Propagation constant

The propagation constant , sometimes also called the propagation constant , coefficient of propagation or degree of propagation , is a quantity that describes the propagation of a wave (e.g. an electromagnetic wave in line theory and electrodynamics ). It depends on the properties of the medium in which the wave is propagating.

In the case of sinusoidal signals and the application of the complex alternating current calculation, it is a complex quantity and can be broken down into real and imaginary parts ( be the imaginary unit ): ${\displaystyle j}$

${\displaystyle \gamma =\alpha +j\beta }$

The real part of the propagation constant is called the damping constant , the imaginary part is called the phase constant . They determine the damping or phase rotation of the shaft and are generally frequency-dependent. The complex wave number is often used as an alternative description quantity (especially for radio and sound waves) : ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle k}$

${\displaystyle \gamma =jk}$

The propagation constant in conduction theory

If the general solution of the line equation is determined in the theory of lines with the help of an operator calculation (e.g. the Laplace transformation ), then the so-called wave parameters in addition to the line impedance also the propagation constant from the line coverings and the complex frequency are defined as ${\displaystyle s}$

${\displaystyle \gamma ={\sqrt {(R'+sL')(G'+sC')}}}$

In the case of sinusoidal signals, the complex frequency can be replaced by the imaginary frequency and the special shape is obtained ${\displaystyle j\omega }$

${\displaystyle \gamma ={\sqrt {(R'+j\omega L')(G'+j\omega C')}}}$

The propagation constant describes the speed, attenuation and distortion of the waves traveling over the line, because they are included in the general solution of the line equations with the factor

${\displaystyle e^{\pm \gamma x}}$

comes in. These three influences become concrete through the speed of propagation

${\displaystyle v={\frac {1}{\sqrt {L'C'}}}}$,

an attenuation measure

${\displaystyle D={\frac {1}{2}}\cdot \left({\frac {R^{\prime }}{L^{\prime }}}+{\frac {G^{\prime }}{C^{\prime }}}\right)}$

and a distortion measure

${\displaystyle V={\frac {1}{2}}\cdot \left({\frac {R^{\prime }}{L^{\prime }}}-{\frac {G^{\prime }}{C^{\prime }}}\right)}$

(which is always positive for real lines). This gives the following form of the propagation constant that is easy to interpret

${\displaystyle \gamma ={\frac {1}{v}}\cdot {\sqrt {\left(s+D\right)^{2}-V^{2}}}}$

which can be used as below to classify the wave propagation on lines.

Lossless leadership

In the case of a lossless line , both and are equal to 0. Then the propagation constant is reduced to ${\displaystyle R'=G'=0}$ ${\displaystyle D}$ ${\displaystyle V}$

${\displaystyle \gamma ={\frac {s}{v}}}$

and the wave is only delayed but not attenuated or distorted because of the expression

${\displaystyle e^{\pm s{\frac {x}{v}}}}$

represents the displacement operator of the Laplace transform .

With sinusoidal signals, the propagation constant is purely imaginary. The delay then means a phase rotation that increases linearly with the frequency.

${\displaystyle \gamma =j\beta ={\frac {j\omega }{v}}={\frac {2\pi j}{\lambda }}}$

Here is the wavelength of the propagating sinusoidal wave. ${\displaystyle \lambda }$

Locus of the propagation constant γ of a line with R '= 10 Ω / km, G' = 1 mS / km, L '= 2 mH / km and C' = 5 nF / km

Distortion-free line

In the case of a lossy but distortion-free line (e.g. a Krarup cable ), the attenuation measure is , but due to the Heaviside condition that applies is the distortion measure . Then the propagation constant appears as ${\displaystyle D>0}$ ${\displaystyle V=0}$

${\displaystyle \gamma ={\frac {s+D}{v}}}$

and the wave is delayed and attenuated but not distorted:

${\displaystyle e^{\pm (s+D){\frac {x}{v}}}=e^{\pm s{\frac {x}{v}}}\cdot e^{\pm D{\frac {x}{v}}}}$

The left term again represents the delay in the line, while the right term represents attenuation of the wave, which, however, does not change its shape.

In the case of sinusoidal signals, the propagation constant becomes

${\displaystyle \gamma ={\frac {j\omega +D}{v}}={\frac {D}{v}}+{\frac {j\omega }{v}}=\alpha +j\beta }$

In addition to the linear frequency-dependent phase shift, there is now frequency-independent damping :

${\displaystyle \alpha ={\frac {D}{v}}={\sqrt {R'\cdot G'}}}$

Distorted line

In the general case, however , the Heaviside condition does not apply. Then a third factor occurs which causes a shape distortion ( dispersion ) of the wave traveling over the line. Its general evaluation is practically only possible with numerical aids.

In the special case of sinusoidal signals, however, an explicit decomposition of the propagation constant into real and imaginary parts can be given:

${\displaystyle \alpha ={\sqrt {{\frac {1}{2}}\cdot {\sqrt {\left(R'^{2}+\omega ^{2}L'^{2}\right)\left(G'^{2}+\omega ^{2}C'^{2}\right)}}+{\frac {1}{2}}\cdot \left(R'G'-\omega ^{2}L'C'\right)}}}$

${\displaystyle \beta ={\sqrt {{\frac {1}{2}}\cdot {\sqrt {\left(R'^{2}+\omega ^{2}L'^{2}\right)\left(G'^{2}+\omega ^{2}C'^{2}\right)}}-{\frac {1}{2}}\cdot \left(R'G'-\omega ^{2}L'C'\right)}}}$

Both components are non-linearly dependent on the frequency. The behavior can be clearly seen from the locus curve of the propagation constant. For the frequency 0 the damping constant assumes its direct current value . For very high frequencies, the behavior of the propagation constant corresponds to the distortion-free line. Theoretically, the damping constant tends towards the frequency-independent value , but in practice it continues to increase with frequency due to the skin effect . Simplified approximation formulas can be found in the literature for the transition range as well as for certain cable types and frequency ranges . ${\displaystyle \alpha _{=}={\sqrt {R'\cdot G'}}}$ ${\displaystyle \alpha _{\infty }={\frac {1}{2}}\cdot \left(R'{\sqrt {\frac {C'}{L'}}}+G'{\sqrt {\frac {L'}{C'}}}\right)}$

Due to the non-linear frequency dependence of the phase constant , a distinction must be made between phase velocity and group velocity of wave propagation. ${\displaystyle \beta }$

literature

• Peter Vielhauer : Theory of transmission on electrical lines . Verlag Technik, Berlin 1970.

Individual evidence

1. ↑ Eugen Philippow : Fundamentals of electrical engineering . Academic publishing company Geest & Portig K.-G., Leipzig 1967.
2. ↑ Heinrich Schröder: Electrical Communication Engineering, Volume I . Publishing house for Radio-Foto-Kinotechnik GmbH, Berlin-Borsigwalde 1966.