Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
(5) chosen set of materials, the results are presented and exemplified
in the following section.
After substitution of into (5) and solving for the The actuation test signal applied to the system has a rising
second derivative of the voltage a differential equation is
edge of , a pulse width of and a trail-
gained, containing the backward coupling, attributed to ing edge of .
, and . represents
the stacked dielectric, e.g. alumina with thickness adjoining
A. Dynamics of Movement
the air gap.
Figure 2 shows the displacement curve of the cantilever
at the position , whereas exhibits
mechanical contact at
. The trigger delay
(6)
of the trailing edge is set to . The velocity depicted
in figure 3 shows two different resonating modes of
the system, one in the down-state position, damped by the
energy absorption by the contact and the squeeze film, and
the second in the up-state position, damped by the squeeze
V. ENERGY BALANCE
film.
The energy balance compares the total energy entering
the system boundary with the sum of the energy
components stored or dissipated in the system. Stored energy
comprises the kinetic and potential component of the
mechanical resonant structure , as well as the energy
stored in the bias line capacitor and the moving
variable capacitor . Dissipation terms are the squeezefilm
damping , the bias line resistor and the absorbed
energy by the contact .
(7)
can be calculated by integration of the absorbed
momentum (31). With the momentum reflection coefficient,
defined as
, and the momentum
absorption coefficient .
(8)
VI. SIMULATION RESULTS
The mathematical model, implemented in Simulink, describes
the switch dynamics by a 1-dimensional model. The
model consists of two coupled subsystems, one describing
the mechanical resonator, and the other representing the
electrical bias line. For a specific switch geometry and a
displacement z(l3,t) [um]
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0 10 20 30 40 50 60 70
time [us]
Figure 2: Displacement versus time of the cantilever contact tip.
velocity dz(l3,t)/dt [um/us]
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0 10 20 30 40 50 60 70
time [us]
Figure 3: Velocity of the cantilever contact tip, showing oscillation in the
down-state and the up-state, including damping caused by the squeezefilm
and the energy absorption by the contact.
130