Page 328 - System on Package_ Miniaturization of the Entire System

302 Cha pte r F i v e

the vertical distance between the beam and the metal pad. This distance is typically 1 to 3 μm.
As this distance is increased, the switching time and actuation voltage also increase. Decreasing
this distance will likewise decrease the switching time and actuation voltage. The speed and
voltage can also be improved by changing the beam material. Very pliable metals, like
aluminum, will switch much easier than stiffer metals, like gold. The stiffness of a material is
denoted by its Young’s modulus, (the higher this value, the stiffer the material).
Equations for predicting the bending of cantilever and double-supported beams
have been around for decades. Unfortunately, trying to apply simplistic equations to
complex MEMS devices can be cumbersome. One of the most important mechanical
parameters of a MEMS switch is the pull-down voltage. This quantity can be estimated
by treating the MEMS switch as a mechanical spring. In order to calculate the pull-
down voltage, one must equate the electrostatic force pulling down on the beam:
ε AV 2
f down = (5.1)
2 g 2

and the force pushing up from the spring (Hooke’s law):
f up =− k g −( o g) (5.2)

In these equations, e is the permittivity, A is the area, V is the voltage, k is the spring
constant, g is the initial gap, and g is the evaluated gap. We can use these simple,
o
spatially independent equations since we know the charge density (and therefore the
force) is uniform across the capacitive region. It is well known that for parallel-plate
electrostatic actuation, when the gap reduces to two-thirds of the original gap, the beam
becomes unstable and experiences a “pull-in” effect. That is, when the gap reaches a
certain threshold, namely two-thirds of the original gap, the switch will snap down.
Magnets experience the same effect. As two magnets of opposite polarity are brought
closer together, the attractive force is barely noticeable until they reach a certain distance
apart. At this point they snap together, and the force between them is great.
Equating Equations (5.1) and (5.2) where the gap (g) is two-thirds of the original gap
(g ) and solving for the pull-down voltage gives
o
8 kg 3
V = o (5.3)
PD 27ε A

In order to reduce the pull-down voltage, design engineers can reduce the spring
constant, reduce the switch gap, or increase the area of the switch.
Comparison of Technologies: Electromechanical versus Solid State
There are a variety of switching elements available on the market today. In order to
choose the right type of switch, one must consider the required performance specifications,
such as frequency, bandwidth, linearity, power handling, power consumption, switching
speed, signal level, and allowable losses. A comparison of a typical RF MEMS, PIN diode,
and FET switching element is summarized in Table 5.4 [67].
PIN diodes are useful switching elements because they are widely available
commercially, have fast switching speeds, are low cost, and are rugged. One of the main
limitations of PIN diodes is the insertion loss. Above a few gigahertz, PIN diodes can start
to have quite a bit of insertion loss. This becomes significant above the X band (8 to 10 GHz)