# Numerical Differential Equation Solution to Circuit Problems

## Abstract

Numerical methods are very powerful and can deal with just about any circuit configuration. They are based on finite difference method as a byproduct of discretizing the various differential equations resulting from doing KVL/KCL around the circuit. The key element here is *Δt* which is the time discretization step. Once that is defined we observe that each of the *RLC* circuit elements contributes its way towards voltage, given a current through it: the *R* scales directly; the *C* furnishes a voltage of the form \(\frac {\varDelta t}{C}\); and lastly the inductor contributes a voltage of the form \(\frac {L}{\varDelta t}\). Numerical methods calculate voltage/current at the next time step knowing those at the prior time step. Generally the smaller *Δt*, the more accurate the resulting solution. We use various example ranging from a 9-branch *RLC* circuit, coupling between two *RC* lines, miniature power plane, simple power delivery problem, and even wave propagation down a transmission line. We also touch on matrix solution to circuits for larger sizes. Throughout the chapter we compare to SPICE and observe excellent match.