generalized scattering matrix modeling of...
TRANSCRIPT
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
CHAPTER 1. INTRODUCTION 3
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]
CHAPTER 1. INTRODUCTION 4
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
CHAPTER 1. INTRODUCTION 5
circuit block by a circuit simulator. Hence any nonlinear (iterative) analysis carried
out will only include one expensive electromagnetic simulation.
Commercial circuit simulation tools such as 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.
CHAPTER 1. INTRODUCTION 6
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.
CHAPTER 1. INTRODUCTION 7
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.
CHAPTER 1. INTRODUCTION 8
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
CHAPTER 1. INTRODUCTION 9
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-
CHAPTER 1. INTRODUCTION 10
lated for all interacting modes (propagating and evanescent) in one step by
considering the incident field to be a summation of waveguide modes instead
of a single mode. This eliminates the need to calculate scattering parameters
for every incident mode separately.
• Implementation of an efficient method of moments formulation for the analysis
of planar conductive and magnetic layers with uniform as well as nonuniform
meshing. The method is based on the extraction of the quasistatic part in the
Green’s function and transforming it into a fast converging series summation
utilizing the fast converging modified Bessel functions of the second kind.
• Theoretical development of a circuit theory to accommodate spatially dis-
tributed circuits allowing conventional harmonic balance and transient simu-
lators to be used. The theory is based on the local reference node concept
introduced in Chapter 4.
• The investigation of the effect of waveguide walls on antenna elements in spa-
tial power combiners. It is 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.
CHAPTER 1. INTRODUCTION 11
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.
CHAPTER 1. INTRODUCTION 12
• 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
CHAPTER 2. LITERATURE REVIEW 15
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.
CHAPTER 2. LITERATURE REVIEW 16
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.
CHAPTER 2. LITERATURE REVIEW 17
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-
CHAPTER 2. LITERATURE REVIEW 18
fier array was 13 dB at 7.5 GHz. A transverse magnetic (TM)-type dielectric slab
with Yagi-Uda slot antennas was introduced in [34]. An amplifier array of 10 GaAs
MMICs was fabricated. The array gain was 11 dB at 8.25 GHz and a 0.65 GHz
3-dB bandwidth was measured.
2.2 Waveguide Power Combiners
Resonant-cavity combiners have been successfully used in oscillator design. The
first design was proposed by Kurokawa and Magalhaes in their 1971 paper [35]. A
12 diode power combiner at X-band was proposed. Each diode was mounted at
one end of a stabilized coaxial line which was coupled to the magnetic field at the
sidewall of a waveguide cavity. The coaxial circuits were located at the magnetic
field maxima and hence spaced one-half wavelength apart as shown in Fig. 2.6. The
oscillator operated at 9.1 GHz and produced 10.5 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
CHAPTER 2. LITERATURE REVIEW 19
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
CHAPTER 2. LITERATURE REVIEW 20
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.
CHAPTER 2. LITERATURE REVIEW 21
Each antenna card accommodates two commercial GaAs MMICs. The results were
obtained with four cards placed in a rectangular waveguide. Fig. 2.9 illustrates
the basic idea, where only two antenna cards are inserted in the waveguide. The
advantages of this system are its wide-band characteristics and reusability. If one of
the antenna elements fail, only 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.
CHAPTER 2. LITERATURE REVIEW 22
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
CHAPTER 2. LITERATURE REVIEW 23
that better designs and higher power levels can be obtained. There are numerous
commercial Computer-Aided Design (CAD) tools in the area of microwave circuits
and antennas. These CAD tools can not simply be combined to analyze spatial
power combiners [41]. The reason is the complex nature of spatial combining sys-
tems. In such systems, many different components are integrated together such as
active devices (diodes, transistors, MMICs, etc.), passive lumped and/or distributed
elements, radiating elements, and cooling elements. In this section we will review
the efforts in developing CAD tools for spatial power combiners.
Many of the approaches developed to model spatial power combiners
assume infinite arrays in free space. 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.
CHAPTER 2. LITERATURE REVIEW 24
To accurately model spatial power combiners of small to moderate sizes,
the unit cell approach is neither practical nor accurate. The edge effects will not
be modeled and consequently the driving point impedances for all unit cells will
not be the same as predicted by the unit cell approach. A full wave analysis of the
whole structure is thus essential. Pioneering work was done here at North Carolina
State University to model open cavity resonators and grid arrays [45–51]. Heron [46]
developed a Green’s function for the open cavity resonator. This Green’s function is
composed of two parts: resonant and nonresonant terms. The fields were represented
using Hermite Gaussian wave-beams. Nuteson [51], implemented the previously
developed Green’s function using the method of moments. He also developed a
dyadic Green’s function for a lens system consisting of two lenses and an array of
active devices as that shown in Fig. 2.11. The Green’s function was derived by
separately considering the paraxial and 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.
CHAPTER 2. LITERATURE REVIEW 25
An integrated electromagnetic and nonlinear circuit simulation environ-
ment for spatial power combining systems was proposed in [52]. This represented the
first result where a full wave analysis and nonlinear circuit simulation were carried
out in the analysis of finite size grid arrays. A 2× 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
CHAPTER 3. MODELING USING GSM 28
voltages and currents) into the GSM formulation. This facilitates the incorporation
of the electromagnetic model of a microwave structure into a nonlinear microwave
circuit simulator as required in computer aided global modeling.
The problem of modeling multilayered structures with ports in a shielded
environment can be analyzed by at least two approaches. In the first, a specific
Green’s function for the proposed structure is constructed and then the method of
moments (MoM) [57] is directly applied to the entire structure. This results in severe
computational and memory demands for electrically large structures. The second
approach, proposed here, is to characterize each layer using a GSM with circuit
ports and then cascade this 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,
CHAPTER 3. MODELING USING GSM 29
input impedance for probe excited waveguides, and waveguide-based spatial power
combiners. From this point on we will refer to a circuit port as just a port, and
an electromagnetic port, which 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
CHAPTER 3. MODELING USING GSM 30
• Perfect conductor interface
An electric current interface is defined as an interface where the con-
ducting portions are small with respect to the dielectric portion. Hence, it is more
efficient to analyze the conducting (electric) than the nonconducting portion. Sim-
ilarly, a magnetic current interface is defined as an interface where the dielectric
portions 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-
CHAPTER 3. MODELING USING GSM 31
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:
CHAPTER 3. MODELING USING GSM 32
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].
CHAPTER 3. MODELING USING GSM 33
The general building block is shown in Fig. 3.3. Here an arbitrarily
shaped metalization is located at the interface of two dielectric media with relative
permitivities 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.
CHAPTER 3. MODELING USING GSM 34
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,
CHAPTER 3. MODELING USING GSM 35
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
CHAPTER 3. MODELING USING GSM 36
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)
CHAPTER 3. MODELING USING GSM 37
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)
CHAPTER 3. MODELING USING GSM 38
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)
CHAPTER 3. MODELING USING GSM 39
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
CHAPTER 3. MODELING USING GSM 40
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 .
CHAPTER 3. MODELING USING GSM 41
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-
CHAPTER 3. MODELING USING GSM 42
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)
CHAPTER 3. MODELING USING GSM 43
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.
CHAPTER 3. MODELING USING GSM 44
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)
CHAPTER 3. MODELING USING GSM 45
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.
CHAPTER 3. MODELING USING GSM 46
As previously stated, to construct the GSM efficiently it is necessary to
assume the incident wave as a summation of 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)
CHAPTER 3. MODELING USING GSM 47
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)
CHAPTER 3. MODELING USING GSM 48
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)
CHAPTER 3. MODELING USING GSM 49
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
CHAPTER 3. MODELING USING GSM 50
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
CHAPTER 3. MODELING USING GSM 51
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)
CHAPTER 3. MODELING USING GSM 52
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] +
CHAPTER 3. MODELING USING GSM 53
([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]
CHAPTER 3. MODELING USING GSM 54
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.
CHAPTER 3. MODELING USING GSM 55
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
CHAPTER 3. MODELING USING GSM 56
demonstrated in the following chapter.
For all other layers, the GSM is computed directly without the need to
call the MoM routine. After a GSM is calculated for a layer, it is then cascaded to the
previously calculated GSMs using the cascade routine. A power conservation check
is then used to check the accuracy of the calculation using the power conservation
routine. The sum of the squares of each column elements for the propagating modes
has to equal 1.
When all layers are computed and a single GSM for the structure is ob-
tained, a new frequency point is calculated. This will continue until a complete sweep
of the frequency range is achieved. The program produces output files containing
the composite scattering 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.
CHAPTER 3. MODELING USING GSM 57
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.
CHAPTER 4. MOM ELEMENT CALCULATION 60
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)
]
CHAPTER 4. MOM ELEMENT CALCULATION 61
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
CHAPTER 4. MOM ELEMENT CALCULATION 62
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)
CHAPTER 4. MOM ELEMENT CALCULATION 63
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)
CHAPTER 4. MOM ELEMENT CALCULATION 64
−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)
CHAPTER 4. MOM ELEMENT CALCULATION 65
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
CHAPTER 4. MOM ELEMENT CALCULATION 66
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)
CHAPTER 4. MOM ELEMENT CALCULATION 67
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)
CHAPTER 4. MOM ELEMENT CALCULATION 68
× 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)
CHAPTER 4. MOM ELEMENT CALCULATION 69
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)
CHAPTER 4. MOM ELEMENT CALCULATION 70
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).
CHAPTER 4. MOM ELEMENT CALCULATION 71
The convergence and percentage error of the impedance element for the
accelerated and direct double series summation are demonstrated in Figs. 4.4 and
4.5, respectively. The relative error is defined as |Zxx−Z∞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
CHAPTER 4. MOM ELEMENT CALCULATION 72
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.
CHAPTER 4. MOM ELEMENT CALCULATION 73
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)
CHAPTER 4. MOM ELEMENT CALCULATION 74
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
CHAPTER 5. LOCAL REFERENCE NODES 77
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.
CHAPTER 5. LOCAL REFERENCE NODES 78
(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
CHAPTER 5. LOCAL REFERENCE NODES 79
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
CHAPTER 5. LOCAL REFERENCE NODES 80
..
.
..
.
..
....
..
.
+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.
CHAPTER 5. LOCAL REFERENCE NODES 81
GROUP 1 GROUP 2
GROUP 3 GROUP 4
Figure 5.3: Grid array showing locally referenced groups.
CHAPTER 5. LOCAL REFERENCE NODES 82
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 .
CHAPTER 5. LOCAL REFERENCE NODES 83
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.
CHAPTER 5. LOCAL REFERENCE NODES 84
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.
CHAPTER 5. LOCAL REFERENCE NODES 85
• 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
CHAPTER 5. LOCAL REFERENCE NODES 86
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.
CHAPTER 5. LOCAL REFERENCE NODES 87
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].
CHAPTER 6. RESULTS 90
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.
CHAPTER 6. RESULTS 91
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.
CHAPTER 6. RESULTS 92
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.
CHAPTER 6. RESULTS 93
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.
CHAPTER 6. RESULTS 94
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.
CHAPTER 6. RESULTS 95
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.
CHAPTER 6. RESULTS 96
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
CHAPTER 6. RESULTS 97
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].
CHAPTER 6. RESULTS 98
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
CHAPTER 6. RESULTS 99
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.
CHAPTER 6. RESULTS 100
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.
CHAPTER 6. RESULTS 101
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
CHAPTER 6. RESULTS 102
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.
CHAPTER 6. RESULTS 103
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.
CHAPTER 6. RESULTS 104
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.
CHAPTER 6. RESULTS 105
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
CHAPTER 6. RESULTS 106
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-
CHAPTER 6. RESULTS 107
ber of steps and the number of modes considered. The number of sections needed
depends on the flaring angle of the transition and on the frequency of operation [19].
In choosing the step size, the 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.
CHAPTER 6. RESULTS 108
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).
CHAPTER 6. RESULTS 109
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.
CHAPTER 6. RESULTS 110
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.
CHAPTER 6. RESULTS 111
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
CHAPTER 6. RESULTS 112
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.
CHAPTER 6. RESULTS 113
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.
CHAPTER 6. RESULTS 114
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
CHAPTER 6. RESULTS 115
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.
CHAPTER 6. RESULTS 116
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
CHAPTER 6. RESULTS 117
of the self impedances are much less away from resonance as shown in Figs 6.39 and
6.40. Operating at 10 GHz is more appealing than operating at resonance since it is
easier to compensate for the imaginary part while designing the amplifier matching
circuit.
When designing an amplifier, the feedback from the output to the input
is very critical. Positive feedback might result in amplifier oscillations. The mutual
coupling between the input and output antennas provides that feedback path and
so it is essential to account for 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.
CHAPTER 6. RESULTS 118
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.
CHAPTER 6. RESULTS 119
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.
CHAPTER 6. RESULTS 120
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.
CHAPTER 6. RESULTS 121
In this example, the impedance of the center cell is calculated. The
magnitude and phase of the input return loss are plotted in Figs. 6.43 and 6.44,
respectively. Resonance is achieved at approximately 31 GHz with −15.7dB return
loss.
Port
0.7 cm
1 cm
1 cm
Figure 6.42: A grid array inside a metal waveguide.
CHAPTER 6. RESULTS 122
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.
CHAPTER 6. RESULTS 123
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
CHAPTER 6. RESULTS 124
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
CHAPTER 6. RESULTS 125
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.
CHAPTER 6. RESULTS 126
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.
CHAPTER 6. RESULTS 127
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
CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH 130
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
CHAPTER 7. CONCLUSIONS AND FUTURE RESEARCH 131
combinations.
Different types of Green’s functions such as the potential Green’s func-
tions and the complex images can be used instead of the electric- and 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.
Bibliography
[1] K. J. Sleger, R. H. Abrams Jr., and R. K. Parker, “Trends in solid-state mi-crowave and millimeter-wave technology,” IEEE MTT-S Newsletter, pp. 11-15,Fall 1990.
[2] Kai Chang and Sun Cheng, “Millimeter-wave power-combining techniques,”IEEE Trans. Microwave Theory and Techniques, vol. 31, February 1983, pp. 91-107.
[3] A. L. Martin, A. Mortazawi, B. C. Deloach Jr., “An eight-device extended-resonance power-combining amplifier,”IEEE Trans. Microwave Theory Tech.,vol. 46, June 1998, pp. 844-850.
[4] H. S. Tsai and R. A. York, “Quasi-Optical amplifier array using direct in-tegration of MMICs and 50 Ω multi-slot antenna,” 1995 IEEE MTT-S Int.Microwave Symp. Dig., May 1995, pp. 593-596.
[5] E. A. Sovero, J. B. Hacker, J. A. Higgins, D. S. Deakin and A. L. Sailer, “AKa-band monolithic quasi-optic amplifier,” 1998 IEEE MTT-S Int. MicrowaveSymp. Dig., June 1998, pp. 1453-1456.
[6] Scott C. Bundy, Wayne A. Shirmoa, and Zoya B. Popovic “Analysis of cascadedquasi-optical grids,” 1995 IEEE MTT-S Int. Microwave Symp. Dig., May 1995,pp. 601-604.
[7] N. S. Cheng, A. Alexanian, M. G. Case, and R. A. York “20 watt spatial powercombiner in waveguide,” 1998 IEEE MTT-S Int. Microwave Symp. Dig., June1998, pp. 1457-1460.
[8] J. Bae, T. Uno, T. Fujii, and K. Mizuno, “Spatial power combining of Gunndiodes using an overmoded waveguide resonator at millimeter wavelengths,”1998 IEEE MTT-S Int. Microwave Symp. Dig., June 1998, pp. 1883-1886.
132
BIBLIOGRAPHY 133
[9] J. Bae, T. Uno, T. Fujii, and K. Mizuno, “Spatial power combining of Gunndiodes using an overmoded-waveguide resonator at millimeter wavelengths,”IEEE Trans. Microwave Theory Tech., Dec.1998, pp. 2289-2294.
[10] Z. B. Popovic, R. M. Weilke, M. Kim and D. B. Rutledge, “A 100-MESFETplanar grid oscillator,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 2,February 1991, pp. 193-200.
[11] J. S. H. Schoenberg, S. C. Bundy and Z. B. Popovic, “Two-level power combin-ing using a lens amplifier,” IEEE Trans. Microwave Theory Tech., Dec. 1994,pp. 2480-2485.
[12] J. B. Hacker, M. P. de Lisio, M. Kim, C. M. Liu, S.-J. Li, S. W. Wedge,and D. B. Rutledge, “A 10-watt X-band grid oscillator,” IEEE MTT-S Int.Microwave Symp. Dig., May 1994, pp. 823-826.
[13] P. L. Heron, G. P. Monahan, J. W. Mink and M. B. Steer, “A dyadic Green’sfunction for the plano-concave quasi-optical resonator,” IEEE Microwave andGuided Wave Letters, August 1993, vol. 3, no. 8, pp. 256-258.
[14] J. Kunisch and Ingo Wolff, “The compression approach: a new technique forthe analysis of distributed circuits containing nonlinear elements,” in IEEEMTT-S Int. Microwave Symp. Workshop WSK, 1992, pp. 16–31.
[15] J. Kunisch and Ingo Wolff, “Steady-state analysis of nonlinear forced and au-tonomous microwave circuits using the compression approach,” InternationalJournal of Microwave and Millimeter-Wave Computer-Aided Engineering, vol.5, No. 4, 1995, pp. 241-225.
[16] P. Ciampolini, P. Mezzanotte, L. Roselli, and R. Sorrentino, “Accurate and effi-cient circuit simulation with lumped-element FDTD technique ,” IEEE Trans.Microwave Theory Tech. vol. 44, Dec. 1996, pp. 2207–2215.
[17] S. M. El-Ghazaly and T. Itoh, “Electromagnetic interfacing of semiconductordevices and circuits,” 1997 IEEE MTT-S International Microwave Symp. Dig.,vol. 1, June 1997, pp. 151-154.
[18] M. A. Ali, S. Ortiz, T. Ivanov and A. Mortazawi, “Analysis and measurementof hard horn feeds for the excitation of quasi-optical amplifiers,” IEEE MTT-SInternational Microwave Symp. Dig., vol. 2, June 1998, pp. 1469-1472.
[19] K. Liu, C. A. Balanis, C. R. Birtcher, and G. C. Barber, “Analysis of pyramidalhorn antennas using moment methods,” IEEE Transactions on Antennas andPropagation, vol. 41, No. 10, October 1993, pp. 1379-1389.
[20] Kenneth J. Russel,“Microwave power combining techniques,” IEEE Trans. Mi-crowave Theory Tech., vol. 27, No. 5, May 1979, pp. 472-478.
BIBLIOGRAPHY 134
[21] E. Wilkinson,“An N-way hybrid power divider,” IRE Trans. Microwave TheoryTech., vol. 8, Jan 1960, pp. 116-118.
[22] M. F. Durkin,“35 GHz active aperture,” IEEE MTT-S International MicrowaveSymp. Dig., June 1981, pp. 425-427.
[23] J. W. Mink,“Quasi-optical power combining of solid-state millimeter-wavesources,”IEEE Trans. on Microwave Theory and Techniques, vol. MTT-34no. 2, February 1986, pp. 273-279.
[24] Z. B. Popovic, M. Kim, and D. B. Rutledge, “Grid oscillators,” Int. JournalInfrared and Millimeter Waves, vol. 9, Nov. 1988, pp. 1003-1010.
[25] M. Kim, J. J. Rosenberg, R. P. Smith, R. M. Weikle II, J. B. Hacker, M. P. DeLisio, and D. B. Rutledge, “A grid amplifier,” IEEE Microwave Guided WaveLett., vol. 1, Nov. 1991, pp. 322-324.
[26] S. Ortiz, T. Ivanov, and A. Mortazawi “A CPW fed microstrip patch quasi-optical amplifier array,” 1998 IEEE MTT-S Int. Microwave Symp. Dig., June1998, pp. 1465-1469.
[27] R. A. York and R. C. Compton, “Quasi-optical power combining using mutualsynchronized oscillator arrays,” IEEE Trans. Microwave Theory Tech., vol. 39,June 1991, pp. 1000-1009.
[28] J. Birkeland and T. Itoh, “A 16 element quasi-optical FET oscillator power com-bining array with external injection locking,” IEEE Trans. Microwave TheoryTech., vol. 40, Mar. 1992, pp. 475-481.
[29] A. Mortazawi, H. D. Folt, and T. Itoh, “A periodic second harmonic spatialpower combining oscillator,” IEEE Trans. Microwave Theory Tech., vol. 40,May 1992, pp. 851-856.
[30] P. Liao and R. A. York, “A high power two-dimensional coupled-oscillator arrayat X-band,” IEEE MTT-S Int. Microwave Symp. Dig., May 1995, pp. 909-912.
[31] M. Rahman, T. Ivanov, and A. Mortazawi, “An extended resonance spatialpower combining oscillator,” IEEE MTT-S Int. Microwave Symp. Dig., June1996, pp. 1263-1266.
[32] J. W. Mink, and F. K. Schwering, “A hybrid dielectric slab-beam waveguide forthe sub-millimeter wave region,” IEEE Trans. Microwave Theory Tech., vol. 41,No. 10, October 1993, pp. 1720-1729.
[33] H. Hwang, G. P. Monahan, M. B. Steer, J. W. Mink, J. Harvey, A. Paol-lea, and F. K. Schwering, “A dielectric slab waveguide with four planar poweramplifiers,” IEEE MTT-S Int. Microwave Symp. Dig., May 1995, pp. 921-924.
BIBLIOGRAPHY 135
[34] A. R. Perkons, and T. Itoh, “A 10-element active lens amplifier on a dielectricslab,” IEEE MTT-S Int. Microwave Symp. Dig., June 1996, pp. 1119-1122.
[35] k. Kurokawa, and F. M. Magalhaes, “An X-band 10-Watt multiple-IMPATToscillator,” Proc. IEEE , Jan. 1971, pp. 102-103.
[36] k. Kurokawa, “The single-cavity multiple device oscillator,” Proc. IEEE , vol.19, October 1971, pp. 793-801.
[37] R. S. Harp and H. L. Stover, “Power combining of X-band IMPATT circuitmodules,” IEEE Int. solid-state circuit conf., 1973.
[38] S. E. Hamilton, “32 diode waveguide power combiner,” IEEE MTT-S Int. Mi-crowave Symp. Dig., 1980, pp. 183-185.
[39] M. Madihian, A. Materka, and S. Mizushina, “A multiple-device cavity os-cillator using both magnetic and electric coupling mechanisms,” IEEE Trans.Microwave Theory Tech., vol. 30, No. 11, November 1982, pp. 1939-1943.
[40] R. Bashirullah, and A. Mortazawi, “A slotted waveguide quasi-optical poweramplifier,” IEEE MTT-S Int. Microwave Symp. Dig., June 1999.
[41] Active antennas and Quasi-Optical Arrays. Edited by A. Mortazawi, T. Itohand J. Harvey. New York, NY: IEEE Press, 1999.
[42] S. C. Bundy and Z. B. Popovic, “A generalized analysis for grid oscillatordesign,” IEEE Trans. Microwave Theory Tech., vol. 42, Dec. 1994, pp. 2486-2491.
[43] L. Epp, P. Perez, R. Smith, “Generalized scattering matrices for unit cell char-acterization of grid amplifiers and device de-embedding,” IEEE AP-S Int. An-tennas and Propagation Symp. Dig., 1995, pp. 1288 - 1291.
[44] L. Epp and R. Smith,“A generalized scattering matrix approach for analysisof quasi-optical grids and de-embedding of device parameters,”IEEE Trans.Microwave Theory Tech., vol. 44, May 1996, pp. 760-769.
[45] P. L. Heron, G. P. Monahan, J. W. Mink, F. K. Schwering, and M. B. Steer,“Impedance matrix of an antenna array in a quasi-optical resonator,” IEEETrans. Microwave Theory Tech., vol. 41, Oct. 1993, pp. 1816-1826.
[46] P. L. Heron, Modeling and Simulation of Coupling Structures for Quasi-OpticalSystems, Ph.D. Dissertation, North Carolina State University, 1993.
[47] G. P. Monahan, P. L. Heron, M. B. Steer, J. W. Mink, and F. K. Schwering,“Mode degeneracy in quasi-optical resonators,” Microwave and Optical Tech-nology Letters, vol. 8, Apr. 1995, pp. 230-232.
BIBLIOGRAPHY 136
[48] T. W. Nuteson, G. P. Monahan, M. B. Steer, K. Naishadham, J. W. Mink, andF. K. Schwering, “Use of the moment method and dyadic Green’s functions inthe analysis of quasi-optical structures,” IEEE MTT-S Int. Microwave Symp.Dig., May 1995, pp. 913-916.
[49] T. W. Nuteson, G. P. Monahan, M. B. Steer, K. Naishadham, J. W. Mink, K.Kojucharow, and J. Harvey, “Full-wave analysis of quasi-optical structures,”IEEE Trans. Microwave Theory Tech. vol. 44, May 1996, pp. 701-710.
[50] T. W. Nuteson, M. B. Steer, K. Naishadham, J. W. Mink, and J. Harvey,“Electromagnetic modeling of finite grid structures in quasi-optical systems,”IEEE MTT-S Int. Microwave Symp. Dig., June 1996, pp. 1251-1254.
[51] T. W. Nuteson, Electromagnetic Modeling of Quasi-Optical Power Combiners,Ph.D. Dissertation, North Carolina State University, 1996.
[52] 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 nonlinearcircuit simulation environment for spatial power combining systems,” IEEEMTT-S Int. Microwave Symp. Dig., vol.3 , June 1998, pp. 1473 - 1476.
[53] A. Alexanian, N. J. Kolias, R. C. Compton, and R. A. York, “Three-dimensionalFDTD analysis of quasi-optical arrays using Floquet boundary conditions andBerenger’s PML,” IEEE Microwave Guided Wave Lett., vol. 6, Mar. 1996,pp. 138-140.
[54] B. Toland, J. Lin, B. Houshmand, and T. Itoh, “FDTD analysis of an activeantenna,” IEEE Microwave Guided Wave Lett., vol. 3, Nov. 1993, pp. 423-425.
[55] G. F. Vanblaricum, Jr. and R. Mittra, “A modified residue-calculus techniquefor solving a class of boundary value problems,” IEEE Trans. Microwave TheoryTech., vol. 17, June 1969, pp. 302-319.
[56] R. Mittra and S. W. Lee, Analytical techniques in the theory of guided waves.New York: Macmillan, 1971, PP. 207-217.
[57] L. Dunleavy and P. Katehi,“A generalized method for analyzing shielded thinmicrostrip discontinuities,”IEEE Trans. Microwave Theory Tech., vol. 36, Dec.1988, pp. 1758-1766.
[58] U. Papziner and F. Arndt, “Field theoretical computer-aided design of rectan-gular iris coupled rectangular or circular waveguide cavity filters,” IEEE Trans.Microwave Theory Tech., vol. 41, March 1993, pp. 462-471.
[59] T. Shibata and T. Itoh,“Generalized-scattering-matrix modeling of waveguidecircuits using FDTD field simulations,”IEEE Trans. Microwave Theory Tech.,vol. 46, Nov. 1998, pp. 1742-1751.
BIBLIOGRAPHY 137
[60] J. Wang, “Analysis of three-dimensional arbitrarily shaped dielectric or biolog-ical body inside a waveguide,”IEEE Trans. Microwave Theory Tech., vol. 26,July 1978, pp. 457–462.
[61] S. H. Yeganeh and C. Birtcher,“Numerical and experimental studies of currentdistribution on thin metallic posts inside rectangular waveguides,” IEEE Trans.Microwave Theory Tech., vol. 22, June 1994, pp. 1063-1068.
[62] H. Auda and R. Harrington,“Inductive posts and diaphragms of arbitrary shapeand number in a rectangular waveguide,”IEEE Trans. Microwave Theory Tech.,vol. 32, June 1984, pp. 606-613.
[63] H. Auda and R. Harrington,“A moment solution for waveguide junction prob-lems,”IEEE Trans. Microwave Theory Tech., vol. 31, July 1983, pp. 515-520.
[64] A. Bhattacharyya,“Multimode moment method formulation for waveguide dis-continuities,”IEEE Trans. Microwave Theory Tech., vol. 42, Aug. 1994, pp.1567-1571.
[65] J. M. Jarem,“A multifilament method-of-moments solution for the inputimpedance of a probe-excited semi-infinite waveguide,”IEEE Trans. MicrowaveTheory Tech., vol. 31, Jan. 1987, pp. 14-19.
[66] R. F. Harrington, Field Computation by Moment Methods. IEEE Press, 1993.
[67] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, 1991.
[68] Y. Rahmat-Samii, “On the question of computation of the dyadic Green’s func-tion at the source region in waveguides and cavities,” IEEE Trans. MicrowaveTheory Tech., vol. 23, Sep. 1975, pp. 762-765.
[69] H. Ghaly, M. Drissi, J. Citerne, and V. Hanna, “Numerical simulation of virtualmatched load for the characterization of planar discontinuities,” 1992 IEEEMTT-S International Microwave Symp. Dig., vol. 2, June 1992, pp. 1119-1122.
[70] C. Wan and J. Encinar,“Efficient computation of generalized scattering matrixfor analyzing multilayered periodic structures,”IEEE Trans. Antennas Propa-gation, vol. 43, Nov. 1995, pp. 1233-1242.
[71] N. Balabanian, T. Bickart, and S. Seshu, Electrical Network Theory. John Wileyand Sons, 1969.
[72] Harrington, R. F., Time-Harmonic electromagnetic fields. McGraw-Hill BookCo., New York 1961, pp. 106-120.
[73] Moment Methods in Antennas and Scattering. Edited by R. C. Hansen. Boston:Artech House, 1990.
BIBLIOGRAPHY 138
[74] P. Overfelt and D. White,“Alternate forms of the generalized composite scat-tering matrix,”IEEE Trans. Microwave Theory Tech., vol. 37, Aug. 1989, pp.1267-1268.
[75] D. G. Fang, F. Ling, and Y. Long,“ Rectangular waveguide Green’s functioninvolving complex images,” Proc. of the 1995 IEEE AP-S, vol. 2, 1995, pp. 1045-1048.
[76] M.-J. Park and S. Nam,“Rapid calculation of the Green’s function in theshielded planar structures,” IEEE Microwave and Guided Wave Let., vol. 7,Oct. 1997, pp. 326-328.
[77] S. Hashemi-Yeganeh,“On the summation of double infinite series field compu-tations inside rectangular cavities,” IEEE Trans. Microwave Theory Tech., vol.43, March 1995, pp. 641-646.
[78] Jie Xu,“Fast Convergent Dyadic Green’s Function in Rectangular Waveguide,”International Journal of Infrared and Millimeter Waves, vol. 14, No.9, 1993,pp. 1789-1800.
[79] G. V. Eleftheriades, J. R. Mosig, and M. Guglielmi,“A fast integral equationtechnique for shielded planar circuits defined on nonuniform meshes,” IEEETrans. Microwave Theory Tech., vol. 44, Dec. 1996, pp. 2293-2296.
[80] K. Knopp,Theory and Application of Infinite Series, New York: Hafner Pub-lishing Company, Inc., 1971.
[81] A. E. Ruehli and H. Heeb “Circuit models for three-dimensional geometriesincluding dielectrics” IEEE Trans. Microwave Theory Tech., vol. 40, July 1992,pp. 1507–1516.
[82] V. Rizzoli, A.Lipparini, and E. Marazzi, “A General-Purpose Program for Non-linear Microwave Circuit Design ,” IEEE Trans. Microwave Theory Tech. vol.31, No. 9, Sep. 1983, pp. 762-769.
[83] H. B. Chu and K. Chang,“Analysis of a wide resonant strip in waveguide,”IEEE Trans. Microwave Theory Tech., vol. 40, March 1992, pp. 495-498.
[84] R. Collin,Foundations for Microwave Engineering, McGraw-Hill, Inc., 1992.
[85] Active and Quasi-Optical Arrays for Solid-State Power Combining, Edited byR. York and Z. Popovic. New York, NY: John Wiley & Sons, 1997.
[86] A. Yakovlev, A. Khalil, C. Hicks, A. Mortazawi, and M. Steer,“ The generalizedscattering matrix of closely spaced strip and slot layers in waveguide,” Submittedto the IEEE Trans. Microwave Theory Tech..
BIBLIOGRAPHY 139
[87] A. Adams, R. Pollard, and C. Snowden, “A method-of-moments study of stripdipole antennas in rectangular waveguide,” IEEE Trans. Microwave TheoryTech., vol. 45, October 1997, pp. 1756-1766.
[88] G. Eleftheriades and J. Mosig,“On the network characterization of planar pas-sive circuits using the method of moments,”IEEE Trans. Microwave TheoryTech., vol. 44, March 1996, pp. 438-445.
[89] A. Adams, R. Pollard, and C. Snowden, “Method of moments and time domainanalyses of waveguide-based hybrid multiple device oscillators,” 1996 IEEEMTT-S International Microwave Symp. Dig., June 1996, pp. 1255-1258.
[90] C. Stratakos, and N. Uzungolu “Analysis of grid array placed inside a waveguidecavity with a rectangular coupling aperture,” International Journal of Numer-ical Modeling: Electronic Networks, Devices and Fields., vol. 10, January 1997,pp. 35-46.
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
APPENDIX A. USAGE OF GSM-MOM CODE 142
”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
APPENDIX A. USAGE OF GSM-MOM CODE 143
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
APPENDIX A. USAGE OF GSM-MOM CODE 144
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
APPENDIX A. USAGE OF GSM-MOM CODE 145
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)
APPENDIX A. USAGE OF GSM-MOM CODE 146
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.