Online proceedings - EDA Publishing Association

IV.
A. Simulation and Sensitivity Analisys
11-13
May 2011, Aix-en-Provence, France
STRESS IDENTIFICATION
With respect to the identification phases a sensitivity
analysis is done for the membrane structure. Parameters
which have to be considered beside the interested ones are
parameters with relevant tolerance ranges. The membrane
thickness is such a parameter – due to technological reasons
the thickness varies within a range of ±5%.
(∂ f 1
/∂ h) ∆ h/f 1
[%]
6
5
4
3
2
1
f 1
[kHz]
60
50
40
30
20
10
0
0.42
Fig. 6: First modal frequency versus membrane thickness and stress
As is apparent from Fig. 6 which show the results of the
two dimensional parameter simulation for the first modal
frequency the most sensitive parameter is the stress. An
approximation of the functional dependency is done with
regard to a quantitative analysis. The default expansion is a
polynomial one. In this case rational functions are used for
the stress motivated by the plate theory [5] on the one hand
and the curve characteristic of Fig. 6 on the other hand The
frequency mode f i,j is given by
with the membrane thickness h and the stress s.
Based on partial derivatives of the approximated course
of the function the sensitivity of the modal frequencies
versus the parameters is determined. Fig. 7 shows the
sensitivity normed on the maximum thickness variation of
5%. In case of a tensile stressed membrane the varying
thickness can be neglected – a relevant sensitivity of the
modal frequencies versus the thickness is given only in case
of a stress-free or compressive stressed membrane.
0
0 2 4 6 8 10 12 14 16 18 20
s [MPa]
Fig. 7: Normed sensitivity of the first modal frequency versus membrane
thickness
B. Measurement Results
Measurements are done at three different wafers at a
pressure range between 0.005 mbar and 0.1 mbar. The
pressure range is determined by the resonance rice on one
hand and a minimal peak width to be detectable by the FFT
on the other hand. The measured quality factors show a
good accordance with the simulated ones given by the
harmonic analysis of the FE model.
0.41
20
0.4
15
10
0.39
5
z [µm] 0.38 0
s [MPa]
10 5 measurement data
simulated data
10 4
10 3
2/1
ji 1,
2
++=
3
),(),(),(
sjipsjipjip f
3/1
+
4
5
6
⋅++
),( shjiphjip sjip 10 2
(2)
10 -5 10 -4 10 -3 10 -2 10 -1 10 0
2/1
3/1
p [mbar]
7
8
),(
⋅+⋅+
shjipshjip
Q-factor
Fig. 8: Q-factor versus ambiance pressure
The first three modal frequencies are used for the
identification of the membrane stress. Mode shapes are
investigated at some samples to guarantee the right
classification of the frequency peaks to the corresponding
modes.
Fig. 9: Measured mode shape f 1,1
108