# A Generalised Multifrequency PWM Strategy for Dual Three-Phase Voltage Source Converters

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## Abstract

**:**

## 1. Introduction

## 2. Modelling of the Dual Three-Phase Voltage Source Converter

## 3. Multifrequency Pulse Width Modulation Algorithm

MOD1: | MOD2: |

${\tau}_{d}$ = ${C}_{\delta}\xb7\left|{v}_{q1}\right|$ = 0.8317 | ${\tau}_{d}$ = ${C}_{\delta}\xb7\left|{v}_{q2}\right|$ = 0.2336 |

${u}^{*}$ = ${C}_{\delta}^{2}\xb7{v}_{d1}+{S}_{\delta}\xb7{\tau}_{d}$= 0.7615 | ${u}^{*}$ = ${C}_{\delta}^{2}\xb7{v}_{d2}+{S}_{\delta}\xb7{\tau}_{d}$ = −0.5593 |

$0<{u}^{*}\le {\tau}_{d}$, then: | $-{C}_{\delta}\le {u}^{*}\le 0$, then: |

${\tau}_{11}={u}^{*}$ = 0.7615 | ${\tau}_{11}$ = 0 |

$a=1-{\tau}_{d}$ = 0.1683 | $a=1+{u}^{*}-{\tau}_{d}$ = 0.2072 |

${t}_{a}={\tau}_{11}+a\xb7{\lambda}_{1}$ = 0.8457 | ${t}_{f}={\tau}_{11}+a\xb7{\lambda}_{2}$ = 0.1036 |

${t}_{b}={t}_{a}-{C}_{\delta}\xb7({C}_{\delta}\xb7{v}_{d1}^{*}-{S}_{\delta}\xb7{v}_{q1}^{*})$= 0.9159 | ${t}_{d}={t}_{f}-{C}_{\delta}\xb7({C}_{\delta}\xb7{v}_{d2}^{*}-{S}_{\delta}\xb7{v}_{q2}^{*})$ = 0.8964 |

${t}_{c}={t}_{b}-{C}_{\delta}\xb7{v}_{q1}^{*}$ = 0.0841 | ${t}_{e}={t}_{d}-{C}_{\delta}\xb7{v}_{q2}^{*}$ = 0.6628 |

## 4. Experimental Validation and Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Generalised Modulation Scheme for Three-Phase VSC

Algorithm A1: Generalised PWM for three-phase inverters |

Inputs: ${v}_{d}^{*}$, ${v}_{q}^{*}$, $\lambda $ |

Constants and variables: ${C}_{\delta}$, ${S}_{\delta}$, ${\tau}_{d}$, ${u}^{*}$, ${\tau}_{11}$, a |

Outputs: ${t}_{a}$, ${t}_{b}$, ${t}_{c}$ |

// Calculate the algorithm constants |

${\tau}_{d}$ = ${C}_{\delta}\xb7\left|{v}_{q}^{*}\right|$ |

${u}^{*}$ = ${C}_{\delta}^{2}\xb7{v}_{d}^{*}+{S}_{\delta}\xb7{\tau}_{d}$ |

// Calculate the values of ${\tau}_{11}$ and a |

if ($-{C}_{\delta}\le {u}^{*}\le 0$) then |

${\tau}_{11}=0$ |

$a={u}^{*}+1-{\tau}_{d}$ |

else if ($0<{u}^{*}\le {\tau}_{d}$) then |

${\tau}_{11}={u}^{*}$ |

$a=1-{\tau}_{d}$ |

else if (${\tau}_{d}<{u}^{*}\le 1$) then |

${\tau}_{11}={u}^{*}$ |

$a=1-{u}^{*}$ |

end if |

//Calculate the duty cycles |

${t}_{a}={\tau}_{11}+a\xb7\lambda $ |

${t}_{b}={t}_{a}-{C}_{\delta}\xb7({C}_{\delta}\xb7{v}_{d}^{*}-{S}_{\delta}\xb7{v}_{q}^{*})$ |

${t}_{c}={t}_{b}-{C}_{\delta}\xb7{v}_{q}^{*}$ |

$\mathit{\lambda}$ | PWM Technique |
---|---|

0 | PWM-Min |

1/2 | SVM |

1 | PWM-Max |

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**Figure 1.**Six-phase voltage source converter (VSC) power-supplying a dual three-phase load with isolated neutrals.

**Figure 4.**Voltage (top) and current (middle) waveforms of the phase a along with the ${V}_{a}$ spectrum (bottom) for the reference voltage ${m}_{1}$ = 1.15 and ${m}_{2}$ = 0.

**Figure 5.**Voltage (top) and current (middle) waveforms of the phase a along with the voltage spectrum (bottom) for the reference voltage ${m}_{1}$ = 0.92 and ${m}_{2}$ = 0.23.

**Figure 6.**Time response of the voltage (upper graph) and current (middle waveform) of the phase a along with the voltage spectrum (bottom) for the reference voltage ${m}_{1}$ = ${m}_{2}$ = 0.57.

**Figure 7.**Voltage (upper graph) and current (middle waveform) of phase a along with the synthesised voltage spectrum (bottom) with a fundamental frequency magnitude of 0.90, while the 5th- and 7th-order harmonics are set to 0.15 and 0.10, respectively.

**Figure 8.**Compound total harmonic distortion (CTHD${}_{v}$) for different operation points within the linear modulation region.

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

Resistance, R | ($\Omega $) | 10 |

Inductance, L | (H) | 10 m |

Dc-bus Voltage, ${V}_{dc}$ | (V) | 100 |

Fundamental frequency, ${f}_{1}$ | (Hz) | 50 |

Frequency index, ${m}_{f}$ | 30 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Riveros, J.A.; Prieto, J.; Rivera, M.; Toledo, S.; Gregor, R.
A Generalised Multifrequency PWM Strategy for Dual Three-Phase Voltage Source Converters. *Energies* **2019**, *12*, 1398.
https://doi.org/10.3390/en12071398

**AMA Style**

Riveros JA, Prieto J, Rivera M, Toledo S, Gregor R.
A Generalised Multifrequency PWM Strategy for Dual Three-Phase Voltage Source Converters. *Energies*. 2019; 12(7):1398.
https://doi.org/10.3390/en12071398

**Chicago/Turabian Style**

Riveros, Jose A., Joel Prieto, Marco Rivera, Sergio Toledo, and Raúl Gregor.
2019. "A Generalised Multifrequency PWM Strategy for Dual Three-Phase Voltage Source Converters" *Energies* 12, no. 7: 1398.
https://doi.org/10.3390/en12071398