About Instrumentation

The principle of these pressure transducers is based on the well-known phenomenon, that when an asymmetrical crystal is elastically deformed along its specific axes, an electrical charge is developed on its sides. The value of this charge is proportional to the force applied to the crystal, and, therefore, to the pressure under measurement.

Fig. 12 shows piezoelectric crystal circuit. An electrical charge developed on the sides of the crystal is converted into a voltage-type signal using a capacitor. This voltage is proportional to the electrical charge developed, and to the pressure to be measured. Piezoelectric sensors cannot measure static pressures for more than a few seconds, but they have a very quick response when measure dynamic pressures.

Figure 12. Piezoelectric pressure sensor with electrical circuit.

Synthetically developed quartz crystals (barium titanate, lead zirconate) have similar properties as natural single crystal (quartz). But natural quartz still is the perfect material for manufacturing piezoelectric sensitive elements, because it has perfect elasticity and stability, it is insensitive to temperature variations and it has high insulation resistance.

These pressure transducers are used for measurements of hydraulic and pneumatic pulsations, flow instabilities, fuel injection, etc.

Let’s develop a differential equation for this sensor.

(60)

. (61)

The dynamic behaviour of a piezoelectric element is analysed from mechanical and electrical points of view.

(62), or using

(63)

we can get the following second order differential equation:

(64)

(65)

Finally, we can re-write (65) with variables in deviation form as follows:

(66)

Equation (66) is a second order linear differential equation for a piezoelectric sensor with variables in deviation form.

After applying the Laplace transform to (66) we get a transfer function for the piezoelectric sensor:

(67)

A_x - cross-sectional area of the piezoelectric sensor in the direction

perpendicular to axis ‘X’, m²;

b - charge sensitivity of the crystal to its deformation in the direction

perpendicular to axis ‘X’, C/m;

F_x - effective force due to pressure in the direction of axis ‘X’, N;

k - stiffness of the crystal is large ≈ 2 x 10⁹, N/m;

K_x - steady state gain of the crystal, in other words it is the charge sensitivity

of the crystal to applied pressure , C/Pa;

m - mass of the crystal, kg;

q_piez - electrical charge developed by the crystal, C;

P - pressure acting on the surface perpendicular to axis ‘X’, Pa;

t - time, s ;

x - deformation of the crystal in the direction of axis ’X’, m;

λ - constant (friction coefficient) for the crystal, N*s / m .

Solution to equation (66) will give us the variation of the electrical charge q_piez , developed on the surfaces of a piezoelement as a function of time for a step change in the input variable – measured pressure, P . Now we should measure this electrical charge. For this purpose two metal electrodes are attached to the opposite sides of a piezoelectric crystal. Thus, a capacitor is formed. The value of capacitance of this capacitor can be evaluated as follows:

(68)

where,

C_piez - electrical capacitance of the piezoelement, F (Farad);

ε₀= 8.85, pF / m - the permittivity of vacuum, 1pF=10^-12 F;

ε - the relative permittivity of the material of the piezoelectric crystal,

this is the dimensionless parameter;

A_x - cross-sectional area of the piezoelectric sensor in the direction,

perpendicular to the axis ‘X’, m²;

d - the thickness of the piezoelectric crystal in the direction,

perpendicular to the axis “X”, m.

The relative permittivity, also called dielectric constant, for various piezoelectric materials is given below :

· for quartz (natural piezoelectric material) ε = 4.5

· for tourmaline (natural piezoelectric material) ε = 6.6;

· for lead-zirconate-titanate (man-made piezoelectric ceramic material) ε = 1500;

· for lead metaniobate (man-made piezoelectric ceramic material) ε = 250.

It is also noted in the above mentioned reference, that natural piezoelectric materials have very low charge to force sensitivity, and therefore man-made piezoelectric ceramic materials are used as sensing elements:

· charge sensitivity to force for quartz 2.3, pC/N;

· charge sensitivity to force for tourmaline 1.9, pC/N or 2.4, pC/N;

· charge sensitivity to force for lead-zirconate-titanate 265, pC/N;

· charge sensitivity to force for lead metaniobate 80, pC/N.

We need to develop an electrical circuit which will allow us to convert variations of the capacitance of the piezoelectric sensor into the variation of an easy measurable electrical signal, voltage, for example. Such equivalent electrical circuit was developed, and is named after Norton (see Figure 13)

Figure 13. Norton equivalent electrical circuit for piezoelectric pressure/force

measurements. 1- piezoelectric element, 2– connecting cable, 3– recorder.

The piezoelectric element can be represented as a current source (or a charge generator) which is connected in parallel with a capacitance C_piez . Then, this element is connected to a voltage recorder via connecting cables, which have the capacitance C_cable. A recorder has a resistive load, R_load. The voltage measured across R_load is equal:

(69)

where,

- the impedance of three resistances connected in parallel,Ohm

According to the definition, the capacitance is equal to the ratio of the charge to the voltage across the capacitor plates, according to:

(70)

Let’s consider capacitance C_piez:

(71)

After differentiating both sides of (71) we can get:

(72) or

(73)

or, according to the Ohm’s Law,

(74)

Similar we can get

(75)

Substitution of (74) and (75) into (69) will give:

(76)

where, s = d/dt - the Laplace operator.

Expressing variables V_load and I_piez in deviation form and applying the Laplace transform to (76) we can get:

(77)

The transfer function for the Norton equivalent electrical circuit for piezoelectric pressure/force measurement system (see Figure 13) is as follows:

(78)

According to the definition:

, (79). Expressing these variables in deviation form and applying the Laplace transform to (79) we can get:

(80)

The transfer function relating current and charge of the piezoelectric sensor is as follows:

(81)

The transfer function relating the voltage V_loadand the measured pressure P can be determined as follows:

. (82)

After substitution of (67), (78) and (80) into (82) we can get an expression for an overall transfer function of the piezoelectric pressure/force measurement system: