# Modeling Dynamic Hysteresis Curves in Amorphous Magnetic Ribbons
## 1 Introduction  
The present paper is a follow up of previous research devoted to modeling hysteresis curves under an increased excitation frequency [1]. The aim of the paper is to demonstrate the usefulness of a viscous-type equation, considered previously by [PERSON] et al. [2], as a means to describe the so-called excess field component. Previously, the equation was applied to describe hysteresis loops of non-oriented electrical steel sheets with a different thickness. In the present paper, the approach is applied to another soft magnetic material, namely the amorphous material Metglas, which has not been considered in the existing literature.  
Amorphous magnetic materials are alloys of either Co, Fe, or Ni metals or their combinations--around 80%--and other chemical elements (B, C, Ge, P, or Si). They are non-crystalline, and exhibit a glass-like structure [3, 4]. Depending on chemical composition, these alloys exhibit different magnetic properties:  
* materials with a high iron (Fe) content achieve high values of saturation polarization and low values of coercive field strength (\(J_{\mathrm{s}}\) = 1.3 T, \(H_{\mathrm{c}}\) = 10 A/m);
* those that include both iron and nickel (Ni) attain high values of initial and maximum permeabilities (\(\mu_{\mathrm{i}}\) = 15 \(\times\) 10\({}^{3}\), \(\mu_{\mathrm{max}}\) = 200-400 \(\times\) 10\({}^{3}\));
* cobalt (Co)-based alloys have extremely high values of initial and maximum permeabilities (\(\mu_{\mathrm{i}}\) = 100 \(\times\) 10\({}^{3}\), \(\mu_{\mathrm{max}}\) = 200-600 \(\times\) 10\({}^{3}\)), and a very low magnetostriction coefficient (\(\lambda_{\mathrm{s}}\) < 0.2 \(\times\) 10\({}^{-6}\)).  
Amorphous magnetic ribbons are produced in the quenching process, and the molten metal is driven onto the surface of a rotating copper drum and subject to cooling [4]. The complex morphology of quenched amorphous materials results in their unique magnetic properties. This issue has been addressed in a number of publications for example in the classical textbooks [5] and review papers [6; 7]. In the present communication, we do not go into the details of processing technologies and underlying physics, instead focusing on macroscopic properties of ready-made ribbon cores available commercially, as their users.  
An example of a commercial magnetic material with an amorphous structure is the alloy with the nominal composition Fe-Si-B, produced by the American company Allied Signal under the brand-name Metglas (abbreviation from \"metallic glass\"). This alloy exhibits saturation induction around 1.3-1.5 T, a coercive field strength of H\({}_{\text{C}}\) < 5 A/m, and a maximum permeability of \(\mu_{\text{max}}\) = 120.000.  
The application scope for amorphous magnetic alloys is quite wide, ranging from energy-saving distribution transformers [4; 8; 9; 10; 11] to sensors of different physical quantities [12; 13; 14; 15]. In the latter case, the sensor output response may become affected by eddy currents induced in the conductive core material. A similar situation occurs during the magnetization of cylinder-shaped (commonly referred to as \"toroid\") magnetic cores, as examined in the present communication. The objective of the paper is to examine whether it is possible to describe the widening of hysteresis loops in a relatively simple way.