Over the past years, MEMS (micro electromechanical system) accelerometers have
developed rapidly and captured the all-important markets consisting of the consumer
electronics, mobile devices, automation, industry, medical, seismometry and
inertial navigation (Perlmutter and Robin, 2012; Je
et al., 2010). Compared to other types of MEMS accelerometers, CMA
(capacitive micromachined accelerometers) have several attractive advantages,
such as low fabrication cost, low power dissipation, low noise, high sensitivity,
high reliability, low drift and low temperature coefficient (Tan
et al., 2011a; Sun et al., 2011;
Chan et al., 2012). Therefore, many research
institutes focus on the design and optimization of CMA.
The main factors which have great effects on the micromachined accelerometer
system include not only the structure and mechanical parameters of the accelerometer,
the noise performance of the interface circuits but also the external environment.
Among many environmental factors, ambient temperature is the most important
one which influences the performance of the accelerometer system significantly,
such as bias stability. At present, there are many research institutes dedicating
to the temperature characteristic research of micromachined accelerometer system.
One of the main approaches is to keep the temperature of the accelerometer system
invariable by temperature control scheme (Lakdawala and Fedder,
2004). The method will induce higher cost, larger size and more complexity.
The other approach is to establish the mathematical expression between the output
signal of the micromachined accelerometer system and the temperature by formula
fitting or model estimation. Based on the mathematical expression, real-time
temperature compensation can be accomplished by using hardware circuit or software
algorithm (Weng et al., 2009; Yu
et al., 2011; Zhang and Chang, 2011). In
addition, several research groups analyzed the effect of the material, structure,
micromachined technology, packaging on the thermal performance of micromachined
accelerometer and put forward some effective optimization methods in structure
design and manufacturing technology (Painter and Shkel,
2001, 2003; Chae et al.,
2005; Yen et al., 2011; Tan
et al., 2011b; Myers et al., 2012).
With the aim of decreasing the temperature coefficient of fence structure CMA
system, a system-level simulation model with temperature built based on theoretical
analysis and experiment is presented in this study. Through the simulation model,
the main factor which has great effect on the temperature performance of the
accelerometer system can be found and an optimizing approach will be put forward.
MATERIALS AND METHODS
Fence structure CMA: The fence structure CMA is composed of a proof-mass
suspended by four U-shape springs and fence structure differential capacitors
used as the sensing elements. A structure diagram of the CMA is shown in Fig.
1a and the fence structure capacitors can be simplified as shown in Fig.
1b (Zheng et al., 2009). The accelerometer
is fabricated by bulk silicon micromachining technology. When an acceleration
is applied along x-axis, the movable proof-mass will have a displacement which
is transferred into the capacitance variance ΔC by changing the overlapped
areas of the differential sensing capacitors, with one sensing capacitor C01
increasing and the other C02 decreasing. The sensing scheme not only
achieves low damping but also eliminates the nonlinear effect.
The dynamic equation of the accelerometer is given by Eq. 1:
where, x is the displacement of the proof-mass from its rest position with
respect to the fixed frame, M is the mass of the proof-mass, kx denotes
the mechanical spring constant along x-axis and c denotes the damping factor.
The input acceleration a can be obtained by measuring ΔC through the interface
Mechanical spring constant: The U-shape spring used in the fence structure
CMA is helpful for residual stress releasing due to the arc part of the beam
as illustrated in Fig. 2.
The mechanical spring constant of U-shape spring along x-axis can be expressed
by Eq. 2 (Chen, 2004):
where Iz = w3t/12 denotes inertia moment of the spring
beam in z-axis, E is the Youngs
modulus of single-crystal silicon for <100> direction, l is the length
of the straight beam, r is the inside radius of the semicircle, w and t represents
the width and thickness of the spring beam separately and ns is the
total beam number.
The representative dimensions of U-shape spring are l = 1050 μm, w = 14
μm, t = 297 μm and r = 22 μm. Meanwhile, ns and E
equals to 4 and 130 GPa, respectively. With those values, kx equals
to 166.28 N m-1 at 25°C calculated using Eq. 2.
||(a) Structure diagram of CMA (capacitive micromachined accelerometer)
and (b) Sketch map of the fence structure differential capacitors
|| Sketch map of U-shape spring
Moreover, the temperature coefficient of kx equals to -72.5 ppm
°C-1 by considering that the temperature coefficient of E is
-75.0 ppm °C-1 and the Thermal Expansion Coefficient (TEC) for
Si is 2.5 ppm °C-1.
Damping factor: The movement of the proof-mass is in X-Y plane and the
fixed electrodes are in parallel with the proof- mass, so the slide-film damping
dominates in the accelerometer (Zheng, 2009). According
to Couette flow model which is effective at low frequency, the slide-film damping
factor is given by Bao (2000):
where, A is the surface area of the plate suffering damping force, h0
is the gap distance between the proof-mass and the fixed electrodes, μeff
represents the effective viscosity coefficient and can be described by Veijola
and Turowski (2001):
where, Kn is the Knudsen number, it is the ratio between the mean
free path of the molecules λ and gap distance h0: Kn
represents the viscosity coefficient,
denotes the average velocity of the gas molecules and ,
the gas density ρ = PMol/RT according to Clapeyron equation,
R is gas constant, Mol is the molar mass of the gas, P is the gas
pressure and T is temperature in Kelvin.
The expression of the mean free path is (Bird, 1983):
where, πd02 represents the molecular collision cross-sectional
area, n is the number density of molecules which can be expressed by n = P/kBT
for ideal gas, kB is Boltzmann constant.
Thus, with Eq. 3-5, the expression of c
can be drawn out:
||The length of the proof-mass
||The width of the proof-mass
||Temperature in °C
In addition, kB = 1.38x10-23 J K-1, d0
= 3.74x10-10 m, R = 8.314 Pa•m3 mol-1•K,
Mol = 29x10-3 kg mol-1 and P = 1.013x105
Pa due to the unsealed package of the accelerometer. The typical dimensions
of the fence structure CMA are h0 = 1.7 μm, LxW = 2200 μmx2360
μm, the thickness of the proof-mass is 297.7 μm, the thickness of
Al electrode is 0.6 μm and the thickness of fixed frame is 300 μm.
All dimensions are affected by thermal expansion of materials certainly. TEC
of Si is 2.5 ppm °C-1 and TEC of Al is 23.6 ppm °C-1.
Combining Eq. 6, it can be got that c equals to 3.3525x10-5
Nm sec-1 at 25°C and its temperature coefficient is 1650.4
EXPERIMENT AND SIMULATION
Experimental test: The macro expression of the spring constant can be
where ω0 is the resonant angular frequency of the accelerometer,
M = 3.619x10-6 Kg represents the mass of the proof-mass.
The macro expression of the damping factor is given by:
where, Δf denotes 3 dB bandwidth.
|| Test condition for initial sensing capacitance
|| Spring constant, damping factor and their temperature coefficients
|| Initial sensing capacitances and their temperature coefficients
Based on Eq. 7, 8, the spring constant
and the damping factor can be calculated by measuring ω0 and
Δf under different temperatures, when an external driving signal with constant
amplitude and variable frequency is applied to the accelerometer. The experimental
results are listed in Table 1.
From Table 1, it can be seen that the spring constants kx
of different accelerometers are not the same due to different beam widths in
design while the temperature coefficients of kx are all in the range
of -60~-50 ppm °C-1 which is agree well with the theoretical
analysis. The damping factors and their temperature coefficients show greater
differences from the theoretical analysis. One of the reasons is due to the
slots on the proof-mass which increases the air damping. The other is the theoretical
error of slide-film damping induced by the simplified Couette flow model.
Initial sensing capacitance: The initial sensing capacitance of the
fence structure CMA can be expressed as:
where, ε is dielectric constant, N is the number of fingers, y0
and x0 denotes the overlap length and overlap width, respectively,
h0 is the gap distance.
In order to measure the temperature coefficients of the two initial sensing
capacitances, the accelerometer was put in the temperature control device and
a constant carrier wave was applied to the sensor. The test condition is shown
in Fig. 3. C-V (capacitance-voltage) conversion circuit can
convert the capacitance of the accelerometer to an amplified voltage signal
proportionally. In the experiment, the amplitudes of the two voltage signals
were measured by Agilent digital multimeter when the temperature was ranging
from 25 to 65°C at intervals of 5°C. Consequently, C01 and
C02 and their temperature coefficients could be obtained. The experimental
results are shown in Table 2.
Table 2 indicates that the two initial sensing capacitance
C01 and C02 in the same accelerometer are not equal or
their temperature coefficients differ from each other either. One reason for
this is that the overlap width x01 differs from x02 due
to the imprecise align of alignment marks and the nonideal packaging for accelerometer
during manufacturing process. On the one hand, temperature influence on the
initial sensing capacitances is caused by thermal expansion and contraction
of materials which change the overlapped area and the gap distance. Furthermore,
there is thermal stress among the layers of the accelerometer which is a function
of temperature because each layer has its own thermal expansion coefficients.
|| System-level model for fence structure CMA system (built
Thermal stress also induces the initial sensing capacitances to change with
temperature. As well as thermal stress, residual stress affects the temperature
characteristic of the initial sensing capacitances directly. MEMS technology
such as high temperature oxidation and high temperature bonding could induce
residual stress among the layers. In addition, residual stress also exists in
the spring beams due to the defects formed during fabrication, although the
circular part of U-shape spring beam is good for stress release to a certain
Establishment and verification of the temperature model: We have reported
a system-level simulation model of fence structure CMA system based on Simulink
platform (Zhang et al., 2010). The model could
simulate noise performance. On this basis, temperature factor was taken into
account. The temperature-dependent parameters in the accelerometer include spring
constant, damping factor and initial sensing capacitances. Furthermore, the
model contains parameters which could be affected by temperature in the interface
circuits, such as amplifier input offset voltage, amplifier input offset current,
gain, gain error, thermal noise, resistance, capacitance and so on. The system-level
simulation model with the mechanical parameters of the accelerometer 0# and
its interface circuits is shown in Fig. 4.
In order to investigate the temperature performance of the accelerometer system,
the accelerometer and its interface circuits were all put in the temperature
control device. The output voltage under zero acceleration was measured with
temperature ranging from 25 to 65°C at intervals of 5°C. Meanwhile,
the simulation was performed under the same conditions. The experimental result
and the simulation result are illustrated in Fig. 5.
As shown in Fig. 5, it is clear that the simulation result
is agree well with the experimental result which not only verifies the correctness
of the simulation model but also best illustrates that the mechanical parameters
such as spring constant, damping factor and initial sensing capacitances and
all circuit parameters took into account in the model can reflect the temperature
performance of the accelerometer system elaborately.
||Output voltage of fence structure CMA system versus temperature
(including experimental results and simulation results)
Meanwhile, Fig. 5 shows that the absolute value of the output
voltage in simulation is a little higher than the experimental result. The reason
is the two input signals of the synchronous demodulation circuit are not in
phase perfectly in the experiment. From the linear fitting curve of the experimental
result, it can be got that the temperature coefficient of the output voltage
is -0.818 mV °C-1 which corresponds to an effective acceleration
variation of -6.245 mg °C-1 with the sensitivity 131 mV g-1.
The sensitivity test of the accelerometer system is performed on a circular
dividing table. By collecting the output dc voltage of the system using Agilent
digital multimeter every 5° and linear data fitting, the system sensitivity
is obtained. The result above shows that the temperature has a significant influence
on the bias stability of the fence structure CMA system.
RESULTS AND DISCUSSION
In order to decrease the temperature coefficient of the fence structure CMA
system, the most important factor influencing the system temperature performance
was found based on the simulation model.
|| Main factors influencing the temperature performance of the
||Temperature coefficient of fence structure CMA system versus
Table 3 illustrates the simulation results under different
situations. It can be observed that the temperature coefficient of the accelerometer
system has been reduced to -0.64 mg °C-1, when the variations
of the two initial sensing capacitances with temperature were set to be the
same accurately in the model. Therefore, the temperature characteristic of C01
and C02 is the most major factor influencing the system temperature
performance. The secondary factor is the temperature characteristic of interface
From Table 2, it can be seen that capacitance variations
of C01 and C02 with temperature differ from each other.
So, we use mismatch degree to calibrate the difference. The expression of the
mismatch degree is shown as:
where TCC01 and TCC02 is the capacitance variation of
C01 and C02 as temperature changing respectively. The
relationship between mismatch degree and system temperature coefficient is illustrated
in Fig. 6 which was obtained from the simulation model.
It is evident that the higher the mismatch degree is, the stronger the temperature
influence on the output voltage of the accelerometer system becomes. In order
to optimize the system temperature performance, the unequal variations of C01
and C02 with temperature should be compensated. The output
dc voltage of the accelerometer system at 25°C can be expressed as:
where, Vcarrier is the amplitude of carrier wave, Cf
is the feedback capacitance of C-V conversion circuit, GIN is the
gain of instrumentation amplifier, GD and GLP represents
the gains of the demodulation circuit and the low-pass filter, respectively.
When the temperature changes Δt, the output dc voltage becomes:
where, KTC2 and KTC3 represent the temperature coefficients
of C01 and C02. KTC5 is the temperature coefficient
of GIN. KTC1, KTC4, KTC6 and KTC7
are the temperature coefficients of Vcarrier, Cf, GD,
GLP, respectively and they are all smaller than 20 ppm °C-1.
Neglecting KTC1, KTC4, KTC6 and KTC7,
Eq. 12 can be simplified as:
Equation 13 shows that choosing KTC5 rationally
could compensate the difference between KTC2 and KTC3
and decrease the temperature coefficient of the accelerometer system. The gain
of the instrumentation amplifier AD8221 is expressed as:
where, RG is the gain resistance. Therefore, KTC5 could
be changed by adjusting the temperature coefficient of RG.
||Output voltage of fence structure CMA system versus temperature
after optimization (experimental results)
In this study, we use a thermal resistor with 1600 ppm °C-1
and resistance 3.6 kΩ at 25°C as the gain resistance of the instrumentation
amplifier to compensate the unequal variations of C01 and C02
with temperature. The optimized accelerometer system was measured with temperature
ranging from 20 to 65°C at intervals of 5°C. The test result is shown
in Fig. 7 and the slope of the linear fitting curve decreases
to -0.275 mV °C-1 which corresponds to -2.1 mg °C-1.
The experimental result after optimization shows that using thermal resistor
with high temperature coefficient as the gain resistance of the instrumentation
amplifier could reduce the system temperature drift drastically. Whats
more, the optimized system is the same as before in size and complexity, owing
to no additional hardware circuit or software algorithm used for temperature
In this study, the relationship between the main physical parameters of the
accelerometer and the ambient temperature was obtained through theoretical analysis
and experiment. The main influencing factor of the temperature performance of
the fence structure capacitive micromachined accelerometer system was found
to be the mismatch between variations of the two initial sensing capacitances
with temperature. The system was optimized by replacing ordinary gain resistance
of the instrumentation amplifier with thermal resistance and the system temperature
coefficient was reduced to -2.1 mg °C-1 which is 33.6% as before.
In the future, a more effective optimization method will be proposed by simulation
model and applied to this accelerometer system.
This study was supported by research foundation project for Young Teachers
of Xian University of Posts
and Telecommunications, Peoples
Republic of China. (101-0477).