Theory and Performance of the Longitudinal Active Damping System for the CERN PS Booster

Longitudinal instabilities have long been trouble-some in the Booster1). They are described by the coupledbunch mode number n = 0 to 4 for five bunches, and by the within-bunch mode number m = 1 for dipole, m = 2 for quadrupole, m = 3 for sextupole, and so on (Fig. 1). The normal beam control system damps the rigid-bunch oscillation (m = 1) when all five bunches move togethier (n = 0)2). The new feedback system3) damps the other coupled-bunch modes n = 1 to 4 for the three lowest orders, m = l to 3. With the damping system off, one can display the evolution of any mode along the cycle, which helps in locating coupling impedances that cause instability. One can also excite the various modes and measure the amplitude-phase response (RF knockout applied to a bunched beam). This gives the frequency spread within the bunch, the coherent frequent shifts, plus the usual stability diagram in the U-V plane.