{"id":3228,"date":"2021-04-15T09:44:22","date_gmt":"2021-04-15T01:44:22","guid":{"rendered":"https:\/\/www.sinosonics.com\/?p=3228"},"modified":"2026-02-16T15:46:38","modified_gmt":"2026-02-16T07:46:38","slug":"piezoelectric-constants","status":"publish","type":"post","link":"https:\/\/www.sinosonics.com\/es\/knowledge-base\/piezoelectric-constants\/","title":{"rendered":"Piezoelectric Constants"},"content":{"rendered":"
\n

Because a\u00a0piezoelectric ceramic\u00a0is 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<\/em>). 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\u00a0piezoelectric charge constant<\/em>, d, the\u00a0piezoelectric voltage constant<\/em>, g, and the\u00a0permittivity<\/em>, e, are temperature dependent factors.<\/p>\n

Figure 1.6 – The direction of positive polarization usually is made to coincide with the Z-axis.<\/em><\/p>\n

Piezoelectric Charge Constant<\/h2>\n

The\u00a0piezoelectric charge constant<\/em>, 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\u00a0piezoelectric material\u00a0per 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.<\/p>\n\n

\n\n\n\n\n\n\n
d33<\/sub><\/strong><\/th>\ninduced polarization in direction 3 (parallel to direction in which ceramic element is polarized) per unit stress applied in direction 3
\nor
\ninduced strain in direction 3 per unit electric field applied in direction 3<\/th>\n<\/tr>\n<\/thead>\n
d31<\/sub><\/strong><\/td>\ninduced 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)
\nor
\n<\/strong>induced strain in direction 1 per unit electric field applied in direction 3<\/td>\n<\/tr>\n
d15<\/sub><\/strong><\/td>\ninduced 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)
\nor
\n<\/strong>induced shear strain about direction 2 per unit electric field applied in direction 1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n
<\/div>\n

Piezoelectric Voltage Constant<\/h2>\n

The\u00a0piezoelectric voltage constant<\/em>, 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\u00a0sensing (sensor) applications.<\/p>\n\n

\n\n\n\n\n\n\n
g33<\/sub><\/strong><\/th>\ninduced electric field in direction 3 (parallel to direction in which ceramic element is polarized) per unit stress applied in direction 3
\nor<\/strong>
\ninduced strain in direction 3 per unit electric displacement applied in direction 3<\/th>\n<\/tr>\n<\/thead>\n
g31<\/sub><\/strong><\/td>\ninduced 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)
\nor<\/strong>
\ninduced strain in direction 1 per unit electric displacement applied in direction 3<\/td>\n<\/tr>\n
g31<\/sub><\/strong><\/td>\ninduced 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)
\nor<\/strong>
\ninduced shear strain about direction 2 per unit electric displacement applied in direction 1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n
<\/div>\n

Permittivity Constant<\/h2>\n

The\u00a0permittivity<\/em>, or\u00a0dielectric constant<\/em>, \u03b5, for a piezoelectric ceramic material is the dielectric displacement per unit electric field. \u03b5T<\/sup>\u00a0is the permittivity at constant stress, \u03b5S<\/sup>\u00a0is the permittivity at constant strain. The first subscript to \u03b5> indicates the direction of the dielectric displacement; the second is the direction of the electric field.<\/p>\n

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).<\/p>\n\n\n\n\n
\u03b5T<\/sup>11<\/sub><\/strong><\/td>\n:permittivity for dielectric displacement and electric field in direction 1 (perpendicular to direction in which ceramic element is polarized), under constant stress<\/td>\n<\/tr>\n
\u03b5S<\/sup>33<\/sub><\/strong><\/td>\n:permittivity for dielectric displacement and electric field in direction 3 (parallel to direction in which ceramic element is polarized), under constant strain<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
<\/div>\n

Elastic And Compliance<\/h2>\n

Elastic compliance<\/em>, 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<\/sup>\u00a0is the compliance under a constant electric displacement; sE<\/sup>\u00a0is the compliance under a constant electric field. The first subscript indicates the direction of strain, the second is the direction of stress.<\/p>\n

\n\n
\n\n\n\n\n\n
sE<\/sup>11<\/sub><\/strong><\/th>\nelastic 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)<\/th>\n<\/tr>\n<\/thead>\n
sD<\/sup>33<\/sub><\/strong><\/td>\nelastic 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)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n<\/div>\n

Young’s Modulus<\/h2>\n

Young’s modulus<\/em>, 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.<\/p>\n

Electromechanical Coupling Factor<\/h2>\n

The\u00a0electromechanical coupling factor<\/em>, 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.<\/p>\n

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.<\/p>\n

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 technology in well designed systems can exhibit efficiencies that exceed 90%.<\/p>\n

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.<\/p>\n\n

\n\n\n\n\n\n\n\n
k33<\/sub><\/strong><\/th>\nfactor for electric field in direction 3 (parallel to direction in which ceramic element is polarized) and longitudinal vibrations in direction 3
\n(ceramic rod, length >10x diameter)<\/th>\n<\/tr>\n<\/thead>\n
kt<\/sub><\/strong><\/td>\nfactor for electric field in direction 3 and vibrations in direction 3
\n(thin disc, surface dimensions large relative to thickness; kt<\/sub>\u00a0< k33<\/sub>)<\/td>\n<\/tr>\n
k31<\/sub><\/strong><\/td>\nfactor 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)
\n(ceramic rod)<\/td>\n<\/tr>\n
kp<\/sub><\/strong><\/td>\nfactor 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)
\n(thin disc)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n

Dielectric Dissipation Factor<\/h2>\n

The \u03b4, tan \u03b4, for a ceramic material is the tangent of the dielectric loss angle. tan \u03b4 is determined by the ratio of effective conductance to effective susceptance in a parallel circuit, measured by using an impedance bridge. Values for tan \u03b4 typically are determined at 1 kHz.<\/p>\n

Piezoelectric Frequency Constant<\/h2>\n

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<\/sub>. The relationship between the\u00a0radial mode resonance frequency constant<\/em>, NP<\/sub>\u00a0, and the diameter of the ceramic element, D\u03a6 , is expressed by:
\nNP<\/sub>\u00a0= fs<\/sub>\u00a0D\u03a6
\nAt higher resonance, another impedance minimum, the\u00a0axial resonance frequency<\/em>, is encountered. The thickness mode frequency constant, NT<\/sub>\u00a0, is related to the thickness of the ceramic element, h, by:
\nNT<\/sub>\u00a0= fs<\/sub>\u00a0h
\nA third frequency constant, the\u00a0longitudinal mode frequency constant<\/em>, is related to the length of the element:
\nNL<\/sub>\u00a0= fs<\/sub>\u00a0l<\/p>\n

Most-Used Constants and Equations<\/h2>\n

Aging Rate
\nAging rate = (Par2<\/sub>\u00a0– Par1<\/sub>) \/ ((Par1<\/sub>) (log t2<\/sub>\u00a0– log t1<\/sub>))
\nBandwidth
\nB \u2261 kfp<\/sub>\u00a0or B \u2261 kfs<\/sub>
\nDielectric Constant (Relative)
\npermittivity of ceramic material \/ permittivity of free space*
\nKT<\/sup>\u00a0= \u03b5T<\/sup>\u00a0\/ <\u03b50<\/sub>
\n*8.85 x 10-12 farad \/ meter
\nDielectric Dissipation Factor (Dielectric Loss Factor)
\nconductance \/ susceptance for parallel circuit equivalent to ceramic element;
\ntangent of loss angle (tan d)
\nmeasure directly, typically at 1 kHz
\nElastic Compliance
\nstrain developed \/ stress applied;
\ninverse of Young’s modulus (elasticity)
\ns = 1 \/ \u03bd2<\/sup>
\nsD<\/sup>33<\/sub>\u00a0= 1 \/ YD<\/sup>33<\/sub>
\nsE<\/sup>33<\/sub>\u00a0= 1 \/ YE<\/sup>33<\/sub>
\nsD<\/sup>11<\/sub>\u00a0= 1 \/ YD<\/sup>11<\/sub>
\nsE<\/sup>11<\/sub>\u00a0= 1 \/ YE<\/sup>11<\/sub>
\nElectromechanical Coupling Factor
\nmechanical energy converted \/ electric energy input
\nor
\nelectric energy converted \/ mechanical energy input
\nStatic \/ low frequencies
\nceramic plate
\nk31<\/sub>2<\/sup>\u00a0= d31<\/sub>2<\/sup>\u00a0\/ (sE<\/sup>11<\/sub>\u03b5T<\/sup>33<\/sub>\u00a0)
\nceramic disc
\nkp<\/sub>2<\/sup>\u00a0= 2d31<\/sub>2<\/sup>\u00a0\/ ((sE<\/sup>11<\/sub>\u00a0+ sE<\/sup>12<\/sub>)\u03b5T<\/sup>33<\/sub>\u00a0)
\nceramic rod
\nk33<\/sub>2<\/sup>\u00a0= d33<\/sub>2<\/sup>\u00a0\/ (sE<\/sup>33<\/sub>\u03b5T<\/sup>33<\/sub>\u00a0)
\nHigher frequencies
\nceramic plate
\n\"Equation\"
\nceramic disc
\nkp<\/sub>\u00a0\u2245 \u221a [(2.51 (fn<\/sub>\u00a0– fm<\/sub>) \/ fn<\/sub>) – ((fn<\/sub>\u00a0– fm<\/sub>) \/ fn<\/sub>)2<\/sup>]\nceramic rod
\nk33<\/sub>2<\/sup>\u00a0= (\u03c0 \/ 2) (fn<\/sub>\u00a0\/ fm<\/sub>) tan [(\u03c0 \/ 2) ((fn<\/sub>\u00a0– fm<\/sub>) \/ fn<\/sub>)]\nany shape
\nkeff<\/sub>2<\/sup>\u00a0= (fn<\/sub>2<\/sup>\u00a0– fm<\/sub>2<\/sup>\u00a0) \/ fn<\/sub>2<\/sup>
\nFrequency Constant
\nresonance frequency o linear dimension governing resonance
\nNL<\/sub>\u00a0(longitudinal mode) = fs<\/sub>\u00a0l
\nNP<\/sub>\u00a0(radial mode) = fs<\/sub>\u00a0D\u03a6
\nNT<\/sub>\u00a0(thickness mode) = fs<\/sub>\u00a0h
\nMechanical Quality Factor
\nreactance \/ resistance for series circuit equivalent to ceramic element
\nQm<\/sub>\u00a0= fn<\/sub>2<\/sup>\u00a0\/ (2\u03c0fm<\/sub>\u00a0C0<\/sub>\u00a0Zm<\/sub>\u00a0(fn<\/sub>2<\/sup>\u00a0– fm<\/sub>2<\/sup>))
\nPiezoelectric Charge Constant
\nelectric field generated by unit area of ceramic \/ stress applied
\nor
\nstrain in ceramic element \/ unit electric field applied
\nd = k\u221a(sE<\/sup>\u03b5T<\/sup>\u00a0)
\nd31<\/sub>\u00a0= k31<\/sub>\u221a(sE<\/sup>11<\/sub>\u03b5T<\/sup>33<\/sub>\u00a0)
\nd33<\/sub>\u00a0= k33<\/sub>\u221a(sE<\/sup>33<\/sub>\u03b5T<\/sup>33<\/sub>\u00a0)
\nd15<\/sub>\u00a0= k15<\/sub>E<\/sup>55<\/sub>\u03b5T<\/sup>11<\/sub>\u00a0)
\nPiezoelectric Voltage Constant
\nelectric field generated \/ stress applied
\nor
\nstrain in ceramic element \/ electric displacement applied
\ng = d \/ \u03b5T
\ng31<\/sub>\u00a0= d31<\/sub>\u00a0\/ \u03b5T<\/sup>33<\/sub>
\ng33<\/sub>\u00a0= d33<\/sub>\u00a0\/ \u03b5T<\/sup>33<\/sub>
\ng15<\/sub>\u00a0= d15<\/sub>\u00a0\/ \u03b5T<\/sup>11<\/sub>
\nYoung’s Modulus
\nstress applied \/ strain developed
\nY = (F \/ A) \/ (\u0394l \/ l) = T \/ S
\nRelationship among d, \u03b5T<\/sup>, and g
\ng = d \/ \u03b5T<\/sup>\u00a0or d = g\u03b5T<\/sup><\/p>\n

Symbols<\/h3>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
A<\/td>\nsurface area of ceramic element (m2<\/sup>\u00a0)<\/td>\n<\/tr>\n
B<\/td>\nbandwidth (frequency)<\/td>\n<\/tr>\n
d<\/td>\npiezoelectric charge constant (C \/ N)<\/td>\n<\/tr>\n
D\u03a6<\/td>\ndiameter of ceramic disc or rod (m)<\/td>\n<\/tr>\n
\u03b50<\/sub><\/td>\npermittivity of free space (8.85 x 10-12<\/sup>\u00a0farad \/ m)<\/td>\n<\/tr>\n
\u03b5T<\/sup><\/td>\npermittivity of ceramic material (farad \/ m) (at constant stress<\/td>\n<\/tr>\n
F<\/td>\nforce<\/td>\n<\/tr>\n
fm<\/sub><\/td>\nminimum impedance frequency (resonance frequency) (Hz)<\/td>\n<\/tr>\n
fn<\/sub><\/td>\nmaximum impedance frequency (anti-resonance frequency) (Hz)<\/td>\n<\/tr>\n
fp<\/sub><\/td>\nparallel resonance frequency (Hz)<\/td>\n<\/tr>\n
fs<\/sub><\/td>\nseries resonance frequency (Hz)<\/td>\n<\/tr>\n
g<\/td>\npiezoelectric voltage constant (Vm \/ N)<\/td>\n<\/tr>\n
h<\/td>\nheight (thickness) of ceramic element (m)<\/td>\n<\/tr>\n
k<\/td>\nelectromechanical coupling factor<\/td>\n<\/tr>\n
keff<\/sub><\/td>\neffective coupling factor<\/td>\n<\/tr>\n
KT<\/sup><\/td>\nrelative dielectric constant (at constant stress)<\/td>\n<\/tr>\n
l<\/td>\ninitial length of ceramic element (m)<\/td>\n<\/tr>\n
N<\/td>\nfrequency constant (Hz*<\/sub>m)<\/td>\n<\/tr>\n
Par1<\/sub><\/td>\nvalue for parameter Par at t1<\/sub>\u00a0(days)<\/td>\n<\/tr>\n
Par2<\/sub><\/td>\nvalue for parameter Par at t2<\/sub>\u00a0(days)<\/td>\n<\/tr>\n
Qm<\/sub><\/td>\nmechanical quality factor<\/td>\n<\/tr>\n
\u03c1<\/td>\ndensity of ceramic (kg \/ m3<\/sup>\u00a0)<\/td>\n<\/tr>\n
s<\/td>\nelastic compliance (m2<\/sup>\u00a0\/ N)<\/td>\n<\/tr>\n
S<\/td>\nstrain<\/td>\n<\/tr>\n
t1<\/sub><\/td>\ntime 1 after polarization (days)<\/td>\n<\/tr>\n
t2<\/sub><\/td>\ntime 2 after polarization (days)<\/td>\n<\/tr>\n
tan \u03b4<\/td>\ndielectric dissipation factor<\/td>\n<\/tr>\n
T<\/td>\nstress<\/td>\n<\/tr>\n
To<\/sup><\/td>\ntemperature<\/td>\n<\/tr>\n
TC<\/sub><\/td>\nCurie point (\u00c2\u00b0C)<\/td>\n<\/tr>\n
\u03bd><\/td>\nvelocity of sound in the ceramic material (m \/ s)<\/td>\n<\/tr>\n
w<\/td>\nwidth of ceramic element (m)<\/td>\n<\/tr>\n
Y<\/td>\nYoung’s modulus (N \/ m2<\/sup>\u00a0)<\/td>\n<\/tr>\n
Zm<\/sub><\/td>\nminimum impedance at fm<\/sub>\u00a0(ohm)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
<\/div>\n<\/div>
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