A Realizability Theory for Smooth Lossless Transmission Lines

This paper concerns the terminal behavior of nonuniform lossless transmission lines of finite length, specifically, the properties of the scattering parameters of such lines. These parameters, normalized to passive source and load impedances, are expressed in terms of solutions to an integral equation in which the electrical length and the local characteristic impedance of the line are the independent parameters. It is shown that the input reflection factor (
S_{11}
, with the normalization used) of a smooth line of known electrical length, terminated in a known passive load, uniquely determines the local characteristic impedance and the remaining three scattering parameters of the line. A formal realizability theory is obtained in terms of the input reflection factor with resistive termination. The technique underlying this theory involves the reduction of the problem to the inversion of a Sturm-Liouville equation; a classic problem for which a well developed theory is available. This latter theory also suggests a possible synthesis procedure.