Lead-Free Piezoelectric Materials. Jing-Feng Li

the laser interferometry should be carried out with great caution and patience, though this type of measurement method is simply based on acquiring the displacement value of the sample surface induced by the corresponding applied voltage.

      1.5.3 Resonance and Anti‐resonance Method

      Both the quasi‐static method and the laser interferometry method are used for the direct measurements of d₃₃/d₃₁. In contrast to the two aforementioned methods, the resonance and anti‐resonance method, or the frequency method can be applied to determine the complete tensor matrix of the material coefficients [21, 36, 37]. These coefficients can be derived by the raw set of measured parameters including resonant frequencies, density, and sample dimensions. The detailed procedure for the determination of these coefficients can be found in the European standard EN 50324‐2:2002 and the world standard CEI/IEC 60483:1976.

      For this method, an accurate impedance analyzer is a crucial requirement as use for the testing of the resonant frequencies f_s and antiresonant frequencies f_a as well as the free capacitance C^T at 1 kHz. The commercially available lineups of impedance analyzers differ in terms of frequency range and sensitivity. The most regularly used one are, for example, Agilent E4294A and Wayne Kerr 65120B, produced by Agilent and Wayne Kerr, respectively. The other thing to notice is that a set of samples in the forms of a disk, a plate, and a cylinder, which have different vibration modes, should be prepared before the measurements are carried out. Obtaining all the needed sample items from a single bulk material of appropriate size is preferred. The sample dimensions should conform to the world standard CEI/IEC 60483:1976.

Material constants such as piezoelectric coefficient d, dielectric coefficient ε, and elastic coefficient s are anisotropic in general. They are usually described by tensor components written in a simplified matrix form according to the point‐group symmetry of the materials. For piezoelectric ceramics, the complete matrix of the coefficients of electromechanical properties can be written as follows:

      (1.21)

Based on the IEEE standards [22], the requirement of the sample dimensions and the complete matrix of piezoelectric constants can be established as illustrated in Figure 1.7 [37].

After acquiring the entry parameters, namely, resonant frequencies, antiresonant frequencies, capacitance, density, and sample dimensions, we can calculate the electromechanical coupling coefficients, according to which elastic coefficients , , , , , and can be derived. As an example, the related formulas for the calculation of the elastic coefficient , the piezoelectric charge coefficient d₃₃, and the piezoelectric voltage coefficient g₃₃ for the longitudinal length mode in the cylinder are deduced based on the IEEE standards as follows:

Figure 1.7 Establishing the complete set of material coefficients defined by IEEE standards [37].

      Source: Reprinted with permission from Fialka and Benes [37]. Copyright 2013, IEEE.

      (1.22)

      (1.23)