Calculation of the magnetic behaviour of electrical steel sheet under two dimensional excitation by means of the reluctance tensor

Abstract

A mathematical description of the non-linear and anisotropic behaviour of electrical steel sheet under two dimensional excitation is presented considering the field penetration into the bulk of the material. This description allows the calculation of the distribution of the magnetic field strength and the magnetic flux density for arbitrary two dimensional flux conditions in t h e cross section of the sheet. The calculation of the dynamic hysteresis loops and the loci of the magnetic field strength, respectively, by means of the proposed mathematical model shows very good agreement with the corresponding measurement results. I. Two DIMENSIONAL EXCITATIONThe components of the mean value of the magnetic flux density in the cross section of the steel sheet sample are controlled to vary sinusoidally with time t at frequency f and the angular velocity w = 2 7~ f . 2 generally has the form of an ellipse: with +cost9 -sin# + s i n @ +cost9 purely rotational flux conditions. The long axis of the ellipse is inclined by the inclination angle 8 towards the rolling direction of the sample. Due t o the non-linear behaviour of electrical steel sheet, this kind of excitation leads to higher harmonics of the magnetic field strength I?, although the components of the magnetic flux density 2 are restricted to the fundamentad harmonic. THE RELUCTANCE TENSORThe reluctance tensor Y" is the link between the harmonics of the magnetic field strength I ? and the magnetic flux density 2 in the sample. The cartesian coordinate system C ( x , y, 2 ) is choosen in such a way, that the sample lies parallel to the (2, y)-plane and that its origin corresponds with the center of the sample. As it is shown in [a], two main axes of the reluctance tensor can be found, both located in the (x,y)-plane of the steel sheet. If the measurements are performed in the system of main axes, the relation between I ? and B' for every set of coordinates r'= (IC, y, z ) can be written in the following form:v" I?(+) = poI? (F,t) (3) and 2 is the amplitude of the long axis of the ellipse of the flux density vector rotating in the mathematically positive ( r = +1) or negative sense ( T = --I),respectively. The amplitude of the short axis is a 2 and 0 5 a 5 1. An axis ratio of a = 0 leads to alternating excitation and the axis ratio of a = 1 means v g ) and v c ' denote the complex reluctances of the n-th order in IC-and y-direction, respcctively, com-bining the fundamental harmonic of 2 with the nth harmonic of I?. The anisotropic behaviour of the steel sheet sample is considered by the absolute values IvP'I and Ivp'I, the lag angle in time and space between the vectors of the magnetic field strength Manuscript received April 30, 1997 M. Birkfeld, phone ++241 807836, fax ++241 8888300, e-majl the magnetic flux density is 'Onsidered by-the birkfeld@i we .rwth-aadien. de arguments arg v p ) and arg v;) ( of the reluctances 121.

Experimental evaluation of magnetic anisotropy in electrical steel sheets

“…There are several good known mathematical models of this non-linear phenomenon: elliptic proposed by di Napoli and Paggi [4], single grain model elaborated by Szpunar and Szpunar [5], numericalfinite element method [6][7][8], coenergy model developed by Pera et al [9], tensor model by Enokizono et al [10] and Birkfeld and Hempel [11]. Magnetic field in electrical sheet steel is often described as two-dimensional ( Fig.…”