Electric Wave Propagation on Non-Uniform Coupled Transmission Lines

The propagation of electric currents and voltages along a pair of wires over a ground plane is studied. The system is assumed to be non-uniform; I. e., the self and mutual inductances and capacitances vary along the wires. The existence and uniqueness of an electric wave having prescribed initial values is shown to follow from recent results of the author on symmetric hyperbolic systems of partial differential equations. A construction for this solution is given in the case of a coupler (i. e., a pair of wires which are non-uniform and coupled over a portion of their length only). This coupler problem is reduced to a two-point boundary value problem, and the latter is reduced to a pair of initial value problems, one of which involves a matrix Riccati equation. A novel feature of the work is an "a priori" estimate which guarantees that a solution of the (non-linear) matrix Riccati equation exists on the whole line.