Semi–analytical method for the identification of inclusions by air–cored coil interaction in ferromagnetic media
Résumé
The magnetostatic vector fields in terms of harmonic scalar potentials scattered by near–surface air inclusion of
arbitrary shape, embedded in a conductive ferromagnetic medium, are investigated. The hollow inclusion is illuminated
by a current–carrying coil, which serves as the primary field. The do-main of interest is separated into homogeneous
subdomains under the assumption of a suitable truncation of the region of magnetostatic activity at a long distance
from the incident source. Therein, the field is considered negligible and consequently a perfect magnetic boundary
condition is implied. On the other hand, the introduced methodology addresses the full coupling between the two
interfaces, i.e. the plane that distinguishes the half–space ferromagnetic material from the open air and the arbitrary
surface among the inclusion and the ferromagnetic region. To this end, continuity conditions are applied in a rigorous
way, while the expected behavior of the fields, either as ascending or as descending are taken into account. The
scattering problem is solved by means of a modal approach, where potentials associated with the half–space are
expanded via cylindrical harmonic eigenfunctions, while those related with the inclusion’s arbitrary geometry admit a
generalized–type formalism, being the key to our method. However, since the transmission conditions involve
potentials with different eigenexpansions, we are obliged to rewrite cylindrical to generalized functions and vice versa,
obtaining handy relationships in terms of easy–to–handle integrals, where orthogonality then would be feasible. Once
done, the calculation of the exact solutions leads to infinite linear algebraic systems, whose solution is achieved
trivially through standard cut–off techniques. Thus, we obtain the implicated fields in a general analytical and compact
fashion, independent of the inclusion’s geometry. In order to demonstrate the efficiency of the analytical model
approach, we assume the degenerate case of a spherical inclusion, whereas the air–cored coil simulation via a
numerical procedure validates our generalized method. The calculation is very fast, rendering it suitable for use with
parametric inversion algorithms.
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