RL877
FULL WAVE ANALYSIS OF MICROSTRIP
SLOT ANTENNAS AND COUPLERS
Norman L. VandenBerg
Linda P.B. Katehi
August, 1991
RL877 = RL877
ACKNOWLEDGEMENTS
Before presenting the material in this document which is a suirmmary of sonime of tlihe
work I have performed over the past four years, I would like to thank and acklnowledge the
many people and organizations who have contributed to and supported this researclh.
First. I would like to express my appreciation to the Radiation Laboratory and (Crtcr
for Space Terahertz Technology for financial support in tlie forml of a Researchl Assistantship. In particular, I thank Professors Val Liepa and ClienTo Tai for support in the
first year of my return to the University, and Professor Linda IKatelmi for the remainder of
my stay.
A special debt of gratitude is owed to Professor Tai for many years of teaching and
encouragement. My first course with Professor Tai was a Junior level course on Vector
Calculus in my undergraduate days and since then I have hlad tlie priviledge of attending
three other courses under his tutelage, including his course on Dyadic Green's Functions
in Electromagnetic Theory. As the reader may note, Professor Tai's approach hlas had a
strong influence on my work and I hope that I have been able to contribute in at least a
small way to the methodology he has pioneered.
I also would like to thank many of my associates who have contributed to this effort.
I especially thank Dr. William Harokopus and Mr. Andrew Engel, not only for their
friendship but their technical contributions as well. I also tliank other members of our
group, including Miss Emilie VanDeventer and Messrs. Nihad Dib and George Eleftheriades
for numerous helpful discussions. In addition, I would like to express my appreciation to
iii
Mr. Jeffrey Collins, Drs. Kamal Sarabandi, Kasra Barkeshli. Jianmning Jin. Mark RicoV'
Leland Pierce and many others too numerous to mention for their contributions through
bothl technical discussions and camaraderie.
A special mention is deserved by Dr. Harry LaFuse at the AlliedBendix Guidance Systems Division  Mishawaka, whose generous contribution of many classic text and reference
books on my departure to return to graduate studies, has made an invaluable contribution
to this work. Also, I am grateful to the AlliedBendix Communications Division for their
support in making many fixtures and measurements for the verification of the slot antenna
models, and to Texas Instruments for their similar support.
Finally I would like to thank my committee members for their helpful suggestions and
careful review of the manuscript. I especially acknowledge my advisor, Dr. Linda Katehi,
for her support an(l encouragement; she has been a constant source of ideas and insights
which lhave greatly contributed to this work.
iv
PREFACE
This dissertation expounds on the thesis that numerical methods and models can be developed which are sufficiently accurate to enable a microwave engineer to design Inicrostrilpfed slot devices, without resorting to the empirical or semiempirical techniques cliaracteristic of more traditional methods. This possibility is of importance since tile tra(itional
approaches are typically timeconsuming and therefore expensive. Tile advantages of numerical models are widely recognized, not only in tile microwave industry, but in nianv
other fields as well. With the rapid development of personal computers and workstatiols
in terms of availability and processing power, computeraided design (CAD) capability is
becoming even more important for the development of technology.
Slots have been used in microwave designs for many years, particularly in waveguidefed
antenna arrays. Stripline, and more recently, microstrip line has also been used frequently,
especially when there are active microwave components involved. It is now recognized that
combinations of these types of structures and devices offer many significant advantages
and will be needed to meet the requirements for advanced, stateoftheart systems whichl
have recently been proposed. Additional changes in technology have created the need
for the ability to analyze structures with multilayered substrates and superstrates and,
increasingly, there has been a push toward higher and higher operating frequencies. For
these reasons, we have developed the analytical and numerical methods to be presented
here. It will be shown that many of these issues can be addressed in terms of computeraided design and a considerable advancement and improvement over previous work in this
v
area hias been achieved.
In Chapter I, we begin by introducing various historical aspects of the use of slots in
microwave antennas. Thlrough an examination of these devices in the context of present
thinking, we will define the types of problems to be analyzed in this work; specifically,
microstripfed slot antenna elements and couplers. The basis for the numerical models will
be formulated in terms of integral equations. Fullwave analysis by means of exact Green's
functions is used with a view toward application of these methods at high frequencies where
other methods generally fail.
Chapter II will present the derivations of the necessary dyadic Green's functions which
will be used throughout the remainder of the work. The approach will use a 'field expansion
method' in terms of vector wave functions which will be explained and defined. The method
of scattering superposition will be used and a method, not previously presented for this
approach, employing impedance boundary conditions and field matching proceedures will
be developed. The entire approach is in contrast to the more widely used, and perhaps
more familiar, 'vector potential method'. Some comparisons to and deficiencies with past
usages of the latter approach will be pointed out. To further illustrate the differences, the
method of scattering superposition with impedance boundary conditions as applied to the
vector potential method will be illustrated by example in the appendix, since even for this
method, there are significant advantages that have not previously been presented.
The application of the method of moments to the integral equations is detailed in
Chapter III. The treatment will be generalized to include variations on the main thrust of
the work, to show howr slots and lines can be modelled with arbitrary orientations relative
to each other and the shielding structure. Although the applications discussed in later
chapters impose some simplifying assumptions on the geometry, the material here lays the
groundwork for further extensions to the work which may be implemented at a later date.
vi
Also included int Chapter III is some discussion of mathematical relationls an(l programminlg
techlniques which have been used to greatly reduce the computation time for generating
the required matrix elements.
Several applications are introduced in Chapter IV. This chapter is devoted to the
discussion of problems which can be reduced to two dimensions. Tile first part demonstrates the procedures for treatment of layered structures through analytical field matching
throughout the layers. Applications discussed include the evaluation of transmission line
parameters for microstrip lines and the visualization of field behavior for both shielded
strip and slot lines. The accuracy of the technique is verified for nicrostrip by comparing
to data available in the literature and a commercial computeraided design package. The
second part deals with the development of a model for scattering from vertical wires in
waveguides. For small diameter wires, the model can be greatly simplified compared to
approaches used in the literature. The validity of the model is verified by comparison to
experimental measurements. The motivation for the work in this chapter is to support the
modelling of applications discussed in Chapter VI.
Chapter V presents the analysis of microstripfed slot couplers. Expressions for the
Sparameters which characterize their behavior are derived based on network analysis of
even and oddmode excitations of the structure. The procedure is referred to as the
'Standing Wave Method' and involves an interpretation of the method of moments solution
for currents on the microstrip lines. This has become a sort of 'standard' approach to a
variety of similar problems but has some drawbacks as will be pointed out. Also presented
in this chapter is a discussion of the fixture design and experimental results which verify
the accuracy and validity of the method.
Chapter VI introduces the radiating slot or antenna element problem by deriving an alternative approach for finding network parameters. The method is based on an application
vii
of the Reciprocity Theorem and is here referred to as thle 'Reaction Tlheorem Mletliod' or
'Reaction Method'. The requirement for a new approach is a consequence of the limitatiolns
of the Standing Wave Method discussed in Chapter V. Historically, the Reaction Nlethod
was introduced many years ago for application to simple waveguidefed slots; however, for
our use, the structures are more complex and as a result, the application of the method
is also. The details of how the technique is applied are discussed in this chapter, together
with experimental results which demonstrate the capability of the numerical methods.
In Chapter VII, a variation of the radiating slot is explored. In this case, a slot of
finite thickness is introduced. Instead of simply making the slot thicker, whichl canll also be
modelled by this approach and has been treated through more approximate methods in the
past, the case where the stripfed slot couples to the radiating slot throughll anll intervelningl,
section of rectangular waveguide is presented. In this case, we find thle important result
that the slots can be detuned to extend the bandwidth of the element.
The dissertation concludes with Chapter VIII which summarizes the techniques developed. The points where the effort is judged to have succeeded are outlined as well as where
the numerical models fail. In the latter cases, the suspected causes for deficiencies in the
approach are discussed along with recommendations for remedies. These issues form the
basis of suggestions for the extension of this work and exploration of related areas. Specifically, the treatment of a 'Tbar' fed slot is discussed. This is a slot fed by a microstrip
line terminated in a Tjunction, whose branch arms are shortcircuited to the cavity side
walls. A similar device has been previously shown to provide extended bandwidth for
cavitybacked aperture antennas.
A few final comments about the mechanics of this work: A number of programs were
developed for the numerical modelling and analysis of the structures discussed as well as for
postprocessing the data. These programs were written almost exclusively in FORTRAN
viii
and run onI a variety of machines. including tile University of MNichigan IBNI 3090/GOOE
mainframe and IBM RSGOO6 workstations for tlhe numerically iltensive o)eratiolls. A
majority of tlie remainintg processinlg was performed onl HP/Apollo workstations. primarilv a DN2500. The manuscript was typeset using LATEX text processing software with
macros developed at thle University of Michigan for dissertation formats. Rectangular twodimensional plots used tllrougllout were produced in PostScript by a menudriven plotting
program developed jointly by the author and Dr. Leland Pierce. Smith Chlart plots were
produced by similar programs developed by tlhe author. Most of thile drawings were generated using eitller XFig or directly with PostScript and incorporated in PostScript form.
ix
TABLE OF CONTENTS
DEDICATION................................ i
ACKNOWLEDGEMENTS....................... iii
PREFACE....................................
LIST OF FIGURES........................... xiii
LIST OF TABLES............................ xvii
LIST OF APPENDICES............................. xviii
CHAPTER
I. INTRODUCTION............................ 1
1.1 Motivation and Background.................. 1
1.2 General Description of Analysis Approach..........1.3 Integral Equations and Notation.......................(
1.3.1 Dyadic Green's Functions for Physical Quantities.....
1.3.2 Dyadic Green's Functions for "Dual" Quantities........
1.3.3 Integral Solutions to Helmholtz Equations using Dyadic
Green's Functions....................... 10
1.4 Formulation by Application of Boundary Conditions......... 12
II. DYADIC GREEN'S FUNCTIONS....................... 18
2.1 Impedance Boundary Conditions for Layered Structures....... 19
2.2 Dyadic Green's Functions for an Infinite Covered Half Space... 22
2.2.1 Magnetic Current....................... 22
2.2.2 Electric Current......................... 31
2.3 Dyadic Green's Functions for Layer Filled Rectangular Waveguides 33
2.3.1 The Parallel Plate Green's Function............ 33
2.3.2 Rectangular Waveguide with Electric Currents....... 37
2.3.3 Rectangular Waveguide with Magnetic Currents...... 39
2.4 Dyadic Green's Functions for Layer Filled Rectangular Cavities.. 40
2.4.1 Homogeneously Filled Rectangular Waveguide: TE and
TM Modes......................... 40
2.4.2 Cavity with Opposing Impedance Walls: Electric Current 43
x
2.4.3 TE and TIM Modes inr Iomogeneously Filled Rectangiulri
Waveguide and Cavities with Opposing Impedance \Walls:
Magnetic Currents.............................. 44
III. METHOD OF MOMENTS FORMULATION................ 7
3.1 Definition of Coordinate Systems and Basis Functions........ 47
3.2 Excitation Models........................... 53
3.3 Expansion of the Dyadic Greenl's Functions for the Cavit... 56
3.4 Identification and Reduction of tile Integrands............61
3.5 Integration................................ 64
3.6 Numerical Evaluation Considerations................ 66
3.6.1 Preconmputation.......................66
3.6.2 Transform at io s........................ 67
3.6.3 Convergence, Algorithms and RunTinme......... 69
IV. ANALYSIS OF TWODIMENSIONAL STRUCTURES........
4.1 Application to (eneral NIMultilayered Shielded Microstrip Structures 78
4.1.1 LSE M odes...........................
4.1.2 LSM Modes...............................83
4.1.2 LSMI MIodes.......................... 83
4.1.3 Cross Ter......................... 4
4.1.4 Integratioii of Power and Reaction terms.......... 85
4.1.5 A pp ts...................... 86
4.2 Application to Multilayered Slotline................ 93
4.2.1 LSE Modes......................... 93
4.2.2 LSM Modes................................. 95
4.2.3 Cross Terms......................................... 96
4.2.4 Application to Finine and Suspended Finline...... 97
4.3 Scattering from Pins in Rectangular W\aveguide........... 99
4.3.1 Reflection Coefficient Formula Derived from the Reciprocity
Theorem............................ 10
4.3.2 Method of Moments Formulation......................102
4.3.3 Validation...........................103
4.4 Summary and Conclusions........................ 113
V. COUPLING THROUGH STRIPFED SLOTS............... 114
5.1 Network Analysis............................................ 114
5.2 Fixture Design........................... 11
5.3 Numerical and Experimental Results...............118
5.4 Summary......................... 126
VI. ANALYSIS OF STRIPFED RADIATING SLOTS................128
6.1 Equivalent Slot Impedance............................. 130
6.2 Numerical Results and Measurements..................... 132
xi
6.3 Sum m ary......................
VII. MODELLING OF THICKSLOTS.........
7.1 Numerical Results for Waveguide Coupled Slots
7.2 Summary and Conclusions............
1 1........... 1 A4 7........... 14 8........... 15 3
VIII. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE
W O R K.....................................
8.1 Summary of Achievements............
8.1.1 TwoDimensional Analysis.......
8.1.2 Couplers.................
8.1.3 Radiating Slots.............
8.1.4 Modelling of Thick Slots........
8.2 Model Limitations.................
8.3 Recommendation for Future Work........
APPENDICES.....................
BIBLIOGRAPHY...............................................................................................
154
155
155
156
157
158
159
160........... 164........... 191
xii
LIST OF FIGURES
Figure
1.1 A hypothetical integrated antenna employing nulmerous slot coupling structures........................................ 4
1.2 Cavity layout and equivalent magnetic currents for a radiating slot... 13
1.3 Electric and equivalent magnetic currents for microstrip)tonliicrostrip) coupler w ith a 'thick' slot............................ 1,
2.1 Impedance boundary condition representation of a mulll ilayered structure. 21
2.2 Layered infinite space as represented by impedance boundary conditions.. 26
2.3 Parallel plate waveguide coordinate system...................... 34
3.1 Cavity (x, y) and Stripfixed (v,v) Coordinate System........... 4
3.2 Current Expansion Functions......................... 49
3.3 Slotfixed Coordinate System (.................... 50
3.4 Convergence behavior for several parameters of a centered shielded microstrip transmission line. Dimensions a = b =.25, w = h.025,
C, = 9.7. Terminal values at 1000 modes were Zo = 49.63Q, (ref f. 6.90,
fix = 12.45A /A................................ 70
3.5 Convergence behavior for peak normalized resistance versus maximum
m ode num ber.................................. 74
3.6 Convergence of resonant slot length as a function of the number of modes. 75
4.1 A shielded stripline, uniform in the y direction with multilayered substrate
and superstrate................................. 79
4.2 Effective relative dielectric constant for shielded microstrip on alumina (a
= b = 250 mils, W = 25 mils, Cr = 9.7 ) compared to measurements [19]
and Touchstone [20]............................... 87
xiii
4.3 C' aracteristic impedance for shielded microst rip on alumina (a = b = 250
nmils, WV = 25 mils, ~r = 9.7 ) compared to Touchstone................ 87
4.4 Multimode propagation constants for the even modes in shielded microstrip (a = b = 12.7 mm, h = 1.27 mm, w =.635 mm r = 8.875).. 89
4.5 Stripline field distribution for the dominant propagating mode........... 90
4.6 Microstrip field distribution for the dominant propagating mode....... 91
4.7 Suspended microstrip field distribution for the dominant propagating mode. 92
4.8 Finline field distribution for the dominant mode........................ 97
4.9 Suspended finline field distribution for the dominant mode........... 98
4.10 A grid of vertical wires in rectangular waveguide.................. 100
4.11 Equivalent circuit for single post in rectangular waveguide...............103
4.12 NMagnitude and phase of reflection coefficient for a single conducting post
in rectangular waveguide. Point match Method of Moments (MoM) results
compared to the data of Marcuvitz [48] (A/a = 1.4)..................104
4.13 Behavior of element values for the equivalent circuit given by Marcuvitz (48] (A/a = 1.4)..................................... 105
4.14 Input impedance as a function of post diameter with a matched load port.
Post diameter increases in the counterclockwise direction from d/a =.005
to d/a =.25 in.005 steps, demonstrating the transition of the input
impedance from inductive to capacitive......................... 106
4.15 Shunt susceptance behavior for up to seven wires placed evenly spaced
across waveguide cross section........................ 107
4.16 Reflection coefficient measurements for a single post at various offsets from
the centerline compared to predictions from the simplified model. Wire
diameters (d =.025,.033,.039 in.) increase in the counterclockwise direction....................................... 109
4.17 Reflection coefficient measurements for pairs of wires at various offsets
from the centerline compared to predicted values. Wire diameters (d =.025,.033,.039 in.) increase in the counterclockwise direction....... 110
4.18 Measured reflection coefficients for centered wire grids of equal spacing and
increasing number. Wire diameters (d =.025,.033,.039 in.) increase in
the counterclockwise direction........................ 111
xiv
4.19 Measured and predicted reflection coefficients for wire grids with a single
pin missing in the sequence. Wire diameters (d =.025,.033,.039 in.)
increase in the counterclockwise direction...................... 112
5.1 Geometry of basic coupler......................... 1 15
5.2 Measurement of location of current peak relative to the slot for standing,
wave calculation................................. 116
5.3 Photograph of coupler fixture assembly.................... 11
5.4 Effect of the line stub length (1) on S21 and Sll magnitudes (s = 0, L =
0.25 inches and f = 12.0 GHz).......................119
5.5 Equivalent circuit for a 2port coupler....................... 120
5.6 Effect of the slot length on S21 and Sll magnitudes (s = 0, I = 0.049 incles
and f = 18.0 GHz)................................. 121
5.7 Comparison of theory and experiment for S21 magnitudes with s = 0,1
0.115 in., L., = 0.250 inches. The error bar indicates the influence on the
high frequency corner of a ~.005 inch change in stub length........ 122
5.8 Comparison of experiment and theory for S21 magnitudes with s = 0,1 =
0.080 in., L:, = 0.250 inches, illustrating the control of the high frequency
corner with the line stub length by comparison to Figure 5.7.............. 121
5.9 Experiment and theory for S21 magnitudes with s = 0,1 = 0.110 in.,
Ls = 0.153 inches, showing the effect of shortening the slot length...... 125
5.10 Comparison showing the effect of line separation with I = 0.115 in., L, =
0.250 in. and s = 0.125 inches............................. 125
5.11 The effect of line separation with shorter line stub length; I = 0.080 in.,
L, = 0.250 in. and s = 0.125 inches..................... 126
6.1 Stripfed cavitybacked slots with couplerfed and seriesfed corporate feed
arrangements........................ 129
6.2 Two cases for application of the Reciprocity Theorem.......... 130
6.3 Normalized resistance as a function of slot offset compared to Shavit's
measurements [67]....................................... 133
6.4 Resonant length as a function of slot offset compared to measurements [67].134
X V
6.5 Phoiograph of slot assembly pieces in various stages of fabrication.... 136
6.6 Uncompensated peak normalized resistance as a function of slot offset compared to measurements for uncovered slots...................... 137
6.7 Predicted resonant length as a function of slot offset for uncovered slots.
Crosses indicate actual test piece dimensions to be compared with the
frequency curve corresponding to the measured resonant frequencies listed
in the table................................................. 139
6.8 Sensitivity of resonant length to cavity length dimension 'a'............. 140
6.9 Compensated peak normalized resistance as a function of slot offset compared to measurements for uncovered slots...................... 142
6.10 Compensated peak normalized resistance as a function of slot offset compared to measulrements for covered slots.......................... 143
7.1 Stripline excitation of waveguidecoupled slots..................... 148
7.2 Typical variation of impedance as a function of slot length for a conventional 'thin' slot with slot offset as a parameter............... 149
7.3 Variation of impedance as a function of slot length with the difference in
top and bottom slot lengths as a parameter................. 149
7.4 Slot coupling (resistance) and resonant length dependence on offset and
waveguide length (Lw)............................. 150
7.5 Apparent bandwidth comparison for identical slots versus slots with different lengths based on an interpretation of the relationship of slot length
to w avelength.................................. 151
7.6 Apparent bandwidth comparison for identical slots versus slots with different lengths based on calculations at selected frequencies............. 152
8.1 TBar fed slot geometry............................................ 161
8.2 Two cases for application of the Reciprocity Theorem for the Tbar fed case. 162
B.1 The derivative of an arbitrary discontinuous function................ 177
xvi
LIST OF TABLES
Table
6.1 Measured resonant frequencies (GHz) and predicted resonant slot lengths
for uncovered slots with compensated cavity length. (Deviation from actual slot length listed in parenthesis in percent.).................. 142
6.2 Measured resonant frequencies (GHz) and predicted resonant slot lengths
for covered slots with compensated cavity length. (Deviation from the
actual slot length listed in parenthesis in percent.)......
144
xvii
LIST OF APPENDICES
Appendix
A. Vector Wave Function Expansions and Relations................ 165
B. On the Use of Vector Potential Functions for the Derivation of Green's Functions............................................... 171
B.1 Potential Theory Solutions for Electric Currents........... 171
B.2 Field Bohavior Electric Current................... 179
B.3 Potenti. Theory Solutions for Magnetic Currents.......... 181
B.4 Field Behavior Magnetic Current.................. 183
B.5 Partitioning............................... 184
C. On the Use of Vector Potentials with the Method of Scattering Superposition 186
xviii
CHAPTER I
INTRODUCTION
1.1 Motivation and Background
Tile use of slots in nicrowave circuits can be traced back at least as far as the 1940's
to the research efforts associated with World War II. This is particularly true in antenna
designs where slots have been used extensively as the primary radiating elements, but also
in feed networks as couplers. W ith the advent of monolithic microwave circuit technology
and also for more traditional construction methods, transitions from microstrip to slotline
are also becoming increasingly important in the design of microwave and millimeterwave
circuit elements. Two such transitions can be combined to form interconnects between
lines and )by using lines on opposite sides of the slot plane, a vertical transition can be
made without the use of viaholes. The basic structure, in a variety of forms, has a wide
range of applications to both broadband and narrowband connections and can be used
as a building block Ifor interconnects [35], phase shifters and inverters [30], directional
couplers [80], filters [50], and many other microwave components [2]. Whether used as a
coupling element between guided wave structures or to free space, both applications require
accurate analysis and design tools in order to minimize costly, timeconsuming empirical
techniques and rework.
Many papers have been published on the design of slotted waveguide antennas over
the past fortyfive years including "classics" by Stevenson [71] and Oliner [57], among
1
4.
2
others [51, 4, 5]. Elliot and his associates have contributed many inmp)ortant works onl tle
analysis and design of slotted rectangular waveguides and arrays [22, 23, 58, 70. 40]. More
recently, slots which are fed by microstrip or stripline have received attention [59. 62. 67]
due to advantages in cost, size, weight and conformability, among others. The development
of variations and new analytical techniques is ongoing [16, 8, 17, 87].
Numerous investigators have also presented approximate analytical techniques to characterize these types of structures with applications to circuit elements [12, 49, 50, 41, 2].
For example, a quasistatic analysis has been provided in [89], however, this may not
be sufficient, particularly for higher frequencies where end effects and higher order mnode
coupling become more significant. A more recent paper [66] presents a transmission line
analysis with excellent results, however, similar shortcomings would be expected. Hybrid
methods which combine twodimensional fullwave analysis with transmission line theory
as in [56], should certainly extend the validity of such models, but still may not account
for all discontinuity effects. Exact methods for microstriptoslotline transitions have only
recently begun to appear, such as the case reported in [91] which has applications to )pen
structures.
For couplers, practical considerations suggest that a shielding structure will almost
always be present. In fact, in many cases, a shielding structure must be introduced in
order to reduce crosstalk and to control undesirable coupling to other structures in tilhe
package (for example, DC control lines in phase shifters have been known to unexpectedly
become part of the microwave circuit). For antennas, cavitybacked slots have been used
for similar reasons; for example, to reduce internal mutual coupling on the feed network
side of an array. This often both simplifies the design process and improves the achievable
performance. For instance, it has been shown that the internal isolation of the slots in
phase steerable antennas can significantly improve scanning performance [47].
3
For these reasons, tile slots studied here will be enclosed by a cat a t least on on ne
side. The dimensions of the cavity can have a strong influence oil the electrical bellavior
of the slot. In some cases, these dimensions are used to control the slot characteristics and
in others they are used to suppress undesirable effects. In practice, slot antennas are often
covered by a protective dielectric sheet. The capability to handle these cases will also be
included in the analysis discussed dhere by employing tile exact Green's function for an infinite covered ground plane. Thlis function will be evaluated by a combination of numerical
and analytical techniques as described in [36], however. the details will not be reiterated
here. Also included is the capability to model multilavered( substrates and superstrates
which is becoming increasingly important for monolithic circuits and for systems which
may combine many circuit functions through three dimensional iltegration over multiple
layers.
An example of a hypothetical structure which employs numerous transitions of tile type
to be analyzed is illustrated in Figure 1.1. Shown is a slot antenna array loosely based
on a conventional waveguide fed slot array. Conventional arrays typically use waveguidetowaveguide slot coupling to feed slotted waveguide branch lines. Iere we assume tlhe
individual slots are fed by microstriptomicrostrip slot couplers as illustrated by the lowest
three layers of Figure 1.1. Each radiating slot is internally isolated by a cavity which may
contain active devices for power generation for example. The feed network may contain
additional active sources coupled by combinations of microstrip and slotlines and mav
be built on multilayered substrates for integration of additional antenna functions, such
as phase control, frequency conversion, detection, etc. This example illustrates just one
application with features drawn from topics being considered in the current literature which
also verifies the need for more accurate and advanced analysis and design tools for the types
of structures to be considered.
4
Radome
Radiating Slots
Cavity Isolation
 cand Cooling Walls
"'  P c=power Generation
'Bslsaa <aBModules
Coupling Slots:and Bias Circuits
Phase Lock,
Frequency and
F ePhase Control
Circuits
F1.1' A hypothetical integrated antenna employing n umero us
Figure 11 ~hp
tures.
1.2 General Description of Analysis Approach
For coupling through slots, the distribution (shape, amplitude and phase) of the field
in the slot is vital to the determination of the slot coupling behavior. For radiating slots.
this distribution has been investigated by Elliott [23] and others [59, 67, 70] )y treatilng tIhe
case of an isolated slot in an infinite ground plane. With the Green's function for a half
space and the fields in rectangular waveguide, they were able to write an integral equation.
The equation was solved by the Method of Moments to determine the field distribution.
Subsequently, the scattering parameters of the slot were found by an application of tlie
Reciprocity Theorem.
The same approach will be used here for radiating slots, referred to here as the lleactioi
Theorem Method' or 'Reaction Method'; although the presence of the strip and. especially
for microstrip, the inhomogeneous filling of the rectangular cross section precludes the use
of simple rectangular waveguide field solutions. In previous works [67], the presence of
the strip has been neglected by assuming a similarity between the strip modes and empty
waveguide modes. This not only places certain restrictions on the position of the strip
between the ground planes but also does not take into account the influence of the strip on
the fields near the slot. Here, the formulation will be generalized to include not only the
case of stripline where the cross section is homogeneously filled, but also the possibility of
multiple layer substrates and superstrates, allowing microstrip or more complex structures
to be analyzed. This will require the solution to the twodimensional 'waveguide' problem
in addition to the threedimensional 'cavity' solution which is the primary focus of this
work. The relative position of the strip within the shield will also be unrestricted which
has not been the case in some of the previous works.
For couplers, the same shielded strip substrate/superstrate capability will be included.
However, in this case, an alternative method will be used to extract the coupling parame
6
ters. This will be referred to as the 'Standing Wave Method' and is based on a more detailed
modelling of the strip current. The scattering parameters are extracted from the positions
of peaks in the strip current standing wave pattern, evaluated for even and odd excitations
of the structure. Although the Reciprocity Method is more general, the Standing Wave
Method is somewhat simpler and therefore may be more convenient.
In many cases thle thickness of the ground plane is very small and the slot can be
considered infinitesimally thin. However, for both types elements it may be necessary to
make a 'thick' slot for structural purposes or heat dissipation, for example. In fact, we
have found that this feature may be used to another advantage which is to broaden the
operating bandwidth of the element. These cases can be treated by the same techniques
and will be included in the formulation.
Highly accurate models are needed to reliably design these types of elements. Therefore, a fullwave spacedomain integral equation approach solved by the Method of Moments
will be used as opposed to quasistatic, modified quasistatic dispersion analysis, or other
similarly limited techniques. This approach is preferred since we can find exact Green's
functions for the structures described which account for all possible electromagnetic interactions. It also allows application of these techniques to problems without restriction as
to the size of dimensions relative to wavelength.
1.3 Integral Equations and Notation
In this section, appropriate integral equations will be derived. One of the primary
purposes of this section is to introduce the notation and conventions used throughout the
text and some of the fundamental equations and relationships which will be needed.
(
1.3.1 Dyadic Green's Functions for Physical Quantities
It is well known that time harmonic electromagnetic fields must satisfy NMaxwsell's Equations ( eJct time convention assumed and suppressed throughout):
V x E = j H (1.1)
V x H= J +jwcE (1.2)
where the constitutive relations D = cE and B = p H have been assumed. Also, J is
defined in terms of the movement of electric charge with time by the Continuity relation:
V.J = jwp (1.3)
Understanding these to be the governing independent equations, by taking the divergence
of Equations (1.1) and (1.2), it is found that the fields also obey
7 E= (1.4)
V H=O (1.5)
That the fields must satisfy the Helmholtz wave equation is readily derived by taking the
curl of Equation (1.1) and substituting (1.2) into the result, yielding
V xx EkE = jw/J (1.6)
where k2 = w2Iec. Similarly, it can be shown that H must satisfy
V x x x H k2H= V x (1.
Dyadic Green's functions can be introduced to represent the solutions to these equations
for infinitesimal current sources. For instance, GeJ will be used to represent the solution
to Equation (1.6) in the form
GeJ = [E ') + + E(z)i] (1.8)
Here, E(V) is the field resulting from an: directed current source
) ( R ) (1.9)
and so forth for the y and 4 components so that
V X V X Gej  k2CeJ IS(R R') (1.10)
w,!,re
l(R. I?') =  L[JX + J()+J ] +(1.11)
Likewise, for the magnetic fields, the dvadic Green's function mlist satisfy
V x v x J k2GJ = V x [I6(R R')] (1.12)
where GmJ represents
GmJ== jm i [.t[I+ TV(W T+ (1.13)
From Equation (1.10) and (1.12) it is seen that Gej is related to GCnj by
V x Gej =G n (1.14)
v x G mi I(R R') + k2Gej (1.15)
The terminology and notation which we use to refer to these functions is that Gej is
the dyadic Green's function of the electric field type (subscript 'e') for an electric current
(subscript 'J'). Similarly, GmJ represents the dyadic Green's function of the magnetic field
type (subscrip: "n') for an electric current.
1.3.2 Dyadic Green's Functions for "Dual" Quantities
The Duality Principle entails the proposal of a system of equations where
V x E=joH K (1.16)
V x H= jwe E (1.17)
9
K representing a fictitious magnetic current. This current in introduced as a matter of
convenience by which we can represent tangential electric fields in terms of equivalent
magnetic currents according to the Equivalence Principle. We then can take advantage of
the duality between these and the "physical" quantity equations to generate solutions to
tile equations [32, pp.98116]. The Helmholtz wave equations for this system are
V x HX7 Hk2  j2 = (1.18)
V x x E — k2E = — V x K (1.19)
Using the representation
GmlK= [HI)i + H +(z)] (1.20)
G mn satisfies
V7 X V GmK  mK= h(R R') (1.21)
where the inhomogeneous term represents
I^(R R') jwe[t) + I('y^ + lI(Z)] (1.22)
with
(= ^ (1.23)
jw(
and so forttl for the i and  components. Similarly, sGei satisfies
V x V x GeK  k2GeK = V x {6(R R') (1.24)
where GeiK represents
GeK =i([E)+ E(r)y+ E( z] (1.25)
Here it is evident that
V x Gm = GeK (1.26)
V x GeKl = I6(R  R') + k2CmK (1.27)
10
The notation and terminology are the same as before with the magnetic current represented
by the subscript 'K'.
1.3.3 Integral Solutions to Helmholtz Equations using Dyadic Green's Functions
Once GeJ, GmJ, GeK and GmK are known for a particular geometry and set of bounldary conditions, it is possible to find the corresponding field quantity for anyl distribution of
electric or magnetic current. The relationship to be used can be derived front the vectordyadic Green's second identity:
[JvPV x \ x Q (V xV x P) Q]dV
= [I I [n x V P]Q +(fx P)V x Q}dS (1.28)
To find the integral solution to
V x xE 2 E =jw J (1.29)
we use the electric dyadic Green's function with an electric current source, GeJ. By letting
P = E and Q = Gej in Equation (1.28), we find that
E( R) = jyJJJ J(R') GeJ( R R)dV'
 JJ {[nii x V'x E(R)] Gej( R,R)
+[nix E(')] V'x Ge( R', R)}dS' (1.30)
Note that in the process of deriving this result, the notation of R and R' has been interchanged in keeping with our conventional usage of R as the source position vector and
R as the observation position vector. (Primes throughout will be used to indicate source
coordinates.) Using the vectordyadic identity of triple products:
a. (b x c) = b (a x c) = (a x b). c
(1.31)
11
Equation (1.30) can be rewritten as
E(R) = J J R R)dlr'
+ {[ x ( R)] [ x Ge( R, R)]
[fix E( R)] [' xGeJ( R, R)]}dS' (1.32)
In our application, GeJ is used in a cavity with impedance walls representing the source
layer. It is assumed that the region outside this layer is bounded by the impedance walls
and a surface which requires either the radiation condition or the Dirichlet condition for
the E and Ge functions so that the [nfx ] terms in Equation (1.32) evaluate to zero on
the boundary. We also assume that this region contains no sources. If such is the case, the
surface integrals of Equation (1.32) disappear and we are left with
E(R) = jw/p J/Gej(R, R) J( R') dV' (1.33)
Here, the symmetry propertv of the dyadic Green's function:
Gej(R, R')= [GeJ( R, R)] (1.34)
has been applied ( 'T' indicating 'the transpose'), which can be shown using dyadicdyadic
Green's second identity as outlined in [76] under the assumptions stated above.
We can find the integral solution to
V xV x k2 H V xJ (1.35)
by returning to Equation (1.28) which with P = H and Q = Gtnj, reduces to
H(R) = G mJ( R )J(Rt) dV' (1.36)
using assumptions similar to those for is the fields with the appropriate radiation or Neumann boundary conditions enclosing the impedance walls. Alternatively, Equation (1.36)
12
can be obtained directly from Equation (1.33) by using
H( R) =  V x E( R) (1.37)
and
V x GeJ = Gmj (1.38)
The procedure for the magnetic currents is exactly the same resulting in
H(R) = jw JGmK( R, R) K(R')dV (1.39)
and
E(R) = J GeK( R, R') K(R')dV' (1.40)
1.4 Formulation by Application of Boundary Conditions
We now derive a set of integral equations by enforcing the boundary conditions for the
problem. To begin, we replace the slot openings in each region by a tangential, conductorbacked equivalent magnetic current in accordance with the Equivalence Principle as illustrated in Figure 1.2 (all conducting walls will be assumed to have perfect conductivity).
The problem is thereby separated into independent regions, coupled together by the magnetic currents as shown in the figure which also illustrates the treatment of finite slot
thickness. Since K = ih x E, using the same current on either side of the slot openings
enforces the continuity of the electric field in the slot.
In addition to the boundary conditions at the cavity walls, slot walls, ground plane
surface and dielectric interfaces which will be satisfied by finding the appropriate Green's
functions, we must also satisfy the following boundary conditions:
h x E (1.41)
13
Cover Slot
cover \
z = d'
z=c
Ecav L
z=d
Strip
Impedance
Boundary
Figure 1.2: Cavity layout and equivalent magnetic currents for a radiating slot.
on the microstrip;
n x T nx i lot = H (1.42)
at the slot opening to the cavity; and
n X Hl1t nx XWI = 7 (1.43)
at the slot opening in the dielectric covered plane. The S function is a vector describing
the source and is normally set to zero except when a gap generator is used to excite the
line'. The 'H function is used in Equation (1.42) when the slot is excited internally2,
or in Equation (1.43) when the source is external3. When ~ is set identically to zero,
Equation (1.41) enforces the boundary condition that the tangential electric field is to be
zero on the strip. Equations (1.42) and (1.43) enforce continuity of the tangential magnetic
fields over the slot openings when 'H is set to zero. The 'u' and 'L' subscripts will be used
'The gap generator is used only for the coupler problem as a mechanism for even and odd excitation of
the lines.
2The right hand side will be set to the incident H field of the dominant strip mode for the Reaction
Method formulation.
3The right hand side can be set to an incident plane wave field for the analysis of the slot as a scatterer
or receiving antenna, however, this problem will not be discussed further in this work.
14
throughout to indicate the upper and lower slot openings for the general case of a 'thick'
slot.
The electric field, E, in the interior of the cavity can be written as an integral in
terms of the electric and magnetic currents in the cavity using the relationships outlined
in Section 1.3.3:
Et= JIc JJt GeJ,int * Js dS' 
i trip is gotL
Similarly, Tnt anywhere in the cavity is given by
GeKint KL dS'
GeK,i~t ' KL dS'
(1.44)
(1.45)
=Ttnt  GmJ,int 'Js dS'  JW(c JJl G mK,int' KL dS'
trp ot
For a thick slot as shown, the H field in the slot can be written as
H = JWes / C
lot i iJs i ot L
The external H field is given by
mnK,slot KL dS'  jw(s f
slotu
Gm nKslot X Ku dS' (1.46)
H'ez = jWJd /I GmKext * Ku dS' (1.47)
slotv
Using these expressions in the boundary conditions given by Equations (1.411.43), we
arrive at the integral equations for the problem:
— strip _.trip
jwxc n X G eJ,int ' Js dS' J n x GeK,int ' KL dS' =
strip J slotL
(1.48)
J ix xG mJint' Js dS'
strip
J= JJ nXGmK<,lofKL dS' +
JJsllot L
SI/
f =slOL 
jWlc lot X G mKsint KL dS'
Jslotu
= H
(1.49)
J / slot L
=KslotU
n x G mK.slot KL dS'
J /slotd u
jw J J slotu
=lotu
n x G mKslot Ku dS'
=sl ot
f X GmK,ext KU dS'=
(1.50)
(1.50)
15
where the superscripts on the dyads imply evaluation of the dyad at the indicated location.
Throughout this work, Equation (1.48) and similar types will be referred to ias an electric
field integral equation (EFIE) since it is derived from a boundary condition oil t he electric
field. Similarly, both Equations (1.49) and (1.50) are magnetic field integral equations
(MFIEs).
Since each of Equations (1.481.50) involve the [ix ] term, from here on it will be
dropped with the understanding that only the tangential components of the various dyads
are used. Equations (1.481.50) can then be written:
f =str:p  =strip.
— jw C I G eJint J dS' G e,int ' KL dS' = c (1.51)
I Istrip Csl3 otL
f slot L s 1 Cf Lf s IotlotL
J1 G mJt int + t s GmK. slot d5
j+ w ] ] GmlIsloti Ku dS'= H (l.)2)
trip 'lotL
= slot L
 Gslotu + __0\lotU lotu m = H
i SlOt Gm otL dS  jm iGmK,[ slot+(dGmK.xt Kt: dS 'H
(1.53)
where all the terms involving i are excluded4. The unknown currents Js, iL and Klr can
have only tangential components on each of their respective structures, we therefore have
six scalar unknowns. Since only the tangential components of G are used, Equations ( 1.51 1.53) represent six scalar integral equations which are sufficient to solve for the unknown
currents.
For convenience of notation, we redefine Equations (1.511.53) as:
i ).Jg dS' +] G L3 dS' C ( 1.5 )
strip slotL
/L (l' JS d5'/"[/ L ~+(13) K 
3 f =(33) J35)_
trIp (J dS'+~] GC KL dS']+ C 1K dS',= ( H (1.55)
 (55)
II CKL dS'+ Ku dS'= H (1.56)
slotL slotu
4Throughout most of this work we will consistently use n = i where n will be orthogonal to the currents
involved. Therefore, no integral equation or Green's function evaluation will involve the ' components
which will subsequently be ignored.
16
where
=(11) =strip
G = ji cG,eJn (1.57)
=(13) =.strtp
G = G eKint (1.58)
=(31) =slot L
G = ^mJ i.nt (15)
=(33) lot s (1.59)OtL
G = j cGmKl,int + s GmK,sljot (1.60)
=(35) =slotL
G = jG mK.slot (1.61)
=(53) =slotu
G = jwsGmI.slot (1.62)
(55) t =slotu
G J= j' jssGm n.slot + d GmI<,ext (1.63)
For the radiating slot problem, it remains to find the various dyadic Green's function
components, solve the equations for the unknown currents, and interpret the results to
obtain parameters which characterize the slot's electrical behavior and properties.
The coupler problem differs only in the final equation. Equation (1.56) which involves
the Green's function for the halfspace is replaced by an EFIE as follows. The field on the
upper slot is now contained by another cavity which may also have a conducting strip as
illustrated in Figure 1.3. Under these circumstances, we then have the following integral
equations:
trtpL JJ etL
j // Gej SL dS'  U GeKtt L K dS' ~ (1.64)
[strt pL slot L
slot  =slot L =slot L _
Gm JL dS'  jw ] LG mK + s3otG mKj KL dS'
tripL lot L
— s lot L
+ Jw(slot G mG' Ku dS' = 7 (1.65)
lotu
=slotu \ =slotu — qllotu
G mJ Ju dS' + jw GmK + t K Ku dS'
tripu lot
ff =slotL u
 JWslot G1 m Kt L dS' = (1.66)
Jlotu
ff =trpU ff.StrTPU
jwl] G SdS' + G Jv dS' + G d = ~ (1.67)
tripu lotut
The subscripts u and L here indicate whether the source is associated with the upper or
17
Impedance
Boundary
Strip
z =d' 
Upper
Cavity
cav,u K
Z — C
z=d
Strip
Impedance
Boundary
Figure 1.3: Electric and equivalent magnetic currents for microstriptomicrostrip coupler
with a 'thick' slot.
lower cavity strip or slot opening.
CHAPTER II
DYADIC GREEN'S FUNCTIONS
In this chapter, the dyadic Green's functions needed to evaluate the integral equations
are derived. The method used in all cases is a 'field expansion method' using Vector
Wave Functions (VW'XFs) as opposed to the perhaps more conventional, 'vector potential
method'. Some of the Green's functions derived will not be needed in the later analysis
but are included in this chapter for completeness.
The use of vector wave functions with dyadic analysis is presented as an alternative.
It has the advantage of producing the complete dyadic Green's function in one solution.
The process also may involve a reduced number of simultaneous algebraic equations which
must be solved for unknown coefficients as compared to the vector potential method. The
disadvantage of the approach is that it requires dyadic analysis which may be unfamiliar.
although it is quite straightforward.
The field expansion approach using VFWs has been extensively detailed by Tai over an
extended period of time [72][79]. Nevertheless, the method is not widely employed which
may be due to several factors including:
1. Early development of the technique involved the use of the vector wave functions
designated L., M and N. As will be shown, the M and N functions have clear, physical
interpretations, however, the interpretation of the L functions is somewhat obscure
18
19
and its handling, at times, somewhat difficult. Consequently, the approach may have
been avoided.
2. An oversight in [72] but corrected in [77], stirred up some controversy wXhich may have
caused some to avoid the method (see [11] also for details and a list of references).
The current method is more mature, having evolved to a stage where the previous difficulties have been eliminated. Problems can now be solved in a methodical and straightforward
manner with no difficulties in physical interpretation (the need for the L functions calln be
avoided).
2.1 Impedance Boundary Conditions for Layered Structures
It can be shown that a plane wave in a homogeneous region exhibits a constant wave
impedance defined by the ratio of a component of the electric field to an orthogonal component of the magnetic field, both transverse to a given direction [6, p.142). A 'plane
wave expansion' of the field is convenient in many problems due to this property. For
many canonical structures, the expansion itself is unnecessary since wave impedances can
be derived directly. For instance, modal wave impedances for homogeneously filled rectangular waveguides are well known. In somewhat more complex structures such as those
treated here, the boundaries are still always planar owing to the rectangular geometry. As
a result, wave impedance surfaces can be chosen to conform to the boundaries and tlhe
wave impedance concept becomes a vehicle through which the boundary conditions can be
applied in a simple way.
The dyadic Green's functions for all of the structures treated here will be derived using
this approach. As the Equivalence Principle states, the fields in a given layer depend
only on the fields at the boundaries and internal sources. We therefore can derive the
Green's function for the source layer alone with the other layers represented by impedance
20
boundary conditions applied at the laver interfaces. In thiis way the Green's function can
be generalized to represent any number of layers above or belo'w the source layer. The
approach greatly simplifies the analysis of the total structure by allowing the fields to be
found first for the source layer alone. Once they are found, expressions for the fields in
the remaining layers can be immediately written in terms of the homogeneous solutions
by matching tangential components on the boundaries. As in [13], this process is greatly
facilitated by expanding the solutions in terms of 'LongitudinalSection Electric' (LSE)
and 'LongitudinalSection Magnetic' (LSMI) fields because the field matching procedure
can then be done on a onetoone basis. (An individual mode on one side of an interface
matches an identical mode on the other side exactly, with an appropriate coefficient.)
The impedance boundary conditions, as used here, are not to be confused with the approximate impedance boundary conditions discussed in [63, 64]. Both usages may be exact
under certain circumstances. The present usage is in the context of the modal impedances
of various structures and is exact under the assumption of perfectly conducting walls where
applicable. For example, the approach is exact for a closed, perfectlyconducting rectangular cavity with uniform side walls. A counterexample is a cavity with perfectly conducting
side walls, but which is open on one end: terminating the open end with the impedance
of freespace as proposed by some, is not exact since this condition is not exact. For all
structures studied in this work, the geometries are such that the representation is exact to
the extent that perfectly conducting walls can be assumed.
In our structures, the impedance boundary conditions require the fields to satisfy
x E
H (2.1)
For electric currents, this becomes
~GeJ j7 x VxGmJ
 or G= jW7 (2.2).Loy G m'Gmj
21
Similarly for the equivalent magnetic current Green's functions, the impedance boundary
conditions are;
x * GeK x: ~ GeK 7jr
= = jwrl or = = (2.3)
Y GmK X V x GeK
For simpler notation, Tie will be the wave impedance associated with the LSE lmodes anid
Trm will be used for the LSM modes. Furthermore, e and rim will denote nornmalizationl to
the intrinsic modal impedance in the layer and are defined as
k~zi lei
where i is an index associated with the ith layer.
WVe can then evaluate the impedance conditions using transmission line analysis and
wave impedances for the various layers as illustrated in Figure 2.1. The impedances on a.......ul
*K 1
given layer's upper boundary are found by the transmission line equation
7i  kzi ie(:l)+jtanFkz(i)I (2.6)
m) [ +tan )() (2.7)
g(il)zive 1 uppr b+ trankz(ision line e ti
^k(ii) 71e(i1+ 'tan k,,i)li I
kz(,_1) I +j tan kz(,_,)l(,_,)J
22
where the index i is iterated from the top layer through successive lower layers to the layer
of interest with L; the thickness of the 1th layer. (We have assumed that Lt = 1 throughout
this work.) Similarly, for the lower layers, since the wave impedance is negative,
k,1  l e(Z+l) tan kz(,+l)l(+l) (2.8)
k z(I ) 1  je (t+l ) tan k('+l1) l(+l)
I kZ(I+l) [ m(+1)  J tan(+l)(+l) ](+I) 1(2.9)
(,+l)k.i 1  J.jm(,+l) tan kz( +I) (i+l) (2.9I
where here the iteration proceeds from the lowest layer upwards.
2.2 Dyadic Green's Functions for an Infinite Covered Half Space
We begin with the derivation of the Green's function for an infinite covered halfspace.
The method of solution closely parallels that of the other cases, except for the boundary
conditions, so that by covering this case in greater detail some of the steps for the later
cases may be omitted. For completeness, we also include the solutions for electric currents
in this section, although this function is not needed for the characterization of the slot. It
is, however, widely used in the analysis of open microstrip and microstrip patch antennas.
2.2.1 Magnetic Current
The dyadic Green's functions for the slot problems use a magnetic current ( K) as the
source. They are the solutions to the dyadic Helmholtz equations:
x V7 x Gml<  k2Gml = I(R  R') (2.10)
V x V x Ge  k2Gei = V' x 6(R  R') (2.11)
The key to eliminating the need for te L vector wave functions, and thus simplifying the
analysis, is in the choice of which of Equations (2.10) or (2.11) to solve first.
The functions L, NM and N form a complete set of solutions to the homogeneous equation
23
\7 x V x F  K2 F = 0. They are defined in the general forms
L = V (2.12)
1
M = V x x, =V x N (2.13)
K
N = V x V x phi =V1 x M (2.14)
K K
where P is a scalar function solution to the equation V2I + K2T = 0 chosen to satisfy the
boundary conditions of the problem; ii is a unit vector called the 'piloting vector'; and K
is the separation constant K2 = k2 + k2 + k2.
Following the OhmRayleigh method as described by Tai [72], we can find the particular
solutions by expanding the right hand sides of (2.102.11) in terms of the eigenfunctions
L, M and N with unknown vector coefficients; deriving the values of the coefficients using
the orthogonality properties of the VWFs; expanding the dyadic in terms of the same
functions with scalar coefficients; and enforcing the equations by performing the derivative
operations. From Equations (2.122.14) it can be seen that only L can have a nonzero
divergence and since the right hand side of (2.11) has no divergence, the L function is not
needed in its solution. We therefore find the solution of (2.11) first. It can also be shown
that GmK and GelK are related by
V x GmK = GeK (2.15)
V x GeK = Ib(RR ) + k GmK (2.16)
so that GmKI can be found from (2.16) once Gel< is known. This is the essence of the
method described in [77].
As with the vector potential method (see Appendix C), we begin as if the space were
infinite and homogeneous. We therefore expand the field in terms of VWF's for free space
defined by
  eJ(kxx+kyy+kzz)
(2.17)
)24
To simplify later application of the boundary conditions, we clioose tile pilotilng vector to
be i. The orthogonality properties of these functions are then
ff, M(k kzN( k,',k ) dl= (2.18)
Iff M(kky, ) M(k,  k )dl
fff N(,...N(.', ,k')dl"
(27r)3(k + k)(  k' )6(k k (k k ) (2.19
where the volume of integration corresponds to the entire space.
To find GeK we first let
=  00 00
V x [i(RR')]= J// dkrdkydk, [M(k 1,ktyk)A + N(kl, k^ )B](2.20
By taking the anterior scalar product of Equation (2.20) with Ml(k', k', k) and
N(k, y,k') respectively, and integrating throughout V, we can determine the unknown vector coefficients A and B through the orthogonality properties. The results are
A Nl'(k, ky, k.) (2.21')
(27r)3(k2 + k2) (221
_ r'(k,, ky,k:)
= M(k, k k (2.22)
(27r)3(k2 + k2)22
in which the primed functions are defined with respect to (x', y', z'), the site of the source
at R = R. Thus,
Vx [t5(kR R')] = 1 JJf j dj dk dk^dkz
^ v ^ (27r)3 C*o (^+^00)
* [M(k, k, k)N'(k,ky, kz) + N(k,, k, kkz)M'(k.,, kz)] (2.23)
Now we let
= 1 t00 00 o00 ndk.dk dkz
eK (2r)13 J J (k2 + k2)
[aKM(k., k,, kz)N'(k,ku,k:) + bN(k:, ky, k2)M'(k,k,,kz)] (2.24)
25
with a and b unknown scalar coefficients, and substitute into Equation (2.11) yielding
1 1
a = b = k2 [2  (k2   k2)] (2.25)
The integration with respect to k, in Equation (2.24) can now be carried out in closed
form by applying Cauchy's theorem:
GCK 8w2 Id k [t4(~kz)N(Tk) + N(~kz)M'(Tkz)]
RCeK 8 j (k 2 + kt2)( ' +' )( )
z<'z (2.26)
where k = k2  k  k2 (note that K becomes k). As indicated, the top sign applies
when > z' and the bottom sign when z < z'. This condition is a result of requiring
the solution to satify the radiation condition at infinity which determines whether the
contour of integration is closed in the upper or lower halfplane. Also, from here on it will
be understood that the primed functions, M' and N', have k, ky or k, arguments,
unless indicated otherwise.
We now can write GmIK by performing the operations indicated by Equation (2.16).
This can be done almost by inspection using the relations between M and N from Equations (2.13) and (2.14) except for the discontinuity which occurs at z = z'. As in [7T], it
can be shown that these relations apply but an additional term is needed to account for
the discontinuity at the source; specifically,
= k r~~ f'2 d0k e0k r   
VxGeK d= jk2 /,ddky '
7 x GeK M(~k,)M(Fk,)+N(~k,)N(:Fkz)
87r2 oo Joo k(k2 + k2 )
+ It(R R') Z<Z' (2.27)
where the transverse idemfactor, It =:H + yy, appears in this case as a result of the
combination of the choice of the piloting vector z and the partitioning of the z dependence.
Notice that the singular terms come from the second derivative of
f f e(kz() z > z'
f(Z)= (2.28)
jk,(zz') Z, Z
26
or the first derivative of
Of(Z) f k z z') >:
g(Z) = = j (2.29)
which is
og(z) _ 02f(Z) _ jk(zz) z >
0f z z 2{
aZ a_  2jk.6(z  z) (2.30)
eJk e (zz') z'
This term has sometimes been overlooked in the potential function method as swell, as
discussed in Appendix B. Thus, Equation (2.16) leads to
^ 1 .D D00 dkI I
GmK =   j ( R2 + 2
k 2 8,2 R') kz(k2 + k2)
*[M(~kz)M (T:Fk) + N(~kz)N (Tkz)] Z<~Z (2.31)
We now divide the infinite space into layers surrounding the source point above and
below. The layered structure can be represented by impedance boundary conditions for
the source layer as previously described and as illustrated in Figure 2.2. We then apply.:c:...;...;:. : :..'. .:...'.':..'...
:'.""'..:........... '.......':'...7..................................................................................................
A+, B+ B *K 9*K
Zd I BI BK
Figure 2.2: Layered infinite space as represented by impedance boundary conditions.
the method of scattering superposition to this case by letting
= GK(P) + S)
Gel< = GeK + GeK (2.32)
27
=(P)
Ge is the particular solution found above for the infinite space which may be referred to
as the primary term, and Ge  the secondary or scattered fields  is made up of solutions
to the homogeneous equation as follows:
d(S) _ jkf. dk dky
7ei J oo.kz(k2 + k)
NI(kz) A+ + M(k:z)A + 4(k)B+ + N(k)nB] (2.33)
where A+, A, B and B are unknown vector coefficients to be determined. The physical
interpretation of this procedure is that the ~ coefficients represent the waves traveling in
the ~i directions as a result of reflections, i.e.. scattering, from the interfaces. In evaluating
the boundary conditions, it is also useful to find GmlK through V x GelK = k2Gm) which
is the sourcefree version of Equation (2.16). Hence,
(S) j [I.Z [ dkdky
mK S2 JJ k,(k^)2 + 2)
[N(k) BA + N(,)A, + M(kz)B + M(k )B ] (2.34)
Note that additional boundary conditions need only be imposed at the newly introduced
interfaces and not at the source, since the primary fields satisfy all boundary conditions at
the source and the secondary terms are continuous there.
Applying the boundary conditions at the top and bottom of the source layer, we derive
the following set of algebraic equations for the unknown coefficients:
(itu  1)ekYcA  (e + I)e)kc A = (feu  1)e1kcN( k,) (2.35)
(NeL  l)e kYdA+  (L+ l)ekYdA = ( AL + 1)ekY N (kz) (2.36)
and
( Im )ekc B + (7mL + 1)ejkYc B = (imu  1)e cM/(kz) (2.37)
(rmL l)ekd B + (imL + l)e)ked B = (1mL + 1)eY M (kz) (2.38)
Here the Lu' and 'L' notation indicates the upper and loxver interface conditions respecttivel.
These four equations decouple into two pairs due to a judicious choice of tlhe pilotingI
vector in the VWFs. Because the piloting vector was chosen as the unit xector normal to
the laver interfaces, the M and N VIWFs appearing in GmIK correspond to the LSMI and
LSE modes of the structure respectively. (In contrast, it should be noted that M and N
correspond to the LSE and LSM modes, respectively, when they appear in GeIK.) As is well
known, the LSE and LSM modes are decoupled on the interfaces, that is, the tangential
components of an individual mode on the interface can be matched by an identical mode
in the adjacent layer, therefore the coefficients are decoupled. That there are only four
equations is a result of the fact that the field in a given layer depends only on the field in
tile adjacent layers.
The solutions are easily found to be
ekYd [(\ u  1)(7ieL + 1)e kcN'(kz) + (ieLt + 1)(7,eL + 1)eJkY c(kz)]
(rieU + 1)( 1eL  l)e)Y(cd)  (7e  l)(7ieL + l)eky(d)
(2.39)
ejkc [(ie  l)(ieL  1)ed( + )(7ieL + 1)eJY'dN'(kz)]
(ieU + 1 )(i1eL  1)ek( ) ( l)(7eL + l)ek (cd)
(2.40)
eiyd [(imU  l)(imL + 1)ecM' k, ) (imU + l)(7imL + 1)eJkcM'(kz)]. 
(ifmU + l)(rmL 1 )ejky(cd)  (irm  1 )( NmL + )eJky(cd)
(2.41)
 ejkyc [(imu  1)(fmL  )eJkydM'(k)  (fmu  1)(7imL + )eJkYdM'(k)]
( 7rm + 1 )(i emL  1)eiky(cd) (imy  1)( mL + l)ejky(cd)
(2.42)
With some algebraic manipulation and use of the relations found in Appendix A, we now
can write
Ge'i = J r dkxdky
Ge 4er2 j (+ )
47 x,.oo kz(k2  k2)
X YO
29
[; 7 lMe[k(Z  c)] jIlo[k(:  C)] [,ieLKe[tz(Z  d)] o d)],
[7eLNe[kz(Z — d))])  jT N[[k(  d)]j  cj [   C)]
(i7eU ?7eL) cos kz(c d) j(7)erL7L  1) sin kz(c  d)
m( mNo[kz(z  c)] + jNel[k((z )]] [1LMO[k z(  d)] + jiN[kz  d)]]
[7m LNo[kz(z  d)] + jNe[kz(z  dJ)] [mU^MI[kz('  C)] + jNl'[(  c)]
(imU  7imL)Cos(kzc d)  j(rmtrimL  1) sin kz(c  d)
for  ' (2.43)
where the Mo and Ne functions are defined by
cos(kz)eJ(kII+ 'y)
=(k:) = (2.44)
sin( kz )e3(kxI +ky)
This expression contains all components of the dyadic Green's functioni separated into LSE
and LSM modes. Note that our convention will be to denote even and odd trigonometric
dependence by the substripts 'e' and 'o'. In order to avoid any ambiguity when these
subscripts are used, the corresponding k,, ky or kz arguments will be shown explicitly in
the same order, sometimes followed by other arguments as appropriate (see Appendix A).
The magnetic field type is given by
 j _, _ dkxdky
G mK k ( R  R )+ 2 k(k2 + k2)
nmuMo[kz(z  c)] + jMe[kz(z c)]] [rmL [kz(z'  d)] + j t[k,(' d)]]
mLMo[k(z d)] + jMe[kz(z  d)]] [ MO[k(z  )] + jM[kz(z'  c)]]
(imuu  imL)coskz(c d)  j(7jmUlmL  1)sin k:(c d)
30
( 7e[;ez.(: c)]  jN,[k(z  C)]] [eLN e[(Z  d)] jNtk '  d)] )
L el[k d)]j No[k(z  d)]] [eUNe[z('  )] j,[ (' c )]]
(ieC 7  7eL ) cos kz(c )  d) 1)sin k(c  d)
for z<z' (2.45)
When applied to the case of a slot on an infinite, perfectly conducting ground plane,
TL is set to zero. For a single dielectric cover layer, on the upper interface we set
k
u  (2.46)
ko
T7mU = rk (2.47)
corresponding to the normalized impedance boundary conditions for freespace above the
slab. For a slot in the ground plane with its axis along i, we take the z > ' terms which
results in
G rm   dk dk ej[k. (I')+ky(Y')]
m, rr 4 1 ^ k,(k2 + k2)
[r k, sin k,(z  d) + jk, cos k,(z  d)l k k 2 k, sin kz(z  d) + jkn cos kz(z  d)f l
Er k, cos kc(c  d) + jkj sin k((c  d)J kos  d)jksin (c  d)J
(2.48)
By transforming the spectral integrals to a cylindrical coordinate system th e double integrations can be replaced by a single radial integral on recognition of the integral representations
of Bessel functions in the angular variable. Through some very tedious algebraic manipulations, the result can be transformed from the present form, which separates the LSE and
LSM modes, to a hybrid form which can be compared to the result in Appendix C or with
previously published forms [38, 36]; although not necessary for numerical evaluation.
31
2.2.2 Electric Current
For the case of the electric current source, the steps are exactly the same as in the
preceding section, except here the Helmholtz equations take the forms:
7 X V 7 x GeJ  k2Gej = Ib(R  R') (2.49)
7 x V x GmJ k2GmJ = V X I(R R') (2.50)
where now we solve for GmJ first. Due to the similarity to the previous set of lielmholtz
equations and because of the way the VWFs have been defined  particularly the symmetry
introduced in the curl relationships  the solution proceeds withli identical equations but
with the following notational replacements:
M N (2.51)
Gel< GmJ (2.52)
GmK o Gej (2.53)
The process is similar to replacements made under the guidance of the Duality Principle,
however, it is important to note the difference. The replacements dictated by the Duality
Principle alone would result in functions which satify 'dual' boundary conditions, i.e., the
electric field dyadics would satisfy the Neumann rather than the Dirichlet conditions on
the conducting boundaries [13, pp. 2939]. By replacing the M and N functions with each
other, the true boundary conditions remain valid since these functions are complementary
with respect to these boundary conditions. Although it would be interesting to more generally state and define this process under a heading such as say, the 'Similarity Principle',
the development and proof is beyond the intended scope of this work, however, we will use
it repeatedly.
32
The final result then is
G  i_ f~D f00 dkrdky
2mJ *j COO kz(k2 + k2)
(f,, [imUo[kz(z  c )] + jek[(  ) + j) [2'  d) ]]
1[mM[kz(zd)]+ [(d)]  d)] + j [ ( 7(  c)] )
(imU  fimL) cos kz(c  d) j(OimUi/mL  1) sin k.(c d)
+( [reUe[k:z( c)]jNo[k(: c)]. [IeLI/T[kz(  d)]jNfo[k(z' d)]]
e,,, d, ]
[7eLNe[kz(Z d)]jNok( d )] [ele[:(Z  c)] j (c)]
(/7L ) cosLk(c  d) j(i'eLVr1L  1) sin k(c  d)
for z>z' (2.54)
Gej can be found by substituting the coefficients into equations similar to Equations (2.32 2.34), but also can be derived from CGmJ directly, applying Equation (1.15) with special
care in performing the derivatives at the source discontinuity. The final expression is
eJ k2 47r2 =00 )o k+ (k2 + k2)
ieUMe[kz(Z c)] jMo[k(  c)] [7eLM[kz(z'  d)] j[ k(' d)]]
7?eLMe[k(.z d)]] jeU/([kz(z  d)] ( )] Me[k(z  c)]j k (z'  )]] i
(i/eU  7eL) cos k(c  l) siJn k((c d)
[/mUNo[kz(z  c)] + jNe[kz(z  c)]] [mLNo[kz(z  d)] + jNe[kz(z  d)]]
[mLNo[kz(2 )] d)] + jNe[kz(z  d)] rmu  c)] [k
(7/mU  imL) cOskz(c  d) j(7imU/mL  1) sin kz(c  d)
for z> z' (2.55)
33
2.3 Dyadic Green's Functions for Layer Filled Rectangular IWaveguides
The electric and magnetic dyadic Green's functions for an electric current source in
rectangular waveguide are needed for the radiating slot problem. As will be shown later,
the electric type is used in an integral equation approach to solve for the propagation
constants of the structure. Once a propagation constant has been determined for a selected
mode, both the electric and magnetic fields on the entire crosssection are required to apply
the Reaction Method to the threedimensional cavity problem. The magnetic current case
is not needed in the main body of this work but will be discussed briefly in Chapter IV.
As in [79], the solution for the multilayered waveguide problem is built upon the
solution for the parallel plate problem. The parallel plate waveguide solution has already
been obtained in the previous section if we set 7ru = Lr = 0 on the source layer boundaries.
However, although the parallel plate solution we need is based on VWFs defined with
respect to a piloting vector z as was used above, the desired planes for the parallel plates
are defined by x = 0 and x = a which do not correspond to the impedance boundaries
used previously. Therefore, the parallel plate solution will first be derived based on the
appropriate conducting planes followed by the layered rectangular waveguide solution.
2.3.1 The Parallel Plate Green's Function
A parallel plate waveguide shown in Figure 2.3 is formed by bounding walls at x = 0
and x = a filled with a uniform dielectric material represented by the wavenumber ki,
where i will ultimately represent the i'h layer of a waveguide containing a source. We now
define the functions
Mo(kx) = V x 'o(kr)i = V x [sin(mrx/a)e (kYy+k )z] (2.56)
Ne(lk) = V X V X Te(kr)i = V X V X [cos(mrx/a)ej(kiY+kZZ)] (2.57)
K K
34
x
'7777777777777777777Y777777777777777777777777A
X =
a        
* (x',z')
x =0
~\\\\~\\\\~\\\\\\~~\\~\\\\\\\~4
z
Figure 2.3: Parallel plate waveguide coordinate system.
which satisfy V x V x F K2 F = 0 where K =  k2 + k2 + + '2 and tlie boundary condition
11an te ou da,co iio
ixx F=0 (i.e., xx E=O)
(2.58)
at x = 0 and x = a. The wavenumbers ky and kz are two continuously distributed
eigenvalues and k, = mwr/a where m is an integer including 711 = 0 for Ne.
The orthogonality properties of these functions are
JJJ M(m, ky, k) NO(m'n, k, /k) ( = 0
(2.59)
for any combination of even and odd functions and for any two sets of eigenvalues ( m, ky, k,)
and (m',k',kz). The volume of integration corresponds to the entire space inside the
parallel plate waveguide. The normalization constants of these functions are stated by the
following relations:
Jj M(m,, kz) Me (m', ky, k') dV
= f NI(m, kyk)N(. m',k,, kZ) dV
(1 + 6,,)22ra(k2 + k)6(k  ky)6(k,  k')
{0 ' 6kzk
17l = 77t'
m = m' = 0, 1,2,...
(2.60)
35
J!N0(m, ky, k.) Nlo(m',  k, ') dV
o=  fyO( m, ky, k,)  IN'( n'  k' ) dVl
0 m mn'
= (2.61)
1 m =7 0
where,b is the Kronecker delta function defined by 6,m = m
0 mo 0
To find Gmj we first let
00 00 00
V x [6(( R  /')] = J k dkydkz [Mo(m, ky, kz) Ao + Ne(m, ky, kz) Be] (2.62)
Joo0 m=O
By taking the anterior scalar product of Equation (2.62) with Mo(m' k, k') and
Ne(m'. k, k' ) respectively, and integrating throughout V, we can determine the vector
coefficients Ao and Be through the orthogonality properties. They are
(2 . )KN'(m,k,,kz)
Ao = m N(m) (2.63)
47r2a(k2 + k2)
e= (2 6)nM,(m, k, kz)
e = (2.64)
47r2a(k + k) (2.64)
In Equations (2.63) and (2.64) the primed functions are defined with respect to (x', y', z'),
the location of the source. Although for m = 0 the function N' vanishes, m = 0 is
included, as implied by the factor (2  6m) in Equation (2.63), to put it in a form similar
to Equation (2.64). Substituting Equations (2.63) and (2.64) into (2.62) we obtain
00 )00 2
V x [T(R R')] = j dkdkz 4r2a(k + k)
[Mo(m, ky, kz)N(m, ky, kz) + Ne(m, k;, kz)M'(m, kf, kz)] (2.65)
Now we let
Gmj (2, )K
Jco dlkdkz mo 47r2a(k2 + k2)
[aM(m, k)o(m, k, k)+ be(mk, k)Mm, k,)] (2.66)
*aao(m, k,,)N.(m, ky, C,)+ bNe(m,;,, k)M'(m,k,.,)] (2.66)
36
Substituting Equations (2.65) and (2.66) into (1.12), and making use the relations givenI
by Equations (2.13) and (2.14), one finds
a = b= 2 (2.67)
as before. The integration with respect to k, can be carried out in closed form by applying
the residue theorem together with the radiation condition yielding
GmJ =  Jdk Z 4Ta(jIM + k2)
[MIo(m, ky, t~k)No(m, ky,:k) + Ne(m, ky, ~zk)M(m, k, Tkz)] z<' (2.68)
where k = k2 k  . Again, the top sign applies to z > z' and the bottom sign to
z < z'. Now applying Equation (1.15), again through use of the relationships given by
Equations (2.13) and (2.14), and taking into account the discontinuity at the source, we
can write
The solutions for tlhe magnetic current follow the same procedure yielding
[MO(r, ky, ~kz)NI(m, k, kz) + Ne(m, ky,~k)Nk(m,  ky, T >)] zz' (2.70)
0ok j(2  b()
GeK 2=  2 'E o,i,)k
e =.jdk Z 47kak,(k (k +k2
[Me(m k, ~kN)lN(m, ku, k) + No(m, k, ~kz)Mom, k^, z) z</ (2.71)
Note again that that tis can also be obtained from the previous case by simply replacing
Gej = Gmli<, GmJ = GeKl, and M < N.
37
2.3.2 Rectangular Waveguide with Electric Currents
We now construct the dyadic Green's function for the source layer of a multilayered
rectangular waveguide (see Figure 2.1) by using the parallel plate Green's function solutions
above and the method of scattering superposition. Let
= (P) (S)
GmJ = Gmj + Gmj (2.72)
(P) =(S)
where G mJ is the parallel plate solution and G mj is defined as
G mJ   dky j a(k2 (2 2m)
GMj AY E 47rak,(k2 + k2)
[N,(m, k, kz) A+ + Ne(m, ky, k) A + Mo(m, ky, kz) B+ + Mo(m, ky, kz) B ] (2.73)
representing fields which are scattered from the dielectric layer interfaces located at z = c
and z = d. The unknown vector coefficients, A, A, B+ and B, can be found by
applying the upper (7OT) and lower (rL) impedance boundary conditions for the layer. This
produces two pairs of equations for the unknown coefficients:
( )eu  l)e( +) A+  (ie 1)ekcA ( )ejklzcM(m, ky, k) (2.74)
(IeL  l)ejkd A+  (feL + l)eikzdA = (f/eL + 1)ejkzdMe(m, ky,k) (2.75)
and
(rmu  1)eJ'kzB+ + (r/m + l)eJkz B = (frnU  1)eiNk'C', k,k,) (2.76)
(f/mL  l)ejk B + (+mL + 1)ekzd B = (fmL + l)eJkzdN (nM k, kz) (2.77)
Notice that these equations are identical to Equations (2.332.35) with the replacements:
MI'(~kz) > M(m, ky, +k) (2.78)
N'(~kf,) = N'(m, ky, ~k) (2.79)
This feature is characteristic of all of the solutions we will be dealing with, and should be
expected since the functions differ only in whether the functional dependence is exponential
38
or trigonometric. As a result, the solutions are identical to Equations (2.392.42) with the
corresponding notational substitutions.
Algebraic manipulation of Equations (2.68,2.72) and (2.73) and use of tlie relation.s
given in Appendix A then leads to
G,, _ r A t ^ 
GmJ m=O 27rak(k + k2)
Mo[7omu; m, kz(z  c)] A'[,mL; m, kz(2  d)]
.o[tmtL; n, kz(z  d)] A^o[imu; m,,(z'  c)]
(7iTn,, /m)cos kz(c d)  7(LmU)imL 1) sin k m(c  d)
AC [eu; m, k,(z  c)] A1' [7eL; m, k(z'  d)]
e[O1eL; m, kz(z  d)] M'[ieeu; m, k(z'  c)]
(fle/U  7eL)cos kz(c d)  j(1ieU1eL ) )sinl k(c  d)
for ><' (2.80)
where we have now defined the new operator functions., Me and NAe, for the sake of compact
notation:
M0o[7;a] = rM(oo[] +jMoe[a] (2.81)
Me[7; a] = 7Mee[a] ]jMeo[a] (2.82)
Aro[7; a] = 77Noo[(] + jNoe[2] (2.83)
Ae[7W; a] = 77Nee[a] jNeo[a] (2.84)
Since a here is [rn, k,(z  c)] and m is associated with k1, we recognize that the 'e' and
'o' subscripts in this case imply trigonometric functions of x and z. Also recall that the
primed functions use ke.
We can find GeJ as we did in the parallel plate case, by performing the derivatives
39
indicated by Equation (1.15):
GeJ  ~ R j(26m)
k2 2m=0 ak,(kI+y
k., k + k2)
I Me[iie;mkz(z  c) M Lkz(z  d)]
Me[TkL, in, k,(  d)] Me/[ijeu; m, kz  c)
(iieLT  J7cL) cos k((c  b) j(ieUfjeL  1) sin kz(c  b)
N' [~ii,,; m, k,  )} A"[i [L; mlk,  d)]
A"[M;M,zd)] AII[fimu; ml k(z'  c)]fo zz'(.5
(fIm7?77L)JcoS k(c b) j(i77mUimL  1) sin kz(c  b)
for z<4, (285)
Both Gej and Cmj involve a spectral integral which can be reduced by Cauchy's Theorem,
once the impedance functions are specified.
2.3.3 Rectangular Waveguide with Magnetic Currents
The magnetic current cases can be found through the same method or 'Similarity
Principle' substitutions (section 2.2.2) to obtain
_ j0 (2 m)
GmK 6( R R ) R +J dky Z
t 01 2rak,(k 2 + ki2
M0 =O
I ~Mo[im; m, k,(z  c)] M Ai'mL; m. k(  d)I
M0[fImL;rnkz(z  d)] M'jiimu; ml k,(z'  c)]
(iin i L) cos k, (C  b)  j(T~mU iTL  1) sin k,(c  b)
C AQ[iJeu; m, kz(z  c)] Ak'~[iL; min, (z'  d)]
A'[lleL; mn, k,(z  d)] A[ij'e; i, k(z'  c)]
(ikUcr  71,L) cos kz(C  b)  ji l  1)fsin (,(2  86
for z> Z/ (2.86)
40
and
=0 j(2 bm )k
G ef= dkv 0 2 + I )
GK = ] dk Z,k(k2 k2
J —0o mo ~Kray~k
M e[eu; m, k(z  c)] AC'[i1eL; m, kz,(  d)]
M 4e[ieL; m, kz(z  d)] Ar[7,u; m, k(2'  c)]
(7  ' 7eL) COS z(c  b)  j('eU7eL  ]) sinl kC(c b)
A/ o[mu; m, k (z  c)] M'[mL; m, k(  d)]
Afo[r1mL; m, kz(z  d)] M'o[imu; m, k(z  c)]
(imUv  7mL) CO kz( b)j m mL 1) sin k(c  )
for >z' (2.87)
2.4 Dyadic Green's Functions for Layer Filled Rectangular Cavities
The scattering superposition approach can now be applied to the result of the previous
section directly to obtain the dyadic Green's functions for the cavity problem by introducing
conducting walls at y = 0 and y = b and applying scattering superposition to the ~y
directed waves. As an alternative, one can take a somewhat simpler approach by first
deriving the dyadic Green's function for a waveguide with its axis along the ^ direction
(ky becomes n7r/b). Then scattering superposition is applied along the z direction with
impedance boundary conditions to obtain the result for the cavity (see also [72, 75, 79]).
This will be the approach demonstrated here since the intermediate Green's functions will
also be needed in Chapter IV.
2.4.1 Homogeneously Filled Rectangular Waveguide: TE and TM Modes
As with the use of the halfspace solution as a buildingblock for the previous solutions,
the preceding modal representations of layered rectangular waveguides are not in a convenient form for the formation of the cavity solution. Again, the VWFs there are defined
41
with the normal to the layer interfaces, resulting in the LSE and LSNI mode representation. What is more convenient here is the solution for a homogeneously filled rectangular
waveguide expressed in terms of NNVWTFs defined to represent the niodes TransverseElectric
(TE) and TransverseMagnetic (TM) to the waveguide axis. We again will first find G mJ
= (w)
from which Gej will follow.
=(W)
Gmj must satisfy the wave equation:
(W)
V x V x Gm  Gm v X[i6(R R) (2.88)
To construct the solution we will need the vector wave functions satisf!ing tile Neumann
boundary conditions which are:
Moo(k1,kc) = V x 4oo(k,ky)^ (2.89)
Nee(kr,k) = V X V X T,(,r,ky) (2.90)
where
cos k:x cos ku y
TI, eoe(kI, ky) = ez (2.91)
sin k:x sin kyy
with k: = km = mr/a and ky = k, = n7r/b.
Following the OhmRayleigh method as before, we expand the source term as
V [16(R R') = dk^z E [1moo(k, ky) A+ Nee(kx,k y) B] (2.92)
00 m=O n=O
The coefficients A and B are found from the properties of the vector wave functions to be
(2  6mn), , ) (2.93)
7rab(k' + k2) N0(k, ky,
B 2 M(k)ky _kz) (2.94)
rab(k2 + k) eM
where the Kronecker delta function 6mn is equal to 1 for m = 0 or n = 0 and 0 otherwise
(the case where both m = 0 and n = 0 is the trivial, zero field solution for this case). Thus,
42
Equation (2.92) can be written as
vx[T oR') [ (2 ,5mn),
VI x 1z(R RIR)] = : E o 2+ )
j 'ab(k2 + k)
*[Koo(kx, ky, kz)Noo(kx, k,, —kz) + Nee(jkx ky,kz)Mee(k:, k, —Z)] (2.95)
= (W)
To find G mj we let
G(W)  (z (2mn)K
GmJ w e z e 2
 m=0O 7rab(k2 + k2 )
[aMoo(k, ky, kz)NoO(kx, ky,kz) + bNee(k, k, kz)Me(kx, ky,kz)] (2.96)
Substituting into Equation (2.88) we find as usual
a = b 2 (2.97)
so that
(Wx m=On=Or(2 b k, ) [l.k2]
[1V/oo(kCky,.)lZ)oNo(kr:k, k:) + ~leei~(kx, kkz)Iee(/' (k, k:)] (2.98)
=(W) r jk2 (2  m n n)
mJ = at2^ (k2 + mk2
NMoo(kx,, ~k )Noo(kx,k,k) + Nee(kz, k, k)z)Me,(kx, k,~k2)]
for z< ' (2.99)
where k2 = t  (k2o + k2). Using Equation (1.15):
=(W) 1  k Z j(2 nm,)
Gej  r ~ r
*= 2 [.o(3 R R ) abky ( b + e )
Ge* 0  n6( Ry  m=O n=O ab (k+ k
[N1oo(kx, ky +~kz)Noo(kx, ky:kz) + Nee(kx, ky,+~kz)M(k,,ky kz)],,,, ~ (~ k~:[:>
for z' (2.100)
43
2.4.2 Cavity with Opposing Impedance Walls: Electric Current
We now introduce impedance walls at z = c and z = d where the added bound;ar!
conditions given by Equation (2.1) are to be enforced. Using the method of scattering
superposition we let
= =(W) (S)
Gmj = Gmj + G (2.101)
where
(S) ~~ ~~ o jki(2 6mn)
mJ On abkz(k2 + k2)
I Moo( kky, k) A+ Moo(k,,k kz ) A + Nee(kr, ky, kz) B+ + Nee(kr, ky, kz ) B3
(2.102)
Evaluating the boundary conditions given by Equations (2.1) we get tlhe same system of
equations as (2.332.35) except this time with
M'(~kz) M'e(k,ky, ~k) (2.103)
N(~kz) = N,(k,, k~kz) (2.104)
Again we already have the solutions for this set by changing the notation of the VW'Fs.
With some algebraic manipulation and use of the relations in Appendix A the results are
mJ ~ 2j ki(2  6mn)
m=0,=0 abk" 2  m
G a bkz(k 2 + k2)
Moo[71mu;, y. kk,(z  c) roo[,mL; ki, ky, kz(  d)]
Moo[mL; k, ky, kz (z  d)] Aoo[1mu; k, k, (z  c)]
(itmu  MmL ) cosk,(c  d)  j(,nuij,,  1)sin kc(c  d)
ANee[ieu; /,Iky, kz(z  )] Me[rL; k, ky k,(z'  d)]
A[ee[jeL; kI, ky kz(z  d)] Me[71u; kx, ky, kz(z'  c)]
(rjeu  1eL)cos kz(c d)  j(ieU/eL  1) sin k,(c  d)
for z>z' (2.105)
.44
and
( R 2j(26 ~mn)
m=0 nk = a0 z(k2 + k )
(Mee[eu; kr, ky, k,( c)] Mee[TeL; kr, k, k,(z  d)]
Me e[?IeL; k, k, k ~(z  d)] M~e[rTeL; r, k, k,(  c)]
(1Ve  reL)cos k:(c  d)  j(tUeL  1) sin k(c  d)
AJroo [ mt; kI y, kty(z  c )] } [ m L; k: k(y, Zt  d)]
(roo[7 mL; k., ky, k(z  d)] A"oo[imL,; k1, k (, k(z  c)]
(jmrLu 7imL)COS k:(c  d)  j( 7mcmL  1) Sin k:(c  d)
for zz' (2.106)
The operator functions are defined by the relations
Moo[7;o a] = IM1N'ooo[a] + jMlooe[a] (2.107)
Me,[ij; o] = rMIeee[] jMeeo[a] (2.108)
Aoo[77; a] = INo,[] +jNoo e[a] (2.109)
Aee[77; o] = Neee[a]  iNeeo[a] (2.110)
2.4.3 TE and TM Modes in Homogeneously Filled Rectangular Waveguide
and Cavities with Opposing Impedance Walls: Magnetic Currents
To model the slots, we also need the cavity dyadic Green's functions of both types
for magnetic currents. The derivation could follow the previous case explicitly, however,
because of the way the functions have been defined, we can take advantage of tle symmetry
of the equations and write the solution by making simple notational replacements. (The
only exception is the treatment of the (2  6,m) term which here is expanded as (2 
bm)(2  6n) since the m = n = 0 case may produce nonzero field components.) Using
this approach we can write the TE and TM solutions for magnetic currents in rectangular
waveguid e as
= (W) 00Cjk (2  5m)(2  b
mbkzk2 0 abk(2)+
* N00kk~ k~ 2(k,ky)F,~k) + Neeo(kx, k, ~kz)NOe(kx ky,TFkz)] Z<>Z` (2.112)
= 2jk,(I  6m(  6 00 '26r)) n
GeKZZaabk~(k +kk)
Jo (k, ee ~kzeM k1, k, Tk(z)  c) kz) z'; 1,k(2k112d).Mei1 L kk, Yz 1 ) A'efieu;k k k ky, ~kze(z', k c)]
(i 0 eu (  i brL)Ck(2  dn) ieieL)sn(cd
AMo[i~mu; kx, ky, kz(z'  c)] MAeeiKL; kx, kykl z  d)].Afoo[l~mL; kx, ky, kz(z  d)] M'O[iimu; k~, ky, kz(z'  c)]
(i~mu  mL) coskz(c d) J(~hnu~irL  1) sin kz(c d)J
for z<z (2.113)
and
GmK( = 'ii6( R R')+ 1: 12j26 )(6
00abkz(k2~ + k2)
j'(ooti~mu; k~, ky, kz(z  c)] M40o[imL; kx, ky, k~(z'  d)]
j \Moo[llmL; kx, k, kz(z  d)] M/ o[imu; kx, k, kz(' C)]
(fi/mU ?jmL) coskz(c  d)  (7jMu~jrL  1) sin k,(c  d)
Are[i/u; kx, kyl kz(z —  C)] A(re,[kL; kx,kyl kz(z'  d)]~
Afee[i/eL; kx, k3,, kz(z  d)] NAer/e;,kzk(z'  c)]) '
(ieu  k/L) cos kz(c  d)  J(i/euk/L  1) sin kz(c  d) J
for z> Z/ (2.114)
46
\W\e now have derived and specified the dyadic Green's functions for all tyI)es of structures to be treated in this work. The integral equations are therefore fully defined and the
task remaining is to solve for the unknown currents in each case and interpret the results.
CHAPTER III
METHOD OF MOMENTS FORMULATION
A general methodology for application to the various problems treated will be presented
in this chapter for the case where two components of current will be allowed on both the
strips and slots. Later, we will restrict our attention to strips and slots which are narrow
so that only the longitudinal component of current need be considered. This assumption
is sufficient to yield accurate results for the experimental cases to be used for verification,
and thus simplifies the numerical implementation without loss of generality.
The solutions to the presented integral equations can be found by choosing basis functions to approximate the various currents. The error in the approximation is minimized
by applying the well known Method of Moments, resulting in highly accurate representations of the currents from which the electrical behavior of the structures can be deduced.
Furthermore, the method of moments formulation will be discussed in the context of the
radiating slot problem only, since this problem contains all the essential elements of the
coupler problem as well.
3.1 Definition of Coordinate Systems and Basis Functions
Let us expand the current on the strip in the following manner. We first define a
stripfixed coordinate system as illustrated in Figure 3.1. The currents on the strip in this
47
448
1)
V I
4)
yo
Il[I
Xo
x
Figure 3.1: Cavity (, y) and Stripfixed (v, v) Coordinate System.
coordinate system can now be expanded as
J, = J,1, + J,v,
J, = (L) E (v,
JV, = () E
(3.1 )
(3.2)
(3.3)
where t are piecewise sinusoidal basis functions defined by
1 k sin kb(a  aq1) for 1o < o  Qq < 0
$(0) sin kbl(n so sin kb(aq+l  a) for 0 < a  aq < 1
and 1, is half the subsection length defined by
(3.4)
lo = [aq+  qn11/2
(3.5)
The subscript q is an index identifying basis functions at various points along the strips
and slots. Actually, kb will always be chosen so that kbla < 7r/2, making the function
basically a triangular pulse. This way, because the basis functions overlap, the current will
essentially be approximated by piecewise 'linear' segments between sample points (see [32]
for an introduction to the method of moments and basis functions). The sampling rate is
determined by field phenomena, phase resolution requirements or numerical limitations, as
49
will be showIn later.. Typically, thie sampling ra.te will be at least 20 samples per wavelength.
often much higher, so we generally set kb 6 k, which is more than sufficient to 'linearize
the basis functions. The motivation for the sinusoidal dependence is to simplify later
integrations and evaluation of the resulting functions.
The v function will be either a 'Maxwellian' distribution or a 'pulse' basis function
defined as
' A faruiellian(3) = 2
jI(3) =  s(/) 1  /3q <_ 13 (3.6)
q pUlse = 1
The Maxwellian function is often used since it closely approximates the true solution for
narrow strips or slots [52. 88]. Tliese expansions are further illustrated by Figure 3.2 where
the sinusoidal functions are exaggerated for clarity.
Longitudinal
1
L.2 piecewise sinusoidal L/2 W/2 W/2
Maxwellian
Transverse
L
I
~ I I I II
/2
W/2 W/2
J2
pulse  rectangular
Li
Figure 3.2: Current Expansion Functions.
In this work, we will deal exclusively with strips and slots which are narrow with respect to wavelength so that only one basis function will be used to represent the narrow
dimension. For wider structures, rooftop functions are commonly used, involving similar
overlapping basis functions in the direction of each component of current, but using the
50
pulse basis function for the transverse dependence. Because our strips and slots are Ilarrow, we will typically only model the longitudinal component of current, however, both
components are discussed here for generality and to form a basis for future efforts. Thle
primary motivation for the use of piecewise functions is that they are very efficient in terms
of changing strip or slot lengths as opposed to entire domain basis functions.
The stripfixed system is related to the cavity coordinate system by
v= (  Xo)cos 4+ (y yo)sin4 (3.7)
(3.8)
v = (x  xo)sin C + (y  y) cos c
We also define a slotfixed coordinate system as illustrated in Figure 3.3 where
y
to
w!
0
I — ON
so
x
Figure 3.3: Slotfixed Coordinate System ((,f).
( = (x  So)cos + (y to)sin
( = (x  So) sin 0 + (y  t) cosG
The slot currents are now written as
KL = A'Lc( + IKL(
KU = KuL(( + IA'KU
(3.9)
(3.10)
(3.11)
(3.12)
where the 'L' subscript represents the slot opening to the cavity and the 't'' subscript
represents the slot interfacing with the dielectric cover. Assuming similar expansions for
these currents,
Ku( =,(11.E )( 3)1':(
A'L~ = sF()ZL~.1
=,
(3.13)
(3.14)
(3.15)'
(3.16)
Equations (1.541l.56) can be written in matrix form
Fl.
zil
Z~21
z31
Z41
0
0
where each term in bold
Z12 Y13 Y14 0 0
Z22 Y23 Y24 0 0
Z32 Y33 Y34 Y3.5 Y36
z42 Y43 Y44 Y45 Y46
o Y53 Y54 Y55 Y56
o Y63 Y64 Y65 Y66
face is a submatrix described by
I V3
'LU
VLCJ
V L3j
vu~J
integrals
r
A7'H
7'
7'H
s h
such as
(3.17)
Z11= J G12VV(x, y, x yI)'I'(v)F(v,)dS,
zi = G(1 1) (X, yx',,y')4,(v )D(v)dSGI G1)(X, yx',y'),V((1)4)(~)dS14 s 0
0
52
or in a more compact form as
(11) (12) 13) G(14)
G(21) G(22) G(23) G(24) 0 0 /(4()I
G(31) G(32) G(33) G(34) (35) G(36) (() VL dx l
s, (41) G(42) t(43) G (44) G(5) G(46) )( dy
 ~ G((53) G(54) G((5) G (56) (
0 0 G(63 G(64) G(65) G( (66) (()('VU j 
o 0oIi G( G^5 G6 ''(j)4(I()Vu~i N
(3.18)
The functions on the right hand sides of Equation (3.17) and (3.18) are discussed in Chapter I and are further defined in section 3.2.
Note that Sj is the jth 'source' segment of the corresponding strip or slot. We also have
a triply mixed coordinate system which must be accommodated using the transformations
given by Equations (3.73.10). For example, computation of G(113) involves both G(13) and
G(. In addition, the derived Green's functions are in the cavityfixed coordinate system.
Consequently, all terms must be transformed to a common coordinate system before the
integrations can be performed. Treatment of these integrations and manipulation of the
various G terms will be discussed in a later section.
The evaluation of the elements of the matrix at the positions where the boundary conditions are being imposed has not yet been discussed. Following the conventional method
of moments formulation, we introduce a weighting function and impose an inner product
to be evaluated at each subsection on the strips or slots. The inner product is defined as
(a,b) = JJ b dS (3.19)
where a = wi, the weighting function, and b will be the vectors represented by Equations (1.541.56). Note that the elements of Equation (1.54) are to be evaluated on the
strip as indicated by (1.57) and (1.58). Similarly, Equation (1.55) is evaluated on the lower
53
slot as indicated by Equations (1.591.61) and Equation (1.56) on the upper slot as imllplied
by (1.62) and (1.63).
Following Galerkin's method we can choose wu to have the same form as the basis
functions used in the expansions of the currents. Thus, for Equation (1.54)
w = i(v) (v)v  + (vl) ( () (3.20)
Similarly, for Equations (1.55) and (1.56)
ws = (0)((C)( + + 4(C0)(k() (3.21)
The elements of the matrix in Equation (3.17) are then changed to
Zl = /Js q(v)4(v)ZjldS,
Z12 = IJJ ( x, ()4(v)Z'2dS,
" = J (f)(QY 3dSi
where the i index represents the 'field' point integration locations.
For certain cases we will use point matching on the slots driven by considerations in the
evaluation of the Sommerfeld integrals of the halfspace Green's function. In this case the
weight function for the transverse dependence of the longitudinal component of magnetic
current on the slot, becomes a delta function resulting in the evaluation of the field at a
point at the center of the slot.
3.2 Excitation Models
The final element to be discussed is the excitation vector represented by the right hand
side of Equation (3.17). As mentioned previously (Section 1.3), the terms represent,
54
nonzero values in the excitation vector corresponding to gap generator locations on tile
strips. Likewise, the 7H terms will be nonzero for the incident H field excitation used with
the Reaction Method. In this case, the incident field must be weighted the same way as
the left hand side so that the right hand side terms become
= I dS (3.22)
For the gap generators, the corresponding field for the gap subsection can be designated
Eg. Application of Galerkin's method then, results in the integral
<b = J iw Eg dSg (3.23)
on the right hand side of Equation (3.17). Eg is an unknown caused by a source at that
location on the microstrip. If we assume Eg = Egv, then we can set  0 in the second
row of Equation (1.54).
In most cases, we can arbitrarily set the integral of (3.23) so that ~ of the first row
of (3.17) becomes a zero column vector except for one element corresponding to the position
of the gap generator of the form
~ =0 * 0 1 0 * — 0 OfT (3.24)
where T denotes the transposition operator. Setting the magnitude of Eg is arbitrary because the Standing Wave Method used with the gap generators uses relative interpretations
of the resulting current, not absolute quantities.
For certain problems, however, we need the use of the gap generator model to determine
absolute values of current on the lines. If we set the field in the gap such that the voltage
over the gap given by
VO=  f Eg dl (3.25)
ap
55
is equal to 1 Volt. the input impedance at the feed point is given by Z2, = 1/Ig to first
order. A similar model can be used for slots with a coaxial feed, as in [21, p.360] for
example, by replacing the coaxial feed with a current source on the slot.
Before moving on, it is worthwhile to examine this process a little more closely in the
context of some of the terminology and physical interpretations of gap generator models
found in the literature. Restricting this discussion to one dimension, at the source our
system of equations represents the enforcement of the boundary condition
w f(x) dx = A' (3.26)
ource
where w is the weight function, f(x) is the field quantity and K is the constant specified
on the right hand side of the matrix equation row corresponding to the location of the
source. It is clear that the only nonzero contribution to the integral can occur on the
domain of w for which lw is nonzero so that any physical interpretation is confined to
that region. It should also be noted that there may be an infinite number of solutions
f(x) satisfying this equation and that this equation does not force further constraints on
what f(x) might be, i.e., the right hand side does not specify how f(x) behaves on a
scale smaller than the domain over which w is nonzero. We then also have no basis for
a physical interpretation which imagines the terminals of the source within this domain,
but rather we should interpret the terminals to be at the domain's endpoints. Obviously,
K will depend on the nature of w. The physical interpretation of the nature of the source
then also depends on wi. For simple cases such as a pulse weight function, the deltagap
physical interpretation is appropriate since Equation (3.26) reduces to a form similar to
Equation (3.25). The point match case, or deltafunction source can be interpreted in
the same manner by taking the limiting case of the pulse weight function, shrinking its
width to an infinitesimal gap while keeping the area constant. For more complex weight
functions, the physical interpretation is unclear except that we can consider the source to
.56
be 'distributed' over the domain of the weight function.
The system of equations represented by Equation (3.17) is now fully specified. Tle
unknown currents J,, IL and Iu can now be found by solving for the matrix elements
and inverting the matrix.
3.3 Expansion of the Dyadic Green's Functions for the Cavity
To evaluate the elements of the matrix, we need to expand the dvadic Green's functions
into the components corresponding to thle electric and magnetic currents which are in xy
planes. All of tile required termns can be separated into trigonometric functions of x,',y
and y' multiplied by a complex coefficient which contains the z and z' dependence. The
Green's functions can then be written in a condensed form as follows. Also, a constant
complex coefficient Cmn can be factored out which appears in all Green's functions for the
cavity and is defined by
2j(2 6o)
Cabm( + k (3.27),,bkz(k2 + k~)
Cavity EFIE  Electric Currents J
The electric field integral equation contribution of the electric currents involves the xy
components of Gej which can be written in the form
00 00
GeJxx = E 3 C0mn eJxx cos kmx sin kny cos kmx' sin kny'
m=O n=O
oo oo
GeJyx = >E E Cmn 5eJyx sin kmx cos knY cos kmx' sin kny'
m=O n=O
GeJxy = > Cmn,GeJxy cos kx sin k, sin kmZ'cos ky'
m=O n=O
oo os
GeJyy = E E Cmn geJyy sin kmX cos kny sin kmX' cos kny (3.28)
m=O n=O
5C7
where k, =, = nT7/a, ky = k, = n7r/b, and
5eJxx = [nee(7e)+ (k) ee(71 ) (3.29)
9eJyy = k2 Zee(e) +  2 2ec(im)] (3.30)
2' 3l /.30)
geJyx = geJxy = kmkn [Zee(,e) ( )2ee(m) (3.31)
The z dependence of the Green's function is contained in the Zee function defined as
[77r cos kz(  C)  j sin k^z  c)] [71L cos k,(z'  d)  j sin kz('  d)]
e() = [L COsk,(z  d)  jsin ki(z  d)] [ru cosk(z'  c)  sin k(z'  c)]
(1  7lL)cos k,(c  d)  j(ilurL  1)sin k(c  d)
for z>z' (3.32)
Here again we use the 'ee' subscript notation to imply the trigonometric dependence of the
function.
Cavity EFIE  Magnetic Currents K
The contribution of the magnetic currents involves the components of GeIK which can
be written as
00 00
Ge<Kxx = Z Cmn geKxx cos kmx sin knY sin kmx' cos ky'
m=O n=O
00 00
GeKyx = Z E Cmn gel<yx sin kmx cos ky sin kmx' cos ky'
m=O n=O
00 00
Gel.x:y =: > Cmn eKlxy cos kmx sin kny cos kmx' sin kny'
m=O n=O
GeKyy = >: Cmn geKlyy sin kmx cos k^y cos kmz'sin ky' (3.33)
m=O n=O
with
GeKyx = k [k Zeo(71e) + kn eo(m)] (3.34)
exy = k^ [keZeo(ie)+ k2Zeo(m)] (3.35)
eKxx =  elKyy = kmknz [Zeo(7e) +,eo(7/m)] (3.36)
5.
Thie 2eo function is defined as
[ritj cos k(z  c) j sin kz(z  c)] [71L sin kz( z'  d) + j cos k:( z'  d)]
eo [L cos k(z  d)  jsin kz(z  d)] [r/u sin k.(z'  c) + cos A:z' c)j
(O  7L) cos k(c  d) j(fi7]iL  1)sin kz(c  d)
for z<' (3.37)
Cavity MFIE  Electric Currents J
The magnetic field integral equation contribution of the electric currents involves the
components of Gmj which can be written in the form
00 00
G mJxx = >3 Cmn Ceixx sin kmX cos kny cos kx'sin ky'
m=O n=O
G mJyx = > > Cmn Gelyx cos kmx sin kny cos k,x'sin kny'
m=O n=O
GmJxy = E E Cmn GeKxy sin kmz cos knY sin kmzx cosky'
m=O n=O
GmJyy = E > Cmn 5eKiyy cos kmix sin kny sin kmx't cos kny' (3.38)
m=O n=O
Notice that the coefficients are the same as for the GelK case. It is not difficult to show
that upon application of Galerkin's method,
K,GmiJ)  J,GeK, ) (3.39)
where the double inner product notation is defined by
(a, c,) = a I  b dS'dS (3.40)
This implies, because of the signs of Equations (1.581.59), that thle submatrix associated
with the electric current contribution to the MFIE is the negative of thle submatrix for the
magnetic current contribution to the EFIE (diagonally opposite for the order given). This
observation reduces the computational effort required since only one of these submatrices
59
needs to be calculated to fill their respective positions in the matrix. However, for tile
radiating slot problem when we use the Maxwellian transverse distribution, evaluation of
the Sommerfeld integral has been accomplished through point matching which does not
produce this symmetry. Hence, in that particular case, we cannot take advantage of this
property.
It can also be shown that the electric current EFIE terms form a submatrix which is
diagonally symmetric as are the terms in the diagonal submatrix representing the MFIE
contribution of the magnetic currents. Therefore, these also can be formed by calculating
only about half of the terms, however, as will be seen later, taking advantage of other
mathematical relations for these terms produces far more significant improvements in the
fill time for these submatrices.
Cavity MFIE  Magnetic Currents K
The cavity magnetic current NIFIE terms are associated with a Green's function which
can be expressed as
00 00
G mIKxx = E E C0mn;mKxx sin kmX cos kny sin km' cos kny'
m=O n=O
00 00
G mKyx = E E Cmn GmKyx cos kmx sin kny sin kmX' cos kny'
m=O n=O
00 00
G rniKxy = Cmn 9mKxy sin kmX cos ky cos kmx sin ky'
m=O n=O
00 00
GmlKyy = E E Cmn jmKyy cos kmX sin ky cos km' sin ky' (3.41)
m=O n=O
where
mmKyy = [2mZoo(ie) + (knkz) oo(m)] (3.43)
grnmlyx = gmlxy kmkn Zoo(e)  ( Zoo(m)] (3.44)
k i
60
The Zoo function is defined as
[77 sin k(z  c)+ jcos c) [ sin 'd) cos k( c [ios k(z'  d)]
z oo [i sin k(z  d) + j cos k2(z  d)] [u sin k(z'  c) + j cos (z'  c)]
(7u  71L) cos kZ(c d)  j(rlL  1) sin k(c  d)
for z>z' (3.45)
Slot MFIE  Magnetic Currents K
For the slot, the transverse components of GmK for a homogeneously filled cavity are
needed which can be written in the forms:
00 00
G mlxx = E E Cmn 9mKx sin kmx cos kny sin km' cos kay'
m=O n=O
oo 00
GmKyx = E > Cmn 5mKyx cos kmx sin ky sin kmx' cos k,y'
m=0 n=O
oo oo
GmlKxy = E E Cmn mKlxysin kmx cos kny cos kxz'sin kny'
m=O n=O
00 00
GmKyy = E E Cmnn mKyycos km sinkn ycos knz'sin knY' (3.46)
m=0 n=O
where
cos k,(z' d) k 2 k2
mKXX sn kz(cd) n k2
mKyy coskz(z'd) [k kWk~]
sin k,(c  d) m +k
Cos kk,(z'  d) k2 k2
sin k,(c  d)  k]
mlNyx = = 9mKxy =kmkn k s) [  k2 (3 47)
These expression are valid for currents on one end of the slot coupling to field points on
the opposite end when z' = c. For currents coupling to the same end, z' is set equal to d.
HalfSpace MFIE  Magnetic Currents K
Although the Green's function for the halfspace was derived in Chapter II, the numerical treatment is quite involved and will not be detailed here. It consist of a real axis
61
Gaussian Quadrature scheme with singularity extraction of the branch point and surface
wave poles and asymptotic evaluation of large arguments. The methodology hlas been
outlined by IKatehi and Alexopoulos in [36]. The numerical implementations used to evaluate the halfspace admittance elements of the matrix Nwere provided by IKatehi for the
Maxwellian transverse distributions and by Harokopus [31] for the rooftop functions.
3.4 Identification and Reduction of the Integrands
The elements of each of the submatrices of Equation (3.17) involve double surface
integrals as shown earlier. The inner surface integrations are over 'source' regions defined
by the current expansion basis functions. The outer surface integrals result from the
application of the weighting functions and cover the 'observation' or 'field' regions of the
problem where the boundary conditions represented by the integral equations (section 1.4)
are being enforced.
Using the condensed notation we can now write expressions for the impedance elements
in a general form which will identify the integrations to be performed for each term. To
illustrate, only the EFIE expansions for the current on the strip will be presented. The
MFIE for the strip and slot currents are handled in an exactly the same manner.
The EFIE:electric current terms can be written as follows:
00 r
z, = E 3 //] ( ),) (V) G P(v)4(^(v,) dSj dS,
m=O n=O J( ) d d
z2 = d Z /7 dS,') o7G^ (v)(i)dSds,
00 0
m=O n=O ' j
s b, the s pts i and j rv (v 2)e (v )(v) dS dSe (3.48)
As before, the superscripts i and j represent the ikh "field" subsection and the jh 'source'
62
subsection. The Green's function terms can be transformed using the relations
= vcosp  vsin O (3.49)
y = vsin +i cos (3.50)
The transverse Green's function,
GeJ = GeJxxH + GeJyxyi + GeJxyiX + GeJyy:y (3.51)
can then be written for the stripfixed coordinate system as
GeJ = G eJvvvv + G eJvtt' + G eJvviv, + G eJvv1il' (3.52)
where
GeJvv = GeJxxcos2 0 + ( GeJyx + GeJxy) sin 5cos 0 + GeJyy sin2
G eJvv = (GeJxx  GeJyy)sinlcos + GeJyx COs2  GeJxysin2
G eJvv = (G eJxx  G eJyy) sin cos GeJyx sin2 + G eJxycos2
G eJvv = GeJxx sin2  ( G eyx + GeJxy) sin 0 cos ~ + GeJyy cos2 f
(3.53)
When the coordinate systems are mixed, such as the case of the magnetic slot current
contribution to the EFIE, the Green's function terms are also mixed. The relations for the
slotfixed coordinate system unit vectors are
x = (coso sin (3.54)
y = (sinO + cos (3.55)
Substituting these into the posterior positions of Equation (3.51) and (3.49,3.50) into the
anterior positions, the Green's function terms for this case become
G eKv( = G exx cos k cos 9 + G eJyx Sill 4 cos 0 + G eJxy cos ~ siI 0 + G eJyy sin k sin 0
63
G ev =  eJxx cos O sin 0  GeJXx sin sin tO + y cos cos + J Sin o cos
GeK/( = G eJxx sin 0 cos S+ GeJyx cos cosS GeJx Sin 0 sin ll + C(eJyy O i
G eK = G eJxjx sin 0 sin 6  G eJyx cos o sin 6  G eJxy sin > cos S+ (;Gjyy COS 0cos O
(3.56)
Returning to the EFIE:electric current case, we can now write the ternms of the associated submatrix terms as (omitting the Cmn constant)
Zj = 6 > geJxxcos2 Ieo(viti)feo(Vjj) + 5eJyxsinl)cos<(Ioc(z'z,,,'l)co( t1 j4)
m=0 n=O
+ 5eJxy sin O(cosdkIeo(vi, l )Ioe(,J L'j) + 'eJyy sill2 /oe( U,. u I, )Ioe( %P )
Z = 1  e (Jxxsin cos OIeo(v v)Io(vj,Vj) + 5eJyx cos2 OIue("I., 1)JIeo>( t,/j] )
m=O n=O
 eJixy sin2 Oleo(i (t V)Ioe(v;j;) + 5eJyy sin ~ Cos </oe(,. (t')le(vj*)
00 00
Z+ = >  eJxx sin )COS Ieo(Vi, vi)Io(l,vj) V eJyx sin 2 )I oe( t,, z )Ieo(, t' )
m=O n=O
+ 9eJxyCos q$IIeo(vivi)Ioe(Jv) 4 seJys in l cosdjoe(Vzl )Io)e('jirj)
z = i 2 1eJxxsin2qJeo(V:vi)Ieo(vj,j)  V eJyxsincos0Ioe(i/z^ )Ieo(,Ijlj)
m=O n=O
 5eJxy sin o'cos >Ieo(vi, vi)Ioe(/vj, v5) + geJyy cos2 0Ioe(Ul, t'i)Io (, 71)
(3.57)
Now it can be seen that the only terms involved in the integration are of the form
(s ) J [ cos kmx sin kny 1
q q JJ Sq sin kmx cos kny] (Cqq (358)
where (aq,,/q) could be either (Vq, Vq) or (Vq, Vq) with dSq = ddi,, for the strip or (cq, 3q)
would be replaced by (C(qq) or (~q,(q) with dSq = d(d, for the slots.
64
3.5 Integration
We will ultimately assume, as is commonly done, that the strip and slot are sufficiently
narrow so that the longitudinal components of current are much greater in magnitude than
the transverse components. The latter can then be neglected which, as will shortly become
apparent, greatly simplifies the bookkeeping required to keep track of various Green's
function and current components, coordinate transformations, crosscoupled terms, etc.
The assumption is further justified in that, at this point, there is no known advantage or
requirement for the crostrip or slot structure to be more complex. For the moment,
however, the complete expansion will be retained so that the numerical model can later be
extended based on these expansions, by evaluating the additional terms of the matrix.
Now the Ieo(cq, 3q) function is still a mixed coordinate system function, therefore, to
perform the integration the function must be transformed to a common system. The simplest approach is to transform the cavityfixed coordinate system functions into the stripfixed system by solving Equations (3.73.8) for x and y and substitute into Equation (3.58)
giving
x = v cos4>  sin + X (3.59)
y = vsink + v cos +y (3.60)
Equation (3.58) then becomes
Ieo (, ) = cos[km(vcos <  v sin k + xO)] sin[k(v sin + v sinl + yO)] }
o (q "Sq sin\[km(vcos > v sin + xo)] cos[kn(vsin ( + v sin < + iy,)]
*4(aq)T)p(3q) dvdu (3.61)
By introducing the notation,;k = km cos f ~ kn sin q (3.62)
to = km sin 4 ~ ikn cos (3.63)
65
V0 = km o (3.64)
'o = kYo (3.65)
we can reduce these integrals to the form
Ie (a q/3) = [sin(kv + vI)cos(kOv  O)  cos(kv + v+ )sin(k'  )
T sin(kv + Vo) cos(k'+v +,,)~ cos(kv + vo) sin(k+> + vo)]
*~(Qq)d(v3q) dvdv (3.66)
It is now clear that there are only two integral forms which must be evaluated:
JJ sin(ka + ao) cos(kop + p,)(o)I(3o) dad/3
JJ cos(ka + ao) sin(kf3 + 0)o(a)() I(/3) dadf (3.67)
The integrals involving 4i(o) are expanded and evaluated in straightforward fashion to give
/ n (k~ ao)] )(o) do si (kq ao) lsi nc [(k+ kb) )] sinc [(k  kb)1]
(3.68)
where sinc(x) = sin(z)/x.
For the case where %P(3) is the Maxwellian distribution,
I = [sin i~B 3), ] (3)d3 = J s(k /3~ )] d (3.69)
J cos Jrrz 1L6 cos 1  (
Using the substitution = sin, we have
I7 = f12 [ sinn ]
(3[ (0/ sin  (k1 d ~n
C cos,/_[/2 ["*"'sin,:~.o,/""2 [2:Cos
7rT J  C/2 OCOS J/2 sn
J'22 [ lsin ( ) sin 7) d(
(3.70)
This form can be reduced to
m 2 = r cOs r~/2 sin (3.71)
f  2[sin(~]00 cos(kflgsin7)d7 (~ — o) Jo(k) (3.71)
dJO COS J
66
For the rooftop functions, the transverse weighting is 1/21, so that
= / [csi ~ 21] J,() [cos [ ]
[s (~0)] sinc(kbl) (3.72)
With these results, the integration of Equation (3.66) is fully specified and we can
proceed to evaluate the terms of the matrix.
3.6 Numerical Evaluation Considerations
Before proceeding to the applications, some comments on the numerical implementation
should be made. At this point we are in a position to go ahead and program the previous
expressions to evaluate the matrix elements as they stand, however, just a brute force
approach, without some consideration of the algorithms to be used, would undoubtedly
result in a very inefficient program which takes much longer to run than need be. In this
section, some 'common sense' features will be pointed out in addition to some mathematical
identities which can be used to significantly improve the convergence rate of the summations
involved.
3.6.1 Precomputation
By writing out the complete expression to be evaluated for the selfimpedance on the
strips, the main points can be illustrated which also apply to other elements of the matrix.
Let us assume that the strip is oriented along the i axis in the cavity with a width of 1W
and is centered at the point (xO, yO). A typical element of the [Z]ll can then be written as
= >1 EE>i 4()(y)JJ Grr 4(x')i'(y') dz dy dx'dy'
= Z Cmn GeJxxIeo(XZiyi)Ieo(X,,y) (3.73)
m n
Assuming the Mlaxwellian transverse dependence, the Ieo terms are evaluated as in the
previous section and Zz, becomes
OZ = i8 E 2 9 geJxx cos[km(z, + zx)] cos[km(j, + x )]
m n kb
sin2 knyoJ2(knly)sinc2[(km + k6)l,]sinc2[(km  kb)II](3.74)
The most basic rule to optimize the speed of the computations is to perform any operation
or function evaluation as few times as possible. Immediately we recognize then, that the
constants should be factored out and the outer loop factors should be removed from the
inner loop:
Z(l) 4fl kb 4[ (+ z)] cos[km(zx + + )]
ab sin2 kbl1m 2
sinc2[(km + kb)l,]sinc2[(km  kb)11]
{Z eJxx sin2 knoJ2(knl )} (3.75)
Now in this expression, although evaluation of the sine and Bessel function J, in each
cycle of the loops is implied, in practice these are computed and multiplied external to
botl loops and stored as a vector dimensioned to include the maximum value of 'n' to be
evaluated. Thus, these functions are evaluated only once per value of 'n'. Similarly, the
(2  6,)/2 factor ( = 1 for m $ 0 ) appears only in the 'm' loop, since for n = 0 there
is no contribution (note also that the conditional statement which tests for m = 0 can be
eliminated by calculating the m = 0 terms separately).
3.6.2 Transformations
Finally it is noted that the 'm' loop implies m x N evaluations of the cosine terms and
with the inner n loop implies m x n x N2 products of these terms and the remaining terms
where N is the number of basis functions on the line. This can be dramatically reduced
68
by making use of the relation
cos A cosB  [cos(A + B) + cos(A  B)] (3.76)
so that our expression becomes
2 2fkl4 ( 2 m (cos[km(xi + x + 2xo)] + cos[km(x,  x)])
I ab sin2 kblx m 2
*sinc2[(km + kb)Ix]sinc2[(km  kb)lx] { geJxx sin2 2ky Ij2(kn1y)} (3.77)
n
We now need at most m x (3N  1) cosine evaluations, on the order of m x n x N products
and NA2 simple additions which greatly improves the computation timle, especially as N
increases. The sum resulting from the application of Equation (3.76) is performed after
the m and n summations are complete. This is the key to the key to the speedup since a factor of Ar
is removed from the number of product evaluations. A factor of A 2 additions are added,
however, this is inconsequential since the additions are simple (two complex numbers)
whereas the previously required product evaluations also involve the calculation of other
nontrivial coefficients. If the matrix elements are to be stored in data files, storage of the
cosine sum and difference terms also greatly reduces the file size  now on the order of
(3N  1) rather than N2.
This technique also applies to the slot selfimpedance terms. For the coupling terms
(slottostrip, etc.), the same principles are used except the sum and difference scheme no
longer applies due to mixing of km and kn wavenumbers. However, in these cases, due to
the separation of the structures in the z direction, sine or cosine factors in the denominators
become hyperbolic for large values of m and n and improve the convergence rate so that
the upper limits of the summations can be reduced.
Other factors can be considered, however, the above points are thought to contribute
the most significant improvements in program efficiency with minimal effort. Of course,
there are other issues which have not been addressed since they often depend on the avail
G9
able facilities; memory limits for example. There are also schemes available to improve
convergence throughl auxiliary series transformations, however, these are often only possible at the sacrifice of' generality. For example, for uniformly filled cavities. e.g.. stripline
problems, techniques such as those found in [13, Appendix A.6] can be applied but then
multilayered cases would require separate treatment. Admittedly, the full extent of these
possibilities have not been adequately explored and may offer further improvements in program efficiency. Rather, the emphasis has been focused on phenomenological exploration
of the applications in tlie discussion to follow.
3.6.3 Convergence, Algorithms and RunTime
As with other eigenfunction expansion methods, consideration of the convergence of
the modal summations is an important process in establishing reliable results. For these
types of solutions it would be desirable to analytically examine the expressions involved
and derive formulas for acceptable upper limits of the summations. Ideally these formulas
would be provided for each type of Green's function and would be functions of all the
relevant geometric and electrical parameters and the desired accuracy. To develop such
a system, however, is a major undertaking in itself and would really only be worthwhile
after a more thorough investigation of possible series transformations alluded to above.
In addition, convergence behavior generally varies depending on which output variable is
sought, further complicating the situation.
Nevertheless, it was necessary to investigate some aspects of this question in order to
produce reasonably efficient programs, the intention here then being to pass along some of'
the information gained to those who may extend the scope of this work. Also it should be
recognized that although it has become common practice to discuss convergence by showing
the behavior of particular parameters for the structure at hand as a function of the number
70
10.0.......,.... 'I. ...
8.0 
0 — Effective Relative Dielectric Constant
6.0 ~   Characteristic Impedance
4.0  \  Transverse Tangential H field on Shield Below Strip: 2.0
0.0 Q , .   
C;, 2.0  4    
2.0 
4.0
6.0 A
8.0  ' 
10.0
0 25 50 75 100 125 150
Maximum Number of Modes
Figure 3.4: Convergence behavior for several parameters of a centered shielded microstrip
transmission line. Dimensions a = b =.25, w = h =.025, Er = 9.7. Terminal
values at 1000 modes were Zo = 49.63Q, Er,ef f. = 6.90, Hx = 12.45A/Ao.
of modes used in their calculation, this information is really of limited value since when
the technique is applied to a different structure, there is usually no guarantee of similar
results. The plots given here are therefore provided only to illustrate some intuitive points
and to give the reader an 'order of magnitude' feeling for the required range of upper limits.
Unless some analytical guidelines are developed and become available for these problems,
similar numerical experiments must be performed for each new application.
To illustrate, let us first look at the convergence rates for several transmission line
characteristics. The theory behind the calculations is presented in Chapter IV. Figure 3.4
shows the convergence behavior for three key transmission line parameters of a shielded
microstrip line. As can be seen, the rate of convergence depends on which characteristic
is to be computed. We reemphasize that these features may vary as a function of the
7 1
geometric and electrical specifications. It should also be mentioned that this application
does not present any practical difficulties since not only does the convergence appear to be
quite rapid but these parameters are computed by a twodimensional formulation involving
only a single summation. The efficiency of evaluating the summation for this problem is
not a significant issue given today's desktop computer capabilities. For example, the data
in the figure were efficiently computed on an Apollo D)N2500 workstation.
This brings us to the consideration of the two dimensional summations required in
the analysis of three dimensional problems of the type discussed in Chapters VVII. The
required number of terms in the double summations depends on their behavior in the
mnplane, where mn and n are the two parameters of summation. Tile real problem is in
predicting or anticipating this behavior apriori so that the corresponding limits can be
set. Some progress in this direction can be achieved biy recognizing that these summations
are similar in many respects to the Fourier series. By drawing on our understanding of this
topic, we can gain some intuitive understanding of how the modal summations behave. For
example, it can easily be shown by numerical experiments that the closer two elements are
in spatial coordinates, the greater will be the extent of the mode spectrum as in Fourier
analysis. In addition, there may be no sign changes for the self coupling terms so that we
can immediately conclude that these terms will display the slowest convergence. Thus, we
could potentially monitor only the self coupling and nearby terms to determine whether
the summations have converged or not. There is also a potential savings in time, if we
can monitor the convergence of the series as a function of one direction (referred to as an
'eigendirection'), the direction of m say, while the other (n) is constant. Thereby we can
potentially eliminate significant fractions of individual rows or even entire rows, depending
'The Apollo DN2500 is quite slow relative to most other engineering workstations, perhaps comparable
in speed to standard Intel 80386 based personal computers. The above data was computed at a rate of
approximately one minute cpu time per 1000 modes. The relation is approximately linear since there is
only a single summation.
72
on the behavior of the functions involved.
While these ideas have an attractive generality to them in that we might be able to
develop an algorithm for monitoring the convergence of the sum, making the convergence
question invisible to the user, there are serious difficulties when we try to put it into
practice. Most significant of the pitfalls involved include the fact that the final result is not
known apriori so it is quite difficult to establish a criteria for convergence. To illustrate,
consider what happens if a 'percent change' criterion is used. Suppose some of the terms of
series at the early stages of summation contribute very large values which are later canceled
by similar terms of the opposite sign. At the early stages then, the sum will be large and
a fixed percentage of the current sum may be quite large compared to the final result.
Thus, truncating the local contributions to the series based on the current percent change
may prevent important terms from making their contributions which in the end leaves a
significant error. Another problem is that, as with Fourier analysis, the mode spectrum
may have extended gaps along the eigendirections so that a summation in that direction
may appear to be converged when in fact there are addition terms farther along which are
needed.
Some time was invested in pursuing an algorithm which takes these factors into account
based on observations of the spectra of various cases. In the end, however, it was found
that while some progress could be made by instituting various monitoring schemes, tilhe
final conclusion was that the overhead required for keeping track of the progress of the sum
and the periodic testing of sum's status, more or less offsets the gains made in reducing the
number of actual terms added to produce the result. However, one scheme that emerged
does bear mention here since it is straightforward. It should be recognized that the cancellation of terms generally occurs along both eigendirections and also the largest terms occur
near the origin. Then, one can readily accumulate the bulk of the sum by making sure
73
to sum the losw index terms near the origin at the onset, thus providing a good estimate
of the final result and alleviating the problem of not having a good estimate of tile final
result in advance. A very simple algorithm accomplishes this scheme by simultaneously
incrementing either mn or n while decrementing the other and successively moving away
from the origin, thereby summing in a direction normal to the diagonal of the n7lup1lane.
However, because of the tradeoff mentioned, the final versions did not use any of tlie developed schemes but simply scanned the mnplane in a straightforward, rasterlike fashion
with terminating conditions set by experience.
Generally, to avoid any question of convergence while other investigations were underway and because sufficient processing power was available, far more terms than necessary
were used anyway, typically on the order of 10001500 modes in the eigendirectionl associated with the directions of the lines or slots, and half as many in the other direction. For
the cases studied here, not nearly as many modes in the one eigendirection are needed as
in the other. This is due to the fact that in the particular cases studied for comparison to
experiments, the strips and slots were always parallel to the side walls of the cavities so
that one coordinate describing the position of the basis functions is constant. This tends
to cause the spectral variations in the corresponding eigenvalue to be similar for all basis
functions, although the variation of the other coordinate prevents this from being strictly
true. If, for example, the y coordinate is constant for all basis functions and n is the
associated eigenvalue, the spectra in n at a fixed value of m will be identical except for a
constant scale factor which depends on x. Truncation error in n then tends to get averaged
out by the variation in x. To illustrate, let the aspect ratio in this example be defined by
NIM, i.e., the denominator is the maximum eigenvalue corresponding to the coordinate
which varies and N is the maximum of the other eigenvalue (N < M). Figures 3.5 and 3.6
demonstrate a typical convergence experiment in which the aspect ratio has been fixed
74
g' 6.0
c t 0 Aspect Ratio =.5
5.0
4.0
3.0
'L 2.0
Z
1.O
j 0.0
0 250 500 750 1000 1250 1500
Number of Modes (mn)
Figure 3.5: Convergence behavior for peak normalized resistance versus maximum mode
number.
at 0.5. The quantities examined are key parameters to be introduced in Chapter VI.
As always, other parameters may converge at different rates and the rates may vary as
geometric or electrical parameters change. The key point here is that at least for this case,
when the maximum m value is set to M = 600, the changes which occur as NIM is reduced
from 1.0 to 0.25 are less than 0.1% for the resistance and 0.03% for the resonant length.
Thus, far fewer modes are needed in the n direction.
For subsectional basis functions, the spatial sampling rate also requires convergence
criteria. Since the quantities we are dealing with, e.g., impedance, are typically highly
sensitive to the behavior of the nearfield and also often depend only on the fields in a
small region, the required spatial sampling rate is generally substantially higher than what
is required for farfield type problems (or nonuniform sampling is needed which is more
complex to implement). Again, the actual requirements depend on the circumstances
7.5
0.000
i .100 
Ce).5 .200
C  Aspect Ratio =.5
.300 I
0 250 500 750 1000 1250 1500
Number of Modes (mn)
Figure 3.6: Convergence of resonant slot length as a function of the number of modes.
and techniques used so it is difficult to provide general statements on these requirements.
Twenty samples per material wavelength is often used as a 'rule of thumb' for farfield
problems which we generally increase to the 3040 samples/wavelength range for sampling
on strips or nonresonant slots. For the resonant slots, we often increase the sampling to
100125 samples per wavelength making sure to overestimate the resonant length of tile
slot by a significant amount. This practice is not driven so much by convergence criterion
as it is by practical matters. It takes far longer to generate the matrix elements than
it does to invert and process the cases studied here. Changes in length or the relative
positions of slots and lines for example, can easily be accomplished by loading a matrix
with longer than needed lengths and scanning the behavior of the structure as a function of
lengths or distances by successively removing the appropriate rows and columns; inverting
and solving for the appropriate parameters at each stage. Thus, the matrix elements
76
need to be generated only once for a given frequency, allowing a wide variety of geometric
variations to be studied as long as the cavity dimensions are not changed. We also use this
technique to provide greater resolution of lengths and relative distances without requiring a
regeneration of the matrix. The upper limit of the spatial sampling rate when the elements
of the matrix are computed with double precision seems to be on the order of about 250
samples per material wavelength. This number, attributable to roundoff and truncation
error, is machine and algorithm dependent which brings us to the final point on this subject.
For problems of these types, it is common practice to quote run times typical for certain
machines, often not very well identified. It is common knowledge that machines vary widely
in their ability to execute codes, especially from machines of one type to another, but even
for the same type of machine with different hardware configurations. In addition, the
information provided can quickly become dated and irrelevant since available computing
facilities are changing rapidly. Not only this, but also the execution time of the same
analysis implemented with different algorithms and different degrees of generality can vary
widely (easily on the order of a 10 to 1 variation, depending on both the abilities and
knowledge of the programmer as well as the time spent in optimizing the algorithms for
speed). Undoubtedly, there are techniques which can be used, in addition to those discussed
above, to improve the run time of codes developed for this analysis. The real issue comes
down to a tradeoff between time spent on optimizing codes versus producing and examining
results. Of course, the outcome of this tradeoff depends on available resources, the objective
in producing the codes, and their intended end use. Suffice it to say that the key element in
the current approach is the generation of the matrix, which for a typical three dimensional
structure at a single frequency, can be generated in 4560 minutes or less on a  25 million
instructions per second machine such as the IBM RS6000/320. This amount of processing
time has been sufficient for our needs.
CHAPTER IV
ANALYSIS OF TWODIMENSIONAL STRUCTURES
The treatment of multiple layers can be illustrated in greater detail by presenting the
methods used to analyze structures which are uniform in one dimension. Complex threedimensional problems are often treated initially in this manner by analyzing their more
fundamental elements  two dimensional transmission lines  with lumped elements added
to represent discontinuities. The total problem can then be treated by network analysis if
the basic properties of the individual structures are known.
The objective here is to show how these fundamental properties can be obtained, for
instance, the propagation constants and characteristic impedances of transmission lines.
These quantities are directly tied to the solutions for the fields in the structure which
thus becomes the main objective of this chapter, that is, to demonstrate the procedure for
matching field components through multiple substrate and superstrate layers as applied to
twodimensional problems. The presentation also serves to illustrate how the technique can
be applied to more general threedimensional problems since the procedure is the same.
Some examples of field solutions are given to demonstrate the utility of the approach. The
work in this chapter is also needed in later chapters which deal with the microstripfed slot
antenna elements.
The scattering of waveguide modes by vertical wires is also studied in this chapter. Thlis
problem reduces to two dimensions for the homogeneouslyfilled waveguide case which
77
78
is relevant to the experimental work on stripline to be discussed later. The problem is
unique in this work in that the currents are normal to directions of the other currents
discussed. The treatment of wires inserted through multiple layers cannot be reduced to a
twodimensional problem and will not be discussed although, as will be seen, the solution to
this problem would be needed to complete the general analysis of waveguides and especially
cavities formed by inserting wires through the ground planes of multilayered parallel plate
waveguides. It is also noted that even for the homogeneouslyfilled case, the problem of
vertical posts or wires has not been extensively treated in the literature, especially for
parallel plate structures, even though they are often used in practice for the suppression
of unwanted higherorder modes.
4.1 Application to General Multilayered Shielded Microstrip Structures
The objective of this section is to show how the fields in the various levels can be calculated from a known form of current density on a single microstrip line in a straightforward
way. Also, although to this point we have considered multiple components of current, the
applications to be discussed will be restricted to narrow strips with one current component
to minimize the complexity of the presentation. There is no restriction on the placement
of the current or the number of strips which can be used, however, a simple case here
will better serve to outline the method. The procedure for the treatment of more complex
multilayer coupled strips can be found in [83], which serves as an example of the use of
potential theory with impedance boundary conditions to generalize multiple layers in a way
similar to what has been done here. This reference also addresses the modelling of strip
conductor loss. In additon, we compare to a much earlier work on this type of analysis
by Yamashita [90], whose method of nonuniform discretization would be appropriate for
wider strips where the form of the current density cannot be assumed.
79
In order to calculate the characteristic impedance. tile definition Z = P/I2 is used
where P is the time average of power propagating along the guide. Thus. if we set I = 1.
the characteristic impedance is simply Z = P which can be computed analytically by
integration of the average Povnting vector on the cross section of the waveguide. Also, in
some cases, the reaction of the fields (R) oni the waveguide cross section is needed which
can be computed in the same way.
For our purposes, let us assume a geometry suclh as shown in Figure 4.1 with a longiz......................................................
z=d
z,,'.:.: .:..:.:.:.:.:.:.:.:.'..:.. .,.................
X x=a
Figure 4.1: A shielded stripline, uniform in the y direction with multilayered substrate
and superstrate.
tudinal current component on a single narrow strip of the form
J(x,y' ') = 26(z'd) ek' zx' jol < I/2 (4.1)
i.e., a Maxwellian transverse variation of the current density which satisfies the edge conditions on an infinitesimally thin narrow strip. The propagation constant k9 represents the
set of characteristic complex phase constants associated with shielded microstrip modes.
80
These complex constants are the eigenvalues of the equation
Et = Je /[/ n x G eJ J d = n x Z, J (4.2)
which enforces the appropriate boundary conditions on the surface of the microstrip line [83].
A particular value for kg must be found first, before the fields of a particular mode can be
calculated.
Using Equation (4.1) in (4.2); the yy component of Gej as derived for layered rectangular waveguide in Chapter II; and the integral representation of the delta function,
ej(gk'dy' = 2rr6(kg  ky) (4.3)
00
we find that the y component of the electric field can be written as
f Jd+00 Z (2  6m)6(kg  ky)
EYY a o / dky E — ( (k s,) sink, xsink kxoJo(k,4:W/2)
a k,(k2m= ( + k2)
{,2 [ _ _je_ eL[)eU COS kz(c  z) + jsin k(c z)]
r  (71eU  7eL) cos kz (c  d) J(reUleL  1) sin k (c d)J,kj [ imLL[mU COS kz(c  z) + j sin k(cz)] (4.
k^ L(mu  fmL) cos kz(c d) j(lmUfimL  1) sin k,(c d)j
where the Fourier integral can be eliminated using the sifting property of the 6 function. Then, using Galerkin's procedure to enforce the boundary condition given by Equation (4.2), we can write
= 0 (26m)
E a E (k m+k) sin2o kXJ( 2
k2 [ f7L[eU coS kz(c  d) +jsin k(c  d)]
[(ieU  7eL)cosk(c  d)  j(74eUeL  1) sin k(c  d)J
kk [ T7mL[flmU coskz(c  d) + j sin kz(c  d)]
k2 L(7mU  mL)cOs k(c d) j(mUmL  1)sin k(c d) (4.5)
We now numerically search, using Muller's method for example [60, p.262], to find the
values of kg which satisfy Equation (4.2) and thus correspond to microstrip modes.
With the propagation constants known, the fields in the ith layer can be easily found.
By inspection of Equations (2.80) and (2.85), the fields can be written in a general form
81
in terms of the homogeneous solutions to Equations (1.10) and (1.12). Thus, the fields in
the i'h layer can be written as
j( 26 m)ki
Hi ~
m=O 2wak,(k2 + k~)
IB [ismoo[kz, kz(z  c)] + jMoe[kx, kz(z  c)]
1. L mLPo,[kx, k,(z  d)] + iMoe[kx, k(z  d)]
~A ([cuI~ee[kxl kz(z  c)]  jNeo[kx, kz(z  c)]]
+t A; (4.6)
1[7IeLNee~kx) kz(z  d)]  jNeo[kx, kz(z  d)]]}(46
and
=  j(26r)
2rakz(k2 + k)
1 ( eUee[kZxkz(z  c)]  jMeo[k,, kz(z  c)]
1 iheLtee[kxl k,(z  d)]  jMeo[k.. k(z d)+B( 77mUNoo[kxI kz(z  c)] + jNoe~[kxl kz(z  c)I]
InB. 4.7)
k 77mLNootkxI kz(z  d)] + jNoe~ki, kz(z d)
which are valid for all layers except the source layer. Some of the leading constants are
preserved for convenience in later notation.
4.1.1 LSE Modes
Expanding the VWFs we then find that the LSE mode fields in each layer can be
written in the form
ELSE = Ai [ O [ kk cosk.xsin kx'  PYkYsink~ xsinkrx']
ak,li(O + k2)
(48)
os cos  z) [ijeui.4 j tan k(1(c  z)] (i 0)(
c cos k,,(z  d2) [ie Li tan kz(z  d)] (i > 0)
82
IILSE t 2  Y
a(k2 + A, j k Sjfl krX' + j kykt COS kjTs jn k~x']
1 cos k,;(c,  z) [7T.1U tan k.,(c,  Z)  i] (t ~ 0)
cos k, (z  d) j[fjLtan kZ(z  d2) +j} (i >0)
+A [ ] IPk, cos kIxsin kIz']
1os J cos k (c1  z)[ieu,U +~jtan k,(c2  z)I (i K 0)
COS k,sd c (z  d) [7eL:  j tan kz,(Z  d1Ifl (i > 0)
(4.9)
where i = 0 for the layer containin'g the strip and
Ao 7Lo(4.10)
(ijeUO  leLo)coskz(CO  d0)  j(i7eUO7je7,o  1) sin k,(co  do)
By matching the tangential components of the field at the interfaces above and below the
strip layer the remaining coupling coefficients are found to be
kzj rreuO+jtankzo(co  do)'
cos k(.i)tanki(ccl  di) 
AA2 kz. 7eU(I~1) >( < 0)
1kz(i+) cos k(,z+l)(c,+l  di+1)1[U, ~j3 tan kz (c d
(4.12)
For computation of power (P), the needed component of Poynting vector is E x
while for reaction (R) we need. BE(ky) x H(ky). For the case of LSE modes, power
density and reaction are given by
2wii Ak~kk ir A"k] cos2 krxx[fsin kr.xl]2
a2 k(k2 + k) k Icosk1(c  d,)12
I7ieU, cos k (c1  z) + j sin k.,(c1  z)12 (i ~ 0)
I7eLt cos kz,(z  d)  j sin kz(z  d)12 (z > 0)
(4.13)
83
_4wi [ Akrk~ 1 [Ak. cos' k~x [f sin k.r]
RE()H(k)a2 [kz(k2 + k2) k~ co2kz2(c,  d1)
4.12 LM2Mde
1 f cosUCo kc z) [ zm) + j'tan kz(c  z] ( 0
C0 k:(t d) o . dz im~ ta k t( ~) i>0
COSkLc, c4oco k z2(z . 2 [dm) ta'sn k,(z  d _)+j]2 ( > 0)
4.1.2 L(4.15)e
HLSM = B, ak2 k) ~k sin k~k osinxSn kx'x rjk~k sco k~x sin k~x']
1 Jcos k c z) [irmui  tan k~(,z) c zj] (I 0
COt~(,d) cos k (z d) [i~mLi tan k z(z  d1)+j] (i > 0)
whrthecefiins r efndb
~B  TmLO
B0 (i~mU0 i~mLOcsin~ d0) s~inmUO~imLO )ik(od) (.
kCkos7m0~ a k (c  )[, a,( —z
B1 =0III (4.18)
koskk LCt j dtakz(Cd
Ioscos z  d1)[iL:j tan k 1(cz  d2)]+ 1)
Th1(o1sos B~ k,(c+z)k i~mui tan kz(c  d)] (i < 0) (.9
84
To calculate the field at a particular point in the Z'h layer, we then calculate only the
coupling coefficients (Ao... A) and (Bo... B.) while evaluating Equations (4.8.4.9) and
(4.15,4.16).
For the LSM modes the required terms for calculating power density and reaction are
given by
P=E,H =
R = E,(k,)H.,(k.) =
22
2wyu [Bk] B.(k*)2 sin2 k~x [f sin k.,x']2
a2 [kj L2 + ~k2)* J cos kZ?(c.  d.)12
fJi~~j~sin k,j(c  z) j'cos k,2(c  Z)12 (i ~ 0)
IlmLisin k,z(z  di) + Jcos k,z(z  d)I1 (i > 0)
(4.20)
4wu [jBk, [ jBk1 sin2 kx [f sin kx']2
a2 LkJ (k2 + k2)J COS2 k,2(c  d)
[im,Uisinkzi(c z  z) jcoskz2(c  Z)]2 (i ~0)
[ijmL sin ki(z  d) + j cos kz(z  d,)]2 (i > 0)
(4.21)
4.1.3 Cross Terms
There are also two sets of crossterms from the ELSM X HLSE product. The power
terms are
P
P
2wy ix A Ikx
H* y /2+ c2 tzt C os kIcx [f sink,,']
a k?(k+ k) Icos kzi(c.  d)12
[7lmci cos kzi(c1  z) + j sin k,,(ci  z)] [Ii~ui cos kz(c  z) + j sin kz(c  z)]*
[?jmli cos k_(z  d ')  j sin kFi (z  di)] [ieLi cos kzi(z  d)  j sin kz2(z  di)]*
2wy Bik1 Ak 1 sin2 kIx [f sin k.x']2
 a2 [k ]? [ k2)* J I cosk2); (c  d.)12
[imUi sin kz2(c  z)  j cos kz2(c2  z)] [?leu sin kbi(c  z)  j cos kz(c  z)]*
[IimLi sin ki( z  d2) + j cos kz(z  d )] [ijeLi sin k,(z  d) + j cos k,2(z  d)]
(i ~ 0)
(i > 0)
(4.22)
(i~< 0)
(i > 0)
(4.23)
85
For the reaction we have
R E (k1 )HJ (k. )= 4w'i [jB kxkykzi [ iA.k' l cos2 kx [f sin krxl2
a2 [k (k + k)J z cos khz(c,  d)
{ [imUi cos kzi(c,  z) + j sin kzi(Ci  z)] [i7eui cos kzi(ci  z) + j sin kzi(ci  z)]
[imLi cos kzi(z  di)  j sin kzi(z  di)] [i7eLi cos kz(z  d)  j sin kzi(z  di)]
R E (k,)H ( k.) 4 FjBiky1 — jA3k 1 sin2 kx[fsinkx']2
a2 L k? [(k2 + k2l)J cos2 kz(c di)
{ [O'mU, sin kz(c,  z)  j cos kz(C  z)] [Rei sin k,(c,  z) j cos k(ci  z)]
[7,mLi sin kzi(z  d,) + j cos kz(z d)] [reL sin kz(z  d,) + j cos kz(z  di)]
(i < 0)
(i > 0)
(4.24)
(i < 0)
(i > 0)
(4.25)
4.1.4 Integration of Power and Reaction terms
The evaluation of total power can be done analytically by integrating the power density
on the waveguide crosssection. Integration of the x dependence is trivial and has been
indicated in the previous expressions. The z dependence appears in two forms, the first of
which is
c [711 cos k(c  z) + j sin k(c z)][/2 cos k(c  z) + j sin k(c  z)]*
IPM = dz
[771 cos k(z  d)  j sin k(z  d)][r72 cos k(z  d)  jsin k(z  d)]*
= 4 (k)sin e(k)(cd) sin2 e(k)(c  d)
1r+ r/~ sinh 2Q~m(k)(c  d) 7F + 2
+ 4,rn() m(k) sinh 2mm(k)(c  d) 2 d) (4.26)
where Re(k) and 'Sm(k) symbolize the real and imaginary parts of k respectively. The
second form is
c [71l sin k(c  z)  j cos k(c z)][r2 sin k(c  z) j cos k(c  z)]*
PN = L dz
[ [r1 sin k(z  d) + j cos k(z  d)][r2 sin k(z  d) + j cos k(z  d)]*
1" — 1in2 e/k)(cd)~j771  772
=  4 e(k)  sin 2 e(k)(c  d)
+ 4h 2m(k)(c)  2d m(k) sinh2 im(k)(c  d) (4.27)
86
For reaction we also have two types of integrations to perform. The first is
c [ cos k(c  )z + j sink(c z)[r2 cosk(c ) + jsin k(c  z)]
IRI = d
[771 cos k(z  d) j sin k(z  d)][72 cos k(z  d) j sin k(z  d)]
= 71772  1 b)+71712 + 1 sin2 c  )sin2 k(c  b) (4.28)
2 4k 2k
Similarly,
t [71 sin k(c  z) j cos k(c  z)][r/2 sin k(c  z) j cos k(c  z)]
IRN = dz
[771 sin k(z  d) + jcos k(z  d)][r2 sin k(z  d) + j cos k(z  d)]
711727771 ~1 712+1 2
= 2  l ( d)  1 +  sin 2(cd) j si2 (cd) (4.29)
2 4k 2k
4.1.5 Applications
With these analytical results, the fields, power flow and reaction, characteristic impedance
and propagation constants can be readily found for shielded strips with any combination
of layered substrates and superstrates. An example is shown in Figure 4.2 where results
of the present technique are compared to a commercial CAD package (Touchstone [20])
and experimental measurements provided by Dunleavy [19]. The relatively large error bars
provided by Dunleavy for this case do not provide for any conclusion on the comparative
accuracy of the fullwave approach since the agreement is excellent. The corresponding
characteristic impedance is shown in Figure 4.3.
We conclude that the accuracy of the fullwave implementation is excellent since the
accuracy of Touchstone is well established and has been further verified by Dunleavy for
this case. One can argue that the fullwave implementation with its greater complexity is
not needed, however, most CAD packages, including Touchstone, are based on approximate
formulas which, although they can be quickly evaluated, generally decrease in accuracy as
frequency is increased. Also they deal almost exclusively with the dominant propagating
' I I I T I * * I I I I ' 1 * I I I 1' ' I' '
t
o
Q
a
0
u
0
C5
Q)
0
>,
LL<
7.40
7.20
7.00
6.80
6.60
6.40
6.20
6.00
0
I  2D Full Wave
  TouchstoAe
Mcasuranents  Dunleavy
I... I... I I I I.& j I..... I.. I L I. I I..I........
).0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
Frequency (GHz)
Figure 4.2: Effective relative dielectric constant for shielded microstrip on alumina (a =
b = 250 mils, W = 25 mils, ~r = 9.7 ) compared to measurements [19] and
Touchstone [20].
55.0
54.0......
0
u
a
0

C.
E
o
CZ
ie
53.0
52.0
51.0
50.0
49.0
48.0
I ' I   I I I I'i I ' ' '....
2D Full Wave
Touchstone.,,. 
47.0 F
46.0
45.C
^
I.. ~ I... I.. I. I... I........ I..... I..
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
Frequency (GHz)
Figure 4.3: Characteristic impedance for shielded microstrip on alumina (a = b = 250 mils,
W = 25 mils, Cr = 9.7 ) compared to Touchstone.
88
modes which sometimes is insufficient. These features are similar to the behavior of "quasistatic" analyses although today's CAD packages generally go beyond 'quasistatics" by
employing higherorder, albeit still approximate techniques. The results, of course, are
very efficient routines for each component in a microwave circuit which make these packages
very powerful and efficient for circuit design.
Fullwave techniques have their greatest promise where the other methods fail, for exanmple, analysis of high frequencies components, systems with multimode interactions, radiation problems and applications with geometries and accuracy requirements not amenable
to simpler analytical techniques, i.e., the type of structures studied here. For instance.
Figure 4.4 shows the multimode propagation constants for the even modes on a shielded
microstrip line produced by a program generalized to handle arbitrary layered structures
as discussed. The simple case of microstrip is shown since there is available data for conilparison and verification. The curves overlay the data supplied by Yamashita [90] with
the exception of the dominant mode. Touchstone results are also shown for the dominant
mode which are in exact agreement with the present method. It is presumed that the
Touchstone results are correct for this mode since the dimensions are not extreme in terms
of wavelengths; thus, quasiTEM assumptions, as used by Touchstone, should be adequate.
Most likely, the discrepancy for Yamashita's approach is attributable to the use of pulse
basis functions at the strip edge which do not satisfy the edge conditions, although his
discretization over the strip improves the capability to approximate the true current. Nevertheless, the form of current assumed with the present method is the exact form for the
static case which should be very close to the true solution and therefore is more likely to
give better accuracy. Since the exact details of the discretization are not supplied in [90],
it is difficult to make a judgement, however, the real point here is the ability to model the
higher order modes. (To represent the odd modes, a higher order expansion of the current
89
3.0. I... 1
LU —) Even Modes  This method
' o Domninant Mode  Touckhsone
0.0
Figure 4.4 Multiode propagation constants for the even modes in shielded microstrip
(a = b = 12.7 mm, h = 1.27 mm, w =.635 mm = 8.875).
10. 15. 20. 25.
Frequency (GHz)
(a = b = 12.7 mm, h = 1.27 mm, w =.635 mm cr = 8.875).
90
is needed such as Yamashita's approach or possibly, expansion by Chebyshev polynomials
as in [83].)
Besides the obvious utility of this type of information for circuit design and the need
for these quantities to be demonstrated in later chapters, the ability to visualize the field
or power distribution often provides important insights into why certain structures behave
as they do, and also, how the behavior might change when the structure is modified. For
example, we later will extensively discuss the special case of a stripline (homogeneously
filled shielded strip) which passes through an aperture in the wall of a cavity. The field
distribution in the vicinity of the strip as shown in Figure 4.5 gives a clear indication of the
constraints which must be placed on the size of the aperture. One can see that the larger
E Field on waveguide cross section
Arrow scaled to logarithm of field magnitude
OdB20 dB 
40 dB........ .... .  .H Field on waveguide cross section.  —— ' 
~ _..=__~.~.. ~o —.
Figure 4.5: Stripline field distribution for the dominant propagating mode.
field magnitudes are tightly confined to the immediate vicinity of the strip as expected
(the tails of the arrows are the field points). For a 'passthru' aperture then, the opening
91
in the wall should be sufficiently large so that minimal energy (proportional to the field)
is intercepted by the wall. From the numerical data, quantitative criteria can easily be
established.
Similar examples for microstrip and suspended microstrip can be found in Figures 4.6
and 4.7, respectively. The upper part of the structure in Figure 4.7 has the same dimenE Field on waveguide cross section H Field on waveguide cross section
Arrow scaled to logarithm of field magnitude Arrow scaled to logarithm of field magnitude
0 dB  0 dB 
20 dB  20 dB 
40 dB  40 dB 
   ,   / v x
t d,,  
   s ~,   ,,, { $ s \,
l I I /
 /   \ \  t
Figure 4.6: Microstrip field distribution for the dominant propagating mode.
sions as the structure in Figure 4.6. By comparing the fields in these two cases, especially
in the substrates, one can appreciate how one structure behaves quite differently from the
other. Consider for example, what would happen if another strip is introduced on the
substrate in these two cases. Clearly, the suspended microstrip field distribution would
be more greatly distubed than the conventional shielded microstrip, and therefore the
coupling between strips would likewise be greater. This type of argument is a 'visual'
application of the Reaction Theorem. Such an ability to visualize the field demonstrates
92
E Field on waveguide cross section
Arrow scaled to logarithm of field magnitude
0 dB20 dB 
40 dB 
H Field on waveguide cross section
Arrow scaled to logarithm of field magnitude
OdB
20 dB
40 dB,,    x '
v / v     \ \
I / v    ' ' \
I /, '   v \ \, /,   s \ \
f / / v  '
\ \
' \ \   '  / 1
' \ '    / I
' \.    / / t
\ \ \    / / J
% \.    I / 1
\ \ ^  ~  / '
\ v      S J
Figure 4.7: Suspended microstrip field distribution for the dominant propagating mode.
93
one of the benefits of this type of analysis: enhancing our intuitive understanding of the
behavior of these structures as well as providing quantitative information.
4.2 Application to Multilayered Slotline
A similar process can be used to evaluate slotlines or coplanar waveguide. Since coplanar waveguide can be treated as coupled slotlines by a simple extension of the method,
only the single slot case will be discussed here. WVe assume the magnetic current is of the
same form as in the electric current case which then is valid for narrow slots:
K(x. y', ') =  26(z' d) jg' x'  xo < W4/2 (4.30)
7rW T 14 4( )
The variable kg here represents the set of characteristic complex phase constants associated
with shielded slotline modes. They are the eigenvalues of the boundary condition equation:
H  H i J [GK  Gm<] K dV' =O (4.31)
which enforces continuity of the tangential magnetic field (Ht) in the slot. As before, a
particular value for kg is found first, before the fields of a particular mode are calculated.
The procedure is the same as before except that the field expressions are changed to reflect
the change in the source terms and complementary boundary conditions for the dual fields.
4.2.1 LSE Modes
Expanding the VWFs we then find that the LSE mode fields in each layer can be
written in the forms:
ELSE = A [a 2) yke cos kzx cos kzx' + yyjkxky sin k x cos k x']
1 J cos ki(c,  z) [flUi + j tan ki(ci  z)] (i < O)
cos (c,  ) cos k (z  di) [jeLi  j tan kzi(z  di)] (i > 0)
(4.32)
94
[(2 6m>Sok~ 1 [2,krk. sin k.,x cosk~x' + ~ Ijk cosk,x cosk2r'1
ILSE = ~A1 ak.(k +ky j
1 f cos kzt(ci  z) [ij,U 'tan k, (ci  z) ii (i ~0)
1 cos k2(z  d;) I'eLi tan k,;(z  d2) + j] (i > 0)
A (2  [ipk cos~ ]ii~~*~ ~
1oJkcosz k,2(c  z) [izIeci + j tan kd(c2  z)] (i ~ 0)
cos kz(Z  d1) [ieta  j tazn kd(z  di)] (i > 0)
(4.33)
where i = 0 for the layer with the Slot so that
A 2b WO [(kco kx oskx'
Coztan k~,o(co  d' ) (4.34)
jak1(11 (4.35)
cos k,  (c di)l dl
By matching the tangential components of the field at the interfaces above and below the
strip layer the remaining coupling coefficients are found to be
A, = Z (i) kz(z  d,) [;  di'l)[ljelitankz (cz  d2) ~ (>
I ~i(i+1kk coskZi(;+l)(C2~l  di+i)[ijeUi tankzI;(ci  d2)  ij 0
(4.36)
The expressions for power density and reaction of the LSE modes are
PE H = AC(2 6) W [
A[(kg+ko)c2 (k)2 j cos k,2(C;  )34
{ Iiieu cos kZ2(c2  z) + j sin kzc(c  z)12 (i ~ 0)
JijeL: cos k,;4Z  d2)  j sin k,2(z  d2)j2 (i > 0)
(4.37)
I?= E,(k;,)H,(kc,) = A 2(2  (5m)2WE0 klcs kxIokx]
2 [(k k)t a k JkJ cos2k2(ci  di)
{t[igteUi cos kap(cs  t ) + j sin ktr(c e  Z)]2 (i a 0)
[kkei cos k _,(C  dt)  j sin kz2(Z  d2)]2 (i > 0)
(4.38)
9 o
4.2.2 LSM Modes
Similarly, the LSM mode fields can be written as
ELSM =Bt [(:~~7;] ~~k, cos k~x cos k.  f~'jk.,k, sin k~x cos k~x']
1 JCos k z) [ii 7,imUi + j tan kd, ( z) ( 0)
cos (c — di) cos k, (z — d) [ '1ijtan kzt2(z d)] (i > 0)
~B9 ( 6) [:pkx sin k~x cos k~x']
1 Jcos kx (c  z) [i~mu i tan kz (c,  z) < ] ( 0)
cos k't1(c.  d)cos k(z? (1kd) [7~zta k2(z d  d) + j (i> 0)
(4.39)
+26L( k ~kk)i k~x cos k~x'  'jk2cos k~x cos k x]
1L~  d2) f o y(,z im:tnk~c )j i~0
cscos k(z — dc [i~mL tan k~z c dz) +j i>0
(4.40.)
where
Bo ___ ~ (4.41)
77muo + tan kzo(co  do)
B1 =(__ I (4.42)
B, = B I ta k(.1) cs di~..)(?.lk, takzc d)+j (i>1
Bt 1 kdi+1 )[fjm~i tnkz(c d) i] <0
t1. k,(:+1) co k(i+1 (c.+1 d — j[~ tan kz( — di, 
(4.43)
For the LSM4 modes tile required terms for calculating power and reaction are
I.2 (2  brn) WC~O [ k] [kxk] ink~x [f cos k~x']
EH22k t[.~J[g coskz2(c.  d2)12
{l'7mi u sin k.,(c  z) jcos k,(c  _Z)12 (I 0
li'l7rL, sin k,(  d.) + j cos k,(z  d )12 (i > 0)
(4.44)
96
2' (2k kk2sn [kxx]F.f 1.osrk,x2
ft Ez(ky)fIx(kl,) B?, ar 2WE0 k____ kz'(k 2I kzX[Jcsk d
a [kzI tk 2(k+ k2)J Cos2 kI(~
{,ju  sin k,4c,  z)  j cos k,4c, <)2 ( 0)
[77Lr snk, l k(z  d i) + j cos k z d 2)]2 (i > 0)
(4.45)
4.2.3 Cross Terms
There are also two sets of crossterms from the ELSM x HLSE product. The power
terms are
P ExH z BZjA t*(2  b)2WE, kX ] k] Cos' k~x [f cos kx
2a2 [(k2g ~k2)qJ[(C 2Jfco
{[rn:cos k,(c, z) ~ j sin k,(c  z)] [7i7eU, COSktct  z) + j sin kz(c  z)]' (i< 0)
[ir77o z( 2 jsnk1zd) ieL' COS (  d) j sin k.,, (z  d) (i> 0)
(4.46)
F  rn )2WEo [k] kk 1si2 kx[ckx]2
B ~ a [k~1 [kz(k~x + (k*)2) JIcos kz2( 
{ [i7rnu: sin kz (c,  z) jcos kz(c  z)I [ieu: sin kz (c,  z)  jcos kz (c,  z)]* i~0
[i7nL1 sin kz (z  d ) + j cos k, (z  d1) [~L sin kz 1(z  d) + j cos kz (z  di) (i > 0)
(4.47)
Similarly, for the reaction we have
 b,25)2WEo 2
R Ex(ky)Hz(ky) = a2A [(2kg [k] Cos k.,x[(co d2)
a2 k2+k2)(,C.)J co2 kz1(c,  d2)
{[mu osi kz1(c  z)  j csi k.,(c  z)] [i~,u osi kz(c  z)  csikn c z](i~0
[i~mLi csi kz(z d) jcosinkz(z d)] [ieLi Csikz(z  d) +jc'sink (z  d1)] (i > 0)
(4.49)
Th ntgaio f hs2trs nte crs2eto novstesm fomsa fr h
97
electric currents which have already been given.
4.2.4 Application to Finline and Suspended Finline
As with strips, the distribution of the field provides insight into the behavior of the
structures. Two examples illustrate by showing the change in the field structure for finline
without a substrate compared to the case whiere a dielectric substrate (,r = 2.2) is added.
These cases are shown in Figures 4.8 and 4.9
E Field on waveguide cross section
Arrow scaled to logarithm of field magnitude
0dB
20 dB 
40 dB 
H Field on waveguide cross section
Arrow scaled to logarithm of field magnitude
0dB
20 dB 
40 dB 
 .u . d ..   
~    ~   
 / / / _/
jHH+t d ~+: \
  \ \ /,
~~~'' — 
.....     i
4 4 4 4 4 4 4 4 4
I I 1 4 I 4 I 4 4
4 4 4444 4 4
* 4 44444414;
4444 X4 4;;
\ \ \ \ 4 4 / / /
\ \ \v \ \ / / / /
"  ./ / / / s \ \ \ \
* /// 44 \\4s
/ / / 4 \ \ \ \ \
* 44 4 1 4 4 4 44X
4 4 4 4 4 4 4 44X
4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4
\  \
Figure 4.8: Finline field distribution for the dominant mode.
98
E Field on waveguide cross section
Arrow scaled to logarithm of field magnitude
OdB
20 dB40 dB 
&  I I I I
 ~  ..      .
w   )          
          
 \ \ l
 ..    
   C    
H Field on waveguide cross section
Arrow scaled to logarithm of field magnitude
OdB 
20 dB 
40 dB 
1 i 1 1 4 J 4 4
\ \ \ 4 4 1 / J 4
\ \ \ \ \ / /
/ / / /4 \ \\
\ i i \ \ \ i \ i
I 11 I .
Figure 4.9: Suspended finline field distribution for the dominant mode.
99
4.3 Scattering from Pins in Rectangular Waveguide
The insertion of wires or pins thlrough tile ground planes is a technique used for many
years to suppress parallel plate waveguide modes in stripline and for structural support. It
is also a convenient way to form waveguides. cavities or isolation walls in stripline circuits.
The experimental work discussed in Chapter VI is an example of the formation of a cavity
where our particular need for this analysis will become evident. Other applications include
the use of posts (or strips) in a rectangular waveguide as reactive elements in filter and
matching networks [9, 10, 68] and for active device mounts and tuning elements [7, 24].
These types of applications have been the focus of most of the theoretical and experimental work appearing in the literature. In contrast, what we are interested in here is the
effectiveness of a 'wall of wires' as a short circuit.
Analytical techniques in early works are primarily based on variational methods [54,
48, 45]. These approaches, however, become impractical for more than a few posts or posts
with irregular spacings and are limited in accuracy at high frequencies. Even later works
focussed primarily the study of three posts at most with emphasis on accuracy for post of
relatively large diameter [43, 44, 3]. The emphasis on a limited number of larger posts is
the opposite of what we require here where we will tend to use wires of small diameter and
of greater number (and density). It is also evident that larger posts require more complex
formulations and numerical treatments than is desirable for our situation1. Image theory
or grating formulations as in [29] or especially [46] would seem to be appropriate for this
problem but appear to be excessively complex and have limited potential for combination
with other structures. The approach developed here will therefore be in spirit of the
approach given in [43] which offers a good balance of simplicity and flexibility. In addition,
'Unless one already has a numerical code to treat the problem at hand, it is always desirable to use
the simplest formulations which can produce results of sufficient accuracy and therefore can be quickly
implemented. Another consideration is whether the resulting numerical model can be efficiently evaluated
which usually, but not always, favors the simpler formulations.
100
the method will be even further simplified for wires of small diameter.
4.3.1 Reflection Coefficient Formula Derived from the Reciprocity Theorem
Consider a grid of wires appearing in the cross section of a rectangular waveguide as
illustrated in Figure 4.10. Let us assume that the diameters of the wires are sufficiently
Figure 4.10: A grid of vertical wires in rectangular waveguide.
small to allow us to represent the current on the wire as an infinitesimal filament located
at the center of the wire, i.e.,
J =.;Ij6(x  x)6(y  yj)
(4.50)
Furthermore, the waveguide dimensions are deemed to be such that all modes but the
dominant one are cutoff. The current on the wires is also assumed to be excited by the
dominant mode field which has no z variation. This fact justifies our assumption that
the current also has no z variation. For multilayered structures this assumption would no
longer be valid and the formulation must be made more general, however, it is sufficient
here since the present need for this effort is restricted to stripline in support of Chapter VI.
101
\WNe can derive a reflection coefficient representing the scattering of the dominant mode
by an application of the Reciprocity Theorem [33] in the form
/f[a x Hb Eb x Ha]. dS = J[Eb a Ea Jbh d (4.51)
The locations of the surfaces defining the volume of integration are coincident with the
side walls of the waveguide and two transverse planes on either side of the grid of wires.
The transverse planes are assumed to be far enough away from the grid wires so that only
the dominant mode has significant field strength. Now let the Ea and IHa fields represent
the normalized fields of the dominant mode, incident from along the y axis. having no
source terms within the volume of integration, i.e., Ja = 0. Then these fields are given by
Ea = Elo= sin(kmx)e3kY (4.52)
Ha = Hlo = xk sin(kmx)eJk (4.53)
WLI
where km = mrr/a, m = 1 for the dominant mode and ky = \/k2 Tk. The Eb and
Hb fields will be produced by the currents on the wires and radiate in both the +y and
y directions. If the currents on the wires, Ij, are those which are excited by the above
incident field, then we can write
Eb = El0 = IF sin(kzmx)e~j (4.54)
Hb = FrH1 = ~i r sin(kmx)e~Jkyy (4.55)
where the top sign is for y < y' and the bottom sign for y > y'. Equation (4.51) now
reduces to
JJI Eo i dV
r =  f E10  d (4.56)
2JJ Elo x Hlo Y dxdz
Note that to obtain this result, the surface integrations over the walls of the waveguide
were found to be zero, since the tangential E fields are zero there for perfectly conducting
102
walls and also the surface integral over the waveguide cross section on the +y side of the
wires evaluates to zero. The remaining integrations are readily performed to give
27tf (L
r  (7a Ij sin(7rxz/a) (.5)
We thus need only determine the unknown currents from which the reflection coefficient is
produced by this simple summation.
4.3.2 Method of Moments Formulation
In the previous section, the assumption of constant current on the z dimension has
already been stated. In the parlance of the Method of Moments this is a 'pulse' basis
function which in effect has reduced the problem from three dimensions to two. For small
diameter wires as treated here, the problem can be further reduced to one dimension, if the
grid of wires are all contained in a transverse plane, however, this produces no significant
advantage.
Since we have previously found the dyadic Green's functions for homogeneously filled
rectangular waveguide (section 2.4), we can use Equation (1.33) to derive the electric fields
in the waveguide. With the current as stated by Equation (4.50), the z component of the
E field produced by the jth current filament is given by:
o0 1
E, =  Ij E sin(kmx)sin(kmxz)eJkylyy1 (4.58)
m=l Y
It remains then, to find the set of Ij's which produce a total electric field that satisfies
the boundary conditions on the wires, specifically E, + z  E1o = 0. The simplest way to
enforce this condition is to use point matching on the surface of the wires which is to use a
delta weight function. Although found to produce identical results, a two point matching
scheme was used to provide symmetry balance in which the field is taken as the average
103
over two points of the wire surface. The resulting expression for the field is
Ez = w1Ij E sin[km(z +~ ri)]sin(k,xzj)6Ji'y^ (4.59)
m=1 l
where i is an index indicating the field evaluation points and r, is the radius of the ith wire.
The ~r, notation is taken to imply the averaging of the field contributions at these points.
The combined field expression can be written in matrix form as
[Z,][Ij] = [ Elo,i]
(4.60)
which is solved by matrix inversion for the unknown currents.
4.3.3 Validation
In order to investigate the validity of this formulation we first compare to the data of
Marcuvitz [48] for single posts. The accuracy of this data is well established for diameter
to waveguide width ratios (dla) of up to 0.25. The equivalent circuit for the single post
is as shown in Figure 4.11. From this equivalent circuit we can convert the values given
Xb
Xb
C0
0
Port
1
Xa
Port
2
Figure 4.11: Equivalent circuit for single post in rectangular waveguide.
by the formula in [48] to a reflection coefficient which can be compared to the results of
Equation (4.57). Since our formulation is a simplified version of the method found in [43],
104
LI —,
I)
3.a
E
0
0.9 
0.8
0.7
0.6 / _
Marcuviz........ Poitn Matled MoM
0.5  l
0.4.... '............ '..*. '.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Normalized Post Diameter (d/a)
0
c)
u
7G
U.L)
a
Qt
170....,.,.,...,...,....,,.
160.
150. 
140.
130.  /
Mrcuvitz
/......... Point Mched MoM
120.
10...,......I.................
110.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Normalized Post Diameter (d/a)
Figure 4.12: Magnitude and phase of reflection coefficient for a single conducting post
in rectangular waveguide. Point match Method of Moments (MoM) results
compared to the data of Marcuvitz [48] (A/a = 1.4).
we would not expect the range of its accuracy to extend as far; thus, we will limit the
range of our discussion to d/a < 0.1. Moreover, we are specifically interested in thin wires
with diameters (d) much smaller 0.la, on the order of 0.02a, the approximate value to
be used later in Chapter VI. The results over this range is given in Figure 4.12 which
shows the magnitude and phase of the reflection coefficient of a single post in rectangular
waveguide as compared to Marcuvitz's data. As can be seen, the comparison is excellent
with a maximum phase error of less than four degrees for the largest wire diameter. Thus,
we have a first indication of the accuracy of the method.
Before proceeding to further results, it is instructive to further examine the single
post case. The behavior of the element values of the equivalent circuit follow the curves
illustrated in Figure 4.13. By evaluating the input impedance for one of the ports with
the second port matched and plotting it on the Smith Chart as in Figure 4.14, we see that
the input impedance progresses from inductive to capacitive as the diameter is increased.
We also note that the single post makes a surprisingly good short circuit at a diameter as
105
0.80....,....,......
0.70 
o 0. \  Shunt Inductive Reactance X., IXa = 1.4. \ ... Series Capacitive Reactance Xb, /a = 1.4.. 0.50
0.4.0 
0.30
i 0.20 
Z 0.10 o.oo..............I.i....
0.00 0.05 0.10 0.15 0.20 0.25
Normalized Diameter ( d/a )
Figure 4.13: Behavior of element values for the equivalent circuit given by Marcuvitz [48]
(A/a = 1.4).
small as d/a = 0.15, the diameter at the 180~ phase shift point.
Not surprisingly, as more pins are added the wires form an even better short circuit
producing ever higher shunt susceptance values. For example, Figure 4.15 shows how
the shunt susceptance value increases rapidly as the number of equally spaced wires is
increased. In this case we have lumped the reactive behavior of the grid into a single
shunt element on the transmission line, i.e., replace the series capacitors of the equivalent
circuit of Figure 4.11 with shorts and the shunt inductor with a general reactive element.
(Note also that all wires in this investigation will be located in the same transverse plane
of the waveguide which is the reference plane for the equivalent circuits.) It is interesting
to observe that the normalized pin diameter (d/a), where the susceptance crosses over
from inductive to capacitive, depends on the number of wires and generally moves toward
smaller post diameters as the number of posts increases. This suggests that for a given
106
Figure 4.14: Input impedance as a function of post diameter with a matched load port.
Post diameter increases in the counterclockwise direction from d/a =.005 to
d/a =.25 in.005 steps, demonstrating the transition of the input impedance
from inductive to capacitive.
107
200.0.. ', '
One Post
150.0....... Three Posts \
100.0  Five Posts
'  — Seven Posts *Ca 50.0 . s.
0.0  . I
U 50.0
3
c 100.0
150.0 ' ' 
200.0
0.000 0.020 0.040 0.060 0.080 0.100
Normalized Pin Diameter ( d/a)
Figure 4.15: Shunt susceptance behavior for up to seven wires placed evenly spaced across
waveguide cross section.
number of wires equally spaced across the guide, there is an optimum post diameter which
most nearly produces an effective short circuit. Similarly, it implies that additional posts
do not necessarily improve the capability to simulate a true short circuit at the reference
plane.
With these large shunt susceptance values, the real part of the input impedance becomes
negligible so that the network can be treated as a one port terminated with this reactive
element. As such, the one port can also be modelled as a true short circuit located at a
distance from the reference plane determined by the calculated value of susceptance and
the transmission line equation:
B = cot(3l) (4.61)
where B is the susceptance, I is the distance from a true short to the reference plane and
/3 = 27r/Ag, with \g the guided mode wavelength. The argument 31 is the phase angle of
the reflection coefficient for the line. The accuracy with which we can determine this angle
108
is a measure of how well the grid of wires is treated by the simplified niodel, thus becomiing'
the focus of the experimental results and comparisons to follow.
To further confirm the accuracy of this method, measurements were made with an
HP8510 Network Analyzer on various combinations of wires in Xband waveguide (0.4 x 0.9
inches) at 12 GHz. A series of seven equally spaced holes where drilled in a line on the
broad wall, transverse to the axis of a section of waveguide. The holes themselves were
small enough in diameter and sufficiently spaced so that they did not significantly perturb
the propagation of the fundamental mode, a fact verified by subsequent measurements.
The reference plane was established and the fixturing deembedded by performing a oneport calibration with two short circuited waveguide sections of different lengths and a
precision waveguide load. To make the measurements, different patterns of wires were
inserted through selected holes and the reflection coefficients recorded. The patterns used
will be denoted by a series of ones and zeroes, for example the case of the single centered
post would be designated by the pattern '0001000'.
The results of one set of measurements are shown in Figure 4.16. In this and the following charts, data marks correspond to the three wire diameters used (d =.025,.033.039
in.), the filledin marks representing the corresponding measured points. As can be seen
by drawing a line from the center of the chart through the various points, the accuracy
of the predicted phase angle is quite good. Similar results are found for wires in pairs as
shown in Figure 4.17.
Figure 4.18 is an experimental demonstration of the case discussed above where the
number of pins is progressively increased. This is perhaps the worst case where errors in
phase angle are on the order of five degrees for some points. Some of these errors are
attributable to experimental error due to variables such as misalignment in the transverse
plane and tilting of the wires in the holes, which necessarily must be larger to accommodate
109
Figure 4.16: Reflection coefficient measurements for a single post at various offsets from the
centerline compared to predictions from the simplified model. Wire diameters
(d =.025,.033,.039 in.) increase in the counterclockwise direction.
110
Figure 4.17: Reflection coefficient measurements for pairs of wires at various offsets
from the centerline compared to predicted values. Wire diameters (d =.025,.033,.039 in.) increase in the counterclockwise direction.
111
Figure 4.18: Measured reflection coefficients for centered wire grids of equal spacing and
increasing number. Wire diameters (d =.025,.033,.039 in.) increase in the
counterclockwise direction.
112
different diameters. Nevertheless, the overall results suggest that the method is capable of
predicting the phase to within a few degrees.
The final plot, Figure 4.19, is an illustration of a case very similar to one which will
Figure 4.19: Measured and predicted reflection coefficients for wire grids with a single pin
missing in the sequence. Wire diameters (d =.025,.033,.039 in.) increase in
the counterclockwise direction.
be encountered later. In this instance, an evenly spaced row of wires were installed across
the waveguide crosssection, however, one pin was removed at various locations in the
sequence. As will be seen in Chapter VI, this type of approach is used to allow a strip to
pass through the pin curtain. The ability to predict the reflection coefficient for this case
1.13
is very important for certain situations as will be seen. Although the comparison for tlin
case looks reasonable, it is pushing the limits of the experimental errors involved with the
fixture. However, a consideration of the previous results leads to the conclusion that thle
results are reliable.
4.4 Summary and Conclusions
In summary, we have seen how problems with uniformity in one dimension can be anlalyzed with a high degree of accuracy. The uniformity allows the structure to be represented
by a single modal series which can be numerically evaluated efficiently.
The use of impedance boundary conditions and a LSE and LSM expansion of tile field.i
greatly simplifies the modelling of multilayer substrates and superstrates. Tllis particularly
facilitates the computation of fields throughout the structure since most of the evaluiat iol
can be done analytically. Application to shielded strip and slot geometries yields botll
reliable quantitative information as well as visual representations of the fields providliIng
insight into the behavior of the structure.
A simple model for scattering from wires in rectangular waveguide has also been developed. The simplicity of the model is in contrast to methods discussed in the literature
which are oriented towards fewer posts with relatively large diameters. The present met hod.
however, emphasizes posts of greater number and smaller diameter. Experimental results
were obtained to verify the modelling ability of the technique. The need for this model in
the context of the present work appears in Chapter VI.
CHAPTER V
COUPLING THROUGH STRIPFED SLOTS
A class of structures which couple from one guiding structure to another through an
aperture is important ill microwave circuits, aside from the radiating slot. In fact, for
antennas, the coupling of waveguide to waveguide through a narrow slot in a common wall
has been used for many years in the design of corporate feed networks for slotted waveguide
arrays. As discussed in Chapter I, the case of coupling between shielded strips in a similar
manner is becoming equally important in current and future systems, including similar feed
networks. In this chapter, the analysis of these types of couplers will be presented along
with experimental verification of the numerical results to demonstrate the applicability of
these techniques to microwave and millimeter wave devices.
5.1 Network Analysis
The basic structure of the coupler to be discussed is as shown in Figure 5.1. Variations
on this geometry include cases with microstrip lines on the same side of the slot; multilayered substrates/superstrates; reverse couplers where the lines exit on the same wall;
additional parallel slots and lines; and 3 and 4port networks, among others, but all can
be analyzed using the same approach.
Integral equations have been formulated for this problem in Chapter I and are solved
as outlined in Chapter III for currents oni the slot and strips. Therefore, in this chapter
114
115..
S Ls! il. 4
Figure 5.1: Geometry of basic coupler.
we need only discuss the interpretation of these currents in order to extract the network
parameters which characterize the coupler's behavior.
Solutions for the currents are found with even and odd gap generator excitations at the
line ends. These are sufficient to characterize the twoport problem since all excitations
can be decomposed into even and odd components. The approach can be generalized using
Nport network analysis [14, pp. 157158] but this aspect is beyond the focus of this work.
From the even and odd currents on the microstrip lines, even and odd impedances
are found by measuring the relative distance (d) from a standing wave maximum to the
location of the slot as illustrated in Figure 5.2. We can also estimate the guided wavelength
for the even and odd modes from the standing wave patterns by measuring the spacings
between minima or maxima. If the materials are lossless, as we typically can assume for
most microwave circuit applications1, the expression for the reflection coefficient referenced
'For slightly lossy structures, the estimation of the attenuation constant can be easily determined from
the ratios of successive current maxima or minima[55]. For larger losses the situation is more complex,
since not only the envelope of the standing wave current but also the spacing between maxima and minima
vary along the line [27, sec. 5.4]. In this case one must resort to a parameter estimation scheme to find the
116
current
magnitude
open
end
slot
position
Figure 5.2: Measurement of location of current peak relative to the slot for standing wave
calculation.
to the slot reduces to
r = e4?d (5.1)
which produces an impedance according to
Z = (5.2)
1 The even and odd impedances are then combined to form the Zparameters which, in the
symmetric case studied here, are given by the simple expressions:
Z = Z= =Z22 (5.3)
2
Z21 = Z12 (5.4)
Finally, the even and odd impedances may be combined to produce Sparameters through
the transformations:
Sil = z2 + 2Za iz e 1 = 5e22 (5.5)
21 = = 21 12 (5.6)
21i i t i i i.
optimum fit to the current with the attenuation and propagation constants as parameters.
117
which are used to characterize the coupling behavior.
Before proceeding, the limitations of this technique should be noted. Transmission line
analysis of the current in this way assumes that there is only one propagating mode on the
line at some distance from the discontinuities. For the shielded structures treated here,
this requires that the crosssectional dimensions of the feeding microstrip lines be such that
all higherorder modes are cutoff or, looked at another way, a given crosssection restricts
the maximum frequency for which circuit parameters can be reliably produced with this
technique. Also, in order to estimate the phase constant, the line must have a minimum
length, typically on the order of the wavelength. Moreover, for good numerical stability of
the results over a wide range of parameters, it has been found useful to average a number
of estimates on a line approximately three wavelength long, discarding the maximum and
minimum values. This restriction demands more unknowns and hence greater computation time, however, the results are quite reliable. The alternative method based on the
Reciprocity Theorem developed in Chapter VI, overcomes these limitations at the expense
of a more complex formulation.
5.2 Fixture Design
To verify the results, we have designed and constructed the fixture shown in Figure 5.3.
Sample substrates with various line and slot dimensions are installed in the fixture in
different combinations to allow frequency response measurements. A number of circuit
boards were made: One set of boards was doublesided with a microstrip line etched to
a certain length relative to a slot etched in the ground plane on the opposite side. Thle
second set was onesided boards designed to be held against the boards of the first set by the
fixture, with microstrip lines of corresponding lengths. The lines are excited by Eisenhart
connectors which provide a reasonable match over a broad frequency range, even when the
contrast in cr is high (cr = 1 for the connector, cr = 10.6 for the substrates). In our case
1.18
Figure 5.3: Photograph of coupler fixture assembly.
this is partly a result of the good match in dimensions between the boards and the tip
to outer conductor spacing of the connector. Another critical factor is the transition fromn
the outer conductor of the connectors to the ground planes of the boards which must he
carefully designed. The match at the connectors was optimized to minimize the launching
discontinuity, allowing us to avoid the necessity for TRL [34] or similar deembedding
schemes which require additional and more complex fixturing.
5.3 Numerical and Experimental Results
A coupler with the geometry of Figure 5.1 was analyzed using the above techniques.
The parameters which can be varied in this design are numerous, consequently, only a few
variations will be presented here. In all cases, although not required in general, symmetric
geometry was maintained to simplify the even and odd mode analysis, as discussed above.
Also, in all cases the crosssection for the cavity was 0.25 x 0.25 inches; the substrates were
0.025 inches thick with ~r = 10.6; and the slot and line widths were 0.025 inches. The
119
cavity length was fixed at 2.0 inches for the measurements and varied for the numerical
results to allow an approximately constant 3A feedline length. This dimension does not
affect the results since for all frequencies considered here, the cavity is below the cutoff
frequency of the higher order microstrip modes and the reference plane was fixed at the
location of the slot.
To illustrate the behavior of the coupler, we first examine the influence of various
parameters at fixed operating frequencies. The effect of the line stub length (1) is shown
in Figure 5.4. It can be seen that the stub is initially too long for an ideal match at this
0.0
3.0
o 6.0
9.0"
915.0 7
/ e  s,, /
18.0
21.0 
0.000 0.025 0.050 0.075 0.100 0.125
Line Stub Length (inches)
Figure 5.4: Effect of the line stub length (1) on S21 and S\ magnitudes (s = 0, Ls = 0.25
inches and f = 12.0 GHz).
frequency. However, as the stub is progressively shortened, a certain length "matches" the
two port coupler and with further shortening the match gets progressively worse. We can
interpret this effect by examining the equivalent circuit shown in Figure 5.5. Variation
of the stub length has the effect of changing the position of the current maxima (virtual
shorts) and minima (virtual opens) on the lines relative to the slot, thus varying the degree
of coupling through the slot represented by the coupling transformers. Consequently, the
120
open / /7 — Port
jX
end 2
Port,// open
1 end
jX
Figure 5.5: Equivalent circuit for a 2port coupler.
peak coupling occurs when line stub length places a current maximum below the slot
or lengths in odd multiples of  A/4. The opposite effect occurs when the line stub is
approximately in multiples of A/2 in length so that there is a virtual open circuit beneath
the slot, in which case there would be very little coupling between the line and slot.
A similar effect is observed for variations in slot length (L,) as illustrated in Figure 5.6.
Again using the transmission line analogy, one can interpret this effect by transforming the
impedances at the ends of the slot to the center. These end impedances are nearly short
circuits, the difference being due to fringing fields which extend beyond the ends of the slot
line, fully accounted for by the fullwave analysis. At the matching length, the resulting
transformed reactances at the center cancel the reactance associated with the junction,
thereby matching the two ports. As the slot becomes very short, the field in the slot is
effectively "short circuited"; thus, coupling is reduced. S21 then tends to zero while 511
approaches unity (since the structure is closed and assumed lossless). All of these effects
would be expected to repeat as the slot length increases in multiples of A, however, for the
case studied here, the maximum slot length is limited by the dimensions of the shielding
121
3.0
6.0.f 
I". 9.0
0 12.0 /,
15.0 
2. . .. I.. t 
18.0
21.0
0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250
Slot Length (inches)
Figure 5.6: Effect of the slot length on S21 and Si, magnitudes (s = 0,1 = 0.049 inches
and f = 18.0 Gllz).
package which have been chosen to allow propagation of only the dominant microstrip
mode.
To generate a frequency response, the programs are run at each frequency of interest
and the slot and line lengths are varied to form a parametric database. The database is
then scanned and interpolated to assemble frequency response plots as functions of the
geometric parameters. Measurement of one of the assemblies is shown in Figure 5.7 in
comparison to corresponding numerical results for a stub length of I = 0.115 inches. The
position of the high frequency corner of the response was found to be very sensitive to the
length of the line stub. As discussed above, this corner is controlled by the length at which
the stub is approximately A/2. Since the effective dielectric constant for the microstrip
is approximately cr,eff = 7.8 at 17.0 GHz, a null is predicted in the coupler response for
that neighborhood, in good agreement with the results shown. The error bar on the high
end indicates the sensitivity of the high frequency corner to a ~5 mil error in line stub
length which is well within the expected tolerance errors for positioning the stubs relative
122
0.0
3.0
6.0  
9.0
12.0
 Measured
15.0 — e Thory
=. =0.115 J.005 inch"e
18.0 
21.0 I
0.0 3.0 6.0 9.0 12.0 15.0 18.0
Frequency (GHz)
Figure 5.7: Comparison of theory and experiment for S21 magnitudes with s = 0,I =
0.115 in., L = 0.250 inches. The error bar indicates the influence on the high
frequency corner of a ~.005 inch change in stub length.
to the slot. In view of the above, we conclude that the theoretical results are in excellent
agreement with the experimental data. In fact, we were able to move the upper board
slightly toward the slot to extend the stub length somewhat, shifting the high corner to a
lower frequency as expected. However, this also created problems with the match at the
Eisenhart microstrip launchers so these results are not shown.
The 'sidelobe' which can be seen at the high frequency end, is also attributed to tolerance errors for the line stub lengths. A difference in lengths would produce multiple nulls
in the response at the high end resulting in undesirable sidelobes in between. Because
of the high sensitivity to line length, owing in part to the high dielectric constant, the
amplitude and span of the sidelobe is a strong function of the relative line stub lengths, a
fact which can be observed when the boards are slightly shifted as described above. The
sidelobes do not appear in the theoretical result since a difference in stub lengths between
the upper and lower lines introduces an asymmetry which has not been included in the
123
current numerical model and contradicts soIme of the assumptions stated for the network
analysis. A more general model can be produced by extending the network analysis to the
nonsymmetric case.
The ripple in all the measurements can be shown to result from mismatch at the microstrip launchers. The measurements are particularly sensitive to this connection because
of the high dielectric constant of the substrate. Ripple occurs to various degrees throughout
the measurements and is also influenced by small air gaps between the connector assembly,
fixture and substrates. It could be removed by more sophisticated deembedding techniques, however, this requires additional fixtures. Nevertheless, the ripple shown in the
results presented here is not substantial and does not significantly interfere with the fundamental behavior of the devices. Also, the broadening of the low frequency response is
typical in the measurements. We were not able to identify a direct cause for this effect,
however, we suspect that it is related to the fixture/connector interface, since we have not
deembedded these transitions. We also postulated that some of the anomalies might be
caused by the sidewall grooves in the fixture which hold the doublesided board in place.
This possibility was eliminated, however, by installing movable sidewall shorts which are
visible in Figure 5.3.
The remaining discrepancy is perhaps a slight additional loss found in some of the
measured results. To deembed the losses for the structure, a through line was measured
and the remaining measurements were postprocessed to compensate for conductor and
dielectric losses on the microstrip lines. This process does not correct for losses associated
with the slot including both conductor and dielectric losses, additional losses on the cavity
walls, and losses due to the added line lengths. The remaining differences are attributed
to these factors together with measurement errors and are judged to be within acceptable
limits.
124
Measuremnents on a different line stub length are shown in Figure 5.8 again showing
0.0 *.
3.0
6.0
 9.0
r) 12.0
15.0  / Measured: — o  Theory
18.0 
21.0, I
0.0 3.0 6.0 9.0 12.0 15.0 18.0
Frequency (GHz)
Figure 5.8: Comparison of experiment and theory for S21 magnitudes with s = 0,1 =
0.080 in., L, = 0.250 inches, illustrating the control of the high frequency corner
with the line stub length by comparison to Figure 5.7.
good agreement with the theoretical results. In this case, the shortening of the stub length
has moved the high frequency corner out of the range of the measurements. What is
interesting to note about this case is that the low frequency corner of the response is quite
insensitive to this change in stub length. Also of particular interest is the wide bandwidth
of this transition.
Figure 5.9 illustrates the effect of shortening the slot length. One consequence is reduced
coupling in the passband which was also demonstrated in Figure 5.6. We also see in this
result, some movement of the high frequency null due to a shortening of the line stub
length.
The final plots. Figures 5.10 and 5.11, show the influence of the line separation parameter (s) on the frequency response. These figures are to be compared to Figures 5.7
and 5.8, respectively, and in general showv a narrowing of the frequency passband as s
125
0.0 ' i** ''*i —' —   i   I '. *3.0 Ma
6.0
9.0
— o  Theory
18.0
21.0
0.0 3.0 6.0 9.0 12.0 15.0 18.0
Frequency (GHz)
Figure 5.9: Experiment and theory for S21 magnitudes with s = 0, = 0.110 in., L, = 0.153
inches, showing the effect of shortening the slot length.
0.0
0.0 3.0 6.0 9.0 12.0 15.0 18.0
Frequency (GHz)
Figure 5.10: Comparison showing the effect of line separation with I = 0.115 in., Ls = 0.250
in. and s = 0.125 inches.
126
0.0 , 
3.0
6.0
9.0
t IZO
15.0 Meaured
  o  Theory
18.0
21.0 /
0.0 3.0 6.0 9.0 12.0 15.0 18.0
Frequency (GHz)
Figure 5.11: The effect of line separation with shorter line stub length; I = 0.080 in.,
Ls = 0.250 in. and s = 0.125 inches.
is increased. This is to be expected, since we have now introduced an additional length
parameter which can influence the response through its relationship to wavelength. Here
again, the numerical model is judged to have correctly predicted the coupler behavior after
the experimental artifacts are considered.
5.4 Summary
The frequency response plots shown demonstrate the utility of the structure as an interconnect. With proper selection of the geometric parameters such as line and slot widths,
lengths, line separation, substrate heights and materials, the frequency response can be
tailored to give the required center frequency, bandwidth, shape, etc. As has been shown,
very wide bandwidths can be achieved which makes the structure very versatile. Avoidance of viahole transitions and their inherent limitations by the use of planar structures
to form vertical interconnects, together with the ability of the model to accurately predict
the coupler behavior as demonstrated by experimental results, are especially important
127
considerations for design of monolithic circuits. In addition, although the approach discussed here uses certain simplifying assumptions about the symmetry of the structures,
the technique can be readily adapted to the general case.
CHAPTER VI
ANALYSIS OF STRIPFED RADIATING SLOTS
Arrays of slots have been widely used in antenna design for nearly fifty years. Thile most
common approach has been to cut slots in the walls of rectangular waveguide forminig a
linear array. An arrangement of linear arrays, fed by additional slotted waveguides, has frequently been used for twodimensional arrays. The design and analysis of these approaches
has proceeded over many years to the point where accurate numerical techniques are now
available so that costly, timeconsuming empirical techniques can be bypassed [23, 61].
More recently, complex systems have imposed new antenna requirements with demands
for compact conformability, reduced weight, and higher order antenna functions such as
electronic beam steering, polarization control and power generation. A suitable approach
employing striplinefed series slots isolated from higher order modes by shorting pins, has
been analyzed by Shavit and Elliott [67]. Their approach, together with an alternative
coupler feed arrangement, is illustrated in Figure 6.1. The isolation of the slots by the
cavity walls, in addition to simplifying the design, has been shown to provide significant
advantages for phase steerable antennas [47]. Other variations on this approach may be
adapted to fit system requirements including the incorporation of active elements and
vertical integration of more complex circuits on multilayered substrates.
As indicated in earlier chapters, we extend the analysis to include strips on multilayered substrates and superstrates, as well as to include the effect of a dielectric cover on
128
129.:e, 
 a*
i,! s
0 0 0 0 0::~ *,:
Figure 6.1: Stripfed cavitybacked slots with couplerfed and seriesfed corporate feed arrangements.
130
the slots. As will be seen the present formulation substantially improves accuracy, to the
point where one could expect to perform the design without empirical methods.
6.1 Equivalent Slot Impedance
In order to produce a slot whose resonant length is less than the width of the cavity, tilhe
cavity dimensions must be such that the first higher order mode in the cavity crosssection
 the TEo10 mode  is not cutoff. In this case, since more than one mode can propagate on
the microstrip line., the method used to extract circuit parameters in Chapter V cannot be
used. In addition, these cavities are typically on the order of onehalf wavelength in length
so that the lines would probably not be long enough to get reliable circuit parameters
using the Standing Wave Method. The scattering parameters for the slot can, however,
be derived through application of the Reciprocity Theorem [33] to the cases illustrated in
Figure 6.2, resulting in
slot
position
I 
. H TEM
a) ETEM T
H TEM 
waveguide
aperture
slot
position
b) E K
HK 
Iwaveguide
aperture
Figure (3.2: Two cases for application of the Reciprocity Theorem.
131
IZ[ ETEM X HA  El X HTEf ] dS = J T } HTEM. 1 dS (6.1)
waveguide slot
where we have assumed the incident wave is the dominant TEMlike mode on tile microstrip
line. The pins forming the walls of the cavity are sufficiently spaced at the microstrip line
so that they do not disturb the dominant mode fields. In this manner, the dominant
microstrip mode can pass through the wall unperturbed, while the higher order modes
are reflected by effectively, a perfectly conducting wall. The fields for the reflected mode,
external to the cavity are then given by
EK = r (TEM  ' ETEM) e1 (6.2)
H = r (hITEM  I iTEMA) (6.3)
where e and h are the field components transverse to the propagation direction, 1.
Rearranging Equation (6.1) we find the reflection coefficient to be
I I! HTEMA K dS
F = s (6.4)
J e ( ETEMA X \xTEAM )(I) dS
waveguide
This is the central equation needed to produce the equivalent impedance of the slot from
which the equivalent scattering parameters can be derived. In order to evaluate this expression, we note that field components from the solutions to the two problems illustrated
in Figure 6.2 are required. The first case  which will be referred to as 'the waveguide
problem'  is a shielded microstrip line with layers matching the cavity structure, and provides the transverse components of the field on the entire crosssection. From these we can
calculate the reaction integral in the denominator of Equation (6.4) which, for the lossless
case, is twice the characteristic impedance of the line. Additionally, we will have the H\TEAl
term in the numerator at the location of the slot which is used as the excitation for the
second case; 'the cavity problem'. In this case we need to find K, a conductorbacked,
equivalent magnetic current, which replaces the tangential electric field in the slot through
the Equivalence Principle [33].
132
W ith assumptions on the symmetry of the field in the slot as in [67], thle cavitybacked
slot can be treated as a twoport series impedance element. Once F is known, we can
determine the equivalent slot impedance (normalized to the microstrip line impedance);
2r
Zs =  (6.5)
1 F
from which tlhe equivalent circuit scattering parameters can be derived by standard network theory. The integral equation formulations for these problems and the solutions for
the currents by means of the method of moments has been described in the previous chapters. We therefore can proceed to tle verification of the theory and numerical model by
presenting the experimental results.
6.2 Numerical Results and Measurements
The solutions for the waveguide problem are obtained as outlined in Chapter IV. The
cavitv problem is solved using conventional Method of Moments techniques, as outlined
in Chapter III with the \\TEM field from the waveguide problem as the incident field on
the slot. Both the strip and the slot are assumed to be sufficiently narrow so that tile
transverse components of current can be neglected. The longitudinal components on the
strip are expanded in terms of piecewise sinusoidal basis functions with the Maxwellian
transverse distribution as used for the waveguide problem. Galerkin's method is used to
enforce the boundary conditions on the strip. For the slot, we have investigated the use
of both the Maxwellian and uniform (rooftop) transverse dependence, using Galerkin's
method for the rooftop functions and point matching for the Maxwellian case. In the
latter case, point matching is required due to numerical considerations in the evaluation
of the Sommerfeld integrals for the halfspace. This requires additional computation since
the matrix loses its symmetry. We would expect the Maxwellian results for narrow slots to
be more accurate since the edge conditions are satisfied, however, in the end the differences
133
1.2  l
0   Shavit's Measurements /
1.0...  — This Theory
0  Shavit's Theory'
' 0.8 .. Cosine Assumption / /
0.6
0.4  / / /
0.2
0.0 
.200 .100 0.000 0.100 0.200
Slot Offset (inches)
Figure 6.3: Normalized resistance as a function of slot offset compared to Slhavit's measurements [67].
are minimal. Therefore, the rooftop functions are preferred and will be used throughout.
with the added benefit of their greater flexibility in modelling wider or more complex slots.
With this discretization of the integral equations, the resulting system of linear equations
is solved for the unknown current distributions by matrix inversion.
To verify the accuracy of the model, we first compare with the theoretical results and
measurements of Shavit [67] shown in Figures 6.3 and 6.4. Figure 6.4 shows the results of
calculations of resonant length which is a key parameter in the design of high performance
antenna arrays. This is a special case involving stripline and no dielectric cover and shows
significantly improved agreement between this work and the experimental measurements.
The reason for the larger discrepancy in Shavit's theory is uncertain, since there are some
similarities with the present method. The most likely factor may be his use of approximate
Green's functions, which also restricts application of his method to stripline only. This
134
2.1.
2.0
M.~ "'. 1
C 9 Shavit's Measurements
c~ 1.9 ',
c 1. .  This Theory
'1) — a  Shavit's Theory... 1.8 .
   
1.7....
.200 .100 0.000 0.100 0.200
Slot Offset (inches)
Figure 6.4: Resonant length as a function of slot offset compared to measurements [67].
leads to certain assumptions about the enforcement of the boundary conditions on the
strip, whereas we have used the exact expressions and enforced the boundary condition
explicitly. The number of modes used in the summations (convergence) can also be an
important factor. High mode numbers are needed, especially since the critical field behavior
occurs in the region where the strip and slot are most closely spaced.
Since making accurate measurements for these types of slots is quite difficult and considering that these measurements were made almost a decade ago, we have performed
extensive measurements on a new set of slots, with and without dielectric covers. In the
process, we have uncovered new factors not previously recognized which can significantly
influence the results. Individual test pieces for several combinations of slot lengths and offsets have been built. The pieces were constructed using two copperclad dielectric boards
(31 mil thick 2 inch square Duroid 5880 ~r = 2.2 with 0.7 mil thick coppercladding). The
135
size of the boards is small enough to minimize material and fixturing costs but large enough
to provide an adequate ground plane and reduce the effects of reflections from the board
edges. The slots and transmission lines for each sample were produced by removing copper
from the top planes of the top and bottom boards, respectively. The slot length (Ls) and
offset of the transmission line from the end of the slot (s) were varied from circuit to circuit
with the slot always in the center of the board. The top and bottom boards were then
bonded together by placing 3M 6700 2 mil thick thermoplastic bonding film at their interface and subjecting them to the heat and pressure profiles specified by the manufacturer. In
addition, to improve accuracy, calibration standards were constructed for deembedding of
the fixtures using the TRL method [34]. Dielectric covers wee made from 125 mil Duroid
(~r = 2.2) bonded to the circuits with Ablebond 550 thermoset in a bonding press at 230~F
for approximately 3 hours. Recessed tracks were milled in the covers to accommodate the
soldered shorting pins which protrude slightly above the top surface of the circuits. Other
dimensions were held constant to the extent possible throughout the experiment and were:
a = 0.425, b = 0.516, W = 0.0279 and Ws = 0.0315 inches. A sample slot is shown in
Figure 6.5 at various stages of fabrication.
Once calibration is established which places the reference plane at the center of the
slot, the Sparameters were measured for each slot. The equivalent impedance of the slot
as a series element can be determined from
Z = 2Zo 1 (6.6)
or
z 2Z=  (6.7)
1  SU
The extent to which Equations (6.6) and (6.7) agree provides a measure of the accuracy of
calibration. In general, however, Equation (6.7) was found to be more accurate for resonant
lengths because the calibration of 521 is more sensitive to dielectric constant and length
136
Figure 6.5: Photograph of slot assembly pieces in various stages of fabrication.
137
variations. In contrast, Equation (6.6) was judged to be more reliable for estimations of tlhe
peak resistance values based on the behavior of S1, with frequency in the few cases where
the results of Equations (6.6) and (6.7) differed. We have also defined resonant length
as corresponding to the peak in the resistance curve, since this definition is less sensitive
to measurement and calibration errors and is more consistent when constant parasitic
reactances are present.
Figure 6.6 shows an example of a comparison between the calculated resistance at
3.00..................  l
* L,=.305 in.  model
2.50  L, =.320in.  model
— 4  L, = 335 in.  model /
* 2.00 L =.305 in.  measured /
L,=.320 in.  measured
3 /f A
L=.335 in. measured /
50 1.50
0.00 ..0/1'.
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
Slot Offset (inches)
Figure 6.6: Uncompensated peak normalized resistance as a function of slot offset compared to measurements for uncovered slots.
resonance and the measured values, normalized to the impedance of the line. The curves
represent the resistance versus offset, the distance from the end of the slot to the centerline
of the strip, at the measured resonant frequencies. The data points shown are the measured
values with error bars indicating the uncertainty due to a combination of repeatability tests
1:38
and an estimate of potential systematic calibration error due to variation in the substrates.
By comparing the measured resistance values with the corresponding points predicted by
the numerical model, it can be seen that virtually all predicted resistance values are losw.
Furthermore, it should be noted that the error bars are conservative, since they do not
account for slot and dielectric cover losses which cannot be calibrated out, or variations
in fabrication such as dimension changes. Also, the plotted offsets have been artificially
staggered ~1 mil for clarity and do not indicate an error estimate. Fabrication variations
are blamed for points which deviate from the trend of the data, for example in this Figure,
the case where s = 0.075 and Ls = 0.305 inches is particularly suspect.
Figure 6.7 displays the predicted resonant lengths for the same frequency range. Here
the crosses indicate the physical parameters for the measured slots. One can then estimate
the error in either resonant length or frequency by correlating the measured resonant
frequencies listed in the table with the curves. A comparison of the resonant lengths shows
that the predicted lengths are all too long. (Equivalently, predicted resonant frequencies
for a given slot length are too high.)
These comparisons, together with other observations made in making the measurements
and numerical parametric studies, led to the important realization that a compensation
must be made for the aperture in the wall which allows the strip to pass through. In the
past it has been assumed that if the pins are sufficiently far from the strip, the dominant,
mode can pass through the wall unperturbed. At the same time, it has been assumed
that if the pin spacing is sufficiently dense, the higher order modes  particularly the TElo
mode  will effectively encounter a perfectly conducting wall allowing us to use the closed
cavity Green's function to model the walls. Although the first assumption is valid, the
second may not be and is of crucial importance because the behavior of the slot is strongly
influenced by the TEio mode, leading to the above discrepancies.
139
c/3
C.
c
a
0
ao
r)
C.
0O
0.370
0.360
0.350
0.340
0.330
0.320
0.310... I"' I... 1.......''i '' 1' I... '......1....1 'I' '''.
3, fff  
. — t'.. +
t.  
+ ++* ~1,
1
13.3 Glz
 13.4 GHz
13.5 GHz
  13.6 G lz
—. 13.7 GHz
13.8 GHz
  13.9 GHz..... 14.0 GHz
 14.1 GHz
— 14.2 Gilz
0.300
1
0.290 0 0,.........l., 06 0.. I.
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
Slot Offset (inches)
MEASURED RESONANT FREQUENCIES (GHz)
Slot Offset Slot Length (inches)
(inches) L, =.305 L, =.320 Ls =.335
s =.030 14.165 13.74 13.39
s =.045 14.145 13.845 13.38
s =.060 14.17 13.80 13.31
s =.075 14.045 13.61 13.285
,
Figure 6.7:
Predicted resonant length as a function of slot offset for uncovered slots.
Crosses indicate actual test piece dimensions to be compared with the frequency curve corresponding to the measured resonant frequencies listed in the
table.
140
The sensitivity of the slot's behavior to the relative position of the walls parallel to the
slot is demonstrated in Figure 6.8. We see that the resonant slot length for the uncovered
0.330 s. I I
~) 0.320 
= 0.310
0.300
0.300   Covered, s =.060, f = 12.0 GHz
 No cover, s=.060 f= 13.8 GHz
0.290.... —AA
0.400 0.410 0.420 0.430 0.440 0.450 0.460 0.470
'a' Dimension (inches)
Figure 6.8: Sensitivity of resonant length to cavity length dimension 'a'.
case depends on the position of these walls on a nearly onetoone basis. Thus, a 1% error
in the position of the walls will produce approximately a 1% error in the resonant length.
Assuming that our average fabrication tolerances are better than that, implying that we
should emphasize the trend of the data, the discrepancy between theory and experiment
can be traced to ignoring the apertures in the walls for the TEo10 mode. (Note, however,
that just the diameter of the wire used to construct the walls of the cavity  10 mils  is on
the order of 1.5% of the electrical length of the cavity.) It is observed that the covered case
is less sensitive to this dimension because the magnitude of the slot field in the covered
case is not as large, implying a smaller excitation of the TEio0 mode relative to the other
141
mechanisms.
In a rectangular waveguide, wall apertures of the type used here can be represented by
a shunt inductance. For a oneport, a shunt inductance can be replaced by an equivalent
length of shorted transmission line. Since for stripline and rectangular waveguide the TE10
modes are identical, variational formulas for calculating this inductance in rectangular
waveguide such as found in [54, pp.165] can be used to estimate a correction to the length
of the cavity, compensating for the affect of the aperture. It is also desirable to suppress
the propagation of energy in this mode through the aperture, by lining the stripline with
shorting pins to cut off all higher order modes outside the cavity as shown in Figure 6.5.
In addition, by modelling the wires explicitly after a manner similar to [43] as discussed in
Chapter IV, and using the 'ruleofthumb' from [54, pp.188189] to account for the change
in width, we have found that the correction may be somewhat smaller than indicated by
variational formulas and has less sensitivity to frequency. The end result, in this case, is an
estimated correction of 12.5 mils for the length of the cavity. The other walls of the cavity
are also made with wires but the slot characteristics are quite insensitive to this dimension.
The model is now reevaluated with the compensated dimension and the predicted resistance and resonant lengths are compared to the measured resistance and actual slot lengths
at the measured resonant frequencies. In Shavit's case, the dimensions used result in only
a 16 mil correction which, for his operating frequency, has minimal influence on the results
shown previously. For our measurements the compensated coupling (resistance) results are
shown in Figure 6.9 and the predicted resonant lengths are listed in Table 6.1. Parametric
curves for resonant lengths for this and all remaining cases are similar to Figure 6.7 except
the family of curves are shifted downward and therefore will not be shown. When the dielectric cover is introduced the same general features are observed. Using the same length
compensation as in the uncovered cases because the cavities are the same, the comparison
142
(.C.)
N
o

Z
d.
cS
3.00
2.50
2.00
1.50
1.00
0.50
0.00 ' ' ' I.... I '*,, I
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
Slot Offset (inches)
Figure 6.9: Compensated peak normalized resistance as a function of slot offset compared
to measurements for uncovered slots.
Slot Length (inches)
Slot Offset L,(actual) =.305 Ls(actual) =.320 L,(actual) =.335
(inches) fres Ls (model) fres Ls (model) fres Ls (model)
s =.030 14.165.3045 (0.2%) 13.74.3234 (+1.1%) 13.39.3409 (+1.8%)
s =.045 14.145.3009 (1.3%) 13.845.3139 (1.9%) 13.38.3361 (+0.3%)
s =.060 14.17.2957 (3.0%) 13.80.3120 (2.5%) 13.31.3351 (+0.0%)
s =.075 14.045.2976 (2.4%) 13.61.317.5 (0.8%) 13.285.3331 (0.67%)
Table 6.1: Measured resonant frequencies (GHz) and predicted resonant slot lengths for
uncovered slots with compensated cavity length. (Deviation from actual slot
length listed in parenthesis in percent.)
143
between theory and experiment for the covered cases can be found in Figure 6.10 and
2.00... 
L, =.305 in.  model
 L =.320 in.  model
1.50  L, =.335 in.  model
m i L =.305 in.  measured
' Ls =.320 in.  measured
7e
Slot Offset (inches)
t1.00 oL.335 in.  measured
0.50
0.00...
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
Slot Offset (inches)
Figure 6.10: Compensated peak normalized resistance as a function of slot offset compared
to measurements for covered slots.
Table 6.2.
The variation of the errors in these tables is attributed primarily to variations in fabrication. It can be seen that the average error for the uncovered slots is on the order of
0.8% which is considered to be excellent. For the covered cases, the average error is approximately 3.3%, however, in this case we would expect an increased uncertainty due to
additional factors including: variations in cover dielectric constants and bonding; a potential increase in the influence of the finite ground plane due to the introduction of surface
waves; and the affect of the milled recesses in the covers, although very shallow, which are
not included in the model. The latter two factors are of particular interest considering the
sensitivity of the resonant length to the position of the internal walls of the cavity parallel
144
Slot Length (inches)
Slot Offset L,(actual) =.305 L,(actual) =.320 L,(actual) =.335
(inches),res L, (model) fe L, (model) fre L, (model)
s =.030 12.40.3123 (+2.4%) 12.00.3318 (+3.7%) 11.75.3453 (+3.1%)
s =.045 12.35.3125 (+2.4%) 12.03.3274 (+2.3%) 11.61.3489 (+4.1%)
s =.060 12.35.3107 (+1.9%) 11.83.3353 (+4.8%) 11.56.3493 (+4.3%)
s =.075 12.35.3092 (+1.4%) 11.81.3348 (+4.6%) 11.54.3487 (+4.1%)
Table 6.2: Measured resonant frequencies (GHz) and predicted resonant slot lengths for
covered slots with compensated cavity length. (Deviation from the actual slot
length listed in parenthesis il percent.)
to the slot. It sliould also be noted that because the resonant length of the slot is controlled
by factors which do not scale identically with frequency, these errors do not indicate the
expected error of predicted resonant frequencies. In fact, for the geometry studied here,
the expected error in resonant frequency for a given slot length, is roughly half of the
prediction error for resonant lengths. Tllis feature can be deduced from Figure 6.7 for the
uncovered slots, and similar parametric plots for the covered case, by calculating the slope
of length over frequency. The case for resonant lengths has been shown in the tables, since
these errors are more relevant to initial antenna designs.
Finally, we point out that the fixtures were built by varying the position of the strip to
achieve different slot offsets while in the numerical model it is more efficient to vary the slot
position relative to the strip. A comparison of numerical results for the actual dimensions
has shown that the relative positions of the strip and slot in the cavity have some influence
on the slot characteristics, however, the variations are within the bounds of the experiment,
indicating the true accuracy of the approach is at least as good as suggested above. In fact,
145
to some extent there is coupling between errors in frequency and resistance values so that
variation of a parameter tends to improve the agreement with experiment for one while
degrading the comparison for the other. For example, in the covered case, evaluating tlhe
numerical model at the measured resonant frequency tends to reduce the error in predicted
resonant length while increasing the discrepancy in resistance values. For the uncovered
slots, the resonant length (frequency) and resistance values are more stable, suggesting not
only that the compensation length is likely to be correct, but also that the introduction of
the covers has introduced an error in the experiment. Nevertheless, the overall agreement
with the experimental data is considered to be excellent to the extent that the variations
can be controlled for this type of construction.
6.3 Summary
In summary, we have presented a fullwave integral equation approach for the analysis
of stripfed cavitybacked slots. The formulation involves the application of the Reciprocity
Theorem which requires the field solutions to two distinct situations; the infinite shielded
line (the 'waveguide problem') and the cavity problem. Methods for deriving the exact
Green's functions for these problems have been described including the incorporation of
multilayered substrate and superstrate capability. Solutions to the integral equations are
then found by the Method of Moments.
Modelling of the strip itself, and accounting for the effect of the openings in the cavity walls, were found to be a key factors not previously recognized for accurate modelling
of these types of stripfed slots. Comparison with measurements for the special case of
stripline has demonstrated that solutions using the exact Green's functions and compensation for the strip aperture lead to significant improvements in accuracy over previous
methods. The analysis described is also exact for more general structures involving multiple layered substrates and superstrates with the exception of determining the compensating
116
length for the feedthrough aperture. Tilis can be found by extension of tile 'waveguide
problem' approach as described above with wires or apertures modelled in a nianner similar
to [43] or [3].
For the type of fabrication techniques described in this experiment, it was found that
the numerical model produces accuracies on the order of that for tile experiments. This
suggests the method has sufficient accuracy to replace many of the empirical steps used in
the design of antenna arrays, which can significantly reduce design and development costs.
CHAPTER VII
MODELLING OF THICKSLOTS
It has been suggested that stripfed radiating slots may play an important role in
the recent trend towards more complex antenna systems. This is particularly true for
monolithic phased array applications where active devices are integrated with the radiating
elements for phase control and in some cases, to generate power. One of the practical
difficulties which arises with the introduction of active devices is the dissipation of heat.
One solution is to increase the thickness of the antenna face for use as a heat sink, which
also adds the possibility of cooling channels. This has an influence on the electrical design.
however, since the slots may no longer be considered infinitely thin as is commonly done.
On the other hand, we can take advantage of the thickness to improve bandwidth, which has
traditionally been a fundamental limitation of slot arrays. This possibility is the primary
topic for this chapter.
While semiempirical and approximate correction factors for finite thickness slots have
been introduced and may be adequate and appropriate in some circumstances [39], it is
desirable to develop exact methods to improve the accuracy and generality of numerical
design tools. The fullwave integral equation technique which has been presented is also
applicable to this class of antennas, allowing not only the analysis of thick slots [84, 85],
but also slots which couple through a section of waveguide as shown in Figure 7.1. This
latter case also provides the mechanism for improved performance. By adjusting the slots
147
148
on the ends of the waveguide section to different lengths, we can essentially 'stagger tune'
each slot to a different frequency, increasing the overall bandwidth.
Radiating Slot
Coupling Slot W
Rectangular
Stripline Waveguide
Figure 7.1: Stripline excitation of waveguidecoupled slots.
7.1 Numerical Results for Waveguide Coupled Slots
To illustrate the behavior of the structure, a particular case at 9.7 GHz is presented
with a 0.60 x 0.60 inch cavity (cr = 2.2) fed by stripline with a ground spacing of 0.125
inches. A strip width of 0.1 inches was assumed which results in a 50Q line and the slot is of
the same width and is assumed to be covered by an ~r = 2.2, 0.0625 inches thick dielectric
sheet. The coupling to the slot can be represented by an equivalent series impedance. As
will be seen, its value is controlled primarily by the length of the waveguide section (Lw)
and the slot offset (s), the distance between the end of the slot and the centerline of the
strip. Note that the waveguide must be dimensioned above cutoff, if substantial lengths
are to be used. Thus, we have assumed halfheight Xband waveguide with transverse
dimensions of 0.900 x 0.225 inches.
Figures 7.2 and 7.3 indicate typical impedance behaviors as functions of the slot length.
Figure 7.2 shows the influence of the slot offset for the case of a conventional 'thin' slot as
treated in Chapter VI which also applies to the 'thick' slot case. Figure 7.3 demonstrates
149
Normalized Resistance R/Zo
Normalized Reactance XI/Z
1.50
1.35
1.20
1.05
0.90
0.75
0.60
0.45
0.30
0.15
nw'
I I i 1
(
/
/ 
/ / /
:. 
0.90 \ s = 05.3 miis
0.70........ s 09.6 amis
 s =13.8 mils
0.10
_____.. s = 47.9 mils
0.10 \ .. '  s.= 56.4 nil
s = 64.9 miIs
0.30
0.30   \^ , = 30.9 mits. .. s = 73.4mils
0.50...1
0.350 0.375 0.400 0.425 0.450
t L
U. ' '
0.350 0.375 0.400 0.425
0.450
Slot Length (inches)
Slot Length (inches)
Figure 7.2: Typical variation of impedance as a function of slot length for a conventional
'thin' slot with slot offset as a parameter.
Normalized Resistance R/ZO
0.30
0.25 7
0.20
ooo ~ i '
0.1 0
0.00
Normalized Reactance X/Z~
0.40....  .............
0.35
0.30
0.25
0.20 \ ',.
0.15
0.10 '.="
0.05 
a  0.0 milm
—... A=43mi
 A = mil
 = 12.8 mils
  A= 17.0 nils.. a 213 miJ~
  = 2_5 nail
a = 29.8 mils
—. =34.0 mil
a = 383 nails
I0.00t.... ..... L
0.350 0.375 0.400 C0.425 0.450 0.475 0.500 0.350 0.375 0.400 0.425 0.450 0.475 0.500
Slot Length (inches)
Slot Length (inches)
Figure 7.3: Variation of impedance as a function of slot length with the difference in top
and bottom slot lengths as a parameter.
150
Peak Normalized Resistance Resonant Slot Length (inches)
1.75 —. —, —. 0.475..
1.50 0.450
1.25 0.425.......
// t^^^o4^  1^.0
1.00  0.400 .....,L, =3 3 iUtk
 L^ =5 imcbS
0.750 0.375.w L.6 i.uxc
0.500 0.350  ~ L=.7inch
0.250  0.325
0.000 0.300
0.000 0.020 0.040 0.060 0.080 0.100 0.000 0.020 0.040 0.060 0.080 0.100
Slot Offset (inches ) Slot Offset (inches )
Figure 7.4: Slot coupling (resistance) and resonant length dependence on offset and waveguide length (Lw).
similar behavior for the 'thick' slot case when the difference in length between the top and
bottom slots (A) is used as a parameter and the slot offset fixed. Notably, the reactive
part does not necessarily cross through zero. This is due to parasitic reactive discontinuities in the geometry which are not strong functions of slot length; for example, the step
discontinuities in the slottowaveguide transitions. Nevertheless, a resonant behavior is
clearly indicated and we will define resonant length by the peak in the resistance curve.
We also see an affect on resonant length which is typical. The influence of slot offset and
waveguide length are summarized in Figure 7.4. In this case the top and bottom slots are
identical in length.
Frequency response can be inferred from the impedance behavior with respect to slot
length, through its relationship to wavelength. Therefore, we can use this parameter as
an indication of apparent bandwidth. Using length as a parameter is preferred because
this data is more easily obtained than direct frequency data. Returning to Figure 7.3
we see that as the difference in length (A) increases with slot offset held constant, the
151
1.00. ' ' ' ' ' I I
0.90
0.80
0.70 d
0.60
0.:50 
~ 0.40
N. 0.30
0.20  7)
o p 80 a m383ms. s81.9nuils
Z
0.10 d. — e  a 0.0 ml, = 39.4mils
0.00!.... I.... I.... I....0.90 0.95 1.00 1.05 1.10 1.15 1.20
Normalized Slot Length L. / LS,
Figure 7.5: Apparent bandwidth comparison for identical slots versus slots with different
lengths based on an interpretation of the relationship of slot length to wavelength.
apparent bandwidth is increased together with a change in resonant length, but the coupling
resistance is simultaneously reduced. Since the resistance is normally a fixed value dictated
by the feed design, we can compensate for this either by increasing the slot offset or
adjusting the waveguide length as indicated by Figure 7.4. However, because apparent
bandwidth reduces in a similar way with increased offset, it is difficult to judge whether any
gains in bandwidth can be produced. To evaluate the improvement in bandwidth for the
largest slot length difference of Figure 7.3, the offset was varied while the waveguide length
was fixed at Loy == 0.5 inches. The normalized resistance curves were then renormalized
to the peak value to remove any remaining discrepancy (R/Zo = 0.294 vs. 0.295). Slot
lengths were also normalized to the resonant lengths to remove variations. A suggestion of
an increase in bandwidth is now obvious as illustrated in Figure 7.5.
The apparent bandwidth interpretations so far have been based on the slot length. As
152
0.35... ............
0.30  
O 0.25
) 0.20 
mo: / d^^. 0.15  
E,i
0.10
~= 81.9, L 4463. Lo = 408 (mils)
0.05 0 s 39.4., L0.  417.7 (mils)
9.4 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2
Frequency (GHz)
Figure 7.6: Apparent bandwidth comparison for identical slots versus slots with different
lengths based on calculations at selected frequencies.
indicated earlier, however, each transition in the structure may have parasitics impedances
which are independent of slot length. These are unlikely to be constant or have the same
variation with frequency, therefore, to rigorously translate the previous figures into a frequency response, it is necessary to run each geometry at an ensemble of frequencies. The
plot in Figure 7.6 shows a comparison of the two cases over a sample of frequencies. It is
evident that the previous estimates based on length calculations were conservative, so that
we may be justified in a preference for the lengthbased results, at least for a parametric
study of the structure. Comparing Figure 7.6 with Figure 7.5 also suggests that although,
strictly speaking, scaling cannot be applied to the length data to derive the frequency response, it does provide a reasonable estimation of the actual response, at least in this case.
Scaling length data could then be used in the initial design stages to increase the efficiency
of the design process.
1.53
7.2 Summary and Conclusions
In conclusion, the thick slot hias been modelled by treating the slot apertures as separate slots coupled through a section of rectangular waveguide. It has been shown through a
parametric study that, based on the theoretical model developed, the apparent bandwidth
of the element can be enhanced by staggertuning the coupled slots. Although this part of
the analysis has not been verified by experimental evidence, the concept is a simple extension of the models verified previously and the behavior of the structure follows intuitive
expectations.
Within the limited range of parameter variations used here, apparent bandwidth increases of 50% are possible. The true bandwidth would depend on the external circuits
actually used to feed and load the slot. It seems likely that with a more detailed study
of the influence of various parameters on the impedance characteristics, even greater improvements in bandwidth could be obtained.
CHAPTER VIII
CONCLUSIONS AND RECOMMENDATIONS FOR
FUTURE WORK
The class of structures treated in this work involves rectangularly shielded microstrip
and slot lines. An integral equation formulation for the analysis of these structures in
various combinations has been shown to provide accurate characterizations of a variety of
elements. The capability to analyze multilayered substrates and superstrates has been
included through the use of impedance boundary conditions and provides an exact representation by modal analysis of the geometries involved.
Solutions to the integral equations require the specification of Green's functions for each
case. A number of Green's functions of different types have been derived by a extending
and generalizing a dyadic analysis technique to incorporate the impedance boundary conditions. This approach greatly simplifies the treatment of multilayered structures through
the use of transmission line analysis. The impedance boundary conditions have thus been
incorporated into the exact dyadic Green's functions making the analysis "fullwave", therefore all electromagnetic interactions are taken into account. For all structures treated, the
derivation of the Green's functions reduces to a straightforward procedure producing all
components of both the electric and magnetic field dyads in one exercise. Because of the
normalization of the functions used for the expansions of the fields, the approach also produces the dual Green's functions by a. simple change in notation, similar to the application
154
155
of the Duality Principle.
The integral equations have been solved through a conventional application of the
Method of Moments. The treatment of basis functions in general coordinate systems has
been shown to reduce to straightforward coordinate transformations. This reduces the
required integrations over source and weight functions to simple forms which have been
evaluated for piecewise sinusoidal basis functions with both constant and Maxwellian transverse dependence. These results lay a foundation for the treatment of coupling between
strips and slots with arbitrary planar orientations within a rectangular cavity.
The remainder of the work is a study of special cases of current interest. Overall, the
method has been shown to produce numerical models with excellent accuracy such that
one could expect to design devices involving these elements with little or no empirical work
or adjustments. This conclusion is evidenced by extensive experimental data provided
throughout. The remainder of this chapter is devoted to discussion of specific findings for
the various applications studied and recommendations for further development of these
techniques.
8.1 Summary of Achievements
8.1.1 TwoDimensional Analysis
A number of geometries have been considered in Chapter IV which can be treated by
a twodimensional analysis. The primary motivation for the studies in Chapter IV was to
support the work in later chapters, particularly Chapter VI. However, the results of this
chapter have some significance themselves.
The benefits of using LSE and LSM modes for the expansion of fields in connection with
a transmission line representation of impedance boundary conditions was demonstrated. It
has been shown that in order to find the propagation constants of microstrip or slot lines,
156
we first need only consider the boundary conditions in the region of the strip or slot. The
remaining layers are represented by the impedance conditions, simplifying the numerical
implementation. Once the propagation constant for a particular mode is known, the fields
in the remaining layers, and hence the characteristic impedance of the line, can be found by
a summation of modes with coefficients specified by analytically matching the modes across
layer interfaces. The matching procedure results in simple expressions as a result of the
use of the LSE and LSM modes. The fields throughout all layers can be efficiently found in
the same manner, allowing a visual representation of the field structure which is useful for
intuitive as well a quantitative information. Comparisons with available data were made
to verify reliability. The presentation also serves as an illustration of the technique when
applied to other structures such as cavities, parallel plate waveguides, or open structures.
We also developed a simple technique for the evaluation of scattering from vertical
wires or pins in a rectangular waveguide. The method is a simplification of more complex
routines discussed in the literature, allowable in our case since we are concerned with
wires of small diameter. This work was needed in support of Chapter VI but also led to
certain observations which have not been widely discussed. The validity of the results was
verified by experimental results and published data. The specialization of this problem to
homogeneously filled rectangular waveguides allows it to be treated as a two dimensional
problem which precludes its use with multilayered structures; however, the extension to
the three dimensions is not difficult.
8.1.2 Couplers
Chapter V contains a study of coupling from microstrip to microstrip through a narrow
slot aperture. The unique feature of the treatment of this problem is the use of the Standing
Wave method with even and odd excitation analysis to extract the circuit parameters. This
1.5T
method has been used by others and thus is only briefly discussed, however some of its
key limitations are pointed out.
The primary significance of this topic is in the numerical and experimental results. It
has been shown that this structure enables one to make very wide band transitions from
one line to the other. The particular structure studied, a transition from one substrate
level to another, has significant implications for vertical interconnects in monolithic circuits
since no viahole is required. Qualitative explanations for the behavior of the structure, as
a function of various parameters, were provided in connection with an equivalent circuit to
develop an intuitive understanding of the device. The numerical analysis has been shown
to produce highly accurate results by comparison with experimental data.
8.1.3 Radiating Slots
The modelling of microstripfed radiating slots is one of the principle topics of this work.
The fundamental results for this element are discussed in Chapter VI. The limitations of
the Standing Wave method developed for the coupler require a different technique to derive
circuit parameters. An alternative method, applied earlier by Shavit [67), was developed
for our use and is here referred to as the 'Reaction Method'. The technique is based on
the Reciprocity Theorem and is applied in a manner similar to that used for waveguidefed
slots developed many years ago. However, due to the presence of the strip, the method is
somewhat more complex and required the development of techniques described in Chapter IV. The explicit treatment of the strip for this type of antenna element, was shown to
be the likely cause for discrepancies found in earlier work.
Here it has been shown that the numerical model is capable of producing accuracies
on the order of experimental work. The model demonstrates much better performance
than earlier work on this type of slot. One of the key discoveries in this area is that the
158
openings in the cavity wall which allow access for the strip cannot necessarily be ignored as
has previously been assumed. A technique has been developed, supported by the models
discussed in Chapter IV, which compensates for the effect of the opening. The resulting
data shows excellent agreement with an extensive experiment.
8.1.4 Modelling of Thick Slots
The motivation for making a thick slot may be for structural purposes or to act as a
heat sink for active devices. The original formulation in Chapter I outlines the modelling
of thick slots by replacing them with a closed cavity and equivalent magnetic currents on
the slots. Thus, the cavity dimensions do no have to correspond to the dimensions of the
slot, allowing greater flexibility in applying the model to different applications.
In Chapter VII, we numerically investigate the behavior of a structure in which the
stripfed slot couples to the radiating slot through an intervening section of rectangular
waveguide, dimensioned to allow the dominant waveguide mode to propagate. The most
significant feature of this chapter is the study of the effect of allowing the stripfed slot and
radiating slot to have different lengths. This detunes the structure by stagger tuning each
slot to a slightly dlifferent frequency, resulting in an apparent increase in bandwidth. In
order to achieve a constant slot impedance, the strip offset and/or waveguide length must
be adjusted simultaneously with the alteration of slot lengths to compensate for a change
in resonant resistance. This is significant, since bandwidth is one of the primary limiting
factors for these types of antenna elements.
Experimental verification of this result is left for later work. However, it is expected to
be correct since it is a relatively straightforward extension of the remainder of the model
which has been validated by experiment. The result is also appealing from our intuitive
understanding of the slot behavior.
159
8.2 Model Limitations
Aside from resource issues such as the matters discussed in section 3.6.3. there are
several technical limitations of the methods used in this work which should be pointed
out. Some have already been discussed regarding the Standing Wave method, however,
the stated limitations of this technique could potentially be circumvented. For example, it
has been stated that this method requires a minimum of about one wavelength of line away
from the discontinuities in order to estimate the guided wavelength. In some cases, this
information can be derived from other methods similar to the twodimensional analyses
derived in Chapter IV. Another restriction is the requirement of only one propagating
mode on the line. This constraint, however, is a result of our simple interpretation of the
standing wave pattern by locating the relative positions of the peaks or nulls. It is possible,
by more sophisticated parameter estimation schemes, to derive the required information
even when more modes are present. Of course, these variations may introduce their own
requirements which may or may not be beneficial.
A more serious limitation involves the twoport microstripfed slots discussed in Chapter VI. It has been shown that the feedthrough apertures, which allow the strip access
to the cavity, can have a significant influence on the behavior of the slot. While we have
demonstrated that this effect can be effectively compensated for in the cases studied, numerical experiments have shown that this technique is limited to strips which are significantly offset from the cavity centerline (perhaps 25% or more). It is suspected that this
is due to certain assumptions made about the behavior of the higher order modes at the
feedthrough apertures. Specifically, the firstorder correction technique we have applied
does not account for potential interactions between the wall aperture, the strip and the
reactive fields of higher order modes. This interaction may have a profound influence on
the amplitude and phase of the TEo10 mode, as well as the dominant mode, which in turn
160
may significantly alter the field on the slot. The effect would be enhanced for a centered
strip since, not only is the TElo mode more strongly excited at the slot, but also it is
more strongly coupled to the feedthrough aperture. Thus, the field on the slot may not
be correctly determined under these conditions and will lead to incorrect slot impedance
values. The rectification of the situation is, of course, to expand the model to include an
explicit representation of the feedthrough apertures, which would fully account for their
influence.
Other limitations of the models are not unique to this particular approach and need not
be discussed in detail. For example, all numerical models involve finite approximations to
reality and therefore involve some errors in their representations. Nevertheless, fullwave
integral equation techniques such as used here, have been proven to provide some of the
most accurate and reliable models available for the simulation of a wide variety of problems.
8.3 Recommendation for Future Work
Several applications should be further investigated based on the formulations presented
and form the basis for extensions to this work. The first instance would employ the transformations derived in Chapter III to study the case discussed in [62]. This case is of interest,
because the sensitivity of the impedance to the strip offset for the geometry of Chapter VI
may place unduly stringent tolerance requirements on the fabrications process. The application of the fullwave techniques to this problem may make possible the elimination of the
semiempirical factors used in [62]. Since the feedthrough apertures could be significantly
offset from the cavity centerline, the apparent problem with their neglect in the present
model may be avoided.
Another case to consider is the 'TBar' fed slot which has the geometry illustrated
in Figure 8.1. The formulation for extracting the impedance of the equivalent impedance
could follow the Reaction Method used in Chapter VI by applying the Reciprocity Theorem
161
Short at walls
~'" Cross Bar
Opening Detail
I I
L
Slot
I,
Input
Port
Figure 8.1: TBar fed slot geometry.
to the two cases shown in Figure 8.2. Here, in both cases, the strip feed line is terminated
by the crossbar within the cavity. We now identify the reflected fields supported by the
currents on the strip in the cavity, case 'b', as
Es = Fs (eTEMA  i ETEM) em (8.1)
Hl/ = rs (hTEMAi HTEM) ejBl (8.2)
These are the fields associated with the dominant stripline mode produced by the scattering
of the slot as before, and it is still assumed that the dominant mode is unaffected by the
required aperture in the cavity wall. The fields supported by the currents in the 'a' case,
however, are now given by
EL = (ej1 + rFLeC)eTEM + (e i'  FLejfP) i ETEM
HL = (eJ'  FLe"jp)hTE + (e ' + rLei') i' H)TEM
(8.3)
(8.4)
where rL is the reflection coefficient of the Tbar line without the slot present, referenced
to the location of the slot on the line, or the reference point for the slot, if not at the plane
162
slot
position;... HM.
E H TEM
a) E TEM:
waveguide
aperture
slot
position
b)
H
*
a 
K K
KI.
K I
waveguide
aperture
Figure 8.2: Two cases for application of the Reciprocity Theorem for the Tbar fed case.
of the slot. The reflection coefficient on the line is now given by
1 ' // eTEA ' K dS,  slot  FL (8.5)
2 / (ETEAI X HTEM) ( dS
J waveguide
We now also need to know FL from which we can find HTEpm which we then use as an
excitation for the cavity problem to find K, as before. Accurately knowing all these
terms, we should be able to produce the reflection coefficient for the slot from which the
slot impedance and circuit parameters can be derived. One difficulty in this approach is
forseen to be the determination of FL. This problem would be straightforward except for
the fact that for the dimensions of the cavity, more than one mode can propagate on the
line. To derive FL for the dominant mode on such a line would require further development
as discussed above. Also the question of the effect of the feedthrough aperture needs to
be resolved which by itself can be viewed as an extension of the treatment of wires in
Chapter IV.
163
Another topic which should be further developed for application to the radiating slots
is the use of a power balance formulation. If we assume that the walls and materials in the
cavity are lossless, which we have done throughout, then the resistive part of the impedance
for the Tbar fed slot is directly proportional to the total radiated power from the slot.
This can be determined by integrating the radiation pattern of the slot field over the entire
half space which can be done in the far field, given the field on the slot. For uncovered
slots, this is a straightforward procedure since the Equivalence Principle can be used to
replace the slot magnetic current by an equivalent magnetic dipole current in free space for
which the total radiated power can easily be derived. For the covered slots, the situation is
a little more complicated, due to the presence of surface waves in the dielectric; however,
the integration can be performed in the farfield, in which case asymptotic expressions can
be used with steepest descent path integration techniques. This method has been applied
to other problems in the past where the slot voltage distribution has an assumed form
at resonance [21, 58, 67, 62], but should be equally applicable with the present technique
where we can determine the actual distribution at resonance.
Finally, it is suggested that the model could be extended to allow arbitrarily shaped slots
and stripline configurations. It is well known that a wider slot has a broader bandwidth and
that loading of the slot ends with different shapes also has some influence. This capability
could be achieved by using rooftop basis functions which approximate the current in a
piecewise fashion in the two planar slot dimensions. As the slot is made wider, however,
one would likely need to model both directions of current as described in Chapter III.
APPENDICES
164
165
APPENDIX A
Vector Wave Function Expansions and Relations
By convention, L, M and N are used throughout the text to denote the Vector W'ave
Functions (VWFs) defined by
L = VT (A.1)
M = Vx i = Vx N (A.2)
K
N 1= Vx x x.i = —V x M (A.3)
K K
These form a complete set of solutions to the homogeneous wave equation V x V x F 
i2 F = 0 when ' is the scalar function solution to the equation V2T + K2%P = 0. The
particular solution for K is chosen to satisfy the boundary conditions of the problem. The
Xi unit vector is called the 'piloting vector' which determines the forms of L, M and N and
K is the separation constant K2 = k2 + k2 + k. In this appendix, the expansions of various
vector wave functions (VWFs) used throughout the text are provided as a convenient
reference. In addition, some identities used to reduce some of the derived expressions to
simpler forms are provided.
By convention, L, M and N are assumed throughout to be formed from the generating
function I = ej(kxzx+kyy+kz) with z as the piloting vector unless indicated otherwise by
additional subscripts, superscripts or in the text. A common superscript to be used will be
a prime (') which will imply both that the x, y and z dependence will be x', y' and z' corre
166
sponding to source coordinates, and that the wavenumbers k1, ky and kZ will be led by a j
instead of a j if the functional dependence is the exponential, e.g., 1, e(k' +kyy'+kz')
Also, a notational convention has been established which should eliminate ambiguity when
subscripts are used to imply trigonometric dependence. Specifically, subscripts 'o' and
'e' are used to indicate sine and cosine variation of the generating function, respectively.
When these are used, the arguments will be provided in onetoone correspondence to indicate the variable to which the trigonometric function applies. For example, the function
Meo[ki,lky, k,(z  c)] implies that the generating function is q'eo[kx, ky, kz(z  c)]. %Ieo must
then have the form [cos krx sin kyyejkz(')], since the first two arguments are k1 and ky,
corresponding to the 'e' and 'o' subscripts, respectively. This example also illustrates the
additional specification of spatial coordinate dependencies as in (z  c) instead of just z.
Parallel Plate:
Te(ki) = coskxe(kyy+kzz) (A.4)
Me(c,) = V X ee(kX
[ijkycosk,:x + Pk, sin kx]ej(kyy+kzz) (A.5)
Ne(k^) = V x V x Iee(k)
K
= [ixjkk, sin kxz  ykyk, cos klx + i(k2 + k) cos kx]e z) (A.6)
=o(k) = sin krxe3j(ky+kzz) (A.7)
Mo(kx) = V x 'oo(kr)= [ijky sin kx + yk cos k;x]ej(kyy+k z) (A.8)
No(k1) = V x V x TO(kl):
K
l [:jkxk cos kx  Pk k, sin kx . (k 2 + k) sin kx]e'(kyy+kz z) (A.9)
KIf
1 6?
Rectangular X\Tavegujde
T'ee(kxrlky)=
Miee(kxrlky)
Ne~e(kxrlky)
Noo(k~r,, Ry:)
Meo(ki, ky)
Neo(k',x, ky)
cos k~xcosk kyeCkzz
V x 4'ee(ki,,ky
[i kycoskzxsink,y+ k~sjlk~xcosk~y~jcJk~Z
1
7 x V x 4Pee(kr,kj)Z
K
[i jk~ksin k~xcos k.y + ' kk cos k~xsin k,y
K
+i(k2 + k 2)cos k~xcos k y] 7kzz
sin k~x sin k~yyc zz
V x Tok y
[~i k~ sin k~x cos k~y  kcos k~x sin k~y]ej'k~z
iV x Moo(kxlky)
[i jk~k cos k~x sin ky  k jkk sin k~x cos ky
+ dz (k2 + k2) sin k~x sin kyyle)k~z
cos k~x sinl kyy6) kzz
V x %~ k, y
[ik.,cos k3,x cos kyy + p k, sin k~x sin kyyjeJ k~z
1
v x V x 'okzvk)i
K
[ijk~kz sin k~xsink~ypj  'kk cos k~xcos ky
+ z k+ k ) sin k~x sin kyyJek.zZ
(A. 1 0)
(All1 )
(A. 12)
(A. 13)
(A. 14)
(A. 15)
(A. 16)
(A. 17)
(A. 18)
(A. 19)
TJ'o(k.r,ky) = sin k~x cos kyyejkzz
Moe(k.ri ky) = V x T oe (kx,k yY~
168
= [xky sin k.x sin kby  Ykh cos k cos kCy}c Y k~
Noe(ki,ky) = V x Moe(k, ky)
= [rjkik: cos kxx cos kty + yljkhk sin k,:x sin ky
+ (2 + 2 )sin krx cos k y]cJkz
(A.20)
(A.21)
Rectangular Cavities:
Meee(kr ky kz)
Meeo(kx, kyI kz)
+
Mooe(k, ky, kz)
Neee(k, kI y k z)
V x [Z cos kx cos k. y cos k, z]
x ky cos k1z sin kyy cos kz + y kr sin kx cos kyy cos kz (A.22)
V x [z cos kx cos kCy sin kz]
x ky cosk xz sin kyy sinkz + y k1 sin kj cos ky sinkz (A.23)
V x [i sin k1x sin ky sin k:z]
+~ kysi sin kx cos ky sin k cos kx sin ky sin k^z (A.24)
V x [ sin k:x sin kyy cos kzz]
+i ky sin k*x cos kyy cos kz y k, cos krx sin kyy cos kz (A.25)
1
V X Mxeee( t,*,2 k )
[+: k^:kz sin klx cos kyy sin k^z + y kykz cos kjz sin kyy sin kz
+i (k 2 + k 2)cos k:x cos kcy cos k.z] (A.26)
1
V X Meeo(kx,ky kz)
[i fkk sin kxz cosk yy cos kzz  y kyk, cos kz sin kyy cos kz
+z; (k2 + ky ) cos kxz coskyy sin kz z] (A.27)
V x Mooo(k. ky,k )
 [+i: kjkz cos kfx sin kyy cos kzz + y kyk, sin kxz cos kyy cos k~z
+5 (k2 + k)sin) krxsin kyy sin kfz] (A.28)
Neeo(kx, kyz) =
N*ooo(k ky2 kz) =
169
Noo~e(kx, kyk)  ~oek~yk~
+ krk cos krx sink~yysiflkzz  v kksin k x cos k yikz
+ Z (k + k 2)sin k~xsink cos kzz] (A.29)
Identities:
Jk CNI(kz) ~C e NkzcMkz)
ek CM(k,)~jkzCM(k,)
e3kzc(k) ~ ekZCN k
e zN('k) ~ e~zN(kz)
&kc~kzk) C jCekxk)
&kcel0k Nkz)~ rekz zcN(k1kz)
e Mrce(kx, k~z) ~ C JkzcMe(ki, k,)
e Mzco(kx, kj) ~ eik7 zc~o(kx, kz)
.M[kz(z  c)] ~M[kz(zv  c)]
= {+2Me[kz(Z  c)]
2j'M0[k,(z  c)]
= M[k,(z  c)] ~ M,[k,(z  C)]
{+2Me[kz(z  c)]
2j'M0[kz(z  c)]
 N[k,(z  c)] ~ N[kz(z  c)j
{+2N,[k,(z  c)]
2jN0.[kz(z  c)]
 N[kz(z  c)] ~N[k,(z  c)]
{+2NI![kz(Z c)]
2j'N.4kz(zY C)]
(A.30)
(A.31)
(A.32)
(A.33)
 Me[kx, kz(z c)]~ Me[kx,, kz(  c)]
(+2Mee[kx, kz(z  c)]
i 2jMeo [kr, kz(z  c)] (.4
M0[kx, kz(z  c)] ~ M0,[k,, k,(z  c)]
+2Moe[kx, kz(z  c)]
(A. 3 5
 2jNMe[kz,k( )~ kz(Z  c)]
{+2Nee[kz, kz(z  c)]
2j'Neo[kx, kz(z  c)]
No[kx, kz( Zr  c)] ~ No[kx, kz(Z  c)]
+2No3e[kx, kz(z  c)]
(A.37)
I. 2jNf4ok1,, k,(z  c)]
170
eikz1Mee(kx, ky I kz) ~  e M21?ee(kx, ky, I~
=Mee[kx, ky, kz(z  c)] ~ Mee[kx, kv, k,(z7  c)]
J+2M~eee[kxl ky, k,(z  c)]
= (A.38)
1.2j'Mee[k1, ky, kz(z  c)]
e kzIMoo(kx, ky, kz) ~  Mozc40(kr, ky, kz)
Mo= xlk, zz c)] ~Moo[kxlky, k,(z c)]
+2Mooe[kx, ky, kz(z  c)]
= (A.39)
l.2jMooo[k., k,, kz(z  c)]
e kzNee(kx, ky, kz) ~  e~zNee(kr, ky, kz)
=Nee[kjrlky, kz(z  c)] ~Nee[ki, ky, k(z c)1
{+2Neee[kxr, ky, kz(z  c)]
(A.40)
= 2j'Neeo[kxl ky, kz( . c)]
ejkzNoo(kx, ky, kz) ~eikcI~oo(kr, ky, kz)
=N04~kxky, kz(z  c)] ~N04~xlky,k kz(z  l
('+2Nooe[kxl ky, kz(z  c)]
= ~(A.41)
1.2jNooo[kxky, kz( Z  c)]
171
APPENDIX B
On the Use of Vector Potential Functions for the
Derivation of Green's Functions
In previous literature there has been considerable discussion on the treatment of sources
in waveguides or cavities, particularly the problem of a longitudinal slot in the broad wall of
a rectangular guide. It has been stated and generally accepted that an additional "mode"
needs to be included in the Green's function for the waveguide to accurately represent the
currents at the slot [86]. However, the origin and derivation of this mode is somewhat
unclear.
The difficulty arises from an incomplete treatment of the potential theory in the source
region for these problems. It is identical to the difficulty associated with the vector wave
function or direct field expansion formulations for which the problem has been adequately
resolved after much discussion in the literature over an extended period of time [81, 65,
42, 15, 77, 11]. Because the use of vector wave functions as used in this work is perhaps
somewhat unfamiliar and since the difficulties for the vector potential method are very
closely related, this appendix presents a discussion of the use of potential theory for source
regions to demonstrate the origin of the required 'additional' terms.
B.1 Potential Theory Solutions for Electric Currents
To examine the method, we will first consider the fields in a rectangular waveguide
excited by a point source of longitudinally directed electric current. The waveguide axis is
172
assumed to lie along the Z direction.
We must find solutions to Maxwell's equations as follows:
V x E= jwuH (B.1)
V x H = jwE + z(z  x')6(y  y')6(z ') (B.2)
where the assumptions stated in section 1.3.1 are employed and in which we have introduced the current source
J = 56(x  x')(y  y')(z z') (B.3)
Since V H = 0 we can represent H by
H = V x A (B.4)
where A is, as yet, an arbitrary unknown vector function. Using Equation (B.4) in (B.1)
we find
V x E= jV x A (B.5)
which we can integrate to get
E= j A + VD (B.6)
where 4 is a scalar function representing the constant of integration.
Using Equation (B.6) in (B.2) we find the relationship between A and 4 must satisfy
V x V x A k2A jweLVF = /J (B.7)
As A is arbitrary, its specification determines 4 through Equation (B.7), to within an
incidental constant. Equivalently, 4 may be arbitrarily specified which in turn defines A.
Both, of course, are not entirely arbitrary since we must consider the resulting boundary
conditions on the E and H fields. Thus, we are free to choose 4 in order to simplify
173
Equation (B.7) to a form for wlhicll the solution is either known or nmay be readily found.
This step is really the quintessential motivation for the use of the vector potential mIetlhod.
since the overall problem is reduced to the solution of a single, presumablMy simple, equation.
The selection of two conditions which specify A or F may be familiar. The first is the
Coulomb condition, or Coulomb 'gauge', in which A is assumed to satisfy V  A = 0. From
Equation (B.7) it can then be shown that ~ satisfies Poisson's equation:
V24 = 2V. J (B.8)
cde
however, this does not lead to a simple equation for A. Thus, this app)roach is not generally
used [53].
The Lorenz gauge [82] is by far more commonly used in which it is assumed that
V  A = jwcU. This reduces Equation (B.7) to (V2 = V x V x +VV )
VA+ k2A = J (B.9)
To satisfy this vector equation requires that we choose A with a component in the same
direction as J, reducing it to a scalar equation. In our case, for example, we set A = A 
reducing Equation (B.9) to
V2A, + 2A, = t6( R  R') (B.10l
the solution of which can be found by separation of variables and is the primary topic for
much of the remainder of this appendix.
Before proceeding however, we must pause to distinguish this approach with other approaches involving potential functions. It is widely recognized that the fields in waveguide
can be expressed in terms of a variety of combinations of potential functions, most notably
in this context, the A and F potentials. For example, the complete set of homogeneous
solutions for rectangular waveguide can be written in terms of A = Az and F = Fz
174
where H = V x A and E = V x F yielding the TM and TE waveguide modes respectively. Similarly, if we use A = Ai: and F = Fi, we produce the LSM and LSE
modes which are TM and TE to the Z direction respectively. Various other combinations
are possible, not restricted to the rectangular waveguide problem, as noted in [25] for the
dielectric half space. We should recognize that these potentials do not proceed from an
attempt to solve the inhomogeneous field equations through the use of potentials, as above,
but rather are solutions to the homogeneous (sourcefree) equations which allow a great
deal more flexibility. Solving the inhomogeneous case by expanding the fields in terms
of homogeneous potentials and enforcing the boundary conditions at the source, amounts
to a field expansion method closely related to the vector wave function method employed
in the main body of this work. As such, they perhaps should not properly be referred
to as 'Potential Theory' methods as is often done. It is in these cases that the functions
themselves can truly be called 'auxiliary potentials', since they are not directly solved for,
but rather are only a vehicle to express the field in a simple way. It is also noted that these
methods often introduce the radiation condition into the potentials which prevents them
from being truly homogeneous solutions as will become evident.
Returning to the solution of Equation (B.10), the particular solution to this inhomogeneous equation can be constructed from the general homogeneous solutions. Once Az has
been found, the fields will be given by
H = V x (Ai)
= A —Z  y (B.11)
i ay 9Ox
r.. VV (Azi)
E = jwAz. +,1 '2A, A, 1 ^A,
[ A [ 2 jwAz] z (B.12)
The hs lui e eiy y ep i ie c ez2
The homogeneous solutions are readily found by separation of variables and can be
175
written in the form
A, = [Al cos kxz + A2 sin kz][B1 cos kty + B2 sin ky][Cle+3z(z') + clk(z  )]
(B.13)
where k2 = k2 + k2 +~ k2. Applying the boundary condition n x E = 0 at the walls of the
waveguide reduces (B.13) to
Az = sinx ksinkyy[Cie+ik(zz') + C2ejk( )] (B.14)
where km = m7r/ar, kn = n7r/b, and (m,n) are any integers. The eigenvalues km and k,
are thus infinite discrete sets while kz has a continuous spectrum.
Before proceeding, it is instructional to verify that this solution satisfies the homogeneous form of Equation (B.10) which can be readily done by direct differentiation. In
rectangular coordinates the V2 operator is V2 =  + v + a which allows t2he r and
y partial derivatives to be performed by inspection leaving
2 Az 2
k2 A,  k Az + z A, =0 (B.15)
Since k2 = k2 + k^2 + k2, Equation (B.15) reduces to
a2AZ k 2Az =0 (3B.16)
Since Az is continuous in z, the second derivative can be performed without difficulty so it
is obvious that (B.16) is satisfied and (B.14) is confirmed as a solution to the homogeneous
equation.
We now proceed as usual and impose the 'radiation condition' on the solution; that is,
E 0, H  0, as Izl . o (B.17)
with finite losses. (Or equivalently, require outward travelling waves outside of all source
regions.) Of course we have anticipated this condition in the form of the solution's z
176
dependence and we now set
Clie:k'(:' z > z'
A = sn sin m sik,y (B.18)
C'2eJk,(z' z < zNotice the implications of imposing the radiation condition. Applying the left hand side of
the scalar wave equation (B.10) we can immediately see that the solution (B.18) remains
homogeneous in the entire domain with respect to the x and y dependence. Applying the
remaining operations  left hand side of Equation (B.16)  we find
AA, jkC(ez( z) > zf
jkzCCle3k(z ') '
sin kmx Sin kny + [Ci  C2] 6(z  z')
jkC2eJk(zz') < z'
(B.19)
Note that the term involving b(z  z') comes from the discontinuity in AZ that may exist
if C1 $ C2. We have performed the derivative by noting that for a function as shown in
Figure B.1 a singularity appears at the discontinuity. Note also that the singularity may
appear and be defined in the sense of a generalized function at z = z' whether or not the
original function is defined at that point [26, pp. 2122].
We can eliminate the singularity in the first derivative by requiring
C1 = C2 = C (B.20)
This is a boundary condition is based on the argument that the potential must approach
the same value from either direction at z = z', i.e., it is continuous [13, p. 160]. Then
Equation (B.19) becomes
DAf A6e)A(Z') z > z'
z = jkzC sin kmx sinky (B.21)
I ek(z ) z < z
However, Equation (B.21) is discontinuous at z = z' and the second derivative is
OA sCJy k(zz') > z'
2' = k2C sin kmx sin kn y e
ej  z) < z'
2jkzCsin kmz sin knY 6(z  z') (B.22)
177
[ f(0)  f(O)](x)
+  +
f(x) f(x) df(x) df(x)
dx dx
df+ (z)
df X > x
df (:r dx + [f +() f (Z)](  X')
dx df(x)
x < x'
dx
Figure B.1: The derivative of an arbitrary discontinuous function.
Using Equation (13.22) in (B.16) we find that Equation (B.18) is indeed a homogeneous
solution for Az except at the point z = z'. Notice that this result was obtained by imposing
the radiation condition alone. The continuity condition was also used but effects only the
nature of the singularity in the second derivative at z = z', that is, whether the delta
function is produced or its first derivative. Thus, as is intuitively obvious, imposing the
radiation condition dictates that the potential must satisfy an inhomogeneous equation
with a driving function (source) located at some location between 00 < z' < 00. Notice
that we arbitrarily chose the form of the z dependence so that the discontinuity would fall
at z = z'. This is, of course, the most reasonable choice, since we ultimately are attempting
to solve the source problem and we do not expect discontinuities in tile fields in the sourcefree regions. However, we have not yet attempted to satisfy the original nonhomogeneous
equation given by Equation (B.10).
17
Equation (B.22) suggests that we can take (3.18) as the homogeneous solutions for
Equation (B.10) if we exclude tile point  = z' from the homogeneous solution's domain.
We thus avoid the inhomogeneous contribution of the singular term in the second derivative.
The complete set of homogeneous solutions of this type (TM) are then
ekz(zz') >
Az = / dkz Cmn sill k sil kny (B.23)
mnn k,(zz') 2< Z1
The homogeneous solutions can be added to the particular solution with arbitrary coefficients without violating Equation (B.10) as long as the thie exclusion of the point z = '
is observed, even (especially) for derivatives of Az. Specifically, this means that the fields
arising from the homogeneous solutions are valid everywhere but the z = z' point. Homogeneous solutions are often added, for example, to satisfy new boundary conditions
introduced by adding obstacles to the waveguide. In fact. this is the basis for the method
of scattering superposition used extensively in this work.
The form of the particular solution is suggested immediately by the singularity encountered in Equation (B.22). Therefore, we assume the particular solution has the form
fe kz(zz'} Z > Z
Az = E Cmn sill kmX sin kny (B.24)
m,n eJk'(zz') z < Z
Note the exclusion of z = z'. In this case, however, we do not exclude z = z' from the
solution domain. As will be seen, this allows the discontinuity to contribute only when the
second derivative at z = z' is evaluated and allows the particular solution to be found. This
type of exclusion is allowed since this potential is a Green's function which is always defined
in the sense of a generalized function. Generally, this means that the function itself has
no useful meaning except in the context of an integral, in which case, nonsingular isolated
points can be removed without effecting the result. Iowever, the point is an essential
point at which the derivatives must be allowed to exist (in the generalized function sense)
in order to satisfy the differential equation.
179
On applying Equation (B.22) then, we are left to find Cn suchI tlhat (B.10) is satisfied.
tlhat is,
2j 3 Crnk, sin lkmX sin ky 6(z  z') =  x')(y  y')b(z  z') (B.25)
m,n
or
E Cmnsin kmx sin knY = 6(x  x')^(y  y') (B.26)
m,n
We can find Cmn by multiplying Equation (B.26) by sin kmx sin kny and integrating over
tlhe cross section of the waveguide getting
Cmn = j2 sin kx y si (B.2 7)
k~ab
Equation (B.26) thus implies
sin kmX sin kny sin kmx' sin k,y' = (z  x')(y  y') (B.28)
mn
or
2
>3sinkm^zsinkcm:Z' = 6(x  x') (B.29)
y bsin kysin ky' = 6(y  y') (B.30)
n
which can be shown independently. The particular solution is then
A2 f jkz(zz')
A =   l sin km sin kn y sin km x' si n ky' (B.31)
rnn abA ekz(zz') z < z!
B.2 Field Behavior  Electric Current
Now that we have the particular solution for the potential, our interest turns to the
behavior of the fields, particularly near the source. The H field is determined by evaluating
Equation (B.4) which yields
^j2k ekz(zz') z > z
H = sin kmx cos kny sin kmx' sin ky' (B.32)
eJkz() < z'
P
( ~
1SO
j2k Ek2^ > '
abk = cos km sin ky sin kmx'sin k, y (B.33)
mn akz (zz') K z'
HIz = 0 (B.34)
The E field can be found from (B.12) with the Lorenz condition and is
j2km (,z > z'
E =   cos kmz sin ky sin km' sin k, y' (B.35)
abe /
m,n abw ek(z') <
j2kn r Ce  Jk:(':') >
EY=  i n sinkmz cos ky sin km' sin ky'{ (B.36)
m,n abwk ek(zz') <
Eaz =
m 3 ki (  j <.]
=  i  jk ) sin kmx sin,ny {sin k,;;' sill ky
m,n abk kz(z') < '
 e  sin kmX sin kny sin kmX' sin n ky' S(z  z' )
]LI m.n ab
s2 k x2 +si > Z'
=  + T ) sin km sin k sin k^y' sin sinn;
m,n kz(z') <
5(x x')6(y y')(z ') (B3)
jwe
Thus, we find that the particular solution contains not only the normal (TM) waveguide
modes but also a singular term in the z component of the E field, that is, the addition of
the term
E 6(x  x')6(y  y')6( ) (3)
E = ; } (B.38)
jwe
This addition corresponds exactly to the singular term previously found by the vector
wave function formulations discussed by many authors (see especially [78]). This term
corresponds to the;z term associated with a dyadic representation.
As was previously pointed out, the function is normally applied to the problem of an
unknown current source with an integral of the type:
F(R) = JJGF( R R)J(R')dV (B.39)
1 IS1
where F is a field or potential and G F is a dyadic Green's function of the appropriate type.
In the case where F is to be a potential, the singular term does not appear directly but
should appear when the E is found by carefully performing the second derivatives indicated
by the VV A operation. Similarly, if F is to be the E field, the correctly constructed
dyadic Green's function will be of the electric field type and, in this case, will have the
form
GE((  ')b(x  y')6(y  z')
G z + Go (B.40)
E `jw
where Go is a modal expressioll.
Finally, we note that the solution found is the complete solution for J = 6( R  R )
in rectangular waveguide which can be seen to excite only the TM modes. The TE modes
can be found by assuming E = IV x F which are excited by currents in the other two
directions. The point also illustrates the benefit of the vector wave function approach with
dyadic analysis. Both the vector potential method and field expansion methods employing
auxiliary potentials, when performed on a vector level, require each component of current
to be considered separately. In contrast, the dvadic analysis produces the entire Green's
function at once. Since all of these methods contain the same pitfalls when the derivatives
at discontinuities are not carefully performed, this is the only fundamental difference. Of
course, depending on the structures involved, when other beneficial techniques such as the
use of impedance boundary conditions and scattering superposition are not employed, the
difficulty of the various derivations may increase dramatically.
B.3 Potential Theory Solutions for Magnetic Currents
The necessity of the singular term is particularly evident in the literature pertaining to
the analysis of apertures in rectangular waveguides and cavities. In these cases, the problem
has a 'dual' nature in that we usually represent the apertures with magnetic currents. The
182
dual problem can then be handled in exactly the same manner as follows.
We take as the dual form of Maxwell's equations:
v x E = jwH  z(x  x')(y  y')b(z  ')
(B.41)
(B.42)
V x H = jwc E
where we have introduced the magnetic current source
1K = ib(x  X'by Y'b z')
(B.43)
Since V E = 0 for the dual case, we set
 1 
E= Q x F
(B.44)
and search for solutions to
V2 F + k2F = e K
(B.45)
Because K has only a component we set
(B.46)
reducing Equation (B.45) to
V2F + k2F, = e6(x  x')6(y y')b(z  z')
When Fz has been found, the fields will be given by
(B.47)
E = Vx(F.)
1 F, 1 Fz,
=   y
H = jwFzz.W iE
1 02 F. 2 Fz
= . x+ r yz
jwp[ ]xz ayaz
(B.48)
J 2FZ jwF.z]
(B.49)
183
Following a procedure identical to thle previous case, thle particular solution is found to be
*c  j( — ')  > 
Fz b km os kXcos kny cos km ' cos ky j (B(.50)
2ab k(
m.,n, ~: z  ) < z~
where cm and Cn are the Neumann numbers defined as
f2 54O
= < (B.51)
1 =0
Note particularly in this case that for m = 0 and n = 0,. the potential function is nonzero,
unlike the previous case, and is the term referred to as the (0, 0) 'mode' in [86]. However, as
can be seen from the field expansion below, this mode does not make the key contribution
at the source, but rather, the additional contribution comes from the singular term.
B.4 Field Behavior  Magnetic Current
The fields produced by the infinitesimal magnetic current source are the following: The
E field is determined by evaluating Equation (B.48) which yields
m 2k ab
Ex =  2 ab cos kmx sill kn y cos kmIz coskn Y (B.52)
mn 2 i a nJk (zz') z< z
>1 s m okycsm'c f k(z) z >z'
EY k os k Y COS km' coS k'jkL(ZZ) (B.53),, 1 2kzabs m kycoskm cosk,
mn e kJ( ') < ' z
Ez = 0
(B.54)
The H field can be found from Equation (B.49) and is
HJ = ^ ~ j e, e, k m y, {  k '(zz'
Hx = 2wnb sin kmx cos kny cos km ' cosknY{
m,n ej k, (zz')
zr jmEnki'n { )k z (z')
Hy=: 2l cos km sin k,y cos kmz' cos kny
m,n (,2w 2k
HZ n  k x cos k, y cos km c' cos k, y'
+. — cos mcos kny cos km:,' cos kny' 6(  z')
' m,I n ab
I
z > '
z < z'
 > z'
2< z
z jk, (ZZ)
{ jk(zz')
(B.55)
(B.56)
z > 2
z < Z'
184
Emm n n
z 2  cos kx cos kny cos km' cos kn
m,n\w a ejk (:Z') z< '
6(x  x')&(y  y')5(  ) (B
Again note that these solutions are complete for the z component of current which excites
only the TE modes in the waveguide. The remaining modes may here be found by setting
H = V x A with A = Az, in a manner similar to the electric current case.
For the problem of a slotted waveguide, the singular component which appears in the
i component of the H field has the form
z6( x')6(y  y')6(z  z')
H =  j (n.58)
jwL
An integral of the type
H(R) = JGH((R, ') K(R')dV' (B.59)
is typically applied to find the H field in the guide for which GH(R, R ) has the form
H, (  x')(y  y')6(  ') +
GH = zz ) ) + Go (B.60)
jwpl
It is obvious that the singular term will significantly alter the numerical value of the 'selfcell' and neighboring cells (with overlapping basis functions) in a Method of Moments
solution to a problem of this type and therefore causes serious consequences when overlooked.
B.5 Partitioning
It has been observed that the singular term appears above in the i component of the
field as a result of partitioning the field about z = z' when we applied the radiation condition. In fact, the occurrence of this term also depends on the choice of modes used to
expand the field. For example, for the multilayered shielded microstrip problem discussed
185
in Chapter IV, it is far more convenient to express thie Green's function in terms of LSE
and LSM modes and partition the problem parallel to thie planes of the dielectric interfaces.
Although the radiation condition was not imposed on that solution, the singular term appeared for the component of current normal to the interface for the normal field. Imposing
the radiation condition in that case, amounts to choosing a contour for the Fourier integral
which does not produce additional singularities. It can thus be seen that the singularity
does not depend soley on the imposition of the radiation condition, but moreso on the
combination of partitions and the manner in which the fields are expressed. Therefore, we
can avoid the appearance of the singularity in certain components if necessary by careful
formulation, now that we understand the origin of the singularity.
These observations, however, raise a possibility which has not been addressed. Throughout this work we have emphasized the simplicity in the application of boundary conditions
which results from the use of the LSE and LSM modes to expand the fields in various
structures. This practice also avoids the singular term in the Green's function, since it
appears in a component which is not needed for the planar circuits treated here. However,
it is not clear that this is necessarily the most desirable form from the point of view of
convergence for the modal sums. It may be that the explicit extraction of the singularity
may improve the convergence of the remaining modal sums for that direction. If this is
the case, it may be more desirable to retain the singularity in the direction of the current
being modelled, even though enforcing the boundary conditions becomes significantly more
difficult analytically. This question should be considered for future work.
186
APPENDIX C
On the Use of Vector Potentials with the Method of
Scattering Superposition
For a magnetic source Maxwell's equations take the form
V x E= jW H  K (C.1)
Vx H=jwcE (C.2)
where the constitutive relations D = cE and B = /H have been assumed. Since from
Equation (C.2) V * E = 0, E can be expressed in terms of a vector potential function F as
E= V xF (C.3)
Using Equation (C.3) in Equation (C.2) and the usual Lorenz condition [82], F must satisfy
V2F k2F = (C.4)
and consequently,
H = jF+ VV. F (C.5)
For an i directected infinitesimal magnetic current in a homogeneously filled infinite space,
K= i6(R R') (C.6)
so that Equation C.4 becomes
V2 F + k2 F = i(6(X x')6(y y')6(z z')
(C.7)
1S7
The particular solution in this case is knowni to be
F = 6k (C.S)
where R = /(  x')2 + (y  yI)2 + (z  z')2. Sommerfeld has derived an integral form for
this solution [69, Ch. 6] which is
~jkR,00 _ AdA o \
R Jo(Ap)elZZ I = J(Ap)e2u,( zz) (C.9)
R u ~ <
where u = /A2  k2 and k2 = 2 /I,.
We now divide the space by introducing an infinite, perfectly conductillng, ground plane
below the source and a dielectric interface above. This introduces new boundary conditions
which must be satisfied. To accomplish this, we are free to add solutions of the homogeneous
form of Equation (C.5) (i.e., 'source free'). These are taken to be in the forms:
f00 AdA
F+ = r A (A)J,(Ap)e" z < h (C.10)
00 AdA
F = 4  A(A)Jo(Ap)eu" <h (C.ll)
Fr =  o A(A)JO(Ap)eo z z > h (C.12)
47r o Uo
in which AA and A' are to be determined so that the new boundary conditions are satisfied.
These terms can be interpreted as producing plane waves scattered from the ground plane
and interface with components travelling in the ~i directions within the slab. The terms
with superscript 'o' corresponding to fields above the interface. Sommerfeld has shown
that we must also include scattered potential functions having. components in order to
satisfy the boundary conditions1. These are taken as
C [00 A2dA
F+ = — coso A+(A)JI(Ap)euz (C.13)
47r Jo U
~ 00 A 2dA
Ft =  cos AA (A)Ji(Ap)euz (C.14)
4w jo 2d
F  cos 4 A (A)JI(Ap)euA2dA (C.15)
47r 4 o Uo
'This choice is not unique as noted in [25]
188
where u, = \/A2  k,, k2 = 2,L7,eo and o is tlle angle p makes with the i axis. It should
be noted that the functional forms have been chosen in anticipation of derivative operators
which will be applied wlhen the fields are evaluated at the boundaries. The solution in the
slab then is
F = )+ (F+ + F)+ F +F;) (C.16)
while in the infinite space above
F = ~F + ^F~ (C.17)
In view of the expansions of Equations (C.3) and (C.5),
 Z)FZ Y' (OF, 3FI"  F(
iE F )(cOF F OF,
E — F  Dy (C.18)
C C)y a:x az Oay
H = J Fr + V.F) +  V F+i kFz+ V.F] (C.19)
We apply the following boundary conditions at the interfaces:
x E = 0 at z= 0 (C.20)
x E = x E at z = h (C.21)
ix H+ =x 2 II at z = h (C.22)
Here the ~ superscripts imply evaluation of the field on either side of the interface (not to
be confused with the ~ waves used earlier). No additional boundary condition is needed
at the source plane, since the particular solution already satisfies the source conditions and
the scattered potentials are solutions to the homogeneous equation.
This leads to the following system of equations:
A++ A = 0 (C.23)
A+  A = eUZ (C.24)
189
l[A+uh + Aeuh 'A0~Ohk.
ZA~ceu h + ,h AOj,u0 0 u~h~ thz') (C.27)
1 uh 1hC u 1u~'
[A~cu + Acu] + [A euh  Auh = A + Ao6uoh C
u1 110 u
(C.28S)
which when solved produce
4+ 
4 
AO
=
=O
E r110 sinh [u(h  z')] + u cosli[u(h  z)
Cr~ Snh~~h z')] + u cosh[u(h —,Z
[C1oSfh1(fl(Cr11,1o, h) JC
11(Cr )cosh(uz')
euoh2Cr Uo cosh(uz')
 euo h2uo0(fr  1) sinh(uh) cosh(uz')
(C.29)
(C.30)
(C.31 )
(C.32)
(C.33)
where
fl(C,U,,u1o,h) = cr1uo cosh(uh) +usinh(uh)
f2(ulu0,h) = ucosh(uh)+uosinh(uh)
(C.34)
(C(.35)
We then find that
cosh(uz') [,cT 10 sinh~u(h  z')} + u cosh[u(h  z')]]
Fx 00 ~ cosh(uz') [,C~u0 sinh[u(h  z')] + u cosh[u(h  z')]] d
F1  J J(p f ('Cr7, uu0, h) d
Z> z' (C(.36)
Fx Er Jo'I A~joAp) cosh(gz') e  uo,(z h) dA z > h ((.37)
z ('Cr 1)fCcos~ j00 A;jl Ap) sinlh,7) cosh(uhI) dA z < h ((.38)
z (C,  )C'cos 0J0 ~~j,(Ap) sinlh;ucosh(uz2')euo(zh )dA z > h ((.39)
190
These expressions are identical to the results in [37] from which the fields can be found by
applying Equations (C.3) and (C.5). Thle fields correspond to the Zi, yx. z components
of the dyadic Green's functions. For y and z directed currents, the process must be repeated
(although the y terms can be obtained by a coordinate rotation due to symmetry).
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191
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