The method of moments:

Capacitance and charge density of metal plates

This section will calculate the capacitance and surface charge density of a square metal plate and a metal disc. It may not seem particularly interesting, but this is hopefully an easy introduction to the important method of moments.

The formula providing the incremental potential is:

dU = σ(x,y) (dx)(dy) [(x-xo)²+(y-yo)²]^-1/2/(4πε_o)

where:

σ(x,y) is the surface charge density at the source point (x,y)
(xo,yo) are the coordinates of the observation point

It is true that some integral equations can be solved in certain coordinate systems, such as the rectangular, spherical and cylindrical. On a more realistic surface though, it is a different matter: The solution, (if at all possible), mostly involves strenuous effort and formidable ingenuity. The moment method, however, allows numerical results to be obtained easily:

The potential on a metal conductor is constant- that's the boundary condition.
A digital computer cannot directly handle continuous signals or surfaces; sampling is needed first: The boundary condition shall be imposed at a limited number of matching points only, not the entire surface. This is formally stated as follows: The testing functions are Dirac- delta.
The solution is no longer determined analytically- its value is only deduced at a corresponding number of points (the basis functions are also Dirac- delta.)

Some numeric integration is needed. Other than that though, an onerous task has been reduced to a set of linear algebraic equations: A programmer's feast!

While the point- matching method only uses Dirac- delta test functions, there are certainly plenty of different choices, particularly for the basis functions:

Step approximation
Piecewise linear model
Polynomial interpolation
Fourier expansion
The method of pertubations: If the solution to a similar problem is known, it can be used as a first- order estimate (plus correction terms.)

Whatever the basis functions though, a numeric calculation does not involve an infinite number of components, nor does it have to (not if the signal is worth transforming.)

The particular instance of the method of moments where the test and basis functions are the same, is called Galerkin's method. This is more or less used in the program for the square metal plate.

Certainly no- one can afford not to take advantage of symmetry when solving problems of potential: One eighth of the metal square plate need only be simulated, though the program is much simpler if actually one quarter is used in the calculations; this redundancy also offers an easy check.

The metal disc only has the boundary condition enforced along a single radius. Symmetric source regions have the same charge density, so the latter only has to be derived once. Thus the program need only solve for the charge density along a single radius, too. (Of course, all symmetric source subareas are involved in the calculations- their distances to a matching point are different.) Curvilinear elements look more convenient for this problem.

It has already been mentioned that Galerkin's method is basically used for these problems, but there is actually a small deviation: Concentrated, rather than distributed, charges are assumed in Galerkin's method, which is, strictly speaking, wrong. (When the source point and observation point are not close, the difference is easily forgotten.) The discrepancy is expected to diminish as the number of lumped charges is increased, but the following technique improves convergence speed: A ten line listing calculates a correction factor. This is the ratio of the total concentrated charge at the corners of a square over the uniform charge in the square giving rise to the same potential at the square center. The value for square elements is about 2.52.

The capacitance value is numerically equal to the charge producing a uniform unity voltage over the plate: A value of about 40 picoFarads for a square metal plate of an area of 1 metre squared is easily verified in the literature. A graph of the approximate distribution of the surface charge density along the diagonal follows: The (constant) metal potential is one Volt. (The distribution for the circular metal plate should not be very different.)

Approximate square metal plate charge density

The program for the metal disc will follow later on: (The compensation factor for curvilinear elements is not constant along the radius.) If the area is the same, the capacitance should be similar.

If a lot of matching points are considered, half of the charge density values (along any direction) can become close to zero, or even negative: This does not matter for the capacitance (the total charge is positive), but it does imply that half the resolution is lost for the distribution of the surface charge density.

All numerical methods should have their convergence verified by performing additional calculations, especially when quantisation of signals or surfaces is taking place, by the user (and, preferably, by the software.) Although the simple program supplied does not do that, the modifications required are minor. It is interesting that the method may converge for a number of samples, and twice that number, but not for one of the numbers in between. The constant for the number of nodes included gives credible results (within about 10% of the true value.)

The following text has useful information, and part of it was instrumental in preparing the material in this section- it is one of the 'oldies but goodies':

IEEE Proc. Vol 55, No 2, Page 136- R.F. Harrington: Matrix Methods For Field Problems.

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