Loading Coil - Campbell Equation

Campbell Equation

The Campbell equation is a relationship due to George Ashley Campbell for predicting the propagation constant of a loaded line. It is stated as;

where,

is the propagation constant of the unloaded line

is the propagation constant of the loaded line

is the interval between coils on the loaded line

is the impedance of a loading coil and

is the characteristic impedance of the unloaded line.

A more engineer friendly rule of thumb is that the approximate requirement for spacing loading coils is ten coils per wavelength of the maximum frequency being transmitted. This approximation can be arrived at by treating the loaded line as a constant k filter and applying image filter theory to it. From basic image filter theory the angular cutoff frequency and the characteristic impedance of a low-pass constant k filter are given by;

and,

where and are the half section element values.

From these basic equations the necessary loading coil inductance and coil spacing can be found;

and,

where C is the capacitance per unit length of the line.

Expressing this in terms of number of coils per cutoff wavelength yields;

where v is the velocity of propagation of the cable in question.