An investigation into the electrostatic force representation for electrically actuated microelectromechanical systems

The ever-increasing demand for microelectromechanical systems (MEMS) in modern electronics has reinforced the need for extremely accurate analytical and reduced-order models to aid in the design of MEMS devices. Many MEMS designs consist of cantilever beams with a tip mass attached at the free end to act as a courter electrode for electrical actuation. One critical modeling aspect of electrically actuated MEMS is the electrostatic force that drives these systems. The two most used representations in the literature approximate the electrostatic force between two electrodes as a point force. In this work, the effects of the representation of the electrostatic force for electrically actuated microelectromechanical systems are investigated. The system under investigation is composed of a beam with an electrode attached to its end. The distributed force, rigid body, and point mass electrostatic force representations are modeled, studied, and their output results are compared qualitatively. Static and frequency analyses are carried out to investigate the influences of the electrostatic force representation on the static pull-in, fundamental natural frequency, and mode shape of the system. A nonlinear distributed-parameter model is then developed in order to determine and characterize the response of electrically actuated systems when considering various representation of the electrostatic forces. The results show that the size of the electrode may strongly affect the natural frequencies and static pull-in when the point mass, rigid body, and plate representations are considered. From nonlinear analysis, it is also proven that the representation may affect the hardening behavior of the system and its dynamic pull-in. This modeling and analysis give guidelines about the usefulness of the electrostatic force representations and possible erroneous assumptions that can be made which may result in inaccurate design and optimal performance detection for electrostatically actuated systems.

     

Closed-form empirical relations to predict the static pull-in parameters of electrostatically actuated microcantilevers having linear width variation

We develop novel closed-form empirical relations to estimate the static pull-in parameters of electrostatically actuated tapered width microcantilever beams. A computationally efficient single degree-of-freedom model is employed in the setting of Ritz energy technique to extract the static pull-in parameters of the distributed electromechanical model that takes into account the effects of fringing field capacitance. The accuracy of this single-dof model together with the variable-width equivalent of the Palmer’s fringing model is established through a comparison with 3D finite element simulations. A unique surface fitting model is proposed to characterize the variations of both the pull-in displacement and pull-in voltage, over a realistically wide range of system parameters. Optimum coefficients of the proposed surface fitting model are obtained using nonlinear regression analysis. Empirical estimates of pull-in parameters are validated against finite element simulations, and available experimental and numerical data. An excellent agreement indicates that the proposed relationships are sufficiently accurate to be safely used for the electromechanical design of tapered microcantilever beams.     

Command-Shaping Techniques for Electrostatic MEMS Actuation: Analysis and Simulation

Precision positioning of microelectromechanical systems (MEMS) structures using electrostatic actuation has been widely used for optical and radio-frequency MEMS. How to achieve fast switching without exciting excessive residual vibration or structural impact is an important issue for these applications. This paper presents the analysis and simulation of applying command-shaping techniques for controlling MEMS electrostatic actuation. According to the nature of application fields, electrostatic actuators are classified into three categories: 1) lateral linear actuation; 2) vertical nonlinear actuation; and 3) pull-in actuation. Their corresponding linear or nonlinear command-shaping schemes are developed and presented. Both lumped element and continuous models of typical MEMS electrostatic actuated structures are simulated using Simulink and the finite-element method, and results indicate that the shaped command would yield a much superior response than that by the unshaped commands. Essential sensitivity studies are also conducted to examine the robustness of these shaping schemes, and results shows that within a certain level of parameter variation, these shapers are robust enough to retain the performance.     

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.     

Complex nonlinear oscillations in electrostatically actuated microstructures

In this paper, new nonlinear dynamic properties of electrostatically actuated microstructures [referred to as electrostatic microelectromechanical systems (MEMS)] observed under superharmonic excitations are presented using numerical simulations. Application of a large dc bias (close to the pull-in voltage of the device) is found to bring the device to a nonlinear state. This nonlinear state (referred to as "dc-symmetry breaking") can be clearly observed from the characteristic change in the phase-plot of the device. Once a steady nonlinear state is reached, application of an ac signal at the Mth superharmonic frequency with an amplitude around "ac-symmetry breaking" gives rise to M oscillations per period or M-cycles in the MEM device. "ac-symmetry breaking" can also be observed by a characteristic change in the phase-plot of the device. On further increasing the ac voltage, a period doubling sequence takes place resulting in the formation of 2/sup n/M-cycles in the system at the Mth superharmonic frequency. An interesting chaotic transition (banded chaos) is observed during the period doubling bifurcations. The nonlinear nature of the electrostatic force acting on the MEM device is found to be responsible for the reported observations. The significance of the mechanical and the fluidic nonlinearities is also studied.     

Dynamic characteristics of voltage induced reciprocated bending in double cantilever configuration of asymmetric comb drive MEMS

The dynamic characteristics of an electrostatically actuated double cantilever beam, often found in asymmetric comb drive microstructures, have been investigated in the present paper. A coupled electromechanical problem is formulated and solved to obtain different performance parameters like pull-in voltage, frequency response etc. Effects of various critical factors on the dynamic pull-in characteristics have been discussed elaborately. It has been further observed through extensive studies that, the dynamic pull-in characteristics differ considerably from the static characteristics for the double beam configuration. Finally, these observations have been supported by experimental results with a fabricated (in SOIMUMPS process) double cantilever based microstructure, using a simple in-house developed low cost test set up. A typical case of design of a closed loop MEMS (microelectromechanical systems) capacitive accelerometer has also been discussed where the present study finds ready applications to predict the dynamic pull-in characteristics more accurately than the conventional lumped model.     

Tilt-angle stabilization of electrostatically actuated micromechanical mirrors beyond the pull-in point

Recently proposed optical subsystems utilizing microelectromechanical system (MEMS) components are being developed for use in optical crossconnects, add-drop multiplexers, and spectral equalizers. Common elements to these subsystems are electrostatically actuated micromechanical mirrors that steer optical beams to implement the subsystem functions. In the past, feedback control methods were used to obtain precise mirror orientations to minimize loss through optical switch fabrics or to stabilize attenuation through spectral equalizers. However, the mirror tilt angle range is limited because of inherent instability beyond a critical tilt angle (pull-in angle), and the usual feedback schemes do not counteract this effect. This work presents a feedback control method to enable operation of electrostatic micromirrors beyond the pull-in angle, yielding advantages including greater scalability of switch arrays and increased dynamic range of optical attenuators. Both static and dynamic tilting behaviors of electrostatic micromirrors under the feedback control are studied. In addition, a practical implementation of the feedback control system by using linear voltage control law is developed. A voltage slightly larger than the pull-in voltage is first applied when the mirror is at small angle positions, and the voltage is then linearly reduced as the mirror approaches the desired position. Experimental measurements, showing that tilt angles beyond the pull-in point can be achieved, are in good agreement with theoretical analysis.     

Static and DC dynamic pull-in analysis of curled microcantilevers with a compliant support

Microsystem Technologies - In the present work, we propose an analytical model for the static and DC dynamic pull-in analysis of electrostatically actuated, initially curled microcantilever beams....     

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.     

Stability analysis of an electrostatically actuated out of plane MEMS structure

In the present research, stability and static analyses of microelectromechanical systems microstructure were investigated by presenting an out-of-plane structure for a lumped mass. The presented model consists of two stationary electrodes in the same plane along with a flexible electrode above and in the middle of the two electrodes. The nonlinear electrostatic force was valuated via numerical methods implemented in COMSOL software where three-dimensional simulations were performed for different gaps. The obtained numerical results were compared to those of previous research works, indicating a good agreement. Continuing with the research, curves of electrostatic and spring forces were demonstrated for different scenarios, with the intersection points (i.e., equilibrium points) further plotted. Also drawn were plots of deflection versus voltage for different cases and phase and time history curves for different values of applied voltage followed by introducing and explaining pull-in and pull-out snap-through voltages in the system for a specific design. It is worth noting that, at voltages between the pull-in and pull-out snap-through voltages, the system was in bi-stable state. Based on the obtained results, it was observed that the gap between the two electrodes and the applied voltage play significant roles in the number and type of the equilibrium points of the system.

     

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.     

VM-TEST: Mechanical property measurement using electrostatically actuated vertical MEMS test structures fabricated in thick metal layers

An efficient method is presented to determine the mechanical properties of thick metal layers using the pull-in voltage of electrostatically actuated structures. To fabricate these high aspect ratio beams without severe deformations, additional features were added, which made existing pull-in voltage equations inaccurate and therefore corrections were necessary. ANSYS Multiphysics was used to analyze the differences between ideal beams and the fabricated beams. To demonstrate the proposed approach, both nickel and gold devices were fabricated. To extract the material property values, a sum of least squares fitting scheme was used. A Young’s modulus of 186.2 and 60.8 GPa was obtained for nickel and gold structures respectively. Both values are significantly smaller than values reported for bulk material, but fall within the range of values reported in the literature.     

Flatness-Based Control of Electrostatically Actuated MEMS With Application to Adaptive Optics: A Simulation Study

Typical adaptive optics (AO) applications require continual measurement and correction of aberrated light and form closed-loop control systems. One of the key components in microelectromechanical system (MEMS) based AO systems is the parallel-plate microactuator. Being electrostatically actuated, this type of devices is inherently instable beyond the pull-in position when they are controlled by a constant voltage. Therefore extending the stable travelling range of such devices forms one of the central topics in the control of MEMS. In addition, though certain control schemes, such as charge control and capacitive feedback, can extend the travelling range to the full gap, the transient behavior of actuators is dominated by their mechanical dynamics. Thus, the performance may be poor if the natural damping of the devices is too low or too high. This paper presents an alternative for the control of parallel-plate electrostatic actuators, which is based on an essential property of nonlinear systems, namely differential flatness, and combines the techniques of trajectory planning and robust nonlinear control. It is, therefore, capable of stabilizing the system at any point in the gap while ensuring desired performances. The proposed control scheme is applied to an AO system and simulation results demonstrate its advantage over constant voltage control.1613     

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.     

M-TEST: A test chip for MEMS material property measurement usingelectrostatically actuated test structures

A set of electrostatically actuated microelectromechanical test structures is presented that meets the emerging need for microelectromechanical systems (MEMS) process monitoring and material property measurement at the wafer level during both process development and manufacturing. When implemented as a test chip or drop-in pattern for MEMS processes, M-Test becomes analogous to the electrical MOSFET test structures (often called E-Test) used for extraction of MOS device parameters. The principle of M-Test is the electrostatic pull-in of three sets of test structures [cantilever beams (CB's), fixed-fixed beams (FB's), and clamped circular diaphragms (CD's)] followed by the extraction of two intermediate quantities (the S and B parameters) that depend on the product of material properties and test structure geometry. The S and B parameters give a direct measure of the process uniformity across an individual wafer and process repeatability between wafers and lots. The extraction of material properties (e.g., Young's modulus, plate modulus, and residual stress) from these S and B parameters is then accomplished using geometric metrology data. Experimental demonstration of M-Test is presented using results from MIT's dielectrically isolated wafer-bonded silicon process. This yielded silicon plate modulus results which agreed with literature values to within ±4%. Guidelines for adapting the method to other MEMS process technologies are presented     

Application of the Generalized Differential Quadrature Method to the Study of Pull-In Phenomena of MEMS Switches

This paper reports on the pull-in behavior of nonlinear microelectromechanical coupled systems. The generalized differential quadrature method has been used as a high-order approximation to discretize the governing nonlinear integro-differential equation, yielding more accurate results with a considerably smaller number of grid points. Various electrostatically actuated microstructures such as cantilever beam-type and fixed-fixed beam-type microelectromechanical systems (MEMS) switches are studied. The proposed models capture the following effects: (1) the intrinsic residual stress from fabrication processes; (2) the fringing effects of the electrical field; and (3) the nonlinear stiffening or axial stress due to beam stretching. The effects of important parameters on the mechanical performance have been studied in detail. These results are expected to be useful in the optimum design of MEMS switches or other actuators. Further, the results obtained are summarized and compared with other existing empirical and analytical models.     

Full characterisation of pull-in in single-sided clamped beams

The most striking characteristic of the voltage-to-deflection curve of an electrostatically actuated beam is pull-in. The actual value of the pull-in voltage depends on: drive mode, temperature dependence and dielectric charging related drift. These aspects have been analysed using structures designed for a 9 V nominal pull-in voltage and fabricated in a commercially available epipoly process. Single-sided clamped beams have been used to avoid any influence of residual stress in the beam on pull-in. Typical results are: less than 5% variation of the pull-in voltage over a wafer, 0.17–1.9 V hysteresis depending on drive mode, a −1 mV/K TC and −12 mV drift during the first 2 weeks of operation.     

Dynamical characteristics of fluid-conveying microbeams actuated by electrostatic force

In this paper, the dynamic characteristics and pull-in instability of electrostatically actuated microbeams which convey internal fluids are investigated. A theoretical model is developed by considering the elastic structure, laminar flow and electrostatic field to characterize the dynamic behavior. In addition, the energy dissipation induced by the fluid viscosity is studied through analyzing the fluid–structure interactions between the laminar fluid flow and oscillating microbeam by comprehensively considering the effects of velocity profile and fluid viscosity. The results indicate that the system is subjected to both the pull-in instability and the fluid-induced instability. It is demonstrated that as the flow velocity increases, both the static pull-in voltage and the dynamic one decrease for clamped–clamped microbeams while increase for clamped-free microbeams. It is also shown that the applied voltage and the steady flow can adjust the resonant frequency. The perturbation viscous flow caused by the vibration of microbeam is manifested to result in energy dissipation. The quality factor decreases with the increment of both the mode order and flow velocity. However, when the oscillating flow dominates, the flow velocity has no obvious effect.     

Application of hybrid differential transformation/finite difference method to nonlinear analysis of micro fixed-fixed beam

Analyzing the dynamic response of electrostatic devices is problematic due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect and the nonlinear electrostatic force. To resolve this problem, this study presents an efficient computational scheme in which the nonlinear governing equation of the electrostatic device is obtained in accordance with Hamilton’s principle and is then solved using a hybrid differential transformation/finite difference method. The feasibility of the proposed approach is demonstrated by modeling the dynamic responses of two micro fixed-fixed beams with lengths of 250 and 350 μm, respectively. The numerical results show that the pull-in voltage reduces as the beam length increases due to a loss in the structural rigidity. Furthermore, it is shown that the present results for the pull-in voltage deviate by no more than 0.75% from those derived in the literature using a variety of different schemes. Overall, the results presented in this study demonstrate that the proposed hybrid method represents a computationally efficient and precise means of obtaining detailed insights into the nonlinear dynamic behavior of micro fixed-fixed beams and similar micro-electro-mechanical systems (MEMS)-based devices.     

Computer-aided generation of nonlinear reduced-order dynamicmacromodels. I. Non-stress-stiffened case

Reduced-order dynamic macromodels are an effective way to capture device behavior for rapid circuit and system simulation. In this paper, we report the successful implementation of a methodology for automatically generating reduced-order nonlinear dynamic macromodels from three-dimensional (3-D) physical simulations for the conservative-energy-domain behavior of electrostatically actuated microelectromechanical systems (MEMS) devices. These models are created with a syntax that is directly usable in circuit- and system-level simulators for complete MEMS system design. This method has been applied to several examples of electrostatically actuated microstructures: a suspended clamped beam, with and without residual stress, using both symmetric and asymmetric positions of the actuation electrode, and an elastically supported plate with an eccentric electrode and unequal springs, producing tilting when actuated. When compared to 3-D simulations, this method proves to be accurate for non-stress-stiffened motions, displacements for which the gradient of the strain energy due to bending is much larger than the corresponding gradient of the strain energy due to stretching of the neutral surface. In typical MEMS structures, this corresponds to displacements less than the element thickness, At larger displacements, the method must be modified to account for stress stiffening, which is the subject of part two of this paper