An approach for directly coupling transient magnetic fields and electric circuits is presented. The circuit can contain arbitrary connected solid conductors located in the magnetic field region. Formulations associated with both nodal method and loop method for coupling fields and circuits are deduced and compared. It is found that the structures of the system equations of the two methods are analogous. The proposed formulations allow the equations in stranded windings and solid conductors to be unified and the coefficient matrix of the system equations to be symmetrical. In order to reduce the solution domain the periodic boundary conditions are still applicable when the solid conductors are involved. The developed modeling technique has been applied to the simulation of electric machines. The first example is to calculate the Input phase current and output torque when a single-phase induction motor with shaded rings is In locked-rotor operation. The second example is to simulate the sudden short-circuit of a synchronous generator with starting cage. The third example Is to study the phase current waveform of an induction motor when the rotor bars are broken. The last example is to Investigate the parasitic capacitive impact of the surge voltage on a winding due to drive switching and cable ring.