# Investigation of Energy Harvesting Capabilities of Metglas 2605 SA1
## 2 Magnetic Behavior
### Result and Discussion
Among the available modeling possibilities and considering the material characteristics (effect of stress and negligible hysteresis among others), we consider a simplified  
Figure 6: Magnetic hysteresis curves under five tensile stresses. anhysteretic model based on a thermodynamic approach presented by [PERSON] in [41] to relate the magnetic behavior of the considered material. This model focuses more on a mathematical formulation rather than a phenomenological approach, which allows describing the physical meaning of each variable as well as highlighting potential interconnections in the different physical mechanisms. Furthermore, due to the soft magnetic behavior of the Metglas 2605 SA1, a biphasic function with an inverse hyperbolic tangent and a linear part is used to describe the sigmoid curves. Therefore, a model that unifies the compressive and tensile excitation is proposed. This biphasic unified model is defined as follows:  
\[B=\left\{\sum_{i=1}^{2}\alpha_{i}\frac{2}{\pi}\text{arctan}[\beta_{i}H((1+ \tan\ln(\gamma_{i}\cdot\sigma))]\right\}+(\mu-\kappa\cdot\sigma)H \tag{3}\]  
where \(\alpha_{i}\) represents the saturation magnetic flux density of a single phase, \(\beta_{i}\) the small-signal slope of the nonlinear part for the considered phase (\(2\alpha_{i}\beta_{i}/\pi\)  thus being the linear permeability of the corresponding nonlinear phase \(i\) under zero stress), \(\mu\) the zero-stress permeability of the linear part, and \(\gamma_{i}\) and \(\kappa\) are the magnetostrictive coefficients of the nonlinear and linear phases, respectively. As Metglas 2605 SA1 shows a soft saturation at a low magnetic field, two functions of \(B\) were combined (biphasic model, \(i=1\ldots\) 2). Each phase can relate different magnetization mechanisms or different magnetization dynamics, for example, arising from domains of different sizes. Based on previous experimental measurements for Metglas 2605 SA1, a curve fitting process was carried out. When separating tension and compression cases, a dedicated parameter set was defined for each case, considering the soft nonlinearity (i.e., saturation) on the magnetic flux variation. This model provides a precise prediction for both cases as shown in Figure 7. The corresponding parameters of the model separating tensile and compressive stress are given in Table 2. However, separating compressive and tensile cases would yield a loss in the physical significance of the parameters and in terms of the simplicity of the model. Therefore, the parameters of another fitting process considering a model taking into account both tension and compression conditions are also given in Table 2 and presented in Figure 7. It can be noticed that both models give very similar results and are in good agreement with experimental measurements. Because of its better physical significance and good agreement with experiments, the unified model is considered in the following. Figure 7: Comparison between anhysteretic curves, common model (Model 1), and the model separating tensile and compressive cases (Model 2) for Metglas 2605 SA1. Considering the equivalent low-field permeability \(\mu_{lin}\) derived from Equation (3) as:  
\[\mu_{lin}=\left\{\frac{2}{\pi}\sum_{i=1}^{2}\alpha_{i}\beta_{i}[1+\tan\ln(\gamma _{i}\cdot\sigma)]\right\}+(\mu-\kappa\cdot\sigma) \tag{4}\]  
the theoretical low field equivalent permeability value was found to be 5900\(\mu_{0}\), 900\(\mu_{0}\), and 10,000\(\mu_{0}\) for the respective cases of stress free, compressive (\(-\)30 MPa), and tensile (30 MPa) stresses, respectively. Such values are in quite good agreement with the previous measurements, preliminarily confirming the relevance of the model. Additionally, the saturation magnetic flux density, given as \(\alpha_{1}+\alpha_{2}\), equals 1.45 T, being also in the right range of magnitude of experimentally observed maximal flux density. Further investigations on the model consisted of assessing the response over the full considered magnetic excitation range.