ELECTRODYNAMICALLY LEVITATED ACTUATOR

Abstract

A microelectromechanical actuator, comprising: a substrate, having a surface; a conductive beam suspended parallel to the substrate, displaceable along an axis normal to the surface of the substrate; a center electrode on the substrate under the beam; a pair of side electrodes on the substrate configured, when charged, to exert an electrostatic force normal to the surface of the substrate on the beam that repulses the beam from the substrate, and exerts a balanced electrostatic force on the beam in a plane of the surface of the substrate, the center conductive electrode being configured to shield the beam from electrostatic forces induced by the side electrodes from beneath the beam, and the center electrode being configured to have a voltage different from a voltage on the beam, to thereby induce an attractive electrostatic force on the beam.

Claims

1. A microelectromechanical actuator, comprising: a substrate, having a surface; an electrostatically displaceable conductive element, suspended over the substrate; a center electrode, provided on the substrate under the electrostatically displaceable conductive element; and a peripheral electrode provided on the substrate; wherein: the center electrode is larger than a projection of the electrostatically displaceable conductive element on the substrate, and is configured to shield the electrostatically displaceable conductive element from electrostatic forces induced by the peripheral electrode from beneath the electrostatically displaceable conductive element, the peripheral electrode is configured to exert an electrostatic force normal to the surface of the substrate on the electrostatically displaceable conductive element that repulses the electrostatically displaceable conductive element from the substrate, and the center electrode is configured to have a voltage different from a voltage on the electrostatically displaceable conductive element, to thereby induce an attractive electrostatic force on the beam.

2. The microelectromechanical actuator according to claim 1, wherein: the electrostatically displaceable conductive element comprises a beam having a conductive portion, freely suspended over the substrate, and having a long axis parallel to the surface of the substrate, and being displaceable along an axis normal to the surface of the substrate; the peripheral electrode comprises a pair of side electrodes, axisymmetric with respect to the beam, provided on the substrate, configured to exert an electrostatic force normal to the surface of the substrate on the beam, that repulses the beam from the substrate, and to exert a balanced electrostatic force on the beam in a plane of the surface of the substrate; and the center electrode is provided on the substrate under the beam, and is configured to shield the beam from electrostatic forces induced by the side electrodes from beneath the beam, and to have a voltage different from a voltage on the beam, to thereby induce an attractive electrostatic force on the beam.

3. The microelectromechanical actuator according to claim 2, wherein the beam has a first state in which the beam is pulled in to the center electrode by the attractive electrostatic force on the beam, and a second state in which the attractive electrostatic force on the beam by the center electrode sufficient to pull in the beam is overcome by the repulsive electrostatic force exerted by the side electrodes.

4. The microelectromechanical actuator according to claim 1, further comprising a triboelectric generator, configured to induce an electrostatic voltage on the peripheral electrode with respect to the center electrode, to thereby selectively overcome the attractive electrostatic force on the electrostatically displaceable conductive element induced by the center electrode, wherein the electrostatically displaceable conductive element has a first state in which the electrostatically displaceable conductive element is pulled in to the center electrode by the attractive electrostatic force on the electrostatically displaceable conductive element, and a second state in which the attractive electrostatic force on the electrostatically displaceable conductive element by the center electrode is overcome by the repulsive electrostatic force exerted by the peripheral electrode from activation of the triboelectric generator.

5. The microelectromechanical actuator according to claim 1, wherein the electrostatically displaceable conductive element has an inertial mass, subject to displacement by inertial forces, and wherein a voltage on at least one of the center electrode and the peripheral electrode with respect to the electrostatically displaceable conductive element is adjusted to control an inertial state which triggers pull-in of the electrostatically displaceable conductive element to the center electrode.

6. A method of actuating a microelectromechanical actuator, comprising: providing the microelectromechanical actuator comprising: a substrate having a surface, a conductive element, suspended over the substrate, and being electrostatically displaceable normal to the surface of the substrate, a center electrode, larger than a projection of the conductive element on the substrate, and is provided on the substrate under the conductive element, and a peripheral electrode, provided on the substrate outside the center electrode, wherein the center electrode is configured to selectively shield the conductive element from electrostatic forces induced by the peripheral electrode from beneath the conductive element, and to permit electrostatic forces induced by the peripheral electrode to interact with the conductive element from above the conductive element; applying an electric potential between the conductive element and the center electrode; and repulsing the conductive element from the substrate by applying an electric potential between the conductive element and the peripheral electrode.

7. The method according to claim 6, wherein: the conductive beam comprises a beam having a conductive portion, freely suspended over the substrate, and having a long axis parallel to the surface of the substrate, and being displaceable along an axis normal to the surface of the substrate; the center electrode is provided on the substrate under the beam; the peripheral electrode comprises a pair of side electrodes, disposed on either side of the center electrode, axisymmetric with respect to the beam and configured to produce an axisymmetric electrostatic force on beam; the center conductive electrode is configured to selectively shield the beam from electrostatic forces induced by the side electrodes from beneath the beam, and not to shield the beam from electrostatic forces induced by the side electrodes from above the beam; the electric potential between the beam and the center electrode is sufficient to cause pull-in; and the electrostatic force on the beam induced by the pair of side electrodes repulses the beam from the substrate.

8. The method according to claim 7, wherein the beam is pulled-in to the center electrode by the center electrode by the electric potential between the beam and the center electrode, and the electrostatic force induced on the beam by the pair of side electrodes releases the beam from pull-in.

9. A method, comprising: providing an electrostatically-repositionable tip of an actuator beam; displacing the actuator beam away from a supporting substrate of the actuator beam, by applying a bias voltage potential with respect to an actuator beam potential between a bias electrode on the supporting substrate beneath the actuator beam, and a side voltage potential between a pair of laterally located electrodes on the supporting substrate and the actuator beam, wherein the side voltage potential produces a force to levitate the actuator beam away from the supporting substrate, toward the surface; and detecting a displacement of the actuator beam.

10. The method according to claim 9, wherein the pair of laterally located electrodes on the supporting substrate exerts a laterally balanced force on the actuator beam in response to the side voltage potential.

11. The method according to claim 9, wherein the actuator beam is displaced to contact a surface.

12. The method according to claim 11, wherein the surface contact is detected by electrical conduction between the actuator beam than the surface.

13. The method according to claim 11, where the tip of the actuator beam is displaced to contact the surface a plurality of times at a plurality of relative positions, further comprising determining a surface profile of a surface.

14. The method according to claim 11, wherein the tip is scanned over a portion of the surface, and the detected displacement of the actuator beam is determined for a plurality of scan positions, to map a surface profile of the surface.

15. The method according to claim 11, wherein contact of the tip with the surface is determined by a piezoelectric transducer.

16. The method according to claim 11, wherein the side voltage potential is increased, to thereby increase an actuator beam deflection, until the tip contacts the surface, and the deflection determined based on the voltage potential at a time of tip contact.

17. The method according to claim 9, wherein the side voltage potential is oscillatory, and temporal characteristics of the actuator beam are determined.

18. The method according to claim 9, further comprising tuning the bias voltage to control a response of the actuator beam to the side voltage.

19. The method according to claim 9, wherein the side voltage is generated by a triboelectric generator.

20. The method according to claim 9, wherein the displacement of the actuator beam is oscillatory, and a dynamic displacement of the actuator beam is detected.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0085] FIG. 1 shows repulsive force electrode configuration with cantilever beam (black), fixed middle electrode (green) and fixed side electrodes (red). The side electrodes are connected together and treated as a single, third electrode.

[0086] FIGS. 2A and 2B show the layout of the impact sensor. Top shows sensor in the initial close state prior to impact. Bottom shows sensor during impact as the switch is opened.

[0087] FIG. 2C shows an optical profiler image of the beam in the neutral state and FIG. 2D shows an optical profiler image of the beam in the levitated state. The images show beam showing pull-in (left) at 2 V.sub.bias and 0 V.sub.side and release (right) at 2 V.sub.bias and 120 V.sub.side. The images were captured with a Wyko NT1100 Optical Profiler.

[0088] FIG. 3A shows the electrostatic force as the gap between electrodes (d) varies at a 2.2 V.sub.bias and 0, 12, and 30 V.sub.side. The green line indicates the rest position of the beam. The results are obtained from Comsol simulations.

[0089] FIG. 3B shows a calculated electrostatic force (N/m) vs. beam-electrode gap (m) for 0 V.sub.bias, 10 V.sub.side; 1 V.sub.bias, 10 V.sub.side; and 1 V.sub.bias, 0 V.sub.side. Electrostatic force on the beam versus the gap distance simulated in COMSOL. The dashed area shows the attractive regime, and the rest is the repulsive regime. The combined force with bias and side voltages behaves similar to the attractive force at low gaps and the repulsive force at large gaps. Applying 10 V on the side electrodes can change the force from attractive to repulsive outside of very small gaps.

[0090] FIG. 3C shows the bias voltage vs. the side voltage at release of pull-in, over a range of bias voltages from 0-4.5V.

[0091] FIG. 4 shows the potential energy of the beam as a function of tip displacement at V.sub.bias=2.2 V and V.sub.side=0V. The red line indicates the smallest possible gap because of the dimples.

[0092] FIG. 5 shows the phase portrait at V.sub.bias=2.2 V and V.sub.side=0V. The red line indicates the smallest possible gap. The marker shows the pulled in position of the beam at 1.25 m and zero velocity

[0093] FIG. 6 shows the potential energy at V.sub.bias=2.2 V and V.sub.side=12V. The red line indicates the smallest possible gap. The pulled in position is the intersection of the red and blue line. The arrows show the beam settling to its new equilibrium position.

[0094] FIG. 7 shows the phase portrait at V.sub.bias=2.2 V and V.sub.side=12V. The inlet shows a zoomed in picture of the intersection at zero velocity and 1.25 m.

[0095] FIG. 8 shows the open circuit generator voltage resulting from a single light impact. The voltage is measured with a Keithley 6514 electrometer. The voltage output can be estimated as a series of step inputs.

[0096] FIG. 9 shows the open circuit generator voltage resulting from a single harder impact. The harder impact results in a higher output voltage. The voltage is measured with a Keithley 6514 electrometer. The voltage output can be estimated as a series of step inputs.

[0097] FIG. 10 shows the beam tip displacement for a side voltage corresponding to the step voltage shown in FIG. 8. The first Voltage step is at time zero and the second voltage step is after 10 ms. The generator voltage is above the threshold and therefore the switch is operating in the open state.

[0098] FIG. 11 shows the beam tip displacement for a side voltage corresponding to the step voltage shown in FIG. 9. The first voltage step is at time zero and the second voltage step is after 28 ms. The generator voltage is above the threshold and therefore the switch is operating in the open state.

[0099] FIG. 12 shows an electrode layout to create electrostatic levitation with electric field lines. The side electrodes (red) are denoted as electrode 1, the middle electrode (green) is denoted as electrode 2, and the beam (blue) is denoted as electrode 3. The geometric parameters (hl, bl, g, etc.) are defined in Table II.

[0100] FIGS. 13a-13c show the capacitance per unit length of .sub.11, .sub.12, and .sub.22 as calculated in COMSOL. FIG. 13d shows the electrostatic force at various voltages showing the comparison between the pure attractive, pure repulsive, and combined forcing cases.

[0101] FIG. 14 shows an optical image of a fabricated beam.

[0102] FIG. 15a shows a first experimental setup.

[0103] FIG. 15b shows a second experimental setup.

[0104] FIG. 15c shows experimental data derived from the second experimental setup.

[0105] FIGS. 16a-16d show a comparison of predicted and measured static results for (FIG. 16a) 0V, (FIG. 16b) 2V, (FIG. 16c) 4V, and (FIG. 16d) 6V bias as a function of side voltage. The results are in very close agreement.

[0106] FIGS. 17a-17d show the nondimensional force at 6 V bias as a function of side voltage when evaluated at the static equilibrium position for a) the total force, b) c.sub.11 component, c) c.sub.12 component, and d) c.sub.22 component. Each component is scaled with its associated voltages as denoted in Equation (22).

[0107] FIGS. 18a-18d shows the natural frequency results for a) 0V, b) 2V, c) 4V, and d) 6V bias as a function of side voltage.

[0108] FIGS. 19a-19d show the frequency response around the first natural frequency for (FIG. 19a) 0V, (FIG. 19b) 2V, (FIG. 19c) 4V, and (FIG. 19d) 6V bias at 170 VDC and 0.5 VAC

[0109] FIG. 20 shows the model displacement frequency response around the first natural frequency for 0V, 2V, 4V, and 6V bias at 170 VDC and 0.5 VAC on the side electrodes.

[0110] FIGS. 21a-21f shows (FIG. 21a) The general schematic of the accelerometer. Base excitation (shown by yellow bi-directional arrows) will be transferred to the microstructure through its supports causing the plate to rotate. (FIG. 21b) A closer view of the configuration of the electrodes. (FIG. 21c) Asymmetric fringing electric field exerts an upward force on the moving electrode (FIG. 21d) Side view of accelerometer and substrate when V.sub.dc=0 (FIG. 21e) when V.sub.dc0 but there is no mechanical load and (FIG. 21f) When V.sub.dc0 and a mechanical load is applied.

[0111] FIG. 22 shows the electrostatic force profile obtained from COMSOL for the two-dimensional unit cell of the sensor.

[0112] FIGS. 23a-23j show the fabrication process (FIG. 23a) 4-inch, 525 m thick bare Si wafer. (FIGS. 23b and 23c) Insulation layers growth and deposition. (FIG. 23d) First polysilicon deposition. (FIG. 23e RIE etch of polysilicon. (FIG. 23f) Sacrificial layer deposition and CMP processing. (FIG. 23g) Anchor etch on a sacrificial layer. (FIG. 23h) Second polysilicon deposition. (FIG. 23i) Polysilicon etch and gold deposition on the pads. (FIG. 23j) Release in HF:HCl solution and CPD.

[0113] FIGS. 24a-24c show (FIG. 24a) SEM image of the device from top. (FIG. 24b) Closer view of the device anchor point. (FIG. 24c) Moving and fixed electrode configuration.

[0114] FIGS. 25a-25c show the frequency response to harmonic excitation for V.sub.dc=60 V, 50 V, and 40 V and acceleration amplitude of 1-4 g.

[0115] FIG. 26 shows simulation results for linearized natural frequency for different DC voltages on side electrodes.

[0116] FIG. 27 shows simulation results for static displacement at the edge of the proof-mass plate for different DC voltages.

[0117] FIG. 28 shows mechanical sensitivity of the sensor for different DC voltages on the side electrodes obtained from experimental results.

[0118] FIG. 29 shows the actual shock in the experiment (25 g and 0.25 ms), simulated shock profile in the mathematical model and relative displacement at the proof-mass edge for the corresponding shock (V.sub.dc=60).

[0119] FIGS. 30a-30d show both simulation and experimental time responses for half sine shocks with different time duration at 60 V bias voltages. (a) T.sub.sh/T.sub.n=1.79, (b) T.sub.sh/T.sub.n=1.18, (c) T.sub.sh/T.sub.n=0.89, (d) T.sub.sh/T.sub.n=0.61.

[0120] FIG. 31 shows both simulation and experimental maximum of the time responses for half sine shocks with time duration (T.sub.sh) equal to 25-300% of the natural period of the microstructure (T.sub.n) at 50 V bias voltages.

[0121] FIG. 32 shows a prior art electrode configuration of Weimin Wang, Qiang Wang, Hao Ren, Wenying Ma, Chuankai Qiu, Zexiang Chen & Bin Fan, Electrostatic repulsive out-of-plane actuator using conductive substrate, Scientific Reports volume 6, Article number: 35118 (2016), FIG. 1c.

[0122] FIG. 33 shows a prior art diagram of a levitating actuator with a uniform ground plane, i.e., FIG. 6 of Liao et al (2004).

[0123] FIG. 34 shows a prior art diagram of a levitating actuator with a central ground plane conductor and two lateral conductors on the substrate, with suspended stationary electrodes, i.e., FIG. 18 of Liao et al (2004).

[0124] FIG. 35 shows a levitating actuator design according to Pallay et al. (2017) with equipotential center electrode and moving electrode.

[0125] FIGS. 36a-36f shows measurements of (FIG. 36a) impact force (FIG. 36b) generator voltage (FIG. 36c) beam velocity using laser vibrometer after the drop of a one-pound weight and (FIG. 36d) impact force (FIG. 36e) generator voltage (FIG. 36f) beam velocity using laser vibrometer after the generator electrodes are shorted. In all cases, the side electrodes are connected to the generator while the beam and the center bottom electrodes are grounded.

[0126] FIG. 37 shows the open circuit voltage profiles as the contact force varies are different when the switch is connected.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Example 1Actuator Embodiment

[0127] FIG. 1 shows repulsive force electrode configuration with cantilever beam (black), fixed middle electrode (green) and fixed side electrodes (red). The side electrodes are connected together and treated as a single, third electrode.

[0128] FIGS. 2a and 2b show the layout of the impact sensor. Top shows sensor in the initial close state prior to impact. Bottom shows sensor during impact as the switch is opened.

[0129] FIG. 2c shows an optical profiler image of the beam in the neutral state and FIG. 2d shows an optical profiler image of the beam in the levitated state. The images show beam showing pull-in (left) at 2 V.sub.bias and 0 V.sub.side and release (right) at 2 V.sub.bias and 120 V.sub.side. The images were captured with a Wyko NT1100 Optical Profiler.

[0130] FIG. 3a shows the electrostatic force as the gap between electrodes (d) varies at a 2.2 V.sub.bias and 0, 12, and 30 V.sub.side. The green line indicates the rest position of the beam. The results are obtained from Comsol simulations.

[0131] FIG. 3b shows a calculated electrostatic force (N/m) vs. beam-electrode gap (m) for 0 V.sub.bias, 10 V.sub.side; 1 V.sub.bias, 10 V.sub.side; and 1 V.sub.bias, 0 V.sub.side. Electrostatic force on the beam versus the gap distance simulated in COMSOL. The dashed area shows the attractive regime, and the rest is the repulsive regime. The combined force with bias and side voltages behaves similar to the attractive force at low gaps and the repulsive force at large gaps. Applying 10 V on the side electrodes can change the force from attractive to repulsive outside of very small gaps.

[0132] FIG. 3c shows the bias voltage vs. the side voltage at release of pull-in, over a range of bias voltages from 0-4.5V.

[0133] To model the actuator, the system can be treated as an Euler-Bernoulli beam with the governing equation of motion defined as,

[00001]$\begin{array}{cc}.Math..Math.A.Math.\frac{{}^2.Math.\hat{w}}{{\hat{t}}^2}+\hat{c}.Math.\frac{\hat{w}}{d.Math.\hat{t}}+E.Math.I.Math.\frac{{}^4.Math.\hat{w}}{{\hat{x}}^4}={\hat{f}}_e\left(\hat{w},\stackrel{.fwdarw.}{V}\right)& \left(1\right)\end{array}$

[0134] where {circumflex over (v)} is the transverse beam deflection (z-direction beam displacement) that depends on the position along the length x of the beam, and time t, I is the moment of inertia, and {circumflex over (f)}.sub.e (, {right arrow over (V)}) is the electrostatic force, which depends on the transverse deflection and applied voltages. Because there are multiple electrodes with different voltages, V is a vector of voltages that are applied to the beam, side electrodes, and center electrode.

[0135] The out of plane deflection governing equation of motion for a clamped-clamped beam is:

[00002]$\begin{array}{cc}.Math..Math.A.Math.\frac{{}^2.Math.\hat{w}}{{\hat{t}}^2}+\hat{c}.Math.\frac{\hat{w}}{d.Math.\hat{t}}+.Math.\frac{E.Math.A}{2.Math.L}.Math.\frac{{}^2.Math.\hat{w}}{{\hat{t}}^2}.Math.{}_0^L.Math.{\left(\frac{\hat{w}}{\hat{t}}\right)}^2.Math.d.Math.x+E.Math.I.Math.\frac{{}^4.Math.\hat{w}}{{\hat{x}}^4}+{\hat{f}}_e\left(\hat{w}\right).Math.V^2=0& \left(1.Math..Math.A\right)\end{array}$

[0136] where indicates the mid-plane stretching effect. is 1 for the clamped-clamped beam and 0 for the cantilever. Because of the small scale of the beams, a nondimensional form of Eq. (1A) may be used. The nondimensional substitutions and parameters are shown below, and the non-dimensional equation of motion is given as:

[00003]$\begin{array}{cc}\frac{{}^2.Math.w}{t^2}+c.Math.\frac{w}{d.Math.t}+.Math.r_1.Math.\frac{{}^2.Math.w}{t^2}.Math.{}_0^1.Math.{\left(\frac{w}{t}\right)}^2.Math.d.Math.x+\frac{{}^4.Math.w}{x^4}+r_2.Math.V^2.Math.\underset{j=0}{\overset{5}{.Math.}}.Math.p_j.Math.h^i.Math.w^j=0& \left(1.Math.B\right)\end{array}$

TABLE-US-00001 Parameter Substitution x-direction position x = {circumflex over (x)}/L y-direction position w = /h Time t = {circumflex over (t)}/T Damping c = L.sup.4/EIT Time constant T = {square root over (AL.sup.4/EI)} Mid-plane stretching constant r.sub.1 = Ah.sup.2/2IL Force constant r.sub.2 = L.sup.4 /EIh

[0137] where p.sub.j are constants from the 5th-order polynomial forcing profile fit. In Table II and Eq. (1B), the nonaccented variables refer to the non-dimensional quantities. Equation (1B) is then discretized using Galerkin's method. The deflection of the beam is approximated as:

[00004]$\begin{array}{cc}w\left(x,t\right)\underset{i=1}{\overset{n}{.Math.}}.Math.q_i\left(t\right).Math.{}_i\left(x\right)& \left(1.Math.C\right)\end{array}$

[0138] where .sub.i(x) are the linear mode shapes of the beam, q.sub.i(t) are the time-dependent generalized coordinates, and n is the number of degrees of freedom (DOF) to be considered. In our case, the mode shapes for a cantilever (CL) and clamped-clamped (CC) microbeam are well known and are of the form .sub.i(.sub.x)=cos h(.sub.i x)cos(.sub.i x).sub.i (sin h(.sub.i x)sin(.sub.i x)), where .sub.i are the square roots of the non-dimensional natural frequencies, and .sub.i are constants determined from the boundary conditions and mode to be considered. Values of .sub.i and .sub.i for the first four modes are obtained from [39]. Once the mode shapes are known, Eq. (1B) becomes a coupled set of ordinary differential equations (ODE) in time for q.sub.i(t). Multiplying through by .sub.k and integrating between 0 and 1 decouples the linear terms because of the orthogonality of the mode shapes and results in a set of ODEs, coupled through the nonlinear forcing terms. See, Pallay, Mark, Meysam Daeichin, and Shahrzad Towfighian. Dynamic behavior of an electrostatic MEMS resonator with repulsive actuation. Nonlinear Dynamics 89, no. 2 (2017): 1525-1538.

[0139] For a classic electrostatic beam with a parallel-plate electrode configuration, {circumflex over (f)}.sub.e can be represented analytically with an inverse polynomial of order two that has a singularity when the gap between electrodes goes to zero. However, in the system examined here, the simple analytical expression for the electrostatic force is not valid because of the different electrode arrangement, and the force must be calculated numerically. One way this can be achieved is to calculate the potential energy of the beam at each position, then take the derivative with respect to the direction of motion (). In this case, the potential energy is calculated from the voltage vector and capacitance matrix, the latter of which is simulated with a 2D finite-element analysis in COMSOL. The potential energy can be represented by,

[00005]$\begin{array}{cc}U=\frac{1}{2}\left[V_{\mathrm{side}}.Math.V_{\mathrm{bias}}.Math.V_{b.Math.e.Math.a.Math.m}\right]\left[\begin{array}{ccc}{\hat{c}}_{1.Math.1}& {\hat{c}}_{1.Math.2}& {\hat{c}}_{1.Math.2}\\ {\hat{c}}_{2.Math.1}& {\hat{c}}_{2.Math.2}& {\hat{c}}_{2.Math.3}\\ {\hat{c}}_{3.Math.1}& {\hat{c}}_{3.Math.2}& {\hat{c}}_{3.Math.3}\end{array}\right]\left[\begin{array}{c}{V}_{\mathrm{side}}\\ V_{\mathrm{bias}}\\ V_{b.Math.e.Math.a.Math.m}\end{array}\right]& \left(2\right)\end{array}$

[0140] where .sub.ij are the capacitances between each pair of electrodes, and V.sub.side, V.sub.bias, and V.sub.beam, are the voltages on the side electrodes (1), center electrode (2), and beam (3) respectively. The capacitance matrix is symmetric with c.sub.ij=c.sub.ji, which results in 6 independent capacitances in the system. Expanding Equation (2), setting V.sub.beam=0 because the beam is assumed to be the reference ground voltage level, and taking the derivative with respect to the transverse beam deflection, yields,

[00006]$\begin{array}{cc}{\hat{f}}_e\left(\hat{w},\stackrel{.fwdarw.}{V}\right)=\frac{1}{2}.Math.\left(V_{s.Math.i.Math.d.Math.e}^2.Math.\frac{c_{1.Math.1}}{\hat{w}}+2.Math.V_{s.Math.i.Math.d.Math.e}.Math.V_{b.Math.i.Math.a.Math.s}.Math.\frac{c_{1.Math.2}}{\hat{w}}+V_{b.Math.i.Math.a.Math.s}^2.Math.\frac{c_{2.Math.2}}{\hat{w}}\right)& \left(3\right)\end{array}$

[0141] In Equation (3), the derivatives of the COMSOL simulated capacitance, .sub.ij, can be estimated with a simple central difference method.

[00007]$\frac{{}_{{\hat{C}}_{1.Math.1}}}{\hat{w}}.Math..Math.\mathrm{and}.Math..Math.\frac{{}_{{\hat{C}}_{1.Math.2}}}{\hat{w}}$

can be fit with 9th order polynomials to create an analytical representation of the numerical data that can be used in Equation (1)

[00008]$\frac{{\hat{c}}_{2.Math.2}}{\hat{w}},$

however, would require upward of a 20th order polynomial for an adequate fit, and therefore requires more consideration regarding its representation in Equation (1).

[0142] FIGS. 13a-13c show the three capacitances from the finite element solution (COMSOL) and the electrostatic force at various side and bias voltages. From Equation (3), when the bias voltage is set to zero, the only remaining force component is

[00009]$\frac{{\hat{c}}_{1.Math.1}}{\hat{w}},$

which represents the electrostatic levitation force. Conversely, if the side voltage is set to zero, only

[00010]$\frac{{\hat{c}}_{2.Math.2}}{\hat{w}}$

remains and the system acts as a parallel-plate. By observing the c.sub.11 and c.sub.22 capacitances, it is clear that c.sub.22 has a much larger nominal capacitance and the magnitude of the slope is greater, especially at small gaps. This shows the parallel-plate force from the bias voltage is by far the dominant force component in this system. To counter this, much larger side voltages are needed to overcome the effect of the bias voltage. FIG. 13d shows the calculated force for several voltage cases. At large gaps, the force is dominated by the levitation force, but as the gap decreases, the total force becomes strongly negative as the behavior is dominated by the attractive parallel-plate force, despite the side voltage being an order of magnitude larger than the bias voltage.

TABLE-US-00002 TABLE I Beam parameters as shown in FIG. 12 Parameter Variable Value Beam Length L 505 m Beam Width b.sub.3 20.5 m Beam Thickness h.sub.3 2 m Beam Anchor Height D 2 m Side Electrode Gap G 20.75 m Middle Electrode Width b.sub.2 32 m Side Electrode Width b.sub.1 28 m Electrode Thickness h.sub.1 0.5 m Dimple Length L.sub.d 0.75 m Elastic Modulus E 160 GPa Density P 2330 kg/m.sup.3 Poisson's Ratio V 0.22

TABLE-US-00003 TABLE II Nondimensional substitutions and forcing coefficients for the governing differential equation Parameter Substitution x-direction position x = {circumflex over (x)}/L z-direction position w = /h.sub.3 Time t = {circumflex over (t)}/T Capacitance c.sub.22 = c.sub.22/c.sub.n Time constant T = {square root over (AL.sup.4/EI)} Damping Constant c* = /EIT Capacitance Constant ALh.sub.3.sup.2/L.sup.2T.sup.2 Force Constant 1 A.sub.i = .sub.ih.sub.3.sup.i1L.sup.4/2EI Force Constant 2 B.sub.i = .sub.ih.sub.3.sup.i1L.sup.4/EI

[0143] After the force is calculated, Equation (3) is plugged into Equation (1) yielding the dimensionalized equation of motion for the cantilever,

[00011]$\begin{array}{cc}.Math..Math.A.Math.\frac{{}^2.Math.\hat{w}}{{\hat{t}}^2}+\hat{c}.Math.\frac{\hat{w}}{d.Math.\hat{t}}+E.Math.I.Math.\frac{{}^4.Math.\hat{w}}{{\hat{x}}^4}=\frac{1}{2}.Math.V_{s.Math.i.Math.d.Math.e}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.{}_i.Math.{\hat{w}}^i+V_{s.Math.i.Math.d.Math.e}.Math.V_{b.Math.i.Math.a.Math.s}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.{}_i.Math.{\hat{w}}^i+\frac{1}{2}.Math.V_{b.Math.i.Math.a.Math.s}^2.Math.\frac{{\hat{c}}_{2.Math.2}}{\hat{w}}& \left(4\right)\end{array}$

[0144] where .sub.i and .sub.i are fitting coefficients for the c.sub.11 and c.sub.12 terms respectively. Equation (4) is then nondimensionalized with the substitutions shown in Table II, which yields the final nondimensional equation of motion shown in Equation (5).

[00012]$\begin{array}{cc}\frac{{}^2.Math.w}{t^2}+c^{*}.Math.\frac{w}{t}+\frac{{}^4.Math.w}{x^4}=V_{\mathrm{side}}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.A_i.Math.w^i+V_{\mathrm{side}}.Math.V_{\mathrm{bias}}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.B_i.Math.w^i+\frac{1}{2}.Math.V_{\mathrm{bias}}^2.Math.\frac{c_{2.Math.2}}{w}& \left(5\right)\end{array}$

[0145] When the time-derivative terms in Equation (5) are set to zero, it becomes a fourth-order ordinary differential equation, with x being the only independent variable. Despite the complicated forcing term, this equation can be solved directly as a boundary value problem. First, however, a form for

[00013]$\frac{{\hat{c}}_{2.Math.2}}{\hat{w}}$

must be determined. Because this force component behaves similarly to the parallel-plate force, a fitting function with a similar form can be used. Because this force component behaves similarly to the parallel-plate force, a fitting function with a similar form can be used.

[00014]$\frac{{\hat{c}}_{2.Math.2}}{\hat{w}}$

is fit with the function shown in Equation (6).

[00015]$\begin{array}{cc}\frac{c_{2.Math.2}}{w}=\frac{-0.9.Math.2.Math.5.Math.2}{{\left(w+\frac{d}{h_3}\right)}^{2.15}}& \left(6\right)\end{array}$

[0146] Plugging this into Equation (5) after dropping the time derivatives gives the static equation of motion that can be solved with the boundary value solver, bvp4c, in MATLAB.

[00016]$\begin{array}{cc}\frac{{}^4.Math.c_{2.Math.2}}{w^4}=V_{s.Math.i.Math.d.Math.e}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.A_i.Math.w^i+V_{s.Math.i.Math.d.Math.e}.Math.V_{b.Math.i.Math.a.Math.s}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.B_i.Math.w^i-V_{b.Math.i.Math.a.Math.s}^2.Math.\frac{0.4.Math.6.Math.2.Math.6}{{\left(w+\frac{d}{h_3}\right)}^{2.15}}& \left(7\right)\end{array}$

[0147] Cantilevers with the electrode arrangement in FIG. 12 may be fabricated with the PolyMUMPs service by MEMSCAP [58] to the dimensions given in Table I. An image of a fabricated beam is shown in FIG. 14. Beam tip displacement may be measured with a Polytec MSA-500 Laser vibrometer. A USB-6366 National Instruments Data Acquisition (DAQ) interfaced with MATLAB supplies the bias and side voltage and records the displacement data from the laser. Because the voltage limitation of the DAQ is +/10V, the side voltage is fed into a Krohn-Hite 7600 Wideband Power Amplifier and stepped up 20 to reach the desired 200V maximum voltage to the side electrodes. To measure the side voltage, the output of the amplifier is also fed into a resistive voltage divider to drop the voltage back down under 10V and is fed back into the DAQ. The actual voltage on the side electrodes can be calculated with the simvoltage to the center electrode and a quasi-static ramp function from 0V to approximately 190V to the side electrodes. The ramp duration is one second, which is four orders of magnitude longer than the natural frequency of the beam to ensure the beam response is quasi-static.

[0148] FIGS. 16a-16d show that the model agrees very well with the experiment, especially at higher bias voltages. To account for support compliance at the fixed end of the cantilever, a corrected length of 505 m is used. The tip displacement reaches almost 14 m at 190V, despite the anchor height of just 2 m. These large travel ranges are not possible in conventional parallel plate actuators and are a distinct advantage of electrostatic levitation. For 4V and 6V bias, the beam pulls in if the side voltage is too low. For example, the beam pulled in below a side voltage of approximately 50V for 4V bias, and 80V for a 6V bias. However, when pull-in occurs, the side voltage can be ramped up to release the beam from its pulled-in position, overcoming both the bias voltage and stiction forces [56]. The model could reach slightly lower voltages before becoming unstable, though this is most likely because the model does not account for the dimples on the underside of the beam. The dimples create an effectively smaller gap between the beam and center electrode, thus increasing the effect of the force from bias voltage, which dominates when the gap is small.

[0149] Another interesting result is the linear relationship between tip deflection and side voltage, especially at higher bias voltages. The slope is approximately 90 nm per volt on the side electrodes. This linear relationship is maintained for more than 10 m of static displacement. (Because high voltages cannot be applied in devices created using the PolyMUMPs fabrication procedure, 190V was not exceeded.) The tip displacement shows no signs of saturating near 190V and the linear voltage relationship most likely extends much further. When there is no bias voltage, an approximately linear relationship exists for deflections above around 4 m. However, when 4V bias is added to the center electrode, the linearity extends all the way down to 0 m. If the bias voltage is too large, such as at 6V, pull-in will occur at small gaps. Therefore, there exists an optimal bias voltage, 4V for the dimensions shown in Table I, that yields a linear response over the entire static range.

[0150] The reason for the linearization between side voltage and tip displacement can be seen by analyzing the force and its associated components. FIGS. 17a-17d shows the nondimensional forcing components at 6V bias as a function of side voltage when evaluated at the static equilibrium points in FIG. 16d. As shown in FIG. 17a, the total force, which is the sum of each forcing component in FIGS. 17b-d, scales linearly with side voltage. Despite Equation (3) showing the force scaling with the square of the side voltage, when considering the increasing gap, the relationship turns out to be linear. Because the force is now effectively a linear function of side voltage, the solution to Equation (7) will also be a linear function of side voltage.

[0151] The reason for the linearization of the force is different when at larger voltages as opposed to smaller voltages. When the side voltage is above 80V, the c.sub.11 and c.sub.22 are negligible compared to the c.sub.11 component. In this case, the system acts like as if there was just a pure levitation force with no bias voltage. When the side voltage is increased, the upward electrostatic force on the beam also increases. If the beam was completely rigid, this force would scale with the square of the side voltage. But the beam is not rigid and deflects upwards when the side voltage is increased. This deflection increases the gap between the beam and fixed electrodes, which in turn causes the upward electrostatic force to decrease. When the side voltage is large, the decrease in the upward electrostatic force from increasing the gap counteracts the nonlinearly increasing force with voltage and the result is an effectively linear scaling of the total force with side voltage. This can be seen in the static position in FIG. 16a, which has no bias voltage, but still shows the linear increase with side voltage. However, when the voltage drops below 80V, FIG. 16a shows a nonlinear relationship with side voltage. This is because the levitation force is inherently weak, and unless the voltage is very large, the force is too low to deflect the beam enough for the increasing gap to counteract the scaling of the force with the square of the side voltage.

[0152] To achieve linearity at low side voltages, a bias voltage on the center electrode must be used. Comparing the upward levitation (c.sub.11) and downward parallel-plate (c.sub.22) components of the force in FIGS. 17b and d shows why. As the side voltage decreases, the levitation c.sub.11 component starts to flatten out around zero. This decrease in total force brings the beam closer to the center electrode and increases the influence of the bias voltage, which pulls the beam downward. The parallel-plate c.sub.22 component is inversely related to the gap (Equation (6)) and therefore the downward force gets stronger as the gap decreases. When combining the forcing terms, the downward bending of the force from the parallel-plate component counteracts the flattening of the force from the levitation component. Because the parallel-plate force scales with the square of the bias voltage, by choosing the appropriate bias, 4V, the two nonlinear effects cancel each other and the result is a nearly linear force that decreases all the way to zero. If the bias voltage is too large the downward bending from the parallel-plate force overcomes the flattening of the levitation force and the system becomes nonlinear in the negative direction before the force can reach zero and the beam becomes unstable. This is the case for 6V bias, where the beam pulls in during the experiment before it can return to its undeflected position. The linearity with side voltage is extraordinarily useful for any actuator that is negatively affected by nonlinearity and requires a large actuating range. Possible applications include micro-mirrors and micro-fluidic devices.

[0153] Natural frequency is an important property of oscillators and sensors that determines their resolution and sensitivity. To calculate the natural frequency, we must return to the dynamic partial differential equation from Equation (5). Now w becomes a function of time and position so it must be reduced to an ordinary differential equation in time before we can calculate the natural frequency. This can be achieved by performing Galerkin's method, which is a modal analysis using separation of variables and integration over x to get a set of second-order ordinary differential equations that are coupled through nonlinear terms. If just a single mode is to be considered, this will yield a single ordinary differential equation that can be used to calculate the natural frequency.

[0154] First, the transverse deflection of the beam is assumed to have separate spatial and time-dependent functions as shown in Equation (8).

w(x,t)=(x)q(t)(8)

[0155] The mode shapes, (x), can be solved from the eigenvalue problem, which depends on the boundary conditions. The mode shapes for a simple cantilever are well known and can be expressed by,

.sub.i(x)=cos h(.sub.ix)cos(.sub.ix).sub.i(sin h(.sub.ix)sin(.sub.ix))(9)

[0156] where .sub.i is the square root of the i.sup.th nondimensional natural frequency, and .sub.i is a constant. For the first mode, .sub.1 (herein referred to as ), .sub.i and .sub.1 are 0.7341 and 1.875 respectively. However, to account for fabrication imperfections the natural frequency may be identified from an experiment and nondimensionalized to use with Equation (9).

[0157] After separation of variables, the governing equation is multiplied by and integrated over x to yield Equation (10), which is a second-order ordinary differential equation that depends only on the time-dependent component of the solution, q,

[00017]$\begin{array}{cc}m.Math.\frac{d^2.Math.q}{d.Math.t^2}+c.Math.\frac{d.Math.q}{d.Math.t}+k.Math.q=V_{\mathrm{side}}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.A_i.Math.{}_i.Math.q^i+V_{\mathrm{side}}.Math.V_{\mathrm{bias}}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.B_i.Math.{}_i.Math.q^i+\frac{1}{2}.Math.V_{\mathrm{bias}}^2.Math.{}_0^1.Math..Math.\frac{c_{2.Math.2}}{\left(.Math..Math.q\right)}.Math.\mathrm{dx}.Math..Math..Math.\mathrm{where}& \left(10\right)\\ .Math.\begin{array}{cc}m={}_0^1.Math.{}^2.Math.\mathrm{dx}& c=c*m\\ k={}_1^4.Math.m& {}_i={}_0^1.Math.{}^{i+1}.Math.\mathrm{dx}\end{array}& \left(11\right)\end{array}$

[0158] The last term in Equation (10) can be problematic if it is not dealt with appropriately. For the static problem, this term could be fitted with the function given in Equation (6). However, the integration on x from the modal analysis will create a singularity at x=0, which will make the differential equation impossible to solve. A standard parallel-plate has a similar problem; however, it can be mitigated by multiplying the entire equation by the denominator, expanding the polynomial, and then performing Galerkin's method. In this case, two complications prevent that from working. First, the order of the polynomial in the denominator of Equation (6) is not an integer, so it cannot be expanded. Second, multiplying the two 9th order polynomials by a noninteger polynomial will be extraordinarily difficult and tedious later in the problem when solving for the natural frequency. Therefore, this term can be treated numerically as a list of numbers, where linear interpolation is used to estimate the values between data points. This is more computationally expensive, but computation time is not an issue for the natural frequency calculation. Before that can be done, the integration on x must be performed.

[0159] The derivative,

[00018]$\frac{c_{2.Math.2}}{\left(.Math..Math.q\right)}$

can be rewritten using the relationship,

[00019]$\begin{array}{cc}\frac{c_{2.Math.2}}{q}=\frac{c_{2.Math.2}}{w}.Math.\frac{w}{q}=\frac{c_{2.Math.2}}{\left(.Math..Math.q\right)}.Math.\frac{\left(.Math..Math.q\right)}{q}=\frac{c_{2.Math.2}}{\left(.Math..Math.q\right)}.Math..Math.\frac{q}{q}.Math..Math.\mathrm{which}.Math..Math.\mathrm{yields},& \left(12\right)\\ \frac{c_{2.Math.2}}{\left(.Math..Math.q\right)}=\frac{1}{}.Math.\frac{c_{2.Math.2}}{q}& \left(13\right)\end{array}$

[0160] When substituting Equation (13) into Equation (10), the s cancel and the integration becomes 1. Now that x has been integrated out of the equation, q represents the deflection of the beam instead of w. The derivative of the numerical c.sub.22 data from COMSOL with respect to the deflection can be substituted in for

[00020]$\frac{c_{2.Math.2}}{q}.$

The final equation becomes,

[00021]$\begin{array}{cc}m.Math.\frac{d^2.Math.q}{d.Math.t^2}+c.Math.\frac{d.Math.q}{d.Math.t}+k.Math.t=V_{s.Math.i.Math.d.Math.e}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.A_j.Math.{}_i.Math.q^i+V_{s.Math.i.Math.d.Math.e}.Math.V_{b.Math.i.Math.a.Math.s}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.B_i.Math.{}_i.Math.q^i+\frac{1}{2}.Math.V_{b.Math.i.Math.a.Math.s}^2.Math.\frac{c_{2.Math.2}}{q}& \left(14\right)\end{array}$

[0161] To find the natural frequency, the eigenvalues of the Jacobian matrix can be calculated by decomposing Equation (14) to two first-order ODEs with y.sub.1=q and

[00022]$y_2=\frac{d.Math.q}{d.Math.t},$

setting the damping coefficient to zero, and plugging the equations into the relationship,

[00023]$\begin{array}{cc}J=\left[\begin{array}{cc}\frac{{\stackrel{.}{y}}_1}{y_1}& \frac{{\stackrel{.}{y}}_1}{y_2}\\ \frac{{\stackrel{.}{y}}_2}{y_1}& \frac{{\stackrel{.}{y}}_2}{y_2}\end{array}\right]& \left(15\right)\end{array}$

[0162] which is the Jacobian matrix. The eigenvalues of Equation (15) are,

[00024]$\begin{array}{cc}=\sqrt{\frac{V_{s.Math.i.Math.d.Math.e}^2}{m}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.i.Math.A_i.Math.{}_i.Math.y_1^{i-1}+\frac{V_{s.Math.i.Math.d.Math.e}.Math.V_{b.Math.i.Math.a.Math.s}}{m}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.i.Math.A_i.Math.{}_i.Math.y_1^{i-1}+\frac{1}{2.Math.m}.Math.V_{b.Math.i.Math.a.Math.s}^2.Math.\frac{{}^2.Math.c_{2.Math.2}}{y_1^2}-{}_1^4}& \left(16\right)\end{array}$

[0163] where the natural frequency is the imaginary part of the eigenvalue. The only variable that is unknown in this equation is y.sub.1, which is the beam deflection. The value of y.sub.1 at each voltage level can be determined from the static model results at the given side and bias voltage, after which the natural frequency can be calculated.

[0164] In order to experimentally measure the values, white noise was superimposed on the DC side voltage while holding the bias voltage constant, and the beam tip velocity was recorded with the laser vibrometer. A fast-Fourier transform (FFT) was performed on the velocity signal to extract the natural frequency. Unlike the static case, the side voltage was not swept up continuously to 190V. Instead, the side and bias voltage were kept constant and the natural frequency was extracted at 10-volt intervals. FIGS. 18a-18d show the natural frequency results for the experiment and the model.

[0165] The model agrees well with the natural frequency experiment. The square root of the natural frequency, .sub.1, was 1.7998. The curling of the beam tip in the fabricated beam needs to be incorporated into the model to obtain a better fit with the experiment. Because the beams are very long, residual stress from fabrication causes them to curl upward when released. This increases the gap between the beam and center electrodes and reduces the total electrostatic force. In this case, the beam tip was curled up approximately 2 m out of plane. Ideally, this would be incorporated into both q and the mode shapes. However, the easiest way of addressing this problem is to simply add 2 m of deflection to q.

[0166] When no bias voltage is applied, FIGS. 18a-18d shows that the natural frequency is relatively constant for side voltages under approximately 50V. However, at higher bias voltages the natural frequency starts to bend down at low side voltages until the system becomes unstable. This agrees with the initial assessment that the bias voltage is dominant at low gaps because a parallel-plate electrostatic force is correlated with a decrease in natural frequency, while electrostatic levitation increases the natural frequency with higher voltage. In the case of 4V and 6V bias, the natural frequency can be tuned from less than 6 kHz up to almost 12 kHz. This is an increase of almost 100%, which gives the beam a tremendous amount of tunability to control where the first resonant peak is located. Many devices, such as sensors, operate near their natural frequency to boost the signal-to-noise ratio. This electrostatic tunability enables accounting for any fabrication imperfection or environment operation changes to ensure a large signal-to-noise ratio and resolution.

[0167] The system can therefore be tuned to capture a wide range of frequencies, which is highly desirable in many sensors.

[0168] At an optimum value of the bias voltage, the highest linearity of natural frequency and the side voltage is achieved. If the bias voltage is set to approximately 2V, the downward bending of the natural frequency curve at low side voltage reaches a point where it matches the slope at higher side voltages. This creates a linear shift in natural frequency with side voltage. Unlike the static solution, the natural frequency is related to the slope of the force at the equilibrium position instead of the magnitude. The slope of the force is the linear stiffness, which is related to the natural frequency through the relationship,

[00025]$\begin{array}{cc}{}_n=\sqrt{\frac{k_{\mathrm{eff}}}{m}}& \left(17\right)\end{array}$

[0169] where .sub.n is the natural frequency, m is the mass, and k.sub.eff is the effective stiffness, which is the sum of the mechanical stiffness of the beam (k.sub.mech) and the added stiffness from the electrostatic force (k.sub.elec), where k.sub.elec is much larger than k.sub.mech. Because m and k.sub.elec are constant, k.sub.elec must be related to the side voltage. If .sub.n is linearly related to side voltage, then k.sub.elec must be related to the square of the side voltage. This is confirmed to be true by analyzing the slope of the force at each static equilibrium position. The linear relationship in frequency is approximately 16 Hz per volt and is maintained for more than 2.9 kHz of frequency shift. The combination of parallel-plate and electrostatic levitation enables a wider range of capabilities that is not possible with each mechanism on its own.

[0170] To obtain the model frequency response, the dynamic equation of motion must be solved. Much of the procedure for calculating the natural frequency of Equation (5) can be used for the frequency response as well. Equation (14) can be integrated directly using linear interpolation to determine

[00026]$\frac{c_{2.Math.2}}{q}.$

However, the computation expense of using linear interpolation becomes problematic and creates very long solve times. To reduce the integration time,

[00027]$\frac{c_{2.Math.2}}{q}$

can be fit with the inverse non-integer polynomial in Equation (6) with q replacing w. This is not problematic, because the substitution is occurring after the modal analysis, which is where the problems were occurring initially. This significantly reduces the computation time so that the frequency response can be calculated. The final dynamic equation is given in Equation (18).

[00028]$\begin{array}{cc}m.Math.\frac{d^2.Math.q}{d.Math.t^2}+c.Math.\frac{d.Math.q}{d.Math.t}+k.Math.q=V_{s.Math.i.Math.d.Math.e}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.A_i.Math.{}_i.Math.q^i+V_{s.Math.i.Math.d.Math.e}.Math.V_{b.Math.i.Math.a.Math.s}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.B_i.Math.{}_i.Math.q^i-V_{b.Math.i.Math.a.Math.s}^2.Math.\frac{0.4.Math.6.Math.2.Math.6}{{\left(q+\frac{d}{h_3}\right)}^{2.1.Math.5}}& \left(18\right)\end{array}$

[0171] The damping coefficient is estimated from the experiment, which was conducted at approximately 300 mTorr, which correlates to a quality factor of 500. The quality factor is used in Equation (18) by the relationship c=.sub.1.sup.2/Q.

[0172] To conduct the experiment, the side voltage is given an AC voltage on top of the DC offset. The DAQ supplies the AC voltage, and the DC offset function of the Krohn-Hite amplifier applies the DC side voltage. The bias was held constant at the same 0V, 2V, 4V, and 6V values used in the static and natural frequency tests. For the frequency response, the tip velocity was measured with the vibrometer. FIGS. 19a-19d show the frequency response around the first natural frequency at 0V, 2V, 4V, and 6V bias, with 170 VDC and 0.5 VAC on the side electrodes.

[0173] As with the static and natural frequency results, the frequency response model agrees with the experiment. In the experiment, the backsweep branch continues to grow until it starts tapping on the center electrode and falls back down to the lower branch. If the bias voltage is high enough, instead of falling back down to the lower branch, it pulls in, as is the case at 6V bias. Before pull-in occurs, the backsweep branch becomes very flat and increases in size. This extension of higher branch oscillation is because of the quadratic nonlinearity that causes excessive softening from electrostatic levitation and parallel-plate schemes. With no bias, the backsweep branch ends around 11 kHz, but with 6V bias the branch travels back to 10 kHz before pulling in. The high amplitude vibration could be useful for the development of high-resolution oscillators. The triggering of dynamic pull-in near the resonant peak can help create combined systems that act as a sensor and a switch, which triggers when the sensed quantity increases past a threshold. The levitation force can be used to release the pulled-in structure and reuse the device.

[0174] To get more insight into the motion of the beam, the model displacement was calculated and is shown in FIG. 20. This dynamic displacement reaches upward of 28 m with no bias voltage. Increasing the bias bends the peak down and to the left, which drops the peak amplitude by about 3 m for 6V bias. Unlike static and natural frequency, the addition of the bias voltage always increases the nonlinearity of the system. This is because levitation and parallel-plate electrostatic forces create softening in the frequency response, therefore the combination of the two yields more softening. Therefore, MEMS oscillators and resonators that require a linear frequency response would not particularly benefit from this method.

[0175] However, other applications that are less hindered by the nonlinear frequency response may be able to exploit other behaviors of the combined system, such as dynamic pull-in shown in FIG. 19d. One possible application could be a pressure sensor that triggers dynamic pull-in at the resonant peak when the pressure drops below a threshold value. As reported in [55], the resonance region of the levitation actuator is heavily dependent on the ambient pressure because the oscillation amplitude is very large and squeeze film damping effects are significant. The voltage at which dynamic pull-in occurs could be related to the ambient pressure to create a threshold pressure sensor that utilized the switching mechanism proposed in [57].

[0176] The relationship between static tip displacement and natural frequency with side voltage can be linearized by choosing an appropriate bias voltage, which can greatly simplify and minimize the electronic circuitry for processing information from sensors and controlling actuator motion. The natural frequency for the design according to Table I can be tuned from 6 kHz up to 12 kHz, increasing tunability, which gives sensors a larger spectrum of detectable frequencies with large signal-to-noise ratios. The softening branch of the frequency response can be flattened and extended by adding a larger bias voltage. The larger bias also triggers dynamic pull-in near the resonant peak, which permits use of the technology for combined sensors and switches that use pull-in to close a circuit. The levitation force enables the system to release from its pulled-in position, allowing the combined sensor and switch system to be reused many times without failure.

Example 2Accelerometer Embodiment

[0177] MEMS sensors and actuators have replaced their macro counterparts in many applications including, for example, pressure sensors, inertial navigation systems, and adaptive optic systems [61-63]. They have enabled consumer electronic products like smartphones, laptops, virtual reality headsets, and health monitoring wearable devices to deliver more enriched functionality. Their reliable performance, low cost, low power consumption, and more importantly, their compatibility with semiconductor fabrication technology have made MEMS technology popular.

[0178] Electrostatic actuation and detection is the most common transduction method in MEMS [64], and almost all the current commercially available electrostatic MEMS devices are based on parallel-plate [65] or interdigitated comb drive configurations [66]. In a parallel-plate configuration, the electrostatic force attracts a movable electrode to the substrate or bottom electrode. One can apply a specific voltage to move an electrode to a desired position in the case of an actuator (like micromirrors), or exert an excitation and sense change in the capacitance between two electrodes in the case of a sensor (like accelerometers).

[0179] The main drawback of this electrostatic force is pull-in instability. Pull-in happens when the restoring force of the microstructure is no longer able to overcome the attractive electrostatic force and as a result, the microstructure collapses to its substrate. Capacitive MEMS transducers are designed to operate far from the pull-in region and this limits the device operation range; for example, it limits the travel range in micro-mirrors [67]. This issue is a significant constraint that needs to be considered in designing the microstructure geometry. Particularly, the initial gap between the movable electrode and the substrate is affected by this design limitation. This initial gap plays a crucial role in squeeze film damping [64] and the magnitude of the electrostatic force. As a result, it has a significant effect on the device sensitivity and performance in general.

[0180] As a new design paradigm, the repulsive capacitive sensing configuration is an alternative to parallel-plate or comb-drive configurations [68-70]. This approach takes advantage of the fringe electrostatic field to generate a repulsive force that pushes the movable structure away from the substrate, which essentially eliminates the possibility of pull-in (FIGS. 21a-21f). This design approach opens a new door to build MEMS sensors and actuators without the limitation on performance imposed by dynamical instability from pull-in. Furthermore, because the movable electrode has the same voltage as the substrate (grounded), the possibility of micro-welding is eliminated. This failure mode happens when two microstructures with different voltages contact each other [71].

[0181] Lee et al. first introduced the repulsive approach [72]. A new generation of repulsive actuators was developed by He and Ben Mrad [73,74,68,69]. The nonlinear dynamics of microstructures under a repulsive force has also been investigated [75-77].

[0182] The repulsive approach is exploited to build a capacitive accelerometer. There are numerous applications for accelerometers. They are main components of inertial navigation systems used in cars, airplanes, smartphones, etc. For example, they are used to deploy airbags in automobiles in an accident [78] or, as a part of wearable health monitoring technologies, to detect the sudden fall of elderly people [79] to provide them with necessary medical attention as soon as possible. For a comprehensive study on accelerometer sensors based on the parallel-plate scheme, see [80].

[0183] The electrostatic force on the moving element in the system examined here differs considerably from that found in conventional parallel-plate electrode designs and cannot be estimated using simple analytical expressions. This, along with the large amplitude vibration of the proof-mass, makes the dynamics of the system fully nonlinear and complicated.

[0184] The general schematic of the structure under base excitation is shown in FIGS. 21a-21f. There are four elements that comprise this design. The first element is the square plate that is attached to its anchors through two torsional beams. The two beams undergo torsional deflection as the proof mass rotates because of applied acceleration and electrostatic forces. The second part is the 200 m long micro-beams that are attached to three sides of the proof mass. The proof-mass and fingers attached to it are all electrically grounded. Under these fingers, there are sets of grounded fingers fixed on the substrate as the third element of the sensor. The fourth part is the sets of electrodes adjacent to the grounded fingers. These fingers carry electrical voltage, the electric field of which exerts electrostatic forces on the beams connected to the proof mass.

[0185] The dimensions of all these beams and their electrostatic field are illustrated in FIGS. 21a-21f. The fringe electrostatic field exerts different forces on the top and bottom surfaces of the moving fingers. The resultant force is an upward force that pushes the fingers and the proof mass away from the substrate. Beyond a certain distance, the electrostatic force becomes attractive and tends to bring the microstructure back into the repulsive region. Therefore, there is an equilibrium point where the electrostatic force on the structure becomes zero. Once a voltage is applied on the side electrodes, the accelerometer plate rotates away from the substrate to an equilibrium position where the electrostatic force is equal to the restoring force of the torsional springs. This equilibrium point can be tuned by changing the voltage on the side electrodes. As will be discussed below, this is advantageous when we want to set the threshold acceleration that the sensor can detect.

[0186] When the base of the plate accelerates, the plate tends to stay at rest because of its inertia and this leads to a relative displacement between the base and the plate. Because of this displacement, the capacitance between the electrodes will change as well. This change in capacitance can be measured and related to the relative motion of the plate. If the load is large enough, the fingers can hit the bottom electrodes. However, because of the repulsive nature of the force, they will bounce back and will remain free to move. This collision can be detected with the aid of an electrical circuit.

[0187] A simple single degree of freedom model [76] may be used as a very powerful tool to predict the response of the system and capture the inherent nonlinearity caused by the electrostatic force. A lumped parameter oscillator is considered:

I{umlaut over ({circumflex over ()})}+c{dot over ({circumflex over ()})}+k{circumflex over ()}=M.sub.es({circumflex over ()})+M.sub.sh(19)

[0188] in which {circumflex over ()} is the rotation of the proof-mass plate about its anchors' axis, I, c and k are the moment of inertia, torsional damping, and torsional stiffness coefficients, respectively. On the right-hand side, M.sub.es and M.sub.sh represent the electrostatic moment and inertial moment because of the base excitation, respectively.

[0189] The electrostatic force has a nonlinear dependence on the position of the moving fingers in the electric field. To obtain an expression for the electrostatic moment, first we calculate the force profile for a unit-cell using a 2D finite element analysis in COMSOL. Such a unit cell consists of the moving electrode, bottom electrode and two side electrodes as shown in FIG. 1. The force profile is shown in FIG. 22. Using a curve fitting technique, the force profile (FIG. 22) can be approximated as a 9th-degree polynomial, in terms of the deflection, z.

[00029]$\begin{array}{cc}{F}_{e.Math.s}=V_{d.Math.c}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.\left(A_i.Math.z^i\right)& \left(20\right)\end{array}$

[0190] where z is the gap between the bottom electrode and moving fingers in each cross section. In order to write the electrostatic force as a function of , we use the following trigonometric equation.

[00030]$\begin{array}{cc}\sin .Math..Math.=\frac{z}{x}& \left(21\right)\end{array}$

[0191] which yields

[00031]$\begin{array}{cc}{F}_{\mathrm{es}.Math.\text{-}.Math.\mathrm{mov}.Math.\text{-}.Math.\mathrm{fing}}\left(x,\right)=V_{d.Math..Math.c}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.{A_i\left(x.Math.\sin .Math..Math.\right)}^i& \left(22\right)\end{array}$

[0192] where z is the gap between the bottom electrode and moving electrode. In order to write the electrostatic force as a function of , we use the following trigonometric equation.

TABLE-US-00004 TABLE III Dimensions for the MEMS accelerometer in FIGS. 21a-21f Parameter Symbol Value Unit Density 2330 kg/m.sup.3 Proof mass length L.sub.s 990 Mm Proof mass width L.sub.w 1000 Mm Finger length L.sub.b 100 Mm Number tip fingers N.sub.t 31 Mm Number side fingers N.sub.s 25 m Voltage fixed finger d.sub.1 8 m width Gap between fixed d.sub.1 8 m fingers Moving finger width d.sub.2 4 m Ground fixed finger d.sub.1 8 m width Electrodes and proof t.sub.1 2 m mass thickness Initial gap h.sub.1 2 m Natural frequency .sub.n 2 1740 rad/s Damping ratio .sub.0 0.016 0 (Vdc = 40) Damping ratio .sub.1 0.033 (Vdc = 40) Damping ratio .sub.0 0.016 1 (Vdc = 50) Damping ratio .sub.1 0.038 (Vdc = 40) Damping ratio .sub.0 0.017 0 (Vdc = 60) Damping ratio .sub.1 0.055 (Vdc = 60) Force constant A.sub.0 3.4079 10.sup.7 N/m Force constant A.sub.1 6.7113 10.sup.2 N/m.sup.2 Force constant A.sub.2 7.5644 10.sup.3 N/m.sup.3 Force constant A.sub.3 8.8555 10.sup.8 N/m.sup.4 Force constant A.sub.4 8.1341 10.sup.13 N/m.sup.5 Force constant A.sub.5 4.6766 10.sup.18 N/m.sup.6 Force constant A.sub.6 1.6139 10.sup.23 N/m.sup.7 Force constant A.sub.7 3.2602 10.sup.27 N/m.sup.8 Force constant A.sub.8 3.5560 10.sup.31 N/m.sup.9 Force constant A.sub.9 1.6175 10.sup.35 N/m.sup.10 Moment of inertia I 1.7954 10.sup.15 kg m.sup.2

[0193] Integrating Eq. (22) over the beam (finger) length results in the total electrostatic force on each beam.

[00032]$\begin{array}{cc}.Math.F_{\mathrm{es}.Math.\text{-}.Math.\mathrm{tip}}\left(\right)={}_{L_s}^{L_s+L_b}.Math.F_{\mathrm{es}.Math.\text{-}.Math.\mathrm{mov}.Math.\text{-}.Math.\mathrm{fing}}.Math.\mathrm{dx}.Math..Math.{}_{L_s}^{L_s+L_b}.Math.V_{d.Math..Math.c}^2\left(\underset{i=0}{\overset{9}{.Math.}}.Math.{A_i\left(x.Math.\sin .Math..Math.\right)}^i\right).Math.\mathrm{dx}=V_{d.Math..Math.c}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.\frac{{A_i\left(x.Math.\sin .Math..Math.\right)}^i}{i+1}.Math.\left({\left(L_s+L_b\right)}^{i+1}-L_s^{i+1}\right)& \left(23\right)\end{array}$

[0194] where the parameters are given in Table III. To calculate the corresponding moment caused by this force, the distance between the acting point of the resultant force and the axis of rotation (x.sub.c) is needed, which can be calculated as follows:

[00033]$\begin{array}{cc}{x}_c=\frac{{}_{L_s}^{L_s+L_b}.Math.x.Math.F_{\mathrm{es}.Math.\text{-}.Math.\mathrm{fing}}.Math.d.Math.x}{{}_{L_s}^{L_s+L_b}.Math.F_{\mathrm{es}.Math.\text{-}.Math.\mathrm{fing}}.Math.\mathrm{dx}}& \left(24\right)\end{array}$

[0195] Therefore, the moment of electrostatic force on the tip fingers about the axis of rotation can be written as

[00034]$\begin{array}{cc}{M}_{\mathrm{es}-\mathrm{tip}}=N_t.Math.F_{\mathrm{es}.Math.\text{-}.Math.\mathrm{tip}}\left(\right).Math.x_c=N_t.Math.V_{d.Math..Math.c}^2.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.\frac{{A_i\left(\sin .Math..Math.\right)}^i}{i+2}.Math.\left({\left(L_s+L_b\right)}^{i+2}-L_s^{i+2}\right)& \left(25\right)\end{array}$

[0196] where N.sub.t is the number of fingers on the tip side of the proof mass. Following a similar approach and assuming that the electrostatic force on the fingers on the sides of the proof-mass does not change over the finger length, we can calculate the moment of the electrostatic force on these fingers as follows:

[00035]$\begin{array}{cc}{M}_{\mathrm{es}.Math.\text{-}.Math.s.Math.i.Math.d.Math.e}=V_{d.Math.c}^2.Math.\underset{j=0}{\overset{N_s}{.Math.}}.Math.\underset{i=0}{\overset{9}{.Math.}}.Math.\left({A_i\left(u_j.Math.\sin .Math..Math.\right)}^i.Math.L_b\right).Math.u_j& \left(26\right)\end{array}$

[0197] where u.sub.j is the distance of the j.sup.th side finger from the axis of rotation and N.sub.s is the number of fingers on each side. So, the total electrostatic moment in Eq. (19) can be written as:

M.sub.es({circumflex over ()})M.sub.es-tip+2.Math.M.sub.es-side(27)

[0198] The factor 2 in Eq. (27) accounts for the fact that there are two sets of side fingers, one on each side of the proof-mass. The moment from acceleration (M.sub.sh) also can be written as:

[00036]$\begin{array}{cc}{M}_{s.Math.h}=\left[m_p.Math.a\left(t\right).Math.\frac{L_s}{2}\right]+\left[N_t.Math.m_b.Math.a\left(t\right).Math.\left(L_s+\frac{L_b}{2}\right)\right]+\left[2.Math.\left(\underset{j=1}{\overset{N_s}{.Math.}}.Math.m_b.Math.a\left(t\right).Math.u_j\right)\right]& \left(28\right)\end{array}$

[0199] where the first bracket is the moment caused by proof mass acceleration, the second shows the moment caused by the tip fingers acceleration, and the third is for the side fingers. In Eq. (28), m.sub.p, m.sub.b, and a(t) are the mass of the rotational plate, mass of the moving electrode, and the acceleration load that is exerted on the microstructure through base excitation. This mechanical load can be modeled as a harmonic base excitation or as a shock load. There are different shock profiles commonly used for modeling shock loads such as square, saw-tooth, half-sine or full-sine shock profiles. The most commonly used shock load is the half-sine with the profile given in Eq. (29), which will be used in the following description of the experiment on the fabricated device,

[00037]$\begin{array}{cc}a\left(t\right)=\mathrm{Asin}\left(.Math..Math.t\right).Math.u\left(\frac{}{}-t\right)& \left(29\right)\end{array}$

[0200] where

[00038]$u\left(\frac{}{}-t\right)$

is the step function. Dividing Eq. (19) by I and using Eqs. (27) and (28) and following non-dimensional parameters (Eq. (30), we can rewrite the governing equation of motion in nondimensional format (Eq. (31).

[00039]$\begin{array}{cc}=\frac{\hat{}}{{}_0},t=\frac{\hat{t}}{T},T=\frac{1}{{}_n},{}_n=\sqrt{\frac{k}{I}}& \left(30\right)\\ \stackrel{\mathrm{.Math.}}{}+2.Math..Math.{}_n.Math.\stackrel{.}{}+{}_n^2.Math.=\frac{T^2}{I.Math.{}_0}.Math.\left(M_{e.Math.s}\left(\right)+M_{s.Math.h}\right)& \left(31\right)\end{array}$

[0201] By applying a DC voltage on the side electrodes, the proof-mass rotates around its anchors' axis and goes to an equilibrium point away from the substrate. To obtain this static rotation, all the time derivative terms in Eq. (31) are set to zero and then the equation is solved for static rotation (.sub.st). The static rotation can be used to solve for the dynamic solution (.sub.dyn) of Eq. (31) in the presence of any time-varying load using the following change of variable.

=.sub.st+.sub.dyn(32)

[0202] The moment of inertia, I, in Eq. (31) can be calculated as

[00040]$\begin{array}{cc}I=\frac{1}{3}.Math.m_p.Math.L_s^2+\mathrm{Nt}.Math.\left(\frac{1}{1.Math.2}.Math.m_b.Math.L_b^2+{m_b\left(\frac{L_b}{2}+L_s\right)}^2\right)+\underset{j=1}{\overset{N_s}{.Math.}}.Math.\left(\frac{1}{1.Math.2}.Math.m_b.Math.b^2+m_b.Math.u_j^2\right)& \left(33\right)\end{array}$

[0203] The natural frequency of the first mode, .sub.n is obtained from the ANSYS finite element package. The first mode is a rotational mode as expected. The corresponding natural frequency for this mode is predicted to be 1320 Hz. After examining subsequent experimental data, this value can be modified for the natural frequency to account for all the fabrication imperfections, residual stresses and mathematical simplification in this 1 DOF model.

[0204] Eq. (31) can be solved numerically using the Runge-Kutta method for various damping conditions. However, it is likely that a simple linear damping does not capture the physics of energy dissipation in this problem. In general, modeling energy dissipation in the repulsive scheme is more challenging compared to the attractive scheme because of two reasons. First, the amplitude of vibration can get very large compared to the initial fabrication gap because the initial static deflection of the structure (from V.sub.dc) moves the structure away from the substrate, providing more space for vibration. Therefore, those damping models available in the literature [81-83] that are based on Reynolds' equation are not valid because the underlying assumptions such as negligible pressure change across the fluid film or small gap to lateral dimension ratio are not valid here. Second, because there is no limitation from pull-in, the device can get very close to the substrate while having a large amplitude vibration. So, in each cycle of vibration, when the moving part is far away from the substrate the dominant damping mechanism is the drag force in free air. However, when the structure gets close to the substrate, the squeeze film damping mechanism starts to play a significant role in energy dissipation. Thus, the damping force is likely to be quite nonlinear.

TABLE-US-00005 TABLE IV Air flow regimes for characteristic length H = 2 m. Pressure (Torr) Regime P > 2200 Continuum 1000 < P < 2200 Slip flow 10 < P < 1000 Transient P < 10 Molecular

[0205] Another challenge in modeling damping in this problem is that the continuity assumption for the air breaks down when the microstructure gets very close to the substrate, especially at low pressures. In this situation, the characteristic length of the problem is even smaller than the initial fabrication gap, and the characteristic length of the problem becomes comparable with the mean free path of air molecules. The Knudson number is the parameter to check to see if the continuity assumption is valid (Eq. (34)). This number is defined as the ratio of the mean free path of air (gas) molecules to the characteristic length of the problem, which is usually the gap between moving and stationary parts of the microstructure. The mean free path of air molecules depends on the air temperature and pressure. Assuming the room temperature condition (25 C.), the dependence of the molecules' mean free path on pressure can be written as in Eq. (34)

[00041]$\begin{array}{cc}{K}_n=\frac{}{H},=\frac{P_0}{P}.Math.{}_0& \left(34\right)\end{array}$

[0206] Using the Knudsen number, the flow is divided into four regimes: continuum flow, slip flow, transitional flow and free molecular flow. Assuming the characteristic length to be equal to the initial gap and using Eqs. (34) we can categorize the flow according to pressure (Table IV). As the experiment is performed at very low pressure on the device (P<350 mTorr), the flow regime would be in the free molecular regime. Also, as the microstructure gets very close to the substrate, the characteristic length of the problem becomes even smaller than 2 m, which leads to higher threshold pressure for the molecular region. The modular damping depends on the ratio of vibration amplitude to the initial gap (initial fabrication gap+static deflection caused by DC voltage). When the vibration amplitude is small compared to the initial gap, the damping is dominated by linear viscous damping. However, as the vibration amplitude becomes comparable with the initial gap, the squeeze film damping increases as the air molecules are trapped between the proof mass and substrate. The equation accounting for this variable damping that modulates itself with the amplitude-gap ratio is given in Eq. (35) [76]. I

[00042]$\begin{array}{cc}={}_0+{}_1\left(\frac{{}_{\mathrm{dyn}}}{{}_{\mathrm{initial}}}\right)& \left(35\right)\end{array}$

[0207] With the description of damping above, all the parameters in Eq. (31) have been defined.

TABLE-US-00006 TABLE V Information related to the thin film depositions. Film Temp. ( C.) Thickness (m) Dep. rate (/min) SiO.sub.2 1100 1 74 Si.sub.3N.sub.4 800 0.2 25 HTO 800 4 12.7 a-Si 570 2 11.7 Annealing 1000

[0208] The process flow of the device fabrication is depicted in FIGS. 23a-21j. It starts with 4-inch standard silicon wafers which are dipped into base and acid baths (RCA Cleaning). This process removes residues such as organic particles and the native oxide on the wafer surface. After the cleaning, we grew 1 m thick oxide as an insulation layer using a low-pressure chemical vapor deposition (LPCVD) furnace. On top of this layer, we deposited 0.2 m thick low-stress nitride. The oxide and nitride layers isolate the device from the substrate. On top of the nitride layer, a 2 m thick phosphorus-doped amorphous silicon (N+a-Si) was deposited in a LPCVD furnace. The deposition rates and the deposition temperatures of the LPCVD films are presented in Table III. This amorphous layer was then formed to polycrystalline silicon (poly-Si) by an annealing process [84,85]. Following the annealing, a 2 m thick positive photoresist was spun, and exposed with an i-line stepper. Exposed wafers were baked at 115 C. for 90 s and developed with 726 MIF developer for 60 s. After the developing step, the first layer of fingers was created via the Bosch etching process. The chamber conditions of the etching process are presented in Table VI. On top of the fixed fingers, a 4 m thick high-temperature oxide (HTO) film was deposited as a sacrificial layer. This layer provides a vertical gap between the first and second poly-Si films which are the fixed and the moving fingers, respectively. Following the deposition process, 2 m of the oxide was planarized using a chemical mechanical polisher (CMP). This process removes the step difference between the proof mass and the fixed fingers. Later, using inductively coupled plasma reactive ion etching (ICP-RIE), the sacrificial oxide was etched to open vias between first and second poly-Si. The chamber condition of this process is presented in Table VI. Upon etching the vias, another layer of 2 m thick phosphorus-doped a-Si was deposited, which is also annealed to form poly-Si as done before. This layer is etched to form the moving fingers that are attached to the rectangular shaped proof mass. After the etching, 20 nm Cr and 150 nm of Au are deposited on the pads using an e-beam evaporation tool. Then the wafers are diced and released in HF:HCl solution. Following the release, the chips are critical point dried (CPD) to avoid stiction. The scanning electron microscope (SEM) images of the fabricated device are presented in FIGS. 24a-24c. The chip is mounted on a printed circuit board and wire-bonded using an aluminum wire wedge-bonder.

TABLE-US-00007 TABLE VI Chamber conditions for dry etching. Etch rates for silicon and oxide are measured around 2 m/min and 150 nm/min, respectively. Bosch process for silicon etching Pressure C.sub.4F.sub.8 SF.sub.6 Ar Dura- Process (mTorr) (sccm) (sccm) (sccm) RIE ICP tion Light 13 35 2 40 40 850 8 Passivation 24 60 2 40 0.1 850 4 Etch 1 23 2 70 40 8 850 2 Etch 2 23 2 100 40 8 850 6 (ICP process for oxide etching Pressure CHF.sub.3 O.sub.2 Ar Dura- Process (mTorr) (sccm) (sccm) (sccm) RIE ICP tion Etch 5 52 2 15 2500

[0209] Experiments were conducted on the fabricated sensor to investigate the accelerometer dynamical performance and verify the mathematical model described above. The sensor was subjected to two different mechanical loads, harmonic excitations and shock loads. For harmonic loading, the dynamic response of the microstructure in the resonance region is considered that sheds light on the effects of inherent nonlinearities in the electrostatic force. Also, the mechanical sensitivity of the accelerometer is obtained, which is an important parameter with a significant contribution to the overall sensitivity of the sensor. This mechanical sensitivity is obtained from the frequency response before reaching the resonance region, because accelerometers are always used at frequencies below their resonance range. The robustness of the accelerometer against mechanical shocks is another important parameter to be analyzed.

[0210] The printed circuit board containing the MEMS chip is mounted to the head of the shaker and placed inside the vacuum chamber at 200 mTorr. A laser vibrometer (Polytec MSA-500) is used to monitor the time-response of the sensor. This set up is shown in FIG. 5. The next part addresses how the shaker acceleration is kept constant while sweeping the frequency.

[0211] The output acceleration of the shaker depends on the amplitude and frequency of the voltage signal given to the shaker. To conduct a frequency sweep on the microstructure, the shaker needs to provide a base acceleration with a constant amplitude at different frequencies. Usually, a closed loop system is used with the shaker where the output acceleration of the shaker is constantly monitored so acceleration amplitude can be kept constant by modifying the amplitude of the voltage signal. Here, the shaker is used in an open loop mode. To identify the required voltage needed to send to the shaker at each frequency (f), the shaker is characterized in a separate setup where a laser vibrometer is used to measure the shaker velocity. For example, if a harmonic base acceleration with 1 g amplitude at 3000 Hz from the shaker is required, voltages with 3000 Hz frequency and different amplitudes are sent to the shaker until a voltage amplitude that results in 1 g1% acceleration is found. Then, the same procedure is repeated for the next frequency step (3010 Hz). To characterize the shaker for half-sine shock excitations with different time duration, the shaker response is measured with a commercial accelerometer (PCB-352A24). If the shock amplitude is within 1% range of the desired amplitude, the voltage will be recorded to be used for the shock experiment.

[0212] After the shaker characterization, the device is tested by applying harmonic base excitation and performing a frequency sweep. Such an analysis is helpful to investigate the accelerometer dynamical performance and verify the mathematical model described above.

[0213] The motion of the sensor proof-mass relative to the substrate is obtained by subtracting the substrate response from the device response for each excitation frequency (f). As the motion of the substrate does not depend on the electrical voltage on the device, for each mechanical loading scenario, the substrate motion was measured by reading its velocity with the laser vibrometer. The device response, on the other hand, depends on the mechanical load and electrical voltage on the side electrodes. For each loading case, the proof-mass response was measured again by reading its velocity using the laser vibrometer. Then, the velocity of the substrate for each mechanical load was subtracted from the device response for the corresponding mechanical load to obtain the relative velocity of the proof-mass. Subsequently, the fast Fourier transform (FFT) of this relative velocity signal was calculated to extract the velocity amplitude in the steady-state region. Dividing this velocity amplitude by 2f results in the displacement amplitude for the corresponding frequency.

[0214] FIGS. 25a-25c show the device response for different voltages on the side electrodes. For each voltage, different harmonic excitations with different amplitudes are applied. These figures show that the resonance frequency increases as the bias DC voltage increases on the side electrode. Furthermore, for DC voltages of 50V and 60V, as the amplitude of vibration increases, a nonlinear softening behavior appears in the response. These two phenomena can be explained by looking at nonlinear terms in the electrostatic force. According to Table III, A.sub.1 (coefficient of x in electrostatic force) is negative, when it is on the right-hand side of Eq. (19). This means that linear stiffness caused by electrostatic force will add to the linear mechanical stiffness of microstructure, which leads to an increase in stiffness and resonance frequency as DC voltage increases. This property of the microstructure can be beneficial to increase the robustness of the sensor against severe mechanical loads. By increasing the DC voltage, the proof-mass plate gets further pushed away from the substrate, hence the maximum mechanical load that is required for the proof-mass to hit the substrate is increased. This means that the voltage on the side electrodes can be increased and the sensor used for more harsh environments in terms of mechanical loads such as mechanical shocks. This is the opposite of what happens to conventional shock sensors that are based on the attractive scheme where an increase in the bias voltage results in early pull-in [80]. Also, one can tune the DC voltage on the side electrodes to obtain the frequency ranges of interest for different applications.

[0215] The softening behavior in the frequency response at a high voltage of 50V and 60V can be explained by the even-order terms in electrostatic force (A.sub.2, A.sub.4, A.sub.6, A.sub.8). Simulation results are presented in FIGS. 25a-25c. These results are obtained when the linear natural frequency of the structure is set to be the measured value of 1740 Hz in the governing equation of motion (Eq. (31). This value is 30% different from what is obtained from finite element analysis (FEA). This discrepancy can be explained by the imperfections in the fabrication process that are not considered in the FEA. These include a slight bending of the plate that happens because of thermal stresses. The residual stress on the supports plays a significant role on the structural stiffness of the sensor. This residual in-plane stress was measured to be approximately 50 MPa in the fabrication process, which causes a slight curvature in the proof mass plate. The deflection due to residual stress was measured to be approximately 6 m per 1 mm length of the plate. Because a static FEA predicted approximately the same deflection as the measured result, we conclude that this deflection is due to in-plane stress only and the variation of residual stress along the thickness may considered to be negligible. Once the estimated linear frequency for the fabricated device is considered in the lumped model, the model captures the experimental results with good accuracy. The slight difference between the simulation results and experimental data, especially at low frequencies, is also attributed to these inevitable fabrication imperfections and residual stresses as well as limitation of approximating a continuous system with a discrete (lumped) model. The damping ratios are adjusted slightly to match the experimental results for each voltage scenario. It is worth mentioning that the squeeze film damping is dependent on the resonance frequency. The resonance frequency changes by varying the DC voltage as shown in FIG. 26. Therefore, the damping coefficients will slightly vary with the DC voltage. For the frequency sweep, the lumped model is solved for the dynamic solution (.sub.dyn) using the shooting method [26]. The static rotation caused by DC voltage is calculated and shown in FIG. 27 as a function of DC voltage.

[0216] The nonlinearity in the device response when the bias voltage is 40V is slightly hardening while the lumped model predicts softening. This discrepancy between the simulation results and experimental data can be reconciled by introducing a cubic nonlinear term for mechanical stiffness for this specific voltage but such a nonlinearity makes the response hardening for higher voltages which does not match the experimental data. Also, because the vibration amplitudes with different voltages for the same mechanical loads are very close to each other (for example, the resonance amplitudes for 40V and 50V when the acceleration amplitude is 3 g), this difference in behavior cannot be attributed to the nonlinearities in the electrostatic force or stiffness caused by air in squeeze film regime.

[0217] Although the model does not catch the nonlinear behavior of the microstructure in the resonance region for low voltages, it matches the experimental data away from this region with good accuracy. Because inertial sensors are generally operated in frequency ranges away from their natural frequency the model still could be used to simulate the device response at low frequencies.

[0218] The sensitivity of the device for different voltages is shown in FIG. 28. The relative displacement of the proof-mass for harmonic base excitations with different amplitudes at 2500 Hz is recorded and shown in FIG. 28. The slopes of these lines give the mechanical sensitivity for each voltage. The sensitivity for lower voltages is higher because the contribution of electrical stiffness in the total stiffness of the sensor (structural and electrical) is smaller. For example, the sensitivity of the accelerometer at 40V is about 0.1 m/g. This is extremely superior to the numbers found in the literature, which are generally in the range of nanometer to femtometer per g [87,88]. In the next step, half-sine shocks will be considered as the mechanical load on the microstructure.

TABLE-US-00008 TABLE VII Threshold shock amplitude that results in proof- mass hitting the substrate (shock duration = 90% of natural period of microstructure for each voltage) and comparing resonance frequencies obtained from experiments (at low g (1 g)) with linearized natural frequencies derived from Jacobian matrix. Threshold Resonance freq. Resonance freq. V.sub.dc (V) shock (g) (Hz) (exp.) (Hz) (sim.) Error (%) 40 200 3000 2981 0.64 50 300 3280 3262 0.55 60 410 3540 3523 0.67

[0219] Robustness of accelerometers against mechanical shocks is an important parameter. Under mechanical shocks, a force is transmitted to the microstructure through its anchors during a short period of time compared to the natural period of the microstructure. These loads are usually characterized by the induced acceleration on the affected structure with three parameters: maximum acceleration (amplitude), time duration, and pulse shape, also called shock profile. Here, half-sine shock loads are used with different amplitudes and time durations to investigate the response of microstructure to a shock load. The actual output of the shaker is measured with the PCB accelerometer, which is mounted on the shaker. FIG. 29 shows this measured acceleration as well as the device response, which is obtained by integrating the velocity of the microstructure measured by the laser vibrometer.

[0220] The time responses of the microstructure for 60 V bias voltage on the side electrode to shock loads with different time durations and amplitudes are given in FIGS. 30a-30d. Similar experiments were conducted for 50V on the side electrodes to obtain the larger displacements of the proof-mass (FIG. 31). These shock experiments are done for shock loads that have time durations close to the natural period of the microstructure. The linearized natural frequency of the microstructure for each voltage (FIG. 26) is obtained from stability and eigenvalue analysis on the Jacobian matrix for Eq. (31). The linearized natural frequency of the microstructure can be extracted from harmonic experiments at low g levels (Table VII). Because the stiffness of the microstructure and hence, the natural period of the microstructure, would change with the DC voltage, the maximum dynamical displacement of the accelerometer would be a function of the voltage.

[0221] The simulation results are in good agreement with the shock-experiment outcomes as shown in FIGS. 31 and 32. The small discrepancies can be attributed to the error in the numerical integration of the velocity signal and the fact that the shock that is actually applied on the device is not a perfect half-sine shock considered in the mathematical model. It should also be mentioned the same damping coefficients that were used for harmonic simulation at 50V and 60V are used for simulating the shock responses. The close agreements of the results show that the damping model can accurately capture the energy dissipation.

[0222] The mechanical sensitivity to shock is an important factor in the performance of shock sensors. According to FIGS. 30a-30d at 60V, the maximum displacement at the proof-mass edge is less than 1.2 m when the shock amplitude is 25 g. If we consider the collision of the microstructure with its substrate as a failure mode or a limit to define the maximum allowable shock load, this 1.2 m is 10% of the initial gap between the proof-mass edge and the substrate (2 m fabrication gap+10 m static gap caused by 60V applied on the side electrodes). However, as discussed above, because there is no pull-in with this repulsive approach, even if the proof-mass hits the substrate, it could bounce off safely. In fact, if this impact could be detected electrically, the device could be employed as a tunable threshold shock sensor. As opposed to conventional parallel-plate shock sensors, the threshold shock could be tuned by applying voltage on the side electrodes. This capability is enabled in the repulsive configuration because of the additional two electrodes on the sides.

[0223] Because of the limitation on the shock amplitudes that could be generated by the shaker, the device could not be tested under more severe shocks. However, the model may be used to predict the threshold shock at different voltages. These threshold shock amplitudes are given in Table VII for different voltages on side electrodes. The sensor resilience against mechanical shock could be improved by increasing the voltage on the side electrodes. This also means the threshold shock the sensor can detect can be tuned by varying the DC voltage on the side electrodes.

[0224] A capacitive electrode system is provided that achieves a repulsive electrostatic force for acceleration-sensing. A single degree of freedom lumped parameter model is described that captures the nonlinear dynamics of the system. As the presented accelerometer does not suffer from pull-in, the DC voltage can be increased to increase the fundamental natural frequency. This capability enables the accelerometer to become tunable. That means the detection range of the accelerometers, which is often below one third of their natural frequency, can be tuned by changing the DC voltage of the side electrodes. This is superior to current commercial accelerometers that often have a fixed resonance frequency, which limits their performance. Here, because the resonance frequency of the accelerometer can be tuned, it has the potential to be used in a wider range of applications. Furthermore, the initial gap between the proof-mass and the substrate is increased by increasing the voltage on the side electrodes, which improves the accelerometer robustness against mechanical shocks without sacrificing its stability. This device also has the potential to be designed and used as a shock sensor. By changing the voltage difference between the side and bottom electrodes, the threshold shock amplitude needed to collide the structure with the substrate can be tuned. Also, the natural period of the sensor can be tuned to use the sensor for shock loads with different time duration.

[0225] A tunable MEMS electrostatic accelerometer is provided that uses a repulsive electrode configuration so that the design is not hampered by capacitive pull-in instability. The repulsive force configuration enables the increase of DC bias voltage without suffering from the pull-in failure mode. This flexibility in increasing voltage can be employed as a tuning parameter to widen the working frequency range and to improve the robustness of the accelerometer.

[0226] A lumped parameter model was developed to simulate the response of the microstructure under a combination of electrostatic and dynamic mechanical loading. The electrostatic force is estimated using a finite element simulation. The nonlinear equations of motion are solved for harmonic base excitations and half-sine shock loads using the shooting and the long-time integration methods, respectively. To validate the model, a sensor is fabricated and characterized under harmonic base excitation and mechanical shocks. A mechanical sensitivity of 0.1 mig is achieved when the bias voltage is 40 V. The experimental data are in good agreement with the simulation results.

[0227] Another flavor of the described accelerometer is an accelerometer with translational degree of freedom. A fabricated MEMS accelerometer with translational degree of freedom is depicted in FIG. 32. The mechanical response of this microstructure to different base excitation at different voltage on side electrodes is shown in FIG. 33. FIG. 33. Shows the frequency response of the microstructuer when the frequency is swept backward. The motion of the proof mass in this accelerometer leads to a change in capacitance. This change in capacitance has been read electrically for the first time for the accelerometer based on levitation electrostatic force. FIG. 34 shows the frequency response of electrical voltage that is read with an electronic circuit. FIG. 35. Shows one of the time responses at a particular frequency while the voltage on side electrode is 78V and the base excitation amplitude is 3 g.

Example 3Switch Characterization

[0228] First, the pull-in voltage for the cantilever is calculated, per Pallay, Mark, Alwathiqbellah I. Ibrahim, and Shahrzad Towfighian. A MEMS Threshold Acceleration Switch Powered by a Triboelectric Generator. In ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V004T08A010-V004T08A010. American Society of Mechanical Engineers, 2018. This is the minimum required voltage for the switch to start in the closed state. Since the cantilever and center electrode act as a simple parallel plate actuator, the pull-in voltage can be calculated with the relationship shown in Equation (36).

[00043]$\begin{array}{cc}{V}_{p.Math.u.Math.l.Math.l}.Math.\sqrt{\frac{8.Math.k_1.Math.d^3}{27.Math.\mathrm{.Math.}.Math..Math.A}}& \left(36\right)\end{array}$

[0229] where is the permittivity of air and A is the area of the underside of the beam. From Equation (36), the pull-in voltage should be approximately 1.9 V. Therefore 2.2 V (V.sub.bias) is applied between the beam and center electrode so that the starting position of the switch is in the closed state. A voltage slightly higher than the pull-in voltage is necessary for the switch to stay closed because the dimples (0.75 m long) do not allow the beam to travel the entire 2 m gap. If the voltage is too low the beam will start to pull-in, but the force will be too weak to hold it in the pulled in position of 1.25 m.

[0230] Next, the threshold generator voltage to open the switch when given an initial bias of 2.2 V is calculated. This can be estimated by calculating the potential energy and phase portrait. If the voltage is too low the beam should have an unstable trajectory in the phase plane. However, when the voltage exceeds a threshold it becomes stable and wants to settle to an equilibrium position somewhere above the substrate. The phase portrait will then show a periodic orbit about the equilibrium point.

[0231] To calculate the potential energy, Equation (37) is integrated. The damping has a small effect on the stability and is set to zero.

[00044]$\begin{array}{cc}{m}_1.Math.{\stackrel{\mathrm{.Math.}}{q}}_1+c_1.Math.{\stackrel{.}{q}}_1+k_1.Math.q_1+\underset{j=0}{\overset{9}{.Math.}}.Math.f_j.Math.q_1^j=0& \left(37\right)\\ m_1.Math.\frac{{\stackrel{.}{q}}_1^2}{2}+k_1.Math.\frac{q_1^2}{2}+\underset{j=0}{\overset{9}{.Math.}}.Math.\frac{f_j}{j+1}.Math.q_1^{j+1}=H& \left(38\right)\end{array}$

[0232] In Equation (38) the first term is the kinetic energy, while the second two terms are the potential energy. H is the total energy of the system, which can be varied to view various trajectories in the phase portrait.

[0233] FIG. 4 shows the potential energy for the case of V.sub.bias=2.2 V and V.sub.side=0V, showing the system will pull in at this voltage. To calculate the phase portrait, Equation (38) is rearranged to solve for {dot over (q)}.sub.1.sup.2, which is a pair of nonlinear function of q.sub.1. The separatrix can be calculated by setting H to be the potential energy (V) at a local peak (if one exists). For the case of FIG. 4, no such peak exists and the system remains unstable everywhere. FIG. 5 shows the phase plane trajectory of the beam starting at rest for a bias voltage of 2.2 V.

[0234] Next, the side voltage is gradually increased until the system becomes stable again. This occurs when the pulled-in position of the beam (indicated in the phase portrait by the intersection of the red line and zero velocity axis) moves inside of a closed loop in the phase portrait. A loop in the phase portrait occurs when a local minimum appears in the potential energy plot. If the side electrode voltage is increased past a threshold, the potential well will grow large enough to allow the beam to oscillate in a stable orbit and the beam will release. FIGS. 6 and 7 show the potential energy and phase portrait respectively when the side electrode voltage is increased to 12 V.

[0235] As can be seen in the phase portrait (FIG. 7), at side voltage of 12 V, the pulled-in position of the beam exists inside a closed loop indicating the beam is now stable. This means, neglecting stiction forces, only 12 V is necessary to open the switch. Contact forces holding the beam in the pulled-in position will inevitably increase this threshold, however it can be difficult to estimate theoretically but its influence may be determined through an experiment. The triboelectric generator is capable of producing voltages over an order of magnitude above the calculated threshold. This allows the bias voltage to be increased significantly to tune the sensitivity of the sensor. It also opens the possibility of miniaturizing the generator to the millimeter or even smaller scale, which is very attractive for MEMS sensors. Both threshold voltages to close and open the switch have been verified through integration of Equation (37).

[0236] Next, the generator output was experimentally extracted using a prototype triboelectric generator from [40]. The generator was struck with a 1 lb. weight to see the open circuit voltage produced from a single impact. FIGS. 8 and 9 show the voltage plots for two separate impact tests.

[0237] In FIGS. 8 and 9 the voltage responds in two steps during the impact. For the smaller impact (FIG. 8) the first step is a negative voltage, while for the larger impact (FIG. 9) the first step is a positive voltage. This difference does not significantly affect the operation of the device. The delay between steps is relatively short at 10 ms and 25 ms respectively. However, the beam response period is less than 0.1 ms and therefore the short delay cannot be ignored. In both cases, the voltage jump occurs in much less than 2 ms (the sample rate of the electrometer) and is modeled as a step input.

[0238] Both cases produced a voltage that was large enough to open the switch. This is a result of the generator size, which has approximately 20 cm.sup.2 of contact area between triboelectric layers. This means that even a light impact will produce enough voltage to open the switch. To tune the output voltage such that it is closer to the threshold voltage level, a new, smaller generator should be made so that lower level impacts will produce small voltages that will not open the switch.

[0239] The voltage from FIGS. 8 and 9 decayed slowly back to approximately zero over the course of a few minutes. This is because the PDMS layer in the generator is an insulator and becomes charged during the impact. The consequence of this is that the switch will stay in the open position for a few minutes until the voltage decays enough for the beam to pull back into the closed state. In this case the charge decay can be beneficial because it allows the sensor to reset itself after being triggered.

[0240] FIG. 10 shows the beam tip displacement for a side voltage corresponding to the step voltage shown in FIG. 8. The first Voltage step is at time zero and the second voltage step is after 10 ms. The generator voltage is above the threshold and therefore the switch is operating in the open state.

[0241] The simplified step voltages are then applied to Equation (37) to see the beam response. The generator output is not applied directly to Equation (37) because the electrostatic force does not scale with the square of the generator voltage. The voltage is approximated as a series of two steps so that the electrostatic force only needs to be simulated at three voltage levels for each plot. If the voltage is assumed to start at 0 V, then only two simulations are necessary to achieve the stepped voltage response. The electrostatic force coefficients in Equation (37)), p.sub.i, become piecewise functions that change suddenly at each jump in FIGS. 8 and 9. For the dynamic beam analysis, the beam will start at its pulled-in position and experience a jump in voltage at time zero corresponding to the first step in FIGS. 8 and 9. Then it will receive a second step to the final voltage after a short delay. The beam responses are shown in FIGS. 10 and 11. In both cases, the beam is released from its initial pulled-in position showing the switch opens from the shock the generator experiences.

[0242] For the smaller impact (FIGS. 8 and 10) the beam first begins to settle around 0.34 m from the first voltage spike, then jumps up to 0.37 m after the second spike, which is its final equilibrium position. The larger impact (FIG. 11) moves the beam to approximately 2 m above the substrate from the first spike, and to over 6 m from the second spike. Because the force is effectively repulsive, there is no voltage limitation on the sensor. This means the sensor should remain mechanically undamaged from a very large voltage spike, which could occur from an unexpectedly large impact.

[0243] Because the delay between each voltage step is much longer than the response period, the beam has time to settle to its static position between the two voltage spikes. If the device is operated at very low pressure, this may not be the case. However, for a quality factor of 70, the beam settles to its equilibrium point in about 10 ms, which is roughly the time between voltage steps in FIG. 8. If the beam does not have time to settle to its equilibrium position prior to the second spike, the operation of the device should not be affected in most cases.

[0244] The sensor uses a triboelectric generator to convert mechanical energy from an impact to an electrical signal that opens a switch when the impact passes a threshold value. The switch consists of a MEMS cantilever beam with a multi-electrode configuration that can generate both attractive and repulsive forces on the beam. The center electrode is given a small bias voltage to initiate pull-in, and the side electrodes receive voltage from the generator. If the impact is large enough, the repulsive force from the side electrodes releases the beam from pull-in and opens the switch. After a few minutes, the charge on the generator decays and the switch resets to its closed state.

[0245] The levitation force is capable of balancing the parallel plate electrostatic force when the beam is in its pulled-in position without tremendously high voltage levels. It has also been experimentally demonstrated that a triboelectric transducer can generate voltages well above this threshold level and will not immediately discharge when the generator electrodes are left as an open circuit.

Example 4Mems Switch

[0246] A microelectromechanical system (MEMS) beam is experimentally released from pull-in using electrostatic levitation. A MEMS cantilever with a parallel plate electrode configuration is pulled-in by applying a voltage above the pull-in threshold. An electrode is fixed to the substrate on each side of the beam to allow electrostatic levitation. Large voltage pulses upwards of 100 V are applied to the side electrodes to release the pulled-in beam. A large voltage is needed to overcome the strong parallel plate electrostatic force and stiction forces, which hold the beam in its pulled-in position. See, Pallay, Mark, and Shahrzad Towfighian. A reliable MEMS switch using electrostatic levitation. Applied Physics Letters 113, no. 21 (2018): 213102.

[0247] One common undesirable phenomenon associated with electrostatic actuation is the pull-in instability. Pull-in failure occurs when the electrostatic force pulling the two electrodes together overcomes the mechanical forces separating them and the structure collapses. In many cases, pull-in results in permanent damage to the device as the electrodes become stuck together and cannot be separated even if the voltage is removed. Stiction forces such as van der Waals become much more significant at the micro-scale, and the parallel plate electrostatic force is only capable of pulling objects together, so release is often impossible. Stiction can be mitigated by placing dimples on the bottom face of the beam or movable plate, thus reducing the contact area and minimizing the force holding the plates together. However, even beams with dimples can frequently become stuck after pull-in, and therefore, many electrostatic devices are designed to avoid pull-in entirely.

[0248] Much effort has been placed in creating electrostatic MEMS designs that do not experience pull-in at all. One method actuates a structure with electrostatic levitation. Electrostatic levitation involves a slightly different electrode configuration than the standard parallel plate design, with two extra electrodes that help induce an effectively repulsive force instead of an attractive one.

[0249] The beam and the fixed middle electrode are kept at similar voltage level, while the fixed side electrodes are supplied with a large voltage. When the beam is close enough to the middle electrode, the electrostatic fringe-field produced by the side electrodes pulls on the top of the beam more than the bottom, resulting in a net force upwards. It is not the case of a purely repulsive force that would occur between two positively charged particles but rather an attractive force that acts in the opposite direction of the substrate and is commonly referred to as repulsive to differentiate it from the attractive parallel plate force. The middle electrode acts as a shield protecting the bottom face of the beam from the electric field and associated electrostatic force. As shown in FIG. 12, some of the electric field lines that would have normally pulled on the bottom face of the beam are now moved to the middle electrode instead. The electric field at the top of the beam is relatively unaffected by the presence of the middle electrode, and therefore, the direction of the net force on the beam becomes upward instead of downward when the beam-electrode gap is small.

[0250] If the beam is held to just one degree of freedom, which is common for thin, wide beams, it will not pull-in at all because the side electrodes are not in the beams' path of motion. The middle electrode will not create attractive electrostatic forces on the beam because they are both at the same voltage potential, and thus, pull-in will not occur.

[0251] A major drawback to electrostatic levitation is that it requires a very high voltage potential because it utilizes the weak fringe fields. To generate an electrostatic levitation force comparable to the one generated by a standard parallel plate configuration, the voltage must be more than an order of magnitude larger than the parallel plate voltage. Voltages upwards of 150 V may be applied to achieve around 10 m of static tip deflection for a 500 m long beam. However, the large voltage potential and elimination of the pull-in instability allow repulsive actuators to move more than an order of magnitude farther than their initial gap, as opposed to parallel plate actuators, which are typically limited to one-third of the initial gap because of pull-in.

[0252] Another advantage of electrostatic levitation is that it can be easily combined with parallel plate electrodes to enable bi-directional actuation. Applying a bias to the middle electrode, along with the voltage on the side electrodes, creates attractive and repulsive forces on the beam. The beam and middle electrode act as parallel plates, while the side electrodes produce the levitation force. As with other bidirectional devices, such as double-sided parallel plates, bidirectional actuation requires multiple voltage inputs with each controlling the magnitude of the force in a single direction.

[0253] According to the present technology, a MEMS beam may be toggled between its pulled-in and released positions using a combination of parallel plate actuation and electrostatic levitation. A bias voltage is applied to the middle electrode to induce pull-in, and release the beam from its pulled-in state. The repulsive force is capable of overcoming the stiction forces holding the beam to the substrate. The capability of recovering from what was once permanent pull-in failure of a MEMS structure is a great advancement and addresses a fundamental issue that has existed since the inception of electrostatically actuated MEMS. This feature can make MEMS devices more reliable and reusable. It also opens the possibility of new applications for electrostatic MEMS by allowing them to use the pulled-in state as a functional element of the device, rather than a limitation. Almost all electrostatic MEMS are designed around pull-in, and by using a combination of attractive and repulsive forcing, this limitation can be relaxed or removed entirely. This attribute has great potential for MEMS switches that will be normally closed, as opposed to current MEMS switches, which are normally open. It also has a promising application in micromechanical memories to read and erase bits as it can switch back and forth between two functional states: pulled-in and released.

[0254] MEMS cantilevers are fabricated using PolyMUMPs standard fabrication by MEMSCAP. Dimples are placed on the bottom of the beam to reduce the contact area and the associated stiction forces. While dimples can aid with release, the beams still suffer from stiction when pulled-in. The cantilevers have the electrode layout shown in FIG. 1.

[0255] The beam can be modeled as an Euler-Bernoulli beam with electrostatic forcing, which can be calculated numerically with a 2D COMSOL simulation. A comparison of pure repulsive, pure attractive, and combined repulsive and attractive forces can be seen in FIG. 3B.

[0256] A schematic for the experimental setup is shown in FIG. 15a. An alternate setup is shown in FIG. 15b. The cantilevers are placed in air, and the tip displacement is measured with a Polytec MSA-500 Laser Vibrometer interfaced with MATLAB through a National Instruments USB 6366 Data Acquisition (DAQ). A B&K Precision 9110 power supply and a Krohn-Hite 7600 Wideband Power Amplifier supply the bias and side electrode voltage, respectively. The bias voltage is measured directly with the DAQ; however, the side voltage is well over the 10 V limitation of the DAQ and is measured with a Keithley 6514 electrometer, which is also controlled with MATLAB.

[0257] In the experiment, a bias voltage is applied to the middle electrode to start the beam in its pulled-in position. The bias is then adjusted to a specified level and held constant before a series of short, high voltage pulses are applied to the side electro des. The beam displacement is observed to determine whether the beam was released during the voltage pulses. A relationship between bias voltage and release side voltage is obtained to demonstrate the working principle of the repulsive switch.

[0258] FIG. 15c shows the recorded switch motion and applied voltages. The bias voltage is initially set at 0 V, increased to the pull-in voltage of 4.5 V, and then held constant. As the bias voltage ramps up, the beam is pulled down slightly before suddenly becoming unstable and collapsing, which can be observed at approximately 1.2 s. Two pulses of 195 V are applied to the sides after the beam is in the pulled-in position. The cantilever releases during both pulses, which can be observed jumping 20 m in the displacement signal. When the side voltage drops to zero, the beam immediately pulls back in and sticks to the substrate. The beam can be toggled to and from pull-in many times without failure or causing noticeable damage to the device by applying and removing a voltage on the side electrodes. The bias voltage determines the minimum side voltage needed to open the switch.

[0259] FIG. 3B shows the electrostatic force on the beam versus the gap distance simulated in COMSOL. The dashed area shows the attractive regime, and the rest is the repulsive regime. The combined force with bias and side voltages behaves similar to the attractive force at low gaps and the repulsive force at large gaps. Applying 10 V on the side electrodes can change the force from attractive to repulsive outside of very small gaps.

[0260] FIG. 15a shows the experimental setup with (a) Krohn-Hite 7600 Wideband Power Amplifier, (b) Keithley 6514 Electrometer, (c) B&K Precision 9110 Power Supply, (d) Polytec MSA-500 Laser Vibrometer, (e) NI USB 6366 Data Acquisition, and (f) the MEMS repulsive switch. The DAQ and electrometer are interfaced with MATLAB.

[0261] The experiment was repeated by adjusting the bias voltage and determining the associated release voltage. FIG. 3C shows the release voltage for various bias levels. For biases that are less than the 4.5 V pull-in voltage, first pull-in is initiated at 4.5 V, and then, the bias is reduced to the specified level. When the bias voltage is removed completely, the beam continues to stick to the middle electrode, and 70 V is required to release the beam. At the pull-in voltage, 195 V is needed for release. Because of limitations with the PolyMUMPs chips, voltages above 200 V were not applied.

[0262] FIG. 3C demonstrates that the release voltage can be adjusted by changing the bias voltage. This is useful for a MEMS switch, which can be tuned to open at different threshold voltage levels. If paired with a transducer that is converting mechanical energy to electrical energy, the entire system can be designed to trigger the opening of a switch when the input passes a threshold.24 In addition to the tunability, it also can act as a normally closed switch, which is not possible with a standard two-electrode parallel plate configuration.

[0263] FIG. 15c bottom, shows measured beam-tip displacement, bias voltage, and side voltage versus time. The beam pulls in as the bias voltage ramps up to 4.5 V and is released by supplying 195 V on the side. The illustrations at the top of the displacement signal roughly show the profile of the beam during each phase of the experiment.

[0264] A MEMS cantilever is experimentally released from its pulled-in position using electrostatic levitation. This method provides a safe and effective way of releasing and reusing pulled-in MEMS beams, which would have otherwise been permanently stuck to the substrate, rendering them unusable.

Example 5Triboelectric Generator

[0265] Several types of transducers have been used to convert mechanical energy to electrical energy including electromagnetic, piezoelectric, electrostatic and triboelectric generators. Recent advances in triboelectric generation show energy conversion efficiency of 60% and higher energy densities compared with other transducers. The prefix tribo comes from the Greek word for rub and more friction generates more electricity. That triboelectricity is contact-induced electrification and has been known for thousands of years, but just recently with the development of micro-patterned surfaces, it has been employed to convert wasted mechanical energy into electrical energy. Despite plenty of research on triboelectric material, there is little research that models the correlation of vibration and triboelectric mechanism. Our team was among the first who discovered the coupled characteristics of triboelectricity and mechanical vibrations. These generators have the advantages of low-cost, high energy density, and flexibility. Two recent studies proposed a triboelectric accelerometer for monitoring the health of railways, in-plane sliding acceleration and motion tracking of a ball.

[0266] The triboelectric generators produce surface charges when two materials with opposite electron affinities contact each other. Although the contacting materials can be two polymers, the system works best when contacting materials consists of a polymer and a conductor. The high output energy density of triboelectric generators is attributed to several factors including: (i) the use of surface patterning to increase contacting areas of the polymer and conductor and to generate more surface charges; (ii) the use of a deformable polymer as one of the triboelectric materials to ensure conformal contact. Continuous contact and separation simultaneously create contact electrification and electrostatic induction, which creates a large potential difference or open circuit voltage.

[0267] The contact surfaces can be made of a polymer, e.g. polydimethylsiloxane (PDMS) layer and a conductor, e.g. aluminum. Another Al layer covers the back of the PDMS layer. The charge generated on the surfaces is directly proportional to the area of the contact. The key to generating more charges is to make micro-textured surfaces. This leads to a larger contact area, more friction at the microsurfaces and consequently more charges and larger output voltage.

[0268] Upon applied mechanical pressure, the Al layer is pressed against the PDMS surface. Because of its flexibility, the PDMS layer fills the gaps between the patterns. The micro-textured surface penetration causes equal but opposite charges to accumulate on both surfaces as electrons are injected into the PDMS from the top Al layer. The triboelectric effect here is not due to macroscopic sliding, but mostly from macroscopic compression, which leads to microscopic sliding. Upon release, the load is removed, and the layers separate as the top surface comes back to its initial position because of the restoring force of an external spring. There will be a large electrical potential difference between the two Al conductors that can directly be used to drive an electrostatic actuator.

[0269] Despite the high voltage that triboelectric generators produce, the current generated by these generators is very low in the nano and micro-amp ranges (high impedance generator). This drawback makes powering an electrical circuit difficult. The high voltage creates a strong electrostatic field that could be useful for running electrostatic devices.

[0270] To overcome the limitations of the typical two-electrode parallel-plate configuration, electrostatic levitation may be employed, where there are four electrodes: a beam above a center electrode and two side electrodes (FIG. 1). The beam and center electrodes are often grounded or have similar voltages, and the side electrodes have voltages. The electrostatic field of the side electrodes on the beam electrode is asymmetric. That means the top of the beam experiences a stronger electrostatic field than the bottom of the beam (FIG. 12). This property creates a force on the beam pushing it away from the substrate (levitation force). This levitation force is in the opposite direction of the substrate, unlike the conventional two parallel-plate electrodes where the electrostatic force is toward the substrate. This attribute eliminates the pull-in instability. Free of instability danger, the beam vibrates at distances away from the substrate, and is not limited to one-third of the capacitor gap, the common limitation of conventional parallel-plate capacitors. Beams with levitation actuation can vibrate up to 24 microns peak to peak despite the nominal capacitor gap of 2 microns. This is possible because when the beam is excited by DC and AC voltage, it can move to its equilibrium position corresponding to the DC voltage that is 12 microns above the substrate. If the AC voltage is adequately increased, the beam vibrates 24 microns peak to peak before hitting the substrate.

[0271] The dynamic behavior of levitation actuators is highly nonlinear.

[0272] The bias voltage on the bottom center electrode determines the minimum side voltage needed to open the switch. When the bias voltage is removed completely, the beam continues to stick to the middle electrode and a relatively low voltage of 70 V is required to release the beam (open the switch).

[0273] FIGS. 36a-36f shows measurements of (FIG. 36a) impact force (FIG. 36b) generator voltage (FIG. 36c) beam velocity using laser vibrometer after the drop of a one-pound weight and (FIG. 36d) impact force (FIG. 36e) generator voltage (FIG. 36f) beam velocity using laser vibrometer after the generator electrodes are shorted. In all cases, the side electrodes are connected to the generator while the beam and the center bottom electrodes are grounded.

[0274] The triboelectric generator controls the levitation actuator. Once the impact occurs, the voltage magnitude steps up quickly and the beam begins to move, see FIGS. 36a-36c. Once the voltage flattens out, the velocity dies down as the beam settles to its new equilibrium position. To verify the change of voltage has moved the beam to a new location, the generator is shorted at approximately 12 s, the voltage drops to zero and the beam experiences a sharp velocity spike corresponding to the beam falling back to its undeflected position, see FIGS. 36d-36f. Such controlling capability is a unique property of the electrostatic levitation and is not possible with conventional two parallel-plate electrodes. If such a voltage was connected to conventional parallel-plate electrodes, the actuator would always collapse from pull-in instability. The high voltage of the generator could permanently damage and burn the MEMS actuator. The levitation mechanism connected to the triboelectric generator enables a sensor to respond directly to impact, which significantly simplifies the sensing circuitry and saves costs.

[0275] The output of the generator changes when connected to an actuator, although they are connected with an open circuit. As shown in FIG. 37, The generator is capable of outputting almost 200 V when disconnected from the beam. As the force increases up to approximately 200 N there is a linear relationship between force and voltage. After 200 N, the voltage begins to saturate. The voltage the generator produces is affected by the presence of the switch. When the switch is connected, the output voltage saturates around 150 V instead of 190 V. This difference changes the performance of the switch and is an important attribute of the gener-actuator because it defines at what impact force the switch opens. This difference may be the result of nonlinear coupling characteristics between the induced charge of the generator and the dynamics of the switch.

[0276] Further embodiments can be envisioned to one of ordinary skill in the art from the specification and figures. In other embodiments, combinations or sub-combinations of the above-disclosed invention can be advantageously made. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. It will, however, be evident that various modifications and changes may be made thereunto without departing from the broader spirit and scope of the invention as set forth in the claims.

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