IEEE zyxwvutsrqponm TRANSACTIONS ON MAGNETICS, zyxwvutsrq VOL. 28, N0.2, MARCH 1992 Coupling of a Nonlinear Diffusive Electromagnetic System to a Linear Electric Circuit L. Corti, G. Miano, C. Visone Dipartimento di Ingegneria Elettrica, Universith di Napoli FEDERICO 11, Italy zyxw Abstract zyxwvutsrqp - A composite electromagnetic system including a non-hysteretic ferromagnetic conducting body and a linear resonant circuit fed by a sinusoidal voltage source is studied. The diffusion of the magnetic field in the iron core “inductor” is described by means of a finite dimensional nonlinear dynamic system obtained by applying the Galerkin method. These equations are coupled with the circuit ones. This physical system might exhibit different nonlinear phenomena: multiple periodic oscillations and even apparently completely disordered aperiodic “chaotic” motions. In this paper multiple harmonic and subharmonic oscillations are investigated by computer-aided analysis using a static method. 1307 solution 111, 121. If the material has very low electrical conductivity, the magnetic field diffuses fully in the body and the considered system reduces to a well known zyx itonlinear resonant circuit. In this case multiple harmonic and subharmonic oscillations have been discovered (see, for example, in [3], [41, zyxwv [5]) and for some values of the circuit parameters the solution apparently does not converge to a periodic or an almost periodic solution: it becomes “chaotic”[~j. From a practical point of view the system describes a series ferroresonant circuit (for example, it can model high voltage measuring transformers in high voltage equipments and power plants). The presence of a second T-periodic oscillation or a subharmonic one, which may have a large amplitude, often cause failures in power systems. ’ LINTRODUCTION PD im, The numerical study of “Eddy Currents” induced in a non-hysteretic iron core is still an open problem. When the magnetizing currents are imposed, the discrete model, obtained by applying the Galerkin method, admits a unique steady state solution. In particular if these currents are T-periodic the discrete model admits a unique T-periodic steady-state solution [l], [2]. Instead a surprising wealth of different nonlinear phenomena may appear when the magnetically non linear conducting body is “coupled” to an electric circuit. In this case the time evolution of the magnetizing currents is determined by the circuit dynamic which strictly depends on the magnetic field diffusion. Here we consider the nonlinear electromagnetic system sketched in Fig.la. The iron core “inductor” is fed by the sinusoidal voltage source e(t)=Ecosot through a capacitor in series with a resistor. The only nonlinear element is the “inductor”. We assume that the magnetic behavior of the material is nonlinear without hysteresis, while its electrical behavior is linear. Furthermore the magnetic and electric properties of the body are supposed to be homogeneous, isotropic and time-invariant. The magnetic characteristic of the iron core is strictly increasing and its differential permeability and reluctivity are bounded. Interest in this problem has arisen from both practical and theoretical reasons. From the theoretical point of view it is the simplest electromagnetic system that does not satisfy all the hypotheses which guarantee uniqueness and global asymptotic stability of the periodic hlnnuscript received July 7, 1991. This work was supported in part by h4inistero della Ricerca Scientifica e Tecnologica. Id b) Fig. 1: a) Studied system; b) one-dimensional model. The purpose of this paper is to examine closely some of these nonlinear behaviors when the conduction phenomena in the iron core are not negligible (i.e., the “skin depth” is about the thickness of the material”). In particular we investigate the multiple harmonic and subharmonic oscillations by computer aided-analysis using a static procedure. II. ‘WE MATHEMATI~AL MODEL In this Section by applying the Galerkin method we derive the finite dimensional dynamic system which describes the magnetic field diffusion in the plate of ferromagnetic material. These equations will be coupled to the circuit ones. The considered iron core shape is a large plate with thickness d, width 11 and length 12 (see Figla). For 0018-9464/92$03.00 zyxwvutsr 0 1992 IEEE