A methodology and model for the pull-in parameters of electrostaticactuators

This paper presents a generalized model for the pull-in phenomenon in electrostatic actuators with a single input, either charge or voltage. The pull-in phenomenon of a general electrostatic actuator with a single input is represented by an algebraic equation referred to as the pull-in equation. This equation directly yields the pull-in parameters, namely, the pull-in voltage or pull-in charge and the pull-in displacement. The model presented here permits the analysis of a wide range of cases, including nonlinear mechanical effects as well as various nonlinear, nonideal, and parasitic electrical effects. In some of the cases, an analytic solution is derived, which provides physical insight into how the pull-in parameters depend upon the design and properties of the actuator. The pull-in equation can also yield rapid numerical solutions, allowing interactive and optimal design. The model is then utilized to analyze analytically the case of a Duffing spring, previously analyzed numerically by Hung and Senturia, and captures the variations of the pull-in parameters in the continuum between a perfectly linear spring and a cubic spring. Several other case studies are described and analyzed using the pull-in equation, including parallel-plate and tilted-plate (torsion) actuators taking into account the fringing field capacitance, feedback and parasitic capacitance, trapped charges, an external force, and large displacements     

Voltage and pull-in time in current drive of electrostaticactuators

The absolute maximum value of the voltage developed across an electrostatic actuator when driven by a current source has been calculated as well as an absolute minimum for the pull-in time. These two results are calculated for a drive using a δ-pulse of current and numerical assessment is given to show that for a nonzero parasitic capacitance, a realistic shape of the current pulse, or a finite value of the damping coefficient do not increase the maximum value of the voltage developed beyond that limit and that the pull-in time is always larger than the analytical minimum. A scaled-up macromodel of an electrostatic actuator has been used to register voltage transients to validate the theoretical predictions     

Pull-in study of an electrostatic torsion microactuator

Pull-in study of an electrostatic microactuator is essential for making the electrostatic actuation more effective. In this paper, pull-in analysis is presented for an electrostatic torsion microactuator. The torsion microactuator can be used as a microtorsion mirror. A polynomial algebraic equation for the pull-in voltage and pull-in angle of a torsion microactuator is derived. Two types of microactuators fabricated using bulk micromachining are presented. Measurements done on the fabricated microactuators are reported, showing deviations within 1% error from the calculations     

Experimental verification of a design methodology for torsion actuators based on a rapid pull-in solver

In this work, an experimental and theoretical study of the effect of various geometrical parameters on the electromechanical response and pull-in parameters of torsion actuators is presented. A lumped two-degrees-of-freedom (L2DOF) pull-in model that takes into account the bending/torsion coupling, previously proposed for cantilever suspended actuators, is tailored for the torsion actuators under study. This model is shown to better capture the measured pull-in parameters than previously proposed lumped single-degree-of-freedom (L1DOF) models. The measurements were conducted on torsion actuators with various shapes, fabricated on silicon-on-insulator (SOI) wafers using deep reactive ion etching (DRIE) and flip-chip bonding. Furthermore, a novel rapid solver, for extracting the pull-in parameters of the L2DOF model of the torsion actuators, is proposed. The proposed solver is based on a Newton-Raphson scheme and the recently presented DIPIE algorithm and is shown to be /spl sim/10 times faster than the prevalent voltage iterations based solvers. The rapid and more accurate pull-in extraction of the proposed approach renders it as a tool for extensive analysis and design optimization of torsion actuators.     

Pull-In Analysis of Torsional Scanners Actuated by Electrostatic Vertical Combdrives

This paper presents pull-in analysis of torsional MEMS scanners actuated by electrostatic vertical combdrives with general comb gap arrangements and cross sections. The analysis is based on a 2-DOF actuator with a single voltage control. Three failure modes of the scanners are identified as in-plane twist, transversal motion, and out-of-plane twist. For each failure mode, analytical expressions of pull-in deflection are obtained by applying 2D analytical capacitance models to the derived pull-in equations. From these, the dominant pull-in mechanism is shown to be in-plane twist for scanners with high-aspect-ratio torsional springs. The analytical calculations for both symmetric and asymmetric capacitances are shown to be in good agreement with simulation results. The optimum scanner design is achieved when the pull-in deflection matches the capacitance maximum angle. The condition can be expressed in terms of the ratio of the comb thickness to the comb gap, which is smaller than the typical aspect ratio of deep reactive ion etching. The optimum tradeoff between the maximum deflection angle and the number of movable combs is achieved by adjusting the overlap of the movable and fixed combs and the distance of the comb sets from the axis of the rotation.     

Charge control of parallel-plate, electrostatic actuators and the tip-in instability

Controlling the charge, rather than the voltage, on a parallel-plate, electrostatic actuator theoretically permits stable operation for all deflections. Practically, we show that, using charge control, the maximum stable deflection is limited by 1) charge pull-in, in which the actuator snaps due to the presence of parasitic capacitance and 2) tip-in, in which the rotation mode becomes unstable. This work presents a circuit that controls the amount of charge on a parallel-plate, electrostatic actuator. This circuit reduces the sensitivity to parasitic capacitance, so that tip-in is the limiting instability. A small-signal model of the actuator is developed and used to determine the circuit bandwidth and gain requirements for stable deflections. Four different parallel-plate actuators have been designed and tested to verify the charge control technique as well as to verify charge pull-in, tip-in, and the bandwidth requirements. One design travels 83% of the gap before tip-in. Another design can only travel 20% of the gap before tip-in, regardless of whether voltage control or charge control is used.     

Analytical approach and numerical /spl alpha/-lines method for pull-in hyper-surface extraction of electrostatic actuators with multiple uncoupled voltage sources

This work presents a systematic analysis of electrostatic actuators driven by multiple uncoupled voltage sources. The use of multiple uncoupled voltage sources has the potential of enriching the electromechanical response of electrostatically actuated deformable elements. This in turn may enable novel MEMS devices with improved and even new capabilities. It is therefore important to develop methods for analyzing this class of actuators. Pull-in is an inherent instability phenomenon that emanates from the nonlinear nature of the electromechanical coupling in electrostatic actuators. The character of pull-in in actuators with multiple uncoupled voltage sources is studied, and new insights regarding pull-in are presented. An analytical method for extracting the pull-in hyper-surface by directly solving the voltage-free K-N pull-in equations derived here, is proposed. Solving simple but interesting example problems illustrate these new insights. In addition, a novel /spl alpha/-lines numerical method for extracting the pull-in hyper-surface of general electrostatic actuators is presented and illustrated. This /spl alpha/-lines method is motivated by new features of pull-in, that are exhibited only in electrostatic actuators with multiple uncoupled voltage sources. This numerical method permits the analysis of electrostatic actuators that could not have been analyzed by using current methods.     

Current drive methods to extend the range of travel ofelectrostatic microactuators beyond the voltage pull-in point

When a voltage source drives an electrostatic parallel plate actuator, the well-known pull-in instability limits the range of displacement to 1/3 of the gap. Different strategies have been reported to overcome this limitation. More recently, experimental results have been presented using a capacitor in series with the actuator. Nevertheless, this strategy requires higher voltage than the pull-in voltage value to achieve full range of travel. In order to reduce the operating voltage, a switched-capacitor configuration has been also proposed. In this paper, two different approaches are introduced to control charge in the actuator by means of current driving. Theoretical equations derived for each method show that full range of travel can be achieved without voltage penalty. Both approaches are based on the use of current pulses injecting the required amount of charge to fix the position of the movable plate. Experimental measurements, showing that displacement beyond the pull-in point can be achieved, are in good agreement with the theoretical and the predicted simulated behavior     

Analytical method of predicating the instabilities of a micro arch-shaped beam under electrostatic loading

An arch-shaped beam with different configurations under electrostatic loading experiences either the direct pull-in instability or the snap-through first and then the pull-in instability. When the pull-in instability occurs, the system collides with the electrode and adheres to it, which usually causes the system failure. When the snap-through instability occurs, the system experiences a discontinuous displacement to flip over without colliding with the electrode. The snap-through instability is an ideal actuation mechanism because of the following reasons: (1) after snap-through the system regains the stability and capability of withstanding further loading; (2) the system flips back when the loading is reduced, i.e. the system can be used repetitively; and (3) when approaching snap-through instability the system effective stiffness reduces toward zero, which leads to a fast flipping-over response. To differentiate these two types of instability responses for an arch-shaped beam is vital for the actuator design. For an arch-shaped beam under electrostatic loading, the nonlinear terms of the mid-plane stretching and the electrostatic loading make the analytical solution extremely difficult if not impossible and the related numerical solution is rather complex. Using the one mode expansion approximation and the truncation of the higher-order terms of the Taylor series, we present an analytical solution here. However, the one mode approximation and the truncation error of the Taylor series can cause serious error in the solution. Therefore, an error-compensating mechanism is also proposed. The analytical results are compared with both the experimental data and the numerical multi-mode analysis. The analytical method presented here offers a simple yet efficient solution approach by retaining good accuracy to analyze the instability of an arch-shaped beam under electrostatic loading.     

An efficient DIPIE algorithm for CAD of electrostatically actuated MEMS devices

Pull-in parameters are important properties of electrostatic actuators. Efficient and accurate analysis tools that can capture these parameters for different design geometries, are therefore essential. Current simulation tools approach the pull-in state by iteratively adjusting the voltage applied across the actuator electrodes. The convergence rate of this scheme gradually deteriorates as the pull-in state is approached. Moreover, the convergence is inconsistent and requires many mesh and accuracy refinements to assure reliable predictions. As a result, the design procedure of electrostatically actuated MEMS devices can be time-consuming. In this paper a novel Displacement Iteration Pull-In Extraction (DIPIE) scheme is presented. The DIPIE scheme is shown to converge consistently and far more rapidly than the Voltage Iterations (VI) scheme (>100 times faster!). The DIPIE scheme requires separate mechanical and electrostatic field solvers. Therefore, it can be easily implemented in existing MOEMS CAD packages. Moreover, using the DIPIE scheme, the pull-in parameters extraction can be performed in a fully automated mode, and no user input for search bounds is required.     

A Model for Electrostatic Actuation in Conducting Liquids

This paper presents a generalized model that describes the behavior of micromachined electrostatic actuators in conducting liquids and provides a guideline for designing electrostatic actuators to operate in aqueous electrolytes such as biological media. The model predicts static actuator displacement as a function of device parameters and applied frequency and potential for the typical case of negligible double-layer impedance and dynamic response. Model results are compared to the experimentally measured displacement of electrostatic comb-drive and parallel-plate actuators and exhibit good qualitative agreement with experimental observations. The model is applied to show that the pull-in instability of a parallel-plate actuator is frequency dependent near the critical frequency for actuation and can be eliminated for any actuator design by tuning the applied frequency. In addition, the model is applied to establish a frequency-dependent theoretical upper bound on the voltage that can be applied across passivated electrodes without electrolysis.     

RMS voltage sensor based on a variable parallel-plate capacitor made of electroplated copper

We present an advanced RMS voltage sensor based on a variable parallel-plate capacitor using the principle of electrostatic force. The device is fabricated in a micromechanical surface process with a high-aspect ratio actuator, reinforced by copper electroplating employing a sacrificial photo-resist layer. Another copper layer with a coplanar waveguide below the actuator provides separated excitation and sensing electrodes. Flip-chip technology is employed for low-loss electrical connectivity. The presented design has a plate area of up to 3 × 3 mm² and an initial gap distance of only 1.5 μm. We present results achieving a pull-in voltage below 1 V at frequencies from DC up to 1 GHz and sensitivities up to 1 fF/mV.     

一种基于静电排斥力的纵向微驱动器研究

为了消除静电塌陷对静电吸引力微驱动器冲程的限制,设计了一种基于静电排斥力的纵向微驱动器实现结构,其可动电极和固定电极在工作过程中互相远离,可以从根本上解决静电塌陷问题,且冲程不受牺牲层厚度的限制;借助Maxwell2D软件,采用数值仿真方法,研究了微驱动器结构参数对静电排斥力大小的影响.数值仿真结果表明,梳齿电极水平间距,固定/可动梳齿电极宽度和两者的比值都是影响静电排斥力大小的关键因素.     

Dynamic pull-in of parallel-plate and torsional electrostatic MEMS actuators

An analysis of the dynamic characteristics of pull-in for parallel-plate and torsional electrostatic actuators is presented. Traditionally, the analysis for pull-in has been done using quasi-static assumptions. However, it was recently shown experimentally that a step input can cause a decrease in the voltage required for pull-in to occur. We propose an energy-based solution for the step voltage required for pull-in that predicts the experimentally observed decrease in the pull-in voltage. We then use similar energy techniques to explore pull-in due to an actuation signal that is modulated depending on the sign of the velocity of the plate (i.e., modulated at the instantaneous mechanical resonant frequency). For this type of actuation signal, significant reductions in the pull-in voltage can theoretically be achieved without changing the stiffness of the structure. This analysis is significant to both parallel-plate and torsional electrostatic microelectromechanical systems (MEMS) switching structures where a reduced operating voltage without sacrificing stiffness is desired, as well as electrostatic MEMS oscillators where pull-in due to dynamic effects needs to be avoided.     

Design, Modeling, and Demonstration of a MEMS Repulsive-Force Out-of-Plane Electrostatic Micro Actuator

An analytical model is developed for a two-layer repulsive-force out-of-plane micro electrostatic actuator by using conformal mapping techniques. The model provides the means to establish the performance characteristics in terms of stroke and generated force of the actuator and is used to develop design and optimization rules for the actuator. Numerical simulations were conducted in order to verify the analytical model. A simple physical model is also presented that explains the mechanism for generating the repulsive force. A Multi-User-MEMS-Processes repulsive-force out-of-plane rotation micromirror is developed to experimentally verify the analytical model and to demonstrate the repulsive-force actuator's capability of driving large-size rotation plates by using surface micromachining technology. Experimental measurements show that the repulsive-force rotation micromirror with a size of 312 mum times 312 mum achieved a mechanical rotation of 0deg-2.1deg at a dc driving voltage of 0-200 V. The micromirror achieved an open-loop settling time of 2.9 ms for a mechanical rotation of 2.3deg and an open-loop bandwidth of 150 Hz (-3 dB).     

Dynamic response of a torsional micromirror to electrostatic force and mechanical shock

In this paper dynamic characteristics of a capacitive torsional micromirror under electrostatic forces and mechanical shocks have been investigated. A 2DOF model considering the torsion and bending stiffness of the micromirror structure has been presented. A set of nonlinear equations have been derived and solved by Runge–Kutta method. The Static pull-in voltage has been calculated by frequency analyzing method, and the dynamic pull-in voltage of the micromirror imposed to a step DC voltage has been derived for different damping ratios. It has been shown that by increasing the damping ratio the dynamic pull-in voltage converges to static one. The effects of linear and torsional shock forces on the mechanical behavior of the electrostatically deflected and undeflected micromirror have been studied. The results have shown that the combined effect of a shock load and an electrostatic actuation makes the instability threshold much lower than the threshold predicted, considering the effect of shock force or electrostatic actuation alone. It has been shown that the torsional shock force has negligible influence on dynamic response of the micromirror in comparison with the linear one. The results have been calculated for linear shocks with different durations, amplitudes, and input times.     

Analytical behavior of rectangular electrostatic torsion actuators with nonlinear spring bending

In this paper, we study the pull-in effect for rectangular electrostatic torsion actuators by using analytical calculations that include the higher order effects of nonlinear spring bending. The calculation approach speeds the design of such systems. The method is found to be suitable for actuators with single long beam springs where the ratio of the resonant frequencies for the torsion and bending modes is up to at least 3.5, in the region where bending dominates torsion. After fitting the theory in this paper to Coventor simulation results with three nonphysical coefficients, the fractional differences between Coventor simulation and analytical calculation results are smaller than 6%. The method is also suitable for at least one class of folded spring designs, with greatly decreased bending mode displacement. The main results are also verified by comparing them with published experimental results.     

Analytical investigation and numerical verification of Casimir effect on electrostatic nano-cantilevers

In this paper, the two-point boundary value problem (BVP) of the nano-cantilever deflection subjected to Casimir and electrostatic forces is investigated using analytical and numerical methods to obtain the instability point of the nano-beam. In the analytical treatment of the BVP, the nonlinear differential equation of the model is transformed into the integral form by using the Green’s function of the cantilever beam. Then, closed-form solutions are obtained by assuming an appropriate shape function for the beam deflection to evaluate the integrals. The pull-in parameters of the beam are computed under the combined effects of electrostatic and Casimir forces. Electrostatic microactuators and freestanding nanoactuators are considered as special cases of our study. The detachment length and the minimum initial gap of freestanding nanocantilevers, which are the basic design parameters for NEMS switches, are determined. The results of the analytical study are verified by numerical solution of the BVP. The centerline of the beam under the effect of electrostatic and Casimir forces at small deflections and at the point of instability is obtained numerically to test the validity of the shape function assumed for the beam deflection in the analytical investigation. Finally, the large deformation theory is applied in numerical simulations to study the effect of the finite kinematics on the pull-in parameters of nano-canilevers.