Highek. Harmonic Rf System For. Aps

The influence of a higher harmonic rf system was considered previously(l) but was treated as a non-accelerating system. This note treats the higher harmonic system more generally and treats the non-linear synchrotron oscillations without the use of a power series expansion. The rf voltage seen by the beam is assumed to be the sum of one or two rf systems with a harmonic relationship to each other, given by the integer n) 1. The frequencies are then f2 = nf 1 • Although this need not be the case (i.e., f1 and f2 need only be harmonically related to f o)' operationally this harmonic relationship makes the rf systems easier to phase lock and assures that all buckets will have similar properties. The voltage provided by this rf system is v (4)) (1) where 4>1 the synchronous phase of the main rf system 4>2 = the synchronous phase of the n-th harmonic rf system V1 = the peak voltage of the main rf system kV 1 the peak voltage of the n-th harmonic rf system.