT-S Int. Microwave Dig., pp. 1195-1198. [35] R. Jackson, "Circuit model for substrate resonance coupling in grounded coplanar waveguide circuits," IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1461-1465, September 1993. [36] C. Tien, C. Tzuang, S.Peng, C. Chang, "Transmission characteristics o f finite-width conductor-backed coplanar waveguides," IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1616-1624, September 1993. [37] Wiltron 36804-25C Universal Test Fixture, September 1990, Wiltron Inc. Morgan Hill, CA. [38] T. Itoh and R. Mittra, "Spectral domain approach for calculating the dispersion characteristics o f microstrip lines," IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 496-499, July 1973. [39] R. Mittra and T. Itoh, "A new technique for the analysis of the dispersion characteristics o f microstrip lines," IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 47-56, January 1971. [40] A. Wexler, "Solution of waveguide discontinuities by modal analysis," IEEE [41] [42] Trans. Microwave Theory Tech., vol. MTT-15, pp.508-517, September 1967. Earl Swokowski, Calculus. Prindle, Weber & Schmidt, Boston, 1992, pp. 463-472. El-Badawy El-Sharawy and R. Jackson, "Coplanar waveguide and slot line on magnetic substrates: analysis and experiment," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 1071-1078, June 1988. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 [43] C. Chang, Y. Wong, C. Chen, "Full-wave analysis o f coplanar waveguides by variational conformal mapping technique," IEEE Tram. Microwave Theory Tech., vol. MTT-38, pp. 1339-1343, September 1990. [44] N. Das, "A study of multi-layered printed antenna structures," Ph.D. Dissertation, University of Massachusetts, Amherst, September 1989. [45] B. Janiczak, "Analysis of coplanar waveguide with finite ground planes," AEU, vol. 38, no. 5, pp. 341-342, 1984. [46] C. Chang, W. Chang, C. Chen, "Full-wave analysis of multilayer coplanar lines," IEEE Tram. Microwave Theory Tech., vol. MTT-39, pp. 747-750, April. 1991. [47] N. Fache and D. De Zutter, "Circuit parameters for single and coupled microstrip lines by a rigorous full-wave space-domain analysis," IEEE Tram. Microwave Theory Tech., vol. MTT-37, pp. 421-425, February 1989. [48] M. Tsuji, H. Shigeswa, A. Oliner, "Printed-circuit waveguides with anisotropic substrates: a new leakage effect," in 1989 IEEE MTT-S Int. Microwave Dig., pp. 783-786. [49] M. Tsuji, H. Shigeswa, A. Oliner, "New interesting leakage behavior on coplanar waveguides of finite and infinite widths," in 1991 IEEE MTT-S Int. Microwave Dig., pp. 563-566. [50] G. Ghione and C. Naldi, "Coplanar waveguides for MMIC applications: effect of upper shielding, conductor backing, finite-extent ground planes, and line-to-line coupling," IEEE Tram. Microwave Theory Tech., vol. MTT-35, pp. 260-267, March 1987. [51] M. Tsuji, H. Shigeswa, A. Oliner, "Dominant mode power leakage from printed-circuit waveguides," Radio Science., vol. 26, pp. 559-564, March-April 1991. [52] D. Kasilingam and D. Rutledge, "Surface-wave losses of coplanar transmission lines," in 1983 IEEE M TT-S Int. Microwave Dig., pp. 113-116. [53] Y.C. Shih and I. Itoh, "Analysis o f conductor-backed coplanar waveguide," Electronic Letters, vol. 18, pp. 538-540, June 1982. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 202 [54] D. Phatak, N. Das, A. P. Defonzo , "Dispersion characteristics of optically excited coplanar striplines: comprehensive full-wave analysis, " IEEE Tram. Microwave Theory Tech., vol. MTT-38, pp. 1719-1730, November 1990. [55] J.R. Pierce, Almost All About Waves. MIT Press, Cambridge, MA, 1974, Chapter 6. [56] L. Carin, G. Slade, K. Webb, A. Oliner, "Packaged printed transmission lines: modal phenomena and relation to leakage," in 1993 IEEE MTT-S Int. Microwave Dig., pp. 1195-1198. [57] C. Chang, W. Chang, C. Chen, "Full-wave analysis of coplanar waveguides by variational conformal mapping technique," IEEE Tram. Microwave Theory Tech., vol. MTT-38, pp. 1339-1344, September 1990. [58] F. Miranda, K. Bhasin, K. Kong, M. Stan,, "Conductor-backed coplanar waveguide resonators o f YBa^UjO^g on LaA103" IEEE Microwave Guided Wave Lett., vol. 2, no. 7, pp. 287-288, 1992. [59] J. Burke and R. Jackson, "A simple circuit model for resonant mode coupling in packaged MMICs," in 1991 IEEE MTT-S Int. Microwave Dig., pp. 1221 -1224. [60] R. Jackson, "Removing package effects from microstrip moment method calculations," in 1992 IEEE MTT-S Int. Microwave Dig., pp. 1225-1228. [61] D. Williams, "Damping of the resonant modes of a rectangular metal package," IEEE Trans. Microwave Theory Tech., vol. MTT-37, pp. 253-256, January 1989. [62] PCAAMT™ , Antenna Design Associates, Inc., Leverett, MA 1990. [63] T. Uwano, "Accurate characterization of microstrip resonator open end with new current expression in spectral domain approach," IEEE Trans. Microwave Theory Tech., vol. MTT-37, pp. 630-633, March 1989. [64] A. Gopinath, "Losses in coplanar waveguides," IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp. 1101-1104, July 1982. [65] T. Edwards, Foundations fo r Microstrip Circuit Design. John Wiley & Sons, 1992, pp. 245-253. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 [66 ] N. Dib, M. Gupta, G. Ponchak, L. Katehi, "Effects of ground equalization on the electrical performance of asymmetric coplanar waveguide shunt stubs," in 1993 IEEE MTT-S Int. Microwave Dig., pp. 701-704. [67] T. Becks and I. Wolff, "Full-wave analysis o f various coplanar bends and T-junctions with respect to different types o f air bridges," in 1993 IEEE MTT-S Int. Microwave Dig., pp. 697-700. [68 ] M. Abramowitz and I. Stegun, Handbook o f Mathematical Functions. Dover Publications, New York, 1972, pp. 369. [69] W. Press, B. Flanners, S. Teukolsky, and W. Vetterling, Numerical Recipes {Fortran). Cambridge University Press, Cambridge, 1989, pp. 121-126. [70] R.W. Hombeck, Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ, 1975, pp. 154-159. [71] W. Press, B. Flanners, S. Teukolsky, and W. Vetterling, Numerical Recipes {Fortran). Cambridge University Press, Cambridge, 1989, pp. 31-39. [72] ,Numerical Recipes {Fortran). Cambridge University Press , Cambridge 1 989, pp. 2 4 0 -2 6 2 . [73] ,Numerical Recipes {Fortran). Cambridge University Press , Cambridge 1 9 8 9 , pp. 5 2 -6 4 . [74] M. Spiegel, Schaum's Outline Series o f Advanced Calculus. McGraw Hill, New York, 1962, pp. 348-361. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 204 APPENDIX A SPECTRAL DOMAIN IMMITTANCE FORMULATION The spectral domain immittance procedure is a powerful technique using the transverse equivalent transmission line model to relate the components of the currents and fields in the Fourier domain and is especially suited to multi-layered structures. The full-wave spectral domain dyadic Green's functions (immittance functions) are obtained using TMX and TEXmodes. These functions are based on the transverse equivalent circuit concept as applied in the spectral domain in conjunction with a simple coordinate transformation. The basic concept can be illustrated by observing the inverse Fourier transform o f the fields from (2.5) and multiplying both sides by the exponential factor and recalling that for an infinite line with variation e'//z, y=k, and GO §(x, y)e~j Yz = J $ (x, k y) e + dky . (A. 1) —00 From (A. 1), all the field components are a superposition o f inhomogeneous (in x) plane waves propagating in the direction of 0 relative to the z-axis as described in Fig. A. 1 which presents the coordinates and components o f the basic TM and TE modes. The decomposition of the spectral waves from the (x,_y,z) coordinate system in to the new system (x,w,v) is written as cos9 - - sin0 = ^y (A.2) and u = zsin0 —_ycos0 v = zcos0 + _ysin0. (A.3) For each 0, waves may be decomposed into TMX(EX, E V, H U) and TEx (HX, E U, H V) in the (x,u,v) system. The current J v creates only the TM fields because it is associated with H u and J u creates the TE fields. The equivalent circuits for the TM and TE fields are found by applying a transverse (x-direction) transmission line analogy as shown in Fig. A.2 for the six-layer structure. For each basic TM and TE mode, one o f the two models Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 205 k y /L. Hx ~ components of TE modes components of TM modes X Fig. A. I. Coordinates and components of the basic TM and TE modes for the SDM immittance method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 Fig. A.2. Equivalent transmission line models of the multi-layered CBCPW for the TM and TE modes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 in Fig. A.2 is applicable and the two models are completely decoupled. For the TM equivalent circuit, the circuit equations become H u3 - H u4 = J V E v-3 = E v4 (A.4) where Y' = y \ + y \ ^ e (A. 5) ^ e and Y3 and Y4 are the driving point wave admittances looking from the location of the current (x=0) by utilizing conventional transmission line theory. Similarly, the circuit equations for the TE fields become H vi —H v4 = - J u E u3 = E U4 (A.6 ) where Y h = YH 3 + Yh4 (A.7) ^ h and Y3 and T4 correspond to the input wave admittances for the TE waves. Once Ye and Yh are determined, the dyadic spectral domain admittance Green's functions can be obtained using the transformations of (A.2-A.3). from the(x,w,v) to the (x,y,z)coordinate The fields are expressed by mapping system. E yandE . are linear combinations o f E u and E v andlikewise J yand J , are those of J u and J v. The results are described as + <A 8 > <A9) K = + ■ (A .io) The input admittance is determined by Y, +Y .tanhK.h, K- - r « r ^Oi i L a ^ i 1 <A11) where YL is the load admittance, hj is the thickness of each layer, and T0; is the wave characteristic admittance defined as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 H > )e ■ i I'.TE, = T r = n r L = i (A.i2) To™ =-%*- = r ^ - = K . TEi Eu / 2 2 2 2 where i refers to the corresponding region and k. = k +fc_ - (A. 13) . For the six-layer waveguide of Fig. A.2, (A. 11) is invoked at each interface. The cover plate conductor and the conductor-backed plane are taken as short circuits. The following relations are then derived & Y$ = Ctg/K^ Y6 = CtgK.g (A. 14) h ~ e r, = , k c t sr 6 + i ^ ^ 6 + C tj K + c tsr 6 ^=<--5-K_.5—H r C t5 + ? 6 (a.is) h , K / t / 5+ l Y4 = -------' Ky4 y 5 + c t 4 ~ e 1 ^y3 = icT y ' JY 1 1 ^ K .+ C t,? 5 r t - K;4 - 2 -------2 - 4 K__4 c t 4 + ? 5 —h l^j = k:3 K_, +CtjKl Hr (A-16) (A17) K_-3Ct 3 + T 2 S 3 y 2 + C t3 ~ e , y, k 2c t , r +1 = r - H - ---y2 ,, v Ky2Y l +Ct 2 ~ e YX =C tj/K j,, K0 + c t , r . r z = * - a — ------- H r ~r. K ^C tj + f -A = C t , K =1 (A. IS) (A. 19) / where Ct; = cothx-hj with i=l,2,...,5,6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 APPENDIX B DERIVATION OF THE FIELD COEFFICIENTS FOR THE SPECTRAL DOMAIN METHOD IMMITTANCE METHOD By imposing the boundary conditions o f (2.48a), (2.48c), (2.49a), (2.49c), (2.50a), (2.51a), (2.51c), (2.52a), (2.52c), (2.57) and (2.58), the following relations are generated TM At x = -h 5-h4 At x = -h. A tx = 0 i] = v D Ky*\p = c4 + E A C6 = C (B .l) B S5 + C C5 = E (B.2 ) S 4] = k y3F L = [ d s t + e .c , - a Jkj+ k\ (B3) At x = h3 k y J [ F C 3 + G e5 3] = Ky2H F S3 + G C3 = / (B.4) At x = h3+h2 ~K# [ H C 2 + / S 2] = H S 2 + I C 2 = J C, (B.5) J Sx TE : h h At x = -h5-h4 A S6 = B k :6A At x = -h. B C5 + C S5 = D h h k.j B S 5 + C C5 C 6 = k .5C (B.6 ) k ,4E h (B.7) A tx = 0 D C4 + E S4 = F Ju = At x = h, f h h S 4 + E C< ) " (B.8) k 24(Z) J k j + k\ h h F S, + G C-, Ka [ *z2J (B.9) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 h h h - k . H hS 2 + I hC2 j = k zXJ -A° ^C,1 H C2 + I S 2 = J S x A t.r = h3+h2 (B. 10) / where C{ = coshKjh; and = sinhKjh; for i=l,2,...,5,6. The goal is to solve the e h — coefficients (TM and TE) in terms of A and A respectively, and recall that J v and J u are known. These equations are derived as TM : B = Krf/Ky5S 6A (B .11) C = C 6A (B. 12) D = E — F = D S A + £*C 4 (Ky6/Ky5C sS 6 + S $ C 6) j A + C s C6~] ^ = D A ~ E A (B.13) (B.14) e F = FA (B .15) , C2 ^2 K-yl^yX £ + q F s2 C, F ~ ^-yl^y\ £ G = ~ [Ky3/Ky 2 V 3 C 3 + / M K y./K ^S ^/^ e H = e i q Ky3/ Ky2 j r 3 /g F /4 = I 2S 3 - C2~r = / /4 = A e J ] _ e £l F\ C, S,H + C ,/ e A • = _F ‘ - ~ I S (B. 16) = H A (B.17) (B. 18) (B. 19) -jJ k fT k f e A A = ^Ky4/Ky3 D C 4 + E S 4 = J, (B.20) /.+71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -jjfcj+kl e A = (B.20) J, TE. h h (B.21) B = Sz A C = D = [K.g/Kjjj’jC g + C 5S 6] A (B.22) k z6/ k z5C 6A h E h = [ k .j / k ^ ( k ^ / k ^ C s C j + iSj.S'g)]^ h h h = h h = E A (B.24) h Ly = K z4 D S d + E F (B.23) =D A Cd h A D C4 + E h = L-,A h (B.25) S. C, L 3 = K jj/k ,! ^ - + 1 Z, 4 C, 5, = k ^ /k .,^ - + r H = ~^-K = ^ K i 2 ^ 2 ^ 4 ^ 3 + ^ 2 ^ i ^ 3 ~ \ k z3J k z2. L .4C ^ h H = h T 2 ^C- 3 - J<s3 — r 6 h I = J J = -1 C rH Si A h L$ .h U A h h (B.26) (B.27) = H A . u L 2S 3 - C 3 t ^su h A Ah _ +L^S~ '3 3 h + S?I -j^^kj+ kl „ = -----— ----- 7-----J u 1 T j. 5 ^ L l + r6 h A h = I A h (B.28) (B.29) (B.30) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX C DERIVATION OF THE POYNTING VECTOR INTEGRAL The derivation of (2.70) is required for evaluation of the Poynting power flow vector for the characteristic impedance calculations. To assist in this process, some mathematical identities are required [18] and presented as f 2 * I |coshKx| cbc = Re(f/sinh jch cosh ich) (C .l) f I |sinhioc| 2cbc = Re(t/coshich sinh * Kh) (C.2) J sinh ice cosh*icxc& = i I/|coshich| + t/*|sinhich| - U (C.3) | cosh ioc sinh*jcc cbc = }■ U* |coshich| + f/|sinhjch| - U* (C4) where U = —i- r + — -[ K+K K -tC and recall if G is a complex function, then G G * (C.5) = \G\ . Let P h| = E^Hyi - E ^ H ^ = P Ai - P gi for i= 1,2,...,5,6. Following this statement, the results for the integrals become PA6 = - j E K . , 6A ^ P B6 = (rjRKytA + jTA ) P A5 R e({/ 6sinhK6h 6cosh*ic6 h 6) Re (C/6coshK 6h 6sinh K6h 6) (C.6) (C.7) KZ6 B_ 5 c c_ ~ «, E) Ji C 51 + U D 52 C 52 + «M D j 2 C 53 + « L)jj C J4 y5 y5 y5 ys Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.8) where £51 = {~jTBe - j R K z5B h) D 52 = {-JTC ~ j R K =5C ) D 53 ={r j R K ySCe + J T C h) D = {- jR K y5B e + J T B h) C 5I = Re({/5coshK5h5sinh*K5h 5) C$2 = Re(f/ 5sinhK5h 5cosh!|‘K5h J) C C 53 -- -<■> f / 5 |coshK5h 5 1 + t/* |sinhK5h 5 1 - f / 5 2 2 t/* |coshK5h 5 1 + U5 |sinhK5h 5 1 - U* D_ — ,uu, D ^41 ^41 + “ ^42 ^42 >'4 *4 ( R B4 E h \ u ~ ^ * ( ^43 4 ) ^41 + * D f •^44 ^"42 + «T V *4 £ “u + ^ 4 2 ^43 “u ^41 ^"44 (CIO) ^4 * D_ u«T ( ^43 C 43 + v ^ 4 ^ ) + M E_ u■T D 44 C 44 Vz 4 y (C .ll) where h \ * d a\ = {~jTD - j R k z4D ) & 42 = - i ^ K: 4 ^ ) { - jT E ^43 = ( - 7 ^ V £ + J T E h) D „ ^ { - J R k^ D + j T D h) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 C 41 = Re(C/4 c o s h ic 4 h 4 sinh*K4 h 4) C 42 = R e ( £ /4 sinhK 4 h 4 c o s h * K 4 h 4) 2 2 + U* |sinhK4h 4 1 - U 'C - 4 3 -- — o ( / 4 |coshK4h 4 | 2 c P A3 ~ "K~£*31 (-31 + ~^~£*32 ^ 3 2 * F B3 ~ *7 kZ3 J £*33 C 31 ( F ^34 -32 + T \ Z3 J V V* 3 ^ £33 U - (C. 12) ~ ^ ~ £ * 31 F 34 * ( F ■3— + + f / 4 |sin h K 4 h 4 | + ~ ^ ~ £ * 3 2 F 33 + ♦ < < f\ 2 - — U* | c o s h K 4h 4 | 44 — 2 C 33 + * G \ 23 £>34 C 34 J (C.13) where D 31 hV {~JTF = ~jR*z3 J ^ ) h\ * £>32 = - j R K :3G D 33 = ( ^ K )l3G e + y7G *) Z) 34 = (-;/? k ,3F (-31 + j T F h) = R e ( t / 3 c o s h K 3 h 3sinh*K 3 h 3) C 32 = R e ( f / 3 sinhK 3 h 3 c o s h * K 3 h 3) 2 C 33 -“ — 2 U 3 |coshK 3h3 | 2 + v \ | sinh k 2 r 34 -- — 2 t / 3 |coshic 3h 3 | e P a2 e % T D 2\ C 2\ = ( P B2 ~ I M 3~ \ Z 2J * £^23 (-21 + J U* e D 22 C 23 + ♦ / / "3— V Z2 2 ■ '2 "2 I -u , + t / 3 |sinhK3h 3 1 - e + { ^ D 22 C 22 + 2 A D 21 C 24 ♦ D 24 C 22 + H \ 2> (C 1 4 ) -^2 # £*23 ( - 23 \ 2y £*24 (-24 ( C l 5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 215 where D 2l = [-JTH D 2i = ( - JR k ^ H e [-JT I h \* - J R k ^ I J D 23 = { - j R K y2i e + j T I h) D 24 = { - j R K y2H C 2l = + j T H h) R e (f/2coshK 2h 2sinh*K 2h 2) C 22 = R e ({ /2sinhK2h 2 cosh *K 2h 2) C ^23 -~ -2 U2 |coshK 2h 2 | + t/^|sinhK 2^2 | i C ^24 -- -2 U* | c o s h K 2 h 2 | P M = 4— [~jTJ + j R K :lj ' j P Bl = (JRKylJ + i U2 | s i n h K 2h 2 1 - U * R e ( { /iS in h K ,h ,c o s h * K ih 1) f M* + j T J ) "yRe ( £ / , co sh K , h,sinh*K ,h1) I 2' e h with the field coefficients A ,A -U 2 (C.16) (C l 7) eh taken from Appendix B. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 APPENDIX D NUMERICAL METHODS WITHIN THE SPECTRAL DOMAIN METHOD A. Bessel Function Approximation The polynomial approximation for the zeroth-order Bessel function of the first kind for the Fourier transformed basis functions of (2.37) and (2.38) is represented as [68 ] for complex argument y as 0 < [y| <3 J 0(y) = 1 - 2.2499997 (y/3) 2 + 1.2656208 (y/3) 4 - .3163866 (y/3) 6 + .0444479 (y/3) 8 - .0039444 (y/3)‘° + .0002100 (y/3) ' 2 (D. 1) 3 < [y| < oo _I •A)(v) = y 2/ 0 cos e0 (D.2 ) where / 0 = .79788456 - .00000077 (3/y) - .00552740(3ly) - .00009512(3ly) + .00137237 (3/y) 4 - .00072805 (3ly) + .00014476 (3/y) 6 and Q0 = y - .78539816 - .04166397 (3/y) - . J0003954 (3/y ) 2 + .00262573 (3/y) 3 -.00054125 (3/y) 4 - .00029333 (3/y) 5 + .00013558 (3/y)6 . B. Numerical Integration For the infinite-width structure of (2.33a)-(2.33d) and Fig. 11, the elements of the matrix [A] require the evaluation of integrals to solve for the complex propagation constant y and these integrals cannot be expressed analytically. Therefore, numerical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 integration (also referred to as quadrature) is necessary to solve for the system of equations [69], The quadraturemethods are based on the summation of anintegrand at sequence of points(abscissas) within therange of integration. a Thegoal is to obtainthe integral as accurately as possible with the smallest number of function evaluations of the integrand. The typical numerical integral of a function is approximated by the sum of the functional values at a set of equally spaced points and multiplied by certain weighting coefficients. By appropriately choosing these coefficients, integration formulas of higher order can be realized which translate to higher accuracy only when the integrand is very smooth or is well-approximated by a polynomial. The Gaussian quadrature methods also allow the selection o f the abscissas at which the function is to be evaluated and provides another degree o f freedom. In other words, given W(x) (some known function), an integer N, a set of weights B i , and abscissas x( , the integral approximation for the function fix) becomes b N J W{x)flx)dx * Z B , A x , ) a (D.3) /=1 and is exact if/(x) is a polynomial and b B, = j W(x) L , dx a (D.4) where Lt (x) is the Lagrange multiplier function which is used for the generation of the interpolating polynomial for unequally spaced arguments. The arguments x,,...,xv are the zeros of the M h degree polynomial PN belonging to a family of functions having the orthogonal property b j JV(x)PN P M dx = 0 M*N. (D.5) a Many different orthogonal polynomials can be employed to obtain Gauss-type quadrature formulas which can very accurately model the integrand. The choice o f the polynomial will depend on the type o f function to be integrated and the limits o f the integral. In the SDM, (2.33) is integrated and by analyzing the functional nature o f the zeroth-order Bessel Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 functions of the first kind from (2.37) and (2.38), the Legendre polynomials are selected from [70], The integrals of (2.33) behave like a* , a which corresponds to Legendre polynomials and the integration formulas obtained are termed Gauss-Legendre quadrature formulas with W(x)= 1. The integration procedure utilized in this dissertation is the Appoint Gauss-Legendre quadrature formula (GAULEG) from [69], The routine scales the range of integration from (a,b) to (-1,1) and provides absciassas xj , weights Bt , and uses a special formula that holds for the Gauss-Legendre case as - ( i = ? f e <D 7) where Ps is the Legendre polynomial of degree N and Newton's Method is employed to determine the derivative (the prime designation) of the Legendre function in (D.7). A recurrence relation also exists among Legendre polynomials of three consecutive degrees (N+ 1)PN+l (x) = (2N + 1)xPN(x) - NPn_, (x) . (D.8) The dominant or truncation error term involves the (27/)th derivative o f J[x) and translates to very high orders. The best method for evaluating the accuracy of the integral determined by Gauss-Legendre quadrature is to compare the results with different values of N. Caution must be employed due to the presence o f one or more singularities in_/(x) or if/(x) is highly oscillatory. Also, with very large values o f N, roundoff errors can be significant to deteriorate the accuracy of the integral. The limits of integration for the computation of the matrix elements o f (2.33) are from ky = 0 to k y = oo via (2.47) and is separated into two integrals with a lower and asymptotic contributions. An example for the following element in the [A] matrix is n A pm = 2 J { 0 00 + 2 J { }dky = 4 , + A 2 . Ni (D.9) The Fourier integrals are usually defined and evaluated along the real spectral axis of ky. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 219 The actual limits of integration for the upper integral range are from k = N x to k=N2. iV, is determined by setting the value initially to 15 and increasing by a factor of 15 until the change in the complex determinant (real and imaginary parts) o f (2.34) is less than 1% (convergence criterion) using a 10-point Gauss-Legendre procedure. The routine locates the integral region where y is no longer a contribution by comparing the determinant of the matrix for two y initial guesses that are vary considerably. In this acceleration approach, the asymptotic integral contribution is only calculated once at eachfrequency. This approach is justified by analyzing (2.9) with ky large as k 2 » ky . (D.10) The lower integral is broken into subregions o f length 5 (with respect to the spectral variable) from k= N x to k=N2 and is solved by using an Appoint Gauss-Legendre integration algorithm. Convergence is obtained by increasing N (the initial iteration sets N= 5) sequentially by ten until the change in y (real and imaginary parts) is less than 1% as part of the iterative root searching procedure. C. Determinant of a Matrix For a given set of linear algebraic equations written as m = i«] (Dio where [C] is a square matrix with as many equations (AO as unknowns with x (j=l,2,...,N). [X] is the column vector of unknowns and [ft] is the right-hand side column C 12 C 1N H C 22 •••• C 2\ *2 _ CNl C N2 •••• CNN 1 r 2 = 1 1 1 1 X C l\ C 21 X 1 i 1 vector written as (D.12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 The formal solution for the unknowns can be found by employing Cramer's Rule. Any arbitrary unknown x is given by d e ,[C j] XJ = (D -13) det[C] where det is the determinant of the matrix and [C] ] is the matrix [C] with its yth column replaced by [/?]. If all o f the elements o f R are zero as in (2.32), then the det [C; ]=0 since one entire column is zero. Then, nontrivial solutions can only be obtained if det [C]=0. Suppose the matrix [C] can be written as a product o f two matrices [L][U\ = [C] (D. 14) where [L\ is lower triangular (has elements only on the diagonal and below) and [U\ is upper triangular (has elements only on the diagonal and above). In other words, (D. 14) can be presented as a li a 21 a 22 a M a ,V2 0 0 am P.l P l 2 0 (322 0 0 IN li 12 ' ' c IN 2N -21 22 ' ' C 2.V A/2 ' •• c .v.v P.w (D. 15) Decomposition is used to solve for the linear equation set. The advantage of breaking one linear set into two successive ones is that the solution o f a triangular set of equations is quite trivial using forward and back substitution methods. Given [C], [Z,] and [£7] are determined using Crout's algorithm which solves for the a's and P's by arranging the equations in a certain order. In other words the right-hand side of (D. 15) is expressed as 3H 11 3h 12 a 21 22 a Wl a .V2 - 3k IN 2N (D. 16) NN and Crout's method fills in the above combined matrix by columns from left to right and within each column from top to bottom. The implementation o f Crout's method includes partial pivoting. Pivoting is the selection o f the appropriate matrix element for the division Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 221 calculations and is essential for numerical stability purposes. largest element of each row. The pivot element is the Only partial pivoting (interchange o f rows) can be implemented efficiently. Actually the matrix [C ] is not decomposed into LU form, but rather it is decomposed into a rowwise permutation o f C. An L U decomposition algorithm from [71] (LUDCMP) is utilized and keeps track of whether the number of row interchanges are even or odd. The determinant o f an L U decomposed matrix is just the product o f the diagonal elements from (D. 16) as N det = I Pr (D.17) D. Root Searching Algorithm The root solutions for the propagation constant y o f the CBCPW are determined by an iterative root search of the determinant equation o f (2.34). Recall again that the propagation constant is a complex quantity. Equation (2.34) can be written in general terms as A x) = 0 (D.18) where x is the desired solution or solutions and / is the dimensional function whose components are the characteristic system of equations. Root finding proceeds by iteration. Starting from some approximate trial solution, a useful algorithm will improve the solution until some predetermined convergence criterion is satisfied. For smoothly varying functions, robust algorithms will always converge provided the initial guess is accurate. The algorithm should also bracket a root between bracketing values and then search to converge to the solution. Consider a point x0 which is not a root of the function A x), but is reasonably close to a root. Expanding A x) *n a Taylor series about x0 becomes A x) = A Xo) + (x ~ x 0) f / (x0) + ^ " 2 °^ • • / / / (* o) + (D. 19) where the single and double prime represents the first and second derivatives, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 222 If7(x) is set to zero, then x must be a root. Unfortunately, the above equation (right-hand side) is a polynomial of infinite degree. However, an approximate value of the root x can be obtained by taking the first two terms o f (D. 19) and solving for x. Now x represents an improved estimate o f the root and can replace x0 onthe next iteration. The general expression for this method (Newton's) can be written as 1*1 - ! . = » * , = - (D-20 ) where the subscript n denotes values obtained on the wth iteration and (n+ \) indicates values found on the (n+ 1,1th iteration. Newton's method will converge to a root for most functions and if it does converge, it will usually do so extremely rapidly. The secant method is essentially a modification o f the conventional Newton's method with the derivative replaced by a difference expression. This is advantageous if the function is difficult to differentiate. Also, it is only necessary to provide a single function subprogram rather than subprograms for both the function and its derivative. Replacing thederivative in (D.20) by a simple difference representationyields l» l - i . = 5*, ------------- ^ ---------- (D.21) where / (x„.,) must be saved in this method. This is the value o f / from two iterations previous to the present one. Since no such value will be available for the first iteration, two different initial guesses for the root must be supplied. For the secant method to be effective, the functions must be smooth near a root since the root does not necessarily remain bracketed. Muller's method [72] generalizes the secant method but uses quadratic interpolation (solving for the zeros of the quadratic) among three points instead of linear interpolation through the two most recently evaluated points as in the secant method. Muller's method is a preferred algorithm for finding complex zeros of analytical functions in the complex plane. Given three previous guesses for the root xn_2 , xn l , xn and the values of the polynomial J(x) at those points, the next approximation x^, is produced by the following * = & <D'22a> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 223 X = q f ( x n) - q( 1 + q ) f ( x n_i) + q 2f ( x ^ 2) (D.22b) Y = (2q + 1) / ( x „ ) - ( l + ^ ) 2 / ( x ll_i) + q 2f ( x n_2) (D.22c) Z = (1 + * ) /( x B) (D.22d) and y± 2Z JY2 - 4 x z (D.23) where the sign in the denominator is chosen to make its absolute value or modulus as large as possible. The method allows the iterations to start with any three values of x such as three equally spaced on the real axis and will locate complex roots. The three initial guesses of the solution for the dominant CBCPW mode complex propagation constant in this dissertation are Yj = * o V maX (e rl ,Sr2>Sr3,er4,Sr5<er6) (D.24a) y 2 - k o ^ (e rl + e r2 + s r3 + e r4 + s r5 + s rt) / 6 (D.24b) Y3 = (Y, + Y2) / 2 (D.24c) where max represents the maximum value o f the relative dielectric constants of the six possible layers of the structure. Muller's method has shown to be a very robust root searching procedure. The algorithm is terminated when the magnitude o f the computed change in the value of the root 5n+, is less than some predetermined quantity p. In this analysis, p is set at 0.1 and the root convergence requirement for the real part of y is (D.25) and is similarly employed for the imaginary part o f y. The above equation is derived from (D.21) with 6^, being the second term on the right-hand side. E. Basis Function Expansion Coefficients Solution After the true value of y is determined from the SDM, the basis function expansion coefficients o f (2.27) are solved from (2.32). In matrix form, this relation is written from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 (2.30) and (2.33) as 11 11 ^ ,1 A 12 11 A 21 A 22 21 AU 21 A N\ 11 12 21 21 21 A N2 21 A 12 12 12 A 22 - a xn 22 22 A NM A VI A A/2 'M 0 dx 0 22 22 A 22 0 12 12 22 22 0 A 2N 22 A ,2 A 21 “ 12 A IN A n MN 22 A 2M 12 A M1 A M2 A 1M A n 21 A 12 a mm 21 A 12 21 ^21 11 A 2M A 21 11 A M2 12 A IM A n 11 11 A Ml 11 0 A 2N 22 A nn 0 4N _ _ (D.26) where this homogeneous system o f equations and the matrix [A] is a (M-*-N) x (M+N) matrix. In this case, while no unique solutions are available, certain relationships exist between the unknowns. The values of the expansion coefficients (cm and dn), which determines the relative amplitude o f the eigenvector distribution, depend on the strength of the excitation of the structure. These coefficients are normalized with respect to the first expansion function coefficient (c,) which is arbitrarily set equal 1.0. This implementation modifies (D.26) as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 11 A 12 11 A IU 11 A 22 11 Am 21 A 12 11 A 2M A 21 A N2 A U im 21 A 2M 21 A NM 12 12 A ,2 A IN 12 12 12 A 21 A 22 A IN 12 A m\ 21 A 2! A 22 11 12 22 ^11 12 A M2 A 12 MN 22 22 A 12 A W 22 22 22 A 21 A 22 A IN 22 22 22 A m A A/2 A NN 11 Am 21 AU 21 I21 21 ^.Vl (D.27) The [A] matrix is not square (M+N) x (M+N-l). In this case, more equations exist than unknowns. A method is required to find the least-squares solution to the overdetermined set of linear equations. Singular value decomposition (SVD) is the method o f choice for solving most linear least-square problems. SVD is a technique for dealing with matrices that are either singular or else numerically very close to singular. SVD methods [73] are based on the following theorem o f linear algebra. For a general M x N matrix [A] whose rows M is greater than or equal to its number of columns N, can be written as the product of an M x N column orthogonal matrix [U\, a x \ N x N diagonal orthogonal matrix [W] with positive or zero elements, and the transpose of an N x N orthogonal matrix [V], These relations can be represented as [A] = [U] [ W \[ V T\ (D.28) The matrices [t/] and [V] are both orthogonal in the sense their columns are orthogonal. The decomposition of (D.28) can always be accomplished. If the matrix [/I] is a square N x N, then [£/], [V], and [W] are all square matrices o f the same size. The matrix inverses Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 226 are trivial to compute. [U\ and [V] are orthogonal so the inverses are equal to the transposes. [W] is diagonal and the inverse is the diagonal matrix whose elements are the reciprocals of the elements w}(j=l,2,....,N). From (D.28), it follows that the inverse of [A] is [A]~l = [V] [diagonal (l/w; )] [ f / 7] . (D.29) The problem with this matrix construction is if one of the wjs is zero or numerically so small, the value o f (D.29) is dominated by roundoff error. Now consider the general set of simultaneous equations [A] [x] = [B] (D.30) where [A] is a square matrix and [5] and [x] are column vectors. Equation (D.30) defines A as a linear mapping from the vector space x to the vector space B. A matrix is singular if the condition number (ratio of the largest of the w's to the smallest of the w's) is infinite and is ill-conditioned if the condition number is too large (approaches the computer floating point precision). If [A] is singular, then there is some subspace of x, called the nullspace, that is mapped to zero as [/4 ][x]=[0 ], There is also some subspace of B which can be related by A , in the sense that there exists some x which is mapped there. This subspace of B is called the range of A. SVD explicitly constructs orthogonal bases for the nullspace and the range of a matrix. Specifically, the columns o f [U\ whose same-numbered elements Wj are nonzero and are an orthogonal set of basis vectors that span the range. Also, the columns o f [V] whose same-numbered elements w. are zero and are orthogonal basis for the nullspace. The important point from (D.30) is whether the vector B on the right-hand side lies in the range of A. If this is true, then the singular set of equations does have a solution [x]. In fact, more than one solution exists since any vector in the nullspace can be added to x in any linear combination. To select one single 2 solution of this set, an appropriate choice would be the one with the smallest length |x| . Using SVD, the Vwjs are replaced by zero if w= 0. The solution is computed working from right to left as [x] = [V\ [diagonal ( 1/w,)] [ l / 7’][£]. (D.31) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 Equation (D.31) will be the solution vector of smallest length. The columns o f [V] are in the nullspace and complete the specification o f the solution set. If B is not in the range of the singular matrix [A], then the set o f equations o f (D.30) has no solution. However, (D.31) can still be used to construct a "solution" vector [x], This vector will not exactly solve (D.30) but among all possible vectors x, the SVD method will find the least-squares compromise solution. This solution is the one that minimizes the residual of (D.30). SVD must be implemented carefully by deciding the threshold to zero the small w's ( 1.0 x 10-6 chosen in the routine). First, the SVD of a matrix [A] is calculated from a call to routine SVDCMP. The solution vector [x] is obtained for a right-hand side vector [B] via (D.31) with a backsubstitution routine SVBKSB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 APPENDIX E PROPAGATION CHARACTERISTICS OF IDEAL PARALLEL PLATE WAVEGUIDE MODES For an ideal homogeneously filled parallel plate waveguide formed by conductors covering the *=0 and x=-h4 in Fig. E. 1 with h5=0, the wave functions for the TEM, TMr , and TEXare TE (E d TM - cosl hf) e (E2) TEM ~ jk TEM vp0 = * Z (E.3) c0 and the propagation phase constant for the TMm and TEm modes with m = 1,2, . . . , o o becomes and that for the TEM TEM k c,0 =k = C O jS re o ^ P o (E.5) . A multi-layered CBCPW configuration proposed in this dissertation can be employed to prevent the leakage effects up to a critical frequency. From Fig. 11 two inhomogeneously filled parallel plate waveguides exist above and below the circuit conductors/ground planes at x=0 in the region outside the cross-sectional circuit area ( [y| >S/2 + W). Consider only the lower ideal parallel plate waveguide here in Fig. E.l. Most of the modes are now hybrid with both E, and H: components. TEM mode does not exist and the characteristic equations for the mode propagation constant of the TM^ and TEr modes are respectively [9] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. E. 1. Parallel plate waveguide representation with two dielectric substrates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with m = 0, 1, 2 , . . . ,oo. The above transcendental equations for the TM and TE modes are solved for the mode propagation constants (kc). Unlike the homogeneously filled waveguide, the TM and TE modes will have different cutoff frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 APPENDIX F DERIVATION OF THE RESIDUE CALCULUS THEOREM The derivation o f the residue calculus terms of (3.12) for a leaky waveguide is presented [74]. Assume fiz) is analytic within and on a simple closed curve C except at a number of poles a,b,c, interior to C and C is traversed in a positive (clockwise) sense as shown in Fig. F. 1 then | J{z) dz = 271/ {sum of residues of f(z) at poles a, b, c, ...}. C (F. 1) By constructing cross cuts from any point on C to any point on C,, C2, C3,.... the following relation can be derived j f[ z) dz = C | ftz)dz + C, j J(z)dz + | J(z)dz + .... C2 (F.2) C3 where J{z) is analytic in the shaded area. If f(z) has a pole of order n at z=a but is analytic at every other point inside and on a circle C with center at a,then (z-a)nfiz) is analytic at all points inside and on C and has a Taylor series about z=a such that <yV Cl—ri+\ y (7 -1 + (^ 5 F + ■+ v / v + a ° * a ' (z ~ a) + a 2 ( z ~ a) ^ + (F .3) and is called the Laurent series forf(z). By integrating (F.3)yields fiz)dz = | I a~\ -dz + | -- - - - - d z + ... + <f ■- -~ dz + I W c (.z-d) I | j a 0 + a , ( r - a ) + a 2( z - a ) 2 + ...Jcfe c = 27ij a_x. (F.4) This is valid from Cauchy's Theorem stating that fora simple closed curve C, if f(z) is analytic within the region bounded byC as well as on C then | j{z) dz = 0. C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (F.5) Fig. F. 1. Illustrative example of the residue calculus theorem with a number o f poles a,b,c interior to a simple closed curve C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 233 The following relationship is also invoked in (F.4) as f _ d z _ = j 27zj if n= 1 1 I (?-a)n j 0 if «=2,3,4,... J where C is a simple closed curve bounding a region having z=a as an interior point. The point a , in (F.4) is the residue o f f{z) at the pole z=a. The above theory can be applied similarly for poles b and c and hence (F. 1) is derived. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 A P PE N D IX G IEEE PERMISSION TO REPRINT COPYRIGHTED MATERIAL IE E E OPERATIONS CENTER T H E I N S T I T U T E O F E L E C T R I C A L A N D E L E C T R O N I C S E N G I N E E R S . INC. ■US HOES l a n E. P O. 9 0 X 1331. PISCATAWAY. NJ 08855-1331. 'J.S.A. TEL. !908) 981-0060 TELEX: 333233 PAX: 1908) 981-0027 D IR E C T NU M BER (908) 562- 3 9 5 5 April 18, 1996 Mr. Mark Magerko 11500 Jollyville Road #1121 Austin, TX 78759 Dear Mr. Magerko: This is in response to your letter of April 10 in which you have requested permission to reprint, in your upcoming dissertation, four of your IEEE copyrighted papers. We are happy to grant this permission. Our only requirement is that the following copyright/credit line appears prominently on the first page of each reprinted paper, with the appropriate details filled in: ® 199x IEEE. Reprinted, with permission, from (full journal name; volume, issue, and page numbers; month/year of issue). Sincerely yours, William J. Hagen, Manager Copyrights and Trademarks WJH:lp Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 235 VITA Mark Alexander Magerko was bom in Colorado Springs, Colorado on August 2, 1962, the son of John Sr. and Mary Ann Magerko. After finishing his work at Kaneland Senior High School in 1980, he entered Northern Illinois University in DeKalb, Illinois to complete pre-engineering courses. Afterward, he transferred to the University o f Illinois at Champaign-Urbana and received his B.S. and M.S. in Electrical Engineering in 1984 and 1987, respectively. From 1984-1995, he was employed with several companies including Magnavox Electric Systems in Fort Wayne, Indiana, Cascade Microtech, in Beaverton, Oregon, Raytheon Research Labs in Lexington, Massachusetts, and Dell Computer in Austin, Texas. He joined the graduate school o f Electrical Engineering at Texas A&M University in 1989 where he worked as a research assistant and assistant lecturer in the Electromagnetics Group under Dr. Kai Chang. He was awarded a Fellowship from the Department of Education - Areas of National Need Program at Texas A&M University. Permanent Address: 138 Neil Road P.O. Box 529 Sugar Grove, Illinois 60554. 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