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This is to certify that the

dissertation entitled

INTERACTION 0F ELECTROMAGNETIC FIELDS WITH
A MATERIAL SAMPLE PLACED WITHIN AN
ENERGIZED CAVITY

presented by
JIANPING ZHANG

has been accepted towards fulfillment
of the requirements for

Ph.D degreeiIELELIRILALENGINEERING

QM

Kun-Mu Chen

Major professor

 

Date July 14,1998

 

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INTERACTION OF ELECTROMAGNETIC FIELDS WITH A MATERIAL
SAMPLE PLACED WITHIN AN ENERGIZED CAVITY

By

J ianping Zhang

A DISSERTATION
Submitted to
Michigan State University

in partial fulfilment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1998

ABSTRACT

INTERACTION OF ELECTROMAGNETIC FIELDS WITH A MATERIAL
SAMPLE PLACED WITHIN AN ENERGIZED CAVITY

By

Jianping Zhang

The investigation of the heating of a material sample in an energized
electromagnetic cavity requires the understanding of the interaction of the electromagnetic
fields with the material sample in a cavity. The key factor for this understanding is to
quantify the distribution of the induced electric field inside the material sample placed in
the cavity.

The goal of this research is to solve Maxwell’s equations in an electromagnetic
cavity in the presence of a material sample based on an Electric Field Integral Equation
(EFIE) or a Magnetic Field Integral Equation (MFIE) and the dyadic Green’s function in
an electromagnetic cavity. In this study, a complete set of vector wave functions which
include both solenoidal and irrotational functions are employed and the electric field (and
magnetic field) integral equation is derived based on the expansion of these vector wave
functions.

When solving the integral equation, due to the slow convergence rate of the dyadic
Green’s function, the infinite triple summation over the cavity eigenfunctions is reduced to
the infinite double summation, and the infinite double summation is then estimated by a

finite double summation plus an infinite single summation using the Poisson summation

formula. For some material samples with specific geometries, a scheme of separating the
material sample into the boundary layer region and the interior region is proposed. This
scheme tends to improve the convergence of numerical results and also to save
computation time. Numerical results agree well with the theoretical estimation using these
methods.

The mode-matching method is also employed to analyze the induced electric field
distribution in homogeneous material samples with simple cylindrical geometries placed
in an energized cylindrical cavity. In this method, the whole cavity is divided into three
waveguide regions and the eigenmodes in the inhomogeneously filled waveguide which
contains the material sample are derived. Numerical calculation shows that the resultant
matrix is sparse and the number of eigenmodes needed in the summation is reduced
considerably compared with the integral equation method. The numerical results of the

mode-matching method are found to be consistent with the corresponding results of the

integral equation method.

 

To my husband Yong Wan and our son Moquan Wan

iv

ACKNOWLEDGMENTS

First and foremost, I would like to express my deep gratitude to my major
professor, Dr. Kun-Mu Chen, for introducing me to Electromagnetics, and his invaluable
help and patience. I owe him no less gratitude for the congenial manner in which he has
overseen my career as a graduate student. In addition, his editorial, technical, and thematic
comments on the rough drafts of this dissertation have improved it more than I would care
to admit. Without his guidance and expert knowledge in Electromagnetics, this
dissertation would not have been possible. I am grateful for the opportunity to learn from
his example as both a researcher and an educator. I would like to thank the other members
of my committee, Dr. Dennis Nyquist, Dr. Edward Rothwell, and Dr. Byron Drachman,
for many helpful comments on this research.

Finally if it were not for the generous and patient nature of my husband, Yong
Wan, our son and family, no one would be reading this today. Grateful acknowledgement
is also due to my former professors, Yong Huang and Dongguang He from Shandong
University, for their continuous concerns during my stay at Michigan State University. I
must also extend thanks to my fellow students for their friendship and valuable
discussions throughout the duration of my program.

This research was supported by the National Science Foundation under Grant NO.

CT S 9526038 and The State of Michigan under a Research Excellence Fund.

TABLE OF CONTENTS

LIST OF TABLES .................................................... ix
LIST OF FIGURES .................................................... x
CHAPTER 1

INTRODUCTION ................................................ 1
CHAPTER 2

INTERACTION OF ELECTROMAGNETIC FIELDS WITH A
MATERIAL SAMPLE PLACED WITHIN A RECTANGULAR
CAVITY ........................................................ 6

2.1 Vector Wave Functions in Rectangular Cavities .................. 9

A A A
2.1.1 Definitions for Vector Wave Functions an1, Mum! and Nnml

in Rectangular Cavities .................................. 9
2.1.2 Expressions for Vector Wave Functions anl , Mnml and Nnml

in Rectangular Cavities ................................. 12
2.1.3 Vector Wave Functions anl. M nml and Nnml Satisfy Vector

Helmholtz Equation . . ................................. 25
2.1.4 Orthggonality of the Vector Wave Functions an1 , M nml

and N nml ............................................ 33
2.1.5 Normalization of the Vector Wave Functions an1, M nml

and N nrnl ............................................ 38
2.1.6 Completeness of the Vector Wave Functions an1 , M nml

and N nml ........................................... 40

2.2 Derivation of Dyadic Green’s Function and Electric Field Integral

Equation (EFIE) in Rectangular Cavities ....................... 44
2.2.1 Maxwell’s Equations in the Material Sample ............... 45

2.2.2 Expansion of E50) and Derivation of the Electric Dyadic

vi

2.3

2.4

CHAPTER 3

Green’s Function ..................................... 46
2.2.3 Derivation of the Integral Equation in the Material Sample . . . .48
2.2.4 Expression of the Dyadic Green’s Function ................ 49

2.2.5 Detailed Expression of GwG’, E) and Comparison with
the Results of Y. Rahmat-Samii [11] ...................... 49

2.2.6 Derivation of the Electrical Field Outside the Material Sample .51

Derivation of the Magnetic Dyadic Green’s Function and Magnetic
Field Integral Equation (MFIE) ............................... 53
Comparison of EFIE with MFIE and Explanation of the Result ...... 57

NUMERICAL TECHNIQUES AND RESULTS ON THE INDUCED
ELECTRIC FIELD IN A MATERIAL SAMPLE PLACED

WITHIN A RECTANGULAR CAVITY ............................ 61

3.1 Applying Galerkin’s Method to EFIE .......................... 62

3.2 Convergence Property of the Dyadic Green’s Function in EFIE ..... 66

3.3 Numerical Examples ....................................... 85

3.4 Some Methods to Increase the Convergence Rate ................ 105
CHAPTER 4

QUANTIFICATION OF THE INDUCED ELECTRIC FIELD IN A
MATERIAL SAMPLE PLACED WITHIN A CYLINDRICAL

CAVITY ..................................................... 125
4.1 Expressions for Vector Wave Functions anl. 117nm! and Nnml in
Cylindrical Cavities ...................................... 126
4.1.1 Expression for Vector Wave Function anz ................ 128
4.1.2 Expression for Vector Wave Function M nml ............... 130
4.1.1 Expression for Vector Wave Function N nml ............... 132
4.2 Normalization of Vector Wave Functions an1 , 117nm! and N run! in
Cylindrical Cavities ...................................... 135
4.2.1 Normalization of Vector Wave Function anz ............. 135
4.2.2 Normalization of Vector Wave Function M nml ............ 137
4.2.3 Normalization of Vector Wave Function N nml ............. 141
4.2.4 Some Field Structures of Vector Wave Functions in
Cylindrical Cavities .................................. 144
4.3 Dyadic Green’s Function in the Cylindrical Cavity .............. 151
4.4 Numerical Examples ...................................... 160

vii

1'1- '

CHAPTER 5
QUANTIFICATION OF THE INDUCED ELECTRIC FIELD IN A
MATERIAL SAMPLE PLACED INSIDE AN EM CAVITY USING

MODE MATCHING METHOD ................................. 182
5.1 Eigenmodes in Different Waveguide Regions ................... 183
5.1.1 Eigenmodes in a Homogeneously Filled Wavguide ......... 183
5.1.2 Eigenmodes in an Inhomogeneously Filled Wavguide ....... 186
5.1.3 Normalization of Homogeneously filled Waveguide
Eigenmodes ........................................ 198
5.2 Electromagnetic Fields in the Three Regions ................... 200
5.3 Numerical Example ....................................... 208
CHAPTER 6
CONCLUSIONS .............................................. 220
APPENDIX A
COMPUTATION OF DYADIC GREEN ’8 FUNCTION IN CAVITIES
BY Y. RAHMAT-SAMII [ll] .................................... 222
APPENIDX B
THE IDENTITY OF
222':an10')an1(?) + Nnmz(?')Nnmz(7-) + Mnmzd'mmmfl] = in; - P)
n m I
IN RECTANGUALR CAVITIES ................................ 227
APPENIDX C
AN ALTERNATIVE REPRESENTATION OF THE ELECTRIC
DYADIC GREEN’S FUNCTION ................................ 232
APPENDIX D
INHOMOGENEOUS DIELECTRIC SPHERE IN UNIFORMELY
APPLIED STATIC FIELD ...................................... 243

BIBLIOGRAPHY .......... 252

viii

Table 3.1

Table 3.2

Table 3.3

Table 5.1

Table 5.2

LIST OF TABLES

Induced electric field inside the 4-mm cubic material sample and its ratio
to the initial electric field for different values of N, where the relative
permittivity of the sample is assumed to be e, = 2.5 ,.the resonant
frequency shift is 5% and the initial electric field is E; = 321.5729.

The geometry of the rectangular cavity is shown in Figure 3.7. . . . . . .90

Induced electric field inside the 4-mm cubic material sample and its ratio to
the initial electric field for different relative permittivities of the material
sample, where the resonant frequency shift is 5% and the initial

electric field is E; = 321.5729 . The geometry of the

rectangular cavity is shown in Figure 3.7. ....................... 91

Induced electric field and the ratio vs. the resonant frequency shift. The
relative permittivity of the 4-mrn cubic materialsample is assumed

to be a, = 2.5 and the initial electric field is E1 = 321.5729. The
geometry of the rectangular cavity is shown in Figure 3.7 ........... 93

Significant modes in the mode-matching method when the dimensions of
the cavity are: a=0.0762m, c=0.15458m and that of the material sample
are: r0=0.004m and h0=0.008m. The operating frequency is 2.45 GHz
and the excitation probe is located at c/4 from the bottom. ......... 210

Number of eigenmodes used in the mode-matching method for the
different geometris of the material sample ...................... 218

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Figure 2.12
Figure 2.13

Figure 3.1

Figure 3.2

LIST OF FIGURES

A rectangular cavity and the designation of the coordinate system . . . . 13
E-field structure of L1“ in the x-y plane with z=c/4. .............. 19
E-field structure of L1,, in the x-z plane with y=b/4. .............. 20
E-field structure of L 1 1 1 in the y-z plane with x=a/4. .............. 21
E-field structure of M22, in the x-y plane with z=c/4. .............. 22
E—field structure of M22] in the x-z plane with y=b/4. .............. 23
E-field structure of M22, in the y-z plane with x=a/4. .............. 24
E-field structure of N22, in the x-y plane with z=c/44. ............. 26
E—field structure of N221 in the x-z plane with y=b/4. .............. 27
E-field structure of N22, in the y-z plane with x=a/4. .............. 28
E-field structure of L22, in the x-y plane with z=c/4. .............. 29
E-field structure of L22, in the x-z plane with y=b/4. .............. 30
E-field structure of L22, in the y-z plane with x=a/4. .............. 31

Integration of the triple summation format G e 0 ”O", E) vs. the number
of summation modes when; = P, ;= [0.033m, 0.014m, 0.0551m]

and Ax = Ay = A2 = 0.002m.The dimensions of the rectangular
cavity areza = 0.072m ,b = 0.034m and c = 0.1163m. ......... 70

Integration of the triple summation format G e 0 ”0', E) vs. the number
of summation modes when i: [0.035m, 0.014m, 0.0551m],
P: [0.033m,0.014m,0.0551m] and Ax = Ay = A2 = 0.002m.

The dimensions of the rectangular cavity are: a = 0.072m ,
b _= 0.034m and c = 0.1163m. ............................. 71

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9a

Figure3.9b

Integration of the triple summation format G e 0xx(i", E) vs. the number
of summation modes when P: [0.035m, 0.016m, 0.0553m],
P: [0.033m,0.014m, 0.0551m] and Ax = Ay = A2 = 0.002m.

The dimensions of the rectangular cavity areza = 0.072m ,
b = 0.034m and c = 0.1163m. ............................. 72

Integration of the double summation format G e 0 “0', E) vs. the number
of summation modes when I" = i" , E: [0.033m, 0.014m, 0.0551m]

and Ax = Ay = A2 = 0.002m. The dimensions of the rectangular
cavity are: a = 0.072m, b = 0.034m and c = 0.1163m. ......... 82

Integration of the double summation format G e 0 ”0', E) vs. the number
of summation modes when P: [0.035m, 0.014m, 0.0551m],

F‘: [0.033m,0.014m, 0.0551m] and Ax = Ay = A2 = 0.002m.

The dimensions of the rectangular cavity are: a = 0.072m ,

b = 0.034m and c = 0.1163m. ............................. 83

Integration of the double summation format G e 0 “0', E) vs. the number
of summation modes when r": [0.035m, 0.016m, 0.0553m] ,

P: [0.033m,0.014m,0.0551m] and Ax = Ay = A2 = 0.002m.

The dimensions of the rectangular cavity are: a = 0.072m ,

b = 0.034m and c = 0.1163m. ............................. 84

Dimensions of the rectangular cavity and the material sample. The
center of the material sample is consistent with the center of the
cavity .................................................... 86

Ratios of E/E),i at different volume cells in the 4-mm cubic material
sample, where the relative permittivity of the material sample is

assumed to be e, = 2.5 . The geometry of the rectangular

cavity is shown in Figure 3.7. The resonant frequency

shift is assumed to be 5%. ................................... 88

The ratios of E/Eyi at different volume cells in the 6-mm cubic material
sample, where the relative permittivity of the material sample is assumed
to be e, = 2.5 . The geometry of the rectangular cavity is shown in
Figure 3.7. The resonant frequency shift is assumed to be 5% ....... 96

The x component of the induced electric field at different volume cells of
the 6-mm cubic material sample, where the relative permittivity of the
material sample is assumed to be a, = 2.5 . The geometry of the
rectangular cavity is shown in Figure 3.7. The resonant

frequency shift is assumed to be 5%. ........................... 97

xi

 

Figure 3.9c

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

The z component of the induced electric field at different volume cells of
the 6-mm cubic material sample, where the relative permittivity of the
material sample is assumed to be e, = 2.5 . The geometry of the
rectangular cavity is shown in Figure 3.7. The resonant

frequency shift is assumed to be 5%. ........................... 98

Ratios of E i varies in the x-direction. Each curve represents this ratio
as a function of x for different locations of y and z. The relative permittivity
of the 2-cm cubic material sample is a, = 2.5 . The geometry of the
rectangular cavity is shown in Figure 3.7. The resonant frequency

shift is assumed to be 8%. ................................... 99

Ratios of E i varies in the x—direction. Each curve represents this ratio as
a function of x for different locations of z. The highest one is for z=z, and
the lowest one for z=z 5. The relative permittivity of the thin square plate
material sample is e, = 2.5 and the geometry of the cavity is shown in
Figure 3.7. The resonant frequency shift is assumed to be 1% ....... 101

Ratios of E /E .i varies as a function of y. The dimensions of the material
sample are: x0=0.002m, y0=0.02m, 20:0. 002m and the geometry of

the cavity is shown in Figure 3.7. The resonant frequency shift is
assumed to be 1%. The upper limit in the double summation is

chosen to be N=200. ...................................... 103

Ratios of E/liyi varies as a function of y for different N. The dimensions
of the material sample are: x0=0.002m, y0=0.02m, 20:0.002m and the
geometry of the cavity is shown in Figure 3.7. The resonant frequency
shift is assumed to be 1%. .................................. 104

Integration of G e 5 “(it E) vs. number of summation modes when
P = P, i: [0.033m,0.014m,0.0551m] and
Ax=Ay=Az=0.002m. ................................ 110

Integration of G e s ”G", E) vs. number of summation modes when
i: [0.035m, 0.014m, 0.0551m], P: [0.033m, 0.014m, 0.0551m]
andAx=Ay=Az=0.002m .............................. 111

Integration of G e ”IO", E) vs. number of summation modes when
i: [0.035m, 0.016m, 0.0553m], P: [0.033m, 0.014m, 0.0551m]
and Ax = Ay = A2 = 0.002m .............................. 112

Numerical results obtained with the scheme of dividing the sample volume
into boundary layer region and interior region. The ratio of E/Ey‘ varies
as a function of y coordinate. The sample dimensions are: x0=0.003m,
y0=0.021m, 20:0.003m and the geometry of the cavity is shown in
Figure 3.7. The resonant frequency shift is assumed to be 1% ....... 114

xii

 

Figure 4.1

Figure 4.2

Figure 4.3

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

A cylindrical cavity and the designation of the coordinate system . . . 127

E-field structure of N012 in the r-z plane with (p = 121°. The dimensions
of the cylindrical cavity are a=0.0762m, c=0.15458m ............. 145

E-field structure of N012 in the MI) plane with z=0.02 71m. The dimensions
of the cylindrical cavity are a=0.0762m, c=0.15458m ............. 146

E-field structure of M I 1 I in the r-z plane with (p = 121°. The dimensions
of the cylindrical cavity are a=0.0762m, c=0.15458m ............ 147

E-field structure of M1“ in the r-¢ plane with z=0.02 71m. The dimensions
of the cylindrical cavity are a=0.0762m, c=0.15458m ............. 148

E—field structure of L112 in the r-z plane with (p = 121° . The dimensions of
the cylindrical cavity are a=0.0762m, c=0.15458m .............. 149

E-field structure of L112 in the Hi) plane‘with z=0. 02 71m. The dimensions
of the cylindrical cavity are a=0.0762m, c=0.15458m ............. 150

Dimensions of the cylindrical cavity and the material sample. The center of
the material sample is consistent with the center of the cavity ....... 162

Ratio of Ez/Ezi varies in the r direction at the different locations of z. The
dimensions of the material sample are d0=0.02m and h0=0.02m with

the relative permittivity of e, = 2.5 . The dimensions of the

cylindrical cavity are shown in Figure 4.8. ..................... 165

Ratio of Ez/Ezi varies in the r direction. The dimensions of the material
sample are d0=0.04m and h0=0.002m with the relative permittivity of

e, = 2.5 .The dimensions of the cylindrical cavity are shown in

Figure 4.8. .............................................. 166

Ratio of Ez/Ezi varies in the z direction at the different locations of r. The
dimensions of the material sample are (10:0. 008m and h0=0.044m with
the relative permittivity of e, = 2.5 . The dimensions of the

cylindrical cavity are shown in Figure 4.8. ..................... 168

Ratio of Ely/Ezi varies in the z direction with the scheme of the separation
of the boundary layer and interior regions. The dimensions of the
material sample are d0=0.008m and h0=0. 044m with the relative
permittivity of e, = 2.5 . The dimensions of the cylindrical

cavity are shown in Figure 4.8. .............................. 170

Ratios of Ez/Ezi varies in the r direction. Each curve represents this ratio as
a function of r for different locations of z in a material sample when the

xiii

 

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Figure 4.18

Figure 4.19

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4 '

material sample has a complex permittivity of e, = 2.5 — j0.5 . The
dimensions of the material sample are: radius r0 = 0.004m and height
ho = 0.008m. The upper graphs are for the real parts of the ratios

and the lower ones are for the imaginary parts of the ratios. ........ 172

Ratios of Ez/Ezi varies in the r direction. Each curve represents this ratio as a
function of r for different locations of z in a material sample when the
material sample has a complex permittivity of e, = 2.5 - j 1.5 . The
dimensions of the material sample are: radius r0 = 0.004m and height

h0 = 0.008m . The upper graphs are for the real parts of the ratios

and the lower ones are for the imaginary parts of the ratios. ........ 173

An inhomogeneous material sample is placed in the center of a
cylindrical cavity .......................................... 175

Ratios of Ez/Ezi in an inhomogeneous material sample with the dimensions
of r1 = 0.004m, h1 = 0.008m, r2 = 0.008m and h2 = 0.016m,
where the relative permittivity in the shadowed region is 81 = 2.5 and
that in the non-shadowed region is 82 = 4.0. The electrostatic
estimations of the ratios are R1 = 0.5818 and R2 = 0.5091 ....... 177

Ratios of Ez/Ezi in the inhomogeneous material sample with the dimensions
of rl = 0.002m, hr = 0.004m, r2 = 0.004m and h2 = 0.008m,
where the relative permittivity in the shadowed region is 81 = 2.5 and
that in the non-shadowed region is 82 = 4.0 . The electrostatic

estimations of the ratios are R1 = 0.5818 and R2 = 0.5091 ...... 178

Geometry of an irregularly shaped material sample placed in the cylindrical
cavity. The material sample is azimuthally symmetrical and the center

of the material sample is consistent with the center of the

cylindrical cavity .......................................... 179

Ratios of E/Ezi in an irregularly shaped material sample. ......... 180

Geometry of the material sample placed in a cylindrical cavity driven by
an excitation probe. ....................................... 184

Geometry of the inhomogeneously filled waveguide ............. 187
Ratio of EZ/Ezi varies in the z direction at r=0.0004m. The dimensions of
the material sample are r0=0.004m and h0=0.008m with the relative
permittivity of 8, = 2.5 . The dimensions of the cylindrical

waveguide are: a=0.0762m and c=0.15458m. The

operating frequency is 2.45 GHz. ............................ 212

Ratio of Ez/Ezi varies in the r direction at the different locations of z. The

xiv

Figure 5.5

Figure 5.6

Figure D.1

dimensions of the material sample are r0=0.01m and h0=0.02m with the
relative permittivity of e, = 2.5 . The dimensions of the cylindrical
waveguide are: a=0.0762m and c=0.15458m The operating

frequency is 2.45 GHz ...................................... 213

Ratio of E/Ezi varies in the r direction. The dimensions of the material
sample are r0=0.02m and h0=0.001m with the relative permittivity of

e, = 2.5 .The dimensions of the cylindrical waveguide are: a=0.0762m
and c=0.15458m. The operating frequency is 2.45 GHz. .......... 215

Ratio of Ez/Ezi varies in the z direction at r=0.0004m. The dimensions of
the material sample are r0=0.004m and h0=0.044m with the relative
permittivity of e, = 2.5 . The dimensions of the cylindrical

waveguide are: a=0.0762m and c=0.15458m.The

Operating frequency is 2.45 GHz. ............................ 216

Geometry of an inhomogeneous dielectric sphere ................ 244

XV

CHAPTER 1

INTRODUCTION

The research reported in this dissertation was motivated by the investigation of
microwave heating of material samples. Microwave heating techniques have been widely
utilized in many industrial process [1]. However, the question of why the microwave
heating is much faster and more efficient than the conventional thermal heating in
promoting the chemical reaction and the heating of materials is still unanswered. Since the
microwave heating of material samples is usually conducted within an energized
electromagnetic cavity, to provide an answer to this question it is essential to study the
interaction of the microwave field with a material sample in an electromagnetic cavity.

To understand the coupling of the microwave energy into molecules of a material
sample, it is necessary to determine the microwave (EM) energy absorption rate (or
dissipated microwave power density) P at any point inside the material sample. To
determine this P, it is essential to quantify accurately the distribution of the induced
electric field at any point inside the material sample. Therefore, the key factor to
understand the heating of a material sample in an energized electromagnetic cavity is to
quantify the induced electric field inside the material sample.

Recently, some studies [3]-[8] on this subject based on the finite difference-time

domain method, the finite element method, or the method of lines have been reported.

However, numerical results of these methods can not provide physical pictures of how the
microwave field interacts with a material sample. The method which gives more physical
pictures is to solve Maxwell’s equations in an electromagnetic cavity in the presence of a
material sample based on an Electric Field Integral Equation (EFIE) or a Magnetic Field
Integral Equation (MFIE) and the dyadic Green’s function in an electromagnetic cavity,
Tai [9]. However, in many studies involving this type of problem [9], [15]-[21], the
unknown induced electric field inside the material sample is expanded in terms of the
normal cavity electric modes which are completely solenoidal. This is not correct for the
following reason. When a material sample is placed in the cavity, the initial cavity electric
field will induce electric charges on the surface of the material sample if it is of finite size
or at the heterogeneity boundaries if it is heterogeneous. Thus, the divergence of the
electric field will not be zero at the location of the induced charges, or the divergence of
the electric field will not vanish at all points in the cavity. Therefore, the normal cavity
electric modes which are solenoidal are not sufficient to represent the unknown induced
electric field inside the material sample. Additional eigenfunctions which are irrotational
will be needed.

In this dissertation, a complete set of vector wave functions which include bo-.'h
solenoidal and irrotational functions are employed and the electric field (and magnetic
field) integral equation is derived based on the expansion of these vector wave functions.
In the solving of the integral equation, the convergence property of the derived dyadic
Green’s function plays a vital role, thus several mathematical methods are explored to
increase the convergence rate of the dyadic Green’s function. The same problem was also

solved by the mode-matching method when the material sample is homogeneous and of

simple cylindrical geometry. The results of this method provide a check for the validity of

that generated by the integral equation method.

A A
In Chapter 2, the properties of the three vector wave functions anI, M nml and

Nam: in a rectangular cavity are introduced. The orthogonality and completeness of these
three vector wave functions are proved. Using these three vector wave functions as a
complete set of eigenfunctions to expand the unknown electric field, we derive the Electric
Field Integral Equation (EFIE). On the other hand, based on the expansion for the
unknown magnetic field, we obtain the Magnetic Field Integral Equation (MFIE). The
EFIE and the MFIE are shown to be consistent even though different approaches for
deriving them are employed.

In Chapter 3, Galerkin’s method is applied to solve the EFIE derived in Chapter 2
and the convergence property of the dyadic Green’s function in the EFIE is studied. Due to
the slow convergence rate of the dyadic Green’s function, the infinite triple summation
over the cavity eigenfunctions is reduced to the infinite double summation, and the infinite
double summation is then estimated by a finite double surmnation plus an infinite single
summation using the Poisson summation formula. Numerical results show that the
electromagnetic fields distribution in the material sample are strongly dependent on the
geometry and the dielectric parameters of the material sample. For some material samples
with specific geometries, a scheme of separating the material sample into the boundary
layer region and the interior region is proposed. This scheme tends to improve the
convergence of numerical results and also to save computation time.

In Chapter 4, the microwave heating of a material sample in a cylindrical cavity is

studied. Theoretical analysis of the induced electric field inside a material sample placed

within an energized cylindrical cavity is more involved than that of a rectangular cavity

A A
case as studied in Chapter 2 and Chapter 3. The vector wave functions anI, M um! and

Nam! in a cylindrical cavity are derived and normalized. The infinite triple and double
summation formats of the dyadic Green’s function in terms of these vector wave functions
are provided. The numerical calculation is conducted for material samples with simple or
complex geometry and homogeneous or heterogeneous composition. Numerical results
agree well with the theoretical estimation.

In Chapter 5, the mode-matching method is employed to analyze the induced
electric field distribution in homogeneous material samples with simple cylindrical
geometries placed in an energized cylindrical cavity. In this method, the whole cavity is
divided into three waveguide regions and the eigenmodes in the inhomogeneously filled
waveguide which contains the material sample are derived. The electromagnetic fields in
each region are then expressed as infinite sums of the eigenmodes, and their tangential
components are matched at the junction surfaces between different regions. Numerical
calculation shows that the resultant matrix is sparse and the number of eigenmodes needed
in the summation is reduced compared with the integral equation method while the
convergence rate is improved. The numerical results of the mode-matching method are
found to be consistent with the corresponding results of the integral equation method
reported in Chapter 4.

Some derivations and proofs that are useful in this dissertation are provided in

Appendices. Appendix A compares the derivation of the dyadic Green’s function with that

 

 

of Rahmat—Samii [11] and explains the discrepancy of the expression in [11]. Appendix B

proves the identity of

222[anl(;0)znml(;) + finml(;o)finml(;) + finml(;0)finml(;)] = 780* — P0) (1.1)

nml

A
which is essential in the proof of the completeness of the vector wave functions an1,

117m! and IVnmz. Appendix C provides a detailed derivation of the infinite double
summation reduced from the infinite triple summation and this reduction is important in
the numerical calculation. Appendix D gives the electric field in an inhomogeneous
dielectric sphere which includes two regions of different dielectric materials induced by a

uniform static electric field and this result is used as a theoretical estimation in Chapter 4.

CHAPTER 2

INTERACTON OF ELECTROMAGNETIC FIELDS
WITH A MATERIAL SAMPLE PLACED WITHIN A
RECTANGUALR CAVITY

In this chapter, the interaction of the electromagnetic field with a material sample
placed in a rectangular cavity is studied. We will consider a material sample of finite
dimensions with dielectric parameters of relative permittivity e = e’ + je" , permeability
it , and conductivity 0' , and assume that a certain electromagnetic mode of a rectangular
cavity has been maintained before a material sample is introduced. Our goal is to
determine the total electromagnetic fields inside the material sample induced by the initial
cavity electromagnetic fields, and the perturbed electromagnetic fields in the vicinity of
the material sample as well.

In many studies involving this type of problem[9], [15]-[20], the unknown induced
electric field inside the material sample is expanded in terms of the normal cavity electric
eigenmodes which are completely solenoidal. This is not correct for the following reason.
When a material sample is placed in the cavity, the initial cavity electric field will induce
electric charges on the surface of the material sample if it is of finite size or at the

heterogeneity boundaries if it is heterogeneous. Thus, the divergence of the electric field

will not be zero at the location of the induced charges, or the divergence of the electric
field will not vanish at all points in the cavity. Therefore, the normal cavity electric
eigenmodes which are solenoidal are not sufficient to represent the unknown induced
electric field inside the material sample. Additional eigenmodes which are irrotational will
be needed. In our study, a complete set of vector wave functions which include both
solenoidal and irrotational functions are employed.

The vector wave functions are the building blocks of the eigenfunction expansions
of various kinds of dyadic Green’s functions [9]. These functions were first introduced by
Hansen [60], [61] and [62] in formulating certain electromagnetic problems. The
effectiveness of these functions was recognized by Stratton [23] who, for example,
reformulated Mie’s theory of the diffraction of a plane electromagnetic wave by a sphere

using the spherical vector wave functions. In his original work [60] Hansen introduced

three kinds of vector wave functions, denoted by Z, A? and N , which are the solutions of

the homogeneous vector Helmholtz equations. Such a presentation was followed by

Stratton [23] and by Morse and Feshbach [42].
In this study, we use the three vector wave functions anI, 117ml and Nnml as the
basis functions to expand the unknown induced electric field inside the cavity. We will

..A A
show that the vector wave function M nml are the normal TE modes, N mm] are the normal

TM modes and anl are the so-called zero frequency modes which are irrotational. Also

A _\

A
the orthogonality and completeness of the vector wave functions anI, M nml and N nml

will be proved to assure that they form a complete and orthogonal set of basis functions.

An Electric Field Integral Equation (EFIE) is constructed when the electric dyadic

A _\ A
Green’s function is derived based on these vector wave functions anI, M nml and N nml.

Although there is a material sample inside the rectangular cavity, the divergence of
the magnetic field vanishes at all points inside the cavity. The solenoidal eigenfunctions
can form a complete set of basis functions within the space of solenoidal vector fields but
not within the space of all vector fields [28]. Thus, we can use the simple cavity magnetic
eigenfunctions which are solenoidal to expand the unknown magnetic field inside the
cavity and the Magnetic Field Integral Equation (MFIE) is obtained after the magnetic
dyadic Green’s function is derived. We will show that the EFIE is equivalent to the MFIE
and we will compare our results with those of Rahmat-Samii [11]. They are almost

identical except a minus sign. After carefully examining the derivation and the results of

[11], we have found an error of a minus sign in [11].

—\ A

The outline of this chapter is as follows: Vector wave functions anI , M and , N nml
and their properties are introduced in Section 2.1. Based on Maxwell’s equations, we
obtain the electric dyadic Green’s function and EFIE in Section 2.2. In Section 2.3, a
MFIE is derived based on the magnetic field expansion and a magnetic dyadic Green’s

function. The results of EFIE and MFIE are compared and the explanation is given in

Section 2.4.

2.1 Vector Wave Functions in Rectangular Cavities

2.1.1 Definitions for Vector Wave Functions anl, 117nm] and IV "ml in
Rectangular Cavities

A A A
The definitions of vector wave functions anI, Mum] and Nnml in rectangular

cavities can be found in [2], [9] and [23] as

 

 

anz = 1 (Vofimp (2.1)
nml
A . M
Mnml = V><(z¢nml) (2.2)
-‘ l A N
Nnml = k VXVX(Z¢nm[) (2-3)
nml

A
where all the scalar wave functions (pm, which yield the vector wave functions anI,

Mnml and Nnml satisfy the scalar Helmholtz equation (V2+k:m,)¢nm, = 0 and the

subscripts n, m, and l are used to identify the eigenmodes in a cavity. The vector wave

A _s A
functions anI, Mnml and Nnml also need to satisfy the boundary conditions on the

perfectly conducting walls of the cavity as:

A

A

n X anz = O (2.4)
a x 117...; = 0 (2.5)
fl X finml = 0 (2.6)

A .A A
Based on the definitions of the vector wave functions anI, M um! and N nml , it is

easy to show that these vector wave functions have the following properties:

V - 117...; = 0 (2.7)
V - 1'17... = 0 (2.8)
VxZ...z = 0 (2.9)

That is, the vector wave functions 117nm! and Nnml are solenoidal and anl is irrotational.

The first complete theory for the spectrum of modes in a cavity was presented by
Kurokawa [12]. Helmholtz’s theorem states that a general vector field has both a
solenoidal and an irrotational part and may be derived from a vector and a scalar potential.
According to Helmholtz’s theorem, the electric field in the interior of a volume V bounded

by a closed surface S can be expressed in the form [2], [12] of

 

.. V .13 ‘ 2 . " *
£0) = -V[ J’Tél‘fldVMV figmdso] (2.10)
V
V OXEISro)d an(r)d
+Vx|:I—OTn—— V0 +§ —4__1tR0d 50]
V

where R = Ii“ - I01 and F2 is the unit inward normal to the surface S. This theorem gives

the conditions for which the electric field is a pure solenoidal or a pure irrotational field.

The pure solenoidal field must satisfy the conditions V - E = 0 in volume V and

A

A
n - E = 0 on the closed surface S, in which case there is no volume or surface charge

associated with the field. In a similar way there are two conditions that must be met in

10

order for a field to be a pure irrotational or lamellar field, namely, V x E = 0 in volume V

and it x E = O on the closed surface S.

For a cavity with perfectly conducting walls the boundary condition fr x E = 0

must hold on the cavity surface S. In general, f1 f E does not vanish, and is not required to
vanish, on S. Hence the electric field in a cavity with perfectly conducting walls is
generally not a pure solenoidal nor a lamellar field. In other words, pure solenoidal and
pure irrotational vector eigenmodes are difficult to find analytically.

In the integral equation method or the moment method, the basis expansion for the
unknown electric field is necessary. That is, we need a set of complete orthogonal basis
functions to expand the unknown electric field and the basis expansion for the unknown
electric field will converge much better if we use the basis functions that satisfy the same
boundary conditions as the unknown electric field we are expanding [1]. From a
mathematical point of view, it really does not matter whether the basis functions are pure

solenoidal or pure irrotational as long as they form a complete set of basis functions.

A _A
Based on the definitions and properties of the vector wave functions anI, Mum! and

A A

A A
Nnml, we can choose the vector wave functions anI, Mnml and Nnml as a set of

expansion basis functions after we prove that they are orthogonal and complete, where

anI will be referred to as pure irrotational modes, while M nml and N nml will be referred
to as solenoidal modes but will not be pure solenoidal modes. All these vector wave

functions can be called the short-circuit modes because they satisfy the boundary

conditions
ft x E = O
on S. In spite of the lack of purity in the short-circuit modes, in many instances non-pure

solenoidal modes turn out to be sufficient to express certain electric field distributions for

A
which n - E is not zero on S.

A

2.1.2 Expressions for Vector Wave Functions in," 1 , M n m l and Nnm I in

Rectangular Cavities

In order to obtain the numerical solution of the unknown induced electric field, we

need to know the expressions for the vector wave functions anI, M and and N nml based
on their definitions given by eqs. (2.1) to (2.6).

The rectangular cavity under consideration has the geometry shown in Figure 2.1.

1. Expression for vector wave function anI.

A
Based on the definition of the vector wave function anI , we have

2...: = LwoL ,) (2.11)
knml nm
(V2+kim1)¢:m1 = 0 (2.12)

Applying the variables separation method to eq. (2.12), we obtain the solution of the scalar

. L
function (pm, as

'H—T‘
t

 

 

 

 

 

 

Figure 2.1 A rectangular cavity and the designation of the coordinate system

13

L cos(kxx) cos(k),y) cos(kzz)
¢.m1 = A...) . . . (2.13)
srn(kxx) srn(kyy) s1n(kzz)
where A m 1 is an unknown constant which will be determined by the normalization of the

. -‘ 2 2
vector wave function anr and kx + ky + kz2= —knml. Then the three components of the

A
vector wave function anI can be expressed as

 

 

Anml -Sin(kxx) COS(kyy) C05(kzZ)
ann = —k,.{ H . H , } (2,14)
knml COS(kxx) srn(kyy) srn(kzz)
Anml COS(kxx) ‘5in(ky)’) C05(kzZ)
anlv = k, . , (2.15)
' k...) ’ {511108.10 H c<>S(k,,y) H sm(kzz)}
Anml COSUCxx) COS(kyy) “Sin(kzZ)
anl = k . . (2.16)
2 km] 3{ srn(kxx) H srn(kyy) H cos(kzz) }

Based on boundary conditions given by eq. (2.4), the vector wave function anI is derived

as

A

"mz[xflcos(fltx)sin(— )sinCZt z)+ m—sin('—’—x)cos(m—ny)
km, a a y y b
(11: J In (rm ) (m1: ) (In H
sin 2 +z—sin —x sin — cos —z
c c a b c

2 2 2
where kim, = (fl—15) + (111?) + (I?) and the expression for the scalar function (Dim, is

 

anl =
(2.17)

given by

L . mr . mrc . lit
q)nml = Anmlsm(7x)Sln(‘5‘)’)sm(?1) (2.18)

2. Expression for vector wave function M nml

Based on the definition of the vector wave function 117ml, we have
-‘ . M
Mnml = VX(Z¢nmI) (2.19)
2 2 M
(V + knm,)¢nm, = 0 (2.20)

Using the variables separation method, the solution of the scalar function (11an 1 is given by

M cos(kxx) cos(kyy) cos(kzz)
M{ 11 )1. 1
s1n(kxx) srn(kyy) srn(kzz)

where Bum, is an unknown constant which will be determined by the normalization of the
. A 2 2 2 2
vector wave functron Mum] and kx +ky -1-kZ = km]. The two components of the

-A
vector wave function M nml can then be expressed as

M B k cos(kxx) —sin(k),y) cos(kzz) 222
"m” _ "'"l 1'{ sin(kxx)11 cos(kyy) 11 sin(kzz)} ( . )
-sin(kxx) cos(k),y) cos(kzz)

Mnmly = —Bnmlkx{ }{ . }{ . } (2'23)
cos(kxx) srn(kyy) srn(kzz)

Based on boundary conditions given by eq. (2.5), the vector wave function Mnml is

derived as

A «m . m . III: Ann . m:
M = B _ 1: (721! j ( 1: J ( J ( j . 4
nml "“11: x—b cos —a x srn —b y srn —Cz +y—a srn _a x (22 )

screamed]

2 2 2
where kim, = (gt) +0551!) +0?) and the expression for the scalar function 4)an, is

given by
(1124,", = Bnmlcos (ng) cos (Piggy) sin (L252) (2.25)

A
3. Expression for vector wave function N nml
I o n A
Based on the definrtron of the vector wave functron N nml , we have

1

am!

A
Nnml =

 

VxVx(2¢,’:’m,) (2.26)

2 2 N
(V + knml)¢nml = O (227)
In a similar way as before, the variables separation method is applied to eq. (2.27), and the

. . N
solution of the scalar functron (pm, becomes

N cos(kxx) cos(k),y) cos(kzz)
$11!"! = Cnml{ - }{ . }{ . } (2-28)
srn(kxx) sm(kyy) srn(kzz)

where C m, is an unknown constant which will be determined by the normalization of the
. -‘ 2 2 2 2
vector wave functron Nnml and kx +ky +kZ = knml. The three components of the

.A .
vector wave function N nml can then be expressed as

16

 

 

Cm, -sin(kxx) cos(kyy) —sin(kzz)
Nnmlx = k kxkz . (2.29)
m.) cos(kxx) srn(kyy) cos(kzz)
Cnml cos(kxx) -sin(k),y) —sin(kzz)
NM, = k kz . (2.30)
y km.) y {srn(kxx)}1 cos(kyy) 11 cos(kzz) 1
Cm] 2 2 cos(kxx) cos(kyy) cos(kzz)
Nnmlz = —(kx+k ) . . . (2.31)
km) y {srn(kxx) 11 srn(kyy) 11 srn(kzz)}

Based on boundary conditions given by eq. (2.6), the vector wave function Nnml is

derived as

 

A C A A ‘
Nnml = "ml[-xn—1-tl—1-tcos(flx)sin(m—n )sin(flz)-yflfltsin(fltx) (2.32)
km, a c a b c b c a

coedsmezrere)’(nimexrme )cosezr

- 2 2 2
2 Mt nut 11: . . N .
where km, = (7 ) + (_b j + (E) and the expressron for the scalar functron om, rs

given by

N . m: . m1: In
¢nml = CnmISID(7X)Sln(—b—y)COS(:Z) (233)
From all of these expressions for the vector wave functions (2.17), (2.24), and

(2.32), we can identify that M m! are the normal TE modes and N nml are the normal TM

modes in a rectangular cavity [10]. We can also identify an1 as the so-called zero-

frequency modes. It is noted that for these three vector eigenfunctions, the eigenvalues

17

.2 -('l’-‘)2+(5"-“)2+(’1)2
run! a b C
are the same for the same indices. This will cause some degenerate modes.

A A

Some field structures of the vector wave functions anI, Mum! and finml which
represent electric fields have been plotted in Figure 2.2 to Figure 2.13.
Figure 2.2 to Figure 2.4 show the electric field structures for the eigenfunction

L1“, where Figure 2.2 depicts for L 1 1 1x and L1,” in the x-y plane with z=c/4, Figure 2.3
depicts L 1 1 1x and L,”Z in the x-z plane with y=b/4 and Figure 2.4 depicts L, 1 U and LHIz

in the y-z plane with x=a/4. From Figure 2.2 to Figure 2.4, we observe that the normal
components of the electric field decrease as the field point moves from the walls of the
cavity towards the center of the cavity. There is a sink point at the center of the cavity for

the eigenfunction L1,].Since the eigenfunction L1 ,1 is irrotational, and

" . 1t . 1t . 1C
V.L111 = —kmAmsrn(;x)srn(5y)srn(zz)

where x e [0, a], y e [0, b] and z e [0, c]. It is obvious that the minimum value of the
divergence of the eigenfunction L 1 1 1 occurs at the center of the cavity and the divergence
of the eigenfunction L 1 1 1 does not vanish at any point inside the cavity.

Figure 2.5-Figure 2.7 show the electric field structures for the eigenfunction M221,
where M221Jr and M221), in the x-y plane with z=c/4, M 22 1x and M2212: in the x-z plane with
y=b/4, and M221), and M221: in the y-z plane with x=a/4 are plotted orderly in Figure 2.5 to
Figure 2.7. Since M2212 is zero, there is only M22” in Figure 2.6 and M221). in Figure 2.7.

Also we can observe that M221x and M221v form a rotational field in Figure 2.5.

ll“ '” _

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Thed

E—field of L111 in the y-z plane when x=a/4

 

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The dimensions of the rectangular cavity are: a=0.072m, b=0.034m, and c=0.1163m.

21

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Figure 2.5 E-field structure of M22, in the x-y plane with z=c/4.
The dimensions of the rectangular cavity are: a=0. 072m, b=0. 034m, and c=0.1163m.

22

=b/4

E-field of M21 in the x-z plane when y

 

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0.1

0.08

0.06

0.04

0.02

E-field structure of M221 in the x-z plane with y=b/4.

Figure 2.6

0. 034m, and c=0.1163m.

=0.072m, b=

ty are: a

f the rectangular cavi

imensrons o

Thed

23

 

Idof M221 in the y—z plane when x—
0.03-:sszzzmrmnm 1;}

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

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Figure 2.8 to Figure 2.10 show the electric field structures for the eigenfunction

N22 1, where N22 1x and N22,). in the x-y plane with z=c/4, N22 1x and N221: in the x-z plane

with y=b/4, and N22” and N221z in the y-z plane with x=a/4 are plotted orderly in Figure

2.8 to Figure 2.10. For the eigenfunction N22,, N22” = 0 at x = a/4 and N221), = O at

y = b/ 4. Because Figure 2.10 is plotted for N221), and N221: at the x = a/ 4 plane, there
are only N22,). and N2212 and they form a rotational field, no sink or source points exist.
Also Figure 2.9 is plotted at the y = b/ 4 plane, N22 Ix and N221z form a rotational field at
this plane.

Figure 2.11 to Figure 2.13 show the electric field structures for the eigenfunction

L221, where L22 1x and L221), in the x-y plane with z=c/4, L22 1x and L2212 in the x-z plane

with y=b/4, and L221), and [.2212 in the y-z plane with x=a/4 are plotted orderly in Figure

2.11 to Figure 2.13. For the eigenfunction L22], L22” = O at x = a/4 and L221), = O

at y = b/4. However, in Figure 2.12 and Figure 2.13, L221): and £2212 in the x-z plane
with y=b/4, and L221), and L221z in the y-z plane with x=a/4 do not form a rotational field.

It looks like there are some sink points and source points in Figure 2.12 and Figure 2.13.

2.1.3 Vector Wave Functions in m 1, 117"," 1 and AA! n m 1 Satisfy Vector
Helmholtz Equation

Since the electric fields satisfy the vector Helmholtz equation, the basis functions
which are used to expand the electric fields should also meet the same requirement. The

vector Helmholtz equation is expressed as

25

E-field of N221 in the x-y plane when z=cl4

 

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0.07

0.06

0.05

0.04

0.03

0.02

0.01

E-field structure of N22, in the x-y plane with z=c/4.

Figure 2.8

0. 034m, and c=0.1 I 63m.

nsions of the rectangular cavity are: a=0.072m, b=

The

26

E—field of N221

in the x-z plane when y=b/4

 

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27

E-field structure of N22, in the x-z plane with y=b/4.
The dimensions of the rectangular cavity are: a=0.072m, b=0. 034m, and c=0.1163m.

E-field of N221 in the y-z plane when x=a/4

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The dimensions of the rectangular cavity are: a=0.072m, b=0. 034m, and c=0.1163m.

28

 

E—field of L221 in the x-y plane when z=cl4

 

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Figure 2.11
The dimensions of the rectangular cavity are: a=0.072m, b=0.034m, and c=0.1163m.

29

E-fleld of L221 in the x-z plane when y=b/4

 

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ons of the rectangular cavity are: a=0.072m, b=0.034m, and c=0.1163m.

Figure 2.12

lmensi

Thed

30

E—fleld of L221 in the y—z plane when x=a/4

 

 

 

 

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f the rectangular cavity are: a=0.072m, b=0. 034m, and c=0.1163m.

unensions o

Thed

31

V22 + kZA = 0 (2.34)

or
V(V.X)—VxVxX+k22=o (2.35)
1. Vector wave function anI satisfies the vector Helmholtz equation
Based on the property of the vector wave function anI (2.9) and using eqs.

(2.11) and (2.12), we have

V(V ' anl) = V(V ' k—l-—V¢nml) = -k:lenml

nm

(2.36)

Therefore,

VZanl + kZanl = V(V ' anl) "" V X V X anl + kilenml = O (2.37)

Namely, the vector wave function an1 satisfies the vector Helmholtz eq. (2.35).
2. Vector wave function M nml satisfies the vector Helmholtz equation

Based on the property of the vector wave function Mum] (2.7) and using eqs.

(2.19) and (2.20), we have

VxVanml = -V2Mnm1 = —2V2Mnm,x— WzMnm,y (2.38)
V2[Mumbc = Vzi‘anm’: 'a—V2¢M"ml= _klv21manmlx (2'39)
3y 3y
2 23 M a 2 M 2
V Mnmly = ‘V 87¢ nml= WV 0) nml)= —knmanmly (2.40)

32

Therefore,

V X V X Mnml = —V2Mnml = kimanml (2.41)

or
2“ 2 A
V Mnml'l'knmanml = O (2.42)

Thus, the vector wave function 117nm: satisfies the vector Helmholtz eq. (2.34).
3. Vector wave function [V nml satisfies the vector Helmholtz equation

Since the vector wave function 07ml has the same property (2.8) as the vector
wave function 07mm has, using eqs. (2.26) and (2.27) and employing the same procedure

_'L
as that used for the vector wave function M nml , we can obtain

V X V X Nnml = —V2Nnml = kimanml (2-43)

i.e.
2A 2 -‘
V Nnml + knmanml = O (2.44)
Therefore, the vector wave function N nml satisfies the vector Helmholtz eq. (2.34).

2.1.4 Orthogonality of the Vector Wave Functions an12 A? n ml and

Nnml

A ._l A
That the vector wave functions anI, M nml and N am] are orthogonal mutually is

necessary for them to form a set of basis functions in order to represent the unknown

33

electric field. We will prove that the vector wave functions are orthogonal for different

indices by themselves and also orthogonal mutually.
1. Vector wave functions finml are orthogonal for different indices.

We know the vector identity of

V-(fimexfin-finxVxll—im) = fin-VxVxfim-fim-VxVxfl—fn (2.45)
So
[(fin'VXVXfim—A—im‘VXVXA—in)dv
= IV‘(fimXVXfin-finXVXfim)dV
" (2.46)

= §fi'(fimXVXA7n—finXVXfim)dS
5

= §{[(fl XA-in) ' (VXfimn—“fi X07171) ' (VX A“2"):|}ds

where the integration region is over the cavity volume V or the surface S of the cavity wall.

Because it x Mn = 0 on the perfectly conducting walls of the cavity, it can be concluded

as

[(fi..VxVxfim—A7m-VxVxA—imv:0 (2.47)

.A
On the other hand, M nml satisfies the vector Helmholtz equation, then

[(fin‘VXVXfim—fim‘VXVXfin)dV

_. _. _. _. _‘ _. (2.48)
= J‘(M.. - kiMm — Mm - kiM.)dv= (k3,, - k3,)j(Mn - Mm)dv
V V

34

Based on eqs. (2.47) and (2.48), we conclude that

1(1)?" - 47..)dv: 0 if m at n (2.49)

That is, the vector wave functions firm: are orthogonal for different indices.
2. Vector wave functions finml are orthogonal for different indices.
The same procedure used for the case of Bum] can be applied to prove the same
property for I? am! .
3. Vector wave functions anI are orthogonal for different indices.
Using the identity of

V . (¢’,;V¢f,) = V01;- V¢,’;+ (afvzq); (2.50)

A
and the properties of the vector wave functions anI, we can prove the orthogonality of

A
the vector wave functions anl for different indices as follows:

1
knkm

 

[(2. ~ Zdevz jwqfi - V¢:]dv

= k 1k 11V - (45%;) -¢,€V2¢;Jdv

 

l (2.51)

l L L -‘ L 2 L
=— V - - V d
4km?” 4.. ds 10km!” 4,. v

 

km L L
— El¢n¢mdv

where the integration region is over the cavity volume Vor the surface S of the cavity wall

35

. . . . L
and the integration over the surface S 18 zero due to the pr0perty of the scalar function on .

Since the scalar function of; has been given in eq. (2.18), it is obvious that they are

orthogonal for different indices. If we assume that the scalar functions (by: have been

normalized, that is

Itftidv = 5... (2.52)

then

V

.s .s k
1(L. - Lm)dv= -k—mj¢:¢;dv = 5.... (2.53)

.8
Therefore, the vector wave functions anI are orthogonal for different indices and

normalized as well.

4. Vector wave functions anI and 07m: are orthogonal.

Using the vector identity of

V-(KXVXB)=VxX-VxB—X-VXVXB (2.54)

we have

V°(ZnXVX;’;m)=VXZn'VXO—l‘m-Zn'VXVXfim (255)
= —Zn'VXVXA-;m= -Zn‘k’2nA—im

A .4
Based on the properties of the vector wave functions anI and M nml . we have

36

kfnjd. . Kimmy: {V - (Z. x V x 47mm;

.. .5. A _~. (2.56)
= 4......vade _§(axL..vXMm)ds=o
S S

A ._\
using the boundary conditions. That is, the vector wave functions anI and Mnml are

orthogonal.

5. Vector wave functions anI and 1V nml are orthogonal

This can be proved if the same preceding procedure is employed.
6. Vector wave functions 11'!A um! and Xinmz are orthogonal.

Based on eqs. (2.41) and (2.43), we know that

((112,. . V x V x 7)}... -10....v x V x 1T2.)dv = (k; — k:)J(fin - 07mm.) (2.57)

V

Using the vector identity of eq. (2.54), we have

J(fin°VXVXfim—fim'VXVXA-';n)dv
V
= JV°(fimXVXA7n-A7nXVXfim)dV
= §(NmXVXMn—MnXVXNm) Eds
S
= (Sulevm) . (V xfi.)—(?zxfi.) - (V XIVdes: 0
S
where the boundary conditions eqs. (2.5) and (2.6) have been employed in the last step.

Therefore,

03,4516}. -1'\7m)dv = 0
V

37

2 2 . A A
When k at k the vector wave functions Mum] and Nnml are ortho onal. For de enerate
m n g g

modes, we can use the Gram-Schmidt orthogonalization procedure to construct a new

subset of orthogonal modes[2].

A _\ A
So far, we have proved that the vector wave functions anI , Mnml and N nml are

orthogonal mutually.
2.1.5 Normalization of the Vector Wave Functions Ln ml , M n ml and
N nml

Up to now there are still three unknown coefficients A m l , BM, and Cm, in the

A A A
expressions for the vector wave functions anI , Mum! and Nnml Which need to be

determined by the normalizations of these vector wave functions.

1. Normalization of anl

A
The normalization of the vector wave function anI is given by [10]

EJ'ZJZanl'anldV = l (2.59)
that is,
m ((1 m.) l". ). (_ )...2(m )...2(e ).(m)2..2(e.)...2(m)
km, b y cz b a b y
1’”) (’le 1"“) 1““) 1’“)
sin —z + -— sin —x sin — cos —z ]dv=1
c a b c

Considering the expression for an1 given in eq. (2.17), we can observe that

 

38

A
anI will be zero if any one of the three indices is zero. Hence,

2 abc

Anml_8— = 1 without n = 0,0r m =0, or I: 0 modes

A
or the normalization coefficient for the vector wave function anI is given by

’ 8
Anml = m (2.60)

2. Normalization of 47m

_\
The normalization of the vector wave function M nml is given by [10]

EEEMnml-Mnmzdv = 1 (2.61)

i.e.
rrrBz [(MTCOST’ExJSinTm—n )sin2(lit )+(n_1t)2
o o o "’"l b a b CZ a
2 n1: 2 m1! 2 11:
sin {—xjcos (—y)sin (—z)]dv = l
a b c

A
Thus, the normalization coefficient for the vector wave function M nml is derived as

8OneOmEOI l
= 1 2.62
Bnml abC (’11:)2 (mfl)2 ( )
_ + —
a b

 

 

 

where

eon = {I if n = O (2.63)

39

3. Normalization of finml

A
The normalization of the vector wave function N nml is given by [10]

EELX/nmz-X/nmzdv = 1 (2.64)

i.e.

 

(C""‘1)2Ja r J‘ (EE)ZCOSZ(ExJSin2(M )sin2(lltz)
km, 0 0 0 a c a b y c
("21:an . 2(mc ) 2(m1t ) . 2(11: )
+ —— sm —x cos — sm —z
b c a b c
+ — + — s1n —x srn — cos —z dv =
a b a b c

A
So the normalization coefficient for the vector wave function N nml is given by

180,180,130! I l
= 2.65
Cnml abc J(nn + m1: ( )

 

 

.A

2.1.6 Completeness of the Vector Wave Functions Zn", 1, M n m 1 and

A

Nnml

As well known, a vector function is uniquely defined only when both the
solenoidal and lamellar or irrotational parts are given. Let the subscript 1 denote the

lamellar part and the subscript r denote the rotational or solenoidal part. For any arbitrary

vector field .6 we have

= 61 + Er (2.66)

CL

40

So in order to represent an unknown electric field which is both solenoidal and lamellar,

we need a set of the basis functions which have both solenoidal and lamellar components.
a n o A A
In the prev1ous sect1ons, we show that the vector wave functlons Mnml and Nnml are

A
solenoidal and anz is lamellar or irrotational.

Conventional proof of the completeness of a set of orthonormal functions can be

found in [2], [42] and [43]. It states: The notion of the completeness for the space of

functions wn(x) defined on the interval 0 Sx S a involves the following: Let fix) be a

piecewise continuous function on 0 S x S a, that is, quadratically integrable with 6(x) as

a weighting function, i.e.

]|f(x)lzo(x)dx < .. (2.67)
0

We assume that 6(x) is always positive. Consider now the approximation

N a

2 cnwn(x) = f(x), c" = ]c(x)f(x)wn(x)dx (2.68)
n = l 0

If the limit as N -—> cc of the integrated square of the error tends to zero, then the functions

\Vn(x) form a complete set. Completeness thus implies that

0

mg

N

f(x)- 2 cnwnoc)

n=1

2
6(x)dx = 0 (2-69)

 

 

" x . . . . . . . .
In our case, E(r) IS a three dlmen51onal vector function Wthh IS p1ecew15e

continuous function in the cavity volume V, i.e., O S x S a , O S y S b and O S 2 S c in the

41

A —A A
rectangular cavity. The vector wave functions anI , M run! and N nml have been defined

in the cavity volume Vbefore. We then consider the approximation

N
30) = 2 [anLn(?)+bnMn(?)+ann(?)]

n=l

where we assume that the vector wave function are normalized and

a
II

n j EGO) 2,100)de
V

cavity

b = j E(?0)-Mn(?o)dv0

cavity

(3
II

I EGO) -1'\7n(?o)dvo

cavity

N
F(?) = 2‘0)— : mum?)+bnMn(?)+ann(?)]

n=l

Substituting eqs. (2.71), (2.72) and (2.73) into eq. (2.74), we have

(2.70)

(2.71)

(2.72)

(2.73)

(2.74)

N
F0) = E(?)- 2 j EGO) . [Ln(?0)Ln(?)+Mn(?o)Mn(?)+Nn(?0)1vn(;)]dv0
" = 1 Vanity
A A N A A .A .A A A
= E(?)- j E(?o)- 2 [Ln(?o)Ln(?)+Mn(?o)Mn(?) +Nn(r0)Nn(;)]dv0
Vcavm- n =1
Then

Nlim 21;) = E(;)- z I E(;O) ' [Zn(;0)Zn(;)+fin(;o)fin(;)+fin(;0)fin(;)]dv0

n=IV

CCVH\

42

In Appendix B we have proved the following identity

2 [Zn(?0)Zn(?) +1i7n(?o)1i7n(?) +1'vn(?o)i\7n(?)] = 780— F0) (2.75)

n=l

Thus, for sufficiently smooth electric field we have

lim 3‘0) = 0

N —-> on
This implies
lim 50) - 2 [anLn(;) + bnMnO) + an..(;)] dv = o (2.76)
N —) 0°

cavrn' n = 1

A A A
Therefore, the vector wave functions anI, Mnml and Nnml form a set of complete
orthonormal basis functions which can be used to expand the unknown electric field.

In case we know that the electric field is solenoidal, we can only use the vector
. .A A
wave functions M nml and N nml to represent the unknown electric field. That is, the vector

wave functions M nml and firm: become complete within the space of solenoidal vector
fields but not within the space of all vector fields [28]. The proof is as follows.

In this special case, we have
V - 20) = 0 (2.77)

Using the complete set of basis functions to expand the unknown electric field, we have

113(7) = 2 [anZn(>) + 6,174?) + cnfvnfin (2.73)

n=l

43

The expansion coefficient a n will be equal to zero in this special case, because

an = j E(>).Zn(?)dv= j E(?)-ki(V¢f)dv (2.79)

cavity cavitv

Using the identity of

V - [($330)] = V¢§-E(?)+¢f;V.E(?) (2.80)

The expansmn coefficrent a n 18 given by

an = 732': I V-[6fE(?)1dv— J’ ¢fv.E(?)dv]=ic1—’;S§ a.¢f;E(?)ds=o (2.81)

cavity Vcavity cavity

based on the boundary condition for ani. That is, the expansion of the solenoidal electric

field can be based only on the solenoidal vector wave functions. In other words, we can

A A
conclude that the vector wave functions M nml and N nml are complete within the space of

solenoidal vector fields.

2.2 Derivation of Dyadic Green’s Function and Electric Field Integral
Equation (EFIE) in Rectangular Cavities

In this section, based on Maxwell’s equations we will investigate the
electromagnetic fields behavior in a rectangular cavity with a non-ionic material sample
placed inside the cavity. The dielectric parameters of the material sample under
consideration are permittivity e = 8' + je”, permeability p.10 and conductivity 0'. We also

suppose that an initial cavity field has been set up before the material sample is placed

inside the cavity.

2.2.1 Maxwell’s Equations in the Material Sample

The curl equations of the Maxwell’s equations in the material sample can be

written as

VxE ‘ = -'(o I? ‘
VXHG) = oE(?)+ jweEG)

where E(;) and H (F) are the unknown electric and magnetic fields in the material

sample we aim to determine.

In the empty cavity, the Maxwell’s equation is given by

V x 3'0) = _jcou0§'(?)
4.- Ai (2.83)
VxH 0) = jweoE (7-)

where E10) and H l(?) are the initial electric and magnetic fields we assumed.
The initial cavity fields will induce electric currents and charges inside the material

sample. These induced electric currents and charges, in turn, will produce the scattered

fields or the secondary fields ESG) and [73(7) . In case the material sample is of finite size

or heterogeneous, there will be induced charges on the sample surface or at the
o a AS 0 s a

heterogeneity boundaries. Thus, V - E Will not be zero at the locations of the induced
"3 s . .

charges. Or E (r) has an irrotational component.

The total electromagnetic fields EO) and E1 (7-) can be expressed as

45

20) 2'0) + 2'0) (2.84)

20) 2'0) + 2'0) (2.85)

Substituting eqs. (2.84) and (2.85) into eqs. (2.82) and (2.83) leads to the equations for the

scattered fields as

Vx2'0) = —jwu021'0) (2.86)
Vx2'0) = 620)+ jme20)— jmeo2'0)= 38.10” jmeo2'0) (2.87)

where
3W0) = [6+ jw(e—eo)]20)= re(?)2(?) (2.88)

is the equivalent current and te(2) = 0+ jco(e—eo) is the equivalent complex

conductivity. Taking curl of eq. (2.86) and using eq. (2.87), we have
V x V x 2'0): — 16.60320) + 1:32:50) (2.89)

2 2 . .
where k0 = 0) 11080. Thus, we have the wave equation for the scattered electric field as

v x v x 2’0) _ 1.32%?) = 46603.4» (2.90)

2.2.2 Expansion of 2'0) and Derivation of the Electric Dyadic Green’s
Function

A A A
The vector wave functions anI , Mnml and Nnml form a complete set of
orthonormal basis functions, satisfy the same boundary conditions as the scattered electric

field does and are the solutions of the homogeneous vector Helmholtz equation with

46

. . . 2 . 2 . . .
particular eigenvalues kind. This km, 18 not equal to k0 appeanng m the inhomogeneous

wave equation of (2.90). However, we can solve eq. (2.90) by expanding E30) in terms of

_.A A

the vector wave functions anI . M um! and N nml . That is,
5’0) = z[a,,L,.0) + bnMnG) + an..0)1 (2.91)
n

where an, bn and on are unknown expansion coefficients. For simplicity, we use one

index n instead of three indices n, m, and I in the summation of eq. (2.91). Substituting eq.

(2.91) into eq. (2.90) gives
v x V x 20,2..0) + 6,67,.0) + cn2n0)] — kgztaninm + but—M?) + cut—Mb]
I! n
= -jmqueq(?>

A A A
Using the properties of the vector wave functions anI , M ntnl and Nnml which

we have derived in Section 2.1, the above equation can be rewritten as

ZI-kéanznfi) + b,,(k,2, - kfifinm + Cn(i'\‘.',21 — k§)X/n0)] = — 16660320) (2.92)

Taking the scalar product of eq. (2.92) with ant , Il—Ifnmt and Nnmt , respectively

and integrating over the volume V, then applying the orthonormal property of the vector

A .A A
wave functions anI , Mum! and MW. we obtain the expressions for the unknown

expansion coefficients as
__ jmllo

an 2 j [Lq0o)-2,.00)1dv0 (2.93)
k0 V

sample

47

 

 

_j (on

b" = 0 I [1.400) Mn(r0)]dv0 (2.94)
k2 _kOV sample
_10)l1

c" = 0 I [1.600) Nn(r0)]dv0 (2.95)
k2 -k(2)V sample

Therefore, the expression for the scattered electric field E50) becomes

V0

2
V (2.96)

E (r) = -J(Dllo l Jeq(ro) Z[ Ln(r:)Ln(r)+ Mn(r0)Mn(:2)+’:Vn(ro)Nn(r)]d
o ‘ 0

sample

=-j0)llo l Jeq(;o)°5e(;02;)d"0
V

sample

where the integration region is over the material sample volume. The electric dyadic

Green’s function is identified as

(2.97)

I!

5800;) = Z[.12,.(r]:,)2,.(r) Mn(r0)Mn(:2)+l:Vn(r0)Nn(r):|
O - 0

2.2.3 Derivation of the Integral Equation in the Material Sample

Based on eq. (2.96) and the definition of the equivalent current J eq(2) given in eq.

(2.88), the expression for the scattered field can be expressed as

5'0) = —jqu j 1,00)E00)-(‘;,00,?)dv0 (2.98)
V

sample

Substituting eq. (2.98) into eq. (2.84) gives the electric field integral equation (EFIE) for

the unknown electric field 27(2) inside the material sample as

48

£0) + 1'qu I 1.091506) - (7.00. 3de = E 0) (2.99)
where G420, 2) is given by eq. (2.97).

2.2.4 Expression of the Dyadic Green’s Function

The identity (2.75) can be applied to the electric dyadic Green’s function of eq.
(2.97) to lead it to an almost identical expression for the electric dyadic Green’s function

derived by Rahmat-Samii [l 1]. Using identity (2.75), eq. (2.97) can be rewritten as

(7,00, F)

 

kzfin0om7n0) + 2n00)2n0) 780 — :0)
2 k2(k2 _ k2) _ k2
- x0 :1 O 0 (2100)
aeoGh'O’ ’8’) " M
k0

where

E;eo(;09 ;) = z

n

 

[kzMn(r0)Mn(r) + Nn(r0)Nn(r)] (2.10,)

n 2 2 2
k0(kn ‘ 1‘0)
Therefore, the EFIE of eq. (2.99) can be rewritten as

A 3 jwl’lOTe . s A x — x s "i s
E(r) 1— k2 + jquJte(ro)E(r0) - 0,000, r)dv0 = E (r) (2.102)
0 v

 

2.2.5 Detailed Expression of (7,000, l) and Comparison with the Results
of Y. Rahmat-Samii [11]

For simplicity, we derive only the coefficients for the different components of the

dyadic Green’s function 580(20, 2) when we give the expressions of (76000, 2) , then we

49

compare the results with those derived by Y. Rahmat-Samii [11].
Substituting the expressions of 117nm! given in eq. (2.24) and Nam! given in eq.

(2.32) into eq. (2.101) and using the normalization constants of 07m: and N nml given in

eqs. (2.62) and (2.65), we obtain the expressions for the coefficients of the nine

components of the dyadic Green’s function Geo(20, 2) as follows:

(1) Coefficient for in? component of 520(20, 2)

 

 

 

 

 

 

32M2+EMZQE2 mttz m2
" b 2 a c — + _
_ 2 kn 8OnfiOmgOl b C
gem kn 2 2 2 = abc 2 2 2 (2.103)
k0(kn — k0) ko(kn — k0)
where B: = C: based on eqs. (2.62) and (2.65).
(2)Coefficient for 9; component of 520(20, 2)
2 1t 2 C2 1: 2 In 2
3(a) +—"('-"-) H (2)2 (9)2
a 2 b c +
2 kn 80718017180! a C 2 04
geow _ kn 2 2 2 = abc 2 2 2 ('1 )
k0(kn — k0) k0(kn — k0)
(3)Coefficient for 22 component of (720(20, 2)
((fl)'+('"—")')' (Sal-’32
2 a b EOHEOMEOI a b
eozz = Cr: - (2.105)

2 2 2 ‘ 2 2 2
“(kn—1(0) “be k0(kn—k0)

(4)Coefficient for if! and 325% component of 520(20, 2)

50

C2 2
2 a b kn a b C EOHEOMEOI ‘1 b
geoxy = kn = (2.106)

2 2 2 2 2
k0(kn-k0) “be k§(k,,—k0)

 

 

(5)Coefficient for 92 and 25) component of Gw(20, 2)

C2
n

 

 

 

 

(’"-")(’-’l
a e e b
eoyz = " 2 2 2 (anX'g): 0:22: 0' 2 2 62 (2'107)
k0(kn — k0) k0(kn — k0)
(6)Coefficient for 3:2 and 25: component of (720(20, 2)
(99(9) "9(1)
2 a b ('11:)(ln) EOREOMEOI a C
eoxz '1 2 2 2 2
k0(kn—k0) a C

abc kgucn _ [(3)

Comparing all these coefficients with the expression (28) of Y. Rahmat-Samii [11],
we find that they are almost the same except there is a minus sign difference. Checking
carefully the results of (28) of Y. Rahmat-Samii [1 l], we found an error occurred in his

expression (28). See Appendix A.

2.2.6 Derivation of the Electrical Field Outside the Material Sample

Outside the material sample, the total electric field can also be expressed as

20) = 2‘0)+E'0) ' (2.109)
where based on eq. (2.96) the scattered field maintained by the induced currents and

charges in the material sample can be expressed as

51

12:50): —jc0u0 j Jeq(?0).c‘;e(?0,?)dv0 (2.110)

sample

Because the field point i' is outside the material sample and the source point ;0 is inside

the material sample, then in the expression of the electric dyadic Green’s function
(LOO, I") given in eq. (2.100) the second term including the unit dyadic is always zero.

, -— 3
That 18, Ge(r0, ;) can be expressed as

6e(;0’;)= aeo(;02;) (2.111)
where 6,060, ;) is given in eq. (2.101). Therefore the electric field outside the material
sample is given by

50) =—jwtto l 1.001560)-6.0(FO,?)dvo+E'(?) (2.112)
V

sample

where we assume that the electric field inside the material sample E ( ;0) has been solved

from the EFIE in the material sample given in eq. (2. 102). ER?) is the initial electric field
we assumed before we place the material sample in the cavity. Therefore, after we obtain
the solution of the electric field inside the material sample, the electromagnetic fields
outside the material sample can be easily calculated based on eq. (2.112). For this reason,

we will only show the electric field inside the material sample in the numerical examples

in Chapter 3.

52

2.3 Derivation of the Magnetic Dyadic Green’s Function and Magnetic
Field Integral Equation (MFIE)

As in Section 2.2, the behaviors of the scattered fields (1?, £3) are described by
eqs. (2.86) and (2.87). Taking curl of eq. (2.87) and using eq. (2.86), we have

V X V X #0) = V x 3.340) + jmeOV x 3350): V x 3.40) + 123330)

Or the wave equation for the scattered magnetic field H 5(2) can be expressed as

VxVxfiS(;)-kgfis(?) .—. VquG) (2.113)

Based on Maxwell’s equations, the magnetic field (total field or scattered field) is

solenoidal inside the material sample or in the cavity. That is,

V - 1730) = 0 (2.114)

So the wave eq. (2.113) can be rewritten as

V2173(?)+ 1:32???) = —V x .74?) (2.115)

Also the solenoidal vector wave functions are complete within the space of the solenoidal
vector fields as discussed in Section 2.1.6. The orthogonality of the cavity magnetic
eigenfunctions is well known [2], [10]. Thus, the cavity magnetic eigenfunctions which

are solenoidal should be sufficient to be employed to expand the magnetic field for solving

eq. (2.115).

0 o A: o o n
The expansmn of the scattered magnetic field H (2) usmg the cav1ty magnetic

eigenfunctions as the basis functions leads to

53

3‘0) = zanfinfi) (2.116)

A
where HA?) is the cavity resonant mode, and it satisfies the homogeneous vector

Helmholtz equation as
V2174?) + kfifinfi) = 0 (2.117)

where k: is the eigenvalue of the nth cavity mode [10].

Substituting eq. (2.116) into eq. (2.115) and using eq. (2.117), we have
zan(k(2, - kfi)Hn(?) = —V x Jeq(;) (2.118)
I:
Since

0 ifn¢m
N ifn=m

n

j fin(?)-§m(?)dv = { (2.119)

Vcavitv

where N n can be found through the normalization of 1:54;), the nth cavity modes of

electric field. That is, we suppose

j En(;).Em(?)dv = {0 ff ”m (2.120)
1 If n = m

vcavuy

Based on Maxwell’s equations for the electromagnetic eigenmodes, we have

 

 

N" = j §n(;)-§n(?)dv=— 212 j VxEn(?).VxEn(?)dv (2.121)
vcavin' n Ovcavity
= _ 212 j [V-(En(?)xVxEn(?))+En(;)-VxVxEn(?)]dv
(on ovcavilr

54

Therefore, based on boundary conditions and the property of the eigenmodes we have

I fin(?)-fi.(?)dv = T810] Em) En(r)dv= 181—0 (2.122)

v 0

cavity ovcavlry

Let’s go back to eq. (2.118). Taking a scalar product of H10) with eq. (2.118),

integrating over the cavity volume, and using the orthogonality of the cavity magnetic

eigenfunctions, we derive the expansion coefficients as

an: _21_2 I Hn(r) VXJeq(r)dV (2.123)
N "(’60- k2 n)V

CCU!"

Substituting eq. (2.123) into eq. (2.116) leads to the expression for the scattered magnetic
A: 3
field H (r) as

I Hn(r0) VXJ¢q(r0)dv0

H30) = 2" Hum ' (2.124)
,. Nn(kn—k0)

 

Using the vector identity as

V . [71.0) x 3.40)] = [V x 3.0)] - 3.60) — [V x 3.60)] - 23.0) (2.125)

we have

I Hn(;0) ‘ V. X Jeq(;0)dvo

cavity

= I V0x§.(;O)-}.q(>o)dv'— j VO'[fin(;0)x;BQ(;0)]dVO (2.126)

V

V

cavm- vca wiry

I V0 X Hn(;0) ' Jeq(;0)dvo

sample

V

55

because J eq(;0) exists only inside the material sample, then J 82,00) = 0 on the walls of

the cavity. Therefore,

I [Hn(r0) V OXJeq(r0)]dv0

As;

 

 

H (r): m Hn(r)
2 N ”(k -k0)
e V0 Hn (2.127)
= 2 J- J 400) X (r0)deHn(r)
vsamplc Nn(kn - k0)

I Jeq(r0) ' 61,100, ?)dv0

vsample

where the magnetic Dyadic Green’s function is identified as

_ , , V x§n(?o)iin(?)
Gm(ro, r) = 2 ° 2 2 (2.128)
n Nn(kn-k0)

 

The equivalent current jer) = 1,0)?50) and based on a Maxwell’s equation,

330) = “—302 (2.129)
04-10»:

the equivalent current can be expressed in terms of the magnetic field as

s VXH(I')

Jeq(r)=1:,r(r)o+jm8 (2.130)

Substituting eq. (2.130) into eq. (2.127) and using eq. (2.85) leads to the magnetic field

integral equation (MFIE) as
.. , VOXHGO). _
H0) J r.r( ro) 6+ij G<ro,r)dv0- H0) ( )

vwmplr

S6

2.4 Comparison of EFIE with MFIE and Explanation of the Result

In MFIE (2.131), the unknown magnetic field appears in the form of the curl under
the integral sign. If we use the pulse functions as the basis and testing functions in the
moment method, there will be singularities occurred when taking curl of the unknown
magnetic field at the boundaries of volume cells. On the other hand, we can solve EFIE
(2.102) using the pulse functions as both the basis and the testing function without any
difficulty because there is no differentiation for the unknown electric field involved.
Actually, we can show that EFIE (2.102) and MFIE (2.131) are exactly the same. If the
pulse functions are to be used as both the basis and testing functions, it is easier to find the
unknown electric fields from the EFIE and then determine the unknown magnetic fields
directly from a Maxwell’s equation. Of course for higher numerical accuracy, we can try
to use the continuous functions as the basis and the testing functions in the moment
method. Then either EFIE or MFIE can be solved directly, and they should give the same
results. We will show that EF IE and MF IE are identical.

Taking curl of both sides of MFIE (2. 131) and using eq. (2.130), we have

Vx§(?)—V x I 3.4%) - Gm(?o,?)dv0 = Vxfi'G) (2.132)

vsamplr

Based on eqs. (2.82) and (2.83), eq. (2.132) can be rewritten as

(6+jme)z'3:(?)— j re(;0)E(;o)-V><G,,,(;o,;)dv0 = jmeOE'G) (2.133)

"sample

Because

57

— x s V Xiizx It; 2
VxGm(rO, r) = sz 0 (JO) 2"“)
n Nn(kn—ko)

 

 

 

 

 

A A 2 2A A (2.134)
_ 2,‘VOXH,.(r0)Van(?) _ ZmnEOE”(;)E"(rO)
‘ 2 2 ‘ - 2 2
n Nn(kn—k0) n Nn(kn-k0)
Substituting eq. (2.134) into eq. (2.133), we have
(6+ '008) 1 (928273“ (i); (i6) '
A s a A s n n n A‘ x
#E(’)+Ee— j Te(ro)E(r0) . 2 ° 2 2 ch», = E (r) (2.135)
1 0 J Ovmmple n Nn(kn _ k0)
Based on eq. (2.122) and “re = o +j(0(8 —80) , eq. (2.135) can be rewritten as
k2 E r E r
[1-2121 (r)]E(r )+{£°5"_° 1: (r0)E(r0) Z" "( ) "( O)dv v0 = E(r) (2.136)
k0 O vsample n"k0

which is exactly the same integral equation as eq. (2.102). That is, even using different
dyadic Green’s functions, we still obtain the consistent results.

At this point, there will be a question raised: When we expand the magnetic field,
we only use the solenoidal eigenfunctions as basis functions. However, we use both the
solenoidal and the irrotational eigenfunctions as basis functions to expand the electric
field. Where do the irrotational eigenfunctions come from if we derive the EFIE from the
magnetic dyadic Green’s function? How can these two different approaches reach the
same result?

As stated before, because of the existence of the material sample of finite size in
the cavity, the divergence of the electric field doesn’t vanish at all points in the cavity, but
the divergence of the magnetic field vanishes at any point inside the cavity. So when we

eXpand the electric field, both the solenoidal and the irrotational eigenfunctions are

58

necessary. However, the solenoidal eigenfunctions are sufficient for expanding the

magnetic field.

When we transform MFIE to EFIE, we take the curl of MFIE. Based on the
definitions of the electric and magnetic dyadic Green’s functions, the electric and

magnetic fields can be expressed in terms of the corresponding dyadic Green’s function as

[20]
E0) = 46100 I j j 6,00, ?) - 300)de (2.137)

H0) = ”[6,,00, ?) 00.021120 (2.138)

The Maxwell’s equations inside the material sample are given as
V x E0) = 46100170) (2.139)

V x 170) = i0) + jmeE0) (2.140)

Combining eqs. (2.137) to (2. 140) we can obtain the relationship between the electric and

magnetic dyadic Green’s functions as

Vx (‘3, = Gm (2.141)

V x 0,, = i130 - i0) + 1226,, (2.142)

That is, taking the curl of the magnetic dyadic Green’s function, we introduce a singularity

at the source point based on eq. (2.142). From eqs. (2.141) and (2.142), we observe that

V . 6,, = 0 (2.143)

and

59

V - (2:40 (2.144)

That is, the electric dyadic Green’s function can have an irrotational component even
though we derive it from the magnetic dyadic Green’s function.

In the previous sections, we have obtained the electric and magnetic dyadic

Green’s functions as

 

 

 

-xs -Zn;2n; fin;fin;+fin;fin;
G,(ro,r)=2[ (0; (1+ (0) (3 2(0) n] (2145)
n k0 kn-ko
and
— x 3 V XI?" 8 I? 3
Gm(r0, r) = 2 ° ('0) "m (2.146)

2 2
N n(kn '— k0)
It can be verified that the electric and magnetic dyadic Green’s functions (2.145) and
(2.146) satisfy the relationship of eqs. (2.141) and (2.142).

After we obtain the appropriate expression for the electric dyadic Green’s function

and EFIE, we will numerically solve this EFIE (2.102) and present some numerical

techniques and results in Chapter 3.

60

CHAPTER 3

NUMERICAL TECHNIQUES AND RESULTS
ON THE INDUCED ELECTRIC FIELD IN A
MATERIAL SAMPLE PLACED WITHIN A
RECTANGULAR CAVITY

In Chapter 2, we have derived the EFIE for a material sample placed inside a
rectangular cavity based on either the electric dyadic Green’s function (2.101) or the
magnetic dyadic Green’s function (2.128). After the appropriate integral eq. (2.102) is
acquired, it will be numerically solved by discretizing the material sample into a large
number of volume cells using Galerkin’s method.

In fact, in the expression of the electric dyadic Green’s function (2.101) there are
nine triple infinite summations which have very poor convergence properties [2]. To
overcome this difficulty, a triple summation over cavity eigenfunctions is reduced to a
double summation using two relations in [2]. For the reduced double summations, several
methods are employed to obtain a faster computation.

The outline of this chapter is as follows: In Section 3.1, Galerkin’s method is '
applied to the integral eq. (2.102). In Section 3.2, the convergence property of the dyadic
Green’s function (2.101) is studied and the results of the double summation and the triple

summation are compared. Some numerical results are presented in Section 3.3. Several

61

methods are investigated to increase the convergence rate of the infinite double

summations in Section 3.4.

3.1 Applying Galerkin’s Method to EFIE

In order to increase the convergence rate of the dyadic Green’s function, Galerkin’s
method is applied to the EFIE (2.102) with the pulse functions as both basis and testing
functions. The material sample is divided into M volume cells where M is assumed to be
large enough in order to generate satisfactory results.

The integral equation we are to solve is eq. (2.102) as

4. 1011110) . .A. _ .. At.
E(r)[1- ——:—2-"——] +JmHoJTe(70)E(r0) - 0.000. r)dv0 = E (r) (3.1)
0 v
and the pulse function is given by

10,0) = { 1 if 'E V" (3.2)

0 otherwise

where v" is the nth cell volume. The unknown electric field E (7') can then be expanded as

M
E0) = 2 Empm0) (3.3)

m=l

where the electric field in each volume cell is considered as a constant. Substituting eq.

(3.3) into eq. (3.1) leads to

62

M A . 10311014?) , . M A . _ . .
2 Empm(r) 1- —2— +](D110 J te(r0) 2 Empm(l"0) ' Ge0(l‘o, r)dv0
m = 1 k0 vramplr m = 1 (3'4)

= 3’0)

Next, applying the testing function (3.2) to the integral eq. (3.4), we have

M . s
. .4 . 1001.1 re(r) . .
JPnU) 2 Eum(r)[1 -—:—2-—]dv + Jpn(r)jo)tto (3.5)
Va "1 =1 0 V"
M _. A1
vsample m = 1 V"

where the initial electric field can be assumed to be a constant within each volume cell and

the integration region is over the nth cell volume vn. Equation (3.5) is integrated with

respect to the variable I" as

A jmlio s s M A x
E. 1—7 [ re(r)dv +ij0 I 1,00) 2 Empm(r0) (3.6)

0 V” vsample m = 1

- [104351.006 ?)dvdv0] = E'0n)Avn

vn

Let’s denote

z”;t'1t’n(;0’;n) = Jae0(;09 ;)dV (37)

V"

where 612,100.31) specifies the integration of (7,000, ;) with respect to ; over the nth

cell volume v". Equation (3.6) can then be rewritten as

63

A 1‘ am
En[1_——OIT(r)d\/)+ 1(1)}.10 [ 2 Empm(r0) I(rO)G,-en(r0, r,,)dv0
k0 vn v-ramplem " 1 (3'8)

= E'0,)AV,,

Because the number of volume cells is assumed to be large enough, in eq. (3.8) the
integration which is over the total material sample volume can be estimated by the

summation of the integrations over each cell volume as:

M
A j (011
En[1--—29JT (r)dv)+j0)uozj 2 Empm(ro) r (r0)Gien(r0’ r,,)]dvo
k0 v I: 1v, =1 (3 9)
= E'0,)Avn
Let’s denote
54.101, a.) = 1 1.001604%, 8.1de (3.10)

V1
where 6:2me 2n) specifies the integration of 18005121200: ;,,) with respect to 20 over
A l
the volume v, and Empm(r0) is not equal to zero in the volume v, only when m = I

based on the definition of the pulse function pm(;) defined in eq. (3.2). Thus, eq. (3.9) can

be rewritten as

En[l —J—fl)J‘t (r)dv]+jtnp.oM 2 E1 Gienl(r,,r r") = E (rn)AVn (3.11)
k0 V» [=1

Also we can express Ethe(?)dv as

Vn

64

M

E. j 1,0)dv = 2 E1 1 1,0)dv - is", (3.12)
V" I: 1 IV,
- . . . 1 tf n = l
where 1 1s the un1tdyad1c and 5", = O f ¢ . Therefore, based on eq. (3.12),
t n

eq. (3.11) can be rewritten as

M .
O) 3 ' — x x A '31. 3
2 [[1_ .Lg—12J16(r)dv]lsnl + jmuOGienKrb rn):| ' E1 = E (Tn)AVn (3.13)

[=1 0 V"

where n=1,2, ...... ,M. Equation (3.13) can be expressed in a matrix format as

[in]... M-[EllMxl = [3.1/14.1 (3.14)
where n, I=I,2,...,M and
— jmll x - . "' 8 A
A", = [1-—5-OJT,(r)dV]15n1+leloTeGieant,Tn) (3.15)
0 V.
8,, = E'0n)Avn (3.16)

If we use the scalar components format instead of the vector format, the matrix eq. (3.14)

can be expressed as
[Anll3Mx3M'[El]3Mx1 = [3113mm (3‘17)

where

T
[E113MX1 = [E1X”'EMXEl)"°'EM_VElz'°'EMZ] (3'18)

65

1811mm IE.’(>1)AVI...E.‘(?M>AVM Ey'<;.)AV1-.-Eyi<;M)AVM (319)
Ezi(2‘1)AVl ...Ezi(;M)AVM]T

[Endxx [Knllxy [XML]
[Anll3Mx3M = [4.11,.14.11,,1K.,1,, (3.20)
[13.1]... 14.11,, 12.11,,

 

 

and the dimensions of each submatrix in (3.20) are M x M .

3.2 Convergence Property of Dyadic Green’s Function in EFIE

In the numerical computation, the most difficult step in solving EFIE is to fill out

the matrix [An,]3Mx 3M in eqs. (3.17) to (3.20). The integrations of the dyadic Green’s

function Ge0(20, 2) at different points in the material sample with respect to both

variables 2 and 20 as specified in eqs. (3.7) and (3.10) need to be carried out in the matrix
composition. However, the convergence property of the integration of the dyadic Green's
function 580(20, 2) is still very poor even though the Galerkin’s method is used.

The dyadic Green’s function Ge0(20, 2) in a triple summation format is given by

eq. (2.101) in Chapter 2 as

_ , , °° °° °° M ((2 )M ((2)44)? ((2 )1? 1(2)
Ge0(r09 r) = Z Z [kind "m 0 nmz 2 m2" 0 "m
0 k0(knml - k0)

:I (3.21)

n=0m=01=

Since eq. (3.21) has a very poor convergence property [2], the integration of it has a poor

convergence property as well. Several numerical results are shown here to illustrate the

slow convergence of the integration of the dyadic Green’s function (760(20, 2).

66

In these numerical results, we only show the convergence property of the

integration of G wnGO, ;) component at the different points and avoid the repetition of
the computation for the other components of the dyadic Green’s function 36000, ;) since
they all have the similar convergence properties. The coefficient of G e ”1‘00, ;) was given

in eq. (2.95) in Chapter 2, and the expression of Geonfio, ;) is then given by

mrt 2 In 2
””0( j +(?) nit nit
Geoxx(r0’ r): 2 2 2 n COS(—x)COS(—x0)
n=0m=ll=labc ko(k,2, -k()) a a (3.22)

sin(’1?-t)sin(m—1t )sinC-Lt )sinGT-t )
b b yo cZ CZO

where so" is defined in eq. (2.63).

 

In the following computations, we assume that the initial cavity mode is TE 101.

then the eigenvalue (wavenumber) of this initial mode is given by

2 2
k3 = (1‘) +(E) (3.23)
a c
and in eq. (3.22) we will change the summation upper limit from 00 to N. We aim to

choose some value of N which makes the integration of the triple summation series (3.22)

converge. The dimensions of the rectangular cavity are a = 0.072m , b = 0.034m and

c = 0.1163m . The initial resonant frequency is then f0=2.4SGHz based on eq. (3.23).

. . 3. k . . . .
The integration of Geon(r0, r) With respect to i” and ;0 in regions v" and v"O is

expressed as

67

x+Axy+Ayz+Azxo+Axy0+Ayzo+Az
3 8
J J I J j j Geon(r0,r)dudvdwdu0dv0dwo

x ) Z In )'0 z0
~ (m)2+(-’l‘lz
- 2 iii 8 ” . ‘
n = 1m = H: labc k(2)(kr21_ k3) (E)2(n1_1t)2({1_t)
a b c

mfl: m1!
—b—y] [COS —b—'(y0 + Ay)

 

2[sing—[Or + Ax)—sinp-1-tx]
a a (3.24)
[sing-rmO + Ax) - sinn—T-chHCOSm—nw + Ay) - cos
a a b
—cos-'-n-1—t ][coslfl( +A )— cost—1E ][coslit(z +Az)- cosy—tz ]
b yo c z z c z c 0 c 0

for n :t O and

x+Axy+Ayz + Azxo + AI)'0+Ayzo+Az

I J I I I J Geonfio,;)dudvdwduodv0dw0 (3.25)
x y 7- 10 )‘o Z0

(anY + (1’5“)2 sz

‘ 2
m. (1.. ko<ki-k3> (Mfffl)
b c

 

mrt mic
2[cos -—b—(y + Ay)-cos Ty]

mtt

mi: In In In In
[cos—b-(y0 + Ay)—cos b y0][cos?(z + Az) — cos Fz][cos:(z0 + Az) — cos :10]

for n = 0.We choose Ax = Ay = A2 = 0.002m in the following computations.

One thing which needs a special attention in the computation is when k: = k: , the
summation term will have a singularity. This occurs because one of the summation modes
is exactly equal to the initial mode, that is, n = no, m = m0 and l = ID, where no, mo
and [0 specify the indices of the initial mode. From the experiments we observed that when

a material sample is placed into the cavity, the resonant frequency of the initial mode will

68

shift down about 1% to 10% depending on the geometry of the material sample (this

resonant frequency shift is also shown in [68]). Based on this experimental observation,

when k: = k: we will make the approximation of

k3, — kg 2 —sk§ (3.26)

where s is the shift rate of the resonant eigenvalue. The followings are the integrations of

waxGO, ;) at different points with the assumed resonant frequency shift rate to be 5 %.
1. When ; = f0 , the integration of Geonfio, ;) is shown in Figure 3.1. The source and

observation points are i = P0: [0.033m, 0.014m, 0.0551m].

2. when ;¢ ;0,

(a) The source and observation points are i: [0.035m,0.014m,0.0551m],

i0: [0.033m, 0.014m, 0.0551 m] , the integration of Geoxxfio, ?) is shown in Figure 3.2.

(b) The source and observation points are i: [0.035m, 0.016m, 0.0553m],
;0= [0.033m, 0.014m, 0.0551m] , the integration of Geonfio, ;) is shown in Figure 3.3.
In Figure 3.1 to Figure 3.3, the horizontal axes are the value of N and the vertical axes are
the integration of G e OHGO, I").

In all of these computations, we varied N from I to 400. These figures show that
when i = h), the convergence rate is slower than those of :‘i’ i0. Also when x #:xo,
y ¢ Y0 and z ¢ 20 , the integration converges fastest. Thus, we can conclude that the farther

the distance between the observation point and the source point is, the faster this

69

 

 

1.8- J

1.6- ..

.A
b

I
l

.L
N
I

l

summation of DGF integration
'?

 

 

 

0.8 - —
0.6 - _
0.4 » .
0.2 - -
0 ' I i 1 1 + I
0 so 100 150 200 250 300 350 400

upper limit of summation N

Figure 3.1 Integration of the triple summation format G e 0 n( i0, ;) vs. the number

of summation modes when ; = :50, i'= [0.033m, 0.014m, 0.0551m] and
Ax = Ay = A2

a

0.002m . The dimensions of the rectangular cavity are:
0.072m , b = 0.034m and c = 0.1163m.

7O

4.5 I I I

 

 

to in3
N U! 0) U!
1 I I I

summation oi DGF integration

.5
0|
I

0.5 P ..

 

 

0 L l 1 l I J l
0 50 100 150 200 250 300 350 400
upper limit of summation N

 

Figure 3.2 Integration of the triple summation format G e 0 ”Go, ;) vs. the number

of summation modes when i: [0.035m, 0.014m, 0.0551m] ,
to: [0.033m,0.014m,0.0551m] and Ax = Ay = A2 = 0.002m.The dimensions of
the rectangular cavity are: a = 0.072m , b = 0.034m and c = 0.1163m.

71

 

 

 

summation 0t DGF integration
I

 

 

 

 

_1 i i i L J 4 i
0 50 100 150 200 250 300 350 400
upper limit of summation N

 

Figure 3.3 Integration of the triple summation format G e 0 x x( F0, ?) vs. the number
of summation modes when F: [0.035m, 0.016m, 0.0553m],
to: [0.033m,0.014m,0.0551m] and Ax = Ay = A2 = 0.002m.The dimensions of

the rectangular cavity are: a = 0.072m, b = 0.034m and c = 0.1163m.

72

. . 8 X . .
integration converges. However, when r = r0 With the parameters chosen in case 1, the

convergence value in Figure 3.1 is about 1.88 x 10’12 and those in Figure 3.2 and Figure

3.3 are about 4.15 X 10.13 and 1.05 x 10‘15 with the parameters chosen in case 2(a) and

case 2(b). The i = 2'0 terms are on the diagonal lines of each submatrix in (3.20) and they
are dominant in this matrix in terms of the numerical results.

Observing these three figures and considering the trade-off between the numerical
accuracy and the computation time, we conclude that when N = 200 we can obtain the
satisfactory convergence results for these three cases. However, this is over 8 million terms
summation! This indicates that the convergence rate of the integration of the dyadic
Green’s function GEOGO, ;) is extremely slow.

To save computation time we can reduce the triple summation in the dyadic

Green’s function to a double summation based on the following two relations:

 

 

. (nit ) . (mt )
co 5m —x Sin —x0
2 a a ' a sin(k (a r ))sin(k x) (3 27)
— ' ml —' b 1 .
8 cos(fltxjcosC-z-Ex )
On a a O _a

 

 

= . cos(k (a—x ))C03(k x) (3.28)
13:0 2(k2-k3) 2kngSIn(kgmla) 8'"! [7 gm] 5

where so" is defined in eq. (2.63) and

2 2 2
k3, = (E) +(m—n) +(l—n) (3.29)
a b c

73

 

2 2

2 mrt In

kgml = Jko- (7;) - (7) (3.30)
It will be convenient to define the following

xb E the greater of x or x0 (3.31)
x5 5 the lesser of x or x0 (3.32)
xb and x 3 will be referred later for the same definitions. Detailed derivation of the last two

relations (3.27) and (3.28) and the representation of Geo in the double summation format
can be found in Appendix C. For simplicity, we will only perform the integration of

G e OMU'O, ?) represented in the double summation format.

In order to obtain the expression for G e 0300, i') in the double summation format,

we can sum over any one of the indices n, m, I in eq. (3.22) using the relations given by

. . mi: 2 In 2 .
eqs. (3.27) and (3.28). However, Since there is a factor of (T) + (2:) in the numerator

of the eq. (3.22), we can obtain the simplest expression for G (7'0, ?) in the double

€0.11

summation format if we sum over the index n by eq. (3.28) in the triple summation (3.22).
Using eq. (C.19), the expression for GwnGO, ;) in the double summation format can be
expressed as
(mn)2 (my
— + _
b C m1:

_ s s 1 °° °° 4 .
Geoxx(rO’ r) = -_ — . g (x,x )Sln—y
kczimgilgibCkgmlsmwkgml) MI 0 b

 

(3.33)

sinr-nlty sinmzsinmz
b 0 c c 0

74

where k W has been defined in eq. (3.30) and

gm,(x, x0) = cos(kgm,(a — xb))cos(kgm,x5) (3.34)
It is noted that there is a factor of kgmlsin(akgm,) in the denominator in eq. (3.33). When

kgml = 0, there exists singularity. For this case, we can not use the double summation

format expression of GGOHGO, i‘) (3.33). We should sum over the index n directly for this

special case taking into account of a slight shift in the resonant frequency. These

summations can be found in Appendix C. From eq. (C.38), for n0 = O , where no is one of

the indices of the initial cavity mode, the summation over index n becomes

 

 

 

nrt nrt
.. Zoos-Execs?“ l 1 1 2 2 a
___ +2- 2 5———i+—(x +x0)+-—xb (3.35)
isko ,_ _ , (r135) asko 2“ 3
a
and for nO it 0 , eq. (C39) gives the summation over index n as:

0° eOn 1 nit m:

— 2 2cos—xcos---x0

~=°“("-") ("L") “ “
a a
4 (non)2
no _—
= 2: - a cosfltxcosflx — l(— a —)2 + -1--(J'f2 + x 2)
. i i (a) (an) . . . .. .

n = _ _ _—

natno a a a

a 2 1 1 "07[ non
+§_xb-E TIE—2+7 COSTXCOS—CI-‘xo

(L) 5k0
a

where s has been defined in eq. (3.26).

75

Substituting eqs. (3.35) and (3.36) into eq. (3.22), we can obtain another

alternative representation of G e 0300’ i). Fortunately, the variables x, y, z are separable

in the summation, thus, we can integrate G e (MOO, P) with respect to them independently.

In the double summation format representation of 080300, ;) given by eq.
(3.33), the factor k gm, may be a real number, eq. (3.30) or an imaginary number for most

cases because we usually assume the initial mode to be a lower order mode. When kgm, is

an imaginary number,

2 2
km = 40%) +(’%‘) —k(2, = ikgm“ (3.37)

where kgmh- is a real number, eq. (3.34) can be rewritten as

 

gm[j(x9x0) = cosh(k (a—xb))cosh(k (3.38)

ngi ngi x5)

and the double summation format representation of G (F0, ;) can be rewritten as

COXX

($.ng m.

sinh(ak gmh-(x, x0)sm-3-y

 

:iMS

2.1

— 1
Geoxx(r0’ r) = I?
kMO

(3.39)

kgmli' gmli)

sinmny sinmzsinmz
b 0 c c O

In the actual computation, we find that sinh(ak ) grows exponentially due to

gmli

akgm“ » l . Based on this condition, sinh(akgm“) can be estimated by

ak

gmli

sinh(ak (3.40)

gmli) E T

76

and eq. (3.38) can then be rewritten as

l k,,,,(a—x) —k,,,,-(a—x) kmx, —k .x
g! b +8 31 b 31 +6

 

gmh.(x,x0) = 2(3 )(e grill: 3)
1 k k 2 (3.41)
E _ a gmli(e- gmltxb + e —kgmli( a- xb))e kgmhx:
4
In eq. (3.41), because akgmh. » 1 , x < a and x0 < a , it can be estimated as
k H
gmnix 1(0):}, a ”"6 ‘"' Ix W (3.42)
Substituting eqs. (3.40) and (3.42) into eq. (3.39) leads to
m1t 2 In 2
1 0° 0° 4 T) + .2- Ix— x '81
G 0' r ;)- = "" — e 4"“ 0 —y
eoxx 0 k3 mgllgl bC 2kgmli nb (3.43)

sinmny sinmzsinmz
b 0 c c 0

Based on these alternative representations of G w ”00, P) for different cases, i.e.,

if kgm, is real or imaginary or if there is any singularity, the integration of eq. (3.33) is

given by

2 2
("—0 (0
b c 1

ZrI-[klsinakml
WU

 

 

1 °° °° 4
-- Z Z 17 gm,(u,u0)dudu0
=1l=1

2
k0",

(3.44)

m1t m1: mit m1:
[cos-70) + Ay)-cos-E- ][cos—b—(y0 + Ay) — cos —b—y0]

[cosLCT—tu + Az) — cos%tz][cos—(zo + Az) - cos {320]

Although in the expression of gm,(x, x0) there are different equations for x>x0 and

77

x < xo , also for the different k gm 1’ the expressions of the gm,(x, xo) involves only sine or
cosine and exponential functions of x and xo as given in eqs. (3.34) and (3.42), the
integrations of gm,(x, xo) with respect to x and xo become easier for x > xo or x < xo.
However, when x = xo , we need to pay a special attention to the integration.

1. When x at xo and kgm, is real, the integration of gm,(x, xo) with respect to x and xo is

 

 

given by
x+Axxo+Ax
I J; kg mlSin1(akgm1)gml(u’ “0)dudu0
°_1 . . (3.45)
= 3 . [sm(kgm,(a-xb—Ax))—sm(kgml(a—xb))]
kgmlsm(akgm,)

[sin(kgm,(x3 + Ax)) - sin(kgm,xs)]

2. When x at xo and k gm , is imaginary, the integration of gm,(x, xo) with respect to x and

xo is given by

x+Axxo+AX

-kgmli u- “(I 1 -krmhx- ()
I I -2———kl l ldu uduoz -—3—-e I xl(1—cosh(k

r0 ”"6 kgmli

mom» (3.46)

3. When k gm 1 is chosen in such a way that there is a singularity occurring, the integrations

of gm,(x, xo) with respect to x and xo are those of eqs. (3.35) and (3.36) with respect to x

and xo.

(1) When no 2 O and x at xo, the integration is given by

78

x+Axxo+Ax
1 1 1 2 2 a
J. J. [_Zm+i—CI(“ +u0 )+§—xb]duduo
1: x0 0

 

 

 

 

2
—-1-A—£-+A—x[(x+Ax)3—x3+(xo+Ax)3-xo3] (3-47)
a 2 6a
sko
an2 1 2 2
3 2
(2) When no at O and x at xo, the integration is given by
_ non 2
X+Axxo+Ax 4no —)
I I 2 a nit nil: l a 2
- 2 cos—ucos—uo—- —
- “(nn)2((nrt 2 (non) ) a a a 7101!
x xo n - l — _ _ __
Lnino a a a
l 2 2 a 2 1 1 "07E "07E
+2304 +uo )+3_xb-E ETC—2+5? cos—a-ucos-Z-uo duduo
_ 0
i . 1
1t 2
4no 2 (’1)
a . nit . nit
_. 2 a [Sin—a—(x+Ax)—Sin—a-x] (3.43)

2:..3, (’%‘l4(("z-“)2-('-’Z-"lzl

 

 

[sinflt(xo+Ax) - singxo]-[ a 2—9]Ax2-%[(xb+Ax)2—x:]Ax
a a (non) 3
+E[(x+Ax)3—x3+(xo+Ax)3—xo3]— 2 1 +-%
60 Sko

2 2
(non) (non)
a — _
a a
. no“ . "on . "on . "on
[Sill-:(X + Ax) - Sin—a—x][sm—(xo + Ax) — SinTxo]
a

4. When x = xo and kgm, is real, the integration of gm,(x, xo) with respect to x and xo is

given by

79

x+Axx+Ax

 

I I kg mlsin1(ak g (t,to)dtdto

 

 

gml)m
x+Ax 1 x+Ax
x gm 1
(3.49)
2' k Ax
1 SH] gnu—2"
km: . k2,...

[sin(kgml%-x)cos(kgml(a — 2x — Ax)) — sin (knga — LEEDI

5. When x = xo and kgm, is imaginary, the integration of gm,(x, xo) with respect to x

and xo is given by

x+Axx+Ax k
— — Ax
I I km“ (Oldtdt0= ——Ax +—1—[e W ~11 (3-50)
2——_k1 mlie [(2 k3
8 gmli gmli

6. When there is a singularity occurring and x = xo, the integration of gm,(x, xo) with
respect to x and xo is given by

(1)for no = O,

x+Axx+Ax

I I I—-—+—(t“ +to2)+-—thdtdto

_ _ 2Ax (x+Ax)2(x—Ax)_)§ 2-1.1. 2
63[(x+Ax) x3][a-1:I+ 2 2+[3 askSAx

 

(2) for no¢0,

8O

 

 

F nor: 2
x+Axx+Ax 4no (_)

I I 2 a nit nrt l( a )2
- cos—tcos—to-- —
.(,,).((,o. ("”121 . . .

x x ’3: — — — —

Inn” a a a
n it n it
+-—1-(t2+to2)+g—tb-Z —l—+-l— cositcosl-to dtdto
2a 3 a nor: 2 skz a a
_ O
i . l
2
4., (M) (3.52)
2 a

 

 

‘Eisner-("°-;))[‘°"“"§‘“A”‘S‘""7"xT-i(.:..2-‘-’l~2

+ (x + Art)3 + 2x3 _ (x + Ax)2(x - Ax)

 

 

+ é-xflx + Ax)3 — x3]
3a

 

6 2
n it n 11: 2
— 2 2 1 +—L [sini(x+Ax)—sin—O—x:I
(non) (non)2 sk2 a a
a —— — 0
a a

In the following computations, we will use eqs. (3.44) to (3.52) to perform the
integration of G eonfio, 2) . In eq. (3.44), we will use a finite number of N instead of 00 as

the summation upper limit and find some value of N which can lead to the converged

results which is consistent with the results of the triple summation. The dimensions of the

cavity, the initial mode, the resonant frequency shift, Ax, Ay, Az and the choices of the

. . 3 3 . . .
source and observation pomts r and r0 remain the same as those for the triple summation
in order to compare the convergence property of the triple and double summations.

Figure 3.4 is the integration of G eonfio, )2) when I“ = 20. Figure 3.5 and Figure

3.6 are the integrations of Geonfio, 2) when 2%: ;O. In Figure 3.4 to Figure 3.6, the

81

 

13l- -i

summation of DGF integration
0 ° .0 .-‘ :-* .-*
is 'm an a N is at
I I I I I I I
L l l l

p
N
I

l

 

 

0 l l l 1 l 1 l
0 50 100 150 200 250 300 350 400
upper limit of summation N

 

Figure 3.4 Integration of the double summation format Geoxxfio, 2') vs. the

number of summation modes when i' = 120, F: [0.033m, 0.014m, 0.0551m] and

Ax = Ay = A2 = 0.002m. The dimensions of the rectangular cavity are:
a = 0.072m, b = 0.034m and c = 0.1163m.

82

 

 

 

 

 

4.5 I i i r r I T
4- -
3.5" -'
S 3- -(
E
a
2
E
u_ 2.5 " -
(9
O
'5
S 2" .4
E
E
31.5“ -'
1- 4
0.5 " _.
0 J_ i i 4 i i i
O 50 100 150 200 250 300 350 400

upper limit at summation N

Figure 3.5 Integration of the double summation format G e 0 J“((20, 2) vs. the

number of summation modes when 2: [0.035 m, 0.014m, 0.0551m] ,
i0: [0.033m,0.014m,0.0551m] and Ax = Ay = A2 = 0.002m.The dimensions of
the rectangular cavity are: a = 0.072m, b = 0.034m and c = 0.1163m.

83

x102

 

 

 

 

 

 

 

3 I1 I I I I I I
C
.0 2 I— -1
‘5’
O
2
.5
§
'5
C
O
2

0 I 1 l J l I I

0 50 1 00 1 50 200 250 300 350 400

upper limit oi summation N

Figure 3.6 Integration of the double summation format G e 0 ”(20, 2) vs. the

number of summation modes when 2: [0.035 m, 0.016m, 0.0553m] ,
i0: [0.033m, 0.014m, 0.0551m] and Ax = Ay = A2 = 0.002m.The dimensions of
the rectangular cavity are: a = 0.072m, b = 0.034m and c = 0.1163m.

84

horizontal axes are the value of N and the vertical axes are the integration of G e ”1:00, i) .

Comparing Figure 3.1 with Figure 3.4, Figure 3.2 with Figure 3.5 and Figure 3.3
with Figure 3.6, which have the same selected parameters values, we observe that the
convergence rate is the same for the cases of the double and triple summations. Also both
cases converge to almost the same value at N = 200. Therefore, in the double

summation, we can set the same upper limit as that in the triple summation. However, the
double summation includes only 40,000 terms instead of 8 millions terms if the triple
summation is used. This drastic simplification is achieved because we used a closed form

evaluation to sum over one of the three indices.

3.3 Numerical Examples

In the following numerical computations, we suppose that a rectangular material
sample is placed in the center of the rectangular cavity and the dimensions of the

rectangular cavity are shown in Figure 3.7. The initial field is assumed to be TE 10] mode

and the resonant frequency of the empty cavity operating at this initial mode is 2.45 GHZ

with the wavelength )L equal to 0.12245m.

In order to quantify the induced electric field inside the material sample, we
uniformly divide the material sample into M = nd x m d x Id volume cells, where nd , m d
and Id are the number of volume cells in the x, y and z directions, respectively. Several

special cases with the selected shape and dimensions of the material sample, which can be

compared with some theoretical approximations, have been studied.

85

 

 

 

 

 

 

 

 

 

\---—--

\
\
I
I
I
I
I
I
N
I
I
I
I
I
I
I

 

,’ 5:410 y / z
_____ ___,
, 0.1163m

 

 

 

 

Figure 3.7 Dimensions of the rectangular cavity and the material sample. The
center of the material sample is consistent with the center of the cavity.

86

1. Cubic case
A cubic material sample, having equal three sides, is placed in the center of the
rectangular cavity. The dimensions of the material sample are set to be x=0.004m,

y=0.004m, z=0.004m and with the relative permittivity of e, = 2.5 and lossless. In the

computation, we chose nd = 2, md = 2 and Id = 2 . The dimensions of each volume

cell are: Ax = 33-, Ay = l- and Az = 5. (xi, yj, zk), i=1,nd,j=1,md and k=1,ld, will be

used to denote the center of the volume cells in the material sample. Based on the

convergence property discussed in Section 3.2 we chose the upper limit in the double

summation of N =200.

Since x/k = 0.0327, which is electrically very small, we may use the static

 

electric field induced inside of a dielectric sphere E = E [14], [38] to estimate the

2+8,

induced electric field in this cubic material sample. We also assume the resonant

frequency shift to be 5% after. placing the material sample in the rectangular cavity. The

numerical results are shown in Figure 3.8 in which the ratios of E v/ E; at the different

volume cells in the material sample are given.

The numerical results are E; = 321.5729 based on eq. (2.24), or the
normalization of the cavity field as discussed in Chapter 2, and E y = 203.9074 obtained

from the moment method. BV and E; are shown to be almost constant in each volume

cell in the material sample. This is expected because the dimensions of the material

87

f y 15)/E; at z=z,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y2 ””””” 0.634 0.634
y1 ----- 0.634 0.634
| l
| l
l I
I |
I l ’
X1 x2 x
+ y E /E iat -
l )' z“Z2
yz ————— 0.634 0.634
)1, ————— 0.634 0.634
I I
l l
l I
| I
I I
I L >
x, X2 x

Figure 3.8 Ratios of Ey/Eyi at different volume cells in the 4-mm cubic material
sample, where the relative permittivity of the material sample is assumed to be
e, = 2.5 . The geometry of the rectangular cavity is shown in Figure 3.7. The

resonant frequency shift is assumed to be 5%.

88

sample are very small compared with the dimensions of the rectangular cavity. The ratio

 

of E/Ey‘ is 0.634 in Figure 3.8. The electrostatic estimation of E y/ E; = gives the

3
2 + e,
approximation of 0.667. The closeness of the numerical result and the electrostatic
estimation gives confidence to the numerical accuracy.
Because the induced charge on the material sample surface and the induced
current in the material sample can maintain a scattered field, the other components of the

electric field are induced to satisfy the boundary conditions. The induced electric field

inside the material sample has Ex and Ez components with the amplitudes of 5.494 and

5.552 in each volume cell, which are very small compared with the y component of the
induced electric field. This shows that the initial mode still dominates inside the material
sample although the other modes are also induced.

In order to assure that the upper limit is chosen properly, we change N, the upper
limit in the double summation, from 160 to 1000 with the same resonant frequency shift of
5%. The results are shown in Table 3.1. As stated before, the induced electric field inside ‘
the 4-mm cubic material sample is almost constant. Since the y component of the induced
electric field dominates, we only compare the results of the y component of the induced

electric field for the different values of N at one volume cell, say (x1, y I,Z1)- From this table

we observe that as we increase N, the ratio of E y/ E; gets closer to 0.667. Considering the

accuracy of the numerical results and the computation time, we choose N=200 as a
compromise and this upper limit will be used in the following computations.
For different relative permittivities, the results of the 4-mm cubic material sample

are shown in Table 3.2. We chose the resonant frequency shift to be 5% in the

89

Table 3.1 Induced electric field inside the 4-mm cubic material sample and its ratio
to the initial electric field for different values of N, where the relative permittivity of

the sample is assumed to be e, = 2.5 , the resonant frequency shift is 5% and the

initial electric field is E 'y = 321.5729 . The geometry of the rectangular cavity is

 

 

 

 

 

 

 

 

 

 

 

shown in Figure 3.7.

N E), 15)/E; aifrcotsgaattitn
160 202.0437 0.628 0.667
200 203.9074 0.634 0.667
300 206.9128 0.643 0.667
400 208.6385 0.649 0.667
500 209.6048 0.652 0.667
600 210.2009 0.654 0.667
700 210.6752 0.655 0.667
800 21 1.0442 0.656 0.667
1000 21 1.5217 0.658 0.667

 

 

 

 

 

90

 

Table 3.2 Induced electric field inside the 4-mm cubic material sample and its ratio
to the initial electric field for different relative permittivities of the material sample,
where the resonant frequency shift is 5% and the initial electric field is

E; = 321.5729 . The geometry of the rectangular cavity is shown in Figure 3.7.

 

 

 

 

 

 

Electrostatic Relative
8, E). E/Eyi =Rc approximation Diflerence
Ra (Ra’Rc)/Ra
2.5 203.9074 0.634 0.667 4.95%
4.0 149.430] 0.4647 0.5 7.06%
7.0 97.4436 0.303 0.333 9.01%
10.0 72.306] 0.225 0.25 10.04%

 

 

 

 

 

 

 

91

computations. Also in Table 3.2, the induced electric field is shown only in one volume

R —R
cell. We define the relative difference as —"T—-‘3 , where R a is the ratio of the induced

a

electric field to the initial electric field based on the electrostatic approximation of

 

3
2+8

r

as shown in the fourth column of Table 3.2, RC is the ratio of the calculated induced

electric field to the initial electric field as shown in the third column of Table 3.2. From

this table we observe that the computational difference tends to increase as the relative
permittivity e, is increased. This is expected because as the relative permittivity e, is

increased, the wavelength in the material sample decreases accordingly and the volume
cell becomes electrically larger if the physical dimensions of the volume cell are kept
constant.

The results shown in Table 3.3 are the calculated induced electric field inside the
material sample with the change of the resonant frequency shift. From experiments the
resonant frequency will shift down about 1% to 10% after a material sample is placed
inside the rectangular cavity. The resonant frequency shift depends on the geometry and
the dielectric parameters of the material sample. In the previous computations, we

assumed the resonant frequency shift to be 5 %. In this table we change the frequency shift
from 1% to 10%. The relative permittivity of the 4-mm cubic material sample is e, = 2.5
and lossless. In this table we only show the y component of the induced electric field in
one volume cell. From this table we observe that the induced electric field inside the

material sample does not change significantly when the resonant frequency shift is

changed from 1% to I 0%. Therefore, it is reasonable to assume the resonant frequency

92

Table 3.3 Induced electric field and the ratio vs. the resonant frequency shift. The
relative permittivity of the 4-mm cubic material sample is assumed to be e, = 2.5

and the initial electric field is E 'y = 321.5729 . The geometry of the rectangular cavity
is shown in Figure 3.7.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“is?” E, 1W
1% 197.1854 0.613
2% 201.3336 0.626
3% 202.8700 0.631
4% 203.0814 0.632
5% 203 .9074 0.634
6% 204.0814 0.635
7% 204.3603 0.636
8% 204.5175 0.636
9% 204.6399 0.636
10% 204.7275 0.637 J

 

93

shift to be 5% in the numerical calculation.
In order to check the stability of the numerical results, we change the dimensions
of the material sample and compute the induced electric field in the material samples.

First, we consider a 5-mm cubic material sample with the dimensions of x=0.005m,

y=0.005m, z=0.005m and nd = 2 . m d = 2 and Id = 2 . The relative permittivity of the

material sample is assumed to be e, = 2.5, and the resonant frequency shift to be 5%.
The computed results are E; = 321.3347 and E), = 204.1318. They are almost constant

in each volume cell. The ratio of E y/ E; is 0.635 which is nearly identical to the case of
4-mm cubic material sample. The x and 2 components E, and E2 are 5.557 and 5.573
which are very small compared with E y. If we consider a 6-mm cubic material sample
with the dimensions of x=0.006m, y=0.006m, z=0.006m and nd = 3, md = 3 and
l d = 3 , the numerical results are shown in Figures 3.9.

Since the material sample is placed in the center of the rectangular cavity and the

initial TE ,0 1 mode is symmetrical with respect to the center of the rectangular cavity, we

expect that the induced electric field will also be symmetric with respect to the center of

the rectangular cavity. Therefore, we only show the electric fields at 2:21 and z=z2 in
Figures 3.9. The values of the initial electric field at each volume cell are 320.3034 at (x 1,
y], 2;), 321.5269 at (x2, y], 2,), 320.7714 at (x,, y1,zz), and 321.9967 at (x2,y1, 22).

Figure 3.9a shows the ratios of the y component of the induced electric field to that
of the initial electric field at different volume cells inside the material sample. Although

the initial electric field is not a function of y, the induced electric field inside the material

94

sample changes as y is varied. Also the induced electric field becomes less uniform
compared with the case of the 4-mm cubic sample because of the increase in the sample
dimensions. Figure 3% and Figure 3.9c show the x and 2 components of the induced
electric field inside the material sample and they are very small compared with the y
component of the induced electric field and can be ignored.

For a 2-cm cubic material sample with the dimensions of material sample as

x=0.02m, y=0.02m, z=0.02m, if we set nd = 10, md = 10 and 1d = 10,0that is, the

volume cell dimensions remain 0.002m at each side, then there will be 1,000 volume cells
for such a material sample and the dimensions of the matrix (3.20) will be 3,000. Due to
the limitation of our present computer resources, we were not able to solve this problem

over half a month of computing time. We were then forced to divide the material sample

with ”d = 6, ma, = 6 and Id = 6. The relative permittivity of the material sample is

chosen as e, = 2.5 and it is lossless. Observing the numerical results, we find that the y

component still dominates the x and 2 components of the induced electric field in the
material sample. Due to the symmetry only a half of the ratios of the y component of the
induced electric field to that of the initial electric field as a function of x for different
locations of y and z are plotted in Figure 3.10. Observing Figure 3.10, the computed y
component of the induced electric field does not change significantly with respect to x and
2 but changes somewhat more with respect to y. The ratios have been reduced from 0.634
for the 4-mm cubic sample to around 0.32~0.36 for the 2-cm cubic sample due to a larger
dimensions of the material sample. There is a possibility that these reduced ratios may be

due to the numerical errors since larger volume cells were used in the calculation.

95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E/Eyi at z=z1
y3 - — — — n0.632 0.610 0.632
Y2 _____ 0.677 0.655 0.677
Y1 _ _ _ _ _0.632 0.610 0.632
1 l I
l I I
I I I
1 L 5 _>
x1 X2 x3 x
I y 15)/E," at z=z2
y3 _ — _ —— 40.615 0.585 0.615
y2 _____ 0.661 0.622 0.661
Y1 _____ 0.615 0.585 0.615
I I I
I I I
l I I ’
X1 X2 x3 x
Figure 3.9a The ratios of E/Eyi at different volume cells in the 6-mm cubic

material sample, where the relative permittivity of the material sample is assumed to
be e, = 2.5 . The geometry of the rectangular cavity is shown in Figure 3.7. The

96

resonant frequency shift is assumed to be 5 %.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ex at 2:21
y3 .. _ _. _ . 5.791 1110‘8 5.791
Y2 ...... . 1*10’9 8*10‘13 1*10‘9
Y1 _____ 5.791 1"‘10'8 5.791
I l l
l l I
4 t i ’
x1 X2 x3 X
A y Exat 2:22
y; _ _ _ _ . 0.269 3110'9 0.269
y, _____ 2*10‘9 8*10‘13 2110'9
y, _ _ _ _ 3 0.269 3*10‘9 0.269
1 i i
I I I
, 'g t w
X1 X2 X3 X

Figure 3.9b The x component of the induced electric field at different volume cells
of the 6-mm cubic material sample, where the relative permittivity of the material

sample is assumed to be e, = 2.5 . The geometry of the rectangular cavity is shown in
Figure 3.7. The resonant frequency shift is assumed to be 5%.

97

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T y Ez at z=zz
_____ .6*10‘9 3*10‘9 6*10‘9
Y3
y, ...... 1*10'1211‘10'12 1*10-12
y, _____ 6*10‘9 3*10'9 6*10'9
I I I
| I |
l 1 1 ’
x1 x2 X3 x
0 y E: at z=22
5.989 0.120 5.989
Y3 ‘‘‘‘
yz ————— 1"‘10'9 2*10'9 1"‘10'9
y] _ _ _ _ 1 5.989 0.120 5.989
| I |
l l I >
x1 x2 x3 x

Figure 3.9c The z component of the induced electric field at different volume cells
of the 6-mm cubic material sample, where the relative permittivity of the material

sample is assumed to be 8, = 2.5 . The geometry of the rectangular cavity is shown in
Figure 3.7. The resonant frequency shift is assumed to be 5%.

98

 

—e—e—e— for y=y2
—t|t—art——¥ for y =y 3
0.38 r , r , . ,

 

 

 

 

 

 

0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046

 

0.38 r r r r r r

 

Ey’ yi

 

 

 

 

 

0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046

 

0.38 1 I I I I T

 

 

 

 

 

 

0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046

Figure 3.10 Ratios of [fl/E),i varies in the x-direction. Each curve represents this
ratio as a function of x for different locations of y and z. The relative permittivity of
the 2-cm cubic material sample is e, = 2.5 . The geometry of the rectangular cavity

is shown in Figure 3.7. The resonant frequency shift is assumed to be 8%.

99

2. Thin square plate case

The material sample with a shape of a thin square plate, having its height much
smaller than its width, is placed in the center of the cavity. The dimensions of the

rectangular cavity are shown in Figure 3.7 and the dimensions of the material sample are:

x=0.02m, y=0.002m, z=0.02m with nd = 10, md = l and Id = 10. The relative

permittivity of the material sample is assumed to be e, = 2.5. Since the initial mode

TE ,0 1 contains only the y component of the electric field and the material sample has a

thin flat geometry, the induced electric field inside the material sample can be estimated by
the boundary condition of E = ( 1/c,)E‘.
The numerical results are shown in Figure 3.11, where only the induced electric

fields in a half of the plate, z=zl to 2:25. are shown due to the symmetry. In Figure 3.11,

ratios of the y component of the induced electric field in the material sample to that of the
initial electric field are plotted as a function of x. Each curve in Figure 3.11 represents this

ratio as a function of x for different locations of z. The highest one is for 2:21 and the
lowest one for z=z5. We observe that the electric field is higher at the edges of plate, an
expected edge effect. The induced electric field inside the material sample is almost

constant. Theoretical estimation of this ratio based on the boundary condition of 1/8,

gives % = 0.4. Our numerical results varies between 0.315 to 0.39 which are in

agreement with this theoretical estimation.

The x and 2 components of the induced electric field in the material sample are

extremely small (1.0*10'8) in all volume cells. This is expected because the sample is very

100

 

0.4 , , ’
0.39
0,37 ,. .. . ..

0.36 ..

 

Ey’ yi

 

0.32

 

 

 

 

0.31

 

 

 

.3
0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046
X

Figure 3.11 Ratios of 15)/E; varies in the x-direction. Each curve represents this
ratio as a function of x for different locations of z. The highest one is for z=zl and the
lowest one for z=z5. The relative permittivity of the thin square plate material sample

is e, = 2.5 and the geometry of the cavity is shown in Figure 3.7. The resonant
frequency shift is assumed to be 1%.

101

thin in the y direction that no'significant x and 2 components of the electric field can be

induced. Therefore, the y component of the induced electric field in the material sample

dominates.

3. Narrow strip case.
We consider next a material sample with a geometry of a narrow strip case. The

dimensions of the material sample are: x=0.002m, y=0.02m, z=0.002m with nd = 1,
m d = 10 and Id = 1 . The relative permittivity of the material sample is assumed to be

e, = 2.5 . Theoretical estimation of the induced electric field in the material sample may

be close to the initial electric field because the initial electric field is tangential to the
major part of the material sample surface, and the continuity of the tangential component
of the electric field at the material sample surface requires this estimation. Based on this
estimation, we expect that more terms will be needed in the computation of the induced
electric field because we have evaluated a delta function out the integration sign when we
derived EFIE (2.102). If we need the induced electric field to be equal to the initial electric
field, then there needs to be another delta function coming out of the integration sign to
cancel the previous delta function. Thus, the convergence rate may be slov. er in this case.
In this computation we assume the resonant frequency shift to be 1%. The
numerical results are shown in Figure 3.12. In this figure, the maximum value of the ratio
of the induced electric field to the initial electric field is 0.82. If we change the upper limit
in the double summation, the numerical results are shown in Figure 3.13. In Figure 3.13
we observe that the ratio becomes closer to 1 as we increase the upper limit of the double
summation; when N=1000, the maximum ratio becomes 0.896; and when N=1500, the

maximum ratio becomes 0.903. Increasing N leads to an increase in the computing time.

102

 

09_,. ........................ . .......... . ........... .... ........ —4
' . a » o

 

 

 

 

 

 

 

\ ' i E .
”I I TE E Z
1' E 3)’
0.6” Ey ...... ..Y... .. .. —
o_5,_ . i .. .......... ..
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024
V

Figure 3.12 Ratios of fir/E},i varies as a function of y. The dimensions of the

material sample are: x=0.002m, y=0.02m, z=0.002m and the geometry of the cavity is
shown in Figure 3.7. The resonant frequency shift is assumed to be 1%. The upper
limit in the double summation is chosen to be N=200.

103

 

o.9s--~--

 

 

 

 

 

 

 

 

...a
X
In
E‘T . ' : t : '
y A ; . —e—;e— forN=200 . 4
0.6.... ....... :L ..... .......... 2 ........ .1
0.55-... . . . _ ..N: 912129111191! .mthegdoublc.summamn ...... _
0.5 1 1 1 i i 1 1 1
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024
V

Figure 3.13 Ratios of Ey/Eyi varies as a function of y for different N. The

dimensions of the material sample are: x=0.002m, y=0.02m, z=0.002m and the
geometry of the cavity is shown in Figure 3.7. The resonant frequency shift is
assumed to be 1%.

104

For such cases, we suggest the scheme of separating the material sample into interior

volume and boundary layer cells as will be discussed later.

3.4 Some Methods to Increase the Convergence Rate

1. Separation of the material sample into the boundary layer and interior regions

In Section 2.1.6 we showed that the vector wave functions 07mm and Nomi are
complete within the space of the solenoidal vector fields. We also explained that the
divergence of the electric field doesn’t vanish at all points in the cavity after placing a
material sample inside. In fact, the divergence of the electric field doesn’t vanish only at
the boundary of a homogeneous material sample of finite size where the induced electric

charges reside. The divergence of the electric field still vanishes in the interior of a

homogeneous material sample.

Based on this observation, in the determination of the induced electric field in the
material sample, we may divide the material sample into two groups of volume cells:

boundary layer and interior volume cells. For the boundary layer volume cells we use the

A —A A
vector wave functions anI , Mum! and Nnml as the complete set of basis functions to

expand the unknown induced electric field. For the interior volume cells only the vector

_A A
wave functions M nml and N nml are used to expand the solenoidal electric field. Thus, the

Electric Field Integral Equation (EFIE) will be quite different for these two groups of
volume cells. The EFIE at the boundary layer is the same as that used before, eq. (2.102).

The EFIE for the interior volume cells is obtained as follows.

In Chapter 2, the scattered electric field is shown to satisfy the Helmholtz eq.

105

(2.90). As we use only the vector wave functions M um! and N nml to expand this scattered

electric field in the interior region of the material sample, the scattered field can be

expressed as

17350) = 2 [anfinanbnfinon (3.53)

n=l

Substituting eq. (3.53) into eq. (2.90) we have

V x V x 2[an1I-'In(;) + 6,117.0” — k32[an1l7n(?)+ 6,1174%]: —jtouo.7.q(?) (354)

Based on the properties of the vector wave functions M nml and Nnml , eq. (3.54) can be

written as
2104/92. - kSWnG) + b,,(k,2, — k§)Nn(?)] = —jtoqu.q(?) (3.55)

Taking the scalar product of eq. (3.55) with 117nm! and Nnml, respectively and integrating

over the cavity volume V, then applying the orthonormal property of the vector wave

_\ _8
functions M nml and N nml . we obtain the expressions for the expansion coefficients as

_jmuo
2
kfi-kov

 

I jeq(;0) ' 07n(;0)dv0 (3.56)

sample

_jwllo
2

 

J. ;eq(;0) ' 1.04%)de (3.57)

sample

where the integration region is over the sample volume. Therefore, the expression for the

scattered field 1.535(2) becomes

106

 

 

., . . . . Hatto)fin<?)+fin(?o)fint?)
E (r) = 0qu I Jeq(r0)'2[ , 2 at»,
14...... " kn ‘ k0 (3.58)
= -j0)110 J Jeq(;0) ' 650:0, 1")de
Vsample
where the dyadic Green’s function in the interior region is identified as
._ , . 17 ‘ t7 " +117 ‘ 1‘0 ‘
o..(ro, r) = 2[ "(’0’ "('2) 2““) "”1 (3.59)
n kn-k0

Finally, the Electric Field Integral Equation (EFIE) for the induced electric field in the

interior region can be derived as
£0) + 1'qu I 1.001506) - €2.00. no, = E (P) (3.60)

Comparing with 58000, I") in the EFIE (2.102) for the boundary layer region, 5,300, P)
can converge faster. Some numerical results are given to show this point.

In the following computations, we only show the convergence property of the
integration of G “A x90, 2) component with respect to I" and 70 in regions V" and VnO'

In Chapter 2, eqs. (2.24) and (2.32) give the expressions of the vector wave

_8 _8
functions M um! and N nml . from which the coefficient of G e ”UGO, 7') can be obtained as

(”)2 (“‘12
_ + ...
EOnEOMEOI b C

abc [(30921 _ kg)

 

(3.61)

X X . .
Thus, G es ”(rm r) can be expressed 1n a double summatron format as

107

m_7c 2 In 2
+ ‘3 g (x x0) g 1 (x x )
mll ’ "12 ’ 0
Ge (r0, r): - [ . — . ]
sxx mérzgtb—cb k3 kgm1181n(kgmlla) kgm125m(kgmlza) (3.62)

sin(’n—1t )sin(m—1t )sincit )sinGJ—t )
by by0 eZ cZO

gm“ is given by eq. (3.30) and

seal-((911%))

gm,p(x, x0) = cos(kgm,p(a-xb))cos(kgm,px3) (3.64)

where k

 

p=1,2 in eq. (3.64). From eqs. (3.63) and (3.30), we find that 1‘ng2 is always imaginary

and kgm” will be approximately close to kgle as m and I increase and this is the reason

why G e S M(20, 2) has a better convergence property.
The integration of 065,370,?) with respect to I" and 70 in regions V" and V"0 is

x+Axy+Ayz+Az t0+Axyn+A)zo+Az

I I I J J I Gesxx(r0,r)dudvdwdu0dv0dwo

x y z x" \ 0 z0
mtt

(may (171:)2
2[cosmfly + Ay) - cos —b—y][cos'l1—(y0 + Ay)

( "£211: (1:) b b (3'65)

_cosm—y0][cos—(z + Az) - coslltz][cos£Z—t(zo + Az) — cos [£20]

 

1°°°°4
=k—222b—
"mt/10::

x+IAxxo+Ax
xJ'OIL gmll(u’ “0) gm12(u’u0)

dudu
kgmllSin(kgmlla) —kgm128in(kgm120)] 0

The details of the integration of

108

x+Axx +Ax

I 0.1 [ gmll(u’u0) 8m12(“’“0)
k

x x0

' 7' - dudu 3.66)
gmltstn(kgm,1a) kgleSln(kgmlza):| o (

for different cases can be carried out in a similar way as that in Section 3.2.
In the following computations, the dimensions of the. rectangular cavity are -

assumed to be the same as that shown in Figure 3.7. We also assume the initial mode to be

T1310]. We choose Ax = Ay = A2 = 0.002m in the following computations. The

integrations of G “”00, 2) are calculated at different positions of 2 and 20 (we will use

the same source and observation points as those in Section 3.2 in order to compare the

results of both integrations) with the assumption of the resonant frequency shift to be 5 %.

Figure 3.14 is for 2 = 20 while Figure 3.15 and Figure 3.16 for 2¢ 20, where the

horizontal axes are the value of N and the vertical axes are the integration of G ““00, 2)
in these three figures. In Figure 3.14 to Figure 3.16, we find that the integrations converge
much faster than those in Figure 3.4 to Figure 3.6 or Figure 3.1 to Figure 3.3. These
integrations converge when N 2 80 for the different positions. This is because as m and I
increase, kgmll becomes approximately equal to kgle' Then the difference in the eq. (3.65)

becomes nearly equal to zero. Thus, the summation terms are greatly reduced leading to

the reduced computation time.

The scheme of the separation of the material sample into boundary layer and
interior region has been successfully applied to the narrow strip case in Section 3.3. In this

case we choose the dimensions of the sample as x=0.003m, y=0.021m, z=0.003m with

nd = l, md = 10 and Id = 1 as the division for the interior region. The dimensions of

109

 

 

 

 

 

4 T I I I T I I W I
u.
(D
0
'5
S 2 — ~
'5
e
03
9.
E

o I l L 1 I l l 1 l
0 20 4o 60 30 100 120 140 160 180 200

Upper limit of summation N

Figure 3.14 Integration of G e 3 x Jr(20, 2) vs. number of summation modes when

t = 7,, t: [0.033m,0.014m,0.0551m] and Ax = Ay = Az = 0.002m.

110

 

 

 

 

 

2.5 I I I I I I I I I
2 -' ..
II.
o 1.5 " "
O
'5
§
1!
.3.“
s 1 - .
0.5 - ..
0 1 1 1 1 1 1 1 1 1
0 20 40 60 80 100 120 140 160 180 200

Upper limit of summation N

Figure 3.15 Integration of G e sxx( 20, 2) vs. number of summation modes when

t: [0.035m, 0.014m, 0.0551m] , to: [0.033m, 0.014m, 0.0551m] and
Ax = Ay = A2 = 0.002m.

lll

11 r r F

 

10'-

 

Integration of DGF

 

 

 

2 I l l I L l l l l
O 20 40 60 80 100 120 140 160 t 80 200
Upper limit of summation N

 

Figure 3.16 Integration of Gesxx(20, 2) vs. number of summation modes when

i: [0.035m, 0.016m, 0.0553m], 20: [0.033m, 0.014m, 0.0551m] and
Ax : A)! = AZ = 0.002m.

112

the volume cells in the boundary layer region are chosen as bx=0.0008m, by=0.0008m,
bz=0.0008m. The dimensions of the interior region are xi=0.0014m, yi=0.0194m,
z,=0.0014m and the dimensions of each volume cell in the interior region are ix=0.0014m,
iy=0.00194m, iz=0.0014m. The number of total volume cells in the boundary layer region
and interior region is (nd + 2) x (md + 2) x (1d + 2) = 108. In the computation, we
suppose the resonant frequency shift to be 1% and choose the upper limits of the double
summation to be N ,- = 150 for the interior region and N b = 400 for the boundary layer
region. The numerical results are shown in Figure 3.17.

Since we are only interested in the induced electric field inside the material
sample, only the solutions for the volume cells in the interior region are presented in
Figure 3.17 while those for the boundary layer region are omitted. It is noted that the
solutions for the volume cells in the boundary layer region are usually not reasonable and
should be discarded. This is justified because the boundary layer region is artificially
created to contain the induced surface charge for a mathematical reason. In Figure 3.17 we
only show the ratio of the y component of the induce electric field in the interior region to
that of the initial mode varies in the y direction. We see that the ratio is 1.04 which is very

close to 1 that is required by the continuity of the tangential electric field at the same
surface. The x and 2 components are around 1.0 x 10‘10 which can be neglected. In this
computation we can also show that it reduces the computation time.

In Section 3.3 the numerical result is 0.903 when N = 1500 for the narrow strip
case. We mentioned that the construction of the matrix (3.20) in the moment method cost

most of the computation time. From the definition of this matrix, it consists of the series

113

 

1.1 , ,

1.05- ...... t; ..... ...... .........

 

5/ y'

035,. ...... fl. _
l- Ex i i 2

0'9 .. ......

 

 

 

 

d
_
0.85_.......... .......... ......... _
L I I L l 1 l I

 

0.8
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024

Figure 3.17 Numerical results obtained with the scheme of separating the material

sample into boundary layer region and interior region. The ratio of E/Eyi varies as a

function of y coordinate. The sample dimensions are: x=0.003m, y=0.021m, z=0.003m
and the geometry of the cavity is shown in Figure 3.7. The resonant frequency shift is
assumed to be 1%.

114

summation, and this summation will be dominant in the computation time. In this case the
number of summations is 10 x 1500 x 1500 = 22, 500, 000. If we use the scheme of

separating the material sample into boundary layer and interior region, the number of the

summations becomes 98 x N: + 10 x N ,2 = 15, 905, 000. Therefore, the computation

time is about 71% of that in Section 3.2 and furthermore we obtain a better result.

Also we observe that in the scheme of separating the material sample into the
boundary layer and interior region for the narrow strip case, the major portion of the
computation time is consumed in the boundary layer region due to the small dimensions of
the material sample. Thus, this scheme will be effective if the number of volume cells in
the interior region is greater than that in the boundary layer region, or for the cases such as
a narrow strip, where a very large number of eigenfunctions are required to produce

accurate results if the sample is not separated into the interior and boundary layer regions
and eq. (2.102) is directly solved.

2. Poisson summation
As stated before, we found that the most difficult step is to fill out the matrix

[An,]3Mx3M in eqs. (3.17) to (3.20) in the numerical computation due to the slow

convergence property of the dyadic Green’s function. Now we will apply the Possion
summation method [2],[32],[42] to accelerate the convergence property of the double

summation in the construction of the matrix (3.20). As before we will consider only the

integration of Gwn(20, 2) component for brevity.

The double summation evaluation of Gwn(20, 2) can be found in eqs. (3.33) to

(3.34) as

115

°° °° 4 ($10032 11:

1 . m
Geoxx(r0.;) = 7:5 2 glakgmsinwkgmflgmfix,x0)srn b y

 

(3.67)
sint-nit sin [32 sinlltz
b yo c c 0

where

 

kg...) = Jko— ((113)2«1'2‘33 . (3.68)
gm,(x,x0) = cos(kgm,(a-xb))cos(kgm,x5) (3.69)

Usually we suppose the initial mode to be a lower order mode. Thus, kgm, will be an

imaginary number as the indices m and I increase in the double summation (3.67) and it

can be approximated by

 

2 2
. mrt In .
kgml E IA/(T) + (?) = lkgmli (3.70)

when m and I become larger. Equation (3.69) can then be expressed by eq. (3.42). Thus,

the double summation (3.67) is rewritten as

m7: 2 In: 2
2 0° 0° (7;) +(:)—k lx— xlsi anI: mtt In:
Geon(r0, r) = —2 2 Z bck e kW ° 11,) —ysin—5-yosin—z
0m=m-l=l

 

k-

gmlr’

i

.t’"—"Y+(’i‘l
C mrr mrt In In
C Sznzbckgmtsmtak )gml‘xxols‘WyC‘WyOC‘“ Zsi" 1'0

(3.71)

 

where m,- and l,- are chosen in such a way that k ml is an imaginary number and can be

approximated by eq. (3.70). In the summation (3.71), the first term is what we need to

116

consider and the second term is a finite summation which doesn’t affect the convergence

property. The first summation is denoted as

run 2 171: 2
2 oo oo ‘3' '1' F _k _|x_x| m1" m“ .
Geoxx1(;0’ ;) = 2 2 e "I“ 0 SID—ySIIl—yo

bckgm“ b b (3.72)

 

Since k gm ,1. can be approximated by (3.70), eq. (3.72) can then be expressed as

 

2 °° °° mtt 2 In: 2 ‘ I?)2*(%t)2"“°'
= — 1H +(-) I
' r0 r betgmgmgi b C e (3.73)

sinmnysinmny sinmzsinmz
b b 0 c c 0

The expression (3.73) can converge faster if x as xO , i.e. off-plane case. However, it will

converge very slow when x = x0 , i.e. on plane case.

The Poisson summation formula [2],[32],[42] can sometimes be used to convert a
slowly converging series into a rapidly converging one by allowing the series to be
summed over in the Fourier transform domain, that is, if a series spreads out, then its
Fourier transform must be concentrated around the origin.

The Poisson summation formula is given by [2],[32],[42] as

Ms

°° _1
2 “MD-a

nz-oo m

F (2%!) (3.74)

—oo

where F is the Fourier transform off. In [32] the Fourier transform of

117

-1.,/x2 + 13,

 

 

f1(x) == e——-——cos(xA0) (3.75)
2 2
x +km
is given as
K0{kaL2 + (65— 4,92} + K0{km,/L2 + (to + 14,02} (3.76)

where K0 is the modified Bessel function of the second kind and it decays rapidly
(KO(0) = 00, K0(0.5) = 0.9244, K0(1) = 0.4210 and K0(1) = 0.1139). Then using

the Poisson summation formula, we have

1: b ..
2 f0?) = E 2 F(2mb) (3.77)

 

2 e cos ($.40)

_,,, rm 2 2
" (‘3) +1Cm (3.78)

=f—tm 2 ”[K0{k lam/L +(2mb— A0) }+KO{ rem/2L +(2mb+A0) }]

 

 

After some algebraic manipulation we have [32]

118

2 5"“ (are)
(are)

.-. 231.: 2 [KO{%J(x—x0)2+(2mb—y+y0)2}

m=—oo

 

 

sint-r-l— 'sin—’
b y b )0

 

(3.79)

 

+ K0{%J(x—xo)2 + (2mb + y —y0)2}

 

_KO{-l§~/(x — x0)2 + (2mb — y — y0)2}

 

_KO{%J(X - x0)2 + (2mb + y + y0)2}:|

Due to the rapid decay of the modified Bessel function of the second kind, the only

significant term in eq. (3.79) is that for m=0. Hence eq. (3.79) can be approximated as

 

m1:2 [1:2
“1““ (7) *1?)
e . m1: . m1:
sm—b—ysm—vo

2 , , .
JV?) *1?) b (3.80)

E %[K0{l‘§al(x " 1’0)2 + (y "' )’0)2}-K0{1n.\/(X " x0)2 + (y + Yo)2}:|

 

 

 

C

In order to employ eq. (3.80) to accelerate the series convergence rate of the dyadic

Green’s function, we rewrite the eq. (3.71) as

ll9

 

J(m1t)2 (an
on on — + ‘—
8 X 2 b C kgmli|x_ 'irlll nm

 

 

 

 

 

_7_1_: mrt
' m7: 2 In 2
I 1 m‘ 1" 4 79— + ? -k Ix-xl
sin-Enzsin?zo——2 'b—c 2 e M“ 0 (3.81)
k0m=ll=1

(mnjz (my

___. + —

b c (x, x ) sin— m“ sinm sinl—nzsinl—T—tz
kgm,sin(akgm,)gm' 0 b y b yo 0

where the first summation can be accelerated by eq. (3.80) and the second term is a finite

summation.

The first summation is denoted as

 

”JV—it): (“trek
+ _
C k1,-»1l1'|'-r “‘0'

mn mu: 17: In
Sll'l —b-y Sll'l Tyo sm —Z S") “—3.0

 

Geoxx2(r0’r) = [7222 Z

k—Om - ll=l (3.82)

In the Galerkin’s method we need to integrate the dyadic Green’s function with

respect to variables ; and r0. For the Geonzfio, I"), whether x = x0 or x¢xo will

determine the convergence rate. Let’s consider the integration of Geonzfio, I“) with

respect to x and x0 for brevity. The integration of Geonzfio, F) when x = x0 is given by

120

x + Axxo + Ax
L 3
I I Geonzvo, r)du0du

x x0

 

 

Q"
h

 

 

...; 311’?)*1?)113—“1(2)1131312“)???

sinmnysinmny sinmzsinmz
b b 0 c c O

 

4” e —l.m1t.m7t.l1t.11t
= — Ax+ sm—ysm—yosm—zsm—zo
bckzmglzll m1: 2 l1: 2 b b c c

0 _ + _
t b c _

and when x at x0 is given by

 

 

j j Geomoo, ?)du0du

1: x0

 

 

 

4—32 28 2
1(7) 1%)

mn . In . In
———-yos1n—zsm—zo
C C

.mn: .
sm—ysm b

b

The first summation in eq. (3.83) is just a delta function 50-150) divided by kg. The

second term in eq. (3.83) and eq. (3.84) can be summed by eq. (3.80). Therefore, the

approximation of eqs. (3.83) and (3.84) by eq. (3.80) can be expressed as

121

x+ AxxO + M
3 I
I I waxzho, r)du0du

.1: x0

=%x5(r—r0)+—‘122[KO{IRJMZ +(y— y0)2}-K0{1§(/Ax2+(y+y0)2} (3-35)

0 "Ck01=1

lit 17: . l7: . l7:
+K0{-C-|y—y0I}-Ko{:(y+yo)}]Sln;ZSIIl-c‘zo

for x = x0 and

 

 

x ... Axxo + Ax
8 k
I J Geonzho, r)du0du

x xo

——24Z|:KO{I—nfx— x0)2 +(y— yo) 2}-K O{l-:‘—tA/(x—xo)2+(y+y0)2}

="Ck01— 1

 

 

(3.86)

 

_K0{l%t./(|x —x0| — Ax)2 + (y -y0)2}/2

 

+ KO{£§~/(|x -x0| — Ax)2 + (y + y0)2}/2]Sin{-CT-czsinl:nzo

for x :2 x0. Comparing eqs. (3.85) and (3.86) with eqs. (3.83) and (3.84), we find that the
double summations in eqs. (3.83) and (3.84) are approximated by the single summations
in eqs. (3.85) and (3.86) when Poisson summation formula (3.77) is used.

In the numerical computation, we set m,- and I, to be 5 in eq. (3.81) and the
summation index I to be from I to 25 in eqs. (3.85) and (3.86). Then we apply eqs. (3.85)
and (3.86) to our numerical examples as discussed in Section 3.3.

The material sample of a 4-mm cubic shape is placed in the center of the

rectangular cavity. The relative permittivity of the material sample is assumed to be

122

8, = 2.5. The material sample is divided as nd = 2, md = 2 and Id = 2. The

numerical results using Poisson summation formula are: 5.09 (5.4949) for the x
component, 193.65 (203. 9074) for the y component and 4.91 (5.552) for the 2 component
of the induced electric field in each volume cell of the material sample, where the values in
the parentheses indicate the numerical results of the double summation for the 4-mm cubic
shape material sample which we obtained before. The ratio of the y component of the
induced electric field to that of the initial electric field is 0.60 while it is 0.634 if the double
summation is applied. Thus, we can conclude that these approximation results are
satisfactory.

If we increase the dimensions of the cubic material sample to have 0.006m in each

side and divide the material sample as n d = 3 , m d = 3 and Id = 3 while keeping the

other parameters the same as previous cases, we found that the computed electric field is
still dominated by the y component and all the components of the induced electric field are
symmetric with respect to the center of the rectangular cavity. The y component of the
induced electric field does not change significantly with respect to the variables x and 2 but
change somewhat more with respect to the variable y. The values of the y component of

the electric field are 194.88 for yzyl, 220.73 for y=y2 and 194.88 for y=y3. The ratios of
the y component of the electric field to that of the initial electric field are 0.61 for y=y1 and
0.69 for y=y2 while the ratios are 0.632 for y=y, and 0.677 for y=y2 if the double

summation is applied to this 6-mm cubic material sample. Comparing these numerical
results we observe that the numerical results with the Poisson summation are very close to
the theoretical approximation and the previous numerical results using double summation.

In the Poisson summation scheme, for each component of the dyadic Green’s

123

function, the summation in each volume cell is only 2 x 5 x 5 + 25: 75 based on eqs.
(3.81), (3.85) and (3.86) while in the double summation there are over 40,000 summation

terms. It is noted that even though we need a numerical integration of the modified Bessel
function with respect to the variables y and yo in this scheme, the overall computation

time is still saved about 60% due to the decrease in the summation terms.

124

CHAPTER 4

QUANTIFICATION OF THE INDUCED ELEC-
TRIC FIELD IN A MATERIAL SAMPLE
PLACED WITHIN A CYLINDRICAL CAVITY

In the microwave heating of material samples, a cylindrical microwave cavity is
more commonly used than a rectangular microwave cavity. Theoretical analysis of the
induced electric field inside a material sample placed within an energized cylindrical
cavity is more involved than that of a rectangular cavity case as studied in Chapter 2 and
Chapter 3.

In this chapter, we will quantify the induced electric fields inside material samples
of various geometries and dielectric parameters which are placed within an energized
cylindrical cavity. The theoretical method used in Chapter 2 can be employed to obtain the

induced electric field inside the material sample placed within a cylindrical cavity. That is,

A _X _l
we will use the vector wave functions anI, M um! and Nnml which will be defined in the

Cylindrical cavity as a complete set of basis functions to determine the induced electric

field inside the material sample.

A ..A
In the cylindrical cavity, the definitions for the vector wave functions anI, M nml

125

 

and Nnml are the same as that in the rectangular cavity as given in eqs. (2.1), (2.2) and
(2.3). These vector wave functions also satisfy the same boundary conditions (2.4), (2.5)

and (2.6) at the perfectly conducting walls of the cavity. We will show that the vector wave

functions an1, gum and finml in a cylindrical cavity are orthogonal and form a
complete set of basis functions and satisfy the vector Helmholtz equations based on the ~
proofs given in Section 2.1.3, Section 2.1.4 and Section 2.1.6 in Chapter 2. However, they
have completely different expressions in a cylindrical cavity.

The outline of this chapter is as follows: In Section 4.1 we will derive the

A __B A
expressions for the vector wave functions anI, M nml and Nnml in the cylindrical cavity.

In Section 4.2 the normalization of these three vector wave functions will be carried out

and some field structures of the vector wave functions an1, Mnml and Nnml will be
plotted. In Section 4.3 the expression for the dyadic Green’s function in the cylindrical

cavity will be obtained. Some numerical examples will be presented in Section 4.4.

4.1 Expressions for Vector Wave Functions an1 , M "m1 and N n ml in

Cylindrical Cavities

The cylindrical cavity under consideration has the geometry shown in Figure 4.1.

126

 

 

 

 

 

 

Figure 4.1 A cylindrical cavity and the designation of the coordinate system

127

4.1.1 Expression for Vector Wave Function an1

Based on the definition and the boundary conditions of the vector wave function

ant given in eqs. (2.1) and (2.4), we have

" l L
anl = k—-(V¢nm1) (4.1)
nml
fiXanl = (4.2)

. . L . .
where the scalar e1genfunction om, satisfies the scalar Helmholtz equation of

(V2+k§m,)¢fm, = o (4.3)

 

2 1 a a 1 a)2 32 . . . . .
and V = ~5— r-a— + 2 2 + 2 1n the cyllndncal coord1nate system. Applymg the
r r r r 31p 32

variables separation method to eq. (4.3), we obtain the solution of the scalar eigenfunction

L
¢nml as

L cos(ncp) COSU‘zZ)
= 4.4
(DWI A"MIJ"(k'r){ sin(n(p) H sin(kzz)} ( )

where J ”(krr) denotes the first kind of the nth order Bessel function, n is an integer and

A n m l is an unknown constant which will be determined by the normalization of the vector

A
Wave function anI. The eigenvalue km, is then expressed as

128

krztml = kr2 + kz2 (4-5)

If n=0, the solution of the scalar eigenfunction (1)8”, 1 is given by

cos(kzz)} (4 6)

L
= A J k

Using the definition (4.1), applying the boundary conditions (4.2) and after some

manipulations, we obtain the expressions for the three components of the vector wave

A
function anI as

 

 

 

 

 

A
anh = —""’lp"mJn'(p"mr) cos(ntp) sin(litz) (4.7)
knml a a 810019) C
A _ .
ankp .-. "'"”lJ,,(&’—'2r) “Mm” sing-152) (4.8)
knmlr a 0050“” C
A
anlz = —"mlE-tln(p"mr) C?S(n(p) cos(ll-tz) (4.9)
knml C a Sln(n(p) C
while the eigenvalue is determined by
2 2
kimz = (pm) +(E) (4.10)
a c

In eq. (4.7), J "'(krr) denotes the derivative of the bessel function J "(krr) with respect to

krr and PM denotes the mth root of the first kind of the nth order Bessel function, that is,

L

J n( pm") = 0. The expression for the scalar eigenfunction (pm, i

s then given by

129

= we)$522314)

A
When n=0, the expression of the vector wave function LOml is given by

AOmIPOm .pOm . 1E

LOmlr — ZO—m-I'TJO (TF)SID(?Z) (4.12)
AOmllTE pOm In:

L0mlz — ®?10(TF)COS(?Z) (4.13)

and Lomup = 0.

4.1.2 Expression for Vector Wave Function 117,, m1

Based on the definition and the boundary conditions of the vector wave function

117nm! given in eqs.(2.2) and (2.5), we have

M...) = Vx(2¢,’§’m,) (4.14)
fiXA—Jnml = O (4.15)

. . M . .
where the scalar eigenfunction om", satisfies the scalar Helmholtz equation of

2 2 M
(V +qnm,)¢nm, = 0 (4.16)
Applying the variables separation method to eq. (4.16), we obtain the solution of

. . M
the scalar eigenfunction om, as

130

M cos(mp) cos(kzz)
= k .
¢nml BnmlJn( rr){ sin(n(p) }{ 510(kzZ)} (4 17)

and the eigenvalue gum, is expressed as

2 2 2
4m; = k, +kz (4.18)

where Bnm, is an unknown constant which can be determined by the normalization of the

_A
vector wave function M nml .

If n=0, the solution of the scalar function 634,", is given by

(4.19)

cos(k 2)
(1,321 = BOmlJO(krr){ Z }

sin (kzz)

Hence, using the definition (4.14), applying the boundary conditions (4.15) and after some

manipulations, we obtain the expressions for the two components of the vector wave

. _A
function M nml as

 

 

 

B . _ .
Mnmlr = nmln1n(pnm I“) sm(n<P) 8111(1—1-tZ) (420)
r a cos(ntp) C
Mm, = —B ,p"'"1'(pi"r) Cf’sm‘p) 3111612) (4.21)
‘p "m a n a sm(ntp) C

while the eigenvalue is determined by

1 2 2
qimz = (p"”') + (’35) (4.22)
a C

 

131

where an' denotes the mth root of the derivative of the nth order Bessel function of the

first kind, that is, Jn'(pnm') = O. The expression for the scalar eigenfunction $21,", is then

given by

(1):", = Bnml}n(l)"_m.r){cos(n¢)}Sin(l7_tz) (4.23)

a sin(n(p) C

.A
When n=0, the expression for the vector wave function M cm: is given by

P ' ,P ' . lit
MOMp = wow—2110 (—:’1r)sm(:z) (4.24)

and MOMII' = 0.

4.1.3 Expression for Vector Wave Function finml

Based on the definition and the boundary conditions of the vector wave function

finml given in eqs. (2.3) and (2.6), we have

1

nml

N

finml = VxVx(2¢nm,) (4.25)

 

fl X finml = O (4.26)

. . N . .
where the scalar eigenfunction (pm, satisfies the scalar Helmholtz equation of

(V2+kfim,)¢fl’m, = 0 (4.27)

Applying the variables separation method to eq. (4.27), we obtain the solution of

132

. . N
the scalar eigenfunction om, as

N _ cos(ntp) 005(kzz)
¢nml - Cnml‘ln(krr){ sin(n(p) }{ sin(kzz)} (4.28)

and the eigenvalue is determined by

2

km, = krz+kz2 (4.29)

where Cum, is an unknown constant which can be determined by the normalization of the

A
vector wave function N nml .

If n=0, the solution of the scalar eigenfunction (1)31", is given by

(4.30)

N cos(kzz)
¢Oml = C0m1J0(krr){ }

sin(k:z)
Using eqs. (4.25) and (4.28) and applying the boundary conditions (4.26), we obtain the

A
expressions for the three components of the vector wave function N nml as

 

 

 

 

 

 

C
Nnmlr = “—nmlpfllfll '(pnmr){cf)8(mp)}sin(l—1Ez) (4.31)
knml a C n a sm(ntp) C
C — .
N 1 = - "mlglit1(p"mr) SIMMP) sinGT—rz) (4.32)
nm (9 knmlr C n a cos(n(p) C
C 2
Nnmlz = _ nml(pnm)1n(pnmr) CoS(n(P) COS('ILIZ) (433)
knml a a Sin(n(p) c

while the eigenvalue is determined by

133

 

pm 2 I 2
k3,", = ( a ) +(l‘) (4.34)

C

. . . N . .
The expreSSion for the scalar eigenfunction (pm, is then given by

¢,,N,,,, = Cnm,Jn(p—;'3 r){ ::::::))}cos(l§z) (4.35)

...8
When n=0, the expression of the vector wave function N Oml is given by

C
NO ,, = mfli’fllitl '(pflrJSmC—“zj (4.36)
'" kom, a c o a c

Com! Pom 2 Pom lit
m

and Nomnp = O.

‘ A A A
In the expressions of the vector wave functions anI, M nml and N nml , we did not
specify their dependences on the variable (p . The variations of these functions with respect

to the variable (p can be determined by the location of the excitation probe and the (p
dependences of the r and 2 components of the electric field. All the vector wave functions

should have the same dependence on the variable (p because they are excited by the same

source.

If the excitation probe is located at (p = O and the r and 2 components are even

A
functions of the variable (p, then the expressions for the vector wave functions anI,

finml and Nnml can be given by

134

 

 

 

A Anml .pnm .pnm . [1!
anl - k [r—Jn (Trjcos(ntp)sm(-c-z)

 

 

 

 

 

 

nml a

(4.38)

.n an . . In In In

-<p;Jn(-a—r)sm(ntp)sm(—c—z)+zC —J "(pm a r)cos(mp)cos(cz)]
Bum! = Emir; J "(fl—r)cos(ntp)sin(L—:z) (4.39)
4‘.)sz J ‘(p'c'lm r) sin(n(p)sin(l§z)]

A Cnml anln pnm . I?! .ant prim

Nnml — ...]:n—m—I-[r a 17],,“ r)cos(ntp)sm(—C-z)—(p;?1n( a r) (4.40)

 

 

. . [TE ,, prim pnm l1:
sm(ntp)sm(?.)+z( a )21n( a r)cos(ntp)cos(?z)]

It is noted that these eqs. (4.38) to (4.40) are valid only for n ¢ 0 case. For n = 0 case,

we have already derived the expressions in eqs. (4.12), (4.13), (4.24), (4.36) and (4.37).

4.2 Normalization of Vector Wave Functions an1, M n m1 and N nml in

Cylindrical Cavities

4.2.1 Normalization of Vector Wave Function in m1

Based on eq. (2.59) the normalization of the vector wave function anI is given by

KKK-[((anl anljrdrdtpdz— - 1 (4.41)

That is,

135

J2]? J; (anl - anzjrdrdtpdz= J2]? J; (1):," 1(1):," ,rdrdcpdz

 

 

2 (4.42)
1‘ 2 P . 11:
= EL: LAnml(Jn( Zmr)cos(n(p)sm(-;z)) rdrdtpdz = l
where we have used the following relation
-‘ -‘ 1 L L
an1 ° anl = 2—V¢nml ' V¢nml
nml
1 L L 1 L 2 L
= TV ° (¢nmlv¢nml) _ 2—(1)nmlV ¢nml (4-43)
nml nml
1 L L L L
= TV ' (¢nmlv¢nm1) + ¢nml¢nml
nml
and the integration of —21—V- (ofleofiml) over the cavity volume is zero using the
nml
boundary conditions. Equation (4.42) can then be simplified as
2 a p
2 TIC 2 nm
A —— J ( rjrdr = 1 (4.44)
nmleoneorg n a

where EOn has been defined by eq. (2.63) in Chapter 2. The identity for the integration of

the Bessel function is given in [64] and [65] as

2

2
ijf,(ax)dx = 3‘2—[1',2,(ax)+[i — —%—E]Ji(ax)] (4.45)
a

x

Hence, using eq. (4.45) and after some manipulations, we may obtain the integration in eq.

(4.44) as

136

 

a 2
j1i(p;mr)rdr = Qfl'iwmo (4.46)

A
The normalization constant for the vector wave function anI is then expressed as

8011801 1
A = / 4.47
"ml nc aJ'n(pnm) ( )

A
When n = O , the normalization of the vector wave function LOml is given by

 

J2]? J2 (Zorn: - Zom1)rdrdtpdz= 2n££¢8ml¢gmlrdrdz

214" f AOm,(JO(p 0'" rjsinC—f zDzrdrdz = 1

(4.48)

A
Based on eq. (4.46), the normalization constant for the vector wave function LOmI is then

expressed as

A = Jim _1_._ (4 49)
Oml ncaJ'O(pOm) .
4.2.2 Normalization of Vector Wave Function M n m 1

._\
Based on eq. (2.61), the normalization of the vector wave function M nml is given

Ejznf)(finml ‘ 117nm!)rdrdtpdz = 1 (450)

Substituting eq. (4.39) into eq. (4.50), we have

137

 

Will? (”2: )......)...(Lg.))2

 

 

 

 

 

(4.51)
pnm . pnm . [TC 2 _
+( a Jn( a r)cos(n(p)sm(cz)) ]rdrdq)dz — 1
After integrating with respect to the variables 2 and (p in eq. (4.51), we obtain that
2 a p p ' 2 p '
Bim— “C ———j[—J§ ( "m H) ( m") Jn'2( "m r)r]dr = l (4.52)
£0n£010ra a a
Using the identities of the Bessel function of
Zn
Jn+1(x) = 71n(x)—Jn_l(x) (4.53)
xJn'(x) = xJn_](x)—an(x) (4.54)
. 1
we may obtain the following relations as
l 2 1
;Jn(x) = 2-71[Jn(ic)1n+1(x)+Jn_l(x)Jn(Jc)] (4.56)
J '2 - 12 3"1 J J 4 57
x n (x) _ x n-1(x)_7 n-1(x) "(x)+2‘]n(x) n+1(x) (‘ )

After changing the integration variable, we can modify the integration in eq. (4.52) as

a l 2 ' pill".

 

 

 

{[11i(p;mr)+ (pam) Jn'2(p:" r)r]dr = g [’LrZ-J§(r)+Jn'2(r)r]dr (4.58)

Substituting eqs. (4.56) and (4.57) into eq. (4.58), we have

138

P...’ p...’

2
J[’ir.1r21(r)+1n.2(r)r]dr= J[n‘ln(r)[‘ln+l(r)_‘ln-l(r)]+J'23-1(r)r]dr
0 0

Based on eq. (4.55), eq. (4.59) can be expressed as

p...‘ p...’

2
I [Er-13(r)+1n'2(r)r]dr = I [Ji—1(r)"2"-’n(’)1n'(r)]dr
0 o

 

= 4P,“ 1'2(pnm') + (I — C)" 0313,- 1(an')] - "13(pnm')

nm

where we have used the integration (4.45). Rearranging eq. (4.54) leads to

Jn_1(x) = 1,;(x) +’;’Jn(x)

then we have

J.._1(an' = J.'<p.m')+;,"—.J.<p.m'> = 55:14pm)

um um

Using identities (4.53) and (4.54), we have

 

 

. —l
J,,_l (x) = "x J,_,(x)-J,,<x)
Hence,
1 n(n—1)—p '2
Jn-l'(pnm’) = n——"Jn-l(pnm')_Jn(pnm')= 2 "m Jn(pnm')
prim pnm'

(4.59)

(4.60)

(4.61)

(4.62)

(4.63)

(4.64)

Substituting eqs. (4.62) and (4.64) into eq. (4.60) and after some manipulations we have

Pm. 2
l [n71.3(r)+1,'2(r)r]d’ = %(p.m'2-n2)J§<an')
0

139

(4.65)

Therefore, based on eqs. (4.52), (4.58) and (4.65), the normalization constant for the

_A
vector wave function M nml is given by

’8 8
Brim! = 01: O, 1 (466)

c , 2 .2 .
pnmAfl _n /pnm Jn(pnm)

 

 

__g 0
When n = 0 , the normalization of the vector wave function M om! is given by

[(1,]?le gmlcp’d’d‘PdZ = 1 (4.67)

because M 0m 1, = 0. Equation (4.67) can then be expressed as

2 2n pOm' .pOm' - [TC 2 _
BOWJZJO J‘o( a JO( a r)sm(cz)) rdrdtpdz — l (4.68)
01'
33m,3—’“jp°"10'2(r)rdr = i (4.69)
801 0

It is noted that J 0'(r) = -J l(r), so based on the integration (4.45), the integration in eq.

 

(4.69) is given by
P I p ' p '2
i .. J 01mm" = I 0” Ji(r)rdr= 0'" 11'2(po,,,') (4.70)
0 o 2
Also using the identities (4.53) and (4.54), we have
Jn+1.(x) = 17100—21 (x) (4.71)
x n + I

that is,

140

Jl'(p0m') = 10(Pom') (4.72)

_\
and the normalization constant for the vector wave function M Oml is given by

 

0”” nepom'lo(P0m') '
4.2.3 Normalization of Vector Wave Function N n m1

Based on the definition of the vector wave function Nnml eq. (2.64), the

..X
normalization of the vector wave function N nml is given by

It .3 .3
rr r(Nnm1-Nnm1)rdrd<pdz = 1 (4.74)
o 0 0

Substituting eq. (4.40) into eq. (4.74), we have

C23,", gr“ J‘ [(127.13 "(Pam r)cos(ntp)sin(l:z))2+ +(glgllp2m r))2 (4.75)

(:in(n(p) sin (lg-ray + ((pzm)21n(p2mr)cos(ntp)cos(%z))2]rdrdtpdz = 1

After integrating with respect to the variables 2 and (p , changing the integration variables,

 

 

 

 

 

 

we can modify eq. (4.75) as

2 pan 2
Cnml 211C

k2 swamg [(lgwrfl +—(l;n1n(r))2++(p"’"1,,(r ))zrdr] =1 (4.76)

nml

 

 

 

Using eq. (4.60), we can derive the integration of the first two terms in eq. (4.76) as

14]

pnm 2

I ["7130) + Jn'2(r)r:|dr

O

 

 

 

 

2 (4.77)
P .2 — 1 2 2 2
= "2'" [1,.-. (p.m>+(1—(’; ) )1..- 1<pnm>]—n1.<p.m>
nm

Using eqs. (4.54) and (4.63), we have the following relations as

1.407,...) = J.’(p.m)+'-;J,,(p,.m) = J,'(p,,,,.) (4.78)
. — 1 — 1 ,

Jn—l (prim) = n Jn-l(pnm)_‘ln(pnm) = n J" (pnm) (4°79)

nm nm

Hence, based on eqs. (4.78) and (4.79), and after some manipulations, we may rewrite eq.

 

(4.77) as
Pm 2 P 2
l ["lr'J§(r)+J,,'2(r)’]d’ = 3m 142(1),...) (430)
0

Based on eq. (4.45), the remaining part of the integration in eq. (4.76) can be derived as

 

pnm 2

pnm .
J rJ:(r)dr = 2 J" 2(an) (4.81)
0

Substituting eqs. (4.80) and (4.81) into eq. (4.76), we have

2 TEC

2 .2
nml—_—pnm 1,. (pm) = 1 (4.82)
EOnEOI

.25
Therefore, the normalization constant for the vector wave function N nml is expressed as

E 8
Cnml = ,/ O" 0' l. (4.83)
M anlnwnm)

142

 

A
When n = O, the normalization constant of the vector wave function N0ml is

given by

fg—ggtiflzfip—zm’? 0(L3m,)...(g.))2
((822) 44104-2 D7444 .

After integrating with respect to the variables 2 and (p , changing the integration variable r,

(4.84)

we have

pOnI

£12122 Konsfwgem(flip—1 (485)
2 c o a 0 _ '

8
k0,", 01 0

Using eqs. (4.70) and (4.71), we have

2
jp°"JO'2(r)rdr = jp°”'1f(r)rdr = 9311mm") (4.86)
0 o 2
Thus, based on eqs. (4.53) and (4.54) we obtain
J. . .(x) = 31.0) —J,,'(x> (4.37)
Hence,
11(pom) = -Jo'(po,,.) (4.88)
and eq. (4.86) can be written as
P P 2
10 "10'2(r)rdr = —02m—Jo'2(p0m) (4.89)

143

Using eq. (4.45) we have

pOm 2

2 P .2
j rJO(r)dr = Jam—JO (p0m) (4.90)
0

Substituting eqs. (4.89) and (4.90) into eq. (4.85), we obtain the normalization constant of

...}
the vector wave function N Oml as

 

C = Jim 1 (491)
OM] TIC p0m10'(p0m) .

4.2.4 Some Field Structures of Vector Wave Functions in Cylindrical

Cavities
In this subsection, several electric field structures are plotted for the vector wave

‘ . .3 ...x _x
funct1ons anz, Mnmz and Nnml.

In the calculations, we assume the dimensions of the cylindrical cavity as: the

radius a=0.0762m and the height c=0.15458m. In Figure 4.2 and Figure 4.3 we plot the

electric field structures for the N012 in the r-z plane with (p = 121° and in the r-q) plane

with z=0.0271m. We can identify that it is just the normal TM012 mode. In Figure 4.4 and

Figure 4.5 the electric field structures for the 117111 are plotted in the r-z plane with

(p = 121° and in the r-tp plane with z=0.0271m. It is noted that this is the normal TE 1 1 1

mode. The electric field structures for the 2112 are plotted in Figure 4.6 and Figure 4.7 in

the r—z plane with (p = 121° and in the r-q) plane with z=0.0271m.

So far we have obtained the normalized expressions for the vector wave functions

144

 

0.15- I I I

”111111111““”
“(111 1\\\\“
’//////1 1\\\\\\‘*
k(///// 11 1 \ \\\\\“‘—’
kk\\\\\ \ '1 1' /' /////v,v._.._.
0'1-K\\\\\\‘ f//////’_..‘
\\\\\X {iv/lb.
11111 11ft,.
,11111 11“\\
-,,///771 11\\\\\.-

1
1
1
1
1
1
1
1
I
f
f
f
1
1
1
1

H—F—fiflfl‘x \ \ \

s
\

I r we've—eh

”T"\\\\ \ \ 1
T“‘\\\\11
\\\\\\\(+

0 \ 1 1 L11 1 1.1
—0.06 —0.04 -—0.02

/ / ///4"""

l
1/////"""
1

J I l / / / I .—
L 11 1 L 1 I f '
0.02 0.04 0.06

 

 

 

Figure 4.2 E-field structure of N012 in the r-z plane with (p = 121° .
The dimensions of the cylindrical cavity are a=0.0762m, c=0.15458m.

145

I’ll/1’
“\\‘\\

 

[l/

01 0‘“

”I

1\\\“ms

{$11111
/

x \ \ \\>‘

 

Figure 4.3 E-field structure of Non in the M) plane with z=0.0271m.
The dimensions of the cylindrical cavity are a=0.0762m, c=0.15458m.

146

 

o—.—o—o—o—o—o—o—o—o—o—o—o—o—o—

...—_-

o—o—o—o—o—o—o—c—o—G—c—hfic—c—o— .— .— .—
O—Q—d—C—Q—fi-d—‘I—d—fi-G—fifi—d—C—C—t—Q—o—

¢—+—<—<-—<-—-<—<—<—<—<-—<—-<——<——<-—<-—4— +— ¢—— <—-
0.1 —

0.05

c— Q— ‘—<—§—4—(—-Q—-4—€—-<—4—d—d—Q—¢— ._ o— «—

t—o—t—t—C—fi-«C—hQ—Q—C—Q—C—C—C—Q—o— .—
“O—O—Q—G—Q—C—C—fi-G—bC—“fifi-O—O—O—
.—h‘—~~~ ..-
.. ..

 

 

—0.06 —0.04 —0.02

 

0.04 0.06

fio~'
.°
ot-
N

Figure 4.4 E-field structure of M1" in the r-z plane with (p = 121°

The dimensions of the cylindrical cavity are a=0.0762m, c=0.15458m.

147

 

c=0.15458m.

a=0.0762m,

cylindrical cavity are

Figure 4.5 E-field structure of M1“ in the MD plane with z=0.0271m.
The dimensions of the

148

 

 

 

 

 

“5311 ‘ 15.531111113:
“\\1 ///*’ \1/ /
*\>\\\1//«"—“"\\ ”’""
M‘\IKWR\ ‘ ’ ”a—H
W”‘\w—<—1-—-4——r I \‘s‘b‘yq‘

0.1f)//;‘\\N""’K// l \ \\““.-
,/// \\\,/ l1 1\ \4

,7) (MW/11 11%.
“ f’ \ r
\\\1 7//,\:11111;,

-‘ M\\11//""‘ ‘
hk"\ ’ ””W\\/’M
‘-’4-—r/ \ \‘W”\Kw—
“N I 1 \\“’”"L’7?'i§'~i‘“

.——D \k
NH 1\\\:’,// M111-
/’//111\\,11 111.
0 [1111. 111L‘ 4 1 1
—0.06 -0.04 -0.02 O 0.02 0.04 0.06
r
Figure 4.6 E-field structure of L112 in the r-z plane with (p =121°

The dimensions of the cylindrical cavity are a=0.0762m, c=0.15458m.

149

 

Figure 4.7 E-field structure of L112 in the r-q) plane with z=0.02 71m.
The dimensions of the cylindrical cavity are a=0.0762m, c=0.15458m.

150

anl, 117ml and 1717 nml in cylindrical cavities. They are orthogonal and form a complete
set of basis functions based on the proofs given in Chapter 2. Therefore, they can be
employed to express any unknown electric field inside the cylindrical cavity.

The Electric Field Integral Equation for a material sample which is placed in the
cylindrical cavity can be obtained in the same way as that used in Chapter 2. We can also

obtain the same EFIE as that expressed in the eq. (2.102) as

A s quote , x A s -— s a. “‘1. at
E(r) 1— k2 +1muOJre(rO)E(r0) - (3,000, r)dv0 = E (r) (4.92)
o v

 

However, due to the different eigenvalues for the vector wave functions 117nm: and finml
in the cylindrical cavity as given in eqs. (4.22) and (4.34), the dyadic Green’s function

61,000, i“) need to be modified from eq. (2.101) as

 

 

(4.93 )

n

_ . . 2117010171?) 2117 010117 1?)
Geo('0”)=2[qn "2 2 n2 +kn "2 2 n2
k0(qn ‘ k0) k0(kn ‘ 1‘0)

for the cylindrical cavity and we need to give the detailed expression for the dyadic

Green’s function (4.93) in the cylindrical cavity in order to solve the EFIE (4.92) using the

Galerkin’s method.

4.3 Dyadic Green’s Function in the Cylindrical Cavity

In this section the detailed expression of the dyadic Green’s function (4.93) in a
cylindrical cavity will be derived. As in Chapter 3, the dyadic Green’s function (4.93) is a

triple summation over the cavity eigenfunctions and we can reduce it to a double

151

summation format based on the relations (3.27) and (3.28) in Chapter 3. Substituting the

expressions for the vector wave functions 117nm! and finml (4.39) and (4.40) and the

normalization constants (4.66) and (4.83) into eq. (4.93), we can obtain the expressions for

the different components of the dyadic Green’s function GCOGO, i) as follows:

1. 9? component of the dyadic Green’s function (76,000, i) in the triple summation

format is given by:

 

 

 

oo oo oo 2 2 ,
a as 1 80:180! qn 1 pnm
Georr<r0’r) = _2 20 m2 2 [ 2 2 2 2 —'J"( r)
[(011 = = ll=1 EC qn_k0(pnm'_ '12)] n(pnm)" a

 

 

 

pn 1 1 11C 2 , pn p
11 :«J 2 2 2 .. (—.—J 11 J 1%)]
kn-kOa Jn (prim)

. 17: . In
cos(mp)cos(ncp0)srn(-c-z)s1n(?zo)

Equation (4.22) gives the eigenvalue qi of the vector wave function 117nm! . and the first

summation term in eq. (4.94) is then split into two parts as:

 

 

co oo oo 2 2 1
i2 2 2%?” q". ‘ 191”” :1

 

 

 

 

 

2 .2 2 2 ,
"0~=0m 9 40(1),... —n )J,,(p,,,,,)" a
1,,(pzmr 0)cos(ntp)cos(ntp0)sin(b—tz)sin(l—:zo)
pm; 2 (5)2 ’ (4.95)
1 0° .. ... 90,1801 17) c 1 n2
_ —2 Z 2 Z 2 2 '1' 2 2 ,2 2 2 , ‘—
k0n=0m=ll=1 KC qn-kO qn_k0 (prim “n )Jn(pnm)rr0

 

     

)cos(ntp)cos(n(p0) sin (15:2) sincgzoj

152

Using the relation (3.27) in Chapter 3, the first term in the summation (4.95) can be

summed over the index I as

 

 

 

 

 

(””12

801 a . (In) . (In ) (an')? f1(Z,ZO)

— s1n —z sm —zO = . (4.96)
l: c q:_k(2) C c a kglsmucglc)
where
an' 2

kgl = Jk3—( a ) (4.97)
f1(z,z0) = sin(kgl(c—zb))sin(kglz5) (4.98)

and it is convenient to define the following (they will be used later for the same definition):
2,, E the greater of z or 20 (4.99)

zssthe lesser of z or 20 (4.100)

Also in [21] it gives

(4.101)

 

2 °° e . k ,
___ a 22 1 25in(g:_cz)sin(ar%)= M_5(z_zo)
Z

sin(kglc)

Thus, based on eq. (4.101), we can obtain the closed form evaluation of the summation

over index I for the second term of the summation (4.95).
The other terms in the summation (4.94) can be summed over the index I in the

same way. So after some algebraic manipulations, we obtain the double summation format

153

 

of the r? component of the dyadic Green’s function 6,00», 2) as

 

 

 

oo oo 2 v I
1 0n 1 pnm pnm
Georr(r0: r) = 1:2)" :0 ml; i;[2)lf1n( a r)1n( 0 r0)

(PM;2 -n22)1,.'(P,,m

."° f (z, ZOJ+__1 J (”nm,)1,v(9_m.,)"__g?f2‘z"0’]

 

 

 

 

 

 

kgls1n(kglc) a 21". 2(1)” m) a a 81n(kgzc)
(4.102)
°° 80!: 1 "2 prim.
cos(ntp)cos(ntp0)—— 2 Z 7t2[ 2 __'Jn( r)
k0n=0m=ln (pram2 _n )Jr21(pnm')rr a
pnm' l prim prim
Jn( r0)+ ————Jn'( r)J '(—r )]cos(n(p)cos(n(po)5(z-zo)
“ 0214711....) a n a 0
where
P m
kg2 = k0 ( —2— ) (4.103)
and
f2(z,zo) = sin(kgz(c-zb))sin(kgzzs) (4.104)

2.1011) component of the dyadic Green’s function 613000,?) in the triple summation

format is given by:

 

 

 

 

 

.. .. .. qz 2(p’m')
x s 1 80,180, qn a punt.
Ge (’9’): [201:121 112 1'1—r)
“NP 2 2 2 . '1
knO = = = nc ’21-k(2)(pnm2 —n )Jn(pnm) a
2
. (’1qu (4.105)

Jn-(Pgmm). ,1 2 2 431,,(94 )J,(&r_~.,) .11....)

kn-kopnm In. (pnm)rr a a

sin(n )sin(!-1—tz)sin(lltz)
(DO 6‘ c O

154

Applying the same procedure used for 5"? component to the summation (4.105), we obtain

the double summation format for the (pfp component as

 

 

 

.2 2 2 .
1" (p... -n)J,.<p,.,,.

(“”32
“ Jn'(£’flr)1n'(p"m
) . a

a

GeO¢¢(r0’ r) = i220 mzeo r0) (4-106)
km -_-0 =

 

 

 

k3f1(z’ 20) 1 n2 pnm pnm k82f2(z9 Z0) .
+ M )J.(— .)—— W)

kglsin(kglc) anzjn'zwnm a a sin(kgzc)

pnm' 2
sin(n(p0—- i2; 28—0— ( a )

.2 2 2
k0n=0m=l 1t "(prim —n )Jn(pnm

+ 2 12 rr' 2:14,)? )J"(pnm
PM. Jn' (PM) “

 

 

 

Jn'(h"_r)1n'(pnm r0)
.) a a

 

 

) sin(nq>)sin(mp0)5(z — 20)

     

3. fl?) component of the dyadic Green’s function (LOGO, F) in the triple summation

format is given by:

ann

oo 00 no q2 .
1 80n:Ol qt: 0 r pm"
Geor<p(r0’r) = _P 20 m2] [11:2 2 2 v 1n( a r) (4107)
k0]; :: = = qnz—k0(p:.21m_n )Jn(pnm)

 

 

 

 

 

 

 

 

' c
J' anr + 1 a 3,] ' anr J anr cos(ncp)sin(mp )
n a 0 2 2 2 .2 r n a n a o o
kn-kOpnm 1n (prim)

The double summation format of the M) component is then expressed as

155

O
ann

W» = -—2 £81 . 0' )J.(anr)x,;("~r~

r 4.108)
2 2 . 0) (
k0n=0m=l TC "(prim -n )Jn(pnm a a

 

 

 

 

2 EMF.

' k f (Z92 )
. + 2 (1 2r Jn'(p"mr)1n(pflro)—g.2—2——O— cos(ntp)
kngln(kgIC) pnm J". (pnm) a a 510(k82C)

 

 

 

 

 

 

 

4 Similarly, the (I)? component of the dyadic Green’s function 6,3000, ;) in the double

summation format is given by:

 

 

 

 

 

 

 

 

 

 

 

 

pnm'p-
s x 1 0° 0° 80:: 0 r0 pnm' pnm'
Geow(r0, r) = —-3 20 2 — 2 2 2 Jn'( r)1n( r0) (4.109)
knO = 171:] n (pnm. _n )Jn(pnm') a a
2 pnmg
W...) . a , 1 (pm); (822. )kuszfz‘z’a’ slum)
kglsm(kglc) pn’n;1n.2(pnm)n a n a srn(k82c)
pnm.£
1 °° °° 8 a r . p ' p '
cos(ncp0)+—22 2 .1032 . 2 02 .Jn( Zmr)1n( 2'" r0)
k0n=0m=l (prim -n )Jn(pnm)

 

 

 

     

5 zz component of the dyadic Green’s function (78000, ;) in the triple summation format

156

is given by

 

an
0.0.204?) = ii i 28°"80’ 1 (—) )1n(&2r)

2 2 2 2 .2
k0n=0m=ll=0 "5 kn-koa 1,. (pm a (4.110)

     

)cos (mp) cos (mpo) cos (%z)cos ($20)

Applying the relation (3.28) in Chapter 3 to the summation (4.110), we obtain the double

summation format for the 22 component as

 

 

 

 

~ ~ (’02
060300, ;) = _kigngom‘gl1;"azjn";(pnm)1.(%r)1,,(P;'"ro) (4 1“)
cos (mp) COS("‘P0)k::i(:;:::C)
where
82(2120) = cos(kgz(c-zb))cos(kgzzs) (4.112)

AA

6. r2: component of the dyadic Green’s function GMGO, ?) in the triple summation

format is given by

 

 

 

(4.113)

In [21], it gives

157

 

 

0
_ 3 0° 801 1 111: In (4114)
_ E2 c k2_k2COS(cz)cos(cz0)
[=0 n 0
1
- ngz(z, Z0)
where
sin(k 2(c—z))cos(k 220) z>zo
fgz(z,zo) = . g g (4.115)
-s1n(kgzz)cos(kgz(c - 20)) z < 20

The double summation format for the $2 component is then given by

8 ;
Georz( rO’ r) =

 

(4.116)

f g2(z, zo)
sin(kgzc)

m

Jn(%r0)cos(nq>)cos(n(po)

7. Similarly, the 2? component of the dyadic Green’s function G800”, ;) in the double

summation format is given by

pnm

8 8 1 0° 0° EOn a pnm
Ge zr(r0’ r) = .2 2 2 _ 2 J( I“)
.o '2 ,1
k0" =0"! =1 1: a In (pnm) a

 

 

(4.117)
fgl(z. zo)

sin(kgzc)

 

Jn'(p;mr0) cos(n(p)cos(n(po)

where

158

—cos(k (c—z))sin(k z ) ' z>z
fg1(z.zo) = { 82 , 82 O 0 (4.118)
cos(kgzz)51n(kgz(c—zo)) z<zO

8. (02 component of the dyadic Green’s function (7,3000, 2) in the triple summat1on

format is given by

 

 

     

 

 

x s 1 80 :01 r C P
Ge0(p2(r0: r) = "-2' 20 ”'12 [11:2 n Jn( Zmr)
knO = =I=k121—k0a2n"] (pnm) (4.119)
pnm . . [It 111:
Jn( a )s1n(ln(p)cos(n(po)s1n(-C-z)cos(-;zo)
The double summation format is then expressed as
oo N :1

180", P m

,. = —-. z z —,.-. . J,( z, ,)
Mo = = Mm) (4.120)

1,,(p2m

9 Sim1larly, the 2(1) component of the dyadic Green’s function GeOGO, ;) in the double

. fgz(z’ Zo)
)sm(n(p)co:(n(po)W

     

summation format is given by

 

 

(4.121)
fgl(z,zo)

nm )cos(n(p) sin ("(PO)W
g

     

After we der1ved the detailed expression for the dyadic Green’s function we can

numer1cally solve the EFIE (4.92) in the material sample by the same Galerkin 8 method

as that used 1n Chapter 3. That is, the material sample is divided into a large number of

159

volume cells and the pulse functions are employed as both the basis and testing functions.
After some similar manipulations as those used in Chapter 3, we can transfer the EFIE
(4.92) to the similar matrices eqs. (3.17) to (3.20) which can be solved to obtain the

unknown electric field. Detailed discussions on the numerical techniques will be presented

in Section 4.4.

4.4 Numerical Examples

In Chapter 3, we have studied the convergence property of the dyadic Green’s
function in both the triple and double summation formats. Based on those results, in the
following numerical examples we will adopt the same volume cell dimensions in the
Galerkin’s method and the same upper limit N=200 in the double summation format of the
dyadic Green’s function as those used in Chapter 3 to assure the convergence of the

numerical results.

In the application of the Galerkin’s method, there will be integrations with respect

to both variables ? and i’ as shown in eqs. (3.7) and (3.10). In the cylindrical cavity, we
deal with the Bessel functions or their derivatives and their numerical integrations in the
formation of the matrix (3.20). These numerical integrations consume much more
computing time than the case of the rectangular cavity. Thus, in general it is very difficult
to find the numerical solution of a EF IE even for a material sample with small dimensions
with our present computer resources. However, for some special cases we can simplify the
expression of the dyadic Green’s function and numerically solve the EFIE in the
cylindrical cavity.

We assume that the material sample, which is azimuthally symmetrical, is placed

160

in the center of a cylindrical cavity. If the initial cavity mode is not a function of variable

(0 , for example TMOIZ, then due to the symmetry property [2],[39] the scattered electric
field induced by this material sample and the total electric field will not be functions of
variable (p. Therefore, in the numerical computation for this special case, the eigenmodes
with n=0 will be sufficient to represent the unknown electric field and the expression for

the dyadic Green’s function in the double summation format over indices n and m will be

further reduced to one summation format over index m.

In the following numerical computations, we assume the dimensions of the

cylindrical cavity to be: the radius a=0.0762m and the height c=0.15458m. The initial
cavity mode is TM012 and the resonant frequency of this mode is f = 2.45 GHZ for

these dimensions of the cavity. This resonant frequency of the cavity will shift slightly
downward when a material sample is placed inside as discussed in Chapter 3. A material

sample with the diameter do and the height ho is placed in the center of the cylindrical
cavity as shown in Figure 4.8. The relative permittivity of the material sample is assumed

to be 8, = 2.5 and it is lossless. Using the Galerkin’s method, the material sample is

uniformly divided into M = n d x [d volume cells, where n d and 1d are the numbers of

the volume cells in the r and 2 directions, respectively. Several numerical calculations are
carried out for the material sample with selected shapes and dimensions.

1. Cubic material sample

A cubic material sample, having the diameter equal to the height, is placed in the

center of the cylindrical cavity. The dimensions of the material sample are chosen as:

diameter d0=0.004m and height h0=0.004m with n d = 1 and Id = 2 . The cavity

161

Z
\

___—___»

 

 

 

 

 

 

Figure 4.8 Dimensions of the cylindrical cavity and the material sample. The center
of the material sample is consistent with the center of the cavity.

162

resonant frequency is assumed to shift down 5% after the material sample is placed inside

the cavity. Since the wavelength A of the initial mode is 0.12245m which is much larger

than the dimensions of the sample (ho/A. = 0.0327 ), the induced electric field inside the

material sample can be estimated by the electrostatic field induced inside of a dielectric

 

sphere as E =

Observing the electric field structures for the initial mode TMOIZ in Figure 4.2 and
Figure 4.3, we find that there is only Ez component of the electric field which is significant

near the center of the cavity. Due to the small dimensions of the material sample, the

numerical results are uniform in each volume cell and the Ez component of the induced
electric field dominates the other two components. The ratio of the EZ component of the

induced electric field to that of the initial electric field in the material sample is found to be

 

0.65 in each volume cell while the electrostatic estimation of E y/ E: = gives the

3
2+8

,
approximation of 0.667. Thus, numerical results and the theoretical estimation are in

satisfactory agreement.

For the stability check of the numerical results, we increase the dimensions of the
cubic material sample to: d0=0.008m and h0=0.008m with n d = 2 and Id = 4. The
cavity resonant frequency is assumed to shift down 5% after the placement of the material
sample. The numerical results show that the Ez component of the induced electric field
still dominates and the ratios of the E: component of the induced electric field to that of

the initial electric field becomes 0.61, 0.64, 0.64 and 0.70 at the different volume cells in

the material sample. These values are still close to the electrostatic estimation of 0. 667.

163

For a larger cubic material sample with the dimensions: d0=0.02m and h0=0.02m
with nd = 5 and Id = 10, the numerical results are shown in Figure 4.9, where the

cavity resonant frequency shift is assumed to be 8% after the placement of the material
sample. Since the 1 component of the induced electric field dominates only the ratios of
the z components of the induced electric field to that of the initial electric field in the
material sample are plotted, in Figure 4.9, as a function of r, for the lower half of the
sample. (Numerical results are symmetrical with respect to the center of the sample.) We
observe that due to the increase in the material sample dimensions, the ratios in Figure 4.9
are now reduced to about 0.51 to 0.62. This indicates that the induced electric field in a
larger sample will be smaller than the value given by the electrostatic estimation.
2. Thin chip case

A material sample with the shape of a thin chip, having its height much smaller

than its diameter, is placed in the center of the cylindrical cavity. The dimensions of the
material sample are h0=0.002m and d0=0.04m with nd = 10 and Id = 1. The cavity
resonant frequency shift is assumed to be 1% after the material sample is placed inside.
Because only the z component of the initial electric field is significant near the center of

the cylindrical cavity and the material sample has a thin chip geometry, theoretically the

induced electric field in the material sample can be estimated by the boundary condition of

E = (1/£,)15’i = 0.4Ei. The numerical results are shown in Figure 4.10.

In Figure 4.10, the ratios of the 2: component of the induced electric field to that of
the initial electric field are plotted as a function of the radial distance, r, of the material

sample. We observe that the numerical results are consistent with the theoretic estimation

164

0.64

 

0.62

0.6

0.58

0.56

Ratio of E/ 1"

0.54

0.52

 

 

 

 

p-
)—

 

0.5 L 1 1 1 l
1

x10

Figure 4.9 Ratio of E,_./Ezi varies in the r direction at the different locations of z.
The dimensions of the material sample are d0=0. 02m and h0=0.02m with the relative

permittivity of e, = 2.5 . The dimensions of the cylindrical cavity are shown in
Figure 4.8.

165

0-5 T I T T l I

 

 

I m t

0.48 F .l
. '- l i 4

0 46 1 E2

0.44 1' E j -

0.42 " E -‘

Ratio of Ell z‘
O
h

 

 

 

0.38 -

0.36 — ..

0.34 — .

0.32 - .
0.3 L 1 1 4 1 1 1 . 1

 

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Figure 4.10 Ratio of Ez/Ezi varies in the r direction. The dimensions of the material
sample are d0=0.04m and h0=0.002m with the relative permittivity of e, = 2.5 .The
dimensions of the cylindrical cavity are shown in Figure 4.8.

166

as expected.
3. Thin pipe case
A material sample with the dimensions: h0=0.044m and the diameter d0=0.008m is

placed in the center of the cylindrical cavity. The material sample is divide with nd = 2

and Id = 22 , the number of the volume cells being 44, and the cavity resonant frequency

shift is assumed to be 1% after the material sample is placed inside. For this case, the
induced electric field inside the material sample should be approximately equal to the
initial electric field because the initial electric field is tangential to the major part of the
material sample surface, and the continuity of the tangential component of the electric
field at the material sample surface requires this estimation. The numerical results are
shown in Figure 4.11.

In Figure 4.11, we observe that the maximum ratio of the z—component of the
induced electric field to that of the initial electric field is around 0.8 which is not very close
to I . This is because when we derived the EFIE (4.92), we have extracted a delta function
out the integration sign and if we expect the induced electric field to be equal to the initial
electric field, another delta function is needed to be generated under the integration sign in
EFIE (4.92) to cancel the former delta function. Thus, we will need much more
summation terms for this special case to meet this requirement just like the narrow strip
case in Chapter 3. If we increase the upper limit in the double summation, the numerical
results become closer to the theoretic estimation but at the expense of increasing the
computation time.

For this thin pipe shaped material sample, if we employ the scheme of separating

the material sample into boundary layer and interior cells as that used in Chapter 3,

167

 

0.8 -

 

0.7 -

Z

 

0.6 r-

 

for fit]
05- — - — forr=r2

Ratio of El!

Ei

0.2 -

 

 

0.1 -

 

 

O l I L l l 1 L l
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08
Z

 

Figure 4.11 Ratio of Ez/Ezi varies in the z direction at the different locations of r.
The dimensions of the material sample are (10:0. 008m and h0=0. 044m with the

relative permittivity of e, = 2.5 . The dimensions of the cylindrical cavity are shown
in Figure 4.8.

168

the results are improved and at the same time the computation time is reduced. In this

computation, the dimensions of the material sample are: d0=0.008m and h0=0.044m with
n d = 1 and Id = 20. The dimensions of the volume cells in the boundary layer region

are: br=0.002m and bz=0.002m in the r and 2 directions, respectively and the dimensions
of each volume cell in the interior region are: ir=0.002m and iz=0.002m in the r and 2
directions, respectively. The number of the total volume cells in the boundary layer and
interior regions is: 2 X 22 = 44 which is the same as the previous computation. The cavity
resonant frequency shift is assumed to be 1% after the material sample is placed inside.
The upper limits in the mode summation are chosen as 200 for the boundary layer region
and 150 for the interior region. The numerical results are shown in Figure 4.12.

Since we are only interested in the induced electric field inside the material
sample, only the solutions for the volume cells in the interior region are plotted in Figure
4.12. Because the numerical results show that the 2 components of the induced electric
field dominate in the material sample, only the ratios for the 2 components of the induced
electric fields to that of the initial electric field in the interior region are plotted as a
function of z in Figure 4.12. We observe that the ratios are now very close to I .

In the scheme of the separating the material sample into the boundary layer and
interior regions for the thin pipe shaped material sample, we keep the number of the
volume cells the same as the previous computation. However, the upper limit of the mode

summation in the interior region is reduced to 150. Hence, the computation time is saved
and at the same time better results are obtained.

4. Lossy material sample

In this numerical example, we will assume that the material sample has a complex

I69

 

 

 

 

 

 

 

1.5 1 1 r —r I T r l
1.4 " "
1.3 - "
1.2 " "
73:
m 1.1 _\ / T
H
O
Q
5 1 _ _
C.
I A
u
0.9 " ‘
E " E
0.8 - z z -
0.7 '- -1
C
0.6 - _
0.5 4 J L L l 1 l l
0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095

2

Figure 4.12 Ratio of Ez/Ezi varies in the z direction with the scheme of separating

the material sample into the boundary layer and interior regions. The dimensions of
the material sample are d0=0. 008m and h0=0. 044m with the relative permittivity of

8, = 2.5 . The dimensions of the cylindrical cavity are shown in Figure 4.8.

170

permittivity of e, = e' — je" ; where 8' is the real permittivity (electrical polarizability) of

the material sample, the magnitude of which reflects its ability to store electric energy; 8"
represents the loss factor of the material sample, and its magnitude determines its ability to

dissipate electric energy into thermal energy [63]. In the computation, we assume
8' = 2.5 and 8" = 0.5 with the dimensions of the material sample as: diameter

d0 = 0.008m and height h0 = 0.008m . The numerical results are shown in Figure 4.13.

In Figure 4.13, we plot the ratios of the 2 components of the induced electric field
to that of the initial electric field inside the material sample. Due to the complex
permittivity of the material sample, the induced electric field has both the real and
imaginary parts. In Figure 4.13 the upper graphs are for the real parts and the lower ones
are for the imaginary parts of the ratios. Comparing the real parts of the ratios in Figure
4.13 (about 0.62 to 0.68) with that for a lossless cubic material sample of the same
dimensions (about 0.61 to 0. 70), we find that the real parts of the ratios remain relatively
unchanged if the material sample has a complex permittivity. Also the numerical results
indicate that if the imaginary part of the relative permittivity is increased, the real parts of
the ratios will decrease while the imaginary parts of the ratios will increase. The numerical
results shown in Figure 4.14 can demonstrate this finding where we assume the complex
relative permittivity of the material sample as e' = 2.5 and e" = 1.5 with the same

sample dimensions as that in Figure 4.13.

In Figure 4.13, where the relative permittivity is 8' = 2.5 and e" = 0.5 , the real

parts of the ratios are about 0.62 to 0.68, while in Figure 4.14, where the relative

permittivity is 8' = 2.5 and 8" = 1.5 , the real parts of the ratios are about 0.54 to 0.63.

171

 

 

 

 

 

 

 

 

 

 

_____ for 2:21
0.7 I I I I I I I I I
0.68- fly _ .. —-.
..N 0.66- ‘ ‘ . ___,...- .
‘14 0.64
In
“5 0.62
1:
a 0.6
'5 0.58 - .
m 0.56 l 1 l l 1 1 1 1 I
1 12 1 4 1 6 1 8 2 2.2 24 26 28 3
' 1110'3
0.085
...N
\
La 0.08
"5
E
=- 0.075
t“
a
g 0.07 J L I l l J L 1 l
.... 12 1 4 16 1 8 2 2.2 24 26 28 3
I

Figure 4.13 Ratios of E/Ezi varies in the r direction. Each curve represents this
ratio as a function of r for different locations of z in a material sample when the
material sample has a complex permittivity of e, = 2.5 — j0.5 . The dimensions of the
material sample are: diameter d0 = 0.008m and height h0 = 0.008m. The upper

graphs are for the real parts of the ratios and the lower ones are for the imaginary
parts of the ratios.

172

0.64 1

 

0.62 1 y . - "’
0.6
0.56
0.56
0.54

 

0.52

 

Real part of E/ 2’

 

0.5 L l L 1 I l 1

 

x 10
0.23 a, T

 

0.225 ~ " ‘ ~-
0.22 — " ' ~.
0215 r- " ' ~.
0.21
0.205
0.2
0.195 . . 1 1 1 1 1
1

 

 

Imaginary part of E/ z1

 

 

Figure 4.14 Ratios of Ez/Ezi varies in the 1' direction. Each curve represents this
ratio as a function of r for different locations of z in a material sample when the
material sample has a complex permittivity of 8, = 2.5 - j1.5 . The dimensions of the
material sample are: diameter do = 0.008m and height ho = 0.008m. The upper

graphs are for the real parts of the ratios and the lower ones are for the imaginary
parts of the ratios.

173

However, the imaginary parts of the ratios in Figure 4.13 are about 0.072 to 0.085 while
that in Figure 4.14 are about 0.2 to 0.23. Thus, a larger imaginary permittivity causes more
power loss or more microwave power dissipated in the material sample.
5. Inhomogeneous material sample

In this numerical example, we assume that the material sample is composed of two

kinds of dielectric materials as shown in Figure 4.15. The relative permittivities of the two
regions of the material sample are denoted as 81 and 82 and their radii and heights
denoted as r 1» r2 and h I, hz, respectively.

If the dimensions of the material sample are electrically small compared with the
operating wavelength, we can estimate the ratios of the 2 components of the induced
electric fields inside the material sample to that of the initial electric field by the

electrostatic approximations, which are given in Appendix D (D45) and (D47), as

-3
E -98 e r
._' = _g 2 1 _3 (4.122)
50 (81+282)(82+280)rl +2(81-82)(82—£o)r2

 

—3 —3

0 (81+ 282K232 + 2.¢:o)r]3 + 2(81- 82)(82 — 6o)r;3

 

(4.123)

In this numerical example, the dimensions of this inhomogeneous material sample

are Chosen as: r1 = 0.004m, h1 = 0.008m, r2 = 0.008m and h2 = 0.016m, and with

the relative permittivities assumed to be 81 = 2.5 and £2 = 4.0. The eqs. (4.122) and

(4.123) give the ratios of the 2 components of the induced electric fields in the

inhomogeneous material sample to that of the initial electric field as R1 = 0.5818 in

174

 

 

 

 

 

 

 

 

 

 

 

Figure 4.15 An inhomogeneous material sample is placed in the center of a
cylindrical cavity.

175

region 1 and R2 == 0.5091 in region 2.

The numerical results are shown in Figure 4.16, where we observe that the
numerical results are smaller than the electrostatic estimations. This is because the

dimensions of the inhomogeneous material sample are not very small compared with the

operating wavelength.

If we change the dimensions of the material sample to r1 = 0.002m,

h1 = 0.004m, r2 = 0.004m and 112 = 0.008m, and remain the relative permittivities of

the material sample unchanged, the numerical results are shown in Figure 4.17. In this

figure, we observe that the numerical results become more consistent with the electrostatic

estimations.
6. Irregularly shaped material sample

In this numerical example, we assume the material sample to have a irregular
shape but keep the material sample azimuthally symmetrical in order to save the

computation time. The shape of the material sample is shown in Figure 4.18. We assume

the dimensions of the material sample as: h1 = 0.004m, h2 : 0.008m, d1 = 0.016m

and d2 = 0.008m , and with the relative permittivity of 81 = 2.5. The numerical results
are shown in Figure 4.19.

After we have quantified the induced electric field inside the material sample, the
.1 2
dissipated power is determined [66] as P = (0808"1El , where e" is the imaginary part of

the permittivity and E is the induced electric field inside the material sample. This _

dissipated power becomes a volumetrically-distributed heat source. The temperature

176

0.0662

0.0642

0.0622

0.0602

0.0582

0.0562

0.0542

0.0522

0.0502

¢z(m)

 

0.2936

0.3132

0.3475

0.3728

 

0.2856

0.3054

0.3722

0.4261

 

0.4492

0.4559

0.4008

0.4506

 

0.4429

A 0.4450

0.4135

0.4612

 

0.4429

0.4450

r—

0.4135

0.4612

 

0.4492

0.4559

0.4008

0.4506

 

0.2856

0.3054

0.3722

0.4261

 

 

0.2936

 

0.3132

0.3475

0.3728

 

 

Ratio of 162/E:

 

 

0.002 0.004 0.006 0.008’r(m)

Figure 4.16 Ratios of Ez/IEZi in an inhomogeneous material sample with the
dimensions of r1 = 0.004m, h1 = 0.008m, r2 = 0.008m and h2 = 0.016m,where

the relative permittivity in the shadowed region is el = 2.5 and that in the non-
shadowed region is 82 = 4.0 . The electrostatic estimations of the ratios are

Rl = 0.5818 and R2 = 0.5091.

177

A z(m)

0.0622

 

0.3642 0.3948 0.4447 0.4531
0.0612

 

0.3551 0.3909 0.4816 0.5142
0.0602

 

.- Ratio of Ez/Ezi
0.5561 0.5958 0.5181 0.5435
0.0592

 

0.5478 0.5864 0.5347 0.5562

 

0.0582
0.5478 0.5864 0.5347 0.5562

 

0.0572
0.5561 0.5958 0.5181 0.5435

 

0.0562

0.3551 0.3909 0.4861 0.5142
0.0552

 

0.3642 0.3948 0.4447 0.4531

 

 

 

 

 

0.0532

 

0.001 0.002 0.003 0.004’r(m)

Figure 4.17 Ratios of Ez/Ezi in the inhomogeneous material sample with the
dimensions of r1 = 0.002m, hl = 0.004m, r2 = 0.004m and h2 = 0.008m,where

the relative permittivity in the shadowed region is 61 = 2.5 and that in the non-
shadowed region is 82 = 4.0 . The electrostatic estimations of the ratios are

Rl = 0.5818 and R2 = 0.5091

178

Az

 

 

 

 

 

 

 

 

0.15458m
/
\ |
l
'd
_ _ hl
d2 1'12
2 _ .. hr
' d1
1
I
I 4 ¢

 

 

 

 

Figure 4.18 Geometry of an irregularly shaped material sample placed in the
cylindrical cavity. The material sample is azimuthally symmetrical and the center of
the material sample is consistent with the center of the cylindrical cavity.

179

42(m)

0.0662

 

0.0642

0.5674

0.5285

0.4687

0.4985

 

0.0622

0.6151

0.5826

0.4498

 

0.4954

 

0.0602

0.6602

0.7054

 

0.0582

0.6915

0.7552

 

0.0562

0.6915

0.7552

 

0.6602

0.7054

 

0.0542

0.0522

0.6151

0.5826

0.4498

0.4954

 

 

0.0502

0.5674

 

0.5285

0.4687

 

0.4985

 

 

Ratio of Ez/Ezi

 

 

0.002 0.004 0.006 0.008 T (m)

Figure 4.19 Ratios of Ez/Ezi in an irregularly shaped material sample.

180

distribution in a material sample due to the microwave radiation is thus governed by the
interaction and absorption of the radiation by the material and the accompanying transport
processes due to the dissipation of electric energy into heat [3]. Detailed discussion about

the heating processing of the material sample are available in [66] to [67].

181

CHAPTER 5

QUANTIFICATION OF THE INDUCED ELEC-
TRIC FIELD IN A MATERIAL SAMPLE
PLACED INSIDE AN EM CAVITY USING
MODE MATCHING METHOD

In the previous chapters, the induced electric field inside a material sample placed
in an energized cavity has been determined based on the numerical solution of an electric
field integral equation (EFIE) or a magnetic field integral equation (MFIE). These integral
equations were obtained after the dyadic Green’s function in an EM cavity was derived. It
has been demonstrated that the integral equation method is a very powerful technique
because it can handle the material samples of arbitrary shapes and heterogeneities. The

only disadvantage of this method is its slow numerical convergence and a large

A _\

computation time. The reason is that we used the vector wave functions, anI, M nml and

..A

N nml, to represent the induced electric field inside the material sample and it requires the

summation a very large number of these vector wave functions to reach a convergent

numerical result.

When the material sample is of a simple cylindrical geometry and homogeneous,

the mode-matching method can be employed to determine the induced electric field inside

182

the material sample placed within an energized cylindrical cavity. This method will be
investigated in this chapter.

Applying the mode-matching method to a homogeneous material sample with a
simple cylindrical geometry placed within a cylindrical cavity, we can divide the cavity
into three waveguide regions as shown in Figure 5.1, where regions I and III are the
normal waveguide regions filled with a homogeneous material or empty and region II is
the inhomogeneously filled waveguide region containing the material sample. The
waveguide eigenmodes in region II are derived first while those in regions I and III are
well known. The EM fields in each region are expressed in terms of its eigenmodes, and
the tangential component of the electric and magnetic fields are matched at the junctions
of the three regions. The equations resulted from the matching of the boundary conditions
are then numerically solved.

The outline of this chapter is as follows: In Section 5.1 the eigenmodes in the
inhomogeneously filled waveguide containing the material sample are derived and those in
the normal waveguide regions are introduced. The mode-matching method is applied to
these three regions in Section 5.2. And the numerical examples are shown and the results

are compared with that of Chapter 4 in Section 5 .3.

5.1 Eigenmodes in Different Waveguide Regions

5.1.1 Eigenmodes in a Homogeneously Filled Waveguide

The eigenmodes in an empty or homogeneously filled waveguide have two
categories [10]: TM and TE mode. They are complete and orthogonal. The expressions for

TM eigenmodes are: (We only show the propagation mode in the plus 2 direction for

183

 

“C

III

 

z2

 

z11

 

I
0 F
excitation
probe

..K’

‘\ D

 

 

 

 

Figure 5.1 Geometry of the material sample placed in a cylindrical cavity driven
by an excitation probe.

184

simplicity.)

 

P 2 —F :
E, = AMI?) Jn(p;mr)cosn06 ""'

’

pnm pnm -rnm‘-
r cosnGe

Er : 'Anmrnm a Jn( a

 

 

 

-r
Eo = Aannm;Jn(p;mr)sinn0e "'"z

 

. p . _rnm
H, = —j(0€Anm'-:Jn( Zmrjsrnnee Z

an

 

. P -I‘,,,,,z
Ho = -J(0£A,,m—;'31n'( r)cosn0e

 

2
2 2 . .
where ( 2'") = 00 118 + I‘m . For n=0, the express1ons for TM e1genmodes are:

(‘1')
II
>
S?
3
"'1
O
3
b
O
S
g.
0.
A
"b
O
5
__/
l
:1
5’
(1

—F 1
Ho - —jw6Aomp—2'1'Jn'(%r)e ”"'

with Eo = 0 and H, = 0. The expressions for TE eigenmodes are:

P '2 —F r.
H = An,n(fl) 1,,(fl'flrjsinn06 ....
a

(I

(\l

p 1 1 . -r 3
Hr = _Anmrnm—nT‘J '(p—er)srnn0e ""'

a n

185

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)

 

nm nm-
r

1 -r z
Hoz—A F n1n(p;mr)cosn0e "'"

- p —rnm
E, = —j(0uAnm’-;Jn(—"5'1r)cosn06 Z

' -r
pm J (sz r)sinn0e M":

£0 = jmp’Anm a n

 

 

2
where (5'27“) = (1)2116 + rim. For n=0, the expressions for TE eigenmodes are:

P '2 P ' —r,,,,
= 1.1401(2). .

P0 ' . P0 ' -l‘o..z
Hr = —A0mrom_'a’_n'1n (%T)e

. 1 p 1 _romz
159 = quAomp%Jn'(—:1r)e

with Ho = 0 and E, = 0.

5.1.2 Eigenmodes in an Inhomogeneously Filled Waveguide

(5.11)

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

The geometry of an inhomogeneously filled waveguide is shown in Figure 5.2

which consists of two sub-regions having the same central axis. The central sub-region is a.

homogeneous material sample and the outer sub-region is empty space. It is noted that if

the material sample has an irregular shape or is heterogeneous, the eigenmodes in such an

inhomogeneously filled waveguides will be difficult, if not impossible, to be determined.

Therefore, we only deal with the homogeneous material samples with simple cylindrical

geometries which are placed in the cavity in this chapter.

186

 

 

 

Figure 5.2 Geometry of the inhomogeneously filled waveguide

The normal eigenmodes in this inhomogeneously filled waveguide are not, in
general, either TE or TM modes, but a combination of an TE and an TM mode, a hybrid
eigenmode. An exception is the case of n=0 modes which will be shown later.

In Figure 5.2, the dielectric parameters of sub-region 1 are: relative permittivity

e, , permeability 111 and conductivity 0', , and its radius is b. The parameters of sub-region
2 are: relative permittivity £2 , permeability 112 and conductivity 0'2 with radius a. Based

on the relations between the longitudinal and the transverse components of the
electromagnetic fields given in [10], we obtain the electromagnetic eigenmodes in these

two sub-regions when n at 0 as follows:

In sub—region 1, the electromagnetic eigenmodes can be expressed as

187

—r
E:l = Anmk311n(kclr)cosn06 "”2

—1‘ z
1L1zl = BnmkfilJn(kclr)sinn0e ""'
- . -rnm:
15,l = [_Jmplanmir'1n(kor)—Anmr,ka,J,,(kclr)]cosn0e

Z

. , n . 4“,...
1391 = [Joulamkcljn (kclr) + Anmrnm;1,(kc,r)]31nn0e

1 . - -rnm
H,l = [-Bnml‘nka1Jn(kclr)—jmeclAnm;Jn(kclr)]smn0e z

rt . , 4‘”:
H61 = [—Bnm1‘nm;1n(kclr)-j(1)£ClAnkalJn (kclr)]cosn0e

In sub-region 2, the electromagnetic eigenmodes can be expressed as

13,, = [CnmJn(k,2r) +0 Y (kczr)]k:2cosn0e-r""z

nmn

H32 : [Enm‘ln(kczr) '1' F Y (kCZr)]k3251nn6e-r"”z

nmn

. n
Er2 = {—jwu2;[EnmJn(kc2r) + anYn(kc2r)]—rnmkc2

-F z
[CnmJn'(k,2r) + Dnm Yn'(kczr)] }cosn0e ""'

E02 = {jwu2kc2[Enm‘ln'(kc2r) + F Y '(kc2r)] + Firm;

nmn

II m N

[Cmnln(k,2r)+D Y (kczr)]}sinnGe-rmz

188

(5.17)

(5.18)

(5.19)

(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)

(5.26)

Hr2 : {_rnmkc2lErtrn‘ln.(k02r) + anYn.(k62r)]—jw€c2; (527)

[CnmJn(k,2r) + Dann(kC2r)]}sinnee_r""'Z

n .
H02 : {—rnm;[EnmJn(k02r) + anYn(kc2r)]_-lm£c2kc2 (5'28)

[CnmJn'(kC2r) + Dnm Yn'(k,2r)]}cosnOe-r”z

where

2
kc, = (021116,, +172", (5.29)
2 2 2
kc2 = (D “28c2+rnm (5'30)

0'.

Applying the boundary conditions to these eigenmodes, we can determine the
unknown coefficients Anm, Bnm, Cm, Dnm, Em and FM. in eqs. (5.17) to (5.28). The
boundary conditions are as follows:

(1) Ezz(r= a): 0
CnmJn(kC2a)+Dann(kcza) = 0 (5.31)

(2) Eo2(r= a): 0

ang[CnmJn(kC2a)+D Y (kcza)] + (5.32)

nmn

jwl'l2kc2[EnmJn'(kc2a) + anYn'UchaH = 0

189

Based on eq. (5.31), eq. (5.32) can be rewritten as
EnmJn'(kC2a) + anYn'(kC2a) = 0

(3) Eol(r= b' ) = Bozo-_- b" )

jwuanmkcl‘ln'(kclb)+Anmrnmgjn(kclb) = jmu2kc21-E J .(kCZb)

nmn

+ anYn'(kc2b)] + anE[CnmJn(kc2b) + Dann(kc2b)]

(4) 5,10: 6‘ ) = 5,20: 6" )

2

2
AnmkclJn(kclb) = [CnmJn(kc2b)+Dann(kc2b)]kc2
(5) H910: b‘ ) = H920: b’ )
-Bnmrnmg‘ln(kclb)_jw€clAnmkclJn'(kclb) = —rnmg[EnmJn(kc2b)

+ Fm, 1909217)] — jmecsz2[CnmJn'(kczb) + D Y (11,219)]

(6) Hon: 6' ) = H320: b“ )

2 2
Bnmkcl‘ln(kclb) = [Enm‘ln(kc2b)+F Y (kc2b)]kc2

nmn

Combining eqs. (5.31) and (5.35), we can express Cum and Dnm in terms of Am

k2
“J k bY k
k_2' n( cl ) n( c261)

C = C2 A
nm Yn(k62b)‘ln(kc2a)_Yn(kc2a)‘]n(kc2b) nm

 

190

(5.33)

(5.34)

(5.35)

(5.36)

(5.37)

as:

(5.38)

k3.

Tjn(kclb)‘ln(kc2a)

D = k” A
nm Yn(kc2b)‘]n(kc2a) - Yn(kc20)‘]n(kc2b) nm

Let’s denote

2
k 1
—;—J,,(kclb)

C2

C :
CDA Yn(kc2b)~’n(kc2a) — Yn(kcza)J,,(kczb)

Equations (5.38) and (5.39) can then be rewritten as
Cnm = _CCDAYn(kCZa)Anm
Dnm = CCDAJn(kc2a)Anm
Using eqs. (5.33) and (5.37), we can express Em and Fun; in terms of Bnm as:
Enm = “CEFBYn'(kcza)Bnm
PM, = CEFBJ,,'(k,2a)B,,m

where C E F3 is denoted as

k2
l
—:-Jn(kclb)
C = C2
E” Yn(k,2b)Jn'(kC2a) - Yn'(kC2a)Jn(kC2b)

Substituting eqs. (5.41) to (5.44) into eqs. (5.34) and (5.36), respectively, we have

191

(5.39)

(5.40)

(5.41)

(5.42)

(5.43)

(5.44)

(5.45)

jwulB m kclJ (k l"mm mrnngnwclb) = jwu2k62[CEFBJn'(kcza) (5.46)

2
k
Yn'(kC2b)-CEFBYn'(kcza)Jn'(kC2b)18m+ "A Jncluc b)

nmb nm Tl
c2

[(21
1J(kb)=nl9—l,.611(kb)

J‘nmb nm 2
CZ

_ jmecch2(_CCDA Yn(kc?.a)‘ln'(kc2b) + CCDA‘In(kc2a)Yn'(kc2b))Anm

_anm nmb ‘Icn(k lb)-jw€cl Anmk c

(5.47)

Equations (5.46) and (5.47) can be rearranged as

2

kC - v I
Aannm%1n(kclb)[l -171] = jw{u2kCZCEFB[—Yn(kcza)1n(kczb) (5.48)
c2

+ Jn'(kc2a)Yn'(kc2b)] — “lkcljn'(kclb) }Bnm

2

k
_I‘Bnm nmb Jn(kc] b1)[ ___C'l'] = jm{8clkcl‘ln'(kclb) (5'49)
c2

_ Ec‘ZkCZCCDA[Jn(kc2a) Yn'(kc2b) _ Yn(kc20)‘]n'(kc2b)] }Anm

Equations (5.48) and (5.49) can then be rewritten in a matrix form as

MIIMIZ Anm ___ O (5.50)
M21 M22 Bnm 0

where

2
k

Mll = M22 =_1“nmEJn(k b)[l—k—:1] (5.51)
CZ

192

M12 = jml‘zkczcEFBl‘Yn'(kcza)Jn'U‘czb) +Jn'(kcza)Yn'(kc2b)l

. ' (5.52)
_jwl‘llkcl‘ln (kclb)
M2] = jwecdeCCD/i[_Yn(kc2a)‘]nl(kc2b) + Jn(kc2a)Ynl(kc2b)]
. ‘ . (5.53)
- Jweclkcl‘ln (kclb)
To have non-trivial solutions for A m and Bnm , it is necessary that
M11M22"M12M21 = 0 (5.54)

i.e. the determinant of the matrix in eq. (5.50) is zero. Therefore, we obtain the

characteristic equation for the eigenmodes as

2 2
kC
c2

— Jn.(kc2a) Yn'(kc2b)] + “Ikcl‘ln'(kclb)}{ec2kc2CCDA[Jn(kc20)Yn'(kc2b)
— Yn(kc2a)‘ln.(kc2b)]_£clkcl‘]n'(kclb)}

Substituting eqs. (5.40) and (5.45) into eq. (5.55) and using the relations between

kcl, kc2 and Fm given in eqs. (5.29) and (5.30), we can numerically obtain the
propagation constant PM for each eigenmode and then determine the corresponding
eigenvalues kcl and ka‘ After that, based on eqs. (5.41) to (5.44) and (5.48) we can

express the other five coefficients in terms of A n m as

193

’1 kil
b k
B = CZ

"m k
jwkc1{lliI‘LLCBAJchlellzjnlkcib)}
C

 

nm

where

_ Jn'(kcza) Yn'(kC2b) — Yn'(kcza)Jn'(kC2b)
BA ’ [Yn(kczb)1n'(kcza) — Yn'(kcza)Jn(kC2b):l

and

C _ Jn(kclb)Yn(kC2a) k2

nm Yn(kc2b)1n(kc2a) - Yn(kc2a)Jn(kc2b)k—:2

 

Jn(kclb)Jn(kC2a) k2

cl

D = —A
"m Yn(kC2b)Jn(kc2a)—Yn(kcza)Jn(kC2b)k:2 "m

 

2

c1

E Jn(kclb)Yn'(kC2a) k

 

nm Yn(kc2b)‘lnl(kc‘2a) — Yn'(kcza)Jn(kc2b)l-CC72 "m

Jn(kclb)Jn'(k62a) k2

 

F = - ' _
nm Yn(kc2b)1n (kcza) - Y" (kcza)Jn(kc2b)k:2

cl
Anm

cl
Bnm

(5.56)

(5.57)

(5.58)

(5.59)

(5.60)

(5.61)

Therefore, we can derive the expressions for all the eigenmodes in the inhomogeneously

filled waveguide if we can find the coefficient Am in region II.

When n=0, the eigenmodes in region 11 can be either TM or TE modes. For TM

modes, the eigenmodes in sub-region 1 can be expressed as

2 -F M:
Ezl = AOmkclJ0(kclr)e 0

194

(5.62)

Er] = _AOmr 10(kclr)erz0m

0m kcl
- o —r0mz
”91 = _Jmeclemkc1J0(kclr)e

The eigenmodes in sub-region 2 can be expressed as
2 .1-‘0mZ
5:2 = [ComJ0(kc2’) + D0mY0(kc‘2r)]kc2e

_r m
Er2 : _FOmkc2[C0mJO’(kc2r) + DOmY0'(kc2r)]e 0 z

-r0mz

H92 = -jm£C2kCZ[COmJO'(kc2r) + DOm Y0'(kczr)le
Applying boundary conditions to eqs. (5.62) to (5.67), we have
C0m10(kcza) + D0mY0(kC2a) = 0
AOmkil‘IOUcclb) = [C0m‘10(kc2b) + Do»: Y0(kC2b)lk:2
EclAOmkcljolkclb) = €c2k62[C0mJO'(k62b) 4' DOm Y0'(kc2b)l
Combining eqs. (5.69) and (5.70), we have the following relation of

2 2
[210(kC2b) + k_c_2 Y____O(kc2b)
omkil‘10(kclb) om—k21‘10(kclb)

 

8(‘2kc2‘l0'(kc2b) + ec2kc2YO'(kc2b)
kc110.(kc1b) 0maclkcl JO.(kclb)

 

 

01'

[kc210(kc2b) ___E_c2]0'(kc c2:)] m[k:2Yo(kc12:)€ ec2Y0(kc2b)
0m

k 12110“ b) e ,‘CJO(k b)

10(k 1?) Ecl‘10(kCl b)

(5.63)

(5.64)

(5.65)

(5.66)

(5.67)

(5.68)

(5.69)

(5.70)

(5.71)

Combining eqs. (5.68) and (5.72), we obtain the following characteristic equation for the

195

TM modes as

kc2 Y_0___(kc2b) _ECZ Y0'(kc2b)
““62“ lk— 10(k b) 2115062)]
C21,,(kC b) e C21,,(kC din] _

Y0(k62“)[17,10(k:b)“e—JoucC b)

and the unknown coefficients can be expressed in terms of C GM as

J0(kcza)

D = -———
0m Y0(kC2a) 0m

 

m - 0m?

2

For TE modes. the eigenmodes in sub-region 1 can be expressed as

HC, = B 10(kC, 0.24““Z

0m kcl

10' (kc ,r)e’r""'z

Omk cl
E61 = jwl'llBomkc1J0(kc1r)eFonz
The eigenmodes in sub-region 2 can be expressed as

2 “FOM:
H 7 : [E0m10(kc2r)+F0mY0(kc2r)]kc28

“r0":

HrZ z _r0mkc2lE0mJO.(kc2r) + FOmY0.(kc2r)]e

E92 ‘ jwl'lzkczlEomjo(kcggr)+1"'0,,,Y0'(kC2r)]er""‘~

Applying boundary conditions to eqs. (5.76) to (5.81), we have

196

(5.73)

(5.74)

(5.75)

(5.76)

(5.77)

(5.78)

(5.79)

(5.80)

(5.81)

ulBOkaIJO'(kC,b) = psz2[EOmJO'(kC2b)+FOmYO'(kC2b)] (5.83)
2
BOInk c1J0(kcl b) = [E0m‘10(k62b)+F0mY0(kc2b)]kc2 (5'84)

Using eqs. (5.83) and (5.84) we have the following relation of

. . 2 2
“2kc210(kc2b) + “zkcz Yo(kc2b) _ ’:c_210(’%2b) + ’fc_2_Yo(kc2b)
omlllkcijo'(kc1b) omllikcljo'(kc1’) 0mk3110(kc1b) omkiflou‘cib)

 

 

(5.85)

01'

[it—210'(kC2b) _k_C_21_0___(kC2b)] OMPYO'CC(k2b)__122_1/,,(k215)]_ o (586)
0'" ”110(kc1b) kcl‘]0(kc b) ”110(kc1b) kc 1cJo(k 1’) .

Combining eqs. (5.82) and (5.86), we obtain the characteristic equation for the TE modes

as

 

Y'k b k Y k b
Jo'(kcza)[l2‘2.‘(—Cz_‘)"—C_2M:l (5,37)
“110(kc1b) kcl‘IO(kclb)
J k b k J k b
_ Yo'(k.aa>[&—‘i(—-C—2-—)-—2—O(—2—,C)] _
‘ JO’C(k b) k 10(kC b)
and the unknown coefficients can be expressed in terms of EOm as
JO'(kC2a)
= ___—___ 5.88
0m YO'(kCZa) 0m ( )
[<32 JO(kC2b) JO'(kC2a)YO(kC2b)
BOm : Orn_2—[— — v ‘- :l (589)
kC,Jo(kc1b) Yo(kcza)10(kc1b)

197

5.1.3 Normalization of Homogeneously filled Waveguide Eigenmodes

The normalization of the homogeneously filled waveguide eigenmodes can be

realized by the relation of

[2.1-221.15 = 1 (5.90)
CS

where 2n1(r,6) is the transverse component of the eigenmode in region I or III and the

integration region is over the cross section of the waveguide.

For TE mode, using eqs. (5.12) and (5.13) the normalization leads to

a21t ,

2 ' ' 2
-(0)p.Anm)2JJ'{|:'—IJR(E£T-r)cosn6] +[flm1n.(an rjsinne] }rdrd9 = 1 (5.91)
00 r a a a

 

when n at 0. Equation (4.65) gives us the integration of

p...’

2
_[ ["71,z,(r)+1,;2(r)r]dr -_- %(pnm'2—n2)li(pnm') (5.92)
0

After integrating over the variable 9 and changing the integration variable r, eq. (5.91) can

be rewritten as

pnmI 2

 

—(qu,Cm)2n j [”7122C(r)+(1n'(r))2r]dr = 1 (5.93)
0
Using eq. (5.92), we have
A... = if 1 (5.94)

It , ’ ,2 2
(0111,2(an) pnm -n

198

When n = O , the normalization is given by eq. (5.16) as

a

—(qu0m) 21:] [@110 (@flrjfrdr = 1 (5.95)

Equations (4.70) and (4.72) give us the integration of

.2

POM. p
[O JO'2(r)rdr = 0; 13(p0m') (5.96)

 

Therefore, the normalization constant is given by

 

. l l
A = f— , , (5.97)

Combining eqs. (5.94) and (5.97), we have

8
Am = ,- fl 1 (5.98)

1C . ,2 2
wu1n(p,,,,, )Jan -n

where 80" is defined in eq. (2.63).

 

For TM mode, using eqs. (5.2) and (5.3) the normalization leads to

 

 

0211:
2 2
(A )2]! anJ, p_,_,,,, rcosnG + ’3] anr sinnG rdrdG = l (5.99)
nm I..nm 00 n a r n a

when n ¢ 0. Equation (4.80) gives us the integration of

”mu 2 2
j [n7li(r)+1n'2(r)r]dr = [L'Z’LJH'RPW (5.100)
0

After integrating over the variable 9, changing the integration variable r and using eq.

I99

(5.100), we obtain the normalization constant A "m as

efl 1
1‘ rnmpnm‘ln'(pnm)

 

(5.101)

nm—

5.2 Electromagnetic Fields in Three Regions

Up to now we have obtained the eigenmodes in each region shown in Figure 5.1.
The electromagnetic fields in each region can then be expressed as the infinite summations

of the eigenmodes in the corresponding region.
Because we assume that there is an excitation probe in region I, the electric field in

region I for 0 S z S zl can be expressed as
310) = j 300) . Z300, ;)dV0 (5.102)
V.
where the dyadic Green’s function is given by [10]

N1
— x ; —enl(r0’90) 4+ x A- x
G(r0, r) .. 212(1—R1nR2n)(1+R‘")Z"( E") (r) +R2nEnl(r)) (5.103)

 

for :. 2 0. 3n.(r0,00) is the transverse component of the eigenmode in region I which is
A: . . .
normalized by eq. (5.90). En 1(r) is the eigenmode propagating in iz direction as
4+ s .5 A —I‘,,z
Enl (r) = [en1(r,0) + zezn1(r,0)]e (5.104)

A- x a. Fz
EnlU’) = [en1(r,0)—Eeznl(r,0)]e n (5.105)

R1" and R2" are the reflection coefficients of the nth mode due to the short-circuit

200

termination at z=—z0 and the discontinuity at z=z1 in region I, respectively. Therefore,

41“
R1" = —e "Z° and R2" is unknown. The wave impedances for the TE and TM modes are

expressed as

 

 

nTE 1""
r"
an _ jme (5.107)

where 1“" is the wave propagation constant of the nth mode and is given by

2 2 2 . . .
F" = kcl ~00 [.1181 With kcl as the e1genvalue of the eigenmode, 1.11 and 81 are the

dielectric parameters of the medium in region I. The upper summation limit N1 is set to

assure a convergent result. The current density on the excitation probe is assumed to have

a sinusoidal distribution as

-‘ s _ ,. sinB(l—a+r)
1(r) — rim sinBl 5(0)5(z) (5.108)

 

where B is the wave number in the medium of region I, l is the length of the excitation

probe.
Rewriting the dyadic Green’s function (5.103) as
NI 5
— x 3 -3 1(r ’6 ) ‘4' ;
G(r0, r) = 2 —1—2—‘l—°(1 +Rln)Zn E,” (r)

,1:

(5.109)

 

"enl( 70980) ->-+ , —x— x
+ ”212(1 -R1nR2n)(l +R1n)R2nZn[R1n E"1(r)+E"1(r)]

and substituting eq. (5.109) into eq. (5.102), we can obtain the electric field in region I as:

201

NI N1
.5 x r" A-1- ; 4“,, A S '3'- x
E1(r) = 2 Vne z‘ En1(r)+ 2 Ane z'[R1n 51:1 (r)+En1(r)]

 

 

n =1 n =1
where
1+ R —r": 5 A 5
v" = — 2 ‘"z,,e ' j 1e..1(r0.eo)-J(ro>1dvo
is known and
1+Rl F Z A A
A = _ n R Z n I . 3
n 2(1—RlnR2n) 2n "8 J[8n1(r0,90) J(f0)]dV0

(5.110)

(5.111)

(5.112)

is unknown due to the unknown reflection coefficient R2". The magnetic field in region I

can also be expressed as

N1 N1

-‘ 3 1‘": —“ .x -F,,:, -‘ ; —"" s.
Hm) = z Vne ' 11:10).» XAne (R1,, H:1(r)+Hn1(r))

"=1 n=l

where
4+ 8 A _r‘nz
Hnl (r) = [lznl(r,0)+2hznl(r,0)]e

F1210) = {—11.100)+2h,,,.<r,e>1er"z

In region II, 21 S z S zz, the electromagnetic fields can be represented as

M
A 3. I'm: 4+ s 4“,": A- s
520) = 2 [Bme ' Em2(r)+Cme 215”,2(r)]
m=l
M
A 5 rmil 4+ 3 -rmzl_‘_ A
H2(r) = 2 [3m Hm2(r)+Cme Hmz(r)]
m=l

202

(5.113)

(5.114)

(5.115)

(5.116)

(5.117)

A: k A: 3 . . . .
where Emz(r) and Hm2(r) denote the electromagnetic eigenmodes propagating in iz

direction derived in Section 5.1.2.

In region 1H, 22 S 2 S 23 , the electromagnetic fields can be represented as

N:

_x x F": A4. 3 A... 3

E30) = 2 one 21 En] (r) +RnEnl(r)] (5.118)
n = 1
N2 r

_x 3 n 2 _\+ 3 .3- s.

H3(r) = Z Dne z[ Hnl (r) +Ran1(r)] (5.119)
n = l

-21‘,,c , . . . .
where Rn = —e is the reflection coeffic1ent at the termination of z = Z3.

After the total electromagnetic fields in the three regions are found, we can express

the transverse components of the electromagnetic fields in each region as:

3110) = 2 Vac-”H" 2,,(r,0)+ (5.120)

n=l
N1

-Fn(:+2:(+z) I‘M-z):
244—62 ‘ ’ +e ')e...<r.e)

n=l

N.
A x —F 2"2] 5
H110) = 2 Vne "‘ ’ h..1(r,0)+ (5.121)

n=l
N1

—l‘,,(; +22. +3) 1“,,(z-z ) A
Z An(—e ' ’ -e ' )hn1(r,0)

n=l

A 3 -F : : F 2‘21 3

52.0) = 213"; "" "+Cme "" ‘)]em2(r,0) (5.122)
m=l

A s -r,,,:-z. rmz-z, -‘

112.0): 213,.6 ( )—C,,,e ‘ ')]hm2(r,0) (5.123)
m=1

203

N.
A 3 - —r z— 7 r Z -2:} A
1331(1) = 2 Dn[e "( 2-’—e "(3" 2 -’]e,,.(r, 0) (5.124)

n = 1
N:
A 3 —r,,( —ZZ r" Z 22 - 23‘) A
H310) = Z Dn[e Z ’+e ‘ I ‘1h..1(r, 0) (5.125)
n = l
The boundary conditions at the junctions of the different regions are the continuity

of the transverse components of the electric and magnetic fields. Therefore, at z = zI , we

have

NI NI
2 vn2n1(r, em 2 An(-— e'zr"“'”"’+ 1)2,,1(r, 0) (5.126)
n =1 n =1 '
M
= 2 [3m + C,ner"'“"z2’]2,,,2(r, 0)
m=l
NI NI -\
2 vni‘znl(r,0)+ 2 An(— e'zr"“'”"’— 1)h,.1(r,0) (5.127)
n =1 n =1
M - A
= 2 [Bm- Cmermh'flfllhmzu, 0)
m =1

At 2 = z2 , the boundary conditions can be expressed as

N2
_r z‘,_ I 5 "Za- S
2 [Bme ”"* “+Cm1em2(r,0)= 20,1142”- z3)]e,,1(r,0) (5.128)

m=1 n=l

N2
‘FM 3 ‘2) A 2r” 21—23 A
2 [Bme (2 ' -Cm]hm2(r,0)= ZDnIHe ‘- )]hn1(r,0) (5.129)

m=1 n=l

Since the electric eigenmodes in the homogeneously filled waveguide are orthonormal.

that is, they satisfy

204

 

 

 

 

, I 1 if n = m
je,1.emlds = , (5.130)
CS 0 1f n¢m
l .
.x A —i If n = m
jhni ~hmids = 2,, (5.131)
CS 0 if n¢m
Thus, eqs. (5.126) to (5.129) can be changed to
M l‘ (’ )
2 [Bm + Cme m ”-22 112121 ' 2mZdS
A _ _ Vn m=l CS (5 132)
n — _zrn(:l +20) -2rn(:l +10) .
l—e l—e
2 M
V" Zn rm(31—32) A A
An = —2Fn(z.+z.,)- -2F,,(z1+zo) 2 [BM—Crne 11h“ ‘hmZdS (5'133)
l + e 1 + e m :1 CS

M
-r,,,(Z -Z) A s
2 [Bme ’ ' +Cmijen1.em2ds

CS

 

 

1),, = 2002-2.) (5.134)
1—e
M —r (~ —-1
2,2, 2 [Bme " -Cm]
_ m=1 " "
Du - 2r.<z2-z.) jhni ~hm2dS (5.135)
1+8 CS

Equations (5.132) and (5.133) are valid for n =

1, 2, ...... ,N1 and eqs. (5.134) and

(5.135) are valid for n = 1, 2, ..., N2. Let’s denote
emn : I 2III ' 2’71st (5136)
CS
hm" = [2,.) . Zmzds (5.137)

CS

205

then based on eqs. (5.132) and (5.133), we have

M 2 RM ‘12)
z emn + Znhmn B + e emn
-2r,(;. + :0) -21’,,(z. + 30) m -Zl‘.(zi + 30)
m = 1 l—e l + e e

 

 

 

 

 

 

 

 

1_
r ) (5.138)
zfie ""2"“ hm 2v"
_ I + e-Zrn(zl + 30) m 1_e-4rn(zl + Z0)
for n = 1, 2, ...... , N1 . Based on eqs. (5.134) and (5.135), we have
M 2
2 emn _ Znhmn e‘rm(32‘~1)B
2rn(32-Z3) 2rn(~’-2‘Zs) m
m =1 1" e I + e
2 (5.139)
[ emn Znhmn :IC J O
+ + =
1 _ ezrn(‘-2—”3) 1 + ezrn(‘-2—Z3) m
for n = 1, 2, ...... , N 2 . Equations (5. 138) and (5.139) are expressed in a matrix form as
[BMllleMlBlMxl + [CMllleM[C]Mxl = [VSllel (5.140)
[BMZIN2XM[B]MXI+[CM21N2XM[C]MXI = [0111/in (5.141)

where the elements in each matrix defined in eqs. (5.140) and (5.141) are expressed as

 

 

 

2
_ em" Znhmn

BMlnm '- -2I‘,,(z. + :0) + —21",,(z( + 3.0) (5142)

[_e I + e

rm(Z|-22 2 rm(zl-ZZ)
CM _ 8 mn _ Zn" hm" (5 143)
lnm — —2r"(zl + 20) -2r,,(zi + 30) .
l—e I + e
2V
_ ’1
VS" — ] —4rn(:l+Z()) (5.144)
-e

206

forn=1,2, ...... ,Nlandm=1,2, ...... ,M.

 

 

 

 

e Zzh 1‘ ( )
__ mn _ n mn ' m Zz’zi
BMan — [ 2F..(z2-~3) 21."(ZZ_23):|e (5.145)
1 -e 1 + e
2
_ emn - Znhmn
CMan — 2Fn(22-23)+ 21“,,(z2—z3) (5.146)
1—e 1 +e
for n = 1,2, ...... ,N2 and m = 1,2, ...... ,M. The unknown coefficients can be
expressed as
T
[3],“x1 = [B1,32, ...... BM] (5.147)
T
[C]MX1 = [C1,C2, ...... CM] (5.148)

If we choose N1 = N2 = M, the matrices [8M1],[CM1], [3M2], and [CMZ] in eqs.

(5.140) and (5.141) are square matrix and the solutions for Em and C m can be expressed

as
[3],”, = [[BMll—[CM,][CM2]-I[BM2]]—1[VS] (5.149)

[C]Mx1 = [[CMI] — [BMI][BM2]_][CM2]]—1[VS] (5.150)

Therefore, we can determine the electromagnetic fields in region 11 after the unknown

coefficients Bm and Cm are obtained from eqs. (5.149) and (5.150). Substituting the
solutions for Bm and Cm into eqs. (5.132) or (5.133) and (5.134) or (5.135), the solutions

for the unknown coefficients An and D" can be found and then the electromagnetic fields

207

in region I and 111 can be detennined.

5.3 Numerical Example

In this section, the numerical results based on the mode-matching method will be
demonstrated. As stated before, we restrict the material samples which are placed in a
cylindrical waveguide to be of simple cylindrical shapes and homogeneous. The numerical
results obtained in this chapter will be compared with the corresponding results shown in
Section 4.4 in Chapter 4.

In the numerical computation, the eigenmodes in region II are derived first after
the dimensions and the dielectric parameters of the material sample are selected. The
integrations of product of the eigenmodes in regions I and II (given in eqs. (5.136) and
(5.137)) are then calculated. After that the unknown coefficients needed in eqs. (5.110).
(5.113) and(5.l 16) to (5.119) are obtained from eqs. (5.149), (5.150), (5.132) and (5.134).
Equations (5.133) and (5.135) can be employed to check the validity of the numerical
solutions for these coefficients. After the total electromagnetic fields are obtained and their

numerical results can be checked further by the boundary conditions at the perfectly

conducting walls and at the junctions of the different regions.

Examining the matrix eqs. (5.140) and (5.141), we find that when VS" = 0 for
some indices n, the solutions for Em and C m become zero also. Thus, we can select the

eigenmodes based on both the values of VS" and the convergence property of the

summation for the electromagnetic fields in each region.
In the following numerical computations, we assume the dimensions of the

cylindrical waveguide shown in Figure 5.1 to be: the radius a=0.0762m and the length

208

c=0.15458m. A cylindrical material sample with the dimensions of radius r0 and length ho

is placed in the center of the waveguide. The position of the excitation probe is at c/4 from

 

the bottom of the waveguide, that is, 20 = g and 23 = 3 I C. The values of b, 2, and 22
. . . . 6 ho
are deterrruned by the dimenSions of the material sample to be b = r0 , 21 = 2- 3- - zo

h
and 22 = g + 30 —— 20. The length of the excitation probe is chosen as the half of the radius

of the waveguide, a/2, and the operating frequency is 2.45 GHz. The relative permittivity
of the material sample is assumed to be e, = 2.5 and it is lossless.

Several numerical calculations are carried out for the material sample with selected

shapes and dimensions.

1. Cubic material sample

A cubic material sample, having the diameter equal to the length, is placed in the
center of the cylindrical waveguide. The dimensions of the material sample are chosen as:

radius r0=0.004m and length h0=0.008m.

In the numerical computation, the number of modes to be summed is set to be 62
based on the non zero values of the right hand side of the eq. (5. 140). However, observing
the numerical results for the solutions of the unknown coefficients A n and D” in regions I
and III, we find that there only exist several waveguide modes with significant magnitudes
which are shown in Table 5.1, and the TM01 mode appears as the dominant mode. This

result is expected because of the choice of the dimensions of the cylindrical waveguide

and the operating frequency of the excitation probe. The computational results show that

209

Table 5.1 Significant modes in the mode-matching method when the dimensions of
the cavity are: a=0.0762m, =0.15458m and that of the material sample are:
r0=0. 004m and h0=0. 008m. The operating frequency is 2.45 GHz, and the excitation

probe is located at c/4 from the bottom.

 

 

 

 

 

 

 

 

 

 

Mode An(real, imaginary) Dn(real, imaginary)
TMO, -79.1583, 11.9465 78.8164, -l4.0405
TM02 0, -0.5485 0, 0.3906
TM03 0, 0.8847 0, -0.8946
TMO4 0,-1.1825 0,1.1827
TM05 0, 1.4291 0, -l.4276
TM06 0, -l.6l62 0,1.6144

T M07 0, 1.7506 0, -1.7483
TM], -O.1651,0.1809 0.1798, 0.1671
TE 1 2 -3.86l8, 2.3057 4.4303, 0.7882

 

 

 

 

 

210

the electromagnetic fields in each region do not vary significantly with respect to the

variable 0 and the 2 component of the induced electric field dominates the other two
components of the electric field near the center of the waveguide.

The validity of the numerical results are checked by the continuity of the tangential
components of the electromagnetic field at the junctions of the different regions and the
boundary conditions at the cavity walls.

Figure 5.3 demonstrates the ratio of the 2 component of the induced electric field in
the material sample to that of the electric field near the material sample in the empty
region of the waveguide as a function of z at r=0.0004m. The ratios are around 0.69 to

3

0.72 which are close to 0.667 given by the electrostatic estimation of Ey/E; = 2—-_1-—t>:_ .
I’

and they are consistent with the results found in Chapter 4.
For a larger cubic material sample with the dimensions: r0=0.01m and h0=0.02m,
the number of the modes is set to be 76. The most significant values for the unknown

coefficients An and D" in regions I and III still belong to the TM01 waveguide mode.

Therefore, the 2 component of the induced electric field dominates near the center of the
waveguide, and Figure 5.4 shows the ratio of the 2 component of the induced electric field
in the material sample to that of the electric field in the empty waveguide near the material
sample varying as a function of r at the different locations of 2. Due to the symmetric
property of the numerical solutions, we only plot the ratios in the lower half of the material
sample in Figure 5.4. Comparing the results of Figure 5.4 with that of Figure 4.9 in

Chapter 4, we can see a good agreement between these two sets of numerical results

generated by two different methods.

211

0.8 T 1

 

0.75 -

0.65 '-

0.6 '-

Ratio of E/ z"

0.55 '-

0.45 -

 

 

l l 1 1 l l l

.4
0.034 0.035 0.036 0.037 0.038 0.039 0.04 0.041 0.042
f

 

Figure 5.3 Ratio of Ez/Ezi varies in the z direction at r=0.0004m. The dimensions of
the material sample are r0=0. 004m and 110:0. 008m with the relative permittivity of

e, = 2.5 . The dimensions of the cylindrical waveguide are: a=0.0762m and
c=0.15458m. The operating frequency is 2.45 GHz.

212

 

0.68 1 1 1 1 1 1 1

0.66

0.64

0.62

0.6

0.58

Ratio of (5,1 1‘

0.56

0.54

0.52 '

 

 

 

0.5 I l 1 l l l 1 l I
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
I'

 

Figure 5.4 Ratio of EZ/Ezi varies in the r direction at the different locations of z.
The dimensions of the material sample are r0=0.01m and h0=0. 02m with the relative

permittivity of e, = 2.5 . The dimensions of the cylindrical waveguide are:
a=0.0762m and c=0.15458m The operating frequency is 2.45 GHz.

213

2. Thin chip case

A material sample with the shape of a thin chip, having its length much smaller
than its diameter; is placed in the center of the cylindrical waveguide. The dimensions of
the material sample are h0=0. 001m and r0=0.02m and the number of the modes which are
involved in the computation is 55. The numerical results are shown in Figure 5.5.

Since the 2 component of the induced electric field dominates near the center of the
waveguide, the ratios of the 2 component of the induced electric field to that of the electric
field near the material sample in the empty region of the waveguide are plotted as a
function of the radial distance, r, in Figure 5.5. We observe that the numerical results are

close to the theoretical estimation given by the boundary condition of
E = (i/e,)E‘ = 045’.
3. Thin pipe case

A material sample with the dimensions: the length h0=0.044m and the radius
r0=0.004m, is placed in the center of the cylindrical waveguide. The number of the modes
involved in the computation is 129 and the numerical result is shown in Figure 5.6.

Examining the numerical results for the solutions of the unknown coefficients A n

and D" in regions I and III, we find that those with the most significant values belong to

the TM01 waveguide mode, that is, the TM01 waveguide mode dominates in the empty

region of the waveguide. For this case, the induced electric field inside the material sample
should be approximately equal to the electric field in the empty region near the material
sample because the electric field in the empty region near the material sample or near the

center of the waveguide is dominated by the 2 component and it is tangential to the major

214

 

0.55 -

0.5

 

0.45

Ratio of Ez/Ez

0.4

 

0.35

 

 

0.3 4 I L 1 I l l l l
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
f

 

Figure 5.5 Ratio of E /E z‘ varies in the r direction. The dimensions of the material
sample are r0=0. 02m and h0=0.001m with the relative permittivity of e, = 2.5 .The

dimensions of the cylindrical waveguide are: a=0.0762m and c=0.15458m. The
operating frequency is 2.45 GHz.

215

 

1.1 e -

 

N
In
0'
'8 0.9 " -1
.9. 3
U
§
0.8 ~ ° 4
E;
Ez
0.7 - -
0.6 e b -

 

 

 

 

l I L I I 1 1 1

0.5
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06
Z

 

Figure 5.6 Ratio of E zIE2i varies in the z direction at r=0.0004m. The dimensions of
the material sample are r0=0. 004m and h0=0. 044m with the relative permittivity of

e, = 2.5 . The dimensions of the cylindrical waveguide are: a=0.0762m and
c=0.15458m. The operating frequency is 2.45 GHz.

216

part of the material sample surface, also the continuity of the tangential component of the
electric field at the material sample surface requires this estimation.

In Figure 5.6, we plot the ratios of the z component of the induced electric field to
that of the electric field near the material sample in the empty region of the waveguide
varying as a function of 2 at r=0.0004m. Most of the ratios are very close to l which is in
agreement with the theoretical estimation.

Considering the numerical accuracy and the computation time, we find that in the
mode-matching method the number of the modes to be summed can be reduced 3 great
deal when compared with the integral equation method. The most important reason for
this finding is that the eigenmodes used in the mode-matching method satisfy the
boundary conditions on the material sample and the cavity wall. (On the other hand, the
vector wave functions used in the integral equation method only satisfy the boundary

conditions at the cavity wall.)

The other reason for this finding is that in the mode-matching method, the angular

dependence of the eigenmodes, 2,”, 31112. It,” and hmz, on 0 is sinn0 or cosn0, and
many of the integrations of the scalar products of these eigenmodes given in eqs. (5.136)
and (5.137) become zero due to the orthogonality of the sinusoidal functions. Therefore,
the matrices [3M1], [CMl], [8M2] , and [CMZ] given in eqs. (5.140) and (5.141) are
sparse and the computation time can be saved greatly. In spite of this advantage for the

mode-matching method, it is not a very general technique because it can not be used to

solve the problem involving material samples with arbitrary shapes or heterogeneous

compositions.

Table 5.2 lists the number of eigenmodes used in the mode-matching method for

217

Table 5.2 Number of eigenmodes used in the mode-matching method for material
samples of different geometries

 

 

 

 

 

 

Geometry of material sample Number of modes
8-mm cubic material sample 62
2-cm cubic material sample 76
Thin chip case 55
Thin pipe case 129

 

 

 

 

218

the material samples of different geometries. From this table, we find that this number is
the largest for the thin pipe case. This confirms our finding that much more summation
terms are needed for the narrow strip case in Chapter 3 and the thin pipe case in Chapter 4
to secure accurate results. To overcome this shortcoming, we suggest the scheme of
separating the material sample into the boundary layer and interior regions to save

computation time while obtain better results.

219

CHAPTER 6

CONCLUSIONS

In this dissertation, both the integral equation method and the mode-matching
method are investigated to quantify the induced electric field in a material sample placed
in an EM cavity. It has been demonstrated that the integral equation method is more
powerful than the mode-matching method because the integral equation method can be
employed to solve the problem involving the material sample with any arbitrary shape or
heterogeneity while the mode-matching method can only handle the case of the
homogeneous material sample with a simple geometry. The only disadvantage of the
integral equation method is its slow numerical convergence and a large computation time
while the mode-matching method is more computational effectiveness. To our best

knowledge, this is the first attempt to solve this type of problem using the integral equation

method.

A
In the integral equation method, a complete set of vector wave functions anI,

M nml and N nml which include both solenoidal and irrotational functions are employed to
expand the unknown electric field in the material sample placed within an energized

cavity. After the electric and magnetic dyadic Green’s functions are obtained both EFIE

220

and MFIE are derived and they are shown to be equivalent.

Increasing the convergence rate of the dyadic Green’s function is a main concern
in solving the EFIE in order to obtain the electromagnetic field distribution in the material
sample. To achieve this goal, the infinite triple summations over the cavity eigenmodes are
reduced to the infinite double summations using the relations given in Collin [2]. The
infinite double summation is further divided into a finite double summation and an infinite
single summation using the well-known Poisson summation formula. However, this
division is only possible for the rectangular cavity case while the infinite double
summation is still used for the cylindrical cavity case because it is difficult to apply the
same division technique in the cylindrical cavity case due to mathematical complexity.

As numerical results demonstrated, the electromagnetic fields in the material
sample are strong functions of the geometry and the dielectric parameters of the material
sample. When the initial cavity electric field is tangential to the major part of the material
sample surface, the induced electric field in the material sample may be close to the initial
cavity electric field as required by the boundary conditions. For this case, the convergence
property of the dyadic Green’s function is very poor. To overcome this difficulty, the
scheme of separating the material sample into the boundary layer and the interior regions
are proposed for this special case. Satisfactory numerical results can be produced with this
scheme at a reduced computation time.

In this dissertation, the mode-matching method is also applied to the case of the
homogeneous material sample with a simple geometry and it is found that the mode-

matching method can save a great deal of computation time attributed to the use of the

well-defined eigenmodes and sparse resultant matrices.

221

APPENDICES

APPENDIX A

APPENDIX A

COMPUTATION OF DYADIC GREEN’S FUNC-
TION IN CAVITIES BY Y. RAHMAT-SAMII [11]

In [11] Y. Rahmat-Samii first obtained the magnetic dyadic Green’s function, then

he used the relation between electric and magnetic dyadic Green’s functions of
1126.00, ?) = V x 51100. 1) — 15(1— 10) (A.l)
to derive the electric dyadic Green’s function. In order to obtain the magnetic dyadic

Green’s function Gm(;0, i“), he introduced another Green’s function based on the

following definition:

(V2+k2)§m(?0, 7) = 45040) (A2)

and the boundary conditions for this Green’s function are expressed as

OP

(A3)

A — s S
n°gm(r02r) =

II
0|!

4 x V x 8.00. 1) (A.4)

on the perfectly conducting walls of a cavity. After applying Green’s theorem, he obtained

the magnetic dyadic Green’s function as

5,..(10, P) = [8,,0'. 1") . V" x 75(10 — ?")dv" (A5)

222

Therefore, in order to derive the expression for the Green’s function gmfio, ?) , we

should first solve eq. (A2) with the boundary conditions (A3) and (A.4) or eq. (11) of [3].

Equation (A.2) can be rewritten as

' xx)

gm 1
( +k )13111 = — 1 5(r—r0) (A6)
I

 

 

22
W .
1. The expression for gfnx is derived as follows:

Based on eq. (A.6), the equation for g: can be expressed as

(V2 + k2)gf,f = -5(?— i0) (A.7)

with the boundary conditions of

g: = 0 when x=0,a

agxx a XX

52'" =§;m=0 when x=0,a

agxx (AB)
55'" =0 when y=0,b

agxx

a—zm =0 when z=0,c

In order to derive the expression for gfnx, we may obtain the eigenmodes H n x(;) which

satisfies the equation,

(V2+k:)Hm(?) = 0 (A9)

223

and also satisfies the same boundary conditions as gfnx does.

Using the variables separation method and applying the boundary conditions

(A.8), the expression for H n Jr(F) can be found as

HMMG) = Anm,sin(Eg-tx)cos('-nb£y)cos(£:—tz) (A.10)

while the normalization factor is given by

8 8 8
Anml = ,/————-°"ag’: 0' (A.11)

and the eigenvalues are expressed as

kim, = (ii—tr + (mgr + Giff (A. 12)

. . . xx
That is, we have obtained a set of orthonormal eigenmodes Hnm,x(r) and gm can be

represented by the linear combination of these eigenmodes H n m ”(7’) as:
g;‘( ?— r0): ZXZamfiWm (A.13)
n m m
Substituting eq. (A.13) into eq. (A.7), we have
2 2 s x s
(V +k )XEXanmlHnth) = —5(r—r0) (A.14)
n m m
or
2 2 a s s
222%ka —knm,)Hnm,x(r) = —6(r— r0) (A.15)

nmm

224

Multiplying H p qrxG') on both sides, integrating over the cavity volume and using the

orthogonality of H pqrxf), we obtain the expression for the unknown expansion

coefficient am, in eq. (A.13) as

s s x —1 3
16(1 — rO)Hnm,x(r)dv = z—E—Hnmuoo) (A.16)

 

nml = 2
_ knle nml

Substituting eq. (A.16) into eq. (A. 13), we obtain the expression for g: as

 

gfnx(;-;0) : 222“___-___.kz1‘1"”!le;())I-InmIJC(;)
— knml
E 8 8
= 222 2 0" 0'" OlsinLEEx)sin(flxO) (A.17)
n m 1112- abc a a
klnm

cos(r—n—T—cv)cos(m )cos(-l£z)cos(l-T—cz )
b - b ’0 c c 0
2. The same procedure can be employed to obtain the expressions for g2: and gfnz. They

are expressed as

 

_222 8OnEOmEOI (n1!) (m: )
cos —x cos —xO
2 abc a a

ik-kzm

 

(A.18)
msin(mv)sin(ET—tv )cos(lit )cosC—Tt )
b ' ‘ b ' O c Z c “0
801150111301 mt m:
=222 2 k2"! abc COS 7x COS 7X0
n k —
’" ’ (A.19)

mcos(’flv)cos(m—-nv )sinCit )sinC-T—t7 )
b ' b '0 c2 c”0

Comparing eqs. (A. 17) to (A. 19) with eq. (26) in [l l], we find that there is a minus

225

sign missing in eq. (26) of [11]. Therefore, we can explain the discrepancy of the

expression for 56,000, P) which is specified in Section 2.2.5.

226

APPENDIX B

APPENDIX B

THE IDENTITY OF

ZZX[ZNMI(;O)Z""II(;)+finml(;0)finml(;)+A7nml(;0)A—;nml(;)] : i8(;_;0) IN
I: m l
RECTANGUALR CAVITIES

The identity which we need to prove is

222[Z"'NI(;O)ZNMI(;)+finml(;0)X/nml(;) +A7nml(;o)finml(;)] = i5(;‘—;0) (B.1)

nml

In Chapter2, we have derived the expressions for anl . M nml and Nnml which

are expressed in eqs. (2.17), (2.24) and (2.32) as:

Z 1 - film—T? nncos(n—x)sin(— )si n?“ z)
"m — knml a ys
+ iT—Tcsin('—11—tx)cos(——y)sin (3;!) (32)
b a C
.IR . 'nn ) . (n11: ) (In: )]
+z—sm —x sm — cos —z
c (a b c

-‘ .mn m: . m1: . l1:
Mnml — Bum/[firi-COS(7X)SID(T)’)SID(:Z)
+;fl..n(£1vx)cos(m_“)si (9%)]
a a by51 c

(3.3)

227

 

" Cnml . nnln mt . mn: . l1:
Nnml = [(-x)——cos —x sm —- 8111 —Z
k a c a b c
mnln . n1: 1: . l1:
-‘——— — —— — B.4
y b c sm(a x)cos( b y)sm(cz) ( )
+ “((EJZ + (at)? sin(flx) sin (m__1t )cos (it )
z a b a b y cz :l

where the normalization constants are given by eqs. (2.60), (2.62) and (2.65) as

E 8 8
Am, = /———°"ag’: 0’ (13.5)

and

 

 

 

 

 

_ _ ’80n80m801 l __ l
Bnml — Cnml— abc n“ 2 mu 2 " Anml n“ 2 m1: 2 (8'6)
H +(—) H +(—)
a b a b

Substituting eqs. (82) to (B.4) into the left hand side (LHS) of eq. (B. l), we can obtain its
nine components as follows:
1. Coefficient for 33c component is expressed as

(“‘vvvflv—v—vfivv m—vfit‘vvvrtv—Wr
km, a "'"l b k a c

nml

(’fl‘lz (MY l
b a c
+ _—

+
“2"“ (’1‘)2+(-"l2 k2 “balm—"Tl
_ a b nml a b _ (3.7)
(nnjzannjz ( 1t 2) 2 (mu)2 (mum)2
— — + — +knm, — + ——
A2 a a b b a c

 

 

 

 

 

 

 

 

228

Thus, the 23: component of the LHS of theeq. (BI) is given by

222/12 cos(flxjcos(flx )sin('—n—ny)sin(my )sin(E—tz)sin(g1—tz
n m l nml a a 0 b b 0 C C 0

'-'-' 6(;-;0)

) (B.8)

based on eq. (19) of [1 l].

2. Coefficients for Sci and if: components are expressed as

(::::Y("-.:—)('-%’-)-vimt%)(%v)+(i:::Y("-;)t-vg-v)(%f
2 (EXmTt) k1 1 ('5')

nml _

a b

 

 

 

2..., (’%‘l2+('%‘)2+vimz(("%‘)2+(’-’Z-‘)zl
.2 ,_, ,_, (afitv-avY-vzmwtaz
”lax” karate")?

Thus, the 5:52 and in? components of the LHS of eq. (3.1) are zero.

 

 

3. Coefficients for 22 and 252 components are expressed as

(2::If'alv'e-(i+::fev)%v(("§f("an
(am—“W
l — = O

a c (nnjz (mnjz
_ + _
L. a b _

Thus, the 522 and 23: components of the LHS of eq. (8.1) are zero.

 

 

 

_ (Anmzfflrgg
‘ k

nml

 

 

4. Coefficients for i2 and Zi'components are expressed as

229

 

 

(Avv>2'v—vv—v-<vava—vvfitef«an
knm, b c k b c a b

nml

(’1‘)2+(’-"1‘)2
_ (Anmljzmngg 1 a b

k

(8.11)

 

7. Kama)”

Thus, the i2 and 2 iv components of the left hand side of eq. (B. 1) are zero.

 

nml

5. Coefficient for iviv component is expressed as

 

 

ll
Lb

 

l .
"vzv'v’ (Hal—JET k3,,((§)2+("'7“)2) 03-12)

Thus, based on eq. (19) of [l 1], the ii component of the LHS of eq.(B.1) is expressed as

EXEAZ Sin({l—1-tXJSin(r_”_tvx )C05(my)cos(’fly )Sin(l-1EZ)SIII(EEZ
n m 1 mm a a O .b b 0 c c 0

= 5(;-;0)

) (3.13)

6. Coefficient for 2‘2 component is expressed as

230

 

1’23::)2(’—:-‘1+(f7"-::Y((%‘Y+('33))
(Ht—W
1"va ("7"Y+('%)2_

nml

 

 

 

 

Thus, the 22 component of the LHS of eq.(B.1) is

222.12 sin(vflxjsm(v21.vx )sin(myjsin(n_vv_vy )COSCEZJCOSCJEZ
n m l nml a a 0 b b 0 C c O

= 6040)

) (B.15)

Therefore, the identity given by eq. (8.1) has been proved after combining eqs. (B.8),

(3.9), (3.10), (3.1 l), (8.13) and (8.15).

APPENDIX C

APPENDIX C

AN ALTERNATIVE REPRESENTATION OF
THE ELECTRIC DYADIC GREEN’S FUNCTION

In Chapter 2, we have obtained the electric dyadic Green’s function (2.100)as

160— P0)
k3

(7.00, b = Geo(?o,?)- ((3.1)

where the detailed expression for 6,300”, ;) is given in Section 2.2.5 as

00

2

’* 15:2,; 2 [11%) +

=OabC(k2 — k0)

. mn . mn: In rm n1: n11:
s1n-b—ys1n-b—yosin— C ZSIII 'l—ZOXX + SIII—a—XSIn-Zl—XO

a

 

"TC
.) ):o._...._.,
a

COSmYCOSmny SIDEZSIIIIZFZ ’ (n ”)2 ++(-:n)2)sinfltx
b' b 0 c 0” a b a

sinmtx sin nysinmny cosmzcos 722 "nmncosnnxsin—nx
a 0 b' b 0 c “0 a b a a 0

sin—ycosmfly Sinlltzsinlnzx .. numnsinnnxcos'fl-tx €08,111; (C.2)
b b 0 C 0 xy (1 b a 0 b

sin v sin Czsin Cz ” m1t11tsinnxsinnnxcosmnysinmn:y
b '0 ny b c a a O b b 0

sin zcoslf: i2 m1t11t nsinnnxsinncfnr sinmn cosmny cosl—Z—cz
c 0 b c a O b y b 0

sinlfz “i nmnsin nxcos" nx sinmlysinmny cosmzsinlitr 25c
0‘ a c a a O b b O c c”0

mtln mr

———cos—xsin——xosin— ysm—yosm—zcos—zoxz
a c a a b b c c

232

and 80" is defined by eq. (2.63).

In Chapter 3, we have demonstrated that this series converges extremely slow. As a
general rule, if one series representation converges very slowly an alternative series

representation usually converges more rapidly.

We know the following relation which is given in [1] as

i cos(nx) = _1__lt_cos((x—n)a)
n=l nZ—az 202 2a sin(na)

. (In ) . (In ) (In) (In )
.. sm —z sm —z0 .... cos —z cos -zo
C C C C

If we evaluate the summations of z and Z
k2 k2 k2 k2
1 = I n — 0 l = l n — 0

 

 

O < x < 2n (C.3)

 

 

into closed form expressions, then the triple summations in eq. (C.2) can be reduced to
double summations to increase the series convergence.
. (In ) . (In )
.. s1n -z s1n —zo
c c

1. Evaluation of 2
=1 k “k0

 

i sin(%z)sin(l%tzo)— 1 i cos(%t(z-z0)) oo cos(é—Rzi-Zm)-
2 ‘ 2

 

 

 

(C.4)
2 2 2
=1 kn‘ko [=1 (13)-]? I: (.15) -k2
c 3 c 3 _
where
2 2
k: = 1.3%?) {Eb-’5) (C.5)

When z > zO, the summations in the right hand side of eq. (C4) are given by

233

1 ... cos(l-gk—ZOU .. COS(%E(z-zo))

_ = l .6. 2
(“1r .1): ,._(.,)z

 

 

 

 

 

_ " 1: 1 _.
c kg n
n c 2
_ ()2 1 . C°vllz<z-ZOv-vlakgl «26>
‘ 2 1": 2'
2(Ek ) 22" sin(gk n)
TC g TI: 8 n g _.

l _C_COS((Z-Zo’c)kg)
4k2 4kg sin(ckg)

 

 

 

g
In
1 a. cos(—C—(z+zo)) _ 1(EYE cos(?(z+zo))
2 __ In 2 2 - 2 TE _ 2 c 2
"‘ (‘5) ’kg "' "(#8) (c7)
1 c cos((z+zO—c)k‘g)

 

‘ 36—2-71}; sin(ckg)

. (In ) . (In )
C C

Substituting eqs. (C6) and (C7) into eq. (C.4), we evaluate 2 2 2 as
k - k
I: l n 0

 

. (In J . (In )
a, 5m -—z sm —z0
c c
2 2 .,

l=l kn—kO
1 _C_cos((z—zo-c)kg) [ 1 C cos((z+zO—c)kg)]

 

 

 

 

 

43:2; 4kg sin(ckg) LEV-17c; sin(ckg) (C.8)
C
— 4kgSIn(Ckg)[COS((~+Z0_c)k8)-COS((‘—ZO—C)kg)]
C . .
_ 2kgsin(ckg)sm(kg(c —.v.))51n(kg~0)

When 2 < 20', the first summation in the right hand side of eq. (C4) is given by

234

 

 

_ n cos (CC-[(20 —- z) — @7212)? (C9)

_ 112) 1 _
- 2 II c 2 C c
._ 2—k ' _
12( kg) n s1n(nk n)

 

 

 

TC

d

1 _C—cos((zO-z—c)kg)
4k: 4kg sin(ckg)

. (In ) . (In )
0.. 3m —z sm —zo
C C

and the evaluation of 2 2 2 leads to
l = 1 kn _ k0

. (In 1 . (In )
co sm —Z Sln _ZO
2 C C

2 2
[=1 kn-kO

1 _c_cos((zo—z-c)kg)_[ 1 I c cos((z+zO-c)kg)]

 

 

 

 

 

 

 

 

 

4k: 4kg Sin(Ckg) 4k: _ 4—kg sin(ckg) (C.10)
c
_ 4kgSin(ckg)[COS((Z + 20 _ CHE) _ cos((zo _ z _ 0kg“
c . . , 7
_ 2kgsin(ckg) s1n(kg.,)s1n(kg(c - ,,O))
Combining eqs. (C8) and (C. 10), we obtain the closed form expression of
. In . In
E s1n c z sm :20 c {sin(kg(c—z))sin(kgzo) z> 10 (C 11)
[___ ki_k(2) _ 2kgsin(kgc) sin(kgz)sin(kg(c—z0)) Z<Zo °
If we define

235

sin(kg(c—z))sin(kgz0) z>zO

fmn(z, 20) = {

 

 

 

 

 

 

 

 

 

 

 

, . (C.12)
s1n(kgz)s1n(kg(c—zo)) z<z0
then eq. (C.11) can be rewritten as
. (In) . (In )
2 c c = C f(zz) (€13)
1:1 ki-ké 2kgsi"(kgc) ’"" ’ ° ‘
(In) (In )
0.. cos ?z cos _c-ZO
2. Evaluation of 2
2 2
[=1 kn_k0
so cos(I—Ezjcos(-IEZO) co cos(lit(z-zo)) co cos(lit(z+zo))
2 C c — 1 2 C + Z 6 (C14)
2 2 ‘ ' 2 2 '
I=l kit—1C0 21:1 ({1}) —k2 [=1 (lit) —k2
c g c 3
where k: is given by eq. (C.5). Based on eqs. (C6), (C7) and (C9), we have
(In) (In )
0., cos —z cos —zo
2 c c
2 2
I: 1 kn _ k0
1 1 c cos((z—zO-cflcg)
2—— .
= )4kg 4kg SI“(C/(g) _1__icos((z+zo-c)kg)
2 .
i__C-cos((zo—z—c)kg) 4kg 4kg s1n(ckg) (015)
(4k: 4kg sin(ckg)

F

 

 

 

1 c
__ . cosk c—z coskz z>z
12": 2kgsm(ckg) ( g( )) ( g 0) o
- i- c cos(k 7)cos(k (C-Z )) Z<Z
12k: 2kgsin(ckg) 8" g 0 0

That is,

236

 

 

 

 

1 c
2 __ = _ 1 g ((2.16)
2 2 2 "m
[=1 kn-ko 2kg 2kgs1n(ckg)
where
_ cos(kg(c—z))cos(kgzo) z>zo (C17)
"m cos(kgz)cos(kg(c—zo)) z<z0 .
Therefore,
(“‘1 (”‘ 1
oo eoncos —2 cos —zo
2 C C = C g (C18)
[:0 2(ki—k3) 2kgSIn(Ckg) m"

After we derived these two closed form expressions (eqs. (C.13) and (C. 18)) of the series

summation, we can apply them to the electric dyadic Green’s function to obtain an

alternative representation.

In order to derive the alternative expressions for the electric dyadic Green’s
function based on eqs. (C.13) and (C.18), we can evaluate the summation over any one of
the three indices 12, m, and I. However, there are different numerators for the different

components in eq. (C.2). Therefore, to derive the simplest expression for the dyadic
Green’s function, we may evaluate the summation over index n for 35:, ii, and 237
components, the summation over index m for iviv, 3:2, and 22 components and the
summation over index 1 for 22 , 3i , and in“: components using eqs. (C.13) and (C.18). The

expression for Cw (D.2) can then be rewritten as

237

 

GwUO’ r) _ —_{m_ 2: ”€71 bckg mlsi4n(akgm1)[((’3blt)2+(l:)2 )gm1(x x0)

sin vsinmnv sinmzsinln7 fric+mmlt f (x,x )(cosm
b“ b’0 c c“0 bcvvv' 0 b

sin’mt sinln7cosmz “2+ sinmnycosmnv cosmzsinmz 257)]
b yo c 3' c 0y b b ' 0 c c 0

+n2112 ackgnISin(bkgn1)[((n—Ea)2 + (LKY )gn,(y, yo) sin 1x

singlgxuosingzsinI—T—tz 70iviv + glint”, y0)(sin'%txcosfltxocosl-C—nz (C. 19)

 

. In M nn . nn . In In M
5m —zozx + cos—xsm —xos1n—z cos —sz2.
C a a C C

°° °° 4 (1:21;)2 (mn)2) . nn
+ . — + — 2,2 )s1n—x
"glmglabkgnmsmkkgnmi a b g""'( 0 a

sinmtr sinr-rL—nysinmn "" +n_nmn
a ' 0 b b —-yozz+

 

nn nn
—f,,m(z, 20) cos—xsin-a—xo
Sinmnycosmn 22+ sinnnxcosnnx cosmn sinmny 372)]

where the parameters

 

vgmz=va-1(%v1+1v-:vn
= jks-((vg)fi(gv)2) m
= I 3—(("%‘Y+('%“))

cos(kgm,(a—x))cos(kgm1x0) x >2:O

 

 

8m1(x,xo) = { (C23)

cos(kgm1x)cos(kgm,(a—x0)) x<x0

238

008(k ,(b-y))COS(k (yo) y>yo

gn,(y,y0) = { g" 8" (C.24)
cos(kgnly)cos(kgn1(b — y0)) y < yO
cos(k (c—z))cos(k 20) z>z

gnm(z,zo) = gm gm 0 (C25)
cos(kgnmz)cos(kgnm(c—20)) z<zO
sin(k (c—z))sin(k n z ) z>z0

fnm(z,zo) = W . g m 0 (C26)
sin(kgnmz)s1n(kgnm(c-20)) z<2:0

fszC xo)= g . 8 (C27)
sin(kgmlx)sm(kgm1(a—x0)) x<x0
sin(k ,(b-y))sin(k m0) y>yo

f,,,(y,.vo) = { . g" . 8 (C28)
51n(kgn,y)s1n(kgnl(b — y0)) y < yo

That is, we have obtained the double summation representation instead of the triple sum-

mation representation for the electric dyadic Green’s function.
So far we have not considered the singularity of the closed-form evaluation. Since

there exists kgmlsin(akgm1) in the denominator of the evaluation (for 25c, i2 , and 29

components of 571,000, I) ), the singularity occurs when kgm, = O or akgm, = pn , where
p is an integer. Since the three sides of the rectangular cavity (a, b, and c) are not in integer
proportion in order to avoid more degenerated modes, the singularity occurs only when
one of the summation modes is exactly equal to the initial mode, that is

k0 = kn (C.29)

for some indices m and I. For this case, we can not use the above closed-form evaluations

to obtain the alternative expressions for the electric dyadic Green’s function. However,

239

based on the analysis given in Chapter 3, we know that the initial resonant frequency will
shift slightly after a material sample is placed within the cavity. Therefore, when the

singularity occurs, we can make the estimation of

k3, — k3, a -../.3 (030)

where s is the shift rate of the resonant eigenvalue and the summation over any one of the

three indices can be obtained as follows. For brevity, we only show the derivation for the

Sci component of 5.9000, ;) when there exists singularity.

When we evaluate the 3:3: component of 6.2000, I"), we obtain a closed form

expression for the summation over index n of

 

°° e

2 fl 2] cosn—1-rxcos'y—tx0 (C31)
a k -k a (1

n=0

n

ON

 

2 m1: 2 In 2 . . .
and k g = kO — ((7) + (27) ). At the smgularzty pomts of k g , we find that they occur

when

2 n It 2
ki—kf, = (’1‘) {it} ((3.32)
a a

where 120 denotes one of the three indices of the initial mode (we assume that the three

sides a, b, and c of the rectangular cavity are not in integer proportion). The summation in

eq. (C31) can be written as

240

 

I111: nit
— 2 2cos —xcos —xo (C33)
a n It a a
,. = 0 (at) _ (L)
a a
There are formulas for summations of [2]

0°

 

 

___—C55"): = __12_-2§-COS,(x-n)a O<x<2n (C34)
":1” —a 2a a sm(1ta)
m cosnx nx n2 2 1tcos(x—1t)a l
2—=__§_——i——— +— O<x<21t (C35)

n=ln2(n2—az) 2a 602 4a2 2a3sin(1ta) 2a,

Using eqs. (C34) and (C35), we can obtain the following summation as

 

 

 

°° 2 2
cosnx x 1: 1tx
2 2 = I + z — E- ((3.36)
n = l n
and based on eq. (C36), we can obtain the following summation of
co cos’l—rcosr—l—r oo cosflt(x+ t )+cos’-1—1—t-x—x|
2 Z a ' a ' O _ l 2 a '1 0 a l 0
2 ’ ’ 2
n = la (’11-!) an = 1 (Pl-t) (C37)
a a
l 2 2 a
= EU +xo )+§—xb

where xb is the greater one of x and xo. In general no is not very large because we
assume a lower order mode as the initial mode.

When no = 0, based on eq. (C37) the summation (C31) or (C33) is given by

 

cosmtrcosmtr
*- "0
’1—l§+ 22 a 2 a = ‘1—13+7i(x2+x02)+€‘xb (C38)
(IS/(6 ":10 (£15) asko _a 3
a

241

When no ¢ 0 , based on eq. (C37) the summation (C33) can be expressed as

 

80n 1 mr nn
2 ‘5' 2 TC 2COS"a—IC087XO
n
”=0 ('35) _(L)
a a

8on 1 m: Mt 2 m: nit
— cos—xcos—xo + X — cos—xcos—xo
a a

n: a (nnjz (non)2 a :1 (fly-[)2 a a
nxno a a a

 

 

 

l 711! mt 2 1 ”o7t "07t
— - 2COS—61"XC087XO - a—ECOSTXCOS-a—zxo
a n1! '
n = (___) Sk0 (C39)
a

 

,
n ("ofij‘
Hf _-
2 a m: rm 1 a 3
= Z - cos—xcos—xo—- —
a

" ‘ (Hillary CORY) a a a
ntno .a a a

l 2 2 a 2 l 1 "()7t "07‘
+—(x +xo )+——xb-— ——+— cos—xcos—xo
2a 3 a n n: 2 2 a a
( 0 ) sko
a

where "or is chosen in such a way that when n > "01' the following approximation is valid

,
("“12 ("“"l"
a a

Equation (C38) or (C39) is another alternative representation for the 3‘)? component of

III

(’33)‘ (C40)

(1

(76000, F) when there exists singularity. The same procedure can be applied to obtain the

alternative expressions for the other components of the dyadic Green’s function when

there exist singularties.

APPENDIX D

APPENDIX D

INHOMOGENEOUS DIELECTRIC SPHERE IN
UNIFORMELY APPLIED STATIC FIELD

In this appendix, the electric field in an inhomogeneous dielectric sphere which
includes two regions of different dielectric materials induced by a uniform static electric

field is determined. The geometry of this inhomogeneous dielectric sphere is shown in

Figure D. 1.

We select the polar axis (6 = O or z-axis) to be in parallel with E0. Using the
spherical coordinate system, the induced electric field inside the dielectric sphere will be

independent of tp [14]. Therefore,

2 = Ema) ‘ (D.l)
inside the dielectric sphere.

In the absence of the sphere, the primary electric field is given by

E” = 502 (D2)

in rectangular coordinate system. The primary potential is given by
v” = -50: (13.3)

in rectangular coordinates. In spherical coordinates, we have

243

E0

 

 

Figure D.l Geometry of an inhomogeneous dielectric sphere

244

z = rcosG (D4)

2 = cosef- sinGé (D5)

So the primary electric field and potential can be expressed in the spherical coordinates as
Ep(r, 9) = Eocosflf—Eosineé (D.6)

V” (r, 0) = —EorcosE) (D7)

2
This Vp(r, 9) also satisfies the Laplace equation V Vp(r, 6) = 0.

The next step is to find the secondary potentials V5(r, 6) which are maintained by

the equivalent induced charges on the spherical surfaces. The total potentials in region 1

and 2 are expressed as:
v,(r,0) = v”(r,0)+v‘,(r,0) (D.8)
V2(r,e) = V”(r,e)+V§(r,0) (D.9)
All the potentials satisfy the Laplace equation V2V(r, 8) = 0.
To determine V5(r, 8), we need to employ the variables separation method to

solve a Laplace equation. The solution of this Laplace equation is given by [14]:

Wm B) = 2 [G,,r" + H,,r“"" "]Pn(cose) (13.10)
n=0

where Pn(cose) denotes the Legendre function of order n and degree 0.

The secondary potentials VS(r, 9) in regions 1 and 2 can be expressed as

v5.0, 0) = Z AnrnPn(cos9) for 0sr<rl
n=0

(r, 9): 2 [En r +Cnr (n+l)]Pn(COSO) for rl_<_r<r2
n=0

and the secondary potential Vs(r, 6) outside the sphere can be expressed as

V3(r,9) = 2 Dnr-(n+l)Pn(cose) for rZr2
n=0

The total potentials in the three regions are then given as:

(l)forOSr<rl

Vl(r,9) = —Eorcose+ 2 AnrnPn(COSG)
n=0

(2) for rl Sr<r2

Vo(r,8) = —Eorcos8+ 2 [Bu/7+Cur—UHI)]P”(COSG)
n=0

(3) for r 2 r2

Vo(r, 6) = —Eorcos9+ 2 Bur—(n+ l)Pn(cose)
n =0

The unknown coefficients A

conditions on r = rl and r = r» as:

246

 

(D.ll)

(D.12)

(D. l 3)

(D. 14)

(D. 15)

(D.l6)

n, B", C" and D" will be determined by the boundary

V1(r],9) = V2(r1,8)
V2(r2,9) = Vo(r2,9)
DM’ve) = D2r(’1’9)

D2r(’2: 9) = Dor(’2’ 9)

Equation (D.17) leads to

2 Anr’fpnmose) = Z [Bu/HCnr;("*”]Pn(cose)
[1:0 "=0

Since Pn(cosfi) are a set of orthogonal functions [14], we have

A = B" + CanQnH)

n

Equation (D. 18) leads to

z [Bnrg + Car?" +1)]Pn(cose) = Dnrgm +1)Pn(cose)
0

n=0 :1:

and

2n+l
Dn=Bnr2 +C

’1

Equation (D. 19) leads to

81{—- EocosO + 2 nml)";— lPn(cose)}
n=0

= eo{—Eocose+ 2 [anrTl —(n+ l)C i'_("+2)]Pn(cose)}

nl
n=0

247

(D. 17)

(D. 18)

(D. 19)

(D20)

(D21)

(D22)

(D23)

(D24)

(D25)

01'

Z [82(ranr'l'_I — (n + l)Can(n+2))—ElnAnr';-1]Pn(cos‘0)
n=0
= (82-81)EoCOSG

Equation (D20) leads to

82{— Eocose + 2 [anr'é— l — (n +1)Cnr;("+2)]Pn(cose)}
n=0

= eo{—Eocose— Z (n +1)Dnr;("+2)Pn(c050)}
n =0

01'

E [82(anrg—1 — (n +1)C"r;(" + 2)) + eo(n +1)Dnr;(n + 2)]Pn(c050)
n =0

= (82 — eo)Eocos(9

Using eqs. (D26) and (D28), we have
€2[anr';-I -(n + l)C r-("+2)]-—€lnAan_ 1= 0

nl

n-l —(n+2) -( .:2)
22023an —(n+1)C,,r2 ]+€o(n+l)Dnr2" = 0

for n=l and
—3
' —2 —3
82(31—2C1r2 )+25001’2 = (32‘50)Eo

Substituting eqs. (D22) and (D24) into eqs. (D29) and (D30), we have

348

(D26)

(D27)

(D28)

(D29)

(D30)

(D31)

(D32)

(82—8,)nr'f'18n—[elrz+ez(n+ 1)]r]("+2)Cn = 0 (D33)

—1

[82n+eo(n+ 1)]r’2’ Bn—(ez—eo)(n+1)r;("+2)Cn = 0 (D.34)

For non-trivial solutions for B" and C n, coefficient matrix needs to be zero. That is, the

determinant of the following matrix is zero.

,,_ _ 2
(82—81)nrl 1 —[81n+€2(n+l)]rl(n+ ) (D35)

[Eon +Eo(n + 1)]r'21-l —(82—Eo)(n +1)r;("+2)

However, the matrix in eq. (D35) is a function of r1, r2, £0, £1, and 82, and it is obvious
that the determinant of this matrix can not be zero. Therefore, the unknown coefficients
8,, and C, are all equal to zero for n at l and the unknown coefficients A" and D" are
also equal to zero based on eqs. (D22) and (D24).

For n = l , the unknown coefficients Al , Bl , C 1 and D1 are to be obtained as
follows:

Substituting eqs. (D22) and (D24) into eqs.(D.3l) and (D32), we have
(81 —82)Bl + (1»:l + 2:52)r]3Cl = (a1 —82)Eo (D36)
—3
(82+230)Bi -2(82—€o)r2 C1 = (ez—eo)Eo (D.37)

Then the solutions forAl , Bl , Cl and D1 can be expressed as

—3 -3
€+28r+2€—€—)r e-eE
3] = I( 1 2) 1 (‘31 _) 2 I( 3 0) 0—3 (13.38)
(81+ 2:52)(e2 + 22o)rl + 2(81 —ez)(e_,_ —eo)r2

 

249

38 e -e E
CI = ”(_‘3 2) 0 _3 (13.39)
(81+ 282)(82 + Zeo)rl + 2(s.,—e2)(22--1-:o)r2

 

2 -3 -3
(288—588+€8+2€ r +22—e e—e r»
A1: 01 0 2 12 -2391 ( 2 0)( 1 3) - 50 (D40)
(81+ 2&2)(e2 + 280% + 2(1-1l —ez)(e2 —eo)r2

 

-3 3
8+2£ e—e r r+e-e 2+284
DI : (1 2)( 2 0)1_32 (1 2)( 0 2.3150 (D.41)
(81+ 282)(82 + Zeo)rl + 2(.<51-13:,_)(s2 - 1»:o)r2

 

Therefore, the potentials in the three regions are given as

—98 8 r_3E
Vl(r,9) = (:32 1 0 _ rcosB (D.42)
(81+ 282)(82 +2.2:o)r1 + 2(81—82)(€2—£o)r2

 

—3eo(e1 + 282)r;3Eo

 

 

 

 

Vo(r,0) = _3 _3rcose
(81+282)(£2+280)rl +2(€l—82)(82—Eo)r2 (D43)
+ 380(81— 82)E0 C050
(81+ 282)(82 + 213o)r]3 + 2(sI -eo)(ez-eo)rg3 r2
-3 3
(e +2c)e —e r r+e-e e +2€)ECOSG
Vo(r,0) = —Eorc030+ ‘ 2( 2 0) [32 ( ‘ 2)( 0 2_3 0 , (D.44)
(£,+2e:2)(32+21:o)rl +2(s,-e2)(t~:2—eo)r2 r“
and the electric field in region 1 is given as
4 —9e 8 P315
51 = 0 2 1 0 2 (13.45)

_3 —3

The ratio of 51/50 is a constant within region 1 of this inhomogeneous sphere. The

electric field in region 2 is given as

250

 

 

 

 

 

 

—3 —3
_t (e —e r -e +284)!“
52 = 38050{ ‘ 2) _3( ‘ ~ ‘ ‘32 (D46)
(81+ 282)(€2 + 2.so)rl + 2(151 — €2)(e2 — eo)r2‘
—3
(E —8-,)r
+ l _3 " _ 3cosG?
(81+ 2.sz)(s2 + 280)r1 + 2(1»:l - ez)(e2 — 90)’2
The 2 component of the electric field in region 2 can be expressed as
(e - e )r'3—(e + 28 )r_3
E2: = 380 1 2 -3 1 2 1 _350 (D47)
It is noted that there are other smaller components of the electric field in region 2.
For the special case of r1 = r2 and 81 = 82, i.e. a homogeneous sphere, based on
eq. (D42) the potential in this region is given by
V1(r, 0) = rcosO (D48)
51 + 280
and the electric field is obtained as
A —38 E
1 = 0 02 (13.49)
8‘ + Zeo

This is the result given in [14]. For another special case of rl —->0, which is also a

homogeneous sphere, based on eq. (D43) the potential in this region is given by

 

E0
rcosG (DSO)

V2(r’8) = 52+ so

and the electric field in this region have the same expression as eq. (D49).

251

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[Ill

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