Telegrapher's equations

The telegrapher's equations (or just telegraph equations) are a set of two coupled, linear equations that predict the voltage and current distributions on a linear electrical transmission line. The equations are important because they allow transmission lines to be analyzed using circuit theory.^[1] The equations and their solutions are applicable from 0 Hz (i.e. direct current) to frequencies at which the transmission line structure can support higher order non-TEM modes.^{[2]: 282–286} The equations can be expressed in both the time domain and the frequency domain. In the time domain the independent variables are distance and time. The resulting time domain equations are partial differential equations of both time and distance. In the frequency domain the independent variables are distance ${\displaystyle x}$ and either frequency, ${\displaystyle \omega }$, or complex frequency, ${\displaystyle s}$. The frequency domain variables can be taken as the Laplace transform or Fourier transform of the time domain variables or they can be taken to be phasors. The resulting frequency domain equations are ordinary differential equations of distance. An advantage of the frequency domain approach is that differential operators in the time domain become algebraic operations in frequency domain.

The equations come from Oliver Heaviside who developed the transmission line model starting with an August 1876 paper, On the Extra Current.^[3] The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line. Originally developed to describe telegraph wires, the theory can also be applied to radio frequency conductors, audio frequency (such as telephone lines), low frequency (such as power lines), and pulses of direct current.