The Paradox of Power Loss in a Lossless Infinite Transmission Line

Abstract

We discuss here the famous paradox of a continuous power drainage from the source at the input of an otherwise lossless infinite transmission line. The solution of the paradox lies in the realization that in an open-circuit finite transmission line/ladder network, there is an incident as well as a reflected wave and the input impedence is determined by the superposition of both waves. It is explicitly shown that the reactive input impedance of even a single block, comprising say a simple LC circuit, is determined at all driving frequencies from the superposition of incident and reflected waves, and that the input impedance remains reactive in nature (i.e., an imaginary value) even when additional blocks are added indefinitely. However in a ladder network or transmission line, taken to be {\em infinite right from the beginning}, there is no reflected wave (assuming the circuit to be ideal with no discontinuities en route). Thus the source while continuously supplying power in the forward direction, does not retrieve it from a reflected wave and unlike in the case of a finite line, there is a net power loss. This apparently lost energy ultimately appears in the electromagnetic fields in the reactive elements (capacitances and inductances which to begin with had no such stored energy), further down the line as the incident wave advances forward. It is also shown that radiation plays absolutely no role in resolving this intriguing paradox.