rflEQBEF
L“- -V _ i, ‘-
. 2 .
uvmzmn.“
. Mum.“ - u

 

3.1%

. V ~
.
hr}.

J»? .I.‘

.2. I
.21.? :us
a .. :2.

.. .4

 

.. ”5‘
91:.
vi...
war...»
.14 z. i

{I

' r

by“ ‘

(.0.

3.8.»??? .
Lu. .r. 2..

Vol nasal-2.1!)

.m , . l>u$¥fllf

 

If)...

L

a“ u I 3 (I15!

.4
£008»

lIBRARY
Michigan State
University

This is to certify that the

dissertation entitled

COUPLING BETWEEN CAVITY-BACKED ANTENNAS
ON AN ELLIPTIC CYLINDER

presented by
Chi -Wei Wu

has been accepted towards fulfillment
of the requirements for

PhD degree in E1ectricai Eng

Major professor

Date [2/03/0/

 

MS U is an Affirmative Action/Equal Opportunity Institution 0.12771

 

 

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJCIRC/Dateoue.p65-p.15

 

 

COUPLING BETWEEN CAVITY-BACKED ANTENNAS ON

AN ELLIPTIC CYLINDER
By

Chi-Wei Wu

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical and Computer Engineering

2001

ABSTRACT

COUPLING BETWEEN CAVITY-BACKED ANTENNAS ON AN
ELLIPTIC CYLINDER

By

Chi-Wei Wu

Radiation by conformal antennas, flush-mounted to surfaces with varying
curvature, is of considerable importance to design engineers. Applications of such
antennas are found in the aerospace, automobile, and watercraft industries. Conformal
antennas are important in these areas due to their relatively low cost, low profile, and
consumer appeal. However, accurate and flexible design methods for such antennas have
not been offered in the literature to date.

In this research, the highly versatile finite element method is combined in a hybrid
formulation with a boundary integral mesh closure scheme to accurately model the fields
within, and in the aperture of, a cavity-backed antenna flush-mounted in a perfectly
conducting infinite elliptic cylinder. For the sake of efficiency, an asymptotically valid
dyadic Green’s function based on the Uniform Theory of Diffraction (UTD) for surface
fields due to a source on a smooth perfectly conducting surface with arbitrary curvature is
used in the boundary integral. This development represents a significant advancement
over prior techniques since surface curvature variation, either across a single element or
across an array of elements, is now accurately included into the antenna model. An

advantage of this approach is the ability to model cavities with curvature varying from

planar to the constant curvature of a circular cylinder. Eigenmodes will be given for
planar-rectangular, circular-rectangular, and elliptic-rectangular cavities recessed in the
cylinder. Furthermore, the input impedance of a conformal cavity-backed patch antenna
will be given. Also, The mutual coupling between microstrip antennas mounted in a

ground plane, a circular and an elliptic cylinder is investigated in this research.

to every member of my family

iv

ACKNOWLEDGEMENTS

In August 2000, I considered quitting my Ph.D. studies when I was informed
that my mother had a serious stroke and had been unconscious for more than one
week after major brain surgery. It would not have been possible for me to
continue this dissertation without the strong support from all my family members,
especially my father, Hwa Jing, who quick from his job after that. I also want to
express my sincere appreciation to my young sister, Soal-Phane, and my two
brothers, Chung-Yi, and Ling-Gate for taking turns caring for mother after normal
working hours. Without their sacrifice and patience, I could not focus on my
studies and complete this research effort while living more than ten thousand
miles away from home. Special thanks go to my loving wife, Kwei-Yin, for
devoting herself to take a good care of my daily life without complaining.

In addition to my family members, those people who guided me academically
have been equally important. I’d like to thank the members of my thesis
committee for their time and insight. Especially, I am grateful to my two advisors,
Dr. Kempel and Dr. Edward Rothwell for their enthusiastic guidance over the
years. I am also grateful to Dr. Dennis N yquist for the valuable advice and
mentoring he has provided. Finally I would like to thanks Dr. Byron Drachman
for his time throughout this study.

Appreciation is extended to Taiwan Department of Defense for providing

financial support in the past four years.

TABLE OF CONTENTS

LIST OF TABLES ...................................................................................... viii
LIST OF FIGURES ...................................................................................... ix
CHAPTER 1
Introduction .......................................................................................... 1
CHAPTER 2
2.1 Introduction ..................................................................................... 8
2.2. FE-BI formulation .......................................................................... 8
2.3. Vector Weight Functions ............................................................. 13
2.4. Finite Element Matrix .................................................................. 17
2.5. Validation of Finite Element Formulation: The Closed
Cavity ........................................................................................ 20
CHAPTER 3
3.1 Introduction .................................................................................. 33
3.2 Eigenfunction expansion method ................................................. 34
3.2.1 Mathieu’s Equation ......................................................... 34
3.2.2 Modified Mathieu’s Equation ......................................... 37
3.2.3 Vector Wave Functions In An Elliptic Cylinder
Coordinate System ......................................................... 39
3.2.4 The Free-Space Dyadic Green’s function ....................... 42
3.2.5 The Electric Dyadic Green’s function Of The First
Kind ................................................................................ 49
3.2.6 The Electric Dyadic Green’s function Of The
Second Kind ................................................................... 51
3.3 The Asymptotic Dyadic Green’s Function ................................... 53
3.4 Boundary Integral Matrix ............................................................. 57
3.5 Excitation: FE-BI .......................................................................... 60
CHAPTER 4
4.1 Introduction ................................................................................... 69
4.2 System Solution ............................................................................ 70
4.3 Input Impedance ............................................................................ 71
4.4 Numerical Results and Discussions .............................................. 74
4.4.1 The Empty Cavity ............................................................ 74

vi

4.4.2 The Slot Antenna ............................................................. 75

4.4.3 The Conformal Patch Antenna ........................................ 77
4.5 Conclusion .................................................................................... 81
CHAPTER 5
5.1 Introduction ................................................................................. 107
5.2 Mutual Coupling ......................................................................... 108
5.3 Numerical Results and Discussions ............................................ 109
5.3.1 Comparisons between FE-BI and Moment Method
for H—Plane Coupling .................................................... 110
5.3.2 Comparisons between FE-BI and Moment Method
for E-Plane Coupling .................................................... 113
5.3.3 Numerical Results and Discussions for H-Plane
Coupling on a Curved Surface ...................................... 114
5.3.4 Numerical Results and Discussions for E-Plane
Coupling on a Curved Surface ....................................... 116
5.3.5 Numerical Results for Different Size of Patch
Antenna .......................................................................... 120
5.4 Conclusion .................................................................................. 122
CHAPTER 6
Conclusion ...................................................................................... 159
BIBLIOGRAPHY ..................................................................................... 164

vii

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 2.6

LIST OF TABLES

The eigenvalues for a rectangular cavity ........................ 23
The eigenvalues for a circular shell cavity ..................... 24
The eigenvalues for a circular shell cavity ..................... 25
The eigenvalues for an elliptic shell cavity .................... 26
Comparison of eigenvalues between a cavity ................ 27

Comparison of eigenvalues between a cavity ................ 28

viii

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 3.1

Figure 3.2
Figure 3.3
Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 4.1

Figure 4.2

Figure 4.3

LIST OF FIGURES

A cross-sectional view of the elliptic coordinate system .......................... 29

The geometry of the cavity-backed antenna embedded in

a metallic elliptic cylinder ......................................................................... 30
Geometry of an elliptic shell element ....................................................... 31
Geometry of a cylindrical shell element ................................................... 32

Contour for converting the eigenfunction expansion into a mode

expansion ................................................................................................... 62
The scattering wave and incident wave for an elliptic cylinder ................ 63
Illustration of unit vectors for a convex surface ........................................ 64
Geodesic path for the creeping wave on an elliptic cylinder .................... 65

Magnitude of the three components of asymptotic dyadic
Green’s function for an elliptic cylinder ................................................... 66

Magnitude of the three components of asymptotic dyadic Green’s
Function for a circular cylinder ................................................................. 67

Cavity-backed probe-fed conformal patch antenna recessed in an infinite,

perfectly conducting elliptic cylinder ....................................................... 68
The geometry of model of source generator ............................................. 80
An empty cavity: 1.5 cm x 6.0 cm x 3.75 cm and its unit cell .................. 81
Input impedance for an empty cavity mounted in the ground ................... 82

ix

 

 

n,

Fig

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Figure 4.18

Figure 4.19

Input impedance for an empty cavity mounted in two circular

cylinders with different radii ..................................................................... 83
Slot antenna: 1.5 cm x 6.0 cm x 3.75 cm and its unit cell ......................... 84
Input impedance for slot antenna embedded in a ground plane ................ 85
Input resistance for slot antenna on cylinders ........................................... 86
Input reactance for slot antenna on cylinders ........................................... 87
Input resistance for slot antenna on cylinder and elliptic cylinder ............ 88
Input reactance for slot antenna on cylinder and elliptic cylinder ........... 91
Fields configurations (modes) for rectangular microstrip patch ............... 92
Input impedance for rectangular patch antenna in a ground plane ........... 93
Input impedance for rectangular patch antenna in a ground plane ........... 94
Input impedance for rectangular patch antenna in a ground plane ........... 95
Input impedance for rectangular patch antenna in a ground plane ........... 96
Patch antenna: 1.5 cm x 6.0 cm x 3.75 cm ................................................ 97
Input impedance for a patch antenna in a ground plane ............................ 98
Resistance for a patch antenna on cylinders ............................................. 99
Reactance for a patch antenna on cylinders ........................................... 100

Figure 4.20

Figure 4.21

Figure 4.22

Figure 4.23

Figure 4.24

Figure 4.25

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Resistance for a patch antenna on cylinders with azimuthal polarization
..................................................................................................... 101

Reactance for a patch antenna on cylinders with azimuthal polarization
..................................................................................................... 102

Resistance for a patch antenna on cylinders with azimuthal polarization

..................................................................................................... 103
Reactance for a patch antenna on cylinders with azimuthal polarization
..................................................................................................... 104
Resistance for a patch antenna on eliptic cylinders with azimuthal
polarization .............................................................................................. 105
Reactance for a patch antenna on elliptic cylinders with azimuthal
polarization .............................................................................................. 106
Experimental arrangement for measurement of microstrip
antenna S-parameter Geometry of an elliptic shell element ................... 125
Measured and calculated mutual coupling between two coax-fed
microstrip antennas ................................................................................. 126
Geometry for patch antennas with H-plane coupling in pseudo-
ground plane ............................................................................................ 127
Mutual coupling for H-plane case ........................................................... 128
Comparison of mutual coupling calculated by FE-BI
with a moment method solution and data by measurements ................... 129
Comparison of mutual coupling using FE-BI method between
different size of cavity ............................................................................. 130

xi

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Geometry for patch antennas with H-plane coupling in pseudo-
ground plane ........................................................................................... 131

Comparison of mutual coupling calculated by FE—BI with a moment
method solution and data by measurements ............................................ 132

Comparison of mutual coupling calculated by FE-BI with a moment
method solution and data by measurements ........................................... 133

Geometry for patch antennas with E-plane coupling in a pseudo-
plane ground ............................................................................................ 134

Comparison of mutual coupling calculated by FE-BI with a moment
method solution and data by measurements ........................................... 135

Comparison of mutual coupling calculated by FE-BI with a moment
method solution and data by measurements ........................................... 136

Geometry for patch antennas mounted in curved surface with H-plane

Coupling .................................................................................................. 137
Mutual resistance for patch antennas with H-plane coupling ................. 138
Mutual reactance for patch antennas with H-plane coupling .................. 139
Mutual Coupling for patch antennas with H-plane coupling .................. 140

Mutual resistance for patch antennas mounted in an elliptic cylinder
with H-plane coupling ............................................................................. 141

Mutual reactance for patch antennas mounted in an elliptic cylinder
with H-plane coupling ............................................................................. 142

Mutual coupling for patch antennas mounted in an elliptic cylinder
with H-plane coupling ............................................................................. 143

xii

Figure 5.20

Figure 5.21

Figure 5.22

Figure 5.23

Figure 5.24

Figure 5.25

Figure 5.26

Figure 5.27

Figure 5.28

Figure 5.29

Figure 5.30

Figure 5.31

Figure 5.32

Figure 5.33

Figure 5.34

Geometry for patch antennas mounted in curved surface with E-plane

coupling ................................................................................................... 144
Mutual resistance for patch antennas with E-plane coupling .................. 145
Mutual reactance for patch antennas with E-plane coupling .................. 146

Mutual Coupling for patch antennas with E-plane coupling ................... 147

Mutual resistance for patch antennas mounted in an elliptic cylinder
with E-plane coupling ............................................................................. 148

Mutual reactance for patch antennas mounted in an elliptic cylinder
with E-plane coupling ............................................................................. 149

Mutual coupling for patch antennas mounted in an elliptic cylinder
with E-plane coupling ............................................................................. 150

Geometry for patch antennas mounted in curved surface
with special E-plane coupling ................................................................. 151

Mutual resistance for patch antennas with special E-plane coupling ...... 152
Mutual reactance for patch antennas with special E-plane coupling ...... 153

Mutual Coupling for patch antennas with special E-plane coupling ...... 154

Geometry for patch antennas with different patch size ........................... 155
Mutual Coupling between antennas ....................................................... '.' 156
Mutual Coupling between antennas ........................................................ 157
Mutual Coupling between antennas ........................................................ 158

xiii

 

CHAPTER 1

INTRODUCTION

Conformal antennas are increasingly being mounted on the surfaces of air vehicles
primarily due to their low volume consumption, low drag, and low cost array properties.
An antenna that has received considerable attention in the literature is the microstrip
patch. This antenna consists of a radiating metallic patch printed on a grounded dielectric
substrate. Typically these antennas are designed using analysis methods developed for
planar apertures. Often such an approach is sufficient for design purposes; however, there
are significant applications where explicit inclusion of surface curvature is necessary. For
example, a characteristic phenomenon of patch antennas conformal to curved platforms is
the dependence of resonant input impedance on surface curvature [1]. During the
previous development of these antennas, due to a lack of rigorous analysis techniques,
antenna designers have had to resort to expensive measurements in order to develop a
conformal array design. This process is very time-consuming since any change in the
antenna geometry will necessitate re-measurement, especially at the resonant frequency,
of the input impedance and mutual coupling properties of the antenna. Due to the narrow
bandwidth of patch antennas, it is important to include variations in the input impedance
attributed to curvature so that the number of prototypes required during the design cycle
can be minimized.

Various theoretical techniques have been employed in the past for the analysis and
design of conformal antennas such as the cavity model [2], integral equation based
methods [3-5], and mode-matching techniques [6]. Many of these techniques were

originally developed for planar surfaces; however, they have also been extended to

Bur
BI I

Cyli

Cyli
unlit
SUll;
al’aj
bou;

C0m

incorporate surface curvature [2]. Each of these methods has advantages and
disadvantages.

The cavity model is computationally inexpensive and offers considerable insight into
the behavior of the antenna. However, it is not amenable to large array simulation since it
ignores mutual interactions amongst array elements. The integral equation-based methods
offer high accuracy through the rigorous inclusion of mutual coupling effects. However,

these are not particularly efficient due to the fully-populated matrix associated with the

pf-n—Ia-flr '

formulation. Highly efficient methods can be developed, particularly for cavity-backed
antennas; however, in doing so the antenna element shape and cavity shape are typically
limited [7].

The finite element-boundary integral (FE-BI) method is successfully employed for
the analysis of large planar arrays of arbitrary composition [8], and this approach has
been extended for aperture antennas conformal to a circular cylinder metallic surface [9].
Both the radiation and scattering problems have been developed in the context of the FE-
BI method. In contrast to the planar aperture array, the implementation of the
cylindrically conformal array requires cylindrical shell elements rather than bricks, and
the required external Green’s function is that of the circular perfectly conducting
cylinder. In its exact form, this Green’s function is an infinite series that imposes
unacceptable computational burden on the method. However, for large radius cylinders,
suitable asymptotic formulas developed from Uniform Theory of Diffraction (UTD) are
available and used for an efficient evaluation of the Green’s function. The finite element-
boundary integral (FE-BI) method provides an alternative approach to modeling

conformal antennas for both planar [8] and curved platforms [9-10].

.5.‘ .

ck
the
af
ca;
dc
ape
oni
chi
ont

ofa

Cont
CUnq
I650“
eStab
IECKH
“sou

aCati

the en

The FE-BI approach is a hybrid method combining the finite element method with a
boundary integral. The finite element method is used to model the volumetric total
electric fields in the cavity as well as the tangential electric fields in the aperture. With
the FE-BI method, the constitutive material parameters are assumed to be constant within
a finite element but are allowed to vary across elements; consequently, this method is
capable of modeling cavity-backed antennas with inhomogeneous loading. Hence, a finite '
element based model is capable of being used to design both geometrically complex L
apertures and apertures with complex material loading. However, as with all second-
order partial differential equation based representations of the wave equation, the finite
element method requires specification of both the tangential electric and magnetic fields
on the boundary of the computational volume. This is accomplished via the introduction
of a boundary integral that includes a dyadic Green’s function to describe the coupling
amongst various portions of the aperture.

In this research, the FE-BI method is extended to model cavity-backed antennas
conformal to a perfectly conducting elliptic cylinder that has a surface with varying
curvature along one principal plane. This hybrid FE-BI method will be used to model the
resonant behavior of cavities recessed in an elliptic cylinder and its validity will be
established by reduction to known results for planar-rectangular and cylindrical-
rectangular cavities. In addition, new results will be presented for the resonance
associated with an elliptic-rectangular cavity and for the input impedance associated with
a cavity-backed patch antenna flush-mounted on an elliptic cylinder.

Vector wave equations in an elliptic cylinder coordinate system are generated when

the elliptic cylinder scalar wave functions are used. Once the orthogonal properties of

they

dyac

equ:
the:
p055
usec
func
Wu]
forn
clerr
clcn

sth

CXpa

aIJJs

\Vhli

Perfo

these functions are known, we can find the eigenfunction expansion of the free-space
dyadic Green function [1 1].

In Chapter 2, the FE-BI formulation will be introduced first using the vector wave
equation in elliptic cylinder coordinates and be extended to cavities that are embedded on
the surface of a metallic elliptic cylinder of infinite extent. Since the elliptic shell element
possess both geometrical fidelity and simplicity for the elliptic-rectangular cavity, it is
used to mesh the elliptic cavity-backed conformal antenna volume. New vector weight
functions for the each edge of the elliptic shell element are presented in this dissertation.
With these vector weight functions, the FE-BI can be written as a matrix equation and the
formulations for each matrix entry are shown in Chapter 2. For validation of the finite
element formulation, a comparison between computed eigenvalues using the finite
element method and analytical values for a closed rectangular cavity and a closed circular
shell cavity is made.

In Chapter 3, the free space dyadic Green function in terms of eigenfunction
expansion is developed. The angular functions, or Mathieu functions, are represented by
a cosine series in the case of even functions and a sine series in the case of odd functions
while the radial functions, or modified Mathieu functions, are expressed in the form of a
series of Bessel functions. Each dyadic component of the dyadic Green function has been
successfully developed in terms of Mathieu functions or modified Mathieu functions in
this chapter; however, the convergence performance of the modified Mathieu functions is
very slow. Hence an asymptotic dyadic Green function that has a good convergence

performance is developed and used.

Hm.‘
I .

' .i'.fi‘ fl-laP- ‘

Theo
Gree
same
elect
the s
Gree
form
form
funcr
com;
pSCUc

ll

Base on the development of an approximate asymptotic solution using the Uniform
Theory of Diffraction (UTD) for surface fields by Pathak and Wang [12], the dyadic
Green function is derived for both a source point and an observation point located on the
same surface of the elliptic cylinder, and thus an approximate asymptotic solution for the
electromagnetic fields that are induced by an infinitesimal magnetic current moment on
the same elliptic surface is generated. In this approach the contributions to the dyadic

Green function for the short path and long path are developed. The reduction of this

I“. .‘
F "‘1'

formulation for the special case of a circular cylinder will be shown to have the same
form as previous results [9]. The numerical results for the magnitude of the dyadic Green
function with respect to the geodesic path will be discussed in Chapter 3. The numerical
comparison will be demonstrated there for a wave traveling on an elliptic cylinder and a
pseudo-circular cylinder.

In Chapter 4, the calculation model for the input impedance of a cavity-backed,
printed antenna is introduced. The input impedance and resonant frequency for an empty
cavity, a slot antenna and a conformal patch antenna embedded on a ground plane are
presented as well as antennas recessed in an elliptic cylinder and a circular cylinder.
From those numerical results, the relationship between the input impedance of different
antennas and the local surface curvature in the vicinity of the antenna mount is found and
discussed. Also, the probe feed for the patch antennas will be located in different
locations to observe effects of the surface curvature with different excited modes.

In Chapter 4, the computation results using FE-BI for antennas mounted in a ground

plane is verified with planar FE-BI results [10] by setting the radius of curvature to be

large. Also, for the empty cavity mounted in a ground plane, the calculated resonant
frequency will be compared with the theoretical value.

In Chapter 5, the mutual coupling between microstrip antennas mounted in a ground
plane, and in a circular and an elliptic cylinder is investigated. A moment method
solution to the microstrip antenna problem was proposed [13] in 1981 and the mutual

coupling between patch antennas embedded on an infinite coated ground plane was

“T” if“ '1 “'

calculated and measured in [14] and [15], respectively. The numerical results using the

FE-BI method are compared to these moment method results.
The mutual coupling between patch antennas embedded in different circular cylinders
with different radii are calculated. The mutual resistance, reactance and coupling

coefficient, $12 , are graphed with respect to frequency to assess the effects of curvature

on coupling. Also the same antenna is mounted in the different portions of the elliptic
cylinder and the computed results are discussed. The field structure with the cavity is
mainly dependent on the position of the probe feed this affects the mutual coupling.
Therefore, the probe feed is relocated and numerical results for coupling are inspected to
determine the influences of the location of the probe feed.

In addition to curvature, the position of the probe feed, the size of the patches and the
separation between the two rectangular patches play an important role in mutual
coupling. In Chapter 5, the different patch sizes are used to analyze mutual coupling.
Also, numerical results are computed for the antenna mounted in circular cylinders with

different radii. For convenience, symmetric patches are used for computation.

1h:

In Chapter 5, a two-port network model is used to determine mutual coupling. Also,
the coupling parameter, S12, is determined from the input impedance and mutual

impedance using conversion between two-port network parameters.

TT-‘T‘f'iV

 

21.11

math:
hnegr
lineh
there
0fek3
contn
Fourh
louro
[0 car
that u

may h

CHAPTER 2
FINITE ELEMENT—BOUNDARY INTEGRAL METHOD

2.1 Introduction

The Finite Element (FE) Method is a computational technique that has been used in
mathematical physics since the 1940’s. It was first coupled with an exact Boundary
Integral (BI) termination condition in an electromagnetics application by Silvester and
Hsieh [l6] and McDonal and Wexler [17] in 1971 and 1972, respectively. In the 1990’s,
there has been renewed interest in the Finite Element-Boundary Integral (FE-BI) method
of electromagnetics principally due to the work of J in and Volakis [18-20]. Their major
contribution was coupling the FE-BI approach with the Biconjugate Gradient-Fast
Fourier Transform (BiCG-FFT) technique, thus allowing high fidelity simulations with
low 0(N) memory and computational demand. In 1994, the FE-BI method was extended
to cavities that are recessed in a metallic circular cylinder of infinite extent [1] and [9], In
that work, the boundary integral utilized uniform zoning and hence the Bi-CG-FFT solver
may be employed to retain low memory and computational burden.

In this chapter the FE-BI method will be extended to cavities that are embedded on
the surface of a metallic elliptic cylinder of infinite extent. The derivation of the FE
method starting with the vector wave equation will be presented first, followed by the
introduction of the boundary integral in the next Chapter.

2.2. FE-BI formulation
Before the scalar wave equation in elliptic cylinder coordinates is discussed, the

parameters of elliptic cylinder coordinate system are described. As shown in Figure 2.1,

ther

anig

“hen

u.)
L.)
~€

theca

dhf

(”a
-—1
P.

manhr

“here

bOUndc
“fight,
Valued

FIBn b

the relations between the rectangular coordinate and elliptic cylinder coordinate systems
are given by

x=ccoshucosv

y=csinhusinv for OSvS2fl', OSu<oo (21)
z = z .
I
where 2c is the distance between the foci of the ellipse. 'F
1:.
Consider a cavity recessed in an infinite metallic elliptic cylinder, shown in Figure i
ii
ii.

2.2. The cavity walls are assumed to coincide with constant u-, v-, and z-surfaces and

the cavity is filled with an inhomogeneous, isotropic material.

The FE-BI formulation is developed by considering the coupled first-order partial
differential equations, the time harmonic Maxwell’s Equations, and performing some
manipulations, to give the vector wave equation,

VXVXE—k2E=-jw/lJ—VXM (2.2)
VxVxH—kZH = -ja)£M +VxJ
where J and M represent, respectively, the electric and magnetic source current.
A unique solution of the wave equation requires the specification of sufficient
boundary conditions. To generate a system of equations from (2.2), the method of
weighted residuals is applied, which results in the symmetric inner product of a vector-

valued weight function and the vector wave equation. The integro-differential equation is

given by

“here
It IS I
source

this in:

the rim:

VxE(u,v, z)

 

J‘V.[Wi(u,v,z).Vx( )—kgwi(u,v,z).grE] dV =

r

M"(u,v.z>

-IV. w,(u,v, z) -Vx[ ]dV (2.3)

r

_jk°Z°.lvi Wi(u,v, z) -Ji(u,v, 2) (N
where W,- (u, v, z) is a subdomain vector-valued weight function to be defined later and
V,- is the ith volume element resulting from a discretization of the cavity. The impressed
sources (J i , Mi) enclosed by the volume V, give rise to the right-hand side of (2.3) and
this interior excitation function is defined by

- Mi(u v 2)
Int __ _ . r 9
f,- _ IV.- w,(u,v,z) Vx[————#r

]dV — jkonIVi W,(u,v, z) -J" (u, v, z)dV (2.4)

Upon application of the first vector Green’s theorem [11],
[V w -Vx(VxE)dV =[V (VxE)-(VxW)dV -[S Wx(VxE) ~ndS (2.5)
the time Harmonic Faraday’s Law for a source free region and the vector identity,
VxE = -ja)uH (2.6)

A'BXC=C-AXB (2.7)

(2.3) can be recognized as the weak form of the wave equation

 

J VxE(u,v, z) -Vx Wi(u,v, z)
V' flr
-—jkoZoI_,. n(u,v, z)xH<u,v.z) . W.(u,v. z)ds = f?”
l

dV 43]“ £,E - w,(u,v, z)dV —
' (2.8)

where n(u,v, 2) indicates the outward pointing normal of the element surface associated

with the ith unknown, Si is the surface area of that element, and H(u,v, z) is the total

magnetic field, evaluated at the test point denoted by the triplet (u,v,z). It can be shown

10

that I
apen
field

the c

ZCl’O

field
equi
kind

that

andS

that the surface integral of (2.8) vanishes for all elements that do not border the cavity

aperture due to the fact that the test function, W,- (u, v, z) , is in fact a tangential electric

field. For this formulation, all metal is assumed to be a perfect electrical conductor and
the only portion of the cavity boundary that is not metal is the aperture. Hence, the non-
zero contribution is limited to the portion of the surface that coincides with the aperture.
The weak form of the wave equation, (2.8), contains unknown electric and magnetic
fields on the surface of the elliptic cylinder. To determine H in (2.8), the surface
equivalence theorem will be introduced with a dyadic Green’s function of the second
kind (e.g. a dyadic Green’s function that satisfies the Neumann boundary condition) such

that the radiated fields due to the equivalent currents can be reduced from [21].
H(r) = [Lam-mamas + jkY ILandS (2.9)
to

H(r) = jkY [Luis-32mm (2.10)

where G, Gez represent the dyadic Green’s function and its second kind, R=| r -—r'|, r

denotes the observation point and r' is the integration point, while the surface on which
the equivalence theorem is applied is denoted by S. Also, Y is the wave admittance in the
medium.

By expanding the unknown electric field in terms of subdomain basis functions as

Nv
is: EjWJ-(u,v,z) (2.11)
j:

u

and substituting (2.11) and (2.10) into (2.8), the resulting system of equations is obtained

11

‘

rum-1- a: '

In ('1
exp;

func

equa

1*her

The r

 

Nv VXW-(u,V,Z)‘wai(u9V,Z)
2 E; { LI 1
j=1 1 fl,
—k§e,wj(u.v.z)-W.-(u.v.z) IdV (2.12)

+k§5,(i)5a( j) I [ w,(u,v.z)-u(u.v.z)><
31' 5i

[n(u9V, Z)ij(u,v, z) 'Ee2(V, Z;V', Zr)]deSI }: flint

In (2.11), the subscript indicates the j'h unknown and W J- (u,v, z) is the same edge-based
expansion function as that used for testing in (2.3); e.g. Galerkin’s procedure is used. The
function 6,, (1)50 (j) is a product of two Kronecker functions and simply indicates that
the boundary integral only contributes when both the test and source unknowns are on the
aperture. The symbol N, in (2.11) denotes the total number of unknowns or the free-

edges space in the mesh.
Expressed in elliptic cylinder coordinates, (2.12) becomes the system of linear

equations given by

 

N VxW.(u,v,z)-V><W,-(U.V.z)
S: Ejl LI 1
=1 ' ”I
—k§£,Wj(u,v.z)-W.-(u,v.2) 1.5szde (2.13)

+k§60(i)6’a(j)j Iw,(u',v',z')-u(v,z;v',z')x
Sj S,‘

J'

[n(u’v’Z)ij(“vV,Z)“31:201.z;V',z')],8',de'dz'dvdz }=fiint

where
fl(u,v, z) = c(cosh2 u -cos2 v)”2 (2 l4)
fl'(u ',v', z ') = C(cosh2 u '— cos2 v ')”2 .
The FE-BI formulation can be written as a matrix equation
[A]{E}-k§[B]{E}+k§[C]{E}={f}. (2.15)

12

Hue

then

 

Aho

where
(1):" _
1,,1 _[Vi[wa,. -waj]dv (216)
(2).. — .
1,, '1 _[Ve[w,. .wj]dv
Also
CIj = ISA‘. [WI '“XISAj UXWJ' 'GeZdS '] d5 (2.17)

where i and j represent the global test and source unknown numbers, respectively.

Also, V,- denotes the element volume associated with the test function 1', while
S At ,SAJ. represent the surface that coincides with the aperture, of the i'h , j'h element,

respectively. The FEM matrix was formed by the matrix sum of [A] and [B], while the
boundary integral matrix was formed by the sub-matrix [C]. The entries in the FEM
matrix are identically zero unless both the test and source edges are within the same
element and hence the FEM matrix is sparely populated. However, the boundary integral
matrix has entries that are nonzero even if the source and test edges are located in

different elements. Hence, the boundary integral sub-matrix, [C], is fully populated.

2.3. Vector Weight Functions

An important factor in choosing the finite elements for gridding the cavity is the
elements’ suitability for satisfying both the mathematical requirements of the formulation
and the physical features of the antenna system. Traditional node-based finite elements

associate the degrees of freedom with the nodal fields. However, these functions have

13

pIO‘
not

gent
viflt

shor

Stmc:

fbrc:

eHipt

elliptj
eIghti
onhfig

$800”.

proven unsatisfactory for three-dimensional electromagnetics applications since they do
not correctly represent the null space of the curl operator and hence spurious modes are
generated [22-23]. Edge-based finite elements, whose degrees of freedom are associated
with the edges of the finite element mesh, have been shown to be free of the above
shortcomings. In the source-free solution region, edge-based finite elements can be
designed to satisfy the divergence-free condition such that the spurious mode can be
removed from the solutions. In addition, edge-based elements avoid explicit specification
of the fields at comers where edge conditions may require a singularity. J in and Volakis
[20] presented edge-based brick elements that are convenient for rectangular-type
structures and cavities. Later, Kempel and Volakis [9] designed the cylinder shell element
for cavities embedded in a circular cylinder. For cavities recessed in an elliptic cylinder,
elliptical shell elements are the natural choice.

Elliptical shell finite elements possess both geometrical fidelity and simplicity for
elliptical-rectangular cavities. Figure 2.3 illustrates a typical shell element, which has
eight nodes connected by twelve edges: four edges aligned along each of the three
orthogonal directions of the elliptic cylinder coordinate system. Each element is

associated with twelve vector shape functions given by

14

where

_ Av] (pa - pub )(Z — Zr)

 

 

 

 

 

 

 

 

 

 

 

 

W14 —v Am =Wv(pu,vez:pubnzu+1)
W23 =—v Av,(Pu *Zsz-zfl : Wv(Pu,V’Z§Puaa3 21,4)
W53 =—v Av[(pu ’21:)(2-210 : W.(p..,v,z:pu,,n zt,_1)
W67 : v Av[(pu _:7u;*)(z_2b)=Wv(PuaV,Z§Pua,',Zbe+1)
W12 =“Aub(v-A:irir(z—ZI) =Wu(Pu,V’Z§',VI,Zn+1)
W43 :_"Aub(v-A:i(z—Z1)= .(p...v.z;-,vnze-1)
W56 ‘1‘“ Aub(v_A:)}iz—Zb) = wu(Pu,VeZth,Zb’-1)
“'78 =“ Aub(V"AV;::Z'Zb) = wu(pu’vrz;°’vr’zb’+l)
W15 :1 (Pu ‘P::r)(v-v1) = WJPmV’ZWubWIV’H)
w26 =—z (Pu ‘P::&’-vz) : Wz(pu,V,Z§Pua’Vl"’—1) (2.18)
W48 : _z (Pu ‘Pr;::(v—vr) = Wz(Puev,Z§Pub,Vrn—1)
W37 _ 100.. *PgtgaXv—vr) : WJPWV’ZmuarvrnH)
p. =%e“
azw—w
h=z, -zb
t = 1)., -p.., =-:-(e“b —e“a) (2.19)

0
A = (cosh2 it —cos2 v)1/‘

Av, = (cosh2 u —cos2 v1)”2

Aub = (cosh2 ub - cos2 v)”2

15

Alsc

each

finit

IIhe

isch

Cirt‘t

R...

Also, Wij is associated with the edge defined by local nodes i and j as shown in Figure

2.3. As seen from (2.18), three fundamental vector weight functions, one associated with
each coordinate axis, are required for the complete representation of the elliptical shell

finite element. They are

§Aub (V- P)(Z — Z)

 

Wu(pu.v,z;-,\7.Z,§)=u

 

 

Aah
5A 00 -P-)(z—Z)
° . ~ " = V] u u
wv(Pu,V,2,pg,,z,s) v Ath (2.20)
Wz(pu.v,z;p,;,i7,-,§)=zs(&¢‘ngV-V)

where the element parameters (ua ,ub, v,,v, , 2,, z, ) are shown Figure 2.3. Each local edge
is distinguished by the parameters 5,1217 and 2 as given in (2.20). The circular cylinder is

a special case of the elliptic cylinder, thus the elliptic cylinder can be reduced to the

circular cylinder and then

¢=v
pie“
2

Nl—e

~
~

1‘
A = (cosh2 u — 0052 v) % (2-21)

t=pub —pua zpb—pa

a=vr—vl z¢r—¢l

Rewriting (2.20) with (2.21), the vector weight function for the circular cylinder [10] can

be written as

16

The

2.4.

Shor
the 3
Sch
in)”
are 5

used

w12(p9¢9 Z) = wp(p9¢9 zi'9¢r9z19+1)9 w43(p9¢9 Z) = wp(p9¢9 zi'9¢[9z(9-1)
w56(p9¢9 Z) = wp(p9¢9 zi9¢r9zb9-1)9 w87(p9¢9z) = wp(p9¢9 Zi9¢l9zb9+l)
wl4(p9¢9 Z) = w¢(p9¢9Z;pr Zt9+l)9 W23(p9¢9 Z) = w¢(p9¢9 Zipa9'9 219-1)

 

 

 

(2.22)
w58(p9¢9z)=w¢(p9¢9z9pb99Zb9-1)9 W67(p9¢9z)=w¢(p9¢9z9pa992b9+1)
w15(p9¢9z)=wz(p9¢szipb9¢r9'9+l)9 W26(p9¢9Z)=Wz(p9¢,Z;par¢,-r‘9’l)
w48(p9¢9z)=wz(p9¢9zipb9¢l9'9—l)9 w37(p9¢9z)=wz(p9¢9zipa9¢l9'9+1)

The three fundamental vector weight functions are
- ~ - - 5p (¢-é>')(z-2)
Wp(p,¢,z;,0, ,z,s)=p b
ahp
- ~ .. - E —" z—"
WZ(p9¢9Z;fi2 ~9295):ZS(p—f;(¢—¢)

Where the element parameters (pa, pb,¢,,¢1 , 2),, 2,) are shown in Figure 2.4. It is noted

that as the radius of the cylinder becomes larger, the curvature of these elements
decreases, resulting in weight functions that are functionally similar to the bricks

presented by J in and Volakis [20].

2.4. Finite Element Matrix

The volumetric mesh of the cavity is formed using the elliptic shell element
shown in Figure 2.3 by meshing the cavity such that all the radius-dividing layers have
the same foci as the surface ellipse, while the meshing along the z and v direction is
subdivided uniformly by the fixed length of the geodesic path, which can be obtained
from the perimeter divided by the total number of nodes in that direction. These elements
are shell elements conformal to the surface of the elliptic cylinder. The exterior functions

used to expand the aperture fields are similar to the volume elements. Applying the

17

re

for

for

fun

I“

111!

[Ill

U1

('1'.

I'll

ll

Ill)

I

vector weight function defined by (2.18)-(2.20) in (2.13), the FE-BI formulation can be
written as a matrix equation as shown in (2.15). The FEM matrix is composed by adding
the matrices [A] and [B] while the boundary integral matrix is formed by the sub-matrix

[C]. Carrying out the required vector operations and organizing each integral in separable

form, as shown in (2.16), six combinations for 13".]. and three combinations

for I 3"? remain. The two auxiliary functions are defined in (2.16). These elliptic auxiliary

functions are

 

 

1(1)=E—E:£1Agczs'sb(v-v)(v—vi)dudv+:2lifzf:lub dudv f”(z—zj)(z-z,-)dz

 

 

 

r A2
(1) b azAhVI
[W =— %-fal [(VA1_2— v )sinv cosv pu A
r “b
u- 17-)
+(v—i7j)sinvcosvcoshusinhu-(—p——€‘—
u AV
[2 I
(p -p-.)A
+AubAwpu + u u' u” coshusinvsinhu]dvdu
VI
c zasijfs'
1515 —h—’I::’A we. -p..,.)(p.. -p..,.)du
b l 1 .
{$721.05 f—[Awpu +pu(pu—pgi)coshusmhu
(p, -p,;. )(pu -p,; . )cosh2 usinh2 u
+pu(pu—pa.)coshusinhu+ ' 2’
J Av]
“.JZ’Rz—zsz—zgdz
bA

Iél’-- — if“; .,(p.— p. ><p.— p. Ma

1’2
"--§
I$’= 35-; f:filpftv—fijxv—fiwpf-p..(p.-.-,.+prj)+pr,prjldudv
(2.24)

18

and

cylir
mm
CerL

CIICL

1(2)_ 62331 WEFAZ b(v— v)—(v v)dvdu Eb (z— z)(z— z,)dz

1(2)_c:621:2:§_-,-23 EMA (p,- ,0.)(.0e p“. )du Eb (z- z- )(z_ Z)dz (2.25)

2]”,
C 115).)" b I .. ..
1:22) $76221; A20,“ -Pr,-)(Pu _p,j)du. l: (v—v,)(v—vj)dv

It should be noted that 1;? = 15:) , and therefore the FEM matrix is symmetric,

and only the upper or lower triangle of the FEM matrix needs to be stored. The
cylindrical auxiliary function, which was developed from the FEM method for the
circular cylinder [24], also can be obtained by reducing the elliptical auxiliary function in
circular cylinder coordinates with the relations in (2.21). The auxiliary functions for the

circular cylinder system are given by

”I”. _, ~ 2
IS} =3f—’,[pih1n(p—:) g (¢—¢.)t¢—¢. )d¢+%<p—’;-1)J:(z—z.)<z-z,-)dz]

I$;=-S;'—— —2’—h§[2pbln(p:)+p,(l- p:)]fb(z— 2,)(2- 2,)dz

1;}; =-—"%’1’- J: (¢-i>'.)(¢—4)d¢

11% “5’1(—(pZ-p.)+—(p.+p.)(p3- p3)+—;p.p.(p§-p.))
+(2(p§- -p3)— 2t<p.+p.)+p.p.1n(::— —))j:b (z- 2,)(2— z.)dz]

I“; = —-’—§—’- 1”” (P-firXP-fifldp

12571—5’[a(-§(pE-p3)-rtp.+p.)+p.p.1n(/’:—:))

_ 2.. 2 ...~ ..~ 2.26
+200. 10.)]: «4 ¢.)(¢ ¢.)d¢] ( >

19

“her

to
('19

Idem
OWL
cam
tom,
[Etta-

IOI a

 

12% g’p” n(— :f) (¢- ¢)(¢— rude]; (z- zxz— 2,)dz

15);):5; —’:—'2a[—(pb_ _pa)+- ;(ps+pr)(pa_ pb)+_ zpspt(pb2- -pa)]

XE) (z-Z,)(z-Z,)dz

(2)_55-h1
Izz— tzh ———-I— (.053- p2)+— ;(p.+i5.)(pf;’— pB)+— :p.p.(pb- -p..)] (2.27)

x134 —¢.)(¢ —¢.)d¢

where each of the unevaluated integrals is of the form

if“; —E,)(§-£,)d§ =%(L2 -U2)(5. +5.)+-;-(U’ —L’)+£,E,(U—L) (2.28)

2.5. Validation of Finite Element Formulation: The Closed Cavity

Finite elements for closed domains can be used for analyzing cavity resonances.
Identification of these resonances is important for understanding and controlling the
operation of many devices including microwave ovens. The eigenvalues for each empty
cavity are computed by solving the generalized eigenvalue problem. The eigenvalues
computed by the finite element method, as well as analytical results for the cylindrical-
rectangular and planar-rectangular cavities, will be discussed in this section. A new result
for an elliptical—rectangle cavity will be presented.

The rectangular cavity is a geometrically simple structure, but is widely used in
complex microwave devices. A comparison between computed eigenvalues using the
finite element method and analytical values is shown in Table 2.1. For a 2cm x 1cm x

1cm rectangular cavity, the average error percentage is less than 1.0% with 520

20

cavn§

r J
[\J
a"!

such 1

unknt

wnh

 

result

deicrr

an CH
Comp
Vafku
ante n
finhr
sash.
P11}.S
gleai

\NhE'

unknowns. Particularly, for the first two most important modes, the numerical results are
very accurate.
In Table 2.2, the total unknowns for a 3cm x 3cm x 3cm quasi-circular shell

cavity mounted on a cylinder of p = 20 cm are 450 and the average error percentage is

2.2%. To have more accurate results, the cavity was subdivided into finer finite elements
such that the variation of electrical field inside the cavity can be represented by more
unknown edges (e.g. degrees of freedom.) In Table 2.3, the error was improved to 1.3%
with 1176 unknowns. It can be concluded that the eigenvalues can be computed with
good accuracy and the accuracy is expected to increase with higher mesh density.
However, the computational cost will rise and several negative trivial eigenvalues may
result. Significantly, these results illustrate the fact that the FE method can be used to
determine the resonance frequency of arbitrary shaped cavities.

The eigenvalues of an elliptic shell cavity mounted in three different locations of
an elliptic cylinder with a major axis of a=50cm and a minor axis of b=20cm were
computed and tabulated in Table 2.4. The eigenvalue for the lowest mode, TE01 1, has less
variation compared to other modes in those cases as one would expect. For the conformal
antenna mounted in the elliptic cylinder starting from the elliptic angle of v=0.02 by
setting the value of the initial angular parameter v0 to 0.02 in computation, it also can be
easily observed that the eigenvalues have larger deviation from the other two cases.
Physically, the cavity embedded in a region of rapidly changing curvature results in
greater variation of field distribution inside the cavity while the field exhibits less change

when the cavity is mounted in a region of surface with little curvature change.

21

elh}
con.

For

resp
equi
'IhC'
surfi
liov

on H

The comparison of eigenvalues for a cavity mounted in different portions of an
elliptic cylinder and the approximately equivalent circular cylinder is of course a major

concern for the design of conformal antenna embedded on a curvature-varying surface.

For a cavity (3cm x 3cm x 3cm) mounted in different positions, v0: 0.02, and M2, of an

elliptic cylinder with a=50cm, b=20cm and the equivalent circular cylinder with radius of
p=200m and p=50cm, the computational results are tabulated in Table 2.5 and 2.6,
respectively. The average deviation of eigenvalues between the elliptic cavity and its
equivalent circular shell cavity is 10.8% in Table 2.5 while it is 1.3% in Table 2.6.
Therefore, the eigenvalues for the elliptic shell cavity mounted in the less curved elliptic
surface can be approximately determined using its equivalent circular shell cavity.
However, it is necessary to accurately model the elliptic shell cavity when it is embedded
on the highly curved area.

From the discussion above it has been verified that the FE method using the new
elliptic shell elements and its vector weight functions successfully compute the
eigenvalues of the rectangular and shell cavities. It remains to develop the boundary

integral and the dyadic Green’s functions so that open problems may be examined.

22

 

 

Tab
cllip

 

Table 2.1 The eigenvalues for a rectangular cavity (2cm x 1cm x 1cm) represented in
elliptic cylinder coordinates as u=2, v=l, z=l(cm), utilizing 520 unknowns.

 

 

 

 

 

 

:Eigenmod Analytical FEM Error (%)
TE011 3.561 3.561 <0.01
TE101 3.561 3.561 <0.01
TM110 4.488 4.521 0.8
TE012 4.487 4.522 0.8
TM112 5.520 5.555 0.6

 

 

 

 

 

 

23

 

T31
6111]

Table 2.2 The eigenvalues for a circular shell cavity (3cm x 3cm x 3cm) represented in
elliptic cylinder coordinates as u= v=z=3(cm), utilizing 520 unknowns.

 

 

 

 

 

 

Eigenmode Analytical FEM Error (%)
TE011 2.195 2.244 2.2
TE111 2.369 2.424 2.3
TM110 2.377 2.433 2.3
TM111 3.474 3.553 2.2
TE121 3.520 3.585 2.3

 

 

 

 

 

 

24

Table 2.3 The eigenvalues for a circular shell cavity (3cm x 3cm x 3cm) represented in
elliptic cylinder coordinates as u= vq=3(cm), utilizing 1176 unknowns.

 

 

 

 

 

 

Eigenmode Analytical FEM Error (%)
TE011 2.195 2.224 1.3
TE 11 2.369 2.399 1.2
TM1 10 2.377 2.409 1.3
TM111 3.474 3.520 1.3
TE121 3.520 3.546 1.2

 

 

 

 

 

 

25

Tabl
ellipt

1'0 :1

 

 

 

Table 2.4 The eigenvalues for an elliptic shell cavity (3cm x 3cm x 3cm) mounted in an
elliptic cylinder with a=50cm, b=20cm, starting from three different angles of v0 =0.02,

v0 =1r/4 and v0 = #2 and utilizing 450 unknowns.

 

 

 

 

 

 

Eigenmode v0 =0.02, v0 =1d4 v0: 1d2
TE011 2.224 2.223 2.244
TE111 2.725 2.280 2.270
TM110 2.817 2.299 2.271
TM111 3.942 3.400 3.393
TE121 4.409 3.409 3.393

 

 

 

 

 

 

26

Tal
an e

app
unit

 

 

Table 2.5 Comparison of eigenvalues between a cavity (3cm x 3cm x 3cm) mounted in
an elliptic cylinder with a=50cm, b=20cm, starting from the angle of v0 =0.02, and its

approximate equivalent circular cylinder with radius of p=20 cm, utilizing 450

 

 

 

 

 

 

unknowns.
Eigenmode Elliptic shell cavity, Circular shell cavity, Deviation (%)
v0 =0.02, p=20 cm

TE011 2.224 2.244 0.90

TE111 2.725 2.424 11.0

TM110 2.817 2.433 13.6

TM111 3.942 3.553 9.9

TE121 4.409 3.585 18.7

 

 

 

 

 

 

27

Table 2.6 Comparison of eigenvalues between a cavity (3cm x 3cm x 3cm) mounted in
an elliptic cylinder with a=50cm, b=20cm, starting from the angle of v0 =1r/2, and its

approximate equivalent circular cylinder with radius of p=50 cm, utilizing 450
unknowns.

 

 

 

 

 

 

Eigenmode Elliptic shell Circular shell cavity, Deviation (%)
cavity, v0 = M2 p=50 cm

TE011 2.244 2.245 <0.05

TE111 2.270 2.312 1.85

TM110 2.271 2.313 1.85

TM111 3.393 3.437 1.30

TE121 3.393 3.438 1.32

 

 

 

 

 

 

28

 

v=7r

 

l"

<~——h'
c I

 

Figure 2.1 A cross-sectional view of the elliptic coordinate system

29

v:

 

Figure 2.2 The geometry of the cavity-backed antenna embedded in a metallic elliptic

cylinder.

30

 

 

 

 

 

A ,
I
I
I
I
I
I
I
—————— I ”—-
______________ L‘ ’4’
"— I ‘~‘ 1
” ‘~ I
a’ I ‘\ I
,’ ‘\ /
z ———————————————— x
I ..... 4“ I x
I "’ ~...‘ I \
I ,v I “ \
/ ” s‘ / \
I ” I v \
I x ' \ \
/ ’ I \ \
I l
I , | 1 \ \
l 1
. u , 1
\ a , t \ ’
\ l
\ I ‘ \ I
\ \ I I
\ \\ I ‘ I
\ I I
\\ \\‘ I ‘ ’
\ “\ ’ \ I
‘ ‘~-__/ - ’I
\\ I —————————————— I
x I b
\ \
-. I _
~- , - v_v
\ \
“ / \s
‘h‘.’ ‘-.-O-p
——————— P,
----- L..-_-_——-"
\
\
....................... ._ v=v
,_— ~4-~ l‘
"- ‘~‘
u” ‘5‘
z’ ‘N
l’ _______ '— ___________ \‘
I '-- ‘4‘- \
I ’—’ ~~~ \
I ” ‘s‘ K
/ ,’ ‘5 \
I ” \ \
l / ‘ \
, , z , ‘ ~
1 I ’ )
‘ \ ’ l
\ ’ I I
\ x I ’
x , ’ l
\ \
\ I I
\ \
“ / I
\ ‘\ I
\ “‘J_ a I
\ , ................ I
‘ a
x‘ I ’
x I
~
\‘ I
‘\‘ /
“~J‘ _.
___________________

 

Figure 2.3 Geometry of an elliptic shell element. The numbers denote the local node

numbering scheme for these elements.

31

v

"1911'

 

 

 

 

 

 

 

Xx? \ \ ~\
by \ \ '\ \ 4 3
, / \(p. \\ /
l‘ 1 <

 

‘%

 

Figure 2.4 Geometry of a cylindrical shell element. The numbers denote the local node

numbers for this element.

32

CHAPTER 3
DYADIC GREEN’S FUNCTION FOR ELLIPTIC CYLINDER

3.1 Introduction

In this chapter the surface dyadic Green’s function for an infinitely long, perfectly
conducting elliptic cylinder is derived [11]. In this approach, vector wave functions
approximate for representing electromagnetic fields in the elliptic cylinder coordinate
system are generated based on the elliptic cylinder scalar wave functions.

Eigenfunction expansion of the required field quantities is the first approach applied
to find the dyadic Green’s function necessary to describe on-surface interactions. In this
chapter, the scalar wave equation is used and its eigenfunctions can be written in terms of
separated angular and radial functions. The radial functions, which are the solution of the
modified Mathieu’s equation and finite at the origin, can be written as a series of Bessel
functions. The angular functions, which are solutions of Mathieu’s equation, are required
to be periodic with respect to the angle of the elliptic cylinder so that the field represented
by these functions is a single-valued function of position.

Since the dyadic Green’s function developed by eigenfunction expansion is very
difficult to evaluate numerically, an approximate asymptotic solution based on the
Uniform Theory of Diffraction (UTD) has also been developed. In this, the surface fields
attributed to a source on a smooth, perfectly conducting surface with arbitrary curvature
are computed using a representation developed by Pathak and Wang [25]. Base on this
development, an approximate asymptotic solution for the electromagnetic fields is found.
These fields are induced by an infinitesimal magnetic current moment on the same

surface. Hence, this solution of the surface magnetic field attributed to the aperture field

33

 

[1

located on the same surface represents the dyadic Green’s function. The superposition of
such aperture field elements represents the total electric field in the aperture of a
conformal antenna. This solution can be employed to calculate mutual coupling between
two or more antenna elements. The volumetric cavity region behind the aperture is
modeled using the finite element method. This hybrid approach allows the simulation of

complex antennas with minimal computational effort. Such information is essential for

f."

designing conformal antenna arrays and for studying the electromagnetic compatibility of
multiple antennas. In this UTD solution, the surface fields that propagate along each ray’s
geodesic path remain uniformly valid within the shadow boundary transition region

including the immediate vicinity of the source. Again, it is noted that time convention

19”" is used through the whole dissertation.

3.2 Eigenfunction expansion method
3.2.1 Mathieu’s Equation

Before the scalar wave equation in elliptic cylinder coordinates is introduced, the
solutions for both Mathieu’s and the modified Mathieu’s equation must be developed. In
the elliptic cylinder coordinate system, the set of coordinates used in this dissertation is
designated by (u, v,z). A cross-sectional view of a plane perpendicular to the z-axis is
shown in Figure 2.1. The relations between rectangular coordinates and elliptic cylinder
coordinates are given in (2.1).

Mathieu’s equation can be written in the form

2
i{igl’l+(a—2qc052v)f(v)=0 (3.1)

dv

34

where a and q are parameters that are usually real number. Also, a is the eigenvalue of

the system and it forms a denumerable set such that the corresponding angular functions
is periodic functions of v

Since the angular solution for Mathieu’s equation should be periodic over the elliptic
cylinder, viz. a = q = 0 , then the periodic solution of Mathieu’s equation is constant, i.e.,
f (v) =c, where c is constant. This is the Mathieu function of order zero, associated with

the eigenvalue a = 0. If a at 0 and q=0, then

2
___—d dfvg") + af(v) = o (3.2)

The solution of (3.2) can be represented as
fl (v) = cos mv or f2 (v) = sin mv (3.3)
where m=1,2,3.
For the general solution of (3.1), where a at 0 and q :1: 0 , the eigenfunction can be

represented in series form as

f(v) = i AJ- cos(jv) + 31- sin(jv) (3.4)
j=0

where A j and B j are the expansion coefficients to be determined. Substituting (3.4) into

(3.1), four different types of corresponding eigenfunctions are obtained:

35

f2km MA F2143], “008(2kv). m = 0,1,2,3...

fzi'if‘m(v)=;o 2;";31 c03(2kV), m=0,l,2,3...
(3.5)

fzimodd (v) = 2 822,2" sin<2kv). m = o, 1, 2,3...
k=0

fzzk'iilodd (V) — Z 1922131 sin(2kv). m = 0,1,2,3...

Here the subscripts even and odd represent the associated eigenfunction expanded in the
base of cosine and sine functions, respectively. The equations above are four different
types of Mathieu function, which are the solutions of Mathieu’s equation associated with
four different eigenvalues: a = a2", (q) , a = a2m+l (q) , a = b2m (q) and a = meH (q) .
These four types of solutions are isolated by odd and even functions and by their cyclic
period of 11: or 211:.

For simplicity, the four solutions in (3.5) are combined as even and odd Mathieu

functions. They are

Sem(v) = 2 D,'," c05(nV), m = 0,1,2,3... (3.6)
n=0

5mm = 2 F,“ sin(nV), m = 0,1,2,3... (3.7)
n=0

where the summation is to be extended over even values of n if m is even and odd values

of n if m is odd. Sem(v) and S0m(v) are also called the even and odd solutions of

Mathieu’s equation, respectively. The coefficients D,'," and Fn'" are normalized, i.e.,

i 0;," =1, i E,“ =1 (3.8)
n=0 n=0

36

 

such that the following relationship between the even and odd solutions of Mathieu’s

equation is established
£2353," (v)dv=- ZISom (v)dv=l (3.9)

Using (3.9), it can be shown that Sem(v) and S0m(v) form a complete orthogonal set. The

orthogonality relationships are

27:
[3,,(v)s,j(v)dv=er(D:;)2-(1+6), for i: j
0 n

27:
ISei(v)Sej (v)dv = 0 for i¢ j
0
2” . (3.10)
I So.(v)S.,-<v)dv=zr2(F,:>2. for i= 1'
o n
27:
[Sm-(v)SOJ-(v)dv=0 for i¢ j
0

where 8:1 if n=0 and 8:0 if n¢0.

3.2.2 Modified Mathieu’s Equation

For convenience, Mathieu’s equation can be rewritten as follows
d—v—wa ——)+(a— 2qcost)f(v)= O (3.11)

The modified Mathieu’s Equation can be obtained by replacing the variable be the

complex form jv [26]. Then (3.1 1) becomes

d2 __f____(J'V)

-(—a 2g cosh 2v) f ( 1v) 0 (3.12)
dv2

37

which is the modified Mathieu’s equation. The solutions of the modified Mathieu’s
equation are found by replacing the variable v with jv in the eigenfunction expressions

(3.5). The resulting expressions for the modified Mathieu functions are

fzk ”new” - 2422;. "’.cosh<2kv> m = 0,1, 2,3...
fzi’iilevenuv): 2A A223? cosh(2k +1)v, m = 0,1, 2,3...
(3.13)
fzi'ii" dd (jv>- — :03221'333‘ sinh(2k +1)v. m = o, 1, 2, 3...
fzi’iézodd (jv1- — 2322,1132 sinh(2k + 2)v, m = o, 1, 2, 3...
2 2

c
Replacing the parameter q and v in (3.12) with L41 and u, respectively, then the

2 2
quantity 2q cosh 2v becomes by (cokp cosh u)2 ——%i , giving

 

‘12 f(“)+(c§k2coh2u— —b)f(u)= o (3.14)

2 2
c k
0 p . The solutions of (3.14), which are equivalent to (3.13), now can be

 

where b = a+

written and expanded in terms of a series of Bessel functions as [27]

Rem/(pol): %Z(1) "In—"AnmJ (COkp coshu)

Rem/(p (u): %Z(J .)m_nA,',nYn (COkp 0081] u)

Rlomkpu‘): g::th-:(j)n —m anJ np(C0k COShu)

R3m1p1u1=f tanth<j)"""nB,'."1/,(cok,, coshu)
n=0

(3.15)

38

 

where superscript 1 represent function 1,,(2) , which is the n'h order Bessel function of

first type, while 2 denotes function Yn, which is the n'h order Bessel function of second
type. Also, the summation is to be extended over even values of n if m is even and odd
values of n if m is odd. It is noted that for these four eigenfunctions, each is associated
with its own eigenvalue in the modified Mathieu’s equation. For simplicity these four

radial eigenfunctions are combined as follows

Mm) g2“ )“‘”D,’,"J,,(c0kp cosh u)
(3.16)
R=omkp(u) £11112 (1)" "'anl ”(cok coshu)

3.2.3 Vector Wave Functions In An Elliptic Cylinder Coordinate System
The scalar wave equation in an elliptic cylinder coordinate system can be written in

the form

2V,
321”+ 82WM:'él+(k/2,+k22)11/=0 (3.17)
Z2

 

i<
,6 Buz +3112

1
where ,6 = c(cosh2 u — cos2 v)2

The variables u, v and z are the radial, angular and axial coordinates in the elliptic
cylinder coordinate system, respectively. The parameter c is again the distance between
the foci of the ellipse. Assuming that f = fl(cosh u) f2(cos v) is the solution of (3.17) and
using it in (3. 17), then the following two equations result

d :—f1——(“) +(c2kf, cosh2 u— b) f1(u) o (3.18)
it

39

d2 f2(V)
dv2

————+ (b— c2k2 cos2 v)f2(v)= o (3.19)
Here (3.18) is the modified Mathieu’s equation, while (3.19) is Mathieu’s equation. The
eigenvalues b form a denumerable set such that the corresponding angular functions is
periodic functions of v, and thus a single-valued function of position for the

electromagnetic fields can be described. Because the wave functions of the elliptic

cylinder can be written as

1v=f(u,v1«f’"zz =f1(u)f2(v)e'j"zz (3.20)
Thus, by applying the solutions of Mathieu’s and the modified Mathieu’s equation, which
have been derived in sections 3.2.1 and 3.2.2, into (3.20), the resulting wave function is

given by

1II=.mkp(v1Romkp(u1e‘fl‘zz (3.211

where

Remkp(u)Semkp(v)= %Z(1)n—m0m cos(nv)J,, (ckp cosh u)

2"" (3.22)

01.1,, <u>Somkp(v) $201" ‘"'F"' sm<nv11,.(ck coshu)

Equation (3.22) is given by Stratton [28] and can be used to represent the standing wave
function inside the elliptic cylinder, while the outgoing wave which is traveling outward
from the cylinder can be represented in the form of Hankel functions of the second kind

as follows

40

 

 

 

COY

\xr

Kl

R(2m)l((u)Semkp (v) 2(1)"me cosan§2)(ckp cosh u)

2“" (3.23)

2
R( "3k p(u)Somkp (v): 325-2300)" "7;" sinan(2 )(ckp coshu)
2n

Since I); is the solution of the scalar wave equation in elliptic cylindrical coordinates, the

corresponding vector wave functions are defined by

M (-kz)=Vx1//

 

 

 

Smkp
(3.24)
(— —k z)=—VxVx1,(/
Nam/(p
where K2 2 k3, + k3. Then these two vector wave functions are
1 asemk 3R,”
M —k =— R 0 p u— —°—P—v e‘szz 3.25
3mkp( 2) fl[ 5’"kp av ngp au ] ( )
BR 8S
1 . g p . g’fl‘p
N —k =—— — k S u— k R
gmkp( Z) kfi[ J Z gntkp Bu 1 z ngp 3v (3.26)

2 —-jk z
+flk R S z]e 2
p gmkp gmkp

The orthogonal properties of these functions are stated by the following equation [11]

fistmkp (—Ic,).N(,)m,k-p (k',)dv =0 (3,27)

mMgmkp(-kz)-Mgm,kb(k',)dv

0 m¢m' (3.28)
= . . ,+
l nzkplgmkpme—k,)6(kp—kp) m=m

 

 

41

 

u'hc

cg“.

an

6‘)

N. <-k.)'N. .-(k;)dv
omkp omkp

0 m ¢ m
(3.29)

 

2 . __ .
7: kplgmkp5(kz‘kz)5(kp‘kp') m—m

 

where

1...), = n2<1+6o><Drr> , 10..., = 2 (51")
"=0 n=0

3.2.4. The Free-Space Dyadic Green’s function
The free-space magnetic dyadic Green’s function satisfies the dyadic differential

equation [11]
VxVxEmo(R,R)—k25mo(R,R) =Vx[75(R,R)] (3.30)

and the radiation condition at infinity. Here 7 is called an idem factor, and its explicit

expression is
7:2“, (3.31)
i
By using the eigenfunction expansion, the right hand side of (3.30) can be written as

Vx 76 R—R = dk dk 0° A —k N —k +13 —k M -k
[ ( )1 g p! zgl gmkp( z) gmkp‘ Z> 5mkp( z) gmkp( z)1

—oo

(3.32)

By taking the anterior scalar product of (3.32) with Nemk (kz) and integrating the
0 9

equation over all space, as a result of the orthogonal relationships given in (3.27),(3.28)

and (3.29), the following relationships are obtained

42

wh

Ni

ei‘.

=——— (k )
emk 2 emk Z
0 I) It kPIgmk o p

p

(3.33)

(3.34)

2 _ 2 2 - - ' '
where K —kp +kz and the superscnpt 1n Mgmkp(kz) and N gmk (k2) represents the

p

source field point. Hence the continuous eigenfunction expansion of Vx[75(R, R ')] is

given by

VIé‘RR=dkp dk ————N
><[( no] I 71:12“ {5MP

—oo

+Memkp (-kz)Ngmkp (kz)}

Now, EMMR, R) in the left hand side of (3.30) may be expanded in terms of

eigenfunction as

—-oo

Emo(R,R)= J'dkp IdeZ[a(kZ)Ne mp1. (k)M k (k,)
0 m=1 0'" gmp

+b(kz )Me (—kz )Nc (kZ )]
omkp omkp

Substituting (3.35) and (3.36) into (3.30), the coefficients are determined as

K“

2k
fl'k (K2 —2k)
plemkp

a(kz)=b(kz)=

 

Thus the expression for EmO(R,R') can be written in the form

43

(3.35)

(3.36)

(3.37)

h-I_

 

Gmo(R R): ojdkp [.11. f;

—kM' k
m_17r2kWI:—(K2—2[k) N" (1)” (Z)

onlkp omk

 

-oo 9
+ M 5""‘19 (—kZ )Ngmkp (kz )]
To simplify the expression in (3.38), Let
K . .
I: —k M k +M —k N
ka ”m:1rr2kpl. (K,_k,)[NO " m1; ) 2.1.2} 1’ amt} z) 2
amp

 

 

 

and
Ne (—kz)M'2 (k2)=T(1)[Jn(ckpcoshu)Jn(ckpcoshu')]
omkp omkp
M2, (—kZ )N'e (k2) = Tmun (ckp cosh u)!" (ckp coshu ')]
omkp omkp
Then (3.39) becomes
I: fdkpi 2 "' 2 2 (Tania);
m_17r kplgmkpw —k )
= [(1) +10)
where
10>- - de " rm
f 32.1%]: I (xi—k2)
p
,(2)__ dkz K Ta)
17 pm:17t2k I (xi—k2)
0 p

(3.38)

22/) (1.2)]

(3.39)

(3.40)

(3.41)

(3.42)

T

For field solutions satisfying the vector wave equation in the elliptic cylinder coordinate
system and for observation within the elliptic cylinder, i.e., u < u ' , the first part of (3.39)

can be rewritten as

 

K‘ .-

,(1)=fkpz 2 2 {T“)[J,,(ckpcoshu)H,‘,"(ckpcoshu)]}

,,, 71' 1.21222 (rz-k )
o p

(3.43)

 

K“ ~(1) (2)

4.5/('02 2 K2- 2 {T [Jn(ckpcoshu)H,, (CkpCOShu')]}

m It kplemkp( k )
0

Here the Bessel function in (3.40) for the source point has been replaced by the
combination of a Hankel function of first kind and second kind, while the Bessel function
for the field point was retained to represent the standing wave inside the elliptic cylinder.

Replacing the variable k p with —kp in the first term of (3.43) and using the
following relationships

H ,9) (ck lac—j” cosh u ') = -e—jMH,(,2) (ckp cosh u ')

 

. . (3.44)
1,,(ckpe'1’r cosh u) = e’”!,, (ckp cosh u)
then (3.43) becomes a principal-value integral as follows
1 rialun (ck cosh u)H,(,2’(ck cosh u ')]
I( ) : Endkp 2 P 2 2 1/2 P 2 2 1/2 (3'45)

P g mk p
where 1,,(ckp cosh u) represents standing wave inside the elliptic cylinder.
The principal-value integral in (3.45) now can be converted to a contour integral along
the contour shown in Figure 3.1. Here e_jkpr is used to represent outward propagating

plane wave in the elm time convention. Initially the medium is assumed to have a small

loss so that the pole at k2, = —(k2 — k3 )” 2 = x0 — jyo, yo > 0 , lies above the contour and

45

 

the pole at k p = (k2 — k3)“ 2 = —x0 + jyO lies below the contour as shown in Figure. 3.1.

When k pr is very large the Bessel functions have the following asymptotic behavior

 

1
2 ."+—-jkr
szrz 2e p
"(p)“ I

,
p (3.46)

2 72' m:
J k r = cos k r————
"( p ) \ln'kpr ( p 4 2 )

Consequently, in the lower half of the complex k p plane the product inside the integral

 

of (3.45) becomes exponentially small for large k pr . The contour can be closed by a

semicircle in the lower half plane, and, when the radius of this circle approaches infinity,

the contribution from this part of the contour is negligible. By applying residue theory to

evaluate the contribution from pole at k p = —(k2 - k 22 )1’ 2 , (3.45) becomes

 

1(1) _ dc K‘imUnkkp coshu)H,(,2)(ckpcoshu')]
_I: pZEZk k k2_k21/2 k _ k2_k21/2
Plgmkp( P+( z) Xp ( z) )
=—27rj-Residue|kp =(,22_1,222)1/2222:0222) +,22221/2 =k
=‘jkfijn(cfl005hu)fl,(.2)(cflooshu')] (3.47)

27177212”,

=—1’£—N(-k2)M(2)'(k2)

Wle

0

 

1
where n=(k2 —k,2)2.

Using the same procedure as above for [(2) in (3.41), gives

_ 'k .
1(2) =—+—M(—k2)N(2) (k2) (3.48)
27"] 15m!)

Combining (3.47) and (3.48), then I is expressed as

46

 

°° K . ,
l= dk N-k M”) k +M -k N”) k

P (3.49)

=._’211‘._[N(—k2)M‘2)'(k2)+ M(-—k2)N‘2)'(k2 )] for u < u'
27:77 I

e
0

For simplicity, the subscripts attached to M and N are omitted here. Consequently,
(3.36) becomes

= 0° ._ k 0 '
Gmo(R,R') = I dkz fi[N(-kz)M(2) (k2)+M(-kz )N‘Z’ (kz )] for u < u'

e
0

(3.50)
For u > u ' , a Hankel function of the first and second kind are used to represent the

traveling waves. Then the first part of (3.39) can be written as

 

°° ~ 1
[(1): k K {T(1)[—H1(ck coshu)J,,(ck coshu')]}
f pm2=17tzk I Mk (Kl—[(2) 2 n P p
o p
(3.51)
I" ~ 1
{T(l)[-2-H3(ckp cosh u)J,,(ckp coshu')]}

 

",4sz 1 m (xi—1(2)
0

Likewise, (3.51) can be transferred to a principal integral by a change of variables, and
then assuming the medium the wave is traveling has a small loss. The evaluation of the

principal integral can proceed as before using the residue theorem to give a closed form
for that integral with respect to k p. The same procedure is applied to [(2) in (3.41) and

then combined. Hence (3.39) for u > 14' becomes

 

°° K’ .
I: k N”) —k M k +M‘2’ —k Nmk
. ”m p (3.52)
=——?,-’1‘—[N‘2’(—kz>M'<k,)+M<2>(—k,)N'(k2)]
27m 12m”

47

'tfr‘l"

 

 

whc

defil

disc.

Whe

field

Thus,

Gmo(R, R): aid}: Z-——[N(2)( —k )M (k )+M(2)(— k )N (k )1 for u>u'

m=l2
o

(3.53)
Consequently, the free-space magnetic dyadic Green’s function, which satisfies the

vector wave equation in an elliptic cylinder coordinate system, can be expressed as

- _ <2>' _ (2)' ,

Gm0(R,R')= Idkzz_;1_:__ [M2 kz)M '(kz)+M( kz)N 2062)] for u<u
‘°° ""121", m[N( )(‘kz)M (kz)+M(2)(-kz)N (19)] for u>u'
(3.54)

where the superscript (2) attached to N (2’ and Mm means that these functions are now
defined with respect to the Hankel function of the second kind. The function :mO is

discontinuous at u = u '. The expression for Geo can be obtained from Gmo as

3.0013) =le—[—uu§(R —R ')+(Vx:;o)U(u —u ')+ (anowm '—u)] (3.55)

where U is the unit step function and the superscript + and — attached to Gmo denote the

field outside and inside the elliptic cylinder, respectively. Inserting (3.54) into (3.55), the
following expression for :30 is obtained

5.0(RR)=—1,-[:n5(R—R)1

527I;zzz.{[M-k>1\4‘2"(I<)+1\I(—k>N‘2"(k)1 for u<u' (356)

__ J
”[1609; [MW—k )M'(k )+N(-k mam. )1 for u>u'

48

3.2.5 The Electric Dyadic Green’s function Of The First Kind
For a perfectly conducting elliptic cylinder as shown in Figure.3.2, the electric
dyadic Green’s function of the first kind must satisfies both the vector wave equation and

Dirichlet boundary condition at the cylinder surface, u = uo.

The dyadic Green’s function of the first kind for a perfectly conducting elliptic
cylinder can be written in terms of the free space dyadic Green’s function that represents
the incident field from source, and a scattering dyadic Green’s function that represents
the scattered field produced by the induced current on the surface of the elliptic cylinder.
To satisfy the Dirichlet boundary condition on the surface of the elliptic cylinder, the

following scattered field dyadic is assumed

= ' °° °° 1 , ,
0.101.139: —§ 1 dk, Z Tfl—tanM‘W—kzm‘” (k,)+fl,,N‘2’<-kz)N‘2’ (kz )1
_oo m=l e

0

(3.57)
and then the total dyadic Green’s function can be represented as the superposition of

(3.56) and (3.57)
5.102.113 = Emma ')+5.1(R.R '>

1 , j°° °° 1
=-k—2-[—uu5(R—R ”—5-”- IdeZ—E—
_oo m=l e
0""?

[M(-kZ ) +a,,M§f’(—kz)]M§,2)'(kz)+[N(—kz) + flnN‘2’(—kz)]N‘2"(k,) for u < u'

[M'(kz ) + o:,,M§,2"(kZ )]M§,2) (-kZ ) + [,6,,N(2"(kz ) + N'(kz )]N(2)(—kz) for u > u'
(3.58)

By the Dirichlet boundary condition,

ux5e1(R,R')=0 for u=u0 (3.59)

49

 

Th

Dir

Flt.-.)

 

 

Ins.

am

for

The two unknown variables, a” and E, in (3.58), are determined by enforcing the

Dirichlet boundary condition. From (3.27) to (3.29), it is observed that N (2) ' and Ma).
are orthogonal. By this orthogonal property, the following relationships are obtained

u><[M(—k,)+cz,,M‘2)(--kz )]u=uo = o

 

 

(3.60)
ux[N(—kz) + ,6,,N(2)(-kz )jmo = 0
From (3.25) and (3.26), the vector wave functions are found to be
3__Rn - k
N(—kz)= k-l—fl-(— jk ZS,,— Bu u—jsz,,-——--v+,6k2 R,,S,, z)e ’ zz
1517 3R1; — 'k
M(—kz)=Z(R’7—5v-u-S”-87v)e ’ 12
(2) 8,,S (3.61)
N‘”(- k )=7(— jk 5,, 1;” u-jkZ 2) Elwfikpkfkn z)e ”‘22
1 35,, if)” .
Ma) -k =__ (2) S e-szz
( z) ,6 (R77 8v Lu 7] Bu
Inserting (3.61) into (3.60), the unknown coefficients are given by
8R,,(uy
_ a _ R1704)
a,, _-—,2)—-‘—‘— ,6" _— (2) (3.62)
8R, (u/ R,, (u)
Bu — “zuo
we
and (3.58) becomes

for u>u'

50

Ge1(R,R)=——[uu5(R— R)]—— jdk 2—21—{[M(k)

m=l e

0""7

M§,’"(—kz)]M§,2’(—kz)

  

Bu u=u0

+[ (2)“) N‘2)'(k,)+N'(k,)]N(2)(—k,)}
R7, (“)qu

for u<u'

Ge1(R, R')=-——[uu6(R— R)]—2i jdk Z— 2
2”...» m=lflllgmfl

Mg”(-kz)]M§,2"(kz)

+[N(—kz)— 1:12;") N"’<—k,)]N""(k,> }
R,7 (mm)

3.2.6 The Electric Dyadic Green’s function Of The Second Kind

 

The electric dyadic Green’s function of the second kind can be determined by

(3.63)

(3.64)

applying the same procedures used for the derivation of the first kind dyadic Green’s

function except that the Neumann condition is enforced on the surface instead of the

Dirichlet condition. The final expression for the electric Dyadic Green’s function is given

by

for u>u'

51

33201.11): —-k1—2[uu5(R - R ')]

‘EjfldkzZ .1 {[M'<k.)-%{ Miz’lkalM‘nZV-kz)
—oo m=1’713m” R” (u)u=uo

(3.65)
3R1,(u)
+[ N'(kz) N""(kz) ]N")<-k,) }

8R5,” my
Bu azuo

 

for u<u'

222(R9R') =-;13[uu6(R-R)]

j 0° 0° 1 R1701) (2) (2)1
_— d" M-k -—' M —k M k
Zfl—i 2,;17121 m{[ ( z) R1(72)(u)u=u0 7) ( 3)] q ( z)

 

a 3 (3.66)
R,(u)
——,——/i“— N‘2’(-k,)]N‘2"(kz>}
dRéNuV
Bu

u=u0

 

For u — u ' = uo , each component of the dyadic Green’s function of second kind can be

simplified as

. °° °° ,R(’(u no)
GW (uO,V',z'|u0,v,z)=% R" o),,S (v)S (v‘)

k2 R(2)l(::0)g’,'n o I _ 3 _ l
__3__:_:—_S (v)S (v') e sz(z Z)

(3.67)
G 200,1», 2 1u0,v,z)=—-j dkzzl

S' (v)S (v)e'j"z(""”
m—ll WWW)”; ” ”

00 k2

' -1 w
G422 (u ,v’,z'|u ,v,z =— alkz E
e2 0 0 ) 212.”! m= -llem :kz R(2)(

Ra) .
77(2)): 550050“) .)e-jkz(z-z')

52

where

2 Di," cosnv, m=1,2,3....
n=0

S,,(v) = 5mm ={ (3.68)

2 F5" sinnv, m=1,2,3....
n=0

and the prime attached to S and R denotes the first derivate with respect with v and u,
respectively, while it represents the source point for the others. Since (3.67) is
computationally prohibitive for m>12 or for a large argument in the Hankel functions, an

asymptotic dyadic Green’s function is developed for practical implementation.

3.3 The Asymptotic Dyadic Green’s Function
The general expression for the magnetic field due to the magnetic source on a
convex surface, shown in Figure 3.3, has been developed by Pathak and Wang [25]. For

convenience, this general expression is given as follows,

_kGoYo . ___J_°____]____~2_J_'_~ ~2_j_~
de(Q|Q)—-—-2flj de(Q){ bbta kt 1.212 To k,)V(§)+To MUG»

. . , (3.69)
m 1(—,f,-+;—,2-t-,-)V(cf>+-,{;17(:)1+(t'b+ b't)1kit(t7(6)—V(§)>Tol 1
where the Z7 (2,“) and W6) are related to the soft and hard Fock functions, respectively.
They are characteristic of on-surface creeping wave interactions and have been
extensively investigated by Logan [29]. The unit vectors in (3.69) associated with surface
coordinates are shown in Figure 3.3. The quantities GO and Y0 are the free-space Green’s

function and admittance, respectively. The quantities k and t refer to the free-space

wavenumber and the surface ray geodesic path from source point Q' to test point Q. The

53

factor To is identified as a ratio of the surface ray torsion and the surface curvature along

the ray direction. It is expressed as

1 1

T2 : [sin2(2§) 1 l _
0 132(2) R. (Q)

4 (Ram—Rue)

 

 

)pg (Q 3] [( )pg(Q)] (3.70)

where R1 and R2 are the principal surface radii of curvature in the b and t direction,
respectively. Also pg denotes the surface radius of curvature and 6 is the angle between

the axial axis and the direction of the geodesic trajectory. For the elliptic cylinder, since
the principal surface radius of curvature in the axial direction, R1, approaches infinity,

(3.70) can be reduced to

7:02 = sin2(26)
4R2(Q)R2(Q3

 

10, (mpg (Q) (3.71)

The asymptotic dyadic Green’s function for the surface interactions on an elliptic
cylinder, as shown in Figure 3.5, can be developed based on the general expression in
(3.69) by using the following relationship between surface and elliptic coordinate systems

u=n, u'=n’
v=tsin6+bcos6, v'=t'sin§+b'cos§ (3.72)
z=tcos§-bsin6, z'=t'cos6—b'sin5

Hence, the components of the asymptotic dyadic Green’s function of the second kind for

the surface of the elliptic cylinder can be expressed as

54

G (IQ Q')=— 2—OG{V(6)[cos2 6+— —(— (1+T02 )cos2 6+sin2 6— —7bsin26)

+2k—217 —cos 225+25in 6)]+(7(6)[7:—(T02 cos 225+sin 5+Tosin25)]}

GZ,(Q1Q) = %{V(6)[—%sin 25+%(sin 26+§T02 sin 26—(eos2 a—sin2 info)

 

 

+ 2:21, sin 26]+U(6)[—li-(—%T02 sin 26+%sin 26+ (eos2 6—sin2 631191} (3'73)
G ,,'(QIQ)= —°G{V(6)[sin2 6+— :(—(1+7~52)sin2 5+eos2 6+Tosin25)
+-I%2-(-sin2 6+2eos2 5)]+r7(;)[;(r'02 sin2 5+eos2 a-r’osinzafl}
where
a“): [2m (Q )an)]3/ZU©’ “5): [2mm )m(Q)§]l/2V© (3 74)

_ In(t)kpg()1/3
and 6- fi—pg(,)dr. mm: [— 1

Here 6 is the Fock parameter and pg (t) denotes the surface radius of curvature. Also in

 

(3.74), tis the geodesic path length I: J( $2 pdv)2 + z2 . From (3.73) and (3.74), it is
Q.

observed that the variation of surface field between the source and test points is primarily

governed by the Fock-type functions, U (6) and V(6). Since reciprocity applies, the

expression for G" is exactly the same as Gvz . For the special case of the circular

cylinder, the various parameters simplify to To=cot 5 , pg (Q) = 108(0) 22 , and

__a__
sin2 5
thereforeU (6) = U (6), W6) = V(6). Accordingly, (3.73) reduces to the dyadic Green’s

function for the circular cylinder as follows,

55

_je-J'k'

 

 

Gw(Q|Q')= 2” kq{V(§)[Sin29+CI(1—q)(2-3Sin29)+q[(U(§)—V(§))Secz9]}
je‘j’“
Gn(Q|Q)= 2” kqsinfiws9{[l-3q(l-q)]V(§)} (3.75)

_ -jk1
Gu(Q|Q)= ’2 kq{[cos26+q(1—qx2—3cos2a)iwé>}

 

where 6=£—6 and q=i.
2 kt

The electric dyadic Green’s function in the surface integral above is expressed in
terms of a rapidly convergent creeping wave series [24] expressed in terms of Fock
functions. As an example, consider an ellipse with major and minor axis of 4 cm and 2

cm, respectively, an angle between axial axis and the direction of creeping wave trace

5 = 80° , and a frequency of 30 GHz. The magnitude of each component of the electric
dyadic Green’s function versus the geodesic path length is shown in Figure 3.5. For
comparison, the dyadic Green’s function on the surface of the circular cylinder with a
radius of 4 cm is shown in Figure 3.6.

In figure 3.5 the creeping waves on the surface of the cylinder are found to have
greater attenuation in regions with larger curvature than those with less curvature. This
can be explained by the fact that the creeping wave energy will rapidly shed away from
the surface as it travels in regions with greater curvature. It also can be observed in
Figure 3.6 that the rate of energy loss for the creeping wave on the surface of a circular
cylinder almost remains constant after traveling three wavelengths along the geodesic
path. This is due to the fact that the curvature of the surface over which the wave travels

does not vary along the geodesic path.

56

3.4 Boundary Integral Matrix

In the FE-BI formulation developed in Chapter 2, the entries of the boundary integral

sub-matrix are given by

6321.3 [j w:(u.v,z)-u(u,v,z>x

51's,

[u(u,v,z)><W,-(u,v,z)-Ee2(u,v,z)],6',6dv'dz'dvdz (3.76)

Substituting the vector weight functions given in Chapter 2 into (3.76), the following
components of (3.76) are obtained

 

ngl=— S' Si H (v— v,')(v— v,- -G) gflfldv'dz'dvdz
Zia} 5,5,-
5 5,-cA
G§J=k§—— ”'15)! (v— v,)(z-Z, )GZ;,6,dv dz dvdz
gi:j:Av;
GE; _k2 ’H (z- 2,) )(—v' v, )G2 "fldv dz dvdz (3.77)
hiaj 5,5,-
5,5, CZA
0:33:16?” (z- z )(z- 2,)Gz22dv'dz'dvdz

where the components of the dyadic Green’s function of the second kind are given in

(3.73) and

,6, = c(cosh2 u, --cos2 v')”2

,6, = c(cosh2 u, —cos2 v)“ 2
i :/2 (3.78)

a
A ,- =(cosh2 u, -cos‘ v,)
V
1

A ,- =(cosh2u1—cos 2v,j)”2
V
1

57

When the elliptic cylinder is reduced to a circular cylinder by the transformation

v = ¢, p z ge“ , the boundary integral contributions for the circular cylinder are given by

55. ~ .. .
BI __ 2 l J _ . o_ . M v 7
Ga — ko‘aa, H (¢ ¢.x¢ ¢,)G,, d¢ dz d¢dz
‘ SjS,‘
023343-551- H (¢—£5,)(z-2-)G¢2d¢'dz'd¢dz
aihj 5.5. 1 Q (3.79)
1 1
5-5- 2 .
GE: =—kg-U—p‘— fl (z—2,)(z'—2,)G:,2d¢'dz'd¢dz

hihj SjS,‘

where the components of the dyadic Green’s functions of the second kind are given in
(3.75) and
‘13} = Pb ' Pa
¢i, j = ¢1i’j — V (3.80)
’27.; = Zf‘j — 21?"
The subscript i and j represent the test and source unknown numbers, respectively, and
the surface integrals have support over the area of test and source elements containing the
test and source edges.

For the non-self—cell contribution, mid-point integration may be used for
computation of (3.76) and the asymptotical dyadic Green’s functions in (3.73) are
applied. Since for the self-cell contribution the source element will coincide with the test
element in evaluating (3.76), the planar Green’s function is used to overcome the singular

integrals in (3.77). The contribution of the planar Green’s function is given by

0,5? = 2(koa)2 H w, ~[uxZo xu] - wsds 'ds (3.81)

sis,-

58

function is applied. Otherwise, the planar dyadic Green’s function is used in the

calculation.

3.5 Excitation: FE-BI

Conformal antenna patches are typically fed by a microstrip line printed along with
the radiator on the surface of the substrate or by a probe from below the patch. The
microstrip lines are in turn fed by a coaxial probe that originates behind the cavity as
shown in Figure 3.7. For convenience, the internal source is located on one of the

unknown edges. The source term ff” in the FE-BI formulation can be expressed for three

probe orientations as follows. For radial probe,

u 6(v—v5)6(z-zs)lo

 

 

 

 

 

J :
fl .. .. ,_ (3.86)
f'int =-jk Z I si(vs —vi)(zs-Zi)(ub —ua)flub,vs
l O 0 0 a h
For azimuthal probe,
J=v5(u—u3)5(z—zs)lo
fl"( )( " )5 (3.87)
’ Si pu _pfi' Zs-Zi u Va
-mt=-'kZI s , 5,,
f. J o o o t-h
For axial probe,
J:25(u—u3)5(v—v3)10
'6 (3.88)

§i(pus _pii,‘ )(vs -9} )flusysh
t-a

 

fiim = ‘jkoZolo

1/2

where flu”, = c(cosh2 u s — cos2 v,) and the Diac delta functions serve to specify the

location of the infinitesimally thin probe.

60

It should be remembered that even though the source edge is shared by four
elements, only one of them is used for computation [10]. Also, it is observed that the
number of non-zero entries in the right hand side of the linear system is equal to the

number of sources.

6]

>‘<

kpsz'D/O kp:'x0+jy0

 

 

 

Figure 3.1 Contour for converting the eigenfunction expansion into a mode expansion.

62

 

u=u0

 

Gls

 

 

 

 

 

C .y

Figure 3.2 The scattering wave and incident wave for an elliptic cylinder.

63

 

Geodesic surface ray path

 

t' NI]
II I) Q

 

(2': Source Point Q: Field Point

Figure 3.3 Illustration of unit vectors for a convex surface.

64

 

 

 

K)
Z
J
. Q v
z' u
t - . -------- - . - - .,"\ .
. . ‘ Geodesm .ath
11 Q v, p

 

Figure 3.4 Geodesic path for the creeping wave on an elliptic cylinder.

65

Dyadic Green Function vs. Geodesic Path Length

 

 

2° 1. T- r F 1
' , E It-ll GVV

_. G
21

 

 

 

Magnitude[dB]

 

 

 

l 1 i 1

0 2 4 6 8 10 12
Geodesic Path Length [lambda]

 

-140

Figure 3.5 Magnitude of the three components of asymptotic dyadic Green’s function
for an elliptic cylinder.

66

a? Ail -. .1. ' ' ' I

Dyadic Green Function vs. Geodesic Path Length
20 l 1 I l r I I l

 

 

 

 

 

—60

—80

Magnitude[dB]

.L
8

l.
8

-140

 

 

 

«on I l 1 L
luv

1 l
0 2 4 6 8 10 12 14 16 18

 

Geodesic Path Length [lambda]

Figure 3.6 Magnitude of the three components of asymptotic dyadic Green’s function
for a circular cylinder.

67

Patch or microstrip line

Cavity

Metallic elliptical cylinder

Coax eed

Figure 3.7 Cavity-backed probe-fed conformal patch antenna recessed in an infinite,
perfectly conducting elliptic cylinder.

68

CHAPTER 4
DRIVING POINT IMPEDANCE RESULTS

4.1 Introduction

The increasing use of microstrip antenna technology requires analysis methods .
capable of accurately predicting the input impedance and mutual coupling between these
antennas. The information generated will provide a useful reference for practicing
engineers and scientists in the design of microstrip antennas and circuits for installation
on curved surfaces and for studying the electromagnetic compatibility of multiple
antennas. There are several methods that have been somewhat successful for calculating
the input impedance and radiation from microstrip antennas such as the transmission line
model, cavity model [2], and moment method. However, those analysis methods only
focus on simple planar or non-planar structures such as cylindrical, spherical and conical
coated surfaces.

The hybrid finite element — boundary integral (FE-BI) method allows the simulation
of complex, cavity-backed antennas with minimal computational effort. The effects of
resonant frequency and input impedance due to the variation of curvature for an elliptic
cylinder can be examined by this approach. In this solution, the surface fields that
propagate along each ray’s geodesic path remain unifonnly valid within the shadow
boundary transition region, including in the immediate vicinity of the source.

In this chapter the calculation model for the input impedance of a cavity-backed,
printed antenna is introduced. The input impedance for an empty cavity, a slot antenna
and a conformal patch antenna embedded on the surface of an elliptic cylinder are
discussed separately. Of course, the elliptic cylinder can be reduced to a circular cylinder.

Results from a previous method appropriate for a circular cylinder structure are compared

69

with the results using this new method. When the radius of curvature of a cylinder
becomes large, the conformal antenna model reduces to a similar method for planar

antennas.

4.2. System Solution

Since the FE-BI method produces a large, sparse linear matrix system, the
biconjugate gradient (BiCG) solver has been chosen as it requires significantly less
memory than is required for a direct method. The BiCG method is also computationally

efficient, since it utilizes only one matrix-vector product per iteration. This operation

represents the bulk of the computational demand of the method and requires 0( N 52 )

complex operations per iteration for the fully populated boundary integral matrix, where

N is the number of aperture unknowns. If the matrix is not fiJlly populated, i.e. it is a

s
sparse matrix, the Compressed Sparse Row (CSR) format may be used to reduce the
memory demand, since only non-zero entries are stored. The FE matrix [A] in (2.14) is
such a sparse matrix. The CSR retains only the non-zero entries of the matrix in one long
data vector with another data vector, the offset vector, which contains the number of non-
zero elements per row of the matrix. An additional long vector, the pointer vector, is
required to indicate the matrix column associated with each matrix entry. Thus the
position of each element in the sparse matrix is uniquely defined. The matrix-vector
product using CSR scheme is carried out by executing the sum

r n]

y[n] = [A]{x} = A[e(n,n')]x[n'] n = 1,2, 3,...N (4.1)

n'=l

70

where r[n] is the number of non-zero entries per row of the matrix and e(n, 71') indicates

which entry of the long data vector is associated with the matrix entry A[e(n,n')]. The

boundary integral matrix-vector product involves the fully populated matrix.

4.3. Input Impedance

To accomplish feedline matching, designers are concerned with the input impedance
of the conformal antenna. The FE approach allows the calculation of the input impedance
of a radiating structure in a rather elegant manner. The input impedance is composed of
two contributions [24]

2,, = 2,, + 2,, (4.2)
where the first term is the self-impedance associated with a finite thickness probe in the
absence of the patch and the second term is the contribution of the patch current to the
total input impedance. In this dissertation, the second term will be the focus of the
computation, since for very thin substrates and thin probe wires, the contribution from the
self-impedance is negligible. To determine the input impedance of a probe feed cavity-
backed conformal antenna, the impressed model is applied to determine the formulation
of the input impedance. The geometry of the impressed model is given in Figure 4.1. In

Figure 4.1 the impressed field maintained by a magnetic surface current K m = (n :;)V

 

is represented by Ei , which is a non-conservative field. The scattered “ coulomb field ”
is expressed as E3 , which also is called the secondary field. For the source generator to

drive current J against the action of E5 through the terminal source region, the following

condition should be satisfied,

71

lE‘l > E2 (4.3)

 

 

Since the material is assumed to be a perfect conductor, then

lim J = lim a,(E" + E) = finite
0', —>oo 0’,-+oo (4.4)
:> E = —E"
Therefore, the total field inside the source generator is zero.

The total field at the conductor surface within the ring in Figure 4.1can be

determined using the integral form of Faraday’s Law,

E=E‘ +E" =—u—V— (4.5)
26

thus

.. 0+6 . __ 0+5 . ,- s- _ ,
fr, u qu— ITO—a“ (E +E )du _v (4.6)

with (4.6), gives

VI(u = u,,) = 2,3,1,2 = — j (u - E)I(u)du
P
z.» Z}, = ——12- u-E(u)I(u)du
1,, r

(4.7)

where E is also the field associated with the feed edge and I (u) is the current at any point
it while 10 is the current at uo on the probe. The integration contour F represents the

path that impressed current flows through.

For a radial post, the impressed current density J int is represented as

5(v -vs)6(z -Z, )10

4.8
fl ( )

Jim(u,v,z) = u

 

72

where ,6 = c(cosh2 u — cos2 v)” 2. Inserting (4.8) into (4.7), the input impedance for the
radial post can be determined by the formula

z:.=— 5"”

0

—c(cosh2 ub— cos2 v,)“ 2(ub—ua) (4.9)

where E (i) is the expansion coefficient of the electric field for the edge associated with
the radial post. This coefficient is numerically determined by the FE-BI program.

Likewise, the impressed current for azimuthal and axial posts can be represented as

5(u —us)6(z — zs)Io

 

 

J int(u, v, z) = v ,3 for azimuthal posts
6 6 I (4.10)
Jim(u,v,z) = z (u —u‘) (V—v‘) O for axial posts
fl
The formulations of the input impedance are
2,9,, =— —(E Il—)c(cosh2 u,,—- cos v,)” 2(v v) for azimuthal probes
1o
(4.11)
Z -" = — gydcoshz ub — cos2 v, )1/ 2h for axial probes
0

Utilizing the same technique that has been used in a previous chapter for reducing
elliptic cylinder coordinate to equivalent circular cylinder coordinate, the input

impedance for a conformal antenna mounted in circular cylinder are given by

2}": —E—(—i) —p,, ln(— p—b)p for radial probes
10

Z,',,- — —-£(—i)- Pb a for azimuthal probes (4. 12)
I0

2,5,, = — £39 pbh for axial probes
0

73

where a = ¢r —¢, and h = z, —zb , and ¢,,¢,,z,,zb are defined in Figure 2.4.

4.4. Numerical Results and Discussions
After the lengthy theoretical development in the previous chapter, the simulation
using this FE-BI model will be applied for the empty cavity, slot and patch conformal

antennas and the numerical results will be discussed later.

4.4.1. The Empty Cavity

For simplification, the empty cavity enclosed on all sides by conducting walls having
infinite conductivity will be discussed first. It involves the computation of finite elements
and the source matrix without the need for the boundary integral matrix. In this case the
empty cavity is embedded in a circular cylinder with a radius of 100 cm, shown in Figure
4.2. The size of the cavity is 6 cm x 3.75 cm x 1.5 cm and it was meshed into 576
elements with 1223 unknowns. Its unit cell is also shown in Figure 4.2. The radial probe
feed is 0.5 cm long and is located at the point 0.9375 cm above the center of the front
surface of cavity. The radial probe feed penetrates the back wall of the cavity and
protrudes into the cavity. The computed result of the input impedance vs. frequency is
shown in Figure 4.3. Since the radius of cylinder is large compared to the arc length of
cavity, the circular shell cavity can be considered a pseudo-rectangular cavity, and thus
the input impedance can be compared with the data computed using the program
LMBRICK(a.k.a. Low Memory Brick) [10]. That program utilizes the optimization for
brick element implementations of the F E-BI method. From the results shown, there is

very good agreement between the results calculated by the FE-BI program and

74

LMBRICK. Since the walls of the cavity are assumed to be a perfect electrical conductor,
there is no loss mechanism associated with the cavity and hence its input impedance is
purely reactive. The computed resonant frequency is 4.725 GHz for the lowest excited

mode. This frequency can be theoretically calculated by the following formulation [30]

 

(an’TE— 1 [(””’)2+(””)2+0332)“2
ya a c

mnp " 27r\/— —b_

(4.13)

where m =1,2,3,..., n =1,2,3,..., p = O,1,2,... for TM modes and
m = 0,1,2,3,..., n = 0,1,2,3,..., p =1,2,...m i n = 0 for TE modes.
The lowest excited mode for this cavity is TE101 and its theoretical resonant

frequency is 4.717 GHz. The deviation between theoretical and computed results is
approximately 0.17%. Agreement can be improved if the sampling frequency step is set
to 0.025 GHz or to a smaller value. If the same conformal antenna is embedded in a
cylinder with a radius of 5 cm, the computation result shown in Figure 4.4 is obtained.
Compared to the previous case it can be observed that the resonant frequency shifts to
4.975 GHz.

From this case the solution involving the computation of the finite element and source
matrices agrees with the theoretical solution when an empty cavity is used. It is noted that
the resonant frequency will be changed with the variation of the geometry of cavity, since

the field distribution inside the cavity is influenced by the boundaries of cavity.

4.4.2 The Slot Antenna
The geometry of a slot conformal antenna and its unit cell are shown in Figure 4.5.

The cavity-backed slot antenna was subdivided into 576 elliptic-shell elements with 1261

75

unknowns. This cavity-backed antenna was embedded in the circular cylinder with very
large radius such that it can be considered to be a rectangular cavity-backed slot antenna.
Figure 4.6 shows the input impedance vs. frequency for the antenna mounted in a circular
cylinder with a radius of 100 cm. Ideally, both the resistance and reactance should exhibit
symmetry about the resonant frequency, and the reactance at resonance should equal zero
[31]. Thus the resonance associated with zero reactance can be determined from the
computed results. In Figure 4.6 it can be observed that the magnitude of the resistance
increases as resonance is approached and it reaches peak value at a frequency slightly
prior to resonance. Physically the energy radiating out of a slot antenna reaches its
maximum at resonance. The reactance is negative across the frequency band, which
implies that this cavity-backed slot antenna can be viewed as an energy-stored antenna,
like a capacitor. To observe the influence of curvature variance on the input impedance,
the slot antenna was mounted in different circular cylinders with radii of 5 cm, 10 cm and
30 cm. Figures 4.7 and 4.8 show that both the resonant frequency and the peak values of
input resistance and reactance increase as the radius of the cylinder is decreased.
Therefore, the resonant input impedance and resonant frequency is curvature-dependent.
For the slot antenna mounted on an elliptic cylinder with major axis a=50 cm and
minor axis b=25 cm, computation results associated with different locations on the

elliptic cylinder are shown in Figures 4.9 and 4.10. From these results, when the antenna
is embedded in the elliptic cylinder starting from v0 z 0 , which is a highly curved region,

the resonant frequency is 4.875 GHz and its resonant input resistance is 584 Q. When the

antenna is moved to a region with little curvature change (i.e. v0 z % ,) f, shifts to 4.825

76

. . . 7r .
GHz, and the resonant input resrstance remains almost unchanged. At v0 z —2— , the quasr-

planar portion of surface, the resonant frequency is 4.80 GHz and the resonant input
resistance decreased to 5050. From the analysis above, it can be observed that the

resonant frequency and input impedance vary in regions of highly changing curvature.

4.4.3. The Conformal Patch Antenna

Cavity model has been used to analyze field structure inside a rectangular patch
antenna with very thin substrate layer very well. Since the height of the substrate is very
small, the fields remain constant along the height. In addition, because of the very small
substrate height, the fringing of the fields along the edges of the patch are also very small
whereby the electric fields is nearly normal to the surface of the patch. Therefore, only
TM mode will be concerned within cavity. In this cavity model the t0p and bottom walls

of the cavity are perfectly electric conducting, the four-side walls will be modeled as

perfectly conducting magnetic walls. The two most important field modes are TMOIO
and T M 00, associated with the azimuthal and axial polarization for the rectangular
microstrip patch antennas. The field structure for T M010 and TM 001 is shown in Figure

4.1 1. For TMmo , the equivalent magnet current due to the electric fields will exist on all

four slot-like walls; however only two walls, referred as radiating slots, that are separated
by the length, L, will radiate power outward. The radiation from the other two side walls
separated by width, W, is small compared to the other two side walls. Therefore, these

two slots are usually referred to as non-radiating slots.

77

Figure 4.12 to Figure 4.15 show the input impedance vs. frequency for the patch
antenna with the azimuthal polarization when the probe feed is removed from the center

of the patch to the edge. Figure 4.12, which corresponds to the probe feed located at the
center of the patch, shows that T M010 is not excited. The input impedance increases as

the probe feed moving along the azimuthal central line and away from the center of the
patch. Figure 4.15 shows the input impedance reaches maximum when the probe feed is
placed right on the edge of the patch.

Figure 4.16 shows the geometry of the patch antenna and its unit cell that are used
for computation using the elliptic cylinder FE—BI and LMBRICK codes. The cavity-
backed patch antenna was meshed into 192 elements with 411 unknowns. For the probe
feed located 0.5 cm left of center, referred as azimuthal polarization, a quasi-planar
surface is considered here. The input impedance vs. frequency is shown in Figure 4.17.
Figure 4.17 exhibits very good agreement between the computed results using the elliptic
cylinder FE-BI method and the planar LMBRICK codes.

For the same patch antenna with the different cavity size of 0.0795 cm x 6.5 cm x
5.5 cm, the cavity-backed patch antenna was mesh into 572 elements with the total
unknowns of 1209 and the unit length of about 1/40/1 in axial and azimuthal direction
for each cell. For the axial polarization, which probe feed is placed at 1.0 cm below the
center of the patch, the input resistance and reactance vs. frequency is plotted as Figure
4.18 and 4.19, respectively. Here the dielectric constant a, =(2.32, 0.0) was used. In
Figure 4.18 and 4.19, it is observed that the input impedance and resonant frequency are

almost independent of curvature while the magnitude of the input impedance very

slightly decreases as the radius of circular decreases. For the axial polarization the

78

TM 00, is excited here, and because the field remains constant along length or the

azimuthal direction, shown in Figure 4.11, it can be observed that the field structure is not
disturbed due to the surface curvature along azimuthal direction. Therefore, for the axial
polarization the input impedance and resonant frequency are almost independent of
curvature. The threshold chosen for using curved dyadic Green’s function or planar
dyadic Green’s function in boundary integral computation is based on the geodesic path
that the wave travels. For a curved surface, the curved dyadic Green’s function is applied
to computation when the wave travels more than half wavelength, while the planar dyadic
Green’s function is used when the distance between the source and test point is less than
half wavelength.

The results for the azimuthal polarization, which the probe feed is placed 1.25 cm to
the left of the center of the patch are shown in Figure 4.20 and 4.21. In Figure 4.20 and
4.21, it can be observed that for the patch antenna with azimuthal polarization, the
resonant frequency is sensitive to the variation of curvature. The input impedance almost
remains unchanged while the resonant frequency shifts to right when the radius of the
cylinder decreased from 500.0 cm to 15.0 cm and 10.0 cm. Since the TMOIO is excited
here, and because the field is varying sinusoidaly along length or along the azimuthal
direction as shown in Figure 4.11, it can be observed that the field structure is easier
disturbed due to the surface curvature along azimuthal direction. Therefore, for the
azimuthal polarization, the resonant frequency is more dependent on curvature compared
to axial polarization. It is also noted that the bandwidth of the patch antenna remained

unchanged no matter axial or azimuthal polarization is applied. Also, the resonant input

79

reactance is approximately zero for both cases, which implies that this cavity-backed
patch antenna is not an energy-stored antenna like the slot antenna.

To ensure that the curvature dependence of the resonant frequency for patch antenna
with azimuthal polarization is dependant of the field mode excited beneath the patch
rather than the geometry of the patch size, now a square patch of 3.0 cm X 3.0 cm with
azimuthal polarization is examined The input resistance and reactance vs. frequency are
shown in Figure 4.22 and 4.23. In Figure 4.22 and 4.23, the similar results of the input
impedance vs. frequency are observed. Therefore, it can be concluded that the field mode
that was excited inside the cavity decides whether the resonant frequency is curvature
dependent or not.

If the patch antenna is flush-mounted on different portions of an elliptic cylinder
with a=30.0 cm and b=15.0 cm and the probe feed is placed 1.0 cm to the lefi of the
center point of patch, similar results are obtained as the previous paragraph. The
numerical results are shown in Figure 4.24 and 4.25. Based on the variation of the surface
curvature for a conformal antenna embedded in an elliptic cylinder, an approximate
equivalent circular cylinder can be determined. It can be concluded that for the conformal
patch antenna mounted on a surface with a high curvature, the input impedance is much
more sensitive to the variation of curvature than in a region of low curvature. That can be
used to explain why the performance of the conformal antenna embedded in a region with
little curvature variation can be approximated by its equivalent circular cylinder, but such
an approximation fails for the case of an antenna embedded in a surface with significant

curvature variation.

80

4.5. Conclusion

In this chapter, from the numerical results and discussion above, it is demonstrated
that the exterior and interior portions of a hybrid finite element-boundary integral
computer program have been validated for an empty cavity, conformal slot antenna, and
conformal patch antenna. In the next chapter, multiple patch antennas embedded on an
elliptic cylinder will be studied to assess the effects of mutual coupling between patch

antennas mounted on surfaces with varying curvature.

81

 

u=u0+6 W

W

If

 

 

 

u=u0-6

 

 

 

Figure 4.1 .The geometry of model of source generator.

82

Km

 

Empty Cavity

 

Unit Cell

“13 SLEVEZ '0

Figure 4.2 An empty cavity: 1.5 cm x 6.0 cm x 3.75 cm and its unit cell.

83

Impedance [Ohms]

 

Impedance vs. Frequency (cavity dimensions: 1.5 cm x 6 cm x 3.75 cm )

 

 

 

 

1000 T r , , r
. 9
-- ' - Resrstance I
----- Reactance by LMBRICK 5
x Resistance 5
500* o Reactance J i: d
{i
i
I
!

 

 

 

 

 

 

_15m r L r r 1
3 3 5 4 4.5 5 5 5 6
Frequency [6111]

Figure 4.3 Input impedance for an empty cavity mounted in the ground.

84

Impedance [Ohms]

Impedance vs. Frequency (cavity dimensions: 1.5 cm x 6 cm x 3.75 cm)

 

 

 

 

 

15m I I I I I
Resistance
----- Reactance for p=100 cm
--- Resistance
1000 H -— Reactance for pa 5 cm , d
I,
l!
I
5m h ..

'—-—-—.-.-._.—

 

 

 

4500 — —

 

 

 

 

Frequency [GHl]

Figure 4.4 Input impedance for an empty cavity mounted in two circular
cylinders with different radii.

85

Slot Antenna

 

Unit Cell

“13 SLEVEZ '0

Figure 4.5 Slot antenna: 1.5 cm x 6.0 cm x 3.75 cm and its unit cell.

86

Impedance [Ohms]

e
s
a
.necooeyoeoozooaoosooa000:0

Impedance vs. Frequency (cavity dimensions: 1.5 cm x 6 cm x 3.75 cm)

 

000000 -

.JanCFC25)Wc>

 

*‘ I
-200- 6" ‘9
' 1
i 6"
«100)- 1: ~
‘1
_m 1 1 1 1 1 1 1 1 1

I r i T I T f I
Resistance
-'-~ Reactance by LMBRICK

 

 

0 Resistance
at Reactance

 

 

 

 

 

4 4.2 4.4 4.6 4.8 5 5.2

Frequency [GHz]

5.4 5.6 5.8 6

Figure 4.6 Input impedance for slot antenna embedded in a ground plane;

£,=1-j0.01,u,=1.0,

87

Impedance [Ohms]

Impedance vs. Frequency( cavity dimensions: 1.5 cm x 6 cm x 3.75 cm )

 

I I I I I I T I I

      

— Resistance for p= 5cm
-------- Resistance for p= 10cm
_._., Resistance for D: 30cm

   
  

 

 

...........................

l l
4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4

Frequency [GHz]

Figure 4.7 Input resistance for slot antenna on cylinders; 5r =1“ 1.0-01: “r =10 .

88

Impedance [Ohms]

Impedance vs. Frequency ( cavity dimensions: 1.5 cm x 6 cm x 3.75 cm with slot)

   

 

   
 

1m 1 x I ' ‘ F
—— Reactance for p- 5 cm
-------- Reactance for p=10 cm
0 _ In —--~ Reactance for p-30 cm

  

 

 

L I I l

I l
4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4

 

Frequency [GI-11]

Figure 4.8 Input reactance for slot antenna on cylinders; 5r =1- j0.01, “r =1-0 .

89

Impedance vs. Frequency ( cavity dimensions: 1.5 an x 6 cm x 3.75 cm with slot)

 

Cl)
C)
C:

l I I

 

........ Vat-W2 I1
_._.. VO~II/6 ,'
___ vo~ra ,-' .
—— cylinder, p=25 cm "'

 

 

 

400

300

Impedance [Ohms]

200

100

 

 

 

 

1 ‘ 1
4.6 4.7 4.8 4.9 5 5.1 5.2

lac:
&
A
(ll

Frequency [GHz]

Figure 4.9 Input resistance for slot antenna on cylinder and elliptic cylinder with
a=50 cm, b=25 cm; s, =l—j0.01, u, =l.0.

90

100

-200

Impedance [Ohms]

—400

-500

I

Impedance vs. Frequency ( cavity dimensions: 1.5 cm x 6 cm x 3.75 cm with slot)

I

 

 

—600
44

 

 

 

..1 Vo~‘n:/2l
_ _u Vo~1U5
II: I, -...- V :19 1'
F]. I I” — cylinder, p-ZS cm

 

 

 

 

4.7 4.9 4 5 5.1 5.2
Frequency [GHz]

Figure 4.10 Input reactance for slot antenna on cylinder and elliptic cylinder with
=50 cm, b=25 cm, 5r =1—fO-01, u, =1.0,

91

TM 010

 

 

 

TM 001

 

 

F' re 4 11 Fields configurations (modes) for rectangular microstrip patch.
Igu -

92

TM 010

 

   

 

 

 

 

 

 

 

h=0.01 1 cm
L=4.0 cm
w=3.0 cm
Impedance vs. Frequency for patch antenna
— Resistance p=500.0 cm
-------- Reactance p=500.0 cm ,
50 ~ _

7

E

.C

9.

0

U

C

(U

U

G)

O.

E

_50 L L L I I 4 I I I
1.8 1.85 1.9 1.95 2.05 2.1 2.15 2.2 2.25 2.3

2
Frequency [GHz]

Figure 4.12 Input impedance for the rectangular patch antenna embedded in a ground
plane.

93

TM 010

 

 

Impedance vs. Frequency for patch antenna

8

 

 

 

 

 

 

 

 

 

I I I I L I r
— Resistance p=500.0 cm
-------- Reactance p=500.0 cm
50 ~ —

'3

E

.2

2

0

0

C

1e

'0

a:

Q.

.E

_50 1 4 1 1 1 1 1
1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency [GHz]

Figure 4-13 Input impedance for the rectangular patch antenna embedded in a ground
plane; 6', = 3.29—j0.01316, ur =10 '

94

m

TM 010

 

 

Impedance vs. Frequency for patch antenna

A
vv I l I I I

_a
>

 

 

— Resistance p=500.0 cm
-------- Reactance p=500.0 cm

 

Impedance [Ohms]

 

 

 

 

I I + I 1 I 1 I l
.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3
Frequency [GHz]

 

_.c:>

Figure 4-14 Input impedance for the rectangular patch antenna embedded in a ground
plane; 8,. = 3.29-j0.013l6, u, = 1.0 .

95

TM 010

 

 

Impedance vs. Frequency for patch antenna

I

 

I I

 

— Resistance p=500.0 cm
-------- Reactance p=500.0 cm

 

Impedance [Ohms]

 

 

 

 

:'
:n 4 1 1 1‘

1.8 1.85 1.9 1.95

 

I l I I I
2 2.05 2.1 2.15 2.2 2.25 2.3
Frequency [GHz]

Figure 4.15 Input impedance for the rectangular patch antenna embedded in a ground
plane; a, =3.29—j0.01316, u, :10,

96

Cavity: 0.1 cm x 6.0 cm x 8.0 cm
Patch: 3.0 cm x 4.0 cm

9 0.0;
r :3”
50...;

 

Unit Cell

 

“135'0

 

 

 

0.5 cm

Figure 4.16 Patch antenna: 1.5 cm x 6.0 cm x 3.75 cm.

97

Impedance vs. Frequency ( cavity dimensions: 0.1 cm x 6 cm x 8 cm)

 

150 I I w 1
— Resistance
------ Reactance by LMBRICK
0 Resistance
x Reactance

 

 

 

 

 

 

 

 

 

 

 

7,
E s -
.fl 2
9 a
s
E i
'3
E‘
i; x
i I!
-50- :8
_1m 1 1 1 1 1
2.5 3 3.5 4 4.5 5 5.5

Frequency [6112]

Figure 4.17 Input impedance for a patch antenna in a ground plane; 5r = 2-0’ “r =1-0.

98

Impedance vs. Frequency for patch antenna

 

 

 

 

 

 

 

 

 

150 T I I T I L I I
— Resistance p-SOO cm
--- Resistance p- 15 cm
0 Resistance p- 10 cm
100 — -
'3‘
E
.C
Q.
0
U
C
'2’.
§
50 — .
o I I I I I I I I
3.02 3.04 3.06 3.08 3.1 3.12 3.14 3.16 3.18 3.2

Frequency [GHz]

Figure 4.18 Resistance for a patch antenna on cylinders; 3r = 2-32 " f 0.0, “r =1-0.

99

Impedance vs. Frequency for patch antenna

 

100 I I I r I I

 

 

Reactance (1:500 cm

 
 

0 Reactancepa 10 cm

 

--- Reactancep- 150m _

 

 

Impedance [Ohms]
o 8

I
M
O

I

l
8

 

 

l

I L I I

 

 

.80 l I I
3.02 3.04 3.06 3.08 3.1 3.12 3.14 3.16 3.18
Frequency [GHz]

Figure 4.19 Reactance for a patch antenna; 5r = 2-32 '1' 0-0. u, =10

100

3.2

Impedance vs. Frequency for patch antenna
I l l

 

350

300

Impedance [Ohms]
8 8
O O

_s
0!
O

  
 

 

-— Resistance p=500 cm
--- Resistance p: 15 cm
' Resistance p= 10 cm H

 

 

 

 

 

 

0 I. ' — I I I
2.3 2.32 2.34 2.36 2.3

 

8 2.4 2.42 2.44 2.46
Frequency [GHz]

Figure 4.20 Resistance for a patch antenna with probe feed located at (-1.25,0.0)

and patch size of 4.0 cm x 3.0 cm; 5r = 2-32-10-00, “r =1-0.

101

Impedance vs. Frequency for patch antenna

 

  
  

150

Impedance [Ohms]
O

 

 

-100

-150

 

 

— Reactance p-SOO cm
--- Reactance p- 15 cm
------ Reactance p- 10 cm “

 

 

 

 

 

2023 2.32 2.34 2.36 2.38
Frequency [GHz]

2.4 2.42 2.44 2.46

Figure 4.21 Reactance for a patch antenna with probe feed located at (-1 25,00)
and patch size of 4.0 cm x 3.0 cm, 5r = 2'32 ‘fO-OO, “r =1-0.

102

Impedance vs. Frequency for patch antenna
250 r 1

 

 

-— Resistance WSOO cm
--- Resistance p= 15 cm
------- Resistance p= 10 cm

 

 

 

 

Impedance [Ohms]

50

 

 

 

3.2 3.25

Figure 4.22 Resistance for a patch antenna with probe feed located at (-1.0, 0.0)
and patch size of3.0 cm x 3.0 cm; 3r = 232—1000, "r =1-0.

103

Impedance vs. Frequency for patch antenna

 

150 1 r

100

Impedance [Ohms]
o 8‘

1
8

-100

 

 

 

__ Reactance p=500 cm
_-- Reactance pa 15 cm
........ Fieamflaflflfliifll 1C)cnn

 

 

 

 

 

 

-150 '
3 1305 31

1115 132 1125

Frequency [GHz]

Figure 4.23 Reactance for a patch antenna with probe feed located at (-1.0, 0.0)
and patch size of 3.0 cm x 3.0 cm; 5r = 2-32 —j0.00, “r =1-0.

104

Impedance vs. Frequency for patch antenna

   
  

 

250
_ eSISIance vo-1.
_ Resistance vo-O.314

________ Resistance vO-O.005

  
 

 

Impedance [Ohms]

 

 

A 1 l L I

 

 

3.1 3.15
Frequency [GHz]

Figure 4.24 Resistance for a patch antenna with patch size of 3.0 cm x 3.0 cm
mounted in an elliptic cylinder; 5r = 2-32‘1'0-0, “r =1-0.

105

Impedance vs. Frequency for patch antenna

 

150

100

o 8

Impedance [Ohms]

I
8

-100

 

 

_ Reacfance vo-i .570
___ Reactance vo-O.314
Reactance vo-ODOS

 

 

 

 

..
.0
-
II
.....
.n
,,,,

I

 

-150

 

 

 

3A 315 112

Frequency [GHz]

1
1105

1125

Figure 4.25 Reactance for a patch antenna with patch size of 3.0 cm x 3.0 cm

mounted in an elliptic cylinder; 5r = 2-32 ‘f 0.0, “r =1-0.

106

CHAPTER 5
MUTUAL COUPLING BETWEEN MICROSTRIP ANTENNA

5.1 Introduction

The mutual coupling between microstrip antennas mounted in a ground plane and in
circular and elliptic cylinders is investigated in this chapter. A moment method solution
to the microstrip antenna problem was proposed [13] in 1981 and the mutual coupling
between patch antennas embedded on the ground plane with infinite extended substrate
was calculated and measured by Pozar [14] and Carver [15], respectively. In this chapter,
the numerical results using FE-BI method are compared with these moment method
results.

The mutual coupling between patch antennas embedded in circular cylinders with

different radii is calculated in this chapter. The mutual resistance, reactance and coupling

coefficient, S12 , are plotted with respect to frequency to analyze the effects of curvature
on coupling. Also, the same antenna is mounted on different portions of the elliptic
cylinder, corresponding to different local curvature, and the computed results are
discussed. The field structure is primarily determined by the position of the probe feed,
and the feed location is found to impact the mutual coupling. Therefore, the probe feed is
relocated and numerical results for coupling for various feed configurations are inspected
to assess the influences of the location of the probe feed on mutual coupling.

In addition to curvature, the position of the probe feed, the size of patches and the
distance between the two rectangular patches play an important role in mutual coupling.
In this chapter, the various patch dimensions are used to analyze the effects of patch size

on mutual coupling. Also, the numerical results are computed for the antenna mounted in

107

circular cylinders with different radii. For convenience, symmetric patches are used in the

examples.

In this chapter, a two-port network model is used to determine mutual coupling. The

coupling parameter S12 is determined from the input impedance and coupled impedance

using conversion between S-parameters and Z-parameters.

5.2. Mutual Coupling
To analyze coupling between the two probe fed microstrip antennas, a two-port

network representation is used. The relation between the port voltages and currents are

[V1]=[Zn 212] [11] (5.1)
V2 221222 I2

The self-input impedance Z” and Z22 can be determined using (4.7), giving

defined as

 

 

Zn '-
15
(5.2)
E”) -J§2)dV
222 = 2
I0

where E“) are the electric fields due to the source current J I” at port one when the
source at port two J $2) is turned off, and E0) is the electric field due to the source

current J $2) at port two when the source at port one is turned off. The coupling

impedance Z2, can be determined by the following relationship,

108

_ ‘1 (2) (1) _ *1 r (1) (1)
22,734,]; -J,. dV—ELm —E )-J, dV
_ ‘1 r (I) 1 (I) (1)
—I—2-LE -J,- dV—I—ZLE -J,. (W (5.3)
O 0
=A-A.
where E' is the total electric field due to both J9) and JEZ) . Also, Z, is the impedance

when both J E” and J 52) are used. Generally, 2,2 = Z21. For simplicity, a unit current is
used for 10 here. It is noted that the self-input impedance of port one is equal to that of
port two (Z,, = Z22) when the two patches are symmetrically located. For microstrip
antennas mounted on an elliptic cylinder, if the two patches are placed in regions with
different surface curvature, then Z1, ¢ Z22 even if the two patches have the same area.

The coupling parameter 5,2 is determined from the following formulation [32]

S _ 221220
1 (Z11 + Zo)(Zzz + 20) — 212221

 

(5.4)

where 20 is 50 (2 here.

5.3. Numerical Results and Discussions

The calculation results of the mutual impedance between two coax-fed microstrip
antennas are shown in this section. Several characteristics of the microstrip antenna are
observed from the presented calculations. The E-plane and H-plane are associated with
the arrangement of the patches and the location of the probe feed and are used here to
facilitate comparison with reference data. The distance between patches is varied to
observe the influence of separation on mutual coupling and resonant frequency. Also, the

effects on coupling due to the surface curvature are checked by applying several

109

scenarios of antennas mounted on differing circular cylinders and on different portions of
the elliptic cylinder.

Since the patch size of the microstrip antenna plays an important role not only on the
strength of the surface wave being excited, but also on the resonant frequency of the
antenna, the computations presented also include several scenarios for observing the

influence due to the patch size.

5.3.1 Comparisons between F E-BI and Moment Method for H-Plane Coupling

For a microstrip antenna embedded on a plane ground coated with substrate, the
mutual coupling between patch antennas has been presented by Pozar [14] in 1982. In
that paper, a moment method solution using the rigorous dielectric slab Green’s function
is presented. Also, the measured results were published in 1981 by Carver [15]. The

geometry of two rectangular microstrip patches is shown in Figure 5.1 and the results for

. s . . . .
mutual couphng vs. — are shown 1n Figure 5.2, where s IS the distance between the

patches. Figure5.2 presents good agreement for mutual coupling (S12) data comparing
measurements and computed results using a moment method solution for two coax-fed
microstrip antennas.

To verify the FE-BI method presented in this dissertation, comparisons are made
with the results shown in Figure 5.2. The microstrip antenna was mounted on a circular
cylinder with a very large radius so that the antenna can be considered mounted on a
ground plane. For H-plane coupling, the geometry in Figure 5.3 is used. The size of
cavity is 0.1588 cm x 35 cm x 24 cm. The size of each rectangular patch is W=10.0 cm

and L=6.0 cm. The cavity—backed antenna was meshed into 420 elements with 1021

110

unknowns using a unit cell with dimensions 0.1588 cm x 1 cm x 2 cm. The dielectric

constant of the cavity was 6, =2.55. The mutual coupling, 512 , vs. 1:: is shown in Figure

5.4. In Figure 5.4 the coupling computed using F E-BI is greater than that using a moment
method solution. Physically since the aperture in FE-BI is not electrically large, there are
interactions between the fields and the cavity walls that impact the mutual coupling. Such
boundary conditions are not present in the moment method model. The size of the cavity
is not sufficiently large so that the antenna fields damp out enough before hitting the
walls of the cavity. The other reason for this deviation may be coming from the position
of the probe feed. If the center point of each patch on the aperture is considered as origin,
then the position of the probe feed in Pozar’s computation is on the central axial line and
probably slightly above the lower edge of the patch. In comparison, the location of the
probe feed for the FE-BI method is on the central line but 1 cm above the lower edge of
the patch.

To improve the agreement illustrated in Figure 5.4, the dimensions of the cavity are
extended to 0.1588 cm x 53 cm x 30 cm while the patch size remains the same. The total
elements and unknowns are increased to 795 and 2104, respectively and each unit cell
remains the same size of 0.1588 cm x 1 cm x 2 cm. The comparison for mutual coupling
using FE-BI with a moment method solution and measured data is shown in Figure 5.5.
In this figure, it is observed that the agreement between the measured data and the FE-BI
computed results have improved.

For convenience, the FE-BI results in Figure 5.5 and 5.4 are presented together in
Figure 5.6. In Figure 5.6, it is observed that mutual coupling contributed from the

standing wave becomes important when the separated distance, 3, increases. This

111

indicates that the reflective fields become the dominant coupling mechanism for coupling
when the antenna separation increases. Meanwhile, the direct fields attributed to coupling
become weaker. On the other hand, when the separation becomes smaller, the deviation
between these two computed results reduces. At that time the direct fields are dominant
for the mutual coupling.

Next, the position of the probe feed is relocated to the central axial line on the lower
edge. An illustration of this geometry is shown in Figure 5 .7. The numerical results are
shown in Figure 5.8. Figure 5.8 illustrates the good agreement between the FE-BI
solution, moment method solution, and measured data.

The resonant frequency found using the FE-BI method is 1.34 GHz, which is less
than the 1.410 GHz computed by a moment method solution. Theoretically, for H-plane
coupling, since the patch length L=6 cm used for FE-BI computation is shorter than
L=6.55 used in a moment method solution, the resonant frequency should be higher than
1.410 GHz. Therefore, there is some accuracy problem arising from cavity meshing for
the FE-BI model. To improve accuracy, a new cavity of 0.1588 cm x 51 cm x 16 cm is
created and meshed finer into 832 elements with 2101 unknowns and with unit cell
dimensions of 0.1588 cm x 1 cm x 1 cm. The computed results are shown in Figure 5.9
and the resonant frequency is 1.430 GHz, which is slightly higher than 1.410 GHz
computed by a moment method solution. For convenience, the results for both cases, in
which the probe feed is placed in the axial central line of the patch and 1 cm above the

lower edge and right on the lower edge, are shown in Figure 5.9. Figure 5.9 shows a good

agreement between the computation results and measured data.

112

5.3.2 Comparisons between FE-BI and Moment Method for E-Plane Coupling

For E-plane coupling, the cavity-backed antenna with a 0.1588 cm x 35 cm x 30 cm
cavity was used as shown in Figure 5.10.It was meshed into 1050 elements with 2749
unknowns with unit cell dimensions of 0.1588 cm x 1 cm x 1 cm. The position of the
probe feed was placed in two different locations. One was on the horizontal central line
of the patch and along the right edge of the patch (i.e. F E-BI case (1) in Figure 5.11)
while the other was on the horizontal central line of the patch and 1 cm left of the right
edge (i.e. F E-BI case (2) in Figure 5.11). The computed results are shown in Figure 5.11.
Good agreement between computed results using the FE-BI model, a moment method

solution, and measured data is achieved except for the case with a very small separation

. s .
between the two rectangular patches, i.e. -— <0.2. Here the resonant frequency Is the

same as H-plane coupling, i.e. f, =1.430 GHz. Also, the resonant frequency is
independent of the position of the probe feed as long as the probe feed is located on the

central line of the patch, since such a feed location excites a single mode. For the case

with i <0.2, the patch antennas is so close to each other such that the fields

dramatically vary with respect to position in the cavity. Hence the finer meshing of the
antenna geometry is required for the FE-BI method to accurately compute the fields
inside the cavity and upon the aperture. A cavity-backed antenna with a 0.1588 cm x 20
cm x 14 cm cavity was meshed into 1120 elements with 2201 unknowns with unit cell
dimensions of 0.1588 cm x 0.5 cm x 0.5 cm. A probe feed was placed on the horizontal

central line of the patch and along the right edge of the patch (i.e. F E-BI case (1) in

113

Figure 5.12) while the other was on the horizontal central line of the patch and 1 cm left
of the right edge (i.e. F E—BI case (2) in Figure 5.12). The computed results are shown in
Figure 5.12. Good agreement between computed results using the FE-BI model, a
moment method solution, and measured data is achieved even for the case with a very
small separation between antennas.

In Figure 5.9 for the H-plane coupling and Figure 5.12 for the E-plane coupling, it is
concluded that the mutual coupling level decreases monotonically with increasing
separation between patches. The difference in the mutual coupling between the E-plane

and H-plane coupling increases as separation increases. This difference in mutual

coupling increases from 3 dB for ji- =0.125 to 11 dB for 7:— =0.75. For E-plane coupling

the mutual coupling is higher than that for the H-plane coupling. Physically, this is
because the surface waves and creeping waves are stronger in the E-plane case.

From the numerical results and discussions above, it is concluded that the cavity size
should be made large enough to ensure that there is no interaction between fields and
wall boundary to contribute to increase mutual coupling. Then the case of a patch on the
top of the infinite extended substrate could be approximated. However, in practice
Operational concerns dictate the smallest cavity possible and hence the need for an FE-BI

model to assess the design trade-offs inherent in such designs. Also, the cavity should be

subdivided into finer elements with an edge length of 3% to achieve greater accuracy.

For E-plane coupling case with ISO— <0.2, the length of unit cell should be less than 410 to

have a reliable computed results.

114

The numerical results from the FE-BI method have shown a very good agreement
with a moment method solution and measured data for E-plane and H-plane coupling
between the microstrip antennas mounted on the ground plane. In the next section a
microstrip antenna will be embedded in a surface with curvature to see the effects on

mutual coupling by the surface curvature.

5.3.3 Numerical Results and Discussions for H-Planc Coupling on a Curved Surface

In section 5.3.2 and 5.3.3 the computed results of E-plane and H-plane coupling for
two microstrip antennas mounted in a ground plane were presented. In this section,
microstrip antennas mounted on surfaces with different curvatures are used to analyze the
variation of the mutual coupling with respect to the surface curvature.

The geometry of a microstrip antenna mounted in a curved surface with two
identical 3 cm x 3 cm patches is shown in Figure 5.13. For this case the two probe feeds
are placed in the locations corresponding to 0.5 cm below the center point of each patch
and along the central line. This cavity-backed patch antenna was meshed with 200
elements consisting of 373 unknowns and with unit cell dimensions of 0.1 cm x 0.5 cm x
0.5 cm. The mutual resistance and reactance versus frequency for an antenna mounted in
different circular cylinders with radii of 25 cm, 50 cm and 100 cm are shown in Figure
5.14 and 5.15, respectively. The mutual coupling (S12) vs. frequency is shown in Figure
5.16. From Figure 5.14 and 5.15, the mutual resistance and reactance have greater
Variation around the resonance when the antenna is mounted on the cylinder with less

Curvature. This is because the surface wave being excited on a surface with less curvature

has little energy shedding away from the surface and more energy can reach the other

115

patch, resulting in greater mutual coupling. In Figure 5.16 it can be observed that the
peak value of mutual coupling occurs at resonance since at this frequency the maximum
energy is radiated from the patch antenna. Also, the difference in the magnitude of
mutual coupling for the antenna mounted in the cylinder with p = 100 cm and p = 25 cm
is about 10 dB. For an antenna mounted in a region with high curvature, the surface wave
has greater energy loss due to the fields shedding away from the surface, thus H-plane
coupling demonstrates lower mutual coupling for the case with p = 25 cm.

For the same microstrip antenna mounted in an elliptic cylinder with major axis
a=50 cm and minor axis b=25 cm, computed results for mutual resistance, reactance and
coupling associated with different locations on the elliptic cylinder are shown in Figure
5.17, 5.18 and 5.19. From these results, when the antenna is embedded in the elliptic
cylinder starting from v0 = 0.02 , which is a highly curved region, the magnitude of the

mutual resistance and reactance are much smaller compared to values for the antenna
mounted in the elliptic cylinder starting from v0 =% or % (e.g., regions with less
curvature). There is little difference in the mutual coupling for the antenna mounted in
the elliptic starting from v0 = :6:- compared to v0 = g- . Therefore, the main variation for

the coupling happens when the antenna is in a region with high curvature. For antennas
mounted on both a circular cylinder and an elliptic cylinder, it can be concluded that
coupling decreases with decreasing radius of curvature for H-plane coupling.
For the H—plane coupling the field in the space between the patches is primarily a TB
mode and there is not as strong a dominant mode surface wave excitation; therefore there

is less coupling between the patches.

116

 

5.3.4 Numerical Results and Discussions for E-Plane Coupling on a Curved Surface
The geometry of a patch antenna mounted on a curved surface with E-plane coupling

between patches is shown in Figure 5.20. Here two coaxial probe feeds are located 0.5

cm to the lefi of the center point of each patch and along the central line of the patch. The

cavity-backed patch antenna was subdivided into 200 elements with 373 unknowns and

mm

with unit cell dimensions of 0.1 cm x 0.5 cm x 0.5 cm. The mutual resistance and

m-

reactance as a function of frequency for an antenna mounted on different circular
cylinders with radii of 25 cm, 50 cm and 100 cm are shown in Figure 5.21 and 5.22. The
mutual coupling vs. frequency for these cases is shown in Figure 5.23.

From Figure 5.21 and 5.22, the absolute value of the mutual resistance and reactance

at resonance have increased with increasing radius from p = 25 cm to 50 cm, then
decreased with increasing radius fiom p = 50 cm to 100 cm. In Figure 5.23, it is observed
that the difference of the magnitude of mutual coupling for the antenna mounted in a
cylinder is 7.42 dB from p =25 cm to p = 50 cm and 4.52 dB from p =50 cm to

p = 100 cm. The total difference is 11.94 dB from p =25 cm to p = 100.0 cm. This is
compared with H-plane coupling in Figure 5.16 that only has a 9.72 dB difference from

p =25 cm to p = 100 cm. Hence E-plane coupling exhibits greater curvature-
dependency. Since for the E-plane arrangement the fields in the space between the
patches are primarily TM, there is a stronger surface wave excitation between the
patches, and the coupling is larger and demonstrates greater curvature-dependency. Also,
the theoretical explanation for the effects of mutual coupling as simply due to the surface

curvature for H-plane coupling is no longer completely satisfied for E-plane coupling.

117

This is because the field structure between the edges of the patches for E-plane coupling
is different from the H-plane coupling and that fields have the least energy loss traveling
at a specific curvature. It can be observed that for the E—plane coupling when the surface
is curved to a specific curvature, the creeping wave travels to the other patch along the
interface has lower loss, resulting in higher coupling.

For the same microstrip antenna mounted in an elliptic cylinder with major axis
a=50 cm and minor axis b=25 cm, computed results for mutual resistance, reactance and
coupling associated with different locations on the elliptic cylinder are shown in Figure
5.24, 5.25 and 5.26, respectively. From these results, when the antenna is embedded in

the elliptic cylinder starting from v0 = 0.02 , which is a surface with significant curvature

variation, the magnitude of mutual resistance and reactance are much smaller compared

. . . . . 7t 7t .
to values for the antenna mounted 1n the elliptic cylmder starting from v0 = g or —2-. It 18

also observed that the mutual coupling has little difference between v0 =-:— and gin an

elliptic cylinder. So the main variation for the coupling is observed when the antenna is
located in the region with high curvature. For a patch antenna mounted in a circular
cylinder, the maximum mutual coupling was observed when the radius of the surface
curvature is around 50 cm, and not for the planar case. Comparing the E-plane coupling
with H-plane coupling at the resonant frequency for the patch antenna mounted in the

elliptic cylinder, it can be observed in Figure 5.19 and 5.26 that the coupling increased

1 1.1 dB for the patch antenna moving from v0 = 0.02 to g in H-plane coupling case

while the coupling increased 8.9 dB for the same antenna moving from v0 = 0.02 to g

118

 

and decreased 1.52 dB from v0 =% to g in E-plane coupling. The total change of

coupling is 10.45 dB for the patch antenna located at v0 = 0.02 as compared to the case

. 7r . . . . .
when the antenna 1S located at v0 = 2 . It Is not obv10us which type of coupling 1S more

dependent on curvature for this antenna mounted on an elliptic cylinder with a=50 cm,
b=25 cm. However, it is expected that if an elliptic cylinder with a=100 cm, b=25 cm, is
used, the E-plane coupling will have a still greater dependence.

Next consider the case where the second probe feed is located 0.5 cm to the right of
the center point of the patch while the first probe feed remains as before. This case is
shown in Figure 5.27. The mutual resistance and reactance versus frequency for the
antenna mounted on circular cylinders with radii of 25 cm, 50 cm and 100 cm is shown in
Figure 5.28 and 5.29, and mutual coupling (S12) vs. frequency is shown in Figure 5.30.
In Figure 5.28, 5.29 and 5.30, the results are generally similar to the previous E-plane
coupling case except that the reactance and resistance now is Opposite to the associated
value in the previous E-plane coupling case. Here the mutual coupling has reached its
maximum value at p = 50 cm and decreases as the radius of curvature increases or
decreases. Comparing the mutual coupling with the value for the E-plane case shown in
Figure 5.23, here the magnitude of the mutual coupling is much higher when the
frequency is less than the resonant frequency, but is of the same order for frequency

greater than the resonant frequency.

119

 

5.3.5 Numerical Results for Various Sizes of Patch Antennas

In this section the microstrip antenna with patch size of 2 cm x 2 cm, 3 cm x 3 cm
and 4 cm x 4 cm are mounted in cylinders with radii of 25 cm, 50 cm and 100 cm, and
the H-plane mutual coupling is calculated to assess the performance of this antenna with
respect to the surface curvature. The geometry of these microstrip antennas is shown in
Figure 5.31. For these cases, the two probe feeds are placed in the locations
corresponding to 0.5 cm below the center point of each patch. These cavity-backed patch
antennas were meshed into 200 elements with 373 unknowns and with unit cell
dimensions of 0.1 cm x 0.5 cm x 0.5 cm.

For the microstrip antenna with the patch size of 4 cm x 4 cm and a separation of
only 1 cm, the mutual coupling vs. frequency is shown in Figure 5.32. In Figure 5.32,
compared with others, the antenna mounted in the cylinder with radius of 25 cm has the

highest coupling at the resonant frequency and the coupling decreases from 13.08 dB for

p =25 cm to 16.48 dB for p r: 100 cm. In this case, since the separation between two
patch antennas is so small compared to the surface wavelength (i- = 0.1 1), the creeping

wave traveling the region between the patches behaves similar to the case of a ground

plane even though the antenna is mounted in the cylinder with the smallest radius of

,0 =25 cm. Therefore, the loss of energy of the creeping wave due to the curvature of
cylinder can be neglected here, and the only factor that causes the decreasing of the
mutual coupling is the wavelength at resonance. For p = 25 cm the resonant frequency is
3 .32 GHz with a resonant wavelength of 9.06 cm while the resonant fiequency is 3.38

GP12 with a resonant wavelength of 8.87 cm for p = 100 cm. Thus the creeping wave

120

with the higher frequency and shorter wavelength has larger energy loss during traveling.
That is the reason for the slightly decreased coupling when the radius of the cylinder is
increased from p =25 cm to 100 cm.

For the microstrip antenna with the patch size of 3 cm x 3 cm and the edge space of
2.0 cm, the mutual coupling vs. frequency is shown in Figure 5.33. In Figure 5.33,
compared with others, the antenna mounted on a cylinder with radius of 100 cm has the
highest coupling value at the resonant frequency and the coupling value decreases from
16.98 dB for p = 100 cm to 26.70 dB for p =25 cm. In this case, since the separation
between the patch antennas has increased, the wave traveling in this region cannot be
treated as if traveling on a ground plane. Therefore, the energy loss of the creeping wave
due to the curvature plays an important role for the mutual coupling of the microstrip
antennas. For an antenna mounted in the cylinder with p = 25 cm, the two patches on the
cylinder body are subtended by a larger angle, which results in attenuation of the space
wave and thus weakens the coupling. The difference of coupling value between p = 25
cm and p = 100 cm is 9.75 dB in this case while it is just 3.55 dB for previous case. In
this case the primary change of the coupling happens when the patch antennas are moved

from a curved area to a less curved region.

For the microstrip antenna with the patch size of 2 cm x 2 cm and a separation of 3
cm, the mutual coupling vs. frequency is shown in Figure 5.34. In Figure 5.34, compared
with others, the antenna mounted in the cylinder with radius of 100 cm has the highest
coupling value at the resonant frequency and the coupling value decreases from 19.3 dB
for p = 100 cm to 29.90 dB for p = 25 cm. In this case, the separation between the patch

antennas is large enough so that the energy loss of the creeping wave due to the curvature

121

 

 

 

 

. _S.”

plays an important role for the mutual coupling of the microstrip antennas. The difference
of coupling value between p = 25 cm and p = 100 cm is 10.6 dB in this case, which is
slightly higher than the case with the patch size of 3 cm x 3 cm. Also, in this case the
primary variation of the mutual coupling appeared when the patch antennas are removed

to a pseudo-ground plane from the circular cylinder with p = 50 cm.

 

In Figure 5.32, 5.33 and 5.34, among all patch antennas with different patch sizes

mounted in a cylinder, the coupling value is the highest for the patch with size of 4 cm x

 

4 cm at resonance. This is not only because the size of patch is the largest but also the
edge space between the patches is the smallest. Also, the shape of the mutual coupling vs.
frequency curve for the antenna with patch size of 4 cm x 4 cm is the sharpest while it is

the broadest for the antenna with patch size of 2 cm x 2 cm.

5.4 Conclusion

In this chapter, the mutual coupling between patch antennas was investigated. For

 

the microstrip antenna mounted in the infinite ground plane, the numerical results agree
with the data provided by measurements and the numerical results using moment method
for both E-plane and H-plane coupling. It should be noted that interactions with the side

walls of the cavity can alter the coupling. Also, the cavity should be meshed into

elements with length less than 510 to have accurate results. For E-plane coupling case

With —S— <0.2, the length of unit cell should be less than 2% to have a reliable computed

results. Physically, for the H-plane coupling, the surface fields in the space between the

patches are primarily TE and there is not as strong a dominant mode surface wave

122

 

 

excitation; therefore there is less coupling observed between the patches. For the E-plane
coupling the fields in the space between the patches are primarily TM, therefore, the
surface wave excitation is stronger between the patches and hence the coupling is greater.

For a microstrip antenna mounted on a circular cylinder and an elliptic cylinder, the
mutual coupling for patch antennas is curvature-dependant. For the H-plane coupling, the
coupling decreases as the radius of curvature increases. Therefore, coupling effects
between patch antennas generally reaches its maximum when it is placed in the ground
plane. Physically, more energy of the surface wave sheds away from the surface in a
region with high curvature, which weakens the antenna coupling. However, for the E-
plane coupling case, the highest coupling occurs at some specific curvature. Generally the
E-plane coupling is more curvature-dependant since there is a stronger surface excitation
between the patches.

However, for the numerical results and discussions for E-plane coupling on a curved
surface, the mutual resistance and reactance as a function of frequency for an antenna
mounted on different circular cylinders shows that the absolute value of the mutual
resistance and reactance at resonance have increased with increasing radius from the
radius of 25 cm to 50 cm, then decreased with increasing radius from the radius of 50 cm
to 100 cm. The numerical results should be further analyzed in future work.

For microstrip antennas with different patch sizes and H-plane coupling, an antenna
with larger patches and smaller separation between patches has greater coupling. The

Variation of the mutual coupling due to the surface curvature is more obvious when

\S > 0.11. The primarily change in the mutual coupling due to the variation of the

,1“

123

 

 

 

surface curvature occurs either for a region with a high curvature or for a region with a

less curvature, depending on the patch size and the separation.

 

 

 

124

 

 

7//

xiii/I‘m”

   
   

   

 

  

\\

nnnnnnnnnnn

 

 

WWW

Port2

 

       

PPPPP

 

E-plane H-plane

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'5 T a
l\ O
V) In
6 O O .n.
v-| . . \0

II 1 u
3 - ...I

— . — I I — l _> I
W=10.57 cm
L=6.55 cm
Mutual coupling between two microstrip antennas
-10 I l I I fi' I
-— Moment Method (Pozar)
are Measured (Carver)
-15 )- _.
is
-20 _ ..
are
E-plane
— _25 -— ..
g are are are
a"; at
g)- -30 I. _.
-35 _ _
H-plane
40 ~ ~
alt
_45 1 1 l 1 1 L ¥
0 0.2 0.4 0.6 0.8 1 1.2 1.4

s/lambda

Figure 5.2 Measured and calculated mutual coupling between two coax-fed
microstrip antennas, for both E-plane and H-plane coupling.
W=10.57 cm, L=6.55 cm, d=0.1588 cm, dielectric constant
=2.55 (David M. 1982).

126

 

 

 

Patch antenna

0.1588 cm

24 cm

 

 

35 cm

Patch size and location of probe feed

 

L=6 cm

 

 

|
1 W=10 cm

Figure 5.3 Geometry for patch antennas with H-plane coupling in pseudo-
ground plane.

127

-1O

Mutual coupling between two microstrip antennas

 

 

 

 

 

 

 

 

f T I F L I
-— Moment Method (Pozar)
III Measured (Carver)
15 -A- FE-BI
4
\\
—20 \ _
\
\
\
A I“
5' -25 \\ ,A’ _
:2. \ ,I’
\
IL"! ‘15
g; _30 H-plane d
-35 _
-4o —
III
_45 1 1 1 1 1 1 t
0 0.2 0.4 0.6 0.0 1 1.2 1.4
sllambda

Figure 5.4 Mutual coupling for H-plane case with cavity of 0.1588. cm x 35 cm

x 24 cm shown in Figure 5.3 for the FE-BI method.

128

' -J
I

 

 

Mutual coupling between two microstrip antennas
'10 I I I T

 

 

 

 

 

 

 

 

 

- Moment Method (Pozar)
a Measured (Carver)
40,- FE-BI
-15 ..
A
\
-20 _ \ _
\
\
13‘
\
u—u .- \\
CD ‘25 A 1
3 ‘x
‘8. \\
_ _30 _ \\\ _
n\
‘\ H-plane
‘A
‘35 1‘ -(
-40 — —
In
_‘5 1 1 1 1 1 L t
0 0 2 0.4 0.6 0 0 1 1 2 1 4
s/lambda

Figure 5.5 Comparison of mutual coupling calculated by FE-BI with a moment
method solution and data by measurements; the size of cavity-backed
antenna in FE-BI calculation is 0.1588 cm x 53 cm x 30 cm.

129

-16

Mutual coupling between two microstrip antennas

 

-18 )—

-2° .—

-24 I.

-26 I-

1812120181

 

I I

I

 

 

44- cavity size: 0.1588 cm x 35 cm x 24 cm
0- cavity size: 0.1588 cm x 53 cm x 30 cm

 

 

H-plane

 

 

~36
0. 1

0.2 0.3

0.4

sllambda

Figure 5.6 Comparison of mutual coupling using FE-BI method between
different size of cavity of 0.1588 cm x 35 cm x 24 cm and 0.1588

cmx 53 cm x 30 cm.

130

 

Patch antenna

 

1r—
E
O
o
('1
AL

   

          

0' ('0 0..

we . .
U“
.0 I0

             

           

       

  
  

 

      

     

              
            

               

S
as” .
new.“ we..." a...
. v.03 . .. . as?
"w... .2" «a “use "w.
ea...” a a.“ Rs... . . .r K. "mu”
| C C . C O Q A O
”0an A “"0900”; . .Noo.
0. fin. god .0 so. “mom“
5... .. ”$52 as... ,. ..

   
  

“’40an O AfiO- .1“ .
fifima

4. Men...”

Q

h .

- .3... maids.
kayaétfimmm

C
sheave... . mmwumm

5n.
@3me .... "saws

      

      

04“

0“ I 0%
. tom WM%M"«.WBW% WM. 1.9% w% r .
V 430 d.“ a. a" a OOOW-HQC toae‘ootooo. I
%%~&%.W¥fie&s a. %
fififimflfififififiwfiwfi

   

   
   

   

     

53 cm

ize and location of probe feed

Patch s

 

 

 

5cm

 

 

I

10 cm

 

Figure 5.7 .Geometry for patch antennas with H-plane coupling in pseudo-

ground plane.

131

Mutual coupling between two microstrip antennas

 

’10 ‘ I fir I f

 

 

 

 

 

 

 

\ --- Moment Method (Pozar)
\ 4* Measured (Carver)
15 ‘\ O FE'B|
‘\
\
l.
\
I?
\
\
\\
a. -25 e ‘0 4
1: ‘3
01": \
_ .30 1' \\\ -l
,9 H-plane
\
\\
*0
’3 l" \\‘J\ "1
‘0--3‘
““*\
40 r ~“* -1
*-
45 1 1 L 1 1 L *
o 0.2 o 4 0.6 0 a 1 1 2 1 4
sllambda

Figure 5.8 Comparison of mutual coupling calculated by FE-BI with a moment
method solution and data by measurements; the size of cavity-backed
antenna in FE-BI calculation is 0.1588 cm x 53 cm x 30 cm, shown in
Figure 5.7.

132

"it I! -

 

|S1ZF[dB]

Mutual coupling between two microstrip antennas

 

 

 

 

 

 

 

 

-10 1 1 r r m I

\ --- Moment Method (Pozar)

\ 4: Measured (Carver)

\ A FE-Blcase(1) ‘
'15" I, o FE-Blcase(2)
\
‘3
-20- sh _.
2K
\A
-25— Q\ —(
\<A
‘9‘ A
40" \\\o "
*\
\\\
‘K H- lane
-35'- 4“ p n
o ‘g‘
0 ~\
-40~- “it ~
41

_45 l l I l 1 l t

0 0.2 0.4 0.5 0.0 1 1.2 1.4

sllambda

Figure 5.9 Comparison of mutual coupling calculated by FE-BI with a moment
method solution and measured data; the size of cavity-backed antenna in
F E-BI calculation is 0.1588 cm x 51 cm x 16 cm and unit cell of 0.1588

cmxlcmxlcm.

133

r

 

 

Pat.h

 

 

P2111

Patch antenna

000

r"
90'.
103030?
I 'v -

 

 

35 cm

Patch size and location of probe feed

1A
I

 

 

5’cm

 

10 cm

 

 

   

 

Figure 5.10 .Geometry for patch antennas with E—plane coupling in a pseudo-
plane ground.

134

30 Cm

 

_meififl

Mutual coupling between two microstrip antennas
-10 I I I T

 

 

 

 

 

 

 

 

A --- Moment Method (Pozar)
o x Measured (Carver)
-12\ A FE-Bl case (1) H
\\ o FE-Blease(2)
1
-14_ \ ..
\\
-16- ‘\ A -(
X\
_ \
in -10— x \\ _
E \
Q. \
93 \\
9?- ’2°” \ E-plane 7
A X \
\
A \\ _1
-22— o %\
O \x
\\
-24"— o 3:\\‘ -J
o \
-26“ o‘\“ X X H
x..‘
-28 1 1 1 1 \“‘* 1 x
0 0.2 0.4 0.0 0.0 1 1.2 1.4
sllambda

Figure 5.11 Comparison of E-plane mutual coupling calculated by FE-BI with
a moment method solution and measured data.

135

R as,
E

 

if "7.
F

Mutual coupling between two microstrip antennas

 

 

 

 

 

 

 

 

‘10 I I I I fl 1
--- Moment Method (Pozar)
x Measured (Carver)
425“ A FE-Bl case (1) ~
0 FE-Bl case(2)
-14. —
\
\\
-10— XX 4
A\\
g -18 h 0" \ E-plane -
'5: A \
N \\
£120- 0 A \ -
X \
O A\\\
-22- %‘ _
O \x
\\
-24~ 0 Q \ J
o \
-20~ o“~, x x _
\“
-281 1 1 1 1 “‘30 1 x
0 02 0.4 0.0 0.0 1 1.2
sllambda

1.4

Figure 5.12 Comparison of E-plane mutual coupling calculated by FE-BI with
a moment method solution and measured data.

136

 

 

 

Pale

“gun

 

Patch antenna

 

 

10cm

Patch size and location of probe feed

 

 

 

 

 

 

 

 

 

 

3cm

 

 

Figure 5.13 Geometry for patch antennas mounted in curved surface with H-plane

Coupling.

137

Mutual coupling between two microstrip antennas

 

10 I T I T

 

 

 

 

 

 

 

 

 

— p- 25 cm

0 _ --- p- 50 cm .

0 — -
.5. I
E
.1:
0 ._
F
U
r:
8
.2 -l
10
0
c
a —4
2
3
2 _

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.0
Frequency[GHz]

Figure 5.14 Mutual resistance for patch antennas with H-plane coupling.

138

 

 

Mutual coupling between two microstrip antennas

 

12 r 1 r 1

 

 

 

 

 

 

 

 

 

 

F I r
— p- 25 cm
--- p- 50 cm
10 __ ........ p-1oo cm
'5‘
E
.c _
o
‘8‘
0
C
a
a
§ -
a:
73
3 «
nu
3
2 \
\
l l l
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7
FrequencinHz]

Figure 5.15 Mutual reactance for patch antennas with H-plane coupling.

139

4.0

 

 

 

 

 

Mutual coupling between two microstrip antennas

 

 

 

 

-15 r 1 I r T 1 T
. -— p- 25 cm
’9..4--“ 0.... --- p. 50 cm

-20 If" \\ o... . p.100 cm _.

 

   

 

 

_50 l 1 L l 1 1 L

 

4 4.1 4.2 4.3 4.4 4.5 4.0 4.7 4.0
FrequencylGHz]

Figure 5.16 Mutual Coupling for patch antennas with H-plane coupling.

 

140

Mutual Resistancethms]

 

 

  

 

 

 

 

 

 

 

 

Mutual coupling between two microstrip antennas
4 I T I I ’—\ I I I
g, \
: l \
:3, .\\
2 — :5" '\'-\\
53': Muffin-fifi‘
3.. ......fiMmM~-- -
0 ...... 3- : m :
i
g l
: l
_5 r
-2 1'
3 i
.\ z I
,1“ 1 _ Vo-0.02
—4 - 32‘“ r: ___ V0- 1116
1‘. ‘\ : ........ v0. III/2
’ \ I
-6 — ' l :r’
1 E l
t 5:
z \ : I
g 1 El
;_ 1 5‘1
-8 ~ ‘r 5! 1
3 l _:l
'-_ I :I
'. \ El
2 \Ei
-10 _ "- I." _
_12 l 1 1 J I 1 1
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
FrequencylGHz]

Figure 5.17 Mutual resistance for patch antennas mounted in an elliptic cylinder

with H-plane coupling.

141

 

 

Mutual coupling between two microstrip antennas

 

 

 

      

 

 

 

‘le I . I I
N
:' ’21.“
.~' ' 2_1
5 ‘ :1
3 ' I
_ I 1 g
E l 1
g : -
E r |
s 1 ‘
3 1 ‘
. . 1
i ' - 1
“a? .' '2.“
E 5 - l '2.“ -
.C l '1 |
O 5 1' '2 1
715' _i 1 ' ‘\
a 1' .
11! 5 1 ‘
H .‘ '
u :‘ 1
a 5 1
d) - r
u: r
a 1’
:3 1
‘5 r
2 ° ____ }
“‘~"‘~1\ l’
'-..\ 1
\\ 1
xx 1
.,\ r
-. \ 1
\ I
\ I
\ I
\ I
‘f
4 l l I'd-.1 I L l 1
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
Frequency[GHz]

Figure 5.18 Mutual reactance for patch antennas mounted in an elliptic cylinder
with H-plane coupling.

142

 

 

Mutual coupling between two microstrip antennas

    

 

 

 

 

 

_55 1 1 1 1 1 1 1
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
Frequency[GHz]

Figure 5.19 Mutual coupling for patch antennas mounted in an elliptic cylinder with
H-plane coupling.

143

 

 

Fl

Patch antenna

Patch size and location of probe feed

 

 

 

 

 

 

 

3cm

 

 

 

3cm

 

Figure 5.20 Geometry for patch antennas mounted in curved surface with E-plane

couphng

 

Mutual coupling between two microstrip antennas

 

 

 

 

 

 

 

 

 

 

 

5 I I I I I I r
'1
° ”i
'_' I
10
E ‘1 1‘
5 1 : _.
15' -5 in. 1' .
o 12 1
C l "._ 0 g
g ‘, l — p- 25 cm
10 1
._ , 1 --- p- 50 cm
é 1: : ........ p_1oo cm
‘3 -10 - I! 1 ~
3 1 l
g 1 :
I I
I I
I l
l 1'
-15 — 1 1' -
I l
l l
\J
_20 l 1 l 1 1 1 L
4 4.1 4.2 4.3 4 4 4.5 4.6 4.7 4.8

FrequencylGHz]

Figure 5.21 Mutual resistance for patch antennas mounted in circular cylinders with
E-plane coupling.

145

Mutual coupling between two microstrip antennas

 

 

 

 

 

 

 

 

 

 

 

 

15 I I I I I I I
— p- 25 cm
4 --- p- 50 cm
I, \‘ ........ p.100 cm
1’ ‘1
10 — : I, _
l l
l t
l i
I 1
'3' 1
E 1
5 1'
0' 5 ’ l .- 1
0 l
C ' :-
(In 1,-
E :5
To 0
3
3
2
-5 _ _
_10 L L
4 4.1 4.2 4.3 4 4 4.5 4.6 4.7 4.8

FrequencylGHz]

Figure 5.22 Mutual reactance for patch antennas mounted in circular cylinders
with E-plane coupling.

146

 

.‘
(II

—20

-25

0

IS Fidel
12g

 

 

Mutual coupling between two microstrip antennas

 

 

l l 1 l I I 1

 

4 4.1 4.2 4.

3 4.4 4.5 4.6 4.7 4.0
Frequency[GHz]

Figure 5.23 Mutual coupling for patch antennas mounted in circular cylinders

with E-plane coupling.

147

 

Mutual coupling between two microstrip antennas

 

 

In

 

 

 

 

Mutual Besietancethms]
l l
61 8

 

 

 

 

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.0
Frequency[GHz]

Figure 5.24 Mutual resistance for patch antennas mounted in an elliptic cylinder
with E-plane coupling.

148

 

 

Mutual coupling between two microstrip antennas

I

1:
Is:

 

    

 

 

 

Mutual Reactance[0hms]

 

 

 

4.1 4.2 4. 4.6 4.7 4.8

Figure 5.25 Mutual reactance for patch antennas mounted in an elliptic cylinder
with E-plane coupling.

149

Mutual coupling between two microstrip antennas

I ’ ‘I

 

 

 

 

 

 

so 1 1 1 l l I l
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
Frequency[GHz]

 

Figure 5.26 Mutual coupling between patch antennas mounted in an elliptic
cylinder with E-plane coupling.

150

Patch antenna

5cm

 

 

 

 

10 cm

Patch size and location of probe feed

 

 

 

 

3 cm 7' 3 cm

Figure 5.27 Geometry for patch antennas mounted in curved surface with special E-plane
coupling

151

 

Mutual coupling between two microstrip antennas

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

20 I I I I I I gl
— p- 25 cm
--- p- 50 cm
In‘ ........ p-1m cm
i I
l I
15 — g l, _
i |
I |
l I
_ 1' 1'
(a
E 1' t
5 : 1
"01° * .' i “
a 1 1
g 1 1 __
E I 5. e... ‘ ..
I: t l
is ’-, I
a 5 ,_ l'.: '1" -( .
a r '1
3 1 1
5 i
i
.5 4 1 l 1 L l 1
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.0
Frequency[GHz]

Figure 5.28 Mutual resistance for patch antennas mounted in circular cylinders with
special E-plane coupling.

152

 

 

 

Mutual coupling between two microstrip antennas
15 I I I I I I

 

 

 

 

 

 

 

   

 

 

 

--p-25cm
--- p-500m 1
........ p-1mcm I
10— 4
E 5~ _
.r:
o
‘6‘
U
1:
Ill
‘5 0
$23
To
a
= - L- -
2 5
-10» 4
_‘5 l 1 1 1 l J l
4.1 4.2 4 3 4 4 4 5 4.6 4.7 4.8

' Frequency[GHz] '

Figure 5.29 Mutual reactance for patch antennas mounted in circular cylinders with
special E-plane coupling.

 

153

 

 

 

Mutual coupling between two microstrip antennas

 

 

 

 

"15 l l I l I f f
’,..-\ — p- 25 cm
1’ \\ '" p- 50 cm
-20 - 1’ \\ """" p-100 cm L-

 

 

 

 

 

 

_55 1 1 1 1 1 1 L
4

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
Frequency[GHz]

Figure 5.30 Mutual coupling for patch antennas mounted in circular cylinders with
special E-plane coupling.

154

cm

 

10cm

 

 

10 cm

 

 

10 cm

Figure 5.31 Geometry for patch antennas with patch sizes of 2 cm x 2 cm,
3cmx3cmand4cmx4cm.

155

 

Mutual coupling between two microstrip antennas with patch size: 4 cm x 4 cm

-.
<3

 

 

~15

 

Isufldal
9

-35

-40

—45

 

 

:n l l l
-w

l l l I
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
Frequency[GHz]

 

Figure 5.32 Mutual coupling between patch antennas with patch size of 4 cm x
4 cm.

156

 

Mutual coupling between two microstrip antennas with patch size: 3 cm x 3 cm

 

'15 I r i r

T l T
‘5'- llllllllll _ p. 25 cm
X7 ‘~\ --- p- 50 cm
-20 — .’. \ -------- p-100cm ~

lsulzldB]

 

 

 

 

 

 

 

 

_50 1 1 L 1 1 1 1
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
Frequency[GHz]

Figure 5.33 Mutual coupling between patch antennas with patch size of 3 cm x

3 cm.

157

 

 

Mutual coupling between two microstrip antennas with patch size: 2 cm x 2 cm
5

_ 1 I

 

-zoL.

 

1312121131

 

 

 

 

 

Frequency[GHz]

Figure 5.34 Mutual coupling between patch antennas with patch size of 2 cm x
2 cm.

158

CHAPTER 6

SUMMARY AND FUTURE WORK

In this dissertation, a hybrid finite element-boundary integral method has been
presented that is appropriate for simulation of conformal antennas and cavities recessed
in an infinite, perfectly conducting, elliptic cylinder. New elliptic-shell elements were
developed that are suitable for discretizing elliptic-rectangular cavities. These shape
functions are surface conforming and divergence-free. These also reduce, for cylinders
whose major and minor axes are identical, to the cylindrical shell elements previously
reported in [33]. The accuracy of the finite element formulation, and in particular the new
elliptic shell shape functions, was demonstrated by comparison with known results for
circular-rectangular and planar-rectangular cavities. Resonances for an elliptic-
rectangular cavity were also presented.

In Chapter 3, the surface dyadic Green’s function for an infinitely long, perfectly
conducting elliptic cylinder was derived. In this approach, vector wave functions
representing electromagnetic fields in the elliptic cylinder coordinate system are
generated based on the elliptic cylinder scalar wave functions. Since the dyadic Green’s
function, developed by eigenfunction expansion, is very difficult to evaluate numerically,
an efficient asymptotic dyadic Green’s function was derived based on a UTD formulation
[25] and this was specialized for elliptic cylinders. The behavior of this Green’s function
as a function of geodesic path length and curvature was demonstrated.

For solution of the linear system, several techniques have been used to reduce the
large memory requirement and improve the efficiency of the solution computation. In the

computer program used to demonstrate the capabilities of this new formulation, the

159

 

:3

 

v...

 

biconjugate gradient (BiCG) solver has been chosen since it requires significantly less
memory than is required for a direct matrix solution method. The BiCG method is also
computationally efficient, since it utilizes only one matrix-vector product per iteration for
symmetric matrices. The Compressed Sparse Row (CSR) [34] storage format is used to
reduce the memory demand for the spare matrix. To increase the computation speed,
several complicated fimctions like the hard and soft type F ock function have been saved
in a data file to provide available data instead of requiring re-computing each time. Also,
the matrix for the FEM is calculated one time only and the result is saved for later use.

The exterior and interior portions of the hybrid finite element-boundary integral
computer program have been validated for the empty cavity, conformal slot antenna and
conformal patch antenna. The input resistance for a typical conformal patch antenna was
presented. The resonance frequency was seen to shift due to location on the elliptic
cylinder and this behavior is attributed to curvature variation. The input impedance and
resonant frequency are sensitive to the variation of the surface curvature for the patch
antennas mounted in a region of high curvature. Therefore, the performance of the
conformal antennas embedded in a region with little curvature variation can be
approximated by using an equivalent circular cylinder. Such an approximation fails for
the case of antennas embedded in a surface with significant curvature variation. Also, the
dependence of the performance of patch antennas on curvature is relative to the excited
mode associated with the location of the probe feed.

In this research the mutual coupling between two patch antennas was investigated.
For the microstrip antenna mounted in an infinite ground plane, the numerical results

have been shown to agree with the data provided by measurements and numerical results

160

 

 

using the moment method for both of E-plane coupling and H-plane coupling. It should

be noted that interactions with the side walls of the cavity can alter the coupling. Also,

the cavity should be meshed into elements with edge length less than % to obtain

accurate results. For patch antennas mounted in a ground plane, the H-plane coupling and
E-plane coupling have been discussed with their associated surface wave mode.
Generally, for H-plane coupling the surface wave mode between the patches is primarily
TE and coupling is less significant between the patches. For the E-plane coupling case,
the surface wave mode is primarily TM; therefore, the surface wave excitation is stronger
between the patches resulting in greater coupling.

For a microstrip antenna mounted on a circular cylinder or an elliptic cylinder, the
mutual coupling for patch antennas is curvature—dependant. For the H-plane
configuration, the coupling decreases as the radius of curvature increases. Therefore,
coupling effects between patch antennas generally reach a maximum when the antennas
are placed in a ground plane. Physically, more energy in the creeping wave is shed away
from the surface in regions with high curvature, and this weakens the antenna coupling.
However, for the E-plane configuration, the greatest coupling occurs at some specific
curvature instead of for the ground plane case. Generally the E-plane coupling is more
curvature-dependant since there is a stronger surface excitation between the patches.

For microstrip antennas with different patch size and with H-plane coupling, the
antenna with larger patches and smaller separation between patches has greater coupling.
The primarily change in the mutual coupling due to the variation of the surface curvature
occurs for a region with either higher curvature or lower curvature, depending on the

patch size and the separation.

161

 

The numerical results presented in this dissertation will serve two purposes. First,
they are used to analyze the performance of antennas with respect to surface curvature.
Second, they can be used as a reference for future developments in this area.

In future work, since the computation of fields near the probe feed or of fields with
higher variation in space requires a fine mesh near the antenna geometry, the total
number of unknowns will dramatically increase, especially when the length of the unit
cell is as small as 1/507t. Therefore, the FE-BI program should be upgraded to compute
the cavity field with higher efficiency and be capable of computing a model case with
more than ten thousand unknowns.

A non-uniform mesh should be developed for the F E-BI program to save
computation time and computer resources. Finer elliptic shell elements should be used to
mesh the regions when the fields have higher variation while coarse elliptic shell
elements are applied to mesh the region where fields have less variation. Distorted elliptic
shell elements should be developed to transition from small to larger elements and they
should retain the property of being divergence free.

For the E-plane coupling case the mutual coupling between antennas mounted on a
cylinder with a specific surface curvature is maximum compared to when they are
mounted on a ground plane, unlike the case for the H-plane coupling. Therefore, the
relation between mutual coupling and the surface curvature should be investigated
further. To achieve more accurate results, the antenna cavity should be meshed into finer
unit cells to better represent variation of the fields inside the cavity and upon the aperture

of the antennas.

162

 

 

The radiation pattern should be analyzed for the patch antennas recessed in different
portions of an elliptic cylinder to realize the influence on the radiation pattern by the
surface curvature. The formulation of asymptotic far-zone dyadic Green’s functions in
the spherical coordinate system transformed from near-zone dyadic Green’s fimction in
the elliptic coordinates system for both the lit and shadow regions should be developed
and applied to the calculation of the fields.

Also, an experiment should be set up to measure the data for mutual coupling
between patch antennas and for the radiation pattern of patch antennas mounted in

different portions of the elliptic cylinder to verify the numerical results.

163

 

 

 

[1]

[2]

[3]

[4]

[5]

[6]

[8]

[9]

[10]

[11]

BIBLIOGRAPHY

L.C. Kempel, J .L. Volakis, and R.J. Sliva, “Radiation by cavity-backed antennas

on a circular cylinder,” IEE Proc.-Microw. Antennas Propag., vol. 142, no. 3,
June 1995.

K-L Wong, Design of nonplanar microstrip antennas and transmission lines, New
York: Wiley, 1999.

J .T. Aberle and F. Zavosh, “Analysis of probe-fed circular microstrip patches
backed by circular cavities,” Electromagnetics, vol. 14, no. 2, pp. 239-258, April-
June 1994.

D.M. Pozar and SM. Voda, “A rigorous analysis of a microstripline fed patch
antenna,” IEEE Antennas Propagat. Mag, vol. 35, pp.1343-l 350, 1987.

J. Ashkenazy, S. Shtrikman, and D Treeves, “Electric surface current model for
the analysis of microstrip antennas on cylindrical bodies,” IEEE Trans. Antennas
Propagat., vol. AP-33, no. 3, pp. 295-300, March 1985.

L. Josefsson and P. Persson, “An Analysis of Mutual Coupling on Doubly Curved
Convex Surfaces, ” 2001 IEEE APS Int. Symp., Dig, vol. 2, Boston, MA, 2001,
pp. 342-345.

J .T. Aberle, D.M. Pozar, and CR. Birtcher, “Evaluation of input impedance and
radar cross section of probe-fed microstrip patch elements using an accurate feed
model,” IEEE Trans. Antennas Propagat., 32, pp. 1691-1696, Dec. 1991.

J-M Jin, The Finite Element Method in Electromagnetics, New York: Wiley-
Interscience, 1993.

LC. Kempel and J .L. Volakis, “Scattering by cavity-backed antennas on a
circular cylinder,” IEEE Trans. Antennas Propagat., 51;, pp. 1268-1279, Sept.
1994.

J.L. Volakis, A. Chatterjee, and LC. Kempel, Finite Element Method for
Engineering, New York: IEEE Press, 1998.

C-T Tai, Dyadic Green Functions in Electromagnetic Theory, 2"d ed., Piscataway,
NJ: IEEE Press, 1994.

164

 

 

 

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]
[22]

[23]

[24]

PH. Pathak and N.N. Wang, “An analysis of the mutual coupling between
antennas on a smooth convex surface,” Final Report 784583-7, Ohio State
University ElectroScience Laboratory, Dept. Elec. Eng, Oct. 1978.

E. H. Newman and P. Tulyathan, “Analysis of microstrip antennas using moment
methods, ” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 47-53, Jan. 1981.

David M. Pozar, “Input impedance and mutual coupling of rectangular microstrip
antennas,” IEEE Trans. Antennas Propagat., vol. AP-30, No.6, Nov. 1982.

R. P. Jedlicka, M. T. Poe, and K. R. Carver, “Measured mutual coupling between R
microstrip antennas,” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 147-149, ‘5-
Jan. 1981.

i
P. Silvester and MS. Hsieh, “ Finite-element solution of 2-dimensional exterior 1
field problem,” Proc. Inst. Elec. Eng, vol. 118, pp. 1743-1747, Dec. 1971. 7

 

B.H. McDonald and A. Wexler, “Finite-element solution of unbounded field
problems,” IEEE Trans. Microwave Theory Tech, vol. 20, pp. 841-847, Dec.
1972.

J-M Jin and J.L. Volakis, “A finite element boundary integral formulation fer
scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas
Propagat., vol. 39, No. 1, pp. 97-104, Jan. 1991.

J-M Jin and J .L. Volakis, “A hybrid finite element method for scattering and
radiation by microstrip patch antennas and arrays residing in a cavity,” IEEE
Trans. Antennas Propagat., vol. 39, No. 11, pp. 1598-1604, Nov. 1991.

J-M Jin and J .L. Volakis, “Electromagnetic scattering by and transmission
through a three-dimensional slot in a thick conducting plane,” IEEE Trans.
Antennas Propagat., vol. 39, No. 4, pp. 543-550, Apr. 1991.

Robert E. Collin, Field Theory of Guided Waves, Oxford: IEEE Press, 1991.

Z.J. Cendes, “Vector finite elements for electromagnetic field computation,”
IEEE Trans. Magnetics,, vol. 27, N0. 5, pp. 3958-3966, Sept. 1991.

A. Chatterjee, J-M. J in, and J .L. Volakis, “Computation of cavity resonances
using edge-based finite elements,” IEEE Trans. Microwave Theory T ech., vol. 40,

No. 11, pp. 2106-2108, Nov. 1992.

LC. Kempel and J.L. Volakis, “Radiation and scattering by cavity-backed
antennas on a circular cylinder,” Technical Report for NASA Grant NAG-1-1478,

O30601-1-T, July 1993.

165

[25]

[26]

[271

[28]

[29]

[30]

[31]

[32]

[33]

[34]

PH. Pathak and N.N. Wang, “A uniform GTD solution for the Radiation fgrom

Sources on a Convex Surface,” IEEE Trans. Antennas Propagat., vol. AP-29, No.

4, July 1981.

Shanjie Zhang, and Jianming Jin, “Computation of Special Functions,” New
York: Wiley-Interscience, l 996.

M. Abramowitz and LA. Stegun, “Hankbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables,” US. Department of Commerce
National Bureau of Standards Applied Mathematics Series 55.

Julius Adams Stratton, “Electromagnetic Theory,” McGraw-Hill, 1941.

NA. Logan, “General research in diffraction theory,” Lockheed Aircraft Corp.,
Missiles and Space Div., vol. 1 and 2, Report LMSD-28808, Dec. 1959.

Constantine A. Balanis, “Advanced Engineering Electromagnetics,” John Wiley
& Sons, 1989.

Constantine A. Balanis, “Antenna theory Analysis and Design,” John Wiley &
Sons, 1997.

David M. Pozar “Microwave Engineering,” John Wiley & Sons, 1998.

LC. Kempel and J .L. Volakis, “A finite element-boundary integral method for
conformal antenna arrays on a circular cylinder,” Technical Report for NASA
Grant NCA2-543, 027723-6-T, July 1992.

R. Mittra, A.W. Peterson, and S.L. Ray, "Computational_Methods_for
Electromagnetics,” New York: IEEE Press, 1998.

166