Page 216 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 203
3.2.2 Sensing

The same device, as has been mentioned previously, can be utilized to
perform motion sensing when the mobile plate is actuated externally. The
gap change between two plates will result in a capacitance change that relates
to a voltage variation of an external circuit comprising the capacitor. As Eq.
(4.17) suggests, the capacitance depends on the distance x, and therefore the
following equation can be written for the capacitance variation:

where the partial derivative of Eq. (4.27) is called sensitivity and is calculated
as:

By analyzing Eqs. (4.27) and (4.28), it is evident that a change in distance
translates in a change in capacitance, on one hand, and, on the other hand,
this relationship is not linear because the sensitivity of Eq. (4.28) is not
constant. The capacitance variation can be related to a voltage variation
because voltage is defined as charge over capacitance:

By assuming that the charge remains constant, one can find the voltage
variation by differentiating Eq. (4.29), namely:

and therefore the voltage change can be related to a capacitance change,
which corresponds to a gap variation, in the form:

Equation (4.31) indicates that the voltage variation, which can be monitored
in an external electric circuit, is inversely proportional to the distance change.
Another form of Eq. (4.31) can be obtained by using Eqs. (4.28) through
(4.30) as: