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Linear-implicit time-step methods of the Rosenbrock type are proposed for the time integration of the nonlinear differential-algebraic systems of equations that arise in transient magnetic field simulations. These methods avoid the iterative solution of the nonlinear systems due to their built-in Newton procedures. Embedded lower order schemes allow an error-controlled adaptive time stepping to take into account the dynamics of the underlying process. Numerical tests show the applicability of these methods for error-controlled adaptive time integration of nonlinear magnetodynamic problems and the comparison to established singly diagonal implicit Runge-Kutta methods shows the benefits and possible problems of these specialized Runge-Kutta methods.