Editing Loading coil (section)

==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;<ref>Brittain, p. 43</ref>

:<math>\cosh \left(\gamma'd\right) = \cosh (\gamma d) + \frac{Z}{2Z_0} \sinh (\gamma d)</math>

where,

:<math>\gamma \!\,</math> is the propagation constant of the unloaded line
:<math>\gamma ' \!\,</math> is the propagation constant of the loaded line
:<math>d \!\,</math> is the interval between coils on the loaded line
:<math>Z \!\,</math> is the impedance of a loading coil and
:<math>Z_0 \!\,</math> 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.<ref>Brittain, p. 42</ref> This approximation can be arrived at by treating the loaded line as a [[constant k filter]] and applying [[image impedance|image filter theory]] to it. From basic image filter theory the angular cutoff frequency and the characteristic impedance of a [[low-pass filter|low-pass]] constant k filter are given by;

:<math>\omega_c = \frac{1}{\sqrt{L_{\frac{1}{2}} C_{\frac{1}{2}}}}</math> &nbsp;and,&nbsp; <math>Z_0 = \sqrt{\frac{L_{\frac{1}{2}}}{C_{\frac{1}{2}}}}</math>

where <math display="inline">L_{\frac{1}{2}}</math> and <math>C_{\frac{1}{2}}</math> are the half section element values.

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

:<math>L = \frac{Z_0}{\omega_c}</math> &nbsp;and,&nbsp; <math>d = \frac{2}{\omega_c Z_0 C}</math>

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

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

:<math>\frac{\lambda_c}{d} = \pi v Z_0 C</math>

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

Since <math display="inline">v = {1 \over Z_0 C}</math> then

:<math>\frac{\lambda_c}{d} = \pi</math>.

Campbell arrived at this expression by analogy with a mechanical line periodically loaded with weights described by Charles Godfrey in 1898 who obtained a similar result. Mechanical loaded lines of this sort were first studied by [[Joseph-Louis Lagrange]] (1736–1813).<ref>Mason, p. 409</ref>

The phenomenon of cutoff whereby frequencies above the cutoff frequency are not transmitted is an undesirable side effect of loading coils (although it proved highly useful in [[Passive analogue filter development|the development of filters]]). Cutoff is avoided by the use of continuous loading since it arises from the lumped nature of the loading coils.<ref>Bakshi & Bakshi, p. 1.56</ref>