## 1 Introduction  
Despite the increasing usage of voltage and current sensors in today's electric power industry, conventional inductive voltage transformers are still widely used. The use of ferromagnetic-core voltage transformers means that ferroresonance in medium-voltage networks is still an ongoing problem. Ferroresonance of (parallel) currents, occurring mostly in medium-voltage networks with an isolated neutral point, is characterised by the appearance of sub-harmonics, harmonics or, in general, non-linear zero-order voltage oscillations. For this reason, the most common methods of detecting ferroresonance in medium-voltage networks are based on the observation of the secondary neutral voltage \(3u_{0}\) of the transformer. This voltage can be measured in an open triangle voltage transformer system or calculated digitally by summing samples of the individual phase waveforms. Nevertheless, there is an alternative method, which consists of observing the waveform of the resultant flux associated with the primary and secondary windings, which induces a voltage in the secondary windings of the voltage transformer. Having the core magnetisation characteristics (obtained, e.g., by measurements) and having information on the shape of the resultant associated flux waveform, it is possible to approximately determine the saturation of the core. The proposed method is an innovative approach, as it is an attempt to estimate the associated magnetic flux in order to determine the state of the transformer core, and is not widely undertaken. The proposed approach is inclined to observe the mechanism causing ferroresonance oscillations and not to observe only its effects--the secondary voltage of the zero sequence. To obtain the waveform of the resultant flux associated with the primary and secondary windings, the time integral must be calculated from the secondary voltage waveform (and then converted to the primary side). A voltage transformer is essentially a transformer in an idle state [1]. According to the simplified equivalent diagram of an ideal transformer shown in Figure 1, we can write the equation describing the current distribution in a voltage transformer:  
\[i_{1}=i_{2}^{\prime}+i_{\mu}=\frac{1}{\vartheta}.i_{2}+i_{\mu} \tag{1}\]  
Due to the relatively high value of the load resistance \(R\) of the transformer, we can write that the secondary current \(i_{2}^{\prime}\), converted to the primary side, is much smaller than the magnetising current \(i_{\mu}\):  
\[i_{2}^{\prime}\ll i_{\mu} \tag{2}\]  
Therefore, Formula (1) takes the form:  
\[i_{\mu}=i_{1}-i_{2}^{\prime} \tag{3}\]  
Assuming that the inequality (2) is valid, we obtain the approximation (4), according to which the magnetising current \(i_{\mu}\) is approximately equal to the primary current \(i_{1}\) of the voltage transformer:  
\[i_{\mu}\approx i_{1} \tag{4}\]  
The individual currents \(i_{1}\), \(i_{2}^{\prime}\) and \(i_{\mu}\) are associated with the flux linkages \(\Psi_{1}\), \(\Psi_{2}^{\prime}\) and \(\Psi_{\mu}=\Psi_{12}\). The linkage in the primary and secondary windings \(\Psi_{12}\) is the resultant linkage according to the following formula:  
\[\Psi_{12}=\Psi_{1}-\Psi_{2}^{\prime}=\Psi_{1}-\vartheta.\Psi_{2} \tag{5}\]  
Relationship (5) is illustrated in Figure 2, which shows the distribution of linkages in the core of a transformer. Figure 1: Simplified equivalent circuit of a transformer (winding leakage resistances and inductances omitted). Similarly to the current in Equation (2), for a magnetic circuit, we can write the inequality resulting from the state of the voltage transformer, which is close to the no-load state:  
\[\Psi_{2}^{\prime}\ll\Psi_{12} \tag{6}\]  
As a consequence, approximation (7) is true, according to which the mutual flux linkage \(\Psi_{12}\) is approximately equal to the flux linkage in the primary winding \(\Psi_{1}\).