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The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models

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Abstract

The mathematical modeling and analysis of electrostatically actuated micro- and nanoelectromechanical systems (MEMS and NEMS) has typically relied upon simplified electrostatic-field approximations to facilitate the analysis. Usually, the small aspect ratio of typical MEMS and NEMS devices is used to simplify Laplace’s equation. Terms small in this aspect ratio are ignored. Unfortunately, such an approximation is not uniformly valid in the spatial variables. Here, this approximation is revisited and a uniformly valid asymptotic theory for a general “drum shaped” electrostatically actuated device is presented. The structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented. The effect of retaining typically ignored terms on the solution set of the standard theory is explored

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References

  1. Crowley J. (1986). Fundamentals of Applied Electrostatics. Laplacian Press, San Jose, 272 pp

    Google Scholar 

  2. Nathanson H.C., Newell W.E., Wickstrom R.A. (1967). The resonant gate transistor. IEEE Trans. on Electron Devices 14: 117–133

    Article  Google Scholar 

  3. Bao M., Wang W. (1996). Future of microelectromechanical systems (MEMS). Sensors and Actuators A 56:135–141

    Article  Google Scholar 

  4. Camon H., Larnaudie F., Rivoirard F., and Jammes B., Analytical simulation of a 1D single crystal electrostatic micromirror. In: Proceedings of Modeling and Simulation of Microsystems (1999)

  5. Chu P.B., Pister K.S.J. Analysis of closed-loop control of parallel-plate electrostatic microgrippers. In: Proc. IEEE Int. Conf. Robotics and Automation (1994) pp. 820–825

  6. Tilmans H.A.C., Elwenspoek M., Fluitman J.H.J. (1992). Micro resonant force gauges. Sensors and Actuators A 30:35–53

    Article  Google Scholar 

  7. Anderson M.J., Hill J.A., Fortunko C.M., Dogan N.S., Moore R.D. (1995). Broadband electrostatic transducers: modeling and experiments. J. Acoust. Soc. Am. 97:262–272

    Article  ADS  Google Scholar 

  8. Saif M.T.A., Alaca B.E., Sehitoglu H. (1999). Analytical modeling of electrostatic membrane actuator micro pumps. J. Microelectromech. Syst. 8:335–344

    Article  Google Scholar 

  9. Pelesko J.A., and Bernstein D. (2003). Modelling MEMS and NEMS. Chapman and Hall/CRC, Boca Raton, FL, 376 pp

    Google Scholar 

  10. Zhao X.P., Abdel-Rahman E.M., Nayfeh A.H. (2004). A reduced-order model for electrically actuated microplates. J. Micromech. Microengng. 14:900–906

    Article  ADS  Google Scholar 

  11. Bochobza-Degabi O., Elata D., Nemirovsky Y. (2002). An efficient DIPIE algorithm for CAD of electrostatically actuated MEMS devices. J. Microelectromech. Syst. 11:612–620

    Article  Google Scholar 

  12. Pelesko J.A., Chen X.Y. (2003). Electrostatically deflected circular elastic membranes. J. Electrostat. 57:1–12

    Article  Google Scholar 

  13. Pelesko J.A. Electrostatic field approximations and implications for MEMS devices. In: Proceedings of ESA 2001 (2001) pp. 126–137

  14. Pelesko J.A., Bernstein D.H., McCuan J., Symmetry and symmetry breaking in electrostatic MEMS. In: Proc. of MSM 2003 (2003) pp. 304–307

  15. Joseph D.D., and Lundgren T.S. (1973). Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49:241–269

    MATH  ADS  MathSciNet  Google Scholar 

  16. Taylor G.I. (1968). The coalescence of closely spaced drops when they are at different electric potentials. Proc. R. Soc. London A 306:423–434

    ADS  Google Scholar 

  17. Ackerberg R.C. (1969). On a nonlinear differential equation of electrohydrodynamics. Proc. R. Soc. London A 312:129–140

    Article  MATH  ADS  Google Scholar 

  18. Fornberg B. (1996). A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, 242 pp

    MATH  Google Scholar 

  19. Trefethen L.N. (2000). Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics, Philadelphia, 184 pp

    MATH  Google Scholar 

  20. McGough J.S. (1998). Numerical continuation and the Gelfand problem. Appl. Math. Comp. 89:225–239

    Article  MATH  MathSciNet  Google Scholar 

  21. Allgower E.L., Georg K. (2003). Introduction to Numerical Continuation Methods. Society for Industrial and Applied Mathematics, Philadelphia, 388 pp

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    John A. Pelesko & Tobin A. Driscoll

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  1. John A. Pelesko
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  2. Tobin A. Driscoll
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Correspondence to John A. Pelesko.

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Pelesko, J.A., Driscoll, T.A. The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models. J Eng Math 53, 239–252 (2005). https://doi.org/10.1007/s10665-005-9013-2

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  • DOI: https://doi.org/10.1007/s10665-005-9013-2

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