‘ .
. ‘ z . . _. . , ‘ ‘ .33.?!33...»
. . ‘ :3.» 3:3.
, . . . . . r. :
. . . . . V .
V .f ., . ‘ . < , ,
.‘ < . 4 . .
. . . . . , K
. ‘ v V . V , 1 . . ,
.. .. . . . ., .
_
.
_
.
n
.
.
.
. .

 

.. __... .uum§d&i§.wA

  

SIRAHTYLIB

Ms 111111111111111111111111111111111111111111111

J 1 3 1293 02058

 

11111111111

 

This is to certify that the

dissertation entitled

FULL-WAVE INTEGRAL-OPERATOR DESCRIPTION OF
PROPAGATION MODES EXCITED ON STRIPLINE STRUCTURES

presented by

David John Infante

has been accepted towards fulfillment
of the requirements for

Ph.D degree in EIectricaI Eng

Major pflfgsor

Date ’0 'IZqI/q?

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

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

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11100 animus-p.14

FULL-WAVE INTEGRAL-OPERATOR DESCRIPTION OF
PROPAGATION MODES EXCITED ON STRIPLINE STRUCTURES
By

David John Infante

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1999

RIL-ll

PROPAG UN

3

--‘ .
‘ c
I.-

I'-
4 ‘
"uw,

’fi'éfrJA‘

the» ‘. ‘..y
s

',"V o

.". O» I.

~., ....‘L 0‘

. .
u'.

~.'

5“!

.‘ ,
“5.

.'r, U‘ "‘

l"

vi.

Ii' ‘ 9-1‘

‘-....17,.L "'If‘»
I “to

rig”

I
~:.‘\§..-‘.‘ _ n
.. I . . ‘
I “~|\‘\_) \ "” ‘
‘ L.
I
T". ~54“
“‘ at g". .,
. I“-‘ ‘
‘F. .‘
'I
k l
k‘mc E" P... ‘
‘¥\\ .
n
a."
0143!. "i" 9. v
I’ -' I ~
. rcsul~ A! '
n ”I
‘I. 1
\“'u "‘-
\~\.¢~. lr‘ -‘,_
.og ‘\\K..."‘
-.K‘
l
i{’~

tit" .
so -
. ’3 ON», A
Inn”;
511
5““ .’ 't.“
k‘ugu'zc.

.4 P...
314‘ .
. \- )‘Ur

ABSTRACT

FULL-WAVE INTEGRAL-OPERATOR DESCRIPTION OF
PROPAGATION MODES EXCITED ON STRIPLINE STRUCTURES

By

David John Infante

Stripline waveguide structures are commonly used in microwave and millimeter-
wave integrated circuit devices, and as field applicators for the electromagnetic
characterization of materials. Despite widespread use as a waveguide device, analysis
of the higher order modes excited on stripline structures has received surprisingly little
attention. This dissertation presents investigations into several topics involving Stripline
waveguide devices.

First, a Hertz potential Green’s function is developed representing a generalized
stripline structure, excited by a three-dimensional volume current source. The associated
Helmholtz equation is solved via a two-dimensional Fourier transform. Pole singularities
and branch points associated with the resulting Sommerfeld integral Green’s function
representations are identified and examined.

The generalized Green’s function is specialized to represent an isolated stripline
structure. Electric field integral equations (EFIES) are developed through enforcement
of appropriate boundary conditions at the conducting center-strip surface. Current
distributions associated with various propagation modes are determined using a Galerkin’s
Moment Method technique employing Chebychev basis functions. Because most stripline
devices are operated in such way that only the fundamental TEM mode is excited, a
TEM specialization of the stripline full-wave theory is developed, leading to an

expression for the characteristic impedance of stripline.

A .‘u I '
"‘ i3.) ”t

-_
5‘1.

“‘4

”a W. ‘1- “1.5 Clap.
’~.~15‘D:IDU Stu

--.

:j ' rww' of but:

. .5 son. «A

.I a. :E"£
»9"' —- '
“\ '65,, .fi» 1 A ‘5
O

r'I "' .fl " 4‘ .-
. \r. i I TI
I...I.. is‘.“no-¢a . a

‘5} 1’5"...»a. i.” ‘O '3
u. “Nil-AAA-tl‘ in K i

bl

120"" Ill "°‘

1U»! 55“

'2’:- . r,”

‘ -. . R
H-W.AIUH*:': '
l
KIT-7" 'L"’ 0‘ v '
tumush die“. “-‘I ‘15.“
V... '
r -
Tij‘.“ “‘6 2'; Fr.
...._ \ m» J.
K
I I

\W”."“ ' n
. | A‘ 9 ..
“‘th “33.3.5 ,9"

.. en" 4
"Uh‘l~'-¥l I...

l
'lu.

I
H
9}“ l‘

'3.“ 't

.. H . I

I kg: in: Irhtro"
‘ u

”1%" “in

I. . ‘ l

r IA. T‘OV
0‘,

“'1.

”4:1"le I

‘9‘ a.
-Atld‘ “|h‘

.t.

x,»

iv?-
“In: L g1~ . y
'1 Lfic‘ i:;I

M

The analysis of isolated stripline is then extended to the case of a structure having
a pair of coupled center strips. Two pairs of coupled EFIES are developed through
enforcement of boundary conditions at the surfaces of the conducting center strips.
These coupled EFIEs are then solved numerically via a Galerkin’s Moment Method. A
TEM specialization of the coupled stripline full-wave theory is also developed, allowing
for determination of the impedance parameters associated with this structure.

In order to reduce the complexity associated with solution of the coupled strip
EFIEs, a full-wave perturbation theory for coupled strips is developed. Here it is
assumed that, for loosely coupled center strips, the current distributions of the coupled
system modes are approximately weighted replicas of the corresponding isolated currents.
Numerical results using this method are compared to those obtained previously.

Finally, a technique for measuring the complex permittivity and permeability of
sample materials placed in a stripline field applicator is presented. Limitations associated
with frequency-domain characterization techniques currently in use can be overcome by
exploiting the inherent time-domain response of the stripline applicator. This time-
domain response is synthesized from frequency-domain measurements made at the
applicator ports. Time windows are then applied to the resulting waveform to remove
unwanted signal components. This technique requires fewer measurements than those

currently in use, leading to increased efficiency and accuracy.

 

for my grandparents,
Rody and Angela Zivkovic
and

Charles and Mary Infante

.
"O ".30. '0"; av- : . I
.o. a... mils nu u A-

h"
i
'.
I
Q
I
“o

’ ’1 ., , 0
m." f | p ‘. '0 F
IN“. .L .‘U I ~

:31: is. 1 :g- I):

.o,

:'\ ‘uka._,‘ .'. ' ,
. 1 l '
.“‘..‘, . b ‘
\ I
s ‘\"“\..

a!~

:3 fly .
u a. ~“. ‘:\\ ET...

[F’si
.4, =0.
A.“.~u‘.r~ 0

 

2a,,
. :
\H 4%. I
“I
"M- 'r»
‘ I.
.
L“
.‘ 'r
‘..N\

ACKNOWLEDGMENTS

It would not have been possible for me to complete this dissertation without the
support of many people. I would first like to thank the members of my thesis committee
for their time and insight. I am most indebted to my advisor Dr. Dennis Nyquist, for
the many years of encouragement and guidance he has provided. I consider myself truly
fortunate to have had the opportunity to study under such a gifted teacher. Special
thanks also go to Dr. Edward Rothwell for the advice, friendship, and mentoring he has
provided over the years. I am also grateful to Dr. K.M. Chen for establishing an
outstanding research group, and leaving a legacy of excellence at MSU. Finally, I would
like to express my sincerest appreciation to Dr. Thomas Livernois for his advice and
support, and to Dr. Byron Drachman for his time and endlessly entertaining stories.

In addition to those who guided me academically, the support of my friends and
family has been equally important. Special thanks go to Daniel and Kathleen Glatz,
Matthew and Michelle Adomaitis, and their families, for their friendship and support
through many difficult years. I am also grateful to my fellow students and friends
Michael Havrilla, Christopher Coleman, Ponniah Illavarasan, and John Ross. Finally,
no amount of thanks are sufficient for the support and encouragement given to me by my
father Daniel, my mother Marilyn, my sister Alicia, and everyone in my extended

family.

 

.s D
9 . ‘ ]. quoflr'\
$1-. .I L’.......-

.fi

- ,
.4 O tfi.‘\' '
\ ‘1 . 53‘

~ bl a I.‘d.0

05s.". 2:1 of A

[T] I I”

I: 1‘ {f

, :r; 5;} "1;
;-‘ "U '1

; :r‘

¥

97
ft
3

OJ IA 0.) C~J OJ

IL." .5. tho IJ ~——-

2
I
S

r1.
.‘4‘:n3‘.3
Girtn'x if
SLFC‘CI‘LIC
31 REP-"CW?"
3.2 Ada] ..,
‘ l: 'ul.
:3 Dav..-
s :
3-1 Hm“, -
‘ . 5
I.) Film-n
s '1’ ‘M
36 Dtgph‘lw
3~ H“~"‘FI’....
II :‘J I.
, .
18Eh7“‘
~-.'.::4
F1411 -,
cl \\ 4",: -\"‘
‘1 GILT-"Ex f
4* I'I'a-y, T
4 3 l
Mtg T'
44 S"I ~
4 “1“" in
5 GIT-”fig
B i “ ’F
{4%}. (
46 [IT‘SL A \.
n 'f' ‘3...”

TABLE OF CONTENTS

Table of Contents ....................................... vi
List of Figures ......................................... ix
Chapter 1
Introduction ....................................... 1
Chapter 2
Overview of Analytical Technique ......................... 10
2.1 Establishment of Helmholtz Equation .................... 10
2.2 Hertz Potential Green’s Function Solution ................. 12
2.3 Electric Dyadic Green’s Function ...................... 14
2.4 Development of Electric Field Integral Equations ............. 15
2.5 Numerical Solution ............................... 17
Chapter 3
Green’s Function for a Generalized Volume Source in a Stripline
Structure ........................................ 18
3.1 Representation of EM Field by Hertz Potential .............. 18
3.2 Application of Fourier Transform ...................... 23
3.3 Decomposition of Hertz Potential ...................... 24
3.4 Homogeneous Solution for Scattered Waves of Hertz Potential ..... 26
3.5 Principal Incident Wave of Hertz Potential in Free Space Region . . . 29
3.6 Determination of Unknown Wave Amplitudes ............... 35
3.7 Hertz Potential Components and Associated Green’s Functions ..... 42
3.8 Examination of Green’s Function Singularities ............... 45
Chapter 4
Full-Wave Analysis of Isolated Stripline Structures ............... 48
4.1 Green’s Function for Homogeneous Stripline Cross-Section ....... 48
4.2 Integral Equation Formulation ........................ 51
4.3 Axial Transform-Domain Integral Equation ................ 54
4.4 Scalar x- and z-Component Integral Equations ............... 56
4.5 Green’s Function Singularities and Axial Transform-Plane
Branch Cuts ................................... 59
4.6 Improper Discrete Stripline Modes ...................... 64
4.7 Method of Moments Numerical Solution .................. 69

vi

F“).
. Ci
" J
2'.

" H {T
o- , ,
' fa, . A r
r,. 5..
N
I

9‘
SI
(7,. .
, .

'l-b

A

’1
I.
p- ._.
u.-
I
r
J

V
I
3
.I

"‘I l)-
r01 p
.1: ‘
{/3

3

p c
I

1)! (I! in «J. i). 1" '1. LI-
J¢J~1JIJA!;JIJI——-

(K) r

u...

w .
‘ l

3

_.

(I7 '/
1) _

......

Raf

4.8 Analysis of Numerical Results ........................ 86

4.9 TEM-Mode Specialization ........................... 88
4.10 Moment-Method Solution for TEM-Mode Specialization ....... 107
4.11 Pole Series Representation of Matrix Elements for TEM Modes . . 110
4.12 Characteristic Impedance of Stripline .................. 119
Chapter 5
Full-Wave Analysis of Coupled Stripline Structures ............. 123
5.1 Coupled Strip Electric Field Integral Equations ............. 123
5.2 Scalar x- and z-Component Integral Equations .............. 127
5.3 Transformation to Local Coordinate Variables .............. 132
5.4 Moment-Method Solution .......................... 135
5.5 Analysis of Green’s Function Singularities ................ 159
5.6 Analysis of Numerical Results ....................... 165
5.7 TEM Specialization for Coupled Stripline ................ 170
5.8 Moment-Method Solution for Coupled Stripline TEM
Specialization ................................. 188
5.9 Pole Series Representation for Matrix Elements ............. 201
5.10 Impedance Parameters of Coupled Stripline ............... 208
Chapter 6
Full-Wave Perturbation Method for Coupled Stripline ............ 218
6.1 Electric Dyadic Green’s Function ..................... 218
6.2 Perturbation Approximation ......................... 223
6.3 Determination of Normalization and Coupling Terms ......... 227
6.4 Transformation to Local Coordinates ................... 233
6.5 Method of Moments Solution ........................ 235
6.6 Analysis of Numerical Results ....................... 242
Chapter 7
Time Domain Technique for Measurement of Material Properties
Using Stripline Field Applicators ......................... 245
7.1 Advantages of Stripline Field Applicators ................. 246
7.2 Frequency Domain Materials Characterization .............. 250
7.3 Disadvantages of Frequency Domain Technique ............. 258
7.4 Time Domain Materials Characterization ................. 259
7.5 Materials Characterization Signal Processing Scheme .......... 261
7.6 Application of Fourier Transforms and Weighting Functions ..... 271

vii

q
'..-u v.‘ V
I

,.I..-.

Lin-wan.“

Irv ‘ii A

..,~..
I I
',o"l ',n"" .
1...; "15....u.
D
',..'... -
l as:

hofi 4 D

' I
1 1' 'F a H
. L _ u.1bu4 ‘

'1 1 B_...a ,,

n o -. unn$~ou14 ‘
S l

‘q “a u --‘v.

c v . .‘chl.

CHM-SC.

’ - .
.. I
3 arm'-
" s “‘.‘
b i ‘

Chapter 8

Conclusion ...................................... 288
Appendix A

Hertz Potential Boundary Conditions at Dielectric and Conducting

Interfaces ....................................... 293

A.1 Boundary Conditions on Hertz Potential at a Dielectric Interface . . . 293

A.2 Boundary Conditions on Hertz Potential at a Conducting Interface . . 303
A3 Application of Hertz Potential Boundary Conditions to Stripline
Cross-Section ................................. 3 10

Bibliography ......................................... 313

viii

. 3....
-

.-

_.c- '5

a ..‘.u
-

’1'?
O
1

J-
!

liv’.

.5“,

tn

(I:

cl‘

:-
5

l‘l

y...

g
‘ T
u .
S Q
‘ S
O u

a-
H

‘5
fl:

(JCS:

u"u
5...

P'-'-
. ...‘

w
1
Ci‘ ...l

51.9....1
\‘ .'. .

C
‘ht.~

.4

9..

\""v~
--A u
‘0
‘A
r,
~al'\\

Figure 1.1
Figure 1.2
Figure 1.3
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4

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

LIST OF FIGURES

General stripline cross-section with volume source current ....... 4
Isolated center strip cross-sectional configuration ............. 5
Coupled strip cross-sectional configuration ................. 7
Generalized Stripline cross-sectional geometry .............. 19
Principal and scattered Hertz potential wave components ....... 25
Complex n-plane inversion contour .................... 33
Poles associated with parallel-plate and surface-wave modes ..... 46

Generalized three-dimensional current source suspended between

parallel conducting ground planes ..................... 50
Stripline structure immersed in layered background environment . . 52
Location of parallel-plate guided-wave poles in complex k-plane . . 62
Location of parallel-plate guided-wave poles in complex E-plane . . 63

Branch-point singularities and inversion-contour deformation in
the complex {-plane; branch cuts appropriate to restrict £-plane
pole migration ................................. 65

Inversion contour in complex £-plane ................... 67

Dispersion characteristics of higher-order, non-leaky modes

supported by stripline (W: 10.1 cm, h=3.67 cm) ........... 89
Cross-sectional current distribution for TEl mode of stripline,
near and well above cutoff (w=10.1 cm, h=3.67 cm) ........ 9O
Cross-sectional current distribution for TE2 mode of stripline,
near and well above cutoff (w=10.1 cm, h=3.67 cm) ........ 91
Dispersion characteristics of higher-order modes supported by
stripline (w=5.0 cm, h=1.75 cm) .................... 92
Cross-sectional current distribution for TE, mode of stripline,
near and well above cutoff (w=5 .0 cm, h=1.75 cm) ......... 93

ix

.’ I‘
.r'0'4
.A-' I.
r 9
1.17-9.3qu
can“. "'
" I
..‘r~’a‘ .
..;¢.- ‘.44
.-

w- . I '
I 'v ‘
13.... fi ‘.
v-

h“ '94 '
'f“ l

pr'

I"
I

i

'1
I“).
J—

C toss-

- "n.-c-
5.... no

‘ i
CON“
1.“ :
I
74.
0‘ “tr r
5"“.
~4‘. I F
I ‘l

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Figure 4.18

Figure 4.19
Figure 4.20
Figure 4.21

Figure 4.22

Figure 5.1
Figure 5.2

Figure 5.3

Figure 5.4

Cross-sectional current distribution for TFG (always leaky)
stripline mode (w=5.0 cm, h=1.75 cm) ................. 94

Dispersion characteristics of first three higher-order modes

supported by stripline (W: 1.07 cm, h=0.37 cm) ........... 95
Dispersion characteristics of first (TEI), second (TE), and fourth

(TM1) higher-order modes supported by stripline (w=1.07 cm,

h=0.37 cm) .................................. 96
Cross-sectional current distribution for TEl mode of stripline,

near and well above cutoff (w=1.07 cm, h=0.37 cm) ........ 97
Cross-sectional current distribution for TE2 mode of stripline,

near and well above cutoff (w=1.07 cm, h=0.37 cm) ........ 98
Cross-sectional current distribution for TB, (always leaky)

mode of stripline, near and well above cutoff (w= 1.07 cm,

h=0.37 cm) .................................. 99
Cross-sectional current distribution for TMl (always leaky)

mode of stripline, near and well above cutoff (w=1.07 cm,

h=0.37 cm) ................................. 100
Poles and integration contours in the complex £—plane ........ 113
Location of additional pole at the origin of complex E-plane . . . . 116
Cross-sectional current distribution for TEM mode of stripline . . . 121
Relationship between stripline characteristic impedance and
cross-sectional dimensions ......................... 122
Cross—section of stripline structure with coupled center strips . . . . 124
Local coordinate variables ......................... 133
Comparison of dispersion characteristics of non-leaky, higher-

order modes of isolated and widely-spaced coupled, coplanar

stripline .................................... 171
Cross-sectional current distribution for TEl mode, on center

conductors of widely separated, coplanar strips (15cm center-

to-center, 5cm width, 1.75cm ground plane half-spacing) ...... 172

Figure 5.5

Figure 5.6

Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10

Figure 5.11

Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15

Figure 5.16
Figure 5.17
Figure 5.18

Figure 5.19

Figure 5.20
Figure 5.21

Figure 5.22

Figure 5.23

Cross-sectional current distribution for TE mode, on center
conductors of widely-separated, coplanar strips (15cm center—

to-center, 5cm width, 1.750m ground plane half-spacing) ......

Dependence of propagation phase constant upon center-strip

spacing for the TEl mode of coupled stripline .............

Strip 1 (left) normalized TE1 symmetric current distribution
Strip 2 (right) normalized TEl symmetric current distribution

Strip 1 (left) normalized TE1 anti-symmetric current distribution

Strip 2 (right) normalized TEl anti-symmetric current distribution .

Dependence of propagation phase constant upon center-strip

spacing for the TE mode of coupled stripline .............

Strip 1 (left) normalized TE anti-symmetric current distribution

Strip 2 (right) normalized TE anti-symmetric current distribution .

Strip 1 (left) normalized TE symmetric current distribution . . . .

Strip 2 (right) normalized TE symmetric current distribution

TEM-mode center strip potentials relative to bottom ground
plane

Comparison of surface current distributions associated with
TEM mode, for isolated stripline, and widely-spaced coupled

center strips .................................

Strip 1 (left) normalized TEM mode current distribution ......

Strip 2 (right) normalized TEM mode current distribution

Relationship between coupled stripline self impedance Z1] and

center-strip spacing .............................

Relationship between coupled Stripline mutual impedance Z12 and

center-strip spacing .............................

Strip 1 (left) normalized TEM mode current distribution ......

Strip 2 (right) normalized TEM mode current distribution

xi

ooooooooooooooooooooooooooooooooooooo

173

174

175

176

177

178

179

180

181

182

183

211

212

214

215

216

..

.2...‘

...o.'
U

'-

b'- .

“~-‘
9

.r1

("I
f
(I

em".

‘I‘.
. 5

(8'

. I.. Q
‘ i..': . I1
“ (1%—
'l‘. \
Tin-1.
“‘lt “ .4
L i
P
if)»; H "
v ”1
U; :5
I
‘N
[‘HAC~ &
If
7‘ ~ ~
‘3‘: I4
‘ Ct-IWH
.. : J
[I
(“‘6‘ .
II.
“54:37 I;
.4 Ci'lh‘h
u“: .2
l"~.~-
'~‘ .3
P.
§~‘
1 . it. Q
t“: . I6
I T.
1h-
"INC“:
U
Shifllp- ~ ‘
\i
‘Z?I'L ~

I
s‘Sk

-l
0‘
O
h.
‘.
t
I

.9“

6:14;"

at f:

t

I
ob'

IF"«\_

I.“

Figure 6.1

Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4

Figure 7.5

Figure 7.6
Figure 7.7

Figure 7.8

Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

Figure 7.13

Figure 7.14

Figure 7.15

Figure 7.16

Figure 7.17

Comparison of propagation constants for symmetric and anti-
symmetric modes obtained using full-wave, and perturbation
methods ...................................
Side view of stripline field applicator ..................
Top view of stripline field applicator (top ground plate removed) .
Placement of sample material in stripline field applicator ......

Cascaded two-port network representation of stripline applicator

Frequency-domain response of short-circuit termination, measured
at front applicator port ...........................

Gaussian Modulated Cosine weighting function ............
Weighted frequency-domain response of short-circuit termination .

Time-domain response of short-circuit termination, measured
at front applicator port ...........................

Windowed time-domain response of short-circuit termination,
measured at front applicator port .....................

Relative permittivity of teflon sample obtained using stripline
time-domain technique ...........................

Relative permeability of teflon sample obtained using stripline
time-domain technique ...........................

Comparison of relative permittivity of plexiglas obtained using
time- and frequency-domain methods ..................

Comparison of relative permeability of plexiglas obtained using
time- and frequency-domain techniques .................

Comparison of relative permittivity of teflon obtained using
time- and frequency-domain methods ..................

Comparison of relative permeability of teflon obtained using
tirne- and frequency-domain methods ..................

Time-domain response of unweighted frequency-domain stripline
short-circuit response ............................

Moderately weighted frequency-domain short-circuit response . . .

xii

247

248

249

252

263

264

265

266

267

272

273

274

275

276

277

280

281

p

o "’v

5““.
.

,.
>"'"J

l»;.-b
v

”Q

m 3
Eliot .1

‘L -."..
‘.A'\. a s I

C ".Q
L's t 5‘

TO~J‘
noao\

“.90...
nl.5, ..

II A..O'1
tot-.5‘ I

G'é'.'
5 Id; .

Figure 7.18 Time-domain response of moderately weighted frequency-domain
short-circuit data .............................. 282

Figure 7.19 Correctly weighted frequency-domain short-circuit response . . . . 283

Figure 7.20 Time-domain response of correctly weighted frequency-domain
short—circuit data .............................. 284

Figure 7.21 Incorrectly weighted frequency-domain short-circuit response . . . 285

Figure 7.22 Time-domain response of stripline applicator, synthesized using
0.15-2.15 GHz frequency-domain data ................. 286

Figure 7.23 Time-domain response of stripline applicator, synthesized using
015-4. 15 GHz frequency-domain data ................. 287

Figure A.1 Interface between free space and non-magnetic dielectric
material .................................... 294

Figure A.2 Generalized stripline cross—sectional structure ............. 312

xiii

 

 

'V 9'.- O-"'
own 4],. ‘ ‘
Ss.1.5u 5 Sol db ‘1!
a
Q ‘ ' ¥" ‘ A,of‘ Il\
. ...._,. 1 \ ht .
.JT.....:1» 15.1. 1 5.11 a

:35 Nit-r: it‘xcifl}. ~

.u.
-__‘¢~

’1 u ' .. LI. -
*h-II‘M) “:15h C\004- t

\In-u

,4... .. . . :
,,....;:i_-n sub 1» \:r1:.:

‘

.r: smplinc st ' -'

Ir
\khcc \

Us for 10w fir-qt”

\tlk ‘

-' . .g and plat-cm;
:1. {LC 'h '
«I d,p1.taiill’. d1 HR
1..

NI: 43115 a

1‘ ., .
ppitalurx \\ It};

.. tailor or such
miles at 11111.11 1: ‘

Chapter 1

INTRODUCTION

Stripline structures represent one of the most widely used types of waveguides.
The stripline has been used in UHF, microwave, and millimeter—wave circuits for many
years. More recently, stripline waveguides have been employed in the production of
materials which exhibit specific reflection and transmission properties over wide
bandwidths. Such materials may be used in microwave circuitry, in broadband
transmission lines, and as coatings for reduced radar cross-section weaponry. Efficient
development and testing of these materials involves measurements made using field
applicators such as Stripline, which are capable of operating over wide frequency ranges.
The stripline structure supports TEM modes and has a zero cutoff frequency, which
allows for low frequency measurements. Stripline also provides good field confinement
at higher frequencies. The layered cross-section of stripline allows for relatively easy
machining and placement of sample materials. Also, sample materials may be clamped
into the applicator, allowing measurements to be made at high temperatures. Designing
stripline applicators with maximum operational bandwidths requires an understanding of
the behavior of such devices over a wide range of frequencies. Specifically, the
frequencies at which higher order modes are excited must be determined.

Due to compatibility with contemporary integrated circuit fabrication techniques,
stripline devices are also used extensively in micro-electronic circuits. As
microprocessor clock speeds increase, the design of associated circuitry requires

knowledge of the high frequency behavior of structures like stripline. In addition, the

‘I
. .g .
N"; LI US}.L'osA\ to A

" t

' " "‘3" ‘1‘ r1,
3:... *3 Lun'.‘l\. ;‘
~vvo34soi, ‘ ._ I
«mun Chm ,

5~5¢n4tu

3*:31'531 11.5."
if?“ 10 d:’.: "-

|At....
‘.‘~
"'\ 3" .. n. '-
u-‘vk.£ rr\1_lu-:u~:

I
it"... PM?» - '
k l‘ altugr UTJS"

V
mu”; .
ecu. . Taxi-Jr‘s

iiwxza'

‘Q- \|
“‘5; "

'w I _
I I." ilcdh““*d'.

T515 dggx... ‘.

‘*§n.

Ki," _
““dl‘l|‘._j ‘H 'Ilo

I * ‘a El

3"“ ,l I

‘M. 4" I

. Mk. Q; 51.,“ w“

r-ulc .
‘04

AL_‘
q
tit; -
K—- 1“. v
.KNImH Irm' '
bIAJ“; [:2
ET! 1
\ ~ ‘ .
Magic 4:“ ~ . .
x it» ”I
P A
‘ \A

decreasing size of micro—electronic circuits has prompted the need for a mathematically
rigorous understanding of coupling effects and leakage mechanisms which may lead to
loss or distortion of signals.

Historically, analysis of the higher order modes associated with stripline devices
has received surprisingly little attention. Although the associated geometry lends itself
well to analysis, no rigorous full wave theory for single and coupled stripline devices
currently exists. Early work in this area relied on conformal mapping techniques and
focused on determining the characteristic impedance of the dominant (TEM) stripline
mode [1]-[3]. Higher order modes were largely ignored by these methods. More recent
efforts to determine the propagation-mode spectrum of stripline have involved the
transverse resonance technique [4], [5]. These approaches have provided information
about higher order mode cutoff frequencies, but not about the center-strip current
distributions associated with the various higher-order modes, or about the excitation of
improper (leaky-wave) modes on stripline structures.

This dissertation investigates the discrete higher order modes, both proper and
leaky, of stripline transmission lines via a full-wave integral operator technique [6].
Full-wave integral operator formulations for both single and coupled stripline
configurations will be presented. A Hertz potential Green’s function is developed for a
generalized stripline cross-sectional geometry, and is then specialized to represent single
and coupled strip structures. The cross-sectional geometry of the devices under
consideration prompts the use of two-dimensional Fourier transtrms to solve the
associated Helmholtz equations. Electric field integral equations (EFIES) representing

the single and coupled strip structures are developed based on the Hertz potential Green’s

 

|
V
‘0'. ‘f' .h' .H ‘h
,3 --.u H‘“

.- ‘. ‘L.I'
t}: ‘oo‘h"". (

. 1 .'
~‘ ‘. x.",5'-.
a: 1\~‘u Fl‘u. ml “ ‘

..-.. .0 1 N '.
2>.\-‘l~vd ' "I; pl

-',_,..-.. J- o "-
ll..-_‘..: ‘ '9' l“

"luflvu "
' . 4‘ L
"”— ' .' a K '°
$4..» UD “-5 l

4..--

‘ .

N » ~,|-.r‘"' "'0‘."

‘M “Lush-u i555“-
.

-+- 4..- fl
TN: Ho‘Qc ton:

u'.‘
I ‘ )fl“! - .
9‘ a ‘ fin'"
kM'TsE i.1\.“.
'

L...“ - .
«mu-D IS d:::.«'
V

s ‘ ‘ c
)Q ”7&1 1m -
W. .«cm
I.
.
Q. '

_-c' ' ..
*.:‘.‘~.?~390 ,. fl
1 ua\Li£ Q r TC\:

f4..._
‘\. H‘v
A‘A“I~APV’|-\ I
«H \ ' v
A p4 i
4., W
”"II
N?!” -
.‘I‘.k.x|P ‘
. 31W.“
. h‘ ‘
‘ NI
5‘-
mg 9:". ,
:¥|.“ .‘. ,
“L-
ie .
.‘4‘-.l
'Lg.
43., '4’
a A;“ ’1‘
\\L ..
.‘\‘ I

function through application of appropriate boundary conditions.

The higher order modes of stripline are investigated through complex analysis in
the axial Fourier-transform plane. Proper, discrete guided-wave modes of stripline are
associated with pole singularities of this Green’s function which occur on the top
Riemann sheet, while improper (leaky-wave) modes are associated with singularities
located on the bottom Riemann sheet. Finally, numerical solutions are implemented
using moment method techniques.

This dissertation is presented in eight chapters. Chapter 2 consists of an overview
of the analysis method which is to be employed. In Chapter 3, a Hertz potential Green’s
function is developed representing a generalized three-dimensional current source
suspended between two infinitely wide, parallel, conducting ground planes [7]-[22]. This
development is presented for the general case in which one of the ground planes is coated
with a simple, non-magnetic dielectric layer (Figure 1.1). Analysis begins with the
establishment of boundary conditions at the dielectric/free space interface, and at the
surfaces of the conducting ground planes [23], [24]. The associated Helmholtz equation
is solved via a two-dimensional Fourier transform, which is prompted by the uniform
cross-sectional geometry of the generalized stripline structure. Pole singularities and
branch points associated with the resulting Sommerfeld integral Green’s function
representations are identified and examined. From this, surface-wave and parallel-plate
propagation modes are also identified.

The generalized Green’s function is specialized in Chapter 4 to the case of an
isolated, perfectly-conducting center strip suspended between two infinitely wide, parallel

conducting ground planes (Figure 1.2). It is assumed that the center strip is immersed

i-
..

 

 

 

 

 

(6.42.)

Figure 1.1 General stripline cross-section with volume source current.

 

 

 

~ y = h
0 : 0° (60’ “0) x
~ <
2h _ >

\

~ x='a X
/

Figure 1.2 Isolated center strip cross-sectional configuration.

...» 'f‘ o .
‘ ...s .‘t a»
I

'1.

. y'- v
;on on ”a. k“. .
un.nvbsn.§.tn a a:

' I .
. -0«. - «v- q. , .

a o
x- vbSu‘ bk node...

"v‘ a ",W‘P...‘ -‘r
-O-A b &‘A¢¢-.U “..

’D-n "OFQ ,30-0.’

.J
.“4 . _
- n

u
s..- 5...‘. ~
'

" R? o,
1n:..15..~;

. . '
N‘.,
I 9" .' F:
. , . r
—‘ MAL ‘k..’.'""
it...‘
‘*-..' .
V . ., O. on - ‘
'..\. , ‘ . .
‘ H..»‘.. k); K
e. .
'3. u":- .
.s; ‘1‘ ".“m‘-.-.
‘5‘..‘.‘.
.
,.
’ ._
."'..\,}c-o.‘.‘ H L
.""‘m I}. '
. ..
h $6..
p
N-

.‘OL -
a‘qobgajt 3"A0.1‘ '
t. 5“. . z 1.
o

TLC as- .9,

‘-u’. \ \
o

)5. -t'H' .
‘r‘mC «a,

W 63.74::
[1" L, ‘ .
- yr 'JI Q'L‘.”v_

"“4. s1

' .p' r-
w.
. “"l"u In.”

D.
“‘A‘ \

.5‘3'.P“. h
-. ., L 1 .
v K
3 L,;3 i
'\‘
x “15.
““ie A
\ d TE
' i \,"
s
. .
“.1; sh-

in free space, and that no dielectric layer exists. EFIEs are developed through
enforcement of appropriate boundary conditions on the tangential electric fields at the
perfectly-conducting center-strip surface. Propagation modes, both proper and leaky, are
then examined and identified [25]-[42]. Current distributions associated with various
propagation modes are determined using a Galerkin’s Moment Method technique
employing entire domain (Chebychev) basis functions [43], [44].

In practice, most stripline transmission devices are operated in such a way that
only the fundamental TEM mode is excited. This prompts the development of a TEM
specialization of the isolated stripline full-wave theory. This TEM specialization allows
for the determination of the stripline characteristic impedance. A pole series
representation which is used in place of the real axis integration implicated in the Green’s
function greatly reduces the amount of computational time required for numerical results.

The analysis of the previous chapter is extended in Chapter 5 to the case of a
stripline structure having two coupled center strips [47]. These center strips are assumed
to be of arbitrary width and placement with respect to both the origin of coordinates and
the ground planes (Figure 1.3). Enforcement of boundary conditions on the tangential
electric fields at the surfaces of the conducting center strips leads to two pairs of coupled
EFIEs. Current distributions are determined through a Galerkin’s Moment Method
solution, and the associated propagation modes (proper and leaky) are examined. An
investigation of the symmetric and anti-symmetric propagation modes which are
supported by this structure is also included. As with the analysis of the isolated strip
structure, a TEM specialization of the full-wave theory for coupled stripline is developed.

This allows for determination of the impedance parameters associated with the coupled

'l/A

’A

Egan

 

\\\

‘<

 

 

I
I,»

Figure 1.3 Coupled strip cross—sectional configuration.

‘<

\\\..

x
..»O.k1'\ ‘
bd~d- on

H I ‘
P "8"" 0'
...........‘..~. 0

. 5.. '. .1 . Ci
L‘Jt‘:.;‘xu trA I

if ht; as 3 Ocr“

It.

‘
'3‘ ‘03-.

.1, . ‘
.v~¢«.b ,Lj‘ ~‘IY.:‘
I

"" 'L.) i"..“’.. ~..
W m» alt-3“» U“

9.- .' .
-L 3"x F’g-,,...y‘;
‘ r~~.\..w\,
'
In C» ,...,
A.“ '
‘3’-J~

Kzfifi I ‘ "1
.kn‘b In. ()1.‘ 1:2‘
1‘
1m;
" I .
.‘ w P ‘ f- .
s r.)‘l‘k\ \\:
I‘ 2']
2w.
M"\:C‘~ ' y"

‘41“ t.
'5‘
Vb. 5"." ,
”.ng In
. '.
use. 1C
!.
\l-‘h_
it . ‘1
“ ‘w‘r
“khcrf

strip structure. Again, a pole series representation is used to reduce the amount of
computational time required for numerical results.

Solution of the coupled strip EFIEs is formidable both analytically and
numerically. To reduce complexity, a full-wave perturbation theory for coupled strips
is developed in Chapter 6 [48]. Here, the eigenmode currents of an isolated center strip
are used as a first approximation for the nearly degenerate eigenmode currents of a
loosely coupled pair of center strips [49]. In addition to reducing the amount of time
required for computation of propagation constants, this method typically provides insight
into the nature of coupling effects. Results obtained using this method will be compared
to those presented in Chapter 5.

In Chapter 7, a measurement technique for determining the constitutive
parameters (complex permittivity and permeability) of sample materials placed in a
stripline field applicator is presented. A frequency domain materials characterization
scheme involving stripline field applicators presently exists and has been shown to yield
good results [58]. However, this technique becomes cumbersome and of questionable
accuracy when used to measure the properties of materials at high temperature. Several
of the limitations associated with the frequency-domain technique can be overcome by
exploiting the inherent time-domain response of the stripline applicator [59]. This time-
domain response is synthesized from frequency-domain measurements made at the
applicator ports. Time windows are then applied to the resulting waveform to remove
unwanted signal components. This technique requires fewer measurements than those
currently in use, leading to increased efficiency and accuracy when knowledge of high

temperature material properties is required.

Finally, conclusions and ideas for future work are presented in Chapter 8.

Results of the previous chapters will be summarized and examined here.

'J‘L" “." I ‘.
r .
“scu.‘¢-$-~l|

.. .
‘a so 9'

. .-‘ W"- "“
5"...l Aaqu§-456~
I
'qw r. I-
“ .oQVV

I x .‘-
‘ '~‘-‘o [noises

.5 _4'

\¢\

-och‘\ ’

-.....'=i)r;r a»...
‘ . Ha-..»-
§
I
‘1. ‘
L \ d...'.
u“

f‘ .
A ‘ _
~ “in. _
A . an ~t‘kktr-§["fi
‘i
2 .. . 1
h ""3. ‘ 4
:‘ ““1 Oirr‘. ‘;.'
‘1
3 llshm
aw - .
' ‘N:.|\i\ h

u. 'l
-, u
s. K~M
51‘s ‘7. I‘.
i‘ I

 

Chapter 2

OVERVIEW OF ANALYTICAL TECHNIQUE

The dominant (TEM) mode of stripline structures has been thoroughly investigated
in the past, and is well understood. Little attention, however, has been given to the
higher order modes of stripline. The earliest treatments of stripline [1] dealt with the
determination of characteristic impedance. Other efforts in the past have determined the
cutoff frequencies of higher order modes through such methods as conformal mapping
[2],[3], fmite-element analysis [4], and the transverse resonance technique [5]. These
techniques have yielded only information about the proper discrete spectrum, and not the
improper, higher order leaky modes supported by stripline. The method employed in this
dissertation identifies the complete (proper) and leaky discrete propagation-mode spectral
components for stripline structures. This is accomplished through a singularity expansion
[6] of center-strip currents in the axial Fourier-transform plane. This chapter provides

a general overview of the technique employed in this dissertation.

2.1 Establishment of Helmholtz Equation

 

Analysis begins by considering a generalized, three-dimensional impressed cutie”

source which is suspended in free space. An appropriately specialized sy Stet“ 0‘

Maxwell’s equations is

vxi = voila}? t0
Vxfi = moi + j. 0‘

   

.fiusqk‘J

. ,- o. .
"’ lA-ucsé.
0‘ ‘

1‘ r,” O.) .c ’

~ ut\ 5.~\‘
9—.
. v
‘
O

vi = £— (3)

(4)

<9
:1
II

o

Equation (4) indicates that H may be represented by the curl of a vector potential

function
ii = jweOfoI (5)

where H is the electric Hertz potential [7]-[l4]. Substitution of this expression into (1)

yields

—0

VxE = -j(op.o(jweonfi) = kf(vxf1) = Vx(k02fi) (6)

2

where k: = poeow is the free space wavenumber. Curl operators on both sides of (6)

may be combined

vx(z?:' -k§fi) = o (7)

and by vector identity the terms of equation (7) in parenthesis may be represented as the

gradient of a scalar

E — kzfi = -v<r>. (8)
leading to an expression for the electric field
E = kffi - vq> (9)

11

--o. a;
- ;* O..I .‘a
I

why-II—
... '
‘. ~

'h 1 ’

1.. -'A“ . ‘

.- ‘ .
la» bwtu‘.‘

‘ '

Qw'. .'3 A '
.4..;i.‘]'._‘. ‘1‘.
- w .

I. 4.1.: “-4 '9’!
5 \l\gu ‘~

in terms of the electric Hertz potential and the electric scalar potential V<I> .

Substitution of the Hertz potential representations for electric and magnetic fields

in equations (5) and (9) respectively into equation (2) gives

 

Vx(jwt—:0Vxfi) = .7, + jweo(k02fl -V<I>) (10)
which results in
V(V-II)—V 11 = , +kOII—V<I>. (11)
10060

The standard choice of the Lorentz condition V-fi = —V<I> leads to the inhomogeneous

Helmholtz equation

 

_. a J.
V21] + kOZH = — _ ' (12)
jweo
and the electric field is then recovered from
E’ = (k,2 + vv-)fi. (13)

2.2 Hertz Potential Green’s Function Solution

The Hertz potential solution to (12) will be determined for the general case of a
three-dimensional impressed current source suspended in a region of free space between
two perfectly-conducting ground planes, one of which is coated with a linear, isotropic,

non-magnetic (u, =1) dielectric layer. The resulting solution will later be specialized

12

5

b. I " '9 ‘n
ugh- L11

—J¢

. . n4
a-.. ,.
L. ;\.\u¢.5-U 5‘

O.- '

-.- I“ O "7* ‘
....~ L-...
l I
¢-¢.—*a-—.- -

.C'_\ IF‘D
Mabfitb-m ~

I: 7' .-'" f- — !
urbg. ‘U \‘. I
‘ A

.
'9- ...
4: X151" 4

4 ".
A. 5‘ 4
M . -u_

In.
3'“-
K} "" 4
‘k‘é‘:§ {awoL
OH.
‘I.
3, w . J
ul‘H‘...‘ '4"
\ '
m...“ g
‘a
".
\ M.
“Li‘dl v F .
‘ ‘ \‘.“h
N
‘u

- i
this 15$ »:
g‘ .
C ~‘..~
in.)
K

‘2
$3,. T‘aI
‘ ‘Q‘, t
.l' :
iv‘I‘.Y:.
.-- a
"31.111
A-} "I
a;
“Tit-5.
U

to the case in which no dielectric layer exists. The ground planes and dielectric layers
are assumed to be infinitely wide and extend to infinity in the longitudinal direction.

This uniformity of the structure along these directions prompts the use of a two-

dimensional Fourier transform

 

= 1 .. " 'X-rz (14)
W) (2102 f fume dA

Wily) = ”mac-flaw (15)

-m

in order to solve the Helmholtz equation (12). This two-dimensional transform reduces
the second order partial differential equation (12) to an ordinary differential equation and
leads to a solution in terms of unknown wave amplitudes. These wave amplitudes are
associated with the various reflected and transmitted wave components present within the
structure and are determined through application of boundary conditions on the remaining
spatial variable (boundary conditions on the Hertz potential at the surfaces of the
perfectly-conducting ground planes and at the dielectric interface are established through
application of boundary conditions on individual electric and magnetic field components)

[23], [24]. This results in space domain Hertz potential solutions to the Helmholtz

equation (12) of the form

110) = [th'IFO-fidv’ (16)
V 10060

where G(f’|?’) is a Sommerfeld-integral representation of the resulting Green’s function.

13

2.3 gm D}Jd1t
The tract. 1".“
“:71 piicfiilai CL“:

!
It (:1?) "' "‘ fit‘d 1'

LIA-n ‘ ‘
|

,-
], v“ ,Q

.m.‘ .
Uasu E . thilin ‘3

'1
(.1

.4 . .
\‘W‘ 3: k‘"; ..fi ’9‘”
.s.... . t 55.1 Li.»

¥ ‘S5sllk “‘4‘

Q:

h

um _.L
“L ‘ ‘ll‘ 15 l:

. “3"!!«4 '
5‘ ~I~
A l..‘tu\" 0. “ H
l“ 19.,

k}...

ifm».,, -
‘Q‘ul‘.
n 15 c“'\r“\ N." ;
f ‘ ~ \. g.

“*6; ;
.‘b‘ : T .
Q Q '
i J, "
ML.
"‘11.:on ‘
. "
“QM”? 4
W.“ g ..,
\. 4‘.“
~¢i‘-'-v-4_
’des

2.3 Electric Dyadic Green’s Function

The direct, full—wave analysis of single and coupled stripline structures involves
a Hertz potential Green’s function, as described above. Due to the form of the testing
operators employed in the perturbation method presented in Chapter 6, an electric dyadic

Green’s function is used in this chapter [48]-[51]. The electric field in the region of the

stripline between the perfectly-conducting ground planes may be represented as

EU) = (k: +VV-)fG‘(F|F§°£)-dv’ (17)
10060
V

with the electric dyadic Green’s function
6‘(?]F’) = P.V.(kj +VV-)G(F|F’) +E5(F-?’). (18)

Here I: = 13)? is known as the depolarizing dyad, and P. V. indicates that integration is

performed in the principal value sense. The transform-domain electric dyadic Green’s

function is expressed

§E<fil6’) = P.V.(k3 +vv-)§c(p'|5’) +Za(5 - (3’). (19)
~ .. . _ , a , a _
Here V = V: +2JC With V: — x— +y— where t—x,z, and
x

(20)

is the transform-domain Hertz potential Green’s function representing a current source

Suspended in free space between parallel conducting plates.

14

14 Dexelopment of

Afar m: Hrr‘.

‘ I i !

' . - t r- ' ~,.‘
lb LITE CMJ-‘z-‘nu-
t~lf ‘5') CL? A
RwAszg hi‘ “A: :1
0 U
“0.4 -f..p‘..§» .7 .
.33:qu )L... ...r\.. 5.1:“
a I -

.
.9. -.Q. a r

. .‘
.“u’t’u' :3 L5“ \W;\ i.

0..~o~., . . V
\ .A .- v-w-r- n ..
vh*>U-¥§ b\§1 .‘

“inlsu .j

.. . L ‘
..»"7 .1 I"
u. 4 J‘.'tu|'.~:|“‘r‘n\u 1"
.
’Y“~-n o; - .
l _‘n' .'
than-all :11 or .‘t. a»
8,.

Ta: perk-.11.

‘v

IOU"? ‘

b i ‘9 .

I4...“ dist: n .o. a

‘- us‘rll.‘\

\.‘."".‘.~
VJg-‘uun 0 Q-
“ - I '- I.-
U ninth“;

'3‘.

““35 311’
' ' K

a). “I
“'I\ IS A 1 1
d (“.111 "1“

“4.;
Vii. ]| .pJ’ fi
i..‘d“ .‘ “:
- 5| ~..
‘5.
O
l
N'
0' -
“a:

2.4 Development of Electric Field Integral Equations

After the Hertz potential Green’s function for the generalized stripline structure
has been established, electric field integral equations (EFIES) are developed by
specializing the three-dimensional volume current to surface currents present within
specific stripline geometries (single and coupled center strips). The general Green’s
function is also specialized to represent structures where no dielectric layer exists. The
structures examined in this dissertation will consist of single and coupled strips immersed
in a homogenous, lossless medium between infinite conducting ground planes. The
examination of structures having center strips over a dielectric layer will be left for
future work.

The perfectly-conducting center strips are modeled as infinitely thin surface
current distributions. This is accomplished through enforcement of the boundary
condition on tangential electric fields at the surface of the strip conductors, which

requires that

{(5“ +5“) = 0 (21)

where f is a unit vector tangent to the strip surface, E i is the impressed field maintained

by ii, and E” is the scattered field maintained by surface current 1?. Enforcement of

this boundary condition leads to fundamental EFIEs of the form

f-(kj+vv-)f G(F|F’)°I?(f’)ds’ = —jmeoz‘-E“(f') (22)
C

Where C represents the integration path over the periphery of the strip conductors.

15

tut". innsz'umiifig U

;.« Linton L) ‘75?" 1‘

NO
pv—

I-D ‘(F - ..' ‘ '.
-.f.'.~ - in)» 51..

Wlo§ofiw 1 F'Wlkffl I)
1 lb ‘

b-...-.-|o ‘4‘ l“*

2.51.3315 of (3:! .‘

z- i’
71...! g
1 .9 ' '
“1. 1‘de 5.13: w‘

- u]: ftfl.’;~
QQ‘I.‘:

i-
v D

‘1' ‘4
AQ‘ " .‘ '
Um»

for I
‘

 

Axially transforming the EFIEs term by term, and applying the differentiation and

convolution theorems leads to

{(1.5%) f§C(F|F’)-E(x’,C)dx’ = -jwe,5°é"(F,C) (23)
C

where 9 = V,+z‘jC with V: = x-ag- + 9%, and §C(Flr") and é"(F,C) are the transform-
x

domain Green’s function and impressed electric field, respectively, with C the axial
transform variable. Discrete guided-wave modes of stripline are associated with the pole
singularities of (23). Near these poles, the strip current can be represented as

136.0 . W (24)
(c — (p)

 

where 12;,(x) is regular at C = (p [25]. Substitution of relationship (24) into the EFIE

(23) yields

13,,(x’)

——dx’ = -' 27““, . (25)
“-9) Jon-:0 e (rC)

f-(kj +W-)fgc(r|r’)-
C

The left hand side of (25) is singular at C = (p, while the right-hand side is analytic

everywhere. Resolving this paradox requires

i-(k2+W-)f§,<r>lfid-E,(x’>dx’ = 0 (26)
C

which is the fundamental EFIE for discrete stripline modes, having non-trivial solutions

that exist only for C = Cp.

16

1.5 Numerical Solu

A nurse: ta! 5

‘.

V1. ‘1 inhxiguc

-

B“,"'"f" 4'” o...

I‘s'filtlll 43% 65‘s. .
r

- ‘

3". -"I . - 0..
..;_;...Jn or 5:4...11

5.41.113 of '31: 112"“

its.

3‘.»- ('

able-4" ‘i are \;

“'i'L u.

M“ mi't'fl] CHEN-
4 ' ‘

:LQ'MO -
”Lt-max [\‘lrx ,

T; _-
fax.» [13.29, -.
M C‘Cnt

|“\

.11..”
~ .
“b... I.» ‘
‘5
1n ?»~-.
:5.‘ ‘ u

5;"

I.

~A

2.5 Numerical Solution

A numerical solution is implemented through a Galerkin’s Method of Moments
(MoM) technique [43]. Chebychev polynomials are used as basis functions for both
expansion and testing [44]. The use of Chebychev polynomials allows for closed form

evaluation of spatial integrals. The EFIEs are thus converted to a homogeneous matrix

equation of the form

[A...<<>l[a.l = 0 (27>

where Am“) are spectral integrals and the an are the unknown amplitudes associated

with current expansion functions. This matrix equation has non-trivial solutions for

discrete-mode poles satisfying the determinantal characteristic equation

det [Amn(C)] = 0. (28)

For the case of a single conducting strip structure, the discrete stripline modes
possess either even or odd symmetry about the origin of coordinates [26]. Therefore,
in order to reduce the time required for computation, even or odd mode symmetry is pre-
selected before solution of the matrix equation. When coupled-strip structures are
examined, in general no such symmetry will exist [47]. The resulting discrete modes

will therefore possess neither even nor odd symmetry about the origin.

17

 

 

GREENS f

d ‘ ~
¢ ,-

! I I ‘1
"—usu. . rum:

. u
o
~ ’I ‘ 1,.
3r. .......
A

I .
I ." h I

l
t“»“

1" (up: 1' fh’.
..~‘“‘v ‘4 §A‘\ .

"as

r RAH! -‘.‘ _
£§>.:\

s-«a..:l,;§ .

"".Iu-

.
' l

I, .. _ I“ ,

"we-mil .11 Ll: .

I
F'A 1. ‘
4*» atlz’” 4 1

. ‘fil‘
it...”L

T‘ :
in p“, “
it“, “Mtg“? .

a.“

u...
'.""7"'!‘~ C, ~

0§~' \ oh .‘ .-

IIII~~;‘..

vi ‘” ‘1'..- '
h” Nl\ van“
0 \‘

‘k .3?! ,' ‘
. ‘W‘ -' r
- Mr 1:
A:
9' 'v.
JA¢l§§vhn . ‘-
s“ , (.‘ h
t ”I PM.
1‘.

Chapter 3
GREEN’S FUNCTION FOR A GENERALIZED VOLUME SOURCE IN A

STRIPLINE STRUCTURE

In this chapter, a Green’s function is established for a generalized three-
dirnensional volume source suspended in the space between two parallel, infinitely-wide,
perfectly-conducting plates, one of which is coated with a non-magnetic dielectric
material (Figure 3.1). This Green’s function will serve as the building block for the
EFIEs which will be determined for specific stripline geometries in later chapters. As
outlined in the preceding chapter, a Helmholtz equation is established for Hertz potential
representations of electric and magnetic fields. Boundary conditions on the Hertz
potential at the surface of the perfect conductors, and the free space/dielectric interface
are determined by examining boundary conditions on the electric and magnetic fields.
The uniformity of the cross-sectional geometry prompts the use of Fourier transforms to
solve the Helmholtz equation. This results in a solution in terms Sommerfeld integral
Green’s function representations [7]-[14]. Pole and branch point singularities associated
with this Green’s function will be identified and examined [15]-[22]. Parallel-plate and

surface-wave modes will also be identified.

3.1 Representation of EM Field by Hertz Potential
For the purpose of this analysis, the electric field will be decomposed into a
primary wave E ” maintained by the source .7 in unbounded space, and scattered waves}? ', E ‘

consisting of reflected and transmitted waves excited at the dielectric and conducting

18

 

 

P,
”gut-E

 

 

 

y = h
G‘v (e0, (1.)

X

y = -d

(6.41.)

Figure 3.1 Generalized stripline cross-sectional geometry.

19

I.
I l 9 )“
".M};a, ||15 L|
unit".-

an"‘

. '
Ra «‘13:??14-1‘ H I1
I.

' fi'r“
Sq u~Tl kc LC"F
“N “a. .

t - ' ‘h
.1”; :0 a H "
lab-.4: ‘

““.¢-

01"“... .'
bsslul'

,,.....l

“,1- '
| \q“ ' 7%,; I .
v.1k..."_1g\ L)“ 5145 ‘1‘

2'3 ”Mk 1' l
”-1. ‘fi- 5 .’7
5L4. ‘5‘ £;\-4 $\

1
4- .
«1:72:13 1

Q. inm TN

I;'\- '
' 1r.“ ' 0‘ '
kodhé 3i uh u.:
h ‘I
Q 4‘» k - '
A“ “J l ‘ I‘. ‘
\I] \RT‘C‘”
1.:
_ In). T"; :\ T}
P“ t l
at “'13.,“
\-;x~
5““; "‘" ‘.
Alarm
KL, I
1'?! '
“‘fi“ “‘1‘ ‘7 . I
“J “1L: ""3
. ‘H\:

interfaces. The electric and magnetic fields may be expressed in terms of the electric

Hertz potential fl maintained by the general volume current .7. As stated in Chapter 2,

these may be represented by
E = szI +V(V'I_i) (1)

17 = jwerfi (2)
leading to a Helmholtz equation for H due to .7
v2fi+k2fi = "—1. (3)

Determination of II from equation (3) requires the application of boundary
conditions on the Hertz potential at the surfaces of the perfectly-conducting ground planes
and at the dielectric/ free space interface. These Hertz potential boundary conditions are
determined from boundary conditions on the electric and magnetic fields. Boundary
conditions at the dielectric/ free space interface will be determined first [23], [24]. These
will then be specialized to determine boundary conditions at the surface of the conducting
ground planes. This procedure is detailed in Appendix A.

The inhomogeneous Helmholtz equation (3) may be decomposed into scalar

components

J
V211“ +k311a = °‘ ...for a =x,y,z (4)

 

which will facilitate the application of boundary conditions. The individual scalar field

20

o
.-
“No-9‘ . ‘
-‘ -.;§-Ofir
.‘w...’

7" ' ‘
"‘;‘" ‘ r. .-
unu'..r...‘. »‘.
w
,
\ It‘t‘l .filypd'.
._ “
|

4 .
“‘k~ ‘ 0“»

'
{-00 ‘ .-
hbie 'ufi.

‘~5.1...)]", 01‘

'1.

I6}. dad la

components may then be expressed in terms of Hertz potentials as

 

 

 

_. an an
E. = k’flnitvm H. = jwe(—Z———>’] <5)
6x 8y dz
_. an an
E. = k’Hw—a—(v-m H =jwe( ._ ) (6)
6y ’ az ax
- an an
E = kzn +-(:(V'H) H = jwe(—’- ") . (7)
Z ‘ 62 Z ax ay

Tangential electric and magnetic fields ( B =x,z) are continuous across the dielectric/ free

space interface located at y=0, therefore

EOB(y=o*) = Em(y=0') (8)

Hop(y=0’) = H,,(y=0-). (9)

Here region 0 is free space, and region 1 is the dielectric. Applying these conditions to

(5), (6), and (7) results in the following relationships

. _ 6 ~ 6 ~
Em(y=o ) = E1x(y=0 ) ~ kfIIOx+$(V-110) = k12111x+5)—C(V'H1) (10)

. _ 6 ~ 6 ~
EOz(y=o ) = Elz(y=0 ) - kozIIOz+a—z(V-IIO) = kllelz+éE(V-III) (11)

 

611 an an an
6y 52 8y dz

21

.
1
C" ,‘gnd’
‘1‘ . . a
4 ‘1 -5.Ab.h4
5

;J-‘._."‘ r... ‘

.-u..suab b-
..

‘ u
" F'. ‘ or
P-‘~:uuidd al ‘4.

.‘l' . A. -
“‘5‘ dnish [h
T)‘ ,.

. boundm U

 

 

+ _ 6110 6110 an an”
H0z(y=0 ) = le(y=0) - eo[ axL ay’) = €1( 1?— ) (13)

For a general 3-D current source .7 = in +ny +z‘Jz suspended in the space above the

dielectric region, (11)-(13) lead to the following boundary conditions on the Hertz

potential at the dielectric/ free space interface at y=0

 

 

 

e
1100‘ = 4111“ ...for a = x,y,z (14)
e
an e 6]]
0" = —1—1—p— ...for B = x,z (15)
6y 50 8y
( c3110y _ 6111),) : (1__€_1_)(6H1x + an”). (16)
6y 8y ea 6x dz

Now the boundary conditions on the Hertz potential at the surfaces of the
perfectly-conducting ground planes, located at y=h and y=-d, must be established. At

the surface of a perfect conductor the tangential components ([3 =x,z) of electric field

must vanish, thus

EOB(y=h) = EIBU=-d) = o. (17)

The boundary conditions on the Hertz potential at the dielectric interface are specialized

such that the material in region 1 becomes perfectly conducting (61" -j°°), leading to

an
11 =II =J=O ...aty=h (18)
02 8y

22

u ‘ ‘
”.". “q l! I... -!I
1'...“ g»? 'u 31 u.» .‘u.

3.2 Application of I”
8.1.3.1.?) ML;

.- V'Jtuwzn J . ,m
- H‘s. ..... ”a Irt‘180 N

.

   
    
   

T: "e:‘ Judy} )t- A " ;
"ML: A kl Xv 9”“

“.42.... '

M-flluif" or 31: \'

I
‘1' ~, .

.uo '~ ‘ '
uuu‘fb “I ""‘\' H‘
V‘ _U‘.‘

5.. . “1‘

t

:7 "1‘ .
115.3): Eh: HC‘TI‘} -,

 

a1.

.4 v
a ..
t \ H, ‘
73ml. J1 ‘

50“,.“
12,“. ‘1
. LA 3'
’lg F0 .

11 =11 =——1’-=O...aty=—d (19)
which apply at the surfaces of the ground planes.

3.2 Application of Fourier Transform

Boundary conditions on the Hertz potential have been established, now TI may
be determined from solution of the Helmholtz equation [24]. Several methods exist for
the solution of second order, inhomogeneous partial differential equations. However,

uniformity of the cross-sectional stripline geometry along the tangential (x- and z-)

directions suggests the use of a two dimensional Fourier transform

m.» = [[11, me 7%de (20)

-oo

 

= 1 m- -' jX-FZ (21)
11,6“) (2n)2£f1ta(l,y)e dA

to solve the Helmholtz equation (4). Here X = JEE +2C, F = £x+yy+2z, d2). = dEdC,
and the symbol " ~ " denotes transform domain quantities. Applying (20) and (21), the

following Fourier transform pairs hold

a . a .
— ~ . — ~ (22)
ax 15 62 1C

Em ~ my), Hm « my) (23)

23

‘ _ d'.)

‘5‘.»

11,0) ~ may). (24)

Transform domain representations for electric and magnetic fields follow from equations

(1) and (2)
é’(X,y) = (k2+W-)fi(i’,y) (25)
Why) = jwerfi(X,y) (26)
where
V4” Vivi“! =(ija+z:io+y*i (27)
ay ay
V2472 = 3:, -(€2+C’). (28)
8y

3.3 Decomposition of Hertz Potential
Determination of 11(1)?) is facilitated by decomposing the Hertz potential in such

a way that

71.012» = 115(7),» + 11:01.32). (29)

Here, 115(7, y) is the principal wave of potential that would be excited by .7“ in an
unbounded medium, in this case a region of free space, and ft: (X, y) represents scattered
waves of potential reflected by the perfectly-conducting ground planes, and reflected and
transmitted at the dielectric interface.

From the cross-sectional geometry of this device (Figure 3.2), it is expected that

24

“31116

 

 

 

Y
Y = h
(e... u.) CDV
wp()t)w;',(1)
X

/ y =6

e1 , O 1 a( h 1 (la)
y = -d

Figure 3.2 Principal and scattered Hertz potential wave components.

25

“.12: 313.2311 p.)

'35-“ 9' 1“
r‘ ‘ F 'jtnn

.....ni-o-.. ns ' ‘
w

. .g .. . 0 ~
;'.;\:.‘t 3? “ o1"¥“

.
9 ~ 9' Q

w‘-\v—‘ ,. 1* ‘
nut.x..-~U .-

e

‘ v o
- o. - o 0'- .-
.‘. .c..'.':] L). ILA}: A

F'._'. '0' . ’9‘
“'4' n“ s) is:
I

7" ’l‘g --.‘.. .
L} m. l‘...‘ .r. ‘

l H

“1.1:; :l'!‘ ‘1 . v
uh‘. id“ '. 0
.‘\‘5M

34 Homogmu

”’th1:

“~-
‘Q) 1 4’
UCgL-U‘AD

the principal potential component will consist of upward and downward traveling waves
emanating from the source in the free space region. Likewise, the scattered potential will
consist of waves reflected at the surface of the perfect conductors and dielectric, and also

a transmitted wave in the dielectric region. The scattered potential is determined by the

solution of the homogeneous Helmholtz equation

Vzfi‘ + kzfi’ = 0 (30)

in the free space and dielectric regions, while the principal potential will be determined

by the solution of the inhomogeneous Helmholtz equation

—.

V211” +kffi” = ‘J (31)
jweo

 

in an unbounded region of free space.

3.4 Homogeneous Solution for Scattered Waves of Hertz Potential

The homogeneous transform-domain Helmholtz equation for the scattered potential

may be decomposed into scalar equations

V21}: +k2fi: = O for a = x,y,z (32)

for tangential and normal components. Applying the two—dimensional Fourier transform

to (32) and expanding the Laplacian operator gives

 

2 2 2 0° -
a— + _a__. + 8— +k2) 1 fffi:(x,y)ejA'Fd2A : 0 (33)
ax2 ay2 622 (2102 ..

26

fl _ 1.6-O-grcstlll [‘FX‘.’

'-
13: J£.l\r

-......|
JAM:-
.

..H' '-n'-
i “‘53 {£5qu in

Cumf‘urvv-w ‘
M4 ' \ l ‘

il“
y _

l."
6

I

l‘l'i’mr Y'4. (.
wt. '1 .
“‘“LQHHHlU‘
.'i

1L m‘. CF56 thu‘h “I

93.
n‘|o—.
5 ”(M

\A

 

 

a...“ r
\l; ”A, : ->

I\ ‘

a,“ I

L‘K :‘K‘I‘a

f“~t and ,
l" -
‘ Tree 5 -
pase

p'll‘] : x

‘ ‘a

 

The differential operators and wavenumber quantity may be passed through the spectral

integral

2 2 2 _,
[fl—a— + a— + (9—2 +k2)fi:(7.y)ej*"’d22 = o (34)

which results in

12(2)? y)

 

—(2£ +c2—k2)fi:(x,y)lefird22 = o. (35)

[la—”—

Comparing (35) to (21), it is apparent that equation (35) represents the inverse

Fourier transform of the bracketed quantity

5211:0100

ayz -(62 + <2 —k2)ft:(X,y) . (36)

The inverse Fourier transform of (36) is zero, therefore by the Fourier transform

theorem

52 ~
[—-p(1°) ) trim,» = o (37)
ayz

where p(X) = V52 + C2 — k2 is referred to as the transform-domain wavenumber. The

free space and dielectric regions possess different transform-domain wavenumbers. In
the free space region, pO(X) = «£2 + C2 - k: and in the dielectric region,

121(1) = V52 + C2 - klz, therefore

27

‘ ‘
"‘ 'mv-s o, l h.
.2‘) sri:3\rjt:\g ( I
I
.‘
-. ‘
.,_ .
3’ '
.. .1}.
~.

-1.. ,. T
“we “5“.) a".

l
g In“

1"" .. .l . .
ink‘h‘flll m w”

t...

4113"!) ,
..,¢l.lgd U336“

".V
Ill.~
&

a:" Q ’
A n, _
.3“ ”54-“ ‘1“:

’au. .

- “W"‘l , .

w. Hal [hr.]“‘-h I'
y .

 

2 —O —o
((9—2 —p0().)2)ft:()t,y) = 0 ...for hsysO (38)

2
(£3 -p1(7)2)fi:(7,y) = O for Osys —d. (39)

Equations (38) and (39) have solutions in terms of exponential functions. The transform-

domain Hertz potentials for the scattered field in the free space and dielectric regions are

thus represented by

113.,(1’3) = Wo’.(X)e”°’ + Wo',,(§l‘)e"°y ...for hs y 30 (40)

fisla(X,y) = W1+a(X)e_p‘y + W1; (2.)(2’)1y for Osys —d (41)

where W02, (7) and Wgad) are upward and downward traveling wave amplitudes
(respectively) in the free space region, and Wfa (X) and W1; (X) are upward and
downward traveling wave amplitudes (respectively) in the dielectric region, where

a =x,y,z (again see Figure 3.2). The space-domain Hertz potentials may now be

recovered through the inverse Fourier transform

er'FdZA ...forhsysO (42)

 

 

s = l °° + -° -pov+ - " Po)’
110,,(F) (2102 £flW0,(2)e W0a().)e

1

115 =
”(a (2102

 

ff [W111 (be‘m + W17“ (X)eP1Y]er-Fd2A fOl' Osys -d, (43)

28

"I"
10‘
.
. 1, ...- .n‘
. a . 1'
. us. u... o..
I
"V' 3" ”'5 ._..
h ‘0.» z u,

33 I ‘
5 “ALTU I“..-
Q’h .-
C4 c
5U; r “"i
‘N‘
'9.
r-hl'
ML, 1‘
“‘ ’. :

JV!

3.5 Principal Incident Wave of Hertz Potential in Free Space Region

The principal component of Hertz potential is determined from solution of the

scalar inhomogeneous Helmholtz equation

2 P 2 P _ a

vn0a+kon0a— .
1006

o

...for a = x,y,z (44)

 

in an unbounded region of free space. In such a region, the Green’s function is defined

by
vzopmw) + marl?) = 6(7-79 (45)

where G”(F|F’) = G”(f' - F”). For simplicity the Green’s function in equation (45) is

evaluated for F’ = 0, therefore

(V2 + k02)GP(F) = 43(7) (46)

and the final result can be shifted to an F’ ¢ 0. The two-dimensional Fourier transform
Y

pair

(”(7)

 

7]:

may)

 

T!

with X = i5 +z‘C may be applied given that the integral representation

29

1:1: For the a:

flu... -

vs}, no" ._ ‘

A4 Nut)” Ll]. ;4~}
"Q

 

N

l szer(F~F/)d2A = 5(x—x/)5(Z-Z/)
(2n) _m

 

holds. For the case of F’ = 0, this becomes

1 .. .~ -
6 2 MO) 2
600 (z) (2n)2_j;fe d A
or
f“) : 6_(__y)2ej1 rdZA
where 6(F) = 6(x)6(y)6(z). As a result

V2+2 p~:_5()’) jX-FZ.
( ko)G (r) (”Wife d 2

Application of (47) lead to

(V2 +kf)

 

lfpra‘ )0er rdZA_ _5___()’) ___ffeji rdz)L

)_(211)2 (21:)2

which may be expressed

”“3175 _(€2+C2'kfllépa,”+5()’)}efi""d2). = o.

30

(49)

(50)

(51)

(52)

(53)

(54)

Fm F.’i"."“
LJS x... -5

, 0
")'fl. .'-‘
.a;.b. a b

I In
,o
«45

1
‘A
(f. ‘.-n’...

I .
L La

‘ iaonq\“‘

0i

.1“
(’1;
LA'

f 1
£7"
(1'

~r1

The Fourier transform theorem requires that the integrand of (54) be equal to zero,

therefore

az_24 ~p~ __ (55)
—2 pom Gt).y>— 66)
6y

where pod) = WE" + C2 — k:.

The ordinary differential equation for GP (7,}1) can be solved by exploiting the

one dimensional transform pair

Epdm) = [GWJV ""ydy (56)
(Wily) = i fanammn. (57>
2n _
This leads to
2 -O m— —o - m .
[J9— -P§(A)]—LIGP(Ln)ern : _ifemydn (58)
ayz 2n -.. 21: ..
or
[{[n2 +P02(5C) 577,71) -1}e"“’dn =0- (59)

—00

Again the Fourier transform theorem requires that the integrand of (59) be equal to zero

31

. l ~
H'2‘ ’ngflt .‘
u.~rg.'

Tilt

{'5'

h, ..
3“- .1; :r

.l ",‘3'.’ u
A. al.4‘. If;
3%..

n >‘<0,

m

112 via) 641,11) — 1 =0. (60)

From this, it is seen that

 

 

EP(X,T]) = ‘77:“ (61)
1'l +P00”
and therefore
- a My m m
G”(A,y)= If e dn=if .e . (In. (62)
(2115) _w "2 +p02 2112 -00 (fl _Jpo)(rl +1p0)

The original inversion contour C which lies along the Re{n} axis for -oo < n, < 00

is shown in Figure 3.3. Because

ejny : e -n,yejn,y (63)

it is clear that C must be closed along C; when y > O, and C must be closed along C;

when y<0. For y>0

 

 

 

€111? ejny . ejny
f (17] +f dn = 21tj———. (64)
_,, n2 +p: C, n2 +p02 (ll +JP0) n =J'po
but
8an
f 2 2"" = 0 (65)
C. 1] +Po

32

figure 33 C127:

 

Figure 3.3

 

Complex n-plane inversion contour.

33

A mI
’/‘___..—-L-.._--““‘
+ x/ \\
Coo /// \ \
\
I, y > 0 \\\
/ \
/ n = ' 7’6) \
/ Jpo Q‘ \\
X \‘
/ C l
i / / i /
l n
l C i Tl r
\ X /
\\ - . I”
\ Y < 0 11 - 'Jpo /’
\\ //l
\\
\ /
\\ /
\\\ // C —
‘\\\ ,/’/ (I)
--‘_-____T__-_“,,,,

rt

-13....

' ll
lime. Ior ) < -

,
”9’.“ 1),“
htry‘lv‘ \

Fr

'31 15.15 n

can he

:7
A...
’.

L

elffme

 

therefore

 

Gp(7,y) = _. for y>O.

Likewise, for y < 0

co

ejny ejny . ejny
f dn + f dn = ’27tJ—__.““
_°° n2 +p: n2 +p02 (1) “1170) fl = "fPo

 

 

C;

but

 

therefore

 

GPOCJ) = _‘ for —oo<y< oo

 

and therefore

jX-F -po(i')ly|
GP(FF =ff——e e g d2).
201021000»)

-N

34

(66)

(67)

(68)

(69)

(70)

(71)

iarf =0. The I:
-.::"..".~' '11: Gwen's!

1L1

Tu~ v— ‘
I77! '0' " ‘
All: \‘k'dl p“‘:\":‘

I =U-AI 1‘ §¢.:'
hL.

“41 ’C ’1'.)

buy“

)3sz
, £(ani::i v~

 

—o

for r’ = 0. The result of equation (71) can be extended to the case of F’ ¢ 0, which
yields the Green’s function G P (FIF’) for the three-dimensional scalar Helmholtz equation

(44)

 

w ji -(F— F’) 70(7) ly -y 'l
GPW’) = He " , dZA. <72)
_. 2(2n)2po(k)

The total potential maintained by Ja is represented by

J F’
11‘3er -- [Gurus-film” (73)
V 1(060

and the associated Hertz potential maintained by the current source embedded in the free

space region is then

 

m J f’ ji-(F-f’) Womb-WI
115.0) = f [M f .4 ’3 e , dv’ (74>
_.., V Jweo 2(21t)2p0(l)

where volume and spectral integrations have been exchanged to facilitate matching of

boundary conditions.

3.6 Determination of Unknown Wave Amplitudes

In order to determine the unknown wave amplitudes W6. (7), W01, (X),
Wfa (X) ,and W1} (7) , boundary conditions must be matched at the surfaces of the perfect
conductors and at the dielectric/free space interface. Boundary conditions on the

tangential components of Hertz potential will be matched first, followed by the normal

35

4‘9““ tfir‘”'§
9.0., . slS||k~
5

“)8 101.13 p

-.: w...) - «~—
u xéuehihb Bk‘lbl.

1:1): in: tuul p

.t'Z’QL‘DCIlIS

‘l n 4'.
B‘lu‘l‘Hdr-‘V ~‘

r“ .
plz’fldi :I“. L4,), ..,.
C

“\Wi

AAJ.‘

 

 

components.

The total potential in the free space region (h sysO) consists of both principal

and scattered components

[I = [[3, +113, (75)

Oct

while the total potential in dielectric region (Osys -d) consists only of scattered

components

(76)

Boundary conditions on the tangential (B =x,z) Hertz potential components were

previously determined to be

 

6. 6H 6 an
HO,3 = An”, , 0‘3 = -—1——13 ...at y = 0 (77)
6. 3y 6. 5y
at the dielectric/free space interface, and
110.: = IIOz = 0 ...at y = h (78)
11” = IIlz = 0 ...at y = -d (79)

at the upper and lower conducting ground planes, respectively. The boundary conditions
at the surface of the perfect conductors are applied first to relate the upward and
downward traveling waves in each region, which greatly simplifies the solution.

At the surface of the lower conductor (y=-d), only the scattered potential exists,

therefore

36

9
...L.-'p‘ v- h..r
.Mlbso. “H“

0 0L _. ',.
A; ...JC )m

t“

a..."’: . \ 93‘
>~»susona 5 .
A

~]: “"5- " '1-
L‘ ha. “Jig...

T‘_;_ _ l
1 '\ "‘ a.
45 bk..;‘n‘[]

3 “4'3
at, ~

Hi9(x,-d,z) = O (80)

which, when applied to equation (43), results in the relationship
Wfpd) = -W1-p(X)e -2p‘d. (81)

At the surface of the upper conducting ground plane both the principal and scattered

potentials exist. According to (78), the total Hertz potential must vanish at the surface

of the conductor, therefore

ng(x,h,z)+1139(x,h,z) = o. (82)

This condition, when applied to (42), results in

 

Wow?) = —e"’°”[V,,(X) + WO*,(X)e'P°"] (83)
where
_, J f‘ —jX-i-°’ moduli-y)
V1410“) : f .B( /)e e dv/ (84)
1‘06. 2190(1)

V

is the value of the principal potential evaluated along the surface of the upper ground
plane (y=h).

The boundary conditions at the dielectric interface

6
ng(x,0,z)+11f)p(x,0,z) = E—IHiB(x,0,z) ...for B = x,z (85)

0

37

m""“~1 v
grab

\

A'nv

.«

U . . U ~ '-
. . . . . u c

On v” . '1' t)... ink rrlxww

'7 u ”I 4“ \IK 4 I
. x. -t n. -.u
v. ... 1. 7.
.PM ~L 9.. 1. Wm.“

C ...m ... a. -

l n 1 h ...WI oust

A (.10 FL

and

 

8115,6802) + an3,(x,o,z) _ i aflip(x,0,z)
By By 6. 6y

...for [3 = x,z (86)

combined with the results above lead to a pair of coupled equations involving the

upward-traveling wave amplitude in the free space region and the downward traveling

wave amplitude in the dielectric region

_. _. _ , - _ e X _ _
V,,,().) = V,,(2)e ”0" + 08(2)[1+e 2”°"]+ ——‘ ’1le +19 4’1"] (87)
6. 100(1)

fl 1010»)
6. p001)

 

de) Vhp(7)e "’0" - W0*,,(X)[1 - e 4’8"] + W‘; [1 - e "”14] (88)

where

_. .«z _ " /
1130’) e-W’e ”0"”

jweo 2p0(A)

(89)

 

14,01) = f

V

is the value of the principal potential evaluated at the dielectric interface (y=0).
Equations (87) and (88) can be solved simultaneously, and combined with expressions
(84) and (89) above, yield the four unknown wave amplitudes for the tangential Hertz

potential components
ngd) = [14(1)- Vh(X)e‘”°"]R,(X) (90)
Wold) = -V.(1')e”°" + [V,.(1')e"’°" - V,(X)]R,(X)e “2%" (91)

38

A.

13‘ .2” _.

.. [Ll
s\|dl‘:~c it.“

In: diti'ic’trt.

W,},(X) = [Vh(X)e‘”°" — K(X)]T,()C)e‘2"l" (92)
W,‘,(X) = [Vim - V,(X)e"’°”]T,(X) (93)

which are expressed in terms of a reflection coefficient

p0(X)(1 - e '29") — p1(7)(1 + e ”2“")

 

 

R,(X) : ~ -2 d -2 0h _. -2 d -2 0h (94)
po(k)(l-e p‘ )(1+e p )+p1(A)(1+e 1" )(1-e p)
and a transmission coefficient
. 2po(X)
71(1) = (95)

3120060 - e "”“’)(I + e "”°”) +12. (1)0 + e '2‘“Xm '- e "2”“)l

0

It is noted that in the case that the conducting ground planes are moved an infinite
distance away from the dielectric interface (h,d ~ co) the expressions above specialize to
the well known results obtained for a source suspended in space above a single dielectric
interface (the downward traveling wave in free space and the upward traveling wave in
the dielectric both vanish, and the reflection and transmission coefficients become those
of a single dielectric interface).

The same process employed above is used to determine the four unknown wave

amplitudes for the normal (y-directed) Hertz potential components. The boundary

conditions on the normal components

6H 6H 6 an 6]]
1 0 1 1 1x 1
0y 1’ 6x 62 a y 0

 

 

I V‘ \ i...
TIES) WC.”

\0‘

.3“.

:0 15...“: mt up-\ ‘1

sethflx 3i Yl'm

xnly \

 

___oy_ : 0 ...at y : (97)
8y

II
6 IV = O ...at y = —d (98)
Gy

are applied to the space domain Hertz potential representations (42), (43), and (74). The
boundary conditions at the surface of the perfect conductors are once again applied first
to relate the upward and downward traveling waves in each region. Then the boundary
conditions at the dielectric interface are applied, yielding a pair of equations which are
solved simultaneously to determine the unknown scattered wave amplitudes for the

normal Hertz potential components

w,;(x) = [Kym + mane 1’0" 112,6) + C,(X){15[V,(X) — V,,.(X)e “’°’]

112(2) — V,z(X)e "’°”]} (1 + e ‘2de )(1 - e ‘29")
(99)

 

+11:

W096) = [V603 + Vhydk-MandV 72M 7’ Cn(X){jE[Vix(x) - th(7)e "9°"
+11] V1.00 - th(X)e”’°"]}(1 + e"P1")(1 - e-Zpld)e ~2poh + Vhy(7)e'p°"

(100)

W801) = [(4,0) + V,,,(1'>e "’°"]T,(X)e ‘2le +—:—1Cn(X){j£[Vix(X) — V...(X)e 1’0"]

0

.jc[V,.,(X) — th(7)e ‘Po”]} (1 + e '24" )(1 — e '2’)“ )e W
(101)

40

ill/.3 3 ‘

T23: we 9::

Ralf.)

TNT)

:Q'r‘g‘l ..

LUL

..\ me

W1;(X) = mm + Vhy(X)e"”°"]Tn(X) + §Cn(X){j€[Vix(X) — th(X)e"’°"]
+jc[ V1212) — may "”01} (1 + e w )(1 — e '2“).
(102)

These wave amplitudes are expressed in terms of reflection and transmission coefficients

pO(X)fl(1 + e ‘2”) —pl(i')(1 — e "2“")
R (X) 6" (103)
P06)?“ + e-zpld)(1_ e-ZPoh) +p1(X)(l _ e—2p1d)(1 + e—ZPO’I)

0

 

and

a 2 X
T,,(A) = e M ) (104)
p0(x)_;l(l + e -2P1d>(1 _ e‘zpoh) +p1(x)(1 _ e ‘2P1d)(1 + e-Zpoh)

0

 

as well as a coupling coefficient

2(3 — 1),,011')
Cad) = 6° 5 (105)
[X000 + X1 (bursa)? + Y1 (X)

0

 

where

Xod) = po(X>(1-e'2Pld)(1+e‘2Poh) (106)

41

oc' 'hc: grew

"'3F“’. \;
“in Iflk~

11‘ .L‘u

. “.
‘ .1 “1 ..,.
‘1“ *1 1.11 w»

('0!

 

X100 = p,(5€)(1 + e ‘2”1")(1 - e‘zpo”) (107)
YOUC) = pO(X)(1 + e 'ZPI")(1 _ e‘ZPo") (108)

Yld) = P1(X)(l - e'ZP‘d)(1 + e_2p°h) (109)

and the expressions for the principal potential evaluated along the dielectric/free space

interface

«I _~”,’-_./ _ X ’
Jy(r) e 11 e PM W dv’ (110)

 

v.00 = f

V jweo 2p0(}t)

and along the surface of the upper ground plane

Jy(i-") e 41°44 e -po(i)(h -y)

 

VhyOC) = f (111)

Again these results show the correct behavior in the case where both conductors are

moved an infinite distance away from the dielectric interface.

3.7 Hertz Potential Components and Associated Green’s Functions

The normal and tangential components of Hertz potential in each region can now
be determined by substituting the wave amplitudes into (42) and (43). The resulting
potentials in each region can be expressed in terms of Green’s functions. Thus, the total

tangential potential in the free space region (h sy 30) can be represented by

42

31:1:

 

G}: Ir

_./
m (,2.le
V 0

 

G”(f‘|f") + GO", (Flf’)]dv’ (112)

 

 

where
m "-F-F’ - (DI ' [I
2010217000
and
°° -* r _ -po(2h-y’-y)
Gorp(Flf./) : ff [Rr(A)AOB(y’y/) e Jer-(F-F’)d2A' (114)
201021900»)
with
Ar (y y/) = [e—poo’w) _e-po(2h+y’-y> _e-po<2h-y’+y> ”wow-WW) (115)
op :

for [3 =x,z. Likewise, the normal component of potential in the free space region is

J (“’)
1W =f{ ,-i.'. l
V 0

M”) _a_ err > a
jweo 6x jweo 62

     

 

 

(FIF’) + Gory(F|F/)] +[ —]Go,(Fl F’)}dv’

(116)

where

2-h
[IJ [R "(1)/1030’ 00],) + e poa y ”J ejx'(7'F/)d21 (117)

0113011") 2(21t)2p (X)

43

e
P
A.

.. «L

0“

.3

Dr.
4“. ~
1

Flu,”

(“4‘1 “
C

.5)
‘s

«a;
i‘~.“

\
hp}
l‘~

with

- ’+ - 211+ ’— - 2h-’+ - 411- ’—
AoryCYyl) = [8 p00 y)+e po( y y)+e po( y y)+e M y y) (118)

 

and
—2dpl _ e-Zpld C
G0y(r|”5 ffC(X)(1+e )(le )AoyU’ysefi‘F‘F’mZA (119)
20702 P0 (X)
with
A06 0 1 : e-p00’+y) _e-po(2h-y’+y) +e-po(2h+y’-y) _e-Po(4h-y'->’) (120)
y .

The total tangential potential in the dielectric region (Osys -d) can be

represented in similar fashion

 

J80”) t ~~/ /
H1501) = f . Glp(r|r )dv for B =x,y (121)
V jwel

where

(122)

-_—./

e-po(2h -y’) _ e_p°yl)(e -p1(2d +y’) _ ear”
ejl (r— r )dZ)‘

 

G‘ (F r") = T151) _,
1" I H 212111211011)

Finally, the total normal potential in the dielectric region is

111:?

 

 

     

J (r’) +2?J( ’) 123
'8'- a —]Gly(r (FIFI) dV/( )
jooel ax jcne1 az

‘,f‘1?’>+—:l[

44

‘ ‘.§
.1-

("\

(’1
> .
‘1

where

°° - ’ — 211- ’ - 211+ *
t «fl, fi(epw +epd y)“ pd 10+emw
Glyelr) = [fun

e (124)

44/

er-(r—r )dzA.

 

21210212011)

and

(e ~p,(2d+y) + ep1y)(e

‘I’o.y _ e ‘Po(2h -y'))

- -, (125)

ejx'("')d21 .

 

ny(f'|f")=ffC,,(X)(1+€ 'WXl ‘3 QM) 20102,, (X)
4” 0

3.8 Examination of Green’s Function Singularities

The wavenumber parameters pod) = (Hz — kc2 and p10.) = ((212 — klz, introduce
apparent branch points at A = 1km. These branch points are removable, because the
associated Green’s functions and are all even in both pO(X) and p10.) The
inhomogeneously loaded stripline structure will support both parallel plate modes (which
propagate in the total region between the parallel ground plates) and surface wave modes
(which propagate only in the dielectric region, and are evanescent in the free space
region). Poles associated with the parallel-plate background structure occur for particular

spatial frequencies AZ 0 = k2 which lead to vanishing of the denominators in

2
11.} 11} ”131

R‘, R", and C". When the operating frequency is sufficiently low, then the AZ 0 are
p

{1}
imaginary, and both the parallel-plate and surface wave modes are evanescent
(Figure 3.4). Propagating parallel plate modes occur when p0 is imaginary ( AP < kc),

and surface wave modes occur for p0 real and p1 imaginary (k0 < AP < k1).

The relationship A: = E: + C2 leads to the expression

45

Q‘V'aWESCe.
modes

figure 34

1i propagating parallel

 

plate modes
M r
x x r x c >
k0 k1
evanescent propagating surface
modes wave mode

Figure 3.4 Poles associated with parallel-plate and surface-wave modes.

46

--.'O1"L. I fi',‘ ‘ 0
M1.“ 1111,1141»

£10532 10 11111111.:
?";'1’V‘- x1 ..

($.3uuui In...”

.1 ,v ’ .

0; ‘1 1s 16);) 11.

.. ‘ I
Ram (11:1 '1

:1: 1113111“

 

5p : -j\/C_AP l/C+}‘p (126)

which implicates branch points at C = i AP in the complex C -plane. Branch cuts are
chosen to maintain Im{ 5p} < O on the top Riemann sheet. Leaky modes occur when the
operating frequency is such that at least one AP lies on the real C-axis, and the real part
of (p is less than AP. In this case, roots are found only on the lower (improper)

Riemann sheet [25], [27]-[42]. This will be discussed in greater detail in the chapters

that follow.

47

HH-

.1-
d n1.

.
‘JYJM'V‘A
ubssinn u

l“

21:55:: of 1
?L’i;:IHC 911.?“-
spm bet-'1 ::r
311.51 “111 1
50131:: 10 :11] u

The EHES 111

Chapter 4

FULL-WAVE ANALYSIS OF ISOLATED STRIPLINE STRUCTURES

In this chapter, the structure consisting of a single perfectly-conducting center
strip suspended in free space between two parallel, infinitely-wide ground planes will be
investigated. The complete propagation mode spectrum for this structure will be
determined through a full-wave integral operator technique involving a singularity
expansion of currents in the axial Fourier-transform plane [6]. The Hertz potentials
established in the previous chapter will first be modified to represent a homogeneous
stripline cross-section consisting of a three—dimensional current source suspended in free
space between parallel conducting ground planes. Electric field integral equations
(EFIES) will be formulated by specializing the generalized three-dimensional current
source to an infinitely-thin surface current through application of boundary conditions.
The EFIEs will be solved using a Galerkin’s Moment Method technique, and current
distributions on the center strip will be determined for various propagation modes. The
discrete higher-order modes, both proper and leaky, which are supported by this single

strip structure will be investigated and identified using complex analysis.

4.1 Green’s Function for Homogeneous Stripline Cross-Section

The stripline structures considered in this and the following chapters consist of
perfectly-conducting center strips which are suspended in free space between parallel
ground planes. Therefore, the Hertz potential functions established in the previous

chapter for a generalized stripline cross-section are specialized for the case where no

48

J."-ro.~ '-'.b1
1.1mm L111.

' l
l h. N " ‘
‘3‘) P‘ T " 1
uamSul‘ aCLN
-

’L‘ O 11 l ‘
2.5 2.11.1111. p”:

1‘

fll'IF v ,. o-
"euuul “‘1‘ Lu

\'~.I_ _ _‘
“arfsth 1n

.Ifm‘ ‘fin’! ~
Amt . “Yup _

dielectric layer is present. This is accomplished by letting the permittivity of the
dielectric region become that of free space (61 ~ ea , which implies kl ~ kc) and placing

the ground planes equidistant from the origin at y: 1h. For this case the transform-

domain wavenumbers in each region are identical
11,1711 = (/£2+r2-k,’ = (12+? -k3 = p011) . (1)

Substituting these conditions into the Hertz potentials for the generalized stripline

cross—section results in reflection coefficients for tangential and normal potential
components given by

—2p0h _. e-ZpOh
R (A) = —— (2)

n

11 - e"‘”’1 (1 - W1

12,011 = '8

transmission coefficients for normal and tangential components

710-) = 71,0.) = m (3)

and normal component coupling coefficient

C,,(X) = o. (4)

The resulting space-domain Hertz potential functions represent a three-dimensional

current source suspended in free space between two parallel ground planes (Figure 4.1).

These can be represented in terms of Green’s functions by

110’ ’)
jwe

 

11m = f 004F114)- dv’ (5)
V

0

Where the associated Green’s function 0011 (Flf’) consists of both principal and scattered

49

‘13.

he”

Y

“Elite 4.1

 

><\/

Figure 4.1 Generalized three-dimensional current source suspended between parallel
conducting ground planes.

50

c
.--u- “1.4"
b‘~]-”:\ tab nut.‘
I

' V '
p.) 9b,. firm
i u.b “13 r u u

-1

*—v

“\1
H
’1
-4
I

components

00.61?) = Go”: (111”) + G: (r111). (61

Here the principal Green’s function is

 

0° -po|y-y’l ji-(F-F’)
GJJFV’) = H e 2" - 11211 (7)
_,, 2(21$) 170(1)

for a=x,y,z, and the scattered Green’s functions are

-(4h+’-) —(4h—’+) —(2h+’+) - 2h-’—)
:20 yy+epo yy_epo yy_epo( yy] 11, (8)

eji-(r-r)d21

 

. 1- _ ..[e
(WWW-Inf 2(21r)2p0(1 -e 4PM)

for the tangential components of Hertz potential B=x,y, and

(9)

- 411+ ’- — 4 - ’ — 211 ’+ - - ’-
Po( 1 y)+e Po( k y +y)+e PM +1 ”+8 Po(2h y 1)] 1.2—4 2
re’ (’ ”d A

 

S —0 —’ _ a [e
GOy(rlr’) ”If 2(2n)2p0(1-e—4p0h)

for normal components.

4.2 Integral Equation Formulation

In order to represent a stripline structure, the generalized three-dimensional
Current source must be further specialized to the surface current on an infmitely—thin,
Perfectly-conducting strip, centered at the origin, which is suspended between parallel
Conducting ground planes (Figure 4.2). This is accomplished through enforcement of
boundary conditions at the center-strip surface. The electric field may be represented in

terms of a Green’s function by

51

‘l\

21

.5.

 

 

2h

y=h
020::
191191 1?
y=-h

 

Figure 4.2 Stripline structure immersed in layered background environment.

52

L. , . ,, 4..
212.51.”).

Thu k
nt¥ \'

NI‘WWQ '1‘. o

I
'1
Ah...» M1111 L4
|
' 7‘ -.
I 1 “ ‘Im'
s I: H-.‘ ““‘.\

1
1‘».
avg
“*1;

“A: JED.C\\
a

93111‘1jv. .'1 CU"

Em = (k:+VV-)f “fifl-fldv’ (10)
V

where the dyadic Green’s function may be expressed as

6061?”) = (1122100, (FIF’) + 1105,0111). 1111

The boundary conditions at the surface of the perfectly-conducting center strip

require that the tangential components of electric field vanish there

f-E = O ...at y=0, for -asxsa. (12)

It is assumed that the current present on the conducting center strip consists of an

induced current 1?, and an impressed, excitatory current .79. In this case the total

electric field is

Em = 3‘1?) + E5111 <13)
Where
5‘0): (113+vv)c’iof(r|r)-Jj—;—(:)dv’ (14)
V 60

is the impressed field maintained by je, and

E(;)— (k§+vv)fGo(r1r’) K——(r/)ds (15)

jweods

is the scattered field maintained by the induced surface current 1?. Enforcement of the

boundary condition at the surface of the center-strip conductor results in
LEW) = —f-E'(F) (16)

53

4.3 Axial

T.;.é.1!'“
.....1..g.

“U.

o. L ‘ .. -
* 3.”

“~15 p :

15 C131"

'1
'.
“11);“.

7’3. .
l 1 ’_
k“5:11

- J“

at y=0, for —asxsa, and f = 12,2. This leads to the fundamental EFIE

111:1{5 -(kj+vv-) f 6001;”) ~I?(r")ds’} = -jo> e0 1im{f ~E"(F)}. (17)
S

y«0 Y*0

4.3 Axial Transform-Domain Integral Equation
The perfectly-conducting center strip is assumed to have zero thickness.

Therefore, the associated surface current has no normal (i-directed) variation, thus

130') = I?(x,z) (18)

and

(30(F|F’) : 60(5|5/;z—z’) (19)

where p = ix +9); is a two-dimensional position vector. The integral operator in (17)

is convolutional with respect to the axial variable 2,
a

[60(7|7/)'12(F/)d8’=fféom'l5’;z-z’)'l?(x’,z’)dz’dx’ (20)
S

Which prompts application of the Fourier transform pair

f(€) = f F(z)e""zdz <21)
= _1_.. NZ (22)
Fm 2“£11m: dc.

Transforming the EFIE (17) term by term, and applying the differentiation and

54

c
‘. fl. -. . v.
.l).,‘l..l..§nu’~

" “39;; T-
‘u~.b '

7

‘
YOVfihfl"
. ., ..5. t
I

I H '
.1~r':

convolution theorems leads to the transform—domain EFIE

fim{f'(k: +\"7v-)f§((p° lfi’) -I€(x’,c)dx’} = -jweo 111:1{5 «am-5,0} (23)

y-0 —a y~0

where {7 = VfiiiC , V, = xai wig, and I; is the surface current residing on the
x

periphery of the center strip. The transform—domain dyadic Green’s function

5545150 = (fi+z‘z‘)g?(6lb") +5ry‘gc’(b‘li5’) (24)

consists of both tangential (B =x,z) and normal components, which in turn are composed

of principal and scattered components

 

 

g.“<5l5’) = gé’wlp") +g§"(5l5’) (25>
gimp") = 85(filfi’)+g§y(i5l6’) (26)
Where
.0 "poly‘y/l J'E(x-X’)
_. _. e e
g{(p|p’) = f _. (27)
_.,., 41tpo(A)
and
°° -Po(4h+y’-y) +e-po(4h-y’+y) _e-po(2h+y’+y) _e-p0(2h—y’—y) (28)
853(5I5I)=f[ _, _4 h Jejflx-Hdg
_,., 4npo()t)(1—e ”0 )
°° [e -po(4h+y’-y) +e ~po(4h—y’+y) +e'po(2h+y’+y) +e ~po(2h -y’-y)] . I (29)
e1£(x_x )dE .

 

sy ~ ~/
gc (PIP )= .. -
41: p0()t)(1-e 4"0”)

Near simple pole singularities for propagation constant Cp, the transform-domain

55

q'",,f‘.. 6‘ 4‘
b‘AAEI“ llsu‘

I
1
H
T"&
h~ Y"
‘K 4P
“a“

L70]

current may be approximated by

 

k (x, C) z p (30)

where kp (x) is the eigenmode current associated with the p‘h propagation mode [25].

Substitution of this approximation into (23) leads to

 

. .. " IE'(x’) . . (31)
1' {lim t-k02+VV- “ " “’- p dx’ }=—' 1' {lim r“ “,C }
(if: Mi ( )£g((plp) (C-Cp) } Jweoclf: y”0’{ e (p )}

It can be seen that as C ~ (p, the left hand side of (31) becomes singular, while the right
hand side remains regular. This apparent paradox is resolved by requiring that the left

hand side of (31) become indeterminant as C ~ Cp, thus

A

t°(k:+V§~)f§c(§|6/)'fp(x’)dx’ = o. (32)

Equation (32) is the fundamental EFIE for the discrete modes of stripline structures with

isolated center strips.

4-4 Scalar x- and z—Component Integral Equations
Because the center strip has no y-directed current, the eigenmode current 13;, (x’)

may be decomposed into tangential components

lax) = ka(x) +z‘kz(x). (33)

The transform-domain Green’s function is
8’46 | 5’) = (“f + 2‘2)[g¢”(fi lb") + 859(5 I69] + fy‘[g¢”(6 lfi’) + 8;”(6 I60] (34)

56

that fl = .t.:

0i

AWN :_
"Hmu‘g it}
5 “A.

i». -
Lia-\Oa
«.le‘l
‘ MW

in.”
d? ..
1 L of 9011;

where [3 = x,z, and therefore

am I 5’) -I€(x’) = 2k,(x’) + 2k,(x’)][g{<5 lb”) +g5"(6 I59] <35)

01'

gm) I 13’) ~E<x’) = 2k,(x’) + z‘k,(x’)]gf(p’ 15’). (36)
Now the scalar x- and z-component equations are given by

12-(k3+ ' VIfgc‘Rrs I a") .120.)de
-a (37)
2 [Hi +k2)gfi(f5 l6/)k (XI) +jC_a-gp(‘—5 Iii/)k (x/)]dx/
3x2 0 C x ax C Z

—a

and

2-(k3 + ' v-Ifgé’as I 6’) -E(x’)dx’
‘0 , (38)

= f [jca—igfm Ifi’)k,(x’) + (k3 — (2)5236 Irs’)k,<x’)]dx’.

'0

Assuming that the conducting center strip lies in the plane of the origin y’ = O, and the

transform-domain Green’s function takes the form

35a) ID”) = gg’(x,ny/,0) <39)
21 pair of coupled integral equations result
f [kx(x’)Kn(x|x’) + kz(x’)sz(x|x’)]dx’ = o (40)

’0

57

'1 5.1.?
*usl

., ,
‘hh
“I? . r
3)}. ‘i
3..., .
“$111111"

f [kx(x’)Ku(xlx’) + kz(x’)Ku(x|x’)]dx’ = 0

‘0

 

where
. 62 2 a
Ku(x|x/) : 11m (_ +ko)g( (x,ylx/,O)
r0 6x2
. . a a
sz(xlx’) = Ku(x|x’) = lunJCa—gdmlx’fl)
y~0 x
Ku(xlx’) = lim (k3 — c2)g;‘(x,ny’,0)
y~0
and
p I °° ejE(x-x’)
x, x,0 = W ,, d
g.( yl ) [41‘ (5):)5
with

(41)

(42)

(43)

(44)

(45)

W(E,y, C) = i{e "Poly! +___1___ [_e "P00" ”0 _ e ‘Po(2h‘)’) + e “Po(4h +y) + e 'Po(4h‘)’)] (46)
(1

Po -e_4p°h)

Substituting these expressions into (40) and (41) gives
a

. 62 + 2)”eje(x-x’)
11.11““; k,f 4n W(€,y,0)da

y*0 -m

 

 

kx(x’)dx’

‘0
a

 

6x w 11'

—a

58

+ [LC—a— f eff-x ) W(E,y,0)d§]kz(xl)dx/} : O

(47)

4.5 Green's
“It It

. ‘ . .
“.1 ‘ .V.
3 (no I 18 5‘

'7‘)
«_3
v
'I‘
fh‘

at»,

‘ \

(I)

1im{ f [it-‘3 f emf ) W<&,y,0)d£ k,(x’)dx’
x a 4n

-a ‘ <48)

 

 

w J'E(x -X’)

 

 

W(E,y,0)d£ kz(x’)dx’} = o.

4.5 Green’s Function Singularities and Axial Transform-Plane Branch Cuts
The transform—domain Green’s function for transverse components of Hertz

potential is given by

e}'€(x-x’)

W(€,y,C)d€ (49>

 

B /
x, x,0 =
g,( yl ) L 11:

Where

W(an,C) : i {e ‘PolYI + —1_4_ [‘8 ‘Po(2h +y) “'8 ‘p0(2h-y) + e —p0(4h+y) +8 —po(4h -y):|}.(50)
po (1 -e M)

For y>0

W(€,y,C) -1— {e W + ——%[ —e "P09" ”9 _ e 72.12}: —y) + 3 100(4). +y) + 8 120(4). —y)]}
po (1 —e p° )
(51)

e “’0’ — e “”0"” ‘y’ _ sinh(po[h -y])

p0(1 + e '2Poh) pocosh(poh) '

 

 

It Can be seen that (51) is even with respect to parameter p0. Likewise, for y < O

59

) .5
W“ h
_ g‘vlf ;
5U
ab. ‘
U
\K

pg...
‘5‘

0!

“IL...“
UJC n '
15 ‘
t

F.
‘Um rl‘
wig. I

M5050 i {8W + ———1_4—[—e "”09" *y’ — e 'poah—y) ..e 720(4); +y) +3 -po(4h—y)]}
p0 (l-e M)

(52)

e —e . 8.11,, [h M)

-2poh) pocosh(poh) °

 

 

p0(1 +e

Equation (52) is also even with respect to parameter p0, and thus the branch point
associated with (50) is removable.

Poles of W(£ , y, C) occur for cosh(p0h) = 0 or for

p0,, = ”£236.14“ (53)

01'

: .(2n-l)1t (54)
p" :1 2h

Where n is a positive integer (n=1,2,3,...). As stated previously,

p3 = 12 - k2 (55)

With AZ = £2 + (2, therefore

2
= )2 _ k2 (56)

P 0 '

(2n - 1)1t
2h

 

 

From this the propagation eigenvalues hp at the poles of W(E, y, C ) are found to be

A: k2_[(2n-l)rc
.I. —2.

These poles are associated with waves which are guided by the parallel ground-plate

 

’ (57)

 

6O

» ~-o
rm“ ,
'|\-._“Nl

. -

"‘ - I
.L‘>..k..1.f

'9'" V a-
l. 4'
““‘s...u.
»

?’ .fi

4 ..A
“Nu-\f
,‘ ‘

I_‘ m.)

‘t‘m‘h

i’ ’
,
.1,"
LA.
.—¢

Film“ ~3-

Or

5‘
'u‘r 9‘
l .

A.\

in:

background structure, having plate separation 2h [26]. When the frequency of operation
is such that k0 < (2n — 1)1t / (2h) it is seen that AP is imaginary, and thus the associated
parallel-plate modes are evanescent. When k: is sufficiently large, )\p is real, and
propagating modes exist which are guided by the background structure. The poles
associated with the guided-wave modes of the background structure migrate from the
imaginary axis to the real axis in the complex k-plane as the operating frequency
increases (Figure 4.3). While an infinite number of discrete poles exist in the complex
A-plane, only a finite number lie along the real—k axis for a given operating frequency.

The location of poles in the complex g-plane is dependent upon the positions of
KP and the value of g“ in the complex {-plane. The relationship between X, and the

location of {p is determined from

£3, + c2 = xi. (58)

From this it is seen that

1:2— 11:)

=_(€ — 1p)“ + A1:)

2 : AZ _ 2
5P P C (59)

 

E, = -j\/C - AM/c + AP. (60)

This expression implicates branch points at C = iAp in the complex {—plane. In
OrdEr to insure solutions that lead to proper modes associated with fields which satisfy
the radiation condition, branch cuts must be chosen to restrict migration of singularities

in the complex E-plane (Figure 4.4). These singularities must be prevented from

61

“gum 4.

 

Figure 4.3

 

lp

Location of parallel-plate guided-wave poles in complex k-plane.

62

FlSUI

:/

 

x<-—x J5“

m X->X
m V

 

Figure 4.4 Location of parallel-plate guided-wave poles in complex E-plane.

63

m.~ol' ”a a

u..'...n.u..
‘ U

I
‘H- 'P ‘_. - ‘
s-ANLA “\NH

‘ I
“A" h .‘m

‘stbtl bed.

r-LI .
t“*~»U‘\
‘3? Sim

r

F? .
}E.II.DN’
A T‘
"a
51“}.
.r 4
‘41“

(I'?

migrating across the Re{E} axis to provide a strip of convergence, leading to fields

which decay in directions away from the guiding structure. Thus Im{ 6p} <0, or

 

Re{ cz—A;}=Re{¢c—ip\/c+)p}>o (61)

is the appropriate criterion. This leads to the conventional Sommerfeld [8] branch cuts

which emanate from the branch points at C = :lp (Figure 4.5).

4.6 Improper Discrete Stripline Modes

The discrete modes of the propagation spectrum identified above are proper
guided-wave modes. In the absence of loss, the fields associated with the proper modes
of stripline give rise to waves which propagate axially without attenuation, and are
confined in directions transverse to the guiding structure. The poles associated with these
proper modes lie on the top Riemann sheet of the multi-sheeted f-plane.

There also exist improper leaky-wave poles which lie on the lower Riemann
sheets [25], [27]-[42]. These modes are associated with waves which attenuate axially,
and grow in directions transverse to the guiding structure. The improper modes lose
power to transversely propagating TM parallel plate modes. Leakage occurs when at
least one 3“,, moves off the Re{C} axis, and becomes complex with Re{C} < AP, and
Im{ C } < O. In this case, roots can no longer be found on the top sheet. If passage is
made to the bottom sheet by violating (passing above and then down across) the {p-related
branch cut, then roots are again found which correspond to the leaky-wave modes.
Passage to the bottom sheet causes the E—plane pole at Ep to migrate across theRe{ E}

axis. The transform—domain Green’s function for transverse components of Hertz

64

fight

Figure 4.5

 

 

Branch-point singularities and inversion-contour deformation in the
complex {-plane; branch cuts appropriate to restrict E-plane pole
migration.

65

l

maxim 149)

m»... 4,, . .
1534—14“ at .“ s

v

«a - .v~
Lkltstfd ill I

“t

I ‘3 " ‘\fi’v
n-

4.
“‘3‘ \k Mu

‘hun

- nun»

G'aan

‘&¥'u

DCiL‘I

E: Icrm I

33d

.‘2‘

h
“R

I)
‘7-1

potential (49) involves integration along the Re{€} axis. To retain physical continuity
of the mode after poles have migrated across the Re{ 5} axis, the integration path must
remain above or below the pole at Ep. The integration path is therefore deformed as
indicated in Figure 4.6 when improper leaky-wave modes exist. This leads to a pole
residue contribution to the real axis integration implicated in the transform-domain

Green’s function (49).

Determination of the pole residue contribution requires the evaluation of

fwmmLzE d5. (62)
Cp P

The term W(E,y, C ) may be expressed

W(€,y,C) = M (63)
po(A)D<£)

Where

NGJKI) = {(1 _ e '4Poh ) e —p0|yl _ e -po(2h +y) _ e -po(2h -y) + e -p0(4h + y) + e -po(4h m} (64)

and
0(5) = 1—e‘4”°". (65)
At the pole
a: = 7W ‘66)
and

66

figure

 

 

 

. . P p
mvers10n contour \ X
__, A. are

 

Figure 4.6 Inversion contour in complex E-plane.

67

"9.; 0L. .

‘ .
nah “USAF

'/
' l
E“
.‘
J 'I
“_o

“I
I‘I
Lu

‘41’0” _

 

 

 

1 — e O
or
e—4p0h = 1
and thus
Np(€,y,C) = {-ew‘flhW)—e—p°(2h—y)+e'p°y+ep°y}.
Near Ep
— dD
D =1-e"”°"zD +— -
(a) (6,) dE Hire a,)
_ dp dp
=—4h(-e ’”°"—°) (a—a )=4h-—° (E-E)
d5 54,, " d5 54,. p
but
1132 _ 3
d8 0
and therefore
4h£
0,,(5) = ”(a — a.)
p
Po

where po" = 513+? —k: . This leads to

P P P
- 2h+ - 2h— —
[-e po( ”-e po( y)+e pox pox]

N,(£) _
poplars 4%“ " 5,.)

 

 

W,(€.y,C) =

68

(67)

(68)

(69)

(70)

(71)

(72)

(73)

. I
'- 1' '
liar ..«L .
de tnw-:u-
-

~..- ‘. . :
‘ v0
L‘h‘u’nusLsu
30‘ 0L

,3"...
“I“ L315.\I(|

h .
”I V .
T ‘ 'I r.
«A. i.‘,¥§
s

)7.

mm.- ,

“mum
“til

, .

*‘fia.

«(E e!

The integral around Cp

 

dE
“—— 74
C (a - 5,) ‘ ’
is evaluated by letting
E = Ep +eej'” (75)
and therefore
dE =jeequr. (76)
This gives
d a," fill
___‘E_ = &d¢ = -2ch (77)
(5 ‘ 5p) 0 6e”
CP
and then
W(E Ode - [_e-pg(2h.y,_e-p,P(2I.—,)+e—p:y+epopy] (78)
s 9 = — TE
f. y . 21..
CP

4.7 Method of Moments Numerical Solution

Equations (47) and (48) are solved using a Galerkin’s Moment Method

formulation [43]. The current functions are first expanded such that

No
kw“) = 2,0365% (79>

Where e; (x’) is a basis set, the 0; represent unknown amplitude coefficients, and

69

'5 ,9
‘u‘ lb

list) let

7-; . I

19?; 99,. _
~““JLL\C\

0%fl. :
Lu». :‘Hrc

iprC/W’jfixldx/ = fiaé’idk’k-ffix’dx’ = "if: afff‘fig) (80)
where
fp"(€) = je;(x’)e'j“/dx’. (81)
Also let
85"(5) = itflxwfiw (82)

‘0

where t6" (x) is a testing function which will be applied later. Application of the partial

derivatives in (40) and (41) gives

 

 

 

 

 

(idea-fl) = ijgejux-x’) : _52ejz(x—x) (83)
6x2 6x
2
therefore —a—3 ~ —52 and 63 -j€ in (47) and (48). These substitutions result in
6x x
.9 2 ejE(x-x’)
lim f f(k,—52) W(e.y.0)d£ -k.(x’)dx’
yaO _°° 41!
‘0 (84)
” ej€(x-x’)
— f [ca 4“ W(a,y.0)d5 ~k.(x’)dx’ = 0
. " °° ej£(x-x’) I /
hm -f fa W(£,y,0)d£ -k,(x)dx
’”° -a -~ 4" (85)

 

 

 

+ [[(k: _ (2)]‘ejjx1gxfi W(€,y,0)d§ 'kz(x’)dx’} ___ O

70

I
E “2", ""fi
turbu.‘
A E

Iirn'

I~C

you tip”.
I I ‘

Expanding the current functions as specified in (79) gives

 

 

 

 

 

a on {(x-x’) N1
1. k2- 2 e] _ n n / /
ymo{.f[f..( ° 5 ) 4n W(E’y’0)d5 "20.6. (x )dx (86)
a on emxfll) N: n n
’ f f“ 4“ W(E,y.0)d€ -Z (11.220.5de z 0
-a "°° n=l
Im{-}[ 0°CE ej£(X"X/) W(E,y,0)dE .E ane "(x/)dx/
y~0 _a w 41: ":1 I x (87)

 

a

+ f [(83 — 6’); 3):.-.) W(&.y.0)d£

‘II

 

 

NZ
2: aznez(x’)dx’} = 0.
n=1

..0 —w

Now applying the testing operator

ft,“(x)ax- (88)

to both sides of (86) and (87), and exchanging the order of integration yields

on N, a a
lim{-4—11;£[(k: - €2)W(£,y,0)ni=:l a," £ex"(x/)e“(Ex/dx’itxm(x)ej5xdx]di

y~0

(89)

on N a a
1 z n n _. x, m .x _
_ E£[CEW(E,%O)"Z=:1 dz £8: (x/)e IE (ix/fat): (X3815 dX]d€} __ 0

on N a a
. 1 x _. / .
1m{-— [cammm a.” e."<x’)e It‘dx’ t.“(x)e1‘*dx]da
y~0 “If "2:1 I; I: (90)

no N; a a
+ 21:“: - C2)fW(§,y,O)Z: aznfez(x/)e-j€x’dx/ftzm(x)ejfixdx]d€} : 0
-co n=l -a a

01'

71

If"). <1. ‘7‘
ALLhALuMH.

Lt»;
I‘ng

m N
. _l- 2._ 2 x n n m
113% 4“ [(k. 5 was»); a.f. (5)8. (ode

<91)
m N
1 z n n m =
74‘; (“W‘W’mg azf. (0g. (E)d£} o
1 °° ”x
l. _— W ’ ,0 xi! xn zm d
fill Ami“ (5y {Ella f (0g (5) a (92)

m N
+ _l_ 2.. 2 z n n m } :
4,,(ko 5 {WWW}; azf. (0g. (0dr 0.

Interchanging the order of summation and integration and multiplying both sides by 41r

results in

NI w
E a."{1im f (k3 - £2)W(€,y,0)f."(£)g.’"(£)d£}
n=l

y~0_w
(93)
N2 w
- z azn{11113[CEW(E,)’,0)fzn(E)8xm(€)d€} = 0
n=1 Y” -w
N; w
— Z) a," {11113 [caW(£,y,0)f."(&)g.’"(€)de}
n=l y -m (94)
NZ .0
+ 2 a." {11m (k3 — E) f W(£,y,0)1;"(£)g.’"(5)d€} -- 0.
n=1 Y*0 _a
Letting
A3" = lh‘gflkf -€2)W(£,y,0)1;"(£)g.'"(€)d€ <95)
Y* -m

72

. ..,.

n1 ‘1 5‘

'HNA‘Ql‘
1

(31»

A2" = ling f:£W(£,y.0)/:"(£)g.'"(£)d£ (96)
Y“ -00

A3" = lim0 fcewct,y,0)):"(a)g;"(£)d£ (97)
Y“ —ao
A3" = 1im(k3 - (2)]W(5,y,0)f.”(a)g.’"(z)da (98>
y~o _m

equations (93) and (94) become

NJIK N2
2 Agnaxn _ Z Aya," z 0 m = 1’2””,NX (99)
"=1 n=l
N; N:
_ Z Agna: + Z: Az’z'maz” : O m :1’2""’Nz’ (100)
"=1 n=1

For a Chebyshev Galerkin’s solution

rpm(x) Tm(x/a)[1 —(x/a)2]11/2 (101)

ep"(x) Tn(x/a)[1 — (Jr/(1)111,2 (102)

H H

where the " + " sign is used for B=x, and the sign is used for 3:2 [44].

In the transform domain, the time-harmonic continuity equation is expressed
("i-12’ = -jw ps(x) (103)
or
6k
—‘ +1’Ck, = —jwp,(x). “04)
5x

73

v
I! ”"n
lbw; \.

FIE IQ;

\" w. v

" ~11 1:1“

1L,
4‘1“?»‘6'.
»‘¥I.'l

It can be seen from (104) that if kz(x) is an even function, and kx(x) is an odd function,
then ps(x) will be an even function, and thus represent an even propagation mode.
Likewise, if k z(x) is an odd function, and kx(x) is an even function, then ps(x) will be
odd, and thus represent an odd mode. Therefore, for even modes ez"(x) is even and

ex"(x) is odd, and

2:"(5) = fe."(x’)e‘j5"dx’ = ~12 e."(x/)smgx'dx/ (105)
_, 0

f."(€) = fez"(x’)e‘j‘"dx’ = 2 f e,"(x’)cosgx’dx’. (106)
_, 0

For odd modes ez"(x) is odd and ex"(x) is even, thus

f."(€) = f e."(x’)e-f€*’dx’ = 2 fe:(x')cosgx/dx/ (107)
_, 0

f."(€) = fe."(x’)e"'“’dx’ = —j2fe,"(x’)s1ngx’dx/. (108)
_, 0

Chebyshev polynomials of even order are even, and those of odd order are odd

such that

T,<-x) = (-1)"T,(x) (109)

therefore, for even modes

e3") = T2n+1(x/a)‘/1—(x/a)2 (110)

74

 

...1 .1
thI ..I.’ 0

LI, .
A3815:

and for odd modes

em) 4.1/1W “12>

n _ T2n+l<x/a) . (113)

Likewise with the testing operator, for even modes tz'"(x) is even and tx'"(x) is odd, thus

g.”(€) = ft,"'(x)e15‘dx = jzft,”(x)sm5xdx (114)
_a 0

gzm(€) = ftzm(x)ej5‘dx = 2ftzm(x)cos5xdx. (115)
-a 0

For odd modes tz'"(x) is odd and tx'"(x) is even, thus

8:“) = ftxm(x)ejgxdx = 2ftxm(x)cosfixdx (116)
-0 0

81(5) = ftz’"(x)e1'5‘dx = 1'2 f tzm(x)sin€xdx. (117)
_,, 0

The 6‘“ component equation is tested with the 18‘“ expansion basis function, or

m

ta = e? , and gflf.) = ifa'"(€) with + (-) for even (odd) current components. It is seen

from expressions (105) through (108), and (114) through (117) that four generic integral

types are implicated [24]. The first of these is

75

IZM(E) = [Moosfixflbn (118)
0 ((1 —(x/a)2

First a change of variables is made, letting x /a = x", and then dx = adf. When x = 0,

i = a, and when x = a, i =1 therefore

 

1
1:75) = ancosGadef. (119)
o 1 ”(5")2
The relationship
1
[Tam ma") = (-1)" £12m) (120)
1 -x2 2

O

is found in [45], therefore (119) may be expressed in closed form

13%) = (—1)"l‘29-Jz,(€a). (121)

The second generic integral type is given by

12"”(5) = [msmgfldx (122)
0 V1 «(x/a)2

which after application of the change of variable becomes

1

Iz"°(£) = a f sz—lm Sin(€ax")d5c'. (123)

0

76

s .0!
‘ :I

. ...a.
-

'Ldv'.‘
an». '5 1

Tu:

Again a relationship found in [45] is applied

1

[TWA x>sm(“x)dx= (— 1)"— " 12,..I(a) (124)

1— —x2

 

0

therefore (123) may be expressed in closed form

1;”(5) = (-1)"-"§“-Jz,..(aa). (125)

The third generic integral type is given by

a

I,”‘(€) = [Tub/(1)11 —(x/a)2cos(5x)dx (126)

O

which becomes

a \2
I."‘<£) f T2,.(X/a)——— HUG; cos<£x)dx. (127)
o (1(x/a)

After the change of variables applied previously, (127) becomes

_ ~ 2
If“): “f Tz.(f)l—£’—)—cos(5ax)di (128)
1

_ x‘)2

which may be expressed

l

I M'(€) - 0]— T2 "<x) cos(Eax)dx— aIHf2f_1008(gaf)dg' (129)
1 _.

1/ (552) 111W

The first term on the right side of (129) may be expressed in closed form through

77

:‘Uli‘filx
L

1; . ‘
lit: )Cu

5' '5'» ‘t
» ..5\\‘.
I

,.
\ :hfi’l"
u Ht

if ‘

Fire‘II‘

 

expression (120), yielding

l

T (f)
1:8“) = “Unite-12$“) - af(i)2——zi——cos(£af)di. (130)
2 0 V1 7 “)2

The second term in (130) must now be examined. In order to obtain a closed form

expression a recursion formula for Chebshev polynomials [46]

Twoc) —2xTn(x) + Tn_l(x) = o (131)

is applied. From this the following relationships are obtained

xTn(x) = arming + Tn_1(x)] (132)
xZTn(x) = %[xTn+l(x) +xTn_1(x)] (133)
xTMoc) = imam + Tn(x)] (134)
xTn_l(x) = %[Tn(x) + Tn_2(x)]. (135)

Therefore

xzrxx) = %{-§[Tn+2(x) + new] + flux) + T,,_2(x>]}
(136)
= 211—{3000 +2Tn(x) + Tn_2(x)}.
Finally

78

1‘fiflr‘,‘

VL'5‘J_
‘1

F' 1'71

?

can ‘
.‘4 L
, . ‘
an“. “11‘

T(=x) —{T T2(m1)(x) + 2T2n(x) + 2(n_1)(x)} (137)
and therefore

af(x‘/_ x)2—-—- 7‘2"“) 008(Eaf)df
1 — (x)2
(138)

waz("+”(x);2T2" "()0 T2“ 1)(x)]cos(€ax")d~x.
l - (x)2

It is seen that when n=0 in (138), a Chebshev polynomial of negative order

appears in this expression. Chebyshev polynomials are defined by [45]

Tn(x) = cos(narccosx) ...for -1 < x < 1. (139)

From this it is seen that

Tn(x) = T_n(x) = Tlnl(x) ...for -1 < x < 1 (140)

and thus

T20; ”(X): T2|n_ ”(1.) ...for -1< x < 1.

Applying this to (138) gives

1 l

af(x')2 1‘2"“) cos( Eax')dx =3 {Imcosfiafldx
/ 4 0 /1 _ (f)2

 

(142)

1 1

f T2,,(f) .. ~ [Em—1|“) ~ ~
+ 2 —cos(£ax)dx + ———cos(Eax)dx .
(/1 — (56)2 (/1 - (102

79

’vv‘ ‘
\ '.
...a-

u

511

am

This

A “gr .

\
IL"

 

Using relationship (120) with the terms in (142) gives

1

T2n(x') ~ ..
2f cos(Eax)dx = (-1)"1r12n(€a)
(/1 - (x')2

 

0

l

[MOOS(€aj)df : (—l)n+l£J2(n+1)(ga)
1w 2

o
and

1

[Moosfiafldj’ =(—1)I"-1|§ 2,,_,,(€a).

0 V1 "(272

Equation (142) may therefore be expressed

af(x)2—§'—(§)—2cos(§ax')df = —:-{(—1)"*1%J2(n+1)(€a)
«1-(1)

+ (—1)"Jz,.(£a) + (-1)""”% 2),.-u<5a>}.

Thus

J2n(Ea) ‘ %a (‘1)n+l %J2(n+l)(ga)

"no

1,, (E) = (-1) —2-

— “T“clisza) — %"(-1>"‘“'§ 21.41““)-

After algebraic manipulation, (147) becomes

M n a 1
1, (a) =(-1) "T[Jz,(£a) + 5(Jz(,.,1,(£a) + J,,,-,.(£a))].

80

(143)

(144)

(145)

(146)

(147)

(148)

n r.
‘\ his“

4"."

“l-\‘

The fourth and final generic integral type is given by

a

1:0(5) = [T2M1(x/a)(/1-(x/a)zsin(Ex)dx

0

which may be written

1;”(5) = szn+1(X/0)M

sin(£x)dx.
2
0 ‘/l - (x/a)
After the change of variables (150) becomes
1 2
no .. 1 _ f . .. ~
I, (E) = afT2ml(x)—(Lsm(5ax)dx
0 V1 " if

01'

1 1
I,"O(E) = afmsinfiaxmx — af(i)2Msm(EMfi.

0V1_(i)2 o ill—()2)2

By relationship (124), this becomes

1

~

1;”(6) = (—1)"12912,,1(£a) — aflifmsinaaodi.

0 V1 ‘1’?)2

Again a recursion formula for Chebyshev polynomials [46]

Tn+1(x) — 2xTn(x) + Tn_l(x) = 0

(149)

(150)

(151)

(152)

(153)

(154)

must be applied to obtain a closed form expression for the second term on the right side

81

(7'
own“!

‘31

of (153). From (154) the following relationships are found

xTn(x) = $11.40) +Tn_l(x)] (155)
2"1(x)— — %T[ 2,. 2(x) +T (x)] (156)
2szoc) = %[xT2n+2(x) +xT2n(x)] (157)
xT2n+2(x) = é-[szm + T2n+l(x)] (158)
7‘2n(x) - %T[ 2,. 1(x)+ Tn_1(x)]. (159)

Combining ( 157), (158), and (159) yields

= éiészmw mi 12 2,, ~01}
(160)
: £{T2n+3(x) + 2TZn+1(x) + 2n—1(x)}°
Therefore
“fl" )——-2 T2" ‘( x)sin(€ai)dx
_ 2
1 (f) (161)

 

0 4(/1 — (12')2
or

82

 

9.1

‘i

l

affirmsimgafldj = g{stm(Eaflfi

o 1—(x")2 0 W
(162)
l l
T " , T *
+ ZIflsmGafc’flix + fiflsmaaiw}.
1 - (x‘)2 (/1 — (:2)2
Applying relationship (124) to the terms in equation (162) gives
1
f T2n+3(x )Sm(€ax)dx_ _ (_1)n+1_12n 3(Ea) (163)
0 (/1 —(x)2
1 ~
2 f Manama? = (—1)"n12,,,1(£a) (164)
0 V1 ’(i)2
1
T -
fflsmaaxw = <-1)"‘1§Jz,_,(£a) (165)

o V1'(f>2

therefore

af(x )2— T2" Sin(€af)dx = —{( 1)" 112tJ2n+3(€a)
0 v1 " (X)2 4 (166)

+ (-1)"fl12,.+1(50) + (-1)"'1-:-Jz,,-1(€a)}-

Thus

130(5) = (-1)"32312,,,(£a) - 270“)" 2Mac)
(167)

— 2t8a(‘1)nHIJZn»,3(£a) — %(—l)n-1J2n-1(€a)

83

which after algebraic manipulation becomes

I,"°(:) = (—1>"149- J,,,,<5a) + —( J+z,,3(€a)

1.2,. 164)]

Summarizing the closed form expressions for the four generic integral types

1222(5) - ( 1)"—Jz,(€a)

1:0(5) = (~1)"-1‘5‘112,,1(€a)

I""(£>- - (— 1)" “a [12441) + l( 2661“") + Jun-11‘5“)”

1:0(a) = H)” T:f’[1+2,,1(5a) %(J+2,,3(aa>

Applying the results above leads to

w 2 f t,m(x)cosgxdx = 21;"(5)
0

g,“(a) = ft,’"<x)ef“dx . a
'2 j2 f1,”(x)sinexdx =j21;"°(1:)
0

2 f t,’"(x)cosgxdx = 21;”‘(0
0

ll

g,m(€) = ftx'"(x)ej“dx

j2f1,"'(x)sin1;xdx =j21;"°(g)
O

84

J—2n $560)]

for even tzm

for odd 1;"

m
for even tJr

for odd t,“

(168)

(169)

(170)

(171)

(172)

(173)

(174)

Zfez"(x/)cos€x’dx’= 21:2(5) for even e,"

fz’m = fez"(x/)e-j5x’dx/ = °

a

(175)

-j2fez"(x’)sin£x’dx’ = -j2[z"0(€) ...for odd e,"
o

and

ma) = fe,"(x’)e "(EX’dx’ =(

 

Thus for even modes

A2" = 1m; 8 f (k3 — £2)W(£,y,0)1,"°(€)1;"°(&)d5
Y‘ -69
A2” = lim 81ft:W(£,y,0)1,"‘(€)1;"°(a>d£
y~0 -m
A3" = lim -8J'fCEW(€,y,0)1xm(€)1§"2(5)d€
y~0 _,

A3" = lim 8(k3 — c2)fW(£,y,0) I,”‘(€)I,’"‘(£)d£

y - o
and for odd modes

:2" = lim 8 f (k3 — £2)W<e,y,0)1,"‘(€)1;"‘(&)d£

85

1 Zfex"(x’)coséx’dx’ = 2132(5) ...for even ex"
0

(176)

,-j2fe,"(x’)sin£x’dx’ = —j21;‘°(5) for odd ex".
0

(177)

(178)

(179)

(180)

(181)

 

 

"‘4
41~

). ' ,
“1‘1
‘u

A2" = ling -81‘ f (£W(e,y,0)1,"°(£)l;"‘(£)d£ (182)
Y" -

A3" = ling 81ft£W(£.y,0)I."‘(£)I,’"°(£)d& (183)
y‘ .0.
AS" = lim 8(k3 - c2)fW(€,y.0>I,"°(5)I,'"°(a)da. (184)
y~0 _m

4.8 Analysis of Numerical Results

The dispersion characteristics of various higher-order modes for several stripline
geometries and the corresponding surface current distributions are presented [25], [26],
[40], [41]. The mode dispersion characteristics of a stripline with dimensions w=10.1
cm, and h=3.67 cm are shown in Figure 4.7. Here the first two higher-order modes are
shown with the 1 =1 TM parallel—plate guided wave pole AP. The first higher-order mode
is the odd TE1 mode, while the second higher-order mode is the even TE; mode. Both
modes are proper non-leaky modes having real propagation constants C> 1p, and exhibit
low frequency cut—off. The poles associated with these modes lie on the top Riemann
sheet. The cross-sectional surface current distributions for the TBI and TF,2 are presented
in Figure 4.8 and Figure 4.9, respectively. It is seen that for both modes the axially-
directed currents possesses the expected edge singularities, while the transverse current
amplitudes are large near cutoff, and decrease far from cutoff.

The dispersion characteristics of the first three higher-order modes of a stripline

with dimensions w=5.0 cm, and h= 1.75 cm are shown in Figure 4.10. The first two

86

o,»
1.»

U. .

+
2L2

w

~41;

a...
l‘ ;
be“

higher-order modes are again the non-leaky TBI and TE) modes. It is noted that the low-
frequency cutoff of each of these modes is higher than the cutoff frequency of the
corresponding modes of the previous stripline structure, which had larger dimensions.
Additionally, the TB; dispersion curve differs to a greater extent from the 1 =1 TM
parallel-plate guided wave pole )xp than the TF4 mode of the stripline with larger
dimensions. The cross—sectional surface current distribution for the even TE2 mode is
shown in Figure 4.11. Again the axially-directed current displays the expected edge
singularity, and the amplitude of the transverse current is large near cutoff.

The third higher-order odd TE3 mode is an improper, non-spectral leaky mode.
It is noted that this mode shows no sharp cutoff, but rather is characterized by an
attenuation constant which becomes large as the phase constant decreases. The cross-
sectional surface current distribution associated with this mode is shown in Figure 4.12.

The dispersion characteristics of the first three higher-order modes supported by
a stripline structure with dimensions w=1.07 cm and h=0.37 cm are shown in
Figure 4.13, while the first, second, and fourth higher-order modes of the same structure
are shown in Figure 4.14. The 1 =1 and 1 =2 TM parallel-plate guided wave poles AP
have been included in both figures. Here the non-leaky TE] and TE; modes display
cutoff at much higher frequencies than the corresponding modes of the stripline structures
with larger dimensions. Also the TE2 is displaced further in frequency from the [:1 TM
parallel-plate guided wave pole than for the structures with larger dimensions. Thus it
is seen that as the dimensions of the device become smaller, the T132 mode begins to
differ significantly from the 1 =1 TM parallel-plate mode. Cross-sectional surface current

distributions for the T131 and TE2 modes are shown in Figure 4.15 and Figure 4.16,

87

5p”

.Trk

4.9

~1.‘1
l

(“r4 ‘

7}.“

121

CA k
A}:

respectively.

The third higher-order mode associated with this structure is the leaky non-
spectral TE3 mode, while the fourth higher-order mode is the leaky, non-spectral TM1
mode. The cross-sectional current distributions associated with these modes are shown
in Figure 4.17 and Figure 4.18. It was observed that the fourth higher-order mode

shows almost no transverse current at all frequencies, and thus is classified as being TM.

4.9 TEM-Mode Specialization

The expressions obtained in the previous sections can be greatly simplified when
only the fundamental TEM mode of stripline operation is considered. The results of this
specialization are valuable because most stripline devices are operated at frequencies such
that only the TEM mode is excited. Also, a characteristic impedance for stripline
devices may be defined for this mode.

For the TEM mode C =k and only axial currents exist on the center strip,

0,

therefore kx(x) = 0. As a result

Ka(xlx’) = 1im(k:-k:)g:(x,y|x’,0) = o (185)

yo

and equations (40) and (41) both reduce to

sz(x’)sz(x|x’)dx’ = o (186)
where
Kn(x|x’) =1im jko—igg(x,y|x’,0). (187)
y-oO ax

88

__ TE1 mode
________ TE2 mode
_.___ hp (parallel—plate) pole

 

0.7

.0
a)

.0
4s

 

.0
(A

w 1
" h=3

 

.0
N

propagation phase constant fr/ko

p O

-* U'l
1iiiliililLJlJlll111L111]lilllllirlllllllllllLUII

 

 

0.0 lllllllll'llllllllllllllllllllllllllIIIITIIIIIIIII

0.0 2.0 4.0 0.0 8.0 10.0
frequency In GHZ

Figure 4.7 Dispersion characteristics of higher-order, non-leaky modes supported by
stripline (w=10.1 cm, h=3.67 cm).

89

 

 

 

kz 2.27 GHZ
H44... kx 2.27 GHZ
-------- k2 10.0 GHz
k, 100 GHz
N 1.0 — i
-% I :
-"~ 0.9 E 5
a I 5
:5 0.8 E E
.t’ - i
_Q_ 3 5
E 0.7 2 1
O 2 :
E 0.6 ‘_‘ t
m : 5
t205: w=101 cnw ;
a 3 h=3.67 cm ;
0.4 j ,'
CD - 1'
0 : I,
E 0.3 : ’
L ’1
3 3
(I) 0.2 : x”,
2 I ...... ”
1g 0.1 -: """"""""""""
F) 2..--.-4”- '1’::;:-*-'*"*"‘*"*--t--*--*--¥--$--$--r--
L- OoO TfilTIIIIIIIIIIITIIIIIIIIIIIIIIIIIIllllllllllllfi
0.0 0.2 0.4 0.6 0.8 1.0

locafion x/a

Figure 4.8 Cross-sectional current distribution for TE1 mode of stripline, near and
well above cutoff (W: 10.1 cm, h=3.67 cm).

90

k2 415 GHz
W k, 415 GHz
-------- k2 10.0 GHz
k, 10.0 GHz

  

_L
O
llllllllllllltllJlllllllllllllJlllllllLllJllllJl]

.0
to

.0
on

 

.0
\1

.0
on

 

.0
.p.

.0
04

.0
N

 

 

relative surface current amplitude kx,kz
O
01

 

0.0 0.2 0.4 . 0.6 0.8 1.0
locahon x/a

Figure 4.9 Cross-sectional current distribution for TF0 mode of stripline, near and
well above cutoff (w=10.1 cm, h=3.67 cm).

91

w=5.0 crn

h=1.75 cm
__ TE1 mode EC“ {1:03
________ TE2 mode fr, (i=0.

0_O_<.>_O_O>\ (parallel—plate) pole
um TEE mode Efr)
{3:31:11 TESS mode —C1)

 

 

 

 

 

 

O -—1
<1.0 Z
MOS-E
+J 3
C 0.8%
O 2
4(7)” 2
C 0.7-5
0 :
0 0.6 '2'
(D I
(I) 0.5:
E j
Q04?
1
C .1
.9 0.3 :
4" Z
0 0.2:
O _
C 2
0.0.1:
8 E ~~*~-'*‘-—*
0.0.0 IIIIIIIIIIIIjTIIIIIIIIII

l | I
1.0 5.0 5.0 7.0 9.0 11.0 15.0 15.0
frequency In GHZ

Figure 4.10 Dispersion characteristics of higher-order modes supported by stripline
(w=5.0 cm, h=1.75 cm).

92

.0 .0 .0 7‘
\1 00 to o

.0
on

relative surface current amplitude kx,kz
.0 .0 .0 .0 O
—* N bl ~i> U'l
llllllllLLlllllllllllllllllllllJlllllLillllllllllll

.0
o

 

.0
o

 

w=5 cm
h=1.75 cm 1

 

'—
.—-

 

[IIIITrlrrIllllll

0.4 0.6 0.8 1.0
locahon x/a

Figure 4.11 Cross-sectional current distribution for TE2 mode of stripline, near and

well above cutoff (w=5.0 cm, h=1.75 cm).

93

Fir.

__ kz w=5 cm
________ kx h=1.75 cm

 

 

 

f=12.0 GHZ
N 1.0 '_'_ r0.0010 x
x : x
.003) 0.9 E - g)
E 0.8 E -0.0008 E
2 a 2
O 0.7“: - g
'E j 'E
8 0.63 -0.0006 8
c3) 0'5 : o
g 0.4-é —0.0004 g
‘t j t
3 0.3 : 3
m : (I)
9 0.2 E -0.0002 (1)
.2 : .2
E 0.1 —‘ 5
93 3 e
0.0 -lllltllllIllIIIIIIIIIIIITIIIII1IIIIIIIIIIIIIIIIII 0.0000

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0
locafion x/a

Figure 4.12 Cross-sectional current distribution for TE3 (always leaky) stripline mode
(w=5.0 cm, h=1.75 cm).

94

____ TE1 stripline mode gt: §r+j0.:; {r
........ TE2 stripline mode {- §,+J'0 6,
11:11.11: 10.. of TM parallel— late mode
TE stripline mode ={',+J'{i):
AAA_A_A 71pm, of TM2 parallel—plate moder

>.<_>$Z<.>£Z< TE; stripline mode: (1

w=1.07 cm, h: 0.37 cm
Z0: 50. 007 ohms

 

 

 

 

 

g 1.0 —; “—0.000

\ E ,v" rwflg’fifi x

«>1 0.9 : TE1 / xr'i‘w'” - x0

E 0.8 E r —-—0. 002},

B 3 ,’

m 0.7 E 1 — 'E

g : II ‘1 O

o 0.6 S ’1 ~—0.004 E
" i

8 0.5 E ,1 1 8

O I

-C 0 4 : 1l ——0. 006 C

0. ° : I g
_ l °-I—J

c .: 1 - o

.9 0‘3 3 : . g

*6 0.2-E : 1' 1' -—0.008 3

g E : 1:1 J(>\pole)2 46

Q 0.1 '2 | : A ’

8 3 ' 5

Q_ 0.0 I T I | T l I J l I I l I l I —0.010
0.0 20.0 40.0 50.0 80.0

frequency in 0H2

Figure 4.13 Dispersion characteristics of first three higher-order modes supported by
stripline (w=1.07 cm, h=0.37 cm).

95

____ TE1 stripline mode E§=§r+10.3: (r
........ TE2 stripline mode (=g‘r+J0.: {r
*_*_* Ape... of TM parallel—plate mode
A_ >10... of TM2 porollel— lote mode
Iii/1, stripline mode: {=§,+J¢i): g,
>.<_>22<.>$2< TM1 stripline mode: {1

w=1.07 cm, h=0.37 cm
20:50.007 ohms

*1
A:

D

 

 

 

 

 

<0 1.0 E _,- vtirfi'x—ODO

.. 3 x” W‘”“ '

101 0.9 —: EE1/ (.11? ——0.10 x0
... / "

E 0.8 -3 r,’ Tsz ——0.20 F;

B I I, {r I"; L- ....)

m 0.7 -§ I ,4 ——0.:50 c

8 - ' 1’ 1- 0
I I ,’

o 0.6 -: 1 .‘I ——O.4O 45
:1 I, I," A ) " g

8 0.5 E .‘ ,1,’( We 1 ——0.50 o

O E 1 1'1 - E

g 0.4 : ; .1‘ -—O.6O O
E ,' 1:1, '- 1‘:

g 0.3 3 , 5, ——0.70 g

._ : I :'J - C

*6 0.2 1 ,' ,1 ——O.80 0

U1 1 I 5' _ fl

8 0.1 E 1 5" ——0.90 U

o : 1 g _

L6— 0.0 u l I l | l I I I l l l | l l l _1.00
0.0 20.0 40.0 60.0 80.0

frequency in GHZ

Figure 4.14 Dispersion characteristics of first (TE1), second (TE), and fourth (TMI)
higher-order modes supported by stripline (w=1.07 cm, h=0.37 cm).

96

 

 

 

 

 

{11:11:11.5 l<x 10.6 GHz
*****1<Z 10.6 GHZ
________ kx 35.0 GHZ
k2 35.0 GHZ
)2: 1.0 T‘i“*‘~*~*\*‘*\
x _ *\
x 0.9 : *\*
g I \‘K.
: *.
.43 0.8 I 311‘
'_ I ‘4:
E” 0.7 j \k
0 3 w=1£W an \*
4—3 0.6 : h=0.37 Cm \*
C I \
g 0.5-: *1.‘
8 E X
0.4 j \
8 E K
E 0.3 '3 “
L — \
:5 : "f
U) 0.2 : ‘
I; 2
t: 01': 1
2 i ............... -
1 ,. ----------------
93 0.0 1 lllllllIIIlllriIrrTIllllllllllllllllllllllllll‘]
0.0 0.2 0.4 0. 6 0.8 1.0

location x/o

Figure 4.15 Cross-sectional current distribution for TE1 mode of stripline, near and
well above cutoff (w=1.07 cm, h=0.37 cm).

97

 

 

 

 

. _ file-'91“
_v:‘ 1 O : ,X *.
;< d I! ‘*\
.x ‘ I \
0.9 ': If *\
a) : #II \\\
'0 ..
3 0.8 : ,’ ‘1;
-I-J _ ’ \
1: -I ,# \\
D. ‘ I ‘
E 0.7 __ ,’ ‘\
_ ;t ”‘5
O I z’ ‘1
_ I \
44 0.6 _ 11’ x
C : I ‘1
8 0 5 ‘ " “
_. I
5 ° : g w=107<wn %
c) : / h=037 an 1
0.4 ': II
<1) - :1
0 : II
1
E 0.3 1 ,1 ,|
L — ’ I
3 : 7 1
m 0.2 - ,’ '1
I ’ '
a; 1 '
‘4: 0.1 _ [I __________________
2 :II’ """""" / ~“\‘
(1) ,,,,,, '\ /‘ ‘.
L 0.0 ‘ IFIIIIJIIIIIIIIII

0.0 0.2 0.4- 0.6 0.8 1.0
locohon x/o

Figure 4.16 Cross-sectional current distribution for TE2 mode of stripline, near and
well above cutoff (w=1.07 cm, h=0.37 cm).

98

relative surface current amplitude kx,kz

'o

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

 

43.0 GHZ
43.0 GHZ
80.0 GHZ
80.0 GHZ

 

 

 

j%*%#%**w

: ‘K

.. Xx

: =11

: ‘*.

E ‘11

E ‘11

73 w=1.07 cm 3..

1 h=0.37 cm

: "i

E ‘11‘

:1 \\

: --------------------- ‘4:

i 1
3 ‘~ I
... \I
0.0 0.2 0.4 0.6 0.8 1.0

location x/a

Figure 4.17 Cross-sectional current distribution for TE3 (always leaky) mode of
stripline, near and well above cutoff (w=1.07 cm, h=0.37 cm).

99

***** k2 35.0 GHZ
k2 80.0 GHZ

 

'o

lllllllllllllllllllllllLllllllllllJlllllLlllllllll

w=1.07 cm
h=0.37 cm

mod(l<x/l<z)<10-4

0.9
0.8
0.7
0.6

 

0.5
0.4
0.3
0.2
0.1

relative surface current amplitude kz

 

0.0
0.0 0.2 0.4 0.6 0.8 1.0

location x/a

Figure 4.18 Cross-sectional current distribution for TMl (always leaky) mode of
stripline, near and well above cutoff (w=1.07 cm, h=0.37 cm).

100

The derivative term a/ax may be removed from the resulting integral expression

 

lim ka(x/)—a_gct(x’ylx/,O)dxl : O (188)
Y"O ‘0 6x
so that
a , a , x / / _ (189)
8; 11m sz(x )g((x,y|x .0)dx _ 0
y~0 _a
and therefore
hm sz(xl)g<t(an1x/,O)dx’ = C (190)
y«o _a

for |x|<a. Here

gg(x,y|x’,0) = g{(x.y|x’,0) +g?(x,y|x’,0). (191)

For C = k0 the transform-domain wavenumber becomes

120(1) =m=m=¢fiz I5" (192)

therefore

iE(x-X’)
lim g”(x.ylx’.0) = fe d5 (193)
y*0 C 41t|E|

 

and

101

lim g“(x.y|x’.0) = f
y~0 C 4n|£|(l-e'“zm

1&(x-x’)
e {—8—2151,’_e_2151h+e-4151h+6-4151h d5

(194)

no

: equ—x’) [8‘2151h_1]d€.
21:15 1(1 -e'4|5|’*)

 

The total transform-domain Green’s function for the TEM mode is then given by

on

 

 

{a
11m g;(x.y1x’,0) =11m{g{(x.ylx’.01+g§"(x.y|x’.0>}= 1" 1110115 (195)
y~0 y~0 4“ 15x1
where
-2|E|h am}.
11(5) = 1-—.—_2" (1 —e-21111) = —__28
1 -6'4151h 1+e-2151h
(196)
— 1 _e-21Elh
- 1+ ——_——W=1anh(lelh)
It is noted that for E < 1
1— 1-2 h 2 h
(1(5) z ( I“ ) z —-—|—l- = |E|h (197)
1 ( 215(1) 2
and therefore
lim oqlail) = h. (198)
5

Thus the integrand of (195) is not singular at £ = O.
The constant C in equation (190) is proportional to the potential difference which

exists between the ground planes and the center conductor. Determination of this

102

potential difference requires an expression for the y-component of electric field which

exists within the stripline device. As stated previously

_. ... 77 -°/
E(f') = (k3 + VV°)fGO(F|F’)--‘{(—r—)dv’ (199)
jweo
V
where
](F’) = urn") + 2120”) (200)

but Jx (F’) = O for the TEM mode, therefore

- U (r )
E(f') = (kf+vv-)f GO(F|F’) [ ]dv (201)
jweo
V
The Green’s function may be expressed
6010?”) = (12‘+ +zz‘ )G.(r|r>+y‘16.’(FIF’) (202)

which then leads to

_./ 77/
E(F)=z 1.165171%); ( —'—)dv’+v{v 210511110 é—e’dv dv} (203)

(7)60

V V

0

and the y—component of electric field is then given by

.. J 7’
5"E(7) =)7'Z kofGoWIFI) 2“ ——)r/dv +Y'V{V 217000 (717’)— 7(7 ) dV’}
jw 0 jcoeo
V V

—~/
=y‘-V {aifGJGIFGj—Zw— (”d v’}
Z (106

0
V

(204)

103

OI'

 

 

62 z a .. 1.07)
Ey(’7) 7 ayaz [Go(r|r/)' . dvl- (205)
V

1(060

Because currents are confined to the surface of the center strip

12(1‘) = Kz(x,z) (206)

and

Go‘tf‘l'r") = Go'(f5|ii’;z -z’) (207)

where p’ = ix + 22 is the transverse position vector. Thus the y-component of electric

field may be expressed

 

)K / /
777(7) 7 7:92]fGo(Plp; z— z) —(—x z) dz.’dx’ (208)

aya

From this it is apparent that the transform-domain electric field is given by

a

     

—. _. _./ xI’ C)dx (209)
ey(paC): jcaaya gg(p 1p )___kZ 7(060
For the TEM mode C = kc, and pod) = , therefore
e y(p,( ) — ——]g 72,5(pl p’)kz(x’,ko)dx’ (210)

with the transform-domain Green’s function given by

104

3.111115) = gar-515’) +gijtalrs’) (211)

 

 

where
p _. q/ : 76-16(x-x’)e-l5lly-y’l (212)
810(919) II... 471151
and
s g a °° eJ'IEKX-X’) _ J .1
.11.. . f 1-. y y)

_ e -|El(2h-y-y’) + e -IEI(41+y—y’) + e -IEI(4h-y+y’) d5.

With the y—component of electric field determined, the potential difference

between the center conductor and the bottom ground plane is given by

 

0
V = -fe,(5.ody
—h
(214)
0 0
k0 a ,, _, a
= -f[-—— gkt(PIP/)kz(xl’ko)dxl]dy
(060 6y_a °
which leads to
k0 a ”I / / / / 7’70 21
V ‘7’7 77—178]: (xaylx 9y )kz(x ’ko)dx ( 5)
meow ° yr"

Expression (215) is most conveniently evaluated at the center of the strip conductor
located at x = O. The strip conductor is assumed to lie equidistant from the ground

planes, therefore y’ = O and

105

 

k a a
V = 0 [ g7]: (09 _h lxlao) kz(x ’9k0)dxl 7' g7; (0,0Ix’,0)kz(x/,ko)dx/ (216)
(DE 0 o

0 -—a -0
but
a

§£0(0,0|x’,0) kz(x’,ko)dx’ = C (217)

-a

from condition (190), and k0 / ((1)620) = (1110/60, therefore

911(0,—hlx’,0)kz(x’,k0)dx/ - c , (218)

’0

“a

60

V:

 

For points along the lower ground plane at y = —h

 

812(01'h1x/10) = gk’:(0,—h|x’,0) +g;:(0,—h|x’,0) (219)
where
gk:(O,-h|x’,0) : iejl5::l-€lélll-hl (220)
and

8:10. -hlx’.0) = f "’77” [11111 — e-Iwh .1 e'|E|3’* . e-111511d51221)
° Minn-.4191)

 

Combining terms leads to

e 7j1£1xl

5‘ _ I z _ -|E|I1 _ -4|Elh (222)
7,770, hlx,O) f4n|€|(1-e—41511)1 e (1 7 >177

 

106

and thus

g.‘,(0. -hlx’,0) = f

411151 4fl|€|

 

 

From this, the potential difference between the center conductor and the top ground plane

V = 1391-0) = ~11.C (224)
6

where no is the intrinsic impedance of the medium between the ground planes (in this

is

case that medium is free space, and '10 e 1201: ). It is noted from expression (189) that

the constant C may be either positive or negative. Thus for convenience

c 1. V (225)
120x

 

is chosen. Normalizing the potential difference V to 1 volt, this becomes

1
120m

 

e 2.653x10‘3 0'1. (226)

4.10 Moment-Method Solution for TEM-Mode Specialization

From the geometry of the isolated stripline structure, it is apparent that the

current distribution for the principle TEM mode is symmetric about the origin, therefore

kZ(-x) = kz(x) (227)

and thus

107

O a
fk,(x’)g£,(x,0lx’.0)dx’+sz(x’)gk:(x,0|x’,o)dx/ = c, (228)
.. 0

Performing a change of variables by letting x’ = -u in the first integral term of (228)

leads to

O a
1’81 ‘“)81‘.(x101 “10”" + sz(x’)gk‘,(x.0Ix’,0)dx’ = c (229)
_, 0

which may be expressed

sz(x/)[gk‘0(X|x/) +gkto(x| _x/)]dx/ : C. (230)
0

The current function is now expanded such that

N
kz(x) = Z anpn(x) (231)
n =1

using square pulse basis functions

1 for xn-78—x < x < xn+E
12,01) = (2321
\ O elsewhere
where
a 1
Ax=2Ax=—, xn=(n-—)Ax. (233)
N 2
Application of this expansion gives
a N
f 2 anpn(x’)[gkto(x|x’) +gk'o(x| —x/)]dx’ = C. (234)
0 n =1

108

Interchanging the order of summation and integration leads to

N a
Z a, fp,(x’)[g,;(x1x’) + gk‘otxl —x’)]dx’ = C
O

01‘

where

m xn+Ax

 

eflEh" 1 / - /
Am" = fqg) f [e-JllilJr +eJ|€lI dx’dé .
1t

4 151

..m _
x”I Ax

Evaluation of the inner integral in (237) gives

xn+Ax

, , jix’- -'5x’
I [e-Jltlx’+ejlfilx’]dx/ = e e 1
1'5 xn-Ax

 

 

xn—E

: 2 (e *J'EJI,I + ejéxn) 811155

therefore

7 - — jE(x,..-x,.) J'E(X,,.+x,.)
A... = 111(5) 8‘“fo " “3 at
E 2n|€|

 

 

: [11(5) sineKE
1rlil 5

 

0

109

[casflxm —xn) + cos€(xm +xn) d5 .

(235)

(236)

(237)

(238)

(239)

4.11 Pole Series Representation of Matrix Elements for TEM Modes
The numerical solution of matrix equation (236) may be greatly accelerated by
replacing the real axis integration involved in (239) with an equivalent pole-series

representation [26]. The matrix elements in (239) may be expressed in terms of complex

exponentials as

no

 

m" = f_l_ q(£) SinEAx ejE(xm-xn) +ej5(xm+xn) d5 (240)
211 lEl E
where
_ -2l€lh
11(1) = ——1 e . (241)
l + 8772151].
It was shown earlier that
lint—(1(5) = h (242)
H1 | I

therefore no pole exists at E = 0, however poles do occur for

1.6411111 z eIEIh +e-IEIh = 0 (243)

01'

ZeoshEh = 0. (244)

At the roots of the matrix elements E = j£ 1., therefore poles occur for

cos 5 1” = O (245)

01'

110

= (2n -1)11:
5" 2h '

The value of E at the poles is thus given by

2 ij(2n-l)1t.

7” 2h

Equation (241) may be expressed

z ALE).
11(6) 0(a)

where N(E) =2 at E = 5p, and

D =l+ 4&th +5!—12 —
(E) 6’ (5,) (1515:5177 Ep)

42111454“, (1 — 1,)

01'

0(1) = 211(1 -1,).

The matrix elements may be expressed

: f__1_ 4(5) 8315758111. dE
211 E E

-oo

run

where L = (xm -xn) or L = (xm +xn), and

E = E, +J'E,

and thus

111

(246)

(247)

(248)

(249)

(250)

(251)

(252)

ejfiL : e 7EiLejErL . (253)

Here L >0 requires that the integration contour be closed in the upper half of the

complex £~plane, and L < 0 requires lower half-plane closure (Figure 4.19).
Cauchy’s integral theorem states

from = [110111 + ff(€)d€ 1» [node = o (254)
C CR . .

C, cp

but it is apparent from (253) that

f 11011 = o (255)

C

therefore the real-axis integration may be expressed as the sum of contributions to the

contour integration around the poles

 

Mod: = {3 [110111. (256)
CR p c;
As E ~ E,
11(1) ~ ——‘—— (257)
h(1 —1,,)
and if
1(5) = 1 “(EA—“‘77 (258)
h 52
the matrix elements may be expressed
112

 

+
CZ) Cp 1 +
p cm
> y >
CR Er
c,Jo —Ep 0
0; 0°

Figure 4.19 Poles and integration contours in the complex g-plane.

 

 

 

1(5)
Amn= 2p: TnfT—Extfi)

Now at points along C;

€= 25p + ac”

and thus
(1?, = jeejq’dtp

therefore

:21:
———I(€) d =1im 55—)- ”d
(1.0111,) 5 16,111“ 1’

I2thI(iEp)

which gives

At the poles 5,, ‘15,).1311

 

 

 

' ' A— inh .K’
1(i6p) = .1 dsm(i{€p' 2 x)675p1L : if}! S (EPZI x)eIEp,-L
h (ijfipi) ‘5“-
leading to
mh A
14"”!le S (gpi x)e IEL .. f01'L<O,L>O.
h p 51211

114

(259)

(260)

(261)

(262)

(263)

(264)

(265)

But

—€piL = -(Epi|L| for L>O
(266)
“rim-L = "EJ011141 for L<0
thus
.Ax _
=1 £7 1711(7’" )e EP‘ILI .. for L at 0. (267)
h p gpi

Now the case when L = 0 must be examined more closely. When this condition
occurs, (258) becomes

1 sin(EAx)_ Elia — e VIE (268)

1(E)- ‘
E2 thEz

 

and

CD on

Amn = fl (1(5) Sin(EK})dE = f—l- (1(5) ejEA—x 7.7757(1);
21: lEl E 2“ |E| 21E
’°° ‘°° (269)

 

 

1 l q(E)[e’:A_" _ 677771115.

35W 1

Expanding the term sin(EA7c) into exponentials creates a pole at the origin (Figure 4.20),

and Cauchy’s integral theorem then states

55110111; = [110111 + M0111; + ff(E)dE+ ff(€)dE+ [110111 =0 (270)
C

o - t t 3
CR CR c, C, Cp

01'

115

 

+ l
C?» C” Q a +
9 Cm
0:
>_ _ >+ >
c:R Ce CR 6,
cm -€p C.

Figure 4.20 Location of additional pole at the origin of complex g-plane.

116

f node + f f(E)d€ = {3 [node - Mods.
' p C‘ C;

CR CR p

As a result

Am. = -2 Mod: - ff(£)d€.
P C: C:

P

Evaluation of the first term on the right side of (272) results in

 

 

 

 

 

 

1 1 81:5 e755
- f(€)d€ = - —.—[ ————d€ - ————-—-Jd5
g! ; 2n12h fem-5p) imam)
_ 1 1 . ”PM . e19“
_ -2513}; -]21t —121r _ 2]
p 5,, ( 5p)
which may be expressed
1 ejgpfi _ 1 e 5"E
—Zp:[f(€)d£ = p 2 £2 - _—h-Zp: 52.
CI, p pi
and thus
A _ 1 e-EP‘E
,,,,. - ~23 — ff(€)d€.
h 2
p 5p: c;

(271)

(272)

(273)

(274)

(275)

Now the second term on the right side of (275) must be evaluated. As E ~ 0,

€1(€) / E ~ h, therefore

117

 

ff(5)d5 = ill—[emu _ e-jEAx1dE

 

 

21' 2n 5 €
‘ (276)
KAI -j€Ax
1 _h_ e ———d 1 h e d5 .
27' 21: E 2j 21: E
CE CE
Letting E = eseje leads to
o o
ij (ee’o) . 'jE(€€je) .
mm = i, i " . reelede — lie—e—jee’ede. (277)
2] 21: eefe 21° 21! ee"6
9 = T! 6 = -n
Examining the behavior of this expression as 5 ~ 0 gives
0
lim f(£)d€ 311,51“; are-211.21t hj d6- — ——[d6———fd0 = —(-1r)-—1t
e- --0
C( n
(278)
therefore
h h h
J=rf(€)d5 :1 Z - “-5- (279)
c

6

The pole-series expression for the matrix elements associated with TEM modes of

stripline when L = 0 is thus

 

(280)

Solution of (236), using pole-series representations for matrix elements (267) and

(280), results in the cross-sectional current distribution associated with the TEM mode

118

of stripline shown in Figure 4.21. It is noted that this distribution displays the expected

edge singularity behavior.

4.12 Characteristic Impedance of Stripline

A unique characteristic impedance can be determined for the TEM mode of

stripline, given by

Z = Z (281)
I

0

where V is the potential difference between the center strip and the ground plates, and

I is the total current flowing on the center strip. In the case of the TEM mode, only

axially-directed currents flow on the center strip, therefore the total current is

1 = sz(x)dx. (282)

The surface current distribution associated with the TEM mode is even about the origin,

thus

jkz(x)dx = 2jkz(x)dx. (283)
-0 0

Expanding the currents using the square pulse basis functions defined in (232) gives

N a N
I = 22 anfpn(x)dx = 2sz an. (284)
n=1 0 n=1

If the potential difference between the center strip and the ground plates is normalized

such that V=1 volt, then the characteristic impedance of the stripline is found from

119

(285)

The characteristic impedance of stripline devices with center strip widths of
w=10.0cm, w=5.0cm, and w=2.5cm are presented in Figure 4.22. This figure shows
the characteristic impedance for fixed center strip widths as the distance between the
center strip and the ground planes (h) is varied. It is noted that for a particular
characteristic impedance, the ratio w/h is the same for all three center strip widths.
Examining the dimensions which produce a 509 characteristic impedance, it is seen that

(10.0cm/3.45cm) = (5.0cm/1.725cm) =(2.5cm/0.8625cm) e 2.9.

120

1.0
0.9 k2 (TEM)
0.8
0.7
0.6

0.5

 

0.4

0.3

0.2

 

0.1

JllLlllllllllllllIllllllllllllllllllllllllllllllll

relative surface current amplitude kZ

 

 

0.0 ljlllrllllllllIIIIIIIIIITIIllllllIIIIIIIIIlllllll

0.0 0.2 0.4 . 0.6 0.8 1.0
Iocabon x/a

Figure 4.21 Cross-sectional current distribution for TEM mode of stripline.

121

 

 

50 Ohms

_C
030‘ '/

 

 

ll
25 rlltllllllllllllIIIIIIIIIIIIlllllllllrrl1llI]

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
stripline half—height (h) in cm

Figure 4.22 Relationship between stripline characteristic impedance and cross-sectional
dimensions.

122

Chapter 5

FULL-WAVE ANALYSIS OF COUPLED STRIPLINE STRUCTURES

The analysis of the previous chapter is now extended to the case of a stripline
structure having a pair of coupled center strips [47]. This structure is of considerable
practical importance because many integrated circuit devices contain layered strip
transmission lines that lie close to one another. Coupling between the center strips may
lead to crosstalk and interference. Additionally, certain combinations of device geometry
and operational frequency may lead to the excitation of leaky-wave modes. The
propagation eigenmodes associated with a structure having widely separated center strips
will approach those of the isolated center strip structure. However, closely spaced center
strips will result in propagation eigenvalues which are different from those of the isolated
strip structure. In this case, separate propagation constants will arise for symmetric and
anti-symmetric current modes. In this chapter, the complete propagation-mode spectrum

of a stripline structure with two conducting center strips will be investigated.

5.1 Coupled Strip Electric Field Integral Equations

Consider a stripline structure having perfectly-conducting center strips of widths

w1 and w, centered at horizontal positions x1 and x2 from the origin, and located at

heights h1 and h2 above or below the origin (Figure 5.1). In the previous chapter, an

expression for the transform-domain electric field was presented

~ ~ " -/
é‘(b‘.C) = (1.3+ Wofgcw Ira-Mair <1)
Jwe
C

0

123

Figure 5.1

o:oo—-—-2__—————

i trim

 

 

 

 

 

y =1
X

(eoallo)
y = -1

Cross-section of stripline structure with coupled center strips.

124

where T = V: +sz with V: = x56— + jig—y. Extending this to a structure having a pair
x

of center strips leads to

 

"‘”‘ . j 6° (2)
E x’,
+ f §¢(x.y|x’.h2)° 2.( de’}
c 1(06

2 0

.. I? x’,
é’(x.h1;C) — lim(k§+VV'){f §c(X.YIX’.h,)- 1:) de,
C

 

 

-- is I,
é‘(x.h2;o -1im(k§+VV-){f §c(x.ylx’.h,)- {(x de’
”‘2 C1 “’60 (3)

.. lax/,0
+ f g¢(x.y|JC’,h2)' 2. dX’}
C 1 e

2 0

 

where C , and C2 represent the peripheries of the center strips. The boundary condition

at the surface of the perfectly-conducting center strips requires

{-53 = -z‘-é"' (4)

4 at y=h,, h2 and for xa ”Wu/2 s x s xa +wa/2, where oz=1,2. This leads to a pair of

coupled EFIEs

11m;(k02+W°){fCl §((xaylxlah1)°g1(x/,C)dx/

M. (5)
+ [C gcmlxchz)-E2(x',c>dx’} = -jwe,z‘-é"<x,h,;<>
2
lim f-(k§+W-){f §C<x.ylx’.h.>-I?,(x’,odx’
M2 0* (6)

+ fc §c(X,}’lx/,h2)'E2(xlaC)dx/} = -jw€0£'Ei(x,h2;C).

125

As with the isolated center-strip structure, as C» (p, near a simple pole

singularity, the current may be approximated as

k,,,(x’.C)
(c - cp)

13,,(x’.C) = (7)

where on: 1,2 and kap(x’,C) is analytic at (p [25]. Substitution of this into (5) and (6)

 

gives
. k(x .C)
1' limt° kj+W-) )[ , h 1 dx
c1511”, ( I g‘w'x’ )(c- (p) (8)

k(x ’4') dx’] . { , . g. }
+ x/h 2 : - - l '
[C §¢(x.ylx . ) (C- C.) dx} (Im{: Jweot e (x.h1.C)

 

 

 

. ~~(117.0
1' lim t° k3+W~ [ , h ‘
(13161,,{Vh2 ( ) [Crgdxylx ) (C Cp)dx (9)
.. E(x/3C) . . “ —.i

The left-hand sides of these expressions are singular at C = (p, but the right-hand sides
remain regular everywhere. This apparent paradox is resolved if the left-hand sides of

(8) and (9) reduce to an indeterminate form at C = (p. A pair of fundamental EFIEs

results

— we
fim{ffl(k:+W)[ f §C(x,y|x’,hl)-Elp(x’)dx’+ f §<(x,y|x’,h2)‘E2p(x’)dx’]} = 0
ygh‘ W1 W2

Jr1‘—' ‘2'—

2 2

(10)

126

W2 W2

 

‘2‘? ‘2“;
lim{f-(k3+W-) f §C(x,y|x’,h1)-Elp(x/)dx’+ f §C(x,y|x’,h2)-h2p(x’)dx’]} = O
2"": w1 W

‘1‘? ‘2"?

(11)

having non-trivial solutions only for C = (p.

5.2 Scalar x- and z-Component Integral Equations

As with the analysis of isolated center-strip structures in the previous chapter, the
coupled perfectly-conducting center strips considered here are assumed to be infinitely
thin. As a result, the surface currents present on the strips have only x- and z-directed

components, and are therefore represented by

EB(x/) = ikflx(x’) + z‘kfiz(x’) (12)

where fi=1,2. The transforrn—domain Green‘s function implicated in the coupled EFIEs
(10) and (11) is associated with principal and scattered waves, having components which

lie transverse and normal to the surface of the center strips

§c(x,y I x’.y’) = (it + z‘z‘)[8{(x.)’|x’.Y’) + g?(x,YIx/,yl)i (13)

+ 599 [g5 (10’ GCy ’) + gammy GCy ’)].

When (13) is operated on by the transversely-directed current functions the result is
§c(x.ylx’,y’)-E,(x’) = [ikax(x’) +z‘k,,(x’)][gc”(x.ylx’.y’) +8?(W|x’,y’)i (14)
= [ikax(x’) + z‘kaz(x’)]g¢'(x.ylx’,y’)

127

where

 

t / / _ m flax-V) / (15)
8c<w|x ,y > - f W(€,y.y )dE.
-.. 4n
Here
W(E:)’a)’ /) = lie IPOIy-yll + —‘_lj—, e TOMB?” + e ‘Po(4I-y+Y’)_ e ‘Po(2z+y+y’)_ e ‘Po(21-Y‘)’/)J
po (1 -e - p0 )
(16)

 

0

and p = (2 + £2 - k2 is the transform-domain wavenumber. Substitutin (16) into the
o g

fundamental EFIEs (10) and (11) gives scalar x- and z-component equations for strip 1

b1

 

 

 

 

 

2
“m i I K3“ *k3)8<‘<x’ylx¢hl>k1.<x’> +jc£gct(x’)’lxl’h1)k1z(xl) fl}
y~h, 6x2 8x (17)
1 b2
+ az+k2'h’hk ._a_. ’k ’dx’=
— o 8((xa [Ix , 2) 21(x/)+JC 8((xahllx shz) 2Z(x) 0
6x2 6x
“2
b1
. . a
111n{ f [165;8§(x,ylx’.h1)kl,(fl +(k02 -Cz)gf(x.YIX’.h1)klz(x5 dx’}
YThl
‘ b, (18)
. a
J [1558509’11 lx”h2)k2.(x’)+(k3“(2)83“. |xCh2>kzz(X’) dx’ = 0
“2

and scalar x- and z-component equations for strip 2

128

b1

 

 

 

 

02 2 r . a t
f (‘7+ko)gc(x,h2|x’.h1)k1,(x5 +JC—g;(x.h2lx’,h1)klz(x5 dx’
8x 6x (19)
“1
b2
1' 69—2 k2 r ’h k _c3_ t / / / _
+ m 2 + o g¢(x.ylx. 2) 2,065 +JC gc(x.ylx.h2)k2,(x) dx —0
y~h2 8x 6x
02

b1

. a
[[chgg(x’h2'xl’hi) k1x(x/) + (k: ’ C2)gct(x,h2|x’,hl)klz(x’) dx’

al (20)
b2

. . a
+1112” [165;8£(x.yIX’.h2)k2,(x’) +(kf-Cz)g{(x,ylx’.h2)k2,(x5
,.

 

 

id...
“2

where

aa=x—-—°‘, b =x+—°‘ (21)

for a=1,2. Applying the Green’s function expression (15) to the scalar x- and z-

component EFIEs (17)-(20) gives

1’1

lim f
YT,“

 

 

2 '5(x—x’)
(i9— +1.3) f 6’4“ W<s.y,h,)d& k1,,(x6

 

 

 

 

 

6x2
a1 -
. a eJ‘EOr-x’)
+ — W , ,h d k x’ dx’
Jcaxf 4” (00511) } 22
b2 ,, ( )
82 2 ejw'xl) ’
+f(3x—£+ko)f 4r: W(E’h1’h2)d€kzx(x)
02 _
. a i£(x-X’)
.1629; e W<g,h,,h,)dek,,<x6dx’=0

 

129

 

. . a eiw'x’) ,
11m { [[JC—BEI 4T! W(E9y9h1)dgklx(x )

 

 

.. 'ax-x’)
+(kf-C2)f €14 W(E,y,h1)dEklz(x’) 6135’}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“It
g: (23)
_ a ejax-x’)
[[Jcaf 41E W(E:hph2)dgk2,(xl)
a2 ..
J'E(x-x’)
+ (k3 — (fif 9 4n W(E,h1,h2)dEk22(x’) dx’ = 0
b1 00
2 x x’)
f [% k3)f " W<£,hz,h>d£k.,(x6
x
E(x- x’)
+1c-— f 8’ W(5,h ,hl)dEk (x’)dx’
(24)
[31540-0
+1111] a—:+k0 W(E’yah2)d€ k2x(x/)
a2 -..,
ejioc x’) /
+1c—f We y, hkzwa (x’) dx =0
bl 0°
. a eiE(x-x’) /
fjcaf 411: W(€ah29hl)d5 k1x(x )
... 'E(x-x’)
+(k3—C2)fej n W(E,h2,h1)dEklz(x’) dx’
"°° b2 (25)

 

 

... 'Ex-x’)
1:3 f e’ n W(a,y,h2)d€k2,(x6

 

.°° w
+(k3'C2)f——- e17? W050” h2)d£k2 (15st }=0

130

Application of the partial derivatives present in the scalar component equations gives

fl ejE(x ~x’) : jgejflx-X’)
6x

and therefore

2
‘3 emx—x’) = fijgejax-x’) : _EZeIE(x~x’).
6x2 6x

As a result, the scalar component equations (22)-(25) become

 

_ .. 2_ 2 emx-x’) I
11m{flf(k, a) 4n W(e.y.h,)d£k,,(x)

Y‘hr
my}

-.., b2 m
'€(x-x’)
+f[ [(kj’fiz) e1 n W(E,hl,h2)d5k2x(x/)

“1

 

 

.. ejEtx—x’) I
- [<5 4“ W(&,y,h,)d€k,,(x)

 

4

 

 

eiEOt-x’)
- [<5 4“ W(5.h,,hz)d£k2,(x’)

—ao

 

b1 u
'«x—x’)
11m { [Hrs 3’ W(£.y.h,)d&k,,(x6
y-ohl -00 41‘

j£(x-x’)

+(kf-C2)fe 4n W(E,y,h1)dEklz(x’) dxl}

 

 

 

b2 u
ejax-x’) /
+f -fc2: 4n W(E.hl,h2)d£k2,(x)
02 -..,

'E(x-X’)
+ (k3 — c2)f 9’4” W(&.h,.h2)d£k,,(x6

—oo

 

 

131

(ix/=0

(ix/=0

(26)

(27)

(28)

(29)

 

. 'E(x-x’)
“I“: ‘52) 6141: W(€.h2.h1)dE k1,,(x5

 

 

ejax-x’) I
- f ca 4“ W(e,h2,h,)d£klz(x6 dx
—~ (30)

b2 on
'E(x-x’)
+ um flf(k: - E2) 6] W(E,y,h2)dE kzx(x/)
1’45 -... 75
02

 

4

 

 

 

 

 

 

.... ejE(x-X’) /
—fce 4n W(€.y,h2>d€k,,<x0dx =0
bl m
ejEOc-X’) /
f -fca 4n W(E,h2.h,)d&k,,(x)
2 ZmeJEOr-X’) /
+(k.-<)f 4n W<£.h,.h,)d£k,,(x6dx
-~ b2 .. <31)
°5(x-x’)
+31% fl-fca e141. W(E.y.h2)d€k2,(x’)
., ..

 

 

... jE(x-X’)
+(k3-c2)f" 4n W(E.y.h2)d5k2,(x5 dx’} =0.
5.3 Transformation to Local Coordinate Variables

In order to facilitate a numerical solution, a change of variables is performed.
Coordinate variables of the scalar component EFIEs presented above are shifted to a
system of local coordinates. This is done in such a way that the shifted coordinate

systems originate at the centers of the strip conductors (Figure 5.2). First let

.2; = x/ <32)

132

 

 

 

 

 

 

 

 

Y
<——x1 x2—>
/ ’ '
. y=l
n1 ’5
X, \L 7N-hz 7]
21 , _ 2 >
I l x
7,’ 7F
1! =-l

Figure 5.2 Local coordinate variables.

133

and then dx’=dfé. It is seen that when x’=xa —wa/2, E; = -wa/2, and when
x’=xa +wa/2, i; =wa/2. Also let

)7 =x-x (33)

and then dx=dfa. In this case, when x=xa—wa/2, fa =—wa/2 and when

x = xa + wa/2 , Eu 2 wa/2. Applying this change of variables to the scalar component
EFIEs (28)-(31) gives

wl/Z

lim{ [ME—£2)

jE(;1—;;)

 

W(£9y1h1)dE k1x(;1/)

dil’}
34
w.12 .. < >

2 ejfi(fl+xl-x1—£2I) _/
, f [f(ko_gz) 4 W(g,h,,h,)dek2,(x2)

Y"h1
-wl/2 -°°

)
W(e,y,h,) d: 1.42:)

 

°° lea, -;,’
e
[<6 4”

 

 

1t
-w2/2 "°°

dfl=o

 

 

elem. «rib _I
— fee 4“ W(§,h,,h2)d&k2,(x.)

w1/2

°° jug—Eh
Im{II-ICEe n W(E.y.h,)dEklx(f{)

Y"h1 4

 

-wl/2 _°°

°° j€(;1 ’fb

+ (k02 - <2 )f e 41: W(€’y,h1)dgklz(fll) dill}

 

 

W2” (35)

6° ej5(;1*xl'X2—;2I) —/
,. f -fqg 4n wtahlmpdakzgxz)

q92/2 ‘°°

 

 

[€1.56] +1‘1 'fi‘g)

 

+ (k3 — c2) W(£.h,,h,>d£ 19,64) d5; = 0

134

 

2_ 2. ej5(;2*‘2‘xl‘ill) —/
[(k, a ) 4n W<£.h2,h,>d&k1,(x,)

-wl/2 -°°

—/
dx1

 

 

exam-xvii) _I
- f ca 4“ W(e.h,,h,)d&k,,(xl)
—~ (36)

w2/2 m

+ . 2_ 2 ej£(f2-i2') —/
11!}: “(k0 E) 4“ W(E,y.h2)dEk2,(x2)

'W2/2 -m

 

(1362’ =0

 

 

 

dag-£26 _I
— ft: 4“ W(€,y,h2)d£k2,(x.)

wI/2 no I
€15(;2+‘2'x1';1) —/
f 'fCE W(€,h2,h1)dE [(1,061)
41:
“VI/2 ‘°°

 

2 2 °° ej5(i2+x2-x1-f{) —/ —/
+ (k0 —C )f 4“ W(€,h2,h1)d€ klz(x1) dx,
_, 7
w./2 .. (3 >

61562-54)
+ lim f -f (a
y—oh2 4T:

-sz ‘°'°

 

 

W(a,y.h2)d5k,,(i2’)

 

jag-£26
+0“? {2” e 4

‘W

 

 

W(£,y.h,)d€ 191(3)

1!

dig}=o.

5.4 Moment-Method Solution

As in the previous chapter, the EFIEs (34)—(37) are solved using a Galerkin’s

Moment Method formulation [43]. The current functions are expanded such that
NO
Mil) = 21 a5. e:p(f.i) (38)

where e551) is the basis set, and the “:11 represent unknown amplitude coefficients for

135

a=1,2, and B=x,z. As a result

Wan I N“ W.” I N,
—/ 7'5?“ —/ —/ 7‘5}, —/
f kaB(xa)e dxa = Z “:11 f e:$(xu)e dxa = ZaIBfanpfi)
_Wa/2 "=1 -wa’2 ":1
where
wa/Z
—/ —j€f; —/
fanfla) : feanfi(xa)e dxa'
-wa/2

In addition, let
wa/2

g.";.(o = f t.:';.(£,)e";“di,
”Wu/2

where rural-Ea) is a testing function which is to be applied later.

(39)

(40)

(41)

Applying the expansion shown in (38) to the current functions in EFIEs (34)-(37)

 

results in
W2 ,,
ejE(x1— ff) Nu n n _/
13f 11k: W(anah1)d62 01x e1x(x1)
n=1
“wt/2 ‘°°

 

 

aha}

emf—if) ”u n n _I
- f ca max/29.152: alz elzoc.)
n=1

 

 

 

(42)
w2/2
eJE(x1-x2’)ej5(x. -x2) ”2: n n _/

+f [(k f(kf- 4n W(5,h,,h2)daza2,ez,(x,)
—w2/2 -°° "=1

°° ejE(il-x2)ei€(x,-x2) N2. n n —/ -/

- [CE 411 W(E,h1,h2)dEEazze22(xz) dx2 = O
_ n=1

136

wl/2

€15th if) M. —1
lim f [- -fce W(&.y.h,)d££a{;el’;(xl)

 

 

 

 

 

 

 

 

 

n=1
-w1/2 ‘°°
2 2 m ejEGl-gll) N1: 1: n -/ —/
+ (k. -< )f W15,y,h1)d£zj a1. e.2(x.) dx,
_. "=1 (43)
w2/2 w
Elfin-@8160! -x2) ”2: _/
I Has W(£.h .Iz2)d££a2’;e2’;(x2)
41E n=l
-w2/2 _°°
2 2 8150. 518150. x2) "2: n n __I _I
+ (k. —c )f In W(£.h,.h2)d£zj a2ze22(x2) d». =0
-00 n=1
”61:02 x2’)e1'5(x2— x.) ”u n n _I
w] [(12)-4n W(E’h2’hl)dgzalxelx(xl)
-w1/2 _°° n=1
ejECEI‘ x{)e1'E(x2- x,) ”I: n n -/ —/
—f ca W(&.h2.h,)d5§: alz e..(x.) dxl
_w n=1
W2 (44)

+m{ “(k3 k-E 24V“

rhz
-w2/2 '°°

1561 a) ”a
W“ y, h2)dEz-:1 az;£2;(x2)

dig} =0

 

 

ejE(x2-x2/) N2: 11 n -/
— fc: 4n W(£,y.h2)d&£a2.e2(x2)
_°° n=1

 

 

 

 

 

w1/2 do
6150‘; 'xz)ej£(xl x2) N11 -/
f {—fce W(E,h2,hl)d52 (11:81,;(1'1)
4“ n=l
-wI/z '°°
2 2 °° 81:62-11) e11<x2-x.) N1. n n _I —1
+ (k0 -c )f 4 W(E,h2,h1)dEE alze,,(x,) dx,
~00 7': n=1 (45)
wzf2 m
. 81:02-31 N2:
+1111?2 f[- “CE W(E y, 9&2 (12.821102)
,.
-w2/2 '°°

°° 1:022 -i2’) N2.

+ (k3 — (2)] e 4 W(€,y,h2)d££ a2: e2’;(i2’) 1152’} = o.
—cn 7: n=1

 

 

137

The testing operator
wa/Z

- f 1,35%)?“ (46)
"Va/2

is now applied to equations (42)-(45), and the order of integration is interchanged, giving

 

 

. °° (kg-£2) N1, nWl/z n -/ 761’ awn/2 m - jEfl _
11m W(E,y,hl)za1x f e,,(x,)e dx, f z,,(x,)e dx, d5
Y“h1 4“ "=1 _w I2 -W/2

-29 l 1

°° N1; w‘l2 wl/2
-f[§: W(E’y’h1)2 “I: [91:(;1/)e-1€xldfl’f tlz(fl)ejgx1d§1]d£}
11:!
-oo 'Wl/Z -w1/2

on N Wl/Z w1/2

_ j£(x1-x2) 2" n n _ _. —/ m _ '35
+ NICE—€68 4n W(E,hl,h2)21 a2, f e2,(x2’)e "Hi; I 2,,(x1)e1E ‘diljdé
-02, n= -w‘/2 -wl/2

 

°° w/2 w/2
N2: 1 l

jE(x1-x2) _. f’ m _ . f
_f[C£ e 41: W(£,h1,h2)nz=:la2: f e2:(5c-2/)e 1E 2d}; f ‘1x(x1)e" 1511111; =0

‘°° -wl’2 -wl/2

 

(47)

 

0° W/2 W/2
N11 1 . -1 l . -
lim -f[—£§-W(E,y,hl)z a]: f e1:(§1/)e7£x‘dfl/ f tl':(3c'l)emr‘d171 dE
yCh‘ 411: "=1 —w/2 -w/2
—oo 1 I
(k2 C2) °° N1 191/2 “’1”
+ 0 f[W(E,y,hl)Z a]: f e1:(fl/)e 75061171, f t1?(§l)e’£x‘dfl d5}
n=l

4n
—wl/2 'W1/2

 

 

... W212 wl/2
' N2: . —/ . _
“Mi—5f. ‘2’ WM»: .2; f em.» 1‘12de f trench...
n=l

_“ -w2/2 —wl/2

d6

 

°° w2/2 191/2

. N71 . -1 . ..
f [81601 x2)W(E,h1,h2)z 02: f e2:(§2/)e szzdle f t1:_'(xl)efl£x‘dx1
n=1
..oo -W2/2 "WI/2

, (123-<2)

d =0
41: E

 

(48)

138

°° N wl/2 192/2

k2 _ 2 . _x Ix _ -' f, - — _ _ ‘-
f[( 04: )e’w‘2 l)W(£,h2,h1)§ 01'; f e1:(x1/)e ’5 ‘dxll f t2:(x2)ejgx2dx2
11:1
-00 -wl/2 ‘W2/2

dE

 

 

on N12 Win 1 WZ/Z
—f[%ej£(xz_x')w(5,h21hl)2“I:f31:51])975x1d’71/ft£(;2)ejmdfz
n=l

’°'° "W1,2 “92/2

 

dE

°° 2 w2/2 w,j2

k — 2 N74 .—1 .—
+1im{f[( ”4 E )W(E,,y,h2)£ a2: f e2';(i2’)e"“2d£2’ f 15(i2)e"‘2¢52 dE

 

 

ya It n =1
’12 -00 “W2” "W212

 

co w/2 W/2
N21 2 . -1 2 . —
— flfimmmzfij a2: f ez’;(i2’)e"“2d§2 f z§('£2)e"‘2d£2 d£}=0
n=l

 

 

-22 “W2/2 ”’92,2
(49)
on Wl/Z W2/2
‘ -x u n n — -'f/ m - '_ -
_f[%61502 1)W(Eah29h1)zalx f elx(xl/)e 151d}; f t22(x2)615x2dx2 d5
-22, ”:1 -wl/2 492/2
co Wl/z 191/2
(kg-C2) jE(x2-xl) N1: 1: n —/ 1'5}: m - 1.552 _
+__4n_ e W(£,h2,hl)2 alz f elz(x,)e ftk(x2)e dx2 d5
-22 ":1 -wl/2 -wl/2
°° N Wz/z W2/2
, (E 2‘ n n —/ —1Ei'2’ -/ m — 15332 -
- f—W(E:y,h2)2 02: f e2x(x2)e dxz f t22(x2)e dxz ‘15
”2 4” "‘1 —w12 -w12

 

 

 

411
_w —w2/2 -w2/2

(k2 - C2) on N21 192/2 / -'E-/ [W2/2 I '5—
. ° f[W(£,y,h2)2 a2: f e,:(x2)e ’ "de2 f tz':(x2)e’ Edi, all: :0.
n=1

(50)

The equations above may be expressed in terms of the definitions given in (40)

and (41). This results in a more compact representation, given by

139

 

on __ 2

Y"h1

—N

 

— f [fimawz altflkogako :5}
n=1
-00 (51)

 

81501-12)
+ [[(kf—Ez )9 4n W“ h1,h2):Z-: 2:“2xf2x(€)81x(5)]d 615

—oo

(I)

em, 12) "2:

—co

 

on N11
Im{ -[ [LE-WGMQZ a11fi2(£)g{:(o] dz
)Hhl 41'! ":1

N12
f [vi/(amp): aflfi:(€)81':(€)]dé}
-00 n=l (52)

2 (lei-:2)
4n

 

N2:
— f [1% ”"1"” W(€,h2,h2)2 a£f£<og{:(o]de
n=1

—@

(IR—m” ~22 2,. 2
n=1

 

(123—£2)

N11
4 emxz —x1)W(€,h2,h1)}: alZfi:(E)g£(€)]dE
1t n=l

 

i ..

Nlz
_f[£_:e J'E(x2 x1)W(§, h2,h1)2 alzflz(€)gyz:(€)]d d5
-~ (53)

. (123-£2)
+1112 f 4 may, h2)2anf£(€)g£(€) d5
y-o

 

 

a. N2:
- flfiMLxMX 02:13:82,:(5) d5} = 0
n=1

—00

140

no

N

-f[%€j£(x2-XI)W(Esh29hl)2 alZfi:(€)gz':(E)
n=l

 

0’5

an

2_ 2
+_—_(k° C>f dE
4n

‘°° (54)

a) N3
n=1
d5} = O .

WI:
Now the order of summation and integration is interchanged, resulting in

N

e"“2“l’W(5,h2,h2>§: al'éfi'z'(€)gz'§(€)
n=l

 

 

N2:

(123—:2) 2 ,. ,2
+ 41: fW(E,y,h2)Z:lazzf2z(§)gzz(§)

 

 

—oo

 

N11

n
Zalx
n=1

1i? (1622—52)W(E,y,h1)f1:(5)81:(5)d€l
,-
N12
‘ 2;] “1:
N2:

+2212:

n=l

lim [CE W(E,y,h1)fi:(5)81’:(€)d€]
’“h‘ -~ (55)

f (k3 -€2)e’“""‘2’ W(£,h1,h2)f£(6)g{2(€)d€]

 

 

N22

2 “2:
=1

n

 

f cae"“l"2’ W(€,h1,h2)f22(€)g$(€)dfi] = o

 

Ni: 00
* Z: 01: Iii? ICEW(€,y,h1)f1:(5)81':(€)d€]
n=1 Y“ 1_0°

 

lim (k: - (2)]-W(E,y,h,)f1:(€)81:'(€)d€]

)th

”1:
II
+ Z alz
n=1
”2:

-2a2’;

n=l

(56)

 

cae’w‘"‘2’W(£,h1,h2)f;;(og{:(£)da]
N2: °°

* X “2'2 [0‘3 - <2) f e’w‘"”Wai.h1,h2)13’;(€>g{;’(£)d5] = 0

n=1

141

N

al’; flkf — £2)e’“"‘1’W(£,h2,h2m:(a)g£(od€J

-ao

N12

‘201:
n=l

le

2

n=1

 

 

[caem”"”W<s,h2,hpffidkflbdt]

 

N2:
+ Z “1:
n=1 y~h2 -2.
N21 °°
‘ Z “2:

n=1

 

PM -2.

[CEem"“"WG,h2,h1)f1§(E)g£(E)d€]

 

le
_ n
"X: alx

Nl °°

Z

+ [(koz C2) )ef ejuxz-XI)W(€9h2sh1)f1:(5)82':(€)d€]

n=1

 

N2: °°
- 2 a2: 11m fcaW<£,y,h2)f2:(€)g2’:<€)d£]
n=1 T432 do

 

N2: °°
+ Z: 02: 111;: (k: - C2”W(E,y,h2)f2:(5)8§(€)d€] = 0
n=1 Y" _°°

Now, by making the following definitions

A233” = 11111 [(113—£2)W(e,y,h1)f12(5)g{;‘<od5
Y" 1

A“’"" =11? (may, hlm'xogfflodz
y

Alf” = [(113-52)e"“’""2’W<€,h2,h2)f2’;(og;:(ode

—W

142

um (k2 —£2)W(&,y,h2)f2’;(og£(a)da

11m <5W(5,y,h2)f2:(:)g£(£)d5 =0

(57)

(58)

(59)

(60)

(61)

14,13“ = Imam”?W(&,h,,h2)f2:(£)g{:(§)d£

A2!” = 11111 C€W(£,y.h1)f12(£)g$(£)d£

Y"h1

Azlzlmn = 111,111“:—C2)fW(E,y,h1)f1:(5)81:(€)d€
Y‘ 1

co

1:” = [cae’w‘l"‘2’W(e,h,,h2)f;;(og;:(od€

—oo

00

AS?” = (k02 — :2) f 81-111, "‘2’W(5,hl,h2)f2’;(og{;'(5)da

-oo

no

1423”" = [(1:3-€2)e""‘2"1’W<£,hz,h1)f1:(£)g£(£)da

—00

A?" = [ca11"“?""’W(E,h2,hl)f1:(£)g§(£)da

Aim" = lim [(kf-£2)W(€,y,h2)f£(€)g£(€)d€

-o

no

1433” = 111112 c5W<5,y,hzm’;(og£(oda
,-

A2!“ = [c5em"1’W<£,h2,hl)fi:(£>g;:(oda

"W

143

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

Azzzmn = (k:“Cafe/£02_xl)W(§,h2,h1)f]:(€)gz':(fi)d§ (72)

—00

A33” =1111h12 cawm, h2)f£(€)g2'§(li)d€ (73)
Y

2133“" = lir:(k3—mfwm,h2)f2;'(&)g;:(£>da (74)
yo

equations (55)—(58) become

N N1 N2,

2a" aAlx A-jjm" Z alt/1,121” + 2 aZXA A11” —§j alt/112“" — o with m =1,2,...,Nx
=1 n=l n=l n=l
(75)
N2; N2:
Allmn Allmn n 12mn n 12mn .
"2:: 012A + Z alZA — 2: (12,4; + 2 aZZAzZ = O w1th m =1,2,...,Nz
n=1 n=l
(76)
N
”A21mnA21nm A22mn_ AZZZmn : 0 th = 1 2 N
Ea" 014” xx '20”.sz +%:az’; N203: ...WI "1 , ,..., x
n=l n= 1 n=1
(77)
N1: N21: N22
- Z aflA A21” + Z aflAilm— Z afiAzzmn+ a2ZAz222m"= O with m = 1,2,...,Nz.
n=1 n=l n=1
(78)

For the Chebyshev Galerkin’s solution, the basis set and testing function are

ea"p(x) = Tn(x/a)[l -(x /a)2]1l/2 (79)

144

1,3411) = Tm(x/a)[1 - (x/a)2]*1’2 (so)

where a=1,2, a: wet/2, and the "+" sign is associated with B=x, and the "-" sign is

associated with [i=2 [44]. In the analysis of the single center-strip structure of the

previous chapter, the time-harmonic, transform-domain continuity equation

6k
—fi +jCk, = —jwp,(x) (81)
fix

was examined to determine the relationship between basis functions ex(x) and ez(x) , and
modal structure. For the isolated stripline, it was observed from equation (81) that if
kz(x) is an even function, and kx(x) is an odd function, then ps(x) is an even function,
representing an even propagation mode [26]. Likewise, if kz(x) is an odd function, and
kx(x) is an even function, then ps(x) is odd, and thus represents an odd mode. It was
therefore concluded that, for even modes ez"(x) is even and ex"(x) is odd, and for odd
modes ez"(x) is odd and ex"(x) is even.

The analysis above is not valid for coupled stripline. Due to interactions between
the conducting strips, in general, the current distribution functions kax(x) and kaz(x) will
be neither even nor odd about the centers of the respective strip conductors. As a result,
the relationships between modal index and basis function structure described above for
the isolated stripline will not exist for the coupled structure. The expanded current
distributions (38) associated with coupled stripline will therefore be sums of both even
and odd basis functions.

In the previous chapter, it was noted that Chebyshev polynomials of even order

are even, and those of odd order are odd such that

145

Tn(—x) = (-1)"Tn(x). (82)

As a result of relationship (82)

wa/2
w/2 2fgoings“()COSEJr’dx’ ...foreap(x’) even
fanp(€) : f e:p(x/)e‘jEx/dx/ : 0 [2 (83)
~wa/2 1‘

-2jfe:fl(x’)sin5x’dx’ f...oreap(x’) odd
0

where oz=1,2, B=x,z. Likewise, with the testing operator

wa/2
w ,2 2 f ta";(x)cosExdx ...for tap(x) even
g,"f,(£) = f tfé(x)ej€xdx = 0,2 (84)
-wa/2 1‘

2jftfl,(x)sin€xdx ..f.ortafi(x) odd
0

for a=1,2, B=x,z.
It is seen from expressions (83) and (84) that four generic integral types are

implicated

[3%) = fmcos(fix)dx (85)
0 ‘/1 -(x/a)2

1520(5) = [Msinflndx (86)
o ‘/1 -(x/a)2

0

13(5) = [TAX/a) 1(1 -(x/a)2 cos(Ex)dx (87)

0

146

0

155(5) = [T2n+l(x/a)‘/1-(x/a)zsin(Ex)dx

0

(88)

where a=wal2 [24]. Following the methods of the previous chapter, closed form

expressions can be obtained for the integrals (85)-(88). As a result, relationships (83)

and (84) may be written as

 

n 21::(5) ...for It even
fux(E) =

-2jI::(E) for n odd

n 21::(E) for u even
faz(€) =

-2jI::(€) for n odd

m 2I$(E) ...for m even
8,45) =

2jlgr°(€) ...for m odd

m 2I$(£) for m even
3.1(5) =

2j1,’,’;°(£) for m odd

where
M n nwa w“
Iaz : (_1) 4 121(67)

no nnwa w,x
Iaz : ("1) TJ2n+1(E—)

147

2

(89)

(90)

(91)

(92)

(93)

(94)

 

1:: = 111111;) + $1sz115-15)+1111—1.(€%))1

= —l<—> 1,1(1,.(.:,11,,_.(.r,))i

Using (89)-(92), the matrix elements (59)-(74) become

1 lim 4 (k,2 -1:2)W(g,y,h,)1{';(§)13‘(g)dg ...for It even, m even
Y"h1

lim —4j (k02 -— £2)W(E,y,h1)lff(£)13'(€)d€ ...for n odd, m even

”’1 -.., (97)
Allmn : < 1

lim 4jf( k: — £2 ),W(€ y,h )Il" :(E)IK°(€)dE ...for It even, m odd
y h _

lim 4 (k3 — :2)W(g,y,h,)1{'f(g)13°(g)dg ...for n odd, m odd
\ y~h‘

 

1 limhl4fC€ W(€,y,hl)11":(£)1{'"x(£)d€ ...for n even, m even

lim -4j CE W(E,y,hl)1;':(£)lfle(€)d€ ...for n odd, m even
Allmn : J )Hhi ‘°° (98)

N

lim 4j CE W(£,y,h1)lf':(E)I$°(€)d€ ...for It even, m odd
Y“h1

lim 4 C5 W(£, y,hl )11" °(€)Ifl°(6)d€ ...for 11 odd, m odd

y «lh

 

148

, 4f(k:—E2)W(E,hl,h2)ejfi(x1‘5)12":(E)Ifle(E)dE for It even, m even

“41]“:‘£2)W(E,hph2)emx1_x2)lff(E)Ifie(E)dE for n odd, m even

 

 

 

Al2mn : ‘ (99)
4jf(k:—E2)W(E,hl,h2)ej€(x1‘8)I;(E)I$°(E)dE ...for It even, m odd
. 4f(kj—£2)W(g,h,,hz)e’“‘*‘1211;f(g)1,’§’,°(£)d§ ...for 11 odd, m odd
r 4fCE W(E,h1,h2)ej£(x1‘7)IZ(E)I§‘(E)dE ...for It even, m even
”41]“W(€,hl,h2)ejé(x1‘5)I;:(E)I£‘(E)dE ...for n odd, m even
12mn -... (100)
An — 1
4ijE W(5,h,,h,)e1“‘l“2112’;‘(g)13°(§)d5 ...for n even, m odd
L4 [(5 W(e,h,,h,)e"“*"211;,°(z)1,",;°(§)d5 ...for n odd, m odd
' linhi 4 CE W(E,y,h1)II":(E)II':e(E)dE for It even, m even
Y” 1 _m
lim -4j CE W(E,y,h1)11":(E)13‘(E)dE ...for n odd, m even
Ann". _ , "h' (101)
11111 4j CE W(E,y,h1)11";(E)Il':°(E)dE ...for It even, m odd
Y”m
L lim 4 CE W(E,y,h1)Il"f(E)Il"z'°(E)dE for n odd, m odd

Y*h1

149

llmn
A

AlZmn :1

U

z 1

( lim 4 (:k —)C2 )fW(E,y,h )II":(E)II"Z"(E)dE ...for n even, m even

y-oh

lim —4j( (3k {—2) )W(E,y,h l)11"z°(E)Ilm(E)dE for n odd, m even
1*” (102)

lim 4j(:k —)c?-)fW(g,y,hl)1;';(£)1{;'°(g)de ...for It even, m odd
y h

 

 

 

. lim «kg-(2)] W(g,y,h,)1,"f(5)1,";°(g)de ...for n odd, m odd

Y"h1

r 4fCE W(E,h1,h2)ejg(x1—x1)I;(E)Il':‘(E)dE ...for n even, m even

*1ij5 W(€,hph2)ej£(x1-72)I§(E)I$‘(E)dE for n odd, m even
(103)

4jfcg W(g,h,,h,)e"“’l"211,";(5)I,";°(g)d5 ...for It even, m odd

4 ft: W(g,h,,h2)e""*"211,:’;’(£)I,",‘°(5)d5 ...for n odd, m odd

4(k3- (2)] W(5,h,,h,)e"“1‘12’1;;(£)I,";‘(5)d§ for It even, m even
-4J'(k: -C2)fW(€,h1,h2)ejE(x12521;0(E)13‘(E)dE for n odd, m even
(104)

4j(kf—c2)f W(z,h,,h2)e""‘l 1211,:‘(5)1,";°(g)d£ ...for It even, m odd

4093—61)] W(z:,h,,h,)e"“l 12’1"°(5)I,"',°(5)d§ for n odd, m odd

150

21mn
14 _

21mn
AXZ

= (

 

l 4 f C a W(£.h,.h,)e11112‘1111r:(o1£‘(5)d5
41'] ca W(&.h,,h,)e’1“2""I{‘;’<£)I£‘(&)da

41'] c5 W(£.h,.h,) e1‘1‘2“‘11r:(5>1£°(5)d£

 

. 4 [cgW(g.h,,h,>e"“"1"‘*’I{',"(£)I;:°(odt

' 111112 4 (k3—52)W<&,y,h2)I;:(oI;:‘(od£
,.

"Q
on

1111}: -4,- (k3 — :2)W(£.y.h,)12".°(£)1;:‘(od5
,.

-Q

1111112 41' f (k3 — £2) W(5,y,h,)1;:(£)1£°(od£
y * -eo

 

1111,12 4 (k3 — :2) W(£.y,h,)I;;’(&)I£°(oda
. ,,

151

—4jf(k§—52)W(E,h2,h,)e1‘1‘2“*1I,"f(z-;)I,";‘(g)dg ...for 12 odd, m even

r 4f(koz—E2)W(E,h2,h1)e25(x1'x121f:(E)IZ";e(E)dE ...for It even, m even

(105)

4} (CZ—52 W(€,h 1h )€j£(x2_x1)IM(E)Im°(E)dE ...for It even, m odd
0 2 1 1x 2h

4 [(113 -g2) W(g,h,,h,)e1‘1"2‘1‘111,"f(g)12";°(5)dg ...for n odd, m odd

for It even, m even

for n odd, m even

(106)
for It even, m odd
for n odd, m odd
for It even, m even
for n odd, m even
(107)

for It even, m odd

for n odd, m odd

lim 4 CE W(E,y,h2)I§(E)Iz"f(E)dE ...for It even, m even
y~h2 _

-W

on

lim —4j CE W(E,y,h2)I£O(E)IZ':‘(E)dE ...for n odd, m even

"’12 (108)

 

 

Affmn ____
lirlnz4j CE W(E,y,h2)I;ze(E)Iz':°(E)dE .f..or n even, m odd
Y" -cn
. lirz4 CE W(E,y,h2)12"z°(E)I£°(E)dE ...for n odd, m odd
’2‘. —oo
r 4fCE W(E,h2,h1)ejz(x1_x121f:(E)Iz':‘(E)dE ...for It even, m even
—4jfEEW(E,h2,h1)ejE1x2-x121ff(E)Iz"z'e(E)dE for n odd, m even
21mn -oo (109)
Au = <
4ijEW(E,h2,h1)ejE(x1-x1211":(E)IZ"zw(E)dE ...for n even, m odd
14 [Ce W(a,h2,h,)e"“2"111ff(5)12",'°(g)d5 ...for n odd, m odd
' 4(kf—c2) f W(£,h2,h,)e"“2‘1‘111,":(5)1,","(g)dg ...for It even, m even
-4j(k,f—(2)]W(g,h2,h,)e"1"2“*1I{‘,°(5)1,’;"(g)d5 ...for n odd, m even
21mn ‘°° (110)
A,z = 1

4j(k:—C2)fW(E,h2,h1)ej£(x1-x1)lfz‘(E)I£°(E)dE ...for n even, m odd

 

, 4(kf -Cz) f W(£,h2,h,)e1“‘2"‘"I{';’(t)1,;:°(5)dg ...for n odd, m odd

152

f

 

on

11111 4 EE W(E,y,h2)I§(E)IZ':‘(E)dE ...for It even, m even
win

lim —4j CE W(E,y,h2)IZ"f(E)12':e(E)dE ...for n odd, m even

Y"h2 -1. (111)
lim 4j CE W(E,y,h2)I2":(E)12':°(E)dE ...for It even, m odd
Y“h2
lim 4 CE W(E, y,h2 )sz(§)12':0(€)d€ ...for n odd, m odd
y h; -...
1 lim 4(k: - (2))fW(E,y,h2)12":(E)12':‘(E)dE ...for It even, m even
y hz
lim —4j(k —-C2)fW(E, y,h )If",(E)I""(E)dE ...for n odd, m even
y k: (112)

lim 4j( (3k - :2) )f W(E, y, h2)1,,(g)12, (5)115 ...for It even, m odd
y ’12

 

lim 4(k -c:2)f W(E, y,h )I;,°(5)I,':°(£)dg ...for n odd, m odd.
PM

It is seen that (k: - E2) and W(E,y,y’) are both even functions of E, therefore

matrix elements (97)-(112) become

llmn
A

= 1

r lim 8f(ko2 -E 2,)W(E y,h )11" :(E)I$‘(E)dE .. for It even, m even

Y”h1 0

O ...for n odd, m even (113)

O for It even, m odd

lim 8 f (k3 - 2:2) W(E,y,h1)11"f(E)Ifl°(E)dE ...for n odd, m odd

L
Y“h

 

O

153

0 for n even, m even

lim -8j CE W(E,y,hl)I{'2°(E)I$‘(E)dE ...for n odd, m even
‘Allmn : Jy—hl 0

lim 81' CE W(E,y,hl)11"2‘(E)I$2°(E)dE ...for It even, m odd

Y"h1 0

 

O ...for n odd, m odd
(114)

00

rzsfllcj»52)”45,112,112)cos[z=,(x1 -x2)]12";(g)13‘(£)dg ...for It even, m even
0

8f(k22—E2)W(E,hl,h2)sin[E(x1—x2)]12"x°(E)I$‘(E)dE ...for n odd, m even

12mn 0
A22 —T

-8 [(1:02 452) W(E,hl,h2)sin[E(xl -x2)]1;(g)13°(5)dg ...for n even, m odd
0

L8f(k:-E2)W(E,h1,h2)cos[E(x1 -x2)]1;,°(g)13°(g)d5 ...for n odd, m odd

0

 

(115)
r 8ijEW(E,h1,h2)sin{E(x1-x2)]12"2‘(E)13e(E)dE ...for n even, m even

0
-8jfcg W(E,h1,h2)cos[E(xl -x2)]12"2°(5)1,"’,‘(g)dg ...for n odd, m even

0

12mn
14 _

8ijE W(E,h2,h2)cos[E(x1 —x2)]12"2‘(5)13°(g)d5 ...for n even, m odd
0

 

, sjfcz W(E,hl,h2)sin[E(xl —x2)]12"2°(E)11':'2°(E)dE ...for n odd, m odd
0

(116)

154

Allmn :2

 

12mn
A :

Y‘h

O for n even, m even

on

lim -8j CE W(E,y,h1)Ilnf(E)Il':e(E)dE ...for n odd, m even

0

W

lim 81' CE W(E,y,h1)11":(E)II"2w(E)dE ...for It even, m odd

Y‘m

0 ...for n odd, m odd

Y“h

 

A

k

Y”h

 

155

lim 8(kj — c2)f W(g,y,h2)1{‘2‘(§)1,"2"(g)dg for It even, m even
0

0 for n odd, m even

0 for n even, m odd

lim 8(k02 - mf W(E,y,hl)Il"2°(E)Il"2'°(E)dE ...for 11 odd, m odd
0

L 8ijE W(E,h1,h2)sin[E(x1 -x2)]I2'_’f(E)II”2'°(E)dE ...for n odd, m odd
0

(117)

(118)

r 8ijEW(E,hl,h2)sin[E(x1-x2)]Iz":(E)Il"2"(E)dE ...for n even, m even
0

-8ijE W(E,h1,h2)cos[E(xl -x2)]1;,°(g)1{’2"(5)d5 ...for n odd, m even
0

8jfcz: W(E,hl,h2)cos[E(x1 -x2)]12’_',‘(5)1{:°(5)d5 ...for I: even, m odd
0

(119)

' 8(k: — (,'2)fW(E,hl,h2)cos[E(xl —x2)]12"2‘(E)11"2"(E)dE ...for n even, m even
0

8(k,2 - (2) f W(E,h1,h2)sin[E(xl —x2)]12"2°(E)11"2"(E)dE for n odd, m even
0

12 run
A _

—8(k: — C2)f W(E,hl,h2)sin[E(x1 —x2)]12"2‘(E)Il':°(E)dE ...for n even, m odd
0

 

2 8(kj — (2)] W(E,h1,h2)cos[E(xl -x2)]12"2°(g)1,"2'°(a)dg ...for n odd, m odd
0

(120)

on

r 8f(ko2 - E2)W(E,h2,hl)cos[E(x2—x1)]11":(E)12":(E)dE ...for It even, m even
0

8 f (k,2 - g2) W(E,h2,h1)sin[E(x2 —x1)]1{‘f(g)12",‘(g)dg ...for n odd, m even

A21mn ___ 2 0

-8f(k: — E2)W(E,h2,h1)sin[E(x2-xl)]11":(E)Iz':°(E)dE ...for It even, m odd
0

L8f(kf — 52)W(g,h2,h2)eos[g(x2—x2)]1,"f(g)12";°(a)dg ...for n odd, m odd

0

 

(121)
r 8ijE W(E,h2,hl)sin[E(x2 —xl)]I;'(E)IZ";e(E)dE ...for n even, m even
0
—8jfEEW(E,h2,h1)cos[E(x2-x1)]11"2°(E)Iz':‘(E)dE ...for n odd, m even
0

Zlmn
A :

8ijEW(E,h2,h1)cos[E(x2—x2)]11":(E)12";°(E)dE ...for It even, m odd
0

 

L BifCE W(E,h2,hl)sin[E(x2—x1)]11":(E)12':°(E)dE ...for n odd, m odd
0

(122)

156

A22mn _

A21mn _

 

L

' lim 811k: — E2)W(E,y,h2)I;:(E)Iz':'(E)dE ...for It even, m even
y~h2 0

O for n odd, m even

0 for It even, m odd

Y""2

O for n even, m even

on

lim -8j CE W(E,y,h2)l;20(E)Iz";e(E)dE for n odd, m even
win; 0

Q

lim 8j CE W(E,y,h2)I;2e(E)Iz':°(E)dE ...for n even, m odd
y~h2 0

 

,

O for n odd, m odd

 

157

lim 8 f (k3 — 52)W(g,y,h2)1;f(£)1,':°(g)dg ...for n odd, m odd
0

L 8ijEW(E,h2,h1)sin[E(x2—x1)]11"f(E)I2"2’°(E)dE for n odd, m odd
0

(123)

(124)

8ijE W(E,h2,h1)sin[E(x2—x1)]11":(E)12"'2‘(E)dE ...for It even, m even
0

—8ijE W(E,h2,h1)cos[E(x2—x1)]11"f(E)Izm2‘(E)dE ...for n odd, m even
o

8455 W(E,h2,hl)cos[E(x2—x1)]I,":(E)12"'2°(E)dE ...for n even, m odd
0

(125)

r 8(k: - E2)f W(E,h2,h1)cos[E(x2—x1)]11"2‘(E)I2':e(E)dE ...for It even, m even
0

8(k3 —— C2)fW(E,h2,hl)sin[E(x2—x1)]I{'2°(E)12':e(E)dE ...for n odd, m even
0

 

 

 

Az2z1nm = 1
43(ka - (2)] W(g,h2,h2)sin[g(x2-x2)]1{‘;(5)12"2‘°(5)dg ...for It even, m odd
0
2 8(kf — (2)] W(E,h2,h1)cos[E(x2~xl)]11"2°(E)I2':°(E)dE ...for n odd, m odd
0
(126)
O for n even, m even
lim —8j CE W(E,y,h2)1;f(E)12"2"(E)dE ...for n odd, m even
Azizmn _ 2 y‘hz 0
lim 8j CE W(E,y,h2)12":(E)12m20(E)dE ...for It even, m odd
Y"h2 o
O ...for n odd, m odd
(127)
' lim 8(k: — C2)fW(E,y,h2)1;2‘(E)I2"2'e(E)dE ...for n even, m even
y~h2 0
O for n odd, m even
Ai2mn ___ 2
O for n even, m odd
111m 3(kf - :2) f W(E,y,h2)12"2°(E)12':°(E)dE ...for n odd, m odd.
y~h2 0
(128)

158

5.5 Analysis of Green’s Function Singularities

For simplicity, the singularities of the Green’s function are first examined for the
case of coplanar center strips which are located in the plane of the origin ( y’ =0). The
transform-domain Green’s function associated with the transverse components of Hertz

potential is

co

gf(x,yIX’,0) = f

J'E(x-x’)
’3 W(&,y,y’)d£ (129)

 

where B=x,z. 1n the region above the strip conductors (y > y’ = O)

1 — l - 21+ ‘ 2" - 41+ _ 41..
W(E,y,0) = —— e p0y+—_ ‘8 p0( y)_e Po( ”+8 Po( ”+8 p0( y)

po (1 -e-4p°l)
(130)

 

e 'Poy -e -Po(2l-y) = Sinh(p0 [l-y])
p0(1 + e '2Pol ) p0 cosh(pol)

 

It can be seen that (130) is even with respect to the transform-domain wavenumber
parameter p0. In the region below the strips (y < y’ = O)

1 1 ' 21+ ‘ 21' — 41+ _ 41-
W(£:y,0) : _ ePOY+—:4__l_[_e po( ”’6 p0( ”+3 p°( ”+8 po( )0]
p0 (l-e ”0)

e ‘POY _ e ”Po(21+)’) = Sinh(po [I +y])
p0(1 + e "21101 ) pocosh(pol)

(131)

 

 

Equation (131) is also even with respect to parameter p0, and thus the branch point

associated with (129) is removable.

For the more general case when the conducting strips are not coplanar

159

W(€ /) = i e ‘PolY”y’l + ____1_ _ e -p0(21+y+y/) _e -p0(21-y-y/) + e -po(4l+y-y’) + e -po(41-y+y’)
any (1 _ ml)

 

P0
(132)
which may be expressed
/ 1 { 4!” ‘POIY‘Y/l -po(21+y+y’)
W(£,y,y) = 22, in —e ")e -e
170(1- °) (133)

-21--’ ~4z+-’ -41-+’
-ep°( yy)+epo( yy)+epo( yy)”.

It is seen that W(E, y, E) is singular at points where

1 _e‘4PoI = €2Pol_e'2PoI : O (134)

01'

2sinh(2pol) = O. (135)

Thus poles of W(E,y, C) occur for

J!“
= t _ (136)
Po 1 21

where n is a positive integer (n: 1,2,3, . . .). It is seen that the poles indicated in equation
(136) differ from those associated with the isolated stripline (examined in the previous

chapter), which occur for cosh(p0h) = 0, or when

= .(2n-1)1t (137)
p" i] 2h

160

where again n is a positive integer (n =1,2,3,...). The poles associated with the isolated
stripline given by (137) are actually a subset of the poles in (136). The difference is due
to the fact that, in the analysis of the isolated stripline, it was assumed that the
conducting strip was centered at the origin, and was equidistant from both ground planes.
As a result, certain modes could not be excited due to the symmetry of the structure.
The additional poles which appear in (136) include those which are introduced when both
conducting strips are not assumed to lie in the plane of the origin.

Returning to the analysis of the coupled stripline structure, the parameter p0 was

previously defined as

p: = 322 -k2 (138)

—(.’£‘.)2 = 12 —k2. (139)

From this the propagation eigenvalues N, at the poles of W(E, y, C ) are seen to be

 

A, =\ kg {32132 . (140)

 

The poles implicated in the expressions above, and the associated propagation
modes for coupled stripline exhibit the same behavior as those associated with the
isolated stripline structure analyzed in Chapter 4. In this case the poles are associated
with waves that are guided by the parallel ground-plate background structure, having

plate separation 21. When the frequency of operation is such that k0 < m: /( 21) it is seen

161

from expression (140) that k2 is imaginary, and the associated parallel-plate modes are
evanescent. When k: becomes sufficiently large, A is real, and propagating modes exist
which are guided by the background structure. During this transition, the poles
associated with the guided—wave modes of the background structure migrate from the
imaginary k-axis to the real Naxis in the manner discussed in the previous chapter.

Also as before, the relationship

gp : 7.)“ — Ap \/C + A1) (141)

 

implicates branch points at C = :22, in the complex {-plane. Solutions which lead to
proper modes (those associated with fields satisfying the radiation condition) are insured
by choosing branch cuts which prevent singularities from migrating across theRe{E}
axis. This creates a strip of convergence, resulting in fields which fall off with distance

in directions away from the guiding structure. Thus requiring that Im{Ep} <0, or

Rewcz 43,} = Re{\/C ‘W5 .12} > o (142)

 

leads to Sommerfeld [20] branch cuts emanating from the branch points at E = Mp.

In addition to the proper guided modes, the coupled stripline structure also
supports improper leaky-wave modes which are associated with poles lying on the lower
Riemann sheets [25], [27]-[42]. As with the isolated stripline, improper modes arise
when at least one {2, moves off the Re{E } axis, and becomes complex with Re{C } < AP,
and Im{ C } < 0. In this case, roots can only be found if passage is made to the bottom
sheet by violating the kp-related branch cut, which causes the E-plane pole at E p to
migrate across the RelE} axis, into the upper-half E-plane.

The integration path must be deformed so as to remain above or below the pole

162

at E p when improper leaky-wave modes exist. This leads to a pole residue contribution

to the real axis integration which occurs in the transform-domain Green’s function. This

residue contribution is determined through evaluation of

fW(5,y,o|E=E da (143)
Cp P

where W(E,y,( ) may be expressed

W(5,y,c> = M)— (144)
whom)

The numerator in expression (144) is given by

N(E,y,C) = {(1_e‘4pol)e'Po|Y“)’/I _e-p0(21+y+y/)
(145)

- 21--’ - (41+-’) - (41-+’)
—ep°( yy)+epo yy +ep° yy

and the denominator is

0(5) = 1-e'4”°’. (146)
At the pole
2: z _j (2 _ 2; (147)
and
0(5) = 1-e‘4”°’ = 0 (148)
or

163

«pot z 1 (149)

and thus

Np(€,y,C) : {(1_e’4pol)e'P01Y"Y/l _ e‘Po(ZI+y+y’)

/ / ’
_e-po(21-y-y ) + e-Po(4l+y-y ) + e ‘PoW‘Y‘Y )} (150)

— ++l — -—l - -/ - ’_
—{-e po(21yy)_epo(21yy)+epo(yy)+epo(y w}.

The denominator term (146) is now expanded about the pole E p, such that

 

 

 

 

0(5) = l-e‘4”°’ z Daip) + ":1? PEG—En)
” (151)
_ dp dP
= -41(— 4”°’——°) — = 41—0 — .
e d2 5:5,“ 5,) d2 MPG 5,)
But
11% = i (152)
dE p0
and therefore
415
02(5) = ”(a—5,) (153)
1%,,

where pop = E; + E2 -k: is the transform-domain wavenumber at the pole. This leads

to

164

_ _/ _ /_ __ 222/ - __/
1’“) [~e poptv y)_e popo' y)+e p0,,(21y y)+e p0,,(21y H]

 

 

 

 

 

N
W,(5,y,5) = = ,(154)
po,D,(5) 415,,(5 — 5,)
The line integral along the inversion contour C2,
d5 (155)
C (5 — 5,)
is evaluated by letting
5 = 5p me” (156)
and therefore
~dE =jeejwdq1. (157)
This gives
(1 4“ - iv
_——§—— = 16", d1]: = -21tj (158)
(E ' 5p) 0 6e”
CP
and then the pole residue contribution is
. [e -po,0-y’) +e-po,(y’-y) _e-Pop(21+y+y’) _e-po,(21-y-y’)]
W,(E,y,C)dE = -JTE (159)
21E p
CP

5.6 Analysis of Numerical Results
The dispersion characteristics of the first two higher-order modes excited on a
coupled stripline structure having widely separated center strips are compared to those

excited on an isolated stripline with similar dimensions in Figure 5.3. The coupled

165

stripline is configured such that both conducting center strips lie in the plane of the origin
(h1=h2=0), the widths of both strips are identical (wl =w2=5.0 cm), the ground plane
half—separation is l=1.75 cm, and the strips are widely separated, having a center-to-
center distance of 15.0 cm. The isolated stripline is such that the center strip has a width
w=5.0 cm and a ground plane half-separation of h=1.75 cm. The 1 =1 TM parallel-
plate guided wave pole A, is also included with the dispersion curves for these structures.

For both the isolated and coupled stripline structures, the first higher-order mode
excited is the TB, mode, while the second higher-order mode is the T132 mode. During
the analysis of the isolated stripline, it was seen that the TEl mode was odd, while the
TB, mode was even. It is noted that, in general due to the lack of symmetry of the
coupled stripline, each of these modes is usually neither even nor odd. The poles
associated with these modes lie on the top Riemann sheet, and thus both are proper non-
leaky modes, having real propagation constants C> AP. Both modes also exhibit a low
frequency cut-off.

It is seen that the dispersion curves for both the TBI and TE2 modes excited on
a coupled stripline, with widely separated center strips, are very nearly the same as those
associated with an isolated stripline (having similar dimensions). It is noted, however,
that for both modes, the slight discrepancy between the curves for the two structures is
greater at lower frequencies than at higher frequencies. This is due to the fact that as
the frequency of operation decreases, the electrical distance between the center strips of
the coupled stripline also decreases, leading to tighter coupling. This stronger coupling
causes a slight change in the modal characteristics of the coupled stripline, as compared

with the isolated stripline.

166

The cross-sectional surface current distributions associated with the TBI and TIE,2
modes excited on the coupled stripline are presented in Figure 5.4 and Figure 5.5,

respectively. The total surface currents which exist on the center strips consist of both

axial and transverse components
E10?) = 2151,62) +z‘k12(§) (160)

E20?) = 21524;) + 215226). (161)

It is seen for both modes that the axially—directed currents show the expected edge
singularities, while the transverse current amplitudes are large for frequencies near
cutoff, and small for frequencies far from cutoff. While it is expected that coupling
between the center strips will cause a perturbation of the current distributions, none is
detected in this case. This is due to the relatively large center-to-center distance between
the conducting strips.

Although the propagation modes excited on the coupled stripline are in general
neither even nor odd, this structure supports modes which are either symmetric or anti-
symmetric. For the case in which the center strips have equal widths, symmetric modes

are denoted (in local coordinates) by

k12(5c') = k22(—§) (162)

and anti-symmetric modes are denoted by

klzo?) = —k22(-i). (163)

When the amount of coupling between the strips is small, the propagation constants for

symmetric and anti-symmetric modes of the same order are nearly identical. This is the

167

case for the dispersion curves presented in Figure 5.3. Due to the wide separation
between the center strips, the amount of coupling is small, the dispersion curves for the
symmetric and anti-symmetric modes are nearly identical, and thus no distinction between
the two is made.

As the amount of coupling increases, however, the degree to which the
propagation constants for the symmetric and anti-symmetric modes differ also increases.
This mode-splitting behavior can be observed in Figure 5.6. Here the propagation
constants associated with the TE1 symmetric and the TE1 anti-symmetric modes at a
single operating frequency are tracked for varying center-to-center strip separations on
a coupled stripline for which h1=h2=0, w1=w2=5.0 cm, and l=1.75 cm. It is seen that
a significant difference between the symmetric and anti-symmetric propagation constants
occurs for center-to-center separations which are less than twice width of the center
strips.

The axial current distributions associated with the TB, symmetric and the TEl
anti-symmetric modes excited on the coupled stripline described above, are presented in
Figure 5.7, Figure 5.8, Figure 5.9, and Figure 5.10. Here the center strips lie in the
plane of the origin (hl =h2=0), with strip 1 lying to the left of the origin, and strip 2
lying to the right of the origin. It is seen that, as expected, the current distributions on
the center strips perturb as the center-to-center distance between the strips is decreased.
The currents associated with the symmetric mode repel. As the distance between the
strips decreases, currents move away from the inner edges of the strips, and begin to
accumulate on the outer edges of the strips. Conversely, the currents associated with the

anti-symmetric mode attract. As the center-to—center distance decreases, currents begin

168

to accumulate on the inner edges of the strips.

A more interesting case is presented in Figure 5.11. Here the propagation
constants associated with the TB, symmetric and TF2 anti-symmetric modes are tracked
for varying center-to-center strip separations on the same coupled stripline structure
examined above, having h1=h2=0, w1=w2=5.0 cm, and l=1.75 cm (it should be noted
that the TB, dispersion curve for the widely separated center strips shown in Figure 5.3
lies close to the [=1 TM parallel-plate guided wave pole k2). At a constant frequency,
as the distance between the center strips is decreased, the propagation constants
associated with the symmetric and anti-symmetric modes are again observed to diverge.
However in this case the dispersion curve for the TB, symmetric mode is observed to
approach, and very nearly cross the hp parallel-plate background pole.

Axial current distributions associated with the anti-symmetric mode are presented
in Figure 5.12 and Figure 5.13. These figures show the expected behavior the T132 anti-
symmetric mode as the distance between the strips decreases. An examination of the
axial current distributions associated with the TB, symmetric mode (shown in Figure 5.14
and Figure 5.15) reveals an abrupt change at the center-to-center separation at which the
T52 dispersion curve very nearly crosses the background pole. For strip separations
greater than the critical value of approximately 7.0 cm, the expected behavior of the TE2
symmetric mode is observed. However, for separations less than the critical value, the
current distributions, which are no longer symmetric, appear to be non-physical. It may
therefore be concluded that as the center strips are moved closer together, the TB,
symmetric mode is excited until some critical center-to-center separation is exceed, at

which point a change in the propagation mode occurs. It is also concluded that the real

169

root found just above the background pole is associated with a spurious, non-physical
mode. It is suspected that the TE2 mode can be tracked below the A, parallel-plate
background pole, although efforts to do this to date have not been successful. After the
TE2 propagation constant crosses the background pole, it is expected that roots associated
with improper leaky wave modes may be found on the lower Riemann sheet. This

investigation requires further attention and will be left for future work.

5.7 TEM Specialization for Coupled Stripline

It was noted previously that stripline devices are often operated at frequencies
such that only the fundamental mode is excited. Here the behavior of the coupled
stripline structure will be examined for the case when only the fundamental TEM mode
is present. As with the isolated stripline, this specialization is also valuable because self
and mutual impedance parameters for coupled stripline structures may be defined for the
TEM mode.

In the analysis above, it was determined that the coupled strip structure is

described by two pairs of coupled integral equations, one pair representing the x— and z-

directed electric field components associated with strip 1, given by

“I

12:11

+1

‘02

 

 

52 . a
{—2 +k3)85t(x,)’|xl,h1)k1,(x’) +1Ca—xgg(x,y|x’,hl)klz(xl)

ex}
5" (164)

dx’=0

 

 

62 . a
(— +k:)g(t(xshllxlsh2)k2x(xl)+1C—'g(t(xah1Ix/9h2)kzz(x/)
8x2 6x

170

------- )xpoi. of TM porolleI—plote mode

_ isoloted strip TE1 mode

ElElElElD isolated strip TE2 mode

AMAA coupled strip TE1 mode Elscm center-to—centerg

QQQQD coupled strip TE2 mode 15cm center—to—center

 

 

 

 

1.0: A

o 2 ‘4
x ..
\L :
M I
1
+1 0.8*
C I
O _.
+J a—t
(I) I
C a
O I
0 0.6-3
<1) 1‘
U) I
O _
-C- 2
Q. q
0.4:
C _
.0 :
+3 :
O .4
0‘ :
U a
Q 0.2:
O -
L ..
O— :

OoO dI'llIIIII'IIIIIIIIIIIIIIIIITTFIIIIITIIT]

2.0 4.0 6.0 8.0 10.0

frequency in GHZ

Figure 5.3. Comparison of dispersion characteristics of non-leaky, higher-order modes
of isolated and widely-spaced coupled, coplanar stripline.

171

 

W x—component 2.22 GHZ

z—component 2.22 GHZ
*-**-** x-component 10.0 GHZ
------- z—component 10.0 GHZ

1.0

.0 .0 .0
4s a) on

current magnitude (normalized)
.0
N

 

"*---*---
,—aie--" \ 2 ”kn-1:-“

 

[11111111141111[trilliilriliiluriililllluilillnj

3k
IIITIIIIIIIIIIIIIIIIITIIIIII]

20.0 0.5 1.0
location x/w

 

.0
O
*-
\

IIIIIIII

I
-1.0 —0.

UI-

Figure 5.4. Cross-sectional current distribution for TEl mode, on center conductors
of widely separated, coplanar strips (15cm center-to-center, 5cm width,
1.75cm ground plane half-spacing).

172

 

 

 

W x—component 4.1 GHz
z—component 4.1 GHZ
HH* x—component 10.0 GHZ
------- z—component 10.0 GHZ
1.0 1: u
..l '
_ : .'
—l' l
A :: i
'0 1 l i
a) — ' '
| l
.5! 0.8 : l i
_ —t I l
O - I, i
E 1 ' '
L _
2 : . 1
— I i
s : = ‘
3 ‘ |‘ i
4: 3 i .'
c .2 ' '
0704 2 )2 '1
O - x 1'
E : ‘ x “““ ~ I’
_ ‘1 ’z’ ‘s\ 'I
-l \ I \ I
0.2 - ‘ x “ '
0L) : \‘\ I, \\ I”
L —. \ [I \\ I
" 1‘- ~ '—.*~‘\ ”'*--‘ ~ I ’\
(3) 1* *~-‘ «viii? \ ' fi‘t“*""*’ *
\ \ I I
- \ I] \\ [I \\ I
: \\/, ‘*\ ’2“ \,I’
0.0 I I I I I I l I l I l I I I T l I I la] I l I I T l I l I I I I l l l 1 IT I I
—1.0 — .5 0.0 0.5 1.0

location x/w

Figure 5.5. Cross-sectional current distribution for TEz mode, on center conductors
of widely-separated, coplanar strips (15cm center-to-center, 5cm width,
1.75cm ground plane half-spacing).

173

.0
co
\J

 

 

 

_\_<°
\ ‘
K; 0.96 ‘
*5 i
\
B 0‘95 \ _ TE1 symmetric mode
2 \\ ._ _ _ TE1 antisymmetric mode
0 \
0 0.94 \\
9 \
O 0.93 _________________
_C
Q.
C 0.92
O
”5 5 0
. cm
8‘ 0’91 .75cm
9 0.90
D.
0.89 |IIIIIIIIIIIIIIIITTIIIIITIIIITITFIIllllIlllllllllII
5.0 7.0 9.0 11.0 13.0 15.0

center—to—center distance (cm)

Figure 5.6. Dependence of propagation phase constant upon center-strip spacing for
the TEl mode of coupled stripline.

174

Normalized strip 1 current distribution
for TE1 symmetric mode

 

 

 

1.00 ‘1
g“ I
+J 0.50 ‘
C '1
(D s
L
L -l
3 Z
0 _
.9 0.00 3
x -l
O I
‘0 -1
<0 I
.‘1‘ :
B - H-e-H 5.5cm center—to—center
E‘O-SO '2 - - - - 6.0cm center—to—center
5 : —— 10.0cm center—to—center
C I
_1.00-‘IIIIITIIIIIIIIIIIIIIIIIFIIIIIIIIIIIIIlll
—1.00 —0.50 0.00 0.50 1.00

location x/w1

Figure 5.7. Strip 1 (left) normalized TE1 symmetric current distribution.

175

Normalized strip 2 current distribution
for TE1 symmetric mode

 

 

 

1.00 3
x“ I
+2 0.50 -‘
C 'l
(D 1
L '1
L _
3 I
0 u
.2 0.00 '1
X
C I
“o I
Q) _
.E Z
‘5 - W 5.5cm center—to—center
5‘050 i - - - - 6.0cm center—to—center
5 j — 10.0cm center—to—center
C I
—1.00—IIIIIIIIIIITIIIIIIIIIIIIIIIIIIIIIIIIIIII
—1.00 -—0.50 0.00 0.50 1.00

location x/w2

Figure 5.8. Strip 2 (right) normalized TEl symmetric current distribution.

176

Normalized strip 1 current distribution
for TE1 antisymmetric mode

 

 

 

1.00 :
N i
x _
+2 0.50 '1 W =
S 1 | =
L.
L -
3 -l
0 I
.9 0.00 3
x —1
o 3 , . . -
.0 3 ,. ’
Q) "‘ .//
.[Z‘ I I
6 - I CHH-e-o 5.5cm center—to—center
5"0'50 '1 I’ - - - - 6.0cm center—to—center
5 3, —— 10.0cm center-to—center
c j!
q
4
_1.00 TI—IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII]
—1.00 —0.50 0.00 0.50 1.00

location x/w1

Figure 5.9 Strip 1 (left) normalized TEl anti-symmetric current distribution.

177

Normalized strip 2 current distribution
for TE1 antisymmetric mode

 

 

 

1.00 3
1 .
_v§ 1 I
*5 0-50 i w = 5.0cm I
<1) 3 | = 1.75cm ,'
t _ ,..
3 ‘ /,,
0 I
.9 0.00 3
X -¢
0 I
‘0 i
Q) ..
.E 1
'6 - W 5.5cm center—to—center
E_O'50 Z - - - - 6.0cm center—to—center
S 1 — 10.0cm center—to—center
C 1
_1.00:IIIIIIIIllllIIITIIIIIIIIIIIIIIIIIIIIIIII
—1.00 —0.50 0.00 0.50 1.00

location x/w2

Figure 5.10 Strip 2 (right) normalized TE1 anti-symmetric current distribution.

178

0.84

 

 

 

 

o _
—V~ ‘ w = 5.0cm
\ . l = 1.75cm
V :x f = 7.0GHZ
_+_) .1 x

C ‘K

o 0.82 3 1‘
-+-J _. \ a.

CD : ‘kw“*

C _ ‘~ —_

8 fl 7* *‘§-¢_-_*—“*——-¢
Q) I

8 0.80 -:
.C 4

C). '1

C 2 """""""""""""""""""""""""""""""""
.0 d
*3 :

cm 0.78 - _

0 : TE2 symmetric mode

O. 9 my: TE2 antisymmetric mode

2 q ------- 190.. of TM parallel-plate mode

0.76 IIIIIIITTTIIIIIITII]IIITIIIIIIIIIIIIIFI]lIIIIIIII]
5.0 7.0 9.0 11.0 13.0 15.0

center—to—center distance (cm)

Figure 5.11 Dependence of propagation phase constant upon center-strip spacing for
the TE2 mode of coupled stripline.

179

Normalized strip 1 current distribution

for TE2 antisymmetric mode
1.00

 

 

 

7.
3 w = 5.0cm
N j | = 1.75cm
x i
+2 0.50 —
C 1
8 4
L 1
8 4
._9 0.00 3
X : \
O - \
‘0 3 \
Q) _.
.E‘ 2
C—0 50 3
E ’ - ...—+4.... 5.5cm center—to—center
5 I - - — 6.0cm center—to—center
c j —— 10.0cm center—to—center
—1.00-iIIIIIIIIIIIIIIIIIIIIIIIIIIIII[IIIIIIIIII
-—1.00 —0.50 0.00 0.50 1.00

location x/w1

Figure 5.12 Strip 1 (left) normalized TE; anti-symmetric current distribution.

180

Normalized strip 2 current distribution
for TE2 antisymmetric mode

 

 

 

1.00 :
3 w = 5.0cm
N j l = 1.75cm
x :
+2 0.50 a
C I
(D _
L
L : \
8 ~ \
_ I \
.9 0.00 -
X —
O I
'0 2
(D _
.'_\_‘ 2‘
0—0 50 3
E ‘ ~ ...—H4 5.5cm center—to—center
5 I - — - 6.0cm center—to—center
c 1 — 10.0cm center—to—center
—1.00 _"'II1IIIIIIIIIIIIIIII[IFIIIIIIIIIIIIIIIIII
-1.00 -0.50 0.00 0.50 1.00

location x/w2

Figure 5.13 Strip 2 (right) normalized TE2 anti-symmetric current distribution.

181

Normalized strip 1 current distribution
for TE2 symmetric mode

 

 

 

1.00 —
-l
4
3 w = 5.0cm
N j l = 1.75cm
__\fi -1
+1 0.50 _
C '1
(D I
L.
L -1
3 '1
0 i
.g 0.00 A ,, \
I \.
O _ I \\
‘0 Z i g \
<1) _
.E I
B ‘ ...—...... 6.0cm center—to—center
E—O'SO‘ ran—«pH 7O
1 . cm center—to—center
L6 - - - - 8.0cm center—to—center
c ; —— 10.0cm center—to—center
.fi
4
—‘
—1.00 ITIIIIITIIIIIIFIIIIIIITIFTHIIIIIITIjI]

—1.00 —0.50 0.00 0.50 1.00
location x/w1

Figure 5.14 Strip 1 (left) normalized TE2 symmetric current distribution.

182

Normalized strip 2 current distribution
for TE2 symmetric mode

 

 

 

1.00 —_
w = 5.0cm
N j l = 1.75cm
x i
+2 0.50 —
C _
(D I
L.
L .1
3 I
0 ..
.9 0.00 3
x -
O I
‘o I
a) +
.t‘ I
6—0 50 _- H+H 6.0cm center—to—center
E ' 3 “...... 7.0cm center—to—center
‘5 1 - - — 8.0cm center—to-center
c 1 —- 10.0cm center—to—center
:l
_1.OO-I—IIIIIIIIIIIIIIIIIIIITIIleIIIIIIIIIIIIVI
—1.00 —0.50 0.00 0.50 1.00

location x/wz

IFigure 5.15 Strip 2 (right) normalized T132 symmetric current distribution.

183

 

a}
(165)

(ix/=0

“1
. . a
11T{ f [ICE-ggc'(x,ylx’,hl)k1x(fl+(k3-C2)g;'(x,ylx’,hl)klz(x’)
“ 1

 

“2
. a :
+f ijcggc(x’h1lx5hz)kz.(x5 +(k3—C2)gc'(x,hl ixl’h2)k2z(xl)
-02
and one pair representing the x- and z-directed electric field components associated with
strip 2, given by

at

 

 

 

 

 

62 . (9
f (—_2 +k:)g(t(xah2ixlshl) k1x(x/) +Jc_g(t(x9h2|x/ah1)k1z(xl) dx/
ax ax (166)
“2
2
+ lim { f [(3:2- +k3)8¢'(x,)’lx’,h2)k2x(X’) +iC-38ct(X,)’lx’,h2)k2z(xl) dx’} = 0
y~h2 6x 6x
#12
. a
f[1€agci(x,h2lx’.hl) 161.05 + (k:-C2)g¢i(x,h2|x’,h1)klz(x’) dx/
(167)
02
. . 8
”ii I [Icagé(x,ylfl.hz)k2.(x’) +(k3-Cz)g;'(x,ylx’,h2)k2.(x’) }dx’=0.
y-
-a2

N o transverse currents exist on the center strips when the TEM mode is excited,

therefore

km(x) a 0 (168)

fOI‘ n=1,2, and also C = k0 in free space. As a result of the TEM assumption

(’9? - Czlgf(x.y lx’,y’)knz(x) = o (169)

184

for n=1,2, and fi=x,z. As a result of (168) and (169), the equations representing the
longitudinal (z-directed) component of electric field (equations (165) and (167)) for both
strip 1 and strip 2 vanish, as expected. Equations (164) and (166), which represent the
transverse (Jr-directed) component of electric field supported by currents on strips 1 and
2, then become

02

mifjk"; —gk ”’0,ny .llh)k (x )dx’}+ +fjka‘: —gko ",(xh Ix ,hh2)k (x )dx’z 0(170)
y~h
-02

“2

ffcingOCJIZlx’,h1)klz(x’)dx’+1im{ fjko—a—gk“(x,y|x/,h2)kzz(xl)dxl} = o. (171)
8x ° y-oh2 112 6x °

-01
The differential operator a/ax acts on field points, while integration occurs over source
points, therefore (170) and (171) may be expressed

al 02
a
Jkoa—{hm 8,. ”£00ny h )k,z(x’)dx’+ fgk‘:(x,h1|x’,h2)kzz(x’)dx/} =0 (172)

-a‘ -02

a2
1" o—x'{a [g]: (x: [12GC h 1)krz(x/)dx/ +£11202 gk‘: 0‘ Yix :zh2)k2 (x’)dx’}0 = - (173)

Integrating both sides of (172)-(173) over x gives

a‘ 02
lim gk°(x,ylx’ .h)klz(x’)dx’+ fg,."(x.h,Ix’,h,)k,z(x’)abc/=c:1 (174)
y-ohl o

1-01 ‘02

185

“1 “2

[8120.112 lx/,hl)k1z(x/)dx’ + In: gk‘:(x,y|x’, 119/(210’)de = Q (175)
y-.
3 ”“2

01

where C, and C2 are constants which are determined by the potential difference existing

between the conducting strips and one of the ground planes.

The Green’s function implicated in (174) and (175) is given by

l3 / / m eJ'E(x-X’)
x, x, =
gko( yl y) f 4“

’1”

W(£,y,y’)d£. (176)

 

As a result of the TEM approximation, the transform-domain wavenumber becomes

p. = 1.2—+1.: +3 = 1? = m “77>

therefore

W(€,y,yl) = —1—{e'|5y”ll + .—__—(1 1 [- e’lEl(21+y+y’)

 

ial -e'4'5“)
(178)
_ e -|El(21-y-y’) + e -lE|(4l+y-y’) + e -lEl(4l-y+y’)]}
OI‘
W(E’y,yl) : 1 {(1 —e'4iEi1)e-iEIIY'y/i _ e-IEI(21+y+y’)
lE|(1-e-4|E|l) (179)

_ e -|£|(21-y-y’) + e -|El(4l+y-y’) , e -l£|(4I-y+y’)}.

It is seen that (179) may be expressed

186

/
W(€.y.y’) = % (180)

where
N(€,y,y/) : (1 —e_4iEil)e-i5iiy-yli _ e'l5|(21+)’+)'/)
(181)
— e “mm—H”) + e —I£|(4z+y-y’) + e -I&I(4z_y.y/,
and
0(8) = l-e'4'5". (182)

In order to determine if the integrand of (176) is singular when E =0, the

behavior of (180) is examined for small values of IE I . It is first noted that

lirnN(E)= {(1— 1)1-1-1+1+1}=o. (183)
E0

For E < 1 the approximation

ele+x+i (184)

Can be made, therefore

 

 

 

Where

187

= 4l€|l, b = IEIIy-y’l. c = l€l(21+y+y’) (186)

and

d = l€l(21-y-y’), e = |€|(4l+y-y’), f= l€l(4l-y+y’). (187)

After a great deal of algebra, it is determined that

ME) = -4l€l21|y-y’l + 4|El212 - 4|El2yy’. (188)
It is also seen that

D(E) == 1-(1—a+(—_:—)3)=4|E|l—8|§|212 (189)

for E < 1, therefore

1m N15) : -4l5|21ly-y’l =,_|y_y,|_1y_’. (190)
M lElD(€) 415.21 1

 

Thus the integrand of (176) is not singular for E = O.

5.8 Moment-Method Solution for Coupled Stripline TEM Specialization
Once again, in order to facilitate a numerical solution, a coordinate shift is
performed. The coordinate variables of the scalar component EFIEs are shifted to a

Sy stem of local coordinates using the same technique applied in section 5.3 (refer again

to Figure 5.2). This is done by letting

_/_ /_
xa —x xa

(191)

and then dx’=d§;. It is seen that when x’=xa —wa/2, in: = —wa/2, and when

188

x’ = xa + wet/2, 27,: = wan/2. The coordinate shift

xa =x—x

a

is also performed, and then dx = d)?“ . In this case, when x = xa - wet/2, fa

and when x =xa +wa/2, 35a =wa/2.

(192)

—wa/2,

Applying this change of variables to the scalar component EFIEs (174) and (175)

gives

 

°° java-ff)
lim {[8 4 W(Esyah1)dg}klz(fll)£ll

112

b

 

4

02 ‘w

)

 

b1 00 _ —/
j€("2‘x1 .
7t

01 “°°

b2

 

y~h2 7‘
(12 —oo

The current functions in (193) and (194) are now expanded such that

N
19.45) = Z ampmfi)
n =1

LlSing square pulse basis functions
l...for J? -x sis} +1?

p.41? ) = {

O elsewhere

189

2 °° jug-E26 .
+f f" N W(E.hl.hz)e’“‘“‘2’de}k2.(fz’)d?2’ = C1

"dug—£4) _ I
+1im f 4 W(E,y,h2)dE k21(x2)df2 = C2.

(193)

(194)

(195)

(196)

where

 

__ 2a _ __
Ax.=2i.= “. =-a 111-31x“. (197)

Here n is a positive integer (n=1,2,3,...), and oz=1,2. Application of this expansion

 

 

 

 

 

 

gives
bl
. ejE(x1 x1/)
13: f 4 We made Za,p,,(x,)dx{
0' 33 (198)
each-£25 My N _/ /
+f f 41: W(E’hl’h2)e ' d5 zaznpzzocfldfz =
n=l
°° jug-32;) . _x N _ _
[{f" 4.. szmoe’“ ”délzczlmlrxbdx:
n=1
(199)
b2
. em: *2)
+1un f Way. lode 2a,,p2,(x2’)df— — C
y~h2 4“
02 -...
Interchanging the order of summation and integration leads to
bl
e}E(x1 11]) _/ _/
Ezlfalnflm 112{sz W(€9y,h1)dE}plz(xl)dxl
“‘ (200)
b2 E( ’1
e} J‘1 12 . _ _
+2 iaZNf{ ["4 W(E,h1,h2)em' ”(141021026115 = C1
n: “2 _m

190

 

N 8183251) jazz-x1) -/ /
2: din 471: W(Esh29h1)e d5 plz(x1)d.;1

 

n=1
1 (201)
b2 ... _ _,
N . ejaxz-xz) _/ I
+ 2a,,fhm f W(E,y,h2)d€ [agenda = C2.
"=1 wk; 4%
02 -00
Now letting
3”” °° '50? -£’>
Am" — 1' e] 1" ' ’ (202)
11 " 1m ——W(E,y,h,)dE dfl
f i y-Jr1 -09 411:
gn+i2 °° , _. _/
mn 615(11m‘xz) 'E(x - ) -/
A12 = f [TW(E,hl,h2)eJ ‘deE (1x2 (203)
E2,1'322 ‘°°
mii‘ °° yeah-17;) °£( _ ) /
A57" . l [Twwphoe’ ‘2 2 d: as (204)
Eln'ft ‘°°
320% °° mam-E26
A2?" = f lim{fe———W(E,y,h2)dE }d5c'2/ (205)
_ _ y~h2 4n
XZn-xZ 3"”
equations (200) and (201) can be expressed
N N
Z All a1" + 2A” 02" : C1 (206)
"=1 n=l
N N
2 A21 all! + 21422 02" : C2 . (207)
n=1 n=1

It is noted that in contrast to the complete propagation mode MoM solution performed

191

previously, the matrix elements in (206) and (207) are determined from a forced solution,

where B = k0 is known. The matrix elements (202)-(205) may be expressed

xln+xl

 

 

 

°° jail»: . —/
An" = Im{fe W(E,y,h1) f evex‘dfl’dE}
y-ohl 411’ __ -
3.” xln—xl
°° 1'53? fhiiz
In . . —/
A3" = [841: W(£’hl,h2)eJE(x1'x2) f e-ngzfidg
on 15x2” 3min
= [e 22,221,221.-.) [...-191.1502
411: _ _
'°° ‘ln'xi

 

0° .5- ‘21-”‘2

AM” — lim gnaw h 'jigdf’d

22 — f 4N (£05 2) e 2 E
7‘2"

and evaluation of the inner integral leads to

 

 

 

I " x
an a. -/ _ - _ _
_/ -j£xa Xan+ x. J35 (xan+ fa) -jE(x¢n- in)
70:, / e _ e — e
f e dim - . _ - .
_ -JE 75
X ' X

e 9532,. e 'J'Ei. _ 8 “15172,. 815132

 

'1' 1i
8 ~15”. (€102. _ e 402,)

1'5

 

2311152“

5

iii-an

As a result

192

(208)

(209)

(210)

(211)

(212)

m j£(x—Im-xln) sin x'
A3" = lim [ETN—W(E,y,hl)——:——ld5 (213)

Y“h1

m

 

 

 

 

 

Mme-aria sinEx'
Am" = fe W ,h,h 2d (214)
12 - 27!: (g 1 2) E 5
Anna on efEszuz-xh-xl) W h h sinEil d (215)
21 — f 21!: (£9 29 1) 5 E
mn 0° ej5(;2m-12,,) sinEx2
A22 = lim ——2—— W(&.y.h2) dE . (216)

24!; -... 7‘

Now the constants C1 and C2 in equations (206) and (207) must be determined.
These constants are proportional to the potential differences that exist between the

conducting strips and the ground planes. In Chapter ‘4, the normal (y-directed)

component of electric field in the region between the ground planes was found to be

J i"
Tie—)dv’. (217)

62 1 s s,
E r = —— G r r -
y(3') ayaz f o( l )
V
Because the center strips are assumed to be infinitely thin
Jz(r”’) = Kz(x’,z’) (218)
and
(35(Fl7’) = Qfilfiflz-z’) (219)

where is = xx + z‘z. Now let the cross-sectional surfaces of strips 1 and 2 be represented

193

by S1 and 52, respectively, then

 

“’ K r"
1650 (FlF’)—— K" )dv v=f Gamm- 14 )ds{
jmeo s jwe

0

V 1 (220)
_._. K2(F’)
+6] o‘trr-I’) 2 (82’.
S

2 jweo

 

The normal component of electric field may therefore be expressed

1m
2 K x/,Z/
E(f')= a f G‘(F|f")'—-———lz( )dz’dx’
ayaz jmeo
al-oo

b2 ...
K x/,Z/
+ ffG‘(F|?’)-——2fi( )dz’dx’
10360
02 -~

and, after application of the convolution theorem, the transform-domain electric field is

 

(221)

given by
b1 b2
.. . a .. 2 2 k (ICC) ,2 _, _, k (x’,C)
e,(p,C) =JC— fg£(plp’)-‘+—dx’ + g¢(plp’)'—2%—dx’ . (222)
6y 1060 1106
For the TEM mode C = k0, therefore
b1 [,2
- k2 a .. - - ..
649,5) = 3:5 g.:(plp’)k2(x’.<:)dx’ + g,;(p|p)k (,x ()de (223)
0 01 (12

The transform-domain Green’s function implicated in (223) is given by

81(plp’) =gk:(b’lfi’)+gi(p|f5’ (224)

194

where

°° e-IE(x-x’)e-l£lly-y’l

 

 

P ~ ~/ = (225)
820(919) I. 41"“
and
s: 2 2, 3 emU—xl) i 1512:.» + ') -lE|(21- - ,
g(plp)= -e 2,. -e 2,)
k" [241: 11: 1(1 - e-W’) (226)

.2 e -lEl(41+y-y’)+ e -|EI(4l-y+y’) d1: .

With the normal component of electric field determined, the potential difference
between the center strips and the ground planes can be found. Let V, be the potential
between a point x, in the center of strip 1, and a point directly below on the nearest
ground plane (Figure 5.16). Then

h1 h1 bl

_ _. '° _ k0 a «I / / /
V1 ‘ — €y(P,C) d1 = " _ (.06 '6; gko(x19yix :h1)k1z(x )dx
-, ‘ _, 0 a, (227)

 

 

b

+ f§£,(x.,ylx’.h2)k22(x’)dx’ dy.
“2

Application of the partial derivative to the outer integral gives

b
k ‘ , "1
Vl = — (9:; f§k0(xl,y|x’,h1)klz(x/)dx’

 

 

-l
“1

(228)
b

2 hi
+ fa: (x1.ylx’.h2)k22(x’)dx’
“2

 

-l

195

 

 

 

 

 

 

 

 

 

Figure 5.16. TEM mode center strip potentials relative to bottom ground plane.

196

01'

 

 

bl
_ k0 «t / / «t / / /
Vl — -(—o-€—— [gko(xl,h1|x ,hl)klz(x ) —gka(x1,-l Ix ,h1)klz(x )]dx
0 “1 (229)
b2
+f gk:(xlah1|x/ah2)k2z(xl) -§k:(x1’—lix/9h )k22(x/)]dx/
02
leading to
k 1"
VI = w: f[§k:(xl,-l1x’,hl) —§k:(x1,hl|x’,hl)]k1z(x’)dx’
o 2, (230)
b2
+ f [92102—1 lx’.h2) - 9:00.21lx’.h2)]k22(x’)dx’ .
“2

But it can be seen that

_"_o_ _ ”Woeo - (82)/go _ h (231)

therefore, the potential which exists between strip 1 and the ground plane located at y=-l

 

 

IS
bl
Po .. ..
V1 2 : [[gk:(x1:_lixlsh1) -gk:(x1ah1 ix/ah1)]k1z(xl)dxl
\ ° 0: (232)
b2
+ f [91102-1 lx’.h2) — 91:01.12. lx’.h2)]k22(x’)dx’ .
“2

In a similar manner, let V2 be the potential between a point x2 in the center of

197

strip 2, and a point directly below on the nearest ground plane (again see Figure 5.16).

This leads to

b

0

 

 

k a
[1.15an
-l

l

1
“1

£1; (xyy ix ,2 h1)k1z(x/)dx/
(233)

b

+ f§.‘o(x2.ylx’,h2)k22(x’)dx’ dy
“2

therefore, the potential which exists between strip 2 and the ground plane located at y=-l

is

“1

t 11
V __°
2 £0

“1

“2

[[91:02—1 Ix’.h,) - 9.102.122Ix’.h1)]k,2(x’)dx'

(234)

+ [1911024 lx’.h2) — 9.102.112lx’,h2)]k22<x’)dx’ * .

“2

It was seen earlier that

bl

«I / / /
fgkoaphl Ix .h,)k,2(x )dx +
bl

at / / /
fgko(x2’h2 ix ah1)k1z(x )dx +

I

therefore (232) and (234) become

“a

V-

“1

b2

E2:(x,,h,|x’,h2)kzz(x’)dx/ = c1 (235)
“2
b2

§£o(x2,h,lx’,h,)k,z(x/)dx/ = c2 (236)
“2

“2

bl
1 _ e {f§k:(x1,-l iX/ah1)k12(x/)dx/ + g’k:(x1,—llx’,h2)k2z(x’)dxl- C1 (237)

02

198

b, b

2
110 ., ..

V2 : : fgk:(x29-l ix/ah1)k1z(x/)dx/ +fgk:(x2,"l Ix/,h2)k2z(x/)dx/- C2 , (238)

“2

“1

The principal and scattered components of the transform-domain Green’s function

implicated in (237) and (238) are given by

 

 

 

°° 150:, -x’) -El-l-h2l j5(x.—x’) —£ll+h.l
rim-I lx’,h2) = f e " dE = f e 6 d5 (239)
° _ 4R5 _ 4nE
and
z , 3 Jim-’6 4 : —£¢ (-Ilh ) (240)
§S(xa,-llx ,ha) = f sne " “dE.
k” _N4n£(1 14");
Here
¢2=21+(—I)+h,=l+h, ¢3=4l+(-l)-ha=3l—ha
(241)
¢2=2l-(-l)-ha=3l-ha ¢4=4l-(-l)+ha=51+ha
and as a result
4
Z: Sr: = [‘19 film“) - 1e 35‘3“”) +1e 35(313’“) + 1e 3561”“)
n=l
= {-18 —£(I+h“)+le {(51%) = (e 'ESI-e'5l)e 3“” (242)

= (e ‘5‘“ - 1)e "E’e 36"“

- 1+1. _
= -e 5' “l(1—e E4’).
From this it is seen that the scattered component of the transform-domain Green’s

199

function is

 

°° 120:. -x’)
f e [_6‘5ll*hal(1_e-4E1)d£

...st /
g (xa,—l Ix ,ha)
k" 41tE(1-e '4“)

-w

(243)

°° 1'60. -x’)

_fe e-Elhhaldg
_ 41:5

and thus the total Green’s function §k'(xa,-l |x’,ha) vanishes

grim-1 Ix’,h2) = §€(xa,-l|x’,ha) + gag-119,112)
fejE(Xa-x’)e —E|l+haldg — fej£(xa_x/)e _E|[+ha|d€ (244)
“NE 4nE

-00 -oo

 

 

As a result, the potentials V1 and V2 can be expressed

| “’0
V1 : :{'C1} = _noCl (245)

V = fi{-C2} = _nocz (246)

where ‘10 :1/110/50 e 1201: in free space. As was noted in Chapter 4 the sign of the

constants C1 and C2 in expressions (174) and (175) can be either positive or negative.

As a result, these constants are chosen for convenience to be

._. Va (247)
“ 120x

 

200

for a =1,2.

5.9 Pole Series Representation for Matrix Elements

As with the isolated strip structure, solution of the coupled matrix equations (206)
and (207) can be greatly accelerated by replacing the real axis integration implicated in
the matrix elements (213)-(216) with an equivalent pole-series representation (refer to

section 4.11). For simplicity, the case of coplanar center strips which lie in the plane

of the origin (h1=h2=0) will be examined. If this specialization is made, (179) becomes

W(€)= 1 (1-e'4lil’)—e'2”5'—e'2’l5|+e'4’l€|+e-4IIEI (248)
IE |(1—e'4i5l1)

which, simplifies to

We) - “3'2”“ (249)
luau-2’19)”

 

From this it is seen that

 

 

1-(1-21151) ___ 21151 22 (250)

W(£) = 2
|e|[1+(1-2z|£|)] ZIEI-ZIlEl

for E < l , and therefore poles occur for

1 +e_2|EP|I ____ 0 (251)

01'

21:21: +1319" = 2cosh(Epl) = o. (252)

201

At the roots E p = jE therefore

pi’
cos(Epil) = O
and thus roots occur for
._ _ 1t _

E221 — (2n 1)? ...for n — 1,2,...
therefore

E2: :K—zw—n— ...for n = 1,2,....

p 21
Now

w(€) - —N(E) = "”2”"

 

_ lama) |E|(1+e'2l5")

and from this it is seen that at E = E p, ME) = 2 and

D(E)=1+ e"2E1 =D(&,) + d—D

22 (a —5,)

(=9

 

= ale-2‘12:E (a 45,)

 

=2“: —&,>

therefore
1imW(E) = 2 = ——1——.
M, 52105-52) £l<£—£,)

As discussed in Chapter 4, Cauchy’s integral theorem states

202

(253)

(254)

(255)

(256)

(257)

(258)

Mode = ff(E)dE + ff(€)dE + ff(€)d€ = o. (259)
C CR . .

C, C2,

The matrix elements (213)-(216) may be generically expressed as

sin x” -
AI? = —1‘W(€)—-§-—peju“"di .f..or a =1,2, B =1,2 (260)
2n E
where
L11 = (Elm -xln) (261)
L12 : (Elm + x1 -x2n -x2) (262)
L21 = (fzm “‘2 -xln _xl) (263)
and
L22 : (52m ' x2n)’ (264)
Because
5 = 5, +135,- (265)
it is apparent that
61.51.” = e "511119815er (266)

and therefore the case when Lap > 0 requires that the integration contour be closed in the

upper half of the complex 1,2-plane, and the case when Lap < 0 requires lower half-plane

closure (refer to Figure 4.19). Thus it is seen that

203

f f(€)d€ = o (267)

C

and the real—axis integration implicated in the matrix element expressions may be

represented as the sum of contributions to the contour integration around the poles

ff(€)d€ = -2 ff(£)dE. (268)
CR p C;

For E 3 E p it was seen that

We) ~ —1—— (269)

we —£,,)

and thus the matrix elements may be written

run 1 I(E)
A2 = — — —dE
’ 2,: hide,» “70’
if
[(5) = ism—“@6859. (271)
l 52
At points along C;
E = 25p + eej‘” (272)
and
dE = jeeji’dtlr (273)
therefore

204

1(5) d5 = lim _I__(E)- Nd
:(€¥5P) e GHZ—1.68am 1|!
= :2rth(:tEp).

As a result, (270) becomes

A2"; = 112105 )

At the poles E p =jEpi, leading to

Sin(ijEp2.x'B )e 22L
(ijgpi)2

 

P

105 >=%

which in turn gives

mu 1 Sinh<£ii) €15.11
Audi :7: 2P1} Pp

P pi

It is noted that

pica :

.. for Lap <0, or Lap >0.

—E22L22 = —E22.|L22| .. for Lap >0
+EP2L22 = —EpiiLaB| .. for Lap <0
thus
A22"; = -:—Z——— ——(——th“ x3e) EP‘IL““| for L22p #0.
p E2222-

(274)

(275)

(276)

(277)

(278)

(279)

It is observed that the generic representation of the matrix elements associated

with the coupled strip structure in (279) closely resembles the expressions for the isolated

stripline presented in Section 4.11. It was determined in this section that the case when

205

Lap =0 must be examined more closely. It can been seen from (261)-(264) that La” =0
only occurs in the self terms AS" and Ag". This is due to the fact that the center strips
cannot overlap, and therefore the distances Ll2 and L2l are never smaller than the center-
to-center distance between the coupled strips (it should be noted that x1 is defined to lie
to the left of the origin, and is thus always negative, while x2 lies to the right of the

origin, and is thus always positive). For the case of Lap =0

 

 

 

 

 

1(a) = 1 31112.22) = 21555., 'e'm’ (280)
1 £2 1’21?
and
mu m 1 Sin(€fp) m 1 ejEiB _e-jEiB
A = —W ——d = —W d
up 2“ (E) 2 E 222 (6) 21.2 E
"°° ‘°° (281)
31 1 ems (”19]
= ——W — .
2j 211 (Di 5 5 d5
0

As noted previously, expanding the term sin(Ex"p) into exponentials creates a pole at the

origin (refer to Figure 4.20), and Cauchy’s integral theorem then states

Mode = f(€)d€+ f f(€)d€+ ff(€)d£+ f f(€)dE+ f f(E)d€=0 (282)
C Ci Cl; C: C: C;

resulting in

Ag"; = {3 f f(E)dE - Mada. (233)
P C; C:

Following the development presented in Section 4.11, it is determined that

206

 

 

1
{3 f nods = ~72: 2 (284)
and
l
ff(€)d€ = —2. (285)
C6
Thus the pole-series expressions for the matrix elements in the case when Lap =0 are
given by
run I 1 e {"1“
A21, = _ - _z . (286)
2 l p {22”.

It is again noted that this special L2223 =0 case only occurs in the self terms A1"? and A2"? .
The coupling terms AS" and A2"? are always given by expression (277).

The cross-sectional surface current distributions associated with the TEM mode
of coupled stripline can now be determined through solution of (206) and (207). In
Figure 5.17 a comparison is made between the current distribution associated with
widely-separated coupled center strips, and that obtained for the isolated stripline using
the method developed in Chapter 4. It is seen here that these two methods produce
identical results. Figure 5.18 and Figure 5.19 present current distributions associated
with a coupled strip structure having identical (w1=w2) coplanar center strips, lying in
the plane of the origin, for the case of VI = 1V, V2 = 1V. It is apparent from these
figures that large center-strip separations result in the expected current distributions,
which are identical to those obtained for the isolated stripline. As the center-to-center
strip separation decreases, however, current begins to accumulate on the outer edges of

the center strips (the edge farthest from the opposing strip). This continues until the

207

center strips come into contact, at which point the current distribution resembles that of

a structure having a single center strip of width w=2w1=2w2.

5.10 Impedance Parameters of Coupled Stripline
The TEM specialization for coupled stripline presented in this chapter allows for
the determination of the total center-strip surface currents I1 and I2 for specific strip

potentials V1 and V2. These total center-strip currents are determined from

van/2 N W../2
IN = f knot—M)? = X a” f pan(f)d§ (287)
-w2/2 " =1 ~w2/2
which becomes
NI
Ia : _A—x-d 2 at!!! (288)
n=1

for a = 1,2, where K}, is defined in (197). The current and potential quantities are

related through the conventional circuit admittance descriptions

I = Y V +Y V
1 11 1 12 2 (289)
I2 = Y21V1 +17221/2-

Here the various admittance parameters are determined by solving the system (206)-(207)

first for Vl = 1V, V2 = 0V and then relating the resulting currents and potentials through

the expressions in (289), which gives

1 ’ Y21 = I2 (290)

208

and then solving the system (206)-(207) again for VI = 0V, V2 = 1V, leading to

Y

12:I

1 ’ Y22 = I2 2° (291)

VI =0,
V2=1 V2=1

Once the admittance parameters have been computed, impedance parameters for the

coupled stripline are found by inverting the admittance matrix, which gives

-1

 

 

 

 

 

 

211212 _ Y11Y12 1 Y22 'le (292)
221 222 Y21 Yzz Y11Y22'Y12Y21 ’Yzl Yu

 

The behavior of impedance parameters associated with a coplanar, coupled
stripline structure is presented in Figure 5.20 and Figure 5.21. For this case, the center
strips lie in the plane of the origin, and have identical dimensions (w1=w2), resulting in
211:222, while Z12=221 due to reciprocity. In Figure 5.20 the variation of Z“, the self
impedance of strip 1, is plotted versus center-strip spacing, for various center-strip
widths. It is seen that, for large center-strip separations, the self impedance of a single
strip is identical to the characteristic impedance obtained for the isolated stripline using
the method outlined in Chapter 4 (refer to Figure 4.21). As the distance between the
center strips decreases, the self impedance is seen to decrease until the strips come into
contact.

Variation of the mutual impedance parameter Z12 is plotted versus center-strip
spacing, for various center-strip widths, in Figure 5.21. Here it is seen that the mutual
impedance that exists between two coupled strips is vanishingly small for large center-
strip separations, but becomes significant as the center—to-center strip separation

decreases.

209

Normalized current distributions associated with the impedance plots discussed
above are also presented. In Figure 5.22, the normalized strip 1 (left) surface current
distribution is presented for the case of VI = 1V, V2 = 0V. Here the expected
distribution, identical to that obtained for the isolated stripline, is displayed when the
center strips are widely separated. As the distance between the center strips decreases,
it is seen that current begins to accumulate on the side of the center strip which is nearest
to strip 2 (right).

From Figure 5.23, it is apparent that, for the case of VI = 1V, V2 = 0V, a current
is induced on strip 2 which accumulates on the side of the center strip nearest to strip 1.
Although the absolute magnitude of this induced current differs with center-strip spacing
(as evidenced by the change in impedance parameters), the normalized distribution is
seen to differ very little as the distance between the center strips changes. A slight shift

of the distribution toward the edge of the center strip is evident in the case of a small

center-to-center strip separation, which is consistent with tighter coupling.

210

Normalized strip current distribution
for TEM mode of stripline

 

 

 

 

 

 

1.00 1
xN E

0.80 —
+4 I
C s
(1) "l
L s
‘- I
3 _
0 0.60 E w = 5.0cm
‘_5_ : l = 1.75cm
°>< ._.
O I
‘O I
q) 0.40 _
.E I
"5 E
E :
1. ._
O 0.20 2
C _

3 ------- coupled strips, 15.0cm center—to-center
E isolated strip
0.00 dIIIIIIIIIIIIIIIIIII]lllIIlIlllIlIIIIITI]
—1.00 —0.50 0.00 0.50 1.00

location x/w

Figure 5.17 Comparison of surface current distributions associated with TEM mode,
for isolated stripline, and widely-separated coupled center strips.

211

Normalized strip 1 current distribution
for TEM mode of stripline

 

 

 

 

 

1.00 j i
3 w = 5.0cm
_XN 0 80 3 l 1: 1.17\5cm 1
. - v = :
.1.) " I
C I = '
m _ v2 1v 1
L — l
L : I
3 - :
0 0.60 i 5
E) E l
x 3 '.
o : ------- 15.0cm center—to—center 1|
: - — - 5.5cm center—to—center 1
8 0-40 3 —— 5.0cm center—to—center 1"
E — I'
'6 3 ,i'l
E : l
L _ 1’
O 0.20 j 2,," /
C _, __________ -_:‘f_ ...— /
0.00 ‘ II I I I II I I I I II I I I II I I I l I II I I I I I I I I I I I I I I]
—1.00 —0.50 0.00 0.50 1.00

location x/w1

Figure 5.18 Strip 1 (left) normalized TEM mode current distribution.

212

Normalized strip 2 current distribution
for TEM mode of stripline

1.00 g
w = 5.0cm
E l = 1.75cm
0.80 i v1 = 1V
i v2 = 1V

0.60

_-

------- 15.0cm center—to—center
- — - 5.5cm center—to—center
— 5.0cm center—to—center

 

—-—

0.40

-..——
f’

_ ‘ -
—

lilllllJJJlllllllLJllIllllllLLLllJllllLllI

0.20

normalized axial current k2

lllllll

 

 

0.00 IIIIIIIII|IIIIIIIII|IIIIIIIIIIIITITITTI]

—1.00 —O.50 9.00 0.50 1.00
location x/wz

Figure 5.19 Strip 2 (right) normalized TEM mode current distribution.

213

self impedance Z11 of coupled stripline

 

 

 

 

55-
: _______ w = 2.50m, l = 1.750m
a _ _ w = 5.0cm, l = 3.5cm
(I) q w = 10.0cm, l = 7.0cm
E _
_C _
O 4 ,’—--------/- ---------------
50— ,I’ /
E “ : /
<1) - i l
O 't : l
C — .
O _ .' l
‘0 — l
(D _ : I
0- - : l
.E 45‘ i l
2: — '
Q) -1
(f) _
4O T7IIITIIIIIIIIIIIWI]IITIIIIIIIIIIIIIIIII
0.0 5.0 10.0 15.0 20.0

center—to—center strip separation in cm

Figure 5.20 Relationship between coupled stripline self impedance Z11 and center-strip
spacing.

214

mutual impedance Z12 of coupled

 

 

 

stripline
20:
q _______ w = 2.5cm, l = 1.75cm
m a _ _ w = 5.0cm, I = 3.5cm
E j w = 10.0cm, | = 7.0cm
-C I
015:
.E : .
_ E l
8 I l l
C I l l
O - l
'O 10- g l
CD -‘ l
Q. j l l
E : l l
: : i l
O - l l
3 5— l \
3 I l \
E : I“ \
I ‘.‘ \
I \ \
.. \\ \ \
O TIIIIIIII‘II‘IIIIIIIII‘ll—7F1I1F11IIIIIIII]
0.0 5.0 10.0 15.0 20.0

center—to—center strip separation in cm

Figure 5.21 Relationship between coupled stripline mutual impedance Zn and center-
strip spacing.

215

Normalized strip 1 current distribution
for TEM mode of stripline

 

 

 

 

 

 

1.00 :
E w = 5.0cm
+2 0.80 j Vl = 1V
C 2 v2 = 0V
0) .—
L _
L— :
3 _
0 0.60 E
E Z
x I
o :|
j; —— 15.0cm center—to—center

8 0-40 z': ------- 5.1cm center—to—center E
_0 El :'
E :‘g .’
L -|
O 0.20 :'
c _

0.00dIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

—1.00 -O.50 0.00 0.50 1.00

location x/w1

Figure 5.22 Strip 1 (left) normalized TEM mode current distribution.

216

Normalized strip 2 current distribution
for TEM mode of stripline

 

 

 

 

 

0.00 3
3

_v_“‘ E
+7020: = 5.0cm
C 3 1.75cm
8 1 = W
L. j _-
8 3 — OV

—0.40 :
E I
{.3 E

3 —— 15.0cm center—to—center

8—0-50 ‘1 ------- 5.1cm center—to—center
N :
6 E
E :
L _
O—O.80 '—
c I

_1.00dIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII—rl

—1.00 —O.5O 0.00 0.50 1.00
location x/wz

Figure 5.23 Strip 2 (right) normalized TEM mode current distribution.

217

Chapter 6

FULL-WAVE PERTURBATION METHOD FOR COUPLED STRIPLINE

In the previous chapter the propagation modes excited on a coupled stripline were
determined through the development and numerical solution of a pair of coupled EFIEs.
Although this method yielded the complete propagation-mode spectrum of the coupled
stripline structure, the solution of the coupled EFIEs is a formidable task both
analytically and numerically. The complexity of this task can be greatly reduced through
implementation of a full—wave perturbation method for the coupled stripline [48]. This
method involves the use of the eigenmode currents of an isolated strip as a first
approximation for the nearly degenerate eigenmode currents of a loosely coupled pair of
center strips [49]. In addition to reducing the amount of time required for computation
of current distributions, this method provides insight into the nature of coupling effects.
The results of this method also provide a convenient first approximation with which to
being searching for the eigenmodes 0f the coupled system using the full-wave EFIE
method of Chapter 5. Results obtained using the perturbation method will be compared
to those obtained using the full-wave EFIE method which were presented in the previous

chapter.

6.1 Electric Dyadic Green’s Function

In the previous analysis of coupled stripline structures, electric field integral
equations were developed based on a Hertz potential Green’s function. Due to the form
of the testing operators which will be employed in the perturbation method, an electric
dyadic Green’s function must be used [50], [51]. It was determined previously that the

218

electric field in the free space region of the stripline which exists between the perfectly-

conducting ground planes may be represented by

~ ~-- f?
£0) = (1.3+ VV->fG(rIr6-.—(—’dv’. (1)
1w e0
v

The Green’s function that appears in this expression contains components associated with
both the principal and scattered Hertz potentials. The component of the Green’s function
associated with the scattered potential is bounded, and remains finite everywhere.
However the Green’s function component associated with the principal Hertz potential

contains a source-point singularity. As a result the term

vv-fc’imm #93 dv’ (2)
V Jon-:0

must be examined more closely. Application of Leibnitz’s rule to equation (2) yields

{:fldw = P.V.fVV-[C3P(FIF’)°7(7’)]dV/“‘Wym' (3)
V

vvofémm.
V JCOEO

Here P. V. indicates that the integration is performed in the principle value sense. The

electric field between the conducting plates may now be expressed as

Em = [mm-flaw (4)
V 1006

0

where the electric field dyadic Green’s function is given by

é‘mf’) = P.V.(k02 +vv-)(;‘(?|F’) +£5(?-?’). (5)

Here the term I: = —yy is known as the depolarizing dyad. The transform-domain

219

electric dyadic Green’s function is expressed

gimp") = P.V.(k3 +VV-)§,<5I6') Jaw-5’) (6)

where v = Vt+z’j(,', and V, = at3 +93.
6x 6y

In the previous chapter, transform-domain electric field expressions for coupled

strips were established

 

 

Ex’, I? ’,
é’(x,hl;C)=lim f§e(x:)’|x,’h1)° l.( del+f§‘(1,)’|xl,h2)‘ 2.(x de’ (7)
Hz, c 10060 c face

I 2

0

 

 

5(x,h235)=fim{f§'(xa>’|x/,h1)° {(x C)dx’+f§‘(x,>’|x/,h2)' 2.0 de’} (8)
y-oh2 [we we

0

o

where Cl and C2 represent the cross-sections of the conducting center strips. Application

of the boundary condition at the surface of the perfectly-conducting center strips

figs : _t‘.é’i (9)

and the approximate form of the current functions near the simple pole singularities

kp(x’)
(C - Cp)

12(x’, c) z (10)

 

leads to a pair of fundamental EFIEs having non-trivial solutions only for C = (p

b b

l 2
f§‘(x,ylx’,h1)°l?1,,(x6dx’+f§‘(x,y|x’,h2)~E2p(x’)dx’]} =0 (11)
al (12

 

lim 5'
Y"h1

220

 

b b2
limt{ afg‘ (x ylx ,h) k1p(x’)dx’+ f§‘(x ylx ,2°h)k gawk/”=0 (12)
Y hz

where an =xa -wa/2 and bu =xa +wa/2.

Now the transverse operator f ~(...) in EFIEs (11) and (12) is replaced with the

testing operator

~:(0)
fdlkmp . (13)

This leads to a pair of modified integral equations

 

f §‘(x,y Ix’,h,) ~12}; (x’)dx’

5311isz 40)“), b1

 

b (14)
+f§‘(x,y|x’,h2)°E2;(x’)dx’]dx}=0
0.2
mn{:f:k*‘°’(x) bf}: (1:,ny h) k (x’)dx’
(15)

b2
+f§e(xaylx/,h2)E2;(xl)dxl]dx} = O.

“2

In expressions (14) and (15), the term 1330) represents the current associated with the p’”
propagation mode which exists along the m’” isolated center strip which has propagation

constant i (:0). This current function satisfies the EFIE for a single center-strip structure

221

b

f-gf (x,”ylx,y;ic,f,°’)k km;(0)dx’-O (16)

an

The transform-domain electric dyadic Green’s function has the reciprocity property
§‘(§l5’;C) = §‘(5’|5;-C). (17)

Applying this property to the modified integral equations (14) and (15) results in

bl bl
. ":1: / we / . . #110) I
111:1{f[klp(x )fg (x ,h1 Ix,y, 1C) klp (x)dx]dx
y 1 al a1 (18)

b2 b
+f[k2p(x )fg‘ (x’ ,Iz,;2|xy :tC) k l;(0)(x )dx]dx’} =0

“2 “1

bl b2
1im{f[kfp (x’) g"(xh1|x y,iC) k ;d2c’(0)(x)dx]
0' (19)

bZaz b2

flkpor’L §e(x :ahzlx y; in k2;(0)(x)dx]dxl} = 0 -

02

These expressions simplify for the case of center strips which lie in the plane of the
origin (h , =h2=0), becoming

b

[g 0" ”leyuC) k;‘°’(x)dx}=0 (20)

“1

b1 b2

ImJHIdx’klux’) +fdx’k2;(x’)
“1 “2

 

bl 1’2

1111 H fax/kl;(x/) + fdx/k2;(x’)
at “2

b2

°f§‘(x’,0lx,y; ¢C)°E2;‘°’(x)dx} =0. (21)
“2

 

222

6.2 Perturbation Approximation

If the geometry of the stripline, and frequency of operation are such that the
center strips are only loosely coupled, then the system propagation mode eigenvalues are
nearly the same as those of a stripline structure with only a single center strip (i.e the
coupling effect is only a perturbation of the isolated system response). This suggests that
a Taylor’s series expansion of the Green’s function be performed about the propagation

constant of the single strip structure C = #332). This results in the expressions

b, b,
lim (){ilfdx ’k’(x’)-fl§‘(x’0|x y 150))
"p 9 , ’ up

3’ " a a, (22)

Cid?) . ---']k ;‘°’(x)dx}=o

 

+ a§e(x/’O|x9y;-C) (
6C Ida)

 

0'

b2

2 II

Y” n=

(23)

a“

a§e(xl,0|x,y; -6)
35

+

 

 

do)“: an,°,3)+ ]°k2;(0)(x) dx} = 0

Due to relationship (16), the leading terms of these expressions vanish when m=n.
Neglecting the second-order and higher terms, and retaining only the leading non-zero

terms leaves

 

b1
/ n(0)( / ag( (x /,,O|xy; C) (0) “1
lynn{fdxk',,(x x) f 22 2221: c p)k1,,(x)dx
an b. (24)

 

+de’k2;‘°’(x’)-g "(x’ O,|x, y;:tC(0)) E,;(x)dx

“1

223

b1 b2
Im{fdx/kltp(o)(x/)-g ”(x ,,O|x y; i((0)) k2;(x)dx
‘0 a (25)

“2

 

b2 b2
+ /*i(0) / , 5§‘(x’,0lx,y;-C ) (0)
[dx k,,, on] 22 Mm, )k;x (x)dx=

“2

Because the center strips are considered to be loosely coupled, the current
distributions associated with the coupled system propagation modes are assumed to be
nearly the same as the current distributions of a single center—strip system modified by

muliplicative amplitude coefficients

k2;(x) .. ank’ngw’. (26)

Here the an are unknown amplitude coefficients. This relationship represents the basic

perturbation approximation. Substituting (26) into expressions (24) and (25), and again

applying the reciprocity property of the dyadic Green’s function yields

 

bl we / .
lyirn{af (c i c‘°,3)flk‘;‘°’(x of ’33 :x’y2': ’h"’)

 

E;‘°’(x x'dx’)dx]
xClp)
(27)

+a2flk2;(0::g)(x) (,xylx,hl;¢C(O))'k1pk.’(o)(x’)dx’]dx}=o

bl b2
lim{aff[k;(o)(x)-g ‘(x ,ylx ,h2;:FC(O)) k2;(x’)dx/]dx
y hz “l “2 (28)

b2
((0) k2P(x ’)dxde }= O

 

+02%; ((0))flk k2;(0)(x)Zag(a (XYIX J12;C)

“2

 

224

Now let

 

 

 

 

b1 bl
~e / .
y~0 a p a 5C xcifi’k
“2 bl
y~0 “2 “l
bl b2
C12 = 1m{f[k°,;‘°’(x)-g (x, ylx, o; nc§°’)k k2;‘°’(x de’] } (31)
y-0 “1 “2
b2 b2 /
C22=1im [[szp(0)(x) fag (XaYIx ,O,C) k2;(0)(x/)dx [de (32)
r0212 0 5C fig)

Assuming that C 1 and C 2 are the isolated propagation eigenvalues of the coupled center

strips, and applying relationships (29)-(32), the following matrix results

C11“ " C1) C12 “1
= 0, (33)

C21 C22 (C ' C2) “2

It is also assumed that a1 $0 and a2 #0, therefore

C11“ " C1) C12
det = 0, (34)

C21 C22“ - (2)

Equation (34) can be solved for C to determine the approximate perturbed propagation

constant in terms of the isolated propagation constants, thus

225

C11“ - C1) C22“ ' C2) ' C21C12 = O
which may be expressed

C C
C2_C(C1+C2)+C1C2+_21—13 :0.

 

 

 

 

 

C11C22
Now letting
C C
a=1, b = (1+9, c=c1c2+ 2‘ ‘2
C11C22
the solution to (36) is seen to be
C C
(n+9): (c,—c,)2 -4(c,c,+ 2‘ ‘2]
N C11C22
C = .

2

However

 

C C 2 2 C21C12
(C1+Cz)2 " 4(CIC2 + 21 12) =(1+ ZCICZ + (2 ' 4(1C2 + 4
C11C22 C11C22

C21C12

=c-c 2+4
(1 2) €11,222

therefore

 

C ___ (C1+C2) + [(C1-C2)J2 + C21C12
2 ' 2 01,022 '

 

(35)

(36)

(37)

(38)

(39)

(40)

Equation (40) simplifies further if both center strips have identical cross-sectional

dimensions (wI =w2=w). In this case C21 = C12, C11 = C22, and C1 = C2 = C0, thus

226

 

:i:

c= 2‘0 5C_’- = c,.(Cn] <41)
2 (C11)

where (0 is the propagation constant of an isolated single center-strip system. Here the

coefficient Cll will be referred to as the normalization term, and the coefficient C12 will

be referred to as the coupling term.

6.3 Determination of Normalization and Coupling Terms

It can be seen that the perturbation propagation constant expression (41) requires
determination of the normalization and coupling coefficients. It is assumed that currents
are confined to the surfaces of the infinitely thin center strips. The coupled stripline

structure supports modes which are either symmetric, or anti-symmetric, therefore

k: p(x)- ‘ xk (x) i zk (x). (42)

apx up:

For simplicity, the center strips are also assumed to be identical (w1=w2=w), therefore

k,p(x) = k2p(x) = kp(x). As a result

 

§‘(6I6’;nc5°’)-k E“°’<x6=[P.V.(k§l‘7V v)g2(p|p/)- yyiw-p -,) )-x(k,5°’(x6 2k,5°’(x6)

= {P.V.(kf +VV) [(22 +z‘z‘)gca(5 | 55 + figcy(5 '55,)

-Y‘Y‘6(6-5’)]}1(xk,§°’(X’) n zk5°’(x6)
(43)

which leads to

§‘(5|5';‘C;0))'k k1(0)(x’)= M” W )[x8c(9|p’,,)k 3’(x’)izg¢(p|p )k(°’(x’)] (44)

227

Substituting (44) into expression (29) for the coupling term gives

bl b2
C12 =1im (xk;0)(x)$ zk(0)(x)) )f{PV VV °)[ng(x,y|x/,O)Ep(g)(x/) (45)
~o
y al “2
12g:(x,y|x’,0)k(o)(x')]}dx]dx.
liere
~ 6 6 .
szA—d-A—i-i (46)
ax y ay 16

represents the transform— domain differential operator, and ((0) =-C0, {(0) =Co, therefore

V-xg:(x,y|x’,0)k(0)(x’)- — aigC (x ylx ,O)k(0)(x’) (47)

~

V-z‘gCa(x,y|x’,0)klfg)(x’) = ng:(x,y|x’,0)k(3 )(x’ ) (48)
and thus

3

a + sz)j:gC (x, ylx’ ,O)k(°)(x/) (50)

~ ~ A a . . a
VV'ng(x,y|x/,O)k]§g)(x/) = (1: “Us;

H+

Applying the relationships above to expression (45) gives

b, b

C12=1im [(x xk;0)(x)- zk(0)(x)) )fP.V.{ {xk :gC(x ,y|x’,0)k(°)(x’)+zk:g Ca(x,y|x’,0)k(°)(x’)

} l

(51)

 

+ (£3— +97% +22ic )[éa—g? 0:,ny 0)k55’ ’(x’) + jog: (x,ny 0)k5° m

228

01'

b, b2

C12 = 11111 f {k(°)(x) f P V[k§gC°‘(x,y Ix’o)k‘°’(x’) + __gC“(x x’,y|x ,o)k‘°’(x’)
.V"0 0 6x2

“1
b2

(5 52
+ k(°)(x) f P V.[— k MgC “x( x’,yyx ,0)k(°)(x’)
‘22
all...

Note that in equations (51) and (52) a direction for the axial current has been chosen in

 

. a a
+ JcoagdeIX’OWO ’(x’ )dx

 

. a a
-Jcogg¢o(x,ylx’0)k,§°’(x’)0g +<:.2,g 2:,(x,ylx’0)k‘° ’(x6

order to simplify the expressions. Because the center strips are identical, the cross terms

integrate to the same values

 

 

 

 

{ f P V[ —gC°‘(x (x’,y|x ,0)kC§°’(x’) dx’}k‘°’(x)dx
0"1 “2 (53)
1%{} P V[jc0: ——gC“(x (x,y|x ,o)k‘° )(x’) dx’}k(0)(x)dx
0‘21 ‘22
and vanish from expression (52), therefore
bl b2
2
C12: lim [{fPVHi +kf)gC°‘(x,y|x’,o)k‘°’(x ) dx’}k‘°’(x)dx
r0 a 6x2 °
‘ (54>
bl b2
- 11m f { f P V[k -c )gC (x,y|x ,O)k(0)(x’) dx’}k‘°’(x)dx.
“1 “2

For the sake of simplicity, only the case of coplanar center strips which lie in the plane

of the origin (h1 =h2=0) will be examined. Thus, for y’ =0

229

jux-x’)
lim gC’:(x,y|x’,O) = e—TdE (55)

 

 

 

 

 

 

 

y~0 41: p00)
and
“2 —p04l -po4l ~p021 -p021
e +e -e —e _ ,
lim gs 3(X,ny/,0) : [ _. 4 I JeJE(x—x)dg
y o .. 4npo(21)(1—e"’°)
(56)
°° ~2p 1 ~2p01
= 28 o (e -1) ejE(x-x’)dg
_m41tp0(X)(l -e ‘42“)
therefore the total transform-domain Green’s function is given by
°° -4p I —2p 1 -2p 1
1-e°+2e °e °-l , ,
Hm gc6(X,YIx/a0) = K ) .. (4 I )]e’E("x)dE
r0 _, 41: p0(1)(1 —e' 2° )
(57)
.. jux-x’) _ -2Pol
= e _. 1 e J d5
-..,47‘POQ) l + e 4%!
62
It is noted that —2 —~ —£2, and therefore the coupling term may be expressed
6x
00 1 -2pol a! (:12
C12 = lim ZL [{(k: - £2) < e 20! fk(°)(x)e’E2‘dx [kpwkx’ )e’JE" dx
.. " ' P
1° 1‘ -.. powwow 1-, -0. (58)
(-1-e 2pd) a1 02
- (ka - (C2,) _2 fk‘°’(x)ei5xdxfk,ff’(x’)e ‘iixdx’}d5.
pO(X)(1+e )2“)

“1'“2

Now the normalization term Cll must be determined. It is seen that

230

 

 

 

a§‘(5|5’,cp) ~ (0) a[ 2 ~ ~
-k* ’=——P.V.k +Vv- “ r r’
6C :c, (x) a: (o )g,0(plp)
wars—p6 (xk‘°’(x’) 2k,52’(x’))
6 ~
= a—C{P V-+(kf VV )[xgfiw'lm k5 ’(xfiizgdplp k5 ’(x’)]}
(59)
therefore
bl bl
C11 =1im (xk;°)(x) zk‘°’(x)) f ac{P.V.(k02+ WV)[52gC°‘(x,y|x’,o)k‘°’(x’)
y-Oa °

“1 “1

:t iga(x,y|x’,0)k5o)(x’)]}dx’]dx

 

 

(0) / / (0)
=1im:f:{:fza(:[P. [15556—322 +k:)gC(x,y|x’,p10)k (x )}dx kp (x)}dx
b1 b2
— limf{f—[P.V.{(k3- -CC, 2)gC (x,y|x’ ,0)k50)(x’)}dx’ k50)(x)}dx
“1 “2
(60)

where again a direction for the axial current has been chosen in order to simplify the

2
expression. Substituting (57) into expression (60) and noting that :2— ~ —€2, gives

 

 

 

x
°° -p,,12
C11 =lim———1fl-m{a[(k2-E2)51 _ 2k k50)(x)e’5’dx:f:k5o)(x’)e"lexdxJ
o 5C p0(X)(1+e 5)”! 25° (61)
-2pol b, b,
_ £[(kf—Cf,) 51-8 ) fk53)(x)e"‘dxfk£)(x’)e“’E‘IdX’] }d5-
33; p0(X)(l +e 4W)“1 a! *9.

 

231

After a great deal of effort it can be seen that

 

 

 

 

 

 

._ -2Pol ([41 0(X)e'2p°5 — 1 -e 4%,]
‘25 P0(M(1+e ”° ) p,(X)[p,(X)(1+e‘2”°’)]
and after an even greater amount of effort, it is determined that
a (k: ‘C2) (1 —e_2p01)
<9_C -+ -2p01
po(A)(1+e ) (63)

(k3—c2)c[4p,(X>ze'2”°’-(1—e "’° )] 2C1» (m e”)
p,(X)[p,(X)(1 +, “W

 

As a result, the normalization term may be expressed

=1im — 4111:]{M11 (C0,E)fk50)(x)e152dek5o)(x’)e "52 dx

 

 

 

(64)
bl bl
+ M;I(CO,E) k50)(x)e1552‘dxfk50)(x/)e"EJr dx ’}d£
where
_ -2pol
M5‘(c,o = ~52— (k3-52) 5.1. e 5 5
6C p0(l)(l +8-21%!)
(65)
- -2p,,z_ _ -4pol
= ([41p0(x)e (1 e 2 )] (1‘34?)
p,(X)[p,(X)(1 +e ’2”°’)]
and

232

 

 

 

 

_ -2pol
M;‘(c,€) = 3- (kf-cz) 5} e 5 J
(9C P00)“ +e‘2Pol>
(66)
(k3 - (2) ([4po(X)le '220’ - (1 -e "P°’)] — 2 (pO(X)2(1 -e ‘4”0’)
p..(X)[p,,(X)(1 +e "’°’)]2
The coupling term may likewise be written
w “1 “2
C12 : “mi ej5(11-12){M:x2(co,£) k;2)(x)ejfixdxfkp(2)(xl)e —j£x/dxl
y~o 411: -.., a
1 “2 (67)
bl b2
+ lez2((0,E)ka53)(x) ejexdxfk£5(x/)e ‘jexldx/}d5
“1 “2
where
_ —2pol
M52(c,£) = (193-:2) (_5, e 5 (68)
P,(A)(1 +e '2’“)
and
_ ~2pol
Mime = (kg—<2) 55 e 5 . (69>

 

po(X)(l +e ’25“)

6.4 Transformation to Local Coordinates
Once again the numerical solution is facilitated by a shift to local coordinates.

As was done previously, this is accomplished through the change of variables

5?; = xI—xa (70)

233

and then dx’ = dig. This leads to a change in the associated limits of integration such
that when x’ =xa - wa/z, i; = -wa/2, and when x’ =xa +wa/2, EC: =wa/2. Also let

fa =x-x (71)

a

and then dx=dxa. In this case, when x=xa-wa/2, fa =—wa/2 and when

x =xa + wa/2, fa = wa/2. This results in

an W1/2 wl/Z
C11 : um i {M11(C0,€) f k;0)(x (x1)e e]£(x1+xl)dx If k(0)(xl/) e~j£(fC+xl)d§l/
)HO -oo “W /2 ~wl/2
1/2 Wl/2
+ M;1(Co,€) f k$)(;l)ejE(XI +x1)d;l f k;0)(x1/)e 'jE(x1+xl)d-1/ }d£
—wl/2 'W1/2
(72)
W1/2 W2/2
_ _limT [{M12(co,£) f( k;0)( xl)ejE(xl +10%] f k(o)(x2/)e -j£(x2/+x2)d-x-2/
4“
Y“ "0 —wl/2 -w2/2 (73)

w1/2 w2/2
+ M52(C,,E) f k,§°’(x le)""‘”"l)d;?, f k‘°’(x,’)e ’E‘fi‘h’dxz’}d5

‘Wl/Z "492/2

but the center strips (w1 =w2=w), and therefore

C11 =1im '4'}; {M;1(CO,E) f k;0)(xl)ej5x5dxl f k;0)(x1/€)-1Ex5dill
y~0
-oo -w/2 "W/2 (74)

W/2 w/2
+ M;‘(co,g) f k5°5(x )e’gx‘dxl f k‘°’(x,’e) ’5"d§{}d£

-wl2 -w/2

234

W/Z w/2
Ctz=1im—1e"“‘ ”’{M‘ztcmo f<k,§"’<x1)t:’"“df1 f k‘°’(x2’)e "We!
-..o -w/2 —w/2 (75)

W/2 w/2
+ M;2(co,g) f k‘°’(x )6“!de f k(0)(x2/)e ’Eizdiz’}d€.

—w/2 —wl2

6.5 Method of Moments Solution

A numerical solution is performed by expanding the current functions such that
Na
kapaof) = 21 ag'p effiaj) (76)

where e £9511) is the basis set, and the 0:9 represent unknown amplitude coefficients for

a=1,2, and B=x,z [44]. As a result

wa/2 Na Mia/2 —/ Na
f kap(f;)e‘15xadfé = Z am"B f e:p(;‘:)ev€xad§; : z: aanpfanp“) (77)
—Wa/2 n = 1 -wa/2 n = l
where
wet/2 _,
ffem = f engage ‘J‘xtdig. (78)
"Wu/2

In addition, let
Wan

ga’zto = f ta"$(fa)ej5f“dfa . (79)
“Va/2

where tampOc-a) is a testing function which is to be applied later. Applying the

relationships above to (74) and (75) results in

235

l m 11 N N
11 : lim—f{Mn(Co’E)Z 01:13:“)2 “1233(5)
-01. n=l m=l

y-oO 411'
(80)
11 N N
M (to, €>Z attflttam al':g{';(5)}d£
"=1 m=1
and
= wife 150:1 -M12){xx 12 ((0,5);afiffl6); 01:81:“)
4 , <81)
12 N N
+ Mzz ((0,5)2 “2322(0): 0138135)}‘15 -
n=l m=1
Now the order of integration and summation are interchanged, to give
1 N1 Nl °°
Ch = lim — {Z 01312 as] M;(c,,£)f;;(£>g{:(od5
y~0 4“ n=1 m=l -.. (82)

ZaZ'fM; 1)(c,,,£ )fi';(€)gf"z(£)d€}

N N2
C12 : lim 04—1-{m=l 2 “1;: "12:1 “2:: fol’zko:€>81x(5)fzx(5)ej£(xl wdg

M (83)
N1 N2 00 .
+ at? at: [Mfkm €)gt';’(£)f2’;(£)e"“‘ “”dé}.
m =1 n =1 _m
For a Chebychev Galerkin’s solution
ea"p(x) = Tn (x / a)[l - (x /a)2]*1/2 (84)

236

and
ta";,(x) = Tm(x/a)[1 -(x /a)2]11/2 (85)

H N

where a=1,2, a = w/ 2, and the " +" sign is associated with 6=x, and the sign is
associated with B=z [45]. Chebyshev polynomials of even order are even, and those of

odd order are odd such that

T,.(-x) = (-1)"T,,(x). (86)
As a result
w/2
w/2 2fea"p(x’)cosEx’dx’ ...for eaB(x/) even
" z n / -'5x’ / = 0 87
121(5) _[lzewtx )e! dx m ( >

—2jfe:p(x’)sin5x’dx’ ...for ew(x’) odd
0

where oz=1,2, B=x,z. Likewise, with the testing operator

w/2
Wu 2 f affix) (:0ng dx for tap“) even
. 0
m = m lax = (88)
gafl(£) _£2tap(x)e dx w/2

2jfta";(x)sin€xdx ...for tafl(x) odd
0

for a=1,2, B=x,z.
It is seen from expressions (87) and (88) that four generic integral types are

implicated. In Chapter 4 these generic integral forms were examined, and it was

determined that

215(5) for It even

flan = (89>
—2jI::(E) ...for 12 odd

237

 

 

21::(5) ...for 12 even
fat“) =
—2jl::(E) ...for n odd
2133(5) ...for m even
83(5) =
2jIZf(E) ...for m odd
and
215(5) ...for m even
813(5) =
2jI£°(E) ...for m odd
where
nnw w
101:: ( 1) 4 _J2n(E2)
Iaz :( l)n—nf’_‘l2n+l(£§)
ne "TE—W w 1 w w
lax: ( 1) 8 .1245?) + 5(J2(n+1)(€3)+J2|n-1I(€E))]
and
no nnw w 1 w w
Iax_ _ (— 1) 8 J2n+l(g 2) + 5(J2n+3(€—2—)+J2n-1(€E))]'
Now let

A: = f 5411160,:

)ft’;(£)g3(£)da

238

(90)

(91)

(92)

(93)

(94)

(95)

(96)

(97)

AA? = fM;‘(c,,e)fl;(e>g{:(odt

(98)
A23 = [MAL2(C,,£)g,':(of;;(oe"“‘l“2&1: (99)
AA? = fMSZ<CAA€)g{:(£)f2’;(£)e"“'“the. (100)
This leads to
l N N N N
Cu =lim—{Z (1122 (ISA; + 2a," :3 4:11;} (101)
Y 0 4113 n- l m=l n= -1 m- 1
1 N N
AAA—{2AM AAA; A A}
Y 1': n=l m=l

where the 004;" terms are determined during the solution of C

It is seen that for even modes

fM;‘(C,,o(-2j)1,"°(o(2j)I:'°(£)d£ = 4fM;‘(co,t)1:°(g)1;'°(g)dg (103)

A? = f M;‘(CAAE)21A'"(€)21370015 = 4 f M;‘(C,AE)I:‘(£)I;"‘(£)d£ (104)

239

Afif : fM;2(c,,£)e""‘* “‘2’12j)1;"°(£)(—2j)I;'°(£)da

(105)
= 4 f M§(c,,&)e"““ '”)I;"°(£)I:°(£)d£
and
A: = f flawed”!"‘2’2I,'"‘(£)21:(£)d5
‘°° (106)

: 4szlz2(Co,E)eJ£(Xl_xz)lzme(€)lzne(€)d€ .

Likewise, for odd modes

AAIAI = [MARCO,E)21£"(€)21;’"(€)d5 =4 f M;‘(co,r)lf‘(g)1;'“(g)dg (107)
Ali = f MA‘A‘(CAA€>(—2j)1:°(€)(2j)1;"°(odt —-— 4 f M;1(Co,£)lz"°(§)lz"'°(£)d5 (108)

A: = fM;2(c,,t)e"‘""‘2)21;""(£)2I;"(od&
'°° (109)

= 4 f M;2(c0,5)e"“"* “2’I§""(£)If‘(£)d£

A: = fM;2(C,AE)e""‘"‘2’(2j)1;"°(o(—2j)2IA"°(£)d£
"°° (110)

= 4 f M;2(C,,E)e’5“"x”Iz’"°(€)Iz"°(£)d£ .

240

Now it is noted

 

 

 

 

 

 

 

 

 

M12(C a) = (kz-EZ) (I 'euzpol) = (kZ-Ez) (Joke—pd)
xx 0’ 0 _. __2 01 0 _, _0
p0(A)(l+e p ) p0()t)(ep°l+e pl) (111)
=(k2—52) sinh(pol)
p0( X ) cosh(pol )
and likewise
-2pol .
M,‘,2(C,,E) = (kf—Cfi) (1'6 ) = (kj-cf, imle) (112)
p,(X)(1 + e '2”°’) po(l)cosh(pol)
c 4lpo(X)e '2?“ - 1 -e '4”°’
MAL‘MAD = [ ( 2 I 2 )] (kf-Ez)
X X 1+ ‘19"
p0( >[p,( >( e )] (113)
moat—£2) _ Mammoth/#52)
[2100(2) cosh(p0 1)] 2 pod) [2po(X) cosh(pol )] 2
2_ 2 -‘ -2pol _ _ -4pol _ -' 2 _ -4pol
145mg): (kA CA)CA[4pA<A>Ie (1 e )] 225017000 (I e )
po<l>lpo<x>tl+e‘”°’>] (114)
Moog-(3)1 _ 2c0(kf—c§)sinh(poz) _ 4cop0(5£)s1nh(pol)

 

 

 

[2p0(X)cosh(pol)]2 p0(X)[2pO(X)cosh(p0l)]2 [2p0(X)cosh(pol)]2.

Through careful examination of the symmetry of expressions (111)—(114), it is determined

that, for even modes

A2: = ling 8 f M;‘(c0,5)1;‘°<£>1;"°(£)dt
Y‘ o

241

(115)

AA? = lim 8 f M;‘<c,,£)1:‘(5)1;"‘(odt (116)
y~0 0

AS = lim SIMLZKME) cos[€(xl -x2)]l;"°(§)1:°(£)dg (117)
Y”0 0

A: = lim 8 f M;2(t:o,g) cos[€(x1 —x2)]lz""(E)Iz"‘(g)d§ (118)
yflo 0

and for odd modes

AA? = 1:3138zM3(C,,£)If‘(E)IF‘(€)dE (119)

AA? = 1:33szMA‘Altc,,£)IA"°(€)I;"°(£)dt (120)

A23 = 8EM;2(C,AE)cos[5(x, —x2)]1;"‘(t)1:‘(g)dg (121)
Ali = lyigrol SEMA‘Ztcwa) cos[5(xl —x,)]1;"°(g)z;°(t)dg . (122)

6.6 Analysis of Numerical Results

Propagation mode constants found using the perturbation method outlined in this
chapter and the full-wave technique developed previously are presented in Figure 6.1.

Here the structure analyzed is a coplanar coupled stripline having w1=w2=5.0cm,

242

l=1.75cm, and operated at 6.0 GHZ (the same structure examined in Chapter 5). The
mode splitting behavior of symmetric and antisymmetric modes is evident as the center-
to-center distance between the center strips is varied. It is seen that the propagation
constants found using the perturbation method differ slightly from those acquired using
the full—wave technique (which are accepted as being more accurate), particularly as the
distance between the center strips becomes small. It must be noted, however, that the
amount of time and computational resources needed to generate the perturbation method
results were several orders of magnitude less than those required for the full-wave
technique.

The perturbation method may also be employed to avoid an additional
shortcoming of the full-wave technique, namely that the associated root search requires
an initial guess for the propagation constant. Due to the complex nature of the
propagation mode spectra of coupled stripline structures, such initial values are
sometimes difficult to determine. This difficulty may be overcome by using the
perturbation method to provide initial guess values for the root search associated with the

full-wave technique.

243

0.98

 

TE1 symmetric mode EGalerkin's MoM)
222*: TE, symmetric mode perturbation)

_ _ _ TE1 antisymmetric mode écalerkin's MOM)
...—._.. TE1 antisymmetric mode perturbation)

_—

0.96

 

0.94

0.92

5.0cm
.75cm
.OGHz

 

propagation phase constant {r/ko
0..

1111414111111111.11111111111111111111111I111111111Litiiiiitll

 

 

0.86 llllllllTllllllTllTlllllllllll

5.0 10.0 15.0 20.0
center—to—center distance (cm)

Figure 6.1 Comparison of propagation constants for symmetric and anti-symmetric
modes obtained using full-wave, and perturbation methods.

244

Chapter 7
TIME DOMAIN TECHNIQUE FOR MEASUREMENT OF MATERIAL

PROPERTIES USING STRIPLINE FIELD APPLICATORS

Stripline devices are widely used in measuring the electromagnetic properties of
materials [52]—[58]. The stripline geometry lends itself well to analytic schemes by
which information about sample materials placed in a field applicator can be extracted
from measurements made at the applicator ports. Such measurements are facilitated by
the fact that, if properly designed, stripline devices functioning in the principal TEM
mode can be operated over wide frequency ranges. Additionally, effective transitions
from the stripline structure to coaxial cable can be manufactured with relative ease. As
a result, several materials characterization techniques employing stripline field applicators
exist, and have been shown to yield good results. However, when used to measure the
properties of materials which are heated to a high temperature, current characterization
methods become cumbersome, and of questionable accuracy. Such measurements are
necessary for the proper design of radar absorbing materials which are placed in areas
subjected to high temperature, such as on the leading edges of aircraft wings, or near
engine intakes, and exhaust ports.

Many of the difficulties associated with using stripline devices to make high
temperature material measurements can be overcome by taking advantage of the inherent
time-domain response of the stripline applicator [59]. The use of time-windowing
techniques can reduce the number of measurements required for materials

characterization, as well as produce insight into the workings of the stripline itself. This

245

chapter describes a time-domain technique used with stripline field applicators for

electromagnetic materials characterization.

7 .1 Advantages of Stripline Field Applicators

Materials characterization is typically performed by interrogating sample materials
either using antennas in a free-field scattering apparatus, or by using some type of
waveguide applicator. Often sample materials are machined and placed into sections of
coaxial line, or within rectangular waveguides which are transitioned to coaxial
connectors. The applicator connectors are then mated to coaxial cables which are
attached to a measurement device such as a sampling oscilloscope, or automatic network
analyzer (ANA).

Stripline field applicators (Figure 7.1, and Figure 7.2) are well suited for making
materials measurements for several reasons. In order to obtain accurate results, sample
materials must be machined so that good electrical contact with the walls of the
waveguide applicator is maintained throughout the measurement process. When coaxial
or closed rectangular waveguides are employed, electrical contact often cannot be
maintained when the applicator and sample are heated, because of differing thermal
expansion coefficients. Due to the planar cross-section of the stripline structure, sample
materials can be firmly clamped into the applicator to prevent air gaps from forming
between the sample and conducting surfaces during high temperature measurements.
Sample materials are machined into simple shapes which interlock above and below the
center conductor (Figure 7.3). Once samples are placed within the applicator they can

be repositioned with relative case.

246

ground plane

 

center conductor

 

coaxial
cable

 

ground plane

 

 

sample
material

tuning
wedges

Figure 7.1 Side view of stripline field applicator.

247

center conductor

ground plane

 

sample
material

tuning
wedges

Figure 7.2. Top view of stripline field applicator (top ground plate removed).

248

sample
material

 

 

 

 

 

 

ground planes

Figure 7 .3. Placement of sample material in stripline field applicator.

249

Stripline applicators provide other advantages over different types of waveguides
used for materials characterization. The stripline structure supports TEM modes with
zero frequency cutoff, and if properly designed, can provide broadband frequency
coverage (spectra of interest typically lie within the range from 100 MHz to 18 GHz).
In addition, the stripline produces unidirectional field polarizations which are useful for
identifying anisotropic behaviors of the sample material. Also, relatively large sample
volumes may be interrogated, so that representative quantities of composite, or layered
materials can be examined. Finally, two complex measurements can be made, because

the applicator has two accessible terminal ports.

7 .2 Frequency Domain Materials Characterization

Electromagnetic materials characterization using stripline field applicators was
originally accomplished at Michigan State University through a straight-forward
application of microwave network theory known as de-embedding [58]. In this process,
the constitutive parameters of sample materials are extracted from frequency domain
measurements made at the stripline ports. This technique accommodates two complex
measurements, as the materials which are measured are likely to have properties which
may be both dielectric and magnetic.

Transmission lines are normally characterized by measuring the response of a
system to some combination of standard terminations such as short circuits, matched
loads, or open circuits. Due to the structure of the stripline, scattering parameters can
only be measured at the applicator ports. Additionally, the only practical type of

standard termination available for characterization is a short circuit, which normally

250

consists of a pair of notched metallic blocks that interlock above and below the center
conductor (again see Figure 7.3). Thus, characterization of the stripline applicator is
accomplished using measurements of short-circuit terminations, and the empty applicator.

As seen in Figure 7.4, the stripline applicator may be modeled as three cascaded
two-port microwave networks. Here, the area from the front coaxial connector part to
the front of the sample material is referred to as transition region a, the area occupied
by sample materials is referred to as the sample region s, and the area extending from
the back of the sample material to the rear coaxial connector port is referred to as
transition region b.

Using the de-embedding method, the forward scattering parameter associated with

a sample material placed within the stripline applicator is determined to be

P- ” 2
Si, = ——S“Q (1)

1 - (5,10)2

and the through scattering parameter of the sample material is given by

 

 

l—SbP
5;, = Q( b“ 2) (2)
1 _ (8110)
where
P : 1‘1 _‘311 2 (3)
(F1 '511)Szz ”'(Szr)
and

251

 

 

 

 

 

 

 

 

 

 

 

 

 

a1 b6
_>_. ——>—
a b
I‘1 [ S l [ S l
e.

—<—J w—<——
b1 as
transition sample transition
region 'a' region region 'b'

Figure 7.4. Cascaded two-port network representation of stripline applicator.

252

(4)

Here the quantities I": and S2"; are measured by the ANA at the stripline applicator ports
(81"; = I": when port 6 is match terminated).

It can be seen from (1)-(4) above that the expressions for the sample region
scattering parameters require determination of unknown quantities associated with
transition regions a, and b, these being S101, $102821 , 52:, Sll’l , and 5201 S5. Three short-
circuit measurements located at the sample material terminal plane (11 = 0), a known
distance in front of the sample plane (12 = d), and a known distance behind the sample

plane (13 = —d), provide calibration reflection coefficients

1‘? = —1e “2179’" (5)

which are used to determine the necessary characteristics of transition region ’a’, given

 

 

by
$0 Sarcl
S,“, = 1‘? - ___” 2: 2d (6)
1-s“1‘°1 1-s“I‘°2
srAsA‘: = (r? -r:‘>( ” 3,” ,, 22 2) m
1"2 ”P2
and
a K" —1
$22 = (8)

IrCI§3-r§2

253

where

Kc __, (11:3 — I‘ll) (F22 - P21) (9)
<1"? - r?) (r? - r?)

 

The remaining transition region characteristics required for de-embedding Slb1 ,

and $20182"1 , are determined from measurements of the empty stripline applicator. An air—

filled sample region is reciprocal and symmetric, thus
SISl = S2“2 = O (10)

and

5132 = 5231 = ejkols (11)

where Is is the length of the sample region. Utilizing condition (11), it is determined
that SI"l is related to the reflection off of the front of the empty sample region, defined

as I“; , modified by a phase shift factor, or

s,”1 = 1‘4 = rgeZ’ko’t (12)

The through measurement of the empty applicator is given by

s“sb _.
S281 : 21 21 e jkol’ (13)

and from equation (13), $20152”1 can be found as

a b e a e 'kols
521521 = 521(17522P2)ej - (14)

Once the sample region S-parameters have been obtained, the material constitutive

254

parameters are determined using the N icholson—Ross-Weir (NRW) technique [60], [61].

According to this technique, the sample region S—parameters are represented by

r[1 -(e""’s)2]
l - (Fe 1131,);

(15)

 

s
511 -

and

5;, = (16)

 

where Is is the length of the sample, [3 is the propagation constant for the fundamental

TEM stripline mode, and I‘ is the interfacial reflection coefficient at the front air/ sample

interface, given by

P: 5 ° (17)

 

where ZS is the impedance of the sample region when a sample material is present, and

Z0 is the impedance of the empty stripline applicator. The expressions for the sample

S-parameters (15) and (16) can be solved simultaneously to yield

.1151: = ($151 + 231) ‘P (18)
1 ' F(Slsl + SZSI)

 

e

and
1‘ = KM/KZ-l (19)

where

255

K = 1 + (5131 + 52‘1)(Sfi “ 5231) . (20)
25131

 

Equating measured values of Bm(w) and I‘m(w) to the corresponding theoretical values
leads to a pair of equations which can be solved for the complex permittivity e and the

complex permeability p.
B(e,p,w)-Bm(w) = o (21)

P(e,p,to)-Pm(w) = O. (22)

It is assumed that measurements are made while the stripline applicator is operated such
that only the fundamental TEM mode is excited. The propagation constant associated

with this mode is given by

Hffi (23)

where e, and u, are the relative permittivity and permeability of the sample region

(respectively), and the TEM impedance of the stripline applicator is

pr
Z : "f3 = no \J'€:fg (24)

where fg is a purely geometrical factor that is dependent on the stripline cross-section.

Applying expressions (23) and (24) to (18) and (19) results in

I‘ = vl‘lr/Gr—1 (25)
‘/ur/er+1

which provides

256

 

3: =(1*I‘)2 = x (26)
G

and

mm = 7131,. <27)
which provides

2
erpr = {inlay} = Y (28)
(01

where

(29)

 

Consequently, the relative permittivity and permeability of the sample are determined to

 

 

be
6,: X, ptzm (30)
X
01'
e = £(1'1‘) (31)
' k0 1+I‘
and
u, = 3(1‘4‘). (32)
k l-I‘

O

257

Thus it is seen that the de-embedding procedure requires seven frequency domain

measurements, these being

1. S11 of the applicator with a sample material present
2. S21 of the applicator with a sample material present
3. S11 of the empty applicator
4. S21 of the empty applicator

5-7. Sll of the applicator with a short located at three different positions.

7.3 Disadvantages of Frequency Domain Technique

The frequency-domain de-embedding process has been used extensively, and has
been shown to provide good results for a variety of sample materials. In spite of this,
the frequency-domain technique has several disadvantages. The first is that the short-
circuit termination standards must be placed with extreme accuracy, in order to obtain
good results. The distances from the sample terminal plane in expression (5) must be
precisely known, or errors will result in the calculated sample permittivity and
permeability. When high temperature measurements are made, the positions of the
standard terminations may be extremely difficult to determine, due to expansion of the
stripline applicator.

The de—embedding procedure is made more difficult by the fact that the short-
circuit terminations must be placed within one half wavelength of the sample terminal
plane (I = :t A/2) in order to provide unique calibration standards. In addition, if any of
the short-circuit terminations are located exactly one-half wavelength apart, the

calibration constant of expression (9) becomes unity, making (8) indeterminate. Thus

258

proper positioning of the required three short-circuit terminations may become nearly
impossible at high frequency.

An additional shortcoming of the de-embedding technique involves the number
of required measurements. The repeated process of positioning the short-circuit
terminations and sample materials can be time consuming. This impacts the accuracy of
the process because the calibration of the ANA may drift in the time required for a
complete set of measurements. High temperature measurements made using the de-
embedding method may be particularly time consuming. After the sample-containing
region of the stripline applicator is positioned inside a furnace, a certain period of time
is required to heat the device to the proper temperature for measurement. Several hours
may then be required for the applicator and sample to cool sufficiently for the next
adjustment to be made. The placement of samples and movement of shorts requires a
minimum of five such heating/cooling cycles.

In addition to limiting the accuracy of measurements, the multiple heating/ cooling
cycles required for high temperature measurements limit the life of the applicator and
sample materials. The repeated heating and cooling eventually deforms the applicator
and sample materials. These deformations result in the formation of air gaps between
the sample and conductive surfaces of the applicator. Deformation of the applicator also
affects the proper positioning of the three short-circuit terminations required for de-

embedding.

7 .4 Time Domain Materials Characterization

An alternate method of measuring the constitutive parameters of sample materials

259

that employs time windowing techniques can Overcome some of the shortcomings of the
frequency domain de-embedding technique. By taking advantage of the inherent time-
domain response of the stripline applicator, the number of measurements required for
materials characterization can be reduced.

Time-domain materials characterization is typically performed by interrogating
sample materials with short duration EM pulses, and then monitoring the response with
a sampling oscilloscope. The technique described in this chapter involves measurements
of the stripline applicator which are first made in the frequency domain using an ANA.
The Inverse Fast Fourier Transform (IFFT) of these measurements is then calculated.
What results is a synthesized, time-domain response representing what would occur if a
pulse of very short duration, containing all the frequencies present in the measured
spectrum were launched into the applicator. This time-domain response consists of
reflections from or transmissions through (depending on the measurement) the various
structures inside the stripline applicator.

Unlike the frequency-domain de—embedding technique, the time-windowing

technique requires only four measurements and, more importantly, only three applicator

adjustments [59]. The required frequency-domain measurements are

1. S11 of the applicator with a sample material present
2. S21 of the applicator with a sample material present
3. S21 of the empty applicator

4. S11 of the applicator with a short located at the sample terminal plane.

To implement this technique, frequency domain measurements are transformed

260

to the time-domain through use of the IFFT. The resulting time-domain responses are
windowed to remove unwanted reflections or transmissions. The processed time-domain
signals are then transformed back to frequency-domain through the Fast Fourier
Transform (FFT). From this, two pairs of equations in two unknowns are established,
which can be solved to determine the scattering parameters associated with the unknown
sample. The complex constitutive parameters (permittivity e, and permeability p) of the
sample material are then calculated from these scattering parameters through the

Nicholson-Ross-Weir method.

7.5 Materials Characterization Signal Processing Scheme

The time-windowing materials measurement scheme begins with making the
required frequency-domain measurements at the stripline applicator ports. An ANA is
calibrated to the ends of the attached coaxial cables, which are in turn connected to the
terminals of the stripline field applicator. These frequency domain measurements are
then processed to obtain the scattering parameters associated with the unknown sample
material.

The first measurement examined is the response at the stripline applicator
terminals of a short-circuit termination (Sf?) placed within the applicator (Figure 7.5).
In order to reduce the magnitude of distortions due to the inverse Fourier transform
process, a weighting function is applied to the raw frequency domain measurement. This
is accomplished by multiplying the raw Sf? data by the weighting function so that the

ends of the resulting frequency-domain response are made to approach zero. Many types

of weighting functions are available. Here a Gaussian Modulated Cosine (GMC) taper

261

[62] weighting function is applied (Figure 7.6), and the weighted frequency-domain

response is shown in Figure 7.7. This GMC weighting function has a spectrum

WGMc(f) = T [e -n(f—f‘)t2 + e 40%sz (33)

where fC is the cosine frequency, and 1: is a user-adjusted shape factor. These
parameters must be defined so as to provide the proper shape for the weighted waveform.
More will be discussed about this process in a later section.

The IFFT of the weighted Sf? is now calculated. What results is the synthesized
time-domain reflection response of the stripline applicator (with the short-circuit
termination in place) to a short duration EM pulse. Several reflections are usually visible
in the resulting time-domain waveform (Figure 7.8). Applying the general rule of thumb
that the speed of light in free space is approximately one foot per nanosecond, these
reflections can easily be correlated with structures within the stripline applicator. A
small reflection from the coax-to-stripline transition at the front of the stripline applicator
is visible first. A much larger primary reflection from the short-circuit termination is
seen several nanoseconds later. Finally, several secondary reflections from the front of
the applicator and the short circuit are usually visible after the primary reflection from
the short circuit.

In order to remove reflections from the front of the applicator, as well as any
secondary reflections, a rectangular time window is applied to the synthesized tirne-
domain response (Figure 7.9). The signal that remains consists only of that part of the
synthesized pulse which travels through transition region (1, reflects off of the short

circuit, and then travels back through transition region a to the front stripline port.

262

Frequency domoin short circuit response.

1.00

 

0.80

.0
o:
o

Mognhude
.0
3

0.20

 

0.00 IVU'IUFIUIjIUIIrI'T'ITTITTITIr]IUTTIIIIUjr

0.00 1.00 2.00 3.00 4.00
Frequency, GHZ

Figure 7 .5. Frequency-domain response of short-circuit termination, measured at front
applicator port.

263

 

 

0'60 : Goussion moduloted cosine toper.
:
0.40 "
a) a
U .4
D '1
.‘t.’
C I
8‘ 4
2 ;
0.20 "
I
-1
'1
J
0000 IIIFUIIlijvll1II1IIIFFTIIIIIYIIIITYYFIVIf
0.00 1.00 2.00 3.00 4.00

Frequency. GHZ

Figure 7 .6. Gaussian Modulated Cosine weighting function.

264

0.60

 

 

': Weighted short circuit response.
0.40 1
q, d
'U .
a d
:‘L’
c d
05 .
o .
5 I
0.20 -
.1
0.00 IIIIIIIIr]Irirrttrtjrtrtvtritltv‘rrrIvvtlr
0.00 1.00 2.00 3.00 4.00

Frequency. GHZ

Figure 7 .7 Weighted frequency-domain response of short-circuit termination.

265

Time domoin short circuit response.

 

 

 

 

 

 

0.50 "l
,
3
A :
U .
Q) .
5’ o 00 J -- We“ ~=~
E ' l
.4
E i
O ..
C 4
V 4
(D .
_O '1
3 .4
r: -o.50 a
C l
0» .
o .
5 .
_1.00 IITIUTTIITTT IIIIIIr‘TIUVIVII'IIII‘FjIrIIUI'It'ITj
0.00 5.00 10.00 15.00 20.00 25.00

Time (ns)

Figure 7 .8. Time-domain response of short-circuit termination, measured at front
applicator port.

266

 

 

050 Windowed time domain short circuit response.

iiiiiitiiJ

 

 

 

.0
O
o
(L

 

Magnitude (normalized)

I
O
U1
Q
LlllllJLJllLJlLLlll

 

 

 

—1.00 IYTYITTTIIYI TITTrT—lIIIrTITITITTTTTT'I[IITY—TII1IT

0.00 5.00 10.00 15.00 20.00 25.00
Time (ms)

Figure 7 .9. Windowed time-domain response of short-circuit termination, measured
at front applicator port.

267

 

 

The FFT of this windowed time-domain response is then calculated, resulting in
a processed frequency-domain response. As a final step, the weighting function is
removed by dividing this processed data by the original GMC. The remaining signal
consists of the forward scattering parameter associated with transition region 0 (S101),
multiplied by the reflection coefficient off of the short-circuit termination, multiplied by
the backward scattering parameter through transition region (1 (S201) . or

a 0 sho a a
R = 512521511 " = "512521 (34)

1

because the ideal response of the short-circuit termination is such that

sff‘“ = -1. (35)

The amplitude of R1 is scaled by factor of -1, and the remaining data is designated as the

reflection calibration

: 51:3. (36)

C 21

R

which will be used to remove the effects of transition region a from the reflection sample
measurement.

The terminal sample Sf?" measurement is now examined. Again, in order to
reduce the distortions caused by the Fourier transform, the GMC taper weighting
function is applied to this data. The IFFT this calibrated sample Sfl" is then calculated.
Once again, a rectangular time window is used to remove reflections from the transition
regions. The FFT of this windowed time-domain data is calculated, and the weighting
function is removed from the resulting frequency-domain response. The remaining data

consists of the frequency response of signals which travel forward through transition

268

 

region a (SS), reflect off of the sample material (8131), and then travel back through

transition region a (55), or
R2 = SlaZSZOISISP (37)

When this reflection measurement is divided by the reflection calibration (36), what

remains is the desired sample region S-parameter S131

’23 = 5102521551 = 5151 . (38)
R 512 521

Now the through scattering parameter of the sample 8231 must be determined. A
weighting function is applied to the terminal frequency domain measurement of the empty
stripline applicator S5, and the IFFT of this weighted data is calculated. A rectangular
time window is applied to the resulting signal to remove any secondary transmissions that
might occur. Then the FFT of this data is calculated, and the weighting function is
removed. The resulting data consists of the frequency response of signals which travel
forward through transition region (1 (S201). experience a phase shift through the empty

sample region (e 77%”), and then travel forward through transition region b (Szbl), or

r. = sass-“o“. (39>

. . . . . . -'kl
This processed transmrssron measurement lS d1v1ded by the phase factor e ’ ° ’ to remove

the effects of the empty sample region. The remaining data is designated the

transmission calibration, given by

269

 

a b -jkl
_ 5215213 0’

T .
e '1 kols

= 5201 S b (40)

21

C

 

and will be used to remove the effects of transition region a, and transition region b from
the terminal sample transmission measurement.

Now the final terminal frequency-domain measurement S2"? is examined. As with
the previous waveforms, the GMC taper is applied to this data, and the IFFT of the
weighted sample S2"? measurement is calculated. A rectangular time window is applied
to remove any secondary transmissions which may be present in the resulting tirne-
domain signal. The FFT of the windowed time-domain waveform is calculated, and the
weighting function is removed from the resulting frequency-domain data. What remains
is the frequency-domain response of signals which travel forward through transition
region a ($201), pr0pagate through the sample material (S27), and then travel forward

through transition region b (52b1 ), given by

a b 3
T2 ._. 521521521- (41)

Now the expression (41) is divided by the transmission calibration factor (40),

giving

T s“sbs‘
_l = 21 21 b2] = S231 (42)
CT 5201 $21
which yields the desired sample—region S-parameter Sf].
The constitutive parameters of the sample material may now be determined from

scattering parameters (38) and (42), using the Nicholson-Ross-Weir method as discussed

in the previous section (see Figure 7.10, Figure 7.11). Comparisons of the complex

270

 

permittivity and permeability of plexiglas and teflon samples obtained using both the de-
embedding and time-windowing techniques are shown in Figure 7.12, Figure 7.13,
Figure 7.14, and Figure 7.15.

In practice, the procedure outlined above can be simplified by combining several
of the steps. The raw sample terminal S13 3'" measurement can be immediately divided by
the reflection calibration factor, and then processed. Also the raw sample through
measurement 52"? can also be immediately divided by the transmission calibration factor
and then processed. In addition to shortening the time required to process the data, this
tends to increase the accuracy of the measurement process, because in each case a
Fourier transform is eliminated (anytime data is transformed information is inevitably

lost).

7.6 Application of Fourier Transforms and Weighting Functions

The materials characterization procedure outlined in the sections above involves
the use of the FFT. The Fourier transform, when used as a signal processing tool, must
always be applied with caution, for data can be easily lost or distorted if
transform/ inverse transform procedures are performed improperly. Of particular
significance is the fact that the Fourier transform assumes that the signal being processed
is infinite in extent. However, the actual data collected using any real measurement
scheme only covers a finite frequency spectrum. The measured data begins abruptly at
some start frequency, and stops abruptly at some end frequency. In the time-windowing
materials characterization method discussed in this chapter, the finite extent of the

measured frequency data results in distortions in the calculated time-domain response.

271

 

 

 

 

 

 

4.00 : Relative permittivity of teflon
: obtained using time domain technique.
3.00 5
d :1
CD 1‘
O :
E :m
"’ 2.00 1 K
6 :
Q) -i
¥ 1
v _
>. 1 00 1 real
.4: 2 ------ imaginary
.2 :
4" -i
t :
E a .......
a) -0.00 1 .....................................................
CL 2
_1oooq’IIIIIIIT‘IIIIT1—TlIUUTWIllIIITI}
0.50 1.00 1.50 2.00

Frequency (GHZ)

Figure 7.10. Relative permittivity of teflon sample obtained using stripline time-domain
technique.

272

 

 

Permeability (real, imag.)

2.00

1.00

—0.00

—1.00

41411111r111Lriiiiilmiiiiiiiil

Relative permeability of tel‘lon
obtained using time domain technique.

 

 

real
------ imaginary

 

 

YTIIIYIFFIVVYIIIY'UIIUUr—UIITII

0.50 1.00 1.50 2.00

Frequency (GHZ)

Figure 7.11. Relative permeability of teflon sample obtained using stripline time-

domain technique.

273

 

L

 

Relative permittivity of plexiglas

 

4.00 3
I
I1
3.00 :
A j
G I
O 3
'— 2.00 -;
_~ 4
O I . .
a) - real (time—domain)
.L, 2 ----- imaginary (time—domqln)
. ...—...... real (frequency—domain)
f} 1 00 2‘ “1“” Imaginary (frequency—domain)
.2 :
.4.) ..
t ..
E 3"- ”— *\
L —O 00 —~ ’\v\vWMvJH*3=r-v¢3:t:r* \‘Q -_-}
Q) ' -l xxvt¢¢p+fjf“4x
Q I
—1.00 IIIIIITTlllrtilllIIIIIIIIIIIIIIIII]

 

 

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Frequency (GHZ)

Figure 7.12 Comparison of relative permittivity of plexiglas obtained using time- and
frequency-domain methods.

274

Relative permeability of plexiglas

 

 

2.00 -
/.\ :
O3 _.
O _.
'6 —i
a) _
L. _
v _
>\ _
._.: d
T: d x) 4‘1
-0 —0.00 ‘b‘,“ ¢=P*¢‘M$**I-r¢4¢t.$4x 1=ht¢**“‘*"‘i
o a
(D _
E ‘ real (time—domain) .
(D j u“... Imaginary (time-domain)
D. ‘ H-t-H real (frequency—domain) .
J ”H... Imaginary (frequency—domain)
—1.00 nit1|IIIIIIrIIIIIIIIIIIIIIIIITIIIII

 

 

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Frequency (GHz)

Figure 7.13 Comparison of relative permeability of plexiglas obtained using time- and
frequency-domain techniques.

275

Relative permittivity of teflon
4.00

3.00

I”
o
o

 

 

real .(time— domain)

----- imaginary (time— —domain)
._.—H... real .(frequency— domain)
...“... imaginary (frequency— domain)

”(R

a- ’*, ' ~ f&*,.-—1.=4=r*‘*‘c‘¢4:bh¢4ws*$w"

I
ll

Permittivity (real, imag.)
SID ._.
8 8

1111111111-1,1111r11111111iiimtlntiitiitiliii1111111

 

_1.00 1111 [TI] IIIIIjjIIIIIII[IIIIIIIII]

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Frequency (GHZ)

 

Figure 7.14 Comparison of relative permittivity of teflon obtained using time- and
frequency-domain methods.

276

 

Relative permeability of teflon

 

2.00 —
1
A _4
O _
O -
E 1.00 —W
6“ 1
a) -
L _
V _
>\ _
._l: —1
z 1 _ X\ -.a’}”:-t1 _fi. ‘
s -O.00 3......“ WWW-a- “M...“
(D 1
E 1 real (time-domain)
(p q ----- Imaginary (time—domain)
D. _ ...—H... real (frequency—domain
J ""4 imaginary (frequency-domain)
_1.00 IIIIIIIIIIIIIIIIIIIIIIIIIIIFIIIIIII

 

 

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Frequency (GHZ)

Figure 7.15 Comparison of relative permeability of teflon obtained using time- and
frequency-domain methods.

277

 

 

These distortions are analogous to the Gibb’s phenomenon associated with discontinuous
time waveforms.

The effects of such distortions can be greatly reduced if the edges of the measured
frequency-domain data are "rolled off", or made to smoothly approach zero in some
systematic fashion which may be easily undone at a later time. This is accomplished by
multiplying raw frequency measurements by a "weighting function" prior to calculating
the IFFT. If the weighting function is carefully selected and applied, it may be removed
after data processing with minimal effect.

Many types of weighting functions exist. The amount and type of "roll off" due
to the weighting function must be carefully adjusted depending on the type of data being
operated upon, and the desired result. The unweighted raw frequency domain response
of the stripline applicator with a short-circuit termination present is shown in Figure 7.5.
The time-domain response of this data is shown in Figure 7.16. Reflections from the
short—circuit standard, and the coax-to-stripline transitions are apparent in this plot.
However, due to the distortions caused by the abrupt data terminations, it is impossible
to resolve where the reflections from one structure end, and those from another begin.
This would make the proper application of a time-window extremely difficult.

The same raw short-circuit data with a moderate weighting function applied is
shown in Figure 7.17. The time domain response of this moderately weighted data is
shown in Figure 7.18. The same reflections are evident in this plot, however the
distortions due to the abrupt data terminations have been reduced. In spite of this, it is
still somewhat difficult to resolve distinct structures in the stripline.

An ideal weighting function has been applied to the raw short-circuit data in

278

Figure 7.19. In the resulting time-domain response, shown in Figure 7.20, reflections
from the short, and coaxial connectors are distinct, making the task of windowing
unambiguous. Great care must be taken to avoid using a weighting function which rolls
off too much of the data, as in Figure 7.21. In this case data at the edges of the
frequency spectrum has been lost. It must be remembered that use of the weighting
function will always cause some loss of data at the edges of the frequency spectrum,
however this can be minimized with careful selection of the weighting function.
Another aspect of the Fourier transform is that if the width of the spectrum in the
frequency domain is increased, resolution in the resulting time-domain response is
increased. This is evident in Figure 7.22, and Figure 7 .23. The short-circuit time-
domain response of a stripline applicator measured from 150 MHz to 2.15 GHz is plotted
in Figure 7.22. In this plot it is difficult to determine where the primary reflection from
the short-circuit termination ends and the secondary reflection from the front coaxial
connection begins. This difficulty is resolved by widening the frequency measurement
spectrum. Figure 7.23 shows the response of the same applicator as measured from 150

Mhz to 4.15 GHz. In this plot all reflections are easily resolvable.

279

 

Time domain short circuit response.

0.00

 

-0.50

Magnitude (normalized)

44miniLJiliimiiiijilitimitjitJ

 

 

 

.71 OO rYTYjV'TYrrTIII,IITTrIIIIVYIV"I‘UVII'YIU'IIUUUFIIII‘
0.00 5.00 10.00 15.00 20.00 25.00
Time (ns)

Figure 7 .16. Time-domain response of unweighted frequency-domain stripline short-
circuit response.

280

 

 

0'60 :- Weighted short circuit response.
'l
0.40 -
Q) -i
.0 -1
3 d
5‘: .
C
D d
O u
2 Z
0.20 -i
1
1
.i
0.00 UIIUIIUUIiTIIUIIIII'UIIIIITIFIVIUUUTT—TfTY

 

0.00 1.00 2.00 3.00 4.00
Frequency, GHZ -

Figure 7.17. Moderately weighted frequency-domain short-circuit response.

I.
..
n
I
t
t
I.
Q
vi“
;

..r
at
E3
91
I!
”i
:
-..'-
,.A
S
-
I.
Q

281

 

0.50

0.00

Magnitude (normalized)
.<'>
U1
Q

11IILJ‘LLIIIJJIIIJLLJ4LIIIl111]

 

 

Time domain short circuit response.

 

 

"1.00 TTTTTFITITII

0.00 5.00

TTrIl—ITIIIY‘YIIUIYUYYYTVIIIIFVTIIIUI

10.00 15.00 20.00 25.00
Time (ns)

Figure 7.18. Time-domain response of moderately weighted frequency-domain short-

circuit data.

282

 

0.60

 

 

'3 Weighted short circuit response.
0.40 d
.l
G) .1
‘0 .
3 .
:‘:.’
c d
o ‘
o d
2 c1
1
0.20 -
1
0.00 ITII‘UIrilrrrrF'IillirfTUIIUVIUUTTTTITVIT
0.00 1.00 2.00 3.00 4.00

Frequency. GHZ

Figure 7 .19. Correctly weighted frequency-domain short-circuit response.

283

Time domain short circuit response.

 

 

 

 

 

0.50 -
7 I

g -4
<0 4
3‘ 0003 .... NM ‘-
6 ' 4 . .
E -l
5 -4
c l
V d
(D ‘1
.0 4
3 1
#:1—0504
C -1
C31
0 1
E 1

-i.OO Tiiirritrrri TTIIIt}titrrrtrtrrtrrTrIUIIIIrTIIIIT]

0.00 5.00 10.00 15.00 20.00 25.00
Time (ns)

Figure 7.20. Time-domain response of correctly weighted frequency-domain short-

circuit data.

284

0.60

1 Weighted short circuit response.
.
1
-l
0.40 -
d
0) 4
U '1
3 ‘
:‘:.’
C d
0'1
O 1
2 -l
0.20 -
1
.i
'1
0.00 ‘jTIUVUUUIrYITUFTT'IUrTTjUYUUIIIUI‘UUFII—r

 

 

0.00 1.00 2.00 3.00 4.00
Frequency, GHZ

Figure 7 .21. Incorrectly weighted frequency-domain short-circuit response.

285

Time domain response of stri line applicator with

1.00 short present, 0.15—2.1 GHz bandwidth.

0.50

 

0.00

 

 

M
*— v v‘v’

|
.0
U!
o

41111111IIJIIIIIUIIJ[11411111111111I11]

 

Magnitude (normalized)

 

 

 

_1.00 IIIITIIIIIITIIIIIIIITIIIIITII

0.00 10.00 . 20.00 30.'00
Time (ns)

Figure 7 .22. Time-domain response of stripline applicator, synthesized using 015—2. 15
GHz frequency-domain data.

286

Time domain response of stripline applicator with

 

 

 

 

 

 

 

1-00 j short present, 0.15—4.15 GHZ bandwidth.

A Z

‘O 0.50 m

a) -

.E I

6 I

E :

L

O I

5 0.00 _‘M We ..

CD I

U I

3 _

:‘L’ _

C _

810.50 :

E -

—1.OOTllllllllllilllIIITWITHIIIIITTj
0.00 10.00 . 20.00 30.00
Time (ns)

Figure 7 .23. Time-domain response of stripline applicator, synthesized using 015-4. 15
GHz frequency-domain data.

287

 

Chapter 8

CONCLUSION

This dissertation has presented investigations into several topics involving stripline
transmission lines. Chapter 2 provided an overview of the analysis techniques which
were employed. In Chapter 3, a Hertz potential Green’s function was developed for a
three-dimensional current source suspended within a generalized stripline structure. In
this case the current source was considered to be suspended between parallel, infinitely-
wide, perfectly-conducting ground planes. A layer composed of a simple, non-magnetic
dielectric material was assumed to coat one of the ground planes. Boundary conditions
at the dielectric/ free space interface and at the surfaces of the perfectly-conducting
ground planes were first established. The use of a two-dimensional Fourier transform
to solve the Helmholtz equation was prompted by the uniform cross-sectional geometry
of the stripline structure. The result was a Hertz potential solution in terms of
Sommerfeld integral Green’s function representations. Pole singularities and branch
points associated with the Green’s function were examined and identified. It was seen
that both surface-wave and parallel-plate propagation modes are supported by this
structure.

In Chapter 4, the generalized Green’s function developed in the previous chapter
was specialized to represent a stripline structure having a single, planar conducting center
strip. For this case it was assumed that the center strip was immersed in free space, and
that no dielectric layer was present between the conducting ground planes. A pair of

EFIEs was established by enforcing the proper boundary conditions on the tangential

288

 

electric fields at the surface of the perfectly-conducting center strip. Current distributions
associated with various higher order modes were determined through application of a
Galerkin’s moment method technique. Here entire domain (Chebychev) basis functions
were employed, leading to closed form expressions for matrix elements. It was seen that
proper, discrete guided-wave modes excited on the stripline structure are associated with
pole singularities of the specialized Green’s function which occur on the top Riemann
sheet. Improper (leaky-wave) modes are associated with singularities located on the
bottom Riemann sheet. It was also noted that current distributions associated with the
discrete modes supported by the isolated stripline possessed either even or odd symmetry
about the longitudinal axis of the center strip.

Most stripline devices are operated in such a way that only the fundamental TEM
mode is excited. This prompted the development of a TEM specialization of the isolated
stripline full-wave theory. The amount of computational time required for the numerical
solution was reduced for this technique by using a pole series representation in place of
the real axis integration implicated in the Green’s function. The TEM specialization
allowed for determination of the isolated stripline characteristic impedance.

The analysis of the isolated stripline was extended in Chapter 5 to the case of a
stripline structure having two coupled center strips. These center strips were assumed
to have arbitrary width and placement within the stripline cross-section. In this case, two
pairs of EFIEs were established through enforcement of the appropriate boundary
conditions at the surfaces of the conducting center strips. Once again the center-strip
current distributions were determined through application of a Galerkin’s Moment

Method solution. It was seen that while the current distributions associated with modes

289

excited on the coupled stripline were in general neither even nor odd (as was the case
with the isolated center strip), both symmetric and anti-symmetric modes are supported
by this structure. When the center strips are widely separated, the propagation constants
associated with the symmetric and anti-symmetric modes are nearly identical. However,
when the center—to—center distance between the strips is decreased, the propagation
constants become significantly different. Several of the discrete, higher-order
propagation modes associated with the coupled stripline structure, both proper and leaky,
were examined and identified. As with the isolated stripline, a TEM specialization of
the full-wave theory describing this structure was performed. A pole series
representation was developed to replace the real axis integration implicated in the matrix
element Green’s functions. From this the associated impedance parameters were
determined.

Solution of the coupled strip EFIEs was seen to be formidable both analytically
and numerically. In order to avoid the complexity of the integral equation solution, a
full-wave perturbation theory for coupled stripline was developed in Chapter 6. For this
method, the eigenmode currents of an isolated center strip were used as a first
approximation for nearly the degenerate eigenmode currents of a loosely coupled system.
The results obtained using this method were compared to those presented in the previous
chapter and seen to agree well for widely separated center strips. This technique was
found to provide a good initial propagation mode for use with the full-wave technique.

In Chapter 7, a technique for measuring the constitutive parameters (complex
permittivity and permeability) of sample materials placed in a stripline field applicator

was presented. For this method, the time domain response of the stripline is synthesized

290

by calculating the IFFT of frequency domain measurements made at the applicator
terminals. Rectangular time-windows are applied to remove unwanted components of the
resulting time-waveform. The scattering parameters associated with the unknown sample
materials are then determined from the FFT of the processed time-domain data using the
Nicholson-Ross—Weir technique. This time-windowing technique requires fewer
measurements for materials characterization than frequency-domain techniques currently
available. This reductions leads to increased accuracy when high temperature
measurements are made.

During the course of the research involved in this dissertation, much was learned
about the behavior of stripline field applicators. In spite of this, much work in this area
remains. The conducting center strips of all structures considered in this dissertation
were modeled as being infinitely thin. In practice, stripline field applicators having very
thin center conductors are often cumbersome, and difficult to use. Proper time-gating
for materials characterization is facilitated by using a very long applicator. This allows
transients excited at the coax-to-stripline transitions to settle before reaching the sample
region, thus insuring single mode propagation required for the Nicholson-Ross-Weir
technique. For long applicators, proper support of the center conductor is a difficulty.
Thin center conductors tend to bow and warp (particularly at high temperature), causing
changes in the characteristic impedance of the applicator. This also causes a change in
the structure of the applicator between when sample materials are present and when the
applicator is empty. As a result it is desirable to use a thicker center conductor when
making materials measurements. However, any departure from a thin strip structure

causes changes in the stripline characteristic impedance, as well as the cutoff frequencies

291

of the higher order modes. Thus a full-wave theory for stripline devices having thick
center strips is needed. The Hertz potential Green’s function specialized for the
homogeneous stripline cross-section which was developed at the beginning of Chapter 4
can be used to represent an extended volume current source. However, a change in how
the EFIEs are established is required, as additional boundary conditions must be satisfied.
Analysis of stripline having a finite thickness center conductor may also prove interesting
because such a structure should support a continuous radiation spectrum, as well as
discrete bound modes.

Finally, the generalized Hertz potential Green’s function which was established
in Chapter 3 can be specialized to represent a thin center strip which lies on top of a
dielectric material, between two parallel conducting ground planes, forming a covered
microstrip. The theory for an isolated center strip can then be extended to describe a
multi-strip structure. In addition to a full wave theory for the coupled structure, a
perturbation method can also be developed. Both single and coupled covered microstrip
are of importance due to widespread use in integrated circuits. An understanding of the
propagation mode spectra of these structures would aid in the design of high frequency

integrated circuits.

292

APPENDIX

Appendix A
HERTZ POTENTIAL BOUNDARY CONDITIONS AT DIELECTRIC AND

CONDUCTING INTERFACES

A.1 Boundary Conditions on Hertz Potential at a Dielectric Interface

Consider an interface between free space and a differing (non-magnetic) medium.
It is assumed that free space occupies the half-space y > 0 (region 0), the dielectric
material occupies the half-space y < 0 (region 1), and electric and magnetic fields are
supported by a generalized current source which is suspended in the free space region
above the material interface at y = 0 (Figure A.1). It was seen in Chapter 3 that the

associated inhomogeneous Helmholtz equation may be decomposed into scalar

components

 

-J
V211 +k0211a = °‘ ...for a=x,y,z. (A-l)

This facilitates the application of boundary conditions, allowing the individual scalar field

components to be expressed in terms of Hertz potentials as

 

 

 

 

_. all an

E = k2II +3(v-H) Hx = jwe( z -_>’) (A.2)
X x ax a)’ 62
s an an

Ey = kZIIy+-§-(V-]1) H = jwe( ‘- Z) (A3)
6y y 62 6x
- all an

E = 1811 +—a—(V-II) H = jcoe(—’-— ) . (A4)
2 z 62 Z 6x 6y

293

/\
Y

free space (medium 0)
(60. Ho)

:1 v

X\

Figure A.1 Interface between free space and non-magnetic, dielectric material.

 

 

 

294

Tangential electric and magnetic fields (6 =x,z) are continuous across the interface at
y = 0, or

EO‘,(y=0*) .-. Elfl(y=0‘) (A5)

H0300) = Hm(y=0‘). (A.6)

These boundary conditions lead to relationships between the various components of Hertz

potential in each region

. _ 6 ~ 6 ~
E0x(y=0)=Elx(y=O) ~ kfIIOx+a(V-IIO)=klzIle+—a;(V-II1) (A.7)

E0z(y=0’)=Elz(y=O") ~ kjnoz+§£(V-fio)wing—$570,) (A.8)

 

= + = =- - album»): (522-9%) (A9)
H0x(y O) H1x(y O) eo( 6y 62 61 by 62

 

511m) = 613111. _ 392), 01.10)
ay

_ + _ _ — z 6H0),—
H°Z@"O )_le(y‘0) £4 ax ax ay

The boundary conditions on the Hertz potential at the dielectric interface due to
the general, three-dimensional current source will be constructed through the linear
superposition of the boundary conditions on Hertz potentials associated with simpler,
one-dimensional current sources. Care must be taken to maintain consistency in the
implementation of these boundary conditions. For simplicity, a vertical current source

is first considered

(A.11)

“41
II
V.
k.

295

The principal outgoing wave for this source in the free space region is given by

fig = 911g (A.12)
and the associated magnetic field is
if; = 211;; + 2H5; (A.13)

As a result, it is conjectured that the total fields are determined by II = ley with

I? = fo + z‘Hz. This conjecture is applied to the boundary conditions at the interface to

 

 

obtain
a an
Ho.0=0*)=H.,0=o-) ~ €0.32“, 1, 01.14)
6 62
. _ all0y an1
HOZ0=0)=H,,0=0) ~ €0—ax =e, (3; (A15)

which are valid for all (x,z). Thus the condition on the Hertz potential which satisfies

the boundary conditions on the tangential magnetic field is

51
Hoy = _e_ Hly (A. 16)
o

for all (x,z) at y = 0. Now from the boundary conditions on the electric field comes

5110 all

E0x(y=0+) : Elx(y-_-O') a §(?y—) = 32(sz (A.17)
6H0 8H

Eoz(y=0"') : E12020—) -o _aa;(7y_) : aiz(_$12) (A.18)

for all (x,z), and thus the condition on the Hertz potential which satisfies the boundary

296

conditions on the tangential electric field is

992 = all; (A.19)
6y 6y

for all (x,z) at y = 0. It is thus seen that the conjecture was correct and the EM field

supported by the vertical current source can be obtained from II = y‘IIy.

Next consider the case of a horizontally oriented current source

J’ = 521 . (A.20)

I

The principal outgoing wave associated with this source in the free space region is given
by

fig = inf; . (A.21)

Because of the previous result for the vertical current source, it is initially tempting to
conjecture that the total fields supported by the horizontal current source are determined
by Ill = XIII. Applying this conjecture, it is apparent that the boundary condition

Hox0’=0+) = H1x(y=O_) (A22)

for all (x,z) is satisfied because Hx E 0 for this case. The remaining boundary condition

on the tangential magnetic field leads to the relationship

 

11
Hoz(y=0’)=le(y=0‘) .. e 9E9. :6 a 1x (A.23)

° ay ‘ 6y
for all (x,z), and the boundary condition on the z-directed electric field gives

+ - 8 Bum) 8(anlx) (A24)
E = =E = ~ —— =—— -
0‘0 O) “(y 0) az( ex 62 ax

297

for all (x,z). It is seen that the conditions on the Hertz potential which satisfy boundary

conditions (A23) and (A24) are

a e 611
110:: = _1 1x, 110x z 11” 01.25)

 

 

for all (x,z). Application of the final boundary condition on the x-directed electric field

yields

62H“, 6211
Eo.0=0*) = E1.(y=0') ~ k311i). + = 1.1211” . 1x (A.26)
6x2 6x2

 

 

which leads to the condition on the Hertz potential

an EU (Azn

0x: 11x

for all (x,z) at y = 0. It is apparent that (A.27) can only be consistent with (A.25) if

ea = e], which is obviously incorrect for this case. It is thus seen that the initial
conjecture was incorrect and II ... 1211]: because the resulting EM field does not satisfy the
boundary conditions at y = O.

The determination of the total fields supported by the horizontal current source
therefore requires a new conjecture, namely

11' = 211 +911 . (A28)

Application of this conjecture to the boundary condition on the z-directed electric field

gives

EOZ(y=O’)=Elz(y=O') ~ -§-Z(V'fio)=%(V°fii) (A29)

298

which requires

 

v.fi0 = V'fil (A30)
or
9E2: -521: = _(anw _ 5%) (A31)
6x 6x 6y 6y

for all (x,z) at y = 0. The remaining boundary condition on the tangential electric field

leads to

E (y=0’)=E (y=0') ~ 1811 +—‘1(V-")=k’11 +i(V-fi) (A32)
0): 1x 0x ax 1 1x ax l

0

but, because of (A.30)

1811 = kfn (A.33)

0 0x 1x
and therefore the resulting condition on the Hertz potential is

H

0x x

= _1111 (A.34)
e

0

for all (x,z) at y = O. The boundary condition on the x-directed magnetic field gives

. _ a an1
Ho.0=0)=H..<y=o> ~ eogzq, azr (A.35)

 

which leads to the condition on the Hertz potential

(A.36)

for all (x,z) at y = 0. Finally, the boundary condition on the z-directed magnetic field

299

 

 

yields the relationship

 

a an 6110 an
HOZ@:O+) = le(yzo_) _. 60—113 — 61 1x = 50—. — €1___1.y. : 0 (A.37)
6x 6y 6x 6):

due to (A.36), therefore

 

 

aHOX e1 aHlx (A38)

and thus

 

 

 

 

anoy 6H1y : _(dIIOx _ 611”) : _(el 6111,: - an”) = (I _ el]6II1x (A.39)

ay ay 6x ax e0 ax ax :0 ax

for all (x,z) at y = 0. It is therefore apparent that the new conjecture was correct, and
the EM fields supported by the horizontal current source (A.20) can be obtained from
it = in, + y‘IIy.

Now consider a second horizontal current source, oriented such that J = ill with

the associated Hertz potential being f1 = yfly + yIIy. Application of this to the boundary

condition on the z-directed electric field gives
E0 (y=0’) = El (y=0') ~ —a—(v-fi0) = 3(V-fil) (A.40)
x x 6x 6x
or

vfio = V-fil (A.41)

which leads to the condition on the Hertz potential

 

 

61102 _ anlz : _(an0y _ an”) (A42)
dz dz 6y 6y

300

 

 

for all (x,z) at y = O. The remaining boundary condition on the tangential electric field

leads to

E02(y=0’)=Elz(y=O') ~ kfn02+ai(v-fi0)=kfnlz+§(v-fil) (A.43)
Z Z

and, because of relationship (A.41), the resulting condition on the Hertz potential is

6
II... = ‘11.. (A44)

67,
for all (x,z) at y = O. The boundary condition on the z-directed magnetic field gives

H
HOZ(y=O’)=le(y=O‘) ~ 6 E19. =e,a 1) (A45)

 

which leads to the condition on the Hertz potential

m

_ 1
110y - E—II1y (A46)

for all (x,z) at y = O. The final boundary condition on the tangential magnetic field

results in the relationship

 

 

an an
Hox(y=0+)=H1x0’=0_) " Ea—F—Oi-el lz efl—el 1’ =0 (A.47)

 

 

° 3y 5) ° <92 <91
due to (A.45), therefore
6110: = fl 6111: (A.48)
ay 60 6y

and then

301

 

 

(EX _fl = _(dIIOz _ $112): _(fldfllz _ 6111:) = [1__e_1)dIIlz (A.49)
8y 8y dz dz ea dz dz e0 dz

for all (x,z) at y = 0.
Now based upon the cases above, the Hertz potential boundary conditions for a
general three—dimensional current source .7 = )EJJr + ny + z’Jz can be constructed through

linear superposition. The boundary conditions on the scalar components of Hertz

potential (Dirichlet boundary conditions) are thus

m

1100‘ = —1-Illa ...for a = x,y,z (A50)

60

for all (x,z) at y = 0. Boundary conditions on the normal derivatives of Hertz potential

(Neumann boundary conditions) are determined from conditions on tangential electric

fields
E (y=0’)=E (y=0') ~ —(—3—(V-f[)=—a—(V-fi) (A.51)
—+— —— a 7-5 7 A52)
EOZ(y—O)-Elz(y—O) ~ —(V-IIO)-—(V-III) ( .
dz dz
which require
v.fi = v.fi (A.53)

for all (x,z) at y = 0, and conditions on tangential magnetic fields

 

_.aI_IOz_=E_IaHu A54
110.0 0) H..0 0) By 60 6y <-)

302

6H0): _ e161-‘le (A.55)

3y any

 

 

Hazo’: 0‘) 2 Hizo’: 0+)

(since 601on = elllly) for all (x,z) at y = 0. As a result of (A.53)

 

 

V'fi = V'fi _, _aEQz _ aHly ___. _(aHOx _ fl)_(a]102 _ 81112). (A.56)
0 1 BY dy dx dx dz dz

Combining (A.56) with (A54) and (A.55) gives

(d110, _ anly) = (I _fl)(anix + 61112) (A.57)
By By ea dx dz °

 

 

Summarizing these results, the boundary conditions on the Hertz potential at the

y = 0 interface between free space and a non-magnetic dielectric material are

m

 

 

 

 

110“ = —1H1a ...for a = x,y,z (A58)
6
an” ___ 2 art”, ...for p = x,z (A59)
dy 50 dy
and
(dIIOy _ 611w) : (1_:1_)(6H1x + anlz). (A.60)
dy dy ea dx dz

A.2 Boundary Conditions on Hertz Potential at a Conducting Interface

At the surface of a perfect conductor, the tangential components of electric field

vanish
Eop(y=0) = 0 (A61)

303

Here the boundary conditions on the Hertz potential at the surface of a perfect conductor
are determined by examining the Hertz potential boundary conditions at a dielectric
interface (A.58)—(A.60) in the limit that medium 1 becomes perfectly conducting
(61 -* -J'°°)-

For this analysis, it is assumed that free space occupies the half-spacey >0
(region 0), and a conducting material occupies the half-space y < 0 (region 1) (see again
Figure A. 1). Suppose a plane wave originates in the free space region and is incident
on the conducting boundary at y = 0. Two incident wave polarizations must be
considered. For the case of a tangentially polarized incident wave, the total potential in

the free space region consists of the superposition of forward (incident) and backward

traveling (reflected) waves, given by

II = A0(e “I” + RteikoY) (A.62)

0):

where R, is a reflection coefficient. The total potential in the conductive material

(medium 1) consists solely of a forward traveling wave given by

11,, = AOTte_jk‘y (A63)

where T t is a transmission coefficient. Applying the boundary conditions on the Hertz

potential at the y = 0 interface

 

60110“ = e111” (A64)
and
811 an
eo___0_x. : 61 1x (A.65)
8y 6y

304

to the wave expressions (A62) and (A63) leads to

eoAOTr= e014004-10)

and

-jkl€1A0Tt= jkoeoAO(--1 + Rt)

respectively. It is seen that (A.66) may be expressed

T= 52m.)

t
l
and from (A.67) comes

kleth= [6060(1 _ Rt) '

Substituting (A.68) into (A.69) yields

k.e.(1-R.> = k.e.[-:—°(1 +10]

1

which simplifies to

k.(1-R.) = ki(1+R.)

01'

 

(A66)

(A67)

(A68)

(A69)

(A.70)

(A.71)

(A . 72)

Thus the reflection coefficient R, takes the expected form. Now substituting (A.72) into

(A.68) gives

305

 

 

T: E[1.ko“kl] = 52[ 21:0 ] (A73)

t
E

y...
p—n

and so the reflection and transmission coefficients associated with a tangentially polarized
plane wave incident from a free space region onto a differing conductive material at

y = O are given by

 

 

R 2 ko‘kl, T = f4 21% ] (A74)
t ko+k1 t 1 ko+kl

The coefficients in (A.74) are now examined in the limit that medium 1 becomes

perfectly conducting (61 ~ -j°°). It is assumed that (00 = (01, and pa = u, , therefore

 

 

R = 0 1 (A.75)
t eo+€1
and thus
Hm R. = -1. (A76)
€1"-j°°
Likewise
T . Ed 2% ] (A77)
1 e1 €o+€1
therefore
11m T. = 0- (A78)
51"!”

Now the conditions on the x-directed scalar components of Hertz potential at the surface

of a perfect conductor are determined from

306

 

11 = A0(1 +Rt) = A0(1—1) = 0 (A79)

0x

and

11,, = AOT, = 210(0) = 0 (A80)

as e -o —joo, for all (x,z) at y = 0. By a similar development

1

II =11 = 0 (A81)

for all (x,z) at y = 0. Thus it is seen that for a tangentially polarized incident wave, the

tangential components of the scalar Hertz potential vanish at the surface of a perfect
conductor.

Now consider the case of a normally polarized plane wave which is incident from
the free space region onto medium 1. As before, the total potential in the free space
region consists of the superposition of forward (incident) and backward traveling

(reflected) waves

II

0y = A0(e 77" +Rnejk°y) (A-82)

where Rn is a reflection coefficient. The total potential in the conductive material

(medium 1) consists a forward traveling wave given by

III)’ : AOTrie-jk1y (A83)

where Tn is a transmission coefficient. From condition (A.58) on the Hertz potential

eofloy = elII1y (151-84)

307

 

for all (x,z) at y = 0, comes the relationship

e0140(I+Rn) : eIAOTn

which may be expressed

E
T=—31R.
.510.)

It was shown above that 1101, = IIOz = O at y = 0, therefore

 

 

As a result of (A87), boundary condition (A.60) yields

(611” - an”) = (1—3)(0)
e0

 

 

 

3y 3y
and thus
dII0y _ dII1y
0y <97 °

Applying condition (A89) to (A82) and (A83) results in

—jk1A0Tn = jkoA0(—1 + Rn)

01'

len= k0(1-Rn).

Now substituting (A.90) into (A.91) gives

308

(A85)

(A86)

(A.87)

(A.88)

(A. 89)

(A . 90)

(A91)

m

k.(1-R.) = k.—°(1+R.> (A92)
61
which yields the reflection coefficient associated with a normally polarized plane wave

incident upon the interface at y = O

= ___elko " eokl (A93)

n .
elko + eokl

Substituting (A.93) into expression (A.86) gives the transmission coefficient associated

with the normally polarized wave

 

e0 e0 elko - 60ch
T. = ——(1+R.>= ——1+——
61 61 61’9: + eokl (A 94)
_ eo[ 261kl ] _ 260k1
t-zl elko + eokl elko + eok1 '
The reflection coefficient (A.93) may be written
: elko - eokl = k0 _kleo/el (A95)
n Elko'teokl k04-kleo/el
and from this it is seen that
lim R, = 1. (A96)
€1'°-j°°
Likewise, from examination of (A94) it is clear that
lim T. = 0- (A97)

61-. __im

Now

309

II.) = AoT.e”°"" = 0 (A98)

for 6 ~ —joo, as expected, and

1

H09 2 "10167.1(“y + Rnejkoy) : 2Ao (A'99)

again for c—:l ~ -joo. It is therefore seen that Roy #0 at the surface of the perfect

conductor (at y = 0). And finally
dII _. .
——°y =A[—jke”‘°’+ije”‘°’] = -jkA(1—1) =0 (A100)
dy 0 o o n o 0

for all (x,z) at y = 0. Thus the boundary conditions on the Hertz potential at the surface

of the perfectly-conducting material are

 

110 = 110 z 61'on : 0 (A101)
for all (x,z) at y = O.

A.3 Application of Hertz Potential Boundary Conditions to Stripline Cross-Section
The boundary conditions established on the Hertz potential in the preceding
sections are now applied to the generalized stripline structure shown in Figure A.2. This

results in the conditions

{[01 = no. z 1;»; = o (A102)

at y = h the free space/perfect conductor boundary,

310

at y = -d the dielectric/perfect conductor boundary, and

e
110“ = III,“ ...for a = x,y,z

e0

 

dIIOI3 _ _€s_ld11”3
6y 6., 3y

...for B = x,z

( 81'on _ an”) _ (1_ e,)(dfl1x + an”)
3y 5y _ a 6x dz

at y = O the free space/dielectric boundary.

311

(A. 103)

(A. 104)

(A. 105)

(A. 106)

___i___y=h

free space (medium 0)

0
J

 

 

oielec Icm rial ediu ) 61,11.
y=-d

Figure A.2 Generalized stripline cross-sectional structure.

312

 

BIBLIOGRAPHY

[1]

[2]

[3]
[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

BIBLIOGRAPHY

S.B. Cohn, "Problems in Strip Transmission Lines," IRE Trans. Microwave
Theory Tech., vol. MTT-3, pp.119-126, March 1955.

A.A Oliner "Equivalent circuits for Discontinuities in Balanced Strip
Transmission Lines," IRE Trans. Microwave Theory Tech. , vol. MTT—3, pp. 134-
143, March 1955.

H. Howe, Jr., Stripline Circuit Design, Artech House, Dedham, Mass., 1974.
S. Tsitsos, A.A.P. Gibson and A.H.I. McCormick, "Higher Order Modes in
Coupled Striplines: Prediction and Measurement," IEEE Trans. Microwave
Theory Tech., vol. 42, 11 November 1994, pp. 2071-2077.

M.H. Burchett, S.R. Pennock and PR. Shepard, "A Rigorous Analysis of
Uniform Stripline of Arbitrary Dimensions," IEEE Trans. Microwave Theory
Tech., vol. 41, 12 December 1993, pp. 2074—2080.

GB. Baum, "The Singularity Expansion Method," Transient Electromagnetic
Fields, L.B. Felson Ed., Springer-Verlag, New York, pp. 130-176, 1975.

RE. Collin, Field Theory of Guided Waves, Second Edition, IEEE Press, New
Jersey, 1991.

A. Sommerfeld, Partial Differential Equations in Physics, New York, Academic
Press, 1949.

L.M. Brekhovskikh, Waves in Layered Media, New York, Academic Press, 1960.

RC. Clemmow, The Plane Wave Representation of Electromagnetic Fields, New
York, Pergamon Press, 1966.

L.B. Felson, and N. Marcuvitz, Radiation and Scattering of Electromagnetic
Waves, New Jersy, Prentice-Hall, 1973.

J .A. Kong, Theory of Electromagnetic Waves, New York, John Wiley & Sons,
1975.

J .A. Kong, Electromagnetic Wave Theory, New York, John Wiley & Sons, 1986.

WC. Chew, Waves and Fields in Inhomogeneous Media, New York,
VanNostrand Reinhold, 1990.

313

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

J .S. Bagby, D.P. Nyquist, and BC. Drachman, "Integral Formulation for
Analysis of Integrated Dielectric Waveguides," IEEE MTT—S Trans. , vol. 33, no.
10. pp.906-915, Oct. 1985.

R.H. Jansen, "The Spectral Domain Approach for Microwave Integrated
Circuits," IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 1043-1056,
1985.

GT. Tai, "Effect of a Grounded Slab on Radiation from a Line Source," Journal
of Applied Physics, vol. 22, pp. 405-414, April 1951.

CT. Tai, Dyadic Green ’s Function in Electromagnetic Theory, New York, Intext
Publishers, 1971.

CT. Tai, "On the Eigenfunction Expansion of Dyadic Green’s Functions," Proc.
IEEE, vol. 61, pp. 480-481, 1973.

J .K. Lee, and J .A. Kong, "Dyadic Green’s Functions for Layered Anisotropic
Medium," Electromagnetics, vol. 3, pp. 111-130, 1983.

WC. Chew, "Some Observations on the Spatial and Eignfunction Representations
of Dyadic Green’s Function," IEEE Trans. on Antennas and Propagation, vol.
AP—37, pp. 1322—1327, 1989.

J. VanBladel, "Some Remarks on Green’s Dyadic for Infinite Space," IEEE
Trans. Antennas and Propagation, vol. AP-9, pp. 563-566, 1961.

M.A. Blischke, Broadband Analysis of Radiating, Receiving and Scattering
Characteristics of Microstrip Antennas and Arrays, Doctoral Dissertation,
Michigan State University, 1989.

DP. Nyquist, EE929A Advanced Topics in Electromagnetics, Planar Waveguides,
and Circuits, Course Notes, Michigan State University, 1989.

D. P. Nyquist, J. M. Grimm, D. J. Infante, and H. Braunisch, "Classification of
the Proper Propagation-Mode Spectrum and Leaky-Wave Modes on Open Planar
Waveguides," Electromagnetics, vol. 17, No. 2, March-April 1997.

D. Infante, and DP. Nyquist, "Analysis of Higher Order Modes of Stripline
Field Applicators," URSI Radio Science Meeting, 1994 URSI Digest, p. 39.

ES. Cassedy, and M. Cohn, "On the Existence of Leaky Waves Due to a Line

Source Above a Grounded Dielectric Slab, " IRE Trans. Microwave Theory Tech. ,
vol. MTT-9, no.3, pp. 243-247, May 1961.

314

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

N. Marcuvitz, "On Field Representations in Terms of Leaky Modes or Eigen
Modes," IRE Trans. on Antennas and Propagation, vol. AP-4, pp. 192-194, July
1956.

A.A. Oliner, and A. Hessel, "Guided Waves on Sinusoidally-Modulated
Reactance Surfaces, " IRE Trans. on Antennas and Propagation, vol. AP-7, pp.
8201-3208, December 1959.

L.O. Goldstone, and A.A. Oliner, "Leaky-Wave Antennas I: Rectangular
Waveguides," IRE Trans. on Antennas and Propagation, pp. 3307-319, July
1959.

A. A. Oliner, " Leakage from Various Waveguides in Millimeter Wave Circuits,"
Radio Science, vol. 22(6), pp. 866-872, 1987.

H.M. Sigesawa, M. Tsuji, and A. A. Oliner, "Dominant Mode Power Leakage
from Printed-Circuit Waveguides, " Proc. URSI Triennial EM Theory Symp., pp.
527—529, Stockholm, Sweden, Aug. 1989.

ST. Peng, and A. A. Oliner, "Guidance and Leakage Properties of a Class of
Open Dielectric Waveguides: Part I -- Mathematical Formulations," IEEE Trans.
Microwave Theory T ech., vol. MIT-29, pp. 843-854, 1981.

A. A. Oliner, S.T. Peng, T. Hsu, and A. Sanchez, "Guidance and Leakage
Properties of a Class of Open Dielectric Waveguides: Part II -- New Physical
Effects," IEEE Trans. Microwave Theory T ech., vol. MIT-29, pp. 855-869,
1981.

Y. Yuan, and DP. Nyquist, "Analysis of dispersion characteristics and Leakage
Properties of Microstrip Transmission Lines, " presented at National Radio
Science Meeting, Boulder, CO, Jan. 1990.

A.A. Oliner, "Microstrip Leaky Wave Strip Antennas," IEEE International
Symposium on Antennas and Propagation, Philadelphia, PA, June 1986.

DR. Jackson, and A.A. Oliner, "A Leaky-Wave Analysis of the High-Gain
Printed Antenna Configuration, " IEEE Trans. on Antennas and Propagation, vol.
36. pp. 905-910, 1988.

A.A. Oliner, K.S. Lee, "The Nature of Leakage from Higher Order Modes on
Microstrip Line," IEEE MTT-S Digest, pp. 57-60, 1986.

A.A. Oliner, "Leakage from Higher Modes on Microstrip Line with Application
to Antennas," Radio Science, vol. 22(6), pp. 907-912, 1987.

315

 

[40]

[41]

[42]

[43]

[441

[45]

[46]

[47]

[48]

[49]

[50]

[51]

DP. Nyquist, and DJ. Infante, "Discrete Higher-Order Leaky-Wave Modes and
Continuous Spectrum of Stripline," IEICE Trans. Electron, vol. E78-C, No. 10,
pp. 1331-1338, October 1995.

D. Infante and DP. Nyquist, "An Analysis of Leaky Modes Excited on Stripline
Structures," Antennas and Propagation Society International Symposium, 1 995
AP—S Digest, vol. 3, p. 1522-1525.

J .M. Grimm, and D. P. Nyquist, "Spectral Analysis Considerations Relevant to
Radiation and Leaky Modes of Microstrip Transmission Line," IEEE MTT-S
Trans, vol. 41, no. 1, pp. 150-153, January 1993.

RF. Harrington, Field Computations by Method of Moments, New York,
Macmillan, 1968.

A. Frenkel, "On Entire-Domain Basis Functions with Square-Root Edge
Singularity," IEEE Trans. Antennas and Propagation, vol. AP-37, pp. 1211-1214,
1989.

Gradshteyn, and Ryszhik, Table of Integrals, Series, and Products, fifth edition,
Academic Press, Harcourt Brace & Company, 1994.

Abramowitz, and Stegun, Handbook of Mathematical Functions, Dover
Publications, Inc.

D. Infante and DP. Nyquist, "Higher Order Modes of Coupled Stripline
Structures," North American Radio Science Meeting, 1997 URSI Digest, p. 214.

D. Infante and DP. Nyquist, "Full-Wave Perturbation Theory for Coupled
Stripline Structures," North American Radio Science Meeting, 1998 URSI Digest,
p. 279.

Y. Yuan, and DP. Nyquist, "Full-Wave Perturbation Theory Upon Electric Field
Integral Equations for Coupled Microstrip Transmission Lines, " IEEE Trans.
Microwave Theory Tech., vol. MTT-38, pp. 1576-1584, 1990.

MS Viola, and DP. Nyquist, "An Observation on the Sommerfeld-Integral
Representation of the Electric Dyadic Green’s Function for Layered Media,"
IEEE Trans. Microwave Theory T ech., vol. MTT-36, pp. 1289-1292, 1988.

J .S. Bagby and DP. Nyquist, "Dyadic Green’s Functions for Integrated

Electronic and Optical Circuits," IEEE Trans. Microwave Theory Tech. , vol.
MTT-35, pp. 206-210, Feb. 1987.

316

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

SS Stuchly and M. Matuszewski, "A Combined Total Reflection-Transmission

Method in Application to Dielectric Spectroscopy," IEEE Trans. Instrum. Meas. ,
vol. IM-27, pp. 285-288, Sept. 1978.

LP. Ligthart, "A Fast Computational Technique for Accurate Permittivity
Determination Using Transmission Line Methods," IEEE Trans. Microwave
Theory Tech., vol. MTT-31, pp.249-254, March 1983.

J. Ness, "Broad-Band Permittivity Measurements Using the Semi-Automatic
Network Analyzer," IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp.
1222-1226, Nov. 1985.

W. Barry, "A Broad-Band, Automated, Stripline Technique for the Simultaneous
Measurement of Complex Permittivity and Permeability" IEEE Trans. Microwave
Theory Tech., vol. MTT-34, pp. 80-84, Jan. 1986.

N. Belhadj-Tahar, A. Fourrier-Lamer, and H. De Chanterac, "Broad-Band
Simultaneous Measurement of Complex Permittivity and Permeability Using A
Coaxial Discontinuity," IEEE Trans. Microwave Theory Tech. , vol. 38, pp. 1-7,
Jan. 1990.

K.M. Fidanboylu, S.M. Riad, and A. Elshabini-Riad, "An New Time-Domain
Approach for Determining the Complex Permittivity Using a Stripline Geometry,"
IEEE Trans. Intrum. Meas., vol. 39, pp. 940-944, Dec. 1990.

G.W. Hanson, J .M. Grimm, and DP. Nyquist, "An Improved De-Embedding
Technique for the Measurement of the Complex Constitutive Parameters of
Materials Using a Stripline Field Applicator," IEEE Trans. Intrum. Meas. , vol.
42, no. 3, pp. 740-745, June 1993.

D. Infante, D.P. Nyquist, J. Ross and M. Havrilla, "Calibration of Microstrip
and Stripline Field Applicators Using Time Domain Techniques," URSI Radio
Science Meeting, 1993 URSI Digest, p. 221.

A.M. Nicolson, and GE Ross, "Measurement of the Intrinsic Properties of
Materials by Time-Domain Techniques," IEEE Trans. Instrum. Meas., vol. IM-
19, pp. 377-382, Nov. 1970

WE. Weir, " Automatic Measurement of Complex Dielectric Constant and
Permeability at Microwave Frequencies," Proc. IEEE, vol. 62, pp. 33-36, Jan.
1974.

J.E. Ross, Application of Transient Electromagnetic Fields to Radar Target
Discrimination, Doctoral Dissertation, Michigan State University, 1992.

317

 

  

          

 

HICHIGAN STRT U

IW M iii

3 293020

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

  

IV. LIBRARIES
I“HI‘HUI’IHIWIWHI
586438

—.
.-