5.6 Green's Function - Integral Equation Methods

As the name indicates, the integral equation methods are based on the solution of an integral equation, rather than a differential equation. The integral equation is derived by means of a suitable Green's function, which constitutes its kernel. There are two integral equation formulations possible [43]. In one, which employs a volume integral equation, the dielectric inhomogeneity (i.e., the DR) is replaced by equivalent polarization currents. In the other, the DR is replaced by equivalent electric and magnetic surface currents, leading to a surface integral equation. The success of either of these methods depends on one's ability to find the suitable Green's function, which is usually a dyadic [44]. The situation is simpler in the case of an open resonator, since only the free-space Green's function is required [49] (see also Ch. 6).

In the case of shielded resonators, the determination of the Green's function tends to be difficult, particularly for more complex resonators. This task can be somewhat simplified by subdividing the resonator into two regular partial regions and by erecting a PMC or PEC wall at the boundary between the regions [45,46]. Magnetic or electric current sheets are then postulated on this wall, maintaining the correct field in both regions. By enforcing the field continuity condition between the partial regions, an integral equation is obtained for the unknown current. This equation is then discretized by the method of moments [1], employing suitable expansion and testing functions. The resonant frequencies are obtained...