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

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