A Realistic Proportional-Integral RF Feedback Model for Longitudinal Beam Dynamics Simulation

Modern fourth-generation storage ring light sources predominantly utilize digital I/Q-based proportional-integral (PI) feedback for their radio-frequency (RF) systems. This paper introduces a dedicated PI feedback model implemented in the STABLE tracking code to enable accurate and fast longitudinal beam dynamics simulations. The model's key innovation lies in its treatment of the continuous generator current, which is discretized into electron-bunch-like charge pulses, while the cavity voltage is refreshed on an RF-cycle basis. This methodology offers a more physically accurate model of the beam-cavity-feedback coupling, providing a versatile tool for precise longitudinal beam dynamics studies in single- and multi-rf configurations.