1. Piezoelectric Equations and Constants
To 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)

2. Piezoelectric Equations and Constants
Directional Dependence
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.

3. Piezoelectric Equations and Constants
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.

4. Piezoelectric Equations and Constants
d33  induced polarization 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 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) or induced 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

5. Piezoelectric Equations and Constants
Piezoelectric Voltage Constant (g)
The electric field generated by a piezoelectric material per unit of mechanical stress applied
alternatively
The 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.

6. Piezoelectric Equations and Constants
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)
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)
induced shear strain about direction 2 per unit electric displacement applied in direction 1

7. d g Definition of the Constants d and g Constant Definition S.I. Units
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
(D = constant)
applied dielectric displacement
Vm/N
m2/C

8. Piezoelectric Equations and Constants
There 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

9. Piezoelectric Equations and Constants
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

10. Modes of Vibration

11. Modes of Vibration

12. Modes of Vibration

13. 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)

14. Piezoelectric Parameters and Measurements
Elastic 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 

15. Piezoelectric Parameters and Measurements
The values of the piezoelectric properties 
Derived from resonance behavior
Suitably Shaped Specimens
The resonance behavior is represented by an equivalent circuit

16. Piezoelectric Parameters and Measurements
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

17. Piezoelectric Parameters and Measurements
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
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

18. Piezoelectric Parameters and Measurements
For a piezoceramic rod ( 6 mm in diameter and 15 mm long) 
Poled along its length and electroded both ends
For resonance condition
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

19. Piezoelectric Parameters and Measurements
The elastic compliance is related to the fp
= density
Superscript D = Open-circuit = constant electric displacement

20. Piezoelectric Parameters and Measurements
For a twin disc of diameter d 
Considering a radial mode resonance
J0, J1 are Bessel functions and  is Poisson’s radio
Curve is very insensitive for  of common piezoceramic 0.28 <  < 0.32

21. Piezoelectric Parameters and Measurements
If minimum impedance |Zm| at resonance is known
Dielectric Q factor = 1/tan 

22. Piezoelectric Parameters and Measurements

23. Piezoelectric Parameters and Measurements
IRE Standards Measurements on Piezoelectric Ceramics (IRE 1961)
fr= Resonant Frequency = frequency at minimum impedance
fa= Anti-resonant Frequency = frequency at maximum impedance

24. Ferroelectric Hysteresis
Sawyer-Tower Circuit
Hysteresis Loop
Hysteresis Loop of PZT

25. 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

26. Measurement Properties of Piezoelectric Ceramics
Test Specimens : Different shapes are required for different constants

27. Equipment for Simple Measurements
The measurements to be performed on the specimen
weight or density
physical dimensions
free capacitance and dissipation factor
frequencies of minimum impedance and maximum impedance
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

28. 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

29. Calculation of Constants
Calculation of Coupling k33
(Applicable for Length Poled Rod)
Calculation of Coupling k31
(Applicable for Long, Slim, Thickness Poled Specimen)

30. Calculation of Constants
Calculation of Coupling kp
(Applicable for Thin Discs)
Determine kp from curve (only for ceramic with Poisson’s Ratio ~ 0.3
BT and PZT have Poisson’s Ration ~ 0.3

31. Calculation of Constants
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)

32. Calculation of Constants
Calculation of Elastic Constant sD11
Calculation of Piezoelectric Constant d33
Calculation of Piezoelectric Constant d33

33. Calculation of Constants
Calculation of Piezoelectric Constant g33
Calculation of Piezoelectric Constant g11
Calculation of Mechanical Q