Transmission Line - 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. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's Equations.

The transmission line model represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

• The distributed resistance of the conductors is represented by a series resistor (expressed in ohms per unit length).
• The distributed inductance (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
• The capacitance between the two conductors is represented by a shunt capacitor C (farads per unit length).
• The conductance 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).

The model consists of an infinite series of the 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., and may also be functions of frequency. An alternative notation is to use, and 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 propagation constant, attenuation constant and phase constant.

The line voltage and the current can be expressed in the frequency domain as

When the elements and are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the and elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.

If and are not neglected, the Telegrapher's equations become:

where

and the characteristic impedance is:

The solutions for and are:

The constants and must be determined from boundary conditions. For a voltage pulse, starting at and moving in the positive -direction, then the transmitted pulse at position can be obtained by computing the Fourier Transform, of, attenuating each frequency component by, advancing its phase by, and taking the inverse Fourier Transform. The real and imaginary parts of can be computed as

where atan2 is the two-parameter arctangent, and

For small losses and high frequencies, to first order in and one obtains

Noting that an advance in phase by is equivalent to a time delay by, can be simply computed as