On the Static Pull-in of Tilting Actuation in Hybrid Levitation Micro-actuator: Theory and Experiment

This work presents the results of the experimental and theoretical study of the static pull-in of tilting actuation executed by a hybrid levitation micro-actuator (HLMA) based on the combination of inductive levitation and electrostatic actuation. A semi-analytical model to study such the pull-in phenomenon is developed, for the first time, as a result of using the qualitative technique based on the Lagrangian approach to analyze inductive contactless suspensions presented in work [1] and a recent progress in the calculation of mutual inductance [2] and force [3] between two circular filaments. The obtained non-linear model, accounting for two degrees of freedom of the actuator, allows us to predict accurately the static pull-in displacement and voltage. The results of modelling were verified experimentally and agree well with measurements.


Introduction
Electromagnetic levitation micro-actuators employing remote ponderomotive forces, in order to act on a target environment or simply compensate a gravity force for holding stably a micro-object at the equilibrium without mechanical attachment, have already found wide applications and demonstrated a new generation of micro-sensors and -actuators with increased operational capabilities and overcoming the domination of friction over inertial forces at the micro-scale.
A wide spectrum of physical principles have been utilized and successfully implemented by using different techniques for micro-fabrication. However, recently developed 3D micro-coil technology [24] together with integration of a polymer magnetic composite material for flux concentration allows announcing inductive levitation micro-actuator systems, firstly, as systems with established micro-fabrication process in comparison to the other levitation actuator systems and, secondly, as high-performance systems. As a results of this progress, our group demonstrated the inductive levitation actuator system with the record lowest current consumption [10] around tens of mA. This permits us to avoid using standard bulky high frequency current amplifiers for exciting the ILMA and to replace them by the integrated control circuit including the signal generator and amplifier and having a size comparable with size of micro-actuator system [25]. Laser spot Figure 1. Experimental test of tilting actuation in hybrid levitation micro-actuator and scheme of electrode set: a) stable levitation of the disc having a diameter of 2.8 mm; b) applying the pull-in voltage to the electrodes, the disc was tilted. The angle of tilting actuation is restricted by the mechanical contact with the surface of the electrode structure: c) the set of electrodes (the energized electrodes are highlighted in red, U is the applied voltage).
HLMAs, in which for instance the inductive levitation micro-actuator system can be joined with a source of electrostatic field, dramatically increase capabilities of levitated micro-systems [26] and demonstrate a wide range of different operation modes such as the linear and angular positioning, bi-stable linear and angular actuation and the adjustment of stiffness components as it was reported in [20,27] and presented by the author at Transducers 2017 [28]. These capabilities open a new very promising perspective to create smart micro-actuator systems with new functional abilities implemented, for instance, by means of the coherent cooperation of distributed microactuators, multi-stable actuation, mechanical and electromagnetic couplings.
Although, the linear pull-in actuation has been comprehensively studied theoretically as well as experimentally [29], however angular pull-in phenomenon of tilting actuation in such HLMAs has not yet been addressed. In this article, in order to fill this gap, a prototype of hybrid levitation micro-actuator was fabricated to experimentally demonstrate and characterise the static pull-in behaviour of tilting actuation. Using the qualitative technique based on the Lagrangian approach [1], a semi-analytical model for mimicking such the static pull-in behaviour based on two circuit approximation of induced eddy current with proof mass is developed. Upon tilting the levitated disc, the geometrical transformation of eddy current circuit is analysed by means of finite element approach proposed in [30]. As a result of this analysis becomes a semi-analytical function for calculation mutual inductance and corresponding electromagnetic force and torque developed by Kalantarov-Zeitlin's method [2]. Thus, the particularity of the developed model is to employ this new derived function for calculation mutual inductance and to account for two degrees of freedom, namely, linear displacement in the vertical direction and angular one of the levitated disc. Through this study we verified successfully the given assumptions for modelling and developed the robust analytical tool to describe the static pull-in actuation of HLMAs.

Fabrication and measurements
In order to comprehensively study the static pull-in of tilting actuation executed by the HLMA, we fabricated a micro-prototype as shown Fig. 1a) by using a similar fabrication process reported in work [20]. Namely, the hybrid actuator consists of two structures fabricated independently, namely, the coil structure and the electrode structure, which were aligned and assembled by flip-chip bonding into one device with the dimensions: 9.4 mm×7.4 mm×1.1 mm as shown in Fig. 1a).
The coil structure includes two coaxial 3D wire bonded microcoils similar to those reported in our previous work [9], namely the stabilization and the levitation coils, fabricated on Pyrex substrate using SU-8 2150. The function of the coil structure is to stably levitate an aluminium disc-shaped proof mass (PM). For this particular device, the height of the coils is 500 µm and the number of windings is 20 and 12 for levitation and stabilization coils, respectively. This coil structure is able to levitate PMs with diameters ranging from 2.7 to 3.3 mm [9].
The top electrode structure was fabricated on a SOI wafer having a device layer of 40 µm, the buried oxide of 2 µm, a handle layer of 600 µm and the resistivity of silicon in a range of 1..30 Ω cm as it was reported in work [31]. Also, the device layer has a 500 nm oxide layer for passivation, on the top of which electrodes are patterned by UV lithography on evaporated Cr/Au layers (20/150 nm). The design of electrode set is schematically shown in Fig. 1c). After etching the handle layer up to the buried oxide by DRIE, the bottom electrode structure was aligned and bonded onto the coil structure. Finally, the fabricated device was glued on a PCB board as shown in Fig. 1a).
The actuator coils were feeded by ac current at the frequency of 10 MHz, with a rms value ranging from 100 mA to 130 mA. This range of changing of ac current allowed us to levitating a disc having a diameter of 2.8 mm within a height from 35 to 190 µm measured from the surface of the electrodes patterned on the silicon substrate. Pull-in of tilting actuation was performed by applying pull-in voltage to electrode "1" and "2" as shown in Fig. 1b) and c). The circular electrode "1" has a diameter of 1 mm corresponding to an area of around ∼0.8 mm 2 . While, the geometry of the sectorial electrode "2" is characterised by the following parameters: an inner radius of 1.1 mm, an outer radius of 1.4 mm and the center angle of π/2 rad, which is corresponding to an area of around ∼0.43 mm 2 . The electrode set was able to generate the tilting torque within a range of 0.7 × 10 −10 N m. The linear displacement of a disc edge was measured by a laser distance sensor directly (see, Fig. 1a), b)). The measurements were performed at two levitation heights. Namely, the disc was levitated at 130 and 150 µm. The pull-in actuation occurred for the applied voltage of 27 and 33 V and the measured pull-in displacements were 34 and 45 µm, respectively. The results of measurements accompanying with results of modeling are summed up in Table 1 below.

Simulation and modeling
The mechanism of stable levitation of the disc shaped proof mass in the framework of two coil design is as follows. The induced eddy currents are distributed along the levitated proof mass in such a way that two circuits having maximum values of eddy current density can be identified. The first circuit corresponds to the eddy current distributed along the edge of disc-shaped PM and the second circuit is defined by the levitation coil. The later one has a circular path with radius equal to the radius of the levitation coil. Hence, this mechanism can be split into two force interactions. The force interaction happens between the current in the stabilization coil and induced eddy current corresponding to the first circuit, which contributes mainly to the lateral stability of the levitated PM. While, the force interaction between the current in the levitation coil and induced eddy current related to the second circuit contributes mainly to the vertical and angular stability of the levitated PM [1].
Upon tilting the PM, the eddy current circuit generated by the levitation coil is transformed form a circular shape into elliptical one. In the section below, this transformation of the shape of eddy current circuit is analyzed by quasi-finite element approach recently developed in [30, 32,33]. As a result of the analysis, formulas for calculation of the mutual inductance and corresponding electromagnetic torque and force acting on the PM are derived by employing Kalantarov-Zeitlin's method [2]. Then the derived formulas are applied for the modeling of the pull-in actuation in the HLMA.

Simulation of induced eddy current within the tilting proof mass
According to the procedure proposed in our previous work [30], the disc is meshed by circular elements as it is shown in Fig. 2a). For the particular case, the levitated micro-disc is meshed by circular elements each of them has the same radius of R e =2.4814 × 10 −5 m. For the disc of a diameter of 2.8mm, a number of elements is n = 2496. 3D scheme of two micro-coils is approximated by a series of circular filaments. The levitation coil is replaced by 20 circular filaments having a diameter of 2.0 mm, while the stabilization coil by 12 circular filaments with a diameter of 3.
containing information about a radius vector and an angular orientation for each element with respect to the coordinate frame {x k } (k = 1, 2, 3). Now a matrix L of self-inductances of circular elements and mutual inductances between them can be formed as follows where E is the (2496 × 2496) unit matrix, M o is the (2496 × 2496) -symmetric hollow matrix whose elements are L o ks (k = s). The self-inductance of the circular element is calculated by the known formula for a circular ring of circular cross-section where µ 0 is the magnetic permeability of free space, ε t = th/(2R e ), th is the thickness of a meshed layer of micro-object (in the particular case, th =13 µm). Accounting for the values of diameters of levitation and stabilization coils, 3D geometrical scheme of the actuator for the eddy current simulation can be build as shown in Fig. 2b). The position of the coordinate frame {x k } (k = 1, 2, 3) with respect to the fixed frame {X k } (k = 1, 2, 3) is defined by the radius vector r cm = [0 0 h l ] T , where the levitation height, h l is to be 100 µm. Then, the position of the s-mesh element with respect to the coordinate frame { (j) z k } (k = 1, 2, 3) assigned to the j-coil filament can be found as (s,j) r = r cm + (s) ρ − (j) r c or in a matrix form as Since all matrices of (j) φ c are zeros. Hence, (j) A zX = E, where E is the (3 × 3) unit matrix and (j) A zx = (j) A zX (j) φ c A Xx (ϕ) = E A Xx (ϕ) = A Xx (ϕ). As an illustrative example, the angle θ is chosen to be 10 • as shown in Fig. 2b). Moreover, the coils are represented by the circular filaments and using the radius vector (s,j) r, the mutual inductance between the jcoil and s-meshed element can be calculated directly by the formula presented in [2]. Thereby, the the (2496 × 32) matrix M c of mutual inductance between coils and finite elements can be formed. The induced eddy current in each circular element is a solution of the following matrix equation  where I is the (2496 × 1) matrix of eddy currents and I c = [I c1 I c2 . . . I cN ] T is the given (32 × 1) matrix of currents in coils. For calculation, dimensionless currents in the levitation coil and stabilization one are introduced by dividing currents on the amplitude of the current in the levitation coil. Noting that the amplitude of the current in both coils are the same. Hence, the input current in the levitation coil filaments is to be one, while in the stabilization coil filaments to be minus one (because of the 180 • phase shift). Now, the induced eddy current in dimensionless values can be calculated [30]. The result of calculation is shown in Fig. 2c) as a 3D plot. The intensity of the color shown by the bar characterizes the value of dimensionless magnitude of the eddy current. Fig. 2d) shows a 2D plot of the distribution of magnitudes of eddy current along the area of the surface of the PM. As it is seen from Figures 2c) and d) that the maximum of eddy current magnitudes are distributed along the edge of the PM and around its centre. Besides, the analysis of Fig. 2d) depicts that a circular shape of maximum magnitude of eddy current induced by the current of the stabilization coil without tilting the PM [30] is transformed into an elliptical shape with shifting its centre as shown in Fig. 2d). The centre of the ellipse is characterized by the following coordinates z c = z c (θ) and y c = y c (θ), which are also functions of the tilting angle θ.

Mutual Inductance between two filaments of circular and elliptic shapes
Approximating the elliptical shape of distribution of the eddy current magnitude around the centre of the PM by a filament, a function for calculation of the mutual inductance between the elliptical filament with the shifted centre and circular filament is derived by using Kalantarov-Zeitlin's method. Scheme with detailed particularities of geometry for calculation is shown in Fig. 3.
Accounting for Fig. 3 and introducing the following dimensionless coordinates: the formula for calculation of the mutual inductance can be written as where r = r(θ) = cos θ (tan 2 (θ) + 1) sin 2 (ϕ) + cos 2 (θ) cos 2 (ϕ) , (7) and K(k) and E(k) are the complete elliptic functions of the first and second kind, respectively, and

Model of static pull-in of tilting actuation
Assuming that the contribution of the force interaction between the current in the stabilization coil and the eddy current circuit flowing along the edge of the PM to the behaviour of the PM in the vertical and angular directions measured by the generalized coordinates z c and θ, respectively, is negligible [30]. Thus, the force interaction between the current in the levitation coil and the eddy current distributed around PM's centre is dominated and determines the actuation of the PM under applying electrostatic force produced by the energized electrodes "1" and "2".
Hence, the Lagrangian of the electromagnetic system under consideration can be written as where m is the mass of PM, J is the moment of inertia of the disc about axis through centre of mass and coinciding with its diameter, g is the gravitational acceleration, M(θ, z c ) is the mutual inductance defined by formula (6), I is the given ac current in the levitation coil (I =Îe jωt , whereÎ is the magnitude, ω is the frequency), i is the induced eddy current, L e is the self inductance of the eddy current circuit, Q 1 and Q 2 are the charges stored by the planar capacitors C 1 and C 2 build on electrodes "1" and "2", respectively. Accounting for linear and angular displacements of the PM, the capacitances corresponding to circular "1" and sectorial "2" electrode can be described, respectively, by the following equations where ϕ 10 = πεε 0 and ϕ 20 = εε 0 π/2, ε is the relative permittivity and ε 0 is the permittivity of free space, R 1 is the radius of the centre electrode, R out and R in are the outer and inner radii of the sectorial electrode, respectively. Noting that the capacitors are connected in the series, due to this fact for charges we can write that Q 1 = Q 2 = Q. The dissipation function can be presented as Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 July 2021 doi:10.20944/preprints202107.0683.v1 where R e is the electrical resistance of the eddy current circuit. Hence, the state of the system can be defined by two coordinates z c and θ, the charge Q and the eddy current i. Adapting the variables and the assumptions introduced above, the Lagrange -Maxwell equations of the system can be written as follows Substituting (12) and (14) into (15) and accounting for the fact that the static problem is considered, so the acceleration and speed of coordinates z c and θ are ignored. Thus, we can finally write the following set of equations describing the static behaviour of the PM [1]: Since the system operated in the air and the air damping supports the stable levitation of the PM, hence we can conclude similar to [1] that i ≈ −MI. Substituting the relation for currents from the later conclusion into (16) and averaging ponderomotive force and torque with respect to the time

Analysis of the derived model
The semi-analytical model (17) to describe the static pull-in actuation of the fabricated prototype is developed by employing the qualitative technique to analyze inductive contactless suspensions, and new formula for calculation of mutual inductance between the circular filament and its projection on a tilted plane derived in Sec. 3.2 based on the result of simulation of induced eddy current within disc shaped PM obtained by means of quasi-FEM approach. The mechanical behavior of the disc is described by two generalized coordinates. They can be represented by dimensionless variables, namely    Table 1).
the spacing between the surface electrode structure and an equilibrium point O; R T = (R out + R in )/2 is the distance between the centre of the disc and applied electrostatic force generated by electrode "2". The electromagnetic part of this system is reduced to the interaction between the current in the levitation coil and the induced eddy current with circuit corresponding to its maximum density within the disc. As a result of numerical analysis, the behavior of the shape of induced maximum density of eddy current within the disc was represented via an analytical function of introduced above two generalized coordinates. This circumstance helps us to exactly define the mutual inductance and then to calculate the magnetic force and torque acting on the disc. A preliminary result of modelling are shown in Fig. 4, which provides an evolution of bifurcation diagrams for equilibrium of the titling PM without linear displacement depending on the applied dimensionless voltage β = ε 0 A 1 U 2 /(2mgh 2 (1 + a)), where A 1 and A 2 are the areas of electrode 1 and 2, respectively, a = A 1 /A 2 and depending on dimensionless parameters characterizing geometrical particularities of the prototype design, namely, κ = h/h l , which is in a range from 0.5 to 0.9, and ξ = h l /(2R l ), which is changing in a range from 0.03 to 0.09. These ranges of parameters κ and ξ are typical for all known designs of HLMA studied in the literature. As it is seen from Fig.  4, increasing a radius of levitation coil (it means that ξ is respectively decreased) and decreasing the space h (it means that the dimensionless parameter κ is also decreased) leads to decreasing pull-in voltage and displacement.
Modelling the experimental measurement discussed in Sec. 2 requires taking into account the angular as well as linear displacement of the disc due to essential contribution of linear force acting along the z axis. Fig. 5 shows the comparison of results of measurements and modelling in normalized values. As a result of the contribution of the linear force, the range of angular displacement is reduced by 40%. Fig. 6 reveals the direct comparison between measurements and modelling. The parameters of the device, particularities of conducted measurements and obtained results are summed up in Table 1. The analysis of Fig. 5, 6 and Table 1 shows that the modelling agrees well with experimental data.  Table 1).

Conclusions
In this article, the static pull-in of tilting actuation in HLMA was studied theoretically as well as experimentally. The semi-analytical model (17) mimicking the static pull-in behaviour of tilting actuation in HLMA is developed by employing the qualitative technique to analyze the dynamics and stability of inductive contactless suspensions, and new formula for calculation of mutual inductance between the circular filament and its projection on a tilted plane. As the result of simulation of induced eddy current within disc shaped PM carried out quasi-FEM approach, which showed the distribution of eddy current around the centre of a conducting disc, this new formula was developed and presented in the integral form based on Kalantarove-Zeitlin's method. The obtained non-linear model, accounting for two degrees of freedom of the actuator, allows us to predict accurately the static pull-in displacement and voltage for the particular prototype of HLMA. The results of modelling were verified experimentally and agree well with measurements. Through this study we verified successfully the given assumptions for modelling and developed the robust analytical tool to describe the static pull-in actuation and to predict pull-in parameters for HLMAs.
Funding: This research was funded by the German Research Foundation grant number KO 1883/37-1 within the priority program SPP2206 "KOMMMA", project "A 2D array of cooperative hybrid levitation micro-actuators".