# Analytical Investigation on Torque of Three-Degree-of-Freedom Electromagnetic Actuator for Image Stabilization

^{1}

^{2}

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## Abstract

**:**

## Featured Application

## Abstract

## 1. Introduction

## 2. Design of Original 3-DOF Spherical Motor

## 3. Design of Proposed 3-DOF Spherical Motor

_{1}, C

_{2}, C

_{3}, and C

_{4}, respectively. As shown, when the rotor is rotating around the X-axis from 0° to 25° in a counter-clockwise direction, the magnetic flux through the coils C

_{2}and C

_{4}is too weak, so the coils C

_{2}and C

_{4}are turned off in the proposed design. Due to the proposed electromagnetic structure with a particular controlling strategy, the proposed 3-DOF spherical motor has higher torque output in the large rotation angle when compared to the original 3-DOF spherical motor [25]. The flowchart designed to actuate the proposed 3-DOF spherical motor for rotating around the X-axis is illustrated in Figure 6. In this paper, we discuss the actuator rotating around the X-axis between 0° and 25°, so we only need to power on the coil C

_{1}and C

_{3}. For another scenario, if we want to let the actuator rotate around the X-axis between 0° and −25°, we only need to power on the coil C

_{2}and C

_{4}. In the next section, the improvement in the static torque of the proposed 3-DOF spherical motor is verified by using the 3D FEM.

## 4. 3D FEM Simulation

^{−7}H and 19.2 Ω, respectively; those of each coil for rotating around the Z-axis are 2.2 × 10

^{−7}H and 6.5 Ω, respectively. In the proposed 3-DOF spherical motor, each coil for rotating around the X-axis, Y-axis, and Z-axis have 450, 450, and 250 turns, respectively. By calculating, the inductance and resistance of each coil for rotating around the X- and Y-axes are 4.0 × 10

^{−7}H and 9.6 Ω, respectively; those of each coil for rotating around the Z-axis are 2.2 × 10

^{−7}H and 6.5 Ω, respectively. The current 0.25 A was selected to pass each coil of the original and proposed 3-DOF spherical motor. Therefore, the current density of the original and proposed structure is 1.4 × 10

^{7}A/m

^{2}. The diameter of the coils is 0.15 mm for both the original 3-DOF spherical motor and proposed 3-DOF spherical motor. It is noted that for the original 3-DOF spherical motor [25] and proposed 3-DOF spherical motor, the main driving range of both motors is around the origin, and the coils are switched at the same high speed for both motors. Here, the power consumption issue for the driving circuit is ignored.

_{o}can be expressed as:

_{o}= 2 I

_{o}

^{2}R

_{o},

_{o}and R

_{o}are the current and the resistance of each coil, respectively, for the original 3-DOF spherical motor. For the proposed 3-DOF spherical motor, in the simulation case for rotating around the X-axis, the power consumption P

_{p}can be expressed as:

_{p}= 2 I

_{p}

^{2}R

_{p},

_{p}and R

_{p}are the current and the resistance of each coil, respectively, for the proposed 3-DOF spherical motor. It is noted that

_{p}= 1/2R

_{o}.

_{o}= P

_{p}, from Equations (1)–(3), the following equation can be obtained:

_{p}= √2I

_{o}.

## 5. Numerical Results of 3D FEM and Discussion

^{7}A/m

^{2}. For easier reading, the results in Table 5 are also shown with step size of 5° except the magnitude of the average values in the bottom. Based on the same design parameters, the improvement percentage of the torque density is the same as that of the torque output in Table 5. Here, the outermost diameter is used to calculate the motor’s volume and the torque density. It is obvious that the greater the rotation angle, the lower the efficiency for the original 3-DOF spherical motor; in contrast, the greater the rotation angle, the higher the efficiency for the proposed 3-DOF spherical motor.

## 6. Conclusions

## 7. Patents

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Detailed components and (

**b**) magnetization of permanent magnets in original 3-DOF spherical motor.

**Figure 4.**Y-Z cross-section of original 3-DOF spherical motor when the rotor is rotating around the X-axis at (

**a**) 0° and (

**b**) 25°, respectively.

**Figure 5.**(

**a**) Basic structure of proposed 3-DOF spherical motor; (

**b**) Y-Z cross-section of proposed 3-DOF spherical motor when the rotor is rotating around the X-axis at 25°.

**Figure 8.**Magnetic flux density distribution of Y-Z cross-section when the rotor is rotating around the X-axis at 0° for (

**a**) original and (

**b**) proposed 3-DOF spherical motor, respectively.

**Figure 9.**Magnetic flux density distribution of Y-Z cross-section when the rotor is rotating around the X-axis at 25° for (

**a**) original and (

**b**) proposed 3-DOF spherical motor, respectively.

**Figure 10.**The vector diagram of the magnetic flux density distribution in the yoke and magnet when the coils of (

**a**) original and (

**b**) proposed 3-DOF spherical motor are powered on and (

**c**) those are not powered on, respectively.

**Figure 11.**Numerical results obtained for the torque output with respect to different rotation angle of, respectively, (

**a**) original and (

**b**) proposed 3-DOF spherical motor rotating from −25° to +25° in a counter-clockwise direction and (

**c**) original and (

**d**) proposed 3-DOF spherical motor rotating from +25° to −25° in a clockwise direction with reverse axis direction.

Type | Voice Coil Motor Type | Stepping Motor Type | Induction Motor Type | Synchronous Motor Type | Reluctance Motor Type |
---|---|---|---|---|---|

Control | Simple | Complex | Complex | Complex | Medium |

Structure | Simple | Complex | Complex | Complex | Simple |

Response | High | Medium | Medium | Medium | Low |

Positioning accuracy | High | Low | Low | High | Low |

Force/Torque | Low | Medium | High | Medium | High |

Cogging effect | No | Yes | No | Yes | No |

Symbol | Corresponding Parameter | Value |
---|---|---|

D_{O1}, D_{O2} | Outermost diameter | 28.0 (mm) |

t_{O1}, t_{O2} | Outer yoke thickness | 2.0 (mm) |

t_{g1} | Air gap length | 2.0 (mm) |

t_{g2} | 2.1 (mm) | |

t_{m1} | Permanent magnet thickness | 3.1 (mm) |

t_{m2} | 2.7 (mm) | |

t_{i} | Inner yoke thickness | 8.0 (mm) |

θ_{m1}, θ_{m2} | Permanent magnet angle | 60.0 (deg) |

θ_{O1}, θ_{O2} | Outer yoke angle | 70.0 (deg) |

w_{O1} | Outer yoke width | 7.5 (mm) |

w_{O2} | 10.0 (mm) | |

w_{m1} | Permanent magnet width | 6.5 (mm) |

w_{m2} | 3.0 (mm) | |

w_{i1} | Inner yoke width | 13.9 (mm) |

w_{i2} | 14.4 (mm) |

Rotation Angle (Deg) | Total Torque Output (mN·m) | Lorentz Torque Output (mN·m) | Attractive Torque Output (mN·m) | Ratio of Lorentz and Attractive Torque Output |
---|---|---|---|---|

Rotate from −25° to +25° in a counter-clockwise direction | ||||

−25 | 6.29 | 5.31 | 0.99 | 5.38 |

−20 | 7.86 | 5.73 | 2.13 | 2.69 |

−15 | 8.60 | 5.99 | 2.61 | 2.30 |

−10 | 8.78 | 6.10 | 2.67 | 2.28 |

−5 | 8.56 | 6.15 | 2.41 | 2.56 |

0 | 8.18 | 6.14 | 2.04 | 3.01 |

5 | 7.78 | 6.09 | 1.69 | 3.60 |

10 | 7.38 | 6.02 | 1.36 | 4.45 |

15 | 7.02 | 5.84 | 1.18 | 4.94 |

20 | 6.65 | 5.54 | 1.12 | 4.95 |

25 | 6.43 | 5.02 | 1.41 | 3.56 |

Rotate from +25° to −25° in a clockwise direction | ||||

25 | −6.23 | −5.29 | −0.94 | 5.65 |

20 | −7.89 | −5.74 | −2.15 | 2.67 |

15 | −8.60 | −6.00 | −2.60 | 2.30 |

10 | −8.68 | −6.11 | −2.57 | 2.38 |

5 | −8.50 | −6.15 | −2.35 | 2.62 |

0 | −8.17 | −6.13 | −2.04 | 3.01 |

−5 | −7.79 | −6.10 | −1.68 | 3.63 |

−10 | −7.33 | −6.01 | −1.32 | 4.56 |

−15 | −7.00 | −5.84 | −1.16 | 5.04 |

−20 | −6.65 | −5.54 | −1.11 | 4.98 |

−25 | −6.38 | −5.04 | −1.34 | 3.76 |

Rotation Angle (Deg) | Total Torque Output (mN·m) | Lorentz Torque Output (mN·m) | Attractive Torque Output (mN·m) | Ratio of Lorentz and Attractive Torque Output |
---|---|---|---|---|

Rotate from −25° to +25° in a counter-clockwise direction | ||||

−25 | 6.64 | 5.72 | 0.93 | 6.18 |

−20 | 7.26 | 5.49 | 1.77 | 3.11 |

−15 | 7.03 | 5.03 | 2.00 | 2.52 |

−10 | 6.22 | 4.39 | 1.84 | 2.38 |

−5 | 5.13 | 3.69 | 1.45 | 2.55 |

0 | 4.09 | 3.23 | 0.86 | 3.77 |

5 | 4.58 | 4.27 | 0.31 | 13.66 |

10 | 4.87 | 4.95 | −0.08 | −61.36 |

15 | 5.24 | 5.52 | −0.29 | −19.33 |

20 | 5.66 | 5.90 | −0.24 | −24.94 |

25 | 6.07 | 6.02 | 0.05 | 123.17 |

Rotate from +25° to −25° in a clockwise direction | ||||

25 | −6.67 | −5.71 | −0.96 | 5.94 |

20 | −7.20 | −5.51 | −1.69 | 3.25 |

15 | −7.04 | −5.02 | −2.02 | 2.49 |

10 | −6.27 | −4.38 | −1.89 | 2.32 |

5 | −5.16 | −3.68 | −1.48 | 2.49 |

0 | −4.11 | −3.23 | −0.87 | 3.71 |

−5 | −4.62 | −4.28 | −0.35 | 12.39 |

−10 | −4.92 | −4.95 | 0.03 | −162.85 |

−15 | −5.25 | −5.53 | 0.28 | −20.01 |

−20 | −5.60 | −5.90 | 0.30 | −19.54 |

−25 | −6.04 | −6.01 | −0.04 | 170.21 |

**Table 5.**Comparison of the torque output and torque density of original and proposed 3-DOF spherical motor.

Rotation Angle (Deg) | Total Torque Output of Original Motor (mN·m) | Total Torque Output of Proposed Motor (mN·m) | Total Torque Density of Original Motor (kN·m/m ^{3}) | Total Torque Density of Proposed Motor (kN·m/m ^{3}) | Improvement Percentage (%) |
---|---|---|---|---|---|

Rotate from −25° to +25° in a counter-clockwise direction | |||||

−25 | 6.29 | 9.01 | 0.55 | 0.78 | 43.13 |

−20 | 7.86 | 9.72 | 0.68 | 0.85 | 23.56 |

−15 | 8.60 | 9.38 | 0.75 | 0.82 | 8.99 |

−10 | 8.78 | 8.34 | 0.76 | 0.73 | −4.95 |

−5 | 8.56 | 6.98 | 0.74 | 0.61 | −18.49 |

0 | 8.18 | 5.76 | 0.71 | 0.50 | −29.64 |

5 | 7.78 | 6.69 | 0.68 | 0.58 | −14.06 |

10 | 7.38 | 7.28 | 0.64 | 0.63 | −1.34 |

15 | 7.02 | 7.88 | 0.61 | 0.69 | 12.22 |

20 | 6.65 | 8.44 | 0.58 | 0.73 | 26.87 |

25 | 6.43 | 8.86 | 0.56 | 0.77 | 37.68 |

Rotate from +25° to −25° in a clockwise direction | |||||

25 | −6.23 | −9.08 | −0.54 | −0.79 | 45.88 |

20 | −7.89 | −9.64 | −0.69 | −0.84 | 22.19 |

15 | −8.60 | −9.38 | −0.75 | −0.82 | 9.14 |

10 | −8.68 | −8.40 | −0.76 | −0.73 | −3.30 |

5 | −8.50 | −6.99 | −0.74 | −0.61 | −17.76 |

0 | −8.17 | −5.78 | −0.71 | −0.50 | −29.30 |

−5 | −7.79 | −6.74 | −0.68 | −0.59 | −13.50 |

−10 | −7.33 | −7.32 | −0.64 | −0.64 | −0.17 |

−15 | −7.00 | −7.91 | −0.61 | −0.69 | 12.95 |

−20 | −6.65 | −8.38 | −0.58 | −0.73 | 25.97 |

−25 | −6.38 | −8.81 | −0.55 | −0.77 | 38.24 |

Average (Magnitude) | 7.69 | 7.98 | 0.67 | 0.69 | 5.22 |

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**MDPI and ACS Style**

Liu, C.-S.; Lin, Y.-H.; Yeh, C.-N.
Analytical Investigation on Torque of Three-Degree-of-Freedom Electromagnetic Actuator for Image Stabilization. *Appl. Sci.* **2021**, *11*, 6872.
https://doi.org/10.3390/app11156872

**AMA Style**

Liu C-S, Lin Y-H, Yeh C-N.
Analytical Investigation on Torque of Three-Degree-of-Freedom Electromagnetic Actuator for Image Stabilization. *Applied Sciences*. 2021; 11(15):6872.
https://doi.org/10.3390/app11156872

**Chicago/Turabian Style**

Liu, Chien-Sheng, Yi-Hsuan Lin, and Chiu-Nung Yeh.
2021. "Analytical Investigation on Torque of Three-Degree-of-Freedom Electromagnetic Actuator for Image Stabilization" *Applied Sciences* 11, no. 15: 6872.
https://doi.org/10.3390/app11156872