# A Position Sensorless Control Strategy for BLDCM Driven by FSTPI Based on Flux-Linkage Function

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{ao,}and u

_{bo}, and the zero crossing point of the voltage function corresponds to the commutation point of the motor. This method does not need to determine the commutation point of the motor by shifting the phase by 30 electrical degrees or 90 electrical degrees. However, this method uses a low-pass filter to eliminate the high-frequency noise signal in the phase voltage. And this method neglects the voltage drop on the resistance and inductance. It leads to inaccurate position detection and bigger errors in the case of low speed, heavy load, and unsatisfactory commutation current. Three error functions are introduced in [24]. Six commutation points are obtained by detecting the zero crossing point of the error function and then delaying 30 electrical degrees. This method needs to collect the nonconducting phase voltage. It can not use when the freewheeling commutation angle exceeds 30 electrical degrees at high speed. And the current of the capacitor middle point connection phase of the FSTPI is uncontrollable. This method does not consider the resistance and inductance voltage drop of the capacitor middle point connection phase, and there is some error. In addition, there are some position sensorless control methods for PMSM driven by FSTPI. In [25], a position sensorless speed control method based on stator feedforward dq-axes voltage control has been proposed for PMSM driven by FSTPI. This method is simpler, more effective, and leads to a lower implementation processing time, but its control is more complex and not suitable for BLDCM, where the control method is simple. In [26], a simple and smooth zero-angle rotor initial position alignment scheme using a low-frequency signal injection that is applied to the d-axis voltage reference is proposed for PMSM driven by FSTPI. However, this method cannot detect the rotor position in real-time during the operation of the motor.

## 2. BLDCM Driven by FSTPI System

_{1}and C

_{2}are the DC-link capacitors.

_{m}, u

_{m}, e

_{m}(m = a, b, c) represent three-phase current, three-phase terminal voltage, and three-phase back EMF, respectively.

^{*}is the expected current amplitude.

_{a}is less than the reference current, control the power transistor T

_{1}to conduct, and the current i

_{a}increases. When the current i

_{a}is greater than the reference current, control the power transistor T

_{2}to conduct, and the current i

_{a}decreases. Similarly, when the current i

_{b}is less than the reference current, control the power transistor T

_{3}to conduct, and the current i

_{b}increases. When the current i

_{b}is greater than the reference current, control the power transistor T

_{4}to conduct, and the current i

_{b}decreases. When the motor operates in other modes, the c-phase is the conducting phase, and the a-phase or b-phase is the nonconducting phase. Therefore, the nonconducting phase current can be controlled to zero by turning off the bridge arm power transistor of the nonconducting phase in a-phase or b-phase. The conducting phase current can be made the reference value by hysteresis control of the bridge arm power transistor of the conducting phase.

## 3. Proposed Position Sensorless Control Method of BLDCM Driven by FSTPI

#### 3.1. Construction and Analysis of Flux-Linkage Function

_{ij}, λ

_{ij}are called the line-to-line back EMF and the line-to-line PM flux linkage.

_{e}is the back EMF coefficient; f

_{ij}is the line-to-line PM flux linkage form function that is a function of rotor position.

_{1}, F

_{2}, and F

_{3}as the ratio of the line-to-line PM flux linkage turns to estimate the rotor-commutation point. The functions F

_{1}, F

_{2}, and F

_{3}as

_{1}, F

_{2}, and F

_{3}can be obtained as

_{1}, F

_{2}, F

_{3}, and the commutation sign. As can be seen from Figure 4, the function F

_{1}, F

_{2}, F

_{3}has the following characteristics:

- (1)
- The jumping of the functions F
_{1}, F_{2}, and F_{3}from positive infinity to negative infinity is only two in each electrical cycle and coincides with the commutation time after a delay of 30 electrical degrees. - (2)
- In an operating mode, the function curve is similar to a hyperbola, and its functional characteristics are theoretically independent of the motor speed. Therefore, the shape of the function is the same in the entire motor speed range.
- (3)
- At the beginning of each operating mode, the value of the function changes slowly. When λ approaches zero, the value of the function changes quickly, and the extremum of the function jumps.

#### 3.2. Position Sensorless Control Strategy Based on Flux-Linkage Function

_{a}is less than the reference current, the power transistor T

_{1}is on, and the current i

_{a}increases. Conversely, the power transistor T

_{2}conducts, and the current i

_{a}decreases. When the current i

_{b}is less than the reference current, the power transistor T

_{3}conducts, and the current i

_{b}increases. Conversely, power transistor T

_{4}conducts, and current i

_{b}decreases. In this current control method, the controller can calculate the terminal voltage based on the switching signal, the bus voltage, and the capacitor voltage. For example, the equivalent circuit diagram of power transistor T

_{1}and T

_{4}conduction is shown in Figure 5.

_{2}. Therefore, the controller can obtain the terminal voltage from the switching signal, the bus voltage, and the capacitor voltage as

_{d}is the DC bus voltage, u

_{C2}is the voltage on capacitor C

_{2}, S

_{a}(t), S

_{b}(t), S

_{c}(t) represent the state of the inverter’s three-phase bridge arm power transistor, S

_{k}(t) = 1 (k = a,b,c) represents the upper bridge arm power transistor open, S

_{k}(t) = 0 (k = a,b,c) represents the lower bridge arm power transistor open.

_{2}to calculate the terminal voltage of each phase winding based on the switching state of the power transistor and then obtain the function F

_{1}, F

_{2}, F

_{3}. This approach simplifies the terminal voltage detection circuit design and also avoids the measurement errors caused by the terminal voltage detection circuit.

_{1}, F

_{2}, and F

_{3}change slowly at the beginning of each operating mode and change quickly as λ approaches zero. This feature is very beneficial for the detection of commutation points. Since the function has two extremum jump moments in each electrical cycle of the motor, each function can provide two commutation signals in one electrical cycle. Since the BLDCM needs six commutation signals during operation, the controller can use the F

_{1}function, F

_{2}function, and F

_{3}function to detect the extremum jump moment in different operating modes. In turn, the controller delays the extremum jump moment by 30 electrical degrees to obtain the next commutation point. According to the different commutation points required in each operating mode, the corresponding flux-linkage functions for each operating mode of the BLDCM driven by FSTPI can be obtained, as shown in Table 2.

_{a}

^{*}and i

_{b}

^{*}is the a-phase expected current and b-phase expected current.

_{c}is the cut-off frequency.

_{s}is the angular frequency.

## 4. Experimental Results and Analysis

_{2}, the three-phase current, the flux-linkage function, the Hall signal, and the commutation signal CP are shown from top to bottom in Figure 10. In the experiment, the controller gets the commutation point by detecting when the flux-linkage function changes from positive to negative polarity and delaying 30 electrical degrees.

## 5. Conclusions

- (1)
- Six phase commutation points are obtained using the speed-independent PM flux linkage without interpolation, and good commutation accuracy is guaranteed at all speeds except for very low speeds.
- (2)
- There is no need to obtain the commutation point by setting the threshold value, which can reduce the commutation error caused by unreasonable threshold setting. And the flux-linkage function increases significantly near the extremum point, which increases the reliability of commutation point detection.
- (3)
- The three-phase current control method suppresses the nonconducting phase current distortion. On this basis, the terminal voltage required to calculate the flux-linkage function by the switching signal avoids the error caused by hardware sampling.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**The position sensorless control system for BLDCM driven by FSTPI based on flux-linkage function.

Operating Mode | Operating Phase | Expected Current | Operating Devices |
---|---|---|---|

I | +a, −b | i_{a} = I^{*}, i_{b} = −I^{*} | T_{1}, T_{4} |

II | +a, −c | i_{a} = I^{*}, i_{c} = −I^{*} | T_{1} |

III | +b, −c | i_{b} = I^{*}, i_{c} = −I^{*} | T_{3} |

IV | +b, −a | i_{b} = I^{*}, i_{a} = −I^{*} | T_{2}, T_{3} |

V | +c, −a | i_{c} = I^{*}, i_{a} = −I^{*} | T_{2} |

VI | +c, −b | i_{c} = I^{*}, i_{b} = −I^{*} | T_{4} |

Operating Mode | Expected Current | Flux-Linkage Function |
---|---|---|

I | i_{a} = I^{*}, i_{b} = −I^{*} | F_{1} |

II | i_{a} = I^{*}, i_{c} = −I^{*} | F_{2} |

III | i_{b} = I^{*}, i_{c} = −I^{*} | F_{3} |

IV | i_{b} = I^{*}, i_{a} = −I^{*} | F_{1} |

V | i_{c} = I^{*}, i_{a} = −I^{*} | F_{2} |

VI | i_{c} = I^{*}, i_{b} = −I^{*} | F_{3} |

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

Rated voltage U_{N} | 24 | V |

Rated current I_{N} | 14 | A |

Rated torque T_{N} | 3.2 | N·m |

Rated speed n_{N} | 600 | r/min |

Phase resistance R | 0.2415 | Ω |

Phase inductance L | 0.387 | mH |

Phase Back EMF coefficient K_{e} | 0.128 | V/(rad/s) |

Poles pairs p | 4 |

Speed | 100 rmp | 200 rmp | 300 rmp | 400 rmp | 500 rmp |
---|---|---|---|---|---|

Commutation error | 4° | 2° | 2° | 2° | 1° |

Flux-linkage function extremum | 80 | 80 | 80 | 77 | 79 |

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**MDPI and ACS Style**

Li, X.; Jiao, G.; Li, Q.; Chen, W.; Zhang, Z.; Zhang, G.
A Position Sensorless Control Strategy for BLDCM Driven by FSTPI Based on Flux-Linkage Function. *World Electr. Veh. J.* **2022**, *13*, 238.
https://doi.org/10.3390/wevj13120238

**AMA Style**

Li X, Jiao G, Li Q, Chen W, Zhang Z, Zhang G.
A Position Sensorless Control Strategy for BLDCM Driven by FSTPI Based on Flux-Linkage Function. *World Electric Vehicle Journal*. 2022; 13(12):238.
https://doi.org/10.3390/wevj13120238

**Chicago/Turabian Style**

Li, Xinmin, Guoqiang Jiao, Qiang Li, Wei Chen, Zhen Zhang, and Guozheng Zhang.
2022. "A Position Sensorless Control Strategy for BLDCM Driven by FSTPI Based on Flux-Linkage Function" *World Electric Vehicle Journal* 13, no. 12: 238.
https://doi.org/10.3390/wevj13120238