Causality study and numerical response of the magnetic permeability as a function of the frequency of ferrites using Kramers–Kronig relations

"In this paper, the numerical treatment of magnetic loss of NiZn, MnZn, Ni2Y, and NiZnCu ferrite and their composites, by using Krameres–Kronig relations, is investigated. The complex magnetic permeability spectra for ferromagnetic materials have been studied.
Due to the principle of causality and time independence in the relation between magnetic induction B and magnetic field H, the real and the imaginary part of the complex magnetic permeability are mutually dependent, and the correlation is given by the Krameres–Kronig equations. Through them, it is possible to measure the real component of the complex magnetic permeability, assuming the real component is given, and by the Hilbert transform, the imaginary part of the magnetic permeability can be calculated. Magnetic circuit model has been studied theoretically, focusing on the model’s poles in the complex plane to verify the principle of causality and the temporary independence."