A note on passivity of nonlinear RL and RC circuits

It is well-known that arbitrary interconnections of passive (possibly nonlinear) resistances (R), inductances (L) and capacitances (C) define passive ports, with port variables the external sources voltages and currents, and storage function the total stored energy. In this brief note we establish some new passivity properties of RC and RL circuits under the more restrictive assumption that the characteristic functions of the capacitances and inductances are nondecreasing. In particular, we prove that for this class of nonlinear RC circuits passivity is preserved even if we take as port variables the sources currents and the voltage derivatives. We also show that nonlinear RL circuits define passive ports with port variables the sources current derivatives and the voltages. The storage function in both cases is the sum of the resistors co-content, which is a function associated to the power dissipation. The proof of these properties is a direct application of Tellegen's theorem. We illustrate the applicability of these results to generate new Lyapunov functions for nonlinear stabilization problems.