Propagation constant
The propagation constant, also sometimes called propagation constant propagation coefficient or propagation measure, is a quantity which (for example, an electromagnetic wave in the transmission line theory and the electrodynamics ) describes the propagation of a wave. It depends on the properties of the medium in which the wave propagates.
For sinusoidal signals and the application of the complex AC circuit analysis, it is a complex quantity and can be decomposed into real and imaginary parts ( is the imaginary unit ):
The real part of the propagation constant is the damping constant, the imaginary phase constant. You determine the attenuation and the phase rotation of the shaft and are generally frequency dependent. As an alternative description size (especially for radio and sound waves) one often uses the complex wave number:
The propagation constant of the transmission line theory
If in theory of the lines, the general solution of the telegraph equation (eg, the Laplace transform ) determined using a operational calculus, then a so-called wave parameters in addition to the characteristic impedance and the propagation constant of the line brake pads and the complex frequency defined as
With sinusoidal signals, one can replace the complex by the imaginary frequency and is given the special form
The propagation constant describes the rate, attenuation and distortion of the current through the line waves, because in the general solution of the transmission line equations with the factor
Is received. Specifically, these three factors are determined by the propagation speed
An attenuation
And a distortion
(which is always positive for real lines ) determined. Thus we obtain the following well- interpretable form of the propagation constant
Which can be used as follows for the classification of wave propagation in pipes.
Lossless line
For a lossless line are due to both equal to 0 Then, the propagation constant is reduced to
And the wave is only delayed but not attenuated or distorted, because the expression
Represents the shift operator of the Laplace transform
For sinusoidal signals, the propagation constant is purely imaginary. The delay is then a linearly increasing with frequency phase rotation.
Here is the wavelength of the propagating wave sinusoidal.
Distortion-free line
In a lossy, but distortion-free line (eg a Krarupkabel ) is the damping factor, but due to the current Heaviside condition is the distortion measure. Then the propagation constant appears as
And the wave is delayed and attenuated, but not distorted:
The left term again represents the delay of the line, while the right-hand term represents the attenuation of the wave that does not change, however, their shape.
For sinusoidal signals from the propagation constant
For linear frequency- dependent phase rotation now comes a frequency-independent damping to:
Distortion Fused line
In the general case the Heaviside condition is not applicable. Then, a third factor occurs which causes a form of distortion (dispersion ) of the current through the line shaft. His general evaluation is practically possible only with numerical tools.
In the special case of sinusoidal signals can be, however, an explicit decomposition of the propagation constant in real and imaginary parts specify:
Both components are non-linearly dependent on the frequency. Clearly one can see the behavior of the locus of the propagation constant. For the frequency 0 is the damping constant assumes its DC value. For very high frequencies the behavior of the propagation constant with the distortion-free line is consistent. Theoretically seeks the damping constant against the frequency-independent value, but practically it is growing due to the skin effect at the frequency further. For the transition region as well as for specific cable types and frequency ranges can be found in the literature simplified approximate formulas.
Due to the nonlinear frequency dependence of the phase constant must be made between phase velocity and group velocity of wave propagation.