Investigation of the permeability tensor of electrical steel sheet

Abstract

The permeability tensor is a mathematical model for the description of the electro-magnetic behavior of electrical silicon iron steel sheet under two-dimensional magnetizing conditions. In this paper, an interpretation of the properties of this tensor is given, the investigation of the tensor elements from measurements under two-dimensional magnetizing conditions is described, and some examples of measurements and the corresponding permeability tensor elements are indicated.Index Terms-Electrical steel sheet, permeability tensor, power loss, two-dimensional measuring conditions.

The Newton‐Raphson method for solving non‐linear and anisotropic time‐harmonic problems

“…In a time‐harmonic context, non‐linear, anisotropic and hysteretic behaviour can be modelled by a complex‐valued reluctivity tensor (Birkfeld, 1997, 1998). This complex‐valued tensor representation is a generalisation of the complex‐valued reluctivity scalar used in Lederer and Kost (1998) and Niemenmaa (1988).…”

“…If α rd = α td ∈]0°,180°[ or α rd ≠ α td ∈[0°,180°], the average value of δ ( t ) is positive, resulting in a positive total loss (Figure 2). As a consequence, a complex‐valued reluctivity tensor allows to model the losses caused by simultaneously alternating and rotating fluxes, though in a simplified way (Birkfeld, 1998). If | ν rd |=| ν td | and α rd = α td ∈[0°,180°], the reluctivity tensor may be replaced by a complex‐valued scalar reluctivity, which models a simplified form of alternating hysteresis (Lederer and Kost, 1998).…”

“…Birkfeld (1998), described how ν rd re , ν rd im , ν td re and ν td im can be determined from measurements in the rolling and transverse direction of grain‐oriented silicon steels, by processing the measured signals in frequency domain. Obviously, the results depend on the shape of the exciting B→ ‐locus.…”

“…Reluctivity tensor entries | ν rd | (top, solid), | ν td | (top, dashed), α rd (bottom, solid) and α td (bottom, dashed) as a function of the magnitude B of a circular flux density (Birkfeld, 1998)…”

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

“…( 7 ) , the measured and t h e calculated signals can be set into relation [3] In the case of the electrical conductivity o = 0, the matrix equals the reluctance tensor. In the case of eddy currents in the sample, (16) must be solved numerically for the first order elements of the reluctance tensor due to t h e non-linear dependence on z and U?)…”