# Virtual Flux Voltage-Oriented Vector Control Method of Wide Frequency Active Rectifiers Based on Dual Low-Pass Filter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PWM Rectifier Model Based on Virtual Flux

#### 2.1. Mathematical Model of Three-Phase PWM Rectifier

_{a}, V

_{b}, V

_{c}are the three-phase AC phase voltages, i

_{a}, i

_{b}, i

_{c}are the three-phase AC currents, S

_{1}-S

_{6}are the six switch signals, V

_{dc}is the rectifier DC side voltage, L

_{f}is the filter inductor, R

_{f}is the sum of the resistance of the filter inductor and the equivalent resistance of the switching losses, and R

_{L}is the load resistance.

_{Ra}, V

_{Rb}, V

_{Rc}indicate the voltage on the side of the PWM rectifier. The parameters in the above equation are transformed into an αβ two-phase static coordinate system by orthogonal mapping, assuming that the line impedance is 0. The mathematical model of the rectifier in the static coordinate system is collated as follows:

_{Rα}and V

_{Rβ}are the projection of the rectifier-side voltage vector in a two-phase static coordinate system, which can be represented by the DC-side voltage and the switching function. The equation is:

_{a}, S

_{b}, and S

_{c}are the switching functions of the three-phase rectifier, respectively. Taking the A phase as an example, S

_{a}= 1 indicates that the upper bridge arm is on and the lower converter arm is off. Likewise, S

_{a}= 0 indicates that the lower bridge arm is on and the upper converter arm is off. The other two-phase bridge arm switching functions are identical.

#### 2.2. Model of Mathematics Based on Virtual Flux

_{α}and ψ

_{β}are components of virtual power grid flux in the two-phase static coordinate system. Substituting the virtual flux component of the α-axis β-axis into Equation (4) gives:

## 3. Wide Frequency Active Rectifier Control Strategy

#### 3.1. Problems with Traditional Virtual Flux Observers

_{e}is the three-phase power supply angular frequency, B is the amount of DC bias error, ϕ is the initial phase angle of the signal and A is the sinusoidal signal maximum.

_{1}and λ

_{2}are as follows:

_{c}. The smaller the cut-off frequency ω

_{c}, the smaller the amplitude and phase error of the observed virtual flux. However, the filter becomes worse at suppressing DC bias errors. The low-pass filter does not fully eliminate the effect of DC errors for virtual flux observations, and the disadvantages of the integrator are not fully solved.

#### 3.2. Dual Low-Pass Filter Design

**V**

_{R}

^{*}is the integral value of the rectifier’s reconfigured voltage. The output of the rectifier-side voltage after the dual low-pass filter is as follows:

**V**

_{Rd}

^{*}is the integral value of the rectifier’s reconfigured voltage using a dual low-pass filter. Combining Equations (13) and (14) gives the following compensation equation:

_{R}

_{α}is always ahead of V

_{R}

_{β}in the time coordinate system by an angle of 90°. The mathematical relationship is as follows:

#### 3.3. Phase Angle Calculation and Instantaneous Phase-Locked Loop Design

## 4. Experimental Results and Analysis

#### 4.1. Start-Up Comparison Experiments

#### 4.2. Comparison Experiments of Steady-State of the System

_{s90}is the observed virtual flux phase angle, and ϕ

_{α}and ϕ

_{β}are the components of the virtual flux in the static coordinate system. Figure 8 shows that the virtual flux phase angle estimation of the method proposed in this paper was more accurate for steady-state conditions at different frequencies because of the added phase compensation. The uncompensated method had a significant deviation in the virtual flux estimation scheme. In this paper, the cut-off frequency used in the traditional strategy takes a larger value, and it can be seen from Equation (9) that there is no DC bias problem when the cut-off frequency is overly large. The graph shows that the virtual flux of the dual low-pass filter control strategy had essentially the same amplitude and there was no DC bias error.

## 5. Conclusions

- The proposed method eliminates the need to calculate the initial value of the virtual flux by using a dual low-pass filter for integration, without the problems of DC bias errors or amplitude-phase error.
- The strategy uses an instantaneous phase-locked loop that can quickly detect the frequency and feedback to the dual low-pass filter. Therefore, the strategy ensures the normal operation of the power system of More Electric Aircraft under wide frequency conditions.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 7.**Experimental comparison of traditional and proposed control strategies at start-up: (

**a**) traditional scheme at 50 Hz power supply frequency; (

**b**) proposed scheme at 50 Hz power supply frequency; (

**c**) traditional scheme at 100 Hz power supply frequency; (

**d**) proposed scheme at 100 Hz power supply frequency; (

**e**) traditional scheme at 200 Hz power supply frequency; (

**f**) proposed scheme at 200 Hz power supply frequency.

**Figure 8.**Phase angle observation and DC bias error experiments: (

**a**) power supply frequency of 50 Hz; (

**b**) Power supply frequency of 100 Hz; (

**c**) power supply frequency of 200 Hz.

Parameter | Symbols | Values |
---|---|---|

DC-side voltage | V_{dc} | 20 V |

AC side phase voltage | V_{a} | 5.5 V |

Filter inductor | L_{f} | 2.1 mH |

Stray resistors | R_{f} | 0.1 Ω |

Resistive load | R_{dc} | 72.9 Ω |

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**MDPI and ACS Style**

Bi, K.; Xu, Y.; Zeng, P.; Chen, W.; Li, X.
Virtual Flux Voltage-Oriented Vector Control Method of Wide Frequency Active Rectifiers Based on Dual Low-Pass Filter. *World Electr. Veh. J.* **2022**, *13*, 35.
https://doi.org/10.3390/wevj13020035

**AMA Style**

Bi K, Xu Y, Zeng P, Chen W, Li X.
Virtual Flux Voltage-Oriented Vector Control Method of Wide Frequency Active Rectifiers Based on Dual Low-Pass Filter. *World Electric Vehicle Journal*. 2022; 13(2):35.
https://doi.org/10.3390/wevj13020035

**Chicago/Turabian Style**

Bi, Kai, Yamei Xu, Pin Zeng, Wei Chen, and Xinmin Li.
2022. "Virtual Flux Voltage-Oriented Vector Control Method of Wide Frequency Active Rectifiers Based on Dual Low-Pass Filter" *World Electric Vehicle Journal* 13, no. 2: 35.
https://doi.org/10.3390/wevj13020035