---
tag: 1HR9NZM8B2
kind: concept
name: Telegrapher's equations
msc: [35L40, 35Q60, 35L05]
sources:
  - url: https://en.wikipedia.org/wiki/Telegrapher%27s_equations
same_as:
  wikidata: Q1156880
  wikipedia: Telegrapher's equations
---

## Summary

The telegrapher's equations are a pair of linear first-order equations for the voltage and the current along an electrical transmission line.
They were developed by Oliver Heaviside to describe signals on telegraph cables.
Eliminating the current gives one second-order equation for the voltage, the telegraph equation, which is a damped [[wave equation|5EHD2XBQDG]].
The equations show how resistance and leakage make a signal both decay and spread out.

## Explanations

### L1 human

A long cable has, per unit length, a resistance, an inductance, a capacitance between its wires, and a small leakage.
A voltage pulse sent along the cable travels like a wave, but it loses strength and changes shape.
The telegrapher's equations describe this with two equations linking how voltage and current change in space and in time.
Without resistance and leakage, they reduce to the wave equation with speed $1/\sqrt{LC}$, and a pulse travels without distortion.