We present an approach for directly coupling transient magnetic fields and electric circuits. The circuit can contain arbitrarily connected solid conductors located in the magnetic field region. A systematic procedure suitable for both nodal method and loop method is used to couple fields and circuits. The structures of the system equations of the two methods are analogous. The 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. Our approach has been applied to the simulation of electric machines. We give four examples: 1) calculation of the input phase current and output torque when a single-phase induction motor with shaded rings is in locked-rotor operation; 2) simulation of the sudden short circuit of a synchronous generator with starting cage; 3) study of the phase current waveform of an induction motor when the rotor bars are broken; and 4) investigation of the parasitic capacitive impact of the surge voltage on a winding due to drive switching and cable ring.