Equivalent area nonlinear static and dynamic analysis of electrostatically actuated microstructures

Nonlinear dynamic investigation of electrostatically actuated micro-electro-mechanical-system (MEMS) microcantilever structures is presented. The nonlinear analysis aims to better quantify, than the linear model, the instability threshold associated with electrostatically actuated MEMS structures, where the pull-in voltage of the microcantilever is determined using a phase portrait analysis of the microsystem. The microcantilever is modeled as a lumped mass-spring system. The nonlinear electrostatic force is incorporated into the lumped microsystem through an equivalent area of the microcantilever for a given electrostatic potential. Electro-mechanical force balance plots are obtained for various electrostatic potentials from which the static equilibrium positions of the microcantilever are obtained and the respective conservative energy values are determined. Subsequently, phase portrait plots are obtained for the corresponding energy values from which the pull-in voltage is estimated for the microsystem. This pull-in voltage value is in good agreement with the previously published results for the same geometric and material parameters. The results obtained for linear electrostatic models are also presented for comparison.     

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.     

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.     

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.

     

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.     

Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams

Conventional continuum theory does not account for contributions from length scale effects which are important in modeling of nano-beams. Failure to include size-dependent contributions can lead to underestimates of deflection, stresses, and pull-in voltage of electrostatic actuated micro and nano-beams. This research aims to use nonlocal and strain gradient elasticity theories to study the static behavior of electrically actuated micro- and nano-beams. To solve the boundary value nonlinear differential equations, analogue equation and Gauss–Seidel iteration methods are used. Both clamped-free and clamped–clamped micro- and nano-beams under electrostatical actuation are considered where mid-plane stretching, axial residual stress and fringing field effect are taken into account for clamped–clamped cases. The accuracy of solution is evaluated by comparison of the pull-in voltages of different micro-electro-mechanical systems with those results previously published in the literature. A main drawback of the previous theoretical researches using nonlocal or strain gradient methods was that they don’t account for effects of the size on the Young modulus of the beam and merely they adjust the length scale parameters for small sizes to fit data with experimental results. In the present study, the experimental voltages for static pull-in of different micro- and nano-beams are used to estimate the silicon Young’s modulus, nonlocal and length scale parameters. Using the estimated parameters, pull-in voltages of silicon micro- and nano-beams based on strain gradient and nonlocal theories are compared with respect to each other and also with the experimental and previous theoretical results. The conducted results demonstrate that the predicted pull-in voltages using proposed strain gradient theory will give the best fit with experimental observations.     

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 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.     

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.     

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.     

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     

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     

Bidirectional Electrostatic Actuator Operated With Charge Control

A method for driving an electrically floating, bidirectional electrostatic actuator using high-speed switching and charge control has been proposed and verified experimentally. A macroscopic experiment is used to model the operation of a surface-micromachined microelectromechanical systems (MEMS) device having the form of two parallel electrodes, one rigid, and the other flexible. The movable, or “free” electrode, is connected to ground via a switching device, and the stationary electrode is connected to a single voltage source. The free electrode is alternately charged by induction, then left floating and subjected to a field produced by the stationary electrode. This charge-control mode of operation overcomes the travel range limits associated with the well-known “snap-through” instability. Our proof-of-concept experiment shows that a periodic charging/driving cycle increases the possible travel range of the free electrode by a factor of two over that possible without charge control. The configuration has the potential for use in a variety of optical MEMS applications involving mirror surfaces that must be exposed to the outside world.hfillhbox[1060]     

The SiOG-based single-crystalline silicon (SCS) RF MEMS switch with uniform characteristics

This paper details single-crystalline silicon (SCS) direct contact radio frequency microelectromechanical systems (RF MEMS) switch designed and fabricated using an SiOG (silicon-on-glass) substrate, so as to obtain higher fabrication and performance uniformity compared with a conventional metal switch. The mechanical and electrical performances of the fabricated silicon switch have been tested. In comparison with a conventional metallic MEMS switch, we can obtain higher productivity and uniformity by using SCS, because it has very low stresses and superior thermal characteristics as a structural material of the switch. Also, by using the SiOG substrate instead of an SOI substrate, fabrication cost can be significantly reduced. The proposed switch is fabricated on a coplanar waveguide (CPW) and actuated by electrostatic force. The designed chip size is 1.05 mm/spl times/0.72 mm. Measured pull-in voltage and actuation voltage were 19 V and 26 V, respectively. Eighteen identical switches taken randomly throughout the wafer showed average and standard deviation of the measured pull-in voltage of 19.1 and 1.5 V, respectively. The RF characteristics of the fabricated switch from dc to 30 GHz have been measured. The isolation and insertion loss measured on the four identical samples were -38 to -39 dB and -0.18 to -0.2 dB at 2 GHz, respectively. Forming damping holes on the upper electrode leads to a relatively fast switching speed. Measured ON and OFF time were 25 and 13 /spl mu/s, respectively. In the switch OFF state, self-actuation does not happen up to the input power of 34 dBm. The measured holding power of the fabricated switch was 31 dBm. Stiction problem was not observed after 10/sup 8/ cycles of repeated actuation, but the contact resistance varied about 0.5-1 /spl Omega/ from the initial value.     

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.     

Snap-Through and Pull-In Instabilities of an Arch-Shaped Beam Under an Electrostatic Loading

The snap-through and pull-in instabilities of the micromachined arch-shaped beams under an electrostatic loading are studied both theoretically and experimentally. The pull-in instability that results in a system collision with an electrode substrate may lead to a system failure and, thus, limits the system maximum displacement. The beam/plate structure with a flat initial configuration under an electrostatic loading can only experience the pull-in instability. With the different arch configurations, the structure may experience either only the pull-in instability or the snap-through and pull-in instabilities together. As shown in our computation and experiment, those arch-shaped beams with the snap-through instability have the larger maximum displacement compared with the arch-shaped beams with only the pull-in stability and those with the flat initial configuration. The snap-through occurs by exerting a fixed load, and the structure experiences a discontinuous displacement jump without consuming power. Furthermore, after the snap-through jump, the structures are demonstrated to have the capacity to withstand further electrostatic loading without pull-in. Those properties of consuming no power and increasing the structure deflection range without pull-in is very useful in microelectromechanical systems design, which can offer better sensitivity and tuning range.     

Two-Dimensional MEMS Scanner for Dual-Axes Confocal Microscopy

In this paper, we present a novel 2-D microelectromechanical systems (MEMS) scanner that enables dual-axes confocal microscopy. Dual-axes confocal microscopy provides high resolution and long working distance, while also being well suited for miniaturization and integration into endoscopes for in vivo imaging. The gimbaled MEMS scanner is fabricated on a double silicon-on-insulator (SOI) wafer (a silicon wafer bonded on a SOI wafer) and is actuated by self-aligned vertical electrostatic combdrives. Maximum optical deflections of plusmn4.8deg and plusmn5.5deg are achieved in static mode for the outer and inner axes, respectively. Torsional resonant frequencies are at 500 Hz and 2.9 kHz for the outer and inner axes, respectively. The imaging capability of the MEMS scanner is successfully demonstrated in a breadboard setup. Reflectance images with a field of view of are achieved at 8 frames/s. The transverse resolutions are 3.94 mum and 6.68 mum for the horizontal and vertical dimensions, respectively.     

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.     

A reduced-order model for electrically actuated microbeam-based MEMS

We present an analytical approach and a reduced-order model (macromodel) to investigate the behavior of electrically actuated microbeam-based MEMS. The macromodel provides an effective and accurate design tool for this class of MEMS devices. The macromodel is obtained by discretizing the distributed-parameter system using a Galerkin procedure into a finite-degree-of-freedom system consisting of ordinary-differential equations in time. The macromodel accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and the nonlinear electric forces generated by the capacitors. A new technique is developed to represent the electric force in the equations of motion. The new approach allows the use of few linear-undamped mode shapes of a microbeam in its straight position as basis functions in a Galerkin procedure. The macromodel is validated by comparing its results with experimental results and finite-element solutions available in the literature. Our approach shows attractive features compared to finite-element softwares used in the literature. It is robust over the whole device operation range up to the instability limit of the device (i.e., pull-in). Moreover, it has low computational cost and allows for an easier understanding of the influence of the various design parameters. As a result, it can be of significant benefit to the development of MEMS design software.