The $\mathbf{A}-\varphi-\mathbf{B}$ magnetodynamic Maxwell system given in its potential and space-time formulation is a popular model considered in the engineering community. It allows to model some phenomena such as eddy current losses in multiple turn winding. Indeed, in some cases, they can significantly alter the performance of the devices, and consequently can no more be neglegted. It turns out that this model is not yet analytically studied, therefore we here consider its well-posedness. First, the existence of strong solutions with the help of the theory of Showalter on degenerated parabolic problems is established. Second, using energy estimates, existence and uniqueness of the weak solution of the $\mathbf{A}-\varphi-\mathbf{B}$ system is deduced.