# Analysing and Computing the Impact of Geometric Asymmetric Coils on Transformer Stray Losses
## 2 Modelling and Considerations
### Modelling Additional Losses  
The measurement of stray losses in the transformer tank walls does not differentiate between hysteresis, eddy, and excess losses. We isolated each of the six walls, treating them as individual mild steel sheets. This simplification facilitates a more detailed analysis of stray losses. The value of \(\left|\mathrm{B_{d}}\right|\) is derived through electromagnetic simulations using the Finite Element Method (FEM). Specifically, for calculating the loss components within the laminations of the ferromagnetic material, the methodology of this study aligns with the conceptual framework introduced in [22]. This strategic separation and focused analysis enhances the precision in evaluating the distinct contributions of hysteresis, eddy and  
Figure 1: Geometry model of the three-phase transformer analysed for stray losses: (**a**) real arrangement, (**b**) 3D view of the components, and (**c**) top view with tank walls nomenclature.  
excess losses in the transformer tank walls. The hysteresis loss density (Watts/m\({}^{3}\)) in each of the tank walls will be given by the Steinmetz equation [23]:  
\[P_{h}=k_{h}B_{d}{}^{\alpha}f, \tag{1}\]  
where \(k_{h}\) is a hysteresis factor that depends on the properties of the material with which the tank wall is made, \(k_{h}=63\) for mild steel commonly used for transformer tanks, \(\alpha\) is the Steinmetz coefficient (the value of Steinmetz coefficient \(\alpha\) is approximately 2 for all modern magnetic materials) and \(f\) is the operating frequency. The additional loss density (Watts/m\({}^{3}\)) due to eddy losses can be determined by [22] the following:  
\[P_{e}=\frac{1}{6}\sigma\pi^{2}B_{d}{}^{2}f^{2}q^{2}, \tag{2}\]  
where \(\sigma\) is the conductivity of the tank wall (In this case, \(\sigma=6.67\times 10^{6}\) S/m for mild steel) and \(q\) is the wall thickness. The additional loss density (Watts/m\({}^{3}\)) due to excess losses can be determined by [22] the following:  
\[P_{x}=8.76\sqrt{\sigma GV_{o}S}B_{d}{}^{1.5}f^{1.5}, \tag{3}\]  
where \(S\) is the cross section of the tank wall and \(G\) and \(Vo\) are dimensionless constants related to the quality of the ferromagnetic material with which the tank wall is made. For example, for M4 (0.27 mm), grain-oriented steel \(G=0.1356\) and \(Vo=0.0110\) at 1.5 T and 60 Hz, the excess losses in the tank walls were not considered in this work; however, it is possible to calculate them with this proposed technique by finding the \(\mid\)B\({}_{d}\)\(\mid\) that reaches the tank wall. Moreover, determining the values of \(G\) and \(Vo\) of the tank wall would be outside the scope and technical feasibility of this work.  
Equations (1)-(3) represent the density of losses per unit volume, so they need to be multiplied by the volume \(V\) of each wall to obtain the losses. The additional plate losses of the magnetic shunts can be calculated in the same way using (1)-(3), taking into account that the values of the coefficients \(k_{h}\), \(\sigma\), \(G\) and \(Vo\) change depending on the material used. Mild steel is used for the tank and yoke beams, while grain-oriented silicon steel (GOES) and amorphous silicon steel were analysed to be used for the magnetic shunts.