Variational integrators for electric circuits

TL;DR

This paper introduces a novel variational integrator for simulating electric circuits, especially effective for stochastic and multiscale systems, offering improved energy and frequency spectrum preservation over traditional methods.

Contribution

The paper develops a geometric, variational formulation for electric circuit simulation that handles constraints and degeneracies, resulting in a new integrator with superior energy and spectral properties.

Findings

Better energy behavior than BDF methods

Preserves frequency spectrum more accurately

Handles constraints and degeneracies effectively

Abstract

In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits. When considering the dynamics of an electrical circuit, one is faced with three special situations: 1. The system involves external (control) forcing through external (controlled) voltage sources and resistors. 2. The system is constrained via the Kirchhoff current (KCL) and voltage laws (KVL). 3. The Lagrangian is degenerate. Based on a geometric setting, an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. A time-discrete variational formulation provides an iteration scheme for the simulation of the electric circuit. Dependent on the discretization, the intrinsic degeneracy of the system can be canceled for the discrete variational scheme. In this way, a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits; in particular, a comparison to BDF methods (which are usually the method of choice for the simulation of electric circuits) shows that even for simple LCR circuits, a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator.