The telegrapher's equations (or telegraph equations) are a set of two coupled, linear partial differential equations that model voltage and current along a linear electrical transmission line. The equations are important because they allow transmission lines to be analyzed using circuit theory. The equations and their solutions are applicable from 0 Hz (i.e. direct current) to frequencies at which the transmission line structure can support higher order non-TEM modes. The equations can be expressed in both the time domain and the frequency domain. In the time domain the independent variables are distance and time. In the frequency domain the independent variables are distance x {\displaystyle x} and either frequency, ω {\displaystyle \omega } , or complex frequency, s {\displaystyle s} . The frequency domain variables can be taken as the Laplace transform or Fourier transform of the time domain variables or they can be taken to be phasors in which case the frequency domain equations can be reduced to ordinary differential equations of distance. An advantage of the frequency domain approach is that differential operators in the time domain become algebraic operations in frequency domain. 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. 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.
Distributed components
The telegrapher's equations result from circuit theory. 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 {\displaystyle R} of the conductors is represented by a series resistor (expressed in ohms per unit length). In practical conductors, at higher frequencies, R {\displaystyle R} increases approximately proportional to the square root of frequency due to the skin effect. The distributed inductance L {\displaystyle 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 {\displaystyle C} between the two conductors is represented by a shunt capacitor C {\displaystyle C} (farads per unit length). The conductance G {\displaystyle 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 R shunt = 1 G Ω {\displaystyle \textstyle R_{\text{shunt}}={\frac {1}{G}}\,\Omega } . G {\displaystyle G} accounts for both bulk conductivity of the dielectric and dielectric loss. If the dielectric is an ideal vacuum, then G ≡ 0 {\displaystyle G\equiv 0} . The model consists of an infinite series of the infinitesimal elements shown in the figure, and 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 ′ {\displaystyle R'} , L ′ {\displaystyle L'} , C ′ {\displaystyle C'} , and G ′ {\displaystyle 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.
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