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This is to certify that the

thesis entitled

Integral Equation Formulation for Natural Modes Of
A Circular Patch Antenna In A Layered Environment

presented by

Eric William Blumbergs

has been accepted towards fulfillment
of the requirements for

Master's EE

degree in

 

 

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Major professor

Date July 17, 1989

 

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TO AVOID FINES return on or More one due.

  

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MSU le An Affirmative ActiorvKuel Opportunity lnetltution
em

 

 

 

INTEGRAL EQUATION FORMULATION FOR NATURAL MODES OF
A CIRCULAR PATCH ANTENNA IN A LAYERED ENVIRONMENT

By
Eric William Blumbergs

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Electrical Engineering

1989

0001403

ABSTRACT

INTEGRAL EQUATION FORMULATION FOR NATURAL MODES OF
A CIRCULAR PATCH ANTENNA IN A LAYERED ENVIRONMENT

By

Eric William Blumbergs

Conventional solutions yielding resonant frequencies of a circular
patch antenna embedded in an integrated circuit environment utilize var-
ious approximations; thus, they produce reasonably accurate results for
only a limited specification of physical parameters. A full wave analy-
sis utilizing integral-operator techniques to determine the resonant
characteristics of the patch circumvents this problem.

Using a polar coordinate Sommerfeld-integral representation of a
Hertzian potential dyadic Green’s function (which accommodates the pos-
sibility of surface wave excitation in the layered background) leads to
coupled electric field integral equations (EFIE’s) describing the cir-
cular disk. Resonant mode solutions satisfy the homogeneous specializa-
tion of these EFIE’s.

For the lowest order resonant modes of the patch, the homogeneous
equations decouple leading to independent IE’s describing resonant modes
supporting radial and circumferential currents. Numerical solutions for

the resonant frequencies and current distributions of these modes are

obtained via Galerkin's method.

Copyright by
ERIC WILLIAM BLUMBERGS
1989

ACKNOWLEDGEMENTS

Great appreciation and thanks are expressed to major professor
Dennis P. Nyquist for his guidance, assistance and patience throughout
this research. Thanks must also be expressed to Dr. Edward J. Rothwell
and Dr. Kun-Mu Chen for their encouragement and support. I would also
like to thank fellow student Michael Blischke for his generous
assistance and advice.

Finally, I am grateful to my mother, Patricia Blumbergs, and my

father, Peter Blumbergs, for their support and understanding throughout

my graduate study.

iv

TABLE OF CONTENTS

LIST OF FIGURES ....................................................

1. INTRODUCTION ..................................................

2. ANALYSIS OF A RESONANT STRUCTURE IN A LAYERED ENVIRONMENT .....
2.1 INTRODUCTION .............................................
2.2 ELECTRIC FIELD IN THE COVER ..............................
2.3 INTEGRAL-OPERATOR DESCRIPTION OF RESONANT STRUCTURES .....
2.4 EXCITATION OF RESONANT MODES .............................

2.5 SUMMARY ..................................................

3. DETAILED ANALYSIS OF A CIRCULAR PATCH .........................
3.1 INTRODUCTION .............................................

3.2 POLAR COMPONENT GREEN’S DYAD .............................

3.3 SPECIALIZATION TO A CIRCULAR PATCH .......................
3.3.1 HERTZ POTENTIAL IN THE COVER ......................

3.3.2 COUPLED POLAR COMPONENT RESONANT IE’S .............

3.4 SUMMARY ..................................................

4. SOLUTION TO DECOUPLED AXIALLY-SYMMETRIC RESONANT IE’S .........

4.1 INTRODUCTION .............................................

4.2 AXIALLY SYMMETRIC DECOUPLING FOR n=0 .....................

4.3 MOM SOLUTION VIA GALERKIN'S METHOD .......................
4.3.1 RADIAL MODE .......................................
4.3.2 CIRCUMFERENTIAL MODE ..............................

vii

11

14

17
17
18
22
24
3O

34

36

36

37

42
43
47

4.4 NUMERICAL EVALUATION .....................................
4.4.1 COMPLEX PLANE SINGULARITIES .......................
4.4.2 PATH OF INTEGRATION ...............................
4.4.3 NUMERICAL RESULTS FOR RADIAL MODE .................

4.5 SUMMARY AND CONCLUSIONS ..................................

5. CONCLUSIONS AND RECOMMENDATIONS ...............................

APPENDIX A .........................................................

APPENDIX B .........................................................

APPENDIX C .........................................................

APPENDIX D .........................................................

APPENDIX E .........................................................

LIST OF REFERENCES .................................................

vi

49
SO
51
57

69

78

80

82

84

88

91

94

LIST OF FIGURES

Figure Page
1. Arbitrarily shaped conducting patch immersed ..................... 4
in the conductor/film/cover environment.
2. Geometry of circular patch (top view) ........................... 23
3. Standard hyperbolic branch cuts along with ...................... 52
TB and TM surface wave pole singularities
in the complex A plane.
4. Migration of singularities across the Ar axis ................... 53
5. Correct path of integration ..................................... SS
6. Deformed contour of integration ................................. 56
7. Normalized complex resonant wavenumber vs ....................... 59
normalized film thickness for the lowest order
radial mode (cc=1.0, cf=1.1).
8. Normalized complex resonant wavenumber vs ....................... 60
normalized film thickness for the lowest order
radial mode (cc=1.0, cf=2.65).
9. Normalized complex resonant wavenumber vs ....................... 61
normalized film thickness for the lowest order
radial mode (ec=1.0, cf=9.6).
10. Current magnitude and phase vs. patch radial .................... 63
coordinate for lowest order radial mode
(8 =1.0, 8 =1.1, t/a=.025).
c f
11. Current magnitude and phase vs. patch radial .................... 64
coordinate for lowest order radial mode
(cc=1.0, cf=i.1, t/a=.3).

vii

12.

13.

14.

15.

16.

17

18.

19.

20

Current magnitude and phase
coordinate for lowest order
(cc=1.0, cf

Current magnitude and phase
coordinate for lowest order
(ec=1.0, cf=2.65, t/a=.3).

Current magnitude and phase
coordinate for lowest order
(cc=1.0. cf=9.6, t/a=.025).

Current magnitude and phase
coordinate for lowest order
(sc=1.0, ef=9.6, t/a=.3).

Current magnitude vs. patch
coordinate for lowest order
mode (ec=1.0, cf

Current magnitude vs. patch
coordinate for lowest order
mode (cc=1.0, cf

Normalized complex resonant

=2.65, t/a=.025).

=2.65, t/a=.

=2.65, t/a=.

vs. patch radial .................... 6S
radial mode

vs. patch radial .................... 66
radial mode

vs. patch radial .................... 67
radial mode

vs. patch radial .................... 68
radial mode

radial .............................. 70
radial

025).

radial .............................. 71
radial

3).

wavenumber vs ....................... 72

normalized film thickness for the first higher

order radial mode (ec=1.0, 8

Current magnitude and phase
coordinate for first higher
(cc=1.0, cf=2.65, t/a=.03).

Current magnitude and phase

f=2.65).

vs. patch radial .................... 73
order radial mode

vs. patch radial .................... 74

coordinate for second higher order radial mode

(cc=1.0, 8 =2.65, t/a=.1).

f

viii

CHAPTER I

INTRODUCTION

The birth of microwave integrated electronics has created a growing
interest in the use of microwave patch devices as resonators or anten-
nas. Obtaining accurate resonant frequencies for patch antennas is of
importance, due to their narrow bandwidth rendering them ineffective
outside the vicinity of their resonant frequencies. For the circular
resonator many approximate methods have been explored, but yield accu-
rate results for only a limited range of physical parameters. This dis-
sertation provides a conceptually exact description of a circular con-
ducting patch immersed in an integrated circuit environment.

Commonly, approximate methods of analysis for the circular patch
have proceeded through a quasi-static approach relying on calculating
the resonant frequencies from the fringing field capacitance. Most of
the approximate methods often ignore or heuristically account for radi-
ation damping. To accommodate a more rigorous formulation prompts ap-
plication of integral-operator techniques. Recently, integral equation
analysis of the circular patch involving Hankel transform techniques has
been investigated [1-3].

This research exploits an integral-operator description of the cir-
cular patch. The text is divided into five chapters. Chapter two pro-
vides the EM theory relevant to the integrated circuit environment of
interest. Once the necessary theory is presented, a general integral

equation (IE) formulation for resonant structures within this

environment is developed. In chapter three, this IE formulation is ap-
plied to the circular patch, resulting in a pair of homogeneous coupled
IE’s describing resonant modes of the patch. For the simplest modes,
chapter four illustrates how the IE’s decouple. These decoupled IE’s,
describing angularly invariant modes of resonance, may be numerically
solved via Galerkin's method.

Some clarification about notation and assumptions used throughout
this dissertation may be useful. Vector and dyadic quantities appear
boldface and are identified with arrows and double bar double arrows,

Jwt

respectively. Harmonic time dependence of e+ is assumed, but sup-
pressed throughout (this sign convention becomes extremely important in
making key observations later on). Finally, all media are linear, homo-

geneous and isotropic.

CHAPTER II

ANALYSIS OF A RESONANT STRUCTURE
IN A LAYERED ENVIRONMENT

2.1 INTRODUCTION

 

This chapter investigates the resonance of an arbitrarily shaped
conducting patch immersed in an integrated conductor/film/cover environ-
ment. Attention is focused on describing the natural resonance of the
device via integral-operator techniques. Once the integral equation for
the patch currents at resonance is obtained, the excitation amplitudes
of the resonant currents and the order of the resonance are determined.

The geometry considered is depicted in Figure 1. A film layer of

thickness t and permittivity e is deposited over a perfectly conducting

f
ground plane (y=-t). The space above the film (y>0) is occupied by a
cover dielectric with permittivity cc. Within the flim/cover interface
(y=0) exists the patch device of infinite conductivity. The unit vector
9, not to be confused with thickness, is tangential to the patch sur-
face. Also, all materials are non-magnetic and infinite in extent.

In section 2.2, the electromagnetics of the layered environment
mentioned above are reviewed. Description of the electric field in the
cover region is expressed in terms of an electric type Hertz potential,

3(3), which is represented by the inner product of a necessary Green's

dyad and an electric source current which sustains the field. The

cover €38 conducting
c
patch

y ‘1Illnllk’
;[———————+ x

/

Z

 

film c=cf ///
/////////////

conductor (ground plane)
can

 

\\
\\\\ e—c+-—+I

Figure 1. Arbitrarily shaped conducting patch immersed in the
conductor/film/cover environment.

Green’s dyad is expressed in terms of Sommerfeld-type integrals. The
general development of a Green’s dyad for tri-layered substrate/film-
/cover dielectric media has been discussed by Nyquist and Bagby [4].
The dyad used in this analysis is the dyad discussed in [4], but spe-
cialized to a perfectly conducting substrate.

In the next section, the integral-operator description for the
patch currents at resonance is pursued. The electric field in the cov-
er, which consists of a scattered field due to surface currents on the
patch and an impressed field exciting the patch, is expressed. Applying
the necessary boundary condition at the patch surface leads to an elec-
tric field integral equation (EFIE), but a broader interpretation of
this equation leads to an integral operator description of the resonant
patch currents.

Finally, by carefully representing the current near resonance and
making various other approximations on the EFIE mentioned previously,
the order of the resonance and the corresponding current excitation am-

plitude coefficient may be obtained.

2.2 ELECTRIC FIELD IN THE COVER

 

This section presents the expressions for obtaining the electric
field in the cover region of the conductor/film/cover environment. This
electric field, as it will soon be seen, can be expressed in terms of a
Hertz potential which, in turn, may be obtained through use of a dyadic
Green’s function. This dyad will be used extensively throughout Chap-
ters II and 111; thus, it requires a somewhat complete explanation. No
details of the dyad’s derivation will be given though. If knowledge of

the derivation is desired, then reference [4) should be consulted.

The electric field maintained by an arbitrary current source in the
cover region of the conductor/film/cover environment can be expressed in

terms of an electric type Hertz potential [5] as

2(3) = (k: + vv-)fi(?) (1)

2 _ 2 _ 9 _
where kc - u use and kc - +w,/uocc. The Hertz potential, fi(r), is ex

pressed as

 

we
v’ J c

9,
fi(?) = J 8(?|?')-3" ’ dv’ (2)

where C(EIE’) is the Hertzian potential Green’s dyad for tri-layered
dielectric media specialized to the conductor/film/cover environment,
and 3(3’) is the source present in the cover region.

The Green’s dyad mentioned above consists of a principal and re-
flected dyad: 8(?|?') = ch(?|?')+ 8’(?|?’). The principal dyad is

represented by the unit dyad weighted by

JI-(F-F') —pcIy-y'u
cp(?l?') = e 2 e dZA (3)
8n p

 

C

which is the two dimensional inverse transform representation of the

scalar Green’s function in an unbounded cover,

a 9
-chlr-r’|

cp(?|?') = e

 

a 9
4nlr-r’l

In eq. (3) the following relationships hold:

A = 52 + :2 dZA = dgdc

2 = $9 + Ca p = A - kc

In all the above relationships, subscript c denotes the cover region.
Also, 8 and C are the Z-d transform variables. This portion of the
Green’s dyad corresponds to a primary wave of potential, flp(?). due to
source currents embedded in an unbounded cover region.

The reflected dyad may be expressed as

r 9 9, _ A rA A 6 rA rA a rA A rA
G (rlr ) — thx + y[5;ch + G y + EEGCZ] + zGtz

where the various scalar components are given in inverse spectral form

as

 

r 9 9,
86"") Rt”) 324%?) -pc(y+y’)

(fall?) = I R (A) e e dZA. (4)
n n 8n2p

a???) " cm °

Rt(A), Rn(A), the reflection coefficients, and C(A), the coupling coef-

ficient, assume the following forms:

pc - pfcoth(pft)

 

 

R (A) =
t Zh(A)
2
Nfcpc - pftanh(pft)
Rn(A) = e
Z (A)

2
2(Nfc - 1)pc

C(A) =

 

2hm zen)

with

“b-
A
>’
V
II

pC + pfcoth(pft)

N
0
A
:v
V
II

2
Nfcpc + pftanh(pft)

n
H: and p£=VA2-k2 for£=c,f.

t

The subscript f denotes the flim region. Also, the relationships for
the transform variable A are identical to the ones used for the princi-
pal dyad. The quantities nf and nC represent the refractive indices of
the film and cover, respectively.

This reflected dyad corresponds to a reflected wave of potential in
the cover, which represents a sum of all waves reflected from the inter-

face regions in Figure 1. It is important to observe that this reflec-

ted dyad is complicated by the coupling terms 92§GZQ and 9g5cga. Physi—

cally, these coupling terms correspond to normal components of reflected
potential due to tangential components of polarization currents at the
interface. Also, this reflected dyad accommodates the possibility of
surface wave excitation in the layered surround. This phenomenon mani-
fests itself in the terms Zh(A) and Ze(A) evident in the reflection and
coupling coefficients described above. When these terms are identically
zero (during inversion integration) they lead to pole singularities and
represent the characteristic equations for TE odd and TM even surface

waves in the layered surround.

2.3 INTEGRAL-OPERATOR DESCRIPTION OF RESONANT STRUCTURES

 

With the help of results presented in the last section, the objec-
tive of this section will be to obtain an integral-operator description
of a resonant conducting patch immersed in the geometry previously de-
scribed (see Figure 1). As mentioned previously, the patch is of arbi-
trary shape, is infinitely thin, and is perfectly conducting.

By properly expressing the fields in the cover region, an EFIE de-
scribing the patch can be constructed. The total electric field within
the cover region may be expressed as Et(?) = E1(?) + 25(3). The field
component E1(?) represents the impressed field exciting the patch (e.g.,
plane wave or microstrip line). The field component ES(?) represents
the scattered field maintained by induced surface currents on the con-
ducting patch. This scattered field may be expressed by specializing
eqs. (2.1) and (2.2) as

.11)
E56?) = — —° (k: + W )L C(P’Ir "')-2(?') ds’ (5)

k
c

1
where -—— = ——
ch cc
c "0 cc

(0

o’flc

Since the patch is perfectly conducting, the total tangential electric
field on the patch surface must vanish. This condition may be stated as
t Es(r) = -t E1(r)|9e es'

leads to the following EFIE:

Using expression (5), this boundary condition

€.(ki + VV-)J C(3I?’)°2(?’) dS' = 'J

k
c MPG?) fies. (e)
I C
S

'1)—

By making suitable modifications, an integral equation describing
natural resonances of the patch can be extracted from eq. (6).. First,
for reasons that will quickly become evident, it is advantageous to use
a broader interpretation of this equation leading to

k
t-(k: + VV-)[ G(?I?’,w)-i(?’,w) ds’ = -J is e-E1(?,w) 3:5 (7)
s’ c
where G(?I?’,w), N(?’,w), and E1(?,w) are the temporal transform domain
representations of C(PIE’), 2(3’), and E1(?). Next, it may be assumed
that temporal resonances of the patch correspond to pole singularities
of H(?’,w) in the complex frequency domain. Thus, near resonance
R(?’,w) may be expressed as

2136" )

2023(0) E ap (8)

(w-wp)

where ap is the excitation current amplitude, (w-w )8 is the required

pole singularity, and N?(?’) is the normalized resonant surface current
on the patch. For convenience, it is assumed the pole is of first order
and the excitation current amplitude equals unity (the order of the pole
singularity and the value of the excitation amplitude are considered in

the next section); therefore, eq. (8) may be specialized as

10

26",»): P . (9)

 

The motivation for representing the current in this manner becomes evi-

dent if it is substituted into eq. (7) and the limit as w approaches up

is considered yielding

 

t-(k: + VV-) I

lim G(?l?’,w)-X (3')ds’
w9w w-w p
P P s,
1‘c i 9 9
= in -J — t-E (r,w) res. (10)
P "C

From this EFIE, it can be seen that as w approaches up, the LHS of the
equation becomes unbounded, while the RHS of the equation remains finite
since E1(?,w) is regular at w=wp. Because this is impossible, the LHS
of eq. (10) is forced to become an indeterminate form which finally

leads to the desired equation for natural resonance of the patch device;

t-(k: + W-)J' 8(?|?',dp)-2p(?')ds' = o ‘r’es. (11)

I
S

2.4 EXCITATION OF RESONANT MODES

To accommodate the determination of l (pole order) and a (excita-
tion amplitude) evident in relation (8). EFIE (7) must be suitably ex-

pressed. First, by passing (k: + VV-) inside the integral, EFIE (7) may

be rearranged as

11

i—

R
ij fie(?|?',u)-2(i~".w) as = -3 c trite...) as. (12)
C

SI
Cé(?l?’,w), an electric Green’s dyad [4,6], is expressed as

acme...) = mm: + w-)C(E-’Ii-",c) + tar-r").

(PV indicates G; is integrated in a principle volume sense [7] by ex-
cluding a “slice" principle volume [8] accommodating the source point
singularity at y=y’, and L=-99 is an associated depolarizing dyad [9].)
Next, since 2p(?) (the p’th resonant current) is tangent to the patch,
i-(argument) may be replaced on both sides of EFIE (12) with
ISR§(?)'(arg.)ds leading to

k
st I ?p(r’)-Ee(?|r",w)-i’(?',w) as, = -J 17° J rpm-tic...) as. (is)
, C

Finally, by using reciprocity of the electric Green’s dyad [S], relation

(13) may be rewritten as

k
2 J 2 (Eb-thaw) ds. (14)
no P

J ds'2(?',m)-J tee! (any?) as .-. -J

I

Near resonance (vamp), modifications on expression (14) may be made
which will inevitably lead to the desired knowledge of the pole order
and the excitation coefficient. First, Gg(?’l?,w) may be expanded in a

Taylor series about the resonant frequency w=wp as

12

tea-”(Kw a Be(?'|?,ap) + 2—0- 8e(?'|E-’,m) ,ifi’”p’ + (15)
p

Also, as it was seen earlier, near resonance i(?’,w) can be represented
by relation (8). Substituting these representations for N(?’,w) and

Gé(?’l?,w) into expression (14) and employing relation (11) yields

a
P Ids/2 (EM-J [g—w ce(?'|?,m)| (d-m ) + m]? (:3) ds
(w-wp) p s w=w p p
1‘c 9 i 9
= -j —— R (r)-E (r,w) ds. (16)
"C P
s

From eq. (16) it may be concluded that as long as

55
s’ S P

, 9, . a 9, 9 _ . 9
I ds Rb(r ) J [ 88(r Ir.w)lw:: wp)] X§(r) ds (17)

is non-vanishing, then b=1 (first order pole) so that the apparent sin-
gularity on the LHS of this equation is annulled when u approaches reso-
nance. With.£ determined, eq. (16) may be rearranged yielding the fol-
lowing expression for the excitation coefficient ap

p nCCp

ID
II

 

I 2§(?)-E1(?,w) ds where

C")
II

p ds’ifi(?’)- [ g; Ge(?’l?,w)l (w-wp)]-2§(?) ds.
w-wp

s’ s
It is important to note that the above expression for ap is valid only

13

for w near up. Since, w is real (the patch will most likely by excited
by a driven element) and ”P is complex, w=wp cannot occur (unless exci-
tation is caused by an indentical resonating patch). Maximum patch exci-
tation amplitude occurs when w=wpr, and frequencies for which w is dis-
tant from wpr (real part of up) are unimportant because the resulting
amplitude is small.

In the previous development, it was assumed the integral specified
by eq. (17) was non-vanishing giving rise to a first order pole. If this
leading term vanishes, integration of higher order terms evident in ex-

pression (15) into the patch surface current must be considered, giving

rise to higher-ordered poles.

2.5 Summary

In the tri-layered conductor/film/cover environment, the scattered
electric field in the cover maintained by currents excited on a perfect-
ly conducting, infinitely-thin patch resonator of arbitrary shape embed-

ded in the film/cover interface may be expressed as

1)
2563) = -J k—C (k: + vqu fi(?|?~”)-2(i~") ds’. (2.5)
C

I

C(PIE’), the Hertzian potential Green’s dyad, consists of a principal
and reflected dyad: C(Elg’) = IGp(?l?') + Gr(?I3’). The principal dyad
corresponds to a primary wave of potential due to currents embedded in
a hypothetical unbounded cover. The reflected dyad corresponds to a re-

flected wave of potential due to polarization currents excited at the

film/cover and conductor/film interfaces.

14

By applying the appropriate boundary conditions at the patch sur-
face and exploiting relation (2.5), the following EFIE can be construct-

ed:

k
-3 e-E1(?,w) ges. (2.7)
"c

t (k: + vv )J C(#I#’.w)-i(?'.m) ds’ = -J

I
5

Using reasonable assumptions near resonance allows 2(3’,w) to be repre-

sented as
a a 2 (3')
2(r',m) s p “B“‘Z . (2.8)
(w-wp)

Properly using this representation in EFIE (2.7) leads to

t (k: + vv )J G(?|?’,wp)-2b(?’)ds’ = o fies. (2.11)

I
S

The above expression may be interpreted as an integral equation which
describes natural resonance of the patch device.
An expression for ap and the value of 2 (order of resonance) in

relation (2.8) can be obtained by rearranging EFIE (2.7) as

k
t-J c§(?(?',c)-2(?',d) ds’ = -J 52 t E1(?,m) res (2.12)

s’ c

where Gé(?l?’,w) now represents an electric Green’s dyad. By substitut-
ing relation (2.8) into EFIE (2.12), applying reciprocity, and expanding

Gé(?’I?,w) in a Taylor series about the resonant frequency up, the value

15

of t can be deduced and an expression for a can be obtained. For the

case i=1, ap was found to be

jk
ap = - n E J 2fi(?)-E1(?,we) ds where
s

 

C P

, 9, . 3 9,19 .
Cp = J ds 2%(r ) J [15; Gé(r Ir,w)| _ ] 2§(?) ds.
w—w

s’ s p

16

CHAPTER III

DETAILED ANALYSIS OF A CIRCULAR PATCH

3.1 INTRODUCTION

 

The last chapter investigated resonance of an arbitrarily shaped
conducting patch in a tri-layered environment. The chapter at hand
focuses attention on a specific structure in the same environment, name-
ly, a circular patch. The goal is to obtain an integral-operator de-
scription of the patch at resonance. The procedure is similar to the
one demonstrated in section 2.3, but more details are carried out since
a specific geometry is implied.

Since the resonant structure has circular symmetry, the rectangular
component Green’s dyad of section 2.2 is transformed to polar form.
First, the Sommerfeld integrals of the scalar component Green’s func-
tions are converted to the appropriate cylindrical coordinates. Then,
the unit vectors of the dyad are transformed.

In section 3.3, the polar component Hertzian potential is obtained.
Because the patch has circular symmetry, the radial and the circum-
ferential components of surface current on the patch (Kp and K9) may be
expanded in an angular Fourier series. Integration of these current
components into the polar Green’s dyad yields the desired expression for
the Hertzian potential in the cover.

Finally, by using the Hertzian potential, the electric field in the

cover, maintained by the patch surface currents, is constructed. With

17

the use of this field, the desired polar component integral equations

for the patch currents at resonance may be derived.

3.2 POLAR COMPONENT GREEN’S DYAD

 

The rectangular component Green’s dyad of section 2.2 can alter-
natively be represented in the following spatial and spectral polar

coordinates:

 

 

 

 

 

e x a #9 E
K ¢
A
C

x = pcose E = Acos¢

z = psine C = Asin¢

3 = Qcose + asine dfidc = AdAd¢

6 = -§sin9 + 90039

Considering first the spectral integral representations of the sca-
lar components of C(Elg’) (eqs. (2.3) and (2.4)), it is seen they take

the followings generic form

9 9,
1635-") =1]. f(A.y|y’)eJx°(r-r ) dgdc.

Geometrically, from the figure depicted on the next page it is obvious

that A-(E—E’) can equivalently be represented as Al3—3’lcos(¢-¢).

18

 

 

 

3 g t X
9.
p
..)
p 3_3. w d
i
2.112.?!) = 2-13-3')
= Al3-3’Icos(¢-¢)
v = A's-3’ ICOS(¢-W)

By applying this result, I(?l?’) may be rewritten as

co 1!
9 9, _
I(?I?') = J AdA f(A,yly’)[ eJA'P'p '°°S(¢ w) d¢.
0

TI

This might seem to be the desired polar representation for the generic

integral, but with some simple manipulations a more elegant form is ob-
l

tainable. Utilizing the substitution u=¢-w and noting the evenness in

the exponential allows I(?l?’) to become

co 1!
9.
I(?|?') = I AdA f(A,y|y’)2J eJAIB'p 'C°S(“) du.
o 0

Further, employing the integral representation of Jo(x), the zeroth or-
der Bessel function of the first kind [10], allows I(?I?’) to ultimately

become

19

N
16’5") = 2n] f(A,y|y’)Jo(AI3-3’l) AdA (1)
O

which is the desired polar representation. Thus, utilizing eq. (1) ena-

bles the spectral integrals in eqs. (2.3) and (2.4) to be expressed in

polar form as

 

 

-p ly-y’l
9 9 " JO(A|3-3’l) e c
Gp(r|r’) = AdA (2a)
41rpc
o

G’(?|?') R (A) -p (y+Y’)

t m t 9 9, c

r 9 9 JO(A|P'P I) e
Gn(rlr ) = Rn(A) 4“pc AdA (2b)
G:(?I?') 0 C(A)

which completes the first portion of the polar transformation.

To finish the transformation, the components of the rectangular

Green’s dyad (e.g. QGXXQ, QGYZQ) must be converted to their appropriate
polar representation (e.g. SGPPS, ycyeé). This is most easily accom-

plished by carrying out the dot product operations given by Gov: Q-G-g,
where a represents the field point components at p, 6, and y, and v rep-
resents source point components at p’,9’, and y’. Before this is done

though, it is advantageous to rewrite the rectangular dyad of section

2.2, given as
C(3IF’) = IGp + 96

in a more applicable form. This is accomplished by noting that the

20

partial derivatives operating on GO in the yx and yz components of the
dyad, are identically equivalent to the transverse gradient operating on

GC. With this observation in mind, C(FIF”) may be rewritten as

FA

C(FIF’) = ICP + 96 t2.

A A rA r A
x + y[Gny + Vth] + zG

Also, noting that G: is a function only of [3'3'l enables Vt to be re-

placed by -V£; therefore, the dyad may alternatively be expressed as

86’5”) = Top + 965‘: + flag? - flog] + ’26

With C(FIP’) expressed in this form, the dot products of C(3I?’)-9 may

be easily calculated as

G(?l?’)°3’ = Q(Gp+Gr)cose’- 9 2—7 Gr + 9(Gp+Gr)sine’
t 8p c t
-) 9I g ’ = —A p r I— A l a r A p r I
8(rlr ) 6 x(G +Gt)sine y #719 cc + z(G +Gt)cose

E(?I?')-9' = 9(GP+G;).

Continuing on with the dot products of 3, a, and 9 into each of the

above yields

3-G(?I?’)-S’ = (Gp+G:)cos(9-9’) QCG’IF’LQ’ = 0
3-6(?|?')-8' = (Gp+G:)sin(9-9’) 9-8(?I?')-3' = -§37 C:
than?!) 9' = o 9-8(E~’IE~") 6' = -1—, $.- GIr

21

6-c(?|?')-3'

-(Gp+G:)sin(e-e’) 9-C(?I?')-9' = cP+c;

a-G(?|?’)°9’ (Gp+G:)cos(a-e’)
which completes the transformation of the unit vectors.
Finally, by using the above results along with expressions (2a) and

(2b). the Green’s dyad may be represented in polar form as

G G G
pp p9 (by
9 9, _

C(rlr ) - G9p G99 G6y where

G G G

YP Y9 YY
G = G cos(9-9’) G = G sin(G-9’) G = 0
pp t (09 t W
G6p = -Gtsin(6-9 ) Gee = Gtcos(9-6 ) G9y = O

- -6 a: --1 2.— =

Gyp - 537 Gc Gye ST 89’ Gc ny Gn

where Gt=Gp+G:, Gn=Gp+G;, and Gc=G:' With C(EIE’) in this form it may
now be applied to an appropriate situation, namely, a circular patch
immersed in the conductor/film/cover environment. This will be the top-

ic of section 3.3.

3.3 SPECIALIZATION TO A CIRCULAR PATCH

The geometry to be analyzed from here on is depicted in Figure 2.
Figure 2 is a top view of Figure 1 with the arbitrarily shaped patch
replaced by a circular patch, and the origin of coordinates shifted to

the center of the circular patch. The circular patch has radius a, is

22

conductor one

////////////

 

film c-cf

 

 

 

\\\\\\\\\\\

 

/
/
/
/
/
/
/
/
/
/
/

 

 

////////////

Figure 2. Geometry of circular patch (top view).

23

perfectly conducting, and infinitely thin. The patch resides in the
film/cover interface (y=0). The polar coordinates for the patch are
identical to the polar coordinates used in section 3.2 and the same re-
lations hold.

Before determining the electric field in the cover due to the patch
currents, it is desirable to obtain the Hertz potential in the simplest
form possible. Taking advantage of the circular symmetry allows the
current to be suitably modified, in turn, enabling the angular integrals
in the Hertz potential to be analytically handled. Once the Hertz po-
tential is in desirable form, the electric field induced by the radiat-
ing patch may be obtained, leading to the polar component IE’s for the

resonant patch currents.

3.3.1 HERTZ POTENTIAL IN THE COVER

 

In general, the Hertzian potential maintained in the cover region
by an infinitely thin conducting patch immersed in the flim/cover inter-

face may be written as

 

m2) = 33’s Jeannie) ds’. (3)
C I

For the situation about to be analyzed, though, it is most advantageous
to represent the surface current as H(?’) = Kp(p’,9’)3’ + K(p’,9’)e’ and

use the polar Green’s dyad found in section 3.2 to express eq. (3) as

 

Hp(?) = chc J [Kp(p',9’)cos(6-9’) + Ke(p’,9’)sin(9-e’)]Gt ds’ (4a)

I
S

24

 

“9(3) 8 J1 J [-Kp(p’,6’)sin(6-6’) + Ke(p’,6’)cos(e-6’)]Gt ds’ (4b)

 

C
S
9 _ 1 _ , , a _ , , 1 a ,
11y(r) - chc I [KPIP ,9 )W K9“) ,9 )Fr W]Gc (IS . (4C)
SI

These representations for the various components of H(?) are now in a
form applicable to a circular patch.

From eqs. (4a). (4b), and (4c) it is obvious that little may be
done to simplify these expressions until more is known about the behav-
ior of the patch surface current. Noting the patch geometry, though,
prompts the following rationale: the patch has circular symmetry; there-
fore, the current will be ZU periodic in e and the various components of

the surface current may be represented in an angular Fourier series ex-

pansion as
Ka(3') = Z Acm(p’ )eJne’. (S)

If relation (5) is used for the various current components, eqs. (4a).

(4b), and (4c) may be expressed as

 

a n
_ 1 , , , Jne’ cos(e-e’) ,
11p - Jwec Z J dp p [iApnm )I e {sin(6-9’ )} Gt d6
9 n 0 n

u
, Jne’ sin(9-e’) ,
+ Aen(p )1 e {cos(6-6’)} Gt d6 ] (6a)
n

25

a u

.._ .a__ ,19_ 3119' ,
J dp p [ Apn(p )ap, + Aen(p )37 39 ]I e Gc d9 . (6b)
0 n

 

:1
‘<
M
if.»
M
0
5M

In expression (6b) it is noted that gar 9 - %5 since Gc is only a func-
tion of (9-9’); thus 2— may be brought outside the integral with re-

’ 89
spect to 9’.

Accounting for the azimuthal variation of the surface current, as
illustrated in relation (5), allows for the analytic evaluation of the
angular integrals evident in eqs. (6a) and (6b). Considering these in-

tegrals, it is evident that they assume the following forms:

fl
, , = jne’ cos(9-e’) ,. _ , , ,
11(p.6.p Iy.y ) I e sin(e-e') Gt(p.p .9 9 ly.y ) d9
u
n
12(p.9.p’ly.y’) = I eJne Gc(p.p’;6-9’ly.y’) d9’.
n

By making the substitution u=9-9’, and using the integral representa-

tions for G and Gc’ I and I may be expressed as

 

 

t 1 2
Q "PCIY’Y'] R (A) ‘Pc(y+y') u ( )
= Jne e + t e -Jnu cos u _ ,
I1 e I AdA 4npc I e {sin(u)}Jo(Alp P I) du
O n
m -pc(y+y’) u
= Jne C(A)e -jnu 9_9,
I2 e I AdA 4npc e Jo(Alp p I) du
O n

 

where IB-B'l = ¢/;2 + p’2 - 2pp’cos(u).

26

The evaluation of the finite angular integrals in the previous expres-

sions for I1 and 12 reduces to the evaluation of

1!

c
s ,o g -Jnu cos(u) 9_9,
fn(p,p ,A) I e {sin(u)} J°(Alp p I) du (7a)
u
u
f:(p,p’;A) = I e'Jm’ JoulB-‘B' I) du (7b)
n

where the superscripts c, s and e denote cosine, sine and empty (no co-
sine or sine), respectively. These evaluations can be readily achieved
by using the summation theorem for Jo (see Appendix A). Applying this

theorem first to eq. (7b) allows f: to be rewritten as

1!

= I e-JnuJo(AV/p2 + p'2 - 2pp’cos(u)) du
n

 

1! m II
= J0(Ap)Jo(Ap’ )I e"Jnu du + 22 Jk(Ap)Jk(Ap’)‘[ e’J‘rm cos(ku) du. (8)
n k=1 n

From eq. (8) it can be seen that the only contribution from the first

term occurs when n=O, yielding
n

I ‘Jnu _ I -
J°(Ap)Jo(Ap )I e du - 2nJ0(Ap)J°(Ap ) for n — 0

TI

and due to orthogonality, the only contribution from the second term

occurs when n=ik, yielding

27

22: Jk(Ap)Jk(Ap’)[ e-Jnu cos(ku) du = Zan(Ap)Jn(Ap’) for n = 11,i2,...
k=1 n

Thus, f: may be evaluated as

e I =
fn - 2an(Ap)Jn(Ap ) for n O,tl,tZ,... (9)

By applying this result, eq. (7a), which with help of Euler’s identity

can be expressed as

I {-3 } (e-J(n—1)u 1 e-J(n+1)u )JO(A|p-p’|) du
u

may also be evaluated as

(no

- 1 I
fn - {-J } n[Jn_1(Ap)Jn_1(Ap )
t Jn+1(Ap)Jn+1(Ap')] for n = 0,11,... (10)

Finally, I1 and I2 may be expressed as

jne I , ,
1 e sn(p.p Iy.y )

H
[I

Jne e , ,
2 - e sn(p.p ly.y )

H
I

where

28

 

s:(p.p’ly.y’) = f:(P.P’;A) dA (11)

c I "’c""'y ' R (A) "’o‘y+y )
e + t e
A
0 C
m -pc(y+y )

e I I _ C(A) e
sn(p.p ly.y ) — I A 4up
O C

 

r:(p.p’:x) dA (12)

c
with f: and f: expressed in eqs. (9) and (10).

Applying the results obtained for I and I the polar components

1 2'
of the Hertz potential (eqs. (6a) and (6b)) become

 

m a
'9 _ 1 Jne , , C I S I
n (r) ch e [p [Apn(p ) gn + Aen(p ) gm] dp (13)
n=-m 0
no a
9 _ 1 Jne , _ , s , c ,
119(r) - chc X e Jp [Apn(p ) gn + Aen(p ) 8n] dp (14)
n=-m 0

e] dp’. (15)

 

Jne , _ , a e , in

e p Apt)“, ) 5"?- Sn 1' A911“) ) P' SD
(The differentiation with respect to 9 evident in eq. (6b) was evaluated
in obtaining relation (15), leading to the Jn term.) With the Hertzian

potential in this relatively simplified form, the electric field in the

cover and inevitably the resonant IE’s for the circular patch can deter-

mined, as will be seen in section 3.3.2.

29

3.3.2 COUPLED POLAR COMPONENT RESONANT IE’S

 

The electric field components E: and E3, maintained in the cover
region by the patch surface currents, may be recovered using the results

of section 3.3.1 along with the relations

5 9 _ 2 9 a . 9

EJN-kc%k)+$vfih) mm
s _ 2 9 1 a . 9

EJE-kc%u)+3fivfihh on

First, using eqs. (13), (14), and (15) from section 3.3.1, the diver-

gence of H(?), given in cylindrical coordinates as

1 a 1 a a
vfim=-—HH+——n+—n
p 89 p p p 66 9 6y y

may be straightforwardly obtained as follows:

a a
- JDG , , 1 a c
—(pII ) - ——c X e Jp [Apn(p ); 5mg“) 1* A9n(p
O

:41
p 69

blH
Q:

'o
"b
c.

E

n

[pg:)] dp’

Il="w

a

on
a _ i Jne , _ , Jn s , Jn c ,
—TI -—Ze JP[Apn(p)_58n+Aen(p)p—gn]dp
O

blH
o:
a:
Q
L.
c
n

n="m

62 e

Jne I_ I__ ,J_na_e ,
e p Apn(p )ap’ay 8n + A6n(p )p’ 6y gn dp

and combining and collecting terms yields

30

2
fifime eJne Jn s a e
Vfi J:p {AP n(p' )- p 16p(p8: )' p 8““ 6p 6y +8 n]

n=-m

en(p’ )-

Jn Jn 8 e ,
p 1gp(pg: )+ p gn+ p, 8y gn]} dp . (18)

Utilizing relations (13), (14), and (18), the expressions for E:(?) and

E:(?) given by eqs. (16) and (17), respectively, may be written as

WIS

m a
s 9 _ _ c Jne k2 8c §_ 1 2_ c _ §_ ;n s
Ep(r) - J c ‘2: 8 JP W{ “(p )[k C 8n + 6p[p 69(pgn)) BP[P 8n]
0

n=-m

_ a e m( )k 8s + a 1 a _( s)
apap’ay 8n p 8p p 6p pgn

8p p apay n
1) °° a 2
s 9 = _ _g Jne , , _ 2 s in Q_ C n_ s
Ee(r) J R X e I p {Apn(p )[ kcgn + 2 p(pgn) 2 3n
C n=-m p p
jn 62 e 2 c Jn a
- p ap’ay 8n] Aen(p )[kc 8n + ;2 -p(pgn)
2 2
-L°-Li.ed:
p2 8n pp’ay 8n] p

A more convenient form for the above equations involving E: and E: may

be expressed as

E

'00)

on
nc I I I I I
(r) 31(— DZ? 9E [Apnm )Kpp(P»P ) + A9n(p )er(p.p )] tip (19)

31

m
S "C I I I I I
Ee(r)= Jr‘— ’12:” 9El} Apn(p )K9p(p.p ) + Aen(p )Keetpm )] dp (20)
where the K , are defined as
«B s
, = , 2 c g_ 1 §_ _ Jn s e
Kpp(P.P .Y) p [kc 3n + ap[p 8p(p8n)] gp[— 8n] 533275; 8n] (21a)
2 s a 1 a s 8 Jn c Jn 82 e
er(PoP IV) = P [kc 8n 4' $[z $(pgn)] '1’ 'a—p P— 8n] + P- 5387 8“] (21b)
2 s Jn 6 c n2 s Jn 62 e
K9p(p.p ,y) = p [—kcgn + ;E 53(pgn) + ;E gn - P— 5375; 8n] (21c)
2 2
._.2c irg__ s-9_<=-n__ie
Kee(p.p .y) - p [kc 3n p2 6p(p8n) p2 8n pp. ay sn]- (21d)

By using the previous components of electric field expressed in
eqs. (19) and (20), the polar component IE’s describing the circular
patch can be obtained. Since the patch is perfectly conducting, the
total electric field tangential to the patch surface (y=0) must vanish,
which may be stated as E:(?) = -E;(?)ly=o and E:(?) = -Eé(?)ly=o where
1 9‘?) is the incident field exciting the patch. With the help of ex-

pressions (19) and (20) these boundary conditions lead to

 

eJne _ kc i _
pn(p’ )Kp p(p. p’ ) + A6 n(p’ )er(p, p’ ) -J —— E (p.e,y-O)
no P
n=-a Y:0
123“" ()xep( )+A()K(::O=-Jk—CE1(9=O)
oA’pnp p.p’ p’ een.’ "c apny .
n=-m

       

32

(It must be remembered that the KaB’s have derivatives with respect to y
present. These derivatives must be calculated before y=0 can be evalat-
ed in the previous two EFIE's.) By invoking the results previously il-
lustrated in section 2.3, the above two equations lead to the following

IE’s for the resonant patch currents:

o 3es (22)

Jne I I I I I
e mj:h n(p )Kpp(P.P ) + A9n(p )er(p.p )] dp

"[‘“’]s

o Bes. (23)

"["’]s

noeJne I} pn(p’)K9p(p,p’) + Aen(p’)Kee(p.p’)] d
0

Although expressions (22) and (23) are in a rather cumbersome series
form, exploting the orthogonality of the exponential permits them to be

simplified as

a n = o,:1,:2,...
[Apn(p )Kpp(P.P ) + Aen(p )er(P.P )] dp - O (24)
pes
0
n = 0,:1,i2,
[Apn(p )Kep(P.P ) + Aen(p )Kee(p.p )] dp - 0 368 (25)

0

which are now coupled IE’s that can be solved independently for each n.
For the special case of n=0, the above equations decouple forming two
IE’s which can be solved independently. Physically, when eqs. (24) and
(25) decouple, they describe resonant modes on the patch in which cur-
rents sustained at resonance are axially symmetric. The solution of

these axially symmetric modes is furnished in the subsequent chapter.

33

3.4 SUMMARY

The Green’s function presented in Chapter II can be alternatively

represented in polar form as

 

 

 

 

I- 1
Gtcos(e-e’) Gtsin(6-6’) O
G (3|?') = -G sin(9-6’) G cos(e-e’) O (26)
«B t t
8 l 8
EB’GC B’EE’GC Gn
)- d
with
-p ly-y’l -p (y+y’)
m a a e c + Rt n(A) e c
ct’n = Jo(Alp-p I) 4np AdA
c
0
w -p (y+y’)
_ a 9, C(A) e c AdA
GC - Jo(Alp p I) 4np .
0 c

This polar representation allows E to be conveniently applied to a cir-
cular patch embedded in the tri-layered medium.

The Hertzian potential maintained in the cover by the radiating
patch can be suitably expressed by implementing relation (26) and ex-
panding 2(3') in an angular Fourier series. Once this is done, fi(?) can
be analytically integrated with respect to 9 leading to a relatively
simplified form. By utilizing this convenient expression for the Hertz
potential (eqs. (3.13), (3.14). (3.15)), the necessary components of
electric field, E: and E3, scattered from the surface the patch can be

constructed (eqs. (3.19) and (3.20)). Once these components of electric

34

field are determined, the boundary condition on the perfectly conducting

patch surface (y=0) leads to IE’s for the resonant patch currents:

Jne

e o 365 (3.22)

n=-m

Ja[Apn(p’)Kpp(p.p’) + Aen(p’)er(p,p')] dp’
0

o Bes. (3.23)

m
z eJn" r[Apn(p’ )Kep(p.p’) + Aen(p’ )Kee(p.p’ )] dp’
0

n=-on

Exploiting the orthogonality of eJnO simplifies the above system to

n = 0.11.12. .
[Apn(p )Kpp(p.p ) + Aen(p )er(p.p )] dp = o 9 (3.24)
0 pes
n = 0.11.12, .
[Apn(p )K9p(p.p ) + Aen(p )Keemm )] dp = 0 _) (3.25)
0 pes

which are now coupled 15’s that may be solved independently for each n

(n representing the various modes of resonances on the patch).

35

CHAPTER IV

SOLUTION TO DECOUPLED AXIALLY-
SYMMETRIC RESONANT IE’S

4.1 INTRODUCTION

 

Pursuing a moment-method solution of the coupled IE’s. eqs. (3.24)
and (3.25), via Galerkin’s method, enables the exact determination of
complex resonant frequencies (which include radiation damping) and cor-
responding resonant currents for the circular patch. The special case
of n=0 causes a decoupling of the IE’s, subsequently providing two inde-
pendent IE’s which describe angularly invariant eigenmodes on the patch.
The solution of these independent IE’s is pursued in this chapter.

Section 4.2 presents the details involved in decoupling the IE’s.
When n equals zero the coupling kernels. er and Kep’ implicated in eqs.
(3.24) and (3.25) vanish, while the self terms. Kpp and K66, simplify
immensely. The expressions for Kpp and K99 may then be expanded leading
to relatively simple forms of the independent IE’s.

Once the IE’s are placed in an appropriate form. Galerkin’s method
may be applied. Tchebychev polynomials, with the expected edge singu-
larities incorporated, provide a suitable choice of basis and testing
functions which may be utilized in both IE’s. Once expansion and test-
ing are carried out, an NxN homogeneous matrix equation results from

each IE. The only nontrivial solution to these matrix equations re-

quires a vanishing determinant. This condition leads to the

36

determination of the desired complex resonant frequencies for the circu-
lar patch.

In section 4.4, the search for the previously mentioned complex
frequencies is pursued. The evaluation of each element involved when
calculating the determinant described above requires numeric evaluation
of a spectral inversion integral in which many subtleties are present.
The spectral integrals all share common branch point and pole singulari-
ties. The proper choice of the branch out and the location of the poles
strongly influence the correct choice of integration path that must be
used. Once the spectral integrals are correctly evaluated the complex
resonant frequencies and corresponding patch currents are obtained.
Numerical results for the resonant mode in which only a radial component

of current exist are presented.

4.2 AXIALLY SYMMETRIC DECOUPLING FOR n=0

 

In order to illustrate the decoupling of (3.24) and (3.25), the

expressions for g:. 3:. and g: , eqs. (3.11) and (3.12), must be spe-

C

cialized for n=0. This task involves first. calculating f3, fn. and f:

when n=0. From expression (3.9) it easily seen that

e I
fo — 2nJo(Ap)JO(Ap ). (1)
Also. expression (3.10) specializes to
S 1
f =»{_J} u[J_1(Ap)J_1(Ap ) 1 J1(Ap)J1(Ap )]

which. with the help of the relation

37

_ __ n
J_n(z) - ( 1) Jn(z) ([10). P.358) (2)

simplifies further to

H,
I

- 2u31(lp)J1(Ap’) (3)

"a
II
D

(4)

Thus. utilizing eqs. (1), (3). and (4), 3:, 3:, and g: simply become

e =JAC(A) "9 y
o

 

c , '
o 2pc e J°(Ap)Jo(Ap ) dA (S)

1 + Rt(A)
0

on
0
ll

g = 0 (7)

where the fact that y=y’=0 has been exercised everywhere except for g2.
This reason for this, as it will soon be seen, is that g; g: must first
be evaluated.

c
With g: and g: conveniently expressed. the K ’5 --represented in

«B
eqs. (3.21a)-(3.21d)-- may be readily evaluated showing the decoupling

and simplification of the IE’s for the mode n=0. Concentrating first on
decoupling the equations, it is easily observed that the expressions for
er and Kep’

2
, _ , 2 s a l a s a Jn c Jn a e
er(P.P ) - p [kc gn + 55(p 55(pgn)] + 5;[;— 3n] + 37 535; 3D (3.21b)

38

2 s Jn nzs _3n 82 e
Kep(P:P ) = p [—kcgn+ F5 a—pc(P8 )+ +?gn “'p— M 8n] (3. 22C)

vanish when n=0 since all terms depend either on n or g:. Thus, IE’s

(3.24) and (3.25) decouple becoming the independent IE’s

I
O

n
F”

JaAe(p’) K99(P.P’) dp’
0

Further, by again applying relations (5), (6), and (7), Kpp and K99 --

expressions (3.21a) and (3.21d)-- become

pp _

c 62 3
-_2.m+-—g+—zg-———. g] (8)

where the superscript 0 denotes n=0.

86
modified further. With the assistance of relation relation (5), the

K0 is already in a suitable form. On the other hand. sz must be

first three terms in eq. (8) may be combined as

m
1 + R (A) 2
t , 2 _ 1 1 a a
[A T J1(AP )[(kC -2-) 4' 5 5-5 +——2 J1(?(p) dA.
0 c 8p

'0

39

Adding and subtracting the quantity A2J1(Ap) leads to

1 + R (A) 2
t , a 1 a 2 _ 1
J A _—§E_—__— J1(Ap )[ ——§ + 5 53 + (A —§)]J1(Ap) dA
o c 60 p

1 + Rt(A) 2 2
0

By virtue of Bessel’s differential equation

2 2

9—— R + R + (T - Z—) R = o ([11), p.364)
2 r 2
6r r

th-
OJIO)

with solutions defined as Bessel functions of the first kind with order
v and argument Tr (R=Jv(Tr)). the first term in expression (9) vanishes
(v=1, T=A). Thus, KO becomes
PP
" 3

1 + R (A)
o _ , t 2 _ 2 . _ a e
Kpp _ p [ J A —-§E;___— [kc A ]J1(Ap)J1(Ap ) dA 535575; go] (10)
0

but this is not yet the desired form; the derivatives on g: still have
to be evaluated in the last term. Continuing on, with the use of rela-

tion (5). this term can be expressed as

3 -p y 2
a e _ AC(A) 8 c 6 ,

Carrying out the derivatives and evaluating at the patch surface (y=0)

leads to

40

3 3

a e _ _ A C(A) ,
0

where the chain rule. %; J1(ax) = «J1'(ax), and the relation J3(z) =

-J1(z) ([10], p.361) were applied in passing from eq. (11) to eq. (12).

Thus, substituting relation (12) back into eq. (10) yields

 

m

1 + R (A) 3

o - . t 2 _ 2 A C(A)
0

C ] J1(Ap)J1(Ap') dA

which is the desired form for sz.
To briefly summarize, for n=0 the IE's represented in eqs. (3.24)
and (3.25) decouple and simplify immensely leading to independent IE’s

expressed as

A ’ K , ’ d ’ = 0 13
[3 p(P ) pp(p p ) p ( )

0

J‘AOW ) Kee(p.p ) dp = 0 (14)

0
where

o 2 w 1 + Rt(A)

0

1 + R (A) 3
sz = p’J [A —————3——— (k: - A2) + §_g£§l] J1(Ap)J1(Ap’) dA (16)
o

41

With a Judicious choice of basis functions representing the radially-
dependent current amplitudes Ap(p’) and A9(p’). Galerkin’s method can be
successfully applied to (13) and (14). This will ultimately lead to
numerical solutions providing both resonant frequencies and current dis-

tributions. as will be shown in the following sections.

4.3 MOM SOLUTION VIA GALERKIN’S METHOD

The IE’s as expressed in eqs. (13) and (14) describe unique reso-
nant modes on the circular patch. EFIE (13) describes modes where the
surface current sustained at resonance has only a radial component with
no azimuthal variation. EFIE (14), on the other hand, describes modes
where the surface current sustained at resonance has only a circumferen-
tial component but is also azimuthally invariant. Applying a moment-
method solution, namely, Galerkin’s method, to IE’s (13) and (14) will
produce a system of equations from which the complex resonant frequen-
cies and corresponding current distributions for these simple modes can
be obtained.

Galerkin’s method, as applied in this situation, involves expanding
the unknown current distributions (represented by amplitudes Ap(p’) and
Ae(p’) in IE’s (13) and (14)) in a set of N basis functions which are
complete. physically realizable, and enable the spatial integration evi-
dent in the IE’s to be evaluated in closed form. Once expansion is com-
pleted. the resulting equations are integrated again over the entire

domain of the patch into the identical set of N functions used for ex-

42

pansion. This last procedure. better known as testing, results in a NxN
linear system of equations for each mode, which in turn can be solved to
yield the desired resonant frequencies and current distributions. The
number of basis functions (and therefore testing functions) to be used
depends on the desired accuracy of the resultant resonant frequency and
current distributions. Theoretically, to yield exact results N should
extend to infinity. Numerically this is impossible and for all practi-
cal purposes the number of basis functions needed for satisfactory con-

vergence is fairly low.

4.3.1 RADIAL MODE

 

Seeking a complete set of basis functions for Ap(p’). which when
multiplied by p’J1(Ap’) can be analytically integrated from p’=0 to
p’=a. leads to the consideration of the Tchebychev polynomials Tm(p’/a)
([10]. p. 778). Although with some minor manipulations these functions
are integrable with p’J1(Ap’). they must be modified to accommodate the
physical behavior of the expected patch current distribution. By naive-

ly representing Ap(p’) as

(I)
Ap(p’) = Z Amem(p'/a) (17)
m=0

it is evident that the known edge condition of vanishing normal current
at an infinitely thin edge (in this case the edge being p=a) is not pre-

sent; therefore, expression (17) must be modified to

43

 

Q
I - I _ I 2 ‘
Ap(p F: Amem(p /a)/1 (p /a) . (18)
m=0

Further. by virtue of the polar coordinate representation 2 = SKP’ there
is the possibility of a discontinuity in the current or it’s derivatives
at p=0. But it has been shown by Blischke [12) that this problem may be
circumvented by retaining only the odd terms in expression (18)
(Blischke has rigorously demonstrated that the even terms in relation

(18) identically equal zero). Thus. Ap(p’) ultimately becomes

I _. I _ I 2 ‘
Ap(p ) - Z Ame2m+1(p /a)/1 (p /a) . (19)
m=0

 

With Ap(p’) represented in this form. the spatial integration in-
volved in IE (13) can conveniently be handled. By utilizing relation

(19) and exploiting the linearity of the summation and integration oper-

ators. IE (13) may be written as

on a

Z Apm I T2mfl(p’/a)/1 - (p’/.11)2 K:p(P.P’) dp' = O. (20)
m=0 0

 

Considering only the spatial integration and the relevant terms in eq.

(20) leads to the required evaluation of

 

I _ I 2‘ I I I
Im1 - JaT2m+1(p /a)¢/1 (p /a) J1(Ap ) p dp
0

Using the substitution u=p’/a and rewriting the square root term leads

44

to

u(1 - uz)

1 - u2

 

J1(Aau) du. (21)

By repeated application of the recursion relation

2uTn(u) = Tn_1(u) + '1'n+ (u) ([13] p.1032)

1

the quantity u(1 - u2)T2m+1(u) evident in eq. (21) can be expressed as

1
§ [T2m(u) + T2m+2(U) + Tl2m-2|(U) + T2m+4(u)]. (22)

(The details omitted in obtaining eq. (22) and an explanation of the
absolute value apparent in the third term are provided in Appendix B.)

Utilizing expression (22) allows eq. (21) to become

1

2
1 = I [12mm + Tmzm) + Tl2m_2|(u)
0

m1

ml”

J1(Aau)
+ T2m+4(u)] ————————— du. (23)

V 1 - 112

With the help of

 

z) y > 0

_ (
£22 2 Re(v) > -n-1

Nl=l
Q

+

5

N

1
T (x) J (yx)
J“ v J (2).)
2

0 1 - x -2—

[13]. the spatial integration is finally evaluated as

45

2
ma

 

 

 

 

Aa Aa Aa Aa
1m " E [J1+2m('2) J1-2m(_2) + J2m+3(—2) Jam-1(7)
2 2 2
Aa Aa Aa Aa
J1+|2m~2I(-2) J1-l2m—2|(—2) + J2m+S(—2) J-2m-3(-§)]
2 2 2
uaz
= '13 Inna). (24)
Using this result, EFIE (20) may be expressed as
N on
2 1 + R (A) 3
na t 2 2 A C(A) _
'13” Z AMI [A —-———2pc (1cc A ) + 2 ] J1(Ap)3m(Aa) dA - o. (25)
m=0 0

Now that current expansion has been completed, testing can be im-

plemented. By multiplying the above with

 

I _ I 2‘
T21+1(p /a) J/i (p /a)

and integrating over the surface of the patch. the following spatial

integral results:

 

- _ 2‘
111 - I T21+1(p/a)¢/1 (p/a) J1(Ap) ds
3

 

- _ 2‘
- 2n T21+1(p/a) 1 (p/a) J1(Ap)p dp.
0

Application of result (24) allows the above integral to be evaluated as

46

Z 2

 

 

 

 

u a Aa Aa Aa Aa
I11 ‘ T [J1+21('2) “'1-21‘7) I J21+3(_2) J-21-1(—2)
2 2 Z 2
Aa Aa Aa Aa
J1+I21-2I(‘2’ Ji-I21-2|(_2) + J21+s(‘2) J-21-3(—2’]
Z 2 2
nza
Thus. EFIE (25) becomes the matrix equation
N
{A B =0 1=0,1.2....,N (27)
pm In
m=0
where
Q
1 + R (A) 3
_ t 2 _ 2 A C(A) ~
31m —J [A ———2pc (kc A ) + 2 ] 3m(Aa)J1(Aa) dA. (28)
0

The difficulty of solving IE (13) has been reduced to solving a linear
system of equations evident in eq. (27). Non-trivial solutions to this
system impose the condition of a vanishing determinant. This condition.
DET[B(u)]=0, enables the numerical extraction of the complex resonant
frequencies. Complications arise though, due to the necessary evalua-
tion of the spectral integrals obvious in the matrix elements B

Im'

4.3.2 CIRCUHFERENTIAL MODE

 

The tedious details involved in applying Galerkin’s method to EFIE
(14) are similar to those implemented in the MoM solution of IE (13).

The same set of basis functions used for Ap(p’) may be used to expand

47

Ae(p’). but the correct edge condition must be incorporated. For the
circumferential mode, the current (which is tangential to the patch

edge) must become appropriately singular at the edge (p=a) which leads
to

 

T (p'/a)
I __ ZHH‘I
Ae(P ) - Z AGE) . (29)

(Again only the odd terms are considered to assure continuity of the
polar coordinate representation 2 = K98 [12].) Substituting expansion

(29) into IE (14) yields

(p ’/a) o
OAOm 2g Keetp.p ) dp = 0. (3o)

/1 - (p’/a)2

 

 

In analogy with section 4.3.1. the spatial integration in IE (30) may be

evaluated creating

N 2 1 + Rt(A) 3
2 Am" kc A 7P:— J1(Ap) u(Aa) dA = o. (31)
m=0 0

Continuing on, and applying the testing operator

 

( eq 31) ds

 

J T2m+1(p’ /a)

5 ¢/;' (p’ /a)2

leads to the following homogeneous matrix equation for the circumferen-

tial mode current amplitudes:

48

N
A C =0 1=0,1.2,....N (32)
6m 1m

n=0

 

where
1 + RM) 3 3
0 c
with
_ Aa Aa Aa Aa _
3p(Aa) - [J1+2p(-§) J1-2p(_2) + J2p+3(-§) J_2p_1(-§)] for p-1.m.
2 2 2

As witnessed in last section, complex resonant frequencies may be ex-
tracted from (32) leading to the construction of Ae(p’). Again problems
arise in the numerical evaluation of the inversion integral. The subtle
difficulties, which are present in both spectral integrals (28) and

(33). are addressed in detail in the subsequent section.

4.4 NUMERICAL EVALUATION

 

To illustrate the complications that arise when evaluating inver-
sion integrals (28) and (33), it is convenient to utilize the expres-

sions for Rt(A) and C(A) in section 2.2 to express Blm and Clm as

 

+ 3 (Aa)3m(Aa) dA (34)

m 2 2 3 2
B = I (kc - A )A A (Nfc- 1)pc
1'" 0 Zh(A) Zh(A)2e(A) 1

49

_ A
C111. - J 31(Aa)3m(Aa) dA (35)

o 2h(A)

 

with Zh(A) = pc + pfcoth(pft) 2°(A) = Nfipc + pftanh(pft).

The real line integration as implicated above must be implemented with
due regard to the branch point and pole singularities present. To ac-
commodate the evaluation of the above integrals it is necessary to in-
voke Cauchey’s theorem for contour integrals [14]. This theorem along
with proper determination of the implicated branch cuts. allows for the

correct numerical evaluation of the above integrals.

4.4.1 COMPLEX PLANE SINGULARITIES

 

Analysis of integral representation (34) suffices to demonstrate
the procedure which can be utilized to evaluate both spectral integrals

(34) and (35).

The integrand in eq. (34) has a complicated functional dependence

 

on the wavenumber parameter pc=VlA - kélle + k;. It is evident that
branch cuts emanating from branch points at A=1kc are implicated by the
square roots in order to render the integrand in eq. (34) analytic.
(Branch cut singularities of pf are not implicated since the integrand
is an even function of this quantity.) Considering the media to be
slightly lossy with kc=ké-Jké’ and kc”>0 leads to the determination of
the standard hyperbolic branch cuts [5]. These branch cuts along with

the surface wave poles. apparent when 2h(A)=0 (TEo d pole) and Ze(A)=0

d

(TMeven pole), are depicted in Figure 3. The branch cuts are chosen to

50

satisfy the physical constraints which require waves to decay and propa-
gate outward from a source point. These conditions. which are consis-
tent with the exponential factors of the form e-pcy implicated through-
out this analysis, require Re{pc}>0 and Im{pc}>0.

When considering the resonance of the circular patch. though, it is
obvious that in order to have fields decay temporally due to radiation.
the imaginary part of w (and therefore kc) must become positive in order

to be consistent with the ert

time dependence. This results in a
"migration" of the branch cuts and poles [1] in Figure 3 across the real
axis. When this "migration“ occurs, the branch cuts to no longer sepa-
rate the proper Riemann sheets which guarantee Re{pc}>0 and the
Im{pc}>0; the natural resonant modes are consequently non-spectral leaky
modes. Although this seems contradictory, similar branch out analysis
has been used by Chew and Kong [1.2.15] which has produced reasonable
numerical results predicting characteristics of resonant structures.
Thus. the accepted and apparently correct configuration of the complex-
plane singularities implicated in the resonant circular patch analysis.
is illustrated in Figure 4. It is noted that the imaginary parts of all
the singularities are very small. These imaginary parts are exaggerat-

ed, and appear to be significant in the figures only for the purpose of

illustration.

4.4.2 PATH OF INTEGRATION

 

Since the integration path implicated in expression (34) passes
from zero to infinity, only the right hand side of the complex A plane

need be considered when determining the proper path of integration.

51

 

‘F
N!
y

tmp tep
+k x x

 

Figure 3. Standard hyperbolic branch cuts along with TB and TM surface
wave pole singularities in the complex A plane.

52

15

Figure 4.

tmp
+k x

 

Migration of singularities across the Ar axis.

53

Atep

\f

Also, the curvature of the hyperbolic branch cuts (see Fig. 4) may be
ignored. The Justification for this approximation relies on the fact
that the imaginary part of RC is small and resides only slightly above
the real axis.

The real line integration path specified in expression (34) must be
slightly deformed above the real axis. as illustrated in Figure 5, so
that it does not violate the migration paths of the singularities and
render the integral undefined [1]. In order to evaluate spectral inte-
gral (34) with the path defined as in Figure 5. it is helpful to con-
sider the path of integration illustrated in Figure 6. Since the inte-
grand in expression (34) is analytic everywhere within the closed con-
tour displayed in Figure 6. Cauchey’s integral theorem may be applied
leading to
Ioriginal + loo + lreal + |TE + ITM + Ibranch = 0 (36)
In the above relation, I‘denotes integration of expression (34) about
corresponding portions C:of the contour specified in Figure 6. In rela-
tion (360, law 0 due to the nature of the integrand (see Appendix C);
therefore, this result may be evaluated as
loriginal g - 'real — ITE - ITM - Ibranch' (37)
As observed in expression (37), evaluation of eq. (34) with the original
path of integration (Fig. 5) is equivalent to it’s evaluation with the

path of integration specified by the RHS of expression (37). The latter

path (HHS) turns out to be numerically superior to the former path

54

 

+k tmp

 

tep

 

 

Figure 5. Correct path of integration.

55

\f

 

Figure 6.

Coriginal

 

real

Deformed contour of integration.

S6

 

(LHS). The reason being that the residues. resulting from '11-: and ITM’
can be analytically evaluated (refer to Appendix D) leaving only the
real line integration and the branch out integration for numerical eval-
uation. This greatly reduces the amount of numerics involving complex
numbers.

The exact procedure employed in the present and the previous sec-
tions may be utilized to evaluate expression (35). The resulting path
of integration obtained for spectral integral (35) is the same as in
relation (37) with the exception of the TM pole contribution. This pole

is of no consequence since it is not implicated in representation (35).

4.4.3 NUMERICAL RESULTS FOR RADIAL MODE

 

Numerical results may be pursued for both the radial and circum-
ferential modes by numerically implementing eqs. (27) and (33). respect—
ively. Although resonant frequencies can be obtained for the radial
modes, none can be determined for the circumferential modes. The fail-
ure to determine resonant frequencies for these modes suggests that
these modes may be purely evanescent, non-resonating modes [1].

A FORTRAN program provides a numerical solution to matrix eq. (27),
yielding resonant frequencies for the radial modes. As stated earlier,
non-trivial solutions to the matrix eq. require a vanishing determinant.
This leads to a polynomial in the complex frequency w, roots of which
correspond to the various modes of resonance. These eigenvalues are
searched for in the complex w plane via secant method (which involves
choice of an initial ”0 that must be in the near proximity of the desir-

ed resonant frequency). For numerical convenience the resonant film

57

wavenumber,

ma 1
kfa = E_ where cf = -——————
f I "ocf

is actually determined.

Computation of the matrix elements involved in calculating the
determinant of eq. (27) is handled in several steps. First. for each
iteration of the secant method, the location of any surface wave poles
(Ap=Apr+JAp1) present must be determined (for all results only the TMo
pole is excited). The surface wave poles lie near the Ar axis in the

region between A=kC and A=kf (refer to Fig. 6). (For the remainder of

this section the cover wavenumber will be represented as kc=kcr+Jk

c1.)
A small region on the Ar axis (of width 26 centered about real part of
the pole) is excluded and integrated analytically (see Appendix E) to
avoid numerical overflow problems. Once the pole is determined. inte-

gration of the integrand in expression (34) along the intervals A=0 to

(C A=k to A=k

real) or ci (Cbranch)’ and A=kcr to A=Ap r.6 (C

real)
handled numerically via Rhomberg integration. The integration of the

integrand along the remaining interval. A=Apr +6 to A=m (Cire a1) is nu-
merically integrated step-by-step in full periods of oscillation of the
integrand. Again. Rhomberg integration is used for each step in this
last interval. Finally. analytic evaluation of the TMO residue com-
pletes the computation of the matrix elements.

Figures 7-9 illustrate plots of the normalized resonant film wave-
number. kfa, as a function of the normalized film thickness, t/a, for

the lowest-order radial mode. In each figure, the permittivity of the

cover is chosen to be that of free space. Film permittivities of cf=1.1

S8

 

    

 

 

3.9 - - 1.0

-i -

‘_ _ -

0 )—

‘g’ 37 1 -

3 ' - -0.8

g " +-

5 1 kno=(w.0/cf) _

3 3.5 - '

+u — _

S - -0.6

C - ..

a - _

on" 3.3 1 L—

_ -l (-

8 '1 )-O.4

a: -l _

.0 3.1 - _

a) - -

.t‘ -

'6 _ .-

E q -O.2

O 2'9 _ kfi0=(CI)IO/Cf> "

Z «I!
: b

2-7 T I I I l I I I T 1 1 1 FT [ I l I I 0-0
0.0 0.1 0.2 0.3 0.4
Normalized Film Thickness (t/a)
Figure 7. Normalized complex resonant wave-

number vs. normalized film thickness

for the lowest order radial mode.
(e.=1.0, 5.=1.1)

59

Normalized Imaginary Resonant Wavenumber

    

 

 

 

3.9 - r- 1.0
‘- _ ..
o n-
‘53 :57 -: ’
g ' - ~o.a
g '; kfr°=(wro/Cf) ’
i.
g _
3.5 — ‘
E - i-
o "* —O.6
C -i ..
g - -
(D 3.3 -*
m -‘ -
E u ~a.4
O: " r-
.O 3.1 - _
a) "l
.u - '
'6 3 b
E .1 -0.2
8‘ 2.9 .4 )-
z : MO" (“10/ Cr) '
fi ..
‘ r
2-7 I I I I l I I I I l I I I 1 l I I I I 0.0
0.0 0.1 0.2 0.3 0.4
Normalized Film Thickness (t/a)
Figure 8. Normalized complex resonant wave—

number vs. normalized film thickness

for the lowest order radial mode.
(ec=1.0, 87=2.65)

6O

Normalized Imaginary Resonant Wavenumber

 

 
    

 

 

3.9 71.0
h _ ..
O.) ..
‘5’ 37 -: -
3 - _008
g - k..a=(w.a/c.) ,
g _ -
3.5 - b
45 - i-
o ' "0.6
C " l-
8 - _
a) 3.3 -l
05 .,
8 - ~0.4
O: “ b-
.O 3.1 -: +-
6 _ )-
E - -O.2
$5 2.9 :1 +-
- kfi°=(wi°/Cf) I
2-7 4m I I I I 0.0
0.0 0.1 0.2 0.3 0.4
Normalized Film Thickness (t/a)
Figure 9. Normalized complex resonant wave-

number vs. normalized film thickness

for the lowest order radial made.
(5510, 559.6)

61

Normalized Imaginary Resonant Wavenumber

(Fig. 7), cf=2.65 (Fig. 8), and cf=9.6 (Fig. 9) are also used. Further,
three basis functions are found to be an adequate number for numerical
convergence.

As expected, in each of the three plots. the real part of the reso-
nant wavenumber (kf?) approaches the patch cavity [16] resonant wave
number (kfa=3.83) as the film thickness approaches zero. Also, the
imaginary part of the resonant wavenumber (kff). which corresponds to
radiation, tends toward zero as the film thickness diminishes. For

thicker film layers. though. the value of k ra decreases appreciably

f
from the value of kfa predicted by the simple patch theory. Also. for
these thicker film layers. it is expected that fields residing under-
neath the patch will radiate more causing kfia to increase. As observed
from the plots, this is indeed true. When larger values of film permit-
tivity are used as opposed to smaller values, the circular patch seems
to radiate less provided the thickness remains fixed.

For the lowest-order mode. Figures 10-15 illustrate convergence
plots of the normalized radial current magnitudes and corresponding
phases as a function of the normalized radial coordinate (p/a) for the
same three ef’s used above. For each cf, two convergence plots are
shown: one for a thin film layer (t/a=.025), and one for a thick film
layer (t/a=.3). As observed from all the plots. it is clear that numer-
ical convergence is in fact reached when three basis functions are used
(as was assumed above). Also, for a given thickness. the current
distributions for the various cf’s seem to remain nearly the same.

Finally, for the lowest-order mode, comparisons of the normalized

current magnitude determined from this analysis (IE solution) and the

current predicted by the patch cavity model (which is given by J1(kfa))

62

Magnfiude

 

 

 

 

 

1.0 "' -~ " \ ”-2.0
-1 \ )-
Z‘ \ I
.. \ F
0 8 4 \ b1.8
4 r-
3 ‘ ‘ '
3 d i b
’5 d ‘ l b I”?
a. 0.6 -+ ‘, ~1.6 g
g , 1 - ‘ - '6
1 ; / Phase \ - E
.8 - I, / \ ‘ I. V
.1: 0 ~ , / ‘ I- a
E a i / ‘E 25
o - ,’ I-
Z q I / _
.. ' / ‘ i
02 _ _ _ 1 Basis Function , l2
' / ..... 2 Basis Functions '
" 00000 3 Basis Functions
" / 4 Basis Functions
- / 3
0-0 I I I I I I I I I 1 I I Fl I I I I I 1-0

0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)

Figure 10. Current ma nitude and phase vs. patch
radial coor mate for lowest order radial
mode (e¢=1.0, e,=1.1, t/o=.025).

63

 

 

Magnflude

 

 

 

 

1.0 “ \ P2.0
.. L
0.8 4 L13
0) - ..
“a - -
£3 '7. 1' 13
5.0.6 w- - - - —1.6 g
o .— — — \-_ "6
2 _ ,1 r' o
1: _ .’ ‘ - 4:'
.g n ’ I... a
'5 0.4 4 ' / HA 0
g . / l. .15
O J " I)-
Z 1 , / _
02 : ' I _ _ 1 Basis Function 12
° , / ..... 2 Basns Functions l °
' DDDDO 3 Basis Functions
1 / 4 Basis Functions
0.0 I I I I I I I I I I T I I I I 1-0
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)
Figure 11. Current ma nitude and phase vs. patch

' radial coor mate for lowest order radial
mode (a¢=1.0. s,=1.1. t/a=.3).

64

 

Magnflude

 

 

 

 

‘LO - " \. -£L0
d \ -
_ \ 2
. \ _
0.8 - \ - 1.8
a) ‘ \ "
D q b
3 .. i A
.. ‘ '- m
§0.6 - _ ; ~. . “_ i 1.5 .5
2 1 / Phase \ r E
8 A I .‘ i )- V
.2 - / ‘~ l” a
O 0-4 'T I, / ‘ “1.4 O
E '1 / 1.. 0C.
1.
O —i I ‘-
2: - . / c
-‘ I, / i
02 _ _ _ 1 Basis Function 12
‘ " / ..... 2 Basns Functions '
‘ 00000 3 Basis Functions
3 / 4 Basis Functions
d / "

0'0 j T Tj T l I I rT T I f l T I 7 l l | 1'0
C10 (AZ (A4 CA6 (A8 1.0
Normalized Radial Coordinate (p/a)

Figure 12. Current ma nitude and phase vs. patch

radial coordinate for lowest order radial
mode (ec=1.0, 89:2.55, t/a=.025).

65

 

 

 

Mognfiude

 

 

 

‘LO - ,,__,. \, r-ZID
- r-
0.8 - ~1.8
q r-
0 .. _.
13 - _
3 A
o- q - w
5.0.6 -- —1.6 g
o -- - -, -- "6
2 d l ' o
I P ‘-
4: q 1’ h. 3
O 04 '4 ' / “-1.4 O
E - w / i. &
L
o - i”
2 fl / -
.. .. , l
02 _ _ _ 1 Basis Function ‘ 12
’ / ..... 2 Basrs Functions °
* DDDDD 3 Basis Functions '
n / 4 Basis Functions ‘
j r
0-0 T I I 1 I l I i I I l T I II | T I 1 1 1-0
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)
Figure 13. Current magnitude and phase vs. patch

radial coordinate for lowest order radial
mode (ec=l.0, 51:42.65, t/a=.3).

66

Magnitude

 

 

 

 

 

1.0 _ s a" \ ”2.0
d ‘ \ L
i \ I
-l \ l-
08 —l \ r- 1.8
l I -
o " '
'0 q \ -
3 - __ A
.- \ m
§O.6 _ _ \\ : \ -1'6 .5
2 - I / Phase ‘\ l' g
I I L.
'D _. I ‘ ‘ l- v
3 - / ‘ - o
._= I ‘ i In
D 0.4 — [I / ‘ "-14 2
E 'I / "' CL
0 "l I, ‘i'
z , / P
j '/ I
. _ _ 1 Basis Function ‘
0‘2 '1 / ..... 2 Basis Functions 1‘2
" D0000 3 Basis Functions
- / 4 Basis Functions
- / .
0-0 I I I I I I I I I I I I I I I I I I T I 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)
Figure 14. Current ma nitude and phase vs. patch

radial coor mate for lowest order radial
mode (s¢=1.0, 51:9.6, t/a=.025).

67

Magnitude

 

 

 

 

 

1.0 — . I’ \ 72.0
08 ‘4 \ - 1.8
'1 r
a, d \ l-
‘O .. \ ..
:3 a _ ”a
5.0.6 \ -1.6 5
O 01 ........ - _ -_ "l— :6
2 - ,’ / Phase ‘ g
B - ’ l V
'3' i x / " lr 3
O 0.4 " , / ‘l—1 4 E
E ' / . ’ 0-
O - 7' Il-
z _ / -
' / 1 B ' F t' I
_. _ OSlS unc Ion
0'2 j , / ..... 2 Basis Functions . 1'2
D0000 3 Basis Functions ‘
‘ / 4 Basis Functions ‘
0'0 I l I I I I I I I I T I I rT I I I I 1'0

 

0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)

Figure 15. Current magnitude and phase vs. patch
radial coordinate for lowest order radial
mode (s¢=1.0, 859.6, t/a=.3).

68

are made for thin and thick film layers. As seen from Figures 16 and
17, the patch cavity current is in good agreement with the current mag-
nitude predicted by the IE solution when the film layer is thin, but the
cavity current deviates appreciably from the IE solution current when
the film layer is thick.

The remaining figures depict results for some of the higher-order
modes. Figure 18 illustrates the resonant wavenumber as a function of
film thickness for the first higher-order mode. For this higher mode,
results converge well when five basis functions are used. Also, the
resonant wavenumber plot for this mode displays similar trends to the
wavenumber plots for the lowest-order mode. The current distribution
shown in Figure 19 demonstrates that five basis functions are sufficient
to yield numerical convergence for this higher-order mode. This seems
reasonable since spatial oscillations of the current have increased.
Figure 20 shows the current distribution for the second higher-order
mode with k a=9.56+J.269. Again, five basis functions are used.

f

4.5 SUMMARY AND CONCLUSIONS

 

The coupled IE’s for the resonant patch currents, developed in
Chapter III, decouple for the case of n=0. These decoupled IE’s, eqs.
(4.13) and (4.14), describe axially symmetric resonant modes. IE (4.13)
describes modes where the current sustained at resonance has only a
radial component (radial modes). IE (4.14) describes modes where the
resonant current has only an azimuthal component (circumferential
modes).

Applying Galerkin’s method to both IE’s leads to the following two

69

1.0

 

 

 

I / \
. / \
_ / \
“ \
0.8 -l / \
q /
~ \
-8 .. ’ \
.43 4 /
8, 0.6 - \
° * \
2 _ /
'o a \
g ~ \
'6 0.4 -— .
E 'l __ _ lE solution \
3 _ Patch cavuty model \
z
I \
0.2 - \
" \
- \
0'0 I I I I I I I I I I I I I I I I I I I I I

0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)

Figure 16. Current ma nitude vs. patch radial
coordinate or lowest order radial
mode (e¢=l.0, 552.65, t/a=.025).

70

1.0

 

 

: / \
d / \
: / \
0.8 a / \
.. / \
l; i / \
.52 - / \
C 0.6 ~
8‘ - I \
2 q / \
B - / \
-'_-“; 7 / \
g 0'4 A __ _ IE solution
B 1 / Patch cavity model \
z : / i
- / \
0.2 q / i
- I
I l
0.0 I I I I I I I I I I I I I I I I I I I I I

 

0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)

Figure 17. Current ma nitude vs. patch radial
coordinate or lowest order radial
mode (s¢=1.0, £52.65. t/a=.3).

71

 

 

 

L
5-, - C
.a 6-9 '2 _
E 3 k,,a=(w,a/cf) —o.s
g 6.7 :i .-
> ..
O .l i—
3 .. ..
H 6.4 1 L
E '3 -0.6
o 6.2 m
(I) - ..
6‘2 3 -
.. 5.9 -l '
8 : ~0.4
O: : ..
a _ -
= Z L-
g 5-4 ': —o.2
2 E ‘
5-2 I kn0=(wi°/Cr) ..
4-9 d f I I I I 1 I I I 1 I I I IT I I T I 0.0
0.0 0.1 0.2 0.3 0.4

Normalized Film Thickness (t/a)

Figure 18. Normalized complex resonant wave-
number vs. normalized film thickness

for the first higher order radial mode.

(8c=1.0. 8I=2.6

72

Normalized Imaginary Resonant Wavenumber

_ _ 3 Basis Function
..... 4 Basis Functions

 

 

 

 

 

 

 

Magnitude __ 5 Basis Functions
1.0 : \ ['20
_ / \\ :
- 1‘ \ \Phase L- 1.5
"1 | A b
0.8 -‘ ‘\‘\ I‘ / \ IL:
I \l i‘ / \ -_-1.0
g - ~\ ' I l :
'3 -i ‘1 | I rs - A
g 0.5 .l ‘1‘ i (’1’ \i \ :0.5 E)
O - l l' l’ ‘~ l : s
2 _ I l | I b D
.g - l E I \ . 8
.6 0.4 -' I l- O
2 - I I‘ I \ l-
- 1:.) I l '__1 0
0.2 -l i \t I '_'
-l \ :
" — —— _ L--1.5
4 l ..
J
0-0 IIITIIFIIIWIIIIIIIII[-2-0

0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)

Figure 19. Current ma nitude and phase vs. patch
radial coor mate for first hi her order
radial mode (sc=l.0, s,=2.6 , t/a=.03).

73

Magnfiude

 

 

‘10 -
l‘ .-
1 \ -2.8
q Phasel P
_ \ ..
l ..
‘l | I
(18 -‘ h
4 | I :13
a) ‘ \ '
'D . I -
8.0.6 -+ \ ~ g
0 - | \ ». :6
z -_ ._ “ r' e
U _ ~ '"’ \ :_ v
g ""l \ I ,. 0.2 a)
T3 04 " 8
' - - .c
-l l—
2 1 i _ -1.2
._ l l ..
0.2 ~ I
-l -2.2
q \ l
\ l
\
\l
0.0 I I I I I I I I I I 1 I I I I I I I I I F-3°2

 

0.0 0.2 0.4 0.6 0.8 1.0
Normalized Radial Coordinate (p/a)

Figure 20. Current ma nitude and phase vs. patch
radial coor mate for second higher order
radial mode (sc=1.0, 81:2.55, t/a=.1.).

74

homogeneous matrix equations

N
ZA B =0 1=0,1,2,...,N (4.27)
pm In
m=0
N
A C = 0 1 = 0,1,2.....N (4.32)
9m 1m
m=0

from which the resonant frequencies and current distributions for the
two types of modes may be obtained. Each matrix element of eqs. (4.27)

and (4.32) contain spectral inversion integrals; given as

 

 

 

i” (k: - A2)A A3lnic- 1) 3 ~
B = + (Aa)J (Aa) dA (4.34)
1'" 0 zhm zhmzem 1 m
m
A
c = 3 (Aa)? (As) dA (4.35)
In JO Zh(A) 1 m

that must be evaluated when solving these matrix equations. The proper
path of integration for matrix elements (4.34) and (4.35) relies on the
location of the surface wave poles and the correct choice of the branch

cut implied by the wavenumber parameter

p = VIA - k‘ VIA + k .
c c c

 

Imposing the physical constraint of a lossy cover region (with
kc=ké-Jké’ and ké’>0) which supports decaying outward propagating waves,

leads to the standard hyperbolic branch cuts emanating downward from

75

A=+kc and upward from A=-kc (refer to Fig. 3). But for temporal reso-
nance, ké’ becomes positive, and in order to sustain a continuous tran-
sition, the branch cuts and pole singularities "migrate" across the real
axis; therefore, the path of integration must be deformed so that it
does not violate the “migration“ paths of the singularities (refer to
Fig. 5).

Once the correct integration paths are chosen, eqs. (4.27) and
(4.32) are implemented numerically. The solutions to eq. (4.26) (radial
modes) yield reasonable resonant frequencies and current distributions.
Resonant frequencies for the circumferential modes, however, can not be
determined. This leads to the conclusion that there exists no pole
singularities corresponding to such modes.

The previously described determination of the branch cuts and inte-
gration contours for spectral integral (4.34) yields reasonable results
but is somewhat contrived. A more rigorous explanation may result from
investigation of the forward transforms of the potential functions used
in the derivation of the resonant IE’s described in Chapter III. For
example, the spatial scalar representation of the principal Green’s
function from Chapter II, given as

-chI?-?' I
c961?! ) = e

 

e 9
4nlr-r’l

may be Fourier transformed in the transverse directions (9 and Q direc-

tions) as

76

9 e
-chlr-r’l

§p(€.C.Y) a H ° {“5" * CZ) dxdz.

a 9
4nIr-r’l

 

For resonance, it has already been observed that kc=ké+3ké'. This al-

lows the above transform to be expressed as

I 'F_ I
-chlr :2 I +1“: I?_?I I
épte.c.y) = e e c e-‘ng * Cz’ dxdz.
_ 4n|?-I~"|

 

It is obvious from the above expression that the forward transform of
the principle Green’s function does not converge due to the growing
exponential factor. It is conjectured that if convergence of the above
forward transform is satisfied, the correct intrgration path will auto—
matically be determined when the inverse transform is taken. This argu-
ment may be extended to all the potential functions (since they have
similar propagation characteristics), in turn, avoiding the previous
difficulties of determining the correct branch cuts and integration

paths.

77

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

Integral-operator techniques provide a conceptually exact descrip-
tion of resonant structures in an integrated circuit environment. Un-
like most other techniques, the EFIE approach allows for the accurate
determination of complex resonant frequencies, which include the effect
of radiation damping. This technique proves useful for the analysis of
a circular patch resonator within the integrated conductor/film/cover
environment.

Chapter two pursued a somewhat general analysis of a arbitrary reso-
nant structure within the integrated environment. A detailed discus-
sion of the electric Green’s dyad for the tri-layered medium was review-
ed. Application of this dyad lead to a homogeneous IE describing the
natural resonance of the structure.

In chapters three and four, the previous Green’s dyad was appro-
priately represented in polar form, enabling it’s convenient application
to a circular patch. This lead to the development of two coupled homo-
geneous IE’s describing the various modes of resonance for the patch.
These IE’s provided analytic angular mode indexing. For the lowest—
order angular mode, the IE’s decoupled leading to independent IE’s
describing axially-symmetric resonant modes. Galerikin’s method with
Tchebychev polynomials lead to successful numeric determination of
resonant frequencies and current distributions for the radial modes.

Investigation of the solution for the first higher-order angular mode,

78

which corresponds to the dominant resonant mode of the circular patch,
is encouraged.

Also, in chapter four it was observed that numeric solution for the
resonant frequencies involved evaluation of spectral inversion inte-
grals. Due to the nature of the resonant structure, determination of
the correct integration path for these integrals became complicated and
lacked a rigorous approach. (Identical problems have also been con-
fronted in determining the leaky modes of an infinite microstrip wave-
guiding structure.) A rigorous determination of the aforementioned

integration path is a topic of research strongly encouraged.

79

APPEND I CBS

APPENDIX A

APPENDIX A
REPRESENTATION FOR Jb VIA SUNHATION THEOREM
The zeroth order Bessel function evident in expressions (3.7a) and

(3.7b).

 

JO(A|3-z'|) = JON/p2 + p’2 - 2pp’cos(u)) (A1)
may be conveniently expressed by invoking the "summation theorem" for
Bessel functions [13]. This theorem is briefly stated as

U
39¢ _ Jk¢
e Zv(mR) — Z Jk(mp)2v+k(mr)e

where

'1
V

O, p > 0, ¢ > O

 

R = Vfrz + p2 - 2pp’cos(u)

p < r, and O < w < E.
The quantities r, p, and R can be considered the sides of a triangle
such that the angle between the sides r and p is equal to p, and the

angle opposite side p is equal to w. With the specializations

80

 

and R = lz-B’l = lez + p’2 - 2pp’cos(u)

JO(AI3-3’I) may be written as

G
JouIB-B' I) = Z Jkupukup' )e

k=-w

3““. (A2)

Finally, by rewriting the exponential and utilizing the relation J_n(z)

= (-1)an(z) [10], eq. (A2) may be ultimately expressed as

O
JOUIB-B’ |) = JO(Ap)J°(Ap’) + Z Jk(Ap)Jk(Ap’ )cos(u)
k=1

where the restriction p’<p (p<r) is no longer necessary. This expres-
sion for JO(AI3-3’l) was implemented in modifying eq. (3.7b) to eq.

(3.8).

81

APPENDIX B

APPENDIX B

TCHEBYCHEV RECURRENCE RELATION

The recursion relation,

2uTn(u) = Tn_1(u) + Tn+1(U) (B1)
presented in section 4.3.1, introduces a potential problem for values of
n less than or equal to zero. For example, when n=O, n-i becomes nega-
tive, but Tchebychev polynomials are not defined for negative orders.
If this recursion relation is to be valid when negative orders are im-

plied, the following constraint must be imposed:

Tn(u) = Tln|(u) for n = -1,-2, ... [12] (82)

When restriction (B2) is implemented, the recursion relation produces

valid results for all integer values of n.
Using recursion relation (Bl) along with restriction (B2) allows

the term u(1 - u2)T2m+1(u), evident in eq. (4.21), to be conveniently

expressed. From this term it is evident that the quantities uT2m+1
3

and u T2m+1(u) must be determined. Applying relation (Bi) leads to

(u)

_ 1
uT2m+1(u) - 5 [T2m (u) + T2 m(u)] (B3)

_ 1
— Z [Tan—1‘“) + 2T2m+1 (u) + T2 MM]

82

3 1
u T2m&1(u) = § [T2m_2(u) + 3T2m(u) + 3T2m+2(u) + T2m+4(u)] (B4)

Combining expressions (B3) and (B4) yields
u(1 - u2)T (u) = l T (u) + T (u) - T (u) - T (u) (35)
Zm+1 8 2m Zm+2 Zm—Z Zm+4

It is observed that since expansion (4.19) begins with m=0, the order
of the Tchebychev in the third term of eq. (BS) may become negative;

therefore, restriction (BZ) must be imposed leading to

2 _1
u(1 - u )T (u) - § [T2m(u) + T

2m+1 (u) + T

2m+2 IZm_2|(u) + T2m+4(u)]. (86)

Expression (B6) is identical result presented by expression (4.22).

83

APPENDIX C

APPENDIX C

ASYHPTOTIC BEHAVIOR OF THE SPECTRAL INTEGRALS

For the radial mode, the matrix elements are given by expression

(4.34) as

 

+ C 3 (Aa)3m(Aa) dA. (4 34)

B =
zh(A) zh(A)ze(A) 1

1m

 

on 2 2 3 Z
J [ (kc - A )A A (Nfc- 1)p
o

By considering the behavior of the above integrand for large values of

A, the following approximations may be made:

(k: — A2)A z —A3

p1 = A - k1 x A

zhm

pc + pfcoth(pft) z 2A

e 2 ~ 2
2 (A) N Cpc + pftanh(pft) ~ MNfc + 1).

f

Also, with the assistance of the asymptotic expansion of Bessel func-

tions Jv with large argument, given by
Jv(z) w J 2/(nz) cos(z - 5; - %) ([10]. p. 364) (c1)

(2 represents a complex quantity) the terms 31 and in may be approx-

imated as

84

31(Aa) w % (sin and cos terms)

3m(Aa) z % (sin and cos terms).
Utilizing all the previous approximations, it is observed that the inte-

grand of expression (4.34) has the following non-convergent asymptotic

behavior for large A:

I z ——§:————— (sin and cos terms).

With this type of behavior it appears that the integral implicated by
(4.34) is divergent. This is not the case, though. A more detailed

inspection of 31 and 3m illustrates that the integral does converge.

The terms 31 and 3m assume the following generic form:

 

Jp(z) = [J1+2p(z) J1_2p(z) + sz+3(z) J_2p_1(z)
2 2 2 2
J1+|2p_2|(z) J1_|2p_2'(z)- sz+5(z) J_ %3(Z)] (C2)
2 2 2 22

where p=1,m. Considering 2 large in the first term and applying rela-

tion (C1) leads to

(Z) J1(Z) x 2- COS(Z - -[l—:—22] - —) cos(z - E[l_:_gg] — E)

J1+2p 12p uz 2 2 2 2 4
2 2

- E. - 32 EB

— nz sin(z 2) sin(z + 2). (C3)

Similarly, for 2 large the second, third, and fourth terms become

85

_ 2. EB - £2
J2p+3(z) J_2p_1(z) z uz cos(z + 2) cos(z 2) (C4)
2 2

2 n u
J1+|2p-Zl(2) J1_|2p_2|(z) z ;E sin(z ZIZP 2|) sin(z + ZlZp 2|) (C5)
2 2

2 - EB EB
J2p+5(z) J_2p_3(z) w E? sin(z 2) sin(z + 2). (C6)
2 2

Utilizing expressions (C3)-(C6). relation (C2) may be rewritten for

large 2 as
” ~ :3 EB - EB
Jp(z) ~ n2 [cos(z + 2) cos(z 2)

+ sin(z - %I2p-2I) sin(z + §|2p-2|)].

The three cases of p=0, p=1, and p>2 must be considered in the above

expression. Considering p=0 leads to

30(2) = %2 [cosz(z) + sin(z - g) sin(z + g)] = 0.
Similarly, p=1 yields
31(2) = fig [cos(z + %) cos(z - g) + sin2(z)] = 0
and p>2 yields
3p(z) = g; [cos(z + 3%) cos(z - Eg)
+ Sm: - 35;. +9.1“. + g- 9] = 0.

Thus, it may be concluded that for all 1 and m, 31(Aa) and 3m(Aa) tend

86

asymptotically to zero for large values of A; therefore, the integrand
in spectral integral (4.34) approaches zero for large values of A.(]m~0)
as was assumed in section 4.3. This renders the numerical integration
convergent.

Similar manipulations may be practiced on inversion integral (4.35).
which represents the matrix elements for the circumferential mode, lead-

ing to the same conclusions.

87

APPENDIX D

APPENDIX D

RESIDUE EVALUATIONS FOR
SURFACE WAVE POLES

Evaluation of spectral integral (4.34) about the TM pole contour

(CTM)’ depicted in Figure 6, demonstrates the procedure used for all

residue calculations for both spectral integrals (4.34) and (4.35). For

convenience, spectral integral (4.34) is expressed as

 

” F2(A)
BM] [w . . ]..
0 z (A)

 

where
k: - A2 ~ I
F (A) = -—————— AJ (Aa) (A3)
1 Zh(A) 1 “
A3(N§C - 1)pc ~
F2(A) = h 31(Aa)Jm(Aa).
z (A)

Considering only the integration about the TMeven pole leads to evalua-

 

tion of
J J FZ(A)
| = F (A) dA + dA. (D1)
T" 1 29(A)
c c
TN TN

Since F1(A) is everywhere analytic within and on CTM , the first term in

88

expression (D1) yields no contribution. Further, assuming CTM to be

limitingly small and F2(A) to be a slowly varying function of A in the

proximity of the pole, integral (D1) may be approximated as

dA

e .
c 2 (A)
TH

 

|TM z F2(Ap) J (DZ)

Considering the leading terms in a Taylor series expansion of Ze(A).

given as

e ~ e _ g_ e
Z (A) ~ 2 (AP) + (A Ap) 8A 2 (A) A=A
P

(where the first term vanishes by definition), allows integral (D2) to

 

become

I z F2(Ap) J d1 (D3)

TM e’ A - A

Z (A ) p
p c
TN

where

ze (A ) = g: Ze(A) .

p A=Ap

Considering the following local polar coordinates

A — A = cer

° is "
A dA = Jce‘w’dup

89

allows integral (DB) to finally be evaluated as

2

F (A ) F (A )
ITM = "—%7—2— I .3de = ZHJ —é'-L-
2 (AP) n 2 (AP)

90

APPENDIX E

APPENDIX E

ANALYTIC REAL LINE INTEGRATION
NEAR SURFACE WAVE POLES

When numerically evaluating spectral integrals (4.34) and (4.35).
the real line integration (refer to Fig.6 ) in the neighborhood of the
surface wave poles leads to potential numerical overflow problems.

These complications arise because the pole values reside very near the
real axis. To overcome this difficulty, the real line integration of
spectral integrals (4.34) and (4.35) in the proximity of the poles may
be analytically evaluated. Analytic evaluation of spectral integral
(4.34) near the TM pole (expressed as Ap=Apr+JApi) illustrates the gen-
eral procedure.

Considering only the real line integration and rearranging the inte-

grand of integral (4.34) yields

 

 

F2(A)
I = - [F (A) + ] dA (51)
real 1 Ze(A)
0
where

xi - A2 3 3
F (A) = A (Aa) (Aa)
1 Zh(A) I ‘"

A3(N§c - 1)pC ~ ~
F2(A) = J1(Aa)Jm(Aa).

Zh(A)

91

By considering a small portion of the real line, with a width of 28 cen-
tered at the real part of the pole, in expression (E1) leads to the eva-

luation of

8

F2(A)
Ip = - [F1(A) + ] dA.
c

 

Ze(A)

Assuming F1 and F2 to be continuous and slowly varying in the region of

interest leads to

8

dA
1 z -[ 28F (A ) + F (A ) ].
p \ 1 pr 2 pr I: Ze(A)

 

Approximating Ze(A) with the first two terms of a Taylor series expan-

sion about Ap as

e ~ e _ a e
2 (A) ~ 2 (Ap) + (A AP) 5A 2 (A)| =A (E2)

permits Ip to be expressed as

8

F (A )
I z -[ 2a? (A ) + —37—EE— d) ] (53)
P 1 pr e A - A

Z (Ap) c

 

P

where the first term in expansion (E2) is zero by definition and

e’ a e
2 (A ) = -— Z (A)
p 6A A=Ap

The remaining integral in expression (E3) is evaluated as

92

C 8

I = I dA = I dA
e A - Ap e(A-Apr) - JAP1

-1
ln[eJ[t' - Ztan (Api/C)]]

 

 

J[(in - 2tan-1(Ap1/c)]. (E4)

The proper choice of sign in expression (E2) is chosen to satisfy the

branch out of the in function (negative real axis). Since Api>0 and

arg(I) must reside between +u and -u, the plus sign is chosen. Thus, Ip

is evaluated as

F2(Apr) _1
Ip z -[ 28F1(Apr) + -—7———— Jln - Ztan (Api/e)) ].

e
2 (AP)

93

[1]

[2)

[3)

[4]

[5)

[6]

[7]

[8]

[9]

[10)

[11]

[12)

LIST OF REFERENCES

W.C. Chew and J.A. Kong, ”Resonance of the axial-symmetric modes
in microstrip disk resonators,” J. Math. Phys., vol. 21, no. 3,
pp. 582-591, Nov. 1979.

W.C. Chew and J.A. Kong, “Resonance of nonaxially symmetric modes
in a circular microstrip disk," J. Math. Phys., vol. 21, no. 10,
pp. 2590-2598, Apr. 1980.

Kiyomichi Araki and Tatsuo Itoh, "Hankel Transform Domain
Analysis of Open Circular Microstrip Radiating Structures,“ IEEE
Trans. Antennas Propogat., vol. AP-29, no. 1, pp. 84-89, Jan. 1981.

J.S. Bagby, D.P. Nyquist, and B.C. Drachman, "Integral formulation
for analysis of integrated dielectric waveguides," IEEE HTT-S
Trans., vol 33, no. 10, pp. 906-915, Oct. 1985.

R.E. Collin, Field Theory of Guided Waves, New York: Mcgraw Hill,
1960.

 

J.S. Bagby and D.P. Nyquist, “Dyadic Green’s functions for
integrated electronic and optical circuits," IEEE MTT-S Trans.,
vol. 35, no. 2, pp. 206-210, Feb 1987.

A.D. Yaghjian, "Electric dyadic Green’s function in the source
region," Proc. IEEE, vol. 68, no. 2, pp. 248-263, Feb. 1980.

M.S. Viola and D.P. Nyquist, "An observation on the Sommerfeld
integral representation of the electric dyadic Green’s function
for layered media," IEEE MTT-S Trans., vol. 36, no. 8, pp. 1289-
1292, Aug. 1988.

D.P. Nyquist, to appear in the proceedings of the URSI Triennial
Electromagnetic Theory Symposium, Stockholm, Sweden, August 14-17,
1989.

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions,
New York: Dover Publications, Inc., 1972.

 

S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields And Waves in
Communication Electronics, New York: John Wiley & Sons, 1984.

 

 

M. Blischke, Broadband Analysis of Radiating, Receiving and
Scattering Characteristics of Microstrip Antennas and Arrays, Ph.d
dissertation, Michigan State University, 1989.

94

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[14)

[15)

[16)

1.5. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and
Products, Orlando: Academic Press, 1980.

 

W.R. Lepage, Complex Variables and the Laplace Transform for
Engineers, New York: Dover Publications, 1961.

Weng Cho Chew, “Resonance Frequency of a Rectangular Microstrip
Patch,” IEEE Trans. Antennas Propogat., vol. 36, no. 8, Aug. 1988.

Y.T. Lo, D. Solomon, and W.F. Richards, "Theory and Experiment on

Microstrip Antennas“ IEEE Trans. Antennas Propagat., AP-27 (1979).
137-46.

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