Page 222 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 209
capacitance change when the relative angle between the fixed and the free
digits varies, according to the equation:

The gaps form an equivalent capacitor whose change in capacitance is the
sum of the individual capacitance changes, so that the total capacitance
variation is:

The total capacitance change can be transformed in voltage by proper
inclusion of the capacitors in an external electric circuit. The voltage
variation is expressed as:

3.4 Out-of-the-Plane Microcantilever-based Transduction

The electrostatic attraction can also be utilized in transduction
applications that are based on out-of-plane relative motion, such as the case
is with microcantilevers. Figure 4.30 illustrates this principle whereby a
microcantilever will bend towards an underlying pad of length either when
the two parts are charged externally with equal and opposite charges, or
when bending of the microcantilever is achieved externally, and the change
in gap between the two conjugate parts is monitored by a variation in
capacitance. In essence, the problem here is one resembling the transverse
principle of transduction, but the major difference, which is also
computationally paramount, consists in the gap not being constant along the
overlapping region. Moreover, determining the basic relationship between
the capacitance change and the gap change, which is fundamental to both
actuation and sensing, means solving an integral-differential equation and
this can only be done by means of numerical methods. This electrostatic
transduction principle will briefly be discussed in the following, together
with a numerical example illustrating the calculation procedure.
When applying external charges on the microcantilever and the pad that
are equal and opposite in sign, the compliant microcantilever will be
attracted by the fixed pad and will bend towards it. In doing so, the gap
between the two parts will vary along the overlapping length according to
the quasistatic equilibrium between actuation forces and elastic properties of
the microcantilever. Thus, the posed problem is not purely an actuation one,
as the elastic features of the microcantilever condition the entire situation,