We want to know the force attracting the electrodes when we apply a voltage. We will obtain this force via the differentiation of the expression of the energy stored in the system.

(1)

The total energy stored in the circuit is given by the energy stored in the capacitor minus the energy taken from the power source to charge it, that is:

(2)

We simplify the computation of the capacitance by considering only the lateral surfaces of neighbor fingers. We neglect the interaction with the part of the finger that is not engaged and with the horizontal surfaces (the fingers are considered infinitely thin). Moreover, we neglect the presence of the other fingers and the effects arising at the comb edge. Using this simplification, we may compute the capacitance of the comb by summing N times the capacitance of one finger. We describe a finger as two planar capacitors in parallel (one per side), and neglecting the fringing fields, we have:

(3)

and:

(4)

where the variable and constant are shown in Figure 2.

Hence the capacitance of the comb is:

C = N C

(6a)

if the two lateral gaps are equal, g

(6b)

(7)

If the two lateral gaps are equal and x = 0, the two lateral force are equal and the resultant net force along the X-axis is nearly zero. However, when x is different from zero, this lateral force may lead to instability when it increases more rapidly than the restoring lateral elastic force, bringing finger into contact. In a similar way, this effect may bend thin and long fingers causing short-circuit. This effect will be described in the next section.

To be complete, if there is only one side of the finger experiencing the electrostatic force, as in a gap-closing actuator, the expression simplifies slightly and yields:

(8)

(9)

where k

_{Example 1:}_{With an air gap of 2 µm width, electrodes height of 2 µm
and 10 µm of engaged electrodes we have:}

F_{Y} = N* 8.85·10^{-12} * 2·10^{-6}
/ 2·10^{-6} = N*8.85·10^{-12} N/V^{2}

thus it is about 0.008 µN by finger for an applied
voltage of 30 V.

For the first application of the movable mirror we need
a displacement of the order of l/2 (~0.5 µm).
For a suspension with a spring constant of 1 N/m and with an applied voltage
of 30V, we need at least 60 fingers. To obtain the displacement of several
µm needed for the second application, we may either increase the
number of finger or decrease the spring constant. With the same actuator
and a spring constant of 0.16 N/m, the actuation range becomes 3 µm.

For this voltage, the force generated by the actuator
will be 0.48 µN. For example this correspond to an acceleration of
100 g applied on a mass of 0.5·10^{-9}
kg. This value is a good estimate of the maximum range obtained when the
actuator is used as a force feedback in an accelerometer, provided
the actuator is not deflected in the reference position. In that latter
case the maximum available force will be decreased by the reaction of the
suspension at the 'rest' position. To account for random direction acceleration,
a two-way actuator should be used.

(8)

Taking the case g_{+} = g_{-} = g, and
e_{r}=1 (air) this relationship becomes:

_{ (9)}

Thus, the maximum voltage, Vmax, that may be applied to the actuator before instabilities occur is:

_{ (10)}

We see that this limit decreases with smaller gap and with larger electrodes total lateral surface (i.e., N h (o + y)).

_{Example 1:}

With an air gap of 2 µm width, electrodes height
of 2 µm and 10 µm of engaged electrodes we have a lateral force
on each surface, determined using (8):

F_{X} = -.5 * 8.85·10^{-12} *
2·10^{-6} *10·10^{-6} / 4·10^{-12}
= 2.21·10^{-11} N/V^{2}

thus it is about 0.02 µN for an applied voltage
of 30 V

For a full rotor finger, with the same gap on both side,
the same force apply on each lateral surface, therefore in the ideal case,
neither a bending moment nor a lateral force should appear as can be computed
from (7). However, this equilibrium is unstable as soon as the voltage
exceeds the critical voltage. To compute this voltage it is necessary to
observe that the lateral stiffness of the suspension is much lower when
the suspension is deflected. Thus for y=0, a typical folded beam suspension
with 300 µm-long beams has a spring constant of k_{X} = 2Ehw/l
= 4533 N/m . However when y>0, the spring constant becomes k_{X}
= 5.5 E h w^{3}/Lx^{2},
thus k_{X} = 503 N/m for a 10 µm deflection and k_{X}
= 125 N/m for a 20 µm deflection. The corresponding critical voltages
for a 60-fingers comb-drive are, respectively:

Vmax = 1306 V, 308 V, and 124 V.

It is apparent that this limitation becomes a real problem
for actuator with large range of displacement.

W. C. Tang, "Electrostatic comb-drive for resonant sensor and actuator applications", PhD thesis, University of California, Berkeley, 1990

R. Legtenberg, A. Groeneveld and M. Elwenspoek, "Comb-drive actuators for large displacements", J. Micromech. Microeng. Vol. 6, pp. 320-329, 1996