(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.
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.
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.
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.
(Color online) Resonant frequency and quality factor measurements. scaling in the resonant frequency is evident.
(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.
Electrical coefficient vs beam length. Theoretically predicted scaling in is verified by measurements.
Optimal bias voltage vs beam length. Theoretically predicted scaling in is verified by measurements.
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.
Motional impedance vs beam length. Weak scaling with is observed. Since this quantity is inversely proportional to the exact dependence is hard to extract.
Maximum power dissipated vs beam length. Theoretically predicted scaling is observed here.
Dimensions and physical properties of the double-ended-tuning-fork resonators.
Analytical models and scaling dependencies on for nonlinear and related properties in electrostatically coupled DETF MEMS resonators.
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