IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010 2915 Inclusion of Eddy Currents in Laminations in Two-Dimensional Finite Element Analysis J. Pippuri, A. Belahcen, E. Dlala, and A. Arkkio Department of Electrical Engineering, School of Science and Technology, Aalto University, FI-00076 Aalto, Finland The inclusion of eddy currents in electrical steel sheets in a two-dimensional (2-D) finite element analysis is studied. For the eddy- current modeling of the sheets a so-called one-dimensional (1-D) approach is applied. Two different techniques for representing the 1-D eddy-current solution within the 2-D field equations are shown. The properties of the two techniques are analyzed by utilizing two simple example geometries. The implementations of the computational algorithm of the coupled 2-D-1-D method are verified by analytical equations. A proper coupling method that accords with the computation results is suggested. Index Terms—Eddy currents, ferromagnetic materials, finite element methods, losses. I. INTRODUCTION A TWO-DIMENSIONAL (2-D) field analysis is com- monly applied to study electromechanical applications. Although such an approach is sufficient in many cases, certain phenomena, such as eddy currents flowing in electrical steel sheets, are by definition excluded. For modeling the eddy currents within the 2-D analysis, a so-called one-dimensional (1-D) approach is one method that has been proposed [1]–[5]. In this work, this approach is adopted. The novelty of this work lies in the careful analysis of the coupling, i.e., how the 1-D eddy-current solution can be reliably included in the 2-D equations. This issue has not been tackled in the literature so far. For the coupling purpose, two approaches are shown. As a measure of the correctness of the approaches, the power balance of the combined 2-D–1-D method is considered. The implementations of the computational algorithm of the com- bined 2-D–1-D method are verified by analytical equations. The results obtained show the scale of errors to which an improper coupling of the two formulations might lead, thus emphasizing the importance of the topic. II. COMBINED 2-D–1-D METHOD A. 1-D Eddy-Current Formulation of Electrical Steel Sheets The 1-D eddy-current model of the electrical steel sheets is developed in the lamination depth. The magnetic vector poten- tial and current density are defined as [1] (1) (2) in which denotes a Cartesian spatial coordinate and and are the unit vectors of the - and -axes, respectively. The magnetic vector potential in the laminations satisfies (3) Manuscript received December 22, 2009; accepted February 16, 2010. Cur- rent version published July 21, 2010. Corresponding author: J. Pippuri (e-mail: jenni.pippuri@tkk.fi). 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.2010.2044490 (4) is the reluctivity given as a spline function of the square of the magnetic flux density, and are the electrical conductivity and half of the thickness of the laminations, respectively, and and are the - and -components of the 2-D magnetic flux density. B. 2-D Field and Winding Equations In the 2-D scheme the magnetic vector potential and current density are given as (5) (6) in which and denote Cartesian spatial coordinates, denotes time, and denotes the unit vector of the -axis. The electromagnetic field in the applied geometries fulfills [6] (7) is the reluctivity given as a spline function of the square of the magnetic flux density, is the component of the magnetic field strength due to the eddy currents, is the number of turns in series, is the current in the coil, and is the cross sectional area of the coil. The voltage in the coil satisfies (8) in which denotes the length of the core of the machine and the DC resistance of the coil. C. Coupling of the 2-D and 1-D Models and Their Solution For the discretization of the coupled problem, the Backward Euler and finite element method are applied. The nonlineari- ties of both 1-D and 2-D equations are handled by the Newton- Raphson technique. In the 2-D scheme, the Newton-Raphson technique is applied in an incomplete manner. Incomplete, as the terms originating from are omitted from the Jacobian matrix. In part resulting from this, under-relaxation is required for convergence. In the derivations, the eddy-current component, , is de- fined as the difference of the 1-D magnetic field strength, , at 0018-9464/$26.00 © 2010 IEEE