# Introduction to the Method of Moments

The MoM is the default solver in Feko. A simple electrostatic example is used to convey the basics of the solver.

## The Charge Distribution of a Straight Wire at a Constant Electric Potential of 1 V.

*a*shown in Figure 1.

According to ^{1}, a linear electric charge distribution
$\rho \left(r\text{'}\right)$
will create an electric potential
$V\left(r\right)$
as follows:

**r**denotes the observation coordinates, $dl\text{'}$ is the path of integration and

*R*is the distance from any point on the source to the observation point which can also be written as

Even though the charge distribution on arbitrarily shaped objects are not generally known, the straight wire example is useful for an introduction to the MoM.

Assume the wire is charged to a constant electric potential of 1 V. For convenience, the wire is oriented parallel to the Z axis. To solve Equation 1 on a computer, the wire is divided into smaller segments and the charge distribution can be approximated as follows:

Therefore Equation 1 can be approximated as follows:

*N*uniform segments where each segment is of length $\Delta =\frac{l}{N}$ .

Since Equation 1 is valid
everywhere, *z* can be chosen to be located at fixed points, *z*_{m}, on
the surface of the wire segments with radii, *a*. This choice simplifies Equation 4 to only a function
of *z*', allowing the calculation of the integral. Furthermore, since the wire was
divided into *N* segments, Equation 4 can be written as one equation with *N* unknowns
(*a*_{n}) as follows:

*N*unknowns requires

*N*equations where each equation stands linearly independent from each other. These

*N*equations can be constructed by selecting the observation points

*z*

_{m}in the centre of each segment of length $\Delta $ as shown in Figure 2.

testingor

sampling$V\left(z,\rho =a\right)$ and the method is referred to as

point-matchingor

collocation.

*N*times reduces Equation 5 to the following:

Equation 6 can be more readily written in matrix form as:

In Equation 7 each
*Z*_{mn} term can be written as:

In addition, we can write the remaining two terms:

*V*

_{m}matrix consists of 1 row and

*N*columns and all entries are equal to $4\pi {\epsilon}_{0}$ . The

*a*

_{n}values are the unknown coefficients for the charge distribution. To solve Equation 7, the matrix requires inversion where

For more complex problems, the integrals cannot be reduced to approximations such as those made here.

^{1}Advanced Engineering Electromagnetics, Second Edition, Constantine A. Balanis, p. 680