View source for Telegrapher's equations

{{Short description|Mathematical descriptions of transmission line voltage and current}}
{{use dmy dates|date=September 2021|cs1-dates=y}}
The '''telegrapher's equations''' (or just '''telegraph equations''') are a set of two coupled, linear equations that predict the [[voltage]] and [[Electric current|current]] distributions on a linear electrical [[transmission line]]. The equations are important because they allow transmission lines to be analyzed using [[circuit theory]].<ref name="Hayt_1989"/> 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 order [[Transverse mode|non-TEM modes]].<ref name="Johnson-Graham-2003"/>{{rp|pp=282–286}} The equations can be expressed in both the [[time domain]] and the [[frequency domain]]. In the time domain the independent variables are distance and time. The resulting time domain equations are partial differential equations of both time and distance. In the frequency domain the independent variables are distance <math> x </math> and either [[frequency]], {{nowrap|<math>\omega</math>,}} or [[complex frequency]], {{nowrap|<math> s </math>.}} The frequency domain variables can be taken as the [[Laplace transform]] or [[Fourier transform]] of the time domain variables or they can be taken to be [[phasors]]. The resulting frequency domain equations are ordinary differential equations of distance. An advantage of the frequency domain approach is that [[differential operator]]s in the time domain become algebraic operations in frequency domain.

The equations come from [[Oliver Heaviside]] who developed the ''[[transmission line model]]'' starting with an August&nbsp;1876 paper, ''On the Extra Current''.<ref name="Hunt-2005">{{cite book | last = Hunt | first = Bruce J. | date = 2005 | title = The Maxwellians | publisher = [[Cornell University Press]] | place = Ithaca, NY, USA | isbn = 0-80148234-8 | pages = 66–67 }}</ref> The model demonstrates that the [[electromagnetic waves]] can be reflected on the wire, and that wave patterns can form along the line. Originally developed to describe [[telegraph]] wires, the theory can also be applied to [[radio frequency]] [[Conductor (material)|conductor]]s, audio frequency (such as [[telephone lines]]), low frequency (such as power lines), and pulses of [[direct current]].

==Distributed components==
[[File:Transmission line element.svg|thumb|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 [[conductor (electricity)|conductor]]s are composed of an infinite series of [[Two-port network|two-port]] elementary components, each representing an [[Infinitesimal|infinitesimally]] short segment of the transmission line:

* The distributed [[Electrical resistance and conductance|resistance]] <math> R </math> of the conductors is represented by a series resistor (expressed in [[ohm]]s per unit length). In practical conductors, at higher frequencies, <math> R </math> increases approximately proportional to the square root of frequency due to the [[skin effect]].
* The distributed [[inductance]] <math> L </math> (due to the [[magnetic field]] around the wires, [[self-inductance]], etc.) is represented by a series [[inductor]] ([[henries]] per unit length).
* The [[capacitance]] <math> C </math> between the two conductors is represented by a [[Shunt (electrical)|shunt]] [[capacitor]] <math> C </math> ([[farad]]s per unit length).
* The [[Electrical resistance and conductance|conductance]] <math> G </math> of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire ([[Siemens (unit)|siemens]] per unit length). This resistor in the model has a resistance of {{nowrap|<math> R_\text{shunt} = \frac{1}{ G }\, \mathrm{ohms}</math>.}} <math>\ G\ </math> accounts for both bulk [[Electrical resistivity and conductivity|conductivity]] of the dielectric and [[dielectric loss]]. If the dielectric is an ideal vacuum, then {{nowrap|<math> G \equiv 0 \,</math>.}}

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 {{nowrap|<math> R' </math>,}} {{nowrap|<math> L' </math>,}} {{nowrap|<math> C' </math>,}} and {{nowrap|<math> G' </math>}} 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 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===
[[File:Transmission line animation3.gif|right|thumb|300px|Schematic showing a wave flowing rightward down a lossless transmission line. Black dots represent [[electron]]s, and the arrows show the electric field.]]
The role of the different components can be visualized based on the animation at right.
; Inductance {{mvar|L}}: The inductance couples current to energy stored in the magnetic field. It 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 {{mvar|L}} makes the wave move more slowly, just as waves travel more slowly down a heavy rope than a light string. Large inductance also ''increases'' the line's [[surge impedance]] (''more'' voltage needed to push the same [[alternating current|{{sc|AC}}]] current through the line).
; Capacitance {{mvar|C}}: 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 the ''other'' 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, {{mvar|C}}, 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'' [[surge impedance]] (''less'' voltage needed to push the same [[alternating current|{{sc|AC}}]] current through the line).
; Resistance {{mvar|R}}: Resistance corresponds to resistance interior to the two lines, combined. That resistance {{mvar|R}} couples current to [[joule heating|ohmic loss]]es that [[Voltage drop|drop]] 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 inductive [[Electrical reactance|reactance]] {{mvar|&omega;L}} 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.
; Conductance {{mvar|G}}: Conductance between the lines represents how well current can "leak" from one line to the other. Conductance couples voltage to [[dielectric loss]] deposited as heat into whatever serves as insulation between the two conductors. {{mvar|G}} reduces propagating current by [[Shunt (electrical)|shunting]] it between the conductors. Generally, wire insulation (including air) is quite good, and the conductance is almost nothing compared to the capacitive [[susceptance]] {{mvar|ωC}}, and for simplicity is treated as if it were zero.

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

===Values of primary parameters for telephone cable===

Representative parameter data for 24&nbsp;[[American wire gauge|gauge]] telephone polyethylene insulated cable (PIC) at 70&nbsp;°F (294&nbsp;K)
:{| class="wikitable" style="text-align:right;"
|-
! Frequency
! colspan="2" | {{mvar|R}}
! colspan="2" | {{mvar|L}}
! colspan="2" | {{mvar|G}}
! colspan="2" | {{mvar|C}}
|-
! [[Hertz|Hz]]
! {{sfrac|[[Ohm (unit)|Ω]]|km}}
! {{sfrac|[[Ohm (unit)|Ω]]|1000&nbsp;[[foot (unit)|ft]]}}
! {{sfrac|[[Henry (unit)|μH]]|km}}
! {{sfrac|[[Henry (unit)|μH]]|1000&nbsp;ft}}
! {{sfrac|[[Siemens (unit)|μS]]|km}}
! {{sfrac|[[Siemens (unit)|μS]]|1000&nbsp;ft}}
! {{sfrac|[[Farad|nF]]|km}}
! {{sfrac|[[Farad|nF]]|1000&nbsp;ft}}
|-
|align=center|    1&nbsp;Hz || 172.24 ||align=right|  52.50 || 612.9 || 186.8 ||align=right|  0.000 ||align=right| 0.000 || 51.57 || 15.72
|-
|align=center|   1&nbsp;kHz || 172.28 ||align=right|  52.51 || 612.5 || 186.7 ||align=right|  0.072 ||align=right| 0.022 || 51.57 || 15.72
|-
|align=center|  10&nbsp;kHz || 172.70 ||align=right|  52.64 || 609.9 || 185.9 ||align=right|  0.531 ||align=right| 0.162 || 51.57 || 15.72
|-
|align=center| 100&nbsp;kHz || 191.63 ||align=right|  58.41 || 580.7 || 177.0 ||align=right|  3.327 ||align=right| 1.197 || 51.57 || 15.72
|-
|align=center|   1&nbsp;MHz || 463.59 ||align=right| 141.30 || 506.2 || 154.3 ||align=right| 29.111 ||align=right| 8.873 || 51.57 || 15.72
|-
|align=center|   2&nbsp;MHz || 643.14 ||align=right| 196.03 || 486.2 || 148.2 ||align=right| 53.205 ||align=right| 16.217 || 51.57 || 15.72
|-
|align=center|   5&nbsp;MHz || 999.41 ||align=right| 304.62 || 467.5 || 142.5 ||align=right| 118.074 ||align=right| 35.989 || 51.57 || 15.72
|}
This data is from {{harvp|Reeve|1995|p=558}}.<ref name=Reeve-1995>{{cite book | last = Reeve | first = Whitman D. | year = 1995 | title = Subscriber Loop Signaling and Transmission Handbook | page = 558 | publisher = [[IEEE Press]] | isbn = 0-7803-0440-3 | url = https://archive.org/details/subscriberloopsi00reev }}</ref> The variation of <math>\ R\ </math> and <math>\ L\ </math> is mainly due to [[skin effect]] and [[proximity effect (electromagnetism)|proximity effect]]. The constancy of the capacitance is a consequence of intentional design.

The variation of <math>\ G\ </math> can be inferred from a statement by {{harvp|Terman|1943|p=112}}:
: "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."<ref name=Terman-1943>{{cite book | last = Terman | first = Frederick Emmons | date = 1943 | title = Radio Engineers' Handbook | edition = 1st | page = 112 | publisher = [[McGraw-Hill]] }}</ref>

A function of the form
<math display="block"> G(f) = G_1 \cdot \left( \frac {f}{f_1}\right)^{\ g_\mathrm{e}} </math>
with <math> g_\mathrm{e} </math> close to {{math|1.0}} would fit Terman's statement. {{harvp|Chen|2004|p=26}} gives an equation of similar form.<ref name=Chen-2004-p26>{{cite book | last = Chen | first = Walter Y. | year = 2004 | title = Home Networking Basics | publisher = [[Prentice Hall]] | isbn = 0-13-016511-5 | page = 26 }}</ref> Where <math>\ G(\cdot)\ </math> is conductivity as a function of frequency, <math>\ G_1,\ f_1\ ,</math> and <math>\ g_e\ </math> are all real constants.

Usually the resistive losses (<math>\ R\ </math>) grow proportionately to <math>\ f^{\ 1/2}\ </math> and dielectric losses grow proportionately to <math>\ f^{\ g_\mathrm{e}}\ </math> with <math>\ g_\mathrm{e} \approx 1\ </math> 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 equation==
===Time domain===
The telegrapher's equations in the time domain are:
<math display="block">\begin{align}
\frac{\partial}{\partial x} V(x,t) &= -L \, \frac{\partial}{\partial t} I(x,t) - R I(x,t) \\[1ex]
\frac{\partial}{\partial x} I(x,t) &= -C \, \frac{\partial}{\partial t} V(x,t) - G V(x,t)
\end{align}</math>

They can be combined to get two partial differential equations, each with only one dependent variable, either <math> V </math> or {{nowrap|<math> I </math>:}}
<math display="block">\begin{align}
\frac{\partial^2}{\partial x^2} V(x,t) - LC \, \frac{\partial^2}{\partial t^2} V(x,t) &= \left( RC + GL \right) \frac{\partial}{\partial t} V(x,t) + GR \, V(x,t) \\[1ex]
\frac{\partial^2}{\partial x^2} I(x,t) - LC \, \frac{\partial^2}{\partial t^2} I(x,t) &= \left( RC + GL \right) \frac{\partial}{\partial t} I(x,t) + GR \, I(x,t)
\end{align}</math>

Except for the dependent variable (<math> V </math> or <math> I </math>) the formulas are identical.

===Frequency domain===
The telegrapher's equations in the frequency domain are developed in similar forms in the following references: Kraus,<ref name="Kraus_1984">{{cite book | last = Kraus | first = John D. | author-link = John Kraus | year = 1984 | title = Electromagnetics | edition = 3rd | publisher = McGraw-Hill | isbn = 0-07-035423-5 | pages = 380–419 }}</ref>
Hayt,<ref name="Hayt_1989">{{cite book | last = Hayt | first = William H. | year = 1989 | title = Engineering Electromagnetics | edition = 5th | publisher= McGraw-Hill | isbn = 0070274061 | url = https://archive.org/details/engineeringelect5thhayt | via = [[Internet Archive]] (archive.org) | pages = 381–392}}</ref>
Marshall,<ref name="Marshall_1987">{{cite book | last1 = Marshall | first1 = Stanley V. | last2 = Skitek | first2 = Gabriel G. | year = 1987 | title = Electromagnetic Concepts and Applications | edition = 2nd | publisher = Prentice-Hall | isbn = 0-13-249004-8 }}</ref>{{rp|pp=59–378}}
Sadiku,<ref name="Sadiku_1989">{{cite book | last = Sadiku | first = Matthew N.O. | year = 1989 | title = Elements of Electromagnetics | edition = 1st | publisher = Saunders College Publishing | isbn = 0-03-013484-6 }}</ref>{{rp|pp=497–505}}
Harrington,<ref name="Harrington_1961">{{cite book | last = Harrington | first = Roger F. | year = 1961 | title = Time-Harmonic Electromagnetic Fields | edition = 1st | publisher = McGraw-Hill | isbn = 0-07-026745-6 | pages = 61–65 }}</ref>
Karakash,<ref name="Karakash_1950">{{cite book | last = Karakash | first = John J. | year = 1950 | title = Transmission lines and Filter Networks | edition = 1st | publisher = Macmillan | pages = 5–14}}</ref>
Metzger.<ref name="Metzger_1969">{{cite book | last1 = Metzger | first1 = Georges | last2 = Vabre | first2 = Jean-Paul | year = 1969 | title = Transmission Lines with Pulse Excitation | edition = 1st | publisher = Academic Press | lccn = 69-18342 | pages = 1–10 }}</ref>
<math display="block">\begin{align}
\frac{\partial}{\partial x} \mathbf{V}_\omega(x) &= -\left( j \omega L_\omega + R_\omega \right) \mathbf{I}_\omega(x) ,
\\[1ex]
\frac{\partial}{\partial x} \mathbf{I}_\omega(x) &= -\left( j \omega C_\omega + G_\omega \right) \mathbf{V}_\omega(x) .
\end{align}</math>
The first equation means that {{nowrap|<math> \mathbf{V}_\omega(x) \,</math>,}} the propagating voltage at point {{nowrap|<math> x </math>,}} is decreased by the voltage loss produced by {{nowrap|<math> \mathbf{I}_\omega(x) \, </math>,}} the current at that point passing through the series [[Electrical impedance|impedance]] {{nowrap|<math> R + j \omega L </math>.}} The second equation means that {{nowrap|<math> \mathbf{I}_\omega(x) \,</math>,}} the propagating current at point {{nowrap|<math> x </math>,}} is decreased by the current loss produced by {{nowrap|<math> \mathbf{V}_\omega(x)\, </math>,}} the voltage at that point appearing across the shunt [[admittance]] {{nowrap|<math> G + j \omega C \,</math>.}}

The subscript ''ω'' indicates possible frequency dependence. <math> \mathbf{I}_\omega(x) </math> and <math> \mathbf{V}_\omega(x) </math> are [[phasors]].

These equations may be combined to produce two, single-variable [[partial differential equation]]s
<math display="block"> \begin{align}
\frac{d^2}{dx^2} \mathbf{V}_\omega(x) &= \gamma^2 \mathbf{V}_\omega(x), \\[1ex]
\frac{d^2}{dx^2} \mathbf{I}_\omega(x) &= \gamma^2 \mathbf{I}_\omega(x),
\end{align} </math>
with
<math display="block"> \gamma \equiv \alpha + j \beta \equiv \sqrt { \left( R_\omega + j \omega L_\omega \right) \left( G_\omega + j \omega C_\omega\right) },</math>
where <math> \alpha </math> is called the [[attenuation constant]] and <math> \beta </math> is called the [[phase constant]].<ref name="Hayt_1989" />{{rp|p=385}}

==== Homogeneous solutions ====
Each of the preceding partial differential equations have two [[Homogeneous differential equation|homogeneous solutions]] in an infinite transmission line.

For the voltage equation
<math display="block"> \begin{align}
\mathbf{V}_{\omega,F}(x) = \mathbf{V}_{\omega,F}(a) \, e^{+\gamma (a-x)} \, ; \\[1ex]
\mathbf{V}_{\omega,R}(x) = \mathbf{V}_{\omega,R}(b) \, e^{-\gamma (b-x)} \, ;
\end{align}</math>
<math display="block"> \mathbf{V}_{\omega}(x) = \mathbf{V}_{\omega,F}(x) + \mathbf{V}_{\omega,R}(x) \, .</math>

For the current equation
<math display="block"> \begin{align}
\mathbf{I}_{\omega,F}(x) = \mathbf{I}_{\omega,F}(a) \, e^{+\gamma (a-x)} \, ; \\
\mathbf{I}_{\omega,R}(x) = \mathbf{I}_{\omega,R}(b) \, e^{-\gamma (b-x)} \, ;
\end{align}</math>
<math display="block"> \mathbf{I}_{\omega}(x) = \mathbf{I}_{\omega,F}(x) - \mathbf{I}_{\omega,R}(x) \, .</math>

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

Note:
<math display="block"> \begin{align}
\mathbf{V}_{\omega,F}(x) = \mathbf{Z}_c \, \mathbf{I}_{\omega,F}(x) , \\[0.4ex]
\mathbf{V}_{\omega,R}(x) = \mathbf{Z}_c \, \mathbf{I}_{\omega,R}(x) ,
\end{align}</math>
<math display="block"> \mathbf{Z}_c = \sqrt{ \frac{ R_\omega + j \omega L_\omega }{ G_\omega + j \omega C_\omega} }\, ,</math>
where the following symbol definitions hold:
:{| class="wikitable"
|+ Symbol definitions
|-
! Symbol !! Definition
|-
| <math> a </math> || point at which the values of the forward waves are known
|-
| <math> b </math> || point at which the values of the reverse waves are known
|-
| <math> \mathbf{V}_{\omega}(x) </math> || value of the total voltage at point {{mvar|x}}
|-
| <math> \mathbf{V}_{\omega,F}(x) </math> || value of the forward voltage wave at point {{mvar|x}}
|-
| <math> \mathbf{V}_{\omega,R}(x) </math> || value of the reverse voltage wave at point {{mvar|x}}
|-
| <math> \mathbf{V}_{\omega,F}(a) </math> || value of the forward voltage wave at point {{mvar|a}}
|-
| <math> \mathbf{V}_{\omega,R}(b) </math> || value of the reverse voltage wave at point {{mvar|b}}
|-
| <math> \mathbf{I}_{\omega}(x) </math> || value of the total current at point {{mvar|x}}
|-
| <math> \mathbf{I}_{\omega,F}(x) </math> || value of the forward current wave at point {{mvar|x}}
|-
| <math> \mathbf{I}_{\omega,R}(x) </math> || value of the reverse current wave at point {{mvar|x}}
|-
| <math> \mathbf{I}_{\omega,F}(a) </math> || value of the forward current wave at point {{mvar|a}}
|-
| <math> \mathbf{I}_{\omega,R}(b) </math> || value of the reverse current wave at point {{mvar|b}}
|-
| <math> \mathbf{Z}_{c} </math> || [[Characteristic impedance]]
|}

====Finite length====
[[File:Coaxial transmission line wih one source and one load.jpg|thumb|right|Coaxial transmission line with one source and one load]]

Johnson gives the following solution,<ref name="Johnson-Graham-2003">{{cite book | last1 = Johnson | first1 = Howard | last2 = Graham | first2 = Martin | year = 2003 | title = High Speed Signal Propagation | edition = 1st | publisher = Prentice-Hall | isbn = 0-13-084408-X }}</ref>{{rp|pp=739–741}}
<!-- {{clear}} -->
<math display="block">
\begin{align}
\frac{\mathbf{V}_\mathsf{L} }{ \mathbf{V}_\mathsf{S} }
&= \left[ \left(\frac{ \mathbf{H}^{-1} +\mathbf{H} }{ 2 }\right) \left( 1 + \frac{ \mathbf{Z}_\mathsf{S} }{\mathbf{Z}_\mathsf{L} }\right) + \left(\frac{\mathbf{H}^{-1} - \mathbf{H} }{ 2 }\right) \left( \frac{ \mathbf{Z}_\mathsf{S} }{ \mathbf{\mathbf{Z}}_\mathsf{C} } + \frac{ \mathbf{Z}_\mathsf{C} }{ \mathbf{Z}_\mathsf{L} }\right) \right]^{-1}
\\[2ex]
&= \frac{ \mathbf{Z}_\mathsf{L} \mathbf{Z}_\mathsf{C} }{ \mathbf{Z}_\mathsf{C} \left( \mathbf{Z}_\mathsf{L} + \mathbf{Z}_\mathsf{S} \right) \cosh\left( \boldsymbol{\gamma} x \right) + \left( \mathbf{Z}_\mathsf{L} \mathbf{Z}_\mathsf{S} + \mathbf{Z}_\mathsf{C}^2 \right) \sinh\left( \boldsymbol{\gamma} x\right) }
\end{align}</math>
where <math> \mathbf{H} \equiv e^{-\boldsymbol{\gamma} x} ,</math> and <math> x </math> is the length of the transmission line.

In the special case where all the impedances are equal, <math> \mathbf{Z}_\mathsf{L} = \mathbf{Z}_\mathsf{S} = \mathbf{Z}_\mathsf{C} ,</math> the solution reduces to {{nowrap|<math> \frac{ \mathbf{V}_\mathsf{L} }{ \mathbf{V}_\mathsf{S} } = \frac{1}{2} e^{-\boldsymbol{\gamma} x} \,</math>.}}

==Lossless transmission==
When <math> \omega L \gg R </math> and {{nowrap|<math> \omega C \gg G </math>,}} 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 the {{mvar|L}} and {{mvar|C}} elements. The telegrapher's equations then describe the relationship between the voltage {{mvar|V}} and the current {{mvar|I}} along the transmission line, each of which is a function of position {{mvar|x}} and time {{mvar|t}}:
<math display="block"> \begin{align}
V &= V(x,t) \\[.5ex]
I &= I(x,t)
\end{align} </math>
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.
<math display="block">
\frac{\partial V}{\partial x} = - L \frac{\partial I}{\partial t}
</math>
<math display="block">
\frac{\partial I}{\partial x} = - C \frac{\partial V}{\partial t}
</math>

These equations may be combined to form two [[wave equation]]s, one for voltage {{nowrap|<math>V</math>,}} the other for current {{nowrap|<math> I</math>:}}
<math display="block">
\begin{align}
\frac{\partial^2 V}{\partial t^2} - {\tilde v}^2 \frac{\partial^2 V}{\partial x^2} &= 0 \\[1ex]
\frac{\partial^2 I}{\partial t^2} - {\tilde v}^2 \frac{\partial^2 I}{\partial x^2} &= 0
\end{align}
</math>
where <math display="block"> {\tilde v} \equiv \frac{ 1 }{ \sqrt{ L C } } </math> is the [[Wave#Wave_velocity|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]].

===Lossless sinusoidal steady-state===
{{see also|Phasor}}
In the case of [[sine wave|sinusoidal]] [[steady-state]] (i.e., when a pure sinusoidal voltage is applied and [[transient state|transients]] have ceased), the voltage and current take the form of single-tone sine waves:<ref name="r208">{{cite book | last=Mooijweer | first=H. | title=Microwave Techniques | publisher=Macmillan Education UK | publication-place=London | date=1971 | isbn=978-1-349-01067-7 | doi=10.1007/978-1-349-01065-3 | pages=74-77}}</ref>
<math display="block"> \begin{align}
V(x,t) &= \operatorname\mathcal{Re} \Bigl\{~ V(x) \, e^{ j \omega t } ~\Bigr\}\ , \\[1ex]
I(x,t) &= \operatorname\mathcal{Re} \Bigl\{~ I(x) \, e^{ j \omega t } ~\Bigr\}\ ,
\end{align}</math>
where <math> \omega </math> is the angular frequency of the steady-state wave. In this case, the telegrapher's equations reduce to
<math display="block">\begin{align}
\frac{\operatorname{d}V}{\operatorname{d}x} &= -j \omega L\ I = - L\ \frac{\operatorname{d}I}{\operatorname{d}t}\ , \\[1ex]
\frac{\operatorname{d}I}{\operatorname{d}x} &= -j \omega C\ V = - C\ \frac{\operatorname{d}V}{\operatorname{d}t} ~.
\end{align}</math>

Likewise, the wave equations reduce to one-dimensional [[Helmholtz equation]]s
<math display="block">\begin{align}
&\frac{\operatorname{d}^2 V}{\operatorname{d}x^2}+ k^2 V = 0\ , \\[1ex]
&\frac{\operatorname{d}^2 I}{\operatorname{d}x^2}+ k^2 I = 0\ , \\[1ex]
\end{align} </math>
where {{mvar|k}} is the [[wave number]]: <math display="block"> k \equiv \omega \sqrt{ LC\ } = \frac{ \omega }{ \tilde v } ~.</math>
In the lossless case, it is possible to show that
<math display="block"> V(x) = V_1 \, e^{ - j k x } + V_2 \, e^{ + j k x }, </math>
and
<math display="block"> I(x) = \frac{ V_1 }{ Z_\mathsf{o} } \, e^{ - j k x } - \frac{ V_2 }{ Z_\mathsf{o} }\, e^{ + j k x }\ ,</math>
where in this special case, <math>\ k\ </math> is a real quantity that may depend on frequency and <math>\ Z_\mathsf{o}\ </math> is the ''[[characteristic impedance]]'' of the transmission line, which, for a lossless line is given by
<math display="block"> Z_\mathsf{o} = \sqrt{ \frac{ L }{ C }\ }\ , </math>
and <math>\ V_1\ </math> and <math>\ V_2\ </math> are arbitrary constants of integration, which are determined by the two [[boundary condition]]s (one for each end of the transmission line).

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

=== Loss-free case, general solution ===
In the loss-free case {{nowrap|(<math> R = G = 0 </math>),}} the general solution of the wave equation for the voltage is the sum of a forward traveling wave and a backward traveling wave:
<math display="block"> V(x,t) = f_1( x - {\tilde v} t ) + f_2( x + {\tilde v} t ) </math>
where
* <math> f_1 </math> and <math> f_2 </math> can be ''any'' two [[analytic function]]s, and
* <math> {\tilde v} \equiv \frac{ 1 }{ \sqrt{ L C } } </math> is the waveform's [[propagation speed]] (also known as ''[[phase velocity]]'').

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

Using the current <math> I </math> and voltage <math> V </math> relations given by the telegrapher's equations, we can write
<math display="block"> I(x,t) = \frac{ 1 }{ Z_\mathsf{o} } \Bigl[ f_1(x - {\tilde v} t) - f_2(x + {\tilde v} t) \Bigr] \,.</math>

==Lossy transmission line==
[[File:Telegrapher equation.gif|right|thumb|300px|In the presence of losses the solution of the telegrapher's equation has both damping and dispersion, as visible when compared with the solution of a lossless wave equation.]]
When the loss elements <math> R </math> and <math> G </math> are too substantial to ignore, the differential equations describing the elementary segment of line are
<math display="block">\begin{align}
\frac{\partial}{\partial x} V(x,t) &= -L \frac{\partial}{\partial t} I(x,t) - R \, I(x,t) \, , \\[6pt]
\frac{\partial}{\partial x} I(x,t) &= -C \frac{\partial}{\partial t} V(x,t) - G \, V(x,t) \, .
\end{align}</math>

By differentiating both equations with respect to {{mvar|x}}, and some algebra, we obtain a pair of [[damped]], [[Dispersion_relation|dispersive]] [[hyperbolic partial differential equation]]s each involving only one unknown:
<math display="block">\begin{align}
\frac{\partial^2}{\partial x^2} V &= L C \frac{\partial^2}{\partial t^2} V + \left( R C + G L \right) \frac{\partial}{\partial t} V + G R V , \\[6pt]
\frac{\partial^2}{\partial x^2} I &= L C \frac{\partial^2}{\partial t^2} I + \left( R C + G L \right) \frac{\partial}{\partial t} I + G R I .
\end{align}</math>

These equations resemble the homogeneous wave equation with extra terms in {{mvar|V}} and {{mvar|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 {{nowrap|(<math> R \ll \omega L </math>}} and {{nowrap|<math> G \ll \omega C </math>),}} signal strength will decay over distance as <math> e^{-\alpha x} </math> where <math> \alpha \approx \frac{R}{ 2 Z_0 } + \frac{ G Z_0 }{2} ~</math>.<ref name="Miano-Maffucci-2001">{{cite book | last1 = Miano | first1 = Giovanni | last2 = Maffucci | first2 = Antonio | date = 2001 | title = Transmission Lines and Lumped Circuits | page = 130 | publisher=[[Academic Press]] | isbn = 0-12-189710-9 }} The book uses the symbol {{math|μ}} instead of {{math|α}}.</ref>

==Solutions of the telegrapher's equations as circuit components==
{{cleanup|section|reason=Poor style|date=June 2012}}

[[File:Transmission Line, Unbalanced, Equivalent Circuit.png|thumb|360px|Equivalent circuit of an unbalanced transmission line (such as coaxial cable) where: {{math|2/''Z''<sub>o</sub>}} is the trans-admittance of VCCS (Voltage Controlled Current Source), {{mvar|x}} is the length of transmission line, {{math|''Z''(''s'') ≡ ''Z''<sub>o</sub>(''s'')}} is the characteristic impedance, {{math|''T''(''s'')}} is the propagation function, {{math|''γ''(''s'')}} is the propagation "constant", {{math|''s'' ≡ ''j'' ''ω''}}, and {{math|''j''<sup>2</sup> ≡ −1}}.]]

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.<ref name="McCammon-2010">{{cite web | last = McCammon | first = Roy | title = SPICE Simulation of Transmission Lines by the Telegrapher's Method | date = June 2010 | url = http://i.cmpnet.com/rfdesignline/2010/06/C0580Pt1edited.pdf | website = cmpnet.com | series = RF Design Line | access-date = 22 Oct 2010 | postscript=;}} also {{cite book | section = Part 1 of 3 | title = SPICE simulation of transmission lines by the telegrapher's method | section-url = https://www.eetimes.com/spice-simulation-of-transmission-lines-by-the-telegraphers-method-part-1-of-3/ | via=[[E.E. Times]] | department = Microwave & RF design}}</ref>

The solution of the telegrapher's equations can be expressed as an [[two-port network#ABCD-parameters|ABCD two-port network]] with the following defining equations<ref name="Karakash_1950"/>{{rp|pp=5–14, 44}}
<math display="block">\begin{align}
 V_1 &= V_2 \cosh( \gamma x) + I_2 Z_\mathsf{o} \sinh(\gamma x) \, , \\[1ex]
 I_1 &=  \frac{V_2}{ Z_\mathsf{o} } \sinh (\gamma x) + I_2 \cosh(\gamma x) \, .
\end{align}</math>
<!-- The symbols <math> E_s, E_L, I_s, I_L, \ell = \text{ ℓ } </math> used in the source book, Karakash (1950) p 44, are replaced by the symbols <math> V_1, V_2, I_1, I_2, x </math> in the preceding two equations. -->
where
<math display="block"> Z_\mathsf{o} \equiv \sqrt{ \frac{ R_\omega + j \omega L_\omega }{ G_\omega + j \omega C_\omega } } ,</math>
and
<math display="block"> \gamma \equiv \sqrt{ \left( R_\omega + j \omega L_\omega \right) \left( G_\omega + j \omega C_\omega \right) } ,</math>
just as in the preceding sections. The line parameters {{mvar|R{{sub|ω}}}}, {{mvar|L{{sub|ω}}}}, {{mvar|G{{sub|ω}}}}, and {{mvar|C{{sub|ω}}}} are subscripted by {{mvar|ω}} to emphasize that they could be functions of frequency.

The ABCD type two-port gives <math> V_1 </math> and <math> I_1 </math> as functions of <math> V_2 </math> and {{nowrap|<math> I_2 \,</math>.}} The voltage and current relations are symmetrical: Both of the equations shown above, when solved for <math> V_1 </math> and <math> I_1 </math> as functions of <math> V_2 </math> and <math> I_2 </math> yield exactly the same relations, merely with subscripts "1" and "2" reversed, and the <math> \sinh </math> 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 the ''shield'', sheath, common, earth, or ground. So every two-wire balanced transmission line has two modes which are nominally called the ''differential mode'' and ''common 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 impedances {{math|''Z''<sub>o</sub>(''s'')}} account for the interaction of the transmission line with the external circuit. This circuit is a useful equivalent for an [[unbalanced line|unbalanced transmission line]] like a [[coaxial cable]].

These are not unique: Other equivalent circuits are possible.

==See also==
* [[Complex power]]
* [[Law of squares]], [[William Thomson, Lord Kelvin|Lord Kelvin's]] preliminary work on this subject
* [[Reflections of signals on conducting lines]]
* [[Smith chart]]

==References==
{{reflist}}

{{Telecommunications}}

{{DEFAULTSORT:Telegrapher's Equations}}
[[Category:Hyperbolic partial differential equations]]
[[Category:Distributed element circuits]]
[[Category:Transmission lines]]