Power exchange with resonant cavities

Power must be coupled into resonant cavities to maintain electromagnetic oscillations when there is resistive damping or a beam load. The feed drives a beam and supplies energy lost to the cavity walls. In this case, power is electrically coupled to the cavity because the current in the power feeds interacts predominantly with electric fields. Although this geometry is never used for driving accelerator cavities, there is a practical application of the inverse process of driving cavity oscillations by a beam. Picture shows a klystron, a microwave generator. An on-axis electron beam is injected across the cavity. The electron beam has time-varying current with a strong Fourier component at the resonant frequency of the cavity, $\omega_0$ . We will consider only this component of the current and represent it as a harmonic current source. The cavity has a finite $Q$ , resulting from wall resistance and extraction of microwave energy. The complete circuit model for the TM$_{010}$ mode is shown in figure . The impedance presented to the component of the driving beam current with frequency $\omega$ is

$$Z=\frac{(j\omega L+R)}{(1-\omega^2 LC)+j\omega RC}$$

Assuming that $\omega=\omega_0 - 1/\sqrt{LC}$ and that the cavity has high $Q$ , it reduces to:

$$Z\approx \frac{L}{RC}=QZ_0$$

The impedance is resistive, the voltage oscillation induced is in phase with the driving current so that energy extraction is maximized. In applications to high-energy accelerators, the aim is to use resonant cavities as step-up transformers. Ideally, power should be inserted at low impedance and coupled to a low-current beam at high voltage. This process is accomplished when energy is magnetically coupled into a cavity. With magnetic coupling, the power input is close to the outer radius of the cavity. therefore, interaction is predominantly through magnetic fields.

Figura: Equivalent circuit for TM$_{010}$ mode