# Theoretical and Experimental Study on Claw-Pole Magnetic Levitation Torque Motor for 2D Valve Using Cogging Torque

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structure and Operating Principle

## 3. Cogging Torque Analysis

#### 3.1. Expression of Cogging Torque

#### 3.2. Influencing Factors of Cogging Torque Waveform

## 4. Sensitivity Analysis of Electromagnetic Torque to Design Parameters

#### 4.1. Electromagnetic Torque Analysis

#### 4.2. Sensitivity Analysis of Design Parameters Based on Orthogonal Test

^{5}= 1024 trials are required. However, each 3D electromagnetic finite element simulation often takes a long time, so conducting too many tests can be computationally consuming and time consuming.

## 5. Experimental Study

#### 5.1. Prepare for the Experiment

#### 5.2. Static Experiment

#### 5.3. Dynamic Experiment

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

EMC | Electro-mechanical converter |

FEM | Finite element Method |

CPMLTM | Claw-pole magnetic levitation torque motor |

MMF | Magnetomotive force |

PM | Permanent magnet |

CPT | Claw pole tooth |

## Nomenclature

$\omega $ | angular displacement |

$W$ | magnetic co-energy |

$\alpha $ | the angle between the centerline of the PM and the centerline of the claw-pole slot |

$I$ | current |

${K}_{n}$ | current factor |

${K}_{m}$ | magnetic spring stiffness factor |

${\beta}_{i}$ | the angle between the side of the claw pole tooth and the bottom of the slot (i = 1,2,3) |

${\mu}_{0}$ | the air permeability |

$\theta $ | Distance from PM centerline in radians |

$\gamma $ | Distance in radians from the centerline of the claw pole slot |

${B}_{r}\left(\theta \right)$ | the distribution of the residual flux density of the PM along the perimeter of the air gap |

${h}_{m}$ | the thickness of the PM along the magnetization direction |

$g\left(\theta ,\alpha \right)$ | the distribution function of the effective air gap length |

$2$ | the number of pole pairs |

$z$ | the slot number |

${B}_{r}$ | the residual flux density of the PM |

${\alpha}_{p}$ | the pole arc coefficient |

${\alpha}_{s}$ | the slot opening coefficient |

${c}_{0}$ | the opening of the claw pole slot in meters |

${\theta}_{0}$ | the slot opening in radians |

${\theta}_{1}$ | the slot pitch in radians |

${\theta}_{2}$ | the PM spacing |

${\theta}_{3}$ | the slot width between PMs |

${\theta}_{4}$ | the overlap angle at neutral |

$g\left(\gamma \right)$ | the effective air gap length within half the claw pole slot distance |

${L}_{Fe}$ | the axial length of the CPT |

${r}_{1}$ | the outer diameter of the claw pole |

${r}_{2}$ | the inner diameter of the shell |

$M$ | the MMF generated by the PM |

$NI$ | the MMF generated by the coil winding |

${G}_{g}$ | the working air gap permeability |

${N}_{s}$ | the number of slots |

${l}_{c}$ | the CPT length |

$g$ | the air gap thickness |

${L}_{n}\left({r}^{m}\right)$ | equal level orthogonal table |

$L$ | the symbol of the orthogonal table |

$n$ | the number of rows of the orthogonal table |

$r$ | the number of factor levels |

$m$ | the number of columns |

$Ki$ | the sum of test results corresponding to a level number i on any column |

$R$ | the extreme difference |

## Appendix A

Factor | ${\mathit{h}}_{\mathit{m}}$ | $\mathit{g}$ | ${\mathbf{\alpha}}_{\mathit{s}}$ | ${\mathbf{\alpha}}_{\mathit{p}}$ | ${\mathit{l}}_{\mathit{c}}$ | Test Index | |
---|---|---|---|---|---|---|---|

Test Number | $\mathit{T}$ | ||||||

1 | 1.3 | 0.3 | 0.1 | 0.9 | 7.7 | 0.1326 | |

2 | 1.3 | 0.4 | 0.2 | 0.8 | 8.7 | 0.1379 | |

3 | 1.3 | 0.5 | 0.3 | 0.7 | 9.7 | 0.1234 | |

4 | 1.3 | 0.6 | 0.4 | 0.6 | 10.7 | 0.092 | |

5 | 1.4 | 0.3 | 0.2 | 0.7 | 10.7 | 0.1264 | |

6 | 1.4 | 0.4 | 0.1 | 0.6 | 9.7 | 0.113 | |

7 | 1.4 | 0.5 | 0.4 | 0.9 | 8.7 | 0.126 | |

8 | 1.4 | 0.6 | 0.3 | 0.8 | 7.7 | 0.1238 | |

9 | 1.5 | 0.3 | 0.3 | 0.6 | 8.7 | 0.1487 | |

10 | 1.5 | 0.4 | 0.4 | 0.7 | 7.7 | 0.1326 | |

11 | 1.5 | 0.5 | 0.1 | 0.8 | 10.7 | 0.0913 | |

12 | 1.5 | 0.6 | 0.2 | 0.9 | 9.7 | 0.113 | |

13 | 1.6 | 0.3 | 0.4 | 0.8 | 9.7 | 0.1364 | |

14 | 1.6 | 0.4 | 0.3 | 0.9 | 10.7 | 0.1177 | |

15 | 1.6 | 0.5 | 0.2 | 0.6 | 7.7 | 0.1274 | |

16 | 1.6 | 0.6 | 0.1 | 0.7 | 8.7 | 0.106 | |

K1 | 0.4859 | 0.5441 | 0.4429 | 0.4893 | 0.5164 | ||

K2 | 0.4892 | 0.5012 | 0.5047 | 0.4894 | 0.5186 | ||

K3 | 0.4856 | 0.4681 | 0.5136 | 0.4884 | 0.4858 | ||

K4 | 0.4875 | 0.4348 | 0.487 | 0.4811 | 0.4274 | ||

R | 0.0036 | 0.1093 | 0.0707 | 0.0083 | 0.0912 |

## Appendix B

## References

- Chao, Q.; Zhang, J.H.; Xu, B.; Huang, H.P.; Pan, M. A Review of High-Speed Electro-Hydrostatic Actuator Pumps in Aerospace Applications: Challenges and Solutions. J. Mech. Des.
**2019**, 141, 050801. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Wang, T.; Gong, G.F.; Gao, R.J. Present Status and Prospect of High-Frequency Electro-hydraulic Vibration Control Technology. Chin. J. Mech. Eng.
**2019**, 32, 93. [Google Scholar] [CrossRef] [Green Version] - Xu, B.; Shen, J.; Liu, S.; Su, Q.; Zhang, J. Research and Development of Electro-hydraulic Control Valves Oriented to Industry 4.0: A Review. Chin. J. Mech. Eng.
**2020**, 33, 13–32. [Google Scholar] [CrossRef] [Green Version] - Zhao, J.A.; Fu, Y.; Ma, J.; Fu, J.; Chao, Q.; Wang, Y. Review of cylinder block/valve plate interface in axial piston pumps: Theoretical models, experimental investigations, and optimal design. Chin. J. Aeronaut.
**2021**, 34, 111–134. [Google Scholar] [CrossRef] - Mao, Z.B.; Asai, Y.; Yamanoi, A.; Seki, Y.; Wiranata, A.; Minaminosono, A. Fluidic rolling robot using voltage-driven oscillating liquid. Smart Mater. Struct.
**2022**, 31, 105006. [Google Scholar] [CrossRef] - Fang, Y.; Zhang, J.H.; Xu, B.; Mao, Z.B.; Li, C.M.; Huang, C.S.; Lyu, F.; Guo, Z.M. Raising the Speed Limit of Axial Piston Pumps by Optimizing the Suction Duct. Chin. J. Mech. Eng.
**2021**, 34, 105. [Google Scholar] [CrossRef] - Elbayomy, K.M.; Jiao, Z.X.; Zhang, H.Q. PID controller optimization by GA and its performances on the electro-hydraulic servo control system. Chin. J. Aeronaut.
**2008**, 21, 378–384. [Google Scholar] [CrossRef] [Green Version] - Zhang, C.C.; Zhu, C.H.; Meng, B.; Li, S. Challenges and Solutions for High-Speed Aviation Piston Pumps: A Review. Aerospace
**2021**, 8, 392. [Google Scholar] [CrossRef] - Amirante, R.; Distaso, E.; Tamburrano, P. Sliding spool design for reducing the actuation forces in direct operated proportional directional valves: Experimental validation. Energy Convers. Manag.
**2016**, 119, 399–410. [Google Scholar] [CrossRef] - Yan, H.; Wang, F.J.; Li, C.C.; Huang, J. Research on the jet characteristics of the deflector-jet mechanism of the servo valve. Chin. Phys. B
**2017**, 26, 044701. [Google Scholar] [CrossRef] - Lu, L.; Long, S.R.; Zhu, K.W. A Numerical Research on Vortex Street Flow Oscillation in the Double Flapper Nozzle Servo Valve. Processes
**2019**, 7, 721. [Google Scholar] [CrossRef] [Green Version] - Li, S.J.; Peng, J.H.; Zhang, S.Z.; McHenya, J.M. Depression of Self-Excited Pressure Oscillations and Noise in the Pilot Stage of a Hydraulic Jet-pipe Servo-Valve using Magnetic Fluids. In Proceedings of the International Conference on Applied Materials and Electronics Engineering (AMEE 2012), Hong Kong, China, 18–19 January 2012. [Google Scholar]
- Wang, B.; Du, Y.H.; Ye, Z.F. Excitation of Piezoelectrically Actuated Nozzle-Flapper Valve and Its Potential for Fuel Flowmeter Dynamic Calibration. IEEE/ASME Trans. Mechatron.
**2020**, 25, 848–858. [Google Scholar] [CrossRef] - Ruan, J.; Burton, R.; Ukrainetz, P.; Xu, Y.M. Two-dimensional pressure control valve. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2001**, 215, 1031–1039. [Google Scholar] [CrossRef] - Ruan, J.; Burton, R.; Ukrainetz, P. An investigation into the characteristics of a two dimensional “2D” flow control valve. J. Dyn. Syst. Meas. Control
**2002**, 124, 214–220. [Google Scholar] [CrossRef] - Ruan, J.; Burton, R.T. An electrohydraulic vibration exciter using a two-dimensional valve. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2009**, 223, 135–147. [Google Scholar] [CrossRef] - Zhang, Q.; Yan, L.; Duan, Z.; Jiao, Z.; Gerada, C.; Chen, I.M. High Torque Density Torque Motor with Hybrid Magnetization Pole Arrays for Jet Pipe Servo Valve. IEEE Trans. Ind. Electron.
**2020**, 67, 2133–2142. [Google Scholar] [CrossRef] - Yu, G.D.; Xu, Y.X.; Lin, T.Y.; Xiao, L.J.; Zou, J.B.; Tan, J.B. Nonlinear EMC Modeling and Analysis of Permanent-Magnet Slotted Limited-Angle Torque Motor. IEEE Trans. Ind. Electron.
**2021**, 68, 8507–8518. [Google Scholar] [CrossRef] - Wu, S.; Zhao, X.Y.; Li, X.; Luk, P.C.K.; Jiao, Z.X. Preliminary Design and Optimization of Toroidally Wound Limited Angle Servo Motor Based on a Generalized Magnetic Circuit Model. IEEE Trans. Magn.
**2016**, 52, 8205209. [Google Scholar] [CrossRef] [Green Version] - Meng, L.K.; Zhu, Y.C.; Ling, J.; Ding, J.J.; Chen, Z.C.; Chen, X.M. Research on mathematical modeling of the servo valve torque motor considering the variation of working air-gaps leakage flux. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2022**, 236, 6347–6362. [Google Scholar] [CrossRef] - Xu, D.; Jiang, X.F.; Tu, Y.X.; Li, N.; Li, Q. Investigation of cogging torque reduction for a 6/10 hybrid axial field flux-switching permanent magnet machine by harmonic field current injection. IET Electr. Power Appl.
**2020**, 14, 2499–2506. [Google Scholar] [CrossRef] - Li, Z.; Yu, X.Z.; Wang, X.T.; Xing, X.X. Optimization and Analysis of Cogging Torque of Permanent Magnet Spherical Motor. IEEE Trans. Appl. Supercond.
**2021**, 31, 5204705. [Google Scholar] [CrossRef] - Simon-Sempere, V.; Simon-Gomez, A.; Burgos-Payan, M.; Cerquides-Bueno, J.R. Optimisation of Magnet Shape for Cogging Torque Reduction in Axial-Flux Permanent-Magnet Motors. IEEE Trans. Energy Convers.
**2021**, 36, 2825–2838. [Google Scholar] [CrossRef] - El-Refaie, A.M.; Jahns, T.M.; Novotny, D.W. Analysis of surface permanent magnet machines with fractional-slot concentrated windings. IEEE Trans. Energy Convers.
**2006**, 21, 34–43. [Google Scholar] [CrossRef] [Green Version] - Xiao, L.Y.; Li, J.; Qu, R.H.; Lu, Y.; Zhang, R.; Li, D.W. Cogging Torque Analysis and Minimization of Axial Flux PM Machines with Combined Rectangle-Shaped Magnet. IEEE Trans. Ind. Appl.
**2017**, 53, 1018–1027. [Google Scholar] [CrossRef] - Koh, C.S.; Seol, J.S. New cogging-torque reduction method for brushless permanent-magnet motors. IEEE Trans. Magn.
**2003**, 39, 3503–3506. [Google Scholar] - Wang, D.H.; Peng, C.; Wang, B.D.; Feng, Z.K.; Li, J.C. Permanent Magnet Synchronous Machines with Nonuniformly Distributed Teeth. IEEE Trans. Ind. Electron.
**2022**, 69, 8705–8715. [Google Scholar] [CrossRef] - Tong, W.M.; Li, S.Q.; Pan, X.L.; Wu, S.N.; Tang, R.Y. Analytical Model for Cogging Torque Calculation in Surface-Mounted Permanent Magnet Motors with Rotor Eccentricity and Magnet Defects. IEEE Trans. Energy Convers.
**2020**, 35, 2191–2200. [Google Scholar] [CrossRef] - Zhu, Z.Q.; Wu, L.J.; Jamil, M.L.M. Influence of Pole and Slot Number Combinations on Cogging Torque in Permanent-Magnet Machines with Static and Rotating Eccentricities. IEEE Trans. Ind. Appl.
**2014**, 50, 3265–3277. [Google Scholar] [CrossRef] - He, T.R.; Zhu, Z.Q.; Xu, F.; Wang, Y.; Hong, B.; Gong, L.M. Influence of Rotor Eccentricity On Electromagnetic Performance of 2-pole/3-slot PM Motors. IEEE Trans. Energy Convers.
**2022**, 37, 696–706. [Google Scholar] [CrossRef] - Li, S.; Ruan, J.; Meng, B. Study on Frequency Response for a 2D Digital Servo Valve. China Mech. Eng.
**2011**, 22, 215–219. [Google Scholar] - Meng, B.; Xu, H.; Ruan, J.; Li, S. Theoretical and experimental investigation on novel 2D maglev servo proportional valve. Chin. J. Aeronaut.
**2021**, 34, 416–431. [Google Scholar] [CrossRef] - Meng, B.; Zhu, C.H.; Xu, H.; Dai, M.Z.; Li, S. Analytical and Experimental Investigations of Novel Maglev Coupling Based on Opposed Halbach Array for a 2D Valve. Actuators
**2021**, 10, 61. [Google Scholar] [CrossRef] - Meng, B.; Dai, M.Z.; Zhu, C.H.; Xu, H.; Jia, W.A.; Li, S. Investigation of Characteristics of a Novel Torque Motor Based on an Annulus Air Gap. Machines
**2021**, 9, 131. [Google Scholar] [CrossRef] - Li, Z.W.; Cao, J.; Chang, L.; Li, S.; Ruan, J. Eddy current loss modelling for torque-motors. IET Power Electron.
**2022**, 15, 368–380. [Google Scholar] [CrossRef] - Wang, X.H.; Yang, Y.B.; Fu, D.J. Study of cogging torque in surface-mounted permanent magnet motors with energy method. J. Magn. Magn. Mater.
**2003**, 267, 80–85. [Google Scholar] [CrossRef] - Meng, B.; Lai, Y.J.; Qiu, X.G. Regulation Method for Torque-Angle Characteristics of Rotary Electric-Mechanical Converter Based on Hybrid Air Gap. Chin. J. Mech. Eng.
**2020**, 33, 35. [Google Scholar] [CrossRef] - Meng, B.; Xu, H.; Liu, B.; Dai, M.Z.; Zhu, C.H.; Li, S. Novel Magnetic Circuit Topology of Linear Force Motor for High Energy Utilization of Permanent Magnet: Analytical Modelling and Experiment. Actuators
**2021**, 10, 32. [Google Scholar] [CrossRef]

**Figure 2.**The structure diagram of CPMLTM. (

**a**) Stator-rotor relative position diagram, (

**b**) Stator structure diagram.

**Figure 21.**The cogging torque angle characteristics of CPMLTM, Circumferential angular displacement_Circumferential resistances.

**Figure 30.**Sine wave following characteristics of CPMLTM. (

**a**) 1 Hz, (

**b**) 10 Hz, (

**c**) 40 Hz, (

**d**) 80 Hz.

Moving Iron Type Torque Motor | CPMLTM | Claw Pole Motor | |
---|---|---|---|

Output angular displacement | Limited rotation angle (−0.5°~0.5°) | Limited rotation angle (−2°~2°) | 360° |

Cogging torque | None | Use of cogging torque | Avoiding cogging torque |

Neutral adjustment | Manual adjustment (inaccurate) | Automatic adjustment (more accurate) | None |

Excitation current | Single Phase | Single Phase | Three phase |

EMC for 2D valves | Yes | Yes | No |

Level | ${\mathit{h}}_{\mathit{m}}$ | $\mathit{g}$ | ${\mathbf{\alpha}}_{\mathit{s}}$ | ${\mathbf{\alpha}}_{\mathit{p}}$ | ${\mathit{l}}_{\mathit{c}}$ |
---|---|---|---|---|---|

1 | 1.3 | 0.3 | 0.1 | 0.9 | 7.7 |

2 | 1.4 | 0.4 | 0.2 | 0.8 | 8.7 |

3 | 1.5 | 0.5 | 0.3 | 0.7 | 9.7 |

4 | 1.6 | 0.6 | 0.4 | 0.6 | 10.7 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huang, J.; Xie, S.; Song, Z.; Li, S.; Ruan, J.
Theoretical and Experimental Study on Claw-Pole Magnetic Levitation Torque Motor for 2D Valve Using Cogging Torque. *Actuators* **2022**, *11*, 301.
https://doi.org/10.3390/act11100301

**AMA Style**

Huang J, Xie S, Song Z, Li S, Ruan J.
Theoretical and Experimental Study on Claw-Pole Magnetic Levitation Torque Motor for 2D Valve Using Cogging Torque. *Actuators*. 2022; 11(10):301.
https://doi.org/10.3390/act11100301

**Chicago/Turabian Style**

Huang, Jiahiu, Shulin Xie, Zhankai Song, Sheng Li, and Jian Ruan.
2022. "Theoretical and Experimental Study on Claw-Pole Magnetic Levitation Torque Motor for 2D Valve Using Cogging Torque" *Actuators* 11, no. 10: 301.
https://doi.org/10.3390/act11100301