A higher harmonic cavity (HHC), used to cause bunch lengthening for an increase in the Touschek lifetime, is a feature of several fourth generation synchrotron light sources. The desired bunch lengthening is complicated by the presence of required gaps in the bunch train. In a recent paper the author and Venturini studied the effect of various fill patterns by calculating the charge densities in the equilibrium state, through coupled Haïssinski equations. We assumed that the only collective force was from the beam loading (wakefield) of the harmonic cavity in its lowest mode. The present paper improves the notation and organization of the equations so as to allow an easy inclusion of multiple resonator wakefields. This allows one to study the effects of beam loading of the main accelerating cavity, higher order modes of the cavities, and short range geometric wakes represented by low-$Q$ resonators. As an example these effects are explored for ALS-U. The compensation of the induced voltage in the main cavity, achieved in practice by a feedback system, is modeled by adjustment of the generator voltage through a new iterative scheme. Except in the case of a complete fill, the compensated main cavity beam loading has a substantial effect on the bunch profiles and the Touschek lifetimes. A $Q=6$ resonator, approximating the effect of a realistic short range wake, is also consequential for the bunch forms.

• Received 16 November 2020
• Accepted 19 January 2021

DOI:https://doi.org/10.1103/PhysRevAccelBeams.24.024401

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Published by the American Physical Society