This letter focuses on the dynamic modeling of synchronous machines and proposes a general exact equation to compute the field current of the rotor winding for transient stability analysis. The proposed approach is based on the semi-implicit formulation of differential-algebraic equations, which allows expressing the filed current in terms of the times derivatives of machine state variables, rather than in terms of the variables themselves. This leads to an expression which is compact, linear, and model independent and, thus, easily implementable in power system software tools. This letter illustrates the derivation of the equation leading to the field current for different machine models and shows, through a 21,177 bus model of the ENTSO-E transmission grid, how the derived equation reduces the computational burden of power system models.