Editing Propagation constant (section)

==Definition==

The propagation constant, symbol {{mvar|γ}}, for a given system is defined by the ratio of the [[Phasor|complex amplitude]] at the source of the wave to the complex amplitude at some distance {{mvar|x}}, such that,

:<math> \frac{A_0}{A_x} = e^{\gamma x} </math>

Inverting the above equation and isolating {{mvar|γ}} results in the quotient of the complex amplitude ratio's [[complex logarithm|natural logarithm]] and the distance {{mvar|x}} traveled:

:<math>\gamma=\ln\left(\frac{A_0}{A_x}\right)/x</math>

Since the propagation constant is a complex quantity we can write:

:<math display="block" qid=Q1434913> \gamma = \alpha + i \beta\ </math>
where
* {{mvar|α}}, the real part, is called the [[#Attenuation constant|''attenuation constant'']]
* {{mvar|β}}, the imaginary part, is called the [[#Phase constant|''phase constant'']]
* <math>i \equiv j \equiv \sqrt{ -1\ }\ ;</math> more often  {{mvar|j}}  is used for electrical circuits. 

That {{mvar|β}} does indeed represent phase can be seen from [[Euler's formula]]:

:<math> e^{i\theta} = \cos{\theta} + i \sin{\theta}\ </math>

which is a sinusoid which varies in phase as {{mvar|θ}} varies but does not vary in amplitude because

:<math> \left| e^{i\theta} \right| = \sqrt{ \cos^2{\theta} + \sin^2{\theta}\;} = 1 </math>

The reason for the use of base {{mvar|e}} is also now made clear. The imaginary phase constant, {{mvar|i β}}, can be added directly to the attenuation constant, {{mvar|α}}, to form a single complex number that can be handled in one mathematical operation provided they are to the same base. Angles measured in radians require base {{mvar|e}}, so the attenuation is likewise in base {{mvar|e}}.

The propagation constant for conducting lines can be calculated from the primary line coefficients by means of the relationship

:<math> \gamma= \sqrt{ Z Y\ }</math>

where

:<math> Z = R + i\ \omega L\ ,</math> the series [[Electrical impedance|impedance]] of the line per unit length and,

:<math> Y = G + i\ \omega C\ ,</math> the shunt [[admittance]] of the line per unit length.

===Plane wave===
The propagation factor of a plane wave traveling in a linear media in the {{mvar|x}} direction is given by
<math display="block"> P = e^{-\gamma x} </math>
where
* <math display="inline">\gamma = \alpha + i\ \beta = \sqrt{i\ \omega\ \mu\ (\sigma + i\ \omega \varepsilon)\ }\ </math><ref name="Jordon&Balman">{{cite book |last1=Jordon |first1=Edward C. |last2=Balman |first2=Keith G. |year=1968 |title=Electromagnetic Waves and Radiating Systems |edition=2nd |publisher=Prentice-Hall }}</ref>{{rp|p=126}}
* <math> x = </math>  distance traveled in the {{mvar|x}} direction
* <math> \alpha =\ </math> [[attenuation constant]] in the units of [[neper]]s/meter
* <math> \beta =\ </math> [[phase constant]] in the units of [[radian]]s/meter
* <math> \omega=\ </math> frequency in radians/second
* <math> \sigma =\ </math> [[Electrical resistivity and conductivity|conductivity]] of the media
* <math>\varepsilon = \varepsilon' - i\ \varepsilon'' \ </math> = [[Permittivity#Complex permittivity|complex permitivity]] of the media
* <math>\mu = \mu' - i\ \mu'' \;</math> = [[Permeability (electromagnetism)#Complex permeability|complex permeability]] of the media
* <math>i \equiv \sqrt{-1\ }</math>
The sign convention is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the {{mvar|x}} direction.

[[Wavelength]], [[phase velocity]], and [[skin effect|skin depth]] have simple relationships to the components of the propagation constant:
<math display="block"> \lambda = \frac {2 \pi}{\beta} \qquad  v_p = \frac{\omega}{\beta}  \qquad   \delta =  \frac{1}{\alpha} </math>