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Monday, 15 April 2019

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 We or 2 Wm.
where We = energy stored in electric field
Wm = 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.

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, Ez = 0 and solution may be derived from Hz component.
For TM mode, Hz = 0 and solution is derived from Ez Component.

TE Mode:

The Hz Component is defined by the equation.
Hz = Ho 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 Hy is defined as

Hy = 1/Kc22Hz/ꝺyꝺz
Kc – cut off value
Hy = 1/Kc2 /y(Hz/z)
      = 1/Kc2 /y[Ho cos (mπx/a) sin (nπy/b) (nπ/b) cos (pπz/d) (pπ/d)]

The component Hx is defined as

Hx = 1/Kc2 2Hz/xz)
Hx = -Ho/Kc2 [sin (mπx/a) cos (nπy/b) cos (pπz/d) (mπ/a) (pπ/d)

For TE Mode, Ez = 0, the component Ey is defined by,
Ey = jωμHo/Kc2 Hz/x
Ey = jωμHo/Kc2 (-mπ/a) Ho [sin (mπx/a) cos (nπy/b) sin (pπz/d)]

The component Ex is defined by,
Ex = -jωμHo/Kc2 Hz/y
Ex = jωμHo/Kc2 nπ/b Ho [cos (mπx/a) sin (nπy/b) sin (pπz/d)]

TM Mode:

The Ez component is defined by the equation,
Ez = Eo sin (mπx/a) sin (nπy/b) cos (pπz/d)]
Ey = Eo/Kc2 (2Ez/ꝺyꝺz)
Ex = Eo/Kc2 (2Ez/ꝺxꝺz)
Hz = 0
Hy = -jωƐEo/Kc2 (Ez/x)
Hx = jωƐEo/Kc2 (Ez/y)

We have,
Ey = Eo/Kc2 (2Ez/ꝺyꝺz)
Ey = -Eo/Kc2 (nπ/b) Eo sin (mπx/a) cos (nπy/b) sin (pπz/d) (pπ/d)
Ex = Eo/Kc2 (2Ez/ꝺxꝺz)
Ex = -Eo2/Kc2 (cos (mπx/a) sin (nπy/b) sin (pπz/d) (mπ/a) (pπ/d)
Hz = 0

The component Hy is defined as
Hy = -jωƐEo/Kc2 (Ez/x)
Hy = -jωƐ/Kc2 Eo2 (mπ/a) cos (mπx/a) sin (nπy/b) cos (pπz/d)
Hx = jωƐ/Kc2 Eo2 (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 TEnmp and TMnmp

TE Mode:

It is described by the equation,
Hz = Ho 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,
Ez = Eo Jn (x’nmp ρ/a) cos nΦ sin(pπz/d)
For the rectangular cavity resonator, the resonant frequency is given by
fr = 1/2(μƐ) √{(m/a)2 + (n/b)2 + (p/d)2}
for circular cavity resonator, the resonant frequency is given by
fr = 1/2π(μƐ) √{( xnmp /a)2 + (pπ/d)2}

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