## 5 Results
### Equivalence of 2D and 3D Models  
The 2D and 3D models were prepared in different coordinate systems, which ultimately lead to differences in the geometry of the obtained objects. While the 3D model strictly corresponded to the geometry of the physical model, the 2D model contained much more ferromagnetic material due to the cylindrical symmetry. Different amounts of the ferromagnetic material in 2D and 3D models caused changes in the value of the reluctance in their magnetic circuits and, as a consequence, in the value of the magnetic flux. The authors assumed that the total magnetic reluctance R in both models were the same. This could be achieved by performing correction of the magnetic permeability in the 2D model. Such an approach is shown in Figure 8 and corresponds to the studies described by [PERSON] (1966). For the considered 2D model, the small signal local permeability was corrected to \(\mu_{\rm r}=214\), taking into account that the skin depth \(\delta\) in the 2D model should have the same value as in the 3D model. In both cases, values of the relative permeability at higher frequencies were assumed according to (6). As mentioned in Section 3 of this paper, the additional column in the 2D model (Figure 5) was used to moderate the value of the magnetic coupling between the analyzed coil and the additional wire. In the 3D model, this magnetic coupling was about 0.3, while in the 2D model, it was 0.14. In order to level up the coupling coefficient in the 2D model, the magnetic permeability value in one of columns was changed. Equalization of the coupling coefficient in the 2D model to the suitable value caused the simulated resonance to arise for the same frequency on the inductance curve. The main issue of the research was to develop a winding model that would allow us to determine the frequency-dependent inductance of a winding while reducing the duration of the FEM analysis. The resonance visible on the inductance characteristic obtained from the FRA measurements was used to verify the conducted simulations. The resonance occurred above the frequency of 1 kHz (Figure 9) and corresponded to the first parallel resonance visible on the FRA results (this phenomenon was described in detail in Section 2). Figure 8: The reluctance network of the 3D model (**a**) and 2D model (**b**). The remaining windings, both in the 2D and 3D model, were simulated as the single wire placed on the outer column of the core (Figures 4 and 5), which provided the magnetic coupling with a modelled coil. Since the first resonance depended in particular on the inter-turn capacitances and capacitances to the ground of the other windings, it was necessary to assign the proper capacitance to the additional turn in order to simulate the resonance. In FEM software, this goal is accomplished by creating the output terminal connected to the additional wire with the capacitance \(C_{\rm HV}\), resulting from the capacitance of the HV-winding. This approach was used both for the 2D and 3D models and is presented in a schematic manner in Figure 5. The value of the capacity of the HV-winding was assumed to be 200 pF; however, because the windings remaining on the transformer core were simulated as a one-turn coil, the value of \(C_{\rm HV}\) had to be recalculated in relation to the single wire. The additional capacitance \(C_{\rm add\_3D}\) connected to the output terminal in the 3D model is given by:  
\[C_{\rm add\_3D}=N_{\rm HV}{}^{2}\cdot C_{\rm HV}, \tag{14}\]  
where \(N_{\rm HV}\) is the number of turns in the HV-winding, which, for a tested transformer, is equal to 1520. Furthermore, due to the halving of the computational area, \(C_{\rm add\_3D}\) also had to refer to half of the model. Accordingly, the final additional capacitance \(C_{\rm add\_3D}\) was equal 240 uF for the 3D model.