# Investigation of the Magnetic Circuit and Performance of Less-Rare-Earth Interior Permanent-Magnet Synchronous Machines Used for Electric Vehicles

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## Abstract

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## 1. Introduction

## 2. No-Load Magnetic Circuit Analysis

_{EM}) of LRE-IPMSMs consists of reluctance torque (T

_{REL}) and PM torque (T

_{PM}). T

_{REL}is generated by the rotor saliency. T

_{PM}is generated by the interaction of magnetic fields produced by the stator current and PMs. T

_{EM}can be expressed as:

_{f}is the no-load flux linkage; Ψ

_{q}and Ψ

_{d}are the q-axis and d-axis flux linkages, respectively; and i

_{q}and i

_{d}are the q-axis and d-axis currents, respectively.

_{REL}accounts for most of the T

_{EM}in LRE-IPMSMs, the q-axis and d-axis in LRE-IPMSMs are defined to be the same as the reluctance machine, namely, the q-axis is aligned with the center of PMs and the d-axis is orthogonal with the q-axis in electrical degree. The basic parameters of the preliminary LRE-IPMSM are shown in Table 1.

#### 2.1. No-Load Magnetic Circuit Model

_{sa}) and a serial-connected reluctance (R

_{sa}) [14]. F

_{sa}and R

_{sa}are expressed as:

_{0}is the permeability of air and L is the lamination stack length. b

_{sa}and h

_{Sa}are the width and length of the saturation area, respectively. B

_{Sa}and μ

_{Sa}can be calculated by the magnetization curve of silicon steel, as shown in Figure 2.

_{g_δi}and R

_{Ba_δi}are the air gap reluctance and flux barrier reluctance in the no-load magnetic circuit, respectively. F

_{PMi}and R

_{PMi}are the equivalent MMF and reluctance of PM, respectively. Φ

_{δi}is the segmental no-load flux that flows through R

_{g_δi}, as shown in Figure 4. F

_{δi}and R

_{δi}are the equivalent MMF and reluctance of the corresponding flux barrier, PM, and bridge parallel circuit, respectively, as shown in Figure 5.

_{δ}) can be expressed as:

#### 2.2. Geometric Parameters of the LRE-IPMSM

_{Bai}), the flux barrier thickness (h

_{Bai}), the PM thickness (h

_{PMi}), the PM width (b

_{PMi}), and the bridge width (b

_{Br}) are shown in Figure 6a,b, respectively.

#### 2.3. Effect of PM Thickness on No-Load Flux

_{B}

_{r}= 0) when the effects of geometric parameters (including h

_{PMi}, b

_{PMi}, h

_{Bai}, and θ

_{Bai}) on Φ

_{δ}are investigated to analyze the no-load magnetic circuit conveniently. The effect of the bridge width on Φ

_{δ}is analyzed in the following subsection.

_{δ}with respect to the PM thickness (h

_{PMi}) is calculated by the EMC method and finite-element analysis (FEA), as shown in Figure 7. It can be seen that Φ

_{δ}has a negligible variation with the increment of h

_{PM}

_{1}and h

_{PM}

_{2}, but Φ

_{δ}increases slightly when h

_{PM}

_{3}increases. This is because the coefficient of F

_{δ}

_{3}is obviously larger than that of F

_{δ}

_{1}and F

_{δ}

_{2}in Equation (7).

#### 2.4. Effect of PM Width on No-Load Flux

_{δ}versus the PM width (b

_{PMi}) is shown in Figure 8. To analyze the effect of b

_{PMi}on Φ

_{δ}simply, the flux barrier thickness is adjusted slightly to keep the reluctance of the flux barrier constant when b

_{PMi}changes. It can be seen that Φ

_{δ}increases linearly with the increment of b

_{PMi}. This is because the flux produced by the PM is proportional to the PM width. By comparing Figure 7 and Figure 8, it can be found that widening the PM to increase Φ

_{δ}is more practical than thickening the PM. However, the design of PM thickness should take the irreversible demagnetization of the PM into consideration.

#### 2.5. Effect of Flux Barrier Thickness on No-Load Flux

_{Bai}is the flux barrier width.

_{δ}because they only change the equivalent reluctance of the flux barrier in the no-load magnetic circuit model. Hence, this paper only analyzes the effect of the flux barrier thickness on Φ

_{δ}because the changing of the flux barrier thickness is more convenient than that of the flux barrier length.

_{δ}with respect to the flux barrier thickness is shown in Figure 9. According to the analysis of the no-load magnetic circuit model, it can be known that the no-load fluxes flowing through R

_{Ba_δ}

_{1}and R

_{Ba_δ}

_{2}are ignorable. Hence the h

_{Ba}

_{1}and h

_{Ba}

_{2}have negligible impacts on Φ

_{δ}, as seen in Figure 9. The flux, which flows through R

_{Ba_δ}

_{3}, becomes the leakage flux instead of flowing through the air gap. Therefore, when h

_{Ba}

_{3}increases, the leakage flux that flows through R

_{Ba_δ}

_{3}decreases and Φ

_{δ}increases, as shown in Figure 9.

#### 2.6. Effect of Flux Barrier Span Angle on No-Load Flux

_{δ}with respect to the flux barrier span angle (θ

_{Bai}). According to the no-load magnetic circuit model, it can be seen that R

_{g_δ}

_{1}decreases and R

_{g_δ}

_{2}increases with the increment of θ

_{Ba}

_{1}. Similarly, R

_{g_δ}

_{2}decreases and R

_{g_δ}

_{3}increases with the increment of θ

_{Ba}

_{2}. This results in the inconsiderable variation of Φ

_{δ}when θ

_{Ba}

_{1}and θ

_{Ba}

_{2}vary, as shown in Figure 10. But when θ

_{Ba}

_{3}increases, R

_{g_δ}

_{3}decreases, and R

_{g_δ}

_{1}, as well as R

_{g_δ}

_{2}, is invariable. Hence Φ

_{δ}increases with the increment of θ

_{Ba}

_{3}, as shown in Figure 10.

#### 2.7. Effect of Bridge Width on No-Load Flux

_{Br}) on Φ

_{δ}. As can be seen, when b

_{Br}increases, Φ

_{δ}decreases because the leakage flux that flows through the bridges increases. In other words, a narrow bridge is conducive to enhance Φ

_{δ}but it may cause rotor fracture failure at the same time. Hence, the selection of b

_{Br}should take the electromagnetic performance and mechanical safety into consideration simultaneously. The saturation level declines when the bridge width increases, which is not taken into consideration in the EMC method (i.e., the bridges are always considered to be saturated when b

_{Br}increases). That leads to an increasing error between the results of the EMC method and FEA with the increment of b

_{Br}, as shown in Figure 11.

## 3. q-Axis Magnetic Circuit Analysis

#### 3.1. q-Axis Magnetic Circuit Model

_{g_qi}and R

_{Ba_qi}are the air gap reluctance and flux barrier reluctance in the q-axis magnetic circuit, respectively. F

_{S_qi}is the MMF produced by the q-axis stator current. F

_{Br_qi}and R

_{Br_qi}are the equivalent MMF and reluctance of the bridge in the q-axis magnetic circuit, respectively. Φ

_{qi}is the segmental q-axis flux that flows through R

_{g_qi}, as shown in Figure 13. F

_{qi}and R

_{qi}are the equivalent MMF and reluctance of the corresponding flux barrier and bridge parallel circuit, respectively, as shown in Figure 14.

_{q}) can be expressed as:

#### 3.2. Effect of PM Thickness on q-Axis Flux

_{PMi}, b

_{PMi}, h

_{Bai}, and θ

_{Bai}) on Φ

_{q}are investigated, the bridges are neglected (i.e., b

_{B}

_{r}=0) to analyze the q-axis magnetic circuit conveniently. The influence of the bridge width on Φ

_{q}is analyzed in the following subsection.

_{q}is shown in Figure 15. In the preliminary LRE-IPMSM, the width of the PM in different layers increases in sequence (i.e., b

_{PMi}> b

_{PMi-}

_{1}). A wider PM means that there will be more q-axis flux flowing through the corresponding PM removed area. Hence the influence of h

_{PMi}on Φ

_{q}is more remarkable than h

_{PMi-}

_{1}, as shown in Figure 15. In addition, as can be seen in Figure 15, Φ

_{q}decreases and the tendency is slowing when h

_{PMi}increases.

#### 3.3. Effect of PM Width on q-Axis Flux

_{q}. For the purpose of analyzing the effect of b

_{PMi}on Φ

_{q}simply, the flux barrier thickness is adjusted slightly to keep the reluctance of the flux barrier invariable when b

_{PMi}changes. The reluctance of the PM removed area decreases when the PM width increases, which results in the reduction of R

_{Ba_qi}as the reluctance of the flux barrier is constant. Hence, Φ

_{q}increases with the increment of b

_{PMi}, as shown in Figure 16.

#### 3.4. Effect of Flux Barrier Thickness on q-Axis Flux

_{q}is shown in Figure 17. In the preliminary LRE-IPMSM, the width of the flux barrier in different layers increases in sequence. Hence, Φ

_{q}is more sensitive to the variation of h

_{Bai}than that of h

_{Bai-1}, as shown in Figure 17. In addition, as can be seen in Figure 17, when h

_{Bai}increases, Φ

_{q}decreases and the tendency is slowing.

#### 3.5. Effect of Flux Barrier Span Angle on q-Axis Flux

_{q}is shown in Figure 18. When θ

_{Bai}increases, there will be more q-axis flux flowing through the corresponding flux barrier, which means that the q-axis reluctance increases. Hence, Φ

_{q}decreases with the increment of θ

_{Bai}, as shown in Figure 18. In addition, it can be seen that when the effect of θ

_{Ba}

_{1}on Φ

_{q}is investigated, there is a considerable error between the results of the EMC method and FEA. This is because more q-axis flux does not flow through the first flux barrier when θ

_{Ba}

_{1}decreases, which results in the saturation of the partial stator teeth and yoke. The saturation area decreases and the error between the results of the EMC method and FEA declines when θ

_{Ba}

_{1}increases, as can be seen in Figure 18. When θ

_{Ba}

_{2}and θ

_{Ba}

_{3}decrease, the stator teeth and yoke are not saturated because the stator MMF F

_{S_q}

_{2}and F

_{S_q}

_{3}are much less than F

_{S_q}

_{1}. Hence, when the effects of θ

_{Ba}

_{2}and θ

_{Ba}

_{3}on Φ

_{q}are investigated, the errors between the results of the EMC method and FEA are negligible.

#### 3.6. Effect of Bridge Width on q-Axis Flux

_{q}with respect to b

_{Br}. When the bridge width increases, more q-axis flux flows through the bridges instead of the flux barriers. That results in the increment of Φ

_{q}as b

_{Br}increases, as seen in Figure 19. The bridges are saturated all the time when b

_{Br}increases. Therefore, when the variation of Φ

_{q}with respect to b

_{Br}is investigated, the error between the results of the EMC method and FEA is negligible.

## 4. d-Axis Magnetic Circuit Analysis

#### 4.1. d-Axis Magnetic Circuit Model

_{Br_di}) and a serial-connected reluctance (R

_{Br_di}) in the d-axis magnetic circuit model.

_{Sy_di}, F

_{St_di}) and serial-connected reluctance (R

_{Sy_di}, R

_{St_di}) in the d-axis magnetic circuit model, respectively. The d-axis EMC model of the investigated LRE-IPMSM is shown in Figure 21 and it is rewritten in Figure 22. To show the d-axis EMC model clearly, the structural dimensions of the machine model are adjusted approximately.

_{d}) can be expressed as:

_{g_di}and R

_{Ba_di}are the air gap reluctance and flux barrier reluctance in the d-axis magnetic circuit, respectively. R

_{σ}is the leakage reluctance. F

_{S_di}is the MMF produced by the d-axis stator current. Φ

_{di}is the segmental d-axis flux that flows through R

_{g_di}, as shown in Figure 22. F

_{di}and R

_{di}are the equivalent MMF and reluctance of the corresponding flux barrier and bridges parallel circuit, respectively, as shown in Figure 23.

#### 4.2. Effect of PM Thickness on d-Axis Flux

_{d}with respect to the geometric parameters (including h

_{PMi}, b

_{PMi}, h

_{Bai}, and θ

_{Bai}). The variation of Φ

_{d}with respect to h

_{PMi}is shown in Figure 24. As mentioned before, most d-axis flux does not flow through the flux barrier and PM removed area. Therefore, Φ

_{d}exhibits negligible variation when the PM thickness increases, as shown in Figure 24.

#### 4.3. Effect of PM Width on d-axis Flux

_{d}with respect to the PM width is shown in Figure 25. To analyze the effect of b

_{PMi}on Φ

_{d}simply, the flux barrier thickness is changed slightly to keep the reluctance of the flux barrier invariable when b

_{PMi}changes. As can be seen, Φ

_{d}is almost invariable as the PM width increases, which is the same as the effect of h

_{PMi}on Φ

_{d}. This can also be explained by the fact that most d-axis flux does not flow through the flux barrier and PM removed area.

#### 4.4. Effect of Flux Barrier Thickness on d-Axis Flux

_{d}with respect to h

_{Bai}. As can be seen, Φ

_{d}calculated by FEA decreases as the flux barrier thickness exceeds about 2.5 mm. This is caused by the rotor saturation as h

_{Bai}increases continuously. The rotor saturation is not taken into consideration in the EMC method and hence Φ

_{d}calculated by the EMC method is almost invariable. When the PM thickness increases to some extent, it also causes the rotor saturation. However, the saturation area of the rotor is relatively small because the PM width is much less than the flux barrier width in the preliminary LRE-IPMSM. Therefore, when h

_{PMi}increases, the rotor saturation has a negligible impact on Φ

_{d}.

#### 4.5. Effect of Flux Barrier Span Angle on d-Axis Flux

_{Bai}on Φ

_{d}is shown in Figure 27. According to the analysis of the d-axis magnetic circuit, it can be easily known that when the i-layer flux barrier span angle changes, Φ

_{d}

_{2i}, as well as the sum of Φ

_{d}

_{2i−1}and Φ

_{d}

_{2i+1}, is constant. Besides, the other segmental d-axis fluxes are scarcely affected by the change of the i-layer flux barrier span angle. Therefore, the influence of θ

_{Bai}on Φ

_{d}is negligible, as shown in Figure 27.

#### 4.6. Effect of Bridge Width on d-Axis Flux

_{d}with respect to b

_{Br}. The saturation level of the bridge end declines as b

_{Br}increases, which inevitably results in the increment of Φ

_{d}. However, Φ

_{d}increases negligibly, as shown in Figure 28, because the saturation area of the bridge end is far smaller than that of the stator teeth and yoke.

## 5. Electromagnetic Performance Analyses

#### 5.1. Torque Characteristics

#### 5.2. Loss and Efficiency Characteristics

## 6. Mechanical Strength Verification

## 7. Conclusions

- (1)
- The PM width has a significant influence on the no-load flux, but the PM thickness does not. Thickening the third flux barrier and increasing the third flux barrier span angle can increase the no-load flux.
- (2)
- Thickening the PM and flux barrier, as well as increasing the flux barrier span angle, can decrease the q-axis flux, while widening the PM increases the q-axis flux.
- (3)
- The PM width and PM thickness, as well as the flux barrier span angle, affect the d-axis flux negligibly. The rotor saturation results in d-axis flux reduction as the flux barrier thickness increases continuously.
- (4)
- Widening the bridge can decrease the no-load flux and increase the q-axis flux obviously. However, the bridge width affects the d-axis flux negligibly because the saturation area of the bridge end is far smaller than that of the stator teeth and yoke.
- (5)
- The analyses of the EMC models can be extended to the rotor structure with a different number of poles and different layers of the flux barriers, as well as various types of materials. However, when the stator current changes, the accuracy of EMC models declines (especially in the d-axis model) because of the variation of the saturation level. Hence, the accuracy of torque calculation by the EMC method declines as the stator current changes.
- (6)
- It is shown by the electromagnetic performance of the optimized LRE-IPMSM that the electromagnetic torque reaches 361.69 Nm and the electromagnetic torque ripple is 5.39% under MTPA control of 371 A. The efficiency reaches the maximum of 96.4% at 6500 rpm. The stress distribution of the optimized LRE-IPMSM at maximum speed is obtained. The result shows that the optimized LRE-IPMSM satisfies the requirement of mechanical strength.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Structure of the preliminary less-rare-earth interior permanent-magnet synchronous machine (LRE-IPMSM); (

**b**) Sketches of flux barriers and bridges.

**Figure 6.**Sketches of geometric parameters: (

**a**) the flux barrier span angle (θ

_{Bai}) and the flux barrier thickness (h

_{Bai}); (

**b**) the PM thickness (h

_{PMi}), the PM width (b

_{PMi}), and the bridge width (b

_{Br}).

**Figure 7.**No-load flux (Φ

_{δ}) versus the PM thickness (h

_{PMi}). EMC: equivalent magnetic circuit. FEA: finite-element analysis.

**Table 1.**Basic parameters of the preliminary less-rare-earth interior permanent-magnet synchronous machine (LRE-IPMSM).

Parameters | Unit | Value |
---|---|---|

Stator outer diameter | mm | 290 |

Rotor outer diameter | mm | 172 |

Air gap length | mm | 1 |

Stack length | mm | 106 |

Maximum speed | rpm | 7000 |

Maximum torque | Nm | 344 |

Maximum efficiency | % | 96 |

© 2017 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**

Zheng, P.; Wang, W.; Wang, M.; Liu, Y.; Fu, Z.
Investigation of the Magnetic Circuit and Performance of Less-Rare-Earth Interior Permanent-Magnet Synchronous Machines Used for Electric Vehicles. *Energies* **2017**, *10*, 2173.
https://doi.org/10.3390/en10122173

**AMA Style**

Zheng P, Wang W, Wang M, Liu Y, Fu Z.
Investigation of the Magnetic Circuit and Performance of Less-Rare-Earth Interior Permanent-Magnet Synchronous Machines Used for Electric Vehicles. *Energies*. 2017; 10(12):2173.
https://doi.org/10.3390/en10122173

**Chicago/Turabian Style**

Zheng, Ping, Weinan Wang, Mingqiao Wang, Yong Liu, and Zhenxing Fu.
2017. "Investigation of the Magnetic Circuit and Performance of Less-Rare-Earth Interior Permanent-Magnet Synchronous Machines Used for Electric Vehicles" *Energies* 10, no. 12: 2173.
https://doi.org/10.3390/en10122173