# Smooth Switching Method for Multi-Mode Synchronous Space Vector Modulation of NPC Three-Level Inverter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{s}of the inverter is only a few hundred hertz [9,10]. However, the range of output fundamental frequency f

_{e}is wide, resulting in a large change in pulse number P = f

_{s}/f

_{e}. To solve the above problem, asynchronous modulation is usually used in the low fundamental frequency region. While in the middle and high fundamental frequency region, multi-mode synchronous modulation is often adopted to ensure the synchronization and symmetry of the output waveform [11].

## 2. Synchronous Space Vector Modulation Method for Three-Level Inverter

_{dc}is the DC-link voltage, the upper and lower capacitors C

_{1}and C

_{2}are capacitors in parallel with the DC bus, each shares half of the DC bus voltage, and the values are defined as v

_{C1}and v

_{C2}, respectively. The AC side is composed of three-phase bridge arms, representing A, B and C phases, each phase has two clamping diodes D

_{X}

_{1}~D

_{X}

_{2}in series and four insulated gate bipolar transistor (IGBT) S

_{X}

_{1}~S

_{X}

_{4}in series, i

_{X}is the three-phase load current, where X∈{A, B, C}.

_{X}. The two clamping diodes are connected to the midpoint O, which can clamp the output of the inverter to the O state, so that the inverter can obtain three working states of P, O and N.

#### 2.1. Basic Principle of Synchronous Space Vector Modulation

^{3}= 27 switching states corresponding to 27 basic voltage vectors. According to the magnitude, the basic voltage vectors can be divided into zero vectors (

**V**

_{0P},

**V**

_{0O},

**V**

_{0N}), small vectors (

**V**

_{1P}~

**V**

_{6P},

**V**

_{1N}~

**V**

_{6N}), medium vectors (

**V**

_{7}~

**V**

_{12}), large vectors (

**V**

_{13}~

**V**

_{18}). The space vector diagram can be divided into six sectors Z

_{I}~Z

_{VI}with the middle vector as the boundary. Connect the summits of the separated middle vectors sequentially, and each sector can further be divided into four triangles ①~④,

**V**

_{ref}is the reference voltage vector, θ is the angle between reference voltage vector and α axis. The basic voltage vector and reference voltage vector corresponding to each switching state are shown in Figure 2.

**V**

_{X},

**V**

_{Y}and

**V**

_{Z}represent the three basic vectors participating in the synthesis of the reference voltage vector

**V**

_{ref}. T

_{X}, T

_{Y}and T

_{Z}represent the action time of each basic voltage vector in a sub-cycle T

_{S}.

#### 2.2. Multi-Mode Synchronous Space Vector Modulation Method

## 3. Stator Flux Trajectories of Synchronous Space Vector Modulation

#### 3.1. The Change of Stator Flux Trajectories during Pulse Pattern Switching

_{I}sector is shown in Figure 4a. Due to the difference of flux trajectories change rate of two pulse numbers, the flux trajectories of SSVM_5 will deviate from the flux trajectories of SSVM_7, and rotate along the new flux circle. The flux trajectories will deviate as shown in Figure 4b. Therefore, in order to eliminate the phenomenon of stator flux trajectories deviation after switching. It is necessary to compensate the phase angle of the reference voltage vector before switching, so that it changes along the original flux trajectories.

#### 3.2. Stator Flux Trajectories Calculation

**ψ**

_{S}and the reference voltage vector

**V**

_{ref}is shown in Equation (2):

_{1}is the angular velocity of the motor during constant rotation, and θ is the angle between the reference voltage and the α axis, Equation (2) can be derived simultaneously on both sides:

_{1}as the stator flux trajectories, and Equation (3) is rewritten in incremental form:

**ψ**

_{S}is the change rate of the flux trajectories, the reference voltage vector and the stator flux vector correspond to each other in the sub-cycle T

_{S}. When the motor works at a constant speed, it can be considered that the amplitude of the stator flux trajectories and the reference voltage is constant, so the flux trajectories’ deviation can be corrected by changing the phase angle of the reference voltage.

_{S}can be obtained. Taking P = 5 as an example, the position of the reference voltage vector ${\mathit{V}}_{\mathrm{r}1}^{5}$ is shown in Figure 5a, ${T}_{\mathrm{S}}^{5}$ = 1/30 f

_{e}, and ${\mathit{\psi}}_{\mathrm{S}}^{5}$ rotates 12° per sub-cycle. The switching sequence used to synthesize ${\mathit{V}}_{\mathrm{r}1}^{5}$ is:

**V**

_{1P}→

**V**

_{7}→

**V**

_{13}, and the flux trajectories of ${\mathit{V}}_{\mathrm{r}1}^{5}$ is shown in Figure 5b. In the figure, the black solid line is the actual flux trajectories change, and the red dotted line is the equivalent flux trajectories change rate.

## 4. Smooth Switching Method

#### 4.1. Switch between Asynchronous Modulation and SSVM_15 (IV)

#### 4.2. Switching between Different Clamping Methods of SSVM

#### 4.3. Switching between Different Pulse Numbers of SSVM

_{e}, ${\mathit{\psi}}_{\mathrm{S}}^{7}$ rotates 10° per sub-cycle. The flux trajectories change rates of the two pulse numbers under the action of the first reference voltage vector in the Z

_{I}sector are shown in Figure 6.

_{e}. According to Equation (4), SSVM_7 flux trajectories change rate Δ${\mathit{\psi}}_{\mathrm{S}}^{7}$ is:

_{7_5}is:

^{j5°}, ${\psi}_{\mathrm{S}}^{5}$e

^{j282°}and ${\psi}_{\mathrm{S}}^{7}$e

^{j270°}indicate that the stator flux vector

**ψ**

_{s}rotates from 270° to 282°. The complex compensation gain k

_{7_5}= e

^{j0.85°}can be obtained by substituting Equations (6) and (7) into Equation (8), since the value of the complex compensation gain k is a constant independent of any parameter, its value can be calculated offline.

_{5_7}can be calculated:

^{j6°}, complex compensation gain k

_{5_7}= e

^{j−0.82°}. Therefore, the reference voltage phase angle of SSVM_5 should be compensated by −0.82° and then switching, and the compensated reference voltage vector phase angle is 60°·n − 54°.

_{e}, ${\mathit{\psi}}_{\mathrm{S}}^{9}$ rotates 8.57° per sub-cycle. The flux trajectories change rates of the two pulse numbers under the action of the first reference voltage vector in the Z

_{I}sector are shown in Figure 7.

_{9_7}is:

^{j4.3°}, complex compensation gain k

_{9_7}= e

^{j−0.6°}. Therefore, the reference voltage phase angle of SSVM_9 should be compensated by −0.6° and then switching, and the compensated reference voltage vector phase angle is 60°·n − 55.7°.

_{7_9}can be calculated:

^{j5°}, complex compensation gain k

_{7_9}= e

^{j0.69°}. Therefore, the reference voltage phase angle of SSVM_9 should be compensated by 0.69° and then switched, and the compensated reference voltage vector phase angle is 60°·n − 55°.

## 5. Experimental Result

^{®}DS1007 rapid prototyping experimental system is used as the controller, the Infineon

^{®}IGBT module F3L75R07W2E3 is used as the power circuit. The experimental system also includes DC power supply, voltage/current sensor, PC and signal conversion circuit. Then, the experimental verification is carried out in a 7.5 kW induction motor. In order to clearly prove that the smooth switching of the motor can be achieved by using only the modulation strategy, the asynchronous motor is directly powered by an open-loop NPC inverter, and the sampling frequency of the dSPACE board is set to 50 kHz. A comparative analysis was conducted, about the phase angle of the reference voltage vector is not compensated and compensated when switching different pulse numbers. The experimental system and experimental parameters are shown in Table 4 and Figure 9.

#### 5.1. Switching Experiment of Asynchronous Modulation and Synchronous Modulation

_{A1}and S

_{A2}, phase voltage v

_{AO}, three-phase current i

_{A}, i

_{B}, i

_{C}. Since there is no deviation in the stator flux trajectories during switching, the three-phase current has no impulse oscillation, achieving smooth switching.

#### 5.2. Switching Experiment of Different Clamping Methods of SSVM

#### 5.3. Switching Experiment of Different Pulse Numbers of SSVM

_{A1}and S

_{A2}, and the position of the switching point can be seen through the waveform when the motor frequency rises in the multi-mode SSVM method. When switching between different pulse numbers, the three-phase current waveform will produce impulse oscillation due to flux trajectories deviation. As the number of pulses decreases during switching, the impact oscillation becomes more and more obvious, as shown in the left experimental results. After using complex compensation gain to compensate the reference voltage vector phase angle before switching, flux trajectories deviation was corrected. From the right experimental results, the three-phase current does not appear impulse oscillation, and smooth switching is achieved. Similarly, the fundamental frequency of the motor decreases, the three-phase current will not also impulse oscillation.

_{THD}as the quality evaluation standard of inverter output waveform, analyze the A-phase output current i

_{A}during switching. It can be seen from the current harmonic content in Figure 12, after the smooth switching method is adopted, the harmonic content and total harmonic distortion rate in the current are significantly reduced. Among them, the amplitude of low-order harmonics mainly by second and fourth harmonics decreases obviously, and more than ten times higher harmonic amplitude has also been reduced, resulting in the inverter having better output waveform quality.

#### 5.4. Output Waveform Quality

_{THD}and A-phase current total harmonic distortion i

_{THD}of the multi-mode SSVM with different pulse numbers are shown in Figure 13. It can be seen that due to the symmetry and synchronization of synchronous space vector modulation, the v

_{THD}and i

_{THD}are reduced at a lower switching frequency. As the number of pulses decreases after the fundamental frequency increases, v

_{THD}and i

_{THD}do not increase significantly.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Switching frequency | f_{s} |

Output fundamental frequency | f_{e} |

Pulse number | P |

Complex compensation gain | k |

Phase current | i_{X}, X∈{A, B, C}. |

Reference voltage vector | V_{ref} |

reference voltage vector phase angle | θ |

Sub-cycle times | T_{S} |

Modulation index | m |

Stator flux vector | ψ_{S} |

Stator flux trajectories change rate | Δψ_{S} |

Voltage total harmonic distortion | v_{THD} |

Current total harmonic distortion | i_{THD} |

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**Figure 4.**Stator flux trajectories and current change during switching. (

**a**) Stator flux trajectories change. (

**b**) Current change.

**Figure 5.**Relationship between space vector and flux trajectories. (

**a**) ${\mathit{V}}_{\mathrm{r}1}^{5}$ in sectors Z

_{I}. (

**b**) Flux trajectories change in subcycle.

**Figure 10.**Experimental results of asynchronous modulation and SSVM_15(IV) switching. (

**a**) Asynchronous modulation→SSVM_15 (IV). (

**b**) SSVM_15 (IV)→asynchronous modulation.

**Figure 11.**Experimental results of different clamping method switching of SSVM. (

**a**) SVM_15(IV)→SSVM_15(I). (

**b**) SSVM_15(I)→SSVM_15(IV). (

**c**) SVM_11(IV)→SSVM_11(I). (

**d**) SSVM_11(I)→SSVM_11(IV).

**Figure 12.**Experimental results of different pulse numbers switching. (

**a**) m = 0.6, SSVM_15(I)→SSVM_13. (

**b**) m = 0.76, SSVM_13→SSVM_11(IV). (

**c**) m = 0.84, SSVM_11(I)→SSVM_9. (

**d**) m = 0.9, SSVM_9→SSVM_7. (

**e**) m = 0.96, SSVM_7→SSVM_5.

**Figure 13.**Experimental results of v

_{THD}and i

_{THD}. (

**a**) m = 0.51, P = 15(IV). (

**b**) m = 0.56, P = 15(I). (

**c**) m = 0.68, P = 13. (

**d**) m = 0.78, P = 11(IV). (

**e**) m = 0.82, P = 11(I). (

**f**) m = 0.86, P = 9. (

**g**) m = 0.93, P = 7. (

**h**) m = 0.96, P = 5.

Power Device | Switch Status | Phase Voltage | |||
---|---|---|---|---|---|

S_{X}_{1} | S_{X}_{2} | S_{X}_{3} | S_{X}_{4} | ||

1 | 1 | 0 | 0 | P | V_{dc}/2 |

0 | 1 | 1 | 0 | O | 0 |

0 | 0 | 1 | 1 | N | −V_{dc}/2 |

Synchronization | Three-Phase Symmetry | Half-Wave Symmetry | ||
---|---|---|---|---|

θ | θ ± 2π | θ + 2/3π | θ − 2/3π | θ ± π |

S_{a} | S_{a} | S_{c} | S_{b} | ${\mathrm{S}}_{\mathrm{A}}^{*}$ |

S_{b} | S_{b} | S_{a} | S_{c} | ${\mathrm{S}}_{\mathrm{B}}^{*}$ |

S_{c} | S_{c} | S_{b} | S_{a} | ${\mathrm{S}}_{\mathrm{C}}^{*}$ |

Pulse Number Change during Switching | Complex Compensation Gain k | The Compensated V_{ref} Phase Angle |
---|---|---|

SSVM_15→SSVM_13 | k_{15_13} = e^{j−2.06°} | 60°·n − 57.7° |

SSVM_13→SSVM_11 | k_{13_11} = e^{j3.14°} | 60°·n − 57.5° |

SSVM_11→SSVM_9 | k_{11_9} = e^{j0.03°} | 60°·n − 56.5° |

SSVM_9→SSVM_7 | k_{9_7} = e^{j−0.6°} | 60°·n − 55.6° |

SSVM_7→SSVM_5 | k_{7_5} = e^{j0.85°} | 60°·n − 55° |

SSVM_5→SSVM_7 | k_{5_7} = e^{j−0.82°} | 60°·n − 54° |

SSVM_7→SSVM_9 | k_{7_9} = e^{j0.69°} | 60°·n − 55° |

SSVM_9→SSVM_11 | k_{9_11} = e^{j−0.06°} | 60°·n − 55.6° |

SSVM_11→SSVM_13 | k_{11_13} = e^{j−4.1°} | 60°·n − 56.5° |

SSVM_13→SSVM_15 | k_{13_15} = e^{j2.27°} | 60°·n − 57.5° |

Parameters | Values |
---|---|

DC-link voltage V_{dc}/V | 200 |

Rated power P/kW | 7.5 |

Rated frequency f_{N}/Hz | 50 |

Rated current I_{N}/A | 17.8 |

Rated speed n_{N}/(r/min) | 720 |

Number of pole pairs p/pairs | 4 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, G.; Zhao, G.; Li, C.; Li, X.; Gu, X.
Smooth Switching Method for Multi-Mode Synchronous Space Vector Modulation of NPC Three-Level Inverter. *World Electr. Veh. J.* **2023**, *14*, 62.
https://doi.org/10.3390/wevj14030062

**AMA Style**

Zhang G, Zhao G, Li C, Li X, Gu X.
Smooth Switching Method for Multi-Mode Synchronous Space Vector Modulation of NPC Three-Level Inverter. *World Electric Vehicle Journal*. 2023; 14(3):62.
https://doi.org/10.3390/wevj14030062

**Chicago/Turabian Style**

Zhang, Guozheng, Guoao Zhao, Chen Li, Xinmin Li, and Xin Gu.
2023. "Smooth Switching Method for Multi-Mode Synchronous Space Vector Modulation of NPC Three-Level Inverter" *World Electric Vehicle Journal* 14, no. 3: 62.
https://doi.org/10.3390/wevj14030062