# An Alternative Formulation of the Harrison Model
## 2 A Brief Introduction to the Harrison Model
### The Reversibility Effect  
An advantage of the original [PERSON] model is that this description considers not just purely irreversible effects but also it explicitly speaks of reversibility, expressed with the anhysteretic curve. As already discussed in [9], there exists a discrepancy between this approach and the Jiles-Atherton (JA) formalism as far as the physical meaning of irreversible and reversible magnetization processes is concerned. [PERSON] and [PERSON] used the so-called effective field, being the sum of the applied field and the bulk magnetization multiplied by a positive constant \(\alpha\) (by analogy to the Weiss mean field), as the argument of the function, which was supposed to describe the anhysteretic curve. [PERSON] et al. criticized the physical assumptions underlying the Jiles-Atherton description, indicating that the summation of irreversible and reversible magnetization components postulated by the model developers leads to the non-physical behavior of modeled curves after field reversals, which is to some extent patched using pseudo-parameter \(\delta_{M}\), whose role is to suppress the negative dynamic susceptibility (\(\mathrm{d}M/\mathrm{d}H<0\)) regions [33]. However, the presence of \(\delta_{M}\) cannot be justified from the derivation of model equations. The source of problems with the physical interpretation of the JA description is the fundamental assumption that the hysteresis loop is obtained by introducing offsets (shifts) from the \"anhysteretic\" curve along the \(M\) (and not along the \(H\)) direction. A counter-example of the correct approach in the irreversible thermodynamics framework is the GRUCAD model [34], also studied by some of the authors of the present contribution [35]. The [PERSON] model bears a striking resemblance to the GRUCAD model in the sense that it distinguishes irreversible and reversible contributions as **field strength** and not magnetization components. If one follows the [PERSON] definition of the effective field, Equation (2), then, in the language of control science, it can be interpreted as positive feedback in the system of magnetic moments. [PERSON] has explicitly stated that positive feedback immediately leads to the occurrence of bistability, and the \(m=m(h)\) dependence becomes S shaped (Figures 1 and 2 resemble rather the letter \"N\", since we consider the mapping \(h=h(m)\)). Even [PERSON] and [PERSON] have noticed that their equation for the \"anhysteretic\" curve, in which the effective field is the argument of the Langevin function, may give rise to the occurrence of a hysteresis loop (what contradicts its physical meaning, cf. Figure 2 in [15]). At this point it seems crucial to remark that, in several recent publications, [PERSON] and [PERSON] [36; 37] argued that the effective field should rather be considered as a demagnetization effect if it were to be considered as the argument to the anhysteretic curve; thus, the feedback would be negative. Regardless of the academic debate on the sign of the mean field parameter, a question arises as to whether one really needs an equation for the anhysteretic curve. A reliable method, which in most cases is sufficient for practical applications, is to recover the anhysteretic curve as the middle curve from the major loop branches [8; 9; 38]. If one accepts that hysteresis loop branches may be described with shifted hyperbolic tangent function, then an equation for the inverse anhysteretic function may be derived in a closed form [39]; thus, there is no need to introduce some complicated functions [40] or their combinations [41]. In the Harrison model, the role of the anhysteretic curve is merely to introduce a skewing of the \(h-m\) coordinate system, which means a virtual contraction of the coordinate related to field strength. This can be appreciated in the example of a point marked with a black dot in Figure 6, which originally has non-zero field value but becomes the remanence point in the irreversible field vs. magnetization coordinate system.