Propagation constant
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the dimensionless change in magnitude or phase per unit length. In the context of 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.
The propagation constant's value is expressed logarithmically, almost universally to the base e, rather than base 10 that is used in telecommunications 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 part being caused by the phase change.
Alternative names
[edit]The term "propagation constant" is somewhat of a misnomer as it usually varies strongly with
Definition
[edit]The propagation constant, symbol
Inverting the above equation and isolating
Since the propagation constant is a complex quantity we can write:
where
α , the real part, is called the attenuation constantβ , the imaginary part, is called the phase constant- more often j is used for electrical circuits.
That
which is a sinusoid which varies in phase as
The reason for the use of base e is also now made clear. The imaginary phase constant, i
The propagation constant for conducting lines can be calculated from the primary line coefficients by means of the relationship
where
- the series impedance of the line per unit length and,
- the shunt admittance of the line per unit length.
Plane wave
[edit]The propagation factor of a plane wave traveling in a linear media in the x direction is given by where
- [1]: 126
- distance traveled in the x direction
- attenuation constant in the units of nepers/meter
- phase constant in the units of radians/meter
- frequency in radians/second
- conductivity of the media
- = complex permitivity of the media
- = complex permeability of the media
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 x direction.
Wavelength, phase velocity, and skin depth have simple relationships to the components of the propagation constant:
Attenuation constant
[edit]In telecommunications, the term attenuation constant, also called attenuation parameter or attenuation coefficient, is the attenuation of an electromagnetic wave propagating through a 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 dB. Attenuation constant can be defined by the amplitude ratio
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 distance x involved:
Conductive lines
[edit]The attenuation constant for conductive lines can be calculated from the primary line coefficients as shown above. For a line meeting the distortionless condition, with a conductance G in the insulator, the attenuation constant is given by
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
Losses in the dielectric depend on the loss tangent (tan
Optical fibre
[edit]The attenuation constant for a particular propagation mode in an optical fiber is the real part of the axial propagation constant.
Phase constant
[edit]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 unit length along the path travelled by the wave at any instant and is equal to the real part of the angular wavenumber of the wave. It is represented by the symbol
From the definition of (angular) wavenumber for transverse electromagnetic (TEM) waves in lossless media,
For a transmission line, the telegrapher's equations 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. Since wave phase velocity is given by
it is proved that
where L and C 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 is not always equivalent to the wavenumber . The relation
applies to the TEM wave, which travels in free space or TEM-devices such as the coaxial cable and two parallel wires transmission lines. Nevertheless, it does not apply to the TE wave (transverse electric wave) and TM wave (transverse magnetic wave). For example,[2] in a hollow waveguide where the TEM wave cannot exist but TE and TM waves can propagate,
Here is the cutoff frequency. In a rectangular waveguide, the cutoff frequency is
where are the mode numbers for the rectangle's sides of length and respectively. For TE modes, (but is not allowed), while for TM modes .
The phase velocity equals
Filters and two-port networks
[edit]The term propagation constant or propagation function is applied to filters and other two-port networks used for signal processing. In these cases, however, the attenuation and phase coefficients are expressed in terms of nepers and radians per network 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 section 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
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/29/Adding_propagation_constant_of_filters.svg/750px-Adding_propagation_constant_of_filters.svg.png)
The ratio of output to input voltage for each network is given by[4]
The terms are impedance scaling terms[5] and their use is explained in the image impedance article.
The overall voltage ratio is given by
Thus for n cascaded sections all having matching impedances facing each other, the overall propagation constant is given by
See also
[edit]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
[edit]- ^ Jordon, Edward C.; Balman, Keith G. (1968). Electromagnetic Waves and Radiating Systems (2nd ed.). Prentice-Hall.
- ^ Pozar, David (2012). Microwave Engineering (4th ed.). John Wiley &Sons. pp. 62–164. ISBN 978-0-470-63155-3.
- ^ Matthaei et al, p49
- ^ Matthaei et al pp51-52
- ^ Matthaei et al pp37-38
References
[edit]This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22..
- Matthaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964.
External links
[edit]- "Propagation constant". Microwave Encyclopedia. 2011. Archived from the original (Online) on July 14, 2014. Retrieved February 2, 2011.
- Paschotta, Dr. Rüdiger (2011). "Propagation Constant" (Online). Encyclopedia of Laser Physics and Technology. Retrieved 2 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. Retrieved 2 February 2011. Free PDF download is available. There is an updated version dated August 6, 2002.