• V. Zhabitsky
  JINR, Dubna

The stability of an ion beam in synchrotrons with digital filters in the feedback loop of a transverse damper is treated. Solving the characteristic equation allows to calculate the achievable damping rates as a function of instability growth rate, feedback gain and parameters of the signal processing. A transverse feedback system (TFS) is required in synchrotrons to stabilize the high intensity ion beams against transverse instabilities and to damp the beam injection errors. The TFS damper kicker (DK) corrects the transverse momentum of a bunch in proportion to its displacement from the closed orbit at the location of the beam position monitor (BPM). The digital signal processing unit in the feedback loop between BPM and DK ensures a condition to achieve optimal damping. Transverse Feedback Systems commonly use digital FIR (finite impulse response) and IIR (infinite impulse response) filters for the signal processing. A notch filter is required to remove the closed orbit content of the signal and correct for the imperfect electric centre of the BPM. Further processing is required to adjust for the betatron phase advance between the beam pick-up (BPM) and the damper kicker (DK). Damping rates of the feedback systems with digital notch, Hilbert and all-pass filters are analysed in comparison with those in an ideal feedback system.