A new two-beam differential resonant micro accelerometer
ABSTRACT A novel uniaxial micro-machined resonant accelerometer is presented. The device working principle is based on the stiffness variations of a beam which is fully clamped to the substrate on one side and clamped to a seismic mass on the other side. A movement of the seismic mass, induced by an external acceleration, causes either a compressive or a tensile stress on the beam, inducing a variation of its stiffness. This variation results in a change of the resonance frequency of the beam. The accelerometer is arranged in a differential structure, with two beams built in such a way that their changes in the resonance frequency have opposite sign. This solution allows obtaining a doubled sensitivity with the same area and allows reducing the non linear behavior. First experimental results show that the device has an overall differential sensitivity Â¿fres/g Â¿ 450 Hz/g in the linear range of operation, with an overall area occupation lower than (500 Â¿m)2.
IEEE Design & Test of Computers. 01/1999; 16:48-56.
[show abstract] [hide abstract]
ABSTRACT: An analysis of the dynamic characteristics of pull-in for parallel-plate and torsional electrostatic actuators is presented. Traditionally, the analysis for pull-in has been done using quasi-static assumptions. However, it was recently shown experimentally that a step input can cause a decrease in the voltage required for pull-in to occur. We propose an energy-based solution for the step voltage required for pull-in that predicts the experimentally observed decrease in the pull-in voltage. We then use similar energy techniques to explore pull-in due to an actuation signal that is modulated depending on the sign of the velocity of the plate (i.e., modulated at the instantaneous mechanical resonant frequency). For this type of actuation signal, significant reductions in the pull-in voltage can theoretically be achieved without changing the stiffness of the structure. This analysis is significant to both parallel-plate and torsional electrostatic microelectromechanical systems (MEMS) switching structures where a reduced operating voltage without sacrificing stiffness is desired, as well as electrostatic MEMS oscillators where pull-in due to dynamic effects needs to be avoidedJournal of Microelectromechanical Systems 09/2006; · 2.10 Impact Factor
Article: Using dynamic voltage drive in a parallel-plate electrostatic actuator for full-gap travel range and positioning[show abstract] [hide abstract]
ABSTRACT: The nonlinear dynamics of the parallel-plate electrostatically driven microstructure have been investigated with the objective of finding a dynamic voltage drive suitable for full-gap operation. Nonlinear dynamic modeling with phase-portrait presentation of both position and velocity of a realistic microstructure demonstrate that instability is avoided by a timely and sufficient reduction of the drive voltage. The simulation results are confirmed by experiments on devices fabricated in an epi-poly process. A 5.5-V peak harmonic drive voltage with frequency higher than 300 Hz allows repetitive microstructure motion up to 70% of gap without position feedback. The results of the analysis have been applied to the design of a new concept for positioning beyond the static pull-in limitation that does include position feedback. The measured instantaneous actuator displacement is compared with the desired displacement setting and, unlike traditional feedback, the voltage applied to the actuator is changed according to the comparison result between two values. The "low" level is below the static pull-in voltage and opposes the motion, thus bringing the structure back into a stable regime, while the "high" level is larger than the static pull-in voltage and will push the structure beyond the static pull-in displacement. Operation is limited only by the position jitter due to the time delay introduced by the readout circuits. Measurements confirm flexible operation up to a mechanical stopper positioned at 2 μm of the 2.25 μm wide gap with a 30 nm ripple.Journal of Microelectromechanical Systems 03/2006; · 2.10 Impact Factor
A new two-beam differential
resonant micro accelerometer
Politecnico di Milano
piazza Leonardo da Vinci 32 - Milano
Electronics and Information
Politecnico di Milano
via Ponzio 34/5 - Milano
via Tolomeo 1, Cornaredo - Milano
Abstract—A novel uniaxial micro-machined resonant ac-
celerometer is presented. The device working principle is based
on the stiffness variations of a beam which is fully clamped to the
substrate on one side and clamped to a seismic mass on the other
side. A movement of the seismic mass, induced by an external
acceleration, causes either a compressive or a tensile stress on
the beam, inducing a variation of its stiffness. This variation
results in a change of the resonance frequency of the beam. The
accelerometer is arranged in a differential structure, with two
beams built in such a way that their changes in the resonance
frequency have opposite sign. This solution allows obtaining a
doubled sensitivity with the same area and allows reducing the
non linear behavior. First experimental results show that the
device has an overall differential sensitivity ∆fres/g ≈ 450 Hz/g
in the linear range of operation, with an overall area occupation
lower than (500 µm)2.
Micro Electro Mechanical Systems (MEMS) represent
the emerging technology for the production of low-cost and
low-power inertial sensors. In the last ten years these devices
have completed the transition phase from the research to
the industrial market –. Most of the available MEMS
accelerometers is based on the movement of a seismic mass
which forms a set of capacitors with suitably designed
constrained parts. An external acceleration directly results in
a capacitance variation, which is the electrical signal to be
read. These kinds of capacitive accelerometers can be split in
two classes depending on how the capacitances are formed,
i.e. by means of parallel plates or by means of comb fingers.
The quest for low-cost products makes it compulsory to
minimize the Silicon area occupied by a device. Both parallel
plates and comb fingers capacitive accelerometers suffer from
this dimension scaling: parallel plates capacitors are prone to
pull-in instability, which gives a limit to the minimum gap
dimension between plates, and thus to the miniaturization
of the accelerometer , ; comb-finger capacitors on the
contrary do not suffer pull-in problems but need a doubled
area to ensure the same sensitivity of parallel plates.
In order to overcome these problems and to design even
smallerdevices, resonating accelerometershavebeen
proposed in the scientific literature –, , . The
working principle is different in that an external acceleration
causes a variation in the resonance frequency of a suitable
micromechanical structure. The change in the resonance
frequency is the signal to be read. Resonant sensing, with
respect to other sensing principles, has the advantage of an
immunity to pull-in instability, a high potential sensitivity
and a large dynamic range. To obtain high sensitivity when
using very small inertial masses, a particular care should be
given to the geometry of the device.
This paper focuses on the analysis and the experimental
results of the new uniaxial accelerometer proposed in ,
fabricated using the ThELMA (Thick Epitaxial Layer for
Microactuators and Accelerometers) surface micro-machining
process of STMicroelectronics . The basic principles
on which this sensor works are similar to those of already
existing resonant accelerometers, but the geometrical setting
and hence the properties of the mechanical part are completely
different. The mechanical behavior of the resonating parts
and of the whole system are described and a geometrical
optimization is performed in order to obtain the highest
sensitivity at a certain proof mass.
The sensitivity of the resonant accelerometers is defined as
the frequency shift produced by an external acceleration of
1 g. The resonant accelerometers obtained through surface
micromachining reported in the literature typically have
sensitivity ranging from 40 Hz/g up to 160 Hz/g, for the
quite big device proposed in . The optimized new design
allows to produce a very small accelerometer: the proof mass
has a square shape of 400 µm x 400 µm. The experimental
results show a high sensitivity of more than 430 Hz/g in a
differential configuration, for devices with a quality factor Q
II. ACCELEROMETER DESIGN
The operating scheme of a linear accelerometer can be
described as follows. An inertial mass m is attached to a
frame by means of a spring of stiffness k and is subject
978-1-4244-5335-1/09/$26.00 ©2009 IEEE158IEEE SENSORS 2009 Conference
to damping (mainly in the free molecular flow regime at
typical MEMS packaging pressures) represented by a damper
of coefficient b. When the reference frame is subjected to an
external acceleration a, the oscillation of the inertial mass is
governed by the dynamic equilibrium equation:
m¨ x + b˙ x + kx = ma
If the frequency Ω of the external acceleration a(t) is well
below the resonance, i.e. if Ω << fres( fres= (1/2π)?k/m
accelerometer response is quasi-static and x(t) = (m/k) · a(t).
The external acceleration turns out to be proportional to the
mass displacement and the sensing can be done by measuring
the mass displacement e.g. via the capacitance variation, as
in capacitive accelerometers.
being the resonance frequency of the accelerometer) the
In resonant accelerometers on the contrary, the input ac-
celeration is detected as a shift in the resonant frequency of
a sensing device coupled to the proof mass. The resonant
part can be the proof mass itself or, more frequently, a beam
connected to it. The scheme corresponding to this second
case is represented in Fig. 1, where the resonant beam is
shown horizontally. The operating principle is based on the
dependence of the resonant characteristic on the axial force
which is present in the resonator. The external acceleration a
produces a force F = ma on the inertial mass m. This force
produces, in turn, an axial force N in a resonating beam. For
a single span beam, the frequency increases in the case of a
tensile load and decreases in the case of a compressive load.
Denoting by f0the fundamental frequency of the beam, when
resonating without any axial load, the resonant frequency f of
the axially loaded beam can be expressed as (cfr e.g. ,
2 · π · L2·
where L, A and I are the length, the cross area and the inertial
momentum of the beam resonator, E is the elastic modulus, ρ
is the material density, while c and α are coefficients which
depend on the boundary conditions of the resonator (in the
approximation of clamped-clamped beam their values are
4.730 and 0.0246 respectively).
Several accelerometers based on this operating principle have
been proposed in the literature through bulk micromachining
,  and surface micromachining technologies –.
These accelerometers have a different geometry (in particular
different position of the resonating beam with respect to the
proof mass) which greatly affects the amplification of the
axial force and hence the sensitivity of the device.
f = f0·
1 + αN · L2
E · I
ρ · A
E · I
In the device proposed in , the proof mass is suspended
by means of two springs which restrain its movement to be
a uniaxial translation. The resonating part of the device is
Simplified scheme of a uniaxial resonant accelerometer.
constituted by two very thin beams, attached to the substrate
at one end and to the springs at the other end [see Fig. 2(a)].
The position of the resonators with respect to the anchor point
of the spring is crucial to obtain high amplification of the axial
force as will be discussed in the following.
Driving and sensing of the resonators is made by two parallel
plate capacitances formed through electrodes attached to the
For zero external acceleration the resonators have the same
nominal frequency f0. When an external acceleration is ap-
plied, one resonator (resonator 1) is subject to a tension and the
other (resonator 2) to a compression of the same magnitude,
N1 = - N2, as shown in Fig. 2(b). The frequency f1 of
resonator 1 increases while the frequency f2 of resonator
2 decreases. Combining the signal of the two resonators,
from Eq. (2) linearized around f0, one obtains the following
f1− f2≈ f0· αN · L2
E · I
Therefore the presence of two resonators undergoing
opposite axial forces has several advantages: (i) the sensitivity
can be doubled by measuring the difference between the
frequencies of the two resonators instead of the variation of
frequency of one resonator, (ii) the linearity of the system is
improved, i.e. the accelerometer response can be linearised
in a wider range of accelerations, (iii) the skew-symmetric
geometry is less sensitive to spurious effects of thermal
loading (indeed an inelastic effect causing pre-stress in
the resonators is cancelled when considering the difference
between the frequencies).
It is important to note that, though the resonator actuation
and readout is performed through parallel plate capacitances,
the resonator displacements are small compared to the
gap between the resonator and the fixed plates. Moreover
the effect of very large external accelerations is only a
displacement of the proof mass, which results in a change of
the resonator stiffness. The driving/sensing capacitances are
not driven to pull-in even in this situation (as on the contrary
may happen for parallel plate capacitive accelerometers).
The relevant parameters of the device tested in this work
are summarized in table 1. To allow for the detachment of
the moving mass from the substrate, the inertial mass has
a pattern of squared holes, so that a reduction coefficient
of 0.63 should be considered when computing its effective
area. Moreover due to the fabrication process, the surfaces in
(a) Scheme of new resonant accelerometer; (b) effect of external
the out of plane direction are waved, thus a resonator width
sligthy reduced with respect to the nominal one should be
The sensitivity of the resonant accelerometer, defined as the
resonator frequency variation produced by an acceleration of
1 g, increases with the dimension of the moving mass but
also strongly depends on the position of the resonating beams
with respect to the anchor point of the spring. In order to
reduce the device size while keeping a high sensitivity, this
position was optimized by means of an analytical approach
and of FEM simulations. Fig. 3 shows the sensitivity of the
accelerometer, given through equation (3), as a function of the
position of the resonator along the spring, normalized with
respect to the spring length. One can observe that the optimal
position for the resonator is very close to the anchor point
of the spring at about one sixtieth of its length. The different
curves correspond to different sizes of the inertial mass: the
intermediate one corresponds to the tested accelerometer, with
a square proof mass with a side of 400 µm. In the following
sections first experimental results both on the mechanical
characterization of the resonators and on the sensivitity of the
accelerometer will be given.
a function of the resonator position. Different curves correspond to the mass
m of the fabricated accelerometer and of double and half the mass of this
Sensitivity of the accelerometer in terms of frequency variation as
RESONANT ACCELEROMETER PARAMETERS
Proof mass area400 x 400
In plane proof mass 1stmode
Out of plane proof mass 2ndmode Hz
III. MECHANICAL CHARACTERIZATION OF THE
The mechanical characterization of the resonant accelerom-
eters has been performed through the low-noise, low-
perturbing readout setup for MEMS capacitances described in
. The setup uses a signal modulation at a frequency fmod
= 1 MHz which is far larger than the resonator mechanical
bandwidth. The amplitude of the test signal has a value of
≈ 250 mVeff. Measurements are controlled through Labview
The devices were wafer-wafer packaged at a pressure p ≈ 1
mbar, and the pads have been wire bonded to an holding board
which can be tilted and which is electrically connected to the
main, fixed, readout board. The actual resonators have a length
L = 400 µm, a thickness t = 15 µm and a nominal width h =
A. Stationary response
Fig. 4 reports experimental results representing the sensed
capacitance variation (measured between the sensing plate and
the resonator) with respect to a stationary voltage applied
between the driving plate and the resonator. As the driving
voltage increases, the beam is deflected towards the actuation
plate and as a consequence the sensing capacitance decreases
Fig. 4. Sensing capacitance vs driving actuator voltage curves for 4 different
resonators belonging to 2 different accelerometers (experimental points and
fitting quadratic polynominial). A certain data spreading is the result of
the high shrinkage used in the resonator width design. Measurements are
performed applying a voltage VAto the driving electrode and measuring the
sensing capacitance through a small-amplitude, high frequency signal VS,
processed through a custom lock-in type readout electronics.
of ≈ 3 fF at 5.4 V. In the simplified assumption of a parallel
plate driving capacitor of area ASand gap x0, the theoretical
relationship between the applied driving voltage VA and the
applied electrostatic force Fel is a function of the resonator
A· ?0· AS
2(x0− x)] =V2
(1 − x/x0)2(4)
being C0the rest capacitance value (for x = 0) and ?0the air
permittivity (taken as that of vacuum). For small displacements
(x → 0) the force can be assumed to be independent from the
The beam displacement x is proportional through the elastic
stiffness kbeamof the beam to the applied force Fel:
x = Fel/kbeam∝ V2
and the measured sensing capacitance variation ∆CS caused
by a displacement x is, in the first order approximation of
x0+ x≈ −C(0)x
being ASthe area of the sensing capacitor (which is the same
as the area of the driving capacitor in our device).
It is worthnoting that in the present case the resonating
beam deforms and therefore the gap variation is non-uniform.
However the above formula can still be used, with a proper
correction coefficient, by interpreting x as the displacement of
the beam mid point.
It turns thus out that the sensing capacitance variation, which
is proportional to the resonator displacement x, has a square
a single resonator of the accelerometer. (b) Corresponding spectral response
derived through the DFT.
(a) Time response to a downward force step (to the rest position) of
dependence from the driving voltage VA for small displace-
ments of the resonator. Indeed parabolic curves in Fig. 4 well
fit the experimental points, as the displacements are small
with respect to the gap between the resonator and the driving
electrode. The results show a non negligible data spreading,
caused by the very reduced width of the resonator, which is
thus affected by an unavoidabe process dispersion.
B. Dynamic response
In this subsection the experimental results representing
the dynamic response of the resonators are reported. The
tests have been carried out in the following way: through
a quasi-stationary voltage VA applied between the driving
plate and the resonator, the beam is slowly deflected until
an established perturbed position. Then the voltage VA
is instantaneously set to zero. As a consequence of this
downward electrostatic force step, the resonator begins to
oscillate around the rest position and through the sensing
plate the value of the sensing capacitance CS(t) is real-time
monitored, as reported in Fig. 5 (a).
From this measurement the spectral response of the device can
be calculated by means of the Discrete Fourier Transform.
Fig. 5 (b) reports a typical result, showing a resonance
frequency f0= 52.21 kHz, with a quality factor Q ≈ 208.
The dashed curve represents the response at 0 g (where a small difference in
the resonance frequency is the result of a process data spreading). The solid
curves represent the spectral response at different external acceleration in the
range ± 1 g.
Peak frequency shift of the resonators of the resonant accelerometer.
IV. EXPERIMENTAL RESPONSE TO EXTERNAL
The response of the device to external accelerations has
been measured in the range ± 1 g. The variation of the
external applied acceleration is simply obtained through
different inclinations of the device. The resonance frequency
is acquired, as described in section III B, at different tilt
angles between −90oand +90o. To simultaneously acquire
the data from both the resonators, the driving and sensing
pads of the two resonators have been short circuited. Fig.
6 reports the expermimental results obtained acquiring
simultaneously the data from the two sensing capacitances.
The spectral response presents a typical double peak, due to
the fact that the resonance frequencies of the two resonators
are not equal, as a consequence of the process spreading
already observed in the CV measurements.The results show
that when a resonator is compressed (and the other one is at
the same time stretched) its resonance frequency decreases
(and that of the other resonator increases).
The device linearity (see Eq. 3) is reported in Fig. 8. Being
∆f0the frequency difference between the resonant peaks of
the two resonators at 0 g, the experimental points represent
the variation of the peak frequency difference ∆f − ∆f0as
a function of the external acceleration in the range ± 1 g.
A good linearity and an overall differential sensitivity larger
than 430 Hz/g (at a resonance frequency of both resonators
around 50 kHz) is obtained.
Fig. 8 reports a SEM picture of an unpackaged device. A
detail of the point where the resonator is attached to the
spring is shown in the close up.
∆f − ∆f0 as a function of the external acceleration in the range ± 1 g.
∆f0corresponds to the peak frequency difference at 0 g. The device shows
a good linearity and a sensitivity higher than 430 Hz/g.
Variation of the peak frequency difference between the resonators
detail shows the point where the resonator is attached to the spring. This point
is chosen to have the maximum device sensitivity.
Scanning Electron Microscope image of an unpackaged device. The
First experimental results on an innovative uniaxial
resonant accelerometer have been presented. The device has
an overall area occupation lower than (500 µm2). Thanks to
an optimization of the geometrical parameters, a very high
differential sensitivity, close to 500 Hz/g (around a resonance
frequency of 50 kHz) has been obtained, with a good linearity
in the range ± 1 g. The work in progress includes the re-
design of the resonating parts of the accelerometers, to reduce
a certain performance spreading caused by an underestimation
of the process overetch, and the coupling of the device to a
suitable readout electronics. A two dimensional accelerometer
based on the same principle is also under development.
Part of the experimental setup used in this work was funded
by Fondazione Cariplo in 2007, within the project Dissipative
and failure phenomena in Micro and Nano Electro Mechanical
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