Page 121 - The Mechatronics Handbook

0066_Frame_C08 Page 12 Wednesday, January 9, 2002 3:48 PM

Deflection
2g 2g
2c 2c 2c
x
A A V V 2d
x > 0 Engaged
<Section A-A>
x < 0 Seperated kV
(a) (b)

FIGURE 8.6 Lateral comb transducers: (a) Dimensions; (b) two orthogonal Si combs.

deflection of the beam. The minimum voltage required to close the gap of a cantilever actuator is known
as the threshold voltage [Petersen 1978], and can be approximated as

( 3
V ≈ 18 IE) eff d 0 (8.45)
th
--------------------------
4
5e 0 L w
where (IE) eff is given by (8.24).
Comb drives also fall in two categories: symmetric and asymmetric. Symmetric comb drive is shown
in Fig. 8.6(a). In this configuration the gaps between the individual fingers are equal. Figure 8.6(b) shows
a pair of asymmetric comb capacitors, used in the force sensor shown in Fig. 8.2 [Enikov 2000a]. In any
case, the force generated between the fingers is equal to the derivative of the total electrostatic energy
with respect to the displacement

F el = n∂C 2 (8.46)
----------V
2 ∂x
where n is the number of fingers. Several authors have given approximate expressions for (8.46). One of
the most accurate calculations of the force between the pair of fingers shown in Fig. 8.6(a) is given by
[Johnson et al. 1995] using Schwartz transforms

 2
 e 0 V    c -- +  2 1 1 + 2g  1+c/g + pd c +  x > ∆ + engaged( )
 
g

 ----------- ln g   1  –   ----- ------ – ----------  ,
c 
p
x
g
el   
F =  (8.47)
 e 0 V  2 c + g) 
(
2
 – ----------- -------------------  , x < – ∆ − separated( )

 p  x 

In the transition region x ∈[−∆ − ; ∆ + ], ∆ +,− ≈ 2g, the force can be approximated with a tangential line
between the two branches described by (8.47).
Electromagnetic Transducers
Electromagnetic force has also been used extensively. It can be generated via planar coil as illustrated in
Fig. 8.7. The cantilever and often the coils are made of soft ferromagnetic material. Using an equivalent
magnetic circuit model, the magnetic force acting on the top cantilever can be estimated as
2n I 2A 2 + A 1 )
(
2 2
F mag = --------------------------------------------- (8.48)
m 0 A 1 A 2 2R 1 + R 2 ) 2
(
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