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Scaling of amplitude-frequency-dependence nonlinearities in electrostatically transduced microresonators
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10.1063/1.2785018
/content/aip/journal/jap/102/7/10.1063/1.2785018
http://aip.metastore.ingenta.com/content/aip/journal/jap/102/7/10.1063/1.2785018

Figures

Image of FIG. 1.
FIG. 1.

(Color online) Schematic showing the double-ended-tuning-fork resonator with length . The resonance mode is also shown, along with a partial SEM cross section of the wafer scale encapsulated resonator. The electrostatic gap size is . is the dc polarization voltage for electrostatic actuation and sensing and is the ac drive voltage.

Image of FIG. 2.
FIG. 2.

Force diagram of a single beam of the DETF resonator. This diagram also illustrates that the motion of the resonator is symmetrical in positive and negative directions.

Image of FIG. 3.
FIG. 3.

Measured effect for a DETF MEMS resonator with and . The plots shows the output current as a function of frequency at (a) low , showing mechanical nonlinearities, (b) at high , showing electrical nonlinearities, and (c) near , where the electrical softening nonlinearities and mechanical stiffening nonlinearities tend to balance out.

Image of FIG. 4.
FIG. 4.

Typical coefficient dependence on . Measurement shown here for a DETF resonator. Such measurements are taken for each device of beam lengths 200, 300, 400, 500, 600, 700, 800, and . The relevant parameters from such measurements (, , and ) are reported in Figs. 6–8.

Image of FIG. 5.
FIG. 5.

(Color online) Resonant frequency and quality factor measurements. scaling in the resonant frequency is evident.

Image of FIG. 6.
FIG. 6.

(Color online) Third order stiffness nonlinearity coefficient vs beam length. The large error bars in the measured data are due to uncertainty in the gap size . Theoretically predicted scaling in is verified by measurements.

Image of FIG. 7.
FIG. 7.

Electrical coefficient vs beam length. Theoretically predicted scaling in is verified by measurements.

Image of FIG. 8.
FIG. 8.

Optimal bias voltage vs beam length. Theoretically predicted scaling in is verified by measurements.

Image of FIG. 9.
FIG. 9.

Maximum output current , or critical bifurcation current at vs beam length. The dashed line uses the expression for and given in Table I and experimentally measured values of . This represents the upper limit of available current ignoring nonlinearity reduction (Refs. 8, 11, and 13). Theoretically predicted scaling in is verified by measurements. Considerable improvement in current handling can be seen for the shortest beams.

Image of FIG. 10.
FIG. 10.

Motional impedance vs beam length. Weak scaling with is observed. Since this quantity is inversely proportional to the exact dependence is hard to extract.

Image of FIG. 11.
FIG. 11.

Maximum power dissipated vs beam length. Theoretically predicted scaling is observed here.

Tables

Generic image for table
Table I.

Dimensions and physical properties of the double-ended-tuning-fork resonators.

Generic image for table
Table II.

Analytical models and scaling dependencies on for nonlinear and related properties in electrostatically coupled DETF MEMS resonators.

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/content/aip/journal/jap/102/7/10.1063/1.2785018
2007-10-02
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Scaling of amplitude-frequency-dependence nonlinearities in electrostatically transduced microresonators
http://aip.metastore.ingenta.com/content/aip/journal/jap/102/7/10.1063/1.2785018
10.1063/1.2785018
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