Thetelegrapher's equations (ortelegraph equations) are a set of two coupled, linearpartial differential equations that modelvoltage andcurrent along a linear electricaltransmission line. The equations are important because they allow transmission lines to be analyzed usingcircuit theory.^[1] The equations and their solutions are applicable from 0 Hz (i.e.direct current) to frequencies at which the transmission line structure can support higher ordernon-TEM modes.^{[2]: 282–286} The equations can be expressed in both thetime domain and thefrequency domain. In the time domain the independent variables are distance and time. In the frequency domain the independent variables are distance${\displaystyle x}$ and eitherfrequency,${\displaystyle \omega }$, orcomplex frequency,${\displaystyle s}$. The frequency domain variables can be taken as theLaplace transform orFourier transform of the time domain variables or they can be taken to bephasors in which case the frequency domain equations can be reduced to ordinary differential equations of distance. An advantage of the frequency domain approach is thatdifferential operators in the time domain become algebraic operations in frequency domain.

The equations come fromOliver Heaviside who developed thetransmission line model starting with an August 1876 paper,On the Extra Current.^[3] The model demonstrates that theelectromagnetic waves can be reflected on the wire, and that wave patterns can form along the line. Originally developed to describetelegraph wires, the theory can also be applied toradio frequencyconductors, audio frequency (such astelephone lines), low frequency (such as power lines), and pulses of direct current.

The telegrapher's equations result fromcircuit theory. In a more practical approach, one assumes that theconductors are composed of an infinite series oftwo-port elementary components, each representing aninfinitesimally short segment of the transmission line:

• The distributedresistance${\displaystyle R}$ of the conductors is represented by a series resistor (expressed inohms per unit length). In practical conductors, at higher frequencies,${\displaystyle R}$ increases approximately proportional to the square root of frequency due to theskin effect.
• The distributedinductance${\displaystyle L}$ (due to themagnetic field around the wires,self-inductance, etc.) is represented by a seriesinductor (henries per unit length).
• Thecapacitance${\displaystyle C}$ between the two conductors is represented by ashuntcapacitor${\displaystyle C}$ (farads per unit length).
• Theconductance${\displaystyle G}$ 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${\displaystyle R_{\text{shunt}}=\frac{1}{G}\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{s}}$.${\displaystyle G}$ accounts for both bulkconductivity of the dielectric anddielectric loss. If the dielectric is an ideal vacuum, then${\displaystyle G\equiv 0}$.

The model consists of aninfinite series of the infinitesimal elements shown in the figure, and the values of the components are specifiedper unit length, so the picture of the component can be misleading. An alternative notation is to use${\displaystyle R^{\prime }}$,${\displaystyle L^{\prime }}$,${\displaystyle C^{\prime }}$, and${\displaystyle G^{\prime }}$ to emphasize that the values are derivatives with respect to length, and that the units of measure combine correctly. These quantities can also be known as theprimary line constants to distinguish from the secondary line constants derived from them, these being thecharacteristic impedance, thepropagation constant,attenuation constant andphase constant. All these constants are constant with respect to time, voltage and current. They may be non-constant functions of frequency.

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

InductanceL

The inductance couples current to energy stored in the magnetic field. It makes it look like the current hasinertia – i.e. with a large inductance, it is difficult to increase or decrease the current flow at any given point. Large inductanceL makes the wave move more slowly, just as waves travel more slowly down a heavy rope than a light string. Large inductance alsoincreases the line'ssurge impedance (more voltage needed to push the sameAC current through the line).

CapacitanceC

The capacitance couples voltage to the energy stored in the electric field. It controls how much the bunched-up electrons within each conductor repel, attract, or divert the electrons in theother conductor. By deflecting some of these bunched up electrons, the speed of the wave and its strength (voltage) are both reduced. With a larger capacitance,C, there is less repulsion, because theother line (which always has the opposite charge) partly cancels out these repulsive forceswithin each conductor. Larger capacitance equals weakerrestoring forces, making the wave move slightly slower, and also gives the transmission line alowersurge impedance (less voltage needed to push the same AC current through the line).

ResistanceR

Resistance corresponds to resistance interior to the two lines, combined. That resistanceR couples current toohmic losses thatdrop a little of the voltage along the line as heat deposited into the conductor, leaving the current unchanged. Generally, the line resistance is very low, compared to inductivereactanceωL at radio frequencies, and for simplicity is treated as if it were zero, with any voltage dissipation or wire heating accounted for as corrections to the "lossless line" calculation, or just ignored.

ConductanceG

Conductance between the lines represents how well current can "leak" from one line to the other. Conductance couples voltage todielectric loss deposited as heat into whatever serves as insulation between the two conductors.G reduces propagating current byshunting it between the conductors. Generally, wire insulation (including air) is quite good, and the conductance is almost nothing compared to the capacitivesusceptanceωC, and for simplicity is treated as if it were zero.

All four parametersL,C,R, andG depend on the material used to build the cable or feedline. All four change with frequency:R, andG tend to increase for higher frequencies, andL andC tend to drop as the frequency goes up.The figure at right shows a lossless transmission line, where bothR andG are zero, which is the simplest and by far most common form of the telegrapher's equations used, but slightly unrealistic (especially regardingR).

The variation ofR andL is mainly due toskin effect andproximity effect. The constancy of the capacitance is a consequence of intentional design.

The variation ofG can be inferred from a statement byFrederick Terman:^[5] "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."

A function of the form$${\displaystyle G(f)=G_1\cdot {\left(\frac{f}{f_1}\right)}^{g_{\mathrm{e}}}}$$withg_e close to1.0 would fit Terman's statement. Walter Chen gives an equation of similar form.^[6] WhereG( ) is conductivity as a function of frequency,G₁,f₁, andg_e are all real constants.

Usually the resistive losses (R) grow proportionately tof^1/2 and dielectric losses grow proportionately tof^g_e withg_e ≈ 1 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 rigidcoaxial 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 telegrapher's equations in the time domain are:

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

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

The telegrapher's equations in the frequency domain are developed in similar forms:^{[7][1][8]: 59–378 [9]: 497–505 [10][11][12]}Here,${\displaystyle {\mathbf{I}}_{\omega }(x)}$ and${\displaystyle {\mathbf{V}}_{\omega }(x)}$ arephasors, with the subscript${\displaystyle \omega }$ indicating the possible frequency-dependence of the parameters.

The first equation means that${\displaystyle {\mathbf{V}}_{\omega }(x)}$, the propagating voltage at point${\displaystyle x}$, is decreased by the voltage loss produced by${\displaystyle {\mathbf{I}}_{\omega }(x)}$, the current at that point passing through the seriesimpedance${\displaystyle R+j\omega L}$. The second equation means that${\displaystyle {\mathbf{I}}_{\omega }(x)}$, the propagating current at point${\displaystyle x}$, is decreased by the current loss produced by${\displaystyle {\mathbf{V}}_{\omega }(x)}$, the voltage at that point appearing across the shuntadmittance${\displaystyle G+j\omega C}$.

These equations may be combined to produce two uncoupled second-orderordinary differential equationswith$${\displaystyle \gamma \equiv \alpha +j\beta \equiv \sqrt{\left(R_{\omega }+j\omega L_{\omega }\right)\left(G_{\omega }+j\omega C_{\omega }\right)},}$$where${\displaystyle \alpha }$ is called theattenuation constant and${\displaystyle \beta }$ is called thephase constant.^[1]: 385

Working in the frequency domain has the benefit of dealing with bothsteady state andtransient problems in a similar fashion.^[13] In case of the latter the frequency${\displaystyle \omega }$ becomes acontinuous variable; a solution can be obtained by first solving the above (homogeneous) second-order ODEs and then applying theFourier inversion theorem.^[14](see§ Lossless sinusoidal steady-state below for an example of solving steady-state problems.)

Each of the preceding differential equations have twohomogeneous solutions in an infinite transmission line.

For the voltage equation

For the current equation

The negative sign in the previous equation indicates that the current in the reverse wave is traveling in the opposite direction.

Note:$${\displaystyle \begin{array}{r}{\mathbf{V}}_{\omega ,F}(x)={\mathbf{Z}}_c{\mathbf{I}}_{\omega ,F}(x),\\ {\mathbf{V}}_{\omega ,R}(x)={\mathbf{Z}}_c{\mathbf{I}}_{\omega ,R}(x),\end{array}}$$$${\displaystyle {\mathbf{Z}}_c=\sqrt{\frac{R_{\omega }+j\omega L_{\omega }}{G_{\omega }+j\omega C_{\omega }}},}$$where the following symbol definitions hold:

Symbol definitions

Symbol	Definition
${\displaystyle a}$	point at which the values of the forward waves are known
${\displaystyle b}$	point at which the values of the reverse waves are known
${\displaystyle {\mathbf{V}}_{\omega }(x)}$	value of the total voltage at pointx
${\displaystyle {\mathbf{V}}_{\omega ,F}(x)}$	value of the forward voltage wave at pointx
${\displaystyle {\mathbf{V}}_{\omega ,R}(x)}$	value of the reverse voltage wave at pointx
${\displaystyle {\mathbf{V}}_{\omega ,F}(a)}$	value of the forward voltage wave at pointa
${\displaystyle {\mathbf{V}}_{\omega ,R}(b)}$	value of the reverse voltage wave at pointb
${\displaystyle {\mathbf{I}}_{\omega }(x)}$	value of the total current at pointx
${\displaystyle {\mathbf{I}}_{\omega ,F}(x)}$	value of the forward current wave at pointx
${\displaystyle {\mathbf{I}}_{\omega ,R}(x)}$	value of the reverse current wave at pointx
${\displaystyle {\mathbf{I}}_{\omega ,F}(a)}$	value of the forward current wave at pointa
${\displaystyle {\mathbf{I}}_{\omega ,R}(b)}$	value of the reverse current wave at pointb
${\displaystyle {\mathbf{Z}}_c}$	characteristic impedance
${\displaystyle \gamma }$	propagation constant

Johnson gives the following solution,^{[2]: 739–741}where${\displaystyle \mathbf{H}\equiv e^{-\mathit{\gamma }x},}$ and${\displaystyle x}$ is the length of the transmission line.

In the special case where all the impedances are equal,${\displaystyle {\mathbf{Z}}_{\mathsf{L}}={\mathbf{Z}}_{\mathsf{S}}={\mathbf{Z}}_{\mathsf{C}},}$ the solution reduces to${\displaystyle \frac{{\mathbf{V}}_{\mathsf{L}}}{{\mathbf{V}}_{\mathsf{S}}}=\frac{1}{2}{e}^{-\mathit{\gamma }x}}$.

When${\displaystyle \omega L\gg R}$ and${\displaystyle \omega C\gg G}$, wire resistance and insulation conductance can be neglected, and the transmission line is considered as an ideal lossless structure. In this case, the model depends only on theL andC elements. The telegrapher's equations then describe the relationship between the voltageV and the currentI along the transmission line, each of which is a function of positionx and timet: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.$${\displaystyle \frac{\mathrm{\partial }V}{\mathrm{\partial }x}=-L\frac{\mathrm{\partial }I}{\mathrm{\partial }t}}$$$${\displaystyle \frac{\mathrm{\partial }I}{\mathrm{\partial }x}=-C\frac{\mathrm{\partial }V}{\mathrm{\partial }t}}$$

These equations may be combined to form twowave equations, one for voltage${\displaystyle V}$, the other for current${\displaystyle I}$:where$${\displaystyle \tilde{v}\equiv \frac{1}{\sqrt{LC}}}$$ is thepropagation 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 thespeed of light.

In the case ofsinusoidalsteady-state (i.e., when a pure sinusoidal voltage is applied andtransients have ceased) theangular frequency${\displaystyle \omega }$ is fixed and the voltage and current take the form of single-tone sine waves:^[15]In this case, the telegrapher's equations reduce to

Likewise, the wave equations reduce to one-dimensionalHelmholtz equationswherek is thewave number:$${\displaystyle k:=\omega \sqrt{LC}=\frac{\omega }{\tilde{v}}.}$$In the lossless case, it is possible to show that$${\displaystyle V(x)=V_1{e}^{-jkx}+V_2{e}^{+jkx},}$$and$${\displaystyle I(x)=\frac{V_1}{Z_{\mathsf{o}}}{e}^{-jkx}-\frac{V_2}{Z_{\mathsf{o}}}{e}^{+jkx},}$$where in this special case,${\displaystyle k}$ is a real quantity that may depend on frequency and${\displaystyle Z_{\mathsf{o}}}$ is thecharacteristic impedance of the transmission line, which, for a lossless line is given by$${\displaystyle Z_{\mathsf{o}}=\sqrt{\frac{L}{C}},}$$and${\displaystyle V_1}$ and${\displaystyle V_2}$ are arbitrary constants of integration, which are determined by the twoboundary conditions (one for each end of the transmission line).

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

In the loss-free case(${\displaystyle R=G=0}$), the general solution of the wave equation for the voltage is the sum of a forward traveling wave and a backward traveling wave:$${\displaystyle V(x,t)=f_1(x-\tilde{v}t)+f_2(x+\tilde{v}t)}$$where

• ${\displaystyle f_1}$ and${\displaystyle f_2}$ can beany twoanalytic functions, and
• ${\displaystyle \tilde{v}\equiv \frac{1}{\sqrt{LC}}}$ is the waveform'spropagation speed (also known asphase velocity).

Here,${\displaystyle f_1}$ represents the amplitude profile of a wave traveling from left to right – in a positive${\displaystyle x}$ direction – whilst${\displaystyle f_2}$ represents the amplitude profile of a wave traveling from right to left. It can be seen that the instantaneous voltage at any point${\displaystyle x}$ on the line is the sum of the voltages due to both waves.

Using the current${\displaystyle I}$ and voltage${\displaystyle V}$ relations given by the telegrapher's equations, we can write$${\displaystyle I(x,t)=\frac{1}{Z_{\mathsf{o}}}[f_1(x-\tilde{v}t)-f_2(x+\tilde{v}t)].}$$

When the loss elements${\displaystyle R}$ and${\displaystyle G}$ are too substantial to ignore, the differential equations describing the elementary segment of line are

By differentiating both equations with respect tox, and some algebra, we obtain a pair ofdamped,dispersivehyperbolic partial differential equations each involving only one unknown:

These equations resemble the homogeneous wave equation with extra terms inV andI 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(${\displaystyle R\ll \omega L}$ and${\displaystyle G\ll \omega C}$), signal strength will decay over distance as${\displaystyle e^{-\alpha x}}$ where${\displaystyle \alpha \approx \frac{R}{2Z_0}+\frac{G{Z}_0}{2}}$.^[16]

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The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the figure implements the solutions of the telegrapher's equations.^[17]

The solution of the telegrapher's equations can be expressed as anABCD two-port network with the following defining equations^{[11]: 5–14, 44}where$${\displaystyle Z_{\mathsf{o}}\equiv \sqrt{\frac{R_{\omega }+j\omega L_{\omega }}{G_{\omega }+j\omega C_{\omega }}},}$$and$${\displaystyle \gamma \equiv \sqrt{\left(R_{\omega }+j\omega L_{\omega }\right)\left(G_{\omega }+j\omega C_{\omega }\right)},}$$just as in the preceding sections. The line parametersR_ω,L_ω,G_ω, andC_ω are subscripted byω to emphasize that they could be functions of frequency.

The ABCD type two-port gives${\displaystyle V_1}$ and${\displaystyle I_1}$ as functions of${\displaystyle V_2}$ and${\displaystyle I_2}$. The voltage and current relations are symmetrical: Both of the equations shown 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 relations, merely with subscripts "1" and "2" reversed, and the${\displaystyle \sinh }$ terms' signs made negative ("1"→"2" direction is reversed "1"←"2", hence the sign change).

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which is called theshield, sheath, common, earth, or ground. So every two-wire balanced transmission line has two modes which are nominally called thedifferential mode andcommon mode. The circuit shown in the bottom diagram only can model the differential mode.

In the top circuit, the voltage doublers, the difference amplifiers, and impedancesZ_o(s) account for the interaction of the transmission line with the external circuit. This circuit is a useful equivalent for anunbalanced transmission line like acoaxial cable.

These are not unique: Other equivalent circuits are possible.

• Complex power
• Law of squares,Lord Kelvin's preliminary work on this subject
• LC circuit
• Reflections of signals on conducting lines
• RLC circuit
• Smith chart

• Hyperbolic partial differential equations
• Distributed element circuits
• Transmission lines

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