**4. Conclusions**

Although many experimental measurements can be at least qualitatively explained by different E-J power law models, for a few decades, some questions about the general validity of the E-J power law remained unsolved. For instance, as the two main criteria for deciding what material law to use in the numerical modelling of a superconducting device are the reproduction of specific experimental evidence (either qualitatively or quantitatively) and the affordability of the computation (due to the usually large demand on memory and processing power that finite element method (FEM)simulations require, especially with the large aspect ratio of geometries such as those implied by the 2G-HTS tapes), it is natural to wonder if the E-J power law invoked by a researcher is a simple artificial function that has been used to match a single piece of experimental evidence. Is the assumed material law model sufficiently valid and well-supported to reproduce all other macroscopical electromagnetic

quantities within a more quantitative perspective? Or, in fact, could we in a practical manner use a simpler material law that is capable of accounting for the different electromagnetic phenomena of type-II superconductors, regardless of whether the superconducting material is known to exhibit magneto-anisotropic properties, as is the case for the majority of 2G-HTS tapes? Thus, in an attempt to answer these questions, in this paper we analysed how the selection of different material laws in the numerical modelling of superconducting coils can strongly influence the accurate estimation of macroscopically measurable physical quantities such as the critical current density per coil turn, the magnetic field near the coil armature, and the accounting of the hysteresis losses per cycle, which were all considered under self-field current conditions.

Four different material laws were considered in this study, where a clear impact of the magneto-anisotropic properties of 2G-HTS tapes was disclosed through the direct comparison between an isotropic critical-state-like model, *CM*, and different versions of the so-called Kim-based models: *KM*1, *KM*2, and *RM* in Table 1. These are amongs<sup>t</sup> the most prevalent E-J power law models for 2G-HTS tapes, all validated up to certain extent under different experimental conditions [20,23,33,39,40,47]. We found that although each of these material laws allows a proper qualitative description of the electromagnetism of superconducting coils, substantial quantitative differences were found between their predictions under common experimental conditions, which ultimately, from a purely computational perspective, can help modellers to make a decision on what material law could be more suitable when time and computing power are both limited. In this sense, we concluded that when the physical quantity to be measured is the critical current density turn-by-turn by I-V measurements, certain caution must be taken when compared with the numerical results (Figure 2), as depending on the positioning of the voltage taps, the local magneto-angular anisotropy of the superconducting tape can lead to deviations of up to 50% between the theoretical and experimental measurements. Moreover, if the *CM* model is assumed, then no difference in *Jc* for the different turns of the superconducting coil could be predicted. On the contrary, if a Kim-based model is invoked, then the local variation of the critical current density across the surface of the superconducting tape could be visualised in good agreemen<sup>t</sup> with the magneto-optical imaging observations reported in [56]. However, if the intensity of the magnetic field at the central axis of the 2G-HTS coil is the physical quantity of focus (Figure 3), then the simplicity and minimum computing time that can be achieved with the *CM* model makes this material law the best option, as no difference was obtained when it was compared with the prognostics of the Kim-based models. Nonetheless, the situation is different if the intensity of the magnetic field near the innermost or outermost turns of the superconducting coil is the desired measurand (Figure 4), as at these locations, local changes in the distribution of current-density profiles can be macroscopically evinced by analysing the *By* behaviour at low, moderate, and high transport currents. In this sense, under this scenario, the simplified Kim's model *KM*1 can be chosen as the most suitable candidate for the numerical modelling of the superconducting properties, as long as a relative tolerance between the experimental and numerical results of ∼25% is accepted, and if the expected increase in the computing time cannot be afforded when a more "tailored" approach such as the *RM* model is invoked.

Remarkably, when comparing the *By* curve of the Kim-based models with the one derived by the *CM* model, over the surface of the innermost turn of the superconducting coil, we obtained an extricable mean for the experimental determination of the magnetic saturation state of the entire coil at self-field conditions and zero transport current (*ω<sup>t</sup>* = <sup>2</sup>*π*), which is shown in the form of a *By*/*BCM* plateau for high amplitudes of the applied transport current. Besides, when the study of the superconducting coil focuses on the measurement or estimation of the hysteresis losses, we found that despite the expected underestimation of the AC losses by the *CM* model, for low intensities of the applied current (*Ia* ≤ 0.4*Ic*0), all the magneto-anisotropic models led to nearly the same results, with a relative difference of maximum twice the losses expected by the *CM* model, which is not necessarily seen to be as large as the hysteretic losses themselves, and can change in orders of magnitude as the intensity of the applied current increases. Nevertheless, for moderate-to-high intensities of the applied current (*Ia* 0.4*Ic*0), the impact of the magneto-angular anisotropy of the superconducting tape was very significant, with differences found between ∼3 to ∼8 times the estimated losses of the isotropic *CM* model. Evidently, if the computing time and power are not a matter of concern for the modeller of superconducting machines, the best option for choosing the material law for quantitative purposes would then be to select one where the majority of the microstructural parameters for the superconducting tape can or have already been determined by experimental measurements, as is the case of the *RM* model. However, if the used 2G-HTS tape is one where the *Jc*(**<sup>B</sup>**, *θ*) is unknown, then we can conclude that the use of the classical Kim's model would be the most advisable choice, if the numerical modellers and experimentalists bear in mind a relative tolerance of ∼28% in the estimation of the AC losses.

**Author Contributions:** B.C.R. contributed to the numerical modelling, simulations, formal analysis, and writing of this paper. B.C.R. and M.U.F. performed post-processing tasks and validation of results. H.S.R. contributed to the supervision, formal analysis, and writing (review and editing) of the paper. All authors have read and approved this manuscript.

**Funding:** This research was funded by the Engineering and Physical Sciences Research Council, EPSRC gran<sup>t</sup> number EP/S025707/1.

**Acknowledgments:** All authors acknowledge the support of the East Midlands Energy Research Accelerator (ERA) and, the High Performance Computing Cluster Facility; ALICE at the University of Leicester. B.C.R. thanks the Scholarship unit of the Niger Delta Development Commission for their funding support, and M.U.F. acknowledges the College of Science and Engineering Scholarship Unit of the University of Leicester.

**Conflicts of Interest:** The authors declare no conflicts of interest.
