experimental and electromagnetic modeling of waveguide

Experimental and ElectromagneticModeling of Waveguide-Based Spatial

Power Combining Systems

by

CHRIS WAYNE HICKS

A dissertation submitted to the Graduate Faculty ofNorth Carolina State University

in partial fulfillment of therequirements for the Degree of

Doctor of Philosophy

ELECTRICAL ENGINEERING

Raleigh

2002

APPROVED BY:

Co-Chair of Advisory Committee

Chair of Advisory Committee

Abstract

HICKS, CHRIS WAYNE. Experimental and Electromagnetic Modeling of Waveguide-Based Spatial Power Combining Systems. (Under the direction of Michael B. Steer.)

Recent technological advancements and demands for high power sources

at microwave and millimeter-wave frequencies have initiated extensive theoretical

and experimental research in the area of quasi-optical and spatial power combining.

The work described here was motivated by the necessity to develop a modeling en-

vironment for the electromagnetic analysis of planar quasi-optical and spatial power

combining systems, in order to understand physical fundamentals and provide a

basis for the design process. Two types of planar quasi-optical (QO) and spatial

power combining systems are investigated.

Propagation in a QO parallel plate system is investigated with the aim of

establishing the mode structure and characteristics of the modes. Theoretical elec-

tromagnetic properties of a Gauss-Hermite beammode expansion was developed, and

verified experimentally, for the prediction of the resonant frequencies of the structure

and beammodes dispersion behavior. The system was designed, fabricated, tested,

and showed good agreement between the experimental and theoretical results. In

addition, a QO parallel-plate stripline-slot amplifier system was designed, tested

and compared to a QO open HDSBW amplifier system with Vivaldi-type antennas.

Experimental results verify that a QO parallel-plate stripline-slot amplifier proposed

in this dissertation can be modeled using Gauss-Hermite beammodes.

A full-wave electromagnetic model is developed and verified for a spa-

tial power combining system consisting of slotted rectangular waveguides coupled

to a strip line. The waveguide-based structure represents a portion of the planar

QO power combiner discussed above. The electromagnetic simulator is developed

to analyze the stripline-to-slot transitions in a waveguide-based environment. The

simulator is based on the method of moments (MoM) technique to model a power

combining array of slotted waveguide modules coupled to a strip line. The simulator

uses Galerkin projection technique with piecewise sinusodial testing and basis func-

tions in the electric and magnetic surface current density expansions. Electric and

magnetic dyadic Green’s functions are developed for an infinite rectangular waveg-

uide in the form of partial expansions over the complete system of eigenfunctions of

a transverse Laplacian operator. Numerical results are obtained and compared with

a commercial microwave simulator for a few representative slot-strip-slot spatial

power combining transitions and arrays.

Dedication

I dedicate this work to my wife, Shirley D. Hicks, my daughter, Christen

A. Hicks, and my mother, Annie B. Hicks. I also dedicate this work to my late

father, Andrew Hicks Jr. who provided me with the vision to dream dreams.

I also dedicate this work to God who gave me the mental and physical

strength to pursue and obtain my Ph.D. degree.

ii

Biographical Summary

Chris Wayne Hicks was born on September 8, 1962 at Kindley Air Force

Base, Bermuda while his father was enlisted in the United States Navy. Chris

attended high school at South Florence High School in Florence, South Carolina. He

received his B.S. degree in Electrical Engineering at the University of South Carolina,

Columbia, South Carolina in May 1985 and a M.S. degree in Electrical Engineering

from North Carolina Agriculture and Technical State University in Greensboro,

North Carolina in May 1994. Since June 1985, he has been employed at the Naval

Air Systems Command (NAVAIR) where he currently works for the RF Sensors

Division in Patuxent River, Maryland. In 1995, he enrolled in North Carolina State

University to pursue his Ph.D. degree in electrical engineering where he worked as

a Research Assistant for the Electronics Research Laboratory in the Electrical and

Computer Engineering Department. Chris received two one-year NAVAIR training

fellowships to pursue his graduate studies. He is a member of the Institute of

Electrical and Electronic Engineers (IEEE) Microwave, Theory Technique Society

and a member of the Antenna and Propagation society. He is also a member of the

Etta Kappa Nu honor society.

iii

Acknowledgements

I would like to thank my academic advisor Dr. Michael B. Steer for

his support and guidance during my graduate studies. I also thank Dr. James

Mink for his wisdom, patience and for serving on my advisory committee. I thank

Dr. Alexander Yakovlev for serving on my advisory committee and teaching me

the fine art of electromagnetics. I also thank Dr. Gianluca Lazzi, Dr. Robert

J. Nemanich, Dr. James F. Kauffman, and Dr. James Harvey for serving on my

advisory committee.

I would like to thank Dr. Huang-Shen Hwang and Dr. Todd Nuteson

for their many discussions on two-dimensional and three-dimensional quasi-optical

power combining research. I would like to thank Dr. Ahmed Khalil and Dr. Mostafa

Abdulla for their discussions on electormagnetics. I would like to thank Mrs. Jaee

Patwardhan-Naik for her moral support and encouragement. I would like to thank

Dr. Mete Ozkar for assisting me with measurements and Dr. Carlos Christofferson

for his computer expertise.

Finally, I would like to thank my wife, Shirley and my daughter, Chris-

ten, for their loving support.

iv

Contents

List of Tables x

List of Figures xi

1 Introduction 1

1.1 Motivation and Objective of This Study . . . . . . . . . . . . . . . . 1

1.2 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Literature Review 12

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Free-Space Quasi-Optical Amplifiers . . . . . . . . . . . . . . 14

2.2.2 Quasi-Optical 2-D Dielectric Power Combining . . . . . . . . . 15

v

2.2.3 Waveguide Spatial Power Combining . . . . . . . . . . . . . . 17

2.3 Numerical Modeling of Spatial Power Combiners . . . . . . . . . . . . 19

2.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Modeling Free-Space Power Combining Systems . . . . . . . . 20

2.3.3 Modeling Waveguide-Based Spatial Power Combining Systems 21

2.3.4 Waveguide Dyadic Green’s Function . . . . . . . . . . . . . . 23

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Two-Dimensional Parallel-Plate Resonator 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Principles of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Beam-Mode Theory In a Closed-Boundary Slab Beam Waveguide . . 28

3.3.1 Orthogonality of Fields . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Power Normalization . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.4 Mode Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.1 Open System Configuration . . . . . . . . . . . . . . . . . . . 40

3.4.2 Vivaldi Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.3 Unit Cell of a Slot Antenna . . . . . . . . . . . . . . . . . . . 43

vi

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Waveguide-Based Slot-Strip-Slot Transitions 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 General Electromagnetic Formulation and Dyadic Green’s Functions . 46

4.3 Dyadic Green’s Functions for a Rectangular Waveguide . . . . . . . . 53

4.3.1 Magnetic Dyadic Green’s Functions . . . . . . . . . . . . . . . 54

4.3.2 Electric Dyadic Green’s Function . . . . . . . . . . . . . . . . 60

4.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Integral Equation Formulation . . . . . . . . . . . . . . . . . . . . . . 63

4.4.1 Geometry Description . . . . . . . . . . . . . . . . . . . . . . 63

4.4.2 Magnetic Currents . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.3 Scattered Electric and Magnetic Fields . . . . . . . . . . . . . 66

4.4.4 Total Electric and Magnetic Fields . . . . . . . . . . . . . . . 68

4.4.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 69

4.4.6 Testing and Basis Functions . . . . . . . . . . . . . . . . . . . 70

4.5 Method of Moment Formulation . . . . . . . . . . . . . . . . . . . . . 73

4.5.1 Incident Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5.2 Unknown Current Coefficient Vector . . . . . . . . . . . . . . 79

4.5.3 Scattering Parameters . . . . . . . . . . . . . . . . . . . . . . 80

vii

5 Simulation and Results 82

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 Waveguide-Based Slot-Strip-Slot Transitions . . . . . . . . . . . . . . 85

5.2.1 Transverse Slot . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.2 Single Slot-Strip-Slot . . . . . . . . . . . . . . . . . . . . . . . 89

5.2.3 Double Slot-Strip-Slot . . . . . . . . . . . . . . . . . . . . . . 89

5.2.4 Double Slot-Strip-Slot with Two Shifted Slots . . . . . . . . . 92

5.2.5 Triple Slot-Strip-Slot . . . . . . . . . . . . . . . . . . . . . . . 92

5.2.6 Single Slot-Strip-Slot with Two Strips . . . . . . . . . . . . . . 96

5.3 Waveguide-Based Slot-Strip-Slot Arrays . . . . . . . . . . . . . . . . 96

5.3.1 Series 1 × 2 Coupler . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.2 2 × 2 Slot-Strip-Slot Array . . . . . . . . . . . . . . . . . . . 104

5.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.4 Computational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Conclusion and Future Research 110

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1.1 Mode Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1.2 Electromagnetic Modeling Technology . . . . . . . . . . . . . 111

6.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

viii

Bibliography 114

A Method of Moments Implementation 124

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.2 Admittance Matrix for Transverse Slots . . . . . . . . . . . . . . . . . 125

A.3 Impedance Matrix for Longitudinal Strips . . . . . . . . . . . . . . . 128

A.3.1 Completely Overlapping Case . . . . . . . . . . . . . . . . . . 129

A.3.2 Non-Overlapping Case . . . . . . . . . . . . . . . . . . . . . . 134

A.3.3 Partially Overlapping Case . . . . . . . . . . . . . . . . . . . . 135

A.4 Strip to Slot Coupling Matrix . . . . . . . . . . . . . . . . . . . . . . 137

A.4.1 Non-Overlapping Case . . . . . . . . . . . . . . . . . . . . . . 138

A.4.2 Partially Overlapping Case . . . . . . . . . . . . . . . . . . . . 139

A.5 Slot-to-Strip Coupling Matrix . . . . . . . . . . . . . . . . . . . . . . 139

ix

List of Tables

3.1 Selected resonance frequencies of the parallel-plate resonator system. 40

x

List of Figures

1.1 RF components sub-area military essential electronics. . . . . . . . . 2

1.2 A 3-D grid quasi-optical power combining system. . . . . . . . . . . . 3

1.3 A 2-D dielectric slab quasi-optical power combining system. . . . . . 4

1.4 A 2-D cascade quasi-optical system. . . . . . . . . . . . . . . . . . . . 6

1.5 Aperture-coupled stripline-to-waveguide transition. . . . . . . . . . . 7

2.1 Free space quasi-optical grid amplifier. . . . . . . . . . . . . . . . . . 15

2.2 Concave and convex lens dielectric slab power combining systems. . . 17

2.3 X-Band waveguide spatial power combining system. . . . . . . . . . . 18

3.1 Passive 2-D quasi-optical power combining system with concave lenses;

(a) open structure and (b) closed structure. . . . . . . . . . . . . . . 26

3.2 The 2-D HDSBW system with convex/concave lenses and 4 × 1 MES-

FET amplifier array. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Input and output coupling antennas on RT/Duriod substrate: (a)

MESFET Vivaldi amplifier; and (b) MMIC stripline-slot amplifier. . . 28

xi

3.4 Electric-field wave model for 2-D power combining system. . . . . . . 29

3.5 Test configuration for the confocal parallel-plate resonator system. . . 36

3.6 Reflection coefficient at the input to a confocal 2-D parallel-plate

resonator system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Electric-field mode profile at 6.898 GHz for the parallel-plate confocal

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.8 A plot of the magnitude of S11 and S21 for selected resonance fre-

quencies of the parallel-plate resonator system. . . . . . . . . . . . . . 39

3.9 The concave-lens system configuration for a unit-cell Vivaldi-based

amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.10 The concave-lens system configuration for a unit-cell amplifier. . . . . 42

3.11 Amplifier gain for a unit-cell MMIC amplifier; (a)Vivaldi cascade

MMICs and (b) single stripline-slot MMIC. . . . . . . . . . . . . . . 43

4.1 Geometry of a closed-boundary waveguiding structure containing aper-

tures and conducting strips in the presence of an impressed electric

current source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 An aperture-coupled electric-magnetic layered waveguide transition. . 53

4.3 Geometry of a rectangular waveguide. . . . . . . . . . . . . . . . . . . 54

4.4 An aperture-coupled stripline-to-waveguide transition. . . . . . . . . . 64

xii

5.1 Full view of waveguide transitions: (a) single slot-strip-slot; (b) dou-

ble slot-strip-slot; (c) double slot-strip-slot one two shifted slots; (d)

triple slot-strip-slot; and (e) single slot-strip-slot with two strips. . . . 83

5.2 Top view of waveguide transitions: (a) single slot-strip-slot; (b) dou-

ble slot-strip-slot; (c) double slot-strip-slot one two shifted slots; (d)

triple slot-strip-slot; and (e) single slot-strip-slot with two strips. . . . 84

5.3 Geometry of a centered transverse slot between two rectangular waveg-

uides: a) full view; b) top view. . . . . . . . . . . . . . . . . . . . . . 86

5.4 Magnitude of S11 versus varying centered transverse slot lengths be-

tween two rectangular waveguides: MoM (solid line), and published

experimental and simulation results (Fig. 2b curves (1) and (3)

in [72]) (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 MoM (solid line) and HFSS (dashed line) comparison of the scattering

parameters at the center of a transverse slot between two rectangular

waveguides: (a) magnitude of S11 and S21; (b) magnitude and phase

of S11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.6 Geometry of an single slot-strip-slot waveguide transition. . . . . . . . 90

5.7 MoM (solid line) and HFSS (dashed line) comparison for the scat-

tering parameters for the single slot-strip-slot waveguide transition.

Magnitude and phase: (a) S11; and (b) S41. . . . . . . . . . . . . . . . 91


tering parameters for the double slot-strip-slot waveguide transition.

(a) Magnitude and (b) phase. . . . . . . . . . . . . . . . . . . . . . . 93

xiii


tering parameters for the double slot-strip-slot waveguide transition

with two shifted slots . Magnitude and phase: (a) S11; and (b) S41. . 94


tering parameters for the triple slot-strip-slot waveguide transition.

Magnitude and phase: (a) S11; and (b) S41. . . . . . . . . . . . . . . . 95


tering parameters for the single slot-strip-slot waveguide transition.

Magnitude and phase (a) S11, and (b) S41. . . . . . . . . . . . . . . . 97

5.12 Full and top view: (a) series 1 × 2 slot-strip-slot coupler array; and

(b) series 1 × 2 slot-strip-slot coupler array with breaks in the strip. . 98

5.13 Input scattering parameter S11 for the series 1 × 2 slot-strip-slot

coupler array; (a) magnitude and (b) phase. MoM (solid line), HFSS

0.001 dB (dashed line) and HFSS 0.005 (dotted line). . . . . . . . . . 99

5.14 MoM results for the magnitude of S11 for the series 1 × 2 slot-strip-

slot coupler array; (a) ε2 = 1.0 (solid line) and ε2 = 2.2 (dotted

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.15 Reverse coupling for the series 1 × 2 slot-strip-slot coupler array,

ε2 = 1.0, MoM (solid line) and HFSS (dotted line). . . . . . . . . . . 101

5.16 Input scattering parameter S11 for the series 1 × 2 slot-strip-slot

coupler array with two strips; (a) magnitude and (b) phase. MoM

(solid line), HFSS 0.001 dB (dashed line). . . . . . . . . . . . . . . . 102

5.17 MoM simulation for the reverse coupling for the series 1×2 slot-strip-

slot coupler array, ε2 = 1.0. . . . . . . . . . . . . . . . . . . . . . . . 103

xiv

5.18 Full and top view of a 2 × 2 slot-strip-slot array . . . . . . . . . . . . 105

5.19 MoM simulation of a 2 × 2 slot-strip-slot waveguide-based array;

ε2 = 1.0 (solid line) and ε2 = 2.2 (dashed line); S11 (a) magnitude,

(b) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.20 Reverse coupling for the series 1 × 2 slot-strip-slot coupler array,ε2 =

1.0, MoM (solid line) and HFSS (dotted line). . . . . . . . . . . . . . 107

A.1 The geometry for: (a) the dzdz′ integration for self-coupling for a

single slot; (b) the dzdz′ integration for the coupling between two

slots; and (c) the dx′ integration for all slots. . . . . . . . . . . . . . . 126

A.2 Testing and basis functions for the longitudinal strip: (a) completely

overlapping case; (b) partially overlapping case; and (c) non-overlapping

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A.3 Longitudinal strip overlapping case (a) z′ < zi, and (b) z′ > zi. . . . . 131

A.4 Longitudinal strip testing and basis functions do not overlap . . . . . 134

A.5 Longitudinal strip test and basis functions partially overlap. . . . . . 136

A.6 Crossed strip and slot testing and basis functions: (a) non-overlapping

case; and (b) overlapping case. . . . . . . . . . . . . . . . . . . . . . . 140

xv

Chapter 1

Introduction

1.1 Motivation and Objective of This Study

Military and civilian applications require significant power at microwave and millimeter-

wave frequencies [1]. Medium-to-high power levels are needed for applications such

as communications, active missile seekers, radar, and millimeter-wave imaging. To

meet this need, klystrons, traveling-wave tubes, and gridded tubes are generally

utilized as shown in Fig 1.1. However, tubes are bulky, costly, require high operat-

ing voltages, and have a short lifetime. As an alternative, solid-state devices offer

several advantages such as lightweight, smaller size, wider bandwidths, and lower

operating voltages. Lower costs also result because systems can be constructed us-

ing planar fabrication techniques. However, as the frequency increases, the output

power of solid-state devices decreases due to their smaller physical size. Therefore,

to achieve sizable power levels that compete with the power levels generated by

vacuum tubes, many solid-state devices must be combined in an array configuration

1

CHAPTER 1. INTRODUCTION 2

utilizing spatial power combining techniques [2]. The analysis, modeling and design

procedure for spatial power combiners is not well developed. While many spatial

power combining topologies have been investigated, it is not clear which topology is

optimum. One form of spatial power combining is quasi-optical (QO) power com-

Potential IVHSPotential

IVHS

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Target Discrimination

Air Defense, Surveillance &

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SHF Satcom

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MICROWAVEPOWER TUBES

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Source: 1998 OSD S&T TARA

RF Components Sub-AreaMilitary Essential Electronics

Figure 1.1: RF components sub-area military essential electronics.

bining. If optical techniques such as diffraction or optical components such as lenses

or reflector mirrors are incorporated into the system, the term quasi-optical power


combining is preferred because the field structure is described in terms of Gaussian

beam modes [3]. Quasi-optical power combining couples the power from an array of

solid-state amplifier or oscillator devices utilizing wavebeam principles. Microwave

lenses are utilized to provide periodic re-focusing of the wavebeams to combine power

in a single paraxial mode over many wavelengths. QO systems are designed to have

cross-section dimensions of 2 to 10 or more wavelengths. Consequently, component

tolerances are greatly relaxed along the transverse and longitudinal directions. As a

result, significant area is made available for numerous solid-state devices and control

components to be included within the structure to achieve the desired output power.

Recent work pioneered at North Carolina State University has demonstrated a vi-

PolarizerInput Output

Polarizer

ArrayLensAmplifier

Lens

Input Horn Far-Field

YX

Z

Figure 1.2: A 3-D grid quasi-optical power combining system.

able two-dimensional (2-D) quasi-optical power combining system [19]. The system

consisted of concave and convex lenses, MESFET or MMIC devices, and Vivaldi-

type antennas operating in an open waveguide configuration as shown in Fig. 1.3. A

4× 1 amplifier array generated 11 dB and 4.5 dB of amplifier gain and system gain

respectively, at 7.12 GHz, and the single MMIC Vivaldi-type antenna produced 24

dB of amplifier gain at 8.4 GHz. The system was tested with the array placed on

the top of the dielectric slab and also with the array located under the dielectric

slab in the bottom ground plane. The electromagnetic field distribution is such

that the electric field is transverse relative to the ground plane and yet the tan-

gential electric at the surface of the top and bottom ground planes is zero. Since


there is no electric field perpendicular to the ground planes, the magnetic field at

the ground planes is also zero. Thus the currents in the ground planes are to a

first approximation are also zero. Consequently, this system has the potential for

very low loss at millimeter-wave frequencies. Open systems have demonstrated the

ability to combine power from a source array. However, using this configuration,

excessive scattering losses were identified. Major losses include radiation losses from

the dielectric top and sidewalls, beam confinement to within the dielectric slab, and

scattering losses of the Vivaldi antenna. With the antenna located at the top inter-

face, the high field distribution causes perturbations and variations which make it

difficult to predict and control the phase distribution of the array. It was determined

that this was due to the scattering of the field which was made more significant as

the field was not strongly guided. Better performance was obtained with the array

in the bottom ground plane. Still fields extend into the region above the slab in this

open system. In an effort to improve the performance of the 2-D open system, an

SIGNAL GENERATOR DETECTOR

LENSES AMPLIFIERS

RECEIVINGHORN ANTENNA

TRANSMITTINGHORN ANTENNA

d1 d2 d3

Figure 1.3: A 2-D dielectric slab quasi-optical power combining system.

experimental and theoretical investigation of a 2-D quasi-optical power combining

system based on a parallel-plate waveguide is undertaken in this dissertation. The

experimental 2-D quasi-optical systems is based on the following conjectures: a) the

use of the transverse magnetic (TM) field distribution in the planar system would


enable better controlled mode behavior; b) separating the input traveling wave and

so eliminating a path matching problems (described below); c) a structure needs

to be developed that better supports development in stages; (there is a need for a

medium-power combining system which will use the staged development); and d) a

structure needs to be chosen that is more amenable to electromagnetic modeling.

Improvements to the open system were made by employing a stripline-slot array in

two closed systems to reduce the losses associated with the open system. The first

closed 2-D quasi-optical parallel-plate waveguide power combining system utilized

two ground planes to minimize the losses associated with the open systems. This

proved to be problematic with two main issues: a) the traveling-wave structure

requires a phase match between the path the signal takes propagating in the dielec-

tric slab and the path through active device. This match was difficult to achieve

except over a very narrow bandwidth. The parallel-plate confines the TM modal

wavebeam, which provides maximum coupling to the source array, located in the

bottom ground plane. Replacing the Vivaldi antenna with a stripline slot antenna

also enhances the system performance. The Vivaldi has greater bandwidth than the

slot antenna. However the Vivaldi has a larger metallic surface area that produces

significant scattered field which leads to loss. Also the Vivaldi has poor isolation

between the input and output antennas because the antennas and amplifiers are

on the same plane and the input and output are in close proximity. The stripline-

slot antenna can be used in a multilayer configuration with the slots and amplifiers

located in different planes, thereby improving isolation between the input and out-

put slots and minimizing scattering loss. The parallel-plate quasi-optical system is

investigated by developing the modal theory for a parallel-plate cavity and conduct-

ing experiments in order to validate the theory. Next, a stripline-slot amplifier is

experimentally compared to a Vivaldi amplifier and tested in a 2-D quasi-optical

parallel-plate environment.


b

c

a

QUASI-OPTICAL 2D POWER COMBINING SYSTEMAMPLIFIER PLANE

AMPLIFIER ARRAYSINPUT COUPLING NETWORK

OUTPUT COUPLINGNETWORK

GROUND PLANE

INPUT PLANE

MICROWAVE ABSORBERPHOTONIC MATERIAL

Pin

PoutOUTPUT PLANE

AMPLIFIER PLANE

Figure 1.4: A 2-D cascade quasi-optical system.

Shown in Fig. 1.4 is the 2-D quasi-optical slot-strip-slot waveguide-based

power combining array. The array consists of three waveguide planes: the input

plane which couples power to the amplifier plane; the amplifier plane which couples

power from the input plane to the output plane; and the output plane which outputs

the power coupled from the amplifier plane. Details of the a single slot-strip-slot ac-

tive transition (or unit cell) is shown in Fig. 1.5. The amplifier located in the middle

of the strip is designed to amplify the signal on the strip. The primary objective

of this dissertation is to understand the principles of operation of the stripline-slot

antenna which is a critical component of the amplifier array. The electromagnetic

model of a rectangular waveguide-based power combining transition can be devel-

oped based on the integral equation formulation for electric and magnetic surface

current density discretized via the method of moments (MoM). In this formulation,

dyadic Green’s functions are obtained in the form of partial expansion over the

complete system of eigenfunctions of a transverse Laplacian operator. Numerical


Port 3 Port 4Output

Port 1Input

Dielectric Between Waveguides

Port 2

Upper Waveguide

MMIC

Bottom Waveguide

Input Slots

Output Slots

εrStripline

Figure 1.5: Aperture-coupled stripline-to-waveguide transition.


results are obtained and compared with a commercial microwave simulator (Agilent

HFSS and Ansoft HFSS) for a few simplified representative structures, including

various configurations of planar arrays of slotted waveguide transitions coupled to

a stripline.

1.2 Dissertation Overview

Chapter 2 presents the literature review for experimental 2-D and 3-D quasi-optical

and waveguide-based spatial power combining systems, and discusses the numerical

and analytical techniques utilized to model quasi-optical and waveguide-based spa-

tial power combining systems.

Chapter 3 presents the Gaussian beammode theory for the 2-D quasi-

optical parallel-plate waveguide and resonator. Next, the beammode theory is

verified by designing and fabricating a 2-D quasi-optical parallel-plate cavity and

stripline-to-slot quasi-optical amplifier. Lastly, the experimental results are pre-

sented and discussed.

In Chapter 4, the electromagnetic model for an aperture-coupled stripline-

to-waveguide transition is investigated. A general formulation for the scattered elec-

tric and magnetic fields inside an arbitrary closed structure is developed and the

dyadic Green’s functions for a rectangular waveguide are derived. Next, an integral

equation formulation for electric and magnetic surface current density discretized

via the MoM is developed for the stripline-to-waveguide transition. Lastly, the MoM

matrix system equations utilized to calculate the unknown current coefficients and

scattering parameters are discussed.

Chapter 5 presents the simulation results of aperture-coupled stripline-

to-waveguide power combining structures. The scattering parameters for single slot-

strip-slot transitions, multiple slot-strip-slot transitions, and slot-strip-slot array


transitions are simulated and compared to a commercial simulator.

Chapter 6 summarizes the work presented in this dissertation and dis-

cusses conclusions and future work.

1.3 Original Contributions

The original contributions for this research are:

• The development of Gaussian beammode theory that predicts the resonant

frequencies for a 2-D QO parallel-plate cavity as described in Section 3.3.

• Successful experiment and design of a 2-D quasi-optical parallel-plate cavity

that agrees with theory as described in Section 3.4.

• Experiment and design of a waveguide 2-D QO dielectric power combining

structure as described in Section 3.4.

• The development of an MoM electromagnetic simulator for a stripline-to-

waveguide transition for a waveguide-based power combining system as de-

scribed in Chapter 4.

• Provide insight in the electromagnetic coupling behavior of a stripline-to-

waveguide transition for a waveguide-based power combining system as de-

scribed in Chapter 6.

1.4 Publications

The material described in this dissertation resulted in the following publications.


• C. W. Hicks, H. Hwang, M. B. Steer, J. W. Mink, J. Harvey, “Spatial power

combining for two dimensional structures,” IEEE Trans. Microwave Theory

Tech., Vol. 46, pp. 784–791, June 1998.

• A. B. Yakovlev, A. I. Khalil, C. W. Hicks, A. Mortazawi, M. B. Steer, “The

generalized scattering matrix of closely spaced strip and slot layers in waveg-

uide,” IEEE Trans. Microwave Theory Tech., Vol. 48, pp. 126–137, Jan.

2000.

• H. S. Hwang, C. W. Hicks, M. B. Steer, J. W. Mink, and J. Harvey, “A quasi-

optical dielectric slab power combiner with a large amplifier array,”IEEE AP-S

International Symp. and USNC/URSI National Radio Science Meeting Dig.,

pp. 482–485, June 1998.

• M. B. Steer, T. W. Nuteson, C. W. Hicks, J. Harvey, and J. W. Mink, “Strate-

gies for handling complicated device-field interactions in microwave systems,”

Proc. PIERS Symp., July 1996.

• J. Harvey, M. B. Steer, H. Hwang, T. W. Nuteson, C. W. Hicks, and J. W.

Mink, “Distributed power combining and signal processing in a 2D quasi-

optical system,”Proc. WRI International Symp. on Directions for the Next

Generation of MMIC Devices and Systems, Edited by N.K. Das and H.L.

Bertoni, Plenum Press: New York, NY, pp. 75–82, September 1997.

• 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 A. B. Yakovlev, “Global modeling of spatially distributed mi-

crowave and millimeter-wave systems,” IEEE Trans. Microwave Theory Tech.,

Vol. 47, pp. 830–839, June 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 application

to spatial power combining systems,” IEEE AP-S International Symp. and

USNC/URSI National Radio Science Meeting Dig., pp. 286–289, July 1999.

• J. W. Mink, H.-S. Hwang, C. W. Hicks, T. W. Nuteson, M. B. Steer, and

J. Harvey, “Spatial power combining for two dimensional structures,” 1997

Topical Symposium on Millimeter Waves, pp. 133–136, July 1997.

Chapter 2

Literature Review

2.1 Introduction

In this chapter, a literature review of quasi-optical and waveguide-based spatial

power combining systems is presented. This literature review consists of two main

sections. The first section reviews the history of quasi-optical power combining

systems and a discussion of experimental free space and waveguide-based spatial

power combining systems is presented. The second section reviews the numerical

modeling techniques that are utilized to model quasi-optical and waveguide-based

spatial power combining systems.

12

CHAPTER 2. LITERATURE REVIEW 13

2.2 Background

Spatial power combining has emerged as a promising technique for combining power

in free space at millimeter and sub-millimeter wave frequencies [3]- [5]. The output

power of individual solid-state devices in a planar array is combined to produce

moderate-to-high power levels. It is desirable to utilize a single solid-state amplifier.

However, as frequency increases, the output power levels become low due to the

1/f 2 fall-off of available power [6]. By utilizing power combining techniques, light-

weight, reliable, and low cost amplifiers and oscillators can be potentially designed

to meet the demand of military and civilian applications. The lack of available

power at millimeter wave frequencies has delayed the development of novel circuits

and systems. If optical techniques such as diffraction and optical elements such as

lenses are incorporated into the system, the term quasi-optical power combining is

preferred. Quasi-optical power combining indicates that the power from an array

of solid-state devices are combined utilizing wavebeam principles. Optical lenses

are utilized to provide periodic refocusing of the beam and to combine power in a

single paraxial mode. The large transverse and longitudinal dimensions of quasi-

optical structures provide significant area for the active MMIC devices and control

components to be included within the structure.

In 1986, Mink published a classic paper that documents quasi-optical

power combiners as a potential source at millimeter-wave frequencies [2]. In the

early 1990’s, the United States (U.S.) Department of Defense Army Research Of-

fice (ARO) and Defense Advanced Research Projects Agency (DARPA) supported

quasi-optical power combining in the U.S. through single investigator projects,

Multi-University Research Initiatives (MURIs) and Microwave and Analog Front

End Technology (MAFET) programs, respectively. The DOD, industry, and uni-

versities teamed together to research, develop, and demonstrate various spatial and


quasi-optical architectures. The military demands high-performance MMICs for

ship, ground, and airborne radars, missile seekers, and satellite communications

links at microwave and millimeter wave frequencies. Other popular schemes uti-

lized for power combining are chip-level and circuit-level power combining [7]. More

specifically, microstrip (or similar transmission line structures) are used to realize

combining either on a chip or in a package. These methods are successful at low

and RF frequencies. However, at millimeter waves these choices become undesirable

because dielectric, radiation, and conductor losses increase rapidly and degrade sys-

tem performance. At the present, chip and circuit level power combining schemes

are not capable of achieving reliable and efficient moderate-to-high power levels at

millimeter wave frequencies. In this literature review, the most successful classes

of systems, 3-D, 2-D quasi-optical and waveguide power combining systems that

appear to be the most promising are reviewed [8].

2.2.1 Free-Space Quasi-Optical Amplifiers

The earliest type of free-space quasi-optical power combiner is the hybrid grid am-

plifier [9]- [12] as shown in Fig. 2.1. The grid is composed of vertical and horizontal

metallic strips in a cross configuration located on a dielectric substrate. Two or

three terminal devices are placed at the intersection of the strips to amplify the

input signal. Input and output polarizers placed orthogonally provide isolation be-

tween the input and output signals. The first experimental grid amplifier utilized

50 metal semiconductor field effect transistors (MESFETs) to design 25 differential

pair amplifiers arranged in a 5 × 5 rectangular array. This system demonstrated

a peak gain of 11 dB at 3.3 GHz [13]. The grid was enhanced by building a 100-

element heterojunction-bipolar-transistor (HBT) differential pair amplifiers fabri-

cated by Rockwell International. The 100-element grid amplifier generated a peak


gain of 10 dB at 10 GHz. The most recent and successful grid amplifier was de-

signed by the California Institute of Technology and Rockwell Science Center. A

1-cm square 16 × 16 monolithic Gallium Arsenide (GaAs) pHEMT grid amplifier

on a single chip was realized and produced 5 watts at 38 GHz [14].

ACTIVE GRID SURFACEOUTPUT POLARIZER

INPUT POLARIZER TUNING SLAB

E

E

INPUTBEAM

OUTPUTBEAM

Figure 2.1: Free space quasi-optical grid amplifier.

2.2.2 Quasi-Optical 2-D Dielectric Power Combining

Two-dimensional (2-D) quasi-optical technology offers an alternative approach that

utilizes a dielectric substrate to combine power from a planar resonator or amplifier

array (Fig. 2.2). The novel planar waveguide structure was first proposed by Mink

and Schwering [15]. The structure was designed to propagate Gauss-Hermite beam

modes so that the structure was termed a Hybrid Dielelectric Slab Beam Waveg-

uide (HDSBW). The advantages of two-dimensional power combining are that it is

more amenable to photolithographic fabrication techniques, more compatible with

MMIC technology, with reduced size and weight, and with improved heat-handling

capability.

North Carolina State University (NCSU) was the first organization to


demonstrate the 2-D QO power combining system. The first experimental system

design was an open cavity dielectric slab resonator based on the transverse electric

(TE) mode of propagation [16], [17]. The resonator incorporated a curved reflector

that was placed at the beam waist of the quasi-optical modes propagating in a

grounded dielectric slab. An oscillator array consisting of four MESFET oscillators

were injection locked to combine the power from the oscillating elements. For the

first time, a resonance frequency was detected at 7.4 GHz with a 3 dB linewidth

of less than 3 kHz [18]. Next, NCSU experimentally designed and demonstrated a

viable 2-D quasi-optical dielectric power combining amplifier system. The system

consisted of concave and convex lenses, MESFET or MMIC devices, and Vivaldi-

type antennas operating in an open waveguide configuration as shown in Fig. 1.3.

The amplifier array was placed underneath the slab with the array placed between

the transmitting and receiving optical lenses. The system was designed to amplify

the quasi-optical TE propagating modes in the dielectric slab. An experimental

4× 1 amplifier array generated 11 dB and 4.5 dB of amplifier gain and system gain,

respectively, at 7.1 GHz, and the single MMIC Vivaldi-type antenna produced 24

dB of amplifier gain at 8.4 GHz [19]. Incorporating a 5 × 4 MMIC 2-D quasi-optical

amplifier array, the system produced 30 dB of amplifier gain, 14 dB of system gain,

and 14.7 dBm of output power at 8.828 GHz [20].

A second 2-D TM-type (transverse magnetic) dielectric quasi-optical

power combiner was proposed and demonstrated by University of California at Los

Angeles (UCLA) [21]. Operating at X-band, the combiner utilized a microstrip-fed

Yagi-Uda slot array antenna to provide a high-efficiency unidirectional excitation

of the dominant TM surface wave propagating inside the grounded dielectric slab.

The TM-type combiner generated 11 dB of amplifier gain at 8.25 GHz with a 3-dB

bandwidth [22].


convex lenses

dielectric slab

ground plane

dielectric slab

ground planeconcave lenses

FET ampilifiers

guided input waves guided output waves

guided output wavesguided input waves

Figure 2.2: Concave and convex lens dielectric slab power combining systems.

2.2.3 Waveguide Spatial Power Combining

The University of California Santa Barbara successfully designed and demonstrated

an X-band waveguide spatial power combining system as shown in Fig. 2.3. The sys-

tem consists of trays of off-the-shelf 6-W GaAs MMICs integrated with tapered slot

antenna arrays. The output power of the system is directly related to the number of

MMICs incorporated on each tray. An experimental waveguide power combiner was

designed with six trays (or cards) and integrated with four input/output tapered-

slot and four MMIC amplifiers. The system produced a maximum output power of

61-W continuous wave (CW) at 8 GHz, and less than ± 1.4 dB power variation,

and between 13% and 31% of power added efficiency from 8 GHz to 12 GHz [23]. A

higher output power level was achieved by adding six trays populated with twenty-

four 6-W MMIC amplifiers [24]. The system produced up to 120 Watts of output

power with ± 1.9 dB power variation from 8 GHz to 11 GHz. A maximum power

of 126 W (CW), gain of 13 dB, and power added efficiency (PAE) over 33% was


obtained at 8.1 GHz. The waveguide spatial power combiner is successful because

it is designed to propagate only the dominant TE10 mode. Eliminating the problem

of higher order modes is regarded as a major reason for the success of this system.

Higher power levels could be achieved with this system by choosing different MMIC

amplifier technologies, such as Silicon Carbide (SiC) or Gallium Nitride (GaN) based

transistors and MMICs as the amplifying devices.

WR42 waveguide opening

Waveguide Opening designed to accommodate the six cards antenna array

Incident Wave

Horn Antenna

Figure 2.3: X-Band waveguide spatial power combining system.

Lockheed Martin-Sanders demonstrated a V-Band spatial power com-

bining system [25]. Based on a tray concept, the system utilized 17 solid-state tray

assemblies with each tray populated with 16 MMICs and radiating dipoles. The 16

× 17 solid-state array consisted of a total of 272 0.1µm Pseudomorphic High Elec-

tron Mobility Transistor (PHEMT) MMICs to produce 36 Watts at 61 GHz with

less than 1 deg/dB of AM/PM distortion. Two waveguide horns and lenses were

designed to transmit and collect the output power from the system. This solid-state

spatial power combiner has produced the highest output power over the V-Band

frequency range of any solid-state source.


2.3 Numerical Modeling of Spatial Power Com-

biners

2.3.1 Background

The fundamental understanding of quasi-optical power combining systems has pri-

marily been investigated experimentally. Several experimental free space, dielectric

quasi-optical power combiners, and waveguide spatial power combiners have been

successful at demonstrating the fundamental concepts of generating usable output

power levels using spatial and quasi-optical techniques [27] - [35]. Although great

strides have been made, to date, quasi-optical/spatial power combining systems have

not yet out-performed conventional power combiners. In order to capture the full

potential of quasi-optical/spatial systems to generate high power levels, numerical

modeling and computer aid engineering tools are needed to fully understand these

systems and to provide the basic tools for design [26]. The development of com-

puter models helps to reduce the cost and time associated with experimental work,

and assist with designing efficient quasi-optical/spatial power combining systems.

Modeling a quasi-optical/spatial power combining system is complex and challeng-

ing [36], [37]. There are several major system components that must be modeled

such as the input and output sources, which are typically waveguide horns with op-

tical lenses inside, the input and output antennas with associated transmission lines

and control components, and the active integrated amplifier circuitry. In addition,

the propagation of the Gaussian wave-beam in free space [38] or in a dielectric slab

must be accurately modeled.

The rapid pace of innovation of microwave and millimeter-wave systems

and the need for shortened research and development times has generated a require-


ment for intuitive schemes for developing customized electromagnetic (EM) analysis

and schemes for reusing electromagnetic models. The alternative is to use a general

purpose package using volumetric girding so that arbitrarily complex structures can

be modeled. These packages include those using the finite element method (FEM)

in the frequency domain and the finite-difference time domain (FDTD) method

in the time domain, to obtain the field through a structure at the nodes of the

grid. However, these volumetric packages are unable to model the electrically large

EM structures as encountered with quasi-optical and waveguide-based spatial power

combiners.

2.3.2 Modeling Free-Space Power Combining Systems

The grid structure has been modeled using the unit-cell approach and by utilizing

full-wave electromagnetic modeling techniques. The unit-cell approach models a

single element of the array subject to special boundary conditions along the unit-

cell edges [39]. The unit-cell approach assumes that all of the elements of an infinite

array are identical. Utilizing this concept, a full-wave electromagnetic model based

on the method of moments was utilized to relate the electric fields to the surface

current density on the electric strips using Galerkin’s method in the spectral domain

[40]. Using this simulator, the driving point impedance of several unit-cells such as

dipoles, cross dipoles, and bow-ties were determined. In this analysis, the grid array

was composed of a dielectric with or without metal on both sides of the dielectric

surfaces.

The unit-cell approach does not take into account the edge effects of

the grid and it assumes the driving point impedance is the same for all the array

elements. To overcome the limitations of the unit-cell approach, North Carolina

State University was the first to use a full-wave electromagnetic modeling approach


to model the entire quasi-optical grid amplifier [41]- [44]. A special dyadic Green’s

function was derived to model input and output horns, lens, polarizers, dielectric

layers, and the quasi-optical amplifier grid [45]. The electric and magnetic fields

derived from Gauss-Hermite wave beams were utilized to develop the dyadic Green’s

function that is composed of two parts; one part that computes the paraxial fields

(quasi-optical modes) and another part that computes the non-paraxial fields. The

Green’s function was used to develop a mixed spectral and spatial domain method

of moment technique. The MoM simulator successfully simulated a 5 × 5 and a

10 × 10 grid array with horns, lenses and polarizers to determine the near and far

fields [46].

2.3.3 Modeling Waveguide-Based Spatial Power Combining

Systems

Waveguide-based structures are playing an increasingly important role in spatial

power combining systems. Numerical techniques and classical electromagnetic tech-

niques can be used to model waveguide-based power combining systems. The electro-

magnetic boundary value problems for adjacent waveguides separated by a coupling

aperture, have been studied for years. Stevenson was one of the first pioneers to

successfully develop Green’s functions and integral equations for slotted waveguides

problems [47]. Stevenson developed analytical expressions to solve for the narrow

and thin longitudinal (shunt) and transverse (series) resonant slots in an infinite

conducting ground plane. Oliner expanded his work to include the rotated, shifted

and the finite thickness of the resonant slots and developed fairly accurate models

based on the variational technique [48].

Khac and Carson were the first to solve an electromagnetic boundary


value problem for a rectangular waveguide coupling into another waveguide or free

space using numerical techniques [49]. They used the method of moments by expand-

ing the basis functions with pulse functions and testing with Dirac-delta functions.

Later, Lyon and Sangester expanded the method of moment analysis by including

the finite thickness of the waveguide walls which results in a thick slot [50]. Si-

nusodial functions were used as the basis and testing functions because they more

accurately represent the electric field in the slot.

In the 1980, Hughes Research Laboratories began investigating a trans-

verse slot array fed by a boxed stripline, which is a slotted waveguide with an

enclosed conducting strip, for military and commercial applications because it is

low cost, lightweight, and compact [51] - [53]. Since the slot element pattern is a

semicircular in the E-plane, an end fire array can be designed for radar and com-

munication systems. In 1980, Park and Elliot investigated a boxed stripline array.

The boxed stripline was designed so that the all waveguide modes were suppressed

except for the dominant TE10 mode. Their design was based on the assumption that

the electric field distribution in the slot was half-cosinusoidal. Their experiments

showed that the array pattern was well formed but the input match was for fully

matched. This was due to the slot electric field distribution assumption. In 1983,

Shavit and Elliott continued the study by adjusting the waveguide width to allow

the TE10 mode to propagate [54]. The adjustment was done to avoid excessive res-

onate slot lengths encountered by Strummwasser et al. To control the slot length,

pin curtains were place near the slots to form a cavity for the TE10 mode and to

eliminate internal higher order mutual coupling. The pin curtains were also designed

to allow the strip transverse electromagnetic (TEM) mode to pass through. Using

the method of moment technique this design showed good agreement with theory. In

1994 Sangster and Smith [55], [56] introduced a method for modeling the geometry

of the boxed stripline by utilizing Green’s functions for a rectangular waveguide, a


rectangular waveguide with H-walls for the narrow walls, and a rectangular cavity

in order to calculate the higher-order modes within the structure.

2.3.4 Waveguide Dyadic Green’s Function

More rapid EM modeling is achieved when surface discretization (discretize the

surface into cells) is employed, as in the MoM, but this requires customized EM

modeling generally the development of a Green’s function describing the relation-

ship between surface currents and fields in an integral form. An integral equation

formulation is commonly used to determine the currents induced on the surface of

electric (strip, patch) and magnetic (slot, aperture) objects enclosed in a shielded

environment [57] - [60]. Discretization of the currents using basis function defined on

geometric cells enables the integral equation to be converted to a matrix equation.

Electric and magnetic Green’s functions provide the necessary relationship between

scattered fields and induced currents serving as kernels of the integral equations.

Dyadic Green’s function for rectangular waveguides and cavities have been studied

by many authors [61] - [66]. A traditional and general way to construct Green’s

functions for a closed-boundary guided-wave structure, semi-infinite waveguide and

cavity is to use the Hansen vector wave function M, N, and L (or only transverse

functions M and N) in a double series expansion. Thus, electric and magnetic dyadic

Green’s functions for uniform infinite and semi-infinite rectangular waveguides were

obtained for a rectangular cavity [67], [68].


2.4 Conclusion

Several 3-D quasi-optical, spatial, and waveguide-based power combining systems

have been experimentally designed and they have produced low-to-moderate power

levels. However, 2-D may have a significant advantage at microwave and millimeter-

wave frequencies due to the low cost of fabrication, and the ability of these struc-

tures to dissipate the heat from the amplifier array. As previously discussed, 2-D

spatial power combiners have been experimentally investigated and these systems

have demonstrated the ability to produce significant power level. However, there

is still a need to gain a more fundamental understanding of the physical aspects of

2-D systems. The next chapters of this dissertation will focus on utilizing analytical

and numerical electromagnetic techniques to investigate 2-D systems.

Chapter 3

Two-Dimensional Parallel-Plate

Resonator

3.1 Introduction

Two-dimensional (2-D) dielectric power combining technology offers significant ad-

vantages. The 2-D hybrid dielectric-slab beam waveguide (HDSBW) is amenable

to photolithographic definition and fabrication, and is compatible with MMIC tech-

nology [15]. The novel 2-D HDSBW has reduced size, weight, and improved heat

removal capability which results in lower costs. Two-dimensional quasi-optical sys-

tems previously fabricated are open planar structures which consist of a ground

plane and a dielectric slab (with geometry shown in Fig. 3.1(a)) [17], [20]. Open

systems have demonstrated the ability to combine power from an amplifier array. In

this section, we document the lessons learned with the open structure, particularly

25

CHAPTER 3. TWO-DIMENSIONAL PARALLEL-PLATE RESONATOR 26

through understanding of radiation mechanisms associated with excessive scattering

losses. The closed 2-D slab beam waveguide configuration shown in Fig. 3.1(b) is

an effort to minimize radiative losses. Beam-mode theory and experimental charac-

terization of this configuration are discussed here.

(b)

(a)

> slabεlensε

zw

sx

yd

phase transformers lensεdielectric slab slab

bottom ground plane

top ground plane

bottom ground plane

ε

Figure 3.1: Passive 2-D quasi-optical power combining system with concave lenses;

(a) open structure and (b) closed structure.

3.2 Principles of Operation

Both open and closed HDSBW systems utilize two distinct waveguiding principles

to guide the electromagnetic wave [15]. For the open system shown in Fig 3.2, the

field distribution in the x-direction is that of a surface-wave mode of the grounded

dielectric slab. The surface wave is guided by the reflection at the air-to-dielectric

interface and system parameters are adjusted such that the energy is transmitted

primarily within the dielectric. In the closed system, the field distribution in the


x-direction is that of a parallel-plate waveguide dominant mode. In both systems

the field distribution in the y-direction corresponds to a wave beam-mode (Gauss-

Hermite), which is guided by the lenses through periodic reconstitution of the cross-

sectional phase distribution. The guided modes are either TE or TM-polarized

modes with respect to the direction of propagation. The 2-D HDSBW principle is

or

E E"E" E""

Convex/Concave Phase Transformer

PortInput

d1 d2

OutputPort

Pout

Amplifier Array

d3

Pin

Figure 3.2: The 2-D HDSBW system with convex/concave lenses and 4 × 1 MES-

FET amplifier array.

used to obtain signal amplification similar to that of a traveling-wave amplifier. An

array of active elements located underneath the dielectric slab is placed in the path

of the wavebeam. Each active element consists of a pair of back-to-back Vivaldi

or slot antennas with an amplifier or MMIC inserted between the two antennas

as shown in Fig. 3.3. Part of the incident signal passes through the dielectric slab

undisturbed and the remaining signal is amplified by the array. The input Vivaldi or

slot antenna couples energy from the incident traveling wavebeam, and the output

antenna reinserts the amplified signal back into the traveling wavebeam. Maximum

coupling to the array occurs when the energy from the first lens focuses energy to the

input of the antennas. The signal is amplified by the MESFETs and is coupled by

the output antennas to the traveling wavebeam where it combines in phase with the


through signal as shown in Fig. 3.4. Consequently, a growing traveling wave-beam

mode is established within the guiding structure resulting in an increased output

power.

MESFET

MMIC

εr

rε

(a)

(b)

Figure 3.3: Input and output coupling antennas on RT/Duriod substrate: (a) MES-

FET Vivaldi amplifier; and (b) MMIC stripline-slot amplifier.

3.3 Beam-Mode Theory In a Closed-Boundary Slab

Beam Waveguide

The closed 2-D power combining systems shown in Fig. 3.1(b) consist of two parallel

conducting planes separated by a dielectric slab of thickness d and with relative

dielectric permittivity εr. The fields in the waveguide are found ¿from Helmholtz

equations and the proper boundary conditions. This leads to solutions for the scalar


Ein,3

Eout,1

Eout,2

Eout,3

Eout,4

Ein Eout

Ein,4

φt

Ein,2

Eth

Ein,1

Ground Plane

Ein

THROUGH WAVES

AMPLIFIED WAVES Eout = Ea aφφ t +

+

φΣn=1

4Ea,n a,n=Ea aφ

Eth

Figure 3.4: Electric-field wave model for 2-D power combining system.

axial fields in the waveguide. Once the axial components are found, the transverse

fields are derived from Maxwell’s equations. By applying orthogonality conditions,

the fields are normalized and the normalized power in the parallel-plate waveguide

is computed. The general solution of the transverse and longitudinal fields for the

guiding structure is obtained from Helmholtz equations in a source field region [62]:

∇2H(r) + k2H(r) = 0, (3.1)

∇2E(r) + k2E(r) = 0

where k = ω/c√

εr with c being the velocity of light in free space. The waves

propagating in the waveguide are classified as Transverse Electric (TE) and Trans-

verse Magnetic (TM) waves with the corresponding transverse and longitudinal field

components. The eigenmodes of electric and magnetic fields can be expressed in the

following form,

H±mn = (±hmn + hzmnaz)e

∓jβmnz, (3.2)


E±mn = (emn ± ezmnaz)e

∓jβmnz

where emn and hmn are transverse vector functions, while ezmn and hzmn are longi-

tudinal scalar functions. The time dependence in the form of ejωt is assumed and

suppressed. The term βmn is the phase constant and m and n are the mode indices

for the x and y mode variations, respectively. The TE-mode solution is obtained

when hzmn = 0 while the TM-mode solution is obtained when ezmn = 0. All the

transverse fields can be expressed in terms of the longitudinal components. Substi-

tuting (3.2) into (3.1) results in the representation of transverse vector functions of

TE modes,

hmn = −jβmn

k2c

∇thzmn, (3.3)

emn = Zhaz × hmn,

and TM modes,

emn = −jβmn

k2c

∇tezmn, (3.4)

hmn = Yeaz × emn

where Ye = jk0Y0/βmn, is the scalar wave admittance of TM modes and Zh =

jk0Z0/βmn is the scalar wave impedance of TE modes; Z0 = (1/Y0) and Y0 are the

intrinsic impedance and admittance of free space. The boundary conditions for the

parallel-plate quasi-optical structure (Fig. 3.1(b)) are

TM−modes : ezmn = 0 at x = 0 and x = d, (3.5)

TE−modes : ∂hzmn/∂y = 0 at y = −∞ and y = +∞. (3.6)

Assuming the axial components for the parallel-plate guiding structure take the

following form, where a functional dependence on (x)-variable can be separated

from (y, z)-dependence,

TM−modes : ezmn = AmnXm(x)Qn(y, z), (3.7)

TE−modes : hzmn = BmnXm(x)Qn(y, z) (3.8)


where Amn and Bmn are the electric and magnetic field normalization factors, re-

spectively, yet to be determined. Utilizing the boundary conditions (3.5), (3.6), a

function Xm(x) is expressed in terms of eigenfunctions of one-dimensional Laplacian

operator having forms of sin(mπx/d) and cos(mπx/d), while the function Qn(z, y)

describes the slow variation in the y-direction and it is defined as [2],

Qn(y, z) =1√

Y n!√

π(1 + ν2

mn)14 ·Hen

(√2y

yzmn

)

· exp

−

(y

yzmn

)2

± exp j

νmn

(y

yzmn

)2

−(n +

1

2

)tan −1(νmn)

(3.9)

where

νmn =z

βmnY

2, yzmn = Y

2(1 + ν2

mn),

and

Y2

=

√(2−D/F )FD

βmn

12

.

In (3.9) D is the distance between the reflecting surfaces and F is the

focal length of the lenses. The function Qn is composed of Hermite polynomials

which form a complete set of orthonormal eigenfunctions of the Fourier transform

operator.

By substituting the axial equations into the scalar wave equations, it is

found that the phase constant βmn is defined as

βmn =√

k2x + k2

y − k2 (3.10)

where k2x = mπ

aand

k2y =

(Q′′

mn(y, z)

Qmn(y, z)

)(βmn + 1) +

(Q′

mn(y, z)

Qmn(y, z)

)βmn. (3.11)


In (3.11),

Q′′mn(y, z) =

∂2Qmn(y, z)

∂y2, Q′

mn(y, z) =∂Qmn(y, z)

∂y.

By substituting the axial fields (3.7) into (3.4), the electric-field compo-

nents of the TM modes are obtained as follows,

ezmn = Amn sin(

mπx

d

)Qn(y, z), (3.12)

exmn = −Amn

(jβ

kc

) (mπ

d

)cos

(mπx

d

)Qn(y, z),

eymn = −Amn

(jβ

kc

)sin

(mπx

d

)∂Qn(y, z)

∂y,

and the magnetic-field transverse components are

hxmn = Yeeymn, (3.13)

hymn = −Yeexmn.

The magnetic-field components of the TE modes are similarly obtained in the fol-

lowing form,

hzmn = Bmn cos(

mπx

d

)Qn(y, z), (3.14)

hxmn = −Bmn

(jβ

kc

) (mπ

d

)sin

(mπx

d

)Qn(y, z),

hymn = −Bmn

(jβ

kc

)cos

(mπx

d

)∂Qn(y, z)

∂y,

and the electric-field transverse components are

exmn = Zhhymn, (3.15)

eymn = −Zhhxmn

where (3.12) through (3.15) are utilized to represent a total electric and magnetic

field in the parallel-plate HDSBW.


3.3.1 Orthogonality of Fields

After the fields for the parallel-plate quasi-optical structure have been determined,

the fields can be normalized by satisfying the orthogonality relationship. The field

orthogonality of TM and TE modes is defined by

∫ d

0

∫ ∞

−∞emn · e∗m′n′dxdy = δmm′δnn′ , (3.16)

∫ d

0

∫ ∞

−∞hmn · h∗m′n′dxdy = δmm′δnn′ .

The normalization factor needed to satisfy the TE and TM orthogonality relations

(3.16) is obtained by evaluating the following integrals

∫ d

0cos

(mπx

d

)cos

(m′πx

d

)dx =

d2, m = m′ 6= 0

d, m = m′ = 0

0, m 6= m′

(3.17)

∫ d

0sin

(mπx

d

)sin

(m′πx

d

)dx =

d2, m = m′ 6= 0

0, m = m′ = 0

0, m 6= m′

and solving ∫ ∞

−∞

(∂Qn(y, z)

∂y

) (∂Qn′(y, z)

∂y

)dy (3.18)

Now substituting the electric and magnetic fields (3.12) through (3.15) and orthog-

onal relations (3.16) into (3.14) the result becomes

∫ d

0

∫ ∞

−∞emn · e∗m′n′dxdy =

∫ d

0

∫ ∞

−∞hmn · h∗m′n′dxdy (3.19)

= Dmnδmm′δnn′

where Dmn, (3.19), is a constant used to determine normalize the electric and mag-

netic fields. By using this relation, the electric and magnetic fields will be normalized


and the TM and TE orthogonality relations will be satisfied. From here it will be

assumed that all the transverse fields are normalized. The electric and magnetic

fields of the mn th mode propagating in the +z-direction are represented as [62]

E+mn = (emn + ezmn) e−jβmnz, (3.20)

H+mn =

(hmn + hzmn

)e−jβmnz, (3.21)

and those for the mnth mode propagating in the −z-direction are

E−mn = (emn − ezmn) ejβmnz, (3.22)

H−mn =

(−hmn + hzmn

)ejβmnz, (3.23)

where E+mn, H+

mn and E−mn, H−

mn represent electric and magnetic fields of forward

and backward traveling waves, respectively. The total electric and magnetic fields

propagating in the +z-direction are then expressed in the form of TE and TM

eigenmode expansion,

E+ =∑m

∑n

amnE+mn, (3.24)

H+ =∑m

∑n

amnH+mn, (3.25)

and in the −z-direction the total fields are

E− =∑m

∑n

bmnE−mn, (3.26)

H− =∑m

∑n

bmnH−mn, (3.27)

where amn and bmn are expansion (amplitude) coefficients which can be determined

from the Lorentz reciprocity theorem solving the excitation problem.

3.3.2 Power Normalization

Power orthogonality between the modal fields means that each mode is independent

of the other modes and it carries its own power and there is no power coupled


between modes. The general expression for the power carried inside a parallel-plate

quasi-optical structure in the +z-direction is defined by

Pmn =1

2Re

∫ d

0

∫ ∞

−∞E+ × H∗+ · azdxdy. (3.28)

Since the fields are orthogonal, the normalized power in the QO structure can be

found from the TM and TE orthogonal relationships. The normalized power prop-

agating in the +z-direction is found to be

pmn = amna∗m′n′δmm′δnn′ (3.29)

where pmn = 2PmnYe for the TE modes and pmn = 2Pmn/Zh for the TM modes.

3.3.3 Verification

The theory of the closed system was verified by testing a confocal parallel-plate

cavity system. The geometry for the parallel-plate resonator is shown in Fig. 3.5.

The width and length are denoted by a = 30.48 cm and b = 30 cm, and the radius

of curvature is denoted by r = 60.96 cm. The upper and lower ground planes are

separated by a dielectric (Rexolite material with the dielectric permittivity εr =

2.57) of thickness t = 1.27 cm. A L-shaped coaxial probe normal to the ground

plane was used to excite the cavity. The resonance frequencies for the parallel-plate

cavity are calculated from

βmz −(n +

1

2

)tan −1νmn = qπ. (3.30)

Solving the above equation for the resonance frequencies of the parallel-plate cavity

structure gives the following result,

fmn =c

2π√

εr

·

qπ + (n + 12) tan

[z/

√(2−D/F )FD

]z

2

+(

mπ

d

)2

12

(3.31)


where m, n, and q are the mode indices for the x, y, and z directions respectively.

Reflector

Curved L-shaped

Antenna

tε

z

x

Planer

Reflector

Ground

Planes

r

b

y

z

a

Figure 3.5: Test configuration for the confocal parallel-plate resonator system.

Measurements were taken using a Hewlett Packard 8510C Vector Network Analyzer

to measure S11 of the resonator. A L-shaped coaxial probe normal to the ground

plane was utilized to excite predominantly the TM modes inside the cavity. The

TMm,n,q modes were selected because m=0 signifies the dominant mode inside a

parallel-plate cavity. Fig 3.6 shows a dispersion behavior of the reflection coefficient

S11. A 1/2 inch dielectric slab has a cutoff frequency of 7.367 GHz for the TM1,0,0

mode which was predicted and measured. Above the cutoff frequency, high order

modes become propagating. The plot shows that the signal increases as frequency

increases and the peak at 6.869 GHz is identified as the TM0,0,22 resonator mode.

Similarly, other higher-order modes have also been predicted. The theory was also

used to predict the frequency spacing s=307 MHz between two adjacent TM modes


as shown in Fig. 3.6.

.

|S11

|

Frequency (GHz)5.0 5.5 6.0 6.5 7.0 7.5 8.0

0

0.2

0.4

0.6

0.8

1

s

Figure 3.6: Reflection coefficient at the input to a confocal 2-D parallel-plate res-

onator system.

3.3.4 Mode Profile

The profile of the electric field distribution was measured by inserting a small vertical

coaxial probe in the top ground plane 15 cm from the planar reflector in the z-

direction. Measurements were conducted at 6.898 GHz (resonance frequency of the

TM0,1,22 mode) by sliding the probe in the y-direction in 5 mm increments. The

reflection coefficient at the input to a confocal 2-D parallel-plate resonator results

in notches in the reflection coefficient as the field in the resonator peaks resulting in

higher loss. The electric field distribution at 6.898 GHz is displayed in Fig. 3.7. Two

sets of measurements were conducted. Fig. 3.8 shows the measured data and the

comparison of the measured data and the numerical results is shown in Table 3.1,


indicating that the theory and measurements are in excellent agreement. The

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140Distance (mm)

TheoryRun #1Run #2

|S11

| (R

efle

ctio

n C

oeff

icie

nt)

Figure 3.7: Electric-field mode profile at 6.898 GHz for the parallel-plate confocal

system.

major errors in the field profile are due to the finite length of the vertical probe, the

conduction and reflection losses on the upper and lower ground planes, dielectric

losses, and leakage from the side walls of the resonator. In order to minimize the

reflections from the sides of the dielectric slab, it was tapered at the edges. The

largest error associated with profile comparisons is at a significant distance from the

center axis but of great accuracy in the paraxial region. The significant departure

between the field profile calculated using the Hermite-Gaussian approximation and

the measured profile is in part due to edge effects but also because Hermite-Gaussian

distribution is approximate. The actual profile could be more accurately determined

using full EM solutions. The measurements, however, indicate that the Hermite-

Gaussian approximation is adequate to describe the modes in the HDSBW. This

validation is essential to the developments in this dissertation.


0

0.2

0.4

0.6

0.8

1

1.2

4.00

0

4.18

8

4.37

5

4.56

3

4.75

0

4.93

8

5.12

5

5.31

3

5.50

0

5.68

8

5.87

5

6.06

3

6.25

0

6.43

8

6.62

5

6.81

3

7.00

0

Frequency

S11

Mag

nitu

de (

dB)

0

0.005

0.01

0.015

0.02

0.025

S21

Mag

nitu

de (

dB)

(1) q=20, n=3, 6.408GHz(2) q=20, n=1, 6.254 GHz(3) q=21, n=1, 6.558GHz(4) q=21, n=3, 6.719GHz(5) q=22, n=0, 6.869GHz

(1)

(2)

(3) (4)

(5)

|S11|

|S21|

Figure 3.8: A plot of the magnitude of S11 and S21 for selected resonance frequencies

of the parallel-plate resonator system.


Table 3.1: Selected resonance frequencies of the parallel-plate resonator system.

Measured Calculated Error n q

Frequency Frequency (MHz)

(GHz) (GHz)

6.869 6.868 0 1 22

6.719 6.715 4.0 3 21

6.558 6.561 3.0 1 21

6.408 6.408 0 3 20

6.254 6.254 0 1 20

3.4 Experimental Results

3.4.1 Open System Configuration

The system configuration for the open TE HDSBW shown in Fig. 3.2 consists of a

rectangular dielectric slab made of Rexolite material with dielectric permittivity εr

= 2.57 and tan δ = 0.0006 placed on a conducting ground plane [20]. Two concave

cylindrical lenses made of Macor material with εr = 5.9 and tan δ = 0.0006 with

the focal length of 28.54 cm were inserted into the dielectric slab waveguide. The

dielectric slab had the following dimensions: length (d1 + d2 + d3), width (w),

and thickness (d) were 62 cm, 27.94 cm, and 1.27 cm, respectively. The Vivaldi-

based antenna MESFET amplifiers were located underneath the dielectric slab in

the ground plane. Each Vivaldi-based antenna was fabricated using RT/Duroid

6010 substrate material with εr = 10.2 and tan δ = 0.0028 with the dimensions of


6.5 cm × 1.5 cm. Two E-plane horns were designed and fabricated to efficiently

transmit and receive the required wavebeam. Two experiments were performed for

the open system. The first experiment utilized a 4 × 1 MESFET Vivaldi amplifier

array as shown in Fig. 3.2, and the second one employed a single MMIC Vivaldi

amplifier located under the dielectric slab (see Fig. 3.9). A measure of the relative

energy coupled to the amplifier array was obtained by switching the amplifier bias

levels off and on while measuring the output power, Pout. The system performance

for the active Vivaldi-based amplifier array was determined by the system gain

and amplifier gain. This provided an indication of the incident signal that passes

through the dielectric as an undisturbed traveling wave. Fig. 3.10 shows the total

Pin Pout

20 cm 20 cm

5.5 cm

Eamp Eth Eamp+Ein

CONCAVE LENS

Eth Ein=~

9.7cm

ME

TA

L W

ALL

Figure 3.9: The concave-lens system configuration for a unit-cell Vivaldi-based am-

plifier.

system performance of the TE MESFET amplifier array at 7.12 GHz using concave

lenses. A plot of Pin versus Pout shows two different amplifier conditions indicated

by AMP OFF and AMP ON, respectively. The input power, Pin, varied from -45

dBm to +10 dBm in +5 dBm increments. The power ratio between Pout and Pin

was relatively constant for the values of Pin less than -15 dBm, however Pout reached

the saturation condition with Pin greater than -15 dBm. The maximum system gain


of 4.5 dB occurred at Pin = -15 dBm while the measured amplifier gain on-to-off

was 11 dB.

-50

-40

-30

-20

-10

0

10

-40 -30 -20 -10 0 10

Po

ut

(dB

m)

Pin (dBm)

CONCAVE-LENS SYSTEM

AMP OFFAMP ON

Figure 3.10: The concave-lens system configuration for a unit-cell amplifier.

3.4.2 Vivaldi Unit Cell

The second experiment for the open TE system was performed with a cascaded

pair of MMIC amplifiers, in order to achieve higher power levels. In Fig. 3.9, the

amplifier gain of the Vivaldi-based amplifier was determined by placing a metal

screen transverse to the Vivaldi structure. The Vivaldi-based amplifier and the


metal wall were placed 5.5 cm and 9.7 cm, respectively, from the input horn. A

concave lens was placed in the middle of a 40 cm dielectric slab. The slit in the

metal wall allowed for only input power of the amplifier to go through the system so

that the amplifier gain could be measured. The amplifier gain was determined by

switching the bias voltage on and off, while measuring the power difference detected

by the receiving horn. The amplifier gain indicated that more than 20 dB of gain

was produced from 7 GHz to 10.5 GHz with a maximum gain of 24 dB at 8.4 GHz.

The gain from 9.5 GHz to 10.5 GHz is shown in Fig. 3.11.

5

10

15

20

25

9.5 9.75 10 10.25 10.5

Am

plifi

er G

ain

(dB

)

Frequency (GHz)

Vivaldi amplifierSlot amplifier #1

Figure 3.11: Amplifier gain for a unit-cell MMIC amplifier; (a)Vivaldi cascade

MMICs and (b) single stripline-slot MMIC.

3.4.3 Unit Cell of a Slot Antenna

A TM unit cell of a slot antenna was also tested and compared to the TE Vivaldi

unit-cell antenna. In this experiment, the slot antenna had only one MMIC and the

lens was not utilized. The TM unit cell was placed in a parallel-plate configuration


in the bottom ground plane under a 1/8 inch Rexolite dielectric slab. In a similar

manner, a metal wall was placed 8 mm from the middle of the input and output

slot antenna where the MMIC was located. Two H-plane horns were designed to

transmit and receive power and to vertically polarize the electric field, in order to

achieve the maximum coupling to the slots. The slots were located λ/4 apart and

the slot width and length were λ/10 and λ/2, respectively.

Fig. 3.11 compares the unit cell Vivaldi-based and slot amplifier gain.

The nominal gain of the MMIC at 10 GHz is 10 dB. Different gains were achieved

because the Vivaldi was used in a cascade configuration while the slot utilized only

one MMIC. Another difference is that the Vivaldi-based amplifier was tested over a

wider frequency range than the slot. The Vivaldi-based amplifier reached 20 dB of

gain whereas the gain of the slot antenna with one MMIC reached 10 dB.

3.5 Conclusions

The electromagnetic model of a QO parallel-plate HDSBW resonator system based

on a Gauss-Hermite beammode expansion is developed and verified experimentally

to predict the resonance frequencies of the structure and beammodes dispersion

behavior. The system was designed, fabricated, and tested, showing a favorable

agreement between the experimental data with the theoretical results. In addi-

tion, a QO parallel-plate stripline-slot amplifier system was designed, tested, and

compared to a QO open HDSBW amplifier system with the previously used Vivaldi-

type antennas. Experimental results verify that a QO parallel-plate stripline-slot

amplifier proposed in the dissertation improves overall system performance.

Chapter 4

Waveguide-Based Slot-Strip-Slot

Transitions

4.1 Introduction

In this chapter, an electromagnetic modeling environment is developed for an aperture-

coupled stripline-to-waveguide transition. This transition is the fundamental build-

ing block for two-dimensional spatial power combining amplifier arrays and in turn

for planar quasi-optical power combining systems. A full-wave electromagnetic

model is developed for a structure that couples a waveguide to a stripline through a

set of slots and from the stripline through another set of slots into a second waveg-

uide. The system modeling is based on an integral equation formulation for the

induced electric and magnetic surface current densities resulting in a coupled set of

integral equations discretized utilizing the method of moments (MoM). The scat-

45

CHAPTER 4. WAVEGUIDE-BASED SLOT-STRIP-SLOT TRANSITIONS 46

tered electric and magnetic fields are expressed in terms of dyadic Green’s functions

and the electric and magnetic surface currents. The surface currents are discretized

by overlapping piecewise sinusodial subdomain basis functions in order to accu-

rately model narrow longitudinal strips and transverse slots. In this formulation, a

MoM matrix includes all possible self and mutual coupling effects between the slots

and strips. The transition is excited with the TE10 dominant waveguide mode and

the scattering parameters are calculated from the forward and backward coupling

coefficients in the waveguide regions.

4.2 General Electromagnetic Formulation and Dyadic

Green’s Functions

A general electromagnetic formulation for a closed-boundary waveguiding structure

containing arbitrarily shaped apertures and conducting strips (see Fig. 4.1) is de-

veloped in this section. The formulation is based on the integral representation

of incident and scattered electric and magnetic fields in terms of dyadic Green’s

functions. Dyadic Green’s functions represent the electric and magnetic fields at

an observation point inside a volume due to an arbitrarily oriented point source.

Fig. 4.1 shows an arbitrary volume V enclosed by the surface S = S ∪ Sm, where S

represents an electric-type boundary surface and Sm represents the surface of aper-

tures (magnetic-type surface). The volume V encloses an impressed electric volume

current source Jimp ⊂ Vimp and an induced electric current source Jind on the surface

of conducting strips Se (electric-type surface). The total electric E(r ) and magnetic

H(r ) fields inside the closed region V are obtained as a superposition of the incident

electric Ei(r ), and, magnetic, Hi(r ), fields due to Jimp and the scattered electric,

Es(r ), and, magnetic, Hs(r ), fields due to Jind in the presence of the surface S.


n

ss HE ,

mSS

impV

impJ

indJeS

ii HE ,

V

Figure 4.1: Geometry of a closed-boundary waveguiding structure containing aper-

tures and conducting strips in the presence of an impressed electric current source.

The total fields due to J = Jimp + Jind inside the homogeneous volume V satisfy the

following vector wave equations [61]:

∇×∇× E(r )− k2E(r ) = −jωµJ(r ), (4.1)

∇×∇× H(r )− k2H(r ) = ∇× J(r ), (4.2)

where ω is the radial frequency, k = ω√

εµ is the wavenumber in the media of volume

V , ε = εrε0 and µ = µrµ0 are the primitivity and permeability of the media with εr

and µr are being the relative primitivity and permeability of the material media, ε0,

µ0 are the primitivity and permeability of free space. The time dependence in the

form of e+jωt is assumed and suppressed.

The corresponding dyadic wave equations are derived by letting=

Ge=

Ew,=

Gm= −jωµHw, and Jw = jωµ=

I δ(r − r ′) where w = x, y, z, and r =

(x, y, z) and r ′ = (x′, y′, z′) are the positions of the observation and the source

point, respectively. Substituting these representations into (4.1) and (4.2) results in

∇×∇× =

Ge (r, r ′)− k2=

Ge (r, r ′) ==

I δ(r − r ′), (4.3)

∇×∇× =

Gm (r, r ′)− k2=

Gm (r, r ′) = ∇× =

I δ(r − r ′), (4.4)


where=

Ge is the electric dyadic Green’s function,=

Gm is the magnetic dyadic Green’s

function, and=

I is the idem-factor (unity dyadic), and δ(r− r ′) is the delta function.

The second vector-dyadic Green’s theorem applied for the electric field

and the corresponding Green’s dyadic allows for the following integral representa-

tion,∫V

E(r ) ·

[∇×∇× =

Ge (r, r ′)]−

[∇×∇× E(r )

]· =

Ge (r, r ′)

dV

= −∮

Sn ·

E(r )×

[∇× =

Ge (r, r ′)]

+[∇× E(r )

]× =

Ge (r, r ′)

dS. (4.5)

where hatn is an outward normal to S. The volume integral on the left-hand side

of (4.5) can be simplified by taking into account (4.1) and (4.3), and the properties

of the delta-function, resulting in,

E(r ′) + jωµ∫

VJ(r )· =

Ge (r, r ′)dV

= −∮

Sn ·

E(r )×

[∇× =

Ge (r, r ′)]

+[−jωµH(r )

]× =

Ge (r, r ′)

dS, (4.6)

where in the surface integral, ∇× E(r ) is substituted by −jωµH(r ) using the curl

Maxwell’s equation (Faraday’s law). Finally, splitting the total electric current into

Jimp and Jind we obtain,

E (r ′) = − jωµ∫

Vimp

Jimp (r ) · =

Ge (r, r ′)dV

− jωµ∫

Se

Jind (r ) · =

Ge (r, r ′)dS

− jωµ∮

SH (r ) ·

[n× =

Ge (r, r ′)]dS

+∮

Sm

[n× E (r )

]·[∇× =

Ge (r, r ′)]dS. (4.7)

By a similar procedure, an integral representation for the total magnetic field can

be obtained as follows,∫V

H(r ) ·

[∇×∇× =

Ge (r, r ′)]−

[∇×∇× H(r )

]· =

Ge (r, r ′)

dV

= −∮

Sn ·

H(r )×

[∇× =

Ge (r, r ′)]

+[∇× H(r )

]× =

Ge (r, r ′)

dS.

(4.8)


The volume integral on the left hand-side of (4.8) is simplified by using (4.2) and

(4.3), and the property of the delta-function,

∫V

H(r ) ·

[∇×∇× =

Ge (r, r ′)]−

[∇×∇× H(r )

]· =

Ge (r, r ′)

dV

= H(r ′)−∫

V

[∇× J(r )

]· =

Ge (r, r ′)dV. (4.9)

Additional simplification of the volume integral in (4.9) results in the following,

H(r ′)−∫

V

[∇× J(r )

]· =

Ge (r, r ′)dV

= H(r ′)−∫

V

∇ ·

[J(r )× =

Ge (r, r ′)]

+ J(r ) ·[∇× =

Ge (r, r ′)]

dV. (4.10)

Utilizing the dyadic divergence theorem for the first term in the volume integral on

the right side of the (4.10) results in the following expression for the volume integral,

H(r ′)−∮

S

[n× J(r )

]· =

Ge (r, r ′)dS −∫

VJ(r ) ·

[∇× =

Ge (r, r ′)]dV. (4.11)

The surface integral in (4.8) can be simplified to the following form,

∮S

[n× J(r )

]· =

Ge (r, r ′)−[n×∇× H(r )

]· =

Ge (r, r ′)

dS. (4.12)

By factoring out=

Ge, the two terms in the surface integral (4.12) can be combined by

using the curl Maxwell’s equation (Ampere’s law), ∇×H− J = jωεE. Substituting

(4.11) and (4.12) into (4.10), a complete expression for the total magnetic field is

found to be

H(r ′) =∫

VJ(r ) ·

[∇× =

Ge (r, r ′)]dV

− jωε∮

S

[n× E(r )

]· =

Ge (r, r ′)dS

+∮

SH(r ) ·

[n×∇× =

Ge (r, r ′)]dS, (4.13)

and splitting the total current J into Jimp and Jind we obtain,

H(r ′) =∫

Vimp

Jimp(r ) ·[∇× =

Ge (r, r ′)]dV


+∫

Se

Jind(r ) ·[∇× =

Ge (r, r ′)]dS

− jωε∮

S

[n× E(r )

]· =

Ge (r, r ′)dS

+∮

SH(r ) ·

[n×∇× =

Ge (r, r ′)]dS. (4.14)

If the electric dyadic Green’s function in (4.7) satisfies the first kind

(Dirichlet-type) boundary condition on the closed surface S, then=

Ge will be de-

noted by=

Ge1, and if the electric Green’s dyadic in (4.14) satisfies the second kind

(Neumann-type) boundary condition on S, then=

Ge≡=

Ge2. The first and second kind

boundary conditions for Green’s dyadics are expressed in the following form,

n× =

Ge1 (r, r ′) = 0, r ∈ S, (4.15)

n×∇× =

Ge2 (r, r ′) = 0, r ∈ S. (4.16)

Note that both=

Ge1 and=

Ge2 are solutions to the dyadic wave equation (4.3). By

applying these boundary conditions to the electric field integral representation (4.7)

and the magnetic field integral representation (4.14), the surface integrals over S

and Sm in both (4.7) and (4.14) vanish. The total electric and magnetic fields are

then rewritten as

E(r ′) = −jωµ∫

Vimp

Jimp(r )· =

Ge1 (r, r ′)dV

− jωµ∫

Se

Jind(r )· =

Ge1 (r, r ′)dS

−∫

Sm

[n× E(r )

]·[∇× =

Ge1 (r, r ′)]dS, (4.17)

H(r ′) =∫

Vimp

Jimp(r ) ·[∇× =

Ge2 (r, r ′)]dV

+∫

Se

Jind(r ) ·[∇× =

Ge2 (r, r ′)]dS

− jωε∫

Sm

[n× E(r )

]· =

Ge2 (r, r ′)dS. (4.18)

By replacing r and r ′ and using the dot product identities in (4.17) and (4.18), we


obtain,

E(r ) = −jωµ∫

Vimp

[=

Ge1 (r ′, r )]T

· Jimp(r′)dV ′

− jωµ∫

Se

[=

Ge1 (r ′, r )]T

· Jind(r′)dS ′

−∫

Sm

[∇′× =

Ge1 (r ′, r )]T

·[n× E(r ′)

]dS ′, (4.19)

H(r ) =∫

Vimp

[∇′× =

Ge2 (r ′, r )]T

· Jimp(r′)dV ′

+∫

Se

[∇′× =

Ge2 (r ′, r )]T

· Jind(r′)dS ′

− jωε∫

Sm

[=

Ge2 (r ′, r )]T

·[n× E(r ′)

]dS ′, (4.20)

where T denotes the transposition operator. Making use of the following identities,[=

Ge1 (r ′, r )]T

==

Ge1 (r, r ′), (4.21)

[∇′× =

Ge1 (r ′, r )]T

= ∇× =

Ge2 (r, r ′), (4.22)

the total electric-field integral representation results in

E(r ) = −jωµ∫

Vimp

=

Ge1 (r, r ′) · Jimp(r′)dV ′

− jωµ∫

Se

=

Ge1 (r, r ′) · Jind(r′)dS ′

−∫

Sm

[∇× =

Ge2 (r, r ′)]·[n× E(r ′)

]dS ′. (4.23)

By utilizing the following two additional identities[∇′× =

Ge2 (r ′, r )]T

= ∇× =

Ge1 (r, r ′), (4.24)

[=

Ge2 (r ′, r )]T

==

Ge2 (r, r ′), (4.25)

the magnetic-field integral representation is obtained as

H(r ) =∫

Vimp

[∇× =

Ge1 (r, r ′)]· Jimp(r

′)dV ′

+∫

Se

[∇× =

Ge1 (r, r ′)]· Jind(r

′)dS ′

− jωε∫

Sm

=

Ge2 (r, r ′) ·[n× E(r ′)

]dS ′. (4.26)


Utilizing notations as discussed in [70], [71], the final expressions for the total electric

and magnetic fields in volume V are summarized as

E(r ) = −jωµ∫

Vimp

=

GEJ (r, r ′) · Jimp(r′)dV ′

− jωµ∫

Se

=

GEJ (r, r ′) · Jind(r′)dS ′

−∫

Sm

=

GEM (r, r ′) · M(r ′)dS ′ (4.27)

H(r ) =∫

Vimp

=

GHJ (r, r ′) · Jimp(r′)dV ′

+∫

Se

=

GHJ (r, r ′) · Jind(r′)dS ′

− jωε∫

Sm

=

GHM (r, r ′) · M(r ′)dS ′ (4.28)

where

=

GEJ (r, r ′) ==

Ge1 (r, r ′), (4.29)

=

GHM (r, r ′) ==

Ge2 (r, r ′), (4.30)

=

GHJ (r, r ′) = ∇× =

GEJ (r, r ′), (4.31)

=

GEM (r, r ′) = ∇× =

GHM (r, r ′). (4.32)

Here the electric-electric dyadic Green’s function,=

GEJ (r, r ′), relates the electric

field in volume V enclosed by surface S to the impressed electric current source

Jimp(r ) ∈ Vimp and the induced electric current Jind(r ) ∈ Se; the electric-magnetic

dyadic Green’s function,=

GEM (r, r ′), relates the electric field in the volume V due

to a magnetic current M(r ) ∈ Sm; the magnetic-magnetic dyadic Green’s function,=

GHM (r, r ′), relates the magnetic field in the volume V to a magnetic current

M(r ) ∈ Sm, and the magnetic-electric dyadic Green’s function,=

GHJ (r, r ′), relates

the magnetic field in the volume V to the impressed electric current source Jimp(r ) ∈Vimp and the induced electric current Jind(r ) ∈ Se. The term M(r ′) = n× E(r ′) is

the magnetic current on the surface Sm, and the unit vector n represents an outward

normal to the surface Sm.


The formulation developed in this section can be applied to solve an

aperture-coupled electric-magnetic layered waveguide power combining array for

scattered electric and magnetic fields due to an impressed source (see Fig. 4.2).

impJ

Volume 1

Volume 2

Volume 3 Se

1

2

3

Sm

Sm

Figure 4.2: An aperture-coupled electric-magnetic layered waveguide transition.

4.3 Dyadic Green’s Functions for a Rectangular

Waveguide

As we discussed in the previous section, four dyadic Green functions,=

GEJ ,=

GHJ ,=

GHM , and=

GEM are required to represent the electric and magnetic fields inside a


closed-boundary domain containing apertures (slots) and conducting strips in the

presence of an impressed electric current source.

In this section, the dyadic Green’s functions=

GEJ , and=

GHM are obtained

for a rectangular waveguide as the solution of dyadic wave equation (4.3) subject

to the boundary conditions of the first and second kind, respectively. The other

two dyadic Green’s functions=

GHJ and=

GEM are obtained from curl equations (4.31)

and (4.32). The solution of boundary value problems for Green’s dyadics is based

on a partial eigenfunction expansion. In this representation, the components of

dyadic Green’s functions are expressed as a double series expansion over a complete

system of eigenfunction of a transverse Laplacian operator. The coefficients in this

expansion are the one-dimensional characteristic Green’s functions in waveguiding

direction determined in a closed form.

a

S

Vbx

y

z

r'r

Figure 4.3: Geometry of a rectangular waveguide.

4.3.1 Magnetic Dyadic Green’s Functions

The magnetic-magnetic dyadic Green’s function=

GHM for a rectangular waveguide

(see Fig. 4.3) is obtained by solving the inhomogeneous dyadic wave equation (4.3)


subject to the boundary conditions of the second kind on the waveguide surface. In

the previous section it was shown that=

Ge2==

GHM . This implies that=

GHM can also

be found from the dyadic wave equation [69],

∇×∇× =

GHM (r, r ′)− k2=

GHM (r, r ′) ==

I δ(r − r ′) r − r ′ ∈ V, (4.33)

subject to the second-kind boundary conditions on the waveguiding surface S, fol-

lowing

∇×∇× =

GHM (r, r ′) = 0 r ∈ S, (4.34)

n· =

GHM (r, r ′) = 0. r ∈ S, (4.35)

where n is a unit normal vector to S. By using the identity, (∇ × ∇ × A) =

∇(∇ · A)−∇2A, (4.33) can be expressed as

∇[∇· =

GHM (r, r ′)]−∇2

=

GHM (r, r ′)− k2=

GHM (r, r ′) ==

I δ (r − r ′) (4.36)

resulting in

(∇2 + k2)=

GHM (r, r ′) = −(

=

I +∇∇k2

)δ(r − r ′). (4.37)

The solution of=

GHM (r, r ′) is obtained ¿from

=

GHM (r, ~r ′) =(

=

I +∇∇k2

)· =gh (~r, ~r ′) (4.38)

where the magnetic potential dyadic Green’s function, gh, is a diagonal dyad defined

as=gh (r, r ′) = xxgxx

h (r, r ′) + yygyyh (r, r ′) + zzgzz

h (r, r ′). (4.39)

The components of=gh (r, r ′) satisfy the following scalar inhomogeneous Helmholtz

equations

∇2gxxh (r, r ′) + k2gxx

h (r, r ′) = −δ(r − r ′), (4.40)

∇2gyyh (r, r ′) + k2gyy

h (r, r ′) = −δ(r − r ′), (4.41)

∇2gzzh (r, r ′) + k2gzz

h (r, r ′) = −δ(r − r ′). (4.42)


By substituting (4.38) into the second-kind boundary conditions (4.35), it is found

that=gh should satisfy the boundary conditions on S,

n×∇× =gh (r, r ′) = 0, (4.43)

n ·(

=

I +∇∇k2

)· =gh (r, r ′) = 0. (4.44)

In this analysis, the scalar components gxxh , gyy

h , and gzzh are assumed to have the

following forms of a partial eigenfunction expansion

gxxh (r, r ′) =

∞∑m=0

∞∑n=0

Φxmn,h(x, y)Φx

mn,h(x′, y′)fx

mn,h(z, z′), (4.45)

gyyh (r, r ′) =

∞∑m=0

∞∑n=0

Φymn,h(x, y)Φy

mn,h(x′, y′)f y

mn,h(z, z′), (4.46)

gzzh (r, r ′) =

∑m=0

∑n=0

Φzmn,h(x, y)Φz

mn,h(x′, y′)f z

mn,h(z, z′), (4.47)

where Φimn,h(x, y) and Φi

mn,h(x′, y′), i = x, y, z are eigenfunctions of the transverse

Laplacian operator at the observation and source points, respectively, and the axial

wave functions, f imn,h(z, z

′), represent one-dimensional characteristic Green’s func-

tions in the waveguiding direction. By substituting (4.39) into the second-kind

boundary conditions (4.43), (4.44) at the waveguide walls results in the following

set of boundary conditions for the components of=gh:

∂gxxh

∂z

∣∣∣∣∣x=0,a

= 0,∂gzz

h

∂x

∣∣∣∣∣x=0,a

= 0,∂gxx

h

∂y

∣∣∣∣∣x=0,a

= 0,∂gyy

h

∂x

∣∣∣∣∣x=0,a

= 0, (4.48)

∂gyyh

∂z

∣∣∣∣∣y=0,b

= 0,∂gzz

h

∂y

∣∣∣∣∣y=0,b

= 0,∂gxx

h

∂y

∣∣∣∣∣y=0,b

= 0,∂gyy

h

∂x

∣∣∣∣∣y=0,b

= 0. (4.49)

Carefully solving (4.48)-(4.49), the solutions of the transverse eigenfunctions are

found to be

Φxmn,h(x, y) =

∞∑m=0

∞∑n=0

√ε0mε0n

absin(kxx) cos(kyy), (4.50)

Φymn,h(x, y) =

∞∑m=0

∞∑n=0

√ε0mε0n

abcos(kxx) sin(kyy), (4.51)

Φzmn,h(x, y) =

∞∑m=0

∞∑n=0

√ε0mε0n

abcos(kxx) cos(kyy), (4.52)


where kx = mπa

, ky = nπb

, and ε0j, j = m,n is the Neumann index defined by

ε0j =

1, if j = 0,

2, if j 6= 0.

(4.53)

The transverse eigenfunctions form a complete set such that,

∞∑m=0

∞∑n=0

Φmn(x, y)Φmn(x′, y′) = δ (x− x′) δ (y − y′) . (4.54)

In addition, the transverse eigenfunctions are also orthonormal and satisfy the or-

thogonality relationship,

∫ a

0

∫ b

0Φmn (x, y) Φps (x, y) dxdy = δmpδns (4.55)

where δmp is the Kronkener delta and is defined as

δmp =

1, if m = p

0, if m 6= p

, (4.56)

and δns is defined in a similar manner. After the solution of the transverse eigen-

functions is known, the scalar inhomogeneous wave equations are solved to find the

solutions for the one-dimensional Green’s functions introduced in (4.45)-(4.47). As

an example, the solution of fxmn,h is obtained by substituting (4.45) into (4.40) and

performing the derivatives to give the following expression,

∑m

∑n

(−k2x − k2

y + k2)Φxmn,h(x, y)Φx

mn,h(x′, y′)fx

mn,h(z, z′)

+∑m

∑n

Φxmn,h(x, y)Φx

mn,h(x′, y′)

∂2fxmn,h(z, z

′)∂z2

= −δ(r − r ′). (4.57)

Now using the completeness relation, (4.54), on the right-hand side of (4.57) reduces

(4.57) tod2fx

mn,h (z, z′)dz2

− Γ2mnf

xmn,h (z, z′) = −δ (z − z′) , (4.58)


where Γmn is the propagation constant defined by

Γmn =

√k2

x + k2y − k2, if k2

x + k2y > k2,

j√

k2 − k2x − k2

y, if k2x + k2

y < k2

(4.59)

and it is imaginary for propagating mode and real for non-propagating mode. The

solution of (4.58) subject to the fitness condition at infinity limiting absorption

principle for waveguides[][nasich] can be obtained by different methods [62], [61],

including the direct method or via Fourier integral transform. Also, it can be im-

mediately represented in a closed form involving a Wronskian of partial solutions of

(4.58) at z = ±∞,

fxmn,h = − 1

∆ (f1, f2)

f2(z)f1(z′), z ≥ z′,

f1(z)f2(z′), z < z′

(4.60)

where f1(z) = eΓmnz and f2(z) = e−Γmnz, and

∆ (f1, f2) =

∣∣∣∣∣∣∣∣∣f1(z), df1(z)

dz

f2(z), df2(z)dz

∣∣∣∣∣∣∣∣∣= −2Γmn. (4.61)

In compact form, (4.60) for an infinite waveguide can be rewritten as follows

fxmn,h =

e−Γmn|z−z′|

2Γmn

. (4.62)

By following a similar procedure, solutions for the characteristic Green’s functions,

f ymn,h and f z

mn,h are also given by (4.62). The final expression for=

GHM (r, r ′) is found

by substituting the complete solution of=gh, (4.39) into (4.38). Carefully evaluating

the derivatives, the complete expression for the magnetic-magnetic dyadic Green’s

function is given as

=

GHM (r, r ′) =∞∑

m=0

∞∑n=0

ε0mε0n

2abk2Γmn

e−Γmn|z−z′|

×[xx(k2 − k2

x) sin(kxx) sin(kxx′) cos(kyy) cos(kyy

′)


− xykxky sin(kxx) cos(kxx′) cos(kyy) sin(kyy

′)

∓ xzΓmnkx sin(kxx) cos(kxx′) cos(kyy) cos(kyy

′)

− yx kxky cos(kxx) sin(kxx′) sin(kyy) cos(kyy

′)

+ yy(k2 − k2y) cos(kxx) cos(kxx

′) sin(kyy) sin(kyy′)

∓ yzΓmnky cos(kxx) sin(kxx′) cos(kyy) cos(kyy

′)

± zxΓmnkx cos(kxx) sin(kxx′) cos(kyy) cos(kyy

′)

± zyΓmnky cos(kxx) cos(kxx′) cos(kyy) sin(kyy

′)]

+ zz

[k2 +

∂2

∂z2

]e−Γmn|z−z′| cos(kxx) cos(kxx

′) cos(kyy) cos(kyy′)

(4.63)

where the zz term can be rewritten as (see Section A.3)

[k2 + Γ2

mn − 2Γmnδ(z − z′)]e−Γmn|z−z′| cos(kxx) cos(kxx

′) cos(kyy) cos(kyy′). (4.64)

In (4.63), the upper sign corresponds to z ≥ z′ and the lower sign is chosen for

z < z′. By taking the curl of=

GHM (r, r ′), as defined in the previous section, the

final representation for=

GEM (r, r ′) is given as

=

GEM (r, r ′) =∞∑

m=0

∞∑n=0

ε0mε0n

2abk2Γmn

e−Γmn|z−z′|

× [∓xyΓmn cos(kxx) cos(kxx′) sin(kyy) sin(kyy

′)

− xzky cos(kxx) cos (kxx′) sin(kyy) cos(kyy

′)

± yxΓmn sin(kxx) sin(kxx′) cos(kyy) cos(kyy

′)

+ yz kx sin(kxx) cos(kxx′) cos(kyy) cos(kyy

′)

+ zxky sin(kyx) sin(kxx′) sin(kyy) cos(kyy

′)

− zykx sin(kxx) cos(kxx′) sin(kyy) sin(kyy

′)] (4.65)

The equations (4.63) and (4.65) are the complete expressions for dyadic Green’s

functions=

GHM (r, r ′) and=

GEM (r, r ′) for an infinite rectangular waveguide due to

an arbitrary oriented point source.


4.3.2 Electric Dyadic Green’s Function

In this section, the complete expressions for electric-electric dyadic Green’s function=

GEJ and the magnetic-electric dyadic Green’s function=

GHJ are developed using a

similar procedure as outlined in the previous section. We begin by noting that=

GEJ

is equivalent to=

Ge1 which satisfies the inhomogeneous dyadic wave equation,

∇×∇× =

GEJ (r, r ′)− k2=

GEJ (r, r ′) ==

I δ(r − r ′), (4.66)

where the boundary condition at the waveguide walls, x = 0, a and y = 0, b is of the

first kind such that,

n× =

GEJ (r, r ′) = 0. (4.67)

In order to solve for the boundary-value problem (4.66), (4.67) we apply the rela-

tionship between electric and potential Green’s dyadics [reference],

=

GEJ (r, r ′) =(

=

I +∇∇k2

)· =ge (r, r ′). (4.68)

It was also shown in the previous section that the potential dyadics Green’s function=ge (r, r ′) represents a diagonal tensor for rectangular waveguides

=ge (r, r ′) = xxgxx

e (r, r ′) + yygyye (r, r ′) + zzgzz

e (r, r ′), (4.69)

where the components of=ge (r, r ′) satisfy the following scalar inhomogeneous Helmholtz

equations

∇2gxxe (r, r ′) + k2gxx

e (r, r ′) = −δ(r − r ′), (4.70)

∇2gyye (r, r ′) + k2gyy

e (r, r ′) = −δ(r − r ′), (4.71)

∇2gzze (r, r ′) + k2gzz

e (r, r ′) = −δ(r − r ′). (4.72)

Substituting (4.68) into (4.67) results in the first-kind boundary condition for=ge

(r, r ′),

n× =ge (r, r ′) = 0. (4.73)


The solutions (4.70)-(4.72)are obtained in the form of a partial eigenfunction expan-

sion over the complete system of eigenfunctions of transverse Laplacian operator,

gxxe (r, r ′) =

∞∑m=0

∞∑n=0

Φxmn,e(x, y)Φx

mn,e(x′, y′)fx

mn,e(z, z′), (4.74)

gyye (r, r ′) =

∞∑m−0

∞∑n=0

Φymn,e(x, y)Φy

mn,e(x′, y′)f y

mn,e(z, z′), (4.75)

gzze (r, r ′) =

∞∑m=0

∞∑n=0

Φzmn,e(x, y)Φz

mn,e(x′, y′)f z

mn,e(z, z′), (4.76)

where the normalized eignfunctions are obtained as follows,

Φxmn,e(x, y) =

∞∑m=0

∞∑n=0

√ε0mε0n

abcos(kxx) sin(kyy), (4.77)

Φymn,e(x, y) =

∞∑m=0

∞∑n=0

√ε0mε0n

absin(kxx) cos(kyy), (4.78)

Φzmn,e(x, y) =

∞∑m=0

∞∑n=0

√ε0mε0n

absin(kxx) sin(kyy). (4.79)

The solution for the one-dimensional characteristic Green’s function f imn,e, i =

x, y, z, is identical to f imn,h as obtained in the previous section. Note that this is

true only for an infinite rectangular waveguide where there no boundary conditions

(except for the conditions at infinity) are imposed on the one-dimensional charac-

teristic Green’s functions, which is the same for the first and second kind Green’s

functions. After substituting the solution of=ge into (4.68), the final representation

for the electric-electric Green’s function=

GEJ (r, r ′) is obtained as

=

GEJ (r, r ′) =∞∑

m=0

∞∑n=0

ε0mε0n

2abk2Γmn

e−Γmn|z−z′|

×[xx(k2 − k2

x) cos(kxx) cos(kxx′) sin(kyy) sin(kyy

′)

− xykxky cos(kxx) sin(kxx′) sin(kyy) cos(kyy

′)

± xzΓmnkx cos(kxx) sin(kxx′) sin(kyy) sin(kyy

′)

− yx kxky sin(kxx) cos(kxx′) cos(kyy) sin(kyy

′)

− yy(k2 − k2y) sin(kxx) sin(kxx

′) cos(kyy) cos(kyy′)

± yzΓmnky sin(kxx) sin(kxx′) cos(kyy) sin(kyy

′)


∓ zxΓmnkx sin(kxx) cos(kxx′) sin(kyy) sin(kyy

′)

∓ zyΓmnky sin(kxx) sin(kxx′) sin(kyy) cos(kyy

′)]

+ zz

[k2 +

∂2

∂z2

]e−Γmn|z−z′| sin(kxx) sin(kxx

′) sin(kyy) sin(kyy′)

(4.80)

where the zz term can be rewritten as (see Section A.3)

[k2 + Γ2

mn − 2Γmnδ(z − z′)]e−Γmn|z−z′| sin(kxx) sin(kxx

′) sin(kyy) sin(kyy′). (4.81)

By using the curl equation ∇× =

GEJ the expression for=

GHJ is given as

=

GHJ (r, r ′) =∞∑

m=0

∞∑n=0

ε0mε0n

2abk2Γmn

e−Γmn|z−z′|

× [∓xyΓmn sin(kxx) cos(kxx′) sin(kyy) cos(kyy

′)

+ xzky sin(kxx) sin(kxx′) cos(kyy) sin(kyy

′)

± yxΓmn cos(kxx) sin(kxx′) cos(kyy) sin(kyy

′)

− yz kx cos(kxx) sin(kxx′) sin(kyy) sin(kyy

′)

− zxky cos(kxx) cos(kxx′) cos(kyy) sin(kyy

′)

− zykx cos(kxx) sin(kxx′) cos(kyy) cos(kyy

′) (4.82)

The equations (4.80) and (4.82) represent the complete expressions for dyadic Green’s

functions=

GEJ (r, r ′) and=

GHJ (r, r ′) for an infinite rectangular waveguide due to

an arbitrary oriented point source.

4.3.3 Summary

In this section, the general electromagnetic formulation for an arbitrary volume with

an impressed electric current source and magnetic (slots) surfaces and the electric

(strips) enclosed in the volume was developed. The total (incident plus scattered)


electric and magnetic fields in the volume are derived by solving vector wave equa-

tions for electric and magnetic fields in conjunction width corresponding dyadic

differential equations for Green’s functions subject to the appropriate boundary

conditions. Using a potential eigenfunction expansion, the electric and magnetic

dyadic Green’s functions for a rectangular waveguide were determined. In the Sec-

tion 4.4, the formulation presented in Sections 4.2 and 4.3 will be applied for the

analysis of the slot-strip-slot waveguide-based transition.

4.4 Integral Equation Formulation

4.4.1 Geometry Description

Fig 4.4 shows the geometry of an aperture-coupled stripline-to-waveguide transi-

tion. The transition consists of three infinite rectangular waveguides. The lower,

middle, and upper waveguide regions are referred to as volume VI , VII , and VIII

respectively. Each of the waveguides is filled with a homogeneous lossless isotropic

medium with the dielectric primitivity of εI , εII and εIII . The lower slots are located

on the surface SI between the lower and middle waveguides and the upper slots are

located on the surface SII between the middle and upper waveguides. The strips are

located inside of the middle waveguide region VII . The lower waveguide ports are

designated as ports 1 and 2 while the upper waveguide ports are designated as ports

3 and 4. The objective of the aperture-coupled stripline-to-waveguide transition is

to efficiently couple energy from the lower waveguide to the upper waveguide. An

incident electromagnetic field is illuminated at port 1 of the lower waveguide. This

signal travels in the lower waveguide and induces magnetic currents on the lower

slots which scatter energy into the lower and middle waveguides. In the middle


1 2

3 4

Iinc

Iinc HE ,

II HE 11 ,

IIII HE 11 ,

IIII HE 22 ,

IIII HE 33 ,

IIIIII HE 22 ,

J

+1M

+2M

IV

IIIV

IIV

0=z

y

z

II

IIII

IIIIII

IS

IIS

1M

2M

Figure 4.4: An aperture-coupled stripline-to-waveguide transition.


waveguide region VII the scattered fields induce surface electric currents and stand-

ing waves are produced on the strips. The scattered field from the strips along with

the scattered field from the lower slots induce magnetic currents on the upper slots.

The magnetic currents on the upper slots causes scattered fields back into the mid-

dle waveguide and into the upper waveguide region. Part of the energy coupled into

the upper waveguide propagates towards port 3 and the other part couples towards

port 4. Optimum performance is achieved by varying the distance between the slots,

adjusting the slot dimensions, rotating the slots, or varying the stripline dimensions.

4.4.2 Magnetic Currents

Utilizing the equivalence theorem, any slot can be replaced with a conducting plane,

where equal and opposite magnetic currents can be postulated on adjacent surfaces

of the slot. The equivalence theorem ensures that the electric field in adjacent

volumes is continuous across the slot surface. At the surface of the lower slot, the

tangential component of the electric field in VI , must be equal to the tangential

component of the electric field in VII . Given these condition, the expressions for the

magnetic currents are written as

M+1 (r ′) = y × EI

1(r′), (4.83)

M−1 (r ′) = −y × EII

1 (r ′), (4.84)

where M+1 (r ′) and (M−

1 (r ′) are the magnetic currents on adjacent sides of the lower

slot surface S1 in adjacent volumes VI and VII respectively. At any point on the slot

surface S1, it is found that (4.83) and 4.84) must be equivalent,

M1(r′) = M+

1 (r ′) = M−1 (r ′), r′ ∈ S1, (4.85)

where M1(r′) is the magnetic current on the lower slot surface S1. Again, using the

equivalence theorem, a similar procedure, expressions can be found for the magnetic


currents on the surface S2 of the upper slot. The magnetic currents are given as

M+2 (r ′) = y × EII

2 (r ′) (4.86)

M−2 (r ′) = −y × EIII

2 (r ′), (4.87)

where the magnetic currents M+2 (r ′) and M−

2 (r ′) are equivalent at the upper slot

surface S2 between the middle and upper waveguides.

M2(r′) = M+

2 (r ′) = M−2 (r ′), r′ ∈ S2, (4.88)

where M2(r′) is the magnetic current on at the middle and upper waveguide surface

S2.

4.4.3 Scattered Electric and Magnetic Fields

In Section 4.1, electric and magnetic field integral equations based on dyadic Green

functions were derived for a general closed-boundary domain containing aperture

(slots) and conducting strips in the presence of an impressed electric current source.

It was shown that the total electric and magnetic fields inside a volume V are given

by (4.25) and (4.26), respectively. For a specific problem of an aperture-coupled

stripline-to-waveguide transition problem similar expressions can be obtained for

the scattered fields inside the three waveguide regions VI , VII , and VIII . In the

lower waveguide region VI , there are no conducting strips, so that Jind = 0. The

induced magnetic current M1 on the surface S1 of the lower slot causes scattered

electric and magnetic fields in the lower waveguide region VI ,

EI1(r ) =

∫S1

=

GI

EM (r, r ′) · M1(r′)dS ′, (4.89)

HI1 (r ) = −jωεI

∫S1

=

GI

HM (r, r ′) · M1(r′)dS ′, (4.90)

where=

GI

EM (r, r ′) and=

GI

HM (r, r ′) are the electric-magnetic and the magnetic-

magnetic dyadic Green’s functions obtained for the region VI as defined in (4.32)


and (4.30) at the end of Section 4.2.

In the middle waveguide region VII , an electric strip is enclosed inside

the volume and the lower and upper slots lie on the boundary surfaces SI and SII ,

respectively. An equal and opposite in sign magnetic current on the lower slot

generates the scattered electric and magnetic fields in VII which are given as

EII1 (r ) = −

∫S1

=

GII

EM (r, r ′) · M1(r′)dS ′, (4.91)

HII1 (r ) = jωεII

∫S1

=

GII

HM (r, r ′) · M1(r′)dS ′, (4.92)

where=

GII

EM (r, r ′) and=

GII

HM (r, r ′) are obtained for the middle waveguide region

VII . The magnetic current M2 on the upper slot also scatters electric and magnetic

fields into VII which are obtained in a similar form,

EII2 (r ) = −

∫S2

=

GII

EM (r, r ′) · M2(r′)dS ′, (4.93)

HII2 (r ) = −jωεII

∫S2

=

GII

HM (r, r ′) · M2(r′)dS ′. (4.94)

The scattered fields inside the middle waveguide region VII due to the surface electric

current J2 induced on the strip results in

EII3 (r ) = −jωµ

∫S2

=

GII

EJ (r, r ′) · J2(r′)dS ′, (4.95)

HII3 (r ) =

∫S2

=

GII

HJ (r, r ′) · J2(r′)dS ′, (4.96)

where=

GII

EJ (r, r ′) and=

GII

HJ (r, r ′) are the electric-electric and the magnetic-electric

dyadic Green’s functions obtained for the middle region VII as defined in (4.29) and

(4.31) at the end of Section 4.2.

Finally, the scattered fields in the upper waveguide region, VIII , are due

solely to the magnetic current, M2 on the surface region S2 of the upper slot. The

scattered electric and magnetic fields into VIII are given as

EIII3 (r) = −

∫S2

=

GIII

EM (r, r ′) · M2(r′)dS ′, (4.97)

HIII3 (r) = −jωεIII

∫S2

=

GIII

HM (r, r ′) · M2(r′)dS ′, (4.98)


where=

GIII

EM (r, r ′) and=

GIII

HM (r, r ′) are obtained for the upper waveguide region

VIII . Equations (4.91)-(4.98) represent the scattered electric and magnetic fields

inside the three waveguide regions due to the induced surface electric or magnetic

currents.

4.4.4 Total Electric and Magnetic Fields

The total electric and magnetic fields at any field point r in each waveguide region

are obtained as a superposition of incident and scattered fields. In lower waveguide

region VI the total fields are the sum of the incident fields generated at port 1 of

the lower waveguide and the scattered fields from the lower slot. The total fields in

VI are given as

EItot(r ) = EI

inc(r ) + EI1(r ), (4.99)

HItot(r ) = HI

inc(r ) + HI1 (r ), (4.100)

where EIinc and HI

inc are the incident electric and magnetic fields generated by an

impressed current Jimp at port 1 of the lower waveguide VI . In middle waveguide

VII , the total fields are the sum of the scattered fields from the lower slot, upper

slot, and strip. The total fields in VII are

EIItot(r ) = EII

1 (r ) + EII2 (r ) + EII

3 (r ), (4.101)

HIItot(r ) = HII

1 (r ) + HII2 (r ) + HII

3 (r ). (4.102)

In upper waveguide VIII , the total fields are the scattered fields due to the upper

slot only. The total fields in VIII are

EIIItot (r ) = EIII

2 (r ), (4.103)

HIIItot (r ′) = HIII

2 (r ). (4.104)


The equations (4.99-4.104) represent the total electric and magnetic fields at an

arbitrary field point r inside the lower, middle, and upper waveguide regions.

4.4.5 Boundary Conditions

In analyzing the aperture-coupled stripline-to-waveguide transition, the magnetic

and electric currents on the lower and upper slots and strip are unknown. In order

to solve for the unknown currents, a set of coupled integral equation must be formu-

lated. This is accomplished by first enforcing two boundary conditions; (1) the total

tangential magnetic field in adjacent volumes is continuous across the lower slot and

the upper slot surfaces, and (2) the total tangential electric field is zero on the strip

surface. Imposing the boundary conditions produces three equations in which y is

the normal unit vector. The first equation is obtained by implementing the first

boundary condition by equating the tangential components of the total magnetic

field in the lower waveguide VI to the tangential component of the total magnetic

field in the upper waveguide VII at the surface of the lower slot,

y × HItot(r ) = y × HII

tot(r ), r ∈ S1. (4.105)

The second equation is obtained by again applying the first boundary condition.

This is accomplished by equating the tangential component of the total magnetic

field in the middle waveguide VII to the tangential component of the total magnetic

field in the upper waveguide VIII at the surface of the upper slot,

y × HIItot(r ) = y × HIII

tot (r ), r ∈ S2. (4.106)

The third equation is obtained by implementing the second boundary condition.

This is done by equating the tangential components of the total electric field in the

middle waveguide VII to zero at the strip surface,

y × EIItot(r ) = 0, r ∈ S3. (4.107)


Now, the total fields (4.99)–(4.104) can be substituted into (4.105)–(4.107) to obtain

y × HIinc(r ) = y ×

[HII

1 (r )− HI1 (r ) + HII

2 (r ) + HII3 (J3)

], (4.108)

0 = y ×[HII

1 (r ) + HII2 (r )− HIII

2 (r ) + HII3 (r )

], (4.109)

0 = y ×[EII

1 (r ) + EII2 (r ) + EII

3 (r )]. (4.110)

4.4.6 Testing and Basis Functions

In (4.108)–(4.110), the three unknown currents, M1, M2, and J3, can be determined

by using Galerkin projection technique by expanding the currents in terms of basis

functions and by testing the three coupled integral equations. All of the transverse

slots and longitudinal strip are narrow and lie in the xz-plane. In a narrow transverse

slot, the transverse component of the electric field Ex is negligible compared to the

longitudinal component of the electric field Ez. From (4.83), it is seen that the z-

directed electric field produces a x-directed magnetic current defined as Mxx = −y×Ez z. Similarly, for the longitudinal strip, the longitudinal component of magnetic

field Hz is negligible compared to the transverse component of magnetic field Hx.

As a result, only the z-directed electric current, Jz z = y×Hxx, is considered on the

strip. The one-dimensional electric and magnetic surface currents can be represented

in terms of basis functions

M(x′) =N∑

i=1

Mxi W x

i (x′)x, (4.111)

J(z′) =N∑

i=1

Jzi W z

i (z′)z, (4.112)

where W xi (x′) and W z

i (z′) are the piecewise sinusodial basis functions, and Mxi and

Jzi are the unknown current coefficients for the slot and strip expansions, respec-

tively. The term N is the number of basis functions on the slot or on the strip. The


expressions for the piecewise sinusodial basis functions are

W xi (x′) =

sin[ks(c−|x′−xi|)]d sin(ksc)

,|x′ − xi| ≤ c

|z′ − zi| ≤ d/2

0, otherwise

, (4.113)

W zi (z′) =

sin[ks(h−|z′−zi|)]w sin(ksh)

,|z′ − zi| ≤ h

|x′ − xi| ≤ w/2

0, otherwise

. (4.114)

The sinusodial basis functions in (4.113) and (4.114) covers two cells where the slot

and strip cell widths are indicated by c and h, respectively, the narrow dimension

of the slot and strip are indicated by d and w, respectively, and the parameter

ks = k0/√

εmax represents the smoothness of the basis function. The application

of Galerkin’s procedure is implemented by testing (4.108)–(4.110). Next, (4.108),

(4.109) and (4.110) are tested by multiplying each equation by the testing functions,

W xj,1(x), W x

j,2(x), and W zj,3(z) and integrating over the lower slot, upper slot, and

strip surfaces, respectively:

∫S1

W xj,1(x)

[y × HI

inc(r )]dS =

∫S1

W xj,1(x)

y ×

[HII

1 (r )− HI1 (r )

+ HII2 (r ) + HII

3 (r )]

dS (4.115)

0 =∫

S2

W xj,2(x)

y ×

[HII

1 (r ) + HII2 (r )− HIII

2 (r ) + HII3 (r )

]dS

(4.116)

0 =∫

S3

W zj,3(z)

y ×

[EII

1 (r ) + EII2 (r ) + EII

3 (J3)]

dS. (4.117)

Equations (4.115)–(4.117) are then rewritten by substituting the integral form of the

scattered fields. These substitutions produce a set of scalar linear equations that


can be solved for the unknown current coefficients. Testing (4.108) at the surface

S1 of the lower slot gives

∫S1

W xj,1(x)HI

inc,x(r )dS = −jωε0

∫S1

∫S′1

W xj,1(x)

[εIG

I,xxHM(r, r ′) + εIIG

II,xxHM (r, r ′)

]

×N1∑i=1

Mxi,1W

xi,1(x

′)dS ′dS

+ jωε0

∫S1

∫S′2

W xj,1(x)εIIG

II,xxHM (r, r ′)

N2∑i=1

Mxi,2W

xi,2(x

′)dS ′dS

−∫

S1

∫S′3

W xj,1(x)GII,xz

HJ (r, r ′)N3∑i=1

Jzi,3W

zi,3(z

′)dS ′dS. (4.118)

Testing (4.109) at the surface S2 of the upper slot surface gives

0 = − jωε0

∫S2

∫S′1

W xj,2(x)εIIG

II,xxHM (r, r ′)

N1∑i=1

Mxi,1W

xi,1(x

′)dS ′dS

+ jωε0

∫S2

∫S′2

W xj,2(x)

[εII

=

GII,xx

HM (r, r ′) + εIII

=

GIII

HM (r, r ′)]

×N2∑i=1

Mxi,2W

xi,2(x

′)dS ′dS

−∫

S2

∫S′3

W xj,2(x)GII,xz

HJ (r, r ′)N3∑i=1

Jzi,3W

zi,3(z

′)dS ′dS. (4.119)

Finally, testing (4.110) at the surface S3 of the strip gives

0 = −∫

S3

∫S′1

W zj,3(z)GII,zx

EM (r, r ′)N1∑i=1

Mxi,1W

xi,1(x

′)dS ′dS

+∫

S3

∫S′2

W zj,3(z)GII,zx

EM (r, r ′)N2∑i=1

Mxi,2W

xi,2(x

′)dS ′dS

− jωµ∫

S3

∫S′3

εIIWzj,3(z)GII,zz

EJ (r, r ′)N3∑i=1

Jzi,3W

zi,3(z

′)dS ′dS. (4.120)

In (4.118)-(4.120), N1, N2, N3 are the number of unknown current coefficients for

Mxi,1, Mx

i,2, and Jzi,3 of the lower and upper slots, and strip, respectively.


4.5 Method of Moment Formulation

In this section, (4.118)–(4.120) are put into matrix form in order to solve for the

unknown current coefficients. First, a general MoM matrix is developed for an

aperture-coupled stripline-to-waveguide transition with an arbitrary number of lower

and upper slots, and strips. A general form of the MoM equation for the waveguide

transition can be rewritten as a matrix equation

E = AX, (4.121)

where A is the MoM matrix that describes the self and mutual coupling between

the strips and slots, E is the vector for the incident magnetic fields, and X is the

vector of unknown electric and magnetic current coefficients. The vectors E and X

will be discussed in the next section. The vector of unknown current X is found by

inverting A and multiplying by the incident vector,

X = A−1E, (4.122)

where the MoM matrix is written as

A =

Y W

U Z

, (4.123)

where the admittance matrix Y describes the magnetic field at the slots due to

magnetic currents on the slots, the impedance matrix Z describes the electric field

on the strips due to electric currents on the strips, the coupling matrix W describes

the magnetic field on the slots due to electric currents on the strips, and the coupling

matrix U describes the electric field on the strips due to magnetic currents on the

slots. In (4.123), the admittance matrix Y can be simplified by separating the

matrix into three parts and summing the results,

Y = YI + YII + YIII , (4.124)


where the matrices YI , YII , and YIII describe the admittance of the lower and

upper slots of the waveguide volumes, VI , VII , and VIII , respectively. The matrices

are defined as

YI =

YI<P×P> 0

0 0

, (4.125)

YII =

YII<P×P> YII

<P×Q

YII<Q×P> YII

<Q×Q>

, (4.126)

YIII =

0 0

0 YIII<Q×Q>

. (4.127)

In (4.125)–(4.127), the total number of lower slots P defines the range p = 1 . . . P ,

where p is the index for the lower slots; the total number of upper slots Q defined by

the range q = 1 . . . Q where q is the index of the upper slots; and the total number

of strips R defines the range r = 1 . . . R where r is the index for the strips.

Using this notation, the YI<P×P> represents a matrix which consists of

< P × P > submatricies that describe the self and mutual admittance between the

lower slots coupling in the lower waveguide region VI ; YIII<Q×Q> consists of < Q×Q >

submatricies that describe the self and mutual admittance between the upper slots

coupling in the upper waveguide region VIII ; the YII<P×P> matrix consists of the self

and mutual admittance between the lower slots coupling in the middle waveguide

region VII ; YII<Q×Q> matrix consists of < Q × Q > submatricies describe the self

and mutual admittance between the upper slots coupling in the middle waveguide

region VII ; the YII<P×Q> matrix consists of < P×Q > submatricies that describe the

mutual admittance from the lower slots to the upper slots coupling in the middle

waveguide region VII ; and YII<Q×P> matrix consists of < Q×P > submatricies that


describe the mutual admittance from the upper slots to the lower slots coupling in

the middle waveguide region VII . The Z matrix in (4.121) is defined as

Z = ZII<R×R>, (4.128)

where ZII<R×R> consists of < R×R > submatricies that describe the self and mutual

impedance between the strips coupling in the middle waveguide region VII . Next,

the W matrix is defined as

W =

WII<P×R>

WII<Q×R>

, (4.129)

where WII<P×R> consists of < P × R > submatricies that describe the mutual

admittance-impedance from the lower slots to the strips coupling through the middle

waveguide region VII and the matrix WII<Q×R> consists of < Q×R > submatricies

that describe the mutual admittance-impedance from the upper slots to the strips

coupling through the middle waveguide volume VII . The last matrix to be defined

U is given as

U =

[UII

<R×P> UII<R×Q>

], (4.130)

where UII<R×P> consists of < R × P > submatricies that describe the mutual

impedance-admittance from the strips to the lower slots coupling in the middle

waveguide region VII and the UII<R×Q> consists of < R×Q > submatricies that de-

scribe the mutual impedance-admittance from the strips to the lower slots coupling

in the middle waveguide region VII . Using (4.125)-(4.128), the MoM matrix A in

(4.123) can be rewritten as

A = AI + AII + AIII , (4.131)


where

AI =

YI<P×P> 0 0

0 0 0

0 0 0

, (4.132)

AII =

YII<P×P> YII

<P×Q> WII<P×R>

YII<Q×P> YII

<Q×Q> WII<Q×P>

UII<R×P> UII

<R×Q> ZII<R×R>

, (4.133)

AIII =

0 0 0

0 YIII<Q×Q> 0

0 0 0

, (4.134)

where AI is the coupling matrix due to the lower slots coupling into the lower

waveguide region VI , AII is the coupling matrix due to the lower and upper slots

and strips coupling into the middle waveguide region VII , and AIII is the coupling

matrix due to the upper slots coupling into the upper waveguide region VIII . Finally,

the elements of A are expressed as

Y xx,αξζ,ji = −jωε0εα

∫Sξ

∫S′

ζ

W xj (x) Gxx,α

HM (x, z; x′, z′) W xi (x′) dSdS ′, (4.135)

W xz,IIξr,ji = −

∫Sξ

∫S′r

W xj (x) Gxz,II

HJ (x, z; x′, z′) W zi (z′) dSdS ′, (4.136)

U zx,IIrζ,ji = −

∫Sr

∫S′

ζ

W zj (z) Gzx,II

EM (x, z; x′, z′) W xi (x′) dSdS ′, (4.137)

Zzz,IIrr,ji = −jωµ0µII

∫Sr

∫S′r

W zj (z) Gzz,II

EJ (x, z; x′, z′) W zi (z′) dSdS ′, (4.138)

where the admittance elements are described by Y xx,αξζ,ji (4.135) which is the admit-

tance coupling between the jth test point on slot ξ and the ith basis magnetic

current on the slot ζ, where both ξ and ζ are be either the lower slot index p or the


upper slot index q in volume α, where α = I, II, or III; the coupling elements are

given by W xz,IIξr,ji , (4.136), which is the mutual coupling between the jth test point on

slot ξ and the ith basis electric current on the strip r; the coupling elements U zx,IIrζ,ji ,

(4.137), is the mutual coupling between the jth test point on the strip ξ and the

ith basis electric current on the strip ζ; and the impedance elements Zzz,IIrr,ji , (4.138),

are the mutual and self coupling between the jth test point on the strip r to the ith

basis electric current on the strip r.

4.5.1 Incident Fields

The incident vector E in (4.121) is obtained by testing the incident magnetic field

HIinc(r ) generated by an impressed current Jimp at port 1 of the lower waveguide. In

this analysis, the slots are considered to be narrow and they lie in the xz-plane. In

this case, only the tangential component of the incident magnetic field HIinc,x(r ) as

described in (4.118) is needed to excite the aperture-coupled stripline-to-waveguide

transition. The incident vector E can now be expressed as

E =

a<1×P>

0<1×Q>

0<1×R>

. (4.139)

The subvector 0<1×Q> is a zero vector where the number of zero elements is given

by the total number of test points Q on the slots located on the lower and middle

waveguide surface S1. The subvector 0<1×R> is defined in a similar manner where

R is the total number of test points on the slots located on the middle and upper

waveguide surface S2. The subvector a<1×P> represents the amplitude coefficients

of the incident magnetic field at the jth test point of the slots located on S1 where

the total number of test points is given by P . The elements of a<1×P> are found by


testing HIinc,x with the test function W x

j to give

aj,p =∫

S1

HIinc,x(x, z)W x

j,p(x)dS, (4.140)

where aj,p are the complex coefficients of the magnetic field incident at test point j

of slot p located at zp. In [4.140], the incident magnetic field is expressed in terms

of normal waveguide TE modes [62]:

HIinc(r ) =

∞∑m=0

∞∑n=0

hTE,±mn (x, y)e∓Γmnz, (4.141)

where hTE,±mn (x, y) is the normalized magnetic transverse vector function. From [62],

the solution for the vector function hTE,±mn (x, y) for a rectangular waveguide is

hTE,±mn (x, y) = ± Γmn

k2Amn

×kx sin

(mπ

ax

)cos

(nπ

by)

x + ky cos(

mπ

ax

)sin

(nπ

by)

y

,

(4.142)

where the amplitude coefficients Amn are used to normalize the magnetic transverse

vector function. The coefficient is found by utilizing the orthogonality property

Zh

2

∫S

hTE,±mn (x, y) · h∗,TE,±

pq (x, y)dxdy = δmpδnp (4.143)

where the wave impedance for TE modes Zh is defined as

Zh =jωµ

Γmn

. (4.144)

By substituting (4.142) into (4.143), the amplitude coefficients are found to be

Amn =kc

Γmn

√ε0mε0n

Zhab. (4.145)

Next plugging the result of Amn into (4.142) and then into (4.141), the final result

for the normalized magnetic field is

HIinc(r ) =

∞∑m=0

∞∑n=0

√√√√ ε0mε0n

Zhab(k2

x + k2y

)e−Γmnz

×[kx sin (kxx) cos (kyy) x +

kc

Γmn

cos (kxx) cos (kyy) y

]. (4.146)


¿From (4.146), the incident magnetic field Hinc,x(r ), at port 1 in the lower waveguide

is defined as

HIinc,x(r ) =

∞∑m=0

∞∑n=0

√√√√ ε0mε0n

Zhab(k2

x + k2y

)e−Γmnzkx sin (kxx) cos (kyy) (4.147)

Substituting (4.147) into (4.140) gives

aj,p =∞∑

m=0

∞∑n=0

√√√√ε0mε0n

Zhab

k2x

k2x + k2

y

∫ zj,p+ d2

zj,p− d2

e−Γmnzdz

×∫ xj,p+ c

2

xj,p− c2

sin (kxx) sin [ks (c− |x− xj|)] dx. (4.148)

where (4.148) represents the amplitude coefficients of the incident field on the slots

located on surface S1 of the lower and middle waveguides.

4.5.2 Unknown Current Coefficient Vector

The vector of unknown electric and magnetic current coefficients X, must be deter-

mine in order to solve the system of equations in (4.122). After inverting the MoM

matrix and multiplying by the incident vector the becomes

X =

M<1×P>

M<1×Q>

J<1×R>

(4.149)

where

M<1×P> =

M1

M2

...

MP

, M<1×Q> =

M1

M2

...

MQ

, J<1×R> =

J1

J2

...

JR

(4.150)


where M<1×P> is a vector that represents the unknown magnetic current coeffi-

cients M1 · · ·MP of the slots located on the lower and middle waveguide surface

S1, M<1×Q> is a vector that represents the unknown magnetic current coefficients

M1 · · ·MQ of the slots located on the lower and middle waveguide surface S2, and

J<1×R> is a vector that represents the unknown electric current coefficients J1 · · · JP

of the strips located on the middle waveguide region VII .

4.5.3 Scattering Parameters

The electromagnetic fields scattered in a waveguide by an aperture (slot) can be

expressed in term of an infinite set of normal waveguide modes. Reference [62] has

shown that the amplitude coupling coefficients of the mnth mode in a waveguide,

excited by a magnetic current Mp on the slot is

amn,p =1

2

∫Sp

H+,I (x′, z′) MpWxi (x′) dx′dz′, (4.151)

bmn,p =1

2

∫Sp

H−,I (x′, z′) MpWxi (x′) dx′dz′, (4.152)

where amn,p, and bmn,p are the forward and backward coupling coefficients in the

+z and −z directions, respectively, due to slot p located at y′ and z′ on the surface

shared by the lower and middle waveguides. Using the scalar component of the x-

directed transverse magnetic function (4.147), the forward and backward magnetic

fields in (4.151) and (4.152) are defined as

H+,I (x′, z′) = hImn(x′)e−Γ(z′−z0), (4.153)

H−,I (x′, z′) = hImn(x′)e+Γ(z0−z′). (4.154)

Following a similar procedure, the forward and backward coefficients of the upper

waveguide region can be obtained

cmn,r = −1

2

∫Sr

H+,III (x′, z′) MrWxi (x′) dx′dz′, (4.155)


dmn,r =1

2

∫Sr

H−,III (x′, z′) MrWxi (x′) dx′dz′, (4.156)

where cmn,r, and dmn,r are the forward and backward coupling coefficients in the

upper waveguide region in the +z and −z directions, respectively, due to a slot r,

located at y′ and z′ on the surface shared by the middle and upper waveguides. The

forward and backward magnetic fields in the upper waveguide region are given as

H+,III (x′, z′) = hIIImn(x′)e−Γ(z′−zr), (4.157)

H−,III (x′, z′) = hIIImn(x′)e+Γ(z′−zr). (4.158)

The scattering parameter S11 for the arbitrary slot-strip-slot waveguide transition

be determined by utilizing (4.151)-(4.154),

S11 =bmn,1e

−Γmn,I(z1) + bmn,2e−Γmn,I(z2) + · · ·+ bmn,P e−Γmn,I(zp)

a1

, (4.159)

where S11 is the scattering parameters at port 1 (z = 0) of the lower waveguide due

to the presence of 1 to P slots located on the lower and middle waveguide surface

S1. A similar expression can be found for S21.

The scattering parameters S31 and S41 for the slot-strip-slot waveguide

transition be determined by utilizing (4.155)-(4.158),

S31 =cmn,1e

−Γmn,I(z1) + cmn,2e−Γmn,I(z2) + · · ·+ cmn,P e−Γmn,I(zp)

a1

, (4.160)

S41 =dmn,1e

−Γmn,I(L−z1) + dmn,2e−Γmn,I(L−z2) + · · ·+ dmn,3e

−Γmn,I(L−zp)

a1

,

(4.161)

where S31 and S41 are the backward and forward scattering parameters at port 3

(z = 0) and port 4 (z = L) of the upper waveguide due to the presence of 1 to R

slots located on the middle and upper waveguide surface S2.

By utilizing (4.159)-(4.161), the scattering parameters for the slot-strip-

slot waveguide transition can be determined for a TE10 mode incident magnetic field

at port 1 of the lower waveguide.

Chapter 5

Simulation and Results

5.1 Introduction

In this chapter, the theory developed in Chapter 4 is verified by simulating various

aperture-coupled slot-strip-slot waveguide transitions and arrays. The MoM simu-

lator is verified by comparing various structures to a commercial simulator, High

Frequency Structure Simulator (HFSS), which is based on the finite element method.

The MoM simulator is utilized to simulate several waveguide transitions: a single

transverse slot between two waveguides; a single slot-strip-slot; double slot-strip-

slot; double slot-strip-slot with two shifted slots; triple slot-strip-slot; and a single

slot-strip-slot with two strips (see Fig. 5.1 and Fig. 5.2). In addition, two series 1

x 2 coupler and a 2 x 2 slot-strip-slot waveguide-based arrays are simulated. The

concept here is that these structures, with an amplifier in the strip, constitute unit

cells of the planar quasi-optical system of Fig. 1.4. The structure can be viewed as

an active directional coupler with multiple slot transitions providing directionality.

82

CHAPTER 5. SIMULATION AND RESULTS 83

(a)

(b) (c)

(d) (e)

Figure 5.1: Full view of waveguide transitions: (a) single slot-strip-slot; (b) double

slot-strip-slot; (c) double slot-strip-slot one two shifted slots; (d) triple slot-strip-

slot; and (e) single slot-strip-slot with two strips.


(b) (c)

(d) (e)

(a)

Figure 5.2: Top view of waveguide transitions: (a) single slot-strip-slot; (b) double

slot-strip-slot; (c) double slot-strip-slot one two shifted slots; (d) triple slot-strip-

slot; and (e) single slot-strip-slot with two strips.


5.2 Waveguide-Based Slot-Strip-Slot Transitions

5.2.1 Transverse Slot

The waveguide-based transverse slot transition is shown in Fig. 5.3. The geometry

consists of a narrow transverse slot centered in the conducting plane between two air-

filled X-band waveguides where the relative dielectric constant in both waveguides is

εr = 1. The incident TE10 mode is utilized to excite port 1 of the lower waveguide.

The results of the MoM simulator are compared against the published experimental

and numerical results obtained in [72] and [73]. Fig. 5.3 plots the magnitude of

S31 versus slot length, Lsl at a frequency of 9.375 GHz. The parameters used in

the comparison are a = 10.16 mm, b = 22.86 mm, and d = 1.58 mm. The MoM

simulation utilized m = n = 75 waveguide modes and the cell width is adjusted to

maintain a 1 mm cell size. As the slot length varies the maximum resonance occurs

when the slot length, Lsl is 14.43 mm. Fig. 5.4 shows that the MoM simulations

and the published results are in excellent agreement.

Next, the MoM simulator and HFSS scattering parameters are compared

for the waveguide-based transverse slot transition. The slot length is fixed at Lsl =

13 mm and the frequency is varied from 8 to 12 GHz. The slot is discretized with

13 basis functions and simulated with m = n = 75 waveguide modes. Fig. 5.5(a)

shows the comparison between the magnitude of S11 and S21 versus frequency for

a transverse slot centered between two waveguides Lwg = 50 mm long. At the

resonance frequency 8.8 GHz, the magnitude of both S11 and S21 are -6.03 dB, which

corresponds to half power. Fig. 5.5(b) shows S11 at the center of the transverse slot.

Both figures show that MoM simulator and HFSS are in excellent agreement for a

waveguide-based transverse slot transition.


(a)

(b)

x

zL wg

a

Slot

d

Lsl

Figure 5.3: Geometry of a centered transverse slot between two rectangular waveg-

uides: a) full view; b) top view.


-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10 12 14 16 18 20 22

Slot Length (mm)

S11

Mag

nitu

de (

dB)

Figure 5.4: Magnitude of S11 versus varying centered transverse slot lengths between

two rectangular waveguides: MoM (solid line), and published experimental and

simulation results (Fig. 2b curves (1) and (3) in [72]) (dashed line).


(a)

(b)

-25

-20

-15

-10

-5

0

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Mag

nitu

de (d

B)

-90

-60

-30

0

30

60

90

S11

Phas

e (D

eg)

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

Mag

nitu

de (d

B)

S11

S21

Figure 5.5: MoM (solid line) and HFSS (dashed line) comparison of the scattering

parameters at the center of a transverse slot between two rectangular waveguides:

(a) magnitude of S11 and S21; (b) magnitude and phase of S11.


5.2.2 Single Slot-Strip-Slot

Fig. 5.1(a), Fig. 5.2(a) and Fig. 5.6 display the geometry of the single slot-strip-slot

transition. The geometry consists of one lower slot, one strip, and one upper slot.

The dimensions for this geometry are Lsp = 30 mm, W = 1 mm, Lsl = 13 mm, d = 1

mm, and S = 19 mm. Along the z-axis, the lower and upper slots are centered at

15.5 mm and 34.5 mm respectively while the strip is centered at 25 mm. Along the

x-axis, the slot and strip are centered at 11.43 mm. The upper and lower X-band

waveguides dimensions are a = 22.86 mm, b1 = b3 = 10.16 mm, ε1 = ε3 = 1.0 while

the middle waveguide dimensions are a = 22.86 mm, b2 = 1.5748 mm (62 mils), and

ε1 = 1.0. In this frequency range, the upper and lower waveguides support only one

propagating mode (TE10). However, the middle waveguide region propagates two

modes, TE10 and TE20 modes. Fig 5.7 compares the MoM simulations with HFSS

and plots the magnitude and phase of S11 and S41 over the frequency range of 8

to 12 GHz. The scattering parameters compare closely with HFSS. However, as a

single transition, the single one slot-strip-slot, does not provide adequate coupling

into the upper waveguide.

5.2.3 Double Slot-Strip-Slot

Fig. 5.1(b) shows the geometry for the double slot-strip-slot transition. The geome-

try is similar to the single slot-strip-slot transition with the addition of a lower and

upper slot. The inter-spacing between a pair of slots is 5 mm. For this geometry,

the MoM scattering parameter results are compared to the Ansoft HFSS (HFSS A)

and the Agilent HFSS (HFSS B). In Fig. 5.8(a), HFSS A is in closer agreement to

MoM results than HFSS B. For the magnitude of S11, the MoM simulator detected

a sharp peak of -14.5 dB at 8.7 GHz. Initially, the HFSS programs did not detect


x

y

b1

b3

b2t

a

x

z

y

t

b2

b1

b3

a

x

z

L wg

wd

L sl

L sp

a

Lower Slot Upper SlotStrip

S

Figure 5.6: Geometry of an single slot-strip-slot waveguide transition.


(a)

(b)

-35

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S11

Mag

nitu

de (d

B)

-180

-135

-90

-45

0

45

90

135

180

S11

Phas

e (D

egre

es)

-40

-35

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S41

Mag

nitu

de (d

B)

-180

-135

-90

-45

0

45

90

135

180

S41

Phas

e (D

egre

es)

Figure 5.7: MoM (solid line) and HFSS (dashed line) comparison for the scattering

parameters for the single slot-strip-slot waveguide transition. Magnitude and phase:

(a) S11; and (b) S41.


this peak. However, by changing the options, these programs may detect the peak

however the simulation times will result in a few days instead of less than an hour

for the MoM simulator. In Fig. 5.8(b), the phase of S11 is in closer agreement to

HFSS A.

5.2.4 Double Slot-Strip-Slot with Two Shifted Slots

Based on the strip-slot-strip waveguide transition, two additional slots are incor-

porated into the geometry, see Fig. 5.1(c). One lower and upper slot overlap the

strip while one lower and upper slot do not overlap the strip; that is the slots are

non-overlapping. The spacing between the lower and upper slots is S = 19 mm

while the inter-spacing between the two lower slots and two upper slots is 10 mm.

The scattering parameters are compared in Fig. 5.9. At approximately 8.4 GHz

and 9.9 GHz, the magnitude of S11 is minimum at -38 dB and maximum at -9.2 dB,

respectively as shown in Fig. 5.9(a). In Fig. 5.9(b), at 9.2 GHz, the magnitude of

S41 peaks at -8.6 dB and provides a 3-dB bandwidth of 1/2 GHz.

5.2.5 Triple Slot-Strip-Slot

Fig. 5.1 (d) shows the configuration for three lower and upper slots that overlap one

strip. This geometry is the similar to the single slot-strip-slot transition except there

are three input and output slots, respectively. The inter-spacing between adjacent

slots is adjusted to 5 mm. The scattering parameters are shown in Fig. 5.10. Both

the magnitude and phase track favorably.


-30

-25

-20

-15

-10

-5

0

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Mag

nitu

de (

dB)

MoM

HFSS A

HFSS B

-180

-135

-90

-45

0

45

90

135

180

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Pha

se (

Deg

)

MoM

HFSS A

HFSS B

(a)

(b)


parameters for the double slot-strip-slot waveguide transition. (a) Magnitude and

(b) phase.


(a)

(b)

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S41

Mag

nitu

de (d

B)

-180

-135

-90

-45

0

45

90

135

180

S41

Phas

e (D

eg)

-40

-35

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (dB)

S11

Mag

nitu

de (d

B)

-180

-135

-90

-45

0

45

90

135

180

S11

Phas

e (D

eg)


parameters for the double slot-strip-slot waveguide transition with two shifted slots

. Magnitude and phase: (a) S11; and (b) S41.


-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S41

Mag

nitu

de (

dB)

-180

-135

-90

-45

0

45

90

135

180

S41

Pha

se (

Deg

)

-35

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S11

Mag

nitu

de (

dB)

-180

-135

-90

-45

0

45

90

135

180

S11

Pha

se (

Deg

)

(a)

(b)


parameters for the triple slot-strip-slot waveguide transition. Magnitude and phase:

(a) S11; and (b) S41.


5.2.6 Single Slot-Strip-Slot with Two Strips

In an effort to improve the coupling, the longitudinal strip is divided into two strips

each Lsr = 10 mm in length. The one-slot, two-strip, one-slot geometry is shown in

Fig. 5.1(e). The length of the lower and upper slots are extended to Lsl = 15 mm. In

the MoM program, the slots and strips are discretized in 1 mm cells and m = n = 50

waveguide modes are utilized in the simulation. Both magnitudes of S11 and S41

reach a peak value of -5.59 dB and -7.26 dB respectively with a 3dB-bandwidth

of approximately 10 GHz (Fig. 5.11). The plot shows that the output coupling

S41 improves with the one lower, one upper slot and two strips waveguide-based

transition.

5.3 Waveguide-Based Slot-Strip-Slot Arrays

5.3.1 Series 1 × 2 Coupler

Fig 5.12 shows the geometry for the series 1 × 2 slot-strip-slot couplers with and

without breaks in the strips. The lengths of the waveguides are extended to Lwg = 90

mm and two transitions are located 10 mm apart in each coupler. The magnitude

and phase of the input scattering parameter S11 are shown in Fig 5.13. The MoM

results are compared with two runs of HFSS where the two runs of HFSS utilized

delta accuracies of 0.001 dB and 0.005 dB for the scattering parameters respectively.

Both HFSS runs took over 12 hours to compute a coupler array while the MoM

simulator took less than an hour to compute. The MoM simulator was simulated

with m = n = 75 waveguide modes and the slots and strips cell sizes set to 1

mm. Fig. 5.13 shows that from 8 to 9 GHz, the magnitude of HFSS tends to move


(a)

(b)

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S11

Mag

nitu

de (d

B)

-180

-135

-90

-45

0

45

90

135

180

S11

Phas

e (D

eg)

-35

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S41

Mag

nitu

de (d

B)

-180

-135

-90

-45

0

45

90

135

180

S41

Phas

e (D

eg)


parameters for the single slot-strip-slot waveguide transition. Magnitude and phase

(a) S11, and (b) S41.


closer to the MoM result for higher HFSS delta accuracy. Next, the series 1 × 2

slot-strip-slot is simulated by changing the middle waveguide dielectric permittivity

to ε2 = 1.0 and ε2 = 2.2. As the dielectric constant increases (see Fig. 5.14), the

minimum magnitude of S11 shifts from -38 dB at 11.3 GHz to -40 dB at 11.5 GHz.

Fig. 5.15 plots the reverse coupling (the magnitude of S41/S31). This parameter

compares the power out of port 4 to the power out of port 3. For this configuration,

at 11 GHz, the power appearing at port 3 is -35 dB below the power appearing at

port 4. Fig. 5.12(b) shows the series 1 × 2 array with a break of 2 mm separating

the strips. Fig. 5.16 and Fig. 5.17 shows the scattering parameters and the reverse

coupling respectively.

(a)

(b)

Figure 5.12: Full and top view: (a) series 1 × 2 slot-strip-slot coupler array; and

(b) series 1 × 2 slot-strip-slot coupler array with breaks in the strip.


-45

-40

-35

-30

-25

-20

-15

-10

-5

0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S11

Mag

nitu

de (

dB)

-180

-135

-90

-45

0

45

90

135

180

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Frequency (GHz)

S11

Pha

se (

Deg

rees

)

(a)

(b)

Figure 5.13: Input scattering parameter S11 for the series 1× 2 slot-strip-slot coupler

array; (a) magnitude and (b) phase. MoM (solid line), HFSS 0.001 dB (dashed line)

and HFSS 0.005 (dotted line).


-45

-40

-35

-30

-25

-20

-15

-10

-5

0

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Mag

nitu

de (

dB)

Figure 5.14: MoM results for the magnitude of S11 for the series 1 × 2 slot-strip-slot

coupler array; (a) ε2 = 1.0 (solid line) and ε2 = 2.2 (dotted line).


-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S41

/S31

Mag

nitu

de (

dB)

Figure 5.15: Reverse coupling for the series 1 × 2 slot-strip-slot coupler array,

ε2 = 1.0, MoM (solid line) and HFSS (dotted line).


-35

-30

-25

-20

-15

-10

-5

0

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Mag

nitu

de (

dB)

-180

-135

-90

-45

0

45

90

135

180

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Pha

se (

dB)

(a)

(b)

Figure 5.16: Input scattering parameter S11 for the series 1× 2 slot-strip-slot coupler

array with two strips; (a) magnitude and (b) phase. MoM (solid line), HFSS 0.001

dB (dashed line).


-40

-35

-30

-25

-20

-15

-10

-5

0

5

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

Mag

nitu

de (

dB)

S41

S31

Reverse Coupling

Figure 5.17: MoM simulation for the reverse coupling for the series 1× 2 slot-strip-

slot coupler array, ε2 = 1.0.


5.3.2 2 × 2 Slot-Strip-Slot Array

The geometry for the 2×2 slot-strip-slot waveguide array is shown in Fig. 5.18. The

lower, middle, and upper waveguide dimensions are a = 46 mm, b1 = b2 = 10.16

mm, b3 = 1.5748 mm, and Lwg = 90 mm. Fig. 5.19(a) compares the magnitude and

phase of S11 where the dielectric constant in the middle waveguide is set to ε2 = 1.0

and ε2 = 2.2. For the 2 × 2 slot-strip-slot array, the maximum coupling into the

middle waveguide occurs over the frequency range of 8.5 GHz to 10 GHz. Maximum

coupling occurs when S11 is a minimum of -41.5 dB at 9.6 GHz. Fig. 5.20 displays

the reverse coupling for ε2 = 1.0. Approximately, -16 dB of isolation is achieved at

9.6 GHz.

5.3.3 Summary

In this section, several configurations of the slot-strip-slot transitions and arrays

are simulated with the MoM simulator. The single slot-strip-slot with two strips,

triple slot-strip-slot, and the double slot-strip-slot with two shifted slots waveguide

transitions provided the strongest coupling to port 4. Each of these transitions

utilized a minimum of two lower and two upper slots. The 1 × 2 slot-strip-slot

coupler array provide much better isolation than the 1× 2 slot-strip-slot with gaps

in the strips. Over a narrow bandwith, the 2 × 2 array provides adquate reverse

isolation. There are endless transition configurations that could be simulated and

adjusted to provide strong coupling. However, the major purpose of this section is

to demonstrate that the MoM simulator can be used to model transitions and arrays

in a fast, efficient, and accurate manner.


Figure 5.18: Full and top view of a 2 × 2 slot-strip-slot array


-45

-40

-35

-30

-25

-20

-15

-10

-5

0

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Mag

nitu

de (

dB)

-180

-135

-90

-45

0

45

90

135

180

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

S11

Pha

se (

Deg

)

(a)

(b)

Figure 5.19: MoM simulation of a 2 × 2 slot-strip-slot waveguide-based array; ε2 =

1.0 (solid line) and ε2 = 2.2 (dashed line); S11 (a) magnitude, (b) phase.


-50

-40

-30

-20

-10

0

10

8 8.5 9 9.5 10 10.5 11 11.5 12

Frequency (GHz)

Mag

nitu

de (

dB)

S31

S41

Reverse Coupling

Figure 5.20: Reverse coupling for the series 1 × 2 slot-strip-slot coupler array,ε2 =

1.0, MoM (solid line) and HFSS (dotted line).


5.4 Computational Analysis

The MoM simulator is written in FORTRAN 77 using a standard Windows-based

personal computer. The main program reads an input geometry file and calculates

the self and mutual couplings between the transverse slots and longitudinal strips.

There are four major routines utilized to calculate coupling and to fill the MoM ma-

trix. The routines determine the self and mutual coupling between transverse slots,

the self and mutual coupling between strips, the mutual coupling between transverse

slots and longitudinal strips, and the mutual coupling between longitudinal strips

to transverse slots. After the MoM matrix has been filled, a routine generates a

magnetic incident vector from an incident TE10 generated at port 1 of the lower

waveguide. Next the MoM matrix is inverted using LU decomposition and then the

unknown electric (strips) and magnetic (slots) currents coefficients are determined.

The unknown coefficients are passed to a routine that solves for the scattering pa-

rameters for the given geometry. The procedure is repeated at all frequencies of

interest.

Concluding remarks:

• The impedance matrix Z includes the m = n = 0 mode.

• Analytical expressions are found for all of the integrals therefore numerical

integrations are not necessary. The use of analytical expressions aided in

reducing the overall computational time.

• HFSS is based on FEM which is based on a volume and surface discretiza-

tion whereas the MoM is based only on surface discretization. Initially HFSS

missed sharp resonates but the resonates are found by increasing the meshing

however this increases the computational time two or three fold. The MoM

simulator detects sharp resonates with as few as 30 to 50 waveguide modes


which takes more than half the time to detect the sharp resonates.

• For the 2× 2 array, the MoM simulator calculated the scattering parameters

at 40 frequencies in 1 to 2 days. However, HFSS did not reach a solution after

two days.

Chapter 6

Conclusion and Future Research

6.1 Conclusion

The purpose of the work described in this dissertation was the development of the

fundamental understanding of planar spatial power combining systems. The major

purpose was the development of electromagnetic modeling tools for planar spatial

power combiners.

6.1.1 Mode Structure

The electromagnetic model of a QO parallel-plate HDSBW resonator system based

on Gauss-Hermite beammode expansion was developed and verified experimentally

to predict the resonance frequencies of the structure and beammodes dispersion

behavior. The system was designed, fabricated, tested, and favorable agreement

110

CHAPTER 6. CONCLUSION AND FUTURE RESEARCH 111

between the experimental data with theoretical results. In addition, a QO parallel-

plate stripline-slot amplifier system was designed, tested and compared to a QO

open HDSBW amplifier system with the previously used Vivaldi-type antennas.

Experimental results verify that a QO parallel-plate stripline-slot amplifier proposed

in the dissertation improves overall system performance.

The key conclusion of this part of the work is that the Gauss-Hermite

beammode expansion is an entirely adequate approximation for dielectric power

combining.

6.1.2 Electromagnetic Modeling Technology

A full-wave method of moment (MoM) electromagnetic simulator was developed

to investigate aperture-coupled stripline waveguide-based transitions and arrays for

spatial power combining systems. The analysis is based on developing a set of cou-

pled integral equations which represent the scattering electric and magnetic fields

and the induced electric and magnetic currents of the transverse strips and longitudi-

nal slots respectively. The scattered fields are expressed in terms of dyadic Greeen’s

functions for rectangular waveguides and surface currents that are discretized uti-

lizing piecewise sinusodial subdomain basis functions. The waveguide-based transi-

tions and arrays are excited with TE10 dominant waveguide mode and the scattering

parameters are calculated from the forward and backward coupling coefficients in

the waveguide regions.


6.2 Future Research

A MoM simulator has been developed to model aperture-coupled stripline waveguide-

based transitions and arrays. This MoM simulator can be utilized to optimize the

coupling from the lower to the upper waveguide. Optimization can be achieve by in-

vestigating transverse slots and longitudinal strips with different lengths and widths,

and offset from the center of the waveguide. In addition dimensions of the lower,

middle, and upper waveguides can be varied for an over-moded waveguide. The

MoM simulator can also be utilized to investigate large arrays that incorporate

different types of transitions.

This work utilizes four different waveguide Green’s functions to calcu-

late the coupling between transverse slot coupling between longitudinally strips,

and the coupling between transverse slots and longitudinal strips. The capability of

the MoM simulator can be expanded by incorporating Green’s functions for trans-

verse strips and longitudinal slots. The MoM can be utilized to calculated the self

and mutual coupling of transverse strips and longitudinal slots. Inclusion of these

Green’s function can expand this work to include patches, spirals, L-shaped bends

for strip (electric layers) and slots (magnetic layers).

This work can also be enhanced by adding vertical posts or strips to kill

or prevent higher order parallel plate modes from propagating inside the waveguides.

Also the inclusion of the edge condition may be needed to improve the accuracy and

convergence.

As the number of strips or slots increase and as the dimensions of the

waveguides increase, the MoM matrix will become large and the computations time

increase. The total computational time can be decreased by using an electromagnetic

acceleration technique such as the Kummer transformation or using a combination


or uniform and non-uniform meshing for the strips and slots. Decrease in computa-

tional time can also be reduced by taking advantage of the symmetry of the MoM

matrix to eliminate redundant calculations. A subroutine can be written to store

and retrieve the calculations.

Also this work can be easily expanded to include multiple waveguide

layers. Presently the MoM simulator runs much faster than HFSS. However the

computational time of the MoM simulator can be greatly enhanced by incorporating

an accelerating technique such as the Kummer transformation.

Another important task is to expand this work so that an active slot-

strip-slot transition that incorporates an active transistor or MMIC device on the

strip can be simulated in a transient simulator. To accomplish this task, circuit ports

should be incorporated within the MoM simulator so that it can interface with the

transient simulator. To verify the simulator, an active slot-strip-slot transition or

array should be experimentally tested.

There are numerous aperture-coupled stripline transitions and arrays

that be be investigated to improved the performance of 2-dimensional quasi-optical

and waveguide-based spatial power combing systems.

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Appendix A

Method of Moments

Implementation

A.1 Introduction

In this section, the matrices Y, W, U, and Z describe the electromagnetic fields

coupling in the aperture-coupled stripline-to-waveguide transition. The integrations

required to evaluate the matrix elements are discussed in this appendix.

124

APPENDIX A. METHOD OF MOMENTS IMPLEMENTATION 125

A.2 Admittance Matrix for Transverse Slots

The upper and lower transverse slots are narrow and lie in the xz-plane. Since the

magnetic currents are x-directed, only the xx-directed component of the magnetic-

magnetic Green’s function, GxxHM , is required to compute the elements of Y,

GxxHM =

∞∑m=0

∞∑n=0

ε0mε0n

k22abΓmn

(k2 − k2x) sin(kxx) sin(kxx

′) cos(kyy) cos(kyy′), (A.1)

Substituting W (x) and Gxx,αHM into (4.135) gives

Y xx,αξζ,ji = − jωε0εα

∞∑m=1

∞∑n=0

k2α − k2

x,α

k2αΓmn,α

ε0mε0n cos(ky,αy) cos(ky,αy′)2aαbα sin(ksc) sin(ksc)didj

×∫ zjξ+ d

2

zjξ− d2

∫ zi,ζ+ d2

ziζ− d2

e−Γmn,α|z−z′|dzdz′

×∫ xiζ+c

xiζ−csin(kx,αx′) sin[ks(c− |x′ − xiζ |)]dx′

×∫ xjξ+c

xjξ−csin(kx,αx) sin[ks(c− |x− xjζ |)]dx (A.2)

where the index α = I, II or III refers to the lower, middle, or upper waveguide

regions, respectively. The integral over dzdz′ in (A.2) is evaluated by considering

two cases. In the first case, shown in Fig A.1(a), the integration for the self-coupling

of a single slot, is performed by fixing z′ in the region zj− d2

< z′ < zj + d2. Breaking

the integral into two parts allows the absolute value in the exponential function to

be removed. Implementing these conditions, the integration of the exponential term

over dz is rewritten as

∫ zj+d2

zj− d2

e−Γmn|z−z′|dz =∫ z′

zj− d2

eΓmn(z−z′)dz +∫ zj+

d2

z′e−Γmn(z−z′)dz. (A.3)


zj+d/ zj z' zj+d/2

(a)

z

zj+d/2 zj zj+d/2

z'

zi+d/2 zi zi+d/2

(b)

z < z' z > z'

x i -c xi x i+c

zj+d/2

zj

zj -d/2

(c)

x ' < xi x ' > xi

Figure A.1: The geometry for: (a) the dzdz′ integration for self-coupling for a single

slot; (b) the dzdz′ integration for the coupling between two slots; and (c) the dx′

integration for all slots.


Next, the result of (A.3) is integrated over dz′ to produce the final expression for

the integral over dzdz′

∫ zi+d2

zi− d2

∫ zj+d2

zj− d2

e−Γmn|z−z′|dzdz′ =∫ zi+

d2

zi− d2

[∫ z′

zj− d2


d2

z′e−Γmn(z−z′)dz

]dz′.

(A.4)

In the second case, shown in Fig A.1(b), the integration of the coupling between

any two slots located at zj and zi, respectively, is performed by noticing that z < z′.

In this case, the exponential absolute value sign is removed and the integration over

dz′ becomes ∫ zj+d2

zj− d2

∫ zi+d2

zi− d2

e−Γmn|z−z′|dz =∫ zj+

d2

zj− d2

e−Γmn(z−z′)dz. (A.5)

Using a similar procedure, the dx′ integration in (A.2) is performed by

breaking the integral over dx′ into two parts, x′ < xi and x′ > xi to remove the

absolute value sign of the sinusodial basis function. The result of the integration

over dx′ is given as

∫ xi+c

xi−csin(kx,αx′) sin[ks(c + |x′ − xi|)]dx′ ;

=∫ x

xi−csin(kx,αx′) sin[ks(c + (x′ − xi))]dx′

+∫ xi+c

xsin(kx,αx′) sin[ks(c− (x′ − xi))]dx′. (A.6)

The integration over dx in (A.2) is performed in a similar manner to (A.6). The

final solution for the admittance elements is obtained by solving (A.4)-(A.6) and

substituting them back into (A.2).


A.3 Impedance Matrix for Longitudinal Strips

The longitudinal strips are narrow and lie in the xz-plane and are enclosed in the

middle waveguide region VII . As a result, only the GzzEJ component of the electric-

electric dyadic Green’s function is required to compute the impedance Z matrix,

GzzEJ =

∞∑m=0

∞∑n=0

ε0mε0n

k22abΓmn

sin(kxx) sin(kxx′) sin(kyy) sin(kyy

′)

(k2 +

d2

dz2

)

× e−Γmn|z−z′|. (A.7)

Notice that the first and second derivatives of the exponential function with respect

to dz are evaluated using distribution theory. The first derivative gives

d

dze−Γmn|z−z′| = [θ(z − z′)− θ(z′ − z)]Γmne

−Γmn|z−z′| = −Γmne−Γmn|z−z′|, (A.8)

and the second derivative results in

d2

dz2e−Γmn|z−z′| =

[Γ2

mn − 2Γmnδ(z − z′)]e−Γmn|z−z′|. (A.9)

By substituting (A.9) into (A.7), GzzEJ is rewritten as

GzzEJ =

∞∑m=0

∞∑n=0

ε0mε0n

k22abΓmn

sin(kxx) sin(kxx′) sin(kyy) sin(kyy

′)

×[k2 + Γ2

mn − 2Γmnδ(z − z′)]e−Γmn|z−z′|. (A.10)

Substituting GzzEJ and Jz

i (z) into (A.11) gives

Zzz,IIrr,ji = −jωµ0µ

M∑m=0

N∑n=0

ε0mε0m

2aIIbII

sin(ky,IIy) sin(ky,IIy′)

wr sin(ksh)wr sin(ksh)

× I


×∫ xj,r+wr

2

xj,r−wr2

sin(kx,IIx)dx

×∫ xi,r+wr

2

xi,r−wr2

sin(kx,IIx′)dx′, (A.11)

where the integral I represents the integration over dzdz′ and is given as

I = I1 + I2. (A.12)

The integrals I1 and I2 are defined as

I1 =(k2

II − Γ2mn,II

)

×∫ zj+h

zj−h

∫ zi+h

zi−he−Γmn,II |z−z′| sin [ks (h− |z′ − zi|)] sin [ks (h− |z − zj|)] dz′dz,

(A.13)

and

I2 = −2Γ2mn,II

×∫ zj+h

zj−h

∫ zi+h

zi−hδ (z − z′) sin [ks (h− |z′ − zi|)] sin [ks (h− |z − zj|)] dz′dz.

In (A.12), the I integral is evaluated by considering three separate cases as shown

in Fig A.2: 1) the testing and basis functions completely overlap the same two cells;

2) the testing function and basis function overlap one cell; and 3) the testing and

basis functions do not overlap.

A.3.1 Completely Overlapping Case

The overlapping case occurs when the testing and basis functions completely overlap

the same two cells, that is, zi = zj. The I integral is evaluated by considering


• • •

Testing Function Basis Function

Testing and Basis Function

Testing Function Basis Function

(a)

(b)

(c)

Figure A.2: Testing and basis functions for the longitudinal strip: (a) completely

overlapping case; (b) partially overlapping case; and (c) non-overlapping case.


zi zi + hzi - hzj zj + hzj - h

z'

z' < z z' > z z' > z

A B S

zi zi + hzi - h

zj zj + hzj - h

z'

z' < z z' > zz' < z

T C D

(a)

(b)

Figure A.3: Longitudinal strip overlapping case (a) z′ < zi, and (b) z′ > zi.


two conditions: 1) zi − h < z′ < zi and z′ < zi as show in Fig. A.3(a), and 2)

zi < z′ < zi + h and z′ > zi as shown in Fig. A.3(b). In both cases, z′ is fixed

for both conditions. The solution of the I integral is given by summing the results

produced by the two conditions. The I1 integral is evaluated by first integrating the

exponential and basis functions over dz′. Applying the first condition to I1, allows

the absolute value signs to be removed. As a result, integrating over dz′ allows the

I1 integral to be separated into three integrals, A, B, and S which are given as

A =∫ z

zi−he−Γmn(z−z′) sin [ks (h + (z′ − zi))] dz′ (A.14)

B =∫ zi

zeΓmn(z−z′) sin [ks (h + (z′ − zi))] dz′ (A.15)

S =∫ zi+h

zi

eΓmn(z−z′) sin [ks (h− (z′ − zi))] dz′. (A.16)

The second condition that is evaluated is shown in Fig A.3(b). Applying this con-

dition and integrating I1 over dz′ produces three integrals C, D, and T which are

given as

C =∫ z

zi

e−Γ(z−z′) sin [ks (h− (z′ − zi))] dz′ (A.17)

D =∫ zi+h

zeΓ(z−z′) sin [ks (h− (z′ − zi))] dz′ (A.18)

T =∫ zi

zi−he−Γ(z−z′) sin [ks (h + (z′ − zi))] dz′. (A.19)

Next, the integral I1 is integrated over dz. The I1 integral can be separated into two

integrals by using the fact that z < zj and z > zj. This enables the absolute value

sign in the sinusodial testing function to be removed. The complete expression for


I1 becomes

I1 =∫ zj

zj−hsin [ks (h + (z′ − zj))] [A + B + S] dz

+∫ zj+h

zj

sin [ks (h− (z′ − zj))] [C + D + T ] dz. (A.20)

Next, I2 integral can be evaluated by using the property of the delta function, that

is, I2 is only valid when z = z′. Integrating I2 over dz′ gives

I2 =∫ zj+h

zj−hsin [ks (h− |z − zj|)] sin [ks (h− |z − zi|)] dz. (A.21)

The I2 integral in (A.21) is evaluated by separating the integral into two parts,

zj − h < z < zj and zj < z < zj + h. Since the testing and basis functions overlap,

zi = zj, removing the absolute value signs in the sinusodial functions and integrating

I2 over gives

I2 =∫ zi

zi−hsin [ks (h + (z − zi))] sin [ks (h + (z − zi))] dz

+∫ zi+h

zi

sin [ks (h− (z − zi))] sin [ks (h− (z − zi))] dz (A.22)

and simplifying

I2 =∫ zi

zi−hsin2 [ks (h + (z − zi))] dz +

∫ zi+h

zi

sin2 [ks (h− (z − zi))] dz (A.23)

The final integral expression for I for the overlapping case is given by using I1 and

I2 as given by (A.20) and (A.23), respectively.


A.3.2 Non-Overlapping Case

The second case, non-overlapping case, occurs when the testing and basis functions

do not overlap. Fig. A.4 shows when the testing function is less than the basis

function, zj < zi, zj + h < zi − h and z < z′. The I1 integral is first evaluated by

integrating over dz′. As a result, the dz′ integral can be separated into two integrals

and the absolute value signs in the exponential and basis function can be removed

and the integration over dz′ becomes

E =∫ zi

zi−heΓmn(z−z′) sin [ks (h + (z′ − zi))] dz′ (A.24)

F =∫ zi+h

zi

eΓmn(z−z′) sin [ks (h− (z′ − zi))] dz′ (A.25)

Next the integral I1 is integrated over dz and testing function is separated into two

zi zi + hzi - hzj zj + hzj - h

Testing function Basis function

Figure A.4: Longitudinal strip testing and basis functions do not overlap

integrals

I1 =∫ zj

zj−hsin [ks (h + (z − zj))] [E + F ] dz +

∫ zj+h

zj

sin [ks (h− (z − zj))] [E + F ] dz.

(A.26)


In the nonoverlapping case, the I2 integral integrates to zero due to the integration

of the delta function. As a result the solution of I for the non-overlapping case is

given by I1 in (A.26).

A.3.3 Partially Overlapping Case

The third case, the partially overlapping case, occurs when the testing and basis

functions partially overlap. This case occurs when zj = zi − h, zi = zj + h, and

zj < zi as shown in Fig. A.5. The I1 integral is evaluated by fixing z′ in the intervals

zj−h < z < zj and zj < z < zj +h, respectively, and then integrating and summing

the results. Fig. A.5(a) shows z′ fixed in the interval zj − h < z < zj. In this

region, the absolute value sign in the testing function is removed, and the testing

function is always less than the basis function. Consequently, the dz′ integration is

separated into two parts to remove the absolute value signs in the exponential and

basis functions. The two integrals over dz′ are

L =∫ zi

zi−heΓmn(z−z′) sin [ks (h + (z′ − zi))] dz′ (A.27)

M =∫ zi+h

zi

eΓmn(z−z′) sin [ks (h− (z′ − zi))] dz′. (A.28)

Fig. A.5(b) shows z′ fixed in the interval zj < z < zj + h. Integrating over this

region produces three integrations over which are identical to A, B, or S integrals

used in the overlapping case. The final result of the integral I1 is

I1 =∫ zj+h

zj

sin [ks (h− (z′ − zi))] [L + M ] dz


A B J

zi zi + hzi - hzj zj + hzj - h z'

L M

(a)

(b)

zi zi + hzi - hzj zj + hzj - h z'



Figure A.5: Longitudinal strip test and basis functions partially overlap.


+∫ zj+h

zj

sin [ks (h− (z′ − zi))] [A + B + S] dz. (A.29)

The I2 integral is evaluated by using the property of the delta function. First

integrating over z′, the exponential function is unity, and the I2 integral is only

valid over the region, zj < zj + h,

I2 =∫ zj

zj−hsin [ks (h− |z − zj|)] sin [ks (h− |z − zi|)] dz. (A.30)

The absolute value signs in (A.30) are removed by using the relation zj < z and

zi > z, respectively.

I2 =∫ zj

zj−hsin [ks (h− (z − zj))] sin [ks (h− (z − zi))] dz. (A.31)

The I integral for the partially overlapping case is given by I1 (A.29) and I2 (A.31).

A.4 Strip to Slot Coupling Matrix

The matrix W describes the magnetic field coupling from the z-directed strips to

the x-directed slots. Both the strip and slot are narrow and lie in the xz-plane.

Therefore only the GxzHJ(r, r′) component is required

=

Gxz

HJ=∞∑

m=1

∞∑n=1

ε0mε0n

2abΓmn

e−Γmn|z−z′|ky sin (kxx) cos (kxx′) sin (kyy) sin (kyy

′) . (A.32)

Substituting GxzHJ and W x

i functions into (A.32), the coupling elements is

W xz,IIξ,r,ji =

∞∑m=1

∞∑n=1

ε0mε0n

2aIIbII

ky,II

Γmn,II

cos (ky,IIy2) sin (ky,IIy′)

wd sin (ksh) sin (ksc)


× I3

×∫ xi+

w2

xi−w2

cos (kx,IIx′) dx′

×∫ xj+c

xj−csin (kx,IIx) sin [ks (c− |x− xj|)] dx (A.33)

where I3 is defined as

I3 =∫ zi+h

zi−h

∫ zj+d2

zj− d2

e−Γmn|z−z′| sin [ks (h− |z′ − zi|)] dzdz′. (A.34)

The integral I3 is carried out by considering two cases as shown in Fig A.6: 1) the

non-overlapping case which occurs when the testing function is always less than or

greater than the basis function; and 2) the partially overlapping case which occurs

when testing function and basis functions partially overlap.

A.4.1 Non-Overlapping Case

The non-overlapping case occurs when then zi + h < zj − d2, zi < zj and z′ < z as

shown in Fig A.6(a). The I3 integral is evaluated by separating the dz′ into two

integrals

I3 =∫ zi

zi−h

[∫ zj+d2

zj− d2

e−Γmn(z−z′)dz

]sin [ks (h + (z′ − zi))] dz′

+∫ zi+h

zi

[∫ zj+d2

zj− d2

e−Γmn(z−z′)dz

]sin [ks (h− (z′ − zi))] dz′. (A.35)


A.4.2 Partially Overlapping Case

The partially overlapping case occurs when zj − d2

= zi, zj + d2

= zi + h as shown in

Fig A.6(b). The I3 integral becomes

I3 =∫ zi

zi−h

∫ zj+d2

zj− d2

e−Γmn(z−z′)dz sin (ks (h + (z′ − zi))) dz′

+∫ zi+h

zi

[∫ z′

zj− d2


d2

z′e−Γmn(z−z′)dz

]sin (ks (h− (z′ − zi))) dz′

(A.36)

A.5 Slot-to-Strip Coupling Matrix

The U matrix describes the magnetic field coupling from the longitudinal strips to

transverse slots. Since both the strip and slot are narrow and lie in the xz-plane,

only the=

Gxx

HJ (r|r′) component is required which is given as

=

Gzx

EM=∞∑

m=1

∞∑n=1

ε0mε0n

2abΓmn

e−Γmn|z−z′|ky,II sin(kx,IIx) sin(kx,IIx′) sin(ky,IIy) cos(ky,IIy

′).

(A.37)

By utilizing GEM and W xj , the elements for the U matrix is given by

U zx,IIr,ζ =

∞∑m=1

∞∑n=0

ε0mε0n

2ab

ky

Γmn,II

sin(ky,IIy) cos(ky,IIy′)

wd sin(ksh) sin(ksc)

× I3

×∫ xi+

w2

xi−w2

sin(kx,IIx′)dx′


zi zi -hzi+h

xi

zj

zj -d/2zj+d/2

xj-c

xj+c

xi

xj -w/2

xj+w/2

zi zi -h

zi+h

xi

zj

zj -d/2zj+d/2

xj-c

xj+c

xi

xj -w/2

xj+w/2

(a)

(b)

Figure A.6: Crossed strip and slot testing and basis functions: (a) non-overlapping

case; and (b) overlapping case.


×∫ xj+c

xj−csin(kx,IIx) sin [ks (c− |x− xj|)] dx (A.38)

By using the techniques in the previous section, the solution to the U elements can

be obtained.

experimental and electromagnetic modeling of waveguide

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