Superconductor Science and Technology

Current numerical simulations of superconducting tapes in electrical applications predominantly utilize the H and T-A formulations within the COMSOL platform. In comparison, the integral method (also known as the J model) can enhance computational speed by one to two orders of magnitude. However, the integral method only focuses on the numerical modelling of the superconducting region, leading to two main drawbacks: it cannot model ferromagnetic materials in space, nor the distribution of external magnetic fields outside the superconducting tape. These limitations hinder the application of the integral method as a fast computation model. This paper innovatively couples the magnetic network model (Φ) with the integral method model (J) which is named J-Φ by using MATLAB. Three critical challenges and their solutions are addressed, including the transmission of external magnetic fields, the calculation of the K matrix, and the selection of the coupling time step. The J-Φ coupled numerical model not only resolves the two mentioned limitations but also maintains a very high computation speed—computation time is only 1% of that of the T-A model. The J-Φ formulation's calculations for magnetic field distribution, current distribution, dynamic loss and AC loss are consistent with those of the T-A and H formulation. The J-Φ model developed here will accelerate the application of the J model in electrical equipment.

The iron-based superconductors are promising candidates for high field magnet applications due to their excellent performance. The hot isostatic pressing (HIP) heat treatment process can significantly improves the mass density of iron-based materials, thereby enhancing their critical performance. This work is the first to apply the HIP process to the fabrication of iron-based double pancake coils, demonstrating a substantial improvement in their critical performance. The results confirm that the HIP process is suitable for preparing high-performance iron-based coils.

The gas–solid method, discovered in recent years, has been shown to have the potential to fabricate dense MgB₂ bulks. In this study, we aimed to investigate the dominant regulation mechanism underlying the superconducting properties and microstructures of MgB₂ bulk obtained from the gas–solid method. To this end, we utilized nano computed tomography (nano CT) and transmission electronic microscopy to examine the structure of the MgB₂ bulk at the micrometer and nanometer scales, respectively. Nano-CT results revealed that the application of a mold during the gas–solid process can significantly reduce the amount of holes present in the sample. Furthermore, the reticular hole structure can be almost completely eliminated through this modification. Such a change can account for the remarkable increase in superconducting connectivity (A_f value) observed in the sample with a mold. Additionally, we found that different B powders can lead to vastly different MgO impurity particle sizes and distributions at the nanometer scale. This finding can explain the differing J_c (B) values and pinning forces observed in the corresponding samples. Overall, our correlation analysis between the structure and superconducting properties provides valuable insights into the possible regulation mechanism of MgB₂ bulk obtained from the gas–solid method. These findings may assist researchers in further improving the performance of MgB₂ bulks.

The critical current of superconducting materials, such as magnesium diboride (MgB₂) bulk superconductors, is a key parameter influencing their performance in various applications, including magnetic field shielding, MRI, and Maglev systems. Spark plasma sintering (SPS) is one of the most efficient methods to fabricate high-quality MgB₂, significantly saving fabrication time and controlling grain growth. The fabrication conditions, including temperature, pressure, and dwell time, can affect the critical current density. Traditional methods for estimating critical currents are time-consuming and costly. This study explores the use of advanced artificial intelligence (AI) techniques to develop accurate and efficient models for predicting the critical current in MgB₂ bulks with respect to 10 different influential fabrication properties and physical conditions. By using AI algorithms such as Gaussian process regression, extremely gradient boosting, and generalized regression neural network (GRNN) an extremely high accuracy in predictions against the actual experimental data was achieved. By defining and studying the extrapolation scenario, this study goes beyond of simple AI-based estimator model that performs well only within the training range of data. The developed AI models not only reduce the need for extensive experimental campaigns but also offer real-time prediction capabilities, paving the way for faster advancements in the research and development of superconducting technology. Overall, GRNN model demonstrated a good performance for both interpolation and extrapolation tasks with an R-squared of 0.999958 and 0.99521, respectively.

AC losses in the high temperature superconducting (HTS) toroidal field (TF) magnets of the Spherical Tokamak for Energy Production (STEP) tokamak are analysed, focusing on the transient electromagnetic response of the centre column to the charging and discharging of the central solenoid (CS) and poloidal field (PF) magnets during a plasma initiation scenario. An innovative H–H₀–Φ simulation method is proposed to quantify the distribution of hysteresis losses and eddy current losses in the TF limb formed from HTS cables. The study examines the impacts of the peak shift present in the HTS tape critical current data and current sharing within the tapes. Furthermore, the magnitude of the coupling losses is estimated both by theoretical analysis and simulation, and the temperature rise caused by the AC losses under adiabatic conditions is evaluated. The results indicate that hysteresis losses dominate, with eddy current losses contributing an additional 40% on top of the hysteresis losses. Consideration of the peak shift in the tape's ${J_{\text{c}}}\left( {B,\theta } \right)$ curve reduces hysteresis losses by 13%, while current sharing in the copper stabiliser has no significant impact. Coupling losses add only about 3% on top of the hysteresis loss. The most notable local temperature rise of approximately 7 K occurs towards the ends of the CS and the S1 PF coil, a plasma shaping coil positioned in close proximity to the centre column. Through detailed simulation and analysis, this study demonstrates the rationality of the AC loss design of the STEP magnet system and provides theoretical support for the application of HTS materials in large fusion magnets, offering insights for future tokamak design and optimization.

The study of irradiation effects in cuprate high temperature superconductors (HTS) has been a topic of interest since their discovery. Enormous progress in the understanding of vortex physics and pinning mechanisms was made in the early 1990s through the irradiation of HTS single crystals with a variety of particles over broad ranges of energies. For YBa₂Cu₃O₇ (YBCO), the overall conclusion was that irradiation could increase the critical current density (J_c) by orders of magnitude. The interpretation of the results was simplified by the fact that the pristine crystals were very clean, with few pinning centers and quite low J_c, thus essentially all pinning in the irradiated crystals could be attributed to the controllably added disorder. The case of the ReBa₂Cu₃O₇ (ReBCO, where Re = Y, a Rare Earth, or combinations of them) epitaxial thin films and coated conductors (CC) is more complex, because the pre-irradiation samples already have high J_c due to the presence of large densities of strong pinning centers, which are fabrication-method and processing dependent. The most popular and efficient method to further increase J_c in CC has been the incorporation of artificial pinning centers (APC) by chemical incorporation of second phases. Efforts by many groups worldwide demonstrated that a diversity of APC can be effective, and it is now clear that mixed pinning landscapes, nanoengineered by the combination of defects of various shapes and sizes, produce the best results. In some cases, particle irradiation is still effective at enhancing J_c in CC, by more modest factors than in the single crystals. Interaction with pre-existing defects cannot be ignored, resulting in both cooperating and competing effects. In this work, I review the vortex pinning generated in YBCO by defects of various geometries (point defects, randomly distributed nanoparticles, aligned or splayed columnar) created either by particle irradiation, incorporation of second phases, or combinations of both routes, and discuss some implications of those results for the design of fusion reactors CC magnets.

The attenuation of magnetic fields is crucial for various application fields, including health, space exploration, and fundamental physics, to name just a few. Superconductors are key materials for addressing this challenge. In this review, we mainly focus on the shielding and screening of quasi-static magnetic fields using superconductor-based passive layouts. After providing a brief overview of the principles of magnetic shielding and screening using superconductors, we outline commonly used procedures for measuring the field attenuation. Next, we give an insight into analytical and numerical models able to reproduce experimental results and predict the performances of new designs. Key challenges and achievements in employing low temperature or high temperature superconducting bulk and tape-based structures for reducing a given applied field are then discussed. Additionally, hybrid designs combining superconducting and ferromagnetic materials, aimed at enhancing the shielding ability or fabricating magnetic cloaks, are described. Finally, we highlight future challenges and potential advancements in this technology.

High-T_c superconducting (HTS) flux pumps are capable of wirelessly powering HTS magnets, and are becoming promising alternatives of driven mode excitation which requires thermally inefficient current leads. HTS transformer-rectifiers, also considered as a type of HTS flux pumps, have drawn broad attention in recent years, since they enabled a number of novel HTS magnet applications. Compared to other types of HTS flux pumps, these devices are clear in physics and circuit topologies, easily controllable, and superior in some key performances. In this work, we aim to give a comprehensive overview on the thriving field of HTS transformer-rectifiers, especially those unconventional ones which do not involve superconducting-to-normal state transition. The work starts with explaining the working principle, including the underlying physics of induction-rectification effect, circuit topologies, and switching methods; followed by introducing design methods and construction considerations for engineering devices; and ends with summarizing research and development status, as well as potential applications of HTS transformer-rectifiers.

The high-temperature superconducting (HTS) closed-loop coil, characterised by shorted coil terminals and the low resistance of HTS conductors, can sustain a persistent DC current with minimal decay. These coils enable the generation of a DC magnetic field without the need for current leads or a power supply during operations, offering several advantages: (i) the development of compact, lightweight and portable DC magnet systems; (ii) the elimination of heat leakage and ohmic losses associated with current leads; and (iii) the removal of magnetic field harmonics caused by current supply. Recent advancements have revealed promising applications for HTS closed-loop coils, including maglev trains, nuclear magnetic resonance, scientific instruments, and energy storage systems. This paper firstly reviews various HTS closed-loop coils constructions, focusing on their distinctive characteristics. Then, the key research aspects of HTS closed-loop coils are overviewed, highlighting the latest advancements in persistent-current joint technologies, excitation methods, current control methods, current decay mechanisms and suppression techniques, simulation models, and quench detection and protection methods. Next, the applications of HTS closed-loop coils are analysed, emphasising their current status and future challenges. A detailed account is provided of our group's progress in developing an electrodynamic suspension train in Changchun, China, where all onboard magnets exclusively utilise HTS closed-loop coils. Finally, suggestions for future research directions are proposed.

The discovery of rare-earth barium copper oxide (REBCO) materials with high critical temperatures, and the continued advancements in the fabrication of REBCO coated conductors with extremely high critical current densities, has enabled the development of ultra-high-field (>20 T) compact and large-scale thermonuclear fusion devices. At present, around a dozen global commercial manufacturers are able to supply high-quality REBCO coated conductors with excellent performance. Significant advancements have been made for high-temperature, low-field applications such as motors, generators, long-length transmission cables, and so on using REBCO coated conductors. Nonetheless, multiple ongoing critical challenges under low-temperature, high-field conditions, such as irreversible degradation of the critical current, along with insufficient mechanical protection and inadequate reduction of AC losses, remain unsolved, collectively hindering their utilization in high-field thermonuclear fusion reactors. This paper provides a comprehensive theoretical and technical review of the current state-of-the-art, associated challenges, and prospects in the research and development (R&D) of REBCO coated conductors, cables, and magnet systems for high-field fusion. It highlights the significant enhancements in current-carrying capacity, mechanical protection, and AC loss reduction achieved over the past decade. The paper delves into detailed analyses of potential cabling solutions that offer exceptional current-carrying capacity while ensuring an optimal inductance balance for toroidal, poloidal, and central solenoid coils in tokamak devices. This work endeavours to lay the groundwork for the R&D of the next-generation REBCO magnets to facilitate the construction of ultra-high-field compact and large-scale tokamak reactors.

Medium and high entropy alloys (MEAs and HEAs) have emerged as promising materials in materials science and engineering, offering exceptional mechanical properties and intriguing low-temperature physical characteristics. In the study of superconducting mechanisms, high-quality, large single crystals are essential for obtaining reliable experimental data. Here, we report the first successful growth of high-quality MEA HfNbZrTi single crystals using the Floating Zone Method and present a comprehensive investigation of their superconducting properties, vortex dynamics, and specific heat. Notably, the specific heat jump extracted from experimental data yields ∆C/γTc = 1.43, while the superconducting gap Δ(0) is 0.92 meV, precisely matching the theoretical prediction of the weak-coupling BCS model. Furthermore, we report the first investigation of magnetic relaxation in MEA single crystals, revealing an exceptionally low relaxation rate (S < 0.02 at 2 K). These findings not only provide strong experimental evidence supporting the application of medium- and high-entropy alloy superconductors but also open new avenues for the development of superconductors with ultralow magnetic relaxation rates.

MR researchers pursue higher and higher B0 field for new discoveries in medical sciences. High temperature superconducting (HTS) magnet has the potential advantages of compactness, cryogen-free operation and ultra-high field (UHF) performance for being applied as the key component in the UHF-MRI system. Intensive researches on the HTS magnets are being conducted in recent years. This review reports the up-to-date HTS magnet techniques and their problems and solutions for the potential application in the UHF-MRI magnet. Two commercial HTS tapes of Bi2223 and REBCO are discussed with their application cases in MRI/NMR magnets and other UHF magnets. The technical problems, especially for the REBCO magnet with the screening current effect, small radial tensile strength, tape uncertainty, etc. are introduced with the potential solutions. No-insulation technique with quench transient behaviour is also introduced. Since there is not an UHF-MRI magnet by HTS in the world by the time of formulating this review, the achievements of some other large-scale UHF-magnet projects by HTS are investigated. Sequential engineering phases with two prototype magnets are thus proposed for developing the UHF-MRI magnet by HTS. The detailed development cost is estimated with a decent price range. Besides, for the development of a first-in-kind UHF-MRI system by HTS magnet, the insight is provided that the UHF-magnet development is a make-or-break component and due a primary and independent project from the entire UHF-MRI system project. . &#xD;Apart from the introduction and the conclusion, this review paper is organized with 8 other sections. The beginning of each section/sub-section is more geared at the UHF-MR researchers with introduction to the basic magnet knowledge, and the remaining and major part is presented with more professional review for the UHF-magnet researchers.&#xD;

Energy production by nuclear fusion can be the breakthrough in the decarbonisation process, and high temperature superconductors (HTS) represent a game changer for the design of compact reactors. However, reduced size implies that the superconducting tapes will be exposed to an intense flux of neutrons and of secondary particles while carrying a high current; in order to employ HTS in compact fusion reactors it is therefore crucial to precisely assess the effects of irradiation on HTS tapes at the working conditions. To achieve this goal, researchers from different fields met at the IREF (IRradiation Effects on HTS for Fusion) workshop to discuss all the aspects of this topic. This roadmap paper, that reflects the common view of the participants, aims at condensing the outcome of the intense and thorough discussion that took place during the conference, providing a path for the investigation of irradiation effects in HTS to assess their limits of operation in a fusion radiation environment.

High-temperature superconducting (HTS) technology enables the realization of compact and efficient designs, with the potential to break through the bottleneck of high-power-density motors. In HTS motors, field windings are crucial components that significantly influence overall performance. The motor's power capacity is generally proportional to the air gap magnetic field (Bg), which is further limited by the critical current of the HTS coils. In this paper, we designed and tested a single pole magnet for the HTS rotor that generated Bg of 3.39 T for an air gap of 3.5 mm and 3.04 T for an air gap of 15 mm at 30 K. These results are promising for high-energy-density motors, especially for HTS motors intended for aircraft. In addition, we established a 2D axisymmetric numerical model accommodating the Jc ( B , θ ) dependence of the critical current of HTS tapes to yield a more accurate estimation of the critical current of the racetrack coils. This methodology has been validated with various HTS tapes possessing different intrinsic flux pinning properties, demonstrating high accuracy and effective capability in simulating HTS coil performance. This work provided valuable guidance for the design and optimization of future high-power-density superconducting motors, while also validating a universal numerical simulation method.&#xD;

The overall, loaded quality factor Q_L quantifies the loss of energy stored in a resonator. Here we discuss on general grounds how Q_L of a planar microwave resonator made of a conventional superconductor should depend on temperature and frequency. We consider contributions to Q_L due to dissipation by thermal quasiparticles (Q_QP), due to residual dissipation (Q_Res), and due to coupling (Q_C). We present experimental data obtained with five superconducting stripline resonators fabricated from lead (Pb), with different center conductor widths and different coupling gaps. We probe the resonators at various harmonics between 0.7 GHz and 6 GHz and at temperatures between 1.5 K and 7 K. We find a strongly frequency- and temperature-dependent Q_L, which we can describe by a lumped-element model. For certain resonators at lowest temperatures we observe a maximum in the frequency-dependent Q_L when Q_Res and Q_C match, and here the measured Q_L can exceed 2 × 10⁵

Current numerical simulations of superconducting tapes in electrical applications predominantly utilize the H and T-A formulations within the COMSOL platform. In comparison, the integral method (also known as the J model) can enhance computational speed by one to two orders of magnitude. However, the integral method only focuses on the numerical modelling of the superconducting region, leading to two main drawbacks: it cannot model ferromagnetic materials in space, nor the distribution of external magnetic fields outside the superconducting tape. These limitations hinder the application of the integral method as a fast computation model. This paper innovatively couples the magnetic network model (Φ) with the integral method model (J) which is named J-Φ by using MATLAB. Three critical challenges and their solutions are addressed, including the transmission of external magnetic fields, the calculation of the K matrix, and the selection of the coupling time step. The J-Φ coupled numerical model not only resolves the two mentioned limitations but also maintains a very high computation speed—computation time is only 1% of that of the T-A model. The J-Φ formulation's calculations for magnetic field distribution, current distribution, dynamic loss and AC loss are consistent with those of the T-A and H formulation. The J-Φ model developed here will accelerate the application of the J model in electrical equipment.

The critical current of superconducting materials, such as magnesium diboride (MgB₂) bulk superconductors, is a key parameter influencing their performance in various applications, including magnetic field shielding, MRI, and Maglev systems. Spark plasma sintering (SPS) is one of the most efficient methods to fabricate high-quality MgB₂, significantly saving fabrication time and controlling grain growth. The fabrication conditions, including temperature, pressure, and dwell time, can affect the critical current density. Traditional methods for estimating critical currents are time-consuming and costly. This study explores the use of advanced artificial intelligence (AI) techniques to develop accurate and efficient models for predicting the critical current in MgB₂ bulks with respect to 10 different influential fabrication properties and physical conditions. By using AI algorithms such as Gaussian process regression, extremely gradient boosting, and generalized regression neural network (GRNN) an extremely high accuracy in predictions against the actual experimental data was achieved. By defining and studying the extrapolation scenario, this study goes beyond of simple AI-based estimator model that performs well only within the training range of data. The developed AI models not only reduce the need for extensive experimental campaigns but also offer real-time prediction capabilities, paving the way for faster advancements in the research and development of superconducting technology. Overall, GRNN model demonstrated a good performance for both interpolation and extrapolation tasks with an R-squared of 0.999958 and 0.99521, respectively.

AC losses in the high temperature superconducting (HTS) toroidal field (TF) magnets of the Spherical Tokamak for Energy Production (STEP) tokamak are analysed, focusing on the transient electromagnetic response of the centre column to the charging and discharging of the central solenoid (CS) and poloidal field (PF) magnets during a plasma initiation scenario. An innovative H–H₀–Φ simulation method is proposed to quantify the distribution of hysteresis losses and eddy current losses in the TF limb formed from HTS cables. The study examines the impacts of the peak shift present in the HTS tape critical current data and current sharing within the tapes. Furthermore, the magnitude of the coupling losses is estimated both by theoretical analysis and simulation, and the temperature rise caused by the AC losses under adiabatic conditions is evaluated. The results indicate that hysteresis losses dominate, with eddy current losses contributing an additional 40% on top of the hysteresis losses. Consideration of the peak shift in the tape's ${J_{\text{c}}}\left( {B,\theta } \right)$ curve reduces hysteresis losses by 13%, while current sharing in the copper stabiliser has no significant impact. Coupling losses add only about 3% on top of the hysteresis loss. The most notable local temperature rise of approximately 7 K occurs towards the ends of the CS and the S1 PF coil, a plasma shaping coil positioned in close proximity to the centre column. Through detailed simulation and analysis, this study demonstrates the rationality of the AC loss design of the STEP magnet system and provides theoretical support for the application of HTS materials in large fusion magnets, offering insights for future tokamak design and optimization.

Rare-earth barium copper oxide (REBCO) superconductors are recognized for their excellent electrical conductivity and thermal stability, making them promising candidates for high-current applications in future fusion reactors. However, their unique structure and anisotropic properties challenge stable operation in low-temperature (4.2 K) and high-field (20 T). To address these issues, the Institute of Plasma Physics, Chinese Academy of Sciences has developed a six-around-one copper-reinforced cable-in-conduit conductor (CICC) configuration using highly flexible REBCO cables (HFRC) for the central solenoid coil in next-generation fusion reactors. This study introduces an optimized design for a six-around-one copper-reinforced CICC and evaluates the feasibility of its compaction and twisting processes. Initially, considering factors such as the CICC's and sub-cable's geometric configuration and dimensions, a statistical analysis was conducted on experimental data concerning variations in outer diameter, inner diameter, and wall thickness of four different copper tube specifications after compaction. The results indicated that the copper tube with an outer diameter of 12.5 mm and a wall thickness of 0.9 mm exhibited minimal deformation during the compaction process by compaction equipment. Consequently, this specification is preferable for sub-cable compaction. Subsequently, three HFRC cables and two conductor-on-round-core cables were compacted within this copper tube, resulting in an outer diameter of 10.86 ± 0.06 mm after compaction. Following this, the wires underwent a six-around-one twisting process with a pitch of 550 mm. The twisting experiments were performed on CICC prototypes wound with sub-cable and dummy copper tubes. Meanwhile, critical current (I_C) tests were conducted on the conductor before and after compaction and twisting at 77 K under self-field conditions. The results show that the I_C of all samples remains above 98% before and after compaction and twisting, which verifies the feasibility and reliability of the compaction and twisting technology. This research provides valuable insights and a theoretical basis for designing and applying high-performance CICC conductors in fusion reactors.

Energy production by nuclear fusion can be the breakthrough in the decarbonisation process, and high temperature superconductors (HTS) represent a game changer for the design of compact reactors. However, reduced size implies that the superconducting tapes will be exposed to an intense flux of neutrons and of secondary particles while carrying a high current; in order to employ HTS in compact fusion reactors it is therefore crucial to precisely assess the effects of irradiation on HTS tapes at the working conditions. To achieve this goal, researchers from different fields met at the IREF (IRradiation Effects on HTS for Fusion) workshop to discuss all the aspects of this topic. This roadmap paper, that reflects the common view of the participants, aims at condensing the outcome of the intense and thorough discussion that took place during the conference, providing a path for the investigation of irradiation effects in HTS to assess their limits of operation in a fusion radiation environment.

The overall, loaded quality factor Q_L quantifies the loss of energy stored in a resonator. Here we discuss on general grounds how Q_L of a planar microwave resonator made of a conventional superconductor should depend on temperature and frequency. We consider contributions to Q_L due to dissipation by thermal quasiparticles (Q_QP), due to residual dissipation (Q_Res), and due to coupling (Q_C). We present experimental data obtained with five superconducting stripline resonators fabricated from lead (Pb), with different center conductor widths and different coupling gaps. We probe the resonators at various harmonics between 0.7 GHz and 6 GHz and at temperatures between 1.5 K and 7 K. We find a strongly frequency- and temperature-dependent Q_L, which we can describe by a lumped-element model. For certain resonators at lowest temperatures we observe a maximum in the frequency-dependent Q_L when Q_Res and Q_C match, and here the measured Q_L can exceed 2 × 10⁵

K-doped BaAs₂Fe₂ (K-Ba122) superconductor is a promising material for applications. However, it has been found challenging to achieve high critical current density (J_c) in untextured bulk sample. In this paper we investigated bulk samples prepared by varying the milling energy density, which affects the grain and grain boundary microstructures, and we investigated their magnetic performance to better understand what causes their different J_c. We found that in our samples, which all have small grain size, T_c does not appear directly correlated to J_c. Moreover, AC susceptibility reveals in at least one case obvious signs of multiscale supercurrents, not caused by granularity but that directly influence the overall J_c performance. Considering the microstructural features and the magnetization response we ascribed the J_c differences to lack of connectivity on a larger scale due to nano-cracks at some grain boundaries, which subdivided the samples into macroscopic regions and inevitably limited the overall performance. We discuss possible routes to overcome those extrinsic current-blocking defects.

Twisted, stacked cable-in-conduit-conductors, including VIPER cables, have emerged as a popular choice for high temperature superconductor fusion applications. The time-varying magnetic fields these cables are exposed to can generate significant AC losses, which are important to quantify for the design of fusion magnets. Although AC loss in twisted tapes and stacks—including VIPER cables—has been the subject of increasing research, many studies focus on higher temperatures and low fields outside the range applicable to fusion, while those considering higher fields tend to provide a less detailed analysis of the loss. This work provides a fundamental examination of the hysteresis loss characteristics of VIPER tapes, stacks and cables under conditions relevant to fusion applications. 3D finite element method models implemented with H-φ formulation are used to simulate the loss at 20, 40 and 77 K, under applied magnetic fields of up to 20 T. Cables with up to 10 tapes per stack are considered. The field-angle dependence of critical current and n-value are accounted for, based on measured data from 4 mm Faraday Factory tape. Results show that hysteresis loss in VIPER strands is independent of pitch length and winding radius, a valuable result for shortening simulation time. A semi-empirical method is proposed to estimate the loss in VIPER cables from 2D simulations of flat stacks, supplementing the established $2/{\pi}$ relationship. It is also shown that hysteresis loss in VIPER geometries can be scaled across temperatures by normalizing with the self-field critical current of a single tape, surprisingly irrespective of cable ${I_{\text{c}}}$.

Upon reviewing the published version of our work, we identified misprints in the equations, which are corrected here. Additionally, we re-evaluated the oxygen concentration depth profiles. This re-evaluation does not affect the validity of the paper's statements or conclusions.

Stochastic computing (SC) is a form of probabilistic computation that encodes information in the probability of a '1' occurring within a finite-length binary sequence. SC has been investigated for applications in various fields that do not require deterministic and precise computation. A superconducting single-flux-quantum (SFQ) circuit is considered a promising candidate for implementing SC hardware due to its high-speed operation and probabilistic behavior. In this study, we propose a novel large fan-out signal splitter to enable large-scale SFQ-based stochastic arithmetic circuits, addressing the issue of computation accuracy degradation caused by correlations between binary sequences. The proposed signal splitter generates uncorrelated output binary sequences by utilizing superconductor random number generators frequency-synchronized to the input binary sequence. The fan-out can be easily increased by simply adding more superconductor random number generators. We implemented a four-output stochastic number signal splitter using the 10 kA cm⁻² Nb four-layer superconducting circuit fabrication process. Its operation was successfully demonstrated by measuring the average voltage at the input and outputs under continuous high-speed binary sequence input. High-speed operation up to 33.2 GHz was confirmed. The proposed signal splitter uniquely leverages the properties of superconducting circuits, where flux quanta determined by fundamental physical constants serve as the information carrier. We believe this development will significantly advance the realization of practical SFQ-based SC systems.