Hyperbolic-Tangent Shocks in a Lossy Nonlinear Transmission Line

Abstract

We consider a lossy transmission line with a nonlinear voltage--charge relation. We derive an equation for a traveling front with the charge approaching constant asymptotic values on both sides of the front and solve the inverse problem for this equation exactly. Starting from a prescribed monotonic front profile and a prescribed front speed, we determine the dimensionless squared local sound speed within the front. This quantity is the central object of our analysis and allows us to determine the voltage--charge relation of the transmission line in which the front propagates. The squared sound speed, averaged uniformly over the charge interval spanned by the front, is equal to the squared front speed. We specifically consider fronts with a hyperbolic-tangent profile. All physically admissible fronts of this form are shocks rather than kinks. The voltage--charge relation of the transmission line in which the shock propagates is expressed in terms of the lower incomplete beta function. We also treat a transmission line with a cubic voltage--charge relation and propose an approximate equation that admits the hyperbolic-tangent shock profile as an exact solution. The results of the approximate approach coincide with the broad-shock approximation of the exact inverse solution.