# Telegrapher's equations

The telegrapher's equations (or just telegraph equations) are a pair of coupled, linear partial differential equations that 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 in the 1880s. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line.

The theory applies to transmission lines of all frequencies including direct current and high-frequency. Originally developed to describe telegraph wires, the theory can also be applied to radio frequency conductors, audio frequency (such as telephone lines), low frequency (such as power lines), and pulses of direct current. It can also be used to electrically model wire radio antennas as truncated single-conductor transmission lines.[1]:7–10 [2]:232

## Distributed components

Schematic representation of the elementary components of a transmission line.

The telegrapher's equations, like all other equations describing electrical phenomena, result from 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 ${\displaystyle R}$ of the conductors is represented by a series resistor (expressed in ohms per unit length).
• The distributed inductance ${\displaystyle L}$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
• The capacitance ${\displaystyle C}$ between the two conductors is represented by a shunt capacitor C (farads per unit length).
• The conductance ${\displaystyle G}$ of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (siemens per unit length). This resistor in the model has a resistance of ${\displaystyle 1/G}$ ohms.

The model consists of an infinite series of the infinitesimal 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 ${\displaystyle R'}$, ${\displaystyle L'}$, ${\displaystyle C'}$, and ${\displaystyle G'}$ 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 characteristic impedance, the propagation constant, attenuation constant and phase constant. All these constants are constant with respect to time, voltage and current. They may be non-constant functions of frequency.

### Role of different components

Schematic showing a wave flowing rightward down a lossless transmission line. Black dots represent electrons, and the arrows show the electric field.

The role of the different components can be visualized based on the animation at right.

• The inductance L makes it look like the current has inertia—i.e. with a large inductance, it is difficult to increase or decrease the current flow at any given point. Large inductance makes the wave move more slowly, just as waves travel more slowly down a heavy rope than a light one. Large inductance also increases the wave impedance (lower current for the same voltage).
• The capacitance C controls how much the bunched-up electrons within each conductor repel the electrons in the other conductor. By absorbing some of these bunched up electrons, the speed of the wave and its strength (voltage) are both reduced. With a larger capacitance, there is less repulsion, because the other line (which always has the opposite charge) partly cancels out these repulsive forces within each conductor. Larger capacitance equals (weaker restoring force)s making the wave move slightly slower, and also gives the transmission line a lower impedance (higher current for the same voltage).
• R corresponds to resistance within each line, and G allows current to flow from one line to the other. The figure at right shows a lossless transmission line, where both R and G are 0.

### Values of primary parameters for telephone cable

Representative parameter data for 24 gauge telephone polyethylene insulated cable (PIC) at 70 °F (294 K)

Frequency R L G C
Hz Ωkm Ω1000 ft mHkm mH1000 ft µSkm µS1000 ft nFkm nF1000 ft
1 Hz 172.24 52.50 0.6129 0.1868 0.000 0.000 51.57 15.72
1 kHz 172.28 52.51 0.6125 0.1867 0.072 0.022 51.57 15.72
10 kHz 172.70 52.64 0.6099 0.1859 0.531 0.162 51.57 15.72
100 kHz 191.63 58.41 0.5807 0.1770 3.327 1.197 51.57 15.72
1 MHz 463.59 141.30 0.5062 0.1543 29.111 8.873 51.57 15.72
2 MHz 643.14 196.03 0.4862 0.1482 53.205 16.217 51.57 15.72
5 MHz 999.41 304.62 0.4675 0.1425 118.074 35.989 51.57 15.72

More extensive tables and tables for other gauges, temperatures and types are available in Reeve.[3] Chen[4] gives the same data in a parameterized form that he states is usable up to 50 MHz.

The variation of ${\displaystyle R}$ and ${\displaystyle L}$ is mainly due to skin effect and proximity effect.

The constancy of the capacitance is a consequence of intentional, careful design.

The variation of G can be inferred from Terman: “The power factor ... tends to be independent of frequency, since the fraction of energy lost during each cycle ... is substantially independent of the number of cycles per second over wide frequency ranges.”[5] A function of the form ${\displaystyle G(f)=G_{1}\left({\frac {f}{f_{1}}}\right)^{g_{\mathrm {e} }}}$ with ${\displaystyle g_{\mathrm {e} }}$ close to 1.0 would fit Terman’s statement. Chen [4] gives an equation of similar form.

G in this table can be modeled well with

${\displaystyle f_{1}=1\;\mathrm {MHz} }$
${\displaystyle G_{1}=29.11\;\mathrm {\mu S/km} =8.873\;\mathrm {\mu S/{1000ft}} }$
${\displaystyle g_{\mathrm {e} }=0.87}$

Usually the resistive losses grow proportionately to ${\displaystyle f^{0.5}\,}$ and dielectric losses grow proportionately to ${\displaystyle f^{g_{\mathrm {e} }}\,}$ with ${\displaystyle g_{\mathrm {e} }>0.5}$ so at a high enough frequency, dielectric losses will exceed resistive losses. In practice, before that point is reached, a transmission line with a better dielectric is used. In long distance rigid coaxial cable, to get very low dielectric losses, the solid dielectric may be replaced by air with plastic spacers at intervals to keep the center conductor on axis.

### The equations

The telegrapher's equations are:

{\displaystyle {\begin{aligned}{\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)\\\end{aligned}}}

They can be combined to get two partial differential equations, each with only one dependent variable, either ${\displaystyle V}$ or ${\displaystyle I\,}$:

{\displaystyle {\begin{aligned}{\frac {~\ \partial ^{2}}{\partial x^{2}}}\ V(x,t)&-LC\ {\frac {~\ \partial ^{2}}{\ \partial t^{2}}}\ V(x,t)&=(RC+GL)\ {\frac {\ \partial }{\partial t}}\ V(x,t)&+GR\ V(x,t)\\{\frac {~\ \partial ^{2}}{\partial x^{2}}}\ I(x,t)&-LC\ {\frac {~\ \partial ^{2}}{\partial t^{2}}}\ I(x,t)&=(RC+GL)\ {\frac {\ \partial }{\partial t}}\ I(x,t)&+GR\ I(x,t)\\\end{aligned}}}

Except for the dependent variable (${\displaystyle \,V}$ or ${\displaystyle I\,}$) the formulas are identical.