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Differences between plate theory and lumped element model in electrostatic analysis of one-sided and two-sided CMUTs with circular microplates

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Abstract

An analytical study of capacitive micromachined ultrasonic transducers (CMUTs) with circular microplates has been carried out. The study comprises one-sided (single electrode back-plate) and two-sided (double electrode back-plate) systems, and derives universal correction factors for pull-in voltage and critical displacement to be used in lumped element model (LEM) analysis. We employ von Karman plate theory and the single-mode Galerkin decomposition method to solve the equations. Consequently, voltage–deflection relations have been derived. By comparing results from plate theory with LEM, it is concluded: (1) for the one-sided CMUT by neglecting geometrical nonlinearity, we find \(\frac{{{V_{\text {Pull in - P}}}}}{{{V_{\text {Pull in - LEM}}}}} = 1.327\) and the ratio of critical displacement derived from plate theory over critical displacement from LEM is always 1.882. (2) For the one-sided CMUT including geometrical nonlinearity \(\frac{{{V_{\text {Pull in} - P}}}}{{{V_{\text {Pull in - LEM}}}}} = 1.45\) and critical displacement from plate theory over critical displacement from LEM is 1.792, for a specific set of parameters. (3) For the two-sided CMUT, there is no differences in using linear nor nonlinear analysis and \(\frac{{{V_{\text {Pull in - P}}}}}{{{V_{\text {Pull in - LEM}}}}} = 1.276\). For all studied cases, finite element (FE) analysis has been performed to validate the analytical outcomes.

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Ahrar Institute of Technology and Higher Education, Rasht, Iran

    Milad Saadatmand

  2. Department of Electrical Engineering, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    Junghwan Kook

  3. GN Audio A/S, Audio Research, Lautrupbjerg 7, 2750, Ballerup, Denmark

    Junghwan Kook

Authors
  1. Milad Saadatmand
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  2. Junghwan Kook
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Correspondence to Milad Saadatmand.

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Technical Editor: Pedro Manuel Calas Lopes Pacheco, D.Sc.

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Appendices

Appendix 1

In Eq. (20) coefficients \({K_{11}}-{K_{16}}\) are:

$$\begin{aligned} {K_{11}}& {}= \frac{{{\alpha _2}\,{V^2}}}{6},\,\,{K_{12}} = \frac{{32}}{3},\,{K_{13}} = - \frac{{64}}{5},\\ {K_{14}}& {}= \frac{{(\nu - 3)}}{{7(\nu - 1)}}{\alpha _1} + \frac{{32}}{7},\,\\ {K_{15}}& {}= - \frac{{8(5\,\nu - 11)\,{\alpha _1}}}{{135(\nu - 1)}},\,{K_{16}} = \frac{{(21\,\nu - 43)\,{\alpha _1}}}{{154(\nu - 1)}}. \end{aligned}$$

Appendix 2

In Eq. (23) coefficients \({K_{21}} - {K_{24}}\) are:

$$\begin{aligned} {K_{21}}& {}= 9.5445-0.3656 {{\alpha _2}\,{V^2}},\nonumber \\ {K_{22}}& {}= 10.4140 + \frac{{-0.1511+0.3518\nu }}{(\nu - 1)},\,\nonumber \\ {K_{23}}& {}= - 3.5780 + \frac{{0.2940-0.55180 \nu }}{(\nu - 1)},\nonumber \\ {K_{24}}& {}= \frac{{(0.2150\nu - 0.1208)}}{{(\nu - 1)}}{\alpha _1}. \end{aligned}$$
(29)

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Saadatmand, M., Kook, J. Differences between plate theory and lumped element model in electrostatic analysis of one-sided and two-sided CMUTs with circular microplates. J Braz. Soc. Mech. Sci. Eng. 42, 468 (2020). https://doi.org/10.1007/s40430-020-02551-8

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  • DOI: https://doi.org/10.1007/s40430-020-02551-8

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