# Effect of Optimal Placement of Permanent Magnets on the Electromagnetic Force in the Horizontal Direction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Outline of the Electromagnetic Levitation System Integrating Permanent Magnets and the Horizontal Positioning Control System

## 3. Control Model

#### 3.1. Electromagnetic Levitation Control System

_{z}is calculated by the state feedback control of Equation (6).

#### 3.2. Horizontal Positioning Control System

## 4. Determination of Optimal Placement

_{x}are 0.3 A and 0.4 A, respectively. The results of this analysis show that the optimal number of permanent magnets at 0.4 A is larger than that at 0.3 A. It is reasonable to assume that the optimal gap at 0.4 A is larger than that at 0.3 A because of the increase of the attractive force from the horizontal electromagnets. Therefore, as the attractive force of each permanent magnet decreases, the placement with the optimal gap requires more permanent magnets.

## 5. Magnetic Levitation Experiment Using Optimal Placement

_{x}was set to 0.3 and 0.4 A. Figure 8 shows the time histories of the measured displacement of the steel plate by the eddy-current sensor installed in the central electromagnet unit and the experimental results with and without permanent magnets. Figure 8a,b show the results when the steady current I

_{x}is 0.3 A and Figure 8c,d show the results when I

_{x}is 0.4 A. For each steady current, when permanent magnets are installed in the optimal placement, the vibration of the levitated steel plate can be suppressed. Figure 9 shows the comparison of the displacement standard deviations for each steady current. In this system, the steel plate deflects where electromagnetic force does not reach the steel plate. It is considered that this bending portion is the cause of the vibration of the steel plate. Furthermore, we consider that bending was suppressed by attaching PM and vibration was suppressed. These results show the validity of searching for the optimal placement of permanent magnets by a GA and the effectiveness of permanent magnets for vibration suppression when the magnetic field generated by the horizontal electromagnets is changed.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

E | Young’s modulus of the thin steel plate (N/m^{2}) |

f | vertical static magnetic force applied to the plate, which is generated by the permanent magnets (N/m^{2}) |

f_{z} | dynamic magnetic force (N) |

F_{x} | magnetic force of the coupled magnets in the equilibrium state (N) |

F_{z} | magnetic force of the coupled magnets in the equilibrium state (N) |

g | acceleration due to gravity (m/s^{2}) |

h | plate thickness (m) |

i_{z} | dynamic current of the coupled electromagnets (A) |

i_{x} | dynamic current of the coupled magnets (A) |

I_{x} | current of the coupled magnets in the equilibrium state (A) |

I_{z} | current of the coupled electromagnets in the equilibrium state (A) |

J_{D} | evaluation function of the maximum deflection (m) |

J_{z} | evaluation function of the average absolute deflection (m) |

L_{lea} | leakage inductance of one magnet coil (H) |

L_{x} | inductance of one magnet coil in the equilibrium state (H) |

L_{xlea} | leakage inductance of one magnet coil (H) |

L_{xeff}/X_{0} | effective inductance of one magnet coil (H) |

L_{z} | inductance of one electromagnet coil in the equilibrium state (H) |

m_{z} | virtually divided steel plate (kg) |

N | the total number of analysis points |

R_{x} | resistance of the coupled magnet coils (Ω) |

R_{z} | resistance of the coupled magnet coils (Ω) |

T_{s} | sampling time (s) |

ν | Poisson ratio |

v_{x} | dynamic voltage of the coupled magnets (V) |

v_{z} | dynamic voltage of the coupled magnets (V) |

x | coordinates in the width direction (m) |

X_{0} | gap between the steel plate and electromagnet in the equilibrium state (m) |

y | coordinates in the longitudinal direction (m) |

z | vertical displacement from the equilibrium state (m) |

z_{i} | displacement at each analysis point on the thin steel plate (m) |

z_{max} | maximum deflection of the thin steel plate (m) |

Z_{0} | gap between the steel plate and electromagnet in the equilibrium state (m) |

ρ | plate density (kg/m^{3}) |

## References

- Onuki, T.; Toda, Y. Optimal design of hybrid magnet in maglev system with both permanent and electromagnets. IEEE Trans. Magn.
**1993**, 29, 1783–1786. [Google Scholar] [CrossRef] - Oka, K.; Higuchi, T. 3 Degree of freedom maglev system with actuators and permanent magnets. IEEJ Trans. Ind. Appl.
**1995**, 115, 294–300. [Google Scholar] [CrossRef] - Morishita, M.; Ito, H. The self-gap-detecting zero power controlled electromagnetic suspension system. IEEJ Trans. Ind. Appl.
**2006**, 126, 1667–1677. [Google Scholar] [CrossRef] - Kamijo, Y.; Ito, H.; Maruyama, Y. Structure and characteristic of magnetic bearing with zero power controlled electromagnetic suspension. Trans. Jpn. Soc. Mech. Eng. Ser. C
**2013**, 79, 4963–4972. [Google Scholar] [CrossRef] - Oshinoya, Y.; Ishibashi, K. Development of electromagnetic levitation control device for a rectangular sheet steel. Trans. Jpn. Soc. Mech. Eng. Ser. C
**2001**, 661, 2855–2862. [Google Scholar] [CrossRef] - Goldberg, D.E. Genetic Algorithms in Search, Optimization, and Machine Learning; Addison-Wesley Professional: Boston, MA, USA, 1989; ISBN 978-0201157675. [Google Scholar]
- Mallik, S.; Mallik, K.; Barman, A.; Maiti, D.; Biswas, S.K.; Deb, N.K.; Basu, S. Efficiency and cost optimized design of an induction motor using genetic algorithm. IEEE Trans. Ind. Electron.
**2017**, 64, 9854–9863. [Google Scholar] [CrossRef] - Okamoto, Y.; Tominaga, Y.; Wakao, S.; Sato, S. Topology optimization of rotor core combined with identification of current phase angle in IPM motor using multistep genetic algorithm. IEEE Trans. Magn.
**2014**, 50, 725–728. [Google Scholar] [CrossRef] - Narita, T.; Hasegawa, S.; Oshinoya, Y. Optimal placement of permanent magnets in a hybrid magnetic levitation system for thin steel plate (experimental consideration on levitation probability). Proc. Sch. Eng. Tokai Univ.
**2013**, 38, 59–66. [Google Scholar] - Narita, T.; Hasegawa, S.; Oshinoya, Y. Hybrid electromagnetic levitation system for thin steel plates using permanent magnets. J. Magn. Soc. Jpn.
**2013**, 37, 29–34. [Google Scholar] [CrossRef] - Ishii, H.; Narita, T.; Kato, H. Hybrid magnetic levitation system for thin steel plate by electromagnets and permanent magnets (basic study on optimal placement search considering the interaction of the magnetic field). J. Jpn. Soc. Appl. Electromagn. Mech.
**2016**, 24, 149–154. [Google Scholar] [CrossRef] - Chiba, A.; Fukao, T.; Ichikawa, O.; Oshima, M.; Takemoto, M.; Dorrell, D.G. Magnetic Bearings and Bearingless Drives; Elsevier: Boston, MA, USA, 2005; p. 20. ISBN 9780750657273. [Google Scholar]
- Timoshenko, S.P.; Woinowsky-Krieger, S. Theory of Plate and Shells; McGraw-Hill Publishing Company: New York, NY, USA, 1959. [Google Scholar]

**Figure 1.**Hybrid electromagnetic levitation system integrating positioning control in the horizontal direction.

**Figure 7.**Optimal placement of permanent magnets considering the horizontal attractive force and distribution of the attractive force acting on the steel plate.

**Figure 9.**Displacement standard deviation of the levitation direction of each horizontal steady current.

Parameters | Values |
---|---|

m | 0.864 kg |

E | 206 GPa |

ν | 0.3 |

Z_{0} | 5 × 10^{−3} m |

R_{n} | 21.0 Ω |

L_{eff} | 2.55 × 10^{−4} Hm |

L_{lea} | 0.090 H |

T_{s} | 0.001 s |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ito, Y.; Oda, Y.; Narita, T.; Kato, H. Effect of Optimal Placement of Permanent Magnets on the Electromagnetic Force in the Horizontal Direction. *Actuators* **2018**, *7*, 54.
https://doi.org/10.3390/act7030054

**AMA Style**

Ito Y, Oda Y, Narita T, Kato H. Effect of Optimal Placement of Permanent Magnets on the Electromagnetic Force in the Horizontal Direction. *Actuators*. 2018; 7(3):54.
https://doi.org/10.3390/act7030054

**Chicago/Turabian Style**

Ito, Yasuaki, Yoshiho Oda, Takayoshi Narita, and Hideaki Kato. 2018. "Effect of Optimal Placement of Permanent Magnets on the Electromagnetic Force in the Horizontal Direction" *Actuators* 7, no. 3: 54.
https://doi.org/10.3390/act7030054