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Research Papers

Integrity Analysis of Electrically Actuated Resonators With Delayed Feedback Controller

[+] Author and Article Information
Fadi Alsaleem

 Microstaq Inc., Austin, TX 78744-1052

Mohammad I. Younis1

Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, NY 13902myounis@binghamton.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 133(3), 031011 (Mar 25, 2011) (8 pages) doi:10.1115/1.4003262 History: Received July 05, 2010; Revised September 12, 2010; Published March 25, 2011; Online March 25, 2011

In this work, we investigate the stability and integrity of parallel-plate microelectromechanical systems resonators using a delayed feedback controller. Two case studies are investigated: a capacitive sensor made of cantilever beams with a proof mass at their tip and a clamped-clamped microbeam. Dover-cliff integrity curves and basin-of-attraction analysis are used for the stability assessment of the frequency response of the resonators for several scenarios of positive and negative gain in the controller. It is found that in the case of a positive gain, a velocity or a displacement feedback controller can be used to effectively enhance the stability of the resonators. This is confirmed by an increase in the area of the basin of attraction of the resonator and in shifting the Dover-cliff curve to higher values. On the other hand, it is shown that a negative gain can significantly weaken the stability and integrity of the resonators. This can be of useful use in MEMS for actuation applications, such as in the case of capacitive switches, to lower the activation voltage of these devices and to ensure their trigger under all initial conditions.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic for the displacement delayed feedback controller

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Figure 2

(a) A top view picture for the tested capacitive sensor (taken apart) (7). (b) A top view picture showing the assembled part. When excited electrostatically, the proof mass oscillates in the out of plane direction (out of the page).

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Figure 3

Simulation results for the frequency response of the capacitive device for Vac=20.1 V and Vdc=40.2 V

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Figure 4

Simulated time history response of the capacitive device when actuated suddenly with Vdc=40.2 V, Vac=20.1 V, and Ω=180 Hz (a) without the delayed feedback controller and (b) with the delayed feedback controller (gain=8 V s/m and time delay T/2)

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Figure 5

The experimental setup used for testing the capacitive accelerometer

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Figure 6

Experimentally measured velocity response of the capacitive device when actuated with Vdc=40.2 V, Vac=20.1 V, and Ω=180 Hz (a) without the delayed feedback controller and (b) with the delayed feedback controller (gain G=8 V s/m and time delay T/2)

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Figure 7

(a) The basin of attraction of the capacitive device at an excitation frequency of 180 Hz without the controller and (b) the corresponding time history at initial conditions (0,0). In the figure, Vdc=40.2 V and Vac=20.1 V.

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Figure 8

(a) The basin of attraction of the capacitive device at an excitation frequency of 180 Hz after introducing the delayed feedback controller and (b) the corresponding time history at initial conditions (0,0). In the figure, Vdc=40.2 V, Vac=20.1 V, and the delayed feedback controller gain is 70 V s/m.

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Figure 9

The frequency response of the capacitive device obtained using the shooting technique along with Dover-cliff curves generated for various values of gain. In the figure, Vdc=40.2 V and Vac=20.1 V. The unit of gain in the legend is V s/m.

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Figure 10

Basin of attractions at an excitation frequency 190 Hz for the case of (a) no control, (b) with control with a positive gain, and (c) with control with a negative gain. (a) G=0 V s/m, (b) G=70 V s/m, and (c) G=−70 V s/m.

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Figure 11

A clamped-clamped microbeam forming a parallel-plate capacitor and showing a dynamic ac and dc actuation

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Figure 12

Frequency response obtained using long time integration assuming zero initial conditions (circles) and the shooting method (solid line: stable, dashed line: unstable). In the figure, Vdc=2 V, Vac=0.6 V, and the quality factor Q=100.

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Figure 13

The normalized displacement response of the clamped-clamped beam when actuated suddenly with Vdc=2 V, Vac=0.6 V, and Ω=23.15 Hz (a) without the delayed feedback controller, (b) with the displacement delayed feedback controller (gain=0.003 V/m and time delay T/2), and (c) the controller output voltage applied in Fig. 1.

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Figure 14

Basin of attractions of the microbeam at Ω=23.15, (a) before control, (b) after the application of the displacement delayed feedback controller with a gain of 0.01 V/m. In the figure, the assumed load is Vdc=2.0 V and Vac=0.6 V.

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Figure 15

Dover-cliff integrity curves for the response of the microbeam at different excitation frequencies before and after adding the delayed feedback controller. In the figure, Vdc=2.0 V and Vac=0.6 V.

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