Last Updated on May 17, 2023 by You Ling

The piezoelectric constant is a parameter that measures the strength of the piezoelectric effect in a material, and it is directly related to the sensitivity of the piezoelectric output. The numerical value quantifying the piezoelectric effect in a material is called the piezoelectric constant. The most important constant for our applications is the piezoelectric charge constant (d), which tells us the material’s ability to convert mechanical stress into electrical charge.

1、What are the piezoelectric-related parameters?

Piezoelectric Charge Constant

Piezoelectric Voltage Constant

Permittivity Constant

Elastic and Compliance

Young’s Modulus

Electromechanical Coupling Factor

Dielectric Dissipation Factor

Piezoelectric Frequency Constant

Curie Temperature

2、What is the piezoelectric charge constant?

The piezoelectric charge constant (d) is the polarization produced per unit mechanical stress (T) applied to the piezoelectric material or the mechanical strain (S) the piezoelectric material experiences per unit applied electric field. The first subscript of d indicates the direction of polarization produced in the material when the electric field E is zero or the direction of the applied field strength. The second subscript is the direction of applied stress or induced strain. Since the strain induced in the piezoelectric material by the applied electric field is the product of the electric field value and the d value, d is an important indicator of whether the material is suitable for strain-related (actuator) applications.

3、What is the piezoelectric voltage constant?

The piezoelectric voltage constant (g) is the electric field generated per unit mechanical stress applied to the piezoelectric material or the mechanical strain experienced by the piezoelectric material per unit applied electric displacement. The first subscript of g indicates the direction of the electric field generated in the material or the direction of applied electric displacement. The second subscript is the direction of applied stress or induced strain. Since the induced electric field strength in the piezoelectric material in response to applied physical stress is the product of the applied stress value and the g value, g is important for assessing the suitability of the material for sensing (sensor) applications.

4、What is the permittivity constant?

The dielectric constant or permittivity (ε) of piezoelectric ceramic materials is the dielectric displacement per unit electric field. ε_T is the dielectric constant under constant stress, and ε_S is the dielectric constant under constant strain. The first subscript of ε indicates the direction of the dielectric displacement, and the second subscript is the direction of the electric field.

The relative dielectric constant (K) is the ratio of the amount of electric charge that can be stored by a component made of ceramic material to the absolute dielectric constant of vacuum (ε0 = 8.85 x 10^-12 Farads/meter) when the same electrodes are separated by a.

5、What are the elastic properties?

Elastic compliance (s) is the strain produced by the piezoelectric material per unit applied stress. For the 11 and 33 directions, it is the reciprocal of the elastic modulus (Young’s modulus, Y). s_D is the compliance under constant electric displacement, and s_E is the compliance under constant electric field. The first subscript indicates the direction of strain, and the second subscript indicates the direction of stress.

6、What is Young’s modulus?

Young’s modulus (Y) is an indicator of the stiffness (elasticity) of ceramic materials. Y is determined by dividing the stress value applied to the material by the strain value produced in the same direction.

7、What is the piezoelectric coupling factor?

The electromechanical coupling factor (k) is an indicator of the effectiveness of a piezoelectric material in converting electrical energy into mechanical energy or converting mechanical energy into electrical energy. The first subscript of k indicates the direction of the applied electrodes, and the second subscript indicates the direction of applied or developed mechanical energy.

8、What is the Curie temperature?

As the working temperature increases, it becomes easier for the piezoelectric material to depolarize. Complete and permanent depolarization occurs at the Curie temperature of the material. Each ceramic has its own Curie point. When the ceramic element is heated above the Curie point, all piezoelectric properties disappear. In practice, the working temperature must be significantly lower than the Curie point.

9、Related symbols

In the world of piezoelectricity, the following symbols are most commonly used:

E = Electric field

D = Electric displacement

T = Mechanical stress

S = Mechanical strain

s = Elastic compliance, reciprocal of Young’s modulus (s=1/Y)

ε = Permittivity constant (sometimes the letter K is used to represent the relative permittivity)

d = Piezoelectric charge constant

g = Piezoelectric voltage constant

k = Coupling factor

10、Most-Used Constants and Equations

Aging Rate

Aging rate = (Par2 – Par1) / ((Par1) (log t2 – log t1))

Bandwidth

B ≡ kfp or B ≡ kfs

Dielectric Constant (Relative)

permittivity of ceramic material / permittivity of free space*

KT = εT / <ε0

*8.85 x 10-12 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 / ν2

sD33 = 1 / YD33

sE33 = 1 / YE33

sD11 = 1 / YD11

sE11 = 1 / YE11

Electromechanical Coupling Factor

mechanical energy converted / electric energy input

or

electric energy converted / mechanical energy input

Static / low frequencies

ceramic plate

k312 = d312 / (sE11εT33 )

ceramic disc

kp2 = 2d312 / ((sE11 + sE12)εT33 )

ceramic rod

k332 = d332 / (sE33εT33 )

Higher frequencies

ceramic plate

Equation

ceramic disc

kp ≅ √ [(2.51 (fn – fm) / fn) – ((fn – fm) / fn)2]

ceramic rod

k332 = (π / 2) (fn / fm) tan [(π / 2) ((fn – fm) / fn)]

any shape

keff2 = (fn2 – fm2 ) / fn2

Frequency Constant

resonance frequency o linear dimension governing resonance

NL (longitudinal mode) = fs l

NP (radial mode) = fs DΦ

NT (thickness mode) = fs h

Mechanical Quality Factor

reactance / resistance for series circuit equivalent to ceramic element

Qm = fn2 / (2πfm C0 Zm (fn2 – fm2))

Piezoelectric Charge Constant

electric field generated by unit area of ceramic / stress applied

or

strain in ceramic element / unit electric field applied

d = k√(sEεT )

d31 = k31√(sE11εT33 )

d33 = k33√(sE33εT33 )

d15 = k15E55εT11 )

Piezoelectric Voltage Constant

electric field generated / stress applied

or

strain in ceramic element / electric displacement applied

g = d / εT

g31 = d31 / εT33

g33 = d33 / εT33

g15 = d15 / εT11

Young’s Modulus

stress applied / strain developed

Y = (F / A) / (Δl / l) = T / S

Relationship among d, εT, and g

g = d / εT or d = gεT

11、Conclusion

This article discusses the parameters of piezoelectric materials and their relationships. The piezoelectric constant is a parameter that measures the strength of the piezoelectric effect in a material. Common piezoelectric parameters include the piezoelectric charge constant, piezoelectric voltage constant, permittivity constant, elasticity and compliance, Young’s modulus, electromechanical coupling factor, dielectric dissipation factor, piezoelectric frequency constant, and Curie temperature. These parameters help us understand the performance and suitability of piezoelectric materials.

The piezoelectric charge constant (d) describes the ability of a piezoelectric material to convert mechanical stress into electrical charge. The piezoelectric voltage constant (g) indicates the sensing capability of the piezoelectric material in response to stress. The dielectric constant (ε) represents the electric displacement per unit electric field. Elastic compliance (s) is the strain produced per unit applied stress and is inversely proportional to Young’s modulus (Y).

The electromechanical coupling factor (k) describes the efficiency of a piezoelectric material in converting electrical energy into mechanical energy or vice versa. The Curie temperature is the critical point at which a piezoelectric material loses its piezoelectric properties when exposed to high temperatures.

In summary, these parameters are helpful in evaluating and selecting suitable piezoelectric materials for efficient energy conversion and sensing applications.