I/Q Model of the SPS 200 MHz Travelling Wave Cavity and Feedforward Design

As part of the LHC injector upgrade project (LIU), that calls for the doubling of the bunch intensity injected from the SPS into the LHC, the SPS Low Level RF (LLRF) is fully upgraded during the LHC Long-Shutdown 2 (LS2). A major issue is the compensation of the beam induced voltage (beam loading) in the 200 MHz Travelling Wave Cavities (TWCs). Uncompensated beam loading limits the intensity threshold above which longitudinal coupled bunch instabilities develop. The SPS beam consists of batches separated by gaps. Given the short cavity filling time, this results in a large transient of beam induced voltage along the batch, with resulting bunch-to-bunch variation of voltage and phase. The phase variation along the batch is a major concern for the alignment of the harmonic cavities used for Landau damping (800 MHz). The LLRF uses two systems for beam loading compensation: The One-Turn delay Feedback (OTFB) and the Feedforward. Both are implemented in I/Q coordinates. These systems are being developed now, for a restart of the SPS in early 2021. This Note derives equations for the I/Q representation of the Generator-Cavity-Beam interaction (TWC case) and then uses this formalism for the design of an optimal Feedforward system.