# Analysis of DC Winding Induced Voltage in Wound-Rotor Synchronous Machines by Using the Air-Gap Field Modulation Principle

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## Abstract

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## 1. Introduction

- (1)
- The analytical MMF-permeance model is established, based on which the spatial–temporal characteristics of the air-gap flux density are analyzed.
- (2)
- The operating mechanism of the DC winding induced voltage in WRSM is analyzed first, and the harmonic order of the DC winding induced voltage is deduced and compared with that predicted by FEA.
- (3)
- Winding configurations of concentrated winding and distributed winding are included in the analysis.

## 2. Magneto-Motive Force-Permeance Modeling for the DC Winding Induced Voltage in WRSM

#### 2.1. Specifications of the Analyzed WRSMs

_{d}= 0 control with fixed 60 W copper loss in both armature windings and DC winding. Both DC winding and armature winding also share the same packing factor k

_{pf}= 0.5.

#### 2.2. MMF-Permeance Modelling

- The steel permeability is infinite.
- Flux lines are perpendicular to the steel surface.
- The effect of finite stack length is negligible.

#### 2.2.1. Open-circuit condition

_{dc}(θ

_{m},t) is the MMF function of the DC winding, W

_{dc}(θ

_{m},t) is the winding function of the DC winding, θ

_{m}is the mechanical degree, ω

_{m}is the mechanical rotor angular speed and t is the time; I

_{f}is the DC winding current.

_{ns}are Fourier transformation coefficient, Λ

_{s}is the stator permeance function, Λ

_{0}is the stator average permeance, θ

_{stip}is the stator tooth-tip angle.

_{s}, 2N

_{s}, 3N

_{s}…. It should be noted that rotor permeance is not modeled in this paper since the DC winding MMF function contained the information on the rotor permeance.

_{goc}can be expressed as

_{g}is the air-gap flux density, μ

_{0}is the vacuum permeability.

_{dcoc}is the open-circuit DC winding flux-linkage.

_{peopen}can be expressed as

_{peopen}can be calculated based on Equation (6), and the results are listed in Table 2.

#### 2.2.2. Armature Reaction Condition

_{i}is the current angle, I

_{A}is the phase current amplitude.

_{aA}, W

_{aB}, and W

_{aC}are armature winding functions, W

_{aA}

_{0}are the average values of the armature winding function, and a

_{nw}are the Fourier transformations for the armature winding. W

_{aA}

_{0}and a

_{nw}for the analyzed 12/4 WRSM and 24/4 WRSM can be expressed as

_{aA}

_{0}and a

_{nw}for specific windings.

_{A}, F

_{B}, and F

_{C}are armature winding MMF for phase A, phase B and phase C.

_{gar}can be expressed as

_{r}is the rotor permeance function, it will be reflected by the DC winding function in the following. Since the armature MMF function contained the information on stator permeance, Equation (13) does not include the stator permeance function. The DC winding flux-linkage ψ

_{dcar}caused by the armature reaction can be expressed as

_{r}is the spatial order introduced by the DC winding function, and np

_{r}is the spatial order introduced by armature winding MMF. By solving Equation (16), only when n = 6k − 1 or 6k + 1, k = 1,2,3… can the corresponding flux density contribute to the DC winding flux-linkage.

_{pear}can be expressed as

_{pear}can be predicted, which is listed in Table 3.

#### 2.2.3. On-Load Condition

_{peol}can be expressed as the least common multiple of N

_{peopen}and N

_{pear}. For the analyzed WRSM, the prediction results of N

_{peol}are listed in Table 4,

## 3. FEA Verification

#### 3.1. Open-Circuit Condition

_{peopen}= 6, which is the same as the prediction result in Table 2. The influence of steel saturation will not influence the harmonic order of the open-circuit DC winding induced voltage. However, the steel saturation will influence the magnitude of the open-circuit DC winding induced voltage. This is evident for the 24/4 WRSM, which has higher amplitude open-circuit DC winding induced voltage. As for the 12/4 WRSM, the saturation will result in a higher magnitude of the 12th harmonic DC winding induced voltage.

#### 3.2. On-Load Condition

_{peol}= 6, which is the same as the prediction result in Table 4. As for the influence of the steel saturation, the on-load DC winding induced voltage has a much higher amplitude for both WRSM machines. However, the steel saturation will not influence the harmonic order of the DC winding induced voltage. The 24/4 WRSM with distributed armature winding has a higher amplitude of DC winding induced voltage than that of 12/4 WRSM with the concentrated armature winding.

## 4. Conclusions

_{peopen}, N

_{pear}and N

_{peal}are deduced analytically. The deduced results are verified by the FEA with linear steel and non-linear steel, which shows that the analytical prediction results agree well with the FEA analysis results. The analytical model in this paper can give a straightforward insight into the physical nature of the DC winding-induced voltage in WRSM. However, the analytical model cannot predict the magnitude and phase angle of the DC winding induced voltage, since the main purpose of this paper is to analyze the operating mechanism of the DC winding-induced voltage in WRSM. It is different from that in the wound field switched flux machine, which was analyzed in [20,27]. Based on the analysis presented in this paper, reduction methods investigation of the DC winding induced voltage for WRSM is the next stage of work for this paper.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Topology of the analyzed WRSM. (Left: 12-stator slot/4-rotor pole pair, concentrated armature winding. Right: 24-stator slot/4-rotor pole pair, distributed armature winding).

**Figure 7.**Open-circuit DC winding induced voltage waveforms and spectra for 12/4 WRSM, (

**a**) Waveforms; (

**b**) Spectra.

**Figure 8.**Open-circuit DC winding induced voltage waveforms and spectra for 24/4 WRSM, (

**a**) Waveforms; (

**b**) Spectra.

**Figure 11.**On-load DC winding induced voltage waveforms and spectra for 12/4 WRSM: (

**a**) Waveforms; (

**b**) Spectra.

**Figure 12.**On-load DC winding induced voltage waveforms and spectra for 24/4 WRSM: (

**a**) Waveforms; (

**b**) Spectra.

Parameters | Unit | WRSM | |
---|---|---|---|

Stator slot number, N_{s} | - | 12 | 24 |

Rotor pole pair number, p_{r} | - | 4 | 4 |

Stator outer radius, r_{so} | mm | 45 | 45 |

Stator yoke radius, r_{sy} | mm | 40.97 | 40.75 |

Stator inner radius, r_{si} | mm | 32.25 | 32.29 |

Slot opening arc, θ_{so} | deg. | 3.196 | 3.018 |

Stator tooth arc, θ_{st} | deg. | 13.15 | 7.133 |

Single side air-gap width, g | mm | 0.5 | 0.5 |

Rotor outer radius, r_{ro} | mm | 31.75 | 31.79 |

Rotor yoke radius, r_{ry} | mm | 15.61 | 16.8 |

Rotor tooth arc, θ_{rt} | deg. | 15.54 | 15.85 |

Rotor tip arc, θ_{rtip} | deg. | 5.71 | 6.39 |

Shaft radius, r_{sh} | mm | 10.4 | 10.4 |

Stack length, l_{stk} | mm | 50 | 50 |

Lamination steel type | - | M270-35 | |

Armature coil turns, N_{ac} | - | 36 | 36 |

DC coil turns, N_{dc} | - | 90 | 90 |

Packing factor, k_{pf} | - | 0.5 | 0.5 |

Rotor speed, n_{r} | rpm | 1000 | 1000 |

Armature winding copper loss, p_{cac} | W | 60 | 60 |

DC winding copper loss, p_{cdc} | W | 60 | 60 |

Parameters | WRSM | |
---|---|---|

Stator-slot/Rotor-pole-pair | 12/4 | 24/4 |

N_{peopen} | 6 | 6 |

Parameters | WRSM | |
---|---|---|

Stator slot/Rotor pole pair | 12/4 | 24/4 |

N_{pear} | 6 | 6 |

Parameters | WRSM | |
---|---|---|

Stator slot/Rotor pole pair | 12/4 | 24/4 |

N_{peol} | 6 | 6 |

Parameters | Linear Steel | Non-Linear Steel | ||
---|---|---|---|---|

Stator slot/Rotor pole pair | 12/4 | 24/4 | 12/4 | 24/4 |

N_{peopen} | 6 | 6 | 6 | 6 |

N_{peol} | 6 | 6 | 6 | 6 |

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**MDPI and ACS Style**

Zhang, W.; Fan, Y.; Zhu, Z.Q.; Wu, Z.; Hua, W.; Cheng, M.
Analysis of DC Winding Induced Voltage in Wound-Rotor Synchronous Machines by Using the Air-Gap Field Modulation Principle. *World Electr. Veh. J.* **2022**, *13*, 215.
https://doi.org/10.3390/wevj13110215

**AMA Style**

Zhang W, Fan Y, Zhu ZQ, Wu Z, Hua W, Cheng M.
Analysis of DC Winding Induced Voltage in Wound-Rotor Synchronous Machines by Using the Air-Gap Field Modulation Principle. *World Electric Vehicle Journal*. 2022; 13(11):215.
https://doi.org/10.3390/wevj13110215

**Chicago/Turabian Style**

Zhang, Wentao, Ying Fan, Z. Q. Zhu, Zhongze Wu, Wei Hua, and Ming Cheng.
2022. "Analysis of DC Winding Induced Voltage in Wound-Rotor Synchronous Machines by Using the Air-Gap Field Modulation Principle" *World Electric Vehicle Journal* 13, no. 11: 215.
https://doi.org/10.3390/wevj13110215