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This is to certify that the

thesis entitled

ANALYSIS OF MICROSTRIP ANTENNAS 0N
SUBSTRATES WITH HIGH PERMEABILITY

presented by

L I LTON NATHAN I EL HUNT

has been accepted towards fulfillment
of the requirements for .

MS ELECTRICAL ENGINEERING

 

degree in

Major professor

 

7/29/02

 

MSUis an Affirmative Action/Equal Opportunity Institution

 

 

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University

 

 

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DATE DUE

 

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6/01 cJCIFICJDateDuopes-sz

 

 

 

 

ANALYSIS OF MICROSTRIP ANTENNAS ON SUBSTRATES
WITH HIGH PERMEABILIT Y

By

Lilton Nathaniel Hunt

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Electrical Engineering and Computer Science

2002

ABSTRACT

ANALYSIS OF MICROSTRIP ANTENNAS ON SUBSTRATES
WITH HIGH PERMEABILITY

By

Lilton N. Hunt

The art of miniaturization has become vital to a number of industries. The term
miniaturization is most synonymous with semiconductor technology. However, it has
become increasingly important for antennas to undergo substantial size reductions as
well, given the need to match the shrinking sizes of wireless devices. The Engineering
Research Center for Wireless Integrated Microsystems (WIMS) seeks such an antenna.
This antenna must be compatible with a sensor whose volume occupies a single cubic
centimeter, and the antenna’s volume must not exceed that of the sensor. In addition, this
antenna is to operate in L-band (1 - 2 GHz).

The most common miniaturization technique is the selection of a material with a
high relative permitivity, 8,. The largest dimension of the antenna can roughly be reduced
by the square root of 8,. However, increasing 8, results in decreased efficiency and
bandwidth. The use of a material that is magnetic for miniaturization is a fairly old idea,
but due to advances in material engineering, it is beginning to gain a new popularity. This
thesis investigates the usefulness of such a technique by way of computational
electromagnetics. The expectation is that increasing the permeability of the substrate will
reduce the necessary size of a loaded antenna without detrimental effects to the antenna’s

bandwidth or radiation properties and patterns.

Copyright by
LILTON N. HUNT
2002

ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. Leo Kempel, for all of his guidance,
support, and friendship. I also extend thanks to Dr. Pierre for all of his encouragement
and support. Thanks also to Dr. Rothwell and Dr. Balasubraniam for serving as
committee members in the defense of this thesis. I would like to thank Dr. Nyquist, and
again, Dr. Kempel and Dr. Rothwell for their great effort and skill in imparting the
knowledge of electromagnetics into their students. Again, thank you, Dr. Kempel and Dr.
Balasubraniam for your helpful suggestions and comments.

I want to give special thanks to my fiance Charisma A. Dixon, for all her support

and assistance throughout the development of this thesis.

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES

CHAPTER 1
INTRODUCTION

CHAPTER 2

CHARACTERISTIC EQUATIONS
2.1 Overview
2.2 Equation Derivation

CHAPTER 3
FEM VS. FDTD, FEM FORMULATION, AND FEM SOFTWARE
3.1 Overview
3.2 FDTD
3.3 FEM
3.4 Comparison FEM & FDTD
3.5 Finite Element Boundary Integral Computer Program
3.5.1 Matrix Entries
3.5.2 Magnetic Field Terms
3.5.3 Electric Field Terms

CHAPTER 4
RESULTS AND ANALYSIS
4.1 Overview of Patch Antenna Characteristics
4.1.1 Substrate Thickness
4.1.2 Patch Width
4.1.3 Substrate Permitivity
4.1.4 Probe Feed Location
4.1.5 Substrate Permeability
4.1.6 Note on Measurements
4.2 Results and Analysis Overview
4.3 1.0 by 1.0 cm Patch Antenna

4.3.1 1.0 by 1.0 cm Patch Antenna with a, = 7 and II, = 20
4.3.2 1.0 by 1.0 cm Patch Antenna with 6,: 7 and u, = 25

4.3.3 1.0 by 1.0 Patch Antenna 8,: 9 and pr = 20
4.4 1.0 by 0.9 cm Patch Antennas

4.4.1 1.0 by 0.9 cm Patch Antenna with E, = 7 and u, = 20
4.4.2 1.0 by 0.9 cm Patch Antenna with E, = 7 and u, = 25
4.4.3 1.0 by 0.9 cm Patch Antenna with E, = 9 and u, = 20

4.5 1.0 by 0.7 cm Patch Antenna

vii

viii

l9
l9
19
23
31
34
39
41
48

53
53
54
54
55
55
57
58
58
59
59
73
83
93
93
105
112
118

4.5.1 1.0 by 0.7 cm Patch Antenna with a, = 7 and u, = 20
4.5.2 1.0 by 0.7 cm Patch Antenna with e, = 7 and u, = 25
4.5.3 1.0 by 0.7 cm Patch Antenna with 8, = 9 and it, = 20

4.6 Conclusion of Analysis
4.6.1 Comparison to an antenna with ur = 1
4.6.2 Feed points and Resonance
4.6.3 Real Impedance Bandwidth and Magnitude
4.6.4 Radiation

CHAPTER 5
CONCLUSION

BIBLIOGRAPHY

vi

118
124
137

143
143
144
145
146

150

153

LIST OF TABLES

Table 3.1 ............................................................................................................................ 38

vii

LIST OF FIGURES

Figure 4.1 ....................................................................................................................... 63
Figure 4.2 ....................................................................................................................... 64
Figure 4.3 ....................................................................................................................... 65
Figure 4.4 ...................................................................................................................... 66
Figure 4.5 ...................................................................................................................... 67
Figure 4.6 ...................................................................................................................... 68
Figure 4.7 ...................................................................................................................... 69
Figure 4.8 ...................................................................................................................... 70
Figure 4.9 ...................................................................................................................... 71
Figure 4.10 ..................................................................................................................... 72
Figure 4.11 ..................................................................................................................... 76
Figure 4.12 ..................................................................................................................... 77
Figure 4.13 ..................................................................................................................... 78
Figure 4.14 ..................................................................................................................... 79
Figure 4.15 ..................................................................................................................... 80
Figure 4.16 ..................................................................................................................... 81
Figure 4.17 ..................................................................................................................... 82
Figure 4.18 ..................................................................................................................... 85
Figure 4.19 ..................................................................................................................... 86
Figure 4.20 ..................................................................................................................... 87
Figure 4.21 ..................................................................................................................... 88
Figure 4.22 ..................................................................................................................... 89
Figure 4.23 ..................................................................................................................... 90
Figure 4.24 ..................................................................................................................... 91
Figure 4.25 ..................................................................................................................... 92
Figure 4.26 ..................................................................................................................... 96
Figure 4.27 ..................................................................................................................... 97
Figure 4.28 ..................................................................................................................... 98
Figure 4.29 ..................................................................................................................... 99
Figure 4.30 ................................................................................................................... 100
Figure 4.31 ................................................................................................................... 101
Figure 4.32 ................................................................................................................... 102
Figure 4.33 ................................................................................................................... 103
Figure 4.34 ................................................................................................................... 104
Figure 4.35 ................................................................................................................... 107
Figure 4.36 ................................................................................................................... 108
Figure 4.37 ................................................................................................................... 109
Figure 4.38 ................................................................................................................... 110
Figure 4.39 ................................................................................................................... 1 11
Figure 4.40 ................................................................................................................... 114
Figure 4.41 ................................................................................................................... 115
Figure 4.42 ................................................................................................................... 116
Figure 4.43 ................................................................................................................... 117

viii

Figure 4.44 ................................................................................................................... 120

Figure 4.45 ................................................................................................................... 121
Figure 4.46 ................................................................................................................... 122
Figure 4.47 ................................................................................................................... 123
Figure 4.48 ................................................................................................................... 127
Figure 4.49 ................................................................................................................... 128
Figure 4.50 ................................................................................................................... 129
Figure 4.51 ................................................................................................................... 130
Figure 4.52 ................................................................................................................... 131
Figure 4.53 ................................................................................................................... 132
Figure 4.54 ................................................................................................................... 133
Figure 4.55 ................................................................................................................... 134
Figure 4.56 ................................................................................................................... 135
Figure 4.57 ................................................................................................................... 136
Figure 4.58 ................................................................................................................... 138
Figure 4.59 ................................................................................................................... 139
Figure 4.60 ................................................................................................................... 140
Figure 4.61 ................................................................................................................... 141
Figure 4.62 ................................................................................................................... 147
Figure 4.63 ................................................................................................................... 148
Figure 4.64 ................................................................................................................... 149

ix

CHAPTER 1

INTRODUCTION

This thesis is divided into five chapters. The first chapter will give an overview of
the motivation for this project and its goals. The second chapter will cover the derivation
of several microstrip patch antenna equations; equations which account for materials with
a permeability constant significantly greater than one. These equations include: electric
and magnetic field equations for TMZ mode on a patch antenna, resonant frequency, the
far-field equations for the antenna, and the Q of the antenna. The third chapter will
explain the choice to use Finite Element Method software and give a brief discussion on
its fundamentals. This chapter will also give a brief overview of the specific FE-IB Prism
software used in this research. The fourth chapter will present the results and analysis of
the data acquired from Prism. Finally, the fifth chapter will conclude the research and its
findings.

The art of miniaturization has become vital to a number of industries, as many
engineers, scientists and their employers have embraced the craft in a race to produce the
finest and tiniest electronic devices. The term miniaturization is most synonymous with
semiconductor technology. However it has become increasingly important for antennas to
undergo substantial size reductions as well, given the need to match the shrinking sizes of
the products that require their service. The wireless revolution is a major source of the
growing demand for small antennas. More and more of the wires we employ on a daily
basis will disappear. The success of this massive transformation depends on the ability to
make antennas that are inconspicuous, efficient, consume little power and function over

various frequency bands.

The Engineering Research Center for Wireless Integrated Microsystems (WIMS)
seeks such an antenna. This antenna must be compatible with a sensor whose volume
occupies a single cubic centimeter. Consequently, the largest dimension of this antenna
must not exceed one cubic centimeter. In addition this antenna is to operate in L-band (1
- 2 GHz, preferably the lower end of L-band. Typically an antenna’s largest dimension is
on the order of one-half a wavelength or larger, as this is conducive to efficient
transformation of a bound wave into a space wave. Therefore applying this guideline in
our scenario would result in antenna whose largest dimension is 15 cm., hence the
challenge represented in this work.

The design of an antenna, which fits the above size constraints and possesses
suitable bandwidth and aperture efficiency, is the goal of this project. The use of
computational electromagnetics to obtain this goal is the focus of this thesis. An antenna
solution will be obtained by way of Finite Element Method (FEM) of analysis.

As stated earlier the antenna solution calls for the most efficient antenna
occupying a volume no greater than one cubic centimeter and operates in L-band; more
specifically at 1 GHz. There are some well-known antenna types at our disposal, antennas
that are generally employed because of their size or flexibility as it pertains to size.
However many of these small structures are notorious for very low gain, low radiation
efficiency and high reactive input impedance as a result of not being resonant. Since size
is a primary factor in our design goals, the complex matching networks that would be
required to match an antenna with a large reactance to a transmission line is undesirable.
The printed class of antennas, which include the microstrip patch antenna, can be built for

resonance, and at resonance there is no reactive input impedance. In addition its size can

be manipulated by several methods; one of which is the selection of a substrate with
certain constitutive parameters. It too has small gain, yet markedly larger than that of the
short dipole.

The most common miniaturization technique is the selection of a material
with a high relative permitivity, 8,. The largest dimension of the antenna can roughly be
reduced by the square root of 8,. However increasing 8, results in decreased efficiency
and bandwidth. The electric fields become increasingly bound to the substrate and thus
the antenna will begin to look more like a resonator than an antenna. The decrease in the
radiation efficiency would also indicate an increase in input impedance. There are a
number of other size reduction techniques, most of which result in considerable
reductions in efficiency.

As a result of new developments within the material science community, the use
of magnetic material for antenna miniaturization may become a reality. The significance
of such a reality stems from the theory that increasing the material's permeability would
reduce the size requirement of an antenna resonant at a certain frequency, without
decreasing the antenna’s bandwidth. The expectation is that the antenna size can be
reduced by a factor of the square root of the material's permeability constant; just as the
antenna size can be reduced by roughly the square root of the material's permitivity
constant. Also the permeability is expected to counter the permitivity’s detrimental

effects on the antenna's radiation properties and impedance.

CHAPTER 2

CHARACTERISTIC EQUATIONS

2.1 Overview

The objective of this document is to analyze an antenna on a magnetic substrate
using FDTD and FEM computational electromagnetic techniques. This study is part of a
project to design an antenna with a volume that'will not exceed one cubic centimeter and
has a resonant frequency of 1 GHz. The number of antenna types and configurations
whose general characteristics lend themselves to the possibility of fitting these
preliminary specifications are few. As discussed in the introduction, a printed antenna of
some type seems the best choice for our goals. A rectangular patch is chosen for the
initial design, since its simplicity will more quickly educate us to the challenges and
difficulties associated with the task at hand.

Having decided on an antenna type, it is necessary to employ mathematics to
paint a preliminary picture of the antennas performance, revealing some of the necessary
adjustments that will have to be made to reach a desirable solution. Equations that model
our patch antenna must be obtained. These equations and the outline of their derivation

are provided in the remaining pages of this chapter.

2.2 Equation Derivation

The Microstrip Patch Antenna (MSA) can be modeled as a dielectric loaded
cavity with four side walls that are ‘Perfect Magnetic Conductors’ (PMC) and a top and
bottom wall that are Perfect Electric Conductors (PEC). The modeling of the sidewalls as

PMCs is made possible by the thinness of the antenna’s substrate. The shorter the

substrate height the smaller the current that is flowing from underneath the patch metal to
the top of the patch metal.. This current flow is nearly zero for thin substrates. Thereby
the tangential magnetic field components created by this current are nearly zero. The
lack of tangential magnetic fields along these peripheral walls is the defining
characteristic of a PMC. [1,3].

Again as a result of the substrate thinness the fields inside the cavity do not vary
much in the direction normal to the patch, the z-direction in this document. Consequently,
only the normal component of the E-field is of any significance. Since the Electric field
has only the 2 component and the Magnetic field has no z-component (by virtue of the

boundary conditions for the PMC walls) we consider only the TMZ mode of operation.
(The Magnetic field does have 52 and 5’ components in the region bound by the PECs).

To derive the field expressions, that are TMz, we may define the magnetic vector
potential such that it has only a component in the z-direction, A = x Az(x, y, z)

The magnetic vector potential must satisfy the scalar wave equation:

VZAZ+ szZ = 0 =>

32 32 32 2
(2.1) —-Az+—A +—Az+k AZ =0
8x2 ay2 2 322

The separation of variables method will be used to determine solutions for the vector

wave equation.

Assume (1) has a solution of the form :

(2.2) A2 (x.y.2) = f (x)g(y)h(z)
and insert (2) into (1), the latter then becomes:

d2 d2 dzh
git-£2: +fh-d—xi£+ngX'—2-+k2fgh=0

Now, both sides are divided by fgh

(the arguments are assumed and suppressed for clarity)

_1_de +1d2g+1d2h
fdxz 3de hdx2

each of the above terms is a function of only one variable. Hence, the

sum of these must total -k2 only if each term is in turn equal to some constant.

Therefore, the original coupled equation becomes three decoupled eqautions:

2 2 2
£_f=_k3 iii=_k2 _l_d_h__ k2

1
(2'3) 7 dx2 8 dx2 y T z

haz—

related by the consistency relationship: k2 + k2 + k2 = -k2
x y z

the solutions to (3) can take on different forms. All the forms that applicable to our

model must represent standing waves in all three spatial coorrdinates. So we choose:

f (x) = Acos(kxx) + B sin(kx)

g(y) = Ccos(kyy) + Dsin(kyy)

h(z) = Ecos(kzz) + F sin(kzz)
therefore (2) becomes:
A2 = [A cos(kxx) + B sin(kx )][C cos(ky y) + Dsin(ky y)][E cos(kzz) + F sin(kZ 2)]

The electric and magnetic fields are related to Az by :

2 2
13,=--j——l a —+k2 A Ey=—j——-l 8A2
was 3Z2 awe ayaz

 

 

2 2
(2.4) E =—j—l—[—a—+k2]Az E =—j——l—a AZ

 

 

60/15 322 y ("#3 ayaz
2 2
H,,=—l-a AZ Hy=--1—a AZ Hz=0
# 3y .11 fix
with BC.

Ey(x,y,z=0)=Ey(x,y.z=h)=0
(2.5) Hx(x,y=0,z)=Hx(x,y=w,z)=0

Hx(x=0.y.z)=Hx(x=L,y,z)=0

Applying boundary condition: Ey (x, y, z = 0) = By (x, y, z = h) = 0

2
with E), =—j—l 8 AZ
(qua aydz

 

we have :

= [A cos(kxx) + Bsin(kx)][- C sin(ky y) + Dcos(ky y)][—E sin(kzz) + F cos(kzz)]

at the bottom of the cavity (2 = 0):

E), =T[-E(0) +F(1)] =

where T = [Acos(kxx) + Bsin(kJr )][— C sin(ky y) + Dcos(l’cy y)]
'. F = 0

and at the top of the cavity (2 = b):

By = T[-Esin(kzh)] = 0

Similarly, applying boundary conditions on Hx and Hy;

3A,,

1
H:—
u 3y

 

Hx = i[Acos(kxx) + B sin(kxx)][—C cos(ky y) + D sin(ky y)] at
,u
o [Ecos( kzz)+ Fsin( kzz)]
At the left wall (y = 0):

Hx = Ill-Tram + D(l)]U
where T = [Acos(kxx) + Bsin(kxx)] and U = [E cos(kzz) + F sin(kzz)]

D=0

and at the right wall (y = w):
Hx = —l-T[-C sin(kyw)]U
,u

-. k =-“-E n=0,l,2,3....

Similarly from the BC. at x = 0 and x = L:

the consistancy relationship kz2 + k )2 + kl2 = k2,

and the values of k2 , kg, and kg found above, we have
2 2 2
(2,6) (M) ,(M) {22) __. k2
h w L
where the operator wavenumber squared is given by

k2 = (02,118

The frequency of resonance for the cavity is given by

2 2 2
1
(2.7) (f,)mnp=2” ”Ex/(11;!!!) +[%) {£5}

The primary interest in this work is for antenna operation in the lowest order mode since

 

 

the minimum physical dimensions are required and the radiation pattern is often nearly
omnidirectional

For TMom, the lowest order mode, we find

(2.8) Az = Aomcos(kxx)

(29) E -——1—- -a—2-+k 2 A cos(k x)-E cos(zr-x)
- z jwpe 82:2 z 001 x o L

where E0 = —ij001

(2.10) Hy =711-5—xA001cos(kxx) = H0 sin(kxx)

and

(2.11) By =13z =11x =Hy =0

To derive the far-field equations we recognize that the four magnetic walls
represent four narrow apertures through which radiation takes place [1]. We apply the
“Equivalence Principle” which states that the fields outside an imaginary closed surface
are obtained by placing over the closed surface suitable electric and magnetic current
densities which satisfies the boundary conditions. The current densities are selected so
that the fields inside the closed surface are zero and outside they are equal to the radiation
produced by the actual sources. Thus the technique can be used to obtain the fields
radiated outside a closed surface by sources enclosed within it [9].

The equivalence principle in general yields equivalent electric and magnetic
source densities Js = n x H and MS = - n x E, where E and H are the fields over the
closed surface. Since the cavity is modeled by magnetic conducting walls the tangential
H-field is ideally zero and therefore ,1, = 0. It is now apparent that the radiation at each
MSA aperture (wall) is due to the magnetic current density Ms. The magnetic current
density is doubled in the MSA since it is flowing above a good conductor, M5 = -2n x E.

Two of the four slots will be the primary source of radiation in the lowest order
mode, namely the slots separated by the greater length. The fields radiated by the other
two slots interfere destructively (total destructive interference) with each other broadside
and the fields they generate in non-principle planes are negligible. Previously we found
the electric field at the radiating slots to be E2 = E0 cos(nIJx) where E0 = -ij001.
Therefore Ms = y2Eo cos(nL/x). The far-fields will now be found, following a method
found in [1]. First we write the electric and magnetic vector potential while utilizing the
far field approximations:

R =— 1' - r'COS(0 for phase variations R = I‘ for amplitude variations.

10

The potentials are then:

 

 

 

 

-ij -ij
.11 e . #6
2.12 A = — ds = N
( ) MUS J3 R 47rr
whereN = 115 Jse'jkr'coswds'
-ij -ij
.0 e . #6
2.13 F = — M d = L
( ) WINS S R s 47tr

where L = ‘HS MSe-jkf'COS¢ds.

but we found earlier that J s = 0 so A and N are immaterial in this
model. So we are left with F and L

L = I IS ( $er + my +2Mz)c'jk"°°S¢

9 = icos 6cos¢ + ycosflsint) - isin t9

'. L6 = “S ( chost9c05¢+ Mycosesin¢+ Mzsin me-jkr'cospds.

We have only an My component

L" 1'1
'. L9 = I‘i". Ii”- (Mycosé’sinrb) e'Jkr'cowdy'dz'
2 2

where h = height & w = width

since

i" = xsinélcostt) + ysinflsin¢ + icosB
and

r'cosgp = r'0 i" = (yy' + 22') 0 (isinflcosq) + ysinasina + icosB)
= y'sint93in¢ + z'cosB

ll

we have :
fit. :1
= 0039811147} I-Zh [3” My e'Jk y'srn651n¢ + z'cost9dy 'dZ'

2 2

L9

. a
:9. . srn(—C)
using Ifc edez = C
— —C
2 t 2 _

 

 

we obtain:

(2.14) L6 = thy [cosOsin¢(Sin YXSi“ 2)]

 

Y Z

where Y = E—E-sinflsing) Z: %cost9

Considering the ¢component:

¢ = -§tsin¢ + ycos¢

..L, = fl {—Mx sin ¢+ Mycos¢] (9"me ds'
S

and once again we have only an My component

h w
L¢ = [3h [3W [My cos ¢] elk" “’5‘” dy dz.
2— T
h z
= COS¢j—2h I3”, [My COS¢1ejkr(y'sm651n¢ + z'cosB)dy'dZ'
T2— T2—

12

. a
E . srn(—c)
again using L2 e’azdz = c

 

9’.
2 2
we obtain:
sinY sinZ
(2.15) L¢= thy[cos¢( Y )( Z )]

Generally the total field is given by:

E=-ij-j——1-V(V- A) - leF
was 8

However in far-field, this expression may be approximated by
E = -ja)A - 77? x H=~ij+jamfo _

The total field is given by

E = jamf- x F

with spherical components

(E199 = -jamF¢

(512%) = ijFB

 

 

 

 

13

and

jke'jk’
4m

 

5;»

La

Substituting in the equations for (14) and (15) found above, we have:

'hWkM 1"“ . ,
(2.16) Ea J ye [cos¢(smY)(smz)]

 

 

 

 

 

 

 

 

4m Y z
and
thkMye'j‘“ . sinY sinZ
(2-17) 13¢ = cosflsrngz)
47rr Y Z
where
2V
Y = gsinflsingb Z = 51156053 My = 2E0: ho
and
since koh << 1 flzl
Z
we have:
. kW . .
- Wk E -jkr srn(—s1nt9$1n (11)
(2.18) E3 = Jh 2° 0e cos¢
7” —sint9$in¢
2
'V e-jkr simflsinfisin ¢)
= .J__£__ cos . _
7! r srn631n¢
and

14

. kW . .
-jkr srn(-— srn 65m (1))

(2.19) 13¢ = J—VO—e— cosasin¢ , ,
fl'I' srnfisrn¢

 

We have solved for the far-field equations for a single radiating slot. Our model consist
of an array of two radiating slots separated by a length L. Thus the total far zone electric

field, with array factor, (to account for the second slot)

(2.20) AF = 2cos(k—;’:sinacos¢)

 

 

is given by:
. kW . .
- V -jkr srn(—s1n081n¢)
(2.21) 1330‘“ = $93— cos¢ 2 . cos(ko—Lsinficos ¢)
7! r srnfisrna 2
. kW . .
. V .jlq srn(——srn6?srn (0)
(2.22) E50”! = i2——°e— cos 63in ¢ 2_ _ cos(filisin 0005(1))
7: r srnflsrn¢ 2

In the principle planes:

E-plane ¢ = 0:

. V -jkr
(2.23) 1330““ = J—2—-<L-cos(1“;—Lsin0)
7! 1'

(2.24) BTW“ = 0
H-plane o = 90:

(2.25) 1350““ = 0

15

simfl sin a)

 

. V -jkr

2.26
( ) 7: r sinfi

Next we will derive the Quality factor (or Q), which will give insight into the frequency
bandwidth of this antenna.

Time-average energy stored at resonant frequency

 

(227) Q = 27! . . . .
Energy drssrpated in one period of resonant frequency
__ 2W,3 _ _f_
PL Af

where Af is the Half-Power Bandwidth

for our antennas dorninat mode, TMOOI , the E-field is (2.9) :

(2.9) E2 = E0 cos(iix)

V
where E0 = -jCUA001 = TO-

The total time average stored electric energy at resonance is:
2
(2.28) w. = 54L] |Ez| dv

We = 1% if i i 13% cos2 (E x) dxdydz

L
= iwhEg [Ax - Lsin(-2—72(—):|
4 2 47: L

0
8r 2 L L. L- }
=—whE —-—srn2fl —O-—s1n0
. all. 4,. < M 4.. “l
=£WhLEg
4

l6

the energy dissipated in one period of resonant frequency is:

(2.29) PL = mod|133|2 +wptan§mIH|2 dV + zfllzs|H|2 ds
V S

where tané‘m is the alternating magnetic loss tangent

The first term in PL is due to conduction and dielectric losses. The second term is due to

magnetic losses and the last term is the loss in the conductors that make up the top and
bottom faces of our cavity.

Integrating the first term:

2

 

 

L
IIJUJ IEIZ = hWO’dI E0 COS(£x) = hWEUdEg
L 2

V 0

Integrating the second term
2 L ,r 2 L
”I wptancsm IHI = thjutané‘mJ' H0 sin(-I:x) = hWEwfltanJmHg
V o

 

 

and the last term:

ZHRSIHIZ ds = WLRSHS
S

(W
20'c

 

where R5 =

Adding the terms we obtain the total power loss:

(2.29) PL = %[h(adr~:g + amamng) + 211mg]

Utilizing the We and PL found above, Q becomes:

2
wehE
(2.30) Q = r 0

 

Mada?) + wptandeg) + 2115113]

17

from which can obtain the bandwidth:

f0
2.31 A = —
( ) f Q

also from [4] we have Input Impedance:

 

—1
(2,32) zin = é = [PrllePd]

d
where V = h - (EZ averaged over the feed segment) = a—t—J‘Giz) dl
_c

and dl is over the feedline with width (d-c)

the characteristic Impedance is given by:

(2.33) 20 = J2
8

The radiation power is a well-known formula:

r- -I

(2.34) P, = Re (13611;, -E¢H;)r2sinad¢ d6

O'——.NI>~1

 

 

18

CHAPTER 3

FEM VS. FDTD, FEM FORMULATION, AND FEM SOFTWARE

3.1 Overview

Both FDTD and FEM electromagnetic computational techniques were considered
for use in this research. The following will provide basic descriptions of each method’s
characteristics, and explain why, FEM was chosen.

Models used to approximate the behavior of a rrricrostrip patch antenna make a
number of simplifying assumptions. These are assumptions make mathematical modeling
possible with closed formed eigenfunctions. The models also provide closed-form
expressions for wall admittances and often use an add-on approach to account for
phenomenon like space and surface wave radiation and mutual coupling as in the case of
Finite Element Method (FEM) combined with the Integral Equation method. However,
because of the simplifying assumptions, there are limitations on what can be modeled
with reasonable accuracy. For instance, when modeling microstrip antennas, there is a
limitation on how thick its substrate can be before accuracy begins to degrade. This is
because the model assumes that the E-field is constant along the height of the structure,
and therefore, thin substrates are needed. The substrate should be less than a tenth of a

wavelength for both Finite Difference Time Domain (FDTD) method and for FEM. [3].

3.2 FDTD
The Finite Difference Time Domain method for Electromagnetic field problems is
useful for solving scattering problems. It is a direct solution of Maxwell’s time-dependent

curl equations and it treats the irradiation as an initial value problem. FDTD algorithms

19

do not require the formulation of integral equations, and relatively complex scatters can
be treated without the inversion of large matrices. It is simple to utilize in analysis of
inhomogeneous conducting or dielectric structures because constitutive parameters can
be assigned at each lattice point.

In order to analyze a structure with FDTD the structure must first be divided into
various regions based on the material properties. Secondly, the unbounded region of the
problem (if any) must be bound by terrrrinating it with an absorbing medium (or other
termination such that reflections do not occur). This is because if the region in which we
wish to compute the E-field is open (unbounded), the amount of computer memory
necessary to store the resulting data collected over an entire domain would be
impractical. Therefore, artificial boundary conditions are enforced to create the numerical
equivalent of a large or infinite space. The new region, smaller and bound with artificial
conditions, is the solution region. Though smaller, the solution region must remain large
enough to enclose the scatterer and enough additional space to facilitate accurate
modeling of the region extending into space. [6] Next, the problem’s physical space is
discretized in the form of cuboids of size AxAyAz. The time domain is also discretized
with interval size At. The structure is then excited by an electromagnetic pulse. Finally,
the wave launched from the structure is studied, and the stabilized time-domain
waveforms are numerically processed to obtain the time-domain and frequency-domain
characteristics of the structure. [3]

Finite Differences Method is based on approximations, which allow
differential equations to be replaced by finite difference equations. These difference

equations are any approximation of a derivative in terms of values at a discrete set of

points. They are algebraic in form and relate the value of the dependent variable at a
point in the solution region to the values at some neighboring points.

One approach in obtaining finite difference approximations is to use the Taylor
Series expansion to approximate a derivative. Using more terms in the Taylor Series
expansion can increase the accuracy of the approximations. [6] As the number of terms
approaches infinity, the approximation approaches the exact result. However the use of
terms in an expansion, which are of higher order than the highest order term in the Partial
Differential Equation (PDE) being approximated, may cause instability and introduce
spurious solutions. In general, the infinite series is truncated after the second order term.
This truncation is good for stability; however it leads to approximation inaccuracies. The
source of these inaccuracies is called truncation error O(Ax).

The three most common sources of errors in computational electromagnetics are
truncation errors (mentioned previously), modeling errors or discretization errors, and
round-off errors. Round-off errors are due to the limited size of registers in the arithmetic
unit of the computer. Computations can only be done with a finite precision on a
computer. This error may be increased by use of a finer mesh, since the number of
arithmetic computations will be increased. Using finer meshes, on the other hand can
reduce truncation and discretization errors, but not indefinitely.

There is optimum mesh fineness for the lowest error in a particular algorithm.

For accurate results, the spatial increment 9, used must be small compared to the

wavelength A (in general g 32/10). This is equivalent to ten cells per wavelength. In

order for a finite difference scheme to be stable, the time increment At must satisfy the

condition: [6]

21

where v max is the maximum phase velocity

Vmax(AI)Sl:-—l-2- + -—1—i— +—-—l—2:l yz
Ax Ay AZ
within the model.

The concern about accuracy leads to a concern about the stability of the finite
difference solution, which is whether it can grow unbounded. A mathematical algorithm
is said to be stable if a small error at any stage produces a smaller cumulative error.
Otherwise, it is unstable. Also, if the algorithm increases the magnitude of the
approximated solution with increases in time, it is not stable.

The FDTD technique is very efficient for finite sized antennas, which are
antennas that do not rest on infinite or effectively infinite ground planes. The effect of
finite size is less severe on microstrip antenna impedance behavior because microstrip
antennas are inherently resonant structures, and the patch primarily determines their
impedance characteristics. The radiation behavior, on the other hand, is influenced
considerably by the finite size of the substrate, primarily due to the launching of surface
waves and their diffraction at the edge of the substrate. Consequently, the theory of
diffractions is occasionally used in conjunction with other methods to improve the
prediction of the radiation pattern. [3] One valuable difference between FDTD and other
numerical techniques is that analytical preprocessing and modeling are nearly nonexistent
in FDTD, allowing antennas of higher complexity to be analyzed. This analytical
approach can be used to include the effects of finite sized substrates and ground planes,
which are very important in the design of many microstrip antennas, especially those
used for consumer electronics applications. Also, interaction between the device and

circuits at the field level can be incorporated when using the FDTD method. [3] It is

FDTD’s direct implementation of Maxwell’s curl equations that makes analytical

22

processing of Maxwell’s equations almost negligible. It uses Maxwell’s differential
equations in Finite Difference form and solves them in the time-domain. That is why the
FDTD method is well suited to take into account the effect of an antenna mounted on a
finite rather than infinite ground plane. It is capable of predicting broadband frequency
response because the analysis is carried out in the time-domain. The characteristics of this
technique make it adept at analyzing complex systems, including wave interaction with
the human body or satellites, nonlinear device simulations, complex antennas, and so on.
[3] Though FDTD is easier to program than FEM, it is not as powerful or versatile when
it comes to handling problems involving complex geometries and inhomogeneous media.
The Finite Difference method represents the solution region by an array of grid points;
therefore its application becomes difficult with problems having irregularly shaped
boundaries (This downside of FDTD would not have been an issue in this research, given

the simple shape of the antenna investigated).

3.3 FEM

The Finite Element Method originated in the field of structural analysis. It was not
applied to EM problems until 1968, yet over the years it has become an integral part of
many engineering disciplines and remains so today.

There are many variations in the formulation of FEM. All electromagnetic
formulations share the same basic characteristics and methods of constructing systems to
be analyzed. In all formulations, the computational domain of the problem is defined
first. Shape functions (a.k.a. expansion functions or interpolation polynomials) are then

chosen. The physical region is subdivided into sub-regions called finite elements (defined

23

by the chosen shape functions), which is a process known as meshing or discretization.
Next, the wave equation (or Laplace’s or Poisson’s equation) is enforced over each
element. At least one of the dependent variables is approximated in functional form over
each element and hence over the whole domain. The parameters of this approximation
subsequently become the unknowns of the problem. The substitution of these
approximations into governing equations (shape functions) yields a set of equations in the
unknown parameters. Next, the boundary conditions are applied and the element matrices
are assembled to form the overall sparse matrix system [A]{x} = {b}. The solution of
these equations yields the parameters and consequently the approximate solution to the
problem.

As mentioned before the Finite Element Method (FEM) is useful for modeling a
wide class of problems, with boundary value problems being one class. The
computational domain is cut into simple shapes: triangles for example. By keeping the
elements small (generally less than a tenth of a wavelength per dimension), the field
interior to the element can be safely approximated by some linear, or if necessary, higher
order expansion. [10] The increased accuracy can be sacrificed to reduce computing cost
by using higher order approximations on each triangle but choosing much larger
triangles. According to Sylvester [8], it is best to subdivide the problem region into the
smallest possible number of large triangles, and to achieve the desired solution accuracy
by the use of hi gh-order polynomial approximations on a very coarse mesh.

Shape functions are used to approximate the unknown functions within each
element. Once the shape functions are chosen, a computer program can be written to

solve complicated geometries while only specifying the shape functions. The element

24

choice is very important. One must consider the shape of the structure in question, the
complexity of the problem, and the available resources in order to properly choose an
element shape. A shape function in N-dimensional space is the minimal possible
nontrivial figure in that space; N+l vertices defines it. A one-dimensional shape function
is a line a two—dimensional shape function is a triangle, and a three-dimensional shape
function is a tetrahedron. The locations of the shape function’s vertices uniquely define
that shape function. [6]

The element shape functions for the point (x,y) inside a triangle on the xy-plane is
(obtained by using Lagrange Interpolation polynomials [10]

a1 x2 y3-x3y2 y2 - y3 x3- x2 1
(3.2) 2A a2 = x3y1 -x1y3 y3- yl x1 -x3 x
a3 xly2 — x2 yl yl — y2 x2 - x1 y

where a's are the shape functions

The triangle is the element shape employed by the software used in this research.
The shape functions are local and independent of any translations or rotations of the
global coordinate system. This characteristic allows the mathematical formulation of
element equations to remain fixed for the triangle element. Therefore, the results for one
triangle element can be applied to any triangle by means of a few simple coordinate
transformation rules. Most structures, no matter how complicated their shapes are, can be
represented as a combination of triangles, which is why the triangle is often employed as
the fundamental element shape.

The approximating functions used with first-order, triangular, finite elements have
two simple properties, which add to the utility of the FEM method. The first is that these

functions can be chosen to assure the continuity of the desired potential or wave functions

across all boundaries between triangles, if the continuity is imposed at the triangles’
vertices. The second is the fact that the approximations obtained are not dependent on the
placement of the triangles with respect to the global x-y coordinate system. This is
because the solution surface is defined locally by the vertex values of the wave function
and is therefore, unchanged if the x and y axes are redefined, even though the
approximating functions on each triangle changes. The above two properties remain true
for higher order approximations, if the shape functions within each triangle are properly
defined. [8]

The collection of these elements and their associated expansion or shape function
facilitate the modeling of arbitrary and rather complex fields in terms of unknown
coefficients which may represent the field values at the nodes (node-based bias) or the
average field values over the edge (edge-based basis). [10]

To construct equations for the unknown coefficients of the shape functions in
FEM, the wave equation is first enforced in a weighted (average) sense over each
element. The next step involves the application of the boundary conditions, leading to a
matrix system of the form [A]{x} = {b}. The choosing of the column matrix, {b} is based
on the boundary conditions or the forced excitation (current source, incident field, etc.).
The matrix [A] is square, very sparse and usually symmetric unless non-reciprocal
material exists in the computational domain. The nonzero entries of matrix [A] show the
relationship between fields or voltages of adjacent elements within the computational
domain. The form of [A] is governed by the problem geometry and the discretization of

the computational domain. [10]

26

Once this set of simultaneous equations, [A]{x} = {b}, from which the solution
can be derived, is formed, an iterative or direct solver is employed to find that solution.
Iterative solvers are primarily used for large systems (systems with a large number of
unknowns, N). These solvers avoid explicit storage of the entire matrix, (only nonzero
entries are stored), thereby saving computer memory and computational resources. As
mentioned before, to construct equations for the unknown coefficients of shape functions,
we enforce the wave equation in a weighted average sense over each element. In Chapter
2, we discussed the wave equation that describes the propagating modes of our cavity
modeled patch antenna. The wave equation is (2.1):

V2AZ+ szZ = 0 ,
where k is the cutoff wave—number. The finite element version of this problem is the
solution of a matrix equation (the trial function plus some linear variational term with a
linear component) subject to boundary conditions.

The approximating equations used with the triangular finite elements can be
written in operator form as
(3.1) Lu — f = 0
subject to appropriate boundary or transition conditions.

The operator L is based on an integral representation of the field as the radiated fields
due to equivalent currents:
E(r) = - [flSCVX(=3(R)O fi’ >< E(r’) dS’

(3.2) +jkoZo[flSC(=3(R)0fi' x H(r’) dS’

where R=|r-r'|

27

The forcing function f is a known excitation while u is the unknown quantity; for our

purposes it is an unknown current density. Very few analytical solutions for (3.1) are
available. Also, the electromagnetic scattering and radiation problems we are concerned
with cannot be solved using analytical methods. We must obtain an approximate
numerical solution, which closely resembles the exact solution. Two methods available
for obtaining such a solution are the Ritz Method and the method of Weighted Residuals.
The software employed by this document uses the latter. [6]

The method of Weighted Residuals was developed as a way of finding simple
functions that were approximate solutions to differential equations found in engineering.
When employed by the Finite Element Method, it treats cases where the undetermined
parameters may be numerous and where the trial function is made of a collection of
different functions covering sub-regions of the solution region. This procedure is most
useful in problems where the partial differential equations and boundary constraints for a
problem are well defined but a corresponding stationary function is not. [8]

The procedure begins with the setting up of an arbitrary but reasonable trial
function in the problem variables, with one or more parameters in the function
undetermined. These parameters are then chosen so that the trial function becomes a ‘best
fit’ to the exact solution by referencing the system’s partial differential equations and
boundary constraints. This is different from the Ritz variational method, where the energy
of a function is made stationary.

According to Sylvester: “To utilize this method it is necessary to define a position
function F(r), also defined for individual elements as F¢(r) (e = 1,2...E), such that there is

a best fit to the problem and boundary constraints. The basic technique is to substitute the

28

inexact trial solution F(r), with its undetermined parameters, back into the system of
equations. This then leads to a function p(r), which does not vanish anywhere on or
inside the boundary, but would have if the correct function f(r) had been used instead of
F(r). Generally the trial function F(r) will be made up of shape functions that span groups
of nodes selected from a total of N nodes. M of these are not constrained, leading to M
degrees of freedom for the trial function F(r). With M weighting functions, Wn(r) are

selected such that:

(3.3) Rn = IQ Wnp d9

The solution region is represented by Q and Rn is a set of weighted residuals, which will
be expressed in terms of M undetermined parameters. Setting each Ru equal to zero gives
a simultaneous set of M equations that can be solved for the unknowns Fm (again Fm is an
approximation to the true set of solutions fm).

The trial function F(x,y) is represented separately in each finite element by means

of a polynomial expansion:

N
(3.4) Fe = Z Feiaei

1:]
where or is again a shape function. The parameters Fe, are determined so that they
represent the best possible approximation to the true nodal values fci. According to the
rules for the weighted residual procedure, for any node on a Dirichlet boundary surface S,
the F-parameter should be set to the known boundary value f in $2. The polynomial
approximation F (x,y) covering all of $2 is now made up of a collection of elements 9,
with common nodes on their mutual boundaries. The global trial function F (x,y) itself

(but not its derivatives) will therefore be continuous across these inter-element

29

boundaries and span (say) M nodes F 1,F2,...., F M in total. Of the M global nodes, Mp will
correspond to prescribed values on S. The finite element process is used to determine best
estimates for the rest of the global nodes, Mf = M-Mp, of the Fm’s. Evidently F (x,y) is
constructed as described above and is a suitable function to substitute into the weighted

residual expression:
__ 2 -
(3.5) Rn— {9(an)o(VU)da+ IQWnUC U g) do

On the boundary surface S, F is equal to fo (the actual solution). Also F maintains the
proper continuity over the elements such that the integration over Q will not be difficult.
The W, are Mf specific weighting functions. By setting each R,I = 0, the set of linear

simultaneous equations
Mf
(3.6) 2 [-Sanm +Tmn(k2Um —G,,, )1 = 0
m=l
may be set up and solved for Fm, There are many ways to obtain proper weighting
functions.” [8]

The Galerkin’s method is the process utilized by the software used in this research
to obtain proper weighting functions. The weighting functions, Wn can be chosen from
the set of interpolation functions 0t(r); when such a procedure is implemented it is called
the Galerkin’s method. According to Sylvester [8], this method usually gives the best
result when compared with all other choices of the weighting functions and ordinarily
gives the same solution as one would have obtained from a variational approach. The
Galerkin’s method chooses the shape functions (1,30) to act as the weighting functions

Wn(r). There is an our) corresponding to each node in every element, yet fewer

weighting functions are required since it is not necessary to have them for the boundary

30

nodes. The interpolation function, ore,(r) is defined only within its triangular element, but
W,(r) must be defined over the entire solution region Q. So, the (1,,(r) that will be used
must be selected carefully. Now we may use the acquired weighting functions in
conjunction with (3.3). The matrix equations (3.4) are established for each element
separately. The next step in FEM would be to connect the elements.

Once the system (3.4) is assembled, its solution can be found with an iterative or
direct solver. Iterative solvers are primarily used for large systems that have a large
number of unknowns. Since these solvers avoid explicit storage of the entire matrix, only
the nonzero entries are stored. The assembly procedure is to take the average of the test
equations from the left and right of the nth node. Application of the boundary conditions

is part of the assembly process. [10]

3.4 Comparison FEM & FDTD

As mentioned earlier there are many varied formulations of the FE method. This
includes hybrid versions like the Finite Element-Absorbing Boundary Condition, Finite
Element Mode Matching, and Finite Element Boundary Integral, the variation employed
in this research. All FE methods combine geometrical adaptability and material
generality. The Finite Element Boundary Integral technique leads to fully rigorous
approaches, which combine the best aspects of volume and surface formulation FEM vs.
FDTD

The electromagnetic computational techniques considered for use in this research
have varying strengths and weaknesses. Their characteristics make one formulation

suitable to one type of problem and less suitable to another. FEM was chosen as it was

31

decided that it best suits the needs of the problem investigated and the logistics of this
research.

For most computational methods it is difficult to model proximity-coupled and
aperture-coupled feeds. Cross-polarized radiation from a patch antenna is also difficult to
predict. According to Garg, these limitations are not a problem in full-wave techniques
like FDTD. [3]

In comparison to FEM, FDTD is more computationally efficient for large
problems, especially when predicting broadband responses. According to Garg, FDTD,
methods are not as powerful, or as versatile a numerical technique when it comes to
handling problems involving complex geometries and inhomogeneous media. FEM’s
superiority at modeling inhomogeneous materials makes it a great tool for modeling
patch antennas, since this antenna is a hybrid of very different materials. The systematic
generality of the method makes it possible to construct general-purpose computer
programs for solving a wide range of problems. Consequently, programs developed for a
particular discipline have been applied successfully to solve problems in a different field
with little or no modification. [6] In addition, this method has low memory requirements.

Like the Finite Difference methods, FEM is useful in solving differential
equations. According to Sadiku, in this method, for any point within an element, the
potential or field can be calculated as long as the potentials at the vertices of the element
are known. This is unlike finite difference analysis, where the potential is known at the
grid points only. As a result, inhomogeneous structures or problems with irregular shaped

boundaries can be handled more easily with the finite element method. [6].

The superiority of FDTD modeling of aperture-coupled feeds and cross
polarization between elements in an array does not come into play, for the antenna
modeled in this research. However FEM excellence at modeling complex geometries and
inhomogeneous media make it very desirable. FEM’s low memory requirements and the
versatility that allows it to be easily combined with other powerful tools, like Boundary
Integral techniques, are also strong reasons for its selection in this research. (The Integral
Equation Techniques though it assumes that the substrate and ground plane are infinite in
lateral dimensions, the formulation of the solution is based on rigorously enforcing the
boundary conditions at the air-dielectric interface. With the application of Green’s
function it includes the effects of dielectric loss, conductor loss, surface wave modes, and
space wave radiation [3]). When dealing with simple non-metallic structure, the
advantages of FEM and its hybrid versions often make it the best choice. [10].

3.5 Finite Element Boundary Integral Computer Program

The remainder of this chapter will be an intense treatise on the anisotropic right
prism finite element-boundary integral (FE-BI) computer program, ”PRISM”, used in this
research. Both the software and the following description are provided, courtesy of Dr.
Kempel.

The development begins with the FE-BI equation, including the option for a
resistive sheet transition condition (or R-card), presents the right prism edge-based
expansion functions, lists the numerous element equations necessary for full tensor

anisotropy, and finally presents validation data.

33

The FE-BI equations are derived by considering Maxwell’s equations for time-
harmonic fields. After some manipulation (see Volakis, Chatterjee, and Kempel for

details [10]), the following FE-BI equations result:

(3.7) Mwa,ft,“-waj]dV—kgjv[w,.f,-wj]dv
+17%st (fiRXWi:(fiRXWj) d5
8

 

 

2 . = . ‘ , _ int ext
—k0ISaJSa[W,--sze2Xz-WdeSdS—f,- +f,-

In this, Wi are vector test functions, Wjare vector source functions (since we are using
Galerkin’s method, these two functions are identical), koand Z0 are the free-space
wavenumber and impedance, respectively, fiRis the outward directed normal to any R-
card surface, Reis the normalized resistivity of that surface, and 5:22 is an appropriated
electric Green’s function of the second kind. In Eq. 3.5, the excitations functions
fiim and fie” correspond to the internal and external excitations (specification of which
is irrelevant to this document). Finally, the tensor permittivity and permeability are given
by:

Ex:

.5311
n

. 0')

r,
m

y:-

 

 

(3.8)

 

 

#3 Izzy ya]

34

Note that for an electric field formulations (such as Eq 3.5), the inverse permeability
tensor is required.

For notational convenience, Eq. 3.5 can be rewritten as:

 

(3.9) 111 —k31° +jk01" -kf,1Bl = ff“ +ff’“
where
I =-l
(3.10) 1'=[V[wa,-rt, ~Vij]dV
e_ =
(3.11) 1 _jV[W,-e,-wj]dv
ii xW- ~r’i xW-
(3.12) 1R=j (R ‘)(R ’) 5
SR Re
and
B] A = . I
(3.13) 1 =[Sajsa[w,-szezxz-wj]d5ds

Before explicitly writing 3.9 to 3.11 (3.12 is available in Volakis, Chatterjee, and Kempel
[10]), we need to define the vector expansion functions, their curls, and the cross product
between the R-card normal and the expansion functions.

The vector edge-based expansion functions used in PRISM are developed by
multiplying the traditional Rao—Wilton-Glisson (RWG) basis function with a function of
the prism height for the transverse (e.g. x- and y-component) functions. The normal
functions are simply the node—based simplex basis function (see for example Zienkiewicz
and Taylor) multiplied by 2. Hence, the transverse basis functions for edges on the top

of the prism are given by

 

(3.14) v.=w,.=[§z-}sk=[fl2‘getix—xay-(y-nfii

Z: 1,2,3
where xis the local edge number and iis the global edge number. Not that the local
edge numbers are defined so that the edge is opposite the local node number as shown in
Figure 3.1

Transverse expansion functions on the bottom of the prism are given by:

_ _AZ—Z _AZ—Z d; _.A_ _..
(3,15) Vi-Mz-[ AZ )Sk'( AZ )2se[(x xrly (y y.)X]

 

 

 

124,5,6

Finally, the normal (z-directed) expansion functions are given by:

(kuYkZ ’kaYkl)+()’kl - Yk2)x+(xk2 -Xk1)y
25"

 

V-=K ="
(3.16) ' I Z
1:7,8,9

where the indices k1 and k2 are given in the table (3.1)

Another quantity of interest is the curls of Eq. 3.13 — 3.15. These are given by:

 

 

d- . . .
(3.16) VxW,=-2Az'Se[(x-Xz)x+(y-yily‘2u]
_ di ~ . *
(3,17) VXM;—ZAZSe[(x—xi)x+(y—yi)y+2(Az-Z)Z]
1 . .
(3.18) (VxK, =;§[(xk2 —xr1)x+(ykz -ykr).v]

Note that the global edge number is being used to index these equations and the vector
symbol indicated the location of the edge (top, bottom, or sides of the prisms). Finally,
the terms required by Eq. (3.7) (on the top surface of a prism since that is the only surface

we are currently interested) are given by

36

(3.19) 2x wi = —wi

With Eq. 3.13-3.19 through, all necessary function are defined for evaluating the matrix

elements. In the following section, we present explicit formulae for these matrix entries.

37

Indices of k1 and k2

 

 

 

 

 

 

 

 

Table 3.1

38

kl k2

 

3.5.1 Matrix Entries
Integrals over Triangles
The following integrals over triangles are valid for triangles whose centroid are centered

at the origin and are taken from Zienkiewicz and Taylor [5].

(3.20) Closed-form Prism Integrals
s" =[Sxds=sy =ijds=0

Se =ISdS =%[x2(y3 -y1)-x1(y3 —y2)-x3(y2 -yl)]= area of triangle
5’“ =[Sx2ds = Se[x12 +x§ +x§]/12
s” =[S yzdS =s"[y,2 + yz2 + y32]/12
Sx'v =sty dS =Se[x1yl +X2y2 +X3y3]/12
(3.21) First Integral
If“ (a,b) =IV[(x-a)(x-b)]dV =AzIS[x2 -(a+b)x+ab]dS =
Az[Sxx—(a+b)Sx+abSe]
ley(a,b)=J'V[(x—a)(y—b)]dV =Azjs[xy-ay-—bx+ab]ds =
Az[s"y -bs-r —aSy +abSe]
1;: (a,b) =IV[(x-a)(z-b)]dV =j:z(z—b)dzjs(x—a)d5 =
A2 e
Az[7—b][Sx—a5 J
11yy (a,b)=jv[(y—a)(y—b)]dv = Az{5[y2 -(a+b)y+ab]d5 =

Ash” —(a+b)sy +abSe:|

39

11y: (a,b)=IV[(y-a)(z—b)]dv =I()AZ(z-b)dzjs(y—a)d5 =

Az[-A2—Z-6][sy —as°’]

Il( (,)ab )=IV[(—z a) (-—z b)]dV= Seal Azz[:z —(a+b)z+ab]dz=

2
AzSe[A: -(a+:)Az+ab]

(3.22) Second Integral
1gp.) =IV(x—a)dV = Azjs(x—a)as =Az[Sx —aSe]

2(“)=IV(y-a)dV =AZIS(y-a)dS =Az[S>’ -058]
12(“)=Iv(2“a)dV=Sej:z(z—a)dz=Azse[Az/2_a]
12 =deV=AzSe

 

(3. 23) Third Integral

:6)={V(x- (x- 6) 22dV= Az3[s”- (6+6)s"+abse]/3
6)({V=— )(—y )z2dv= Az3[Sxy— —be—aSy+abSe]/3
13”(a 6)=({Vy— )(—y —)6 z2dV= Az3[5yy— (a+b)Sy+abSe]/3

(3.24) Fourth Integral
[f(a,6)=jv(x—a)(x—6)Z(Az—z)dv =Az3[S’“ —(a+b)Sx +abSe]/6
)(y-b)z(Az—z)dv =Az3[Sxy —be -aSy +abSeJ/6

Az3[Syy—(a+b)Sy+abSe]/6

N
Q
A
Q
G“
V
11
<
A
5.,
I
Q

AN
92
A
Q
a.
V
II
<
A
‘C
I
S
v
A
"<
I
6“
v
N
E
l
N
V
‘1.
'<
11

(3.25) Fifth Integral
I; (a,b,c,d) = Iv[z(x—a)(b+cx+dy)]dV =

212.2 [65” +61st (b—ac)Sx —adSy —abSe]/2

40

15y (a,b,c,d) = {V[z(y—a)(6+cx+dy)]dv =

A22 [615” +cs’>’ (b—ad)Sy —acs" —abSe]/2

(3.26) Sixth Integral

13(66): jv(y—a)(y—6)(Az—z)2 av = 1,W(a,6)

(3.27) Seventh Integral
I;(a,6,c,d)= 1;(a,6,c,d)
I7y (a,b,c,d)= I,"’(a,b,c,d)

(3.28) Eighth Integral

[é (a,b,c,d,e,f) = IV[(a+bx+cy)(d +ex+fy):]dV

= Az[6es’“r +613” +(bf +66)st +(ae+6d)s-‘ +(af +cd)Sy +adSe]

3.5.2 Magnetic Field Terms

The magnetic field terms are of the form:
(3.29) 011;,=jv[(p-va,)(a.vaJ-)]dv {p,q}e{x,y,z}

where (i, j) are the global edge numbers and (p,q) denote the component of the inverse

permeability tensor. The magnetic field matrix entries are divided into nine terms
presented next. These terms are identified by the location of the test and source edges as:
top, bottom, and sides. These correspond to the surfaces of the prism. In the following

sections, the first label denotes the location of the test edge while the second label

41

indicates the location of the source edge (e.g. Top-Bottom refers to the case where the
test edge is on the top of the prism and the source edge is on the bottom of the prism).

Note, only non-zero entries are presented.

(3.30) Top-Top

012,- J 2 V[(x-x,)(x—xj)]dV= l j 211n(x,-,xj)
( .) (we)
”I; did} ZIV[(x x,)(y-yj)]dV= didj [f(xiyj)
(2 .) (M)
011;. ' J 2{V[(x x,)2z]dV-—2 ' J 211"=(x,.,o)
(ms) (w)
11132: did] 11%") y!)
(2AzSe)
-d 3d-
gli'y‘ d' J zjV[(1-n)(y-y))]dV= d, 211W(y,,yj)
(2... (M)
dd 61
{/th l J 2IV[(y—y,)22]dV--2 l :2 lyz(yi’0)
(2 l (2&5 l
”I; —2 M" 211"=(.rJ-,0)
(2AzSe)
U 51,- 2 didj zlfvz(yj’0)
(Me)
0.1; 4 d'd’ 2[flat/=4 ’dj 2113(00)

42

(3.31) Top-Bottom

 

 

 

 

 

 

 

 

 

 

 

 

d-d- dd-
,JI£x=— I J ZJV[(x—x,)(x—xj)]dv=— ' J zlln’(xi,xj)
(2W) (2W)
,JIfY—_ didj V[(x—x,)(y yj)]dV=— did} [F(x,yj)
(M) (W)
”1:2: didj ZIVUX x)2(Az—z)]dV=2 didj 21,“(x, AZ)
(M) (M)
.4. .160. y.)
(2AzSe)
. . d-d-
“1 ,I,[0—y.-)(y—y,)]dV=- ' , 2160.6.)
(2 .) (2315‘)
11132 " did] 2Iv[(y’yt)22]dV=2 did} zllyzUr AZ)
(2AzSe) (2Azs‘)
616-2 M" 160.0)
(2W)
ijlgy 2 didj 2132010)
(We)
0.15:4 did] I,“(0.Az)
(2W)

. . d; xj —xj
2 jv[(x—x,-)(x,{2 -xl{12)]dV = — :;:Se):l)1§(xr')

 

 

43

 

 

 

,1" __ di
J U Az(ZSe)2
'jII’Ix_ di 2
A4258)
111:0: Az(:;e)2 jv[()’ Yr

 

 

 

 

 

:1";— did} 21;“(xj,x,-)
(2625‘?)
(Id-

27153 ="—"J—2’i’y(xj’yil
(2AzSe)

.4. = 2—“1' ,1.“ (m)
(We)
d-d-

Ulgxz— I j 21:0(2‘1'3’1)
(2652’)
d-d-

Wily:- l j 211w(yj’yi)

(265")

 

 

 

 

 

(3.34) Bottom-Bottom

d-d-
1712x=——l—:—2’Iiu(xirxj)
(2Azs )
d-d- .
.7156 =—"l—J;31f’(xi,yj)
(2Azs )
1}]!ch _2 did] 211xz(xl A~)
(2:12:56)
d-d-
(11:11! I J [F(x} Yr)
(2AzSe)
dd-
111% I] 11W(yi,yj)
(2W)
ad- .
ij i'z 2 I J Irv’Ui’AZ)
(2AzSe)
”I; —2 d’dj 2If"~'(xJ-,Az)
(zazse)

45

(2625f 2
612-? did:) [F(AZAZ)
2AZS

(3.35) Bottom-Sides

. J' _ 1'
dl(xk2 xkl)

 

 

 

 

 

 

ijIxx = 2 I; (xi)
Az(2s°’)
d- y} _yj)
h ‘( k2 kl
1711?: 2 [2(xi)
Az(2se)
di(x1{2‘x1{1)
tjlyzx = 2 13%”)
Az(2s")
d. y} _yj)
h ‘( k2 kl
ijlyy = 2 1201')
Az(25")
di(x/{2”‘I{1) ,
tjlzx :T' 2(AZ)
A4258)
di(yI{2 —yI{l)
ijILI'ly :_ [22 (AZ)
Az(2$")

46

(3.36) Sides-Top

r i
d} (xkz ‘2‘“)

 

 

 

 

 

 

 

 

U12:- 2 12(le
132(25")

,"Ih __dj(x;22-x;tl) 2y(y)
J 20’ A4256) 1
til-(Jri x )
615.2% I ’2 kl 12(0)
Az(2Se)

1,1,? (11012-21) §( )
J y Az(25")2 j
”'1ti _"dj(y;;2 23:1) 2y(yj)
Az(2s")
i-Ih-- d}(y;;2 yil)12z(0)

J y” Az(2$")

(3.37) Sides-Bottom

,4; _ d,- (42 4221),; 6,)
Az(2$")

01;; zdj(x;’2-:;‘1)I§V(yj)

A4256)

47

r 1'
dj (xkz -xkl)

 

 

 

 

611'. =-2 A4258), 15(Az)
.14: = 25:51)” ) 1: (.,)
,4. = 1,:5:5:),)12y (6-)
61?. = -2 (117225;?) If (Az)

Sides-Sides (3.38)

ijlxx " """—2 [(222 —x;(1)(xk2 ”91”]?—
23")

A

 

h _ l ,i _ i j _ j r
ij’yx' [(2k2 ykl)(xk2 xk1)]12

 

2
(25")

up" 2 yk2 ykl yk2 ykl
(25’)

3.5.3 Electric Field Terms

The electric field terms are of the form:

48

(3.39) .)IZq=lv[(fi-V.~)(é-Vj)]dv {p,q}e{x,y.z}

where (i, j)are the global edge numbers and (p,q) denote the component of the

permittivity tensor. The following sections present the various matrix entries associated

with the electric field portion of Eq 3.5. Note that only non-zero entries are listed.

(3.40) Top-Top

 

 

 

 

 

 

.)1§x= didj zjv[(y-yr)(y-yj)zz]dV= did]. 21?(yi,yj)
(we) (w)
171;); =__fli7jv[(y-yi)(x—xj)zz]dV=- did] 2 1?,“ij
(M) (M)
015x z-(2:;)2 I? (xi’yj)
01;), ={i‘3‘i—33IV[(x—xi)(x-xj)zz]dV = (2::)2 I? (xhxj)
(3.41) Top-Bottom
.712. =-—d""—"71,[(y-y.)(y-y,)z(Az-z)]dv92242-1206»)
(M) (M)
)][ny=__diii_zjv[(y—yi)(x—xj)z(Az-z)]dv=- didj 21?(xj.yi)
(2A2?) (21329")
til-5x "(23:21? (x,,yj)

49

171;), =-(;:—E:—)7_[V|:(x—xi)(x—xj)z(Az-z)]dV =(—2:—::—)2—ijJr (xhxj)

(3.42) Top-Sides

.i j _ I j
(xklykz xkzykl)+

C O I O dv
J _ J J _ J
(ykl yk2)x+(xk2 xk1)y

6-1; =-——1V 20%)

1° j j j
di (xklykz -xk2ykl)+
=——JV z x-xi ( dV

i _ 1° f _ j
ykl yk2)x+(xk2 xkrjy

Note that in this case, the notation Ig‘y (a, j) means that the “a” parameter is either xi or

y 1 as appropriate and the remaining three parameters (“b”,”c”, and”d”) are given by the

quantrtres 1n the parentheses for the j (or source) edge. That rs, b = (le yzz — x62 >711) .

c = (3’21 — ygz), and d = (Jr/{2 —x,{l). Later, we will see cased where the i‘h (or test)

edges are used to specify these parameters.

50

(3.43) Bottom-Top

 

 

 

 

 

 

 

 

 

d-d- -
012x: 1 J 2jv[(Y‘yi)(y—Yj)Z(AZ‘2)]dV= I J 2132(yhyj)
(2665:) (245)
171:), =——;di—d—J:_§.J‘V[(y—yi)(x—xj°)Z(AZ"Z)]dV =- did 2 I? (xjvyt)
(2W) (2W)
0'15):— did] 21400”! yr)
(26259)
e didj did} xx
ijIyy -- 2IV[(x—xi)(X—Xj)Z(AZ"Z)]dV= 214 (XI-,1?!)
(M...) (Me)
(3.44) Bottom-Bottom
ijIJea=———didj 2jv[(y-y;)(y-y,~)(Az-z)2]dv= didj 213(Yid’j)
(2W) (M...)
d-d- . .
.-,-z:.=- ' I , i,[6-y.->(x-x,-)<Az-zr]dv=— “”1 21:40.3.)
(2AzSe) (2AzSe)
”I; didj 2 lg)’ (x, y!)
(2AzSe
d~d~ . .
”1;: I J 2IV[(x—xi)(x-xj)z2:|dV=Ji31gx(xi,xj)
(w) (M)

(3.45) Bottom-Sides

J' J' J' 1'
d. (xklykz ‘xkzyk1)
515:?“ I 2Iv (Az"z)(Y"Yi) . . dV
1 J
(xk2’xk1)y

Az(ZSe) +(yl{1—y]{2)x+

 

51

=- 17(y.1')
Az(2Se)
0152: d2)V( (-AZ 2) (—x xi)
Az()2Se
217 (xii!)
:2‘42‘)
(3.46) Sides-Top
d.
111:: —-AZ(2SC)2 1:01”)
d.
”lg—AAZS‘ I;(xj,t)

(3.47) Sides-Bottom

d.
"1”: Az(2:9e)2"i(ypi)
t d] x .
'11:)? =WI7 (xj,l)

(3.48) Sides-Sides

r

 

X

 

i i i i
(xklykz —xk2ykl)+

i _ i j __ J'
(Ykr yk2)"+(2"k2 xkl)y

.i j_.i i

(xklykZ xkzykl)+

j_ j j _ j
l (ykl yk2)x+(xk2 29(1)?

1' 1' _ J" 1'
)(xklykz xkzykl)

+(3’it 'y1{2)x+(xiz —x,{1)y

X

 

 

52

W

CHAPTER 4

RESULTS AND ANALYSIS

4.1 Overview of Patch Antenna Characteristics

The purpose of this research was to analyze a patch antenna with a magnetic
substrate and to evaluate the prospect of using such a substrate for antenna
miniaturization. Several microstrip patch antennas with varied dimensions and substrate
material properties were analyzed. The dimensions of each antenna were varied to
determine how they alter the antenna’s performance, independent of the substrate
material properties. This distinction is necessary so as to properly ascertain the effects of
the permeability on the antenna’s characteristics; ultimately, this would provide insight
into the usefulness of magnetic. maten'als in the production of acceptable miniaturized
antennas. An acceptable antenna is defined as a structure with the typical patch antenna
radiation pattern, an impedance bandwidth of at least 10 MHz, and an input impedance of
significant magnitude.

In the following paragraphs, the properties of the patch antenna and the
properties’ individual affects on the antenna’s performance are discussed. The theorized
and expected affects of the antenna substrate’s permeability are also discussed.
Additionally, after this, the results from the software used, FE_BI Prism, are presented
and analyzed and compared to the expectations and known patch antenna behaviors that

are outlined in sections 4.1.1 to 4.1.5 of this chapter.

53

4.1.1 Substrate Thickness

The patch antennas studied were on substrates, which ranged in thickness from
0.20 cm to 0.4 cm. The thickness was varied in hopes of obtaining the optimal thickness
dimension for use with this antenna, as well as to determine its effect on the antenna’s
performance. Generally, a thicker substrate would result in a decreased resonant
frequency, increased radiated power and improved impedance bandwidth. However, such
a substrate would also require a longer probe, increasing the feed inductance, which
would cause an increase in waves trapped along the surface (surface waves). These
surface waves give rise to spurious radiation. This undesired radiation would increase the
side-lobe level and the cross polarization amplitude, causing the degradation of the
desired radiation pattern. [1-2]. One must be mindful of the effects of varying thickness

when attempting to discover the independent effects of varying permeability.

4.1.2 Patch Width

While the length was kept constant at 1.0 cm for all configurations, the patch
widths investigated ranged from 0.7 cm to 1.0 cm. The length was kept constant because
the maximum allowed dimension for the patch is one centimeter, and a decrease in length
would only increase the frequency of operation. The patch width was varied in an effort
to acquire desired input impedance values as well as to determine how different values of
permeability may offset these values. Decreasing the patch width results in an increase in

patch input impedance. However, given the limited range of practical patch widths, the

54

variation in material characteristics and feed point locations dominate the magnitude of

the input impedance.

4.1.3 Substrate Permitivity

Permitivity values of seven and nine were used for all the patch antenna
variations. As mentioned in the previous chapters, using a substrate with high permitivity
is the most common method used to reduce the size of a microstrip antenna. Using a
substrate with a high permitivity constant 6, decreases the guided wavelength, which
means that a smaller patch length can be used for a fixed frequency of operation.
Increasing permitivity decreases the antenna’s impedance bandwidth. An increase in
permitivity also results in a decrease in radiated power due to the fields becoming
increasingly bound within the substrate. These two affects are the opposite of what
accompanies an increase in substrate thickness. Surface wave power is increased with
increased permitivity as it is with increased thickness; again, this results in poor radiation
efficiency and pattern degradation. Thus, it can be seen that increasing thickness can
counterbalance an increase in permitivity as it pertains to impedance bandwidth and
radiated power. However, increasing either thickness or permitivity is enough to increase
surface wave power. Therefore, increasing both may compound the increase in spurious
radiation causing pattern distortion, and negate the increase in radiated power that

accompanies (limited) increases in thickness.

4.1.4 Probe Feed Location

The probe is a simple feed mechanism, and the point at which it makes contact

55

with the patch has an effect on the magnitude of the input impedance. In general, the
probe is fed along the center line (parallel to the length) of the patch to avoid the
excitation of extra modes, which could cause distorted radiation patterns and unwanted
input impedance fluctuations over a frequency band. The input impedance changes
rapidly as the probe is moved from the edge of the patch along its center line to the center
of the patch; the input impedance decreases in magnitude as the probe is moved away
from the edge of the patch.

We are particularly interested in the input impedance value at resonance, at which
point the total impedance is real and the need to match the feed to an imaginary
component is eliminated. The probe feed adds an inductance to the impedance profile,
which causes a slight shift in the frequency at which resonance occurs. The inductance
added by the probe is increased as the thickness is increased, since a longer probe is
necessary. If the added inductance becomes too problematic, it can be countered by
adding a capacitance in series. Garg offers a method to add this capacitance without
increasing the size of the patch; this method is to cut a circular slot (concentric with the
probe) into the patch.

The antenna in this research was probe fed at four to eight points along the
centerline (in order not to excite extra modes) of the antenna, depending on the
configuration. This was to determine the best and most stable place to feed the antenna,
as well as to get an idea of the modal behavior of the antenna. The number of feed points
range from four to eight because in FEM the antenna models were discretized into

elements of equal size, which results in two columns of center-line feed points for an

56

antenna whose width is an even number and a single column of feed points for an antenna

whose width is an odd number.

4.1.5 Substrate Permeability

Permeability values of twenty and twenty-five were used for all configurations.

From eq. (2.7), increasing the magnitude of the permeability u, will reduce the resonant
length of the antenna by a factor (Lu, . Increased substrate permeability is expected to

result in increased radiation efficiency for the antenna. This prediction is supported by the

radiation efficiency equation provided by Balanis:

e = 3‘— where Qt is the total quality factor and

Qrad
(4.1) Qr is the quality factor due to radiation

losses
and the equation for Qr provided by Das:

2 L
_1- = JIL— where Pr is the total power radiated and L is

Q, P.

(4.2) the effective inductance and is proportional

lg
to A (,u,£ of the substrate).

The permeability of the material is also expected to reduce the total quality factor
as shown in eq. (2.30). This results in increased bandwidth. Increased patch antenna
bandwidth and reduced resonant length are two of the foremost benefits that increased

substrate permeability is hoped to possess.

57

4.1.6 Note on Measurements

The impedance profiles for several feed points are compared according to their

maximum values and their bandwidths. The impedance bandwidth is taken to be the

range of frequency for which the impedance is greater than X5 times its maximum

value. The minimal acceptable impedance bandwidth is 10 MHz and the minimal

acceptable peak impedance is 7.9 (so that the minimum impedance contained within the

bandwidth is 59).

The principle planes are the E-plane (d) = 90°) and H-plane (¢=0°). These assignments
are different from many texts, because the width of the antenna modeled is along the x-

axis rather than the y-axis.

4.2 Results and Analysis Overview

As mentioned before, several microstrip patch antennas with varying dimension
and material pr0perties were analyzed. Some, but not all, of the results obtained from the
Prism software will be presented in this thesis. The results are chosen according to their
relevance. For many probe feed locations, the antenna did not reach resonance, or it did
not reach resonance at a point with significant real impedance. These locations will be
mentioned, but the graphs of their results will not be displayed for brevity. Each antenna
section will be preceded by a two-dimensional, top-down view of the patch antenna
discretized by Prism. These figures (Fig. 4.1, 4.27, 4.45) will also show the location of
the probe feed points. After a presentation of the individual results, a comparison of the

results will be made to determine trends and to confirm some expectations.

58

4.3 1.0 by 1.0 cm Patch Antenna
The 1.0 by 1.0 cm patch antenna utilized the maximum surface area available for

the goals of this project. The three antennas of this section each employ a substrate with
different combinations of permeability and permitivity: 8,=7 and it, = 20, 8,=7 and p, =
25, and er=9 and p, = 20. Substrate thicknesses of 0.025 cm, 0.030 cm and 0.035 cm were
investigated in conjunction with the aforementioned parameters. The probe feed map
(from the FEM meshing of the patch geometry) is shown in Figure 4.1. The lighter
portion of the map represents the patch metallization, and the darker portion represents
the substrate the patch rests on. This antenna model and all its substrate variations were
probe fed in FE-BI Prism at eight feed locations 85, 86, 104, 105, 123, 124, 142 and, 143

(see Figure 4.1).

4.3.1 1.0 by 1.0 cm Patch Antenna with e, = 7 and ti, = 20
Thickness 0.025 cm

The thickness of 0.025 cm yielded no desirable impedance characteristics.

Thickness 0.03 cm

First, a thickness of 0.03 cm will be discussed. For only two of the eight feed
locations modeled, the antenna reaches resonance in a frequency range with a useable
real input impedance and impedance bandwidth. These locations are 105 and 123 and
their impedance profiles are display in Figures 4.2 and 4.3.

For a probe feed at position 105, there are two points of resonance separated by

59

just over 100 MHz. The second resonance occurs at the peak of a 30 MHz band, which is
a band with a 20 MHz plateau of approximately 652 (an exception was made for this
antenna and feed point in regards to the 752 minimal peak, because of the shape of its
impedance graph). Feed position 123 has a resonance point that occurs at 1.2 GHz, the
impedance bandwidth is now just over 10 MHz, and there is no longer a plateau of
impedance but a narrow peak of 752.

Feed position 105 was chosen as the best place to feed the antenna, because of its
bandwidth and nearly flat impedance band around the resonant frequency f, = 1.19 GHz.
The patterns of the two components are shown in Figures 4.4 and 4.5 for the E-plane. In
both the principle planes, 13¢. and E9 terms have the standard pattern shape and form that .
one has come to expect from a microstrip patch antenna. In the E—plane, the maximum
gain is E9 = —21 dB, and the E4, gain maximum is —26 dB, though it is expected to be non-
existent (mathematically) in this plane. Likewise, the E, gain maximum is —21 dB in the
H-plane, and the E9 maximum gain is —26dB when it is expected to be nonexistent in this
plane. Neither of the components show gains above 0 dB in either plane. This indicates
the power radiated in the maximum direction of radiation is less than the power density
that would be radiated if all the power accepted by the antenna were radiated
isotropically. E4, has such high gain in the E-plane and Be in the H-plane possibly because
surface waves are increasing the cross polarization level. The primary reason for the low
gains is the fact that the fields are strongly trapped within the substrate and not radiating.
So, this antenna is in fact behaving more like a resonator than an antenna.

Thickness 0.035 cm

Now, an antenna with the same material properties and the same dimensions

60

(except thickness) as the antenna in the previous section will be discussed. A thickness of
0.035 cm replaces the 0.030 cm substrate height discussed above. As in the previous
section, this antenna was fed at eight different feed locations and only three of these
locations produced suitable results: feed locations 85, 105 and 123. Feed points 105 and
123 remain viable despite the change in thickness. Given that the feed positions remain
relatively unchanged, one can assume that the 0.005 cm increase did not create surface
waves of such magnitude as to change the modes that were excited or upset the resonant
characteristic of the antenna. The impedance characteristics corresponding to these feed
points are presented in Figures 4.6 - 4.8.

The impedance profile for position 85 has one resonant point at about 1.265 GHz.
The resonance occurs at the peak input impedance value of 14.552, and the corresponding
impedance bandwidth is around 20 MHz.

For the impedance resulting from a probe feed at point 105 there are two points
of resonance about 70 MHz apart. One is at 1.15 GHz, a lower frequency of resonance
than in the case of the thinner substrate of thickness 0.030 cm. This decrease in resonant
frequency is expected for increased thickness. The other point of resonance at 1.21 GHz
is unexpected. The first resonant point occurs in a band whose peak input impedance is
approximately 5582 with a 15 MHz bandwidth. The second resonant point has a peak
impedance of 8552 and a bandwidth of 40 MHz. The first resonant point could be quite

useful, given that it meets our 10 MHz bandwidth requirement and has an impedance

near 50S2.
The final feed position (123) results show a single resonant point at 1.15 GHz (a

lower frequency for this feed in comparison to the previous substrate thickness, which is

61

expected for increased thickness). The resonant point comes at the peak of a long
impedance spike. The peak of that spike is 3752 and has a bandwidth of about 10 MHz.
There is an adjacent impedance plateau averaging 1052 and spanning 100 MHz.
Depending on the application and its bandwidth requirements, it may better to move the
resonant point into this 1052 band. One way to do this would be to slightly decrease the
length of the patch antenna. Altering the length slightly may alter the impedance profile,
but not significantly change the radiation properties aside from the frequency shift.

The radiation data was collected using feed position 123, and the radiation
patterns are shown in Figures 4.9 and 4.10. These patterns were taken at feed point 123
because its point of resonance is centered at the peak of the real impedance band. This
antenna has the standard radiation patterns for a patch antenna, except the E9 component
is somewhat misshapen for 0 < 0° in both principle planes (only the E- plane is shown.)
The maximum gain of the antenna in the E-plane is around - 30 dB for E9 and for E9 it is

-3ldB. In the H-plane the maximum gain for E9 is -30dB and for E9 it is -31 dB.

One must assume that the increase in thickness for this configuration has had a
profound effect on the radiation properties of this antenna, given the low gain values at
resonance. Again, the gain is very low because the fields are trapped in the substrate,
even more so than in the previous section with the lower thickness. Increases in thickness
generally decrease the degree to which fields are trapped within the substrate, allowing
for better radiation. However, the intensifying of surface wave activity that accompanies

thickness increases seems to dominate for the 1.0 by 1.0 cm antennas.

62

Probe Feed Location Map For a 1.0 by 1.0 cm Patch

5952 253 254 255-25813258 2582 251" "
Q 234 235235237238 238 2482415242 7'
57414 215 218 21721842118228.22212225223:
85 185187 188188 288 281 282 283 284

78 177178 178188.1811821831845185 “

1;: 57 158 158 158151152 153154155155 ‘
38 138 148141 1423143 144145 148 147
18128121 122123 124 125125127128 :2
88 181 182 183 184 185 185 187188188 5;

 

Figure 4.1

63

Impedance at Feed
Polnt 105

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm
Patch with e, 8 7, up 20 and Thickness I 0.03 cm

 

——Ro-xzm
-~---muzi~)

 

.2 a)"

42

4:

 

9i

 

Frequency (GHz)

Figure 4.2

 

135'

 

Impedance at Feed

Polnt 123

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm

Patch with e, = 7. u, = 20 and Thickness = 0.03 cm

 

 

— ReeKZln)
- ----- mam

 

 

1 .05

 

 

1.25 ‘

Frequency (GHz)

Figure 4.3

65

1.3.

1.35 *

 

Radiation Pattern for E ¢ vs. Theta with Constant Phi = 90
0

330

f
//

 

 

 

 

Figure 4.4

66

Radiation Pattern for Ea vs. Theta with Constant Phi = 90

330

 

\ x .. ,2, 2° 1°

67

 

Impedance at Feed

Point 85

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm

 

Patch with e, = 7, u, = 20 and Thlckness = 0.035 cm

 

 

 

 

 

 

— Real(Zin)
------- lmag(Zin)
1s -
1o 5
5 a
o I 4",“ I I I
In '- ID N to co to
0. .- ". .—‘ N ..' r2
-5 4
-10 i
-15 J
Frequency (GHz)
Figure 4.6

68

Impedance at Feed

Poln‘t 105

so.
701

504

48 .

30‘
201
101

 

am-

Real Part oi Impedance vs. Frequency for 1.0 by 1.0 cm

 

Patch with e' = 7, u, = 20 and Thlckness 8 0.035 cm

 

—— ReaKZln)
~ ----- mama)

 

 

Figure 4.7

69

 

impedance at Feed

Point 123

35-1

25« ..

151

10‘
5.

 

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm

Patch with e, =- 7. u, =- 20 and Thickness I 0.035 cm

 

 

— ReeKZin)
------- Win)

 

 

 

 

1.1 ‘

.1

12‘
1.25i

Frequency (6112)

Figure 4.8

70

Radiation Pattern for E ¢ vs. Theta with Constant Phi = 90

 

Figure 4.9

71

.5}

(45—15—25—20— 9°

150

// 120

Radiation Pattern for E9 vs. Theta with Constant Phi = 90
c

K
(

7
my

 

\
24o ' \
\ \
\E
218

Figure 4.10

72

4.3.2 1.0 by 1.0 cm Patch Antenna with a, = 7 and u, = 25

An antenna with the same patch dimensions as in section 4.3.1 but with different
material properties will now be considered. Feed points 161 and 162 were added to the
investigation for the antennas of this section. Substrate thicknesses of 0.025 cm, 0.030
cm, and 0.035 cm were investigated in conjunction with the aforementioned parameters.
The thickness of 0.025 cm yielded very few points of resonance and very few feed
positions occurring within real impedance bands of significant magnitude and width; for
these reasons, the results will not be shown in this document. Overall, this version of the
patch antenna is of slightly less quality than the preceding antenna, in that the Q of the
antenna is obviously higher, creating more narrow impedance bandwidths with higher
impedance peaks. One may surmise from this section and the section before it (4.3.1)
that the increase in permeability is actually decreasing the bandwidth, instead of
increasing it as expected. The comparison between the antennas is hard to make because
of the very low values of impedance shown in the previous section, and its surface wave

distortion in both thickness cases, particularly the 0.035 cm case.

Thickness 0.025 cm

Thickness 0.025 cm yielded no desirable impedance characteristics.

Thickness 0.030 cm
For a thickness of 0.030 cm, only one of the ten feed points (feed point 104)
investigated had resonance points coinciding with real impedance bandwidths of

significant magnitude and bandwidth. The impedance profile of feed location 104 is

73

shown in Figures 4.11. The resonant point of 104 occurs at about 1.055 GHz on the
declining side of its impedance peak of 1152. The bandwidth of that peak is about 15
MHz.

The radiation patterns were collected using feed 104. The radiation patterns in the
E- and H-planes were of the shapes expected for a patch antenna. In Figures 4.12 and
4.13, the patterns for E9 and E9 in the E-plane are shown. The gain of E9 in the E- plane
peaks around -23 dB, and for the E9, it is around —29dB. The cross-polarization levels are
still significant, as E9 is expected to be negligible in this plane. In the H-plane, the gain of
E9 peaks around -23 dB, and for the E9, it is around —29dB. The gain results revealed that
most of the power is still trapped in the substrate at resonance. The radiation and gain
plots are shown only for the E-plane because the pattern shapes for E9 and E9 in the H-

plane are very similar

Thickness 0.035 cm

A substrate height of 0.035 cm was also investigated for use with this antenna.
This 0.005 increase in thickness yields more favorable impedance results. This is
opposite the scenario of the 1.0 by 1.0 cm antenna with u, = 20, discussed in section
4.3.1. Recall that in section 4.3.1, that the 0.030 cm substrate had better impedance
bandwidths and much better radiation characteristics than the 0.035 cm case.

Two out of ten probe feed locations reached resonance in acceptably wide
frequency ranges with suitable real impedance levels: positions 123 and 162. Their
impedance characteristics are shown in Figures 4.14 and 4.15. Figure 4.14 shows that for

feed 123, the first resonant point at 1.105 GHz isn’t quite aligned with the peak of the

74

real impedance band. The second resonant point for feed 123 also occurs on the declining
slope of a real impedance band with a higher peak and larger width. Figure 4.15 reveals
the most desirable feed point location (162) for this version of the patch antenna. At this

location, resonance occurs at about 1.105 GHz and is at the center of a band slightly
greater than 10 MHz with a peak impedance just over 952.

The radiation data was collected using feed 162. The electric fields in the E-plane
corresponding to feed point 162 are shown in Figures 4.16 to 4.17. The E9 component is
well shaped in both principle planes. The E9 component is misshaped in both planes for -
45° < 0 < 60°. In the E-plane, the E9 component has a maximum gain of -24 dB and the
E9 component has a maximum gain of -22 dB. In the H-plane, E, has a gain of -24 dB,
and E9 has a gain of —22dB. Both components have the higher gain in the planes in which

they are expected to be negligible (assuming this antenna is not circular polarized).

75

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm
Patch with e, I 7, u, =- 25 and Thickness I 0.03 cm

 

 

 

 

 

 

 

——Red(2in)
------- Imain)
7o 4 _.
so 4 '.
B
o 60 ‘1 'I
u. 3
g ' 40 " . “"1"
o E . --------------
_ 30 « ....-
5 o
8 e 20 .
‘5
— 10 .
O I —\ I I A
m In 0’ In P Q- '0
.10 ‘D g Q g g I'- :3
Frequency (6112
Figure 4.11

76

Radiation Pattern for E 0 vs. Theta with Constant Phi = 90

270

 

 

 

 

180

Figure 4.12

77

Radiation Pattern for E5 vs. Theta with Consatnt Phi = 90

Q

_, ,, _, ,.
>42)
X / / 12°
)/

  

 

24o \ .
\
21\0\J 150

180

 

Figure 4.13

Impedance at Feed

Point 123

301
251
201
151

10‘

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm

Patch with er = 7, ur = 25 and Thickness = 0.035 cm

 

— ReaI(Zin)
------- lmeg(Zin)

 

 

 

 

8.95

 

1.11

Frequency (6112)

Figure 4.14

79

1.15

1.2‘

1 .25

 

Impedance at Feed

Polnt 162

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm

Patch with er = 7, u, = 25 and Thickness = 0.035 cm

 

 

 

 

 

 

 

— ReaI(Zin)
------- Imag(ZIn)
9 l
7 a
5 4
3 4
.. /\ .
I A I I I I g I U I
.1 . g m. g; .— g «.310 N. g
D . O . . v-1 '1 I- .
O o '- 1 '— v-
-3 4
-5 1
Frequency (GHz)
Figure 4.15

80

Radiation Pattern for E 1 vs. Theta with Constant Phi

 

\ :8 W“ 9°
\fi/C/ .2.

Figure 4.16

 

81

=90

Radiation Pattern for E6 vs. Theta with Constant Phi = 90

 

(‘ ~40 -30 -20 -10 0
240 ' 120
\ . x. /
\\£\._
210 \j 150

 

82

4.3.3 1.0 by 1.0 Patch Antenna with 8r : 9 and Pr = 20

Thickness 0. 025

The thickness of 0.025 cm yielded no desirable impedance characteristics.

Thickness 0.030 cm

The substrate thickness of 0.030 cm had only one feed point out of ten that was
viable: feed position 162. the impedance characteristic for this feed is shown in Figure
4.18. The resonant frequency f, = 1.0 GHz occurred at the center of a real impedance
band with a width just over 10 MHz. The peak of that band is 1852.

In both planes, the E9 radiation pattern has the shape of the standard patch
antenna. In the E-plane E, has a maximum gain of -15dB and E9 has a maximum gain of
—16 dB. The E9 pattern is misshapen in the E-plane and slightly misshapen in the H-
plane. In the H-plane, E9 has a maximum gain of —15 and E, has a maximum gain of —
lSdB. Though in this plane it is expected to have a value much lower than that of the E9
component. The E9 gain levels are not as low as the antenna in section 4.3.2 whose
substrate properties were 8, = 7 and u, = 25. The gain results show that the fields are still
very confined to the substrate at resonance. The radiation patterns for E9 and E9 are

shown in Figures 4.19 and 4.20.

Thickness 0.035 cm

The antenna with the above parameters, except a thickness increased to 0.035 cm,

83

has two out of ten satisfactory feed positions: 104 and105. The impedance characteristics
corresponding to these feed positions are shown in Figures 4.21 to 4.22. For both cases
with a permittivity of nine, the thicker scenarios proved superior in producing suitable
impedance characteristics. Feed point 104 has a real impedance bandwidth of nearly 20
MHz, 8 maximum of nearly 752 and a resonant frequency f, z 1.005 GHz. Feed point 105
has a bandwidth of 20 MHz, a peak of 2352 and f, = 1.115 GHz, (f, not quite aligned
with the real impedance peak). The thickness 0.030 in the last section had only one
desirable feed location to compare the feed points in this section to, but by using that
single feed point for comparison one can see that the increase in thickness in this scenario
did not decrease the frequency as usual.

The radiation data was collected using feed position 104. The pattern for E9 is
well shaped in both planes. The E9 component is nearly well shaped in the E- plane, but it
is very severely misshaped for 0 < 60° in the H-plane. In the E-plane E9 has a maximum
gain of -20 dB and the E9 component has —28dB. In the H-plane, E9 has a maximum gain
of -20 dB and E9 has a maximum gain of —28dB. The radiation pattern for E9 in the E-
plane is shown in Figure 4.23. The radiation patterns for E9 in the E- plane and in the H-

plane are shown in Figures 4.24 and 4.25.

84

Impedance at Feed

Point 162

20-

15‘

101

-5.

 

-10 -‘

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm

Patch with e, I 9, u, = 20 and Thickness I 0.03 cm

1.1q

I
1.0
O.
P

Frequency (GHz)

Figure 4.18

85

 

1.15-

 

—— Reel(Zin)
------- lmag(Zin)

 

 

 

M
“1
F

1.25 q

Radiation Pattern for E ¢ vs. Theta with Constant Phi
o

330/ “

 

 

180

(""40

9

Figure 4.19

86

-30

=90

30

“_:,

-20

/0

120

150

Radiation Pattern for E9 vs. Theta with Constant Phi = 90
o

330 30

’f 60
7/

_ \ ,
240 \\i/ j 120
/150

180

 

 

 

Figure 4.20

87

Impedance at Feed

Point 104

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm

Patch with er I 9, u, = 20 and Thickness I 0.035 cm

 

 

— ReeI(Zin)
------- inn-91211)

 

 

 

 

0.85 2

0.9 3
0.95 ‘
1.05 ‘

 

Frequency (GI-I2)

Figure 4.21

88

1.1"

1.15‘

1.2«

1.25 *\l

Impedance at Feed

Polnt 105

Real Part of Impedance vs. Frequency for 1.0 by 1.0 cm
Patch with cr = 9. u, =3 25 and Thickness = 0.035 cm

25-

201

151

10-

0.95 ‘

0.85

.5.

 

-1o 4 - :
Frequency (GHz)

Figure 4.22

89

I
ID
'1
F

1.2'

 

 

— ReeKZin)
------- Imag(Zin)

 

 

1.25 "

 

Radiation Pattern for E 0 vs. Theta with Constant Phi = 90
0

 

2,,”

240

 

\
\ \
\b
210 a. ‘

150

180

Figure 4.23 '

90

120

Radiation Pattern for E9 vs. Theta with Constant Phi = 90
o

330 ' 30

so. " ’. co

ml 9’: .._... 9°

\
210 J

180

 

 

Figure 4.24

91

Radiation Pattern for Ea vs. Theta with Constant Phi = 0
0

330

,,,/

270

‘AT‘TFFfi—‘o 9°
0 / /

 

150

180

Figure 4.25

92

4.4 1.0 by 0.9 cm Patch Antennas

In the previous section, square-shaped patch antennas were modeled. Now, we
model an antenna that is rectangular. An antenna whose width is shorter than its length is
best for the excitation of the structure’s lowest order mode. For this reason and more, this
section’s version of the patch antenna was expected to yield better results than that of the
square patch. The following are more reasons that this section’s antenna was expected to
surpass the square patch’s performance: The feed points of the 1.0 by 0.9 cm antennas
are unmistakably on the antenna’s center line. Secondly, the width is decreased, which
should result in impedance bands with higher values. In addition, a decreased width
means that the width to height ratio is decreased which should allow the electric field
lines to concentrate less in the substrate and thus resulting in improved radiation.

For this antenna four substrate parameter combinations were investigated, as well
as four thickness levels. The substrate material property combinations modeled are as
follows:, 8, = 7 and 84:20, 5, = 7 and “1:25, 8, = 9 and “”20, and 8, = 9 and ”=25. The
thickness modeled were again 0.02 thru 0.035 cm in steps of .005 cm. The feed locations
employed were 81, 99, 117 and 135, and the feed map for this antenna is shown in Figure

4.26

4.4.1 1.0 by 0.9 cm Patch Antenna with a, = 7 and Pr = 2

Thicknesses 0. 02 cm

The thickness 0.02 cm produced no desirable impedance characteristics.

93

Thicknesses 0. 025 cm

The thickness 0.025 cm produced no desirable impedance characteristics.

Thickness 0.030 cm

Feed point 117 has a resonant frequency f, = 1.125 GHz and is nearly aligned with
a real impedance peak of 3352. The width of this real impedance band is 30 MHz. The
impedance characteristics can be seen in Figure 4.27

Only one of the four feed points produced desirable impedance characteristics
(feed position 117), and therefore, only one qualified to be used in the radiation studies
for this thickness and antenna version. The radiation patterns for E9 and E9 in the E-plane
are shown in Figure 4.28 and 4.29. The E9 patterns in the principle planes have the
expected shapes. The E9 component pattern is severely misshapen for 0 < 0° in the H-
plane but well shaped in the E-plane. The maximum gain of E9 in the E-plane is around -
47 dB. The maximum gain of E9 is around -38 dB; this is low but higher than that of E9,
which is expected, since E9 is expected to be negligible in this plane. The fields are
strongly trapped within the substrate; this structure is not resonating. In the H-plane the
gains of both components peak at around —24 dB; illustrating that cross-polarization is a
problem, since in each plane one component should be significantly lower than the other.
Note the gain of the components change drastically from one principle plane to the other.
Thickness 0.035 cm

Position 117 is the only viable point for this thickness. The first resonant point
occurred around 1.050 GHz. The reactance barely' crossed from inductive to capacitive

which makes it nearly unnoticeable. In addition the first resonance point does not occur

94

within a band with significant real impedance. Therefore, the second resonant point is
what would have to work with for this antenna, substrate and feed position. The second
resonance occurs at about 1.165 GHz and is not quite aligned with the 10552 peak of the
impedance. The impedance band is about 20 MHz wide. The impedance peaks is much
higher than desired. The impedance characteristics are shown in Figure 4.30.

The radiation data was collected using feed position 117. The radiation patterns
for E-plane are shown in Figures 4.31 and 4.32. In the E-plane, E9 of the Electric field
has a slightly misshapen pattern; the E9 pattern is misshaped, though not as noticeably. In
the H—plane, E9 is well shaped, but E9 is severely misshaped and has a ldb minimum near
0=0°. Since the patterns are so radically varied, the H- plane patterns are also shown for
this antenna in Figures 4.33 and 4.34. The gain in the E-plane peaks at -32 dB for the E9
component and —34 dB for the E9 component. In the H—plane, E9 peaks at -24 dB and E9
peaks at -15 dB. The gains vary significantly from one principle plane to the other. The

antenna radiates better in the H- plane, but it radiates poorly nevertheless.

95

Probe Feed Location Map For a 1.0 by 0.9 cm Patch

188 187 1885188 118T 71

" 150 151 152 133.154 158
134 138 138 137 138 138
118419 120 121 122 123

. 182 103 .184 185-188,187

as 87 88 as so 91

 

Figure 4.26

96

Impedance at Feed

Polnt 117

Real Part of Impedance vs. Frequency fat 1.0 by 0.9 cm
Patch with e, I 7. u, - 20 and Thlckness I 0.03 cm

 

 

 

 

 

 

1.18“

I
O
p

128‘
1.3‘

 

Ftequency (GHz)

Figure 4.27

97

Radiation Pattern for E ¢ vs. Theta with Constant Phi = 90

m I

 

Figure 4.28

98

150

Radiation Pattern for E9 vs. Theta with Constant Phi
0

330 30

‘50. ,7

/

21o ____/ 150

180

 

 

Figure 4.29

99

=90

Impedance at Feed

PoInt 117

Real Part of Impedance vs. Frequency for 1.0 by 0.9 cm
Patch with e, - 7. u, . 20 and ThIcItnees I 0.035 cm

 

 

 

 

 

 

 

 

— Redaln)
------- Wt.)
118
90 1
7o «
so 4
3° ‘ ,3" :E ,‘ xxx
18 1 3. j /\
1o '- 18 1‘... '3 I: In '3
Frequency (GHz)
Figure 4.30

100

Radiation Pattern for E ¢ vs. Theta with Constant Phi = 90
o

 

 

 

Radiation Pattern for E6 vs. Theta with Constant Phi = 90
0

 

 

Radiation Pattern for E ¢ vs. Theta with Constant Phi = 0

330 _ 3O
/ ‘\

300 V
' ‘«)‘
\_ 4’

240 / 120

\Cg/i/

270

 

210

 

180

Figure 4.33

103

Radiation Pattern for E9 vs. Theta with Constant Phi = 0

 

Figure 4.34

4.4.2 1.0 by 0.9 cm Patch Antenna with a, = 7 and u, = 25
Thicknesses 0.02 cm

The thickness 0.02 cm produced no desirable impedance characteristics.

Thicknesses 0.025 cm

The thickness 0.025 cm produced no desirable impedance characteristics.

Thickness 0.030

Two of five feed positions, 117 and 135, have desirable impedance
characteristics. Their impedance profiles are shown in Figures 4.35 and 4.36. Feeding at
position 117 results in an antenna resonant at f, x 1.085 GHz with a real impedance
bandwidth around 70 MHz and peak of 852. The point of resonance is aligned with the
89 peak. Position 135 has two points of resonance occurring at around 1.045 GHz and
1.085 GHz. The first resonant point occurs at an unexpected frequency and does coincide
with a real impedance band. The second resonant point occurs within a real impedance
band that peaks at 209 and has a width of about lSMHz.

Position 117 was chosen over 135 to collect the radiation data, because 135 has a
extra resonance point that is less than 50 MHz away. After comparing the two points, it
seemed that position 117 was least likely to have been effected by extraneous radiation
and cross-polarization interference since it had only one point of resonance (meaning

extra modes were more likely to be of negligible strength). The radiation patterns E9 and

E¢ in the E- plane for this antenna and feed are shown in Figures 4.37 and 4.38. The E,

105

has the pattern shape expected in both principle planes. The E9 pattern is slightly
misshaped in the principle planes: in the E-plane it is disfigured for 0 < 0° and in the
other plane it is disfigured for 0 > 0°. E; for the H-plane is shown in Figure 4.39. The
gain of E9 in the E-plane peaks around - 23 dB, illustrating that the E-fields are trapped
within the substrate. In the E-plane, the gain of E9 is only one dB higher than that of 13,,
indicating that surface waves and other factors are still boosting the cross-polarization
power. In the H-plane, the E, gain peaks around —23dB and the E9 is about two dB lower,

indicating the same possibility.

Thickness 0.035 cm

Thickness 0.035 cm produced no desirable impedance characteristics.

106

Impedance at Feed
PoInt 117

Real Part of Impedance vs. Frequency for 1.0 by 0.9 cm
Patch wlth er = 7, u, = 25 and ThIckness = 0.03 cm

 

— ReaI(Zin)
------- Imasmn)

 

 

 

 

 

-1..

89
8.95‘
1.85“

11‘
1.15‘

12‘
1.25‘

1
z
0
I

Frequency (GI-I2)

-34

 

-5.

Figure 4.35

107

Impedance at Feed

Polnt 135

251

151

101

0.8

 

-5.

Real Part of Impedance vs. Frequency for 1.0 by 0.9 cm
Patch wlth e, = 7, u, = 25 and Thlcknees =- 0.03 cm

 

 

 

 

wy..r'oo-uu .- 8... . -.-u. .8-
' "~.o--'..
'0 .o. .

 

 

 

 

 

0.85 "
1.25 '

Frequency (GI-I2)

Figure 4.36

108

Radiation Pattern for E 0 vs. Theta with Constant Phi = 90
0

330 _ 30

"s

- / \ .
8?)“ w
v’\’ W"

240 120

/

210 150
180

 

270

 

 

Figure 4.37

109

Radiation Pattern for E9 vs. Theta with Constant Phi = 90

330

§

\

J"—

270 \
240

\\\<, \

 

210 \\_

 

150

180

Figure 4.

110

38

Radiation Pattern for E9 vs. Theta with Constant Phi = 0
0

330

270 \

240

210

\

I

”
’ __ ..

 

\
\H

..a-I—F",

180

Figure 4.39

111

4.4.3 1.0 by 0.9 cm Patch Antenna with a, = 9 and u, = 20
Thickness 0.020 cm

Thickness 0.020 cm produced no desirable impedance characteristics.

Thickness 0. 025 cm

Of the five feed positions only position 99 qualified. Feed point 99 had one
resonant point around 0.915 GHz that occurred near a dip between two real impedance
peaks. The real impedance’s bandwidth is around 40 MHz, and peaks at about 1252; the
height of the dip is about 752. This feed point is the only qualified feed, though it is not an
optimum position since its f, occurs within an impedance dip that is 25% lower than the
lowest of the neighboring peaks. The impedance characteristics for position 99 is shown
in Figure 4.40.

The radiation patterns were collected using feed point 99. The radiation patterns
for E, are well shaped in both principle planes; the pattern for E.» in E- plane is shown in
Figure 4.41. The pattern for E9 in E- plane is slightly misshaped for 0° < 0 < 90° (Figure
4.42) and in the other plane it is slightly misshaped for -90° < 0 < 0° (Figure 4.43). The
gain for E9 in the E- plane peaks around -30 dB which is —5dB higher than B, in the
same plane. In the H- plane, 13., peaks around —30dB and is about -5dB higher than E9.
The gain is still much too low, showing that the fields are still very much trapped in the

substrate.

Thickness 0.030 cm

Thickness 0.030 cm produced no desirable impedance characteristics.

112

Thickness 0.035 cm

Thickness 0.035 cm produced no desirable impedance characteristics.

113

Impedance at Feed

PoInt 99

25-

201

15‘

10‘

0.8

 

-sd

Real Part of Impedance vs. Frequency for 1.0 by 0.9 cm

Patch wIth er = 9, ur = 20 and Thlckness = 0.025 cm

0.85 ‘

v
a!
O

 

0.95 ‘

Frequency (GHz)

Figure 4.40

114

 

— ReeI(Zin)
------- Imeg(Zin)

 

 

 

0

Radiation Pattern for E1 vs. Theta with Constant Phi = 90

O

\30

\.
o v

‘Ab—Tsrsz—ffi—o 9°

 

Figure 4.41

115

Radiation Pattern for Ea vs. Theta with Constant Phi = 90

 

 

 

 

180

Figure 4.42

116

Radiation Pattern for E9 vs. Theta with Constant Phi = 0

0

1"
”°\

210 \\
21 o\__

\
(

 

180

.0

Figure 4.43

117

 

 

4.5 1.0 by 0.7 cm Patch Antenna

In the previous section a 1.0 by 0.9 cm patch antenna was modeled. Now we
model an antenna that has an even shorter width. An antenna whose width is shorter than
its length is best for the excitation of the structure’s lowest order mode. This is one
reason why the 1.0 by 0.7 cm patch antenna was expected to yield better results than that
of patch whose width was 0.9 cm. The following are more reasons that this sections
antenna was expected to surpass the previous patch antenna’s performance: Just as in the
previous section, the feed points of the 1.0 by 0.7 cm antenna are unmistakably on the
antenna’s center line; however because the width is decreased the impedance bands
should peak at higher values. In addition, the decrease in width means that the width to
height ratio is decreased, which should allow the electric field lines to concentrate less in
the substrate resulting in improved radiation.

Feed points 72,88,104,120 and 136 were investigated in conjunction with the

antennas of this section. The Feed map for this antenna is shown in figure 4.44.

4.5.1 1.0 by 0.7 cm Patch Antenna with e, = 7 and u, = 20

Thickness 0. 020 cm

Thickness 0.020 cm produced no desirable impedance characteristics.

Thickness 0. 025 cm

Thickness 0.025 cm produced no desirable impedance characteristics.

118

Thickness 0.030 cm

Of the five feed positions only position 88 qualified. Feed position 88 had single
point of resonance very close to a 3052 peak within a 15 MHz band. This point of
resonance occurred at 1.135 GHz, this antenna was expected to be resonant at 1.0 GHz.
The impedance characteristic for this feed position is shown in figure 4.66.

Feed position 88 was used to acquire the radiation data. The radiation patterns foe
E9 and E9 in both of the principle planes are well shaped. In the E-plane E9 has a
maximum gain of about -0.5dB dB and E, has a maximum gain of about -3.9 dB.
However in the H-plane, E9 has a maximum gain of about —-0.6 dB and E0 has a
maximum gain of about —3.9 dB. So although the pattern shapes are fine, their spatial
power distribution is somewhat warped. The gain of this antenna far surpasses those seen
previously. The patterns for E9 and E, for the E- plane are shown in figures 4.67 and

4.68.

Thickness 0.035 cm

Thickness 0.035 cm produced no desirable impedance characteristics.

119

Probe Feed Location Map For a 1.0 by 0.7 cm Patch

 

Figure 4.44

120

Impedance at Feed

Polnt 88

Real Part of Impedance vs. Frequency for 1.0 by 0.7 cm

Patch wIth e, = 7, u, = 20 and Thlckness = 0.030 cm

 

 

 

Frequency (GI-I2)

Figure 4.45

121

 

 

 

-— Reel(ZIn)
------ Imam»

 

.__ ooooo
u...“ ........
.9...-
.‘

 

Radiation Pattern for E ¢ vs. Theta with Constant Phi = 90

330/ 30

<1. 2“

— so
I \ ‘Wso -28'—-18'—o

,//>/

\ \\__.—~ \ /
X\\ ____//>1/50

180

”\§

2C

 

Figure 4.46

122

 

Radiation Pattern for E9 vs. Theta with Constant Phi = 90
o

330 30

m ’w 90
\

0
248 \
>\ ..

180

 

Figure 4.47

123

 

4.5.2 1.0 by 0.7 cm Patch Antenna with e, = 7 and "r = 25
Thickness 0. 020 cm
Of the five feed positions, only position 136 qualified. Feed position 136 has two
resonance points, the first of which occurs at a frequency 200 MHz lower than expected.
The second occurs 100MHz lower than the expected frequency at 1.015 GHz. This
second resonant point falls within a real impedance band with a width of about 15 MHz
and an impedance peak of 1552. The impedance characteristics are shown in Figure 4.48.
The radiation patterns were collected using feed point 136, as it was the only
viable feed position for these antenna dimensions and substrate properties.
The radiation patterns for both electric field components were well shaped in both
principle planes. The gain for the E9 component peaks around —25dB in the E- plane, and
E9 peaks around —38dB in the same plant, it is expected to be negligible in this plane. In
the H-plane, the E9 component peaks at around -26dB and E9 around —38dB and it is
expected to be zero. The electric field patterns in the E- plane are shown in Figures 4.49

and 4.50.

Thickness 0.025 cm

Thickness 0.025 cm produced no desirable impedance characteristics.

Thickness 0.030 cm
Of the five feed positions, only position 72 qualified. Feed position 72 had three
points of resonance occurring approximately at frequencies 0.965, 1.095 and 1.170 GHz.

The first resonant frequency does not occur within a real impedance band, the other two

124

points occur within two separate and desirable real impedance bands. The first resonance
occurs within a band that peaks at 3852 and has a width just over 10 MHz. The second
point occurs within a band that peaks at 5952 and has a width of about 15 MHz. These
points occur equidistant from the 1.133 GHz the resonant frequency that was expected.
These impedance characteristics are shown in Figure 4.51.

The radiation results occurring at the third point of resonance for feed 72 will be
presented since they have the highest gain. The E9 pattern is slightly misshaped in the E-
plane for —90° < 0 < 0° and for 0°<0<90° in the other principle plane. E9 is well shaped
in both principle planes. In the E- plane, the gain E9 reached a maximum of about —19 dB
the E9 reached a maximum of -24 dB. In the other plane, E9 reached gains of -l9dB and
E9 reached a maximum of —24 dB. The patterns for the electric field components in the E-

plane are shown in Figures 4.52 and 4.53.

Thickness 0.035 cm

Of the five feed positions investigated, positions 88 and 136 qualified. Feed
position 88 has two resonance points. The first occurred at f, 2 1.015 GHz occurred
within a band that peaks as 1252 with a width of 20 MHz. The second point occurs at
1.235 GHz on the declining side of an 1152 peak hitting 652 on the downward slope. Feed
position 136 has one resonant point at frequency 1.115 GHz occurring just off of a 1352
peak with a bandwidth just over 30 MHz. The impedance characteristics for feeds 88 and
136 are shown in Figures 4.54 and 4.55

The radiation characteristics were collected using feed point 136. The E9 patterns

are misshapen for -45° < 0 < 90° in both principle planes. The E9 component patterns are

125

well shaped in both principle plains. In E-plane, the E9 component has a maximum gain
of -25 dB and the E9 component reaches a gain of -28 dB. In the other principle plane,
E9 reaches a maximum gain of -24 dB and the E9 component a maximum gain of -28
dB. The radiation patterns for the components in the E- plane are shown in Figures 4.56

and 4.57

126

Impedance at Feed
Point 136

Real Part of Impedance vs. Frequency for 1.0 by 0.7 cm
Patch wIth e, = 7, u, = 25 and Thlckness = 0.02 cm

 

—- Reelflh)
------- lmeg(Zln)

 

 

 

  

 

 
  

'0“...

i: 1;?
.M

d
V V
-1 Q i i '- '- '1
. o : : 5 r 0. v-
: t . .' 1-
.3 : : S f
: 3 i :'
.5 . :

Frequency (GHz)

 

1.154
12‘
125'

Figure 4.48

127

Radiation Pattern for E ¢ vs. Theta with Constant Phi = 90

'Qc

270

 

fi'

\4
I

 

k

Figure 4.49

128

Radiation Pattern for E9 vs. Theta with Constant Phi = 90
o

330

300

30

 

2“ \\ 4

210

 

150

180

Figure 4.50

129

Impedance at Feed

Polnt 72

so«
50‘
4o-

30~'

20-

10-

 

‘ 0.9

Real Fan of Impedance vs. Frequency for 1.0 by 0.7 cm

Patch wIth er = 7, u, = 25 and Thlckness = 0.030 cm

0.95 ‘

I
to
O.
F

Frequency (GI-I2)

Figure 4.51

130

 

 

— Reel(Zin)
------- |m89(Zin)

 

 

 

1.25 I

 

Radiation Pattern for E I» vs. Theta with Constant Phi = 90

3O

"‘\

,M ”

 

270 \

210

425—730 -26—-'16—o 9°

/
/ 120

 

/ 150

180

Figure 4.52

131

Radiation Pattern for E9 vs. Theta with Constant Phi = 90

§

270

 

IX / 120
/

Figure 4.53

132

Impedance at Feed

Polnt 88

15-
13‘
11‘

5.
31

Real Part of Impedance vs. Frequency for 1.0 by 0.7 cm

Patch with e, a 7, u, I 20 and Thlckness 8 0.035 cm

 

 

 

 

— ReeKZin)
....... Inn-win)

 

 

.1 E
-3 4D.

.5.

 

0.9 ‘

0.95 4

. fit r
v- : : In '-
l . C
o ._
’

Frequency (GHz)

Figure 4.54

133

U
ID
F

I
F

I—
N.
P

 

Impedance at Feed
Polnt 136

Real Part of Impedance vs. Frequency for 1.0 by 0.7 cm

 

Patch wIth e, = 7. u, = 25 and Thlckness = 0.035 cm
— ReeI(Z'n)
------- Imagmn)

 

 

 

30}
20‘
15'

mi

 

‘03
0.95 ‘
1.05 ‘
1.1 ‘
1.2 ‘
1.25 ‘

o
I

e

I

l

I C
o

e
In.
F:
'0
F:
D

 

Frequency (GHz)

Figure 4.55

134

Radiation Pattern for E ¢ vs. Iheta with Constant Phi = 90
5W3 , ‘
m \ ' '9 ,4

 

 

135

Radiation Pattern for E9 vs. Theta with Constant Phi = 90

270

 

 

120

 

150

180

Figure 4.57

136

4.5.3 1.0 by 0.7 cm Patch Antenna with a, = 9 and u, = 20

Thickness 0.020 cm

Thickness 0.020 cm produced nod desirable impedance characteristics.

Thickness 0. 025 cm

Of the five feed position investigated, only position 72 qualified. Feed position 72
had two points of resonance, the first occurring at 0.935 GHz within a 20 MHz band.
This point of resonance occurs very close to the band’s 4352 peak. The second point of
resonance occurs at a 1052 trough between two impedance peaks at about 0.985 GHz. The
impedance characteristic for feed position 72 is shown in Figure 4.58.

The radiation data was collected with the use of feed 72 at the first point of
resonance. The radiation patterns for both components of the electric field are shown in
Figures 4.59 and 4.60. The E9 component pattern in the H-plane is shown in Figure 4.61.
In the E-plane, both components of the electric field have the expected patterns. In this
plane, the E9 component reaches a maximum of -23 dB, and the E9 component reaches a
maximum gain of -30 dB. In the H-plane, E9 has the expected shape and a maximum gain

of -23 dB. In this plane, E9 is slightly misshapen and has a maximum gain of —30 dB.

Thickness 0.030 cm

Thickness 0.03 cm produced no desirable impedance characteristics.

Thickness 0.035 cm

Thickness 0.03 cm produced no desirable impedance characteristics.

137

Impedance at Feed

Polnt 72

Real Part of Impedance vs. Frequency for 1.0 by 0.7 cm
Patch wIth e, I 9, u, 8 20 and Thlckness I 0.025 cm

40‘

 

 

Frequency (GI-I2)

Figure 4.58

138

.......

0"".

 

 

— ReeKZIn)
-~--~ Miami)

 

1.25 "

 

Radiation Pattern for E 0 vs. Theta with Constant Phi = 90

270

 

139

150

Radiation Pattern for Ea vs. Theta with Constant Phi = 90
o

 

 

 

 

«at
2" ("° mm”
/
- 4 ' /
. J //

180
Figure 4.60

140

Radiation Pattern for E9 vs. Theta with Constant Phi = 0

”“«§ ,0

A40 -30 xii—4: o
O

330

270 I

 

240 120

 

210 150

141

4.6 Conclusion of Analysis

4.6.1 Comparison to an antenna with u, = 1

When considering a patch antenna with a typical loading of 8, = 9 and u, =1 and a
length of 1.0 cm, the resonant frequency of the lowest order mode would be 5 GHz.
Keeping the substrate material parameters constant, the length would have to be
increased by a factor of five to achieve a one GHz frequency of operation. Alumina is a
popular substrate material with which to load patch antennas. It has the material
properties above, 8, = 9 and u, =1. An antenna on an alumina substrate was modeled with
the goal of achieving the smallest possible antenna that would function in the L-band.
This antenna was modeled with two different substrate heights: 0.025 cm and 0.030 cm.
For thickness 0.025 cm, only three feed points of eight came close to qualifying, and no
points were useful with a thickness of 0.030 cm. Of the three feed points, 136 had the
highest bandwidth and impedance peak. For feed 136, the resonance frequency occurred
at about 1.60 GHz within an impedance band that peaked at 4.552 and had a bandwidth: 5
MHz < BW < 10 MHz. The Impedance profile is shown in Figure 4.62.

The radiation data was collected at feed 136 for thickness 0.025 cm. Both
components of the E-field had the expected shapes in both principle planes. In the E-
plane, E9 had a maximum gain of 5dB, and E9 had a maximum gain of -1 dB. In the H-
plane, E9 had a maximum gain of 5dB, and E9 had a maximum gain of -1 dB. The
radiation pattems for both components in both principle planes are well shaped and
shown in Figures 4.63 and 4.64.

From all the data collected by way of the FE-BI Prism software, it is clear that

permeability does indeed reduce the resonant frequency of these structures by a factor

142

equal to its square root. The antenna gains were low for most of the antennas modeled.
Only one version of the patches investigated produced gains close to 0 dB, and none
produced gains above that.

The impedance profiles for most versions of this antenna surpassed those of the
alumina case. This is in terms of producing one or more feed locations that would result
in a suitable impedance bandwidth and peak impedance values. However none of the
antennas modeled surpassed the alumina model in terms of power gain. In regards to the
radiation patterns, many of the antennas modeled and the antenna on the alumina

substrate possessed radiation patterns that were well shaped.

4.6.2 Feed points and Resonance

In many cases the point of resonance missed the peak of a desirable band of
impedance by a few MHz or a few dozen MHz. Moving the resonant frequency to the
center of an impedance peak, without changing the input impedance of the antenna, could
make several of the patch antennas investigated more desirable. This could be achieved
by altering the antenna’s length with the only stipulation being that the goal of this
research requires that the antenna length not exceed 1.0 cm.

The maximum impedance should be decreasing as the probe moves closer to the
center. Often, we see in these figures that instead, the impedance increases as we move to
feed positions closer to the center. The antenna has a length of 1 cm and therefore a small
range (0.0 to 0.5 cm) in which to choose feed locations. Therefore, the modes
propagating on the structure and the material properties may have more effect on the

impedance characteristics than the patch metal properties and the probe feed’s proximity

143

to the antenna center. Note that the feed point has a strong effect on what modes are

excited and their strength.

4.6.3 Real Impedance Bandwidth and Magnitude

For the 1.0 by 1.0 cm case, thicknesses less than 0.030 cm proved too thin to
produce desirable impedance characteristics. Increases in the substrate thickness seemed
to increase the impedance bandwidths of the antennas with respect to the qualifying feed
points. Though the qualifying feed point positions changed to adjacent positions with
increases in thickness, the bandwidths increased consistently with increased thicknesses
for all substrate materials. There was no detectable link between permeability and
bandwidth.

For the 1.0 by 0.9 cm case, thicknesses less than 0.25 cm proved too thin to
produce desirable impedance characteristics. It seems that the decrease in width has
relaxed the constraints on how thin the substrate could be in order to produce desired
impedance profiles. This is to be expected since a decreased width allows for less
trapping of the fields in the substrate in the same way increased thickness does. However,
for the antenna of this shape, only one substrate (£,=7 and u,=20) produced useful feed
locations for more than one thickness; therefore, there was not much data to draw on for
comparison. Furthermore, the impedance bandwidth decreased slightly for an increased
thickness for that single substrate.

For the 1.0 by 0.7 cm case, all thicknesses produced useful feed points, depending
on the material properties of the substrate used. Only one substrate (8.=7 and tip-25)

produced useful feed points for more than one thickness, and this one supports the link

144

between increased thickness and bandwidth.

Overall, for all antenna versions, there was no detectable link between increased
permeability and increased bandwidth. In comparison to the alumina case, there is a
strong connection between improved bandwidths and large permeability values.
Considering the comparison between the antennas on magnetic substrates, it may be that
a permeability increase of 25% is not enough to detect its affect on impedance
bandwidth, or there may be an upper limit on the values of permeability that would
improve the impedance bandwidths.

There was a weak link between impedance peak levels and increased permeability
between antennas having permeability values greater than unity. Perhaps the extra modal
behavior in many of the versions of this patch hid any connections by shifting or
eliminating expected resonance points. The fact that the feed positions changed often also
adds to the difficulty in forming any conclusions about the effects of permeability on the

magnitude of the real impedance bands.

4.6.4 Radiation
The 1.0 by 1.0 cm cases do not support the idea that permeability would improve
the radiation levels of an antenna. However, all the gain levels for the 1.0 by 1.0 cm

antennas were —20 dB and lower. Occasionally, the pattern of the E9 components was

distorted in one of the principle planes.

The 1.0 by 0.9 cm cases do support the idea that permeability improves the
radiation properties of the antenna. The substrate with the highest permeability (£F7 and

11,:25) averaged gains 10 dB higher than the other two substrates (8,=7 and “1:20) and

145

(er-:9 and u,=20). The support for this idea remains weak, however, in that the highest
gains for this antenna were on the order of -20 dB.

The 1.0 by 0.7 cm case essentially strongly opposed the idea of improved
radiation with increased permeability. The substrate with £,=7 and ”1:20 had the best
gains for these antennas and the best of all antenna configurations investigated. Its gains
nearly reached 0 dB, which is far better than the -20 dB cases above. A 25 % increase in
permeability may not be enough to see improved radiation or the benefits of permeability
may be hidden in the general radiation problems these antenna have had.

One limiting factor in the gains of our antennas is the fact that only the gain levels
corresponding to frequencies within real impedance bands are observed. For many
configurations FE-BI Prism does show higher gains at frequency points for which
resonance is not reached or where resonance is reached and the real impedance is nearly
zero.

Overall, the antennas we have investigated have behaved more like resonators
than antennas. The effects of permeability on power gain may be better assessed with

more resonating antenna types to compare too.

146

Impedance at Feed

Polnt 136

N U & 01
n l l I

Real Part of Impedance vs. Frequency for 3.0 by 3.0 c

Alumlna Case

Patch wIth er = 9, u, = 1 and Thlckness = 0.025 cm

 

II

 

 

— Ree|(Zin)
------- lmeg(2in)

 

 

-1-

-2-
~31

 

-5.

1.5

1.55

(D m

.—' «2

Frequency (GHz)
Figure 4.62

147

.....

 

Radiation Pattern for E vs. Theta with Constant Phi

=90

 

 

s
Wk ‘ _.
\ '(ix _2°— "° °
2.. \ J / m
210 Kin/150

Figure 4.

148

63

Radiation Pattern for E9 vs. Theta with Constant Phi = 90

 

 

24o \ ’ 120

 

210 150
180

Figure 4.64

149

CHAPTER 5

CONCLUSION

In this thesis research, microstrip patch antennas on magnetic substrates were
analyzed with the use of the Finite Element Boundary Integral (FE-BI) hybrid software.
The Engineering Research Center for Wireless Integrated Microsystems (WIMS)
provided the motivation for this research; as a result of their quest for a very small
antenna that would be resonant in the lower portion of L-band. The goal of this thesis as
discussed in Chapter 1, was to investigate the use of magnetic substrates for antenna
miniaturization. The expectation was that a material with a permeability constant greater
than one, would reduce the resonant size of an antenna while improving its performance
in some areas. The results obtained from the FE-BI software confirms that the resonant
frequency of the antenna can be reduced. Also, as discussed in Chapter 4, the impedance
bandwidth of the antenna is improved over the use of a material with only a high
permitivity constant. However, the antennas ability to radiate energy did not seem to
improve as a result of the magnetic substrate and may have worsened.

Equations were derived to aid in predicting some of the changes in antenna
performance that a permeability constant greater than one would produce. These
equations and some of their derivations are presented in Chapter 2. Descriptions of FEM
and Finite Difference Time Domain (FDTD) computational techniques were provided in
Chapter 3, as well as reasons for the selection of FEM. Also provided in Chapter 3 is a
brief description of the theoretical formulation for the Prism Finite Element-Boundary

Integral software that provided the results presented in Chapter 4.

150

Lastly, the performance of several microstrip patch antennas with varying
dimensions and material properties were presented in Chapter 4. The performance of an
antenna miniaturized via an increased permitivity constant (with a permeability constant
equal to one) was also presented as a comparison. The antennas on magnetic substrates
were compared with like-kind and analyzed in order to isolate the contribution of the
magnetic properties of the substrate. The relationship between permeability, bandwidth
and impedance were visible to some extent. The relationship between permeability and
radiation needs more study.

The goal of this research was reached; more research would yield deeper insight
into the affects of permeability on antenna radiation and perhaps provide a better link
between permeability and impedance bandwidth. Use of other computational techniques
like Method of Moments and Finite Difference Time Domain in conjunction with the FE-
BI hybrid technique used could provide more insight into the problems investigated.

Future research could involve additional variations in the constitutive properties
of the substrates modeled, including a wider range of the permitivity and permeability
values. Also more antenna patch dimensions can be modeled, to further isolate the affects
of the materials used from the patch parameters. More software tools can be employed as
mentioned above. Finally, if the necessary materials are acquired and are cost effective,
experiments involving tangible antennas may provide many answers to the problems at
hand.

The art of antenna miniaturization is expanding and reaching new heights, as
industry and consumer demand for wireless technology fuels the need for small antennas.

Research that produces the techniques necessary to fulfill those needs is a valuable

151

commodity. Consumer electronics, medical devices/technology, and the military alike
can benefit from the growth and development of wireless technology. This thesis is a part

of a large body of research that aims to and is needed to nurture such growth.

152

[l]

[2]

[3]

[4]

[5]

[6]

17]

[8]

[9]

[10]

BIBLIOGRAPHY

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Das, N. and Chowdhury, S. K. “Microstrip Rectangular Resonators on
Ferrimagnetic Substrates.” Electronic Letters, vol. 16, no. 21, pp. 817-818, 9
October 1980.

Garg, Ramesh, et al. Microstrip Antenna Design Handbook. Boston: Artech
House, 2001.

Lo Y. T. et. al. ”Theory and Experiment on Microstrip Antennas.” IEEE Antennas
and Propagation March 1979: 137-144.

O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, 4th Ed., New
York: McGraw-Hill, 1989.

Sadiku, Matthew N. 0. Elements of Electromagnetics. Philadelphia, PA:
Saunders College Publishing / Harcourt Brace College Publishing, 1994, 1989.

Sadiku, Matthew N. 0. Numerical Techniques in Electromagnetics. Second
Edition. Boca Raton: CRC Press LLC, 2001.

Silvester, P. P. and Ferrari, R. L. Finite Elements for Electrical Engineers.
Cambridge: Cambridge University Press, 1983, 1990.

Stutzman, Warren L. and Thiele, Gary A. Antenna Theory and Design, Second
Edition. New York: John Wiley & Sons, Inc., 1998.

Volakis, John L., et al. Finite Element Method for Electromagnetics. New York:
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