Stability analysis of rf accelerating mode with feedback loops under heavy beam loading in SuperKEKB

In SuperKEKB high-current beam operation, achieving the stability of the zero mode associated with the accelerating mode of the rf system is an important concern because of heavy beam loading. The issue is a more serious problem in the high-energy ring (HER) compared to the low-energy ring (LER) because of the high impedance of superconducting cavities operating in the HER. As a countermeasure, the rf system is equipped with a direct rf feedback system (DRFB) to effectively reduce the impedance of the cavities and a zero mode oscillation damper. The effect of these mitigations on improving the stability while coexisting with other control loops was quantitatively studied using two independent methods, namely, analysis with a characteristic equation and simulation in the time domain. For the analysis, we extended the transfer functions in the Pedersen model to include the DRFB to analyze the rf system with beam loading, while coexisting with the other control loops such as amplitude and phase controls. The consistency between both methods was confirmed by comparing the results obtained from them for different cases. Furthermore, simulation results were compared with measurements conducted during beam operation; they were consistent with each other. Having confirmed the validity of these methods, they were applied to extensively study the stability for future higher beam current operations in SuperKEKB. Particularly, several operation parameter sets, including possible system modifications, were identified for the HER and LER with considerably good margins to compensate for possible deficiencies in the performance of cavities or control functions at higher beam currents. The results can be used as guidelines for future beam operations with beam current increasing in a stepwise manner.