Editing Propagation constant (section)

==Filters and two-port networks==
The term propagation constant or propagation function is applied to [[Electronic filters|filters]] and other [[two-port network]]s used for [[signal processing]].  In these cases, however, the attenuation and phase coefficients are expressed in terms of nepers and radians per [[Electronic filter topology#Passive topologies|network section]] rather than per unit length.  Some authors<ref>Matthaei et al, p49</ref> make a distinction between per unit length measures (for which "constant" is used) and per section measures (for which "function" is used).

The propagation constant is a useful concept in filter design which invariably uses a cascaded section [[Electronic filter topology|topology]].  In a cascaded topology, the propagation constant, attenuation constant and phase constant of individual sections may be simply added to find the total propagation constant etc.

===Cascaded networks===
[[Image:Adding propagation constant of filters.svg|center|thumb|750px|Three networks with arbitrary propagation constants and impedances connected in cascade.  The ''Z<sub>i</sub>'' terms represent [[image impedance]] and it is assumed that connections are between matching image impedances.]]

The ratio of output to input voltage for each network is given by<ref>Matthaei et al pp51-52</ref>

:<math>\frac{V_1}{V_2}=\sqrt{\frac{Z_{I1}}{Z_{I2}}}e^{\gamma_1}</math>
:<math>\frac{V_2}{V_3}=\sqrt{\frac{Z_{I2}}{Z_{I3}}}e^{\gamma_2}</math>
:<math>\frac{V_3}{V_4}=\sqrt{\frac{Z_{I3}}{Z_{I4}}}e^{\gamma_3}</math>

The terms <math>\sqrt{\frac{Z_{In}}{Z_{Im}}}</math> are impedance scaling terms<ref>Matthaei et al pp37-38</ref> and their use is explained in the [[Image impedance#Transfer function|image impedance]] article.

The overall voltage ratio is given by

:<math>\frac{V_1}{V_4}=\frac{V_1}{V_2}\cdot\frac{V_2}{V_3}\cdot\frac{V_3}{V_4}=\sqrt{\frac{Z_{I1}}{Z_{I4}}}e^{\gamma_1+\gamma_2+\gamma_3}</math>

Thus for ''n'' cascaded sections all having matching impedances facing each other, the overall propagation constant is given by

:<math>\gamma_\mathrm{total}=\gamma_1 + \gamma_2 + \gamma_3 + \cdots + \gamma_n</math>