An efficient finite element formulation for transient eddy current calculations in thin conductors is presented and corroborated numerically. In an attempt to develop a simple yet robust finite element formulation, the magnetic field vector is directly chosen as the state variable and decomposed into two parts (the incident and reactive fields respectively) and a set of suitable boundary conditions is constructed. Finite-element implementation of the formulation is described. A penalty method is utilized to enforce the divergence-free condition for the magnetic flux. Since only the conducting solid needs to be discretized and lower-order polynomials are utilized, the proposed direct field formulation is computationally economical and versatile, as demonstrated in numerical examples. The numerical results of example problems are compared to known analytical solutions or experimental results.