# Generalized scattering matrix modeling of distributed microwave and millimeter -wave systems

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GENERALIZED SCATTERING MATRIX MODELING OF DISTRIBUTED MICROWAVE AND MILLIMETER-WAVE SYSTEMS by AH M ED IBR A H IM KHALIL A dissertation subm itted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy ELECTRICAL EN G IN EER IN G Raleigh 1999 A PPR O V ED BY: Chair of Advisory Com m ittee Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9946430 UMI Microform 9946430 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract KHALIL, AHMED IBRAHIM. Generalized Scattering M atrix Modeling of Dis tributed Microwave and Millimeter-Wave Systems. (Under the direction of Michael B. Steer.) A full-wave electromagnetic simulator is developed for the analysis of transverse multilayered shielded structures as well as waveguide-based spatial powercombining systems. The electromagnetic simulator employs the method of moments (MoM) in conjunction with the generalized scattering m atrix (GSM) approach. The Kummer transformation is applied to accelerate slowly converging double series ex pansions of Green’s functions th at occur in evaluating the impedance (or adm it tance) m atrix elements. In this transformation the quasi-static part is extracted and evaluated to speed up the solution process resulting in a dramatic reduction of terms in a double series summation. The formulation incorporates electrical ports as an integral part of the GSM formulation so th at the resulting model can be integrated with circuit analysis. The GSM-MoM method produces a scattering m atrix th at represents the relationship between waveguide modes and device ports. The scattering m atrix can then be converted to port-based adm ittance or impedance matrix. This allows the modeling of a waveguide structure that can support multiple electromagnetic modes by a circuit with defined coupling between the modes. Since port-based representations are not suited for most circuit simulation tools, a circuit theory based on th e local reference node concept, is developed. The theory adapts modified nodal analysis to accommodate spatially distributed circuits allowing conventional harmonic balance and transient simulators to be used. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To show the flexibility of the modeling technique, results are obtained for general shielded microwave and millimeter-wave structures as well as vaxious spatial power combining systems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Biographical Summary Ahmed Ibrahim Khalil was born in Cairo, Egypt, on November 15, 1969. He received the B.S. (with honors) and M.S. degrees from Cairo University, Giza, Egypt, both in electronics and communications engineering, in 1992 and 1996, re spectively. From 1992 to 1996 he worked at Cairo University as a Research and Teaching Assistant. While working towards his Ph.D. degree in electrical engi neering at North Carolina State University, since 1996, he held a Research Assistantship with the Electronics Research Laboratory in the Department of Electrical and Com puter Engineering. Interests include numerical modeling of microwave and millimeter-wave passive and active circuits, MMIC design, quasi-optical power com bining, and waveguide discontinuities. He is a student member of the Institute of Electrical and Electronic Engineers and the honor society Phi Kappa Phi. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments This dissertation would have never been finished without the will and blessing of God, the most gracious, the most merciful. AL HAMDU LELLAH. I would like to express my gratitude to my advisor Dr. Michael Steer for his support and guidance during my graduate studies. I would also like to express my sincere appreciation to Dr. Jam es Mink, Dr. Frank Kauffman, and Dr. Pierre Gremaud for showing an interest in my research and serving on my Ph.D. committee and to Dr. Amir Mortazawi for helping m e with the measurements and many useful discussions. A very big thanks go to my colleagues, Mr. Mostafa N. Abdulla for many useful suggestions regarding my work, Mr. Mete Ozkar for working with me on the excitation horn, Mr. Carlos E. Christoffersen for his computer skills which came in handy many times, Dr. Todd W. Nuteson for his encouragement while starting my PhD. degree, Mr. Satoshi Nakazawa for sharing the same cubical, Dr. Hector Gutierrez for many useful advice, Mr. Usman Mughal, Mr. Rizwan Bashirullah, Mr. Adam M artin, Mr. Chris W. Hicks, and Dr. Huan-sheng Hwang. Also, I would like to thank m y professors and colleagues at Cairo Uni versity, Egypt, for the part they played in my academic career. They are truly outstanding. And finally, I wish to thank my wife and two sons Om ar and Ali for their support, understanding and encouragement and my parents whom without their total love, guidance, and dedication I would not have made it this far. m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents List o f Figures 1 2 3 viii Introduction 1 1.1 Motivation For and Objective of This S t u d y ......................................... 1 1.2 Dissertation Overview ................................................................................ 8 1.3 Original C o n trib u tio n s ................................................................................ 9 1.4 P u b lic atio n s........................................................................................................11 Literature R eview 13 2.1 B a c k g ro u n d ........................................................................................................13 2.2 Waveguide Power C om biners.......................................................................... 18 2.3 Numerical Modeling and C A D ....................................................................... 22 M odeling U sing GSM 26 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 In tro d u ctio n ....................................................................................................... 26 3.2 GSM-MoM W ith P o r t s ....................................................................................27 3.3 Electric Current Interface 3.4 ............................................................................. 32 3.3.1 Mode to mode sc a tte rin g ..................................................................... 34 3.3.2 Mode to port s c a t te r i n g ..................................................................... 39 3.3.3 Port to port s c a tte rin g ........................................................................ 41 Magnetic Current Interface............................................................................. 41 3.4.1 Mode to mode sc a tte rin g ..................................................................... 41 3.4.2 Mode to port s c a tte r in g ..................................................................... 47 3.4.3 Port to port s c a tte rin g ........................................................................ 48 3.5 Dielectric and Conductor In te rfa c e s ............................................................. 48 3.6 Cascade Connection 3.7 Program D e s c rip tio n ....................................................................................... 54 ....................................................................................... 49 3.7.1 Geometry-layout and input file 3.7.2 Electromagnetic s im u la to r ..................................................................55 ........................................................ 54 4 M oM Elem ent Calculation 4.1 58 In tro d u ctio n ....................................................................................................... 58 4.1.1 Uniform d is c re tiz a tio n ........................................................................ 59 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.2 4.2 Nonuniform d is c re tiz a tio n ...................................................................61 Acceleration of MoM M atrix E le m e n ts ....................................................... 65 4.2.1 Acceleration of impedance m atrix elem ents..................................... 66 4.2.2 Acceleration of adm ittance m atrix e le m e n ts .................................. 71 5 Local Reference N odes 75 5.1 In tro d u c tio n ........................................................................................................ 75 5.2 Nodal-Based Circuit S im u la tio n .................................................................... 77 5.3 Spatially Distributed C i r c u i t s ........................................................................78 5.3.1 Port rep re se n tatio n ............................................................................... 78 5.3.2 Port to local-node r e p re s e n ta tio n ..................................................... 83 5.4 Representation of Nodally DefinedC irc u its .................................................. 84 5.5 Augmented A dm ittance M a t r i x .................................................................... 85 5.6 S u m m a r y ............................................ ...........................................................87 6 R esults 88 6.1 In tro d u c tio n ........................................................................................................88 6.2 Analysis of General S tru c tu re s....................................................................... 89 6.2.1 Wide resonant s t r i p ............................................................................... 89 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3 6.4 6.2.2 Resonant patch a r r a y ............................................................................ 91 6.2.3 Strip-slot transition m o d u l e ............................................................... 93 6.2.4 Shielded dipole a n t e n n a ..................................................................... 96 6.2.5 Shielded microstrip f il t e r ......................................................................98 Patch-Slot-Patch A rra y .................................................................................. 105 6.3.1 Array s im u la tio n ................................................................................. 105 6.3.2 Horn sim u lation.................................................................................... 106 6.3.3 Numerical r e s u lts ................................................................................. 108 CPW A r r a y ......................................................................................................I l l 6.4.1 Folded slot a n te n n a ..............................................................................I l l 6.4.2 Five slot a n te n n a ................................................................................. 114 6.4.3 Slot antenna a r r a y ..............................................................................116 6.5 Grid A rra y .........................................................................................................120 6.6 Cavity O s c illa to r .............................................................................................. 123 6.6.1 Single d ip o le...........................................................................................123 6.6.2 A 3 x 1 dipole antenna a r r a y ...........................................................126 7 Conclusions and Future Research 7.1 128 C o n c lu sio n s......................................................................................................128 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.2 Future R esearch.............................................................................................. 130 References 132 A Usage o f GSM -M oM Code 140 A .l E x a m p le ........................................................................................................... 141 A.1.1 Input file ................................................................................................141 A. 1.2 Geometry f i l e ...................................................................................... 142 A. 1.3 O utput file ......................................................................................... 144 A.2 M a k e file ........................................................................................................... 145 A.3 Program D e sc rip tio n .....................................................................................146 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Figures 1.1 Power capacities of microwave and millimeter-wave devices: solid line, tube devices; dashed line, solid state devices. After Sieger et al. . . . 1.2 Spatial power combiners: (a) grid power combiner, (b) cavity oscillator. 3 4 1.3 Waveguide-based power combining showing active arrays, feeding and receiving horns.................................................................................................. 6 1.4 Typical unit cells: (a) CPW unit cell, (b) grid unit cell........................... 7 2.1 Multiple-level combiner....................................................................................... 14 2.2 A quasi-optical power combiner configuration for an open resonator. 15 2.3 A spatial grid oscillator....................................................................................... 16 2.4 A spatial grid amplifier....................................................................................... 16 2.5 Dielectric slab beam waveguide with lenses.................................................... 17 2.6 Kurokawa waveguide combiner.......................................................................... 19 2.7 Overmoded-waveguide oscillator with Gunn diodes...................................... 20 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.8 Slotted waveguide spatial combiner............................................................21 2.9 Waveguide spatial combiner......................................................................... 21 2.10 Rockwell’s waveguide spatial power combiner: (a) schematic of the array unit cell, (b) rectangular waveguide test fixture.......................... 22 2.11 Quasi-optical lens system configuration with a centered amplifier/oscillator a rra y ................................................................................................................24 3.1 A multilayer structure in metal waveguide showing cascaded blocks . 29 3.2 Definition of electric and magnetic layers................................................. 30 3.3 Geometry of the j th electric layer. The four vertical walls are metal. 33 3.4 Geometry of x directed basis functions......................................................36 3.5 Geometry of the j th magnetic layer. The four vertical walls are metal. 42 3.6 Cross section of a slot in a waveguide : (a) slot in a conducting plane, (b) equivalent magnetic currents................................................................43 3.7 Block diagram for cascading building blocks............................................50 3.8 Rectangular patch showing x and y directed currents............................55 3.9 A flow chart for cascading multilayers.......................................................57 4.1 Geometry of uniform basis functions in the x and y directions........... 59 4.2 Geometry of nonuniform basis functions in the x and y directions. . . x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 4.3 Integral of K 0.........................................................................................................69 4.4 Convergence of Zxx matrix elements................................................................ 72 4.5 Percentage error in the convergence of Zxx m atrix elements....................... 72 5.1 Nodal circuits: (a) general nodal circuit definition (b) conventional global reference node; and (c) local reference node proposed here. . . 78 5.2 Port defined system connected to nodal defined circuit............................... 80 5.3 Grid array showing locally referenced groups................................................. 81 6.1 Wide resonant strip in waveguide, o = 1.016 cm, 6 = 2.286 cm, w = 0.7112 cm, £ = 0.9271 cm, yc = 6/2......................................................... 90 6.2 Normalized susceptance of a wide resonant strip in waveguide.................. 90 6.3 Geometry of patch array supported by dielectric slab in a rectangular waveguide: a = 1.0287 cm, 6 = 2.286 cm, I = 2.5 cm, er = 2.33, d = 0.4572 cm, c = 0.3429 cm, tx = 0.1143 cm, tv = 0.2286 cm........................91 6.4 M agnitude of S n and S2 1 for th e patch array embedded in a waveguide. 92 6.5 Phase of S n and S2 1 for the patch array embedded in a waveguide. . 92 6.6 Slot-strip transition module in rectangular waveguide: a = 22.86 mm, b = 10.16 mm, r = 2.5 m m ................................................................................ 93 6.7 M agnitude of S n for the strip-slot transition module...................................94 6.8 Phase of S n for the strip-slot transition module........................................... 94 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.9 Magnitude of S2i for the strip-slot transition module......................... 95 6.10 Phase of S2i for the strip-slot transition module..................................95 6.11 Center fed dipole antenna inside rectangular waveguide.....................97 6.12 Comparison of Real and Imaginary parts of input impedance. GSMMoM (developed here), M o M ................................................................... 97 6.13 Calculated input impedance for centered and off-centered positions. . 98 6.14 Geometry of a microstrip stub filter showing the triangular basis func tions used. Shaded basis indicate port locations....................................99 6.15 Three dimensional view illustrating the layers of the stub filter. . . . 100 6.16 Port definition using half basis functions..............................................100 6.17 Block diagram for the GSM-MoM analysis of shielded stub filter.. . . 101 6.18 Scattering param eter S n : solid line GSM-MoM, dotted line from . . 102 6.19 Scattering param eter 5 2i: solid line GSM-MoM; dotted line from . . . 103 6.20 Propagation constant: solid lines for air, dashed lines for dielectric. . 103 6.21 Various cascading modes showing convergence of Su ....................... 104 6.22 Various cascading modes showing convergence of 5 2 1 ....................... 104 6.23 A patch-slot-patch waveguide-based spatial power combiner........... 105 6.24 Geometry of the patch-slot-patch unit cell, all dimensions are in mils. 106 6.25 K a band to X band transition................................................................107 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.26 M agnitude of transmission coefficient S 2 1 .....................................................109 6.27 Angle of transmission coefficient S2 1 ..............................................................109 6.28 A two by two patch-slot-patch array in metal waveguide......................... 110 6.29 M agnitude of transmission coefficient S 2 1 .....................................................110 6.30 Geometry of the folded slot in a waveguide................................................. 112 6.31 Real part of the input impedance for folded slot........................................ 113 6.32 Imaginary part of the input impedance for folded slot.............................. 113 6.33 Five-slot a n t e n n a .............................................................................................114 6.34 M agnitude of input return loss for 5 folded slots........................................ 115 6.35 Phase of input return loss for 5 folded slots................................................. 115 6.36 A 3 x 3 slot antenna array shielded by rectangular waveguide............... 117 6.37 Real part of self impedances............................................................................118 6.38 Imaginary part of self impedances..................................................................118 6.39 Real part of self impedances............................................................................ 119 6.40 Imaginary part of self impedances..................................................................119 6.41 Real and imaginary parts for the m utual impedance Z s,u........................120 6.42 A grid array inside a m etal waveguide.......................................................... 121 6.43 M agnitude of input return loss........................................................................122 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.44 Angle of input return loss......................................................... 122 6.45 Geometry of a dipole array cavity oscillator................................................123 6.46 Input impedance of a dipole antenna inside a cavity.................................124 6.47 Block diagram for the GSM-MoM analysis of cavity oscillator................ 125 6.48 Dipole antenna array in a c a v i t y . ...............................................................126 6.49 Magnitude of scattering coefficients for a dipole antenna array inside a c a v ity ..............................................................................................................127 6.50 Phase of scattering coefficients for a dipole antenna array inside a c a v ity ................................................................................................................. 127 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction 1.1 Motivation For and Objective of This Study There is an increasing demand for efficient power sources at microwave and millimeterwave frequencies. These power sources are utilized in commercial and military ap plications such as beam steering, neax vehicle detection radar, sm art antenna arrays, high-resolution radar image system, satellite cross links, and active missile seekers. T he main sources of high power at microwave and millimeter-wave frequencies axe still traveling wave tubes (TW T) and Klystrons. Although these devices axe capable of producing high power levels at high frequencies, they suffer from large size and short life tim e. For these reasons, solid-state devices which have none of these prob lems axe more appealing to use than tube devices. The power levels for both tube devices and solid-state devices axe shown in Fig. 1.1 [1]. It is obvious that a single solid-state device (PHEM T, M ESFET, IMP ATT, etc..) has limited output power at 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R 1. INTRODUCTION 2 the frequency bands of interest with respect to TW Ts. To reach comparable power levels, many solid-state devices must be combined together. Four basic power combining strategies are used in conjunction with solid state technology. These are chip level combining, circuit level combining, spatial combining, and combinations of these three [2]. For chip level power combiners, large transistors with multiple fingers are used to produce higher output powers. In circuit level combiners, Wilkinson power combiner has been used extensively along with newly developed circuit ideas such as the extended resonance method [3]. The limitations imposed by the various technologies such as breakdown voltages, lossy substrates, maximum current densities, and thermal handling capabilities set an upper bound on achievable power levels using either chip level or circuit level combining. Hence, a system level approach th at merges both chip and circuit level is much desired to overcome these drawbacks. Spatial power combining systems have recently received considerable attention [4-7] to combine power from solid-state devices or Monolithic Microwave Integrated Circuits (MMICs) in either waveguides [7,8] or free space [10-13]. Two types of three-dimensional spatial power combiners are shown in Fig. 1.2, a grid-type power amplifier and a cavity-type oscillator. Conceptually spatial power combiners are low loss systems as power is combined in space and hence high efficiencies should be achievable. However, the design of such systems is much more complicated than that of circuit level combiners. The main difficulty is the field-circuit interaction th a t can not be ignored as done often in the circuit level combiners [14,15]. This interaction forces th e integration of electromagnetic and circuit analysis to accurately model the system. Transmission Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 CHAPTER 1. INTRO D U CTIO N ■i " i i 1 1 n r i - 10 r i i i i i ii; i i i 11 h i i i i i n i| _ i Klystrons I Gyrotrons; Gridded ! Tubes , TWT’s VFET FreeElectron ■ Laser- i SiBJT MESFET I -i 10 -2 10 0.1 \ IMPATT j PH EM Y ' \ i i i i mi. ■ fc 1 \ _____ ■ \\ ^ - G u n n ! \V \ 10 100 1000 FREQUENCY (GHz) Figure 1.1: Power capacities of microwave and millimeter-wave devices: solid line, tube devices; dashed line, solid state devices. After Sieger et al. [1] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R l. INTRO D UCTIO N 4 PARTIALLY TRANSPARENT SPHERICAL REFLECTOR OUTPUT BEAM Figure 1.2: Spatial power combiners: (a) grid power combiner, (b) cavity oscillator. line models, unit cell approaches and equivalent lumped elements fall far short in its analysis. The integration of electromagnetic and circuit simulators is inevitable. The question is at which level? Some researchers approached the problem from the electromagnetic point of view by incorporating lumped and nonlinear models into the Finite Difference Time Domain (FDTD) electromagnetic simulators [16], while others solved the semiconductor equations along with the wave equation to arrive at a unique solution th at satisfies both systems of equations [17]. Although this type of analysis takes into account almost all the physical aspects of the circuit behavior, it is very tim e consuming and requires considerable computing resources for a relatively simple circuit. In our view, the most suitable solution is to take advantage of the readily available powerful circuit simulation techniques and integrate it with an efficient electromagnetic simulator. The electromagnetic simulator should produce circuit port parameters th at are converted into nodal parameters and read as a linear Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R l. INTRODUCTION 5 circuit block by a circuit simulator. Hence any nonlinear (iterative) analysis carried out will only include one expensive electromagnetic simulation. Commercial circuit simulation tools such as L ib r a ^ ^ have an integrated Method of Moments (MoM) electromagnetic simulator in this case called Momen tum . However, although efficient, M om entum -^^ works only for structures designed in free space and not waveguides. Dedicated electromagnetic simulators based on the Finite Element Method (FEM), such as the High Frequency Structure Simulator (HFSS), produce output circuit-port files compatible with both Libra and Touch stone. Since it is based on the FEM method it is very general and can, in theory, be applied to any structure. The limiting factor in its effectiveness is the neces sity to discretize the whole three dimensional space. This renders it impractical for electrically large systems such as spatial power combiners. In this dissertation, the focus is on the electromagnetic analysis of waveguidebased spatial power combiners such as th at shown in Fig. 1.3. Active arrays, po larizers, tuning slabs, reflectors and cooling substrates are all placed in transverse planes inside an oversized rectangular waveguide that can accommodate many prop agating modes. The structure is fed using horns or step transformers. The active arrays can be described in general as active transm itting/receiving antenna arrays. The incident wave is detected by a receiving antenna and then amplified by an active element (MMIC). The output of the amplifier feeds a transm itting antenna which radiates into the waveguide. Power combining occurs when the individual signals coalesce into a single propagating waveguide mode. Typical unit cells are shown in Fig. 1.4. The Coplanar Waveguide (CPW ) unit cell incorporates a single ended MMIC amplifier while the grid unit cell uses a balanced differential pair amplifier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 1. INTRO D UCTION 6 B’ B Receiving Herd Horn Transmitting Hard Hom B V .B’ Active Array Figure 1.3: Waveguide-based power combining showing active arrays, feeding and receiving horns. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R 1. INTRO DUCTION 7 Receiving Dipole__« rransmitting Dipole HOIDED SLOTS t I A M P L I H E K Differential Pair (a) (b) Figure 1.4: Typical unit cells: (a) CPW unit cell, (b) grid unit cell. The strategy is to develop a flexible and efficient methodology to electromagnetically model waveguide-based power combining systems and interface it to commercial circuit simulators. For each part of the system there is an opti mum numerical field analysis method. For example, the feeding and receiving horns have been efficiently analyzed using the Mode Matching technique (MM) [18,19]. The planar active antenna arrays are best modeled using the M ethod of Moments. This avoids the unnecessary discretization of the whole volume and limits the dis cretization to planar surfaces. To integrate the two quite different techniques, the Generalized Scattering M atrix (GSM) with circuit ports is introduced. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 1. INTRO DU CTION 1.2 8 Dissertation Overview The dissertation is organized as follows: Chapter 2 presents a review of the various spatial power combining techniques. In this Chapter two and three dimensional spatial power combiners axe also reviewed. Various waveguide power combiners, the focus of this dissertation, are reviewed. Experimental results as well as numerical modeling techniques are discussed. C hapter 3 contains the theoretical developments of the MoM for electric current and magnetic current on dielectric interfaces. The electric and magnetic type Green’s functions are presented as well as the derivations for the GSM for both electric and magnetic current interfaces. The GSMs are computed without calculating the induced current as an interm ediate step. Each GSM is calculated for all modes in one step by assuming the incident field to be a summation of all modes. The GSM also includes the device ports as an integral part of its representation. Finally two cascading formulas are derived to cascade the individual GSMs. C hapter 4 investigates an efficient acceleration technique to speed up the double series summations involved in MoM m atrix element computations. The technique is based on extracting the quasistatic term and applying Kummer trans formation. The impedance and adm ittance m atrix elements are derived for uniform and nonuniform elements. C hapter 5 presents a new circuit theory for interfacing Spatially Dis tributed Linear Circuits (SDLC), with no global reference node, with circuit simula tors. These circuit simulators use the modified nodal adm ittance representation in its implementation and hence the SDLC is transformed from port representation to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R 1. INTRODUCTION 9 nodal representation by means of the local reference node concept introduced in this Chapter. W ith this development the techniques developed in the previous chapters axe made available for integrated and circuit analysis. Chapter 6 contains the results obtained for two classes of problems. The first one is a general class such as waveguide filters, a shielded microstrip notch filter, and a shielded dipole antenna. The second is the waveguide spatial power combiner class. Various examples are given such as patch-slot-patch, CPW, and grid arrays as well as a cavity oscillator. Chapter 7 is a summary of the work presented in this dissertation along with conclusions and future work. 1.3 Original Contributions The original contributions presented in this dissertation are: • Derivation and implementation of the generalized scattering m atrix with de vice ports for shielded electric layers. Device ports are an integral part of the GSM and hence this permits the analysis of grid arrays and strip-like structures containing active elements. • Derivation and implementation of the generalized scattering m atrix with de vice ports for shielded magnetic layers. This permits the analysis of CPW array structures containing active elements. • Efficient calculation of the generalized scattering m atrix based on the method of moments for interacting discontinuities in waveguides. The GSM is calcu- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 1. INTRO D U CTIO N 10 lated for all interacting modes (propagating and evanescent) in one step by considering the incident field to be a summation of waveguide modes instead of a single mode. This eliminates the need to calculate scattering parameters for every incident mode separately. • Implementation of an efficient method of moments formulation for the analysis of planar conductive and magnetic layers with uniform as well as nonuniform meshing. The method is based on the extraction of the quasistatic part in the Green’s function and transforming it into a fast converging series summation utilizing the fast converging modified Bessel functions of the second kind. • Theoretical development of a circuit theory to accommodate spatially dis tributed circuits allowing conventional harmonic balance and transient simu lators to be used. The theory is based on the local reference node concept introduced in Chapter 4. • The investigation of the effect of waveguide walls on antenna elements in spa tial power combiners. It is dem onstrated th at the input impedances of the antenna elements vary considerably when placed inside shielded environment. • Network characterization of strip-slot-strip, grid, CPW , and cavity oscillator arrays in waveguide. The impedance m atrix is calculated for all cases, includ ing self and m utual coupling among array elements. This demonstrates the flexibility of the modeling technique proposed in this dissertation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 1. INTRO D U CTIO N 1.4 11 Publications The work associated with this dissertation resulted in the following Publications: • A. I. Khalil and M. B. Steer, “Circuit Theory for Spatially Distributed Mi crowave Circuits,” IEEE Transactions on Microwave Theory and Techniques, vol. 46, No. 10, Oct. 1998, pp. 1500-1502. • A. I. Khalil and M. B. Steer, “A Generalized Scattering M atrix Method using the Method of Moments for Electromagnetic Analysis of Multilayered Struc tures in Waveguide,” IEE E Transactions on Microwave Theory and Tech niques, In Press. • A. I. Khalil, A.B. Yakovlev and M. B. Steer, “Efficient M ethod of Moments Formulation for the Modeling of Planar Conductive Layers in a Shielded Guided-Wave Structure,” IEE E Transactions on Microwave Theory and Tech niques, Sep. 1999. • M. B. Steer, J. F. Harvey, J. W. Mink, M. N. Abdulla, C. E. Christoffersen, H. M. Gutierrez, P. L. Heron, C. W. Hicks, A. I. Khalil, U. A. Mughal, S. Nakazawa, T. W. Nuteson, J. Patwardhan, S. G. Skaggs, M. A. Summers, S. Wang, and B. Yakovlev, “Global Modeling of Spatially Distributed Microwave and Millimeter-Wave Systems,” IE E E Transactions on Microwave Theory and Techniques, vol. 47, June 1999, pp. 830-839. • A. I. Khalil, A.B. Yakovlev and M. B. Steer, “Efficient MoM-Based Gener alized Scattering M atrix M ethod for the Integrated Circuit and Multilayered Structures in Waveguide,” 1999 IE E E M T T -S International Microwave Sym posium Digest, June 1999. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTE R 1. INTRO D U CTIO N 12 • A. I. Khalil, M. Ozkar, A. Mortazawi and M. B. Steer, “Modeling of WaveguideBaaed Spatial Power Combining Systems,” 1999 IEE E A P -S International Antennas and Propagations Symposium Digest, July 1999. • A. I. Khalil, A.B. Yakovlev and M. B. Steer , “Analysis of Shielded CPW Spa tial Power Combiners,” 1999 IE E E A P S International Antennas and Propa gations Symposium Digest, July 1999. • A. B. Yakovlev, A. I. Khalil, C. W. Hicks, and M. B. Steer, “Electromagnetic Modeling of a Waveguide-Based Strip-to-Slot transition Module for Applica tion to Spatial Power Combining Systems,” 1999 IEEE A P S International Antennas and Propagations Symposium Digest, July 1999. • M. A. Summers, C. E. Christoffersen, A. I. Khalil, S.Nakazawa, T . W. Nuteson, M. B. Steer, and J. W. Mink, “An Integrated Electromagnetic and Nonlin ear Circuit Simulation Environment for Spatial Power Combining Systems,” 1998 IEE E M T T S International Microwave Symposium Digest, June 1998, pp. 1473-1476. • M. B. Steer, M.N.Abdulla, C.E.Christofersen, M.Summers, S. Nakazawa, A.Khalil and J.Harvey, “Integrated Electromagnetic and Circuit Modeling of Large Mi crowave and MillimeterWave Structures,” Proc. o f the 1998 IE E E A P S In ternational Antennas and Propagations Symposium, June 1998, pp. 478-481. • A. Yakovlev, A. Khalil, C. Hicks, A. Mortazawi, and M. Steer “T he general ized scattering m atrix of closely spaced strip and slot layers in waveguide,” Submitted to the IEEE Transactions on Microwave Theory and Techniques. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Literature Review 2.1 Background Combining microwave and millimeter-wave power from solid state sources is an ac tive research area. High power at high frequency is a m ajor goal. A single solid-state device cannot meet this goal. The device size is inversely proportional to the operat ing frequency and so is its power handling capability. Hence, novel power combining techniques th at minimize loss and allow higher device-packing density are a must. Russel [20], in an invited paper, reviews various techniques for coherently combin ing power from two or more sources using circuit techniques. These approaches are separated into two general categories, iV-way combiners and corporate (or chain) combiners. Fundamental as well as practical limitations with circuit level power combiners were discussed. It was dem onstrated th at the number of combined de vices is limited by th e lossy components used in the case of the corporate or th e chain 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 2. L ITE R A T U R E R E V IE W 14 structures. To illustrate this, Russel calculated the efficiencies for various corporate power combiners. He varied the amount of loss introduced by the adders used in each combining stage. For a loss of 0.5 dB in each adder, a 16-device corporate combiner has 65% efficiency. A similar argument was made for the chain structure. The N -way power combiner has less accumulated loss since it has only one stage. The drawback of such a scheme is its realization and bandwidth. For example, the Wilkinson power combiner [21] can not provide sufficient isolation between the N ports at high frequencies when N is greater than 2. In a broader definition, Chang [2] classified power combiners in four classes. These axe chip-level, circuit-level, spatial, and combination of all three. Chang proposed a logical sequence of multiple-level combining with spatial power combining being at a higher hierarchical level as illustrated in Fig. 2.1. SPATIAL COMBINERS Figure 2.1: Multiple-level combiner. Spatial power combining gained attention in the past two decades. Early research focused on experimental investigation of spatial combiners [22]. It was not until 1986 when Mink presented detailed analysis of th e theory of solid state power combining through the application of quasi-optical techniques [23]. In [23], a plano-concave open resonator as that shown in Fig. 2.2 was investigated. An array of sources on a planar reflecting surface was studied by modeling it as current filaments. The driving point resistance of each source in th e presence of all other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 2. L IT E R A T U R E R E V IE W 15 excited sources was calculated. Mink showed th at efficient power transfer, between the array and the wave beam, was obtainable with appropriate spacing between elements. PLANAR REFLECTOR SOURCE ARRAY PARTiALLY TRANSPARENT SPHERICAL REFLECTOR Figure ‘2.2: A quasi-optical power combiner configuration for an open resonator. Other structures used for spatial combining include grid arrays as os cillators or amplifiers [24,25] are shown in Figs. 2.3 and 2.4. The m irror used in the oscillator grid provided the required feedback. The amplifier grid used two or thogonal polarizations at the input and output for isolation. A polarizer at both the input and output were used for th at purpose. The active grid used could be populated with either two or three terminal solid state devices. Patch arrays had been used in spatial power combiners for higher efficiencies and better input/output isolation [26-30]. Slot antennas had also been introduced for spatial combining [31]. Microwave and millimeter power sources will be utilized in, for example, active missiles. This renders spatial power combining systems operating in free space impractical since they occupy large area, difficult to align and are not properly shielded. For these reasons two dimensional (2D) versions of spatial power combiners as well as waveguide power combiners are under investigation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. L ITE R A TU R E R E V IE W ACTIVE GRID SURFACE OUTPUT BEAM MIRROR TURING SLAB Figure 2.3: A spatial grid oscillator. OUTPUT POLARIZER ACTIVE GRID SURFACE \E INPUT BEAM OUTPUT \ BEAM u INPUT POLARIZER TUNING SLAB Figure 2.4: A spatial grid amplifier. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. LITE R A T U R E R E V IE W 17 An im portant part of the 2D structure is the dielectric slab-beam waveg uide presented in [32], shown in Fig. 2.5. In the dielectric waveguide, two waveguiding principals are used. Guided fields in the normal direction to the slab are considered trapped surface waves and largely confined to the dielectric. In the lat eral direction the field has a Gaussian distribution and is guided by the lenses to form a wavebeam that is iterated with the lens spacing. A second paper implemented the ^ l e n s > f c sla b S dielectric slab £ slab phase transformers 8 lens ground plane ^lens < ^slab Figure 2.5: Dielectric slab beam waveguide with lenses. dielectric slab-beam waveguide concept to build, for the first time, a four MESFET amplifier employing quasi-optical techniques [33]. The active antennas used were Vivaldi-type broadband antennas which are gate-receiver and drain-radiators. Three different configurations of the active antennas were studied. In all configurations, the antennas were coupled at the beam waist of th e transverse electric (TE)-type slab mode to provide power combining. The maximum power gain for the ampli Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. LITE R A T U R E R E V IE W 18 fier array was 13 dB at 7.5 GHz. A transverse magnetic (TM )-type dielectric slab with Yagi-Uda slot antennas was introduced in [34]. An amplifier array of 10 GaAs MMICs was fabricated. The array gain was 11 dB at 8.25 GHz and a 0.65 GHz 3-dB bandwidth was measured. 2.2 Waveguide Power Combiners Resonant-cavity combiners have been successfully used in oscillator design. The first design was proposed by Kurokawa and Magalhaes in their 1971 paper [35]. A 12 diode power combiner at X-band was proposed. Each diode was mounted at one end of a stabilized coaxial line which was coupled to the magnetic field at the sidewall of a waveguide cavity. The coaxial circuits were located at the magnetic field maxima and hence spaced one-half wavelength apart as shown in Fig. 2.6. The oscillator operated at 9.1 GHz and produced 10.5 W atts. The circuit configuration was stable and the oscillation theory was developed by Kurokawa in a following paper [36]. Another cavity combining technique was introduced utilizing more solid state diodes placed in a circle inside a cylindrical cavity [37]. W ith this technique, no m inim um spacing (half wavelength in Kurokawa’s model) was required. In an effort to increase th e number of active devices used in Kurokawa’s model, Hamilton modified the design to accommodate twice the number of diodes [38]. This was achieved by placing two coaxial lines on either side of the magnetic field maxima. Using electric field as well as magnetic field coupling, Madihian was able to increase the number of active devices per half guide wavelength from 2 to 3 in a cavity [39]. More recent results were obtained for spatial power combining using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 2. L ITE R A T U R E R E V IE W 19 MAGNETIC FIELD 7 COAXIAL LINE SHORT CIRCUIT Figure 2.6: Kurokawa waveguide combiner. overmoded waveguide resonator [8,9]. An array (N x M ) of T F’lo-mode waveguides containing Gunn diodes was used as the active oscillator array. The resonator con sisted of an overmoded rectangular waveguide with sliding short circuit for tuning as shown in Fig 2.7. The (N x M ) T E \o-mode waveguides coupled energy into the T E n o-mode in the overmoded waveguide through the horn couplers with conversion efficiency of 100%. All other modes in the resonator were suppressed because of th e perfect field distribution m atch between the horn arrays and the overmoded waveguide. A 3 x 3 array was built and tested. The overall efficiency at 61.4 GHz was 83% and an output power of 1.5 W (CW) with a C /N ratio of —95.8 dBc/Hz at 100kHz offset was measured. A power amplifier array using slotted waveguide power divider/combiner was proposed at North Carolina State University [40]. In this work, two waveguides were used as shown in Fig. 2.8. One distributing the input signals and th e other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. LITE R A T U R E R E V IE W SLIDING SHORT 20 N xMWAVEGUIDE ARRAY OUTPITI .GUNN DIODE Figure 2.7: Overmoded-waveguide oscillator with Gunn diodes. combining the amplified signals. The waveguides have longitudinal slots, one-half guide wavelength spaced, th at couple to microstrip lines. An array of eight active devices (FLM0910 2 -W att internally matched GaAs MESFETs) and a passive 8 -way divider/combiner was designed. The power combiner operated at 9.9 GHz with 6.7 dB of gain and 14 W atts of power. The advantage of such a design is its simplicity and good heat sinking. The power devices were mounted on the metal waveguide directly, and so a natural heat sink was provided. If transm itting patch antennas are used, the structure can be used to combine power in free space instead of waveguide combining. The most significant result obtained for spatial power combining to date has occurred at the University of California, Santa Barbara. An X-band waveguide based amplifier has produced a CW output of 40 W peak power with 30% power added efficiency [7]. The combiner is a 2D array of tapered slotline sections (antenna cards). The dominant mode T E \ q is received, amplified, and then retransm itted using slotline antennas. The received signal is amplified using GaAs MMIC devices. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. L ITE R A TU R E R E V IE W 21 Each antenna card accommodates two commercial GaAs MMICs. The results were obtained with four cards placed in a rectangular waveguide. Fig. 2.9 illustrates the basic idea, where only two antenna cards are inserted in the waveguide. The advantages of this system are its wide-band characteristics and reusability. If one of the antenna elements fail, only th at caxd needs to be replaced. This opens the door to modular spatial power combining design. The limitation of the existing design is the axea of the waveguide cross section. It has to be small enough to accommodate only the dominant mode. Amplifiers Microstrip Lines Waveguide Power in Power OUT Figure 2.8: Slotted waveguide spatial combiner. ANTENNA RECTANGULAR WAVEGUIDE RECTANGULAR WAVEGUIDE Figure 2.9: Waveguide spatial combiner. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. L IT E R A T U R E R E V IE W 22 Targeting Ka-band, a Rockwell group, designed a monolithic quasioptical amplifier [5]. The amplifier was packaged in a waveguide that is both compact and suitable as a drop-in replacement for systems th at are designed to use a conven tional waveguide tube-type amplifier. A 2D array of 112 PHEMTs was fabricated and measured. The amplifiers coupled to individual input and output slot antennas, with orthogonal polarizations. The array provided a peak gain of 9 dB at 38.6 GHz and 29 dBm maximum output power. The unit cell used in the array design as well as the waveguide package are illustrated in Fig. 2.10. OUTPUT FIELD i i INPUT FIELD (•) (b) Figure 2.10: Rockwell’s waveguide spatial power combiner: (a) schematic of the array unit cell, (b) rectangular waveguide test fixture. 2.3 Num erical M odeling and CAD Experimental results were obtained for various spatial power combining topologies, but the output power levels are still much smaller than expected. W ith the help of a dedicated Computer-Aided Engineering (CAE) environment, it is anticipated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 2. L IT E R A T U R E R E V IE W 23 that better designs and higher power levels can be obtained. There are numerous commercial Computer-Aided Design (CAD) tools in the area of microwave circuits and antennas. These CAD tools can not simply be combined to analyze spatial power combiners [41]. The reason is the complex nature of spatial combining sys tems. In such systems, many different components are integrated together such as active devices (diodes, transistors, MMICs, etc.), passive lumped and/or distributed elements, radiating elements, and cooling elements. In this section we will review the efforts in developing CAD tools for spatial power combiners. Many of the approaches developed to model spatial power combiners assume infinite arrays in free space. W ith these assumptions, the analysis is greatly simplified. Using a simple equivalent circuit approach, Popovic et al. separated the equivalent circuit of the grid from that of the active circuitry [10]. In this analysis only a unit cell was considered assuming an infinite periodic array. Further more, electric and magnetic walls were used restricting the input to only incident TEM wave [42]. Still assuming an infinite array, Epp et al. of JPL , presented a novel approach to model quasi-opticai grids [43,44]. They decomposed the incident and scattered fields into a summation of Floquet modes. The modes interacted with the device ports. To account for till interactions, they characterized the unit cell using a generalized scattering m atrix method with device ports. This allowed a general representation of the incident field. Also, the interaction of various quasioptical components such as polarizers, lenses, and feeding horns can be used if their appro priate GSMs are computed. In implementation the surface currents were calculated using a spectral domain m ethod of moments formulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. LITE R A TU R E R E V IE W 24 To accurately model spatial power combiners of small to moderate sizes, the unit cell approach is neither practical nor accurate. The edge effects will not be modeled and consequently the driving point impedances for all unit cells will not be the same as predicted by the unit cell approach. A full wave analysis of the whole structure is thus essential. Pioneering work was done here at North Carolina State University to model open cavity resonators and grid arrays [45-51]. Heron [46] developed a Green’s function for the open cavity resonator. This Green’s function is composed of two parts: resonant and nonresonant terms. The fields were represented using Hermite Gaussian wave-beams. Nuteson [51], implemented the previously developed Green’s function using the method of moments. He also developed a dyadic Green’s function for a lens system consisting of two lenses and an array of active devices as that shown in Fig. 2.11. The Green’s function was derived by separately considering the paraxial and nonpar axial fields. A combination of spatial and spectral domain techniques was used in computing the method of moments m atrix elements. Amplifier / Oscillator Array Lens Transmitting Horn Horn z=-D Figure 2.11: z=0 z=D Quasi-optical lens system configuration with a centered ampli fier/oscillator array. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2. L IT E R A T U R E R E V IE W 25 An integrated electromagnetic and nonlinear circuit simulation environ ment for spatial power combining systems was proposed in [52]. This represented the first result where a full wave analysis and nonlinear circuit simulation were carried out in the analysis of finite size grid arrays. A 2 x 2 grid array was fabricated using differential pair transistor units. A harmonic balance, nonlinear circuit simulator, was used to simulate the system’s nonlinear behavior. Three-dimensional electromagnetic analysis was also applied to spatial power combiners [53] and active antennas [54]. The main disadvantages of such techniques are their large memory demand as well as computational resources. The power of such methods reside in its flexibility to model complicated structures. However, spatial power combiners are planar in shape and that enables the MoM to be applied efficiently in the analysis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 M odeling Using GSM 3.1 Introduction A typical waveguide-based spatial power-combining system of the transverse type, such as that shown in Fig. 1.3, consists of passive components (horns, polarizers, lenses, etc.) and active components (grid arrays, patch arrays, etc.). Electromag netic modeling of such systems can be memory demanding and very time consuming. The main reason for this is their electrically large sizes. The most efficient and flex ible way to model these systems is to partition them into blocks. Each block is modeled separately and characterized by its own GSM [55,56]. This ensures that propagating mode coupling is accounted for as well as evanescent mode coupling. Also, since each block is considered separately, the GSMs are computed using the most efficient EM technique for each particular block (eg. MoM, mode matching, and FEM). Cascading all blocks leads to a m atrix describing the entire linear sys- 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 3. M ODELING USING GSM 27 tem response. Since the feeding horns have been analyzed elsewhere [18], we will focus on the body of the waveguide spatial power combiner, such as polarizers, grid arrays, patch arrays, CPW arrays, etc.. The GSM is derived for the fundamental building blocks. These are the electric current interface, the magnetic current interface, the dielectric interface and the short circuit. W ith these fundamental blocks, almost all multilayer transverse active arrays can be modeled. Two different formulas are derived to cascade the GSMs of individual blocks. 3.2 GSM-MoM W ith Ports The Generalized Scattering M atrix (GSM) m ethod has been widely used to charac terize waveguide junctions and discontinuities. The GSM is a m atrix of coefficients of forward and backward traveling modes and describes all self and mutual interactions of scattering characteristics, including contributions from propagating and evanes cent modes. Thus structures of multiple discontinuities are modeled by cascading a num ber of GSMs. The GSM is adapted here to globally model waveguide-based spatial power combining systems. In such systems a large number of active cells radiate signals into a waveguide, and power is combined when the individual signals coalesce into a single propagating waveguide mode. Most spatial power combiners can be viewed as m ultiple arbitrarily layers of electric or magnetic currents arranged in planes transverse to the longitudinal direction of a metal waveguide. Active de vices are inserted at ports in some of the metalized or magnetic transverse planes. In this Chapter we efficiently derive the GSM and introduce circuit ports (ports with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTE R 3. MODELING USING GSM 28 voltages and currents) into the GSM formulation. This facilitates the incorporation of the electromagnetic model of a microwave structure into a nonlinear microwave circuit simulator as required in computer aided global modeling. The problem of modeling multilayered structures with ports in a shielded environment can be analyzed by at least two approaches. In the first, a specific Green’s function for the proposed structure is constructed and then the method of moments (MoM) [57] is directly applied to the entire structure. This results in severe computational and memory demands for electrically large structures. The second approach, proposed here, is to characterize each layer using a GSM with circuit ports and then cascade this m atrix with its neighbors to obtain the composite GSM of the complete system such as th at shown in Fig. 3.1. Various formulations have been used in developing the standard GSM (without circuit ports). The mode matching technique is the most widely used for waveguide junctions and discontinuities of relatively simple geometries [58]. The FDTD has been introduced to calculate the GSM for complex waveguide circuits [59]. The MoM has also been used in developing the GSM of arbitrarily shaped dielectric discontinuities [60], metallic posts [61,62], waveguide junctions [63,64], and waveguide problems with probe excitation [65]. In its common implementation, the MoM uses subdomain basis functions of current. This implementation is used here to compute a port impedance m atrix in the solution process [6 6 ]. As well as using subdomain current basis functions on the metalization, the MoM formulation implemented here uses delta gap voltages and so the MoM characterization yields port voltage and current variables. The ports are explicitly defined in the GSM and they are accessible after cascading. The m ethod can address a wide class of problems such as a variety of shielded multilayered structures, iris coupled filters, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 3. MODELING USING GSM 29 input impedance for probe excited waveguides, and waveguide-based spatial power combiners. From this point on we will refer to a circuit port as just a port, and an electromagnetic port, which axe defined for incident and scattered modes, as a mode. DIELECTRIC INTERFACE MAGNETIC INTERFACE ELECTRIC INTERFACE CONDUCTOR INTERFACE Figure 3.1: A multilayer structure in metal waveguide showing cascaded blocks The key concept in the method developed here is formulation of a GSM for one transverse layer at a time, and the GSM of individual blocks are cascaded to model a multilayer structure. The general building blocks considered here are • Electric current interface with ports • Magnetic current interface with ports • Dielectric interface Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 3. M ODELING USING GSM 30 • Perfect conductor interface An electric current interface is defined as an interface where the con ducting portions are small with respect to the dielectric portion. Hence, it is more efficient to analyze the conducting (electric) than the nonconducting portion. Sim ilarly, a magnetic current interface is defined as an interface where the dielectric portions axe small with respect to the conducting portion. Hence, it is more effi cient to analyze the dielectric (magnetic) than the conducting portion. The electric and magnetic current interfaces axe shown in Fig. 3.2. H, Hi Electric Layer Magnetic Layer Figure 3.2: Definition of electric and magnetic layers. The electric current interface with ports is used in th e analysis of mi crostrip, grid and stripline structures, while the magnetic current interface with ports is used for CPW structures. The analysis begins by expressing th e phasors of the electric and mag Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 3. M ODELING USING GSM 31 netic field vectors in terms of their eigenmode expansions [67] E +{ x ,y ,z ) = '5 2 a {E ? (x ,y ,z) /=i (3.1) H + ( x ,y ,z ) = ' £ a{ H f( x ,y ,z ) i=i (3.2) 00 ~ E ~ (x ,y ,z) = ^ ,b { E ^ ( x ,y ,z ) i=i (3.3) ~ H ~ (x,y,z) = '52b{T T [(x ,y ,z) (3.4) /= i where the individual electric and magnetic eigenmodes are !?,*(*, y ,z) = { e i ± e zt) e x p ^ IY r) H f i x ^ z ) — (± h i + h zi) e x p (^ r jz) The propagation constant of the I th mode is defined as: r, = \]kli - k) , kj < kct (3.5) j \ J kj - k cl , k cl < ki with kd = yjkh + kyi, kj = UyjyotQtj , kxi = rrnr/a and kyt = n x/b . For simplicity the index pair (m ,n) has been replaced by a single index L Note th a t all Transverse Electric (TE) and Transverse Magnetic (TM) waveguide modes are considered. The amplitude coefficients of mode / are denoted as a/ and 6 ; for waves propagating in the positive and negative z directions, respectively. The “± ” sign indicates propagation in the positive and negative z directions, respectively. The electric and magnetic mode functions, e; and hi, are normalized using the normalization condition J Jjei x hi] • (z) ds = 1 resulting in th e following expressions for T E and TM modes: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3. MODELING USING GSM 32 TE-inodes: ex = C ky \[Zh cos(kxx) sin(Aryy ), ey = — C kx \[Zh sin(kxx) cos(kyy), hx = kx C - 7 == sin(A:xa:) cos(kyy), hy = k C —7 == cos(fcxa:) sin{kyy) y/Zh (3.6) TM-modes: ex = — C kx\fZ c cos(kxx) sin{kyy), ey = — C ky\ f z e sin(&xx) cos(Arvy), ky C - 7 = sin(fcxx) cos(Aryt/), y/Ze kx hy — C - 7 == cos(Arxar) s\u{kyy) y/Ze hx = - (3.7) where ^ 1 ‘ [iomtOn V r, _ JUf*0 ^ " — ’ Ze - ; W ’ and A is the waveguide cross section 3.3 Electric Current Interface The concept behind the procedure th at follows is th at distinct waveguide modes are coupled by irregular distributions of conductors at the dielectric/dielectric interface. The regions at the interface th at are not metalized do not couple modes. The characterization of the metalized interface is developed by separately considering mode to mode, port to port, and port to mode interactions [44]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 3. M ODELING USING GSM 33 The general building block is shown in Fig. 3.3. Here an arbitrarily shaped metalization is located at the interface of two dielectric media with relative permitivities ej and C(j+i), respectively. For illustration purposes an internal port is specified to show the location of a device and an excitation port is defined in connection with the source or load although the number of circuit ports is arbitrary. The vector of coefficients [<z{] represents the coefficients of modes incident from medium j into medium j + 1 , [aj] represent the coefficients of modes incident from medium j + 1 into medium j , and [0 3 ] represents all coefficients of power waves incident from the circuit ports. Similarly [6 ^] are the vectors of reflected mode or power wave coefficients corresponding to [af] , i = 1,2,3. The relations between [&•'] and [a1-] will be determ ined in this section and the m atrix relationship is the GSM. ITATlOft PORT LOAI Figure 3.3: Geometry of the j th electric layer. The four vertical walls are meteil. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 C H APTE R 3. MODELING USING GSM 3.3.1 M ode to mode scattering In this section, only the layer at the interface of the dielectric media is considered and the m atrix model developed relates the variables at the ports to the coefficients of the modes (in each dielectric medium) th at axe incident and reflected at that layer. First, the MoM is applied to the problem and then the GSM is calculated. The electric field integral equation formulation is obtained by enforcing the following impedance boundary condition on the metal surface: T ( r ) + E a(r) = Z ,J (r) (3.8) where 1T* denotes the tangential incident field, ~E“ the tangential scattered field, Z, the surface impedance, and J is the unknown surface current density. Later on, the surface impedance will be used to represent the lumped load impedances of the ports. The first step in the MoM formulation is to express the scattered field in terms of the electric dyadic Green’s function (Ge): E a(r) = J f sp t { r , r ) - 7 ( r ) ds (3.9) Here prim ed coordinates denote the source location while unprimed coordinates denote the observation location. In general the electric dyadic Green’s function has nine components G’J, where i , j represent the Cartesian coordinates x,y, and z [6 8 ]. In our case we axe only concerned by the transverse components which limit the required dyadic Green’s function to four components. G ^ ( x ,y ; x ',y ') = £ £ m=0 n= 0 y )< m n (X'» »')/«.»»»» m=0n=0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTE R 3. M ODELING USING GSM G f ( x ,y , x ', y ') = £ 35 X > " mn( x , y ) < mn(x',y')/le>mn, m =0 n=0 G f (x, y, x', S/') = £ £ V’e.mnC^ ^K m n • (3.10) m =0 n=0 The functions <p%mn{x,y) and y) represent a complete set of orthonormal eigenfunctions satisfying appropriate boundary conditions on the surface of the metal waveguide: < m n ( x >y) = J ^ j ^ c o s ( k xx)sin(kyy) , = y j ^ ^ s m ( k sx)c°s(kyy) (3.11) with conn con being Newman indexes such that, too = 1, and com = 2 , m ^ O . Fi nally, the one-dimensional Green’s functions f e,mn(z, -s'), h^mn(z, s'), and ge,mn{z, z') are calculated on the interface at z = s ' = e.mn 0 : (*; - t p r , + (fcg - t ; i r , (Ti + I ^ W ^ e i + T ^ ) , t c,m n — J ^(^ei + rie2)’ = (fc; - fe;)ra+ (fcj - feg)r, J (r, + r2Mrj£, + rlti) ' In solving for the scattered electric field £ * (r), the surface current density J { r ) is expanded as a set of subdomain two-dimensional basis functions: 7(r‘) = Z l i % ( r ) (3.13) 1=1 where B i is the i th basis function and /,- is the unknown current am plitude at the i th basis. Each basis corresponds to one of N ports. Typical sinusoidal basis functions in the x direction are shown in Fig. 3.4. Using the current expansion formula (3.13) and the integral representa tion for the scattered electric field (3.9), th e impedance boundary condition (3.8) is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 3. MODELING USING GSM x-c 36 x„+c Figure 3.4: Geometry of x directed basis functions. w ritten in terms of the Green’s function as f E i(r) = - ' £ l i [ G e( r , r ) • 5 ,( r ') ds + ZsJ{r) ;=i J Js' (3.14) A Galerkin procedure yields the discretization of the integral equation (3.14): J = - i t I, j j s j Js i B ,(r ).!5 , ( r , r t).B ,(r')d id 3 + t ‘i ( j=l J JS (3.15) leading to a m atrix system for the unknown current coefficients / = [/i •••/,••• • I^]T: [2 + Z L][I] = [V] (3.16) where the j i th element of the impedance m atrix [Z\ is Zji = ~ j f s j f s ,'B j { r ) G '( r 1r ) • B i( r ) d s d s (3.17) the j th port voltage vi = f f s B i(r) J ' M * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.18) CH APTER 3. M ODELING USING GSM 37 and the load impedance Z li [^ ] = (3.19) Z Li •• • 0 0 • • • Z ln with Z n being the loading impedance at port i. If port i is not loaded then its corresponding entry is zero [69]. Conventionally, the GSM is constructed one column at a time. This is achieved by exciting the structure by a single mode. The excitation mode generates reflected and transm itted modes. The computed coefficients of these modes fill a single column in the GSM. This filling process continues until the whole m atrix is completely filled. It is obvious that for a large GSM the conventional approach is very time consuming. In order to construct the GSM efficiently, it is essential to treat the incident field as being composed of a summation of waveguide modes rather than considering a single mode one at a time [70]. For an incident field propagating in the positive z direction from medium . £*(r ) = 1 into medium 2 at the interface £max i=i al (1 + R i) ef e r p ( - r ,xs) (3.20) where T*, ef axe the propagation constant and the electric mode function of mode I corresponding to medium 1, respectively. Ri is the reflection coefficient of mode /, defined so th at the transverse electric and magnetic mode reflection coefficients are R i* = I7 +r? Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.21) C H APTER 3. MODELING USING GSM dTM R ‘ _ ~ F /« l 38 — r?£, + rft, (3'22) The incidentelectric field defined in (3.20) consists of two parts, incident and re flected waves. This is im portant to account for the dielectric discontinuity at the interface. Using this expression for the incident field, the port voltages of an electric current layer located at z = 0 is given by Lm ax ___ y* a Vi = £ a< (! + f t ) / L * lB j( r ) d s i=i J JS (3.23) Hence the m atrix form, (3.16), can be w ritten as [Z + Z Lm = [W'\[U + R)[a\] (3.24) Where the current vector [/] is written in terms of the modal vector [<■{] = W as [/] = [r][W'1][CT+i!)K) (3.25) the adm ittance m atrix [V] = [Z + Z L] - 1 the elements of the [W?] m atrix is given by is U is the identity m atrix and R is a diagonal m atrix with diagonal elements being the modal reflection coefficients. Scattering from both the metalization and the dielectric interface leads to scattered fields with mode coefficients 6i = j j f j ^ ds + Rial i = L.Tmox (3.26) Using the current density expansion (3.13) the coefficients of the scattered modes £] can be w ritten as [‘il = 4 [ t f + * l [ » T M + [ * M Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.27) CH APTER 3. M O DELING USING GSM 39 where T indicates the transpose m atrix operation. Substituting the expression for the electric current (3.25) into (3.27) results in the following representation: Ml = ( - j [ V + + R] + [*])[»;] (3.28) Since [b\] = [5|J[aJ], we can readily write [Si'll = - \ { U + R)[W'}r {Y]{W')[U + R) + [fll (3.29) and [Sj,] = -i[C \[W '\ t [ Y \ { W 'W + fi) + [C] (3.30) where [C] is a diagonal m atrix representing the transmission coefficients. The obtained expressions (3.20) to (3.30) is for an incident field traveling in the positive z direction from layer 1 into layer 2. By symmetry, when the incident field is propagating in the negative z direction from layer 2 into layer 1 , we can write [5{J = ~ \ [ V - R\[W2\t [ Y ] [ W \ U - R) - [*1 - fll + [C ] (Si'J = (3.31) (3.32) Equations (3.29)-(3.32) are a full representation of scattered modes due to incident modes on a loaded scatterer residing on the interface of two adjacent dielectrics inside a meted waveguide. 3.3.2 Mode to port scattering The interaction between an incident mode and a port can be described using the concept of generalized power waves [44j. First assume th at port k is term inated by an arbitrary impedance Since the scattering param eters are normally given Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 3. MODELING USING GSM 40 with reference to a 50 (I system it is appropriate to set Z ik to R q = 50 fi. The generalized power waves at the ports are then given by [71] Vi = + (3.33) V, = l- (Vt - R o h ) (3.34) “* = 7 E ( 3 ' 3 5 ) bk = 7S; (3-36) Where VJ and VT are the incident and reflected voltage waves. When there is no excitation at port fc, K = 0 and VT = —R ah . Hence the scattered power wave coefficient at port k due to mode excitation is bk = —v ^ o h> Thus the scattering coefficients at the ports due to incident modes from medium 1 can be written in a m atrix form as t e i = -[* » ]■ m ( 3 .3 7 ) Substituting for the current using (3.25) and recalling that [6 3 ] = [5 ^][a]] the scat tering submatrix [SJJ = -[flo]*[11[W',][tf + R ] (3-38) Similarly, the scattering coefficients at the ports due to incident modes from medium 2 can be w ritten as [Sl2] = - m * [ Y ] [ W 2][U - R 1 (3.39) By reciprocity the scattering m atrix of modes due to port excitation is readily ob tained as [ S y = [5^]r and [S]3] = [ S |jT. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 3. MODELING USING GSM 3.3.3 41 Port to port scattering Port quantities axe related by a scattering m atrix which relates port to port scat tering [71]: P s J = [*bl*[Zp + Ro]~l [ZP - j y [Jio]-* (3-40) where Zp is the port impedance matrix, obtained by selecting the appropriate rows and columns from the MoM impedance matrix [Z]. 3.4 M agnetic Current Interface A similar analysis to the electric current interface is carried out for the magnetic current interface in this section. Distinct waveguide modes are coupled by irregular distributions of magnetic current at the dielectric/dielectric interface. The charac terization of the magnetic interface is developed by separately considering mode to mode, port to port, and port to mode interactions. The general building block for the magnetic interface is shown in Fig. 3.5. Here a CPW structure is located a t the interface of two dielectric media with relative permitivities e,- and respectively. A three terminal device is explicitly drawn to illustrate the location of the device ports. 3.4.1 Mode to mode scattering In this section the mode to mode coupling for a magnetic layer is calculated. Let us consider an aperture in a conducting plane transverse to the direction of propa- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3. M ODELING USING GSM 42 Figure 3.5: Geometry of the j th magnetic layer. The four vertical walls axe metal. gation at the interface of two adjacent dielectrics with relative permitivities Ci and £2 as shown in Fig. 3.6. The equivalence principal isapplied representation for the field in region 1 to obtain separate (Z < 0) and region 2 (Z > 0) [72,73] by short circuiting the aperture (covering the aperture by an electric conductor). Assuming a propagating wave TF is incident from region 1 into region 2. The field in region 1 is determined by the incident field and the equivalent magnetic current M over the aperture area produced by the tangential electric field on the aperture E t. The field in region 2 is determined only by the equivalent magnetic current —M only. The equivalent magnetic currents M and —M ensure the continuity of the tangential components of the electric field across the aperture. I f = az x region 1 (3.41) [z = 0 “ ) H \ =TTt + H \{M ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.42) CH APTE R 3. M ODELING USING GSM region 2 [z = 0 43 +) (3.43) Where H\ is the total tangential component of the magnetic field on the aperture in region j’, j = 1,2. H{(M ) is the tangential component of the magnetic field on the aperture due to the magnetic current M in region j . Equating the tangential magnetic fields on both sides of the aperture given by (3.42) and (3.43): (3.44) - H i = H]{M ) + H f ( M) It should be noted th at due to the presence of the perfect conductor the incident magnetic field is doubled. Also, the image theory can be applied and the magnetic currents M and —M are doubled as well when calculating the fields in regions 1 and 2, respectively. The magnetic fields H i[M ) can be expressed in terms of the integral equation W{M) = j r , r ) ■ M ( r ) is' (3.45) W here G ^ (r, r ) is the magnetic Green’s function in region j . The incident magnetic 1 Y (•) (b) Figure 3.6: Cross section of a slot in a waveguide : (a) slot in a conducting plane, (b) equivalent magnetic currents. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 3. MODELING USING GSM 44 field defined by (3.44) is then written in its integral form using (3.45) as -Hi = I J &m( r , r ) ■ M ( r ) da (3.46) Where Gm(r, r')=Gm(r, r ) + G ^ ( r ,/ ) . with transverse components GX* { x ,y \x ',y ') = £ m =0n=0 m =0 n=0 m =0 n=0 G»f(x, y; y') = £ £ ¥&,«»(*, «»(*'» • (3.47) m =0 n=0 The functions tr Jm )m n ( x i2 / ) represent a complete set of orthonor mal eigenfunctions satisfying appropriate boundary conditions on the surface of the metal waveguide: v C m » (* » y ) = y ^ j r sin( k x ) cos( M ) > y) = cos( ^ x ) sin(Aryy) with Com? eon being Newman indexes such that, eoo = 0. (3.48) 1,and Com = 2, m ^ Finally, the one-dimensional Green’s functions f m,mn(z,z'), h,m,mn(z, ~), and Sm,mn(2 | 2 ;) are calculated on the interface at z = z' = 0 : ,• r( * ? - g ) , { k i - k D , c r, _ 9 m ,m n — r2 j^cky r 1 1’ 1 , m,mn uiu Tj. r2J’ y (fc?-fcv2) (3 -* * ), Uffl [ p it + p J (3.49) 1 2 To solve the integral equation (3.46) for the unknown magnetic current vector M , M is expanded as a set of subdomain basis functions: M ( r ) = £ K-^-(r') i= 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.50) 45 C H APTER 3. MODELING USING GSM where 5,- is the i th basis function and K is the unknown magnetic current amplitude at the i th basis. Each basis corresponds to one of N ports. A Galerkin procedure yields the discretization of the integral equation (3.46): - j Js'Bj(r).lf{r)ds = ? V5/ I s l (3.51) leading to a m atrix system for the unknown voltage coefficients [V] = [V1---V i ---VN]T: (3.52) m m = [/] where the j i th element of the adm ittance m atrix [y] is Y„ = - j j j j f i 3 J(r).5m(r,r') ■%(/)di'is (3.53) and the j th port current / i = - / / * i(r) J f (r)d» (3.54) The linear system of equations (3.52) is for an unloaded aperture. If the aperture contains device ports, the loaded aperture has an adm ittance m atrix of [Y + Yl \. Where Yu. ini = * 0 0 (3.55) YLi 0 0 • ** I'LiV where Y u is the load adm ittance at port i. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 3. MODELING USING GSM 46 As previously stated, to construct the GSM efficiently it is necessary to assume the incident wave as a summation of waveguidemodes. For magnetic field propagating in the positive z direction from medium 1 an incident into medium 2 at the interface (Z = 0) , Lmax 7T(r) = £ al A,' (3.56) /= 1 Using the expression for the incident magnetic field given above, the electric current defined by (3.54) is written as Lmax r t i __ ‘i = E ° ‘ / / 1=1 J JS (3-57) which leads to the m atrix representation [/] = [ W V J (3.58) where w Ji = J j V i . B j ds (3.59) W ith this expression for the current, the voltage vector [V] is written as [V] = [ZHH'MM] (3.60) where the impedance m atrix [Z] = [K + The modal amplitude b\ representing the scattered mode /, due to the magnetic current M , is written in terms of the induced magnetic current = \ f f2MH)ds (3.61) expanding the magnetic current M as given by (3.50)leads to the following repre sentation of th e modal coefficients ft] = [M 'T ft] = IWYmW1}^) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.62) C H APTER 3. M ODELING USING GSM 47 Similaxly, the amplitudes of the transm itted modes are [6>] = \ W * n Z ) [ W ' ][<.{] (3.63) hence the scattering submatrices for the reflected (due to M and reflection from the conductor) and transm itted modes axe [Sn] = [W']T[Z][W'] - [£/] (3.64) [S«] = {W*]T[ZHW1} (3.65) When the field is incident from region 2 into region 1 the reflected and transm itted submatrices, [£2 2 ] and [S2 1 ] are similaxly derived and given by [&»] = [W’RZN W '2] - [V] (3.66) [Sid = [H",1T[Z ][»'2] (3.67) where [17] is the identity m atrix. 3.4.2 Mode to port scattering Referring to (3.33)-(3.36), when there is no excitation at port k, Vi = 0 and Vr = V*. Hence the scattered power wave coefficient at port k due to mode excitation is bk = R ^ 2Vk- Thus the scattering coefficients at the ports due to incident modes from medium 1 can be w ritten in m atrix form = [«o]-»[V] (3.68) Substituting for th e voltage vector [V] using (3.60) and recalling that [6 3 ] = [5^] [a]] the scattering subm atrix [Sk] = m * [Z][Wl] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.69) C H APTE R 3. MODELING USING GSM 48 Similarly, the scattering coefficients at the ports due to incident modes from medium 2 can be written as I 3 J = [Bo]-^\Z)\W*] (3.70) By reciprocity the scattering m atrix of modes due to port excitation is readily ob tained as [Si3] = [S^]T and [Sj3] = [S^2]T. 3.4.3 Port to port scattering Port quantities are related by a scattering m atrix which relates port to port scat tering [71]: [S*\ = [Ro]*[Zp + Ro]~l [Zp - /2o][/2o]“ ? (3.71) where Zp is the port impedance m atrix. 3.5 Dielectric and Conductor Interfaces In the absence of metalization there is no coupling of modes at the dielectric in terface. Hence the scattering m atrix is diagonal. For a dielectric interface between medium 1 and medium 2 with, relative permitivities ei and e2i respectively. The scattering parameters are given by = diag (Ri~.Ri...RLmax) [Si2] = diag(Ci...C7...Cx max) [Sn] [S1 2 } [5n] = [S22] = d i a g ( - RLmax) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3. MODELING USING GSM 49 where V rjr? ' r,1+ r? ftTM ' r;e2+ r?e, As expected (R f E)2 + ( C f E ) 2 = 1 and ( R f M)2 + (C?M)2 = 1 indicating conservation of power. For a perfect conductor interface, the reflection coefficient is simply —1. Hence its scattering m atrix is diagonal with 3.6 — 1 as its diagonal element. Cascade Connection The technique discussed in the previous sections develops a GSM for a single inter face at a transverse plane (with respect to the direction of propagation) in a metal waveguide. A multilayer structure such as th at shown in Fig. 3.1 is modeled by cas cading the GSMs of individual layers and propagation matrices. Each propagation m atrix describes translation of the mode coefficients horn one transverse plane to another through a homogeneous medium. Several cascading formulas axe found [74] for cascading two port networks. Two cascading formulas of three port networks (involving modes and device ports) are derived in the following section. The modeling of a two layer structure with the layers separated by a waveguide section is illustrated in Fig. 3.7. The analysis proceeds by computing the GSM of the first layer [S ^ ] and then evaluating a propagating m atrix [P\ describing the waveguide section. Finally, computation of the GSM of the second layer [5 ^ ] enables cascading of [S*1*], [P] and [S ^ ] to obtain the composite GSM [S ^]. Each Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 3. MODELING USING GSM 50 INPUT MOOES OUTPUT MODES PORTS PORTS Figure 3.7: Block diagram for cascading building blocks, block is represented by [(.‘I = [ s i ' M , i = 1,2 (3.72) where Qi J ll [S (i|] = S[2 $13 C« C« *->21 S 22 °23 C« °31 S 32 (3.73) C* °33 In calculating the composite GSM the internal wave coefficients [aj], [6 3 ], [a*], and [6 f] must be translated through the waveguide section. This is achieved using the propagation m atrix [P] = diag(exp(-rid)...exp(-r/d)...exp(-r£max</)) where d is the waveguide section separating the two layers and [P] is a diagonal m atrix as the modes do not couple in the waveguide section and each is translated by its exponential propagation constant. So the internal mode coefficients are related Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 3. MODELING USING GSM 51 by [b\] = i / r i [«2 i (3.74) [%}= (3.75) The coefficients [6 |] and [6 f] can be written using (3.72) as (3.76) [® = [ S k M + [S&[<# + [S&[a& f t] = [ S I M + [5?2 ][a22] + [S 23 ][a2] (3.77) Thus the internal mode coefficients, [a2] and [a*], can be written in terms of the modes at the external interfaces: [-5] = [ * |( [ s ? li m [ s j j [ « a + [ s j M s y M + [s m + (s*j[*^j), (3.78) and [«3 = [ff.]([sjj[«i] + p i M S M + [S{J[J’H^J[4I + ( s j « ) (s-™) Here the matrices [Hi] and [Hi] are given by and m = ([£ /i- r a s m s y r ' t i i Combining (3.74)-(3.79) yields the composite scattering m atrix Cc ° ll Cc ° 1 2 Cc *-'13 Cc °14 Cc °23 Cc ‘-'24 Cc Cc ° 2 1 * ^ 2 2 Cc °31 Cc °32 °3 3 Cc J 34 CC Cc *-, 42 cc J 43 Cc C4 4 [5(c)] = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.80) CH APTER 3. M ODELING USING GSM 52 with submatrices [5,M [Sfd = [Sti[Ht}[SiH [*^13 ] = [^3] + [5112][^2][5121][P][5213] [■S'u} = [S'n][Hi)[Sti [•$21 ] = [S%\[Hx][Stt [Sc22] [*^23 ] = [5U [ s im r a = [5 |3] + [5221][^ ][5 212][P][5123] = P i] + [‘S’3 2 ] = [^33 ] = [ ^ ] + [^ ][/f2 ][5 21][P][5213] [S5J = P ilM t f J = p iH fr jp i] KJ is „ c ] = [•?322] + [5 |1][^ i][5212][P][522] [S5J = [S&[Bi\[S& [SI4 ] = [ ^ ] + [521][^ ][5 212][P][523] This representation involves two inverted matrices [Hi] and [Hi]. An alternative representation that involves only one inverted m atrix is also derived below. The internal mode coefficients, [a2l and [aj], can be written in an alternative form as [a i] = [^l([*^2i][a i] + + [*^22! [-^1[-^13[“ 3] + [*5*23] [**3])» and [<41 = m ([5f,im [5j,i[«ii+([s?,im [5y [ i >i[ s ? j + K !i)[<.ii+ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3-81) C H APTER 3. MODELING USING GSM ( [ s m s m s y + [s 53 u m + Where [H] = ([£/] - m 2][/>][SIM)-'[/>] and the composite scattering can then be derived as [s.M = [SfJ = [*^13 ] = [su + isu m su m su [SU + [S112][P}[SU[H}[SU [Scu ] = + [ s m x m is m s f ti [*^2l] = [su m su [SU = [5|2] + [5 |1][^ ][5212][P][5 i22] [sy = [S5J = KJ = [su m su [su + isu im isu n su [^ 1] + [5'312][/3][5121][^ ][5211] [‘S’laJ = [SU = [^ + [s& m sm is& [*^34 ] = [ ^ [ P i r a + [5U [P )[S?JO T 5y [F][52J [54M = [& m sy [Sc„ } = [SU+[SU[H][SU[P}[SU [5SJ = [sy = [*^31][P^][*^23] [SU + [ s i m [ s U [ P ] [ s U Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.82) 54 C H APTER 3. MODELING USING GSM 3.7 Program Description A computer program was developed based on the GSM-MoM derived for the building blocks in the previous sections. In this section, we will highlight the main steps involved in analyzing multilayered structures using this program. The program consists of two main parts. A graphical user interface (GUI) and an electromagnetic simulator engine. 3.7.1 Geometry-layout and input file A layout of each layer geometry is drawn using the GUI of CADENCE tools (icfb). Each geometry is discretized into rectangular cells. The x- and y- directed currents axe evaluated at the intersections of neighboring cells. A typical geometry for a patch, with current directions, is shown in Fig. 3.8. The circuit ports locations are distinguished by using labels, provided by CADENCE. The layout is then extracted to a CIF file format. A parser (written in C) transforms the CIF file into a compatible form that is read by the program. The input file contains the frequency range (start frequency, stop fre quency and number of points), waveguide dimensions, number of layers, type of layers, dielectric constants, layer separations, and output file names (impedance m atrix, scattering parameters, etc.). Also a symmetry flag is included in the input file to indicate which layers are repeated if any. By setting this flag, identical layers are computed only once and unnecessary redundant analysis of similar layers are avoided. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3. MODELING USING GSM 55 Y Figure 3.8: Rectangular patch showing x and y directed currents. 3.7.2 Electromagnetic simulator An electromagnetic simulator written in FORTRAN is developed to handle the analysis of multilayered structures. The simulator is composed of five routines. These axe the m ain routine, MoM calculation, GSM calculation, cascade of GSMs, and power conservation check. The m ain routine reads in the input and the geometry files and controls all other routines. It carries out the analysis in two main loops, frequency loop and layer loop with th e frequency loop being the outer loop. For each frequency point, the type of layer is checked. If the layer is magnetic or electric, the MoM is calculated using the MoM routine. Then the GSM is computed using the GSM calculation routine. To speed up the element calculation, an acceleration technique is used. This technique is based on the extraction of the quasi-static term of the Green’s function. A detailed analysis for the acceleration procedure is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3. MODELING USING GSM 56 demonstrated in the following chapter. For all other layers, the GSM is computed directly without the need to call the MoM routine. After a GSM is calculated for a layer, it is then cascaded to the previously calculated GSMs using the cascade routine. A power conservation check is then used to check the accuracy of the calculation using the power conservation routine. The sum of the squares of each column elements for the propagating modes has to equal 1 . When all layers are computed and a single GSM for the structure is ob tained, a new frequency point is calculated. This will continue until a complete sweep of the frequency range is achieved. The program produces output files containing the composite scattering m atrix of the whole structure, the impedance m atrix of the whole structure, the circuit scattering parameters, and the circuit impedance parameters. A flow chart illustrating the algorithm for the analysis of multilayered structures described above is shown in Fig. 3.9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 3. MODELING USING GSM READ l/P DATA GEOMETRY FILES LAYER I MAGNETIC -OR ELECTRIC? NO YES OF LAYER (I) SCATTERING MATRIX OF LAYER ( I ) NO YES CASCADE LAYERS 0,1-1) YES (I < MAX_LAYER) NO GENERATE Q/P FILES (F<FMop) FaF + df Figure 3.9: A flow chart for cascading multilayers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 MoM Elem ent Calculation 4.1 Introduction The most tim e consuming process in the GSM-MoM technique is the impedance or adm ittance m atrix element calculation. This is specially true for large waveguide dimensions and small cell discretizations with respect to the guide wavelength. An acceleration procedure is adopted in this work to speed up the element calculation. The impedance elements defined in (3.17) and the adm ittance elements given in (3.53) are derived in this section. These elements involve quadruple integrals of the form: - 1 JJ - ~Bi{r)dsds The integration is carried out for the electric or magnetic type Green’s functions over both the source and test basis functions. The two-dimensional space 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 4. M OM ELEM EN T CALCULATION 59 is discretized using rectangular cells. Each, basis function spreads over two adja cent cells. In our implementation, the basis functions are chosen to be subdomain sinusoidal functions. We use two discretization schemes for both the electric and magnetic currents, uniform and nonuniform. Uniform griding is more suitable for relatively simple geometries since the element computation time is less than in the nonuniform case. However, nonuniform griding enables the modeling of structures with adjustable spatial resolution to account for complex geometrical details, hence reducing the total number of unknowns with respect to the uniform case. 4.1.1 Uniform discretization Uniform discretization implies equal cell dimensions for all cells constructing the grid. The cells are rectangular in shape as shown in Fig. 4.1. The grid is uniform in the x and y directions with cell sizes c and d, respectively. The x directed sinusoidal Y x.* V+d *1 X'+C Figure 4.1: Geometry of uniform basis functions in the x and y directions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 4. M O M ELEM EN T CALCU LATIO N 60 basis function B f centered at (x,-, y3) is given by sin[fca ( c —|x —x,|)] £ f(x ) = |x —x,| < c d sin (k3c) 0 (4.1) |y - y < | < d/2 otherwise , and for a y directed sinusoidal basis function * sin[fc,(rf— ly - y d ) } Iy - y i \ < d * E f (y) — ' c sin(fcad) 0 (4.2) |x —xt| < c/ 2 , otherwise . where k3 = Uy/fi0eot3. Using the above expressions for the basis functions, the impedance ma trix elements given in (3.17) axe obtained in closed form expressions as follows r /x x ij ~ X ' m^O K ' ^O co m conn x ox ox dx dx nL m=0 n=0 yyy _ ‘ ‘i j — V ' V ' fom£on Z~t Z-j m =0 n=0 yxy ij ~ \ ' \ ' Z^ „L ab cv cv p v p v 9 e ,m n ‘->e,iJ e ,jr l e,iI l e,j £0m £ 0 n » o r qy px pV L m=0 n=0 where S * . = ok cos(k x ) [ ^ ( M - c o s t k c ) ' =’* “ * C°S{kxXt} [(** _ £2 ) d ^ ksC) S^ - O k c o s ( k v ) r cos( ^ - cos( M ) ' e’‘ * ( [ { k * - k * ) c sin(k3d) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.3) CH APTER 4. M O M ELE M E N T CALCU LATIO N 61 sin(fcyy,)sin(fcy| ) K i = 2 sin(fcr x,) sin(fcg|) Rh = 2 (4.4) Similarly, the adm ittance elements are obtained using (3.53) and the uniform basis function expressions -y x x *ij oo oo - c0mc0n / ^ ^ Onk * nx qx nx nx y-yy _ _ Y ' Y ' ^Om^On l ij — / / . Onh m=0 n = 0 qy qy qy qy \y x y er 1ij ~ Z-r m = 0n = 0 ^Om^On L o 02/ ox DU (4.5) 2ab where Sm,i = 2k» sia{kxXi) cos(kac) —cos(fcr c) {k* — k*) d sin(fcsc) •5m,,' = 2A:, sin(Aryy,) cos(kad) — cos(kyd) (k* - kj) c sin(kad) cos{kyyi) sin(fcy| ) 4.1.2 K ,i = 2 ?y _ 0 cos(ArgXt) sin(Arg| ) (4.6) Nonuniform discretization Nonimiform discretization implies unequal cell dimensions. For row i in the x direc tion, all cells have th e same width but variable length as shown in Fig. 4.2. Different Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 4. M O M ELEM EN T CALCULATION 62 rows can have different widths. The same is true for the columns in the y direction. The x directed sinusoidal basis function B f centered at (x,-, y.) is given by x,- X.+ -r- 1 Figure 4.2: Geometry of nonuniform basis functions in the x and y directions. sin[fca(ci —Xj + g)] d sin(A:sCi) , x,- —Ci < x < x,- sin[fca(c2 —x + x,)] (4.7) d sin(AraC2 ) , x^ < x < x,• + c2 , otherwise Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 4. M OM ELE M E N T CALCULATION 63 and for a y directed sinusoidal basis function is given by sin ft^di - yj + y)] , Vj ~ di < y < yj c s in (M i) sin[fca(d 2 - y + yj)} (4.8) B yj ( y ) ) = c sm(k3d2) 0 , yj < y < yj + d 2 , otherwise Using the above expressions for the basis functions in (3.17) and integrating results in closed form expressions for the impedance elements given bellow. = - £ £ = - £ £ + Q la m ji + m =0 n=0 m m =0 n=0 ab + Q h t W l i i + Q’j 2 ) R l R l n=0 n = 0 where 1 Q U = (fc* - **) d 8 in(fc.ca) [k3 cos(Arr (xI- —Cfi)) — k3 cos(kr X{) cos(Ar,ctl) —kx sin(A:r x,) sin(Ar,c,i)] Q U = (fc? _ *2 ) d sin(Arsc,-2) [k3 cos(kx(x{ + c,-2)) — k3 cos(kxX{) cos(Ar,c,-2) +A:r sin(Arr x{) sin(A:sCj2)] Q l n = (A£ - fe) c sin (M ti) [ks cos(A^(yt- —da)) — At, cos(Arvy,) cos(Ar,dtl) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 4. M OM E LE M E N T CALCULATION 64 —ky sin {kyi/i) sin(fcsdii)] Qve,i2 = Til— , 2x 1 ■ , , , ~ k ‘ ) c sm{kadi2 ) cos(^(y,- + di2)) - ka cos(Aryy.) cos{ksdi2) +ky sin(kyyi)sin (kadi 2 )] (4.10) Similarly, the adm ittance elements are obtained using (3.53) and the nonuniform basis function expressions n r = - £ £ T j r V - w - . W U + Q k i M Q l j 1+ 0 1 ^ ) A A i m=0 n= 0 “ O 0 n r = - £ £ n=0 n=0 + «*») oo oo m=0 n= 0 “ao where = ( y ^ ) 7 1 t o ( t e ) 1<:- sin(*-(zi “ c“ )) ~ nsm(A*x,)cos(A,Ci,) + k x cos (fcrx.) sin(fc,c,i)] Qm.,-2 = Tli— Mx ! . „ , [ka sin(fcj(x{ + c*)) - Ar,sin(Arr x f)cos(ArsCf2) («; — «*) d sin(«4c,-2) —A:r cos(A;I x,) sin(fcJct-2)] Qm.,1 = (fe2 _ fc2 ) c s i n ( M , i ) ^ jSin^fcy^ t ~~ ^ ” kasm{kyyi)cos(kadn) + ky cos(kyyi) sin(fc,dt-i)] Qm,i2 = (fc2 1 fc2 ) c sin(Ar4dt2) ^ sin^ ^ ‘ + ^ 2)) ~ k» sin(fcyy.) cos(kadi2) —ky cos (kyyi) sin(fcsd£2)] (4.12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 4. M OM E LE M E N T CALCU LATIO N 4.2 65 Acceleration of MoM M atrix Elements The MoM impedance matrix elements appealing in (3.17) and the MoM adm it tance m atrix elements Yij defined in (3.53) involve the integration of the electric and magnetic type Green’s functions, respectively. The Green’s functions are twodimensional infinite series. To evaluate the Green’s functions, at a given source and observation points, the double series summation must converge to a stable value. The simplest technique to compute the m atrix elements is the direct summation technique, where a term-by-term summation is carried out while checking for con vergence at progressive intervals. However these double series summations are slow to converge which, as well as leading to time-consuming computations, can result in numerical instabilities and imprecision in determining when the summations should be truncated. Also, the number of summation terms is directly proportional to the waveguide size and inversely proportional to the dimension of the basis functions used to discretize the integral equation. Several methods were employed to accelerate the computation of the waveguide Green’s function. A rectangular waveguide Green’s function involving complex images was proposed in [75], where the real images are replaced by the fullwave complex discrete images. T he resulting Green’s function is fast convergent. Park and Nam [76], in considering a shielded planar multilayered structure, trans formed a scalar Green’s function into a static image series which was evaluated using the Ewald method. It was pointed out th at the final form of the Green’s function converges rapidly with a small number of terms in a series summation. Transfor m ation of a double series expansion into a contour complex integral to which the residue theorem was applied was developed by Hashemi-Yeganeh [77]. This method Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 4. M OM E LE M E N T C ALCU LATIO N 66 leads to the computation of a few single summations of fast converging series. Al ternative fast converging formulas for th e dyadic Green’s function in a rectangular waveguide by way of the Poisson summation technique was developed in [78]. By far the most widely used technique to accelerate waveguide Green’s functions is the quasistatic extraction m ethod. In this method the Green’s function is partitioned into an asymptotic static (frequency-independent) part and a dynamic (frequency-dependent) part. The asymptotic part needs to be evaluated once per frequency scan. The dynamic part is now fast converging owing to the extraction of the slowly converging static part. Eleftheriades, Mosig, and Guglielmi [79] pio neered a procedure th at partitions a potential Green’s function into an asymptotic (frequency-independent) part and a dynamic part, where the asymptotic part was converted to a rapidly converging series summ ation. We found that this technique is most suitable for our problem and implemented it to the electric and magnetic type Green’s functions. 4.2.1 Acceleration o f impedance matrix elements It is known th at a double series expansion of Green’s function components is slowly convergent due to the presence of the quasi-static part. An efficient technique based on the Rum m er transform ation [80] has been applied to accelerate slowly convergent series [79]. This technique is applied here to the Green’s function components, (3.10), =Q 5 leading to their transform ation so that a quasi-static part (Ge ) is extracted. The Green’s function is then + G?S Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.13) CH APTER 4. M OM E LE M E N T CALCULATION 67 A zsdQS where Ge = Ge — Gc , Ge is the electric type Green’s function, and Gt is the qua sistatic electric type Green’s function. To compute the quasistatic Green’s function, (3.11) and (3.12) are calculated for large m and n and are given by 2 Vlmn(x iy) = ^ = c o s (fc r x)sin(fcj,y) , <«»»(*» y) = ^ 2 sin(fcr x) cos(fcj,y) k2 *------- f n = Je,m — 7J j ^e,ran (4.14) + £2 ) ’ kxky + £2 ) ’ *— - S2) (4-15) The quasistatic components of the electric type Green’s function are derived using (3.10), (4.14) and (4.15): x c o s(^ A , ) f • (4.16) ,/« • + (* )* G % ( x , r , x ’, y n = - r r j — abu(ei + e2)r: £^ C v%b Y c o s { l T y ) rn r A ^ 4 s i n ( ^ x ) s in (= f* ') x « * (t» 0 E .2 / b m=1 1=1 v W +0f) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4-17) CH APTER 4. M OM E LE M E N T CALCULATION 68 ,nw x ~ 4 ( ^ ) c o s ( ^ x ) s i n ( ^ x ') f« 0 £ x« ‘ i, V . + ,( f V ) 1 ■ <4-18> It can be seen that the quasistatic part is inversely proportional to u. However, it needs to be calculated only once per frequency scan since the summations are frequency independent. Still the expressions obtained above axe slowly converging, following the procedure described in [79] the second infinite summation in (4.16) and (4.17) can be transformed into a fast converging series of A'o, the modified Bessel functions of the second kind, thus » * s jn ( y » ) « in ( y » ') = 5 3 n = —o o - K q a x 1 ’+ M * {tfo ( — {y - y ' + 2 n 6 )) (— (y + y' + 2 n 6 )) } , a ” £ 5 3 i/M 2 6 4 sin f— x) sin f— xz) 9a A= — x ,/ ( ? ) + (T )’ (4.20) 1 {Ao(*r~(a: - x ' + 2 m o)) - K 0( ^ - { x + x' + 2 m a ) ) | , m = —oo Only a few terms of the series on the right hand side of equations (4.19) and (4.20) axe required to reach convergence due to the exponential decay of the modified Bessel functions. This property together with the frequency independence of the summations axe the key attributes leading to computational speed-up. To compute the accelerated impedance m atrix elements a Galerkin MoM procedure is applied to (4.13) resulting in the following representation: Z MoM(ui) = [ZA M + Z QS] (4.21) where J L J L Bj(r).Ge ( r , r ) *£?,-(/)dsds, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.22) C H APTER 4. M OM ELE M E N T CALCULATION 69 and # ~ J U L B j { r ) ^ S(r, r ) • B i ( r ) d s d s (4.23) The integrations of the modified Bessel functions in (4.23) axe easily converted using change of variables to a standard integral / Ko(v)dv Jo (4.24) This integral is shown in Fig. 4.3 and is shown to be fast convergent as the argument 1.4 1.2 o _ 0.8 2 o> c 0.6 ® 0.4 0.2 V Figure 4.3: Integral of K 0. V increases. For example, the x x impedance m atrix element Zjjs is written as J J x j - c J y j - d /2 J n - C Jy i-d f2 1 (x, y; x'y')Bf(x')dx dy dx'dy' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.25) C H APTE R 4. M OM E LE M E N T CALCU LATIO N 70 with B j ( x ) and Bf(x') being piecewise sinusoidal functions defined by (4.1). For simplicity uniform basis functions are considered: o o 9 f rmr\2 o o £ = J ^ S) ( 4 -2 6 ) Thus the problem of evaluating Z f is reduced to the calculation of a double integral over the y-domain: nc i f 3 a rVj+d/2 r r sz L(y-yi-d/2+2nb) = - ± - / {/ K o(v)dv - / TUT J y j - d / 2 '•Jo J0 /,:r I(y+v«+<i/2+2ni) + Jo rr*r{v+yi-d/2 +2 nb) Ko(v)dv-JQ +d/2+2nb) K o(v)dv . Ko{v) dv} dy (4.27) The inner integrals are of the same form as (4.24). This standard integral is numer ically computed only once and stored in a table. The outer integrals are calculated numerically by means of Gaussian quadrature using the data of the previous inte gration. The second series appearing in (4.26) is now very fast convergent. This is due to the fast converging nature of the integrals when the index n gets larger. Typically only three terms of n need to be evaluated (from — 1 to 1 ) to achieve very small errors. So the double series sum m ation involved in the quasistatic impedance element calculation is effectively converted to a single series summation. Similar expressions can be derived for the xy, yx, and yy impedance m atrix elements. As an example, th e convergence and accuracy of the speed up procedure discussed is demonstrated by a comparison to the direct summ ation case for the x-directed current element placed inside a WR-90 waveguide with x p = a / 2 and yp = 6 /2 (the unit cell shown in Fig. 3.4 has dimensions c=0.2318 cm and d=0.2371 cm). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 4. M OM ELE M EN T CALCU LATIO N 71 The convergence and percentage error of the impedance element for the accelerated and direct double series summation are demonstrated in Figs. 4.4 and 4.5, respectively. The relative error is defined as \ZXX —Z ~ |/ |Z ” | x 100, where Zxx represents the impedance matrix element either calculated as a direct summation or using the proposed acceleration technique, and Z “ is the value of Zxx obtained for a large number of summation terms m°° and n°° (using direct summation). To generate the results shown in Fig. 4.5 we used m°° = n°° = 1500 summation terms resulting in a value of equal to 10.928331 —^163.503317. It is shown th at the error of 0.5% is obtained for 200 terms used in the accelerated summation procedure in comparison with 2500 terms required in the direct double series summation to reach the same error. The com putation tim e is almost directly proportional to the number of terms in the summation. Also, it can be difficult to determine when sufficient terms have been used with the direct method. 4.2.2 Acceleration of admittance matrix elements The same procedure for accelerating the impedance m atrix elements is followed for the adm ittance m atrix elements. The magnetic type Green’s function is now written as: 3™ = 3 ^ + 5 ^ —A rs =QS = t (4.28) rsQ 5 where Gm = Gm — Gm , Gm is the magnetic type Green’s function, and Gm is the quasistatic magnetic type Green’s function. To compute the quasistatic Green’s function equations (3.48) and (3.49) are calculated for large m and n and are given Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 4. M OM E LE M E N T CALCULATION 200 190 - ...f~... .......|...... ----- Accelerated -----Direct summation 180 i.... ;...... 170 V 160 ,*T i -----;----150 i !..... 140 I 1 130 * 11 120 1 1 110 11.... . I1 100 0 500 1000 ----- ..... -...~"i...... ...... ..... T.. -T ---- 1500 2000 2500 3000 Number of terms Figure 4.4: Convergence of Zxx m atrix elements. 5.0 Accelerated Direct summation 4.5 4.0 | 3.5 « 3.0 & 3 IS | 2.0 £ 1.5 1.0 0.0 500 1000 1500 2000 2500 3000 Number of terms Figure 4.5: Percentage error in the convergence of Z xx m atrix elements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 4. M OM ELE M E N T CALCULATION 73 by: < m n ( 2 .y ) = J - ^ s m ( k xx)cos{kyy) , C n t 1 - y) = ]J^ cos(*xx) sin(fcyy) (4.29) fc2 fm ,m n = 2 j km,mn — 2 j Uflkc k X k y uykc k2 9m,mn = 2 ; - ^ - (4.30) Equations (4.29) and (4.30) when combined with (3.47) result in the quasistatic Green’s function components. x s in ( ^ ) f M r » H M , (4.31) \/(~) + (x )2 ° / n x v ~ 4 cos f — x) cos f — x') * » ( t »9 s J * ■ ( 4 -3 2 ) ~ ( t ») (4.33) " =1 7 ( x ) + ( f )’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 C H APTE R 4. M OM E L E M E N T CALCU LATIO N The second infinite summation in (4.31) and (4.32) can be transformed into a fast converging series of Ko, the modified Bessel functions of the second kind, thus £ (4.34) - + W * mx x { K 0( — (y + y' + 2 n b ) ) + K 0( — ( y - y ' + 2nb))}\ , £ n = —o o ® » y* -■ ® 4 c o s (^ x ) c o s f^ x ') Va i/ ( t 5 3 {#o ("T-fa + / ) + M Va . ; = 26 r2a ’ "x x (4.35) + 2 m a)) + AT0 (-t~ (x - *' + 2 m a)) }] , 0 m = —o o To compute the accelerated adm ittance m atrix elements, a Galerkin procedure is applied to (4.28) resulting in the following representation: Y MoM{uj) = [Ka M + Y q s ] (4.36) * ■ ‘ - n i L - * ,(r).Z ^ (r, r ) • Bi{r)ds'ds, (4.37) i(r).Zr^ (r , r ) ' B i ( r ' ) d s ' d s (4.38) where and Yji — The quasistatic adm ittance elements are evaluated in a similar m anner as the qua sistatic impedance elements. The problem is reduced into integrals involving the Bessel functions of the second kind which are fast convergent. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Local Reference N odes 5.1 Introduction The GSM-MoM method described in Chapters 3 and 4 produces a scattering m atrix that represents the relationship between modes and ports. The scattering m atrix can then be converted to port-based adm ittance or impedance matrix. This allows the modeling of a waveguide structure th at can support multiple electromagnetic modes by a circuit with defined coupling between the modes. However, port-based representations are not suited for most circuit simulation tools. Nodal analysis is the m ainstay of circuit simulation. The basis of the technique is relating nodal voltages, voltages at nodes referenced to a single common reference node, to th e currents entering the nodes of a circuit. T he art of modeling is then, generally, to develop a current/nodal-voltage approximation of the physical characteristics of a device or structure. W ith spatially distributed structures a reasonable approximation can 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 5. LO CAL REFERENCE NODES 76 sometimes be difficult to achieve. The essence of the problem is th at a global ref erence node cannot reasonably be defined for two spatially separated nodes when the electromagnetic field is transient or alternating. In this situation, the electric field is nonconservative and the voltage between any two points is dependent on the path of integration and hence voltage is undefined. This includes the situation of two separated points on an ideal conductor. P ut in a time-domain context, it takes a finite tim e for the state at one of the points on the ideal conductor to af fect the state at the other point. In the case of waveforms on digital interconnects this phenomenon has become known as retardation [81]. W ith high-speed digital circuits, it is common to model ground planes by inductor networks so that inter connects are modeled by extensive RLC meshes. Consequently no two separated points are instantaneously coupled. In transient analysis of distributed microwave structures, lumped circuit elements can be embedded in the mesh of a tim e dis cretized electromagnetic field solver such as a finite difference time domain (FDTD) field modeler [16,17]. The temporal separation of spatially distributed points is then inherent to the discretization of the mesh. W ith a frequency domain electromagnetic field simulator, ports are de fined and so a port-based representation of the linear distributed circuit is produced. W ith ports, a global reference node is not required. Instead a local reference node, one of the terminals of the two terminal port, is implied. The beginnings of a circuit theory incorporating ports in circuit simulation has been described and term ed the compression m atrix approach [14,15]. This milestone work presented a technology for integrating port-based electromagnetic field models with nonlinear devices. Cir cuit simulation using port representation has been reported in [82]. This requires the representation of nodally defined circuits in its port equivalent by a general-purpose Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 5. LOCAL REFERENCE NODES 77 linear m ultiport routine. Hence, the advantage of accessing information at all nodes as in nodal analysis is lost. This Chapter extends the circuit theory beyond the compression ma trix approach to general purpose circuit simulators based on nodal analysis. In particular, we present the concept of local reference nodes that enables port-based network characterization to be used with nodally defined circuits in the develop ment, by inspection (the preferred approach), of what is termed a locally referenced nodal adm ittance matrix. A procedure for handling and moving the local reference nodes is described along with circuit reduction techniques that facilitate efficient simulation of nonlinear microwave circuits. 5.2 Nodal-Based Circuit Simulation The most popular method for circuit analysis in the frequency domain is the nodal adm ittance m atrix method. In the nodal formulation of the network equations, a m atrix equation is developed that relates the unknown node voltages to the external currents using th e model shown in Fig. 5.1. All node voltages are then defined with respect to an arbitrarily chosen node called th e global reference node. Eliminating the row and column associated with the global reference node leads to a definite adm ittance m atrix and then the solution for the node voltages is straight forward. In this type of analysis only one reference node can exist. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 5. LO CAL REFERENC E NODES 78 S7 (b) N+l GLOBAL REFERENCE NODE (a) 6 (c) Figure 5.1: Nodal circuits: (a) general nodal circuit definition (b) conventional global reference node; and (c) local reference node proposed here. 5.3 Spatially Distributed Circuits 5.3.1 Port representation Electromagnetic structures can only be strictly analyzed using port excitations. The spatially distributed linear circuit (SDLC) consists of groups with each group having a local reference node. The scattering parameters are the most natural parameters to use with ports and their local reference nodes. They can be converted to port adm ittance m atrix using th e following equation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 5. LOCAL REFEREN C E NODES 79 Y = Y0( l - Yo 1/2SY i/2)(l + Y o 1l/aSY j/2) - 1 (5.1) This is the most convenient form to use in circuit simulators. Before continuing, a distinction is required between the global reference node and the local reference nodes, with the symbols shown in Fig. 5.1 axe adopted here. A general circuit with local reference nodes required with an SDLC and nonlinearities is shown in Fig. 5.2. For the specific case of power combiners the SDLC can be illustrated by the 2 x 2 grid array shown in Fig. 5.3. Here the grid array is composed of four locally referenced groups with each group having a differential pair as its active components. Referring to Fig. 5.2, the SDLC is the electromagnetic port representation of the grid array, the linear subcircuits axe the linear elements in the equivalent circuit model of the differential pair, and the nonlinear subcircuits are the nonlinear elements associated with the active device model. Figure 5.2 depicts the essential circuit analysis issue: integrating the representation of an SDLC with a circuit defined in a conventional nodal manner, to obtain an augmented nodal based description. The problem is how to handle the additional redundancy introduced by the local reference nodes. For locally refer enced group number m there are E m terminals, Em —1 locally referenced ports and one local reference node designated Em. The port-based system may be expressed as tm V ] = W W here the port-based adm ittance m atrix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.2) CH APTER 5. LOCAL REFERENCE NODES LOCALLY REFERENCED GROUP : N o n lin ear S u b c irc u it L in ear S u b c irc u it E n & N o n lin ea r S u b c irc u it 80 I L in ear S u b c irc u it PORT BASED SPATIALLY DISTRIBUTED LINEAR CIRCUIT (SDLC) N o n lin ea r S u b c irc u it L in e ar S u b c irc u it A u g m e n te d N odal B a s e d Figure 5.2: Port defined system connected to nodal defined circuit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 5. LOCAL REFERENCE NODES GROUP 1 GROUP 3 I GROUP 2 |I GROUP 4 Figure 5.3: Grid axray showing locally referenced groups. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 5. LO CAL REFE RE N C E NODES PY i,m pYi,i „Y = bY i ,m p * m ,m pYm ,i 82 "' * ••• pYm ,1 p * m ,M »Y m ,m the port-based voltage vector 0V = v Pv 1 ... P m v " • P1 ™ ... P„V„, T M and the port-based current vector 0I = P1 ! " • P1 iM The subm atrix pY ij is the m utual port adm ittance m atrix (of dimension £,■ Ej — 1 ) between groups i and j of the SDLC, 1 x = [/i, I2i ... I e, - i ] is the current vector of group i of the SDLC, and pVj = [(V^ —Ve, ) {V2] — Ve} ) ... (V^-i - Ve} ) ] is the port voltage vector of group j of the SDLC. Defining the total number of ports for groups 1 to j of the SDLC as: (5.3) t= l Then PY is square of dimension tim x tim - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 5 . LOCAL REFERENC E NODES 5.3.2 83 Port to local-node representation In order to use nodal analysis, the port-based system must be formulated in a nodal adm ittance form. Since there are M localized reference nodes, another redundant M rows and M columns can be added to the port adm ittance m atrix such that: (5.4) [nY][nV] = [nI] Where the nodal adm ittance m atrix PY Yx xY = (5.5) Y2 y 3 (n \f+ A f) x(njvf+ M ) The elements of each submatrix are given by ne Y i( r ,c ) = - £ pY (r,j), r = l..n M, c = 1..M i = " ( c - i ) + i Y 2 (r,c ) PY (i,c ), r = l..M ,c = l..nM = » = ” ( r - l ) + l Y 3 (r,c ) = - £ Y i(i,c ), r = l ..M ,c = 1..AT. *‘= " < r - t ) + l with no = 0, nr ,n e,n (c_i), and »(r-i) are given by (5.3). The nodal voltage vector nY — [Vjj... Vjj... Vjjj... Vgjj-i and the branch current vector nI — [/^ .../g j - i /ij ...J c j-i... h r . . 1 ^ —1lE\ ***IE\[ ] The adm ittance m atrix [nY] is a nodal m atrix and has M dependent rows and M dependent columns. Hence, it is an M fold indefinite nodal adm ittance m atrix corresponding to the M local reference nodes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 5. LOCAL REFEREN C E NODES 5.4 84 Representation of Nodally Defined Circuits Since there are no connections between the linear circuits at each group, the linear circuit at group i will have no m utual coupling with the linear circuit at group j. The only coupling th at can exist between different locally referenced groups is accounted for in the description of the SDLC. Hence, for the linear subcircuits (as in Fig. 5.2) all the entries in the adm ittance m atrix are zero except those relating the node parameters at the same group. Defining interfacing nodes as the nodes between lumped linear circuits and nonlinear circuits, a lumped linear circuit embedded at group m can be represented as [Y][V] = [I] 0 0 0 0 » » 0 0 0 0 (5.6) ••• 0 0 0 0 nV, " n il • • • 0 0 0 0 {Vi ill 0 0 nVm • Ym(i,i) Ym(1,2) • • • Ym(2,l) Ym(2,2) nlm 0 0 t‘V m t'lm nViVf nljVf » » • . t 0 0 • • • 0 0 0 0 0 0 • • • 0 0 0 0 • > «Iat • where „ /m = [I\m l 2 m ... /^m ] th e current vector at group m of the SDLC. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.7) CH APTER 5. LOCAL REFERENCE NODES 85 • nVm = [Vlm V2 m ... Vsm ] is the node voltage vector at group m of the SDLC. • ,Im is the branch current vector (the currents flow into the linear network) of interfacing nodes and linear subcircuit nodes at group m. • ,V m the node voltage vector of interfacing nodes and linear subcircuit nodes at group m. is the conventional indefinite nodal admittance • Ym = Y m(2,l) Y m(2,2) m atrix of the linear sub-circuit. Thus, the indefinite nodal adm ittance matrix of all of the linear sub circuits combined is a block diagonal m atrix Y 5.5 l — D ia g ( Y i, . . , Y m, . . , Y m ) (5.8) Augmented Admittance Matrix To combine the linear circuits (lumped and distributed) in an augmented adm ittance m atrix as shown in Fig. 5.2, (5.4) is expanded in the full set of voltages [V] yielding: [Y„]tV] = pq with [Y e ] being the expanded nodal representation of the SDLC and given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.9) C H APTE R 5. LO CAL REFERENC E NO DES 86 nYi,x 0 ••• 0 nYi,m 0 0 nYi,M 0 0 0 ••• 0 0 0 0 0 0 0 0 0 0 0 0 n Y iu .m 0 0 nYm,M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n Y n i.m 0 0 hYm,M 0 0 0 0 0 0 0 0 n Y m .x 0 0 ••• 0 0 0 h Y m .X 0 0 0 ••• 0 ••• 0 Equations (5.8) and (5.9) are added together to form the overall linear circuit. Ya =Y e +Y l (5.10) In microwave nonlinear circuit analysis, the network parameters of the linear circuit are reduced to just include th e interfacing nodes. Standard m atrix reduction techniques can be used to obtain this reduced circuit. An interfacing node is assigned to be the local reference node at each group, hence eliminating the corresponding rows and columns. The resulting system of equations is definite and represents the augmented linear circuit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 5. LOCAL REFERENC E NODES 5.6 87 Summary The scheme for the augmentation of a nodal adm ittance m atrix by a port-based m atrix with a number of local reference nodes perm its field derived models to be incorporated in a general purpose circuit simulator based on nodal formulation. The method is immediately applicable to modified nodal adm ittance (MNA) anal ysis as the additional rows and columns of the MNA m atrix are unaffected by the augmentation. This work is being used in the simulation of spatial power combiners (in both free space and waveguide) which are electrically large and do not have a global reference node or perfect ground plane. This is demonstrated in [52] where a free space 2 x 2 active grid array is simulated using the local reference node concept presented in this Chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Results 6.1 Introduction To illustrate the flexibility and generality of the GSM-MoM method developed in Chapter 3, the GSM-MoM method is applied to the analysis of spatial power com bining elements and arrays in addition to general structures. Although originally the m ethod was developed to simulate spatial power combining structures, it can handle, in general, any transverse structure in a waveguide such as waveguide filters, input impedance of probe excited waveguides, and shielded multilayered structures. In this Chapter, we will consider two categories of examples. These are general structures and spatial power combining structures. The spatial power combining structures include patch-slot-patch arrays, CPW arrays, grid arrays, and cavity os cillators. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESU LTS 6.2 89 Analysis of General Structures In this section several common structures such as wide strip in a waveguide, patches on a dielectric slab, and strip-slot transition module are simulated. The obtained results (modal scattering parameters) are compared to either measured results or other analysis techniques. To dem onstrate the validity and accuracy of the GSM-MoM with ports, a completely shielded microstrip notch filter, in a cavity, is simulated. The results (two port scattering parameters) compare favorably to measurements. This represents an extreme test to the method. 6.2.1 Wide resonant strip To illustrate the calculation of the Generalized Scattering M atrix procedure pro posed in Chapters 3 and 4, respectively, a wide resonant strip structure embedded in an X-band rectangular waveguide (with geometry shown in Fig. 6.1), is investi gated numerically. The strip current is discretized using rectangular meshing and sinusoidal basis functions. It should be noted that to accurately model the current on the strip, the continuity of the edge current is accounted for by half basis func tions as shown in Fig. 6.1. Numerical calculation for the normalized susceptance (of the dominant T E w mode) show th at the structure goes through resonance at approximately 11 GHz which agrees well with the measured data provided in [83] as shown in Fig. 6.2. Such a wide strip can be used as a section of waveguide filters [84]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESU LTS Figure 6.1: Wide resonant strip in waveguide, a — 1.016 cm, 6 = 2.286 cm, tv 0.7112 cm, i = 0.9271 cm, yc = 6/2. *------ Simulation + Maasurod S » IB 4> >a.io •20 -30 Frequency (GHz) Figure 6 .2 : Normalized susceptance of a wide resonant strip in waveguide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 6.2.2 91 Resonant patch array As another example, a resonant patch array consisting of six metal patches and supported by a dielectric slab in a rectangular waveguide (Fig. 6.3) is analyzed for application in high-frequency electromagnetic and quasi-optical transm itting and receiving systems [41,85]. Results are obtained for the frequency band 8-12 GHz. In this frequency range the air-filled portions of the waveguide support only one propagating mode (TEio) while the dielectric slab accommodates multimodes. The magnitude of the reflection coefficient for the dominant mode is —26 dB as shown in Fig. 6.4. The phase angle is given in Fig. 6.5 showing the resonant properties of the structure. Figure 6.3: Geometry of patch array supported by dielectric slab in a rectangular waveguide: a = 1.0287 cm, b = 2.286 cm, £ = 2.5 cm, eP = 2.33, d = 0.4572 cm, c = 0.3429 cm, rx = 0.1143 cm, ry = 0.2286 cm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 92 09 ■o ■o 3 -25 -30 7.0 8.0 9.0 10.0 Frequency (GHz) Figure 6.4: Magnitude of S n and S 21 for the patch array embedded in a waveguide. 200 150 100 1 & 7.0 7.5 8.0 8.5 9.0 9.5 10.0 Frequency (GHz) Figure 6.5: Phase of S n and S 2 1 for the patch array embedded in a waveguide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESU LTS 6.2.3 93 Strip-slot transition module To verify both GSMs for strips and slots, cascaded together, a strip-slot transition module is analyzed using the GSM-MoM technique described in the previous chap ters. The results obtained for the transmission and reflection coefficients for the dominant TE\o mode are compared with two other techniques based on the FEM and MoM methods. A commercial High Frequency Structure Simulator (HFSS) based on the FEM is used for comparison as well as an inhouse MoM program utilizing a Green’s function for the composite structure (strip-slot) [8 6 ]. Figure 6 .6 : Slot-strip transition module in rectangular waveguide: a = 22.86 mm, b = 10.16 mm, r = 2.5 mm. The structure consists of two layers with electric (strip) and magnetic (slot) interfaces as shown in Fig. . . The strip and slot dimensions are 0.6 mm 6 6 x 5.4 mm and 5.4 mm x 0.6 mm, respectively. The relative permitivities of the dielectric materials used are ei = 1 , 62 = 6 , and £3 = 1 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 6. RESU LTS Cfi S. CA 0) •O 3 |CQd-4 S 18.5 MoM-GSM MoM HFSS 18.8 19.1 19.4 19.7 Frequency (GHz) 20.0 20J Figure 6.7: Magnitude of S u for the strip-slot transition module. 200 160 120 Jf •o I £ MoM-GSM MoM HFSS -80 -120 -160 -200 18.5 18.8 19.1 19.4 19.7 Frequency (GHz) 20.0 20.3 Figure 6 .8 : Phase of S u for the strip-slot transition module. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 6. RESULTS 0 ■5 -15 MoM-GSM MoM HFSS -30 18.5 18.8 19.1 19.4 19.7 Frequency (GHz) 20.0 20.3 Figure 6.9: Magnitude of S 21 for the strip-slot transition module. 200 160 120 MoM-GSM MoM HFSS 2 V v. -40 • -80 -120 - 1(0 -200 18.5 18.8 19.1 19.4 19.7 Frequency (GHz) 20.0 20.3 Figure 6.10: Phase of S2 1 for the strip-slot transition module. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESU LTS 96 Very good agreement is obtained for all three methods as shown in Figs. 6.7, . , 6.9, and 6.10 representing the magnitude and phase of the reflection and 6 8 transmission coefficients. A minimum reflection coefficient of -5.5 dB is achieved at 19.64 GHz. The number of modes considered in the cascading process for the GSM-MoM technique is 128 modes. Note th at eventhough the dispersion behavior of the scattering param eters is shown for the dominant T E W mode, the X-band waveguide is overmoded in th e frequency range (18.5-20.3 GHz), specially in region 2 where the dielectric constant is high. 6.2.4 Shielded dipole antenna In this section a dipole antenna (Fig. 6.11) of length (L) 8 m m and width (W ) 1 mm is investigated. The antenna is placed in the center (XI = X i — ^ ) of a hollow rectangular waveguide WR-90 with both waveguide ports perfectly matched (no reflections). This antenna has been investigated by Adams et al. in [87]. In [87] a finite gap excitation was assumed to account for the gap capacitance. In our implementation the input impedance of the antenna is calculated using a delta gap voltage model. It is shown that good agreement is obtained for both the reed and imaginary parts of the input impedance (Fig 6.12) for the frequency range 8.0 to 12.5 GHz. The input impedance is shown to be capacitive specially at the lower end of th e frequency range, indicating coupling to evanescent TM modes instead of evanescent TE modes. To investigate th e effect of the antenna position within the waveguide Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 CH APTER 6. RESU LTS Y X X Figure 6.11: Center fed dipole antenna inside rectangular waveguide. -so g-100 .-150 - -250 O ♦ -300. 10 10.5 R e a l- P a r t (G SM -M oM ) I m a a in a iy - P a r t (G SM -M oM ) R M P -P art(M o M ) Im a g in a ry -P a rt (MoM) 11.5 frequency (GHz) Figure 6.12: Comparison of Real and Imaginary parts of input impedance. GSMMoM (developed here), MoM [87]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 CH APTER 6. RESU LTS -50 -150 R e a l- P a r t (c e n te re d ) -250 I m a g fn a iy -P a rt (o ff-cen tered ; -300. 8.5 9.5 10 11.5 10.5 frequency (GHz) Figure 6.13: Calculated input impedance for centered and off-centered positions. on its input impedance, the antenna is placed at X \ = 3 mm away from the vertical waveguide wall. Considerable variation in the input impedance is observed in Fig. 6.13 due to the close proximity to the waveguide wall. 6.2.5 Shielded microstrip filter The GSM-MoM m ethod can also be applied to completely shielded microwave and millimeter wave structures. Numerical results have been obtained for the specific example of the shielded microstrip filter shown in Fig. 6.14. The filter is contained in a box of dimensions 92 x 92 x 11.4 mm (a x 6 x c). The substrate height is 1.57 mm and it has a relative perm itivity of 2.33. In analysis, the structure is decomposed into three layers as shown in Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. C H APTE R 6. RESULTS 99 Fig. 6.15, with layers 1 and 3 being the top and bottom covers, respectively. The covers axe perfect conductors and hence their GSMs are diagonal matrices with —1 as diagonal elements. Layer 2 is a metal layer with ports. 4.6 mm 1 4.6 mm 23 mm 92 mm Figure 6.14: Geometry of a microstrip stub filter showing the triangular basis func tions used. Shaded basis indicate port locations. The excitation ports are modeled by the delta-gap voltage model pro posed by Eleftheriades and Mosig [8 8 ] (the current basis functions for the excitation ports axe shaded in Fig. 6.14). This serves two purposes, to ensure the current continuity at the edges and to allow the direct computation of network parameters without the need to extend the line beyond its physical length. It should be noted th at these half-basis functions can only be used, for direct port computation as de scribed in [8 8 ], at the microstrip-wall intersection. The equivalent circuit model of th e port representation using half basis function is shown in Fig. 6.16. The voltage source V is the delta gap voltage source accompanying the half basis function. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 100 TOP COVER METAL LAYER BOTTOM COVER Figure 6.15: Three dimensional view illustrating the layers of the stub filter. -V+ rO Figure 6.16: Port definition using half basis functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 CH APTER 6. RESU LTS The GSM of layer 2 is computed using the method described in Chapters 3 and 4. The number of modes considered in the GSM for layers 1 and 3 is 287. Layer 2 has 289 ports, 287 modes and 2 circuit ports. After cascading the three layers the modes are augmented. The final scattering m atrix has rank two representing the circuit ports of the filter. This is illustrated in Fig. 6.17. SH O R T CIRCUIT WAVEGUIDE SECTION M ICRO STRIP +1 WAVEGUIDE SECTION SHORT CIRCUIT - V1 CASCADING Vi SHIELDED Figure 6.17: Block diagram for the GSM-MoM analysis of shielded stub filter. The reflection and transmission coefficients S u and S2 1 axe calculated in Figs. 6.18 and 6.19, respectively. The transmission from port 1 to port 2 is approximately —37 dB at 2.7 GHz and compares favorably with previously reported results [8 8 ]. To explain the box effect appearing in the reflection and transmission coefficients, a plot of the propagation constant diagram is shown in Fig. 6.20. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 CH APTER 6. RESU LTS solid and dashed curves represent the air filled and the dielectric substrate regions, respectively. It is observed that the notches in the S u and 621 curves (at 2.2 and 3.4 GHz) correspond to the cut off frequencies of certain modes in the dielectric substrate. ~ ■* •10 -12J Figure 6.18: Scattering param eter Su.: solid line GSM-MoM, dotted line from [8 8 ]. Convergence curves for the scattering parameters are shown for various numbers of modes in Figs. 6.21 and 6.22. As desired, convergence to a result is asymptotically approached as the number of modes considered increases. The need for a large number of modes is in intuitive agreement since dimensions axe small compared to the guide wavelength and so evanescent mode coupling should dominate. This example represents an extrem e test of the method developed here and it also verifies th e calculation of the GSM with circuit ports technique. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESU LTS 103 -10 -15 i n (O -25 -30 •35 -45 S3--F ra q u a n c y (GHz) Figure 6.19: Scattering param eter 621 : solid line GSM-MoM; dotted line from [8 8 ]. 15 FREQUENCY (GHz) Figure 6.20: Propagation constant: solid lines for air, dashed lines for dielectric. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 'S . V *. ••* + x •' / ,* / .* / y 't / / • 1 // / i\ V 2 • */ . 1/ ■'// ;» / ill \ ------ 190 modas If If ----- 127 modas ........ 71 modaa IS i 2i Fraquancy (GHz) I 3S 4 Figure 6.21: Various cascading modes showing convergence of S n •10 -15 ■ 3. .20 •25 -30 -35 190 - - 127 -45. Figure 6.22: Various cascading modes showing convergence of S2 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESU LTS 6.3 105 Patch-Slot-Patch Array In this section a spatial power combiner structure is simulated and measured. The system shown in Fig. 6.23 is divided into three blocks, transm itting horn, receiving horn and a double layer array. Each block is simulated by a separate EM routine. To reduce the coupling between the receiving and the transm itting patch antennas, a strip-slot-strip transition is designed to couple energy from one patch to the other. The patch antenna used is shown in Fig. 6.24 along with the amplifying unit. Figure 6.23: A patch-slot-patch waveguide-based spatial power combiner. 6.3.1 Array simulation The double layer array consists of three interfaces (patch-slot-patch). Each interface is modeled separately using the Generalized Scattering Matrix-Method of Moment technique. The m ethod first calculates the MoM impedance m atrix for an interface from which a GSM m atrix is calculated directly without the intermediate step of current calculation. This enables the modeling of arbitrary shaped structures and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 106 99 66 44 5 32.5 46 AMPLIFIER 130 Figure 6.24: Geometry of the patch-slot-patch unit cell, all dimensions are in mils. the calculation of large number of modes needed in the cascade to obtain the required accuracy. The nonuniform meshing scheme described in Chapter 4 is used here to reduce the number of basis functions required in case of the uniform meshing scheme. 6.3.2 Horn simulation The GSMs for the transm itting and receiving horns are calculated using the mode matching technique [18]. The mode matching technique is known to be an efficient method for calculating the GSM of horn antennas. For horns used in this study the length of the Ka to X band waveguide transition is 16.51 cm (Fig. 6.25). This long transition is to insure minimum higher order mode excitations. The GSM of th e waveguide transition is obtained using the mode matching technique program described in [18]. The two im portant parameters in using the program are the num- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6 . RESULTS 107 ber of steps and the number of modes considered. The number of sections needed depends on the flaring angle of the transition and on the frequency of operation [19]. In choosing the step size, the A/32 criteria can be used. 16.51 cm Ka-band W aveguide X-band W aveguide i x. Figure 6.25: K a band to X band transition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESU LTS 108 A typical double step plane junction section is shown in Fig. 6.25. The smaller waveguide dimensions are X \ and Y\ , and the larger waveguide dimensions are X 2 and Y^. A t the double plane step discontinuity, incident and reflected waves for all modes (evanescent and propagating) are excited, thus the total field can be expressed as a superposition of an infinite number of modes. The total power in all modes on both sides of the junction is matched according to the mode matching technique. The GSM for the whole waveguide transition is obtained by cascading the GSMs for all sections. 6.3.3 Numerical results Numerical results are obtained for two cases a single cell and a 2 x 2 array. In the first example a single unit cell (Fig. 6.24) is centered in an X-band waveguide. The circuit was fabricated on a 0.381-mm-thick Duroid substrate with relative permitivity e = 6.15. The GSM, for each layer, is calculated for 512 modes. The horns are simulated using 80 modes. The calculated magnitude and phase of the transmission coefficient 52 i for the dominant T E w mode is shown in Figs. 6.26 and 6.27, respectively. It is shown that a transmission of approximately —13 dB is achieved at 32.25 GHz. The second example is a two by two patch array. The same number of modes is considered as in the first example. The results for the transmission coefficient 52 i is shown in Fig. 6.29. The maximum transmission obtained is —6 dB at 32.5 GHz and agrees well with our measured results. Again, this example is for an overmoded waveguide, where many modes can be excited (approximately 18 modes in the air-filled sections of th e X-band waveguide). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 6. RESULTS -13 -14 -15 -16 .-17 -18 CM -19 -20 -21 -22 II.5 32.5 33 33.5 FREQUENCY (GHz) Figure 6.26: Magnitude of transmission coefficient S 2 1 • 200 150 100 CM -50 Q. -100 -150 11.5 32.5 33 33.5 34 FREQUENCY (GHz) Figure 6.27: Angle of transmission coefficient S 2 1 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESULTS Figure 6.28: A two by two patch-slot-patch array in metal waveguide. measured simulated -a -7 -a « -1 0 -11 -1 2 32.5 FREQUENCY 0 H z Figure 6.29: Magnitude of transmission coefficient S2 1 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESU LTS 6.4 111 CPW Array Three examples, based on the magnetic current interface, are numerically investi gated in this section. The first two examples concentrate on the effect of waveguide walls on the antenna input impedance. Two antennas are proposed, a folded slot and a five slot antenna. The third example is a 3 x 3 slot antenna array for the use in spatial waveguide power combiners. The self and mutual impedances of the array elements are calculated. 6.4.1 Folded slot antenna The input impedance of a folded slot antenna shielded in a waveguide, shown in Fig. 6.30, is calculated. The dimensions of the waveguide are a = 6 = 20 cm and the antenna dimensions are c = 7.8 cm and d = 0.9 cm. The dielectric constant is 2.2 and of thickness 0.0813 cm. The structure is decomposed into two layers. These are a magnetic layer with ports, and a dielectric interface. The GSM of the magnetic layer with ports is calculated using the GSM-MoM method and cascaded with the dielectric interface to give the composite GSM. The results are compared with an algorithm utilizing only MoM calcu lation by using the Green’s function for the composite structure (slot backed by dielectric slab). In implementation of the MoM scheme piecewise testing and basis functions are used [8 6 ]. The port impedance in this case is claculated directly from the MoM impedance m atrix. The real and imaginary parts of the input impedance at the center of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 112 folded slot axe shown in Figs. 6.31 and 6.32, respectively. The antenna resonates at 1.5 GHz and has input resistance of 365 Q at the resonant frequency. It can be seen th at the GSM-MoM solution agrees favorably with the MoM port calculation. This verifies the cascading of the GSM for a magnetic layer with ports. The high input resistance at resonance makes the design for a 50 H match more challenging. For this reason York et al. suggested the use of multiple slot antenna configurations for spatial power combining applications. The following example is a demonstration of this idea. Y b -------------------------------------------- a a 2 Figure 6.30: Geometry of th e folded slot in a waveguide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 400 G S M -M o M MoM 350 300 E 250 100 2.5 1.5 Frequency (GHz) Figure 6.31: Real part of the input impedance for folded slot. 250 G S M -M o M MoM 200 150 100 -50 -100 -150 -200 Frequency (GHz) Figure 6.32: Imaginary part of th e input impedance for folded slot. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESU LTS 6.4.2 114 Five slot antenna In an effort to design a CPW antenna system matched to 50 Q, York [4] suggested a five slot configuration shown in Fig. 6.33. Since the input impedance is inversely proportional to the square of the number of turns, increasing the number of slots will automatically reduce the input impedance. Free space measurements [4] show th at an input return loss of —28 dB is observed at 10.5 GHz for the 5-slot antenna shown in Fig. 6.33. Figure 6.33: Five-slot antenna [4]. The GSM-MoM technique is used to calculate the input impedance of the five-slot antenna inside a square waveguide. This gives an insight on the change of the input impedance of th e antenna when operating inside shielded environment. In analysis, the CPW cell is composed of two layers. A magnetic layer with ports and a dielectric interface. The scattering parameters for the magnetic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTE R 6. RESULTS 0 -2 -3 " -6 -8 ■7 -8 8 8.5 9 9.5 10 10.5 11 11.5 12 FREQUENCY (GHz) Figure 6.34: Magnitude of input return loss for 5 folded slots. 200 150 100 -5 0 -ISO -200 11.5 FREQUENCY (GHz) Figure 6.35: Phase of input return loss for 5 folded slots. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 116 layer axe computed as described in Chapter 3. The dielectric interface has a diagonal scattering submatrices representing the transmission and reflection coefficients. The antenna is placed in the center of a square waveguide of dimensions 22.86 x 22.86 mm. The geometry and dimensions of the antenna are shown in Fig. 6.33. The dielectric thickness is 0.635 mm and its dielectric constant er is 9.8. The simulated input returned loss and its phase are shown in Figs. 6.34 and 6.35, respectively. The input return loss has in fact increased from —28 in free space to —8 dB when placed in the waveguide. This might result in less achievable gain when using matched MMIC devices (to 50 Q). 6.4.3 Slot antenna array A 3 x 3 slot antenna array fed by CPW transmission lines is shown in Fig. 6.36. The array consists of nine unit cells. Each unit cell is composed of two orthogonal slot antennas, one for receiving and the other for transm itting. The amplifying unit is a single ended amplifier. To properly design the amplifiers, it is essential to calculate th e driving point impedances of each antenna. This impedance depends on self as well as m utual coupling between the antennas. The array is placed in a square waveguide (a = 6 = 4 cm) and the antenna length is 0.72 cm. The self impedance m atrix (18 x 18) is calculated for the slot array for the frequency range 8-12 GHz. T he reed and imaginary parts of the self impedances of the antenna elements 1, 2, and 5 axe shown in Figs. 6.37 and 6.38, respectively. Resonance is achieved at 9.25 GHz for the self impedances. The impedance at resonance is very high (1700 12) which make it difficult to m atch to 50 12. The value Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESU LTS 117 of the self impedances are much less away from resonance as shown in Figs 6.39 and 6.40. Operating at 10 GHz is more appealing than operating at resonance since it is easier to compensate for the imaginary part while designing the amplifier matching circuit. When designing an amplifier, the feedback from the output to the input is very critical. Positive feedback might result in amplifier oscillations. The mutual coupling between the input and output antennas provides that feedback path and so it is essential to account for th at kind of coupling. The mutual impedance for the center unit cell is shown in Fig. 6.41. Ideally the coupling should be zero. To minimize the coupling, the antennas should be at right angles. 16 10 15 Figure 6.36: A 3 x 3 slot antenna array shielded by rectangular waveguide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESU LTS 1800 1600 1400 '1200 OC 600 400 200 8.5 9.5 10.5 11.5 Frequency (GHz) Figure 6.37: Real part of self impedances. 1500 1000 500 a * -500 -1000 -1500. 8.5 9.5 10.5 11.5 Frequency (GHz) Figure 6.38: Imaginary part of self impedances. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESULTS 400 3SO ’55 300 250 0 150 100 10.5 11 11.5 Frequency (GHz) Figure 6.39: Real part of self impedances. -100 -200 J-300 01 <o-600 •700 -600 10.5 11 11.5 Frequency (GHz) Figure 6.40: Imaginary part of self impedances. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTE R 6. RESU LTS 120 Real 60 • '5,14 oT SO • 5Z. 40 IV TS 30 ■ = 20 - 10. 8.5 10.5 11.5 Frequency (GHz) Figure 6.41: Real and imaginary parts for the m utual impedance Z 5 4 4 . 6.5 Grid Array Perhaps most of the early design efforts for spatial power combiners have been ori ented towards grid structures. The grid array systems are easy to build and fabricate. Analysis and design techniques have emerged specifically for these structures, all for free space case. In this section we will investigate grid arrays when constructed in a shielded environment. A 3 x 3 grid array is shown in Fig. 6.42. The array is composed of nine unit cells. Each unit cell consists of two perpendicular dipole antennas, one for receiving and the other for transm itting. The grid structure uses a differential pair amplifying unit as th at shown in Fig. 1.4. To accurately design the differential pair, the driving point impedance of the antennas must be accurately calculated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 6. RESULTS 121 In this example, the impedance of the center cell is calculated. The magnitude and phase of the input return loss are plotted in Figs. 6.43 and 6.44, respectively. Resonance is achieved at approximately 31 GHz with —15.7dB return loss. 1 cir 1 cm Figure 6.42: A grid array inside a m etal waveguide. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS -12 -14 FREQUENCY (GHz) Figure 6.43: Magnitude of input return loss. 200 150 100 (0 i -10 0 >150 -200 27 30 31 34 FREQUENCY (GHz) Figure 6.44: Angle of input return loss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P TE R 6. RESULTS 6.6 123 Cavity Oscillator A m ultiple device oscillator using dipole arrays was proposed in [89,90]. In both referenced papers, a dedicated Green’s function was developed to model the cavity oscillator and predict the coupling effects. In this section we will analyze a cavity oscillator of the type described in [90] and shown in Fig 6.45. DipoleArray Figure 6.45: Geometry of a dipole array cavity oscillator. 6.6.1 Single dipole The first example is a single dipole antenna inside a cavity. The cavity dimensions are 22.6 x 10.2 x 5.0 mm (a x 5 x c) and the patch is centered in the transverse plane. The dipole length and width are 6 mm and 1 mm, respectively. The frequency of operation is chosen to be from 30 to 33.5 GHz. This means th a t the X band waveguide is overmoded. The calculated input impedance of the dipole is shown in Fig. 6.46. The dipole goes through resonance at 32.25 GHz. Below resonance it is capacitive and above resonance it becomes inductive. A negative resistance diode can be placed at the center of the dipole antenna by properly choosing the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 124 800 ■ 600 | Real-Part Imaginary-Parl / 400 / / / y o 200 S C jo 3CL ✓ y o -200 rf*. . . ™-400 ............... / -600 / y 30.5 31 31.5 32 32.5 33 33.5 Frequency (GHz) Figure 6.46: Input impedance of a dipole antenna inside a cavity. resistive part. Also, since the diode is usually capacitive in nature, an inductive input impedance might be chosen for the dipole antenna. In the analysis, the structure is decompsed into three layers. These are short-circuit, electric current interface with ports (dipole), and magnetic current interface (patch). A block diagram illustrating the modeling process using the GSMMoM technique is shown in Fig. 6.47. After cascading all GSMs, the composit GSM with ports will describe the relationship between th e device ports and the output modes. The composit GSM can be represented in terms of a scattering matrix, impedance m atrix, or an adm itta n c e m atrix. Any of these forms may be employed in nonlinear analysis using a nonlinear frequency-domain circuit simulator. It is interesting to note that if a multilayer array (more than one trans verse active dipole array) of the same structure is used, the modeling scheme will Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 CH APTER 6. RESULTS only require the analysis of one of these arrays. The analysis would then proceed with cascading all sections to obtain the composite GSM. This is in comparison with the direct MoM technique, where the coupling between all arrays m ust be accounted for numerically. Hence the number of elements and the size of th e MoM m atrix are increased. _ SHORT CIRCUIT -- WAVEGUIDE — WAVEGUIDE • — SECTION » - r • • DIPOLE * — SECTION PATCH i P MODES ‘ t 11 (ciRCurr-PORT) i CASCADING COMPOSITE GSM WITH PORTS MOOES (CRCUIT-PORT) Figure 6.47: Block diagram for th e GSM-MoM analysis of cavity oscillator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 6.6.2 126 A 3 x 1 dipole antenna array As a second example a 3 x 1 dipole antenna array is placed inside a similar cavity of the one described in the previous example. The antennas are shown in Fig. 6.48. The m utual and self scattering coefficients are calculated when all antennas are of same lengths and width (6 x 1 mm) and the separations X i = X 2 . Due to symmetry there are only four distinct scattering coefficients (S u, 5 i 2 , S 1 3 , and S 2 2 ). The magnitude and phase of the self and m utual scattering coefficients axe shown in Figs. 6.49 and 6.50, respectively. It is observed that the scattering coefficient S 22 has changed considerably, from the previouse example, due to coupling to the other two antennas. This is illustrated by the nonresonant behaviour of S 2 2 which is now purely capacitive within the frequency range. The port scattering coefficients calculated in this example is essential for designing an active array oscillator. Y Figure 6.48: Dipole antenna array in a cavity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 6. RESULTS 127 0.9 S: 0.7 0.4 0.3 0.V 30.5 31.S 32 32.5 33.5 frequency (GHz) Figure 6.49: Magnitude of scattering coefficients for a dipole antenna array inside a cavity. 150 ■g 100 ° - 1 00 30 30.5 31 32 32.5 33 frequency (GHz) Figure 6.50: Phase of scattering coefficients for a dipole antenna array inside a cavity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 Conclusions and Future Research 7.1 Conclusions A generalized scattering m atrix technique is developed based on a method of mo ments formulation to model multilayer structures with circuit ports. Four general building blocks axe considered. These axe electric interface with ports, magnetic in terface with ports, dielectric interface, and perfect conductor. W ith these blocks the method is applicable to almost all shielded transverse active or passive structures. The GSM for each block is derived separately. The method explicitly incorporates device ports and circuit ports in the formulation of both the electric and magnetic current interfaces. The scattering parameters axe derived for all modes in a single step without the need to calculate the current distribution as an intermediate step. Two cascading formulas axe presented to calculate the composite scat tering m atrix of a multilayer structure. This m atrix is a complete description of 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P TE R 7. CONCLUSIONS AN D FU TU RE R E SEA RC H 129 the structure. The technique can be applied to general structures as well as to waveguide-based spatial power combiners. Various general type structures are sim ulated. These are a wide strip in waveguide, patch array on a dielectric slab, a strip-slot transition module, shielded dipole antenna, and a shielded microstrip stub filter. Spatial power combiners such as patch-slot-patch array, CPW array, grid array, and cavity oscillator array are also simulated. Results are verified by either comparisons to measurements or to other numerical techniques. The interaction of layers is handled using a GSM method where an evolv ing composite GSM m atrix m ust be stored to which only the GSM of one layer at a tim e is evaluated and then cascaded. Thus the com putation increases approximately linearly as the number of layers increases. Memory requirements are determined by the num ber of modes and so is independent of the number of layers. The result ing composite m atrix can be reduced in rank to the number of circuit ports to be interfaced to a circuit simulator. An acceleration procedure based on the Kummer transformation is im plemented to speed up the MoM m atrix elements. The quasistatic terms are ex tracted and evaluated using fast decaying modified Bessel functions of the second kind. T he convergence as well as the accuracy of the acceleration scheme axe demon strated. In implementation two discretization schemes are used, uniform and nonuni form. Although simple, uniform discretization can not accurately represent struc tures with high aspect ratios nor can it capture fine geometrical details without a gross increase in the number of elements. W ith the nonuniform scheme, finer resolutions can be obtained for selective areas as desired. The port param eters obtained from th e electromagnetic simulator are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 7. CONCLUSIONS AN D FU TU RE RESEARC H 130 converted to nodal parameters using the localized reference node concept described in Chapter 5. A scheme for the augmentation of a nodal adm ittance m atrix by a port-based m atrix with a number of local reference nodes permits field derived models to be incorporated in a general purpose circuit simulator based on nodal formulation. The method is immediately applicable to modified nodal adm ittance (MNA) analysis as the additional rows and columns of the MNA m atrix are unaf fected by the augmentation. 7.2 Future Research There axe still many new ideas to be explored in the modeling of waveguide based spatial power combiners. One feature that can be added to the current program is the implementation of nonuniform triangular basis functions. This will enable the modeling of geometrical curves and bends with much better accuracy. Another feature is to include the losses due to dielectrics and metal portions. It is well known that as the separation between layers decreases, in terms of guide wavelength, the number of modes required in the GSM representation will increase to achieve the required accuracy. This might render the procedure impractical for very small separations (less than 0.01 As). In this case it is more efficient to construct a separate analysis module based on the MoM th at takes into account both layers in the Green’s function. Then a GSM is constructed for the closely spaced layer using the calculated MoM m atrix. We adopted this methodology and implemented it for strip and slot layers [86}. There axe other combinations to be considered such as strip and strip, slot and slot layers, and even three layer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H APTER 7. CONCLUSIONS AND FU TU RE RESEA RC H 131 combinations. Different types of Green’s functions such as the potential Green’s func tions and the complex images can be used instead of the electric- and magnetictype Green’s functions used here. This may reduce the CPU time, eventhough an acceleration procedure might still be necessary. Diakoptics in conjunction with the GSM-MoM scheme is another area to be investigated. If the structure can be decomposed in the transverse plane into separate structures related by a matrix then a three dimensional segmentation is achieved (GSM and Diakoptics). Furthermore, when the waveguide dimensions are several wavelengths, then the number of modes involved in the modal expansion of the electromagnetic fields becomes very large and approaches the free space case. It would be interesting to see when the free space solution approaches the waveguide solution and if a hybrid analysis can be employed. In term s of applications to spatial power combining design, the structures to be modeled are endless. Many novel designs can be thought of and perhaps achieve the desired power combining efficiencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] K. J. Sieger, R. H. Abrams Jr., and R. K. 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Microwave Theory Tech., vol. 40, July 1992, pp. 1507-1516. [82] V. Rizzoli, A.Lipparini, and E. Marazzi, “A General-Purpose Program for Non linear Microwave Circuit Design ,” IE E E Trans. Microwave Theory Tech. vol. 31, No. 9, Sep. 1983, pp. 762-769. [83] H. B. Chu and K. Chang, “Analysis of a wide resonant strip in waveguide,” IE E E Trans. Microwave Theory Tech., vol. 40, March 1992, pp. 495-498. [84] R. Collin,Foundations fo r Microwave Engineering, McGraw-Hill, Inc., 1992. [85] Active and Quasi-Optical Arrays fo r Solid-State Power Combining, Edited by R. York and Z. Popovic. New York, NY: John Wiley & Sons, 1997. [86] A. Yakovlev, A. Khalil, C. Hicks, A. Mortazawi, and M. Steer, “ The generalized scattering m atrix of closely spaced strip and slot layers in waveguide,” Submitted to the IEE E Trans. Microwave Theory Tech.. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIB LIO G R A P H Y 139 [87] A. Adams, R. Pollard, and C. Snowden, “A method-of-moments study of strip dipole antennas in rectangular waveguide,” IEE E Trans. Microwave Theory Tech., vol. 45, October 1997, pp. 1756-1766. [88] G. Eleftheriades and J. Mosig,“On the network characterization of planar pas sive circuits using the method of moments,” IE E E Trans. Microwave Theory Tech., vol. 44, March 1996, pp. 438-445. [89] A. Adams, R. Pollard, and C. Snowden, “Method of moments and time domain analyses of waveguide-based hybrid multiple device oscillators,” 1996 IEEE M T T -S International Microwave Symp. Dig., June 1996, pp. 1255-1258. [90] C. Stratakos, and N. Uzungolu “Analysis of grid array placed inside a waveguide cavity with a rectangular coupling aperture,” International Journal o f Numer ical Modeling: Electronic Networks, Devices and Fields., vol. 10, January 1997, pp. 35-46. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ppendix A Usage o f GSM -M oM Code Two steps axe involved to run GSM-MoM: • Converting layout CIF into input geometry file • Running GSM-MoM with input param eter file To convert the CIF file into the geometry file standard format a program called yomoma is used. The command is yomoma 'file.cif' a.d This will cause yomoma to convert the ’file.cif’ into a geometry file and give it the name ’geometry’. The input param eter file contains the necessary information to run GSM-MoM. These information axe: • Frequency range (start-stop-number of points) • Waveguide dimensions • Number and type of layers • Separation between Layers • Dielectric constants 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P P E N D IX A. USAGE OF GSM -M OM CODE 141 • Input geometry files • Number of cascading modes • Number of modes involved in the MoM m atrix element calculation • O utput file names A .l Example In this section a sample run of GSM-MoM is illustrated. The input file, geometry file, and output file axe listed bellow. A.1.1 Input file The input file used in the simulation is described as follows: "FREQUENCY:” n___________________ n "S tart at Frequency:” l.d9 "Stop at Frequency:” 2.5d9 "Num ber of Frequency Points:” 31 "GREEN:” r>____________n ”m max:” 450 ”n max:" 450 "WAVEGUIDE” » n "a:(x direction):” 20.d0 ”b:(y direction):” 20.d0 ”xm ax:(maximum x-dimensions)” 2.5d-l "num ber of units in m axim um x-dimension:” 1 n » "LAYERS:” n ____________ r> "N um ber of Layers:” 2 "Type of Layer 1:" 2 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A P PEN D IX A. USAGE OF GSM-MOM CODE "Type of Layer 1:” 0 0 "Geometry File of Layer 1:” "cpw2.dat” "Geometry File of Layer 2:” ”" "epson 1:” l.dO "epson 2:” 2.2d0 "epson 3:” l.dO "Normalizing Resistance (Ohms):” 50.d0 "separation:” 0.0813 "CASCADING PARAMETERS:” ”m scatter max:” 10 ”n scatter max:” 10 w rt "QUASISTATIC PARAMETERS:” "nqs max :” 450 ”m qsmax :” 450 ”m k u m m er:” 150 ”n k u m m er:” 150 y> r) "power conservation flag:” 0 "O U TPU T FILES” ” ” ” ” ” ” ” 1 ports-s: (port S-parameters) ” "ports-s.dat” 2 ports-z: (port z-parameters) ” "ports-z.dat” 3 modes-s: (modes s-parameters) ” "modes-s.dat” 4 modes-z: (modes z-parameters) ” "modes-z.dat” 5 modes-ports-s: (both modes and ports s-parameters) ” "modes-ports-s.dat” 6 modes-ports-z: (both modes and ports z-parameters) ” "z.dat" 7 power-conservation: (power conservation check) ” "conservation.dat” A .1.2 Geometry file The geometry file ’cpw2.dat’ is constructed as follows X-center 3.94e-02 4.06e-02 Y-center c l c2 2.39e-02 1.2e-03 1.2e-03 2.39e-02 1.2e-03 1.2e-03 dl d2 direction 1.2e-03 1.2e-03 1 1.2e-03 1.2e-03 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 A P PEN D IX A. USAGE OF GSM-MOM CODE 3.94e-02 4.06e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 3.88e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 5.39e-02 5.39e-02 5.33e-02 5.21e*02 5.09e-02 4.97e-02 4.85e-02 4.73e-02 4.61e-02 4.49e-02 4.37e-02 4.25e-02 4.13e-02 4.01e-02 3.89e-02 3.77e-02 3.65e-02 3.53e-02 3.41e-02 3.29e-02 3.17e-02 3.05e-02 2.93e-02 2.81e-02 2.69e-02 2.57e-02 2.45e-02 5.33e-02 5.21e-02 5.09e-02 4.97e-02 4.85e-02 4.73e-02 4.61e-02 4.49e-02 4.37e-02 4.25e-02 4.13e-02 4.01e-02 3.89e-02 3.77e-02 3.65e-02 3.53e-02 3.41e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 L2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 L2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPEND IX A. USAGE OF GSM-MOM CODE 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 4.12e-02 3.29e-02 3.17e-02 3.05e-02 2.93e-02 2.81e-02 2.69e-02 2.57e-02 2.45e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1.2e-03 144 3 3 3 3 3 3 3 3 Where X-center and Y-center are the center coordinates for the basis functions, c,- and d,- are the x and y dimensions of the basis function described in Chapter 4, direction is either 1 or 2 representing either r or y direction. A. 1.3 Output file A sample of the output file ports-s.dat is shown bellow Frequency Row Column Real+ imaginary -0.91289578394812+i* -7.1111105317860D-02 1.0000000000000 5 5 9 -0.59945553696502+i* 0.60133418779286 5 1.0500000000000 5 1.1500000000000 9 9 -0.32376060701267+i* 0.78198669048933 -3.9945984045278D-02+i* 0.84052863832245 9 1.2000000000000 9 0.21275834500019+i* 0.80899515276673 1.2500000000000 9 9 0.42130121767699+i* 0.71580412291173 1.3000000000000 9 9 0.58405064072347+i* 0.57934969403024 1.3500000000000 9 9 0.70158107310522+1* 0.40810140007813 1.4000000000000 9 9 0.77028276979754+i* 0.20204238977448 1.4500000000000 9 9 0.77401948182578+i* -4.6523796388117D-02 1.5000000000000 13 13 0.66256671113527+i* -0.34821472284445 1.5500000000000 13 13 0.28048687899227+i* -0.67340488907707 1.6000000000000 13 13 1.6500000000000 -0.71241823369701+i* -0.43141063223245 13 13 -7.4014430138632D-03+i* 0.43601024646452 1.7000000000000 13 13 21 0.15342478578630+i* 0.30093744513284 1.7500000000000 21 21 1.8000000000000 21 0.20419668859631 +i* 0.23153735363350 21 0.21291162875193+i* 0.18126065668905 1.8500000000000 21 1.9000000000000 21 21 0.19584893946645+i* 0.14532729621506 0.16038858904577+i* 0.12508973667232 21 1.9500000000000 21 21 0.11212046082019+i* 0.12381333047817 2.0000000000000 21 5.7196728080260D-02+i* 0.14523453183630 2.0500000000000 21 21 1.9448436254146D-03+i* 0.19301324044885 2.1000000000000 21 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AP PE N D IX A. USAGE OF GSM-MOM CODE 2.1500000000000 2.2000000000000 2.2500000000000 2.3000000000000 2.3500000000000 2.4000000000000 2.4500000000000 2.5000000000000 A .2 21 25 25 29 29 29 37 37 21 25 25 29 29 29 37 37 -2.9969462383864D-02+i* 0.26404303848613 -4.9908826387294D-02+i* 0.35390399938845 -4.0663996216727D-02+i* 0.46077415705524 1.6785089962526D-O2+i* 0.55806133230530 0.11020052940192+i* 0.64268899167752 0.15458275296470+i* 0.64501106357930 0.24259495341224+i* 0.70028699244470 0.33768340295208+i* 0.72427841760270 Makefile # GSM-MoM MAKEFILE FOR SUN ULTRAS # FC=f77 -fast LEX=flex YACC=bison # # #C FL A G S= -g3 CFLAGS= LDFLAGS= -L/ncsu/gnu/lib # linker flags # FORTRAN SOURCE FILES FSRCS = scatter_main.f m om Jayer.f em pty .guide.f m atrix.f scatterJayer.f scatter_dielectricjconductor.f cascade.f conservation.f circuit_parameters.f zqsjempty.f constants.f gama.f OBJS = { F S R C S : . / = .o) # L I N K F O R T R A N O B J E C T F I L E S T O C R E A T E E X E C U T A B L E F IL E G S M - M o M :(OBJS) f77 -fast $(OBJS) $(LDFLAGS) # Flex and Bison stuff # # -n n -f $(OBJS) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 A P PEN D IX A. USAGE OF GSM-MOM CODE A.3 146 Program Description The program consists of the following subroutines: • scatter_main.f: This subroutine is the main program. It reads in the geometry files and the input d ata file and calls all other programs. • momJayer.f: This is the MoM calculation subroutine. It calls the approperiate functions to calculate the MoM impedance and adm ittance m atrix elements. • em pty .guide.f: Contains functions used for the MoM m atrix element calcula tions. • matrix.f: Contains the m ath routine for m atrix inversion. • scatterJayer.f: calculates scattering parameters for each layer. • scatter.dielectricjconductor.f: calculates scattering parameters for dielectric and conductor layers. • cascade.f: Cascades ail layers. • conservation.f: Checks the power conservation of individual as well as cascaded layers. • circuit-parameters.f: Calculates the circuit parameters. • zqs.empty.f: Calculates the quasistatic part of the MoM m atrix • constants.f: Calculates constants used by ail routines. • gama.f: Calculates the propagation constants of modes used in the GSM. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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