AN ABSTRACT OF THE THESIS OF
Ashraf
Mohammed
Sultan
for
the
degree
of
Doctor
of
Philosophy in Electrical and Computer Engineering presenAlril 28, 1986
ted on
Title:
THE RADIATION CHARACTERISTICS OF OPEN- AND CLOSEDRING MICROSTRIP ANTENNAS
Redacted for privacy
Abstract approved:
Dr. Vijai°K. Tripath1-
The purpose of this study is to investigate the
radiation characteristics of open- and closed-ring micro-
strip structures for applications as microwave antenna
elements.
The expressions for the radiation fields of
these structures are obtained from the aperture fields of
the ideal cavity model associated with the microstrip
structure.
The expressions for the radiation fields are
then used to calculate the radiation characteristics of
these microstrip structures.
The radiation fields of the ideal gap structure (gap
angle
0) are derived in Chapter 3.
For this case the
aperture fields can be expressed either in terms of the
spherical Bessel functions (odd-modes) or in terms of the
Bessel functions of integer order (even-modes) which is
shown to result in a convergent series or closed form
expressions respectively for the radiation fields.
study of the radiation patterns for the various modes of
the ideal gap open-ring structures reveals that the first
radial TM12 mode can potentially be an efficient useful
mode for applications of this structure as a radiating
element.
The radiation fields of the general annular and cir-
cular sectors are numerically examined in the following
chapter in terms of the various physical parameters of
these structures.
The derived expressions for the radia-
tion fields are used to study the radiation patterns,
radiation:peak in the broadside direction and the beam-
width of these structures for various sector angles,
This includes the
widths and the modes of excitation.
special cases of quarter, half, three-quarters and ideal
gap open-ring structures.
It is shown that the radiation
properties of these structures are comparable with other
microstrip antennas and should result in the applications
of such sectors as useful radiating elements.
An important part of this work
is
the study of
closed-ring microstrip structures which is presented in
Chapter 5.
It is shown that all of the useful properties
of such structures for various modes of excitation can be
derived from the radiation fields.
The expressions for
the antenna characteristics such as total radiated power,
directivity, bandwidth and input impedance are derived
from the expressions for the radiation fields.
These
parameters are then evaluated for the useful TM12 and TM11
modes for typical structures.
The results should be help-
ful in the design of such structures for applications as
radiating elements.
Copyright by Mohammed Ashraf Sultan
April 28, 1986
All Rights Reserved
THE RADIATION CHARACTERISTICS OF OPEN- AND CLOSED-RING
MICROSTRIP ANTENNAS
by
Mohammed Ashraf Sultan
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
DOCTOR OF PHILOSOPHY
Completed
April 28, 1986
Commencement June 1986
APPROVED:
Redacted for privacy
Professor of Electrical and Computer Engineering in Charge
of Major
Redacted for privacy
Head of the Electrical airld Computer Engineering Department
Redacted for privacy
School 1
Dean of Gradu
J
April 28, 1986
Date thesis is presented
Typed by Admin. HQ
for
Mohammed Ashraf Sultan
DEDICATION
looking
For
everyone
who
is
For
everyone
who
wants
to
for love
know
how
and
peace.
fair god is.
For my mother for her love, patience and determination
to help me walk through this life with a smile.
For my father for his financial support of my mother's
goal.
ACKNOWLEDGEMENTS
Thank
this
God
for the health and energy I found
course of study,
assuring him that I'll go
during
towards
achieving the second dream regardless of lack of health or
problems I may forsee.
Thanks
is extended to include all of the
faculty members,
department
staff and students for their cooperation
and encouragement during this course of study,
especially
Dr. Tripathi for his patience.
Thanks to Dr.
Oberhettinger for his cooperation
via
mail.
Thanks to Admin.
HQ,
especially Ms. Celeste Correia
for diligence in typing this thesis.
TABLE OF CONTENTS
1.
2.
3.
4.
ORIENTATION OF THE STUDY
1.1 Microstrip Antennas
Statement of the Problem
1.2
REVIEW OF RELATED LITERATURE
2.1 The Radiation Characteristics
2.2 Analysis of Microstrip Antenna
Principle of Field Equivalence
2.3
2.4 The Radiation Fields
2.5 Classification and Discussion of Other Analysis Methods
2.5.1 Magnetic (or Electric) Wall Methods
2.5.2 Patch Current Distribution Methods
2.5.3 Spectral Domain Methods
2.6 Comparison of Cavity Model with Other Analysis Methods
2.7
Concluding Remarks
1
3
6
7
7
10
15
18
21
22
25
27
28
31
ANALYSIS OF THE OPEN-RING MICROSTRIP ANTENNA
3.1 Antenna Analysis
3.2 Formulation of the Integral Equation
3.3 Method of Solution for Radiation Fields
Results and Discussion
3.4
3.5 Concluding Remark6
32
33
38
THE RADIATION BEHAVIOR OF AN ANNULAR SECTOR MICROSTRIP ANTENNA
4.1 Review of Resonant Frequencies and Field Distribution
4.2 Derivation of Radiation Fields
4.2.1 Radiation Fields from the Curved Aper-
51
40
45
50
52
56
58
tures
4.2.2
Radiation Fields from the Linear Apertures
4.3
4.4
The Gap Effects and the Radiation Peak
Results and Discussion
4.4.1 Quarter and Three-Quarters Ring Anten-
61
62
65
67
nas
4.5
5.
4.4.2 Half-Ring Antenna
4.4.3 Ideal Gap Open-Ring Antenna
4.4.4 General Annular Sector Antenna
Concluding Remarks
THE RADIATION CHARACTERISTICS OF A CLOSED-RING
MICROSTRIP ANTENNA
Review of the Radiation Fields
5.1
Method of SoluFormulation of the Problems,
5.2
tions and Results
5.2.1 Total Energy Stored
5.2.2 Radiated Power, Losses and Q-Factor
5.2.3 Resonance and Input Impedance
71
76
80
88
89
90
91
93
94
106
5.3
5.4
6.
5.2.4 Radiation Resistance
5.2.5 Bandwidth
5.2.6 Efficiency, Directivity and Gain
Typical Example
Concluding Remarks
125
SUMMARY AND CONCLUSIONS
127
BIBLIOGRAPHY
APPENDICES
Appendix A - Vector Transformations
Appendix B - Power Radiated from a
Microstrip Antenna
113
116
119
123
124
133
Closed-Ring
138
LIST OF FIGURES
1.1
General configuration of a microstrip antenna
1.2
Some of the investigated microstrip antenna configurations
2.1
A microstrip-fed rectangular patch
Probe or coax feed
(a)
Line feed
(b)
11
2.2
The cavity model of microstrip patch antenna
11
2.3
The choice of selecting a closed surface
17
2.4
Block diagram for computing radiated fields from
known M sources
18
2.5
Example to calculate the phase delay between any
point in the source and the observation point
19
2.6
The concept of segmentation and
techniques
desegmentation
23
2.7
Wire-grid model of microstrip patch antenna
3.1
The open-ring microstrip antenna
view of the magnetic wall model
3.2
Field analysis for axial and radial modes of
ideal gap open-ring antenna
(a)
Normalized E z versus p
(b)
Normalized E versus
Field distribution
(c)
3.3
Radiation patterns of an
antenna for axial TM11,
(excluding gap fields)
3
26
top
33
an
37
open-ring
TM31 modes
48
3.4
Radiation patterns of an ideal gap open-ring
antenna for radial modes (excluding gap fields)
TM12 mode
(a)
TM22 mode
(b)
49
4.1
Normalized
resonant frequency versus sector
angle for TM11 and TM12 modes for a typical case
53
4.2
Normalized Ez of annular sector for various secfor angles
as a function of p
(a)
as a function of 4
(b)
54
ideal
and
gap
TM21 and
the
4.3
Equivalent magnetic current sources of an annular sector antenna
56
4.4
Radiation patterns for a quarter-ring antenna
for TM11 mode with different widths
(a)
(b)
for TM11 and TM21 modes
for TM12 mode
(c)
68
4.5
Radiation patterns for a three-quarters-ring
antenna
(a)
for TM11 mode with different widths
for TM11 and TM21 modes
(b)
(c)
for TM12 mode
69
4.6
Radiation pattern of a half-ring antenna excited
in TM11 mode
72
4.7
Radiation patterns for the half-ring antenna
excited in TMil mode
for different widths
(a)
for the limiting case as 'a' becomes small
(b)
73
4.8
Radiation patterns for the half-ring antenna for
TM11, TM21 and TM31 modes
E-plane (b) H-plane
(a)
75
4.9
Radiation patterns for the
excited in TM12 mode
antenna
75
4.10 Gap effect on the radiation patterns of an ideal
gap open-ring antenna excited in TM11 mode
77
4.11 Radiation patterns for the ideal gap open-ring
antenna excited in TM11 mode
for different widths
(a)
for the limiting case as 'a' becomes small
(b)
78
open-ring
79
4.13 Gap effect of the radiation patterns of an ideal
gap open-ring antenna excited in TM12 mode
79
4.14 Radiation patterns for annular sector antenna
excited in TM11 mode for different sector angle
81
4.15 Radiation patterns for annular sector
excited in TMil mode
a=135° (d)
(c)
a=85°
a=45°
(67
(a)
a=315° (h)
a=225° (g)
a=210° (f)
(e)
antenna
83
sector
antenna
half-ring
4.12 Radiation patterns for the ideal gap
antenna for TM11, TM21 and TM31 modes
4.16 Radiation patterns for annular
excited in TM12 mode
(a)
a=60°
(b)
a=120°
a=170°
a=355°
87
5.1
The closed-ring microstrip antenna and its equivalent cavity model
90
5.2
Radiated power vs h for different widths and Er
for a closed-ring antenna excited in TM11 mode
102
5.3
Radiated power vs frequency for different widths
and h for a closed-ring antenna excited in TM11
mode
103
5.4
Q-factors vs h for different widths of a closedring antenna excited in Tn.]. mode
107
5.5
Q-factors vs h for different Er for
ring antenna excited in TM11 mode
a closed-
107
5.6
Resonant input impedance vs h for different
widths and er for a closed-ring antenna excited
in TM11 mode, outer-edge fed point
109
5.7
Input impedance vs frequency for different h for
a closed-ring antenna excited in TM11 mode
(a)
h = 160 mils
(b)
h = 80 mils
111
5.8
Input
impedance vs frequency for different Er
for a closed-ring antenna excited in TM11 mode,
112
outer-edge fed point
Radiation resistance vs h for different widths
and Er for a closed-ring antenna excited in TM11
mode, outer-edge fed point
114
5.10 Radiation resistance vs h for different er for a
closed-ring antenna excited in TMil mode, inneredge fed point
115
5.11 Radiation resistance vs frequency for different
b/a ratios and h for a closed-ring antenna excited in TM11 mode, outer-edge fed point
115
5.12 Bandwidth vs h for different widths and Er for a
closed-ring antenna excited in TMil mode
118
5.13 Antenna loss vs h for different widths and E r
for a closed-ring antenna excited in TM11 mode
120
5.14 Directivity vs h for different widths and Er for
a closed-ring antenna excited in TMil mode
122
5.15 Gain vs h for different widths and Er for a
closed-ring antenna excited in Tn.]. mode
122
5.9
LIST OF TABLES
geometries of micro-
14
2.1
Field solution for various
strip patch antenna
2.2
Summary of results for typical simple
structures
30
4.1
Comparison of the radiation pattern of a rectangular patch with an annular sector with large
radius of curvature and small angle
66
4.2
beamwidth and
Calculated resonant frequency,
total radiation peak of the 90° and 270° antennas excited in TM11 mode for different widths
70
4.3
Effect of changing the annular width on
radiation behavior of a half-ring antenna
the
74
4.4
Effect of changing the annular
radiation behavior of an ideal
antenna
width on the
gap open-ring
80
4.5
Relative broadside radiation and resonant frequency compared to a closed-ring for different
structures for TM11 and TM12 modes
82
THE RADIATION CHARACTERISTICS OF OPEN- AND CLOSED-RING
MICROSTRIP ANTENNAS
CHAPTER 1
ORIENTATION OF THE STUDY
The
use
of
microstrip to construct antennas
relatively recent development.
is
a
The first proposal for its
use in the microwave telecommunication in the modern sense
of
the word is traced back to the 1953's
Deschamps
and Bhartia 1980) introduced the concept.
(Bahl
receive
not
when
It
did
enough attention until the early 1970's
be-
cause of the low gain and narrow bandwidth associated with
this type of antenna.
conformal
spacecraft
With progress in the technology of
antennas,
interest
antennas has increased in recent years.
in
microstrip
Significant proand
has been made in developing various techniques
gress
introducing new models which are important in the analysis
of the radiation characteristics of such antennas.
The
first practical antenna was patented
Munson
by
(1974) who introduced the transmission line model and used
it
to
patches.
analyze
In
basic rectangular and
1975,
square
eight square elements were
microstrip
by
used
and Klein to develop one of the first
microstrip
phase scanned arrays (Bahl and Bhartia, 1980).
The anten-
Sanford
na
operated at 1.5435 to 1.5585 GHZ and was used to
pro-
vide direct communications between aircraft and ground.
2
mathematical
the attention to other
this stage,
At
models sufficient to analyze other microstrip patch shapes
was increased.
In 1976,
wire-grid model,
modified it in Nov. 1977, and used it to
In
rectangular and circular microstrip patches.
analyze
Lo
1977,
Agrawal and Bailey reported the
introduced the
et al.
cavity
modal-expansion
model technique and used it to analyze rectangular, circuand
semicircular,
lar,
and Mink 1981).
(Carver
strip
and
Carver (1979) used the model to
demonstrated theoretically and
validity to analyze circular and
patches.
patches
circular
and
In 1979, Lo et al. reported a cavity
microstrip patches.
its
microstrip
design equations for rectangular
formulate
model
triangular
experimentally
rectangular
micro-
similar report was also published
A
by
Derneryd (1978).
The
antennas are thin planar
microstrip
that are lightweight,
low cost,
and easy to manufacture.
They can be conformal with the surface of
kets,
also
aircraft,
structures
roc-
missiles,
vehicles and satellites because they are
low flat profile antennas (Gabriel 1976;
Garvin
et
al. 1977; Mailloux 1977; Ball Aerospace Systems Division Data
for
Sheet n.d.).
used
These structures have also been
other applications at microwave
These
frequencies.
include their use as a single element radiator for biomedical
and
applications and measurements (Bahl et
in
microwave
circuits and devices
as
al.,
1980),
filters
resonators (Wu and Rosenbaum 1973; Lo et al. 1979).
and
3
1.1
Microstrip Antennas
The term "microstrip" is the standby name of a micro-
wave circuit configuration which is constructed by printed
circuit
used
technology.
together
to
microstrip antenna.
There are three terms which can
be
definition
of
constitute the precise
These terms are:
microstrip device,
printed antenna and flush-mounted antenna, which imply the
following:
a.
Term represents the nature of its construction.
b.
Term
manufacturing
represents the nature of its
process.
c.
Term
represents
the
nature of
its
main
use,
because it can be conformal with the surface of a
mechanism, or of a vehicle respectively.
Figure 1.1 shows a microstrip antenna in its
simplest configuration.
general
It consists of a radiating patch
constructed on a thin dielectric sheet of thickness h over
Radiating Patch
-r
h
Ground Plane
Dielectric Substrate
Fig. 1.1
General Configuration of a Microstrip Antenna
4
a
ground
etching techniques.
photo-
and
using printed-circuit board
plane
The dual-copper-coated Teflon-fiber-
glass is a commonly used board because it is flexible
and
allows the antenna to be curved to conform to the mounting
The
surface.
circular
conventional shapes (such as rectangular or
but
their
of
are commonly used because of simplicity
discs)
shape,
antenna patch conductor can be any
analysis and easiness of their fabrication.
Typically, three basic categories are
used to
classify the class of microstrip antennas.
These categor-
ies are:
Microstrip Patch Antennas
1.1.1
A
earl-
microstrip patch antenna is the one defined
There
which
antenna
These
include rectangular,
are
patches
many shapes of conducting
ier.
radiation performances can
square and
be
for
evaluated.
circular
patches
which have been investigated in detail for their radiation
properties.
is
The bandwidth associated with these antennas
usually less than a few percent.
However,
it can be
improved either by increasing the thickness of the dielectric
substrate or using a lower value of dielectric
stant which can be obtained by using composite
con-
materials.
There are also two other methods whereby the bandwidth can
be improved (Munson, 1974).
These methods are:
5
Increase the patch inductance by cutting holes or
a.
slots in it.
Add
b.
reactive
components
to improve
the
match
the patch and the feed line or simply to
between
reduce the VSWR.
Microstrip Traveling-Wave Antennas
1.1.2
A microstrip traveling-wave antenna is an open structure which guides the electromagnetic
by radiation into space.
patch
antenna
that
(e.g.
Comb
traveling-wave
or
Line and Rampart Line)
conductor in the form of a'chain (e.g.
concentric circles of different radii).
a
periodic
rectangular chain,
The structures of
type of microstrip antenna can be designed such that
this
the
the
can be either in a form of an ordinary long TEM
structure
line
accompanied
Its construction is similar to a
differs in
but
waves,
beam lies in any direction
main
from
broadside
to
endfire when the antenna is terminated in a matched resistive load.
either
reduces
For the limiting case when the matched load is
open or short circuit,
the traveling-wave antenna
to a standing-wave antenna with the main beam
the broadside direction.
of the previous one.
in
This category is a special case
6
Microstrip Slot Antenna
1.1.3
microstrip slot antenna can be a radiating element
A
by cutting a slot in the ground plane of a
formed
strip
element and fed with a microstrip line.
can be rectangular (wide or narrow),
nular ring.
circular,
micro-
The
slot
or an an-
The main feature of this category of antenna
is the ability to produce unidirectional or
bidirectional
radiation patterns.
1.2
Statement of the Problem
The purpose of this study is to analyze the radiation
behavior of open- and closed-ring microstrip patch
nas
including the case of ideal gap open-ring
anten-
structure.
This family of structures are shown in Figure 1.2.
Disk
Three-Quarters
Disk
Ring
Three-Quarters
Ring
Fig. 1.2
Semi Disk
Semi Ring
Disk Sector
Open Disk
Ring Sector
Open Ring
Some of the Investigated Microstrip Antenna
Configurations
Our objective is to formulate a general technique
compute
to
the radiation fields and the radiation character-
istics of all of these structures.
7
CHAPTER 2
REVIEW OF RELATED LITERATURE
Numerous
used
char-
over the recent years to deduce the radiation
acteristics
of various microstrip antenna configurations.
In this chapter,
the method based on the cavity model and
principle
Huygens'
methods
discussed along
is
analysis
of
reviewed.
been
analytical and numerical methods have
of
other
the
briefly
and
classified
which are
Comparisons
with
these methods and
concluding
remarks are made at the end of the chapter.
2.1
The Radiation Characteristics
In general,
is
any microstrip antenna configuration
completely characterized in terms of the following:
a) Antenna Radiation Pattern:
of
E/H plane pattern.
It is defined in terms
For a Linearly
polarized
antenna, we have:
E-plane:
A plane which contains the E vector and
direction
the
pattern
is
of maximum
radiation,
a plot of R(e) for
= 0°
and
its
180°
or
where
R(e) =
H-plane:
the
(2.1)
1E6112 + 1E012
A plane which contains the H vector and
direction
of maximum
radiation,
and
its
pattern is a plot of R(e) for 0 = 90° or 270°.
8
The frequency
b) Antenna Bandwidth:
range
within
the performance of the antenna with respect
which
to some characteristics,
conforms to a
specified
It is calculated from
standard.
S - 1
B.W =
(2.2)
QT(S)1/2
where S is the VSWR (typically 1:2), and QT is the
total quality factor.
c) Antenna Input Impedance:
It denotes the impedance
presented by the antenna at its terminals
V2
(2.3)
Zi
2PT
where Vo is the terminal voltage at $ = 0°, and PT
is the total power fed into the antenna.
The ratio of the total power
d) Antenna Efficiency:
radiated
(Pr)
to
net power
the
fed
into
the
antenna
n
(2.4)
x 100
% =
PT
e)
Antenna
The ratio of the
Directivity:
radiation
intensity
to
the
average
maximum
radiation
intensity
D =
1/2 Re (E H *
0 $
Pr/41Tr2
E $ H *)
(2.5)
9
f)
G = D x
g)
It is defined as
Antenna Gain:
The
Beamwidth:
Antenna
which
(2.6)
n
is
equal
to the
half
beamwidth
power
width
angular
between
directions where the gain decreases by 3db or the
radiated
field reduces to 2-1/2
of the
maximum
value.
h)
It includes the total power
Antenna Power Loss:
radiated and the power dissipated in the radiator
conductors
and
the
imperfect
sub-
dielectric
strate, i.e.
(2.7)
PT = Pr + Pc + Pd
Antenna Q-Factor:
It is defined as
Wm
(2.8)
QT = 21Tf
PT
where
within
the
necessary for calculating
the
W T is the total energy stored
antenna.
It
is
impedance
input
at
frequencies
removed
from
the
total
resonance.
j)
Radiation
Resistance:
The ratio of
radiated to the square of the rms
power
antenna
current reffered to specific point, i.e.
2
Rr
Vo
2Pr
(2.9)
10
As a first step toward the determination of these
and underneath
the
The following section outlines the procedure
for
tics, we have to know the field
microstrip patch.
characteris-
structure
on
determining the fields underneath the patch by utilizing the
cavity
model.
2.2
Analysis of Microstrip Antenna
The general configuration of microstrip patch antenna
was illustrated in Figure 1.1.
thickness
the dielectric
Normally,
is very small compared to the wavelength on the
microstrip and substrate permittivity er is low to enhance
the
fields necessary for the radiation.
fed
either
plane
patch,
or
by
is
through
the
and terminates on the upper surface of
the
by a probe (or a coaxial
ground
The antenna
line)
a microstrip line printed on
top
of
the
dielectric substrate as shown in Figure 2.1 for an example
of a rectangular patch.
will
As a result of this,
the energy
transport along the feeding tool to the feed
point.
It spreads out into the region underneath the patch;
of
it will radiate into the space through the
leading to a complex boundary value problem.
radiation
some
substrate,
However, the
pattern of such structures can be evaluated
by
using equivalent sources on the boundary, if the fields on
the boundary can be determined.
11
(b)
(a)
Fig. 2.1 A Microstrip-Fed Rectangular Patch
(a) Probe or Coax Feed (b) Line Feed
This problem can be visualized easily, if we consider
the similarity between the region underneath the patch and
a parallel plate transmission line.
feed
point,
When waves leave the
they see an approximate open circuit as they
approach the patch perimeter.
This open circuit condition
the
microstrip
(high
impedance condition) suggests that
patch
behaves like a cavity because most of the energy is
reflected back.
model
top
This suggestion led Lo et al.
the microstrip antenna as a cavity bounded
(1979) to
on
and bottom by electric walls and on its perimeter
magnetic
walls as shown in Figure 2.2 with the
its
by
following
boundary conditions.
Electric Wall
Magnetic Wall
Fig. 2.2 The Cavity Model of Microstrip Patch Antenna
12
an x E = 0
;
on electric walls
an x H = 0
;
on magnetic walls
(2.10)
The
is the unit
an
where
model is based on the assumption that the
not vary along the z-direction since h << A0,
with
aE/az
wall.
cavity
vector normal to the
fields
i.e.
modes
= aH/az = o need only to be considered
(TMnm
This assumption together with (2.10) leads to
modes).
(2.11)
Ex = Ey = Hz = 0
The
ving
do
other field components can now be determined by
sol-
the scalar Helmholtz wave equation for Ez subject to
the boundary condition of the cavity wall, i.e.
(v2
with
the
magnetic
the
k2)
(2.12a)
0
boundary
conditions that aEz/a5n =
0
on
walls where k = w(11001/2 is the wave number
dielectric medium.
the
in
The magnetic field in the cavity
is then given by
H =
The
give
v x Ea z
(2.12b)
value
problem
resonant frequencies of the cavity for
various
eigenvalues knm of the above boundary
the
modes and lead to the expressions for the field
tion
underneath
exactly
ary
the patch.
The solution of
distribu(2.12)
is
the dual of resonance TE mode fields of an ordin-
waveguide whose metal boundary has the same shape
as
13
Table 2.1 shows some exam-
the patch effective boundary.
pies related to annular and circular microstrip structures
where
integer n refers to the mode number,
the
corresponds
to the order of the Bessel function
From this solution,
the
and
the equivalent sources on the
characteris-
boundary can be determined and the radiation
tics
v
m represents the mth zero of the eigenvalue equa-
integer
tion.
or
n
It should also be noted that the
can be calculated.
usually,
the radiated fields are determined in
in
and
coordinates
is
components.
specified
cylindrical
solution
spherical
Therefore, we must use vector transformations
which are summarized in Appendix A.
The radiation pattern
and the input impedance have been evaluated experimentally
and
They were able to obtain good correlation
workers (1979).
between both results for a rectangular,
semicircular
introduced
Effective
shapes.
patch
a circular, and a
dimensions
their analysis to account for
in
co-
based on this model by Lo and his
theoretically
the
were
fields
fringing outside the physical dimensions of the patches as
suggested by Schneider (1972).
The
microstrip
(Troughton 1969;
Luypaert
1973;
permittivity
and
line
is also dispersive
in
nature
Itoh and Mittra 1973; Van de Capella and
Kompa and Mehran
1975).
The
effective
the equivalent microstrip width can
be
expressed by the following empirical equations (Hammerstad
and Jensen 1980):
14
0
cos n4
Ez = Eo Jn(knmp) [
sin n.
JA(kmma) = 0
Disk
a
a
E z = E0 Jv(kvmp) cosy*
nw
J1(k
vma) = 0
v
=
v
;
a
Disk Sector
Ez = E0 Jv(kvmP) cosv4
n
Ideal Gap
Open-Disk
j.)(kvma) = 0
Ez
Ring
;
Eo[Jn(knmp)
JA(ka)
mm
Ys(k
n
nma) Yn(kn
Ez = Eo[Jv(kvmP)
0
Ideal Gap
Open-Ring
3
cos n4
sin n4
JZ,(kvma)
JZ,(kvinb)
Y(kvma)
YZ,(kvmb)
[Jv(kvmP)
JZ)(kvma)
v
YA(kvma)
nit
=
0
'i(kvirtb)
;
v = -a
,J(kmma)
Y v nb)
m
,..T(kvmb)
p)] cosv.
Y (k
Y1(knmb)
v
-
Ring Sector
P
a z 2w
JA(kmma) - JA(knmb)
YA(kmmb)
Y11-(kmma)
JZ)(kmma)
0 .
v = 7 ;
=
nmP)) cosv*
v
2....
=
T
Table 2.1 Field Solution for Various Geometries of
Microstrip Patch Antenna (Lo et al. 1979)
2n
15
120wh
We(f) =
Zo(f)[cre(f)]1/2
cr
re(f) = cr
ere(0)
1 + G(f/fp)2
where the subscript "e" refers to the effective values
at
the specified frequency, G and fp are empirical parameters
and ere(0),
and Zo(f) can be obtained from Hammer-
G,
stad and Jensen (1980).
to
Khilla (1984) used these formulas
analyze a closed-ring antenna by utilizing the
model
discussed
above
for Mil mode.
He gave
cavity
a
good
correlation between experimental and theoretical results.
The
iation
sources
next
step toward the determination of the
pattern
is
by
achieved
calculating
along the boundary of the patch.
section outlines this procedure in terms of
field
The
rad-
equivalent
following
Equivalence
and
Huygens' principles.
2.3
Principle of Field Equivalence
The
field equivalence is a principle theorem whereby
actual sources within a region, are replaced by equivalent
"fictitious"
sources,
such
that the latter produce
same fields within that region.
the
16
The equivalence theorem states
"The field in a source-free region bounded by a
(S) could be produced by a distribution
surface
of electric and magnetic currents on this sur-
face and in this sense the actual source distribution can be replaced by an "equivalent" distribution" (Schelkunoff 1936; 1943).
can be expressed by the vector Huygens' principle as
This
given by (Sommerfeld 1954):
(ds x
3koR
E =
1 vxj
47
[
S
(ds x E)e
E = 41
vxfs(-J;
R
R
+
3 vxvxf
we
s
ibe
jkoR
jkoR
x E)e
(2.15)
R
-
3
vxVxj (ds x E)e
s
we
1
R
(2.16)
[
where R is the distance from the observation to the source
point.
be
Notice that an exact solution of the E and H could
obtained if the exact boundary values Et (the
electric
tial
field)
and Ht
(the
tangential
tangen-
magnetic
field) (or more correctly Et or Ht) were known (Sommerfeld
1954).
As a consequence of the above theorem,
the
the fields in
source-free region can be determined once the tangen-
tial components of the electric and magnetic fields on
imaginary closed surface are known.
an
This can be achieved
by placing equivalent electric and magnetic current densities over the closed surface as given by,
J = an x H
(2.17)
17
Figure
surface
shows the choice of selecting
2.3
to bound the volume underneath
the
closed
a
The
patch.
upper and lower faces of S lie inside the conducting parts
Patch
t-
h
::
.
r
-.Ground Plane
Fig. 2.3
of
The Choice of Selecting a Closed Surface
the patch and the ground plane,
As
respectively.
a
result of this choice, no equivalent magnetic sources will
appear
parts
on
and
these faces of S since E t is zero
over
these
no equivalent electric sources will appear
the 'perimeter
faces since Ht = 0
along
the
on
perimeter.
This reduces the equivalent sources necessary to calculate
the actual fields outside S to:
a.
Electric
currents
on the upper surface
of
the
patch.
b.
Electric currents in the ground plane.
c.
Magnetic currents on the perimeter faces of S.
d.
Volume
polarization
currents (bound sources) in
the dielectric material outside S.
The
bound
sources can be treated as dipoles composed
of
positive and negative charges which make a minor contribution to radiated field because h is small, the permittivity
is
low and the electric field polarizing
the
medium
18
The equivalent electric and/or magnetic current
outside S is small.
sources can not be used to evaluate
radiation
of
the
formulate
the
problems in terms of the equivalent magnetic current sources on
the
structures (Elliott 1981; Balanis
magnetic wall.
the
1982).
We
will
fields
i.e.,
M =-251.1 x E
;
along the perimeter where 2
stands
for replacing the
ground plane by the mirror
image of the original M
sources.
2.4
The Radiation Fields
The
potential
known
(2.18)
auxiliary function F,
known as electric
be used to determine the fields
can
M sources as shown in Figure 2.4.
M --Kntegrator-4 F
Differentiator-*
vector
from
the
In the form
-4
of
Multiplier
(Trio)
Fig. 2.4
Block Diagram for Computing Radiated Fields from
Known R Sources
the Helmholtz wave equation,
F can be expressed in
terms
of M as follows:
V2F
kT
0
= _e
0"
(2.19)
19
where ko and eo are the free space wave number and permittivity respectively.
The solution of (2.19) can be given
by taking into account the phase delay due to the distance
R
between
any
point in the source and
the
observation
point as
-jkoR
F=
R e
I
ds'
(2.20)
411
S
For
example,
for
a circular aperture mounted on an
plane (Figure 2.5), R can be expressed as
Fig. 2.5 Example to Calculate the Phase Delay
Between Any Point in the Source and the
Observation Point
x-y
20
2rp' cos* )1/2
R = (r2 + pl2
-p
r
COS 41
1 - 2
coss)1/2
r
;
for phase variations
;
for amplitude variations
(2.21a)
r
where
* is the angle between the vectors r and r'
Ph.
and
cos* = ap,
ar
(ax sine cos, +
= (ax cosy' + ay sins')
+ay sine sin* + az cose)
= sine cos(*-*')
(2.21b)
Substituting (2.21a) into (2.20) reduces it to
2
F =
e
-jk o r
R e-3
p' cos *ds
,
(2.22)
r
S
Specifying
the sources in
cylindrical
coordinates,
and
use of (2.22) and (A.9) the spherical F components
making
can be written as follows:
Fe = C
[M,1 cose cos(*-*')+M4), cose sin(*-4)')S
-Mz, sine]
-jkop'cos*dsi
(2.23)
= C
F
sin(*-**)+Dy cos(4,-,01)] e-jkoplcos*ds'
I
-Mp
C
= co e-jkor/41Tr,
(1)
where
and
ds' = p' dp' d4'.
can be written as:
cos*
is
given
by
(2.21b),
Finally, the components of H and E
21
He '-iwo
;
H40 =-jwo F4
(2.24)
Ee = no H
where no,
to 120w.
ture
=-j
kn
co
#
FA
E
ko
-- Fe
=-n 0 H 9 =
co
'
the intrinsic impedance in free space is
equal
It should also be noted that the radiating aper-
can be mounted on an y-z or z-x ground
plane.
For
these cases, the analytical forms for the fields would not
be the same,
whereas the computed values will be the same
because
physical problem is identical in all
the
cases.
The only differences in the analysis will be in the formulation of R (Eq.
At
2.21) and ds' defined with (2.23).
this stage,
the radiation pattern and the
radiation characteristics can be deduced.
other
However, it may
be of some interest to briefly review some of the other useful tech-
niques along with the models which are used
analyze
to
microstrip
antenna configurations.
2.5
Classification and Discussion of Other Analysis
Methods
In the previous discussion, the reviewed
was based on three basic assumptions.
analytical
technique
These are:
a.
Modeling the antenna as a cavity.
b.
Introducing effective patch dimensions to account
for the fringing field.
c.
Ignoring the bound sources in the dielectric substrate outside the model.
22
Many other analytical and numerical methods have been used to
the properties of microstrip antennas.
study
methods
Some of these
that
have been used successfully in the analysis of microstrip structures
are discussed below.
It should be noted that the transmission
line
model (Munson 1974) is omitted from this review because of its
sim-
plicity and its limited applications to the rectangular
and
square
Other methods which approximate the patch boundaries by
magne-
microstrip patches.
2.5.1
Other Maznetic (or Electric) Wall Methods
tic (or eledtric) wall include the Green's function approach (Chadha
and Gupta
1980;
1981)
and
the
segmentation
and
(Chadha and Gupta 1981; Sharma and Gupta 1981; 1984;
desegmentation
Okoshi
1985).
For arbitrary patch shapes where formulation of Green's function
an analytical form may be difficult, numerical methods such as
tour integral (Okoshi and Miyoshi 1971) or a finite
(Silvester 1973) can be used.
boundaries imposed by
Moreover,
impedance
B.C.'s
the
have
modal
been
element
in
con-
method
solutions
for
introduced by
Carver (1979) in terms of "a modal-expansion cavity model."
The Green's function corresponding to the cavity model
source can be expressed in terms of the inhomogeneous wave
with
a
equation
as,
(v2
k2)
= -jwph 6(r\ro)
(2.25)
.
23
where
r and ro denote the field and source point
The method of images or the expansion method
tively.
Green's
respec-
function
in terms of eigen functions (Morse
Feshbach 1953) can be used to solve Eq.
(2.25).
of
and
The pat-
ches for which the solution can be constructed include:
rectangle,
an
a triangle (a 30° - 60° right-angled triangle,
equilateral
triangle),
a
triangle,
a circle,
and a
right-angled
isosceles
an annular-ring and a circular
and
annular sector.
The segmentation and desegmentation methods have been
developed (Okoshi et al.
determine
can
1976;
Gupta and Sharma 1981) to
the Green's function of patches whose
geometry
be expressed as a superposition of patches for
which
the Green's function is known as shown in Figure 2.6.
continuity
of
current and voltages on the
segmented
The
or
desegmented lines when expressed in a discrete form enable
one
to write the Green's function of the patch by utiliz-
Patch to be analyzed
Seg.
Fig. 2.6
Deseg.
The Concept of Segmentation and Desegmentation
Techniques
24
basic
ing
concepts from circuit theory.
both
In
tech-
niques, the perimeter of the patch is divided into a large
number
of
ports which are used in formulating the
impe-
dance matrix from which the Green's function and hence the
fields at the boundary of the cavity model are determined.
Numerical
arbi-
methods may also be used to analyze
trary patches. For example, the contour integral method is
based
on the relation between the field inside
volume
and its value along the enclosing surface (Green's
theorem).
This
integration
approaches
includes
used
the formulation of
to express the RF voltage
the
at
current
the integration is replaced
summation over these sections.
The voltage distribu-
along the periphery can be determined by
total current flowing through each
solving
point
a
By dividing the periphery into N
sections of arbitrary widths,
tion
contour
a
the periphery in terms of voltage and
all along the periphery.
by
closed
a
specifying
and
section,
by
the z-matrix necessary to calculate the section's
voltage.
In the finite element method, the given boundary value problem is reduced to two boundary value problems.
homogeneous
wave
equation with inhomogeneous
B.C.'s
The
is
decomposed to Laplace's equation with inhomogeneous B.C.'s
and
inhomogeneous wave equation with homogeneous
The
equivalent
certain
basis
field solution is expressed in
functions and integrated over
B.C.'s.
terms
the
patch by dividing it into a large number of ports.
of
entire
25
modal-expansion cavity model is similar
The
discussed earlier except for
model
cavity
B.C.'s
impedance
the
at all of the radiating walls.
internal fields from the
effects
of
exterior
infinite
the
external
It
The
fields.
region
the stored and radiated energy in the
complex
to the patch are considered as a finite
wall admittance Yw.
is
separation
based on the concept of edge admittance in the
of
the
to
The wall conductance corresponds
to
power radiated into a half-space and the wall suscep-
the
tance
corresponds
to the energy stored in
fringing
the
fields and can be used to account for the patch
dimensions.
effective
Until now, no exact solution for Yw has been
found, but Wiener Hopf method can be used for its computation
(Carver
and Mink 1981).
circular
Rectangular and
More-
patches have been analyzed by utilizing this model.
over,
Green's
function approach can be used for
patches
wall.
The ad-
mittance wall Green's function technique has been
applied
with boundaries approximated by admittance
to
a circular patch to analyze its input impedance
(Yano
and Ishimaru 1981).
2.5.2
Patch Current Distribution Methods
The methods that attempt to find the radiation fields in
terms
of source currents on the patch are primarily based on treating
the
problem as if the dielectric sheet was not present and
the
wire-grid model (Agrawal and Bailey
(Newmann and Tulyathan 1971).
1977)
and
the
include
moment
method
26
the antenna is modeled by a
In the wire-grid method,
as shown in Figure 2.7 and fully immerged in
fine-grid
homogeneous dielectric medium.
Richmond's reaction theor-
is used to formulate the current on each of
em
grid
segments.
program
wire
the
computer
A standard wire-grid modelling
any
is used to calculate these currents whereby,
antenna property of interest can be determined.
The
sults are then modified for the true structure by
Wire Grid
Model of
the Patch
a
re-
scaling
AmprAmarrAirr
IralrA1111211/
2V0
.1111, .1 .11110.
INN.
INIMD
==. 1111. IMOIO MM.
111110110
Ground Plane
AIVAINFAIVAIIV
AMMMOMMI
AMOIMMOMMOr
Fig. 2.7
factors
layers
Image Plane
Wire-Grid Model of Microstrip Patch Antenna
obtained
by
loading the antenna
various thicknesses.
of
This
dielectric
by
has
method
been
applied to circular and square patches.
In the method of moment,
and
replaced
the ground plane is removed
by the image of the patch and
feed
probe.
The dielectric sheet is removed and replaced by free space
equivalent
currents
volume
polarization currents.
on
the microstrip patches
The electric
and
the
surface
equivalent
polarization currents are used to model the anten-
27
na.
patches
An integral of unknown currents on the microstrip
and
on wire feed lines is formulated and solved by using the
method
of
moments (Harrington 1968).
analyze
a
This method has been used to
rectangular patch.
2.5.3
Spectral Domain Methods
The methods which consider relationships between field
distri-
butions and current sources in the presence of a ground plane
dielectric substrate are:
(Uzunoglu
with
TEM-mode transmission line current method
Al. 1979), basis current mode
expansion method
(Itoh
and Menzel 1981) and orthogonal current mode expansion method
(Wood
1981).
These methods are based on the evaluation of the
tribution on the patch and the
plane of the patch.
electric
field
current
dis-
on
everywhere
the
This is accomplished in the spectral domain by
expanding the unknown current distribution in
terms
of
set
a
suitable basis function and numerically determining the current
desired moment
fields at the plane of the patch by using a
such as the Galerkin's method.
The choice of
basis
of
and
method
functions
de-
pends on the patch shape.
The orthogonal current mode expansion method has been used
analyze a circular patch microstrip antenna (Wood 1981).
Here,
to
the
fields in the air and dielectric regions are first obtained by solving the wave equation and
applying
Maxwell's
equations
and
then
28
formulated by the Hankel transform representation.
The relationship
between the field and current components are obtained by utilizing
the continuity relations at z
- El)
(E2
h,
x Az = 0
(2.26)
A
z
x
2
-
)
1
= I
The electric field is obtained by assuming the current
on the patch in a form of orthogonal
amplitudes.
modes
series
distribution
with
arbitrary
The series are expressed in terms of cylindrical
tions and obtained from the analysis of the cavity model.
func-
The elec-
tric field is used with each of the current modes to set up a matrix
equation.
It is solved by using the Gauss
elimination method
and
used to determine the input impedance and other antenna characteristics.
2.6
Comparison of Cavity Model With Other Analysis. Methods
The methods of microstrip antenna analysis are
classified and
briefly discussed without referring to the relative accuracy of
corresponding solutions.
the
In Table 2.2, the basis techniques used in
the analysis of simple patch antennas are listed together
type of results obtained from these techniques.
with
All of the
methods
listed are based on some judicious approximations so that the
lem can be solved and the answers are accurate enough
solutions useful in an engineering sense.
the
prob-
make
the
The technique chosen
for
to
29
a specific patch configuration depends on the
the accuracy and the type of
results
desired.
patch
For
geometry,
and
example,
the
rectangular and circular patch antennas can be tackled quite
ately by the spectral domain
technique
to
compute
the
accur-
radiation
pattern, but the technique cannot be easily applied to evaluate
input impedance which is also an equally important parameter.
the
Structure
Category
Method of analysis
Results available in referenced literature
A
Cavity model
A
Modal-expansion
Radiation pattern, input impedance,
resonant frequency
Input impedance
cavity. model
Rectangular
The method of
moments
Input impedance
Basis current modeexpansion
Radiation pattern
C
TEM-mode transmission line
current
Radiation pattern, input impedance,
surface-wave/free-space power ratio
A
Cavity model
Input impedance
B
Wire-grid model
Radiation pattern, input impedance
Cavity model
Radiation pattern, input impedance
Modal-expansion
cavity model
Input impedance, efficiency
B
Wire-grid model
Radiation pattern, input impedance
C
Orthogonal current
mode-expansion
Resonant frequency, Q-factor, radiation
pattern, surface-wave/free-space power
ratio
B
Narrow rectangular strip
Square
Circular
Table 2.2
ummary of Results for Typical Simple Structures
31
2.7
Concluding Remarks
In this chapter,
microstrip
antenna
the techniques for determining
characteristics have been
the
introduced
with a primary emphasis on the simple yet versatile cavity
model method.
The cavity model and other related methods
that use the equivalence principle were reviewed.
It
is
seen that all of these methods can be used to evaluate the
radiation
fields
closed- and
and
estimate the
input
impedance
open-ring microstrip structures with
of
varying
degrees of accuracy.
It is seen that the cavity model method that utilizes
discrete
eigen modes to find the aperture field distribution is a conceptually simple method which is also helpful in understanding the
characteristics from a first principle analysis.
method can be used to estimate all of
the
In
antenna
antenna
addition,
characteristics
including radiation pattern, directivity and input impedance of
structure.
Therefore, even though some or all of the
sented in the thesis can be obtained more
accurately
results
cavity model because it is conceptually simple, versatile and
operation
and
the
pre-
by utilizing
more sophisticated numerical methods, we have chosen to utilize
us a physical insight into the
the
properties
of
the
gives
such
open- and closed-ring microstrip structures for various useful modes
of excitation.
32
CHAPTER 3
ANALYSIS OF THE OPEN-RING MICROSTRIP ANTENNA
open-ring microstrip structure has been
The
in
recent
years for various
applications
circuits and as antenna element.
studied
microwave
in
Even though a consider-
able
amount of analytical and experimental work has
done
on the resonant frequency,
distribution
the
for various mode,
the corresponding
structure,
work
on the radiation characteristics such as the
tion
pattern
limited (Lo et al.,
and Tripathi,
the
directivity
1979;
field
the Green's function
input impedence of such an open-ring
and
been
have
been
Chadha and Gupta,
and
the
radia-
somewhat
1981; Wolff
1984; Tripathi and Wolff, 1984; Richards et
al., 1984).
In
this chapter,
the special case of an
open-ring antenna is considered.
its
tures
ideal
General expressions for
radiation fields produced by the semi-circular
are
derived by using the
gap
cavity
model,
aper-
Huygens'
principle and the properties of the cylindrical functions.
These
are used to study the radiation patterns (excluding
the gap fields) for various modes of excitation.
33
3.1
Antenna Analysis
The
open-ring microstrip antenna and its
cavity
model
(Wolff
Figure
3.1.
The antenna consists of a planar
microstrip
radius b,
element
and Tripathi,
1984) are
having an inner radius
a gap angle 2R-a.
a,
equivalent
shown
in
open-ring
an
outer
The ring is separated from a
ground plane by a thin dielectric substrate of thickness h
and permittivity sr.
The corresponding cavity model con-
sists
of a prefectly conducting open-ring with
inner
and outer radii rie and rae
effective
respectively,
perfect
magnetic walls between the edges and the ground planes and
the cavity is filled with a medium having frequency-depen-
dent effective permittivity ere( f) given by
effective
radii are given by (Wolff and
(2.16).
Tripathi,
The
1984;
Khilla, 1984):
R = (a+b)/2
h
Ground P ane
Fig. 3.1
The Open-Ring Microstrip Antenna and The Top
View of the Magnetic Wall Model
34
W e (f)-W a
rie = a -
-2
= b + We(f)-w
r ae
where
2
We(f)
is given by (2.15) and the
Since h is small compared to A0,
fields
width
W=b-a.
the electromagnetic
are assumed to be independent of the
direction
z
and then it is seen that only TMnm modes need to be considered
(James
equations
et al.,
for
the
1981).
A
solution
of
field components can be
Maxwell's
obtained
by
utilizing the solution of the wave equation that satisfies
the boundary conditions.
These are tabulated in Table 2.1
for Ez and lead to:
Ez = E o EJ v(k P)
vm
i(kvrarie)
y)(kv- r. )
v
H
1
= j
wPop
= -j
H
wpop
1
=-j
0
m le
(k
v
v"
cosv
to
aEz
a(t)
E 0 EJ v (k vm p) -
3.3)
J:)(kvmrie)
Y'(k
vmr.ie
v
)
Y v (k vm p)] sinvp
aE z
wPo aP
k
vm E0 ELT(kvm
Wuo
(3.2)
3.4)
,J)(kvmrie ) -y.(kvmp)] cosv0
.Y 1(kvmrie)
35
where J
v
kind (Bessel and Newmann
second
and
and Y v are the cylindrical functions of first and
of
order v.
respectively
functions)
J' and Y,, are the derivatives
the
of
functions with respect to the total argument (kvmp).
v is
dependent on the gap angle and is given by
nit
=
n = 0,1,2,3,
,
(3.5)
a
The
equivalent
wave number kvm is
the solution
the
eigenvalue equation given by
J(k
v
vmr ae )Y(k
v
v
The
)
- Ji(k
vm r.le )Y1(k vmr ae
v
)
= 0
(3.6)
above eigenvalue equation is obtained from the
boun-
dary conditions that the azimuth H(0- component become zero
at the outer edge of the ring i.e. at
value
imate
attained
strip
of k vM for axial modes (m =
by
line
p =rae.
1)can
An approxalso
assuming that the mean length of the
forming the radiator is a
multiple
of
be
microhalf
wavelength of a wave on the microstrip line as is the case
for closed-rings (Bahl et al. 1980),
a+b
a
or
k
-7
= = n
2 nIT/a
vm
a+b
= n
i.
W-vm
(3.7)
2v
a+b
Then, the resonant frequency can be determined from
(3.8)
36
c k
fr =
vm
21r[ere(
( 3 . 9
)
f))112
where c is the velocity of light.
field
The
distribution associated with
the
cavity
model can now be used to evaluate the radiation fields for
various
modes
outlined
of excitation by utilizing
in the previous chapter.
the
procedure
The case of ideal gap
open-ring structure leads to solutions in terms of integer
order Bessel functions or spherical Bessel functions
well
defined
properties and is treated in this
with
chapter.
The
variations of
and
the azimuth angle to are shown in Figure 3.2 for typi-
Ez as a function of the
coordinate
p
The
cor-
responding field distributions are also illustrated.
The
cal axial TM11 and TM21 and radial TM12 modes.
equivalent
and
are
modes,
magnetic
also indicated in Figure 3.2c.
the
polarity.
n
current are in the direction E
For
inner and outer ring sources are of
the
axial
opposite
For the radial modes, the inner and outer ring
sources are of the same polarity for even m and the resulting radiation can be larger making such modes potentially useful in practice.
This is similar to closed-ring structures where TM12 modes have been
found to be efficient and have been investigated in recent years for
possible applications as an antenna element.
37
TM12
1.
Scale 1:8 for TM12 mode
TM11
TM22
.8
TM21
.6
.
Ez
E
°
n
R
A
7
= 2w
crirrr
1
TM3
4
Axial Mode
Radial Mode
.2
.0
ie5P.Srae
_.2
-.4
v \
W
_.6
_.8
= 1.
cm
W/R = 2/3
h
= .159 cm
er
= 2.32
R/A
\
A
-... -- .55
\
\
-......535
(a)
Scale 1:8 for TM12 mode
Axial Mode
Radial Mode
360
(deg.)
I
TM22
TM13,'"
= 1.
W
W/R = 2/3
.159 cm
=
h
= 2.32
er
o
Fig. 3.2
(a)
(b)
(c)
= rae
(b)
Field Analysis for Axial and Radial Modes
of an Ideal Gap Open-Ring Antenna
Normalized Ez versus p
Normalized E versus
Field Distribution (Wolff and Tripathi 1984)
(0
38
(c)
Fig. 3.2
(a)
(b)
(c)
Field Analysis for Axial and Radial Modes
of an Ideal Gap Open-Ring Antenna (continued)
Normalized Ez versus p
Normalized E, versus 0
Field DistriBution (Wolff and Tripathi 1984)
To analyze the ideal gap open-ring,
as follows:
a)
Formulate the integral expression neces-
sary to calculate the radiation field.
radiation
c)
we shall proceed
b)
Determine the
field for the even-and-odd modes
individually.
Examine
and plot the radiation patterns for
various
modes of excitation.
3.2
Formulation of the Integral Equation
As
analysis
outlined
is
the
in
Chapter 2,
knowledge of
the first step
the
equivalent
in
the
magnetic
radiation
current
sources
fields.
Excluding the gap fields, the M sources reduce to
Si
= -2anxE =
M necessary to determine
= 2E z a
,
at P=rae
0<0<27r
M2 =-2Eza,
,
at p=rie
0<co<21T
0
,
otherwise
the
(3.10)
39
where
a n is a unit vector normal to the surface and E z is
The radiation field can be derived
given by (3.2).
from
electric vector potential F and is given by:
k
= -3
E
0
0 F
co
=
E
0
.
3
0
k
° F
0
Co
(3.11)
where
-jk r
°
0 e
F =
e-ikoP '
f
sine cos(0-.1) ds'
(3.12)
471
and,
ds' = pldo"
The
spherical
= h
components of F can be obtained by
making
use of (A.9), since the sources are specified in cylindrical coordinates.
i.e.
2n
Fe= Chcose J
M(p',01)sin(0-0')e
-jkop' sine cos(0-4).)
do'
0
(3.13)
2ff
j
F = Ch j
M(P1,01)cos(0-().)e
p' sine cos(4,
)-1)
p d
4)
0
where C =
e
°
4n
e
-jkor
,
as defined earlier with Eq. (2.23).
40
3.3
Method of Solution for Radiation Fields
above
The
integral
given
expressions as
(3.13) can be simplified by making use of Eqs.
by
Eqs.
and
(3.2)
It is seen that the integration procedure for the
(3.10).
expressions
radiaion fields is different
for
modes
for
with
Bessel function of integer order as opposed to these
with
Bessel functions of fractional order.
classified
This can
as even-and-odd modes respectively as given by
n
,
n = 0,1,2,3,...
;
for even modes
n/2
,
n = 1,3,5,7,...
;
for odd modes
v =
since
(3.14)
the fields with respect to the axis have
odd symmetry for the two cases.
Bessel
for
shown
be
For the even modes,
functions are of integer order and the
radiation
in
integrals
field can be written in a closed
Khilla
even-and-
(1984) whereas for the
odd
can be represented in the form of a
the
expression
form
modes
as
the
converging
series as shown in the following sections.
3.3.1
Even Modes Solution
Using
the
following
expressions
(Abramowitz
and
Stegun, 1970)
21.
Jn(x)
=
77-
cos nC e
J
0
jxcosci
dC
(3.15)
41
J_n(x) = Jn(-x) = (-1)n Jn(x) = cos nil. Jn(x)
(3.16)
along with the trigonometric identities and the recurrence
formulas for Bessel functions which are
n Jn(x)/x = [Jn-1(x)
Jn+1(x)]/2
(3.17)
JA(x) = [Jn-1(
Jn+1(x)7 /2
)
it can be shown that
27r
cos no' cos(0-0') e
j
jxcos(0-01)
do'
0
= jn-121- cos no JA(x)
2n
Jcos no' sin(0-0') e
(3.18)
jxcos(0-0')
do'
0
= jn-1 2
sin no n Jn(x)/x
2n
j sin
n
'
cos(0-0') e
(3.19)
jxcos(0-0')
do'
0
= jn-1 2ir sin no JA(x)
2n
sin n
f
sin(0-(01) e
(3.20)
jxcos(0-0')
c141
0
=-jn-1 21. cos no n Jn(x)/x
where x is a dummy variable.
(3.21)
42
This
and
set of equations is valid for all values
for an integer n.
to
of
x
The components of the radiation
field for the even modes are derived using the appropriate
integral of these equations with Eqs.
(3.12) and
(3.13).
In a closed form expression, the results are
E0 =-Ce cos no [KiBi(aae)
E
K2B1(
(3.22a)
)]
= Ce n cose sin n(1) [K1B2(aae) - K2B2(aie)]
(3.22b)
with
Ce = jnhk0E0e-jkor/r
,
the subscript "e" denotes even
K2 = rie An(rie)
K1 = rae An(rae)
JA(knmrie)
YA(knmrie)
An(x) = Jn(knmx) - CiYn(knmx)
Cg
,
C1 =
a ae =
,
aie = ko rie sine
rae sine
j (x)
n Jn(x)
B (x) =
B1(x) = Jn-1(x)
where
r ae
and rie are defined in Eq.
argument x refers to rae,
n
x
(3.1) and the dummy
rie, aae, or aie.
Notice
that
K1 and K2 are dependent on the value of the electric field
at
each
the equivalent outer and inner radii respectively
mode of excitation.
B1(x) and B2(x) are
dependent on the resonant frequency,
and
the observation angle 0.
for
functions
the patch dimensions
Finally,
Ce is a constant
directly proportional to the substrate thickness h and the
resonant frequency.
43
3.3.2
Odd Modes Solution
Using
the
following
expressions
(Abramowitz
and
Stegun 1970)
ejxcos& = cos(xcos0+j sin(xcost)
OD
=
X Na Ja(x) cos
(3.23)
ca
along with the trigonometric identities,
it can be
shown
that
2w
cos(4-4') e
j
J
jxcos(0-4')
d41
0
(3.24)
= X N4
q=0
q+1
q-1
sin(q+1)4 +
sin(q-1)4Dq(x)
(q...1)2-v2
(q+1)2-v2
2w
cosv4' sin(v-4,') e
j
jxcos(4-(0')
d()
(3.25)
0
CO
=
q+1
q =0
j
q -1
cos(q+1)4 -
X
2w sin
(q+1)2-v2
cos(
cos(q- 1)4]Jq(x)
(q-1)2-v2
4)
)
e
(3.26)
0
CO
=-X Nq[
v
cos(q+1)(1) +
cos(q-1)4]Jci(x)
q=0
(q+1)2-v2
(q-1)2-v2
44
21T
e
jxcos(0-0')
(1,0
0
(3.27)
CO
= I Nq[
sin(q+1)0 sin(q-1)0]Jq(x)
q=0
(q+1)2-v2
(q-1)2-v2
where x is a dummy variable,
Jci(x) is the Bessel's
func-
tion of the first kind and of interger order q and Nca, the
Neumann's number is given by
q = 0
1
N
=
Ljci
This
and
q > 1
,
set of equations is valid for all values
with v a half odd integer.
of
x
The radiation field for
odd modes are derived following the same procedure as that
for
even modes case.
In a form of a converging
series,
the results are
CO
q+1
Ee=-00 1 N4
(q+1)2-v2
q=0
q-1
sin(q+1)0 +
sin(q-1)0]*
(4-1)2...v2
*[K1Jq(aae) - K2Jq(aie)]
CO
N4
E =C cose
0
o
q=0
q+1
cos(q+1)0 (q+1)2-v2
(3.28a)
q-I
cos(q-1)0]*
(q-1)2-v2
*[K1Jq(aae) - K2Jq(aie)]
(3.28b)
45
where
odd,
C
o
=
jhk0E0e-jkor/2wr with subscript
"o"
denotes
aie, K1 and K2 are defined with Eq. (3.22)
and aae,
as if v is used instead of n.
Now,
calculate
can make use of Eqs.
we
(3.22) and (3.28)
to
the radiation pattern of an ideal gap open-ring
microstrip antenna for various modes of excitation.
3.4
Results and Discussion
The radiation pattern R(e) is defined in terms of the
magnitude
given
squared
by Eq.
of the radiation field
(2.1).
For the case of
as
components
even
modes,
the
following properties are drawn by examining Eq. (3.22).
a.
The TMOm modes possess nulls in the normal direcThis is obvious from Eq. (3.22) which with
tion.
n=0 at 0=0, reduces it to E =0 and E0 =0.
e
b.
The TM2m modes (which correspond to TMim modes of
the
pro-
closed-rings) are the only modes which
radiation in the
duce
normal
With
direction.
e=0, Eq. (3.22) reduces to
E0 =-Ce coso (Ki-K2)/2
(3.29a)
= Ce sing) (K1 -K2)/2
(3.29b)
E
for
n=1,
otherwise
Ee = E
= 0.
The
corres-
0
ponding radiation peak R(0) is
R(0) = Ce2 (K1- K2)2/4.
(3.30)
It increases with increase in h and fr due to Ce,
and
increases also with the annular
width.
It
should be noted that fr is inversely proportional
46
to the effective permittivity.
This implies that
R(0) increases also with decrease in Cr.
c.
The
TM8m ...
TM4m,
TM 2m$
modes of the closed-rings) pro-
TM4m
duce the same
planes.
modes (which correspond to
E-and-H
radiation patterns in the
This is
because E
is usually equal to
zero and 'Eel is the same in both planes.
In
addition,
the
following properties of
the
odd
modes are drawn by examining Eq. (3.28).
a.
The
radiation pattern produced by any of the odd
modes
is symmetric with respect to the
tion angle e.
observa-
This is obvious from the nature of
the derived equations together with Eq. (3.16).
b.
All
odd
modes produce
direction.
radiation in the
In the normal direction,
normal
the fields
are
(K1-K2)/(1-v 2
Ee =-2C0 sin4,
E
N
/
(3.31a)
= 2c0 COS4 (K1-K2)/(1-v2)
(3.31b)
/
Hence, the radiation peak R(0) is given by
2
R(0)= 4CO2 [K1-K2
1-v2
v=1/2,3/2,5/2,...
It depends on v and also increases with
(3.32)
increase
in h, fr and annular width.
The
radiation
antenna
are
various
modes
patterns of the ideal
computed
for some
of excitation.
typical
gap
open-ring
parameters
For the axial
modes
for
the
47
resonant
are obtained using
frequencies
utilizing
Richards et al.
Eq.
and
(3.9)
(1984) results for the
radial
modes.
The
same
results for the first three axial modes for
structure
shown in Figure
are
Tripathi,
1985).
radiation
patterns.
The
(3.3)
and
(Sultan
curves indicate the nature of the
are
The radiation field magnitudes
different for different modes since the resonant
cies
the
frequen-
and aperture field amplitudes are different for each
case.
The
radiation peak should also increase with
in-
creasing the mode number, since R(0) increases with fr for
the same antenna.
The radiation patterns for the radial modes are shown
in Figure (3.4).
the
axial
patterns.
radiation
due
They have narrow beamwidth compared
This
reflects the fact
fields are reinforced in the
that
sources
that
for the same antenna,
which have the same polarity.
It is
the side lobe level
seen
with
pattern and that their resonant frequencies are
proximately
mode
cur-
produced
with TM12 pattern is lower compared to that produced
TM 22
the
direction
normal
to the inner and outer ring equivalent magnetic
rent
to
of
an
closed-ring.
the same.
It should be noted that the
ideal gap open-ring is the TM12
mode
ap-
TM22
of
a
Eo
=
h
=
=
1.
W
= 1.
W/R = 2/3
Er
R
=
V/m
cm
E-plane
H-plane
.159 cm
2.32
.35n
a
a=21.
;
-70.00
1
-50.00
+-
-30.00
20
i
I
-10.00
10.00
0
.
-
30.00
50.00
Fig. 3.3 Radiation Patterns of an Ideal Gap OpenRing Antenna for Axial TM11, TM21 and TM31
Modes (Excluding Gap Fields)
70.00
90.00
49
E0
W
W/R
h
er
=
=
=
=
=
V/m
1.
1.
cm
2/3
.159 cm
2.32
E-plane
H-plane
R/A = .535
50.00
70.00
90.00
e
(a)
/
/
/ \
\
\
/
1
/
\
/
E0
= 1.
V/m
cm
= 1.
W
W/R = 2/3
.--1
/
1
/
2
I
/
/
.159 CM
h
=
cr
= 2.32
PI
.
1
-,
Ix
I
4
/
--80.00
-70.00
----
6
/
1
1
\
\
\
10
,/
-60.00
1
1
/
/
1
1
/
/
\
1
3
----.
11
/
1
1
o
/
R/A = .55
E-plane
H-plane
1
-30.00
2
-10.00
10.00
30.00
\
641.00
70.00
MAIO
e
(b)
Fig. 3.4 Radiation Patterns of an Ideal Gap OpenRing Antenna for Radial Modes (Excluding Gap Fields)
(b) TM22 Mode
(a) TM12 Mode
50
the open-ring microstrip antenna offers a
short,
In
viable alternative to a closed-ring structure with superior
radiation
exactly
properties when excited in
TM12
mode
and
the same properties when excited in the even TM22
mode.
3.5
Concluding Remarks
this chapter,
In
the analysis of an ideal gap open-
ring microstrip antenna based on the cavity model has been
presented.
radiation
General
fields
distribution
in
expressions for the
curved
been derived by using
have
the cavity
Huygens'
model,
aperture
principle,
trigonometric identities and the properties of the
drical
functions.
The
radiation patterns
field
the
for
cylinvarious
geometries and modes have been computed from these expressions.
The
open-ring
solutions
microstrip
for the even modes of the
structures
closed-ring microstrip structures.
for
an
are
the
ideal
gap
as
the
same
The radiation patterns
be
odd modes reveals that an open-ring structure can
efficient
antenna element when excited in TM12
With the characteristic of its radiation
properties,
mode.
the
open-ring structure should become a new useful element for
applications as an antenna element or in antenna arrays.
51
CHAPTER 4
THE RADIATION BEHAVIOR OF AN ANNULAR SECTOR
MICROSTRIP ANTENNA
In
the
this chapter we analyze the radiation behavior of
open-ring microstrip antenna
angle.
gap
The effect of the gap field for structures having
different
In
arbitrary
with an
angles and widths is included in the
addition,
quency,
analysis.
the relationship between the resonant
fre-
the mode number and the angle of such antennas is
also reviewed in this chapter.
Expressions
case
are
for the total radiation field of such
outlined
derived (following the same procedure
earlier) by utilizing the cavity model.
a
It is seen that a
simplified expression can be obtained for e = 0 giving the
radiation peak in the normal direction.
expression
sectors
simplified
This
can be used to compare the radiation peaks
with
different angles and widths.
All of
of
these
expressions are formulated in terms of the antenna parameters and are valid for any arbitrary gap angle.
The results for the radiation patterns along with the
other characteristics (e.g. resonant frequency, beamwidth,
radiation peak...
tigated
for
etc.) of such a general case are inves-
various modes of
excitation.
The
special
cases of quarter, half, three-quarters and ideal gap openring microstrip antennas with their limiting case of small
inner radius (a 4 0) are included.
52
Review of Resonant Frequencies and Field Distribution
4.1
outlined in the preceding chapter,
As
frequency
resonant
the
can be determined assuming that the mean length
of the microstrip line forming the radiator is a
of
wavelength
half
of a wave on
the
multiple
microstrip
Referring to Figure (3.1) and substituting Eq.
line.
(3.8) into
Eq. (3.9) reduces it to
nc
v c
f
=
r
(4.1)
1(a+b)Cere(f)]1/2
a(a+b )Ecre(f)]1/2
where all the symbols are defined earlier.
This equation
represents an approximate relationships between the resonant
frequency,
the axial modes.
the mode number and the sector angle
for
The resonant frequency is directly pro-
portional to the mode number for the same antenna.
There-
fore, we have
fr of TM11 = 1/2 fr of TM21 = 1/3 fr of TM31... etc. (4.2)
The
relationship between the resonant frequency
gap
angle
and
the
is shown in Figure 4.1 for a typical case
for
the TM11 and TM12 modes.
electric
of
The variations of the normalized
field for different sector angles as a
the coordinate p and the azimuth angle
Figure
field
4.2 for these modes.
amplitude
(I)
function
are shown
The difference between
at the cavity edges decreases
with
in
the
the
increase in a for the TM11 mode and increases for the TM12
mode.
53
TM12
E0
W
W/R
h
Er
=
=
=
=
=
V/m
1.
1.
cm
2/3
.159 cm
2.32
TM 11
I
0
l
60
1
120
240
180
Sector Angle,
300
360
a
Fig. 4.1 Normalized Resonant Frequency versus Sector
Angle for TM11 and TM12 Modes for a Typical Case
54
I.0
a=360°
Scale 1:2 for TM12 of 270°
Scale 1:8 for TM 12 of 360°
270°
\
1
.8
tt
a=360°
\ 1t
270°
180°
\
tt
11
180°
A5
120°
ttl
120°
Ez
Eo
.35n;
a
90°
111
60° \
904'
60°
A
.2
.0
3
I%
-
le p<r ae
r.
2
R/x
.675
.595
.57
-.4
.55
6
_.8
TM11 Mode
TM12 Mode
W
= 1.
cm
W/R = 2/3
h
=
.159 cm
cr
= 2.32
.54
.535
(a)
Fig. 4.2
Normalized Ez of Annular Sector
for Various Sector Angles
(a) as a function of p
(b) as a function of 0
n=1
Scale 1:2 for TM 12 of 270°
Scale 1:8 for TM 12 of 360°
TM11 Mode
TM12 Mode
a=360°
270°
180°
,180
240
300
360 t(deg.)
W
= 1.
cm
W/R = 2/3
h
=
.159 cm
Er
= 2.32
P
= r ae
(b)
Fig. 4.2
Normalized Ez of Annular Sector for Various Sector Angles (continued)
(a) as a function of p
(b) as a function of
(1)
56
At
this stage,
we shall analyze the annular
sector
microstrip antenna, having an arbitrary gap angle
radiation
fields
for its
by using the cavity model discussed
in
Chapter 2.
4.2
Derivation of Radiation Fields
The
the
main step in the analysis consists
integral
radiation
magnetic
deriving
equations necessary to determine the
field.
Fig. 4.3
of
Figure 4.3 shows the top view
total
the
of
Equivalent Magnetic Current Sources of an
Annular Sector Antenna
wall
model of the annular sector antenna
with the equivalent magnetic current sources.
along
The sources
are obtained using Eq. (2.18) and expressed as follows:
M =-2anxE =
Ml = 2Ezao
;
at P=rae,
0 14)
M2 =-2Ezaco
;
at p=rie,
0 ioi a
M3 = 2Ezap
;
at 0=0
M4 =-2Ezap
;
at 0=a
0
;
otherwise
-
,
a
r ielf3Xae
rie<P<rae
(4.3)
57
where
Ez is given by Eq.
M1 and M2 represent the
(3.2).
outer and inner equivalent magnetic current sources
with the circular apertures.
ciated
the
asso-
M3 and M4 represent
linear aperture equivalent sources which are
ignored
previously with the special case of an ideal gap open-ring
3.10).
(Eq.
cal
In a form of integral equations, the spheri-
components
of the vector potential F
are
expressed
using Eqs. (2.20) and (A.9), as follows:
N
Fe = C f [ Mpicos(*-*1)+isin(0-')]cose e
jkopscos*
d
4)
(4.4)
S
F(0 = C J
[- Mpisin(0- *') +Nicos(4)- (01)]ejkop
cos*
ds'
S
where C is defined with Eq.
(2.23), cos* = sine cos(*-*')
and ds' is given by
p' d4' dz'
;
with M 1 and M 2
dp' dz'
;
with M3 and M4
ds' =
The total radiation field is then the superposition of the
fields
produced by all four radiating edges and is
given
by:
E0 =-j
k,
4
F(0 (due to Mi)
o i=1
(4.5)
4
E4) =
j -2 1 Fe (due to M.)
o i=1
58
4.2.1
Radiation Fields from the Curved Apertures
The
sources
fields
produced
from M1 and
are derived following the same
trated in the preceding chapter.
M2
semi-circular
procedure
illus-
Using Eq. (3.23), it can
be shown that
a
cosv(0' cos(0-01) e
j
pccos(0-0')
d'
(4.6)
O
CO
= 1/4 1 Nci[Ri(g+1,v)+Ri(q+1,-v)+Ri(q-1,v)+Ri(q-1,v)]Jci(x)
q=0
a
j cosvo' sin(0-0') e
jxcos(0-0')
d'
O
(4.7)
= 1/4 1 Ncp2(1+q,v)+R2(1+q,-v)+R2(1-g,v)+R2(1-q,-v)]J(4(x)
q=0
a
sinv(01 cos(o-o') e
jxcos(0-4)1)
do'
(4.8)
O
03
=-1/4
N4R2(q+1,v)-R2(q+1,-v)+R2(q-1,v)-R2(q-1,-v)]Jc4(x)
q=0
a(
sin(o-0'
sin
0
do
(4.9)
OD
1/4
Nci[R1(1+q,v)-R1(1+q,-v)+Ri(1-q,v)-R1(1-q,-v)]Jci(x)
q=0
59
where
Jq is a Bessel function of the first kind and inte-
ger order q, x is a dummy argument and Ng is the Neumann's
number
defined earlier with Eq.
functions dependent on v,
Ri(q,v) =
1
R1 and R2
(3.23).
a and 0 and are given by,
[sin((q+v)a-q0)+sinq0]
(71+7
R2(q,v) =
1
q+v
are
(4.10a)
[cos((q+v)a-q0)-cosq0]
It is also important to note that the sum of the arguments
of
R1 or R2 can take the value of zero,
q=0 and v=1 or if q=1 and v=2 etc.
e.g.
q+1-v=0 if
For these cases,
the
results of R1 and R2 are found to be:
Ri(q+1,-v)=R1(q-1,±v)=R1(1-q,±v)=acosvo
R2(q+1,-v)=asinv0
(4.10b)
R2(q-1,±v)=R2(1-q,±v)=Tasinvo
evaluate
Equations (4.6)...(4.10) can now be used to
the
radiation fields of annular sectors.
They are valid
for any a, since v is an unrestricted number as defined by
Eq.
For instance,
(3.5).
equations
outlined
in the
the other two special sets of
previous
(3.21) and (3.24)(3.27) can
it.
leads
curved
be
(3.18)...
chapter,
easily
obtained
from
Substitution of these equations into (4.4) and (4.5)
to
the
expressions for radiation
apertures
given by,
fields
in the form of a converging
of
series
the
as
60
CO
E
191,2
=-C IN [R1(q+1,v)+121(q+1,-v)+Ri(q-1,v)+Ri(q-1.-v)]*
s
q
q=0
(4.11a)
*EK1Jq(aae)-K2Jq(aie)]
CO
E
4)1,2
=C coseIN [112(1+q,v)+R2(1+q,-v)+R2(1-q,v)+R2(1-c1,-v)]*
s
q=0
*EK1Jq(aae)-K2Jq(aie)]
where
Cs = j hk0E0e-jkor/8ffr,
sector
(4.11b)
the subscripts "s" denotes
and "1,2" refers to the fields produced
sources M1 and M2 respectively.
K1,
the same as those defined with Eq.
from
the
K2, aae and aie are
(3.22) as if v is used
instead of n.
Eq. (4.11) represents the general expressions for the
spherical
components of the curved aperture fields.
expressions
are derived without the need to classify
The
the
modes as even-and-odd modes and formulated in terms of the
antenna parameters.
They are valid for any mode of exci-
tation and for any gap angle.
For examples, by utilizing
the properties of the cylindrical functions and with
some
manipulations, it can be shown that
a. With a = 21r and v = n,
(4.11) reduces to Eq.
Eq.
corresponding to the case of a closed-ring
microstrip antenna.
(3.22)
b. With
= 21r and v takes the values of half odd
Eq. (4.11) reduces to Eq. (3.28) corresponding to the case of the odd modes of an ideal
gap open-ring microstrip antenna. Notice that this
case was analyzed without considering the
effect
of
the gap fields on the total antenna performance.
a
integer,
61
4.2.2
Radiation Fields from the Linear Apertures
ponents of the fields produced from the linear
current
com-
mathematical expressions for the spherical
The
equivalent
sources M3 and M4 representing the fields in
gap are obtained using Eqs. (4.3) and (4.4).
E03= 4Cesin0
Ez(01)10=oe
J
j
jkop'sine cosh
the
These are:
,
(4.12a)
rie
E03= cose cot0 Ee3
(4.12b)
r
E
04
=-4C s cosnwsin(0-a)
ae
Ez(p1)1
j
rie
jkop'sine cos(0-a)
0=0
e
(4.13a)
(4.13b)
E04= cose cot(0-a) Ee4
where
subscripts
the
,
"3" and "4" refer
to
the
fields
produced from the sources M3 and M4 respectively, and
Ez(ps)I0=0 =
(k mPt)
v
v
JI(k
mrie) Yv(kvm
v
v
Y1(kvmrie
)
These integrals are evaluated numerically since efforts to
integrate
them analytically for the given field distribu-
tion were not successful.
point rule (Carnahan et al.
The local form of Bode's
four
1969) is used and has made it
possible to reach our goal successfully.
The Bode's rule
is
x4
f(x) dx
J
x
X4- X
=
9U
0 [7f(x0)+32f(x1)+12f(x2)+32f(x3)+7f(x4)]
o
where xl = (x4-x0)/4, x2 = 2x1 and x3 = 3x1
(4.14)
62
Finally, the general solution for the total radiation
field
It can be constituted from the
can be determined.
summation
of the curved aperture fields analytical
solu-
tion and the linear aperture fields numerical solution
called for in Eq.
as
This can be simply written in a
(4.5).
form of mathematical expressions as follows:
= E
E
E
The
+ E
G1,2
+ E
4)1,2
+ E
(4.15a)
G4
+ E
4)3
radiation
squared
03
pattern
(4.15b)
4)4
R(e) in terms
of
the
magnitude
of these components can now be computed by
which
the other characteristics (Chapter 2) can be extracted.
4.3
The Gap Effects and the Radiation Peak
contribution
The
radiated
of
the gap fields
to
the
depends on the equivalent magnetic current
tribution in the linear apertures associated with the
and
hence
the mode of excitation.
angle
structure
axial
(TMnis
(TM nl,
For ideal
disgap
zero gap
the equivalent currents add for the
n odd) modes and cancel for the even
n even) modes.
power
odd
axial
For other angles such as 90°, 180°
or 270°,
this contribution can be determined in a similar
manner.
For
example
for a 90°
sector
current sources are given as
R3
R4
=
mo(P) ap
at
=
mo(P)
at
= 90°
0°
the
equivalent
63
for the odd and even modes of excitation.
This is similar
to two linear dipoles carrying equal and opposite currents
located
in
the respective places making an angle of
90°
with respect to each other.
The effect of the gap fields can also be examined for
any
structure having an arbitrary gap angle by
Eqs. (4.11), (4.12) and (4.13).
can
width
These derived expressions
used to determine the effects of
be
and gap angle on the radiation
such structures.
utilizing
the
microstrip
characteristics
of
In the normal direction where e = 0, the
expressions are simplified giving the radiation peak.
The
results are:
R(0)=(1E6112 + IE(012)10=0
ae
Tr-K2
Cs
Ez(p.)1
1-v2
=0
dp'
(1±cosa)
(4.16)
r.
le
where
161.2C2s(K1 -K2)2;
v=1 for a=360° with TM2m modes (4.17)
41.2C(K1 -K2)27
v=1 for a=180° with TMim modes (4.18)
the
positive and negative signs refer to the
and-even modes, respectively.
the
(4.17) refers also
to
case of a closed-ring excited in the TMim mode and it
is the same as Eq.
the
Eq.
odd-
radiation
(3.29).
It is
important to note that
peak due to the curved aperture fields
given by the first term of (4.16) whereas that due to
linear aperture fields is given by the second term.
is
the
64
These
expressions depend on the
including
annular
the
width.
They are valid
arbitrary a and for any mode of excitation.
modes where K1 is usually greater than K2,
gap
the
K1
negative
is
fields
>
any
For the axial
the effect
of
the
1 and decreases it as long
For the radial TM 12 and other modes for
1.
<
for
fields is to increase the power radiated in
normal direction as long as v
as v
parameters
antenna
and K2 is positive,
the effect
which
of
is to increase the radiation in the normal
gap
direc-
tion. Furthermore, there are two special cases for which v
= 1.
These cases are
TM lm
mode and an ideal gap open-ring excited in the
mode.
R(0)
a half-ring antenna excited in the
of the former antenna is one fourth that
TM2m
of
the latter antenna having the same parameters. Additionally,
the
fields produce no effect on
gap
their
R(0)'s,
they cancel out in the normal direction as outlined
since
earlier.
In
short,
the derived expressions for the radiation
peak can demonstrate the dependence of the radiation
perties
on
results
for the radiation patterns in the following
tion
are
the annular width and the sector
computed as a function of
parameters of the sectors.
these
angle.
two
proThe
sec-
physical
65
4.4
Results and Discussion
A
computer program was written first to compute
rdiation pattern of the sectors.
two
main steps.
the
The program consists of
radiation
The first was to compute the
fields due to curved and linear apertures as given by Eqs.
4.12 and 4.13) and the second was to compute their
(4.11,
sum and the radiation patterns.
In order to validate the
accuracy of the program, we computed the radiation pattern
small
for
this
angle a and large radius of
curvatures.
case the sector almost becomes a rectangle
For
and
the
results obtained from the computed program are found to be
excellent agreement with known results for the rectan-
in
gular
patch (James et al.,
should
It
also
1981) as shown in Table
be noted that
the
general
4.1.
expressions
(James et al., 1981) for the radiation fields of a rectangular
patch can be extracted from that of a sector
(Eqs.
4.11-4.13).
This
can be
patch
obtained assuming
that:
a.
E
01,2
= E
=
0
(because a is small and v is large)
=
1
(to constitute for the equality of
the field distribution underneath
the patch)
4)1,2
b.
Ez(p1)1(0=0
c.
rae
- rie
=
the equivalent width of the
rectangular
patch.
For example, under these assumptions the radiation peak in
the normal direction (Eq.
found equal to
4.16) for the dominant mode
is
R(0), dB
0 = 0°
0 = 90°
e
(deg.) Sector Rectangular Sector Rectangular
0
-74.851
-74.851
-74.851
-74.851
10
-75.053
-75.054
-75.001
-75.003
20
-75.669
-75.665
-75.46
-75.451
V/m
.159 cm
= 2.32
Eo = 1
h
=
30
-76.691
-76.69
-76.18
-76.171
40
-78.152
-78.149
-77.115
-77.113
For sector patch
50
-80.099
-80.101
-78.187
-78.203
a = 1.146 m
b = 1.166 m
60
-82.689
-82.292
-79.314
-79.321
70
-86.326
-86.331
-80.293
-80.334
80
-92.435
-92.444
-80.982
-81.042
-
-81.290
-81.299
r
a = 10
90
-117.35
For rectangular
patch
aeq = 2.3877 cm
b
= 2.0242 cm
Table 4.1 Comparison of the Radiation Pattern of a Rectangular
Patch with an Annular Sector with Large Radius of Curvature
and Small Angle
67
R(0) = j e-jkor
h ko (rae - rie)/wr
(4.19)
which is the same as that for a rectangular patch.
As a consequence of this validity the results for the
patterns
radiation
of
various typical
computed for various modes of excitation.
the
special cases of quarter,
ideal
gap
half,
were
structures
This
includes
and
three-quarters
open-ring microstrip antennas.
The
computed
results for structures having the same resonance frequency
but
different withs and structures having the same
but
different
gap angles are included in
the
width
following
sections.
4.4.1
Quarter and Three-Quarters-Ring Antennas
micro-
quarter-ring and the three-quarters-ring
The
strip antennas have the same radiation patterns in the
and-H
planes.
equivalent
This
of
is because of the symmetry
magnetic current sources with respect to
Ethe
both
planes.
Furthermore, the resonant frequency of the 90° (quarter-ring) antenna excited in the TM11 mode is approximately three times that of a 270° (three-quarters-ring) antenna having the same parameters, as called for in Eq. (4.1).
The
results
excited
for the radiation patterns of such
antennas
in this mode are shown in Figures 4.4a and
respectively,
for different widths.
4.5a,
Table 4.2 indicates
68
R/
R
= .35/a
= 1.5 cm
W = 2.8 cm
W = 1.
cm
W = .2 cm
0
E0 = 1.
Vim
h =
.159 cm
= 2.32
o°
o°
(a)
o°
W
W/R
= 1.
= 2/3
cm
R/A = .675
o°
(c)
Fig. 4.4
Radiation Patterns for a Quarter-Ring Antenna
for TM11 mode with Different Widths
for Tn.]. and TM21 modes
for TM12 mode
(a)
(b)
(c)
69
R/A
R
=
=
W =
W =
--- W =
.35/a
1.5 cm
2.8 cm
1.
cm
.2 cm
00
E0 = 1.
V/m
h =
.159 cm
er = 2.32
0
0
(a)
R/X = n/2afeTe
W
= 1.
cm
W/R = 2/3
TM11 mode
TM21 mode
0°
(c)
Fig. 4.5
Radiation Patterns for a Three-Quarters-Ring
Antenna
(a) for TM11 mode with Different Widths
(b) for TM11 and TM21 modes
(c) for TM12 mode
70
90° antenna
b
a
fr
Case
(cm)
(cm)
(GHZ)
270° antenna
beamwidth
R(0)
(deg.)
(dB)
Wi 0.1
2.9
4.31
79.08
w2 1.0
2.0
4.43
92.92
W3 1.4
1.6
4.65
94
fr
(GHZ)
beamwidth
R(0)
(deg.)
(dB)
1.44
92
-3.032
1.48
94.72
-7.603
-9.268
1.55
93.86
-15.742
0
0
Table 4.2 Calculated Resonant Frequency, Beamwidth and
Total Radiation Peak of the 90° and 270° Antennas
Excited in Tn.]. Mode for Different Widths.
the
effect of changing the annular width on their
tion performances.
radia-
There is some effect on the shift
in
their resonant frequency due to the change of their effecpermittivity
tive
radiation
peaks
with width.
Most
their
importantly
increase with increase in their
annular
width.
In the limiting case as a 4- 0, the annular sectors
become
circular
normal
direction is maximum for both cases.
sectors and the radiation power
in
also
It is
seen
that the 90° sector has a narrow beamwidth than
270°
sector.
Similar comparisons can be made for
the
the
other
axial modes of excitation, e.g. TM21 mode by utilizing the
radiation patterns illustrated in Figures 4.4b and 4.5b.
In addition, the beamwidth of the 90° antenna excited
in the radial TM12 mode is wider compared to a 270° antenna as shown in Figures 4.4c and 4.5c,
of
respectively.
the former antenna is 29 dB below that of
the
R(0)
latter
71
The physical reason
antenna, as called for in Eq. (4.16).
for
this
increase in power is that for
TM12
mode,
the
power radiated from the 270° structure is due to two semicircular
current
sources
that are in
phase
their
and
length is three times the length of the equivalent sources
the
for
structure.
90°
This implies a
radiated
more
power, a wider bandwidth and thus a lower Q-factor for the
270° antenna.
4.4.2
Half-Ring Antenna
The
shown
The
radiation
antenna
patterns of a half-ring
mode.
in Figures 4.6 and 4.7 for the dominant Tmil
direction
dependent
of the radiation peak em is
upon
the sector width.
As the
are
found
to
be
width
sector
increases, the radiation pattern increments and Om and the
difference between R(em) and R(0) decrease.
the
possibility
As a result,
of producing more radiation can
be
tained from the limiting case of a + 0 and b 9 3 cm,
when
the
half-ring reduces to a half-disk,
Figure 4.7b for a small inner radius.
as shown
ati.e.
in
Table 4.3 indicates
this behavior for some typical cases along with the effect
of
changing
frequency
case
the annular width on the shift
of such antennas.
resonant
Notice that R(0) of the
is found to be 5.44 dB below that of the
limited case.
in
W1
considered
E0 = 1.
V/m
W
= 1.
cm
W/R = 2/3
h
=
.159 cm
Er
= 2.32
E-plane
H-plane
R/A = .35/a
3
_4
Including gap fields
6
Excluding gap fields
/
10
-90.00
i
-70.00
i
-50.00
Fig. 4.6
20
-30.00
-10.00
10.00
30.00
f
50.00
t
70.00
Radiation Pattern of a Half-Ring Antenna Excited
in TM 11 Mode
r
50.00
73
E0
Wl
W2
W3
R
h
=
1.
=
=
1'
.5
.2
V/m
cm
cm
cm
E-plane
H-plane
= 1.5
cm
=
.159 cm
cr = 2.32
R/X = .35/a
Eo
W
W/R
h
Er
=
=
=
=
=
E-plane
H-plane
V/m
1.
2.8
cm
1.867
.159 cm
2.32
R/A = .35/a
-2g
.1111.11111
ICAO
.1.11.111
e
30.00
51.01)
1
711.01)
(b)
Fig. 4.7
Radiation Patterns for the Half-Ring Antenna
Excited in TM 11 Mode
(a) for Different Widths
(b) for the Limiting Case as 'a' becomes small
74
b
a
f
Case
em
R(em)-R(0)
(deg.)
(dB)
r
(cm)
(cm)
(GHZ)
.10
2.90
2.16
20
.103
W1
1.00
2.00
2.22
27
.354
W2
1.25
1.75
2.27
29
.493
W3
1.40
1.60
2.33
30
.574
Limited
Table 4.3 Effect of Changing the Annular Width on
the Radiation Behavior of a Half-Ring Antenna
In
addition,
some
for
the results for the radiation
typical
axial and radial modes
Figures 4.8 and 4.9, respectively.
radiation
mately
the
are
shown
in
For the TM1m modes.the
patterns of the half-ring antenna are
same as that of a
patterns
closed-ring
approxi-
antenna
with
6.021
dB lower in the normal direction.
This is because
their
resonant frequencies are the same,
they
same field
radiating
the
distributions underneath their patches and the
area of the circular aperture for this case
exactly half that of a closed-ring.
no
have
The linear sides made
contribution to the radiation in the normal
for
this mode.
the
TM21
direction
The computed results indicate also
mode produce more power than the TM11 mode
the case considered,
where the TM31,
duce nulls in the normal direction.
is
that
for
TM51 ... modes pro-
/
E
o
W
W/R
1.
Eo
V/m
h
= 1.
CM
= 2/3
=
.159 cm
er
= 2.32
TM11
TM21
TM31
W
W/R
cr
.159 cm /
= 2.32
R/ A
=
Eplane
1
cm /
H plane
1
.55
...-
-90.00
,
V/m/
= 1.
= 1.
= 2/3
-70.00
....-
-...
-20
-50.00
-36.00
-10.00
16.00
30.00
56.00
76.60
9
Fig. 4.8 Radiation Patterns for the
Half-Ring Antenna for TM11, TM21
and TM31 Modes
(a) E-Plane (b) H-Plane
Fig. 4.9
Radiation patterns for the Half-Ring
Antenna Excited in TM12 Mode
76
4.4.3
Ideal Gap Open-Ring Antenna
effect of the gap fields on the
The
terns
of an ideal gap open-ring antenna was not
in the previous chapter.
direction
of
included
The power radiated in the normal
is decreased for the TM 11 mode when the
the gap fields is included in the
example,
for
the
pat-
radiation
effect
computations.
case considered here the
For
decrease
is
approximately 11.27 dB as shown in Figure 4.10. For the Hplane pattern the beam remains in the broadside
and
shifts
angle
em
slightly for E-plane pattern.
tion
gap open-disk antenna (Fig.
ideal
for
closed-rings (Chew 1982).
numerical
of
4.11a.
wider bandwidth for the limiting case
a
This
the
of
4.11b) as is the
Table 4.4
the annular width on the shift
case
indicates
results of this behavior along with the
changing
de-
R(0)
whereas the radia-
patterns increment as shown in Figure
implies
shifting
The
and the difference between R(em) and
crease with increase in annular width,
direction
in
the
effect
resonant
frequency of such antennas.
addition,
In
for
some
typical
the results for the radiation patterns
axial and radial modes
Figures 4.12 and 4.13,
respectively.
are
shown
The linear aperture
fields
produce
modes,
but they affect the patterns of the odd ones.
no
in
effect on the patterns
of
the
the
TM 11 mode the pattern is declined as indicated
and
raised (by 6.04 dB for the case considered)
even
For
above
in
the
Excluding
gap fields
2
E-plane
H-plane
E0
= 1.
V/m
W
= 1.
cm
W/R = 2/3
h
=
.159 cm
Er
co
3
= 2.32
ao
R/X = .35/a
6
10
90.00
70.00
I
50.00
o
i
30.00
s
Including gap fields
20
I
10.00
10.00
30.00
50.00
70.00
0
Fig. 4.10 Gap Effect on the Radiation Patterns of an
Ideal Gap Open-Ring Antenna Excited in TM11 Mode
90.00
78
V/m
cm
cm
E0 = 1.
W1 = 1.
W2 = .5
W3 =
.1
R = 1.5
h = .159
Er = 2.32
E-plane
H-plane
cm
cm
cm
R/A = .35/a
_20
-Nate
-70.N
e
lo:ot
70.tit
30.o0
SLOD
(a)
Eo
\
= 1.
V/m
cm
W
= 2.8
W/R = 1.867
h
=
.159 cm/
Er
= 2.32
//
\
/
\
/
E-plane
H-plane
R/X = .35/a
-MN
,-90.1110
-MN
(b)
Fig. 4.11
Radiation Patterns for the Ideal Gap Open-Ring
Antenna Excited in TM 11 Mode
(a) for Different Widths
(b) for the Limiting Case as 'a' becomes small
79
Eo
W
W/R
h
er
= 1.
= 1.
= 2/3
.159
=
V/m
TM31
cm
E-plane
H-plane
cm
= 2.32
.35n
a
R/ X
-2
_4
\ TM 2 1
-6
-10
TM,
20
-90-00
-70.00
-50.00
-30.00
-10.00
10.00
30.00
50.00
70.00
90.00
0
Radiation Patterns for the Ideal Gap Open-Ring
Antenna for TM11, TM21 and TM31 Modes
Fig. 4.12
E-plane
E-Plane
V/m
E0 = 1.
cm
= 1.
W
W/R = 2/3
=
h
.159 cm
E
r.
=
2.32
-2
R/ X = .535
1 _Including gap fields
-3
Excluding gap fields
-90.00
-70.00
10.00
-10.00
e
Fig. 4.13 Gap Effect on the Radiation Patterns of an
Ideal Gap Open-Ring Antenna Excited in TM12 Mode
80
b
a
Case
(cm)
Limited
8m
R(em) -R(0)
(deg.)
(dB)
fr
(GHZ)
.10
2.90
1.08
0
W1
1.00
2.00
1.11
30
.559
W2
1.25
1.75
1.13
32
.809
W3
1.45
1.55
1.18
34
1.041
0
Table 4.4 Effect of Changing the Annular Width on
the Radiation Behavior of an Ideal Gap Open-Ring Antenna
direction for the TM31 mode as compared to
normal
3.3
or as in Eqs.
(3.32) and (4.16).
Figure
The effect of gap
fields on the radiation pattern of the TM12 mode, however,
is negligible.
In the case of considered parameters,
H-plane pattern is unaffected,
width is slightly increased.
the
the
beam-
whereas the E-plane
Therefore, the TM12 mode of
open-ring structure is a potentially useful mode com-
pared
to that of a closed-ring structure as
demonstrated
in Chapter 3.
4.4.4
General Annular Sector Antenna
radiation patterns for a large number of
The
antennas,
angle,
the
same annular width
different
and
are computed for various modes of excitation.
results
the
having
sector
peak
are shown in Figure 4.14 for the TM11 mode
decreases with increase in
sector
where
angle.
results for other gap angle antennas are shown in
The
The
Figures
a = 30/
/
/
/
Eo
W
W/R
h
er
b
/
=
=
=
=
=
1.
1.
V/m
/
cm
2/3
.159 cm
2.32
E-plane
H-plane
-
/
0
...
/
c4
/
60 '1-
/
R/A = .35/a
/
/
/
/
/
/
/
.-
\
/
\
N
--*
-90.00
-70.00
-50.00
-30.00
-10.00
10.00
30.00
50.00
70.00
Fig. 4.14 Radiation Patterns for
Annular Sector Antenna
Excited in TM 11 Mode for Different
Sector Angle
90.00
82
4.15
and
4.16.
radiation
They all exhibit a
behavior
similar to that of the other cases discussed earlier.
The
an
in-
sector angle and is maximum for the ideal
gap
power
radiated
crease
in
for the TM12 mode increases with
structure when the sector angle is equivalent to 211. These
cases
results are tabulated in Table 4.5 for some typical
and
compared to a closed-ring for which R(0) of the
mode
TM11
is found to be 19.39 dB below that of the TM12 mode.
TM11 Mode
Tm12 Mode
Sector Angle
Structure
(deg.)
Closed-Ring
R(0)
fr
(GHZ)
(dB)
2.22
fr
(GHZ)
0
11.0
R(0)
(dB)
0
Annular Sector
60
6.65
-
.22
13.5
-30.79
Quarter-Ring
90
4.43
- 1.64
11.9
-21.54
Annular Sector
120
3.32
- 3.05
11.4
-15.68
Half-Ring
180
2.22
- 6.02
11.0
- 6.02
Three-Quarter
Ring
270
1.48
-11.08
10.8
7.46
Ideal Gap
Open-Ring
360
1.11
-17.47
10.7
33.95
Table 4.5 Relative Broadside Radiation and Resonant
Frequency Compared to a Closed Ring for Different
Structures for TM11 and TM12 Modes
83
E0
W
W/R
h
er
= 1.
= 1.
= 2/3
=
.159
= 2.32
V/m
cm
E-plane
H-plane
/
a=45°
cm
R/A = .35/a
1114-0C
.070.00
lz.oc
30.0;
7:.0C
3;.IN
e
(a)
/
/
E0 = 1.
V/m
W
= 1.
cm
W/R = 2/3
h
=
.159 cm
Er
= 2.32
E-plane
H-plane
a=85°
/
R/A = .35/a
/
/
\
m
\
\
/
\
/
\
/
\
/
\
/
\
/
\
/
\
51140
70.00
90.80
(b)
Fig. 4.15
Radiation Patterns for Annular Sector Antenna
Excited in TM11 Mode
(a) a=45°
(b) a =85°
(c) a=135° (d) a=170°
(e) a=210° (f) a=225° (g) a=315° (h) a=355°
84
h
V/m
= 1.
cm
= 1.
= 2/3
.159 cm
=
er
= 2.32
E0
W
W/R
E-plane
H plane
0=135°
R/A = .35/a
2
/
/
/
/
/
3
/
/
.410.00
-70.DC
.-1504/1)
30.
SRN
*-10.19
30.00
e
50.E
70410
90410
(c)
/'
= 1.
V/m
cm
= 1.
W
W/R = 2/3
.159 cm
h
=
er
= 2.32
E0
E-plane
H-plane
a=170°
-2
R/A = .35/a
/
/
4
6
10
.1110.0C
- 3041C
e
(d)
Fig. 4.15
(continued)
1GAC
3:4;
MCC
70.0C
90.00
85
E0
= 1.
h
=
cr
= 2.32
E-plane
H-plane
V/m
W
= 1.
W/R = 2/3
Cm
.159 cm
/
R/A = .35/a
/
/
6
8lam
/
-1110.00
.430.00
-711.00
-30.00
.40.00
30.00
70.00
50.30
00.00
(e)
/
/
/
E0
\
/
V/m
= 1.
= 1.
-1
/
W
cm
W/R = 2/3
h
=
.159 cm
er
= 2.32
/
/
/
\
\
co
\
.....
1:4
/
/
\
,- 2
/
/
\
.
/
RiX = .35/a
\\
m
rci
/
/
E-plane
H-plane
\
\
/
_,-.5
/
//
.
\
\
_4
\
/
6
.410.00
-70.00
Fig. 4.15
-30.00
(continued)
-10.00
10.00
(f)
30.00
50.00
70.00
30.00
86.
E0
V/m
= 1.
W
= 1.
W/R = 2/3
h
/
R/A
=
a=315°
\
=
.159 cm
= 2.32
er
E-plane
H-plane
\
cm
/
.35/a
2
0
3
4
-5
6
9C-00
7t.00
sc.00
UAL
7C.CIC
(g)
E0 = 1.
V/m
W
= 1.
cm
W/R = 2/3
.159 cm
e
=
2.32
r
R/A
1
E-plane
H-plane
a=355°
-2
.35/a
6
10
70.SC
70AP0
(h)
Fig. 4.15
Radiation Patterns for Annular Sector Antenna
Excited in TM11 Mode (continued)
(a) a=45°
(b) a=85°
(c) a=135° (d) a=170°
(e) a=210°
a=225° (g) a=315° (h) a=355°
87
E0
= 1.
E-plane
H-plane
W
= 1.
W/R = 2/3
h
=
.159
er
= 2.32
R/x = .675
(a)
E0
W
W/R
h
er
=
=
=
=
=
1.
1.
E-plane
H-plane
V/m
cm
2/3
.159 cm
2.32
R/A = .57
(b)
Fig. 4.16
Radiation Patterns for Annular Sector Antenna
Excited in TM.1.) Mode
(a) a=60°
(67 a=120°
88
Concluding Remarks
4.5
In this chapter,
to
evaluate
the
analytical and numerical techniques
radiation fields of an
annular
sector
microstrip antenna having an arbitrary gap angle, has been
The techniques are then used to compute
formulated.
radiation
characteristics of various structures and it is
shown that some of these antennas can be efficient
ting
angles
the
elements.
(e.g.
This
includes
the sectors
with
< 90°) excited in TMil mode and
open-ring
structure excited in Tm12 mode.
mentioned
that
radiasmall
gap
ideal
It should
from a practical point of view the
be
ideal
gap structures can be physically realized with a gap angle
of approximately 5° (Wolff and Tripathi,
sults
strate
1984).
presented in this chapter for various cases
The redemon-
that such structures can join the family of
patch antennas with useful radiation characteristics.
other
89
CHAPTER 5
THE RADIATION CHARACTERISTICS OF A CLOSED-RING
MICROSTRIP ANTENNA
A considerable amount of work has been done in recent
years
on the resonant behavior and the radiation
teristics
of closed-ring microstrip structures (Wolff and
Knoppik 1971;
al.
1984;
radiation
of
Wu and Rosenbaum 1973;
Dahele
1980;
Khilla
The
fields have been evaluated by utilizing a
host
(Mink
1984;
Khilla
1984;
ranging from the use of
1980;
a
simple
Bahal and Bahartia 1980;
1984)
to the use of
the
cavity
Das et
spectral
use of the method of
the
expansion (Chew 1982).
teristics
matched
al.
domain
in Fourier-Hankel transform domain (Ali et
and
cavity
1984;
al.
1985).
model
technique
Richards et
and Lee 1982;
Bahl et
Mink 1980;
Bhattacharyya and Gary
techniques
1982)
charac-
al.
asymptotic
A study on the radiation
an annular array of elements based
characon
the
model has also been reported by Bhattacharyya
and
of
Gary (1985).
this chapter,
In
the radiation characteristics of a
closed-ring
microstrip antenna
expressions
for
power,
presented.
General
the antenna properties such as
radiated
total energy stored,
rectivity
are
are
radiation resistance and di-
derived by using the expressions
radiation fields presented in Chapter 3.
for
the
These are accom-
90
plished
and
by
Gauss'
utilizing the properties of both
hypergeometric
Euler's transformation.
functions and
by
cylindrical
using
The expressions are formulated in
terms of the antenna parameters and used to develop
radiation characteristics.
efficiency,
results
bandwidth,
for
the
other
This includes input impedance,
losses and Q-factors.
Numerical
typical structures excited in TM11 and
TM12
modes are presented.
5.1
Review of the Radiation Fields
closed-ring
The
model
are
scalar
microstrip antenna and
shown in Figure (5.1).
cavity
its
The solution
of
Helmholtz wave equation for its electric field
the
in
Ring
Conductor
Ground
Plane
Magnetic
Wall
Fig. 5.1
the
The Closed-Ring Microstrip Antenna and
its Equivalent Cavity Model
cylindrical
coordinate
system which
satisfies
magnetic wall boundary conditions was given in Table
The
spherical
components of the radiation field
of
the
2.1.
the
91
closed-ring
by
given
antenna
Eq.
were reviewed in Chapter 3
These expressions
3.22.
are
and
are
reproduced
below:
Ee =-Ce cos of [K1B1(aae)-K2B1(aie)]
E
(5.1a)
= Ce n cose sin of EK1B2(aae)-K2B2(aie)]
(5.1b)
and
Ce = jnhk0E0 e
jk,r
r
/,
K1 = rae An(rae)
K2 = rie An(rie)
An(x) = J (knmx)-Cgn(knmx)
aae = kd rae sine
B1(x) = Jn_1(x) -
Cl
Jn(knmrie)
Yn(knmrie)
aie = 1(0 rie sine
;
J.,(x)
nJn(x)
;
B2(x)
where rae and rie are defined by Eq.
(3.1) and the
dummy
variable x refers to rae, rie, aae or ale.
The
radiation characteristics of
closed-ring
can be derived by using the above
expressions
basic
structures
and are presented in the following sections.
5.2
Formulation of the Problems, Methods of Solutions and
Results
A microstrip antenna can be characterized in terms of
its radiation pattern,
radiation resistance,
and gain.
losses, Q-factor, input impedance,
bandwidth,
efficiency, directivity
The radiation pattern and the input
impedance
92
are
of the basic antenna properties from
two
which
the
other characteristics can be calculated once conductor and
dielectric losses are known.
The
radiation properties are dependent on the
amplitudes
at
the walls of the
cavity
field
We
model.
can
either normalize the electric field amplitude at a convenient location to 1 V/m or use field amplitudes that normalize
the total energy stored in the cavity.
say,
finding E0 in Eq. (5.1)makes the total energy stored
That is
to
equal to 1 (joule) for a given mode.
To
link
the antenna characteristics with the
energy stored (WT),
by the antenna.
we have to utilize the power absorbed
This includes the power radiated into the
far
field,
and
dielectric medium and the power loss associated
the power dissipated in the conducting
the generation of substrate surface waves.
be
1981) with small loss tangent.
walls
with
The latter can
negligible for thin dielectric substrate
Mink
total
(Carver
The absorbed
and
power
and the corresponding total antenna Q-factor are given by
PT = Pr + Pc + Pd
1
=
QT
1
+
Qr
1
Qc
+
(5.2)
1
(5.3)
Qd
where subscripts r, c, and d refer to radiation, conductor
and dielectric,
respectively.
The main link that relates
the terms of Eq. (5.3) to that of (5.2) is WT.
tion (James et al. 1981), we have
By defini-
93
WT
Qx = 2wfr Px
(5.4)
with the subscript r,
The
c
d,
or T takes the place of
x.
are
Q-factors due to conductor and dielectric losses
independent of the field distribution underneath the patch
as shown in (James et
__ al.,
mination
1981).
Therefore, the deter-
of WT and Pr are of great concern
other terms can be evaluated.
determined,
the
In fact, when WT and Pr are
other radiation
the
whereby,
characteristics
stated
above can be deduced in a straight forward manner, as will
be outlined in the following sections.
Total Energy Stored
5.2.1
sum of the time average electric (We) and magne-
The
(Wm) energies stored within the antenna at
tic
resonance
is constant and equal to
WT = We + Wm
=
At
4
f
V (Eli12
resonance,
WT
where
1
=
u
2) dv
We is equal to Wm and reduces Eq.
eh f s
1E
z 12 ds
(5.5)
(5.5) to
(5.6)
the surface of integration S is the planar area
the patch.
i.e.
of
94
1
Wm =
2
ae
jn(knmp)
ehnEo
2
JA(kniurie)
(5.7)
pdp
YA(knmrie)Yn(knmP)
rie
This integral can be solved using the identity (Erdelyi et
al. 1953)
2
2Zp (ax )Bp (ax )-Zp+i ( ax)Bp...1 ( ax)-
xZp ( ax ) Bp ( ax )dx=21-
-Zp_i ( ax)Ep+1 (
(5.8)
x)j
where Zp(ax) and Bp(ax) are any Bessel function of the
first or second kind and of order p.
After some manipula-
tions, closed form expression defines at resonance WT as a
function of the antenna geometry is attained.
2
2
2
2
This is
WT = C2 [rae(An(rae)-An4.1(rae)An_1(rae))(5.9)
rie(An(rie )-An+1 (rie)An_1(rie))]
where C2 = EhirE02/4,
with
Eq.
using
(5.1).
c
=
cocre(fr) and An(x) is defined
Notice that the determination of WT by
the magnetic fields also leads to the
same
result
with different analytical form.
5.2.2
Radiated Power, Losses and Q-Factor
Radiated Power and Qr-Factor
A.
The
total power radiated into the far field
can
be
determined by solving the surface integral of the Pointing
vector
over
a closed spherical surface.
diated is given by,
The power
ra-
95
P
=
r
1 1
R
2 2
e
x il*)
(E
ds'
S
w
2ff
f
r
(E012) r2 sine do de
klE1312
4no
This is because 110 = Eo/no
,
(5.10)
He =-Es/no and the factor 1/2
is due to the radiation of the power through the upper
Making use of Eq.
half space only.
(5.1)
reduces Eq.
(5.10) to
2 2
h kowE
Pr =
"o
2
a
[
°
2
K1
(Bi(aae)+n2 82(aae)cos
)
sine de +
0
w
2
(
K2
J
2
(Bi(aie)+n28 22(aie)cos2e) sine de 0
2K1K2
(Bi(aae)B1(aie)+
0
]+n2B2(aae)B
(
ie)cos20) sine de
(5.11)
The solution for these integrals can be attained by
expanding each one in the form of a converging series in
terms of Gauss' hypergeometric function.
tity (Gradshteyn and Ryzhik 1980)
Using the iden-
96
n
F(- q,- m- q;n +l;b2 /a2)
Jm(ax)Jn(bx)=
where
n1
F(a,0;y;z),
(m+q)!
q=0
(5.12a)
the Gauss' hypergeometric function
is
given by
aB
F = 1 +
z +
a(a+1)8(0+1)
y(y+1)2
z2
a(a+1)(a+2)8(B+1)(0+2)
y(y+1)(y+2)31
3
z
+
along with the identity (James et al. 1981)
Jw/2
22q(q1)2
sin2q1-10 de =
0
(2q+1)1
(5.12b)
it can be shown that
2 2 2
h k En
P
Pr
=
240
2
Lr
K1/1
2
K2/2 - 2K1K213]
with
=
(_1)q u2n-2+2q
1
(2n -2 +q)! ql (2n-1+2q)
2
q =0
(_1)q u2n-2+2q
-n21
(2n+q)1 q1 (2n+1+2q)
q=0
co
1
2
q=0
(_1)cl u2n+2+2q
(2n+2+q)1 ql (2n+3+2q)
(5.13)
97
vn-'
13 =
2 (n-1)1
w
2
1
q=0
,
q =0
vn+1
2 (n+1)1
,
ql (2n+1+2q)!
(n -1)!
1
,
2
2
/
04 2a4(n+q)I F(-q,-n-q;n+1;rie/rae)
k-li
vn-1
-n
+
2
cl
(-1)u2q(n-l+q)1
F(-q,-n+l-q;n;rie/rae)
ql (2n-1+2q)1
2
,
q=0
(-1) u 2 cl(n+1+q)! F(-q,-n-l-q;n+2;rie
q! (2n+3+2q)!
2
ae)
2
where u
ko rae, v = ko rae rie and the expression for 12
is the same as that for I1 if rae is replaced by rie.
Eq. (5.13) in terms of the antenna parameters and the
mode number, represents the general solution for the power
radiated by the antenna.
excitation.
simplify
the
There
It is valid for all the modes of
are two methods which can be used
above equation because of
the
to
alternating
series nature of I1, 12 and 13.
a.
Expansion Method
After some manipulations for TMnm modes for the three
terms
13,
of the Gauss' hypergeometric functions defined with
it is seen that all of these functions are equal to 1
for q=0 and 1+(rie/rae)2 for q=1.
written in a matrix form as follows:
For q=2,
they can
be
98
TMim
Tm3m
TM2m
F1
F2
F1
F3
F2
F1
F3
F2
F1
=
F2
L- F3
1
4
1
1
3
1
(r1e/rae)2 (5.14)
1
8/3
1
(rie/rae)4
1
5/2
1
1 12/3
1
1
1
1
%MEM
where the subscripts 1,
which
and 3 stand for the order
the F functions are presented in 13
elements
hand
The
formula.
of the second column of the first matrix
in
(right
side) have magnitude equal to 2(1+1)/i where i takes
values of n,n+1 and n+2,
the
noted
that
negligible
the values of I 1,
for
1
2
and I 3 for q
This simplifies Eq.
>
3
are
which
(5.13) for its
in the determination of the radiated power for
modes.
also
It is
respectively.
a class of structure and modes for
the argument is < 1.
use
2,
these
the radiated power in the form of a
For example,
series for n=1 is found to be:
2 2 2
h koE0
P =
r
960
2 4
8
Ki[ 3
2 4
K2[3
4
(korae) 2 +
15
11
10
(k r ae)4...] +
8
Ts. (k o rl e)2
105
(k0 rl. e )4
4
-2K1K2[7 - rs_((korae)2+(korie)-9 )
16
2
7u((korae
+ 7_(koraerie)2 + (korie)4)...]
(5.15)
99
This
equation
circular
power
can be
case
of
and
1981),
For other values of
is found to be the same.
radiated
the
where K2=0 (James et al.
disk
result
checked for the available
is
obtianed
by
utilizing
a
the
n,
Eq.
(5.14).
Direct Transformation Method
b.
derived
The
converges
This
given
power series as
by
Eq.
(5.13)
(absolutely) for all values of u (Appendix
implies
that its radius of convergence
is
B).
finite.
Euler's transformation is the most convenient method which
applied directly to such an
be
can
Each
alternating
series.
series in u2 can be transformed into another one
in
the variable
u2
(5.16)
=
E
1+u
2
as outlined in Appendix B.
For example, the series Il for
the TMlm modes is found equal to:
-
E
1
1
1-
T
2
which
1
+
149
1680
547
1
(5.17)
4--)+1126 240 787- 18144(2
is valid for any value of u.
obtained
For u=1,
using
the
11=.231106
previous
compared
to
method.
Following the same procedure for 12 and 13,
.2261905
by
radiated power for such modes is found to be:
the
100
h2 22
1
Pr= 24(3 ° K1
{
u
2
2
149 2
1 1
c
+K2
149 2
1
(
(7+5"1680C ...)+11
547
1
1
c+
+
181440
240 280
2
547
1
1
...) +
2
1814401)
2(3 +3t4-1680
ui
1
1
[C.
2
-2K1K2 --(---(19-4z)+
2 3 60
2
+u E(
1
1680
(342-199z+6z2)0-
1
+
208
1
240 3360
(13-z)C+
120960
(433-
3
z+z-0
)t--1
(5.18)
2
2
2
2
wherell.=,/u,C-7zuill+u.and z = rie/rae
The first
term of each series is the effective one for u < 1 whereas
the second term is the most effective for u >
For u =
1.
1 all of the terms must be considered for better accuracy.
Similarly, the radiated power for the TM2m modes can
be obtained and is given by
2 2 2
=h koE0 K2[1:(14 4r4. 2483r2
r
240
)4.
u4
1241 2
/14.1
1 5-'2V71-18144' .." 12608-7t.126/2
2 1
/1
4
ui
2483 2
4
1
c
...I +
1241 2
1
+-v2 5Cie21ti+18144t ..") 1-1260Ci(e71 1-12672i...)
-2K 1 K 2
5
1
28579
1
0(1+42(37-5z)t+42504(32677"
v2
+
504
17
(1+
1z+z
+ 1 (205- -71(17-z)777
3
2jN
z+197z 2)t 2
2
..))
18
(5.19)
where u, ui,
c,
ci, v and z are defined earlier.
+
101
results for the radiated power as a function
The
of
some typical microstrip parameters of a closed-ring antenna
excited in the TM11 mode,
and
are shown in Figures
(5.2)
In the former figure the mean radius of
(5.3).
the
patch is kept fixed, and the inner and outer radii changed
whereas, in the second one, the b/a ratio is kept constant
and
that
The results are computed assuming
fr changed.
E0-1 V/m and are found to be in agreement with similar results
for
microstrip
other
radiated
antenna.
We note also
power increases with increase in
the
that
annular
width
region
especially
at
where
the
increase is more significant.
mode,
the radiated power of an antenna having a = 1 cm, b
the lower frequencies of microwave
For
TM 12
the
= 2 cm and Er = 2.32 is found to be equal to .09326 x
watts
h = 10 mils and 2.0063 x 10-8 watts for h
for
10-8
=
60
mils.
The
now
Qr-factor associated with the radiation term can
be determined in a straight forward manner using
Eq.
(5.4), i.e.
= 2fffr WT
Pr
where WT and Pr are given by Eqs.
pectively.
(5.20)
(5.9) and (5.13),
res-
Qr is in contrast to Pr where it is inversely
proportional to h and fr.
102
0
50
100
150
200
250
300
350
h, mils
Fig. 5.2 Radiated Power vs h for Different Widths
and Cr for a Closed-Ring Antenna Excited in TMil Mode
103
7
10,
V/m
E o = 1.
er = 2.32
h = 160 mils
h = 80 mils
108
4-1
3
---- -
a
3._ _ - -
-
9
-
L5_
0
2
4
8
6
10
12
f, GHZ
Fig. 5.3 Radiated Power vs Frequency for Different
Widths and h for a Closed-Ring Antenna
Excited in TM11 Mode
14
104
The
factor
total power absorbed by the antenna and the
be determined
now
can
taking into
QTthe
account
conductor and dielectric losses.
The corresponding Qc and
Qd-factors are usually constant,
independent of the patch
shape
can be obtained from different
and
Carver
and Mink 1981).
reproduce
However,
it is
sources
(e.g.
interesting
their simple derivations to complete this
to
part
of the study.
Conductor Loss and Qc-Factor
B.
conductor
The
currents
from
Pc = 2 2a j
are
obtained
at
It is given by
lal2 ds
,
with J = an x H
(5.21)
is the magnitude of the current density on
1JI
electric
surface
the
tangential components of the magnetic field
the surface.
model
in the microstrips which
flowing
the
where
loss is determined from
surfaces
wall
and
R 9,
(top and bottom) of
the conductor surface
the
resistivity
the
cavity
as
a
function of the conductivity a is given by
R
Making
rf
(
use
r
1/2
(5.22)
)
of Eq.
(5.5) and
substituting
(5.22)
into
(5.21) reduces it to
Pc = 2
1/2 WT
nf
(
11
a
(5.23)
105
The
quality factor Qc associated with the
conductor
loss can be determined using Eq. (5.4), i.e.
Qc = h (wfrpa)1/2
(5.24)
It
is dependent on the resonant frequency and independent
of
the antenna geometry.
The term of
( wfrua)1/2 is also
the inverse of the skin depth associated with the
conduc-
tor.
C.
Dielectric Loss and Qd-Factor
knowledge
The dielectric loss is determined from the
of electric field underneath the patch.
Pd = 2wfr tans 2
elEzI
J
2
It is given by
dv
(5.25)
V
or simply
Pd = 2wfr tans WT
where tans,
for
most
(5.26)
the loss tangent is typically less than
substrates used in microstrip
antenna
.001
designs
(Carver and Mink, 1981).
The quality factor Qd associated with the
dielectric
loss can be written by using Eq. (5.4), i.e.
Qd = 1 /tans
It
(5.27)
is independent of the antenna geometry and is equal to
the reciprocal of the loss tangent.
106
the total power absorbed by the antenna can
Now,
determined as the sum of Eqs.
(5.13),
be
(5.23) and (5.26),
i.e.
= p
P
T
whereas
+ o-l%--wfr)1/2
r
u
WT + 2wfr tants WT
total antenna Q-factor which
the
(5.28)
a
specifies
its
frequency selectivity can be written as
1
P
1
r
QT = [ 2wfr WT
hOrfr401/2
where WT and Pr are given by Eqs.
+ tandri
(5.29)
(5.9) and (5.13),
res-
pectively.
The
antenna
results for typical parameters of a
excited
in
in
Figures
The variation of Q-factors as a function
(5.4) and (5.5).
of
the TM 11 mode are shown
closed-ring
substrate thickness for different annular
widths
are
illustrated in the former figure and for different dielectric
constants in the second figure.
assuming
that tand = .0005 and the microstrip
tion is copper.
computed
They are
metalliza-
The results are similar to those obtained
for rectangular and circular microstrip antennas (Bahl and
Bhartia 1980; Carver and Mink 1981).
Resonance and Input Impedance
5.2.3
At resonance,
can
be
antenna.
the input impedance is nonreactive and
determined from the total power absorbed
It is defined by Eq. (2.3) as
by
the
4
10
10
=1.
\* \
\
er = 2.32
er = 6.8
er = 9.8
103
QT
Qr
Qc
N
Qd
\
\ , .
\\
\
\
\\.
N.
I.
.
`.-
,
.
""'"
21
z
=1. cm
=2. cm
tand= .0005
a
=5.8x107 U/m
=4wx10-7 H/m
P
a
b
=2.32
tand= .0005
er
a
p
=5.8x107 tg/m
=4/rX10-7 H/M
ii.11
10
0
50
IOD
200
h,
mils
10-
1
.
I
150
250
300
250
Fig. 5.4 Q-Factors vs h for Different
Widths of a Closed-Ring Antenna
Excited in TM 11 Mode
1
0
1111
50
100
I
150
200
250
300
350
h, mils
Fig. 5.5 Q-Factor vs h for Different Cr
for a Closed- Ring Antenna Excited
in TM 11 Mode
108
2
V,
Ro =
(5.30)
2PT
where PT is given by Eq.
(5.28) and Vo, the resonant mode
voltage at the feed point at 4)=0 is given by
(k
Vo = hEz10=0= hEo [Jn(knm
n
nm
p)]
(5.31)
13)
The resonant mode voltage which acts between the patch and
ground
the
input impedance at resonance as
the
for
results
The
the same manner as Ez does (Figure 3.2).
in
point
plane varies as a function of the feed
function
a
of
microstrip parameters for the TM11 mode are shown
typical
in Figure (5.6) for an outer-edge fed element.
The input impedance as a function of frequency can be
equiva-
easily determined by utilizing the antenna simple
other
parallel tuned RLC circuit as is the case for
lent
shapes
of
microstrip patch antennas (Long et
Richards et al.
1981; Krowne 1981).
1978;
al.
The equivalent
RLC
parameters are determined from the total antenna Q-factor,
resonance input impedance and the resonant frequency.
the
These are:
= Ro
CI =
(5.32)
1
QT
2fff-rRI
1
=
(21rfr)201
(f r =
f-ff
(or LI =
0'
'')1/2, QT
(LC
RI
2fr
= R1( --)
1/2
)
(5.33)
L'
(5.34)
109
0
50
100
150
200
250
300
350
h, mils
Fig. 5.6 Resonant Input Impedance vs h for Different
Widths and r for a Closed-Ring Antenna Excited
in TM 11 Mode, Outer-Edge Fed Point
110
They can also be obtained by representing the antenna as a
lumped
In such a case,
element.
the equivalent capaci-
tance can be expressed as
2WM
C' =
(5.35)
2
Vo
This leads to the same results,
by Eqs.
dance
(5.9) and (5.31),
where WT and Vo are given
respectively.
The input impe-
as a function of frequency can now be expressed
in
terms of these parameters as follows:
Z.
1
= Re + j Im =
1
1
-1
(5.36)
+ jwC' + jwL
and this leads to the following well known identity
Zi = Ro [1 + j QT(,
1r
where
fr Ni-1
--/J
(5.37)
f
QT and Ro are given by (5.29) and
(5.30),
respec-
tively.
The results for the input impedance versus
frequency
for typical parameters of a closed-ring antenna excited in
the TM11 are illustrated in Figures (5.7) and (5.8).
former
figure
shows
the input impedance
for
The
different
substrate thickness for an outer and inner-edge feed point
The second figure shows the input impedance for
dielectric
constants.
that a = lcm,
different
The results are computed assuming
b = 2cm with tans = .0005 and copper is the
conductor material.
They indicate the effect of changing
the substrate thickness on the shift in resonant frequency
111
911/6.100
Outer fed
Inner fed
700.00
500.00 -
= 1.
= 2.
a
b
31111.00
r
0
.,-p
cm
cm
= 2.32
1110.00
-100.011
N
-300.0.7
-500.00
-7110.011
- 7100.00 -
f, GHZ
(a)
1700.00 /000.00
Outer fed
Inner fed
-
000.0)) GM00
400.1111
cm
= 1.
cm
b = 2.
Er = 2.32
a
J.
200.110
23.
11+0
2i5
.00
23
N -200.00
.1.
-100.00 -(700.00
-000.00 -1000.00 -1700.00 -
f, GHZ
(b)
Fig. 5.7 Input Impedance vs Frequency for Different h
for a Closed-Ring Antenna Excited in TM11 Mode
(b) h = 80 mils
(a) h = 160 mils
900.00 -
700.00
II
II
11
500.00 -
I
Re
300.00
III II
40
'r+
/ Ill
100.00
er = 2.32
= 9.8
r
1. cm
a =
2. cm
b =
h = 160. mils
e
2.5
-100.00
I'
-300.00
1,
Im
I I
-500-00 -
II
II
II
-700.00 -
II
II
-900.00
f, GHZ
Fig. 5.8 Input Impedance vs Frequency for Different c
for a Closed-Ring Antenna Excited in TM11 Mode,
Outer-Edge Fed Point
113
and the reduction in the input impedance as the feed point
is moved toward the annular inner circumference.
In con-
trast to a circular disk (Bahl and Bhartia 1980),
we note
that
real part of the input impedance
the
decreases
by
choosing a thicker substrate.
5.2.4
Radiation Resistance
radiation resistance can be considered as a spe-
The
cial
case
the input impedance at
of
if
one
It is
de-
resonance
neglects the conductor and dielectric losses.
fined by Eq. (2.9) as
V2
2Pr
Rr
(5.38)
The results for the radiation resistance are shown in
Figures
strip
(5.9),
(5.10) and (5.11) as a function of micro-
geometrical
parameters for a
closed-ring
antenna
The
excited in the TM11 mode (Sultan and Tripathi, 1985).
radiation
the outer to the inner-edge.
from
increase
point
resistance decreases as we move the feed
in
frequency
It decreases with
as is the case of
a
an
rectangular
patch.
For
radiation
and
34.1
the
TM 12 mode,
the calculated
values
of
the
resistance is equal to 11.5 0 for an outer feed
for an inner
feed closed-ring antenna having
a = 1 cm, b = 2 cm, h = 60 mils and Cr = 2.32.
114
= 2.32
r
Er = 6.8
= 9.8
e
0
E
0
1:4
10-
102
1
0
50
1
1
100
1
1
150
I
1
200
I
1
250
1
1
300
t
350
h, mils
Radiation Resistance vs h for Different Widths
Fig. 5.9
and er for a Closed-Ring Antenna Excited in TMil Mode,
Outer-Edge Fed Point
115
E
50
100
150
200
250
300
350
h, mils
Fig. 5.10 Radiation Resistance vs h for Different Cr
for a Closed-Ring Antenna Excited in TM11 Mode,
Inner-Edge Fed Point
4
10
er = 2.32
h = 160 mils
h = 80 mils
E
0
3
I0
2
10
0
2
4
6
8
I0
12
14
f, GHZ
Fig. 5.11 Radiation Resistance vs Frequency for
Different b/a ratios and h for a Closed-Ring
Antenna Excited in TM11 Mode, Outer-Edge Fed Point
116
5.2.5
Bandwidth
The
bandwidth
expressed
(half
power or -3dB
can
width)
be
the
in terms of the total antenna Q-factor and
resonant frequency as follows:
B.W =
fr
(5.39)
QT
This
usual
expression is not
extremely
because
useful
there is an impedance matching network between the antenna
feed
point and its radiating element which must be
into
consideration.
from
VSWR
1979;
The determination of the
bandwidth
measurements is commonly used (Derneryd
Derneryd and Link 1979;
James et al.
taken
1978;
1981; Carver
and Mink 1981) because this ratio represents the
critical
parameter which limits the antenna performance.
This can
be
obtained from the input impedance in the region
to the resonant frequency.
close
Substituting f = fr + of into
Eq. (5.37) reduces it to
Zi = Ro [1 + j2 QT
A
f -1
(5.40)
fr
the reflection coefficient can be obtained
At
resonance,
by
matching the antenna to the feed line,
and it can
be
expressed as follows:
r = Zi-Rn
zi+Ro
=
jQTAf/fr
1+jQTAf/fr
(5.41)
117
reflection
The
coefficient can now be used to
percen-
the standard expression of the
the VSWR whereby,
bandwidth (Eq.
tage
determine
2.2) in terms of QT can be obtained,
i.e.
100(S-1)
B.W =
(5.42)
QT(s)1/2
expression represents the bandwidth as the
This
frequencies
value,
The
i.e.
where
the
band
input VSWR is less than
given
a
VSWR < S.
results
substrate
for the bandwidth versus the
thickness for typical parameters of a closed-ring
antenna
excited in the TM11 mode are shown in Figure (5.12).
are
computed
assuming that S = 3 with tans =
copper as the conductor material.
results
for
lower value of dielectric constant.
results
indicate
annular
width.
the increase of B.W
The
and
They conform to similar
microstrip antennas and
other
They
.0005
confirm
increase of B.W by choosing a thicker substrate and
a
of
the
using
In addition,
with
increase
results are also in agreement
those obtained by Chew (1982) whose analysis was based
the method of matched asymptotic expansion.
the
in
with
on
118
102
Er = 2.32
er = 9.8
= 3.
tans = .0005
S
U/m
= 5.8x107
413(10-7
H/m
=
a
10
J
1
0
I
50
1
I
100
1
I
I
1
150
200
i
1
250
I
1
300
350
h, mils
and er
Fig. 5.12 Bandwidth vs h for Different Widths Mode
for a Closed-Ring Antenna Excited in TMil
119
Efficiency, Directivity and Gain
5.2.6
Efficiency
A.
As indicated in Chapter 2, the antenna efficiency can
be obtained from the ratio between the radiated power
the
total power absorbed by the antenna.
Making use
and
of
Eq. (5.4), the antenna efficiency can also be expressed in
terms of Q-factors, i.e.
n = Pr = QT
PT
The
(5.43)
Qr
results for the antenna loss (Antenna Loss =
log(1/71)
dB) versus the substrate thickness for a closed-
ring antenna excited in the TM11 mode are shown in
(5.13)
10
for
different dielectric constants and
Figure
different
annular widths.
Directivity
B.
The directivity is defined by Eq. (2.5) as "the ratio
of
the
maximum power intensity in the main beam
average radiated power intensity."
to
the
For the TMim mode
the
directivity can be written as
1
D
Tr
where
Re(1E012+1Et12)10=0
5.44)
Pr/4wr2
the numerator represents the radiation peak in
normal direction and can be obtained from Eq.
no = 120r.
the
(3.30), and
Thus, the directivity can be written as
O
1-3
rt
0
Up
O
O
1\.) r"fx)
///
/
1/11///
/
//7/7///
/ /7/7-7/
,:'
---
-
i
Ill,
////"
4///
,....
'67,6
./
(*)
I
I
I
II
N
II
1-1
CO CO (0
1.0
II
mmm
I
I
O
O
,.
...." .
/
/
//
y
(D
In
0
01
_
v
,,
//.
I
0
CA
<
cn
(1)
0
a
(D
0
u,
0
1-
m
rt.
U0
(D
a
(1)
0 rt
ti
0
1-h LP
O
Antenna loss, dB
121
h
1
D =
2
ko2 E02
240
(Ki-K2)2
(5.45)
Pr
where K1 and K2 are defined with Eq. (5.1) and Pr which is
given
by Eq.
(5.13).
(5.45)
Eq.
Substituting Pr into
indicates that the directivity is independent of substrate
thickness,
but
indeed,
This
it varies slowly with it.
little
variation
fields
on the antenna dimensions and the resulting reson-
is due to the effect
of
fringing
the
ant frequency.
The results for the directivity (DdB = 10 Log D) verthe substrate thickness for a typical parameter of
sus
closed-ring antenna excited in the TM11 mode are shown
2 cm,
=
er = 2.32 and excited in the TM12
calculated
in
For a closed-ring antenna having a = 1 cm,
Figure (5.14).
b
a
the
mode,
values of the directivity is equal to 8.55
dB
for h = 10 mils and 8.36 dB for h = 60 mils.
Gain
C.
ciency
effi-
gain of an antenna takes into account its
The
its directional capabilities as given by
and
Eq.
(2.6). For the TMlin modes, this is given as
2
G =
Eq.
ko2 E02
240
where PT,
by
h
1
(Ki-K2)2
(5.46)
PT
is given
the total power fed into the antenna,
(5.28).
directivity
Notice
that the gain is equal
if one neglects the conductor and
to
the
dielectric
8.-
W(cm)
1.5;
7.5-
7.-
_ --
Er = 2.32
= 9.8
6------ 2
5.5-
0
50
100
150
200
250
300
350
50
h, mils
Fig. 5.14 Directivity vs h for Different
Widths and Er for a Closed-Ring Antenna
Excited in TM11 Mode
100
ISO
200
250
300
350
h, mils
Gain vs h for Different
Widths and Er for a Closed-Ring
Antenna Excited in TM11 Mode
Fig. 5.15
123
losses.
Typical results for the antenna gain versus sub-
strate thickness are shown in Figure (5.15) for the TM11
mode.
5.3
Example
The data presented earlier can be helpful in the
design purposes of such antennas.
For example,
for b/a =
3 cm and W = 2 cm, i.e. b = 3 cm and a = 1 cm, together
with h = 160 mils and er = 2.32, we can
determine the
radiation characteristics of this antenna excited in the
TM11 mode.
Assuming that tand = .0005 and copper is the
conductor material of the antenna, the characteristics are
found to be:
QT
= 30,
Rr
= 410 ohm
Qr = 31,
Qc = 2900,
B.W = 3.9% (for VSWR = 3)
Antenna Loss =
.11 dB
Directivity
=
7.76 dB
Gain
=
7.65 dB
Qd = 2000
124
5.4
Concluding Remarks
In this chapter, the radiation characteristics of a closed-ring
microstrip antenna have been evaluated for the
axial modes.
radial
General closed form expression for
as
the
well
total
as
energy
stored has been derived by utilizing the properties of the cylindrical functions.
Known closed
form
expressions
for
field based on the cavity model (Eq. 3.22) have also
its
radiation
been used
derive general expression for the radiated power in a
series
to
form.
This has been accomplished by utilizing the properties of the Gauss'
hypergeometric functions.
By using
Euler's
transformation
it
shown that the power series converges for all of the modes of
tation.
is
exci-
The expressions have been used to set the relation between
efficiency and bandwidth and to formulate the other radiation
acteristics.
This includes input impedance,
losses, Q-factors, directivity and gain.
cused on the dominant TM
11
radiation
resistance,
Much of the data
mode, presented in
a
char-
was
systematic
fo-
manner
and covered by a host of typical results in a form of graphs.
In short, a simple method to calculate
all
of
the
radiation
characteristics together with typical results in the form of
and formulas for closed-ring antennas has been presented.
sults should be helpful in the design of such antennas.
graphs
The
re-
125
CHAPTER 6
SUMMARY AND CONCLUSIONS
The purpose of this study was to analyze the radiation behavior
of open- and closed-ring microstrip structures including the
limit-
ing case of an ideal gap open-ring structure.
In Chapter 3, the analysis of an ideal
was presented based on the cavity model.
antenna
open-ring
gap
The cavity model
the effect of microstrip curvature, fringing fields and
includes
dispersion.
It is seen that the results for the radiation fields can be
expres-
sed in terms of the spherical Bessel functions for the odd modes and
Bessel functions of integer order for the even modes.
solution is of course the same as that for
a
seen that the field distribution in the cavity
The even mode
closed-ring.
model
It
is
can help
us
envision the radiation characteristics for various modes.
puted results for various modes were presented and it is
the TM
12
mode of an open-ring structure is potentially an
The
com-
seen
that
efficient
useful mode for applications as an antenna element.
In Chapter 4, the general case of an annular sector
microstrip
antenna having an arbitrary gap angle was analyzed including the gap
fields.
The radiation characteristics of various typical cases were
computed by utilizing the derived general expression for the
tion fields.
radia-
In addition, closed form expressions for the radiation
peak for the modes that produce radiation in
the
normal
direction
126
were derived.
The results obtained for various sector angles can be
of
used to determine the useful axial and radial modes
It was shown that the
ture.
radiation
the
struc-
associated
characteristics
with some of these modes are similar to other useful patch antennas.
terms
The sector provides additional degrees of freedom in
of
its
shape, resonant frequency and modification in the radiation pattern
due to the relative flexibility in having the radiating apertures at
desired locations.
In Chapter 5, the special case of
then considered in order to analyze its
taking into account conductor and
antenna was
closed-ring
a
radiation
dielectric
characteristics,
closed
The
losses.
form expressions of its radiation field (the derived expressions
the even-modes case of the ideal gap open-ring) have
been used
This includes a general
closed
form
expression
to
para-
derive the expressions for the useful antenna characteristics
meters.
of
for
the
total energy stored, and an expression for the radiated power in the
form of a converging series which is valid for any mode
tion and for any antenna dimensions.
An equivalent
of
In addition, expressions for other radiation
teristics such as radiation resistance, bandwidth,
rectivity and gain were derived to
Numerical results for
properties.
TM
11
and TM
12
modes were presented.
study
typical
the
function
charac-
efficiency,
closed-ring
structures
di-
antenna
excited
The results should be
in the design of microstrip ring antennas.
tuned
parallel
RLC circuit was used to determine the input impedance as a
of frequency.
excita-
in
helpful
127
BIBLIOGRAPHY
M. and Stegun, I.A., Handbook of Mathematical
Abramowitz,
Functions, New York: Dover Inc., 1970.
Agrawal, P.K. and Bailey, M.C., "An Analysis Technique for
Microstrip Antennas,"
IEEE Trans. Antennas Propagat.
AP-25, Nov. 1977, pp. 756-759.
Ali,
S.A.,
"Vector Hankel
Chew,
W.C.
and Kong,
J.A.,
Transform Analysis of Annular-Ring Microstrip Antenna," IEEE Trans. Antennas Propagat. AP-30, July 1982,
pp. 637-644.
Bahl, I.J. and Bhartia, P., Microstrip Antennas, MassachuArtech House Inc., 1980.
setts:
Bahl,
M.A.,
"A New
I.J.,
Stuchly,
S.S.
and Stuchly,
Microstrip Radiator for Medical Applications," IEEE
Trans. Microwave Theory Tech. MTT-28, Dec. 1980, pp.
1464-1468.
Balanis,
C.A.,
Antenna Theory Analysis and Design,
York:
Harper and Row, 1982.
New
Ball Aerospace Systems Division, Ball Microstrip Antenna
Data Sheets, Colorado: Boulder, n.d.
Bhattacharyya, A.K.
"Input Impedance of
and Garg,
R.,
Annular-Ring Microstrip Antenna Using Circuit Theory
Approach,"
IEEE Trans.
Antennas Propagat. AP-33,
April 1985, pp. 369-374.
Admittance
Mutual
and
"Self
Between Two Concentric Coplanar Circular Radiating
Current Sources," IEE Proc. 131, No. 3, June 1984,
pp. 217-220.
Concentric
"A Microstrip Array of
Annular Rings," IEEE Trans. Antennas Propagat. AP-33,
June 1985, pp. 655-659.
,
Carnahan,
B.,
Luther,
Numerical Methods,
H.A.
New
and Wilkes, J.0., Applied
John Wiley & Sons,
York:
1969.
Carver, K.R., "A Modal Expansion Theory for the Microstrip
Antenna," IEEE AP-S Int. Symp. Digest, Seattle, WA,
June 1979, pp. 101-104.
128
"Microstrip Antenna
J.W.,
Carver,
K.R.
and Mink,
Propagat. AP-29,
Trans.
Antennas
IEEE
Technology,"
Jan. 1981, pp. 2-24.
"Green's Functions for
in Planar Microwave Circuits,"
IEEE Trans. Microwave Theory Tech. MTT-28, Oct. 1980,
pp. 1139-1143.
Chadha,
R.
and Gupta,
Triangular Segments
K.C.,
"Green's Functions for Circular
Sectors, Annular-Rings, and Annular Sectors in Planar
Microwave
Circuits,"
IEEE Trans. Microwave Theory
Tech. MTT-29, Jan. 1981, pp. 68-71.
"Segmentation Method Using Impedance
Microwave
Planar
Matrices
for
Analysis
of
Circuits," IEEE Trans. Microwave Theory Tech. MTT-29,
Jan. 1981, pp. 71-74.
,
Chew,
Microstrip
W.C.,
"A
Broad-Band Annular-Ring
Antenna," IEEE Trans. Antennas Propagat. AP-30, Sept.
1982, pp. 918-922.
Collin,
R.E.,
Field Theory of Guided Waves,
McGraw-Hill, 1960.
New
York:
K.F., "Characteristics of AnnularDahel, J.S.
and Lee,
Nov.
Electron Lett. 18,
Ring Microstrip Antenna,"
1982, pp. 1051-1052.
Das,
"Radiation
S.P.,
Mathur,
Microstrip
in
Characteristics of Higher-Order Mode
Ring Antenna," IEE Proc. 131, April 1984, pp. 102A.,
Das,
S.K.
and
106.
Derneryd,
A.G.,
"Circular and Rectangular Microstrip
Antenna Elements," Ericsson Technics No. 3, 1978, pp.
162-177.
the Microstrip Disk
"Analysis of
Antenna Element," IEEE Trans. Antennas Propagat. AP27, Sept. 1979, pp. 660-664.
"Extended Analysis of
and Lind,
A.G.,
A.G.
Derneryd,
Rectangular Resonator Antennas," IEEE Trans. Antennas Propagat. AP-27, Nov. 1979, pp. 846-849.
Antenna Theory and Design,
Elliott,
R.S.,
Prentice-Hall Inc., 1981.
New
Jersey:
A.,
Erdelyi,
Magnus, W., Oberhettinger, F. and Tricomi,
New York:
Higher Transcendental Functions,
F.G.,
McGraw-Hill Inc., Vol. 2, 1953.
129
"Adaptive Arrays - An Introduction," IEEE
W.F.,
Gabriel,
Proc. 64, 1976, pp. 239-272.
Munson, R.E., Ostwald, L.T. and Schroeder,
Garvin,
C.W.,
"Missile Base Mounted Microstrip Antennas,"
K.G.,
IEEE Trans. Antennas Propagat. AP-25, Sept. 1977, pp.
604-610.
Gradshteyn,
Table of Integrals,
I.S.
I.M.,
and Ryzhik,
Academic Press,
Series, and Products,
New York:
1980.
K.C.
"Segmentation and
and
Sharma,
P.C.,
Desegmentation Techniques for Analysis of Planar
Microstrip Antennas,"
IEEE AP-S Int. Symp. Digest,
Los Angeles, CA, June 1981, pp. 19-22.
Gupta,
"Accurate Models for
Hammerstad,
and Jensen,
O.,
O.E.
Microstrip-Computer Aided Design," IEEE MTT-S Int.
Symp. Digest, Washington 1980, pp. 407-409.
Field Computation by Moment Methods,
Harrington,
R.F.,
New York: MacMillan & Co., 1968.
Itoh,
T. and Menzel, W., "A Full-Wave Analysis Method for
Open Microstrip Structures,"
IEEE Trans. Antennas
Propagat. AP-29, Jan. 1981, pp. 63-67.
Itoh,
T.
"Spectral Domain Approach for
and Mittra,
R.,
of
Characteristics
the
Dispersion
Calculating
Microstrip Lines," IEEE Trans. Microwave Theory Tech.
MTT-21, July 1973, pp. 496-499.
James,
J.R.,
Hall, P.S. and Wood, C., Microstrip Antenna
Peter Pereginus Ltd.,
New York:
Theory and Design,
1981.
A.M., "Ring and Disk Resonator CAD Model," Microwave J. 27, Nov. 1984, pp. 91-105.
Khilla,
R., "Planar Waveguide Model for
Mehran,
Calculating Microstrip Components," Electron Lett.
11, Sept. 1975, pp. 459-460.
Kompa,
G.
and
C.M., "Cylindrical-Rectangular Microstrip Antenna
Radiation Efficiency Based on Cavity Q-Factor," IEEE
AP-S Int. Symp. Digest, Los Angeles, California, riEg
1981, pp. 11-14.
Krowne,
Lo,
W.F., "Theory and
Solomon, D.
and Richards,
IEEE Trans.
Experiment
on Microstrip Antennas,"
Antennas Propagat. AP-27, March 1979, pp. 137-145.
Y.T.,
130
Long,
S.A., Shen, L.C., Walton, M.D. and Allerding, M.R.,
"Impedance of a Circular-Disc Printed-Circuit Antenna," Electron Lett. 14, Oct. 1978, pp. 684-686.
Mailloux,
R.J.,
"Phased Array
Satellite Communications,"
1977, pp. 38-42.
Mink,
Aircraft Antennas for
Oct.
Microwave J. 20,
J.W., "Circular Ring Microstrip Elements," IEEE APS Int. Symp. Digest, Canada, June 1980, pp. 286-289.
Morse,
P.M. and Feshbach, H., Methods of Theoretical PhyNew York:
McGraw-Hill,
1953, ch. 7, p. 812,
sics,
820.
Munson,
R.E.,
"Conformal Microstrip Antennas and Microstrip Phased Arrays," IEEE Trans. Antennas Propagat.
AP-22, Jan. 1974, pp. 74-78.
Newman,
E.H.
Antennas
and Tulyathan,
P., "Analysis of Microstrip
Using Moment Methods," IEEE Trans. Antennas
Propagat. AP-29, Jan. 1981, pp. 47-53.
Okoshi,
T.
and Miyoshi,
T.,
"The Planar Circuit - An
Approach to Microwave Integrated Circuitry," IEEE
Trans. Microwave Theory Tech. MTT-20, April 1972, pp.
245-252.
Okoshi, T., Uehara, Y. and Takeuchi, T., "The Segmentation
Method - An Approach to the Analysis of Microwave
Planar Circuits," IEEE Trans. Microwave Theory Tech.
MTT-24, Oct. 1976, pp. 662-668.
Okoshi, T., Planar Circuits for Microwaves and Lightwaves,
Springer Series in Electrophysics-18, 1985.
Richards,
W.F., Lo, Y.T. and Harrison, D.D., "An Improved
Theory for Microstrip Antennas and Applications,"
IEEE Trans. Antennas Propagat. AP-29, Jan. 1981, pp.
38-46.
Richards,
W.F.,
Ou,
J.D. and Long, S.A., "A Theoretical
and Experimental Investigation of Annuar, Annular
Sector,
and a Circular Sector Microstrip Antenna,"
IEEE Trans. Antennas Propagat. AP-32, Aug. 1984, pp.
864-867.
Sanford, G.
and Klein, L.,
"Development and Test of a
Conformal Microstrip Airborne Phased Array for Use
with
the
ATS-6
Satellite,"
IEE Int. Conf. on
Antennas for Aircraft and Spacecraft," 1975, pp. 115122.
131
Some Equivalence Theorems of ElectroSchelkunoff,
S.A.,
magnetics and Their Application to Radiation ProbVol. 15, 1936, pp. 92J.,
Bell System Tech.
lems,
112.
Electromagnetic Waves, New York:
Van Nostrand Company, 1943, pp. 158-159.
,
D.
Schneider,
M.V.,
"Microstrip Dispersion," Proc. IEEE
Vol.
(Special Issue on Computer in Design) (Lett.),
60, Jan. 1972, pp. 144-146.
P.C.
and Gupta, K.C., "Desegmentation Method for
Circuits,"
Analysis of Two-Dimensional Microwave
IEEE Trans. Microwave Theory Tech. MTT-29, Oct. 1981,
pp. 1094-1098.
Sharma,
for
Alternative Procedure
"An
Implementing the Desegmentation Method," IEEE Trans.
Microwave Theory Tech. MTT-32, Jan. 1984, pp. 1-4.
Planar
Silvester,
P.,
"Finite Element Analysis of
Microwave
IEEE Trans. Microwave Theory
Network,"
Tech. MTT-21, Feb. 1973, pp. 104-108.
Sommerfeld, A.,
Optics,
1954, pp. 325-328.
New York:
Academic Press Inc.,
Sultan, M.A. and Tripathi, V.K., "The Open-Ring Microstrip
Vancouver
Antenna,"
IEEE AP-S Int. Symp. Digest,
B.C., Canada, June 1985, pp. 417-421.
Annular
Digest,
of
Characteristics
"Radiation
Microstrip Antenna," IEEE AP-S Int. Symp.
Canada, June 1985, pp. 421Vancouver B.C.,
424.
Tripathi, V.K.
and Wolff, I., "Perturbation Analysis and
Design Equations for Open-and-Closed Ring Microstrip
Resonators," IEEE Trans. Microwave Theory Tech. MTT32, April 1984, pp. 405-410.
Troughton,
P.,
"Measurement Techniques in
Electron Lett. 5, Jan. 1969, pp. 25-26.
Microstrip,"
J.G.,
and Fikioris,
Uzunoglu,
N.K.,
Alexopoulos,
N.G.
"Radiation Properties of Microstrip Dipoles," IEEE
Trans. Antennas Propagat. AP-27, Nov. 1979, pp. 853858.
Van de Capelle, A.R. and Luypaert, P.J., "Fundamental and
in Open Microstrip Lines," Electron
Higher Order
Lett. 9, July 1973, pp. 345-346.
132
Wolff,
I. and Knoppik, N., "Microstrip Ring Resonator and
Dispersion Measurements on Microstrip Lines," Electron Lett. 7, Dec. 1971, pp. 779-781.
Wolff,
I.
and Tripathi,
V.K., "The Microstrip Open-Ting
Resonator,"
IEEE Trans. Microwave Theory Tech. MTT32, Jan. 1984, pp. 102-107.
Wood,
C.,
"Analysis of Microstrip Circular Patch Antennas," IEE Proc. 128,
Pt.
H.
No. 2, April 1981, pp.
69-76.
Wu,
Y.S.
Yano,
and Ishimaru, A.,
"A Theoretical Study of the
Impedance of a Circular Microstrip
Disk
Antenna," IEEE Trans. Antennas Propagat. AP-29, Jan.
1981, pp. 77783.
and Rosenbaum, F.J., "Mode Chart for Microstrip
Ring
Resonators,"
IEEE Trans. Microwave Theory
Tech. MTT-21, July 1973, pp. 487-489.
S.
Input
APPENDICES
133
Appendix A
VECTOR TRANSFORMATIONS
134
This
appendix describes the
from cylindrical-to-spherical.
vector
transformations
The source points, in this
thesis, are represented in cylindrical components and usu-
ally, the radiated fields are determined in spherical components.
The
problem
can be resolved by utilizing
the
vector transformations from rectangular-to-cylindrical and
from
and
rectangular-to-spherical.
Referring to Figure
A.1
designating primed (p1,01,z1) for the source coordin-
ates, unprimed (r,8,0) for the observation coordinates and
(x,y,z) for rectangular coordinates,
the vector transfor-
mations can be given as follows:
z
X
(a) Rectangular
(b) Cylindrical
(c) Spherical
Fig. A.1
The Three Different Coordinate Systems
135
A.1
Rectangular-to-Cylindrical (and Vice-Versa)
For any vector T, the transformation from the rectan-
gular-to-cylindrical
coordinates can be obtained from the
matrix [T]rc:
T
p
T
,
0
cos0'
sin01
0
Tx
-sin01
cos,'
0
Ty
1
Tz
0
0
(A.1)
Awn.
Since [T]rc is an orthonormal matrix (its inverse is equal
to its transpose), we can write at once the transformation
matrix [TJcr for cylindrical-to-rectangular coordinates as
follows:
_
Tx
cosh'
-sin.'
Ty
sin4'
cos.'
0
Tz
0
0
1
Tpl
(A.2)
T4,u
Tz'
IWO&
where the coordinate systems are related by
A.2
x
= p' coso'
p'
=
(X4y2)1/2
y
= p' sine'
CO'
=
tan-1 y/x
z = Z
z'
= z
(A.3)
(A.4)
Rectangular-to-Spherical (and Vice-Versa)
For any vector T, the transformation from the rectangular-to-spherical
matrix [T] rs.
coordinates
can be obtained from
the
136
Tr
sine cos0
sine sink
cose
T
cose cos$
cose sin0
-sine
Ty
0
Tz
e
cos,
-sin,
T
(A.5)
The [T]rs is also an orthonormal matrix so that the transformation matrix [T] sr for spherical-to-rectangular
coor-
dinates can be written at once as follows:
Tx
sine cos0
cose cos0
-sin0
Tr
Ty
sine sin$
cose sin$
cos4
Te
Tz
-sine
cose
(A.6)
T
0
4)
where the coordinate systems are related by
x = r sine cos,
y = r sine sing)
z = r cose
(A.7)
r = (x2+1,21_z2)1/2
e = cos-1 z/r
0 = tan-1 y/x
(A.8)
Cylindrical-to-Spherical (and Vice-Versa)
A.3
For any vector T, the transformation matrix [T]cs for
cylindrical-to-spherical
coordinates can be
obtained
by
utilizing (A.2) and (A.5) and can be written as follows:
cose
Tr
sine cos(0-0')
sine sin(0-0')
T
cose cos(0-0')
cose sin(0-0') -sine
e
-sin(0 -0')
cos(0 -0')
0
T
1
Too
(A.9)
Te
_
-
The [T]cs is also an orthonormal matrix so that the transformation
matrix [T] sc for spherical-to-cylindrical coor-
dinates can be written at once as follows:
137
IMO
T
T
=
sine cos(0-0')
case cos(0-0') -sin(0-0')
sine sin(0-0')
cose sin(0-0')
-sine
case
Tz,
cos(0-0')
0
Tr
T e (A.10)
T
(A.9) and (A.10) can be simplified as follows:
For 0' = 0,
T
Tr
sine
0
cose
T
case
0
-sine
T0
1
0
Tz,
0
- Tr
1
Te
0
T 01
e
TO
0
case
sine
T
1
I
(A.11)
p
T
0
1
case
Tz,
0
-sine
(A.12)
4mm
where for this case we have
It
p'
= r sine
r
=
(1)14z12)1/2
= T
0
e
= tan-1 p' /z'
also be noted that (A.11) and (A.12) are
should
if T
z' = r case
,
(A.13)
(A.14)
valid
otherwise we have to use (A.9) and (A.10).
138
Appendix B
POWER RADIATED FROM A CLOSED-RING
MICROSTRIP ANTENNA
139
power
derived
appendix describes the behavior of the
This
5.13) on the circle of convergence and
series (Eq.
its rapidly convergence for any mode of excitation.
The
three
terms of Eq.
alternating
consist
of
Each of these series can
be
(5.13) II,
series.
12 and 13
written in the form of
f(u) =
(-1)q
aq u
+ 2q
(B.1)
q=0
where
*
is
on
a constant function depending
mode
the
Therefore, we have
number.
a
(-1)- aq
f(u) = u *
(B.2)
u2c1
q=0
The convergence nature of such a series has been
using
the ratio test' and the equivalent of aq as defined
in Il,
As a
12 or 13 for different modes of excitation.
result,
of u,
examined
Eq.
(5.13) converges (absolutely) for all values
and we say that it converges for
lently,
we
can
lul
that the radius of
say
<
Equiva-
convergence
is
infinite or the circle of convergence is infinite.
1
If lim
f(u
=
c, then f(u) converges (absolutely)
if c < 1 and diverges if c >
1.
If C = 1, the tests fail.
140
Let us now consider Euler's transformation (Morse and
Feshbach, 1953) and apply it directly to (B.2).
Multiply-
ing both sides by (1 +u2), one obtains
co
(l+u2) f(u) = u* (1 +u2)
(-1)q aq u2q
q=0
CO
OD
= u*
1 (-1)q aq u2q + u2 1 (-1)q aq
(B.3)
q=0
q=0
[
Now, we have
CO
q=0
u4 - a3 u6 + a4 u8 -
(-1)q aq u2q = a o - alu2
CO
= ao -
u2
(-1)q aq+l ,2q
1
(B.4)
q=0
Substituting (B.4) into (B.3) leads to
2
(1 +u
)
f(u) = u*
ao-u
2
(-1)q ag+1 u
2q
co
+u
2
(-1)
q
aq u
2]
q=0
q=0
OD
[= u*
-aq+1) u21
ao+u2 1 (-1)q
q=0
(B.5)
Hence (B.2) can be written as follows:
co
f(u) = u*
a° +
1+u2
[
1
(-1)q sag
q=0
where daq = aq - aq+1 and
u2
=
1+u2
(B.6)
141
Applying the procedure to the coefficient of E reduces
(B.6) to
co
f(u) = u*
y
a° + E 6a°
1+u2
[ 1+u2
(...1)q 62aq u2q
(B.7)
o
(B.8)
q=0
where 62aq = 6(6aq) = 6aq - 6aq+1
Continuing on in this way we get
u*
f(u) =
ao
2a
2
6ao
6
1+u2
Y
where
6Ya
=
q
)
(-1) x
x=0
)
/Ixf
aq+x and
Y
(x ),
the
binomial
coefficients is given by
(Y)
x
Yi
(B.9)
xi(y-x)1
an alternating series defined by (B.2) is converted
Thus,
into
=
power
a
series which converge rapidly due
to
the
nature of E.
To demonstrate the Euler's transformation method, let
us consider the following example
CO
f(u) =
1
(-1)q
--q=0
which
Eq.
represents
(5.13).
cients 6qao.
The
1
(2n-2+q)1 ql (2n-1+2q)
u 2n-2+2q
(B.10)
the first series of I1 as defined
aim is usually to compute the
For example, for n = 1 we have
with
coeffi-
142
sa
2a
2
23
7
bU
1
17
299
7
EU
126U
1
29
'2U
630
q
a
0
1
1
q
q
2
63a
q
46
315
1
3
252
Thus, the transformed series of (B.10) for such a case can
be written at once in the form of (B.8) as follows:
f(u)1
=
n=1
can
E
4. 23
E2 4.
60
46 E3... (B.11)
315
The transformed series can also be obtained
any value of n.
(5.13)
1 + 2
1+u
where F < 1.
for
1
Following the same
procedure,
be transformed and the results were given
Eq.
in
Chapter 5 for the TMim and TM2m modes.
In
short,
Eq.
(5.13) is a converging power series,
valid for any mode of excitation.