# Waveguide Parameters

Updated on 2017/08/09 17:28

## Syllabus

• Waveguide Parameters

## Phase constant of TE and TM mode

For both TEmn and TMmn modes the modal phase constant β is given by:

$\beta =\sqrt{\omega ^{2}\mu \epsilon -(\frac{m\pi }{a})^{2}-(\frac{\eta \pi }{b})^{2}}$

• For the mode to be travelling β has to be a real quantity.
• If β becomes imaginary then the fields no more remain travelling but become exponentially decaying.

## Cut-off Frequency of TE and TM mode

The frequency at which β changes from real to imaginary is called the cut-off frequency of the mode.
At cut-off frequency, therefore β = 0 gives,

$\omega _{c} =\frac{1}{\sqrt{\mu \epsilon }}\sqrt{(\frac{m\pi }{a})^{2}+(\frac{n\pi }{b})^{2}}$

which further gives, $f _{c} =\frac{1}{2\pi \sqrt{\mu \epsilon }}\sqrt{(\frac{m\pi }{a})^{2}+(\frac{n\pi }{b})^{2}}$

The cut-off frequencies for lowest TM and TE modes i.e. TM11, TE10 and TE01 can be obtained as

$f _{cTM_{11}} =\frac{1}{2\pi \sqrt{\mu \epsilon }}\sqrt{(\frac{\pi }{a})^{2}+(\frac{\pi }{b})^{2}}$

$f _{cTE_{10}} =\frac{1}{2\pi \sqrt{\mu \epsilon }}\frac{\pi }{a}=\frac{1}{2a\sqrt{\mu \epsilon }}$

$f _{cTE_{01}} =\frac{1}{2\pi \sqrt{\mu \epsilon }}\frac{\pi }{b}=\frac{1}{2b\sqrt{\mu \epsilon }}$

Since by definition we have  a > b we get the frequencies as

$f _{cTE_{10}} < f _{cTE_{01}}

• We can make an important observation that, if at all the electromagnetic energy travels on a rectangular waveguide its frequency has to be more than the lowest cut-off frequency i.e. fc of TE10 mode.
• As the order of the mode increases the cut-off frequency also increases.

### Cut-off  wavelength of TE and TM mode

The very first mode that propagates on the rectangular waveguide is TE10 mode and therefore this mode is called the dominant mode of the rectangular waveguide.
The cut-off wavelength  is given by
$\lambda _{C}=\frac{2}{\sqrt{(\frac{m}{a})^{2}+(\frac{n}{b})^{2}}}$

• For dominant mode , λc=2a.
• For propagation of wave  in the waveguide

λ < λc    or    f > fc

## Waveguide Parameters

### Guide Wavelength

Definition:-It is defined as the distance travelled by the wave in order to undergo a phase shift of 2π radians.

• It is related to phase constant by the relation
•  λg  = 2π / β
• $\lambda _{g}=\frac{\lambda}{\sqrt{1-(\frac{f_{c}}{f})^{2}}}$

### Wave Impedance

Definition:-It is defined as ratio of strength of electric field in one transverse direction to the strength of magnetic field along other transverse direction.

• $Z_{TE}=\frac{\eta }{\sqrt{1-(\frac{f_{c}}{f})^{2}}}$
• $Z_{TM}={\eta }{\sqrt{1-(\frac{f_{c}}{f})^{2}}}$

### Phase Velocity

Definition:-The phase velocity is defined as the velocity with which the wave changes phase in terms of the guide wavelength

• Vp = λg * f
• $V _{P}=\frac{C}{\sqrt{1-(\frac{f_{c}}{f})^{2}}}$

### Group Velocity

Definition: The group velocity of a wave is defined as the rate at which the wave propagates through the waveguide.

• $V _{g}=\frac{d\omega }{d\beta }$
• $V _{P}={C}{\sqrt{1-(\frac{f_{c}}{f})^{2}}}$

The product of phase and group velocities is equal to square of the velocity of light. i.e.  $V _{p}.V _{g}=C^{2}$

## References

• WikiNote Foundation
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Created by Vishal E on 2017/07/22 18:26

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