# Modeling Dynamic Hysteresis Curves in Amorphous Magnetic Ribbons
## 3 The Viscous-Type Equation  
The issue of loss separation in soft magnetic materials may be tackled from different perspectives, and a recent review on the subject can be found in [1]. Here, we follow the so-called two-term separation scheme, in which field strength at an increased excitation frequency is computed from  
\[H(t)=H_{\text{stat}}(B)+\left[\frac{1}{r(B)}\frac{\text{d}B}{\text{d}t}\right]^{ 1/\
u} \tag{3}\]  
in which \(H_{\text{stat}}(B)\) corresponds to the material response under slow driving (quasi-static excitation); the exponent \(1/\
u\) is fractional, and it accounts for viscous effects mostly due to eddy currents in different time and spatial scales in the conductive material, whereas \(r(B)=K\left[1-(B/J_{\text{sat}})^{2}\right]\), where \(J_{\text{sat}}\) is the saturation polarization and the proportionality coefficient \(K\) is found to vary both on induction and its rate, \(K=K(B,\text{d}B/\text{d}t)\)[1]. We chose the two-component loss separation scheme as the preferred one, since it is naturally rooted in Maxwell equations [18; 19; 20; 21]. As noticed by [PERSON] [22], it is a common practice to neglect displacement currents in the calculation of eddy currents in conducting media. Thus, from the Poynting theorem (which considers energy dissipated in a unit volume of a ferromagnetic material), it follows that there would be just two components, which we interpret as related to two distinct physical phenomena, namely hysteresis and eddy currents generated in the conductive material in different time and spatial scales [19]. This interpretation is in the spirit of the statement found in the classical paper by [PERSON]: \"The physical origin of losses in conducting ferromagnetic materials is Joule heating caused by the flow of eddy currents around moving domain walls\" [23]. It should be remarked that more than 40 years ago, in Japan, there were successful attempts to describe dynamic losses in amorphous materials under an increased excitation frequency using fractional-type relationships [24; 25]. The consideration of both loss terms related to eddy currents flowing in bulk (macroscale) and around moving domain walls (the so-called excess or abnormal losses) as a single quantity in which the dependence on excitation frequency is given as a fractional allows one to avoid a number of interpretational problems, particularly as far as the quasi-static regime is considered [19; 26; 27]. Fractional  
Figure 3: A family of measured minor hysteresis loops at different excitation frequencies for the Metglas sample. power laws related to anomalous diffusion in ferromagnets have recently been the subject of considerable research, and [28; 29; 30] can be mentioned as representative examples. As indicated in the previous study [1], the starting point for the analysis is the dependence of coercive field strength vs. frequency. Figure 4 depicts the experimental dependence \(H_{\rm c}=H_{\rm c}(f)\) for the examined Metglas core. The dependencies \(H_{\rm c}=H_{\rm c}(f)\) were fitted to a relationship, power law + free term, and thus \(H_{\rm c}=H_{\rm c0}+c\,f^{1/\
u}\). It was found that the value of exponent \(1/\
u\) varied depending on induction amplitude; however, in a quite wide range of induction amplitude, it could be set and fixed to \(1/\
u\) = 0.65 (apart from the technical saturation region). A different (significantly smaller) value of exponent \(1/\
u\) for \(B_{\rm m}=1.2\) T may suggest the onset of some other dissipation mechanism observed for high induction levels. The dependence of the \(1/\
u\) value vs. induction amplitude is depicted in Figure 5.  
Figure 4: A family of measured dependencies (coercive field vs. frequency) at different excitation amplitudes for the Metglas sample. Figure 5: The dependence of the value of exponent \(1/\
u\) vs. induction. Figure 6 depicts the dependence of the free term \(H_{\text{c0}}\) vs. induction amplitude. It can be noticed that in a wide range of induction values, this dependence may be described with a straight line.