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The coupling between a three-dimensional modified magnetic vector potential formulation discretized by the finite integration technique and an electrical circuit containing switches is presented. The changes in the states of the switching elements are introduced as changes in the topology of the circuit which results in a structural change of the differential-algebraic system describing the coupled field-circuit problem. This transient formulation is integrated by embedded Runge-Kutta methods. As soon as state changes are detected, the evaluated stage values of the corresponding Runge-Kutta step are used to build the interpolation polynomial with its roots approximating the switching time instants.