1. Slide 6.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Lecture 6 Analysis of electrostatically actuated micro devices Some features of nonlinearly coupled electrostatic and elasto-static and dynamic governing equations of electrostatic MEMS.

2. Slide 6.2 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Contents Why is electrostatic actuation popular in MEMS? Nonlinearity explained with examples Computing the electrostatic force –Parallel-plate capacitor –General Effects of nonlinearity –Pull-in, pull-up, hysteresis, etc. Squeezed-film damping –Iso-thermal Reynolds equation Design challenges –Shape optimization –Topology optimization?

3. Slide 6.3 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Why is electrostatic actuation popular in MEMS? Ease of fabrication Ease of actuation Energy-efficient High frequency (MHz and even GHz) Scalability Easy sensing mechanism (capacitance- based) Some inconveniences -- high voltages for large displacements -- small forces (they move themselves mostly; suited for sensors) -- charging and stiction

4. Slide 6.4 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Micro-mechanical filters C.T.-C. Nguyen, “Micromechanical Components for Miniaturized Low-Power Communications,” Proc. 1999 IEEE MTT-S Int. Microwave Symposium, RF-MEMS Workshop, Anaheim, CA, June 18, 1999, pp. 48-77. Electrostatic actuation and sensing is the key to this.

5. Slide 6.5 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Why micro-mechanical filters? Why mechanical filters? Narrow bandwidth (high selectivity) Low loss (high Q, 10,000 to 25,000) Good stability with temperature variation Passive (no power and clock required) Why micromechanical filters? Better performance and cost Low power consumption Smaller size (more applications: e.g., cellular phones)

6. Slide 6.6 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Working principle of mechanical filters Frequency Transmission (dB) Frequency

7. Slide 6.7 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Schematics of Nguyen’s micro-mechanical filter and a high-Q disk resonators J. R. Clark, W.-T. Hsu, and C.T.-C. Nguyen, “Measurement Techniques for Capacitively-transduced VHF-to-UHF Micromechanical Resonators, Proc. of Transducers, 2001, Munich, June 10-14, 2001, pp. 1118-1121. Side view Contour disk resonator “Free-free” beam resonator Top view

8. Slide 6.8 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh A bi-directional pump V Diaphragm Passive inlet valvePassive outlet valve Frequency Flow rate R. Zengerle, J. Ulrich, S. Kluge, M. Richter, and A. Richter, “A Bidirectional Silicon Micropump”, Sensors and Actuators, A 50, 1995, pp. 81-86.

9. Slide 6.9 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Electrostatic comb-drive—the prime mover for MEMS today anchor Shuttle mass Folded-beam suspension Moving combs Fixed combs Misaligned parallel-plate capacitor W. C. Tang, T.-C. H. Nguyen, M. W. Judy and R. T. Howe, “Electrostatic-comb drive of lateral polysilicon resonators,” Sensors and Actuators, Vol. A 21-23, pp. 328 – 381, 1990.

10. Slide 6.10 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Computing the electrostatic force in the parallel-plate capacitor Electrostatic energy Force in the length direction Force in the width direction Force in the gap direction = permittivity of free space = applied voltage = capacitance

11. Slide 6.11 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Computing the electrostatic force in general 3-D problems Electric potential = Electric field = Electrostatic force = Dieletric constant of the intervening medium Charge density = charge per unit area Surface normal Conductor 1 Conductor 2 It is a surface force (traction).

12. Slide 6.12 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Computing the electrostatic force (contd.) On the conductors In the intervening medium Plus, potentials on the conductors are specified. Governing equations to solve for the charge density in the differential equation form: This is suited for FEM but sufficient intervening medium also needs to be meshed along with the interior of the conductors. Governing equations to solve for the charge density in the integral equation form: This is suited for BEM because only conductor boundaries need to be meshed.

13. Slide 6.13 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Static equilibrium of an elastic structure under electrostatic force V + + + + + + + ++ + + - - - - - - - - - - - - + + + + + + + + + + + - - - - - - - - - - - - Charge distribution causes electrostatic force of attraction between conductors Electrostatic force deforms conductors Deformation of conductors causes charges to re- distribute Electrostatic force =

14. Slide 6.14 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Coupled governing equations of electro- and elasto- statics A self-consistent solution is needed!

15. Slide 6.15 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Start with 1-dof lumped model… V = plate area = permittivity of free space Static equilibrium A cubic equation!

16. Slide 6.16 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Lumped 1-dof modeling of coupled electro- and elasto- static behavior Forces Three solutions Two stable; one unstable; And, one infeasible Potential energy Stable Unstable Use to test stability.

17. Slide 6.17 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Pull-in phenomenon Potential energy < < Condition for critical stability

18. Slide 6.18 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh With a dielectric layer: pull-up and hysteresis Dielectric layer Pull-up voltage is found by equating the forces of spring and electrostatics at. Gilbert, J. R., Ananthasuresh, G. K., and Senturia, S. D., “3-D Modeling and Simulation of Contact Problems and Hysteresis in Coupled Electromechanics,” presented at the IEEE-MEMS- 96 Workshop, San Diego, CA, Feb. 11-15, 1996.

19. Slide 6.19 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh V +++++++++ V + + + + + + + + + Distributed modeling of the electrostatically actuated beam Include the effects of residual stress as well: FEM or FDM could be used to solve the nonlinear equation: Finite element method Finite difference method A correction due to fringing field (edge and corner effects) is also included.

20. Slide 6.20 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Solving the general 3-D problem Boundary element method for the integral equation of electrostatics Potential on k th panel Charge on k th panel Area of k th panel Discretize the boundary surfaces into n panels. Assemble to get: Finite element method for the differential equation of elastostatics

21. Slide 6.21 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Solution approaches Relaxation -- iterate between the elastic and electrostatic domains. -- converges except in the vicinity of pull-in voltage; but slow. Surface Newton -- compute sensitivities of surface nodes. -- use a Newton step to update those nodes. -- then, re-compute electrostatic force and internal deformations. Direct Newton -- compute all derivatives to update charges and deformations. Residuals in mechanical and electrical domains For example, see: G. Li and N. R. Aluru, “Linear, non-linear, and mixed-regime analysis of electrostatic MEMS,” Sensors and Actuators, A 91, 2001, pp. 279-291, and references therein.

22. Slide 6.22 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh What about dynamic behavior? Potential energy = dynamic pull-in voltage Lumped 1-dof model Beam model Frequency = Will contain a term! So, the response will show two resonance at two frequencies.

23. Slide 6.23 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Damping: squeezed film effects V Squeezed-film damping Use isothermal, compressible, narrow gap Reynolds equation to model the film of air beneath the beam/plate/membrane. It is widely used in lubrication theory. By analyzing this equation, we can extract the essence of damping as a lumped parameter – the so called “macromodeling”. Lumped 1-dof modelBeam model How do you obtain ?

24. Slide 6.24 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Modeling squeezed film effects: isothermal Reynolds equation Pressure distribution in the 2-D x-y plane Gap varies in the x-y plane for a deformable structure (beam,plate, membrane) Viscosity of air For lumped 1-dof modeling, we have a rigid plate. So, does not depend on. Assume further that pressure distribution is the same along the length of the plate so that it becomes a one dimensional problem. Assumed pressure distribution S. D. Senturia, Microsystems Design, Kluwer, 2001.

25. Slide 6.25 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Behavior with small displacements Linearizearound : Separation of spatial and temporal components: Also, use non-dimensional variables: width Assume a sudden velocity impulse to the plate. Then, for t > 0, this term is zero. (with displacement )

26. Slide 6.26 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Behavior with small displacements (contd.) Boundary conditions and velocity-impulse assumption give: Force on the plate = Take the Laplace transform (continued on the next slide).

27. Slide 6.27 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Finally, getting to lumped approximation… For only. Damping coefficient Cut-off frequency Transfer function for general displacement input!

28. Slide 6.28 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh What does it mean mechanically? Thus, squeezed film effect creates two effects: Viscous damping + “air-spring” Further analysis indicates that at low frequencies, damping dominates, and air-spring at high frequencies. See S. D. Senturia, Microsystems Design, Kluwer, 2001, for details.

29. Slide 6.29 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Move up to beam modeling… Note that this is still a parallel-plate approxmation! Solve these two coupled equations. An approach Use FDM for pressure equation and FEM or FDM for discretizing the dynamic equation, and integrate in time using the Runge-Kutta method.

30. Slide 6.30 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh A typical response The transverse deflection of the mid-point of a fixed- fixed beam under (Vdc+Vac) voltage input under the squeezed film effect:

31. Slide 6.31 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh What about this problem now? V Diaphragm Passive inlet valvePassive outlet valve Frequency Flow rate A problem involving three energy domains that are strongly coupled. Furthermore, the fluids part is non-trivial.

32. Slide 6.32 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Shape optimization: example of electrostatic comb-drive Straight-finger comb-drive Variable force comb-drives with curved stationary fingers Linear Quadratic Cubic Need to compensate the non- linearities caused by the folded- beam suspension. (Figures provided by W. Ye) (Made with Cornell’s SCREAM process) (Ye and Mukherjee, Cornell) W. Ye and S. Mukherjee, “Optimal design of three-dimensional MEMS with applications to electrostatic comb drives,” International Journal for Numerical Methods in Engineering, Vol. 45, pp. 175-194, 1999.

33. Slide 6.33 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Synthesis with electrostatic actuation (Ye and Mukherjee, Cornell) (Figures provided by W. Ye) Shape-optimized comb-fingers to compensate suspension’s nonlinearities. (Made with Cornell’s SCREAM process) Ye and Mukherjee used BEM for discretizing both the electrostatic and elasto-static governing equations.

34. Slide 6.34 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh How about topology optimization? Introducing new holes (i.e., topology variation) certainly helps elastic behavior but it complicates the electrostatics problem. To use the “smoothening” (between 0 and 1) approach, every spatial point should be able to assume the states of empty space, a conductor, or a dielectric. Fringing field effect becomes harder to deal with as new holes get introduced. Can be done but there are issues to be resolved…

35. Slide 6.35 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Main points Electrostatic force is THE most widely used actuation in MEMS – for many good reasons. Very interesting nonlinear behaviors. Analysis of couple electrostatic and elastostatics (and dynamics) is non-trivial. Squeezed film effect causes damping as well as air-spring stiffness. Shape optimization makes more sense but topology optimization can certainly be attempted.