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The application of the finite element method to the calculation of nonlinear periodic eddy current problems is discussed. Special attention is paid to the case when specific conditions on the current or voltage of several conductors have to be fulfilled. An efficient iterative procedure has been developed to satisfy these conditions in 2D problems solved by a single component magnetic vector potential with the nonlinearity taken into account. The total current condition can be fulfilled in 3D with a noniterative scheme, but voltage conditions still require a multistep approach. Additionally, a possibility of accelerating the time stepping algorithm in reaching steady state is presented for a 3D eddy current problem involving a single conductor.