Thepropagation constant of a sinusoidalelectromagnetic wave is a measure of the change undergone by theamplitude andphase of the wave as itpropagates in a given direction. The quantity being measured can be thevoltage, thecurrent in acircuit, or a field vector such aselectric field strength orflux density. The propagation constant itself measures thedimensionless change in magnitude or phaseper unit length. In the context oftwo-port networks and their cascades,propagation constantmeasures the change undergone by the source quantity as it propagates from one port to the next.

The propagation constant's value is expressedlogarithmically, almost universally to the basee, rather than base 10 that is used intelecommunications in other situations. The quantity measured, such as voltage, is expressed as a sinusoidalphasor. The phase of the sinusoid varies with distance which results in the propagation constant being acomplex number, theimaginary part being caused by the phase change.

The term "propagation constant" is somewhat of a misnomer as it usually varies strongly withω. 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 includetransmission parameter,transmission function,propagation parameter,propagation coefficient andtransmission constant. If the plural is used, it suggests thatα andβ are being referenced separately but collectively as intransmission parameters,propagation parameters, etc. In transmission line theory,α andβ are counted among the "secondary coefficients", the termsecondary being used to contrast to theprimary line coefficients. The primary coefficients are the physical properties of the line, namelyR,C,L andG, from which the secondary coefficients may be derived using thetelegrapher's equation. In the field of transmission lines, the termtransmission coefficient has a different meaning despite the similarity of name: it is the companion of thereflection coefficient.

The propagation constant, symbolγ, for a given system is defined by the ratio of thecomplex amplitude at the source of the wave to the complex amplitude at some distancex, such that

${\displaystyle \frac{A_0}{A_x}=e^{\gamma x}.}$

Inverting the above equation and isolatingγ results in the quotient of the complex amplitude ratio'snatural logarithm and the distancex traveled:

${\displaystyle \gamma =\ln \left(\frac{A_0}{A_x}\right)/x}$

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

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

where

• α, the real part, is called theattenuation constant
• β, the imaginary part, is called thephase constant
• ${\displaystyle i\equiv j\equiv \sqrt{-1}}$; the symbolj is more typical in electrical and electronic engineering.

Thatβ does indeed represent phase can be seen fromEuler's formula:

${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$

which is a sinusoid which varies in phase asθ varies but does not vary in amplitude because

${\displaystyle \left|e^{i\theta }\right|=\sqrt{{\cos }^2\theta +{\sin }^2\theta }=1}$

The reason for the use of basee 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. To arrive at radians requires the basee, and likewise to arrive at nepers for attenuation requires the basee.

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

${\displaystyle \gamma =\sqrt{ZY}}$

where

${\displaystyle Z=R+i\omega L}$ is the seriesimpedance of the line per unit length and,

${\displaystyle Y=G+i\omega C}$ is the shuntadmittance of the line per unit length.

The propagation factor of a plane wave traveling in a linear media in thex direction is given by$${\displaystyle P=e^{-\gamma x}}$$where

• $\gamma =\alpha +i\beta =\sqrt{i\omega \mu (\sigma +i\omega \epsilon )}$^[1]: 126
• ${\displaystyle x}$: distance traveled in thex direction
• ${\displaystyle \alpha }$:attenuation constant (SI the unit:neper/metre)
• ${\displaystyle \beta }$:phase constant (SI unit:radian/metre)
• ${\displaystyle \omega }$:angular frequency (SI unit: radian/second)
• ${\displaystyle \sigma }$:conductivity of the media
• ${\displaystyle \epsilon ={\epsilon }^{\prime }-i{\epsilon }^{″}}$:complex permitivity of the media
• ${\displaystyle \mu ={\mu }^{\prime }-i{\mu }^{″}}$:complex permeability of the media
• ${\displaystyle i}$: theimaginary unit

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 thex direction.

Wavelength,phase velocity, andskin depth have simple relationships to the components of the propagation constant:$${\displaystyle \lambda =\frac{2\pi }{\beta }{v}_p=\frac{\omega }{\beta }\delta =\frac{1}{\alpha }}$$

Intelecommunications, the termattenuation constant, also calledattenuation parameter orattenuation coefficient, is the attenuation of an electromagnetic wave propagating through amedium per unit distance from the source. It is the real part of the propagation constant and is measured using the unitneper per metre. A neper is approximately 8.7 dB. Attenuation constant can be defined by the amplitude ratio

${\displaystyle \left|\frac{A_0}{A_x}\right|=e^{\alpha x}}$

The propagation constant per unit length is defined as the natural logarithm of the ratio of the sending end current or voltage to the receiving end current or voltage, divided by the distancex involved:

${\displaystyle \alpha =\ln \left(\left|\frac{A_0}{A_x}\right|\right)/x}$

The attenuation constant for conductive lines can be calculated from the primary line coefficients as shown above. For a line meeting thedistortionless condition, with a conductanceG in the insulator, the attenuation constant is given by

${\displaystyle \alpha =\sqrt{RG}.}$

However, a real line is unlikely to meet this condition without the addition ofloading 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 theskin effect inside a conductor. The skin effect causes R along the conductor to be approximately dependent on frequency according to

${\displaystyle R\propto \sqrt{\omega }}$

Losses in the dielectric depend on theloss tangent (tan δ) of the material divided by the wavelength of the signal. Thus they are directly proportional to the frequency.

${\displaystyle {\alpha }_d=\frac{\pi \sqrt{{\epsilon }_r}}{\lambda }\tan \delta }$

The attenuation constant for a particularpropagation mode in anoptical fibre is the real part of the axial propagation constant.

Inelectromagnetic theory, thephase constant, also calledphase change constant,parameter orcoefficient is the imaginary component of the propagation constant for a plane wave. It represents the change in phase per unit length along the path traveled by the wave at any instant and is equal to thereal part of theangular wavenumber of the wave. It is represented by the symbolβ (SI unit: radians per metre).

From the definition of (angular) wavenumber fortransverse electromagnetic (TEM) waves in lossless media,

${\displaystyle k=\frac{2\pi }{\lambda }=\beta }$

For atransmission line, thetelegrapher's equations tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in thetime 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 thewaveform, 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 agroup. Since wavephase velocity is given by

${\displaystyle v_p=\frac{\lambda }{T}=\frac{f}{\tilde{\nu }}=\frac{\omega }{\beta },}$

it is proved thatβ is required to be proportional toω. In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition

${\displaystyle \beta =\omega \sqrt{LC},}$

whereL andC are, respectively, the inductance and capacitance per unit length of the line. However, practical lines can only be expected to approximately meet this condition over a limited frequency band.

In particular, the phase constant${\displaystyle \beta }$ is not always equivalent to thewavenumber${\displaystyle k}$. The relation

${\displaystyle \beta =k}$

applies to the TEM wave, which travels in free space or TEM-devices such as thecoaxial cable andtwo parallel wires transmission lines. Nevertheless, it does not apply to theTE wave (transverse electric wave) andTM wave (transverse magnetic wave). For example,^[2] in a hollowwaveguide where the TEM wave cannot exist but TE and TM waves can propagate,

${\displaystyle k=\frac{\omega }{c}}$

${\displaystyle \beta =k\sqrt{1-\frac{{\omega }_c^2}{{\omega }^2}}}$

Here${\displaystyle {\omega }_c}$ is thecutoff frequency. In a rectangular waveguide, the cutoff frequency is

${\displaystyle {\omega }_c=c\sqrt{{\left(\frac{n\pi }{a}\right)}^2+{\left(\frac{m\pi }{b}\right)}^2},}$

where${\displaystyle m,n\ge 0}$ are the mode numbers for the rectangle's sides of length${\displaystyle a}$ and${\displaystyle b}$ respectively. For TE modes,${\displaystyle m,n\ge 0}$ (but${\displaystyle m=n=0}$ is not allowed), while for TM modes${\displaystyle m,n\ge 1}$.

The phase velocity equals

${\displaystyle v_p=\frac{\omega }{\beta }=\frac{c}{\sqrt{1-\frac{{\omega }_{\mathrm{c}}^2}{{\omega }^2}}}>c}$

The term propagation constant or propagation function is applied tofilters and othertwo-port networks used forsignal processing. In these cases, however, the attenuation and phase coefficients are expressed in terms of nepers and radians pernetwork section rather than per unit length. Some authors^[3] 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 sectiontopology. 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.

The ratio of output to input voltage for each network is given by^[4]

${\displaystyle \frac{V_1}{V_2}=\sqrt{\frac{Z_{I1}}{Z_{I2}}}{e}^{{\gamma }_1}}$

${\displaystyle \frac{V_2}{V_3}=\sqrt{\frac{Z_{I2}}{Z_{I3}}}{e}^{{\gamma }_2}}$

${\displaystyle \frac{V_3}{V_4}=\sqrt{\frac{Z_{I3}}{Z_{I4}}}{e}^{{\gamma }_3}}$

The terms${\displaystyle \sqrt{\frac{Z_{In}}{Z_{Im}}}}$ are impedance scaling terms^[5] and their use is explained in theimage impedance article.

The overall voltage ratio is given by

${\displaystyle \frac{V_1}{V_4}=\frac{V_1}{V_2}\cdot \frac{V_2}{V_3}\cdot \frac{V_3}{V_4}=\sqrt{\frac{Z_{I1}}{Z_{I4}}}{e}^{{\gamma }_1+{\gamma }_2+{\gamma }_3}}$

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

${\displaystyle {\gamma }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\gamma }_1+{\gamma }_2+{\gamma }_3+\cdots +{\gamma }_n}$

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 articleMathematical descriptions of opacity.

• Propagation speed

• "Propagation constant". Microwave Encyclopedia. 2011. Archived fromthe original(Online) on July 14, 2014. RetrievedFebruary 2, 2011.
• Paschotta, Dr. Rüdiger (2011)."Propagation Constant"(Online). Encyclopedia of Laser Physics and Technology. Retrieved2 February 2011.
• Janezic, Michael D.; Jeffrey A. Jargon (February 1999)."Complex Permittivity determination from Propagation Constant measurements"(PDF).IEEE Microwave and Guided Wave Letters.9 (2):76–78.doi:10.1109/75.755052. Retrieved2 February 2011. Free PDF download is available. There is an updated version dated August 6, 2002.

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