# Combined Electromagnetic and Mechanical Design Optimization of Interior Permanent Magnet Rotors for Electric Vehicle Drivetrains

^{*}

*World Electr. Veh. J.*

**2024**,

*15*(1), 4; https://doi.org/10.3390/wevj15010004 (registering DOI)

## Abstract

**:**

## 1. Introduction

## 2. Design Study Baseline Machine Dimensions and Performance

## 3. Material Properties and Mechanical Properties

#### 3.1. Magnetic Material Properties

#### 3.2. Influence of Interference Fitting of Shaft

^{3}), the resulting radial growth at 12,000 rpm is a mere 0.02 µm. The difference in radial growth between the shaft and the bore of the rotor core in effect reduces the net interference at the maximum speed from the interference set at standstill. The interference at standstill must be sufficient to accommodate the differential radial growth of the rotor core while still maintaining sufficient inward radial pressure on the shaft to transmit the rated torque with an appropriate safety margin. A well-established method for calculating the contact pressure and resulting hoop stresses between interfering concentric cylinders is given in [10]. For the case of a solid shaft, the interference, ${\delta}_{i}$, required to achieve a contact pressure between the shaft and the rotor core of ${P}_{i}$ is given from [10] by

#### 3.3. Modeling

_{1}values from 0.4 mm to 2.5 mm in 0.1 mm steps and h

_{2}values from 0.8 mm to 2 mm in 0.2 mm steps. An example of the resulting von Mises stress distribution in the vicinity of the rotor magnets is shown in Figure 3; this case is a 135 mm diameter rotor at a maximum speed of 12,000 rpm with h

_{1}= 2 mm and h

_{2}= 1 mm. Although much of the rotor operates at stress levels below 50 MPa, as would be expected, there are regions of stress concentration at the outer tips of the magnets and in the gap between the innermost regions of the magnets. In this case, the peak localized stress is 230 MPa, which is just within the design stress limit of 240 MPa set for the NO20 rotor material.

_{1}and h

_{2}for the 154 designs considered is shown in Figure 4. In this case, 40 combinations of h

_{1}and h

_{2}result in localized stress levels greater than the 240 MPa design stress limit set for this study.

_{1}and h

_{2}that yield viable designs from a mechanical stress threshold perspective, the electromagnetic torque is calculated using a two-dimensional, magneto-static, non-linear finite element analysis for the specific case of a stator rms current density of 10 A/mm

^{2}(at an assumed slot fill factor of 0.45) with a current advance angle of 45° (electrical).

_{1}(0.4 mm) and h

_{2}(0.8 mm), although this results in a peak localized stress of 290 MPa. The highest predicted torque per unit length for a design that also results in a von Mises stress that falls within the design limit of 240 MPa is 1468 Nm/m, which is achieved for an h

_{1}value of 0.5 mm and an h

_{2}value of 1.4 mm. Scaling this torque per unit length to meet the torque specification set out in Table 1 yields an axial length of 162 mm for this rotor diameter of 135 mm, in turn yielding a predicted mass for this design, including an estimate of end-winding mass of 50.6 kg, which corresponds to ~2 kW/kg.

^{2}(at an assumed slot fill factor of 0.45).

_{1}and h

_{2}to maintain the localized stress within the rotor below the 240 MPa limit. Thickening up the rotor core beyond the end of the magnets tends to promote increased leakage flux within the rotor. It is apparent from the results presented in Table 4 that there is an optimum rotor diameter of 165 mm, albeit that this is specific to this combination of rotational speed, rotor core maximum stress limit, machine split ratio and current density constraints. Without mechanical stress considerations, the torque density tends to continually increase with increasing rotor diameter because of the nature of scaling with the diameter of the electric loading (Ampere turns per unit of airgap periphery). There is a further tendency for the predicted torque density to increase with diameter due to the limitations of two-dimensional finite element modeling. A two-dimensional finite element electromagnetic analysis does not account for the influence of end effects in short-axial-length machines and, hence, tends to overestimate the torque produced by machines with short axial lengths relative to diameters. The presence of an optimum in Table 4 is a consequence of the electromagnetic penalty, which is increasingly incurred with the need to thicken up the regions of the rotor core adjacent to the airgap to ensure that the entire rotor core remains with the specified design mechanical stress.

## 4. Influence of Hub Diameter

_{1}and h

_{2}in Table 4 as a function of the hub diameter. As is apparent, there is the expected increase in the peak stress as the hub diameter is increased. Since the original design with a shaft diameter of 30.5 mm is just under the limiting design stress of 240 MPa, it is to be expected that almost all designs with a larger hub diameter will exceed the stress limit. The calculated stress in Figure 10 is shown at four different shaft-to-rotor-core interference levels, from the original 25 µm up to 60 µm. Since the radial growth of the core increases as the hub diameter is increased, there is an upper limit on the hub diameter for a given interference, beyond which the rotor core separates from the hub; e.g., for the original 25 µm, the rotor core lifts of the hub at 12,000 rpm for shaft diameters greater than ~58 mm. Even if the stress could be accommodated for in the 165 mm diameter rotor, it would be necessary to increase the interference to at least 35 µm. This additional interference would further exacerbate the stress levels in the rotor core.

## 5. Influence of Magnet Shape

_{1}and h

_{2}in Table 4 was adopted. The mechanical properties of the non-magnetic filler used in the rotor in Figure 13b will have some influence on the overall stress distribution within the rotor core. To cover the likely range of physical properties exhibited by various loaded resins and putties, a series of simulations were performed assuming that the non-magnetic filler was assigned inclusive combinations of elastic moduli between 2 GPa and 8 GPa and densities between 1200 and 1800 kg/m

^{3}. The resulting range of stresses for these different combination of non-magnetic filler properties was rather narrow, with, as expected, the marginally lowest stress of 245 MPa encountered with the lowest density and highest modulus filler and the highest stress of 261 MPa for the highest density and lowest filler.

_{2}in Figure 2. The additional leakage flux through this wider region more than offsets the reduced flux leakage through the outer bridge regions, thus reducing the overall torque.

_{2}in Figure 2, is maintained at its original value of 4.6 mm, and the magnet is simply extended into the region near the outer surface rather than displacing the fixed magnet size. This extension to the rectangular magnet block increases its mass, and, hence, it would be expected that the amount of extension that can be applied before reaching the 240 MPa design stress limit will be less than the displacement of 1.06 mm established previously with the fixed magnet size. For the same stress limit of 240 MPa, the magnet can be extended in the direction shown by 0.53 mm. Figure 15 and Figure 16 show the finite element predicted flux density distribution and von Mises stress distribution at 12,000 rpm for this case of an extended rectangular magnet block, respectively. This extension of the magnet block results in an electromagnetic torque of 218 Nm, which is still 21 Nm short of the 239 Nm achieved by the profiled magnets in Figure 1. Hence, the intricate detailing around the edge of the profiled magnets in Figure 1 results in a ~10% higher torque on a like-for-like basis compared to a plain rectangular block. Bringing the design with the rectangular blocks up to the same torque rating could be achieved with a 10% increase in the stack length and, hence, the mass of the active materials. Further detailed cost modeling would be required to establish whether, in volume production, the intricately profiled magnet could be manufactured within the cost saving margin associated with the 10% reduction in the volume of the active material required.

## 6. Conclusions

^{2}rms. The selection of a higher-strength electrical steel for the rotor would improve the torque density due to the ability to accommodate thinner sections within the rotor geometry and, hence, reduce flux leakage within the IPM rotor. However, this might require the use of different electrical steels in the rotor and stator since increased mechanical properties are usually obtained at the expense of reduced magnetic performance, particularly core loss. The 10 A/mm

^{2}rated current density employed throughout this study is representative of many higher-performance machines that employ some level of in-direct cooling of the winding, e.g., liquid-cooled casings or high-performance air cooling. The adoption of higher current densities that are representative of say direct liquid-cooled coils is likely to have a limited influence on the optimal rotor diameter, as it will only affect the armature reaction field contribution to the flux in the bridges within the rotor IPM structure. Although the optimal rotor diameter and resulting torque density would change with the use of higher-strength electrical steels in the rotor and higher current densities in the stator winding, it is likely to be the case that there will be an optimal IPM rotor diameter for a given set of constraints and that establishing this optimum requires consideration of both electromagnetic and mechanical aspects of behavior.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bianchi, N.; Bolognani, S.; Luise, F. Potentials and limits of high-speed PM motors. IEEE Trans. Ind. Appl.
**2004**, 40, 1570–1578. [Google Scholar] [CrossRef] - Yu, A.; Jewell, G.W. Systematic design study into the influence of rotational speed on the torque density of surface-mounted permanent magnet machines. J. Eng.
**2019**, 2019, 4595–4600. [Google Scholar] [CrossRef] - Binder, A.; Schneider, T. High-speed inverter-fed AC drives. In Proceedings of the 2007 International Aegean Conference on Electrical Machines and Power Electronics, Bodrum, Turkey, 10–12 September 2007; pp. 9–16. [Google Scholar] [CrossRef]
- Lovelace, E.C.; Jahns, T.M.; Keim, T.A.; Lang, J.H. Mechanical design considerations for conventionally laminated, high-speed, interior PM synchronous machine rotors. IEEE Trans. Ind. Appl.
**2004**, 40, 806–812. [Google Scholar] [CrossRef] - Binder, A.; Schneider, T.; Klohr, M. Fixation of buried and surface-mounted magnets in high-speed permanent-magnet synchronous machines. IEEE Trans. Ind. Appl.
**2006**, 42, 1031–1037. [Google Scholar] [CrossRef] - Wang, A.; Jia, Y.; Soong, W.L. Comparison of Five Topologies for an Interior Permanent-Magnet Machine for a Hybrid Electric Vehicle. IEEE Trans. Magn.
**2011**, 47, 3606–3609. [Google Scholar] [CrossRef] - Han, Z.; Yang, H.; Chen, Y. Investigation of the rotor mechanical stresses of various interior permanent magnet motors. In Proceedings of the 12th International Conference on Electrical Machines and Systems, ICEMS 2009, Tokyo, Japan, 15–18 November 2009. [Google Scholar] [CrossRef]
- Ma, J.; Zhu, Z.Q. Optimal split ratio in small high speed PM machines considering both stator and rotor loss limitations. CES Trans. Electr. Mach. Syst.
**2019**, 3, 3–11. [Google Scholar] [CrossRef] - Leuning, N.; Schauerte, B.; Hameyer, K. Interrelation of mechanical properties and magneto-mechanical coupling of non-oriented electrical steel. J. Magn. Magn. Mater.
**2023**, 567, 170322. [Google Scholar] [CrossRef] - Qiu, J.; Zhou, M. Analytical solution for interference fit for multi-layer thick-walled cylinders and the application in crankshaft bearing design. Appl. Sci.
**2016**, 6, 167. [Google Scholar] [CrossRef] - Budynas, R.; Nisbett, J.K. Shigley’s Mechanical Engineering Design, 10th ed.; McGraw-Hill Education: New York, NY, USA, 2014; ISBN 9813151005. [Google Scholar]

**Figure 3.**IPM von Mises stress distribution for a 135 mm diameter rotor at 12,000 rpm (h

_{1}= 2 mm; h

_{2}= 1 mm).

**Figure 4.**Variation in peak localized stress in a series of 135 mm outer diameter rotor designs at 12,000 rpm (core-to-shaft diametrical interference is 25 µm in each case).

**Figure 5.**Variation in electromagnetic torque in a 135 mm diameter rotor for various combinations of h

_{1}and h

_{2}.

**Figure 6.**Predicted flux density distribution for the 165 mm design in Table 4.

**Figure 7.**Predicted von Mises stress distribution for the 165 mm design in Table 4.

**Figure 8.**Predicted torque for a 165 mm diameter rotor as a function of rotor core inner diameter at a current density of 10 A/mm

^{2}rms (slot fill factor of 0.45).

**Figure 9.**Finite element predicted flux density distribution in a 165 mm diameter rotor with a shaft diameter of 120 mm at full load torque of 253 Nm.

**Figure 10.**Variation in the finite element predicted peak localized stress in the 165 mm diameter rotor core at 12,000 rpm as a function of hub diameter for different diametrical interference fits from 25 µm to 60 µm (240 MPa design stress limit shown as dashed line).

**Figure 11.**Variation in the maximum allowable speed for a 165 mm diameter rotor core to remain within a 240 MPa design stress limit for four levels of interference.

**Figure 12.**Close-up of finite element predicted peak stress distribution in the 165 mm diameter rotor core at 12,000 rpm for a hub diameter of 78 mm.

**Figure 13.**Alternative arrangement of magnet poles. (

**a**) Void from profiled end incorporated into rotor core. (

**b**) Void from profiled end filled with non-magnetic filler.

**Figure 15.**Predicted flux density distribution for the 165 mm rotor design with an extended rectangular magnet block.

**Figure 16.**Predicted von Mises stress distribution for the 165 mm rotor design with an extended rectangular magnet block.

Feature | Value |
---|---|

Rated power at base speed | 100 kW |

Base speed | 4000 rpm |

Maximum speed | 12,000 rpm |

Rated torque at base speed | 239 Nm |

Feature | Value |
---|---|

Rotor outer diameter | 1.000 |

Stator outer diameter | 1.556 |

Magnet thickness (in direction of magnetization) | 0.0296 |

Stator tooth body width | 0.0504 |

Shaft diameter | 0.185 |

Feature | NO20-1200H | NdFeB |
---|---|---|

Strength | 400 MPa (0.2% Yield) | 80 MPa (UTS) |

Young’s modulus | 205 GPa | 160 GPa |

Poisson’s ratio | 0.3 | 0.3 |

Density | 7650 kg/m^{3} | 7500 kg/m^{3} |

D120 | D135 | D150 | D165 | D180 | |
---|---|---|---|---|---|

h_{1} (mm) | 0.2 | 0.5 | 0.7 | 1.2 | 3.2 |

h_{2} (mm) | 0.7 | 1.4 | 2.9 | 4.6 | 6.9 |

Core axial length (mm) | 224 | 162 | 125 | 103 | 99 |

Rotor volume (dm^{3}) | 2.87 | 2.32 | 2.21 | 2.20 | 2.52 |

Machine mass (kg) | 60.5 | 50.6 | 44.9 | 42.3 | 46.3 |

Torque density (Nm/kg) | 3.93 | 4.70 | 5.30 | 5.63 | 5.14 |

Power density (kW/kg) at 4000 rpm | 1.65 | 1.97 | 2.22 | 2.36 | 2.15 |

**Table 5.**Comparison of predicted electromagnetic torque and maximum localized stress for different magnet profiles.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, G.; Jewell, G.W.
Combined Electromagnetic and Mechanical Design Optimization of Interior Permanent Magnet Rotors for Electric Vehicle Drivetrains. *World Electr. Veh. J.* **2024**, *15*, 4.
https://doi.org/10.3390/wevj15010004

**AMA Style**

Zhang G, Jewell GW.
Combined Electromagnetic and Mechanical Design Optimization of Interior Permanent Magnet Rotors for Electric Vehicle Drivetrains. *World Electric Vehicle Journal*. 2024; 15(1):4.
https://doi.org/10.3390/wevj15010004

**Chicago/Turabian Style**

Zhang, Guanhua, and Geraint Wyn Jewell.
2024. "Combined Electromagnetic and Mechanical Design Optimization of Interior Permanent Magnet Rotors for Electric Vehicle Drivetrains" *World Electric Vehicle Journal* 15, no. 1: 4.
https://doi.org/10.3390/wevj15010004