Effects of an rf phase feedback system on the coherent tunes and damping rates

One possible feedback system, designed mainly for damping the longitudinal dipole oscillations, utilizes a beam position monitor and an rf cavity. The horizontal displacement of the beam is measured at the monitor and the measurement is sent to the rf cavity. The phase of the cavity voltage is then adjusted so that an electron changes its energy by the additional amount of Δ/var epsilon/ = /zeta/E/sub o/x/sub monitor/. This FB system introduces damping or anti-damping to the horizontal betatron oscillation and the longitudinal synchrotron oscillation. Although approximate expressions for the associated damping constants α/sub x,s/ can be obtained by elementary considerations, it is perhaps constructive to have an exact calculation available as well. In the following, we will describe the exact calculation; obtain approximate expressions of α/sub x,s/ from the exact calculation; obtain approximate expressions of Δν/sub x,s/, the coherent tune shifts caused by the FB systems; and numerically compare the exact and approximate results under various conditions. We assume that there is only one active rf cavity in the storage ring and that the monitor signal reaches the rf cavity before the beam completes one turn. 5 refs., 6 figs