GENERALIZED SCATTERING MATRIX MODELING OF … · GENERALIZED SCATTERING MATRIX MODELING OF...

GENERALIZED SCATTERING MATRIX

MODELING OF DISTRIBUTED MICROWAVE

AND MILLIMETER-WAVE SYSTEMS

by

AHMED IBRAHIM KHALIL

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

1999

APPROVED BY:

Chair of Advisory Committee

Abstract

KHALIL, AHMED IBRAHIM. Generalized Scattering Matrix Modeling of Dis-tributed Microwave and Millimeter-Wave Systems. (Under the direction of MichaelB. Steer.)

A full-wave electromagnetic simulator is developed for the analysis of

transverse multilayered shielded structures as well as waveguide-based spatial power-

combining systems. The electromagnetic simulator employs the method of moments

(MoM) in conjunction with the generalized scattering matrix (GSM) approach. The

Kummer transformation is applied to accelerate slowly converging double series ex-

pansions of Green’s functions that occur in evaluating the impedance (or admit-

tance) matrix elements. In this transformation the quasi-static part is extracted

and evaluated to speed up the solution process resulting in a dramatic reduction of

terms in a double series summation. The formulation incorporates electrical ports

as an integral part of the GSM formulation so that the resulting model can be

integrated with circuit analysis.

The GSM-MoM method produces a scattering matrix that represents

the relationship between waveguide modes and device ports. The scattering matrix

can then be converted to port-based admittance or impedance matrix. This allows

the modeling of a waveguide structure that can support multiple electromagnetic

modes by a circuit with defined coupling between the modes. Since port-based

representations are not suited for most circuit simulation tools, a circuit theory

based on the local reference node concept, is developed. The theory adapts modified

nodal analysis to accommodate spatially distributed circuits allowing conventional

harmonic balance and transient simulators to be used.

To show the flexibility of the modeling technique, results are obtained

for general shielded microwave and millimeter-wave structures as well as various

spatial power combining systems.

Biographical Summary

Ahmed Ibrahim Khalil was born in Cairo, Egypt, on November 15, 1969.

He received the B.S. (with honors) and M.S. degrees from Cairo University, Giza,

Egypt, both in electronics and communications engineering, in 1992 and 1996, re-

spectively. From 1992 to 1996 he worked at Cairo University as a Research and

Teaching Assistant. While working towards his Ph.D. degree in electrical engi-

neering at North Carolina State University, since 1996, he held a Research Assis-

tantship with the Electronics Research Laboratory in the Department of Electrical

and Computer Engineering. Interests include numerical modeling of microwave and

millimeter-wave passive and active circuits, MMIC design, quasi-optical power com-

bining, and waveguide discontinuities. He is a student member of the Institute of

Electrical and Electronic Engineers and the honor society Phi Kappa Phi.

ii

Acknowledgments

This dissertation would have never been finished without the will and

blessing of God, the most gracious, the most merciful. AL HAMDU LELLAH.

I would like to express my gratitude to my advisor Dr. Michael Steer for

his support and guidance during my graduate studies. I would also like to express

my sincere appreciation to Dr. James Mink, Dr. Frank Kauffman, and Dr. Pierre

Gremaud for showing an interest in my research and serving on my Ph.D. committee

and to Dr. Amir Mortazawi for helping me with the measurements and many useful

discussions.

A very big thanks go to my colleagues, Mr. Mostafa N. Abdulla for many

useful suggestions regarding my work, Mr. Mete Ozkar for working with me on the

excitation horn, Mr. Carlos E. Christoffersen for his computer skills which came

in handy many times, Dr. Todd W. Nuteson for his encouragement while starting

my PhD. degree, Mr. Satoshi Nakazawa for sharing the same cubical, Dr. Hector

Gutierrez for many useful advice, Mr. Usman Mughal, Mr. Rizwan Bashirullah,

Mr. Adam Martin, Mr. Chris W. Hicks, and Dr. Huan-sheng Hwang.

Also, I would like to thank my professors and colleagues at Cairo Uni-

versity, Egypt, for the part they played in my academic career. They are truly

outstanding.

And finally, I wish to thank my wife and two sons Omar and Ali for

their support, understanding and encouragement and my parents whom without

their total love, guidance, and dedication I would not have made it this far.

iii

Contents

List of Figures viii

1 Introduction 1

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

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

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

1.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Literature Review 13

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Waveguide Power Combiners . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Numerical Modeling and CAD . . . . . . . . . . . . . . . . . . . . . . 22

3 Modeling Using GSM 26

iv

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 GSM-MoM With Ports . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Electric Current Interface . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Mode to mode scattering . . . . . . . . . . . . . . . . . . . . . 34

3.3.2 Mode to port scattering . . . . . . . . . . . . . . . . . . . . . 39

3.3.3 Port to port scattering . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Magnetic Current Interface . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.1 Mode to mode scattering . . . . . . . . . . . . . . . . . . . . . 41

3.4.2 Mode to port scattering . . . . . . . . . . . . . . . . . . . . . 47

3.4.3 Port to port scattering . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Dielectric and Conductor Interfaces . . . . . . . . . . . . . . . . . . . 48

3.6 Cascade Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.7 Program Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7.1 Geometry-layout and input file . . . . . . . . . . . . . . . . . 54

3.7.2 Electromagnetic simulator . . . . . . . . . . . . . . . . . . . . 55

4 MoM Element Calculation 58

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.1 Uniform discretization . . . . . . . . . . . . . . . . . . . . . . 59

v

4.1.2 Nonuniform discretization . . . . . . . . . . . . . . . . . . . . 61

4.2 Acceleration of MoM Matrix Elements . . . . . . . . . . . . . . . . . 65

4.2.1 Acceleration of impedance matrix elements . . . . . . . . . . . 66

4.2.2 Acceleration of admittance matrix elements . . . . . . . . . . 71

5 Local Reference Nodes 75

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Nodal-Based Circuit Simulation . . . . . . . . . . . . . . . . . . . . . 77

5.3 Spatially Distributed Circuits . . . . . . . . . . . . . . . . . . . . . . 78

5.3.1 Port representation . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.2 Port to local-node representation . . . . . . . . . . . . . . . . 83

5.4 Representation of Nodally Defined Circuits . . . . . . . . . . . . . . . 84

5.5 Augmented Admittance Matrix . . . . . . . . . . . . . . . . . . . . . 85

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 Results 88

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2 Analysis of General Structures . . . . . . . . . . . . . . . . . . . . . . 89

6.2.1 Wide resonant strip . . . . . . . . . . . . . . . . . . . . . . . . 89

vi

6.2.2 Resonant patch array . . . . . . . . . . . . . . . . . . . . . . . 91

6.2.3 Strip-slot transition module . . . . . . . . . . . . . . . . . . . 93

6.2.4 Shielded dipole antenna . . . . . . . . . . . . . . . . . . . . . 96

6.2.5 Shielded microstrip filter . . . . . . . . . . . . . . . . . . . . . 98

6.3 Patch-Slot-Patch Array . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3.1 Array simulation . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3.2 Horn simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.4 CPW Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4.1 Folded slot antenna . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4.2 Five slot antenna . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4.3 Slot antenna array . . . . . . . . . . . . . . . . . . . . . . . . 116

6.5 Grid Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.6 Cavity Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.6.1 Single dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.6.2 A 3× 1 dipole antenna array . . . . . . . . . . . . . . . . . . 126

7 Conclusions and Future Research 128

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

vii

7.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

References 132

A Usage of GSM-MoM Code 140

A.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.1.1 Input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.1.2 Geometry file . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.1.3 Output file . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A.2 Makefile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

A.3 Program Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

viii

List of Figures

1.1 Power capacities of microwave and millimeter-wave devices: solid line,

tube devices; dashed line, solid state devices. After Sleger et al. . . . 3

1.2 Spatial power combiners: (a) grid power combiner, (b) cavity oscillator. 4

1.3 Waveguide-based power combining showing active arrays, feeding and

receiving horns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Typical unit cells: (a) CPW unit cell, (b) grid unit cell. . . . . . . . . 7

2.1 Multiple-level combiner. . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 A quasi-optical power combiner configuration for an open resonator. 15

2.3 A spatial grid oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 A spatial grid amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Dielectric slab beam waveguide with lenses. . . . . . . . . . . . . . . 17

2.6 Kurokawa waveguide combiner. . . . . . . . . . . . . . . . . . . . . . 19

2.7 Overmoded-waveguide oscillator with Gunn diodes. . . . . . . . . . . 20

ix

2.8 Slotted waveguide spatial combiner. . . . . . . . . . . . . . . . . . . . 21

2.9 Waveguide spatial combiner. . . . . . . . . . . . . . . . . . . . . . . . 21

2.10 Rockwell’s waveguide spatial power combiner: (a) schematic of the

array unit cell, (b) rectangular waveguide test fixture. . . . . . . . . . 22

2.11 Quasi-optical lens system configuration with a centered amplifier/oscillator

array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 A multilayer structure in metal waveguide showing cascaded blocks . 29

3.2 Definition of electric and magnetic layers. . . . . . . . . . . . . . . . . 30

3.3 Geometry of the j th electric layer. The four vertical walls are metal. 33

3.4 Geometry of x directed basis functions. . . . . . . . . . . . . . . . . . 36

3.5 Geometry of the j th magnetic layer. The four vertical walls are metal. 42

3.6 Cross section of a slot in a waveguide : (a) slot in a conducting plane,

(b) equivalent magnetic currents. . . . . . . . . . . . . . . . . . . . . 43

3.7 Block diagram for cascading building blocks. . . . . . . . . . . . . . . 50

3.8 Rectangular patch showing x and y directed currents. . . . . . . . . . 55

3.9 A flow chart for cascading multilayers. . . . . . . . . . . . . . . . . . 57

4.1 Geometry of uniform basis functions in the x and y directions. . . . . 59

4.2 Geometry of nonuniform basis functions in the x and y directions. . . 62

x

4.3 Integral of K0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Convergence of Zxx matrix elements. . . . . . . . . . . . . . . . . . . 72

4.5 Percentage error in the convergence of Zxx matrix elements. . . . . . 72

5.1 Nodal circuits: (a) general nodal circuit definition (b) conventional

global reference node; and (c) local reference node proposed here. . . 78

5.2 Port defined system connected to nodal defined circuit. . . . . . . . . 80

5.3 Grid array showing locally referenced groups. . . . . . . . . . . . . . . 81

6.1 Wide resonant strip in waveguide, a = 1.016 cm, b = 2.286 cm,

w = 0.7112 cm, ` = 0.9271 cm, yc = b/2. . . . . . . . . . . . . . . . . 90

6.2 Normalized susceptance of a wide resonant strip in waveguide. . . . . 90

6.3 Geometry of patch array supported by dielectric slab in a rectangular

waveguide: a = 1.0287 cm, b = 2.286 cm, ` = 2.5 cm, εr = 2.33, d =

0.4572 cm, c = 0.3429 cm, τx = 0.1143 cm, τy = 0.2286 cm. . . . . . . 91

6.4 Magnitude of S11 and S21 for the patch array embedded in a waveguide. 92

6.5 Phase of S11 and S21 for the patch array embedded in a waveguide. . 92

6.6 Slot-strip transition module in rectangular waveguide: a = 22.86 mm,

b = 10.16 mm, τ = 2.5 mm. . . . . . . . . . . . . . . . . . . . . . . . 93

6.7 Magnitude of S11 for the strip-slot transition module. . . . . . . . . . 94

6.8 Phase of S11 for the strip-slot transition module. . . . . . . . . . . . . 94

xi

6.9 Magnitude of S21 for the strip-slot transition module. . . . . . . . . 95

6.10 Phase of S21 for the strip-slot transition module. . . . . . . . . . . . 95

6.11 Center fed dipole antenna inside rectangular waveguide. . . . . . . . . 97

6.12 Comparison of Real and Imaginary parts of input impedance. GSM-

MoM (developed here), MoM . . . . . . . . . . . . . . . . . . . . . . . 97

6.13 Calculated input impedance for centered and off-centered positions. . 98

6.14 Geometry of a microstrip stub filter showing the triangular basis func-

tions used. Shaded basis indicate port locations. . . . . . . . . . . . . 99

6.15 Three dimensional view illustrating the layers of the stub filter. . . . 100

6.16 Port definition using half basis functions. . . . . . . . . . . . . . . . . 100

6.17 Block diagram for the GSM-MoM analysis of shielded stub filter. . . . 101

6.18 Scattering parameter S11: solid line GSM-MoM, dotted line from . . 102

6.19 Scattering parameter S21: solid line GSM-MoM; dotted line from . . . 103

6.20 Propagation constant: solid lines for air, dashed lines for dielectric. . 103

6.21 Various cascading modes showing convergence of S11. . . . . . . . . . 104

6.22 Various cascading modes showing convergence of S21. . . . . . . . . . 104

6.23 A patch-slot-patch waveguide-based spatial power combiner. . . . . . 105

6.24 Geometry of the patch-slot-patch unit cell, all dimensions are in mils. 106

6.25 Ka band to X band transition. . . . . . . . . . . . . . . . . . . . . . 107

xii

6.26 Magnitude of transmission coefficient S21. . . . . . . . . . . . . . . . 109

6.27 Angle of transmission coefficient S21. . . . . . . . . . . . . . . . . . . 109

6.28 A two by two patch-slot-patch array in metal waveguide. . . . . . . . 110

6.29 Magnitude of transmission coefficient S21. . . . . . . . . . . . . . . . 110

6.30 Geometry of the folded slot in a waveguide. . . . . . . . . . . . . . . 112

6.31 Real part of the input impedance for folded slot. . . . . . . . . . . . 113

6.32 Imaginary part of the input impedance for folded slot. . . . . . . . . . 113

6.33 Five-slot antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.34 Magnitude of input return loss for 5 folded slots. . . . . . . . . . . . 115

6.35 Phase of input return loss for 5 folded slots. . . . . . . . . . . . . . . 115

6.36 A 3 × 3 slot antenna array shielded by rectangular waveguide. . . . . 117

6.37 Real part of self impedances. . . . . . . . . . . . . . . . . . . . . . . . 118

6.38 Imaginary part of self impedances. . . . . . . . . . . . . . . . . . . . 118

6.39 Real part of self impedances. . . . . . . . . . . . . . . . . . . . . . . . 119

6.40 Imaginary part of self impedances. . . . . . . . . . . . . . . . . . . . 119

6.41 Real and imaginary parts for the mutual impedance Z5,14. . . . . . . 120

6.42 A grid array inside a metal waveguide. . . . . . . . . . . . . . . . . . 121

6.43 Magnitude of input return loss. . . . . . . . . . . . . . . . . . . . . . 122

xiii

6.44 Angle of input return loss. . . . . . . . . . . . . . . . . . . . . . . . . 122

6.45 Geometry of a dipole array cavity oscillator. . . . . . . . . . . . . . . 123

6.46 Input impedance of a dipole antenna inside a cavity. . . . . . . . . . . 124

6.47 Block diagram for the GSM-MoM analysis of cavity oscillator. . . . . 125

6.48 Dipole antenna array in a cavity. . . . . . . . . . . . . . . . . . . . . 126

6.49 Magnitude of scattering coefficients for a dipole antenna array inside

a cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.50 Phase of scattering coefficients for a dipole antenna array inside a

cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xiv

Chapter 1

Introduction

1.1 Motivation For and Objective of This Study

There is an increasing demand for efficient power sources at microwave and millimeter-

wave frequencies. These power sources are utilized in commercial and military ap-

plications such as beam steering, near vehicle detection radar, smart antenna arrays,

high-resolution radar image system, satellite cross links, and active missile seekers.

The main sources of high power at microwave and millimeter-wave frequencies are

still traveling wave tubes (TWT) and Klystrons. Although these devices are capable

of producing high power levels at high frequencies, they suffer from large size and

short life time. For these reasons, solid-state devices which have none of these prob-

lems are more appealing to use than tube devices. The power levels for both tube

devices and solid-state devices are shown in Fig. 1.1 [1]. It is obvious that a single

solid-state device (PHEMT, MESFET, IMPATT, etc..) has limited output power at

1

CHAPTER 1. INTRODUCTION 2

the frequency bands of interest with respect to TWTs. To reach comparable power

levels, many solid-state devices must be combined together.

Four basic power combining strategies are used in conjunction with solid

state technology. These are chip level combining, circuit level combining, spatial

combining, and combinations of these three [2]. For chip level power combiners,

large transistors with multiple fingers are used to produce higher output powers.

In circuit level combiners, Wilkinson power combiner has been used extensively

along with newly developed circuit ideas such as the extended resonance method [3].

The limitations imposed by the various technologies such as breakdown voltages,

lossy substrates, maximum current densities, and thermal handling capabilities set

an upper bound on achievable power levels using either chip level or circuit level

combining.

Hence, a system level approach that merges both chip and circuit level is

much desired to overcome these drawbacks. Spatial power combining systems have

recently received considerable attention [4–7] to combine power from solid-state

devices or Monolithic Microwave Integrated Circuits (MMICs) in either waveguides

[7, 8] or free space [10–13]. Two types of three-dimensional spatial power combiners

are shown in Fig. 1.2, a grid-type power amplifier and a cavity-type oscillator.

Conceptually spatial power combiners are low loss systems as power is

combined in space and hence high efficiencies should be achievable. However, the

design of such systems is much more complicated than that of circuit level combiners.

The main difficulty is the field-circuit interaction that can not be ignored as done

often in the circuit level combiners [14,15]. This interaction forces the integration of

electromagnetic and circuit analysis to accurately model the system. Transmission


0.1 1 10 100 100010

10

1

10

10

10

10

10

10

10

−2

−1

1

2

3

4

5

6

7

Klystrons

Gyrotrons

TWT’s

Free−ElectronLaser

VFET

Si BJT

MESFET

IMPATTPHEMT

Gunn

FREQUENCY (GHz)

GriddedTubes

OU

TP

UT

PO

WE

R (

W)

Figure 1.1: Power capacities of microwave and millimeter-wave devices: solid line,

tube devices; dashed line, solid state devices. After Sleger et al. [1]


ACTIVE GRID SURFACEOUTPUT POLARIZER

INPUT POLARIZER TUNING SLAB

E

E

INPUTBEAM

OUTPUTBEAM

PARTIALLYTRANSPARENTSPHERICALREFLECTOR

(a) (b)

Figure 1.2: Spatial power combiners: (a) grid power combiner, (b) cavity oscillator.

line models, unit cell approaches and equivalent lumped elements fall far short in

its analysis.

The integration of electromagnetic and circuit simulators is inevitable.

The question is at which level? Some researchers approached the problem from the

electromagnetic point of view by incorporating lumped and nonlinear models into

the Finite Difference Time Domain (FDTD) electromagnetic simulators [16], while

others solved the semiconductor equations along with the wave equation to arrive

at a unique solution that satisfies both systems of equations [17]. Although this

type of analysis takes into account almost all the physical aspects of the circuit

behavior, it is very time consuming and requires considerable computing resources

for a relatively simple circuit.

In our view, the most suitable solution is to take advantage of the readily

available powerful circuit simulation techniques and integrate it with an efficient

electromagnetic simulator. The electromagnetic simulator should produce circuit

port parameters that are converted into nodal parameters and read as a linear


circuit block by a circuit simulator. Hence any nonlinear (iterative) analysis carried

out will only include one expensive electromagnetic simulation.

Commercial circuit simulation tools such as LibraTM have an integrated

Method of Moments (MoM) electromagnetic simulator in this case called Momen-

tum. However, although efficient, MomentumTM works only for structures designed

in free space and not waveguides. Dedicated electromagnetic simulators based on

the Finite Element Method (FEM), such as the High Frequency Structure Simulator

(HFSS), produce output circuit-port files compatible with both Libra and Touch-

stone. Since it is based on the FEM method it is very general and can, in theory,

be applied to any structure. The limiting factor in its effectiveness is the neces-

sity to discretize the whole three dimensional space. This renders it impractical for

electrically large systems such as spatial power combiners.

In this dissertation, the focus is on the electromagnetic analysis of waveguide-

based spatial power combiners such as that shown in Fig. 1.3. Active arrays, po-

larizers, tuning slabs, reflectors and cooling substrates are all placed in transverse

planes inside an oversized rectangular waveguide that can accommodate many prop-

agating modes. The structure is fed using horns or step transformers. The active

arrays can be described in general as active transmitting/receiving antenna arrays.

The incident wave is detected by a receiving antenna and then amplified by an active

element (MMIC). The output of the amplifier feeds a transmitting antenna which

radiates into the waveguide. Power combining occurs when the individual signals

coalesce into a single propagating waveguide mode. Typical unit cells are shown

in Fig. 1.4. The Coplanar Waveguide (CPW) unit cell incorporates a single ended

MMIC amplifier while the grid unit cell uses a balanced differential pair amplifier.


Active Array

A

A’

TransmittingHard Horn

Receiving Hard Horn

A

B

A’

B’

B’

B

Figure 1.3: Waveguide-based power combining showing active arrays, feeding and

receiving horns.


ReceivingDipole

TransmittingDipole

DifferentialPair

(a)

FOLDED SLOTS

AMPLIFIER

(b)

Figure 1.4: Typical unit cells: (a) CPW unit cell, (b) grid unit cell.

The strategy is to develop a flexible and efficient methodology to elec-

tromagnetically model waveguide-based power combining systems and interface it

to commercial circuit simulators. For each part of the system there is an opti-

mum numerical field analysis method. For example, the feeding and receiving horns

have been efficiently analyzed using the Mode Matching technique (MM) [18, 19].

The planar active antenna arrays are best modeled using the Method of Moments.

This avoids the unnecessary discretization of the whole volume and limits the dis-

cretization to planar surfaces. To integrate the two quite different techniques, the

Generalized Scattering Matrix (GSM) with circuit ports is introduced.


1.2 Dissertation Overview

The dissertation is organized as follows:

Chapter 2 presents a review of the various spatial power combining techniques. In

this Chapter two and three dimensional spatial power combiners are also reviewed.

Various waveguide power combiners, the focus of this dissertation, are reviewed.

Experimental results as well as numerical modeling techniques are discussed.

Chapter 3 contains the theoretical developments of the MoM for electric

current and magnetic current on dielectric interfaces. The electric and magnetic

type Green’s functions are presented as well as the derivations for the GSM for

both electric and magnetic current interfaces. The GSMs are computed without

calculating the induced current as an intermediate step. Each GSM is calculated for

all modes in one step by assuming the incident field to be a summation of all modes.

The GSM also includes the device ports as an integral part of its representation.

Finally two cascading formulas are derived to cascade the individual GSMs.

Chapter 4 investigates an efficient acceleration technique to speed up

the double series summations involved in MoM matrix element computations. The

technique is based on extracting the quasistatic term and applying Kummer trans-

formation. The impedance and admittance matrix elements are derived for uniform

and nonuniform elements.

Chapter 5 presents a new circuit theory for interfacing Spatially Dis-

tributed Linear Circuits (SDLC), with no global reference node, with circuit simula-

tors. These circuit simulators use the modified nodal admittance representation in

its implementation and hence the SDLC is transformed from port representation to


nodal representation by means of the local reference node concept introduced in this

Chapter. With this development the techniques developed in the previous chapters

are made available for integrated and circuit analysis.

Chapter 6 contains the results obtained for two classes of problems. The

first one is a general class such as waveguide filters, a shielded microstrip notch filter,

and a shielded dipole antenna. The second is the waveguide spatial power combiner

class. Various examples are given such as patch-slot-patch, CPW, and grid arrays

as well as a cavity oscillator.

Chapter 7 is a summary of the work presented in this dissertation along

with conclusions and future work.

1.3 Original Contributions

The original contributions presented in this dissertation are:

• Derivation and implementation of the generalized scattering matrix with de-

vice ports for shielded electric layers. Device ports are an integral part of

the GSM and hence this permits the analysis of grid arrays and strip-like

structures containing active elements.

• Derivation and implementation of the generalized scattering matrix with de-

vice ports for shielded magnetic layers. This permits the analysis of CPW

array structures containing active elements.

• Efficient calculation of the generalized scattering matrix based on the method

of moments for interacting discontinuities in waveguides. The GSM is calcu-


lated for all interacting modes (propagating and evanescent) in one step by

considering the incident field to be a summation of waveguide modes instead

of a single mode. This eliminates the need to calculate scattering parameters

for every incident mode separately.

• Implementation of an efficient method of moments formulation for the analysis

of planar conductive and magnetic layers with uniform as well as nonuniform

meshing. The method is based on the extraction of the quasistatic part in the

Green’s function and transforming it into a fast converging series summation

utilizing the fast converging modified Bessel functions of the second kind.

• Theoretical development of a circuit theory to accommodate spatially dis-

tributed circuits allowing conventional harmonic balance and transient simu-

lators to be used. The theory is based on the local reference node concept

introduced in Chapter 4.

• The investigation of the effect of waveguide walls on antenna elements in spa-

tial power combiners. It is demonstrated that the input impedances of the

antenna elements vary considerably when placed inside shielded environment.

• Network characterization of strip-slot-strip, grid, CPW, and cavity oscillator

arrays in waveguide. The impedance matrix is calculated for all cases, includ-

ing self and mutual coupling among array elements. This demonstrates the

flexibility of the modeling technique proposed in this dissertation.


1.4 Publications

The work associated with this dissertation resulted in the following Publications:

• A. I. Khalil and M. B. Steer, “Circuit Theory for Spatially Distributed Mi-

crowave Circuits,” IEEE Transactions on Microwave Theory and Techniques,

vol. 46, No. 10, Oct. 1998, pp. 1500-1502.

• A. I. Khalil and M. B. Steer, “A Generalized Scattering Matrix Method using

the Method of Moments for Electromagnetic Analysis of Multilayered Struc-

tures in Waveguide,” IEEE Transactions on Microwave Theory and Tech-

niques, In Press.

• A. I. Khalil, A.B. Yakovlev and M. B. Steer, “Efficient Method of Moments

Formulation for the Modeling of Planar Conductive Layers in a Shielded

Guided-Wave Structure,” IEEE Transactions on Microwave Theory and Tech-

niques, Sep. 1999.

• M. B. Steer, J. F. Harvey, J. W. Mink, M. N. Abdulla, C. E. Christoffersen,

H. M. Gutierrez, P. L. Heron, C. W. Hicks, A. I. Khalil, U. A. Mughal, S.

Nakazawa, T. W. Nuteson, J. Patwardhan, S. G. Skaggs, M. A. Summers, S.

Wang, and B. Yakovlev, “Global Modeling of Spatially Distributed Microwave

and Millimeter-Wave Systems,” IEEE Transactions on Microwave Theory and

Techniques, vol. 47, June 1999, pp. 830–839.

• A. I. Khalil, A.B. Yakovlev and M. B. Steer, “Efficient MoM-Based Gener-

alized Scattering Matrix Method for the Integrated Circuit and Multilayered

Structures in Waveguide,” 1999 IEEE MTT-S International Microwave Sym-

posium Digest, June 1999.


• A. I. Khalil, M. Ozkar, A. Mortazawi and M. B. Steer, “Modeling of Waveguide-

Based Spatial Power Combining Systems,” 1999 IEEE AP-S International

Antennas and Propagations Symposium Digest, July 1999.

• A. I. Khalil, A.B. Yakovlev and M. B. Steer , “Analysis of Shielded CPW Spa-

tial Power Combiners,” 1999 IEEE AP-S International Antennas and Propa-

gations Symposium Digest, July 1999.

• A. B. Yakovlev, A. I. Khalil, C. W. Hicks, and M. B. Steer, “Electromagnetic

Modeling of a Waveguide-Based Strip-to-Slot transition Module for Applica-

tion to Spatial Power Combining Systems,” 1999 IEEE AP-S International

Antennas and Propagations Symposium Digest, July 1999.

• M. A. Summers, C. E. Christoffersen, A. I. Khalil, S.Nakazawa, T. W. Nuteson,

M. B. Steer, and J. W. Mink, “An Integrated Electromagnetic and Nonlin-

ear Circuit Simulation Environment for Spatial Power Combining Systems,”

1998 IEEE MTT-S International Microwave Symposium Digest, June 1998,

pp. 1473-1476.

• M. B. Steer, M.N.Abdulla, C.E.Christofersen, M.Summers, S. Nakazawa, A.Khalil

and J.Harvey, “Integrated Electromagnetic and Circuit Modeling of Large Mi-

crowave and MillimeterWave Structures,” Proc. of the 1998 IEEE AP-S In-

ternational Antennas and Propagations Symposium, June 1998, pp. 478-481.

• A. Yakovlev, A. Khalil, C. Hicks, A. Mortazawi, and M. Steer “The general-

ized scattering matrix of closely spaced strip and slot layers in waveguide,”

Submitted to the IEEE Transactions on Microwave Theory and Techniques.

Chapter 2

Literature Review

2.1 Background

Combining microwave and millimeter-wave power from solid state sources is an ac-

tive research area. High power at high frequency is a major goal. A single solid-state

device cannot meet this goal. The device size is inversely proportional to the operat-

ing frequency and so is its power handling capability. Hence, novel power combining

techniques that minimize loss and allow higher device-packing density are a must.

Russel [20], in an invited paper, reviews various techniques for coherently combin-

ing power from two or more sources using circuit techniques. These approaches are

separated into two general categories, N-way combiners and corporate (or chain)

combiners. Fundamental as well as practical limitations with circuit level power

combiners were discussed. It was demonstrated that the number of combined de-

vices is limited by the lossy components used in the case of the corporate or the chain

13

CHAPTER 2. LITERATURE REVIEW 14

structures. To illustrate this, Russel calculated the efficiencies for various corporate

power combiners. He varied the amount of loss introduced by the adders used in

each combining stage. For a loss of 0.5 dB in each adder, a 16-device corporate

combiner has 65% efficiency. A similar argument was made for the chain structure.

The N-way power combiner has less accumulated loss since it has only one stage.

The drawback of such a scheme is its realization and bandwidth. For example, the

Wilkinson power combiner [21] can not provide sufficient isolation between the N

ports at high frequencies when N is greater than 2.

In a broader definition, Chang [2] classified power combiners in four

classes. These are chip-level, circuit-level, spatial, and combination of all three.

Chang proposed a logical sequence of multiple-level combining with spatial power

combining being at a higher hierarchical level as illustrated in Fig. 2.1.

CHIP LEVEL

COMBINERS

NONRESONANT

COMBINERS

RESONANT

COMBINERS

SPATIAL

COMBINERS

Figure 2.1: Multiple-level combiner.

Spatial power combining gained attention in the past two decades. Early

research focused on experimental investigation of spatial combiners [22]. It was

not until 1986 when Mink presented detailed analysis of the theory of solid state

power combining through the application of quasi-optical techniques [23]. In [23],

a plano-concave open resonator as that shown in Fig. 2.2 was investigated. An

array of sources on a planar reflecting surface was studied by modeling it as current

filaments. The driving point resistance of each source in the presence of all other


excited sources was calculated. Mink showed that efficient power transfer, between

the array and the wave beam, was obtainable with appropriate spacing between

elements.

Dd

ax

ya

za

SOURCEARRAY

PLANARREFLECTOR

PARTIALLYTRANSPARENTSPHERICALREFLECTOR

Figure 2.2: A quasi-optical power combiner configuration for an open resonator.

Other structures used for spatial combining include grid arrays as os-

cillators or amplifiers [24, 25] are shown in Figs. 2.3 and 2.4. The mirror used in

the oscillator grid provided the required feedback. The amplifier grid used two or-

thogonal polarizations at the input and output for isolation. A polarizer at both

the input and output were used for that purpose. The active grid used could be

populated with either two or three terminal solid state devices. Patch arrays had

been used in spatial power combiners for higher efficiencies and better input/output

isolation [26–30]. Slot antennas had also been introduced for spatial combining [31].

Microwave and millimeter power sources will be utilized in, for example,

active missiles. This renders spatial power combining systems operating in free

space impractical since they occupy large area, difficult to align and are not properly

shielded. For these reasons two dimensional (2D) versions of spatial power combiners

as well as waveguide power combiners are under investigation.


TUNING SLAB

E

OUTPUTBEAM

ACTIVE GRID SURFACE

qMIRROR

Figure 2.3: A spatial grid oscillator.

ACTIVE GRID SURFACEOUTPUT POLARIZER

INPUT POLARIZER TUNING SLAB

E

E

INPUTBEAM

OUTPUTBEAM

Figure 2.4: A spatial grid amplifier.


An important part of the 2D structure is the dielectric slab-beam waveg-

uide presented in [32], shown in Fig. 2.5. In the dielectric waveguide, two waveg-

uiding principals are used. Guided fields in the normal direction to the slab are

considered trapped surface waves and largely confined to the dielectric. In the lat-

eral direction the field has a Gaussian distribution and is guided by the lenses to form

a wavebeam that is iterated with the lens spacing. A second paper implemented the

ground plane

ε lens εslab>

ground plane

dielectric slab εslab

ε lens εslab<

ε lens phase transformers

z

z

w

s

y

x

x

y

d

Figure 2.5: Dielectric slab beam waveguide with lenses.

dielectric slab-beam waveguide concept to build, for the first time, a four MESFET

amplifier employing quasi-optical techniques [33]. The active antennas used were

Vivaldi-type broadband antennas which are gate-receiver and drain-radiators. Three

different configurations of the active antennas were studied. In all configurations,

the antennas were coupled at the beam waist of the transverse electric (TE)-type

slab mode to provide power combining. The maximum power gain for the ampli-


fier array was 13 dB at 7.5 GHz. A transverse magnetic (TM)-type dielectric slab

with Yagi-Uda slot antennas was introduced in [34]. An amplifier array of 10 GaAs

MMICs was fabricated. The array gain was 11 dB at 8.25 GHz and a 0.65 GHz

3-dB bandwidth was measured.

2.2 Waveguide Power Combiners

Resonant-cavity combiners have been successfully used in oscillator design. The

first design was proposed by Kurokawa and Magalhaes in their 1971 paper [35]. A

12 diode power combiner at X-band was proposed. Each diode was mounted at

one end of a stabilized coaxial line which was coupled to the magnetic field at the

sidewall of a waveguide cavity. The coaxial circuits were located at the magnetic

field maxima and hence spaced one-half wavelength apart as shown in Fig. 2.6. The

oscillator operated at 9.1 GHz and produced 10.5 Watts. The circuit configuration

was stable and the oscillation theory was developed by Kurokawa in a following

paper [36]. Another cavity combining technique was introduced utilizing more solid

state diodes placed in a circle inside a cylindrical cavity [37]. With this technique,

no minimum spacing (half wavelength in Kurokawa’s model) was required. In an

effort to increase the number of active devices used in Kurokawa’s model, Hamilton

modified the design to accommodate twice the number of diodes [38]. This was

achieved by placing two coaxial lines on either side of the magnetic field maxima.

Using electric field as well as magnetic field coupling, Madihian was able to increase

the number of active devices per half guide wavelength from 2 to 3 in a cavity [39].

More recent results were obtained for spatial power combining using


COAXIALLINE

g

2

g

4

MAGNETICFIELD

SHORTCIRCUIT

Figure 2.6: Kurokawa waveguide combiner.

overmoded waveguide resonator [8,9]. An array (N ×M) of TE10-mode waveguides

containing Gunn diodes was used as the active oscillator array. The resonator con-

sisted of an overmoded rectangular waveguide with sliding short circuit for tuning

as shown in Fig 2.7. The (N ×M) TE10-mode waveguides coupled energy into the

TEN0-mode in the overmoded waveguide through the horn couplers with conversion

efficiency of 100%. All other modes in the resonator were suppressed because of

the perfect field distribution match between the horn arrays and the overmoded

waveguide. A 3× 3 array was built and tested. The overall efficiency at 61.4 GHz

was 83% and an output power of 1.5 W (CW) with a C/N ratio of −95.8 dBc/Hz

at 100kHz offset was measured.

A power amplifier array using slotted waveguide power divider/combiner

was proposed at North Carolina State University [40]. In this work, two waveguides

were used as shown in Fig. 2.8. One distributing the input signals and the other


OVERMODEDWAVEGUIDERESONATOR

SLIDINGSHORT

N xM WAVEGUIDEARRAY

GUNNDIODE

OUTPUT

Figure 2.7: Overmoded-waveguide oscillator with Gunn diodes.

combining the amplified signals. The waveguides have longitudinal slots, one-half

guide wavelength spaced, that couple to microstrip lines. An array of eight active

devices (FLM0910 2-Watt internally matched GaAs MESFETs) and a passive 8-way

divider/combiner was designed. The power combiner operated at 9.9 GHz with 6.7

dB of gain and 14 Watts of power. The advantage of such a design is its simplicity

and good heat sinking. The power devices were mounted on the metal waveguide

directly, and so a natural heat sink was provided. If transmitting patch antennas are

used, the structure can be used to combine power in free space instead of waveguide

combining.

The most significant result obtained for spatial power combining to date

has occurred at the University of California, Santa Barbara. An X-band waveguide

based amplifier has produced a CW output of 40 W peak power with 30% power

added efficiency [7]. The combiner is a 2D array of tapered slotline sections (antenna

cards). The dominant mode TE10 is received, amplified, and then retransmitted

using slotline antennas. The received signal is amplified using GaAs MMIC devices.


Each antenna card accommodates two commercial GaAs MMICs. The results were

obtained with four cards placed in a rectangular waveguide. Fig. 2.9 illustrates

the basic idea, where only two antenna cards are inserted in the waveguide. The

advantages of this system are its wide-band characteristics and reusability. If one of

the antenna elements fail, only that card needs to be replaced. This opens the door

to modular spatial power combining design. The limitation of the existing design is

the area of the waveguide cross section. It has to be small enough to accommodate

only the dominant mode.

Screws

MicrostripLines

Dielectric

Amplifiers Slots

Waveguide

Power IN Power OUT

Figure 2.8: Slotted waveguide spatial combiner.

AMP

AMP

AMPINPUTPOWER

OUTPUTPOWER

WAVEGUIDERECTANGULAR RECTANGULAR

WAVEGUIDE

ANTENNACARD

Figure 2.9: Waveguide spatial combiner.


Targeting Ka-band, a Rockwell group, designed a monolithic quasi-

optical amplifier [5]. The amplifier was packaged in a waveguide that is both compact

and suitable as a drop-in replacement for systems that are designed to use a conven-

tional waveguide tube-type amplifier. A 2D array of 112 PHEMTs was fabricated

and measured. The amplifiers coupled to individual input and output slot antennas,

with orthogonal polarizations. The array provided a peak gain of 9 dB at 38.6 GHz

and 29 dBm maximum output power. The unit cell used in the array design as well

as the waveguide package are illustrated in Fig. 2.10.

INPUT FIELD

OUTPUT

FIELD

(a) (b)

Figure 2.10: Rockwell’s waveguide spatial power combiner: (a) schematic of the

array unit cell, (b) rectangular waveguide test fixture.

2.3 Numerical Modeling and CAD

Experimental results were obtained for various spatial power combining topologies,

but the output power levels are still much smaller than expected. With the help

of a dedicated Computer-Aided Engineering (CAE) environment, it is anticipated


that better designs and higher power levels can be obtained. There are numerous

commercial Computer-Aided Design (CAD) tools in the area of microwave circuits

and antennas. These CAD tools can not simply be combined to analyze spatial

power combiners [41]. The reason is the complex nature of spatial combining sys-

tems. In such systems, many different components are integrated together such as

active devices (diodes, transistors, MMICs, etc.), passive lumped and/or distributed

elements, radiating elements, and cooling elements. In this section we will review

the efforts in developing CAD tools for spatial power combiners.

Many of the approaches developed to model spatial power combiners

assume infinite arrays in free space. With these assumptions, the analysis is greatly

simplified. Using a simple equivalent circuit approach, Popovic et al. separated the

equivalent circuit of the grid from that of the active circuitry [10]. In this analysis

only a unit cell was considered assuming an infinite periodic array. Further more,

electric and magnetic walls were used restricting the input to only incident TEM

wave [42].

Still assuming an infinite array, Epp et al. of JPL, presented a novel

approach to model quasi-optical grids [43, 44]. They decomposed the incident and

scattered fields into a summation of Floquet modes. The modes interacted with the

device ports. To account for all interactions, they characterized the unit cell using

a generalized scattering matrix method with device ports. This allowed a general

representation of the incident field. Also, the interaction of various quasioptical

components such as polarizers, lenses, and feeding horns can be used if their appro-

priate GSMs are computed. In implementation the surface currents were calculated

using a spectral domain method of moments formulation.


To accurately model spatial power combiners of small to moderate sizes,

the unit cell approach is neither practical nor accurate. The edge effects will not

be modeled and consequently the driving point impedances for all unit cells will

not be the same as predicted by the unit cell approach. A full wave analysis of the

whole structure is thus essential. Pioneering work was done here at North Carolina

State University to model open cavity resonators and grid arrays [45–51]. Heron [46]

developed a Green’s function for the open cavity resonator. This Green’s function is

composed of two parts: resonant and nonresonant terms. The fields were represented

using Hermite Gaussian wave-beams. Nuteson [51], implemented the previously

developed Green’s function using the method of moments. He also developed a

dyadic Green’s function for a lens system consisting of two lenses and an array of

active devices as that shown in Fig. 2.11. The Green’s function was derived by

separately considering the paraxial and nonparaxial fields. A combination of spatial

and spectral domain techniques was used in computing the method of moments

matrix elements.

z=−D z=0 z=D

a x

ay

az

LensTransmitting

Horn

Lens

Amplifier Oscillator/Array

Receiving

Horn

Figure 2.11: Quasi-optical lens system configuration with a centered ampli-

fier/oscillator array.


An integrated electromagnetic and nonlinear circuit simulation environ-

ment for spatial power combining systems was proposed in [52]. This represented the

first result where a full wave analysis and nonlinear circuit simulation were carried

out in the analysis of finite size grid arrays. A 2× 2 grid array was fabricated using

differential pair transistor units. A harmonic balance, nonlinear circuit simulator,

was used to simulate the system’s nonlinear behavior.

Three-dimensional electromagnetic analysis was also applied to spatial

power combiners [53] and active antennas [54]. The main disadvantages of such

techniques are their large memory demand as well as computational resources. The

power of such methods reside in its flexibility to model complicated structures.

However, spatial power combiners are planar in shape and that enables the MoM

to be applied efficiently in the analysis.

Chapter 3

Modeling Using GSM

3.1 Introduction

A typical waveguide-based spatial power-combining system of the transverse type,

such as that shown in Fig. 1.3, consists of passive components (horns, polarizers,

lenses, etc.) and active components (grid arrays, patch arrays, etc.). Electromag-

netic modeling of such systems can be memory demanding and very time consuming.

The main reason for this is their electrically large sizes. The most efficient and flex-

ible way to model these systems is to partition them into blocks. Each block is

modeled separately and characterized by its own GSM [55, 56]. This ensures that

propagating mode coupling is accounted for as well as evanescent mode coupling.

Also, since each block is considered separately, the GSMs are computed using the

most efficient EM technique for each particular block (eg. MoM, mode matching,

and FEM). Cascading all blocks leads to a matrix describing the entire linear sys-

26

CHAPTER 3. MODELING USING GSM 27

tem response. Since the feeding horns have been analyzed elsewhere [18], we will

focus on the body of the waveguide spatial power combiner, such as polarizers, grid

arrays, patch arrays, CPW arrays, etc..

The GSM is derived for the fundamental building blocks. These are the

electric current interface, the magnetic current interface, the dielectric interface and

the short circuit. With these fundamental blocks, almost all multilayer transverse

active arrays can be modeled. Two different formulas are derived to cascade the

GSMs of individual blocks.

3.2 GSM-MoM With Ports

The Generalized Scattering Matrix (GSM) method has been widely used to charac-

terize waveguide junctions and discontinuities. The GSM is a matrix of coefficients of

forward and backward traveling modes and describes all self and mutual interactions

of scattering characteristics, including contributions from propagating and evanes-

cent modes. Thus structures of multiple discontinuities are modeled by cascading

a number of GSMs. The GSM is adapted here to globally model waveguide-based

spatial power combining systems. In such systems a large number of active cells

radiate signals into a waveguide, and power is combined when the individual signals

coalesce into a single propagating waveguide mode. Most spatial power combiners

can be viewed as multiple arbitrarily layers of electric or magnetic currents arranged

in planes transverse to the longitudinal direction of a metal waveguide. Active de-

vices are inserted at ports in some of the metalized or magnetic transverse planes. In

this Chapter we efficiently derive the GSM and introduce circuit ports (ports with


voltages and currents) into the GSM formulation. This facilitates the incorporation

of the electromagnetic model of a microwave structure into a nonlinear microwave

circuit simulator as required in computer aided global modeling.

The problem of modeling multilayered structures with ports in a shielded

environment can be analyzed by at least two approaches. In the first, a specific

Green’s function for the proposed structure is constructed and then the method of

moments (MoM) [57] is directly applied to the entire structure. This results in severe

computational and memory demands for electrically large structures. The second

approach, proposed here, is to characterize each layer using a GSM with circuit

ports and then cascade this matrix with its neighbors to obtain the composite GSM

of the complete system such as that shown in Fig. 3.1.

Various formulations have been used in developing the standard GSM

(without circuit ports). The mode matching technique is the most widely used for

waveguide junctions and discontinuities of relatively simple geometries [58]. The

FDTD has been introduced to calculate the GSM for complex waveguide circuits

[59]. The MoM has also been used in developing the GSM of arbitrarily shaped

dielectric discontinuities [60], metallic posts [61, 62], waveguide junctions [63, 64],

and waveguide problems with probe excitation [65]. In its common implementation,

the MoM uses subdomain basis functions of current. This implementation is used

here to compute a port impedance matrix in the solution process [66]. As well as

using subdomain current basis functions on the metalization, the MoM formulation

implemented here uses delta gap voltages and so the MoM characterization yields

port voltage and current variables. The ports are explicitly defined in the GSM

and they are accessible after cascading. The method can address a wide class of

problems such as a variety of shielded multilayered structures, iris coupled filters,


input impedance for probe excited waveguides, and waveguide-based spatial power

combiners. From this point on we will refer to a circuit port as just a port, and

an electromagnetic port, which are defined for incident and scattered modes, as a

mode.

DIELECTRICINTERFACE

ELECTRICINTERFACE

CONDUCTOR

MAGNETICINTERFACE

INTERFACE

Figure 3.1: A multilayer structure in metal waveguide showing cascaded blocks

The key concept in the method developed here is formulation of a GSM

for one transverse layer at a time, and the GSM of individual blocks are cascaded

to model a multilayer structure. The general building blocks considered here are

• Electric current interface with ports

• Magnetic current interface with ports

• Dielectric interface


• Perfect conductor interface

An electric current interface is defined as an interface where the con-

ducting portions are small with respect to the dielectric portion. Hence, it is more

efficient to analyze the conducting (electric) than the nonconducting portion. Sim-

ilarly, a magnetic current interface is defined as an interface where the dielectric

portions are small with respect to the conducting portion. Hence, it is more effi-

cient to analyze the dielectric (magnetic) than the conducting portion. The electric

and magnetic current interfaces are shown in Fig. 3.2.

E , H

E , H

E , H

1

S S

1

11

I I

2 2

S S

Electric Layer

E , H

E , H

E , H

1

S S

1

11

I I

2 2

S S

Magnetic Layer

Figure 3.2: Definition of electric and magnetic layers.

The electric current interface with ports is used in the analysis of mi-

crostrip, grid and stripline structures, while the magnetic current interface with

ports is used for CPW structures.

The analysis begins by expressing the phasors of the electric and mag-


netic field vectors in terms of their eigenmode expansions [67]

E+

(x, y, z) =∞∑l=1

alE+l (x, y, z) (3.1)

H+

(x, y, z) =∞∑l=1

alH+l (x, y, z) (3.2)

E−

(x, y, z) =∞∑l=1

blE−l (x, y, z) (3.3)

H−

(x, y, z) =∞∑l=1

blH−l (x, y, z) (3.4)

where the individual electric and magnetic eigenmodes are

E±l (x, y, z) = (el ± ezl) exp(∓Γlz)

H±l (x, y, z) = (±hl + hzl) exp(∓Γlz)

The propagation constant of the l th mode is defined as:

Γl =

√k2cl − k2

j , kj < kcl

j√k2j − k2

cl , kcl < kj

(3.5)

with kcl =√k2xl + k2

yl, kj = ω√µ0ε0εj , kxl = mπ/a and kyl = nπ/b. For simplicity

the index pair (m,n) has been replaced by a single index l. Note that all Transverse

Electric (TE) and Transverse Magnetic (TM) waveguide modes are considered. The

amplitude coefficients of mode l are denoted as al and bl for waves propagating in the

positive and negative z directions, respectively. The “±” sign indicates propagation

in the positive and negative z directions, respectively. The electric and magnetic

mode functions, el and hl, are normalized using the normalization condition

∫ ∫A

[el × hl] · (z) ds = 1

resulting in the following expressions for TE and TM modes:


TE-modes:

ex = C ky√Zh cos(kxx) sin(kyy),

ey = − C kx√Zh sin(kxx) cos(kyy),

hx = Ckx√Zh

sin(kxx) cos(kyy),

hy = Cky√Zh

cos(kxx) sin(kyy) (3.6)

TM-modes:

ex = − C kx√Ze cos(kxx) sin(kyy),

ey = − C ky√Ze sin(kxx) cos(kyy),

hx = − Cky√Ze

sin(kxx) cos(kyy),

hy = Ckx√Ze

cos(kxx) sin(kyy) (3.7)

where

C =1√

k2x + k2

y

√ε0mε0nab

, Zh =jωµ0

Γ, Ze =

Γ

jωε0ε,

and A is the waveguide cross section

3.3 Electric Current Interface

The concept behind the procedure that follows is that distinct waveguide modes are

coupled by irregular distributions of conductors at the dielectric/dielectric interface.

The regions at the interface that are not metalized do not couple modes. The

characterization of the metalized interface is developed by separately considering

mode to mode, port to port, and port to mode interactions [44].


The general building block is shown in Fig. 3.3. Here an arbitrarily

shaped metalization is located at the interface of two dielectric media with relative

permitivities εj and ε(j+1), respectively. For illustration purposes an internal port

is specified to show the location of a device and an excitation port is defined in

connection with the source or load although the number of circuit ports is arbitrary.

The vector of coefficients [aj1] represents the coefficients of modes incident from

medium j into medium j + 1, [aj2] represent the coefficients of modes incident from

medium j + 1 into medium j, and [aj3] represents all coefficients of power waves

incident from the circuit ports. Similarly [bji ] are the vectors of reflected mode or

power wave coefficients corresponding to [aji ] , i = 1, 2, 3. The relations between [bji ]

and [aji ] will be determined in this section and the matrix relationship is the GSM.

LOADPORT

EXCITATIONPORT

a1

(j)b

1

(j)

a2(j) b

2

(j)

a3

(j)[ ] b

3

(j)[ ]

a

b

ε

Y

X

Z

ε

j+1

j

DEVICE PORT

[ [ ]]

[ [ ]]

Figure 3.3: Geometry of the j th electric layer. The four vertical walls are metal.


3.3.1 Mode to mode scattering

In this section, only the layer at the interface of the dielectric media is considered

and the matrix model developed relates the variables at the ports to the coefficients

of the modes (in each dielectric medium) that are incident and reflected at that

layer. First, the MoM is applied to the problem and then the GSM is calculated.

The electric field integral equation formulation is obtained by enforcing the following

impedance boundary condition on the metal surface:

Ei(r) + E

s(r) = ZsJ(r) (3.8)

where Ei

denotes the tangential incident field, Es

the tangential scattered field, Zs

the surface impedance, and J is the unknown surface current density. Later on,

the surface impedance will be used to represent the lumped load impedances of the

ports. The first step in the MoM formulation is to express the scattered field in

terms of the electric dyadic Green’s function (Ge):

Es(r) =

∫ ∫S′Ge(r, r

′) · J(r

′) ds

′(3.9)

Here primed coordinates denote the source location while unprimed coordinates

denote the observation location.

In general the electric dyadic Green’s function has nine components Gije ,

where i, j represent the Cartesian coordinates x,y, and z [68]. In our case we are only

concerned by the transverse components which limit the required dyadic Green’s

function to four components.

Gxxe (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕxe,mn(x, y)ϕxe,mn(x′, y′)fe,mn,

Gxye (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕxe,mn(x, y)ϕye,mn(x′, y′)he,mn,


Gyxe (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕye,mn(x, y)ϕxe,mn(x′, y′)he,mn,

Gyye (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕye,mn(x, y)ϕye,mn(x′, y′)ge,mn . (3.10)

The functions ϕxe,mn(x, y) and ϕye,mn(x, y) represent a complete set of orthonormal

eigenfunctions satisfying appropriate boundary conditions on the surface of the

metal waveguide:

ϕxe,mn(x, y) =

√ε0mε0nab

cos(kxx) sin(kyy) ,

ϕye,mn(x, y) =

√ε0mε0nab

sin(kxx) cos(kyy) (3.11)

with ε0m, ε0n being Newman indexes such that, ε00 = 1, and ε0m = 2, m 6= 0. Fi-

nally, the one-dimensional Green’s functions fe,mn(z, z′), he,mn(z, z′), and ge,mn(z, z′)

are calculated on the interface at z = z′ = 0:

fe,mn = −j (k21 − k2

x)Γ2 + (k22 − k2

x)Γ1

(Γ1 + Γ2)ω(Γ2ε1 + Γ1ε2),

he,mn = jkxky

ω(Γ2ε1 + Γ1ε2),

ge,mn = −j(k2

1 − k2y)Γ2 + (k2

2 − k2y)Γ1

(Γ1 + Γ2)ω(Γ2ε1 + Γ1ε2)(3.12)

In solving for the scattered electric field Es(r), the surface current density J(r

′) is

expanded as a set of subdomain two-dimensional basis functions:

J(r′) =

N∑i=1

IiBi(r′) (3.13)

where Bi is the i th basis function and Ii is the unknown current amplitude at

the i th basis. Each basis corresponds to one of N ports. Typical sinusoidal basis

functions in the x direction are shown in Fig. 3.4.

Using the current expansion formula (3.13) and the integral representa-

tion for the scattered electric field (3.9), the impedance boundary condition (3.8) is


y

x xxy

y

y

p pp

p

p

p

-c +c

-d/2

+d/2

xx +cp+1

Figure 3.4: Geometry of x directed basis functions.

written in terms of the Green’s function as

Ei(r) = −

N∑i=1

Ii

∫ ∫S′Ge(r, r

′) ·Bi(r

′) ds

′+ ZsJ(r) (3.14)

A Galerkin procedure yields the discretization of the integral equation (3.14):

∫ ∫SBj(r).E

i(r)ds =

−N∑i=1

Ii

∫ ∫S

∫ ∫S′Bj(r).Ge(r, r

′).Bi(r

′)ds

′ds

+N∑j=1

Ij

∫ ∫SZs(r)Bj(r).Bj(r)ds (3.15)

leading to a matrix system for the unknown current coefficients I = [I1 · · · Ii · · · IN ]T :

[Z + ZL][I ] = [V ] (3.16)

where the ji th element of the impedance matrix [Z] is

Zji = −∫ ∫

S

∫ ∫S′Bj(r).Ge(r, r

′) ·Bi(r

′)ds

′ds (3.17)

the j th port voltage

Vj =∫ ∫

SBj(r).E

i(r)ds (3.18)


and the load impedance

[ZL] =

ZL1 · · · 0 · · · 0

.... . .

.... . .

...

0 · · · ZLi · · · 0

.... . .

.... . .

...

0 · · · 0 · · · ZLN

, (3.19)

with ZLi being the loading impedance at port i. If port i is not loaded then its

corresponding entry is zero [69].

Conventionally, the GSM is constructed one column at a time. This is

achieved by exciting the structure by a single mode. The excitation mode generates

reflected and transmitted modes. The computed coefficients of these modes fill a

single column in the GSM. This filling process continues until the whole matrix is

completely filled. It is obvious that for a large GSM the conventional approach is

very time consuming.

In order to construct the GSM efficiently, it is essential to treat the

incident field as being composed of a summation of waveguide modes rather than

considering a single mode one at a time [70]. For an incident field propagating in

the positive z direction from medium 1 into medium 2 at the interface

Ei(r) =

Lmax∑l=1

a1l (1 +Rl) e

1l exp(−Γ1

l z) (3.20)

where Γ1l , e

1l are the propagation constant and the electric mode function of mode

l corresponding to medium 1, respectively. Rl is the reflection coefficient of mode l,

defined so that the transverse electric and magnetic mode reflection coefficients are

RTEl =

Γ1l − Γ2

l

Γ1l + Γ2

l

(3.21)


RTMl =

Γ2l ε1 − Γ1

l ε2Γ2l ε1 + Γ1

l ε2(3.22)

The incident electric field defined in (3.20) consists of two parts, incident and re-

flected waves. This is important to account for the dielectric discontinuity at the

interface. Using this expression for the incident field, the port voltages of an electric

current layer located at z = 0 is given by

Vj =Lmax∑l=1

a1l (1 +Rl)

∫ ∫Se1l .Bj(r)ds (3.23)

Hence the matrix form, (3.16), can be written as

[Z + ZL][I ] = [W 1][U +R][a11] (3.24)

Where the current vector [I ] is written in terms of the modal vector

[a11] = [a1

1 · · · a1l · · · a1

Lmax]T as

[I ] = [Y ][W 1][U +R][a11] (3.25)

the admittance matrix

[Y ] = [Z + ZL]−1

the elements of the [W q] matrix is given by

W qji =

∫ ∫Seqi . Bj ds

U is the identity matrix and R is a diagonal matrix with diagonal elements being

the modal reflection coefficients. Scattering from both the metalization and the

dielectric interface leads to scattered fields with mode coefficients

b1l = −(1 +Rl)

2

∫ ∫SJ.E

+l ds+Rla

1l , l = 1..Lmax (3.26)

Using the current density expansion (3.13) the coefficients of the scattered modes

[b11] can be written as

[b11] = −1

2[U +R][W 1]T [I ] + [R][a1

1] (3.27)


where T indicates the transpose matrix operation. Substituting the expression for

the electric current (3.25) into (3.27) results in the following representation:

[b11] = (−1

2[U +R][W 1]T [Y ][W 1][U +R] + [R])[a1

1] (3.28)

Since [b11] = [S1

11][a11], we can readily write

[S111] = −1

2[U +R][W 1]T [Y ][W 1][U +R] + [R] (3.29)

and

[S121] = −1

2[C][W 1]T [Y ][W 1][U +R] + [C] (3.30)

where [C] is a diagonal matrix representing the transmission coefficients.

The obtained expressions (3.20) to (3.30) is for an incident field traveling

in the positive z direction from layer 1 into layer 2. By symmetry, when the incident

field is propagating in the negative z direction from layer 2 into layer 1, we can write

[S122] = −1

2[U −R][W 2]T [Y ][W 2][U −R]− [R] (3.31)

[S112] = −1

2[C][W 2]T [Y ][W 2][U −R] + [C] (3.32)

Equations (3.29)-(3.32) are a full representation of scattered modes due to incident

modes on a loaded scatterer residing on the interface of two adjacent dielectrics

inside a metal waveguide.

3.3.2 Mode to port scattering

The interaction between an incident mode and a port can be described using the

concept of generalized power waves [44]. First assume that port k is terminated by

an arbitrary impedance ZLk. Since the scattering parameters are normally given


with reference to a 50 Ω system it is appropriate to set ZLk to R0 = 50 Ω. The

generalized power waves at the ports are then given by [71]

Vi =1

2(Vk +R0Ik) (3.33)

Vr =1

2(Vk −R0Ik) (3.34)

ak =Vi√R0

(3.35)

bk =Vr√R0

(3.36)

Where Vi and Vr are the incident and reflected voltage waves. When there is no

excitation at port k, Vi = 0 and Vr = −R0Ik. Hence the scattered power wave

coefficient at port k due to mode excitation is bk = −√R0Ik. Thus the scattering

coefficients at the ports due to incident modes from medium 1 can be written in a

matrix form as

[b13] = −[R0]

12 [I ] (3.37)

Substituting for the current using (3.25) and recalling that [b13] = [S1

31][a11] the scat-

tering submatrix

[S131] = −[R0]

12 [Y ][W 1][U +R] (3.38)

Similarly, the scattering coefficients at the ports due to incident modes from medium

2 can be written as

[S132] = −[R0]

12 [Y ][W 2][U −R] (3.39)

By reciprocity the scattering matrix of modes due to port excitation is readily ob-

tained as [S113] = [S1

31]T and [S1

23] = [S132]

T .


3.3.3 Port to port scattering

Port quantities are related by a scattering matrix which relates port to port scat-

tering [71]:

[S133] = [R0]

12 [Zp +R0]−1[Zp −R0][R0]

−12 (3.40)

where Zp is the port impedance matrix, obtained by selecting the appropriate rows

and columns from the MoM impedance matrix [Z].

3.4 Magnetic Current Interface

A similar analysis to the electric current interface is carried out for the magnetic

current interface in this section. Distinct waveguide modes are coupled by irregular

distributions of magnetic current at the dielectric/dielectric interface. The charac-

terization of the magnetic interface is developed by separately considering mode to

mode, port to port, and port to mode interactions.

The general building block for the magnetic interface is shown in Fig.

3.5. Here a CPW structure is located at the interface of two dielectric media with

relative permitivities εj and ε(j+1), respectively. A three terminal device is explicitly

drawn to illustrate the location of the device ports.

3.4.1 Mode to mode scattering

In this section the mode to mode coupling for a magnetic layer is calculated. Let

us consider an aperture in a conducting plane transverse to the direction of propa-


a1

(j)b

1

(j)

a2(j) b

2

(j)

a

b

ε

Y

X

Z

ε

j+1

j

[ [ ]]

[ [ ]]

εj+1

Figure 3.5: Geometry of the j th magnetic layer. The four vertical walls are metal.

gation at the interface of two adjacent dielectrics with relative permitivities ε1 and

ε2 as shown in Fig. 3.6. The equivalence principal is applied to obtain separate

representation for the field in region 1 (Z < 0) and region 2 (Z > 0) [72,73] by short

circuiting the aperture (covering the aperture by an electric conductor).

Assuming a propagating wave Hi

is incident from region 1 into region

2. The field in region 1 is determined by the incident field and the equivalent

magnetic current M over the aperture area produced by the tangential electric field

on the aperture Et. The field in region 2 is determined only by the equivalent

magnetic current −M only. The equivalent magnetic currents M and −M ensure

the continuity of the tangential components of the electric field across the aperture.

M = az × Et|z=0 (3.41)

region 1 (z = 0−)

H1

t = Hi

t +H1

t (M) (3.42)


region 2 (z = 0+)

H2t = H

2t (−M) = −H2

t (M) (3.43)

Where Hjt is the total tangential component of the magnetic field on the aperture

in region j, j = 1, 2. Hjt (M) is the tangential component of the magnetic field on

the aperture due to the magnetic current M in region j. Equating the tangential

magnetic fields on both sides of the aperture given by (3.42) and (3.43):

−Hit = H1

t (M) +H2t (M) (3.44)

It should be noted that due to the presence of the perfect conductor the incident

magnetic field is doubled. Also, the image theory can be applied and the magnetic

currents M and −M are doubled as well when calculating the fields in regions 1

and 2, respectively. The magnetic fields Hjt (M) can be expressed in terms of the

integral equation

Hjt (M) =

∫ ∫ApGj

m(r, r′) · M(r

′) ds

′(3.45)

Where Gj

m(r, r′) is the magnetic Green’s function in region j. The incident magnetic

Z

Y

M -M

(b)

1 2

Y

Z

(a)

1 2

Figure 3.6: Cross section of a slot in a waveguide : (a) slot in a conducting plane,

(b) equivalent magnetic currents.


field defined by (3.44) is then written in its integral form using (3.45) as

−Hit =

∫ ∫ApGm(r, r

′) · M(r

′) ds

′(3.46)

Where Gm(r, r′)=G

1

m(r, r′) +G

2

m(r, r′). with transverse components

Gxxm (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕxm,mn(x, y)ϕxm,mn(x′, y′)fm,mn,

Gxym (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕxm,mn(x, y)ϕym,mn(x′, y′)hm,mn,

Gyxm (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕym,mn(x, y)ϕxm,mn(x′, y′)hm,mn,

Gyym (x, y; x′, y′) =

∞∑m=0

∞∑n=0

ϕym,mn(x, y)ϕym,mn(x′, y′)gm,mn . (3.47)

The functions ϕxm,mn(x, y) and ϕym,mn(x, y) represent a complete set of orthonor-

mal eigenfunctions satisfying appropriate boundary conditions on the surface of the

metal waveguide:

ϕxm,mn(x, y) =

√ε0mε0n

2absin(kxx) cos(kyy) ,

ϕym,mn(x, y) =

√ε0mε0n

2abcos(kxx) sin(kyy) (3.48)

with ε0m, ε0n being Newman indexes such that, ε00 = 1, and ε0m = 2, m 6=

0. Finally, the one-dimensional Green’s functions fm,mn(z, z′), hm,mn(z, z′), and

gm,mn(z, z′) are calculated on the interface at z = z′ = 0:

fm,mn = − j

ωµ[(k2

1 − k2x)

Γ1+

(k22 − k2

x)

Γ2] ,

hm,mn =jkxkyωµ

[1

Γ1+

1

Γ2] ,

gm,mn = − j

ωµ[(k2

1 − k2y)

Γ1+

(k22 − k2

y)

Γ2] (3.49)

To solve the integral equation (3.46) for the unknown magnetic current vector M ,

M is expanded as a set of subdomain basis functions:

M(r′) =

N∑i=1

ViBi(r′) (3.50)


where Bi is the i th basis function and Vi is the unknown magnetic current amplitude

at the i th basis. Each basis corresponds to one of N ports. A Galerkin procedure

yields the discretization of the integral equation (3.46):

−∫ ∫

SBj(r).H

i(r)ds =

N∑i=1

Vi

∫ ∫S

∫ ∫S′Bj(r).Gm(r, r

′).Bi(r

′)ds

′ds

(3.51)

leading to a matrix system for the unknown voltage coefficients

[V ] = [V1 · · ·Vi · · ·VN ]T :

[Y ][V ] = [I ] (3.52)

where the ji th element of the admittance matrix [Y ] is

Yji = −∫ ∫

S

∫ ∫S′Bj(r).Gm(r, r

′) ·Bi(r

′)ds

′ds (3.53)

and the j th port current

Ij = −∫ ∫

SBj(r).H

i(r)ds (3.54)

The linear system of equations (3.52) is for an unloaded aperture. If the aperture

contains device ports, the loaded aperture has an admittance matrix of [Y + YL].

Where

[YL] =

YL1 · · · 0 · · · 0

.... . .

.... . .

...

0 · · · YLi · · · 0

.... . .

.... . .

...

0 · · · 0 · · · YLN

, (3.55)

where YLi is the load admittance at port i.


As previously stated, to construct the GSM efficiently it is necessary to

assume the incident wave as a summation of waveguide modes. For an incident

magnetic field propagating in the positive z direction from medium 1 into medium

2 at the interface (Z = 0)

Hi(r) =

Lmax∑l=1

a1l h

1

l (3.56)

Using the expression for the incident magnetic field given above, the electric current

defined by (3.54) is written as

Ij =Lmax∑l=1

a1l

∫ ∫Sh

1

l .Bj(r)ds (3.57)

which leads to the matrix representation

[I ] = [W 1][a11] (3.58)

where

W qji =

∫ ∫shq

i . Bj ds (3.59)

With this expression for the current, the voltage vector [V ] is written as

[V ] = [Z][W 1][a11] (3.60)

where the impedance matrix [Z] = [Y + YL]−1.

The modal amplitude b1l representing the scattered mode l, due to the

magnetic current M , is written in terms of the induced magnetic current

b1l =

1

2

∫ ∫s2M.H

1l ds (3.61)

expanding the magnetic current M as given by (3.50) leads to the following repre-

sentation of the modal coefficients

[b11] = [W 1]T [V ] = [W 1]T [Z][W 1][a1

1] (3.62)


Similarly, the amplitudes of the transmitted modes are

[b12] = [W 2]T [Z][W 1][a1

1] (3.63)

hence the scattering submatrices for the reflected (due to M and reflection from the

conductor) and transmitted modes are

[S11] = [W 1]T [Z][W 1]− [U ] (3.64)

[S21] = [W 2]T [Z][W 1] (3.65)

When the field is incident from region 2 into region 1 the reflected and transmitted

submatrices, [S22] and [S21] are similarly derived and given by

[S22] = [W 2]T [Z][W 2]− [U ] (3.66)

[S12] = [W 1]T [Z][W 2] (3.67)

where [U ] is the identity matrix.

3.4.2 Mode to port scattering

Referring to (3.33)–(3.36), when there is no excitation at port k, Vi = 0 and Vr = Vk.

Hence the scattered power wave coefficient at port k due to mode excitation is

bk = R−1

20 Vk. Thus the scattering coefficients at the ports due to incident modes

from medium 1 can be written in matrix form

[b13] = [R0]

−12 [V ] (3.68)

Substituting for the voltage vector [V ] using (3.60) and recalling that [b13] = [S1

31][a11]

the scattering submatrix

[S131] = [R0]

−12 [Z][W 1] (3.69)


Similarly, the scattering coefficients at the ports due to incident modes from medium

2 can be written as

[S132] = [R0]

−12 [Z][W 2] (3.70)

By reciprocity the scattering matrix of modes due to port excitation is readily ob-

tained as [S113] = [S1

31]T and [S1

23] = [S132]

T .

3.4.3 Port to port scattering

Port quantities are related by a scattering matrix which relates port to port scat-

tering [71]:

[S133] = [R0]

12 [Zp +R0]−1[Zp −R0][R0]

−12 (3.71)

where Zp is the port impedance matrix.

3.5 Dielectric and Conductor Interfaces

In the absence of metalization there is no coupling of modes at the dielectric in-

terface. Hence the scattering matrix is diagonal. For a dielectric interface between

medium 1 and medium 2 with relative permitivities ε1 and ε2, respectively. The

scattering parameters are given by

[S11] = diag(R1...Rl...RLmax)

[S12] = diag(C1...Cl...CLmax)

[S21] = [S12]

[S22] = diag(−R1...−Rl...−RLmax)


where

CTEl =

2√

Γ1l Γ

2l

Γ1l + Γ2

l

CTMl =

2√

Γ1l ε2Γ

2l ε1

Γ1l ε2 + Γ2

l ε1

As expected (RTEl )2 +(CTE

l )2 = 1 and (RTMl )2 +(CTM

l )2 = 1 indicating conservation

of power.

For a perfect conductor interface, the reflection coefficient is simply −1.

Hence its scattering matrix is diagonal with −1 as its diagonal element.

3.6 Cascade Connection

The technique discussed in the previous sections develops a GSM for a single inter-

face at a transverse plane (with respect to the direction of propagation) in a metal

waveguide. A multilayer structure such as that shown in Fig. 3.1 is modeled by cas-

cading the GSMs of individual layers and propagation matrices. Each propagation

matrix describes translation of the mode coefficients from one transverse plane to

another through a homogeneous medium. Several cascading formulas are found [74]

for cascading two port networks. Two cascading formulas of three port networks

(involving modes and device ports) are derived in the following section.

The modeling of a two layer structure with the layers separated by a

waveguide section is illustrated in Fig. 3.7. The analysis proceeds by computing the

GSM of the first layer [S(1)] and then evaluating a propagating matrix [P ] describing

the waveguide section. Finally, computation of the GSM of the second layer [S(2)]

enables cascading of [S(1)], [P ] and [S(2)] to obtain the composite GSM [S(c)]. Each


S(1)

][

b(1)3

S ][(2)

b(1)

1

a (1)3

b (2)3 a (2)

3

a(2)2

b 2

(2)

b 3(C) b (C)

4 a (C)4a

3(C)

a 1

(C)a

(1)

1 a

(1)

b(1)

2

2

[P]

a 1

b 1

(2)

(2)b 1

b 2

a 2

(C) (C)

(C)

PORTS PORTS

INPUT OUTPUTMODES MODES

Figure 3.7: Block diagram for cascading building blocks.

block is represented by

[bi] = [S(i)][ai], i = 1, 2 (3.72)

where

[S(i)] =

Si11 Si12 Si13

Si21 Si22 Si23

Si31 Si32 Si33

(3.73)

In calculating the composite GSM the internal wave coefficients [a12], [b

12], [a

21], and

[b21] must be translated through the waveguide section. This is achieved using the

propagation matrix

[P ] = diag(exp(−Γ1d)...exp(−Γld)...exp(−ΓLmaxd))

where d is the waveguide section separating the two layers and [P ] is a diagonal

matrix as the modes do not couple in the waveguide section and each is translated

by its exponential propagation constant. So the internal mode coefficients are related


by

[b12] = [P ]−1[a2

1] (3.74)

[b21] = [P ]−1[a1

2] (3.75)

The coefficients [b12] and [b2

1] can be written using (3.72) as

[b12] = [S1

21][a11] + [S1

22][a12] + [S1

23][a13] (3.76)

[b21] = [S2

11][a21] + [S2

12][a22] + [S2

13][a23] (3.77)

Thus the internal mode coefficients, [a12] and [a2

1], can be written in terms of the

modes at the external interfaces:

[a12] = [H2]([S

211][P ][S1

21][a11] + [S2

11][P ][S123][a

13] + [S2

12][a22] + [S2

13][a23]), (3.78)

and

[a21] = [H1]([S1

21][a11] + [S1

22][P ][S212][a

22] + [S1

22][P ][S213][a

23] + [S1

23][a13]) (3.79)

Here the matrices [H1] and [H2] are given by

[H1] = ([U ]− [P ][S122][P ][S2

11])−1[P ]

and

[H2] = ([U ]− [P ][S211][P ][S1

22])−1[P ]

Combining (3.74)-(3.79) yields the composite scattering matrix

[S(c)] =

Sc11 Sc12 Sc13 Sc14

Sc21 Sc22 Sc23 Sc24

Sc31 Sc32 Sc33 Sc34

Sc41 Sc42 Sc43 Sc44

(3.80)


with submatrices

[Sc11] = [S111] + [S1

12][H2][S211][P ][S1

21]

[Sc12] = [S112][H2][S2

12]

[Sc13] = [S113] + [S1

12][H2][S211][P ][S1

23]

[Sc14] = [S112][H2][S2

13]

[Sc21] = [S221][H1][S1

21]

[Sc22] = [S222] + [S2

21][H1][S122][P ][S2

12]

[Sc23] = [S221][H1][S1

23]

[Sc24] = [S223] + [S2

21][H1][S122][P ][S2

13]

[Sc31] = [S131] + [S1

32][H2][S211][P ][S1

21]

[Sc32] = [S132][H2][S2

12]

[Sc33] = [S133] + [S1

32][H2][S211][P ][S1

23]

[Sc34] = [S132][H2][S2

13]

[Sc41] = [S231][H1][S1

21]

[Sc42] = [S232] + [S2

31][H1][S122][P ][S2

12]

[Sc43] = [S231][H1][S1

23]

[Sc44] = [S233] + [S2

31][H1][S122][P ][S2

13]

This representation involves two inverted matrices [H1] and [H2]. An alternative

representation that involves only one inverted matrix is also derived below. The

internal mode coefficients, [a12] and [a2

1], can be written in an alternative form as

[a21] = [H]([S1

21][a11] + [S1

22][P ][S212][a

22] + [S1

22][P ][S213[a

23] + [S1

23][a13]), (3.81)

and

[a12] = [P ]([S2

11][H][S121][a

11] + ([S2

11][H][S122][P ][S2

12] + [S212])[a

22] +


([S211][H][S1

22][P ][S213] + [S2

13])[a23] + [S2

11][H][S123][a

13]) (3.82)

Where

[H] = ([U ]− [P ][S122][P ][S2

11])−1[P ]

and the composite scattering can then be derived as

[Sc11] = [S111] + [S1

12][P ][S211][H][S1

21]

[Sc12] = [S112][P ][S2

12] + [S112][P ][S2

11][H][S122][P ][S2

12]

[Sc13] = [S113] + [S1

12][P ][S211][H][S1

23]

[Sc14] = [S112][P ][S2

13] + [S112][P ][S2

11][H][S122][P ][S2

13]

[Sc21] = [S221][H][S1

21]

[Sc22] = [S222] + [S2

21][H][S122][P ][S2

12]

[Sc23] = [S221][H][S1

23]

[Sc24] = [S223] + [S2

21][H][S122][P ][S2

13]

[Sc31] = [S131] + [S1

32][P ][S211][H][S1

21]

[Sc32] = [S132][P ][S2

12] + [S132][P ][S2

11][H][S122][P ][S2

12]

[Sc33] = [S133] + [S1

32][P ][S211][H][S1

23]

[Sc34] = [S132][P ][S2

13] + [S132][P ][S2

11][H][S122][P ][S2

13]

[Sc41] = [S231][H][S1

21]

[Sc42] = [S232] + [S2

31][H][S122][P ][S2

12]

[Sc43] = [S231][H][S1

23]

[Sc44] = [S233] + [S2

31][H][S122][P ][S2

13]


3.7 Program Description

A computer program was developed based on the GSM-MoM derived for the building

blocks in the previous sections. In this section, we will highlight the main steps

involved in analyzing multilayered structures using this program. The program

consists of two main parts. A graphical user interface (GUI) and an electromagnetic

simulator engine.

3.7.1 Geometry-layout and input file

A layout of each layer geometry is drawn using the GUI of CADENCE tools (icfb).

Each geometry is discretized into rectangular cells. The x- and y- directed currents

are evaluated at the intersections of neighboring cells. A typical geometry for a

patch, with current directions, is shown in Fig. 3.8. The circuit ports locations are

distinguished by using labels, provided by CADENCE. The layout is then extracted

to a CIF file format. A parser (written in C) transforms the CIF file into a compatible

form that is read by the program.

The input file contains the frequency range (start frequency, stop fre-

quency and number of points), waveguide dimensions, number of layers, type of

layers, dielectric constants, layer separations, and output file names (impedance

matrix, scattering parameters, etc.). Also a symmetry flag is included in the input

file to indicate which layers are repeated if any. By setting this flag, identical layers

are computed only once and unnecessary redundant analysis of similar layers are

avoided.


Y

X

Figure 3.8: Rectangular patch showing x and y directed currents.

3.7.2 Electromagnetic simulator

An electromagnetic simulator written in FORTRAN is developed to handle the

analysis of multilayered structures. The simulator is composed of five routines.

These are the main routine, MoM calculation, GSM calculation, cascade of GSMs,

and power conservation check. The main routine reads in the input and the geometry

files and controls all other routines. It carries out the analysis in two main loops,

frequency loop and layer loop with the frequency loop being the outer loop. For

each frequency point, the type of layer is checked. If the layer is magnetic or electric,

the MoM is calculated using the MoM routine. Then the GSM is computed using

the GSM calculation routine. To speed up the element calculation, an acceleration

technique is used. This technique is based on the extraction of the quasi-static

term of the Green’s function. A detailed analysis for the acceleration procedure is


demonstrated in the following chapter.

For all other layers, the GSM is computed directly without the need to

call the MoM routine. After a GSM is calculated for a layer, it is then cascaded to the

previously calculated GSMs using the cascade routine. A power conservation check

is then used to check the accuracy of the calculation using the power conservation

routine. The sum of the squares of each column elements for the propagating modes

has to equal 1.

When all layers are computed and a single GSM for the structure is ob-

tained, a new frequency point is calculated. This will continue until a complete sweep

of the frequency range is achieved. The program produces output files containing

the composite scattering matrix of the whole structure, the impedance matrix of

the whole structure, the circuit scattering parameters, and the circuit impedance

parameters.

A flow chart illustrating the algorithm for the analysis of multilayered

structures described above is shown in Fig. 3.9.


SET (i = 1 )

GENERATE O/P FILES

IS (F < Fstop) F= F + df

NO

YES

GEOMETRY FILES

OR ELECTRIC?IS LAYER i MAGNETIC

IS (i > 1)

i = i + 1

NO

NO

NO

YES

YES

YES

&

SET (F = Fstart)

READ I/P DATA

SCATTERING MATRIX

CASCADE LAYERS (i , i-1)

IS (i < MAX_LAYER)

MoM OF LAYER ( i )

OF LAYER ( i )

Figure 3.9: A flow chart for cascading multilayers.

Chapter 4

MoM Element Calculation

4.1 Introduction

The most time consuming process in the GSM-MoM technique is the impedance or

admittance matrix element calculation. This is specially true for large waveguide

dimensions and small cell discretizations with respect to the guide wavelength. An

acceleration procedure is adopted in this work to speed up the element calculation.

The impedance elements defined in (3.17) and the admittance elements given in

(3.53) are derived in this section. These elements involve quadruple integrals of the

form:

−∫ ∫

S

∫ ∫S′Bj(r).G(r, r

′) ·Bi(r

′)ds

′ds

The integration is carried out for the electric or magnetic type Green’s

functions over both the source and test basis functions. The two-dimensional space

58

CHAPTER 4. MOM ELEMENT CALCULATION 59

is discretized using rectangular cells. Each basis function spreads over two adja-

cent cells. In our implementation, the basis functions are chosen to be subdomain

sinusoidal functions. We use two discretization schemes for both the electric and

magnetic currents, uniform and nonuniform. Uniform griding is more suitable for

relatively simple geometries since the element computation time is less than in the

nonuniform case. However, nonuniform griding enables the modeling of structures

with adjustable spatial resolution to account for complex geometrical details, hence

reducing the total number of unknowns with respect to the uniform case.

4.1.1 Uniform discretization

Uniform discretization implies equal cell dimensions for all cells constructing the

grid. The cells are rectangular in shape as shown in Fig. 4.1. The grid is uniform in

the x and y directions with cell sizes c and d, respectively. The x directed sinusoidal

x -ci

xi x +ci

yi

y +2d

i

y -2d

i

yj

xj

y +dj

y -dj

c2j

x +c2j

x -

X

Y

Figure 4.1: Geometry of uniform basis functions in the x and y directions.


basis function Bxi centered at (xi, yi) is given by

Bxi (x) =

sin [ks (c− |x− xi|)]

d sin (ksc)

,|x− xi| ≤ c

|y − yi| ≤ d/2

0 , otherwise

(4.1)

and for a y directed sinusoidal basis function

Byi (y) =

sin [ks (d− |y − yi|)]

c sin (ksd)

,|y − yi| ≤ d

|x− xi| ≤ c/2

0 , otherwise .

(4.2)

where ks = ω√µ0ε0εs.

Using the above expressions for the basis functions, the impedance ma-

trix elements given in (3.17) are obtained in closed form expressions as follows

Zxxij = −

∞∑m=0

∞∑n=0

ε0mε0nab

fe,mnSxe,iS

xe,jR

xe,iR

xe,j

Zyyij = −

∞∑m=0

∞∑n=0

ε0mε0nab

ge,mnSye,iS

ye,jR

ye,iR

ye,j

Zxyij = −

∞∑m=0

∞∑n=0

ε0mε0nab

he,mnSxe,iS

ye,jR

xe,iR

ye,j (4.3)

where

Sxe,i = 2ks cos(kxxi)

[cos(ksc)− cos(kxc)

(k2x − k2

s) d sin(ksc)

]

Sye,i = 2ks cos(kyyi)

[cos(ksd) − cos(kyd)

(k2y − k2

s) c sin(ksd)

]


Rxe,i = 2

sin(kyyi) sin(kyd2)

ky

Rye,i = 2

sin(kxxi) sin(kxc2)

kx(4.4)

Similarly, the admittance elements are obtained using (3.53) and the uniform basis

function expressions

Y xxij = −

∞∑m=0

∞∑n=0

ε0mε0n2ab

fm,mnSxm,iS

xm,jR

xm,iR

xm,j

Y yyij = −

∞∑m=0

∞∑n=0

ε0mε0n2ab

gm,mnSym,iS

ym,jR

ym,iR

ym,j

Y xyij = −

∞∑m=0

∞∑n=0

ε0mε0n2ab

hm,mnSxm,iS

ym,jR

xm,iR

ym,j (4.5)

where

Sxm,i = 2ks sin(kxxi)

[cos(ksc)− cos(kxc)

(k2x − k2

s) d sin(ksc)

]

Sym,i = 2ks sin(kyyi)

[cos(ksd)− cos(kyd)

(k2y − k2

s) c sin(ksd)

]

Rxm,i = 2

cos(kyyi) sin(kyd2)

ky

Rym,i = 2

cos(kxxi) sin(kxc2)

kx(4.6)

4.1.2 Nonuniform discretization

Nonuniform discretization implies unequal cell dimensions. For row i in the x direc-

tion, all cells have the same width but variable length as shown in Fig. 4.2. Different


rows can have different widths. The same is true for the columns in the y direction.

The x directed sinusoidal basis function Bxi centered at (xi, yi) is given by

x -ci

xi x +ci

yi

y +2d

i

y -2d

i

yj

xj

y +dj

y -dj

c2j

x +c2j

x -

X

Y

2

1

21

Figure 4.2: Geometry of nonuniform basis functions in the x and y directions.

Bxi (x)) =

sin[ks(c1 − xi + x)]

d sin(ksc1) , xi − c1 ≤ x ≤ xi

sin[ks(c2 − x+ xi)]

d sin(ksc2) , xi ≤ x ≤ xi + c2

0 , otherwise

(4.7)


and for a y directed sinusoidal basis function is given by

Byj (y)) =

sin[ks(d1 − yj + y)]

c sin(ksd1) , yj − d1 ≤ y ≤ yj

sin[ks(d2 − y + yj)]

c sin(ksd2) , yj ≤ y ≤ yj + d2

0 , otherwise

(4.8)

Using the above expressions for the basis functions in (3.17) and integrating results

in closed form expressions for the impedance elements given bellow.

Zxxij = −

∞∑m=0

∞∑n=0

ε0mε0nab

fe,mn(Qxe,i1 +Qx

e,i2)(Qxe,j1 +Qx

e,j2)Rxe,iR

xe,j

Zyyij = −

∞∑m=0

∞∑n=0

ε0mε0nab

ge,mn(Qye,i1 +Qy

e,i2)(Qye,j1 +Qy

e,j2)Rye,iR

ye,j

Zxyij = −

∞∑m=0

∞∑n=0

ε0mε0nab

he,mn(Qxe,i1 +Qx

e,i2)(Qye,j1 +Qy

e,j2)Rxe,iR

ye,j (4.9)

where

Qxe,i1 =

1

(k2s − k2

x) d sin(ksci1)[ks cos(kx(xi − ci1))− ks cos(kxxi) cos(ksci1)

−kx sin(kxxi) sin(ksci1)]

Qxe,i2 =

1

(k2s − k2

x) d sin(ksci2)[ks cos(kx(xi + ci2))− ks cos(kxxi) cos(ksci2)

+kx sin(kxxi) sin(ksci2)]

Qye,i1 =

1

(k2s − k2

y) c sin(ksdi1)[ks cos(ky(yi − di1))− ks cos(kyyi) cos(ksdi1)


−ky sin(kyyi) sin(ksdi1)]

Qye,i2 =

1

(k2s − k2

y) c sin(ksdi2)[ks cos(ky(yi + di2))− ks cos(kyyi) cos(ksdi2)

+ky sin(kyyi) sin(ksdi2)] (4.10)

Similarly, the admittance elements are obtained using (3.53) and the nonuniform

basis function expressions

Y xxij = −

∞∑m=0

∞∑n=0

ε0mε0n2ab

fm,mn(Qxm,i1 +Qx

m,i2)(Qxm,j1 +Qx

m,j2)Rxm,iR

xm,j

Y yyij = −

∞∑m=0

∞∑n=0

ε0mε0n2ab

gm,mn(Qym,i1 +Qy

m,i2)(Qym,j1 +Qy

m,j2)Rym,iR

ym,j

Y xyij = −

∞∑m=0

∞∑n=0

ε0mε0n2ab

hm,mn(Qxm,i1 +Qx

m,i2)(Qym,j1 +Qy

m,j2)Rxm,iR

ym,j (4.11)

where

Qxm,i1 =

1

(k2s − k2

x) d sin(ksci1)[ks sin(kx(xi − ci1))− ks sin(kxxi) cos(ksci1)

+kx cos(kxxi) sin(ksci1)]

Qxm,i2 =

1

(k2s − k2

x) d sin(ksci2)[ks sin(kx(xi + ci2))− ks sin(kxxi) cos(ksci2)

−kx cos(kxxi) sin(ksci2)]

Qym,i1 =

1

(k2s − k2

y) c sin(ksdi1)[ks sin(ky(yi − di1))− ks sin(kyyi) cos(ksdi1)

+ky cos(kyyi) sin(ksdi1)]

Qym,i2 =

1

(k2s − k2

y) c sin(ksdi2)[ks sin(ky(yi + di2))− ks sin(kyyi) cos(ksdi2)

−ky cos(kyyi) sin(ksdi2)] (4.12)


4.2 Acceleration of MoM Matrix Elements

The MoM impedance matrix elements Zij appearing in (3.17) and the MoM admit-

tance matrix elements Yij defined in (3.53) involve the integration of the electric

and magnetic type Green’s functions, respectively. The Green’s functions are two-

dimensional infinite series. To evaluate the Green’s functions, at a given source and

observation points, the double series summation must converge to a stable value.

The simplest technique to compute the matrix elements is the direct summation

technique, where a term-by-term summation is carried out while checking for con-

vergence at progressive intervals. However these double series summations are slow

to converge which, as well as leading to time-consuming computations, can result in

numerical instabilities and imprecision in determining when the summations should

be truncated. Also, the number of summation terms is directly proportional to the

waveguide size and inversely proportional to the dimension of the basis functions

used to discretize the integral equation.

Several methods were employed to accelerate the computation of the

waveguide Green’s function. A rectangular waveguide Green’s function involving

complex images was proposed in [75], where the real images are replaced by the full-

wave complex discrete images. The resulting Green’s function is fast convergent.

Park and Nam [76], in considering a shielded planar multilayered structure, trans-

formed a scalar Green’s function into a static image series which was evaluated using

the Ewald method. It was pointed out that the final form of the Green’s function

converges rapidly with a small number of terms in a series summation. Transfor-

mation of a double series expansion into a contour complex integral to which the

residue theorem was applied was developed by Hashemi-Yeganeh [77]. This method


leads to the computation of a few single summations of fast converging series. Al-

ternative fast converging formulas for the dyadic Green’s function in a rectangular

waveguide by way of the Poisson summation technique was developed in [78].

By far the most widely used technique to accelerate waveguide Green’s

functions is the quasistatic extraction method. In this method the Green’s function

is partitioned into an asymptotic static (frequency-independent) part and a dynamic

(frequency-dependent) part. The asymptotic part needs to be evaluated once per

frequency scan. The dynamic part is now fast converging owing to the extraction

of the slowly converging static part. Eleftheriades, Mosig, and Guglielmi [79] pio-

neered a procedure that partitions a potential Green’s function into an asymptotic

(frequency-independent) part and a dynamic part, where the asymptotic part was

converted to a rapidly converging series summation. We found that this technique

is most suitable for our problem and implemented it to the electric and magnetic

type Green’s functions.

4.2.1 Acceleration of impedance matrix elements

It is known that a double series expansion of Green’s function components is slowly

convergent due to the presence of the quasi-static part. An efficient technique based

on the Kummer transformation [80] has been applied to accelerate slowly convergent

series [79]. This technique is applied here to the Green’s function components, (3.10),

leading to their transformation so that a quasi-static part (GQS

e ) is extracted. The

Green’s function is then

Ge = G∆

e +GQS

e (4.13)


where G∆

e = Ge−GQS

e , Ge is the electric type Green’s function, and GQS

e is the qua-

sistatic electric type Green’s function. To compute the quasistatic Green’s function,

(3.11) and (3.12) are calculated for large m and n and are given by

ϕxe,mn(x, y) =2√ab

cos(kxx) sin(kyy) ,

ϕye,mn(x, y) =2√ab

sin(kxx) cos(kyy) (4.14)

fe,mn = jk2x

ωkc(ε1 + ε2),

he,mn = jkxky

ωkc(ε1 + ε2),

ge,mn = jk2y

ωkc(ε1 + ε2)(4.15)

The quasistatic components of the electric type Green’s function are derived using

(3.10), (4.14) and (4.15):

GQSxx (x, y; x′, y′) =

j

abω(ε1 + ε2)

∞∑m=1

(mπa

)2cos(mπax)

× cos(mπax′) ∞∑n=1

4 sin(nπby)

sin(nπby′)

√(mπa

)2+(nπb

)2, (4.16)

GQSyy (x, y; x′, y′) =

j

abω(ε1 + ε2)

∞∑n=1

(nπb

)2cos(nπby)

× cos(nπby′) ∞∑m=1

4 sin(mπax)

sin(mπax′)

√(mπa

)2+(nπb

)2, (4.17)

and

GQSxy (x, y; x′, y′) =

j

abω(ε1 + ε2)

∞∑n=1

(nπb

)sin(nπby)


× cos(nπby′) ∞∑m=1

4(mπa

)cos(mπax)

sin(mπax′)

√(mπa

)2+(nπb

)2. (4.18)

It can be seen that the quasistatic part is inversely proportional to ω. However,

it needs to be calculated only once per frequency scan since the summations are

frequency independent. Still the expressions obtained above are slowly converging,

following the procedure described in [79] the second infinite summation in (4.16) and

(4.17) can be transformed into a fast converging series of K0, the modified Bessel

functions of the second kind, thus

∞∑n=1

4 sin(nπby)

sin(nπby′)

√(mπa

)2+(nπb

)2=

2b

π× (4.19)

∞∑n=−∞

K0

(mπa

(y − y′ + 2nb))−K0

(mπa

(y + y′ + 2nb))

,

∞∑m=1

4 sin(mπax)

sin(mπax′)

√(mπa

)2+(nπb

)2=

2a

π× (4.20)

∞∑m=−∞

K0

(nπb

(x− x′ + 2ma))−K0

(nπb

(x+ x′ + 2ma))

,

Only a few terms of the series on the right hand side of equations (4.19) and (4.20)

are required to reach convergence due to the exponential decay of the modified

Bessel functions. This property together with the frequency independence of the

summations are the key attributes leading to computational speed-up.

To compute the accelerated impedance matrix elements a Galerkin MoM

procedure is applied to (4.13) resulting in the following representation:

ZMoM(ω) = [Z∆(ω) + ZQS] (4.21)

where

Z∆ji = −

∫ ∫S

∫ ∫S′Bj(r).G

∆

e (r, r′) ·Bi(r

′)ds

′ds, (4.22)


and

ZQSji = −

∫ ∫S

∫ ∫S′Bj(r).G

QS

e (r, r′) ·Bi(r

′)ds

′ds (4.23)

The integrations of the modified Bessel functions in (4.23) are easily converted using

change of variables to a standard integral

∫ x

0K0(v)dv (4.24)

This integral is shown in Fig. 4.3 and is shown to be fast convergent as the argument

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

V

Inte

gral

of K

0(V)

Figure 4.3: Integral of K0.

V increases. For example, the xx impedance matrix element ZQSji is written as

ZQSji =

∫ xj+c

xj−c

∫ yj+d/2

yj−d/2

∫ xi+c

xi−c

∫ yi+d/2

yi−d/2Bxj (x)

×GQSxx (x, y; x′y′)Bx

i (x′)dx dy dx′dy′ (4.25)


with Bxj (x) and Bx

i (x′) being piecewise sinusoidal functions defined by (4.1). For

simplicity uniform basis functions are considered:

ZQSji = −j

∞∑m=1

2 (mπa

)2

aπω(ε1 + ε2)Sxe,iS

xe,j

∞∑n=−∞

(IQSji ) (4.26)

Thus the problem of evaluating ZQSji is reduced to the calculation of a

double integral over the y-domain:

IQSji = − a

mπ

∫ yj+d/2

yj−d/2

∫ mπa

(y−yi−d/2+2nb)

0K0(v)dv −

∫ mπa

(y−yi+d/2+2nb)

0K0(v)dv

+∫ mπ

a(y+yi+d/2+2nb)

0K0(v)dv −

∫ mπa

(y+yi−d/2+2nb)

0K0(v)dv

dy (4.27)

The inner integrals are of the same form as (4.24). This standard integral is numer-

ically computed only once and stored in a table. The outer integrals are calculated

numerically by means of Gaussian quadrature using the data of the previous inte-

gration.

The second series appearing in (4.26) is now very fast convergent. This

is due to the fast converging nature of the integrals when the index n gets larger.

Typically only three terms of n need to be evaluated (from −1 to 1) to achieve very

small errors. So the double series summation involved in the quasistatic impedance

element calculation is effectively converted to a single series summation. Similar

expressions can be derived for the xy, yx, and yy impedance matrix elements.

As an example, the convergence and accuracy of the speed up procedure

discussed is demonstrated by a comparison to the direct summation case for the

x-directed current element placed inside a WR-90 waveguide with xp = a/2 and

yp = b/2 (the unit cell shown in Fig. 3.4 has dimensions c=0.2318 cm and d=0.2371

cm).


The convergence and percentage error of the impedance element for the

accelerated and direct double series summation are demonstrated in Figs. 4.4 and

4.5, respectively. The relative error is defined as |Zxx−Z∞xx|/|Z∞xx| × 100, where Zxx

represents the impedance matrix element either calculated as a direct summation

or using the proposed acceleration technique, and Z∞xx is the value of Zxx obtained

for a large number of summation terms m∞ and n∞ (using direct summation).

To generate the results shown in Fig. 4.5 we used m∞ = n∞ = 1500

summation terms resulting in a value of Z∞xx equal to 10.928331 − 163.503317. It

is shown that the error of 0.5% is obtained for 200 terms used in the accelerated

summation procedure in comparison with 2500 terms required in the direct double

series summation to reach the same error. The computation time is almost directly

proportional to the number of terms in the summation. Also, it can be difficult to

determine when sufficient terms have been used with the direct method.

4.2.2 Acceleration of admittance matrix elements

The same procedure for accelerating the impedance matrix elements is followed for

the admittance matrix elements. The magnetic type Green’s function is now written

as:

Gm = G∆

m +GQS

m (4.28)

where G∆

m = Gm − GQS

m , Gm is the magnetic type Green’s function, and GQS

m is

the quasistatic magnetic type Green’s function. To compute the quasistatic Green’s

function equations (3.48) and (3.49) are calculated for large m and n and are given


0 500 1000 1500 2000 2500 3000Number of terms

100

110

120

130

140

150

160

170

180

190

200

|Zxx

|

Accelerated Direct summation

Figure 4.4: Convergence of Zxx matrix elements.

0 500 1000 1500 2000 2500 3000Number of terms

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Per

cent

age

erro

r

Accelerated Direct summation

Figure 4.5: Percentage error in the convergence of Zxx matrix elements.


by:

ϕxm,mn(x, y) =

√2

absin(kxx) cos(kyy) ,

ϕym,mn(x, y) =

√2

abcos(kxx) sin(kyy) (4.29)

fm,mn = 2jk2x

ωµkc,

hm,mn = 2jkxkyωµkc

,

gm,mn = 2jk2y

ωµkc(4.30)

Equations (4.29) and (4.30) when combined with (3.47) result in the quasistatic

Green’s function components.

GQSxx (x, y; x′, y′) =

j

abωµ

∞∑m=1

(mπa

)2sin(mπax)

× sin(mπax′) ∞∑n=1

4 cos(nπby)

cos(nπby′)

√(mπa

)2+(nπb

)2, (4.31)

GQSyy (x, y; x′, y′) =

j

abωµ

∞∑n=1

(nπb

)2sin(nπby)

× sin(nπby′) ∞∑m=1

4 cos(mπax)

cos(mπax′)

√(mπa

)2+(nπb

)2, (4.32)

and

GQSxy (x, y; x′, y′) =

j

abωµ

∞∑n=1

(nπb

)cos

(nπby)

× sin(nπby′) ∞∑m=1

4(mπa

)sin(mπax)

cos(mπax′)

√(mπa

)2+(nπb

)2. (4.33)


The second infinite summation in (4.31) and (4.32) can be transformed into a fast

converging series of K0, the modified Bessel functions of the second kind, thus

∞∑n=1

4 cos(nπby)

cos(nπby′)

√(mπa

)2+(nπb

)2= − 2a

mπ+[2bπ× (4.34)

∞∑n=−∞

K0

(mπa

(y + y′ + 2nb))

+K0

(mπa

(y − y′ + 2nb))]

,

∞∑m=1

4 cos(mπax)

cos(mπax′)

√(nπb

)2+(mπa

)2= − 2b

nπ+[2aπ× (4.35)

∞∑m=−∞

K0

(nπb

(x+ x′ + 2ma))

+K0

(nπb

(x− x′ + 2ma))]

,

To compute the accelerated admittance matrix elements, a Galerkin procedure is

applied to (4.28) resulting in the following representation:

Y MoM(ω) = [Y ∆(ω) + Y QS] (4.36)

where

Y ∆ji = −

∫ ∫S

∫ ∫S′Bj(r).G

∆

m(r, r′) ·Bi(r

′)ds

′ds, (4.37)

and

Y QSji = −

∫ ∫S

∫ ∫S′Bj(r).G

QS

m (r, r′) ·Bi(r

′)ds

′ds (4.38)

The quasistatic admittance elements are evaluated in a similar manner as the qua-

sistatic impedance elements. The problem is reduced into integrals involving the

Bessel functions of the second kind which are fast convergent.

Chapter 5

Local Reference Nodes

5.1 Introduction

The GSM-MoM method described in Chapters 3 and 4 produces a scattering matrix

that represents the relationship between modes and ports. The scattering matrix

can then be converted to port-based admittance or impedance matrix. This allows

the modeling of a waveguide structure that can support multiple electromagnetic

modes by a circuit with defined coupling between the modes. However, port-based

representations are not suited for most circuit simulation tools. Nodal analysis is the

mainstay of circuit simulation. The basis of the technique is relating nodal voltages,

voltages at nodes referenced to a single common reference node, to the currents

entering the nodes of a circuit. The art of modeling is then, generally, to develop

a current/nodal-voltage approximation of the physical characteristics of a device

or structure. With spatially distributed structures a reasonable approximation can

75

CHAPTER 5. LOCAL REFERENCE NODES 76

sometimes be difficult to achieve. The essence of the problem is that a global ref-

erence node cannot reasonably be defined for two spatially separated nodes when

the electromagnetic field is transient or alternating. In this situation, the electric

field is nonconservative and the voltage between any two points is dependent on

the path of integration and hence voltage is undefined. This includes the situation

of two separated points on an ideal conductor. Put in a time-domain context, it

takes a finite time for the state at one of the points on the ideal conductor to af-

fect the state at the other point. In the case of waveforms on digital interconnects

this phenomenon has become known as retardation [81]. With high-speed digital

circuits, it is common to model ground planes by inductor networks so that inter-

connects are modeled by extensive RLC meshes. Consequently no two separated

points are instantaneously coupled. In transient analysis of distributed microwave

structures, lumped circuit elements can be embedded in the mesh of a time dis-

cretized electromagnetic field solver such as a finite difference time domain (FDTD)

field modeler [16,17]. The temporal separation of spatially distributed points is then

inherent to the discretization of the mesh.

With a frequency domain electromagnetic field simulator, ports are de-

fined and so a port-based representation of the linear distributed circuit is produced.

With ports, a global reference node is not required. Instead a local reference node,

one of the terminals of the two terminal port, is implied. The beginnings of a circuit

theory incorporating ports in circuit simulation has been described and termed the

compression matrix approach [14, 15]. This milestone work presented a technology

for integrating port-based electromagnetic field models with nonlinear devices. Cir-

cuit simulation using port representation has been reported in [82]. This requires the

representation of nodally defined circuits in its port equivalent by a general-purpose


linear multiport routine. Hence, the advantage of accessing information at all nodes

as in nodal analysis is lost.

This Chapter extends the circuit theory beyond the compression ma-

trix approach to general purpose circuit simulators based on nodal analysis. In

particular, we present the concept of local reference nodes that enables port-based

network characterization to be used with nodally defined circuits in the develop-

ment, by inspection (the preferred approach), of what is termed a locally referenced

nodal admittance matrix. A procedure for handling and moving the local reference

nodes is described along with circuit reduction techniques that facilitate efficient

simulation of nonlinear microwave circuits.

5.2 Nodal-Based Circuit Simulation

The most popular method for circuit analysis in the frequency domain is the nodal

admittance matrix method. In the nodal formulation of the network equations, a

matrix equation is developed that relates the unknown node voltages to the external

currents using the model shown in Fig. 5.1. All node voltages are then defined with

respect to an arbitrarily chosen node called the global reference node. Eliminating

the row and column associated with the global reference node leads to a definite

admittance matrix and then the solution for the node voltages is straight forward.

In this type of analysis only one reference node can exist.


(c)

(b)

N +1

vN

vk

v

v

2

1

Jk

JJ

21

NJ

GLOBALREFERENCE NODE

(a)

Figure 5.1: Nodal circuits: (a) general nodal circuit definition (b) conventional

global reference node; and (c) local reference node proposed here.

5.3 Spatially Distributed Circuits

5.3.1 Port representation

Electromagnetic structures can only be strictly analyzed using port excitations. The

spatially distributed linear circuit (SDLC) consists of groups with each group having

a local reference node. The scattering parameters are the most natural parameters

to use with ports and their local reference nodes. They can be converted to port

admittance matrix using the following equation


Y = Y0(1−Y−1/20 SY1/2

0 )(1 + Y−1/20 SY1/2

0 )−1 (5.1)

This is the most convenient form to use in circuit simulators. Before continuing,

a distinction is required between the global reference node and the local reference

nodes, with the symbols shown in Fig. 5.1 are adopted here. A general circuit with

local reference nodes required with an SDLC and nonlinearities is shown in Fig. 5.2.

For the specific case of power combiners the SDLC can be illustrated by the 2×2 grid

array shown in Fig. 5.3. Here the grid array is composed of four locally referenced

groups with each group having a differential pair as its active components. Referring

to Fig. 5.2, the SDLC is the electromagnetic port representation of the grid array,

the linear subcircuits are the linear elements in the equivalent circuit model of the

differential pair, and the nonlinear subcircuits are the nonlinear elements associated

with the active device model.

Figure 5.2 depicts the essential circuit analysis issue: integrating the

representation of an SDLC with a circuit defined in a conventional nodal manner,

to obtain an augmented nodal based description. The problem is how to handle the

additional redundancy introduced by the local reference nodes. For locally refer-

enced group number m there are Em terminals, Em− 1 locally referenced ports and

one local reference node designated Em. The port-based system may be expressed

as

[pY][pV] = [pI] (5.2)

Where the port-based admittance matrix


..

.

..

.

..

....

..

.

+1

+2

1 1

..

.

..

.

..

.1

E

F1

E M1

2 m

m1

E 1

11+1

M

F

Em

Em

mf

+1

Fm

F -1m

me

Em -1

mE

MEM

m

M

Linear

M M

m

1

m

Augmented Nodal Based

GROUP

LOCALLY

NonlinearSubcircuit

LinearSubcircuit

LinearSubcircuit

NonlinearSubcircuit

NonlinearSubcircuit Subcircuit

REFERENCED

PORT BASEDSPATIALLY

DISTRIBUTEDLINEAR

CIRCUIT

(SDLC)

Figure 5.2: Port defined system connected to nodal defined circuit.


GROUP 1 GROUP 2

GROUP 3 GROUP 4

Figure 5.3: Grid array showing locally referenced groups.


pY =

pY1,1 · · · pY1,m · · · pY1,M

.... . .

.... . .

...

pYm,1 · · · pYm,m · · · pYm,M

.... . .

.... . .

...

pYM,1 · · · pYM,m · · · pYM,M

,

the port-based voltage vector

pV =

[pV1 · · · pVm · · · pVM

]T,

and the port-based current vector

pI =

[pI1 · · · pIm · · · pIM

]T.

The submatrix pYi,j is the mutual port admittance matrix (of dimension Ei − 1×

Ej − 1) between groups i and j of the SDLC, Ii = [I1i I2i ... IEi−1 ] is the current

vector of group i of the SDLC, and pVj = [(V1j − VEj ) (V2j − VEj ) ... (VEj−1 − VEj )

] is the port voltage vector of group j of the SDLC. Defining the total number of

ports for groups 1 to j of the SDLC as:

nj =j∑i=1

(Ei − 1) (5.3)

Then pY is square of dimension nM × nM .


5.3.2 Port to local-node representation

In order to use nodal analysis, the port-based system must be formulated in a nodal

admittance form. Since there are M localized reference nodes, another redundant

M rows and M columns can be added to the port admittance matrix such that:

[nY][nV] = [nI] (5.4)

Where the nodal admittance matrix

nY =

pY Y1

Y2 Y3

(nM+M)×(nM+M)

(5.5)

The elements of each submatrix are given by

Y1(r, c) = −nc∑

j=n(c−1)+1

pY(r, j), r = 1..nM , c = 1..M

Y2(r, c) = −nr∑

i=n(r−1)+1

pY(i, c), r = 1..M, c = 1..nM

Y3(r, c) = −nr∑

i=n(r−1)+1

Y1(i, c), r = 1..M, c = 1..M.

with n0 = 0, nr, nc, n(c−1), and n(r−1) are given by (5.3). The nodal voltage vector

nV = [V11...VE1−1V12 ...VE2−1...V1M ...VEM−1VE1VE2 ...VEM ]T

and the branch current vector

nI = [I11...IE1−1I12...IE2−1...I1M ...IEM−1IE1IE2...IEM ]T

The admittance matrix [nY] is a nodal matrix and has M dependent

rows and M dependent columns. Hence, it is an M fold indefinite nodal admittance

matrix corresponding to the M local reference nodes.


5.4 Representation of Nodally Defined Circuits

Since there are no connections between the linear circuits at each group, the linear

circuit at group i will have no mutual coupling with the linear circuit at group j. The

only coupling that can exist between different locally referenced groups is accounted

for in the description of the SDLC. Hence, for the linear subcircuits (as in Fig. 5.2)

all the entries in the admittance matrix are zero except those relating the node

parameters at the same group. Defining interfacing nodes as the nodes between

lumped linear circuits and nonlinear circuits, a lumped linear circuit embedded at

group m can be represented as

[Y][V] = [I] (5.6)

0 0 · · · 0 0 · · · 0 0

0 0 · · · 0 0 · · · 0 0

......

. . ....

.... . .

...

0 0 · · · Ym(1,1) Ym(1,2) · · · 0 0

0 0 · · · Ym(2,1) Ym(2,2) · · · 0 0

......

. . ....

.... . .

...

0 0 · · · 0 0 · · · 0 0

0 0 · · · 0 0 · · · 0 0

nV1

iV1

...

nVm

iVm

...

nVM

iVM

=

−nI1

iI1

...

−nIm

iIm

...

−nIM

iIM

(5.7)

where

• nIm = [I1m I2m ... IEm ] is the current vector at group m of the SDLC.


• nVm = [V1m V2m ... VEm ] is the node voltage vector at group m of the SDLC.

• iIm is the branch current vector (the currents flow into the linear network) of

interfacing nodes and linear subcircuit nodes at group m.

• iVm the node voltage vector of interfacing nodes and linear subcircuit nodes

at group m.

• Ym =

Ym(1,1) Ym(1,2)

Ym(2,1) Ym(2,2)

is the conventional indefinite nodal admittance

matrix of the linear sub-circuit.

Thus, the indefinite nodal admittance matrix of all of the linear sub-

circuits combined is a block diagonal matrix

YL = Diag(Y1, ..,Ym, ..,YM) (5.8)

5.5 Augmented Admittance Matrix

To combine the linear circuits (lumped and distributed) in an augmented admittance

matrix as shown in Fig. 5.2, (5.4) is expanded in the full set of voltages [V] yielding:

[YE][V] = [I] (5.9)

with [YE] being the expanded nodal representation of the SDLC and given by


nY1,1 0 · · · 0 nY1,m 0 · · · 0 nY1,M 0

0 0 · · · 0 0 0 · · · 0 0 0

......

......

......

.........

...

0 0 · · · 0 0 0 · · · 0 0 0

nYm,1 0 · · · 0 nYm,m 0 · · · 0 nYm,M 0

0 0 · · · 0 0 0 · · · 0 0 0

......

......

......

.........

...

0 0 · · · 0 0 0 · · · 0 0 0

nYM,1 0 · · · 0 nYM,m 0 · · · 0 nYM,M 0

0 0 · · · 0 0 0 · · · 0 0 0

Equations (5.8) and (5.9) are added together to form the overall linear circuit.

YA = YE + YL (5.10)

In microwave nonlinear circuit analysis, the network parameters of the

linear circuit are reduced to just include the interfacing nodes. Standard matrix

reduction techniques can be used to obtain this reduced circuit. An interfacing

node is assigned to be the local reference node at each group, hence eliminating the

corresponding rows and columns. The resulting system of equations is definite and

represents the augmented linear circuit.


5.6 Summary

The scheme for the augmentation of a nodal admittance matrix by a port-based

matrix with a number of local reference nodes permits field derived models to be

incorporated in a general purpose circuit simulator based on nodal formulation.

The method is immediately applicable to modified nodal admittance (MNA) anal-

ysis as the additional rows and columns of the MNA matrix are unaffected by the

augmentation.

This work is being used in the simulation of spatial power combiners (in

both free space and waveguide) which are electrically large and do not have a global

reference node or perfect ground plane. This is demonstrated in [52] where a free

space 2 × 2 active grid array is simulated using the local reference node concept

presented in this Chapter.

Chapter 6

Results

6.1 Introduction

To illustrate the flexibility and generality of the GSM-MoM method developed in

Chapter 3, the GSM-MoM method is applied to the analysis of spatial power com-

bining elements and arrays in addition to general structures. Although originally

the method was developed to simulate spatial power combining structures, it can

handle, in general, any transverse structure in a waveguide such as waveguide filters,

input impedance of probe excited waveguides, and shielded multilayered structures.

In this Chapter, we will consider two categories of examples. These are general

structures and spatial power combining structures. The spatial power combining

structures include patch-slot-patch arrays, CPW arrays, grid arrays, and cavity os-

cillators.

88

CHAPTER 6. RESULTS 89

6.2 Analysis of General Structures

In this section several common structures such as wide strip in a waveguide, patches

on a dielectric slab, and strip-slot transition module are simulated. The obtained

results (modal scattering parameters) are compared to either measured results or

other analysis techniques.

To demonstrate the validity and accuracy of the GSM-MoM with ports, a

completely shielded microstrip notch filter, in a cavity, is simulated. The results (two

port scattering parameters) compare favorably to measurements. This represents

an extreme test to the method.

6.2.1 Wide resonant strip

To illustrate the calculation of the Generalized Scattering Matrix procedure pro-

posed in Chapters 3 and 4, respectively, a wide resonant strip structure embedded

in an X-band rectangular waveguide (with geometry shown in Fig. 6.1), is investi-

gated numerically. The strip current is discretized using rectangular meshing and

sinusoidal basis functions. It should be noted that to accurately model the current

on the strip, the continuity of the edge current is accounted for by half basis func-

tions as shown in Fig. 6.1. Numerical calculation for the normalized susceptance

(of the dominant TE10 mode) show that the structure goes through resonance at

approximately 11 GHz which agrees well with the measured data provided in [83]

as shown in Fig. 6.2. Such a wide strip can be used as a section of waveguide

filters [84].


y

x

b

aw

a-l

ycz

Figure 6.1: Wide resonant strip in waveguide, a = 1.016 cm, b = 2.286 cm, w =

0.7112 cm, ` = 0.9271 cm, yc = b/2.

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

30

20

10

0

-10

-20

-30Nor

mal

ized

Sus

cept

ance

Frequency (GHz)

MeasuredSimulation

Figure 6.2: Normalized susceptance of a wide resonant strip in waveguide.


6.2.2 Resonant patch array

As another example, a resonant patch array consisting of six metal patches and

supported by a dielectric slab in a rectangular waveguide (Fig. 6.3) is analyzed for

application in high-frequency electromagnetic and quasi-optical transmitting and

receiving systems [41,85].

Results are obtained for the frequency band 8–12 GHz. In this frequency

range the air-filled portions of the waveguide support only one propagating mode

(TE10) while the dielectric slab accommodates multimodes. The magnitude of the

reflection coefficient for the dominant mode is −26 dB as shown in Fig. 6.4. The

phase angle is given in Fig. 6.5 showing the resonant properties of the structure.

z

x

y 0

al εr

b

d τ

τ

ττ

c

y

x

y

x

Figure 6.3: Geometry of patch array supported by dielectric slab in a rectangular

waveguide: a = 1.0287 cm, b = 2.286 cm, ` = 2.5 cm, εr = 2.33, d = 0.4572 cm, c =

0.3429 cm, τx = 0.1143 cm, τy = 0.2286 cm.


7.0 7.5 8.0 8.5 9.0 9.5 10.0Frequency (GHz)

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de S

(dB

)

S

S11

21

Figure 6.4: Magnitude of S11 and S21 for the patch array embedded in a waveguide.

7.0 7.5 8.0 8.5 9.0 9.5 10.0Frequency (GHz)

−200

−150

−100

−50

0

50

100

150

200

Pha

se S

(de

gree

s)

S

S

11

21

Figure 6.5: Phase of S11 and S21 for the patch array embedded in a waveguide.


6.2.3 Strip-slot transition module

To verify both GSMs for strips and slots, cascaded together, a strip-slot transition

module is analyzed using the GSM-MoM technique described in the previous chap-

ters. The results obtained for the transmission and reflection coefficients for the

dominant TE10 mode are compared with two other techniques based on the FEM

and MoM methods. A commercial High Frequency Structure Simulator (HFSS)

based on the FEM is used for comparison as well as an inhouse MoM program

utilizing a Green’s function for the composite structure (strip-slot) [86].

0 τ

ε1 ε 2 ε3

zx

y

ba

Figure 6.6: Slot-strip transition module in rectangular waveguide: a = 22.86 mm,

b = 10.16 mm, τ = 2.5 mm.

The structure consists of two layers with electric (strip) and magnetic

(slot) interfaces as shown in Fig. 6.6. The strip and slot dimensions are 0.6 mm

× 5.4 mm and 5.4 mm × 0.6 mm, respectively. The relative permitivities of the

dielectric materials used are ε1 = 1, ε2 = 6, and ε3 = 1.


18.5 18.8 19.1 19.4 19.7 20.0 20.3Frequency (GHz)

−6

−5

−4

−3

−2

−1

0

Mag

nitu

de S

11 (

dB)

MoM−GSM MoM HFSS

Figure 6.7: Magnitude of S11 for the strip-slot transition module.

18.5 18.8 19.1 19.4 19.7 20.0 20.3Frequency (GHz)

−200

−160

−120

−80

−40

0

40

80

120

160

200

Pha

se S

11 (

degr

ees) MoM−GSM

MoM HFSS

Figure 6.8: Phase of S11 for the strip-slot transition module.


18.5 18.8 19.1 19.4 19.7 20.0 20.3Frequency (GHz)

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de S

21 (

dB)

MoM−GSM MoM HFSS

Figure 6.9: Magnitude of S21 for the strip-slot transition module.

18.5 18.8 19.1 19.4 19.7 20.0 20.3Frequency (GHz)

−200

−160

−120

−80

−40

0

40

80

120

160

200

Pha

se S

21 (

degr

ees) MoM−GSM

MoM HFSS

Figure 6.10: Phase of S21 for the strip-slot transition module.


Very good agreement is obtained for all three methods as shown in Figs.

6.7, 6.8, 6.9, and 6.10 representing the magnitude and phase of the reflection and

transmission coefficients. A minimum reflection coefficient of -5.5 dB is achieved

at 19.64 GHz. The number of modes considered in the cascading process for the

GSM-MoM technique is 128 modes.

Note that eventhough the dispersion behavior of the scattering param-

eters is shown for the dominant TE10 mode, the X-band waveguide is overmoded

in the frequency range (18.5–20.3 GHz), specially in region 2 where the dielectric

constant is high.

6.2.4 Shielded dipole antenna

In this section a dipole antenna (Fig. 6.11) of length (L) 8 mm and width (W )

1 mm is investigated. The antenna is placed in the center (X1 = X2 = a2

) of a

hollow rectangular waveguide WR-90 with both waveguide ports perfectly matched

(no reflections). This antenna has been investigated by Adams et al. in [87]. In [87]

a finite gap excitation was assumed to account for the gap capacitance. In our

implementation the input impedance of the antenna is calculated using a delta gap

voltage model. It is shown that good agreement is obtained for both the real and

imaginary parts of the input impedance (Fig 6.12) for the frequency range 8.0 to

12.5 GHz. The input impedance is shown to be capacitive specially at the lower

end of the frequency range, indicating coupling to evanescent TM modes instead of

evanescent TE modes.

To investigate the effect of the antenna position within the waveguide


W

L V

X

Y

b

aX X1 2

Figure 6.11: Center fed dipole antenna inside rectangular waveguide.

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5−300

−250

−200

−150

−100

−50

0

50

frequency (GHz)

Inpu

t Im

peda

nce

(Ohm

s)

Real−Part (GSM−MoM) Imaginary−Part (GSM−MoM) Real−Part (MoM) Imaginary−Part (MoM)

Figure 6.12: Comparison of Real and Imaginary parts of input impedance. GSM-

MoM (developed here), MoM [87].


8 8.5 9 9.5 10 10.5 11 11.5 12 12.5−300

−250

−200

−150

−100

−50

0

50

frequency (GHz)

Inpu

t Im

peda

nce

(Ohm

s)

Real−Part (centered) Imaginary−Part (centered) Real−Part(off−centered) Imaginary−Part (off−centered)

Figure 6.13: Calculated input impedance for centered and off-centered positions.

on its input impedance, the antenna is placed at X1 = 3 mm away from the vertical

waveguide wall. Considerable variation in the input impedance is observed in Fig.

6.13 due to the close proximity to the waveguide wall.

6.2.5 Shielded microstrip filter

The GSM-MoM method can also be applied to completely shielded microwave and

millimeter wave structures. Numerical results have been obtained for the specific

example of the shielded microstrip filter shown in Fig. 6.14. The filter is contained

in a box of dimensions 92× 92× 11.4 mm (a× b× c). The substrate height is 1.57

mm and it has a relative permitivity of 2.33.

In analysis, the structure is decomposed into three layers as shown in


Fig. 6.15, with layers 1 and 3 being the top and bottom covers, respectively. The

covers are perfect conductors and hence their GSMs are diagonal matrices with −1

as diagonal elements. Layer 2 is a metal layer with ports.

Port 1

Port 2

23 mm

92 m

m

92 mm

18.4

mm

4.6 mm

4.6 mm

Figure 6.14: Geometry of a microstrip stub filter showing the triangular basis func-

tions used. Shaded basis indicate port locations.

The excitation ports are modeled by the delta-gap voltage model pro-

posed by Eleftheriades and Mosig [88] (the current basis functions for the excitation

ports are shaded in Fig. 6.14). This serves two purposes, to ensure the current

continuity at the edges and to allow the direct computation of network parameters

without the need to extend the line beyond its physical length. It should be noted

that these half-basis functions can only be used, for direct port computation as de-

scribed in [88], at the microstrip-wall intersection. The equivalent circuit model of

the port representation using half basis function is shown in Fig. 6.16. The voltage

source V is the delta gap voltage source accompanying the half basis function.


TOP COVER

METAL LAYER

BOTTOM COVER

Figure 6.15: Three dimensional view illustrating the layers of the stub filter.

-V+

Figure 6.16: Port definition using half basis functions.


The GSM of layer 2 is computed using the method described in Chapters

3 and 4. The number of modes considered in the GSM for layers 1 and 3 is 287. Layer

2 has 289 ports, 287 modes and 2 circuit ports. After cascading the three layers the

modes are augmented. The final scattering matrix has rank two representing the

circuit ports of the filter. This is illustrated in Fig. 6.17.

SHORTCIRCUIT

WAVEGUIDE SECTION

..

. ... WAVEGUIDE

SECTION... ..

. SHORTCIRCUIT

V1 V2+ - + -

MICROSTRIP

V1 V2

+

-

+

-FILTERSHIELDED

CASCADING

Figure 6.17: Block diagram for the GSM-MoM analysis of shielded stub filter.

The reflection and transmission coefficients S11 and S21 are calculated

in Figs. 6.18 and 6.19, respectively. The transmission from port 1 to port 2 is

approximately −37 dB at 2.7 GHz and compares favorably with previously reported

results [88].

To explain the box effect appearing in the reflection and transmission

coefficients, a plot of the propagation constant diagram is shown in Fig. 6.20. The


solid and dashed curves represent the air filled and the dielectric substrate regions,

respectively. It is observed that the notches in the S11 and S21 curves (at 2.2 and

3.4 GHz) correspond to the cut off frequencies of certain modes in the dielectric

substrate.11

S

1.5 2 2.5 3 3.5 4

Frequency (GHz)

(dB

)

-10

-12

-8

-6

-4

-2

0

1

Figure 6.18: Scattering parameter S11: solid line GSM-MoM, dotted line from [88].

Convergence curves for the scattering parameters are shown for various

numbers of modes in Figs. 6.21 and 6.22. As desired, convergence to a result

is asymptotically approached as the number of modes considered increases. The

need for a large number of modes is in intuitive agreement since dimensions are

small compared to the guide wavelength and so evanescent mode coupling should

dominate. This example represents an extreme test of the method developed here

and it also verifies the calculation of the GSM with circuit ports technique.


21(d

B)

S

1.5 2 2.5 3 3.5 4Frequency (GHz)

0

-5

-10

-15

-20

-25

-30

-35

-40

-451

Figure 6.19: Scattering parameter S21: solid line GSM-MoM; dotted line from [88].

1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

FREQUENCY (GHz)

RO

PA

GA

TIO

N C

ON

ST

AN

T (

Bet

a)

Figure 6.20: Propagation constant: solid lines for air, dashed lines for dielectric.


71 modes

127 modes

(dB

)11

199 modes

S

1.5 2 2.5 3 3.5 4-14

-12

-10

-8

-6

-4

-2

0

Frequency (GHz)1

Figure 6.21: Various cascading modes showing convergence of S11.

21(d

B)

S

1 1.5 2 2.5 3 3.5 4

-5

-10

-15

-20

-25

-30

-35

-40

-45

0

Frequency (GHz)

199 modes

127 modes

71 modes

Figure 6.22: Various cascading modes showing convergence of S21.


6.3 Patch-Slot-Patch Array

In this section a spatial power combiner structure is simulated and measured. The

system shown in Fig. 6.23 is divided into three blocks, transmitting horn, receiving

horn and a double layer array. Each block is simulated by a separate EM routine.

To reduce the coupling between the receiving and the transmitting patch antennas,

a strip-slot-strip transition is designed to couple energy from one patch to the other.

The patch antenna used is shown in Fig. 6.24 along with the amplifying unit.

Figure 6.23: A patch-slot-patch waveguide-based spatial power combiner.

6.3.1 Array simulation

The double layer array consists of three interfaces (patch-slot-patch). Each interface

is modeled separately using the Generalized Scattering Matrix-Method of Moment

technique. The method first calculates the MoM impedance matrix for an interface

from which a GSM matrix is calculated directly without the intermediate step of

current calculation. This enables the modeling of arbitrary shaped structures and


18

4632.5

75130

99

66

44

5

AMPLIFIER

Figure 6.24: Geometry of the patch-slot-patch unit cell, all dimensions are in mils.

the calculation of large number of modes needed in the cascade to obtain the required

accuracy. The nonuniform meshing scheme described in Chapter 4 is used here to

reduce the number of basis functions required in case of the uniform meshing scheme.

6.3.2 Horn simulation

The GSMs for the transmitting and receiving horns are calculated using the mode

matching technique [18]. The mode matching technique is known to be an efficient

method for calculating the GSM of horn antennas. For horns used in this study

the length of the Ka to X band waveguide transition is 16.51 cm (Fig. 6.25). This

long transition is to insure minimum higher order mode excitations. The GSM of

the waveguide transition is obtained using the mode matching technique program

described in [18]. The two important parameters in using the program are the num-


ber of steps and the number of modes considered. The number of sections needed

depends on the flaring angle of the transition and on the frequency of operation [19].

In choosing the step size, the λ/32 criteria can be used.

Y

XY

X

2

1

1

2

16.51 cm

Ka-bandWaveguide

X-bandWaveguide

Figure 6.25: Ka band to X band transition.


A typical double step plane junction section is shown in Fig. 6.25. The

smaller waveguide dimensions are X1 and Y1, and the larger waveguide dimensions

are X2 and Y2. At the double plane step discontinuity, incident and reflected waves

for all modes (evanescent and propagating) are excited, thus the total field can be

expressed as a superposition of an infinite number of modes. The total power in

all modes on both sides of the junction is matched according to the mode matching

technique. The GSM for the whole waveguide transition is obtained by cascading

the GSMs for all sections.

6.3.3 Numerical results

Numerical results are obtained for two cases a single cell and a 2×2 array. In the first

example a single unit cell (Fig. 6.24) is centered in an X-band waveguide. The circuit

was fabricated on a 0.381-mm-thick Duroid substrate with relative permitivity ε =

6.15. The GSM, for each layer, is calculated for 512 modes. The horns are simulated

using 80 modes. The calculated magnitude and phase of the transmission coefficient

S21 for the dominant TE10 mode is shown in Figs. 6.26 and 6.27, respectively. It is

shown that a transmission of approximately −13 dB is achieved at 32.25 GHz.

The second example is a two by two patch array. The same number

of modes is considered as in the first example. The results for the transmission

coefficient S21 is shown in Fig. 6.29. The maximum transmission obtained is −6

dB at 32.5 GHz and agrees well with our measured results. Again, this example is

for an overmoded waveguide, where many modes can be excited (approximately 18

modes in the air-filled sections of the X-band waveguide).


31.5 32 32.5 33 33.5 34−23

−22

−21

−20

−19

−18

−17

−16

−15

−14

−13

S21

(dB

)

FREQUENCY (GHz)

Figure 6.26: Magnitude of transmission coefficient S21.

31.5 32 32.5 33 33.5 34−200

−150

−100

−50

0

50

100

150

200

FREQUENCY (GHz)

PH

AS

E S

21 (

degr

ees)

Figure 6.27: Angle of transmission coefficient S21.


Figure 6.28: A two by two patch-slot-patch array in metal waveguide.

31.5 32 32.5 33 33.5 34−14

−13

−12

−11

−10

−9

−8

−7

−6

−5

FREQUENCY GHz

S21

dB

measured simulated

Figure 6.29: Magnitude of transmission coefficient S21.


6.4 CPW Array

Three examples, based on the magnetic current interface, are numerically investi-

gated in this section. The first two examples concentrate on the effect of waveguide

walls on the antenna input impedance. Two antennas are proposed, a folded slot

and a five slot antenna. The third example is a 3× 3 slot antenna array for the use

in spatial waveguide power combiners. The self and mutual impedances of the array

elements are calculated.

6.4.1 Folded slot antenna

The input impedance of a folded slot antenna shielded in a waveguide, shown in Fig.

6.30, is calculated. The dimensions of the waveguide are a = b = 20 cm and the

antenna dimensions are c = 7.8 cm and d = 0.9 cm. The dielectric constant is 2.2

and of thickness 0.0813 cm. The structure is decomposed into two layers. These are

a magnetic layer with ports, and a dielectric interface. The GSM of the magnetic

layer with ports is calculated using the GSM-MoM method and cascaded with the

dielectric interface to give the composite GSM.

The results are compared with an algorithm utilizing only MoM calcu-

lation by using the Green’s function for the composite structure (slot backed by

dielectric slab). In implementation of the MoM scheme piecewise testing and basis

functions are used [86]. The port impedance in this case is claculated directly from

the MoM impedance matrix.

The real and imaginary parts of the input impedance at the center of the


folded slot are shown in Figs. 6.31 and 6.32, respectively. The antenna resonates

at 1.5 GHz and has input resistance of 365 Ω at the resonant frequency. It can be

seen that the GSM-MoM solution agrees favorably with the MoM port calculation.

This verifies the cascading of the GSM for a magnetic layer with ports.

The high input resistance at resonance makes the design for a 50 Ω

match more challenging. For this reason York et al. suggested the use of multiple

slot antenna configurations for spatial power combining applications. The following

example is a demonstration of this idea.

2

2

X

Y

a

b

b

a

c

d

Figure 6.30: Geometry of the folded slot in a waveguide.


1 1.5 2 2.50

50

100

150

200

250

300

350

400

Frequency (GHz)

Rea

l−pa

rt (

Ohm

s)

GSM−MoM MoM

Figure 6.31: Real part of the input impedance for folded slot.

1 1.5 2 2.5−200

−150

−100

−50

0

50

100

150

200

250

Frequency (GHz)

Imag

inar

y−pa

rt (

Ohm

s)

GSM−MoM MoM

Figure 6.32: Imaginary part of the input impedance for folded slot.


6.4.2 Five slot antenna

In an effort to design a CPW antenna system matched to 50 Ω, York [4] suggested

a five slot configuration shown in Fig. 6.33. Since the input impedance is inversely

proportional to the square of the number of turns, increasing the number of slots

will automatically reduce the input impedance. Free space measurements [4] show

that an input return loss of −28 dB is observed at 10.5 GHz for the 5-slot antenna

shown in Fig. 6.33.

7.2 mm

2.7

mm 0.3mm

Figure 6.33: Five-slot antenna [4].

The GSM-MoM technique is used to calculate the input impedance of

the five-slot antenna inside a square waveguide. This gives an insight on the change

of the input impedance of the antenna when operating inside shielded environment.

In analysis, the CPW cell is composed of two layers. A magnetic layer

with ports and a dielectric interface. The scattering parameters for the magnetic


8 8.5 9 9.5 10 10.5 11 11.5 12−9

−8

−7

−6

−5

−4

−3

−2

−1

0

FREQUENCY (GHz)

S11

(dB

)

Figure 6.34: Magnitude of input return loss for 5 folded slots.

8 8.5 9 9.5 10 10.5 11 11.5 12−200

−150

−100

−50

0

50

100

150

200

FREQUENCY (GHz)

S11

(de

gree

s)

Figure 6.35: Phase of input return loss for 5 folded slots.


layer are computed as described in Chapter 3. The dielectric interface has a diagonal

scattering submatrices representing the transmission and reflection coefficients.

The antenna is placed in the center of a square waveguide of dimensions

22.86 × 22.86 mm. The geometry and dimensions of the antenna are shown in

Fig. 6.33. The dielectric thickness is 0.635 mm and its dielectric constant εr is 9.8.

The simulated input returned loss and its phase are shown in Figs. 6.34 and 6.35,

respectively. The input return loss has in fact increased from −28 in free space to

−8 dB when placed in the waveguide. This might result in less achievable gain when

using matched MMIC devices (to 50 Ω).

6.4.3 Slot antenna array

A 3 × 3 slot antenna array fed by CPW transmission lines is shown in Fig. 6.36.

The array consists of nine unit cells. Each unit cell is composed of two orthogonal

slot antennas, one for receiving and the other for transmitting. The amplifying

unit is a single ended amplifier. To properly design the amplifiers, it is essential to

calculate the driving point impedances of each antenna. This impedance depends

on self as well as mutual coupling between the antennas. The array is placed in a

square waveguide (a = b = 4 cm) and the antenna length is 0.72 cm.

The self impedance matrix (18 × 18) is calculated for the slot array for

the frequency range 8–12 GHz. The real and imaginary parts of the self impedances

of the antenna elements 1, 2, and 5 are shown in Figs. 6.37 and 6.38, respectively.

Resonance is achieved at 9.25 GHz for the self impedances. The impedance at

resonance is very high (1700 Ω) which make it difficult to match to 50 Ω. The value


of the self impedances are much less away from resonance as shown in Figs 6.39 and

6.40. Operating at 10 GHz is more appealing than operating at resonance since it is

easier to compensate for the imaginary part while designing the amplifier matching

circuit.

When designing an amplifier, the feedback from the output to the input

is very critical. Positive feedback might result in amplifier oscillations. The mutual

coupling between the input and output antennas provides that feedback path and

so it is essential to account for that kind of coupling. The mutual impedance for

the center unit cell is shown in Fig. 6.41. Ideally the coupling should be zero. To

minimize the coupling, the antennas should be at right angles.

1 2 3

4 5 6

7 8 9

10

11

12

13

14

15

16

17

18

Figure 6.36: A 3 × 3 slot antenna array shielded by rectangular waveguide.


8 8.5 9 9.5 10 10.5 11 11.5 120

200

400

600

800

1000

1200

1400

1600

1800

Frequency (GHz)

Rea

l−pa

rt (

Ohm

s)

Z11

Z22

Z55

Figure 6.37: Real part of self impedances.

8 8.5 9 9.5 10 10.5 11 11.5 12−1500

−1000

−500

0

500

1000

1500

Frequency (GHz)

Imag

inar

y−pa

rt (

Ohm

s)

Z11

Z22

Z55

Figure 6.38: Imaginary part of self impedances.


9.5 10 10.5 11 11.5 120

50

100

150

200

250

300

350

400

Frequency (GHz)

Rea

l−pa

rt (

Ohm

s)

Z11

Z22

Z55

Figure 6.39: Real part of self impedances.

9.5 10 10.5 11 11.5 12−900

−800

−700

−600

−500

−400

−300

−200

−100

0

Frequency (GHz)

Imag

inar

y−pa

rt (

Ohm

s)

Z11

Z22

Z55

Figure 6.40: Imaginary part of self impedances.


8 8.5 9 9.5 10 10.5 11 11.5 12−10

0

10

20

30

40

50

60

70

Frequency (GHz)

Mut

ual I

mpe

danc

e (O

hms)

Real Z5,14

Imaginary Z

5,14

Figure 6.41: Real and imaginary parts for the mutual impedance Z5,14.

6.5 Grid Array

Perhaps most of the early design efforts for spatial power combiners have been ori-

ented towards grid structures. The grid array systems are easy to build and fabricate.

Analysis and design techniques have emerged specifically for these structures, all for

free space case. In this section we will investigate grid arrays when constructed in

a shielded environment.

A 3 × 3 grid array is shown in Fig. 6.42. The array is composed of

nine unit cells. Each unit cell consists of two perpendicular dipole antennas, one

for receiving and the other for transmitting. The grid structure uses a differential

pair amplifying unit as that shown in Fig. 1.4. To accurately design the differential

pair, the driving point impedance of the antennas must be accurately calculated.


In this example, the impedance of the center cell is calculated. The

magnitude and phase of the input return loss are plotted in Figs. 6.43 and 6.44,

respectively. Resonance is achieved at approximately 31 GHz with −15.7dB return

loss.

Port

0.7 cm

1 cm

1 cm

Figure 6.42: A grid array inside a metal waveguide.


27 28 29 30 31 32 33 34−16

−14

−12

−10

−8

−6

−4

−2

0

FREQUENCY (GHz)

S11

(dB

)

Figure 6.43: Magnitude of input return loss.

27 28 29 30 31 32 33 34−200

−150

−100

−50

0

50

100

150

200

FREQUENCY (GHz)

Pha

se S

11

Figure 6.44: Angle of input return loss.


6.6 Cavity Oscillator

A multiple device oscillator using dipole arrays was proposed in [89, 90]. In both

referenced papers, a dedicated Green’s function was developed to model the cavity

oscillator and predict the coupling effects. In this section we will analyze a cavity

oscillator of the type described in [90] and shown in Fig 6.45.

b

c

a

B

A

Dipole Array

Figure 6.45: Geometry of a dipole array cavity oscillator.

6.6.1 Single dipole

The first example is a single dipole antenna inside a cavity. The cavity dimensions

are 22.6×10.2×5.0 mm (a×b×c) and the patch is centered in the transverse plane.

The dipole length and width are 6 mm and 1 mm, respectively. The frequency of

operation is chosen to be from 30 to 33.5 GHz. This means that the X band

waveguide is overmoded. The calculated input impedance of the dipole is shown

in Fig. 6.46. The dipole goes through resonance at 32.25 GHz. Below resonance

it is capacitive and above resonance it becomes inductive. A negative resistance

diode can be placed at the center of the dipole antenna by properly choosing the


30 30.5 31 31.5 32 32.5 33 33.5−800

−600

−400

−200

0

200

400

600

800

Frequency (GHz)

Inpu

t Im

peda

nce

(Ohm

s)

Real−Part Imaginary−Part

Figure 6.46: Input impedance of a dipole antenna inside a cavity.

resistive part. Also, since the diode is usually capacitive in nature, an inductive

input impedance might be chosen for the dipole antenna.

In the analysis, the structure is decompsed into three layers. These are

short-circuit, electric current interface with ports (dipole), and magnetic current

interface (patch). A block diagram illustrating the modeling process using the GSM-

MoM technique is shown in Fig. 6.47. After cascading all GSMs, the composit GSM

with ports will describe the relationship between the device ports and the output

modes. The composit GSM can be represented in terms of a scattering matrix,

impedance matrix, or an admittance matrix. Any of these forms may be employed

in nonlinear analysis using a nonlinear frequency-domain circuit simulator.

It is interesting to note that if a multilayer array (more than one trans-

verse active dipole array) of the same structure is used, the modeling scheme will


only require the analysis of one of these arrays. The analysis would then proceed

with cascading all sections to obtain the composite GSM. This is in comparison with

the direct MoM technique, where the coupling between all arrays must be accounted

for numerically. Hence the number of elements and the size of the MoM matrix are

increased.

SHORT

CIRCUIT

WAVEGUIDE

SECTION

.

.

.

.

.

.

.

.

. DIPOLE

WAVEGUIDE

SECTION

.

.

.PATCH

.

.

.

+ -V

MODES

(CIRCUIT-PORT)

.

.

.

+ -V

CASCADING

MODES

(CIRCUIT-PORT)

COMPOSITEGSM WITH

PORTS

Figure 6.47: Block diagram for the GSM-MoM analysis of cavity oscillator.


6.6.2 A 3× 1 dipole antenna array

As a second example a 3 × 1 dipole antenna array is placed inside a similar cavity

of the one described in the previous example. The antennas are shown in Fig.

6.48. The mutual and self scattering coefficients are calculated when all antennas

are of same lengths and width (6 × 1 mm) and the separations X1 = X2. Due to

symmetry there are only four distinct scattering coefficients (S11, S12, S13, and S22).

The magnitude and phase of the self and mutual scattering coefficients are shown

in Figs. 6.49 and 6.50, respectively. It is observed that the scattering coefficient

S22 has changed considerably, from the previouse example, due to coupling to the

other two antennas. This is illustrated by the nonresonant behaviour of S22 which is

now purely capacitive within the frequency range. The port scattering coefficients

calculated in this example is essential for designing an active array oscillator.

V1 V2 V3

X X1 2

X

Y

a

b

1 2 3

Figure 6.48: Dipole antenna array in a cavity.


30 30.5 31 31.5 32 32.5 33 33.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

frequency (GHz)

Mag

nitu

de o

f Sca

tterin

g C

oeffi

cien

ts

S11

S12

S13

S22

Figure 6.49: Magnitude of scattering coefficients for a dipole antenna array inside a

cavity.

30 30.5 31 31.5 32 32.5 33 33.5

−150

−100

−50

0

50

100

150

frequency (GHz)

Pha

se o

f Sca

tterin

g C

oeffi

cien

ts (

degr

ees)

S11

S12

S13

S22

Figure 6.50: Phase of scattering coefficients for a dipole antenna array inside a

cavity.

Chapter 7

Conclusions and Future Research

7.1 Conclusions

A generalized scattering matrix technique is developed based on a method of mo-

ments formulation to model multilayer structures with circuit ports. Four general

building blocks are considered. These are electric interface with ports, magnetic in-

terface with ports, dielectric interface, and perfect conductor. With these blocks the

method is applicable to almost all shielded transverse active or passive structures.

The GSM for each block is derived separately. The method explicitly incorporates

device ports and circuit ports in the formulation of both the electric and magnetic

current interfaces. The scattering parameters are derived for all modes in a single

step without the need to calculate the current distribution as an intermediate step.

Two cascading formulas are presented to calculate the composite scat-

tering matrix of a multilayer structure. This matrix is a complete description of

128

CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH 129

the structure. The technique can be applied to general structures as well as to

waveguide-based spatial power combiners. Various general type structures are sim-

ulated. These are a wide strip in waveguide, patch array on a dielectric slab, a

strip-slot transition module, shielded dipole antenna, and a shielded microstrip stub

filter. Spatial power combiners such as patch-slot-patch array, CPW array, grid

array, and cavity oscillator array are also simulated. Results are verified by either

comparisons to measurements or to other numerical techniques.

The interaction of layers is handled using a GSM method where an evolv-

ing composite GSM matrix must be stored to which only the GSM of one layer at a

time is evaluated and then cascaded. Thus the computation increases approximately

linearly as the number of layers increases. Memory requirements are determined by

the number of modes and so is independent of the number of layers. The result-

ing composite matrix can be reduced in rank to the number of circuit ports to be

interfaced to a circuit simulator.

An acceleration procedure based on the Kummer transformation is im-

plemented to speed up the MoM matrix elements. The quasistatic terms are ex-

tracted and evaluated using fast decaying modified Bessel functions of the second

kind. The convergence as well as the accuracy of the acceleration scheme are demon-

strated. In implementation two discretization schemes are used, uniform and nonuni-

form. Although simple, uniform discretization can not accurately represent struc-

tures with high aspect ratios nor can it capture fine geometrical details without

a gross increase in the number of elements. With the nonuniform scheme, finer

resolutions can be obtained for selective areas as desired.

The port parameters obtained from the electromagnetic simulator are


converted to nodal parameters using the localized reference node concept described

in Chapter 5. A scheme for the augmentation of a nodal admittance matrix by

a port-based matrix with a number of local reference nodes permits field derived

models to be incorporated in a general purpose circuit simulator based on nodal

formulation. The method is immediately applicable to modified nodal admittance

(MNA) analysis as the additional rows and columns of the MNA matrix are unaf-

fected by the augmentation.

7.2 Future Research

There are still many new ideas to be explored in the modeling of waveguide based

spatial power combiners. One feature that can be added to the current program

is the implementation of nonuniform triangular basis functions. This will enable

the modeling of geometrical curves and bends with much better accuracy. Another

feature is to include the losses due to dielectrics and metal portions.

It is well known that as the separation between layers decreases, in terms

of guide wavelength, the number of modes required in the GSM representation

will increase to achieve the required accuracy. This might render the procedure

impractical for very small separations (less than 0.01 λg). In this case it is more

efficient to construct a separate analysis module based on the MoM that takes into

account both layers in the Green’s function. Then a GSM is constructed for the

closely spaced layer using the calculated MoM matrix. We adopted this methodology

and implemented it for strip and slot layers [86]. There are other combinations to

be considered such as strip and strip, slot and slot layers, and even three layer


combinations.

Different types of Green’s functions such as the potential Green’s func-

tions and the complex images can be used instead of the electric- and magnetic-

type Green’s functions used here. This may reduce the CPU time, eventhough an

acceleration procedure might still be necessary.

Diakoptics in conjunction with the GSM-MoM scheme is another area

to be investigated. If the structure can be decomposed in the transverse plane into

separate structures related by a matrix then a three dimensional segmentation is

achieved (GSM and Diakoptics).

Furthermore, when the waveguide dimensions are several wavelengths,

then the number of modes involved in the modal expansion of the electromagnetic

fields becomes very large and approaches the free space case. It would be interesting

to see when the free space solution approaches the waveguide solution and if a hybrid

analysis can be employed.

In terms of applications to spatial power combining design, the structures

to be modeled are endless. Many novel designs can be thought of and perhaps

achieve the desired power combining efficiencies.

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Appendix A

Usage of GSM-MoM Code

Two steps are involved to run GSM-MoM:

• Converting layout CIF into input geometry file

• Running GSM-MoM with input parameter file

To convert the CIF file into the geometry file standard format a program calledyomoma is used. The command is

yomoma ’file.cif’ a.d

This will cause yomoma to convert the ’file.cif’ into a geometry file and give it thename ’geometry’. The input parameter file contains the necessary information torun GSM-MoM. These information are:

• Frequency range (start-stop-number of points)

• Waveguide dimensions

• Number and type of layers

• Separation between Layers

• Dielectric constants

140

APPENDIX A. USAGE OF GSM-MOM CODE 141

• Input geometry files

• Number of cascading modes

• Number of modes involved in the MoM matrix element calculation

• Output file names

A.1 Example

In this section a sample run of GSM-MoM is illustrated. The input file, geometryfile, and output file are listed bellow.

A.1.1 Input file

The input file used in the simulation is described as follows:

”FREQUENCY:””———————””Start at Frequency:” 1.d9”Stop at Frequency:” 2.5d9”Number of Frequency Points:” 31”GREEN:””————””m max:” 450”n max:” 450” ””WAVEGUIDE””——————-””a:(x direction):” 20.d0”b:(y direction):” 20.d0”xmax:(maximum x-dimensions)” 2.5d-1”number of units in maximum x-dimension:” 1” ””LAYERS:””————-””Number of Layers:” 2”Type of Layer 1:” 2 0


”Type of Layer 1:” 0 0”Geometry File of Layer 1:” ”cpw2.dat””Geometry File of Layer 2:” ”””epson 1:” 1.d0”epson 2:” 2.2d0”epson 3:” 1.d0”Normalizing Resistance (Ohms):” 50.d0”separation:” 0.0813” ””CASCADING PARAMETERS:””——————————————””m scatter max:” 10”n scatter max:” 10” ””QUASISTATIC PARAMETERS:””——————————————–””nqs max :” 450”m qsmax :” 450”m kummer :” 150”n kummer :” 150” ””power conservation flag:” 0”OUTPUT FILES””———————–”” 1 ports-s: (port S-parameters) ” ”ports-s.dat”” 2 ports-z: (port z-parameters) ” ”ports-z.dat”” 3 modes-s: (modes s-parameters) ” ”modes-s.dat”” 4 modes-z: (modes z-parameters) ” ”modes-z.dat”” 5 modes-ports-s: (both modes and ports s-parameters) ” ”modes-ports-s.dat”” 6 modes-ports-z: (both modes and ports z-parameters) ” ”z.dat”” 7 power-conservation: (power conservation check) ” ”conservation.dat”

A.1.2 Geometry file

The geometry file ’cpw2.dat’ is constructed as follows

X-center Y-center c1 c2 d1 d2 direction3.94e-02 2.39e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 14.06e-02 2.39e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 1


3.94e-02 5.39e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 14.06e-02 5.39e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 13.88e-02 5.33e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 5.21e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 5.09e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.97e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.85e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.73e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.61e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.49e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.37e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.25e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.13e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 4.01e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.89e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.77e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.65e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.53e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.41e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.29e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.17e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 3.05e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 2.93e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 2.81e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 2.69e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 2.57e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 33.88e-02 2.45e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 5.33e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 5.21e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 5.09e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.97e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.85e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.73e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.61e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.49e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.37e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.25e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.13e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 4.01e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 3.89e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 3.77e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 3.65e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 3.53e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 3.41e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 3


4.12e-02 3.29e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 3.17e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 3.05e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 2.93e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 2.81e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 2.69e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 2.57e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 34.12e-02 2.45e-02 1.2e-03 1.2e-03 1.2e-03 1.2e-03 3

Where X-center and Y-center are the center coordinates for the basisfunctions, ci and di are the x and y dimensions of the basis function described inChapter 4, direction is either 1 or 2 representing either x or y direction.

A.1.3 Output file

A sample of the output file ports-s.dat is shown bellow

Frequency Row Column Real+ imaginary1.0000000000000 5 5 -0.91289578394812+i* -7.1111105317860D-021.0500000000000 5 5 9 -0.59945553696502+i* 0.601334187792861.1500000000000 9 9 -0.32376060701267+i* 0.781986690489331.2000000000000 9 9 -3.9945984045278D-02+i* 0.840528638322451.2500000000000 9 9 0.21275834500019+i* 0.808995152766731.3000000000000 9 9 0.42130121767699+i* 0.715804122911731.3500000000000 9 9 0.58405064072347+i* 0.579349694030241.4000000000000 9 9 0.70158107310522+i* 0.408101400078131.4500000000000 9 9 0.77028276979754+i* 0.202042389774481.5000000000000 13 13 0.77401948182578+i* -4.6523796388117D-021.5500000000000 13 13 0.66256671113527+i* -0.348214722844451.6000000000000 13 13 0.28048687899227+i* -0.673404889077071.6500000000000 13 13 -0.71241823369701+i* -0.431410632232451.7000000000000 13 13 -7.4014430138632D-03+i* 0.436010246464521.7500000000000 21 21 0.15342478578630+i* 0.300937445132841.8000000000000 21 21 0.20419668859631+i* 0.231537353633501.8500000000000 21 21 0.21291162875193+i* 0.181260656689051.9000000000000 21 21 0.19584893946645+i* 0.145327296215061.9500000000000 21 21 0.16038858904577+i* 0.125089736672322.0000000000000 21 21 0.11212046082019+i* 0.123813330478172.0500000000000 21 21 5.7196728080260D-02+i* 0.145234531836302.1000000000000 21 21 1.9448436254146D-03+i* 0.19301324044885


2.1500000000000 21 21 -2.9969462383864D-02+i* 0.264043038486132.2000000000000 25 25 -4.9908826387294D-02+i* 0.353903999388452.2500000000000 25 25 -4.0663996216727D-02+i* 0.460774157055242.3000000000000 29 29 1.6785089962526D-02+i* 0.558061332305302.3500000000000 29 29 0.11020052940192+i* 0.642688991677522.4000000000000 29 29 0.15458275296470+i* 0.645011063579302.4500000000000 37 37 0.24259495341224+i* 0.700286992444702.5000000000000 37 37 0.33768340295208+i* 0.72427841760270

A.2 Makefile

# GSM-MoM MAKEFILE FOR SUN ULTRAS#FC=f77 -fastLEX=flexYACC=bison###CFLAGS= -g3CFLAGS=LDFLAGS= -L/ncsu/gnu/lib # linker flags

# FORTRAN SOURCE FILESFSRCS = scatter main.f mom layer.f empty guide.f matrix.fscatter layer.f scatter dielectric conductor.f cascade.fconservation.f circuit parameters.f zqs empty.f constants.fgama.f

OBJS = (FSRCS : .f = .o)

#LINKFORTRANOBJECTFILESTOCREATEEXECUTABLEFILEGSM −MoM :(OBJS)f77 -fast $(OBJS) $(LDFLAGS)

# Flex and Bison stuff#

# -rm -f $(OBJS)


A.3 Program Description

The program consists of the following subroutines:

• scatter main.f: This subroutine is the main program. It reads in the geometryfiles and the input data file and calls all other programs.

• mom layer.f: This is the MoM calculation subroutine. It calls the approperiatefunctions to calculate the MoM impedance and admittance matrix elements.

• empty guide.f: Contains functions used for the MoM matrix element calcula-tions.

• matrix.f: Contains the math routine for matrix inversion.

• scatter layer.f: calculates scattering parameters for each layer.

• scatter dielectric conductor.f: calculates scattering parameters for dielectricand conductor layers.

• cascade.f: Cascades all layers.

• conservation.f: Checks the power conservation of individual as well as cascadedlayers.

• circuit parameters.f: Calculates the circuit parameters.

• zqs empty.f: Calculates the quasistatic part of the MoM matrix

• constants.f: Calculates constants used by all routines.

• gama.f: Calculates the propagation constants of modes used in the GSM.

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