# Coil Number Impact on Performance of 4-Phase Low Speed Toothed Doubly Salient Permanent Magnet Motors

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## Abstract

**:**

## 1. Introduction

## 2. Machines Design and Modeling

- Rotor and stator teeth depth ${h}_{r}$ and ${h}_{s}$;
- Teeth cyclic ratio ${\alpha}_{r1}$, ${\alpha}_{r2}$, ${\alpha}_{s1}$, and ${\alpha}_{s2}$.

#### 2.1. Average Torque

#### 2.2. EM Calculation Method

- In the conductor part$$\frac{1}{{\mu}_{0}}\left(\frac{\partial}{\partial x}\frac{\partial {A}_{z}}{\partial x}+\frac{\partial}{\partial y}\frac{\partial {A}_{z}}{\partial y}\right)={J}_{z}$$
- In the iron part$$\frac{1}{\mu}\left(\frac{\partial}{\partial x}\frac{\partial {A}_{z}}{\partial x}+\frac{\partial}{\partial y}\frac{\partial {A}_{z}}{\partial y}\right)=0$$
- In the air gap part$$\frac{1}{{\mu}_{0}}\left(\frac{\partial}{\partial x}\frac{\partial {A}_{z}}{\partial x}+\frac{\partial}{\partial y}\frac{\partial {A}_{z}}{\partial y}\right)=0$$
- In the PM part$$\frac{1}{{\mu}_{pm}}\left(\frac{\partial}{\partial x}\frac{\partial {A}_{z}}{\partial x}+\frac{\partial}{\partial y}\frac{\partial {A}_{z}}{\partial y}\right)={J}_{pm}$$

## 3. Design Process and Optimization

## 4. Results and Discussion

#### 4.1. Objective Function and Torque Evolution

#### 4.2. Efficiency

#### 4.3. Basic Characteristics and Performance Comparison

#### 4.3.1. PM-Flux

#### 4.3.2. Air-Gap Flux Density

#### 4.3.3. Ripple Torque and Energy Ratio

## 5. Comparison with Reference PMSM

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

- Define the objective function, fixed parameters, geometrical and thermal constraints.
- For i = 1: N do
- (a)
- Initialize randomly a particle (machine).
- (b)
- If machine is not feasible or thermal constraints is not satisfied return to step 2a.
- (c)
- End

- Calculate and evaluate the objective function of each particle in the swarm using FEMM software (2D-FEA)
- Update local and global best (pbest and gbest )
- For i = 1: N do
- (a)
- Produce new particle (displacement of each particle) using (15)
- (b)
- If machine is not feasible or thermal constraints are not satisfied return to step 5a.
- (c)
- End

- Check if the stop criterion is satisfied; if not return to step 3.
- Output optimal results.
- Stop.

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**Figure 7.**PSO algorithm flowchart (see also Appendix A).

**Figure 8.**Objective function and torque evolution as a function of generations number: (

**a**) TDSPM with two coils/phase structures, (

**b**) TDSPM with four coils/phase structures.

**Figure 11.**Air-gap flux density of 4-phase TDSPM with 2 and four coils/phase optimal structures versus rotor position.

**Table 1.**TDSPM design specifications and constraints [4].

Specifications | |

Power output (MW) | 2 |

Rated torque (kNm) | 191 |

Nominal frequency (Hz) | 50 |

Speed (rpm) | 100 |

Geometrical and thermal constraints | |

Stator outer diameter ${D}_{o}$ (m) | 1.83 |

Active axial length ${L}_{m}$ (m) | 1.125 |

Air gap g (mm) | 5 |

Thermal product (RMS) (A${}^{2}$/m${}^{3}$) | 22.52 × 10${}^{10}$ |

Integer Design Parameters | 2-Coils/Phase | 4-Coils/Phase |
---|---|---|

Number of magnet pairs, ${P}_{pm}$ | 2 | 4 |

Number of teeth in the stator, ${N}_{s}$ | 24 | 32 |

Number of teeth in the rotor, ${N}_{r}$ | 30 | 28 |

Number of statoric salient poles, ${N}_{ps}$ | 8 | 16 |

Number of teeth per statoric pole, $\left({N}_{dp}\right)$ | 3 | 2 |

Search Variable (Vector ${\mathit{x}}^{*}$) | Lower Limit | Upper Limit |
---|---|---|

Stator yoke thickness, ${E}_{s}$ | 10 mm | 150 mm |

Stator and rotor teeth depth, ${h}_{s}$, ${h}_{r}$ | 3 mm | 50 mm |

Magnet thickness, ${E}_{m}$ | 5 mm | 100 mm |

Stator and rotor teeth cyclic ratios, ${\alpha}_{s1}$, ${\alpha}_{r1}$ | 0.15 | 0.50 |

Stator and rotor teeth cyclic ratios, ${\alpha}_{s2}$, ${\alpha}_{r2}$ | 0.15 | 0.50 |

Coil height, ${h}_{b}$ | 10 mm | 150 mm |

Slot radius, ${R}_{a}$ | ${R}_{o}$/3 | $0.9\times {R}_{o}$ |

Angular pole opening, $\beta $ | 3${}^{\circ}$ | 11.5${}^{\circ}$ |

Angular slot opening, ${\beta}_{a}$ | 3${}^{\circ}$ | 11.5${}^{\circ}$ |

Rotor yoke thickness, ${E}_{r}$ | 10 mm | 150 mm |

Current density, J | 1 A/m${}^{2}$ | 15 A/m${}^{2}$ |

${\mathit{x}}_{\mathit{o}\mathit{p}\mathit{t}}$ | 4 Coils/Phase | 2 Coils/Phase |
---|---|---|

${E}_{s}$ | 99.50 mm | 104.39 mm |

${E}_{r}$ | 84.83 mm | 107.97 mm |

${h}_{b}$ | 78.64 mm | 75.77 mm |

$\beta $ | 8.26${}^{\circ}$ | 13.10${}^{\circ}$ |

${\beta}_{a}$ | 10.32${}^{\circ}$ | 10.32${}^{\circ}$ |

${R}_{a}$ | 814.82 mm | 740.26 mm |

${\alpha}_{s1}$ | 0.36 | 0.2 |

${\alpha}_{r1}$ | 0.28 | 0.31 |

${\alpha}_{s2}$ | 0.47 | 0.2 |

${\alpha}_{r2}$ | 0.49 | 0.48 |

${E}_{m}$ | 25.90 mm | 30 mm |

${h}_{s}$ | 32.04 mm | 35 mm |

${h}_{r}$ | 30.30 mm | 35 mm |

Iron weight | 11,002.60 kg | 10,993.34 kg |

Copper weight | 511.68 kg | 990.13 kg |

Magnet weight | 85.80 kg | 54.92 kg |

Performance | 2 Coils/Phase | 4 Coils/Phase |
---|---|---|

${\mathsf{\Gamma}}_{max}$ (kNm) | 125.387 | 215.41 |

${\mathsf{\Gamma}}_{min}$ (kNm) | 85.408 | 167.698 |

${\mathsf{\Gamma}}_{mean}$(kNm) | 100.00 | 190.85 |

RT | 40% | 25% |

ER | 62% | 70% |

Mass–torque ratio (kNm/kg) | 10.44 | 17.42 |

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**MDPI and ACS Style**

Guerroudj, C.; Charpentier, J.-F.; Saou, R.; Karnavas, Y.L.; Bracikowski, N.; Zaïm, M.E.-H.
Coil Number Impact on Performance of 4-Phase Low Speed Toothed Doubly Salient Permanent Magnet Motors. *Machines* **2021**, *9*, 137.
https://doi.org/10.3390/machines9070137

**AMA Style**

Guerroudj C, Charpentier J-F, Saou R, Karnavas YL, Bracikowski N, Zaïm ME-H.
Coil Number Impact on Performance of 4-Phase Low Speed Toothed Doubly Salient Permanent Magnet Motors. *Machines*. 2021; 9(7):137.
https://doi.org/10.3390/machines9070137

**Chicago/Turabian Style**

Guerroudj, Cherif, Jean-Frederic Charpentier, Rachid Saou, Yannis L. Karnavas, Nicolas Bracikowski, and Mohammed El-Hadi Zaïm.
2021. "Coil Number Impact on Performance of 4-Phase Low Speed Toothed Doubly Salient Permanent Magnet Motors" *Machines* 9, no. 7: 137.
https://doi.org/10.3390/machines9070137