Tuning of 3-tap Bandpass Filter During Acceleration for Longitudinal Beam Stabilization at FAIR

During acceleration in the heavy-ion synchrotrons SIS18/SIS100 at GSI/FAIR longitudinal beam oscillations are expected to occur. To reduce longitudinal emittance blowup, dedicated LLRF beam feedback systems are planned. To date, damping of longitudinal beam oscillations has been demonstrated in SIS18 machine experiments with a 3-tap filter controller (e.g. [1]), which is robust in regard to control parameters and also to noise. On acceleration ramps the control parameters have to be adjusted to the varying synchrotron frequency. Previous results from beam experiments at GSI indicate that a proportional tuning rule for one parameter and an inversely proportional tuning rule for a second parameter is feasible, but the obtained damping rate may not be optimal for all synchrotron frequencies during the ramp. In this work, macro-particle simulations are performed to evaluate, whether it is sufficient to adjust the control parameters proportionally (inversely proportionally) to the change in the linear synchrotron frequency, or if it is necessary to take more parameters, such as bunch-length and synchronous phase, into account to achieve stability and a considerable high damping rate for excited longitudinal dipole beam oscillations. This is done for single- and dual-harmonic acceleration ramps.