Hello,
I got confused when I've been face with different explanations about wave number and phase constant.
Propagation constant=alpha+j Beta (Many textbooks, such as Balanis -advanced engineering electromagnetic book)
where Beta is phase constant. Regarding These web sites:
https://physics.stackexchange.com/questions/19994/relation-between-wavenumber-and-propagation-constant
https://sci.physics.electromag.narkive.com/9vGZGIpc/propagation-constant-beta-and-wave-numbr-k
Sometimes Beta is complex where is meaningless when we use Propagation constant=alpha+j Beta even alpha is negligible. Sometimes Beta is wave-number, and Sometimes we have complex wave number(k) and Propagation constant is equal to jk. And sometimes, wave number is for plane wave and Beta is for wave guide structures. Balanis in his advanced engineering electromagnetic book, declared beta as phase constant in "sloving helmholtz wave equation" and in "sloving helmholtz wave equation in wave-guide " he mentioned Beta as wave number.
I will be grateful if you could kindly help me on this issue.
John Jin A wave solution in free-space leads to wave number k, which is for TEM mode of propagation. Here k is a vector along the wave direction that makes it the phase constant (as relates to phase velocity). If k has real and imaginary terms, the imaginary part refers to attenuation due to losses.
In transmission lines, the term beta is used. If the transmission mode is TEM, then both are identical. If the transmission line has loss, we have attenuation constant alpha also.
In a waveguide there is no TEM mode, and kz is used to denote the direction of propagation (z-axis is normally used propagation direction; x-axis and y-axis are bounded by the structure walls). Note that it is only one component of k, the others being kx, ky. In that scenario, the phase velocity is determined by kz and not k. Many books use the term Kz or beta interchangeably, to refer to the phase constant and related phase velocity.
Since kz is expected to be real for propagation, under some conditions, it may cease to be real. This is the evanescent wave scenario and causes the cutoff frequency limits in waveguides.
I hope I have addressed some of the confusion.
Lastly, the relationship of beta and wave number... I leave it for you to derive and verify :)
Regards
Usually the propagation constant is gamma = alpha plus j beta. Beta is the same as k. If people make beta complex they are using it instead of gamma, and rolling alpha into it. In that case they may also take j out and then the complex part becomes negative, because gamma = j( beta -j alpha). Just look at where the parts end up, in exp(j (w t - beta z)-alpha z) to decide which is which. Alpha is the attenuation and beta is the rate of phase change with distance.
Watch out for sign changes, some people use -z instead of z because their wave is travelling in the opposite direction.
One reason why people use complex beta is because it is the product of the free-space wavenumber and the complex refractive index. j ko ( n'+ jn") = j (beta - j alpha). The complex refractive index is the square root of the complex relative permittivity (usually) sqrt(e' - j e") = sqrt( e' (1 - j tan( delta ))) = n' - j n" = n' (1 - j tan(delta/2))
A typical unatenuated traveling wave is expressed as say along x direction
A cos(kx- omega t + phi)
The magnitude of wavevector is wavenumber k, the phase is phi.
As you see they are different
k=2 pi/lambda
lambda is wavelength
The phase phi is an angle between 0 and 2 pi
In 3D
A cos(k.r -omega t + phi)
where wavector is dotted into position vector r
A is amplitude of wave.
It can return to wave equation, propagation constant is came from physical properties of medium; Epsilon r and Mu r. Yes Beta is a phase constant that equal 2pi/Lambda, but Epsilon r some time is a complex and for human body, and it affect the wave length of propagation. Attenuation constant some time is neglected when taking about far field of propagation and it have been talking about channel loss instead.
Hi John,
As an EM signal propagates there is a sinusoidal variation in field strength with distance and coupled to this therefore a sinusoidal variation in field strength in time. The phase constant (beta) captures the variation in phase with distance and is given as the number of radians per metre. As Malcolm White has pointed out above, beta is the imaginary component of the propagation constant (gamma); the real part of the propagation constant describes the variation in amplitude with distance. The wave number (K) is the number of wavelengths per metre. For TEM modes there is no distinction between the wave number and the phase constant. (An interesting consequence of this is that the cut-off frequency of TEM modes is zero.) However, TEM modes are not possible in transmission lines consisting of ONE conductor, e.g. waveguides, and consequently K is NOT equal to beta (and there is a cut-off frequency > 0).
Just to complete the story, the variation in phase with time is captured by the angular frequency (omega) and is given as the number of radians per second.
Othe respondants above have supplied the appropriate equations.
Another good source to consult is "Microwave Engineering" by David M Pozar.
Hope this helps.
Clive
A bout TEM yes it was impossible in waveguide as it was reading before, it neads for electromagnetic field surrounding a line, thus the mode of coaxial cable is TEM.
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