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Subdomain method for brushless double-rotor flux-switching permanent magnet machines with yokeless stator

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Abstract

This paper presents a two-dimensional analytical model for brushless double-rotor flux-switching permanent magnet machines (BDRPFSPMM) with yokeless stator recognized as a type of partitioned-structure flux-switching permanent magnet that is an appropriate machine to be employed as the electric motor in industry. The most salient advantages of the proposed machine are low iron loss due to volume of iron in stator and high space of stator slot for winding. The radial and tangential components of the magnetic flux density due to permanent magnets and armature currents in each active domain of the machine, electromagnetic torque, self- and mutual inductance, unbalanced magnetic force (UMF), and local traction are calculated based on the subdomain method. Teethes on both rotor and stator structures are effective on magnetic quantities, so interactions of rotor and stator saliency are considered. The method is general for the magnetic field calculation with any combination of rotor- and stator-pole number. To validate the proposed model, the analytical outputs are compared with those obtained from the finite element method (FEM).

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No datasets were generated or analyzed during the current study.

Abbreviations

A:

Magnetic vector potential (V.s/m)

B:

Magnetic flux density vector (T)

B rem :

Permanent magnet residual flux density (T)

Br :

Radial component of B (T)

B t :

Tangential component of B (T)

H:

Magnetic field intensity vector (A/m)

H t :

Tangential component of H (T)

J:

Armature current density vector (A/m2)

M:

Magnetization vector (A/m).

μ 0 :

Free space permeability (H/m)

μ r :

Relative permeability

μ rpm :

Relative permeability of permanent magnet

N ss :

Number of stator slots

N irs :

Number of inner rotor slots

N pm :

Number of permanent magnets

N ors :

Number of outer rotor slots

α i :

Central angle of ith slot of inner rotor

β i :

Central angle of ith slot of stator

λ i :

Central angle of ith PM

σ i :

Central angle of ith slot of outer rotor

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Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Author information

Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Bojnourd University, Bojnourd, Iran

    Ehsan Shirzad, Hassan Mohammai Pirouz & Mohammad Taghi Shirzad

Authors
  1. Ehsan Shirzad
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  2. Hassan Mohammai Pirouz
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  3. Mohammad Taghi Shirzad
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Contributions

ES that simulation and novelty has been done by him and initial writing and sentencing is on him. HMP and MTS only works on correcting sentencing.

Corresponding author

Correspondence to Ehsan Shirzad.

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The authors declare no competing interests.

Conflict of interests

This research is sponsored by [Bojnourd University] and may lead to development of products.

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I hereby declare that this thesis represents my own work which has been done after studying at university of Bojnourd and has not been previously included in a thesis or dissertation submitted to this or any other institution for a degree, diploma, or other qualifications. I have read the research ethics guidelines and accept responsibility for the conduct of the procedures in accordance with Springer journal. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing, we confirm that we have followed the regulations of our institutions concerning intellectual property. We further confirm any aspect of the work covered in this manuscript that not has involved either experimental animals or human patients. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author.

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Appendix

Appendix

The expressions for calculating the coefficients are as follows:

$$ a_{irs,0} = \frac{1}{{w_{irs} }}\mathop \int \limits_{{\alpha_{i} - w_{irs} /2}}^{{\alpha_{i} + w_{irs} /2}} A_{z}^{ia} \left( {R_{irs} ,\theta } \right)d\theta $$
(40)
$$ a_{irs,n} f\left( {n,R_{irs} } \right) = \frac{2}{{w_{irs} }}\mathop \int \limits_{{\alpha_{i} - w_{irs} /2}}^{{\alpha_{i} + w_{irs} /2}} A_{z}^{ia} \left( {R_{irs} ,\theta } \right)\cos \left( {\frac{n\pi }{{w_{irs} }}\left( {\begin{array}{*{20}c} {\theta - \alpha_{i} } \\ { + \frac{{w_{irs} }}{2}} \\ \end{array} } \right)} \right)d\theta $$
(41)
$$ m( - b_{ia,1,m} R_{irs}^{m - 1} + b_{ia,1,m} R_{irs}^{ - m - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{irs} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\alpha_{i} - w_{irs} /2}}^{{\alpha_{i} + w_{irs} /2}} H_{\theta }^{irs,i} \left( {R_{irs} ,\theta } \right)\sin \left( {m\theta } \right)d\theta $$
(42)
$$ m ( - a_{ia,1,m} R_{irs}^{m - 1} + a_{ia,1,m} R_{irs}^{ - m - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{irs} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\alpha_{i} - w_{irs} /2}}^{{\alpha_{i} + w_{irs} /2}} H_{\theta }^{irs,i} \left( {R_{irs} ,\theta } \right)\cos \left( {m\theta } \right)d\theta $$
(43)
$$ \left( {a_{\iota ss,0} + b_{\iota ss,0} \ln \left( {R_{is} } \right) - \frac{1}{4}\mu_{0} J_{ss,0} R_{is}^{2} } \right) = \frac{1}{{w_{ss} }}\mathop \int \limits_{{\beta_{i} - w_{ss} /2}}^{{\beta_{i} + w_{ss} /2}} A_{z}^{ia} \left( {R_{is} ,\theta } \right)d\theta $$
(44)
$$ \begin{gathered}\frac{2}{n\pi }\sin \left( { \frac{n\pi d}{{w_{ss} }}} \right)\left( { - \frac{1}{4}\mu_{0} R_{is}^{2} } \right)(J_{ss,1} +J_{ss,2} \left( { - 1} \right)^{n} ))\\\quad \times(a_{ss,n} R_{is}^{{\frac{ - n\pi }{{w_{ss} }}}} + b_{ss,n} R_{is}^{{\frac{n\pi }{{w_{ss} }}}} ) \hfill \\\quad = \frac{2}{{w_{ss} }}\mathop \int \limits_{{\beta_{i} - w_{ss} /2}}^{{\beta_{i} + w_{ss} /2}} A_{z}^{ia} \left( {R_{is} ,\theta } \right)\cos \left( {\frac{n\pi }{{w_{ss} }}\left( {\begin{array}{*{20}c} {\theta - \beta_{i} } \\ { + \frac{{w_{ss} }}{2}} \\ \end{array} } \right)} \right)d\theta \hfill \\\end{gathered}$$
(45)
$$ n( - b_{ia,1,m} R_{is}^{m - 1} + b_{ia,1,m} R_{is}^{ - m - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{ss} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\beta_{i} - w_{ss} /2}}^{{\beta_{i} + w_{ss} /2}} H_{\theta }^{ss,i} \left( {R_{is} ,\theta } \right)\sin \left( {m\theta } \right)d\theta + \mathop \sum \limits_{i = 1}^{{N_{pm} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\lambda_{i} - w_{pm} /2}}^{{\lambda_{i} + w_{pm} /2}} H_{\theta }^{pm,i} \left( {R_{is} ,\theta } \right)\sin \left( {m\theta } \right)d\theta $$
(46)
$$ m ( - a_{ia,1,m} R_{is}^{m - 1} + a_{ia,1,m} R_{is}^{ - m - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{ss} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\beta_{i} - w_{ss} /2}}^{{\beta_{i} + w_{ss} /2}} H_{\theta }^{ss,i} \left( {R_{is} ,\theta } \right)\cos \left( {m\theta } \right)d\theta + \mathop \sum \limits_{i = 1}^{{N_{pm} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\lambda_{i} - w_{pm} /2}}^{{\lambda_{i} + w_{pm} /2}} H_{\theta }^{pm,i} \left( {R_{ss} ,\theta } \right)\cos \left( {m\theta } \right)d\theta $$
(47)
$$ a_{pm,0} + b_{pm,0} \ln \left( {R_{is} } \right) - \mu_{0} M_{pm} R_{is} = \frac{1}{{w_{pm} }}\mathop \int \limits_{{\lambda_{i} - w_{pm} /2}}^{{\lambda_{i} + w_{pm} /2}} A_{z}^{ia} \left( {R_{is} ,\theta } \right)d\theta $$
(48)
$$ (a_{ipm,n} R_{is}^{{\frac{ - n\pi }{{w_{pm} }}}} + b_{ipm,n} R_{iss}^{{\frac{n\pi }{{w_{ipm} }}}} ) = \frac{2}{{w_{pm} }}\mathop \int \limits_{{\lambda_{i} - w_{pm} /2}}^{{\lambda_{i} + w_{pm} /2}} {\text{A}}_{{\text{z}}}^{ia} \left( {R_{is} ,\theta } \right)\cos \left( {\frac{n\pi }{{w_{pm} }}\left( {\begin{array}{*{20}c} {\theta - \lambda_{i} } \\ { + \frac{{w_{pm} }}{2}} \\ \end{array} } \right)} \right)d\theta $$
(49)
$$ a_{ors,0} = \frac{1}{{w_{ors} }}\mathop \int \limits_{{\sigma_{i} - w_{ors} /2}}^{{\sigma_{i} + w_{ors} /2}} {\text{A}}_{{\text{z}}}^{oa} \left( {R_{ors} ,\theta } \right)d\theta $$
(50)
$$ a_{ors,n} f\left( {n,R_{ors} } \right) = \frac{2}{{w_{ors} }}\mathop \int \limits_{{\sigma_{i} - \frac{{w_{ors} }}{2}}}^{{\sigma_{i} + \frac{{w_{ors} }}{2}}} {\text{A}}_{{\text{z}}}^{oa} \left( {R_{ors} ,\theta } \right)\cos \left( {\frac{n\pi }{{w_{ors} }}\left( {\begin{array}{*{20}c} {\theta - \sigma_{i} } \\ { + \frac{{w_{ors} }}{2}} \\ \end{array} } \right)} \right)d\theta $$
(51)
$$ m ( - b_{oa,1,m} R_{ors}^{m - 1} + b_{oa,1,m} R_{ors}^{ - m - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{ors} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\sigma_{i} - w_{ors} /2}}^{{\sigma_{i} - w_{ors} /2}} H_{\theta }^{ors,i} \left( {R_{ors} ,\theta } \right)\sin \left( {n\theta } \right)d\theta $$
(52)
$$ n ( - a_{oa,1,n} R_{ors}^{n - 1} + a_{oa,1,n} R_{ors}^{ - n - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{ors} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\sigma_{i} - w_{ors} /2}}^{{\sigma_{i} + w_{ors} /2}} H_{\theta }^{ors,i} \left( {R_{ors} ,\theta } \right)\cos \left( {n\theta } \right)d\theta $$
(53)
$$ \left( {a_{ss,0} + b_{ss,0} \ln \left( {R_{os} } \right) - \frac{1}{4}\mu_{0} J_{0,ss} R_{os}^{2} } \right) = \frac{1}{{w_{ss} }}\mathop \int \limits_{{\beta_{i} - w_{ss} /2}}^{{\beta_{i} + w_{ss} /2}} A_{z}^{oa} \left( {R_{ss} ,\theta } \right)d\theta $$
(54)
$$ \frac{2}{n\pi }\sin \left( {\frac{n\pi d}{{w_{ss} }}} \right)\left( { - \frac{1}{4}\mu_{0} R_{os}^{2} } \right)(J_{ss,1} + J_{ss,2} \left( { - 1} \right)^{n} )(a_{ss,n} R_{os}^{{\frac{ - n\pi }{{w_{ss} }}}} + b_{ss,n} R_{os}^{{\frac{n\pi }{{w_{ss} }}}} ) = \frac{2}{{w_{ss} }}\mathop \int \limits_{{\beta_{i} - \frac{{w_{ss} }}{2}}}^{{\beta_{i} + \frac{{w_{ss} }}{2}}} {\text{A}}_{{\text{z}}}^{oa} \left( {R_{os} ,\theta } \right)\cos \left( {\frac{n\pi }{{w_{iss} }}\left( {\begin{array}{*{20}c} {\theta - \beta_{i} } \\ { + \frac{{w_{ss} }}{2}} \\ \end{array} } \right)} \right)d\theta $$
(55)
$$ m ( - b_{oa,1,m} R_{os}^{m - 1} + b_{oa,1,m} R_{os}^{ - m - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{ss} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\beta_{i} - w_{oss} /2}}^{{\beta_{i} + w_{oss} /2}} H_{\theta }^{ss,i} \left( {R_{os} ,\theta } \right)\sin \left( {m\theta } \right)d\theta + \mathop \sum \limits_{i = 1}^{{N_{pm} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\lambda_{i} - w_{pm} /2}}^{{\lambda_{i} + w_{pm} /2}} H_{\theta }^{pm,i} \left( {R_{os} ,\theta } \right)\sin \left( {m\theta } \right)d\theta $$
(56)
$$ m ( - a_{oa,1,m} R_{os}^{m - 1} + a_{oa,1,m} R_{os}^{ - m - 1} ) = \mathop \sum \limits_{i = 1}^{{N_{ss} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\beta_{i} - w_{ss} /2}}^{{\beta_{i} + w_{ss} /2}} H_{\theta }^{ss,i} \left( {R_{os} ,\theta } \right)\cos \left( {m\theta } \right)d\theta + \mathop \sum \limits_{i = 1}^{{N_{pm} }} \frac{{\mu_{0} }}{\pi }\mathop \int \limits_{{\lambda_{i} - w_{opm} /2}}^{{\lambda_{i} + w_{opm} /2}} H_{\theta }^{pm,i} \left( {R_{os} ,\theta } \right)\cos \left( {m\theta } \right)d\theta $$
(57)
$$ a_{pm,0} + b_{pm,0} \ln \left( {R_{os} } \right) - \mu_{0} M_{pm} R_{os} = \frac{1}{{w_{pm} }}\mathop \int \limits_{{\lambda_{i} - w_{pm} /2}}^{{\lambda_{i} + w_{pm} /2}} A_{z}^{oa} \left( {R_{os} ,\theta } \right)d\theta $$
(58)
$$ (a_{pm,n} r^{{\frac{ - n\pi }{{w_{pm} }}}} + b_{pm,n} r^{{\frac{n\pi }{{w_{pm} }}}} ) = \frac{2}{{w_{pm} }}\mathop \int \limits_{{\lambda_{i} - \frac{{w_{pm} }}{2}}}^{{\lambda_{i} + \frac{{w_{pm} }}{2}}} {\text{A}}_{{\text{z}}}^{oa} \left( {R_{os} ,\theta } \right)\cos \left( {\frac{n\pi }{{w_{pm} }}\left( {\begin{array}{*{20}c} {\theta - \lambda_{i} } \\ { + \frac{{w_{pm} }}{2}} \\ \end{array} } \right)} \right)d\theta $$
(59)

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Shirzad, E., Pirouz, H.M. & Shirzad, M.T. Subdomain method for brushless double-rotor flux-switching permanent magnet machines with yokeless stator. Electr Eng 106, 1475–1485 (2024). https://doi.org/10.1007/s00202-023-01795-6

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