Robinson instability and beam loading

The Robinson instability problem is developed in three stages. The first step is to derive the synchrotron oscillation equations in the absence of beam loading (unloaded case). Next, the equations are evaluated in the presence of beam loading at the fundamental RF frequency (statically loaded case). Finally, the system is redeveloped taking into account beam loading at the synchrotron sidebands (dynamically loaded case). Following the theoretical development, the results are applied to calculate the synchrotron frequency in the presence of beam loading, automatic gain control, and automatic tune control. The results of this calculation are compared with data from the NSLS vuv-ring, a 750 MeV electron storage ring.