Objective Functions for Optimizing Resonant Mass Sensor Performance
Michele H. Miller and Chengzhang Li
Introduction
Model System
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.
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.
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.
Illustration of the Tradeoff Between Q and f/f/m
Three Objective Functions for Comparison
Q
wn
O1 =w n
N
O2 =RMSD% =
( M
i=1
N
1
i
- M
0 2
i
)
100%
0 2
i
( M )
i=1
O3 =
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.
dC
e 2 AsAaVo
1
=
3
2
dV ( go - xo ) ( go - 3xo ) ms + bs+ k
Plate size
Plate thickness
Gap, go
Silicon density,
Air pressure, Pa
DC offset voltage, Vo
Stiffness of supporting legs, kleg
Material damping, bmat'l
1.862x10-5 N-s/m2
200 m x 200 m
3 m
4 m
2330 kg/m3
101325 Pa
10 V
50 N/m
1x10-5 N-s/m
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:
x1 - x2
2
2
s1 s 2
+
n1 n2
n x1 - x2
=
>1.96
2 s
(M
1
n0
- M
0 2
n0
0 2
( M n0
0 2
n1
) +( M - M )
) +( M )
1
n1
0
n1
2
100%
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.
Effect of Plate Hole Size on the Three Objective Functions