Telegrapher's equations

The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who developed the "transmission line model" which is described in this article. The theory applies to high-frequency transmission lines (such as telegraph wires and radio frequency conductors) but is also important for designing high-voltage energy transmission lines. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can appear along the line.

The equations

The telegrapher's equations can be understood as a simplified case of Maxwell's equations. In a more practical approach, one assumes that the conductors are composed of an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

* The distributed resistance $R$ of the conductors is represented by a series resistor (expressed in ohms per unit length).
* The distributed inductance $L$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
* The capacitance $C$ between the two conductors is represented by a shunt capacitor C (farads per unit length).
* The conductance $G$ of the dielectric material separating the two conductors is represented by a conductance G shunted between the signal wire and the return wire (siemens per unit length).

It should be repeated for clarity that the model consists of an "infinite series" of the elements shown in the figure, and that the values of the components are specified "per unit length" so the picture of the component can be misleading. An alternative notation is to use $R\text{'}$, $L\text{'}$, $C\text{'}$ and $G\text{'}$ to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant.

When the elements "R" and "G" are very small, their effects can be neglected, and the transmission line is considered as an idealized, lossless, structure. In this case, the model depends only on the "L" and "C" elements, and we obtain a pair of first-order partial differential equations, one function describing the voltage "V" along the line and the other the current "I", both function of position "x" and time "t":

:$frac\{partial\}\{partial\; x\}\; V(x,t)\; =-L\; frac\{partial\}\{partial\; t\}\; I(x,t)$

:$frac\{partial\}\{partial\; x\}\; I(x,t)\; =-C\; frac\{partial\}\{partial\; t\}\; V(x,t)$

These equations may be combined to form either of two exact wave equations:

:$frac\{partial^2\}$partial t}^2} V =frac{1}{LC} frac{partial^2}partial x}^2} V

:$frac\{partial^2\}$partial t}^2} I =frac{1}{LC} frac{partial^2}partial x}^2} I

In the steady-state case (assuming a sinusoidal wave $E=E\_\{o\}.e^\{-jomega\; (\; frac\{x\}\{c\}\; -\; t)\}$, these reduce to

:$frac\{partial^2V(x)\}\{partial\; x^2\}+\; omega^2\; LCcdot\; V(x)=0$

:$frac\{partial^2I(x)\}\{partial\; x^2\}\; +\; omega^2\; LCcdot\; I(x)=0$

:where $omega$ is the frequency of the steady-state wave

If the line has infinite length or when it is terminated with its characteristic impedance, these equations indicate the presence of a wave, travelling with a speed $v\; =\; frac\{1\}\{sqrt\{LC$.

(Note that this propagation speed applies to the wave phenomenon on the line and has nothing to do with the electron drift velocity. In other words, the electrical impulse travels very close to the speed of light, although the electrons themselves travel only a few centimeters per second.) For a coaxial transmission line, made of perfect conductors with vacuum between them, it can be shown that this speed is equal to the speed of light.

Lossy transmission line

When the loss elements "R" and "G" are not negligible, the original differential equations describing the elementary segment of line become

:$frac\{partial\}\{partial\; x\}\; V(x,t)\; =-L\; frac\{partial\}\{partial\; t\}\; I(x,t)\; -\; R\; I(x,t)$

:$frac\{partial\}\{partial\; x\}\; I(x,t)\; =-C\; frac\{partial\}\{partial\; t\}\; V(x,t)\; -\; G\; V(x,t)$

By differentiating the first equation with respect to "x" and the second with respect to "t", and some algebraic manipulation, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:

:$frac\{partial^2\}$partial x}^2} V =L C frac{partial^2}partial t}^2} V +(R C + G L) frac{partial}{partial t} V + G R V

:$frac\{partial^2\}$partial x}^2} I =L C frac{partial^2}partial t}^2} I +(R C + G L) frac{partial}{partial t} I + G R I

Note that these equations resemble the homogeneous wave equation with extra terms in "V" and "I" and their first derivatives. These extra terms cause the signal to decay and spread out with time and distance. If the transmission line is slightly lossy (small "R" and "G" = 0), signal strength will decay over distance as e^-α"x", where α = "R"/2Z₀

Direction of signal propagations

The wave equations above indicate that there are two solutions for the travelling wave: one forward and one reverse. Assuming a simplification of being lossless (requiring both "R"=0 and "G"=0) the solution can be represented as:

:$V(x,t)\; =\; \{\; f\_1(omega\; t\; -\; kx)\; +\; f\_2(omega\; t\; +\; kx)\}$

where::$k\; =\; omega\; sqrt\{LC\}\; =\; \{omega\; over\; v\}$

:$k$ is called the wavenumber and has units of radians per meter,

:ω is the angular frequency (in radians per second),:$f\_1$ and $f\_2$ can be "any" functions whatsoever, and

:$v\; =\; \{\; frac\{1\}\{sqrt\{LC$ }

:is the waveform's speed of propagation.

"f"₁ represents a wave traveling from left to right in a positive x direction whilst "f"₂ represents a wave traveling from right to left. It can be seen that the instantaneous voltage at any point x on the line is the sum of the voltages due to both waves.

Since the current "I" is related to the voltage "V" by the telegrapher's equations, we can write

:$I(x,t)\; =\; \{\; f\_1(omega\; t-kx)\; over\; Z\_0\; \}\; -\; \{\; f\_2(omega\; t+kx)\; over\; Z\_0\; \}$

where $Z\_0$ is the characteristic impedance of the transmission line, which, for a lossless line is given by

:$Z\_0\; =\; sqrt\; \{\; \{L\; over\; C$

Signal pattern examples

Depending on the parameters of the telegraph equation, the changes of the signal level distribution along the length of the single-dimensional transmission media may take the shape of the simple wave, wave with decrement, or the diffusion-like pattern of the telegraph equation. The shape of the diffusion-like pattern is caused by the effect of the shunt capacity.