788 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 38, NO. 3, MAY/JUNE 2002 Evaluation of Different Analytical and Semi-Analytical Methods for the Design of ELF Magnetic Field Shields Aldo Canova, Alessandra Manzin, and Michele Tartaglia Abstract—The aim of this paper is the evaluation of different analytical methods devoted to the study of shielding of extremely low-frequency magnetic fields with conducting and ferromagnetic materials. In this paper, an extensive analysis of some approaches is presented, by comparing these methods to a numerical one based on a hybrid finite-element method/boundary-element method for- mulation. The study wants to underline the accuracy of the dif- ferent techniques, evaluating their limits of validity, depending on physical and geometrical shield parameters. Index Terms—Analytical methods, electromagnetic field, ex- tremely low frequency, shielding. I. INTRODUCTION T HE reduction of extremely low-frequency (ELF) magnetic fields produced by electric systems in public areas or in in- dustrial environments is nowadays a very actual problem, both concerning the human exposure and the disturbances for elec- tronic equipment. The importance of this problem is provided by the increasing interest of the scientific community and of the organizations which define the standards. In this direction, na- tional governments are making laws that impose very low limits on electric and magnetic fields, in order to safeguard the health of human beings. The reduction of magnetic field can be obtained by using conductive or ferromagnetic shields. In the former, the induced eddy currents produce an additional magnetic flux, which par- tially cancels those created by the source. In the latter, the high permeability of ferromagnetic materials creates a preferential path, so the magnetic flux is diverted away from the region to be shielded. The design of magnetic shields, which depends on their ge- ometry, their position with respect to the sources and their ma- terials, needs the solution of a complex field problem, which in- volves partial differential equations. Numerical codes allow us Paper ICPSD 01–28–2, presented at the 2001 Industry Applications Society Annual Meeting, Chicago, IL, September 30–October 5, and approved for pub- lication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Power Systems Engineering Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 15, 2001 and released for publication February 25, 2002. A. Canova and M. Tartaglia are with the Dipartimento di Ingegneria Elettrica Industriale, Politecnico di Torino, I-10129 Turin, Italy (e-mail: canova@polel1.polito.it; tarmich@athena.polito.it). A. Manzin is with the Dipartimento di Ingegneria Elettrica Industriale, Po- litecnico di Torino, I-10129 Turin, Italy, and also with the Istituto Elettrotecnico Nazionale “Galileo Ferraris,” I-10135 Turin, Italy (e-mail: manzin@ien.it). Publisher Item Identifier S 0093-9994(02)04522-X. to reach this objective, but for some frequent configurations it is possible to reduce the complexity of the problem, and very fast analytical methods can be employed. For example, a power line system, where the conductors are parallel to each other, can be shielded by a ferrous or nonferrous but conducting plate, parallel to the line system and invariant along its length. In this case, the electromagnetic field analysis regards a two-dimensional (2-D) problem, under the hypothesis that magnetic field lies in the ( ) plane and the current flows along the axis. This problem can be solved by using analytical methods, which provide a closed-form solution of Maxwell’s equations or by transforming the field problem into the solution of a set of well-known more simple ones. In the first category of methods, two analytical techniques are the most popular in the ELF shielding: the method of variable separation (VSM) and the conformal transformation method (CTM), while in the second type of approach, the multiconductor method (MC) represents a simple and powerful way to rapidly get the field solution. Numerical techniques are used to validate the proposed analytical approaches, by comparing both the magnetic den- sity field distribution and the shielding effectiveness. Since the field problem investigated is an open boundary one, a numerical method coupling finite-element method (FEM) with boundary-element method (BEM) is employed, as suggested in [3]. Thus, the domain is divided into two regions: the inner one, including all the conductors, the shield, and a portion of air, is handled by FEM; the external one, homogeneous and infinite, is treated using BEM. II. VSM A general 2-D model with planar layers is shown in Fig. 1. Each layer is labeled by an integer index (where ) and it is characterized by a relative permeability , a conduc- tivity , and a thickness . In the analyzed domain, it is pos- sible to define three regions: a source region where current fila- ments are located (indicated with index 0), a shield region, and a shielded region [indicated with index ( )]. The number of conductors is arbitrary and their location is denoted by ( ), while their current magnitude is indicated with , where . Magnetic field distribution is obtained by considering the magnetic vector potential , which has only one component along the axis (orthogonal to the – plane). For the generic 0093-9994/02$17.00 © 2002 IEEE