Propagation constant
The propagation constant itself measures the dimensionless change in magnitude or phase per unit length.
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.
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 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 propagation factor of a plane wave traveling in a linear media in the x direction is given by
Wavelength, phase velocity, and skin depth have simple relationships to the components of the propagation constant:
It is the real part of the propagation constant and is measured in nepers per metre.
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: 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.
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 δ) of the material divided by the wavelength of the signal.
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.
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.
However, practical lines can only be expected to approximately meet this condition over a limited frequency band.
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.
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.
The propagation constant is a useful concept in filter design which invariably uses a cascaded section topology.
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 The concept of penetration depth is one of many ways to describe the absorption of electromagnetic waves.
