piezoelectric equations and constants to a good approximation, the interaction between the...

Piezoelectric Equations and ConstantsTo a good approximation, the interaction between the electrical and

mechanical behavior of the piezoelectric medium can be described by the following relationships:

S = sET + dE : D = dT + TE

E = -gT + ( T)-1D : S = sDT + gD

E = Field (Vm-1) : T = Stress (Nm-2) : S = Strain (dimensionless) D = Dielectric Displacement (Cm-2)

The superscripted permittivity and compliance s denotes the quantity kept constant under boundary conditions (e.g T is the permittivity under constant stress)

"d" and "g" are piezoelectric constants:

d = r og

r = relative permittivity (or dielectric constant)

o = permittivity of free space ( 8.85x10-12 F/m)

Piezoelectric Equations and Constants

Because poled piezoelectric ceramics are anisotropic and the direction of polarizing may be freely chosen, a method of identifying the axes of a component is necessary in order to specify its parameters.

The direction of polarization is conventionally taken as the 3 axis, with axes 1 and 2 perpendicular to this. The terms 4, 5 and 6 refer to shear stains associated with the 1, 2 and 3 directions.

Directional Dependence


Piezoelectric Charge Constant (d)

The polarization generated per unit of mechanical stress applied to a piezoelectric material

alternatively The mechanical strain experienced by a piezoelectric material

per unit of electric field applied

The first subscript indicates the direction of polarization generated in the material when the electric field, E, is zero or, alternatively, is the direction of the applied

field strength.

The second subscript is the direction of the applied stress or the induced strain, respectively.

d is an important indicator of a material's suitability for strain-dependent

(actuator) applications.


d33 induced polarization in direction 3 (parallel to direction in which

ceramic element is polarized) per unit stress applied in direction 3or

induced strain in direction 3 per unit electric field applied in direction 3

d31 induced polarization in direction 3 (parallel to direction in which

ceramic element is polarized) per unit stress applied in direction 1 (perpendicular to direction in which ceramic element is polarized)

orinduced strain in direction 1 per unit electric field applied in direction 3

d15 induced polarization in direction 1 (perpendicular to direction in

which ceramic element is polarized) per unit shear stress applied about direction 2 (direction 2 perpendicular to direction in which ceramic element

is polarized)or

induced shear strain about direction 2 per unit electric field applied in direction 1


Piezoelectric Voltage Constant (g)

The electric field generated by a piezoelectric material per unit of mechanical stress applied

alternativelyThe mechanical strain experienced by a piezoelectric material per unit of electric displacement applied.

The first subscript to g indicates the direction of the electric field generated in the material, or the direction of the applied electric displacement.

The second subscript is the direction of the applied stress or the induced

strain, respectively.

g is important for assessing a material's suitability for sensing (sensor) applications.


g33 induced electric field in direction 3 (parallel to direction in which ceramic

element is polarized) per unit stress applied in direction 3

or

induced strain in direction 3 per unit electric displacement applied in direction 3

g31 induced electric field in direction 3 (parallel to direction in which ceramic

element is polarized) per unit stress applied in direction 1 (perpendicular to direction in which ceramic element is polarized)

or

induced strain in direction 1 per unit electric displacement applied in direction 3

g15 induced electric field in direction 1 (perpendicular to direction in which

ceramic element is polarized) per unit shear stress applied about direction 2 (direction 2 perpendicular to direction in which ceramic element is polarized)

or

induced shear strain about direction 2 per unit electric displacement applied in direction 1

Definition of the Constants d and g

Constant Definition S.I. Units

d

dielectric displacement developed

applied mechanical stress

(E = constant)

strain developed

applied field

(T = constant)

coulomb/meter2

Pa

meter/meter

volt/meter

C/N

m/V

g

field developed

applied mechanical stress

(D = constant)

strain developed

applied dielectric displacement

(T = constant)

volt/meter

Pa

meter/meter

coulomb/meter2

Vm/N

m2/C

Piezoelectric Equations and ConstantsThere are other parameters to be considered which characterize a

piezoelectric material; of prime importance are the coupling coefficient, loss factor, the mechanical quality factor, and the dielectric permittivity.

The Electromechanical Coupling Coefficient (k)

This parameter determines the efficiency of energy conversion in the component (but not the overall efficiency of the ceramic as a

transducer) and is defined as follows:

(i) For an electrically stressed component

k2 = stored mechanical energy total stored energy

(ii) For a mechanically stressed component

k2 = stored electrical energy total stored energy


The electromechanical coupling factor (k)

An indicator of the effectiveness with which a piezoelectric material converts electrical energy into mechanical energy, or converts mechanical energy

into electrical energy

The first subscript to k denotes the direction along which the electrodes are applied

The second denotes the direction along which the mechanical

energy is applied, or developed

Modes of Vibration

Piezoelectric Parameters and Measurements

The direct and converse effects d constant

D = dX + T E Direct Effect

S = sDX + dE Converse Effect

S = elastic compliance

Ferroelectric ceramics have non-linear properties

D = d X + T E

x = sE X + d E

These coefficients are not all independent

(D =ex and E = hx)

x

iiijij

x

iiijij

he

gd

/

/

Piezoelectric Parameters and MeasurementsElastic behavior can be expressed in terms of

sij = elastic compliance

cij = elastic stiffness

cij Sij

For poled ceramics sjk = skj and cjk = ckj

Only six terms are needed s11, s12, s13, s33, s44, s66 or c11, c12, c13, c33, c44, c66

Short circuit

Open circuit

The Poisson ratio E

E

E

E

ss

ss

33

1313

11

1212

;

D

ij

D

ij

E

ij

E

ij

cs

cs

:

:


The values of the piezoelectric properties

Derived from resonance behavior

Suitably Shaped Specimens

The resonance behavior is represented by an equivalent circuit


fr and fa : resonant and anti-resonant frequencies

when reactance (Xe) is zero

fs : frequency at which the series arm has zero reactance (X1 = 0)

fp : frequency when resistive component Re is maximum

fm and fn : frequencies for the minimum and maximum impedance Z


An important parameter for piezoelectric specimen

The effective electromechanical coupling coefficient keff

keff is related to c0, c1 and fp, fs, fa, fr, fm, and fn

2

22

2

22

2

22

10

12

n

mn

a

ra

p

sp

eff f

ff

f

ff

f

ff

cc

ck

Values for fn and fm are measured by a suitable bridge

(approximation is good if Q of the resonator > 100)

d and g coefficients can be determined from k

Piezoelectric Parameters and MeasurementsFor a piezoceramic rod ( 6 mm in diameter and 15 mm long)

Poled along its length and electroded both ends

For resonance condition

p

sp

p

s

f

ff

f

fk

2tan

233

Dielectric Permittivity can be determined from capacitance C at a frequency well below resonance

A = cross-sectional area of the rod

l = length of the rod

x

33

A

cx

33


The elastic compliance is related to the fp

Ds33

22

33

41

lfs pD

= density

Superscript D = Open-circuit = constant electric displacement

)1(

1

2

333333

33

33

33

21

33333333

2

33

33

33

k

dg

skd

k

ss

XX

X

EX

D

E

Piezoelectric Parameters and MeasurementsFor a twin disc of diameter d

Considering a radial mode resonance

s

sp

p

p

f

ffJJf

k

k,,

1 102

2

J0, J1 are Bessel functions and is Poisson’s radio

Curve is very insensitive for of common piezoceramic 0.28 < < 0.32


X

EX

sE

p

dg

Skd

fd

s

kk

33

3131

21

11333131

21

2222

11

2231

)(

)2~()1(1

2

1

If minimum impedance |Zm| at resonance is known

Dielectric Q factor = 1/tan

)(41

0 ccZfQ mm

Piezoelectric Parameters and MeasurementsIRE Standards Measurements on Piezoelectric Ceramics (IRE 1961)

fr= Resonant Frequency = frequency at minimum impedance

fa= Anti-resonant Frequency = frequency at maximum impedance

Ferroelectric Hysteresis

Sawyer-Tower Circuit Hysteresis Loop

Hysteresis Loop of PZT

Procedures for Measurement Properties of Piezoelectric Ceramics

Constants to be measured

Coupling Factors: k33 k31 kp

Free Relative Dielectric Constant :

Dissipation Factor: D

Elastic-Compliances:

Piezoelectric d and g constants: d33 g33 d31 g31

Mechanical Factor: Qm

Test Specimens

Different shapes are required for different constants

TK3

EDED SSSS 11113333

Measurement Properties of Piezoelectric CeramicsTest Specimens : Different shapes are required for different constants

Equipment for Simple Measurements

The measurements to be performed on the specimen

1. weight or density

2. physical dimensions

3. free capacitance and dissipation factor

4. frequencies of minimum impedance and maximum impedance

5. The magnitude of the minimum impedance

Equipment required to measure the data

Balance, Micrometer, Capacitance Bridge (capable of 10 pF-10,000 pF)

Oscillator (up to 200 kHz), Frequency counter,

Sensitive electronic voltmeter (200 KHz), variable resistor

Determination of Frequency and Impedance

fm= Meter Peak at the the frequency at minimum impedance

fn= Meter Null at the frequency at maximum impedance

Zm = The magnitude resistance at the frequency of minimum impedance

Calculation of Coupling k33

(Applicable for Length Poled Rod)

Calculation of Coupling k31

(Applicable for Long, Slim, Thickness Poled Specimen)

Calculation of Constants

Determine kp from curve (only for ceramic with Poisson’s Ratio ~ 0.3BT and PZT have Poisson’s Ration ~ 0.3

Calculation of Coupling kp

(Applicable for Thin Discs)


Calculation of Elastic Constant sD33

(Applicable for Length Poled Rod)

Calculation of Elastic Constant sE33

Calculation of Elastic Constant sE11

(Applicable for Long, Slim, Thickness Poled Specimens)


Calculation of Elastic Constant sD11

Calculation of Piezoelectric Constant d33

Calculation of Piezoelectric Constant d33


Calculation of Piezoelectric Constant g33

Calculation of Piezoelectric Constant g11

Calculation of Mechanical Q
