1. Sensors and Measurements Penderia & Pengukuran ENT 164 Piezoelectric Sensors
Hema C.R.
School of Mechatronics Engineering
Northern Malaysia University College of Engineering
Perlis , Malaysia
Contact no:

2. General Structure of Measurement System
ENT 164 Sensors & Measurements
General Structure of Measurement System
SIGNAL CONDITIONING
ELEMENT
SIGNAL PROCESSING
ELEMENT
SENSING
ELEMENT
INPUT
TRUE VALUE
DATA PRESENTATION
ELEMENT
Piezo-electric
Hall effect
OUTPUT
MEASURED VALUE

3. Sensing Elements Resistive Capacitive Inductive Thermo Electric
temperature & strain
Capacitive
Pressure, level ,strain & humidity
Inductive
strain
Thermo Electric
temperature
Piezoelectric
vibration , force & acceleration
Electro Chemical
gas composition & ionic concentration
Hall Effect Sensor
Magnetic field
silicon
pressure
temperature O2
Flow

4. Piezoelectric Sensing Elements

5. ENT 164 Sensors & Measurements
The word piezo is derived from the Greek piezein, which means to squeeze or press.
The effect known as piezoelectricity was discovered by brothers Pierre and Jacques Curie in 1880.
Crystals which acquire a charge when compressed, twisted or distorted are said to be piezoelectric.
Piezoelectric materials also show the opposite effect, called converse piezoelectricity, where the application of an electrical field creates mechanical deformation in the crystal.
Further Reading : Crystal classes & Piezoelectric crystal classes

6. Crystals
Crystals are naturally occurring material that can be induced to resonate or vibrate at an exact frequency.
Crystals are anisotropic materials
physical properties depend on the direction
Quartz, a piezoelectric crystal that provides excellent mechanical and electrical stability, acquires a charge when compressed, twisted, or distorted.
Quartz crystals are used as active elements in oscillators
A Quartz "Crystal"
Isotropic materials have same physical properties in all directions

7. Piezoelectric Materials
Quartz (SiO2)
Barium Titanate (BaTiO3)
Gallium Orthophosphate (GaPO4),
Polymer materials like rubber, wool, wood and silk exhibit piezoelectricity to some extent
Applications
Microphones, guitars, sonar, motors microbalances, clocks and vibration sensors.

8. Piezoelectric Effect

9. Displacement x is proportional to applied force F (1)
When force is applied to a crystal , the crystal atoms are displaced from their normal positions
Displacement x is proportional to applied force F
(1)
where k is the stiffness in the order of
The displacement can be summarised using a transfer function

10. Transfer Function of an Element
When input signal of an element is changed suddenly the output signal will not change instantaneously. The way in which an element responds to sudden input changes are termed its dynamic characteristics, which can be conveniently summarised using a transfer function
Element Transfer Function:
Transfer function of an output signal is the product of element transfer function and transfer function of the input signal

11. Transfer Function Of Second Order Elements
Sensor converts force into displacement , diagram shows the conceptual model which has a mass m kg, a spring of stiffness k N/m and a damper constant Ns/m.
The system is initially at rest at time t =0- so that the initial velocity and the initial acceleration
. The initial input force F(0-) is balanced by the spring force at the initial displacement x(0-)
Spring
Damper
Mass m F x
x=0 kx
Model of an Elastic force sensor an analogous system to a piezo force sensor

12. If input force is suddenly increased at t = 0, then element is no longer in a steady state and its dynamic behavior is described by Newton ‘s second law
resultant force = mass x acceleration
and
Defining and to be deviations in F and x
(i)
(ii)

13. (iii)
The differential equation now becomes
Which using equation (i) reduces to
(iv)
Second-order Linear Differential Equation

14. Undamped natural frequency rad /s
If we define
Undamped natural frequency
rad /s
and
Damping ratio
(v)
then
Eqn.(iv) can be expressed in standard form
(vi)
Second-order Linear Differential Equation
(xi )

15. Laplace Transform of Time functions f(t)
To find transfer function of the element we use Laplace transform of equation (vi)
(vii)
Since
and
Equation (vii) reduces to

16. (viii)
Thus
Where 1/k =steady-state sensitivity K
(ix)
Transfer Function for a second–order element

17. Transfer Function of a Piezoelectric Element
Using transfer function for a second order element
x and F can be represented by the second order transfer function
Where natural frequency is large
= 10 to 100 kHz and damping ratio
= 0.01
(2)
Further Reading : Page 56 - Bentley

18. From equation (1) and (3) we get
This deformation of crystal lattice results in crystal acquiring a charge q , proportional to x
q = Kx (3)
From equation (1) and (3) we get
(4)
where is the charge sensitivity to force
Direct Piezoelectric Effect

19. inverse effect is used in ultrasonic transmitters is identical with
A piezoelectric crystal gives a direct electrical output, proportional to applied force, so that a secondary displacement sensor is not required.
Piezoelectric crystals also produce an inverse effect where an voltage applied to the crystal causes a mechanical displacement.
(5)
inverse effect is used in ultrasonic transmitters
is identical with
Inverse Piezoelectric Effect

20. ENT 164 Sensors & Measurements
Measuring ‘q’
Metal electrodes are deposited on opposite faces of the crystal to form a capacitor to measure the charge q
Capacitance of the parallel plate capacitor formed
(6)
Metal Plate
Piezoelectric crystal t
Permittivity of free space (vacuum)
Relative permittivity or dielectric constant of the insulating material (here the piezo )
A Area of plate
Further Reading : Page Bentley

21. or a Norton equivalent circuit
The crystal can be represented as charge generator q in parallel with a capacitance
or a Norton equivalent circuit
consisting of current source in parallel with
Magnitude of is
(7)
Further Reading : Page 82 - Bentley

22. F and x are constant with time
transfer function form of
(8)
where d/dt is replaced by the Laplace operator s
For steady force F ,
F and x are constant with time
Such that dx/dt and are zero.
Further Reading:

23. Piezoelectric Force Measurement System

24. Circuit of a force measurement system
Piezoelectric Crystal
Recorder
Capacitive Cable
Figure 1. Piezoelectric Force measurement system
Consider a piezoelectric crystal connected to a recorder where
is a pure resistive load
is pure capacitance of the cable
is the recorder voltage

25. Transfer function relating to and is (9)
Overall system transfer function relating recorder voltage to input force is
(10)
Further Reading : Page 84 - Bentley

26. (11) From equation (2),(8) and (9) we get where
Transfer Function for basic Piezoelectric force measurement system
(11)
(Tau )

27. Disadvantages of the basic piezoelectric system
1.Steady state sensitivity is equal to Thus the system sensitivity depends on the cable capacitance i.e. length and type of cable.
2.The dynamic part of the system transfer function is (ignoring recorder dynamics)
(12)
The second term is characteristic of all elastic elements and cannot be avoided , however it causes no problem if the highest signal frequency is well below
(Tau )

28. The first term indicates that system cannot be used for measuring d. c
The first term indicates that system cannot be used for measuring d.c. and slow varying forces.
Illustration
Consider a frequency response
characteristics plot for and arg
of a typical measurement system
Piezoelectric Crystal
Recorder
Capacitive Cable
Figure 1. Piezoelectric Force measurement system

29. (13) Amplitude Ratio Phase difference arg
Figure 2: Approximate Frequency Response Characteristics Piezoelectric Measurement System with charge amplifier

30. The term causes a low frequency
roll-off so that at and system cannot be used for frequencies much below
These disadvantages can be overcome by introducing a charge amplifier into the system as shown in Figure 2

31. This system gives an output proportional to
i.e. an output proportional to charge q .
Since the system gives a non zero
output for steady force input. From Figure 3 we
Have
and charge on feedback capacitor is
For an ideal operational amplifier we have
and
In this case we have and
so that
(14)
(15)
(16)

32. Since the potential drop across and is zero
From equation and we have
From equation , and the overall transfer function for force measurement system is
(17)
(16)
(17)
(18)
Transfer Characteristic Of Ideal Charge Amplifier
(18)
(3)
(2)
(19)
Transfer Function for Piezoelectric system with Ideal Charge Amplifier

33. Common Piezoelectric materials
The steady state sensitivity is now i.e. it depends only on the capacitance of the charge amplifier and is independent of transducer and cable capacitance
Common Piezoelectric materials
Quartz
Lead zirconium titanate (PZT)
Barium titanate (BaTi2O3)
PolyVinylidine DiFluoride (PVDT)

34. Piezoresistive Sensing Elements

35. Piezoresistivity is defined as the change in resistivity of a material with applied mechanical strain and is represented by the term in the equation (20)
Silicon doped with small amounts of n type or p type materials exhibits a large piezoresistive effect and is used to manufacture strain gauges.
Gauge factor of Strain gauge
(20)
v is Poisson’s Ratio
Further Reading : Page Bentley

36. Poisson’s Ratio
When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. Poisson's ratio (v) is a measure of this tendency. It is defined as the ratio of the strain in the direction of the applied load to the strain normal to the load. For a perfectly incompressible material, the Poisson's ratio would be exactly 0.5. Most practical engineering materials have v between 0.0 and 0.5..

37. Reference
‘Principles of Measurement Systems’ by John P Bentley. [Text book]
‘Piezoelectric Transducers and Applications’ by Antonio Arnau