# Optimized Synchronous SPWM Modulation Strategy for Traction Inverters Based on Non-Equally Spaced Carriers

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Synchronous SPWM

_{dc}is the DC bus voltage, C is the filter capacitor, S1–S6 is the power device on the three-phase bridge arm of A, B, and C, i

_{A}, i

_{B}, and i

_{C}are the corresponding three-phase AC currents, and M is the motor. To satisfy the symmetry condition, similar to the case of synchronous SVPWM, where the fundamental voltage vector is sectorized in one fundamental cycle, the entire carrier is zoned in synchronous SPWM.

_{A}, V

_{B}, and V

_{C}are the reference voltage fundamental, and one fundamental period of phase A is divided into six regions. In contrast, the control pulses generated by comparing the three-phase fundamental period with the carrier wave are given. To facilitate the observation of the carrier waveform of each region, take phase A as an example to introduce the carrier waveform with the carrier wave peak point over 90°. Phase B and C are the same. The horizontal axis represents the angle, and the vertical axis represents the three-phase voltage amplitude. The dotted line in the middle of area I is a 90° dotted line, and the base point θ

_{1}Set at 90°.

_{1}, i.e., have to meet even symmetry. To keep the carrier wave evenly symmetric at both sides of the base point θ

_{1}, the carrier wave must be peak or trough at the 90° moment.

- The angle of each region is 60°.
- The carrier waveform in each adjacent region maintains odd symmetry centered on the boundary.
- The carrier waveform centered on the centerline maintains even symmetry in each region.

## 3. Optimal Synchronous SPWM for Unequally Spaced Carriers

#### 3.1. Optimized Synchronous SPWM Carrier Waveform Design Principles for Non-Equally Spaced Carriers

_{1}and occupies an angle of α/2 at 30°. The carrier wave of another width is close to the base point θ

_{1}and occupies an angle of β at 30° The angle between two adjacent red lines in the region I is 30°, and under the 30° principle, the angle occupied by 30° is taken as γ. γ is expressed as

_{1}and K

_{2}represent the half wave of two carrier widths, respectively and the number of α and β in the 30° range is:

#### 3.2. V_{WTHD} for Different Modulation Strategies

_{WTHD}) is usually used as the inverter output waveform quality measure to compare the advantages and disadvantages of different modulation strategies. The definition is as follows:

_{1}and V

_{n}are the effective values of the fundamental and sub-harmonic voltage of the line voltage waveform, respectively; therefore, whichever modulation strategy has a low line voltage weighted total harmonic distortion coefficient (V

_{WTHD}) displays good performance.

_{1}= 0.5, and K

_{2}= 1, where the width ratio 1:1 is the conventional strategy.

_{1}and K

_{2}will lead to different strategies. To distinguish strategies more clearly, the following provisions are made:

- In each cycle, the carrier wave at the 90° position is noted as H if it is a peak and L if it is a trough.
- Depending on the clamp type, 60° clamp, and 30° clamp are recorded as S and T.

_{1–2}; the same for other strategies. Table 1 shows all modulation strategies with different carrier ratios and width ratios 1–1, 1–2, and 2–1.

_{WTHD}when N = 5 is shown in Figure 5. When 0 ≤ M ≤ 1, the chosen modulation strategy is K

_{1}= 0.5, K

_{2}= 1 (LT

_{2–1}).

_{WTHD}when N = 7 is shown in Figure 6. When 0 ≤ M ≤ 0.55, the chosen modulation strategy is K

_{1}= 0.5, K

_{2}= 1 (HS

_{2–1}). When 0.55 ≤ M ≤ 0.82, the chosen modulation strategy is K

_{1}= 0.5, K

_{2}= 1 (HS

_{1–1}).When 0.82 ≤ M ≤ 1, the modulation strategy selected is K

_{1}= 0.5, K

_{2}= 1 (HS

_{1–2}).

_{WTHD}when N = 9 is shown in Figure 7. When 0 ≤ M ≤ 0.7, the chosen modulation strategy is K

_{1}= 0.5, K

_{2}= 1 (L

_{1–1}). When 0.7 ≤ M ≤ 1, the modulation strategy selected is K

_{1}= 0.5, K

_{2}= 1 (H

_{1–2}). When the width ratio of the two carriers is fixed, the output harmonic content of the inverter can be changed by changing the modulation index. Normally, the output harmonic content of the inverter decreases with the increase of the modulation index. However, this relationship is not necessarily the case in some modulation index ranges, because the output harmonic content of the inverter is calculated by the value of the line voltage-weighted total harmonic distortion. Therefore, the green line in Figure 7 shows an upward trend at the modulation index of 0.65 to 0.75.

_{WTHD}when N = 11 is shown in Figure 8. When 0 ≤ M ≤ 0.9, the chosen modulation strategy is K

_{1}= 0.5, K

_{2}= 2 (HT

_{2–1}). When 0.9 ≤ M ≤ 1, the modulation strategy selected is K

_{1}= 0.5, K

_{2}= 2 (LT

_{1–2}).

_{WTHD}when N = 13 is shown in Figure 9. When 0 ≤ M ≤ 0.85, the chosen modulation strategy is K

_{1}= 0.5, K

_{2}= 2 (HS

_{2–1}). When 0.85 ≤ M ≤ 1, the modulation strategy selected is K

_{1}= 0.5, K

_{2}= 2 (HS

_{1–1}).

_{WTHD}when N = 15 is shown in Figure 10. When 0 ≤ M ≤ 0.65, the chosen modulation strategy is K

_{1}= 0.5, K

_{2}= 2 (L

_{1–1}). When 0.65 ≤ M ≤ 1, the modulation strategy selected is K

_{1}= 0.5, K

_{2}= 2 (L

_{1–2}).

#### 3.3. Multi-Mode Segmented Synchronous Modulation Strategy

_{WTHD}in the whole modulation range according to the comparison principle in Section 3.2. Table 2 shows the optimal modulation strategy patterns under different carrier ratios.

## 4. Experimental Results and Analysis

_{THD}) is used as the evaluation index for evaluating the PWM output waveform in the data processing:

_{1}and I

_{n}are the RMS values of the fundamental and sub-harmonic currents of the current waveform, respectively. Voltage and current signals are collected by the voltage and current probe and displayed on the oscilloscope, and then stored in the oscilloscope and input to the upper computer through the u disk. The sampled signal is processed by sampling and filtering circuits. Yokogawa brand voltage and current probe and oscilloscope are used in the experiment.

_{AB}) and A-phase phase current (i

_{A}) are compared for different carrier ratios (5, 9, 11, 13, 15) and modulation index M = 0.8 under the optimal strategy in the conventional synchronous SPWM and the optimal strategy in the optimized synchronous SPWM.

_{2–1}) I

_{THD}= 14.33%, and the optimized synchronous SPWM (LT

_{2–1}) I

_{THD}= 12.10% with K

_{1}= 0.5 and K

_{2}= 1.

_{1–1}) I

_{THD}= 10.08%, and the optimized synchronous SPWM (H

_{1–2}) I

_{THD}= 9.92% with K

_{1}= 0.5 and K

_{2}= 1.

_{1–1}) I

_{THD}= 8.40%, and the optimized synchronous SPWM (HT

_{2–1}) I

_{THD}= 8.22% with K

_{1}= 0.5 and K

_{2}= 2.

_{1–1}) I

_{THD}= 9.05%, and the optimized synchronous SPWM (HS

_{2–1}) I

_{THD}= 8.62% with K

_{1}= 0.5 and K

_{2}= 2.

_{1–1}) I

_{THD}= 6.25%, and the optimized synchronous SPWM (L

_{1–2}) I

_{THD}= 6.19% with K

_{1}= 0.5 and K

_{2}= 2.

## 5. Conclusions

_{WTHD}. However, in practical application, the design of non-equally spaced carriers is more complex than that of traditional equally spaced carriers.

- Applying the strategy to the closed-loop control of the motor system.
- Combining the modulation algorithm and advanced control strategy proposed in this paper.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 11.**Variation curve of inverter switching frequency with fundamental frequency and modulation index.

**Figure 13.**Line voltage waveform and phase current waveform at carrier wave ratio 5: (

**a**) traditional synchronous SPWM (LS

_{2–1}); (

**b**) optimized synchronous SPWM (LT

_{2–1}).

**Figure 14.**Line voltage waveform and phase current waveform at carrier wave ratio 9: (

**a**) traditional synchronous SPWM (L

_{1–1}); (

**b**) optimized synchronous SPWM (H

_{1–2}).

**Figure 15.**Line voltage waveform and phase current waveform at carrier wave ratio 11: (

**a**) traditional synchronous SPWM (HT

_{1–1}); (

**b**) optimized synchronous SPWM (HT

_{2–1}).

**Figure 16.**Line voltage waveform and phase current waveform at carrier wave ratio 13: (

**a**) traditional synchronous SPWM (HS

_{1–1}); (

**b**) optimized synchronous SPWM (HS

_{2–1}).

**Figure 17.**Line voltage waveform and phase current waveform at carrier wave ratio 15: (

**a**) traditional synchronous SPWM (L

_{1–1}); (

**b**) optimized synchronous SPWM (L

_{1–2}).

**Figure 18.**Smooth switching between adjacent different carrier ratios: (

**a**) carrier ratio 15 to 13; (

**b**) carrier ratio 13 to 11; (

**c**) carrier ratio 11 to 9; (

**d**) carrier ratio 9 to 7; (

**e**) carrier ratio 7 to 5.

**Table 1.**All modulation strategies with width ratios of 1–1, 1–2, and 2–1 under different carrier ratios.

N | K_{1}, K_{2} | Modulation Strategy |
---|---|---|

5 | K_{1} = 0.5, K_{2} = 1 | LT_{1–1}, LT_{1–2}, LT_{2–1} |

7 | K_{1} = 0.5, K_{2} = 1 | HS_{1–1}, HS_{1–2}, HS_{2–1} |

9 | K_{1} = 0.5, K_{2} = 1 | H_{1–1}, L_{1–1}, LS_{1–1}, HT_{1–1}, H_{1–2}, L_{1–2}, LS_{1–2}, HT_{1–2}, H_{2–1}, L_{2–1}, LS_{2–1}, HT_{2–1} |

11 | K_{1} = 0.5, K_{2} = 2 | LS_{1–1}, HT_{1–1}, LT_{1–1}, LS_{1–2}, HT_{1–2}, LT_{1–2}, LS_{2–1}, HT_{2–1}, LT_{2–1} |

K_{1} = 1.5, K_{2} = 1 | LS_{1–2}, HT_{1–2}, LT_{1–2}, LS_{2–1}, HT_{2–1}, LT_{2–1} | |

13 | K_{1} = 0.5, K_{2} = 2 | HS_{1–1}, HS_{1–2}, HS_{2–1} |

K_{1} = 1.5, K_{2} = 1 | HS_{1–2}, HS_{2–1} | |

15 | K_{1} = 0.5, K_{2} = 2 | H_{1–1}, L_{1–1}, H_{1–2}, L_{1–2}, H_{2–1}, L_{2–1} |

K_{1} = 1.5, K_{2} = 1 | H_{1–2}, L_{1–2}, H_{2–1}, L_{2–1} |

N | K_{1}, K_{2} | Optimal Modulation Strategy |
---|---|---|

5 | K_{1} = 0.5, K_{2} = 1 | LT_{2–1} |

7 | K_{1} = 0.5, K_{2} = 1 | HS_{1–1} |

9 | K_{1} = 0.5, K_{2} = 1 | L_{1–1} |

11 | K_{1} = 0.5, K_{2} = 2 | HT_{2–1} |

13 | K_{1} = 0.5, K_{2} = 2 | HS_{2–1} |

15 | K_{1} = 0.5, K_{2} = 2 | L_{1–1} |

Parameter | Values | Unit |
---|---|---|

DC-side capacitance/C | 780 | μF |

DC-link voltage/V_{dc} | 150 | V |

Load resistance/R | 10 | Ω |

Load inductance/L | 20 | mH |

Fundamental frequency/f | 50 | Hz |

Dead-time/t | 600 | ns |

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## Share and Cite

**MDPI and ACS Style**

Jin, X.; Li, S.; Sun, W.; Chen, W.; Gu, X.; Zhang, G.
Optimized Synchronous SPWM Modulation Strategy for Traction Inverters Based on Non-Equally Spaced Carriers. *World Electr. Veh. J.* **2023**, *14*, 157.
https://doi.org/10.3390/wevj14060157

**AMA Style**

Jin X, Li S, Sun W, Chen W, Gu X, Zhang G.
Optimized Synchronous SPWM Modulation Strategy for Traction Inverters Based on Non-Equally Spaced Carriers. *World Electric Vehicle Journal*. 2023; 14(6):157.
https://doi.org/10.3390/wevj14060157

**Chicago/Turabian Style**

Jin, Xuefeng, Shiwei Li, Wenbo Sun, Wei Chen, Xin Gu, and Guozheng Zhang.
2023. "Optimized Synchronous SPWM Modulation Strategy for Traction Inverters Based on Non-Equally Spaced Carriers" *World Electric Vehicle Journal* 14, no. 6: 157.
https://doi.org/10.3390/wevj14060157