NUMERICAL MODELING OF VERY THIN DIELECTRIC MATERIALS
Except where reference is made to the work of others, the work described in this
thesis is my own or was done in collaboration with my advisory committee. This
thesis does not include proprietary or classified information.
Tyler Norton Killian
Certificate of Approval:
Michael E. Baginski
Associate Professor
Electrical Engineering
Sadasiva M. Rao, Chair
Professor
Electrical Engineering
Lloyd S. Riggs
Professor
Electrical Engineering
Joe F. Pittman
Interim Dean
Graduate School
NUMERICAL MODELING OF VERY THIN DIELECTRIC MATERIALS
Tyler Norton Killian
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
August 9, 2008
NUMERICAL MODELING OF VERY THIN DIELECTRIC MATERIALS
Tyler Norton Killian
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Tyler N. Killian, son of John and Robin Killian, was born on March 9, 1983,
in Gadsden, Alabama. He attended high school at Sand Rock High School in Sand
Rock, Alabama and graduated in 2001. He graduated with a Bachelor of Electrical
Engineering degree from Auburn University, Auburn, Alabama in 2005. In Fall 2005,
he entered graduate school at Auburn University and plans to obtain a Ph.D. in
Electrical Engineering.
iv
Thesis Abstract
NUMERICAL MODELING OF VERY THIN DIELECTRIC MATERIALS
Tyler Norton Killian
Master of Science, August 9, 2008
(B.E.E., Auburn University, 2005)
46 Typed Pages
Directed by Sadasiva M. Rao
In this work, a Method of Moments (MoM) formulation is presented for the
numerical solution of very thin dielectric materials in the frequency domain. The
dielectric material is represented by a triangular mesh and a parameter controlling the
thickness of the dielectric. The triangular mesh can represent bodies with arbitrary
curvature and results in significantly fewer unknowns than volume formulations.
The dielectric material is first replaced with an equivalent set of currents. An
integral equation is then developed to relate the currents in the dielectric material
to an incident excitation wave. Currents flowing tangentially to the surface of the
dielectric are represented by a set of RWG functions and half-RWG functions at the
boundary edges in order to account for charges at the edge of the dielectric. Currents
normal to the surface are modeled by pulse functions which also account for surface
charges on the dielectric sheet. The MoM procedure is then applied resulting in a
linear system which can be easily solved by matrix inversion.
Finally, the dielectric is coupled with a perfect electrical conductor solution al-
lowing us to solve systems involving both conducting and thin dielectric materials.
v
Furthermore, the similarity between the dielectric and conductor basis functions al-
lows one to easily add support for conductors once the dielectric code has been im-
plemented or vice versa.
vi
Acknowledgments
To the author?s parents, he expresses his appreciation for their love and encour-
agement throughout life.
To his advisor, Dr. Sadasiva M. Rao, he thanks for his patience and top-notch
guidance. His expertise has been invaluable for this work.
To Dr. Michael Baginski, he thanks for his encouragement and interesting tech-
nical discussions.
To the NASA Langley Research Center, he expresses his appreciation for the
fellowship which made this work possible.
vii
Style manual orjournal used Journal ofApproximation Theory (together with the
style known as ?aums?). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically
LATEX) together with the departmental style-file aums.sty.
viii
Table of Contents
List of Figures x
1 Introduction 1
1.1 Background and Objectives . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Thin Dielectric 5
2.1 Electric Field Integral Equation - EFIE . . . . . . . . . . . . . . . . . 6
2.2 Current Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Combined PEC/Thin Dielectric 21
3.1 Electric Field Integral Equation - EFIE . . . . . . . . . . . . . . . . . 21
3.2 Current Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Conclusions 29
Bibliography 31
A Numerical Results 32
A.1 RCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A.2 Antenna Input Impedance . . . . . . . . . . . . . . . . . . . . . . . . 33
A.3 A Note On Dielectric Meshing Requirements . . . . . . . . . . . . . . 34
ix
List of Figures
1.1 Illustration of dielectric sheet with thickness ?. Each triangle can be
assigned a relative permittivity. . . . . . . . . . . . . . . . . . . . . . 3
2.1 Thin dielectric sheet with incident field Ei and observation point r. . 6
2.2 Basis for tangential currents in dielectric. . . . . . . . . . . . . . . . . 11
2.3 Basis for normal currents in dielectric. . . . . . . . . . . . . . . . . . 11
2.4 Quantities used when tangential basis functions overlap. . . . . . . . 14
2.5 Bistatic RCS at ? = 00 by 0.5 m x 0.02 m x 0.001 m dielectric strip
with ?r = 2.6 and an incident wave from (?,?) = (1800,00) at 0.2 Ghz
with E? = ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Bistatic RCS at ? = 00 by 0.5 m x 0.02 m x 0.001 m dielectric strip
with ?r = 2.6 and an incident wave from (?,?) = (1800,900) at 0.2 Ghz
with E? = ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Inhomogeneous dielectric square of dimension 1.0 m x 1.0 m x 0.02 m. 18
2.8 Bistatic RCS at ? = 00 by 1.0 m x 1.0 m x 0.02 m inhomogeneous
square with ?r = 2.2,4.1,5.7,7.3 and an incident wave from (?,?) =
(1200,300) at 0.1 Ghz with E? = ?? . . . . . . . . . . . . . . . . . . . 18
2.9 Bistatic RCS at ? = 00 by a dielectric sphere with radius = 1.0 m
and thickness = 0.05 m with ?r = 2.6 and an incident wave from
(?,?) = (1800,00) at 0.2 Ghz with E? = ? . . . . . . . . . . . . . . . . 19
2.10 Bistatic RCS at ? = 900 by a dielectric sphere with radius = 1.0 m
and thickness = 0.05 m with ?r = 2.6 and an incident wave from
(?,?) = (1800,00) at 0.2 Ghz with E? = ? . . . . . . . . . . . . . . . . 19
2.11 Monostatic RCS at ? = 00 by a homogeneous dielectric square with
dimension 10.16 cm x 10.16 cm x 0.00635 cm with ?r = 4 and an 2.4
GHz incident wave with E? = ? (TE polarization) . . . . . . . . . . . 20
x
2.12 Monostatic RCS at ? = 00 by a homogeneous dielectric square with
dimension 10.16 cm x 10.16 cm x 0.00635 cm with ?r = 4 and an 2.4
GHz incident wave with H? = 1 (TM polarization) . . . . . . . . . . 20
3.1 Geometry for patch antenna. . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Return loss by patch antenna with homogeneous dielectric with ?r =
5.7 and thickness = 0.15 in. The ground plane has dimensions 3.0 in
x 2.4 in. The patch has dimensions 1.8 in x 1.2 in. . . . . . . . . . . . 28
xi
Chapter 1
Introduction
1.1 Background and Objectives
Thin dielectrics appear in various practical situations such as microstrip anten-
nas, radomes, and aircraft built with composite materials. Typical numerical formula-
tions based on either surface or volume integral equations suffer from large numbers of
unknowns as well as meshing difficulties. In this work, a Method of Moments (MoM)
[4] solution is presented which models a thin dielectric with constant thickness as a
two-dimensional sheet with a triangular mesh. This technique not only increases the
numerical efficiency, but also eases the meshing process.
Using the equivalence principle, the dielectric is replaced by an equivalent set of
currents. Since the dielectric is thin, equivalent currents both tangential and normal
to the surface can be assumed constant throughout the thickness of the dielectric.
These currents are then represented by two sets of basis functions. A set of RWG [1]
bases represents tangential currents in the dielectric. In order to account for charge at
the outer edges of the dielectric, a half-RWG basis is used so the current flowing into
the edge is non-zero. Pulse functions represent currents flowing normal to the surface.
The inclusion of this normal component ensures accuracy for problems involving graz-
ing incident waves. Furthermore, it allows us to model problems where the current is
dominated by the normal component. For example, a conductor coated with a thin
dielectric will have a near-zero electric field near the surface of the conductor and
1
consequently very little tangential current in the dielectric. Microstrip problems with
thin dielectric substrates also have a strong field in the normal direction. Allowing
the pulse functions to extend to the surface of the dielectric accounts for any charge
accumulating on the surface. Although the normal component can be modeled as an
equivalent magnetic current as in [6], the electric current equivalent seems to be more
intuitive. Also, since we replace the dielectric with a set of currents in free space, we
no longer have to explicitly enforce boundary conditions for the fields at the bound-
ary of the dielectric - they are satisfied automatically. This is one advantage over the
D-field formulation as in [2] and [7]. Furthermore, in the current equivalent model,
the boundary conditions between mesh elements are enforced directly in the basis
function.
The dielectric sheet is meshed by a two-dimensional triangular mesh along with a
single parameter to control the thickness of the dielectric. Triangular meshing makes
it possible to accurately model complex geometries since triangular patches can easily
conform to curves and support refinement where a denser mesh is required. Further-
more, typical volume formulations, such as [8], involve using a three-dimensional
tetrahedral mesh. Triangular meshes are much easier to generate especially for com-
plex geometries. For very thin dielectrics, tetrahedral elements may also be severely
skewed causing numerical errors and meshing difficulty. Finally, both homogeneous
and heterogeneous dielectrics can be simulated by assigning a relative permittivity to
each triangle as illustrated in Figure 1.1.
Finally, the dielectric code can be easily integrated with existing moment codes
for simulating conductors. Due to the thin sheet approximation, many of the moment
2
terms are similar to those found in PEC formulations using RWG functions. Conse-
quently, one can add support for combined conductor and dielectric problems with
little extra programming effort. The resulting code supports microstrip problems
with thin dielectrics as well as conductors with thin material coatings.
Figure 1.1: Illustration of dielectric sheet with thickness ?. Each triangle can be
assigned a relative permittivity.
1.2 Outline
In Chapter 2 we develop the integral equation for the thin dielectric surface.
The MoM code is then developed through a straightforward expansion and testing
process. This results in a linear system which can be easily solved by matrix inversion.
Numerical results are then presented to verify the accuracy of the method.
The combined PEC/thin dielectric code is developed in Chapter 3 by expanding
the dielectric code. A pair of coupled integral equations is used to generate a new
linear system. The dielectric portion is seen to be a submatrix in the new system.
Also, due to the choice of basis functions and the surface formulation used for the
3
dielectric, the conductor coupling terms are compared to those for the dielectric in
order to aid the programming process.
4
Chapter 2
Thin Dielectric
In this chapter, we develop the integral equation formulation and the MoM pro-
cedure for the thin dielectric. In section 2.1, we use the current equivalent formulation
to replace the dielectric material with a set of currents in free space. We then develop
an integral equation which relates the currents to the incident electric field providing
us with the governing equation for the system.
In section 2.2 we replace the currents with two sets of basis functions. One set
approximates the currents tangential to the surface of the dielectric while the other
set represents currents flowing normal to the surface. We then expand the governing
equation in terms of these functions. Due to the dielectric being very thin, we can
reduce all the volume integrals to surface integrals.
Next, in section 2.3, we generate a matrix system by testing the integral equation
using the Galerkin method. This system can be solved by a standard linear equation
solver providing us with the currents in the dielectric.
5
Figure 2.1: Thin dielectric sheet with incident field Ei and observation point r.
2.1 Electric Field Integral Equation - EFIE
Consider a dielectric sheet illuminated by an incident electric field as shown in
Figure 2.1. First, we replace the dielectric structure with an equivalent set of currents.
If ?r(r) is the relative permittivity at a point r in the dielectric then the equivalent
current J(r) is related to the total electric field Et(r) by:
J(r) = j??o(?r(r)?1)Et(r)
= ?(r)Et(r) where ?(r) = j??o(?r(r)?1) (2.1)
6
Here, ?o is the permittivity of free space and ? = 2?f where f is the frequency of
interest. Equation (2.1) effectively acts as the boundary condition for the electric
field. The total electric field is the sum of the incident field Ei and the scattered field
Es:
Et(r) = Ei(r) +Es(r) (2.2)
The scattered field is related to the current by:
Es(r) = ?j?A(r)???(r) (2.3)
where the magnetic vector potential is
A(r) = ?
integraldisplay
Vd
J(r?)G(r,r?)dv? (2.4)
and the scalar potential is given by:
?(r) = ?1j??
integraldisplay
Vd
??J(r?)G(r,r?)dv? (2.5)
where Vd is the dielectric volume and
G(r,r?) = e
?jkR
4?R and R = |r?r
?| (2.6)
Substituting for the total electric field and rewriting the scattered field in terms
of A and ? provides us with the governing equation:
Ei(r) = J?(r)+ j?A(r)+??(r), r ? Vd (2.7)
7
This equation is suitable for use in the MoM formulation.
2.2 Current Expansion
The total current can be broken up into a tangential part Jp and a normal part
Jq. Each current component is then approximated by a set of basis functions. Let
Pn for n = 1,...,Np and Qn for n = 1,...,Nq be the basis for the tangential and
normal components of the current. Then
J = Jp +Jq ?=
Npsummationdisplay
n=1
?nPn +
Nqsummationdisplay
n=1
?nQn (2.8)
Since the mesh is triangular, we can use a slightly modified RWG basis for the
tangential current. For nonboundary edges, the definition is similar to that used for
PEC surfaces [1]:
Pn,nonboundary(r) =
?
???
???
?
???
???
?
parenleftbigg
?+r,n?1
?+r,n
parenrightbigg
ln
2A+n ?
+
n, r ? T
+
n
parenleftbigg
??r,n?1
??r,n
parenrightbigg
ln
2A?n ?
?
n, r ? T
?
n
0, otherwise
(2.9)
with the exception that each side is multiplied by
parenleftBig??
r ?1
??r
parenrightBig
to compensate for varying
relative permittivities. This multiplier ensures that the electric flux density D normal
to the edge corresponding to the basis is continuous. In terms of the current equivalent
model, this means charge will collect between two adjacent triangles with different
permittivities. The quantities are shown in Figure 2.2. Also, in the dielectric case, the
basis is defined on a pair of prisms rather than a pair of triangles. Since the dielectric
8
is thin, the tangential current is assumed constant throughout the thickness of the
dielectric. Also, since we need to be able to support charge accumulating at the edge
of the dielectric, we use a half-basis for boundary edges:
Pn,boundary(r) =
??
??
???
parenleftBig?
r,n?1
?r,n
parenrightBig l
n
2An?n, r ? Tn
0, otherwise
(2.10)
This basis is the same as the RWG basis with the exception that it is defined on a
single triangle.
For the current component normal to the surface, a pulse basis is used:
Qn(r) =
?
???
???
?nn, r ? Tn
0, otherwise
(2.11)
where ?nn is a unit vector normal to the surface as illustrated in Figure 2.3. The pulse
extends to each side of the dielectric sheet allowing for any charge accumulation on
the surface.
Substituting (2.8) into (2.7) we get:
Ei =
Npsummationdisplay
n=1
?n(Pn? + j?Ap,n +??p,n) +
Nqsummationdisplay
n=1
?n(Qn? + j?Aq,n +??q,n) (2.12)
where
Ap,n(r) = ?
integraldisplay
Vn
Pn(r?)G(r,r?)dv? (2.13)
?p,n(r) = ?1j??
integraldisplay
Vn
??Pn(r?)G(r,r?)dv? (2.14)
Aq,n(r) = ?
integraldisplay
Vn
Qn(r?)G(r,r?)dv? (2.15)
9
?q,n(r) = ?1j??
integraldisplay
Vn
??Qn(r?)G(r,r?)dv? (2.16)
We let dv = dSdn where n is positive in the direction of ?nn. Since the dielectric
is thin, we can approximate the integrands in (2.13), (2.14), and (2.15) as being
constant throughout the thickness of the dielectric. Also, we note that ? ? Qn =
??(n = ?) + ?(n = 0). We can then rewrite (2.13-2.16) as
Ap,n(r) = ??
integraldisplay
Sn
Pn(r?)G(r,r?)dS? (2.17)
?p,n(r) = ??j??
integraldisplay
Sn
??Pn(r?)G(r,r?)dS? (2.18)
Aq,n(r) = ??
integraldisplay
Sn
Qn(r?)G(r,r?)dS? (2.19)
?q,n(r) = 1j??
bracketleftbiggintegraldisplay
S+n
G(r,r?)dS? ?
integraldisplay
S?n
G(r,r?)dS?
bracketrightbigg
(2.20)
where S+n is the top surface (pointed to by ?nn) and S?n is the bottom surface of Tn.
Sn is the surface located at the center of the prism element. Thus, all integrals are
reduced to surface integrals rather than volume integrals.
Some useful quantities are the following:
??Pn =
?
???
???
?
???
???
?
parenleftbigg
?+r,n?1
?+r,n
parenrightbigg
ln
A+n , r ? T
+
n
parenleftbigg
??r,n?1
??r,n
parenrightbigg
?ln
A?n , r ? T
?
n
0, otherwise
(2.21)
10
Figure 2.2: Basis for tangential currents in dielectric.
Figure 2.3: Basis for normal currents in dielectric.
11
To determine the surface charge onthe dielectric we apply the continuity equation
??J = ?j??s to get:
?s,n = ??nj?
parenleftBigg?
r,n ?1
?r,n
parenrightBigg
(2.22)
for a half-rwg function at a boundary edge where the plus sign is taken for a basis
directed out of the triangle and the minus sign if the basis is directed into the triangle.
For the top and bottom surface we have
?s,n = ??nj? (2.23)
for the pulse functions where the positive sign is taken for the side pointed to by ?nn
and the negative sign taken otherwise. Note that equal charges with opposite signs
accumulate on the top and bottom of the dielectric sheet.
2.3 Testing Procedure
We use Galerkin?s procedure to generate a set of linear equations. To do this we
test both sides of (2.12) with each of the previously defined basis functions using the
following inner product:
?f,g??
integraldisplay
f ?gdv (2.24)
This gives us a matrix equation of the form:
?
??
?
Zpp Zpq
Zqp Zqq
?
??
?
?
??
?
Ip
Iq
?
??
? =
?
??
?
Vp
Vq
?
??
? (2.25)
12
The submatrices Zpp, Zqq, Zpq, and Zqp represent the coupling between the basis
functions. Vp and Vq are the excitations from the tangential and normal incident
fields. Ip and Iq are the values to be solved for representing the magnitudes of the
expansion functions.
The coupling terms are given by:
Zpp,mn = 1? ?Pn,Pm?+ j??Ap,n,Pm?+???p,n,Pm? (2.26)
Zpq,mn = j??Aq,n,Pm?+???q,n,Pm? (2.27)
Zqp,mn = j??Ap,n,Qm?+???p,n,Qm? (2.28)
Zqq,mn = 1? ?Qn,Qm?+ j??Aq,n,Qm?+???q,n,Qm? (2.29)
and the excitation vectors by:
Vp,m =
angbracketleftBig
Ei,Pm
angbracketrightBig
(2.30)
Vq,m =
angbracketleftBig
Ei,Qm
angbracketrightBig
(2.31)
The testing terms are evaluated in the following. If Pm and Pn overlap on some
triangle T with permittivity ?r, let rct, r1, r2, and r3 represent the position vectors
of the centroid of T and each of the three vertices as shown in Figure 2.4.
?Pn,Pm?T = ?lmln?4A
m
parenleftbigg?
r ?1
?r
parenrightbigg2bracketleftBigg3|rct|2
4 +
|r1|2 +|r2|2 +|r3|2
12
?(rm +rn)?rct +rm ?rn
bracketrightBig
(2.32)
13
where rm and rn are the position vectors corresponding to vertices associated with
Pm and Pn. If m = n and Pm is associtted with a nonboundary edge, then both T+m
and T?m will contribute to ?Pn,Pm?. Also note that the sign is positive if both bases
are directed into or out of T, and is negative otherwise [5].
Figure 2.4: Quantities used when tangential basis functions overlap.
The testing terms for the RWG functions are similar to those for the PEC solution
in [1] with the exception that they are multiplied by ? to compensate for the thickness
of the dielectric.
angbracketleftBigg
?
???
???
Ei
Ap/q,n
?
???
???,Pm
angbracketrightBigg
= lm2 ?
?
??
?
parenleftBigg?+
r,m ?1
?+r,m
parenrightBigg
?
???
???
Ei(rc+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
?
??
? (2.33)
14
angbracketleftBig
??p/q,n,Pm
angbracketrightBig
= ?lm?
bracketleftBiggparenleftBigg?+
r,m ?1
?+r,m
parenrightBigg
?p/q,n(rc+m )?
parenleftBigg??
r,m ?1
??r,m
parenrightBigg
?p/q,n(rc?m )
bracketrightBigg
(2.34)
where rct+m and rct?m are the centroids of T+m and T?m. If Pm is a half-basis, then the
first summation terms in (2.33) and (2.34) are used if the basis is directed out of Tm
and the second terms are used otherwise.
The testing terms for the pulse functions are as follows:
?Qn,Qm? =
?
???
???
?Am
? , if n = m
0, otherwise
(2.35)
angbracketleftBig
Ap/q,n,Qm
angbracketrightBig
= ?AmAp/q,n(rctm)? ?nm (2.36)
angbracketleftBig
??p/q,n,Qm
angbracketrightBig
= ?Am
parenleftBigg?
p/q,n(rctm + 0.25??nm)??p/q,n(rctm ?0.25??nm)
0.5?
parenrightBigg
(2.37)
The term (2.37) comes from the fact that ???Qm = ???n. This derivative is approxi-
mated at the centroid of Tm and is assumed constant throughout the prism volume.
At this point, a MoM formulation has been developed to handle both homo-
geneous and heterogeneous thin dielectric sheets. In the next chapter, this code is
expanded to include arbitrary conductors as well.
15
2.4 Numerical Results
In this section, numerical results are presented to demonstrate the accuracy of
the dielectric code. In Figures 2.5 and 2.6 a bistatic RCS is presented for a 0.5 m x
0.02 m x 0.001 m homogeneous rectangular strip with ?r = 2.6. The incident wave is
directed perpendicular to the strip surface with the electric field oriented horizontally
in Figure 2.5 and vertically in Figure 2.6. The results are compared with those found
in [2].
In Figure 2.8 a bistatic RCS is given for an inhomogeneous square plate with
dimensions 1.0 m x 1.0 m x 0.02 m. The plate is divided up into four equally sized
smaller squares and each is assigned a different ?r as shown in Figure 2.7. This plate
is illuminated with a 0.1 GHz incident wave from (?,?) = (1200,300). The values are
compared to those in [2] and are found to be in agreement.
Figures 2.9 and 2.10 show the bistatic RCS from a spherical shell with a radius
of 1 m and thickness of 0.05 m. The dielectric is homogeneous with ?r = 2.6. The
excitation is a 0.2 GHz incident wave at (?,?) = (1800,00). Figure 2.9 shows the RCS
at ? = 00 and Figure 2.10 gives the result at ? = 900. The results are also compared
to those in [2].
Figures 2.11 and 2.12 show the monostatic RCS at ? = 00 for a square plate with
dimensions 10.16 cm x 10.16 cm x 0.00635 cm. The dielectric is homogeneous with
?r = 4. The incident wave has a frequency of 2.4 Ghz. Figure 2.11 shows the RCS
for a TE polarized wave. Figure 2.12 shows the TM case. The results are compared
with those found in [7].
16
-88
-87.5
-87
-86.5
-86
-85.5
-85
-84.5
-84
-83.5
-83
 0  20  40  60  80  100  120  140  160  180
RCS (dB)
? (deg)
Computed
Chiang and Chew
Figure 2.5: Bistatic RCS at ? = 00 by 0.5 m x 0.02 m x 0.001 m dielectric strip with
?r = 2.6 and an incident wave from (?,?) = (1800,00) at 0.2 Ghz with E? = ?
-260
-240
-220
-200
-180
-160
-140
-120
-100
-80
 0  20  40  60  80  100  120  140  160  180
RCS (dB)
? (deg)
Computed
Chiang and Chew
Figure 2.6: Bistatic RCS at ? = 00 by 0.5 m x 0.02 m x 0.001 m dielectric strip with
?r = 2.6 and an incident wave from (?,?) = (1800,900) at 0.2 Ghz with E? = ?
17
Figure 2.7: Inhomogeneous dielectric square of dimension 1.0 m x 1.0 m x 0.02 m.
-38
-36
-34
-32
-30
-28
-26
 0  20  40  60  80  100  120  140  160  180
RCS (dB)
? (deg)
Computed
Chiang and Chew
Figure 2.8: Bistatic RCS at ? = 00 by 1.0 m x 1.0 m x 0.02 m inhomogeneous square
with ?r = 2.2,4.1,5.7,7.3 and an incident wave from (?,?) = (1200,300) at 0.1 Ghz
with E? = ??
18
-30
-25
-20
-15
-10
-5
 0
 5
 10
 15
 0  20  40  60  80  100  120  140  160  180
RCS (dB)
? (deg)
Computed
Chiang and Chew
Figure 2.9: Bistatic RCS at ? = 00 by a dielectric sphere with radius = 1.0 m and
thickness = 0.05 m with ?r = 2.6 and an incident wave from (?,?) = (1800,00) at 0.2
Ghz with E? = ?
-50
-40
-30
-20
-10
 0
 10
 20
 0  20  40  60  80  100  120  140  160  180
RCS (dB)
? (deg)
Computed
Chiang and Chew
Figure 2.10: Bistatic RCS at ? = 900 by a dielectric sphere with radius = 1.0 m and
thickness = 0.05 m with ?r = 2.6 and an incident wave from (?,?) = (1800,00) at 0.2
Ghz with E? = ?
19
-70
-60
-50
-40
-30
-20
-10
 0  10  20  30  40  50  60  70  80  90
RCS (dB)
? (deg)
Computed
Khayat
Figure 2.11: Monostatic RCS at ? = 00 by a homogeneous dielectric square with
dimension 10.16 cm x 10.16 cm x 0.00635 cm with ?r = 4 and an 2.4 GHz incident
wave with E? = ? (TE polarization)
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
 0  10  20  30  40  50  60  70  80  90
RCS (dB)
? (deg)
Computed
Khayat
Figure 2.12: Monostatic RCS at ? = 00 by a homogeneous dielectric square with
dimension 10.16 cm x 10.16 cm x 0.00635 cm with ?r = 4 and an 2.4 GHz incident
wave with H? = 1 (TM polarization)
20
Chapter 3
Combined PEC/Thin Dielectric
In Chapter 2 a MoM formulation was developed for thin dielectric sheets. In
this chapter, we will add support for perfect electric conductors (PECs) of arbitrary
shape. In section 3.1, we apply the boundary conditions for the electric field on the
conducting surface to obtain an integral equation for the PEC. Using this result and
the integral equation developed in Chapter 2, we arrive at a pair of coupled integral
equations for the combined system.
In sections 3.2 and 3.3, we apply the MoM procedure to develop a linear system
where the dielectric terms from Chapter 2 appear as a submatrix to the new system.
Furthermore, the coupling terms that arise from including the PEC turn out to be
very similar to terms in the dielectric formulation. Thus, the codes can be combined
with little extra effort.
3.1 Electric Field Integral Equation - EFIE
The dielectric code can easily be expanded to incorporate perfect electrical con-
ductors (PECs). Again, the scattered field is related to the scalar and vector poten-
tials by:
Es(r) = ?j?A(r)???(r) (3.1)
21
For a PEC, the total electric field tangential to the surface is zero. This leads to
the boundary condition for the PEC:
Eitan(r) = ?Estan(r), r ? Sc (3.2)
where Sc is the conductor surface. Substituting for the scattered field we get the
desired integral equation:
Ei(r)tan = [j?A(r)+??(r)]tan (3.3)
Combining this with the boundary conditions for the dielectric, we arrive at a
pair of coupled equations.
Ei(r)tan = [j?A(r)+??(r)]tan, r ? Sc
Ei(r) = Jd? (r) + j?A(r)+??(r), r ? Vd (3.4)
where Sc represents the conductor surface and Vd the volume of the dielectric. Here
the magnetic vector potentialA is generated by the currents on boththe dielectric and
conductor. Let Ac be the vector potential due to the PEC, Ap be due to the currents
tangential to the surface of the dielectric, and Aq the potential due to currents normal
to the surface of the dielectric. The total vector potential is then:
A = Ac +Ap +Aq (3.5)
22
The total scalar potential is defined similarly:
? = ?c +?p +?q (3.6)
The PEC magnetic vector potential Ac and scalar potential ?c are given by
Ac(r) = ?
integraldisplay
Sc
Jc(r?)G(r,r?)dS? (3.7)
?c(r) = ?1j??
integraldisplay
Sc
??Jc(r?)G(r,r?)dS? (3.8)
and the potentials due to the currents in the dielectric are the same as those given in
Chapter 2.
3.2 Current Expansion
We define a set of basis functions to approximate the current on the PEC. Let
fn for n = 1,...,Nc be basis defined on the surface of the conductor. The total current
on the conductor is then given by:
Jc =
Ncsummationdisplay
n=1
?nfn(r) (3.9)
We use RWG basis functions for the conductor. They are defined as:
fn(r) =
?
???
???
?
???
???
?
ln
2A+n ?
+
n, r ? T
+
n
ln
2A?n ?
?
n, r ? T
?
n
0, otherwise
(3.10)
23
Substituting (3.9) into (3.5) and (3.6) we get:
A =
Npsummationdisplay
n=1
?nAp,n +
Nqsummationdisplay
n=1
?nAq,n +
Ncsummationdisplay
n=1
?nAc,n (3.11)
and
? =
Npsummationdisplay
n=1
?n?p,n +
Nqsummationdisplay
n=1
?n?q,n +
Ncsummationdisplay
n=1
?n?c,n (3.12)
where
Ac,n(r) = ?
integraldisplay
Sc
fn(r?)G(r,r?)dS? (3.13)
?c,n(r) = ?1j??
integraldisplay
Sc
??fn(r?)G(r,r?)dS? (3.14)
and the remaining terms are the same as those given in Chapter 2.
3.3 Testing Procedure
Both sides of the equations in (3.4) are tested using the Galerkin procedure. For
testing on the PEC structure we use the inner product
?f,g??
integraldisplay
f ?gdS (3.15)
in order to generate a matrix equation
?
??
??
??
??
Zcc Zcp Zcq
Zpc Zpp Zpq
Zqc Zqp Zqq
?
??
??
??
??
?
??
??
??
??
Ic
Ip
Iq
?
??
??
??
??
=
?
??
??
??
??
Vc
Vp
Vq
?
??
??
??
??
(3.16)
24
which can be solved by matrix inversion. Zpp, Zpq, Zqp, and Zqq are the same as those
given in chapter 2. The remaining submatrices are given by:
Zcc,mn = j??Ac,n,fm?+???c,n,fm? (3.17)
Zcp,mn = j??Ap,n,fm?+???p,n,fm? (3.18)
Zcq,mn = j??Aq,n,fm?+???q,n,fm? (3.19)
Zpc,mn = j??Ac,n,Pm?+???c,n,Pm? (3.20)
Zqc,mn = j??Ac,n,Qm?+???c,n,Qm? (3.21)
and the excitation vector by:
Vc,m =
angbracketleftBig
Ei,fm
angbracketrightBig
(3.22)
Now we evaluate terms given by testing on the conductor:
angbracketleftBigbraceleftBig
Ac/p/q,n
bracerightBig
,fm
angbracketrightBig
= lm2
bracketleftBigbraceleftBig
Ac/p/q,n(rct+m )
bracerightBig
??ct+m
+
braceleftBig
Ac/p/q,n(rct?m )
bracerightBig
??ct?m
bracketrightBig
(3.23)
angbracketleftBig
??c/p/q,n,fm
angbracketrightBig
= ?lm
bracketleftBig
?c/p/q,n(rc+m )??c/p/q,n(rc?m )
bracketrightBig
(3.24)
25
For the dielectric testing, conductor source terms we have
?{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
bracketrightBig
(3.25)
???c,n,Pm? = ?lm?
bracketleftBiggparenleftBigg?+
r,m ?1
?+r,m
parenrightBigg
?c,n(rc+m )?
parenleftBigg??
r,m ?1
??r,m
parenrightBigg
?c,n(rc?m )
bracketrightBigg
(3.26)
where we have tested using the tangential currents on the dielectric. For normal
component testing on the dielectric we have
?Ac,n,Qm? = ?AmAc,n(rctm)? ?nm (3.27)
???c,n,Qm? = ?Am
parenleftBigg?
c,n(rctm + 0.25??nm)??c,n(rctm ?0.25??nm)
0.5?
parenrightBigg
(3.28)
The coupling terms due to conductor source/testing are of the same form (within
a constant multiple) of the terms in the dielectric formulation. They correspond as
follows:
Zcc ?? Zpp
Zcp ?? Zpc ?? Zpp
Zqc ?? Zqp
Zcq ?? Zpq (3.29)
These relationships can be very helpful when programming the combined code.
26
3.4 Numerical Results
In this section, the return loss is given for a patch antenna with a homogeneous
dielectric substrate. The antenna has a 3.0 in x 2.4 in ground plane and a 1.8 in x 1.2
in patch. The dielectric has a thickness of 0.15 in and ?r = 5.7. The excitation pin
is given by a flat conducting strip with a width of 0.03 in. The geometry is shown in
Figure 3.1. The return loss is calculated by using the procedure in Appendix A.2 to
solve for the input impedance and is matched to a 50 ? feed.
27
Figure 3.1: Geometry for patch antenna.
-12
-10
-8
-6
-4
-2
 0
 1.6  1.8  2  2.2  2.4  2.6  2.8  3  3.2  3.4
Return Loss (dB)
Frequency (GHz)
Computed
HFSS
Figure 3.2: Return loss by patch antenna with homogeneous dielectric with ?r = 5.7
and thickness = 0.15 in. The ground plane has dimensions 3.0 in x 2.4 in. The patch
has dimensions 1.8 in x 1.2 in.
28
Chapter 4
Conclusions
In this work, an efficient numerical technique for modeling thin dielectric sheets
in the frequency domain was developed. The dielectric is a thin sheet with constant
thickness and arbitrary curvature. Futhermore, it may be either homogeneous or
inhomogeneous. It was represented by a two-dimensional triangular mesh resulting
in much fewer unknowns and an easier meshing process than volume formulations.
In Chapter 2, the dielectric was replaced by a set of equivalent currents. These
currents were then related to the incident field using an EFIE approach. Next, the
tangential currents were represented by a set of RWG basis functions while the normal
component was represented by a set of pulse functions. These bases allow accurate
modeling of currents within the dielectric as well as charges accumulating on the
surface. Furthermore, due to the dielectric being thin, all integrals in the formulation
were reduced to surface integrals greatly increasing the numerical efficiency.
In Chapter 3, the dielectric code was combined with an arbitrarily shaped con-
ductor formulation resulting in a pair of coupled integral equations. Due to the
similarity in the formulations, the conducting code was shown to be easily incorpo-
rated with the dielectric code. The validity of this formulation was demonstrated by
calculating the resonances for a patch antenna.
29
Further research may be done in this area to extend the formulation?s capabilities
to multiple dielectric layers. This analysis could prove useful for thicker dielectrics
where volume formulations become much more inefficient.
30
Bibliography
[1] S.M. Rao, D.R. Wilton, and A.W. Glisson, ?Electromagnetic scattering by sur-
faces or arbitrary shape,? IEEE Trans. Antennas Propag., vol. 30, pg 409-418,
May 1982.
[2] I.T. Chiang and W.C. Chew, ?Thin dielectric sheet simulation by surface integral
equation using modified RWG and pulse bases,? IEEE Trans. Antennas Propag.,
vol. 54, pg 1927-1934, Jul. 2006.
[3] I.T. Chiang and W.C. Chew, ?A Coupled PEC-TDS Surface Integral Equa-
tion Approach for Electromagnetic Scattering and Radiation From Composite
Metallic and Thin Dielectric Objects,? IEEE Trans. Antennas Propag., vol. 54,
pg 3511, Jul. 2006.
[4] R.F. Harrington, Field Computation by Moment Methods. New York: IEEE
Press, 1993.
[5] S.M. Rao, ?Electromagnetic Scattering and Radiation of Arbitrarily-Shaped Sur-
faces by Triangular Patch Modeling,? Ph.D. Dissertation University of Missis-
sippi, Aug. 1980.
[6] Shian-Uei Hwu, ?Electromagnetic Modeling of Conducting and Dielectric Coated
Wire, Surface, and Junction Configurations,? Ph.D. Dissertation University of
Houston, May 1990.
[7] Michael A. Khayat, ?Numerical Modeling of Thin Materials in Electromagnetic
Scattering Problems,? Ph.D. Dissertation University of Houston, Dec. 2003.
[8] D. Schaubert, D. Wilton, and A. Glisson, ?A tetrahedral modeling method for
electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bod-
ies,? IEEE Trans. Antennas Propag., vol. 32, pg 77-85, Jan. 1984.
31
Appendix A
Numerical Results
A.1 RCS
Once we have solved for the currents on a given structure, we may calculate its
radar cross section (RCS) given by the quantity
? = limr??4?r2|E
s|2
|Ei|2 (A.1)
where Ei is the incident field and Es is the scattered field calculated in the far-field
in the direction of
?r = sin?cos??x+ sin?sin??y + cos??z (A.2)
The far field is given by:
Es = ?j?A = ?j??
integraldisplay
Je
?jkR
4?R dv
? (A.3)
If we let r? be the integration coordinates, then as r ? ? we can approximate
R as
R = (r?r
?)?(r?r?)
|r?r?| ? r??r?r
? (A.4)
giving us
Es ? ?j??4?r e?jkr
integraldisplay
Jejk?r?r?dv? (A.5)
32
Using this approximation we can calculate the far-field contributions given by the
currents on the dielectric (Esp and Esq) as well as the conductor (Esc). We approximate
each basis function at the centroid of the corresponding triangle(s). For the tangential
component on the dielectric sheet we have
Esp = ?j???8?r e?jkr
Npsummationdisplay
n=1
?nln
bracketleftBiggparenleftBigg?+
r,n ?1
?+r,n
parenrightBigg
?ct+n ejk?r?rct+n (A.6)
+
parenleftBigg??
r,n ?1
??r,n
parenrightBigg
?ct?n ejk?r?rct?n
bracketrightBigg
(A.7)
and for the normal component:
Esq = ?j???4?r e?jkr
Nqsummationdisplay
n=1
?nAnejk?r?rctn (A.8)
Similarly the scattered field from the conductor is given by:
Esc = ?j??8?r e?jkr
Ncsummationdisplay
n=1
?nln
bracketleftBig
?ct+n ejk?r?rct+n + ?ct?n ejk?r?rct?n
bracketrightBig
(A.9)
where ?n, ?n, and ?n are the magnitudes for Pn, Qn, and fn respectively.
A.2 Antenna Input Impedance
Another problem of practical importance is calculating the input impedance of
an antenna. In order to do this we use the delta-gap feed model. An E-field is applied
to a junction between two triangles such that a constant voltage is maintained across
an infinitesimal gap where the triangles join. We define Ei = V across the gap and
33
Ei = 0 otherwise. The field is oriented in the same direction as the basis fm connecting
the two triangles. Testing provides us with the excitation
Vc,m =
integraldisplay
Ei ?fmdS = V
integraldisplay
dl = V lm (A.10)
If Im is the coefficient for fm,the total current across the gap is lmIm. The input
impedance is then calculated by
Zin = Vl
mIm
(A.11)
For a 50 ? feed line, the reflection coefficient is given by:
?L = Zin ?50Z
in + 50
(A.12)
The return loss is then calculated by:
RL = 20log|?L| (A.13)
A.3 A Note On Dielectric Meshing Requirements
It should be noted that increasing the relative permittivity of a dielectric also
increases the electrical size of the material. The typical meshing recommendation for
a surface is 100 divisions/square wavelength (defining electrically small as less than
34
a tenth of a wavelength). The wavelength in a dielectric is given by
?G = cof??
r
(A.14)
where co is the speed of light in free space and ?r is the relative permittivity of the
dielectric material. This implies ?2G = ?2o?r where ?o is the wavelength in free space for
the frequency of interest. In other words, the meshing requirement increases linearly
with the relative permittivity of the material. This results in a large increase in
unknowns for problems involving dielectrics with large ?r.
35