Three families of transmission conditions of different order are proposed for thin conducting sheets in the eddy current model. Resolving the thin sheet by a finite element mesh is often not possible. With these transmission conditions only the middle curve, but not the thin sheet itself, has not to be resolved by a finite element mesh. The families of transmission conditions are derived by an asymptotic expansion for small sheet thicknesses $\eps$, where each family results from a different asymptotic framework. In the first asymptotic framework the conductivity remains constant, scales with $1/\eps$ in the second and with $1/\eps^2$ in the third. The different asymptotics lead to different limit conditions, namely the vanishing sheet, a non-trivial borderline case, and the impermeable sheet, as well as different transmission conditions of higher orders. We investigated the stability, the convergence of the transmission conditions as well as their robustness. We call transmission conditions robust, if they provide accurate approximation for a wide range of sheet thicknesses and conductivities. We introduce an ordering of transmission conditions for the same sheet with respect to the robustness, and observe that the condition derived for the $1/\eps$ asymptotics is the most robust limit condition, contrary to order 1 and higher, where the transmission conditions derived for the $1/\eps^2$ asymptotics turn out to be most robust.
Resolving thin conducting sheets for shielding or even skin layers inside by the mesh of numerical methods like the finite element method (FEM) can be avoided by using impedance transmission conditions (ITCs). Those ITCs shall provide an accurate approximation for small sheet thicknesses $d$, where the accuracy is best possible independent of the conductivity or the frequency being small or large -- this we will call robustness. We investigate the accuracy and robustness of popular and recently developed ITCs, and propose robust ITCs which are accurate up to $O(d^2)$.