Equivalence of Sturm-Liouville-Type Distributed-Parameter Systems

Abstract

This paper analyzes equivalence properties of linear distributed-parameter systems characterized by Sturm-Liouville-type differential equations. The approach taken here may be regarded as an application of the method of independent-variable transformation to those fundamental equations. In accord with the meaning of equivalence used for lumped-parameter systems, distributed-parameter systems are considered here as equivalent when they possess the same terminal behavior. The principal result is an equivalence theorem which establishes a general method of equivalent transformation of Sturm-Liouville systems. The fact that we can generate any number of equivalent parameter distributions from a given one is not only of theoretical interest but also of practical importance, because it offers a flexibility in the design of distributed-parameter structures. The derivation of the transformation function which plays a central role in the equivalent transformation is briefly described. Discussion is also given to the quantities which are invariant under the equivalent transformation. In the later part of the paper applications of the general theory to several problems of practical interest are considered, which include guiding systems for electromagnetic and space-charge waves such as nonuniform transmission lines, lens-like media, and nonuniform drift regions. The physical meanings of the transformation function, the parameters introduced, and various transformation invariants are discussed in detail. A simple example of equivalent systems is also illustrated in the case of nonuniform transmission lines.