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.