In this section, the scattering of a plane electromagnetic wave by a 3-D perfectly conducting body with a dielectric material coating is studied by the parametric method of moments described in the previous chapter with an impedance boundary condition. The problem can be formulated in terms of various integral equations derived from the Leontovich [58] impedance boundary condition (IBC). There are many papers on IBC [63][62][61][60][59]. This approximation makes integral equation formulations of the problem nearly as simple as those for perfectly conducting bodies without coating. When using IBC, one can take advantage of the derivation in the previous chapter for using IBC.
Let S denote the closed surface of a three-dimensional perfectly conducting
body with an infinitely thin layer of lossy dielectric material coating shown
in Figure 3.1 where
is the outward unit vector normal on
and
(
) are incident fields which are produced by some
impressed sources
in the absence of the
scatterer. The material coating on the perfectly conducting body is
characterized
by a pair of complex parameters
, where
is the
permittivity and
is the permeability. The space
outside the scatterer is filled with the homogeneous material with
permeability
and permittivity
.
In solving the problem, it is often useful to apply the equivalence
principle (Chapter 3, [47]) using equivalent
electric and magnetic
surface currents to represent the scatterer.
If the task is to find the exterior field only, an exterior equivalent
problem can be shown in Figure 3.2,
where the fields external to
the scatterer can be considered equivalent (to those of the original problem)
due to the electric ()
and magnetic (
) surface current densities on
which are given
by
where is the external surface of the dielectric coating.
Although the exterior fields must be unique, there are many sets of
equivalent currents and interior fields which will give rise to the correct
exterior fields in general scatterers. It is natural to let the interior
field be the null field, since the perfectly conducting
body is inside
.
The Leontovich impedance boundary condition on
implies that only the electric and magnetic fields external to the
scatterer are relevant and their relationship is a function of the material
constitution (here, surface impedance) of the scatterer. As shown in Figure 3.2,
the electric and magnetic fields are zero inside
, and the
electric and magnetic fields outside
are related by [63]
The dual form of the IBC is
where is the intrinsic impedance of free space which is given by
;
is the
relative surface impedance.
The total electric field is the vector sum of the incident electric field
and the scattered electric field.
The scattered electric field produced by the surface currents can be
expressed in terms of the magnetic and electric vector potentials and
the electric scalar potential, and the total electric field is given by
where and
are the magnetic vector
potential and the electric scalar potential, respectively, given by (
)
and (
) in
the previous chapter.
is the electric vector potential
given by
where is the free space Green's function given
in the previous chapter. Substituting (
) and
(
) into (
) yields
Either of the boundary conditions in () and (
)
the following simple
relationship between the electric surface current and the magnetic surface
current
With given by (
), (
)
is the
so-called electric field integral
equation when the impedance boundary condition exists. In the next section,
we will apply
the parametric method of moments to solve for the electric current.