Transcript are piezoelectric constants

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)
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
d
g
Definition
S.I. Units
dielectric displacement developed
applied mechanical stress
(E = constant)
coulomb/meter2
Pa
C/N
strain developed
applied field
(T = constant)
meter/meter
volt/meter
m/V
field developed
applied mechanical stress
(D = constant)
volt/meter
Pa
Vm/N
strain developed
applied dielectric displacement
(T = constant)
meter/meter
coulomb/meter2
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
total stored 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 ij / g ij   iix
eij / hij   iix
(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
sijE : cijE
 Short circuit
sijD : cijD
 Open circuit
The Poisson ratio 
 12 
 s12E
s11E
;  13 
 s13E
s33E
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
2
2
2
2
2
2
f

f
c
f

f
f

f
keff2  1  p 2 s  a 2 r  n 2 m
c0  c1
fp
fa
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
  f p  fs 

k33 
tan 

2 fp
 2 fp 
 fs
Dielectric Permittivity  33 can be determined from capacitance C at a
frequency well below resonance
x
c
 33 
A
x
A = cross-sectional area of the rod
l = length of the rod
Piezoelectric Parameters and Measurements
The elastic compliance
s33D
is related to the fp
1
2 2

4

f
l
p
D
s33
= density
Superscript D = Open-circuit = constant electric displacement
s33D
s33 
1  k 332
E
d 33  k 33  33 s33  2
X
g 33 
d 33
E
1
 33X
 33X   33X (1  k332 )
Piezoelectric Parameters and Measurements
For a twin disc of diameter d

Considering a radial mode resonance
k p2
f p  fs 


 f  J 0 , J 1 ,
2
1 kp
fs 

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
1 2
k 
kp
2
1  2 d 2 f s2 (1   2 ) 

( ~ 2)
E
2
s11
1
2
31
d 31  k31 (
g 31 
X
33
E
11
S )
1
2
d 31
 33X
If minimum impedance |Zm| at resonance is known
1
 4 f Z m (c0  c)
Qm
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
T
 Free Relative Dielectric Constant : K 3
 Dissipation Factor: D
Elastic-Compliances: S33D S33E S11D S11E
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
1.
2.
3.
4.
weight or density
physical dimensions
free capacitance and dissipation factor
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 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