# Telegrapher's equations

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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.

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named volt. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for volt uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by V, but more often simply as V, for instance in the context of Ohm's or Kirchhoff's circuit laws.

An electric current is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

## Contents

The theory applies to transmission lines of frequencies from direct current through high-frequency transmission lines. Examples famously include telegraph wires, as well as 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 transmission lines.

Direct current (DC) is the unidirectional flow of an electric charge. A battery is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even through a vacuum as in electron or ion beams. The electric current flows in a constant direction, distinguishing it from alternating current (AC). A term formerly used for this type of current was galvanic current.

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around twenty thousand times per second to around three hundred billion times per second. This is roughly between the upper limit of audio frequencies and the lower limit of infrared frequencies; these are the frequencies at which energy from an oscillating current can radiate off a conductor into space as radio waves. Different sources specify different upper and lower bounds for the frequency range.

## Distributed components

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:

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

• 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 electrical resistance of an object is a measure of its opposition to the flow of electric current. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S).

The ohm is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm. Although several empirically derived standard units for expressing electrical resistance were developed in connection with early telegraphy practice, the British Association for the Advancement of Science proposed a unit derived from existing units of mass, length and time and of a convenient size for practical work as early as 1861. The definition of the ohm was revised several times. Today, the definition of the ohm is expressed from the quantum Hall effect.

In electromagnetism and electronics, inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it.

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.

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively and equivalently it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

### Role of different components

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.

Inertia is the resistance of any physical object to any change in its velocity. This includes changes to the object's speed, or direction of motion. An aspect of this property is the tendency of objects to keep moving in a straight line at a constant speed, when no forces act upon them.

Restoring force, in a physics context, is a force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system. The restoring force is often referred to in simple harmonic motion. The force which is responsible to restore original size and shape is called restoring force.

### Values of primary parameters for telephone cable

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

FrequencyRLGC
Hz Ω kmΩ1000  ft mH kmmH1000 ft µS kmµS1000 ft nF kmnF1000 ft
1 Hz172.2452.500.61290.18680.0000.00051.5715.72
1 kHz172.2852.510.61250.18670.0720.02251.5715.72
10 kHz172.7052.640.60990.18590.5310.16251.5715.72
100 kHz191.6358.410.58070.17703.3271.19751.5715.72
1 MHz463.59141.300.50620.154329.1118.87351.5715.72
2 MHz643.14196.030.48620.148253.20516.21751.5715.72
5 MHz999.41304.620.46750.1425118.07435.98951.5715.72

More extensive tables and tables for other gauges, temperatures and types are available in Reeve. [1] Chen [2] 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.” [3] 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 [2] 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.

## Lossless transmission

When ωL >> R and ωC >> G, resistance can be neglected, and the transmission line is considered as an ideal lossless structure. In this case, the model depends only on the L and C elements. The Telegrapher's Equations then describe the relationship between the voltage V and the current I along the transmission line, each of which is a function of position x and time t:

${\displaystyle V=V(x,t)}$
${\displaystyle I=I(x,t)}$

### The equations for lossless transmission lines

The equations themselves consist of a pair of coupled, first-order, partial differential equations. The first equation shows that the induced voltage is related to the time rate-of-change of the current through the cable inductance, while the second shows, similarly, that the current drawn by the cable capacitance is related to the time rate-of-change of the voltage.

${\displaystyle {\frac {\partial V}{\partial x}}=-L{\frac {\partial I}{\partial t}}}$
${\displaystyle {\frac {\partial I}{\partial x}}=-C{\frac {\partial V}{\partial t}}}$

The Telegrapher's Equations are developed in similar forms in the following references: Kraus, [4] Hayt, [5] Marshall, [6] Sadiku, [7] Harrington, [8] Karakash, [9] and Metzger. [10]

These equations may be combined to form two exact wave equations, one for voltage V, the other for current I:

${\displaystyle {\frac {\partial ^{2}V}{{\partial t}^{2}}}-u^{2}{\frac {\partial ^{2}V}{{\partial x}^{2}}}=0}$
${\displaystyle {\frac {\partial ^{2}I}{{\partial t}^{2}}}-u^{2}{\frac {\partial ^{2}I}{{\partial x}^{2}}}=0}$

where

${\displaystyle u={\frac {1}{\sqrt {LC}}}}$

is the propagation speed of waves traveling through the transmission line. For transmission lines made of parallel perfect conductors with vacuum between them, this speed is equal to the speed of light.

In the case of sinusoidal steady-state, the voltage and current take the form of single-tone sine waves:

${\displaystyle V(x,t)=\mathrm {Re} \{V(x)\cdot e^{j\omega t}\}}$
${\displaystyle I(x,t)=\mathrm {Re} \{I(x)\cdot e^{j\omega t}\}}$,

where ${\displaystyle \omega }$ is the angular frequency of the steady-state wave. In this case, the Telegrapher's equations reduce to

${\displaystyle {\frac {dV}{dx}}=-j\omega LI}$
${\displaystyle {\frac {dI}{dx}}=-j\omega CV}$

Likewise, the wave equations reduce to

${\displaystyle {\frac {d^{2}V}{dx^{2}}}+k^{2}V=0}$
${\displaystyle {\frac {d^{2}I}{dx^{2}}}+k^{2}I=0}$

where k is the wave number:

${\displaystyle k=\omega {\sqrt {LC}}={\omega \over u}.}$

Each of these two equations is in the form of the one-dimensional Helmholtz equation.

In the lossless case, it is possible to show that

${\displaystyle V(x)=V_{1}e^{-jkx}+V_{2}e^{+jkx}}$

and

${\displaystyle I(x)={V_{1} \over Z_{0}}e^{-jkx}-{V_{2} \over Z_{0}}e^{+jkx}}$

where ${\displaystyle k}$ is a real quantity that may depend on frequency and ${\displaystyle Z_{0}}$ is the characteristic impedance of the transmission line, which, for a lossless line is given by

${\displaystyle Z_{0}={\sqrt {L \over C}}}$

and ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$ are arbitrary constants of integration, which are determined by the two boundary conditions (one for each end of the transmission line).

This impedance does not change along the length of the line since L and C are constant at any point on the line, provided that the cross-sectional geometry of the line remains constant.

The lossless line and distortionless line are discussed in Sadiku, [11] and Marshall, [12]

### General solution

The general solution of the wave equation for the voltage is the sum of a forward traveling wave and a backward traveling wave:

${\displaystyle V(x,t)\ =\ f_{1}(x-ut)+f_{2}(x+ut)}$

where

• ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ can be any functions whatsoever, and
• ${\displaystyle u={\frac {1}{\sqrt {LC}}}}$ is the waveform's propagation speed (also known as phase velocity).

f1 represents a wave traveling from left to right in a positive x direction whilst f2 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

${\displaystyle I(x,t)\ =\ {\frac {f_{1}(x-ut)}{Z_{0}}}-{\frac {f_{2}(x+ut)}{Z_{0}}}}$

## Lossy transmission line

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

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

By differentiating both equations with respect to x, and some algebraic manipulation, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:

{\displaystyle {\begin{aligned}{\frac {\partial ^{2}}{{\partial x}^{2}}}V&=LC{\frac {\partial ^{2}}{{\partial t}^{2}}}V+(RC+GL){\frac {\partial }{\partial t}}V+GRV\\{\frac {\partial ^{2}}{{\partial x}^{2}}}I&=LC{\frac {\partial ^{2}}{{\partial t}^{2}}}I+(RC+GL){\frac {\partial }{\partial t}}I+GRI\\\end{aligned}}}

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 only slightly lossy (R << ωL and G << ωC), signal strength will decay over distance as e−αx, where α ≈ R/2Z0 + GZ0/2. [13] :130

## 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 medium 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 capacitance.

## Solutions of the telegrapher's equations as circuit components

The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the top figure implements the solutions of the telegrapher's equations. [14]

The bottom circuit is derived from the top circuit by source transformations. [15] It also implements the solutions of the telegrapher's equations.

The solution of the telegrapher's equations can be expressed as an ABCD type two-port network with the following defining equations [16]

{\displaystyle {\begin{aligned}V_{1}&=V_{2}\cosh(\gamma x)+I_{2}Z\sinh(\gamma x)\\I_{1}&=V_{2}{\frac {1}{Z}}\sinh(\gamma x)+I_{2}\cosh(\gamma x).\\\end{aligned}}}

The ABCD type two-port gives ${\displaystyle V_{1}\,}$ and ${\displaystyle I_{1}\,}$ as functions of ${\displaystyle V_{2}\,}$ and ${\displaystyle I_{2}\,}$. Both of the circuits above, when solved for ${\displaystyle V_{1}\,}$ and ${\displaystyle I_{1}\,}$ as functions of ${\displaystyle V_{2}\,}$ and ${\displaystyle I_{2}\,}$ yield exactly the same equations.

In the bottom circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the forward wave and the other carries the backward wave. The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from ${\displaystyle V_{1}\,}$ to ${\displaystyle V_{2}\,}$ in the sense that ${\displaystyle V_{1}\,}$, ${\displaystyle V_{2}\,}$, ${\displaystyle I_{1}\,}$ and ${\displaystyle I_{2}}$ would be same whether this circuit or an actual transmission line was connected between ${\displaystyle V_{1}\,}$ and ${\displaystyle V_{2}\,}$. There is no implication that there are actually amplifiers inside the transmission line.

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the bottom only models the differential mode.

In the top circuit, the voltage doublers, the difference amplifiers and impedances Z(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable or a microstrip line.

These are not the only possible equivalent circuits.

## Notes

1. Reeve 1995 , p. 558
2. Chen 2004 , p. 26
3. Terman 1943 , p. 112
4. Kraus 1989 , pp. 380–419
5. Hayt 1989 , pp. 382–392
6. Marshall 1987 , pp. 359–378
7. Sadiku 1989 , pp. 497–505
8. Harrington 1961 , pp. 61–65
9. Karakash 1950 , pp. 5–14
10. Metzger 1969 , pp. 1–10
11. Sadiku 1989 , pp. 501–503
12. Marshall 1987 , pp. 369–372
13. Miano, Giovanni; Maffucci, Antonio (2001). Transmission Lines and Lumped Circuits. Academic Press. ISBN   0-12-189710-9. This book uses μ instead of α.
14. Hayt 1971 , pp. 7377
15. Karakash 1950 , p. 44

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