## 3 Voltage Transformer Model
 \tag{32}\]  
By performing the substitution of Equation (31) into Equation (30), the flux \(\Psi\) was obtained, which is a first-order non-linear differential equation of the following form:  
\[\frac{d\Psi}{dt}=R_{Fe}(i_{Fe})\Big{[}i_{1}-\Big{(}a_{1}\Psi+a_{5}\Psi^{5}+a_ {7}\Psi^{7}\Big{)}-i_{2}\Big{]} \tag{33}\]  
Figure 4 shows the non-linear magnetising current characteristics of a non-linear ferromagnetic core coil \(i_{\Psi}(\Psi)\), depending on the magnetic flux associated with the primary winding \(\Psi\). The characteristic shown in Figure 4 was approximated by a folded function with two functions: for \(i_{\text{Fe}}\) between 0 and \(I_{\text{RFe}\_1}\)  
\[R_{\text{Fe}}=\frac{R_{\text{Fe\_max}}+R_{\text{Fe\_min}}}{2}-\frac{R_{\text{ Fe\_max}}-R_{\text{Fe\_min}}}{2}\cdot\cos\biggl{(}\frac{\pi\cdot|i_{\text{Fe}}|}{I_{ \text{RFe\_max}}}\biggr{)} \tag{34}\]  
and for \(i_{\text{Fe}}\) greater than \(I_{\text{RFe}2}\)  
\[R_{\text{Fe}}=a\cdot|i_{\text{Fe}}|+b \tag{35}\]  
where, for the individual constants appearing in Equation (34), values were used as follows:  
\[ \begin{array}{l}R_{\text{Fe\_2}}=574000\ \Omega\\ R_{\text{Fe\_min}}=400000\ \Omega\\ I_{\text{RFe\_max}}=0.0107\ \text{A}\\ I_{\text{RFe\_1}}=0.0116\ \text{A}\\ I_{\text{RFe\_2}}=0.0212\ \text{A}\\ I_{\text{RFe\_2}}=715700\ \text{A}\end{array}  \tag{36}\]  
The value of the constant \(R_{\text{Fe\_1}}\), which defines the range for the first component of the folding function, is calculated from the following formula:  
\[R_{\text{Fe\_1}}=\frac{R_{\text{Fe\_max}}+R_{\text{Fe\_min}}}{2}-\frac{R_{ \text{Fe\_max}}-R_{\text{Fe\_min}}}{2}\cos\biggl{(}\frac{\pi\cdot I_{\text{ RFe\_1}}}{I_{\text{RFe\_max}}}\biggr{)} \tag{37}\]  
Figure 4: Nonlinear function describing: (**a**) nonlinear coil with ferromagnetic core, (**b**) nonlinear resistance representing total losses in the core. The constants \(a\) and \(b\), appearing in Equation (35), were calculated from the following equation:  
\[ \begin{array}{c}a=(R_{\text{Fe\_2}}-R_{\text{Fe\_1}})/(I_{\text{RF\_2}}-I_{ \text{RF\_1}})\\ b=-(a\cdot I_{\text{RF\_1}})+R_{\text{Fe\_1}}\end{array}  \tag{38}\]  
On the right-hand side of Figure 4, one can see the non-linear characteristic of the resistance of the total losses in the core of the voltage transformer \(R_{\text{Fe}}\) depending on the active current of the losses in the core \(i_{\text{Fe}}\). Equations (34)-(38) were developed by the authors of the paper and represent a proposed approximation of the non-linear resistance function \(R_{\text{Fe}}\). The composite of the non-linear characteristics described by Equations (31), (34) and (35) (shown in Figure 4) together form an electrical representation of the static hysteresis loops. The equations allow the shape of the static hysteresis loops to be reproduced for the entire range over which they were measured. The static hysteresis loops obtained by modelling are shown in Figure 5.  
It is worth noting at this point that the modelled static hysteresis loops are, in fact, a composite of the averaged trajectories along which the peak value point of the current and associated flux waveforms move. The resulting static loops are, therefore, only an electrical representation of the active and reactive components of the current drawn by the magnetising branch. This has consequences because, in dynamic states, the proposed hysteresis loop model will behave differently from what the real waveforms would suggest. Figure 5: Static hysteresis loops modelled by nonlinear equations: static loop in the deep saturation state of the modelled voltage transformer (below), superimposed static hysteresis loops for different values of the measured voltage (on top). The inconsistency of the simulated waveforms with the real waveforms will be smaller the deeper the core saturation is simulated. In the case of modelling ferroresonance oscillations, the transformers operate in a deeply saturated state, which is conducive to reducing the relative error of the simulation results.