On the dynamic equations of linear multiconductor transmission lines with terminal nonlinear multiport resistors

Distributed circuits composed of linear multiconductor transmission lines and terminated with nonlinear weakly active multiport resistors are considered. The line is represented as a linear dynamic multiport through recursive convolution relations and special considerations are given to some general properties of the line impulse responses. The convolution technique allows the mathematical description of these distributed circuits by means of a set of nonlinear algebraic-integral equations of Volterra type for the terminal voltages and currents. The conditions under which these governing equations can be reformulated as a set of Volterra integral equations of second kind in normal form are given with the explicit means for doing so. These conditions also assure the existence and the uniqueness of the solution. In particular if the terminal multiport resistors are strictly locally passive, then the normal form exists and the solution is unique. Transmission lines with terminal multiport resistors that are locally active may not admit a normal form for the governing equations, and hence, several solutions that have the same initial conditions are possible. In these cases a simple method is presented for revising the original network model so that the normal form exists, and hence, the uniqueness of solution is assured, under mild restrictions.