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{{Redirect|Transmission parameter|ABCD transmission parameters|two-port network#ABCD-parameters|scattering transfer parameters|scattering parameters#Scattering transfer parameters}}
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The '''propagation constant''' of an [[electromagnetic wave]] is a measure of the change undergone by the amplitude of the wave as it [[wave propagation|propagates]] in a given direction. The quantity being measured can be the [[voltage]] or [[electric current|current]] in a circuit or a field vector such as [[electric field strength]] or [[flux density]]. The propagation constant itself measures change per metre but is otherwise dimensionless. In the context of [[Two port networks|two-port networks]] and their cascades, '''propagation constant '''measures the change undergone by the source quantity as it propagates from one port to the next.
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The propagation constant is expressed logarithmically, almost universally to the base e, rather than the more usual base 10 used in [[telecommunication]]s in other situations.  The quantity measured, such as voltage, is expressed as a sinusoidal [[phasor]].  The phase of the sinusoid varies with distance which results in the propagation constant being a [[complex number]], the [[imaginary number|imaginary]] part being caused by the phase change.
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__TOC__
 
==Alternative names==
 
The term propagation constant is somewhat of a misnomer as it usually varies strongly with ''ω''.  It is probably the most widely used term but there are a large variety of alternative names used by various authors for this quantity.  These include, '''transmission parameter''', '''transmission function''', '''propagation parameter''', '''propagation coefficient''' and '''transmission constant'''. In plural, it is usually implied that ''α'' and ''β'' are being referenced separately but collectively as in '''transmission parameters''', '''propagation parameters''', '''propagation coefficients''', '''transmission constants''' and '''secondary coefficients'''. This last occurs in transmission line theory, the term ''secondary'' being used to contrast to the ''[[primary line coefficients]]''. The primary coefficients being the physical properties of the line; R,C,L and G, from which the secondary coefficients may be derived using the [[telegrapher's equation]]. Note that, at least in the field of transmission lines, the term [[transmission coefficient]] has a different meaning despite the similarity of name.  Here it is the corollary of [[reflection coefficient]].
 
==Definition==
 
The propagation constant, symbol γ, for a given system is defined by the ratio of the amplitude at the source of the wave to the amplitude at some distance ''x'', such that,
 
:<math>\frac{A_0}{A_x}=e^{\gamma x}</math>
 
Since the propagation constant is a complex quantity we can write:
 
:<math>\gamma = \alpha +i \beta \,</math>
where
:&alpha;, the real part, is called the [[#Attenuation constant|attenuation constant]]
:&beta;, the imaginary part, is called the [[#Phase constant|phase constant]]
 
That 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 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 e is also now made clear. The imaginary phase constant, iβ, can be added directly to the attenuation constant, 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 e, so the attenuation is likewise in base e.
 
The propagation constant for copper (or any other conductor) lines can be calculated from the primary line coefficients by means of the relationship;
 
:<math>\gamma=\sqrt{ZY}</math>
 
where;
 
:<math>Z=R+i\omega L\,\!</math>, the series [[Electrical impedance|impedance]] of the line per metre and,
 
:<math>Y=G+i\omega C\,\!</math>, the shunt [[admittance]] of the line per metre.
 
==Attenuation constant==
 
In [[telecommunication]]s, the term '''attenuation constant''', also called '''attenuation parameter''' or '''attenuation coefficient''', is the attenuation of an electromagnetic wave propagating through a [[Transmission medium|medium]] per unit distance from the source. It is the real part of the propagation constant and is measured in [[nepers]] per metre.  A neper is approximately 8.7[[decibel|dB]].  Attenuation constant can be defined by the amplitude ratio;
 
:<math>\left|\frac{A_0}{A_x}\right|=e^{\alpha x}</math>
 
The propagation constant per unit length is defined as the natural logarithmic of ratio of the sending end current or voltage to the receiving end current or voltage.
 
===Copper lines===
 
The attenuation constant for copper lines (or ones made of any other conductor) can be calculated from the primary line coefficients as shown above. For a line meeting the [[Heaviside condition|distortionless condition]], with a conductance G in the insulator, the attenuation constant is given by;
 
:<math>\alpha=\sqrt{RG}\,\!</math>
 
however, a real line is unlikely to meet this condition without the addition of [[loading coils]] and, furthermore, there are some frequency dependent effects operating on the primary "constants" which cause a frequency dependence of the loss. There are two main components to these losses, the metal loss and the dielectric loss.
 
The loss of most transmission lines are dominated by the metal loss, which causes a frequency dependency due to finite conductivity of metals, and the [[skin effect]] inside a conductor. The skin effect causes R along the conductor to be approximately dependent on frequency according to;
 
:<math>R \propto \sqrt{\omega}</math>
 
Losses in the dielectric depend on the [[loss tangent]] (<math>{\tan \delta}</math>) of the material, which depends inversely on the wavelength of the signal and is directly proportional to the frequency.
 
:<math>\alpha_d={{\pi}\sqrt{\varepsilon_r}\over{\lambda}}{\tan \delta}</math>
 
===Optical fibre===
 
The attenuation constant for a particular [[propagation mode]] in an [[optical fiber]], the real part of the [[axial propagation constant]].
 
==Phase constant==
In [[electromagnetic theory]], the '''phase constant''', also called '''phase change constant''', '''parameter''' or '''coefficient''' is the imaginary component of the propagation constant for a plane wave. It represents the change in phase per metre along the path travelled by the wave at any instant and is equal to [[real part]] of the [[wavenumber#In wave equations|angular wavenumber]] of the wave.  It is represented by the symbol β and is measured in units of radians per metre.
 
From the definition of (angular) wavenumber;
 
:<math>k = \frac{2\pi}{\lambda} = \beta</math>
 
For a [[transmission line]], the [[Heaviside condition]] of the [[telegrapher's equation]] tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in the [[time domain]]. This includes, but is not limited to, the ideal case of a lossless line. The reason for this condition can be seen by considering that a useful signal is composed of many different wavelengths in the frequency domain.  For there to be no distortion of the [[waveform]], all these waves must travel at the same velocity so that they arrive at the far end of the line at the same time as a [[group velocity|group]].  Since wave [[phase velocity]] is given by;
 
:<math>v_p=\frac{\lambda}{T}=\frac{f}{\tilde{\nu}}=\frac{\omega}{\beta}</math>
 
it is proved that β is required to be proportional to ω. In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition;
 
:<math>\beta = \omega \sqrt{LC}</math>
 
However, practical lines can only be expected to approximately meet this condition over a limited frequency band.
 
==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 metre.  Some authors<ref>Matthaei et al, p49</ref> make a distinction between per metre 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_{Tot}=\gamma_1 + \gamma_2 + \gamma_3 + \cdots + \gamma_n</math>
 
==See also==
The concept of penetration depth is one of many ways to describe the absorption of electromagnetic waves. For the others, and their interrelationships, see the article: [[Mathematical descriptions of opacity]].
 
==Notes==
{{reflist}}
 
==References==
<div class="references-small">
*{{FS1037C}}.
*Matthaei, Young, Jones ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'' McGraw-Hill 1964.
</div>
 
==External links==
*{{Cite web|last =|title =Propagation constant|publisher =Microwave Encyclopedia|year = 2011|url =http://www.microwaves101.com/encyclopedia/propagation.cfm|format =Online
  | accessdate =February 2, 2011}}
*{{Cite web| last =Paschotta| first =Dr. Rüdiger| title =Propagation Constant
  | publisher =Encyclopedia of Laser Physics and Technology| year = 2011
  | url =http://www.rp-photonics.com/propagation_constant.html
  | format =Online| accessdate =February 2011}}
*{{Cite journal
  | last =Janezic
  | first = Michael D.
  | coauthors =Jeffrey A. Jargon
  | title = Complex Permittivity determination from Propagation Constant measurements
  | journal =[[Microwave and Guided Wave Letters, IEEE]]
  | volume = 9
  | format =
  | pages = 76–78
  | date =February 1999
  | url =http://www.eeel.nist.gov/dylan_papers/MGWL99.pdf
  | doi = 10.1109/75.755052
  | accessdate =February 2011}} Free PDF download is available. There is an updated version dated August 6, 2002.
 
[[Category:Physical quantities]]
[[Category:Telecommunication theory]]
[[Category:Electromagnetism]]
[[Category:Electromagnetic radiation]]
[[Category:Analog circuits]]
[[Category:Filter theory]]
[[Category:Image impedance filters]]

Revision as of 23:05, 4 March 2014

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