Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
damping and negligible spring effect in the frequency range of
Air gap spring coefficient
interest. So, a non-staggered configuration with 5 x 5 µm²
The air gap spring coefficient, k airgap , can be calculated
square holes and 10 µm pitch is used.
from (5):
Air gap damping coefficient
Fstiff
+ iner
k
airgap
= (11)
The damping coefficient, R airgap , can be determined from
ha
(4):
In a similar way, we compare FEM and lumped-model
Fdamp
results for various configurations, but there were important
R
airgap
= (10)
h
errors (more than 80%). Even if this model is not accurate
aω
enough to calculate the air gap spring coefficient, according to
First, we compare FEM and lumped-model results for
the simulation result for the microphone case, in the audio
various configurations. For simulations, we use CoventorWare
frequency range, the air gap spring coefficient,
which provides the damping force and coefficient for squeezefilm.
Next, we apply the lumped-model on the chosen
k airgap = 0.006 N/m (at 20 kHz), is very low comparing to the
diaphragm spring coefficient k mem (Table II). Therefore, the air
configuration. Considering possible device applications,
gap stiffness can be neglected. Indeed, according to the
simulations were performed up to 100 kHz.
obtained air gap spring value, we can suppose a condition of
Table III shows the air gap damping coefficient error
incompressible fluid. This can be also confirmed by the
between the FEM and the lumped-model results for different
squeeze number σ estimation, which characterizes the
diaphragm sizes and thus for different number of holes. We
compressibility effect for a perforated diaphragm:
have compared several diaphragm configurations. In this table,
2
12 μω r0
the air gap value similar to that of the designed microphone
σ = (12)
2
was fixed and we have used the non-staggered (matrix) hole
hP
aa
configuration with 10 µm pitch for each diaphragm.
TABLE III
MODEL/SIMULATION ERRORS
Diaphragm size Damping coefficient (kg/s)
Air gap (µm²) (number
Error (%)
Simulation Analytical
of holes)
100x100 (100) 2.1x10 -6 2.6x10 -6 22.2
200x200 (400) 9.4x10 -6 1.0x10 -6 10.7
300x300 (900) 2.1x10 -5 2.3x10 -5 7.2
2 µm 400x400 (1600) 3.9x10 -5 4.1x10 -5 5.6
500x500 (2500) 6.2x10 -5 6.5x10 -5 4.6
600x600 (3600) 9.0x10 -5 9.3x10 -5 3.9
700x700 (4900) 1.2x10 -4 1.2x10 -4 3.5
In the considered frequency range, the damping coefficient
R airgap is constant and the agreement between simulated and
calculated values varies from 22 % to 3.5 %.
We have found similar results when using the analytical
model of the squeeze-film for the microphone as when
performing the FEA with CoventorWare. Fig. 6 shows the
damping force simulated with CoventorWare and using (10).
We have obtained R airgap = 5.4x10 -5 kg/s, which is within 10 %
of the simulated value (6.1x10 -5 kg/s).
Where μ is the fluid viscosity, h a is the air gap thickness and ω
is the pulsation. If σ