Formulas for the potentials due to uniform and Linearly varying source distributions defined on simply shaped domains are systematically developed and presented. Domains considered are infinite planar strips, infinite cylinders of polygonal cross sections, planar surfaces with polygonal boundaries, and volumetric regions with polyhedral boundaries. The expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated.

Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains

Numerical procedures, based on a method of moments approach, are given for computing the magnetic and electric polarizabilities of electrically small apertures of arbitrary shape. The magnetic polarizability density is determined through the use of pulse expansion functions defined over quadrilateral subdomains, while the electric polarizability density is obtained by using basis functions, each of which consists of a piecewise arrangement of simple linear functions defined over triangles and having an area coordinate representation. All the subdomains are generated automatically by applying either the Gordon-Hall or the Zienkiewicz-Phillips subdivision techniques. Computed results are obtained for several aperture shapes, including the circle and the ellipse. The calculations for the last two cases are in excellent agreement with the exact values.

The Polarizabilities of Electrically Small Apertures of Arbitrary Shape

In this paper an exact expression is derived for the potential at an arbitrary point due to a linear distribution of source over a triangular domain. The integral which gives the potential is evaluated by means of an extension of an analytical procedure previously evolved for a constant distribution. Numerical values of the potential are obtained for several field points by evaluating the above analytical expression, and are compared with those calculated by using triangle quadratures.

The potential integral for a linear distribution over a triangular domain

An exact expression is derived for the potential due to a uniform source distribution over a triangular domain. This expression, which is valid for arbitrary field points, is important for the numerical solution of electromagnetic field problems. Numerical values of the potential are calculated by using the expression obtained, and are compared with the results derived by numerical integration.

The potential due to a uniform source distribution over a triangular domain

Exact expressions are presented for the potentials, at an arbitrary point in space, due to two quadratic distributions in a triangular disc. These expressions are obtained by means of extensions of the analytical techniques previously developed for evaluating the potential integrals for constant and linear distributions. It is shown that the potential due to any quadratic distribution in a triangle may be obtained by using the two expressions derived here, together with an expression for the potential corresponding to a linear distribution. This latter expression was presented in an earlier paper.1 Numerical values obtained through the use of the two expressions are compared with those which result from the numerical evaluation of the potential integrals by triangle quadratures.

Potential integrals associated with quadratic distributions in a triangular domain

A method of moments procedure is presented for solving the integral equation satisfied by the electrostatic charge distribution over arbitrary planar domains. This procedure is based on a representation of such domains as a collection of quadrilateral subdomains, using the Gordon-Hall and Zienkiewicz-Phillips subdivision techniques. All integrals are effectively evaluated over triangles, and rely on an expression for the potential at a point due to a constant charge distribution over a triangular domain. As an illustration, the computational procedure is applied to a variety of discs. The computed results are in excellent agreement with the exact values for the circle and the ellipse, and are compared with known numerical values for certain other configurations.

R. F. Harrington

Papers

A method of computing the capacitance of flat discs of arbitrary shape