Equipotential shells for efficient inductance extraction

To make three-dimensional (3-D) on-chip interconnect inductance extraction tractable, it is necessary to ignore parasitic couplings without compromisi ng critical properties of the interconnect system. It is demonstrated that simply discarding faraway mutual inductance couplings can lead to an unstab le approximate inductance matrix. In this paper, we describe an equipotenti al shell methodology, which generates a partial inductance matrix that is s parse yet stable and symmetric. We prove the positive definiteness of the r esulting approximate inductance matrix when the equipotential shells are pr operly defined. Importantly, the equipotential shell approach also provably preserves the inductance of loops if they are enclosed entirely within the shells of their segments. Methods for sizing the shells to control the acc uracy are presented. To demonstrate the overall efficacy for on-chip extrac tion, ellipsoid shells, which are a special case of the general equipotenti al shell approach, are presented and demonstrated for both on-chip and syst em-level extraction examples.