A theory for deriving exactly solvable nonuniform transmission lines systematically

A general idea is given for a systematic and recursive technique to derive exactly solvable LC lines one after another. It consists of three simple operations, i.e., the Liouville transformation (LT) applied to the Telegrapher´s equation, its inverse, and LC exchange (LCX). The differential equation obtained as a result of the LT is unique for a specific original line and is called a Liouville normal form (LNF). In contrast, an LNF can generate through the inverse LT an infinite number of lines which are all exactly solvable provided the original line is so. Meanwhile, LCX serves effectively for finding out a new LNF from which we can derive further exactly solvable lines. Starting from a known exactly solvable line (typically, a uniform line), the technique proceeds by executing the operations alternately to yield more and more complicated exactly solvable lines endlessly.