# Power Transformer Inrush Current Analysis: Simulation, Measurement and Effects
## 2 Theoretical Concepts  
The inrush current of a power transformer can be influenced by the following categories of parameters [1; 7; 25]:  
* power source parameters: RMS voltage values, frequency, initial phase of supply voltage;
* network parameters in the vicinity of the transformer: the appropriate connection inductance/resistance of the electrical network together with it short-circuit power at the common connection point;
* magnetic core characteristics: magnetization curve, hysteresis loop of the core material, residual core flux and its polarity at the time of connection. The influence of the magnetic materials (from which the magnetic core is made) and the constructive parameters of the transformer, on the inrush waveform is known/studied in the specialized literature. The models found in the literature approximate the magnetization curve through a hyperbolic function or determined either by experimental or numerical methods, in our case the magnetization curve was determined by calculation starting from the magnetization characteristics of the sheet from which the transformer coils are made. By identifying the coordinates of the points corresponding to the magnetization curve of the magnetic core, i.e., the values corresponding to the magnetic induction (B) on the horizontal axis and the magnetic field strength (H) on the vertical axis, we can determine the magnetization curve of the magnetic circuit, \(\emptyset=\text{f}(\text{l})\). The equation of the magnetic flux is as follows (1) [7; 25]:  
\[\phi[\text{Wb}]=B[\text{T}]\cdot S_{\text{c}}\Big{[}\text{m}^{2}\Big{]} \tag{1}\]  
where \(\text{S}_{\text{c}}\) is the cross-section of the magnetic core column. To determine the current that occurs when the transformer is connected to the mains, we use the magnetic field strength values obtained from the magnetization curve. The no-load current values are obtained from the relationship (2) [7; 25],  
\[I_{0}=(H\cdot l_{med})/w_{1} \tag{2}\]  
where: \(l_{med}\) is the mean length of the magnetic field line; H--magnetic field strength; w\({}_{1}\)--number of turns in the primary [7; 25]. \[l_{med}=H\big{(}3\cdot l_{c}+2\cdot l_{j}\big{)} \tag{3}\]  
where: \(l_{j}\) is the length of the yoke, \(l_{c}\)--column length. Determination of transformer parameters  
* the line current according to the Equation (4) [7; 25]: \[I_{1N}=\frac{S_{N}}{\sqrt{3}U_{1}}\] (4)
* no-load current [7; 25]: \[I_{0}=k[\%]\cdot I_{1N}\] (5)  
If the primary winding connection is Y\({}_{N}\), the phase current is equal to the line current, and if the primary winding connection is \(\Delta\), the current through the sides of the triangle is given by the relationship (6) [7; 25]. \[I_{Nf}=\frac{I_{1N}}{\sqrt{3}} \tag{6}\]  
where I\({}_{Nf}\) is the current flowing through the winding of one phase. * \(\phi_{0}\) magnetic flux for no-load transformer operation [7; 25]: \[\phi_{0}=f(I_{0})\] (7)
* leakage reactance [7; 25] \[X_{\sigma}=\frac{U_{k}[\%]\cdot S_{N}}{100\cdot I_{1N}^{2}}\] (8)
* the leakage inductance is calculated using the relationship (9) [7; 25] \[L_{s}=\frac{X_{\sigma}/3}{2\pi f}\] (9)
* magnetizing inductance [7; 25] \[L_{m}=\mu_{0}\cdot\mu_{r}\cdot\frac{S_{c}}{l_{med}}\cdot w\] (10)  
where: \(\mu_{0}\)--is the absolute magnetic permeability of the vacuum; \(\mu_{r}\)--relative permeability; \(S_{c}\)--cross sectional area of the magnetic core. The magnitude of the magnetizing inductance varies over time with the variation of the magnetic flux and current through the transformer winding. For each flux value \(\mathcal{O}_{1}\) there is a relative permeability value \(\mu_{ri}\) corresponding to the time ti of the transient process. \[I_{0}=(H\cdot l_{med})/w_{1} \tag{2}\]0For the moment ti, the transient value of the inductance L\({}_{\rm m}\) is given by the equation [7; 25]:  
\[I_{0}=(H\cdot l_{med})/w_{1} \tag{2}\]1  
- core loss resistance  
The power P\({}_{0}\) drawn by the transformer from the mains is practically equal to the core losses (\(P_{0}\cong P_{Fe}\)) [7; 25].