Analytical formulation for magnetic shields taking into account hysteresis effects in the Rayleigh region

Peter Sergeant (Ghent University, Ghent, Belgium)
Luc Dupré (Ghent University, Ghent, Belgium)
Lode Vandenbossche (Ghent University, Ghent, Belgium)
Jan Melkebeek (Ghent University, Ghent, Belgium)



To study the magnetic shielding and the losses of non‐linear, hysteretic multilayered shields by using fast to evaluate analytical expressions.


In order to evaluate the shield in the frequency domain, the non‐linear shield is divided into a sufficient number of piecewise linear sublayers. Each sublayer has a permeability that is constant (space independent) and complex (to model hysteresis). This expression for the permeability is found from the Preisach model by a Fourier transform. Once H is known in the entire shield, analytical expressions calculate the eddy current losses and hysteresis losses in the material. The validity of the analytical expressions is verified by numerical experiments.


In the Rayleigh region, the shielding factor of perfectly linear material is better than the one of non‐linear metal sheets, but also the eddy current losses are higher. The results of the optimization show that steel is only a useful shielding material at low frequencies.

Research limitations/implications

The analytical method is valid for infinitely long shields and for weak imposed fields in the Rayleigh region.

Practical implications

As the analytical expressions can be evaluated very fast (in comparison with slow finite elements models), many magnetic shields can be compared in parametric studies.


Analytical expressions exist for the shielding factor and the losses of linear materials. In this paper, the method is extended for non‐linear hysteretic materials. The effects of several parameters (material parameters, incident fields parameters) on the shielding and the losses are shown.



Sergeant, P., Dupré, L., Vandenbossche, L. and Melkebeek, J. (2005), "Analytical formulation for magnetic shields taking into account hysteresis effects in the Rayleigh region", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 24 No. 4, pp. 1470-1491.

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