1. Objective Functions for Optimizing Resonant Mass Sensor Performance
Michele H. Miller and Chengzhang Li
Introduction
Resonant mass sensors have proven to have high potential for sensing very small quantities of gas vapors. Micro-fabrication technologies provide a way to realize sensitive and inexpensive devices. In particular, MEMS (micro-electrical mechanical systems) offer great promise as chemical sensors. An electrostatically actuated microcantilever with an appropriate coating can serve as a resonant mass sensor for detecting chemical vapors. The ideal resonant mass sensor experiences a large change in resonant frequency in the presence of a small gas vapor concentration.
A number of design variables can be adjusted in an attempt to improve the performance of an electrostatically actuated resonant mass sensor. These include cantilever size, shape and material, gap between cantilever and substrate, actuation voltage, areas and locations of actuation and sensing pads, selection of resonance mode to monitor. During the design optimization process, an appropriate objective function must be selected. The function should capture what good performance is. Because the number of possibilities is so vast, shape optimization can involve extensive computation time. Genetic algorithms are one approach for realizing shapes that improve performance. Given the computation requirements of this type of optimization approach, the objective function needs to be very fast to calculate. In this paper, we evaluate performance measures for resonant mass sensors in terms of their ability to identify the “best” sensor and their computational efficiency.
Conclusion
This poster describes the evaluation of three objective functions that could be used in the optimization of a resonant mass sensor design. A simple problem was to determine the optimum hole size in a resonator design involving a square plate and square hole. Two of the objective functions (the RMSD% and two point RMSD%) identified an optimum hole size that matched a reference measure. Both would be computationally efficient if used in a design optimization algorithm.
Three Objective Functions for Comparison
M0 and M1 are the magnitude ratio curves for the original system and the one with additional mass, respectively. n0 and n1 are the resonant frequencies of the original system and the one with additional mass, respectively.
Model System
We considered the system shown below. It is an electrostatically actuated resonator that is a flat plate supported by four leg springs. Half of the plate area is used for electrostatic actuation and half is used for capacitive sensing. The plate is coated with a polymer coating that will accumulate mass when in the presence of a gas of interest. The hole in the center is for the purpose of reducing squeeze film damping.
This system was modeled as a lumped parameter system in which the squeeze film affects the system stiffness and damping. The plate is modeled as eight rectangular sub-elements. The corner elements have venting on adjacent sides while the side elements have venting on opposite sides. The models of Darling, Hivick, Xu (1998) were used to analytically determine stiffness and damping constants for the squeeze film for the two types of elements.
Effect of Plate Hole Size on the Three Objective Functions μ
1.862x10-5 N-s/m2
Plate size
200 μm x 200 μm
Plate thickness
3 μm
Gap, go
4 μm
Silicon density, ρ
2330 kg/m3
Air pressure, Pa Pa
DC offset voltage, Vo
10 V
Stiffness of supporting legs, kleg
50 N/m
Material damping, bmat'l
1x10-5 N-s/m
Illustration of the Tradeoff Between Q and δf/δm
Illustration of Detectability in the Presence of Uncertainty
Let x1 and x2 be measures of the resonant frequency before and after mass is added. Assuming multiple measures of x1 and x2 are normally distributed, then a hypothesis test as follows would indicate with 95% confidence that x1 and x2 are different: