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first one is due to the coupling of the EM field with the
bridge, which senses a force induced by the RF voltage, and
the second one is due to the electrical mismatch along the
line, and it is almost independent of the presence of the
bridge in a given location, for beams far at least 2 μm ca.
from the central conductor of the CPW. On the other hand,
for an almost perfectly matched line we can assume that the
last contribution is negligible.
The frequency of resonance for the bridge (or cantilever,
or any other mechanical structure) is given by the well
known equation:
ω
M
k
m
1 k
== π
M
;2
ff
M
=
(2)
2π
m
i.e. the angular frequency ω M is defined by means of the
spring constant k and the mass m, eventually modified in an
effective value m eff with respect to the nominal one because
of additional contributions (gas damping, holes, …) to be
included and considered in the structure.
Frequencies due to the longitudinal excitation modes have
vlong
to be also included. As well established flong
= ,
λlong
where λ long is the wavelength of the longitudinal mode, and
v long is the longitudinal velocity of the oscillating structure.
For a double clamped configuration, we have λ long = 2L
for the fundamental mode, where L is the full length of the
T
bridge, and v long = , where T is the tension and µ is
μ
the mass per unitary length. T is related to the constraints at
the ends. For this reason, strain and residual stress on the
beam due to the manufacturing process should play a
dominant role. For µ, after some algebra, we can get
μ eff = ρ eff LtA
, being A eff the effective area of the bridge
accounting for the presence of holes, t the thickness and ρ
the density. Because of the above considerations we can
write the following formula for f long :
11-13
May 2011, Aix-en-Provence, France
power [7],[9]. The force sensed by the beam will be the
result of the composition of voltage contributions coming
from all of the above effects.
By using the relation between the power and the energy
P = ωE , and considering that the power processed by the
MEMS has to be lower with respect to the threshold value
needed for the actuation of the switch, we can also write:
RF
8 k
2
0 in
=