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To find the induced currents in conductive objects, the circuit method (CM) replaces these objects by a set of magnetically (mutually) coupled filaments. The unknown induced currents are obtained by solving an electrical circuit and the magnetic field is found by the Biot-Savart law. In the literature, the CM is described as a fast and easy-to-implement alternative for the finite-element method (FEM), with however some limitations: the objects to be modeled in the CM should be nonferromagnetic and much thinner than the penetration depth. In this paper, the CM is extended. It also models ferromagnetic behavior and objects thicker than the penetration depth, by adding extra filaments carrying proper currents. To validate the CM, an axisymmetric shielding problem is solved both with CM and FEM for a conductive shield, a nonlinear hysteretic ferromagnetic shield, and a double (multilayer) shield.