Fast extraction of resistance and inductance in complex 3D interconnects using surface-volume-surface electric field integral equation

Abstract

The solution of magneto-quasi-static problem is foundational to characterization of current flow in 3D wires of arbitrary cross-sections electrically small extent. Such problems typically arise in signal and power integrity analysis when impedance matrix of electrically short but topologically complex interconnects is required. Traditionally, current flow distribution in such 3D wires is obtained via volumetric Method of Moment (MoM) discretization of the Volume Electric Field Integral Equation (V-EFIE) under quasi-static approximation (Kamon, et.al., IEEE T-MTT, vol. 42, no. 9, pp. 1750–1758, Sept. 1994). Such solution while effective at low frequencies becomes computationally very demanding when the skin-depth becomes much smaller than the size of conductor cross-sections. To alleviate computational complexity associated with MoM solution of the V-EFIE we recently proposed a novel surface formulation of the Electric Field Integral Equation (EFIE) in which only a single surface current has to be determined in order to obtain the complete volumetric distribution of the current throughout interconnect's volume (Menshov and Okhmatovski, IEEE T-MTT, vol. 61, no. 1, pp. 341–350, Jan. 2013). This formulation was termed the Volume-Surface-Volume EFIE (SVS-EFIE) due to the field translations from conductor surfaces to their volumes and back to the surfaces featured in the formulation. Due to surface localization of the unknown currents the MoM solution of SVS features substantially lower number of unknowns than MoM solution of it's V-EFIE counterpart. The unknown count in the MoM solution still remains high, however, when terminal impedance matrix in topologically complex interconnects is sought. In this work we consider iterative and direct matrix-implicit strategies for acceleration of the MoM solution of the SVS-EFIE. The iterative acceleration scheme is based on the Fast Multipole Method utilizing spherical harmonic based expansions (Aronsson and Okhmatovski, IEEE AWPL, vol. 10, pp. 532–535, 2011). The direct matrix implicit solution is based on hierarchical-LU (H-LU) factorization utilizing H-matrix strategy for storage, addition, and multiplication of the matrix blocks in the H-LU matrix decomposition.