Coupling interaction between the power coupler and the third harmonic superconducting cavity

Fermilab has developed a third harmonic superconducting cavity operating at the frequency of 3.9 GHz to improve the beam performance for the FLASH user facility at DESY. It is interesting to investigate the coupling interaction between the SRF cavity and the power coupler with or without beam loading. The coupling of the power coupler to the cavity needs to be determined to minimize the power consumption and guarantee the best performance for a given beam current. In this paper, we build and analyze an equivalent circuit model containing a series of lumped elements to represent the resonant system. An analytic solution of the required power from the generator as a function of the system parameters has also been given based on a vector diagram.