Stability of Algorithms for Electro-MagneticTransient Simulation of Networks with Switches and Non-linear Inductors

This paper extends the analysis of the stability of electromagnetic transient simulation algorithms to non-linear systems with switching elements and non-linear inductor branches. A theoretical analysis based on common quadratic Lyapunov function (CQLF) theory is used to investigate the stability of numerical algorithms for the simulation of lumped strictly passive switched circuits (LSPSC). It is proved that only when certain fundamental physical properties, i.e., passivity and invariance of Lyapunov energy function are satisfied, does the widely used trapezoidal method result in stable simulations of such networks for any time-step size. This is different from the simulation of linear time invariant (LTI) systems where any real world stable system has a stable simulation if an A-stable integration method (e.g., trapezoidal rule) is used. Subsequently, it is shown that the problem of simulating a piecewise linear inductor can be equivalent to simulating a LSPSC; and ergo its simulation with the trapezoidal rule is also stable.