An electronic calibration technique forcapacitive MEMS accelerometers based on themeasurement of pull-in voltages is described. Acombination of pull-in voltages and resonance frequencymeasurements can be used for the estimation of processinducedvariations in device dimensions from layout anddeviations in material properties from nominal value,which enables auto-calibration. Measurements onfabricated devices confirm the validity of the proposedtechnique and electronic calibration is experimentallydemonstrated

Edmond Cretu

José Machado da Silva

L. A. Rocha

Lukas Mol

Reinoud F. Wolffenbuttel

Open Repository of the University of Porto

 AUTO-CALIBRATED CAPACITIVE MEMS ACCELEROMETER  
L.A. Rocha1, L. Mol2, E. Cretu3, R.F. Wolffenbuttel2 and J. Machado da Silva1 
1Dept. of Electrical and Computer Eng., Fac. of Eng. of the U. of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal 
2Dept. Microelectronics, Fac. EEMCS, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands  
3U. of British Columbia, Dept. Electrical and Computer Eng., 2332 Main Mall, Vancouver, BC, V6T 1Z4 Canada 
 
Abstract ⎯  An electronic calibration technique for 
capacitive MEMS accelerometers based on the 
measurement of pull-in voltages is described. A 
combination of pull-in voltages and resonance frequency 
measurements can be used for the estimation of process-
induced variations in device dimensions from layout and 
deviations in material properties from nominal value, 
which enables auto-calibration. Measurements on 
fabricated devices confirm the validity of the proposed 
technique and electronic calibration is experimentally 
demonstrated. 
Key Words: MEMS accelerometers, pull-in, calibration 
I. INTRODUCTION 
Accelerometers and pressure sensors are products based on 
microelectromechanical systems (MEMS) technology with 
a high commercial success and impact [1]. Several 
accelerometer designs are commercially available, from 
simple single axis to more complex three axes devices. 
Despite the evolution of acceleration microsensors in the 
last 30 years, MEMS based accelerometers need to be 
calibrated for use in high performance applications. 
Accelerometer calibration is usually done by measuring the 
response of the sensor to a sequence of +1g followed by 
-1g static acceleration test signals; this is accomplished by 
aligning the sensitive axis parallel with the gravitational 
field, and switching between the two possible orientations. 
This calibration step is often done manually, and is 
therefore an inconvenient alternative for remotely placed 
sensors or for sensors difficult to access. Auto-calibration 
capabilities would be preferred instead, since they allow 
the inertial sensors to be integrated in autonomous self-
calibrated systems. 
This paper introduces an electronic calibration technique 
for capacitive MEMS accelerometers based on the 
measurement of pull-in voltages. Pull-in [2] is a unique 
feature of gap-varying capacitive MEMS devices, and can 
provide detailed information about their characteristics. 
Since the attractive force due to an electrostatic field is 
inversely proportional to the square of the deflection, while 
the restoring elastic force is (to a first approximation) linear 
with deflection, an unstable system results in case of a 
deflection, v, beyond a critical value, vcrit. The pull-in 
voltage, Vpi, is defined as the voltage that is required to 
obtain this critical deflection and depends mainly on 
geometry, residual stress level and material properties. This 
dependence makes it ideal for characterizing structural 
materials in surface micromachining processes [3,4]. 
Unlike the case of the comb drive, which is based on area-
varying capacitors, the design of most electrostatic 
actuators relies on gap-width varying capacitors and the 
pull-in phenomenon has to be considered [5]. Pull-in causes 
the displacement range due to electrostatic force to be limited 
to 1/3 of the gap between the electrodes, in case of a motion 
of the movable capacitor plate perpendicular to the plates. 
When pull-in voltage measurements are combined with the 
measurement of the resonance frequency (a single 
measurement is needed), fabrication process non-idealities 
like over-etching and process asymmetries can be 
accurately estimated [6]. The parameters of an accurate 
device model could be extracted from this measurement 
setup, providing in fact an equivalent calibration capability 
as the application of an external inertial force of ±1g, but 
using only electronic excitation. 
II. PULL-IN VOLTAGE AND NON-
IDEALITIES ESTIMATION 
The simplest symmetric micromechanical system suitable 
for studying the pull-in voltage is composed of three 
electrodes. Two of them have a fixed position on the rigid 
supporting substrate, while the middle one is movable and 
connected to an elastic suspension with the spring constant 
k (Fig. 1a). This is often the case of capacitive 
accelerometers with separate electrodes for sensing and 
actuation. The static balance between the elastic and 
electrostatic forces defines the equilibrium position of the 
movable plate for a given actuation voltage. Its stability is 
given by the rate of variations of the two forces for small 
perturbations around the equilibrium point, that is, by the 
second derivative of the global potential energy. For a 
stable equilibrium, the second derivative of the potential 
energy of the system with respect to deflection should be 
positive: 2 2 0pU x∂ ∂ > ; thus the pull-in voltage (Vpi) 
results as the solution of the two equations corresponding 
to 0pU x∂ ∂ =  (force equilibrium) and 2 2 0pU x =∂ ∂  
(margin of stability). Pull-in voltage is determined by the 
elastic flexure (beam) material and dimensions, residual 
stress, and the geometry of the electrodes.  The beam should 
be suspended using folded tethers at each end, in order for the 
residual stress to not affect Vpi[7]. This approach ensures that 
the built-in strain energy component caused by longitudinal 
stress is negligible. The geometry of the structure allows the 
definition of two pull-in voltages, as shown in Fig. 1a: 
asymmetric-right (Vpr), and asymmetric-left (Vpl). They will 
be equal in ideal conditions (perfectly symmetric structure), 
as can be shown by the theoretical analysis [2]: 
 
3
0
0
8
27pr pl
d kV V
wlε= = , (1) 
Here d0 is the capacitor initial gap, k is the mechanical 
spring constant, ε0=8.8546x10-12 is the air permittivity and 
w and l are the capacitor plate width and length, 
respectively.  
 
Figure 1. Sketch of the basic device with a) ideal 
conditions and b) with over-etch and asymmetries 
 
If non-ideal process conditions are now considered (Fig. 
1b) like over-etching [8], capacitor gap mismatch and 
Young’s Modulus (E) value deviations, the pull-in voltage 
values will vary. They become suitable, easy to perform 
measurements (with simple electronics) for estimating 
technological and other non-idealities and for use as 
diagnostic mechanism.  
The method proposed in [6] uses the pull-in voltage as a 
test mechanism for microelectromechanical systems, for 
instance to identify process-induced variations in the real 
device geometry. Fabricated devices often exhibit actual 
dimensions smaller than the designed on the mask layout 
(due to over-etching). Over-etching can be considered 
uniform at the scale of one microfabricated device [8], 
which means that all layout dimensions will be affected by 
the same parameter α. This will have a uniform effect on 
both pull-in voltages. Besides over-etching effects, small 
gap mismatches (a few nm in misalignment) are also 
observed in fabricated devices. As the gap mismatch (β) 
will affect differently the pull-in voltages, it becomes easy 
to estimate β from the differences between Vpl and Vpr. The 
parameter α is more difficult to estimate, because there is 
an extra unknown parameter: the Young’s Modulus (its 
average value is known, but it can show large deviations). 
If we introduce a new measurement, the resonance 
frequency, both α and E can now be estimated and the 
uncertain parameters of the mechanical device are 
identified. Therefore, as the sensitivity of the device is 
known, test signals can now be applied to the actuation 
capacitors to calibrate the full system (device plus readout 
electronics). 
One disadvantage of the proposed technique is that it relies 
on using accurate device models to predict the device 
electro-mechanical behavior. These models have to 
incorporate all the non-idealities existing in the micro-
domain, like capacitor fringe fields and residual stress, 
which makes the modeling one of the critical parts of the 
proposed calibration method. 
III. EXPERIMENTAL RESULTS 
Accelerometers fabricated using Bosch epi-poly 
technological process[9] were used to evaluate the auto-
calibration method. A simplified drawing of the devices 
used is depicted in Fig. 2. 
 
Figure 2. Drawing of the accelerometer 
III.1 FABRICATED ACCELEROMETERS 
The fabricated accelerometers (Fig. 3.) are composed of 
four folded springs, 340 μm long and 3 μm wide (layout 
dimensions), connected to two rigid central bars of about 
1mm long. Parallel-plate capacitors with a 2 μm gap are 
used for actuation. The displacement measurement 
involves sensing the changes of various sets of differential 
capacitors. The main device layout parameters and bulk 
material properties are shown in Table 1. 
Table 1. Main nominal parameters of the device 
(layout dimensions and bulk material mean values)  
Parameter Value 
Spring length (l) 340 μm  
Spring width (b) 3 μm 
Mechanical layer thickness (h) 10.6 μm 
Capacitor length (l) 282 μm 
Capacitor width (w) 10.6 μm 
Capacitor gap (d) 2 μm 
Young’s Modulus (E) 163 GPa (Poly-Si) 
Density (ρ) 2.5 g cm-3 
 
 Figure 3. Fabricated device 
III.2 ACCELEROMETER ESTIMATED MODEL 
A MEMS capacitive accelerometer is a second order 
mechanical system and can be described mathematically in 
Laplace complex domain by: 
 
2
( ) 1
( )
x s
b ka s s s
m m
=
+ +
  (2) 
where x is the displacement of the proof mass, a is the 
external acceleration, b is the damping coefficient, k is the 
mechanical spring constant and m is the mass of the proof 
mass. The mechanical system has a resonance frequency 
given by n k mω = , and the (DC) sensitivity S m k= .  
For a typical calibration procedure the mechanical spring 
constant and the mass of the accelerometer are the 
parameters of interest, while the dynamic behavior is less 
important (calibration is mainly performed in static mode, 
such that it needs only simple electronic circuits). Using the 
method described in [6], pull-in and resonance frequency 
measurements were performed on a fabricated 
accelerometer, and the estimated model parameters are 
shown in Table 2.  
As the dynamic behavior can be neglected in static 
equilibrium, the balance of forces on the accelerometer can 
be written as: 
   (3) ( )elect extk x F V m a⋅ = + ⋅
where Felect(V) is the electrostatic force applied to the 
movable structure for a given voltage. Equation 3 shows 
that there are two alternative ways to achieve a certain 
displacement x: by controlling the applied voltage, while 
maintaining a zero external acceleration or by applying an 
external acceleration with no voltage applied.  
According to the estimated structure parameters, the 
fabricated accelerometer has a sensitivity of 
(
Similarly, if one uses the estimated accelerometer model, 
the voltage necessary to achieve a displacement of ±33nm 
can be computed. Considering that a positive displacement 
occurs when a voltage is applied to the left electrode and a 
negative one, for a right electrode actuation, the following 
voltages levels are found: 
   (4) 1
-1g
1.14V
1.11V
gV
V
+ =
=
Table 2. Measurements and estimated accelerometer 
parameters using a pull-in based test method [6] 
Measurements Value 
Pull-In voltage left (Vpl) 3.942 V 
Pull-In voltage right (Vpr) 3.788 V 
Resonance frequency 2740 Hz 
Estimated technological parameters Value 
Over-etching (α) 255 nm 
Mismatch (β) 34.5 nm 
Young’s Modulus (E) 147.8 GPa 
Estimated accelerometer parameters Value 
Capacitor gap rigth (dright=d+2α-β) 2475.5 nm 
Capacitor gap left (dleftt=d+2α+β) 2544.5 nm 
Mass (m) 3.978 μg 
Mechanical spring (k) 1.18 N.m 
III.3 EXPERIMENTAL RESULTS 
Two different experiments were made in order to 
experimentally check the proposed calibration method. The 
response of the accelerometer to a ±1g external 
acceleration was firstly recorded. The measured output 
voltage is shown in Fig. 4. 
 
Figure 4. Response to a ±1g external acceleration 
Subsequently, the accelerometer response to an increasing 
voltage on the electrodes, in the absence of an external 
acceleration, was verified. The measured results are 
presented in Fig.5. 
III.4 DISCUSSION 
The results presented in Figures 4 and 5 are very promising 
and clearly indicate that electric calibration is feasible in 
capacitive accelerometers. In fact, the results show that the 
output voltage to +1g external acceleration is equivalent to 
the response when a voltage of 1.14V is applied to the right 
electrode (61mV output voltage), while the response to an 
)93.3702 10S Kg−= × ⋅N m . An acceleration of ±1g will 
generate therefore a displacement of ±33nm.  
actuation voltage of 1.11V on the left electrode 
corresponds to output voltage for an applied external 
acceleration of -1g.  
The very good agreement between measured and computed 
values using the estimated model validates again the 
calibration approach.  
These results also prove that the estimated model offers a 
good description of the real behavior of the accelerometer. 
To further verify model validity, extra measurements were 
performed: left and right pull-in voltages were measured in 
the presence of a ±1g external acceleration and compared 
with the pull-in voltages given by the models for this 
situation. Since a ±1g generate a displacement of ±33nm, 
this can be modeled as a new β (mismatch). Therefore, a 
+1g gives a β of 67.5nm and a -1g gives a β of 1.5nm. The 
experimental and computed results are shown in Table 3. 
IV. CONCLUSIONS AND FUTURE 
WORK 
A pull-in based solution for MEMS capacitive 
accelerometer auto-calibration was presented. The 
technique was validated through experimental 
measurements and their comparison with the theoretical 
models.    
Experimental results showed that the response to a ±1g 
external acceleration can be replaced by an equivalent 
asymmetric actuation voltage, approach that allows a pure 
electronic calibration of the sensor. The technique can be 
used in capacitive inertial sensors (e.g. accelerometers and 
angular rate sensors) and allows their integration in 
autonomous self-calibrated systems. 
a) 
 
REFERENCES 
[1] S.D. Senturia, “Perspective on MEMS Past and Future: 
The Tortuous Pathway from Bright Ideas to Real 
Products, Proceedings Tranducers’03, pp.10-15, 2003 
[2] L.A. Rocha, E. Cretu and R.F. Wolffenbuttel, 
“Analysis and Analytical Modeling of Static Pull-In 
with Application to MEMS-Based Voltage Reference 
and Process Monitoring”, J. Microelectromech. Syst., 
vol. 13,pp. 342-354, 2004. b) 
 
[3] S. T. Cho, K. Najafi, and K. D. Wise, “Internal stress 
compensation and scaling in ultra-sensitive silicon 
pressure sensors,” IEE Tr. ED, vol. 39, pp. 836–842, 
1992. 
[4] P. M. Osterberg and S. D. Senturia, “M-TEST: a test 
chip for MEMS material property measurement using 
electrostatically actuated test structures,” J. 
Microelectromech. Syst., vol. 6, pp. 107–118, 1997. 
[5] H. A. C. Tilmans and R. Legtenberg, “Electrostatically 
driven vacuum encapsulated polysilicon resonators, 
part 2, theory and performance,” Sens. Actuators, vol. 
A45, pp. 67–84, 1994. 
[6] L. A. Rocha, L. Mol, E. Cretu, R. F. Wolffenbuttel, J. 
Machado da Silva and J.S. Matos, “A Test Mechanism 
for Device Diagnostics and Process Characterization”, 
in Proceedings DCIS‘07, pp. 597-602, 2007 
Figure 5. Response to an applied voltage in a)left 
electrode and b) right electrode. 
Table 3. Comparative values between measured and 
estimated pull-in voltages for different external 
accelerations 
[7] W. C. Tang, T.-C. H. Nguyen, and R. T. Howe, 
“Laterally driven polysilicon microstructures,” Sens. 
Actuators, vol. A20, pp. 25–32, 1990. 
[8] J.V. Clark, D. Garmire, M. Last, J. Demmel and S. 
Govindjee, “Practical Techniques for Measuring 
MEMS Properties”, Proceedings Nanotech 2004, vol. 
1, pp. 402-405, 2004. 
 Measured 
values 
Computed 
values 
Pull-In voltage left  3.942 V 3.943 V 
0g 
Pull-In voltage right  3.788 V 3.786 V 
Pull-In voltage left  4.019 V 4.018 V 
+1g 
Pull-In voltage right  3.712 V 3.712 V 
Pull-In voltage left  3.867 V 3.868 V 
-1g 
Pull-In voltage right  3.862 V 3.861 V 
[9] http://www.bosch-sensortec.com. 


English

Auto-calibrated capacitive MEMS accelerometer

An electronic calibration technique for
capacitive MEMS accelerometers based on the
measurement of pull-in voltages is described. A
combination of pull-in voltages and resonance frequency
measurements can be used for the estimation of processinduced
variations in device dimensions from layout and
deviations in material properties from nominal value,
which enables auto-calibration. Measurements on
fabricated devices confirm the validity of the proposed
technique and electronic calibration is experimentally
demonstrated

Repositório Aberto da Universidade do Porto

 AUTO-CALIBRATED CAPACITIVE MEMS ACCELEROMETER  L.A. Rocha1, L. Mol2, E. Cretu3, R.F. Wolffenbuttel2 and J. Machado da Silva1 1Dept. of Electrical and Computer Eng., Fac. of Eng. of the U. of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal 2Dept. Microelectronics, Fac. EEMCS, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands  3U. of British Columbia, Dept. Electrical and Computer Eng., 2332 Main Mall, Vancouver, BC, V6T 1Z4 Canada  Abstract ⎯  An electronic calibration technique for capacitive MEMS accelerometers based on the measurement of pull-in voltages is described. A combination of pull-in voltages and resonance frequency measurements can be used for the estimation of process-induced variations in device dimensions from layout and deviations in material properties from nominal value, which enables auto-calibration. Measurements on fabricated devices confirm the validity of the proposed technique and electronic calibration is experimentally demonstrated. Key Words: MEMS accelerometers, pull-in, calibration I. INTRODUCTION Accelerometers and pressure sensors are products based on microelectromechanical systems (MEMS) technology with a high commercial success and impact [1]. Several accelerometer designs are commercially available, from simple single axis to more complex three axes devices. Despite the evolution of acceleration microsensors in the last 30 years, MEMS based accelerometers need to be calibrated for use in high performance applications. Accelerometer calibration is usually done by measuring the response of the sensor to a sequence of +1g followed by -1g static acceleration test signals; this is accomplished by aligning the sensitive axis parallel with the gravitational field, and switching between the two possible orientations. This calibration step is often done manually, and is therefore an inconvenient alternative for remotely placed sensors or for sensors difficult to access. Auto-calibration capabilities would be preferred instead, since they allow the inertial sensors to be integrated in autonomous self-calibrated systems. This paper introduces an electronic calibration technique for capacitive MEMS accelerometers based on the measurement of pull-in voltages. Pull-in [2] is a unique feature of gap-varying capacitive MEMS devices, and can provide detailed information about their characteristics. Since the attractive force due to an electrostatic field is inversely proportional to the square of the deflection, while the restoring elastic force is (to a first approximation) linear with deflection, an unstable system results in case of a deflection, v, beyond a critical value, vcrit. The pull-in voltage, Vpi, is defined as the voltage that is required to obtain this critical deflection and depends mainly on geometry, residual stress level and material properties. This dependence makes it ideal for characterizing structural materials in surface micromachining processes [3,4]. Unlike the case of the comb drive, which is based on area-varying capacitors, the design of most electrostatic actuators relies on gap-width varying capacitors and the pull-in phenomenon has to be considered [5]. Pull-in causes the displacement range due to electrostatic force to be limited to 1/3 of the gap between the electrodes, in case of a motion of the movable capacitor plate perpendicular to the plates. When pull-in voltage measurements are combined with the measurement of the resonance frequency (a single measurement is needed), fabrication process non-idealities like over-etching and process asymmetries can be accurately estimated [6]. The parameters of an accurate device model could be extracted from this measurement setup, providing in fact an equivalent calibration capability as the application of an external inertial force of ±1g, but using only electronic excitation. II. PULL-IN VOLTAGE AND NON-IDEALITIES ESTIMATION The simplest symmetric micromechanical system suitable for studying the pull-in voltage is composed of three electrodes. Two of them have a fixed position on the rigid supporting substrate, while the middle one is movable and connected to an elastic suspension with the spring constant k (Fig. 1a). This is often the case of capacitive accelerometers with separate electrodes for sensing and actuation. The static balance between the elastic and electrostatic forces defines the equilibrium position of the movable plate for a given actuation voltage. Its stability is given by the rate of variations of the two forces for small perturbations around the equilibrium point, that is, by the second derivative of the global potential energy. For a stable equilibrium, the second derivative of the potential energy of the system with respect to deflection should be positive: 2 2 0pU x∂ ∂ > ; thus the pull-in voltage (Vpi) results as the solution of the two equations corresponding to 0pU x∂ ∂ =  (force equilibrium) and 2 2 0pU x =∂ ∂  (margin of stability). Pull-in voltage is determined by the elastic flexure (beam) material and dimensions, residual stress, and the geometry of the electrodes.  The beam should be suspended using folded tethers at each end, in order for the residual stress to not affect Vpi[7]. This approach ensures that the built-in strain energy component caused by longitudinal stress is negligible. The geometry of the structure allows the definition of two pull-in voltages, as shown in Fig. 1a: asymmetric-right (Vpr), and asymmetric-left (Vpl). They will be equal in ideal conditions (perfectly symmetric structure), as can be shown by the theoretical analysis [2]:  300827pr pld kV Vwlε= = , (1) Here d0 is the capacitor initial gap, k is the mechanical spring constant, ε0=8.8546x10-12 is the air permittivity and w and l are the capacitor plate width and length, respectively.   Figure 1. Sketch of the basic device with a) ideal conditions and b) with over-etch and asymmetries  If non-ideal process conditions are now considered (Fig. 1b) like over-etching [8], capacitor gap mismatch and Young’s Modulus (E) value deviations, the pull-in voltage values will vary. They become suitable, easy to perform measurements (with simple electronics) for estimating technological and other non-idealities and for use as diagnostic mechanism.  The method proposed in [6] uses the pull-in voltage as a test mechanism for microelectromechanical systems, for instance to identify process-induced variations in the real device geometry. Fabricated devices often exhibit actual dimensions smaller than the designed on the mask layout (due to over-etching). Over-etching can be considered uniform at the scale of one microfabricated device [8], which means that all layout dimensions will be affected by the same parameter α. This will have a uniform effect on both pull-in voltages. Besides over-etching effects, small gap mismatches (a few nm in misalignment) are also observed in fabricated devices. As the gap mismatch (β) will affect differently the pull-in voltages, it becomes easy to estimate β from the differences between Vpl and Vpr. The parameter α is more difficult to estimate, because there is an extra unknown parameter: the Young’s Modulus (its average value is known, but it can show large deviations). If we introduce a new measurement, the resonance frequency, both α and E can now be estimated and the uncertain parameters of the mechanical device are identified. Therefore, as the sensitivity of the device is known, test signals can now be applied to the actuation capacitors to calibrate the full system (device plus readout electronics). One disadvantage of the proposed technique is that it relies on using accurate device models to predict the device electro-mechanical behavior. These models have to incorporate all the non-idealities existing in the micro-domain, like capacitor fringe fields and residual stress, which makes the modeling one of the critical parts of the proposed calibration method. III. EXPERIMENTAL RESULTS Accelerometers fabricated using Bosch epi-poly technological process[9] were used to evaluate the auto-calibration method. A simplified drawing of the devices used is depicted in Fig. 2.  Figure 2. Drawing of the accelerometer III.1 FABRICATED ACCELEROMETERS The fabricated accelerometers (Fig. 3.) are composed of four folded springs, 340 μm long and 3 μm wide (layout dimensions), connected to two rigid central bars of about 1mm long. Parallel-plate capacitors with a 2 μm gap are used for actuation. The displacement measurement involves sensing the changes of various sets of differential capacitors. The main device layout parameters and bulk material properties are shown in Table 1. Table 1. Main nominal parameters of the device (layout dimensions and bulk material mean values)  Parameter Value Spring length (l) 340 μm  Spring width (b) 3 μm Mechanical layer thickness (h) 10.6 μm Capacitor length (l) 282 μm Capacitor width (w) 10.6 μm Capacitor gap (d) 2 μm Young’s Modulus (E) 163 GPa (Poly-Si) Density (ρ) 2.5 g cm-3   Figure 3. Fabricated device III.2 ACCELEROMETER ESTIMATED MODEL A MEMS capacitive accelerometer is a second order mechanical system and can be described mathematically in Laplace complex domain by:  2( ) 1( )x sb ka s s sm m=+ +  (2) where x is the displacement of the proof mass, a is the external acceleration, b is the damping coefficient, k is the mechanical spring constant and m is the mass of the proof mass. The mechanical system has a resonance frequency given by n k mω = , and the (DC) sensitivity S m k= .  For a typical calibration procedure the mechanical spring constant and the mass of the accelerometer are the parameters of interest, while the dynamic behavior is less important (calibration is mainly performed in static mode, such that it needs only simple electronic circuits). Using the method described in [6], pull-in and resonance frequency measurements were performed on a fabricated accelerometer, and the estimated model parameters are shown in Table 2.  As the dynamic behavior can be neglected in static equilibrium, the balance of forces on the accelerometer can be written as:    (3) ( )elect extk x F V m a⋅ = + ⋅where Felect(V) is the electrostatic force applied to the movable structure for a given voltage. Equation 3 shows that there are two alternative ways to achieve a certain displacement x: by controlling the applied voltage, while maintaining a zero external acceleration or by applying an external acceleration with no voltage applied.  According to the estimated structure parameters, the fabricated accelerometer has a sensitivity of (Similarly, if one uses the estimated accelerometer model, the voltage necessary to achieve a displacement of ±33nm can be computed. Considering that a positive displacement occurs when a voltage is applied to the left electrode and a negative one, for a right electrode actuation, the following voltages levels are found:    (4) 1-1g1.14V1.11VgVV+ ==Table 2. Measurements and estimated accelerometer parameters using a pull-in based test method [6] Measurements Value Pull-In voltage left (Vpl) 3.942 V Pull-In voltage right (Vpr) 3.788 V Resonance frequency 2740 Hz Estimated technological parameters Value Over-etching (α) 255 nm Mismatch (β) 34.5 nm Young’s Modulus (E) 147.8 GPa Estimated accelerometer parameters Value Capacitor gap rigth (dright=d+2α-β) 2475.5 nm Capacitor gap left (dleftt=d+2α+β) 2544.5 nm Mass (m) 3.978 μg Mechanical spring (k) 1.18 N.m III.3 EXPERIMENTAL RESULTS Two different experiments were made in order to experimentally check the proposed calibration method. The response of the accelerometer to a ±1g external acceleration was firstly recorded. The measured output voltage is shown in Fig. 4.  Figure 4. Response to a ±1g external acceleration Subsequently, the accelerometer response to an increasing voltage on the electrodes, in the absence of an external acceleration, was verified. The measured results are presented in Fig.5. III.4 DISCUSSION The results presented in Figures 4 and 5 are very promising and clearly indicate that electric calibration is feasible in capacitive accelerometers. In fact, the results show that the output voltage to +1g external acceleration is equivalent to the response when a voltage of 1.14V is applied to the right electrode (61mV output voltage), while the response to an )93.3702 10S Kg−= × ⋅N m . An acceleration of ±1g will generate therefore a displacement of ±33nm.  actuation voltage of 1.11V on the left electrode corresponds to output voltage for an applied external acceleration of -1g.  The very good agreement between measured and computed values using the estimated model validates again the calibration approach.  These results also prove that the estimated model offers a good description of the real behavior of the accelerometer. To further verify model validity, extra measurements were performed: left and right pull-in voltages were measured in the presence of a ±1g external acceleration and compared with the pull-in voltages given by the models for this situation. Since a ±1g generate a displacement of ±33nm, this can be modeled as a new β (mismatch). Therefore, a +1g gives a β of 67.5nm and a -1g gives a β of 1.5nm. The experimental and computed results are shown in Table 3. IV. CONCLUSIONS AND FUTURE WORK A pull-in based solution for MEMS capacitive accelerometer auto-calibration was presented. The technique was validated through experimental measurements and their comparison with the theoretical models.    Experimental results showed that the response to a ±1g external acceleration can be replaced by an equivalent asymmetric actuation voltage, approach that allows a pure electronic calibration of the sensor. The technique can be used in capacitive inertial sensors (e.g. accelerometers and angular rate sensors) and allows their integration in autonomous self-calibrated systems. a)  REFERENCES [1] S.D. Senturia, “Perspective on MEMS Past and Future: The Tortuous Pathway from Bright Ideas to Real Products, Proceedings Tranducers’03, pp.10-15, 2003 [2] L.A. Rocha, E. Cretu and R.F. Wolffenbuttel, “Analysis and Analytical Modeling of Static Pull-In with Application to MEMS-Based Voltage Reference and Process Monitoring”, J. Microelectromech. Syst., vol. 13,pp. 342-354, 2004. b)  [3] S. T. Cho, K. Najafi, and K. D. Wise, “Internal stress compensation and scaling in ultra-sensitive silicon pressure sensors,” IEE Tr. ED, vol. 39, pp. 836–842, 1992. [4] P. M. Osterberg and S. D. Senturia, “M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures,” J. Microelectromech. Syst., vol. 6, pp. 107–118, 1997. [5] H. A. C. Tilmans and R. Legtenberg, “Electrostatically driven vacuum encapsulated polysilicon resonators, part 2, theory and performance,” Sens. Actuators, vol. A45, pp. 67–84, 1994. [6] L. A. Rocha, L. Mol, E. Cretu, R. F. Wolffenbuttel, J. Machado da Silva and J.S. Matos, “A Test Mechanism for Device Diagnostics and Process Characterization”, in Proceedings DCIS‘07, pp. 597-602, 2007 Figure 5. Response to an applied voltage in a)left electrode and b) right electrode. Table 3. Comparative values between measured and estimated pull-in voltages for different external accelerations [7] W. C. Tang, T.-C. H. Nguyen, and R. T. Howe, “Laterally driven polysilicon microstructures,” Sens. Actuators, vol. A20, pp. 25–32, 1990. [8] J.V. Clark, D. Garmire, M. Last, J. Demmel and S. Govindjee, “Practical Techniques for Measuring MEMS Properties”, Proceedings Nanotech 2004, vol. 1, pp. 402-405, 2004.  Measured values Computed values Pull-In voltage left  3.942 V 3.943 V 0g Pull-In voltage right  3.788 V 3.786 V Pull-In voltage left  4.019 V 4.018 V +1g Pull-In voltage right  3.712 V 3.712 V Pull-In voltage left  3.867 V 3.868 V -1g Pull-In voltage right  3.862 V 3.861 V [9] http://www.bosch-sensortec.com. 

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Auto-calibrated capacitive MEMS accelerometer

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