# Considerations in the simulation of large monolithic microwave integrated circuits enclosed in a conducting package

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University of Massachusetts, 1993 C opyright © 1993 b y B urke, Joh n Josep h . A ll rig h ts reserved. UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONSIDERATIONS IN THE SIMULATION OF LARGE MONOLITHIC MICROW AVE INTEGRATED CIRCUITS ENCLOSED IN A CONDUCTING PACKAGE A Dissertation Presented by JOHNJ. BURKE Submitted to the Graduate School of the University of Massachusetts in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY February 1993 Department of Electrical and Computer Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © Copyright by John J. Burke 1993 All Rights Reversed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONSIDERATIONS IN THE SIM ULATION OF LARGE M ONOLITHIC M ICROW AVE INTEGRATED CIRCUITS ENCLOSED IN A CONDUCTING PACKAGE A Dissertation Presented by JO H N J. BURKE Approved as to content and style by: Robert W. Jackson, Chair Daniel H. Schaubert, Member / s j f-x Q j jCt— Robert E. McIntosh, Member Donald F. St. Mary, Member .____ Lewis E. Franks, Department Head Department of Electrical and Computer Engineering College of Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This dissertation is dedicated to the memory of my grandfathers, John J. Burke and Joseph P. Bambera. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I wish to express my sincere thanks to Dr. Robert W. Jackson for his guidance, support and encouragement throughout the preparation of this dissertation. I would like to thank Dr. Robert E. McIntosh for the teaching experience I gained while acting as a TA for his Microwave Engineering course, and for his input to the research. Professor Daniel H. Schaubert for the suggestions and insights he contributed to my work at the monthly seminars and Professor Donald F. St. Mary for serving on my dissertation committee. Many thanks to Dr. El-Badawy El-Sharawy, Jason Gerber and the other members of LAMMDA for their assistance and friendship. Jean Sliz is also greatly acknowledged for her friendship and assistance. I would like to thank the great game of hockey and my teammates for helping me keep my sanity, especially over the past two years. I would like to thank my parents, sisters and in-laws for their support and encouragement over what seemed like an everlasting endeavor. Last but not least, I would like to thank my wife, Angela, for her love, support and perseverance throughout this long process. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A B ST R A C T CO NSIDERATIONS IN TH E SIM ULATIO N OF LARGE M ONOLITHIC M ICROW AVE INTEG RATED CIRCUITS ENC LO SED IN A CO ND UCTING PACK AG E FEBRUARY 1993 JOHN J. BURKE B.S., NORTHEASTERN UNIVERSITY M.S., UNIVERSITY OF CALIFORNIA AT LOS ANGELES Ph.D., UNIVERSITY OF MASSACHUSETTS Directed by: Professor Robert W. Jackson This thesis presents a numerical and experimental analysis of MMIC circuits enclosed in a conducting package. Three different simulation techniques are developed: a full-wave method o f moments (MOM) procedure, a simple circuit model and the enhanced diakoptic method. In the design of MMICs, the effect of enclosing the circuit is typically assumed to be negligible. However, as the electrical size of enclosures increase package resonances are possible. If the system operates at frequencies near one o f these resonances, coupling between the fundamental microstrip mode and the resonant mode is possible. This phenomena is referred to as parasitic coupling to a resonant mode and its importance is emphasized in this thesis. vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. An experimentally verified full-wave MOM procedure is used to examine some of the fundamental aspects o f resonant mode coupling. In addition, methods of reducing this coupling will also be investigated. For example, both the addition of loss to an enclosure or layout modifications can be used to reduce resonant mode coupling. Since a full-wave analysis, although rigorous, is also very complex to implement, a simple circuit model is developed to describe resonant mode coupling. Simple analytical expressions for the entire model are easily evaluated, making this is a very attractive feature for implementation into a CAD package. In addition, it requires several orders of magnitude less CPU time than the MOM. As the size o f MMIC circuits increase they become too complicated to analyze using a straightforward full-wave approach. A full-wave analysis of a typical MMIC of moderate complexity may require the solution of a large system of equations. Therefore, as second alternative to the MOM the diakoptic method is modified to analyze an MMIC in an enclosure. For very large circuits significant CPU savings result. A new spectral filtering technique, called the enhanced diakoptic method, is developed to improve accuracy. vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page ACKNOWLEDGMENTS........................................................................................................... v ABSTRACT................................................................................................................................. vi LIST OF TABLES...................................................................................................................... xi LIST OF FIGURES.................................................................................................................... xii CHAPTER 1. INTRODUCTION..............................................................................................................1 Chapter 1 References...................................................................................................... 5 2. FULL-WAVE ANALYSIS OF PACKAGED M M ICs.............................................. 8 2.1 2.2 2.3 Introduction......................................................................................................... 8 Spectral Green's Function....................................................................................8 Enclosure Resonances........................................................................................20 2.3.1 2.3.2 2.3.3 2.4 2.5 2.6 2.7 Method of Moments Solution.......................................................................... 31 Efficient Computation of the Impedance Matrix Elements......................... 32 Network Parameters...........................................................................................37 Expansion Functions..........................................................................................40 2.7.1 2.7.2 2.7.3 2.8 Introduction......................................................................................... 20 Evaluation of the Resonant Frequencies......................................... 22 Summary o f the Enclosure Resonances Determination................ 30 Region 1-Discontinuities................................................................... 42 Region 2-Transmission Lines........................................................... 44 Region 3 -Sources.............................................................................. 44 Conclusion........................................................................................................... 56 Chapter 2 R eferences...................................................................................................57 3. RESONANT MODE COUPLING IN PACKAGED MMICs: A THEORETICAL AND EXPERIMENTAL INVESTIGATION.......................... 60 3.1 3.2 Introduction......................................................................................................... 60 Experimental Verification of the MOM for Circuits in Resonant Enclosures..........................................................................................................61 3.2.1 Introduction......................................................................................... 61 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2.2 Large Gap.............................................................................................63 3.2.3 Shunt Stub............................................................................................74 3.3 Parasitic Coupling to Resonant M odes..........................................................99 3.3.1 Large Gap........................................................................................... 102 3.3.2 Shunt Stub.......................................................................................... 108 3.4 C onclusioa.......................................................................................................119 Chapter 3 R eferences................................................................................................120 4. A SIMPLE CIRCUIT MODEL FOR RESONANT MODE COUPLING IN PACKAGED M M ICs.......................................................................................... 122 4.1 4.2 Introduction......................................................................................................122 Development of the Circuit M odel...............................................................123 4.2.1 Circuit Mutual Impedance................................................................123 4.2.2 Full-wave Mutual Impedance.......................................................... 125 4.2.3 Comparison of the Mutual Impedances.........................................129 4.3 4.4 Numerical Evaluation of Circuit components............................................. 130 Results.............................................................................................................. 132 4.4.1 4.4.2 4.4.3 4.4.4 4.5 Small Gap........................................................................................... 132 Shunt Stub.......................................................................................... 136 Shunt Stub in a Larger Box..............................................................140 Band Pass Filter................................................................................. 146 Conclusioa.......................................................................................................150 Chapter 4 R eferences................................................................................................152 5. ANALYSIS OF MMICs IN RESONANT ENCLOSURES WITH THE DIAKOPTIC M ETHOD........................................................................................... 153 5.1 5.2 Introduction ................................................................................................. 153 Diakoptic Method............................................................................................154 5.2.1 Theory................................................................................................ 154 5.2.2 The Relationship Between the Diakoptic and Moment Methods..............................................................................................157 5.2.3 Results................................................................................................ 161 5.3 Enhanced Diakoptic Method......................................................................... 168 5.3.1 Theory of Spectral Filtering............................................................ 168 5.3.2 Determination of the Cutoff Mode Numbers................................ 170 5.3.3 Results................................................................................................ 174 5.4 Conclusioa.......................................................................................................189 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 R eferences.................................................................................................193 6. CONCLUSION............................................................................................................. 194 Chapter 6 R eferences.................................................................................................198 APPENDICES A. B. C. D. DERIVATION OF dYv /d co.......................................................................... 199 LARGE ARGUMENT SPECTRAL GREEN’S FUNCTION.................. 201 THE SCATTERING M ATRIX .................................................................... 203 THE ELECTRICAL CHARACTERISTICS OF THE DIELECTRIC ABSORBER........................................................................ 206 BIBLIOGRAPHY.................................................................................................................... 209 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table Page 2.1 Summary of the resonant frequencies and Q's for various enclosures............31 2.2 Summary of the transition models employed for all substrates used in this dissertation...................................................................................................56 3.1 The theoretical resonant frequency and Q for each mode in cavity B ........... 63 3.2 The resonant frequency and Q for each mode in the low 2-dielectric cover cavity B......................................................................................................... 92 3.3 The resonant frequency o f each mode in the low 2 -S i cover cavity B.......... 92 3.4 The ratio of the TM i io mode powers to the incident power for the large gap................................................................................................................ 108 3.5 The ratio of the TM i io mode powers to the incident power for the stub.... 118 4.1 Transformer turns ratios for the transmission lines on both sides of the gap....................................................................................................................133 4.2 Transformer turns ratios for the transmission lines on both sides of the stub...................................................................................................................136 4.3 Transformer turns ratios for the stub..................................................................138 4.4 The lumped elements of the resonant circuits for each mode in the high Q package......................................................................................................142 4.5 Transformer turns ratios for the transmission lines on both sides of the stub...................................................................................................................145 4.6 Transformer turns ratio for the stu b ...................................................................145 4.7 The lumped elements o f the resonant circuits for each mode in the low Q package.......................................................................................................145 4.8 The lumped elements of the resonant circuits for each mode in the high Q package......................................................................................................146 4.9 Transformer turns ratios for the bandpass filter...............................................148 5.1 kc for 5.2 Comparison of the operation counts for the M OM and enhanced diakoptic m ethods................................................................................................ 191 a few representative substrates................................................................ 172 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Figure Page 2.1 Geometry of the MMIC package used in the derivation of the Green's function...................................................................................................... 9 2.2 The transmission line equivalent circuit for the geometry shown in F ig u re 2 .1 ................................................................................................... 16 2.3 The transmission line equivalent circuit for determining Y[$ (i = 1, k ) ........................................................................................................17 2.4 The transmission line equivalent circuit for determining Yil), (i = Jc+1, ..., N ) ................................................................................................. 18 2.5 Im ( I ^ ) and Im(Qtm ) versus frequency for the lossless enclosure with the mode numbers, n and m , are equal to 1....................................23 2.6 Im (} ^ ) and Im(<2rAf) versus frequency for tiie high Q enclosure with the m ode numbers, n and m , are equal to 1.............................................25 2.7 Im (J ^ ) and QTM versus frequency for the moderate Q enclosure with the m ode numbers, n and m, are equal to 1 .............................................28 2.8 Im(YM) and Qtm versus frequency for the low Q enclosure with the m ode numbers, n and m, are equal to 1.............................................29 2.9 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located at x c = 7.5 mm (a/2), has a length L = 1.8 mm and is attached to a transmission line, located at y c = 12 mm (b!2) of width w = 1.4 mm.............................................................36 2.10 Comparison of the predicted transmission response of the stub using three different values of a and s A; a = 4 and e A = 0 .0 1 , a = 4 and s A = 0.2, and a = 2 and £A = 0.2.................................................................... 38 2.11 Schematic of an port microstrip circuit with a voltage generator and a terminating impedance connected to each external port.................... 39 2.12 A simple microstrip circuit consisting o f a step embedded between two transmission lines.................................................................................................. 41 2.13 The arrangement of the rectangular cells used near a discontinuity..................43 2.14 The arrangement of the rectangular cells used along a uniform section of transmission line.................................................................................................... 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.15 Geometry of a transmission line with a gap in the center. The transmission line, located at y c = 12 mm (b!2), has a width of w = 1.4 mm and a gap of g = 10.5 m m ................................................................................................47 2.16 Comparison of the predicted transmission response of the large gap in the high Q package using the MOM without the region 3 configuration versus the m easured resu lts...................................................................... 49 2.17 The region 3 configuration of basis functions used to simulate the coaxial to microstrip transition......................................................................................... 50 2.18 Comparison of the predicted transmission response of the large gap in the high Q package using the MOM with the region 3 configuration versus the m easured results.......................................................................51 2.19 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located a tx c = 15 mm (a/2), has a length L = 1.9 mm and is attached to a transmission line, located aty c = 24 mm (b/2), o f width w = 1.4 m m ................................................................................ 52 2.20 Comparison of the predicted transmission response of the stub in the high Q package using the MOM without the region 3 configuration versus the measured results............................................................................................. 54 2.21 Comparison of the predicted transmission response of the stub in the high Q package using the MOM with the region 3 configuration versus the measured results............................................................................................. 55 3.1 Basic geometry of the brass cavity with the following inner dimensions: a , b and c ...........................................................................................................62 3.2 Geometry of a transmission line with a gap in the center. The transmission line, located at y c, has a width of w = 1.4 mm and a gap of g = 10.5 m m ................................................................................................ 64 3.3 Comparison of the calculated and measured transmission coefficient of the large gap in the high Q cavity A for two different locations: y c = 12 mm and yc = 5 mm................................................................................ 6 6 3.4 Comparison of the calculated and measured transmission coefficient of the large gap located at y c = 5 mm in the high Q cavity A. The MOM simulation was performed with the circuit located at y c = 5.35 mm.............. 6 8 3.5 Comparison of the calculated and measured transmission coefficient of the large gap in the low Q cavity A for two different locations: y c = 12 mm and yc = 5 mm................................................................................ 70 3.6 Comparison of the calculated and measured transmission coefficient of the large gap located at yc = 5 mm in the low Q cavity A. The MOM simulation was performed with the circuit located aty c = 5.35 mm.............. 72 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.7 Computed 77 for the large gap in the low Q cavity A for two different locations: yc = 12 mm and y c = 5 mm............................................................... 73 3.8 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located at x c (a/2), has a length L = 1.9 mm and is attached to a transmission line, located at y c, of width w = 1.4 mm..............75 3.9 Ideal theoretical response of the shunt open circuit stub................................... 76 3.10 Comparison of the calculated and measured transmission coefficient of the shunt stub in the high Q cavity A for two different locations: y c = 12 mm and y c = 5 mm.................................................................................77 3.11 Comparison of the calculated and measured transmission coefficient of the shunt stub circuit of length L = 1.9 mm located aty c = 1 2 mm in the high Q cavity A .......................................................................................... 80 3.12 Comparison of the calculated and measured transmission coefficient of the shunt stub in the high Q cavity A for two different locations: y c = 12 mm and y c = 5 mm. For the circuit positioned at y c = 12 mm, the MOM simulation was performed with y c = 11.45 mm, L = 1.8 mm and er =10.75. For the circuit positioned at y c = 5 mm, the MOM simulation was performed with y c = 5.35 mm, L = 1.9 mm and er =10.75........................................................................................................ 82 3.13 Comparison of the calculated and measured transmission coefficient of the shunt stub in the low Q cavity A for two different locations: y c = 12 mm and y c = 5 mm....................................................................................................... 84 3.14 Computed 77 for the shunt stub in the low Q cavity A for two different locations: yc = 12 mm and yc = 5 mm. The MOM simulation was perform ed with L = 1.9 mm and er = 1 0 .5 .................................................87 3.15 Comparison of the calculated and measured 77 for the shunt stub in the low Q cavity A for two different locations: y c = 12 mm and y c = 5 mm.............................................................................................................. 8 8 3.16 Comparison o f the calculated and measured transmission coefficient of the shunt stub in the high Q cavity B............................................................. 91 3.17 Comparison o f the calculated and measured transmission coefficient of the shunt stub in the low Q-dielectric cover cavity B ...................................93 3.18 Comparison of the calculated and measured transmission coefficient of the shunt stub in the low Q-Si cover cavity B...............................................95 3.19 Computed 77 for the shunt stub enclosed in the dielectric and Si cover cavities....................................................................................................................96 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.20 Comparison of the calculated and measured 77 for the shunt stub enclosed in the: low Q dielectric cover cavity B and low <2-Si cover cavity B................................................................................................................... 97 3.21 Summary of the calculated transmission coefficient of the large gap in the high Q cavity A for two different locations: yc = 12 mm and y c = 5 mm............................................................................................................. 103 3.22 Current, at/ = 10.8129 GHz, on the two strips of the large gap in the high Q cavity A . ............................................................................................ 104 3.23 Power lost per unit length to the T M i 1 0 mode via the x component o f the electric field a t / = 10.8129 GHz for the large gap in the high Q cavity A..................................................................................................................104 3.24 Summary of the calculated transmission coefficient of the large gap in the low Q cavity A for two different locations: yc = 12 mm and yc = 5 mm............................................................................................................. 106 3.25 Current, at/ = 10.8129 GHz, on the two strips of the large gap in the low Q cavity A................................................................................................107 3.26 Power lost per unit length to the T M i 1 0 mode via the x component of the electric field at/ = 10.8129 GHz for the large gap in the low Q cavity A ..................................................................................................................107 3.27 Ideal currents on the: transmission line and stub.............................................. 109 3.28 Summary of the calculated transmission coefficient of the shunt stub in the high Q cavity A for two different locations: yc = 12 mm and y c = 5 mm ............................................................................................................. I l l 3.29 Currents, at/ = 11.1 GHz, on the: transmission line and stub. The circuit is enclosed in the high Q cavity A.................................................. 112 3.30 Power lost per unit length along the transmission to the T M i 10 mode via the x component o f the electric field at/ = 11.1 GHz. The circuit is enclosed in the low Q cavity A .......................................................................... 113 3.31 Summary of the calculated transmission coefficient of the shunt stub in the low Q cavity A for two different locations: yc = 12 mm and yc = 5 mm ............................................................................................................. 115 3.32 Currents, at/ = 11.1 GHz, on the: transmission line and stub. The circuit is enclosed in the low Q cavity A .................................................. 116 3.33 Power lost per unit length to the T M i 1 0 mode via the x component of the electric field, at / = 11.1 GHz, along the transmission line and along the stub. The circuit is enclosed in the low Q cavity A....................... 117 xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 Schematic of a small length of a microstrip transmission line and the resulting circuit m odel........................................................................ 124 4.2 Geometry of the MMIC package used to determine the full-wave mutual impedance between a u directed current element located at ) and a v directed current element located at ( x j , y j , Z k ) ............................................126 4.3 Geometry o f a transmission line with a gap in the center. The transmission line, located at y c - 1.55 mm (b/2), has a width of w = 0.1 mm and a gap of g = 0.1 mm...................................................................................................... 134 4.4 Comparison of the predicted transmission response of the small gap in the high Q package using the circuit model versus the conventional MOM.....................................................................................................................135 4.5 Geomeuy of a transmission line with a single shunt open circuit stub attached. The stub, located at xc (a/2), has a length L = 1.9 mm and is attached to a transmission line, located at y c (b/2), of width w = 1.4 mm..........................................................................................................137 4.6 Comparison of the predicted transmission response of the stub in the high Q package using the circuit model versus the conventional MOM 139 4.7 Comparison of the predicted transmission response of the stub in the low Q enclosure using the proposed circuit model versus the conventional MOM..................................................................................................................... 141 4.8 Comparison of the predicted transmission response of the stub in the larger high Q package using the circuit model versus the conventional M OM .................................................................................................................... 143 4.9 Comparison of the predicted transmission response of the stub in the larger low Q package using the circuit model versus the conventional MOM.......................................................... :......................................................... 144 4.10 Geometry of a two resonator coupled line bandpass filter. The width and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively. The spacing of the resonators are = 0.13 mm and S2 - 0.64 mm............147 4.11 Comparison of the predicted transmission response of the bandpass filter in the high Q package using the circuit model versus the conventional MOM..................................................................................................................... 149 5.1 Schematic of center fed dipole and the resulting diakopted circuit................ 155 5.2 Geometry of a center fed dipole located at xc = 7.0 mm and y c = 2.85 mm.......................................................................................................162 5.3 Comparison o f the computed input reactance of the dipole in the high Q package using the diakoptic method versus the input reactance computed using the conventional MOM............................................................................. 163 xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4 Comparison of the predicted input reactance of the dipole in the high Q package using the diakoptic method versus the conventional MOM 165 5.5 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located at xc (a/2), has a length L = 1.9 mm and is attached to a transmission line, located atyc (b/2), of width w = 1.4 mm.........................................................................................................166 5.6 Comparison of the predicted transmission response of the stub in the high Q package using the diakoptic method versus the conventional MOM 167 5.7 Comparison of the predicted input reactance of the dipole in a high Q package using the enhanced diakoptic method for different values of Kc versus the conventional M OM ............................................................. 171 5.8 Comparison of the computed input reactance of the dipole using the enhanced diakoptic method with the optimum value of k c ( 9.625) versus the conventional MOM.......................................................................................173 5.9 Comparison of the predicted transmission response of the stub in the high Q package using the enhanced diakoptic method ( k c = 13.5) versus the conventional MOM............................................................................................. 175 5.10 Comparison of the predicted transmission response of the stub in the low Q package using the enhanced diakoptic method versus the conventional MOM....................................................................................................................177 5.11 Comparison of the predicted transmission response of the stub in the larger high Q package using the enhanced diakoptic method ( k c = 13.5) versus the conventional M OM .................................................................. 179 5.12 Comparison of the predicted transmission response of the stub in the larger low Q package using the enhanced diakoptic method versus the conventional MOM............................................................................................. 180 5.13 Geometry of a two resonator coupled line bandpass filter. The width and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively. The spacing of the resonators are s j = 0.13 mm and S2 = 0.64 mm 182 5.14 Comparison of the predicted transmission response of the bandpass filter in the high Q package using the enhanced diakoptic method versus the conventional M O M ............................................................................................183 5.15 Comparison of the predicted transmission response of the bandpass filter in the low Q package using the enhanced diakoptic method ( k c = 14.25) versus the conventional M OM ...............................................................184 5.16 Geometry of a transmission line with a gap in the center. The transmission line, located at yc = 1.55 mm (b/2), has a width of w = 0.1 mm and a gap of g = 0 .1 mm..................................................................................................... 186 xvii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.17 Comparison of the predicted transmission response of the small gap in the high Q package using the enhanced diakoptic method ( k c = 15.75) versus the conventional M OM ............................................................... 187 5.18 Comparison of the predicted transmission response of the stub in the high Q package using the enhanced diakoptic method ( kc = 9.625) versus the conventional MOM..............................................................................................190 D. 1 Geometry of a transmission line with a gap in the center. The transmission line, located at y c = 12 mm (b/2), has a width of w = 1.4 mm and a gap of g = 10.5 m m ...............................................................................................207 D.2 Comparison of the calculated and measured transmission coefficient of the large gap located at y c = 12 mm in the low Q cavity A......................208 xviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 1 IN T R O D U C T IO N Large microwave systems are generally made up of smaller modules, each of which contains smaller sub-modules or individual circuits. These sub-modules are usually enclosed in a conducting package to reduce outside electromagnetic interference and to isolate one sub-module or circuit from another. In recent years, GaAs technology has matured to the point where many submodules or even whole modules are being replaced by a single monolithic microwave circuit (MMIC). MMICs are potentially low cost circuits because they are fabricated using batch processing. MMICs offer improved reliability and reproducibility through minimization o f wire bonds and have the added benefit of small size and weight. Because circuit tuning of MMICs is difficult and expensive, there is a critical need for accurate microwave CAD packages. In the design of MMICs, the effect of enclosing the circuit is typically assumed to be negligible. This has not been considered a very serious problem in current designs because circuit packages have not been electrically large. As the level of integration and/or the frequency of operation increases, the electrical size of the enclosure will also increase. For a moderately sized enclosure, one or two packaged resonances are possible. In a large enclosure, many resonances may occur in the frequency band of operation. If the system operates at frequencies near one of these resonances, coupling between the fundamental microstrip mode and the resonant mode is possible. This phenomena is referred to as parasitic coupling to a resonant mode. Resonant mode coupling can result in catastrophic coupling between different elements of a circuit that might otherwise have been isolated. Note that it differs from proximity coupling where two or more areas of a circuit in close proximity couple to each other. An example of proximity coupling is a section of parallel coupled microstrip transmission lines. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A number of full-wave techniques have been developed for the very accurate modeling of microstrip structures [1] - [13]. Many of these techniques neglect the parasitic coupling of widely spaced circuit elements and also neglect the coupling between the circuit and the package that encloses them. Although [6 ], [7], [11] model a microstrip circuit in an enclosure and analyze at least one circuit with a resonant mode, none of them specifically examine the causes of resonant mode coupling and ways to reduce i t The first goal o f this dissertation is to examine some of the fundamental aspects of resonant mode coupling. There is still much to be learned about the way a circuit interacts with the resonant modes of an enclosure. In a microstrip circuit with no cover plate and side walls, it is usually assumed that most of the power lost to radiation and surface waves occurs at a discontinuity. But Lewin [14] has shown that the interaction of circuit with space wave radiation and surface waves can occur at more than a guided wavelength from the discontinuity. For a circuit in an enclosure a similar effect has been observed [15]. Using a full-wave analysis it will be shown that the largest coupling to a resonant mode occurs where large current standing waves are present in regions where the co-polarized mode field is largest For example, an area of a circuit with a large x-directed current will couple strongly to a resonant mode in the areas where the Ex mode field is large. In addition to examining the fundamental aspects of resonant mode coupling, methods o f reducing this coupling will also be investigated. For example, the addition of loss to an enclosure may reduce resonant mode coupling, but may not eliminate it [15], [16], [17]. Repositioning a circuit in an enclosure to reduce resonant mode coupling is another technique that will be investigated in this dissertation. Little work has been done on the effect that circuit layout has on resonant mode coupling with the exception o f [15] and [18]. However, this technique is best applied to a relatively simple circuit consisting of only a few discontinuities in a moderately sized enclosure. Since most current MMICs 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. contain more than a few discontinuities, repositioning the circuit in an enclosure is often not practical. The next two goals of this dissertation are to develop alternatives to the MOM for analyzing a circuit in an enclosure. A full-wave analysis although rigorous it is also very complex to implement. In addition, as the size of MMIC circuits increase they become too complicated to analyze using a straightforward full-wave approach. A full-wave analysis of a typical MMIC of moderate complexity may require the solution of a large system of equations to accurately model the complete circuit. The need for new ways to analyze MMICs in an enclosure is therefore apparent The second goal of this dissertation is to develop a circuit model to describe resonant mode coupling for use on commercially available CAD packages. Commercially available full-wave CAD packages model each discontinuity in isolation, and then obtain the overall performance by combining the models using a circuit CAD program such as SUPERCOMPACT. Parasitic coupling is neglected using this technique. Toward that end, Jansen and Wiener [18] have developed a simple circuit theory model to describe coupling of circuit junctions to a resonant mode. The details of their formulation are not completely clear, but their model is based on the assumption that resonant mode coupling occurs only at a discontinuity. As previously discussed, this picture of resonant mode coupling is incomplete. Therefore, the circuit model developed in this thesis will incorporate the theory of resonant mode coupling derived from the rigorous analysis described in the first part of the thesis. Implementing this circuit model in a CAD package for a complex MMIC in an enclosure may be very tedious. Consequently, this circuit model is suited for MMIC circuits of moderate complexity. For complex MMIC circuits in an enclosure, a different modeling technique is necessary. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The third goal of this dissertation is to develop a method to analyze a complex MMIC circuit in an enclosure. The diakoptic method will be used to analyze complex MMIC circuits. Goubau et al [19] developed the diakoptic method to analyze complex multi-element antennas. Until very recently [20] the diakoptic method had not been applied to MMIC problems and even now has not been reported in sufficient detail for an assessment of it to be made. Also, it has not been applied to MMIC circuits in an enclosure. Therefore, the proposed solution is to modify the diakoptic method to analyze MMIC circuits in an enclosure by introducing the concept o f spectral filtering. The topics which must be addressed to study an MMIC circuit in a conducting enclosure are treated in the Chapters that follow. Chapter 2 is devoted to the full-wave analysis of a packaged MMIC. Although many of the techniques presented in Chapter 2 are not new, they provide an essential basis for the remaining Chapters of the dissertation. In Chapter 3, the full-wave analysis developed in Chapter 2 is experimentally verified. In addition, some of the fundamental aspects of resonant mode coupling and methods of reducing this coupling by relocating a circuit in an enclosure are discussed. Chapter 4 discusses the development and implementation of a simple circuit model on a commercially available CAD package that describes resonant mode coupling. In Chapter 5, the diakoptic method is modified to analyze a complex MMIC in an enclosure. The thesis is concluded in Chapter 6 . 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 References [1] R.H. Jansen, "Hybrid M ode Analysis of End Effects of Planar Microwave and Millimeter W ave Transmission Lines," Proc. Inst. Elec. Eng., Vol. 128, pt. H. pp. 77-86, April 1981. [2] J. Boukamp and R.H. Jansen, "The High Frequency Behavior of Microstrip Open Ends in Microwave Integrated Circuits Including Energy Leakage," 14th European M icrowave Conf. Proc., pp. 142-147, 1984. [3] R.W. Jackson and D.M. Pozar, "Full-Wave Analysis of Microstrip Open-End and Gap Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1036-1042, October 1985. [4] P.B. Katehi and N.G. Alexopoulos, "Frequency-Dependent Characteristics of Microstrip Discontinuities in Millimeter-Wave Integrated Circuits," IEEE Trans. M icrowave Theory Tech., Vol. MTT-33, pp. 1029-1035, October 1985. [5] R.H. Jansen, "The Spectral-Domain Approach for Microwave Integrated Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1043-1056, October 1985. [6 ] J.C. Rautio and R.F Harrington, "An Electromagnetic Time-Harmonic Analysis of Shielded Microstrip Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT35, pp. 726-730, August 1987. [7] R.H. Jansen, "Modular Source-Type 3D Analysis of Scattering Parameters for General Discontinuities, Components and Coupling Effects in (M)MICs," 17th European Microwave Conf. Proc., pp. 427-432, 1987. [8 ] J.R. Mosig, "Arbitrarily Shaped Microstrip Structures and their Analysis with a Mixed Potential Integral Equation," IEEE Trans. Microwave Theory Tech., Vol. MTT-36, pp. 314-323, February 1988. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [9] N.H.L. Koster and R.H. Jansen, "The Microstrip Step Discontinuity: A Revised Description," IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 213-223, February 1986. [10] W.P. Harokopus and P.B. Katehi, "Characteristics of Microstrip Discontinuities on Multilayer Dielectric Substrates Including Radiation Losses," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 2058-2066, December 1989. [11] L.P. Dunleavy and P.B. Katehi, "A Generalized Method for Analyzing Shielded Thin Microstrip Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 1758-1766, December 1988. [ 12] R.W. Jackson, "Full-Wave, Finite Element Analysis of Irregular Microstrip Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 81-89, January 1989. [13] H.Y. Yang and N.G. Alexopoulos, "A Dynamic Model for Microstrip-Slotline Transition on Related Structures," IEEE Trans. Microwave Theory Tech., Vol. MTT-36, pp. 286-293, February 1988. [14] L. Lewin, "Spurious Radiation From Microstrip," Proc. IEE, Vol. 125, No. 7, pp. 633-642, July 1978. [15] J.J. Burke and R.W. Jackson, "Reduction of Parasitic Coupling in packaged MMICs," IEEE MTT-S Int. Microwave Symp. Dig., pp. 255-258, May 1990. [16] D.F. Williams, "Damping of the resonant modes of a rectangular metal package," IEEE Trans. Microwave Theory and Tech., Vol. MTT-37, pp. 253-256, January 1989. [17] A.F. Armstrong and P.D. Cooper, "Techniques for Investigating Spurious Propagation in Enclosed Microstrip," The Radio and Electronic Engineer, Vol. 48, No. 1/2, pp. 64-72, Jan/Feb, 1978. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [18] R.H. Jansen and L. Wiemer, "Full-Wave Theory Based Development of MM-wave Circuit Models for Microstrip Open End, Gap, Step, Bend and Tee," IEEE MTT-S Int. Microwave Symp. Dig., pp. 779-782, June 1989. [19] Goubau, G., Puri, N.N. and Schwering, F.K., "Diakoptic theory for multielement antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-30, No. 1, pp. 15-26, January 1982. [20] Howard, G.E. and Chow, Y.L., "A high level compiler for the electromagnetic modeling of complex circuits by geometrical partitioning," IEEE MTT-S International Microwave Symposium Digest, pp. 1095-1098,1991. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2 FULL-W AVE A N A LY SIS OF PACK AG ED M M ICs 2.1 Introduction The full-wave analysis of packaged MMICs is developed in this chapter. First, the derivation of the spectral Green's function is presented. Following this is a discussion of locating the resonances of a dielectric loaded cavity. The next section is a review of the method of moments (MOM) which is followed by the development of an acceleration technique for the efficient evaluation of MOM impedance matrices. The solution for the unknown current distribution under the condition that each port is terminated in its characteristic impedance is then given. Lastly, the different basis functions used to expand the currents are discussed. 2.2 Spectral Green's Function Figure 2.1 shows the geometry of the MMIC package for which a Green's function will be derived. There are N dielectric layers of thickness dj, relative permittivity Erf and relative permeability (in'- The MMIC circuitry is located at z = zjc. The side walls of the enclosure are assumed to be perfect conductors. The surface impedance of the top cover is designated as Z s j and the bottom cover as Z5 5 . An electric cunrent source is located at z = zk J (x ,y ,z ) = Js ( x ,y ) 8 ( z - z k ) (2 . 1 ) The resulting electromagnetic fields in general can be expressed as the superposition of TM and TE fields. For the TM and TE fields in each layer let: A(i)= z ^ (2 .2 a) 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.1 Geometry o f the MMIC package used in the derivation of the Green's function. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2 . 2 b) Fi i ) =z<P^ where i = 1, N The potential functions are the solutions of the scalar Helmholtz equation: (2.3) V 24>tu + kf ’Ptu = 0 where kf = £ ^ 1 $ U = E otM The total electric and magnetic field components in each layer are given by: E ji\ x , y , z ) = ^ - V t JEriko £ ? (W dz + z x v r0 g ( 32 ) = - ^ - — + t2 ■ JSrik0 & 2 H ^ ( x ,y ,z ) = - z x V M + 1 M r i k olo tf f( W 2 where E ^ and (2.4a) (2.4b) V, d$TE dz ( ) = —. . J _ [ df zj + k f 0% jf^rik0^lo / (2.4c) (2.4d) are the tangential components of the electric and magnetic fields, respectively. The boundary conditions for equation (2.3) are given as follows. At z = zi (i = 1, ..., k -1, £+1,..., AM), the tangential components of the electric and magnetic fields are continuous: Et(0(x,y,Z i) = £?'+1)(x,y,Zi) (2.5a) H ^ \ x , y , Zi) = H ji+l\ x , y , Zi) (2.5b) The tangential component of the electric field at z = zk is also continuous. However, there is a jump discontinuity in the tangential magnetic field. These two boundary conditions can be expressed as: % k)(x ,y ,z k ) = E ^ +1\ x , y , z k ) 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.6a) Js(x,y) = z x (H <*+1 W , ^ ) - £ ,(*W , ^ ) } (2.6b) The tangential component of the electric field must vanish on the side walls of the enclosure because they are assumed to be perfect conductors. These boundary conditions can be expressed by the following: y - % i>(x,y) = 0 fo rx = 0 ,a (2.7a) fory = 0 ,b (2.7b) M.i\ x , y ) = 0 x -E ^ i\ x , y ) = 0 £ f ( x ,y ) = 0 The solution to equation (2.3) is obtained by the separation of variables, in general the solution can be written as: (2.8) Substituting equation (2.8) into (2.3) yields the following pair of equations: V,2* ® + **?(*> = o (2.9) d 2mH L + k2P ^ - f t 2 zi TU ~ u (2. 10) dz where —p 11 a.2_i 2 Kz i ~ c r ir LriK0 Im (£,,)< 0 p Vf = V - l - j oz (2 . 11) (2. 12) The solution to equation (2.9) can be obtained by enforcing the boundary conditions given by equation (2.7) (*’>') = Ctm s^n (kxnx)sin(kymy) (2.13a) ^ T E ^ y ) = CTE cos(kxnx)cos(kymy) (2.13b) where , niz kxn= — a , _ mn ym~~b~ k 2p = kln+ k$m (2.14a) (2.14b) (2.14c) 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The total solution to equation (2.3) can be expressed as a summation over all possible modes such th a t: oo oo <1>%(x,y,z) = X m=0n=0 (2.15) Substituting equation (2.15) into (2.4), the electric and magnetic field components in each layer can be expressed by the following: 771=077=0 LJ vi 0 + />^ )(z)[£xVr'Fr£ (x,y)]} oo (2' 16) oo 771=0 7J=0 (2.17) +~ 7 ^ - [-V ,^ fe 7 )]} jH rikQTiQ dz JJ 4 W( W ) = X X - T T - , S f e > ,I'rM fc )’) m-On-QJ8^ OO oo £ f ( x ,y ,z ) = X n=Qn=QJ£rik0 m'2 oo oo (2.18) H ? H x,y,z) = X X ^ " T *@ « )* ie(* * 3 0 £> £> M ri*orio (2-19) In what follows it will be convenient to express the transverse fields in each region by the following [ 1 ], [2 ]: oo oo E ^ \ x ,y ,z ) = X X { Vr M ( ^ ™ ( ^ > 0 + (z)?te(*30} (2.20) 771=0 71=0 OO H ^ \ x ,y ,z ) = OO X X ^ l^ ^ r M ^ y J +z ia fe U y )} ( 2 .2 1 ) 771=0 77=0 Where eTM \eTE) is the transverse electric field mode vector, V j^ 0 ® ) the mode voltage and I jm {Ij e } the mode current for the TMnm (TEnm) mode. Comparing equation (2.20) to (2.16) and (2.21) to (2.17), one may write ® 7 ) = i§ fe ) 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.22) _ ~Hn d^TM _ " ^ fi WSfe) = T * j £ r ik 0 (2.23) * J £ rik 0 for the TM mode (2.24) V&(z> = P & (z) jHrikoTlo dz -1 dVj£ jUrikoVo dz (2.25) and for the TE mode. The mode vectors eTM and eTE are given by: £TM(x,y) = - ^ t ^ m ( x , y ) = CTM{ - x k xnTx { x ,y ) - y k ymTy {x,y)] (2.26) eTE(x ,y ) = z x V j'FreC ^y) = Cr e {-xA:ym7;(x,y) + y^„Ty(x,y)} (2.27) where = c o s(^ „ x )sin (^ TO>-) (2.28a) Ty(x,y) = sm (kxnx)cos(kymy) (2.28b) In addition the mode vectors are normalized such that: ba ba rr- JJ 00 e TUnm j \ e m -eTEd xdy = 0 oo ri f l nn-- 1 and m = k ' e TUlk * ^ ~ ) q .^ lo r m ^ k (2.29) (2.30) ‘ where U = E orM . Substituting equation (2.26) into (2.30) yields: 1 CTM - \ k , 4s s 1 -n^ ab m and n * 0 0 m or n = 0 (2.31) and substituting equation (2.27) into (2.30) yields: — Cte - m^ Oo r n ^ O aft 0 m = 0 and n = 0 where 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.32) r 0 .5 P =o 1 .0 P* o e^ " t After some algebraic manipulation, the normal components of the electric and magnetic fields, Ez and Hz , can be expressed as: oo oo E ^ ( x ,y ,z ) = X X ^ T - /(2 ^ ) { V • * T M ^ y )} n=0 m = oo (2.33) oo Hz \ x , y , z ) = £ X “— 7-----V ^(z){V -[zx?r£(x,j)]} (2.34) m=0n=QJf1ri'c0rl0 Substituting equations (2.20), (2.21), (2.33) and (2.34) into Maxwell's equations and applying equations (2.29) and (2.30), yields the equivalent transmission-line equations and the modal current, 1 $ , [2 ]: in the ^-direction for the modal voltage, ( 2 ' 3 5 ) - f - =- ^ r ^ ( z ) dz (2.36) where U = E orM . and the wave admittances for the TM and TE modes are defined as: (2.37) ^O^Zi 4 E = -rzh r (2-38) Note that equations (2.35) and (2.25) are equivalent if U = E. Likewise, equations (2.36) and (2.23) are equivalent if U = M. The boundary conditions for the solution of equations (2.35) and (2.36) are obtained by substituting (2.20) and (2.21) into (2.5) and (2.6). Applying the orthonormal properties of the mode vectors, the boundary conditions can be expressed as: for z = zi (i = 1 ,..., Vm(Zi) = v w l)(Zi) (2.39a) ^ffi(Zi) = 4 u 1)(z«) (2.39b) k -1, £+1, ..., N - 1), and 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V T V = V « \z k ) = V&+1\ z k ) (2.40a) lTU - I tu (zk ) _ ^TUl)(zk ) (2.40b) h u = j p i (*>>’)• e-ru (x ,y )d x dy (2.41) for z = zk, where We can now draw an equivalent circuit for the TU mode (£/ = £ or A/) as shown in Figure 2.2 [2], [3]. By using transmission line theory the current, i j y , and the voltage, vTU, at z = Zk are related via P -42) xu where Yu is the driving point admittance at z = Zk. From the equivalent circuit shown in Figure 2.2, Yu, is given by: y _ y ( * ) . y ( * + l) (2.43) *U ~ *LU + *RU where y ( i ) * LU ^ + J ^ T V ^ y(i') _ ~ I t U (z i ) _ ^T U +J ^ L U tan &i L U ~ v & ( Zi) (2.44) for / = 0 -SB y(0 Yr U1'*+ jY jV ^ for i = k + l,...,N y ( 0 _ I r u ( Zi - 1) _ (2.45) for i = N + \ -ST d i= k zidi (2.46a) di = zi - z i. i (2.46b) Figure 2.3 shows the equivalent circuit for determining Y[y (i = 1 ,..., k) and Figure 2.4 shows the equivalent circuit for determining Y^u (i = k+1, ..., N ). 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. + + t? tv? + q? =S? j?a "^T ts? gS) CB* «?I N The transmission line equivalent circuit for the geometry shown in Figure 2.1 it Figure 2.2 CM ig 'S ? •4^ TS? 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (/ = k+1, The transmission line equivalent circuit for determining Figure 2.4 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the subsequent method of moments solution, the tangential component of the electric field at z = zjt due to a surface current on the same surface will be needed. Substituting equation (2.42) into (2.20) yields: oo oo (2.47) El(* \x ,y ,z Jt) = X X j - S M-h M ^x ,y) + - ^ E-eTE^x,y)\ w=o«=ol £ J Expressing the surface current, J s (x ,y ), as J s(x,y)= xJx (x,y) + yJy (x,y) (2.48) and substituting equations (2.26), (2.27) and (2.48) into (2.41) yields: 4s„£m I km -f L J x ab j 1tm ~ ~ kyn ^ x n , k y m ) + - f - J y { k x n , k ym) _ _ l^ £n£m ^ym r 1 ^ ^x(kxnikym) ab 1 kp lTE ~ a/ ^ •^yikxn,kym) , (2.49) (2.50) where Jx(kxn->kym) —I f Jx (x,y)Tx (x,y)d xd y s (2.51a) Jy^xn'kym ) ~ I I Jy(x,y)Ty (x,y)d xd y (2.51b) Substituting equations (2.26), (2.27), (2.49) and (2.50) into (2.47), the tangential component of the electric field at z = zk after some algebraic manipulation can be represented as: oo oo Elk\ x , y , z k )= X X . - ^ T (x ^ & k ^ k y J -Z ik ^ k y j] (2.52) 772=0/1=0 where Js(kxn1kym ) = xJx (kxn,kym) + yJy (kxn,kym) 'Tx (x,y) 0 T(x,y) = 0 Ty (x,y) Q(kxn,kym) = 'x x Q n ik ^ k y J xyQ ^k^kyJ yxQ y X (k x rrkym ) y y Q y y ik ^ k y J Qxx (kxn, kym) — -jfQ rM + ^rrQ T E v p v (2.53) (2.54) (2.55) (2.56a) 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Q x y ^ k y J = QyX(kxn,kym) = J ^ y a L ( Q m - Qt e ) "P (2.56b) (2.56c) V P P (2.57) U = E or M. The poles of the spectral Green's function, Q(kxn,kym), correspond to surface waves (no enclosure), parallel-plate waves (with cover and no side walls) or cavity modes (in an enclosure). Since all the circuits analyzed in this dissertation are in an enclosure, the frequency at which a pole occurs is referred to as a resonant frequency of the cavity. Locating poles of the spectral Green's will be discussed in the next section. 2.3 Enclosure Resonances 2.3.1 Introduction In this section locating the poles of the spectral Green's function, Q(kxn,kym), will be discussed. The frequency at which a pole occurs is referred to as a resonant frequency of the cavity. A pole of Qm (QTE) corresponds to a TM (TE) resonant mode of the cavity. These poles are located by finding the zeros of Yu and Ye . For a typical enclosure housing an MMIC chip the cover height is low enough that only TM modes are resonant over the frequency of operation. For this reason, locating only the zeros of YM will be discussed. However, everything discussed in this section is also applicable to TE modes. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. After determining the resonant frequencies of an enclosure, a useful figure of merit is the quality factor or Q. In what follows three definitions for the Q of a lossy enclosure are developed. The first expression for the Q is given by [4], [5]: 2 tonW ™ <2 -58> n 'T’JUf where the average electric energy stored in the resonant TM mode, We , is given by: W ™ = i - R e ^ J J J |f :™ ( x , y , z f dxdyckj (2.59) the average power loss to the resonant TM mode, P ™ , is given by: = ± R e ( J J E ™ (x ,y ) ■J* (x,y) dxdyj (2.60) and (Qq is the real resonant frequency. The solution of the real resonant frequency will be discussed shortly. An alternative expression for the Q can be found from the equivalent circuit for the TM mode (Figure 2.2) developed in section 2.2. By using transmission line theory the current, iTM, and the voltage, vm , at z = zk are related via % = (2.61) XM where YM is the driving point admittance at z = Zjfc. From the equivalent circuit shown in Figure 2.2, it i is easily shown that the average power loss to the resonant TM mode, P ™ , is given by: r™ = j M 2 Re[r«(a»o>] ( 2 -6 2 ) Applying Foster's reactance theorem [1], [4], [6 ],[7] to the equivalent circuit, the average electric energy stored in the resonant TM mode is given by: (2.63) CO=COq Substituting equations (2.62) and (2.63) into equation (2.58) yields a second expression for the Q: 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Q)0 Im <2i = d(0 0-0 o (2.64) 2Re[rM(©0)] where dYM/do) is given by equation (A.1). The third expression for the Q of an enclosure is given by [1], [4]: Re(<pr ) 2 2 Im(<or ) where (Or is the complex resonant frequency of the enclosure. The solution of the complex resonant frequency will be discussed shortly. 2.3.2 2.3.2.1 Evaluation of the Resonant Frequencies Lossless For a lossless cavity, QTM and YM are purely imaginary and (0r is purely real. The poles o f Qtm are located by finding the zeros of Ym - The zeros of YM can be located by searching for the real frequency at which I m (I^ ) = 0. The poles of QTM and the zeros of Ym are simple. Conversely, the zeros of Qtm and the poles o f YM are simple. Furthermore, the poles and zeros of QfM alternate along the co axis and the poles and zeros of Ym alternate along the 6 ) axis. Because the poles and zeros alternate, the slopes of QfM and Ym versus frequency must be positive [1], [4], [6 ]. For example, consider a cavity of the following dimensions: a = 15 mm, b = 24 mm and c = 10 mm. There are three dielectric layers o f thickness d \ - 1.27 mm, d j = 7.967 mm, and d 3 = 0.762 mm with relative permittivities of er l = 10.5, er2 = 1-0, and er3 = 11.7. Figure 2.5a shows I m ( l^ ) versus frequency and Figure 2.5b shows Im(<2rM) versus frequency. For Figure 2.5, the mode numbers, n and m, are equal to 1. Note that the slopes of QTM and YM are positive. In the band 8-13 GHz, there is one 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lossless 2.50e-2 > 0.00e+0 -5.00e-2 8 9 10 11 12 13 F (GHz) (a) Lossless 2000 1000 S H 0 1 -1000 - -2000 8 9 10 11 12 13 F (GHz) (b) Figure 2.5 (a) Im(yM) and (b) ]m(QTM) versus frequency for the lossless enclosure with the mode numbers, n and m, are equal to 1. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequency, f 0, for which ~kn.(YM) = 0. This frequency has been determined to be f 0 = 10.3106 GHz. 23.2.2 High O If loss is present in the cavity, YM is complex and the resonant frequency of the enclosure, cor, is also complex. The zeros of YM are located by searching for a complex CD where the real and imaginary parts of YM simultaneously equal zero. Searching for the real CD where Im(yM) = 0 may result in a solution which is not an actual zero of Ym (i.e. R e ( i^ ) * 0). Therefore, generally the zeros should not be located by searching for a frequency where only one part of YM equals zero. However, for cavities with small to moderate loss, the real part of the resonant frequency can be determined with good accuracy if the approximate solution satisfies both of the following conditions: =0 (2.66) (2.67) The condition imposed by equation (2.67) may not seem obvious at first. In order to better understand this condition, let us reexamine the example given above with a small amount of loss added to the enclosure. Loss is introduced to the enclosure through a complex permittivity £r l = 10.5(l-j0.0023). This cavity is referred to as the high Q or low loss enclosure. Figure 2.6a shows lm(YM) versus frequency and Figure 2.6b shows Im (Qtm ) versus frequency for the high Q enclosure. For Figure 2.6, the mode numbers, n and m, are equal to 1. Comparing Figure 2.5b to Figure 2.6b shows that adding loss to the cavity results in not becoming infinite for any real a>. The poles of QTM have been 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q S.OOe-2 2.50e-2 > 0.00e+0 -5.00e-2 8 9 10 11 12 13 F (GHz) (a) High Q 2000 1000 S H O' w S -1000 -2000 8 9 10 11 12 13 F (GHz) (b) Figure 2.6 (a) Im(yM) and (b) Im(QrM) versus frequency for the high Q enclosure with the mode numbers, n and m, are equal to 1. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shifted a small amount from the real co axis to points slightly above it. In another words, the poles have a small imaginary part. Examination of Figure 2.6a shows that there are two solutions where I m ( l^ ) = 0 for a real co. The first zero of Im(yM), 10.3106 GHz, occurs at the same frequency as the resonant frequency of the lossless enclosure. This was expected because the addition of a small amount of loss will result in the addition of a small imaginary part to the resonant frequency. The second zero of Im(7M) , 11.15 GHz, occurs at the same frequency as the pole of I m ( l^ ) for the lossless enclosure. Both solutions satisfy equation (2.66). However, only the first solution satisfies equation (2.67). Therefore, in the band 8-13 GHz, there is one real frequency, / 0, that satisfies equations (2.66) and (2.67). This frequency has been determined to be / 0 = 10.3106 GHz. To verify that / 0 is equal to the real part of the complex resonant, a complex root finder was employed. In the band 8-13 GHz, there is one complex frequency, f r , for which 1 ^ = 0. This frequency has been determined to be f r = 10.3106+j0.0103 GHz. The real part o f the complex resonant frequency, f r, does equal f 0 ; therefore, equations (2.66) and (2.67) can be used to find the real part of the resonant frequency for high Q enclosures. The Q for the high Q enclosure was calculated using equations (2.58), (2.64) and (2.65). All three equations give the same value o f 5013.6. 2.3.2.3 Moderate O To examine the effect of increasing the loss has on the resonant frequency and Q, the permittivity of layer 3 for the example above was changed to £r3 = 11.7(l-j0.5). This cavity is referred to as the moderate Q or moderate loss enclosure. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.7a shows Im (I^ ) versus frequency and Figure 2.7b shows and Re.(QTM) versus frequency for the moderate Q enclosure. For Figure 2.7, the mode numbers, n and m, are equal to 1. Comparing Figure 2.6b (high Q) to Figure 2.7b shows that adding more loss to the cavity results in becoming smaller in the vicinity of the pole. In the band 8-13 GHz, there is one real frequency, f 0, that satisfies equations (2.66) and (2.67). This frequency has been determined to be f 0 = 10.3076 GHz. In the band 8-13 GHz, there is also one complex frequency, f r, for which YM = 0. This frequency has been determined to be f r = 10.3052+j0.0369 GHz. The real resonant frequency, / 0, and the real part of the complex resonant frequency, f r, differ by only 0.4 MHz; therefore, equations (2.66) and (2.67) can be used to find the real part of the resonant frequency for moderate Q enclosures. The Q for the moderate Q enclosure was calculated using equations (2.58), (2.64) and (2.65). Equation (2.58) gives Qq = 139.65, equation (2.64) gives Q1 = 138.64, and equation (2.65) gives Q2 = 139.67. The Q's calculated by equations (2.58) and (2.65) give essentially the same result, while equation (2.64) is slightly different 2.3.2.4 Low O To examine the effect of a large loss has on the resonant frequency and Q, the permittivity of layer 3 for the example above was changed to £r3 = 11.7(l-j 10.0). This cavity is referred to as the low Q or large loss enclosure. Figure 2.8a shows Im (YM) versus frequency and Figure 2.8b shows lm (Q TM) and Re(<2rAf) versus frequency for the low Q enclosure. For Figure 2.8, the mode numbers, n and m, are equal to 1. Comparing Figure 2.6b (high Q) to Figure 2.8b shows that a large loss to the cavity results in Im(<2rM) becoming much smaller in the vicinity of 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Medium Q 5.00e-2 2.50e-2 /•“•s E b 0.00e+0 E -2.50e-2 -5.00e-2 9 8 10 11 12 1? F (GHz) (a) Medium Q 2000 Im(QTM) Re(QTM) 3.00e+3 -1000 1.00e+3 Rc(QTM) 1000 0.00e+0 -2000 8 9 10 11 12 13 F (GHz) (b) Figure 2.7 (a) Im(7M) and (b) QTM versus frequency for the moderate Q enclosure with the mode numbers, n and m, are equal to 1. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q b 0.00e+0 -2.50e-2 -5.00e-2 9 8 10 11 12 13 F (GHz) (a) Low Q 600 800 Im(QTM) Re(QTM) 400 600 200 400 E 200 —■ —^~l*‘ -200 8 9 10 11 12 13 F (GHz) (b) Figure 2.8 (a) Im(YM) and (b) QTM versus frequency for the low Q enclosure with the mode numbers, n and m, are equal to 1. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the pole. Comparing Figure 2.7b (moderate Q) to Figure 2.8b shows that adding more loss to the cavity results in Re(QTM) spreading out versus frequency. In the band 8-13 GHz, there is one real frequency, / 0, that satisfies equations (2.66) and (2.67). This frequency has been determined to be / 0 = 10.5091 GHz. In the band 8-13 GHz, there is also one complex frequency, f r, for which YM = 0. This frequency has been determined to be f r = 10.4810+j0.1394 GHz. The real resonant frequency, / 0, and the real part of the complex resonant frequency, f r, differ by only 28.1 MHz; therefore, equations (2.66) and (2.67) can be used to find the real part of the resonant frequency for low Q enclosures. The <2 for the low Q enclosure was calculated using equations (2.58), (2.64) and (2.65). Equation (2.58) gives <2b = 38.784, equation (2.64) gives Ql = 35.467, and equation (2.65) gives Q2 = 37.587. The Q calculated by equation (2.58) results in a value about 3.2 % higher than equation (2.65). The Q calculated by equation (2.64) results in a value about 5.6 % higher than equation (2.65). 2.3.3 Summary of Enclosure Resonance Determination In this section, the location of the real and complex resonant frequencies was discussed. Comparing the real resonant frequencies to the real part of the complex resonant frequency shows that difference is on the order of tens of mega-Hertz for Q's as low as 40. Three equations for the Q of an enclosure were developed. Comparing the results of the three equations shows that difference between any of the two is less than 10 % for Q's as low as 40. Table 2.1 summarizes the results of this section for an enclosure containing different amounts of loss. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 2.1 Summary of the resonant frequencies and Q's for various enclosures. Loss fo (GHz) fr (GHz) None Small Moderate Large 10.3106 10.3106 10.3076 10.5091 10.3106 10.3106+i0.0103 10.3052+i0.0369 10.4810+i0.1394 2.4 Qo a q2 Eq. (2.58) Eq. (2.64) Eq. (2.65) oo OO OO 5013.6 5013.6 5013.6 138.64 139.65 139.67 38.784 35.467 37.587 Method of Moments Solution The electric field integral equation (EFIE) representing the boundary condition that the total tangential electric field, E\ot, must vanish on the microstrip line can be written as E \ ° \x ,y ) = Einc(x ,y ) + Estcat{x,y) = 0 (2.68) where y) = £<*>(W i ) = S -7, m=0n=0 aD and E f10 is the incident electric field and Js is the surface current on the microstrip line. The surface current on the microstrip line is expanded into a set of basis functions as follows: Nx Ny Js (x,y) = x £ l xiJxi (x, y) + IyjJyj (x,y) (2.69) j =l /= i where Ixi and Iyj are the unknown current coefficients. Following Galerkin's method, equation (2.68) is tested with Jxi and Jyj which results in a set of algebraic equations for the unknown current coefficients, such that v* = N* «=i + Ny, ^ .Iyj j =i k = l,...,N x J 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.70a) The elements of the excitation vector are given by [8], [9], [10], [11], [12]: Vx k = j j E ‘xnc(x,y)Jxk(x ,y )d x d y Vy l = \\E ™ { x ,y )J yl(x ,y )d x d y Wr [0 if (**Ofc) = ( * f o ^ ) otherwise Vrp if( x k ,yk ) = ( x ? ,y { ) 0 otherwise (2.71a) (2.71b) In the above the expressions, ( x f , y f ) is the position of the rth excitation and r = 1 Np, where Np is the total number of ports. The superscript P is used to designate a port. A typical element of the impedance matrix is in the form: J a _ __rv m=0n=0 where Jpifcxn’kym) = JJ Jpi(x,y)Tp (x ,y )d x d y (2.73) Si p ,q = x o r y 2.5 Efficient Computation of the Impedance Matrix Elements For an MMIC with no side walls, efficient techniques have been developed to calculate the elements of the impedance matrix [13], [14], [15]. When the MMIC is contained is a conducting enclosure, the evaluation of the impedance matrix is numerically intensive. Hill and Tripathi [16] developed an efficient technique using the two dimensional fast Fourier transform to determine the elements of the impedance matrix. However, the fast Fourier transform requires the circuit to be discretized using a uniform grid. The use of a uniform grid can lead to a large number of unknowns for a relatively simple circuit that requires a high spatial resolution in a particular region. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In this section an efficient technique for the evaluation of the impedance matrix elements is developed using Kummar's transform [17]. Jansen and Sauer [18] independently developed a similar technique using the spectral operator method [19]. The slowly converging series in equation (2.72) is decomposed into two series. The first term is a rapidly converging series that is evaluated at each frequency. The second term is a slowly converging series; however, it is only evaluated at a single frequency. In order to implement equation (2.72) on a computer, the summations over n and m are truncated at N°° and M°° such that m=0n=0 The choice o f N°° and M°° and the convergence behavior of equation (2.74) will be discussed shortly. From the discussion in Appendix B, the spectral Green's Function for large kp can be represented as: (2.75) As a result, we can rewrite equation (2.74) as: (2.76) where (Z m \ j = ^ ^ m=0n=0 ab Ob &pq^xn.kym)Jpi(kxn,kym ) 7 ^ (Icxn,kym) (2.77a) (2.77b) m=0n=0 Zpq).. decays more rapidly for increasing kp than does therefore, the limits for the summations in equation (2.77a) are truncated at N A and M A where: Na « N°° 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MA « M°° The choice of N A and M A and the convergence behavior of equation (2.77a) will be discussed shortly. The frequency dependence of equation (2.78) can be separated into two parts by substituting equation (B. 10) into (2.78). The first term is inversely proportional to frequency, while the second term is proportional to frequency. Equation (2.78) can now be expressed as: (2 ™ (Z 7 9 ) where k k AT . ( z j f ( /)).. = £ 1 "T" rJ £ m = 0n= 0 a° M°° N°° . „ ( z ™ ( /) )., = £ £ J m = 0n= 0 y aD <2-80> Kp jr h W. 2P * e g i W J p l ^ . k y n V ^ . k y J (2 .8 1 ) kP _ \ k xn ^ P = X _f+l if /7= y if p = q [-1 if p * q pq Although the evaluation of equations (2.80) and (2.81) still require intensive computational effort, they need only be evaluated at a single frequency ( f R ). After evaluating equations (2.80) and (2.81) at f R, ( z ^ ) .. as a function of frequency is simply given by equation (2.79). As stated previously, ZA is a rapidly converging series. In order for this to be true, QA(kxn,kym) must be approach zero for small to moderate values of kxn and kym. In another words, Qq s approaches Q(kxn,kym) as k ^ and kym increase. A useful figure of merit used to determine how well Q q s (kxn,kym) approximates Q(kxn,kym) is defined as: 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e4 = (2.82) Q fcm ’kyn,) In order to determine N A and M A for a given e A, first assume kA = kx„ + kym and kjm = Km- Next, kA is varied until equation (2.82) is satisfied. After determining kA, N A can be found by substituting kA into equation (2.14) which yields: (2.83a) II o Cl , =~K Tt 3 and M II 1' b . (2.83b) ii o tT n Note that as e A becomes smaller, N A and M A become larger. In order to implement equations (2.72) of (2.76) on a computer, The summations in Z and Z°° over n and m are truncated at N°° and M “ . Dunleavy [11] investigated the convergence of Z and determined the optimum truncation of the series to be: at , roo oca N =— A, a t =— where 1.5 < a < 4.0 and A x is the length of the x-subsection and (2.84a) (2.84b) is the length of the y- subsection. A^ and A y are described in more detail in section 2.7. To examine the effect of varying a and s A has on the solution obtained using the method of moments, consider a transmission line with a single shunt open circuit stub attached. The stub is located atx c = 7.5 mm (a/2) and has a length of 1.8 mm. The transmission line is located aty c = 12 mm (bJ2). The width of the transmission line and the stub is 1.4 mm (Figure 2.9). The circuit is enclosed in a cavity of the following dimensions: a = 15 mm, b = 24 mm and c = 12.7 mm. The substrate thickness is d\ = 1.27 mm and the relative permittivity is er i = 10.5(l-y0.0023). An enclosure of this size has only one resonant mode, the T M no (10.8129 GHz), in the band 9-12 GHz. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b Figure 2.9 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located at x c - 7.5 mm (a/2), has a length L = 1.8 mm and is attached to a transmission line, located at y c = 12 mm (jb/2), of width w = 1.4 mm. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.10 compares the predicted transmission response of the stub using three different values of a and e A; a = 4 and e A = 0.01, a = 4 and e A = 0.2, and a = 2 and e A = 0.2. The MOM was performed using 123 expansion functions. Agreement between the three different values of a and e A is excellent; therefore, a = 2 and e A = 0.2 can be used and still obtain excellent convergence. 2.6 Network Parameters In this section, the unknown current coefficients will be determined under the conditions that each port is terminated in its characteristic impedance. After finding the unknown current coefficients, the s-parameters are then determined. In order to determine the current coefficients with the ports terminated in its characteristic impedance, equation (2.72) will have to be modified to include the port terminations. Before doing this modification, equation (2.70) is written in a more compact form: V = ZI (2.85) The z-matrix in equation (2.85) can be viewed as the open circuit impedance matrix for a - N p terminals are internal nodes. Equation (2.85) can be rewritten as: _ z IP N 1 z pn i ’z pp i i v i 1 the remaining 1---- ---- 1 terminals (N^, = N x + Ny ). Np o f the terminals are external ports and i— network with where the superscript P denotes a port and the superscript / denotes an internal node. Examining equation (2.71), it is easily seen that the matrix V7 is equal to the null matrix. A voltage generator u f with an internal impedance Z0l- is connected to each external port as illustrated in Figure 2.11. The voltage, V p , and the current, i f , at each port are related by: (2.87) 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm -10 -20 -30 - Ar = 2 / Er = 0.20 - Ar = 4 / Er = 0.20 ‘ Ar = 4 /E r = 0.01 -40 9 10 11 12 F (G H z) Figure 2.10 Comparison of the predicted transmission response of the stub using three different values of a and e A; a = 4 and e A = 0 .0 1 , a = 4 and e A = 0.2, and a = 2 and s A = 0.2. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.11 Schematic of an Np port microstrip circuit with a voltage generator and a terminating impedance connected to each external port. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 .1 'zpp z pr ZIP z n_ (2 . 88 ) I --- 1 Substituting equation (2.87) into (2.86) and simplifying yields: where yPP ij _ j ZPP+ZQi for i —j for i * j [Z f Equation (2.88) is then solved for the unknown current coefficients, l p and I 7. This yields the current under the conditions that each port is terminated in its characteristic impedance. The network described by equation (2.88) is designated as an augmented network. The short circuit admittance matrix of the augmented network is given by: jP Ya — ‘ ~ UP rrP 1 U {= 0 (2.89) for k * j where Ya is the short circuit admittance matrix of the augmented network. The relationship between the augmented y-matrix and the s-matrix is given by (see Appendix C): S = U - r 1/2Y V /2 (2.90) where U is the identity matrix and r7i/2 C01 rV2 = (2.91) z l/2 Z0Ar 2.7 Expansion Functions Figure 2.12 shows a simple microstrip circuit consisting of a step embedded between two transmission lines. The circuit is divide into three regions. Different types of basis function are used to expand the current in each of the three regions [20]. Rooftop functions are used for expansion functions in areas of discontinuities (region 1). Functions 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.12 A simple microstrip circuit consisting of a step embedded between two transmission lines. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that satisfy the edge conditions are used for expansion functions in uniform sections of transmission lines (region 2). For a source near the side o f the enclosure (region 3), rooftop functions are used in a special configuration. 2.7.1 Region 1 - Discontinuities The microstrip conductor near a discontinuity is divided into rectangular cells along the x and y axes as shown in Figure 2.13. Although the cells in this formulation do not have to be of the same size, typically the same cell size occurs over small sections. Mosig and Gardiol [21] recommend that the linear size of a cell should not exceed one tenth of a guided wavelength. The functions used to expand the current near a discontinuity are rooftop functions which are described by: J Xi ( x , y ) = f xt i(x)gPi{y) (2.92a) Jyj(x,y) = gPj(x)f$j(y) (2.92b) where 1 + ~ n r ^ u ~ uk ) Ui-A^^u^Uk A uk / 4 («)= 1 R 1 - 7 7 r ( “ ~ uk ) Uk < u < uk + A,* 0 otherwise 1 (2.93) A vk (2.94) Sv*(v) = - \ k 0 otherwise The center of the x-directed currents are marked with a cross and the center of the ydirected currents are marked with a circle. The x-directed currents overlap each other in the x-direction, but not in the y-direction. For y-directed currents the reverse is true. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r- XI Figure 2.13 XI The arrangement of the rectangular cells used near a discontinuity. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7.2 Region 2 - Transmission Lines The microstrip conductor along a uniform section of transmission line is also divided into rectangular cells. The arrangement of these cells is shown in Figure 2.14. The center of the x-directed currents are marked with a cross and the center o f the ydirected currents are marked with a circle. Functions that satisfy the edge condition are along the width of the transmission line. The functions used to expand the current along uniform sections of transmission lines are described by: JXi(x,y) = fxi(x)g]i(y) (2.95a) Jyj(x,y) = gPj(x)f^(y) (2.95b) where /*,(x) and g^-(x) are given by equations (2.93) and (2.94), respectively and /iw=- -wJy l - (2.96) 0 otherwise -.21-V2 Itw' 1- (v —V;.) w u 0 2.7.3 |v -v ,|sf (2.97) otherwise Region 3 - Sources For a source near the side of the enclosure (region 3), the current could be expanded with the basis functions used in region 1 or 2. However, if the results are to be compared to measured data, the effect of the coaxial to microstrip transition must be either removed or simulated. The method most often used to remove the effect of the coaxial to microstrip transition is called de-embedding. Several de-embedding techniques have been developed to remove the effect of the coaxial to microstrip transition from the measured sparameters. Two of the most popular de-embedding techniques are the time domain [22] 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C w ;) t w 0 X 0 I XI Figure 2.14 iR . *Xl The arrangement of the rectangular cells used along a uniform section of transmission line. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the thru-short-delay [23] methods. Similar techniques are used to remove the effect of the source from the simulated s-parameters [24]. However, in order for these de embedding algorithms to work, the enclosure containing the MMIC can not have any package resonances. In Chapter 3, the effect that cavity resonances have on packaged MMICs is investigated. Since the s-parameters of these circuits can not be de-embedded, an effort will be made in this section to simulate the physical coaxial to microstrip transition used in the circuits presented in Chapter 3. 2.7.3.1 Transition Model for 1.27 mm Thick Duroid A new arrangement of the basis functions used to model the current in the vicinity o f the source region is developed for a 1.27 mm thick Duroid 6010 substrate (er = 10.5) with SMA coax to microstrip launches (Omni Spectra OSM 2052-1215-00). This new arrangement will be referred to as the region 3 configuration. Two circuits will be simulated with and without the region 3 configuration and the results will be compared to measurements. These measurements are discussed in more detail in Chapter 3; therefore, the measured results will for now only be presented without discussion. For a circuit simulated using the region 3 configuration, region 1 and 2 basis functions are used in conjunction with the region 3 configuration as illustrated in Figure 2.12. For a circuit simulated without using the region 3 configuration, region 2 basis functions are used over the entire source region in lieu of the region 3 configuration. As a first example, consider a transmission line with a large gap in the center as shown in Figure 2.15. The transmission line, located aty c = 12 mm, has a width of w = 1.4 mm and a gap of g = 10.5 mm as shown in Figure 2.15. The circuit is enclosed in a cavity of the following dimensions: a = 15 mm, b = 24 mm and c = 12.7 mm. The substrate thickness is d\ = 1.27 mm and the relative permittivity is er l = 10.5(l-y'0.0023). 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b Figure 2.15 Geometry of a transmission line with a gap in the center. The transmission line, located at yc = 12 mm (b/2), has a width of w = 1.4 mm and a gap of g = 10.5 mm. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since this cavity contains only a small amount of loss it will be referred to as the high Q package. Figure 2.16 compares the transmission response of the large gap in the high Q package using the MOM without the region 3 configuration versus the measured results. The MOM was performed using 144 expansion functions. The response of the large gap predicted by the MOM has the same characteristics and is qualitatively similar to the measured results between 10.5 GHz and 11.5 GHz. However, below 10.5 GHz and above 11.5 GHz, the slope of the MOM solution is different than the measured response. The discrepancy between the MOM simulation and the measured response is as large as 7 dB at 9 GHz. Further investigation indicates that a better arrangement of the basis functions is needed to simulate the coaxial to microstrip transition. This new arrangement, shown in Figure 2.17, will be referred to as the region 3 configuration. By using the region 3 configuration in the vicinity of the source, it has been found that very good agreement between the simulated and measured results can be obtained. For a 1.27 mm thick Duroid 6010 substrate, a 1.4 mm wide transmission line near the source should be divided up into = 4 sub-sections along its length and Nys = 5 sub-sections along its width. Rooftop functions are used to expand the current on the shaded areas shown in Figure 2.17. However, in the cross-hatched regions, the current is not modeled. Thus, no expansion functions are necessary. The optimum length of the source region, Ls, was determined by trial and error to be 1.5 mm. Figure 2.18 compares the transmission response of the large gap in the high Q package using the MOM with the region 3 configuration versus the measured results. The MOM was performed using 56 expansion functions. Agreement between the MOM with the region 3 configuration and the measurements is excellent As a second example, consider a transmission line with a single shunt open circuit stub attached as shown in Figure 2.19. The stub, located at xc = 15 mm (a/2), has a length 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm •20 Without region 3 Measured ■30 -40 9 11 10 12 F (G H z) Figure 2.16 Comparison of the predicted transmission response of the large gap in the high Q package using the MOM without the region 3 configuration versus the measured results. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.17 The region 3 configuration of basis functions used to simulate the coaxial to microstrip transition. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm o -10 -20 With region 3 Measured -30 -40 9 10 11 12 F (G H z) Figure 2.18 Comparison of the predicted transmission response of the large gap in the high Q package using the MOM with the region 3 configuration versus the measured results. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b Figure 2.19 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located at x c = 15 mm (a/2), has a length L = 1.9 mm and is attached to a transmission line, located a ty c = 24 mm (b/2), of width w = 1.4 mm. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L = 1.9 mm and is attached to a transmission, located a ty c = 24 mm (fc/2), of width w = 1.4 mm. The stub is enclosed in a cavity o f the following dimensions: a = 30 mm, b = 48 mm and c = 10.0 mm. The substrate thickness is d i = 1.27 mm and the relative permittivity is £r l = 10.5(l-_/0.0023). Since this cavity contains only a small amount of loss it will be referred to as the high Q package. Figure 2.20 compares the transmission response of the stub in the high Q package using the MOM without the region 3 configuration versus the measured results. The MOM was performed using 227 expansion functions. Agreement between the MOM and the measured is terrible. Next, the region 3 configuration that was developed for the large gap (Ls = 1 .5 mm, = 4 and Nys = 5) is employed to better model the coaxial to microstrip transition. Figure 2.21 compares the transmission response of the stub in the high Q package using the MOM with the region 3 configuration versus the measured results. The MOM was performed using 179 expansion functions. Agreement between the MOM with the region 3 configuration and the measured is very good. 2 .13 .2 Transition Model For The Other Substrates Used in This Dissertation When developing the transition model for the 1.27 mm thick Duroid 6010 it was determined that good agreement between the simulated and measured results was obtained with = Xg/ 25, Ays ~ A ^/20, = 4 and Nys = 5 where Ls = N ^ A ^ and Ays = W / N ys. However, it was discovered that satisfactory agreement between the simulated and measured results can be obtained with only necessary to use = 2 and A ^ ~ A.g/15. It was - 4 and A ^ = A ^/25 to fine tune the simulated results to the measured results. However, for the remaining substrates used in this dissertation no measured results are available. Therefore, for these substrates the following criteria will be used: = 2, A ^ = A^/15 and A ys ~ A ^/20. Using the above criteria, the transition 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 24 mm -10 -20 -30 ■ Without region 3 ‘ Measured 40 9 10 11 12 13 F (G H z) Figure 2.20 Comparison of the predicted transmission response of the stub in the high Q package using the MOM without the region 3 configuration versus the measured results. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 24 mm -10 a ■c -20 -30 " With region 3 ‘ Measured ^0 9 10 11 12 13 F (G H z) Figure 2.21 Comparison of the predicted transmission response of the stub in the high package using the MOM with the region 3 configuration versus the measured results. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. models employed for those substrates are given in Table 2.2. Table 2.2 also summarizes the model used for 1.27 mm thick Duroid 6010. Note, only the transition model for the 1.27 mm thick Duroid 6010 substrate has been verified experimentally. Table 2.2 Summary of the transition models employed for all substrates used in this dissertation. 2.8 Material £r Thickness (mm) Ls (mm) W (mm) Duroid 10.5 1.27 1.5 1.4 4 5 Duroid 10.5 0.635 1.5 0.64 2 3 Quartz 4.5 0.127 0.498 0.24 2 3 GaAs 12.9 0.1 0.18 0.1 2 3 Nys Conclusion In this chapter the full-wave analysis of packaged MMICs was discussed. The derivation of the spectral Green's function using and equivalent transmission line was presented. Following this was a discussion of locating the resonances o f a dielectric loaded cavity. Next a review of the method of moments (MOM) was given and an acceleration technique for the efficient evaluation of MOM impedance matrices was presented. The solution for the unknown current distribution under the condition that each port is terminated in its characteristic impedance was also given. Finally, the various basis functions used to expand the currents were discussed. A special basis function arrangement was developed to more accurately model the coaxial to microstrip transition. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 References [1] Harrington, R.F., Time harmonic electromagnetic fields. McGraw-Hill Book Company, New York, 1961. [2] Felsen, L.B. and Marcuvitz, N., Radiation and scattering of waves. Prentice-Hall, Inc., New Jersey, Chapter 2, 1973. [3] Itoh, T., "Spectral domain immitance approach for dispersion characteristics of generalized printed transmission lines," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-28, pp. 33-736, July 1980 [4] Collin, R.E., Foundations for microwave engineering. McGraw-Hill Book Company, New York, 1966. [5] Ramo, S., Whinnery, J. and Van Duzer, T., Fields and waves in communications electronics. John Wiley and Sons, New York, 1984. [6] Beringer, R. "Resonant cavities as microwave circuit elements". In Montgomery, C.G., Dicke, R.H., and Purcell, E.M. (eds) Principles of microwave circuits. MIT Radiation Laboratory Series, Vol. 8., McGraw-Hill Book Company, Inc., New York, pp. 207-239, 1948. [7] Kajfez, D., Notes on Microwave Circuits Volume 1. Kajfez Consulting, University, MS, 1984. [8] Rautio, J.C. and Harrington, R.F., "An electromagnetic time-harmonic analysis of shielded microstrip circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-35, pp. 726-730, August 1987. [9] Dunleavy, L.P. and Katehi, P.B., "A generalized method for analyzing shielded thin microstrip discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT37, pp. 1758-1766, December 1988. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [10] Rautio, J.C., "A time-harmonic electromagnetic analysis of shielded microstrip circuits," Ph.D. Thesis, Syracuse University, Syracuse, NY, 1986. [11] Dunleavy, L.P., "Discontinuity characterization in shielded microstrip: a theoretical and experimental study," Ph.D. Thesis, University of Michigan, 1988. [12] Katehi, P.B., "Radiation losses in mm-wave open microstrip filters," Electromagnetics, Vol. 7, pp. 137-152,1987. [13] Jackson, R.W. and Pozar, D.M., "Full-wave analysis of microstrip open-end and gap discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1036-1042, October 1985. [15] Katehi, P.B. and Alexopoulos, N.G., "Frequency-dependent characteristics of microstrip discontinuities in millimeter-wave integrated circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1029-1035, October 1985. [16] Hill, A. and Tripathi, V.K., "An efficient algorithm for three-dimensional analysis of passive microstrip components and discontinuities for microwave and millimeterwave integrated circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-39, pp. 83-91, January 1991. [17] Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions. Dover., New York, 1972. [18] Jansen, R.H. and Sauer, J., "High-speed 3D electromagnetic simulation for MIC/MMIC cad using the spectral operator expansion (soe) technique," 1991 IEEE MTT-S Digest, pp. 1087-1090, 1991. [19] Jansen, R.H., "Recent advances in the full-wave analysis of transmission lines for the application in MIC and MMIC design," 1987 SBMO International Microwave Symposium Digest, pp. 467-475, 1987. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [20] Jansen, R.H., "Modular source-type 3D analysis of scattering parameters for general discontinuities, components and coupling effects in (M)MICs," 17th European Microwave Conf. Proc., pp. 427-432, 1987. [21] Mosig, J.R.and Gardiol, F.E., "General integral equation formulations for microstrip antennas and scatters," Proc.Inst. Elec. Eng., Part H, Vol. 132, pp. 425-432, December 1985. [22] Stinehelfer, H.E., "Discussion of de-embedding techniques using time-domain analysis," IEEE Proceedings, Vol. 74, No. 1, pp. 90-94, Jan. 1986. [23] Franzen, N.R. and Speciale, R.A., "A new procedure for system calibration and error removal in automated s-parameter measurements," 5th European Microwave Conf. Proc., pp. 69-73, 1975. [24] Rautio, J.C., "A de-embedding algorithm for electromagnetics," International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, Vol. 1, No. 3, pp. 282-287, July 1991. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3 RESO NANT M ODE CO UPLING IN PACK AG ED M M IC s: A TH EO R ETIC A L A N D EX PER IM EN TA L IN V E STIG A TIO N 3.1 Introduction A number of full-wave techniques have been developed for the very accurate modeling of microstrip structures [1] - [13]. However, many o f these techniques neglect the effect o f enclosing the circuit in a package. This has not been a very serious problem in current designs because circuit packages have not been electrically large. For sufficiently large enclosures package resonances are possible. If the system operates at frequencies near one of these resonances, catastrophic coupling can occur between different elements of a circuit. Although [6], [7], [11] model a microstrip circuit in an enclosure and analyze at least one circuit with a resonant mode, none of them specifically examine the causes of resonant mode coupling and ways to reduce i t In this chapter the coupling of power to resonant modes and its effect on a circuits' performance will be discussed. In the first section, the accuracy of the full-wave method of moment (MOM) procedure developed in Chapter 2 will be verified. Two groups of circuits are fabricated, enclosed in a brass cavity, and measured. The measured results are compared to those obtained with the MOM procedure. Following this, the MOM procedure is used to examine some of the fundamental aspects of resonant mode coupling. In addition, methods of reducing this coupling will also be investigated. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 Experimental Verification of the MOM for Circuits in Resonant Enclosures 3.2.1 Introduction To verify the accuracy of the full-wave method of moment (MOM) procedure developed in Chapter 2, two groups of circuits are fabricated and measured while enclosed in a brass cavity with at least one resonant mode occurring in the operating bandwidth. The effect of adding loss to the enclosure is also examined. The measured results are compared to those obtained with the MOM procedure. The first group consists of a large gap (~ Xg) in a transmission line. The second group consists of a shunt open circuit stub attached to a transmission line. In addition, the second group was measured in two different size enclosures. The smaller enclosure has one resonant mode, while the larger cavity has five resonant modes. The circuits were fabricated on a 1.27 mm thick Duroid 6010 substrate. Duroid 6010 has a relative permittivity o f 10.5 ± 0.25 and a loss tangent o f 0.0023. The circuits were enclosed in two different size cavities milled out of solid brass. The first cavity, cavity A, has the following inner dimensions: a = 15 mm, b - 2 4 mm and c = 12.7 mm (Figure 3.1). An enclosure o f this size has only one resonant mode in the 9-12 GHz band. This mode, the T M no, was theoretically determined to occur at 10.8129 GHz and has a Q of 4196 (see Chapter 2). The second cavity, cavity B, has the following inner dimensions: a = 30 mm, b = 48 mm and c = 10 mm. An enclosure of this size has five resonant modes in the 9-13 GHz band. The resonant frequency and Q for each mode of the high Q enclosure is listed in Table 3.1. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T c Figure 3.1 Basic geometry of the brass cavity with the following inner dimensions: a, b and c. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.1 The theoretical resonant frequency and Q for each mode in cavity B. Mode TM130 TM]40 TM210 TM220 TM230 fr (GHz) 9.7043+10.0089 11.9517+i0.0227 9.5793+i0.0084 10.6661+10.0132 12.1289+i0.0245 Q 5480 2634 5692 4049 2478 Attaching the substrate to the inside of the cavity proved to be very difficult. First, the substrate had to be carefully cut to size in order to fit into the cavity. Cutting the substrate too small will leave a large of a gap between the sides of the substrate and cavity. If the substrate is cut too large it will not lay flat on the bottom of the cavity. This trimming could result in the circuit not being positioned exactly in the desired location within the enclosure. The exact location of the circuit in the enclosure was difficult to determine to within ± 1 mm. In order to insure that a proper connection is achieved between the ground plane of the substrate and cavity, the cavity is heated on a hot plate and the solder is allowed to flow while applying pressure to the substrate. In the experiments that follow, the measured data was obtained using a Hewlett Packard 8510 network analyzer. A 50-Ohm system is assumed throughout. S M A coaxto microstrip launches (Omni Spectra OSM 2052-1215-00) were utilized to connect the circuits inside the cavities to the network analyzer. 3.2.2 Large Gap As a first example, consider a transmission line with a large gap in the center as shown in Figure 3.2. A large gap in a transmission line is chosen because of its simplicity 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b Figure 3.2 Geometry of a transmission line with a gap in the center. The transmission line, located at y c, has a width o f w = 1.4 mm and a gap o f g = 10.5 mm. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and because the ideal response has a very small transmission coefficient (IS211)- However, it will be shown that the cavity resonances can significantly increase the transmission coefficient of the circuit. The transmission line, located at yc, has a width of w = 1.4 mm and a gap o f g = 10.5 mm. The circuit is enclosed in cavity A which was previously determined to have one resonant mode in the band 9-12 GHz. Since this cavity contains only a small amount of loss it will be referred to as the high Q cavity A. Figure 3.3 compares the calculated and measured transmission coefficient (IS2 1 O of the large gap for two different locations, yc = 12 mm and 5 mm, in the high Q cavity A. The MOM simulation was performed using 56 expansion functions. When the circuit is located aty c = 12 mm, agreement between the calculated and measured results is very good across the entire bandwidth except for a slight shift in the peak of IS2 1 I as shown in Figure 3.3a. When the circuit is located aty c = 5 mm, agreement between the calculated and measured results also exhibits a slight shift in the peak o f IS2 1 I as shown in Figure 3.3b. However, above 11.25 GHz the two curves start to deviate from one another. This discrepancy, as large as 4 dB at 13 GHz, may be due to the circuit not being positioned exactly aty c = 5 mm. It was mentioned above that the exact location of the circuit in the enclosure was difficult to determine to within ± 1 mm. Assuming that the circuit is located 0.35 mm closer to the center of the enclosure is not unreasonable under these circumstances. Therefore, the MOM simulation was repeated with the circuit located at y c = 5.35 mm instead of y c = 5 mm with the results shown in Figure 3.4. Above 11.7 GHz the two curves start to deviate from one another. However, moving the circuit from y c = 5 mm to yc = 5.35 mm resulted in better agreement between the simulated and measured results. It also indicates the sensitivity of the measurement. To reduce the effect of the resonant mode, a 1.27 mm thick microwave absorbing layer, characterized by Br = 60(1-70.12) and |ir = 7.3(l-7'0.3), was attached to the cover of the enclosure. The characterization of the absorbing material is described in Appendix C. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm -10 oa ■o w -20 MOM Measured -30 -40 9 11 10 12 F (G H z) (a) Figure 3.3 Comparison of the calculated and measured transmission coefficient of the large gap in the high Q cavity A for two different locations: (a) y c = 12 mm and (b) yc = 5 mm. Continued, next page. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 5 mm •20 MOM Measured -40 9 10 11 12 F (G H z) (b) Figure 3.3 Continued. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 5.35 mm o -10 -20 -30 MOM Measured -40 9 11 10 12 F (G H z) Figure 3.4 Comparison o f the calculated and measured transmission coefficient o f the large gap located at y c = 5 mm in the high Q cavity A. The MOM simulation was performed with the circuit located at y c = 5.35 mm. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cavity A with this absorbing layer attached to the cover again has only one resonant mode, the T M no, which was theoretically found to occur at 10.888 + y'0.285 GHz and has a Q of 19. Since the Q of this cavity is very small, it will be referred to as the low Q cavity A. Figure 3.5 compares the calculated and measured transmission coefficient (IS2 1 O of the large gap for two different locations, yc = 12 mm and 5 mm, in the low Q cavity A. When the circuit is located a ty c = 12 mm, Figure 3.5a shows that agreement between the calculated and measured results is excellent across the entire bandwidth. When the circuit is located aty c = 5 mm, Figure 3.5b shows that the calculated and measured results again exhibits about a 1 dB deviation across the entire bandwidth. As noted above, the sensitivity was tested by repeating the MOM simulation with the circuit located at yc = 5.35 mm instead of y c = 5 mm. Figure 3.6 compares the calculated and measured transmission coefficient (IS2 1 O of the large gap. Agreement between the simulated and measured results is excellent. Power lost to the package can be very significant for low Q enclosures. The fraction of the incident power lost to the enclosure is defined to be T = i W 4 , s = l - I S „ l 2 - I S 2 ll2 (3.1) In Figure 3.7, 77 is plotted for two different locations of the circuit, y c = 12 mm and y c = 5 mm. The figure shows that when the circuit is located at y c = 12 mm a significant amount of power is lost to the enclosure in vicinity of 11 GHz. Repositioning the circuit in the enclosure from yc = 12 mm to y c = 5 mm reduces the power lost to the package by a factor 2 at 11 GHz. To summarize, the large gap was located at two different positions in cavity A, y c = 12 mm and 5 mm. The resonant mode of the high Q enclosure has a drastic effect on IS2 1 1 of the circuit located at either position. Both circuits have a large transmission coefficient (IS2 1 I - 0 dB) in the vicinity of 10.8 GHz. Next, a lossy dielectric layer was attached to the cover of the enclosure. When the circuit is located a ty c = 12 mm in the low Q 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 12 mm -10 /•“k aa T3 -20 ' -30 MOM Measured -40 9 11 10 12 F (G H z) (a) Figure 3.5 Comparison of the calculated and measured transmission coefficient of the large gap in the low Q cavity A for two different locations: (a) y c = 12 mm and (b) yc = 5 mm. Continued, next page. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 5 mm -10 -20 * MOM ' Measured -30 -40 9 11 10 12 F (G Hz) (b) Figure 3.5 Continued. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 5.35 mm -10 /-V a TS -20 N V.1 - MOM ■ Measured -30 -40 9 10 11 12 F (G H z) Figure 3.6 Comparison of the calculated and measured transmission coefficient of the large gap located at yc = 5 mm in the low Q cavity A. The MOM simulation was performed with the circuit located aty c = 5.35 mm. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover 1.0 0.8 u B ’ Y c= 12 mm • Yc = 5 mm 0.6 C A M i 0.4 0.2 0.0 9 11 10 12 F (G H z) Figure 3.7 Computed rj for the large gap in the low Q cavity A for two different locations: y c = 12 mm and y c = 5 mm. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. enclosure, the effect of the resonant mode is much smaller than in the high Q enclosure. Repositioning the circuit from y c = 12 mm to yc = 5 mm results in a 7 dB improvement in the isolation over most of the bandwidth. In addition to a smaller transmission coefficient, the circuit located atyc = 5 mm has less power lost to the package than the circuit located at yc = 12 mm. 3.2.3 Shunt Stub As a second example, consider a transmission line with a single shunt open circuit stub attached as shown in Figure 3.8. The stub, located at xc = a ll , has a length L = 1.9 mm and is attached to a transmission line, located atyc, of width w = 1.4 mm. A shunt stub is chosen because it is a typical building block used in more complicated circuits. The ideal theoretical response for this circuit has a single null in IS2 1 I at 11.1 GHz as shown in Figure 3.9. The ideal response was obtained by enclosing the circuit in cavity with no resonant modes below 13 GHz. The shunt stub was measured in the enclosures cavity A and cavity B. 3.2.3.1 Cavity A To examine the effect that one mode has on the response of the shunt stub, the circuit is located in cavity A at two different positions (yc = 12 mm and 5 mm). Figure 3.10 compares the calculated and measured transmission coefficient (IS2 1 O of the circuit in the high Q cavity A. The MOM simulation was performed using 123 expansion functions. When the circuit is located aty c = 12 mm, the response of the stub predicted by the MOM has the same characteristics and is qualitatively similar to the measured results (Figure 3.10a). The first null in IS2 1 I is shifted down in frequency by about 0.2 GHz. When the 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b Figure 3.8 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located at x c {all), has a length L = 1.9 mm and is attached to a transmission line, located aty c, of width w = 1.4 mm. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Id eal o -10 -20 -30 -40 9 11 10 12 F (G H z) Figure 3.9 Ideal theoretical response of the shunt open circuit stub. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm IS21I (dB) -10 -20 -30 - MOM • Measured -40 9 11 10 12 F (GH z) (a) Figure 3.10 Comparison of the calculated and measured transmission coefficient of the shunt stub in the high Q cavity A for two different locations: (a) yc = 12 mm and (b) y c = 5 mm. Continued, next page. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 5 mm -10 09 ■O -20 M cc ' -30 MOM Measured -40 9 10 11 12 F (G H z) (b) Figure 3.10 Continued. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. circuit is located at yc = 5 mm, agreement between the calculated and measured results also exhibits a slight shift of the null in IS2 1 I (Figure 3.10b). In addition to the frequency shift o f IS2 1 I, the simulated response does not exhibit the rapid variation of IS2 1 I in the vicinity of 10.9 GHz that the measured response does. The discrepancy between the simulated and measured results may be due to one or more of the following: 1) The uncertainty in circuit dimensions. 2) The uncertainty in dielectric constant 3) The uncertainty in determining the exact location of the circuit in the enclosure. To determine if the desired circuit dimensions were correct, a careful measurement to within ± 0.025 mm, for each circuit was performed. For the circuit located a ty c = 12 mm, the actual length of the stub was measured to be 1.8 mm instead of 1.9 mm. If the length of the stub was altered in the etching of the circuit one would also expect for other the circuit dimensions to be effected. However, all of the other dimensions were unchanged. Careful examination of the photographic mask used in the etching process, revealed that the error in the length of the stub occurred in the making of the mask and not in the etching. For the circuit located at yc = 5 mm, this error was not made in the photographic mask and all circuit dimensions were measured to be correct. Figure 3.11 compares the calculated and measured transmission coefficient (IS2 1 O of the shunt stub circuit located aty c = 12 mm in the high Q cavity A. The calculated response shown in Figure 3.11 was performed for a shunt stub of length 1.8 mm located a ty c = 12 mm. The response of the stub predicted by the MOM has the same characteristics and is qualitatively similar to the measured results. However, the second null in IS2 1 I is shifted up in frequency by about 0.1 GHz. Comparing Figure 3.11 to 3.10a shows that reducing the length of stub in the simulation gives slightly better results overall. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm -10 ap) iiz s i -20 - " MOM • Measured -30 ^0 9 11 10 12 F (G H z) Figure 3.11 Comparison of the calculated and measured transmission coefficient of the shunt stub circuit of length L - 1.9 mm located aty c = 12 mm in the high Q cavity A. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As previously stated, the second source of error may be due to the uncertainty in the permittivity. The manufacture's quoted value for the permittivity of Duroid 6010 is 10.5 ± 0.25. Therefore, the MOM simulations were repeated with the permittivity varying from 10.25 to 10.75. The optimal value of the permittivity was determined to be 10.75; however, this resulted in only a slight improvement. In the previous example (large gap), it was shown that the MOM simulations are very sensitive as to the exact location o f the circuit in the enclosure. It has been determined that the best agreement between the simulated and measured results occurs when the location of the circuit in the enclosure is slightly different than the stated value of yc (12 —> 11.45 mm and 5 —» 5.35 mm). A slight improvement was shown by relocating the circuit Using the knowledge gained from examining the three sources of errors, the MOM simulations were repeated. For the circuit positioned at y c = 12 mm, the MOM simulation was performed with yc = 11.45 mm, L = 1.8 mm and s r =10.75. As shown in Figure 3 .12a, agreement between the calculated and measured results is very good. For the circuit positioned a iy c = 5 mm, the MOM simulation was performed with yc = 5.35 mm, L = 1.9 mm and £r =10.75. Figure 3.12b shows that agreement between the simulated and measured results is very good. The MOM solution no longer exhibits a slight shift of the null in IS2ll and it does exhibit the rapid variation of IS2 1 I in the vicinity of 10.9 GHz. In the previous section it was shown that attaching a 1.27 mm thick microwave absorbing layer to the cover of the enclosure can reduce the effect of the resonant mode. Figure 3.13 compares the calculated and measured transmission coefficient (IS2 1 O of the shunt stub circuit for two different locations, yc = 12 mm and 5 mm, in the low Q cavity A. For the circuit positioned aty c = 12 mm, the MOM simulation was performed with y c = 11.45 mm, L = 1.8 mm and er =10.75. As shown in Figure 3 .13a, agreement between the simulated and measured results is very good except in the vicinity of the first null, where the simulated response is much deeper. For the circuit positioned at y c = 5 mm, the MOM 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 11.45 mm IS21I (dB) -10 -20 ' -30 MOM Measured -10 9 11 10 12 F (GHz) (a) Figure 3.12 Comparison of the calculated and measured transmission coefficient of the shunt stub in the high Q cavity A for two different locations: (a) yc = 12 mm and (b) yc = 5 mm. For the circuit positioned at yc = 12 mm, the MOM simulation was performed with yc = 11.45 mm, L = 1.8 mm and er =10.75. For the circuit positioned at yc = 5 mm, the MOM simulation was performed with yc = 5.35 mm, L = 1.9 mm and er =10.75. Continued, next page. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 5.35 mm -10 - -20 ' -30 MOM Measured -40 9 11 10 12 F (G H z) (b) Figure 3.12 Continued. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 11.45 mm -10 -20 MOM Measured -30 -40 9 10 11 12 F (G H z) (a) Figure 3.13 Comparison of the calculated and measured transmission coefficient of the shunt stub in the low Q cavity A for two different locations: (a) y c = 12 mm and (b) y c = 5 mm. Continued, next page. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 5.35 mm -10 aa -20 </} - MOM ■ Measured -30 -40 9 11 10 12 F (G H z) (b) Figure 3.13 Continued. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. simulation was performed with y c = 5.35 mm, L = 1.9 mm and &r =10.75. Figure 3.13b shows that agreement between the simulated and measured results is very good except in the vicinity of the null, where the simulated response is again much deeper. This discrepancy may be due to the finite conductivity of the microstrip line not being included in the MOM calculation. Power lost to the package can be very significant for low Q enclosures. In Figure 3.14, 77 is plotted for two different locations of the circuit, y c = 12 mm and y c = 5 mm. The MOM simulation was performed with L = 1.9 mm and er =10.5. The figure shows that when the circuit is located at yc = 12 mm a significant amount of power is lost to the enclosure from 10.5 GHz to 11.75 GHz. Repositioning the circuit in the enclosure from yc = 12 mm to yc = 5 mm significantly reduces the power lost to the package. In order to verify Figure 3.14, reflection measurements were made for the circuit located aty c = 12 mm and 5 mm. Combining the reflection measurements with the transmission measurements done earlier, the fraction of incident power lost to the enclosure was determined. Figures 3.15a and 3.15b plots 77 calculated from these measurements and the MOM simulation for the circuit located at yc = 12 mm and yc = 5 mm, respectively. For the circuit positioned at y c = 12 mm, the MOM simulation was performed with y c = 11.45 mm, L = 1.8 mm and £r =10.75. For the circuit positioned at y c = 5 mm, the MOM simulation was performed with y c = 5.35 mm, L = 1.9 mm and er =10.75. W hen the circuit is positioned atyc = 12 mm, agreement between the MOM simulation and the measurements is very good. When the circuit is positioned at yc = 5 mm, agreement between the MOM simulation and the measurements is very good up to 10.8 GHz. Above 10.8 GHz the two curves start to deviate from another. Since 77 is relatively small and there is a large standing wave current, this discrepancy may again be due to the finite conductivity of the microstrip line not being included in the MOM calculation. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover 1.0 0.8 u s (A o eu ■ Yc = 12 mm ' Yc = 5 nun 0.6 C/3 0.4 0.2 0.0 9 10 11 12 F (G H z) Figure 3.14 Computed T] for the shunt stub in the low Q cavity A for two different locations: y c = 12 mm and yc = 5 mm. The MOM simulation was performed with L = 1.9 mm and er =10.5 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 11.45 mm 1.0 Ploss/Pinc 0.8 - MOM ’ Measured 0.6 0.4 0.2 - 0.0 9 11 10 12 F (G H z) (a) Figure 3.15 Comparison of the calculated and measured tj for the shunt stub in the low Q cavity A for two different locations: yc = 12 mm and y c = 5 mm. Continued, next page. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 5.35 mm 1.0 0.8 w s 2 V i VI £ - MOM • Measured 0.6 0.4 0.2 0.0 9 10 11 12 F (G H z) (b) Figure 3.15 Continued. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To summarize, when the shunt open circuit stub circuit was located aty c = 12 mm in the high Q cavity A, the resonant mode had a drastic effect on IS2 1 I. Repositioning the circuit from yc = 12 mm to yc = 5 mm significantly reduced the effect of the resonant mode. Next, a lossy dielectric layer was attached to the cover of the enclosure. For the circuit located at yc = 12 mm in the low Q enclosure, the effect of the resonant mode was much smaller than in the high Q enclosure. However, the resonant mode of the enclosure still had a large effect on IS2 1 I. The circuit located atyc = 5 mm had very little power lost to the package in addition to a transmission coefficient similar to the ideal response. 3.2.3.2 Cavity B To examine the effect that several resonant modes have on the response o f the shunt stub, the circuit is located in cavity B aty c = 24 mm. Figure 3.16 compares the calculated and measured transmission coefficient (IS2 1 I). The MOM simulation was performed using 179 expansion functions. Very good agreement is obtained except for minor discrepancies in the vicinity of the resonant modes. To reduce the effect of the resonant modes two different materials were used to lower the Q of the enclosure. The first material employed is a microwave absorbing layer attached to the cover of the enclosure. The absorbing layer is 0.762 mm thick with a permittivity of er = 60(l-_/0.12) and permeability of (ir = 7.3(l-y'0.3). Cavity B with this absorbing layer attached to the cover will be referred to as the low 2-dielectric cover cavity. The resonant frequency and Q for each mode of the low Q-dielectric cover is listed in Table 3.2. 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 24 mm IS21I (dB) -10 -20 - MOM • Measured -30 ^0 9 10 11 12 13 F (GH z) Figure 3.16 Comparison of the calculated and measured transmission coefficient of the shunt stub in the high Q cavity B. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.2 The resonant frequency and Q for each mode in the low 2-dielectric cover cavity B. Mode TM 140 TM230 f r (GHz) 11.845 l+j0.5578 12.0632+i0.5730 Q 1 0 .6 10.5 Figure 3.17 compares the calculated and measured transmission coefficient (IS2 1 O of the shunt stub circuit in the low 2-dielectric cover cavity B. The figure shows that reasonable agreement is obtained between the calculated and measured results. The second material employed to reduce the effect of the resonant modes is a doped silicon layer attached to the cover of the enclosure. The silicon layer is 16 mil (0.4064 mm) thick with a resistivity o f 1.45 £2-cm. The resistivity of the silicon wafer was measured using the 4 point probe method. Cavity B with this doped silicon layer attached to the cover will be referred to as the low 2 -S i cover cavity. The resonant frequency and Q for each mode of the low 2 -S i cover cavity is listed in Table 3.3. Table 3.3 The resonant frequency of each mode in the low 2-S i cover cavity B. Mode TM 130 TM 140 TM 210 TM 220 TM230 fr (GHz) 9.5383+i0.0438 11.8080+i0.0477 9.4145+i0.0423 10.5052+10.0460 11.9908+i0.0451 Q 109.0 123.9 111.4 114.3 132.8 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 24 mm IS21I (dB) -10 -20 - MOM • Measured -30 -40 9 10 11 12 13 F (G H z) Figure 3.17 Comparison of the calculated and measured transmission coefficient of the shunt stub in the low (7-dielectric cover cavity B. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.18 compares the calculated and measured transmission coefficient (IS2 1 I) of the shunt stub circuit in the low g-S i cover cavity B. The figure shows that good agreement is obtained between the calculated and measured results. As mentioned previously, 77 is a very important figure-of-merit for circuits enclosed in low g packages. Figures 3.19 plots 77 for the circuit enclosed in the dielectric and Si cover cavities. A significant amount of power for the circuit enclosed in the low g dielectric cover cavity. When the circuit is enclosed in the low g-Si cover cavity a significant amount of power is lost to the enclosure in the vicinity of the resonant modes only. In order to verify Figure 3.19, reflection measurements were made for the circuit enclosed in both low g cavities. Combining the reflection measurements with the transmission measurements, the fraction of power lost to the enclosure was determined. Figure 3.20 plots 77 calculated from these measurements and the MOM simulation. When the circuit is enclosed in the low g dielectric cover cavity B, the agreement between the simulated and measured results is satisfactory as shown in Figure 3.20a. Figure 3.20b shows that when the circuit is enclosed in the low g-Si cover cavity B reasonable agreement exists between the simulated and measured results. To summarize, when the circuit is enclosed in the high g cavity B the resonant mode of the enclosure had a significant effect on iS2il. A similar effect on IS2 1 I was also observed when the circuit was enclosed in cavity A. The lossy dielectric layer appeared to significantly reduce the effect of the resonant modes on the transmission coefficient. However, examining 77, shows that a significant amount of power is lost to the enclosure over a large portion of the bandwidth. Examining 77 for the doped silicon layer, shows that a significant amount of power is lost to the enclosure in the vicinity of the resonant modes only. However, the resonant modes of the enclosure still has a large effect on IS2 1 I- 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Si cover Yc = 24 mm -10 /■“•s as ■o -20 ce ' -30 MOM Measured -40 9 10 11 12 13 F (G H z) Figure 3.18 Comparison o f the calculated and measured transmission coefficient of the shunt stub in the low <2-Si cover cavity B. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Yc = 24 mm 1.0 Dielectric cover Si cover 0.8 u e 0* W £ 0.6 0.4 0.2 0.0 9 10 11 12 13 F (GH z) Figure 3.19 Computed r\ for the shunt stub enclosed in the dielectric and Si cover cavities. 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric cover Yc = 24 mm 1.0 - MOM • Measured 0.8 u s 0.6 0.4 0.2 0.0 9 10 11 12 13 F (G H z) (a) Figure 3.20 Comparison of the calculated and measured 77 for the shunt stub enclosed in the: (a) low Q dielectric cover cavity B and (b) low Q-Si cover cavity B. Continued, next page. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Si cover Yc = 24 mm 1.0 0.8 MOM Measured 0.6 0.4 0.2 0.0 9 10 11 12 13 F (G H z) (b) Figure 3.20 Continued. 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Parasitic Coupling to Resonant Modes In the previous section it was observed, experimentally and theoretically, that the location of a circuit in an enclosure can effect the coupling of power to a resonant mode. In order to gain a better understanding of this phenomenon, the effect that resonant mode coupling has a discontinuity in a circuit on will be investigated. At a discontinuity, power is radiated in the form of surface waves. The surface waves propagate parallel to the substrate away from the circuit over a wide range of angles [1], [2]. This radiated power is reflected at the side walls of the enclosure which can then recombine with the circuit at another location. This effect can be catastrophic because it creates additional links between parts of a circuit that were not meant to be coupled to one another [1], [2], [3]. In addition, the reflected power will modify the standing wave current which will in turn modify the power radiated in the form of surface waves and so on. At frequencies in the vicinity of a package resonance, the fields that comprise the resonant mode become extremely large. The larger the Q of the enclosure, the larger the fields associated with a resonance become. Therefore, the resonant mode can drastically alter the standing wave current of the circuit which can result in a drastic change in the scattering parameters of the circuit. An important quantity for characterizing microwave circuits in a package is the amount o f power lost to a resonant mode. The power lost to a resonant mode can be expressed as: C ° de = - | R e / j £ rmode(x,y)-7;(x,y)d!xrfy (3.2) where /)™°de is the power lost to the resonant mode, £ rmode is the tangential component of the resonant mode electric field and Js is the surface current on the microstrip line. The surface current on the microstrip line is determined by the full-wave MOM procedure developed in Chapter 2. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In a typical enclosure housing an MMIC chip, the cover height is low enough that only the TM nmfl modes are resonant over the frequency of operation. For this reason, only the TM mode will be considered in the following analysis. However, the analysis could easily be applied to the TE mode. For a TM mode, the tangential component of the resonant mode electric field is given by E ? odeU y ) = E ™ (x ,y ) = x E ™ (x ,y ) + y E ™ (x ,y ) (3.3) Substituting equation (3.3) into equation (3.2) yields n m o d e pT M pTM , pTM H oss ~ H oss ~ r x + ry W -* ) where P™ = “ Re j j E ™ (x ,y )J * (x ,y )d x d y (3.5a) P ™ = - j R e JJ E ™ (x,y)Jy (x ,y )d x d y (3.5b) Equation (3.4) describes the total power lost to a resonant mode and equation (3.5) describes the power lost to a resonant mode via the components of the resonant mode field. However, equations (3.4)and (3.5) do not describe what regions o f a circuit couple more strongly to the resonant mode.In addition, a circuit in a high Q enclosure will only lose a small amount o f power to a resonant mode even though the resonant mode has a drastic effect on the circuit Therefore, an additional measure o f resonant mode coupling is needed. Equation (3.5) can be rewritten as: P ™ = jp ™ ( x ) d x (3.6a) P ™ = ]p ™ (y )d y (3.6b) P™ (x) = - i - R e J E™{x,y)J*x (x ,y )d y (3.7a) P ™ 0 0 = ~ R e J E™(.x,y)J*y {x,y)d x (3.7b) where In Chapter 2 it was shown that the tangential component of the electric field can be expressed as: 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E ™ (jc, y) = A ™ cos(kxnx)sin(kymy) (3.8a) E ™ (x,y) = A$m sin(kxnx)cos(kymy) (3.8b) where A ™ = ~V b ^ t ^ kxnJx{kxn' kym) + )} A ™ = (3.9a) ( 3 -9 b ) ( k xrt ’ k ym ) — (x, y) c o s ik ^ x ) sin (kymy) d xd y (3.1 Oa) (x, y) sin(kx„x) cos (kymy) d xd y (3.1 Ob) s J y ( k xn ’ k ym ) s n and m correspond to the indices of the resonant mode and YM is defined by equation (2.43). Inspection of equations (3.7), (3.8), (3.9) and (3.10) indicates that p ™ (x ) and/or p ™ (y) can reduced by locating areas of high current in areas of low electric field and visaversa. For example, consider a circuit where the current is predominantly x-directed and centered at y = yc, it can be shown that J x ( k ^ , kym) is given by Jx(kxn’kym) = Fx (kxn, kym) sin(kymyc) (3.11) Substituting equations (3.8a), (3.9a) and (3.11) into equation (3.7a), p ™ (x) can be written as: P ™ M - (p ™ )max sin2{kymyc) (3.12) Thus p ™ (x) can be reduced by changing y c- Note, reducing p ™ (x ) will ultimately result in a reductionin P™ . Positioning the circuit in the center of the box will result in a maximum power loss for the T M n o (kyi = n/b) resonance and a minimum power loss for the TM 120 (ky2 = 2iz/b) resonance. To develop a better understanding of resonant mode coupling, the two circuits discussed in the previous section will be re-examined. The first circuit consists o f a large gap (~ Xg) in a transmission line. The second group consists of a shunt open circuit stub 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. attached to a transmission line. Both circuits are enclosed in cavity A which was described in the previous section. 3.3.1 Large Qap As a first example, again consider the transmission line with a large gap in the center as shown in Figure 3.2. The transmission line, located at yc, has a width o f w = 1.4 mm and a gap of g = 10.5 mm. The circuit is located in cavity A at two different positions (yc = 12 mm and 5 mm). The ideal response for the large gap is a very small transmission coefficient. As result, the ideal current on the strip should exhibit a large standing wave on the first strip and the current on the second strip should be very small. Also, the dominate current on the circuit is x directed. Thus, power is radiated into the TM n o mode via the x component of the electric field. Inspection of equations (3.6a), (3.7a) and (3.8a) indicates that moving the circuit from yc = 12 mm to yc = 5 mm should reduce p ™ (x) and P ™ . Figure 3.21 summarizes the computed transmission coefficient (IS2 1 Oof the large gap for two different locations, yc = 12 mm and 5 mm, in the high Q cavity A. The resonant mode of the high Q enclosure has a drastic effect on IS2 1 I of the circuit located at either position. Both locations result in a large transmission coefficient (IS2 1 1~ 0 dB) in the vicinity of 10.8 GHz. To gain a better understanding on how the coupling of power to the TM i 10 mode effects the operation of the large gap, the current on the two strips at/ = 10.8129 GHz is illustrated in Figure 3.22. Note that the first strip extends from x = 0 mm to x = 2.25 mm and the second strip extends from x = 12.75 mm to x = 15 mm. The figure shows that a large and nearly equal current is on both strips for the circuit located in either position. Figure 3.23 plots p ™ (x), the power lost per unit length to the T M i 10 mode via the x component of the electric field, at/ = 10.8129 GHz. If p ™ (x) is positive, 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q ca ■B ■20 Yc = 12 mm Yc = 5 mm -40 9 10 11 12 F (G H z) Figure 3.21 Summary of the calculated transmission coefficient of the large gap in the high Q cavity A for two different locations: yc = 12 mm and y c = 5 mm. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q 20 15 10 Yc= 12 mm Yc = 5 mm 5 0 0 10 5 15 x (mm) Figure 3.22 Current, a t / = 10.8129 GHz, on the two strips of the large gap in the high Q cavity A. High Q 500 300 Yc = 12mm Yc = 5 mm '© 100 X C ar > -100 E -300 -500 0 10 5 15 x (mm) Figure 3.23 Power lost per unit length to the T M i io mode via the x component of the electric field a t / = 10.8129 GHz for the large gap in the high Q cavity A. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. this indicates that power is being radiated from the T M i 10 mode and if p ™ (x) is negative, this indicates that power is being absorbed into the T M i 10 mode. The figure TM TJUt shows that p x (x) on the first strip is nearly equal and opposite in magnitude to p x (x) Tldf TKA on the second strip. Thus, Px which is equal to the area under p x (x) is approximately zero. Inspection of equations (3.6a), (3.7a) and (3.8a) indicates that moving the circuit TKA TKA from yc = 12 mm to yc = 5 mm will reduce px (x) and Px . Since the circuit is being operating at the resonant frequency of the high Q cavity A, the TM i io mode electric field is nearly infinite. Therefore, a reduction in the coupling of power to the T M i io mode is not observed. However, for frequencies in the vicinity o f the resonant frequency, the reduction in the coupling of power to the TM i io mode is due to the unloading of the Q of the resonator. To reduce the effect of the resonant mode, the large gap is enclosed in the low Q cavity A. Figure 3.24 summarizes the computed transmission coefficient (IS2 1 Oof the large gap for two different locations, y c = 12 mm and 5 mm, in the low Q cavity A. Repositioning the circuit from yc = 12 mm to yc = 5 mm results in a 7 dB improvement in IS2 1 I over most of the bandwidth. The current on the two strips at/ = 10.8129 GHz is illustrated in Figure 3.25. The figure shows that when the circuit is located at yc = 12 mm, the standing wave current on the second strip is larger than if the circuit is located at yc = 5 mm. Therefore, when the circuit is located at yc = 5 mm, the standing wave current on the circuit is closer to the ideal current. Figure 3.26 plots p ™ (x) a t / = 10.8129 GHz. As predicted by equations (3.6a), (3.7a) and (3.8a), moving the circuit from y c = 12 mm to yc = 5 mm reduced p ™ (x) and PXM. An exact calculation of P ™ shown in Table 3.4 verifies this. The third column in Table 3.4 is the fraction of the incident power lost to the enclosure, /}oss//^nc, including all fields. 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover -10 n -20 -30 " Yc = 12 mm ’ Yc = 5 mm -40 9 10 11 12 F (G H z) Figure 3.24 Summary of the calculated transmission coefficient of the large gap in the low Q cavity A for two different locations: y c = 12 mm and yc = 5 mm. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q 20 15 10 Yc = 12 mm Yc = 5 mm 5 0 0 10 5 15 x (mm) Figure 3.25 Current, at/ = 10.8129 GHz, on the two strips of the large gap in the low Q cavity A. Low Q 500 r' 300 Yc= 12 mm Yc = 5mm rl 100 - © X gS -100 - -300 -500 0 10 15 x (mm) Figure 3.26 Power lost per unit length to the TM i io mode via the x component o f the electric field at/ = 10.8129 GHz for the large gap in the low Q cavity A. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.4 The ratio o f the T M n o mode powers to the incident power for the large gap. y c (mm) 5 12 p T M /p 1x / 1 me 0.210 0.366 F \oss! Fine 0.229 0.395 3.3.2 Shunt Stub As a second example, again consider the transmission line with a single shunt open circuit stub attached as shown in Figure 3.8. The stub, located at xc = a ll, has a length L = 1.9 mm and is attached to a transmission line, located at yc, of width w = 1.4 mm. The shunt stub is located in cavity A at two different positions (yc = 12 mm and 5 mm). The ideal response for this circuit has a single null in IS2 1 I at 11.1 GHz as shown in Figure 3.9. The dominate currents on the transmission line and on the stub are, respectively, x and y-directed as illustrated in Figure 3.27. Thus they radiate power into the T M i 10 mode via different field components. As a result, moving the circuit will change the power radiated from the stub in a different way than it changes the power radiated from the transmission line. Since the current on the transmission is predominately x-directed and centered a ty c, equation (3.10a) can be written as Fx (^xn ’kym) sm(&.y;j,yc) The current on the stub is predominately y-directed and centered at x c\ thus, equation (3 .10b) can be written as Jy(kxn,kym ) —Fy (kxn, kym) sin(/rj,nxc) Examining Figure 3.27 shows that for the majority the of circuit the current is x-directed; therefore, the dominate mechanism for power being radiated into the T M i 1 0 mode is via 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ideal 25 20 15 10 5 0 0 15 10 5 x (mm) (a) Ideal 30 25 20 15 10 5 0 0.0 0.5 1.0 1.5 2.0 y-yl (mm) (b) Figure 3.27 Ideal currents on the: (a) transmission line and (b) stub. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E ™ . Thus, locating areas of high current along the transmission line in areas o f low E ™ and visa-versa will have a significant effect on the coupling of power to the TM i 10 mode. Inspection of equations (3.6a), (3.7a) and (3.8a) indicates that moving the transmission line from yc = 12 mm to yc = 5 mm will reduce p ™ (x) and P jM. On the other hand, the same shift will only have a small effect on p ™ (y) and P ™ . The net power lost to the T M i io mode, P ™ plus P ™ , decreases as the circuit is moved toward the side wall. Figure 3.28 summarizes the computed transmission coefficient (IS2 1 I) of the shunt stub circuit for two different locations, yc = 12 mm and 5 mm, in the high Q cavity A. The figure shows that when the shunt open circuit stub circuit was located aty c = 12 mm in the high Q cavity A, the resonant mode had a drastic effect on IS2 1 I- Repositioning the circuit from y c = 12 mm to y c = 5 mm significantly reduced the effect of the resonant mode. To gain a better understanding on how the coupling of power to the TM i 1 0 mode effects the operation of the shunt stub, the current on the circuit a t / = 11.1 GHz is illustrated in Figure 3.29. Figure 3.29a shows that when the circuit is located at yc = 12 mm, the standing wave current on the transmission line is significantly different than the ideal current When the circuit is located aty c = 5 mm, the standing wave current on the transmission line is nearly identical to the ideal current. A similar effect is observed for the standing wave current on the stub as illustrated in Figure 3.29b. Figure 3.30 plots p ™ (x), the power lost per unit length along the transmission line to the TM i 1 0 mode via the x component of the electric field, at/ = 11.1 GHz, for the circuit located at y c = 12 mm. The figure shows that the region in the vicinity of x = 2.5 mm is dominated by power being radiated into the T M n o mode and the region in the vicinity of x = 12.5 mm is dominated by power being absorbed from the TM i 1 0 mode. This occurs because a large current standing wave peak is present at that these points and because the x-directed mode field is reasonably large there. The figure also shows that the area under p ™ (x ), P ™ , is approximately zero. p ™ (x ) for the circuit located at yc = 5 mm and p ™ (y ) for both locations are not shown 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q IS21I (dB -10 -20 -30 ‘ Yc = 12 mm • Yc = 5 mm -40 9 11 10 12 F (G H z) Figure 3.28 Summary of the calculated transmission coefficient of the shunt stub in the high Q cavity A for two different locations: yc = 12 mm and yc = 5 mm. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 min Yc = 5 mm 0 5 10 15 x (m m ) (a) High Q Yc = 12 mm Yc = 5 mm 1.0 y -y l (m m ) (b) Figure 3.29 Currents, a t / = 11.1 GHz, on the: (a) transmission line and (b) stub. The circuit is enclosed in the high Q cavity A. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q 250 £ £ Yc = 12 mm 150 o X c H a. -50 -150 -250 0 5 10 15 x (mm) Figure 3.30 Power lost per unit length along the transmission line to the T M i 10 mode via the x component of the electric field a t / = 11.1 G H z . The circuit is enclosed in the high Q cavity A. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. because they are approximately zero. Equations (3.6), (3.7) and (3.8) predict that moving the circuit from y c - 12 mm to yc = 5 mm will reduce the coupling of power to the TM i io mode. Since the circuit is not being operating at the resonant frequency o f the high Q cavity, this reduction is observed in 1S21I and in the standing wave current on the circuit. As in the previous section, a 1.27 mm thick microwave absorbing layer was attached to the cover of the enclosure to reduce the effect of the resonant mode. Figure 3.31 summarizes the computed transmission coefficient (IS2 1 O of the shunt stub circuit for two different locations, y c = 12 mm and 5 mm, in the low Q cavity A. For the circuit located at yc = 12 mm in the low Q enclosure, the effect of the resonant mode was much smaller than in the high Q enclosure. However, the resonant mode of the enclosure still had a large effect on IS2 1 I. The current on the circuit at/ = 11.1 GHz is illustrated in Figure 3.32. The figure shows that when the circuit is located at yc = 12 mm, the standing wave current on the transmission line and the stub is much different than the ideal current. Again, when the circuit is located at yc = 5 mm, the standing wave current on the transmission line and the stub is nearly identical to the ideal current. Figure 3.33a plots 'T'JJf Px (x ) , the power lost per unit length along the transmission line to the TM i 1 0 mode via the x component of the electric field, a t / = 11.1 GHz. The figure shows that the region in the vicinity of x = 2.5 mm is dominated by power being radiated into the T M i 1 0 mode and the region in the vicinity of x = 12.5 mm is dominated by power being absorbed from the T'Uf T M i 1 0 mode. The figure also shows that the area under p x (x) is not zero for the circuit located aty c = 12 mm, indicating net power lost to the T M i 1 0 mode. For the circuit T'ljt located at yc = 5 mm, px (jc) is approximately zero along the transmission line. Figure 'T'JJf ___ 3.33b plots p y (y ), the power lost per unit length along the stub to the TM i 1 0 mode via the y component of the electric field, a t / = 11.1 GHz. The figure shows that for either location of the circuit, power is being radiated into the T M i 10 mode. The figure also shows that the area under p ™ (y) for the circuit located at y c = 12 mm is larger than the 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover -10 as "O -20 -30 - ■ Yc = 12 mm • Yc = 5 mm -40 9 10 11 12 F (G H z) Figure 3.31 Summary of the calculated transmission coefficient of the shunt stub in the low Q cavity A for two different locations: yc = 12 mm and y c = 5 mm. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q 25 20 Yc = 12 mm Yc = 5 mm 15 10 5 0 0 5 10 15 x (mm) (a) Low Q 25 < E 10 Yc= 12 mm Yc = 5 mm 0.0 0.5 1.0 2.0 y-yl (mm) (b) Figure 3.32 Currents, a t / = 11.1 GHz, on the: (a) transmission line and (b) stub. The circuit is enclosed in the low Q cavity A. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q 250 i s £ 150 Yc = 12 mm Yc = 5 mm •250 0 5 15 10 x (mm) (a) Low Q 25 i £ £ 20 «*> © Yc = 12 mm Yc = 5 mm X O 0.0 0.5 1.0 1.5 2.0 y-yl (mm) (b) Figure 3.33 Power lost per unit length to the TM i io mode via the x component of the electric field, at/ = 11.1 GHz, (a) along the transmission line and (b) along the stub. The circuit is enclosed in the low Q cavity A. 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. circuit located at yc = 5 mm. Equations (3.6a), (3.7a) and (3.8a) predict that moving the transmission line from y c = 12 mm to yc = 5 mm will reduce P ™ . An exact calculation of P ™ shown in Table 3.5 verifies this. In addition, the same shift also decreased P ™ . The fifth column in Table 3.5 is the fraction of the incident power lost to the enclosure, P\oss/Pinc ’ including all fields. Note that when the circuit strongly couples to the T M no mode, the total loss is almost equal to P ™ plus P ™ . For weak coupling the total loss is not dominated by the resonant mode loss. Table 3.5 The ratio of the TM i io mode powers to the incident power for the stub. yc (mm) 5 12 p1 x™ lIp. r mc -0.0166 0.436 pl y™ /I 1pme 0.0173 0.0316 p1 ™ II 1pme 0.007 0.468 P \oss! Pine 0.077 0.516 The analysis shows that package resonances can have a very significant effect on circuit operation even at frequencies which are not very close to resonance. These effects, can be reduced by including lossy material in the enclosure. A further reduction in the coupling of power to resonant modes can be obtained by repositioning the circuit Locating areas of high current in areas of low electric field in the enclosure reduces the power lost to these resonant modes. This principle would find most application in moderately sized enclosures where the resonant frequencies are not closely spaced and the mode field structure is relatively simple. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 Conclusion In this chapter the full-wave method of moment (MOM) procedure developed in Chapter 2 was experimentally verified. Two groups o f circuits were fabricated, enclosed in a brass cavity with at least one resonant mode occurring in the operating bandwidth and measured. The effect of adding loss to the enclosure was also examined. The measured results for the above circuits were compared to those obtained with the MOM procedure. Agreement between the measured and calculated results were reasonable for all circuits. After verifying the MOM procedure developed in Chapter 2, a discussion on the coupling of power to resonant modes was presented. It was shown that the coupling of power to resonant modes can be reduced by repositioning the circuit. Locating areas of high current in areas of low electric field in the enclosure reduces the power lost to these resonant modes. This principle would find most application in moderately sized enclosures where the resonant frequencies are not closely spaced and the mode field structure is relatively simple. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 References [1] R.H. Jansen, "Hybrid Mode Analysis of End Effects of Planar Microwave and M illimeter Wave Transmission Lines," Proc. Inst. Elec. Eng., Vol. 128, pt. H. pp. 77-86, April 1981. [2] J. Boukamp and R.H. Jansen, "The High Frequency Behavior of Microstrip Open Ends in Microwave Integrated Circuits Including Energy Leakage," 14th European Microwave Conf.Proc., pp. 142-147, 1984. [3] R.W. Jackson and D.M. Pozar, "Full-Wave Analysis of Microstrip Open-End and Gap Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1036-1042, October 1985. [4] P.B. Katehi and N.G. Alexopoulos, "Frequency-Dependent Characteristics of Microstrip Discontinuities in Millimeter-Wave Integrated Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1029-1035, October 1985. [5] R.H. Jansen, "The Spectral-Domain Approach for Microwave Integrated Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-33, pp. 1043-1056, October 1985. [6] J.C. Rautio and R.F Harrington, "An Electromagnetic Time-Harmonic Analysis of Shielded Microstrip Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT35, pp. 726-730, August 1987. [7] R.H. Jansen, "Modular Source-Type 3D Analysis of Scattering Parameters for General Discontinuities, Components and Coupling Effects in (M)MICs,” 17th European Microwave Conf. Proc., pp. 427-432, 1987. [8] J.R. Mosig, "Arbitrarily Shaped Microstrip Structures and their Analysis with a Mixed Potential Integral Equation," IEEE Trans. Microwave Theory Tech., Vol. MTT-36, pp. 314-323, February 1988. 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [9] N.H.L. Koster and R.H. Jansen, "The Microstrip Step Discontinuity: A Revised Description," IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 213-223, February 1986. [10] W.P. Harokopus and P.B. Katehi, "Characteristics of Microstrip Discontinuities on Multilayer Dielectric Substrates Including Radiation Losses," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 2058-2066, December 1989. [11] L.P. Dunleavy and P.B. Katehi, "A Generalized Method for Analyzing Shielded Thin Microstrip Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 1758-1766, December 1988. [ 12] R.W. Jackson, "Full-Wave, Finite Element Analysis of Irregular Microstrip Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-37, pp. 81-89, January 1989. [13] H.Y. Yang and N.G. Alexopoulos, "A Dynamic Model for Microstrip-Slotline Transition on Related Structures," IEEE Trans. Microwave Theory Tech., Vol. MTT-36, pp. 286-293, February 1988. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 A SIM PLE CIRCUIT M ODEL FOR RESO NANT M ODE COUPLING IN PACKAGED M M IC s 4.1 Introduction A variety of full-wave techniques have been developed to analyze MMIC circuits in an enclosure [1], [2], [3], [4], A full-wave analysis is a numerically rigorous method which includes parasitic coupling to resonant modes, but the cost of this accuracy is increased CPU time and complexity. Consequently a simpler approach is required. Toward that end, Jansen and Wiemer [5] have developed a simple circuit theory model to describe coupling of circuit junctions via an enclosure resonance. Their model appears to be based on the assumption that the resonant mode coupling occurs only at a discontinuity. In contrast, Lewin [6] has shown that the interaction of a circuit with space wave radiation and surface waves can occur at more than a guide wavelength from a discontinuity. Although this phenomena is for a circuit with no cover plate and side walls, it was shown in Chapter 3 that a similar effect occurs for a circuit in an enclosure. These results suggest that resonant mode coupling can be modeled in a way which does not strictly tie it to discontinuities. In this chapter a simple circuit model is developed to accurately describe resonant mode coupling between circuit elements within an MMIC enclosure. This model is easily implemented on commercially available CAD packages and requires several orders of magnitude less CPU time than a full-wave technique. 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Development of the Circuit Model To implement the resonant mode circuit model, a circuit is divided into several segments. In the center of each segment, the primaiy of a coupling transformer is inserted. The secondaries of all the transformers for a particular mode are all connected in shunt with a single shunt RLC tank circuit. In an enclosure with multiple modes, a separate set of transformers and a tank circuit are required for each mode. The turns ratio of each transformer is a function of the corresponding segment's location within the package, its orientation, and the mode being modeled. To illustrate the use of the resonant mode coupling circuit consider a small length of a microstrip transmission line. The transmission line is divided into N segments of length Axi (i = 1,..., N) and a coupling transformer is inserted in the center of each segment (marked by a circle). The resulting circuit is shown in Figure 4.1. Assuming there is only one resonant mode, the one tank circuit shown is the only one needed. The characteristics of the tank circuit are independent of the number of transformers. An important characteristic of this model is that all circuit components R, L, C and n can be determined analytically as described in the next section. 4.2.1 Circuit Mutual Impedance For the circuit model in Figure 4.1, the mutual impedance, , between the z'th and jth current elements of the transmission line can be written as: ’m odel (4.1) where Z fjr is the mutual impedance between the two circuit elements if there were no resonant mode coupling present, n/ is the turns ratio for the z'th coupling transformer and 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Transmission line (-*; ’ yi) H ♦ (•^i+l’^i+l) I » |— A xi - *|* h Axi+1— | (a ) Resonant Mode Coupling Circuit Model resonant circuit A «/2 .. A */2 Aj.j+i/2 coupling transformer #i A„-+1/2 coupling transformer # i+ l (b) Figure 4.1 Schematic of (a) a small length of a microstrip transmission line and (b) the resulting circuit model. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Yr is the admittance o f the RLC tank circuit. The turns ratio,«/, is defined as the ratio of the number o f turns in the primary to the number of turns in the secondary. The resonant frequency, co0, is given by: o>02 = ^ (4.2) Analytic expressions for the turns ratio of the coupling transformers and the R, L, and C components of the tank circuit are determined via a comparison between the rigorous full wave mutual impedance, 4.2.2 and the circuit model mutual impedance, Z™odel. Full-wave Mutual Impedance For the structure illustrated in Figure 4.2, the full-wave mutual impedance between a u directed current element located at 2 *) and a v directed current element located at (X j,y j,zk ) is given by (see Chapter 2): (z " l r ( z ™ l j + (z % l j (43) where m °° N ° ° * , = X X , “2 Q m ( k v l ‘y J J A . k , M j ( k x„,kym-) (4.4) M “ jV“ (z » )„ = X X ^ f X" ^ K e ^ „ , k ym) j ^ k „ , k „ ) j vj(kx„ k ym) Jui^xn'kym) ~ a Jui(x,y)Tu(x ,y )d x d y Si (4.5) (4.6) Qtw = 7 xw (4.7) Yw = Y $ + Y i& l) (4.8) Tx (x,y) = cos(kxnx)sm (kymy) Ty (x,y) = sin(kxnx)cos(kymy) “ [Km Xu =x [*ym if« = X 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11^*1111111111 Figure 4.2 Geometry of the MMIC package used to determine the full-wave mutual impedance between a u directed current element located at (x/,y;,z*) and a v directed current element located at ( x j,y j,z k). 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. uv +1 if u = v -1 if u & v ro.5 k= £k ~ [ 1 .0 o k*0 W = E oxM u ,v = x ,y Yu# and Y ^ l) are given by equations (2.44) and (2.45), respectively. If the current elements in the above equations are assumed to be infinitesimal elements such that, Jui(x,y) = AUI<5(xI )5(yI ), equation (4.6) can be written as: (4.9) J u i ^ k y m ) = A uTu(x i^yi ) Combining equations (4.4) and (4.5) with (4.9), (z ™ ).. and ).. can now be written as: 4 ■ y tifa tU x j.] '] ) ] a t* <4.10) In a typical enclosure housing an MMIC chip, the cover height is low enough that only the TMnmO modes are resonant over the frequency of operation. Therefore, only the terms of the summations in equation (4.10) that correspond to a resonant TMnmO mode will significantly contribute to the mutual impedance. For this reason, only the TMnmO modes will be considered in the modeling procedure developed below. However, this modeling procedure could easily be applied to a TMnml or TEnml ( 1 = 1 , 2 , ...) modes. Near a resonance of the enclosure, the term in the summation (z™).. that corresponds to the resonant mode (TMnmO) is much larger than the remaining terms. Therefore, can be written as: (4.12) 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the resonant part of ( z ™ where (-Z™).. is the non-resonant part of ., ( Z ™ ) f 5, is given by: (z™)** =Ae’al" Qr^kx,,kym^ , lTM,yi)\K,Uxi,yl)} (4.13) An expression can now be written for the full-wave mutual impedance by substituting equation (4.13) into equation (4.3) ( 4 - 14> where the non-iesonant part o f ( z ^ j .. is given by (zn?-(isr£+{i2\ and the resonant part of { z ™ <4.i5) is given by: y( z ^ r =(z™f5 (4.16) In Chapter 2 it was noted that there are an infinite number of poles and zeros of Qtm which alternate along the co axis. A function that exhibits these characteristics can be approximated by a rational fraction of the form [7]: n _ A m where 0 ^ 0 )3 , ( 6 ) 2 - CO!2 ) ( 6 ) 2 - - 6/ n)32? )\ .-.(. 6( <v>2 ? 2 -_ t/'ii? a L . !. )'I 77tf ^ ^ 2 _ m | ) ( © 2 _ G)2y_^(02 _ and 0 , 6 )2 >®4 >'” >®2 n- 2 ^ (4.17) die complex zeros and poles of Qtm -> respectively. For frequencies in the vicinity of the pole that corresponds to the TMnmO resonant mode, cor, QTM can be approximated by the following [7] n ~j2C0 Tm (co2 -c o l -jlcocofj 0 3 II 3 I dco ) (4.18) 1 where the enclosure is assumed to have a small to moderate amount of loss and cor = co'r + jco" =co0 + jco" (4.19) It should be emphasized that when the enclosure contains loss, the resonant frequency becomes complex. The solution for cor will be discussed shortly. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2.3 Comparison of the Mutual Impedances In the limit as A ui becomes small, equations (4.1) and (4.14) are approximately equal. By comparing like terms in each equation one may write: ( 7 f w \ n r _ 7 cir V m hj * (4.20) (4.21) where n; and Yr can be identified as: n‘= ^ ^ r [AMXi’yi)] 4 £„£m kxn [Axi cos(kxnXi) sin(fcyTOy,-)] ab k„ for an x directed current sin(/:x„xi) cos(kymyi)j for an y directed current (4.22) W 2) = Q™ {[0)) 0) = 1 +^ 2 . R J 0 )K (4.23) Substituting equation (4.18) into the right side of equation (4.23) and solving fori? and C yields: i ( 2 vM ..\ 2 dco R= (4.24) 0)=0)Q 1 2co"C (4.25) The inductance, L, of the tank circuit is given by equation (4.2). Calculating R with equation (4.25) requires that the imaginary part of the complex resonant frequency, co", be determined. An alternative expression for R may be derived by equating equation (4.7) and the left side o f equation (4.23) with co = C0o. Solving for R yields R = Qtm (°>o) (4-26) In summary equations (4.22), (4.24) and (4.26) determine n/, C and R in terms of the circuit location and the electrical characteristics of the enclosure. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Numerical Evaluation of Circuit Components The initial step in the numerical evaluation of the circuit components is to determine the resonant frequencies of the enclosure. The zeros of (equation 4.8) correspond to the resonant frequencies of the cavity. For a lossless cavity, YM is purely imaginary and av is purely real. The zeros of YM can be located by searching for the frequency at which Im (I^ ) = 0. If loss is present in the cavity, YM is complex and the resonant frequency Ct>r is also complex. The zeros of YM are located by searching for a complex co where the real and imaginary parts of YM simultaneously equal zero. Searching for the real co, C0o , where Im(yM) = 0 may result in a solution which is not an actual zero of YM (i.e. R e (J ^ ) ^ 0 ). However, for cavities with small to moderate loss, the real part of (Or can be determined with good accuracy from co0 if co0 satisfies both of the following conditions (see Chapter 2): (4.27) (4.28) After finding the real frequency, C0q, under these conditions, the components of the RLC tank circuit can be determined by equations (4.2), (4.24) and (4.26). For example, consider a cavity of the following dimensions: a = 15 mm, b = 24 mm and c = 12.7 mm. The substrate thickness is d \ = 1.27 mm and the relative permittivity is £r i = 10.5(1/0.0023). In the band 9-12 GHz, there is one real frequency for which Im(TM) = 0. The resonant frequency (f0 ) and Q of this enclosure have been determined to be 10.8129 GHz and 10^, respectively. Substituting f 0 into equations (4.2), (4.24) and (4.26) yields R = 0.1 M fi, C = 0.615 pF and L = 0.352 nH. Solving for the complex resonant frequency,//-, yields 10.8129/0.00129 GHz. Comparing the real resonant frequency, f 0, to the real part of the complex resonant frequency,//-, shows that excellent agreement is obtained for high Q enclosures. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Next, a dielectric layer of thickness ch, = 0.381 mm and relative permittivity of er3 = 11.9(1-7*25) is attached to the cover of the enclosure. The resonant frequency (f0 ) and Q of this enclosure have been determined to be 10.7417 GHz and 123, respectively. Substituting f 0 into equations (4.2), (4.24) and (4.26) yields R = 2.786 KQ, C = 0.655 pF and L = 0.335 nH. Solving for the complex resonant frequency, f r, in the conventional manner yields 10.738 l-y‘0.0439 GHz. Comparing the real resonant frequency, f Q, to the real part of the complex resonant frequency, f r, shows that good agreement is obtained for Q's on the order of 100. Lastly, the dielectric layer attached to the cover of the enclosure is replaced with a microwave absorbing layer of thickness ds = 1.27 mm, relative permittivity of £r 3 = 60(1y0.12) and relative permeability of M-/-3 = 7.3(1-j0.3). In the band 9-12 GHz, there is one real frequency for which Im (J^ ) = 0. The resonant frequency (fQ ) and Q of this enclosure have been determined to be 11.0467 GHz and 19, respectively. Substituting/0 into equations (4.2), (4.24) and (4.26) yields R = 443 Q , C = 0.383 pF and L = 0.542 nH. Solving for the complex resonant frequency, f r, in the conventional manner yields 10.8883-y0.2849. Comparing the real resonant frequency, f 0, to the real part of the complex resonant frequency, f r, shows that they differ by about 1.4%. Since finding the real resonant frequency is much easier than finding the complex resonant frequency and the error in obtaining the resonant frequency in this manner is not too large, this method will be employed for low Q enclosures also. After determining the tank circuit components for each resonant mode of the enclosure, the circuit is divided into segments of length Aw-. In the center of each segment, the primary of a coupling transformer is inserted. The turns ratio,«/, for a coupling transformer located at (x/,y/) is given by equation (4.14). The secondaries of all the transformers for a particular mode are connected in shunt to a RLC tank circuit. For 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. optimal results, a sufficient number of segments should be used so that the length of each segment, AH-, is less than a quarter of a guided wavelength. 4.4 Results To verify the accuracy of the proposed circuit model, the following circuits are analyzed: 1. Small gap in a transmission line 2. Shunt open circuit stub 3. Stub in a larger enclosure 4 . Coupled line bandpass filter The scattering parameters for each circuit are determined using the proposed circuit model and compared to the results using the conventional MOM outlined in Chapters 2 and 3. 4.4.1 Small Gap To examine the validity of using the circuit model to simulate a circuit operating in the millimeter-wave band, consider a transmission line with a gap in the center. The transmission line, located at y c = 1-55 mm, has a width of w = 0.1 mm and a gap of g = 0.1 mm. The circuit is enclosed in a cavity of the following dimensions: a = 3.2 mm, b = 3.1 mm and c - 0.6 mm. The substrate thickness is d \ = 0.1 mm and the relative permittivity is er l = 12.9(l-j0.0016). An enclosure of this size has only one resonant mode, the T M n o (61.471 GHz), in the band 50-70 GHz. The lumped elements R, C and L were found to be 0.163 M£2, 0.333 pF and 0.0201 nH. 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The circuit is divided into 8 segments as illustrated in Figure 4.3. Eight transformers are inserted with turns ratios listed in Table 4.1. The gap discontinuity itself is modeled using the GAP element in a typical CAD program. Table 4.1 Transformer turns ratios for the transmission lines on both sides of the gap. 1 1 2 3 4 5 6 7 8 xi (mm) 0.000 0.440 0.880 1.325 1.875 2.320 2.760 3.200 yi (mm) 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 A xi (mm) 0.22 0.44 0.44 0.45 0.45 0.44 0.44 0.22 ni (TMno) 9.72x10-2 1.90x10-1 1.37x10-1 1.53x10-2 -1.53x10-^ -1.37x10-1 -1.90x10-1 -9.72x10-2 Figure 4.4 compares the predicted transmission response of the small gap in the high Q package using the circuit model versus the conventional MOM. The conventional MOM is performed using 70 expansion functions. Over the entire bandwidth, the difference between the circuit model and the conventional MOM is less than 3 dB. The circuit model very accurately predicts the resonant mode coupling at the package resonance. This example showed the use of the circuit model for a simple circuit enclosed in a cavity with one resonant mode. In addition, this example also showed that the circuit model can be used for circuits operating in the millimeter-wave band. 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T -H w Figure 4.3 A xi h - Geometry of a transmission line with a gap in the center. The transmission line, located a ty c = 1.55 mm (b/2), has a width of w = 0.1 mm and a gap o f g = 0.1 mm. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gap - High Q - MOM ‘ Model -10 -20 -30 r* CO -40 -50 -60 50 55 60 65 70 F (G H z) Figure 4.4 Comparison of the predicted transmission response o f the small gap in the high Q package using the circuit model versus the conventional MOM. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.2 Shunt Stub As a second example, the circuit model will be used to simulate a transmission line with a single shunt open circuit stub attached. A shunt stub is chosen because it is a typical building block used in more complicated circuits. The stub, located at x c = 7.5 mm (a/2), has a length L = 1.9 mm and is attached to a transmission, located at yc = 12 mm (b/2), of width w = 1.4 mm. The stub is enclosed in a cavity of the following dimensions: a = 15 mm, b - 24 mm and c = 12.7 mm. The substrate thickness is d \ = 1.27 mm and the relative permittivity is £r l = 10.5(l-j0.0023). An enclosure of this size has only one resonant mode, the T M n o (10.8129 GHz), in the band 9-12 GHz. The lumped elements R, C and 7. were found to be 0.1 M£2, 0.615 pF and 0.352 nH. The transmission line on either side o f the stub is divided into 4 segments with 4 transformers and the stub is divided into 2 segments with 2 transformers as illustrated in Figure 4.5. The turns ratio of the transformers for the transmission line are listed in Table 4.2 and the stub in Table 4.3. The Tee-junction itself is modeled using the Tee element in a typical CAD program. Table 4.2 Transformer turns ratios for the transmission lines on both sides of the stub. 1 1 2 3 4 5 6 / 8 xi (mm) 0.00 2.44 4.88 6.80 8.20 10.12 12.56 15.00 yi (mm) 12 12 12 12 12 12 12 12 A xi (mm) 1.22 2.44 2.44 1.40 1.40 2.44 2.44 1.22 ni CIMiio) l.lO x lO -i 1.90x10-1 1.11x10-1 1.83x10-4 -1.83x10-2 -1.11x10-1 -1.90x10-1 -1.10x10-1 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i » '0111^ iT * -H Figure 4.5 a *; i T h - w Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located at xc (a/2), has a length L = 1.9 mm and is attached to a transmission line, located a ty c (b/2), of width w = 1.4 mm. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.3 Transformer turns ratios for the stub. 1 9 10 xi (mm) 7.5 7.5 yi (mm) 12.7 14.0 A xi (mm) 0.70 1.55 ni (TMno) -3.58x10-3 -1.32x10-3 Figure 4.6 compares the predicted transmission response of the stub in the high Q package using the circuit model versus the conventional MOM. The conventional MOM is performed using 123 expansion functions. Below 11.5 GHz, the agreement between the circuit model and the conventional MOM is good. The difference between the circuit model and the conventional MOM is no more than 3 dB. Above 11.5 GHz, the agreement between the two methods deteriorates. A major source of the discrepancy between the circuit model and the conventional MOM may be due to the coaxial to microstrip transition. The MOM solution employs special basis functions to better simulate the coaxial to microstrip transition, whereas the circuit model does n o t The ideal response for this circuit with no enclosure has a single null in IS2 1 I at 11.1 GHz. However, the resonant mode of the enclosure has a drastic effect on the circuit's performance and the circuit model does in fact simulate this effect qualitatively. To reduce the effect of the resonant mode, a microwave absorbing layer is attached to the cover of the enclosure. The thickness of this layer is d$ = 1.27 mm, the relative permittivity is e r 3 = 60(1-/). 12) and the relative permeability is M7 3 = 7.3(l-/).3). This cavity was also analyzed in the previous section and was found to have one resonant mode, the T M n o (11.13 GHz). The lumped elements R, C and L were found to be 443 £2,0.383 pF and 0.542 nH. 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm SQ ■S3 CO MOM Model -40 9 10 11 12 F (GH z) Figure 4.6 Comparison of the predicted transmission response of the stub in the high Q package using the circuit model versus the conventional MOM. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.7 compares the predicted transmission response of the stub in the low Q enclosure using the proposed circuit model versus the conventional MOM. Below 11.5 GHz, the agreement between the circuit model and the conventional MOM is good. The difference between the circuit model and the conventional MOM is no more than 3 dB. Above 11.5 GHz, the agreement between the two methods deteriorates. Again a major source of the discrepancy between the circuit model and the conventional MOM may be due to the coaxial to microstrip transition. This example showed the use of the circuit model for a typical circuit enclosed in a cavity with one resonant mode. The circuit model was very easy to implement on a commercially available CAD package and gives reasonably good agreement with a more rigorous analysis even in cases where a low Q resonance is involved. 4.4.3 Shunt Stub in a Larger Box Next, the circuit model is used to simulate the shunt stub enclosed in a cavity with several resonant modes. The shunt stub, of the previous example, is enclosed in a cavity with the following dimensions: a = 30 mm, b = 48 mm and c = 10.0 mm. An enclosure of this size has 5 resonant modes in the 9 to 13 GHz bandwidth. The resonant frequencies, Q's and the lumped elements for each mode are listed in Table 4.4. In this new enclosure, the stub is located atx c = 15 mm (a/2) and the transmission line is located a ty c = 24 mm (b/2). The transmission line on either side of the stub is divided into 8 segments and the stub is divided into 2 segments. Since the transmission line is located aty c = 24 mm (b/2) the turns ratios for the TM220 and TM 1 4 0 transformers are zero. Therefore, the transmission line on either side of the stub requires only 24 transformers instead of 40 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 12 mm -10 /—N aa TJ S21 -20 — v -30 MOM Model ^0 9 10 11 12 F (G H z) Figure 4.7 Comparison o f the predicted transmission response of the stub in the low Q enclosure using the proposed circuit model versus the conventional MOM. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.4 The lumped elements of the resonant circuits for each mode in the high Q package. Mode TM130 TM140 TM210 TM220 TM23Q fo (GHz) Q i?(M O ) C (pF ) L (nH) 9.7043 11.9517 9.5793 10.6661 12.1289 5.48x103 2.63x103 5.69x103 4.05x103 2.48x103 0.105 0.092 0.106 0.100 0.091 0.853 0.380 0.892 0.604 0.357 0.315 0.467 0.310 0.369 0.483 transformers. Since the stub is located at xc = 15 mm (a/2) the turns ratios for the TM 2 1 0 . TM 2 2 O and TM 2 3 0 transformers are zero. Therefore, the stub requires only 4 transformers instead o f 10 transformers. The turns ratio of the transformers for the transmission line are listed in Table 4.5 and the stub in Table 4.6. The Tee-junction itself is modeled using the Tee element in a typical CAD program. Figure 4.8 compares the predicted transmission response of the stub in the high Q package using the circuit model versus the conventional MOM. The conventional MOM is performed using 179 expansion functions. Agreement between the circuit model and the conventional MOM is good except in the vicinity o f 12.2 GHz. To reduce the effect of the resonant modes, a 16 mil (0.4064 mm) thick silicon layer with a resistivity of 1.45 Q-cm is attached to the cover of the enclosure. The resonant frequencies, Q's and the lumped elements of the resonant circuits for each mode are listed in Table 4.7. Figure 4.9 compares the predicted transmission response of the stub in the low Q package using the circuit model versus the conventional MOM. Agreement between the circuit model and the conventional MOM is good. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Si cover Yc = 24 mm -10 -20 * MOM Model -30 ^0 9 10 11 12 13 F (G H z) Figure 4.8 Comparison of the predicted transmission response of the stub in the larger high Q package using the circuit model versus the conventional MOM. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 24 mm -10 sa IS21I T3 -20 MOM Model -30 -40 9 10 11 12 13 F (G H z) Figure 4.9 Comparison of the predicted transmission response of the stub in the larger low Q package using the circuit model versus the conventional MOM. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.5 Transformer turns ratios for the transmission lines on both sides of the stub. 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 xi (mm) 0.0 2.2 4.4 6.6 8.8 11.0 12.85 14.3 15.7 17.15 19.0 21.2 23.4 25.6 27.8 30.0 yi (mm) 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 A xi (mm) 1.1 2.2 2.2 2.2 2.2 2.2 1.5 1.4 1.4 1.5 2.2 2.2 2.2 2.2 2.2 1.1 ni (TM130) -2.73x10-2 -5.31x10-2 -4.89x10-2 -4.20x10-2 -3.30x10-2 -2.22x10-2 -8.31x10-3 -2.54x10-3 2.54x10-3 8.31x10-3 2.22x10-2 3.30x10-2 4.20x10-2 4.89x10-2 5.31x10-2 2.73x10-2 ni (TM210 ) 5.53x10-2 9.91x10-2 6.69x10-2 2.07x10-2 -2.98x10-2 -7.41x10-2 -6.79x10-2 -6.97x10-2 -6.97x10-2 -6.79x10-2 -7.41x10-2 -2.98x10-2 2.07x10-2 6.69x10-2 9.91x10-2 5.53x10-2 ni (TM230 ) -4.23x10-2 -7.58x10-2 -5.11x10-2 -1.59x10-2 2.27x10-2 5.66x10-2 5.19x10-2 5.33x10-2 5.33x10-2 5.19x10-2 5.66x10-2 2.27x10-2 -1.59x10-2 -5.11x10-2 -7.58x10-2 -4.23x10-2 Table 4.6 Transformer turns ratios for the stub. 1 17 18 xi (mm) 15.0 15.0 yi (mm) 24.7 26.0 ni CIM130) 8.92x10-3 2.14x10-2 A xi (mm) 1.4 1.2 ni CIM140) 6.74x10-2 5.09x10-2 Table 4.7 The lumped elements of the resonant circuits for each mode in the low Q package. Mode TM130 TM140 TM210 TM220 TM230 fo (GHz) 9.5419 11.8119 9.4191 10.5051 11.9960 R (k£2) 1.914 4.072 1.953 2.719 4.884 Q 109 127 115 121 145 C (pF) 0.952 0.421 0.992 0.671 0.393 L (nH) 0.292 0.481 0.288 0.341 0.448 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This example showed the use of the circuit model for a circuit enclosed in a cavity with several resonant modes. Since there are several resonant modes, a large number of transformers are needed to model this circuit. Thus, the circuit model was very tedious to implement on a commercially available CAD package. 4.4.4 Band Pass Filter To examine the use of the circuit model to simulate a circuit containing coupled lines, consider a two resonator coupled line bandpass filter. The width and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively. The spacing of the resonators are 5] = 0.13 mm and S2 = 0.64 mm. The circuit is enclosed in a cavity of the following dimensions: a = 24 mm, b = 15 mm and c = 6.35 mm. The substrate thickness is d \ = 0.635 mm and the relative permittivity is £r l = 10.5(l-y‘0.0023). The resonant frequencies, Q's and the lumped elements of the resonant circuits for each mode are listed in Table 4.8. Table 4.8 The lumped elements of the resonant circuits for each mode in the high Q package. Mode TMno TM210 So (GHz) 11.1654 15.0319 Q 2.01x104 1.22x104 R (MQ) 0.115 0.116 C (pF ) 2.499 1.111 L (nH) 0.081 0.101 The circuit is divided into 14 segments with 28 transformers as illustrated in Figure 4.10. The turns ratio of the transformers for the bandpass filter are listed in Table 4.9. The coupled line sections are modeled using the CPL element in a typical CAD program. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T i L a ~ r XI W Figure 4.10 Geometry of a two resonator coupled line bandpass filter. The width and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively. The spacing of the resonators are s i = 0.13 mm and S2 = 0.64 mm. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.11 compares the predicted transmission response of the bandpass filter in the high Q package using the circuit model versus the conventional MOM. The conventional MOM is performed using 104 expansion functions. The response of the bandpass filter predicted by the circuit model and the conventional MOM have similar shapes, qualitatively speaking. Between 7 and 15 GHz, the difference is no more than 10 dB; however, a significant discrepancy exists at the lower and higher frequencies. The discrepancy may be attributed to the coupled line circuit model in the CAD program. Dunleavy [8] has also observed a discrepancy between a full wave method and CAD packages when modeling a bandpass filter in an enclosure. Table 4.9 Transformer turns ratios for the bandpass filter. 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x[ (mm) yi (mm) 0.0 2.0 4.5 7.0 9.5 9.5 12.0 12.0 14.5 14.5 17.0 19.5 22.0 24.0 6.09 6.09 6.09 6.09 6.09 6.86 6.86 8.14 8.14 8.91 8.91 8.91 8.91 8.91 A xi (mm) 1.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 1.5 ni (TMno) 4.01x10-4 1.29x10-1 1.11x10-1 8.13x10-4 4.30x10-4 4.45x10-2 0 0 -4.45x10-2 -4.3x10-4 -8.13x10-4 -i.iixio-i -1.29x10-1 -4.01x10-4 ni (TM210 ) 5.91x10-4 1.70x10-1 7.53x10-4 -5.1x10-2 -1.56x10-1 -1.62x10-1 -2.04x10-1 -2.04x10-1 -1.62x10-1 -1.56x10-2 -5.1x10-4 7.53x10-2 1.70x10-4 5.91x10-4 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q -10 -30 -40 MOM Model -50 -60 6 8 10 12 14 16 F (G H z) Figure 4.11 Comparison of the predicted transmission response of the bandpass filter in the high Q package using the circuit model versus die conventional MOM. 149 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. This example showed the use of the circuit model for a more complicated microwave circuit The circuit was enclosed in a cavity with two resonant modes, one occurring in the passband and the other in the stopband. 4.5 Conclusion A circuit model has been developed that models resonant mode coupling. It has good accuracy for enclosures with a Q of over 100 and is useful for Q's as low as 20. For lower Q's, the full wave spectral summation, which serves as the starting point for this model, no longer has a single spectral term which produces a dominant resonance. The circuit model can be used on commercially available software packages. Simple analytical expressions for the entire model are easily evaluated. This is a very attractive feature of the circuit model for implementation into a CAD package. In addition, it requires several orders of magnitude less CPU time than the MOM. However, the circuit model for an MMIC circuit in a large enclosure (one with more than a few resonances) may require a large number of transformers. The large number of transformers makes the implementation of the circuit model into a CAD package very tedious. Consequently, this circuit model may be best suited for MMIC circuits in moderately sized enclosures. A potential solution to this inconvenience is an automated procedure for inputting the transformers. To test the feasibility of such an automated procedure, a simple FORTRAN program was written to calculate the response of a circuit in an enclosure. The program uses discontinuity models {e.g. a TEE element) from a commercially available CAD package which, if needed, are read into the program as data files in the form of sparameters. The program automatically determines the number of resonances, thus the number of transformers per location, the characteristics of each transformer and the 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. components o f each resonant circuit Using the automated procedure significantly reduced the time and complexity o f entering the model into a CAD package; which allows for the utilization o f the circuit model for any size enclosure. 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 References [1] Rautio, J.C. and Harrington, R.F., "An electromagnetic time-harmonic analysis of shielded microstrip circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-35, pp. 726-730, August 1987. [2] Dunleavy, L.P. and Katehi, P.B., "A generalized method for analyzing shielded thin microstrip discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT37, pp. 1758-1766, December 1988. [3] Hill, A. and Tripathi, V.K., "An efficient algorithm for three-dimensional analysis of passive microstrip components and discontinuities for microwave and millimeterwave integrated circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-39, pp. 83-91, January 1991. [4] Jansen, R.H. and Sauer, J., "High-speed 3D electromagnetic simulation for MIC/MMIC cad using the spectral operator expansion (soe) technique," 1991 IEEE MTT-S Digest, pp. 1087-1090, 1991. [5] R.H. Jansen and L. Wiemer, "Full-wave theory based development of mm-wave circuit models for microstrip open end, gap, step, bend and tee," IEEE MTT-S Int. Microwave Symp. Dig., pp. 779-782, June 1989. [6] L. Lewin, "Spurious radiation from microstrip," Proc. IEE, Vol. 125, No. 7, pp. 633-642, July 1978. [7] Beringer, R. "Resonant cavities as microwave circuit elements". In Montgomery, C.G., Dicke, R.H., and Purcell, E.M. (eds) Principles of microwave circuits. MIT Radiation Laboratory Series, Vol. 8., McGraw-Hill Book Company, Inc., New York, pp. 207-239, 1948. [8] Dunleavy, L.P., "Discontinuity characterization in shielded microstrip: a theoretical and experimental study," Ph.D. Thesis, University of Michigan, 1988. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 ANALYSIS O F M M ICs IN RESONANT ENCLOSURES W ITH THE DIAKOPTIC M ETHOD 5.1 Introduction In the procedure developed in Chapters 2 and 3, circuits are divided into small cells. The current is expanded over each cell using a set of known basis functions. The unknown current coefficients are then determined by the method of moments (MOM). For a small circuit, such as a Tee-junction, the number of unknowns is on the order of a few hundred. With larger circuits, the number of unknowns is on the order of a few thousand and for an entire MMIC chip, the number of unknowns can exceed tens of thousands. The majority of analysis time using the MOM can attributed to two sources: the time required to fill the matrix (fill time) and the time required to solve the linear system of equations (solve time). The solve time represents a large portion of the analysis time when the number of unknowns is very large. A simpler approach that reduces the solve time will be developed in this chapter. In order to reduce the solve time, Goubau et al [1] developed the diakoptic method. The diakoptic method and the modified diakoptic method [2] were originally developed to analyze complex multi-element antennas. Until very recently [3] the diakoptic method had not been applied to MMIC problems and even now has not been reported in sufficient detail for an assessment of it to be made. In what follows, the diakoptic method is used to analyze MMICs in resonant enclosures. Results obtained using the diakoptic method agree well with those obtained using the MOM (Chapters 2 and 3) for a few simple circuits. However, a significant discrepancy exists between the two methods for more complicated circuits. A new filtering technique is presented which significantly reduces this discrepancy. 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 Diakoptic Method 5.2.1 Theory In this section, the diakoptic method is reviewed. Due to the similarity of the diakoptic and modified diakoptic methods, only the modified diakoptic theory will be discussed. To avoid confusion with the new method developed later in this chapter, we will refer to the modified diakoptic method simply as the diakoptic method. The first step in the diakoptic theory is to divide (diakopt) a circuit into smaller elements. However, the size of each element is not nearly as small as the cells used in the conventional MOM. As a result of diakopting a circuit, artificial ports are introduced. At the terminals of each port, a voltage and current can be defined. For example, consider the center fed dipole shown in Figure 5.1a. The port for the original structure is designated as port i. The dipole is divided or diakopted into 4 elements as shown in Figure 5.1b. The diakopted circuit has 3 ports. Port 2 is the port for the original dipole (port i), while the other 2 ports are artificially introduced. The two elements meeting at the Ath port will be referred to as structure k. For example, structure 2 is made up of elements B and C which meet at port 2. The second step in the diakoptic method is the characterization of the diakopted circuit by an impedance matrix. If the diakopted circuit has N DK ports, the relationship between the port currents and the port voltages is given by: (5.1) /=i where V[DK and l f K are the voltage and current at the /th port and /i=o The elements of the impedance matrix are determined by [4], [5]: 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Port i (a) D B Port 1 Port 2 Port 3 (b) Figure 5.1 Schematic of (a) center fed dipole and (b) the resulting diakopted circuit. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z kiK = - J J [ ^ ( ^ , y ) ] r • JkDK(x ,y )d x d y (5.3) sk Jk is the normalized current distribution on the entire circuit due to an excitation at port k with the other ports open circuited. E ? K is the electric field generated by the normalized current distribution, J p K, and can be represented by: E ? K(x,y) = £ ' t ^ !Ln x , y ^ Q i k xn,kym) ■J ? K(kxn,kym)\ (5.4) m=0n=0 where JlDK(kxn,kym) = j \ T ( x , y ) J lDK(x ,y )d x d y T(x,y) = Sl c o ^ k xnx)sm (kymy ) 0 0 s in ( ^ nx)cos(/:>TO>-) Q(kxn,kym) = X xQ xx& xn ’kym ) ^Q xy^xn ^yJ yxQyx(kxrl ’kym ) yyQyy(kxn,kym) j The elements of the impedance matrix, calculated by equation (5.3), are stationary about the exact current; therefore, a first-order error in the current will produce only a second-order error in Z ^ K . The excitation of each element is generated by two different sources. The first source is the current entering the element at its terminals and is referred to as current coupling. The second source of excitation is called field coupling. Field coupling is generated by the currents on all the other elements. If the length of structure k is below resonance, current coupling is more dominant than field coupling; therefore, the current on the two elements of structure k is much larger than on all the other elements. Since current coupling dominates field coupling and the expression for Zk[ is variational, the current distribution on the structure being excited can be approximated by assuming the remaining elements are not present In Figure 5.1 for example, J®K is approximated by exciting elements A and B at port 1 and assuming the currents on C and D are zero. After approximately determining the normalized current distributions, JtDK (1=1, 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N d k ), under isolated-structure conditions, the diakoptic impedance matrix is determined by equation (5.3). Equation (5.1) is solved for the port currents, l f K, assuming all the port voltages, VjPK, are zero except the actual driven port, port 2 in Figure 5.1. 5.2.2 The Relationship Between the Diakoptic and Moment Methods The use of the diakoptic method to analyze a circuit can be thought of as a double application of the moment method [5], [6]. In the first application of the moment method, the current on the &th structure, , is expanded as follows: = x X 4 f ^ )U ,y ) + y i=i j =i W ) (5.5) where the number of x-directed and y-directed expansion function on the kth structure are and , respectively. Since the size of a "structure" is much smaller than the entire circuit, the number of expansion functions needed is small. Following Galerkin's method, the electric field integral equation (EFTE) is tested with and which results in a set of algebraic equations for the unknown current coefficients, such that: y(A’) = Z (k)j(.k) where the elements of the excitation vector are given by: n-'i rr (V v V = j E ^ , y - > J pi(X,y^ { 0 at a port and a typical element of the impedance matrix is given by: M °° N ~ (4 * 1 = J I I m = 0n= 0 a ^ Q Pq^ k ym) j f { k m ,kym) J ^ \ k m ,kym) (5.7) aD p ,q = x o r y 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. After determining the unknown current coefficients, and i f f , a new expansion function or "super mode", Jj?K, is formed from equation (5.5). This super mode can be represented as: 3t“ W Mik) Myk) i=l j =1 = =7 lk (5.8) T)K where lj? is the port current and /(* ) (5.9a) lk /(*) < ’ =7f e lk (5.9b) This procedure is repeated until a super mode is found for each structure. For a diakopted circuit with N DK ports, N DK super modes are required. In the second application of the moment method, the surface current on the entire circuit is expanded into a set of super modes as follows: n dk J s(x ,y )= J , l F Kj P Kix ,y ) z=i (5.10) Following Galerkin's method, the electric field integral equation (EFIE) is tested with jj?K which results in a set of algebraic equations for the unknown current coefficients: WD K = Z DKI DK (5n) with the elements of Z DK determined by substituting equation (5.4) and equation (5.3): oo ©o & =I (5.12) m = 0n=0 a° where the superscript T refers to the transpose of a matrix and M[k) (5-(5) i= l y=l The elements of the excitation vector are given by: sk 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The expression for Z $ K can be simplified by substituting equation (5.13) into (5.12): M °° N °° . z™= X 1 = ab^n^m‘ { ^ k\ kxn^m)[Qxxikxn^ yJF^l\k ^ ,k yJ + Q ^{k^kym) l f \ k xn,kym)\ (5.14) + F ^ ' \ k xn , k y m '^Q yX(Jcxn,k y m ')F^ ^(k xn >k ym ) Q yy(k x n ’k ym ) ^(^OT’^ym)]j' where M(xk) <5-15a> i= l M(yk) Fik\ k m ,k ,m) = j , w V j « j \ k m ,kym) (5.15b) 1=1 Substituting equation (5.15) into (5.14), Z $ K can also be written as: \f(k) fij(l) zSK= 1=1 j =1 f ^ k ) ftf(l) ± i u +x x i= l J 7=1 7 xx 1=1 7=1 J +x x (5.16) i= l 7=1 where (z"0M) = 7 7Vf~ at~ X J d- ^ ^ Q pq( k „ ,k ym) j f { k „ , k ym) j " h k xn,k ,m) m = 0n= 0 (5.17) ab p , q = x o ry Equations (5.14) and (5.16) represent two different ways of calculating the diakoptic impedance matrix. The disadvantage of using equation (5.14) is that the series can not be summed using the technique developed in Chapter 2, since the coefficients W^P and W^P are frequency dependent. The series is equation (5.16) can be summed using the technique developed in Chapter 2. However, equation (5.14) will probably be more efficient in terms of the number of operations needed to fill the diakoptic impedance matrix and thus lead to lower CPU times if a technique can be applied to enhance the convergence of the series in equation (5.14). Recently, S. Singh and R. Singh [7] have used Shank's 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transform to enhance the convergence of a free-space periodic Green's function. Application of Shank’s transform to summations in equation (5.14) should make it more attractive to use. For this research, however, equation (5.16) is used to calculate the diakoptic impedance matrix. As described previously, the diakoptic method above is a two step moment method process. In the first step, N DK super modes are formulated for the circuit. The number of operations that are required to form all N DK super modes is on the order o f l/6 (N DKM ^,), where M ^ is the number of expansion functions that make up the kth super mode. One operation is defined as one multiplication and one addition. For simplicity it is assumed that = M ^, for each of the pieces of the diakopted circuit. In the second step, the super modes are used to expand the current on the entire circuit The total number of operations required using the diakoptic method is approximately l/6 ^N DKMly + operations; whereas the conventional MOM requires l/6 (A ^ 0Af) operations, where N MOM represents the number of unknowns. For example, consider a circuit that requires 1000 subsections when using the conventional MOM. Assume that the circuit is diakopted into 100 structures with each structure consisting of 20 subsections. Solving the linear system using the conventional MOM requires approximately 1.7x10^ operations, whereas the diakoptic method requires 3.3x10^ operations. Using the diakoptic method can result in the dramatic reduction of analysis time. The example given above serves as a rough comparison between the diakoptic method and the conventional MOM. Numerous factors will effect runtimes such as the number of diakoptic ports or the number of subsections used to expand the current on a diakoptic structure. In order to determine the number of operations required for the solution of the linear system of equations, it was assumed that a direct solver such as Gauss's elimination was used. However, for large matrices, an iterative solver (e.g. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. conjugate gradient method) would likely be used; this may reduce the number of operations required. 5.2.3 Results The diakoptic method described above is used to analyze three microstrip circuits. The first two circuits are a half-wave dipole on a GaAs substrate and a quartz substrate, respectively. The third circuit is a single shunt open circuit stub attached midway between the input and output connectors. 5.2.3.1 Dipole on GaAs As a first example, consider a center fed dipole o f length, L = 1.55 mm and width of, w = 0.2 mm. The dipole is located at xc - 7.0 mm and yc = 2.85 mm in a cavity (Figure 5.2) of the following dimensions: a = 10 mm, b = 6 mm and c = 0.5 mm. The substrate thickness is d \ = 0.1 mm and the relative permittivity is er l = 12.9(l-j0.0016). An enclosure of this size has three resonant modes in the band 25-45 GHz. The dipole is diakopted into 4 elements with 3 ports as illustrated in Figure 5.1. Figure 5.3 compares the computed input reactance of the dipole in the high Q package using the diakoptic method versus the input reactance computed using the conventional MOM. The conventional MOM is performed using 11 expansion functions while the diakoptic method uses 3 super modes. The results obtained using the diakoptic method are offset from the conventional MOM by no more than 5 £2 over the whole bandwidth. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y t a k — L x Figure 5.2 Geometry of a center fed dipole located at x c = 7.0 mm and y c = 2.85 mm. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ipole 50 30 MOM Diakoptic 10 10 ■30 •50 25 30 35 40 45 F (G H z) Figure 5.3 Comparison of the computed input reactance of the dipole in the high Q package using the diakoptic method versus the input reactance computed using the conventional MOM. 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2.3.2 Dipole on Quartz Next, consider a center fed dipole o f length, L = 1.319 mm and width of, w = 0.24 mm. The dipole is located at x c - 7.0 mm and y c = 9.4 mm in a cavity (Figure 5.2) of the following dimensions: a = 14 mm, b = 18.8 mm and c = 0.635 mm. The substrate thickness is d\ = 0.127 mm and the relative permittivity is £/-l = 4.5(l-j0.0001). An enclosure of this size has seven resonant modes in the band 58-62 GHz. The dipole is diakopted into 4 elements with 3 ports as illustrated in Figure 5.1. Figure 5.4 compares the predicted input reactance of the dipole in the high Q package using the diakoptic method versus the conventional MOM. The conventional MOM is performed using 11 expansion functions while the diakoptic method uses 3 super modes. The results obtained using the diakoptic method are offset from the conventional MOM by about 8 Q over the whole bandwidth. 5.2.3.3 Shunt Stub As a third example illustrating the application of the diakoptic method, consider a transmission line with a single shunt open circuit stub attached. A shunt stub is chosen because it is a typical building block used in more complicated circuits. The stub, located at x c = 7.5 mm (a/2), has a length L = 1.9 mm and is attached to a transmission, located at yc = 12 mm (b/2), of width w = 1.4 mm. The stub is enclosed in a cavity of the following dimensions: a = 15 mm, b = 24 mm and c = 12.7 mm. The substrate thickness is d \ = 1.27 mm and the relative permittivity is er l = 10.5(l-j0.0023). An enclosure of this size has only one resonant mode, the T M n o (10.8129 GHz), in the band 9-12 GHz. The circuit is diakopted into 15 elements with 16 ports as illustrated in Figure 5.5. Figure 5.6 compares the predicted transmission response of the stub in the high Q package 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ipole 50 MOM Diakoptic 25 0 58 60 61 62 F (GHz) Figure 5.4 Comparison of the predicted input reactance of the dipole in the high Q package using the diakoptic method versus the conventional MOM. 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i J l T w Figure 5.5 Geometry of a transmission line with a single shunt open circuit stub attached. The stub, located atx c (a/2), has a length L = 1.9 mm and is attached to a transmission line, located aty c (b/2), of width w = 1.4 mm. 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm -10 ea ■o w -20 -30 ^0 - MOM * Diakoptic 9 10 11 12 F (G H z) Figure 5.6 Comparison o f the predicted transmission response of the stub in the hig package using the diakoptic method versus the conventional MOM. 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. using the diakoptic method versus the conventional MOM. The conventional MOM is performed using 123 expansion functions while the diakoptic method uses 16 super modes. Agreement between the diakoptic method and the conventional MOM is very poor. 5.3 Enhanced Diakoptic Method In the previous section, the diakoptic method was applied to a shunt open circuit stub enclosed in a conducting package. A significant discrepancy existed between the results obtained using the conventional MOM (Chapters 2 and 3) and the diakoptic method. Jackson [8] has also observed a significant discrepancy between the results obtained using the conventional MOM [9] and the diakoptic method for circuits with no enclosures. Howard and Chow [3], [10] indicated that they have also encountered a problem when applying the diakoptic method. They reported that this problem can be avoided by removing the source fringe effects; however, the details of this removal were not given. The accuracy and efficiency of the diakoptic method depends upon the quality of the super mode expansion functions, how well they resemble the final currents on the entire structure. In this section, spectral filtering will be investigated as a way of improving the super mode quality. 5.3.1 Theory of Spectral Filtering The diakoptic method has been previously described as a two step moment method process. In the first step, N DK super modes are formulated for the circuit. In the second step, the super modes are used to expand the current on the entire circuit When solving for the super modes, jjPK ( k = l , ..., N DK), in step one, the current on the M i structure is expanded in a set of known basis functions with coefficients I (fc). 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As described in section 5.2.2 these coefficients are determined by solving equation (5.6). -»ni7 Jk , is then formed from equation (5.5). However, experience has shown that removing the lower order spectral terms (small kp ) in equation (5.7) will produce a superior super mode. The removal of the lower order spectral terms can be thought as passing the components of equation (5.7) through a high pass spectral filter. This high pass filtering can be expressed by the following relationship: Z (k)---------------------------- >Z(fc) High pass spectral filter (5.18) Instead of equation (5.6), the unknown current coefficient on the fcth structure, I (i), is now determined by solving the following equation: y ( k ) = 2 l(k)1(k) (5 .19) A typical element of Z ^ is given by: Af°° N °° _ 4 4 =2 ( *% (5.20) m = 0n= N ' where +1 “ jo m<Mc m >Mc and Nc and M c are defined as the cutoff mode numbers. Although numerical values for the cutoff mode numbers have not yet been defined, it will be shown shortly that N c and Mc are large. Since N c and M c are large, kp will be also be large. From the discussion in Chapter 2 and Appendix B, the spectral Green’s Function for large kp can be represented as: Qpqikm ,kym) = i^ k y j As a result, equation (5.20) can be expressed as: M °° N °° . (3 ? )„ ' I I --- 7f-<$ 3 m = 0n= N' (5.21) a0 In the second application of the moment method, the surface current on the entire circuit is expanded into a set of super modes, J[DK (I = 1 ,..., N DK). The unknown 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. current coefficients of the super inodes, 1DK, are determined by solving equation (5.11). Note, when calculating the elements of the Z DK with equation (5.12), the total spectrum is used. In summary, the super modes are determined using the higher order spectral terms as described by equation (5.21). The super modes are used to expand the current on the entire circuit When calculating the elements of the diakoptic impedance matrix, the total spectrum is used as described by equation (5.12). 5.3.2 Determination of the Cutoff Mode Numbers Before determining the cutoff mode numbers, Nc and Mc, it will be convenient to define cutoff wave numbers as: C5'22a) (5.22b) where Pc = and kc (5-23) is a frequency independent constant. Substituting equation (5.23) into equations (5.22a) and (5.22b) yields the cutoff mode numbers, N c and M c: NC=^ 2 x M c = bK£ko (5'24a) V2 tt To examine the effect of varying kc has on the solution obtained using the enhanced diakoptic method, again consider the dipole on 0.125 mm thick quartz. The dipole is diakopted into 4 elements with 3 ports as illustrated in Figure 5.1. Figure 5.7 compares the predicted input reactance of the dipole in a high Q package using the enhanced diakoptic method for different values of k c versus the conventional MOM. The 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ipole 1 CA E js O * Mrn t X • fT MOM I f -----■— Kc=7.625 Ef 0 Kr=8.625 f # -----• — kc=9.625 ----- °— Kr=10.625 Fi l[ -----*— v r = 11.625 . .1 -5 58 59 60 61 62 F (G H z) Figure 5.7 Comparison of the predicted input reactance of the dipole in a high Q package using the enhanced diakoptic method for different values of versus the conventional MOM. 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. conventional MOM is performed using 11 expansion functions. Below 59 GHz, agreement between the enhanced diakoptic method with kc = 9.625 and 10.625 and the conventional MOM is excellent. Above 59 GHz, the enhanced diakoptic method with kc = 9.625 gives the same values as the conventional MOM. The enhanced diakoptic method with Kc = 8.625 and 10.625 deviates from the conventional MOM by no than 1 Cl. Based on these observations, the optimum value of Kc for 0.125 mm quartz is 9.625. For comparison, the same calculation using an unfiltered supermode resulted in an offset from the conventional MOM by about 8 £2 over the whole bandwidth. Figure 5.8 summarizes the comparison of the computed input reactance of the dipole using the enhanced diakoptic method with the optimum value of Kc (9.625) versus the conventional MOM. The above procedure is repeated for a dipole printed on different substrates. Table 5.1 lists the optimum kc for these substrates. Table 5.1 Kc for a few representative substrates. S u b s tra te £r T h ick n ess (m m ) Kc Duroid 6010 Duroid 6010 Quartz GaAs 10.5 10.5 4.5 12.9 1.27 0.635 0.127 0.1 13.5 14.25 9.625 15.75 Unfortunately, kc in Table 5.1 is different for different substrates. In order for the enhanced diakoptic method to be a useful tool for the simulation of microstrip circuits, a procedure must be prescribed for finding kc for a particular substrate. Experience indicates that an error function ec defined as 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ipole 50 ' 25 MOM Enhanced Diakoptic X ll 0 -25 -5 0 58 59 60 61 62 F (G H z) Figure 5.8 Comparison of the computed input reactance of the dipole using the enhanced diakoptic method with the optimum value of kc (9.625) versus the conventional MOM. 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. has a numerical value of 0.03 when optimum value o f kc 0.03. Substituting kc is optimum. Thus, in order to determine the for a particular substrate, kc kc in equation (5.25) is varied until £c = into equation (5.24) yields the cutoff mode numbers, N c and M c. One important feature of Therefore, once kc kc is that it is not dependent on the circuit topology. is determined for a given substrate it does not need to be recalculated for different circuits. 5.3.3 Results The enhanced diakoptic method described above is used to analyze five microstrip circuits. The circuit types are as follows: 1. Shunt open circuit stub 2. Stub in a larger enclosure 3. Coupled line bandpass filter 4. Small gap in a transmission line 5. 60 GHz shunt stub 5.3.3.1 Shunt Stub As the first example using the enhanced diakoptic method, again consider the shunt open circuit stub discussed in the previous section. The circuit is diakopted into 15 elements with 16 ports as illustrated in Figure 5.5. Figure 5.9 compares the predicted transmission response of the stub in the high Q package using the enhanced diakoptic 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 12 mm IS21I (dB) -10 -20 -30 ' MOM Enhanced Diakoptic -40 9 10 11 F (G H z) Figure 5.9 Comparison o f the predicted transmission response of the stub in the high Q package using the enhanced diakoptic method ( k c = 13.5) versus the conventional MOM. 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. method ( kc = 13.5) versus the conventional MOM. Agreement between the enhanced diakoptic method and the conventional MOM is excellent The ideal response for a single shunt open circuit stub is a single null in IS2 1 I. However, the resonant mode of the enclosure has a drastic effect on the circuit's performance and the enhanced diakoptic method does in fact simulate this effect very well. To reduce the effect of the resonant mode, a microwave absorbing layer is attached to the cover of the enclosure. The thickness of this layer is J 3 = 1.27 mm, the relative permittivity is e r 3 = 60(l-_/0.12) and the relative permeability is Mr3 = 7.3(1-j'0.3). Figure 5.10 compares the predicted transmission response of the stub in the low Q package using the enhanced diakoptic method versus the conventional MOM. The results obtained using the enhanced diakoptic method deviate from the conventional MOM by no more than a 1 dB over the whole bandwidth. The shunt stub described above, the conventional MOM was performed using N mom = 123 expansion functions while the diakoptic method used N DK = 16 super modes. The total number of expansion functions used to create the &th super mode is where k = 1,.., N DK. is different for each super mode because basis functions of varying size and type (e.g. functions that satisfy the edge conditions) are used in different regions of the circuit. The average number of expansion functions used to j Ndk create the super modes was M ^ vg^ = ------------------- = 14.4. Solving the linear system of N d k k= i equations using the conventional MOM required approximately N ^ qm ~ 1/ 6( 3x l0 3 A = operations, whereas the enhanced diakoptic method required ndkV =1 / 6 j = 3 x l ° 4 operations. This example showed the use of the enhanced diakoptic method for a typical circuit enclosed in a cavity with one resonant mode. Compared to the MOM, the enhanced diakoptic method resulted in a reduction of the number operations needed to solve for the 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 12 mm -10 IS21 -20 MOM Enhanced Diakoptic -30 -40 9 10 11 12 F (G H z) Figure 5.10 Comparison of the predicted transmission response of the stub in the low Q package using the enhanced diakoptic method versus the conventional MOM. 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. unknown current distribution by a factor o f 10. The enhanced diakoptic method gives excellent agreement with the MOM even in cases where a low Q resonance is involved. 5 .33.2 Shunt Stub in Larger Box Next, the enhanced diakoptic method will be used to simulate the shunt stub enclosed in a cavity with several resonant modes. The shunt stub, of the previous example, is enclosed in a cavity with the following dimensions: a = 30 mm, b = 48 mm and c = 10.0 mm. An enclosure of this size has five resonant modes in the 9 to 13 GHz bandwidth. In this new enclosure, the stub is located at x c = 15 mm (a/2) and the transmission line is located at y c = 24 mm (b/2). The circuit is diakopted into 25 elements with 26 ports. Figure 5.11 compares the predicted transmission response of the stub in the high Q package using the enhanced diakoptic method ( kc = 13.5) versus the conventional MOM. Agreement between the enhanced diakoptic method and the conventional MOM is very good. To reduce the effect of the resonant modes, a 16 mil (0.4064 mm) thick silicon layer with a resistivity of 1.45 O-cm is attached to the cover of the enclosure. Figure 5.12 compares the predicted transmission response of the stub in the low Q package using the enhanced diakoptic method versus the conventional MOM. Again, the agreement between the enhanced diakoptic method and the conventional MOM is very good. Simulating the shunt stub in the larger box with the conventional MOM required N Mom = 179 expansion functions while the enhanced diakoptic method used N DK = 26 super modes. The average number of expansion functions used to create the super modes was = 12.8. Solving the linear system of equations using the conventional MOM required approximately Njfiom ~ 9.5x105 operations, whereas the enhanced diakoptic method required N ^ LV ~ 3 .2 x l0 4 operations. 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q Yc = 24 mm IS21I ((IB) -10 -20 -30 ' MOM Enhanced Diakoptic -40 9 10 11 12 13 F (G H z) Figure 5.11 Comparison of the predicted transmission response of the stub in the larger high Q package using the enhanced diakoptic method ( Kc = 13.5) versus the conventional MOM. 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Si cover Yc = 24 mm IS21I (dB -10 -20 -30 - MOM • Enhanced Diakoptic -40 9 10 11 12 13 F (G H z) Figure 5.12 Comparison of the predicted transmission response of the stub in the larger low Q package using the enhanced diakoptic method versus the conventional MOM. 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This example showed the use of the circuit model for a circuit enclosed in a cavity with several resonant modes. Compared to the MOM, the enhanced diakoptic method resulted in a reduction of the number operations needed to solve for the unknown current distribution by a factor of 30. Agreement between the enhanced diakoptic method and the MOM was excellent even in the vicinity of the resonant modes. 5.3.3.3 Coupled Line Bandpass Filter To examine the use of the enhanced diakoptic method to simulate a circuit containing coupled lines, consider a two resonator coupled line bandpass filter. The width and length of the resonators are w = 0.64 mm L = 5.0 mm, respectively. The spacing of the resonators are s i = 0.13 mm and S2 = 0.64 mm. The circuit is enclosed in a cavity of the following dimensions: a = 24 mm, b = 15 mm and c = 6.35 mm. The substrate thickness is d \ = 0.635 mm and the relative permittivity is £r l = 10.5(l-y'0.0023). An enclosure of this size has two resonant modes in the band 6-16 GHz. The circuit is diakopted into 24 elements with 22 ports as illustrated in Figure 5.13. Figure 5.14 compares the predicted transmission response of the bandpass filter in the high Q package using the enhanced diakoptic method versus the conventional MOM. The results obtained using the enhanced diakoptic method deviate from the conventional MOM by no more than a 2.5 dB over the most of the bandwidth except less than 7 GHz. To reduce the effect of the resonant modes, a 16 mil (0.4064 mm) thick silicon layer with a resistivity of 1.45 Q-cm is attached to the cover of the enclosure. Figure 5.15 compares the predicted transmission response of the bandpass filter in the low Q package using the enhanced diakoptic method ( kc = 14.25) versus the conventional MOM. Agreement between the enhanced diakoptic method and the conventional MOM is excellent. 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.13 Geometry of a two resonator coupled line bandpass filter. The width and length of the resonators are w = 0.64 mm L —5.0 mm, respectively. The spacing of the resonators are s \ = 0.13 mm and S2 = 0.64 mm. 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q -10 -20 -30 -40 MOM Enhanced Diakoptic -50 -60 6 8 10 12 14 16 F (G H z) Figure 5.14 Comparison of the predicted transmission response o f the bandpass filter the high Q package using the enhanced diakoptic method versus the conventional MOM. 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Si cover o -10 -20 -30 -40 MOM Enhanced Diakoptic -50 -60 6 8 10 12 14 16 F (GHz) Figure 5.15 Comparison of the predicted transmission response of the bandpass filter the low Q package using the enhanced diakoptic method ( k c = 14.25) versus the conventional MOM. 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The results obtained using the enhanced diakoptic method deviate from the conventional MOM by no more than a 2.5 dB over the most of the bandwidth except less than 7 GHz. For the coupled line bandpass filter, the conventional MOM was performed using N MOm = 104 expansion functions while the enhanced diakoptic method used N DK = 22 super modes. The average number of expansion functions used to create the super modes was M ^ vg) = 7.9. Solving the linear system of equations using the conventional MOM required approximately N mqm ~ 1-8x10^ operations, whereas the enhanced diakoptic method required N ^ LV ~ 1.8x10^ operations. This example showed the use of the enhanced diakoptic method for a more complicated microwave circuit The circuit was enclosed in a cavity with two resonant modes, one occurring in the passband and the other in the stopband. Compared to the MOM, the enhanced diakoptic method resulted in a reduction of the number operations needed to solve for the unknown current distribution by a factor of 100. 5.3.3.4 Small Gap To examine the validity of using the enhanced diakoptic method to simulate a circuit operating in the millimeter-wave band, consider a transmission line with a gap in the center. The transmission line, located a ty c = 1-55 mm, has a width of w = 0.1 mm and a gap of g = 0.1 mm. The circuit is enclosed in a cavity of the following dimensions: a = 3.2 mm, b = 3.1 mm and c = 0.6 mm. The substrate thickness is d\ = 0.1 mm and the relative permittivity is £r l = 12.9(l-j0.0016). An enclosure of this size has only one resonant mode in the band 50-70 GHz. The circuit is diakopted into 14 elements with 14 ports as illustrated in Figure 5.16. Figure 5.17 compares the predicted transmission response of the small gap in the high Q package using the enhanced diakoptic method versus the conventional MOM. Agreement 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A A ---1-----f Jr b J_... 1----i----1----1 t -M s W 3>c Figure 5.16 Geometry of a transmission line with a gap in the center. The transmission line, located at y c = 1.55 mm (b/2), has a width of w = 0.1 mm and a gap of g = 0.1 mm. 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gap - High Q o MOM Enhanced Diakoptic IS21I (dB) -10 -20 -30 ■40 -50 -60 50 55 60 65 70 F (GH z) Figure 5.17 Comparison of the predicted transmission response of the small gap in the high Q package using the enhanced diakoptic method ( k c = 15.75) versus the conventional MOM. 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between the enhanced diakoptic method ( kc = 15.75) and the conventional MOM is excellent above 60 GHz. Below 60 GHz, the difference between the enhanced diakoptic method and the conventional MOM is about 2.5 dB at the -30 dB level. Simulating the small gap with the conventional MOM required N MOM = 70 expansion functions while the enhanced diakoptic method used N DK = 14 super modes. The average number of expansion functions used to create the super modes was = 8.7. Solving the linear system o f equations using the conventional MOM required approximately = 5.7x10^ operations, whereas the enhanced diakoptic method required N ^ v = 1.5x10^ operations. This example showed the use of the enhanced diakoptic method for a simple circuit operating in the millimeter-wave band. In addition, the circuit was enclosed in a cavity with one resonant mode. Compared to the MOM, the enhanced diakoptic method resulted in a reduction of the number operations needed to solve for the unknown current distribution by a factor of 370. 5.3.3.S 60 GHz Shunt Stub The final example using the enhanced diakoptic method is a transmission line with a single shunt open circuit stub attached operating in the millimeter-wave band. The shunt stub is a more practical circuit than the small gap from the previous example. The stub, located a tx c = 7.0 mm (a/2), has a length L = 0.62 mm and is attached to a transmission, located at yc = 9.4 mm (bl2), o f width w = 0.28 mm. The circuit is enclosed in a cavity of the following dimensions: a = 14 mm, b = 18.8 mm and c = 0.635 mm. The substrate thickness is d \ = 0.127 mm and the relative permittivity is £r l = 4.5(l-j0.0001). An enclosure of this size has seven resonant modes in the band 58-62 GHz. Since this cavity contains only a small amount of loss it will be referred to as the high Q package. 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The circuit is diakopted into 47 elements with 48 ports. Figure 5.18 compares the predicted transmission response of the stub in the high Q package using the enhanced diakoptic method ( k c = 9.625) versus the conventional MOM. Agreement between the enhanced diakoptic method and the conventional MOM is veiy good except around 59.5 and 62 GHz where the difference is about 2.5 dB. For the circuit described above, the conventional MOM was performed using N MOm = 213 expansion functions while the enhanced diakoptic method used N DK = 48 super modes. The average number of expansion functions used to create the super modes was = 7.8. Solving the linear system of equations using the conventional MOM required approximately N mqm = 1-6x10^ operations, whereas the enhanced diakoptic method required N ^ v ~ 4.3x10^ operations. This example showed the use of the enhanced diakoptic method for a more complicated circuit operating in the millimeter-wave band. In addition, the circuit was enclosed in a cavity with several resonant modes. Compared to the MOM, the enhanced diakoptic method resulted in a reduction of the number operations needed to solve for the unknown current distribution by a factor of 370. 5.4 Conclusion In this chapter the diakoptic method was used to analyze a complex MMICs in an enclosures. In order to reduce the number of operations required to analyze complex circuits, Goubau eta l [1] developed the diakoptic m ethod Results obtained using the diakoptic method agreed well with those obtained using the MOM for a few simple circuits. However, a significant discrepancy existed between the two methods for more complicated circuits. A new filtering technique was presented in this chapter which significantly reduced this discrepancy. The new technique was called the enhanced diakoptic method. 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. High Q o IS21I (dB ) -10 -20 -30 -40 MOM Enhanced Diakoptic -50 58 59 60 61 62 F (G H z) Figure 5.18 Comparison of the predicted transmission response of the stub in the high Q package using the enhanced diakoptic method ( k c = 9.625) versus the conventional MOM. 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As described above, the main advantage of using the enhanced diakoptic method instead o f the conventional MOM is the reduction in the number of operations required to analyze a complex circuit. Table 5.2 summarizes the number o f operations, N ^ qm , needed to solve the linear system o f equations associated with the conventional MOM and the enhanced diakoptic method, respectively. Compared to the MOM, the enhanced diakoptic method resulted in a reduction o f the number operations needed to solve for the unknown current distribution by a factor of 10 to 370. It is expected that for larger circuits this reduction will be even greater. Table 5.2 Comparison of the operation counts for the MOM and enhanced diakoptic methods. w s o l v / n solv n mom / n dk Circuit N m om n s° l v ^ MOM N dk Stub 123 3 x l0 5 16 14.4 3 x l0 4 10 Stub in larger box 179 9 .5 x l0 5 26 12.8 3 .2 x l0 4 30 Bandpass filter 104 1.8x10s 22 7.9 1.8x10s 100 Small gap 70 5.7X104 14 8.7 1 .5 x l0 2 370 60 GHz stub 213 1.6x10s 48 7.8 4.3x10s 370 < V5) n solv n dk No previous mention has been made of the CPU time required to analyze a circuit employing the MOM or the enhanced diakoptic method. It was previously stated that the majority o f analysis time using the MOM can attributed to two sources: the fill time and the solve time. For the conventional MOM, filling the matrix requires approximately operations and solving the matrix requires approximately N f^ M operations where 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n mom = N ^ M a N mom and N s^om = V 6 ( n \ io m )- N A and M A are defined in Chapter 2. The times required to perform these two operations are T^o.m and T ^ om where Tmom = g N moM' Tmqm = < xN mom and a is a machine dependent constant For a small circuits such as those presented in this chapter, Tm$ m < T^/o m - Thus, employing the diakoptic method to reduce Tmom f°r a small circuit will reduce the total solution time (T mom + Tm o m ) only by a small amount. For example, consider the stub in the larger box. Employing the enhanced diakoptic method reduced the solve time by a factor of 28. However, the solve time represented only 6% o f the total solution time. Since all of the circuits presented in this chapter were relatively small in terms of the number of unknowns, the overall savings in total CPU time was typically around 5 to 10%. For large circuits, the solution of the linear system of equations represents a large portion of the analysis time. Therefore, employing the diakoptic method will result in a large reduction in the total CPU time. 192 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 References [1] Goubau, G., Puri, N.N. and Schwering, F.K., "Diakoptic theory for multielement antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-30, No. 1, pp. 15-26, January 1982. [2] Schwering, F, Puri, N.N. and Butler, C.M., "Modified diakoptic theory of antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-34, No. 11, pp. 1273-1281, November 1986. [3] Howard, G.E. and Chow, Y.L., "A high level compiler for the electromagnetic modeling of complex circuits by geometrical partitioning," IEEE MTT-S International Microwave Symposium Digest, pp. 1095-1098,1991. [4] Harrington, R.F., Time harmonic electromagnetic fields. McGraw-Hill Book Company, New York, 1961. [5] Butler, C.M., "Diakoptic theory and the moment method," 1990 IEEE AP-S International Symposium Digest, pp. 72-75,1990. [6] Howard, G.E. and Chow, Y.L., "Diakoptic theory for microstripline structures," 1990 IEEE AP-S International Symposium Digest, pp. 1079-1082,1990. [7] Singh, S. and Singh, R., "Efficient computation of the free-space periodic Green's function," IEEE Transactions on Microwave Theory and Techniques, Vol. 39, No. 7, pp. 1226-1229, July 1991. [8] Jackson, R.W., Personal Communication. [9] Jackson, R.W., "Full-wave, finite element analysis of irregular microstrip discontinuities," IEEE Transactions on Microwave Theory and Techniques, Vol. 37, No. 1, pp. 81-89, January 1989. [10] Howard, G.E. and Chow, Y.L., "Removal of source fringe effects in the diakoptic theory," 1991 IEEE AP-S International Symposium Digest, p. 99, 1991. 193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6 C O N C LU SIO N In this dissertation the analysis of an MMIC in an enclosure was presented. Several new techniques were developed and the results obtained were compared to a full-wave method of moments (MOM) analysis. In view of its importance, the full-wave analysis of packaged MMICs was discussed and a review of the MOM was given at the beginning of the dissertation. The spectral Green's function was derived using an equivalent transmission line. In addition, an acceleration technique for the efficient evaluation of MOM impedance matrices was presented and a special basis function arrangement was developed to better model the coaxial to microstrip transition. Also, a detailed discussion of locating the resonances of a dielectric loaded cavity was presented. To verify the accuracy of the full-wave MOM procedure several circuits were fabricated and measured while enclosed in a brass cavity. Agreement between the measured and calculated results was reasonable for all circuits. The first goal of this dissertation was to examine some of the fundamental aspects of resonant mode coupling. Using a full-wave analysis, it was shown that package resonances can have a very significant effect on circuit operation even at frequencies which are not very close to resonance. The addition of loss to an enclosure reduced resonant mode coupling, but often did not totally eliminate it. A further reduction in the coupling of power to resonant modes was obtained by repositioning the circuit Locating areas of high current in areas of low electric field in the enclosure reduces the power lost to these resonant modes. However, layout modifications such as these are best applied to a relatively simple circuit consisting of only a few discontinuities in a moderately sized enclosure. Since most current MMICs contain more than a few discontinuities, repositioning the circuit in an enclosure is sometimes not practical. 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The next two goals of this dissertation were to develop alternatives to the MOM for analyzing a circuit in an enclosure. A simpler alternative is o f interest because a full-wave analysis, although rigorous, is also very complex to implement. In addition, the MOM is difficult to use for circuit diagnosis or to develop an intuitive understanding of package enhanced coupling. Thus, a second goal of this dissertation was to develop a circuit model to describe resonant mode coupling for use on commercially available CAD packages. The model which was developed has good accuracy for enclosures with a Q o f over 100 and is useful for Q's as low as 20. Simple analytical expressions for the entire model are easily evaluated, making this is a very attractive feature of the circuit model for implementation into a CAD package. In addition, it requires several orders of magnitude less CPU time than the MOM. However, implementing the circuit model into a CAD package for a complicated MMIC in a large enclosure is very tedious. Consequently, this circuit model may be best suited for MMIC circuits in moderately sized enclosures. A solution to this inconvenience is an automated procedure for implementing the circuit model. To test the feasibility of such an automated procedure, a simple CAD program was written. Using the automated procedure significantly reduced the time and complexity of entering the model into a CAD package; which allows for the utilization of the circuit model for any size enclosure. Although the circuit model has good accuracy and requires several orders of magnitude less CPU time than the MOM, it does have several limitations. The first limitation, implementing the circuit model, was discussed above. The second limitation is that accuracy of the circuit model is dependent on the availability of discontinuity models (e.g. a TEE element) in CAD package. Some discontinuity models, such as the proximity coupling between two bends, do not exist in CAD packages. In addition, as the size of MMIC circuits increase they become too complicated to analyze using a straightforward 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. full-wave approach. A full-wave analysis of a typical MMIC of moderate complexity may require the solution of a large system of equations (1000 x 1000 matrix) to accurately model the complete circuit The need for new ways to analyze MMICs in an enclosure is therefore apparent Therefore, a second alternative to the MOM is sought for more complicated MMICs. The third goal of this dissertation was to develop a method to analyze a complex MMIC circuit in an enclosure. The diakoptic method [1], [2] was employed in order to achieve this goal. Results obtained using the diakoptic method agreed well with those obtained using the MOM for a few simple circuits. However, a significant discrepancy existed between the two methods for more complicated circuits. A new filtering technique was developed which significandy reduced this discrepancy. The new technique was called the enhanced diakoptic method. The main advantage of using the enhanced diakoptic method compared to the conventional MOM is that the enhanced diakoptic method will result in a reduction of the CPU time. The majority of analysis time using the MOM can be attributed to two sources: the time required to fill the matrix (fill time) and the time required to solve the linear system of equations (solve time). For circuits of small complexity, the matrix fill time will be much larger than the solve time. For circuits of moderate complexity, the matrix fill time and solve time are approximately of the same order and for circuits of large complexity, the matrix solve time will be much larger than the fill time. For the circuits presented in this dissertation, the enhanced diakoptic method resulted in a reduction of the solve time by a factor of 10 to 370 when compared to the MOM. However, the total CPU time required was only reduced by about 10%. Since all of the circuits presented were relatively simple, the majority of CPU time was dominated by the fill time. It is was shown that for more complex circuits, the total CPU time will also be reduced. 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A desirable extension of the enhanced diakoptic method would be to develop a more efficient technique for filling the matrix [3]. This would reduce the fill time for a circuit of any complexity. After developing an efficient "fill" technique, the next logical step will be to incorporate the enhanced diakoptic method into an existing a commercially available MOM based electromagnetic simulator. The computer program developed in this dissertation for the enhanced diakoptic method utilized many of the subroutines used in the MOM computer program. The enhanced diakoptic method computer program also required several new routines; however, they were generally very simple to im plem ent Therefore, this author is of the opinion that a straightforward implementation of the enhanced diakoptic method into a current electromagnetic simulator could easily be obtained. 197 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 References [1] Goubau, G., Puri, N.N. and Schwering, F.K., "Diakoptic theory for multielement antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-30, No. 1, pp. 15-26, January 1982. [2] Schwering, F, Puri, N.N. and Butler, C.M., "Modified diakoptic theory of antennas," IEEE Transactions on Antennas and Propagation, Vol. AP-34, No. 11, pp. 1273-1281, November 1986. [3] Singh, S. and Singh, R., "Efficient computation of the free-space periodic Green's function," IEEE Transactions on Microwave Theory and Techniques, Vol. 39, No. 7, pp. 1226-1229, July 1991. 198 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P PE N D IX A D ER IV A TIO N O F dYv /da) In this appendix, an analytic expression for dYv /dco is developed. In order to formulate an analytic expression for dYv Jdco, it will be convenient to express the solution as. S l - U S ul dco vc dk0 (A.1) y _ y(k) . y ( * + l ) XU ~ XLU + XRU (A.2) where CO= VckQ vc = velocity of light in free space U = E o rM Yj^j and Y $ y l'i are given by equations (2.44) and (2.45), respectively. Substituting equations (2.44) and (2.45) into equation (A.2) and differentiating with respect to k0 yields: dYv _ d Y $ ) dkQ dk0 d YRU $ 7 l) dkQ (A.3) where d yLU _ YLU dYrU + y{k)Mk) *0 ~y% TU t o 0 + T tu G lu -\y{k+l) y(k+1) 3y(A+l) dXRU _ XRU dXTU , v ( * + l) / - ( A + l) RU ~ ^ - Y ^ - - d k ~ ' TU (k) OILU M ir " “ dkp rrr v(k-1) s u W dkn (A.4) (A.5) ■*£,[/ ljLU \^TU cos &k + JY& ^ sin 199 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.6) rJ k+1) RU ~ - U 2) ^ +i){[>^+i)]2- [ 4 i +2)f } + ^ +1) ^ Rdkn ~ ypfc+2) ZRU dkg (A.7) [ l $ +1) cos 0M + j Y & 2) sin 0M ] p(i) _ Sri^rihdj (A. 8 ) hi d^TM _ ~ h y(i) dkQ *o& ™ Z-2 dY$ # 7 *7 2- 1*TE dkQ k0kZi 200 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.9) (A. 10) A PPEN D IX B LARG E ARGUM ENT SPECTRAL G R E E N ’S FU N C TIO N In this appendix, an asymptotic form of the spectral Green’s function for large values of kp is developed. When kp » [(£r/O m ax]*o» where (er^r)max = m ax(£„//n ), the following approximations can be made kz i ~ - j k p for i = (B .l) tan 6t = tan (kzidi) = —j tanh(/:p4 ) (B.2) As a result of equations (B .l) and (B.2), the large argument spectral Green's function derived in this appendix is equivalent to the quasi-static Green's function. Substituting equation (B.2) into (2.44) and (2.45) yields: |«> r„,Ki(.> tv 1(0 J S 1 (0 for i = l,.„,A: (B.3) for i = k + l,...,N (B.4) J(:> + \ y & ](,_1) tanh(^pfifj) [Y g^ + lY & ftm h jkpd;) TU where U = E or M and i(0 _ j£ rik0 _ j2 jrfs ri TM ' 1(0 TE 1 no kp -jk p Vrikono nokpvc - j k pvc (B.5) (B.6) 27tfHriT]0 vc = velocity of light in free space The quasi-static driving point admittance at z = z/c for the equivalent circuit shown in Figure 2.2 is given by: yv isl(fc) S = \ i lu ,( * + ! ) (B.7) Examining equations (B.3) - (B.6) it is easily seen that Y$s is proportional to frequency and Y ^s is inversely proportional to frequency; therefore, Y$s ( f ) and Y$s ( f ) can be written as: 201 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.8a) JR y^s( f ) = j Y i s(fR) (B.8b) where f R is a reference frequency. The frequency dependence of the asymptotic spectral Green's function can be found by substituting equation (B.8) into (2.57) such that: kxn QtM ^ r ) , Qgs(kxn,kym) =n QS(k k k \- iZxy \ Kxn’*ym J ~ '*lyx \ Kx r f Kym> ~ (B.9a) K fR/f f/fR V*P Q n f lf e ) kxnkym f Qt m Wr) ,2 r,f I Q t£ W r ) f/fR f (B.9b) if f R/ f kyn Qtm Wr ) , ^xn QtE (fR) k f R/ f \ Kp f / f R (B.9c) where o S (/)= - Y$s ( f ) l < 2 f(/) = YgS{ f ) (B.lOa) (B.lOb) 202 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A PPEN D IX C THE SC ATTER ING M ATRIX In this appendix, the relationship between the augmented y-matrix and the scattering matrix (s-matrix) is developed. To formulate the solution of the s-matrix for a network with Np ports, it will be convenient to express the solution in terms of wave quantities. The two quantities needed are the incident and reflected waves. The normalized incident, a, and reflected, b, waves are defined as [1], [2]: a = ^r-V2[Z + r]I (C.l) b = -|r _1/2[Z -r ]I (C.2) where Zoi r= (C.3) Z0N„ and Z is the open circuit impedance matrix. The reflected wave is related to the incident wave by: b = Sa (C.4) where S is defined as the scattering matrix or the s-matrix. Substituting equations (C.l) and (C.2) into (C.4) and solving for S yields: S = r -1/2[Z -r][Z + r ] - y /2 = r_1/2[Z + r - 2r][Z + r]_1r1/2 = r-1^2{ u - 2r[Z + r]-1}r^2 (C-5) S = U - 2 r V2[Z + r]_1r1/2 The matrix, [ Z + r ] , is the open circuit impedance matrix of the augmented network. The inverse. [Z + r]-1, is the short circuit admittance matrix of the augmented network, which is refened to as Ya. The augmented network is defined as a network that combines in 203 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. series the original network and the terminating impedance of each port [3]. In terms of Ya , the s-matrix is given by: S = U - r 1/2Yar 1/2 204 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.6) Appendix C References [1] Gonzalez, G., Microwave transistor amplifiers. Prentice-Hall, Inc., New Jersey, Chapter 1, 1984. [2] Ha, T.T., Solid-state microwave amplifier design. John W iley & Sons, Inc., New York, Chapter 2, 1981. [3] Balabanian, N. and Bickart, T.A., Electrical network theory. Robert E. Krieger Publishing Company, Florida, Chapter 8, 1983. 205 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A PPE N D IX D THE ELECTRICAL CHARACTERISTICS OF THE D IELECTR IC ABSO R BER When measuring the performance of the circuits discussed in Chapter 3, a lossy dielectric material (absorber) was attached to the cover of the enclosure in an effort to reduce the effects of the resonant modes. However, the electrical characteristics of the absorber were not known. In this appendix, a simple circuit will be used to characterize the lossy dielectric material. In order to determine the electrical properties of the absorber, the relative permittivity and permeability of the absorber were varied until the agreement between the simulated and measured results were excellent The simple circuit consists of a transmission line with a large gap in the center as shown in Figure D .l. The transmission line, located a ty c = 12 mm, has a width o f w = 1.4 mm and a gap of g = 10.5 mm. The circuit is enclosed in cavity A which was determined in Chapter 3 to have one resonant mode in the band 9-12 GHz. Since this cavity contains only a small amount of loss it will be referred to as the high Q cavity A The optimum relative permittivity and permeability of this absorber were determined to be 60(l-y'0.12) and 7.3(l-;'0.3), respectively. Figure D.2 compares the calculated using the optimum parameters of the absorber to the measured transmission coefficient (IS2 1 I) of the large gap. Agreement between the measured and calculated results is excellent 206 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b W 4 Figure D .l Geometry of a transmission line with a gap in the center. The transmission line, located atyc = 12 mm (b/2), has a width of w = 1.4 mm and a gap of g = 10.5 mm. 207 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Low Q - Dielectric Cover Yc = 12 mm -10 CQ -20 - MOM * Measured -30 -40 9 11 10 12 F (GHz) Figure D.2 Comparison of the calculated and measured transmission coefficient of the large gap located at y c = 12 mm in the low Q cavity A. 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / BIBLIO G R A PH Y Balabanian, N, and Bickart, T.A. Electrical Network Theory. Florida: Robert E. Frieger Publishing Company, 1983. Balanis, Constantine, A. Advanced Engineering Electromagnetics. New York: John Wiley & Sons, 1989. Collin, Robert E. Field Theory of Guided Waves. New York: IEEE Press, 1991. Gonzalez, G. Microwave Transistor Amplifiers. New Jersey: Prentice Hall Inc, 1984. Ha, T.T. Solid -State Microwave Amplifier design. New York: John Wiley & Sons, 1981. Harrington, Roger F. Field Computation by Moment Methods. Florida: Robert E. Frieger Publishing Company, 1968. Itoh, Tatsuo. Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. New York: John W iley & Sons, 1989. 209 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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