We extract the lumped-parameter model of a wound-rotor synchronous machine from its physics-based magnetic-equivalent circuit model. Model extraction is formulated as a weighted least square optimization with nonlinear constraints in which time-domain trajectories of flux linkages, currents, and the electromagnetic torque are used as input data to obtain the parameters of the qd0 model of the machine. The resulting problem is non-convex and cannot be solved using standard methods. The optimization problem is, therefore, convexified using a cone programming relaxation. The solution to the relaxed problem is used as an initial point for the interior-point method, leading to a reliable framework. Accurate estimations on stator resistance, leakage and mutual inductances in stator and rotor, rotor speed, effective turns-ratio between the field and stator windings, and the number of poles are obtained. Estimated parameters are validated against measured and estimated values reported in literature, and are used to develop a behavioral qd0 macromodel of the machine.