A.A. Shvachko – Ph.D.(Eng.), 

Department «Electronic Devices and Systems Engineering», Yuri Gagarin State Technical University of Saratov E-mail: alexandr1899@gmail.com

A.A. Zakharov – Dr.Sc.(Eng.), Professor, 

Department «Electronic Devices and Systems Engineering», Yuri Gagarin State Technical University of Saratov E-mail: zaharov@sstu.ru

The existing mathematical models of the magnetic field in periodic magnetic focusing system do not allow one to estimate the effect of a separate magnet on a ruzing magnetic field on the axis of the system. Created by the authors mathematical model primarily addresses this problem. The mathematical model is based on the expression of the magnetic field of a separate ring magnet, and the magnetic field on the axis of the periodic magnetic focusing system is created due to the superposition of the magnetic fields of individual magnets and pole pieces. Expressions taking into account the influence of pole piece are based on the charge model. According to this model, the pole piece is represented as a set of charged planes. Where the charged plane considered to tip the plane of contact with the magnet. Pole piece is divided into 3 parts and each made up its expression using known techniques. The resulting expression for the pole pieces into account the magnitude and the direction of magnetization of each magnet in contact with them, as well as its own geometric parameters. The developed mathematical model allows to take into account the impact of each magnet in the resulting magnetic field on the axis of the periodic magnetic focusing system, while taking into account the pole pieces mounted between the magnets.

1. Chang Kern K.N. Optimum Design of Periodic Magnet Structures For Electron Beam Focusing. RCA Rew. 1955. V. 6. № 1. S. 65−81.
2. Hu Q. Asymptotic boundary conditions for the finite element modeling of axisymmetric period permanent magnet structures. IEEE Eighteenth International Vacuum Electronics Conference (IVEC-2017). 2017. S. 1−2. DOI: 10.1109/IVEC.2017.8289507.
3. Tsarev V.A., Spiridonov R.V. Magnitnye fokusiruyushchie sistemy elektrovakuumnykh mikrovolnovykh priborov O-tipa: Ucheb. posobie. Saratov: Izd-vo «Novyi veter». 2010. (in Russian)
4. Kogen-Danilin V.V., Komarov E.V. Raschet i ispytaniya sistem s postoyannymi magnitami. M.: Energiya. 1977. (in Russian)
5. Chen W. et al. 3D symmetric computation algorithm with finite element method for permanent magnet focusing system. IEEE International Vacuum Electronics Conference (IVEC-2016). IEEE 2016. S. 1−2. DOI: 10.1109/IVEC.2016.7561932.
6. Bittanti S., Colaneri P. Periodic systems: filtering and control. Springer Science & Business Media. 2009. P. 407. DOI: 10.1007/978-184800-911-0.
7. Shvachko A.A., Zakharov A.A., Afonin I.N., Turkin Ya.V. Matematicheskaya model nastroiki MPFS metodom perestanovki magnitov. Radiotekhnika. 2016. № 10. S. 203−206. (in Russian)