# Limits in MoM calculations

Dr. Edward P. Sayre ([email protected])
Thu, 30 May 1996 16:26:21 -0400

The question of accuracy and errors in Method of Moments (MoM) calculations
is very subtle since it involves both the representation of the unknown
vectors (pulses, triangles, impulses) as well as the representation of the
Green's functions. The formation of the matrix which represents the Green's
functions involves an integral operation involving "testing" functions. The
integral operation usually involves the statement of a field equivalence
principle such as those stated in Harrington "Time Harmonic Fields", Sect
3-5, McGraw Hill or Colin "Field Theory of Guided Waves" 2nd Ed,Sects 1.8 -
1.10, IEEE press. From a function theoretic point of view, such integral
operators posses properties related to best mean square approximations when
the testing functions are adjoint functions to the expansion functions and
the testing operation (which is really a projection operator) is the adjoint
Green's function. For many frequency domain formulations, the Green's
function is self-adjoint. (The name MoM comes from the similarity of the
method used in mechanical engineering to compute mechanical moments. For
those with historical interests, the method is also known as Garlekin's method.)

That being said, what about accuracy? When the expansion functions are
applied to approximate the unknown vectors and the testing operation is
applied, the continuous problem is reduced to a matrix operator applied to a
vector of unknowns which equal some known vector. The question of accuracy
comes down to how well does the chosen expansion function approximate the
continuous unknown AND how well does the "testing" operation result in a
discrete Green's function which well approximates the integral operator. So,
the question of small segments (1/10 wavelength etc. is really a question of
Green's function representation and accuracy of expansion functions. As an
example where making the problem subsections too small, take the problem of
scattering from a wire object. If the axial approximation is used for the
current, and if the subsections are on the order of the wire radius, most
MoM solutions will be incorrect no matter what the the ratio of subsection
length to wavelength. On the other end of the scale, MoM is wrong when the
sub-sections become too coarse. There appears to be a "sweet region" where
this formulation is accurate and convenient.

>From a classical fields perspective, the use of orthogonal functions (sines,
cosines, Bessel & sperical functions of one sort or another) as the
expansion functions to represent fields is a special case of MoM where the
orthogonality of the functions is preserved by the "testing" operator due to
the geometry fit. In such cases it is easy to show that the problem reduces
to a diagonal matrix operator operating on the expansion coefficients of the
orthogonal series.

In summary, there is no single answer to the question of accuracy and error.
It is all tied up in the field representation in terms of expansion
functions and the application of the "testing functions" and the definition
of a proper operation for testing with known function theoretic implications.

By the way, the comments expressed above should be considered very carefully
when one is using canned programs to predict package inductances and
capacitances. They are usually electrically small and blind application of
2-D MoM field theory to deduce SPICE circuit elements is very dangerous.
Don't bet the ranch on the answers.

Sincerely,

Edward P. Sayre

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