A ferromagnetic material is composed of microscopic domains, with uniform internal magnetization.
Below the Curie temperature, alignment of domains is random, to minimize internal energy.
When subjected to a magnetic field, domains rotate to align with the field, creating friction and heat between domain walls. When the magnetic field is removed, the domains return to their original position, except for a few, locked in defects of the crystal lattice, resulting in a remaining magnetic field (saturation remanence). Magnetostriction causes the material to heat up and also change dimension, creating a low buzzing noise in transformer cores.
When subjected to an alternating magnetic field, extra energy is required to dislodge these aligned domains every time the field is changing.
Soft ferromagnetic materials have less defects in their lattices and are thus less subject to magnetization (low coercivity).
The relationship between the magnetic field and core magnetization is thus not linear. The hysteresis loop shows the relationship for one cycle of the alternating magnetic field. The area comprised between the curves indicates the energy lost for each cycle.
