Understanding the magnetic polarizability tensor

Abstract

The aim of this paper is provide new insights into the properties of the rank 2 polarizability tensor } } M proposed in (P.D. Ledger and W.R.B. Lionheart Characterising the shape and material properties of hidden targets from magnetic induction data, IMA Journal of Applied Mathematics, doi: 10.1093/imamat/hxv015) for describing the perturbation in the magnetic field caused by the presence of a conducting object in the eddy current regime. In particular, we explore its connection with the magnetic polarizability tensor and the Pólya-Szegö tensor and how, by introducing new splittings of } } M, they form a family of rank 2 tensors for describing the response from different categories of conducting (permeable) objects. We include new bounds on the invariants of the Pólya-Szegö tensor and expressions for the low frequency and high conductivity limiting coefficients of } } M. We show, for the high conductivity case (and for frequencies at the limit of the quasi-static approximation), that it is important to consider whether the object is simply or multiply connected but, for the low frequency case, the coefficients are independent of the connectedness of the object. Furthermore, we explore the frequency response of the coefficients of } } M for a range of simply and multiply connected objects.

Generalised magnetic polarizability tensors

“…Explicit formulae for the calculation of polarizability tensors have been found in a range of different electromagnetic applications. These include the leading order term in an expansion of the perturbed magnetic field ( H α − H 0 )( x ) as α →0 for a (multiply connected) permeable object with β 0 ( B )=1, β 1 ( B )≥0, β 2 ( B )≥0 in magnetostatics and expansions of ( H α − H 0 )( x ), and of the perturbed electric field ( E α − E 0 )( x ) as α →0 in electromagnetic scattering by simply connected dielectric, permeable, or conducting objects with β 0 ( B )=1, β 1 ( B )= β 2 ( B )=0 . In the aforementioned cases, is a suitably parameterised rank 2 Póyla‐Szegö tensor, and its coefficients can be computed by solving 3 scalar transmission problems or through the solution of 3 integral equations for given α , B , and material contrast c .…”

“…This result also provides a solid mathematical footing for denoting as the rank 2 magnetic polarizability tensor (MPT), which the electrical engineering community predict for describing the characteristics of a conducting object in metal detection, eg, previous studies . In Ledger and Lionheart, we have obtained further results that relate the coefficients of to in the limiting case of ω →0, and, independent of the value of β 1 ( B ). For the limiting case of σ ∗ → ∞ , we have also shown that the coefficients of tend to those of if β 1 ( B )=0.…”

“…These results allow permeable and nonpermeable objects to be distinguished and some topology information to be extracted. Furthermore, we have computed for a range of simply and multiply connected objects using a h p ‐finite element approach and explored how their coefficients vary over a range of frequencies, within the validity of the eddy current model …”

Classification of materials for conducting spheroids based on the first order polarization tensor

“…In applied mathematics and analysis, previous studies by [1,14,15,16,17,18,19] have shown that the perturbation due to the presence of the conducting objects can be represented by asymptotic formulas and PT is actually the dominant term in the formula. The asymptotic formula and the corresponding PT are generally not the same for different systems.…”

“…This will increase the chance of correctly identifying the target. Previous studies comparing the PT obtained by using explicit formula and field measurements were done by [5,18,24].…”