Telegrapher's equations

The telegrapher's equations are the two coupled differential equations that govern how voltage and current change along a transmission line, both in space and in time. They were derived in the 19th century when engineers struggled to understand why telegraph pulses on long undersea cables came out smeared and delayed — hence the name. The equations model the line as an infinite ladder of tiny series resistors and inductors (R, L) with shunt capacitors and conductances (C, G).

Solve them and a wave equation falls out: signals propagate as forward- and backward-travelling waves whose speed is 1/√(LC) and whose impedance ratio is √(L/C). For a lossless line the pulse keeps its shape; add the loss terms R and G and the high frequencies attenuate faster, which is exactly why a sharp digital edge arrives rounded at the end of a long cable. These equations are the foundation on which characteristic impedance, reflection and standing waves all rest.