## 1 Introduction  
The electromechanical coupling coefficient indicates the degree to which the piezoelectric element is able to convert the input energy \(W_{in}\) into the output energy \(W_{out}\). When the element is in actuator mode, the input energy is electrical energy, while the output energy is mechanical energy, and vice versa in sensor mode. The magnitude of the electromechanical coupling coefficient can be determined either theoretically based on knowledge of the material constants (static value), or calculated from the measured series and parallel resonance frequencies (dynamic value) [1]. In a simple way, the properties of a resonator can be described by an equivalent electrical circuit with lumped parameters. It describes the resistor impedance in the vicinity of resonance and antiresonance frequencies [2]. From the equivalent electrical circuit of a piezoelectric resonator, the effective electromechanical coupling coefficient \(k_{eff}\), [2], can be determined. This coefficient may be expressed independently of the mode of oscillation as \[k_{eff}^{2}=\frac{f_{P}^{2}-f_{S}^{2}}{f_{P}^{2}}\approx\frac{f_{a}^{2}-f_{r}^{2}} {f_{a}^{2}}, \tag{1}\]  
where \(f_{S}\), \(f_{P}\) are the series and parallel resonance frequencies expressed from the electrical equivalent circuit, and \(f_{r}\) and \(f_{a}\) are the resonance and antiresonance frequencies meeting the requirement of zero imaginary impedance component. It is possible to show the relationship between the effective electromechanical coupling coefficient and the electromechanical coupling coefficients of certain types of oscillations of resonators of certain shapes [2], an example being the thickness mode of oscillation of circular fully electroded resonators. In a general case, however, the dependence cannot be simply defined and one must make do with the general relation (1). This is also the case for circular resonators with wrap-around electrodes. Because of the axially asymmetrical electrodes, the magnitude of the electromechanical coupling coefficient cannot be expressed analytically. As shown in [3], the wrap-around electrodes cause inhomogeneous polarization in the resonator; moreover, in the area where the electrodes are wrapped-around, polarization is not achieved at all. It can be therefore expected that the electromechanical coupling coefficient will be lower for resonators with wrap-around electrodes than for resonators of the same dimensions with fully electroded bases. In the presented work, we will verify this hypothesis computationally and validate it by comparison with experimental results. In order for the obtained results for resonators with fully electroded bases to be comparable with the results calculated for resonators with wrap-around electrodes, we will use the relation (1) to calculate the coefficient of electromechanical coupling. It will therefore be necessary to determine the resonance and antiresonance frequencies of the oscillation for which we determine \(k_{eff}\). The vibration to be studied will be the fundamental thickness oscillation. The study of the resonance characteristics of piezoelectric resonators with incomplete or asymmetric electrodes is already the subject of [4; 5], or more, recently [6; 7]. In the publications [8; 9], the analytical solution is compared with theexperimental results. However, in all these publications, the authors focus on resonators where a certain form of symmetry can be seen in the electrode surfaces, e.g., circular electrodes with different diameters but a common axis of circles, or rectangular electrodes on a resonator of rectangular plan, where two planes of symmetry can be found. In the available literature, the author has not found a relevant study concerning circular, partially electroded resonators and effects of this feature on the resonance frequency of the thickness oscillation. Neither for resonators electroded symmetrically according to the axis of symmetry of the disc, nor for asymmetrical electroding similar to the wrap-around electrodes.