# Nonlinear Varying-Network Magnetic Circuit Analysis of Consequent-Pole Permanent-Magnet Motor for Electric Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. NVNMC Model

_{TR}, P

_{YR}, P

_{CR}, P

_{TS1}, P

_{TS}, P

_{YS}, and P

_{CS}are the permeances of rotor tooth, rotor yoke, rotor consequent-yoke, stator inner tooth, stator outer tooth, stator yoke, and stator consequent-yoke, respectively, which vary with the nonlinear saturation in the corresponding magnetic paths, while P

_{PM}, P

_{LR}, and P

_{LS}are the permeances of PM, rotor tooth-to-PM leakage flux, and stator tooth-to-tooth leakage flux, respectively, which are of constant permeability [18].

_{A}module drawn in front of each stator tooth in the model, it can be considered as each stator tooth having two P

_{A}modules connecting the PM and rotor tooth. So, there is a total of 24 P

_{A}modules in this model, and apparently, the value of some P

_{A}modules could become zero when the rotor is rotating. Besides, the connection lines between the outer P

_{A}and inner PM, as well as rotor teeth, are omitted in this figure.

_{PM}, Φ

_{A}, Φ

_{B}, Φ

_{C}and Φ

_{DC}represent the magnetic flux source (MFS) of the PM, phase A winding, phase B winding, phase C winding and field winding, respectively. It should be noted that the armature MFS supplied by phase A, phase B, and phase C winding is represented by eight Φ

_{A}, Φ

_{B}, and Φ

_{C}modules separately, while the MFS provided by DC winding is represented by twelve Φ

_{DC}modules.

## 3. Model Analysis

_{0}and μ

_{iron}are the permeability of air and iron region, while b, h, and l are the width, height, and length of the element.

#### 3.1. The Permeance of Air-Gap

_{1}, Region 2 θ

_{1}< θ ≤ θ

_{2}, …, and Region 7 θ

_{6}< θ, where θ is the angle between the central lines of rotor and stator teeth. As depicted in Figure 3, θ

_{1}to θ

_{6}can be calculated as θ

_{1}= 1/2 × (β

_{r}− τ

_{s}), θ

_{2}= 1/2 × (β

_{r}− β

_{s}), θ

_{3}= 1/2 × (τ

_{r}− β

_{s}), θ

_{4}= 1/2 × (β

_{r}+ β

_{s}), θ

_{5}= 1/2 × (β

_{r}+ τ

_{s}), and θ

_{6}= 1/2 × (τ

_{r}+ τ

_{s}), where τ

_{r}and τ

_{s}are the pole pitches of rotor and stator, and β

_{r}and β

_{s}are the tooth arc of rotor and stator.

_{A}between PM and stator pole. In this paper, P

_{A}-θ curve, the relationship between air-gap permeance and rotor angle, is shown in Figure 4.

#### 3.2. The Permeance of Tooth, Yoke, and Consequent-Yoke

_{TS1}and P

_{TS}), stator yoke, and stator consequent-yoke, so that the corresponding magnetic field distributions can be seen as radial, circumferential, and axial directions, respectively [22]. Similarly, the magnetic field distributions in the rotor tooth, rotor yoke, and rotor consequent-yoke are separated in accordance with the relevant parts of the stator.

_{CR}and P

_{CS}, as well as the lengths of P

_{YR}and P

_{YS}, are specified based on FEM [18], and change dynamically with field current. But it should be mentioned that the FEM simulation only needs to run a few angles to get the widths of P

_{CR}and P

_{CS}, thus the time can be saved. Then, as the magnetic field distribution in each element is simplified as one-directional [23], (1) can be employed to calculate the P

_{TR}, P

_{YR}, P

_{CR}, P

_{TS1}, P

_{TS}, P

_{YS}, and P

_{CS}.

#### 3.3. The Magnetic Flux Source

_{m}and B

_{r}is the thickness and the remanence of PM, separately.

_{PM}, can be acquired by

_{PM}is the permeance of PM, which can be obtained through (1).

_{C}and I

_{C}are the number of turns and the current of phase C winding, respectively.

_{DC}and I

_{DC}are the numbers of turns and the current of DC windings, separately.

## 4. Magnetic Circuit Equations

#### 4.1. Establishment of the Magnetic Circuit Equations

- n number of nodes;
- P(i, j) for i, j = 1, 2, …, n − 1, is the permeance of branch, which connects node i and j;
- F(i) for i = 1, 2, …, n − 1, is the node magnetic potential;
- Φs(i) for i = 1, 2, …, n − 1, is the node magnetic flux source.

#### 4.2. Calculation of the Magnetic Circuit Equations

_{A}− θ curve (depicted in Figure 4), the precise permeance of the iron region can only be obtained by solving the NVNMC model iteratively [17]. In detail, by employing a reliable initial value of iron permeability, such as 3500 μ

_{0}, the magnetic flux in each iron element can be acquired through (10), as well as the magnetic flux density can be obtained. Therefore, the permeability of this region can be updated through the material’s B-H curve.

^{k−1}

_{iron}would be relatively small, such as 100 μ

_{0}, then with this value, the flux density acquired in kth step could be much lower, such as 1T, and the corresponding permeability, namely μ

^{k}

_{iron}, could be 4000 μ

_{0}, so the flux density acquired in (k + 1)th step could be 2T again. Hence, to avoid this problem, the permeability for kth iteration can be obtained by [24]

^{k−1}

_{iron}and μ

^{k}

_{iron}is tolerable, the iteration can be finished, and the precise solution of this NVNMC model can be acquired [23].

## 5. Results and Discussion

#### 5.1. Coil Flux Linkage

_{A}, namely the coil flux linkage of phase A. Meanwhile, the strengthening and weakening actions of the armature current are tested under different field winding currents with the value of 0 A, 10 A and −10 A. As aforementioned, hybrid excitation machines can regulate the field flux by varying the current of field winding, as shown in Figure 10. It can be seen that more flux flows at circumferential direction with 10 A field winding current, thus the width of P

_{CR}and P

_{CS}should be smaller; however, when the system is weakening the field flux, more flux flows at axial direction; thus, the width of P

_{CR}and P

_{CS}should be larger. But it should be mentioned that the width of P

_{CR}and P

_{CS}are not linearly related to the field current because of the saturation. Therefore, based on FEM, the widths of P

_{CR}and P

_{CS}are set as 40%, 25% and 55% of the stator and rotor pitches, when the field current is 0 A, 10 A and −10 A, respectively. As can be seen in Figure 9, a good agreement can be obtained between NVNMC and 3-D FEM.

#### 5.2. Back-EMF

_{A}, the back-EMF of phase A, versus rotor angle under different I

_{DC}and phase current, which can be acquired through [14]

_{5}< θ), thus when the rotor turns around 30 and 110 degrees, the results of these two methods are different.

#### 5.3. Power

#### 5.4. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Rotor positions at different regions: (

**a**) Region one; (

**b**) Region two; (

**c**) Region three; (

**d**) Region four; (

**e**) Region five; (

**f**) Region six.

**Figure 10.**Magnetic vector plot of CPPM machine. (

**a**) With 10A field current. (

**b**) With −10A field current.

**Figure 11.**Back-EMF of phase A with different DC currents: (

**a**) No phase current; (

**b**) Feeding 3 phase AC current with the peak of 3 A.

Method | Application Cases | Feature |
---|---|---|

Finite element method | - Accurate
- Available for 2-D and 3-D machines
- Magnetic field diagram
- Time-consuming
| |

Fourier modeling method | - Generally for 2-D
- Suitable for periodic structures [17]
- Time-saving
| |

Magnetic circuit analysis |

Item | Value |
---|---|

Number of phases | 3 |

Rotor inner radius (R_{IR}) | 35 mm |

Rotor outer radius (R_{OR}) | 63.5 mm |

Rotor tooth height (L_{IP}) | 2.5 mm |

Rotor tooth arc (β_{r}) | 80° |

PM thickness (L_{PM}) | 2.5 mm |

Air-gap length (L_{A}) | 0.5 mm |

The thickness of PM (L_{PM}) | 2.5 mm |

Stator inner radius (R_{IS}) | 64 mm |

Stator outer radius (R_{OS}) | 100 mm |

Stator tooth height (L_{TS}) | 16.5 mm |

Stator tooth arc (β_{s}_{1}, β_{s}_{2}) | 10°, 7.5° |

Height of motor (H_{M}) | 46 mm |

Height of DC winding (H_{DC}) | 6 mm |

DC winding turns (N_{DC}) | 35 |

Winding turns per phase (N_{A}, N_{B}, N_{C}) | 130, 130, 130 |

Residual magnetism of PM | 1.4 T |

Rotor speed | 1800 rpm |

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**MDPI and ACS Style**

Wang, H.; Chau, K.T.; Lee, C.H.T.; Chan, C.C.; Yang, T.
Nonlinear Varying-Network Magnetic Circuit Analysis of Consequent-Pole Permanent-Magnet Motor for Electric Vehicles. *World Electr. Veh. J.* **2021**, *12*, 254.
https://doi.org/10.3390/wevj12040254

**AMA Style**

Wang H, Chau KT, Lee CHT, Chan CC, Yang T.
Nonlinear Varying-Network Magnetic Circuit Analysis of Consequent-Pole Permanent-Magnet Motor for Electric Vehicles. *World Electric Vehicle Journal*. 2021; 12(4):254.
https://doi.org/10.3390/wevj12040254

**Chicago/Turabian Style**

Wang, Hui, Kwok Tong Chau, Christopher H. T. Lee, C. C. Chan, and Tengbo Yang.
2021. "Nonlinear Varying-Network Magnetic Circuit Analysis of Consequent-Pole Permanent-Magnet Motor for Electric Vehicles" *World Electric Vehicle Journal* 12, no. 4: 254.
https://doi.org/10.3390/wevj12040254