Reduced-order modeling of weakly nonlinear MEMS devices with Taylor-series expansion and Arnoldi approach

In this paper, we present a new technique by combining the Taylor series expansion with the Arnoldi method to automatically develop reduced-order models for coupled energy domain nonlinear microelectromechanical devices. An electrostatically actuated fixed-fixed beam structure with squeeze-film damping effect is examined to illustrate the model-order reduction method. Simulation results show that the reduced-order nonlinear models can accurately capture the device dynamic behavior over a much larger range of device deformation than the conventional linearized model. Compared with the fully meshed finite-difference method, the model reduction method provides accurate models using orders of magnitude less computation. The reduced MEMS device models are represented by a small number of differential and algebraic equations and thus can be conveniently inserted into a circuit simulator for fast and efficient system-level simulation.     

MEMS薄膜平均应力梯度及等效弹性模量在线提取方法

薄膜沿厚度方向的平均应力梯度及薄膜的弹性模量对器件性能有重要影响.提出了一种利用静电作用下悬臂梁的吸合电压提取薄膜沿厚度方向的平均应力梯度及等效弹性模量的方法,该方法的关键在于实现悬臂梁吸合电压的快速精确计算.考虑了悬臂梁由应力梯度引起的沿宽度方向的弯曲及实现其固定端接近理想固支的方法,提高了吸合电压的计算精度.实际模拟表明该测量方法计算速度快、精度高,能够应用于实际工艺过程中材料参数的在线测量.     

Generating efficient dynamical models for microelectromechanicalsystems from a few finite-element simulation runs

In this paper, we demonstrate how efficient low-order dynamical models for micromechanical devices can be constructed using data from a few runs of fully meshed but slow numerical models such as those created by the finite-element method (FEM). These reduced-order macromodels are generated by extracting global basis functions from the fully meshed model runs in order to parameterize solutions with far fewer degrees of freedom. The macromodels may be used for subsequent simulations of the time-dependent behavior of nonlinear devices in order to rapidly explore the design space of the device. As an example, the method is used to capture the behavior of a pressure sensor based on the pull-in time of an electrostatically actuated microbeam, including the effects of squeeze-film damping due to ambient air under the beam. Results show that the reduced-order model decreases simulation time by at least a factor of 37 with less than 2% error. More complicated simulation problems show significantly higher speedup factors. The simulations also show good agreement with experimental data     

Design and optimization of RF ICs with embedded linear macromodels of multiport MEMS devices

In this paper, we demonstrate efficient modeling approach for simulation, analysis, design, and optimization of multiport radio frequency microelectromechanical systems (RF MEMS) resonating structures embedded in RF circuits. An in‐house finite element method (FEM) solver is utilized to develop accurate and efficient macromodels that capture all the essential characteristics of the device. Using the datasets generated from the FEM simulations, the artificial neural network models are trained for two‐way mapping between the physical input and electrical output parameters. Realized model is implemented in a circuit simulator, enabling a simple yet accurate circuit simulator compatible modeling and optimization procedure instead of memory and time demanding FEM analysis. The derivation of dynamic macromodels with preserved electromechanical behavior of the multiport resonating structures is also presented. Capabilities of the proposed approach are demonstrated with several examples featuring capacitively actuated MEMS resonating structures: a clamped–clamped beam, a free–free beam, and a coupled clamped–clamped beam. © 2007 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2007.     

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.

     

Stroboscopic Imaging Interferometer for MEMS Performance Measurement

The insertion of microelectromechanical systems (MEMS) components into aerospace systems requires advanced testing to characterize performance in a space environment. Here, we report a novel stroboscopic interferometer test system that measures nanometer-scale displacements of moving MEMS devices. By combining video imagery and phase-shift interferometry with an environmental chamber, rapid visualization of the dynamic device motion under the actual operational conditions can be achieved. The utility of this system is further enhanced by integrating the interferometer onto the chamber window, allowing for robust interferometric testing in a noisy environment without requiring a floating optical table. To demonstrate these unique capabilities, we present the time-resolved images of an electrostatically actuated MEMS cantilevered beam showing the first-order to sixth-order plate modes under vacuum.     

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.     

Electronically probed measurements of MEMS geometries

Measurement of microelectromechanical systems (MEMS) geometry is critical for device design and simulation, material property extraction, and post-fabrication trimming. In this paper electrostatically driven laterally resonant comb-drive test structures with prescribed changes in spring width are used to ascertain systematic variations in process offsets (edge biases) and sidewall angles. The technique is both in situ and nondestructive. An analytical model for the resonant frequency, tuned with three-dimensional (3-D) simulations using MEMCAD, includes effects of a distributed mass, residual stress, and compliant supports. The model is corroborated by 3-D numerical simulations to validate the extraction approach. Fits of this model to experimental data determine the offset and sidewall angle of polysilicon devices fabricated by the Multi-User MEMS Processes of the Microelectronics Center of North Carolina, Research Triangle Park, NC     

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.     

Computer-aided generation of nonlinear reduced-order dynamicmacromodels. II. Stress-stiffened case

For pt. I see ibid., vol. 9, no. 2, p. 262-9 (2000). Reduced-order dynamic macromodels to describe the behavior of microelectromechanical system structures with stress stiffening are presented in this paper. The approach is based on potential and kinetic energy representations of selected fundamental modes of motion, modified to take account of stress stiffening. Energy data are calculated by several finite-element runs, fitted to polynomial functions, and used to develop the equations of motion according to Lagrangian mechanics. The accuracy and restrictions of these macromodels are shown     

Numerical simulation and analysis of electrically actuated microbeam-based MEMS capacitive switch

The MEMS capacitive switch based on fixed-fixed microbeam has garnered significant attention due to their geometric simplicity and broad applicability. The accurate model which describes the multiphysical coupled-field of MEMS capacitive switch should be developed to predict their electromechanical behaviors. The improved macromodel of the fixed-fixed microbeam-based MEMS capacitive switch is presented to investigate the behavior of electrically actuated MEMS capacitive switch in this paper, the macromodel provides an effective and accurate design tool for this class of MEMS devices because of taking account into some effects simultaneously including fringing field effect, midplane stretching effect, residual stress and multiphysical coupled-field effect. The numerical analysis of mechanical characterizations of electrically actuated microbeam-based MEMS capacitive switch are performed by the finite element Newmark method, and the performances of static and dynamic of MEMS capacitive switch are obtained. The numerical results show that, with only a few nodes used in the computation, the FEM-Newmark gives the identical results to other numerical methods, such as the shooting method and experiments. Moreover, the proposed model can offer proper and convenient approach for numerical calculations, and promote design of MEMS devices.     

基于受力分析的MEMS薄膜系统级模拟方法的研究

介绍一种MEMS系统级模拟方法.该方法直接分析运动物体的受力情况,并结合能量原理,利用受控源的反馈实现耦合作用,建立机电耦合MEMS薄膜的等效电路模型.利用该等效电路实现对薄膜动态行为的系统级模拟,并将Spice和有限元法,Saber的模拟结果进行对比,验证了该模拟方法应用于二维情况分析的有效性.     

Investigation of the internal stress effects on static and dynamic characteristics of an electrostatically actuated beam for MEMS and NEMS application

Internal stress is often encountered in fixed–fixed beam based devices with micron or sub-micron length scales during device fabrication or operation. In this paper, we have investigated the effects of internal stress on static and dynamic characteristics of an electrostatically actuated cylindrical beam. The beam has been modelled using Euler–Bernoulli theory including the nonlinearities due to beam stretching and electrostatic forcing. The analysis has been carried out by solving the governing differential equations using a Galerkin based multi-modal reduced order modelling technique. A standard collocation based numerical scheme has also been used to confirm the results of the reduced order method. Our study shows that internal stress significantly influences the static and dynamic characteristics of the beam. We also find that, when compressive internal stress is high, it is important to include higher modes in the reduced order model. A design technique to achieve high resonant frequency stability under temperature variation, for electrostatically actuated beam oscillators, has also been proposed as a result of this investigation.     

Characterization of contact electromechanics throughcapacitance-voltage measurements and simulations

Electrostatically actuated polysilicon beams fabricated in the multiuser MEMS process (MUMPs) are studied, with an emphasis on the behavior when the beam is in contact with an underlying silicon nitride dielectric layer. Detailed two-dimensional (2-D) electromechanical simulations, including the mechanical effects of stepups, stress-stiffening and contact, as well as the electrical effects of fringing fields and finite beam thickness, are performed. Comparisons are made to quasi-2-D and three-dimensional simulations. Pull-in voltage and capacitance-voltage measurements together with 2-D simulations are used to extract material properties. The electromechanical system is used to monitor charge buildup in the nitride which is modeled by a charge trapping model. Surface effects are included in the simulation using a compressible-contact-surface model. Monte Carlo simulations reveal the limits of simulation accuracy due to the limited resolution of input parameters     

An Efficient Macromodeling Methodology for Lateral Air Damping Effects

In this paper, a macromodeling methodology for lateral air damping effect is presented. This methodology employs a simplified governing equation, the Quasi-3D (Q3D) Stokes equation , and an Arnoldi-based model-order-reduction algorithm. A finite-difference (FDM) solver based on the Q3D-Stokes equation is implemented, and then the Arnoldi-based algorithm is used to create macromodels from the system matrices generated by the solver. This methodology can also be realized by using commercial MEMS packages for solid-model generation, and by using commercial finite-element (FEM) thermal packages for system-matrix generation. The generated macromodels are compatible with system-level modeling simulators, such as SPICE, Saber, or Simulink for fast transient and frequency analyses. It is demonstrated that the macromodels are at least 600 times more efficient than the FDM Q3D Stokes solver, while are still capable of capturing the three-dimensional (3-D) effect that usually requires very expensive 3-D FEM Stokes-flow calculations. Experimental results of comb-drive devices show that the error of the macromodel is less than 10%, which is a significant improvement when compared with the results by widely used 1-D analytical approaches. Finally, the guidelines of using this macromodeling methodology for typical MEMS devices are also provided.hfillhbox[1262]     

基于模态分析的静电驱动圆薄板宏模型建立方法

为了研究静电致动圆板静态和动态特性,在模态分析法的基础上,利用多维非线性函数的Levenberg-Marquardt拟合方法,将板的动能、弹性能和电容写成以模态坐标表示的解析式.结合Hamilton原理,导出静电致动微圆板的动力学特性方程的宏模型.以此为基础研究了板的静态特性及其三角波、方波信号激励下的动态响应.结果表明,模态分析方法能够考虑到残余应力的影响,所建立的动态宏模型不仅大大地减少了计算费用,而且具有足够的仿真精度.     

Identification of nonlinear dynamic models of electrostatically actuated MEMS

This paper focuses on the identification of nonlinear dynamic models for physical systems such as electrostatically actuated micro-electro-mechanical systems (MEMS). The proposed approach consists in transforming, by means of suitable global operations, the input–output differential model in such a way that the new equivalent formulation is well adapted to the identification problem, thanks to the following properties: first, the linearity with respect to the parameters to be identified is preserved, second, the continuous dependence on noise measurements is restored. Consequently, a simple least-square resolution can be used, in such a way that some of the difficulties classically encountered with identification methods are by-passed. The method is implemented on real measurement data from a physical system.     

The strain gradient effect in microelectromechanical systems (MEMS)

Metallic materials display strong size effect when the characteristic length of deformation is of the order of microns. The theory of mechanism-based strain gradient (MSG) plasticity established from the Taylor dislocation model has captured this size dependence of material behavior at the micron scale very well. The strain gradient effect in microelectromechanical systems (MEMS) is investigated in this paper via the MSG plasticity theory since the typical size of MEMS is of the order of microns (comparable to the internal material length in MSG plasticity). Through an example of a digital micromirror device (DMD), it is shown that the strain gradient effect significantly increases the mechanical strain energy in the DMD, and reduces the rotation time of the micromirror. However, the strain gradient has no effect on the critical bias voltage governing the fast rotation of the micromirror     

Simulation model for micromechanical angular rate sensor

A simulation model for an angular rate sensor, a gyroscope, is presented. The device is based on a micromechanical dual torsional mass system which is actuated electrostatically and sensed capacitively. Model equations describing a dynamic, non-linear system are first presented and then realized as an electrical equivalent circuit. The vibrational modes of the system are modelled with coupled resonator circuits. The electrostatic and Coriolis forces as well as variable capacitances in the small air gaps are modelled with non-linear controlled current sources. External forces, torques and electrical actuation can act as inputs to the device. The model presented allows numerical sensor simulations concurrently with the interfacing electronics in the time and frequency domains. The model is verified by comparing its simulation results to measured frequency responses and capacitance-voltage characteristics.