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High-speed induction machines equipped with a solid steel rotor are capable of achieving a high-rotating speed than other types of machines, because of their simpler and more robust rotor structure. At the same time, however, the eddy-current losses in the solid rotor may be critical, because of the high conductivity of the rotor material, which makes it easy for axial eddy currents to travel in the solid rotor. To efficiently mitigate the rotor eddy-current losses, it is important to accurately determine the rotor eddycurrent losses induced by a particular harmonic in advance. In this paper, an improved virtual permanent magnet harmonic machine (VPMHM) model equipped with a sinusoidally magnetized virtual magnet based on the finite element method (FEM) is proposed for determination of the rotor eddy-current harmonic losses. The 2-D fast Fourier transform was used to accurately analyze the time-spatial air-gap flux density harmonics. The VPMHM model was enhanced to ensure that it was able to exactly produce the required flux density harmonics in the air gap. Two algorithms for the improved VPMHM models with different hybrid excitations were proposed to determine the harmonic losses together with the other important harmonic behavior. The model was further investigated to separate the electromagnetic transients from different harmonics. Finally, the simulation time for the harmonic losses required by the enhanced VPMHM model was significantly reduced by separating the harmonic transients. All the results and conclusions presented in this paper are based on the FEM analysis. INDEX TERMS Solid-rotor high-speed induction machine (IM), finite element method (FEM), 2-D fast Fourier transform, eddy-current losses, improved virtual permanent magnet harmonic machine (VPMHM).

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