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Model of Five-Phase Permanent Magnet Synchronous Machines Including the Third Harmonic of the Airgap Induction

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Abstract

We present an analytical model of permanent-magnet five-phase synchronous machines including the third space harmonic of the airgap induction, which is generally disregarded by most authors, despite its relevance for the estimation of the machine behavior. Firstly, we address the numerical estimation of the self and mutual inductances required by the model using the machine dimensions and winding characteristics. Afterward, we derive a model in the stator reference frame, which is subsequently simplified through complex coordinate transformations, thus resulting in a set of simpler, decoupled equations expressed in the stator reference frame. Then, to test and validate the model, it was applied to two prototype machines available in our laboratory. We designed the rotor of these two machines so that each has a different amplitude for the third harmonic of the airgap induction. Finally, to further validate the model, analytical results were compared with corresponding results obtained through finite element analyses and also through practical experiments, from which an excellent agreement emerged between the three types of results. The results obtained with the finite element method and through experiments also proved not only that the model is consistent and valid, but also that it can be used to predict the machine behavior under normal and faulty operation.

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Acknowledgements

This study was financed in part by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. This study was also supported by Foundation for Research Support of the State of Rio Grande do Sul (FAPERGS), (Program Pesquisador Gaúcho—2022) under the grant number 21/2551-0002155-1. Finally, the authors also thank the National Council for Scientific and Technological Development (CNPq) for the financial support under grants numbers 306126/2019-2, 404491/2021-9, and 305036/2022-0, and WEG Equipamentos Elétricos (Jaraguá do Sul, Brazil) for the technical support.

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  1. Gabriel Garcia Castelo Branco, Guilherme Nicol and Rodrigo Borges Tavares have contributed equally to this work.

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  1. PPGEE, Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha, 103, Porto Alegre, RS, 90035190, Brazil

    Luís Alberto Pereira, Gabriel Garcia Castelo Branco, Guilherme Nicol & Rodrigo Borges Tavares

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  1. Luís Alberto Pereira
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Appendices

Appendix A: Main Dimensions of the Prototype Machines

The main dimensions of the two prototype machines of 1.5 KW designed and currently available in our laboratory at UFRGS can be seen in Figs 19 and 20, in which Fig. 19 illustrates the stator dimensions and details of the stator slots, while Fig. 20 details the rotor dimensions. Note also that both machines have been used to test and validate the proposed model.

Fig. 19
figure 19

Main dimensions of stator (a) and stator slot (b)

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Fig. 20
figure 20

Main dimensions a of the rotor, b rotor slots where the permanent magnets are located, and c permanent magnets

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Appendix B: Iron and Permanent Magnet Characteristics

In Fig. 21, we present the main magnetic characteristic (induction B versus magnetic field H) of the iron parts of the two prototypes used in this paper for numerical and experimental validation of the proposed analytical model. In addition, Fig. 22 shows the demagnetization curve of permanent magnets (N38UH) at a temperature of \(22\,^\circ \)C, where the remanent induction and the coercive force are in red.

Fig. 21
figure 21

Curve of magnetization of the iron parts used in the prototypes

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Fig. 22
figure 22

Second quadrant curve of the permanent magnets

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Pereira, L.A., Branco, G.G.C., Nicol, G. et al. Model of Five-Phase Permanent Magnet Synchronous Machines Including the Third Harmonic of the Airgap Induction. J Control Autom Electr Syst 35, 191–211 (2024). https://doi.org/10.1007/s40313-023-01056-8

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