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A new, systematic way of coupling low-frequency finite-element (FE) models with circuit systems is proposed using a modified nodal analysis framework. Solid and filamentary conductors can be equally treated as magnetic vector potential-controlled voltage sources. Their matrix contributions to the coupled system are basically the same, leading to a natural way of adding conductors to arbitrary circuit topologies. Algebraic equations of the same type describe the coupling between FE and circuit models, for both massive and filamentary conductors. Any class of independent and dependent source can be considered as well as resistances, inductances, capacitances, transformers, auto-transformers and magnetically-coupled branches. The resulting matrix for the coupled problem is symmetric if the circuit configuration does not contain voltage- or current-controlled sources. The methodology is developed for the general nonlinear time-harmonic problem, but it can be extended to the transient case. An induction motor test case is employed to show the power of the new coupling approach. Comparison of the results against a well-known, validated commercial software (Flux2D) shows that the code developed in this work performs efficiently.