Telegrapher's equations

The telegrapher’s equations are the coupled PDEs governing voltage $v(z,t)$ and current $i(z,t)$ along a Transmission line:

$-\frac{\partial v}{\partial z}=R^{\prime }i+L^{\prime }\frac{\partial i}{\partial t},$ $-\frac{\partial i}{\partial z}=G^{\prime }v+C^{\prime }\frac{\partial v}{\partial t}.$

Where $R^{\prime },L^{\prime },G^{\prime },C^{\prime }$ are the per-unit-length parameters of the line (resistance, inductance, conductance, capacitance). Their physical meanings are described in Transmission line.

The name dates from the 1870s — Oliver Heaviside derived these equations while working on long-distance telegraph cables, which were the first practical setting where transmission line effects mattered.

Derivation

Consider a differential section of length $\Delta z$ modeled as a series RL section plus a shunt GC. At one end the voltage is $v(z,t)$ and current is $i(z,t)$; at the other, $v(z+\Delta z,t)$ and $i(z+\Delta z,t)$.

Apply KVL around the loop:

$v(z,t)-R^{\prime }\Delta zi(z,t)-L^{\prime }\Delta z\frac{\partial i}{\partial t}-v(z+\Delta z,t)=0.$

Rearrange and divide by $\Delta z$:

$\frac{v(z+\Delta z,t)-v(z,t)}{\Delta z}=-R^{\prime }i-L^{\prime }\frac{\partial i}{\partial t}.$

Take $\Delta z\to 0$: this is the first telegrapher’s equation.

Apply KCL at the right node (current in = current out + shunt current to ground):

$i(z,t)-G^{\prime }\Delta zv(z+\Delta z,t)-C^{\prime }\Delta z\frac{\partial v}{\partial t}-i(z+\Delta z,t)=0.$

Rearrange and take $\Delta z\to 0$: this is the second telegrapher’s equation.

Decoupling: the wave equation

Differentiate the first equation with respect to $z$ and substitute $\partial i/\partial z$ from the second. After cancellation:

$\frac{{\partial }^{2}v}{\partial z^2}=R^{\prime }{G}^{\prime }v+(R^{\prime }{C}^{\prime }+L^{\prime }{G}^{\prime })\frac{\partial v}{\partial t}+L^{\prime }{C}^{\prime }\frac{{\partial }^{2}v}{\partial t^2}.$

Same kind of PDE for $i$.

For a lossless line ($R^{\prime }=G^{\prime }=0$), this reduces to the wave equation:

$\frac{{\partial }^{2}v}{\partial z^2}=L^{\prime }{C}^{\prime }\frac{{\partial }^{2}v}{\partial t^2},$

with wave speed $u_p=1/\sqrt{L^{\prime }{C}^{\prime }}$. Voltage on a lossless line is a wave that propagates without distortion.

Phasor form

For sinusoidal steady state $v(z,t)=\text{Re}[\tilde{V}(z)e^{j\omega t}]$, replace ${\partial }_t\to j\omega$:

$-\frac{d\tilde{V}}{dz}=(R^{\prime }+j\omega L^{\prime })\tilde{I},$ $-\frac{d\tilde{I}}{dz}=(G^{\prime }+j\omega C^{\prime })\tilde{V}.$

Differentiating gives decoupled second-order ODEs:

$\frac{d^2\tilde{V}}{d{z}^2}-{\gamma }^2\tilde{V}=0,$

with the propagation constant:

$\gamma =\sqrt{(R^{\prime }+j\omega L^{\prime })(G^{\prime }+j\omega C^{\prime })}=\alpha +j\beta .$

General solution:

$\tilde{V}(z)=V_0^{+}{e}^{-\gamma z}+V_0^{-}{e}^{+\gamma z}.$

The two terms are forward- and backward-propagating waves. $V_0^{+}$ and $V_0^{-}$ are complex amplitudes, fixed by source and load conditions.

For the current:

$\tilde{I}(z)=\frac{V_0^{+}}{Z_0}{e}^{-\gamma z}-\frac{V_0^{-}}{Z_0}{e}^{+\gamma z},$

where $Z_0=\sqrt{(R^{\prime }+j\omega L^{\prime })/(G^{\prime }+j\omega C^{\prime })}$ is the Characteristic impedance. Note the sign on the backward current: $Z_0=-V_0^{-}/I_0^{-}$, not $+V_0^{-}/I_0^{-}$. This is the convention that defines $Z_0$ as the ratio for forward propagation only.

Time-domain back from phasor

For lossless ($\gamma =j\beta$, $Z_0$ real), $V_0^{+}=|V_0^{+}|e^{j{\varphi }^{+}}$:

$v(z,t)=|V_0^{+}|\cos (\omega t-\beta z+{\varphi }^{+})+|V_0^{-}|\cos (\omega t+\beta z+{\varphi }^{-}).$

Two traveling waves: $(\omega t-\beta z)$ moves in $+z$ with speed $\omega /\beta$; $(\omega t+\beta z)$ moves in $-z$ with the same speed.

For lossy ($\gamma =\alpha +j\beta$), add exponential envelope:

$v(z,t)=|V_0^{+}|e^{-\alpha z}\cos (\omega t-\beta z+{\varphi }^{+})+|V_0^{-}|e^{+\alpha z}\cos (\omega t+\beta z+{\varphi }^{-}).$

Forward wave decays with $+z$; backward wave decays with $-z$. Energy is dissipated in $R^{\prime }$ and $G^{\prime }$.

Why “telegrapher’s”

The historical context: 19th-century telegraph engineers were trying to figure out why messages on transatlantic cables came through distorted and slow. Heaviside derived these equations (with later contributions from many others), and used them to design distortionless lines by satisfying $R^{\prime }/L^{\prime }=G^{\prime }/C^{\prime }$ — the Heaviside condition. This was a turning point in long-distance signaling.

Today, the equations underpin every PCB design, every RF system, every fiber-optic cable. They’re the bridge between Maxwell’s equations (which describe TEM waves in space) and lumped circuit theory (which describes voltages and currents in components). The transmission line is the one-dimensional connection between fields and circuits.