Telegrapher's equations

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'' starting with an August 1876 paper, ''On the Extra Current''. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line.

The theory applies to transmission lines of all frequencies including direct current and high-frequency. Originally developed to describe telegraph wires, the theory can also be applied to 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 single-conductor transmission lines.

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:

* The distributed resistance $R$ of the conductors is represented by a series resistor (expressed in ohms per unit length). In practical conductors, at higher frequencies, $R$ increases approximately proportional to the square root of frequency due to the skin effect.
* The distributed inductance $L$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor ( henries per unit length).
* The capacitance $C$ between the two conductors is represented by a shunt capacitor C (farads per unit length).
* The conductance $G$ 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 $1/G$ ohms. $G$ accounts for both bulk conductivity of the dielectric and dielectric loss. If the dielectric is an ideal vacuum, then $G \equiv 0$.

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 $R'$, $L'$, $C'$, and $G'$ 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

The role of the different components can be visualized based on the animation at right.
; Inductance : The inductance 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 string. Large inductance also ''increases'' the line's surge impedance (''more'' voltage needed to push the same current through the line).
; Capacitance : The capacitance 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, , 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 current through the line).
; Resistance : Resistance corresponds to resistance interior to the two lines, combined. That resistance dissipates 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 at radio frequencies, and for simplicity is treated as if it were zero, with any voltage dissipation or wire heating accounted for as an afterthought, with slight corrections to the "lossless line" calculation deducted later, or just ignored.
; Conductance : Conductance between the lines represents how well current can "leak" from one line to the other, and higher dissipates more current as heat, deposited in whatever serves as insulation between the two conductors. Generally, wire insulation (including air) is quite good, and the conductance is almost nothing compared to the capacitive susceptance , and for simplicity is treated as if it were zero; the caveat is that materials that are good insulation at low frequencies are often "leaky" at very high frequencies.

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

Values of primary parameters for telephone cable

Representative parameter data for 24-gauge telephone polyethylene insulated cable (PIC) at 70 °F (294 K)
:
More extensive tables and tables for other wire gauges, operating temperatures, and insulation are available in Reeve (1995). Chen (2004) gives the same data in a parameterized form, that he reports is usable up to 50 MHz.

The variation of $~R~$ and $~L~$ is mainly due to skin effect and proximity effect.

The constancy of the capacitance is a consequence of intentional, careful design.

The variation of 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."
A function of the form
$G(f) = G_1 \cdot \left( \frac \right)^$
with $~g_\mathrm~$ close to 1.0 would fit Terman's statement. Chen gives an equation of similar form. Whereas (·) is conductivity as a function of frequency, $~G_1, ~ f_1~,$ and $~ g_e ~$ are all real constants.

(·) in this table can be modeled well with
:$f_1 = 1\;\mathrm$
:$G_1 = 29.11\;\mathrm = 8.873\;\mathrm$
:$g_\mathrm = 0.87$

Usually the resistive losses grow proportionately to $~ f^ ~$ and dielectric losses grow proportionately to $~ f^ ~$ with $~ g_\mathrm > \tfrac ~$ 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:

:\begin
\frac\ V(x,t) &= -L\ \frac \ I(x,t) - R\ I(x,t) \\ pt\frac\ I(x,t) &= -C\ \frac V(x,t) - G\ V(x,t)
\end

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

:\begin
\frac\ V(x,t) - LC\ \frac\ V(x,t) &= ( RC + GL )\ \frac\ V(x,t) + GR\ V(x,t) \\ pt\frac\ I(x,t) - LC\ \frac\ I(x,t) &= (RC + GL)\ \frac\ I(x,t) + GR\ I(x,t) \end

Except for the dependent variable ($~V~$ or $~I~$) the formulas are identical.

General solution for terminated lines of finite length

Let
:$\hat_\mathsf(\omega) \equiv \frac\,\int\limits_^ V_\mathsf(t)\; e^\;\mathrmt$
be the Fourier transform of the input voltage $\, V_\mathsf(t) \, ,$ then the general solutions for voltage and current are
:$V(x,t) = \frac\,\int\limits_^ H(\omega,x) \cdot \hat_\mathsf(\omega)\,e^\, \mathrm\omega$
and
:$I(x,t) = - \frac\,\int\limits_^ \frac\;\frac \, \hat_\mathsf(\omega)\;e^ \;\mathrm\omega$
with $\, \hat_\mathsf(\omega) \,$ being the Fourier transform of the input current $\, I_\mathsf(t) \, ,$ similar to $\, \hat_\mathsf(\omega) \,$ and
:$H(\omega,x) \equiv \frac$
being the transfer function of the line,
:$Z' \equiv R + j\,\omega\,L$
the series impedance per unit length, and
:$Y' \equiv G + j\,\omega\,C$
the shunt admittance per unit length (reciprocal of the shunt impedance). The parameter $~\ell~$ represents the total length of the line, and $\, x \,$ locates an arbitrary intermediate position along the line. $\,Z_\mathsf\,$ is the impedance of the electrical termination.

With no termination (broken line), $~Z_\mathsf~$ is infinite, and the $~\sinh~$ terms vanish from the numerator and denominator of the transfer function, $~ H(\omega,x)~.$ If the end is perfectly grounded (shorted line), $\,Z_\mathsf\,$ is zero and the $~\cosh~$ terms vanish.

;Remarks on notation:
As in other sections of this article, the formulas can be made somewhat more compact by using the secondary parameters
:$Z_\mathsf \equiv \sqrt \qquad \mathsf \qquad \gamma \equiv \sqrt ~,$
replacing the product and ratio square root factors written out explicitly in the definition of $~ H(\omega,x)~.$

The Fourier transforms used above are the symmetric versions, with the same factor of $\, \tfrac \,$ multiplying the integrals of both the forward and inverse transforms. This is not essential; other versions of the Fourier transform can be used, with appropriate juggling of the coefficients that ensures their product remains $~ \tfrac ~ .$

Lossless transmission

When $~ \omega L \gg R ~$ and $~ \omega C \gg G ~,$ 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 the and elements. The telegrapher's equations then describe the relationship between the voltage and the current along the transmission line, each of which is a function of position and time :

:$V = V(x,t)$
:$I = I(x,t)$

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.

:$\frac = -L \frac$

:$\frac = -C \frac$

The telegrapher's equations are developed in similar forms in the following references:
Kraus (1989), Hayt (1989), Marshall (1987), Sadiku (1989), Harrington (1961), Karakash (1950) and
Metzger & Vabre (1969).

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

:$~ \frac - u^2 \frac = 0 ~$

:$~ \frac - u^2 \frac = 0 ~$

where
:$~ u = \frac ~$

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.

Sinusoidal steady-state

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:

:$V(x,t) ~ = ~ \mathcal \; \Bigl\ ~$

:$I(x,t) ~ = ~ \mathcal \; \Bigl\~,$

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

:$\frac = -j \omega L I = -L \frac$

:$\frac = -j \omega C V = -C \frac$

Likewise, the wave equations reduce to

:$\frac+ k^2 V = 0 ~$

:$\frac + k^2 I = 0 ~$

where is the wave number:

:$~ k = \omega \sqrt = ~.$

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
:$V(x) = V_1 e^ + V_2 e^$
and
:$I(x) = \, e^ - \,e^$

where $~ k ~$ is a real quantity that may depend on frequency and $~ Z_0 ~$ is the ''characteristic impedance'' of the transmission line, which, for a lossless line is given by

:$~ Z_0 = \sqrt ~$

and $~ V_1 ~$ and $~V_2~$ 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 and 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) and Marshall (1987).

Loss-free case, general solution

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

:$~ V(x,t) \, = \, f_1(x - u\,t) + f_2(x + u\,t) ~$

where
*$\,f_1\,$ and $\,f_2\,$ can be ''any'' two analytic functions, and
*$\,u = 1/\sqrt\,$ is the waveform's propagation speed (also known as ''phase velocity'').

represents the amplitude profile of a wave traveling from left to right in a positive direction whilst
represents the amplitude profile of a wave traveling from right to left. It can be seen that the instantaneous voltage at any point on the line is the sum of the voltages due to both waves.

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

:$I(x,t) \, = \, \frac - \frac ~.$

Lossy transmission line

When the loss elements $~ R ~$ and $~ G ~$ are too substantial to neglect, the differential equations describing the elementary segment of line are

:\begin
\frac V(x,t) &~=~ -L\,\frac I(x,t) - R\,I(x,t) ~,\\ pt\frac I(x,t) &~=~ -C\,\frac V(x,t) - G\,V(x,t) ~.
\end

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

:\begin
\frac V &~=~ L\,C\,\frac V + \left(\, R \, C + G \, L \,\right) \, \frac V + G \, R \, V ~,\\ pt\frac I &~=~ L \, C \, \frac I + \left(\, R \, C + G \, L \,\right) \, \frac I + G \, R \,I ~.\\
\end

These equations resemble the homogeneous wave equation with extra terms in and 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 ($~ R \ll \omega L ~$ and $~ G \ll \omega C ~$), signal strength will decay over distance as $~ e^~$ where $~ \alpha ~\approx~ \frac + \frac ~$

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.

Antennas

:In the first approximation, the current in a thin antenna is distributed
exactly as in a transmission line. — Schelkunoff & Friis (1952)

Because the conductor of an antenna element closely approximates a single-conductor transmission line, the telegrapher's equations can be used to analyze antenna currents, as was commonly done in the first half of the 20th century, before the common availability of computing equipment.

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.

The bottom circuit is derived from the top circuit by source transformations. 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

:$\begin V_1 &= V_2 \cosh ( \gamma x) + I_2 Z_\mathsf \sinh (\gamma x)~, \\ I_1 &= V_2 \frac \sinh (\gamma x) + I_2 \cosh(\gamma x)~. \\ \end$

where
:$Z_\mathsf \equiv \sqrt \; ,$
and
:$\gamma \equiv \sqrt \; ,$
just as in the preceding sections. The line parameters are subscripted by to emphasize that they could be functions of frequency.

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

In the bottom circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have connections to ground not shown. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedance _o(), 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 () blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the blocks marked "()" carries the ''forward wave'' and the other carries the ''backward wave''. The depicted circuit is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from $\, V_1 \,$ to $\, V_2 \,$ in the sense that $\, V_1 \,$, $\, V_2 \,$, $\,I_1 \,$ and $\, I_2 \,$ would be same whether this circuit or an actual transmission line was connected between $\, V_1 \,$ and $\, V_2 \; .$ 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 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 _o() 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 unique: Other equivalent circuits are possible.

See also

* Reflections of signals on conducting lines
* Law of squares, Lord Kelvin's preliminary work on this subject

Notes

References

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