Propagation constant
The propagation constant of a sinusoidal
The propagation constant's value is expressed
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. If the plural is used, it suggests that α and β are being referenced separately but collectively as in transmission parameters, propagation parameters, etc. In transmission line theory, α and β are counted among the "secondary coefficients", the term secondary being used to contrast to the
Definition
The propagation constant, symbol γ, for a given system is defined by the ratio of the complex amplitude at the source of the wave to the complex amplitude at some distance x, such that,
Inverting the above equation and isolating γ results in the quotient of the complex amplitude ratio's natural logarithm and the distance x traveled:
Since the propagation constant is a complex quantity we can write:
where
- α, the real part, is called the attenuation constant
- β, the imaginary part, is called the phase constant
- more often j is used for electrical circuits.
That β does indeed represent phase can be seen from Euler's formula:
which is a sinusoid which varies in phase as θ varies but does not vary in amplitude because
The reason for the use of base e is also now made clear. The imaginary phase constant, i β, 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 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
The propagation factor of a plane wave traveling in a linear media in the x direction is given by
- [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
In
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
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
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
Optical fibre
The attenuation constant for a particular
Phase constant
In
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 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
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
The term propagation constant or propagation function is applied to
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
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
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.
- Propagation speed
Notes
- ^ Jordon, Edward C.; Balman, Keith G. (1968). Electromagnetic Waves and Radiating Systems (2nd ed.). Prentice-Hall.
- ISBN 978-0-470-63155-3.
- ^ Matthaei et al, p49
- ^ Matthaei et al pp51-52
- ^ Matthaei et al pp37-38
References
- 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
- "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). . Retrieved 2 February 2011. Free PDF download is available. There is an updated version dated August 6, 2002.