# A Reference Voltage Self-Correction Method for Capacitor Voltage Offset Suppression of Three-Phase Four-Switch Inverter-Fed PMSM Drives

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling for the TPFS Inverter-Fed PMSM Drive System

_{dc}is the system input DC source, C

_{dc}is the DC-link capacitor, C

_{1}and C

_{2}are the two capacitors on the capacitor bridge arm, whose capacitor value is C

_{1}= C

_{2}= C, and S

_{b}and S

_{c}indicate the controllable power switch devices on the bridge arm of B and C phases.

_{d}and u

_{q}are the stator voltage components of the motor in the d–q frame; i

_{d}and i

_{q}are currents in the d–q frame; R

_{s}is the stator resistance; L

_{d}and L

_{q}are the d-axis stator inductance and q-axis stator inductance; ω

_{e}is the electrical angular velocity; φ

_{f}is the stator flux linkage of the permanent magnet; p is the difference operator.

_{d}and i

_{q}are the projections of the PMSM stator current vector I

_{s}on the d–q axis, respectively, which can be expressed as

_{s}is the stator current vector amplitude, and β is the angle between the current vector and the q-axis.

_{M}is the current vector angle at the MTPA operating point; according to the literature [16], this angle varies between 0 and 0.25π.

_{1}and C

_{2}are V

_{dc1}and V

_{dc2}, respectively, the voltages u

_{AN}, u

_{BN}, and u

_{CN}at the three-phase winding endpoints A, B, and C of the motor relative to the motor midpoint N at different switching states of the TPFS inverter are

_{b}and S

_{c}are the switching states of the B and C phase bridge arms, respectively. When S

_{b}and S

_{c}are 0, it means the state in which the upper switch of the B and C phase bridge arm is not conducting, and the lower switch is conducting; when S

_{b}and S

_{c}are 1, it means the state in which the upper switch of the B and C phase bridge arm is conducting, and the lower switch is not conducting. According to the states of the power switch devices, there are four switching states of the TPFS inverter—namely, S

_{00}, S

_{10}, S

_{11}, and S

_{01}.

## 3. Capacitor Voltage Offset Suppression Strategy

#### 3.1. The Influence of TPFS Inverter Capacitor Voltage Offset

_{1}and C

_{2}are V

_{dc1}= V

_{dc}/2 + ΔV and V

_{dc2}= V

_{dc}/2 − ΔV, respectively. Substituting V

_{dc1}and V

_{dc2}into Equation (4), we can obtain the relationship between the three-phase winding voltage and the switching state when the capacitor voltage is unbalanced as

_{α}is offset by −2ΔV/3 after the Clark transformation, while the β-axis reference voltage u

_{β}is not offset.

_{α}and u

_{β}, as αβ-axis reference voltages, determine whether the correct PWM pulse signal can be output. According to Equation (5), the reference voltage u

_{α}is offset due to the capacitor voltage offset, which causes the basic voltage vector of the TPFS inverter to shift along the α-axis. The reference voltages u

_{α}and u

_{β}that generate the offset cannot be synthesized into the correct reference voltage vector according to the SVPWM strategy of the TPFS inverter. The wrong PWM pulse signal will affect the control performance of the motor.

_{00}, V

_{10}, V

_{11}, and V

_{01}. With the capacitor voltage offset taken into account, the distribution of the basic voltage vectors of the TPFS inverter is shown in Figure 2.

#### 3.2. The TPFS Inverter Capacitor Voltage Offset

#### 3.3. Reference Voltage Self-Correction

_{α}. Therefore, the correct PWM pulse signal can be guaranteed to be output by correcting the α-axis reference voltage to give good control performance of the motor.

_{dc1}and V

_{dc2}across the capacitors varying with the motor rotation as follows:

_{e}is the angular velocity of the motor, and ω

_{e}t is the rotor position angle of the motor. ΔV

_{dc}represents the DC voltage offset across the capacitors of the three-phase four-switch inverter, which is negligible, due to the equal capacitance of the two capacitors on the bridge arm of the selected capacitor in the TPFS inverter. Therefore, the voltage offset across the capacitors can be expressed as

_{α}generates an offset of −2ΔV/3 and the reference voltage u

_{β}remains unchanged, so only 2ΔV/3 needs to be compensated on the reference voltage u

_{α}to obtain the corrected α-axis reference voltage. The corrected αβ-axis reference voltage expresses as

## 4. Simulation and Experimental Results

#### 4.1. Introduction to the Experimental Platform

#### 4.2. Experimental Results for Comparision of Control Performance

_{e}, three-phase current i

_{a}i

_{b}, i

_{c}, capacitor voltage V

_{dc1}and V

_{dc2}, the amplification waveforms of three-phase current, and the amplification waveforms of capacitor voltage are shown, respectively. Figure 5a shows the experimental waveforms without capacitor voltage offset suppression. The waveforms show that the torque ripple reaches 21.5 Nm, and the three-phase current and the capacitor voltage of the TPFS inverter are unbalanced, where the capacitor voltage offset is 30 V. Figure 5b shows the experimental waveforms with the proposed method. Since the proposed method corrects the reference voltage so that the TPFS inverter synthesizes the correct reference voltage vector, the torque ripple of the proposed method is 9 Nm, compared with the experimental waveform without capacitor voltage offset suppression, which effectively reduces the torque ripple caused by the capacitor voltage offset. The three-phase current and the capacitor voltage of the TPFS inverter are balanced by the proposed method, where the capacitor voltage offset is reduced to 28 V, which effectively improves the motor control performance.

#### 4.3. Simulation Results of the Capacitor Value Variation

_{a}i

_{b}, i

_{c}is the three-phase current, and T

_{e}is the motor torque. Figure 11a–c are the simulation results for capacitor values of 2000 μF, 1500 μF, and 1000 μF, respectively. It can be seen from the simulation results that the method proposed in this paper still maintains the three-phase current balanced, and torque ripple is almost constant when the capacitor value variations.

#### 4.4. Simulation Results of Speed Transient

_{a}, i

_{b}, i

_{c}is the three-phase current, and T

_{e}is the motor torque. It can be seen from the simulation results that the motor will generate current and torque strikes but will soon return to normal when the motor speed is transient, and the motor actual speed can quickly track the reference speed. After the motor speed is stabilized, the three-phase current returns to balance, and the torque ripple returns to normal.

## 5. Conclusions

- (1).
- The proposed method effectively reduces the torque ripple while keeping the three-phase current and capacitor voltage in balance. Load increase and speed decrease have little effect on the proposed method;
- (2).
- The proposed method does not need voltage sensors or filters to extract the offset components, nor does it require complex parameter adjustment, as the algorithm is simple and easy to implement;
- (3).
- The proposed method will not affect the control performance of the motor when the capacitor value variation and the motor speed are transient.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Zhu, Y.; Tao, B.; Xiao, M.; Yang, G.; Zhang, X.; Lu, K. Luenberger Position Observer Based on Deadbeat-Current Predictive Control for Sensorless PMSM. Electronics
**2020**, 9, 1325. [Google Scholar] [CrossRef] - Lin, Z.; Li, X.; Wang, Z.; Shi, T.; Xia, C. Minimization of additional high-frequency torque ripple for square-wave voltage injection IPMSM sensorless drives. IEEE Trans. Power Electron.
**2020**, 35, 13345–13355. [Google Scholar] [CrossRef] - Chen, W.; Zeng, S.; Zhang, G.; Shi, T.; Xia, C. A modified double vectors model predictive torque control of permanent magnet synchronous motor. IEEE Trans. Power Electron.
**2019**, 34, 11419–11428. [Google Scholar] [CrossRef] - Kivanc, O.; Ozturk, S. Low-cost position sensorless speed control of PMSM drive using four-switch inverter. Energies
**2019**, 12, 741. [Google Scholar] [CrossRef] [Green Version] - Ni, K.; Hu, Y.; Liu, Y.; Gan, C. Performance analysis of a four-switch three-phase grid-side converter with modulation simplification in a doubly-fed induction generator-based wind turbine (DFIG-WT) with different external disturbances. Energies
**2017**, 10, 706. [Google Scholar] [CrossRef] - Lu, J.; Hu, Y.; Zhang, X.; Wang, Z.; Liu, J.; Gan, C. High-frequency voltage injection sensorless control technique for IPMSMs fed by a three-phase four-switch inverter with a single current sensor. IEEE Trans. Mech.
**2018**, 23, 758–768. [Google Scholar] [CrossRef] - Zeng, Z.; Zhu, C.; Jin, X.; Shi, W.; Zhao, R. Hybrid space vector modulation strategy for torque ripple minimization in three-phase four-switch inverter-fed PMSM drives. IEEE Trans. Ind. Electron.
**2017**, 64, 2122–2134. [Google Scholar] [CrossRef] - Zhu, C.; Zeng, Z.; Zhao, R. Comprehensive analysis and reduction of torque ripples in three-phase four-switch inverter-fed PMSM drives using space vector pulse-width modulation. IEEE Trans. Power Electron.
**2017**, 32, 5411–5424. [Google Scholar] [CrossRef] - Zhou, D.; Zhao, J.; Liu, Y. Predictive torque control scheme for three-phase four-switch inverter-fed induction motor drives with DC-link voltages offset suppression. IEEE Trans. Power Electron.
**2015**, 30, 3309–3318. [Google Scholar] [CrossRef] - Hang, J.; Zhang, J.; Wu, H.; Ding, S. Model predictive control with fixed weighting factor for three-phase four-switch inverter-fed PMSM drives considering capacitor voltage offset suppression. IET Electric. Power Applica
**2020**, 14, 2697–2706. [Google Scholar] [CrossRef] - Sun, D.; Su, J.; Sun, C.; Nian, H. A simplified MPFC with capacitor voltage offset suppression for the four-switch three-phase inverter-fed PMSM drive. IEEE Trans. Ind. Electron.
**2019**, 66, 7633–7642. [Google Scholar] [CrossRef] - Zeng, Z.; Zheng, W.; Zhao, R.; Zhu, C.; Yuan, Q. Modeling, modulation, and control of the three-phase four-switch PWM rectifier under balanced voltage. IEEE Trans. Power Electron.
**2016**, 31, 4892–4905. [Google Scholar] [CrossRef] - Zhu, C.; Zeng, Z.; Zhao, R. Adaptive suppression method for DC-link voltage offset in three-phase four-switch inverter-fed PMSM drives. Electron. Lett.
**2016**, 52, 1442–1444. [Google Scholar] [CrossRef] - Yuan, Q.; Zhao, R. DC-link capacitor voltage offset suppression with no filters for three-phase four-switch inverter fed PMSM drives. Electron. Lett.
**2017**, 53, 751–752. [Google Scholar] [CrossRef] - Kim, B.; Lee, D. Sensorless control of PMSM by a four-switch inverter with compensation of voltage distortion and adjustment of position estimation gain. J. Electr. Eng. Technol.
**2017**, 12, 100–109. [Google Scholar] [CrossRef] [Green Version] - Bolognani, S.; Petrella, R.; Prearo, A.; Sgarbossa, L. Automatic tracking of MTPA trajectory in IPM motor drives based on AC current injection. IEEE Trans. Ind. Appl.
**2011**, 47, 105–114. [Google Scholar] [CrossRef]

**Figure 5.**Experimental results at 2500 r/min and 10 Nm: (

**a**) without capacitor voltage offset suppression; (

**b**) proposed method.

**Figure 6.**Experimental results at 2500 r/min and 20 Nm: (

**a**) without capacitor voltage offset suppression; (

**b**) proposed method.

**Figure 7.**Experimental results at 2500 r/min and 30 Nm: (

**a**) without capacitor voltage offset suppression; (

**b**) proposed method.

**Figure 8.**Experimental results at 1500 r/min and 10 Nm: (

**a**) without capacitor voltage offset suppression; (

**b**) proposed method.

**Figure 9.**Experimental results at 1500 r/min and 20 Nm: (

**a**) without capacitor voltage offset suppression; (

**b**) proposed method.

**Figure 10.**Experimental results at 1500 r/min and 30 Nm: (

**a**) without capacitor voltage offset suppression; (

**b**) proposed method.

**Figure 11.**Simulation results of the capacitor value variation at 2500 r/min and 30 Nm: (

**a**) 2000 μF, (

**b**) 1500 μF, and (

**c**) 1000 μF.

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

Rated voltage | U_{N} | 320 V |

Rated current | I_{N} | 150 A |

d-axis inductance | L_{d} | 0.158 mH |

q-axis inductance | L_{q} | 0.292 mH |

Stator resistance | R_{s} | 7.34 mΩ |

Permanent magnet flux | ψ_{f} | 0.067 Wb |

Rated speed | n_{N} | 3000 r/min |

Rated torque | T_{N} | 64 Nm |

Pairs of poles | p | 4 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, W.; Wang, S.; Li, X.; Zhang, G.
A Reference Voltage Self-Correction Method for Capacitor Voltage Offset Suppression of Three-Phase Four-Switch Inverter-Fed PMSM Drives. *World Electr. Veh. J.* **2022**, *13*, 24.
https://doi.org/10.3390/wevj13020024

**AMA Style**

Chen W, Wang S, Li X, Zhang G.
A Reference Voltage Self-Correction Method for Capacitor Voltage Offset Suppression of Three-Phase Four-Switch Inverter-Fed PMSM Drives. *World Electric Vehicle Journal*. 2022; 13(2):24.
https://doi.org/10.3390/wevj13020024

**Chicago/Turabian Style**

Chen, Wei, Sai Wang, Xinmin Li, and Guozheng Zhang.
2022. "A Reference Voltage Self-Correction Method for Capacitor Voltage Offset Suppression of Three-Phase Four-Switch Inverter-Fed PMSM Drives" *World Electric Vehicle Journal* 13, no. 2: 24.
https://doi.org/10.3390/wevj13020024