# Design of Permanent Magnet-Assisted Synchronous Reluctance Motor with Low Torque Ripple

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis of Motor Structure and Cogging Torque

#### 2.1. Motor Structure and Basic Parameters

#### 2.2. Cogging Torque Generation Principle

_{0}is the vacuum permeability and V is the air gap volume. The expression of the air gap magnetic density B(θ,α) is as follows:

_{r}(θ) is the circumferential distribution of the remanent magnetic field of the permanent magnet; G(θ,α) is the circumferential distribution of the effective air gap’s relative permeability; θ is the electrical angle of the rotor; and α is the relative position angle between the fixed rotors.

_{s}is the axial length; R

_{1}is the inner radius of the stator; R

_{2}is the outer radius of the rotor; n is the harmonic frequency of the air gap flux density and can make nz/2p an integer; and G

_{n}is the permeability of the motor to the n order of the air gap flux density. The specific expression of B

_{r(nz/2p)}is as follows:

_{p}is the polar arc coefficient.

#### 2.3. Cogging Torque Reduction Principle

#### 2.3.1. Magnetic Pole Migration Principle

_{cog}of the total cogging torque fundamental, which is generated by two symmetrically arranged magnetic poles, is T

_{cog1}of the cogging torque fundamental of magnetic pole 1, and is T

_{cog2}of the cogging torque fundamental of magnetic pole 2 in phase superposition, as shown in Formula (6).

_{2pz}is the least common multiple of the number of poles and slots of the motor.

#### 2.3.2. Principle of Stator Auxiliary Slot

## 3. Motor Structure Design

#### 3.1. Magnetic Pole Migration Structure Design

_{c1}of cogging torque, a single magnetic pole should be offset at 5°, as shown in Figure 2b. In order to weaken the second harmonic T

_{c2}of cogging torque, the relative deviation angle of a single magnetic pole is 2.5°. In addition, in order to determine the direction of magnetic pole offset, in addition to the known magnetic pole offset angle, it is necessary to determine which magnetic poles are offset on the basis of the original symmetry. For the 4-pole 36-slot motor, the total cogging torque is the cogging torque generated by four magnetic poles in the same phase superposition; here, two magnetic poles are considered as a group of peak and valley elimination factors, which can be divided into two groups. Therefore, it is necessary to rotate both magnetic poles counterclockwise at the same time. At once, any two of the four magnetic poles are rotated, applying any of the 12 rotation modes; this involves, namely, rotating magnetic poles 12, 13, 14, 23, 24, 34, 21, 31, 41, 32, 42, 43. Because the order of the poles in each rotation does not affect the effect achieved, there are 6 rotation modes, namely 12, 13, 14, 23, 24, and 34. Among them, fixed magnetic pole 1 and 4, and rotating magnetic pole 2 and 3 have the same effect as fixed magnetic pole 2 and 3, and rotating magnetic pole 1 and 4; fixed pole 1 and 3 and rotating pole 2 and 4 have the same effect as fixed pole 2 and 4 and rotating pole 1 and 3; fixed magnetic pole 1 and 2 and rotating magnetic pole 3 and 4 have the same effect as fixed magnetic pole 3 and 4 and rotating magnetic pole 1 and 2. Therefore, after removing the repetitive mode, there are three rotation modes: mode 1: rotating magnetic pole 2 and 3 counterclockwise; mode 2: rotating magnetic pole 2 and 4 counterclockwise; and mode 3: rotating magnetic pole 3 and 4 counterclockwise. In addition, for the sake of expression, the original symmetric magnetic pole structure is named mode 0 here.

_{c}, cogging torque fundamental harmonic T

_{c1}, cogging torque second harmonic T

_{c2}, electromagnetic torque T, torque ripple T

_{r}and the unbalanced magnetic tension F of the motor under no-load and rated load conditions (phase current effective value 42 A and current angle 45.9°). The specific waveform comparison of each performance is shown in Figure 3. The specific values of each performance are shown in Table 2.

#### 3.2. Stator Auxiliary Groove Design

#### 3.2.1. Optimize the Selection of Variables

_{16}(4

^{5}) is established, as shown in Table 6.

#### 3.2.2. Test Results

_{r}, which correspond to the 16 groups of orthogonal tests, are here analyzed. The specific calculation results are shown in Table 7.

#### 3.2.3. Results Processing

_{r}, and their relative importance, it is necessary to carry out a mean value analysis and variance analysis on the results of the 16 groups of orthogonal tests.

- Mean analysis
- (1).
- Population mean analysis

_{i}is the test result of the i test in a column in Table 7, and n is the number of tests.

_{T}of the mean value of electromagnetic torque is shown in Equation (11).

_{T}

_{r}for the overall mean value of torque ripple is shown in Equation (12).

- (2).
- Average value analysis of all variables at all levels

_{TA}

_{(1)}, which is the mean value of electromagnetic torque T under level 1 of factor A, is as follows:

- 2.
- Variance Analysis

_{A}is the variance in an objective function under factor A; Q is the number of levels taken by each factor; m

_{A}

_{(j)}is the average value of an objective function under factor A level j; and m is the overall average value of the objective function.

_{TA}is as follows.

_{STx}is the proportion of variance and S

_{Tx}in the total variance.

_{ST}

_{rx}is the proportion of variance and S

_{T}

_{rx}in the total variance.

## 4. Motor Performance Analysis

#### 4.1. No-Load Performance Analysis

#### 4.2. On-Load Performance Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Performance comparison diagram of each mode. (

**a**) Cogging torque waveform. (

**b**) Cogging torque harmonic distribution. (

**c**) Electromagnetic torque waveform. (

**d**) Unbalance magnetic tension waveform.

**Figure 4.**Performance comparison diagram from different angles. (

**a**) Cogging torque waveform. (

**b**) Cogging torque harmonic distribution. (

**c**) Electromagnetic torque waveform. (

**d**) Unbalance magnetic tension waveform.

**Figure 8.**Cogging torque comparison diagram of each model. (

**a**) Waveform contrast diagram. (

**b**) Harmonic distribution diagram.

**Figure 9.**Comparison of A-phase performance of each model. (

**a**) Back EMF waveform. (

**b**) Flux linkage waveform.

**Figure 11.**Performance comparison diagram of each model. (

**a**) Unbalance magnetic tension waveform. (

**b**) Core loss comparison diagram.

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

Rated speed | n_{N} | 3000 | r/min |

Rated current | I_{N} | 42 | A |

Number of pole Pairs | p | 2 | - |

Number of slots | z | 36 | - |

Radius of the stator inner surface | R_{1} | 90.1 | mm |

Radius of rotor outer surface | R_{2} | 89.6 | mm |

Radius of the stator outer surface | R_{s} | 132.6 | mm |

Length of motor | l_{s} | 155 | mm |

Mode | 0 | 1 | 2 | 3 | |
---|---|---|---|---|---|

Performance | |||||

T_{c} (Nm) | 13.2 | 6.2 | 5.9 | 6.2 | |

T_{c1} (Nm) | 4.6 | 4 | 4.3 | 4 | |

T_{c2} (Nm) | 4.65 | 0.05 | 0.19 | 0.04 | |

T (Nm) | 77.6 | 74.5 | 74.34 | 74.54 | |

T_{r} (%) | 8.9 | 5.8 | 6.4 | 5.7 | |

F (N) | 14 | 179 | 3 | 184 |

Offset Angle | 0° | 2.5° | 5° | |
---|---|---|---|---|

Performance | ||||

T_{c} (Nm) | 13.2 | 5.9 | 6.87 | |

T_{c1} (Nm) | 4.6 | 4.3 | 0.13 | |

T_{c2} (Nm) | 4.65 | 0.19 | 4.21 | |

T (Nm) | 77.6 | 74.34 | 73.32 | |

T_{r} (%) | 8.9 | 6.4 | 12.4 | |

F (N) | 14 | 3 | 2.4 |

Interval | a | b | c | d | |
---|---|---|---|---|---|

Number | |||||

1 | 0° | 0° | 0° | 0° | |

2 | 1° | 2° | 3° | 4° | |

3 | 0.5° | 1° | 1.5° | 2° | |

4 | 0.25° | 0.5° | 0.75° | 1° |

Factor | A (°) | B (mm) | C | D | E | |
---|---|---|---|---|---|---|

Level | ||||||

1 | 0.35 | −0.2 | 1 | Rectangle | a | |

2 | 0.7 | −0.1 | 2 | Triangle | b | |

3 | 1.05 | 0.1 | 3 | Arc | c | |

4 | 1.4 | 0.2 | 4 | Trapezoid | d |

Test Time | A | B | C | D | E |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 2 | 2 |

3 | 1 | 3 | 3 | 3 | 3 |

4 | 1 | 4 | 4 | 4 | 4 |

5 | 2 | 1 | 2 | 3 | 4 |

6 | 2 | 2 | 1 | 4 | 3 |

7 | 2 | 3 | 4 | 1 | 2 |

8 | 2 | 4 | 3 | 2 | 1 |

9 | 3 | 1 | 3 | 4 | 2 |

10 | 3 | 2 | 4 | 3 | 1 |

11 | 3 | 3 | 1 | 2 | 4 |

12 | 3 | 4 | 2 | 1 | 3 |

13 | 4 | 1 | 4 | 2 | 3 |

14 | 4 | 2 | 3 | 1 | 4 |

15 | 4 | 3 | 2 | 4 | 1 |

16 | 4 | 4 | 1 | 3 | 2 |

Test Time | T (Nm) | T_{r} (%) |
---|---|---|

1 | 74.46 | 6.5 |

2 | 75.74 | 7.8 |

3 | 74.14 | 7.56 |

4 | 73.92 | 7.36 |

5 | 76.34 | 8.6 |

6 | 76.16 | 7.32 |

7 | 74.88 | 6.54 |

8 | 75.36 | 7.6 |

9 | 76.7 | 7.27 |

10 | 76.23 | 5.57 |

11 | 74.02 | 7.8 |

12 | 75.64 | 6.5 |

13 | 76.64 | 7.3 |

14 | 76.88 | 5.7 |

15 | 75.34 | 5.47 |

16 | 75.48 | 6.9 |

Factor | Level | m_{T} (Nm) | m_{T}_{r} (%) |
---|---|---|---|

A | 1 | 74.57 | 7.3 |

2 | 75.69 | 7.5 | |

3 | 75.65 | 6.78 | |

4 | 76.1 | 6.3 | |

B | 1 | 76 | 7.4 |

2 | 76.25 | 6.6 | |

3 | 74.6 | 6.8 | |

4 | 75.1 | 7.1 | |

C | 1 | 75 | 7.13 |

2 | 75.76 | 7.09 | |

3 | 75.77 | 7.03 | |

4 | 75.4 | 6.7 | |

D | 1 | 75.46 | 6.3 |

2 | 75.44 | 7.6 | |

3 | 75.55 | 7.16 | |

4 | 75.5 | 6.86 | |

E | 1 | 75.35 | 6.285 |

2 | 75.7 | 7.13 | |

3 | 75.6 | 7.17 | |

4 | 75.3 | 7.365 |

Factor | S_{T} | S_{T}_{r} |
---|---|---|

A | 0.32 | 0.22 |

B | 0.45 | 0.1 |

C | 0.1 | 0.02 |

D | 0.001 | 0.22 |

E | 0.03 | 0.17 |

Factor | K_{STx} (%) | K_{ST}_{rx} (%) |
---|---|---|

A | 35.5 | 30.1 |

B | 50 | 13.7 |

C | 11.1 | 27.4 |

D | 0.1 | 30.13 |

E | 3.3 | 23.3 |

Model | 0 | 1 | 2 | |
---|---|---|---|---|

Performance | ||||

Back EMF | fundamental amplitude | 170 V | 169.6 V | 169.9 V |

THD | 22.03% | 21.25% | 20.96% | |

Flux linkage | fundamental amplitude | 0.27 Wb | 0.27 Wb | 0.27 Wb |

THD | 7.29% | 7.26% | 7.17% |

Model | 0 | 1 | 2 | |
---|---|---|---|---|

Performance | ||||

Electromagnetic torque | 77.6 Nm | 74.34 Nm | 76.32 Nm | |

Torque ripple | 8.9% | 6.4% | 5.5% |

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## Share and Cite

**MDPI and ACS Style**

Li, X.; Sun, Z.; Sun, W.; Guo, L.; Wang, H.
Design of Permanent Magnet-Assisted Synchronous Reluctance Motor with Low Torque Ripple. *World Electr. Veh. J.* **2023**, *14*, 82.
https://doi.org/10.3390/wevj14040082

**AMA Style**

Li X, Sun Z, Sun W, Guo L, Wang H.
Design of Permanent Magnet-Assisted Synchronous Reluctance Motor with Low Torque Ripple. *World Electric Vehicle Journal*. 2023; 14(4):82.
https://doi.org/10.3390/wevj14040082

**Chicago/Turabian Style**

Li, Xinmin, Zihan Sun, Wenbo Sun, Liyan Guo, and Huimin Wang.
2023. "Design of Permanent Magnet-Assisted Synchronous Reluctance Motor with Low Torque Ripple" *World Electric Vehicle Journal* 14, no. 4: 82.
https://doi.org/10.3390/wevj14040082