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


The resonance method The automatic iterative method Programs to Download Matrix characterization Recent advances and applications


The resonance method

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to the classical literature in the matter (“IEEE Standard on piezoelectricity”, ANSI/IEEE Std. 176-1987 (status withdrawn), and “Piezoelectric properties of ceramic materials and components.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

A practical reference on standard methods of calculation of a poled piezoceramic material parameters from analysis of impedance curves at resonance is:
J. Fialka and P. Beneš, “Measurement of piezoelectric ceramic parameters – A characterization of the elastic, dielectric and piezoelectric properties of NCE51 PZT,” in Proceedings of the 13th International Carpathian Control Conference (ICCC), High Tatras, Slovakia, 2012, pp. 147-152, doi: 10.1109/CarpathianCC.2012.6228632.

The automatic iterative method

The freeware that can be dowloaded in this site must be handle with previous knowledge of the basis of the piezoceramic parameters determination from analysis of impedance curves at resonance, as it only gives valid results when analyses valid resonance curves.

This freeware offers solutions to the limitations of the Standard calculation methods of a poled piezoceramic material parameters from analysis of impedance curves in general and, specifically, concerning characterization of high loss and low sensitivity poled ferro-piezoelectric ceramic materials, as well as, other types of piezoelectric materials with 6mm symmetry, as some poled piezoelectric composites and polymers.

Practical reference on advantages and application of the automatic iterative method:
L. Pardo, M. Alguero and K. Brebøl “Iterative method in the characterization of piezoceramics of industrial interest” Advances in Science and Technology, vol. 45, pp. 2448-2458 (2006).

L. Pardo , Á. García, F. Schubert, A. Kynast, T. Scholehwar, A. Jacas and J. F. Bartolomé.“Determination of the PIC700 Ceramic’s Complex Piezo-Dielectric and Elastic Matrices from Manageable Aspect Ratio Resonators”. Materials, 14 (15), 4076 (2021).

The set of four programs here avaliable allows to determine all material coefficients (piezoelectric elastic and dielectric) in complex form, thus including losses, of a piezoceramic from the measurement of the frequency dependence of the complex impedance/admittance at the electromechanical resonance modes of material samples with regular geometries, so as to provide unidimensional models, and apropriated aspect ratios, intended to avoid coupling of the resonances under study with other undesired resonances.

The material coefficients are 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. Such analytical solutions are valid for sample geometries with given aspect ratios, which allows exciting uncopled modes.

This set of equations is established for as many frequencies, which are automatically selected by the programs, 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),


    • 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).

Regular sample geometries are compulsory 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 impedance spectra using the above mentioned analytical expression of each mode of resonance and the material parameters obtained. This reconstruction is a built-in quality criterium for the calculated arameters. It is carried out as R and G versus frequency plots, where Y=G+iB=1/Z = 1/(R+iX). Such (R,G) vs. frequency plots are here used as an alternative to the usual complex impedance representation by modulus and phase of the complez impedance Z*= /Z/ e) . 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.

Besides, the method can be applied to obtain effective properties of device resonators (A.M.Gonzalez, J. de Frutos and M.C.Duro. “Procedure for the Characterization of Piezolectric Samples in Non-standard Resonant Modes” J. Eur. Ceram. Soc. 19, 1285-1288 (1999).

Download programs

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
      • Thickness extensional mode of thin plates or disks, poled and excited along their thickness
      • Radial extensional mode of thin disk, poled and excited along its thickness
      • Shear mode of a thickness-poled thin plate, excited along its length

Each downloaded compressed folder contains the program, an example file and a read-me file with brief explanations on input and outputs  of the program, the specific analytical expression of the resonance used and the specifically determined parameters. For doubts of use , bug reports of this freeware or for general questions on the method, please contact here.

It will be greatly acknowledged that upon contacting us you provide your name, affiliation and topic of research interest.

(*)Notes on the installation of the software that can be downloaded from this page

Matrix characterization

It is worth noting that programs for the analysis of the 4 resonance modes of the 3 samples specified above are enough to get the full set of 10 independent parameters in complex form, thus including all losses. These are 2 dielectric (εT11 and εT33), 3 piezoelectric (d33, d31 and d15) and 5 elastic (sE11, sE12, sE13, sE33 and sE44) parameters , which characterizes a ferro-piezoelectric poled ceramic material, as well as, other types of piezoelectric materials with 6mm symmetry.

A consistent and accurate full matrix characterization of ferro-piezoceramics and similar materials can be obtained from four resonances mentioned above using only three sample shapes: (1) the thickness-poled thin disk, (2) the thickness-poled shear plate – the best alternative to the in-plane poled shear plate suggested by the Standards – and (3) the long rod or bar. The thickness poled long bar and its length extensional resonance is not needed, as the corresponding parameters can be obtained in an alternative way from the impedance data measured  in the thickness poled thin disk. For this purpose, both impedance at the fundamental radial resonance and the first overtone of this must be known.

This is an advantage with respect to the 4 samples (5 resonances) required for the purpose according to the Standards methods (see section “The resonance method” in this web page).

Another advantage of the matrix characterization in this way is the use of the thickness poled shar plate, instead of the standard in-plane poled one. This is an easily poled sample, which provides higher consistency of the results of the set of 3 samples. It provides a higher accuracy in the determination of the piezoelectric shear cofficients as it has a more homogeneous mechanical vibration mode at the electromechanical shear resonance  when  decoupled from spureous modes. Decoupling can be achieved even in samples with moderate aspect ratio (L/t <10) between the length for electrical excitation (L) and the thickness of the plate (t). This is due to the periodicity of the coupling-decoupling phenomena as the thickness of the poled and re-electroded sample for measurement is reduced in steps of 0.02mm.  

Concerning the proper use of the thickness-poled shear plate see the section “The iterative automatic method” in this web page and :

    • 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 piezoceramic shear geometry: A discussion based on Finite Element Analysis” Journal de Physique IV France 128, 207-211 (2005)

Recent advances and applications

These programs have been tested throughoutly from their development 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(Bi0.5Na0.5)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(Mg1/3Nb2/3)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).

Notes on the installation of the software that can be downloaded from this page

This software is valid in computers with Labview 8.5. installed and with Windows Operating Systems.
It is not valid for Windows Operating Systems above Windows 7, 32 bits version (W7, 32 bits OS).

In order to use these programs in other computers, we suggest the following:

(a) Install a virtual machine with W7, 32 bits OS or a previous one (e.g. Windows XP) For this, visit the page: (Please, read the terms of use for this page, in particular on: 3. Use of Materials You may download, store, display on your computer, view, listen to, play and print Materials that Oracle publishes or broadcasts on the Site or makes available for download through the Site subject to the following: (a) the Materials may be used solely for your personal, informational, noncommercial purposes; (b) the Materials may not be modified or altered in any way; and (c) the Materials may not be redistributed.)

(b) In case that you do not have Labview 8.5. installed in your computer or virtual machine, the appropiated NI Runtime shall be installed in your virtual machine to run the piezoceramics characterization programs.


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: May 2023

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