Driving-Point Synthesis of Resistor-Terminated Cascades Composed of Lumped Lossless Passive 2-ports and Commensurate TEM Lines

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Abstract
It has long been appreciated that microwave filters incorporating cascades of lumped reactive 2-ports and equi-delay ideal TEM lines offer many practical advantages over those designed with lines alone. To develop an insertion-loss design theory for these fiters comparable in precision to the one now available in the lumped case, it is first necessary to solve the intermediate problem of discovering necessary and sufficient conditions for a driving-point function to represent the input impedance of a resistor-terminated cascade of generic type. A simple decisive solution with the aid of the 2-variable positive-real concept is offered by 1) introducing the 1-variable real polynomial "resistivity matrix" associated with the driving-point function and 2) demonstrating that it must satisfy a fundamental structure constraint. Lastly, in addition to elucidating several important corollaries, the class of characteristic polynomials of all such microwave filters is given a description which reveals the difficulties to be overcome before an exact filter design procedure can be achieved. We wish to emphasize that the necessary and sufficient conditions for realizability presented in this paper are not algorithmic in character but explicit. To make the results accessible to as wide an audience as possible we have adopted a leisurely tutorial style with considerable attention paid to some of the more relevant background material.