# A Methodology for Applying Skew in an Automotive Interior Permanent Magnet Rotor for Robust Electromagnetic and Noise, Vibration and Harshness Performance

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## Abstract

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## 1. Introduction

- Pole–slot number variation is described in [2], where with the help of the Maxwell stress tensor and equivalent magnetizing current method, the authors study two different motors with similar performances but different pole–slot combinations.
- Two-dimensional geometry optimization is applied with the aim of reducing the mechanical sources of torque ripple and electromagnetic forces. This methodology, outlined in [3], also manipulates structural resonances to augment the overall stiffness characteristics.
- Utilization of asymmetric slot design allows for non-symmetrical magnetic reluctance of the rotor around the d-axis. As illustrated in reference [4], this approach can be advantageous under certain working conditions but poses manufacturing challenges for automotive applications.
- Skewing of the rotor or stator is a widely adopted method to minimize NVH. However, determining the optimal skew parameters is challenging and may result in suboptimal designs. The authors of [7] analyze rotor skewing for short-length and highly saturated machines, emphasizing the importance of the fringe effect and challenging the validity of 2D multi-slice modeling. The authors of [8] investigate the effects of skewing for different magnet shapes and pole–slot combinations yet neglect 3D effects such as axial force and its relation to the order of rotor stacks also referred to as slices. Reference [9] focuses on structural analysis coupled with electromagnetic FEA to enhance excitation force harmonics, aiming to reduce motor core deformations that lead to vibration and noise.

## 2. Analytical Modeling of Torque Ripple, Cogging Torque and Axial Forces

#### 2.1. Torque Ripple

#### 2.2. Cogging Torque

_{s}is the motor stack length, ${r}_{stat}$ and ${r}_{rot}$ are the radii of the stator and rotor, and ${G}_{n{N}_{cog}}$ and ${B}_{n{N}_{cog}}$ are the Fourier coefficients of the airgap permeance function and flux density. A method for calculating the Fourier coefficients analytically was proposed by [14] and given as

#### 2.3. Axial Force

## 3. Two-Dimensional FEA Modeling

#### 3.1. Torque Ripple Excitations without Skew

#### 3.2. Impact of Stator Skew

#### 3.3. Impact of Rotor Skew

#### 3.4. Impact of Skew on Back-EMF Harmonics

## 4. Three-Dimensional Finite Element Modeling of Axial Forces in IPM Rotors

## 5. NVH Structural Analysis

## 6. Impact of Skew on Rotor Manufacture

## 7. NVH Optimization Workflow

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Skewing examples: (

**a**) stator skewing (continuous); (

**b**) IPM rotor with partial V-skew (discrete).

**Figure 2.**Magnets from two consecutive axial segments: (

**a**) non-skewed consecutive stacks (orange); (

**b**) discrete skewing (magnets from stack 1 in orange and stack 2 in red); (

**c**) continuous skewing in yellow overlapping case (

**b**) for analytical approximation of axial force.

**Figure 5.**(

**a**) Total peak-to-peak torque ripple magnitude; (

**b**) peak-to-peak torque ripple magnitudes of the 36th harmonic; (

**c**) peak-to-peak torque ripple magnitudes of the 72nd harmonic for a motor without skew.

**Figure 8.**Impact of skew angles on the normalized torque ripple and peak torque using stator skew at different rotor speeds: (

**a**) 150 A; (

**b**) 350 A.

**Figure 9.**Impact of skew angles on the total torque ripple of the motor using a 4-step rotor skew: (

**a**) 150 A; (

**b**) 350 A.

**Figure 10.**Impact of skew angles on the 36th and 72nd harmonics of torque ripple of the motor using a 4-step rotor skew for 350 A: (

**a**) 1000 rpm; (

**b**) 5000 rpm; (

**c**) 8000 rpm.

**Figure 11.**(

**a**) A 3D Ansys Maxwell model of a single pole section showing skewing pattern. (

**b**) Mesh elements used for axial force calculation.

**Figure 12.**Stepwise skew arrangement for 2, 3 and 4 steps with linear or V-skew: (

**a**) 2-slice linear skew; (

**b**) 2-slice V-skew; (

**c**) 3-slice linear skew; (

**d**) 3-slice V-skew; (

**e**) 4-slice linear skew; (

**f**) 4-slice V-skew.

**Figure 13.**Partial V-skew options for 3 and 4 steps: (

**a**) 3-slice partial V-skew; (

**b**) 4-slice partial V-skew option 1; (

**c**) 4-slice partial V-skew option 2; (

**d**) 4-slice partial V-skew option 3.

**Figure 15.**Axial forces in forward vs. reverse direction at different speeds using linear and partial V-skew: (

**a**) 350 A; (

**b**) 150 A.

**Figure 16.**Average normal surface normal mean squared velocity with and without skew: (

**a**) 36th-order excitation; (

**b**) 72nd-order excitation.

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

Number of slots—Q | 72 | - |

Number of poles—2p | 12 | - |

Motor torque | 400 | Nm |

Motor maximum speed | 8700 | rpm |

Maximum Voltage | 400 | V |

Maximum Power | 155 | kW |

Maximum Current | 400 | A |

Order | Harmonic | Excitation |
---|---|---|

1 | 36 | Torque ripple |

2 | 72 | Cogging + Torque ripple |

3 | 108 | Torque ripple |

4 | 144 | Cogging + Torque ripple |

Skew Method | Harmonic Distortion Back-EMF Line-Line Voltage (%) |
---|---|

Non-skewed | 9.7 |

5° rotor skew (2 slices) | 4.5 |

5° rotor skew (4 slices) | 4.1 |

5° stator skew | 4.0 |

10° stator skew | 1.3 |

Total Axial Force (N) | ||||
---|---|---|---|---|

Linear Skew | Partial V-Skew (Opt. 1) | Partial V-Skew (Opt. 2) | Partial V-Skew (Opt. 3) | |

Analytical Approximation | −29.9 | −19.9 | −9.9 | 10.0 |

3D FEA Simulation | −31.9 | −19.7 | −9.1 | 8.6 |

Error | 6.4% | −1.0% | −9.1% | −15.7% |

**Table 5.**Number of rotor stacks and its impact on active magnet material and manufacturing complexity.

Number of Stacks | Magnet Insertions | Equivalent Axial Length Reduction |
---|---|---|

( ) | ( ) | (%) |

2 | 48 | 0.19 |

3 | 72 | 0.29 |

4 | 96 | 0.39 |

6 | 144 | 0.58 |

8 | 192 | 0.77 |

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## Share and Cite

**MDPI and ACS Style**

Cawkwell, T.; Haris, A.; Gonzalez, J.M.; Rodrigues, L.K.; Shirokov, V.
A Methodology for Applying Skew in an Automotive Interior Permanent Magnet Rotor for Robust Electromagnetic and Noise, Vibration and Harshness Performance. *World Electr. Veh. J.* **2023**, *14*, 350.
https://doi.org/10.3390/wevj14120350

**AMA Style**

Cawkwell T, Haris A, Gonzalez JM, Rodrigues LK, Shirokov V.
A Methodology for Applying Skew in an Automotive Interior Permanent Magnet Rotor for Robust Electromagnetic and Noise, Vibration and Harshness Performance. *World Electric Vehicle Journal*. 2023; 14(12):350.
https://doi.org/10.3390/wevj14120350

**Chicago/Turabian Style**

Cawkwell, Thomas, Ahmed Haris, Juan Manuel Gonzalez, Leon Kevin Rodrigues, and Vladimir Shirokov.
2023. "A Methodology for Applying Skew in an Automotive Interior Permanent Magnet Rotor for Robust Electromagnetic and Noise, Vibration and Harshness Performance" *World Electric Vehicle Journal* 14, no. 12: 350.
https://doi.org/10.3390/wevj14120350