# Study of Winding Structure to Reduce Harmonic Currents in Dual Three-Phase Motor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Coordinate System of Dual Three-Phase Motors

#### 2.1. Stationary Orthogonal Subspaces

- k = 1, 13, 25, …, 12 m + 1 (m = 0, 1, 2, …):

- 2.
- k = 11, 23, 35, …, 12 m + 11 (m = 0, 1, 2, …):

- 3.
- k = 5, 17, 29, …, 12 m + 5 (m = 0, 1, 2, …):

- 4.
- k = 7, 19, 31, …, 12 m + 7 (m = 0, 1, 2, …):

#### 2.2. Rotational Orthogonal Subspaces

## 3. Mathematical Model of Dual Three-Phase IPMSM

#### 3.1. Model with Inductance Saliency

#### 3.2. Model Ignores Inductance Saliency

_{2}= 0, M

_{2}= 0, and M

_{2′}= 0 in (16a–c). In this case, transforming (9) to the stationary orthogonal subspaces using (1) yields:

## 4. Design Method for Inductance of Dual Three-Phase Motors

#### 4.1. Definition of Winding Distribution Function

_{wn}is the winding factor for nth spatial harmonics. Furthermore, the winding factor k

_{wn}is:

_{pn}is the pitch factor for nth spatial harmonics and k

_{dn}is the distribution factor for nth spatial harmonics. In this paper, the number of slots per pole per phase q is assumed to be an integer. Figure 3 shows the waveform of the winding distribution function $R\left({\theta}_{e}\right)$ when ${k}_{dn}=1$ and only the pitch factor k

_{pn}is considered. From Figure 3, the MMF distribution of the winding with the coil pitch γ can be seen.

#### 4.2. Magnetomotive Force Distribution for Harmonic Currents

- k = 1, 13, 25, …, 12 m + 1 (m = 0, 1, 2, …):

- 2.
- k = 11, 23, 35, …, 12 m + 11 (m = 0, 1, 2, …):

- 3.
- k = 5, 17, 29, …, 12 m + 5 (m = 0, 1, 2, …):

- 4.
- k = 7, 19, 31, …, 12 m + 7 (m = 0, 1, 2, …):

_{e}in the positive direction and the latter with speed kω

_{e}in the negative direction.

_{e}) as a dotted line. From Figure 4a,b, it can be seen that the amplitude of the MMF is large when the energizing currents are in the α-β subspace. On the other hand, Figure 4c,d shows that the amplitude of the MMF is small when the energizing currents are in the z1-z2 subspace. The reason for this can be explained by focusing on the coefficient k

_{wn}/n in (26a–d). Since |k

_{wn}| is always less than one, the larger n is, the smaller the nth spatial harmonic component of the MMF becomes. Here, the minimum n in (26a,b) is 1, while the minimum n in (26c,d) is 5. Therefore, the amplitude of the synthetic MMF is smaller in the latter case.

#### 4.3. Calculation of Inductance

- k = 1, 13, 25, …, 12 m + 1 (m = 0, 1, 2, …):

- 2.
- k = 11, 23, 35, …, 12 m + 11 (m = 0, 1, 2, …):

- 3.
- k = 5, 17, 29, …, 12 m + 5 (m = 0, 1, 2, …):

- 4.
- k = 7, 19, 31, …, 12 m + 7 (m = 0, 1, 2, …):

_{αβ}and L

_{z}

_{1z2}in the α-β and z1-z2 subspaces are obtained as follows:

_{αβ}and L

_{z}

_{1z2}for the coil pitch γ and the number of slots per pole per phase q are normalized by L and shown in Figure 6. From Figure 6, it can be confirmed that L

_{z}

_{1z2}is basically smaller than L

_{αβ}regardless of the values of γ and q. The reason for this is that Equation (34a,b) are heavily influenced by the inverse of n

^{2}, with the former dominated by the term at n = 1 and the latter by the terms at n = 5 and 7.

_{αβ}depends on the winding factor k

_{w}

_{1}for the first spatial order, and the magnitude of L

_{z}

_{1z2}depends on the winding factors k

_{w}

_{5}and k

_{w}

_{7}for the fifth and seventh spatial order. Figure 7 shows the pitch factor k

_{pn}for the coil pitch γ and Figure 8 shows the distribution factor k

_{dn}for the number of slots per pole per phase q. From Figure 6a, we can see that L

_{αβ}is larger when the coil pitch γ is larger. From Figure 7, it can be seen that the trend of L

_{αβ}is similar to that of the pitch factor k

_{p}

_{1}for the 1st spatial order. The fact that L

_{αβ}does not show much dependence on the number of slots per pole per phase q is also similar to that of the distribution factor k

_{d}

_{1}for the 1st spatial order in Figure 8. On the other hand, Figure 6b shows that the magnitude of L

_{z}

_{1z2}has valleys around the coil pitch γ of 2/6 (=0.333) and 5/6 (=0.833) and peaks around 1/6 (=0.167), 4/6 (=0.667), and 1. From Figure 7, it can be seen that this trend is influenced by the pitch factors k

_{p}

_{5}and k

_{p}

_{7}for the 5th and 7th spatial order. Furthermore, the larger the number of slots per pole per phase q, the smaller L

_{z}

_{1z2}is, and this tendency is also similar to the distribution factors k

_{d}

_{5}and k

_{d}

_{7}for the 5th and 7th spatial order in Figure 8.

_{z}

_{1z2}in the z1-z2 subspace must be designed to be large. Figure 6b shows that L

_{z}

_{1z2}can be maximized when the coil pitch γ = 1 and the number of slots per pole per phase q = 1. On the other hand, the short-pitch winding with the coil pitch γ = 5/6, which is often used in conventional 3-phase motors, has small pitch factors k

_{p}

_{5}and k

_{p}

_{7}, which means that the 5th and 7th order components of the back-EMF can be reduced when applied to a dual 3-phase motor, but at the same time L

_{z}

_{1z2}is also reduced.

## 5. Verification by Finite Element Analysis

_{d}, L

_{q}, L

_{dz}, and L

_{qz}are obtained for the interior magnet model with the saliency in Figure 9a, and L

_{αβ}and L

_{z}

_{1z2}are obtained for the disk model without the saliency in Figure 9b. The purpose of this is to confirm that the parameters L

_{d}, L

_{q}, L

_{dz}, and L

_{qz}, which take into account the inductance saliency, show the same trend as L

_{αβ}and L

_{z}

_{1z2}, which ignore the saliency. The analysis was performed for the cases of the coil pitch γ = 1/6, 2/6, 3/6, 4/6, 5/6, and 1, with a two-layer structure where two coils fit in one slot, as shown in Figure 9.

_{αβ}and L

_{z}

_{1z2}in Figure 10 are multiplied by the correction factors K

_{αβ}and K

_{z}

_{1z2}, respectively, to the values calculated by (34a,b). From Figure 10, the theoretical values of L

_{αβ}and L

_{z}

_{1z2}are in good agreement with the FEA values. Figure 10a also shows that L

_{αβ}and L

_{q}are close to each other. This is because L

_{αβ}obtained using the model in Figure 9b is the ideal q-axis inductance. On the other hand, L

_{d}is smaller than L

_{αβ}because the magnetic resistance of the d-axis magnetic path is larger due to the presence of the magnet region. Figure 10b also shows that L

_{dz}and L

_{qz}are not affected by the saliency and are consistent with L

_{z}

_{1z2}. These results confirm the validity of the theoretical inductance equations presented in (34a,b). It was also confirmed that the inductance values L

_{d}, L

_{q}, L

_{dz}, and L

_{qz}, which take into account the saliency, show the same tendency as the inductance values L

_{αβ}and L

_{z}

_{1z2}, which ignore the saliency.

## 6. Verification by Circuit Simulation

_{αβ}and L

_{z}

_{1z2}in the α-β and z1-z2 subspaces.

#### 6.1. Simulation Method

_{z}

_{1}* = 0 and v

_{z}

_{2}* = 0. The drive conditions are shown in Table 3.

#### 6.2. Simulation Results

_{c}± 2f and 2f

_{c}± f [Hz] components indicate the carrier harmonic sideband components. From Figure 12 and Figure 13, it can be seen that the phase currents contain large harmonic components for both coil pitch γ = 5/6 and 1. For the coil pitch γ = 1, the phase currents contain mainly 5th and 7th harmonics. At coil pitch γ = 5/6, in addition to the 5th and 7th harmonics, the phase currents also contain large f

_{c}± 2f [Hz] components. Figure 14 shows that these large harmonic components are decomposed into the z1-z2 subspace by the orthogonal coordinate transformation.

#### 6.3. Consideration of the Relationship between the Magnitude of Harmonic Currents and Inductance Value

_{αβ}(ω) and Z

_{z}

_{1z2}(ω). From Figure 15, it can be seen that the impedance Z

_{z}

_{1z2}(ω) is small even in the high frequency band, basically due to the inductance L

_{z}

_{1z2}being smaller than L

_{αβ}, regardless of the coil pitch γ. This results in large harmonic currents in the z1-z2 subspace even when the harmonic components in the voltage input of the z1-z2 subspace are small.

_{c}± 2f [Hz] component) in the z1-z2 subspace for γ = 5/6 in Figure 14a. The FFT results of the applied voltage by the inverter are shown in Figure 16. From Figure 16, we see that for both γ = 5/6 and 1, the f

_{c}± 2f [Hz] component is included in the z1-z2 subspace. On the other hand, Figure 15a shows that the impedance Z

_{z}

_{1z2}(ω) around f

_{c}± 2f [Hz] for γ = 5/6 is very small, about 1 [Ω], despite the high frequency. This is thought to cause large harmonic currents at the corresponding frequencies.

_{p}

_{5}/5| and |k

_{p}

_{7}/7| for the winding with the coil pitch γ, and, as shown in Figure 7, the pitch factors k

_{p}

_{5}and k

_{p}

_{7}are particularly small when γ = 5/6. On the other hand, from Figure 15, the impedance Z

_{z}

_{1z2}(ω) for γ = 5/6 is about 1/10 of that for γ = 1 in the 5th and 7th harmonic frequency bands. This is because, from (34b), the magnitude of the inductance L

_{z}

_{1z2}is strongly influenced by (k

_{p}

_{5}/5)

^{2}and (k

_{p}

_{7}/7)

^{2}. Therefore, for the coil pitch γ = 5/6, even though the 5th and 7th harmonic components of the no-load back-EMF can be smaller than for γ = 1, the inductance L

_{z}

_{1z2}decrease is more dominant, and large 5th and 7th harmonic currents are generated.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**MMF distribution for kth harmonic currents: (

**a**) k = 12 m + 1 (α-β/positive sequence); (

**b**) k = 12 m + 11 (α-β/negative sequence); (

**c**) k = 12 m + 5 (z1-z2/positive sequence); (

**d**) k = 12 m + 7 (z1-z2/negative sequence).

**Figure 6.**Theoretical value of inductance for coil pitch γ, number of slots per pole per phase q: (

**a**) in α-β subspace; (

**b**) in z1-z2 subspace.

**Figure 9.**Model used for FEA: (

**a**) interior magnet rotor with the saliency; (

**b**) disk rotor without the saliency.

**Figure 10.**Comparison of theoretical and FEA values of inductance: (

**a**) in α-β subspace; (

**b**) in z1-z2 subspace.

**Figure 14.**FFT results of α-axis and z1-axis currents: (

**a**) Coil pitch γ = 5/6; (

**b**) Coil pitch γ = 1.

**Figure 15.**Frequency characteristic of impedance in α-β and z1-z2 subspaces: (

**a**) Coil pitch γ = 5/6; (

**b**) Coil pitch γ = 1.

**Figure 16.**FFT results of applied voltage by the inverter: (

**a**) Coil pitch γ = 5/6; (

**b**) Coil pitch γ = 1.

Harmonic Order k | 1 | 3 | 5 | 7 | 9 | 11 | 13 | … |
---|---|---|---|---|---|---|---|---|

Conventional3-phase motor | α-β Positive sequence | Zero sequence | α-β Negative sequence | α-β Positive sequence | Zero sequence | α-β Negative sequence | α-β Positive sequence | … |

Dual3-phase motor | α-β Positive sequence | o1-o2 Zero sequence | z1-z2 Positive sequence | z1-z2 Negative sequence | o1-o2 Zero sequence | α-β Negative sequence | α-β Positive sequence | … |

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

Number of poles | 8 |

Number of slots | 48 |

Stator inner diameter r [mm] | 131 |

Stack length l [mm] | 141 |

Air gap length δ [mm] | 0.5 |

Number of turns N | 4 |

Number of parallels in circuit b | 2 |

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

Reference current [Apk] | 100 |

Fundamental electrical frequency f [Hz] | 66.67 |

Carrier frequency f_{c} [Hz] | 10,000 |

DC voltage [V] | 365 |

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**MDPI and ACS Style**

Yoshida, A.; Akatsu, K.
Study of Winding Structure to Reduce Harmonic Currents in Dual Three-Phase Motor. *World Electr. Veh. J.* **2023**, *14*, 100.
https://doi.org/10.3390/wevj14040100

**AMA Style**

Yoshida A, Akatsu K.
Study of Winding Structure to Reduce Harmonic Currents in Dual Three-Phase Motor. *World Electric Vehicle Journal*. 2023; 14(4):100.
https://doi.org/10.3390/wevj14040100

**Chicago/Turabian Style**

Yoshida, Akito, and Kan Akatsu.
2023. "Study of Winding Structure to Reduce Harmonic Currents in Dual Three-Phase Motor" *World Electric Vehicle Journal* 14, no. 4: 100.
https://doi.org/10.3390/wevj14040100