Exact determination of HOM frequency in storage ring RF cavity

A formula for the determination of higher-order mode (HOM) frequency in an active storage ring RF cavity is derived, which utilizes the frequency tuning mechanism of the low-level RF feedback system. It helps us to understand the behavior of coupled bunch instabilities in a storage ring whenever we change the operating beam current, the cavity gap voltage, and the ring RF frequency as well. This analysis includes the fundamental mode frequency detuning for beam loading and Robinson damping. At any operating conditions of the RF cavity, the HOM frequencies can be precisely calculated with the low-power measurement data of the frequency shift as a function of the tuner position. The formula shows a good agreement with the measurement results.