Equivalent circuit study of beam-loading using a moment method

In this work, we present a formalism by considering the perturbations in the moments of a bunched beam for the equivalent circuit model to include all harmonics of the synchrotron oscillation in a beam-cavity interaction system. The linear coupling among all longitudinal modes under the influence of a narrow-band impedance can be naturally incorporated in this new approach. We used this method to re-examine the coupling between the dipole and the quadrupole modes. The dispersion relation obtained by this new method was compared with that derived from the linearized Vlasov equation up to the second harmonic of the synchrotron motion. We found excellent qualitative agreements between these two approaches.