Piezoelectric Constant

Forces of Piezoelectric ConstantsBecause 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.

**Piezoelectric Charge Constant**

**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 Voltage Constant**

**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

The permittivity, or dielectric constant,

The relative dielectric constant,

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.

**sD33** 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,

**Electromechanical Coupling Factor**

A high

The dimensions of a ceramic element can dictate unique expressions of

**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)

**kt** factor for electric field in direction 3 and vibrations in direction 3

** ** (thin disc, surface dimensions large relative to thickness; **kt** < k33)

**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)

**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)

The

**Frequency Constant**

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:

A third frequency constant, the longitudinal mode frequency constant, is related to the length of the element:

Most-Used Constants and Equations

**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

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