Mathematical Modeling the Electrical Impedance of the Piezoceramic Disk Oscillating in a Wide Frequency Range (Part 1. Low Frequencies)




piezoelectric transducer, acoustoelectronics, mathematical modeling, impedance, disk element


The article presents the results of mathematical modeling and analysis of the electrical impedance of a piezoceramic disk that undergoes oscillations at low frequencies, i.e., when the length of the elastic wave significantly (by an order of magnitude or more) exceeds the radial size of the disk. Thus, the proposed mathematical model of disk-shaped ceramic elements of piezoelectric transducers, which are an important component of modern communication devices, environmental sensors, precision equipment, medical devices, etc. A key characteristic of the mathematical model described in the article is its ability to determine analytical dependencies that allow estimating such fundamental electrical properties of the piezoceramic disk element as electrical impedance and quasi-static electrical capacitance, thereby significantly simplifying the calculation of such an element already at the design stage.
The static dielectric permittivity of a piezoceramic disk vibrating at low frequencies has been investigated. The calculated value of this parameter, based on the physical constants’ characteristic of the piezoceramic of the PZT (lead zirconate titanate) type, is 1.844 times higher compared to the high-frequency (dynamic) dielectric permittivity.

It has been found that in the low-frequency range, when the mechanical stresses in the vibrating piezoceramic disk approach zero and the direct piezoelectric effect is almost negligible, the electrical impedance of such a disk can be described as the reactive resistance of a capacitor with electrical capacitance equivalent to the quasi-stationary capacitance of the disk. This is confirmed by a high degree of convergence between theoretical data and experimental results, with discrepancies not exceeding 6%.

The results obtained in the article can be valuable for scientific research in the fields of precision instrument engineering and radio equipment manufacturing. Additionally, they have practical applications in the development and production of high-tech equipment.



Piezoelectric Ceramics Market Size, Changing Dynamics and Future Growth Trend 2022-2029. Market Intelligence, date of access: 17 November, 2023.

Piezoelectric Devices: From Bulk to Thin-Film 2019. Yolegroup, date of access: 17 November, 2023.

Francisco A., Marcos G., Leonardo B.-R., Muhlen S. (2014). Electric Impedance of Piezoelectric Ceramics under Acoustic Loads. Transactions on Electrical Engineering, Vol. 12, Iss. 2, pp. 48-54. DOI: 10.37936/ecti-eec.2014122.170819.

Ogbonna V. E., Popoola A. P. I., Popoola O. M. (2022). Piezoelectric ceramic materials on transducer technology for energy harvesting: A review. Front. Energy Res., Vol. 10, doi: 10.3389/fenrg.2022.1051081.

Aladwan I. M., Bazilo C., Faure E. (2022). Modelling and Development of Multisectional Disk Piezoelectric Transducers for Critical Application Systems. Jordan Journal of Mechanical and Industrial Engineering, Vol. 16, No. 2, pp. 275–282.

Ishchuk V., Kuzenko D., Sobolev V. (2018). Piezoelectric and functional properties of materials with coexisting ferroelectric and antiferroelectric phases. AIMS Materials Science, Vol. 5, Iss. 4. pp. 711-741. DOI: 10.3934/matersci.2018.4.711.

Kirilyuk V. S., Levchuk O. I. (2018). Mathematical modeling of the electrostressed state in the orthotropic piezoelectric space with an arbitrary orientated circle crack under uniaxial tension. System research and information technologies, No. 3. DOI: 10.20535/SRIT.2308-8893.2018.3.06.

Cao P., Zhang S., Wang Z., Zhou K. (2023). Damage identification using piezoelectric electromechanical impedance: a brief review from a numerical framework perspective. Structures, Vol. 50, pp. 1906-1921. doi: 10.1016/j.istruc.2023.03.017.

Gogoi N., Chen J., Kirchner J., Fischer G. (2022). Dependence of Piezoelectric Discs Electrical Impedance on Mechanical Loading Condition. Sensors, Vol. 22, Iss. 5, 1710. DOI: 10.3390/s22051710.

Zhou K., Zhang Y., Tang J. (2022). Order-Reduced Modeling-Based Multi-Level Damage Identification Using Piezoelectric Impedance Measurement. IFAC-PapersOnLine, Vol. 55, Iss. 27, pp. 341-346. doi: 10.1016/j.ifacol.2022.10.536.

Han H., Cheng C., Xiong X-G., Su J., Dai J-X., Wang H., Yin G., Huai P. (2015). Piezoelectric, Mechanical and Acoustic Properties of KNaNbOF5 from First-Principles Calculations. Materials, Vol. 8, Iss. 12, pp. 8578-8589. DOI: 10.3390/ma8125477.

Nguyen T. T., Hoang N. D., Nguyen T. H., Huynh T. C. (2022). Analytical impedance model for piezoelectric-based smart Strand and its feasibility for prestress force prediction. Structural Control and Health Monitoring, Vol. 29, Iss. 11, e3061. doi: 10.1002/stc.3061.

Kenji Uchino (2017). The Development of Piezoelectric Materials and the New Perspective. Chapter 1 In Advanced Piezoelectric Materials, pp. 1-92. DOI: 10.1016/B978-0-08-102135-4.00001-1.

Brissaud M. (2022). Modeling and characterization of thick piezoelectric disk including radial, longitudinal and shear modes. Ferroelectrics, Vol. 600, Iss. 1, pp. 46-58. DOI: 10.1080/00150193.2022.2115796.

Antonyuk V. S., Bondarenko M. A., Bondarenko Yu. Yu. (2012). Studies of thin wear-resistant carbon coatings and structures formed by thermal evaporation in a vacuum on piezoceramic materials. Journal of Superhard Materials, Vol. 34, No. 4, pp. 248-255. doi: 10.3103/S1063457612040065.

Rathod V. T. (2019). A Review of Electric Impedance Matching Techniques for Piezoelectric Sensors, Actuators and Transducers. Electronics, Vol. 8, Iss. 2, 169. DOI: 10.3390/electronics8020169.

Bazilo C. V. (2017). Principles of electrical impedance calculating of oscillating piezoceramic disk in the area of medium frequencies. Radio Electronics, Computer Science, Control, Vol. 4, pp. 15–25. DOI: 10.15588/1607-3274-2017-4-2.

Kathavate V. S., Prasad K. E., Kiran M. S. R. N., Zhu Y. (2022). Mechanical characterization of piezoelectric materials: A perspective on deformation behavior across different microstructural length scales. J. Appl. Phys., Vol. 132, Iss. 12, 121103. DOI: 10.1063/5.0099161.

Petrishchev O. N. (2012). Harmonic vibrations of piezoceramic elements. Part 1. Harmonic vibrations of piezoceramic elements in vacuum and the method of resonance-antiresonance. Avers, Kiev, Ukraine.

Bazilo, C. V. (2018). Principles and methods of the calculation of transfer characteristics of disk piezoelectric transformers. Radio Electronics, Computer Science, Control, Vol. 4, pp. 7-22. DOI: 10.15588/1607-3274-2018-4-1.




How to Cite

Bazilo, C. V., Trembovetska, R. V., Usyk, L. M., Faure, E. V. and Chorniy, A. M. (2023) “Mathematical Modeling the Electrical Impedance of the Piezoceramic Disk Oscillating in a Wide Frequency Range (Part 1. Low Frequencies)”, Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia, (94), pp. 41-48. doi: 10.20535/RADAP.2023.94.41-48.



Telecommunication, navigation, radar systems, radiooptics and electroacoustics

Most read articles by the same author(s)