Low-level RF - Part I: Longitudinal dynamics and beam-based loops in synchrotrons

The low-level RF system (LLRF) generates the drive sent to the high-power equipment. In synchrotrons, it uses signals from beam pick-ups (radial and longitudinal) to minimize the beam losses and provide a beam with reproducible parameters (intensity, bunch length, average momentum and momentum spread) for either the next accelerator or the physicists. This presentation is the first of three: it considers synchrotrons in the lowintensity regime where the voltage in the RF cavity is not influenced by the beam. As the author is in charge of the LHC LLRF and currently commissioning it, much material is particularly relevant to hadron machines. A section is concerned with radiation damping in lepton machines.