The method of moments can solve other types of electromagnetic problems: for example, electrostatic problems involving charges and dielectrics. These solutions can determine the characteristic impedance of transmission lines useful in the design of antenna feeders. All moment method solutions are found from the solution of integral equations over boundary conditions. The boundary conditions can be either the tangential electric field (EFIE) or magnetic field (MFIE) conditions given by Eq. (1a,b) or a combination applied using an integral scalar product. We need a combination for closed bodies near an internal resonance frequency (resonant cavity) because the solutions exhibit resonances that make the solution invalid over a narrow frequency range. The method of moments can be applied to dielectric bodies when we use the constitutive relations of Eqs. (1) and (2), where the formulations for dielectric bodies use either volume or surface integrals [9].
 (1a,b) (2)Consider the use of the electric field integral equation (EFIE) with metal surfaces. We expand the currents on the objects using basis functions with coefficients  :
 (3)
The basis functions can be applied over a limited range of the structure in piecewise linear functions, which can be staircase pulses, overlapping triangular functions, or sinusoidal basis functions, whereas multiple functions can be applied over the whole or part of the structure for entire domain basis functions. For example, these could be a sum of sinusoidal functions which form a Fourier series representation.
On a PEC surface the tangential electric field vanishes [Eq. (4)].
 (4)At field point r along the surface S,
 (5)We can only satisfy Eq. (5) using a finite sum in the average sense of an integral. Since the integral and summation operate on a linear function, we can interchange them. We introduce weighting (or testing) vector functions tangent to the surface Wn(r) and take the scalar (dot) product of this vector with the sum of electric fields. This limits the result to the tangential component of the electric field:
 (6)We identify the weighted integral of the incident field with the source and weighted integral of the field radiated by the basis functions (scattered field) as the impedance matrix terms. The integrals over the boundaries are one form of scalar product represented by <·> notation. Using unity current on each basis function, we calculate the matrix terms by using the scalar product:
 (7) (8)The combination of Eqs. (7) and (8) when integrated over each portion of the source gives a matrix equation:
 (9)
The weighting functions could be as simple as pulse functions, overlapping triangular functions on lines or surfaces (rooftop), piecewise sinusoidal functions, or others. The type of basis functions determines the convergence more than the weighting (testing) functions, which only determine the averaging. Realize that the moment method converges to the exact solution when we increase the number of basis functions, but it is a matter of engineering judgment to determine how many terms give acceptable answers.
Equation (8) defines the source voltage occurring over a segment when the formulation uses a piecewise function expansion. The incident voltage is the weighted integral of the incident electric field. For example, the NEC formulation applies an excitation voltage across one segment. The modeling of sources is an important part of the art in the method of moments.
The expansion of Eq. (5) is only one possible moment method solution. We could use the boundary condition on the magnetic field, a combination of the electric and magnetic field conditions on a PEC. If the surface has finite conductivity, the boundary conditions are modified. The moment method is a general method that computes approximate solutions to the currents. Unlike physical optics, the currents do not have to be assumed beforehand but are found as a finite series approximation.
Antenna designers discover that adequate codes are available for most problems. Moment method solutions are typically limited to objects only one or two wavelengths in size, although any method can be stretched. Analysis of large structures becomes intractable because of the large amount of computer memory required and the length of time needed to calculate the solution. Coarse models may not give totally accurate results but can be useful in determining trends. Given these ideas, remember that physical models can be built that solve the electromagnetic problem instantaneously. We found that it takes considerable time to learn any code, and a new code has to offer considerable advantages or solve problems that the present one cannot solve before we invest our time. |
