# Limitations and Constraints of Eddy-Current Loss Models for Interior Permanent-Magnet Motors with Fractional-Slot Concentrated Windings

^{*}

## Abstract

**:**

## 1. Introduction

#### Contributions and Outline of the Paper

## 2. Review of Eddy-Current Loss Models

#### 2.1. Model A: Assumed Eddy-Current Paths

**Remark**

**1.**

#### 2.2. Model B: Solving the Helmholtz Equation with the Imposed Source Term

**Remark**

**3.**

#### 2.3. Model C: Solving the Helmholtz Equation Prescribing Boundary Surface Currents

## 3. Analysis and Evaluation

#### 3.1. Loss-Model Constraints When Applied to IPMs

#### 3.2. Limits for Model A Due to Eddy-Current Reaction Fields

#### Approximation of ${\epsilon}_{\mathrm{A}\mid \mathrm{B}}$

#### 3.3. 3DFEM-Evaluation

**Remark**

**5.**

#### 3.3.1. Negligible Eddy-Current Reaction Fields

#### 3.3.2. Non-Negligible Eddy-Current Reaction Fields

**Remark**

**6.**

#### 3.3.3. Impact of Non-Uniform Flux-Density Variation

#### 3.4. Visualization of the Resulting Eddy-Current Distribution

## 4. PM Losses in Automotive Applications

#### 4.1. Thermal Impact

#### 4.2. Losses for p and ${Q}_{s}$ Common in Automotive Applications

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. FSCW Fundamentals

#### Appendix A.1. Preliminaries

#### Appendix A.2. Air-Gap MMF Distribution Due to Stator Current

#### Appendix A.3. Winding Factor for Harmonic ν

#### Appendix A.4. PM Flux Density Variations

**Remark**

**A1.**

**Figure A1.**Magnet dimensions and definition of rotor-cap coefficient ${\alpha}_{p}$ (note that for the coordinate system fixed to PMs, the y-axis is directed in the paper): (

**a**) V-shaped interior PMs; (

**b**) straight interior PMs.

**Remark**

**A2.**

## Appendix B. IPM Parameters

Parameter | Value | Unit |
---|---|---|

# poles (p) | 8 | - |

# stator slots (${Q}_{s}$) | 12 | - |

# turns per slot (${n}_{s}$) | 16 | - |

Rated current (I) | 97 | A (rms) |

Rotor radius (${r}_{r}$) | $69.25$ | mm |

Air-gap length (δ) | $0.75$ | mm |

Magnet height (${h}_{m}$) | $7.51$ | mm |

Magnet conductivity (${\sigma}_{m}$) | 694 | kS/m |

Magnet relative permeability (${\mu}_{r}$) | $1.04$ | - |

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**Figure 1.**Sample fractional-slot concentrated winding (FSCW)-interior permanent-magnet motor (IPM) with ${Q}_{s}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}12$ slots and $p\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8$ poles.

**Figure 2.**Assumed eddy-current paths adopted in Model A (redrawn from [19]).

**Figure 3.**Contours corresponding to ${\epsilon}_{\mathrm{A}\mid \mathrm{B}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.2$ (red curves) for ${\omega}_{{\nu}_{m}}/(2\pi )\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}300$ Hz to ${\omega}_{{\nu}_{m}}/(2\pi )\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3000$ Hz in steps of 300 Hz assuming ${\mu}_{r}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.04$ and ${\sigma}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}694$ kS/m. The arrow denotes the direction of increasing ${\omega}_{{\nu}_{m}}$. Note that the contours ${\epsilon}_{\mathrm{A}\mid \mathrm{C}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.2$ are essentially identical.

**Figure 4.**Contours corresponding to ${\epsilon}_{\mathrm{A}\mid \mathrm{B}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.2$ (red curves) using the approximate formulation of ${\epsilon}_{\mathrm{A}\mid \mathrm{B}}$ given by (26) for ${\omega}_{{\nu}_{m}}/(2\pi )\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}300$ Hz to ${\omega}_{{\nu}_{m}}/(2\pi )\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3000$ Hz in steps of 300 Hz assuming ${\mu}_{r}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.04$ and ${\sigma}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}694$ kS/m. The arrow denotes the direction of increasing ${\omega}_{{\nu}_{m}}$.

**Figure 5.**IPM 3D-FEM models: (

**a**) V-shaped interior PMs (${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}15$ mm); (

**b**) V-shaped interior PMs (${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}30$ mm); and (

**c**) Straight interior PMs (${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}40$ mm).

**Figure 6.**Predicted losses per magnet segment ${P}_{m}$ as a function of rotor speed and for different magnet lengths ${l}_{m}$ in V-shaped interior PMs (${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}15$ mm) for Model A (blue line), Model B (red line), Model C (green line) and corresponding 3D-FEM results (diamonds “⋄"): (

**a**) ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}10$ mm; (

**b**) ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}30$ mm; and (

**c**) ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}100$ mm.

**Figure 7.**Predicted losses per magnet segment ${P}_{m}$ as a function of rotor speed and for magnet lengths ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}60$ mm in V-shaped interior PMs (${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}30$ mm) for Model A (blue lines), Model A, but compensated as ${P}_{m}/({\epsilon}_{\mathrm{A}\mid \mathrm{B}}+1)$ (dashed blue line), Model B (red line), Model C (green line) and corresponding 3D-FEM results (circles “∘").

**Figure 8.**Predicted losses per magnet segment ${P}_{m}$ as a function of rotor speed for ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}60$ mm for the straight IPM depicted in Figure 5c (${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}40$ mm) for Model A (blue line), Model B (red line), Model C (green line) and corresponding 3D-FEM results (squares “□”).

**Figure 9.**Resulting eddy current distribution using Model B (red contours) and Model C (green contours) evaluated at ${\omega}_{{\nu}_{m}}t\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\pi /4$ with a magnet length ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}60$ mm at two different speeds: (

**a**) 1000 rpm; (

**b**) 15,000 rpm. Note that the green contours are almost completely covered by the red contours, which indicates that the two models predict essentially the same eddy current distribution.

**Figure 10.**Resulting eddy current distribution from the 3D-FEM model evaluated at ${\omega}_{{\nu}_{m}}t\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\pi /4$ with a magnet length ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}60$ mm at two different speeds: (

**a**) 1000 rpm; (

**b**) 15,000 rpm.

**Figure 11.**$|\mathbf{J}|$ evaluated at 15,000 rpm for Model A (blue lines), Model B (red lines), Model C (green lines) and 3D-FEM model (black lines) at selected points: (

**a**) ‘Pa’; (

**b**) ‘Pb’.

**Figure 12.**Implemented 3D-FEM thermal model with ${p}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.1$ W/cm${}^{3}$ at rated current and 9000 rpm (winding impregnation not shown).

**Figure 13.**Resulting PM average temperature as a function of PM volumetric loss density ${p}_{m}$ at rated current and 9000 rpm. The considered low, medium and high temperature ranges have been indicated using green, orange and red colors, respectively.

**Figure 14.**Rotor geometries (shaft excluded) for: (

**a**) $p\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8$, ${\alpha}_{p}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3/4$, ${h}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5$ mm, ${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}14.2$ mm; (

**b**) $p\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}14$, ${\alpha}_{p}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3/4$, ${h}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5$ mm, ${w}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8.1$ mm.

**Table 1.**${p}_{m}$ (W/cm${}^{3}$) for ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}10$ mm at 9000 rpm.

${\mathit{Q}}_{\mathit{s}}$\p | 8 | 10 | 12 | 14 |
---|---|---|---|---|

6 | 4.0 | 4.7 | N.F. | 4.1 |

9 | N.F. | N.F. | 6.3 | N.F. |

12 | 0.8 | 2.0 | N.F. | 6.2 |

15 | N.F. | 1.0 | N.F. | N.F. |

18 | 0.5 | 0.5 | 1.2 | 4.6 |

21 | N.F | N.F. | N.F. | 1.3 |

24 | ${q}_{s}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$ | 5.6 | N.F. | 7.7 |

27 | N.F. | 0.8 | N.F. | |

30 | ${q}_{s}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$ | N.F. | 0.9 | |

N.F. | Not feasible/unbalanced winding | |||

Distributed windings | ||||

Low losses | ||||

Medium losses | ||||

High losses |

**Table 2.**${p}_{m}$ (W/cm${}^{3}$) for ${l}_{m}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}30$ mm at 9000 rpm.

${\mathit{Q}}_{\mathit{s}}$\p | 8 | 10 | 12 | 14 |
---|---|---|---|---|

6 | 9.8 | 9.8 | N.F. | 6.7 |

9 | N.F. | N.F. | 11.3 | N.F. |

12 | 1.9 | 4.0 | N.F. | 9.1 |

15 | N.F. | 2.0 | N.F. | N.F. |

18 | 1.2 | 1.0 | 2.1 | 7.4 |

21 | N.F | N.F. | N.F. | 2.2 |

24 | ${q}_{s}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$ | 11.5 | N.F. | 12.5 |

27 | N.F. | 1.4 | N.F. | |

30 | ${q}_{s}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$ | N.F. | 1.5 | |

N.F. | Not feasible/unbalanced winding | |||

Distributed windings | ||||

Low losses | ||||

Medium losses | ||||

High losses |

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## Share and Cite

**MDPI and ACS Style**

Zhang, H.; Wallmark, O. Limitations and Constraints of Eddy-Current Loss Models for Interior Permanent-Magnet Motors with Fractional-Slot Concentrated Windings. *Energies* **2017**, *10*, 379.
https://doi.org/10.3390/en10030379

**AMA Style**

Zhang H, Wallmark O. Limitations and Constraints of Eddy-Current Loss Models for Interior Permanent-Magnet Motors with Fractional-Slot Concentrated Windings. *Energies*. 2017; 10(3):379.
https://doi.org/10.3390/en10030379

**Chicago/Turabian Style**

Zhang, Hui, and Oskar Wallmark. 2017. "Limitations and Constraints of Eddy-Current Loss Models for Interior Permanent-Magnet Motors with Fractional-Slot Concentrated Windings" *Energies* 10, no. 3: 379.
https://doi.org/10.3390/en10030379