Synchrotron beam-loading stability with a higher RF harmonic

This work extends Robinson´s stability studies (1956) to include a higher RF harmonic. Using an equivalent circuit model, the stability of the “0-mode” coherent dipole oscillation of bunched beams is studied for synchrotrons or storage rings with RF systems operated at the fundamental and a higher harmonic, i.e., a second or third harmonic. Analytical expressions of the stability criteria are derived from the linearized circuit equations. Numerical solutions of the fully nonlinear equations are provided to compare with the analytical results. A simple feedback model for stabilization is discussed