# On the Static Pull-In of Tilting Actuation in Electromagnetically Levitating Hybrid Micro-Actuator: Theory and Experiment

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## Abstract

**:**

## 1. Introduction

## 2. Fabrication and Measurements

## 3. Simulation and Modeling

#### 3.1. Simulation of Induced Eddy Current within the Tilting Proof Mass

#### 3.2. Mutual Inductance between Two Filaments of Circular and Elliptic Shapes

#### 3.3. Model of Static Pull-In of Tilting Actuation

## 4. Analysis of the Derived Model

## 5. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

ILMA | Inductive Levitation Micro-Actuator |

HLMA | Hybrid Levitation Micro-Actuator |

PM | Proof Mass |

MLMA | Magnetic Levitation Micro-Actuator |

ELMA | Electric Levitation Micro-Actuator |

## Appendix A. Nomenclature

- ${A}_{1}$
- area of the electrode “1” (${\mathrm{m}}^{2}$)
- ${A}_{2}$
- area of the electrode “2” (${\mathrm{m}}^{2}$)
- a
- dimensionless parameter ${A}_{1}/{A}_{2}$
- ${C}_{1}$
- capacitance of capacitor build on electrode “1” ($\mathrm{F}$)
- ${C}_{2}$
- capacitance of capacitor build on electrode “2” ($\mathrm{F}$)
- E
- the complete elliptic function of the second kind
- ${F}_{l}$
- generalized force ($l=1,2,3$) ($\mathrm{N}$)
- $\mathit{g}$
- gravity acceleration vector ($\mathrm{m}/{\mathrm{s}}^{2}$)
- ${h}_{l}$
- height of levitation ($\mathrm{m}$)
- h
- space between the electrode surface and cm of levitated disc ($\mathrm{m}$)
- i
- induced eddy current ($\mathrm{A}$)
- I
- AC current in the levitation coil ($\mathrm{A}$)
- $\widehat{I}$
- magnitude of AC current in the levitation coil ($\mathrm{A}$)
- j
- imaginary unit
- K
- complete elliptic function of the first kind
- N
- number of wire loops
- n
- number of finite elements
- L
- Lagrange function ($\mathrm{J}$)
- ${L}_{e}$
- self-inductance of the eddy current circuit ($\mathrm{H}$)
- ${L}_{jj}^{c}$
- self-inductance of the j-wire loop ($\mathrm{H}$)
- ${L}^{o}$
- self-inductance of the finite circular element ($\mathrm{H}$)
- ${L}_{ks}^{c}$
- mutual inductance between k- and s-finite circular elements ($\mathrm{H}$)
- M
- between two filaments of circular and elliptic shapes ($\mathrm{H}$)
- m
- mass of levitated object ($\mathrm{k}\mathrm{g}$)
- Q
- electric charges ($\mathrm{C}$)
- ${R}_{eddy}$
- electrical resistance of the eddy current circuit ($\Omega $)
- ${R}_{e}$
- radius of circular element ($\mathrm{m}$)
- ${R}_{in}$
- inner radius of sector electrode ($\mathrm{m}$)
- ${R}_{l}$
- radius of levitation coil ($\mathrm{m}$)
- ${R}_{out}$
- outer radius of sector electrode ($\mathrm{m}$)
- ${R}_{T}$
- mean distance ($\mathrm{m}$)
- $th$
- thickness of micro-object ($\mathrm{m}$)
- U
- voltage ($\mathrm{V}$)
- ${y}_{c}$
- coordinate of the centre of the ellipse along the y-axis ($\mathrm{m}$)
- ${z}_{c}$
- coordinate of the centre of the ellipse along the z-axis ($\mathrm{m}$)

**Matrices**

- $\underline{E}$
- unit matrix of size $(n\times n)$
- $\underline{I}$
- matrix of eddy currents of size $(n\times 1)$ ($\mathrm{A}$)
- ${\underline{I}}_{c}$
- matrix of coil currents of size $(N\times 1)$ ($\mathrm{A}$)
- ${\underline{M}}^{o}$
- symmetric hollow matrix of size $(n\times n)$ whose elements are ${L}_{ks}^{o}$ ($k\ne s$) ($\mathrm{H}$)
- ${\underline{M}}_{c}$
- mutual inductance between coils and finite elements of size $(n\times N)$ ($\mathrm{H}$)

**Greek symbols**

- $\beta $
- dimensionless square voltage
- $\theta $
- angular displacement of the levitated disc ($\mathrm{rad}$)
- $\overline{\theta}$
- dimensionless angle ${R}_{T}\theta /h$
- $\kappa $
- dimensionless parameter $h/{h}_{l}$
- $\lambda $
- dimensionless displacement ${z}_{c}/h$
- $\xi $
- dimensionless parameter ${h}_{l}/\left(2{R}_{l}\right)$
- $\Psi $
- dissipation function ($\mathrm{W}$)
- $\omega $
- frequency of AC current ($\mathrm{Hz}$)

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**Figure 1.**The prototype of micro-actuator-performed tilting pull-in actuation: (

**a**) the fabricated prototype of the micro-actuator glued and wire-bonded on a PCB board and levitated stably micro-disc with a diameter of 2.8 mm; (

**b**) the set of electrodes fabricated on the top of the silicon structure: the energized electrodes generating the electrostatic forces executing the pull-in actuation are highlighted in red, where U is the applied voltage; (

**c**) video frames demonstrating actual tilting pull-in actuation: the top figure shows the original flat state of the micro-disc and the bottom figure shows its tilted state after applying the pull-in voltage.

**Figure 2.**Simulation of eddy current: (

**a**) the disc is meshed by circular elements $n=2496$, {${x}_{k}$} ($k=1,2,3$) is the coordinate frame assigned to the center of disc; (

**b**) 3D schematic diagram of the actuator; (

**c**) 3D plot of the distribution of dimensionless magnitude of the eddy current along the surface of the disc; (

**d**) 2D plot of the distribution of magnitudes of eddy current.

**Figure 3.**Reduced scheme for modeling electromagnetic interaction between the levitation coil and the tilt-disc: ${h}_{l}$ is the levitation height between a plane of coils and equilibrium point; $\theta $ is the tilting angle; ${i}_{e}$ is the induced eddy current corresponding to the maximum current density within the disc; ${R}_{l}$ is the radius of the levitation coil; ${z}_{c}={z}_{c}\left(\theta \right)$ and ${y}_{c}={y}_{c}(\theta ,{z}_{c})$ are the coordinates of the center of the ellipse as functions of generalized coordinates; $b=b\left(\theta \right)$ is the length of minor axis of the ellipse.

**Figure 4.**Stable and unstable angular equilibrium of the tilt-disc and its evolution depending on the parameters of device such as $\xi ={h}_{l}/2{R}_{l}$ and $\kappa =h/{h}_{l}$, where h is the space between an electrode plane and equilibrium point of the disc, ${R}_{l}$ is the radius of levitation coil.

**Figure 5.**The results of modeling together with measured data in the normalized values: (

**a**) measurement I; (

**c**) measurement II. Applied voltage vs. the linear displacement of the disc: (

**b**) measurement I; (

**d**) measurement II (other details are shown in Table 1).

**Figure 6.**Measured data and results of model=ling in absolute values: (

**a**) for measurement I; (

**b**) for measurement II (other details are shown in Table 1).

Measurement I | Measurement II | ||
---|---|---|---|

Measured | Levitation height, ${h}_{l}$ | 130 $\mathsf{\mu}\mathrm{m}$ | 150 $\mathsf{\mu}\mathrm{m}$ |

parameters | Spacing, h | 100 $\mathsf{\mu}\mathrm{m}$ | 120 $\mathsf{\mu}\mathrm{m}$ |

Results of medelling | Pull-in displacement | 34$\mathsf{\mu}\mathrm{m}$ | 45$\mathsf{\mu}\mathrm{m}$ |

Pull-in voltage,U | 27$\mathrm{V}$ | 33$\mathrm{V}$ | |

Parameters of medelling | $\xi ={h}_{l}/{d}_{l}$ | 0.065 | 0.075 |

$\kappa =h/{h}_{l}$ | 0.7692 | 0.8 | |

Results of medelling | Pull-in displacement | 38$\mathsf{\mu}\mathrm{m}$ | 48$\mathsf{\mu}\mathrm{m}$ |

Pull-in voltage,U | 28$\mathrm{V}$ | 33$\mathrm{V}$ | |

Device design | Diameter of levitation coil, ${d}_{l}$ | 2 $\mathrm{m}\mathrm{m}$ | |

Area of electrodes, ${A}_{1}$ and ${A}_{2}$ | 0.8 and 0.43 $\mathrm{m}{\mathrm{m}}^{2}$ |

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**MDPI and ACS Style**

Poletkin, K.
On the Static Pull-In of Tilting Actuation in Electromagnetically Levitating Hybrid Micro-Actuator: Theory and Experiment. *Actuators* **2021**, *10*, 256.
https://doi.org/10.3390/act10100256

**AMA Style**

Poletkin K.
On the Static Pull-In of Tilting Actuation in Electromagnetically Levitating Hybrid Micro-Actuator: Theory and Experiment. *Actuators*. 2021; 10(10):256.
https://doi.org/10.3390/act10100256

**Chicago/Turabian Style**

Poletkin, Kirill.
2021. "On the Static Pull-In of Tilting Actuation in Electromagnetically Levitating Hybrid Micro-Actuator: Theory and Experiment" *Actuators* 10, no. 10: 256.
https://doi.org/10.3390/act10100256