Page 407 - The Mechatronics Handbook

0066_frame_C19 Page 29 Wednesday, January 9, 2002 5:17 PM

of oscillations over the range of variation of the capacitance determined by the applied acceleration. The
value of the capacitance C for each of the variable capacitors is given by

C = ekS (19.32)
--------
h

where k is the dielectric constant, ε is the permittivity of free space, S is the area of the electrode, and h
is the variable gap.
Denoting the magnitude of the gap for zero acceleration as h 0 , the value of h in the presence of
acceleration a may be written as

h = h 0 + ma (19.33)
-------
K
where m is the value of the proof mass and K is the spring constant. Thus,

ekS
C = ------------------------------ (19.34)
h 0 + ( ma/K)

For example, the frequency of oscillation of the resistance-capacitance type circuit is given by the expression

6
f = -------------- (19.35)
2pRC
Substituting this value of C in Eq. (19.34) gives

6 h 0 +[ ( ma/K)]
f = ------------------------------------------ (19.36)
2pRekS

Denote the constant quantity 6/(2pRekS) as B and rewrite Eq. (19.36) as

Bma
f = Bh 0 + ----------- (19.37)
K
The first term on the right-hand side expresses the fixed bias frequency f 0 and the second term denotes
the change in frequency resulting from acceleration, so that the expression may be written as

f = f 0 + f a (19.38)

If the output frequency is compared with an independent source of a constant frequency of f 0 , then f a
can be determined easily.
A commonly used capacitive-type accelerometer is based on a thin diaphragm with spiral flexures that
provide the spring, proof mass, and moving plate necessary for the differential capacitor. Plate motion
between the electrodes pumps air parallel to the plate surface and through holes in the plate to provide
squeeze film damping. Since air viscosity is less temperature sensitive than oil, the desired damping ratio
of 0.7 hardly changes more than 15%. A family of such instruments are easily available with full-scale
ranges from ±0.2g (4 Hz flat response) to ±1000g (3000 Hz), a cross-axis sensitivity less than 1%, and a
3
full-scale output of ±1.5 V. The size of a typical device is about 25 mm with a mass of 50 g.

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