Q factor for cilindrical resonating cavities

Resonant cavities are usually constructed from copper or copper-plated steel for the highest conductivity. Nonetheless, effects of resistivity are significant because of the large reactive current. Resistive energy loss from the flow of real current in the walls is concentrated in the inductive regions of the cavity; current penetrates into the wall a distance equal to the skin depth $\delta$ . Power loss clearly depends on mode structure through the distribution of magnetic fields. The $Q$ value for the TM$_010$ mode of a cylindrical resonant cavity is

$$Q=\frac{d/\delta}{1+d/R_0}$$

where the skin depth $\delta$ is a function of the frequency and wall material. In a copper cavity oscillating at $f = 1$ GHz, the skin depth is only $2 \mu m$ . This means that the inner wall of the cavity must be carefully plated or polished; otherwise, current flow will be severely perturbed by surface irregularities lowering the cavity $Q$ . With a skin depth of $2 \mu m$ Q value is $3 \times 10^4$ in a cylindrical resonant cavity of radius $R_0=12$ cm and length $4 cm$ . This is a very high value compared to resonant circuits composed of lumped elements. The bandwidth for exciting a resonance

$$\frac{\Delta f}{f_0}\approx \frac{1}{Q}=3 \times 10^{-5}$$

Figura: Meaning of the $Q$ factor in terms of resonance frequency.

This last result means that the rf generator must drive the resonant cavity in a very stable range, i.e. for $f=1 GHz$ the maximum frequency range to keep TM$_{010}$ excited is less than $33 KHz$ . If that frequency is varying in time, the cavity can support higher but undesiderable modes. These higher modes could produce beam deflection, particle loss or great energy waste. In general is possible to insert metallic pieces in order to prevent mode degeneracy and power coupling between modes.