# Analytical solution to the transient beam loading effects of a superconducting cavity

• Transient beam loading is one of the key issues in any high beam current intensity superconducting accelerators, and needs to be carefully investigated. The core problem in the analysis is to obtain the time evolution of effective cavity voltage under transient beam loading. To simplify the problem, the second order ordinary differential equation describing the behavior of the effective cavity voltage is intuitively simplified to a first order one, with the aid of two critical approximations which lack proof of their validity. In this paper, the validity is examined mathematically in some specific cases, resulting in a criterion for the simplification. It is popular to solve the approximate equation for the effective cavity voltage numerically, while this paper shows that it can also be solved analytically under the step function approximation for the driven term. With the analytical solution to the effective cavity voltage, the transient reflected power from the cavity and the energy gain of the central particle in the bunch can also be calculated analytically. The validity of the step function approximation for the driven term is examined by direct evaluations. After that, the analytical results are compared with the numerical ones.
PCAS:
•  [1] Y. He et al, in Proceedings of IPAC 2011 (Spain:San Sebastian, 2011) p. 2613-2615 [2] Z. J. Wang, Y. He, Y. Liu et al, Chinese Physics C, 36:256-260(2012) [3] S. H. Liu, Z. J. Wang, W. M. Yue et al, Chinese Physics C, 38:117006(2014) [4] P. Wilson, SLAC-PUB-2884(Revised), (1991) [5] W. M. Yue et al, in Proceedings of Linac 2012 (Israel:Tel-Aviv 2012) [6] H. Padamsee, J. Knobloch, T. Hays, RF superconductivity for accelerators 2nd ed., (Weinheim:Wiley-VCH, 2008) [7] S. H. Kim, M. Doleans, RF/Microwave interaction and beam loading in SRF cavity (USPAS Course Materials, Duke University, 2013) [8] T. P. Wangler, RF linear accelerators (2nd completely rev. and enl. ed., Wiley-VCH, Weinheim, 2008) [9] E. Fehlberg, NASA Technical Report R-315, (1969) [10] P. F. Hsieh, Y. Sibuya, Basic theory of ordinary differential equations (New York:Springer, 1999)

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Ran Huang, Yuan He, Zhi-Jun Wang, Wei-Ming Yue, An-Dong Wu, Yue Tao, Qiong Yang, Cong Zhang, Hong-Wei Zhao and Zhi-Hui Li. Analytical solution to the transient beam loading effects of a superconducting cavity[J]. Chinese Physics C, 2017, 41(10): 107001. doi: 10.1088/1674-1137/41/10/107001
Ran Huang, Yuan He, Zhi-Jun Wang, Wei-Ming Yue, An-Dong Wu, Yue Tao, Qiong Yang, Cong Zhang, Hong-Wei Zhao and Zhi-Hui Li. Analytical solution to the transient beam loading effects of a superconducting cavity[J]. Chinese Physics C, 2017, 41(10): 107001.
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Revised: 2017-06-24
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Supported by National Natural Science Foundation of China (11525523, 91426303)

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## Analytical solution to the transient beam loading effects of a superconducting cavity

###### Corresponding author: Yuan He,
• 1. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
• 2. University of Chinese Academy of Sciences, Beijing 100049, China
• 3. Sichuan University, Chengdu 610064, China
• 4.  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
• 5.  Sichuan University, Chengdu 610064, China
Fund Project:  Supported by National Natural Science Foundation of China (11525523, 91426303)

Abstract: Transient beam loading is one of the key issues in any high beam current intensity superconducting accelerators, and needs to be carefully investigated. The core problem in the analysis is to obtain the time evolution of effective cavity voltage under transient beam loading. To simplify the problem, the second order ordinary differential equation describing the behavior of the effective cavity voltage is intuitively simplified to a first order one, with the aid of two critical approximations which lack proof of their validity. In this paper, the validity is examined mathematically in some specific cases, resulting in a criterion for the simplification. It is popular to solve the approximate equation for the effective cavity voltage numerically, while this paper shows that it can also be solved analytically under the step function approximation for the driven term. With the analytical solution to the effective cavity voltage, the transient reflected power from the cavity and the energy gain of the central particle in the bunch can also be calculated analytically. The validity of the step function approximation for the driven term is examined by direct evaluations. After that, the analytical results are compared with the numerical ones.

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