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## Rectangular and Circular Cavity Resonators

### Microwave Cavity Resonators:

A cavity resonator is a metallic enclosure that confines Electromagnetic energy. The stored electric and magnetic energy inside the cavity determine inductance and capacitance. The energy inside the cavity determine inductance and capacitance. The energy dissipated by the conductivity of the walls of the cavity determines the resistance. The parameter which describe the performance of a resonator are,

1. Resonant frequency: It is the frequency at which the energy in the cavity attain a maximum value of 2 W

_{e}or 2 W_{m}.
where W

_{e}= energy stored in electric field
W

_{m}= Energy stored in the Magnetic field.
2. Quality Factor, Q : It is the measure of frequency selectivity of the cavity.

Q = 2 π x max energy stored/ Energy dissipated per cycle

3. Modes of the Cavity: A given resonator has infinite number of modes and each mode correspondes to a definite frequency. When the frequency of the signal is equal to resonant frequency a maximum amplitude of wave occurs and the energies stored in electric and magnetic fields are equal. The mode having lowest resonant frequency is known as dominant mode.

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**Rectangular Cavity Resonator:**

Rectangular Cavity Resonators |

There are 2 modes of propagation possible inside a rectangular cavity resonator. They are TE mode in which electric mode is transverse and TM mode in which magnetic mode is transverse.

For TE mode, E

_{z}= 0 and solution may be derived from H_{z}component.
For TM mode, H

_{z}= 0 and solution is derived from E_{z}Component.**TE Mode:**

The Hz Component is defined by the equation.

H

_{z}= H_{o}cos (mπx/a) cos (nπy/b) sin (pπz/d)
Where, Ho = amp of magnetic field

m = number of waves in x direction

n = number of waves in y direction

p = number of waves in z direction

The Component H

_{y}is defined as
H

_{y}= 1/K_{c}^{2}ꝺ^{2}Hz/ꝺyꝺz
K

_{c}– cut off value
H

_{y}= 1/K_{c}^{2}ꝺ/ꝺy(ꝺH_{z}/ꝺz)
= 1/K

_{c}^{2}ꝺ/ꝺy[H_{o}cos (mπx/a) sin (nπy/b) (nπ/b) cos (pπz/d) (pπ/d)]
The component H

_{x}is defined as
H

_{x }= 1/K_{c}^{2}ꝺ^{2}H_{z}/ꝺxꝺz)
H

_{x}= -H_{o}/K_{c}^{2}[sin (mπx/a) cos (nπy/b) cos (pπz/d) (mπ/a) (pπ/d)
For TE Mode, E

_{z}= 0, the component E_{y}is defined by,
E

_{y}= jωμH_{o}/K_{c}^{2}ꝺH_{z}/ꝺx
E

_{y}= jωμH_{o}/K_{c}^{2 }(-mπ/a) H_{o}[sin (mπx/a) cos (nπy/b) sin (pπz/d)]
The component Ex is defined by,

Ex = -jωμH

_{o}/K_{c}^{2}ꝺH_{z}/ꝺy
Ex = jωμH

_{o}/K_{c}^{2}nπ/b H_{o}[cos (mπx/a) sin (nπy/b) sin (pπz/d)]**TM Mode:**

The E

_{z}component is defined by the equation,
E

_{z}= E_{o}sin (mπx/a) sin (nπy/b) cos (pπz/d)]
E

_{y}= E_{o}/K_{c}^{2}(ꝺ^{2}E_{z}/ꝺyꝺz)
E

_{x}= E_{o}/K_{c}^{2}(ꝺ^{2}E_{z}/ꝺxꝺz)
H

_{z}= 0
H

_{y}= -jωƐE_{o}/K_{c}^{2}(ꝺE_{z}/ꝺx)
H

_{x}= jωƐE_{o}/K_{c}^{2}(ꝺE_{z}/ꝺy)
We have,

E

_{y}= E_{o}/K_{c}^{2}(ꝺ^{2}E_{z}/ꝺyꝺz)
E

_{y}= -E_{o}/K_{c}^{2}(nπ/b) E_{o}sin (mπx/a) cos (nπy/b) sin (pπz/d) (pπ/d)
E

_{x}= E_{o}/K_{c}^{2}(ꝺ^{2}E_{z}/ꝺxꝺz)
E

_{x}= -E_{o}^{2}/K_{c}^{2}(cos (mπx/a) sin (nπy/b) sin (pπz/d) (mπ/a) (pπ/d)
H

_{z}= 0
The component H

_{y }is defined as
H

_{y}= -jωƐE_{o}/K_{c}^{2}(ꝺE_{z}/ꝺx)
H

_{y}= -jωƐ/K_{c}^{2}E_{o}^{2}(mπ/a) cos (mπx/a) sin (nπy/b) cos (pπz/d)
H

_{x}= jωƐ/K_{c}^{2}E_{o}^{2}(nπ/b) sin (mπx/a) cos (nπy/b) cos (pπz/d)###
**Circular Cavity Resonators**

Circular Cavity Resonator |

A circular cavity resonator is a circular waveguide with two ends closed by a metallic wall. The field components inside the cavity are described as TE

_{nmp }and TM_{nmp}**TE Mode:**

It is described by the equation,

H

_{z}= H_{o}Jn (x’_{nmp}ρ/a) cos nΦ sin(pπz/d)
Where a, ρ and Φ are the cylindrical coordinates

**TM Mode:**

It is defined by the Equation,

E

_{z}= E_{o}Jn (x’_{nmp}ρ/a) cos nΦ sin(pπz/d)
For the rectangular cavity resonator, the resonant frequency is given by

f

_{r}= 1/2√(μƐ) √{(m/a)^{2}+ (n/b)^{2}+ (p/d)^{2}}
for circular cavity resonator, the resonant frequency is given by

f

_{r}= 1/2π√(μƐ) √{( x_{nmp}/a)^{2}+ (pπ/d)^{2}}### This post was written by: Sreejith Hrishikeshan

Sreejith Hrishikesan Nair is a M-Tech graduate in Communication Systems. He completed B-tech Degree in Electronics and Communication.He is a person who wants to implement new ideas in the field of Technology.

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