Unified Scaling Law for flux pinning in practical superconductors: III. Minimum datasets, core parameters, and...

In Part 2 of these articles, an extensive analysis of pinning-force curves and raw scaling data was used to derive the Extrapolative Scaling Expression (ESE). This is a parameterization of the Unified Scaling Law (USL) that has the extrapolation capability of fundamental unified scaling, coupled with the application ease of a simple fitting equation. Here in Part 3, the accuracy of the ESE relation to interpolate and extrapolate limited critical-current data to obtain complete $I_{\text{c}}(B,T,\epsilon )$ datasets is evaluated and compared with present fitting equations. Accuracy is analyzed in terms of root mean square (RMS) error and fractional deviation statistics. Highlights from 92 test cases are condensed and summarized, covering most fitting protocols and proposed parameterizations of the USL. The results show that ESE reliably extrapolates critical currents at fields $B$ , temperatures $T$ , and strains $\epsilon$ that are remarkably different from the fitted minimum dataset. Depending on whether the conductor is moderate- $J_{\text{c}}$ or high- $J_{\text{c}}$ , effective RMS extrapolation errors for ESE are in the range 2-5 A at 12 T, which approaches the $I_{\text{c}}$ measurement error (1-2%). The minimum dataset for extrapolating full $I_{\text{c}}(B,T,\epsilon )$ characteristics is also determined from raw scaling data. It consists of one set of $I_{\text{c}}(B,\epsilon )$ data at a fixed temperature (e.g., liquid helium temperature), and one set of $I_{\text{c}}(B,T)$ data at a fixed strain (e.g., zero applied strain). Error analysis of extrapolations from the minimum dataset with different fitting equations shows that ESE reduces the percentage extrapolation errors at individual data points at high fields, temperatures, and compressive strains down to 1/10th to 1/40th the size of those for extrapolations with present fitting equations. Depending on the conductor, percentage fitting errors for interpolations are also reduced to as little as 1/15th the size. The extrapolation accuracy of the ESE relation offers the prospect of straightforward implementation of the USL in several new areas: (l) A five-fold reduction in the measurement space for unified temperature-strain apparatuses through extrapolation of minimum datasets; (2) Combination of data from separate temperature and strain apparatuses, which provides flexibility and productive use of more limited data; and (3) Full conductor characterization from as little as a single $I_{\text{c}}\left(B\right)$ curve when a few core parameters have been measured in a similar conductor. Default core scaling parameter values are also given, based on analysis of a wide range of practical Nb₃Sn conductors.