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Analysis of a hemispherical dielectric resonator antenna with very high
permittivity (e e r=169) using WIPL-D Analysis of a hemispherical dielectric resonator antenna with very high
permittivity ( r
=169) using WIPL-D

Seong-man Jang , Branko Kolundzija*, Tapan K. Sarkar

Department of Electrical Engineering and Computer Science
Syracuse University
121 Link Hall
Syracuse, NY 13244-1240
Tel : (315)443-3775
Email:
sejang@mailbox.syr.edu
:
tksarkar@mailbox.syr.edu


*Department of Electrical Engineering
University of Belgrade
P.O.Box 35-54, 11120 Belgrade, Yugoslavia
Email: KOL@kiklop.etf.bg.ac.yu
Abstract -
The analysis of very high permittivity ( r
=169) structure using WIPL-D is presented.
As an example , a microstrip patch hat antenna is treated. Network parameters, radiation pattern
and current distributions on the surface are analyzed.

I.
Introduction
The analysis of an antenna or a scatterer, which has very high dielectric constant, is difficult
because it is easy to change the performance with small variation of the dimensions. Because of
material discontinuity between metal and high permittivity dielectric substrate electromagnetic
field varies abruptly in the vicinity of the boundary. It is difficult to model that situation correctly
and also difficult to check the quality of the solutions. In this paper, the characteristics of
hemispherical resonator with very high permittivity ( r
=169) are analyzed using the WIPL-D,
which adapted the Method of Moment (MoM)[1].
WIPL-D is a software package for electromagnetic modeling of composite metallic and
dielectric structures. In WIPL-D, electric and magnetic surface currents are approximated by an
entire domain polynomial expansion, and the unknown expansion coefficients are determined by
applying the Galerkin method to the corresponding surface integral equation. All other quantities
of interest such as radiation field and near field are obtained using the calculated current
distribution.

II.
Method of Analysis
1. Electric field integral equation
The currents on a plate and a wire as represented in Fig. 1 are modeled by an electric field
integral equation for the perfectly conducting body in a vacuum. For the surface current
distribution, electric field produced by the currents and charges induced over the structure surface
can be represented by surface current density J
s
and free space Green Function g(R) as shown in
Eq. 1.






+ =
S
R
S
S
S
S
dS
dR
R
dg
div
dS
R
g
j
)
(
1
)
(
2
0
i
J
J
E µ


(1) where J
s
is a surface current density, R is the distance between the field point and the source
point, i
R
(=R/R) is the unit vector from the source point to the field point, and g(R) is a free space
Green Function. The free space Green Function is given by
R
e
R
g
R
j
4
)
( =
;
0
0
µ
=
;
2
2
2
)
'
(
)
'
(
)
'
(
z
z
y
y
x
x
R + + =

(2)
The integral equation is obtained by substituting Eq. 1 into the boundary condition of Eq. 3
where the tangential component of the total electric field equals zero on the surface. In other
words,
(E+Ei)
tan
= 0







(3)
where E is electric field produced by the currents and charges induced over the structure surface,
E
i
is an incident field. In the case of wires, boundary condition is simpler, i.e. the axial
component of the total electric field is zero along the wire, e.g. local z-axis. Thus,
0
=
+
iz
z
E
E








(4)
The electric field E is expressed as Eq. 4 in terms of the total current intensity I(s) flowing along
the wire.






+ =
2
1
2
1
2
0
)
(
)
(
)
(
1
)
(
)
(
cos
s
s
s
s
e
e
e
e
z
ds
dR
R
dg
R
grad
ds
s
dI
ds
R
g
s
I
j
µ
i
E

(5)

The current is approximated by a finite sum of known functions multiplied by unknown
coefficients. These coefficients are calc ulated by applying the Method of Moment (MoM) to the
electric field integral equations of (1) and (5).
Next, lets consider an arbitrary shaped body made of isotropic linear homogeneous dielectric
situated in vacuum and excited by a time-harmonic electromagnetic field. According to the
equivalence theorem, determination of total electromagnetic field inside and outside can be
facilitated if the problem is decomposed into two parts, as shown in Fig. 2. The field outside the
body is uniquely determined by sources situated outside the body and equivalent electric and
magnetic currents, J</b>s and M</b>s, placed over the surface of the body. These equivalent currents are
expressed as

tan
tan
E
n
M
H
n
J
× =
×
=
s
s





(6)
where n is outward normal vector to the surface of the body and H
tan
and E
tan
are components of
total field vectors tangential at body surface. If the impressed electric and magnetic fields are
given in both domains E
i(1)
, E
i(2)
, H
i(1)
and H
i(2)
, equivalent electric and magnetic currents placed
over the dielectric boundary surface are uniquely determined from the boundary conditions for
tangential field components, that is, from
[
] [
]
[
] [
]
tan
)
2
(
)
2
(
tan
)
1
(
)
1
(
tan
)
2
(
)
2
(
tan
)
1
(
)
1
(
i
i
i
i
H
H
H
H
E
E
E
E
+
=
+
+
=
+

(7)
Electric and magnetic field vectors inside the first and second domains, E
(1)
, E
(2)
, H
(1)
and H
(2)
are produced by equivalent current. E
(1)
and

H
(1)
are expressed in terms of the electric and
magnetic vector potentials, A
e(1)
and A
m(1)
, and electric and magnetic scalar potentials V
e(1)
and
V
m(1)
, as,
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
1
e
e
m
V
jw
A
A
E
× =





(8)
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
1
m
m
e
V
jw
A
A
H
× =
µ






(9) where (1)
and

µ
(1)
are electric and magnetic constant of the first domain, and is angular
frequency. The potentials are expressed in terms of the equivalent electric and magnetic surface
current density vectors, J
s
and M
s
, and corresponding surface charge densities, s
and s
, as
dS
R
g
V
dS
R
g
V
s s
m
s
s
e
)
(
1
)
(
1
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
=
= µ



(10)
dS
R
g
dS
R
g
s
s
m
s
s
e
)
(
)
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
=
=
M
A
J
A µ



(11)
where g
(1)
(R) is a free space Green function for the first domain and R is the distance between the
field point and the source point. The free space Green Function is same as Eq. 2. The surface
current density vectors and surface charge densities are related by the continuity equations
s
s
s
s
j
j =
= M
J






(12)
By substituting Eq 10, 11 and 12 into Eq 8 and 9, electric and magnetic field vectors E
(1)
and
H
(1)
are expressed in terms of unknown equivalent currents J
s
and M
s
. The same expressions are
valid for electric and magnetic fiels vectors E
(1)
and H
(1)
, except that index (1) should be replaced
by index (2) and equivalent currents J
s
and M
s
should be used with a minus sign. After replacing
the expressions for vectors E
(1)
, E
(2)
, H
(1)
and H
(2)
into Eq 7 , a system of two surface integral
equations in terms of unknown equivalent currents is obtained. Using the Galerkin method solves
this system of surface integral equations.


2. Approximation of current

The currents along wires and plates are approximated using polynomial expansions for the
single element and modified in such a manner that final current expansions are satisfied the
continuity equation including enforcing, at element interconnections ant free ends. The starting
current along the wire is given by eq. (13).
[
]
[
]
[
]
[
]
L
s
L
s
a
L
s
a
L
s
I
L
s
I
s
I
i
n
i
n
i
i
i
i
/
)
/
(
1
)
/
(
/
1
2
1
/
1
2
1
)
(
,...
4
,
2
,...
5
,
3
2
1 + +
+
+ =
=
=

(13)

where, -L s L, 2L is the total length of the wire along the s-coordinate, n is the limit of the
polynomial sum, and
i
a
, i = 1,2,n are the unknown coefficients that should be determined.
Basis functions that correspond to the unknown
i
a
are equal to zero at the wire ends.
The surface current over the plate is decomposed into its local p and s-components, that is,
s
ss
p
sp
s
J
J
i
i
J
+
=









(14)

If we consider only s-component of the current over a bilinear surface, Jss(p,s) is approximated
by a finite double polynomial sum in the form

=
=


=
p
s
n
j
j
n
i
i
ij
ps
p
ss
s
p
p
s
a
s
p
s
e
s
p
J
0
0
1
1
,
1
1
)
,
(
sin
)
(
1
)
,
(
(15)

where, n
p
and n
s
are the degree of the polynomial approximations along p and s local coordinates,
ij
a
, i=1,2 n
p
and j=1,2, n
s
are unknown coefficients, e
p
(s) is the Lame coefficient and ps
(p,s) is the angle between local p and s-coordinate lines. In the case of a bilinear surface of
Fig.3, Lame coefficient e
p
(s) is a function of the s-coordinate only and continuity equation at the
interconnection can be expressed only in terms of the s-current component of these bilinear
surfaces. So, continuity equation for th