# Design of an Observer-Based Architecture and Non-Linear Control Algorithm for Cogging Torque Reduction in Synchronous Motors

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## Abstract

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## 1. Introduction

## 2. Cogging Torque Modeling and Sensorless Power Drive System

## 3. State-of-Art on Cogging Torque Reduction

## 4. Design of the Control System

## 5. Simulation Results

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the magnetic interaction points in the internal structure of a synchronous motor.

**Figure 4.**Schematic representation of the sensor-less architecture with extended Kalman filter (EKF) designed in three-phase axes.

**Figure 7.**Transitory phase of the $\vartheta $ trend with the variation of initial conditions in a limited range.

**Figure 8.**Error in desired position tracking and zoom of the transitory phase in the time interval 0–5 s.

**Figure 9.**Result of the angular speed estimation in closed loop and zoom in the time interval 15–20 s.

**Figure 10.**Result of the speed estimation error with details in the interval time 0–5 s and 15–20 s.

Model Parameter | Values |
---|---|

${R}_{s}\left(statorresistance\right)$ | $3.3\mathrm{Ohm}$ |

${L}_{eq}\left(statorinductance\right)$ | $50\mathrm{mH}$ |

${k}_{\phi}\left(pmfluxintensity\right)$ | $0.5\mathrm{Wb}$ |

${J}_{m}\left(rotorinertia\right)$ | $0.02{\mathrm{Kgm}}^{2}$ |

$p\left(polepair\right)$ | $3$ |

$\beta \left(frictioncoefficient\right)$ | $0.01\mathrm{Ns}/\mathrm{m}$ |

Data Sheet Parameters | Values |
---|---|

${P}_{n}$ (Nominal Power) | $15\mathrm{k}\mathrm{W}$ |

${\omega}_{n}$ (Nominal Speed) | $1500\mathrm{r}\mathrm{p}\mathrm{m}$ |

${\omega}_{max}$ (Maximum Speed) | $2000\mathrm{r}\mathrm{p}\mathrm{m}$ |

${T}_{n}$ (Nominal Torque) | $95.5\mathrm{N}\mathrm{m}$ |

${T}_{s}$ (Stall Torque) | $230\mathrm{N}\mathrm{m}$ |

Harmonic Expansion Coefficients | Values |
---|---|

${T}_{1}$ | $4.85\mathrm{N}$ |

${T}_{2}$ | $2.04\mathrm{N}$ |

${T}_{3}$ | $0.3\mathrm{N}$ |

${T}_{4}$ | $0.006\mathrm{N}$ |

${\alpha}_{1}$ | $0.009\mathrm{rad}$ |

${\alpha}_{2}$ | $0.01\mathrm{rad}$ |

${\alpha}_{3}$ | $0.017\mathrm{rad}$ |

${\alpha}_{4}$ | ${\alpha}_{3}$ |

Summary Profiler Field | Value |
---|---|

${T}_{rt}$ | $2741.70\mathrm{s}$ |

${N}_{bm}$ | $918$ |

${N}_{im}$ | $9$ |

${N}_{mm}$ | $13$ |

${N}_{nsm}$ | $46$ |

${C}_{p}$ | $30\mathrm{ns}$ |

${C}_{s}$ | $3800\mathrm{MHz}$ |

Functional Block | Time | Calls | Time/Call | Self-Time | ||
---|---|---|---|---|---|---|

Non-Linear Control | 43.8672 | 1.6% | 400036 | $~$0.0001184 | 30158.7 | 1.1% |

Kalman Filter | 265,047 | 9.7% | 2800221 | $~$0.0000947 | 137.578 | 5.0% |

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**MDPI and ACS Style**

Dini, P.; Saponara, S.
Design of an Observer-Based Architecture and Non-Linear Control Algorithm for Cogging Torque Reduction in Synchronous Motors. *Energies* **2020**, *13*, 2077.
https://doi.org/10.3390/en13082077

**AMA Style**

Dini P, Saponara S.
Design of an Observer-Based Architecture and Non-Linear Control Algorithm for Cogging Torque Reduction in Synchronous Motors. *Energies*. 2020; 13(8):2077.
https://doi.org/10.3390/en13082077

**Chicago/Turabian Style**

Dini, Pierpaolo, and Sergio Saponara.
2020. "Design of an Observer-Based Architecture and Non-Linear Control Algorithm for Cogging Torque Reduction in Synchronous Motors" *Energies* 13, no. 8: 2077.
https://doi.org/10.3390/en13082077