General Amplitude Relations for Transmission Lines with Unrestricted Line Parameters, Terminal Impedances, and Driving Point

Subject to no other restrictions and approximations than those actually implied in the derivation of the transmission-line equations and their application to a terminated line, general expressions in simple form are derived for the current amplitude in, and the potential difference between, the two conductors of a uniform transmission line at any point along its extension. The terminal impedances are assumed to be perfectly general, and the line may be driven by a loosely coupled generator which maintains a sine-symmetrical, a cosine-symmetrical, or an unsymmetrical field along any section of the line. General resonance conditions are obtained and expressions written down for extreme values of the amplitudes. The special cases of low line damping and low over-all damping are considered.