# Torque Ripple Reduction of DTC Based on an Analytical Model of PMSM

^{*}

## Abstract

**:**

## 1. Introduction

## 2. SVPWM-DTC

#### 2.1. Principle of SVPWM-DTC

**,**and ${\psi}_{s}$ are stator voltage, current, and flux linkage vector, respectively; R represents the stator resistance. When the control period is small enough, the voltage equation can be discretized as:

#### 2.2. Framework of SVPWM-DTC

**.**The key of SVPWM-DTC is to calculate the voltage that is required for the next control cycle by flux linkage in present control cycle. The maximum torque per ampere (MTPA) is a look-up table that transfers the reference torque ${t}_{e}^{*}$ into reference current ${i}_{d}^{*}$ and ${i}_{q}^{*}$, and then ${\psi}_{s}^{*}$ could be obtained by (4). The difference between the reference torque and the real torque feedback by torque observer is the input of PI controller, which outputs the increment of load angle $\mathsf{\Delta}{\delta}_{\mathrm{sf}}$ and $\rho \left(k+1\right)$ could then be calculated by (5). The expected flux linkage ${\psi}_{s}\left(k+1\right)$ is then obtained. The flux linkage observer calculates the stator flux linkage ${\psi}_{\mathsf{\alpha}}$, ${\psi}_{\beta}$ and feedback to the expected voltage calculation module. The torque and flux linkage observers are significant to DTC. A precise torque and flux linkage observer based on the analytical model of PMSM was applied in order to verify the proposed algorithm better. The observer takes the nonlinear characteristic of PMSM into account and requires not complex look-up tables, but simplified matrix operations. The voltage that is required for next control cycle ${u}_{s}\left(k+1\right)$ could be obtained by (3). The inverter outputs the required voltage vector by the SVPWM algorithm in the next control cycle and the torque control of PMSM is realized.

## 3. Feedforward Controller Based on Analytical Model of PMSM

#### 3.1. Analytical Model of PMSM

#### 3.2. Calculation of Optimal Stator Flux Linkage

#### 3.3. Feedword Controller

## 4. Simulation and Experiment Result

#### 4.1. Simulation Result

#### 4.2. Applicability Analysis

#### 4.3. Experiment Result

^{−4}. CAN realized the communication between the controller and the computer. The working point is the same as the simulation. The control algorithms with and without feedforward controller are separately tested under the same condition. The torque response was measured by the torque transducer and Figure 8 and Figure 9 show the results.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Simulation results without feedforward controller. (

**a**) Torque wave; (

**b**) Fast Fourier transform (FFT) analysis of torque wave.

**Figure 4.**Simulation results with feedforward controller. (

**a**) Torque wave; (

**b**) FFT analysis of torque wave.

**Figure 8.**Experiment results without feedforward controller. (

**a**) Torque wave; and, (

**b**) FFT analysis of torque wave.

**Figure 9.**Experiment results with feedforward controller. (

**a**) Torque wave; and, (

**b**) FFT analysis of torque wave.

**Table 1.**Parameters of finite element analysis permanent magnet synchronous motors (FEA PMSM) model.

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

Maximum power | 60 kW | Rated torque | 50 Nm |

Rated speed | 6000 rpm | Pole pairs | 3 |

Stator outer diameter | 180 mm | Stator inner diameter | 105 mm |

Rotor outer diameter | 103.6 mm | Rotor inner diameter | 50 mm |

PM width | 35 mm | PM thickness | 10 mm |

Parameter | $\mathbf{Angular}\text{}\mathbf{Velocity}\text{}{\mathit{\omega}}_{\mathit{r}}$ | Period T | $\mathbf{Torque}\text{}{\mathit{t}}_{\mathit{e}}^{*}$ |
---|---|---|---|

Value | 500 rpm | 1 × 10^{−4}s | 5 Nm |

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## Share and Cite

**MDPI and ACS Style**

Zhong, Z.; You, J.; Zhou, S. Torque Ripple Reduction of DTC Based on an Analytical Model of PMSM. *World Electr. Veh. J.* **2020**, *11*, 28.
https://doi.org/10.3390/wevj11010028

**AMA Style**

Zhong Z, You J, Zhou S. Torque Ripple Reduction of DTC Based on an Analytical Model of PMSM. *World Electric Vehicle Journal*. 2020; 11(1):28.
https://doi.org/10.3390/wevj11010028

**Chicago/Turabian Style**

Zhong, Zaimin, Junming You, and Shuihua Zhou. 2020. "Torque Ripple Reduction of DTC Based on an Analytical Model of PMSM" *World Electric Vehicle Journal* 11, no. 1: 28.
https://doi.org/10.3390/wevj11010028