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Improved Mathematical Model and Modeling of Permanent Magnet Synchronous Motors Considering Saturation, Spatial Harmonics, Iron Loss and Deadtime Effect

  • Research Article-Electrical Engineering
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Abstract

A machine model of permanent magnet synchronous motors (PMSMs), which combines the improved mathematical model, electrical circuit model and the data-driven method, is proposed to simulate the real nonlinear behavior of PMSMs in this paper. A PMSM used for the electric compressor of electric vehicles (EVs) is analyzed and tested, and the proposed model can be used for PMSMs of various applications. Firstly, the improved mathematical model of PMSMs considering the magnetic saturation, spatial harmonics and iron loss is established. In particular, the torque formulas of PMSMs are analytically derived by using the energy conservation, which predicts the average torque and torque ripple more accurately. Secondly, based on the proposed mathematical model and the data derived from finite-element analysis, the high-fidelity reduced-order model (ROM) with flux linkage as state variable is built, wherein the radial basis function interpolation method is used to interpolate the discrete current points. To consider the deadtime effect, the electrical circuit model of PMSMs is further incorporated into the ROM. Finally, the model-in-the-loop environment is utilized for simulation validation, which along with experiment shows good accuracy of the proposed mathematical model and ROM.

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Abbreviations

\(u_{A} ,u_{B} ,u_{C}\) :

Three-phase voltage

\(e_{A} ,e_{B} ,e_{C}\) :

Three-phase induced electromotive force (EMF)

\(i_{A} ,i_{B} ,i_{C}\) :

Three-phase current

\(i_{{A{\text{m}}}} ,i_{{B{\text{m}}}} ,i_{{C{\text{m}}}}\) :

Three-phase magnetizing current

\(\psi_{A} ,\psi_{B} ,\psi_{C}\) :

Three-phase flux linkage

\(u_{d} ,u_{q}\) :

d- And q-axis voltages

\(e_{d} ,e_{q}\) :

d- And q-axis induced voltages

\(i_{d} ,i_{q}\) :

d- And q-axis currents

\(i_{{d{\text{m}}}} ,i_{{q{\text{m}}}}\) :

d- And q-axis magnetizing currents

\(i_{{d{\text{i}}}} ,i_{{q{\text{i}}}}\) :

d- And q-axis iron loss currents

\(\psi_{d} ,\psi_{q}\) :

d- And q-axis flux linkages

\(\psi_{{\text{f}}}\) :

Permanent magnet flux linkage

\(\psi_{{{\text{f}}d}} ,\psi_{{{\text{f}}q}}\) :

d- And q-axis components of permanent magnet flux linkage

\(R\) :

Phase resistance

\(R_{{\text{i}}}\) :

Iron loss resistance

\(\omega_{{\text{e}}} ,\omega_{{\text{m}}}\) :

Electrical and mechanical angular speeds

\(\theta_{{\text{e}}} ,\theta_{{\text{m}}}\) :

Rotor electrical and mechanical positions

\(f_{{\text{e}}}\) :

Electrical frequency

\(p\) :

Number of pole pairs

\(L_{d} ,L_{q}\) :

d- And q-axis inductances

\(T_{{\text{e}}}\) :

Electromagnetic torque

\(T_{{\text{L}}}\) :

Load torque

\(t_{{{\text{cog}}}}\) :

Motor cogging torque

\(J\) :

Moment of inertia

\(B\) :

Viscous damping coefficient

\(W_{{\text{m}}}\) :

Motor stored energy

\(W_{{\text{m}}}^{\prime }\) :

Motor co-energy

\(P_{{{\text{Fe}}}}\) :

Core iron loss

References

  1. Balasubramanian, S.; Henke, M.: Performance evaluation of a high-speed permanent magnet synchronous machine with hairpin winding technology. In: 2020 International Conference on Electrical Machines (ICEM). pp. 2445–2451. IEEE (2020)

  2. Polat, M.; Akyun, Y.; Nory, H.: Minimizing the influence of cogging torque on motor performance of PM synchronous machines for elevator applications. Arab. J. Sci. Eng. (2022). https://doi.org/10.1007/s13369-021-06385-x

    Article  Google Scholar 

  3. Sharma, T.; Bhattacharya, A.: Performance analysis of encoderless DTC of IPMSM for wide operating range. Arab. J. Sci. Eng. 45, 6501–6515 (2020). https://doi.org/10.1007/s13369-020-04550-2

    Article  Google Scholar 

  4. Sun, K.; Tian, S.: Multiobjective optimization of IPMSM with FSCW applying rotor notch design for torque performance improvement. IEEE Trans. Magn. 58, 1–9 (2022). https://doi.org/10.1109/TMAG.2022.3155269

    Article  Google Scholar 

  5. Kim, S.-I.; Lee, G.-H.; Hong, J.-P.; Jung, T.-U.: Design process of interior PM synchronous motor for 42-V electric air-conditioner system in hybrid electric vehicle. IEEE Trans. Magn. 44, 1590–1593 (2008). https://doi.org/10.1109/TMAG.2007.916136

    Article  Google Scholar 

  6. Dhulipati, H.; Mukundan, S.; Lai, C.; Mukherjee, K.; Tjong, J.; Kar, N.C.: Multiple reference frame-based extended concentrated wound PMSM model considering PM flux linkage and inductance harmonics. IEEE Trans. Energy Convers. 34, 731–740 (2019). https://doi.org/10.1109/TEC.2018.2880869

    Article  Google Scholar 

  7. Reduced Order Modeling, https://ww2.mathworks.cn/discovery/reduced-order-modeling.html#reduced

  8. Kano, Y.; Watanabe, K.; Kosaka, T.; Matsui, N.: A novel approach for circuit-field-coupled time-stepping electromagnetic analysis of saturated interior PM motors. IEEE Trans. Ind. Appl. 45, 1325–1333 (2009). https://doi.org/10.1109/TIA.2009.2023391

    Article  Google Scholar 

  9. Cai, H.; Hu, D.: On PMSM model fidelity and its implementation in simulation. In: 2018 IEEE Energy Conversion Congress and Exposition (ECCE). pp. 1674–1681. IEEE (2018)

  10. Luo, G.; Zhang, R.; Chen, Z.; Tu, W.; Zhang, S.; Kennel, R.: A novel nonlinear modeling method for permanent-magnet synchronous motors. IEEE Trans. Industr. Electron. 63, 6490–6498 (2016). https://doi.org/10.1109/TIE.2016.2578839

    Article  Google Scholar 

  11. Hu, D.; Alsmadi, Y.M.; Xu, L.: High-fidelity nonlinear IPM modeling based on measured stator winding flux linkage. IEEE Trans. Ind. Appl. 51, 3012–3019 (2015). https://doi.org/10.1109/TIA.2015.2407864

    Article  Google Scholar 

  12. Qiu, J.; Wang, Z.; Zheng, L.; Cao, C.; Fu, Z.: Development of Model-in-the-Loop Environment for Permanent Magnet Synchronous Motors. Presented at the December 14 (2020)

  13. Chauvicourt, F.; Ciceo, S.; van der Auweraer, H.: On the use of vibration synthesis to ease electric machine powertrain design. In: 2019 IEEE International Electric Machines & Drives Conference (IEMDC). pp. 1118–1125. IEEE (2019)

  14. Pinto, D.E.; Pop, A.-C.; Myrria, G.; Kempkes, J.; Gyselinck, J.J.C.: Vibration analysis of delta-connected PMSMs using lookup table-based models—influence of the 0-sequence component. IEEE Trans. Industr. Electron. 69, 6561–6571 (2022). https://doi.org/10.1109/TIE.2021.3097607

    Article  Google Scholar 

  15. Wang, M.: A Novel mathematical nonlinear PMSM realization method for electric machine emulator. IEEE J. Emerg. Sel. Top Power Electron. 10, 4171–4181 (2022). https://doi.org/10.1109/JESTPE.2022.3152429

    Article  Google Scholar 

  16. Scheer, R.; Bergheim, Y.; Heintges, D.; Rahner, N.; Gries, R.; Andert, J.: An FPGA-based real-time spatial harmonics model of a PMSM considering iron losses and the thermal impact. IEEE Transact. Transport. Electrif. (2021). https://doi.org/10.1109/TTE.2021.3119460

    Article  Google Scholar 

  17. Chen, X.; Wang, J.; Griffo, A.: A high-fidelity and computationally Efficient electrothermally coupled model for interior permanent-magnet machines in electric vehicle traction applications. IEEE Transact. Transport. Electrifi. 1, 336–347 (2015). https://doi.org/10.1109/TTE.2015.2478257

    Article  Google Scholar 

  18. Uygun, E.; Hecquet, M.; Tounzi, A.; Depernet, D.; Lanfranchi, V.; Bruno, S.; Tollance, T.: Influence of the load angle on magnetic radial forces and torque ripple of a low power permanent magnet synchronous machine. Math Comput. Simul. 184, 153–164 (2021). https://doi.org/10.1016/j.matcom.2020.05.020

    Article  MathSciNet  MATH  Google Scholar 

  19. Wu, Z.; Zuo, S.; Hu, S.; Hu, X.; Chen, S.; Liu, C.: Electromagnetic torque prediction and harmonic reduction of interior permanent magnet synchronous motor for electric vehicles using an analytical method. Proceed. Instit. Mech. Eng., Part D: J. Automob. Eng. (2021). https://doi.org/10.1177/09544070211058258

    Article  Google Scholar 

  20. Yi, P.; Sun, Z.; Wang, X.: Research on PMSM harmonic coupling models based on magnetic co-energy. IET Electr. Power Appl. 13, 571–579 (2019). https://doi.org/10.1049/iet-epa.2018.5196

    Article  Google Scholar 

  21. Feng, G.; Lai, C.; Tan, X.; Wang, B.; Kar, N.C.: Optimal current modeling and identification for fast and efficient torque ripple minimization of PMSM using theoretical and experimental models. IEEE Trans. Industr. Electron. 68, 11806–11816 (2021). https://doi.org/10.1109/TIE.2020.3047058

    Article  Google Scholar 

  22. Fitzgerald, A E.; Kingsley, C Jr.; Umans S D: Electric Machinery. McGraw-Hill (2013)

  23. Weidenholzer G.; Silber S.; Jungmayr, G.; Bramerdorfer, G.; Grabner, H.; Amrhein, W.: A flux-based PMSM motor model using RBF interpolation for time-stepping simulations. In: 2013 International Electric Machines & Drives Conference. pp. 1418–1423. IEEE (2013)

  24. Ciceo, S.; Chauvicourt, F.; Gyselinck, J.; Martis, C.: A comparative study of system-level PMSM models with either current or flux-linkage state variables used for vibro-acoustic computation. In: 2019 IEEE International Electric Machines & Drives Conference (IEMDC). pp. 1881–1888. IEEE (2019)

  25. Hang, J.; Wu, H.; Ding, S.; Huang, Y.; Hua, W.: Improved loss minimization control for IPMSM using equivalent conversion method. IEEE Trans Power Electron. 36, 1931–1940 (2021). https://doi.org/10.1109/TPEL.2020.3012018

    Article  Google Scholar 

  26. Shutta, Y.; Takahashi, Y.; Fujiwara, K.: Behavioral modeling of permanent magnet synchronous motor fed by PWM inverters considering iron losses due to carrier harmonics. Electrical Eng. Jpn. 210, 37–46 (2020). https://doi.org/10.1002/eej.23265

    Article  Google Scholar 

  27. Sun, X.; Xiao, X.: Precise non-linear flux linkage model for permanent magnet synchronous motors based on current injection and bivariate function approximation. IET Electr Power Appl. 14, 2044–2050 (2020). https://doi.org/10.1049/iet-epa.2020.0137

    Article  Google Scholar 

  28. Queval, L.; Ohsaki, H.: Nonlinear abc-model for electrical machines using N-D lookup tables. IEEE Trans.. Energy Convers. 30, 316–322 (2015). https://doi.org/10.1109/TEC.2014.2358854

    Article  Google Scholar 

  29. Dimas, I.D.; Lourenço, P.R.; Rebelo, T.; Rocha, H.: Maximizing learning through cohesion: contributions from a nonlinear approach. Small Group Res. 52, 162–188 (2021). https://doi.org/10.1177/1046496420944488

    Article  Google Scholar 

  30. Skala, V.: RBF interpolation with CSRBF of large data sets. Procedia Comput Sci. 108, 2433–2437 (2017). https://doi.org/10.1016/j.procs.2017.05.081

    Article  Google Scholar 

  31. Liu, G.; Liu, M.; Zhang, Y.; Wang, H.; Gerada, C.: High-speed permanent magnet synchronous motor iron loss calculation method considering multiphysics factors. IEEE Trans. Industr. Electron. 67, 5360–5368 (2020). https://doi.org/10.1109/TIE.2019.2934075

    Article  Google Scholar 

  32. Boglietti, A.; Cavagnino, A.; Lazzari, M.; Pastorelli, M.: Predicting iron losses in soft magnetic materials with arbitrary voltage supply: an engineering approach. IEEE Trans Magn. 39, 981–989 (2003). https://doi.org/10.1109/TMAG.2003.808599

    Article  Google Scholar 

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Acknowledgements

This work was supported by a Grant (Project 51875410) from the National Natural Science Foundation of China.

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Authors and Affiliations

  1. School of Automotive Studies, Tongji University, Shanghai, 201804, China

    Shuguang Zuo, Zhiyong Huang, Zhipeng Wu & Chang Liu

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Zuo, S., Huang, Z., Wu, Z. et al. Improved Mathematical Model and Modeling of Permanent Magnet Synchronous Motors Considering Saturation, Spatial Harmonics, Iron Loss and Deadtime Effect. Arab J Sci Eng 48, 6939–6955 (2023). https://doi.org/10.1007/s13369-022-07507-9

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