# FEM Analysis of Piezoelectric Resonator Polarization Process
## 4 Discussion  
The result of a quantitative nature, which indicates the conditions during polarization, is the graph in the Figure 8.  
Figure 6: Calculated electrical potential for \(E_{app}=1\) kV \(\cdot\) mm\({}^{-1}\), sections A-A\({}^{\prime}\), B-B\({}^{\prime}\), (**a**) in the shape of circular segment, (**b**) in the shape of an intersection of circles. Figure 7: Calculated electric field for \(E_{app}=1\) kV \(\cdot\) mm\({}^{-1}\), (**a**) in the shape of circular segment, (**b**) in the shape of an intersection of circles. The graph shows the dependence of the polarized volume of the transducer on the magnitude of the polarization electric field for both variants of the analyzed transducers. A significant increase in the polarized volume occurs when the magnitude of the coercive field in the area between the opposite electrodes is reached. The polarized volume of the transducer is about 1.2% higher in the case of variant (b) compared to variant (a). This difference can be explained by the difference in the volume occupied by the wrap-around area, including the area below the gap between the electrodes. This difference is about 1% higher in the case of variant (a) compared to variant (b). The further increase after exceeding the coercive electric field between the opposite electrodes occurs very slowly, in the vicinity of the wrapped-around electrode. The graph in Figure 9 shows the dependence of the volume polarized in the thickness direction of the transducer on the magnitude of the polarization field \(E_{app}\). In the case of variants (a) and (b), the volume polarized in the thickness direction is clearly lower than the total polarized volume of the transducers. An interesting result can be seen in the interval (0.8-0.9) kV/mm of the polarization field. In this interval, an increase in the polarized total volume is evident from the graph in Figure 8, whereas in the graph in Figure 9, which shows the dependence of the volume polarized in the thickness direction, this increase is not apparent. We can therefore hypothesize that polarization occurs in the interval (0.8-0.9) kV/mm in directions different from the direction perpendicular to the thickness. This hypothesis  
Figure 8: Dependence graph of the polarized volume of the transducer on \(E_{app}\), (a) in the shape of circular segment, (b) in the shape of an intersection of circles. Figure 9: Dependence graph of the volume of the transducer polarized in the thickness direction \(E_{app}\), (a) in the shape of circular segment, (b) in the shape of an intersection of circles. is consistent with the fact that the gap between the electrodes is slightly smaller than the thickness of the transducer, see Table 1. The polarization of the material in the area of the gap between the electrodes will therefore occur at a lower voltage difference between the electrodes, and thus in a direction different from the direction perpendicular to the thickness than in the direction of the transducer thickness, between the opposite electrodes. It stems from the relation (9); the magnitude of the electric field intensity is inversely proportional to the distance between the electrodes. The macroscopically piezoelectric area, which is effective in terms of electromechanical response, is precisely the area polarized in the direction of the transducer thickness. That is, as would be achieved with a circular resonator that is fully electroded on both sides. It is clear from the graph in the Figure 9 that  
* 88% resonator volume in configuration (a),
* 90% resonator volume in configuration (b)  
is effective in terms of electromechanical transformation, and therefore also in terms of the coefficient of electromechanical coupling. The remaining 10% or 12% respectively of the resonator are polarized either in a direction different from the direction perpendicular to the circular surfaces or not polarized at all. The most prominent area that is not polarized is the part covered by the wrapped-around electrode.