Explicit solution for the synthesis of two-variable transmission-line networks

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Abstract
Using the properties of polynomials orthogonal on the unit circle, an explicit solution is derived for the synthesis of resistively terminated one- or two-variable cascaded transmission-line networks. In the two-variable case, in addition to the cascade of ideal commensurate transmission lines, passive lossless lumped two-ports are allowed to exist between the junctions of adjacent lines. For this case, the explicit solution form enables the test for two-variable positive reality to be discarded in favor of a matrix factorization condition. In the onevariable case, due to the intimate relationship between the synthesis of a cascade of transmission lines and the generation of a sequence of polynomials orthogonal on the unit circle, Richards' theorem is not required for the explicit-form solution. Initially, the main theorem describing the explicit solution for the one- and two-variable cases is presented. After the formulation of the proofs, two nontrivial examples are cited to illustrate the use of the explicit-form solution in the two-variable case.