MODIFIED ELECTRODE SHAPE FOR THE IMPROVED DETERMINATION OF PIEZOELECTRIC MATERIAL PARAMETERS

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1 MODIFIED ELECTRODE SHAPE FOR THE IMPROVED DETERMINATION OF PIEZOELECTRIC MATERIAL PARAMETERS CARSTEN UNVERZAGT, JENS RAUTENBERG, BERND HENNING University of Paderborn, Measurement Engineering Group, Warburger Strasse 1, 3398 Paderborn, Germany KSHITIJ KULSHRESHTHA University Paderborn, Institute of Mathematics, Warburger Strasse 1, 3398 Paderborn, Germany ABSTRACT: Piezoceramic materials are widely used in ultrasonic devices. To reduce the development time of piezoelectric sensors or actuators numerical methods like the finite element method (FEM) are often applied (e.g. Lahmer et al. (28)). Therefore a precise knowledge of the piezoelectric material parameters is required to obtain a realistic simulative representation of the real piezoelectric material behavior. There are several inverse identification methods using a comparison between simulated and measured electrical impedance characteristics in order to determine and to optimize the piezoelectric material parameter set (e.g. Kybartas (22), Rupitsch (29)). It was shown that the electrical impedance curve is influenced differently by all these material parameters. For example the electrical impedance is less sensitive to some of the material parameters (e.g. c44e) if the used specimen has symmetrical properties or dimensions (e.g. Rautenberg et al. (2)). Hence the goal of this work is the modification of the electrode shape in order to increase the influence of all material parameters on the electrical impedance characteristic to get more accurate values for these less sensitive and therefore critical material parameters. To achieve this goal a variable FEM-model for the piezoceramic disc is used. The influence of the electrode shape on the sensitivity of material parameters on the electrical impedance curve is investigated. An adapted electrode shape is determined to increase the sensitivity of the electrical impedance curve particularly concerning the selected critical material parameters. INTRODUCTION The investigated piezoelectric ceramic material can be represented by the material parameter matrices shown in Eq. (1). There is a clearly visible symmetry in the matrices which reduces the material parameters to be determined. In the stiffness matrix C there are five independent mechanical parameters, the piezoelectric coupling parameters e contain three and the dielectric terms ε consist of two different parameters, summarized there are ten independent parameters. The density ρ and the Rayleigh damping parameters α M und α K are not considered in this contribution. The influence of critical piezoelectric material parameters on the electrical impedance curve is increased by changing the electrode shape. The changed electrode topology (Fig. (1a)) in conjunction with a suitable electrical impedance (Fig. (1b)) leads to stronger radial electric field components and therefore to an increased sensitivity to the material parameters which usually have little influence on the electrical impedance curve for discs with full-surface electrodes. The mentioned impedance Z leads to a desired voltage ratio for the different electrodes. The shown Edited by Gan Woon Siong, Lim Siak Piang and Khoo Boo Cheong. Copyright c 213 ICU Organisers. All rights reserved. Published by: Research Publishing ISBN: :: doi:1.385/ P

2 results are derived with a PIC255 rotationally symmetrical piezoceramic disc with a diameter of D = 7, 4 mm and a thickness of t= 2,1 mm. ª c «c «12 «c C = «13 ««««c12 c13 c c13 c13 c33 c44 c44 º»» ª» ; e = «e31»» c66»¼ e15 e31 e33 e15 º º ªε» ; ε = «ε»» ¼ ε 33»¼ (1) Concentric ring electrodes are used as modified electrodes to show the different influences of the piezoelectric material parameters on the electrical impedance curves. The chosen dimensions are indicated in Fig. (1a). Using this approach a two-dimensional rotationally symmetrical simulation model can be created to calculate the impedance curve of the piezoceramic disc with the finite element method (FEM). The used model is described in the following chapter. Figure 1. (a) Dimensions of the piezoceramic disc and their concentric ring electrodes (bottom side: full-surface electrode) with an illustration of the spatial discretization for the frequency f max. (b) Electrical circuit for the measurement of impedance curves. SIMULATION MODEL As mentioned above, the ring electrodes are chosen rotationally symmetrical and a simplified twodimensional FEM model can be used to calculate the electrical impedance for the piezoceramic disc. The geometry of the variable simulation model as well as the simulation parameters and boundary conditions are created within a Matlab script. For this contribution, the influence of varied material parameters on the electric impedance curve is investigated for two different electrode configurations to show the increased sensitivity. As an example the comparison between a full-surface electrode and a ring electrode configuration is shown in the chapter results and discussion. The spatial discretization of the FEM model is set to 16 points per wavelength (see Fig. 1a). The impedance simulations are accomplished for 25 frequencies in the range from 5 khz to 2,5 MHz as a compromise between the calculation time and the precision of the impedance curve. For each frequency reference point, a harmonic simulation is calculated to get the complex impedance 759

3 value. The impedance curves in the results chapter only show the absolute values of these curves in combination with the derived distance d, which is described in the following chapter. ANALYSIS ALGORITHM The variation of four out of the above mentioned ten material parameters is chosen to demonstrate the change in sensitivity to the impedance curve exemplary: c 33, c 44, e 15 and ε. An excerpt of such a parameter variation is shown in Fig. (3). For these five impedance curves, one of the four parameters is varied between 9 % and % in 5 % steps of the initial value. In general there are two effects on the impedance curve, a frequency dependent shift in the direction of the frequency axis f and another one in the direction of impedance Z. For an automated analysis of these sensitivities and to obtain a scalar representation as a function of frequency, an approximation of derivatives of discretely sampled data is applied with the use of first-order directional derivative kernels. These are optimized for the multi-dimensional differentiation of two-dimensional discrete points. The approximated derivatives are shown in Eq. (2) where x denotes one of the four parameters. Δ Z Z f f Δ Z u u = ; v = = Δx = x x x Δ Z Δ Z vh Δf u 1 1 = v v h h vh 8 vh sign 1 u + u u (2) (3) Figure 3. Exemplary impedance curves for a parameter variation of x with simulated (red) and predicted (cyan) points (top) and the calculated distance d between both (bottom). The first summand in the denominator in Eq. (3) is used to bypass the occurring zero division. The 76

4 approximated derivations are used to predict the impedance curve for a given change of x as a verification. In Fig. (3) the red dots represent the simulated impedance values with the initial parameter set and the cyan points are the predicted points for a 5 % change of x. They lie on the cyan simulated curve for that parameter change in most cases and can be used to express the sensitivity as a function of frequency. To get a scalar value the distance d is defined as the magnitude of the prediction vector. The distance d is visualized in the bottom of Fig. (3). RESULTS AND DISCUSSION Figure 4. Impedance curves for full-surface electrodes (left) and ring electrodes (right) for the variation (9%, 95%, 1%, 15% and % of the initial value) of the exemplary mechanical material parameters c 33 ((a) and (b)) and c 44.((c) and (d)). Below each impedance curve the distance d is presented as a criterion for the sensitivity of the material parameter on the impedance curve. 761

5 In this chapter the results for the four parameters c 33, c 44, e 15 and ε are shown. Each subfigure consists of the simulated impedance curves (top) and the derived distance d (bottom) for a parameter variation of 5 %. The left subfigures are the results for the full-surface electrodes and on the right there are the simulation results for the ring electrodes. Two different parameters are taken from the stiffness matrix ( c 33 and c 44 ) and the results are visualized in Fig. (4). The results for the piezoelectric coupling parameter e 15 are presented in Fig. (5) and the results for the dielectric parameter ε are shown in Fig. (6). Figure 5. Impedance curves (top) and distance d (bottom) for a full-surface electrode (a) and the ring electrodes (b) for the variation of the piezoelectric coupling parameter e 15. Figure 6. Impedance curves (top) and distance d (bottom) for a full-surface electrode (a) and the ring electrodes (b) for the variation of the dielectric parameter ε. 762

6 The distance d as the magnitude of the prediction vector for each simulated frequency is a measure for the sensitivity of different material parameters on the impedance curve. The comparison between piezoceramic discs with full-surface electrodes and ring electrodes shows an increase in the sensitivity for the above mentioned material parameters. For example there is a significant sensitivity only in the frequency range between,5 MHz and 1,5 MHz for the parameter c 44 at piezoceramic discs with full-surface electrodes (Fig. 4c). If the electrodes are modified and the ring electrodes are used (Fig. 4d), the sensitivity spread more evenly over the whole considered frequency range from 5 khz up to 2,5 MHz. This behavior can be observed for all four simulated parameter variations (Fig. 4a-b, 5). Especially the very small sensitivity for the parameter ε (Fig. 6a) can be increased in the desired frequency range. Using this knowledge and an inverse parameter optimization algorithm, a more accurate parameter set can be calculated. OUTLOOK The next step will be an optimization of the electrode shapes to reach a high sensitivity for every searched material parameter without the loss of sensitivity for well determinable parameters. Besides a high sensitivity the evenness of the sensitivity as a function of the frequency will be of interest. Further developments will use the increased and optimized sensitivity and the new derived parameter distance d for the improvement of the inverse algorithms for the piezoelectric material parameter identification. The determination of the previously inaccurate material parameters will be improved and a more representative piezoelectric material parameter set will be determined. REFERENCES Kybartas, D.; Lukosevicius, A. (22). Determination of piezoceramics parameters by the use of mode interaction and fitting of impedance characteristics; In: Ultragarsas 45 (4), S Lahmer, T., Kaltenbacher, M., Kaltenbacher, B., Lerch, R., et al. (28). FEM-Based Determination of Real and Complex Elastic, Dielectric, and Piezoelectric Moduli in Piezoceramic Materials; IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control: vol. 55, no. 2 Rautenberg, J., Rupitsch, S., Henning, B., Lerch, R. (2). Utilizing an Analytical Approximation for c44e to Enhance the Inverse Method for Material Parameter Identification of Piezoceramics; 7th International Workshop on Direct and Inverse Problems in Piezoelectricity, , Duisburg Rupitsch, S., Lerch, R. (29). Inverse Method to estimate material parameters for piezoceramic disc actuators; Appl. Phys. A, vol. 97, no. 4, pp