Thin sheets of resistive or dielectric material are commonly encountered in radar cross section calculations. Analysis of such sheets is simplified by using sheet impedances. It is shown that sheet impedances can be modeled easily and accurately using Finite Difference Time Domain (FDTD) methods. These sheets are characterized by a discontinuity in the tangential magnetic field on either side of the sheet but no discontinuity in tangential electric field. This continuity, or single valued behavior of the electric field, allows the sheet current to be expressed in terms of an impedance multiplying this electric field

Kunz, Karl

Luebbers, Raymond

English

NASA Technical Reports Server

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FDTD Modeling
of
Thin Impedance Sheets
i
/
by
Raymond Luebbers and Karl Kunz
Department of Electrical Engineering
The Pennsylvania State University
University Park, PA 16802
(814) 865-2362
Abstract
Thin sheets of resistive or dielectric material are commonly
encountered in radar cross section calculations. Analysis of
such sheets is simplified by using sheet impedances. In this
paper it is shown that sheet impedances can be modeled easily and
accurately using Finite Difference Time Domain (FDTD) methods.
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requires the impedance sheet to be small compared with the free
space wavelength. In most FDTD calculations the FDTD cell size
(Yee [3] cells are used here) must be on the order of i/i0
wavelength or less for reasonably accurate results. Scattering
from an infinitesimally thin perfectly conducting plate was has
been calculated by approximating the plate as being one FDTD cell
thick with good results [4]. If it is assumed that the same
approach can be applied to infinitesimally thin impedance sheets,
then the plate thickness T in (i) merely becomes the thickness of
the FDTD cell, and the conductivity and/or relative permittivity
to be used in the FDTD calculations are merely adjusted in
accordance with (I) to give the desired sheet impedance. Note
that the FDTD cell dimension need not correspond to the thickness
of the actual physical plate. The FDTD cell thickness is used
only to determine the conductivity and relative permittivity of
the FDTD electric field location so that the desired sheet
impedance is approximated. Note also that, even if the
wavelength in the material forming the impedance sheet is much
smaller than a free space wavelength, the FDTD cell size need not
be correspondingly reduced.
Demonstration
The first demonstration will consist of calculating the far
zone backscatter from a 29 x 29 cm flat plate of sheet impedance
Zs = 500 n. The FDTD calculations will use cubical Yee cells
with 1 cm edges. Using T = 1 cm, the corresponding FDTD
conductivity is a = 0.2 S/m. The FDTD calculations shown in
Figures 1-8 are all made with the plate modeled by setting the
conductivity to 0.2 S/m for x and y polarized electric field
locations corresponding to single z dimension index over a range
of x and y dimension indices to model the plate. The FDTD
approach used and the transformation to the far zone is described
in [4]. The problem space size, orientation and position of the
plate, incident Gaussian pulse plane wave, and time step size are
also consistent with those in [4].
Figure 1 shows the far zone backscattered electric field for
a Gaussian pulsed plane wave normally incident on the plate. In
Figure 2 this result is Fourier transformed, converted to cross
section, and compared with results using the Method of Moments
[2]. The agreement is quite good, with the approximately 20 dB
reduction in radar cross section relative to a perfectly
conducting plate of the same size [4] consistently predicted by
both methods.
In Figures 3-8 the same plate geometry and composition is
considered but for non-normal incidence. The plate is
perpendicular to the z axis, with edges parallel to the x and y
axes, and the plane wave is incident from 8=45, _=30 degrees.
Figures 3-5 show the co-polarized backscatter far zone electric
field for _ and 8 polarizations and the cross-polarized
backscatter as well. In Figures 6-8 these time domain results
are Fourier transformed and converted to radar cross section for
comparison with Moment Method [2] results. Again the agreement
is quite good, except at the highest frequencies considered.
These results indicate that perhaps 12 cells/wavelength are
required for good accuracy for off-normal incidence. Comparing
the results in Figure 6 with those in Figure 5 of [4], it is
clear that changing from a perfectly to a finitely conducting
plate changes the scattering level and frequency behavior, and
that the FDTD and Moment Method results agree quite well on these
effects.
In Figure 9 both FDTD and Moment Method [4] results for
scattering by a plate with a complex sheet impedance are shown.
The sheet impedance is determined by applying eqs. (1,2) with
conductivity 0.25 S/m, relative permittivity 3.0, and thickness 1
cm., corresponding to the FDTD parameters used. Again the plane
wave is a Gaussian pulse incident from 8=45, _=30 degrees. The
FDTD results agree with the Moment Results for frequencies up to
about 12 cells/wavelength.
The final result is for a plate with edge treatment. For
this demonstration a 21 x 21 cm thin perfectly conducting plate
is given a 4 cm border of sheet impedance Zs = 500 n, resulting
in a square plate 29 x 29 cm. This edged plate is modeled in
FDTD by setting x and y polarized electric field locations for a
single z dimension index as being either perfect conductor for
the central portion of the plate or with a conductivity of 0.2
S/m for the edges. The ESP4 calculations were made with a
central perfectly conducting plate surrounded by 4 plates of
sheet impedance Zs = 500 _ attached to the central plate using
overlap modes. The results are compared in Figure i0 with
excellent agreement between the two methods, both showing a
significant difference due to the edge treatment when compared
with the results of Figure 6.
Conclusions
The ability of the FDTD method to easily and accurately
model scattering by sheet impedances was demonstrated by
comparing FDTD results for scattering from flat plates modeled
using sheet impedances with Method of Moment results. The
approach described here is directly applicable to the Yee cell,
and demonstrated good accuracy for frequencies up to
approximately 12 cells per wavelength.
5
References
i.
.
,
•
T B. A. Senior,• "Approximate Boundary Conditions," IEEE
Transactions on Antennas and Propaqation, vol. AP-29, pp
826-829, September 1981.
E. Newman, "A user's manual for the electromagnetic surface
patch code: ESP Version IV," The Ohio State University
Research Foundation, ElectroScience Laboratory, Department
of Electrical Engineering, Columbus, Ohio 43212.
K. S. Yee, "Numerical solution of initial boundary value
problems involving Maxwell's equations in isotropic media,"
IEEE Transactions on Antennas and Propaqation, vol. AP-14,
pp 302-307, May 1966.
R. Luebbers, K. Kunz, M. Schneider, F. Hunsberger,
Difference Time Domain Near-Zone to Far-Zone
Transformation", IEEE Transactions on Antennas and
Propaqation, vol. AP-39, pp 429-433, April 1991.
"A Finite
Fiqure Titles
i • Co-Polarized far zone electric field vs time scattered by a
29 x 29 cm flat plate of sheet impedance 500 ohms for a 8-
polarized normally incident Gaussian pulse plane wave
computed using FDTD.
• Radar cross section for a 29 x 29 cm flat plate of sheet
impedance 500 ohms, normal incidence, obtained from FDTD
results of Figure 1 compared with Moment Method [2] results.
• Co-Polarized far zone electric field vs time scattered by a
29 x 29 cm flat plate of sheet impedance 500 ohms for a _-
polarized incident Gaussian pulse plane wave from 8=45, _=30
degrees computed using FDTD.
• Co-Polarized far zone electric field vs time scattered by a
29 x 29 cm flat plate of sheet impedance 500 ohms for a 8-
polarized incident Gaussian pulse plane wave from 8=45, _=30
degrees computed using FDTD.
So Cross-Polarized far zone electric field vs time scattered by
a 29 x 29 cm flat plate of sheet impedance 500 ohms for a _-
polarized incident Gaussian pulse plane wave from 8=45, _=30
degrees computed using FDTD.
• Co-Polarized radar cross section for a 29 x 29 cm flat plate
of sheet impedance 500 ohms, 8=45, _=30 degree incidence, _-
polarized, obtained from FDTD results of Figure 3 compared
with Moment Method [2] results.
. Co-Polarized radar cross section for a 29 x 29 cm flat plate
of sheet impedance 500 ohms, 8=45, _=30 degree incidence, 8-
polarized, obtained from FDTD results of Figure 4 compared
with Moment Method [2] results.
• Cross-Polarized radar cross section for a 29 x 29 cm flat
plate of sheet impedance 500 ohms, 8=45, _=30 degree
incidence, obtained from FDTD results of Figure 5 compared
with Moment Method [2] results.
• Co-Polarized radar cross section for a 29 x 29 cm flat plate
of sheet impedance corresponding to conductivity of 0.25,
relative permittivity of 3.0, and thickness 1 cm, for 8=45,
_=30 degree _-polarized incident plane wave calculated using
FDTD and compared with Method of Moments [2].
i0. Co-Polarized radar cross section for a 21 x 21 cm perfectly
conducting flat plate with a 4 cm 500 ohm edge treatment on
all sides (total plate size 29 x 29 cm) for 8=45, _=30
degree _-polarized incident plane wave calculated using FDTD
and compared with Method of Moments [2].
7
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FDTD modeling of thin impedance sheets

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