# Instrumentation & Measurement Magazine 24-5 - 79

```mechanical stresses, that generate a deviation of the behavior
of the clamped membrane from the expected one.
The above model (2) is valid under the hypothesis of a
linear relationship between the pressure and the generated
strain. The latter condition, with a maximum error of 0.3%,
is assured in cases where displacement in the middle of the
membrane, Y, is lower or equal to a quarter of the membrane's
thickness t. In such a case (Y ≤ t/4), the displacement at the center
of the membrane is given by [29]:
31
Y 
PR 
16tE
42
3
(3)
To fix the thickness of the membrane to satisfy the condition
of linearity and provide a suitable sensitivity, model (3) was
simulated in Mathworks Matlab®
. The theoretical displacement
of the central point of the membrane along the direction
normal to the membrane surface (the direction of the applied
pressure) for pressures in the range 0 bar to 10 bar and different
thickness of the membrane in the range 300 μm to 700 μm,
is shown in Fig. 3a.
Fig. 2. Section of the pressure sensor's body.
in the range of pressure to be measured, a preliminary study
was conducted. The analytical model linking the unknown
water pressure P to be measured and the strain ε generated at
the center of a circular membrane clamped rigidly around its
periphery is [29]:
31
ε 
PR 
8tE
22
2
 k
(2)
where R is the radius of the active part of the steel membrane
exposed to the water pressure, ν is the Poisson's
ratio, t is the membrane's thickness, E is Young's modulus
of elasticity, and the constant k is used to take into account
the strain contribution due to all the exogenous quantities,
such as the temperature, the humidity and the residual
The simulations show that for a given pressure the maximum
displacement decreases as the thickness of the membrane
increases. In other words, as expected, thicker membranes
show a lower sensitivity. Furthermore, membranes with a
thickness less than 450 μm do not meet the linearity condition
in the range of pressures of interest, since the displacements
are higher than a quarter of the membrane's thickness. It was
chosen to adopt a membrane with a thickness of 500 μm, that in
addition to the linearity condition in a larger range of pressure
(with respect to the specifics), shows strains compatible with
the FBG sensor and the slot of wavelengths assigned in the optical
acquisition system.
The strain in the center of a 500 μm thick stainless steel
membrane, estimated by model (2), is shown in Fig. 3b. The
strain, across the range of pressures of interest and over is
lower than the maximum strain used in Table 3 for the design
Fig. 3. (a) Displacement of the central point of the membrane for different values of the pressure in the range 0 bar to 10 bar and of the thickness in the range
300 μm to 700 μm; (b) The strain at the center of the membrane with a thickness of 500 μm.
August 2021
IEEE Instrumentation & Measurement Magazine
79
```

# Instrumentation & Measurement Magazine 24-5

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