Solving low-frequency EM-CKT problems using the PEEC method

The partial element equivalent circuit (PEEC) formulation is an integral equation based approach for the solution of combined electromagnetic and circuit (EM-CKT) problems. In this paper, the low-frequency behavior of the PEEC matrix is investigated. Traditional EM solution methods, like the method of moments, suffer from singularity of the system matrix due to the decoupling of the charge and currents at low frequencies. Remedial techniques for this problem, like loop-star decomposition, require detection of loops and therefore present a complicated problem with nonlinear time scaling for practical geometries with holes and handles. Furthermore, for an adaptive mesh of an electrically large structure, the low-frequency problem may still occur at certain finely meshed regions. A widespread application of loop-star basis functions for the entire mesh is counterproductive to the matrix conditioning. Therefore, it is necessary to preidentify regions of low-frequency ill conditioning, which in itself represents a complex problem. In contrast, the charge and current basis functions are separated in the PEEC formulation and the system matrix is formulated accordingly. The incorporation of the resistive loss (R) for conductors and dielectric loss (G) for the surrounding medium leads to better system matrix conditioning throughout the entire frequency spectrum, and it also leads to a clean dc solution. We demonstrate that the system matrix is well behaved from a fullwave solution at high frequencies to a pure resistive circuit solution at dc, thereby enabling dc-to-daylight simulations. Finally, these techniques are applied to remedy the low-frequency conditioning of the electric field integral equation matrix.