**1. Introduction**

One of the advantages of non-insulation (NI) high-temperature superconducting (HTS) coils is the capability to operate under the fault (typically overcurrent and hot-spot quench) conditions with a much lower risk of burnout [1]. The high stability against fault conditions [2,3] puts NI coils forward as a promising option for the large-scale application of DC high-temperature superconducting magnets, such as in maglev trains [4], motors [5,6], TOKAMAK systems [7], and high-field NMR [8]. Overcurrent conditions with an operating current higher than the critical current sometimes occur and play an important role in achieving a maximum field and in the elimination of screening currents [7].

Several studies have focused on the performance of NI–HTS coils operating under overcurrent conditions. Experimentally, a saturation of the magnetic field was observed in various overcurrent tests [7,9–11], and burnout [11] of both the innermost and outermost turns close to the electrodes occurred [12]. Similar phenomena were also observed during overcurrent tests of HTS coils with turn-to-turn metal insulation [13,14]. Numerically, the partial element equivalent circuit (PEEC) [15] or equivalent circuit grid (ECG) [16], coupled with a thermal model, which was proposed to numerically analyze the behaviors of NI coils during overcurrent conditions, suggested that a considerable amount of Joule heat is generated near the outer electrode, initiating the quench propagation and ascribing the magnetic field saturation to a local decrease in critical current [17]. Overcurrent test results in [18,19] also demonstrate the capability of NI coils to operate steadily in the saturated phase without leading to thermally induced degradation.

To further investigate the evolution of current distributions, understand the more quantitative characteristics during the transition to the magnetic-field saturation phase,

**Citation:** Wu, W.; Gao, Y.; Jin, Z. Magnetic Field Saturation of Non-Insulation High-Temperature Superconducting Coils during Overcurrent. *Electronics* **2021**, *10*, 2789. https://doi.org/10.3390/ electronics10222789

Academic Editors: Alon Kuperman and Alessandro Lampasi

Received: 14 October 2021 Accepted: 12 November 2021 Published: 14 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and clarify the potential risks of NI coils during overcurrent excitation, this study adopted an equivalent circuit grid model [15–17] coupled with a magnetic field the *E*–*J* power law of superconductors, calculated by the method mentioned in [20], to realize a real-time circuit-field simulation. In addition, a double pancake NI test coil was wound and charged with exquisite excitation procedures to validate the model. The responses of both the magnetic field and coil voltage were recorded and compared with the numerical results. The main contributions of this paper are as follows: (1) Detailed current distributions inside the coil during overcurrent charging are presented and discussed. Potential quenching risks were found to be at the innermost and outermost turn near the electrodes, as well as at the pancake-to-pancake connection part. (2) Magnetic field saturation, which is a unique phenomenon in non-insulation superconducting coils during overcurrent charging, was studied in detail and first quantitatively defined by a new concept "converged load factor". Its relationship with turn-to-turn resistivity was revealed.
