Leakage flux effect on a long time-constant of transient eddy current in a solid magnet

Leakage of a magnetic flux of an electromagnet is considered to shorten a magnet gap length effectively which dominates strength of a magnetic flux density in an iron core. Because a transient eddy current occurs inside a core, its time constant becomes longer when the leakage is larger. In addition, the leakage is larger when the permeability of an iron core is larger. On the other hand, Faraday's law which induces an eddy current includes a time derivative of a flux density. Therefore an incremental permeability which is a field derivative of the flux density is considered to dominate a time constant of a transient eddy current. Consequently the time constant can have various values depending on excitation field levels due to hysteresis characteristics. The analytical expression for a transient eddy current is formulated by two-dimensional simultaneous Laplace's transformation for space and time. This expression makes clear that a transient eddy current causes a spatial distribution of a flux density both at a gap and inside a core with a long time constant.