IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 11, NOVEMBER 2015 7002904 Modeling the Field Inside a Soft-Magnetic Boundary Using Surface Charge Modeling D. T. E. H. van Casteren, J. J. H. Paulides, and E. A. Lomonova Eindhoven University of Technology, Eindhoven 5612AZ, The Netherlands Recent advances have been made into modeling the soft-magnetic materials using the surface charge model. Until now, the model has only been used to model the field outside the soft-magnetic material. To include the saturation in the future, the magnetic field inside the material should be known. In this paper, the model is used to calculate the field inside the material when three different shapes of boundaries are considered. Comparing the results with Finite Element Method shows that the differences in the air region are <1 mT and those in the soft-magnetic region are <4 mT. Larger discrepancies can, however, occur near the corner of the material. Index Terms— Electromagnetic, relative permeability modeling, surface charge. I. I NTRODUCTION M AGNETS in free space can be modeled using the surface charge model [1]. This model assumes that a magnet can be represented as a layer of positive charges on the top surface of the magnet and a layer of negative charges on the bottom of the magnet. In the case of cuboidal magnets, this results in analytical field and force [2], [3] equations. Other shapes are also possible, such as cylindrical [4], [5] and spherical [6], although this does not always result in full analytical solutions. Until recently, the surface charge model was only capable of modeling the relative permeability of a material by implement- ing the method of images [7]. This assumes that the material is infinitely long, such that the boundary is also infinitely long. Furthermore, the field is not always valid inside the material and the angle between two boundaries should always be equal to (π/ n), where n = 1, 2,... To overcome these limitations, in [8]–[10], the equations of the charge model have been improved, and the relative per- meability can be included for cuboidal, soft-magnetic blocks. This is obtained by creating a position-dependent charge on the surfaces of the block. These equations are, however, only tested in the region outside the soft-magnetic material. In this paper, the model is used to describe the field inside a soft- magnetic boundary, which is the first step toward modeling the nonlinear effects, such as saturation and hysteresis. In this paper, first, the equations for the position-dependent charge are explained. Second, the three different boundaries used to verify the model are described, and the results of the model are compared using the Finite Element Method (FEM). Finally, the conclusions are given. II. EQUIVALENT SURFACE CHARGE The standard magnetic charge model uses a uniform charge distribution to represent a magnet. These charges are located Manuscript received March 20, 2015; revised May 8, 2015; accepted June 8, 2015. Date of publication June 10, 2015; date of current version October 22, 2015. Corresponding author: D. T. E. H. van Casteren (e-mail: d.t.e.h.v.casteren@tue.nl). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2015.2443177 Fig. 1. Boundary between the two regions. on the surface of the magnetic material. The method assumes a relative permeability, μ r , equals to one. In [11], it is shown that materials with μ r  = 1 could also be modeled using the surface charge model. The charge density, σ , is calculated by σ = ⇀ B r μ 0 μ r · ⇀ n +  μ r - 1 μ 0 μ r  ⇀ B · ⇀ n (1) where n is the normal vector. The charge of a material is now not only dependent on the remanent magnetization, B r , but also dependent on the flux density, B , on that surface. To explain how this equation works, the situation described in Fig. 1 is used. Here, the boundary between the two regions, Region I and Region II, is shown. Each region has its own relative permeability, μ I r and μ II r , and surface charge density, σ I and σ II , respectively. For Region I, the charge density of (1) is written as σ I = ⇀ B I r μ 0 μ I r · ⇀ n I +  μ I r - 1 μ 0 μ I r  ⇀ B I · ⇀ n I (2) where magnetic flux density in Region I can be described as ⇀ B I = μ 0 μ I r ⇀ H I + ⇀ B I r . (3) This results in σ I = ⇀ B I r μ 0 μ I r · ⇀ n I +  μ I r - 1 μ 0 μ I r  ( μ 0 μ I r ⇀ H I + ⇀ B I r ) · ⇀ n I (4) = ⇀ B I r μ 0 · ⇀ n I + ( μ I r - 1 ) ⇀ H I · ⇀ n I (5) = ⇀ B I r μ 0 · ⇀ n I + ( μ I r - 1 )( ⇀ H I ext + ⇀ H s + ⇀ H I (σ II ) ) · ⇀ n I (6) 0018-9464 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.