Piezoelectric Equations and Constants

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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)
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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.
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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.








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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 3orinduced 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)orinduced shear strain
about direction 2 per unit electric field applied
in direction 1








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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.








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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) 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








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Definition of the Constants d and g
Constant Definition S.I. Units 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
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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 total stored energy
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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
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Modes of Vibration








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Modes of Vibration








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Modes of Vibration








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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)

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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 ?
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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
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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
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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 gt
  100)
• d and g coefficients can be determined from k

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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
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Piezoelectric Parameters and Measurements
The elastic compliance is related to the
fp

• density
• Superscript D Open-circuit constant electric
  displacement

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Piezoelectric Parameters and Measurements
J0, J1 are Bessel functions and ? is Poissons
radio
Curve is very insensitive for ? of common
piezoceramic 0.28 lt ? lt 0.32
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Piezoelectric Parameters and Measurements
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Piezoelectric Parameters and Measurements
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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
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Ferroelectric Hysteresis
Sawyer-Tower Circuit
Hysteresis Loop
Hysteresis Loop of PZT
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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

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

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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
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Calculation of Constants
Calculation of Coupling k33 (Applicable for
Length Poled Rod)
Calculation of Coupling k31 (Applicable for
Long, Slim, Thickness Poled Specimen)
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Calculation of Constants
Calculation of Coupling kp (Applicable for Thin
Discs)
Determine kp from curve (only for ceramic with
Poissons Ratio 0.3 BT and PZT have Poissons
Ration 0.3
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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)
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Calculation of Constants
Calculation of Elastic Constant sD11
Calculation of Piezoelectric Constant d33
Calculation of Piezoelectric Constant d33
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Calculation of Constants
Calculation of Piezoelectric Constant g33
Calculation of Piezoelectric Constant g11
Calculation of Mechanical Q