Page 137 - MEMS Mechanical Sensors

126 Pressure Sensors

diaphragm pressure sensors typically requires the use of electromechanical transduc-
ers rather than mechanical linkages. Electromechanical effects can be used to meas-
ure displacement directly or to measure the stress/strain induced in the diaphragm
material. Therefore, it is also useful to provide an analysis of the stress distribution
across a pressurized diaphragm.
The stress distribution will vary both across the radius and through the thick-
ness of the diaphragm. For example, the neutral axis [shown in Figure 6.11(a)] expe-
riences zero stress while the maximum stress occurs at the outer faces. At any given
distance r from the center of the diaphragm, one face will experience tensile stress
while the other experiences compressive stress. There are two stress components
associated with a circular diaphragm: radial and tangential. The radial stress, σ ,at
r
distance r from the center of the diaphragm is given by (6.11). The maximum radial
stress that occurs at the diaphragm edge (r = a) is given by (6.12).

2
3 Pa  r 2 
σ =±  3 ) ν − 1 ) ν  (6.11)
( −
( +
r
2
8 h  a 2 
3 Pa 2
σ =± ( + ) ν (6.12)
1
4 h
r max 2
1/2
Radial stress is equal to zero at a value of r given by a((1 + ν )(3 + ν )) (shown in
Figure 6.12). This equals 0.628 if ν = 0.3.
The tangential stress, σ , at distance r from the center of the diaphragm is given
t
by (6.13). The maximum tangential stress that occurs at the diaphragm center (r =0)
is given by (6.14).
2
3 Pa  r 2 
( +
σ =±  3 ) − 1 ) ν  (6.13)
( ν+1
t 2 2
8 h  a 
3 Pa 2
1
σ =± ( + ) ν (6.14)
8 h
t max 2
The inflection circle for tangential stress is removed from that of radial stress
1/2
and is given by a((1 + ν )(3ν + 1)) . This equals 0.827 if ν = 0.3.
In the case of simply supported diaphragms [as shown in Figure 6.12(a)], for a
round diaphragm under a uniform pressure P, the deflection y at radial distance r is
given by (6.15). The maximum deflection occurs at the diaphragm center and,
assuming ν = 0.3, is given by (6.16).
3 1 2 ) ( 2 r 2 )5 +ν 
( −ν
Pa −
2
y =  a − r  (6.15)
2
16 Eh 3 1 +ν 
0695 Pa 4
.
y = (6.16)
0
Eh 3
The deflection of a simply supported diaphragm is shown in Figure 6.12(b). The
radial is given by (6.17). The maximum radial stress that occurs at the diaphragm
center (r = 0) is given by (6.18).