Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
Residual stress in the suspended membrane induced by the
Here, q denotes the vector of modal amplitudes, φ
pad , n
the
fabrication process has been taken into account by calibrating
the fundamental eigenfrequency to the measured one.
averaged modal shape factor for the n-th contact pad,
The submodel for the electrostatic forces exerted by the
ground electrode is derived in two steps. First, the electrostatic
energy, which is stored between a single electrode finger and
the membrane, is determined in terms of the modal amplitudes.
Second, Lagrangian energy functionals are calculated for each
eigenmode and are included as electrostatic actuation term in
the respective eigenmode equation of the mechanical model.
In order to take into account the viscous damping forces the
mixed-level approach as presented in [4] is applied. It is based
on the Reynolds equation, which is evaluated by applying a
fluidic Kirchhoffian network distributed over the device
geometry. At perforations and outer boundaries lumped
physics-based fluidic resistances are added accounting for the
additional pressure drops at these locations (see fig. 5).
Consequently, this mixed-level model does not constitute a
pure lumped element model and – depending on the granularity
of the finite network – still exhibits a rather larger number of
degrees of freedom. However, the advantage of this approach is
that it can be tailored to the topography of the real structure, i.e.
take into account all perforations and – in the case of the
considered switch – also locally varying gap heights which
occur due to the elevated contact pads and electrode fingers.
ambient pressure
P 0
moving
plate
{
finite network
R O
R C
R T
fixed plate
moving
plate
{
finite network
Figure 5. Illustration of the mixed-level model. Models for holes are
embedded in the finite network solving the Reynolds equation. The resistor R T
models the region, where the fluid enters the channel. The resistor R C models
the channel resistance; R O models the orifice flow [5,6].
The mechanical contact that occurs, when the switch is
closed, is included into the mixed-level model by adding
contact forces at the respective locations above the contact
pads. The modal formulation of these forces reads as follows:
12
⎧
⎪
( )
∑ φpad , n
⋅kcontact , n
⋅gn ( q) if gn
( q)
≤0
Fcontact, total,
i
q = ⎨ (1)
n=
1
⎪⎩ 0 else
k
contact,
n
the lumped contact stiffness of the n-th pad and gn
( )
q the
locally averaged displacement at the n-th pad.
This model enables to simulate also bouncing during the
landing phase of the membrane. In order to avoid numerically
undesired discontinuities resulting from the if-then-else
construct proposed in previous work [7], we now use a
function Θ
n
based on the tanh-function instead, in order to
implement a more stable transition into the contact state:
⎛ ⎛ gn
Θ
n ( q)
= 0.5⋅⎜1−tanh
⎜
⎜ ⎜ β
⎝ ⎝
( q)
⎞⎞
⎟⎟
⎟⎟
⎠⎠
β denotes a parameter controlling the smoothness of this
transition. The complete contact formulation then reads:
12
( ) = Θ ( ) ⋅φ
⋅ ⋅ ( )
F q ∑ q k g q (3)
contact , total , i n pad , n contact , n n
n=
1
The compact model of the entire switch is then assembled
by formulating all submodels in terms of the modal amplitudes
and combining them with the mechanical submodel:
2 7
2 Vb
∂Ck( q)
T
i
+ ωi i
= ∑ + θi 2 k = 1 ∂qi
ext, i
0
F el
q
q F ( qq , , p)
Here, q i and ω i denote the amplitude and the frequency of
the i-th eigenmode, θ i
denotes the vector of the respective
discretized mode shape, Ck
( q ) stands for the capacitance of
the k-th electrode finger and V k for the respective applied
voltage. F ext represents the vector of external forces comprising
in this case the models for damping and contact forces.
Finally, the derived macromodels of the subsystems are
formulated in terms of conjugated variables (”across”- and
”through”-variables) and interlinked to form a generalized
Kirchhoffian network, which inherently governs the exchange
of energy and other physical quantities through Kirchhoff’s
laws and can be implemented easily in any standard system
simulator of an IC framework (in this work: Spectre from the
Cadence IC design suite).
IV. EXPERIMENTAL VALIDATION OF SIMULATED RESULTS
The macromodel was evaluated w.r.t. measurements
performed with a laser Doppler vibrometer (LV) and a white
light interferometer (WLI). A pressure chamber as depicted in
fig. 3 was used to enable measurements at different pressure
levels.
First, the combined electro-mechanical model was validated
against the quasi-statically measured pull-in/pull-out
characteristic of the membrane (see Fig. 6). It shows good
agreement with the pull-in characteristic, but yields an
incorrect pull-out voltage. Since the electromechanical model
works quite accurately for the pull-in curve, this discrepancy is
(2)
(4)
10