#
Performance Enhancement of Direct Torque and Rotor Flux Control (DTRFC) of a Three-Phase Induction Motor over the Entire Speed Range: Experimental Validation^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Theories and Principles

## 3. The Effect of VVs at Low Speeds in the DTRFC Algorithm

## 4. The Effect of VVs at High Speeds in DTRFC Algorithm

## 5. Determination of UCAs Values at Low and High Speeds

## 6. The Proposed 18 SSs DTRFC Strategy for Medium High Speeds

- For the first sub-sector, shown in ${\mathrm{SS}}_{1}$ Figure 7a, it can easily be seen that the four effective vectors $\left({\mathrm{V}}_{2},\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3},\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5},\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}\right)$ achieve the optimum condition for the two controlled functions ${\mathrm{dS}}_{\mathsf{\Phi}\mathrm{r}}\phantom{\rule{3.33333pt}{0ex}}$ and $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{dS}}_{\mathrm{Tem}}$ because their sign does not reverse over the entire ${\mathrm{SS}}_{1}$.
- For the second sub-sector, shown in ${\mathrm{SS}}_{2}$ Figure 7b, the choice of the two vector $\left({\mathrm{V}}_{5},\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}\right)$ achieve the optimum condition of the control requirements for the two controlled functions ${\mathrm{dS}}_{\mathsf{\Phi}\mathrm{r}}\phantom{\rule{3.33333pt}{0ex}}$ and $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{dS}}_{\mathrm{Tem}}$ in the first and second quadrants, respectively. However, there is only one choice in the third and fourth quadrants, which is the vector $\left({\mathrm{V}}_{3}\right)$. The latter vector fulfils the optimum control requirements for the torque without achieving that for the rotor flux as it causes UCA.
- For the third sub-sector, shown in ${\mathrm{SS}}_{3}$ Figure 7c, it can easily be seen that the four effective VVs $\left({\mathrm{V}}_{3},\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4},\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6},\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}\right)$ achieve the optimum condition for the two controlled functions ${\mathrm{dS}}_{\mathsf{\Phi}\mathrm{r}}\phantom{\rule{3.33333pt}{0ex}}$ and $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{dS}}_{\mathrm{Tem}}$ as long as they maintain their sign over the entire ${\mathrm{SS}}_{3}$.

- 1.
- ${\mathrm{SS}}_{1},{\mathrm{SS}}_{4},{\mathrm{SS}}_{7},{\mathrm{SS}}_{10},{\mathrm{SS}}_{13},{\mathrm{SS}}_{16}.$
- 2.
- ${\mathrm{SS}}_{2},{\mathrm{SS}}_{5},{\mathrm{SS}}_{8},{\mathrm{SS}}_{11},{\mathrm{SS}}_{14},{\mathrm{SS}}_{17}.$
- 3.
- ${\mathrm{SS}}_{3},{\mathrm{SS}}_{6},{\mathrm{SS}}_{9},{\mathrm{SS}}_{12},{\mathrm{SS}}_{15},{\mathrm{SS}}_{18}.$

## 7. Determination of the Transition Speed ${\mathbf{\omega}}_{\mathbf{T}}$ between the Conventional and Improved Strategy

## 8. Comparison of Current Case Studies toward the Proposed 18-SS DTRFC

## 9. Simulation Results

## 10. Hardware Experimental Results

## 11. Conclusions

- The use of a normal 6-sector DTRFC technique allows for simplicity at low speeds.
- High performance at medium and high speeds is due to the elimination of the uncontrollable angles of the torque response.
- Accurate analytical determination of the transition speed between traditional and improved strategy.
- The switching table is based on the analysis’s findings, not on designer experience.
- There is no increase in the switching frequency due to the increased number of sectors.
- Robustness to the parameters variations.
- Lower chattering of the torque at high speed when compared with classical strategy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Abbreviation | Definition |
---|---|

PWM | Pulse Width Modulation |

FOC | Field-Oriented Control |

DTC | Direct Torque Control |

DTRFC | Direct Torque and Rotor Flux Control |

UCAs | Uncontrollable angles |

IM | Induction Motor |

PMSM | Permanent Magnet Synchronous Motor |

CSFC | Constant Switching Frequency Controller |

VVs | Voltage Vectors |

VSI | Voltage Source Inverter |

HCs | Hysteresis Controllers |

LUT | Lookup Table |

SS | Sub-Sector |

## Appendix B

Variable | Unit Value |
---|---|

Nominal voltage | 230/400 V |

Phase resistance stator | ${\mathrm{R}}_{\mathrm{s}}=45.83\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

Phase resistance rotor | ${\mathrm{R}}_{\mathrm{r}}=31\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ |

Phase inductance stator | ${\mathrm{L}}_{\mathrm{s}}=1.24\phantom{\rule{3.33333pt}{0ex}}\mathrm{H}$ |

Phase inductance rotor | ${\mathrm{L}}_{\mathrm{r}}=1.11\phantom{\rule{3.33333pt}{0ex}}\mathrm{H}$ |

Mutual inductance | ${\mathrm{L}}_{\mathrm{m}}=1.05\phantom{\rule{3.33333pt}{0ex}}\mathrm{H}$ |

Inertia | $\mathrm{J}$ = 0.006 kg·m${}^{2}$ |

Friction factor | $\mathrm{F}$ = 0.001 N·m.s/rad |

Number of poles pairs | $\mathrm{P}$ = 2 |

Nominal stator flux | ${\mathsf{\Phi}}_{\mathrm{s}}$ = 1.14 Wb |

Nominal rotor flux | ${\mathsf{\Phi}}_{\mathrm{r}}$ = 0.945 Wb |

Nominal power | ${\mathrm{P}}_{\mathrm{n}}$ = 0.25 kW |

Nominal frequency | $\mathrm{F}$ = 50 Hz |

Nominal speed | ${\omega}_{\mathrm{n}}$ = 282 rad/s |

Nominal torque | ${\mathrm{T}}_{\mathrm{em}}$ = 1.76 N$\xb7\mathrm{m}$ |

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**Figure 2.**Values of ${\mathrm{dS}}_{\mathsf{\Phi}\mathrm{r}}\phantom{\rule{3.33333pt}{0ex}}$ and $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{dS}}_{\mathrm{Tem}}$ for low positive speed $\left(20\%\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}\right)$.

**Figure 3.**Values of ${\mathrm{dS}}_{\mathsf{\Phi}\mathrm{r}}\phantom{\rule{3.33333pt}{0ex}}$ and $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{dS}}_{\mathrm{Tem}}$ for positive high speed $\left(75\%\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}\right)$.

**Figure 4.**UCA of ${\mathrm{V}}_{\mathrm{i}+1}$ or ${\mathrm{V}}_{\mathrm{i}-1}$ on ${\mathrm{dS}}_{\mathsf{\Phi}r}$ for low speed $\left(20\%\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}\right)$.

**Figure 5.**UCA of ${\mathrm{V}}_{\mathrm{i}+2}$ or ${\mathrm{V}}_{\mathrm{i}-2}$ on ${\mathrm{dS}}_{\mathsf{\Phi}r}$ for high speed $\left(75\%\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}\right)$.

**Figure 6.**UCA of ${\mathrm{V}}_{\mathrm{i}+2}$ or ${\mathrm{V}}_{\mathrm{i}+1}$ on ${\mathrm{dS}}_{\mathrm{Tem}}$ for high speed $\left(75\%\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}\right)$.

**Figure 7.**The errors derivatives of the rotor flux and torque for the proposed strategy at high speed $\left(75\%\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}\right)$: (

**a**) SS1, (

**b**) SS2, (

**c**) SS3.

**Figure 8.**Block diagram of the conventional six sectors and proposed 18 SSs DTRFC strategies operating over the entire speed range.

**Figure 9.**Change of errors of the rotor flux and torque for the range speed (0:63%$\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}$).

**Figure 19.**The torque response of both the classical and improved strategies at high speed is equal to the nominal speed.

**Figure 24.**The switching frequency of the conventional and proposed strategies over the speed range.

**Figure 26.**Experimental results of 6-sector DTRFC strategy: (

**a**) torque response; and (

**b**) three-phase currents.

**Figure 27.**Flux at high speed ($75\%\phantom{\rule{3.33333pt}{0ex}}{\omega}_{n}$): (

**a**) Stator flux components, (

**b**) Rotor flux components.

**Figure 29.**(

**a**) Torque response towards VVs for the two strategies and (

**b**) zoomed torque for 18-SS strategy.

**Figure 30.**Experimental results of the two strategies: (

**a**): stator currents; (

**b**): rotor flux response.

**Figure 31.**The components in synchronous reference frame: (

**a**) stator and rotor fluxes responses; (

**b**) stator currents responses.

**Figure 34.**The DTC strategy with the adaptive estimator: (

**a**) Speed response, (

**b**) Torque response, (

**c**) Stator flux, (

**d**) Stator currents, (

**e**) Stator resistance.

${\mathit{S}}_{\mathit{Tem}}$ | ${\mathit{S}}_{{\mathbf{\Phi}}_{\mathit{r}}}$ | 6-Sector Index | |||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

0 | 0 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ |

1 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | |

1 | 0 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ |

1 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ |

Sub-Sectors Index | ||
---|---|---|

${\mathbf{SS}}_{\mathbf{1}}$ | ${\mathbf{SS}}_{\mathbf{2}}$ | ${\mathbf{SS}}_{\mathbf{3}}$ |

0${}^{\circ}$:15${}^{\circ}$ | 15${}^{\circ}$:45${}^{\circ}$ | 45${}^{\circ}$:60${}^{\circ}$ |

$\vdots \vdots $ | $\vdots \vdots $ | $\vdots \vdots $ |

${\mathbf{SS}}_{\mathbf{1}}\mathbf{6}$ | ${\mathbf{SS}}_{\mathbf{1}}\mathbf{7}$ | ${\mathbf{SS}}_{\mathbf{1}}\mathbf{8}$ |

300${}^{\circ}$:315${}^{\circ}$ | 315${}^{\circ}$:345${}^{\circ}$ | 345${}^{\circ}$:360${}^{\circ}$ |

${\mathit{S}}_{\mathit{Tem}}$ | ${\mathit{S}}_{{\mathsf{\Phi}}_{\mathit{r}}}$ | Sub-Sectors Index | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | ||

0 | 0 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ |

1 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | |

1 | 0 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ |

1 | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{3}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{4}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{5}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{6}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{1}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ | $\phantom{\rule{3.33333pt}{0ex}}{\mathrm{V}}_{2}$ |

Operating Parameters Values | Classical 6-Sector DTRFC | Improved 18-SS DTRFC |
---|---|---|

Torque ripples (low and medium speed) | 0.55 N.m | 0.55 N.m |

Torque ripples (High speed) | 0.75 N.m | 0.55 N.m |

Flux ripples | 0.01 Wb | 0.013 Wb |

Switching frequency (low and medium speed) | 6 to 7.8 kHz | 6 to 7.8 kHz |

Switching frequency (nominal speed) | 3 kHz | 2.5 kHz |

Parameters variations | Weak | Strong |

Lookup Table construction | Designer’s knowledge | Analytically |

UCAs | Exist | Fully eliminated |

Computational time | 18.5 $\mathsf{\mu}$s | 22.3 $\mathsf{\mu}$s |

Speed Range (rad /s) | Classical DTC | Proposed DTC |
---|---|---|

100 | 6100 Hz | 6000 Hz |

150 | 5350 Hz | 5100 Hz |

200 | 4215 Hz | 3900 Hz |

250 | 3250 Hz | 2850 Hz |

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## Share and Cite

**MDPI and ACS Style**

Alshbib, M.M.; Elgbaily, M.M.; Alsofyani, I.M.; Anayi, F.
Performance Enhancement of Direct Torque and Rotor Flux Control (DTRFC) of a Three-Phase Induction Motor over the Entire Speed Range: Experimental Validation. *Machines* **2023**, *11*, 22.
https://doi.org/10.3390/machines11010022

**AMA Style**

Alshbib MM, Elgbaily MM, Alsofyani IM, Anayi F.
Performance Enhancement of Direct Torque and Rotor Flux Control (DTRFC) of a Three-Phase Induction Motor over the Entire Speed Range: Experimental Validation. *Machines*. 2023; 11(1):22.
https://doi.org/10.3390/machines11010022

**Chicago/Turabian Style**

Alshbib, Mussaab M., Mohamed Mussa Elgbaily, Ibrahim Mohd Alsofyani, and Fatih Anayi.
2023. "Performance Enhancement of Direct Torque and Rotor Flux Control (DTRFC) of a Three-Phase Induction Motor over the Entire Speed Range: Experimental Validation" *Machines* 11, no. 1: 22.
https://doi.org/10.3390/machines11010022