The theorem about the transformer excitation current waveform mapping into the dynamic hysteresis loop branch for the sinusoidal magnetic flux case

This paper analyses aspects of the approximation theory application on the certain subsets of the measured samples of the transformer excitation current and the sinusoidal magnetic flux. The presented analysis is performed for single-phase transformer case, Epstein frame case and toroidal core case. In the paper the theorem of direct mapping the transformer excitation current in the stationary regime is proposed. The excitation current is mapped to the dynamic hysteresis loop branch (in further text DHLB) by an appropriate cosine transformation. This theorem provides the necessary and satisfactory conditions for above described mapping. The theorem highlights that the transformer excitation current under the sinusoidal magnetic flux has qualitatively equivalent information about magnetic core properties as the DHLB. Furthermore, the theorem establishes direct relationship between the number of the transformer excitation current harmonics and their coefficients with the degree of the DHLB interpolation polynomial and its coefficients. The DHLB interpolation polynomial is calculated over the measured subsets of samples representing Chebyshev nodes of the first and the second kind. These nonequidistant Chebyshev nodes provides uniform convergence of the interpolation polynomial to the experimentally obtained DHLB with an excellent approximation accuracy and are applicable on the approximation of the static hysteresis loops and the DC magnetization curves as well.