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Determination of piezoelectric, dielectric and elastic complex coefficients in the linear range from the electromechanical resonance modes of poled ferroelectric ceramics

Automatic iterative method

Programs to Download

Matrix characterization

For the fundamentals and definitions of the resonance method for piezoceramic characterization in the linear range the user of this freeware is addressed to the Standars in the matter (“IEEE Standard on piezoelectricity”. ANSI/IEEE Std. 176-1987, and “Piezoelectric properties of ceramic materials and components. Part 2: methods of measurement – Low power”. European Standard CENELEC, EN 50324-2.).

The freeware that can be dowloaded in this site offers a solution to the limitations of the Standard calculation method concerning characterization of high loss and low sensitivity ferro-piezoelectric ceramic materials.

These programs allows to determine a number of material coefficients in complex form, thus including losses, from the measurement of the frequency dependence of the complex impedance at the electromechanical resonance modes of ferro-piezoelectric ceramics with given geometries.

The material data is here determined by solving a set of non-linear equations that results when experimental impedance data at a number of frequencies are introduced into the appropriate analytical solution of the wave equation for each electromechanical resonance mode. This set of equations is established for as many frequencies, which are automatically selected by the program, as unknown coefficients. Solution is carried out by an iterative numerical method described in:

• C. Alemany, L. Pardo, B. Jiménez, F. Carmona, J. Mendiola and A.M. González. "Automatic iterative evaluation of complex material constants in piezoelectric ceramics". J. Phys. D: Appl. Phys. 27, 148-155 (1994),
  

and

• C. Alemany, A.M. González, L. Pardo, B. Jiménez, F. Carmona and J. Mendiola. “Automatic determination of complex constants of piezoelectric lossy materials in the radial mode”. J. Phys. D: Appl.Phys. 28, 945-956 (1995).
  

Such analytical solutions are valid for sample geometries with given aspect ratios, which allows exciting uncopled modes. Regular sample geometries are recommended for the use of this characterization method and extreme care shall be taken to determine accurately the dimensions and density of the samples, required for the accurate determination of the complex material parameters. For error sensitivity calculations, see: M. Algueró, C. Alemany, L. Pardo and A.M.Gonzalez. “Method for obtaining the full set of linear electric, mechanical and electromechanical coefficients and all related losses of a piezoelectric ceramic”. J. Am. Ceram. Soc. 87(2) 209 (2004).

A loss factor is calculated and displayed for each material complex parameter P*= P´- i P´´, defined as Qi(P) = P´/P´´(i=p for piezoelectric, i=m for elastic and i=e for dielectric coefficients).

This software also carries on the reconstruction of the spectra (R and G versus frequency curves, where Y=G+iB=1/Z = 1/(R+iX)) using the above mentioned analytical expression and the material parameters obtained. Reconstructed curves are plotted together with the experimental ones as an accuracy test of the final set of calculated coefficients. This accuracy is also quantitatively characterized by the regression factor (R2) of such reconstruction to the experimental data.

In order to test the frequency dependence of the material parameters, this software allows calculation of the coefficients, not only for the fundamental resonance, but also for the overtones, taking place at odd multiples of the fundamental frequency (A.M. Gonzalez and C. Alemany.”Determination of the frequency dependence of characteristic constants in lossy piezoelectric materials”. J. Phys. D: Appl.Phys. 29, 2476-2482 (1996)). To do so, information on the measured overtone is asked when the data file used for the calculation is created.

Though developed also for other geometries, here we made avaliable the programs made by C. Alemany et al. for the following four resonance modes:

• Length extensional mode of long rods of bar, poled and excited along their length
  download
• Thickness extensional mode of thin plates or disks, poled and excited along their thickness
  download
• Radial extensional mode of thin disk, poled and excited along its thickness
  download
• Shear mode of a thickness-poled thin plate, excited along its length
  download
• Labview runtime needed to run these programs

For doubts of use or bug repporting of this software or for general questions, please contact here. It will be greatly acknowledged that upon contacting you provide us your name, affiliation and topic of research interest.

It is worth noting these four resonance modes are enough to get the full set of independent parameters (2 dielectric (ε^S₁₁ and ε^S₃₃ ), 3 piezoelectric (d₃₃ d₃₁ and d₁₅)and 5 elastic (s^E₁₁, s^E₁₂, s^E₃₃, s^E₁₃, s^E₄₄)) that characterizes a ferro-piezoelectric poled ceramic material (6mm symmetry).

As an advantage with respect to the five samples required according to the Standards for the full matrix characterization, the four resonances mentioned above can be measured using three sample shapes (thickness-poled thin disk, thickness-poled shear plate and long rod or bar) and they provide a consistent and accurate matrix characterization of ferro-piezoceramics. Concerning the proper use of the thickness-poled shear sample,see:

• L. Pardo, A. García, F. Montero De Espinosa and K. Brebøl. “Shear Resonance Mode Decoupling to Determine the Characteristic Matrix of Piezoceramics For 3-D Modelling.” EEE Trans. UFFC, 58 (3), 646-657 (2011).
• L. Pardo, F. Montero de Espinosa, A. García and K. Brebøl."Choosing the best geometries for the linear characterization of lossy piezoceramics: study of the thickness poled shear plate" .Applied Physics Letters 92, 172907 (2008).
  
• L. Pardo, M. Algueró and K. Brebøl .“A Non-Standard Shear Sample for the Matrix Characterization of Piezoceramics and its Validation Study by Finite Element analysis”. J. Phys. D: Appl. Phys. 40 , 2162–2169 (2007).
  
• L. Pardo, M. Algueró and K. Brebøl "Resonance modes in Standard Characterization of Piezoceramics: A discussion based on Finite Element Analysis" .Ferroelectrics 336, 181-190 (2006).
  

These programs have been tested throughoutly in the past years for a wide number of ceramics at ICMM-CSIC and other laboratories. See for example:

• L. Pardo, A. García, K. Brebøl, E. Mercadelli and C. Galassi. “Enhanced properties for ultrasonic transduction, phase transitions and thermal depoling in 0.96(Bi_0.5Na_0.5)TiO₃-0.04BaTiO₃ submicron structured ceramic”. J. Phys. D: Appl. Phys 44 , 335404 (2011).
  
  
  
• A. Moure, T. Hungría, A. Castro and L. Pardo.“Microstructural effects on the phase transitions and the thermal evolution of elastic and piezoelectric properties in highly dense, submicron structured NaNbO₃ ceramics”. Journal of Materials Science 45 , 1211–1219 (2010).
  


• M. Algueró, C. Alemany, L. Pardo and M.P. Thi. “Piezoelectric Resonances, Linear Coefficients and Losses of Morphotropic Phase Boundary Pb(Mg_1/3Nb_2/3)O₃-PbTiO₃ Ceramics” . J. Am. Cer. Soc. 88(10), 2780-2787 (2005).


• L. Pardo, A. Castro, P. Millán, C. Alemany, R. Jiménez and B. Jiménez. “(Bi₃TiNbO₉)_x(SrBi₂Nb₂O₉)_1-x Aurivillius type structure piezoelectric ceramics obtained from mechanochemically activated oxides”. Acta Materialia 48(9), 2421-2428 (2000).


• L. Pardo, P.Duran-Martín, J.P. Mercurio, L. Nibou and B. Jiménez. “Temperature behaviour of structural, dielectric and piezoelectric properties of sol-gel processed ceramics of the system LiNbO₃-NaNbO₃". J. Phys. and Chem. Solids 58(9), 1335 (1997).


• J. Ricote, C. Alemany and L. Pardo. “Microstructural effects on dielectric and piezoelectric behaviour of calcium modified lead titanate ceramics”. J. Mater. Res.10(12), 3194 (1995).

The LABVIEW 8.5. interfaces to the original programs in BASIC of C. Alemany et al. were made by Alvaro García under supervision of Lorena Pardo. MIND NoE (FP6 515757-2 CE contract) and CSIC project 201060E069 funding support is acknowledged. ICMM-CSIC.

ICMM-CSIC. Madrid, 23 February, 2009.
Revised version of 10 October, 2011

Last Update December 2011

Background: coloured image of SEM of the microstructure of a (Bi_0.5Na_0.5)TiO₃-BaTiO₃ ceramic obtained by hot-pressing and recrystallization (A. García, J.L. Millán and L.Pardo. ICMM-CSIC(ES)-E. Mercadelli and C. Galassi ISTEC-CNR (IT)).