**Subject :**Microwave Engineering

## Q Factor of a Cavity Resonator

The quality factor Q is a measure of the frequency selectivity of a resonant or antiresonant circuit, and it is defined as

..(1)

where W is the maximum stored energy and P is the average power loss.

At resonant frequency, the electric and magnetic energies are equal and in time quadrature. When the electric energy is maximum, the magnetic energy is zero and vice versa. The total energy stored in the resonator is obtained by integrating the energy density over the volume of the resonator.

..(2)

where IEI and IHI are the peak values of the field intensities.

The average power loss in the resonator can be evaluated by :

...(3)

where H_{t} is the peak value of the tangential magnetic intensity and R_{s} is the surface resistance of the resonator. Fromequations (1) ,(2) and (3), we can write :

...(4)

Since the peak value of the magnetic intensity is related to its tangential and normal components by :

...(5)

where H; is the peak value of the normal magnetic intensity, the value of I HI12 at the resonator walls is approximately twice the value of IH 12averaged over the volume. So the Q of a cavity resonator as shown in Eq. (4) can be expressed approximately by

....(6)

An unloaded resonator can be represented by either a series or a parallel resonant circuit. The resonant frequency and the unloaded Qo of a cavity resonator are

.......(7)

If the cavity is coupled by means of an ideal N: I transformer and a series inductance L, to a generator having internal impedance Zg then the coupling circuit and its equivalent are as shown in figure :

**Figure :Cavity coupled to a generator. (a) Coupling circuit. (b) Equivalent circuit**

The loaded Q_{l}of the system is given by :

....(8)