Beam Stability in Synchrotrons with Notch and All-Pass Filters in the Feedback Loop of a Transverse Damper

The stability of a beam in synchrotrons with digital filters in the feedback loop of a transverse damper is treated. A transverse feedback system (TFS) is required in synchrotrons to stabilize the high intensity 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 processor in the feedback loop between BPM and DK ensures the adjustment of the phase advance and the correction of the time of flight for optimum damping. Digital FIR (finite impulse response) and IIR (infinite impulse response) filters are used commonly for the signal processing. A notch filter with zeros at the revolution frequency 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 BPM and DK. The phase shift introduced by the notch filter can be compensated by an all-pass filter with constant frequency-response magnitude but variable, adjustable, phase advance. The presented analysis of the feedback system with digital notch and all-pass filters shows that in the low gain regime the damping characteristics coincide with those in an ideal feedback system.