# Rotor Eddy Current Loss Calculation of a 2DoF Direct-Drive Induction Motor

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## Abstract

**:**

## 1. Introduction

## 2. Structure and Parameters

## 3. Rotor Calculation

#### 3.1. Linear Analytical Method of ECLCL and ECLRC

#### 3.1.1. Magnetic Field Analysis

- (1)
- The curvature effect is disregarded. The rotor and stator are expanded into a semi-infinite flat model [13].
- (2)
- The phenomenon wherein the no-load speed exceeds the synchronous speed caused by the longitudinal edge effect is disregarded [14]. The synchronous speed is ${n}_{1}={60f}_{1}/p.$
- (3)
- The primary core has infinite permeability, and the conductivity is zero.
- (4)
- The end, hysteresis, and saturation effects are ignored. The rotor material is isotropic, and permeability and conductivity are constant.
- (5)
- The displacement current and influence of asymmetric three-phase current are ignored. The inducted current in the rotor only includes the z-component.
- (6)
- Each field includes a fundamental component, and the time variations of each field are sinusoidal.

_{1}, y, z) is fixed on the stator, in which the origins of the coordinates are taken at the center of the rotor. Here, x, y, and z express the circumferential, radial, and axial directions of the motor, respectively. The relationship between x

_{1}and x is ${\omega}_{1}{t-ax}_{1}{=s\omega}_{1}t-ax$, where s indicates the slip, ${\omega}_{1}$ denotes the primary angular frequency, t is time, $a=\pi /\tau $, and $\tau $ is the pole pitch.

_{e}as follows [16]:

_{dp}represents the coefficient of fundamental winding, N

_{1}indicates the armature winding number, I

_{1}denotes the root mean square (RMS) of the stator phase current, ${m}_{1}$ is the number of the phases; $-{L}_{e}/2\le z\le {L}_{e}/2$, and L

_{e}is the stator axial length.

_{gn}, D

_{n}, C

_{cn}, D

_{cn}and C

_{2n}are undetermined coefficients.

#### 3.1.2. Mathematical Model of ECLCL

_{2}expresses the linear velocity of the rotor relative to the fundamental magnetic field.

#### 3.1.3. Mathematical Model of ECLRC

#### 3.2. Nonlinear Analytical Method of ECLCL and ECLRC

- (1)
- Select the calculating point and divide the time period into 2m segments. Owing to the symmetry and parity of $\dot{{\mathit{J}}_{sz}}$, calculating ECLCL and ECLRC in a quarter of the period is sufficient.
- (2)
- Use ${J=J}_{0}\mathrm{sin}(\frac{(m-2k)\pi}{2m})$ as an amplitude to establish the expression of $\dot{{\mathit{J}}_{sz}}$, $\dot{{\mathit{J}}_{sz}}={\sum}_{n}\frac{4}{\pi}{Je}^{{j(\omega}_{1}{t-ax}_{1})}\frac{1}{n}\mathrm{sin}\frac{n\pi}{{L}_{e}}z$, $0\le k\le \frac{m}{2}$ (when m is even), $0\le k\le \frac{m-1}{2}$ (when m is odd); k is an integer.
- (3)
- Assume the initial relative permeability of the calculating point ${\mu}_{r}$.
- (4)
- Count the vector magnetic potentials by using ${\mu}_{r}$.
- (5)
- On the basis of the relationship between vector magnetic potential and magnetic field strength ($\dot{\mathit{H}}=\frac{1}{\mu}\mathrm{rot}\dot{\mathit{A}}$), obtain the magnetic field strength of the rotor core (H
_{x}, H_{y}, H_{z}). - (6)
- Compute the magnetic field strength $H=\sqrt{{\left|{H}_{x}\right|}^{2}+{\left|{H}_{y}\right|}^{2}+{\left|{H}_{z}\right|}^{2}}$ and obtain the relative permeability ${\mu}_{r}^{\prime}$ corresponding to H by checking the B-H curve.
- (7)
- Calculate the error. If $\left|\frac{{\mu}_{r}{-\mu}_{r}^{\prime}}{{\mu}_{r}}\right|>\epsilon $, ${\mu}_{r}{=\mu}_{r}^{\prime}$, return to step 4; otherwise, proceed to the following steps.
- (8)
- Calculate the inducted current density $\dot{\mathit{J}}$ and the copper layer and rotor core loss powers in one segment by using Equations (15) to (17).
- (9)
- Multiply the obtained loss powers by the corresponding volumes to obtain ECLCL and ECLRC in one segment.

- ${K}_{eK}$ was proposed by Kesavamurthy [23]:$${K}_{eK}=(1+\frac{\tau}{{L}_{e}}\left)\right(1+\frac{s}{4}\frac{{L}_{e}}{\tau})$$
- ${K}_{ey}$ was proposed by Yee [24]:$${K}_{ey}=1+\frac{\frac{2}{\pi}\frac{\tau}{{L}_{e}}}{1+\mathrm{coth}\left(\frac{\pi}{2}\frac{{L}_{e}}{\tau}\right)-\frac{2}{\pi}\frac{\tau}{{L}_{e}}}$$
- ${K}_{em}$ was proposed by Mokirinihoph [25]:$${K}_{em}=\frac{1}{1-\frac{2\tau}{{\pi L}_{e}}\frac{\mathrm{ch}(\frac{\pi}{2}\frac{{L}_{e}}{\tau}\left)\mathrm{sh}\right(\frac{\pi}{2}\frac{{L}_{e}}{\tau})}{\mathrm{ch}(\frac{\pi}{2}\frac{{L}_{e}{+D}_{2}}{\tau})}}$$
- ${K}_{eu}$ was proposed by Lee Tesor [26]:$${K}_{eu}=1+\frac{{D}_{2}}{{L}_{e}}\mathrm{sin}\frac{\pi}{2p}$$
- ${K}_{eF}$ was proposed by Fengli Fu and Jinming Lin [27]:$${K}_{eK}=(1+\frac{\tau}{{L}_{e}}\left)\right(\frac{{a}^{2}\delta +\frac{1}{{\mu}_{r}\u2206}}{{\lambda}_{1}^{2}+\frac{1}{{\mu}_{r}\u2206}})$$

#### 3.3. Analytical Model of SECLCL and SECLRC

- (1)
- The materials of the rotor and stator are isotropic. The primary cores have infinite permeability and resistance.
- (2)
- The curvature effect is neglected. The currents only include the z-component.
- (3)
- The time variation of each harmonic field is sinusoidal.

_{vm}denotes the amplitude of the electric field intensity generated by the v-th order harmonic. When the edge effect is ignored, the Equation (22) is simplified as [10]:

## 4. Results Comparison and Analysis

#### 4.1. Copper Layer TECL

- (1)
- With the increase in slip, the saturation of the rotor core increases. When the saturation is ignored, the relative permeability of the rotor core is much higher than the actual value, and the magnetic density and analysis value of the loss are large. The new method, which considers the nonlinearity of the rotor core, can improve the accuracy of the rotor core’s relative permeability. Thus, the rotor core’s relative permeability decreases, and the loss analysis value decreases.
- (2)
- The radial inducted current is disregarded. The FEM can consider the loss caused by the radial inducted current.
- (3)
- Owing to the stator circumferential breaking structure, the longitudinal edge effect is formed at the stator end. The end magnetic field and the end induced current caused by the longitudinal edge effect increase the loss in the rotor. According to Assumptions 2 in Section 3.1, the edge effect is ignored in the analysis calculation, so the error increases.
- (4)
- In this work, the period of surface current density $\dot{{\mathit{J}}_{sz}}$ is divided into 22 segments (m = 11); m is not large enough and influences the accuracy.

#### 4.2. Rotor Core TECL

## 5. Conclusions

- (1)
- For TECL in the copper layer, most errors are less than 6%, except for $s$ = 0.1–0.2. For the rotor core, the errors of the corrected nonlinear analysis results are less than $7.3\%$. Hence, the new analytical method is valid.
- (2)
- In this paper, the time period of $\dot{{\mathit{J}}_{sz}}$ is divided into 2m = 22 segments. The value of m influences calculation accuracy. Calculation accuracy and computation time increase with the increase of m.
- (3)
- Induced current is observed in the radial direction, and the edge effect occurs at the stator end. Hence, the calculation values are lower than those determined in the simulation, as shown in Figure 17 and Figure 18. The new nonlinear analytical method requires modifications to consider the edge effect and radial induced current.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Structure of the two-degree-of-freedom direct-drive induction motor (2DoFDDIM). (

**a**) Rotary motion arc-shape stator; (

**b**) Linear motion arc-shape stator; (

**c**) Rotor; (

**d**) Assembly of 2DoFDDIM.

Item | Parameter |
---|---|

rated voltage | 220 V (Y) |

stator outer diameter | 155 mm |

stator inter diameter | 98 mm |

stator axial length | 156 mm |

slot number | 12 |

phase | 3 |

pole pair | 2 |

thickness of air gap | 1 mm |

thickness of copper layer | 1.5 mm |

frequency | 50 Hz |

Slip | FEM (W) | Linear (W) | Error (%) | Nonlinear (W) | Error (%) |
---|---|---|---|---|---|

0.1 | 717.33 | 602.037 | 16.07 | 592.333 | 17.43 |

0.2 | 769.09 | 748.601 | 2.664 | 690.507 | 10.22 |

0.3 | 867.74 | 983.466 | 13.34 | 820.599 | 5.433 |

0.4 | 1004.3 | 1264.13 | 25.87 | 976.030 | 2.818 |

0.5 | 1169.9 | 1563.02 | 33.60 | 1150.22 | 1.684 |

0.6 | 1355.6 | 1863.41 | 37.46 | 1336.60 | 1.399 |

0.7 | 1552.3 | 2155.96 | 38.89 | 1528.57 | 1.529 |

0.8 | 1751.2 | 2435.95 | 39.10 | 1719.57 | 1.806 |

0.9 | 1943.3 | 2701.24 | 39.04 | 1903.02 | 2.074 |

1 | 2119.7 | 2950.84 | 39.21 | 2072.34 | 2.235 |

Slip | FEM (W) | Linear (W) | Error (%) | Nonlinear (W) | Error (%) |
---|---|---|---|---|---|

0.1 | 4.837 | 5.719 | 18.24 | 4.808 | 0.593 |

0.2 | 6.028 | 16.75 | 177.8 | 6.158 | 2.167 |

0.3 | 7.100 | 28.14 | 296.3 | 6.702 | 5.596 |

0.4 | 8.078 | 39.44 | 388.3 | 7.587 | 6.068 |

0.5 | 8.985 | 50.19 | 458.7 | 8.942 | 0.474 |

0.6 | 9.844 | 59.94 | 508.9 | 10.29 | 4.546 |

0.7 | 10.68 | 68.22 | 538.7 | 11.28 | 5.591 |

0.8 | 11.52 | 74.59 | 547.6 | 12.04 | 4.551 |

0.9 | 12.38 | 78.57 | 534.7 | 13.09 | 5.738 |

1 | 13.29 | 79.72 | 499.8 | 14.25 | 7.233 |

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## Share and Cite

**MDPI and ACS Style**

Wu, W.; Si, J.; Feng, H.; Cheng, Z.; Hu, Y.; Gan, C.
Rotor Eddy Current Loss Calculation of a 2DoF Direct-Drive Induction Motor. *Energies* **2019**, *12*, 1134.
https://doi.org/10.3390/en12061134

**AMA Style**

Wu W, Si J, Feng H, Cheng Z, Hu Y, Gan C.
Rotor Eddy Current Loss Calculation of a 2DoF Direct-Drive Induction Motor. *Energies*. 2019; 12(6):1134.
https://doi.org/10.3390/en12061134

**Chicago/Turabian Style**

Wu, Wei, Jikai Si, Haichao Feng, Zhiping Cheng, Yihua Hu, and Chun Gan.
2019. "Rotor Eddy Current Loss Calculation of a 2DoF Direct-Drive Induction Motor" *Energies* 12, no. 6: 1134.
https://doi.org/10.3390/en12061134