Analysis of Gigahertz TEM Cell using Finite-Difference Time-Domain(FDTD) Method

Abstract

In this paper, an analysis of Gigahertz TEM cell using finite-difference time-domain method (FDTD) is described. Some facts about the implementation of the algorithm are presented. Finally, the obtained results are discussed and validated.

Full text

Page 1

ANALYSIS OF GIGAHERTZ TEM CELL USING
FINITE-DIFFERENCE TIME-DOMAIN (FDTD) METHOD

Ondřej Franek, Vlastimil Navrátil, Zbyněk Raida
Dept. of Radio Electronics, Brno University of Technology,
Purkyňova 118, 612 00 Brno, phone: +420 5 4114 9140,
e-mail: franek@feec.vutbr.cz

Abstract. In this paper, an analysis of Gigahertz TEM cell using finite-difference time-domain
method (FDTD) is described. Some facts about the implementation of the algorithm are
presented. Finally, the obtained results are discussed and validated.
Introduction
Recently, every newly developed electronic equipment has to satisfy certain conditions of
electromagnetic compatibility, as the electromagnetic interference increases. One of these
conditions is low susceptibility to the incident electromagnetic field. The transversal
electromagnetic (TEM) cells are constructed to generate homogeneous electromagnetic fields,
in which the designed equipment is tested for susceptibility. The Gigahertz TEM (GTEM)
cells are modified TEM cells up to range of several gigahertz. When designing the GTEM
cell, preliminary numerical analysis is usually needed. Here, the homogeneity of the
generated field is primarily observed.
The finite-difference time-domain (FDTD) method was chosen for analysis of the designed
GTEM cell. This method is one of the most exploited methods for wideband modeling of
microwave structures. Both the implementation of the method and the interpretation of the
results are straightforward.
The FDTD Method
The method originates directly from the Maxwell’s equations, which are modified into
discrete form by differencing. Three components of electric intensity E and three components
of magnetic intensity H are involved. These six components are distributed in space according
to the Yee grid [1]. This means that the field component can be evaluated on the basis of the
component preceding in time and the surrounding components, in every time step [2]:
(
kji
kji
kji
kji
kji
,,
,,
,,
,,
,,
(
i
z
kji
kji
kji
kji
/ 1
,,
,,
,,
,,
(
ji
x
kji
kji
kji
kji
/ 1,
,,
,,
,,
,,
+
(
, 2/ 1,
,,
,,
,,
,,
+
kji
kji
kji
kji
kji
(
/ 1,,
,,
,,
,,
,,
kji
kji
kji
kji
kji
(
,, 2/ 1
,,
,,
,,
,,
+
kji
kji
kji
kji
kji
)
)
)
n
kji
z
n
kji
z
n
kji
y
n
yHb
n
xHa
n
x
EEEEDHDH
xx
, 2/ 1
+
,, 2/ 1
−
,
2/ 1
−
,,2/ 1
+
,
2/ 1
−
,
2/ 1
+
−+−+=
(1a)
n
kji
x
n
kji
x
n
kji
z
n
kj
Hb
n
yHa
n
y
EEEEDHDH
yy
2/ 1
+
,,2/ 1
−
,,,, 2/ 1
−
,, 2
,
2/ 1
−
,
2/ 1
+
+
−+−+=
(1b)
n
kji
y
n
kji
y
n
kji
x
n
k
Hb
n
zHa
n
z
EEEEDHDH
zz
,, 2/ 1
+
,, 2/ 1
−
, 2/ 1
−
,, 2
,
2/ 1
−
,
2/ 1
+
−+−+=
(1c)
)
)
)
2/ 1
+
2/ 1
+
,,
2/ 1
+
2/ 1
−
,,
2/ 1
+
−
j
, 2/ 1,
2/ 1
+
,
,
1
+
−+−+=
n
kji
y
n
kji
y
n
ki
z
n
zEb
n
xEa
n
x
HHHHCECE
xx
(2a)
2/ 1
+
+
,, 2 / 1
2/ 1
+
−
,, 2/ 1
2
−
/ 1
+
2/ 1,,
2
+
/ 1
+
2
,
,
1
+
−+−+=
n
kji
z
n
kji
z
n
kji
x
n
xEb
n
yEa
n
y
HHHHCECE
yy
(2b)
2/ 1
+
+
j, 2/ 1,
2/ 1
+
−
j
, 2/ 1,
2/ 1
+
,, 2/ 1
−
2/ 1
+
,
,
1
+
−+−+=
n
ki
x
n
ki
x
n
kji
y
n
yEb
n
zEa
n
z
HHHHCECE
zz
(2c)

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The material constants Ca, Cb, Da, Db depend on permittivity, permeability and conductivity
of the media. The indices i, j, k are three-dimensional Cartesian coordinates and n is a number
of the time step.
In case the lattice is cubic, the stability condition

2
≥
∆
∆
tc
(3)
applies [2]. Here, ∆ is the dimension of elementary cell, or the space increment, ∆ is the time
increment and c is the velocity of light.
t
The equations (1) and (2) form a cycle producing the field values in time. The cycle has to be
repeated until the excitation of the structure diminishes. Moreover, the equation
1
=∆∆
t
Nft (4)
must be satisfied, where N is the total number of time steps and ∆ is the frequency
resolution in the spectrum of the response.
t
f
The GTEM Cell Model
The shape and dimensions of the analyzed GTEM cell are shown in Fig. 1. Here, L is the cell
length, A is the width and at the same time height of the cell (cross-section of the cell is
square), B is the distance from the upper wall to the septum (i. e. the inner conductor) and
W is the width of the septum. The dimension P represents the depth of the PML layer (see
below) and R are the positions of field cross-sections from the back wall.

P
L
R
W
A
A
B

PML
z
x
z
y

Fig. 1. Proportions of the analyzed GTEM cell
The excitation pulse enters the structure at the vertex of the pyramid, propagates through the
body towards the back wall and vanishes in the absorbing layer. The homogeneous field
occurs in the area under the septum, at the position indicated by the dashed lines. Actually,
these lines define the cross sections, which should be obtained from the simulation.
The Implementation
Modeling the designed GTEM cell, the orthogonal FDTD lattice was employed for simplicity.
Therefore, the cell walls had to be approximated by stairstepped surfaces of perfect electric

Page 3

conductor (PEC). The solution accuracy was determined by this fact to a great extent. This
type of boundary also brings some difficulties at the feeding point. A lumped resistive source
with R
, connected between the septum and the cell shield, was utilized here, instead
of more complex microwave port. The stairstepping is also the cause of cutting the cell at the
vertex, as the source cannot be positioned in the narrowest part of the cell. The length of the
model is then slightly smaller than the real length.
Ω= 50
s
Further, the real absorbers on the back wall of the GTEM cell are replaced by the artificial
perfectly matched layer (PML). This anisotropic material is nearly ideally reflectionless in
wide range of frequencies, so the septum need not be terminated by the 50
necessary with real GTEM cells). The conductivity of the implemented PML has a quadratic
dependence.
load (which is
Ω
The modeled structure is excited by pulse, whose frequency range extends from zero to the
maximum required frequency. This wideband pulse is weighted by Gaussian window in time
domain in order that the input is causal.
Concerning the output, we obtain the time behavior of the field components (e. g. electric
intensity E) in the given cross-sections. This time response is Fourier transformed into
frequency domain, where particular frequencies can be processed.
The Results
The FDTD algorithm was implemented in MATLAB v6.1 environment. The computation was
run with parameters presented in Tab. 1 and took 3 hrs 8 min 45 sec on Athlon 1.2 GHz,
256 MB RAM.
Tab. 1. Computation parameters
A [m]
0.65
B [m]
0.22
L [m]
1.3
P [m]
0.25
R [m]W [m] ∆ [m]
0.56
∆f [GHz]
0.01
f [GHz]
(0.4, 0.5, 0.6)0.01(0.1, 0.5, 1, 5, 10)

The specified space increment and frequency resolution lead in 5996 time steps. The ratio
was used with respect to sufficient stability. The retrieved cross-sections were
limited to the rectangle with dimensions 80 × 80 mm, 50 mm far from the bottom wall.
2/
=∆∆
tc
The Cartesian components of the electric field E ,
cross-sections of the GTEM cell. Introducing similar phase angles they could be added up
into the module

x
EE
+=
, were expressed in the obtained
x
y
E
z
E
2
z
2
y
2
EE
+
. (5)
For every desired cross-section and every desired frequency the maximum and minimum
values were determined. The ratio of maximum and minimum values in dB can be seen in
Tab. 2.
Tab. 2. Ratio of maximum and minimum values of electric intensity module
f = 100 MHzf = 500 MHzf = 1 GHzf = 5 GHzf = 10 GHz
0.4 m
0.5 m
0.6 m
4.1
3.3
2.9
4.6
3.4
2.1
3.3
3.3
4.5
4.6
2.8
2.0
5.0
2.9
2.0
section
Emax/Emin [dB]

Page 4

Conclusion
The absolute value of z-component of electric field in cross-section with R
is shown in Fig. 2. The z-component has a small, nonzero, quite uniform
magnitude in the vicinity of the bottom wall, where the tested object will be placed.
Considering the homogeneity of the field, the optimal location of the object appears to be
farther from the septum, as the near field is deformed due to the septum edges.
and
m 4 . 0
=
GHz 1
=
f
Concerning Tab. 2, the homogeneity of the field is unstable across the parameters, which can
be explained by the inaccuracy of the stairstepping FDTD model. Hence, the current model is
planned to be replaced with the contour-path FDTD technique [2] in the future. Refining the
mesh is also expected to improve the performance especially on the high frequencies,
although the computer burden increases significantly in such cases.


Fig. 2. The distribution of the z-component of the electric field, for R = 40 cm, f = 1 GHz
Acknowledgement
Research described in the paper was financially supported by the research programs
MSM 262200011 and MSM 262200022, by the grants of the Czech Grant Agency
no. 102/01/0571 and 102/01/0573, and by the grant of the Czech Ministry of Education
no. 1953/2002.
References
[1] YEE, K. S. Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s
Equations in Isotropic Media. In IEEE Transactions on Antennas and Propagation,
vol. 14, No. 3, May 1966, pp. 302–307.
[2] TAFLOVE, A. Computational Electrodynamics: The Finite-Difference Time-Domain
Method. Boston: Artech House, 1995.