**Subject :**Microwave Engineering

## Rectangular Cavity Resonator

The electromagnetic field inside the cavity should satisfy Maxwell's equations, subject to the boundary conditions that the electric field tangential to and the magnetic field normal to the metal walls must vanish. The geometry of a rectangular cavity is illustrated in Figure :

**Figure : Coordinates of a rectangular cavity.
**

The wave equations in the rectangular resonator should satisfy the boundary condition of the zero tangential E at four of the walls. It is merely necessary to choose the harmonic functions in z to satisfy this condition at the remaining two end walls. These functions can be found if

...(1)

where m = 0, I, 2, 3, ... represents the number of the half-wave periodicity in the x direction

n = 0, I, 2, 3, ... represents the number of the half-wave periodicity in the y direction

p = I, 2, 3, 4, ... represents the number of the half-wave periodicity in the Z direction

and

....(2)

where m = I, 2, 3, 4, ..

n = 1, 2, 3, 4, ..

P = 0, 1, 2, 3, ..

The separation equation for both TE and TM modes is given by :

...(3)

For a lossless dielectric, k^{2} = ω^{2}με; therefore, the resonant frequency is expressed by :

...(4)

For a > b < d, the dominant mode is the TE_{101 }mode.

In general, a straight-wire probe inserted at the position of maximum electric intensity is used to excite a desired mode, and the loop coupling placed at the position of maximum magnetic intensity is utilized to launch a specific mode. Figure shows the methods of excitation for the rectangular resonator. The maximum amplitude of the standing wave occurs when the frequency of the impressed signal is equal to the resonant frequency :

**Figure : Methodsof exciting wave modes in a resonator**