In this paper, a multiscale induction machine model for the simulation of diverse transients in power systems is developed in the dq0 domain. The proposed model includes main flux saturation. The equations that describe the electrical performance of the induction machine are represented through analytic signals whose Fourier spectrum can be shifted. In addition to the time-step size, the shift frequency appears as a novel simulation parameter. When the shift frequency is set to zero, then the model processes instantaneous signals for the representation of natural waveforms. When the shift frequency is set to the frequency of the stator electrical quantities in the dq reference frame, the Fourier spectra of the analytic signals are shifted to become dynamic phasors that allow for efficient envelope tracking. The setting of the shift frequency is adapted to the transients observed during simulation. Test cases verify the proposed multiscale saturable induction machine model and show that it supports accurate and efficient simulation of both electromagnetic and electromechanical transients within a simulation run.