# An In-Phase Unit Slot-Opening Shift Method for Cogging Torque Reduction in Interior Permanent Magnet Machine

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed IPU-Based Slot-Opening Shift Method

#### 2.1. The Introduction of IPU

#### 2.2. The Proposed IPU Slot-Opening Shift

_{1}. Each unit within a group is to be shifted by a certain angle to cancel out one order of harmonic in the total IPU cogging torque. In the second iteration, each group in the preceding iteration becomes a basic unit for the second round of grouping. Again, the adjacent units are grouped up, and the number of units in each group in the second iteration is factor f

_{2}. Each unit (with corresponding slot openings) within a group is to be shifted by a certain angle to cancel out the next order of harmonic in the total IPU cogging torque. The iterations are performed x times.

_{1}different angles (increment step of θ

_{shift_n}), the other units also have slot openings with the same set of different angles. Therefore, in the second iteration, a group has f

_{2}units, each unit has one slot opening with the same angle as one slot opening in the other units, so that there will be f

_{2}slot openings with the same angle. Thereafter, if the units are shifted to cancel out a new order of harmonic, the shift angle (incremental step) can be similarly obtained with the derivation in (6)–(9), and the same Equation (10) applies for subsequent iterations.

_{shift_n}, starting from the first one in the sequence. However, in each iteration, the number of units in each group (f

_{1}, f

_{2}…or f

_{x}) can be either two, or another prime number. Therefore, there are two different ways of shifting angle assignments along the unit sequence within a group.

_{shift_n}/2.

_{shift_n}, 0, and θ

_{shift_n}, respectively. As for the case with k units per group, the shift angles of the k units are −(k − 1) θ

_{shift_n}/2…, −θ

_{shift_n}, 0, θ

_{shift_n},…(k−1) θ

_{shift_n}/2, respectively. Note that the shift angle of the unit in the middle of the sequence is zero, in order to keep symmetry.

#### 2.3. Screening for the Right Harmonic Orders to Cancel

#### 2.4. The Complete Cogging Torque Reduction Process

#### 2.4.1. Initial Pole-Arc Coefficient Optimization

#### 2.4.2. Final Pole-Arc Coefficient Optimization

## 3. Numerical Analysis Case Studies

#### 3.1. The Cogging Torque Reduction

#### 3.2. Back-EMF Compensation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The no-load magnetic field distribution for IPM machine: (

**a**) the normal slot opening, (

**b**) the IPU slot-opening shift.

**Figure 9.**Back-EMF compensation of the 8P12S machine: (

**a**) waveforms, (

**b**) FFT, and (

**c**) cogging torque after back-EMF compensation.

**Figure 10.**Back-EMF compensation of the 10P12S machine: (

**a**) waveforms, (

**b**) FFT, and (

**c**) cogging torque after back-EMF compensation.

Parameters | 8P12S | 10P12S |
---|---|---|

Outer diameter of stator (mm), ${D}_{1}$ | 269.24 | 269.24 |

Inner diameter of stator (mm), ${D}_{i1}$ | 161.9 | 161.9 |

Outer diameter of rotor (mm), ${D}_{2}$ | 160.4 | 160.4 |

Inner diameter of rotor (mm), ${D}_{i2}$ | 110.64 | 110.64 |

Machine active length (mm), ${L}_{ef}$ | 83.82 | 83.82 |

Rated/peak power (kW) @1200 rpm, $P$ | 33/50 | 33/50 |

Rated/peak torque (Nm) with MTPA, $T$ | 264/400 | 264/400 |

Rated/peak current RMS vaule (A), $I$ | 127/212 | 127/212 |

Maximum speed (r/min), ${n}_{r}$ | 6000 | 6000 |

Air-gap length (mm), $\delta $ | 0.75 | 0.75 |

Magnet width (mm), ${\omega}_{m}$ | 19 | 14 |

Magnet thickness (mm), ${h}_{m}$ | 6.48 | 6.48 |

Pole-arc coefficient, ${\alpha}_{p}$ | 0.67 | 0.58 |

Slot-opening coefficient, ${\alpha}_{s}$ | 0.19 | 0.09 |

Magnet remanence (T), ${B}_{r}$ | 1.191 | 1.191 |

Magnet relative permeability, ${\mu}_{r}$ | 1.03 | 1.03 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Lu, S.; Chen, Y.; Wang, S.
An In-Phase Unit Slot-Opening Shift Method for Cogging Torque Reduction in Interior Permanent Magnet Machine. *Mathematics* **2023**, *11*, 1735.
https://doi.org/10.3390/math11071735

**AMA Style**

Wang L, Lu S, Chen Y, Wang S.
An In-Phase Unit Slot-Opening Shift Method for Cogging Torque Reduction in Interior Permanent Magnet Machine. *Mathematics*. 2023; 11(7):1735.
https://doi.org/10.3390/math11071735

**Chicago/Turabian Style**

Wang, Linwei, Shuai Lu, Yangming Chen, and Shiya Wang.
2023. "An In-Phase Unit Slot-Opening Shift Method for Cogging Torque Reduction in Interior Permanent Magnet Machine" *Mathematics* 11, no. 7: 1735.
https://doi.org/10.3390/math11071735