Longitudinal Microwave Instability in a Multi-RF System

The longitudinal microwave instability is observed as a fast increase of the bunch length above some threshold intensity. Recently, this type of instability was seen for a single proton bunch at high energies in the CERN SPS and is proven to be one of the limitations for an intensity increase required by the HL-LHC project. In this paper a theoretical approach to the analysis of the microwave instability is verified by particle simulations. The study is applied to the SPS and is based on the current SPS impedance model. Finally, the effect of the 4th harmonic RF system on the microwave instability threshold is investigated as well.