The resistive limit response of an ellipsoidal conductor: a magnetostatic formulation

Abstract

Interpreting transient electromagnetic (TEM) anomalies in terms of conductive rectangular plates is effective in many situations. However, not all conductors are thin and planar. Triaxial ellipsoid conductors are an attractive alternative: geometrically simple (corner-free), mathematically tractable at early and late time limits, and able to encompass shapes ranging from discs to elongate lenses to equi-dimensional pods. Accordingly a fast magnetostatic algorithm has been developed to compute the resistive limit (RL) response of a ellipsoidal conductor, which may also be permeable. The algorithm has been validated against new analytic resistive limit solutions for spherical and spheroidal conductors and against 3D multigrid finite difference modelling for a triaxial ellipsoidal conductor. A uniformly conductive ellipsoid supports three fundamental current modes in the resistive limit, an independent mode for excitation parallel to each of the principal axes. The RL current density increases linearly with radial distance from the ellipsoid centre. A formula for the time constant of an oblate spheroid has been derived for excitation parallel to its rotational axis, namely ${\tau }_3\approx \sigma {\mu }_{0}bc/(4+6c/b)$, where $\sigma$ is the conductivity and $c$ and $b$ are respectively the minor and major radii.

KEYWORDS:

• Ellipsoid
• resistive limit
• electromagnetics
• modelling

Acknowledgements

I am indebted to the late Dr. Yves Lamontagne (Lamontagne Geophysics) who kindly modelled triaxial ellipsoid RL responses using his MGEM 3D multigrid finite difference program and who provided physical insights. Dr. David Clark (CSIRO/Integrated Magnetics LLC) assisted with theoretical aspects, and in particular confirmed my derivation of equation (22) for the RL field inside a spherical conductor.

Disclosure statement

No potential conflict of interest was reported by the author(s).