Control Instabilities in a Pulsed Multi-Cavity RF System with Vector Sum Feedback (A Mathematical Analysis)

Upcoming projects relying on pulsed linear accelerators intend to use superconducting RF systems. Cost reasons suggest driving several cavities by a common transmitter, controlled over a vector sum feedback system, possibly supported by a feed forward system. Numerical simulations hint that such a system may become uncontrollable under certain conditions. In the present paper, for a model very close to reality, we will present a mathematical proof that in fact spontaneous symmetry braking is possible for these configurations, defining also the precise conditions under which it will take place. These can be used as an estimate for the real RF system stability limits. The listing of a small program demonstrating the mechanism numerically for two cavities is attached.