The next generation of synchrotron light murces and particle accelerators will require active feedback systems to control multi-bunch instabilities [1,2,3]. Stabilizing hundreds or thousands of potentially unstable modes in these accelerator designs presents many technical challenges.Feedback system to stabiliee' coupled-bunch instabilities may be understood in the frequency domain (modeb a d f d b a c k ) or in the time domain (bunch-by-bunch feedback). In both approaches an external amplifier system is wed to create damping fields that prevent coupledbunch d a t i o n s from growing without bound. The system requirements for transverse (betatron) and longitudinal (synchrotron) feedback are presented, and possible implementation options developed. Feedback system designs based on digital signal-processing techniques are described. Experimental results are shown from a synchrotron oecillation damper in the SSRL/SLAC storage ring SPEAR that uses digital signal-processing techniques. I. A CLASSICAL ANALOGYThe dynamics of coupled-bunch motion can be illustrated by the-mechanical analog of coupled pendulums. In Figure 1 this aniogy is applied to the charged particle bunches in a storage Sng, with each pendulum representing the oscillatory motion (synchrotron or betatron) of a bunch. The coupling springs represent the impedances of the accelerating cavities and vacuum structures. Bunchj+i and subsequent bunches are driven from the excitations of bunchi, much as pendulumj drives pendulumsj+k through the coupling springs [4]. xi' hl 1+2 k3 695 -' 7U?& Figure 1. Coupled pendulum analogy. * Work supported by Department of Energy contract DE-AC03-76s F005 1 5.In a storage ring with many bunches and many external higher-order mode resonators, the resulting motion can be found by coherently summing the driving terms and considering the periodic excitation due to the orbit of the particles [5,6]. Unstable, growing Oscillatory motion can result, in which the motion of a few bunches can excite an unstable normal mode. Thee instabilities can be controlled by reducing the magnitude and number of external, parasitic higher-order modes, carefully controlling the resonant frequenciea of the parssitic resonators to avoid coupling to the beam, and by adding damping to the motion of each bunch.External beam-feedback systems do the latter. In the analogy of Figure 1, they act to add dashpots to each pendulum. Each bunch can be thought of aa a harmonic cecillator obeying the equation of motion where wo is the bunch synchrotron (longitudinal) or betatron (transverse) frequency, f(t) is an external driving term and y is a damping term. An external feedback system acts on the beam, contributing to this damping term, and allowing control of external disturbances ,f(t) driving the beam. 11. TIME DOMAIN VS. FREQUENCY DOMAIN PROCESSINGThe action of the feedback system can be understood in either the time or frequency domains [7]. If each unstable normal-mode frequency is identified, a single narrowband feedback channel for each mode can be impl...

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The proposed next generation accelerator and synchrotron light facilities will require active feedback systems to control multi-bunch instabilities. These feedback systems must operate in machines with thousands of circulating bunches and with short (2-4 ns) interbunch intervals. The functional requirements for transverse (betatron) and longitudinal (synchrotron) feedback systems are presented. Several possible implementation options are discussed and system requirements developed. Results are presented from a digital signal processing based synchrotron oscillation damper operating at the SSRL/SLAC SPEAR storage ring.