Calculations of the magnetic field distribution in the given sheet sample is possible by means of an appropriate model of the magnetization process, which can allow us to take into account anisotropic properties of the dynamo sheets. These properties can be considered in a simplified way assuming that the sheet grains have only one easy magnetization axis. However, the iron crystals have three axes of the easy magnetization, so changes of the flux density depend also on the field strength direction [8, 9].

Different approaches which allow us to take into account the anisotropic properties of the dynamo sheets are presented in scientific papers, e.g. the so-called elliptical model of the magnetic anisotropy [10] and the model based on the co-energy density are described in [11], however, these proposals refer to non-hysteresis materials. The magnetic anisotropy of the dynamo sheets can also be considered using the reluctivity or permeability tensor [12]. It is necessary to stress that these methods do not take into account the texture types of the dynamo sheets. It should be emphasized, as it is pointed in [13], that the anisotropic properties of the dynamo sheets should be considered taking into account a real crystallographic structure of the tested dynamo sheet.

The magnetization processes in electrical steel sheets are usually considered as two-dimensional processes. However, the magnetization in the iron crystals (iron grains) should be treated as three-dimensional processes because these processes in iron crystals occur along all three easy magnetization axes (edges of the cubic shape). The exception here is the magnetization in grains creating the cube-on-face texture {100}; in this case two easy magnetization axes are parallel to the sheet plane. When the direction of the field strength in an elementary segment of the tested sheet sample is not parallel to any easy magnetization axis, then during increase of the field strength value the flux density value increases along the easy magnetization axis which is the closest to the field strength direction. This problem is qualitatively described in [14], where physical mechanism of this magnetization process is explained; similar comments of this process are contained in [15]. Due to changes of the magnetic field strength in the given sheet sample, reconstructions of the domain structures occur; it means that apart displacements of the 180° walls there occur also changes of the 90° walls.

Considerations were carried out with the assumption that rotations of the flux density vectors towards the direction of the field strength vectors in elementary subsegments can be neglected. So, the flux densities can change only along the easy magnetization axes of the individual iron crystals. In order to determine the flux densities in each easy magnetization axis, the values of the field strength in these axes have to be calculated with respect to the field strength occurring in the two-dimensional space of each elementary segment of the tested dynamo sheet. In this purpose we have to save the field strengths in these axes as functions of the field strength components, which are associated with the branches of the elementary segments (Figure 2). Considering e.g. the cube-on-edge texture {110} we can write the following relationships for all elementary segments (Figure 3)

Figure 3 Determination of the field strength values in easy magnetization axes as dependences on the field strength components *H*_{m}, *H*_{p}

$$\begin{array}{}{\displaystyle {\mathbf{H}}_{\mathbf{g}\mathbf{1}}={f}_{hg1m}{\mathbf{M}}_{\mathbf{h}\mathbf{m}}{\mathbf{H}}_{\mathbf{m}}+{f}_{hg1p}{\mathbf{M}}_{\mathbf{h}\mathbf{p}}{\mathbf{H}}_{\mathbf{p}}}\end{array}$$(3a)

$$\begin{array}{}{\displaystyle {\mathbf{H}}_{\mathbf{g}\mathbf{2}}={f}_{hg2m}{\mathbf{M}}_{\mathbf{h}\mathbf{m}}{\mathbf{H}}_{\mathbf{m}}+{f}_{hg2p}{\mathbf{M}}_{\mathbf{h}\mathbf{p}}{\mathbf{H}}_{\mathbf{p}}}\end{array}$$(3b)

$$\begin{array}{}{\displaystyle {\mathbf{H}}_{\mathbf{g}\mathbf{3}}={f}_{hg3m}{\mathbf{M}}_{\mathbf{h}\mathbf{m}}{\mathbf{H}}_{\mathbf{m}}+{f}_{hg3p}{\mathbf{M}}_{\mathbf{h}\mathbf{p}}{\mathbf{H}}_{\mathbf{p}}}\end{array}$$(3c)

where **H**_{g1}, **H**_{g2}, **H**_{g3} are column vectors of the field strengths values *H*_{g1}, *H*_{g2}, *H*_{g3} of the first, second, and third easy magnetization axis of the considered texture, respectively, **M**_{hm}, **M**_{hp} denote matrixes that assign components *H*_{g1}, *H*_{g2}, *H*_{g3} to the appropriate components *H*_{m}, *H*_{p}, *f*_{hg} are appropriate trigonometric functions.

The field strengths in easy magnetization axes of the remaining textures are written in similar form as it is proposed. Changes of the flux density along the easy magnetization axes of the considered texture are described as follows (Figure 4)

Figure 4 Determination of the flux density components *B*_{bx}, *B*_{by} of a chosen subsegments as dependences on the flux densities *B*_{g1}, *B*_{g2}, *B*_{g3}

$$\begin{array}{}{\displaystyle {\mathbf{B}}_{\mathbf{g}\mathbf{1}}={f}_{n}({\mathbf{H}}_{\mathbf{g}\mathbf{1}}({\mathbf{H}}_{\mathbf{m}},{\mathbf{H}}_{\mathbf{p}}))}\end{array}$$(4a)

$$\begin{array}{}{\displaystyle {\mathbf{B}}_{\mathbf{g}\mathbf{2}}={f}_{n}({\mathbf{H}}_{\mathbf{g}\mathbf{2}}({\mathbf{H}}_{\mathbf{m}},{\mathbf{H}}_{\mathbf{p}}))}\end{array}$$(4b)

$$\begin{array}{}{\displaystyle {\mathbf{B}}_{\mathbf{g}\mathbf{3}}={f}_{n}({\mathbf{H}}_{\mathbf{g}\mathbf{3}}({\mathbf{H}}_{\mathbf{m}},{\mathbf{H}}_{\mathbf{p}}))}\end{array}$$(4c)

where **B**_{g1}, **B**_{g2}, **B**_{g3} are column vectors of the flux density values *B*_{g1}, *B*_{g2}, *B*_{g3} on the first, second, and third easy magnetization axis of the considered texture, respectively, *f*_{n} denotes nonlinear relationship between the field strength and the flux density in the form *B* = *B*_{sat}f_{n}(*H*), where *B*_{sat} denotes the saturation flux density of the given axis of the easy magnetization, *f*_{n}(*H*) is a nonlinear function of the field strength.

It is necessary to stress that the value of the saturation flux density which refers to each easy magnetization axis, does not have a constant value in contrary to proposals which have treated iron crystals as particles with only one easy magnetization axis. The saturation flux density depends on, among other, the value and direction of the field strength vector in particular elementary segments.

In order to save vectors **B**_{bx}, **B**_{by}, **B**_{tx}, **B**_{ty} in (2) as dependences on the column vectors **H**_{m}, **H**_{p}, the flux densities occurring in particular easy magnetization axes of all textures have to be projected on the sheet plane. Considering the chosen cube-on-edge texture we can write the following relationships

$$\begin{array}{}{\displaystyle {\mathbf{B}}_{\mathbf{b}\mathbf{x}}={f}_{bxg1}{\mathbf{M}}_{\mathbf{b}\mathbf{x}\mathbf{g}\mathbf{1}}{\mathbf{B}}_{\mathbf{g}\mathbf{1}}+{f}_{bxg2}{\mathbf{M}}_{\mathbf{b}\mathbf{x}\mathbf{g}\mathbf{2}}{\mathbf{B}}_{\mathbf{g}\mathbf{2}}+{f}_{bxg3}{\mathbf{M}}_{\mathbf{b}\mathbf{x}\mathbf{g}\mathbf{3}}{\mathbf{B}}_{\mathbf{g}\mathbf{3}}}\end{array}$$(5a)

$$\begin{array}{}{\displaystyle {\mathbf{B}}_{\mathbf{b}\mathbf{y}}={f}_{byg1}{\mathbf{M}}_{\mathbf{b}\mathbf{y}\mathbf{g}\mathbf{1}}{\mathbf{B}}_{\mathbf{g}\mathbf{1}}+{f}_{byg2}{\mathbf{M}}_{\mathbf{b}\mathbf{y}\mathbf{g}\mathbf{2}}{\mathbf{B}}_{\mathbf{g}\mathbf{2}}+{f}_{byg3}{\mathbf{M}}_{\mathbf{b}\mathbf{y}\mathbf{g}\mathbf{3}}{\mathbf{B}}_{\mathbf{g}\mathbf{3}}}\end{array}$$(5b)

$$\begin{array}{}{\displaystyle {\mathbf{B}}_{\mathbf{t}\mathbf{x}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}{f}_{txg1}{\mathbf{M}}_{\mathbf{t}\mathbf{x}\mathbf{g}\mathbf{1}}{\mathbf{B}}_{\mathbf{g}\mathbf{1}}+{f}_{txg2}{\mathbf{M}}_{\mathbf{t}\mathbf{x}\mathbf{g}\mathbf{2}}{\mathbf{B}}_{\mathbf{g}\mathbf{2}}+{f}_{txg3}{\mathbf{M}}_{\mathbf{t}\mathbf{x}\mathbf{g}\mathbf{3}}{\mathbf{B}}_{\mathbf{g}\mathbf{3}}}\end{array}$$(5c)

$$\begin{array}{}{\displaystyle {\mathbf{B}}_{\mathbf{t}\mathbf{y}}={f}_{tyg1}{\mathbf{M}}_{\mathbf{t}\mathbf{y}\mathbf{g}\mathbf{1}}{\mathbf{B}}_{\mathbf{g}\mathbf{1}}+{f}_{tyg2}{\mathbf{M}}_{\mathbf{t}\mathbf{y}\mathbf{g}\mathbf{2}}{\mathbf{B}}_{\mathbf{g}\mathbf{2}}+{f}_{tyg3}{\mathbf{M}}_{\mathbf{t}\mathbf{y}\mathbf{g}\mathbf{3}}{\mathbf{B}}_{\mathbf{g}\mathbf{3}}}\end{array}$$(5d)

where **M** are matrixes that assign components *B*_{bx}, *B*_{by}, *B*_{gx}, *B*_{gy} to the appropriate components *B*_{g1}, *B*_{g2}, *B*_{g3}, *f* are appropriate trigonometric functions.

This described algorithm allows us to transform (2) to the form that contains the functions of the column vectors **H**_{m}, **H**_{p}. Using (1) we can eliminate the vector **H**_{m} and finally we obtain one nonlinear, matrix equation, where only the column vector **H**_{p} is unknown. In order to solve this equation the Newton-Raphson method is used for assumed values of the external currents as sources of the magnetic field.

The experimental system for validation of the correctness of the formulated equation, which takes into account the textures occurring in the dynamo sheets, is shown in Figure 5. The magnetic field was induced in the pack of electrically insulated dynamo sheets by two mutually perpendicular coils. For assumed value of the sinusoidal current, the voltages of the measurement coils were stored. Tests were carried out for two cases; in the first one the both coils were supplied but the currents were shifted in phase by 90 degree, in the second case the sinusoidal current was flowing only through the coil which is parallel to the transverse direction. These voltages were compared with the corresponding voltages calculated numerically for the same conditions, as in the experiment; this comparison is presented in Figure 6.

Figure 5 Measurement system for indirect validation of the correctness of the equation describing the magnetic field distribution taking into account the textures, RD, TD – the rolling and transverse direction, respectively

Figure 6 Voltages of the measurement coils in the rolling direction of the dynamo sheet M400-50A (Russia): a) both coils were supplied, b) only coil parallel to the transverse coil was supplied; continuous lines – measured waveforms, dashed lines – calculated waveforms

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