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.     

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.     

静电驱动微梁的节点分析法

采用加权残余法建立了静电驱动梁单元的节点法模型,在HSPICE中构建了相应的等效电路模型.以静电驱动的双端固支梁为例用ANSYS模拟软件对模型进行了验证,结果表明模型具有较高的精度.     

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.

     

A nodal analysis method for electromechanical behavior simulation of bow-tie shaped microbeams

In order to perform mixed-domain simulation of electrically actuated bow-tie shaped doubly-fixed microbeams, a nodal model for the trapeziform beam and corresponding equivalent circuit are developed by HSPICE. The nonlinearities including mid-plane stretching and electrostatic forcing are considered. In the modeling, the Galerkin residual method is used to discretize the partial differential equations of the trapeziform beam. It proves accurate in comparison with FEA simulation results and available experimental data. In particular, as a lumped behavioral model, it is executed much faster than FEM program. The proposed model could be expected to be useful in the optimum design of related MEMS devices.     

Design and modeling of a continuously variable piezoelectric RF MEMS switch

A structure for a piezoelectrically actuated capacitive RF MEMS switch that is continuously variable between the ON state and the OFF state has been proposed. The device is based on variable capacitance using a cantilever fixed at both ends that is actuated using a lead zirconate titanate thin film. Because the device is contactless, the reliability issues common in contact-type RF MEMS switches can be avoided. A comprehensive mathematical model has been developed in order to study the performance of the device, and allow for design optimization. Electrical measurements on test structures have been compared with the performance predicted by the model, and the results used to design a prototype RF MEMS switch. The model and simulations indicate the proposed switch structure can provide an insertion loss better than 0.7 dB and an isolation of more than 10 dB between 6 and 14 GHz with an actuation voltage of 22.4 V. The state of the device is continuously variable between the ON state and the OFF state, with a tunable range of capacitance of more than 15\(\times \).     

多晶硅双端固支梁高周循环下疲劳特性的分析

多晶硅固支梁是MEMS器件中较常见的可动部件,通过静电激励的方式对其进行疲劳振动加载;所用结构为面外运动结构,为了测试样品的加速疲劳特性,通过在固支梁面内引入缺陷的方式来增大应力水平值;器件在经历了1.72×1011次循环之后,微梁的谐振频率、振动幅度发生了较大偏移,其谐振频率的偏移量达到14.531 kHz,器件性能发生了严重的退化.研究结果表明,利用谐振频率的改变来表征材料性能的退化是一种准确、可行的方法,同时本文进一步分析指出,器件上引入凹槽缺陷的方法确实可起到加速疲劳的作用;可利用此方法制作不同应力水平幅度的结构进行振动载荷疲劳加载实验,从而得到固支梁结构疲劳加速因子.     

基于有限元法的MEMS加速度计热应力分析

残余应力对MEMS器件的力学性能、可靠性和寿命都有较大的影响。基于MEMS电容式加速度计,采用热结构耦合场分析方法,对热应力的影响进行了仿真分析,并和实验结果进行比较。结果表明热应力不是影响器件性能的主要因素,而仿真模型中未考虑到的一些物理因素和工艺误差则可能是主要因素。针对课题组加速度计的制备过程从工艺的角度提出了相应的改进措施,为加速度计温度补偿模型的完善以及改版设计提供参考。     

Design and simulation of a thermally actuated MEMS switch for microwave circuits

The design, modeling, and optimization of a novel, thermally actuated CMOS‐MEMS switch are presented in this article. This series capacitive MEMS switch solves the substrate loss and down‐state capacitance degradation problems commonly plaguing MEMS switches. The switch uses finger structure for capacitive coupling. The vertical bending characteristic of bimorph cantilever beams under different temperatures is utilized to turn the switch on and off. A set of electrical, mechanical, and thermal models is established, and cross‐domain electro‐thermo‐mechanical simulations are performed to optimize the design parameters of the switch. The fabrication of the switch is completely CMOS‐process compatible. The design is fabricated using the AMI 0.6 μm CMOS process and a maskless reactive‐ion etching process. The measured results show the insertion loss and isolation are 1.67 and 33 dB, respectively, at 5.4 GHz, and 0.36 and 23 dB at 10 GHz. The actuation voltage is 25 V and the power consumption is 480 mW. This switch has a vast number of applications in the RF/microwave field, such as configurable voltage control oscillators, filters, and configurable matching networks. © 2009 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2009.     

A new in situ residual stress measurement method for a MEMS thin fixed-fixed beam structure

A new method is described to measure the in situ residual stress state in a thin fixed-fixed beam structure used in microelectromechanical systems (MEMS). The methodology can be applied to devices at the anticipated operational and environmental temperatures. The new technique makes use of differences in the thermal expansion coefficient between the thin beam and the substrate. The residual stress distribution is determined by matching the thermal deflections from a finite element model (FEM) to measured deflections of the beam. All previous residual stress measurement methods for MEMS suspended structures reported a uniformly distributed residual stress. Experimental data coupled with the new analytical method suggests that this may not be adequate for the case of a suspended thin structure with nonplanar surface topology. A stress gradient through the thickness must be included in the determination of the stress state of the beam. The new method indicates a spatially varying residual stress distribution and is capable of de-coupling the mean stress and the stress gradient through the thickness. It was found through an extensive literature review that the quantification of the stress gradient in a thin suspended structure has never been reported. The de-coupling makes the prediction of the stress state at different temperature points possible. Details of the new method are demonstrated and discussed by the use of a capacitive radio frequency (RF) MEMS switch.     

Electromechanical model of RF MEMS switches

With the recent rapid growth of Radio Frequency Micro-Electro-Mechanical Systems (RF MEMS) switches, there has developed an emergent requirement for more accurate theoretical models to predict their electromechanical behaviors. Many parameters exist in the analysis of the behavior of the switch, and it is inconvenient for further study. In this paper, an improved model is introduced, considering simultaneously axial stress, residual stress, and fringing-field effect of the fixed-fixed bridge structure. To avoid any unnecessary repetitive model tests and numerical simulation for RF MEMS switches, some dimensionless numbers are derived by making governing equation dimensionless. The electromechanical behavior of the fixed-fixed bridge structure of RF MEMS switches is totally determined by these dimensionless numbers.The supports from the Key Project from the Chinese Academy of Sciences (No. KJCX2-SW-L2), projects from the NSFC (Nos 19928205, No. 50131160739 and No. 10072068), and the National 973 Project (Grant No. G1999033103) are gratefully acknowledged. The authors would like to thanks Prof. Yilong Hao of Institute of Microelectronics, Peking University, for helpful discussion.     

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.     

串联电容式RF MEMS开关设计与制造研究

介绍了一种串联电容式RF MEMS开关的设计与制造。所设计的串联电容式RF MEMS开关利用薄膜淀积中产生的内应力使MEMS桥膜向上发生翘曲,从而提高所设计的开关的隔离度,克服了串联电容式RF MEMS开关通常只有在1GHz以下才能获得较高隔离度的缺点。其工艺与并联电容式RF MEMS开关完全相同,解决了并联电容式RF MEMS开关不能应用于低频段(<10GHz)的问题。其插入损耗为-0.88dB@3GHz,在6GHz以上,插入损耗为-0.5dB;隔离度为-33.5dB@900MHz、-24dB@3GH和-20dB@5GHz,适合于3~5GHz频段的应用。     

Reliability design of thermally actuated MEMS switches based on V-shape beams

This paper presents the design and fabrication of the thermally actuated MEMS switches based on out-of-plane V-beams. The purpose of this research is to analyze the mechanical response of a V-thermal actuator fabricated from aluminum in order to improve the accuracy in response and to increase the switch lifetime. The actuation of this kind of switches is based on the thermal displacement of the mobile electrode under thermal load that is generated when the actuation voltage is applied. It can be used either as a capacitive switch or as a metal-to-metal one. The displacement of the mobile electrode for a given temperature is analytically calculated and validated both numerically and experimentally. Experimental investigations are performed on a macro-scale sample using a 3D digital image correlation measuring system, a heating source and a thermal camera for temperature monitoring. The first fabrication steps of the MEMS switch based on the V-beam thermal actuator are presented. The out-of-plane V-beams thermal MEMS switches can be monolithically integrated in RF applications.

     

梳齿谐振器宏模型的系统级应用

宏模型的获取是由Krylov子空间投影法结合器件级有限元分析的宏建模方法来实现,利用此模型和基于可重用IP设计思想的系统级组件建模方法建立由VHDL-AMS语言描述的梳齿谐振器系统级模型,在华大VDE仿真平台上搭建出了梳齿谐振器的系统模型,并进行一系列的仿真,包括时域阶跃、交流电压信号仿真和频域交流小信号仿真,仿真结果与商用仿真工具ANSYS仿真结果保持良好的一致.     

An improved analytical model for static pull-in voltage of a flexured MEMS switch

Microsystem Technologies - This paper presents the design of low-k meander based MEMS shunt capacitive switch with beam perforations. A closed form model to accurately calculate the pull-in voltage...     

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     

A comparative analysis of efficiency and reliability of capacitive micro-switches with initially curved electrodes

Enhancement of both efficiency and reliability of MEMS structures has always been an interesting and even essential issue for research community. This paper provides a comparative investigation in this field focusing on the role of initially curved electrodes of capacitive micro-switches. Four models have been introduced by appliance of curved microbeams as either upper or lower electrodes of a capacitive MEMS switch, as well as the conventional base model with straight both electrodes. By introducing a mathematical model and appropriate numerical procedure, the contact area between two electrodes, which has direct effect on the reliability has been estimated using Hertz relation for all models. The electromechanical coupling factor which is related to the efficiency of the switch has been calculated considering the stored mechanical and electrical energy of the system. The results have shown that by appliance of an initial curvature to the both electrodes, the estimated contact area can increase up to 279% compared to the conventional switches. Also, a switch with straight moveable electrode and curved substrate exhibits an increase in coupling factor up to 24% compared to the base model, while increasing the pull-in voltage of the switch.

     

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.