## 11. what is waveguide cutoff frequency , guide wavelength , phase velocity , group velocity , propagation constant

Lets explain cutoff frequency , guide wavelength , group velocity , phase velocity and propagation constant of a waveguide.

Cutoff frequency is the frequency below which attenuation occurs and above which propagation takes place.Each mode have a specific cutoff frequency.

For TE

\[f_{c}=\frac{1}{2\sqrt{\mu \varepsilon}}\sqrt{{(\frac{m}{a})}^{2}+{(\frac{n}{b})}^{2}}\]

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.

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 ⋋

☞ When ⋋

☞ When ⋋

Where ⋋

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

\[v_{p}=\frac{c}{\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}\Big)^{2}}}\]

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

\[v_{g}=c\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}\Big)^{2}}\]

\[v_{p}v_{g}=c^{2}\]

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

\[\beta=\sqrt{\mu\epsilon}\sqrt{\omega^{2}-\omega_c^2}\]

**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 TE

_{mn}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 waveguideWhere ⋋

_{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 TE**

_{mn}mode it is given by

\[\beta=\sqrt{\mu\epsilon}\sqrt{\omega^{2}-\omega_c^2}\]

Amazing!

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