Hybrid Method Analysis of Electromagnetic Transmission through
Apertures of Arbitrary Shape in a Thick Conducting Screen
ABUNGU N ODERO1, DOMINIC B O KONDITI1
ALFRED V OTIENO2
1
Jomo Kenyatta University of Agriculture and Technology
2
University of Nairobi
Abstract – In this paper a hybrid numerical technique is presented, suitable for analyzing transmission
properties of an arbitrarily shaped slot in a thick conducting plane. The slot is excited by an
electromagnetic source of arbitrary orientation. The analysis of the problem is based on the "generalized
network formulation" for aperture problems. The problem is solved using the method of moments(MOM)
and the finite element method(FEM) in a hybrid format. The finite element method is applicable to
inhomogeneously filled slots of arbitrary shape while the method of moments is used for solving the
electromagnetic fields in unbounded regions of the slot. The cavity region has been subdivided into
tetrahedral elements resulting in triangular elements on the surfaces of the apertures. Validation results
for rectangular slots are presented. Close agreement between our data and published results is observed.
Thereafter, new data has been generated for cross-shaped, H-shaped and circular apertures.
Key – words: Finite–Element Method, Method of Moments, Perfect Electric Conductor (PEC)
1
T
Introduction
HE problem of electromagnetic transmission
through apertures has been extensively investigated
[1-6]. Many specific applications, such as apertures in a
conducting screen, waveguide-fed apertures, cavity-fed
apertures,
waveguide-to-waveguide
coupling,
waveguide-to-cavity coupling, and cavity-to-cavity
coupling have been investigated in the literature.
Both intentional and inadvertent apertures of various
shapes are encountered in the many applications of
today's technology especially due to increased scale of
utilization of microwave and millimeter-wave bands for
communications, radar, industrial and domestic
applications. Examples of undesirable electromagnetic
coupling are leakage from microwave ovens and printed
circuit boards, electromagnetic penetration into
vehicles, and electromagnetic pulse interaction with
shielded electronic equipment. These lead to problems
of Electromagnetic Compatibility (EMC) and
Electromagnetic Interference (EMI) which should be
minimized or eliminated. Desirable coupling exists in
some practical systems such as slotted antenna arrays,
micro-strip patch antennas, directional couplers and
cavity resonators.
For the purposes of evaluating electromagnetic
coupling through or radiation from apertures, it is
desirable to develop efficient evaluation schemes. The
past seventeen years have witnessed an increasing
reliance on computational methods for the
characterization
of
electromagnetic
problems.
Traditional integral-equation methods continue to be
used for many applications. However, in recent times,
the greatest progress in computational electromagnetics
has been in the development and application of hybrid
techniques, such as FEM/MOM and (finite-difference
time-domain)FDTM/MOM .
The FDTM requires a fine subdivision of the
computational domain for good resolution and is quite
computationally intensive. The method of moments is
an integral equation method which handles unbounded
problems very effectively but also becomes
computationally
intensive
when
complex
inhomogeneities
are
present.
In
contrast,
inhomogeneities are easily handled by the finite element
method, which requires less computer time and storage
capacity because of its sparse and banded matrix. The
matrix filling time is also negligible when simple basis
functions are used [3]. However, it is most suitable for
1
boundary value problems. Since the investigation to be
carried out encompasses unbounded problems as well as
bounded problems with complex inhomogeneities, a
procedure combining the finite element method and the
method of moments would be effective.
In this paper, we discuss the hybrid formulation of
the aperture problem followed by a presentation and
discussion of results for the various apertures.
Rectangular, circular, cross-shaped and H-shaped
apertures are some of the apertures studied.
2
Problem Formulation
Consider the 3–dimensional ( aperture–cavity–
aperture ) structure illustrated in Fig.1.
region A and that below the plane ( z < -d ) as region B.
Also, the volume occupied by the cavity ( -d < z < 0 )
will be referred to as region C. It is assumed that the
cavity is filled with an inhomogeneous material having
a relative permittivity  c r  and relative permeability
 c r  .
Using an approach similar to that presented in [8],
one can replace the tangential electric fields in the
aperture planes with equivalent magnetic currents M 1 in
the z  0 aperture plane and M 2 in the z   d 
aperture plane. In the interior region, the equivalent
magnetic currents  M 1 and M 2 in the z  0 and the
z   d  aperture
planes, respectively.
Perfect
conductors can be placed in the z  0 S1  and the
z   d S 2  planes, as illustrated in Fig. 2, without
altering the fields. The problem geometry is then
decomposed into 3 regions as illustrated in Fig. 3.
y
cavity wall
Region A ( z > 0 )
upper aperture
o
x
S1
M1(x,y)
x
-M1(x,y)
Region C
lower aperture
d
M2(x,y)
-M2(x,y)
(a)
Ji, Mi
Region A
z
y
o
S2
Region B ( z < -d )
x
Fig. 2: Equivalent currents introduced in the aperture
planes.
z
 A, A
Ji, Mi
Region C
S1
x
conductor
Region B
M1(x,y)
conductor
z =0
(b)
S1
Fig. 1. Problem geometry: (a) Top view (b) Cross-sectional view
The specific configuration consists of a cavity in a
thick conducting plane having an aperture at the top
surface and another one at the bottom surface. The free
space region above the plane ( z > 0 ) is denoted as
-M1(x,y)
M2(x,y)
 C , C
S2
2
the S1 and S 2 planes, respectively. Taking the inner

product of equation (3) with W1m x, z  0 and taking
advantage of the linearity of the operators results in
 
W1m , H1sct  
N
V
1n
  V
 

W1m , H1tot
t M 1n 
n1
x = -d

S2




nˆ  H tot  nˆ  H sc  nˆ  H scat
(1)
In the interior region C, the closed cavity is
perturbed by the equivalent currents M 1 and M 2 , as
illustrated by Fig. 3. The total tangential magnetic field
that is produced on S 1 is expressed as the superposition
of the fields produced by each current





nˆ  H1tot  M 1  H1tot  M 2

(2)
with a similar expression on S 2 . To ensure the
uniqueness of the solution, the tangential magnetic field
must be continuous across the aperture planes. To this
end, continuity of the total tangential magnetic fields
across the aperture planes is enforced. Therefore, on S1


 







nˆ  H1sc  nˆ  H1tot  M 1  H1tot  M 2  H1scat M 1
(3)
At this point, the equivalent currents are still
unknown. Their approximate solution is derived using
the method of moments. To this end, the equivalent
magnetic currents are expanded into a series of known
basis functions weighed by unknown coefficients.
N
V

1n M 1n
x, y 
n 1

M 2 x, y, z  d  
P

V2n M 2n x, y 

P
V
2n
  V



W2 m , H 2tott M 2 n 
P
V
2n
N
1n
 



W2 m , H 2tott M 1n
n 1
(7)
 



W2 m , H 2scat
M 2n
t
n 1
The total magnetic field in regions A and B can be
expressed as a superposition of the short-circuited

magnetic field H sc ( i.e., the incident magnetic field )

plus the specular magnetic field H scat , produced by the
equivalent surface currents. The tangential components
of the total magnetic field in the aperture planes are
then expressed as

M 1 x, y, z  0 
(6)
 
n 1
Fig. 3: Problem geometry decomposed into
three regions; A, B, and C.

n1
 

V1n W1m , H1scat
M 1n
t


W2 m , H 2sct 
-z
 
 
 

W1m , H1tot
t M 2n
where, a similar expression is derived on S 2
 B , B


2n
n1
-M2(x,y)
 
N
P
where the subscript t denotes the tangential fields. By
introducing N  P vector testing functions, equations
(6) and (7) provide N  P linear equations from which
the unknown coefficients V1n and V2 n can be obtained.
The first two inner products of equations (6) and (7) are
referred to as the aperture admittance matrices for
region C. These matrices relate the tangential magnetic
fields in the aperture to the equivalent currents, which
are equivalent to the tangential electric fields. The last
inner products appearing in equations (6) and (7) are
referred to as the interior aperture admittance matrix for
regions A and B respectively. In terms of the aperture
admittance matrix, the coupled equations (6) and (7) are
expressed as
  Y  V
  Y  V

 H 1sc   Y11C
  sc    C
 H 2   Y21
C
12
C
22
 
 Y A

 0
2n 
1n
0
YB
 V1n 
 
 V2 n 
 
(8)
The advantage of using this function is that the
admittance matrices of the interior and the exterior
regions can be constructed independently. The aperture
admittance matrices Y A  and Y B  are computed by
solving the problem of the equivalent magnetic currents
radiating into a half-space. The aperture admittance
matrix Y C  is computed by solving the interior cavity
problem for each equivalent magnetic current basis
function to compute in the aperture planes, and then
taking the inner products with the appropriate testing
functions. The following section describes the solution
of this interior problem using the FEM.
(4)
3
Interior Admittance Matrix - FEM
(5)
n 1
The approximate equivalent currents are substituted
into equation (3). Two sets of vector testing functions


are also introduced: W1m x, z  0 and W2m x, z  d  in
3.1
INTERIOR PROBLEM SOLUTION
Within the cavity region, the magnetic fields must
satisfy the vector Helmholtz equation
3
 
1
r


  H  k 2 r H  0
(9)

H
where
is the total magnetic field. The cavity region
is defined as a closed space confined by PEC walls,
which will be referred to as the domain V . A functional
described by the inner product of the vector wave

equation and the vectors testing function, H , is
expressed as
 


 

1
F H    H * .    H  k 2  r H * .H dV
r




(14)

k  
1 *
H e .nˆ    H e  j 0 H e* .M
r
is defined. The solution of the boundary-value problem
is then found variationally, by solving for the magnetic
field at a stationary point of the functional via the first
variation
 
 

Also
(10)

F H  0
 *

*   
 1
2
    H e .   H e  k0 r H e .H e dVe 

  r
 
 F H  



1 *

e 1 

ˆ

H
.
n



H
dS
e
e
    e

 r



Ne
0
implying contribution from only regions of implied
magnetic current sources, i.e, at the apertures only.

M = surface magnetic current source.
So that
Ne

 *

  
 1

 F H       H e .   H e  k 02  r H e* .H e dVe 
e 1 
 r


Nb
k  
   j 0 H b* .M n dS
 
(11)
The functional has a stationary point ( which occurs at
an extremum ) and equations (9) and (10) can be used to
derive a unique solution for the magnetic field. For an
arbitrary V the solution of the above variational
expression cannot be found analytically and is derived
via the FEM. To this end, the domain V is discretized
into a finite number of subdomains, or element
domains, Ve . It is assumed that each element domain
Ve has a finite volume and that the material profile
within this volume is constant. Within each element

domain the vector magnetic field is expressed as H e . If
there are N e element domains within V , the functional
can be expressed in terms of the approximate magnetic
fields as
Ne


  
 

1
F H     H e* .    H e  k 02  r H e* .H e dVe 
r
e 1 



 
 
Ne


  

1
F H     H e* .    H e  k 02  r H e* .H e dVe
r
e 1


(15)



0
b 1
(16)
The approximate vector field in each element domain is
represented as
m


H   ejWej
j 1
(17)

with the spacial vector Wej being the Whitney basis
function defined by

Wij  i  j   j i
(18)
and i is the barycentric function of node i expressed
as


i r  
 
1 r  rb . Ai

4
3Ve
(19)

(12)
where Ai is the inwardly directed vectorial area of the
Applying the vector identity
tetrahedron face opposite to node i , Ve the element


   1
 

  1
1
H e* .    H e  . H e*     H e      H e .   H e*
r


 r
  r




(13)
and the divergence theorem, (12 ) can be written as


volume, and r the position vector. Also, rb is the
position vector of the barycenter of the tetrahedron
defined as
   
 r0  r1  r2  r3 
(20)
rb 
4

in which ri is the position vector of node i .
Using equations (17) in (16) leads to:
4
Ne
m



  
m

1
F H e     ej*   ei     Wej .   Wei  k 2Wej .Wei  dVe 
e 1  j 1
i 1
 r


 

 k
   j 0
b 1
 0


4
 

 Wb .M n dS 
Nb
*
b
Now
(21)
Letting





  
1
N ji      Wej .   Wei  k 2Wej .Wei  dVe
 r

(22)
and
Pnb  j
k0
0
 
 Wb .M n dS
(23)
equation (21 ) becomes
Ne
m

m
 Nb
F H e    ej*  ei N ji    b* Pnb
e 1  j 1
i 1
 b 1
 
N eb   e   0 

N bb   b   Pnb 
 
 



[Y A ]  [W1m , H1scat
M n  ] NxN
t



[Y B ]  [ W2 m , H 2scat
M p  ] PxP
t



I i  [  W1m , H1sct ] N 1
4.1
(25)
(28)
MATRIX ELEMENTS EVALUATION
As explained in [12], following Galerkin procedure

( Tm  M m ), a typical matrix element for the rth region is
given by
r
Ymn
 2  M m , H t r (M n ) 


= 2
elements lying in the aperture plane S1 and S 2 .  e are
coefficients weighting the remaining edge elements in
V.
= 2
COMPUTING Y C 
(27)
The discretization of the MOM computational domain
is based on triangular patch modeling scheme as
proposed by Rao et al [9] and implemented by Konditi
and Sinha [ 6 ].
where :  b are coefficients weighting the edge
3.2
(26)
(24)
By enforcing the continuity of the magnetic field, a
global number scheme can be introduced.
The
following symmetric and highly sparse matrix results
 N ee
N
 be
Computing Exterior Admittance
Matrices- MOM

 Tm
M m  H t r ( M n ) ds 
 M
Tm
m
 H t r ( M n ) ds
 
  are then
(29)
Tm
 (
where the notation
The computation of the nth column of Y C in ( 6) and
(7) is performed by perturbing the cavity with an
equivalent current basis function M n . Then, with the
use of ( 21 ), the interior tangential magnetic fields in
the aperture planes S1 and S2 are computed. The first



M m  H t r ( M n ) ds 

) ds
for compactness.
In terms of the electric vector potential F (r ) and the
magnetic scalar potential  (r ) , the magnetic field
H r ( M n ) can be written as
N row elements of the nth column of Y C
H r (M n )   j Fn (r )  n (r )
computed by taking the inner product of the aperture
field on S1 successively with the N testing functions
where Fn (r ) =  r

Tm . The next P row elements of the nth column are
computed by taking the inner product of the aperture

field on S 2 with the P testing functions Tq . Similarly,
 
has been introduced
Tm
 G (r / r ' )  M
(30)
(31)
n ( r ' ) ds '
T
n (r ) 
  Fn (r )
 j  r  r
(32)
the remaining P columns of Y C can be computed by
perturbing the cavity with equivalent current basis
functions M p .
5
In Eqn. (31), G ( r / r ' ) denotes the dyadic Green's
function of the half space. Substitution of Eqn. (30)
in Eqn. (29) and use of two-dimensional divergence
theorem leads to
r
Ymn
  2 j
 M
m  Fn
ds  2
Tm
where
mm 
 m
m n
(33)
ds
Tm
 Mm
 j
(34)
Eqn. (33) contains quadruple integrals; a double
integral over the field triangles Tm and a double
integral over the source triangles Tn involved in the
computation of Fn (r ) and n (r ) . In order to reduce
the numerical computations, the integrals over Tm
can be approximated by the values of integrals at the
centroids of the triangles. This procedure yields
r
Ymn


 mc
 mc 
c
c
 j  Fn (rm )  2  Fn (rm )  2 
  2lm 



c
c
 n (rm )  n (rm )
where
Fn (rmc )   r
 G (r
Tn
n (rmc ) 
1
j  r
c





| r ' )  M n (r ' ) ds'
(35)
(36)

   G (r
c

| r ' )  M n (r ' ) ds'
(37)
Tn
In (35),  mc  are the local position vectors to the
centroids of Tm and rmc  = ( rm1  rm2   rm3 ) / 3 are the
position vectors of centroids of Tm with respect to
the global coordinate system.
Similarly, as explained in [6], using the
centroid approximation in Eqn. (29), an element of
excitation
vector
can
be
written
as

 c
 c
I mi   lm  H t i (rmc  )  m  H t i (rmc  )  m
2
2




regions on opposite sides of the conducting screen,
for the sake of computational simplicity, we consider
here two identical half spaces separated by an
aperture-perforated
conducting
screen.
The
formulation has first been validated (Figure 4) by
considering a rectangular aperture for which results
by Auckland and Harrington[8] are available in the
literature; the incident electric field amplitude is unity
at the center of S1 for the validation case. The results
are seen to be in good agreement. A y-polarized
plane wave (Hx) of unit amplitude has been assumed
to be incident normally on the apertures for the rest
of the results presented.
Thereafter, a number of aperture shapes have
been studied which include circular, cross and H. In
each case magnetic current distributions and
transmission coefficients were calculated. The
number of expansion functions employed was based
on the results of Konditi and Sinha [6].
It is observed that for very thin screens (0.001λ ), the
magnitudes and phases of the x-directed magnetic
currents on the upper (M1) and the lower(M2)
apertures were equal. As the thickness of the screen
is increased, the magnitudes of the magnetic current
M2 becomes progressively smaller than that of M1;
the phase of M2 becomes progressively larger than
that of M1. As the screen thickness increases the
transmission cross-section and the magnitudes of M1
and M2 decrease. Roughly the same trend is observed
in the case of circular, cross and H-shaped slots, only
the profile of the magnitudes and phases of the
magnetic currents differ.
(38)
Following the procedure as outlined in [6],
transmission coefficient and transmission crosssection are evaluated for the aperture problem.
5 Results and Discussion
Based on the preceding formulation, a general
computer program has been written which can be
used to analyze apertures of arbitrary shape in an
infinite conducting screen of finite thickness.
Although the formulation is valid for dissimilar
6
1.6
y
1.4
y
Aw
1.2
Aw
|Mx|
1.0
h
0.8
M2-This Method
-x
x
h
M1-This Method
0.6
x
M1-Method in Harrington [12]
0.4
M2-Method in Harrington [12]
0.2
0.0
0.2
0.4
0.6
0.8
L
L
0.0
1.0
-y
y/w
Fig. 5 : Equivalent Surface Magnetic Current Distributions for
a rectangular slot λ-wide by 5λ-long in a λ/4-thick Conducting Screen
Fig. 4 : Equivalent Surface Magnetic Current
Distributions for a rectangular slot λ-wide by 5λ-long
in a λ/4-thick Conducting Screen
Fig. 5 :H-shaped slot dimensions
Fig. 6 :Cross-shaped slot dimensions
14.0
90
14.0
90
14.0
90
14.0
90
12.0
12.0
60
60
12.0
12.0
60
60
10.0
10.0
0
Magnitude
0
M1, M2-Phase
-30
M2-Phase
4.0
M1-Phase
4.0
M1-Phase
-30
-30
2.0
2.0
M1
M2
-60
2.0
M1
M2
M1
M2
-60
2.0
M1
M2
0.0
0.0
0.2
0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
x/L
-90
0.0
-90
0.0
90
12.0
90
12.0
M2
M2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
x/L
-90
90
M1
M2
M1
M2
90
M2-Phase
60
10.0
60
10.0
60
30
8.0
30
60
M2-Phase
10.0
8.0
M2-Phase
30
0
6.0
0
M1-Phase
-30
M1-Phase
-30
8.0
30
Magnitude
x
|Mx| |M |
M2-Phase
x
Phase
Phase
of Mxof M
8.0
4.0
0.4
0.2
x/L
Magnitude
6.0
0.2
0.0
(b) t = 0.01
M1
Magnitude
10.0
0.0
(b) t = 0.01
M1
12.0
-60
-90
(a) t = 0.001
14.0
12.0
-60
(a) t =x/L
0.001
14.0
|Mx| |Mx|
M2-Phase
6.0
4.0
0.0
0
6.0
-30
M1, M2-Phase
4.0
0
6.0
0
Magnitude
6.0
x
x
PhasePhase
of M of M
6.0
30
Magnitude
8.0
x
x
PhasePhase
of M of M
Magnitude
Magnitude
8.0
x
|Mx| |M |
8.0
6.0
10.0
x
x
Phase
Phase
of M of M
30
8.0
x
| Mx| | M |
30
30
10.0
0
M1-Phase
4.0
-30
M1-Phase
4.0
-30
2.0
-60
4.0
-60
2.0
-60
2.0
0.0
-90
0.0
0.2
0.4
0.0
0.2
0.4
0.0
0.6
0.8
1.0
0.6
0.8
1.0
-90
x/L
x/L
(c) t = 0.1
(c) t = 0.1
2.0
-60
0.0
-90
0.0
0.2
0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
x/L
0.0
-90
x/L
(d) t = 0.25
(d) t = 0.25
Fig. 7 : x-directed Surface Magnetic Current Distributions for /20-wide by λ/2-long Rectangular
Fig. : x-directed
Surface
Magnetic
Currentt. Distributions for /20-wide by
Slots in Conducting
Screens
of various
thickness
λ/2-long
Slots in
a Conducting
Screen.
Fig. : Rectangular
x-directed Surface
Magnetic
Current
Distributions for /20-wide by
λ/2-long Rectangular Slots in a Conducting Screen.
7
40
1.4
40
1.4
M2-Phase
M2-Phase
35
1.2
35
1.2
M1-Phase
M1-Phase
30
30
1.0
1.0
15
0.8
Magnitude
x
|M |
x
|M|
20
Magnitude
0.6
25
Phase of M x(degrees)
25
0.8
20
0.6
15
0.4
0.4
10
10
0.2
M1
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
5
M2
0
0.0
0
0.0
-0.5
M1
0.2
5
M2
-0.5
0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
x/d
x/d
(a) t = 0.001
(b) t = 0.01
80
1.2
140
1.2
M2-Phase
70
1.0
120
1.0
M2-Phase
60
100
0.8
0.8
50
x
|M |
80
Magnitude
40
0.6
|Mx|
Magnitude
0.6
60
30
0.4
0.4
M1-Phase
40
20
0.2
M1-Phase
0.2
M2
M2
0
0.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
x/d
(c) t = 0.1
0.2
0.3
20
M1
10
M1
0.4
0.5
0
0.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
x/d
(d) t = 0.25
Fig. 3.4 : x-directed equivalent Surface Magnetic Current Distributions M x
at y/d = -0.0833 for Circular Apertures of diameter D/ = 0.5 in Conducting
various
thickness
Fig. 8 Screens
: x-directedof
equivalent
Surface
Magnetict.Current Distributions Mx at y/d = -0.0833 for Circular Apertures of
diameter D/ = 0.5 in
8
3.5
80
100
80
M2
Method in [6]
20
M1-Phase
1.5
M2-phase
Phase in [6]
0.5
-20
0.0
-40
0.1
0.2
0.3
0.4
60
M2
Method in [6]
x
2.0
1.5
0
1.0
M1-Phase
M2-phase
40
Phase in [6]
20
1.0
0
0.5
-20
0.0
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
0.5
-40
0.1
0.2
0.3
0.4
0.5
y/h
x/L
(b) x = 0.0, t = 0.25
(a) y/h = -0.1667, t = 0.25
4.0
4.0
160
40
M1
Method in [6]
M1-Phase 120
30
3.0
M2-phase
3.0
2.5
M1
M2
2.0
Method in [6] 10
M1-Phase
1.5
M2-phase
Phase in [6]
0
|Mx|
80
60
2.0
40
1.5
20
|M x|
1.0
1.0
0
-10
0.5
0.5
0.0
0.1
0.3
0.4
0.5
x/L
4.0
-40
0.1
0.2
0.3
0.4
0.5
y/h
(d) x = 0.0, t = 0.1
2
4.0
10
3.5
5
3.0
0
0
3.5
M1
M1-Phase
-4
M2-phase
Phase in [6]
|Mx|
2.5
-6
2.0
-8
-10
1.5
-12
M1
2.5
-5
M2
Method in [6]
2.0
-10
M1-Phase
M2-phase
Phase in [6]
1.5
1.0
-20
0.5
-25
|Mx|
1.0
-15
x
3.0
Phase of M (degrees)
M2
-2
Method in [6]
Phase of M (degrees)
(c) y/h = -0.1667, t = 0.1
0.2
-20
0.0
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
-20
0.0
x
-0.5 -0.4 -0.3 -0.2 -0.1
100
Phase in [6]
2.5
Phase of Mx(degrees)
20
140
M2
3.5
3.5
Phase of Mx(degrees)
0.0
2.5
|M |
40
M1
Phase of M x(degrees)
M1
2.0
-0.2 -0.1
3.5
60
2.5
-0.4 -0.3
120
3.0
3.0
-0.5
4.0
Phase of Mx(degrees)
100
| Mx|
4.0
-14
0.5
-16
0.0
0.0
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
-18
0.1
(e) y/h = -0.1667, t = 0.01 x/L
0.2
0.3
0.4
0.5
-0.5
-0.4
-0.3
-0.2
-0.1
(f) x = 0.0, t = 0.01
-30
0.0
0.1
0.2
0.3
0.4
0.5
y/h
Fig. 9 : Equivalent Surface Magnetic Current Distributions for Cross-shaped slots for arm width Aw = 2L/3 in Conducting
Screens of various thickness t..
Fig. 3.4 : Equivalent Surface Magnetic Current Distributions for Cross-shaped
slots for arm width Aw = 2L/3 in Conducting Screens of various thickness t.
9
200
| Mx|
1.2
1.4
160
1.2
140
M1
M1-Phase
120
1.0
M2-phase
100
100
x
Phase in [6]
0.6
Phase of M
0.8
50
0.4
0
0.2
0.8
80
0.6
60
M1
-50
0.4
M2
Method in [6]
40
0.2
M1-Phase
M2-phase
20
Phase in [6]
0.0
-0.3
-0.2
0.1
0.2
0.3
0.4
-0.5
-0.4
-0.3
-0.2
-0.1
0.5
0.0
0.1
0.2
0.3
0.4
0
0.5
y/h
x/L
(b) x/L = 0.3 for t = 0.25
(a) y/h = -0.1 for t = 0.25
1.2
M2
120
Method in [6]
1.0
160
1.2
140
100
M1-Phase
120
1.0
Phase in [6]
60
0.8
Phase of Mx
80
|Mx|
M2-phase
0.8
1.4
140
M1
0.6
100
80
40
0.6
60
20
0.4
0.2
M1-Phase
|Mx|
-40
-0.5
-0.4
-0.3
-0.2
0.0
-0.1 0.0
(c) y/h = -0.1 for t = 0.1
0.2
0.3
0.4
-0.5
0.5
-0.4
-0.3
-0.2
(d) x/L = 0.3 for t = 0.1
120
M1
1.0
M2
Method in [6]
100
M1-Phase
M2-phase
80
Phase in [6]
0.1
0.2
0.3
0.4
0
0.5
y/h
1.4
70
1.2
60
1.0
50
0.8
40
60
40
0.6
20
0.6
30
M1
0
0.4
-20
Method in [6]
M1-Phase
0.2
0.2
20
M2
|Mx|
0.4
Phase of M
x
|Mx|
0.8
Phase in [6]
0.0
-0.1 0.0
x/L
1.2
20
M2-phase
-60
0.1
40
M2
Method in [6]
-20
0.2
M1
0.4
0
10
M2-phase
-40
Phase in [6]
0.0
0.0
-0.5
-0.4
-0.3
-0.2
-0.1
Phase of Mx
-0.4
-100
-60
0.0
(e) y/h = -0.1 for t = 0.01
0.1
x/L
0.2
0.3
0.4
0.5
Phase of Mx
-0.5
0.0
-0.1 0.0
x
150
Method in [6]
|Mx|
1.0
Phase of M
M2
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.0
(f) x/L = 0.3 for t = 0.01
0.1
0.2
0.3
0.4
0.5
y/h
Fig.
3.4 : Equivalent
SurfaceCurrent
Magnetic
Current for
Distributions
forfor
H-shaped
Fig. 10
: Equivalent
Surface Magnetic
Distributions
H-shaped slots
arm width Aw = 0.4 in Conducting
Screens
of various
thickness
t..w = 0.4 in Conducting Screens of various thickness t.
slots
for arm
width A
10
0.40
1.4
 Lh = 2/3
t = 0.0λ, Method in [6]
0.35
1.2
t = 0.25λ
t = 0.1λ
t = 0.01λ
0.30
xx/2
1.0
t = 0.001λ
0.25
t = 0.0λ, Method in [6]
t = 0.25λ
0.8
t = 0.1λ
0.20
xx/
2
0.15
y/2
0.4
0.2
-60
-30
0.10
0.05
y/2
0.00
0.0
-90
t = 0.01λ
t = 0.001λ
0.6
0
30
60
-90.0
90
-60.0
-30.0
0.0
30.0
60.0
90.0
Angle(degrees)
Angle(degrees)
2 for
Fig. 11 : Transmission Cross-sections τθy/λ2 and2τxx/λ2 for 2
Fig.
12 : :Transmission
Cross-sections
τθy/λ2 and
τxx2/λand
Fig.
Transmission
Cross-sections
y/
xx/ 2 for/20Fig.
:
Transmission
Cross-sections

/
and

/
forCross-shaped
y
xx
Cross-shaped slots, L = h = 2/3, in Conducting Screens of
λ/20-wide
by
/2-long
Rectangular
Slots
in
conducting
screens
wide
by λ/2-long Rectangular Slots in Conducting Screens of
slots, thickness
L = h = 2/3,
in Conducting Screens of various thickness
t.
various
t.
of various
thickness t.
v
a
r
i o
u
s
t h
i c
k
n
e
3.5
0.7
t = 0.25λ
 Lh = 
t = 0.1λ
t = 0.1λ
t = 0.01λ
3.0
0.6
t = 0.001λ
t = 0.001λ
xx/
2.5
2
0.5
2.0
0.4
1.5
0.3
1.0
0.2
0.5
-60
-30
y/2
0.0
0
30
60
-90
90
-60
: Transmission Cross-sections
-30
0
30
60
90
Angle(degrees)
Angle(degrees)
Fig. 13
xx/2
0.1
y/2
0.0
-90
t .
t = 0.0λ, Method in [6]
t = 0.0λ, Method in [6]
t = 0.01λ
s
0.8
4.0
t = 0.25λ
s
τθy/λ2
and
τxx/λ2
for
Fig. 14
: Transmission Cross-sections τθy/λ2 and
τxx/λ2 for2
2
2
Fig.
:slots
Transmission
Cross-sections
yConducting
/ and xx/ forCircular
Fig.
: Transmission
y=/0.4
andinxx/2 forlong H-shaped
of
 long
H-shaped slotsCross-sections
of arm width Aw
Circular
Apertures
of diameter
D/ = 0.5 in
arm
width AwScreens
= 0.4 in
Screens
Conducting
of Conducting
various thickness
t. of various thickness t.Apertures
Screens of of
various
thickness
diameter
D/ t.= 0.5 in Conducting Screens of various
thickness t.
11
6
Conclusion
In this paper, a numerical study based on a hybrid
FEM/MOM formulation for aperture problems has been
presented for apertures of arbitrary shape in a thick
conducting screen of infinite extent separating two half
spaces. The cavity domain has been discretized into
tetrahedral elements which on the apertures resulted in
triangular patch modeling with appropriately defined basis
functions.
Validation of computer code has been achieved
by using data available in the literature for rectangular
apertures and for circular, cross-shaped, and H-shaped
apertures in thin conducting screens Thereafter new data
has been generated for circular, cross-shaped, and Hshaped apertures in thick conducting screens which to our
knowledge do not exist in the open literature. Some of the
parameters computed include transmission cross-section
and equivalent magnetic currents.
In all the problems investigated, transmission
cross-section and magnetic currents magnitudes decrease
with increase in thickness of the conducting screen.
[9] Rao, S.M., Wilton, D.R. and Glisson, A.W.:
"Electromagnetic scattering by surfaces of arbitrary
shape," IEEE Trans. Antennas Propagat., Vol. AP-30,
pp.409-418, 1982
[10] Collin, R.E.: Field Theory of Guided Waves, New
York: McGraw-Hill, 1960
[11] Harrington, R.F.: Time-harmonic Electromagnetic
Fields, New York : McGraw-Hill, 1961
[12] N. Amitay. V. Galindo. and C. P. Wu, Theory and
Analysis of Phased Array Antennas. New York:
Wiley-Interscience, 1972
[13] W. F. Croswell. "Antennas and wave propagation,"
Electronics Engineers' Handbook, D. G. Fink and A.
A. McKenzie, Eds. New York: McGraw-Hill, 1975,
pp. 18-26.
References:
[1] R.F. Harrington, J.R. Mautz, and D.T. Auckland:
"Electromagnetic coupling through apertures," Dept.
Elec. Comput. Eng., Syracuse Univ., Syracuse NY,
Rep. TR-81-4, Aug. 1981.
[2] C. M. Butler. Y. Rahmat-Samii. and R. Mittra.
"Electromagnetic penetration through apertures in
conducting surfaces," IEEE Trans. Electromagn. . vol.
EMC-20. pp. 82-93, Feb. 1978.
[3] Xingchao Yuan, “Three-dimensional electromagnetic
scattering from inhomogeneous objects by the hybrid
moment and the finite element method”, IEEE Trans
Microwave Theory Tech., Vol. MTT-38, pp. 10531058, 1990.
[4] Harrington, R.F. and Mautz, J.R. : "A generalized
network formulation for aperture problems," IEEE
Trans. Antennas Propagat., Vol. AP-24, pp. 870-873,
1976
[5] S. K. Jeng, "Scattering from a cavity-backed slit in a
ground plane TE case," IEEE Tram. Antennas
Propagat., vol. 38, pp. 1523-1529, 9, Oct. 1990.
[6] Konditi, D.B.O. and Sinha, S. N. : “Electromagnetic
Transmission through Apertures of Arbitrary Shape in
a Conducting Screen,” IETE Technical Review, vol.
18, Nos. 2&3, March-June 2001, pp. 177-190.
[7] Mautz, J.R. and Harington, R.F.: "Electromagnetic
transmission through a rectangular aperture in a
perfectly conducting plane," Tech. Rept. TR-76-1,
Electrical and Computer Engg. Deptt., Syracuse Univ.,
New York, 13210, 1976.
[8] David T. Auckland and Harington, R.F.:
"Electromagnetic transmission through a filled slit in a
conducting plane of finite thickness, TE case," IEEE
Microwave Theory Tech., Vol.26, 499-505, July 1978
12