A discrete-time approach to the steady-state and stability analysis of distributed nonlinear autonomous circuits

Abstract

We present a direct method for the steady-state and stability analysis of autonomous circuits with transmission lines and generic non- linear elements. With the discretization of the equations that describe the circuit in the time domain, we obtain a nonlinear algebraic formulation where the unknowns to determine are the samples of the variables directly in the steady state, along with the oscillation period, the main unknown in autonomous circuits.An efficient scheme to buildtheJacobian matrix with exact partial derivatives with respect to the oscillation period and with re- spect to the samples of the unknowns is described. Without any modifica- tion in the analysis method, the stability of the solution can be computed a posteriori constructing an implicit map, where the last sample is viewed as a function of the previous samples. The application of this technique to the time-delayed Chua's circuit (TDCC) allows us to investigate the stability of the periodic solutions and to locate the period-doubling bifurcations.