On the uniqueness of the numerical solutions of nonlinearly loaded lossy transmission lines

The purpose of this paper is to investigate the uniqueness of the solution of lossy lines with frequency-dependent parameters terminated with non-linear resistors. Several solutions that satisfy the same initial conditions may exist if the terminal resistors are locally active. In these cases the uniqueness of solution is assured adding parasitic capacitances in parallel to the voltage controlled resistors and parasitic inductances in series to the current controlled resistors. In this way, among all the possible solutions, the only one that assures the time continuity of the current and voltage waveforms at the ends of the line is captured. In the light of these results, the properties of numerical models of these distributed circuits based on convolution techniques have been studied, and conditions assuring the uniqueness of the numerical solution have been found. Numerical simulations, when based on qualitative information of this type, enable us to obtain the quantitative properties in an efficient manner. In particular, a simple numerical method that enforces artificially the time continuity of the solution is proposed to circumvent the need of adding parasitics.