# Design for Reliability: The Case of Fractional-Slot Surface Permanent-Magnet Machines

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## Abstract

**:**

## 1. Introduction

## 2. The Slot–Pole Combination

## 3. Machine Optimization

## 4. Performance of the Healthy Combinations

## 5. Analysis of Robustness to Manufacturing Defects

#### 5.1. Eccentricity

#### 5.2. Permanent-Magnet Defects

#### 5.3. Machine Performances vs. Severity of Manufacturing Defects

## 6. Design for Reliability: Robustness Analysis towards Manufacturing Defects as a Function of Slot–Pole Combinations

## 7. Conclusions

- Static and dynamic eccentricities induce new components in the spectra of mechanical quantities. Static eccentricity induces a frequency component proportional to the number of poles $2p$, while dynamic eccentricity induce a frequency component proportional to the number of stator teeth Q.
- Radial force is the most sensitive performance benchmark to manufacturing defects. Specifically, FEA results show how the radial force is deeply affected both by mechanical and by magnetic defects.
- The 12-10DL and 12-14DL were the best performing machines in both healthy and faulty conditions and they showed the lowest difference in terms of TPI.
- According to the $\Delta $ TPI, the 9-8DL was the slot–pole combination mostly affected by manufacturing imperfections.
- Finally, the slot–pole combination with lowest performance at healthy conditions (6-4DL) showed the best improvement in term of performance at defective condition.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

SPM | Surface Permanent-Magnet |

FEA | Finite-Element Analysis |

MMF | Magneto-Motive Force |

B-EMF | Back-Electro-Motive Force |

HCF | Highest Common Factor |

FFT | Fast Fourier Transformation |

THD | Total Harmonic Distortion |

DL | Double Layer |

SL | Single Layer |

TPI | Total Performance Index |

## References

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**Figure 1.**Magneto-Motive Flux (MMF) spectra for different double layer slot–pole combinations: (

**a**) MMF of 6-4DL; (

**b**) MMF of 9-8DL; (

**c**) MMF of 9-10DL; (

**d**) MMF of 12-10DL; and (

**e**) MMF of 12-14.

**Figure 2.**Distribution of magnetic flux density of optimized machines: (

**a**) 6-4; (

**b**) 9-8; (

**c**) 9-10; (

**d**) 12-10; and (

**e**) 12-14.

**Figure 3.**Examples of a machine: without eccentricity (

**a**); static eccentricity (

**b**); and dynamic eccentricity (

**c**).

**Figure 4.**FFT of the radial forces: 9-8DL (

**a**); 6-4DL and 12-14DL slot-pole combination without static eccentricity (

**b**); 6-4DL (

**c**); 9-8DL (

**d**); 12-14DL slot-pole combinations with static eccentricity of 0.2 mm (

**e**).

**Figure 5.**Cogging torque comparison over a B-EMF period of 9-8DL (

**a**) and 9-10DL (

**b**) with a static eccentricity of $0.2$ mm.

**Figure 6.**FFT of the mean torque of 9-8DL and 9-10DL without and with dynamic eccentricity of $0.2$ mm. (

**a**) 9-8DL without defect; (

**b**) 9-8DL with dynamic eccentricity; (

**c**) 9-10DL without imperfection; and (

**d**) 9-10DL with imperfection.

**Figure 9.**Cogging torque FFT of 12-14DL: in healthy condition (

**a**); with magnets dislocation (

**b**); and with radial force FFT with magnet dislocation (

**c**).

**Figure 10.**Correlation between severity of magnetic defects and machine performances of slot–pole combination 12-14DL: (

**a**) trend of torque ripple; (

**b**) trend of cogging torque; and (

**c**) trend of radial force for increasing values of magnetic defects.

**Figure 11.**Correlation between severity of mechanical defects and machine performances of slot–pole combinations 9-8DL and 12-14DL: (

**a**) trend of torque ripple; (

**b**) trend of cogging torque; and (

**c**) trend of radial force for increasing values of mechanical defects.

**Figure 12.**Kiviat diagrams of healthy machines for different slot–pole combinations (

**a**) and of faulty machines (

**b**); and slot–pole legend (

**c**).

Benchmarks | Unit |
---|---|

Mean Torque | Nm |

Torque Ripple | % |

Cogging Torque | Nm |

Radial Force | N |

THD of B-EMF | % |

Slot–Pole Combinations | |||||||
---|---|---|---|---|---|---|---|

6-4 | 9-8 | 9-10 | |||||

Parameters | Symbol | SL | DL | SL | DL | SL | DL |

Poles | $2p$ | 4 | \ | 8 | \ | 10 | |

Stator Slots | Q | 6 | \ | 9 | \ | 9 | |

Highest Common Factor | $HCF$ | 2 | \ | 1 | \ | ||

Cogging Periods vs. Slot Pitch Rotation | $Np$ | 2 | \ | 8 | \ | 10 | |

Slot per Pole per Phase | $SPP$ | 0.5 | \ | 0.38 | \ | 0.3 | |

Winding Factor | ${k}_{w}$ | 0.866 | \ | 0.945 | \ | 0.945 | |

MMF Total Harmonic Distortion related to Torque Ripple | $TH{D}_{MMFripple}$ | 0.302 | 0.302 | \ | 0.101 | \ | 0.102 |

MMF Total Harmonic Distortion | $TH{D}_{MMFtot}$ | 0.745 | 0.66 | \ | 0.977 | \ | 0.702 |

Slot–Pole Combinations | |||||
---|---|---|---|---|---|

12-10 | 12-14 | ||||

Parameters | Symbol | SL | DL | SL | DL |

Poles | $2p$ | 10 | 14 | ||

Stator Slots | Q | 12 | 12 | ||

Highest Common Factor | $HCF$ | 2 | 2 | ||

Cogging Periods vs. Slot Pitch Rotation | $Np$ | 5 | 7 | ||

Slot per Pole per Phase | $SPP$ | 0.4 | 0.3 | ||

Winding Factor | ${k}_{w}$ | 0.966 | 0.933 | 0.966 | 0.966 |

MMF Total Harmonic Distortion related to Torque Ripple | $TH{D}_{MMFripple}$ | 0.160 | 0.145 | 0.162 | 0.146 |

MMF Total Harmonic Distortion | $TH{D}_{MMFtot}$ | 0.887 | 0.867 | 0.736 | 0.689 |

Independent Variables | Symbol |
---|---|

Inner Stator Diameter | ${D}_{i}$ |

Slot Opening at Pole Pitch | ${w}_{so}$ |

Pole Pitch Thickness at Slot Opening | ${h}_{so}$ |

Angular Distance between two magnets | ${w}_{im}$ |

Slot-Pole Combinations | ||||||
---|---|---|---|---|---|---|

Parameter | Symbol | 6-4 | 9-8 | 9-10 | 12-10 | 12-14 |

Stator ExternalDiameter [mm] | ${D}_{e}$ | 92 | 92 | 92 | 92 | 92 |

Stator InternalDiameter [mm] | ${D}_{i}$ | 40 | 46 | 48 | 50 | 47 |

Air-gap thickness [mm] | g | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 |

Tooth Width [mm] | ${w}_{t}$ | 10 | 6.5 | 5 | 5 | 4 |

Stator Ring Height [mm] | ${h}_{bi}$ | 5.5 | 4.5 | 4.5 | 4.5 | 4.5 |

Polar ShoeThickness [mm] | ${h}_{wed}$ | 3.5 | 2.5 | 2.5 | 2 | 2 |

Polar ShoeSlot Opening [mm] | ${h}_{so}$ | 1 | 1 | 1 | 1 | 1 |

Slot Opening Width [mm] | ${w}_{so}$ | 3 | 4 | 4 | 4 | 4 |

Magnet Height [mm] | ${h}_{m}$ | 3 | 3 | 3 | 3 | 3 |

Angular Distancebetween Magnets [deg] | ${w}_{im}$ | 17 | 3 | 3 | 4 | 3 |

Motor Lenght [mm] | ${L}_{stk}$ | 100 | 100 | 100 | 100 | 100 |

Force and Torque | Mass of Materials | |||||||
---|---|---|---|---|---|---|---|---|

Slot–Pole Combinations | Mean Torque [Nm] | Torque Ripple [Nm] | Torque Ripple [%] | Radial Force [N] | Cogging Torque [Nm] | Weight Iron Core [kg] | Weight Copper [kg] | Weight Magnets [kg] |

6-4 | 6.3909 | 1.0152 | 15.8857 | 0 | 0.884 | 2.1745 | 1.1951 | 0.2069 |

9-8 | 8.3917 | 0.3507 | 4.1794 | 347 | 0.048 | 1.8492 | 1.114 | 0.2782 |

9-10 | 8.5726 | 0.4717 | 5.5019 | 35 | 0.0718 | 1.6473 | 1.1568 | 0.2864 |

12-10 | 7.918 | 0.3132 | 3.9562 | 0 | 0.1989 | 1.767 | 0.9801 | 0.2904 |

12-14 | 8.5541 | 0.2685 | 3.139 | 0 | 0.0914 | 1.6737 | 1.1131 | 0.2696 |

Electrical Coefficients | |||
---|---|---|---|

Slot–Pole Combinations | Winding Factor ${\mathit{k}}_{\mathit{w}}$ | THD B-EMF [%] | MMF f1/ Torque Mean |

6-4 | 0.866025 | 4.4483 | 0.129387097 |

9-8 | 0.945214 | 5.6768 | 0.08065112 |

9-10 | 0.945214 | 7.7553 | 0.063143037 |

12-10 | 0.933013 | 2.2574 | 0.089984845 |

12-14 | 0.933013 | 5.6987 | 0.059468559 |

Force and Torque | Mass of Materials | |||||||
---|---|---|---|---|---|---|---|---|

Slot–Pole Combinations | Mean Torque [Nm] | Torque Ripple [Nm] | Torque Ripple [%] | Radial Force [N] | Cogging Torque [Nm] | Weight Iron Core [kg] | Weight Copper [kg] | Weight Magnets [kg] |

6-4 | 6.0022 | 1.3717 | 22.8529 | 187 | 0.884 | 2.1745 | 1.3468 | 0.2069 |

12-10 | 7.5987 | 1.1069 | 14.5676 | 0 | 0.1989 | 1.767 | 1.0601 | 0.2904 |

12-14 | 8.723 | 0.4348 | 4.9843 | 0 | 0.0971 | 1.6737 | 1.2165 | 0.2696 |

Electrical Coefficients | |||
---|---|---|---|

Slot–Pole Combinations | Winding Factor ${\mathit{k}}_{\mathit{w}}$ | THD B-EMF [%] | MMF f1 /Torque Mean |

6-4 | 0.866025 | 4.4618 | 0.068891406 |

12-10 | 0.965926 | 6.4866 | 0.048534618 |

12-14 | 0.965926 | 8.0146 | 0.03018457 |

PM Imperfections | Mean Value | Standard Deviation |
---|---|---|

Demagnetization | 883310$\frac{A}{m}$ | 0.025 |

Magnetic Axis Deviation | 0 | 0.6 |

Dislocation | 0 | 0.2 |

Defect | Value |
---|---|

Standard Deviation of Magnetic Axis Deviation | $\pm 1$ deg |

Standard Deviation of Coercive Field | $\pm 5$% |

Permanent-Magnets Dislocation | 0.25 mm |

Static Eccentricity | 0.2 mm |

Dynamic Eccentricity | 0.2 mm |

Healthy TPI | Defective TPI | $\Delta $ TPI | |
---|---|---|---|

6-4 DL | −4.38 | −2.56 | −1.82 |

9-8 DL | −0.53 | −1.94 | 1.41 |

9-10 DL | 0.37 | −0.01 | 0.38 |

12-10 DL | 2.55 | 2.53 | 0.02 |

12-14 DL | 1.99 | 1.98 | 0.01 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Torreggiani, A.; Bianchini, C.; Davoli, M.; Bellini, A. Design for Reliability: The Case of Fractional-Slot Surface Permanent-Magnet Machines. *Energies* **2019**, *12*, 1691.
https://doi.org/10.3390/en12091691

**AMA Style**

Torreggiani A, Bianchini C, Davoli M, Bellini A. Design for Reliability: The Case of Fractional-Slot Surface Permanent-Magnet Machines. *Energies*. 2019; 12(9):1691.
https://doi.org/10.3390/en12091691

**Chicago/Turabian Style**

Torreggiani, Ambra, Claudio Bianchini, Matteo Davoli, and Alberto Bellini. 2019. "Design for Reliability: The Case of Fractional-Slot Surface Permanent-Magnet Machines" *Energies* 12, no. 9: 1691.
https://doi.org/10.3390/en12091691