Cagniard–DeHoop technique-based computation of retarded zero-thickness partial elements

Integral-equation based methods for the solution of Maxwell's equations require that the physical system under analysis be meshed into elementary volumes and surfaces. This happens for the method of moments (MoM) and the partial element equivalent circuit (PEEC) method. Then, interaction integrals describing the electric and magnetic field coupling between these elementary regions need to be computed. This is typically done in the frequency domain by resorting to numerical quadrature schemes. In the time domain (TD), brute approximations are typically done leading to simplified schemes which lack the accuracy especially for electrically large problems when propagation delays are important. Such approximations are quite poor also for close elementary regions which are very strong and, thus very important to the overall solution. Hence, in the perspective of developing an accurate time domain solver, it is desirable to have time domain analytical or quasi-analytical forms of the interaction integrals. In this work, we shall derive quasi-closed-form expressions for retarded coefficients as they appear in the partial element equivalent circuit (PEEC) method. To this aim, the Cagniard–DeHoop (CdH) technique exploiting pertinent integration path deformation in the complex-domain leads to semi-analytical forms. The analysis is carried out for a pair of parallel (but non-coplanar) and orthogonal fundamental surface elements as they occur in the modeling of the electric field coupling due to free or bound charges on the surface of conductors and dielectrics. The same results hold also for the magnetic field coupling assuming that currents flow within thin conductors. The accuracy of the proposed approach is tested for representative parallel and orthogonal patches.