## 2 Materials and Methods
### Task Assignment  
We calculate \(k_{eff}\) of the fundamental thickness oscillation for resonators with wrap-around electrodes, where we expect inhomogeneous polarization due to electrodes being wrapped around. We compare the result with the calculation of \(k_{eff}\) thickness oscillation of a resonator of identical dimensions but fully electroded on both circular surfaces, where homogeneous polarization can be expected throughout the resonator volume. By comparing the calculated \(k_{eff}\) of the resonators with wrap-around electrodes with the \(k_{eff}\) of the fully electroded resonators, we conclude the effect of polarization on the electromechanical coupling coefficient.  
This approach is applied when computing \(k_{eff}\) of two types of piezoelectric transducers, labeled (a) and (b), shown in Figure 1. The transducers differ in the shape of the electrode wrap-around. In case of (a), the overlay is in the shape of a circular segment, in case of (b), it is in the shape of intersecting circles.  
To make the results comparable, we will consider both piezoelectric transducers of the same size. The criterion of equivalence for the wrap-around electrodes needs to be established. We will consider equivalent the electrodes that will have the same area of wrap-around electrode in both cases.  
Let \(\alpha=d/t\), where \(d\) is the diameter of the disc and \(t\) its thickness. We will consider a disc with a dimensional ratio of \(\alpha\gg 1\), where the basic thickness mode of the oscillation is significantly distinct. In the monograph [1] (chapter 5.3), the frequency spectrum of a resonator with entire electrodes on both circular surfaces with ratio \(\alpha=23.7\), made of NCE51 piezoelectric ceramics, is analyzed. We will thus work with a resonator that maintains the dimensions of resonator from chapter 5.3 of the monograph [1], except that it will not be fully electroded on both circular surfaces. A diagram of the dimensions of the transducers is shown in Figure 2. The dimensions of the transducers are given in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \(d\) (mm) & 47.9 & \(t\) (mm) & 2.02 \\ \(s_{1}\) (mm) & 4.66 & \(r_{1}\) (mm) & 9.50 \\ \(s_{2}\) (mm) & 2.00 & \(r_{2}\) (mm) & 7.50 \\ \hline \end{tabular}
\end{table}
Table 1: Dimensions of the transducers.  
Figure 1: Piezoelectric transducers with wrap-around electrodes. Wrapping of the electrode from the bottom to the top surface is apparent, (**a**) wrapping in the shape of a circular segment; (**b**) wrapping in the shape of an intersection of circles.  
Figure 2: Diagram of the dimensions of the analyzed transducers, (**a**) in the shape of circular segment, (**b**) in the shape of an intersection of circles.  
The dimensions of the wrap-around are chosen so that the area of the wrapped around electrodes is identical in both cases. The ratio of the wrap-around to the total circular area is approximately 1/25.  
We consider the transducers made of NCE51 piezoelectric ceramics, again in accordance with Chapter 5.3 in [1]. The values of the elements of the above-mentioned tensors are given in Table 2 and are taken from [1,20]. In the table, permittivity is given as relative permittivity.  
The polarization of the above-mentioned resonators with a wrap-around electrode is analyzed in the work by [3]. We will consider the polarization directions as calculated in [3]. Thus, we know the polarization directions of piezoelectric ceramics and consequently the macroscopic piezoelectric properties that are determined by polarization.