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)

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.

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.

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

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.

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

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

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

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

Modes of Vibration

Modes of Vibration

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)

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 

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

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

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

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

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

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

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

Piezoelectric Parameters and Measurements

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

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

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

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

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

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

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)

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

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