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).Comment: Submitted on behalf of EDA Publishing Association
  (http://irevues.inist.fr/EDA-Publishing

Coste, Ph.

Dupret, A.

Gilles, J. -P.

Mathias, H.

Megherbi, S.

Parrain, F.

Zhang, M.

English

arXiv

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)

Gilles, J.-P.

I-Revues

 Stresa, Italy, 25-27 April 2007 
 
ARCHITECTURE FOR INTEGRATED MEMS RESONATORS QUALITY FACTOR 
MEASUREMENT 
 
Hervé Mathias, Fabien Parrain, Jean-Paul Gilles, Souhil Megherbi, Ming Zhang, Philippe Coste and 
Antoine Dupret 
 
Institut d’Electronique Fondamentale UMR 8622 - Université Paris-Sud 11 - Bât. 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
0 0
( )
11
AH f
f fj
Q f f
= ⎛ ⎞+ − ⎜ ⎟⎝ ⎠
D.U.T.
0
0
2
0 0
( )
11
fjA
f
H f
f fj
Q f f
= ⎛ ⎞+ − ⎜ ⎟⎝ ⎠
1 1 1 1
meas mech elec couplQ Q Q Q
= + +
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 (V0), the obtained damped signal is as follows:  
 
0-  t
0 2Q 0
0
4Q²
4Q²Q²
ω 1V(t)= V  cos  t 1- e ω
1 1sin  t 1-ω
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 V0 down to a fixed voltage V0/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
Tm = n T0'
k fixed
0
0
1
T'
11
4 ²
f
Q
=
−
 
Figure 2: Fixed Voltage Interval measurement Principle 
From equation (1) we find:  
0
2
Tm
Qk e
ω⎛ ⎞⎜ ⎟⎝ ⎠=   (2) 
 
From which we obtain the measured value of Q, using the 
pseudo-period equation given figure 2: 
 
( ) ( )( )meas
f
Q
π= +0 m 21 4 π² n²= 1Tln 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
R
el
at
iv
e 
er
ro
ro
n 
Q
 in
 %
Value of Q to be measured
k=2
k=12
k=6
R
el
at
iv
e 
er
ro
ro
n 
Q
 in
 %
 
Figure 3: Theoretical Measurement Error 
 
The ripples observed on the curves on figure 3 
correspond to the fact that the final V0/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 
V0 and V0/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 (f0 < 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 < f0 < 50 kHz), this error compensates 
the offsets impact and a very small error can be 
obtained. Then the offsets impact dominates (50 kHz< f0 
< 1 MHz). For higher frequencies (1 MHz < f0), 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              


ARCHITECTURE FOR INTEGRATED MEMS RESONATORS QUALITY FACTOR MEASUREMENT

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Architecture for Integrated Mems Resonators Quality Factor Measurement

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