Nonlinear transmission line: shock waves and the simple wave approximation

TL;DR

This paper analyzes nonlinear transmission lines with nonlinear inductors and capacitors, deriving shock wave solutions with dissipation and a simple wave approximation for lossless lines, advancing understanding of wave propagation in such systems.

Contribution

It provides analytic shock wave profiles for dissipative nonlinear transmission lines and introduces a simple wave approximation for lossless lines, bridging the understanding of wave dynamics in these systems.

Findings

Analytic shock wave solutions for strong dissipation cases.

Stationary dispersive shock waves in weak dissipation.

A simple wave approximation for lossless transmission lines.

Abstract

The transmission lines we consider are constructed from the nonlinear inductors and the nonlinear capacitors. In the first part of the paper we additionally include linear ohmic resistors. Thus, the dissipation being taken into account leads to the existence of \mbox{shocks -- the} travelling waves with different asymptotically constant values of the voltage (the capacitor charge) and the current before and after the front of the wave. For the particular values of ohmic resistances (corresponding to strong dissipation) we obtain the analytic solution for the profile of a shock wave. Both the charge on a capacitor and current through the inductor are obtained as the functions of the time and space coordinate. In the case of weak dissipation, we obtain the stationary dispersive shock waves. In the second part of the paper we consider the nonlinear lossless transmission line. We formulate a simple wave approximation for such transmission line, which decouples left/right-going waves. The approximation can also be used for the lossy transmission line, which is considered in the first part of the paper, to describe the formation of the shock wave (but, of course, not the shock wave itself).