Jing-Feng Li

Lead-Free Piezoelectric Materials

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alt="d equals k StartRoot epsilon 0 k Superscript normal upper T Baseline s Superscript normal upper E Baseline EndRoot left-parenthesis normal upper C slash normal upper N right-parenthesis"/>

      where k is electro‐mechanical coupling coefficient, kT denotes relative dielectric constant at a constant stress, and sE is elastic compliance (10 m/N) at a constant electrical field.

      There are two important d constants:

      (1.5)d 31 equals k 31 StartRoot epsilon 0 k 3 Superscript normal upper T Baseline s 11 Superscript normal upper E Baseline EndRoot left-parenthesis normal upper C slash normal upper N right-parenthesis

      (1.6)d 33 equals k 33 StartRoot epsilon 0 k 3 Superscript normal upper T Baseline s 33 Superscript normal upper E Baseline EndRoot left-parenthesis normal upper C slash normal upper N right-parenthesis

      (1.7)d 33 almost-equals negative 2.5 dot d 31

      1.4.1.2 Piezoelectric Voltage Coefficient (G‐constant)

      The piezoelectric voltage coefficient is also called voltage output constant, which is defined as the ratio of the electric field produced to the mechanical stress applied and is expressed in the unit of voltage‐meter/Newton (Vm/N).

      (1.8)g equals StartFraction Strain developed Over Applied charge density EndFraction equals StartFraction Field developed Over Applied mechanical stress EndFraction

      The g‐constants are calculated from the piezoelectric charge (strain) constant (d) and relative permittivity (ε) from the equation:

      (1.9)g equals StartFraction d Over epsilon EndFraction left-parenthesis upper V m slash normal upper N right-parenthesis

      Depending on the type of relative directions, the g constant can be categorized as g33, g31, or g15, corresponding to d33, d31, or d15, respectively.

      1.4.2 Piezoelectric Coupling Coefficient

      The piezoelectric coupling coefficient (sometimes referred as the electromechanical coupling coefficient) is defined as the ratio of the mechanical energy accumulated in response to an electrical input or vice versa. It also corresponds to the fraction of electrical energy that can be converted into mechanical energy and vice versa. Thus, the piezoelectric coupling coefficient can be expressed by the following equation:

      (1.10)x equals StartRoot StartFraction Mechanical energy stored Over Electrical energy applied EndFraction EndRoot equals StartRoot StartFraction Electrical energy stored Over Mechanical energy applied EndFraction EndRoot

      The coupling factor can be calculated based on the measured resonance and anti‐resonance frequencies of a piezoelectric element, depending on the vibration mode at which the element is excited. The most used coupling factors are kp and kt for the vibration along the radial and thickness directions in a circle‐shaped disk, respectively. In general, a useful parameter keff is frequently used to express the effective coupling coefficient of an resonator with an arbitrary shape, either at its fundamental resonance or at any overtone modes, and is expressed as follows:

      (1.11)k Subscript e f f Superscript 2 Baseline equals 1 minus left-parenthesis StartFraction f Subscript normal r Baseline Over f Subscript normal a Baseline EndFraction right-parenthesis squared

      where fr and fa stand for resonating frequency and anti‐resonating frequency, respectively. The coupling coefficients can be calculated for the various modes of vibration from the following equations:

      (1.12)StartFraction k Subscript normal p Superscript 2 Baseline Over 1 minus k Subscript normal p Superscript 2 Baseline EndFraction equals StartStartFraction left-parenthesis 1 minus sigma Superscript normal upper E Baseline right-parenthesis upper J 1 left-bracket eta 1 left-parenthesis 1 plus StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction right-parenthesis right-bracket minus eta 1 left-parenthesis 1 plus StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction right-parenthesis upper J 0 left-bracket eta 1 left-parenthesis 1 plus StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction right-parenthesis right-bracket OverOver left-parenthesis 1 plus sigma Superscript normal upper E Baseline right-parenthesis upper J 1 left-bracket eta 1 left-parenthesis 1 plus StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction right-parenthesis right-bracket EndEndFraction

J Bessel function of the first kind and zero order
J 1 Bessel function of the first kind and first order
σ E Poisson's ratio
η 1 Lowest positive root of (1 + σE)·J1η = ηJ0(η)
F r Resonance frequency (Hz)
F a Anti‐resonance frequency (Hz)
ΔF = Fa − Fr (Hz)

      Assuming that σE = 0.31 for PZT ceramics and η1 = 2.05, the following simplified equations holds:

      (1.13)k 33 squared equals StartStartFraction StartFraction normal pi Over 2 EndFraction OverOver 1 plus StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction EndEndFraction tangent StartStartFraction StartFraction normal pi Over 2 EndFraction StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction OverOver 1 plus StartFraction normal upper Delta upper F Over upper F Subscript normal r Baseline EndFraction EndEndFraction

      (1.14)StartFraction k 31 squared Over 1 </p> </div><hr> <div class= Скачать книгу