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.