A Novel 3-D Analytical Modeling Method of Trapezoidal Shape Permanent Magnet Halbach Array for Multi-objective Optimization

  • Jiaheng Duan
  • Song XiaoEmail author
  • Kunlun Zhang
  • Yongzhi Jing
Original Article


In this paper, a novel three-dimension magnetic flux density analytical model of linear permanent magnet (PM) Halbach arrays with trapezoidal shape PMs, is established based on the theory of magnetic charge. The proposed analytical model is numerically efficient and can be utilized to evaluate the effects on motor performance caused by varying their configurations. Furthermore, the configurations of PMs for both single and double-sided structures are optimized exploiting a surrogate model assisted by a multi-objective optimization algorithm, where the surrogated model is constructed by the kriging method with expected improvement utility function. The validity and accuracy of this proposed analytical model is verified via finite element modeling, the optimization results with different schemes of weight distribution show the further improvement of magnetic flux density distribution.


Analytical method Halbach array Kriging surrogate model Permanent magnet 



This work was supported by the National Natural Science Foundation of China (Grant no. 51707166).


  1. 1.
    Compter JC, Lomonova EA (2003) Direct 3-D method for performance prediction of a linear moving coil actuator with various topologies. IEE Proc Sci 150(4):183–191Google Scholar
  2. 2.
    You YM, Lin H (2012) Optimal design of a distributed winding type axial flux permanent magnet synchronous generator. J Electr Eng Technol 7(1):69–74CrossRefGoogle Scholar
  3. 3.
    Jang SM, Lee SH (2002) Comparison of two types of PM linear synchronous servo and miniature motor with air-cored film coil. IEEE Trans Magnetics 38(5):3264–3266CrossRefGoogle Scholar
  4. 4.
    Zhang YQ, Yu MH (2010) Field and thrust analysis of linear servo motor with permanent-magnet of different shapes. In: Proceedings of IEEE ICEM conference, Kuala Lumpur, MalaysiaGoogle Scholar
  5. 5.
    Zhang YQ, Yu MH (2011) Analysis and design of double-sided air core linear servo motor with trapezoidal permanent magnets. IEEE Trans Magnetics 47(10):3236–3239CrossRefGoogle Scholar
  6. 6.
    Meessen KJ, Gysen BL (2008) Halbach permanent magnet shape selection for slotless tubular actuators. IEEE Trans Magnetics 44(11):4305–4308CrossRefGoogle Scholar
  7. 7.
    Paul S, Bobba D (2012) Source field modeling in air using magnetic charge sheets. IEEE Trans Magnetics 48(11):3879–3882CrossRefGoogle Scholar
  8. 8.
    Akoun G, Yonnet JP (1984) 3D analytical calculation of the forces exerted between two cuboidal magnets. IEEE Trans Magnetics 20(5):1962–1964CrossRefGoogle Scholar
  9. 9.
    Barba PD (2009) Multi-objective shape design in electricity and magnetism. Springer Science & Business Media, New York, pp 63–69Google Scholar
  10. 10.
    Xiao S, Rotaru M, Sykulski JK (2013) Adaptive weighted expected improvement with rewards approach in kriging assisted electromagnetic design. IEEE Trans Magnetics 49(5):2057–2060CrossRefGoogle Scholar
  11. 11.
    Ren ZY, Zhang DH, Koh CS (2013) A new sensitivity-based reliability calculation algorithm in the optimal design of electromagnetic devices. J Electr Eng Technol 8(2):331–338CrossRefGoogle Scholar
  12. 12.
    Xia B, Ren ZY, Zhang YL, Koh CS (2014) An adaptive optimization algorithm based on kriging interpolation with spherical model and its application to optimal design of switched reluctance motor. J Electr Eng Technol 9(5):1544–1550CrossRefGoogle Scholar
  13. 13.
    Xiao S, Rotaru M, Sykulski JK (2012) Exploration versus exploitation using kriging surrogate modelling in electromagnetic design. COMPEL 31(5):1541–1551CrossRefGoogle Scholar

Copyright information

© The Korean Institute of Electrical Engineers 2019

Authors and Affiliations

  • Jiaheng Duan
    • 1
  • Song Xiao
    • 1
    Email author
  • Kunlun Zhang
    • 1
    • 2
  • Yongzhi Jing
    • 1
    • 2
  1. 1.School of Electrical EngineeringSouthwest Jiaotong UniversityChengduChina
  2. 2.Key Laboratory of Magnetic Suspension Technology and Maglev VehicleMinistry of EducationChengduChina

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