Coupling to Wires in Cavity Enclosure using Iterative Algorithm
T. Yang1 and J. L. Volakis1,2
1
Radiation Lab., EECS Dept., The University of Michigan, Ann Arbor, MI 48109
2
ElectroScience Lab., The Ohio State University, Columbus, OH 43212
{taesiky, volakis}@eecs.umich.edu
Introduction
Electromagnetic field coupling to a wire placed inside a cavity enclosure through apertures has
been studied by various numerical methods such as finite element method (FEM) [1], and
method of moment (MoM) [2, 3]. Work done before is mainly limited to analysis of a small hole
in cavity. In this paper, we deal with a cavity having apertures of wavelength comparable size.
This problem can be analyzed using frequency domain integral equation via MoM. However
this is a CPU intensive method and moreover it leads to ill-conditioned matrices. Thus, an
alternative approach is proposed that decomposes the problem in different computational
components which are integrated for a final solution. Aperture coupling in cavity structure is
modeled by cavity modal Green’s function and wire is analyzed by usual free space Green’s
function. Then the interactions between decomposed structures can be handled by iterative
coupling approach. In this paper we use the electric field shielding (EFS) defined as the ratio of
total electric field measured in the presence of the structure and in the absence of the structure,
respectively. To validate our method we compare it with the well-validated full wave multilevel
fast multipole moment method (MLFMM) [4, 5].
Aperture coupling
We begin by applying the surface equivalence principle where aperture fields are replaced by
equivalent magnetic currents over the aperture.
M  E  zˆ
where M is equivalent magnetic currents and E is aperture field. These currents act as equivalent
sources for both interior and exterior of the cavity. Inside the cavity, the modal Green’s function
is employed to account for the mode field interactions. For the cavity exterior, an infinite ground
plane is assumed and the half free space dyadic Green’s function is applied. This is an
approximation because it does not include the effects from the far away exterior edges of the
cavity. However, such effects are of little importance since the slot fields are mostly controlled
by local phenomena (slot resonance and cavity resonances). Fig. 1. demonstrates the validity of
the assumption. By enforcing continuity of tangential magnetic fields across the aperture,
coupled integral equations for the aperture fields are formulated.
zˆ  [ H a ( M )  H i ]  zˆ  H b (M )
where Hi is the incident field, Ha is the external field radiated by M, and Hb is the internal field
due to -M. The unknown aperture currents are represented by rooftop basis functions, and by
means of Galerkin’s method, the system of integral equations is reduced to an admittance matrix
system whose solution gives the aperture fields. More details are available in [6].
Iterative field coupling
Consider a rectangular cavity with wire inside under external plane wave illumination in fig. 3.
The wire is modeled by free space Green’s function whereas the cavity structure is analyzed
using cavity modal Green’s function. On the wire zero tangential electric field is applied and
over the aperture the continuity of tangential magnetic field is enforced. The coupling between
the wire and cavity structure is obtained by iterative algorithm. At the first iteration, field is
computed on the location of the wire inside the empty cavity in the absence of the wire (Fig. 2.
(a)). Then the field induces electric current on the wire and generate scattered field from wire in
free space (Fig. 2. (b)). At the next iteration, both incident field from the exterior and radiated
field from the internal wire excite the aperture and the internally scattered field from the
aperture induces again the electric current on the wire. This iteration procedure is repeated till
the convergence of the electric current on the wire is achieved.
Validation
To validate the iterative method, we observe the convergence of electric current on the wire in
terms of iteration for the cavity in fig. 3. As can be seen in fig. 4 the current value converges
rapidly in 9th iteration at 0.7 GHz and 0.8 GHz, respectively. Fig. 5 shows comparison validation
with MLFMM. The EFS is measured at (15, 15, -10) for the cavity in fig. 3. The result using
iterative method is in good agreement with reference data in broad frequency range. Each
iteration takes just a few seconds and it is rapidly convergent typically in 10 iterations. Hence
overall computation time can be reduced a lot compared to full wave method.
References
[1] W. P. Carpes, Jr., L. Pichon, and A. Razek, "Analysis of the coupling of an incident wave
with a wire inside a cavity using an FEM in frequency and time domains," Electromagnetic
Compatibility, IEEE Transactions on, vol. 44, pp. 470-475, 2002.
[2] D. B. Seidel, "Aperture Excitation of a Wire in a Rectangular Cavity," Microwave Theory
and Techniques, IEEE Transactions on, vol. 26, pp. 908-914, 1978.
[3] D.Lecointe, W.Tabbara, and J. L. Lasserre, “Aperture coupling of electromagnetic energy to
a wire inside a rectangular metallic cavity,” in Dig. IEEE AP-S Antennas Propagat. Soc. Int.
Symp., vol. 3, 1992, pp. 1571-1574.
[4] K. Sertel and J.L. Volakis, "Incomplete ILU Preconditioner for Fast Multipole Method
(FMM)", Microwave and Optical Tech. Letters, Vol. 28, pp. 265-267, Aug. 20, 2000.
[5] K. Sertel and J.L. Volakis, “Multilevel Fast Multipole Method Implementation Using
Parametric Surface Modeling”, IEEE A P-S Conference Digest, Vol.4, pp. 1852-1855, Utah,
2000.
[6] E. Siah, T. Yang, Y. Erdemli, J. Volakis, and V. Liepa, "6th quarterly GM report on EMC
studies", University of Michigan, May 2002
open cavity
30
MODAL
MLFMM
25
20
15
EFS (dB)
10
y
5
30 cm
30 cm
E
0
-5
z
-10
k
x
-15
-20
0.2
0.4
0.6
0.8
1
1.2
Freq (GHz)
1.4
1.6
1.8
2
Fig. 1. EFS measured in the middle of open cavity : modal solution vs. MLFMM
(a)
(b)
Fig. 2. (a) contribution from the structure (b) contribution from the wire
12 cm
y
Cavity : 30x30x20 (cm)
Aperture : 20x3 (cm)
Wire length : 20 (cm)
Wire location : x = 15, z = -12, y = 5~25
Excitation : normal incident plane wave
Ey
kz
z
0
x
Fig. 3. Set-up for cavity with wire
-3
8
-3
0.7 GHz
x 10
7
0th
1st
2nd
7th
8th
9th
7
6
0.8 GHz
x 10
0th
1st
2nd
7th
8th
9th
6
5
5
|I| (A)
3
3
2
2
1
1
0
0
2
4
6
8
10
12
wire (cm)
14
16
18
20
0
0
2
4
6
8
10
12
wire (cm)
14
16
18
Fig. 4. Convergence of current on the wire with iterations
50
Iterative
MLFMM
40
30
20
EFS (dB)
|I| (A)
4
4
10
0
-10
-20
-30
0.2
0.4
0.6
0.8
1
1.2
Freq (GHz)
1.4
1.6
1.8
Fig. 5. Validation of Iterative method with. MLFMM using EFS
2
20