Considerations on the equivalence between non-uniform transmission lines

Abstract

We investigate the relation of equivalence between non-uniform lossless transmission lines (LC lines or lines, for short) with sectionally smooth distributed parameters (series inductance L(x) and shunt capacitance C(x) per unit length, x being the spatial variable). The equivalence condition is derived by heuristic and well-grounded inference. The condition provides us with the basis on which we can generate by rote a line equivalent to a fixed original line according to a specified change of spatial variable (elasticity function). The concept of intermediary lines between a pair of equivalent lines is introduced to enhance precise understanding of the condition and the procedure to generate equivalent lines. In addition, fused parameters are defined as a generalization of distributed parameters, in terms of which the argument and formulas are developed in a brief and inclusive manner. It is shown to be possible, on generating equivalent lines, to preassign a fused parameter of the equivalent line or one of the associated intermediary lines instead of specifying the elasticity function. Equivalent lines thus obtained are called conditioned equivalent lines. We discuss how to get conditioned equivalent lines as well as equivalency tests of arbitrarily given pair of LC lines to present some comprehensible and promising methods involving a versatile, graphical technique. We also refer to such interesting topics as a normalization of LC lines, a kind of conservation law of fused parameters between equivalent lines and the scope of realizable characteristics by LC lines conditioned in an arbitrary way. Some formerly known equivalence conditions are located appropriately in the context of our theory. Further, the equivalence condition is shown to be not only sufficient but also necessary for LC lines whose distributed parameters may have a finite number of points of discontinuity by invoking a theorem for an inverse scattering problem. Copyright © 2005 John Wiley & Sons, Ltd.