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Properties of generalized magnetic polarizability tensors

Ledger, Paul

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We present new alternative complete asymptotic expansions for the time harmonic low‐frequency magnetic field perturbation caused by the presence of a conducting permeable object as its size tends to zero for the eddy current approximation of the Maxwell system. Our new alternative formulations enable a natural extension of the well‐known rank 2 magnetic polarizability tensor (MPT) object characterization to higher order tensor descriptions by introducing generalized MPTs (GMPTs) using multi‐indices. In particular, we identify the magnetostatic contribution, provide new results on the symmetries of GMPTs, derive explicit formulae for the real and imaginary parts of GMPT coefficients and also describe the spectral behavior of GMPT coefficients. We also introduce the concept of harmonic GMPTs (HGMPTs) that have fewer coefficients than other GMPT descriptions of the same order. We describe the scaling, translation and rotational properties of HGMPTs and describe an approach for obtaining those HGMPT coefficients that are invariant under the action of a symmetry group. Such an approach is one candidate for selecting features in object classification for hidden object identification using HGMPTs.


Ledger, P., & Lionheart, W. R. B. (2023). Properties of generalized magnetic polarizability tensors. Mathematical Methods in the Applied Sciences, 46(5), 5604-5631.

Journal Article Type Article
Acceptance Date Oct 19, 2022
Online Publication Date Nov 19, 2022
Publication Date Mar 30, 2023
Publicly Available Date May 30, 2023
Journal Mathematical Methods in the Applied Sciences
Print ISSN 0170-4214
Electronic ISSN 1099-1476
Publisher Wiley
Peer Reviewed Peer Reviewed
Volume 46
Issue 5
Pages 5604-5631
Keywords asymptotic analysis; eddy current; inverse problems; magnetic polarizability tensor; metal detection; spectral problems
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