# Optimal Design of Permanent Magnet Synchronous Machine Based on Random Walk Method and Semi 3D Magnetic Equivalent Circuit Considering Overhang Effect

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## Abstract

**:**

## 1. Introduction

## 2. Magnetic Equivalent Circuit Model of PMSM

#### 2.1. Simplification of MEC

_{g}, R

_{m}, R

_{s}, R

_{r}, R

_{ls}, and Φ

_{r}are the air gap reluctance, reluctance by the permanent magnet, reluctance of rotor, stator yoke, leakage flux of permanent magnet, and flux of permanent magnet, respectively. Equations (1) and (2) are used to calculate R

_{g}and R

_{m}, respectively. Moreover, g′, μ

_{0}, A

_{g}, T

_{m}, and μ

_{r}denote the effective air gap length, the permeability of vacuum, the axial cross-sectional area of the air gap, the thickness of the permanent magnet, and relative permeability, respectively.

_{r}means the reluctance coefficient of the iron core [4].

#### 2.2. Slotting Effect Using Carter’s Coefficient

_{t}in Equation (6). In Equation (4), γ is obtained by substituting the B

_{s0}with the width of the slot opening. g′ denotes the air gap length of the slotless model [5]. In Equation (5), D

_{si}and N

_{s}are the inner radius and number of slots of the stator, respectively. τ

_{t}can be calculated. In Equation (7), the effective air gap length can be obtained using the Carter coefficient. Therefore, the slot effect can be analyzed in the slotless model [6].

#### 2.3. D MEC Considering Overhang Structure

_{m_3D}, H

_{m_3D}, and V

_{_3D}mean the magnetic flux density, magnetic field strength, and permanent magnet volume of the 3D overhang. B

_{m_2D}, H

_{m_2D}, and V

_{_2D}represent the magnetic flux density, magnetic field strength, and permanent magnet volume of the two-dimensional (2D) permanent magnet. In Equation (9), PC represents the permeance coefficient. To compensate for the magnetic energy according to the overhang structure, the magnetic flux density and magnetic field strength of the 2D permanent magnet are increased. With the former, a 3D permanent magnet with an overhang can be made equivalent to a 2D permanent magnet, as shown in Equation (10) [7,8].

#### 2.4. Electromagnetic Analysis Based on MEC

_{r}and ϕ

_{m}are the remanence magnetic flux of the permanent magnet with overhang and the amount of half-pole magnetic flux, respectively. The axial cross-sectional area of the permanent magnet with overhang is denoted by A

_{m}. In Equation (12), k

_{r}is the reluctance coefficient. In Equation (13), ϕ

_{g}and k

_{ls}represent the air-gap flux per pole of the permanent magnet with overhang and the leakage coefficient between two poles, respectively. Moreover, by doubling the magnetic flux magnitude of the overhang half-pole in Equation (13), the magnetic flux magnitude of ϕ

_{g}can be regarded as the air-gap flux per pole [11].

_{g}obtained in Equation (13) into the identity equation of the magnetic flux density, the air gap magnetic flux density B

_{g}of the overhang can be derived using Equation (14).

_{coil_pitch}denotes the coil pitch. In Equation (15), N

_{ph}is the number of equivalent series turns; L is the stack; q is the number of pole pairs; B

_{n}is the air-gap flux density of the nth harmonic; ω

_{m}is the mechanical angular velocity; and t is the period.

## 3. Optimal Design of PMSM Using RWA

#### 3.1. Random Walk Algorithm’s Method

_{n}to Z

_{n+1}by the size of the Walk in the direction of the random number. Substitute Z

_{n+1}into the Object function, and repeat the previous step using the current value if it is less than the previous value. After repeating these steps n times, the search range of the Walk is reduced by a certain percentage. Eventually, the global minimum will be reached. Furthermore, if it exceeds the set number of repetitions n, the RWA is terminated, and the current minimum value is considered as the minimum value of the Object function.

- Z
_{n}: variable value - Walk: search range
- Radom: A random number between −1 and 1

#### 3.2. Optimal Design Application of PMSM

_{si}, e

_{m}, T

_{m}, and L denote the rotor outer diameter, pole arc ratio, permanent magnet thickness, and stack length, respectively [13].

_{dc}and i

_{e}are the input voltage and inverter efficiency, respectively.

_{rms}value using the number of turns obtained by Equation (19), the area per conductor and copper loss can be derived according to Equations (20) and (21), respectively.

_{z}, C

_{slot_area}, j, R

_{ph}, and Layer denote the area per conductor, slot area, maximum current density, phase resistance, and number of layers, respectively. The layer value of PMSM using concentrated winding is 2 [15].

_{w}can be obtained according to the tooth saturation limit value B

_{tm}of the stator in Equation (22). In general, the saturation limit of electrical steel is 1.6~1.7 T. Considering the armature reaction, the saturation limit of the stator teeth was selected as 1.5 T. In Equation (22), F

_{gap}, N

_{sm}, L, and lf represent the air gap flux of one pole, number of slots per pole, stack length, and fill factor.

_{stw}is the thickness of the stator teeth, and t

_{sc}, t

_{rc}are the lengths of the rotor and stator core [16].

#### 3.3. PMSM Optimal Design Results

^{3}and 7.56 kg, as shown in Figure 8a,b.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Structures of PMSM: (

**a**) 3D FEM and (

**b**) magnetic equivalent circuit considering only half of each pole.

**Figure 2.**Simplification of MEC model: (

**a**) sum of air gap reluctance; (

**b**) MEC with leakage factor applied; (

**c**) MEC with iron core’s reluctance factor applied.

**Figure 3.**Operating point conversion through magnetic energy compensation of PMSM with overhang structure: (

**a**) operating point of permanent magnet with overhang structure, (

**b**) PMSM model with overhang structure.

**Figure 4.**Characteristic analysis of overhang structure according to RPM: (

**a**) flux linkage, (

**b**) back EMF.

**Figure 5.**Design parameters used in optimization algorithm: (

**a**) RWA’s global solution exploration process and (

**b**) optimal design parameters of the rotor.

**Figure 7.**Comparison of 3D model through optimization algorithm: (

**a**) initial 3D model, (

**b**) optimized 3D model.

**Figure 8.**PMSM volume and weight optimization results according to the iteration of the optimization algorithm: (

**a**) PMSM volume minimization process, (

**b**) PMSM weight minimization process.

Slot Model FEM | MEC | Error | |
---|---|---|---|

Br | 1.407 [T] | 1.392 [T] | 1 [%] |

Slotless Model FEM | Error | ||

Br | 1.415 [T] | 1.6 [%] |

3D Overhang Model | 2D Equivalent Model | Unit | ||
---|---|---|---|---|

B_{r_3D} | 1.2 | B_{r_2D} | 1.267 | [T] |

H_{c_2D} | −954,984 | H_{c_2D} | −1,008,724 | [A/m] |

B_{m_3D} | 1.13T | B_{m_2D} | 1.19T | [T] |

H_{m_3D} | −54,576 | H_{m_2D} | −57,648 | [A/m] |

Parameter | Initial Model | Optimal Model | Unit |
---|---|---|---|

Stator outer/inner dia. | 319.4/197 | 369.2/249.4 | (mm) |

Rotor outer/inner dia. | 196/135.8 | 248.4/184.4 | (mm) |

Air gap | 2 | 2 | (mm) |

Pole/Slot | 32/27 | 32/27 | − |

PM thickness | 8 | 8 | (mm) |

PM ratio | 0.9 | 0.73 | (%) |

Stator/rotor stack | 120/130 | 86.4/96.4 | (mm) |

Weight | 62.15 | 54.59 | (kg) |

Volume | 7900 | 7000 | (m^{3}) |

Efficiency | 93 | 93 | (%) |

Parameter | FEM | MEC | Unit |
---|---|---|---|

CPU utilization (Clock: 3.7 GHz) | 80–90 | 45–60 | (%) |

Analysis time (Count: 4000) | 133 | 10 | (hour) |

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**MDPI and ACS Style**

Kim, S.-m.; Jung, W.-S.; Kim, W.-H.; Bang, T.-K.; Lee, D.-H.; Kim, Y.-J.; Choi, J.-Y.
Optimal Design of Permanent Magnet Synchronous Machine Based on Random Walk Method and Semi 3D Magnetic Equivalent Circuit Considering Overhang Effect. *Energies* **2022**, *15*, 7852.
https://doi.org/10.3390/en15217852

**AMA Style**

Kim S-m, Jung W-S, Kim W-H, Bang T-K, Lee D-H, Kim Y-J, Choi J-Y.
Optimal Design of Permanent Magnet Synchronous Machine Based on Random Walk Method and Semi 3D Magnetic Equivalent Circuit Considering Overhang Effect. *Energies*. 2022; 15(21):7852.
https://doi.org/10.3390/en15217852

**Chicago/Turabian Style**

Kim, Su-min, Woo-Sung Jung, Woo-Hyeon Kim, Tae-Kyoung Bang, Dae-Hyun Lee, Yong-Joo Kim, and Jang-Young Choi.
2022. "Optimal Design of Permanent Magnet Synchronous Machine Based on Random Walk Method and Semi 3D Magnetic Equivalent Circuit Considering Overhang Effect" *Energies* 15, no. 21: 7852.
https://doi.org/10.3390/en15217852