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Sensors and MeasurementsPenderia & Pengukuran

ENT 164Piezoelectric Sensors

Hema C.R.School of Mechatronics Engineering

Northern Malaysia University College of EngineeringPerlis , Malaysia

Contact no: 04 9798442Email: hema@kukum.edu.my

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General Structure of Measurement System

SENSING ELEMENT

SIGNAL CONDITIONING

ELEMENT

SIGNAL PROCESSING

ELEMENT

DATA PRESENTATION

ELEMENT

INPUT

TRUE VALUE

OUTPUT

MEASURED VALUE

Piezo-electric

Hall effect

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Sensing ElementsResistive

temperature & strain

Capacitive Pressure, level ,strain & humidity

Inductivestrain

Thermo Electrictemperature

Piezoelectricvibration , force & acceleration

Electro Chemicalgas composition & ionic concentration

Hall Effect SensorMagnetic field

silicon

pressure temperature

Flow

O2

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Piezoelectric Sensing

Elements

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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

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CrystalsCrystals 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

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Piezoelectric MaterialsQuartz (SiO2)

Barium Titanate (BaTiO3)

Gallium Orthophosphate (GaPO4),

Polymer materials like rubber, wool, wood and silk exhibit piezoelectricity to some extent

ApplicationsMicrophones, guitars, sonar, motors microbalances, clocks and vibration sensors.

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Piezoelectric Effect

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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

Fk

x1

19 Nm102

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Transfer Function of an Element

sf

sfsG

i

0

sfsGsf i0

Element Transfer Function:

Transfer function of an output signal is the product of element transfer function and transfer function of the input

signal

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

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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 k

Damper

Mass

m

F

x

x=0

kx

.

xk

Model of an Elastic force sensor an analogous system to a piezo force sensor

00

x

00

x

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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

)0()0( kxF

Fkxxxm

xmxkxF

F x

(i)

(ii)

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Fk

xdt

xd

kdt

xd

k

m

ei

Fxkxxm

FFxkkxxxm

xxxx

xxxFFF

1

.

)0()0(

,

)0(),0(

2

2

The differential equation now becomes

Which using equation (i) reduces to

(iii)

Second-order Linear Differential Equation

(iv)

Fkxxxm

14

nn

n

kkm

km

m

k

/2/,/1/

22

If we define

Undamped natural frequency rad /s

and

Damping ratio

then

(v)

Eqn.(iv) can be expressed in standard form

Fk

xdt

xd

dt

xd

nn

1212

2

2

Second-order Linear Differential Equation

(vi)

(xi )

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)0()0()()(

)0()()(

22

2

fsfsfstfdt

d

fssftfdt

dLaplace Transform of Time functions f(t)

)(1

)()]0()([2

)]0()0()([1 2

2

sFk

sxxsxsxxssxsnn

To find transfer function of the element we use Laplace transform of equation (vi)

(vii)

0)0(

0)0()0(

x

xxSince

and

Equation (vii) reduces to

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)(1

)(

)(

)(1

)(121 2

2

sGk

sF

sx

sFk

sxssnn

Thus

Where 1/k =steady-state sensitivity K

(viii)

1

21

1)(

22 ss

sG

nn

Transfer Function for a second–order element

(ix)

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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

121

1

2

s)s(

k)s(F

x

nn

(2)

nn f 2

nf

Further Reading : Page 56 - Bentley

121

1

22

ss

)s(G

nn

Transfer Function of a Piezoelectric Element

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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

dFFk

Kq

Direct Piezoelectric Effect

1 CNk

Kd

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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

1CN 1mV

dVx Inverse Piezoelectric Effect

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Further Reading : Page 160 - Bentley

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 PlatePiezoelectric crystal

t

Measuring ‘q’

t

ACN

0

0 Permittivity of free space (vacuum)

Relative permittivity or dielectric constant of the insulating material (here the piezo )

A Area of plate

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Further Reading : Page 82 - Bentley

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)

NC

NCNi

dt

dxK

dt

dqiN

Ni

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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.

Ks)s(x

iN

Ni

Further Reading: http://en.wikipedia.org/wiki/Laplace_transform#Formal_definition

Ni

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Piezoelectric Force

Measurement System

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Circuit of a 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

Piezoelectric Crystal

Recorder

Capacitive Cable

Ni NC CC LRLV

LR

CC

LV

Figure 1. Piezoelectric Force measurement system

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Transfer function relating to and is

(9)

Overall system transfer function relating recorder voltage to input force is

(10)

LV Ni

sCCR

R

)s(i

)s(V

CNL

L

N

L

1

LV F

F

x

x

i

i

Vs

F

V N

N

LL

Further Reading : Page 84 - Bentley

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From equation (2),(8) and (9) we get

where

121

1

1 22

ss

kKssCCR

Rs

F

V

nn

CNL

LL

1

211

1

1

22

sssCCR

sCCR

CCk

K

nn

CNL

CNL

CN

121

1

1 22 ss

s

s

CC

ds

F

V

nn

CN

L

Transfer Function for basic Piezoelectric force measurement system

(11)

CNL CCR

121

1

2

s)s(

k)s(F

x

nn

Ks)s(x

iN

sCCR

R

)s(i

)s(V

CNL

L

N

L

1

(Tau )

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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

CN CC/d

CC

121

1

1)(

22 ss

s

ssG

nn

MAXn

(Tau )

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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

1s/s

jG jG

Piezoelectric Crystal

Recorder

Capacitive Cable

Ni NC CC LRLV

Figure 1. Piezoelectric Force measurement system

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2

22

2

222

41

1

1

nn

jG

Amplitude Ratio

Phase difference arg

2

2

110

1

290

n

ntantanjG

Figure 2: Approximate Frequency Response Characteristics Piezoelectric Measurement System with charge amplifier

(13)

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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

1s

s

0jG 0

1

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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

dtiN

dtdqiN

iii F1 (14)

FC

OUTFF VVCq (15)

0 ii VV

0 VV dtdqi F

F

dt

dVCdt

dqii OUTF

FF 1 (16)

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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

NC CC

dt

dqii N 1 (17)

(16) (17)

(18)

FOUT

F

OUT

C

qVei

dt

dq

Cdt

dV ..

1

Transfer Characteristic Of Ideal Charge Amplifier

(18) (3) (2)

1

21

1)(

222

ssC

ds

F

V

n

F

OUT

Transfer Function for Piezoelectric system with Ideal Charge Amplifier

(19)

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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 materialsQuartz

Lead zirconium titanate (PZT)

Barium titanate (BaTi2O3)

PolyVinylidine DiFluoride (PVDT)

FCd

FC

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Piezoresistive Sensing

Elements

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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.

e

e1

v is Poisson’s Ratio

evG 121

Gauge factor of Strain gauge

(20)

Further Reading : Page 158 - Bentley

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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..

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Reference

1. ‘Principles of Measurement Systems’ by John P Bentley. [Text book]

2.http://www.resonancepub.com/piezoele.htm3.http://hyperphysics.phy-astr.gsu.edu/hbase/solids/

piezo.html 4. ‘Piezoelectric Transducers and Applications’ by

Antonio Arnau