Finite Element Modeling and Simulation of Electrostatic Micro Actuators
T Murali1*, P Shiva Kishore Reddy2, P Dasthagiri3
1 Department of Mechanical Engineering, YSR Engineering College of Yogi Vemana University, Andhra Pradesh, India
2,3 Department of Mechanical Engineering, Siddartha Institute of Engineering and Technology, Hyderabad, India
Abstract— Electrostatic actuators are widely used in the MEMS devices because of their simple construction and compatibility with micro fabrication processes. However, the interaction of non-linear electrostatic force with linear elastic force results in a phenomenon called pull-in instability, which imposes serious limits on the travelling range of the micro actuator. There are two types of pull-in instabilities: Static and Dynamic. Accurate prediction of static and dynamic pull-in parameters (pull-in voltage and pull-in displacement) is hence essential in the design of such micro devices. The present work deals with the analysis of both the pull-in instabilities in Electro Static Micro Cantilever (ESMC) actuator. Static and dynamic analysis is carried out using a simulation technique based on finite element method using COMSOL multiphysics. The prismatic configuration of ESMC actuators has been taken for the analysis.
The actuator is modeled and simulated in COMSOL. Static and dynamic pull-in parameters of the prismatic ESMC are obtained through COMSOL. The results of static and dynamic analysis are compared with the results of Ritz energy technique. They are good in agreement.
Keywords – Pull-in instability, Prismatic micro cantilever, COMSOL, Finite Element Method
1. INTRODUCTION
Micro-Electro-Mechanical Systems (MEMS) is the integration of mechanical elements, sensors, actuators, and electronics ranging in size from a few micrometers to millimeters on a common silicon substrate through micro fabrication technology.
MEMS is a rapidly expanding field of multi- disciplinary technology, which takes the advantage of semiconductor fabrication processes to produce micro-scale mechanical, electrical, optical, and other devices. MEMS devices are emerging as products in both commercial and defense markets such as automotive, aerospace, medical, industrial process
control, electronic instrumentation, and telecommunications.
2.PRINCIPLE OF ELECTROSTATIC ACTUATION AND PULL-IN INSTABILITY An actuator typically is a mechanical device that takes energy created by air, electricity, or liquid, and converts that into some kind of motion. Various physical properties of different materials and their interactions are used to achieve the desired sensing and actuation in the micro domain. The electromechanical operation of an electrostatic micro actuator can be best understood by studying the one dimensional parallel plate model as shown in Fig. 1 [1]. It consists of a pair of rigid electrodes called as movable and fixed electrodes, which are maintained at different electric potentials and separated by a nominal gap g0.
Fig. 1 Parallel Plate Model of an Electrostatic Actuator [1]
This configuration forms a capacitor having capacitance C given by,
0
0
A C Q
V g
where, Q is the charge, V is the applied voltage, ԑ0 is the permittivity of free space, A is the area of overlap and g0 is the distance between the two parallel plates (electrodes). A small increase in the potential difference leads to the generation of the electrostatic force of attraction that pulls the movable electrode in the downward direction. Thus by controlling the potential difference between the two capacitor
electrodes, one can have control over the displacement of the movable electrode.
While the electrostatic force of attraction is in action, the stiffness of the movable electrode offers the required restoring force to maintain the system in equilibrium. The electrostatic force being a surface force, varies as per the inverse square law, i.e. the force of attraction between the two electrodes is inversely proportional to the square of the distance between them. It thus explains the nonlinear nature of the electrostatic force. However, the mechanical restoring force offered by the movable electrode is linearly proportional to its displacement. The interaction between this linear and nonlinear pair of forces leads to an interesting phenomenon termed as the pull-in instability. The force equilibrium relates the applied voltage to the displacement of movable plate. When the differentiation of this equation with respect to displacement is equated to zero, it leads to the point of pull-in instability.
Fig. 2 depicts the voltage - displacement characteristics of a typical parallel plate actuator. The normalized voltage is referred on abscissa, while the normalized displacement is referred on the ordinate.
Fig. 2 Voltage - Displacement Characteristics of the Parallel Plate Model
The following non-dimensional quantities are defined in the analysis
*
0
u u
g
,2
* 0
3 0
V AV
g
The stable range and pull-in voltage were found out to be
max 0
3 x g
3 0 0
8
PI
27 V g
A
where α denotes the stiffness of the linear suspension associated with the movable electrode. Fig. 2 shows a distinct point of bifurcation in voltage-displacement
response of the system, at which the slope of the curve becomes infinity. It is called as the static pull-in point. Each state before this point is a state of stable equilibrium and is shown by a thick solid line in Fig.
2. Every state after the pull-in point is an unstable state and is not practically possible. This is shown by a dotted curve in Fig. 2. Thus, pull-in can be perceived as the threshold between the stable and the unstable equilibrium states. The analysis indicates that the maximum travel range of the parallel plate model is limited to 1/3rd of the initial gap, leaving nearly 66.67% of the maximum available travel range unused.
Different researchers proposed different closed form empirical models to estimate the static pull-in parameters of micro beams having simple forms of width variation. In this regards, Osterberg and Senturia [3] first proposed the closed-form empirical models (pull-in models) to estimate the pull-in voltage of prismatic micro beams (i.e. constant width along the entire beam length). The closed-form relations for static pull-in voltage proposed by Weber and Wang [4] pertain to the cantilever beams, whose width increase towards the free-end. However, such a configuration assists the pull-in instability, and hence results in a reduced travel range than that of a prismatic beam. In addition, the formulae developed by Weber and Wang [4] have limited applicability since they are derived for fixed values of the beam- thickness and the initial gap between the two electrodes forming an actuator.
3.DYNAMIC ANALYSIS OF PARALLEL PLATE ACTUATOR
The quasi-static analysis assumes an extremely slow rate of application of the external force (voltage in the static analysis done above), so that equilibrium is allowed to exist at every step. However, in many real time situations, the force is not applied quasi- statically. Many a times, e.g. in micro switches, the actuators are driven by sudden application of a voltage signal, which can be closely approximated by a step voltage signal. In a step-loaded case, the pull-in instability occurs at voltages lower than the static pull-in voltage of the device [2]. Also, the pull-in range in such a case is greater than the static pull-in range. It is thus important to analyze the response of the micro actuators in the dynamic regime. It is observed that, the response is periodic up to a certain value of the voltage, and even a small increase in that value results in a periodic response of the system.
This critical value of the applied voltage is referred as
the dynamic pull-in voltage (Vpd) and the critical displacement corresponding to this voltage is referred as the dynamic pull-in displacement (upd). A comparison between the normalized values of the static and the dynamic pull-in parameters is shown in Table 1.
Table 1 Static and Dynamic pull-in Parameters of Parallel Plate Actuator
Non-dimensional parameters
u* V*
Static parameters 0.3333 0.5433 Dynamic parameters 0.4999 0.5000
Apart from the immediate consequence of pull-in i.e. the snapping of two electrodes causing the short circuit, it can physically damage the system. It limits the range of stable equilibrium states of the system, and in this respect limits the performance of the actuator.
4.PROBLEM DESCRIPTION
Here, the static and dynamic pull-in analysis of cantilever beams is obtained by Finite element analysis using COMSOL Multi physics. It is a powerful interactive environment for modeling and solving all kinds of scientific and engineering problems based on partial differential equations.
The dimensions and material properties of prismatic ESMC actuator are given in Table 2.
Table 2 Parameters of Prismatic Actuator
L b h g0 E ν
100 µm 50 µm 3 µm 1 µm 169 GPa 0.06
where, L is the length of micro beam, b and h are the width and thickness of the beam, g0 is the gap between movable and fixed electrode. E and ν are the Young's modulus and Poisson's ratio of the micro beam material. The material used for fabrication of actuator is poly silicon.
4.1 Modeling in COMSOL Multiphysics
The COMSOL multiphysics model requires three application modes: one for the structural deformation, which uses the reference frame and is only active in the beam; one for the electrostatic field, which exists everywhere and uses the spatial frame; and one moving mesh (ALE-Arbitrary Lagrangian Eulerian)
application mode, which defines the relation between the spatial frame and the reference frame.
Geometry off the Prismatic actuator is shown in the Fig.3. The beam is placed in an air-filled chamber that is electrically insulated. However, the lower side of the chamber has a grounded electrode. The model considers a layer of air 20 µm thick both above and to the sides of the beam, and the air gap between the bottom of the beam and the grounded layer is initially 1 µm.
Fig. 3 Geometry of Prismatic Actuator 4.2 Meshing in COMSOL Multiphysics
The electrostatic and structural domain has been meshed with quadratic tetrahedral elements.
Tetrahedral elements generated in the beam having ten nodes and each node having three degrees of freedom, translations in x, y and z directions. The elements generated in the chamber having one degree of freedom i.e. electric potential. The prismatic ESMC actuator after meshing is shown in Fig. 4.
Fig. 4 Prismatic Actuator after Meshing The mesh statistics have been given below.
Number of tetrahedral elements generated:
4378
Number of boundary(triangular) elements generated: 2044
Number of degrees of freedom: 44032 4.3 Steps for Dynamic Analysis in COMSOL
The dimensions and material properties of the actuators are same as given in Table 2. The density of the material (ρ) is 2231 Kg/m3. For the dynamic analysis in COMSOL, the steps for modeling,
meshing and applying boundary conditions are similar to the static analysis which is explained in the section IV (A) and IV (B).
For the dynamic analysis also COMSOL using Arbitrary Lagrangian Eulerian (ALE) method [5] to account for geometry changes associated with the deformation in this analysis, first the modal analysis of the prismatic cantilever beam has been done in COMSOL and extracted first three transverse natural frequencies. In the time-history response of the tip of the micro beam, time (µs) will be taking on the x-axis and the tip deflection (µm) will be taking on the y- axis. First the fundamental period of each cross section has been determined which is given by 1/f1, where f1 is the first mode natural frequency. Initial time on the x-axis is taken as 0 and final time is taken as approximately 3 times of the fundamental period in order to get one complete cycle of periodic response of the micro beam. The response of the micro beam is periodic up to a critical value of the applied voltage known as the dynamic pull-in voltage. When the magnitude of the applied voltage exceeds this critical value, the movable electrode performs a non-periodic motion and strikes the bottom fixed electrode. The critical applied voltage and the corresponding amplitude of oscillations are together known as the dynamic pull-in parameters of the micro actuator system.
Here, the voltage applied to the movable electrode is step voltage. Applying voltage suddenly at 0 seconds is hypothetical, so in order to get the response of the beam from 0 sec, 50 values from 0 to 5x10-10 sec have been given as the starting time range and after that the response of the beam from 5x10-10 sec to the final time by taking the step size as 1/500 f1 for the analysis. The final times and step size are given in Table 3.
Table 3 Details of Final Time and Time Step Size of ESMC Actuator
Configuration Prismatic beam
Fundamental frequency (Hz) 421951.915
Final time (µs) 7.110
Step size (ns) 4.739
By using these final times and step sizes, the dynamic analysis of the configuration has been done in COMSOL. It requires the computation time of the order of few hours to obtain the accurate estimates of dynamic pull-in parameters.
5. RESULTS AND DISCUSSIONS 5.1 Comparison of Static Pull-in Parameter
Finite element modeling and analysis of ESMC actuators having prismatic cross section has been discussed in section IV. The static pull-in parameters of prismatic actuator (obtained from COMSOL) are compared with the results obtained through Ritz energy technique is shown in Table 4. Percentage difference between two results is less than 1%.
Table 4 Comparison of Static Pull-In Parameters Obtained from COMSOL with Ritz Energy Results
for Prismatic Actuator Static pull-in
parameters
COMSOL results
Ritz energy results
Difference (%) Pull-in voltage
(V ) 38.376 38.063 0.81
Pull-in displacement
(µm)
0.449 0.448 0.22
Fig. 5 shows the boundary plot for the static pull-in displacement (µm) at pull-in voltage (volts).
Fig. 5 Boundary Plot for Prismatic Actuator at Static Pull-in Voltage
Fig. 6 represents a graph plotted with Voltage (volts) on X-axis and Tip deflection (micron) on Y- axis using the results obtained from both COMSOL and Ritz energy technique. Graph shows that both results are in good agreement.
Fig. 6 Comparison of COMSOL and Ritz Energy Results for Prismatic Actuator.
5.2 Comparison of Dynamic Pull-in Parameter The steps for dynamic analysis in COMSOL for the ESMC actuators have been explained in section IV (C). Here, time-history responses of the tip of the prismatic actuator are given. Fig. 7 represents the time history response of the tip of the prismatic actuator, for applied voltages on either side of the dynamic pull-in voltage.
Fig. 7 Time-History Response of the Tip of the Prismatic Actuator
We can observe that the response of the micro beam is periodic for all voltages up to 34.9 volts and becomes non-periodic for 34.95 volts, which is identified as the dynamic pull-in voltage of prismatic actuator. The response of the micro beam is extremely sensitive near the dynamic pull-in voltage that's why it is difficult to extract a highly precise estimate of dynamic pull-in displacement using finite element simulations. Table 5 gives the comparison of dynamic pull-in voltage with pull-in voltage obtained from Ritz energy technique.
Table 5 Static and Dynamic Pull-in Parameters of Parallel Plate Actuator
Dynamic pull- in parameters
COMS OL results
Ritz energy results
Difference (%)
Pull-in voltage
(V ) 34.95 34.45 1.43
5.3 Comparison of Static and Dynamic Pull-in Parameters
Here, we are going to compare static and dynamic pull-in parameters of the prismatic beam. Table 6 represents the comparison of static and dynamic pull- in parameters of prismatic beam obtained from COMSOL. The dynamic pull-in displacement indicated in this table is obtained from Ritz energy technique.
Table 6 Comparison of Static and Dynamic Pull-in Parameters
% decrease in pull-in voltage
% decrease in pull-in displacement Static pull-
in parameters
Vps 38.376
8.93 31.55
ups 0.449 Dynamic
pull-in parameters
Vpd 34.950 upd 0.656
From the Table 6, we can say that static pull-in voltage is more compared to the dynamic pull-in voltage, it is because of the inertial effects which are considered in dynamic analysis and the static pull-in displacement is less than the dynamic pull-in displacement. If we compare the static and dynamic pull-in parameters, the percentage decrease in pull-in voltage for the prismatic cross section is observed as 8.93 and the percentage increase in pull-in displacement is 31.55.
6.CONCLUSION
In this work, a simulation technique based on finite element method has been introduced in order to predict the static and dynamic pull-in parameters of electro statically actuated micro-cantilever beams having prismatic cross-section. COMSOL Multi physics is used for modeling and simulation of prismatic ESMC actuator. The static and dynamic pull-in parameters obtained through COMSOL are compared with the results obtained from Ritz energy technique. The variation in results is approximately less than 2%. It can be concluded that finite element method is a good technique to determine the static and dynamic pull-in parameters of ESMC actuators.
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