How to interpret the telegrapher’s equations for voltage and current on a transmission line

Key concept: These two coupled partial differential equations describe how voltage and current vary along a transmission line. The constants $R$, $L$, $C$, and $G$ represent the line’s distributed resistance, inductance, capacitance, and conductance, so the equations are really a compact model of signal propagation with loss.

Step by step

The pair of equations

$\frac{\partial}{\partial x}V(x,t)=-L\frac{\partial}{\partial t}I(x,t)-RI(x,t)$ $\frac{\partial}{\partial x}I(x,t)=-C\frac{\partial}{\partial t}V(x,t)-GV(x,t)$

is the telegrapher’s equations written in a first-order form.

1) What the variables mean

• $V(x,t)$: voltage at position $x$ and time $t$
• $I(x,t)$: current at position $x$ and time $t$
• $R$: resistance per unit length
• $L$: inductance per unit length
• $C$: capacitance per unit length
• $G$: conductance per unit length

So these are not ordinary circuit equations for a single lumped component. They describe a distributed system: every tiny piece of the line contributes a little resistance, inductance, capacitance, and leakage.

2) Why the signs are negative

The negative signs come from the direction of energy flow and the way voltage/current change along the line. If you move a small distance $dx$ forward, part of the signal is lost to resistance and leakage, while the inductive and capacitive terms control how the wave evolves in time.

3) Physical reading of each equation

• In the first equation, the spatial change of voltage depends on:
  • the time change of current through the inductive term $L\,\partial I/\partial t$
  • the ohmic loss term $RI$
• In the second equation, the spatial change of current depends on:
  • the time change of voltage through the capacitive term $C\,\partial V/\partial t$
  • the leakage term $GV$

4) If you want the wave equation

You can combine them to get second-order equations. For example, differentiate the first with respect to $x$ and substitute the second where needed. In a lossless line ($R=G=0$), this leads to the familiar wave form:

$\frac{\partial^2 V}{\partial x^2}=LC\frac{\partial^2 V}{\partial t^2}, \qquad \frac{\partial^2 I}{\partial x^2}=LC\frac{\partial^2 I}{\partial t^2}$

That means voltage and current travel as waves with speed

$v=\frac{1}{\sqrt{LC}}.$

5) Practical meaning

This model is used for:

• transmission lines
• coaxial cables
• printed circuit traces at high frequency
• signal integrity analysis

At low frequency, a wire often behaves almost like a simple conductor. At high frequency, these distributed effects matter a lot, and the telegrapher’s equations become the right tool.

Pitfall alert

A common mistake is treating $R$, $L$, $C$, and $G$ as if they were total component values for the whole wire. They are per-unit-length quantities. Another trap is dropping the time-derivative terms too early; that only works in special steady-state or low-frequency approximations. If you ignore $G$ or $R$ without checking the problem statement, you can end up with the wrong propagation speed or attenuation.

Try different conditions

If the line is ideal, meaning $R=0$ and $G=0$, the equations simplify to a pure wave model. If the line is low-loss but not ideal, you still get wave propagation, but the amplitude decays with distance. In sinusoidal steady state, you can also switch to phasor form and write the line equations using complex impedance and admittance, which is often easier for solving practical circuit problems.

Further reading

telegrapher's equations, transmission line, distributed parameters