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A Modified Iterative Automatic Method for Characterization at Shear Resonance: Case Study of Ba_{0.85}Ca_{0.15}Ti_{0.90}Zr_{0.10}O_{3} Eco-Piezoceramics

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

_{15}, the elastic compliance s

_{55}

^{E}and the dielectric permittivity component ε

_{11}

^{S}for a piezoelectric ceramic can be determined, including all losses, using the automatic iterative method of analysis of the complex impedance curves for the shear mode of an appropriated resonator. This is the non-standard, thickness-poled and longitudinally excited, shear plate. In this paper, the automatic iterative method is modified. The purpose is to be able to deal with the analysis of the impedance curves of the non-standard plate as the periodic phenomena of coupling and decoupling of the main shear resonance and other resonances takes place. This happens when the thickness of the plate is reduced, and its aspect ratio (width of the excitation (w)/thickness for poling (t)) is increased. In this process, the frequency of the shear resonance also increases and meets those of other plate modes periodically. We aim to obtain the best approach for the shear properties of near coupling and to reveal both their validity and the limitations of the thus-obtained information. Finally, we use a plate of a Ba

_{0.85}Ca

_{0.15}Ti

_{0.90}Zr

_{0.10}O

_{3}eco-piezoceramic as a case study.

## 1. Introduction

_{1−x}Ca

_{x}Ti

_{0.90}Zr

_{0·10}O

_{3}(BCZT), where x = 0.10–0.18 [17]. To this aim, the automatic iterative method of analysis of the complex impedance spectra at the resonance [18,19,20] of such a non-standard shear plate was implemented. Since the decoupling of the fundamental shear mode and other undesired natural resonance modes of the non-standard plate can be effectively achieved, the required spectrum to analyze was obtained. Indeed, a number of uncoupled shear resonances of the plate, amenable for material characterization, including all losses, were effectively obtained in the range of the aspect ratios (length for excitation (w)/thickness for poling (t)) between nine and 15 for PZT and 5.5 and 12.5 for BCZT. It must be noted that the standard measurement methods recommend to use plates with w/t and L/t > 20 [10] and w/t and L/t > 32 [8], to minimize the effect of the undesired modes on the shear mode.

_{0.85}Ca

_{0.15}Ti

_{0.90}Zr

_{0.10}O

_{3}eco-piezoceramic are analyzed as the coupling of shear and lateral modes evolves with the change in the aspect ratio (w/t) of the plate. The evolution of the spectrum (R and G peaks) of the shear mode of the non-standard shear ceramic item was studied by measuring resonators of different aspect ratios and obtained by the progressive reduction in their thickness.

## 2. Material and Methods

#### 2.1. Material

_{0.85}Ca

_{0.15}Ti

_{0.90}Zr

_{0.10}O

_{3}piezoceramic, with an initial thickness for polarization t (distance between electrodes for poling) = 1.09 mm, lateral dimensions of L = 8.18 mm and w (distance between electrodes for electrical excitation of the resonance) = 6.20 mm and a density of 5.62 g.cm

^{−3}, was fabricated by conventional solid state route, as explained elsewhere [24], by sintering at 1400 °C for 2 h in air. Silver paint electrodes of area L × w were painted on both major faces and annealed at 600 °C for 30 min. The ceramics were thickness-poled at room temperature under 3 kV.mm

^{−1}for 30 min. After poling, the electrodes were removed by fine polishing and new electrodes of area L × t were attached for the longitudinal electrical excitation of the electromechanical resonances and for the impedance measurements. Thickness was reduced in steps of 0.01 mm to a final value of t = 0.5 mm, in order to change the aspect ratio (w/t), and the impedance spectrum was measured again at each step.

#### 2.2. The Automatic Iterative Method for Analysis of Impedance Curves

_{15}and the complex elastic compliance s

_{55}

^{E}, as well as the complex dielectric permittivity at the resonance frequency (ε

_{11}

^{S}), were directly calculated using the automatic iterative method of analysis of the impedance measurements [16,18] and were previously reported [17]. They will be considered in this work for the sake of comparison with the results here obtained using the modified method.

_{1}and f

_{2}, are those of maximum piezoelectric energy and are also automatically calculated at each iteration [18]. Experimental G and R peaks (symbols) for the BCZT plate are shown in Figure 1 together with the reconstructed spectra (continuous lines), which was obtained after the material parameters were determined. The software computes the shear electromechanical coupling factor, k

_{15}, from [7]:

_{15}), fs and fp. Figure 4 shows the spectra for two aspect ratios with low ${\Re}^{2}$ and high coupling, marked with (*) in Figure 3a.

_{15}correspond to the lowest values of ${\Re}^{2}$. This was also observed in the study of a PZT ceramic [16]. However, coupling produces a transfer of energy from the excited mode to the spurious one, with the consequence of a lower amount of energy transduction in the shear mode. Therefore, such high values of k

_{15}lack physical meaning. Let us comment on the origin of this result as follows.

_{15}coupling coefficient (Figure 3b) is not much different than that measured for the previous aspect ratios and follows the trend of decreasing as the aspect ratio and the coupling between modes gets higher and, consequently, as the transfer of energy to the mechanical vibration of the spurious mode takes place.

_{15}coupling coefficient is also overestimated and other coefficients also have anomalies [17].

_{15}(Figure 3b) and other material parameters in the intervals of aspect ratio mentioned above is mainly a consequence of the incorrect selection of the frequencies for the shear mode used to solve Equation (4). This would also affect the calculation by any other method [7,8]. To avoid this, we here propose a modification of the analysis method.

#### 2.3. The Modified Automatic Iterative Method

_{15}), fs and fp.

_{15}are not observed in Figure 7b, where there is a quasi-parallel evolution of the two characteristic frequencies of the shear resonance, except in critical points corresponding to the “jump” of the frequencies of the shear mode over those of each lateral mode.

## 3. Results

#### 3.1. Elastic Properties

_{55}

^{E}of the material depends mainly on the frequency of the resonance, as can be obtained from the following relationships [7]:

_{55}

^{E}with the aspect ratio (w/t). Abrupt jumps in the recalculated compliance (real part) from a maximum to a local minimum are shown in Figure 8b. These jumps take place in parallel to the jumps in the frequencies of the shear mode over those in each lateral mode, at the minima of the ${\Re}^{2}$ periods. From each jump to the next, the recalculated compliance s

_{55}

^{E}increases as the frequency of the shear mode increases. Since the material coefficient s

_{55}

^{E}′ is inversely proportional to the fp of the shear mode, this is the result of the overall modification of the spectrum near coupling (Figure 6), which limits the use of Equation (4). Unless we know if the aspect ratio of the measured plate is moving away from or coming closer to the nearest one for a coupled mode at ${\Re}^{2}$ minima, it is not possible to know if the calculation is underestimating or overestimating s

_{55}

^{E}′. The deviation from the material coefficient, calculated at ${\Re}^{2}$ maxima, can be estimated as ±4% by looking at the second period in Figure 8b (7 < w/t < 8).

#### 3.2. Dielectric Properties

_{11}

^{S}obtained by the iterative method (Figure 9a) and the recalculated permittivity (Figure 9b) depend on the values of the admittance at the four frequencies used for the calculation. Therefore, the complex ε

_{11}

^{S}is strongly affected by the coupling of modes that determines the shape of the spectrum near coupling (Figure 5).

_{11}

^{S}(Figure 9b), but still obtain a broad dispersion of the real part of the permittivity value, due to the changes in the shape of R and G curves in the shear mode, when dispersion takes place near the lateral modes (Figure 6).

_{55}

^{E}, jumps in the recalculated permittivity (real part) with anomalously large values (up to 40% higher than the permittivity of the material) can also be observed at the minima of the ${\Re}^{2}$ periods (Figure 9b). From there until the next jump, the recalculated ε

_{11}

^{S}slightly decreases as the frequency increases.

_{11}

^{S}shows anomalies with minimum values at the mentioned frequency jumps, meaning an increase in the dielectric loss tangent. As with the recalculated s

_{55}

^{E}″, the dispersion of ε

_{11}

^{S}″ in the first semi period, marked with red squares in Figure 9, decreases when recalculated. Even so, for the best case, the dispersion is of ±10% of the imaginary part of the material coefficient calculated for ${\Re}^{2}$ maxima.

#### 3.3. Piezoelectric Properties

_{15}′ by the automatic iterative method (Figure 10a), was overestimated due to the erroneous selection of the frequencies of the shear mode near the coupling with a lateral mode, similarly to what happened with the coupling factor (Figure 3b). Recent FEA modeling [25] shows that, under coupling, the mode of movement of the non-standard shear plate undergoes a dynamic clamping and the plate vibrates inhomogeneously. Therefore, the resonator is not amenable for characterization and those calculated coefficients for very low ${\Re}^{2}$ lack physical meaning. Similar to the coupling factor (Figure 7b), after the recalculation with the modified method, e

_{15}′ exhibits minima in the aspect ratio with the highest coupling (Figure 10b) and lowest ${\Re}^{2}$, which is meaningful. This also takes place in narrower ranges of w/t. Moreover, the deviation from the material coefficient of the recalculated e

_{15}′ is lower. Superimposed with this periodic evolution, in a similar manner to the coupling factor (Figure 3b), a continuous decrease in the recalculated e

_{15}′ as a function of the aspect ratio and frequency (Figure 10b) is observed. This decrease as w/t increases is characteristic of BCZT [17], and was not observed for PZT [16], which is out of the scope of this work.

_{15}″ became wide peaks when recalculated by the modified method. These peaks are wider than for any other parameter analyzed here and, again, their maxima took place together with the ${\Re}^{2}$ minima (Figure 10b). The deviations from the material e

_{15}″ are higher than for any of the other studied parameters. To consider only those calculations for ${\Re}^{2}$ > 0.80, it is necessary to reduce the dispersion of e

_{15}″ to close to 100% of the material coefficient that is calculated for the maxima of ${\Re}^{2}$. Consequently, a small deviation from this condition in the maxima of ${\Re}^{2}$ is critical for the determination of the shear electromechanical losses of the material.

## 4. Conclusions

_{0.85}Ca

_{0.15}Ti

_{0.90}Zr

_{0.10}O

_{3}plates were considered in this analysis, which required measurements on samples of different dimensions to properly identify the resonance modes. For this, impedance curves were obtained as a function of the increasing aspect ratio (length for excitation (w)/thickness for poling (t)), by reducing the plate thickness. It is concluded that the origin of the anomalies in the material parameters at the intervals of very low ${\Re}^{2}$ is mainly a consequence of the incorrect selection of the frequencies for the shear mode used to solve Equation (4). This would also affect any other calculation based on such frequencies [7,8]. To avoid this, the iterative method was here modified to allow the expert analysis of shear modes when coupling with the periodic lateral modes takes place. The modified software allows graphically determining the R or G, or even both, values for the maxima when double peaks are measured and, therefore, gives the operator the choice of the proper frequencies that will be used in the calculation using Equation (4) and the iterative procedure.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The original ceramic disk; (

**b**) the shear plate cut from the disk and then thickness (t) poled (after poling electrodes were removed and new ones attached); (

**c**) the shear plate excited in the length between electrodes (w) of area txL.

**Figure 2.**Impedance spectra for the shear mode of the Ba

_{1−x}Ca

_{x}Ti

_{0.90}Zr

_{0·10}O

_{3}(BCZT) plate for aspect ratios (

**a**) w/t = 5.68; (

**b**) w/t = 7.47. Measured G and R peaks (symbols) and reconstructed peaks (solid lines) are displayed.

**Figure 3.**Evolution as a function of the aspect ratio of the BCZT plate: (

**a**) of the reconstructed to the experimental resonance spectra regression factor, ${\Re}^{2}$, (the points marked with (*) correspond to the spectra of Figure 2, which are examples of uncoupled shear modes with high ${\Re}^{2}$, and Figure 4, examples of coupled modes with low ${\Re}^{2}$) and (

**b**) of the frequencies fp and fs and the electromechanical coupling factor, k

_{15}; squares indicate anomalies in the difference (fp − fs).

**Figure 4.**Impedance spectra showing coupled modes (low ${\Re}^{2}$) for the BCZT plate for aspect ratios (

**a**) w/t = 6.52; (

**b**) w/t = 8.61. Experimental G and R peaks (symbols) are displayed, as well as reconstructed peaks (solid lines).

**Figure 5.**Impedance spectra showing coupled modes (very low ${\Re}^{2}$) for the BCZT plate for aspect ratios (

**a**) w/t = 6.60; (

**b**) w/t = 6.89. Experimental G and R peaks (symbols) are displayed, as well as reconstructed peaks (solid lines).

**Figure 6.**Impedance spectra showing coupled modes for the BCZT plate for aspect ratios (

**a**) w/t = 6.60; (

**b**) w/t = 6.89. Experimental G and R peaks (symbols), also shown in Figure 4, together with the reconstructed peaks using the material coefficients determined by the modified automatic iterative method (solid lines), are displayed.

**Figure 7.**Evolution as a function of the aspect ratio of the BCZT plate: (

**a**) of the regression factor, ${\Re}^{2}$, reconstructed to the experimental shear resonance spectra (the points marked with (*) correspond to the spectra of Figure 4 and Figure 6, which are examples of coupled modes with low (black) and very low (red) ${\Re}^{2}$, respectively) and (

**b**) of the frequencies fp and fs and the electromechanical coupling factor, k

_{15}; all parameters of coupled modes were recalculated by the modified automatic iterative method.

**Figure 8.**Evolution of the complex elastic compliance s

_{55}

^{E}as a function of the aspect ratio of the BCZT plate; the real and the imaginary parts (s

_{55}

^{E}′ and s

_{55}

^{E}″) are both plotted for: (

**a**) calculation by the automatic iterative method; (

**b**) recalculation by the modified method. The evolution of ${\Re}^{2}$ is shown as a reference.

**Figure 9.**Evolution of the complex dielectric permittivity ε

_{11}

^{S}as a function of the aspect ratio of the BCZT plate; the real and the imaginary parts (ε

_{11}

^{S}′ and ε

_{11}

^{S}″) are both plotted: (

**a**) as-calculated by the automatic iterative method; (

**b**) as-recalculated by the modified automatic iterative method. The evolution of ${\Re}^{2}$ is shown as a reference.

**Figure 10.**Evolution of the complex piezoelectric coefficient e

_{15}as a function of the aspect ratio of the BCZT plate; the real and the imaginary parts (e

_{15}′ and e

_{15}″) are both plotted: (

**a**) calculated by the automatic iterative method; (

**b**) recalculated by the modified automatic iterative method. The evolution of ${\Re}^{2}$ is shown as a reference.

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**MDPI and ACS Style**

Pardo, L.; Reyes-Montero, A.; García, Á.; Jacas-Rodríguez, A.; Ochoa, P.; González, A.M.; Jiménez, F.J.; Vázquez-Rodríguez, M.; Villafuerte-Castrejón, M.E.
A Modified Iterative Automatic Method for Characterization at Shear Resonance: Case Study of Ba_{0.85}Ca_{0.15}Ti_{0.90}Zr_{0.10}O_{3} Eco-Piezoceramics. *Materials* **2020**, *13*, 1666.
https://doi.org/10.3390/ma13071666

**AMA Style**

Pardo L, Reyes-Montero A, García Á, Jacas-Rodríguez A, Ochoa P, González AM, Jiménez FJ, Vázquez-Rodríguez M, Villafuerte-Castrejón ME.
A Modified Iterative Automatic Method for Characterization at Shear Resonance: Case Study of Ba_{0.85}Ca_{0.15}Ti_{0.90}Zr_{0.10}O_{3} Eco-Piezoceramics. *Materials*. 2020; 13(7):1666.
https://doi.org/10.3390/ma13071666

**Chicago/Turabian Style**

Pardo, Lorena, Armando Reyes-Montero, Álvaro García, Alfredo Jacas-Rodríguez, Pilar Ochoa, Amador M. González, Francisco J. Jiménez, Manuel Vázquez-Rodríguez, and María E. Villafuerte-Castrejón.
2020. "A Modified Iterative Automatic Method for Characterization at Shear Resonance: Case Study of Ba_{0.85}Ca_{0.15}Ti_{0.90}Zr_{0.10}O_{3} Eco-Piezoceramics" *Materials* 13, no. 7: 1666.
https://doi.org/10.3390/ma13071666