RF Design and Analysis of a Novel Elliptical Reduced Surface Wave Antenna
for Wireless Applications
by
Joshua Roy Jacobs
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
May 14, 2010
Keywords: Shorted Annular Ring Antenna, Shorted Annular Elliptical Patch Antenna,
Method of Moments
Copyright 2010 by Joshua Roy Jacobs
Approved by:
Michael E. Baginski, Chair, Associate Professor of Electrical and Computer Engineering
Stuart M. Wentworth, Associate Professor of Electrical and Computer Engineering
Lloyd S. Riggs, Professor of Electrical and Computer Engineering
Abstract
The Reduced Surface Wave (RSW) antenna is a relatively new class of microstrip patch
antenna. An RSW antenna has the ability to drastically reduce both surface and lateral wave
radiation, which leads to an increase in antenna efficiency, a reduction in mutual coupling,
and a reduction of edge diffraction and scattering. One specific type of antenna displaying
RSW characteristics is the Shorted Annular Elliptical Patch (SAEP) antenna. As a result,
the SAEP antenna has shown a great deal of promise for wireless applications and as array
elements.
In this research, a novel enhanced bandwidth Shorted Annular Elliptical Patch (SAEP)
structure specifically designed for wireless applications is presented. A Method of Moments
(MoM) code serves as the basis for the antenna design and analysis. The MoM formulation
is based on the Electric Field Integral Equation (EFIE) and Rao-Wilton-Glisson (RWG)
basis functions. Bandwidth of the SAEP antenna was enhanced by designing the antenna
to resonate over two overlapping frequency bands resulting in a broader operating frequency
range.
The prototype design was fabricated on Rogers Duroid board. Input impedance mea-
surements were made using a vector network analyzer, and radiation pattern measurements
were done in a compact antenna range.
ii
Acknowledgments
First of all, I would like to express my love to my mother, Deborah, and sister, Lauren.
Thanks for your support and encouragement throughout all of my endeavors.
To my advisor, Dr. Michael E. Baginski, thanks for your patience and guidance in the
creation of this thesis.
To Dr. Tyler N. Killian and Dr. Sadasiva M. Rao, thanks for your technical advice and
willingness to help with my MoM issues. This would have never been completed without
your expertise.
To Veneela Ammula, thanks for your help with the FEKO simulation software and
antenna measurements.
To Calvin Cutshaw, Joe Haggerty, and Dr. David Beale, thanks for your help with
fabrication of the antennas.
To the SMART Fellowship and American Society of Engineering Education, thanks for
the financial support that made this graduate work possible.
To my soon-to-be wife Shelley, thanks for patiently listening to all of my complaints
during the development of this thesis. You helped me keep my sanity. I love you.
This work is dedicated to my late father Walter Jacobs. I thank God for giving me such
a great father. I am sure you would have loved reading this.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline/Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 MoM Code Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Combined PEC/Thin Dielectric Code . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 PEC Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Thin Dielectric Formulation . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 PEC and Thin Dielectric Combination . . . . . . . . . . . . . . . . . 15
2.1.4 Meshing Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.5 Modeling the Probe Feed . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.6 Feed Point and Shorting Ring Connections . . . . . . . . . . . . . . . 19
2.1.7 Calculating Input Impedance . . . . . . . . . . . . . . . . . . . . . . 20
2.1.8 Far Field Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Design and Characteristics of Enhanced Bandwidth SAEP . . . . . . . . . . . . 33
3.1 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
iv
List of Figures
2.1 RWG basis function description. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Pulse basis function description. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Quantities for when triangles overlap. . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Illustration of aspect ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Model of the probe feed as a thin strip. . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Three triangle junction edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Rectangular microstrip patch antenna dimensions. . . . . . . . . . . . . . . . . . 23
2.8 Computed microstrip patch antenna return loss. . . . . . . . . . . . . . . . . . . 24
2.9 Radiation patterns of rectangular microstrip patch antenna at resonance. . . . . 25
2.10 SAR-RSW dimensions (top view). . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.11 SAR cross section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.12 Computed SAR input impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.13 Radiation patterns of SAR antenna at resonance. . . . . . . . . . . . . . . . . . 28
2.14 SAEP dimensions (top view). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.15 Computed SAEP input impedance. . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.16 Radiation patterns of SAEP antenna at lower resonance. . . . . . . . . . . . . . 31
2.17 Radiation patterns of SAEP antenna at upper resonance. . . . . . . . . . . . . . 32
3.1 SAR design input impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 SAR design radiation patterns at 2.09 GHz. . . . . . . . . . . . . . . . . . . . . 36
3.3 SAEP design input impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
v
3.4 Computed radiation patterns of enhanced bandwidth SAEP at lower resonance
(f = 2.09 GHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Computed radiation patterns of enhanced bandwidth SAEP at upper resonance
(f = 2.15 GHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Computed VSWR of SAR and SAEP antenna designs. . . . . . . . . . . . . . . 41
3.7 Fabricated SAR antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Fabricated SAEP antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.9 Measured and computed VSWR of SAR and SAEP antenna designs. . . . . . . 43
3.10 Measured radiation patterns of enhanced bandwidth SAEP at lower resonance. . 45
3.11 Measured radiation patterns of enhanced bandwidth SAEP at upper resonance. 46
vi
Chapter 1
Introduction
1.1 Background and Objectives
A relatively new class of microstrip patch antennas, known as Reduced Surface Wave
(RSW) antennas, has shown a great deal of promise for wireless applications and as array
elements [8]. The key feature of the RSW antenna is the ability to drastically reduce both
surface and lateral wave radiation. This leads to an increase in antenna efficiency and a
reduction in both mutual coupling and edge diffraction and scattering. Therefore, the RSW
antenna is ideally suited for situations where a small ground plane or antenna substrate is
essential and for array applications.
One favorable RSW design is the Shorted Annular Ring Reduced Surface Wave (SAR-
RSW) antenna [4], [11]. The inner radius of the ring determines the operating frequency of
the antenna and is shorted to ground; therefore, the inner edge does not radiate. The outer
radius is chosen by determining the value at which surface and lateral waves are minimized
and functions as a radiating edge. The SAR-RSW antenna operates over a single frequency
band whereas the Shorted Annular Elliptical Patch (SAEP) antenna is a dual band structure
[7]. Like the SAR-RSW, the inner radius of the patch determines the operating frequency
and is shorted to ground, and the outer radius is chosen to minimize surface and lateral wave
radiation. The elliptical shape of the inner and outer radii allows the SAEP to resonate on
two frequency bands.
In this research a novel enhanced bandwidth SAEP structure specifically designed for
wireless applications is presented. A Method of Moments (MoM) code served as the basis
for the antenna design and analysis [5]. Bandwidth of the SAEP antenna was enhanced
by designing the antenna to resonate over two overlapping frequency bands resulting in a
1
broader operating frequency range. The MoM formulation is based on the Electric Field
Integral Equation (EFIE) and Rao-Wilton-Glisson (RWG) basis functions. A commercially
available MoM code (FEKO) [13] was also used in the design process to further refine the
design and confirm the antenna characteristics. The prototype antenna was fabricated on
Rogers Duroid [12] board and radiation pattern measurements taken in an anechoic chamber.
1.2 Outline/Organization of Thesis
The thesis is organized into four chapters. Following the introduction, Chapter 2 dis-
cusses the MoM formulation used for the solution of surface current densities on a Perfect
Electric Conductor (PEC)/thin dielectric structure. The integral equations for the PEC and
thin dielectric are developed, along with integral equations that couple the PEC and thin
dielectric. The expansion and testing processes are described and the final form of the re-
sulting linear system is given in matrix form. Several examples of solutions are also included
that validate the code?s accuracy.
Chapter 3 describes the entire design procedure used for the enhanced bandwidth SAEP
antenna. The MoM code is then used to determine the antenna?s dimensions and estimate
the input impedance and radiation pattern. The chapter continues with a description of
the machining techniques used for the antenna fabrication, followed by discussion of the
measurement methodology. Measurement results are then presented and compared to the
computer simulations.
2
Chapter 2
MoM Code Development
This chapter provides a discussion of the development of the integral equations that are
applicable to a PEC/thin dielectric antenna structure. Solutions for the current densities are
developed using a MoM formulation resulting in a linear system of equations in matrix form.
Section 2.1 describes the procedure for computing the currents on a PEC/thin dielectric
structure. Several test cases are given in Section 2.2 that compare the code?s accuracy to
known solutions for three types of patch antennas. The examples shown are as follows:
a microstrip patch antenna above a finite ground plane, an SAR-RSW antenna above an
infinite ground plane, and an SAEP antenna with an infinite ground plane.
2.1 Combined PEC/Thin Dielectric Code
2.1.1 PEC Formulation
Consider a perfect electric conductor (PEC) illuminated by an incident electric field.
The total electric field (Et) is the sum of the incident electric field (Ei) and scattered electric
field (Es).
Et(r) = Ei(r) +Es(r) (2.1)
Boundary conditions for a PEC state that the total electric field tangential to the surface
is zero, therefore
Eitan(r) = ?Estan(r), r ? Sc (2.2)
3
where Sc is the surface of the conductor, and the vector r is a position vector to observation
points from a global coordinate origin.
The scattered field is then related to the PEC magnetic vector potential Ac and scalar
electric potential ?c by
Es(r) = ?j?Ac(r)???c(r) (2.3)
where the magnetic vector potential Ac and scalar electric potential ?c are defined as
Ac(r) = ?
integraldisplay
Sc
Jc(r?)G(r,r?)dS? (2.4)
?c(r) = ? 1j??
integraldisplay
Sc
??Jc(r)G(r,r?)dS? (2.5)
and
G(r,r?) = e
?jkR
4piR where R = |r?r
?| (2.6)
is the free space Green?s function. The vector r? is a position vector to source points from a
global coordinate origin.
Substituting (2.3) for the scattered field in (2.2) yields
Eitan(r) = [j?Ac(r) +??c(r)]tan , r ? Sc (2.7)
In order to approximate the current on the PEC, we define a set of basis functions. Each
basis function is to be valid only on a non-boundary edge. fn is the basis function defined
over the n = 1,...,Nc non-boundary edges on the conductor. If ?n is the unknown coefficient
for the current component on each of the conductor non-boundary edges, the total current
on the conductor is given as
4
Jc =
Ncsummationdisplay
n=1
?nfn(r) (2.8)
RWG basis functions with triangular patch modeling are used for the conductor. The
triangular mesh consists of two planar triangles, T+n and T?n , and is illustrated in Fig. 2.1.
The edge shared by the two triangles, the nth edge, is referred to as a non-boundary edge,
and edges not associated with the nth edge are defined as boundary edges. The vertex of a
triangle not associated with the nth edge is referred to as the triangle?s free point. The basis
function associated with the nth edge is defined as
fn(r) =
?
????
?
????
?
ln
2A+n ?
+
n, r ? T
+
n
? ln2A?
n
??n, r ? T?n
0, otherwise
(2.9)
and the divergence of the basis function is defined as
??fn(r) =
?
????
?
????
?
ln
A+n , r ? T
+
n
? lnA?
n
, r ? T?n
0, otherwise
(2.10)
where ln is the length of the nth edge, A+n is the surface area of triangle T+n , and A?n is the
surface area of triangle T?n . Vector ?+n extends from the triangle T+n free point to point r?,
and vector ??n extends from point r? to the triangle T?n free point. Vector rct?n is defined from
a global coordinate origin to the centroid of triangle T?n , and vector ?ct?n is defined from the
free point to the centroid of triangle T?n .
Substituting (2.8) into (2.7) results in
Ei(r) =
Ncsummationdisplay
n=1
?n [j?Ac,n(r) +??c,n(r)] (2.11)
5
?nct+
?nct-
Tn+
Tn-rnct+
rnct-
ln
?n+
?n-
r?
r?
Figure 2.1: RWG basis function description.
where
Ac,n(r) = ?
integraldisplay
Sc
fn(r?)G(r,r?)dS? (2.12)
?c,n(r) = ?1j??
integraldisplay
Sc
??fn(r?)G(r,r?)dS? (2.13)
The testing procedure is carried out by testing both sides of (2.11) with the same basis
function defined in (2.9) that was used for the expansion. This method is known as the
Galerkin procedure [2]. The testing procedure uses the inner product as shown below
?f,g? ?
integraldisplay
f ?gdS (2.14)
and results in the expression
angbracketleftbigEi,f
m
angbracketrightbig = Ncsummationdisplay
n=1
?n [j??Ac,n,fm?+???c,n,fm?] (2.15)
6
This will generate a matrix equation of the form
[Zcc][Ic] = [Vc] (2.16)
where Zcc is a N ?N impedance matrix of moment method elements, Ic is a column vector
containing unknown current coefficients, and Vc is a column vector consisting of exitations
values for each non-boundary edge defined in the triangular mesh.
The currents on the PEC can be solved for by matrix inversion of (2.16). After testing,
the Z matrix is given by
Zcc,mn = j??Ac,n,fm?+???c,n,fm? (2.17)
The terms defined by testing on the PEC are as follows:
?Ac,n,fm? = lm2 bracketleftbigAc,n(rct+m )??ct+m +Ac,n(rct?m )??ct?m bracketrightbig (2.18)
???c,n,fm? = ?lmbracketleftbig?c,n(rct+m )??c,n(rct?m )bracketrightbig (2.19)
The V matrix is given by
Vc,m = angbracketleftbigEi,fmangbracketrightbig (2.20)
where
angbracketleftbigEi,f
m
angbracketrightbig = lm
2
bracketleftbigEi(rct+
m )??
+
m +E
i(rct?
m )??
?
m
bracketrightbig (2.21)
This formulation allows one to solve for the currents on any arbitrarily shaped conductor.
7
2.1.2 Thin Dielectric Formulation
We now consider a dielectric sheet illuminated by an incident electric field. As in the
case of the PEC, the total electric field is the sum of the incident electric field and scattered
electric field (equation 2.1) and the scattered field is related to the magnetic and scalar
potentials by
Es(r) = ?j?A(r)???(r) (2.22)
where the dielectric magnetic and scalar potentials are
A(r) = ?
integraldisplay
Vd
J(r?)G(r,r?)dv? (2.23)
?(r) = ?1j??
integraldisplay
Vd
??J(r?)G(r,r?)dv? (2.24)
and Vd is the volume of the dielectric.
The dielectric structure can be represented as a set of currents and has a relative per-
mittivity ?r(r) at a point r in the dielectric. The set of polarization currents can then be
related to the total electric field by
J(r) = j??o(?r(r)?1)Et(r) = ?(r)Et(r) (2.25)
where ?(r) = j??o(?r(r)?1).
Substituting (2.25) and (2.22) into (2.1) and writing in terms of A and ? yields
Ei(r) = J?(r) + j?A(r) +??(r), r ? Vd (2.26)
The total current in the dieletric can be divided into tangential Jp and normal Jq
components. The tangential and normal currents are with respect to the surface of the
8
dielectric. There are two different basis functions are used to represent the tangential and
normal current components. Pn is the basis function defined over the n = 1,...,Np non-
boundary and boundary edges in the dielectric, and Qn is the basis function over the n =
1,...,Nq triangles in the dielectric. Pn and Qn represent the basis functions for the tangential
and normal components of the dielectric currents, respectively. The total current on the
dielectric is now defined as
J = Jp +Jq ?=
Npsummationdisplay
n=1
?nPn +
Nqsummationdisplay
n=1
?nQn (2.27)
where ?n is the unknown coefficient for the current component on the nth edge in the dielec-
tric, and ?n is the unknown coefficient for the current component in the nth triangle in the
dielectric.
The mesh for the dielectric, like that of the PEC, is triangular, so modified RWG basis
functions can be used for the tangential current. For non-boundary and boundary edges, the
definition of the basis function is as follows:
Pn,non?boundary(r) =
?
????
?
???
??
parenleftBig?+
r,n?1
?+r,n
parenrightBig
ln
2A+n ?
+
n, r ? T
+
nparenleftBig
??r,n?1
??r,n
parenrightBig
ln
2A?n ?
?
n, r ? T
?
n
0, otherwise
(2.28)
Pn,boundary(r) =
??
?
??
parenleftBig?
r,n?1
?r,n
parenrightBig
ln
2An?n, r ? Tn
0, otherwise
(2.29)
and the divergence of the non-boundary basis function is
??Pn =
?
????
?
????
?
parenleftBig?+
r,n?1
?+r,n
parenrightBig
ln
A+n , r ? T
+
n
?
parenleftBig??
r,n?1
??r,n
parenrightBig
ln
A?n , r ? T
?
n
0, otherwise
(2.30)
9
?r,n
Tn
n^n
?
Figure 2.2: Pulse basis function description.
The pulse basis function (illustrated in Fig. 2.2) used for the normal current component
is given by
Qn(r) =
?
??
??
?nn, r ? Tn
0, otherwise
(2.31)
where ?nn is the unit vector normal to the surface of the nth triangle of the dielectric.
Substituting (2.27) into (2.26) yields
Ei =
Npsummationdisplay
n=1
?n
parenleftbiggP
n
? + j?Ap,n +??p,n
parenrightbigg
+
Nqsummationdisplay
n=1
?n
parenleftbiggQ
n
? + j?Aq,n +??q,n
parenrightbigg
(2.32)
where
Ap,n(r) = ?
integraldisplay
Vn
Pn(r?)G(r,r?)dv? (2.33)
10
?p,n(r) = ?1j??
integraldisplay
Vn
??Pn(r?)G(r,r?)dv? (2.34)
Aq,n(r) = ?
integraldisplay
Vn
Qn(r?)G(r,r?)dv? (2.35)
?q,n(r) = ?1j??
integraldisplay
Vn
??Qn(r?)G(r,r?)dv? (2.36)
Since the dielectric is thin, the integral equations (2.33), (2.34), and (2.35) are approximated
as constant throughout the thickness of the dielectric
Ap,n(r) = ??
integraldisplay
Sn
Pn(r?)G(r,r?)dS? (2.37)
?p,n(r) = ??j??
integraldisplay
Sn
??Pn(r?)G(r,r?)dS? (2.38)
Aq,n(r) = ??
integraldisplay
Sn
Qn(r?)G(r,r?)dS? (2.39)
?q,n(r) = 1j??
?
??integraldisplay
S+n
G(r,r?)dS? ?
integraldisplay
S?n
G(r,r?)dS?
?
?? (2.40)
where S+n and S?n are the area of the top (in the direction of ?nn) and bottom surfaces,
respectively, of triangle Tn in the dielectric.
Galerkin?s method is used to complete the testing procedure. This results in a set of
linear equations. Both sides of (2.32) are tested with the RWG and pulse basis functions
using the inner product as shown
?f,g? ?
integraldisplay
f ?gdS (2.41)
11
and testing yields the following expressions:
angbracketleftbigEi,P
m
angbracketrightbig = Npsummationdisplay
n=1
?n
bracketleftbigg1
? ?Pn,Pm?+ j??Ap,n,Pm?+???p,n,Pm?
bracketrightbigg
+
Nqsummationdisplay
n=1
?n
bracketleftbigg1
? ?Qn,Pm?+ j??Aq,n,Pm?+???q,n,Pm?
bracketrightbigg
(2.42)
angbracketleftbigEi,Q
m
angbracketrightbig = Npsummationdisplay
n=1
?n
bracketleftbigg1
? ?Pn,Qm?+ j??Ap,n,Qm?+???p,n,Qm?
bracketrightbigg
+
Nqsummationdisplay
n=1
?n
bracketleftbigg1
? ?Qn,Qm?+ j??Aq,n,Qm?+???q,n,Qm?
bracketrightbigg
(2.43)
As a result, the following matrix equation is generated
?
?? Zpp Zpq
Zqp Zqq
?
??
?
?? Ip
Iq
?
?? =
?
?? Vp
Vq
?
?? (2.44)
The subscripts in the Zpp, Zpq, Zqp, and Zqq submatrices each denote the coupling
between the RWG and pulse basis functions. The exitation from the tangential and normal
components in the dielectric are represented by Vp and Vq, respectively. Similarly, Ip and Iq
symbolize the tangential and normal components of the current in the dielectric volume.
The Z submatrices are defined as
Zpp,mn = 1? ?Pn,Pm?+ j??Ap,n,Pm?+???p,n,Pm? (2.45)
Zpq,mn = j??Aq,nPm?+???q,n,Pm? (2.46)
12
Zqp,mn = j??Ap,n,Qm?+???p,n,Qm? (2.47)
Zqq,mn = 1? ?Qn,Qm?+ j??Aq,n,Qm?+???q,n,Qm? (2.48)
The V submatrices are defined as
Vp,m = angbracketleftbigEi,Pmangbracketrightbig (2.49)
Vq,m = angbracketleftbigEi,Qmangbracketrightbig (2.50)
Each of the testing terms in the above equations will now be evaluated. If Pm and
Pn overlap on triangle T with a relative permittivity ?r, the following testing term must be
evaluated
?Pn,Pm?T = ?lnlm?4A
m
parenleftbigg?
r ?1
?r
parenrightbiggbracketleftBigg3|rct|2
4 +
|r1|2 +|r2|2 +|r3|2
12
?(rm +rn)?rct +rm ?rn
bracketrightbigg
(2.51)
where rct is the centroid vector and r1, r2, r3 are the vectors of the vertices of triangle T
with respect to a global coordinate origin. rm and rn are the vectors of the free points of Pm
and Pn with respect to a global coordinate origin. If m = n then Pm and Pn are associated
with an interior, or non-boundary, edge, both T+m and T?m will contribute to ?Pn,Pm?. Note
that if both basis functions are pointing in the same direction over T, then the sign will be
positive. If the basis functions are pointing in opposite directions over T, then the sign will
be negative. The quantities in (2.51) are illustrated in Fig. 2.3.
Testing terms for the RWG basis functions are as follows:
13
edge n
edge mr-
1
r-m
r-2
r-n
r-3r-ct
T
?r
Figure 2.3: Quantities for when triangles overlap.
angbracketleftBigg???
??
Ei
Ap/q,n
??
?
??,Pm
angbracketrightBigg
= lm2
bracketleftbiggparenleftbigg?+
r,m ?1
?+r,m
parenrightbigg
??
?
??
Ei(rct+m )
Ap/q,n(rct+m )
??
?
????
ct+
m
+
parenleftbigg??
r,m ?1
??r,m
parenrightbigg
??
?
??
Ei(rct?m )
Ap/q,n(rct?m )
??
?
????
ct?
m
?
?? (2.52)
angbracketleftbig?
p/q,n,Pm
angbracketrightbig = ?l
m?
bracketleftbiggparenleftbigg?+
r,m ?1
?+r,m
parenrightbigg
?p/q,n(rct+m ) +
parenleftbigg??
r,m ?1
??r,m
parenrightbigg
?p/q,n(rct?m )
bracketrightbigg
(2.53)
If testing is occuring over a boundary edge, then Pm is a half RWG basis. A half RWG basis
is defined over only one triangle, unlike a normal RWG basis function, which is defined over
two triangles. In this situation, the ?+? term of the two summation terms in (2.52) and
(2.53) are used if the basis function is directed out of Tm, and the ?-? term is used otherwise.
The pulse basis function testing terms are defined as
14
?Qn,Qm? =
??
?
??
?Am
? , if n = m
0, otherwise
(2.54)
angbracketleftbigA
p/q,n,Qm
angbracketrightbig = ?A
mAp/q,n(r
ct
m)? ?nm (2.55)
angbracketleftbig??
p/q,n,Qm
angbracketrightbig = ?A
m
parenleftbigg?
p/q,n (rctm + 0.25??nm)??p/q,n (rctm ?0.25??nm)
0.5?
parenrightbigg
(2.56)
angbracketleftbigEi,Q
m
angbracketrightbig = ?A
mE
i(rct
m)? ?nm (2.57)
This concludes the MoM formulation suitable to calculate currents in a homogeneous
or heterogeneous thin dielectric sheet.
2.1.3 PEC and Thin Dielectric Combination
A PEC/thin dielectric structure is now considered. The total magnetic vector potential
for the structure is now the sum of the magnetic vector potentials due to the currents on
conductor and the tangential and normal current components of the dielectric. The total
magnetic vector potential is
A = Ac +Ap +Aq (2.58)
where Ac, Ap, and Aq are magnetic vector potentials due to currents on the conductor and
the tangential and normal components of the currents in the dielectric, respectively.
Similarly, the total scalar potential is
? = ?c + ?p + ?q (2.59)
15
where ?c, ?p , and ?q are scalar potentials due to currents on the conductor and the
tangential and normal components of the currents in the dielectric, respectively.
Substituting (2.8) and (2.27) into (2.58) and (2.59) yields
A =
Npsummationdisplay
n=1
?nAp,n+
Nqsummationdisplay
n=1
?nAq,n+
Ncsummationdisplay
n=1
?nAc,n (2.60)
and
? =
Npsummationdisplay
n=1
?n?p,n+
Nqsummationdisplay
n=1
?n?q,n+
Ncsummationdisplay
n=1
?n?c,n (2.61)
Now, when (2.26) is tested using the Galerkin procedure and the inner product is used,
the following matrix equation is generated
?
??
??
?
Zcc Zcp Zcq
Zpc Zpp Zpq
Zqc Zqp Zqq
?
??
??
?
?
??
??
?
Ic
Ip
Iq
?
??
??
?
=
?
??
??
?
Vc
Vp
Vq
?
??
??
?
(2.62)
and can be solved by matrix inversion. Z submatrices Zcc, Zpp, Zpq, Zqp, and Zqq have
already been defined. The remaining Z submatrices result from the coupling between the
conductor RWG basis functions and the RWG and pulse basis functions in the dielectric are
defined as
Zcp = j??Ap,n,fm?+??p,n,fm? (2.63)
Zcq = j??Aq,n,fm?+??q,n,fm? (2.64)
Zpc = j??Ac,n,Pm?+??c,n,Pm? (2.65)
Zqc = j??Ac,n,Qm?+??c,n,Qm? (2.66)
16
where
angbracketleftbigA
p/q,n,fm
angbracketrightbig = lm
2
bracketleftbigbraceleftbigA
p/q,n(rct+m )
bracerightbig??ct+
m +
braceleftbigA
p/q,n(rct?m )
bracerightbig??ct?
m
bracketrightbig (2.67)
angbracketleftbig??
p/q,n,fm
angbracketrightbig = ?l
m
bracketleftbig?
p/q,n(rct+m )??p/q,n(rct?m )
bracketrightbig (2.68)
?{Ac,n,Pm}? = lm2
bracketleftbiggparenleftbigg?+
r,m ?1
?+r,m
parenrightbiggbraceleftbig
Ac,n(rct+m )bracerightbig??ct+m
+
parenleftbigg??
r,m ?1
??r,m
parenrightbiggbraceleftbig
Ac,n(rct?m )bracerightbig??ct?m
bracketrightbigg
(2.69)
???c,n,Pm? = ?lm?
bracketleftbiggparenleftbigg?+
r,m ?1
?+r,m
parenrightbigg
?c,n(rct+m )?
parenleftbigg??
r,m ?1
??r,m
parenrightbigg
?c,n(rct?m )
bracketrightbigg
(2.70)
?Ac,n,Qm? = ?AmAc,n(rctm)? ?nm (2.71)
???c,n,Qm? = ?Am
parenleftbigg?
c,n (rctm + 0.25??nm)??c,n (rctm ?0.25??nm)
0.5?
parenrightbigg
(2.72)
2.1.4 Meshing Considerations
In order to obtain accurate results, a three dimensional triangular surface mesh must
be constructed with maximum edge length limited to ?g/10 where
?g = cf??
r
(2.73)
where ?g is the wavelength in the dielectric, c is the speed of light, and ?r is the relative
permittivity of the dielectric. If the mesh contains edges that are greater than ?g/10, a poorly
17
conditioned moment method matrix could result. Therefore, when creating a triangular
surface mesh for an antenna including a PEC and dielectric, care must be taken to ensure
that no edge length exceeds ?g/10 [5].
Another mesh consideration is aspect ratio [2]. A triangle with a poor aspect ratio
contains small internal angles, or more simply put, the triangles look ?thin.? A mesh that
consists of triangles that have a poor aspect ratio can cause the moment method matrix to
be more poorly conditioned. A mesh that is ideally constructed will consist of triangles that
are all equilateral. The programmer should attempt to create a mesh that contains triangles
with the best aspect ratio possible; however, this is not a realistic possibility. Figures 2.4(a)
and 2.4(b) show examples of poor and good aspect ratios.
(a) Poor Aspect Ratio (b) Good Aspect Ratio
Figure 2.4: Illustration of aspect ratio.
2.1.5 Modeling the Probe Feed
One option of feeding a microstrip patch antenna is by using the coaxial probe feed
method. Here the outer coaxial conductor is soldered to the back side of the ground plane
and the inner cylindrical conductor is fed through the dielectric material and attached to
the conducting patch. Since triangular meshing is used, it is convenient to model the inner
conductor as a thin strip [1] as illustrated in Fig. 2.5. The width of the strip is given as
18
a = 0.25s (2.74)
where a is the radius of the conducting probe, and s is the width of the strip.
Conducting Patch
Feed
Ground Plane
Figure 2.5: Model of the probe feed as a thin strip.
2.1.6 Feed Point and Shorting Ring Connections
An edge connected to three triangles can be associated with three RWG basis func-
tion edges. This situation occurs at the ground plane/probe feed junction, conducting
patch/probe feed junction, and, in the case of an SAR or SAEP antenna, at the ground
plane/shorting ring junction. Despite this, only two basis function edges need to be associ-
ated with the junction edge. As illustrated in Fig. 2.6, triangles 1, 2, and 3 are all connected
to the junction edge. In this orientation, triangles 1 and 2 represent triangles of the ground
plane, and triangle 3 can represent a triangle of the probe feed or shorting ring. An RWG
edge element can be created at the junction edge between triangles 1 and 2, between trian-
gles 1 and 3, and between triangles 2 and 3. Since only two edge elements need be used at
this junction, only the RWG edge elements sharing triangles 1 and 3 and triangles 2 and 3
need to be utilized. The edge element sharing triangles 1 and 2 can be ignored.
19
1
2
junction edge
three triangle
3
Figure 2.6: Three triangle junction edge.
2.1.7 Calculating Input Impedance
When calculating input impedance, a gap voltage induces the currents on the antenna
instead of an incident wave. The delta-gap feed model will be used to supply the voltage
applied at the mth edge of the conductor and is given as
Vc,m = V lm (2.75)
where lm is the length of the edge that is being driven. The other entries in V are set to
zero.
The input impedance is defined as
Zin = Vl
mIc,m
(2.76)
where Ic,m is the current flowing on the mth edge of the conductor.
Equation (2.76) is suitable for a situation where there is only one RWG element present
at the feed edge. In the case of a probe-fed microstrip antenna, the driving edge could consist
of two RWG elements as described in Section 2.1.6. The exitation is now
Vc,m1 = Vc,m2 = V lm (2.77)
20
where the subscripts 1 and 2 denote the two RWG elements associated with the driving edge.
Again, the other entries in V are set to zero.
As a result, the input impedance is given as
Zin = Vl
m (Im1 + Im2)
(2.78)
Thereflectioncoefficientisameasureoftheimpedancemismatchbetweenatransmission
line and its load and is calculated by
?L = Zin ?ZoZ
in + Zo
(2.79)
where Zo is the characteristic impedance of the feed line.
The return loss given in decibels becomes
RL = 20log|?L| (2.80)
The voltage standing wave ratio (VSWR), which is a ratio of the maximum to the
minimum voltage amplitudes, is related to the reflection coefficient by
V SWR = 1 +|?L|1?|?
L|
(2.81)
2.1.8 Far Field Radiation Pattern
Determining the currents on the antenna structure allows the scattered far field radiation
pattern to be calculated as shown below
Es = ?j?A = ?j??
integraldisplay
Je
?jkR
4piR dv
? (2.82)
where R can be approximated as
21
R = r??r?r? (2.83)
and ?r is the unit vector to observation points from the global point origin. The scattered
electric field can now be rewritten as
Es ? ?j??4pir e?jkr
integraldisplay
Jejk?r?r?dv? (2.84)
Equation (2.84) can be used to calculate the far-field contributions from the currents
on the conductor Esc as well as the tangential and normal current contributions from the
dielectric Esp and Esq respectively as shown below.
Esc = ?j??8pir e?jkr
Ncsummationdisplay
n=1
?nln
bracketleftBig
?ct+n ejk?r?rct+n + ?ct?n ejk?r?rct?n
bracketrightBig
(2.85)
Esp = ?j???8pir e?jkr
Npsummationdisplay
n=1
?nln
bracketleftbiggparenleftbigg?+
r,n ?1
?+r,n
parenrightbigg
?ct+n ejk?r?rct+n +
parenleftbigg?+
r,n ?1
?+r,n
parenrightbigg
?ct?n ejk?r?rct?n
bracketrightbigg
(2.86)
Esq = ?j???8pir e?jkr
Nqsummationdisplay
n=1
?nAnejk?r?rct+n ? ?nn (2.87)
The total far field radiation Estotal is given as
Estotal = Esc +Esp +Esq (2.88)
The total radiated electric field Erad is alway perpendicular to the direction of propa-
gation and is given as
Erad = Estotal ?(Estotal ??r)?r (2.89)
22
7 mm
31 mm 4.5 mm
22 mm
3.5 mm
15.5 mm
31 mm
40 mm
Figure 2.7: Rectangular microstrip patch antenna dimensions.
2.2 Test Cases
This section contains numerical results that demonstrate the accuracy of the PEC/thin
dielectric code. The three test case antennas were chosen as follows: a rectangular microstrip
patch antenna [6], an SAR-RSW antenna [4], and an SAEP antenna [7].
The rectangular microstrip patch dimensions are shown in Fig. 2.7. The dielectric sub-
strate has a relative permittivity of 11.9 and a thickness of 1.5 mm. Figures 2.8, 2.9(a), and
2.9(b) illustrate the simulated return loss and E and H plane radiation patterns, respec-
tively, and the results are compared with those given by Nakar [6]. The computed results of
the PEC/thin dielectric code show an approximately 5% difference with respect to the re-
sults computed by the Zeland IE3D software; however, the computed results show sufficient
agreement. The differences in resonant frequency can be possibly explained by differences
in the two codes, which could be, but are not limited to basis function selection, meshing
strategies, or integration method. The radiation pattern comparison shows good agreement.
The top and cross-sectional view of the SAR-RSW antenna is shown in Fig. 2.10 and
Fig. 2.11. The SAR-RSW dimensions are rin = 2.417 cm, rout = 4.392 cm, and ?feed = 2.95
23
1.6 1.7 1.8 1.9 2 2.1 2.2 2.3?35
?30
?25
?20
?15
?10
?5
0
Frequency (GHz)
Return Loss (dB)
Computed
Zeland IE3D
Figure 2.8: Computed microstrip patch antenna return loss.
cm. The probe feed radius is 0.0635 cm. The substrate dielectric constant is 2.94 and
thickness is 0.1524 cm. The antenna is simulated on a finite circular ground plane with a
radius of 6.588 cm. The antenna in [4] was measured on a 1 m circular ground plane. A
comparison of the computed return loss is shown in Fig. 2.12. Fig. 2.13 compares E and H
plane radiation patterns to the results given in [4]. The return loss comparison demonstrates
good agreement. The radiation pattern comparison shows good agreement in the main lobe,
but poor agreement in the backlobe. The power in the backlobe of the antenna measured in
[4] is much lower because the antenna is built on a much larger ground plane. The simulation
performed by the PEC/thin dielectric code restricted the ground plane to smaller dimensions
because a larger ground plane resulted in to many unknowns.
The geometry for a SAEP antenna is shown in Fig. 2.14. Fig. 2.11 shows a cross
section view of the antenna. The SAEP dimensions are ain = 11.2 mm, aout = 1.515 cm,
bin = 11.1 mm, and bout = 1.438 cm. The input impedance measurements were taken with
the feed located at ?feed = 12.85 mm and ? = 45? with an unspecified ground plane size. The
probe feed radius for the input impedance measurements was 0.15 mm. This antenna was
24
?18
?18
?16
?16
?14
?14
?12
?12
?10
?10
?8
?8
?6
?6
?4
?4
?2
?2
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Zeland IE3D
Computed
(a) E-plane
?18
?18
?16
?16
?14
?14
?12
?12
?10
?10
?8
?8
?6
?6
?4
?4
?2
?2
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Computed
Zeland IE3D
(b) H-plane
Figure 2.9: Radiation patterns of rectangular microstrip patch antenna at resonance.
25
Figure 2.10: SAR-RSW dimensions (top view).
feed?
r
?
a
?feed
b
Figure 2.11: SAR cross section.
26
1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5?40
?20
0
20
40
60
80
100
Frequency (GHz)
Input Impedance (
?)
Real, Computed
Real, Davis et al.
Imaginary, Computed
Imaginary, Davis et al.
Figure 2.12: Computed SAR input impedance.
simulated on a finite elliptical ground plane where the major axis and minor axis are 1.67
cm and 1.58 cm, respectively. Fig. 2.15 compares the simulated input impedance with the
measured input impedance given in [7]. The upper frequency resonance locations differ by
approximately 2.2%, the lower frequency resonance locations differ by approximately 1.6%,
and the computed input impedance values are higher than those given in [7]. Fabrication
error could be the reason for the difference input impedance values because a slight error
in feed location can change input impedance values. In [7], the relative permittivity of the
dielectric is given, but more detailed information on the substrate is not provided. Perhaps
this substrate performs differently at the frequencies studied, resulting in a different relative
permittivity. This could change the location of the resonance frequencies. Despite these
conditions, the PEC/thin dielectric simulation demonstrates sufficient agreement with the
measured results in [7].
27
?45
?45
?40
?40
?35
?35
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Computed
Davis et al.
(a) E-plane
?45
?45
?40
?40
?35
?35
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Computed
Davis et al.
(b) H-plane
Figure 2.13: Radiation patterns of SAR antenna at resonance.
28
Fig. 2.16 compares the computed E and H plane radiation patterns of the lower res-
onance with those illustrated in [7]. This measurement was taken with the feed located at
?feed = 13.1 mm and ? = 0? in order to excite only the lower frequency resonance of the
antenna. As a result the E-plane pattern is measured at ? = 0? ? 360? and ? = 0?, and
the H-plane pattern is measured at ? = 0? ? 360? and ? = 90?. Fig. 2.17 compares the
computed E and H plane radiation patterns of the upper resonance with those illustrated
in [7]. This measurement was taken with the feed located at ?feed = 12.95 mm and ? = 90?
in order to excite only the lower resonance of the antenna. As a result the E-plane pat-
tern is measured at ? = 0? ? 360? and ? = 90?, and the H-plane pattern is measured at
? = 0? ? 360? and ? = 0?. The probe feed radius for the pattern measurements was 1 mm.
For the substrate, ?r = 10, and the dielectric thickness is 0.605 mm. The two antennas in
[7] were built on a 13 cm ? 30 cm grounded dielectric slab and were placed on a metallic
plane with a 40 cm radius during measurement. The antennas simulated with the PEC/thin
dielectric code was constructed on a finite elliptical ground plane where the major axis and
minor axis were 1.67 cm and 1.58 cm, respectively. The computed patterns in Figures 2.16
and 2.17 demonstrate sufficient agreement in the main lobes but poor agreement in the back-
lobes. As in the SAR-RSW test case, the power in the backlobes of the antennas measured
in [7] is much lower because the antenna is built on a much larger ground plane. The sim-
ulation performed by the PEC/thin dielectric code restricted the ground plane to smaller
dimensions because a larger ground plane resulted in too many unknowns. In addition, large
?bumps? are observed at ?90? in Fig. 2.16(a) and at 90? in Fig. 2.17(a). This is likely
due to the fact that the inner conductor radius is not given in [7]. Only the coaxial probe
thickness is given, so that value was used to determine the feed probe radius used in the
PEC/thin dielectric code. As a result, the feed probe is likely thicker in the simulation than
in the fabricated antenna in [7], which would explain why the computed magnitudes of the
?bumps? are much larger.
29
bmax
amax
Figure 2.14: SAEP dimensions (top view).
5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3?50
0
50
100
150
200
Input Impedance (
?)
Frequency (GHz)
Real, Amendola et al.
Imaginary, Amendola et al.
Real, Computed
Imaginary, Computed
Figure 2.15: Computed SAEP input impedance.
30
?30
?30
?20
?20
?10
?10
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Amendola et al.
Computed
(a) E-plane
?30
?30
?20
?20
?10
?10
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Amendola et al.
Computed
(b) H-plane
Figure 2.16: Radiation patterns of SAEP antenna at lower resonance.
31
?30
?30
?20
?20
?10
?10
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Amendola et al.
Computed
(a) E-plane
?30
?30
?20
?20
?10
?10
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
Amendola et al.
Computed
(b) H-plane
Figure 2.17: Radiation patterns of SAEP antenna at upper resonance.
32
Chapter 3
Design and Characteristics of Enhanced Bandwidth SAEP
This chapter presents the design for an enhanced bandwidth SAEP antenna. In sec-
tion 3.1, design equations for the SAEP are defined and used to design an SAR antenna
only resonating at one frequency. The enhanced bandwidth SAEP antenna is designed by
altering the dimensions of the SAR antenna in order to produce two overlapping resonances.
Computational results are used to analyze the bandwith of both antennas to demonstrate
the bandwidth improvement of the SAEP antenna design.
Section 3.2 describes the fabrication process of the SAR and SAEP antenna designs.
The radiation patterns of both antennas are measured in the Auburn University anechoic
chamber, and the bandwidth analysis was done by measuring the Voltage Standing Wave
Ratio(VSWR) with an HP 8753C vector network analyzer. The results are compared to the
PEC/thin dielectric computational results from Section 3.1.
3.1 Design Procedure
In order to design a Shorted Annular Elliptical Patch antenna, we must determine
the inner radii for the major and minor axes (amin and bmin, respectively) and outer radii
for the major and minor axes (amax and bmax, respectively) as illustrated in Fig. 2.14.
Davis et al. and Amendola et al. state that the inner radius of a shorted annular patch
antenna determines the resonant frequency of the antenna, and the outer radius is designed
to minimize surface wave radiation.
The entire set of design equations for the SAEP are given in [7], but they were not used
because of their complexity. Approximate design equations given in [9] were used as the
33
design starting point and are equivalent to the design equations presented for the Shorted
Annular Ring antenna [4] and are found by solving the following:
amax = 1.814?
TM0
(3.1)
and
J?1(kamax)Y1(kamin)?Y ?1(kamax)J1(kamin) = 0 (3.2)
where ?TM0 is the surface wave propagation constant, k is the wavenumber in the dielectric,
and J1 and Y1 are Bessel functions of the first and second kind, respectively, with order
v = 1. J?1 and Y ?1 are derivatives of the aforementioned Bessel functions.
In order to demonstrate that the SAEP design enhances bandwidth relative to the SAR
antenna, an SAR antenna was designed first. Since the SAEP antenna resonances must
overlap in order to enhance bandwidth, the center frequency of the overlapping resonances
is set to be 2 GHz. For a beginning estimate, the lower resonance was set to be 1.975 GHz,
and the upper resonance was set at 2.025 GHz. The substrate constitutive parameters were
based on Rogers Corporation RT/duroid c?5880 high frequency laminate with ?r = 2.2 and
the substrate thickness is 3.175 mm. Here only the SAR case is considered, so the antenna
will be designed to resonate on just the lower frequency. Equations (3.1) and (3.2) are
solved to determine the inner and outer radii of the SAR antenna illustrated in Fig. 2.10
where amin and amax can be replaced with rin and rout, respectively. The design procedure
yields rin = 23.24 mm and rout = 43.85 mm and the distance from the feed to the origin
and feed radius are chosen to be ?feed = 27.888 mm and 0.635 mm, respectively. The
design is simulated using both the PEC/thin dielectric code and FEKO simulation software.
The PEC/thin dielectric code simulates the antenna over a finite circular ground plane of
radius 65.775 mm, but the FEKO simulation is performed using an infinite substrate and
ground plane. A comparison of the input impedance results is presented in Fig. 3.1, and
a comparison of the E and H plane radiation patterns is presented in Figures 3.2(a) and
34
1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3?10
0
10
20
30
40
50
60
Frequency (GHz)
Input Impedance (
?)
Real, PEC/thin dielectric
Imaginary, PEC/thin dielectric
Real, FEKO
Imaginary, FEKO
Figure 3.1: SAR design input impedance
3.2(b), respectively. The input impedance comparison shows very good agreement. The
radiation pattern comparisons show good agreement in the main lobes, but there are no
backlobes shown for the FEKO simulations. This is because the FEKO simulation assumes
an infinite ground plane, and the PEC/thin dielectric code can only simulate a finite ground
plane. Also note that the design equations yield a resonance near 2.1 GHz instead of the
desired resonance frequency of 1.975 GHz.
The SAEP design begins by using the same radii used for the SAR design for amin
and amax. This will set the lower frequency. In order to enhance the bandwitch of the
SAR antenna, the SAEP antenna resonances must be designed to be close enough to have
overlapping bandwidth. Voltage Standing Wave Ratio (VSWR), defined in (2.81), is used
to analyze the bandwidth of the antennas. Bandwidth will be defined as V SWR ? 2 : 1
or better, which is a commonly accepted figure [10]. Rewritting (3.1) and (3.2) in terms of
bmin and bmax allows for the determination of the inner and outer radii of the minor axis of
35
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
FEKO
PEC/thin dielectric
(a) E-plane
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
FEKO
PEC/thin dielectric
(b) H-plane
Figure 3.2: SAR design radiation patterns at 2.09 GHz.
36
the ellipse; however, this estimate was revised using FEKO simulation software to optimize
the antenna in order to achieve the enhanced bandwidth design criteria because of the noted
inaccuracy in the design equations.
The lower resonance of the SAEP will be at the same frequency as the resonance in
the SAR, since amin and amax in the SAEP design are equal to rin and rout, respectively,
for the SAR design. Using FEKO?s Particle Swarm Algorithm (PSO) optimization setting,
the upper resonance frequency location was specified, and FEKO determined bmin and bmax
for this criteria. The new SAEP design was simulated to determine if the upper resonance
is close enough to the lower resonance to allow for an enhanced bandwidth. After several
iterations, the optimization procedure yielded bmin = 21.215 mm and bmax = 41.403 mm.
?feed is chosen to be 28.926 mm, and the feed radius is also chosen to be 0.635 mm. The design
was simulated using both the PEC/thin dielectric code and FEKO simulation software. The
PEC/thin dielectric code simulates the antenna over a finite elliptical ground plane with
a major axis and minor axis of length 65.775 mm and 62.1 mm, respectively. The FEKO
simulation is performed using an infinite substrate and ground plane. A comparison of the
input impedance calculated using the PEC/thin dielectric code and FEKO is presented in
Fig. 3.3, and the results show good agreement. The E-plane and H-plane results for the
lower resonance (2.09 GHz) are compared in Figures 3.4(a) and 3.4(b), respectively. The
E-plane and H-plane results for the upper resonance (2.15 GHz) are compared in Figures
3.5(a) and 3.5(b), respectively. All radiation pattern comparisons show good agreement in
the main lobes, but there are no backlobes shown in the FEKO simulations. This is because
the FEKO simulation assumes an infinite ground plane, and the PEC/thin dielectric code
simulates the antenna over a finite ground plane.
To determine whether the SAEP design enhances the bandwidth relative to the SAR
design, a VSWR comparison is performed. This comparison is shown in Fig. 3.6. The
relative bandwidth of the antenna is given as
37
1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35?20
?10
0
10
20
30
40
50
60
70
Frequency (GHz)
Input Impedance (
?)
Imaginary, PEC/thin dielectric
Real, PEC/thin dielectric
Imaginary, FEKO
Real, FEKO
Figure 3.3: SAEP design input impedance.
%BW = ?ff
0
(3.3)
where ?f is the bandwidth over which V SWR ? 2 : 1 occurs and f0 is the center frequency.
For the SAR, ?f ? 2.11 GHz - 2.08 GHz = 0.03 GHz and f0 = 2.095 GHz. Using (3.3),
percent bandwidth for the SAR is calculated to be 1.43%. For the SAEP, ?f ? 2.17 GHz
- 2.09 GHz = 0.08 GHz and f0 = 2.125 GHz. Using (3.3), percent bandwitch for the SAEP
is calculated to be 3.76%. Therefore, the SAEP design enhances bandwidth with respect to
the SAR design by a remarkable 163%.
38
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
FEKO
PEC/thin dielectric
(a) E-plane
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
FEKO
PEC/thin dielectric
(b) H-plane
Figure 3.4: Computed radiation patterns of enhanced bandwidth SAEP at lower resonance
(f = 2.09 GHz).
39
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
FEKO
PEC/thin dielectric
(a) E-plane
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
FEKO
PEC/thin dielectric
(b) H-plane
Figure 3.5: Computed radiation patterns of enhanced bandwidth SAEP at upper resonance
(f = 2.15 GHz).
40
2.05 2.1 2.15 2.21
1.5
2
2.5
3
3.5
4
4.5
Frequency (GHz)
VSWR
SAEP Design
SAR Design
Figure 3.6: Computed VSWR of SAR and SAEP antenna designs.
3.2 Experimental Results
In order to validate the accuracy of the design results of the PEC/thin dielectric code?s
simulations, the SAR and optimized SAEP antennas designed in the previous section were
fabricated. Both antennas were built using Rogers Corporation RT/duroid c?5880 high fre-
quency laminate. The area inside the inner radius was drilled completely through the board,
and only the top layer of copper was removed beyond the area outside the outer radius. This
procedure was performed in the Auburn University Electrical Engineering machine shop for
the SAR antenna, and the SAEP antenna was milled by Precision Prototype in Opelika, AL.
A hole was drilled at the feed point for each antenna, and an SMA connector was attached.
The top conducting patch was shorted to ground using a strip of adhesive copper foil. Fi-
nally, a piece of copper foil was soldered to the ground plane in order to cover the hole cut
in the ground plane. The SAR antenna was built on a 6 in. x 4.5 in. rectangular ground
41
Figure 3.7: Fabricated SAR antenna.
plane, and the SAEP was built on a 150 mm x 150 mm square ground plane. A picture of
the fabricated SAR and SAEP antennas is shown in Figures 3.7 and 3.8, respectively.
The VSWR of the fabricated SAR and SAEP antennas were analyzed with an HP
8753C vector network analyzer. These results are compared with the simulations conducted
with the PEC/thin dielectric code for the SAR and SAEP antennas in Fig. 3.9. Only
VSWR results in the operating band of the antennas is shown in Fig. 3.9 to compare their
bandwidths. The comparison shows good agreement, although the location of the fabricated
SAEP resonance location differs from the simulation results by approximately 2.4%. This
small difference is likely due to fabrication error. Using (3.3), the percent bandwidth for
the fabricated SAR antenna is 2.39%, and the percent bandwidth for the fabricated SAEP
antenna is 5.1%. Therefore, the fabricated SAEP design enhances bandwidth with respect
to the fabricated SAR design by 113%.
The radiation patterns of the fabricated SAEP antenna were measured in the Auburn
University anechoic chamber. An L-band Scientific Atlanta standard gain horn (SGH 1.7)
42
Figure 3.8: Fabricated SAEP antenna.
2 2.02 2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18 2.21
1.5
2
2.5
3
3.5
4
Frequency (GHz)
VSWR
SAR Computed
SAEP Computed
SAR Measured
SAEP Measured
Figure 3.9: Measured and computed VSWR of SAR and SAEP antenna designs.
43
was used as the source antenna. The source antenna was excited with an HP 8753C vector
network analyzer at the lower (2.05 GHz) and upper (2.1 GHz) resonant frequencies of the
SAEP antenna. The E-plane (? = 0? ? 360?, ? = 45?) and H-plane (? = 0? ? 360?,
? = 135?) pattern cuts were measured using Diamond Engineering?s Desktop Antenna Mea-
surement System (DAMS). The radiation patterns for both resonant frequencies are com-
pared with the radiation patterns obtained from the PEC/thin dielectric code computations
in Figures 3.10 and 3.11. The radiation pattern comparisons display some differences, which
are likely due to different sized ground planes used in the simulations and measurements
and fabrication error. Despite these differences, the measurements show sufficient agreement
with the simulations.
44
?45
?45
?40
?40
?35
?35
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
PEC/thin dielectric
Measured
(a) E-plane
?45
?45
?40
?40
?35
?35
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
PEC/thin dielectric
Measured
(b) H-plane
Figure 3.10: Measured radiation patterns of enhanced bandwidth SAEP at lower resonance.
45
?45
?45
?40
?40
?35
?35
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
PEC/thin dielectric
Measured
(a) E-plane
?45
?45
?40
?40
?35
?35
?30
?30
?25
?25
?20
?20
?15
?15
?10
?10
?5
?5
0 dB
0 dB
90o
60o
30o
0o
?30o
?60o
?90o
?120o
?150o
180o
150o
120o
PEC/thin dielectric
Measured
(b) H-plane
Figure 3.11: Measured radiation patterns of enhanced bandwidth SAEP at upper resonance.
46
Chapter 4
Conclusion
In this work, an enhanced bandwidth Shorted Annular Elliptical Patch antenna was
presented. Reduced surface and lateral wave radiation is a key feature of the Shorted Annular
Ring antenna, but this geometry only allows for operation at one frequency band. The SAEP
design allows for operation in two frequency bands, and this research demonstrates that the
two frequency bands can be overlapped so that the bandwidth of the antenna increased in
relation to the SAR case by greater than 100%.
In Chapter 2, a computational electromagnetic code was developed to design the en-
hanced bandwidth SAEP. This Method of Moments code utilized the Electric Field Integral
Equation and RWG basis functions to simulate a perfect electric conductor on a thin di-
electric sheet. The accuracy of this code was confirmed by comparing the code?s computed
results with the documented results of a rectangular microstrip patch, SAR, and SAEP
antennas. The compared results included return loss, input impedance, and E-plane and
H-plane radiation patterns.
The enhanced bandwidth SAEP antenna was then designed in Chapter 3. The de-
sign equations for the SAEP were inaccurate, so because both resonance frequencies for the
enhanced bandwidth SAEP needed to be precisely located, FEKO?s Particle Swarm Algo-
rithm was used to obtain a more refined and accurate solution. The PEC/thin dielectric
code results from the optimized design were compared with FEKO?s results and showed
good agreement; therefore, the enhanced bandwidth SAEP design was confirmed. Both the
SAR and SAEP antennas were built and measurements were taken and compared with the
47
computational results. The bandwidth of the SAEP design was significantly increased com-
pared to the SAR, and the computed and measured radiation patterns displayed good overall
agreement.
Future work on this topic would likely focus on improving the PEC/thin dielectric
code radiation pattern results. In order to more accurately predict radiation patterns, the
code needs to accurately model antennas with large or infinite ground planes. Modeling
of an infinite ground plane can be acheived by using image theory. With image theory,
the ground plane need not be included in the mesh, resulting in a drastic reduction in the
number of unknowns. Also, a more efficient evaluation of the impedance matrix elements
by computation of the potential integrals by face-pair combinations, instead of the direct
computation of each impedance matrix element used in this work, will reduce the number of
integrations required by the code [3]. This improvement would allow faster evaluation of the
impedance matrix and reduced memory usage, resulting in an ability for the code to handle
problems with a larger number of unknowns.
48
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[3] S.M. Rao, D.R. Wilton, and A.W. Glisson, ?Electromagnetic Scattering by Surfaces
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[4] V. B. Davis, M. A. Khayat, S. Jiang, and D. R. Jackson, ?Characteristics of the Shorted-
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[5] T. N. Killian, ?Numerical Modeling of Very Thin Dielectric Materials,? Master of Sci-
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[6] P.S. Nakar, ?Design of a Compact Microstrip Patch Antenna for Use in Wire-
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[12] http://www.rogerscorp.com/
[13] http://www.feko.info/
49