There are two equations for two unknowns. The surface currents will be
obtained after solving the equations together.
Using ( )
the curl of the electric vector potential (
) can be expressed
as
where is the divergence of the Green's
function with respect to the primed coordinates. Since
is on
,
the integral on the right-hand side of the above
equation must be interpreted in the Cauchy principal
value sense.
The surface current will be computed using the parametric method of moments described in the previous
chapter. To do so, first the surface is decomposed
into a set of curved triangular patches. Secondly, the unknown current on S
is approximated as a linear combination of the basis functions
in (
) with a set of unknown coefficients
in (
).
Using the symmetric product defined
in (
) to test the result of substituting
(
) into (
) with
gives
where is defined in (
) for
. The electric charge is obtained from the electric current
according to
the continuity equation (
). Substituting the current
expansion (
)
into (
) yields the following matrix equation for the unknown electric
current coefficients
where and
are column vectors, and
is
an
square matrix. The
element of the vector
is
, the unknown coefficient associated with the
expansion
function. The
element of the vector
is given by
The element of the row and the
column of the matrix
is given by:
where and
are given in (
) and
(
), respectively.
is the magnetic current related to
the
electric current expansion by (
):
Applying the same technique as stated in Chapter 2, we intend to compute
sequentially by faces all the vector and scalar potential integrals associated
with each observation-face and source-face combination, to avoid the
costly and inefficient recomputation of identical integrals which would
result if the elements of were computed sequentially by edges. The
numerical integration for a triangular region in [49] is
applied to evaluate the integrals in (
) after
transforming coordinates to a local system of area coordinates. To
implement this idea, it is necessary to rewrite the element of
in (
) in terms of each pair of faces as
where is the contribution from
testing over
the electric field due to the parts of
and
on
, and p and
q are
either + or
signs.
can be further divided into two parts, as
where , the part of
originally contributed by the
electric surface current, arises from the second and third terms on the
right-hand side of equation (
).
It is given by
where and
are given in (
) and
(
).
Similarly,
, the part of
originally contributed
by the
magnetic surface current, arises from the first and fourth terms on the
right-hand side of equation
. It is given by
Comparing the expression of in (
) with the
expression in (
), they are exactly the same. Therefore, the
numerical evaluation of
in (
) is going to
be exactly the same as that of (
) described in Chapter 2.
Therefore, we will only use the
result produced in Chapter 2 and leave the derivation out here. The
comparison between the expression of
in (
)
and that of
in (
), shows that there is only a
little bit of difference in the inner integral of the second term on the
right-hand side of them. Here, we only focus on the numerical evaluation
of the second term on the
right-hand side of (
).
As with MFIE for a perfect conductor in the previous chapter, let
be the inner integral of the second term on the right
hand side of (
). It can be written in terms of
the
coordinates, as:
where is the Jacobian defined in (
), and
the vector function
is defined as
where
and
where denotes the area of the triangle
,
is a
+ or
sign,
is
if
is + and
if
is
(see Fig.
). If we
transform the parametric coordinates to a local system of area coordinates
within in
, (
) can be
rewritten as:
where and
are given in
(
), and the details of the coordinates transformation are
in Appendix A.
Directly applying the numerical integration technique
for a triangular region in
[49] to () gives
where is the total number of integration points on the triangle
,
and
is the weight corresponding to the integration point
.
,
,
, and
are given in
(Table 8.2, [49]).
When , all the integrals in
are
well-behaved, so
can be numerically evaluated as
where the vector function and the scalar function
are
defined in (
), the vectors
and
are given by (
) and (
),
respectively,
is given in (
),
and
and
are given in
(
).
For the case of ,
and
have a
singularity at
.
To remove the
singularity of
in
, the treatment described in Section
2.2.2 of Chapter 2 will be applied. The singularity of
in
will be treated by the method used in Section 2.3 of Chapter
2.
When , numerical formulation of
is given by
where and
are defined in (
), and
Furthermore, ,
,
, and
are given in
(
), (
), (
), and
(
), respectively.
,
,
and
are defined in (
) with the integral
domain
,
, and
, respectively.