# 3-D Integral Formulation for Thin Electromagnetic Shells Coupled with an External Circuit
## 1 Introduction  
All of the electromagnetic phenomena occurring in the electrical systems are described by [PERSON]'s equations, together with the constitutive material laws. It is a set of partial differential equations associated with the relationships of electromagnetic field (\(\mathbf{E}\), \(\mathbf{H}\)) distributing into space and varying into time, the distribution of currents and charges (\(\mathbf{J}\), \(\mathbf{\rho}\)) and material properties (\(\mathbf{\mu}\), \(\mathbf{\sigma}\)) [1,2]. In order to compute the electromagnetic field (i.e., solutions to [PERSON]'s equations), an analytical method or a numerical method can be used. For these devices with simple geometries, a correct analytical solution can be identified. However, in some general cases, especially for the complex structure of electrical devices, the numerical methods are used as a sole solution. The numeric methods applied in modelling of electromagnetic fields can be divided into two categories: finite methods like FEM (Finite Element Method), FVM (Finite Volume Method) and the numerical integration methods such as BEM (Boundary Elements method), MoM (Methods of Moments), PEEC method (Partial Element Equivalent Circuit) [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The choice of an appropriate method totally depends on the physical phenomena that need modelling such as the physical phenomena in high or low frequencies, with or without magnetic material, considering the effect of inductance or capacitance, external excitation source. However, there is no universal, optimal method for these problems, and the choice of the most suitable method depends on the nature of the electrical devices and their operation range. Generally, the electromagnetic modelling of structures including thin shell model and thin wires is a complex problem in the fields of electrical engineering. Its geometry is characterized by a high ratio of the length and the thickness. Thus, the use of a volume mesh leads to a large number of elements and/that makes this method expensive and time consuming to apply to practical devices. Furthermore, due to the skin effect, when the skin depth \(\delta\)is thinner than the thickness \(e\), the size of the studied meshes must besmaller than that of the skin depth. This leads to the difficulties when the thickness of some parts of the structures is very small in comparison with the overall size of the actual devices. The shell element formulations have recently been developed by many authors in order to cope with difficulties in thin plate models, e.g., with the boundary element method (BEM) [3], with the finite element method (FEM) [5] and with the integral methods [11; 14; 15; 16]. In [11], we presented a coupling method between the Partial Element Equivalent Circuit (PEEC) and an integro-differential method to model the devices that include thin electromagnetic shells and complicated conductor systems. However, this coupling formulation cannot be applied to problems when the skin depth is low in comparison with the thickness of the thin shell. In [14], the authors presented a hybrid of volume and surface integral formulations for the eddy current solution of the conductive regions with arbitrary geometry. However, this formulation cannot be used for the magnetic material and as in [11], the skin effect is not yet considered. In [16], we developed a general shell element formulation for modelling of thin magnetic and conductive regions. The thin shell is modelled by an integral method to avoid meshing the air region. As in [3; 5; 15], the field variation through the thickness is considered. Based on a simple discretization of the shell averaged surface, the number of unknowns is greatly reduced. However, this formulation is not suitable for model systems with complex conductive structures.