Telegrapher's equations

The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who in the 1880s developed the transmission line model which is described in this article. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can appear along the line. The theory applies to transmission lines of all frequencies including high-frequency transmission lines (such as telegraph wires and radio frequency conductors), audio frequency (such as telephone lines), low frequency (such as power lines) and direct current.

1 The equations
2 Signal pattern examples
3 Solutions of the Telegrapher's Equations as Circuit Components
4 External links
5 See also
6 Notes
7 References

The equations

The telegrapher's equations, like all other equations describing electrical phenomena, can be held to 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).
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.

It should be repeated for clarity that 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. 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.

The Telegrapher's Equations are developed in similar forms in the following references: Kraus,^[1] Hayt,^[2] Marshall,^[3] Sadiku,^[4] Harrington,^[5] Karakash,^[6] Metzger,^[7]

Values of Primary Parameters for Telephone Cable

Representative parameter data for 24 gauge PIC telephone cable at 70°F

Frequency	R	L	G	C
Hz	Ω/kft	mH/kft	µS/kft	nF/kft
1	52.50	0.1868	0.000	15.72
1k	52.51	0.1867	0.022	15.72
10k	52.64	0.1859	0.162	15.72
100k	58.41	0.1770	1.197	15.72
1M	141.30	0.1543	8.873	15.72
2M	196.03	0.1482	16.217	15.72
5M	304.62	0.1425	35.989	15.72

More extensive tables and tables for other gauges, temperatures and types are available in Reeve.^[8] Chen^[9] gives the same data in a parameterized form that he states 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 design.

The variation of G can be inferred from Terman^[10] "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 $G(f) = G_1 \left( \frac {f}{f_1}\right)^{ge}$ with ge close to 1.0 would fit the statement from Terman. Chen ^[9] gives an equation of similar form.

G in this table can be modeled well with

$f_1 \,$ = 1MHz

$G_1 = 8.873 \mu$ S/kft

ge = 0.87

Usually the resistive losses grow proportionately to $f^{0.5} \,$ and dielectric losses grow proportionately to $f^{ge} \,$ with ge > 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. The dielectric can be reduced down to air with an occasional plastic spacer.

Lossless transmission

When the elements R and G are very small, their effects can be neglected, and the transmission line is considered as an idealized, lossless, structure. In this case, the model depends only on the L and C elements, and we obtain a pair of first-order partial differential equations, one function describing the voltage V along the line and the other the current I, both function of position x and time t:

$\frac{\partial}{\partial x} V(x,t) = -L \frac{\partial}{\partial t} I(x,t)$

$\frac{\partial}{\partial x} I(x,t) = -C \frac{\partial}{\partial t} V(x,t)$

These equations may be combined to form either of two exact wave equations:

$\frac{\partial^2}{{\partial t}^2} V = \frac{1}{LC} \frac{\partial^2}{{\partial x}^2} V$

$\frac{\partial^2}{{\partial t}^2} I = \frac{1}{LC} \frac{\partial^2}{{\partial x}^2} I$

In the steady-state case (assuming a sinusoidal wave $E=E_{o}\cdot e^{-j\omega ( \frac{x}{c} - t)}$ , these reduce to

$\frac{\partial^2V(x)}{\partial x^2}%2B \omega^2 LC\cdot V(x)=0$

$\frac{\partial^2I(x)}{\partial x^2} %2B \omega^2 LC\cdot I(x)=0$

where $\omega$ is the frequency of the steady-state wave

If the line has infinite length or when it is terminated with its characteristic impedance, these equations indicate the presence of a wave, travelling with a speed $v = \frac{1}{\sqrt{LC}}$ .

(Note that this propagation speed applies to the wave phenomenon on the line and has nothing to do with the electron drift velocity. In other words, the electrical impulse travels very close to the speed of light, although the electrons themselves travel only a few centimeters per second.) For a coaxial transmission line, made of perfect conductors with vacuum between them, it can be shown that this speed is equal to the speed of light.

The Lossless line and Distortionless line are discussed in Sadiku,^[11] and Marshall,^[12]

Lossy transmission line

When the loss elements R and G are not negligible, the original differential equations describing the elementary segment of line become

$\frac{\partial}{\partial x} V(x,t) = -L \frac{\partial}{\partial t} I(x,t) - R I(x,t)$

$\frac{\partial}{\partial x} I(x,t) = -C \frac{\partial}{\partial t} V(x,t) - G V(x,t)$

By differentiating the first equation with respect to x and the second with respect to t, and some algebraic manipulation, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:

$\frac{\partial^2}{{\partial x}^2} V = L C \frac{\partial^2}{{\partial t}^2} V %2B (R C %2B G L) \frac{\partial}{\partial t} V %2B G R V$

$\frac{\partial^2}{{\partial x}^2} I = L C \frac{\partial^2}{{\partial t}^2} I %2B (R C %2B G L) \frac{\partial}{\partial t} I %2B G R I$

Note that these equations resemble the homogeneous wave equation with extra terms in V and 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 (small R and G = 0), signal strength will decay over distance as e^-αx, where α = R/2Z₀

Direction of signal propagations

The wave equations above indicate that there are two solutions for the travelling wave: one forward and one reverse. Assuming a simplification of being lossless (requiring both R=0 and G=0) the solution can be represented as:

$V(x,t) \ = \ { f_1(\omega t - kx) %2B f_2(\omega t %2B kx)} \$

where:

$k = \omega \sqrt{LC} = {\omega \over v} \$

$k$ is called the wavenumber and has units of radians per meter,

ω is the angular frequency (in radians per second),

$f_1$ and $f_2$ can be any functions whatsoever, and

$v = { \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

$I(x,t) \ = \ { f_1(\omega t-kx) \over Z_0 } - { f_2(\omega t%2Bkx) \over Z_0 }$

where $Z_0$ is the characteristic impedance of the transmission line, which, for a lossless line is given by

$Z_0 = \sqrt { {L \over C}}$

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 media 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 capacity.

Solutions of the Telegrapher's Equations as Circuit Components

The solutions of the telegrapher's equations can be inserted into a circuit as components of an equivalent sub-circuit as shown the figure. As drawn, all voltages are with respect to ground and all amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be an unbalanced transmission line such as a strip line on a circuit board. The impedance Z(s), the voltage doubler (the triangle with the number "2") and the difference amplifier (the triangle with the number "1") account for the interaction of the transmission line with the rest of the circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the "forward wave" and the other carries the "backward wave". The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from V1 to V2 in the sense that V1, V2, I1 and I2 would be same whether this circuit or an actual transmission line was connected between V1 and V2. There is no implication that there are actually amplifiers inside the transmission line.

This is not the only possible equivalent circuit. Voltage amplifiers and sensors can be replaced with current, transimpedance or transconductance amplifiers. Series impedances can be replaced with shunt admittances. The circuit can be augmented to account for different "grounds" at each end. The circuit can be made fully differential.

External links

SPICE Simulation of Transmission Lines

Notes

^ Kraus (1989, pp. 380–419)
^ Hayt (1989, pp. 382–392)
^ Marshall (1987, pp. 359–378)
^ Sadiku (1989, pp. 497–505)
^ Harrington (1961, pp. 61–65)
^ Karakash (1950, pp. 5–14)
^ Metzger (1969, pp. 1–10)
^ Reeve (1995, p. 558)
^ ^a ^b Chen (2004, p. 26)
^ Terman (1943, p. 112)
^ Sadiku (1989, pp. 501–503)
^ Marshall (198y, pp. 369–372)

References

Chen, Walter Y. (2004), Home Networking Basics, Prentice Hall, ISBN 0130165115
Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill
Hayt, William (1989), Engineering Electromagnetics (5th ed.), McGraw-Hill, ISBN 0070274061
Karakash, John J. (1950), Transmission Lines and Filter Networks (1st ed.), Macmillan
Kraus, John D. (1984), Electromagnetics (3rd ed.), McGraw-Hill, ISBN 0070354235
Marshall, Stanley V. (1987), Electromagnetic Concepts & Applications (1st ed.), Prentice-Hall, ISBN 0132490048
Metzger, Georges; Vabre, Jean-Paul (1969), Transmission Lines with Pulse Excitation, Academic Press
Reeve, Whitman D. (1995), Subscriber Loop Signaling and Transmission Handbook, IEEE Press, ISBN 0780304403
Sadiku, Matthew N. O. (1989), Elements of Electromagnetics (1st ed.), Saunders College Publishing, ISBN 993013846
Terman, Frederick Emmons (1943), Radio Engineers' Handbook (1st ed.), McGraw-Hill