Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 19, ICA 213 Montreal Montreal, Canada 2-7 June 213 Engineering Acoustics Session 3aEA: Computational Methods in Transducer Design, Modeling, Simulation, and Optimization II 3aEA9. Accurate determination of piezoelectric ceramic constants using a broadband approach Nicolas Perez, Marco Aurelio B. Andrade, Ronny C. Carbonari, Julio C. Adamowski and Flavio Buiochi* *Corresponding author's address: Mechatronics Engineering, University of Sao Paulo, Sao Paulo, 558-3, SP, Brazil, fbuiochi@usp.br Piezoceramic property values are required for modeling piezoelectric transducers. Most datasheets present large variations in such values. For precise simulations, adjustments are necessary. Recently the authors presented a methodology to obtain the real part of ten material constants of piezoelectric disks. It comprises four steps: experimental measurements, identification of vibration modes and their sensitivity to material constants, preliminary identification algorithm, and final refinement of the constants using an optimization algorithm. Given an experimental electrical impedance curve of a piezoceramic and a first estimate for the material constants, the objective is to find the constants that minimize the difference between the experimental and numerical curves. Using a new finite element method routine implemented in Matlab, the original methodology was extended to obtain the corresponding imaginary part of all the material constants. Results of sensitivity analysis for the imaginary part and the guidelines to construct an algorithm are presented. This complex model allows adjusting the amplitude over a wide frequency range, as opposed to the models described in the literature. It is applied to 1-MHz APC85 disks with diameters of 1 and 2mm. The methodology was validated by comparing the numerical displacement profiles with the displacements measured by a laser Doppler vibrometer. Published by the Acoustical Society of America through the American Institute of Physics 213 Acoustical Society of America [DOI: / ] Received 22 Jan 213; published 2 Jun 213 Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 1

2 INTRODUCTION The knowledge of the material constants of piezoelectric materials is required for the accurate modeling of piezoelectric transducers and devices. A 6mm symmetry class piezoceramic, polarized in a given direction, can be described by five elastic, three piezoelectric and two dielectric constants. The experimental determination of these constants requires the fabrication of special test samples [1], thus being expensive and time consuming. The model described in [1] assumes a linear behavior and a lossless material. This model well established to determine real constants, but not to determine complex constants. One of the numerical techniques used by several researchers to determine the material constants of piezoelectric ceramics consists in solving an inverse problem. Given an experimental electrical impedance curve of a piezoceramic and a first estimate for the material constants, the objective is to find the material constants that minimize the difference between the experimental and numerical electrical impedance curves. The Finite Element Method (FEM) model can be used as part of an iterative method to reach the values of the constants, until the calculated electrical impedance spectrum match the measured one [2]-[5]. In a previous work [4], an algorithm to minimize the error that takes into account a sensitivity analysis was implemented. The purpose of the sensitivity analysis is to determine the influence of each material parameter over all vibrational modes. This powerful tool was recently used by other researchers to evaluate the influence of the parameters in each frequency range of interest [6]. The commercial FEM software Ansys was used in [4] to obtain the numerical electrical impedance curve. In the Ansys software, only the real part of the material constants is considered and the material damping is calculated by using the classical Rayleigh damping model. Because differences in the amplitude of an individual mode can still exist, the previous method is now extended by including complex parameters in the model. In this work, the problem consists in determining twenty independent constants, comprising the real and imaginary parts of the ten material properties. A new FEM routine, that takes into account the complex material constants, has been fully implemented using the software Matlab. This complex model allows adjusting the amplitude over a wide frequency range, from the first radial mode to the main thickness mode. The constants predicted by the FEM model are input into a new model to obtain the theoretical displacements in the center of a ceramic. A laser vibrometer is used to experimentally measure these displacements, just so the model can be validated. METHODOLOGY Before performing simulations, the convergence of the FEM results is evaluated over the desired frequency range. The number of elements in the FEM simulations is selected to obtain a convergence deviation less than.1% in that frequency range. Considering a 2-mm-thick, 2-mm-diameter APC85 piezoceramic disk, Table 1 shows the results of the convergence analysis for five different resonance frequencies: the two first radial modes R1 (98 khz) and R2 (254 khz), the edge mode E (62 khz), the first thickness mode TH1 (1. MHz) and a coupled mode C (965 khz) near the thickness mode. The first column presents the number of elements in the thickness of the sample. The second column presents the processing time normalized to the time of the first line (which is the smallest processing time). The other columns (percentage variations of the resonant frequencies) present values relative to the last line (which is the most precise). From these results, the 25 element/thickness is selected to achieve a convergence in the order of.1%. Due to the computational cost, only axisymmetric simulations are implemented in the present methodology. TABLE 1. Convergence analysis considering a 2-mm-thick, 2-mm-diameter APC85 piezoceramic. Elements/thickness Time ΔR1 % ΔR2 % ΔE % ΔC % ΔTH1 % / Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 2

3 The methodology to determine the material properties of a piezoelectric disk is based on the one described by [4]. It consists of measuring the electrical impedance of a piezoelectric disk and using an inverse algorithm based on the FEM to minimize the error between the theoretical (FEM analysis) and experimental electrical impedance curves. This methodology was applied to obtain only the real part of the material properties. Here this methodology is first applied to obtain the real part of the material constants and then a similar strategy is applied to determine the imaginary part, which results in 2 independent material constants. This new finite element method routine was implemented in Matlab. The proposed methodology was applied to 1 MHz (thickness mode), 1- and 2-mmdiameter disk samples of APC85 (American Piezoceramics, Mackeyville, PA). The methodology can be divided into four steps. First, the electrical impedance was measured using an HP4194A impedance analyzer. In this case, 1 equally spaced points were used in the frequency range of 13kHz to 1.3MHz. The selection of the frequency range was done by taking two decades of frequencies starting bellow the first radial mode and ending above the thickness resonance frequency. The electrical impedance in an intermediate value between the first and the third harmonic of the thickness frequency (assumed as a capacitance) was used to estimate the start value of the dielectric constant ε 33, according to the IEEE standard [1]. The geometrical dimensions (diameter and thickness) and the density were also measured. As a start point for simulating the material constants of APC85, mean values for the constants were used, as shown in Table 2. These mean values were presented in a previous work [7]. At that point, the real part of each constant was identified using six different geometries to give a mean value for each constant. TABLE 2. APC85 initial material constants [7]. c E N/m 2 c E N/m 2 c E N/m 2 c E N/m 2 c E N/m 2 e 31 C/m 2 e 15 C/m 2 e 33 C/m 2 ε S 11/ ε ε S 33/ ε Second, the vibration modes were identified and their sensitivity in relation to material constants was calculated. For the sensitivity analysis, the properties were varied one at a time, while all the other parameters were kept constant. The properties of the model were changed over a wide range of values (± 25% from a start value) and the intensities (G-curve or R-curve values in decibels) were plotted as a function of frequency. The maximum of the conductance G and the resistance R correspond, respectively, to the resonance and antiresonance frequencies. Then, a map was plotted for each parameter. In the map, each horizontal line is the result of a single FEM simulation, and corresponds to one value of the parameter under study. Figure 1 presents the sensitivity analysis for three properties usually applied to one-dimensional models: c 33, e 33 and ε 33. The top map (G-curve) shows that c 33 is most sensitive around the thickness frequency (1. MHz), so this property is used to adjust around this frequency the theoretical electrical impedance curve to the one obtained experimentally. Similarly, the middle and bottom maps (R-curves) show that e 33 and ε 33 are sensitive in the antiresonance of the thickness mode. Table 3 presents a summary of the results of the sensitivity analysis, which are the guidelines to construct a first algorithm to adjust the resonances. FIGURE 1. Sensitivity analysis of the 2-mm-thick, 2-mm-diam. APC85 piezoceramic. (A) electrical conductance G as a function of frequency for c 33, (B) and (C) electrical resistance R as a function of frequency for e 33 and ε 33, respectively. Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 3

4 Third, a preliminary identification algorithm was developed using the sensitivity analysis results. The algorithm is based in the minimization of the distance between the resonant frequencies. It starts with a preliminary set of material constants, which can be obtained from the manufacturer or from a piezoelectric material with similar characteristics. The solution is approximate and only adjusts the most sensitive parameters, but the convergence is robust and gives a very good starting point to other refinement method. This algorithm was constructed to obtain a solution close to the real solution avoiding local minimums [4]. TABLE 3. Sensitivity analysis results for the 2-mm-thick, 2-mm-diameter APC85 piezoceramic. Parameter Radial mode Coupled mode Thickness mode c 11 High, + slope High, + slope No influence c 12 Low, + slope No Influence No influence c 13 High, slope High, slope No influence c 33 High, + slope High, + slope High, + slope c 44 No influence High, + slope Low, + slope e 13 Low, + slope Low, + slope No influence e 15 No influence High, slope No influence e 33 Low, + slope Low, + slope High, + slope 11 No influence Low, slope No influence 33 Low, slope Low, slope High, slope Last, the Nelder-Mead optimization algorithm [8], implemented in Matlab through the fminsearch function, was used to achieve a final refinement of the material constants. The initial conditions to start this optimization are the parameters obtained using the preliminary identification algorithm described above. Figure 2 shows the results obtained after the final refinement of the material properties, considering only the real part of the constants. All resonances in the theoretical and in the experimental curves agree in the selected frequency range but the amplitudes do not match due to the absence of dumping in the model. This problem can be solved by using the imaginary part of the complex constants. 1 5 Z [ ] 1 R1 R2 E C TH1 1 Phase [Deg] FIGURE 2. Final refinement results for the 2-mm-thick, 2-mm-diam. APC85 piezoceramic. Adjustment of the electrical impedance using only the real part of the complex constants. Theoretical data is plotted in continuous line and experimental data is plotted in dotted line. To adjust the imaginary part of the model the strategy is similar to the implemented for the real part: making a sensitivity analysis, implementing a first algorithm to adjust the most sensitive parameters and refining the parameters using a standard algorithm like the Nelder-Mead. However, the sensitivity analysis for the imaginary part is quite different from that shown in Figure 1, due to the fact that the frequency of the modes remains unchanged. In a mass-spring-dumper system, the resonance frequency depends on the parameter of the dumper, but in the case of the piezoceramic model, the dumper term is negligible in the practical range of values. To give a global idea of the sensitivity for one parameter, the results for c 33 are plotted in Figure 3. In this figure, the maximum of each resonance follows a vertical straight line, verifying the hypothesis that the frequency does not change. The reference values (% change) are the initial parameters (only the imaginary part). These parameters are manually selected by trial and error, such that the solution is close to the experimental data. Changing one parameter at a time (in this Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 4

5 case, the c 33 ), one is able to identify the most sensitive modes. This graphical image can be complemented plotting the evolution of each maximum of the G curve against the variation of the selected parameter, as shown in Figure 4. Figure 4 clearly shows if the variation of the parameter increases or decreases the local maximum of the G-curve. TH1 1-1 G [S] 1-2 R1 R2 E C 1-3 FIGURE 3. Sensitivity analysis of the imaginary part of the complex constant c 33. G-curves simulated considering the 2-mmthick, 2-mm-diam. APC85 piezoceramic. In this example, only one constant (c 33 ) is changed from -8% to 2% in steps of 2%. Each curve corresponds to one value of the parameter and the local maximums are indicated with dots. 2 1 R1 G [%] C R2 E c 33 [%] FIGURE 4. Evolution of maximums of the electrical conductance G in the sensitivity analysis of c 33 considering the 2-mm-thick, 2-mm-diam. APC85 piezoceramic. The results are expressed as a percentage of the first value of each curve. For a first approach to the solution an iterative algorithm based in this sensitivity analysis is constructed. Table 4 summarizes the sensitivity result for the imaginary part of the ten constants relative to five different resonance frequencies (already listed in Table 1, and presented in Figures 2 and 3). The maximum percentage change is listed for each parameter and for the five modes considered in this example. Also, the evolution of the maximum of G is indicated with the slope (+ means that the maximum of G at the resonance increases with the parameter, means that it decreases and means that the slope changes its sign). TABLE 4. Sensitivity analysis of the imaginary part of the complex constants considering the 2-mm-thick, 2-mm-diameter APC85 piezoceramic. Parameter ΔR1 % ΔR2 % ΔE % ΔC % ΔTH1 % c ( slope) 68 ( slope) 72 ( slope) 55 ( slope) 25 ( slope) c 12 6 (+ slope) 2 ( slope) 1 ( slope) <.1 ( slope) <.1 ( slope) c 13 5 ( slope) 55 (+ slope) 72 (+ slope) 27 (+ slope) 7 (+ slope) c ( slope) 36 ( slope) 52 ( slope) 21 ( slope) 62 ( slope) c 44 <.1 ( slope) <.1 ( slope) 52 ( slope) 36 ( slope) 29 ( slope) e 31.6 (+ slope).2 (+ slope) 2.5 (+ slope) 4 (+ slope) 1.6 (+ slope) e 33 <.1 (+ slope) <.1 (+ slope) <.1 (+ slope) <.1 (+ slope) <.1 (+ slope) e 15 5 ( slope) 9 (+ slope) 54 (+ slope) ε 11 <.1 ( slope) 17 ( slope) 46 ( slope) 47 ( slope) ε 33 1 ( slope) 2 ( slope) 5 ( slope) TH1 Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 5

6 Using these guidelines, a closed loop to iteratively adjust the imaginary part of the elastic constants is implemented. In this algorithm, the parameters c 11, c 12, c 13, c 44, and c 33 adjust, respectively, the resonance frequency modes R2, R1, C, E and TH1. In each step k the next value of a parameter p is calculated as ( +1) = ( ) +, (1) where α is a coefficient based on the sensitivity analysis and Error is the difference in the phase curve around the selected resonance. During the optimization process, the parameters approach to an equilibrium value and the error between the numerical and the experimental data decreases. The global error, calculated as the difference in absolute value between the experimental and the numerical phase, decreases monotonically. Once the properties were refined, new simulations (harmonic analysis) were carried out resulting in numerical displacement profiles. The validation of the methodology was done by comparing the displacement profiles obtained by FEM with the displacements measured by a laser Doppler vibrometer (Polytec, Waldbronn, Germany). The Polytec vibrometer system consists of OFV-5 vibrometer controller, DD-9 digital displacement decoder, and OFV-534 compact sensor head. An impedance analyzer was used to sweep the frequency over a range from 13kHz until 1.3MHz. The output of the impedance analyzer was amplified by a broadband power amplifier AR 15A1A (Souderton, PA), using the minimum gain to protect the system. The experimental setup (Figure 5) allows for measuring the displacement of the ceramic surface as a function of frequency. The displacement was measured in the central point of both the 1-mm-diameter and the 2-mm-diameter ceramics. The experimental displacements were normalized by dividing the absolute value measured (reference channel of the impedance analyzer) by the voltage applied (test channel of the impedance analyzer). This normalization is necessary because the harmonic analysis considers a driving voltage of 1V applied to the ceramic. All the data was stored in a computer for further processing in Matlab. Computer HP4194A Impedance/Gain-Phase Analyzer output Ref. Test Polytec Vibrometer System Laser Sensor Head Power Amplifier AR 15A1A Piezoelectric disk (APC85) FIGURE 5. Experimental setup for measuring the displacements of the piezoelectric ceramic surfaces as a function of frequency. RESULTS Using the Nelder-Mead algorithm, the parameters e 31, e 33, e 15, ε 11 and ε 33 are refined in the frequency range from the E mode to the TH1 mode. Table 5 presents the complete set of parameters obtained for the two 2-mm-thick APC85 samples (diameters of 2 and 1 mm). TABLE 5. Real and imaginary parts of the ten complex constants adjusted for the APC85 piezoceramic samples (diameters of 2 and 1 mm). Parameter Real (D = 2 mm) Imaginary (D = 2 mm) Real (D = 1 mm) Imaginary (D = 1 mm) c 11 (N/m 2 ) x x x x 1 8 c 12 (N/m 2 ) 8.26 x x x x 1 8 c 13 (N/m 2 ) 8.72 x x x x 1 7 c 33 (N/m 2 ) x x x x 1 8 c 44 (N/m 2 ) 2. x x x x 1 8 e 31 (C/m 2 ) x x 1-3 e 33 (C/m 2 ) x x 1-2 e 15 (C/m 2 ) x x 1-2 ε 11 (F/m) 1.44 x x x x 1-11 ε 33 (F/m) 6.99 x x x x 1-12 Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 6

7 Figures 6 and 7 show the electrical impedance results in the two selected samples (2-mm-diam. and 1-mmdiam., respectively). Theoretical and experimental curves are superposed for both magnitude and phase. The electrical impedance curves are quite well adjusted in the whole frequency range. 1 5 Z [ ] 1 Phase [Deg] FIGURE 6. Optimization for the 2-mm-diameter APC85 ceramic disk. The numerical results for magnitude (top) and phase (bottom) of the electrical impedance are presented in continuous line and the experimental results are presented in contiguous circles. 1 5 Z [ ] Phase [Deg] FIGURE 7. Optimization for the 1-mm-diameter APC85 ceramic disk. The numerical results for magnitude (top) and phase (bottom) of the electrical impedance are presented in continuous line and the experimental results are presented in contiguous circles. In order to verify the validity of the numerical solution, the theoretical and experimental axial displacements in a central point of the piezoceramic disk are compared. Both the theoretical and the experimental displacements were acquired over a frequency range from 13kHz until 1.3MHz. Figures 8 and 9 show the comparison in the selected frequency range for the 2-mm-diameter and the 1-mm-diameter samples. The comparisons between the numerical and experimental displacement profiles show a reasonable agreement. Some situations might be preventing the occurrence of a better agreement: (a) the difficulty of manually adjusting the laser beam focusing exactly the center of the ceramic for the experimental measurements; (b) the difficulty of perfectly reproducing the boundary conditions of the theoretical modeling in the experimental measurements (e.g. the ceramic was held during the experiments whereas the model considered a free border); (c) the difference in the values obtained for the imaginary parts of the two samples (Table 5). At the moment, these situations are under investigation. Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 7

8 3 25 Amplitude [nm] FIGURE 8. Experimental verification using laser vibrometer for D =2mm, thickness = 2mm. The numerical results of the displacements in the center of the piezoceramic are presented in continuous line and the experimental results are presented in contiguous circles Amplitude [nm] FIGURE 9. Experimental verification using laser vibrometer for D =1mm, thickness = 2mm. The numerical results of the displacements in the center of the piezoceramic are presented in continuous line and the experimental results are presented in contiguous circles. CONCLUSION The complex constants (real and imaginary parts) of the APC85 piezoceramic were determined. An algorithm that was able to establish only the real part of the complex constant was improved to find the imaginary part as well. This was done by implementing a new FEM routine into the Matlab software. Here the methodology was applied to axisymmetric simulations. However, other bi-dimensional models (like plane strain or plane stress) and even threedimensional models can be used to determine the properties of the material to be modeled. The electrical impedance of the piezoceramic was measured experimentally and an initial set of material properties were input into the algorithm. The theoretical electrical impedance curves generated by the algorithm and the experimental data showed quite good agreement in the whole frequency range. Thus, the constants seem to be well adjusted. The methodology was validated by comparing the theoretical displacement profiles obtained by the FEM with the displacements measured by the laser vibrometer. This comparison showed a satisfactory agreement. This way the results were verified in an independent experiment. Differences were found in the values determined for the two samples concerning the imaginary parts. These differences might be attributable to the fact that the algorithm still requires improvement. The authors are currently working for the algorithm to reach a better performance. Moreover, the experimental setup needs to be rearranged in order to better match the theoretical assumptions. Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 8

9 ACKNOWLEDGMENTS The authors thank the Brazilian funding agencies FAPESP and PETROBRAS/ANP, and the Uruguayan funding agencies PEDECIBA, CSIC and ANII. The authors also thank APC International for providing the samples used in this work. REFERENCES 1. ANSI/IEEE Standard on Piezoelectricity (ANSI/IEEE Standard ), IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, (1996). 2. H.-W. Joo, C.-H. Lee, J.S. Rho, and H.-K. Jung, Identification of material constants for piezoelectric transformers by threedimensional, finite-element method and a design-sensitivity method, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 5, (23). 3. T. Lahmer, M. Kaltenbacher, B. Kaltenbacher, R. Lerch and E. Leder, FEM-based determination of real and complex elastic, dielectric, and piezoelectric moduli in piezoceramic materials, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, (28). 4. N. Pérez, M. A. B. Andrade, F. Buiochi, and J. C. Adamowski, Identification of elastic, dielectric, and piezoelectric constants in piezoceramic disks, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57, (21). 5. U. G. Jonsson, B. M. Andersson, and O. A. Lindahl, A FEM-based method using harmonic overtones to determine the effective elastic, dielectric, and piezoelectric parameters of freely vibrating thick piezoelectric disks, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 6, (213). 6. S. J. Rupitsch, F. Wolf, A. Sutor, and R. Lerch, Reliable modeling of piezoceramic materials utilized in sensors and actuators, Acta Mech 223, (212). 7. N. Pérez, F. Buiochi, M. A. B. Andrade, and J. C. Adamowski, Numerical characterization of soft piezoelectric ceramics, AIP Conf. Proc. 1433, (212). 8. J. A. Nelder and R. Mead, A simplex-method for function minimization, Comput. J. 7, (1965). Proceedings of Meetings on Acoustics, Vol. 19, 371 (213) Page 9