  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 characterisation to higher order tensor descriptions by introducing
generalised 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 behaviour 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.
