handbook of modern sensors

344 10 Pressure Sensors
For the membrane, the lowest natural frequency can be calculated from [4]
√
f 0 = 1.2 S
πr ρg , (10.7)
where ρ is the membrane material density. If the thickness of the membrane is not
negligibly small (r/g ratio is 100 or less), the membrane is called a thin plate (Fig.
10.3A). If the plate is compressed between some kind of clamping rings, it exhibits a
noticeable hysteresis due to friction between the thin plate and the clamping rings. A
much better arrangement is a one-piece structure where the plate and the supporting
components are fabricated of a single bulk of material.
For a plate, the maximum deflection is also linearly related to pressure:
z max = 3(1 − v2 )r 4 p
16Eg 3 , (10.8)
where E is Young’s modulus (N/m 2 ) and v is Poisson’s ratio. The maximum stress at
the circumference is also a linear function of pressure:
σ max ≈ 3r2 p
4g 2 . (10.9)
Equations (10.8) and (10.9) suggest that a pressure sensor can be designed by exploiting
the membrane and thin plate deflections. The next question is: What physical
effect should be used for the conversion of the deflection into an electrical signal
There are several options which we discuss in the following sections.
10.5 Piezoresistive Sensors
To make a pressure sensor, two essential components are required. They are the plate
(membrane) having known area A and a detector which responds to applied force
F [Eq. (10.1)]. Both of these components can be fabricated of silicon. A silicondiaphragm
pressure sensor consists of a thin silicon diaphragm as an elastic material
[5] and a piezoresistive gauge resistors made by diffusive impurities into the
diaphragm. Because of single-crystal silicon’s superior elastic characteristics, virtually
no creep and no hysteresis occur, even under strong static pressure. The gauge
factor of silicon is many times stronger than that of thin metal conductors [6]. It is
customary to fabricate strain gauge resistors connected as the Wheatstone bridge. The
full-scale output of such a circuit is on the order of several hundred millivolts; thus, a
signal conditioner is required for bringing the output to an acceptable format. Further,
silicon resistors exhibit quite strong temperature sensitivity; therefore, a conditioning
circuit should include temperature compensation.
When stress is applied to a semiconductor resistor, having initial resistance R,
piezoresistive effect results in change in the resistance R [7]:
R
R = π 1σ 1 + π t σ t , (10.10)