Transformation of the trapped flux in a SC disc under electromagnetic exposure

Superconducting permanent magnets (SCs) with trapped magnetic flux are used in technical devices (motors, generators, etc.). These magnets endure repeated magnetic “shocks” during operation, which can affect their performance. In this work, we investigated the dynamic behavior of magnetic induction in the trapped flux in an SC disk when exposed to stepwise changes in the external magnetic field, simulating these operational shocks. Our results reveal a direct correlation between the stepwise changes in the magnetic field and the trapped flux response, with each increase or decrease in the field inducing a corresponding 40-50% change in trapped flux for a 600 G field step at temperature of 5 K. The magnitude of these changes depends on the external parameters and their dynamics could lead to additional energy dissipation and potential heating, which may affect the reliability of SC magnets in applications. A scaling analysis of the induction flux profiles, revealing roughness exponents in the range of 0.435 to 0.475 was performed as well, and we determined the Hausdorff dimension of the surface structure.

💡 Research Summary

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This paper investigates how the trapped magnetic flux in a NbTi superconducting (SC) disk responds to stepwise changes in an external magnetic field, a situation that mimics the magnetic “shocks” experienced by SC permanent magnets in real devices such as motors and generators. The authors used magneto‑optical (MO) imaging to visualize the normal component of magnetic induction (Bz) on the surface of a thin (0.1 mm) NbTi disk (12 mm diameter) while applying external fields in discrete steps of 50 G, 100 G, 200 G, 400 G, and 600 G at temperatures between 5 K and 8 K. Two sample states were examined: (i) a disk fabricated by extrusion only, and (ii) the same material after a single long‑duration heat‑treatment (three 80‑hour stages at 420 °C, a protocol known to improve pinning).

Key experimental observations are:

1. Large‑scale flux variation – When the external field is increased by 600 G, the trapped flux inside the disk decreases by roughly 40–50 %; decreasing the external field by the same amount restores the trapped flux by a comparable percentage. This demonstrates a strong, quasi‑linear coupling between external field steps and the internal critical state.

2. Effect of thermomechanical processing – The extruded‑only sample shows a highly irregular flux front with two distinct roughness scales: a large‑amplitude, non‑periodic waviness superimposed with fine ripples. After heat‑treatment, the flux front becomes markedly smoother; the depth of flux penetration is reduced by a factor of 2–3, indicating a 2–3‑fold increase in the critical current density (jc). The improvement is attributed to the refinement and homogenization of pinning centers, as the heat‑treatment eliminates large deformation‑induced defects that act as strong, irregular pinning sites.

3. Fractal analysis of the flux front – By fitting the flux‑front profiles to a power‑law correlation function, the authors extracted roughness exponents (α) ranging from 0.435 to 0.475. These values correspond to a Hausdorff dimension D_H = 1 + α ≈ 1.44–1.48, confirming that the flux front possesses a fractal geometry rather than a simple Euclidean line. The exponent variation reflects the degree of pinning‑center inhomogeneity and the interplay between vortex elasticity and the underlying material disorder.

4. Flux jumps (avalanches) and dynamic losses – In the annealed sample, at high external fields (≥ 500 G) a localized “gate” region appears where the current streamlines form small‑radius semicircles. This geometry concentrates magnetic pressure, triggering a rapid flux avalanche that propagates as a narrow finger. Such events are accompanied by sudden local heating and increased dissipation, which, if repeated in rotating machinery, could raise the overall dynamic resistance and reduce device efficiency.

5. Practical implications for SC magnet design – The study highlights three crucial considerations for reliable SC permanent magnets: (a) limiting the amplitude and frequency of external field variations to avoid large trapped‑flux excursions; (b) optimizing extrusion and heat‑treatment schedules to produce a uniform pinning landscape, thereby reducing flux‑front roughness and suppressing avalanches; and (c) employing MO imaging as a non‑destructive quality‑control tool to assess pinning homogeneity before device integration.

Overall, the work provides a comprehensive experimental framework linking material processing, vortex‑front morphology, and macroscopic magnetic performance under realistic electromagnetic perturbations. By quantifying the 40–50 % trapped‑flux change per 600 G step and revealing the fractal nature of the flux front, the authors deliver actionable insights for engineers seeking to improve the efficiency, stability, and lifetime of superconducting permanent‑magnet technologies.