Piezoelectric Constants
Because a piezoelectric ceramic is anisotropic, physical constants relate to both the direction of the applied mechanical or electric force and the directions perpendicular to the applied force. Consequently, each constant generally has two subscripts that indicate the directions of the two related quantities, such as stress (force on the ceramic element / surface area of the element) and strain (change in length of element / original length of element) for elasticity. The direction of positive polarization usually is made to coincide with the Zaxis of a rectangular system of X, Y, and Z axes (Figure 1.6). Direction X, Y, or Z is represented by the subscript 1, 2, or 3, respectively, and shear about one of these axes is represented by the subscript 4, 5, or 6, respectively. Definitions of the most frequently used constants, and equations for determining and interrelating these constants, are summarized here. The piezoelectric charge constant, d, the piezoelectric voltage constant, g, and the permittivity, e, are temperature dependent factors.




Figure 1.6  The direction of positive polarization usually is made to coincide with the Zaxis.
Piezoelectric Charge Constant
The piezoelectric charge constant, d, is the polarization generated per unit of mechanical stress (T) applied to a piezoelectric material or, alternatively, is the mechanical strain (S) experienced by a piezoelectric material per unit of electric field applied. The first subscript to d 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. Because the strain induced in a piezoelectric material by an applied electric field is the product of the value for the electric field and the value for d, d is an important indicator of a material's suitability for straindependent (actuator) applications.
d_{33} 
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 
d_{31} 
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 
d_{15} 
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
The piezoelectric voltage constant, g, is the electric field generated by a piezoelectric material per unit of mechanical stress applied or, alternatively, is 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. Because the strength of the induced electric field produced by a piezoelectric material in response to an applied physical stress is the product of the value for the applied stress and the value for g, g is important for assessing a material's suitability for sensing (sensor) applications.
g_{33} 
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 
g_{31} 
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 
g_{15} 
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 
Permittivity
The permittivity, or dielectric constant, , for a
piezoelectric ceramic material is the dielectric displacement
per unit electric field. ^{T}
is the permittivity at constant
stress, ^{S}
is the permittivity at constant strain. The first
subscript to indicates the direction of the dielectric
displacement; the second is the direction of the electric field.
The relative dielectric constant, K, is the ratio of , the
amount of charge that an element constructed from the ceramic
material can store, relative to the absolute dielectric
constant, _{0} , the charge that can be stored by the same
electrodes when separated by a vacuum, at equal voltage (_{0} =
8.85 x 10^{12} farad / meter).
^{T}_{11} 
permittivity
for dielectric displacement and electric field in
direction 1 (perpendicular to direction in which ceramic
element is polarized), under constant stress 
^{S}_{33} 
permittivity
for dielectric displacement and electric field in
direction 3 (parallel to direction in which ceramic
element is polarized), under constant strain 
Elastic Compliance
Elastic compliance, s, is the strain produced in a piezoelectric
material per unit of stress applied and, for the 11 and 33
directions, is the reciprocal of the modulus of elasticity
(Young's modulus, Y). sD is the compliance under a constant
electric displacement; sE is the compliance under a constant
electric field. The first subscript indicates the direction of
strain, the second is the direction of stress.
s^{E}_{11} 
elastic compliance for stress in direction 1 (perpendicular
to direction in which ceramic element is polarized) and
accompanying strain in direction 1, under constant electric
field (short circuit) 
s^{D}_{33} 
elastic compliance for stress in direction 3 (parallel to
direction in which ceramic element is polarized) and
accompanying strain in direction 3, under constant electric
displacement (open circuit) 
Young's Modulus
Young's modulus, Y, is an indicator of the stiffness
(elasticity) of a ceramic material. Y is determined from the
value for the stress applied to the material divided by the
value for the resulting strain in the same direction.
Electromechanical Coupling Factor
The electromechanical coupling factor, k, is 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.
k values quoted in ceramic suppliers' specifications typically
are theoretical maximum values. At low input frequencies, a
typical piezoelectric ceramic can convert 30  75% of the energy
delivered to it in one form into the other form, depending on
the formulation of the ceramic and the directions of the forces
involved.
A high k usually is desirable for efficient energy conversion,
but k does not account for dielectric losses or mechanical
losses, nor for recovery of unconverted energy. The accurate
measure of efficiency is the ratio of converted, useable energy
delivered by the piezoelectric element to the total energy taken
up by the element. By this measure, piezoelectric ceramic
elements in well designed systems can exhibit efficiencies that
exceed 90%.
The dimensions of a ceramic element can dictate unique
expressions of k. For a thin disc of piezoelectric ceramic the
planar coupling factor, k_{p} , expresses radial coupling  the
coupling between an electric field parallel to the direction in
which the ceramic element is polarized (direction 3) and
mechanical effects that produce radial vibrations, relative to
the direction of polarization (direction 1 and direction 2). For
a disc or plate of material whose surface dimensions are large
relative to its thickness, the thickness coupling factor, k_{t} , a
unique expression of k_{33} , expresses the coupling between an
electric field in direction 3 and mechanical vibrations in the
same direction. The resonance frequency for the thickness
dimension of an element of this shape is much higher than the
resonance frequency for the transverse dimensions. At the same
time, strongly attenuated transverse vibrations at this higher
resonance frequency, a result of the transverse contraction /
expansion that accompanies the expansion / contraction in
thickness, make k_{t} lower than k_{33} , the corresponding factor for
longitudinal vibrations of a thin rod of the same material, for
which a much lower longitudinal resonance frequency more closely
matches the transverse resonance frequency.
k_{33} 
factor for electric field in direction 3 (parallel to
direction in which ceramic element is polarized) and
longitudinal vibrations in direction 3
(ceramic rod, length >10x diameter) 
k_{t} 
factor for electric field in direction 3 and vibrations in
direction 3
(thin disc, surface dimensions large relative to thickness; k_{t}
< k_{33}) 
k_{31} 
factor for electric field in direction 3 (parallel to
direction in which ceramic element is polarized) and
longitudinal vibrations in direction 1 (perpendicular to
direction in which ceramic element is polarized)
(ceramic rod) 
k_{p} 
factor for electric field in direction 3 (parallel to
direction in which ceramic element is polarized) and radial
vibrations in direction 1 and direction 2 (both perpendicular to
direction in which ceramic element is polarized)
(thin disc) 
Dielectric Dissipation Factor
The dielectric dissipation factor (dielectric loss factor), tan
, for a ceramic material is the tangent of the dielectric loss
angle. tan is determined by the ratio of effective conductance
to effective susceptance in a parallel circuit, measured by
using an impedance bridge. Values for tan
typically are
determined at 1 kHz.
Frequency Constant
When an unrestrained piezoelectric ceramic element is exposed to
a high frequency alternating electric field, an impedance
minimum, the planar or radial resonance frequency, coincides
with the series resonance frequency, f_{s}. The relationship
between the radial mode resonance frequency constant, N_{P} , and
the diameter of the ceramic element, D , is expressed by:
N_{P} = f_{s} D
At higher resonance, another impedance minimum, the axial
resonance frequency, is encountered. The thickness mode
frequency constant, N_{T} , is related to the thickness of the
ceramic element, h, by:
N_{T} = f_{s} h
A third frequency constant, the longitudinal mode frequency
constant, is related to the length of the element:
N_{L} = f_{s} l
MostUsed Constants and Equations
Aging Rate
Aging rate = (Par_{2}  Par_{1}) /
((Par_{1}) (log t_{2}  log t_{1}))
Bandwidth
B
kf_{p} or B kf_{s}
Dielectric Constant (Relative)
permittivity of ceramic material / permittivity of free space*
K^{T} =
^{T} /
_{0}
*8.85 x 1012 farad / meter
Dielectric Dissipation Factor (Dielectric Loss Factor)
conductance / susceptance for parallel circuit equivalent to
ceramic element;
tangent of loss angle (tan d)
measure directly, typically at 1 kHz
Elastic Compliance
strain developed / stress applied;
inverse of Young's modulus (elasticity)
s = 1 / ^{2}
s^{D}_{33} = 1 / Y^{D}_{33}
s^{E}_{33} = 1 / Y^{E}_{33}
s^{D}_{11} = 1 / Y^{D}_{11}
s^{E}_{11} = 1 / Y^{E}_{11}
Electromechanical Coupling Factor
mechanical energy converted / electric energy input
or
electric energy converted / mechanical energy input
Static / low frequencies
ceramic plate
k_{31}^{2} =
d_{31}^{2} / (s^{E}_{11}^{T}_{33} )
ceramic disc
k_{p}^{2} =
2d_{31}^{2} / ((s^{E}_{11} + s^{E}_{12})^{T}_{33} )
ceramic rod
k_{33}^{2} =
d_{33}^{2} / (s^{E}_{33}^{T}_{33} )
Higher frequencies
ceramic plate
ceramic disc
ceramic rod
any shape
k_{eff}^{2} =
(f_{n}^{2}  f_{m}^{2} ) / f_{n}^{2}
Frequency Constant
resonance frequency o linear dimension governing resonance
N_{L} (longitudinal mode) =
f_{s} l
N_{P} (radial mode) =
f_{s} D
N_{T} (thickness mode) =
f_{s} h
Mechanical Quality Factor
reactance / resistance for series circuit equivalent to ceramic
element
Piezoelectric Charge Constant
electric field generated by unit area of ceramic / stress
applied
or
strain in ceramic element / unit electric field applied
d = k(s^{E}^{T} )
d_{31} = k_{31}(s^{E}_{11}^{T}_{33} )
d_{33} = k_{33}(s^{E}_{33}^{T}_{33} )
d_{15} = k_{15}(s^{E}_{55}^{T}_{11} )
Piezoelectric Voltage Constant
electric field generated / stress applied
or
strain in ceramic element / electric displacement applied
g = d / T
g_{31} =
d_{31} / ^{T}_{33}
g_{33} =
d_{33} / ^{T}_{33}
g_{15} =
d_{15} / ^{T}_{11}
Young's Modulus
stress applied / strain developed
Y = (F / A) /
(l / l) = T / S
Relationship among d, ^{T}, and g
g = d / ^{T}
or d = g^{T}
Symbols
A 
surface area of ceramic element
(m^{2} ) 
B 
bandwidth (frequency) 
d 
piezoelectric charge constant (C / N) 
D 
diameter of ceramic disc or rod (m) 
_{0} 
permittivity of free space (8.85 x
10^{12} farad / m) 
^{T} 
permittivity of ceramic material (farad / m) (at constant
stress 
F 
force 
f_{m} 
minimum impedance frequency (resonance frequency) (Hz) 
f_{n} 
maximum impedance frequency (antiresonance frequency) (Hz) 
f_{p} 
parallel resonance frequency (Hz) 
f_{s} 
series resonance frequency (Hz) 
g 
piezoelectric voltage constant
(Vm / N) 
h 
height (thickness) of ceramic element (m) 
k 
electromechanical coupling factor 
k_{eff} 
effective coupling factor 
K^{T} 
relative dielectric constant (at constant stress) 
l 
initial length of ceramic element (m) 
N 
frequency constant
(Hz_{*}m) 
Par_{1} 
value for parameter Par at
t_{1} (days) 
Par_{2} 
value for parameter Par at
t_{2} (days) 
Q_{m} 
mechanical quality factor 

density of ceramic (kg /
m^{3} ) 
s 
elastic compliance
(m^{2} / N) 
S 
strain 
t_{1} 
time 1 after polarization (days) 
t_{2} 
time 2 after polarization (days) 
tan 
dielectric dissipation factor 
T 
stress 
T^{o} 
temperature 
T_{C} 
Curie point (°C) 

velocity of sound in the ceramic material (m / s) 
w 
width of ceramic element (m) 
Y 
Young's modulus (N /
m^{2} ) 
Z_{m} 
minimum impedance at
f_{m} (ohm) 
