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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 Z-axis 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. |
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Figure 1.6 - The direction of positive polarization
usually is made to coincide with the Z-axis. |
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Piezoelectric Charge Constant |
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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 strain-dependent (actuator)
applications. |
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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 |
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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 |
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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 |
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Piezoelectric
Voltage Constant
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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. |
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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 |
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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 |
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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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Permittivity |
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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). |
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 T11 |
permittivity for dielectric displacement and
electric field in direction 1 (perpendicular to
direction in which ceramic element is
polarized), under constant stress |
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S33 |
permittivity for dielectric displacement and
electric field in direction 3 (parallel to
direction in which ceramic element is
polarized), under constant strain |
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Electromechanical
Coupling Factor |
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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, kp ,
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, kt , a unique
expression of k33 , 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 kt lower than
k33 , 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. |
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k33 |
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) |
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kt |
factor for electric field in direction 3 and
vibrations in direction 3
(thin disc, surface dimensions large relative to
thickness; kt < k33) |
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k31 |
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) |
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kp |
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) |
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Dielectric Dissipation Factor |
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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. |
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Frequency
Constant |
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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, fs. The relationship between the
radial mode resonance frequency constant, NP
, and the diameter of the ceramic element, D
, is expressed by:
NP = fs D
At higher resonance, another impedance minimum, the
axial resonance frequency, is encountered. The
thickness mode frequency constant, NT , is
related to the thickness of the ceramic element, h, by:
NT = fs h
A third frequency constant, the longitudinal mode
frequency constant, is related to the length of the
element:
NL = fs l |
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Curie temperature |
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Temperature at which the permittivity of ferroelectric
ceramics reaches its peak. Above this temperature the
ceramic material will not exhibit piezoelectric
properties. |
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Mechanical quality factor |
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Amplitude magnification of oscillating
piezoelectric parts in a resonant state. This is a
non-dimensional factor indicating the mechanical loss of
the component under dynamic operating conditions. |
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