Journal of Engineering Mathematics

, Volume 53, Issue 3–4, pp 239–252 | Cite as

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

  • John A. PeleskoEmail author
  • Tobin A. Driscoll


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


continuation method microelectromechanical systems MEMS nanoelectromechanical systems NEMS nonlinear elliptic problem 


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  1. 1.
    Crowley J. (1986). Fundamentals of Applied Electrostatics. Laplacian Press, San Jose, 272 ppGoogle Scholar
  2. 2.
    Nathanson H.C., Newell W.E., Wickstrom R.A. (1967). The resonant gate transistor. IEEE Trans. on Electron Devices 14: 117–133CrossRefGoogle Scholar
  3. 3.
    Bao M., Wang W. (1996). Future of microelectromechanical systems (MEMS). Sensors and Actuators A 56:135–141CrossRefGoogle Scholar
  4. 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)Google Scholar
  5. 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. 820–825Google Scholar
  6. 6.
    Tilmans H.A.C., Elwenspoek M., Fluitman J.H.J. (1992). Micro resonant force gauges. Sensors and Actuators A 30:35–53CrossRefGoogle Scholar
  7. 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. 97:262–272CrossRefADSGoogle Scholar
  8. 8.
    Saif M.T.A., Alaca B.E., Sehitoglu H. (1999). Analytical modeling of electrostatic membrane actuator micro pumps. J. Microelectromech. Syst. 8:335–344CrossRefGoogle Scholar
  9. 9.
    Pelesko J.A., and Bernstein D. (2003). Modelling MEMS and NEMS. Chapman and Hall/CRC, Boca Raton, FL, 376 ppGoogle Scholar
  10. 10.
    Zhao X.P., Abdel-Rahman E.M., Nayfeh A.H. (2004). A reduced-order model for electrically actuated microplates. J. Micromech. Microengng. 14:900–906CrossRefADSGoogle Scholar
  11. 11.
    Bochobza-Degabi O., Elata D., Nemirovsky Y. (2002). An efficient DIPIE algorithm for CAD of electrostatically actuated MEMS devices. J. Microelectromech. Syst. 11:612–620CrossRefGoogle Scholar
  12. 12.
    Pelesko J.A., Chen X.Y. (2003). Electrostatically deflected circular elastic membranes. J. Electrostat. 57:1–12CrossRefGoogle Scholar
  13. 13.
    Pelesko J.A. Electrostatic field approximations and implications for MEMS devices. In: Proceedings of ESA 2001 (2001) pp. 126–137Google Scholar
  14. 14.
    Pelesko J.A., Bernstein D.H., McCuan J., Symmetry and symmetry breaking in electrostatic MEMS. In: Proc. of MSM 2003 (2003) pp. 304–307Google Scholar
  15. 15.
    Joseph D.D., and Lundgren T.S. (1973). Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49:241–269zbMATHADSMathSciNetGoogle Scholar
  16. 16.
    Taylor G.I. (1968). The coalescence of closely spaced drops when they are at different electric potentials. Proc. R. Soc. London A 306:423–434ADSGoogle Scholar
  17. 17.
    Ackerberg R.C. (1969). On a nonlinear differential equation of electrohydrodynamics. Proc. R. Soc. London A 312:129–140zbMATHADSCrossRefGoogle Scholar
  18. 18.
    Fornberg B. (1996). A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, 242 ppzbMATHGoogle Scholar
  19. 19.
    Trefethen L.N. (2000). Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics, Philadelphia, 184 ppzbMATHGoogle Scholar
  20. 20.
    McGough J.S. (1998). Numerical continuation and the Gelfand problem. Appl. Math. Comp. 89:225–239CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Allgower E.L., Georg K. (2003). Introduction to Numerical Continuation Methods. Society for Industrial and Applied Mathematics, Philadelphia, 388 ppzbMATHGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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