The longitudinal beam dynamics in circular accelerators is mainly defined by the interaction of the beam current with the accelerating Radio Frequency (RF) stations. For stable operation, Low Level RF (LLRF) feedback systems are employed to reduce coherent instabilities and regulate the accelerating voltage. The LLRF system design has implications for the dynamics and stability of the closed-loop RF systems as well as for the particle beam, and is very sensitive to the operating range of accelerator currents and energies. Stability of the RF loop and the beam are necessary conditions for reliable machine operation. This dissertation describes theoretical formalisms and models that determine the longitudinal beam dynamics based on the LLRF implementation, time domain simulations that capture the dynamic behavior of the RF station-beam interaction, and measurements from the Positron-Electron Project (PEP-II) and the Large Hadron Collider (LHC) that validate the models and simulations. These models and simulations are structured to capture the technical characteristics of the system (noise contributions, non-linear elements, and more). As such, they provide useful results and insight for the development and design of future LLRF feedback systems. They also provide the opportunity to study diverse longitudinal beam dynamics effects such as coupled-bunch impedance driven instabilities and single bunch longitudinal emittance growth. Coupled-bunch instabilities and RF station power were the performance limiting effects for PEP-II. The sensitivity of the instabilities to individual LLRF parameters, the effectiveness of alternative operational algorithms, and the possible tradeoffs between RF loop and beam stability were studied. New algorithms were implemented, with significant performance improvement leading to a world record current during the last PEP-II run of 3212 mA for the Low Energy Ring. Longitudinal beam emittance growth due to RF noise is a major concern for LHC. Simulations studies and measurements were conducted that clearly show the correlation between RF noise and longitudinal bunch emittance, identify the major LLRF noise contributions, and determine the RF component dominating this effect. With these results, LHC upgrades and alternative algorithms are evaluated to reduce longitudinal emittance growth during operations. The applications of this work are described with regard to future machines and analysis of new technical implementations, as well as to possible future work which would continue the directions of this dissertation.

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