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Integral Technique for Capacitance Calculation Disclosure Number: IPCOM000107419D
Original Publication Date: 1992-Feb-01
Included in the Prior Art Database: 2005-Mar-21
Document File: 4 page(s) / 133K

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An efficient algorithm was devised that reduced the CPU time and storage requirement in applying the integral equation method to compute capacitance.

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Integral Technique for Capacitance Calculation

       An efficient algorithm was devised that reduced the CPU
time and storage requirement in applying the integral equation method
to compute capacitance.

      The accurate characterization of capacitance in computer
components is essential in that the capacitance can affect impedance,
delay, and noises.  The method of moments has been used in the past
(1,2,3) to computer the capacitance of three-dimensional (3-D)
conductors.  However, the use of rectangular patches in (1) hinders
their ability to model arbitrary 3-D structures.  Although triangular
patches were adopted in (2,3), the less efficient collocation method,
as opposed to the Galerkin's method, was used.  In this article, we
demonstrate that the Galerkin's method, using triangular patches to
model the conductor surfaces, can be implemented.  An efficient
algorithm was also devised to reduce the CPU time in the computation
of matrix elements.

      The Galerkin's method, in its simplest form, uses pulses as
both expansion and testing functions, and requires that the following
integral be evaluated.

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      Here, S' and S are the source and testing "sub-areas", and r is
the distance between S' and S.  The simplest sub-areas that allow one
to model an arbitrary conductor realistically are the triangular
patches.  The above integral, however, cannot be easily evaluated
over two triangular areas.  It is noted that analytic formulae exist
for the first area integral (4).  In this article, it is shown that
the second integral can be evaluated numerically using a quadrature
formula for a triangular area (5,6). Then, the Galerkin's method can
be efficiently applied to compute the capacitance of arbitrary 3-D
conductors.  As opposed to the collocation method, the Galerkin's
method, besides creating a symmetric matrix, needs fewer patches to
achieve the same accuracy.  Thus, the CPU time and the storage
requirement can be significantly reduced with the Galerkin's method.

      We note that even with a quadrature formula, the numerical
integration over a triangular area can become expensive.  A 3rd- and
4th-order closed-type Newton-Cotes formula would require that 10 and
12 integration points be evaluated, respectively, and it appears
that, at least, a 3rd-order formula is needed for reasonable
numerical accuracy.  In this article, it is further shown that a
short-cut can be taken in doing this numerical integration.

      Note that the diagonal terms of system matrix, as a result of
Galerkin's method, are merely the above double-area integral with S =
S'.  The closed-type Newton-Cotes formula tends to underestimate
these diagonal terms and more weights are given to the smaller side
of a non-linear positive function.  The underestimation of the
diagonal terms will contribute to larger charge densities and, hence,
larger capacitance...