Exact Solutions to Some Deterministic and Random Transmission Line Problems

Abstract

A special class of transmission lines is considered, in which the modes decompose into two noninteracting sets. Both a single transmission line with constant characteristic impedance and variable propagation factor, and two transmission lines with equal propagation factors and variable coupling, in which the forward modes do not interact with the backward modes, are investigated. Exact expressions are obtained for the reflection and transmission coefficients when a section of such a transmission system connects two semi-infinite transmission systems consisting of constant impedance and admittance lines. These results hold for arbitrarily varying propagation factors and coupling; and while they are of independent interest in the case of deterministic variations, we make an application of them here in the case of stochastic variations. Exact results are obtained for the ensemble averages of the transmission coefficient and transmitted power, and their variances, for the inserted section of single line, when the variable propagation factor is a random function involving either a Gaussian process or the random telegraph process. Asymptotic results are also obtained in the general case of weak fluctuations and long inserted sections. Analogous results may be obtained for the inserted section of two lines when they are randomly coupled, and the results are given in the case of matched lines, for which no reflections occur. Finally, some of the time domain statistics for lossless lines are considered, and expressions are derived for the ensemble averages of the transmitted pulse, due to pulses incident on the inserted section.