General Analytic Solution of the Telegrapher’s Equations and the Resulting Consequences for Electrically Short Transmission Lines

Abstract

Based on classical circuit theory, this article develops a general analytic solution of the telegrapher’s equations, in which the length of the cable is explicitly contained as a freely adjustable parameter. For this reason, the solution is also applicable to electrically short cables. Such a model has become indispensable because a few months ago, it was experimentally shown that voltage fluctuations in ordinary but electrically short copper lines move at signal velocities that are significantly higher than the speed of light in a vacuum. This finding contradicts the statements of the special theory of relativity but not, as is shown here, the fundamental principles of electrical engineering. Based on the general transfer function of a transmission line, the article shows mathematically that an unterminated, electrically short cable has the characteristics of an ideal delay element, meaning that an input signal appears at the output with a slight delay but remains otherwise unchanged. Even for conventional cables, the time constants can be so small that the corresponding signal velocities can significantly exceed the speed of light in a vacuum. The article also analyses the technical means with which this effect can be conveyed to very long cables.