Experimental Study and Simulation of Pull-In Behavior in Hybrid Levitation Microactuator for Square-Shaped Proof Masses

Abstract

This paper presents the results of a comprehensive study of the pull-in phenomenon in the hybrid levitation microactuator (HLMA), in which square-shaped proof masses (PMs) of different sizes, namely, length sides of 2.8 and 3.2 mm and thicknesses of 25 and 10 μm were electromagnetically levitated. The pull-in actuation of the square-shaped PMs was performed by the electrostatic force generated by the set of energized electrodes and acting on the bottom surface of the PMs along the vertical direction. The pull-in parameters, such as pull-in displacements and the corresponding applied pull-in voltages, were measured with the developed setup. The experimental measurements showed that the pull-in actuation is nonlinearly dependent on the size and mass of the PMs and a levitation height. In particular, it was found that PMs levitated within a height range from 140 to 170 μm can be stably displaced within a range of 30 μm. The results of measurements were extensively simulated with the developed analytical model by means of the quasi-FEM method. The direct comparison of the results of simulation and measurements showed a very good agreement between the theory and experiments.

1. Introduction

Employing electromagnetic levitation in microactuator systems eliminates mechanical attachments in their moving parts and, consequently, mechanical friction and wear in these systems, which are further referenced as levitation microactuators (LMAs). In fact, the control of the friction and flexibility in the motion from remote actuation provided by electromagnetic levitation at the microscale has already found wide applications in microsystem devices and, in particular, led to the creation of a new generation of microactuators with increased operational capabilities and performance [1].

Electromagnetic levitation in LMA is implemented through sources of electric and magnetic force fields and their combination. According to the sources of the force fields, LMAs can be classified as electric levitation (ELMA), magnetic levitation (MLMA), and hybrid levitation (HLMA) microactuators. In particular, ELMAs were successfully used as linear transporters [2] and in microinertial sensors [3,4,5]. MLAMs can be further split into inductive (ILMA), diamagnetic (DLMA), and superconducting microactuators, which have found applications in microbearings [6,7,8], micromirrors [9], microgyroscopes [10,11], microaccelerometers [12], bistable switches [13], nanoforce sensors [14], inclination sensors [15], gas flow meters [16], microrobots [17], energy harvesters [18] and ultra-high-Q resonators [19].

Combining different force fields in one LMA, for instance, magneto- and electro-static, variable magnetic and electro-static, or magneto-static and variable magnetic fields, is an HLMA. HLMAs have found applications, for instance, in micromotors [20,21], micromirrors [22], and microaccelerators [23]. A wide range of different operation modes, such as linear and angular positioning, bistable linear and angular actuations, and the adjustment of stiffness components of a levitated microdisc, were developed and experimentally studied in a prototype of an HLMA [24]. In this prototype, the stiffness components were adjusted by changing the equilibrium position of the inductively levitated disc along the vertical axis. Recently, an HLMA was presented in which the electrostatic forces acting on the bottom and top surfaces of the inductively levitated microdisc maintained their equilibrium position while the vertical component of stiffness was decreased by means of increasing the strength of electrostatic field [25,26]. Thus, HLMAs are a very promising direction for further improvement in the performance of microsystem actuators.

In general, analytical and numerical analysis of LMAs for their further development and for proposing new designs for advanced applications requires applying the Maxwell equations. Although field equations are universally applicable, their application even for simple designs is a difficult task [27]. Even though these designs have been numerically studied by using commercially available software [28,29,30,31], this task is still a challenge, as not all aspects of LMA behavior, including the stability and linear and nonlinear dynamical response, cannot be covered.

Fortunately, LMA can be analyzed by using a certain analogy [32]. Upon holding the condition of quasi-stationarity for the force field ([33], page 493), the electromagnetic part of an LMA can be approximated by an equivalent circuit consisting of a set of lumped components, such as inductance, resistance, and capacitance. Considering a levitated object in LMA as rigid, the methods of analytical mechanics based on Lagrangian formalism become applicable. In particular, Okress et al. proposed replacing a levitated sphere by an alternating magnetic dipole [34]. Using this analogy, which can be known as one electric current circuit approximation (one-CCA), the levitating force acting on the sphere was studied. The same analogy was successfully applied to study the stability and dynamics in a number of different designs of a micromachined inductive contactless suspension, in which disc- and ring-shaped proof masses were levitated [7,35,36,37,38]. Moreover, the one-CCA analogy was used to examine the pull-in phenomenon in an HLMA, in which a levitated disc-shaped PM was actuated by the set of electrodes with a flouting electric ground [26,39,40].

The analogy of one-CCA was further generalized and extended to the approximation of the induced eddy current within a levitated proof mass by n number of electric current circuits [41]. This analogy is known as the n electric current circuit approximation (n-CCA). It helps to establish the necessary and sufficient conditions for stable levitation of an electromagnetic contactless suspension and, in particular, to provide general guidelines for designing LMAs. Due to the recent progress in the calculations of mutual inductance between filament loops [42,43,44], forces [45,46], and magnetic stiffness [47] between current-carrying filaments, the simulation approach for LMAs based on quasi-FEM was formulated [48]. In the formulated quasi-FEM method, the calculation of the induced eddy current within PMs is performed by means of the n-CCA analogy, but an each current circuit has a circular shape [49]. This makes this calculation applicable for any shape of a levitated PM and a coil wire carrying an excitation A/C current generating the force field. In particular, this method was successfully applied to simulate the pull-in actuation of a disc-shaped PM in an HLMA [48].

This paper presents the results of the comprehensive study of the pull-in phenomenon in an HLMA, in which square-shaped PMs of different sizes, namely, length sides of $2.8$ and $3.2$ $\mathrm{m}$$\mathrm{m}$ and thicknesses of 25 and 10 μm, were electromagnetically levitated. The pull-in actuation of the square-shaped PMs was performed by the electrostatic force generated by the set of energized electrodes and acting on the bottom surface of the PMs along the vertical direction. The pull-in parameters, such as pull-in displacements and the corresponding applied pull-in voltages, were measured by means of the developed setup. The experimental measurements showed that the pull-in actuation is nonlinearly dependent on the size and mass of the PM and the levitation height. In particular, it was found that PMs levitated within a height range of 140 to 170 μm can be stably displaced within a range of 30 μm. The results of measurements were extensively simulated with the developed analytical model by means of the quasi-FEM method based on the n-CCA analogy. The direct comparison of the results of simulation and measurements showed a very good agreement between the theory and experimental results. Note that this study is an extended version of our results presented at the ACTUATOR 2022 conference and published in the conference proceedings [50].

2. Hybrid Levitation Microactuators

The prototype of an HLMA levitating a square-shaped proof mass with a side length of $2.8$ $\mathrm{m}$$\mathrm{m}$ and at a height of 70 μm is shown in Figure 1A. The device consists of two independently fabricated structures: the two-coil structure and electrode structure, which were aligned and assembled by flip-chip bonding into one device with the dimensions of 9.4 × 7.4 × 1.1 mm, as shown in Figure 1A [24,51].

The coil structure included two coaxial 3D wire-bonded microcoils, similar to those reported in our previous study [7]: the stabilization and levitation coil have radii of $1.9$ $\mathrm{m}$$\mathrm{m}$ and $1.0$ $\mathrm{m}$$\mathrm{m}$, respectively, which were fabricated on Pyrex substrate using SU-8 2150. The function of this structure is to stably levitate an aluminum (Al) square-shaped PM. For this particular device, the height of the coils was 500 μm, with 20 and 12 windings for the levitation and stabilization coil, respectively. This coil structure is able to levitate PMs with a side-length ranging from 2.7 to 3.3 $\mathrm{m}\mathrm{m}$.

The electrode structure was fabricated on an SOI wafer having a device layer of 40 μm, a buried oxide of 2 μm, a handle layer of 600 μm, and silicon resistivity in the range of 1.30 $\Omega \mathrm{c}\mathrm{m}$, as was reported in our study [26]. Additionally, the device layer had a 500 nm oxide layer for passivation, on the top of which electrodes were patterned by UV lithography on evaporated Cr/Au layers (20/150 nm). The fabricated electrode set, which is hidden by the levitated PM in Figure 1A, is shown in Figure 1B. After etching the handle layer up to the buried oxide by DRIE, the electrode structure was aligned and bonded onto the coil structure. Finally, the fabricated device was glued and wire-bonded on a PCB board.

In order to carry out the pull-in actuation of the levitated PM, electrodes “1” and “2”, which had the same area, $A_e$, of $8.0\times {10}^{-7}$ m${}^2$ as shown in Figure 1B, were energized so that the squared PM was moved toward the electrode surface. The corresponding electrostatic force F could be estimated by the well-known formula,

$$F=\frac{\epsilon {\epsilon }_0{A}_e{U}^2}{4h^2},$$

(1)

where ${\epsilon }_0$ is the vacuum permittivity (${\epsilon }_0=8.85\times {10}^{-12}$ F/m), $\epsilon$ is the relative permittivity (for air $\epsilon \approx 1$), U is the applied voltage, and h is the levitation height.

3. Experimental Setup and Measurements

In order to measure the pull-in parameters of the prototype, the experimental setup was developed. A schematic of the setup is shown in Figure 2. The PCB board with the device was mounted on an optical table, as shown in Figure 2. The device coils were fed by the square-wave A/C current provided by the control circuit developed by our group. It had dimensions of $60\times 60\times 25$ $\mathrm{m}\mathrm{m}$ and consisted of a generator based on high-speed flip-flop components and a current amplifier built on an H-bridge configuration. The circuit was able to generate an A/C current with a squared waveform in a frequency range of 8 to 43 $\mathrm{M}\mathrm{Hz}$ and with a peak-to-peak amplitude of up to 420 $\mathrm{m}\mathrm{A}$ [49]. An electrically conductive proof mass was levitated by excitation of the nested 3D microcoils with the A/C current. Electrically conductive Al films were used for PMs, and they were cut with an Acsys Piranha II (ACSYS Lasertechnik GmbH, Kornwestheim, Germany) operating at 355 $\mathrm{n}$$\mathrm{m}$ wavelength. This excitation induced the eddy current in the PM, and the interaction between the magnetic field generated by this eddy current and the coil current gave rise to the levitation force. The outer stabilization coil of the nested 3D coil structure prevented the proof mass from sliding off when the current was applied and provided lateral stability. The inner levitation coil of the nested 3D coil structure supported the stable levitation of the PM along the vertical direction. The levitated PM was moved down by the electrostatic force (1) generated by energizing the electrodes “1” and “2” and acting on the bottom PM surface. The linear displacements of the PM along the vertical direction at different points were measured by a laser distance sensor (Panasonic HL-G103-S-J) having a resolution of $0.5$ μm, and the corresponding voltages applied to electrodes “1” and “2” were recorded at the same time. Additionally, the USB microscope was mounted on an optical table, as shown in Figure 2, to control the stable levitation of the PM under the pull-in actuation visually. The data generated by the laser sensor and the USB microscope were collected by a computer for further treatment.

The results of measurements for three squared proof masses of different sizes are jointly shown in Figure 3. At the beginning of each measurement, a PM was located on the surface of the electrode structure, and the output of the laser sensor was set to null. Then, the coils of the actuator were excited by the control circuit with an A/C electric current of 120 $\mathrm{m}$$\mathrm{A}$ rms; the levitation height, h, from the electrode surface was measured. Keeping the value of the electric current in the coils, a PM was moved down by electrostatic force. The difference between the initial and actual levitation height was estimated at each point, as shown in Figure 3. The levitation heights for two squared PMs having the same side length of $2.8$ $\mathrm{m}$$\mathrm{m}$ but different thicknesses of 25 μm and 10 μm were measured as 70 μm and 95 μm, respectively. The PM with a side length of $3.2$ $\mathrm{m}$$\mathrm{m}$ and a thickness of 10 μm was levitated at a height of 94 μm.

The following pull-in parameters were observed. For the square PM having a side length of $2.8$ $\mathrm{m}$$\mathrm{m}$ and a thickness of 25 μm, the pull-in displacement and voltage were 23 μm and 30 $\mathrm{V}$, respectively. For the square PM having the same side length but a thickness of 10 μm, the pull-in displacement and voltage were 30 μm and 40 $\mathrm{V}$, respectively. For the square PM having a side length of $3.2$ $\mathrm{m}$$\mathrm{m}$ and a thickness of 10 μm, the pull-in displacement and voltage were 30 μm and 45 $\mathrm{V}$, respectively.

4. Simulation

For the simulation of the pull-in actuation, the quasi-FEM based on the n-CCA analogy was used [48]. The method involves two main steps. In the first step, the eddy current induced within the PM is calculated; in the second step, knowing the eddy current, the calculation of the electromagnetic force between the PM and coils is performed. According to the procedure [48], the following bases are introduced (the notations for vectors, bases, and matrices were adopted from the book of J. Wittenburg [52]). The fixed base ${\underline{\mathbf{e}}}^0={\left[{\mathbf{e}}_1^0{\mathbf{e}}_2^0{\mathbf{e}}_3^0\right]}^T$, where ${\mathbf{e}}_k^0$ ($k=1,2,3$) are the unit vectors of ${\underline{\mathbf{e}}}^0$, and the exponent ${}^0$ denotes the base number, as shown in Figure 4. The origin O of the fixed base ${\underline{\mathbf{e}}}^0$ is assigned to the center of the circular filament corresponding to the first top winding of the levitation coil, and the unit vectors ${\mathbf{e}}_1^0$ and ${\mathbf{e}}_2^0$ are lying on the filament plane, as shown in Figure 4. The base ${\underline{\mathbf{e}}}^1$ and corresponding unit vectors ${\mathbf{e}}_k^1$ ($k=1,2,3$) are rigidly attached to the square-shaped PM so that its origin is located at the center of mass of the PM, and the unit vectors ${\mathbf{e}}_1^1$ and ${\mathbf{e}}_2^1$ are lying on the PM plane. The position of the origin of base ${\underline{\mathbf{e}}}^1$ with respect to the fixed base ${\underline{\mathbf{e}}}^0$ is defined by the radius vector ${\mathbf{r}}_{cm}$, and its coordinates in base ${\underline{\mathbf{e}}}^0$ are ${\underline{r}}_{cm}^0={\left[00h_l\right]}^T$, as shown in Figure 4. The position of each meshing element of the PM is defined by vector $\rho$ with respect to the base ${\underline{\mathbf{e}}}^1$. Because the PM is a plane, the position of the sth meshing element has the following coordinates ${}^{\left(s\right)}{\underline{\rho }}^1={{[}^{\left(s\right)}{\rho }_1{}^{\left(s\right)}{\rho }_{2}0]}^T$. The 3D geometry of two microcoils was approximated by a series of circular filaments, the number of which depends on the number of turns of the coil windings. Hence, the levitation coil was replaced by 20 circular filaments, while the stabilization coil was replaced by 12 circular filaments. Thus, the total number of circular filaments, N, was 32. The linear position of the jth circular filament of the coils can be defined by vector ${}^{\left(j\right)}{\mathbf{r}}_c$. Its coordinates for the jth winging of the levitation coil are ${}^{\left(j\right)}{\underline{r}}_c={[00(1-j)·p]}^T$, $(j=1,\dots ,20)$, where p is the pitch, equaling to 25 μm, as shown in Figure 4. The same is applicable for the stabilization coil, ${}^{\left(j\right)}{\underline{r}}_c={[00(21-j)·p]}^T$, with the difference being the index j being varied from 21 to 32.

4.1. Eddy Current Calculation

The PM with a side length of $2.8$ $\mathrm{m}$$\mathrm{m}$ was homogenously meshed by $n=$ 2500 circular elements of the same radius, $R_e=$ 2.8025 × 10⁻⁵ m, as shown in Figure 5A. The result of meshing was a list of elements $\underline{C}={\{}^{\left(s\right)}\underline{\rho }\}$ ($s=1,\dots ,n$) containing information about a radius vector for each element with respect to the base ${\underline{\mathbf{e}}}^1$. A position of the meshed PM relative to the circular filaments replacing the helix levitation and stabilization coil is shown in Figure 5B.

To calculate the eddy current within the PM, the following matrix of inductances was formed:

$$\underline{L}=L_{}^o\underline{E}+{\underline{M}}^o,$$

(2)

where $\underline{E}$ is the $(n\times n)$ unit matrix, ${\underline{M}}^o$ is the $(n\times n)$-symmetric hollow matrix whose elements are the mutual inductance $L_{ks}^o$ ($k\ne s$) between the sth and kth meshing element, and $L^o$ is the self-inductance of the circular element, which is calculated by the known formula for a circular ring with a circular cross-section

$$L_{}^o={\mu }_0{R}_e\left[ln8/\epsilon -7/4+{\epsilon }^2/8\left(ln8/\epsilon +1/3\right)\right],$$

(3)

where ${\mu }_0$ is the magnetic permeability of free space, $\epsilon =th/\left(2R_e\right)$; $th$ is the thickness of a mashed layer of the PM. The distance between the kth and sth meshing element is characterized by vector ${}^{(k,s)}\rho {=}^{\left(k\right)}\rho {-}^{\left(s\right)}\rho$, as shown in Figure 4. Hence, the mutual inductance $L_{ks}^o$ can be calculated as follows [43]:

$$L_{ks}^o=\frac{{\mu }_0{R}_e}{\pi }{\int }_0^{2\pi }\frac{1{+}^{(k,s)}{\overline{\rho }}_1·cos\phi {+}^{(k,s)}{\overline{\rho }}_2·sin\phi }{{\overline{\sigma }}^{1.5}}·\Phi \left(k\right)d\phi ,$$

(4)

where ${}^{(k,s)}{\overline{\rho }}_1{=}^{(k,s)}{\rho }_1/R_e$ and ${}^{(k,s)}{\overline{\rho }}_2{=}^{(k,s)}{\rho }_2/R_e$ are the dimensionless coordinates, and

$$\overline{\sigma }=\sqrt{1+2{(}^{(k,s)}{\overline{\rho }}_1·cos\phi {+}^{(k,s)}{\overline{\rho }}_2·sin\phi ){+}^{(k,s)}{\overline{\rho }}_1^2{+}^{(k,s)}{\overline{\rho }}_2^2},$$

(5)

$$\Phi \left(k\right)=\frac{1}{k}\left[\left(1-\frac{k^2}{2}\right)K\left(k\right)-E\left(k\right)\right],k^2=\frac{4\overline{\sigma }}{{(\overline{\sigma }+1)}^2},$$

(6)

$K\left(k\right)$ and $E\left(k\right)$ are the complete elliptic functions of the first and second kind, respectively.

Knowing the radius vector ${}^{(s,j)}\mathbf{r}={\mathbf{r}}_{cm}{+}^{\left(s\right)}\rho {-}^{\left(j\right)}{\mathbf{r}}_c$ (see Figure 4), the mutual inductance between the s-meshing element and j-coil can be calculated similar to (4) as follows:

$$M_{sj}=\frac{{\mu }_0{R}_{cj}}{\pi }{\int }_0^{2\pi }\frac{1{+}^{(s,j)}{\overline{r}}_1·cos\phi {+}^{(s,j)}{\overline{r}}_2·sin\phi }{{\overline{\sigma }}^{1.5}}·\Phi \left(k\right)d\phi ,$$

(7)

where

$$\overline{\sigma }=\sqrt{1+2{(}^{(s,j)}{\overline{r}}_1·cos\phi {+}^{(s,j)}{\overline{r}}_2·sin\phi ){+}^{(s,j)}{\overline{r}}_1^2{+}^{(s,j)}{\overline{r}}_2^2},$$

(8)

$$k^2=\frac{4{\nu }_j\overline{\sigma }}{{({\nu }_j\overline{\sigma }+1)}^2+{\nu }_j^2{}^{(s,j)}{\overline{r}}_3^2},$$

(9)

where ${\nu }_j=R_e/R_{cj}$, $R_{cj}$ is the radius of the j-coil filament; ${}^{(s,j)}{\overline{r}}_1{=}^{(s,j)}{r}_1/R_e$, ${}^{(s,j)}{\overline{r}}_2{=}^{(s,j)}{r}_2/R_e$, and ${}^{(s,j)}{\overline{r}}_3{=}^{(s,j)}{r}_3/R_e$ are the dimensionless components of the radius vector ${}^{(s,j)}\mathbf{r}$ in base ${\underline{\mathbf{e}}}^0$. Using Equation (7), the $(n\times N)$ matrix ${\underline{M}}_c$ consisting of elements of the mutual inductance $M_{sj}$ is formed. Hence, the induced eddy current in each circular element is calculated as a solution of the following matrix equation:

$$\underline{I}={\underline{L}}^{-1}{\underline{M}}_c{\underline{I}}_c,$$

(10)

where $\underline{I}$ is the $(2500\times 1)$ matrix of eddy currents $\underline{I}={[I_1{I}_2\dots I_s\dots I_n]}^T$, and ${\underline{I}}_c={[I_{c1}{I}_{c2}\dots I_{cj}\dots I_{c32}]}^T$ is the given $(32\times 1)$ matrix of currents in coils. It is convenient to present the result of calculation in dimensionless form. For this reason, the dimensionless currents in the levitation and stabilization coils are introduced by dividing the currents on the amplitude of the current in the levitation coil because the amplitudes of the current in both coil are the same. Hence, the input current in the levitation coil filaments is one and that in the stabilization coil filaments is minus one (because of the 180° phase shift).

In order to illustratively present the calculation result, the obtained ($2500\times 1$) eddy current matrix, $\underline{I}$, was transformed into a ($50\times 50$) 2D matrix, $\underline{I}$. The data in this ($50\times 50$) 2D matrix are located similarly to the structure corresponding to Figure 5A. Then, the distribution along the PM surface of the induced eddy current in the mesh circular elements is shown in Figure 6A. The analysis of Figure 6A shows that in a central area of the PM, corresponding to the area of the circular cross-section of the levitation coil, the eddy current has a negative sign (meaning that the direction of the induced eddy current flow is opposite to the flow direction in the levitation coil) due to the significant contribution of the A/C magnetic field generated by this coil. Outside of this area, the sign becomes positive due to the A/C magnetic field of the stabilization coil.

The result in vector form through unit vectors ${\mathbf{e}}_1^1$ and ${\mathbf{e}}_2^1$ of the base ${\underline{\mathbf{e}}}^1$ is shown in Figure 6B. Taking the numerical gradient of the ($50\times 50$) 2D matrix $\underline{I}$ with respect to the rows and columns, the components in the form of the ($50\times 50$) 2D matrixes of ${\underline{I}}_1$ and ${\underline{I}}_2$ relative to the unit vectors ${\mathbf{e}}_1^1$ and ${\mathbf{e}}_2^1$ were calculated, respectively. Then, the ($50\times 50$) 2D matrix of magnitudes of the eddy current for each mesh point was estimated by $\sqrt{{{\underline{I}}_1}^2+{{\underline{I}}_2}^2}$. The result of estimation is shown in Figure 6B. Figure 6B depicts that the maximum magnitudes of the eddy current are concentrated along the edge of the PM and in its central part along the circle having the same diameter as the levitation coil.

4.2. Pull-In Modeling

The linear displacement of the PM along the ${\mathbf{e}}_3^0$ axis can be characterized by the generalized coordinate $q_3$, similar to what was previously reported [48]. Hence, the quasi-FEM model can be written as

$$m\frac{d^2{q}_3}{d{t}^2}+mg+{\underline{I}}^T\frac{\partial {\underline{M}}_c}{\partial q_3}{\underline{I}}_c+\frac{A_e}{4}\frac{U^2}{{(h+q_3)}^2}=0.$$

(11)

Then, Equation (11) is rewritten in dimensionless form:

$$\begin{array}{c}{\displaystyle \frac{d^2\lambda }{d{\tau }^2}+1+{\eta }_0{F}_m\left(\lambda \right)+\frac{\beta }{{(1+\lambda )}^2}=0};\hfill \\ {\displaystyle F_m\left(\lambda \right)=\sum _{s=1}^n\sum _{j=1}^N{\eta }_{sj}\frac{\partial {\overline{M}}_{sj}{(}^{(s,j)}{\overline{r}}_1,{}^{(s,j)}{\overline{r}}_2,(1+\lambda \kappa )\chi )}{\partial \lambda }},\hfill \end{array}$$

(12)

where $\tau =\sqrt{g/h}t$ is the dimensionless time, $\lambda =q_3/h$ is the displacement, $\beta =A_e{U}^2/\left(4{\mathrm{mgh}}^2\right)$ is the squared voltage, ${\eta }_0={\mu }_0{I}_{c1}^2\sqrt{R_{c1}{R}_e}/\left(mg{R}_e\right)$, $R_{c1}$ is the radius of the first winding of the levitation coil, ${\eta }_{sj}={\overline{I}}_s{\overline{I}}_{cj}\sqrt{{\overline{R}}_{cj}}/\chi$, ${\overline{I}}_s=I_s/I_{c1}$ and ${\overline{I}}_{cj}=I_{cj}/I_{c1}$ are the dimensionless currents, ${\overline{R}}_{cj}=R_{cj}/R_{c1}$, $\chi =h_l/R_e$ is the scaling factor, and $\kappa =h/h_l$ is the spacing constant. Note that the spacing constant $\kappa$ is responsible for the redistribution of electro-magnetic energy stored by the electrical field of the capacitors and the magnetic field of the coils and levitated PM, which occurs by changing the location of the electrodes along the ${\mathbf{e}}_3^0$ axis. The derivative of the dimensionless mutual inductance with respect to $\lambda$ [46] is

$$\frac{\partial {\overline{M}}_{sj}}{\partial \lambda }=\frac{1}{\pi }{\int }_0^{2\pi }\frac{1{+}^{(s,j)}{\overline{r}}_1·cos\phi {+}^{(s,j)}{\overline{r}}_2·sin\phi }{{\overline{\sigma }}^{1.5}}\frac{d\Phi \left(k\right)}{dk}\frac{dk}{d\lambda }d\phi ,$$

(13)

where

$$\frac{d\Phi \left(k\right)}{dk}=\frac{1}{k^2}\left(\frac{2-k^2}{2(1-k^2)}E\left(k\right)-K\left(k\right)\right),$$

(14)

$$\frac{dk}{d\lambda }=-\frac{{\nu }_j^2\lambda {\kappa }^2{\chi }^2\sqrt{4{\nu }_j\overline{\sigma }}}{{\left({(1+{\nu }_j\overline{\sigma })}^2+{\nu }_j^2{\left(\lambda \kappa \chi \right)}^2\right)}^{3/2}}.$$

(15)

Accounting for (12), the static pull-in model based on quasi-FEM becomes

$${\displaystyle \beta =-{(1+\lambda )}^2\left(1+{\eta }_0{F}_m\left(\lambda \right)\right)},$$

(16)

where the ${\eta }_0$ constant is defined at the equilibrium state as follows:

$${\displaystyle {\eta }_0=-1/F_m\left(0\right).}$$

(17)

4.3. Square-Shapef Proof Mass with Side Length of 2.8 mm and Thickness of 10 μm

The PM was meshed by 2500 elements, as shown in Figure 5A. For the simulation, the 3D geometrical scheme, as shown in Figure 5B, with same dimensions for coils was used. The PM was levitated at a height of $h=$ 95 $\mathsf{\mu }$$\mathrm{m}$ measured from the surface of electrode structure, while the levitation height was estimated from the top first winding of the coils as $h_l=$ 175 $\mathsf{\mu }$$\mathrm{m}$. Hence, the constants $\kappa$ and $\chi$ were equal to $0.5429$ and $6.25$, respectively. The result of the simulation is shown in Figure 7. Figure 7A shows the result of the experimental measurements (cross points) and simulation (solid line) in terms of normalized voltage and dimensionless displacement $\lambda$. Figure 7B depicts a direct comparison between the experimental data and simulation results in absolute values. The model predicted the following pull-in parameters (see the circular solid point in Figure 7): the pull-in voltage is 39 $\mathrm{V}$, and the pull-in displacement is 34 $\mathsf{\mu }$$\mathrm{m}$.

4.4. Square-Shapes Proof Mass with Side Length of 2.8 mm and Thickness of 25 μm

The PM was meshed by 2500 elements, as shown in Figure 5A, and the same 3D geometrical scheme shown in Figure 5B was used. The height of levitation was measured as $h=$ 70 $\mathsf{\mu }$$\mathrm{m}$ from the surface of the electrode structure, while the levitation height $h_l$ was 150 $\mathsf{\mu }$$\mathrm{m}$. Hence, the constants $\kappa$ and $\chi$ were $0.4667$ and $5.3571$, respectively. The result of the simulation is shown in Figure 8. Figure 8A shows the result of the experimental measurements (cross points) and simulation (solid line) in terms of normalized voltage and dimensionless displacement $\lambda$. Figure 8B depicts a direct comparison between the experimental data and simulation in absolute values. The model predicted the following pull-in parameters (see the circular solid point in Figure 8): the pull-in voltage is 30 $\mathrm{V}$, and the pull-in displacement is 25 $\mathsf{\mu }$$\mathrm{m}$.

4.5. Square-Shaped Proof Mass with Side Length of 3.2 mm and Thickness of 10 μm

The PM was meshed by 2500 elements, while the radius of the meshing element was $R_e=$ 3.2229 × 10⁻⁵ m. The height of levitation was $h=$ 94 $\mathsf{\mu }$$\mathrm{m}$ measured from the surface of the electrode structure, while the levitation height $h_l$ was $h_l=$ 174 $\mathsf{\mu }$$\mathrm{m}$. The constants $\kappa$ and $\chi$ were equal to $0.5402$ and $5.4375$, respectively. The result of the simulation is shown in Figure 9. Figure 9A shows the result of the experimental measurements (cross points) and simulation (solid line) in terms of normalized voltage and dimensionless displacement $\lambda$. Figure 9B depicts the direct comparison between the experimental data and simulation in absolute values. The model predicted the following pull-in parameters (the circular solid point in Figure 9): the pull-in voltage is 46 $\mathrm{V}$, and the pull-in displacement is 35 $\mathsf{\mu }$$\mathrm{m}$.

The results of the simulation and the measurements of pull-in parameters for the three PMs are summarized and tabulated in

Table 1. Results of measurements and modeling of the static pull-in actuation.

Measured parameter	Side length of PM	2.8 mm	2.8 mm	3.2 mm
	Thickness of PM	10 μm	25 μm	10 μm
	Levitation height, $h_l$	175 μm	150 μm	174 μm
	Spacing, h	95 μm	70 μm	94 μm
Calculated parameter	$\chi =h_l/R_e$	6.25	5.3571	5.4375
	$\kappa =h/h_l$	0.5429	0.4667	0.5402
Measured pull-in	Displacement	30 μm	23 μm	30 μm
parameters	Voltage	40 V	30.0 V	45 V
Pull-in parameters	Displacement	34 μm	25 μm	35 μm
simulated by Equation (11)	Voltage	39 V	30 V	46 V

. The analysis of Table 1, Figure 7, Figure 8 and Figure 9 reveals a very good agreement between the proposed quasi-FEM model and measurement. Indeed, the developed model mimics the behaviors of the PMs displaced by electrostatic force as well as accurately predicts the parameters of pull-in actuation.

5. Conclusions

The pull-in actuation in a prototype of an HLMA with different sizes of square-shaped PMs was experimentally as well as theoretically studied. We experimentally demonstrated that the prototype using two coaxial 3D wire-bonded microcoils, namely, the stabilization and levitation coil with radii of $1.9$ $\mathrm{m}$$\mathrm{m}$ and $1.0$ $\mathrm{m}$$\mathrm{m}$, was able stably levitate PMs with a side-length ranging from 2.7 to 3.3 $\mathrm{m}\mathrm{m}$. The pull-in actuation of the square-shaped PMs was performed by the electrostatic force generated by the set of energized electrodes and acting on the bottom surface of the PMs along the vertical direction. The experimental measurements showed that the pull-in actuation is nonlinearly dependent on the size and mass of the PM and the levitation height. In particular, we found that the PMs levitated within a range of a height from 140 to 170 $\mathsf{\mu }$$\mathrm{m}$ can be displaced stably within a range of 30 $\mathsf{\mu }$$\mathrm{m}$. The results of measurements were extensively simulated by the analytical model Equation (11) that we developed by means of the quasi-FEM method based on the n-CCA analogy. The direct comparison of the results of the simulation and measurements showed a very good agreement between the theory and experimental results. Thus, the quasi-FEM approach combining the finite element method to calculate the induced eddy current and the differential equations describing the behavior of the mechanical part of the electromagnetic levitation system was shown to be a very efficient mathematical tool for the analysis and simulation of LMAs, including the stability, linear, and nonlinear dynamical responses.