Generalized transmission line equations

Abstract

Generalized transmission line equations (GTLEs) are derived by circuit theory, and equation parameters are determined by the method of moments (MoM). In comparison with conventional transmission line equations (CTLEs), the new equations have added two new terms expressed by dependent series voltage and shunt current sources. For an infinite-length uniform transmission line, the GTLEs are the same as the CTLEs since two coefficients for the two added terms in the GTLEs are found to be zero. For a finite-length uniform transmission line or nonuniform transmission line, the GTLEs, however, are quite different from the CTLEs since two coefficients for the two added terms in the GTLEs are found to be nonzero. In words, the GTLEs are modifications to the CTLEs. Introduction It is known that the derivation of conventional transmission line equations (CTLEs) is based on such an assumption of an infinite-length transmission line and the CTLEs are extended into an infinite-length nonuniform transmission line without any mathematical derivation. Unfortunately, practical transmission lines are finite-length. When the CTLEs are used in a finite-length unmatched uniform transmission line or arbitrary length nonuniform transmission line, the description of the CTLEs for such line discontinuities needs further scrutiny. The reason is that when the nonuniform transmission line (including continuously varying transmission line) is generally treated as a cascading of many short uniform transmission lines, the discontinuities between any two neighbouring segments are not only generate reflections, but also produce radiations. Although the radiation from the sharp discontinuities is observed for a long time, no one has given us the transmission line equations that can take account of both reflections and radiations. In this paper, based on the finite-length line concept, we derive generalized transmission line equations (GTLEs) by using circuit theory. However, the coefficients of the GTLEs need to be determined by numerical methods, such as moment of methods (MoM). In comparison with the CTLEs, the CTLEs have added two new terms that express dependent series voltage and shunt current sources, respectively. For an infinite-length uniform transmission line, the GTLEs are the same as the CTLEs since two coefficients for the two added terms in the GTLEs are found to be zero. For a finite-length uniform transmission line or nonuniform transmission line, the GTLEs, however, are quite different from the CTLEs since two coefficients for the two added terms in the GTLEs are found to be nonzero. In words, the GTLEs are modifications to the CTLEs. Generalized Transmission Line Equations For simplicity, we start with a 1-D finte-length nonuniform transmission line. As mentioned above, the nonuniform transmission line could be regarded as the cascading of infinitely short segments of the uniform transmission line with different characteristic parameters. For each segment, the perunit length series impedance Z and per-unit length shunt admittance Y are different. For an infinitelength nonuniform transmission line, the CTLEs are ) ( ) ( ) ( ) ( ) ( ) ( l v l Y dl l di l i l Z l l v