Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
The model is represented by the mechanical and the The transformation factor is primarily valid for the
electrical equations, concatenated by the coupling terms, static case. Nevertheless, is an appropriate approximation
and returns the time-dependent solutions for any physical in the time-dependent case, as long the cantilever bending
quantity of interest. Figure 1 shows the underlying model moment and shape is not significantly altered by the damping
of the electro-mechanical transducer. The RF switch discussed
action.
in this paper consists of a free-standing cantilever At the time , when the contact surfaces at the tip
made of gold. The bias line impedance comprises a touch each other, the velocity at the tip changes its sign, reproducing
series resistor and a series capacitor , whereas the
the bouncing effects, well-known from mechani-
variable capacitance is formed by the two electrodes,
cal relays. Attention has paid to these changing boundary
of the moveable cantilever, as one electrode, and the conditions while solving (1). The non-harmonic parametric
bias line electrode used to actuate the switch, separated by conditions (3) for the differential equation (1), describing
the gas and eventually a dielectric thin film. A voltage drop the movement of the tip, are used to reproduce bouncing, in
across the gap
causes the lever to bend case when a vibration of the cantilever itself after contact is
downward, and enables this way mechanical and electrical not relevant. This turns out to be true after the calculations,
contact at . During the switch off, a current is fed because the vibration frequency is much lower than the
back to the bias line.
bouncing frequency.
III. THE MECHANICAL SUBSYSTEM
For the modelling of the time-dependence of the MEMS
switch, a 1-dimensional model with concentrated quantities
is aimed. Therefore a representative point was defined.
In order to calculate the lumped coefficients ,
, and the lumped force , an analytical
approach was applied. References on this topic can be
found throughout various literature [8] [9] [10] [11]. From
the derived lumped parameters , , and
the force , it becomes possible to formulate the
differential equation of motion (1).
(1)
The solutions of (1) for returns the timedependent
displacement of the cantilever at the point . In
order to determine the displacement at the cantilever tip,
has to be scaled to . Hence, we define a transformation
factor (2), which links the displacement and
velocity along the -direction at to the displacement and
velocity at the tip of the cantilever and vice versa.
(2)
(3a)
(3b)
In case of a vibrating cantilever, bouncing is not expected,
and the cantilevers will resonate around the steadystate
equilibrium position.
IV. THE ELECTRICAL SUBSYSTEM
Besides its mechanical part, the RF switch consists also
of an electrical counterpart. The complete modelling of
both parts, including forward and backward coupling, allows
studying the influence of different actuation waveforms
in more detail. The bias line treated in this model
comprises the two serial components: a bias line resistor
and a capacitor . The voltage at the actuation electrode
is given by .
Whereas the current in the bias line is the sum of two
components (4), on one hand introduced by the reduction of
the gap
and thus, by the change of the capacitance
, and on the other hand by a change of the electrode
potential .
(4)
129