The telegrapher's equations (or just telegraph equations) are a pair of coupled, linear differential equations that 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.
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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 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
Role of different components
The role of the different components can be visualized based on the animation at right.
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 gauges, temperatures and types are available in Reeve. Chen 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 "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 in this table can be modeled well with
Usually the resistive losses grow proportionately to
Lossless transmission
When the elements R and G are very small, their effects can be neglected, and the transmission line is considered as an ideal lossless structure. In this case, the model depends only on the L and C elements. The Telegrapher's Equations then describe the relationship between the voltage V and the current I along the transmission line, each of which is a function of position x and time t:
The equations
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.
The Telegrapher's Equations are developed in similar forms in the following references: Kraus, Hayt, Marshall, Sadiku, Harrington, Karakash, and Metzger.
These equations may be combined to form two exact wave equations, one for voltage V, the other for current I:
where
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, the voltage and current take the form of single-tone sine waves:
where
Likewise, the wave equations reduce to
where k is the wave number:
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
and
where
and
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, and Marshall,
General solution
The general solution of the wave equation for the voltage is the sum of a forward traveling wave and a backward traveling wave:
where
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
Lossy transmission line
When the loss elements R and G are not negligible, the differential equations describing the elementary segment of line are
By differentiating both equations with respect to x, and some algebraic manipulation, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:
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 (R << ωL and G << ωC), signal strength will decay over distance as e−αx, where α ≈ R/2Z0 + GZ0/2.
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.
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
The ABCD type two-port gives
In the bottom circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), 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 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
Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the bottom only models the differential mode.
In the top circuit, the voltage doublers, the difference amplifiers and impedances Z(s) 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 micro strip line.
These are not the only possible equivalent circuits.