## 2 Problem Definition
### Eddy Currents in an Open-Type Core  
The focus of the paper is on the open-type transformer core, which belongs to a set of simply connected laminated media, such as the one shown in Figure 1. The problem of eddy currents in an open-type transformer core is characterized by certain specifications which are not necessarily present in the case of a closed-type transformer core. In the open-type core, eddy currents of density \(\vec{J}\) are mostly concentrated in the outer lamination sheets. Furthermore, in the outer lamination sheets of the open-type core, eddy currents \(\vec{J}\) are predominantly induced by the perpendicular component of the magnetic induction \(\vec{B}\), with respect to the lamination sheets. On the other hand, towards the interior of the open-type core, eddy currents \(\vec{J}\) are predominantly induced by the tangential component of the magnetic induction \(\vec{B}\), with respect to the lamination sheets. It is then useful to separate the normal and tangential components of magnetic induction \(\vec{B}\) and the eddy current density \(\vec{J}\) in the calculation itself. Therefore, in the local orthogonal \(\alpha\beta\gamma\)-coordinate system of a single lamination sheet shown in Figure 1, induction \(\vec{B}\) can be written as the sum of the normal component \(\vec{B}_{\gamma}\) and the tangential component \(\vec{B}_{\alpha\beta}\) with respect to the lamination sheet, that is  
\[\vec{B}=\vec{B}_{\gamma}+\vec{B}_{\alpha\beta}, \tag{6}\]  
In accordance with (6), two corresponding equations follow from (1):  
\[\
abla\times\rho\vec{J}_{\alpha\beta} =-\partial_{t}\vec{B}_{\gamma}, \tag{7}\] \[\
abla\times\rho\vec{J}_{\alpha\beta\gamma} =-\partial_{t}\vec{B}_{\alpha\beta}, \tag{8}\]  
where \(\vec{J}=\vec{J}_{\alpha\beta}+\vec{J}_{\alpha\beta\gamma}\) holds. According to (7), the component of the magnetic induction perpendicular to the lamination sheets induces the current density \(\vec{J}_{\alpha\beta}\), which only has tangential components with respect to the lamination sheets. On the other hand, the tangential component of magnetic induction, according to (8), induces eddy currents of density \(\vec{J}_{\alpha\beta\gamma}\) which generally have all three spatial components, as shown in Figure 1.  
Although the \(\vec{J}_{\alpha\beta\gamma}\) and \(\vec{J}_{\alpha\beta}\) are coupled, it is a good approximation to simulate them separately [11]. Because the value of magnetic permeability of the core is significantly lower in the perpendicular than in the tangential direction (with respect to the lamination sheets), \(\vec{J}_{\alpha\beta\gamma}\) has a negligible effect on \(\vec{B}_{\gamma}\), and consequently (according to (7)) a negligible effect on \(\vec{J}_{\alpha\beta}\). Therefore, it is possible to ignore \(\vec{J}_{\alpha\beta\gamma}\) when calculating \(\vec{J}_{\alpha\beta}\); the reverse is not valid because the influence of \(\vec{J}_{\alpha\beta}\) on \(\vec{B}_{\alpha\beta}\) is not always negligible. Because the open-type transformer core is surrounded by a medium of low magnetic permeability (air), a large proportion of the total magnetic flux enters the core perpendicular to the lamination sheets; thus, \(\vec{J}_{\alpha\beta}\) is usually several times larger than \(\vec{J}_{\alpha\beta\gamma}\) in the outer lamination sheets, which further reduces the importance of considering \(\vec{J}_{\alpha\beta\gamma}\) when calculating \(\vec{J}_{\alpha\beta}\). Therefore, in contrast to the closed-type core, the losses due to \(\vec{J}_{\alpha\beta}\) usually represent a major part of the total eddy current losses in the open-type core [12; 13]. In this paper, the emphasis is therefore on the method of calculating the eddy current density \(\vec{J}_{\alpha\beta}\), i.e., losses due to \(\vec{J}_{\alpha\beta}\).