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.

• 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.

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

More extensive tables and tables for other gauges, temperatures and types are available in Reeve.[4] Chen[5] 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.”[6] 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 [5] 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 telegrapher's equations are:

${\displaystyle {\begin{aligned}{\frac {\ \partial }{\partial x}}\ V(x,t)&=-L\ {\frac {\ \partial }{\partial t}}\ I(x,t)-R\ I(x,t)\\[6pt]{\frac {\ \partial }{\partial x}}\ I(x,t)&=-C\ {\frac {\ \partial }{\partial t}}V(x,t)-G\ V(x,t)\end{aligned}}}$

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

${\displaystyle {\begin{aligned}{\frac {~\ \partial ^{2}}{\partial x^{2}}}\ V(x,t)-LC\ {\frac {~\ \partial ^{2}}{\ \partial t^{2}}}\ V(x,t)&=(RC+GL)\ {\frac {\ \partial }{\partial t}}\ V(x,t)+GR\ V(x,t)\\[6pt]{\frac {~\ \partial ^{2}}{\partial x^{2}}}\ I(x,t)-LC\ {\frac {~\ \partial ^{2}}{\partial t^{2}}}\ I(x,t)&=(RC+GL)\ {\frac {\ \partial }{\partial t}}\ I(x,t)+GR\ I(x,t)\end{aligned}}}$

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

Let

${\displaystyle {\hat {V}}_{in}(\omega )={\frac {1}{\sqrt {2\,\pi }}}\,\int \limits _{-\infty }^{\infty }V_{in}(t)e^{-j\,\omega \,t}\mathrm {d} t}$

be the Fourier transform of the input voltage ${\displaystyle V_{in}(t)}$, then the general solutions for voltage and current are

${\displaystyle V(x,t)={\frac {1}{\sqrt {2\,\pi }}}\,\int \limits _{-\infty }^{\infty }H(\omega ,x)\cdot {\hat {V}}_{in}(\omega )\,e^{j\,\omega \,t}\mathrm {d} \omega }$

and

${\displaystyle I(x,t)=-{\frac {1}{\sqrt {2\,\pi }}}\,\int \limits _{-\infty }^{\infty }{\frac {1}{Z_{L}'}}\,{\frac {\partial H(\omega ,x)}{\partial x}}\,{\hat {U}}_{in}(\omega )\,e^{j\,\omega \,t}\mathrm {d} \omega }$

with

${\displaystyle H(\omega ,x)={\frac {\mathrm {cosh} \left((l-x)\,{\sqrt {\frac {Z_{L}'}{Z_{Q}'}}}\right)+{\frac {\sqrt {Z_{L}'\,Z_{Q}'}}{Z_{T}}}\,\mathrm {sinh} \left((l-x)\,{\sqrt {\frac {Z_{L}'}{Z_{Q}'}}}\right)}{\mathrm {cosh} \left(l\,{\sqrt {\frac {Z_{L}'}{Z_{Q}'}}}\right)+{\frac {\sqrt {Z_{L}'\,Z_{Q}'}}{Z_{T}}}\,\mathrm {sinh} \left(l\,{\sqrt {\frac {Z_{L}'}{Z_{Q}'}}}\right)}}}$

being the transfer function of the line,

${\displaystyle Z_{L}'=j\,\omega \,L+R}$

the series impedance, and

${\displaystyle Z_{Q}'={\frac {1}{j\,\omega \,C+G}}}$

the shunt impedance. The parameter ${\displaystyle l}$ represents the total length of the line. ${\displaystyle Z_{T}}$ is the impedance of the electrical termination. Without termination, ${\displaystyle Z_{T}}$ is infinite.

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,[7] Hayt,[8] Marshall,[9] Sadiku,[10] Harrington,[11] Karakash,[12] and Metzger.[13]

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:

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

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

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

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

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

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:

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

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,[14] and Marshall,[15]

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

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