On the basis of damping principle of vibration, the domain wall processes is investigated and modeled at low-to-medium frequencies in electrical steel sheets. Due to the energy dissipation chiefly descends from a micro-vortex current caused by domain wall motion, the coupled Landau-Lifshitz-Gilbert (LLG) and Maxwell electromagnetic diffusion equations are thus considered to describe the high-frequency characteristics. The overall core losses are eventually deduced in terms of separate contributions by domain wall processes and classical eddy current. Moreover, the calculation model can be extended to rotational excitation pattern. Hence, taking the typical electrical steel sheets as example, the novel core losses calculation model is analysed and compared with the total alternating core losses supplied by electrical steel sheets manufacturers and the 3-D rotating experimental core losses of sheets specimens which are carried out by using a 3-D magnetic properties testing system, and also achieve some beneficial conclusions. 1. Since the domain wall processes is essentially the magnetic moment rotation, then domain wall motion can be interpreted by solving LLG equation, that is [1], [2]:\begin{align*} \partial J/ \partial t \quad =- xJ \times [H_{eff}+(\alpha /J_{s}) J\times H_{eff}], \vert J \vert = \quad J_{s}(1) \end{align*}where, $J \quad = \mu _{0}M$ represents the magnetic polarization vector inside the domain, $M$ is the magnetization vector, $\mu _{0}$ is the permeability of vacuum, $ \chi =|e| / m_{e} \quad = 1.76 \times 10 ^{11} \mathrm {T}^{-1} \mathrm {s}^{-1}$ is the absolute value of the electron gyromagnetic ratio [3]; $H_{eff}$ represents the overall effective field, which affects on the magnetic moments; $\alpha = \eta J_{s}$ is the qualitative dimensionless damping constant (Landau-Lifshitz damping coefficient), $eta$ is the damping coefficient of domain wall motion, $J_{s}$ is the saturation polarization. The first term on the right side of Eq. (1) describes the magnetic moment precession around the effective field direction, while the second term is the damping motion towards the effective field. 2. As regard to the right sheet with the walls in Fig. 1, in effect of the high-frequency excitation field $H$, the wall moves to make on both sides of the domains contraction and expansion. Now, by applying Maxwell electromagnetic diffusion equation:\begin{equation*} \partial ^{2}H_{vor}/ \partial y^{2} \quad =\sigma \mu _{0}[ \partial (M+H_{vor}+H)/\partial t](2)\end{equation*}where, $\sigma $ is the electrical conductivity of magnetic materials and the vortex current field $H_{vor}$ is directed to $y -$axis. At this point, such a response can be described in general terms by the solution of the coupled LLG (Eq. (1)) and Maxwell electromagnetic diffusion (Eq. (2)) equations. 3. According to the above derivation, the overall core losses per unit mass of electrical steel sheets are eventually deduced in terms of separate contributions by classical eddy current and domain wall processes, as follows:\begin{equation*} P_{cl} \quad = P_{cel}+P_{mel}^{-} \quad = \left({ \sigma \pi ^{2}d^{2}/ 3 \rho }\right) f^{2}(B_{m})^{2}+\left({\sigma \pi ^{2}d^{2}/ 4 \rho }\right) f^{2}(B- \mathrm {m}) ^{2}=\left({ 7 \sigma \pi ^{2}d^{2}/ 12 \rho }\right) f^{2}(B_{m})^{2}[\mathrm {W} /kg](3)\end{equation*}where, $P_{cl}$ is the overall core losses per unit mass of electrical steel sheets; $P_{cel}$ is the classical eddy current loss per unit mass; $P_{mel}^{-}$ is the mean micro-vortex current losses per unit mass; $d$ is the thickness of the magnetic sheet; $ \rho$ is the mass density of magnetic materials; $f$ is the excitation frequency; $B_{m}$ is the peak flux density. 4. In order to validate the precision of the calculation model, the total alternating core losses supplied by electrical steel sheets manufacturers are used to compare with the calculation of Eq. (3), including cold-rolled GO electrical steel, NO electrical steel and hot-rolled electrical steel over wide range of excitation frequency [4], [5]. In addition, the model can be extended to rotational excitation pattern, so the 3-D rotational experimental core losses of typical electrical steel sheets carried out by using a 3-D magnetic properties testing system are also considered in the validity of the calculation model [6], [7]. The typical comparison result is shown in Fig. 2.
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Apr 1, 2024
Juanmian Lei + 3
Open Access