Nonlinear Pull-In Study of Electrostatically Actuated MEMS Structures Rakesh Kalyanaraman 1, # Muthukumaran Packirisamy 2, Rama B Bhat 3 Micromechatronics Laboratory, Concave Research Centre, Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd. Ouest, Montreal, QC H3GJM8, Canada. 'Graduate Research Assistant, Email:
[email protected], 2Assistant Professor, Ph: 1-514-8482424 #7973, Email:
[email protected] 3 Professor, Email:
[email protected]
Abstract
The pull-in voltage of an electrostatically actuated Micro Electro Mechanical System (MEMS) is determined using the phase portrait analysis of the system. The velocity equation for a simple mass-spring model is derived and the phase portraits are presented. A continuous cantilever system is modelled as a lumped mass-spring system and the method of estimation of its equivalent stiffness, equivalent mass and equivalent areas are explained. The equilibrium positions are obtained from the force balance plot for different voltages and their corresponding conservative energy values are determined. The phase portraits are given for different voltages with their respective energy values from which the pull-in voltage is obtainedfor this system. This pull-in voltage value is in very close agreement with the previously published results for the same geometric and material parameters. 1. INTRODUCTION
mechanism which is an effective way of preventing the instability. A "voltage control algorithm" has been applied by Chu et al. [6] which is based on controlling the voltage according to a feedback loop that exactly prevents the pull-in. The effect of nonlinearity has to be taken into account for getting more exact solutions to this instability. The use of numerical methods to solve the nonlinear Jacobian matrix is explained [7] as a solution to achieve greater accuracy of predicting the pull-in. The nonlinear dynamics of a MEMS structure with time varying capacitors [8] is studied using the elliptic integral method and a more exact solution for predicting the pull-in and frequency responses is given. This nonlinearity in the electrostatic MEMS structures can be compared with the nonlineanrty in the case of a simple pendulum [9]. When the initial displacement is comparatively small, the system can be considered linear. But when the initial displacement is large enough, the system becomes dominantly nonlinear and the approximations will not yield good results. Thus, the concept of phase portrait becomes useful and can clearly show the nonlinear behaviour of the system.
Micro Electro Mechanical System (MEMS) actuators have numerous advantages because of size, cost and the feasibility for bulk microfabrication. Different actuation methods such In the present study, a simple mass-spring model is taken and as piezoelectric, thermal, electromagnetic and electrostatic its phase portrait is obtained. Then, a continuous cantilever principles [13] have been employed for MEMS devices. The system [1] is converted into an equivalent lumped model with electrostatic actuation is one of the most commonly used its equivalent stifffiess, equivalent mass and equivalent areas. The pull-in voltage of the lumped system is first found by methods because of its simplicity and ease of operation. means of the force balance plot. Then it is also obtained from The MEMS devices must operate in the safe working range the phase plot and both are compared with the published and away from instabilities. One of the main drawbacks with results [1]. the electrostatically actuated MEMS structures is the 2. MODELLING OF A MASS-SPRING SYSTEM phenomenon of pull-in instability. In 1967, Nathanson et al. [4] conducted experiments on a simplified mass-spring model system and analysed its static equilibrium. When the top A simple rigid moving plate [4] with stiffness 'k' and mass moving plate is at one-third of the initial gap, it snaps to the 'm' shown in Fig. l is subjected to an electrostatic force 'Fei' bottom and he termed it as a push-down or a pull-in which is given by phenomenon. Different methods have been employed so far Fel =60rAV to control this phenomenon. Joseph et al. [5] showed that the (1) 2(d -x)2 pull-in can be avoided by the simple addition of a series capacitance. This method is called as capacitive control
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where a closed loop and it separates. This demonstrates the nonlinearity existing in the system. This nonlinearity is = 8.854 x 10 '2F / m , permittivity of vacuum influenced by conservative energy value Eo and also the £r= 1, relative permittivity for air voltage applied. A, area of the plate, m2 V49 V, voltage applied between the moving plate and the bottom plate, volts d, initial gap between the top and the bottom plate, m V=02Y x, displacement of the moving plate, m 05
7
k
Tx v
2
d
15
-1
-05
a
0.6
displacement (m)
1
115
2
X'
Fig. 2: Example of a phase portrait for a constant Eo =O for different voltages.
In Fig.2, xl , x2 and x3 represent the points where the velocity becomes zero. The first equilibrium position, called the static Fig. 1: A simple mass-spring model under an electrostatic force equilibrium position, is between the points xl and x2 while The equation of motion of the system under the electrostatic the second unstable equilibrium position is between x2 force can be written as and x3 . There is a particular voltage at which all these three MA'9 k:x = F I (2) points become a single point and is found to be the pull-in voltage. It is explained clearly later in Section 4. It is The equation can be rewritten by integrating the above interesting to note that the velocity reaches a maximum value equation with respect to time as at the static equilibrium position for each particular voltage. This can be clearly seen for higher voltages where the peak velocity occurs somewhat away from the zero displacement # kX2 .60rA VV2 value, which corresponds to the static equilibrium value for 0 d-x that voltage.
where Eo is the integration constant which represents a constant conservative energy available for the system. Hence the velocity of the system can be obtained as
+ CO-CrA kX;72 | E 2&b ±V
m
(4)
Thus the velocity can be plotted against the displacement for different voltages and also for different Eo values. For example, as shown in the Fig.2, the velocity is plotted for different voltages at a particular Eo value. The mass and stiffness of this system are 10-3 kg and 0.5N/m. There is an initial gap of 2 microns between the top and the bottom plate. Length and width of the plate are taken to be 0.5mm and 0.2mm. It can be seen that as the value of voltage is increased from 0.1 to 0.9V, the variation of the velocity changes from a circle to an ellipse or oval shape. Beyond a particular voltage (between 0.8 and 0.9V, in Fig.2), the velocity no longer forms
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3. CONVERSION OF A CANTILEVER SYSTEM INTO A LUMPED MODEL
An electrostatically actuated micro cantilever beam is a continuous system which is complex to solve for its velocity, phase plot and also its pull-in phenomenon directly. In this section, the distributed system is converted into an equivalent rigid plate-spring model similar to the one in the previous section. The geometric and material properties which are used for this cantilever beam are taken from Hu et al. [1]. _b
d Fig. 3: A cantilever beam model under an electrostatic force
The deflection of the beam, as shown in Fig.3, is assumed as
w(x) = a(6x2 - 4x3 +x4) w(x) = a.-(x)
and
C4
=-
(q(x))2dx. From Equations 9, 11, 14 and 16,
20
(5)
the equivalent stiffness and mass are obtained as follows: or which satisfies the boundary conditions of the cantilever K (17) beam. Further, C3
q=C
w"(x) = 12a(1 - 2x + x2) or w'(x) = a.q$(x) (7) where a is an arbitrary constant. The potential energy of the beam is given by
Ub = 2EI (Wf(x))2dx
(8)
0
The above equation is reduced to the form Ub = Cl.a2
where C1
(9)
r1
1 -EJ Ei(0(x))2dX
2
M qeq
dx U I6 = jI8bV2 Uel.be 2 0£rb i l(d -_ Wd,())
The kinetic energy of the beam is known to be
-2
(18)
Hence the mechanical part of a continuous system is converted into an equivalent lumped system and these values will be used in the Section 4. Similar to the mechanical conversion, the equivalent electrostatic values can be obtained as shown below. The electrostatic potential energy for a voltage of Vi is given by
k
Tb b =-PAO(2(W(X))2dx
C2
C4
where 'i' represents a particular voltage
Vi. Also,
Wsti (x) = astj (X)
(10)
(19) (20)
Similar to the Equation 9, kinetic energy also can be reduced is the corresponding static deflection and ast is obtained from to the force balance plot for that voltage Vi. Hence, Equation 19 (1 Tb is numerically integrated which results in a constant value =C2Oa 1 r 2~~2 such that where C2 =-pA dx. (21) el.be C5 2 0 The equivalent electrostatic potential energy for a lumped The corresponding equations for an equivalent mass-spring system can now be expressed as system are given below. The equivalent potential energy for a UoSrAeqi Vi rigid plate would be
I(O(X))2
Urig =2Keq (w2)
Uel.rig
-2(d-
.
(X))
(22)
(12)
Here, it should be noted that the equivalent area of the rigid plate changes when the voltage changes due to the where Keq is the equivalent stiffness of the lumped system nature and hence the term Aeq . Similar to the electrostatic and mechanical part, there is an average deflection which is a (13) constant value for a voltage. (W =! Jw2(X)dX
1)
0
is the mean square value of the deflection. Hence, Equ 12 can be written in the form 2 14) U.rig C3Kqa eq
(x)dx :iWsti 0 The potential energy in Equation 22 thus reduces to
where C3
where
=-J(b(x))2 dx .
20
The
equivalent
of the system would be
Trigng=-M eq co2 w2
kinetic eniergy
Wavst =.-
Uel.rig =C6Aeq
C6
6
=
(23) (24)
2(d -Wav.st (X))
Hence, the equivalent area for the lumped system is obtained (15) from Equations 21 and 24 as
( 2 Cs C5 AeQ (25) where Meq is the equivalent mass of the lumped system. The eqi = C6i above equation reduces to 16 This equivalent area varies with the voltage. This means that, Tr.g =C4Mieqo2a2 (16) when the voltage is increased, the equivalent area also
197
increases since it no longer has the same area as that of the beam. This is clearly shown below in Fig.4. When the voltage increases, area slowly increases and when it is around the pull-in voltage, the equivalent area increases rapidly. I or 4n
105
4-.E 1o4 O
103 c
102
r 101
wu
100
*---*
6
Fig. 5: Force balance plot of the lumped model 99
0
20
40 Voltage (V)
60
80
Fig. 4: Variation of the equivalent area with the voltage Thus, the cantilever system is effectively converted into a rigid plate-spring model with its equivalent values to study the pull-in voltage. 4.
PULL-IN INVESTIGATION OF THE EQUIVALENT SYSTEM
The equivalent values acquired in the previous section are used in Equation 2 to obtain the relation
MeqkKeq A-KX = Meq
2(dr.eq -x)2V;2
6rAeqi
E+
0
d-x
')-KeX2 e
Meq
(27)
Thus different phase plots are drawn depending on the values of Eo and voltage. A specific conservative energy value (EO=O) is taken and the phase sketch is done for varying voltages as shown in the Fig. 6.
(26) V.
and can be called as the equivalent equation of motion. Force Balance A.
\
From Equation 26, for the static case, the force balance 0.X diagram is plotted between the equivalent mechanical and the electrostatic forces which are shown in Fig.5. As the voltage increases, the electrostatic force increases and when it reaches a particular voltage, called the pull-in voltage, the linear mechanical force becomes tangential to the nonlinear 3 4 .6 * 0 2 -4 .v *2 electrostatic curve. The pull-in voltage of the system is obtained from the force balance as shown and its value is Fig. 6: Phase portrait for a energy Eo= 0 and varying voltages V=68.4 V which is almost equal to the experimental value of 68.5 V obtained in [1]. Here xl , x2 and x3 are the points where the velocity B. Phase Portrait becomes zero for a particular energy value and a particular When the equation of motion, as given in Equation 26, is voltage. It can be noted that for a particular voltage (40 V as integrated and solved as explained in Section 2, the velocity in Fig.6), the first equilibrium position lies between the points equation is obtained from which the phase portrait can be xl and x2 while the second equilibrium position lies drawn. between x2 and x3. When the value of the voltage is increased (49.2V as in Fig.6), both x2 and x3 become a single point which is the second equilibrium position.
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For a particular voltage, an energy value E0 is calculated by taking the second equilibrium point from the force balance plot and by putting the velocity as zero, in Equation 27. Similarly, different energy values are calculated for different voltages and the velocities are plotted as shown in Fig.7. 0.25 0.2,-
v Vtov
V 30v
A
V -20v
Vv40v
0.15
Then, a continuous cantilever system is converted into an equivalent rigid plate-spring model and the equivalent stiffness, mass and areas of the lumped system are calculated. Based on these parameters, the phase portraits are plotted for
different energy levels, which are obtained from the equilibrium positions for different voltages. The pull-in voltage obtained from both the phase portrait and force balance seemed to be in very close agreement with the previously published results.
0.1
REFERENCES
0.05
>
[1] Y.C. Hu, C.M. Chang, S.C. Huang, "Some design considerations on the electrostatically actuated microstructures", Sensors and Actuators A, vol. 112, pp 155161, Dec 2003.
-.00-
-0.15 V
50v
-0V2
Pul-hn V
-0.25
-8
-6
-2
-4
0
2
displacemert x (m)
4
-
68.4v 6
8
0xl
Fig. 7: Phase portrait for different Eo corresponding to second equilibrium position and for varying voltages.
When the voltage increases, the velocity converges towards a particular point. When the voltage corresponds to 68.4 V as in this case which is shown in the Fig.8, the velocity converges to a point which is the pull-in voltage and the value of the xaxis for that point is one-third of the initial gap. 100
-
80
I
--x2,x3
if
20 O
0
-20 A .40
20
[3] Y. Lai, J. McDonald, M. Kujath, T. Hubbard, "Force, deflection and power measurements of toggled microthermal actuators," Joumal of Micromechanics and Microengineering, vol. 14, pp. 49-56,2004. [4] H.C. Nathanson, W.E. Newell, R.A. Wickstrom, Jr. J. Davis, "The Resonant Gate Transistor," Transactions on Electron Devices, ED-14(3), pp. 117-133, 1967.
[5] J.I. Seeger, S.B. Crary, "Stabilization of electrostatically actuated MEMS devices", Intemational Conference on SolidState Sensors and Actuators, pp. 1133-1136, 1997.
40
EF
[2] B. Piekarski, M. Dubey, D. DeVoe, E. Zakar, R. Zeto, J.Conrad, R. Piekarz, M. Ervin, "Fabrication of suspended piezoelectric microresonators," Integrated Ferroelectrics, Vol. 24, pp 147-154, 1999.
0
[6] P.B. Chu, K.S.J. Pister, "Analysis of closed-loop control of parallel-plate electrostatic microgrippers," International Conf Robotics and Automation, pp. 820-825, May 1994.
-.
40
[7] S. Taschini, H. Baltes, J.G. Korvink, " Non linear analysis of electrostatic actuation in MEMS with arbritary geometry," International Conference on Modeling and Simulation of Microsystems, pp.485-488, 2000.
.1*0 o
Fig. 8: Plot of Xi , X2 and X3 values with varying voltages.
It can be inferred from Fig.8 that as the voltage increases, the xi value increases and values of x2 and x3 (which is same as x2 ) decrease. All the three points converge to a single point at a particular voltage which is at the pull-in voltage of the system. Thus, the nonlinear phenomenon can be studied by varying the energy level from positive to negative for different voltages and hence, the stable operation range of the system can be achieved. 5. CONCLUSION
The phase portrait analysis is presented for a simple massspring model and the nonlinearity in the system is studied.
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[8] A.C.J. Luo, F.Y. Wang, "Nonlinear dynamics of a MicroElectro-Mechanical Systems with time-varying capacitors", Journal of Vibration and Acoustics, vol. 126, pp. 77, Jan 2004.
[9] P. Hagedom, W. Stadler (trans. And ed.), Non-Linear oscillations, Second Edition, Oxford: Clarendon Press, 1982
