Modeling and non-linear responses of MEMS capacitive accelerometer

A theoretical investigation of an electrically actuated beam has been illustrated when the electrostatic-ally actuated micro-cantilever beam is separated from the electrode by a moderately large gap for two distinct types of geometric configurations of MEMS accelerometer. Higher order nonlinear terms have been taken into account for studying the pull in voltage analysis. A nonlinear model of gas film squeezing damping, another source of nonlinearity in MEMS devices is included in obtaining the dynamic responses. Moreover, in the present work, the possible source of nonlinearities while formulating the mathematical model of a MEMS accelerometer and their influences on the dynamic responses have been investigated. The theoretical results obtained by using MATLAB has been verified with the results obtained in FE software and has been found in good agreement. Criterion towards stable micro size accelerometer for each configuration has been investigated. This investigation clearly provides an understanding of nonlinear static and dynamics characteristics of electrostatically micro cantilever based device in MEMS.


Introduction
Technology has been improved in such a way that we can build products so small that it cannot be seen by our Human eye.In our 21 st century, MEMS has been identified as one of the emerging and promising technology and has the ability to revolutionize both Industrial and consumer products by combining Silicon based microelectronics with micromachining technology [1].Recently, MEMS devices are used almost in all fields with some well-known examples of MEMS devices applications in our daily life include micro actuators, transducers, Pressure and flow sensors [2].Today, this fast developing Technology has emerged in such a way that a new set of applications have been opened up.For example, if we consider Medical field, we are now able to make medical and biomedical devices so small that they can be directly injected into our blood stream.These intelligently monitor blood substances and release drugs wherever necessary, thus selectively killing sick cells or germs [3,5].Most common MEMS device now days are MEMS accelerometer..Normally accelerometer consists of a simple inertial mass which is suspended by springs, which is acted upon by acceleration forces which causes the mass to deflect from its initial position.Now this deflection is converted into an electrical signal, which is appeared at sensor output.MEMS technology has been applied to accelerometer, which is relatively new development.These type of transducers are typically constructed with a deformable diaphragm, beams with both ends fixed (commonly known as fixed-fixed beams) and cantilever beams (single side fixed beams).Their geometries are separated from a fixed ground plane by an air-gap of suitable thickness.To drive these devices we need drive mechanism, which includes constant voltage source or constant current source [10,11].In most of the cases, most electrostatic based MEMS devices are of constant voltage drive [12].There are many types of accelerometers developed and reported in the literature.The vast majority is based on piezoelectric crystals, but they are too big and clumsy.People tried to develop something smaller that could increase capability and started searching in the field of microelectronics.They developed MEMS (Micro electromechanical systems) accelerometers.The electrostatic force developed with the constant voltage drive mode is non-linear and this gives rise to a phenomenon known as Pull-in voltage.This is a phenomenon which causes an electrostatically actuated beam to collapse on ground plane if the drive voltage exceeds permissible limit depending upon the geometry of our device.But we observe that accurate determination of pull in voltage is critical in our design process to determine sensitivity, instability [11].We also have methods to determine pull in voltage of electrostatically fixed-fixed actuators.Another important which accounts for consideration is squeezed film effects in MEMS devices to perform squeeze-film damping measurements for different film thicknesses.In spite of a rigorous research has carried out in this area, very limited authors have considered combine effects of nonlinear electrostatic actuation and squeeze-film damping in their dynamic model.Hence, nonlinear dynamics of electrostatic systems still remains unexplored.Therefore, an attempt has been made to study the nonlinear characteristics of electrically actuated micro beams for two different set of geometric configurations.While the static analysis mainly focuses pull in voltage analysis for studying the structural instability, dynamics analysis has been carried out for investigating the transient behaviors at the dynamic pull in voltage.We investigate the linear and nonlinear characteristics of micro beam which is actuated under applied voltages and solving the boundary value problem numerically is discussed.Specifically, the effect of nonlinearities on both static and dynamics analysis has been investigated thoroughly.Results obtained numerically using MATLAB has been compared with results obtained in FE software.

Model description
. Now in order to analyze the static and dynamic behavior of the beam, reduced order model based on Galerkin's decomposition with undamped Linear Eigen modes as base functions is developed.The deflection of the beam is given as: Here, ‫ݍ‬ ሺ‫ݐ‬ሻ is time varying generalized coordinates and ߮ ሺ‫ݔ‬ሻ are eigenmodes of the associated linear undamped homogeneous equation.In our case, we consider first mode so the equation becomes: ( , ) ( ) ( ) w x x q τ = ϕ τ

Results and discussion
The above differential equation is solved using MATLAB to obtain numerical solution.The equation includes a nonlinear term so the system can have multiple solutions and the solution depends on the initial guess we took and the accurate solution is obtained for absolute tolerance of 10 -6 .Commands such as BVP4 and BVP5 have been used to implement in MATLAB.The micro beam was modeled using following specifications: Air gap was modeling using relative permittivity value 1.
The beam resides in an air filled chamber that is electrically insulated.The upper side of the chamber has a grounded electrode.Polysilicon is assumed to be heavily doped so that electric field penetration into the structure can be neglected.The micro beam with length 300 μ m, width 20 μ m, thickness 2 μ m has been taken.Meshing was done using with maximum element size = 4 and minimum element size = 0.064 and using free Quad elements.

Case 1: Clamped-free configuration
Figure 2 show the deflection of the micro beam for the cantilever and double clamped beams actuated under applied voltage where red color shows the deflection with maximum amplitude obtained by using ES software.Figures 3 and 4 depict the variation of deflection of the clamped-free beam and subsequent end gaps under electrostatic actuation.It is observed that deflection slope is sharply increased for high value of applied voltage.The deflection of free end of the microcantilever under the influences of applied voltage has been illustrated in Fig. 4 and it has been observed that the pull in condition is anticipated when both stable and unstable solution coalesces at a certain value.Here, while upper branch solution represents the unstable solution, lower branch solution is indicating the stable solution.The pull-in results obtained in MATLAB has been compared with the same obtained in ES software and has been found very good agreement.The time history for the tip deflection obtained in Fig. 5 predicts the dynamic pull-in voltage.It is observed that for certain voltage above the critical value, the motion exposes to a divergent motion and the beam abruptly collapses to on the ground electrode.The qualitative phase plot for the corresponding time history is illustrated in Fig. 6.

Movable electrode
Movable electrode It is observed that the dynamic pull-in of the micro beam took place at an applied voltage is well below than that of static counterpart.

Conclusions
Electrically actuated micro beam has been analyzed theoretically when the electrostatic-ally actuated microcantilever beam is separated from the electrode by a moderately larger gap.Two distinct types of geometric configurations of MEMS accelerometer are analyzed.For each configuration, both static and dynamic pull in analysis have been carried out and a possible comparison between the configurations has been illustrated.Higher order nonlinear terms have been taken into account in the dynamic model in addition to nonlinear of gas film squeezing damping effect.The theoretical results obtained by using Engineering Simulation software has been verified with the results obtained in MATLAB and found to be in good agreement.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, Fig.1a: Clamped-free configuration

Fig 6 :Fig. 7 :
Fig 6: Phase plot of the model of the microcantilever excited by various applied DC voltages