IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 535 A New Efficient Technique for Harmonic-Balance Finite-Element Analysis of Saturated Electromagnetic Devices O. Deblecker and J. Lobry Electrical Engineering Division, Faculté Polytechnique de Mons, 7000 Mons, Belgium The paper presents a new efficient technique for the harmonic-balance finite-element modeling of magnetically saturated electromag- netic devices. It is based upon the transmission-line modeling method that leads to a numerical scheme where the field-circuit coupled problem harmonic solutions are obtained iteratively. A superposition principle applies due to the substitution of a fictitious linear mate- rial to the nonlinear one. This amounts to significant savings both in computation time and storage requirements. The method is validated by applying it to a two-dimensional voltage-driven model of a three-phase inductor. Index Terms—Finite-element methods, harmonic analysis, iterative methods, magnetic saturation. I. INTRODUCTION S TEADY-STATEfinite-element (FE) analysis is a vital tool in the design of saturable electromagnetic devices. Unlike time-stepping methods, the harmonic-balance (HB) approach [1], [2], solves the FE magnetic field problem in the frequency domain. Yet, as it requires the cumbersome assembly and expen- sive solution (in term of CPU time) of a much larger and denser system of nonlinear equations, its use is not widespread. In this paper, a novel approach to the HBFE modeling of electromag- netic devices, based on the transmission-line modeling (TLM) technique [3], is presented that removes these drawbacks. The nonlinear material is replaced by a fictitious linear one which allows to apply a superposition principle. Thence, the field-cir- cuit coupled problem solutions at all harmonic frequencies are obtained separately in an iterative fashion from a much smaller system matrix. The presented HB approach is validated by ap- plying it to a two-dimensional (2-D) voltage driven model of a three-phase inductor. II. FIELD-CIRCUIT COUPLED PROBLEM To formulate the field-circuit coupled problem, we consider a 2-D domain partitioned into electrically conducting and nonconducting regions. The conduction region of is the union of cross sections of stranded conductors and the (saturable) nonconducting region is denoted . All electromagnetic quantities are assumed to vary periodically in time with fundamental angular frequency . With each stranded conductor, we associate the current and the voltage drop . The number of windings and the area of the stranded conductors are given by and , respectively. Finally, we denote by the magnetic reluctivity that is generally field-dependent . Digital Object Identifier 10.1109/TMAG.2006.870930 Fig. 1. Electric analog circuit of magnetic field problem. The 2-D magnetic field problem is formulated using the component of the magnetic vector potential . By in- troducing the notation , the problem can be stated as on and on (1) supplied with appropriate boundary conditions. As in [4], it can be demonstrated that (1) is a differential for- mulation of Kirchhoff’s current law of an electric analog cir- cuit, as shown in Fig. 1, made of distributed nonlinear resis- tors connected in a Cartesian topology above a ground refer- ence plane. Indeed, the partial derivatives of may be seen as the voltage drops across the resistors per unit length along the coordinate axes. Thence, and give the com- ponents of the currents flowing through the nonlinear resistors per unit width. According to the Pouillet’s law, the conductance value of those elements is either along the axis or along the axis. In this equivalent representation, boundary conditions appear as nodes with fixed voltage (with respect to the ground) for the Dirichlet condition or open-cir- cuited nodes for the Neumann condition, whereas the electrical excitation results in locally injected currents. 0018-9464/$20.00 © 2006 IEEE