Mathematical Models of Feedback Systems for Control of Intra-Bunch Instabilities Driven by E-Clouds and TMCI

The feedback control of intrabunch instabilities driven by E-Clouds or strong head-tail coupling (TMCI) requires sufficient bandwidth to sense the vertical position and drive multiple sections of a nanosecond scale bunch. These requirements impose challenges and limits in the design and implementation of the feedback system. This paper presents models for the feedback subsystems: receiver, processing channel, amplifier and kicker, that take into account their frequency response and limits. These models are included in multiparticle simulation codes (WARP/CMAD/Head-Tail) and reduced mathematical models of the bunch dynamics to evaluate the impact of subsystem limitations in the bunch stabilization and emittance improvement. With this realistic model of the hardware, it is possible to analyze and design the feedback system. This research is crucial to evaluate the performance boundary of the feedback control system due to cost and technological limitations. These models define the impact of spurious perturbations, noise and parameter variations or mismatching in the performance of the feedback system. The models are validated with simulation codes and measurements of lab prototypes.