## 3 Voltage Transformer Model  
This chapter presents the developed electrical model of a conventional (inductive) voltage transformer. The description of the prepared simulation model has been narrowed down only to the model of the voltage transformer, given its key role in the problem of testing simulation systems with ferroresonance oscillations in medium voltage networks. Figure 3 shows an electrical equivalent model of a voltage transformer in the form of a T-type quadrupole with two secondary windings, whose terminals are labelled a-n and da-dn [24]. The primary winding, with terminals marked a-n, is used to obtain a secondary three-phase system connected in a full star arrangement. The secondary winding, with terminals marked da-dn, is designed to be connected in a secondary open-triangle arrangement to measure the zero-sequence voltage (triple zero voltage \(3u_{0}\)). The equivalent circuit diagram shown is for a voltage transformer in one phase only (phase L1) [25; 26]. The developed mathematical model includes voltage transformers operating on all three phases. This procedure is necessary because the phenomenon of parallel ferroresonance (currents) occurs in medium-voltage networks only as a disturbance of a three-phase nature (which, it should be noted, is not necessarily of a symmetrical nature). The primary circuit of a single-phase voltage transformer for phase L1 (terminals on the left side of the figure marked A-N) is described by the voltage equation:  
\[u_{1}=u_{\text{L1}}+u_{\text{R1}}+u_{\text{\mu 1}} \tag{13}\]  
The flow of currents in the primary circuit, on the other hand, is described by following the equation:  
\[i_{1}=i_{\text{\mu}}+i_{2}=(i_{\Psi}(\Psi)+i_{\text{Fe}})+i_{2} \tag{14}\]  
Assuming that the two secondary circuits are connected to the secondary circuit via an idealised air transformer with winding ratios \(\theta_{1-21}\) and \(\theta_{1-22}\), the current, \(i_{2}\), can be described by the following equation:  
\[i_{2}=\frac{i_{21}}{\theta_{1-21}}+\frac{i_{22}}{\theta_{1-22}} \tag{15}\]  
Current \(i_{2}\) is the total load current of the voltage transformer, fed to the primary side of the transformer, and resulting from the individual values of secondary currents \(i_{21}\) and  
Figure 3: Equivalent diagram of the T-type voltage transformer (phase L1) with secondary circuits (terminals da-dn connected in an open delta configuration). \(i_{22}\). According to the equivalent diagram shown in Figure 3, Equation (13) can be expanded to the form of the following equation:  
\[u_{1}=L_{1}\frac{\mathrm{d}i_{1}}{\mathrm{d}t}+R_{1}i_{1}+R_{\mathrm{Fe}}(i_{ \mathrm{Fe}})i_{\mathrm{Fe}} \tag{16}\]  
After substituting Equation (14) into Equation (16), the following equation was obtained:  
\[u_{1}=L_{1}\frac{\mathrm{d}i_{1}}{\mathrm{d}t}+R_{1}i_{1}+R_{\mathrm{Fe}}(i_{ \mathrm{Fe}})(i_{1}-i_{\mathrm{\Psi}}(\mathrm{\Psi})-i_{2}) \tag{17}\]  
Equation (17) can then be transformed to form (18) which is a first-order non-linear differential equation. \[\frac{\mathrm{d}i_{1}}{\mathrm{d}t}=\frac{1}{L_{1}}(u_{1}-R_{1}i_{1}-R_{ \mathrm{Fe}}(i_{\mathrm{Fe}})(i_{1}-i_{\mathrm{\Psi}}(\mathrm{\Psi})-i_{2})) \tag{18}\]  
The non-linearity of the differential Equation (18) results from the non-linearity of the function describing the magnetisation curve of the non-linear coil (in the form of the dependence of the magnetising current on the associated flux \(i_{\mathrm{\Psi}}(\mathrm{\Psi})\)) and from the non-linearity of the function describing the total core loss curve (in the form of the dependence of the loss resistance on the core loss current \(R_{\mathrm{Fe}}(i_{\mathrm{Fe}})\)).