# FEM Analysis of Piezoelectric Resonator Polarization Process
## 1 Introduction  
Solid solutions based on PbTiO\({}_{3}-\)PbZrO\({}_{3}\), referred to as PZT ceramics, or solutions based on BaTiO\({}_{3}\) are primarily used for the production of piezoelectric transducers. The piezoelectric properties of these materials are conditioned by the existence of spontaneous polarization. However, these samples do not show macroscopic piezoelectric properties after sintering because the directions of polarization in individual grains of the ceramics are different. Only after polarization, i.e., after application of a sufficiently strong DC electric field, and usually at increased temperatures, these macroscopically observable piezoelectric properties appear in the transducers. The efficiency of electromechanical conversion in piezoelectric transducers is assessed by the magnitude of the electromechanical coupling coefficient, which can be generally expressed as  
\[k=\sqrt{\frac{W_{out}}{W_{in}}}. \tag{1}\]  
Its square expresses the ratio of the fraction of energy \(W_{out}\) that changes from electric to mechanical or vice versa to the total energy received \(W_{in}\), e.g., [1]. An important factor affecting the magnitude of the electromechanical coupling coefficient is the polarization process. When an electric field of higher intensity than the so-called coercive electric field \(E_{c}\)[2], is applied, the polarization of the piezoceramics changes in the direction of the applied electric field. The direction of the applied electric field is influenced by the shape of the piezoelectric element and the placement of the electrodes. In the case of circular resonators, the polarization field is homogeneous due to the solid electrode and the high permittivity of the piezoelectric ceramics. As a consequence, the polarization is homogeneous over the entire volume of the piezoceramic resonator. However, there are applications, such as ultrasonic generators, where it is necessary for the supply leads to be connected to the piezoelectric element from only one side. The freeside is then glued to a flexible membrane which transmits the vibrations of the piezoelectric transducer to the surroundings. In such a case, transducers with a so-called wrap-around electrode are used, as can be seen in the Figure 1.  
As a result of the electrodes being wrapped around, an inhomogeneous electric field is created in the transducer. The field forms even during the polarization process. Thus, the transducer will not be polarized homogeneously in its entire volume. Some parts of the transducer will be polarized in a different direction than desired, while some parts of the transducer will not be polarized at all. This fact causes different operating characteristics of the piezoelectric element than expected, e.g., a lower electromechanical coupling factor. Relevant publications dealing with similar issues include [3; 4; 5; 6; 7; 8; 9; 10; 11]. The effect of polarization is studied in these works, either by analytical methods due to the significantly simpler geometry [9], experimentally [11], or using the finite element method [3; 4; 5; 6; 7; 8], however, always in a configuration or application different from the one presented here. In piezoelectric ceramics, nonlinear phenomena are also present in the elastic, piezoelectric and dielectric spheres. As an example, phenomena on domain walls can be mentioned. For simple configurations of several domain walls, these are studied analytically, e.g., in [12; 13], and experimentally, e.g., in [14]. Phenomena at domain walls have been studied by the author in [15; 16]. Neglecting of nonlinearities, lattice effects, grain boundaries, self-polarization, etc. would not be possible in the case of analysis of polarization phenomena on a microscopic scale, at the level of individual domain interfaces. On a macroscopic scale, in terms of the macroscopic properties of the piezoelectric element, a linear model is used in publications [3; 4; 5; 6; 7; 8] to determine the direction of polarization in the piezoelectric ceramic.