# Discontinuous Space Vector PWM Strategy for Three-Phase Three-Level Electric Vehicle Traction Inverter Fed Two-Phase Load

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## Abstract

**:**

## 1. Introduction

## 2. Discontinuous Pulse Width Modulation (DPWM) under Three-Phase Load Condition

^{3}= 27 switching state combinations for a three-phase three-level inverter. According to Equation (1), each switching state combination corresponds to one basic vector in the space vector diagram:

**V**

_{0}, small vector

**V**

_{1}–

**V**

_{6}, medium vector

**V**

_{7}–

**V**

_{12}and large vector

**V**

_{13}–

**V**

_{18}) according to their amplitudes. There are three switching state combinations corresponding to zero vector, and there are two switching state combinations corresponding to each small vector. Taking large vectors as the boundaries, the space vector diagram can be divided into six sectors Z

_{I}–Z

_{VI}, and each sector can be further divided into six small triangles ①–⑥. The nearest three basic vectors are usually used to synthesize the reference vector

**V**

_{ref}. Basic vectors used to synthesize

**V**

_{ref}in each triangle of sector Z

_{I}are listed in Table 2. The situation in other sectors can be obtained in the same manner.

**V**

_{ref}in sector Z

_{I}are shown in Table 3. The switching sequences of other sectors can be obtained according to sector and vector symmetry. By the use of the volt-second balance principle, the duty cycles of the basic vectors synthesizing the reference vector can be achieved. If the reference vector

**V**

_{ref}is located in triangle ① of sector Z

_{I}, then

**V**

_{1},

**V**

_{7}, and

**V**

_{13}will be used to synthesize

**V**

_{ref}, yielding:

_{1}, d

_{2}, and d

_{0}are duty cycles of basic vectors

**V**

_{1},

**V**

_{7}, and

**V**

_{13}, respectively. The duty cycles of the nearest three vectors used to synthesize

**V**

_{ref}in each triangle of sector Z

_{I}are shown in Table 4.

_{ref}/V

_{dc}. As shown in Table 3, in the first half or the second half of sector Z

_{I}, the switching state of one of the three phases is always P or N. This phase is defined as the clamping phase and the switching state is defined as the clamping state. For example, the switching state of phase A is P in sector Z

_{I}when DPWM1 is adopted, therefore the clamping phase is phase A and the clamping state is P in Z

_{I}. The clamping phase and clamping state of DPWM0–DPWM3 are listed in Table 5. For example, A [P] means that the clamping phase is A and the clamping state is P, and C [N] means that the clamping phase is C and the clamping state is N.

## 3. DPWM under Two-Phase Load Condition

_{I}and Z

_{IV}are π/2, while intervals of sector Z

_{II}, Z

_{III}, Z

_{V,}and Z

_{VI}are π/4. Each sector can also be divided into six triangles.

**V**

_{ref}in each triangle of sector Z

_{I}need to be recalculated, which are listed in Table 6.

_{ref}/V

_{dc}. The four types of discontinuous modulation strategies under two-phase load conditions are named as IDPWM0–IDPWM3. The switching sequences of IDPWM0–IDPWM3 are consistent with DPWM0–DPWM3 (Table 3). However, clamping intervals are changed with the variation of basic vectors. The modulation waves of phase A for DPWM0–DPWM3 and IDPWM0–IDPWM3 are shown in Figure 5a,b, respectively.

- 1)
- Sector and triangle judgement: m’ and θ are used to determine in which sector (Z
_{I}–Z_{VI}) and triangle (①–⑥) the reference vector**V**_{ref}is located. - 2)
- Switching sequence selection: The switching sequence used to synthesize reference vector
**V**_{ref}can be selected according to Table 3. - 3)
- Duty cycle calculation: According to Equation (2), the duty cycles d
_{1}, d_{2}, and d_{0}corresponding to three basic vectors can be determined. - 4)
- PWM generation: The switching signal of each phase could be generated according to the switching sequence and duty cycles.

## 4. Experimental Verification

#### 4.1. Output Waveform Quality

_{AO}, output line voltage v

_{AB}, output current i

_{A}and neutral-point voltage ripple v

_{O}for DPWM0–DPWM3 and IDPWM0–IDPWM3 under two-phase load conditions when m’ = 0.3 and m’ = 0.8 are presented in Figure 7, Figure 8, Figure 9 and Figure 10, respectively.

_{THD}) in the whole modulation range can be calculated on the basis of experimental results. Then variations of I

_{THD}with the change of modulation index m’ for DPWM0–DPWM3 and IDPWM0–IDPWM3 can be obtained, which are illustrated in Figure 11. For IDPWM1, I

_{THD}is always considerably lower than that of DPWM1 in the whole modulation range, and I

_{THD}is decreased by about 50% at most. While for IDPWM0, IDPWM2, and IDPWM3, the improvement is significant in the lower (m’ ≤ 0.5) and higher (m’ ≥ 0.8) modulation range.

#### 4.2. Neutral-Point Voltage Ripple

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The topology of the neutral-point-clamped (NPC) three-level inverter fed three-phase Resistance-Inductance (RL) load. (Insulated Gate Bipolar Transistor (IGBT)).

**Figure 5.**Modulation signals of DPWM0–DPWM3 and IDPWM0–IDPWM3: (

**a**) DPWM0–DPWM3 (

**b**) IDPWM0–IDPWM3. (Note: IDPWM corresponds to improved DPWM).

**Figure 6.**The experimental prototype of the neutral-point-clamped three-level inverter: (

**a**) the overall experimental prototype; (

**b**) the main components of the NPC three-level inverter.

**Figure 7.**Experimental results of DPWM0–DPWM3 under two-phase load conditions when m’ = 0.3: (

**a**) DPWM0; (

**b**) DPWM1; (

**c**) DPWM2 (

**d**) DPWM3.

**Figure 8.**Experimental results of DPWM0–DPWM3 under two-phase load conditions when m’ = 0.8: (

**a**) DPWM0; (

**b**) DPWM1; (

**c**) DPWM2 (

**d**) DPWM3.

**Figure 9.**Experimental results of IDPWM0–IDPWM3 under two-phase load conditions when m’ = 0.3: (

**a**) IDPWM0; (

**b**) IDPWM1; (

**c**) IDPWM2 (

**d**) IDPWM3.

**Figure 10.**Experimental results of IDPWM0–IDPWM3 under two-phase load conditions when m’ = 0.8: (

**a**) IDPWM0; (

**b**) IDPWM1; (

**c**) IDPWM2 (

**d**) IDPWM3.

**Figure 11.**Variations of I

_{THD}with the change of m’ for DPWM0–DPWM3 and IDPWM0–IDPWM3: (

**a**) DPWM0 and IDPWM0; (

**b**) DPWM1 and IDPWM1; (

**c**) DPWM2 and IDPWM2; (

**d**) DPWM3 and IDPWM3.

Switching State | S_{A1} | S_{A2} | S_{A3} | S_{A4} |
---|---|---|---|---|

P | 1 | 1 | 0 | 0 |

O | 0 | 1 | 1 | 0 |

N | 0 | 0 | 1 | 1 |

Small Triangle | Basic Voltage Vector |
---|---|

① | V_{1} [POO/ONN], V_{7} [PON], V_{13} [PNN] |

② and ⑤ | V_{1} [POO/ONN], V_{2} [PPO/OON], V_{7} [PON] |

③ and ④ | V_{0} [PPP/OOO/NNN], V_{1} [POO/ONN], V_{2} [PPO/OON] |

⑥ | V_{2} [PPO/OON], V_{7} [PON], V_{14} [PPN] |

**Table 3.**Switching sequences of discontinuous pulse width modulation (DPWM): DPWM0–DPWM3 in sector Z

_{I}Reproduced with permission from [19].

θ | Triangle | DPWM0 | DPWM1 | DPWM2 | DPWM3 |
---|---|---|---|---|---|

[0, π/6) | ① | POO↔PON↔PNN | POO↔PON↔PNN | ONN↔PNN↔PON | ONN↔PNN↔PON |

② | PPO↔POO↔PON | PPO↔POO↔PON | ONN↔OON↔PON | ONN↔OON↔PON | |

③ | POO↔PPO↔PPP | POO↔PPO↔PPP | OON↔ONN↔NNN | OON↔ONN↔NNN | |

[π/6, π/3) | ④ | OON↔ONN↔NNN | POO→PPO→PPP | OON↔ONN↔NNN | POO→PPO→PPP |

⑤ | ONN↔OON↔PON | PPO↔POO↔PON | ONN↔OON↔PON | PPO↔POO↔PON | |

⑥ | OON↔PON↔PPN | PPO→PPN→PON | OON↔PON↔PPN | PPO→PPN→PON |

Triangle | d_{1} | d_{2} | d_{0} |
---|---|---|---|

① | V_{13}: 2msin (π/3 − θ) − 1 | V_{7}: 2msinθ | V_{1}: 2[1 – msin (π/3 + θ)] |

② and ⑤ | V_{1}: 1 − 2msinθ | V_{2}: 2msin (θ − π/3)+1 | V_{7}: 2msin (π/3 + θ) − 1 |

③ and ④ | V_{1}: 2msin (π/3 − θ) | V_{2}: 2msinθ | V_{0}: 1 − 2msin (π/3 + θ) |

⑥ | V_{14}: 2msinθ − 1 | V_{7}: 2msin (π/3 − θ) | V_{2}: 2[1 – msin (π/3 + θ)] |

Sector | θ | DPWM0 | DPWM1 | DPWM2 | DPWM3 |
---|---|---|---|---|---|

Z_{I} | [0, π/6) | A [P] | A [P] | C [N] | C [N] |

[π/6, π/3) | C [N] | A [P] | C [N] | A [P] | |

Z_{II} | [π/3, π/2) | C [N] | C [N] | B [P] | B [P] |

[π/2, 2π/3) | B [P] | C [N] | B [P] | C [N] | |

Z_{III} | [2π/3, 5π/6) | B [P] | B [P] | A [N] | A [N] |

[5π/6, π) | A [N] | B [P] | A [N] | B [P] | |

Z_{IV} | [π, 7π/6) | A [N] | A [N] | C [P] | C [P] |

[7π/6, 4π/3) | C [P] | A [N] | C [P] | A [N] | |

Z_{V} | [4π/3, 3π/2) | C [P] | C [P] | B [N] | B [N] |

[3π/2, 5π/3) | B [N] | C [P] | B [N] | C [P] | |

Z_{VI} | [5π/3, 11π/6) | B [N] | B [N] | A [P] | A [P] |

[11π/6, 2π) | A [P] | B [N] | A [P] | B [N] |

Triangle | d_{1} | d_{2} | d_{0} |
---|---|---|---|

① | V_{13}: $\sqrt{\mathrm{,2}}$m’cosθ − 1 | V_{7}: $\sqrt{\mathrm{,2}}$m’sinθ | V_{1}: 2 − 2m’cos (θ − π/4) |

② and ⑤ | V_{1}: 1 − $\sqrt{\mathrm{,2}}$m’sinθ | V_{2}: 1 − $\sqrt{\mathrm{,2}}$m’cosθ | V_{7}: 2m’cos (θ − π/4) − 1 |

③ and ④ | V_{1}: $\sqrt{\mathrm{,2}}$m’cosθ | V_{2}: $\sqrt{\mathrm{,2}}$m’sinθ | V_{0}: 1 − 2m’cos (θ − π/4) |

⑥ | V_{14}: $\sqrt{\mathrm{,2}}$m’sinθ − 1 | V_{7}: $\sqrt{\mathrm{,2}}$m’cosθ | V_{2}: 2 − 2m’cos (θ − π/4) |

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

dc-link voltage V_{dc} | V | 80 |

dc-link capacitor C_{1},C_{2} | μF | 1000 |

switching frequency f_{sw} | kHz | 2 |

Load resistance R | Ω | 10 |

Load inductance L | mH | 40 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, G.; Wan, Y.; Wang, Z.; Gao, L.; Zhou, Z.; Geng, Q.
Discontinuous Space Vector PWM Strategy for Three-Phase Three-Level Electric Vehicle Traction Inverter Fed Two-Phase Load. *World Electr. Veh. J.* **2020**, *11*, 27.
https://doi.org/10.3390/wevj11010027

**AMA Style**

Zhang G, Wan Y, Wang Z, Gao L, Zhou Z, Geng Q.
Discontinuous Space Vector PWM Strategy for Three-Phase Three-Level Electric Vehicle Traction Inverter Fed Two-Phase Load. *World Electric Vehicle Journal*. 2020; 11(1):27.
https://doi.org/10.3390/wevj11010027

**Chicago/Turabian Style**

Zhang, Guozheng, Yuwei Wan, Zhixin Wang, Le Gao, Zhanqing Zhou, and Qiang Geng.
2020. "Discontinuous Space Vector PWM Strategy for Three-Phase Three-Level Electric Vehicle Traction Inverter Fed Two-Phase Load" *World Electric Vehicle Journal* 11, no. 1: 27.
https://doi.org/10.3390/wevj11010027