On Certain Two-Variable Generalizations of Circuit Theory, with Applications to Networks of Transmission Lines and Lumped Reactances

Abstract
This paper presents two results concerning the two-variable generalizations of circuit theory concepts discussed by Ozaki and Kasami [8], and indicates applications to finite networks combining commensurate delay transmission lines with lumped reactances. The first of the two contributions to this two-variable circuit theory is a necessary and sufficient test of the two-variable reactance property of any given nonzero two-variable real rational expression. The test requires a finite number of arithmetic computations. The second is a set of necessary and sufficient conditions on the driving-point impedance function for two-variable lossless one-port realizability. The formulation is in terms of the existence of a two-variable polynomial decomposition with certain properties. (A synthesis based on such a formulation would require, at least, the development of techniques of finding the relevant polynomial decomposition, whose existence is assured by network realizability.) An application of each of these results to the transmission linelumped reactance networks, is noted. In connection with the first result, it is observed how the formal replacement of the exponential portions of the driving-point impedance expression, by expressions introducing an independent variable, defines a function of two complex variables with the two-variable reactance property. The first result thus provides a test of a property necessary for drivingpoint impedance realizability using commensurate-delay transmission lines and lumped reactances. The second result provides a realizability formulation (of the type described above) for these networks.