Telegrapher's equations

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.

Inductance 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 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 AC current through the line).

Capacitance 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, 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 forces, making the wave move slightly slower, and also gives the transmission line a lower surge impedance (less voltage needed to push the same AC current through the line).

Resistance R

Resistance corresponds to resistance interior to the two lines, combined. That resistance R couples current to ohmic losses that 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 reactance ωL 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 G

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

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

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

Frequency	R		L		G		C
Hz	Ω⁄km	Ω⁄1000 ft	μH⁄km	μH⁄1000 ft	μS⁄km	μS⁄1000 ft	nF⁄km	nF⁄1000 ft
1 Hz	172.24	52.50	612.9	186.8	0.000	0.000	51.57	15.72
1 kHz	172.28	52.51	612.5	186.7	0.072	0.022	51.57	15.72
10 kHz	172.70	52.64	609.9	185.9	0.531	0.162	51.57	15.72
100 kHz	191.63	58.41	580.7	177.0	3.327	1.197	51.57	15.72
1 MHz	463.59	141.30	506.2	154.3	29.111	8.873	51.57	15.72
2 MHz	643.14	196.03	486.2	148.2	53.205	16.217	51.57	15.72
5 MHz	999.41	304.62	467.5	142.5	118.074	35.989	51.57	15.72

This data is from Reeve (1995).[4] 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 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 ${\displaystyle G(f)=G_{1}\cdot \left({\frac {f}{f_{1}}}\right)^{g_{\mathrm {e} }}}$ with ${\displaystyle ~g_{\mathrm {e} }~}$ close to 1.0 would fit Terman's statement. Chen[6] gives an equation of similar form. Whereas G(·) is conductivity as a function of frequency, ${\displaystyle ~G_{1},~f_{1}~,}$ and ${\displaystyle ~g_{e}~}$ are all real constants.

Usually the resistive losses grow proportionately to ${\displaystyle ~f^{1/2}~}$ and dielectric losses grow proportionately to ${\displaystyle ~f^{g_{\mathrm {e} }}~}$ with ${\displaystyle ~g_{\mathrm {e} }\approx 1~}$ 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 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 telegrapher's equations in the time domain 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.

The telegrapher's equations in the frequency domain are developed in similar forms in the following references: Kraus,[7]^{: 380–419} Hayt,[1]^{: 381–392} Marshall,[8]^: 59–378 Sadiku,[9]^{: 497–505} Harrington,[10]^: 61–65 Karakash,[11]^: 5–14 Metzger.[12]^: 1–10

${\displaystyle \ {\frac {\ \partial }{\partial x}}\ \mathbf {V} _{\omega }(x)=-(\ j\omega L_{\omega }+R_{\omega }\ )\mathbf {I} _{\omega }(x)\ }$

${\displaystyle \ {\frac {\ \partial }{\partial x}}\ \mathbf {I} _{\omega }(x)=-(\ j\omega C_{\omega }+G_{\omega }\ )\mathbf {V} _{\omega }(x)\ }$

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 series impedance ${\displaystyle \ (R+j\omega L)~.}$ 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 shunt admittance ${\displaystyle \ (G+j\omega C)~.}$

The subscript ω indicates possible frequency dependence. ${\displaystyle \ \mathbf {I} _{\omega }(x)\ }$ and ${\displaystyle \ \mathbf {V} _{\omega }(x)\ }$ are phasors.

These equations may be combined to produce two single variable partial differential equations.

${\displaystyle \ {\frac {d^{2}}{dx^{2}}}\mathbf {V} _{\omega }(x)=\gamma ^{2}\mathbf {V} _{\omega }(x)\ }$

${\displaystyle \ {\frac {d^{2}}{dx^{2}}}\mathbf {I} _{\omega }(x)=\gamma ^{2}\mathbf {I} _{\omega }(x)\ }$

where ${\displaystyle \ \gamma \equiv {\sqrt {(R_{\omega }+j\omega L_{\omega })(G_{\omega }+j\omega C_{\omega })\;}}=\alpha +j\beta ~}$[1]^: 385
${\displaystyle \ \alpha \ }$ is called the attenuation constant and ${\displaystyle \ \beta \ }$ is called the phase constant.

Each of the preceding partial differential equations have two homogeneous solutions in an infinite transmission line.

For the voltage equation

${\displaystyle \ \mathbf {V} _{\omega ,F}(x)=\mathbf {V} _{\omega ,F}(a)\ e^{-\gamma (a+x)}~.}$

${\displaystyle \ \mathbf {V} _{\omega ,R}(x)=\mathbf {V} _{\omega ,R}(b)\ e^{-\gamma (a+x)}~.}$

${\displaystyle \ \mathbf {V} _{\omega }(x)=\mathbf {V} _{\omega ,F}(x)+\mathbf {V} _{\omega ,R}(x)~.}$

For the current equation

${\displaystyle \ \mathbf {I} _{\omega ,F}(x)=\mathbf {I} _{\omega ,F}(a)\ e^{-\gamma x}~.}$

${\displaystyle \ \mathbf {I} _{\omega ,R}(x)=\mathbf {I} _{\omega ,R}(b)\ e^{-\gamma (b-x)}~.}$

${\displaystyle \ \mathbf {I} _{\omega }(x)=\mathbf {I} _{\omega ,F}(x)-\mathbf {I} _{\omega ,R}(x)\ }$ The negative sign indicates that the current in the reverse wave is traveling in the opposite direction.

Note:

${\displaystyle \ \mathbf {V} _{\omega ,F}(x)=\mathbf {Z} _{c}\ \mathbf {I} _{\omega ,F}(x)\ ,}$

${\displaystyle \ \mathbf {V} _{\omega ,R}(x)=\mathbf {Z} _{c}\ \mathbf {I} _{\omega ,R}(x)\ ,}$

${\displaystyle \ \mathbf {Z} _{c}={\sqrt {{\frac {\ R_{\omega }+j\omega L_{\omega }\ }{\ G_{\omega }+j\omega C_{\omega }}}\ }}\ ,}$

where

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 point x
${\displaystyle \ \mathbf {V} _{\omega ,F}(x)\ }$	value of the forward voltage wave at point x
${\displaystyle \ \mathbf {V} _{\omega ,R}(x)\ }$	value of the reverse voltage wave at point x
${\displaystyle \ \mathbf {V} _{\omega ,F}(a)\ }$	value of the forward voltage wave at point a
${\displaystyle \ \mathbf {V} _{\omega ,R}(b)\ }$	value of the reverse voltage wave at point b
${\displaystyle \ \mathbf {I} _{\omega }(x)\ }$	value of the total current at point x
${\displaystyle \ \mathbf {I} _{\omega ,F}(x)\ }$	value of the forward current wave at point x
${\displaystyle \ \mathbf {I} _{\omega ,R}(x)\ }$	value of the reverse current wave at point x
${\displaystyle \ \mathbf {I} _{\omega ,F}(a)\ }$	value of the forward current wave at point a
${\displaystyle \ \mathbf {I} _{\omega ,R}(b)\ }$	value of the reverse current wave at point b
${\displaystyle \ \mathbf {Z} _{c}\ }$	Characteristic impedance

Coaxial transmission line with one source and one load

Johnson gives the following solution,[2]^{: 739–741}

${\displaystyle \ {\frac {\mathbf {V} _{L}}{\mathbf {V} _{S}}}={[({\frac {\mathbf {H} ^{-1}+\mathbf {H} }{2}})(1+{\frac {\mathbf {Z} _{S}}{\mathbf {Z} _{L}}})+({\frac {\mathbf {H} ^{-1}-\mathbf {H} }{2}})({\frac {\mathbf {Z} _{S}}{\mathbf {\mathbf {Z} } _{C}}}+{\frac {\mathbf {Z} _{C}}{\mathbf {Z} _{L}}})]}^{-1}={\frac {\mathbf {Z} _{L}\mathbf {Z} _{C}}{\mathbf {Z} _{C}(\mathbf {Z} _{L}+\mathbf {Z} _{S})\cosh {{\boldsymbol {\gamma }}x}+({\mathbf {Z} _{L}\mathbf {Z} _{S}}+{\mathbf {Z} _{C}}^{2})\sinh {{\boldsymbol {\gamma }}x}}}\ }$

where

${\displaystyle \ \mathbf {H} =e^{-{\boldsymbol {\gamma }}x},\ x\ =\ }$ length of the transmission line.

In the special case of ${\displaystyle \ \mathbf {Z} _{L}=\mathbf {Z} _{S}=\mathbf {Z} _{C}\ }$ the solution reduces to ${\displaystyle \ {\frac {\mathbf {V} _{L}}{\mathbf {V} _{S}}}={\frac {1}{2}}e^{-{\boldsymbol {\gamma }}x}\ }$

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.

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

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

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 (i.e., when a pure sinusoidal voltage is applied and transients have ceased), the voltage and current take the form of single-tone sine waves:

${\displaystyle \ V(x,t)\ =\ {\mathcal {R_{e}}}\ {\Bigl \{}\ V(x)\cdot e^{j\omega t}\ {\Bigr \}}\ ,}$

${\displaystyle \ I(x,t)\ =\ {\mathcal {R_{e}}}\ {\Bigl \{}\ I(x)\cdot e^{j\omega t}\ {\Bigr \}}\ ,}$

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

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

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

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:

${\displaystyle \ k=\omega {\sqrt {LC\;}}={\frac {\ \omega \ }{u}}~.}$

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

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

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 (1989)[13]^{: 501–503} and Marshall (1987).[14]^{: 369–372}

In the loss-free case (${\displaystyle \ R=G=0\ }$), the most general solution of the wave equation for the voltage is the sum of a forward traveling wave and a backward traveling wave:

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

where

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

f₁ represents the amplitude profile of a wave traveling from left to right in a positive x direction whilst f₂ represents the amplitude profile of 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.

Using the current I and voltage V relations given by the telegrapher's equations, we can write

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