# On the Analysis and Torque Enhancement of Flux-Switching Permanent Magnet Machines in Electric Power Steering Systems

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Formulation of FSPMM Features

#### 2.1. PM Air Gap Flux Density

- ${B}_{r}$, ${\mu}_{PM}$, and ${W}_{PM}$ are the magnet remanence, relative permeability, and width, respectively;
- ${\alpha}_{PM}$ is the angular PM width at the air-gap stator inner border;
- g is the air gap thickness, with $g={R}_{ints}-{R}_{extr}$.

#### 2.2. Permeance Functions

#### 2.2.1. Stator Permeance Functions

#### Permeance Function Accounting for the Slotting Effect

#### Permeance Function Accounting for the PM Flux Concentrating Arrangement

#### 2.2.2. Rotor Permeance Function

#### 2.3. Improvement of the Accuracy of the PM Air Gap Flux Density Prediction

#### 2.3.1. Correction Allied to the Rotor Position Variation

#### 2.3.2. Saturation Correction Function

- The nonlinear behavior of the iron due to the magnetic saturation which is commonly accounted for using the B–H curve,
- The deformation of the flux tube within the air gap due to the fringing effect, as illustrated in Figure 3.

#### 2.4. Armature Magnetic Reaction

#### 2.5. Torque Formulation

- ${B}_{m}$ is the equivalent no-load air gap flux density;
- ${l}_{a}$ is the active length of the machine;
- ${N}_{p}$ is the number of coils per phase;
- ${k}_{w}$ is the winding factor;
- ${k}_{a}$ and ${\sigma}_{0}$ are the permeance correction and leakage flux factors, respectively, [18];
- ${i}_{q}$ is the armature current quadrature component.

## 3. Case Study

#### 3.1. Analysis Assuming a Linear Magnetic Circuit

#### 3.2. Analysis with the Magnetic Saturation Taken into Account

#### 3.2.1. PM Air Gap Flux Density

#### 3.2.2. Armature Magnetic Reaction

#### 3.2.3. Torque Production Enhancement

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Flux tube surface in the air gap (hatched in blue).

**Legend:**(

**a**) Case of a slotless rotor, (

**b**) position of minimum flux linkage (PM in the stator facing the middle of a slot in the rotor), (

**c**) position of maximum flux linkage (PM in the stator facing the middle of a rotor tooth).

**Figure 4.**PM flux density spatial repartition with the stator slotting effect taken into account, assuming a slottless rotor, and without correction.

**Figure 5.**PM flux density spatial repartition with the stator and rotor slotting effects taken into account.

**Figure 6.**PM flux density spatial repartitions for different rotor positions, with the stator and rotor permeance functions and the rotor position correction function taken into account.

**Legend:**(

**a**) ${\theta}_{r}$ = 0°, (

**b**) ${\theta}_{r}$ = 5.4°, (

**c**) ${\theta}_{r}$ = 12.6°, (

**d**) ${\theta}_{r}$ = 18°, (

**e**) ${\theta}_{r}$ = 23.4°, (

**f**) ${\theta}_{r}$ = 30.6°.

**Figure 7.**PM flux density spatial repartitions with the stator and rotor slotting effects and the rotor position correction taken into account: (analytic) without saturation correction, (FEA) B–H curve implemented.

**Figure 8.**PM flux density spatial repartitions for different rotor positions, with the stator and rotor permeance functions and the rotor position and saturation corrections taken into account.

**Legend:**(

**a**) ${\theta}_{r}$ = 0°, (

**b**) ${\theta}_{r}$ = 5.40°, (

**c**) ${\theta}_{r}$ = 12.6°, (

**d**) ${\theta}_{r}$ = 18°, (

**e**) ${\theta}_{r}$ = 23.4°, (

**f**) ${\theta}_{r}$ = 30.6°.

**Figure 9.**Spectra corresponding to the FFT of the analytically- and FEA-predicted ${B}_{PM}\left(\theta \right)$ shown in Figure 8a.

**Figure 10.**Analytically- and FEA-predicted spatial repartitions of the armature magnetic reaction for ${\theta}_{r}$ = 0°.

**Legend:**(

**a**) ${I}_{max}$ = 3.6 A, (

**b**) ${I}_{max}$ = 4.8 A.

**Figure 11.**Spectrum of the spatial repartition of the analytically-predicted flux density at ${\theta}_{r}$ = 0°. (

**a**) Flux density created by the PMs, (

**b**) armature magnetic reaction.

**Figure 12.**Phases of the dominant harmonics of the spatial repartition of the analytically-predicted flux density versus the rotor position. (

**a**) Harmonics of the flux density created by the PMs, (

**b**) harmonics of the armature magnetic reaction.

**Figure 13.**Mapping of the mean total torque developed following the injection of an harmonic current of rank 8, plotted in the plane (8th harmonic initial phase, 8th harmonic amplitude to fundamental amplitude ratio).

**Figure 14.**Mapping of the ripple of the total torque developed following the injection of an harmonic current of rank 8, plotted in the same plane as Figure 13.

**Figure 15.**Mapping of the ripple to mean value ratio of the total torque developed following the injection of an harmonic current of rank 8, plotted in the same plane as Figure 13.

**Figure 16.**Torque-angle characteristics for a peak fundamental current of 3.6 A.

**Legend 1:**(continuous line) armature current supply limited to the fundamental of rank 4; (interrupted line) armature current supply including both the fundamental of rank 4 and the harmonic of rank 8, such that the 8th harmonic initial phase is equal to $13\pi /8$ and the 8th harmonic amplitude to fundamental amplitude ratio is equal to 5%.

**Legend 2:**(red) analytical results, (bleu) FEA results.

Parameter | Symbol | Value |
---|---|---|

Number of phases | q | 3 |

Number of stator slots | ${N}_{s}$ | 12 |

Number of rotor slots | ${N}_{r}$ | 10 |

Number of coils per phase | ${N}_{p}$ | 4 |

Number of turns per coil | ${N}_{c}$ | 120 |

PM remanence | ${B}_{r}$ | 1.1 T |

PM relative permeability | ${\mu}_{PM}$ | 1.04457 |

PM width | ${W}_{PM}$ | 2.5 mm |

Stator outer radius | ${R}_{exts}$ | 67 mm |

Stator inner radius | ${R}_{ints}$ | 35.8 mm |

Stator slot radius | ${R}_{slot}$ | 59.7 mm |

Rotor outer radius | ${R}_{extr}$ | 35.3 mm |

Rotor slot radius | ${R}_{extr}^{{}^{\prime}}$ | 34.8 mm |

Stack length | ${l}_{a}$ | 100 mm |

Stator slot pitch | ${\tau}_{s}$ | 30° |

Rotor slot pitch | ${\tau}_{r}$ | 36° |

Stator tooth opening | ${\alpha}_{th}$ | 25.2° |

Stator slot opening | ${\alpha}_{s}$ | 4.8° |

Stator PM opening | ${\alpha}_{PM}$ | 4° |

Rotor tooth opening | ${\alpha}_{r}$ | 16.92° |

RMF Dominant Harmonic Rank | PM Harmonic RMF Amplitude (T) | Rotation Direction | Armature Harmonic RMF Amplitude (T) | Rotation Direction |
---|---|---|---|---|

4 | 0.33 | forward | 0.36 | forward |

6 | 1.01 | constant | 0.07 | static |

8 | 0.17 | backward | 0.23 | backward |

14 | 0.14 | forward | 0.18 | forward |

16 | 0.42 | forward | 0.14 | forward |

18 | 0.48 | constant | 0.1 | static |

28 | 0.22 | forward | 0.052 | forward |

**Table 3.**Comparison of the analytically- and FEA-predicted torque-angle characteristics with and without injection of harmonic currents of rank 8.

Analytic Torque | FEA Torque | |||
---|---|---|---|---|

Mean (Nm) | Ripple (%) | Mean (Nm) | Ripple (%) | |

with injection of 8th harmonic | 15.2 | 6.9 | 15 | 11.9 |

without injection of 8th harmonic | 13.9 | 7.5 | 13.8 | 10.8 |

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**MDPI and ACS Style**

Abdelkefi, A.; Souissi, A.; Abdennadher, I.; Masmoudi, A.
On the Analysis and Torque Enhancement of Flux-Switching Permanent Magnet Machines in Electric Power Steering Systems. *World Electr. Veh. J.* **2022**, *13*, 64.
https://doi.org/10.3390/wevj13040064

**AMA Style**

Abdelkefi A, Souissi A, Abdennadher I, Masmoudi A.
On the Analysis and Torque Enhancement of Flux-Switching Permanent Magnet Machines in Electric Power Steering Systems. *World Electric Vehicle Journal*. 2022; 13(4):64.
https://doi.org/10.3390/wevj13040064

**Chicago/Turabian Style**

Abdelkefi, Anis, Amal Souissi, Imen Abdennadher, and Ahmed Masmoudi.
2022. "On the Analysis and Torque Enhancement of Flux-Switching Permanent Magnet Machines in Electric Power Steering Systems" *World Electric Vehicle Journal* 13, no. 4: 64.
https://doi.org/10.3390/wevj13040064