# Instantaneous Electromagnetic Torque Components in Synchronous Motors Fed by Load-Commutated Inverters

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## Abstract

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## 1. Introduction

#### 1.1. Classification of Variable Frequency Drive (VFD) Applications: Power Ratings

#### 1.2. Motivation, Purpose and Organization of This Paper

#### 1.3. Notes on Alternative Method to Evaluate Air-Gap Torque

## 2. LCI Systems for Synchronous Motors

#### 2.1. Operating Range of LCIs in O&G Industry

- Liquefied Natural Gas plants, with a speed between 3000 and 3600 rpm, and a power range between 10 and 80 MW.
- Pipeline recompression and decompression, with speed between 4000 and 8000 rpm, and a power range between 10 and 40 MW.
- Storage recompression and decompression, with speed between 8000 and 18,000 rpm, and a power range between 15 and 20 MW.
- Gas refrigeration with a speed between 15,000 and 20,000 rpm, and a power range between 0.75 and 4 MW.

#### 2.2. Engineering Challenges of LCIs

#### 2.2.1. Input Power Quality

#### 2.2.2. Machine Compatibility: Induction vs. Synchronous Machines

#### 2.2.3. Reliability

#### 2.2.4. Torsional Excitation and Low-Torque Ripple

#### 2.3. LCI System Configurations

#### 2.4. Advantages and Drawbacks of LCIs

## 3. Instantaneous Electromagnetic Torque Harmonics in LCI Systems

#### 3.1. Assumptions

#### 3.2. LCI Current Harmonic Performance

#### 3.3. LCI Current Harmonic Performance

#### 3.4. Torque Harmonic Families for a 6/6-Pulse LCI System

#### 3.5. Torque Harmonic Families for a 12/12-Pulse LCI System

#### 3.6. Torque Harmonic Families for a 24/24-Pulse LCI System

#### 3.7. Campbell Diagrams of LCI Systems

## 4. Numerical Validation of the Theoretical Results

#### 4.1. Simulated Systems

#### 4.2. Validation Method

#### 4.2.1. Principle of Validation

#### 4.2.2. Frequency Domain Validation

#### 4.2.3. Confirmation of the Robustness of the Results

#### 4.3. Selected Simulation Results of a 6/6-Pulse LCI System

#### 4.3.1. Simulation Results in Time and Frequency Domains

#### 4.3.2. Detailed Analysis of Harmonic Families

- Baseband harmonic family of stator current are located at $\left(0,{N}_{i}\right)$: (i) $200Hz\to \left(0,5\right)$ and $280\to \left(0,7\right)$; (ii) $440Hz\to \left(0,11\right)$ and $520\to \left(0,13\right)$; (iii) $680Hz\to \left(0,17\right)$ and $760\to \left(0,19\right)$.
- Sideband harmonic family of stator current located around $1\times 6{f}_{g}$ are given in the form of $\left(6,{N}_{i}\right)$: (i) $20Hz\to \left(6,-7\right)$ and $100Hz\to \left(6,-5\right)$; (ii) $140Hz\to \left(6,-11\right)$ and $220Hz\to \left(6,-13\right)$; (iii) $260Hz\to \left(6,-1\right)$ and $340Hz\to \left(6,1\right)$; (iv) $380Hz\to \left(6,-17\right)$ and $460Hz\to \left(6,-19\right)$; (v) $500Hz\to \left(6,5\right)$ and $580Hz\to \left(6,7\right)$; (vi) $740Hz\to \left(6,11\right)$ and $820Hz\to \left(6,13\right)$; (vii) $980Hz\to \left(6,17\right)$ and $1060Hz\to \left(6,19\right)$.
- Sideband harmonic family of stator current located around $2\times 6{f}_{g}$ are given in the form of $\left(12,{N}_{i}\right)$: (i) $80Hz\to \left(12,-13\right)$ and $160Hz\to \left(12,-11\right)$; (ii) $320Hz\to \left(12,-7\right)$ and $400Hz\to \left(12,-5\right)$; (iii) $560Hz\to \left(12,-1\right)$ and $640Hz\to \left(12,1\right)$; (iv) $800Hz\to \left(12,5\right)$ and $880Hz\to \left(12,7\right)$; (v) $1040Hz\to \left(12,11\right)$ and $1120Hz\to \left(12,13\right)$.
- Sideband harmonic family of stator current located around $3\times 6{f}_{g}$ are given in the form of $\left(18,{N}_{i}\right)$: (i) $860Hz\to \left(18,-1\right)$ and $940Hz\to \left(18,1\right)$.

- Baseband harmonic family of torque current are located at $\left(0,{N}_{i}\pm 1\right)$: (i) $240Hz\to \left(0,6\right)$; (ii) $480Hz\to \left(0,12\right)$; (iii) $720Hz\to \left(0,18\right)$.
- Sideband harmonic family of torque current located around $1\times 6{f}_{g}$ are given in the form of $\left(6,{N}_{i}\pm 1\right)$: (i) $60Hz\to \left(6,-6\right)$; (ii) $180Hz\to \left(6,-12\right)$; (iii) $300Hz\to \left(6,0\right)$; (iv) $420Hz\to \left(6,-18\right)$; (v) $540Hz\to \left(6,6\right)$; (vi) $780Hz\to \left(6,12\right)$; (vii) $1020Hz\to \left(6,18\right)$.
- Sideband harmonic family of torque current located $2\times 6{f}_{g}$ are given in the form of $\left(12,{N}_{i}\pm 1\right)$: (i) $120Hz\to \left(12,-18\right)$; (ii) $360Hz\to \left(12,-6\right)$; (iii) $600Hz\to \left(12,0\right)$; (iv) $840Hz\to \left(12,6\right)$; (v) $1080Hz\to \left(12,12\right)$.
- Sideband harmonic family of stator current located around $3\times 6{f}_{g}$ are given in the form of $\left(18,{N}_{i}\pm 1\right)$: (i) $900Hz\to \left(18,0\right)$.

#### 4.3.3. Reconstruction of the Instantaneous Electromagnetic Air-Gap Torque Waveform

#### 4.3.4. Campbell Diagram and Relevant Air-Gap Torque Harmonic Families and Their Magnitude

#### 4.4. Selected Simulation Results of a 12/12-Pulse LCI System

#### 4.4.1. 12/12-Pulse LCI System Operation at 40 Hz

#### 4.4.2. Detailed Analysis of Harmonic Families

- Baseband harmonic family of stator current are located at $\left(0,{N}_{i}\right)$: (i) $440Hz\to \left(0,11\right)$ and $520Hz\to \left(0,13\right)$; (ii) $920Hz\to \left(0,23\right)$ and $1000Hz\to \left(0,25\right)$; (iii) $1400Hz\to \left(0,35\right)$ and $1480Hz\to \left(0,37\right)$.
- Sideband harmonic family of stator current located around $1\times 12{f}_{g}$ are given in the form of $\left(12,{N}_{i}\right)$: (i) $80Hz\to \left(12,-13\right)$ and $160Hz\to \left(12,-11\right)$; (ii) $320Hz\to \left(12,-23\right)$ and $400Hz\to \left(12,-25\right)$; (iii) $560Hz\to \left(12,-1\right)$ and $640Hz\to \left(12,1\right)$.
- Sideband harmonic family of stator current located $2\times 12{f}_{g}$ are given in the form of $\left(24,{N}_{i}\right)$: (i) $200Hz\to \left(24,-25\right)$ and $280Hz\to \left(24,-23\right)$; (ii) $680Hz\to \left(24,-13\right)$ and $760Hz\to \left(24,-11\right)$; (iii) $1160Hz\to \left(24,-1\right)$ and $1240Hz\to \left(24,1\right)$.

- Baseband harmonic family of torque current are located at $\left(0,{N}_{i}\pm 1\right)$: (i) $480Hz\to \left(0,12\right)$; (ii) $960Hz\to \left(0,24\right)$; (iii) $1440Hz\to \left(0,36\right)$.
- Sideband harmonic family of torque current located around $1\times 12{f}_{g}$ are given in the form of $\left(12,{N}_{i}\pm 1\right)$: (i) $120Hz\to \left(12,-12\right)$; (ii) $360Hz\to \left(12,-24\right)$; (iii) $600Hz\to \left(12,0\right)$.
- Sideband harmonic family of torque current located $2\times 12{f}_{g}$ are given in the form of $\left(24,{N}_{i}\pm 1\right)$: (i) $240Hz\to \left(24,-24\right)$; (ii) $720Hz\to \left(24,-12\right)$; (iii) $1200Hz\to \left(24,0\right)$.

#### 4.4.3. Reconstruction of the Instantaneous Air-Gap Torque Waveform

#### 4.4.4. Summary: Campbell Diagram and Relevant Air-Gap Torque Harmonic Families and Their Magnitudes

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Simplified representation of current source and voltage source VFDs: (

**a**) current source system, where the DC tank is an inductance and the AC/DC stage is controlled; (

**b**) voltage source system, where the DC tank is a capacitor and the control of the AC/DC stage is optional.

**Figure 4.**LCI simplified modeling for electromechanical harmonic analysis purpose: (

**a**) rectifier side model with its firing angle and line impedance and (

**b**) inverter side model with its firing angle and motor impedance.

**Figure 11.**Campbell diagram of air-gap torque harmonics of a synchronous motor fed by (

**a**) 6/6-pulse LCI system, (

**b**) 12/12-pulse LCI system and (

**c**) 24/24-pulse LCI system.

**Figure 12.**Simplified representation of the simulated 6/6-pulse LCI system with a 3-phase synchronous motor.

**Figure 13.**Simplified representation of the simulated 12/12-pulse LCI system with a 6-phase synchronous motor.

**Figure 14.**Sample simulation results of a 6/6-pulse LCI system at 40 Hz: (

**a**) grid voltages, (

**b**) spectrum of grid voltage, (

**c**) grid currents, (

**d**) spectrum of grid current, (

**e**) DC-link current, (

**f**) spectrum of DC-link current, (

**g**) grid active power and reactive power and (

**h**) motor active power and reactive power.

**Figure 15.**Sample simulation results of a 6/6-pulse LCI system at 40 Hz: (

**a**) motor voltages, (

**b**) spectrum of motor voltage, (

**c**) motor currents, (

**d**) spectrum of motor current, (

**e**) motor electromagnetic torque, (

**f**) spectrum of motor electromagnetic torque, (

**g**) shaft speed and (

**h**) spectrum of shaft speed.

**Figure 16.**Sample simulation results of a 6/6-pulse LCI system at 40 Hz in alpha–beta reference frame: (

**a**) motor current, (

**b**) spectrum of motor current, (

**c**) phasor of motor current, (

**d**) motor voltage, (

**e**) spectrum of motor voltage, (

**f**) phasor of motor voltage, (

**g**) stator flux, (

**h**) spectrum of stator flux and (

**i**) phasor of stator flux.

**Figure 17.**Detailed analysis of motor current and air-gap torque harmonic families of a 6/6-pulse LCI system at 40 Hz: (

**a**) motor current spectrum and (

**b**) air-gap torque spectrum.

**Figure 18.**Reconstructed instantaneous electromagnetic torque waveform: (

**a**) reconstructed motor current waveform and (

**b**) reconstructed torque waveform.

**Figure 19.**Summary of simulation results of a 6/6-pulse LCI system: (

**a**) Campbell diagram with specific parameters $\left(m,n\right)$ to localize the simulated torque harmonics at 40 Hz, (

**b**) relevant air-gap torque harmonic families and (

**c**) magnitudes of relevant air-gap torque components for each harmonic family.

**Figure 20.**Sample simulation results of a 12/12-pulse LCI system at 40 Hz: (

**a**) grid voltages, (

**b**) spectrum of grid voltage, (

**c**) grid currents, (

**d**) spectrum of grid current, (

**e**) DC-link current, (

**f**) spectrum of DC-link current, (

**g**) grid active power and reactive power and (

**h**) motor active power and reactive power.

**Figure 21.**Sample simulation results of a 12/12-pulse LCI system at 40 Hz: (

**a**) motor voltages, (

**b**) spectrum of motor voltage, (

**c**) motor currents, (

**d**) spectrum of motor current, (

**e**) motor electromagnetic torque, (

**f**) spectrum of motor electromagnetic torque, (

**g**) shaft speed and (

**h**) spectrum of shaft speed.

**Figure 22.**Sample simulation results of a 12/12-pulse LCI system at 40 Hz in alpha–beta reference frame: (

**a**) motor current, (

**b**) spectrum of motor voltage, (

**c**) phasor of motor current, (

**d**) motor voltage, (

**e**) spectrum of motor voltage, (

**f**) phasor of motor voltage, (

**g**) stator flux, (

**h**) spectrum of stator flux and (

**i**) phasor of stator flux.

**Figure 23.**Detailed analysis of motor current and air-gap torque harmonic families of a 6/6-pulse LCI system at 40 Hz: (

**a**) motor current spectrum and (

**b**) air-gap torque spectrum.

**Figure 24.**Reconstructed instantaneous electromagnetic torque waveform for a 12/12-pulse LCI system at 40 Hz: (

**a**) reconstructed waveform of motor current set 1, (

**b**) reconstructed waveform of motor current set 2, and (

**c**) reconstructed torque waveform.

**Figure 25.**Summary of simulation results of a 12/12-pulse LCI system: (

**a**) Campbell diagram with specific parameters $\left(m,n\right)$ to localize the simulated torque harmonics at 40 Hz, (

**b**) relevant air-gap torque harmonic families and (

**c**) magnitudes of relevant air-gap torque components for each harmonic family.

Parameters | 3-Phase (SM) | 6-Phase (SM) |
---|---|---|

P_{n} | 1.85 MW | 1.85 MW |

V_{llrms} | 6000 V | 6000 V |

f_{n} | 50 Hz | 50 Hz |

poles | 3 | 3 |

Power Factor | 0.85 | 0.85 |

R_{s} | 0.0085 pu | 0.0085 pu |

L_{ls} | 0.13 pu | 0.13 pu |

R_{f} | 0.00218 | 0.005 pu |

Ll_{fd} | 0.307 pu | 0.307 pu |

R_{kd} | 0.0915 pu | 0.0915 pu |

L_{lkd} | 0.208 pu | 0.208 pu |

R_{kq} | 0.079 pu | 0.079 pu |

L_{lkq} | 0.185 pu | 0.185 pu |

L_{md} | 1.26 pu | 1.29 pu |

L_{mq} | 0.695 pu | 0.695 pu |

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## Share and Cite

**MDPI and ACS Style**

Ekemb, G.; Slaoui-Hasnaoui, F.; Song-Manguelle, J.; Lingom, P.M.; Fofana, I.
Instantaneous Electromagnetic Torque Components in Synchronous Motors Fed by Load-Commutated Inverters. *Energies* **2021**, *14*, 3223.
https://doi.org/10.3390/en14113223

**AMA Style**

Ekemb G, Slaoui-Hasnaoui F, Song-Manguelle J, Lingom PM, Fofana I.
Instantaneous Electromagnetic Torque Components in Synchronous Motors Fed by Load-Commutated Inverters. *Energies*. 2021; 14(11):3223.
https://doi.org/10.3390/en14113223

**Chicago/Turabian Style**

Ekemb, Gabriel, Fouad Slaoui-Hasnaoui, Joseph Song-Manguelle, P. M. Lingom, and Issouf Fofana.
2021. "Instantaneous Electromagnetic Torque Components in Synchronous Motors Fed by Load-Commutated Inverters" *Energies* 14, no. 11: 3223.
https://doi.org/10.3390/en14113223