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  • Open Access

Periodic transient beam loading effect in triple radio-frequency systems

Jincheng Xiao, Tianlong He*, and Weimin Li†

  • *Contact author: htlong@ustc.edu.cn
  • †Contact author: lwm@ustc.edu.cn
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Phys. Rev. Accel. Beams 28, 104401 – Published 3 October, 2025

DOI: https://doi.org/10.1103/zd54-9t48

Abstract

For fourth-generation synchrotron light sources, the triple rf system has been proposed to achieve further bunch lengthening compared to the commonly used double rf system, as well as to meet specific requirements for longitudinal injection. In a previous study for the double rf system [T. He et al., Periodic transient beam loading effect with passive harmonic cavities in electron storage rings, Phys. Rev. Accel. Beams 25, 024401 (2022)], it was shown that the periodic transient beam loading (PTBL) effect, also referred to as mode 1 instability, could limit the maximum bunch lengthening. Intuitively, in the triple rf system, the harmonic cavity (HC) operating at a higher harmonic of the fundamental rf frequency is expected to help mitigate the mode 1 instability driven by the lower-order HC, as the two HCs are tuned in opposite directions. However, contrary to this intuition, both tracking simulations and semianalytical calculations presented in this paper show that the two HCs of the triple rf system together actually strengthen the PTBL effect. To better understand this unexpected behavior, we introduce a novel concept of steady-state response matrix of cavity voltage, which simplifies the semianalytical algorithm and directly links cavity voltage perturbations to the complex bunch form factor perturbations. Additionally, we propose a modified semianalytical method that efficiently and accurately determines the PTBL threshold for the triple rf system. Using the parameters of the Hefei Advanced Light Facility storage ring as a case study, we apply this modified method to explore the relationship between the threshold R/Q values of the two HCs under different bunch lengthening scenarios. All results are well validated through tracking simulations. Our findings show that the R/Q values of the two HCs must be sufficiently low to avoid the PTBL effect under the desired optimum bunch lengthening condition. This result provides valuable guidance for the design of the triple rf system.

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Phys. Rev. Accel. Beams 28, 104401– Published 3 October, 2025
  • Received 7 January 2025
  • Accepted 20 June 2025
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