March 05, 2014
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Lets explain cutoff frequency , guide wavelength , group velocity , phase velocity and propagation constant of a waveguide.
Cutoff frequency :
Cutoff frequency is the frequency below which attenuation occurs and above which propagation takes place.Each mode have a specific cutoff frequency.
For TEmn modes the cutoff frequency is given by
\[f_{c}=\frac{1}{2\sqrt{\mu \varepsilon}}\sqrt{{(\frac{m}{a})}^{2}+{(\frac{n}{b})}^{2}}\]
Guide Wavelength :
It is the distance traveled by the wave in order to undergo a phase shift of 2π radians.
It is related to propagation constant β as
\[\lambda_{b}=\frac{2\pi}{\beta}\]
Wavelength in waveguide is different from wavelength in free space.
Relation between cutoff frequency and guide wavelength :
The relationship between two is as follows
\[\frac{1}{\lambda_{0}^{2}}=\frac{1}{\lambda_{g}^{2}}+\frac{1}{\lambda_{c}^{2}}\]
Also it can be written as
\[\lambda_{g}=\frac{\lambda_{0}}{\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}}\Big)^{2}}\]
☞ When ⋋0 ≪ ⋋c , then ⋋g= ⋋0
☞ When ⋋0 = ⋋c , then ⋋g becomes ∞
☞ When ⋋0 > ⋋c , then ⋋g becomes imaginary ,that means no propagation in the waveguide
Where ⋋0 is the free space wavelength.
Phase velocity :
The phase velocity of a wave is the rate at which the phase of the wave propagates in space.
The phase velocity is given by
\[v_{p}=\frac{\omega}{k}\]
Where k = wave number
Also it is given as
\[v_{p}=\frac{c}{\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}\Big)^{2}}}\]
Group velocity :
If there is modulation in carrier, the modulation envelope travels at a velocity slower than the carrier. This velocity of the modulation envelope is called as group velocity.
Or in other words
The group velocity of a wave is the velocity with which the overall shape of the waves' amplitudes — known as the modulation or envelope of the wave — propagates through space.
It is given as
\[v_{g}=\frac{\partial \omega}{\partial k}\]
Also it is given as
\[v_{g}=c\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}\Big)^{2}}\]
Relationship between group velocity and phase velocity :
\[v_{p}v_{g}=c^{2}\]
Propagation constant :
The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction
For a TEmn mode it is given by
\[\beta=\sqrt{\mu\epsilon}\sqrt{\omega^{2}-\omega_c^2}\]
