Stresa, Italy, 25-27 April 2007
ARCHITECTURE FOR INTEGRATED MEMS RESONATORS QUALITY FACTOR
MEASUREMENT
Herv? Mathias, Fabien Parrain, Jean-Paul Gilles, Souhil Megherbi, Ming Zhang, Philippe Coste and
Antoine Dupret
Institut d?Electronique Fondamentale UMR 8622 - Universit? Paris-Sud 11 - B?t. 220
F. 91405 Orsay Cedex ? France
ABSTRACT
In this paper, an architecture designed for electrical
measurement of the quality factor of MEMS resonators is
proposed. An estimation of the measurement performance
is made using PSPICE simulations taking into account the
component's non-idealities. An error on the measured Q
value of only several percent is achievable, at a small
integration cost, for sufficiently high quality factor values
(Q > 100).
1. INTRODUCTION
Quality Factor (Q) is one of the most important
characteristics of MEMS resonators, especially for
vibrating structures where the resonant frequency
variation is monitored. For these applications, the higher
the Q value, the better the obtained frequency resolution.
Quality factor measurement is in these cases very
important during testing, since it defines the sensitivity of
the corresponding micro-system [1-3].
Within the PATENT DfMM European Network of
Excellence, we studied new BIST strategies that would
allow the measurement of the MEMS apparent Q factor
[4]. As shown on figure 1, the apparent Q factor includes
the contributions of the actuation and read-out systems
but also the effects due to the packaging or the
environment, which play an important role on the
effective mechanical quality factor. The aimed Quality
Factor Measurement (QFM) techniques could thus be
useful at different levels of the component's lifetime:
process control monitoring, end-of-line testing; vacuum
packaging monitoring, on-line self-test or even auto-
calibration, for the aforementioned micro-systems.
In these previous studies, the considered MEMS
device has been modeled either as a low-pass 2nd order
filter or a band-pass filter, depending on the principle of
the movement detection (cf. figure 1). Two promising
measurement principles have been identified: Transfer
Function Measurement and Step Response Analysis. It
has been shown that in terms of accuracy, the second
principle is better than the first, which is more suited to
the monitoring of Q factor variations [4].
This paper presents the best architecture, among those
studied, in terms of accuracy and cost to perform Step
Response Analysis. We first present the corresponding
measurement principle and the expected performance,
then the proposed architecture. Finally, we estimate the
effective performance taking into account the components
non-idealities.
Capacitance or piezo-resistive detection
0
2
00
()
1
1
A
Hf
ff
j
Q ff
=
??
+?
??
??
D.U.T.
0
0
2
00
()
1
1
f
jA
f
Hf
ff
j
Q ff
=
??
+?
??
??
1111
meas mech elec coupl
QQQQ
=++
Vin
Vout
Capacitive MEMS with current
detection
Figure 1: MEMS devices considered
2. MEASUREMENT PRINCIPLE
The Step Response Analysis method consists in
applying a step actuation on the MEMS structure (like
opening the loop in the case of an integrated MEMS-
based oscillator) and measuring the amplitude variation of
the damped oscillation response at the MEMS output. If
the step takes place at t=0s when the oscillator signal is
maximal (V
0
), the obtained damped signal is as follows:
0
- t
0 2Q
0
0
4Q?
4Q?
Q?
? 1
V(t)= V cos t 1-
e ?
11
sin t 1-
?
4-1
??
??
??
? ??
? ??
?
???
???
+ ???
?
???
(1)
The measurement's principle consists in counting, in
terms of number (n) of elapsed pseudo-periods T'
0
, the
time necessary for the response envelop to move from its
initial value V
0
down to a fixed voltage V
0
/k. It is shown
on figure 2.
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
H. Mathias, F. Parrain, J.-P. Gilles, S. Megherbi, M. Zhang, P. Coste and A. Dupret
ARCHITECTURE FOR INTEGRATED MEMS RESONATORS QUALITY FACTOR MEASUREMENT
0V
k
0V
mT
T
m
= n T
0
'
k fixed
0
0
1
T'
1
1
4?
f
Q
=
?
Figure 2: Fixed Voltage Interval measurement Principle
From equation (1) we find:
0
2
Tm
Qk
e
?
??
??
??
=
(2)
From which we obtain the measured value of Q, using the
pseudo-period equation given figure 2:
()
()()
meas
f
Q
?
=+
0
m
2
14 ?? n?
= 1
T
ln k 2
ln k
(3)
The decaying signal monitoring is performed using a
peak detector that measures the values of the signal's
successive maxima until the desired value is reached.
This method is preferable to the use of an envelope
detector. The output of the latter indeed presents ripples
that stop the measurement too early, resulting to a
decrease in accuracy. The measurement error also
depends on the MEMS resonant frequency, which is not
the case with the chosen method, at least for a given
range of frequencies.
Figure 3 shows the theoretical error obtained with respect
to the to-be measured Q value for different values of the
division factor k.
10 20 50 100 200 500 1000
0
5
10
15
20
25
30
35
Rel
a
ti
ve erro
r
o
n
Q in
%
Value of Q to be measured
k=2
k=12
k=6
Rel
a
ti
ve erro
r
o
n
Q in
%
Figure 3: Theoretical Measurement Error
The ripples observed on the curves on figure 3
correspond to the fact that the final V
0
/k value may not
correspond exactly to a maximum of the decaying signal.
The number of elapsed pseudo-periods is then an
approximation of the effective decaying duration between
V
0
and V
0
/k and the maximum corresponding error on n
is 1. For large k values, the number of elapsed pseudo-
periods is also large and this error becomes negligible, as
can be seen on figure 3. For high Q factors, the
theoretical obtained error can be less than 1%.
3. PROPOSED ARCHITECTURE
The proposed architecture to perform the integrated Fixed
Voltage Interval Measurement is shown on Figure 4. It
has been chosen to implement a discrete electronics
prototype. The stages used in the final integrated version
will probably be different.
Provided its amplitude remains large enough, the
decaying signal is used, via a comparator, to generate the
clock signal necessary to the control block. The clock
signal is also used directly to drive the switch SW4 that
resets the peak detector monitoring the decaying signal
(C2 capacitor). This way, no a priori knowledge of the
device's resonant frequency is necessary.
The control block is simple and cheap to implement
within an ASIC: it is mainly composed of a counter used
to count the desired numbers of pseudo periods. The latter
is roughly of the same order as the to-be measured Q
values. The control block is also used to drive switches 1
to 3, to launch the step actuation at the beginning of the
testing procedure and to compute the effective Q value
from the counter output.
The used peak detector requires two wideband opamps
allowing the cancellation of the D1 diode threshold while
ensuring that the X1 opamp doesn't enter saturation while
D1 is off. The components used in simulation have a
GBW of 45 MHz and a static voltage gain of 90 dB. The
diodes used are small-signal fast diodes. A practical
compromise between Gain Bandwidth and input bias
current has to be found in the choice of the discrete
opamps used: wideband opamps generally have a bipolar
input stage that requires a non negligible input bias
current that affects the stored voltage on the C2 capacitor.
In the ASIC version, the design of a specific wide band
OTA should solve this problem. A similar compromise
between speed and reverse current has to be found for the
diodes.
The voltage divider stage features one low noise, low
offset, high input impedance opamp driving a resistance
bridge. A comparator is then used to provide the count
enable signal to the counter. The counting is stopped as
soon as the last detected maximum becomes inferior to
the desired threshold.
Figure 5 shows PSPICE simulation results using
commercially available components models. The quality
factor value is 300 and the k factor is 6, for a resonant
frequency of 50 kHz.
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
H. Mathias, F. Parrain, J.-P. Gilles, S. Megherbi, M. Zhang, P. Coste and A. Dupret
ARCHITECTURE FOR INTEGRATED MEMS RESONATORS QUALITY FACTOR MEASUREMENT
Figure 4: Prototype architecture for step response measurement
Figure 5: Prototype architecture PSPICE simulation results
The measured number of pseudo-periods is 171,
corresponding to a measured Q value of 299,8 and thus
an error of 0.07%. In this simulation, however, the most
important non-idealities (opamps and comparator offsets
and noises, technology dispersion impact on k, leakage
currents on the capacitors) are not taken into account.
Their impact will be estimated and taken into account in
the next part.
4. PERFORMANCE ESTIMATION
In order to correctly choose the fixed parameters in the
architecture (mainly k) and to evaluate the effective
performance to be expected, a high level model
(Matlab/Simulink) of the architecture has been
developed in which the two main sources of
measurement error are taken into account: the X5
comparator offset and the error on k.
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
H. Mathias, F. Parrain, J.-P. Gilles, S. Megherbi, M. Zhang, P. Coste and A. Dupret
ARCHITECTURE FOR INTEGRATED MEMS RESONATORS QUALITY FACTOR MEASUREMENT
Figure 6: Worst case errors with combined non-idealities
Figure 7: Measurement error with respect to input signal frequency (Q=300)
We chose pessimistic values for these error sources: 1%
error on k due to technology dispersions (0.1% is
achieved in literature) and 10mV offset for the
comparator. Simulations have been performed over a
wide range of k and Q values. The worst-case errors are
obtained when these two sources induce a threshold
variation in the same direction. The corresponding
curves are shown Figure 6.
These curves show that, depending on the Q values to
be measured, there is an optimal value for k. A choice
between 4 and 8 seems a good compromise to be able to
accurately measure a wide range of quality factors. For
our prototype, we have chosen k=6 to minimize the
measurement error. Another value of k would be of
interest and will probably be preferred for the ASIC:
k=4.81. It is still in the range of k values with small
errors and presents the very interesting advantage that
the post-processing computation phase is reduced in this
case to a simple multiplication by 2 of the counter value.
It would thus require no additional hardware.
Further PSPICE simulations have been made taking into
account the maximum values of offsets for all the used
opamps and comparators and an error on the division
ratio of 1% with an input signal with the same quality
factor and amplitude as before. The signs of the
different offsets have been chosen so as to get the
worst-case effects. Different input signal frequencies
have been tested to get an insight of the useable
frequency range for this architecture. The corresponding
measurement errors found are shown on figure 7. They
are slightly higher than what was expected with the
Matlab/Simulink simulations. It is certainly due to the
fact that the offsets of the peak detector and voltage
divider opamps cause a further error on the effective k
value. A more thourough analysis shows that for low
frequencies (f
0
< 2 kHz), the main error source is the
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
H. Mathias, F. Parrain, J.-P. Gilles, S. Megherbi, M. Zhang, P. Coste and A. Dupret
ARCHITECTURE FOR INTEGRATED MEMS RESONATORS QUALITY FACTOR MEASUREMENT
leakage current at the capacitors. As the frequency
increases (2 kHz < f
0
< 50 kHz), this error compensates
the offsets impact and a very small error can be
obtained. Then the offsets impact dominates (50 kHz< f
0
< 1 MHz). For higher frequencies (1 MHz < f
0
), the
peak detector fails at canceling the diodes threshold and
the error increases greatly. The obtained error remains
nonetheless reasonable over a wide range of
frequencies: errors of the order of a few percents can be
achieved over 3 decades. The results should be
improved, both in terms of accuracy and frequency
range, with an adapted design of the integrated
architecture. Noise has not been taken into account. It
should have much less impact than the offsets, being
typically 2 orders of magnitude lower.
5. CONCLUSION
A very interesting architecture to accurately measure the
quality factor of MEMS resonators at low extra cost has
been proposed. Its performance taking into account the
non-idealities of the components has been estimated and
has been shown to be quite good: with pessimistic
configurations, the error level is limited to a few
percents. A discrete electronics prototype is under
development and will soon be used to perform Q factor
characterizations. Together with the analysis of error
sources performed in this paper, this prototype will give
us an important practical feedback in order to properly
design the integrated architecture, which will be the
next step of this study.
6. REFERENCES
[1] N. Deb, R.D. Blanton, ?Built-In Self-Test of CMOS-
MEMS Accelerometers?, International Test
Conference, IEEE, Baltimore, pp. 1075-1084, October
7-10, 2002.
[2] X. Xiong, Y. L. Wu and W. B. Jone, ?A Dual-Mode
Built-In Self-Test Technique for Capacitive MEMS
Devices?, VLSI Test Symposium, IEEE, Napa Valley,
April 25-29, 2004.
[3] V. Beroulle, Y. Bertrand, L. Latorre and P. Nouet,
?Evaluation of the oscillation-based test methodology
for micro-electro-mechanical systems?, VLSI Test
Symposium, IEEE, Monterey, pp 439-444, April 28-
May 2, 2002.
[4] Mathias H., Parrain P., Gilles J.-P., Megherbi S. and
Dupret A., ?Quality factor measurement and reliability
for MEMS resonnators?, DTIP of MEMS and MOEMS,
IEEE, Montreux, Switzerland, 01-03 June 2005.
Acknowledgements: this work has been funded by the
European PATENT ? Design for Micro and Nano
Manufacture Network of Excellence.
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3