Page 296 - Mechanics of Microelectromechanical Systems

5. Static response of MEMS 283
stable operation point. Both aspects will be studied next. A similar analysis is
given by Sattler et al. [2] for the particular case where and
(width of the actuation plate is equal to half-width R of the mirror plate).
Figure 5.17 (b) shows the central plate in a rotated position. An
elementary electrostatic force acts on the mobile central (mirror) plate
over a length dr (not drawn in Fig. 5.17 (b)), namely:

where, under the small-displacement modeling assumption, the variable gap
g(r) depends on the initial gap and the rotation angle as:

This force produces an elementary active torque:

The total active (electrostatic) torque is the sum of all the elementary torques
of Eq. (5.48), and is determined by integration between the limits
and as:

Equation (5.49) gives the actuation moment that is produced under
conditions of voltage control. Another possibility is to control the
electrostatic model by means of charge, the so-called charge-control problem.
The elementary charge corresponding to the area determined by and dr is:

The total charge q can be found by integrating Eq. (5.50) between the limits
and The voltage U can be expressed in terms of
capacitance C and charge q as:

and therefore the charge-control actuation equation – the counterpart of the
voltage control Eq. (5.49) – is: