Summary: The \({\mathbf A}\)-\(\phi\oplus {\mathbf A}\) schemes for 3D eddy current model based on ungauged potentials are presented. The finite element error estimates of this method in finite time \(T\) are given. And it is verified that provided the time-stepsize \(\tau\) is sufficiently small, the proposed algorithm yields for finite time \(T\) an error of \({\mathcal O}(\sqrt{\tau^2+h^2})\) on coupled \({\mathbf A}\)-\(\phi \oplus{\mathbf A}\) scheme in the \(L^2\)-norm for the electric field \({\mathbf E}\) in the eddy-current domain \(\Omega_1\) and the magnetic field \({\mathbf H}\) in the whole domain \(\Omega\), and of \({\mathcal O}(\sqrt{\tau+\tau^{-1}h^4+h^2})\) on two decoupled \({\mathbf A}\)-\(\phi\oplus{\mathbf A}\) scheme for the electric field \({\mathbf E}\) in the eddy-current domain \(\Omega_1\) and the magnetic field \({\mathbf H}\) in the whole domain \(\Omega\), where \(h\) is the mesh size.