Superconductor Science and Technology Supercond. Sci. Technol. 37 (2024) 123005 (42pp) https://doi.org/10.1088/1361-6668/ad8bf8 Topical Review Review on high-temperature superconducting trapped field magnets Qi Wang1, Hongye Zhang2, Luning Hao1,∗ and Tim Coombs1 1 Department of Engineering, University of Cambridge, Cambridge CB3 0FA, United Kingdom 2 Institute for Energy Systems, School of Engineering, The University of Edinburgh, Edinburgh EH9 3FB, United Kingdom E-mail: lh699@cam.ac.uk, qw273@cam.ac.uk, hongye.zhang@ed.ac.uk and tac1000@cam.ac.uk Received 30 April 2024, revised 26 August 2024 Accepted for publication 27 October 2024 Published 25 November 2024 Abstract Superconducting (SC) magnets can generate exceptionally high magnetic fields and can be employed in various applications to enhance system power density. In contrast to conventional coil-based SC magnets, high-temperature superconducting (HTS) trapped field magnets (TFMs), namely HTS trapped field bulks (TFBs) and trapped field stacks (TFSs), can eliminate the need for continuous power supply or current leads during operation and thus can function as super permanent magnets. TFMs can potentially trap very high magnetic fields, with the highest recorded trapped field reaching 17.89 T, achieved by TFSs. TFMs find application across diverse fields, including rotating machinery, magnetic bearings, energy storage flywheels, and magnetic resonance imaging. However, a systematic review of the advancement of TFMs over the last decade remains lacking, which is urgently needed by industry, especially in response to the global net zero target. This paper provides a comprehensive overview of various aspects of TFMs, including simulation methods, experimental studies, fabrication techniques, magnetisation processes, applications, and demagnetisation issues. Several respects have been elucidated in detail to enhance the understanding of TFMs, encompassing the formation of TFBs and TFSs, trapped field patterns, enhancement of trapped field strength through pulsed field magnetisation, as well as their applications such as SC rotating machines, levitation, and Halbach arrays. Challenges such as demagnetisation, mechanical failure, and thermal instability have been illuminated, along with proposed mitigation measures. The different roles of ferromagnetic materials in improving the trapped field during magnetisation and in reducing demagnetisation have also been summarised. It is believed that this review article can provide a useful reference for the theoretical analysis, manufacturing, and applications of TFMs within various domains such as materials science, power engineering, and clean energy conversion. ∗ Author to whom any correspondence should be addressed. Original content from this workmay be used under the terms of the Creative Commons Attribution 4.0 licence. Any fur- ther distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1 © 2024 The Author(s). Published by IOP Publishing Ltd https://doi.org/10.1088/1361-6668/ad8bf8 https://orcid.org/0000-0002-9265-7573 https://orcid.org/0000-0002-8960-4614 https://orcid.org/0000-0003-1116-1400 https://orcid.org/0000-0003-0308-1347 mailto:lh699@cam.ac.uk mailto:qw273@cam.ac.uk mailto:hongye.zhang@ed.ac.uk mailto:tac1000@cam.ac.uk http://crossmark.crossref.org/dialog/?doi=10.1088/1361-6668/ad8bf8&domain=pdf&date_stamp=2024-11-25 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Keywords: high-temperature superconductors, trapped field magnets, trapped field bulks, trapped field stacks, magnetisation, demagnetisation Contents 1. Introduction 3 1.1. Comparison between TFSs and TFBs 4 1.1.1. Geometrical flexibility 4 1.1.2. Homogenous SC properties 4 1.1.3. Enhanced mechanical and thermal properties 4 1.1.4. Superior performance under cross-field (CF) demagnetisation 5 1.2. Comparison between REBCO bulks and MgB2 bulks 5 2. Modelling methods 5 2.1. E–J power law and Jc dependence 5 2.2. H and T–A formulations 6 2.2.1. H-formulation 6 2.2.2. T–A formulation 7 2.3. Multi-layered and homogenisation models 7 3. Fabrication and macroscopic structures 8 3.1. HTS TFBs 8 3.1.1. HTS bulk fabrication 8 3.1.2. Seeds for melt growth approaches 8 3.1.3. Large-dimension bulks and bulk composites 9 3.1.4. Hybrid bulk/FM configuration 9 3.2. HTS TFSs 10 3.2.1. Hybrid magnets 10 3.2.2. Sectioned stacks 11 3.2.3. Angled stacks and patterned stacks 11 3.2.4. Self-supporting stacks 12 3.2.5. Ring-shape tape stack magnets 12 4. Magnetisation 13 4.1. Characteristics of trapped field 13 4.1.1. Trapped field profile 13 4.1.2. Influence factors for trapped field strength 14 4.1.3. Highest trapped fields of TFMs 15 4.2. Magnetisation methods 16 4.2.1. FC and ZFC 16 4.2.2. PFM 17 4.2.3. Flux pumping 21 5. Applications 22 5.1. Electrical machines 22 5.1.1. TFB machines 23 5.1.2. TFS machines 24 5.2. SC levitation 24 5.3. Halbach arrays 24 5.4. MRI/NMR 25 5.5. Other applications 25 5.5.1. SC undulators (SCUs) 25 5.5.2. Electromagnetic launchers as linear actuators 26 5.5.3. Magnetic separation systems 26 6. Challenges and solutions 26 6.1. Demagnetisation 26 6.1.1. Flux creep 26 6.1.2. Rotating magnetic field demagnetisation 27 6.1.3. CF demagnetisation 27 6.1.4. Methods to suppress demagnetisation 28 6.2. Mechanical strength 30 6.3. Thermal stability 30 7. Conclusions 31 Data availability statement 32 References 32 Acronyms and symbols 2D two-dimensional 3D three-dimensional AC alternating current AMSC American Superconductor ASCEND Advanced Superconducting and Cryogenic Experimental powertraiN Demonstrator ASuMED Advanced Superconducting Motor Experimental Demonstrator project BSCCO bismuth strontium calcium copper oxide BA-TSIG buffer-assisted top-seeded infiltration and growth CC coated conductor CMDC controlled magnetic density distribution coil CF cross-field DC direct current ETT evacuated tube transport FC field cooling FEM finite element method FM ferromagnetic FSCW fractional slot concentrated winding GSB growth sector boundary GSR growth sector region GdBCO gadolinium barium copper oxide HTS high-temperature superconducting IBAD ion beam assisted deposition IGBT insulated gate bipolar transistor IMRA iteratively magnetising pulsed field method with reducing amplitude LTS low-temperature superconducting Maglev magnetic levitation MCU microcontroller unit MOCVD metal-organic chemical-vapour deposition MRI magnetic resonance imaging MgB2 magnesium diboride MMEV minimum magnetic energy variation MEMEP minimum electromagnetic entropy variation MMPSC modified multi-pulse technique with stepwise cooling MPSC multi-pulse technique with stepwise cooling MST multi-seeding technique MUSLE multi-seeded seamless NMR nuclear magnetic resonance 2 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review PFM pulsed field magnetisation PM permanent magnet PMG permanent magnet guideway PST partially slit tape QMG quench melt growth RABiTS rolling-assisted biaxially textured substrate REBCO rare earth barium copper oxide SC superconducting SCU superconducting undulator SDMG single-direction melt growth SMB superconducting magnetic bearing SPA sequential pulsed field application TAM thermally-actuated magnetisation TFB trapped field bulk TFM trapped field magnet TFS trapped field stack TSIG top-seeded infiltration growth TSMG top-seeded melt growth WCPM waveform control pulsed magnetisation YBCO yttrium barium copper oxide ZFC zero field cooling B∥ parallel magnetic fields to the HTS superconductors B⊥ perpendicular magnetic fields to the HTS superconductors Bap(t) applied pulsed field Bap0 magnitude of the applied pulsed field Bm magnetisation field Bm,p the magnetisation field precisely sufficient to fully magnetise TFMs Btrap trapped field BT,max maximum trapped field Cp specific heat capacity E0 characteristic electric field with E0 = 10−4 V∙m−1 FL Lorentz force Fp pinning force Fv viscous force Jc critical current density Jc0(T) temperature-dependent critical current density with self-field Je engineering critical current density k thermal conductivity M(t) magnetisation of superconductors n exponential index of the power law Q heat generation power density Qp pinning losses Qv viscous losses RTFM TFM’s radius S relaxation rate/creep rate τ rise time of the applied pulsed field ∆T temperature rise Tc critical temperature Ts operating temperature U effective activation energy µ0 free space permeability µr relative permeability of the studied material v flux propagation velocity σB tensile strength ρm mass density 1. Introduction Superconducting (SC) magnets, designed to produce high magnetic fields, have seen global development over several decades [1–5]. Their applications are extensive, spanning fields but not limited to accelerator magnets, magnetic res- onance imaging (MRI), nuclear magnetic resonance (NMR), fusion magnets, etc [1, 6]. Traditionally, these magnets are constructed using wound SC wires or tapes shaped into coil forms. Alternatively, high magnetic fields can be generated by employing the concept of high-temperature superconducting (HTS) trapped field magnet (TFM). Before their application, magnetisation serves as the prerequisite step, enabling TFMs to function as super permanent magnets (PMs). Operating in ‘persistent mode’ [7] by capturing persistent currents from the magnetisation process, the most significant advantage of TFMs compared to the traditional solenoid SC coils is the elimination of the need for bulky current sources, current leads, and SC joints [7] during operation. This results in min- imisation of thermal and electrical losses in the entire magnet system. One example is the application of TFMs in rotary elec- tric machines, where the utilisation of brushes and slip rings can be eliminated. This contrasts with wound-field machine topologies that require continuous current sources and current leads. Due to the high current-carrying capacity with higher crit- ical temperatures, Tcs, and critical magnetic fields, compared to low-temperature superconducting (LTS) materials [8], HTS materials are more commonly employed in large-scale applic- ations, including HTS TFMs, which is the focus of this article. Conventional magnets, constrained by the saturation curve of ferromagnetic (FM) materials, are generally limited to 1.8 T [9]. Thanks to their high critical current densities at cryogenic temperatures, HTSTFMs demonstrate the remarkable capabil- ity to exceed the maximum magnetic field strengths attainable with conventional magnets by nearly tenfold [7, 10–13]. HTS materials are commercially available mainly in two types: conductors in the form of multifilamentary wires or multilayer tapes, and bulk samples. Currently, there are three main types of HTS materials available: bismuth strontium cal- cium copper oxide (BSCCO), rare earth barium copper oxide (REBCO), and magnesium diboride (MgB2). BSCCO, known as the first-generation HTS material, has a Tc ranging from 90 K to 110 K, depending on its SC phase [14]. Most rare earth elements can be used to fabricate REBCO, with popular ele- ments including yttrium (Y, [15]), gadolinium (Gd [16],), neo- dymium (Nd, [17]), europium (Eu, [18]), and samarium (Sm, [19]). Notably, yttrium barium copper oxide (YBCO) was the first HTS material discovered by Wu et al [15], with a Tc of 93K. REBCO features a Tc between 89K and 96K, depending on the rare earth elements used, such as DyBCO (89 K [20]), EuBCO (93K [18]), GdBCO (94K [16]), NdBCO (95K [17]). In contrast, MgB2 exhibits a Tc of 39 K. Among these materials, REBCO stands out as the pre- ferred choice for power applications and magnet production [21], including TFMs, due to its significantly higher in-field 3 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review critical current density, Jc, compared to BSCCO [22], as well as its higher Tc and Jc compared to MgB2. Additionally, the manufacturing process based on high-strength substrates enables REBCO conductors to withstand higher electromag- netic forces than the other two HTS candidates [22]. The primary forms of a TFM are either stacked HTS coated conductors (CCs) or HTS bulk superconductors [23]. The adoption of TFMs based on HTS materials carries signific- ant implications for a wide range of engineering applications, encompassing domains such as rotating machinery [24, 25], magnetic bearings [26, 27], levitation [28, 29], energy storage flywheels [30],MRI/NMR technology [31, 32], andmore [33]. Generally, there are various applications for HTS CCs and bulk superconductors, including flux shielding [34, 35], flux pinning [36, 37], and flux trapping [7, 10]. In the current work, the primary emphasis will be focused on applications related to flux trapping. Upon magnetisation, bulks or stacked HTS CCs can capture magnetic flux and transform into TFMs. Stacked HTS CCs and HTS bulk samples with trapped field are designated as trapped field stacks (TFSs) and trapped field bulks (TFBs), respectively. The TFMs can confine elev- ated magnetic fields at low temperatures, making them well- suited for various applications in the domain of PM technology [38]. However, HTS tape stacks and bulks differ in various aspects, including their manufacturing process, mechanical and thermal properties, SC and magnetic properties. 1.1. Comparison between TFSs and TFBs HTS bulk samples typically contain 100%HTSmaterial, facil- itating a much higher engineering critical current density, Je, compared to TFSs. However, their brittleness results in low mechanical strength, necessitating external reinforcement to withstand Lorentz forces, FLs [39]. Additionally, the field trapping capability of bulk superconductors is also limited by their poor thermal conductivity [40]. In contrast, SC tape stacks outperform SC bulks as TFMs due to their higher mech- anical strength and shape flexibility [13]. Stacked supercon- ductors feature a substantial metallic content and a layered structure, offering numerous advantages for TFSs compared with TFBs [13]. All of these characteristics make HTS tape stacks outperform HTS bulks in the following respects: 1.1.1. Geometrical flexibility. The multilayer structure of HTS CCs provides flexibility in geometry, allowing for ver- satile shaping through easy machining. HTS CCs can be con- figured in various ways to form TFMs, such as hybrid config- urations combing different tapes to build magnets with higher field strength and wider pole surfaces [13], stack arrangements with tilt angles [41], and crisscross structures (a series of nar- row tapes arranged in parallel in the vertical direction in odd layers and an equal number of tapes arranged in parallel in the horizontal direction in even layers, as shown in section 3.2.3) [42] to obtain uniform fields. Additionally, stack configurations are scalable, enabling the addition of more layers in the thickness direction for a higher field strength or expansion of the field surface in the width direction [43, 44], as long as the magnetisation method and HTS materials are adequate [41]. Furthermore, stacked super- conductors can be customised for various applications. For instance, they can be bent to align with the curved, spherical contours of rotors and seamlessly integrated into the narrow airgap of motors by adjusting the curvature [45] and thickness (i.e. the layer number) of the SC stacks [46]. 1.1.2. Homogenous SC properties. The HTS tape stacks exhibit a more homogeneous current density across the stack’s volume compared to HTS bulks due to the multi-layer struc- ture of HTSCCs. This homogeneous structure helps to smooth anomalies in the stacks [47], ensuring consistent performance among different stacks from the same tape batch. A suboptimal area or a minor weak point in individual layers does not impair the overall performance of the entire stack assembly. In con- trast, the field-trapping performance of HTS bulks varies from sample to sample due to numerous voids and cracks within the sample microstructure generated during the top-seeded melt growth (TSMG) process. In [11], it was noted that the TFB which achieved the record field strength of 17.6 T, failed upon subsequent remagnetisation. Interestingly, two other TFBs, which were fabricated identically to the one achieving 17.6 T, could only trap 10 T and 15.4 T, respectively. This variabil- ity among supposedly ‘standard’ TFBs underscores the chal- lenges and inconsistencies in the fabrication process. Furthermore, stacks of HTS CCs display a relatively uni- form Jc both radially and axially. This is evident from the comparison between a stack of 2800 thin YBCO square hol- low CCs and 10 YBCO bulk annuli (ring-shaped bulks) in [48] and [49]. It has been shown that TFSs made of CCs exhibit superior spatial homogeneity and temporal stability compared to their bulk counterparts. 1.1.3. Enhanced mechanical and thermal properties. Substantial FL will be generated during magnetisation, lead- ing to fractures in the brittle ceramic HTS materials, and mechanical failure of HTS bulks. Therefore, reinforcement measures are required for bulks to trap a high magnetic field due to the low mechanical stability. In contrast, TFSs do not need external mechanical reinforcement thanks to the robust superalloy substrates, which exhibit heightened tensile strength and account for more than 85% of the volume [50]. The metallic substrate, typically Hastelloy or nickel-tungsten (Ni-W) alloys, contributes to superior mechanical strength in TFSs compared to TFBs and eliminates the need for external mechanical reinforcement against FL at high-trapped fields. The penetration of magnetic flux into superconductors dur- ing magnetisation generates significant heat due to the high Jc of HTS materials. Inhomogeneous Jc within HTS bulks results in large local temperature increases due to excessive heat dissipation [51], which can cause magnetisation decay at a high magnetisation field, Bm. In contrast, more homogen- eous heat dissipation is expected within HTS tape stacks due to their more uniform Jc, preventing the suppression of mag- netic induction from local overheating and enhancing the field- trapping capability of TFSs. Additionally, metallic layers like 4 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review the silver stabiliser layer in HTS stacks enhance thermal stabil- ity by dissipating heat and suppressing flux jumps (an unres- trained flux movement within SC materials, induced by sub- stantial thermal generation during magnetisation) in the HTS layer, providing TFSs with higher thermal stability compared to TFBs. 1.1.4. Superior performance under cross-field (CF) demagnet- isation. CF demagnetisation occurs when the trapped field of TFMs decreases in response to an external alternating current (AC) field which is orthogonal to the initial magnetisation dir- ection. TFSs are more robust to CFs or transverse fields com- pared to TFBs due to their large aspect ratios of stacked HTS tapes and the ultra-thin thickness in their SC layers [52, 53]. 1.2. Comparison between REBCO bulks and MgB2 bulks Same as REBCO bulks, MgB2 bulks can also be magnetised and utilised as TFMs. The first reported utilisation of MgB2 as a superconductor was in 2001 [54]. Compared to REBCO, MgB2 has a much lower Tc (39 K), necessitating operation at low temperatures (e.g. at 20 K) [55]. This requirement results in a more complex cryogenic system and an increased like- lihood of flux jumps due to the thermal instability of the SC material. However, in contrast to REBCO,MgB2 exhibits amore uni- form and isotropic distribution of Jc [56, 57], and the pro- duction of large-grain polycrystalline single-domain MgB2 bulks is easier [58], thanks to their longer coherence lengths. Additionally, MgB2 offers advantages such as a simpler fab- rication process, lighter weight due to lower mass density, and lower cost without the need for rare earth elements [59, 60]. Furthermore, MgB2 can be manufactured into fully machin- able bulks, exhibiting a near-net shape [61], which can meet the demand for customised shapes [62]. The ability to fabric- ate sizeable and customised shapes enables the application of MgB2 for high-performance magnetic shielding [62]. Compared to REBCO bulks, MgB2 bulks face challenges related to thermomagnetic stability, which is attributed to their small specific heat, large thermal conductivity, and narrow temperature margin against operating temperature, Ts [63]. This compromise leads to flux jumps during magnetisation, and the increased vulnerability to flux jumps can significantly impact the trapped magnetic field [64]. The most recent recor- ded trapped field in MgB2 bulk superconductors was 5.4 T at 12K in 2013 [65], which is much lower than the record trapped field of REBCO bulks (17.6 T). Given the advantages of REBCO over MgB2, such as high Jc and high Ts, REBCO becomes more prevalent in large- scale applications. Therefore, this paper will concentrate on REBCO bulks in terms of HTS bulks. In the following, section 2 will cover various model- ling methods, including the most volatile formulations for modelling TFMs. In section 3, detailed descriptions of fab- rication methods to produce large HTS bulks and various approaches to assemble HTS stacks for higher trapped field and trapped flux will be provided. Section 4 will focus on the magnetisation aspects of TFMs, including trapped field characteristics for various cases and common magnetisa- tion methods to achieve better magnetisation. Section 5 will discuss applications for TFMs such as SC motors, levita- tion systems, and MRI/NMR. Finally, section 6 will address the challenges that TFMs are facing and their correspond- ing solutions. In the final section, a comprehensive sum- mary of all key aspects of TFMs will be provided, fol- lowed by an outlook on their prospects, including their potential for broader applications in real-world large-scale scenarios. 2. Modelling methods As for modelling superconductors, there exist multiple meth- ods developed such as the finite difference method [66], finite volume method [67], variational principle [68], integral methods [69], and finite element method (FEM) [70], among which FEM stands as the predominant approach for address- ing challenges related to SC materials with high non-linearity, including TFMs. These techniques can be applied to various formulations with different state variables when simulating the nonlinear superconductors, e.g. A–V formulation [71, 72], T– Ω formulation [73], E-formulation [74], Campbell’s equation [75, 76], H-formulation [77–80], T–A formulation [81, 82], MMEV method [83], MEMEP [84, 85], H-φ formulation [86, 87], H–A Formulation [88]. Additionally, other derived approaches, such as the densification method, homogenisa- tion, and multi-scaling methods, have been developed for both the H-formulation and T–A formulation [89]. For the simulation, there are commercial software pack- ages, e.g. COMSOL Multiphysics [90, 91], ANSYS [92], Opera-3D [93], Flux2D [94], Flux3D [95], and FlexPDE [96, 97]. Self-programmed codes e.g. MEMEP based on C++ code [98, 99], and open-source coding such as GetDP [100], and FreeFem++ [101] can also be employed. In this section, various numerical methods for modelling TFMs are introduced. The mathematical descriptions for the highly nonlinear critical current densities are summarised under various conditions, such as external magnetic fields, temperature variations, and angular orientations. Additionally, the most employed modelling formulations and homogenisa- tion approaches are introduced. 2.1. E–J power law and Jc dependence TheE–J power law as per [102] is used to model the nonlinear electrical behaviour of the SC materials through: ∥E∥= E0 ( ∥J∥ Jc )n (1) where the exponential n is the exponential index of the power law, and n represents the steepness of the transition from SC state to normal state, E0 represents the characteristic electric field with E0 = 10−4 V∙m−1. 5 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review The two parameters in the E–J power law, n and Jc, are essential for accurately modelling superconductors in real- world scenarios. They are normally determined by experi- mental results and provided as self-field values by manufac- turers for specific superconductors. Both parameters can be influenced by the local temperature T and local magnetic field B. However, for rapid estimations, they can be simplified. For instance, in the critical state model, Jc is independent of B [103]. In simulations, the exponential index n is typically assumed to be a constant within the range from 5 to 50 (corres- ponding to strong flux creep and critical state model, respect- ively) [104], to achieve fast and converged simulation results. It was observed by Zou et al that a higher trapped field can be obtained with a larger n, as smaller n values cause more over- critical current and heat generation [104]. This is because, with a smaller n value, superconductors transition to the mixed state more easily, resulting in larger heat generation during magnet- isation. Jc typically increases at lower Ts and with a smaller B perpendicular to the a–b plane of the SC materials. Consequently, various combinations can be explored with different values of n and Jc, which can be treated as constants or extended to include dependencies on magnetic field (n(B), Jc(B)) [105], temperature (n(T), Jc(T)), or both (n(B, T), Jc(B, T)) [106, 107]. Additionally, n and Jc can vary depending on the angle of the applied magnetic field. Studies have been con- ducted on the magneto-angular dependence of n and Jc in the form of Jc (B, θ) [108, 109] and n(B, θ) [108], through exper- imental measurements and modelling work. Data on Jc(B) measurements are commonly available from manufacturers’ datasheets. Many studies have also provided those data in published works, such as for the AMSC 46 mm wide tapes [13, 110], and for bulk superconductors [111, 112]. A performance dataset, including the Jc and the exponents for various HTS tapes, is accessible online at www.victoria.ac. nz/robinson/hts-wire-database [113]. Recently, comprehens- ive measurements have been conducted using transport cur- rent measurements of thin REBCO bulk slices, and a dataset of Ic(T, B, θ) and n(T, B, θ) was obtained [16]. For the mathematical description of Jc, measurement res- ults from experiments or the empirical model introduced by Kim [114, 115] can be employed. Jc (B,T) = Jc0 (T) · B0 (B0 + ∥B∥) (2) where B0 is a constant associated with the SC materials and Jc0(T) is the temperature-dependent critical current density for ||B|| = 0, defined as: Jc0 (T) = α(T) B0 (3) where α(T) is a temperature-dependent parameter and charac- terised by the definition: α(T) = 1 d (a− bT), where d repres- ents a constant influenced by the specific physical microstruc- ture of the superconductor and a/b⩽ Tc. There is another way to describe Jc0(T) [116]: Jc0 = α { 1− ( T Tc )2 } 3 2 (4) where Tc is the critical temperature (92 K for YBCO) at zero local magnetic fields and α is a constant. When considering only the dependency of HTS supercon- ductors on the magnetic field, there is another commonly employed form of the Kim model, considering the anisotropy of the HTS materials [117, 118]: Jc (B) = Jc0( 1+ √ (k0∥B∥∥)2+∥B⊥∥2 Bc )b (5) where B∥ and B⊥ denote the parallel and perpendicular mag- netic fields to the HTS superconductors respectively. Bc, k0 and b are derived from curve fitting of experimentally meas- ured data. 2.2. H and T–A formulations The H-formulation and T–A formulation are widely recog- nised as the primary modelling approaches for simulating SC magnets with various geometries, whether in the form of coils or TFMs [89, 119]. These formulations are also more read- ily integrated into commercially available software, such as COMSOL Multiphysics. In contrast, other methods such as MEMEP and MMEV, while offering faster calculation speed and time-saving benefits [84], often rely on self-programmed codes [120] and are less commonly supported by widely used commercial software, making them challenging to implement for a broad range of users. Consequently, this section will focus solely on outlining the fundamental equations associated with H and T–A formulations. 2.2.1. H-formulation. The H-formulation stands as a widely employed approach for the modelling of diverse HTS config- urations, including TFMs [90, 121]. The essence of this formu- lation is to utilise magnetic field components as state variables. Its notable advantages lie in accuracy, robust convergence, and reasonable computational efficiency [119]. According to the Faraday’s law, a time-varying magnetic field is invariably associated with a spatially varying electric field: ∇×E=−∂B ∂t . (6) The constitutive relation between magnetic flux density B and magnetic field H is defined as B= µ0µrH (7) where µ0 and µr represent the free space permeability and the relative permeability of the studied material, respectively. The 6 http://www.victoria.ac.nz/robinson/hts-wire-database http://www.victoria.ac.nz/robinson/hts-wire-database Supercond. Sci. Technol. 37 (2024) 123005 Topical Review current density within the superconductors is defined by the Ampere’s law: J=∇×H. (8) The electric field within the superconductors is defined by Ohm’s law: E= ρJ (9) where ρ is the material resistivity. From the equations (6)–(9), the governing equation for the H-formulation can be derived as: µ0µr ∂H ∂t +∇× ρ(∇×H) = 0. (10) 2.2.2. T–A formulation. In the T–A formulation, the T- formulation handles the SC domains, while the A-formulation handles the non-SC domains. It is worth noting that T- formulation can also incorporate non-SC materials, as demon- strated in [122]. However, due to efficiency concerns with the integral method and the challenges with standard implement- ation, the T–A formulation, which is a coupled formulation built upon the T-formulation, has been developed to enhance the simulation efficiency and performance [82]. The SC tapes are usually regarded as infinite-thin sheets in the T–A formulation. By neglecting the thickness of the SC layer, the thin-sheet model can yield satisfactory simulation results while also reducing the complexity of the mesh and enhancing computational efficiency [81]. The current density vector J is defined as the curl of the current vector potential T, which is computed only in the SC domain: J=∇×T. (11) While the current density vector is constrained to flow within the SC sheets, the current vector potential is oriented perpendicular to the wide surface of the conductor at every point, J=∇× (T ·n) (12) where T is the amplitude of T, and n represents the unit normal vector of the sheet plane. The T-formulation can be derived from the equations (1), (6), and (12) as: ∇× ( E0 ( ∇× (T ·n) Jc )n) =−∂B ∂t . (13) The magnetic flux density B is calculated from the A- formulation. It is calculated as the curl of the magnetic poten- tial vector A: B=∇×A. (14) The magnetic potential vector A is used for the calculation of the source-free region. According to Ampere’s law: ∇×∇×A= 0. (15) The equations (13) and (15) are called the T-formulation and A-formulation, respectively. The essential boundary con- dition required for the solution of equation (15) involves the surface current density within the HTS tapes, which corres- ponds to the J term in equation (12). This linkage effectively integrates the T-formulation and the A-formulation. From the equations (13) and (14), the coupled T–A formulation is: ∇× ( E0 ( ∇× (T ·n) Jc )n) =−∂ (∇×A) ∂t . (16) 2.3. Multi-layered and homogenisation models Due to the symmetric geometry, HTS bulks are typically modelled as two-dimensional (2D) axisymmetric models [104, 123]. For simulating HTS tape stacks, two preval- ent approaches are typically employed. The first involves a detailed structural model, which can be called the multi- layered model, encompassing all metallic and SC layers with their original thickness [124]. Conversely, the second approach, known as the homogenisation method, proposed by Zermeno et al [125], treats the entire structure as an equivalent bulk. In this method, HTS stacks with intricate tape structures are replaced by equivalent bulks exhibiting the same overall electromagnetic performance. The multilayer structural model offers insights into indi- vidual layers of CCs, elucidating aspects such as heating and heat transfer between layers during magnetisation [46, 124] and magnetisation losses during CF demagnetisation [126]. However, the multi-layered model demands an extensive num- ber of mesh nodes for convergence, resulting in a substan- tial increase in computational time, especially when simulat- ing electromagnetic-thermal coupled models for TFMs. The multitude of domains and elements in the simulation model induces software instability and necessitates a large memory volume of computational devices. Conversely, the homogenised model allows to quickly model tape stacks and can yield very similar results to the multi-layered model. It is noteworthy that the homogenisa- tion approach represents stacked CCs as uniform bulks, with the prerequisite that the Jc of the bulk material be adjusted to the Je based on the proportion of SC materials in the entire tape [43, 45]. Nevertheless, Zhang et al first proved that in high-frequency environments, the electromagnetic character- istics of HTS stacks, specifically the electromagnetic interac- tions between the normal conducting and SC layers within the TFSs, become more intricate and complex, requiring the con- sideration of the multi-layer structure of HTS CCs [46, 122, 126, 127]. Between the multi-layered structure and the homogen- ised structure, a semi-homogenised structure has also been employed by modelling a tape with the SC layer and a single layer measuring the total thickness of the other metallic lay- ers, encompassing the substrate layer, the silver layer, and the copper layer. This was proposed in a study conducted byWang et al [46], where the semi-homogenised structural model was utilised for the HTS tapes produced by AMSC, for purposes 7 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 1. Illustration of different model structures with the example of an HTS stack made of AMSC tape (a) multi-layered model (b) semi-homogenised model (c) homogenised model. of model validation and motor operation, as the reasonable partial homogenisation minimally affected magnetisation and motor performance and accelerated the simulation process. Figure 1 demonstrates the multi-layered structure, the semi- homogenised, and the homogenised bulk model for a stack made of AMSC tapes [46]. 3. Fabrication and macroscopic structures Large-grain HTS bulks in the RE–Ba–Cu–O form (with RE representing a rare-earth element) have been investigated for a long time while the study for stacked-tape magnets has just started in recent years [50]. REBCO bulks are gener- ally produced through TSMG and TSIG techniques, while the stacked tape magnets can be created by stacking slices of CCs. For HTS bulks, advanced fabrication techniques, such as multi-seeding and multi-grain approaches, can be used to produce bulks with larger sizes. HTS CCs can be stacked into various shapes and combinations for enhanced trapped field strength, increased trapped flux, enhanced field uniformity, and improved mechanical strength. Additionally, the utilisa- tion of other materials like FM materials can further enhance the performance of TFMs. In this section, various fabrication methods for pro- ducing HTS bulk samples are presented, including the approaches for creating large-dimensional HTS bulks. The hybrid SC/FM configurations for bulk materials are also dis- cussed. Additionally, the multi-layer structure of HTS CCs is introduced, which is the foundation to understand the beha- viour of stacked CC magnets. The different configurations of TFSs have been summarised, which aim at achieving higher trapped field strength, enhanced trapped flux, trapped field with higher uniformity, and improved mechanical strength. 3.1. HTS TFBs REBCO bulks are normally cultivated from seed crystals, which are usually positioned on the top of seed disks with their c-axis vertical to the surfaces of the disks. The growth of the seed crystals was completed with the following heat treatment. Figure 2. Various shapes of HTS bulks. HTS bulks are typically crafted into various forms and shapes such as cylinders, disks, rectangles, rings (annuli), or ring segments, as shown in figure 2. 3.1.1. HTS bulk fabrication. Research on the fabrication pro- cess of ceramic REBCO materials initially employed sinter- ing technique, a common method used for producing ceram- ics. However, the sintered body contains numerous crystal grains, often resulting in high-angle grain boundaries, which weaken the connections between grains and reduce the Jc of the specimens. To address this issue, melt-growth tech- niques such as TSMG [12], QMG [128, 129], melt powder melt growth (MPMG) [130, 131], and oxygen-controlled melt growth (OCMG) [132, 133] have been employed to produce REBCO bulks with highly aligned grains with large domain sizes, resulting in higher Jc values and improved field-trapping capabilities. For instance, TSMG, which is the most com- monly used approach, has been shown to reliably produceHTS bulks with Jc values exceeding 10 kA∙cm−2 at 77 K [7]. TSIG, an advanced enhancement to TSMG, allows for the creation of dense single-grain microstructures with near-net shape pro- cessing. The two-step BA-TSIG process, representing a fur- ther advancement over TSIG, has successfully produced HTS bulks with Jc values ranging from 20 to 50 kA∙cm−2 at 77 K, significantly surpassing those obtained by conventional TSMG and TSIG methods [134]. Table 1 lists a few examples of the record trapped field values of TFBs made with different techniques. Bulk samples produced by the TSMG process often exhibit several GSBs, with the areas between these boundaries referred to as GSRs. This results in inhomogeneous SC prop- erties within the fabricated bulk samples. For instance, the Jc is approximately four times higher in GSBs compared to in GSRs [51]. Consequently, the SDMG method is emerging as a superior alternative to the TSMG and TSIG, as it produces REBCO bulks with higher homogeneity [137, 138]. 3.1.2. Seeds for melt growth approaches. In melt growth approaches, such as TSMG and TSIG, seeds are crucial for obtaining high-quality bulk samples. Small seed crystals are typically placed atop pre-sintered REBCO pellets, either before (cold seeding method) or during (hot seeding method) 8 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Table 1. Examples of record field values for TFBs made with different techniques. Fabrication method Bulk sample Btrap (T) Size (Diameter/mm × thickness/mm) Ts (K) Reference TSMG 2 GdBCO bulks 17.6 24.15 × 30 26 [7] BA-TSIG 2 YBCO bulks 14.3 25 × 10.2 28 [135] QMG 1 GdBCO bulk 10 60 × 2 20 [129, 136] the process [139], to initialise the large grain growth and control its orientation [140]. These seeds, such as SmBCO, NdBCO or MgO, are structurally compatible (same crystal structure) with YBCO materials and possess higher melting points than the bulk pellet, with a useful temperature differ- ence being over 30 K [139]. In addition to crystal seeds, RE-123 thin film seeds can also be employed for seeding REBCO bulks. YBCO, SmBCO, and NdBCO bulk samples have been seeded using NdBCO/MgO thin film seeds [141–143]. Moreover, various REBCO/MgO thin films have been employed for multi-seeded growth [144] and batch production [145] of REBCO bulks. 3.1.3. Large-dimension bulks and bulk composites. The production of large-dimension REBCO bulk superconductors on a centimetric scale, particularly those exceeding 100 mm [146], is essential for large-scale applications like electric motors for electric aircraft. However, their fabrication poses a challenge due to the uncontrollable and slow crystal growth rate associated with single-crystal growth techniques such as TSMG. This results in a lack of homogeneity due to grain mis- alignment and a time-consuming fabrication process [136]. To address this issue, various approaches involving HTS bulks have been proposed: a large-area-seed method [147], an interior seeding approach [148, 149], and a MST [150–152]. Among those approaches, the MST is well-suited for produ- cing large samples as it effectively reduces the bulk growth time. The utilisation of the multiple seeding technique has been extensively implemented to augment the size of YBCO bulks [147–151]. The growth process involves placing multiple seeds on top of a YBCO compact and simultaneously melt- processing the compact [150]. This allows for straightforward heat treatment, resulting in the fabrication of large samples at a higher speed [152]. For instance, a multi-seeded seamless (MUSLE) bulk technique was introduced in [147], leading to the successful production of REBCO bulk samples with dia- meters of up to 100 mm. Figure 3 demonstrates the single- and multi-seeded samples with various sizes. A similar graph depicting samples with various seed numbers and sizes can be found in [26]. However, grain boundaries may be present in the multi-seeded sample, increasing the misalignment of grains and diminishing the overall homogeneity of the sample [150]. As a result, the trapped field distribution of bulk super- conductors produced by the MST may exhibit several peaks, attributed to the presence of the excluded non-SC phases at grain boundaries. In addition to the techniques to produce larger-sized bulk samples, various composites of bulks have been investigated. Figure 3. Single- and multi-seeded melt-textured REBCO bulk samples. Reproduced from [6]. CC BY 4.0. A hybrid structure composed of a REBCO disk bulk enclosed by a MgB2 ring bulk was proposed by Naito et al [153]. The magnetisation was conducted at 20 K and at this temperat- ure, the Jc of the GdBCO bulk is about two orders larger than that of the MgB2 ring bulk under the same conditions. Enhancement of trapped magnetic field and flux was observed by experimental measurements and numerical simulations. Furthermore, the trapped magnetic flux can be enhanced by arranging HTS bulks perpendicular [11, 154–156] or parallel [157] to the ab-plane of the SC material, forming an array of bulk samples. This method has been proven to effectively increase the trapped field and flux. 3.1.4. Hybrid bulk/FM configuration. The integration of FM materials with TFMs and magnetisation coils serves mul- tiple functions, including structural reinforcement, reduction of demagnetisation decay rates, and overall enhancement of superconductor performance. The comprehensive functions of FM materials for TFMs, involving distinct application scen- arios, can be summarised as follows: (1) Augmentation of the trapped magnetic field strength in both TFSs and TFBs. (2) Modification of the trapped field shape. (3) Mitigation of demagnetisation effects. Details in section 6.1.4. Philippe et al demonstrated that placing a plate of soft FM material underneath a bulk REBCO superconductor can increase the measured trapped field above the superconductor 9 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review [158]. The authors observed that the FM material functions as a magnetic shield on the side to which it is attached, while the flux density and its gradient are amplified on the opposite face [158]. Furthermore, the shape of FM materials can be used to customise the magnetic field profile [158]. In [159], placing a ferromagnet on top of bulks also increased magnetic flux density. The augmentation increases linearly with the thick- ness of the thickness of the FM layer, as observed in [160]. This has been further confirmed by Mitchell-Williams et al in [41], where the inclusion of an FM plate beneath a stacked sample of HTS CCs enhanced the trapped field magnitude. The field trapping capabilities of TFMs have been shown to be improved not only by macroscopic hybrid structures combining SC bulks with FM materials but also by impreg- nating the SC materials with FM powders in drilled holes. In [161], Lousberg et al examined the effect of drilled struc- tures on HTS bulk samples by creating holes in the bulk and filling them with FM powders. They observed an improve- ment in the trapped field due to the concentration of flux lines inside the holes. This enhancement wasmore pronounced with higher volume ratios of FM material. Additionally, the study predicted that the enhancement in field trapping would fur- ther increase with a higher relative permeability of the FM material. However, it was noted that beyond the saturation field of the FM material, the improvement in magnetisation is limited due to the small effective permeability of the FM powders. 3.2. HTS TFSs REBCO CCs consist of multiple layers, typically including a SC layer with a thickness ranging from 1 to 5 µm, a sub- strate layer with a thickness between 30 and 150 µm, a buffer layer with a thickness of 0.1–0.25 µm, and silver stabilising layers with a thickness of 1–2 µm on one side [162]. Most CCs also feature two copper stabiliser capping layers, typic- ally ranging in thickness from 5 to 50 µm, on one side [163, 164]. An example of the multilayer structure of the HTS CC is depicted in figure 4(b). The substrate layer, typically made of a magnetic material such as Ni–W alloy or a non-magnetic magneticmaterial likeHastelloy, significantly enhancesmech- anical strength. Nowadays, most HTS CCs are produced with Hastelloy substrate [165]. In the multilayer structure of CCs, a stack of buffer layers will then be deposited on the Hastelloy or a stainless metal substrate by employing the IBAD method. The SC layer is then deposited on the buffer layer [162]. Additionally, a silver overlayer is commonly applied for sta- bilisation atop the SC layer, allowing for current flow in case the SC layer transitions to a normal state while carrying cur- rent. Many applications also necessitate additional stabiliser layers, often in the form of copper layers. However, HTS CCs with magnetic substrate, typically fab- ricated based on a rolling assisted biaxially textured substrate (RABiTS) process, have also found various applications. For instance, they have been utilised as part of the hybrid mag- net to trap the 17.7 T field [13], to make tape annuli for levitation [167], and to serve as rotor poles [45, 168]. The tapes Figure 4. Structure of a TFS (a) stacked HTS CCs (b) the layer composition of a REBCO CC. Reproduced from [166]. CC BY 4.0. containing the Ni–W substrate exhibit a magnetic permeabil- ity that varies with the external magnetic field perpendicular to the substrate surface. This variation can be described by an empiric law with a fitting function determined by Nguyen et al based on experimental measurements [169]. Simply by placing HTS tapes on top of each other, as shown in figure 4(a), HTS stacks present a more straightfor- ward manufacturing process, compared to bulk SC materi- als. This simplicity arises from the capacity to cut, bend, and bond tapes together without compromising their electromag- netic characteristics [24]. HTS stacks offer flexible geometry and reasonably predictable SC characteristics, making them highly engineerable. Figure 5 displays various configurations and arrangements of stacked CCs, which can serve as TFMs after magnetisation. The configurations of TFMs significantly impact the distribution of trapped fields. Various designs have been proposed to enhance different aspects of the performance of HTS stacks for diverse application scenarios: (1) Hybrid stacks—higher trapped field strength (2) Sectioned stacks—higher trapped flux (3) Angled stacks and patterned stacks—improved field uni- formity (4) Self-supporting stacks—enhanced mechanical strength (5) Ring-shape tape stack magnets—multifaceted advantages 3.2.1. Hybrid magnets. Hybrid magnet configurations are composed of various parts, either stacks from different tapes [13], stacks + PMs [170], or stacks + FM materials 10 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 5. Various arrangements of HTS tape stacks and tape fragments showcasing a range of shapes and sizes applicable for magnetic levitation and TFMs. Reproduced from [167]. CC BY 4.0. Figure 6. Composition and geometry of the hybrid stack consisting of a square stack made of SuperPower tapes inside a larger cylindrical stack made of AMSC tapes. Reproduced from [13]. CC BY 4.0. [50, 110, 171, 172]. Those configurations are developed for distinct reasons, either aiming for a higher trapped field mag- nitude, or reduction of the consumption of the HTS materials, or to reduce the susceptibility to external fields. To harness the benefits of both the high Je and larger width, a hybrid design was employed [13], incorporating a high Je stackmade of SuperPower tapes and a lower Je but larger-sized stack crafted from 46mmwideAMSC tape. The 12mm square SuperPower stack is embedded inside the larger 34.4 mm dia- meter AMSC stack. Figure 6 illustrates the geometry and com- position of this hybrid stack. Hao et al introduced an innov- ative hybrid magnet configuration comprising an HTS tape stack and a PM. This design aims for simplified magnet- isation, reduced consumption of HTS materials for achiev- ing a comparable trapped field, and enhanced safety during operation [170]. Similarly to HTS bulks, FM materials have been demon- strated to enhance the trapped field of HTS stacks in SC/FM configurations [50, 110, 171, 172]. Notably, the introduction of new materials reduces the effective Jc of the stacks, which should be compensated by the increase in the trapped field obtained by the newmaterials [50]. By sandwiching FM layers (NiFe) between HTS CC layers, an increase in trapped mag- netic field and flux was observed. For example, increases were observed in a 90-layer stack and a 145-layer stack at 10 K in [50], and a 2-layer stack at 77 K in [110]. Interestingly, it was reported in [110] that inserting NiFe with a smaller area than the HTS layers resulted in a greater increase in trapped flux compared to the NiFe with the same area as the HTS layers. 3.2.2. Sectioned stacks. Increasing trapped flux is crucial for various applications such as higher torque and output power for electric motors or wind turbines, necessitating lar- ger TFMs. Utilising wide HTS tapes as motor poles is advant- ageous due to their ability to trap a higher magnetic flux in comparison to narrower tapes. Customised wide HTS CCs, such as 46 mm [173] and 40 mm [47] variants have been developed by AMSC and Bruker HTS, respectively. However, these tapes are typically produced based on specific orders and are not widely available. Consequently, the notion of sub- stituting wide tapes with narrower counterparts has emerged naturally [168, 174]. Wide stacks demonstrate proficiency in capturing a mag- netic field over a broad expanse, increasing the trapped flux. Conversely, sectioned stacks exhibit a trapped field pattern characterised by alternating positive and negative pole peaks, as scrutinised in [44], where an investigation was conducted into the impact of various stack architectures for TFSs on CF demagnetisation across a significant number of cycles. This alternating pattern bears relevance to the practical applications of TFMs, particularly when deployed as motor field poles, introducing potentially disruptive high-order harmonics in the transverse direction within the motor air gap, which will cause CF demagnetisation of TMFs mounted on the rotor surface. Regarding the peak values of trapped flux, Smara et al elucid- ated in [43] that segmenting wide stacks into several narrower components could yield higher trapped flux levels, particu- larly when subjected to a relatively low applied Bm, suitable for motor applications. 3.2.3. Angled stacks and patterned stacks. Efforts have been made to make the trapped field more uniform, which is essential for applications such as NMR/MRI. In the pursuit of achieving a more uniform trapped field, Mitchell-Williams et al, built stacks composed of HTS tape utilising leftover Roebel cable offcuts to explore diverse stacking configura- tions, to create a uniformly trapped field [175]. They dis- covered that a novel angled stacking arrangement exhibited 11 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 7. Angled stack configurations with HTS CCs (a) different configurations of layered stacks (b) trapped field profiles. Reproduced from [41]. CC BY 4.0. Figure 8. The crisscross configuration with HTS CCs (a) the stack arrangement with three 12 mm-width tapes per layer (b) the trapped magnetic field profile of a 74-layer stack with crisscross-arranged CCs. Reproduced from [42]. © IOP Publishing Ltd. All rights reserved. the flattest and most uniform field among various overlapping stacking configurations and demonstrated scalability poten- tial. The research group extended their investigation in [41], employing self-supporting angled stacks of HTS CCs, which generate highly uniform trapped field profiles, as depicted in figure 7. Notably, the uniformity and magnitude of trapped fields were further enhanced by stacking multiple layers of the stacks and introducing an FM plate beneath the samples. Furthermore, Selva et al as detailed in [42] and [176], investigated a crisscross arrangement of stacked CCs. This configuration was observed to yield a more uniform trapped- field profile than a straight tape arrangement, as illustrated in figure 8. The thermal behaviour of stacks at various tilt angles was investigated by a numerical model in [177]. 3.2.4. Self-supporting stacks. To integrate TFSs into real- world applications, e.g. SC motors or levitation devices, it is crucial to ensure a high geometric tolerance required for engin- eering applications. In [178], Baskys et al detailed a method involving the consolidation of individual layers of HTS tapes into a self-supporting block through soldering. This process entailed the soldering of 12mmwideHTSCCs from SuperOx. Evaluation of the tape performance before and after soldering with Pb–Sn solder revealed that the trapped field of a soldered 100-layer stack after PFM increased to 1.56 T at 10 K, reflect- ing a 4% enhancement attributed to the elevated Je achieved through soldering. Self-supporting stacks employing the same solder plat- ing approach have been utilised in [57] for magnetic levita- tion and in [179] as field poles in a synchronous motor. In [57], Patel et al demonstrated stable levitation between a cyl- indrical PM and a 30 mm square slab composed of stacked and soldered HTS tapes. By employing the soldered self- supporting stack, the uniformity of geometry and trapped field can also be increased. In [179], Patel et al examined the same self-supporting stacks as solid composite SC bulks with a high geometric tolerance. The employed long stacks demonstrated high uniformity and well-defined field trapping ability, mak- ing them suitable for being used as rotor field poles in motor applications. Stycast 1266 epoxy, in addition to solder, serves as an alternative for creating self-supported stacks. A study [180] involved impregnating 40 mm-wide SC tapes from Deutsche Nanoschicht with Stycast 1266 epoxy, forming planar HTS stacks. These stacks were employed to investigate the impact of angled fields on the magnetisation for TFMs. Beyond planar configurations, the geometric flexibility of TFMs is essential, particularly when they are required as sub- stitutions for rotor PMs (whether surface-mounted or interior), to fit into narrow air gaps or rotor slots. This is exemplified by C-shaped stacks in the ASuMEDmotor [24], where the stacks are inserted in the rotor core to mitigate CF demagnetisation from airgap harmonics [181]. The hot press bending method, as demonstrated in [182], was employed to shape epoxy-glued stacks to the desired curvature. This approach offers flexibil- ity in shaping HTS stacks, albeit with observable performance degradation, including lower current density and trapped flux, showcasing the superior adaptability of HTS stacks compared to HTS bulks. 3.2.5. Ring-shape tape stack magnets. The previous con- figurations focus primarily on the spatial arrangements of HTS CCs. Another avenue for creating an innovative magnet vari- ant involves the modification of the tape itself by slitting tapes longitudinally and shaping them into a closed racetrack shape, facilitating the trapping of a magnetic field through induced persistent currents. Sheng et al proposed a ring-shaped HTS TFM in 2017, emerging as a potential alternative to PMs [183], offering a high degree of configurational flexibility for large- scale industrial applications. With the trapped field aligning parallel to the a-b plane of individual tapes, this configura- tion exhibits a heightened Jc compared to conventional TFMs. With field cooling at 25 K [184], Ali et al reported a trapped field of 4.6 T produced by an HTS ring magnet composed of 200 sheets of HTS slit tapes impregnated with paraffin wax, as shown in figure 9(a). To augment the maximum trapped field, BT,max, while enhancing central uniformity, Sheng et al introduced an innov- ative hybrid ring magnet architecture incorporating ‘wind- and-flip’ coils, integrating HTS stacks within the central con- fines of a ring-shape magnet [187], leading to enhanced spa- tial utilisation. Owing to the inherent spatial asymmetry char- acteristic of ring-shape magnets, the trapped magnetic field 12 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 9. Ring-shape magnet configurations with (a) a brass holder. Reproduced from [184]. CC BY 4.0. (b) an HTS stack. Reproduced from [185]. CC BY 4.0. (c) HTS bulks in the central space. Reproduced from [186]. CC BY 4.0. commonly manifests a non-negligible magnetic declination. Shi et al proffered an efficacious compensatory strategy [188], introducing a novel hybrid variant featuring HTS bulk samples strategically placed within the central space of the ring-shaped magnet. The precise adjustment of bulk placement within this central space constitutes a pivotal facet of this compensatory approach. Liao et al investigated the magnetisation characteristics of two hybrid configurations of HTS-stacked ring magnets with HTS stacks [185] and bulk samples [186] in the central area, respectively, as shown in figures 9(b) and (c). The magnetisa- tion mechanism of the former variant, which was magnetised to 7.35 T at 25 K, was identified as a two-stage process, and a single criterion for the penetration of the HTS-stacked ring magnet was established to discern themagnetisation speed dif- ference between these two stages [186]. In the latter configur- ation, they observed that appropriately designing and magnet- ising allowed trapping a final field higher than theBm with field cooling magnetisation [185]. The outcomes of these investig- ations bear substantial implications for the design and utilisa- tion of SC magnets. 4. Magnetisation Magnetisation is essential before TFMs can be utilised in practical applications. TFMs can capture magnetic fields after being magnetised due to persistent SC currents circu- lating within the superconductors. The characteristics of the trapped field of TFMs based on Bean’s model have been analysed and the various factors influencing the strength of the TFMs’ trapped field were also demonstrated in this section. Furthermore, various magnetisation methods and approaches to enhance TFMs’ trapped field have been discussed. 4.1. Characteristics of trapped field Describing a trapped field involves considering two aspects: its shape and strength. Fully and partially magnetised TFMs exhibit distinct patterns in their trapped field profiles. In applications such as motors or generators, the shape of the trapped field is crucial, in addition to its peak value. This is because the generated electromagnetic torque and power depend on the average value of the trapped field, which is determined by both its peak value and shape. Additionally, deformities in the trapped field can introduce high-order har- monics, as discussed in [189]. To evaluate the strength of the trapped field, conventional methods often rely on measuring the magnetic flux dens- ity at the centre point. However, this approach may offer an incomplete understanding of the overall magnetisation of TFMs since the central point may not reflect the entire pic- ture of the trapped field. For saturated TFMs, the peak value of the trapped field is found at the centre point, resulting in an average trapped field magnitude along the surface above the superconductors being half of the peak value. However, in the case of unsaturated TFMs, the location of the peak trapped field cannot be pinpointed, as the trapped field pro- file exhibits an M pattern—in other words, two field peaks are present. Consequently, two quantities being more informative in reflecting the complete trapped field are the average value of the trapped field above the TFMs’ surface and the total mag- netic flux. 4.1.1. Trapped field profile. Due to the significant anisotropy of the electrical conductivity along the a–b plane and the c-axis of REBCO, the supercurrents induced by Bm are predomin- antly restricted to the a–b plane [190]. Consequently, the dir- ection of the trapped field maintains uniformity, perpendicular to the a–b plane. According to the findings of Brandt [191], external mag- netic flux enters the superconductors from both edges towards the centre of an HTS sample during magnetisation. As a result, when the sample is fully magnetised during FC or ZFC mag- netisation with a sufficiently strong Bm, the trapped field dis- tribution above the surface of a fully magnetised TFM shows high non-uniformity, exhibiting a convex profile (conical or pyramidal shape) as predicted by Beanmodel [192, 193] indic- ating saturated magnetisation [43, 46]. This is attributed to the relationship between magnetic field density and current dens- ity, which can be derived from equations (7) and (8): ∇×B= µ0J. (17) For a 2D slab with a length of 2a in the x direction and infinite length in the y, and z directions, with a homogeneous 13 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 10. The trapped magnetic field and the current density by ZFC and FC techniques using the Bean model approximation for the magnetisation of TFMs. external Bm applied in the z direction, equation (17) can be simplified as: dB dx = µ0Jc. (18) The constant gradient of the field distribution extending towards the periphery of the TFM results in the peak field dens- ity existing at the sample’s centre, as observed in [194]. This aligns with the observation that persistent macroscopic cur- rents captured by TFMs during magnetisation circulate within the samples, which leads to conical field patterns [195]. This can be observed in figure 10, which illustrates the ZFC and FC magnetisation processes for a general TFM with the Bean model. The magnetisation field that is precisely sufficient to fully magnetise the HTS sample is denoted as Bm,p. At the end of both magnetisation methods, the TFM is fully magnetised, achieving the maximum trapped field, with a supercurrent cir- culating within the magnet at an amplitude reaching critical current density. Conversely, when Bm is insufficient to fully penetrate the entire sample due to the strong pinning force, Fp, in the superconductors [196], the central area remains unmagnetised with no circulating currents, as the magnetisation is related to applied field intensity. In such cases, the field profile above the specimen’s surface exhibits a concave pattern (M-shape), indicating an undersaturated magnetised TFM [197], which is also predicted in the Bean model. The trapped field pro- files of fully and partially magnetised TFMs are illustrated in figure 11, which depicts three cases with varying magnitudes of magnetisation field. The trapped field profiles for case A exhibit triangular shapes, corresponding to the maximum flux trapping through full penetration. In contrast, the trapped field profiles for cases B and C display M shapes due to partial magnetisation. Figure 11. Evolution of trapped field flux density from the partial to the full magnetisation. The M-shaped profiles correspond to partially magnetised TFMs, while the triangular-shaped profile represents fully magnetised TFMs. (RTFM is the radius of a TFM; 0 < a < b < 1). Moreover, trapped field profiles are influenced by TFMs’ shapes. Typically, TFMs have a cylindrical shape for HTS bulks or a cubic shape for stacked HTS CCs, resulting in a conical or pyramidal shape of the trapped field profile, respect- ively. However, TFMs can also exhibit diverse shapes, like ring-shaped bulks or annuli-shaped stacks, affecting the shape of their trapped fields. The ring-shaped TFMs, for instance, exhibit a hollow shape of their trapped fields caused by the hollow structure in the centre [198]. The trapped field profiles for different shapes of TFMs (e.g. cylindrical, hexagonal, ring sector, and ring-shaped TFMs) can be found in [199]. Compared with HTS bulks, stacked HTS CCs offer greater flexibility in geometry, enabling the creation of tailored field shapes. An example is given in [47], where a stack of CC tapes was cut into an octagonal shape, resulting in a highly symmet- rical octagonal field pattern when projected onto the a–b plane of the stack. 4.1.2. Influence factors for trapped field strength. The trapped field strength of TFMs can be impacted by various factors. Besides the mechanical strength and thermal stability of HTS bulks, which set the upper limit of the trapped field, other factors include the Jc and volume of the SC materials, the number of layers and multi-layer structure of HTS CCs for TFSs, the angle at which the SC samples are tilted to the mag- netisation coils during magnetisation, and the activation field. 4.1.2.1. Jc and volume of SC materials. For TFMs, BT,max can be expressed as follows [200]: BT,max = Aµ0JcRTFM (19) 14 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Table 2. Overview of the magnetisation experiments with trapped field over 17 Tesla. Magnet [12] [13] [7] [11] [10] Btrap (T) 17.89 17.7 17.6 17.6 17.24 Type TFS TFS TFB TFB TFB Number of CCs/bulks 200 EuBCO CCs 822 REBCO CCs (416 AMSC + 406 SuperPower) 2 GdBCO bulks 10 GdBCO bulk disks 2 YBCO bulks Magnet size (including reinforcement) 13 × 12 × 11.7 mm3 Dia. 34.4 mm, th. 37.75 mm Dia.: 24.15 mm, th.: 30 mm Dia.: 30 mm, th.: 30 mm Dia.: 26.5 mm, th.:15 mm Magnetisation method FC FC FC FC FC Ts (K) 6.5 8 26 22.5 29 Ramp rate (mT min−1) ⩾ 40 (Reduced with external field) 15.5 15 15 150 Field source (T) 18 18 18 18 18 Dia. = diameter; th. = thickness. where A represents a geometrical constant, µ0 is the permeab- ility of a vacuum, and RTFM signifies the radius of the TFM. It can be deduced from equation (19) that for the fully mag- netised TFMs, the magnitude of the trapped field is directly proportional to both the Jc and the presence of sufficiently large current loops, i.e. the superconductor’s volume [140]. Consequently, enhancing the trapped field in TFMs neces- sitates an increase in either the size or the critical current densities. The former can be achieved by employing bulk superconductors with a larger dimension or by increasing the layer number and tape width of HTS stacks, while the lat- ter can be accomplished by employing superconductors with higher Jc or lowering the Ts for specific superconductors [196]. Introducing artificial pinning centres, which are defects intentionally created within the superconductors using chem- ical methods [201] or irradiation techniques [202], can also enhance the Jc [173]. In addition to the factors mentioned above, several other factors can also influence the trapped field strength. These include the presence of metal layers within the multilayer structure of CCs, the tilt angle of the TFMs relative to the mag- netisation field, the tape number of HTS tape stacks, and the activation field for the TFMs to be fully magnetised. 4.1.2.2. CCs’ multilayer structure. The presence of metal layers within the multilayer structure, apart from the SC layer, can influence the field trapping capability. The AC losses, especially the hysteresis loss, of HTS CCs can be increased by the Ni-based FM magnetic substrate [203, 204] compared to the Hastelloy substrate, which exhibits a constant permeability of one. This can be attributed to the higher magnetic permeab- ility of magnetic substrates, which varies with the external magnetic field and can alter the magnetisation in TFMs [203, 205, 206]. In terms of stabilisers, such as the silver layers, the impact is minimal, according to Page et al [124]. Their find- ings revealed that the trapped field and flux exhibited relatively low sensitivity to the thickness of the stabiliser layers. 4.1.2.3. Tilt angle during magnetisation. TFMs used as field poles in SC machines may encounter issues such as misalignment from the central axis due to vibration. Studies indicate that theBT,max of bulk superconductors [207] and HTS stacks [180, 208] decreases as the tilt angle of the TFM upper surface with the c-axis increases. However, Ida et al [207] reported that a significant amount of flux can still be attained by a bulk superconductor, nearly equivalent to the flux trapped with the applied field parallel to the c-axis, even when the bulk was magnetised with a field inclination up to 30◦. This discov- ery implies that there is greater flexibility in the design space for the assembly of TFMs in HTS machines. 4.1.2.4. TFSs’ tape numbers and activation field. Theoretically, there is a linear relationship between the trapped flux density and the number of layers in HTS stacks as per equation (19), a phenomenon experimentally observed [42]. However, according to Rudnev et al [209], this linear cor- relation is applicable only within a specific range of layer numbers, regardless of the operating temperature. Beyond this threshold, the trapped field attains an asymptotic value, indicating a saturation behaviour, which was also observed by Baskys et al [108]. This saturation tendency is presumably associated with the shielding effect of the outer layers on the inner layers of the stack [209]. Similarly, the BT,max for HTS bulks exhibits an initial increase followed by a decrease as the applied Bm increases during PFM [210]. This threshold field, also known as the activation field [91], is the minimum field required to fully magnetise a TFM. Beyond the activation field, there is no sig- nificant increase and even a reduction in the trapped field due to substantial heat generated by the rapid flux motion during pulsed magnetisation [91, 211, 212]. The value of the activa- tion field increases when the Ts decreases, due to the increased current density of the TFMs. 4.1.3. Highest trapped fields of TFMs. In this section, the experimental results ever achieved at trapped magnetic fields exceeding 17 Tesla are presented. The strongest TFM achieved thus far was 17.89 T at 6.5 K [12] by TFSs, while TFBs reached 17.6 T at 26 K [7]. An overview of the five magnets is summarised in table 2. 15 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Table 3. Overview of magnetisation experiments with record trapped field values for the four magnetisation methods. Magnetisation method Btrap (T) Sample T (K) Source field (T) Operation approach Reference FC 17.89 TFS 6.5 18 Ramp rate ⩾ 40 mT·min−1 (reduced with external field) [12] ZFC 10.7 TFS 4.2 14 Ramp rate = 250 mT·min−1 [222] PFM 5.3 TFB 30 7 Flux jumps assisted PFM [223] Flux pumping 0.2 TFB 77 ∼0.5 Thermally actuated magnetisation (TAM) [224] 4.1.3.1. TFSs. The current record trapped field by a TFS is 17.89 T, reported by Suyama et al. This trapped field was effectively captured at 6.5 K within the nucleus of a com- pact CC stack over 75 min and achieved through flux jump suppression [12]. In [12], the stack comprises 200 sheets of EuBa2Cu3O7 CCs with incorporated BaHfO3 nanorods to increase Jc at low temperatures and high fields. For improved thermomagnetic stability, the central 50 CCs are coated with a 1 µm thick layer of Pb, offering substantial specific heat at reduced temperatures. The previous record field of trapped-field magnets was reported by Patel et al [13], which is a field of 17.7 T at 8 K captured in a hybrid stack composed of two kinds of CCs, with a 12 mm square SuperPower stack embedded inside a larger 34.4 mm diameter stack from AMSC tapes. 4.1.3.2. TFBs. Durrell et al trapped a magnetic field of 17.6 T at a temperature of 26 K using a 24.14 mm diameter GdBCO bulk with added Ag, as reported in [7]. The reinforce- ment uses a shrink-fit steel ring, making it a relatively practical approach for real-world applications. Huang et al [11] reported the sequential trapping of mag- netic fields, achieving 17.6 T at 22.5 K after an initial trapped field of 16.8 T at 26 K. This was accomplished using a double stack composed of two Ag-doped GdBCO bulk supercon- ductor composites, each with a radius of 12 mm. The two stacks have been reinforced with (1) stainless-steel lamina- tions, and (2) shrink-fit stainless-steel rings. A trapped field measuring 17.24 T at a temperature of 29 K was reported by Tomita and Murakami in [10]. The configur- ation comprised two samples of YBa2Cu3O7−δ (YBCO) with a diameter of 26 mm. These samples were impregnated with Bi–Pb–Sn–Cd alloy and resin, further reinforced with carbon fibre. 4.2. Magnetisation methods To magnetise an HTS TFM, four methods can be utilised: FC [213–215], ZFC [106, 216], PFM [43, 45, 104], and flux pumping [195, 217–219]. In the FC process, a high-magnitude DC magnetic field is applied until the HTS sample reaches its Tc, gradually diminishing afterwards. In ZFC and PFM, the TFMs are cooled below their Tc before being exposed to a magnetic field that rises incrementally and then recedes. FC and ZFC employ a linear field alteration by SC coils, while PFM employs a pulsed waveform via a capacitor bank discharge [170]. Flux pumping involves sweeping a smaller magnetic field across the superconductors in a wave, inducing a consistent current direction, leading to higher induced mag- netic fields with successive waves [220, 221]. FC and ZFC normally need expensive fixtures (SC coils) to generate high magnetic fields, while less expensive fixtures are required for PFM and flux pumps. For the former, reinforced copper coils with suitable cooling can be utilised instead of SC coils as they work in a very short pulse time, and the latter requires a small Bm. Table 3 provides an overview of experiments for the four magnetisationmethods, detailing the respective record trapped field values achieved to date. In practical applications such as electrical machines, mag- netisation is a prerequisite before the operation, preferably conducted within the machines themselves. Despite the heat generation in superconductors affecting flux trapping, PFM stands out as the most practical in-situ magnetisation method for a compact winding configuration [157, 189], eliminating the necessity for a substantial current source akin to FC and ZFC. While FC and ZFC have been employed for magnetisa- tion in electrical machines [110, 214, 215, 225] the majority of these instances have only been simulated, in contrast to the successful implementation of PFM demonstrated in practical electrical machines [45, 226–228]. 4.2.1. FC and ZFC. FC and ZFC are the common quasi- static magnetisation methods for the application of TFMs. In both instances, for the optimal utilisation of the sample’s flux- trapping capability, the applied Bm needs to be at least so large as the targeted trapped field for FC and twice so large for ZFC, as per the Bean model [192, 193]. Therefore, the trapped field records of HTS stacks and bulks over 17 T have all been achieved through field cooling magnetisation [7, 10–13], as shown in table 2. FC and ZFC processes usually involve slow and exten- ded procedures to incrementally raise and lower the applied Bm, resulting in no significant temperature rise, ∆T, within the TFM samples. Therefore, FC and ZFC are considered as the methods for estimating the maximum trapped field capability of TFM samples. Typically, the trapped field obtained employing these two approaches serves as a baseline for comparing results obtained with other magnetisation techniques, such as PFM and flux pumping. 16 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review The Bm for FC and ZFC is applied often at small rates to reduce the possibility of flux jumps, leading to that both FC and ZFC are time-consuming compared with PFM. For example in [7, 11, 13], 15 mT∙min−1 was taken for the sweep rate for FC magnetisation. As a result, the heat transfer pro- cess has negligible effects on the trapped field capability, and it is generally not considered during both magnetisation processes. Normally the ramp rates for lower temperatures should be lower [222]. Remarkably, in [10], higher sweep rates were applied following customised treatments to enhance the thermal conductivity of the samples. 4.2.2. PFM. In contrast to the alternative magnetisation approaches, PFMwith a pulsedmagnetic field of milliseconds, offers notable advantages such as compact fixture, mobil- ity, and cost-effectiveness for magnetising SC samples. This method necessitates a smaller and less complex magnetisa- tion fixture [106] and significantly reduces magnetisation time [111]. Nonetheless, due to the rapid and dynamicmovement of magnetic flux within the superconductor during PFM, a con- siderable temperature increase occurs, leading to the limitation that PFM is generally capable of trapping a smaller magnetic field compared to the slower ZFC or FC techniques [229], par- ticularly at a lower Ts [106]. The heat generation during PFM is cumulated from pinning losses, Qp, associated with trapped flux and viscous losses, Qv, linked to flux movement [230, 231], resulting in a sig- nificant ∆T. This temperature elevation stems from dynamic flux motion counteracting vortex pinning force, Fp and vis- cous force, Fv [232–234]. In the PFM process, the motion of flux is governed by a force equilibrium, where the total of Fp and Fv must balance FL [213]. The FL results from the cross product of the current density vector and themagnetic flux density vector. In the Beanmodel, Fp is defined as the product of the Jc and the magnitude of the flux density. Conversely, Fv is directly proportional to the flux velocity, which decreases rapidly as fluxes move towards the centre of the sample and becomes zero at the centre [234]. The flux velocity increases with higher applied fields [234], result- ing in larger temperature increases due to increasedQp andQv. Additionally, operating at lower temperatures enhances heat generation due to increased Fp and reduced heat capacity of the samples [233]. Figure 12 demonstrates a typical magnetisation circuit for PFM, comprising several key components: a voltage source, and a capacitor bank to supply high voltage; an IGBT serving as a switch to discharge power from the capacitor bank and isolate the magnetisation coils and the SC sample during the charging process. To better understand the heat transfer dynamics in the field trapping process, electromagnetic-thermally coupled models have been developed to include temperature factors into the Jc of the superconductors [46, 104, 166]. A bi-directional connection is built between the electromagnetic model and the thermal model. The thermal model provides real-time temperature data to the electromagnetic model, which, in turn, Figure 12. A typical PFM circuit for magnetising TFMs. delivers the current density, J, and electrical field, E, to the thermal model, as expressed in the following equation: ρmCp ∂T ∂t −∇ · k∇T= Q(T) (20) where ρm is the mass density, Cp denotes the specific heat capacity, k is the thermal conductivity, andQ is the heat gener- ation power density. The heat generated in the superconduct- ors is defined as the product of E and J in the superconductors: Q = E∙J. However, HTS tapes exhibit anisotropic thermal conduct- ivity in both longitudinal and transverse directions. Thus, for thermal modelling, it is imperative to devise a methodology, particularly for the homogenisation approach, that signific- antly expedites the calculation of heat transfer within an aniso- tropic stack structure. For that, Tomków et al [177] developed a computational procedure to determine anisotropic thermal conductivity components for a homogenised thermal model. This involved generating coefficients for polynomial fits that are readily applicable in numerical models, contingent upon the composition of the considered tape and the geometry of the stack. This method yields accurate results promptly, com- parable to experimental outcomes, thereby facilitating thermal modelling for TFSs with diverse shapes and dimensions. In the PFM modelling, the best way to describe the shape of the applied pulsed field, Bap(t), is by using the experimental data from measurements, such as pulse waveforms presented in [45, 168], and [228]. Alternatively, mathematical equations can be utilised [91, 104, 196]. One example of the widely adopted mathematical approximation is [196]: Bap (t) = Bap0 t τ exp ( 1− t τ ) (21) where Bap0 and τ represent the magnitude and the rise time of the applied pulsed field, respectively. The duration of pulses ranges from a few milliseconds to several tens of milliseconds, comprising both the rise time and fall time. The influence of the rise time and the pulse width on the trapped field has been investigated, demonstrating that the trapped field magnitude can be increased by the extension of 17 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review pulse width and rise time [196, 235]. A larger ascending time can reduce the flux propagation velocity, v, leading to reduced heat generation, specifically Qv, as Qv is proportional to v. Additionally, it is observed that the activation field for bulk superconductors decreases with the rise time [196]. Factors affecting the length of the rise time include the type, induct- ance, and resistance of the magnetising coil [235], capacitance of the capacitor bank [123], as well as the presence of other materials such as FM materials [236] or copper plates [58]. The pulse width extension can also be achieved by incorpor- ating soft iron yokes [236, 237]. Because of the heat produced within the TFM, the trapped field and acquired flux through PFM are typically lower compared to FC or ZFC methods, particularly at lower temperatures [238, 239]. Consequently, there have been endeavours to optimise the PFM process for enhancing the trapped field. The PFM process can be improved generally through the following approaches: (1) Employing multiple pulses (2) Modifying coil configuration (3) Incorporating an SC/FM structure (4) Mitigating and utilising flux jumps (5) Utilising active waveform control 4.2.2.1. PFM optimisation through multiple pulses. Several multi-pulse techniques, which could optimise the field trap capability, have been proposed in the literature: SPA method [229], IMRA [213, 240], MPSC [178, 241, 242], andMMPSC [230, 243]. There are also combinations of different tech- niques, e.g., MMPSC+ IMRA [229]. The multi-pulse applic- ation maximises the trapped field by compensating for the lost flux from the periphery of TFMs during initial pulses, caused by excessive heating, and it can also broaden the conical trapped field profile resulting from pulses at a fixed temperature [213, 244]. The multi-pulse technique operates on the principle that the presence of trapped flux lines within the superconductors reduces the amount of flux moving in the samples during applying subsequent pulsed fields, thereby minimising heat generation [240]. Typically, ∆T decreases with increasing pulse number during multi-pulse applications. SPA involves applying several pulses with the same magnitude sequentially to TFMs [229], while IMRA is an approach that iteratively applies magnetic pulses with reduced magnitude [213]. The SPA capitalises on the phenomenon whereby ∆T during consecutive pulses is mitigated by the presence of supercurrents induced by preceding pulses. The IMRA method was proposed on the basis that the presence of trapped flux, obtained after the first pulse, reduces the amount of flux moving within the TFM. This reduction in flux move- ment results in diminished heat generation during the applica- tion of subsequent pulses. While SPA and IMRA methods are applied at a fixed temperature, MPSC and MMPSC involve applying multiple pulses stepwise at varying temperatures. In MMPSC, the trapped field at the final stage with a lower temperature is significantly influenced by the pattern and magnitude of the trapped field at the initial stage. An optimal ‘M-shaped’ pro- file, characterised by a moderate number of trapped magnetic fluxes at the first stage, can effectively promote flux intru- sion and trapping, thereby maximising the final trapped field [197, 230]. In [243], the record trapped field of 5.2 T for PFM was attained by implementing the MMPSC on a Gd–Ba–Cu– O bulk superconductor. This was achieved through a two-stage magnetisation process involving various pulse magnitudes at various temperatures. In [229], a combination of MMPSC and IMRA was conducted, with the first four pulses in MMPSC (first two pulses at 70 K, next two pulses at 30 K) followed by nine pulses in IMRA at 30 K. This combined approach effect- ively enhanced both the trapped magnetic field and the trapped magnetic flux. The majority of multi-pulse methodologies have historic- ally been proposed and implemented for bulk magnets. Recent investigations have extended these approaches to explore their applicability to HTS stacks. In [178], MPSC was applied to a 100-layer stack of HTS CCs, with the temperature progress- ively reduced from 70 K to 10 K, resulting in the attainment of a trapped field of 1.56 T. Another investigation, in [50], util- ised IMRA to magnetise an HTS stack consisting of 90-layer 12 mm-square CCs at 30 K, achieving a trapped field of 2 T. In [38], Zou et al conducted a comprehensive investigation of an HTS stack magnetised by SPA and IMRA. The research out- lines the evolving characteristics of the trapped field and flux under various pulse sequences, and the resulting patterns have been thoroughly analysed, leading to a recommended magnet- isation strategy by applying a high pulse to oversaturate a stack and using successive lower pulses to achieve higher trapped field and flux. 4.2.2.2. Magnetisation coil configurations. The magnetisa- tion coils impact the trapped field distribution depending on its type and shape. Typical magnetisation coils are solenoid coils [196, 245], shown in figure 13(a), vertex-type coils (also called split-type vortex coils [123, 246, 247], and split coils [111]), shown in figure 13(b). Another type of magnetisation coil is called CMDC [246, 248, 249], comprising of an inner vortex coil surrounded by an additional winding to form an outer air-cored solenoid, as shown in figure 13(c). A solenoid, characterised by a hollow structure, is the most conventional type of magnetising coil, exhibiting a homogen- eous hollow pattern. In contrast, the magnetic field distribu- tion of vortex-type coils differs from that of solenoid coils, exhibiting a conical shape along the central axis [247], lead- ing to a higher applied field in the centre compared to the periphery of the SC material being magnetised. This con- figuration arises from the winding of wires from the edges towards the centre in vortex-type coils, leaving no mid-air space in the centre [194]. Moreover, the split-type geometry of the vortex coils combines the vectors of the generated mag- netic field from the upper and lower coils, resulting in them being parallel to the c-axis of the bulk superconductors [251]. Typically, SC materials are positioned inside the centre hole of the solenoid while sandwiched between a pair of vortex 18 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 13. Structures of different magnetising coils (a) solenoid coil (b) vortex-type coil (c) CMDC with two different modes. Reproduced from [246]. CC BY 4.0. Reproduced from [248]. CC BY 4.0. Reproduced from [250]. CC BY 4.0. The insets in (a) and (b) show a concave and a convex trapped field profiles of a bulk magnetised by a small applied field using a solenoid coil and a vortex-type coil. coils. Consequently, when magnetising with vortex-type coils, the magnetic flux predominantly penetrates the bulk from the upper and lower surfaces. Conversely, with solenoid coils, the magnetic flux enters the bulk from the periphery [123]. This effect can be visualised in the insets of figures 13(a) and (b), which show a concave and a convex trapped field profiles of a bulk magnetised by a small applied field using a solenoid coil and a vortex-type coil, respectively [250]. The different pat- terns arise from the distinct flux penetrationmechanisms of the two configurations. With a small magnetisation field, which cannot fully penetrate theHTS sample, the trapped field shapes resemble the generated field shapes, a phenomenon known as the field memory effect [250]. When a sufficiently large applied field or multi-pulse field is employed, both coils can fully magnetise SC samples and exploit the maximum potential of the trapped field of the SC sample. However, vortex-type coils are advantageous to enhance the trapped field and its distribution shape by minim- ising the heat generated on the periphery of SC samples during magnetisation [247]. The cooling efficiency for the vortex coil configuration could be more effective when applied from the side of the SC samples, given the larger thermal conductivity along the ab-plane compared to the c-axis of the SC samples [252]. The vortex coils can also serve as stator windings [226, 253], and their arrangement may be more suitable for practical motor applications [247]. To optimise the design of vortex-type coils, it is crucial to consider the radial dimension of the coils, as it directly impacts the distribution of the trapped magnetic flux. When the radial dimension of the coils exceeds that of the HTS bulks, the dis- tribution of the trapped magnetic flux may deviate from the ideal conical shape. Therefore, to achieve the desired conical trapped flux distribution, the outer radius of the vortex coils should be smaller than that of the HTS bulks [194]. While the selection of coil type depends on specific applic- ation scenarios, vortex coils are typically preferred for PFM due to their smaller geometrical size compared to solen- oids. Additionally, vortex coils help minimise heat generation and ∆T during PFM, making them advantageous for various applications [250]. However, as a result of the smaller dimen- sions of the vortex coils, the applied magnetic field density is also reduced compared to that of solenoid coils, resulting in a lower applied magnetic field density. Consequently, the trapped magnetic field is smaller when using vortex coils com- pared to solenoid coils [251]. In figure 13(c), two modes of the CMDC were implemen- ted for PFM: mode A and mode B [248, 251]. In mode A, the pulsed current was applied to both the solenoid and split- type vortex coils in series, while mode B utilised only the inner vortex coil [248]. The CMDC was used in conjunction with a multi-pulse approach to magnetise a large bulk with a diameter of 140 mm. An effective combination of mode A and mode B was employed. Initially, mode A was utilised to increase the total magnetic flux, followed by mode B to lever- age the vortex coil and achieve a conical shape of the mag- netic field, ensuring complete magnetisation with a BT,max of the bulk superconductor. By injecting currents with reversed directions into the inner vortex coils and the outer solenoid coils of a CMDC respect- ively, Zou et al operated the CMDC with a novel approach [249]. This configuration led to an enhancement of the trapped field compared to a solenoid coil configuration. The improve- ment is attributed to lower heat generation and ∆T associated with CMDCs, which apply a gradient Bm with the peak field located at the sample centre. 4.2.2.3. PFM optimisation through FM structure and copper plates. Iron yokes enhance the trapped field strength of TFMs by attracting magnetic flux, thereby retaining it within TFMs, owing to the high permeability of iron yokes [29]. Specifically, in the case of PFM, the presence of FM mater- ials assists in improving flux penetration during the pulse rise time, and substantially reduces flux exiting the bulk after reaching the pulse peak. Consequently, this leads to a higher peak trapped field and an increased overall trapped flux [111]. Moreover, FM materials aid in increasing the trapped field by facilitating heat removal, thereby slightly reducing ∆T [111, 254]. The size (diameter and thickness) and shape of iron yokes employed in magnetisation systems play a crucial role in increasing the trapped field and suppressing ∆T. Yokoyama et al reported that both the duration of magnetic field expos- ure and the magnitude of the trapped field increased with the size of the iron yoke [236]. In addition, ∆T decreased with increasing yoke size. A 5% suppression of ∆T was observed for a 64 mm diameter iron yoke compared to an 80 mm dia- meter one [236]. Moreover, the trapped field increases with 19 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 14. Structure of the 2D axisymmetric model for optimising the PFM technique, including the split coil magnetisation fixture and iron yokes. Reproduced from [256]. CC BY 4.0. the thickness of the iron yoke, while ∆T decreases [231]. In [255], three different shapes of iron yokes were investigated: disk-, ring-, and cross-shaped. It was noted that the trapped field of the magnetisation system with the cross-shape yoke was the highest, while the ring-shape yoke magnetisation sys- tem trapped the least amount of magnetic field. Hybrid configurations of iron yokes with both solenoid coils and split coils have been constructed and tested for PFM. Experimental data demonstrates that combinations of iron yokes with split coils yield significantly higher trapped fields compared to solenoid coils [111]. In addition to the enhancement of trapped field strength due to flux attraction by soft iron yokes, an extended rise time and pulse duration can further increase the trapped field, facilit- ated by the presence of copper. This can be attributed to the generated eddy currents. Sandwich structures that insert cop- per and FM layers between REBCO tape layers can effectively combine the functionalities of both copper and soft iron yokes. To enhance the trapped field in a MgB2 bulk sample, several methods were utilised [58], including a combination of exten- ded magnetic pulse application facilitated by sandwiching the MgB2 ring bulks by thin copper plates, the augmentation of the effective applied field through the insertion of a soft iron yoke, and the application of a double pulse using a split-type coil for PFM, as depicted in figure 14. This led to a record-high trapped field of 1.61 T for MgB2 bulk composites at 20 K with PFM. 4.2.2.4. Flux jump mitigation and utilisation. Flux jumps refer to an avalanche-like flux motion phenomenon occurring within SC materials during magnetisation (typically during PFM), which generates significant heat due to the substantial flux movement. The heat generated during flux jump can lead to a reduction in the local Jc, further exacerbating heat- ing generation [257]. This can result in an uncontrolled flux motion, leading to the quenching of the SC material and mag- netisation failure. Flux jumps can be explained as a phenomenon where the local temperature of the SC specimen exceeds its Tc. Therefore, flux jumps can be attributed to factors such as the low specific heat, and high thermal conductivity of HTS samples [254]. Additionally, when the temperature difference between the Ts and the SC transition temperature is minimal, a flux jump may occur [254]. Moreover, it was noted that flux jumps occur when the Bm exceeds a critical value, which varies for specific samples, and when there is a high sweep rate of the applied Bm, dBm/dt [258]. Both factors contribute to significant heat generation, resulting in a decrease in Jc. However, if the heat can be effectively removed through suf- ficient cooling or redistributed within the TFMs, the decrease in Jc can be mitigated, helping to prevent flux jumps [259]. Additionally, a longer pulsed field can be employed to pre- vent uncontrolled heat generation, thus suppressing flux jumps [64]. Furthermore, it can also be avoided through the insertion of iron yokes in split coils during magnetisation [254]. Alternatively, flux jumps can also serve as a means to enhance magnetisation by allowing magnetic flux for a full penetration to the HTS samples’ centre, with a smaller Bm than predicted by the Bean model, as observed by Weinstein et al [260, 261] and Zhou et al [258]. A magnetic field of exceeding 3 T has been trapped in a single-grain GdBCO bulk supercon- ductor through the exploitation of flux jumps [258]. Figure 15 displays the occurrence of flux jumps during PFM, where flux jumps can be observed in the first two of the three phases of the application of Bm. In real experiments, flux jumps are char- acterised by an abrupt increase in the trapped field, which res- ults from the swift and rapid penetration of the magnetisation field into SC samples. In addition, Ainslie et al reproduced the phenomenon of the flux jump assisted PFM employing a 2D H-formulation-based FEM model [121]. Furthermore, they observed that the magnetic flux was driven into the sample by a substantial FL during the PFM process, which is responsible for the occurrence of the ‘giant field leaps’. 4.2.2.5. Active waveform control. In studies [263] and [264], Ida et al introduced the WCPM method to alter the waveform of magnetising pulses. This approach utilised an IGBT as a high-speed switch, providing control over the switching frequency in the pulse-generating electric circuit to modify the magnetising pulse waveform. The active con- trol of the rise time of the magnetisation pulse serves to sup- press magnetic flux motion [207], which is associated with heat generation and ∆T within TFMs, consequently enhancing 20 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 15. Schematic depiction illustrating the three distinct phases of the flux jump-assisted PFM, including flux ingress, flux trapping, and flux creep. Reproduced from [262]. CC BY 4.0. Figure 16. Schematic of the pulse magnetisation equipment with a feedback circuit. Reproduced from [264]. CC BY 4.0. magnetisation efficiency. Unlike multi-pulse methods, which require a long time for the magnetisation to complete, the WCPM approach is more efficient for practical applications such as HTS motors with only one single pulse being applied. As depicted in figure 16, the central element of the WCPM is a magnetiser equipped with a feedback circuit responsible for regulating the magnetic flux density within the bulk. The control circuit receives feedback signals comprising the mag- netising current and the penetrated magnetic flux, measured by Hall elements at the HTS bulk surface. When the flux density surpasses a predefined threshold at the bulk centre, the MCU decreases the duty ratio of the gate pulse transmit- ted to the IGBT, resulting in a lower magnitude of the pulse discharge [207]. Various temperatures have been investigated: 77 K using liquid nitrogen [265], 70 K [266], and 60 K [267] with a Gifford–McMahon cryocooler. The BT,max at the centre of a 45 mm diameter and 19 mm thickness bulk superconductor increased from 1.63 T to over 1.79 T and further to 2.17 T at different temperatures, respectively. Additionally, negative feedback [268] and flux jump [265] have been identified as effective methods for enhancing the trapped magnetic field when employing WCPM technology. In addition to conventional bulk discs, an HTS ring bulk has been magnetised through the WCPM method (without feed- back) [269]. It was observed that optimising the magnetising pulse waveform employed in the PFM method can signific- antly enhance the trapped field and the thermomagnetic sta- bility of an SC ring bulk. 4.2.3. Flux pumping. The flux pumping technique, conven- tionally devised for energising SC coils [270, 271], exhibit adaptability in magnetising stacks of HTS tapes and bulk superconductors. The underlying principle of flux pumping involves the cyclic application of a modest magnetic field to induce magnetisation in SC coils [272, 273]. This prompts a persistent flux flowwithin a closed SC loop, culminating in the accrual of DC within the loop and, consequently, a substan- tially augmented magnetic field confined within the SC coils [217]. Diverse flux pump variants tailored for charging SC coils have been advanced to date [270]. Three flux-pumping methods have been effectively employed for magnetising HTS stacks and bulks: (1) Thermal method: TAM [272, 274] (2) Electrical approach: transformer-rectifier flux pumps based on AC field-triggered flux flow [218] (3) Mechanical method: mechanical flux pumping variant utilising rotating PMs [221, 275]. The TAM method [274, 276] entails the induction of a travelling heat wave within a thermally actuated medium, such as Gadolinium, strategically positioned between PMs and an HTS bulk, as shown in figure 17(a). Activation of the thermal wave is achieved through the state switch- ing on and off by cooling exceeding and below its Curie point (290 K), this generates a quasi-travelling magnetic wave as the heat wave traverses the material, propagat- ing towards the centre. The thermal wave, concurrently serving as a magnetic wave, iteratively traverses the gad- olinium material during repeated thermal cycles. This pro- cess accumulates multiple magnetic waves, progressively magnetising the YBCO bulk through incremental flux dens- ity accrual on its surface, ultimately yielding a remnant field. The electrical approach, proposed by Zhang et al [220], employs the transformer-rectifier flux pumping approach [218] to magnetise modified HTS stacks, as demonstrated in figure 17(c). The modification entails cutting slits in the middle of tapes longitudinally in each layer. These slits serve as insulating barriers, creating prescribed current flowing paths. The rectifier-type flux pump requires two slits for each tape—a long slit and a short slit, separated by a gap area form- ing a bridge between the two current loops. The short slit forms a charging loop, while the long slit forms a load loop. By leveraging the concept of accumulatedmagnetic flux over time in HTS stacks, this rectifier-type flux pumping method can 21 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 17. Principles of the three types of flux pumps: (a) thermal method. Reproduced with permission from [276]. (b) mechanical method (c) electrical approach. Reproduced with permission from [220]. effectivelymagnetise HTS stacks. However, only less than half of the fully penetrated field can be trapped due to reduced cur- rent capacity caused by slits and the mutual couplings between loops and electromagnets, generating significant loss [220]. The mechanical approach, based on the mechanical flux pump [277], relies on a magnetic disk embedded with PMs rotating atop a modified HTS stack sample, producing the necessary travelling magnetic fields through the PMs’ circu- lar motion. Zhang et al investigated this approach for magnet- ising stacked PSTs [221]. The magnetisation process is illus- trated in figure 17(b). By cutting only one slit in the middle of each tape longitudinally between two tape ends, the tapes are modified into PSTs. In this type of flux pump, the bridge region is positioned at one end of the stack, perpendicular to the path of the travelling PMs. These fields induce a continu- ous DC voltage in the SC load, facilitating the charging of the SC load. However, the rotating magnets variant can only trap a small amount of field due to uncontrollable DC flux com- ponents and large magnitude oscillation caused by the small inductance of the stacks [221]. Overall, the three flux pump types mentioned are effect- ive for magnetising TFMs but have different applications. The TAM method is suitable for TFBs through a travelling thermal pulse, while the other two are effective for magnet- ising long HTS stacks by applying the field to a small section of TFSs, allowing for a more compact magnetisation pro- cess compared to FC or ZFC methods. However, they have a smaller trapped field capacity compared to FC and ZFC methods. Additionally, the relatively low system efficiency of HTS flux pumps remains a significant factor restricting their applications. 5. Applications TFMs are utilised across a wide range of applications, includ- ing SC machines (motors and generators) [24, 25], magnetic bearings [26, 27], levitation systems [28, 29], NMR and MRI [31, 32], Halbach arrays [278, 279], and more [32, 33]. These applications span diverse fields such as electric aircraft, ship- ping, railway transportation, medical devices, magnetic separ- ation systems, and beyond. 5.1. Electrical machines SC machines represent fundamental components in the elec- trical propulsion system of large-scale mobility applications, including electric ships and electric aircraft. In conven- tional electrical machines, rotor assemblies typically feature field winding coils or PMs. However, wound field coils are burdened by the limitations imposed by brushes and slip rings [280, 281], and PMs are constrained by their remanent flux density. TFMs stand out as the most promising candidates for achieving the requisite high-power density demanded by large-scale propulsion systems, primarily due to their excep- tional magnetic field trapping capabilities. Both bulk super- conductors and HTS stacks can be employed as field-poles on the rotor for brushless rotating machines. TFMs have thus far found application as replacements for PMs in two types of HTS synchronous machines, char- acterised by the support material in the machine structures: iron-cored machines and air-cored (with non-magnetic cores) machines. In the former type, the HTSmachine design is mod- ified based on conventional structures, featuring iron yokes in both the stator and rotor. In contrast, air-cored machines util- ise non-FM materials in the stator and rotor to support stator windings and rotor magnets. Air-cored machines can maxim- ise the potential utilisation of superconductors, as the mag- netic loading capacity in iron-cored machines is constrained by the saturation level of FM materials. However, TFMs in air-cored machines experience elevated CF demagnetisation due to airgap harmonics induced by stator windings, while TFMs in iron-cored machines may exhibit a lower level of trapped decay due to the shielding effect of iron materials on the harmonics. Other machine types, aside from those utilising TFMs in the rotors, include hysteresis motors (which exploit the flux pinning property of HTS materials) [282], reluctance motors (which leverage the high diamagnetism of super- conductors to create a larger difference in magnetic per- meability on the direct and quadrature axes) [283–285], and flux modulation motors (which utilise the diamagnetic prop- erties of HTS materials for flux concentration) [286–289]. These machine types are all promising candidates for achiev- ing a higher power-to-weight ratio compared to conventional motors. However, they are beyond the scope of this paper. 22 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 18. A fully HTS machine developed at the University of Cambridge with TFBs as rotor field poles. Reproduced from [298]. CC BY 4.0. 5.1.1. TFB machines. Numerous SC machines employing TFBs have been proposed or demonstrated. They can be in the form of radial flux type [227, 290], or axial flux type [190, 291, 292]. Since 2005, a fully SC PM motor, which was designed as a radial flux motor, has been built and analysed at the University of Cambridge [25, 227, 293–295], as shown in figure 18. The stator of the motor consists of six air-cored HTS armature windings. AC losses of the armature windings were analysed in [296]. A total of 75 bulk superconductors were mounted on the rotor surface to serve as four-pole super- magnets upon PFM. Xian et al utilised the SPA technique— multiple pulse PFM with the same magnitude—to magnet- ise the rotor bulks and obtained a BT,max of 375 mT after 19 pulses [227]. Huang et al implemented a modified variable- voltage, variable-frequency control scheme on a digital signal processor for the motor control in [294] and conducted trial tests for the motor in [297] at 77 K, with a speed of 150 rpm. Following PFM, an average magnetic field of 305.2 mT was trapped and an overall efficiency of 78.8% was measured dur- ing the motor operation. Since 2003, the Tokyo University of Marine Science and Technology (TUMSAT) has started to develop an axial-type synchronous machine using TFBs on the rotor [226, 253, 299]. The axial flux configuration is a more compact design compared to radial-flux cylindrical machines, thus enabling a higher power density [190]. The vortex-type copper coils in the armature windings were employed to magnetise bulks to function as rotor field poles. By incorporating WCPM within the motor, the GdBCO bulk achieved a BT,max that is 40% higher compared with the conventional magnetisation approach employing a passive pulsed magnetic field. This compact, low-speed, high-torque rotating machine has two Figure 19. An axial-type synchronous machine constructed by TUMSAT employing TFBs as rotor field poles. Reproduced from [291]. CC BY 4.0. variants, featuring a single and twin rotor. The twin-rotor type outperforms the single-rotor variant by achieving twice the magnitude of the trapped magnetic flux and a smoother sinus- oidal waveform of magnetic flux density in the air gap. During the operation of the twin-rotor variant, as shown in figure 19, a BT,max of up to 0.7 T was achieved at 720 rpm [226]. Attempts have been undertaken to substitute turbine engines in aircraft with electric propulsion by incorporating HTS motors. This aims to achieve a higher power density, res- ulting in a more compact and lighter design. In [190], Masson et al proposed a 450 kW SC motor design with an axial flux configuration. The motor’s rotor incorporates six TFMs in the form of YBCO bulk plates. Field coils made of Bi-2223 super- conductors are wrapped around the motor, serving as mag- netising coils to activate the TFMs employing field cooling, achieving a trapped field of 6 T. The cooling system utilises liquid hydrogen as a coolant, maintaining an operation tem- perature of 20–30 K, which also aids in reducing the size and power losses of power electronics. In [300], Masson et al introduced a new configuration that integrates SC pancake coils with bulk materials to construct a rotor for an eight-pole radial-flux motor with a 150 kW output power at 2700 RPM. The HTS coils serve dual purposes: activating the YBCO plates as magnetisation coils and functioning as field wind- ings for four poles alongside the YBCO plates. Furthermore, this design offers scalability by enabling the addition of more coil-plate combinations, leading to increased torque and power output [301]. In addition to HTS motors, HTS bulks have also found application as TFMs in generators. In [302], melt-textured YBCO bulks were utilised as field poles in the rotor to con- struct an 8-pole axial-flux configuration operating at 77 K. Both FC and PFM methods were employed to magnetise the bulk samples separately for two test runs. Following magnet- isation, the generator was operated at 2000 rpm, resulting in a maximum output power of 33W and 100W, respectively. Hull et al [303] investigated the use of HTS bulks as rotor magnets for a brushless generator. A key aspect of this application 23 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 20. Magnetic field distribution of (a) the air-borne stator and (b) the C-shaped HTS stack interior rotor of the ASuMED motor. Reproduced from [24]. CC BY 4.0. involves the utilisation of dysprosium (Dy), which has a much higher saturation level than conventional FM materials and can effectively concentrate flux, aiding in magnetisation by enhancing the trapped flux density. The magnetisation results demonstrated that a trapped field exceeding 4 T was achieved. After magnetisation, the rotor spins to induce voltages in the stator coils, enabling generator operation of the machine. 5.1.2. TFS machines. Up to now, there have been several types of SC machines employing HTS stacks: • surface-mounted radial-flux synchronous motor • interior magnet radial-flux synchronous motor • linear motor A radial-flux synchronous SC machine, featuring a FSCW design with HTS stacks affixed to the rotor surface, under- went testing as a generator for demagnetisation investigation subsequent to the stacks’ magnetisation in studies [45] and [228]. The investigation into demagnetisation under practical machine conditions involved the injection of DC currents into one of the stator windings [228] or the connection of the three- phase stator windings to an external load [45]. An interior magnet synchronous motor structure has been developed by the ASuMED project, which aimed to devise the first prototype of a fully SC motor for potential applica- tion in future large civil aircraft and aimed for a power dens- ity of 20 kW∙kg−1 [24]. This innovative motor design features a unique rotor magnet arrangement utilising HTS tape stacks configured in a curved C-shape inserted within the rotor’s iron yoke [181], as demonstrated in figure 20. This rotor arrange- ment not only facilitates efficient magnetisation through inter- action with the stator but also provides protective measures for the stacks against demagnetisation risks from demagnetisation risks [24]. An innovative linear motor, incorporating stacked HTS CCs, was devised and experimentally assessed by Sotelo et al [304]. This distinctive design, featuring a stator equipped with Nd-Fe-B magnets and a flat nine-layer HTS tape stack, achieved an impressive force density of 0.5 kN∙kg−1—an unprecedented accomplishment for a linear motor at that time. The findings from this study validate the viability of employ- ing HTS tape stacks in SC linear machines. 5.2. SC levitation Both HTS bulks and stacked CCs can be employed in mag- netic levitation (Maglev) systems and SMB applications. In a magnetic bearing system, the rotating component is suppor- ted through magnetic levitation without physical contact. FC and ZFC are typical magnetisation methods for TFMs before the levitation measurements [57, 305]. Halbach PM arrays are utilised to produce the levitation force with TFMs. HTS bulks have found applications in an SC bearing system developed at the University of Cambridge for energy storage flywheels [30]. A flywheel is a mechanical device that stores rotational energy by conserving the angular momentum of its rotating mass, such as wheels. This SC bearing system under- went testing at rotation speeds of up to 12 Hz, where a reson- ance issue was observed between 2 and 3 Hz. HTS bulks have also been utilised in maglev systems, including the full-scale HTS magnetic levitation train prototype ‘Maglev–Cobra’ in Brazil in 2014 [306]. Additionally, the first HTS Maglev test system employing the ETT technique was developed in 2014 [307]. In these systems, the SC levitation is achieved through stable levitation and lateral guidance forces generated by the interaction between the PMG and the TFMs beneath the maglev vehicles [308]. For maglev vehicle applications, both vertical levitation and lateral forces are crucial, especially for navigating curves where lateral displacements are expected. TFSs have been integrated into the development of SMB, characterised by the broad loops of persistent currents flowing through narrow tapes [309]. Compared with HTS bulks, the consistent SC properties, geometrical flexibility, and reprodu- cibility of HTS CCs make TFSs a preferred candidate for SC levitation [57, 310, 311]. In [57], stable levitation with up to 18 N between a cylindrical PM and a slab of soldered HTS tape stack was achieved for the axial force test. In [28], an SMB incorporating TFSs as a passive levitator was investig- ated and compared with a three-seed YBCO bulk sample on the Maglev Cobra PMG for vertical and lateral displacement tests. The levitation forces exerted by TFSs exhibit signific- ant anisotropy, showcasing varied performance under differ- ent magnetic field orientations of the PMG. The lateral forces produced by TFSs are also lower than those exhibited by HTS bulks during the experiments. However, it is important to notice that TFSs, despite having the same volume as HTS bulks, contain a small amount of SC material. Quéval et al conducted a comprehensive review of mod- elling techniques developed for SMB systems [27]. They also developed a comprehensive solution for 2D and 3D FEMmod- elling using H-formulation for both tape stack and bulk bear- ings with moving magnets. 5.3. Halbach arrays A Halbach array is a configuration of magnets arranged in such a way that the magnetic field is intensified on one side 24 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review of the array while cancelled on the other side, leading to a spatially rotating magnetisation pattern [312]. Halbach arrays, where the magnetisation directions of adjacent magnets are perpendicular to each other, can generate the large magnitude and gradient of the magnetic field necessary to produce the forces required for applications such as magnetic drug target- ing systems [313, 314]. TFMs offer a superior alternative to conventional PMs for constructing a Halbach array due to their superior field-trapping capability. To our knowledge, one of the earliest attempts to employ HTS Halbach arrays involved replacing PMs with TFBs to create Halbach-array internal- dipole magnet [315]. Hull et al examined the feasibility of a circular SC Halbach array utilising SC coils [315]. In [278], experimental and ana- lytical investigations were conducted on a linear SC Halbach array consisting of five large-grain melt-textured YBCO bulks in cuboid shape, each with a boundary length of 14 mm at 77 K. Prior to the assembly in the Halbach configuration, the bulks were magnetised using FC. The results of the measure- ments revealed a partial demagnetisation ofmagnetised TFMs, resulting in a 13% reduction in the trapped field. This was attributed to the changes in the induced supercurrents within the TFMs caused by neighbouring ones during the assembly process. To tackle this problem, two approaches were pro- posed in [279]. The first involved replacing the HTS bulks with TFSs, showing no significant improvement to the demagnet- isation problem. The second method involved adding an addi- tional TFM sample atop the central one during magnetisation and removing it after the assembly process in a three-TFM Halbach arrangement. The latter method led to an enhance- ment of the BT,max by 5% and 11% for the TFMs in the form of TFBs and TFSs, respectively, which was attributed to the remagnetisation induced by the removal of the additional cent- ral sample. 5.4. MRI/NMR At MIT, researchers have used stacked hollowed annuli made of HTS CCs to build compact 300 MHz/38 mm SC magnets with a centre bore for micro-NMR spectroscopy application over a long time [31, 49, 316–318]. A homogeneous field can be trapped in the centre bore by employing these magnets. The researchers have developed a range of annulus magnets by stacking different layers of YBCO CCs annuli delivered by AMSC. Each CC measured 40 mm square and 0.8 mm thick, with a 25 mm centre hole. These magnets were named after their respective layer numbers: YP500 with 500 CC annuli [49], YP750with 750 CCs [31], YP1070with 1070 CCs [317], and YP2800 with 2800 CCs [317]. For all the magnets, FC magnetisation was employed. In general, both the magnitude and uniformity of the trapped field increase as the number of CCs in stacked annulus magnets increases [317]. However, when combining YBCO bulks and the YBCO CC annuli, while field strength improves, field homogeneity decreases. A comparison conducted in [48] between 2800 YBCO thin square CCs annuli (YP2800) and 10 YBCO thick bulk annuli (YB10) revealed the superior performance of YP2800, not Figure 21. MRI systems employing TFBs (a) the setup of an HTS bulk magnet system tailored for an MRI comprising stacked ring-shaped HTS bulks (b) six c-axis-oriented bulk EuBa2Cu3Oy superconductors. Reproduced from [32]. CC BY 4.0. only in terms of spatial homogeneity but also temporal sta- bility. The field homogeneity of YP2800 even surpassed that of the magnetising field. HTS bulks have also been found in applications in NMR systems [319, 320]. In [320], Nakamura et al achieved a mag- netic field of 4.74 T for high-resolution NMR spectroscopy, as depicted in figure 21. They utilised an SC bulk magnet composed of six vertically stacked annular HTS bulks with a 60 mm outer diameter. These bulk samples were fabricated from single-domain c-axis-oriented EuBCO crystals. To pre- vent mechanical failure, the bulks were encased into 5 mm thick aluminium rings to alleviate hoop stress. 5.5. Other applications 5.5.1. SC undulators (SCUs). An undulator consists of a long channel through which an electron beam passes, and numerous dipole magnets arranged in a periodic repeating pat- tern. The magnets create a periodic magnetic field that forces the electron beam to oscillate transversely along the beam path, resulting in energy emission. Undulators are normally used as insertion devices in free electron lasers and synchro- tron radiation facilities [156]. Conventional undulators employ PMs while SCUs utilise HTS bulks to obtain enhanced tunab- ility of radiation wavelength and increased magnetic fields to improve the performance of undulators [321]. 25 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review One of the earliest applications of SCUs was a helical design with LTS material, NbTi, proposed by Elias et al [322]. In addition to HTS coils [323], HTS bulks can also be employed in SCUs. Calvi et al designed a staggered-array bulk HTS undulator (BHTSU) to reduce the period length and increase the magnetic field [324]. Recently, Zhang et al achieved a record field of 2.1 T in a 10 mm-period BHTSU [321]. 5.5.2. Electromagnetic launchers as linear actuators. In [325], an innovative linear propulsion system is introduced, utilising YBCO bulk superconductors to generate thrust forces. The propulsionmechanism is grounded in theMeissner effect and the proposed HTS launcher design, YBCO bulks serve as a diamagnetic mover. 5.5.3. Magnetic separation systems. HTS bulks have been employed in magnetic separation systems [326–328] to remove waste containing metal elements such as FM compounds [326, 328] and caesium (Cs) contamination [327] for waste fluid purification and the recovery of spilled oil [329]. This is attributed to the high magnitude and sharp gradient [326] of the magnetic field generated by bulk superconductors. 6. Challenges and solutions TFMs face several challenges in achieving higher trapped fields, including demagnetisation (for both TFSs and TFBs), poor mechanical strength and thermal instability (for TFBs), and flux jumps (for both TFSs and TFBs). 6.1. Demagnetisation While TFMs hold the potential to replace traditional rotor- mounted PMs in electrical machines, resulting in augmented magnetic flux density within the airgap, they are susceptible to CF effects within airgaps. This leads to reduced magnetisa- tionwhen subjected to alternating transversemagnetic fields in machine airgaps [52], particularly in FSCW machines, where unique winding configurations lead to additional sub- and super-space harmonic components [330]. The reduced trapped field of TFMs results in comparatively lower power output [331, 332]. Notwithstanding extensive research endeavours focused on comprehending the demagnetisation impact and the concerted attempts aimed at alleviating this phenomenon, the demag- netisation phenomenon persists as a central obstacle requir- ing resolution to facilitate the viable implementation of TFM applications. Overall, there are three types of physical causes leading to the decay of initially magnetised HTS TFMs: (1) Thermally activated flux creep [333] (2) Rotating magnetic field demagnetisation [172, 334] (3) Cross magnetic field demagnetisation [332] Figure 22. Decomposition of the external magnetic field applied to a TFM in the operation of a motor. Reproduced from [337]. CC BY 4.0. Flux creep, describing a phenomenon in Type-II supercon- ductors according to the Anderson–Kim theory [335], arises from the thermally activated motion of fluxoids overcom- ing pinning barriers. The latter two types of demagnetisation mostly occur in motors and generators, where the decay is typ- ically induced by higher-order harmonic waves within the air gap [126, 336], stemming from the stator’s alternating arrange- ment of slots and teeth. Figure 22 illustrates the decomposition of the external field experienced by an SC magnet in electrical machines [337]. The environmental harmonics can be broken down into AC and DC magnetic fields in horizontal and ver- tical directions relative to the TFM. The rotatingmagnetic field comprises the AC field in both directions, and the transverse AC field is referred to as the CF. 6.1.1. Flux creep. For TFMs, when the persistent currents trapped during the magnetisation process fall below Jc, the trapped flux lines remain in potential energy wells, effectively pinned at pinning centres. In this scenario, the FL, resulting from the interaction of currents and flux lines, is smaller than Fp, and no flux motion is expected. However, flux lines can still overcome the pinning energy barriers when the temperat- ure is above 0 K [338]. Thus, flux bundles can be thermally activated and jump from one pinning centre to another over time, driven by FL, which leads to energy dissipation [339]. This phenomenon is called flux creep, wherein the flux lines are depinned from the potential energy wells by thermal activ- ation, leading to reduced field gradients as flux leaks out of the TFMs and a logarithmic decay in the trapped field. As per Anderson and Kim, the probability of a pinned flux escaping a pinning potential well is described by e−U/kT [333]. Here, U(B, J, T) represents the effective activation energy, U, dependent on the magnetic field, current density, and temper- ature, and can be expressed as: U(B,J,T) = U0 (B,T)(1− J/Jc0) (22) where Jc0 is the critical current density at zero external field. The time-dependent decay of magnetisation in supercon- ductors due to thermally-activated flux creep as per [340] is described by: dM M0 = −kBT U dln t (23) 26 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review where U is the pinning potential, M(t) is the magnetisation at time t, M0 is the magnetisation at t = 0, T is the absolute tem- perature of the superconductor, and kB is Boltzmann’s con- stant. Equation (23) indicates that the magnetisation follows a logarithmic decay with time, and the rate of decay diminishes with an increase in pinning potential. It is important to note that equation (23) holds only if the entire HTS sample exhib- its a homogeneous current density. A relaxation rate/creep rate S can be defined as [209]: S= kBT U . (24) Experimental measurements of flux creep after magnetisa- tion were conducted in for example [8, 170, 179, 222], and [341]. Typically, S was determined by fitting the magnetisa- tion decay over an extended period, such as 2 h [342]. Due to the logarithmic nature of the field decay, S becomes negligible after a few hours. In [341], it was estimated that over 90% of the trapped field was maintained even for one day following magnetisation. According to [13], S for bulks and stacks can broadly be considered similar. For TFSs, the rate of decay in the trapped magnetic field due to flux creep decreases with an increas- ing number of layers of tape [176]. Additionally, accord- ing to equation (24), flux creep rate reduces with decreased temperature [13, 343], leading to that HTS motors are recom- mended to operate at lower temperatures below the magnet- isation temperature [344, 345], which can also increase Jc and effectively eliminate the flux creep effect. At low temperat- ures, for instance, at and below the temperature of 20 K, the magnetisation is assumed to be permanent in [190], since the flux creep of HTS bulks under this condition is almost negli- gible. Furthermore, flux pinning can reduce the flux creep rate by increasing the pinning energy U, achieved by introducing pinning centres into superconductors [346]. 6.1.2. Rotating magnetic field demagnetisation. The demagnetisation of the trapped field for HTS bulk materials [334] and stacks [172] caused by rotating magnetic fields has been investigated through both experiments and simulations. A rotatingmagnetic field can be decomposed of two sets of AC fields in vertical and horizontal directions, respectively, which is a common method employed in the relevant investigation [172, 334]. Consequently, the effect of rotating magnetic fields is also the combination of the decay caused by the AC field parallel and perpendicular (CF demagnetisation) to the c-axis of the TFMs. Therefore, the decay under the condition of a rotating magnetic field is more pronounced than that of the CF demagnetisation. However, as observed in [347], where the two cases were separately investigated, the decay of the magnetisation of TFMs caused by the AC field applied in the perpendicular is smaller than that caused by the AC field applied in the parallel direction. The AC field parallel to the c-axis generate heat in the TFMs, leading to a decrease in Jc, which in turn results in a deeper penetration. Figure 23. A typical setup of measurement configuration for the demagnetisation study of TFMs, which are magnetised before being exposed to transverse AC fields. Reproduced from [337]. CC BY 4.0. 6.1.3. CF demagnetisation. The decay in the trapped mag- netic field due to the transverse field applied orthogonally to the initial magnetisation direction is known as a phenomenon of magnetic moment collapse [332, 348] and is termed as CF demagnetisation, which has been extensively documented [52, 99, 337, 349–351]. The CF demagnetisation effect is attributed to a recognised dissipative effect referred to as dynamic magneto-resistance (DMR) [352, 353]. Upon magnetisation, a persistent current is induced in HTS tapes or bulk superconductors. When a trans- verse ACmagnetic field, which is applied to a TFM, surpasses a particular threshold value, it induces a DC voltage in the TFM. This DC voltage opposes the prevailing DC transport current. Vanderbemden et al also demonstrated that the com- plex adjustment of the macroscopic current distribution within the TFMs results in demagnetisation [332], which was previ- ously predicted by Brandt and Mikitik [53, 354]. To investigate the demagnetisation effect, SC samples need firstly to be magnetised along the c-axis and then demag- netised by a series of AC magnetic fields parallel to the ab-axis [22]. Figure 23 illustrates a standard configuration for CF demagnetisation measurement [337]. Initially, the TFM sample undergoes magnetisation using an electromag- net, applying the magnetised field parallel to the c-axis of the sample. Subsequently, an AC external field is applied parallel to the sample’s ab-plane. The decay rate of CF demagnetisation in HTS bulk is influ- enced by several factors, including the magnitude and fre- quency of the applied crossed-field, temperature, and the size of the bulk samples. Similar factors impact the decay rate of CF demagnetisation in TFSs, encompassing the magnitude and frequency of the CF, temperature, tape width, tape thick- ness, and the layer number in the stack. Those factors have been extensively investigated in previous studies [46, 99, 126, 337, 349, 350]. The demagnetisation rate decreases with the layer number in the stack [44] and with the tape width [44], respectively. 27 https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review The dependence of the decay and magnetisation loss on the amplitude and frequency of the applied CF is of great import- ance to be investigated before any other aspects. In [99], it was found by Dadhich et al that for a high number of cycles, as long as the ripple field amplitudes remain below the parallel penetration field of a single tape in the stack, both SC bulks and stacks exhibit non-zero stationary values for the trapped field. As per the established slab model [192, 355], with the height far less than the width, the parallel penetration field of an individual CC, Bp||ab, can be computed via: Bp∥ab = µ0 Jc0d 2 (25) where Jc0 and d are the critical current density in self-field and thickness of the slab, respectively. Numerous studies have investigated demagnetisation phenomena in both TFBs and TFSs. Below, some representative examples are introduced. 6.1.3.1. Demagnetisation of bulk superconductors. Hong et al conducted a numerical analysis of the demagnetisation phenomenon in bulk superconductors exposed to a magnetic field perpendicular to the initially trapped flux direction [336]. They simulated the scenario where an SC sample is subjected to a transverse field of varying intensity, closely resembling the operational conditions within a synchronous machine. In their work [99], Dadhich et al developed a numerical model based on DMR, offering significantly faster simulations than conventional E–J power law models. This accelerated model facilitates the simulation of high numbers of cycles for TFMs. Utilising the DMR model, the demagnetisation beha- viour of SC stacks, consisting of 10–100 tapes, was simulated for up to 2 million cycles of applied ripple field. Notably, bulks maintain substantial stationary values for much higher ripple field amplitudes than stacks in scenarios involving a high num- ber of cycles. However, for a low number of cycles, stacks experience considerably less magnetisation loss compared to bulks. 6.1.3.2. Demagnetisation of TFSs. Baghdadi et al presen- ted the crossed-magnetic-field effect on the demagnetisation factor of stacked HTS CCs [351]. It was found that the demag- netisation factor increases linearly with the amplitude of the AC transverse magnetic field. Zhang et al employed a 3D numerical model to explore the electromagnetic properties of curved HTS TFSs within high- speed rotating machines [126]. Their research reveals the lim- itations of 2D-axisymmetric models due to the introduction of the new electromagnetic criss-cross concept. It also identified potential issues associated with high-frequency ripple fields, which can induce current towards the periphery of the HTS TFS due to the skin effect. This phenomenon could lead to a rapid increase in AC loss and potentially irreversible stack demagnetisation. Dadhich et al investigated the CF demagnetisation of HTS stacks of tapes through a comprehensive examination of the relaxation time constant τ , with which the decay of CF demag- netisation over numerous cycles can be estimated [356]. With the MEMEP modelling method, key parameters which can influence the demagnetisation decay rate, have been examined, including the amplitude and frequency of the CF, tape width, tape thickness (ranging from 1 to 20 µm), and the tape num- ber (up to 20). It has been found that when the CF amp- litude exceeds the penetration field of a single tape, the entire stack experiences full demagnetisation, characterised by an exponential decay. The introduction of more tapes primar- ily extends the relaxation time, with an increase in the self- inductance of the magnetisation currents being a contributing factor. In numerous investigations, including the above- mentioned, into the demagnetisation of SC bulks or stacked tape-based CCs, the AC fields often take the sinusoidal [52, 126, 350] or triangle-shaped [332, 336, 337] shape to replic- ate the harmonics existing in the machine’s air gap. However, these studies oversimplified the complexity of harmonics in real machines, where harmonics normally consist of multiple waves with varying magnitudes and directions. Moreover, these works primarily focused on elucidating the physics underlying demagnetisation, overlooking practical scenarios in real electric machines [331]. However, the practical scen- arios where PMs in conventional motors commonly interact with abundant harmonics also apply to situations where TFMs serve as motor poles. Taking this into account, Wang et al proposed a comprehensive methodology for systematically examining the magnetisation and CF demagnetisation beha- viour of HTS TFSs within practical SC machines [46]. They investigated the interaction of harmonics within a real electric machine environment involving CCs and extracted these har- monics as CF components to examine the demagnetisation in a practical situation. 6.1.4. Methods to suppress demagnetisation. To tackle the demagnetisation problems, there have been various measures proposed based on various principles: (1) Shielding with HTS tapes/bulks (2) Soldering tapes at ends for TFSs (3) Remagnetisation (4) Incorporating the SC/FM materials structure (5) Doping and impregnation 6.1.4.1. Shielding with HTS tapes/bulks. Superconductors’ diamagnetic properties have found utility in passive shielding applications, driven by the generation of induced macroscopic shielding currents along the samples’ outer perimeter [357]. For example, the modification of SC tapes to an ‘eye-shaped’ cross-section has found applications in SC screens designed to shield against DC magnetic fields [358]. Additionally, bulk superconductors have also been utilised in diverse configura- tions of semi-closed SC shields in [357] and their interaction with magnetic fields in the directions along and perpendicular of the shield sample’s axial direction was analysed. Moreover, 28 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Figure 24. A CF-shielding configuration with side tapes as the passive shielding on both sides of an HTS CC stack. Reproduced from [337]. CC BY 4.0. a hybrid configuration combining eye-shaped CCs and disk- shaped bulks can be employed to achieve enhanced screen- ing ability [359]. In a recent study, Rotheudt et al systemat- ically examined the magnetic field distribution and shielding efficiency of various hybrid configurations [360]. HTS CCs can also be employed as preventive shielding layers placed on the side of HTS stacks, perpendicularly to their ab-planes, as shown in figure 24. With this configura- tion, Baskys et al demonstrated that these side tapes can par- tially shield the magnetised HTS from external CFs [337]. Additionally, Baghdadi et al presented a shielding approach where the tape stack was enclosed either by orthogonal lay- ers of HTS CCs or by a material with high magnetic per- meability, specifically bulk metallic glass made of FM Fe– Cr–Ni–Ga–P–Si–C alloys containing 3% Ni [361], to func- tion as magnetic shields against varying transverse fields [52]. Additionally, tests were conducted on HTS CC annuli with an FM substrate Ni-5at.%W to assess their effectiveness as shields against external AC fields at both room temperature and liquid nitrogen temperature [362]. The results showed that the stack annuli could shield up to 0.93 T axial field. 6.1.4.2. Soldering tapes at ends for TFSs. In [98], Li and Pardo proposed a novel method to suppress the CF demag- netisation effect of coupled stacks, of which the tape ends were soldered together with normal conductors, as shown in figure 25. The reduction in decay was observed to be depend- ent on the resistance between the tapes, which was introduced by the soldering materials, and the coupling conditions within the stack—whether fully or partially coupled. These findings suggest that altering the resistance values and stack coupling can offer diverse design capabilities. 6.1.4.3. Remagnetisation. Remagnetising the demagnetised TFMs before the next operation is necessary to ensure proper operational function, which is acceptable even for electric air- craft between the two flights. Vanderbemden et al conducted experimental and numerical analyses of the remagnetisation process in an HTS bulk, which underwent partial CF demag- netisation after initial magnetisation [334]. It was observed Figure 25. Reduction of demagnetisation by soldering the tape ends together. Reproduced from [98]. CC BY 4.0. that rotating the applied field by 90◦ or 180◦ between the two transverse field directions was more efficient in remagnetising the sample compared to inserting a remagnetising field pulse of the same amplitude between the transverse cycles. Notably, a field amplitude on the order of the full-penetration field was deemed sufficient for achieving a satisfactory remagnetisation of the sample. 6.1.4.4. Incorporating the SC/FM material structure. FM materials have been utilised to mitigate the CF demagnetisa- tion by various studies for TFMs. For HTS tape stacks, FM materials can be inserted between the layers. The configura- tion has shown that the magnetisation decay can be reduced compared to conventional TFSs [172]. Fagnard et al [160] investigated various configurations of the SC/FM structure to mitigate the magnetisation decay in TFBs by attaching FM on the top, both the top and bottom, and lateral sides of a bulk sample. They found that attach- ing FM materials on both the top and bottom sides of a bulk sample (FM/SC/FM configuration) led to a lower reduction in the trapped field than the SC/FM configuration with only one single FM layer attached. However, FM materials also exhib- ited a shielding effect, resulting in a less effective magnetisa- tion. When FM materials were placed on the sides of the bulk sample (an FM ring wrapping around the bulk), not only was the crossed field demagnetisation reduced, but the magnetisa- tion was also enhanced. 6.1.4.5. Doping and impregnation. HTS materials doped with amorphous magnetic particles were found to have a lower decay rate when exposed to theACmagnetic field environment in electric motors, which can be attributed to the higher Jc of the doped samples. GdBCO bulks with and without Fe–B–Si– Nb–Cr–Cu amorphous magnetic particles were assembled on the rotor as field poles for an eight-field-pole motor [363, 364]. After a 5 h synchronous rotation at 40 K, the decay of the trapped field in the GdBCO bulk without amorphous magnetic 29 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review particle doping is 7.2%. In contrast, for the doped case, the decay is reduced to 4.1%. The applied AC stator field aver- aged 36 mT, with approximately 6% of the field trapped in the rotor bulks. 6.2. Mechanical strength The interplay between the trapped field and the persistent cur- rent inside TFMs results in an outward pressure directly in proportion with the trapped field. Consequently, the tensile strength σB establishes the BT,max [365], approximated by: σB = 0.282B2 T,max (26) where σB and BT,max are in MPa and T, respectively. The rela- tionship between the tensile strength of TFMs and the BT,max suggests that sufficient mechanical strength is essential for capturing a higher magnetic field. Consequently, the mechan- ical properties of HTS bulks become one of the limiting factors in applications requiring magnetic fields exceeding 10 Tesla. The fragility stems from inherent small-scale defects, such as pores and micro-cracks, which are characteristic of the bulk production processes and contribute to the diminished tensile strength in HTS bulks. Effective reinforcement techniques, whether internal through material adjustments or external via high-strength metal rings, are essential for addressing these challenges [32]. There have been numerous post-melt-processing treatments investigated for HTS bulks to improve their mechanical prop- erties, including reinforcement with stainless steel, impregna- tion with drilled structure, and addition of silver. Encapsulation of HTS bulk samples with stainless-steel rings/tubes as shown in figure 26, can avoid the cracking prob- lem by generating compressive stress on HTS bulks. This is facilitated by the higher thermal expansion coefficient of steel compared to SC materials [366]. Morita et al proposed a composite structure consisting of bulk disks and stainless- steel discs in [129], which was utilised in [11] to capture a field of 17.6 T at 22.5 K, as depicted in figure 26(b). This approach showcased that by minimising mechanical strains on the superconductors and preventing crack formation, the field- trapping capacity can be substantially improved. Additionally, artificial arrays of drilled holes filled with alloys or high-strength resins in SC bulks have been shown to increase the mechanical strength of the SC samples [367]. In [10], a YBCO monolith was treated with resin impregna- tion and wrapped in carbon fibre to improve its mechanical strength. Furthermore, a Bi–Pb–Sn–Cd alloy was impregnated into a small, drilled hole at the centre of the YBCO bulk super- conductor, supported by an aluminium wire. Additionally, silver impregnation in SC materials has been shown to reduce microcracks, leading to improved mechanical proper- ties and tensile strength [368–370]. This enhancement res- ults in increased trapped field capabilities of HTS samples [366, 371]. Mechanical reinforcement measures are often combined to achieve optimal results. In [366], Gruss et al achieved a Figure 26. Examples of mechanical reinforcement (a) an assembled stack of two GdBCO bulk samples, which are reinforced with shrink-fit steel rings and trapped a field of 17.6 T at 26 K. Reproduced from [7]. CC BY 4.0. (b) a TFB made of a two-sample stack, which trapped a field of 17.6 T at 22.5 K and has a composite structure comprised of layers of bulk superconductor and stainless steel. Reproduced from [11]. CC BY 4.0. trapped field measuring 16 T at 24 K by doping a YBCO sample with Zr to improve flux pinning and impregnating it with silver. The sample was enclosed in a reinforcing stainless- steel tube. Table 4 lists various reinforcement methods along with their record trapped field values to date. 6.3. Thermal stability REBCO materials typically exhibit low thermal conductivity. The generation of heat during flux movement in magnetisation can result in catastrophic thermal instabilities and flux jumps [11]. This occurs because the heat generated in the supercon- ductor cannot be effectively transferred to the cooling device. REBCO bulks suffer from flux jumps typically at 20–30 K [222]. While tape stacked magnets exhibit improved thermal stability compared to bulk materials, they remain susceptible to flux jumps below 10 K [12]. There are several ways to improve the thermal conductiv- ity of TFMs. In the study outlined in [372], thermal conduct- ivity was found effectively enhanced by the impregnation of a modified GdBCO bulk sample with a Bi–Sn–Cd alloy. In the centre of the sample, a central hole with a 1 mm diameter was drilled, following by the insertion of an aluminium wire into the hole. Subsequent to the alloy impregnation, a temperature- rise suppression of 4 K and a 25% increase in the trapped field 30 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ Supercond. Sci. Technol. 37 (2024) 123005 Topical Review Table 4. A summary of reinforcement methods for TFBs and their record trapped field values. Reinforcement methods Record trapped field values (T) Temperature (K) References Encapsulation with stainless-steel rings/tubes 17.6 26 [7] Composite structure with stainless steel 17.6 22.5 [11] Drilled holes filled with alloys or resins 17.24 29 [10] Zr/Ag-doping + steel tube encapsulation 16 24 [366] at 44 K was observed. Furthermore, Patel and Glowacki [245] reported a practical method to increase the local conductivit- ies in the ab- and the c-planes of TFBs by embedding copper discs with high thermal conductivity in bulk configurations. This adjustment affects the speed of heat flow into or out of the bulks. As a result, a 30% increase in the trapped field and flux was observed with this innovative structure. For HTS stacks, introducing copper foils as interlayers between HTS CC layers in stacks, specifically incorporating one copper layer for every second tape layer, was shown to enhance the thermal conductivity of HTS stacks, leading to an improvement in trapped field [50]. Additionally, reducing the rate of field ramping is another method to mitigate flux jumps [341, 343, 373], which could be employed for both TFBs and TFSs. Additionally, flux jumps in TFSs can be triggered by flux jets, a phenomenon where the trapped magnetic field profiles become strongly distorted due to cutting defects in sliced CCs utilised for stacked magnets [341]. This distortion is further amplified by the strong non-linear behaviour of the superconductors, as defined by their E–J characteristics [341]. Suyama et al [341] suggest that suppressing flux jets by using cuttingmethods that minimise defects can lead to a higher field below 10 K. 7. Conclusions This paper offers a comprehensive review of various aspects of HTS TFMs, encompassing modelling methodologies and fabrication techniques, magnetisation approaches, real-world applications, as well as the challenges encountered, and poten- tial solutions proposed. Generally, TFMs, including TFBs and TFSs, can offer promising solutions to real-world applica- tions requiring high-magnitude and uniform magnetic fields. However, challenges remain for continuous and stable applic- ation of TFMs without failure or degradation. Overall, the key conclusions and some insightful discussions from the entire paper can be drawn as follows. (a) Compared to conventional coil-based SC magnets, the advantage of HTS TFMs can be summarised as: firstly, TFMs operate in persistent current mode after magnet- isation, eliminating the need for continuous power supply and current leads; and secondly, free from the influence of the minimum bending radius affecting the current carry- ing thus magnetic field generation capabilities [14], TFMs exhibit a higher Jc under the same size, leading to a greater magnetic field trapping capability. However, SC magnets composed of HTS coils can achieve much higher fields than TFMs. It should be noted that the magnets utilised for achieving the world records of TFMs are all 18 T SC coil magnets from Japan and the US. Additionally, all the state- of-the-art ultra-high-field magnets above 30 T are fabric- ated with REBCO coils [3–5]. Generally, TFMs are more suitable for applications that do not require bulky current sources and electrical contacts. Specifically, rotary elec- tric machines utilising TFMs are superior to wound-field topologies, which require slip rings and brushes and con- sequently incur higher thermal and electrical losses. (b) TFSs outperform TFBs in various aspects, including geo- metrical flexibility, improved mechanical and thermal properties, homogeneous SC properties, and higher res- istance to CF demagnetisation. However, TFBs offer two distinct advantages over TFSs in real-word applications. Firstly, they possess a higher content of SC materials, enabling them to exhibit a higher Je. This characteristic makes them attractive for diverse large-scale applications, as they can be operated at a higher Ts. Secondly, bulk superconductors can grow into much larger-sized samples compared to stacked HTS CCs. This is crucial for applica- tions that require a large pole area, necessitating a magnet with a continuous and substantial amount of SC materi- als. TFBs excel in this regard because bulk superconduct- ors can be fabricated with diameters exceeding 100 mm by employing methods such as the MST. In contrast, few companies produce wide CCs and although wide TFSs can be assembled using narrower HTS CCs, the electromag- netic and trapped field characteristics are not continuous or uniform, as shown in [44]. In contrast, the trapped field of large TFBs exhibits greater continuity of the trapped field and field gradients, which is essential for producing substantial electromagnetic forces [314]. (c) There are many challenges associated with the bulk fab- rication methods e.g., TSMG and TSIG, including time- consuming processes, low growth speeds, and lack of con- trollability during the growth process, making the pro- duction of large-sized HTS bulk superconductors even more challenging. Consequently, novel TFB concepts have emerged, such as stacking bulk samples to form an array, to increase the trapped flux and thereby output power for applications like SC motors. Inspired by the idea of TFSs, these concepts involve stacking bulk cylinders or 31 Supercond. Sci. Technol. 37 (2024) 123005 Topical Review disks with their ab-planes aligned parallel, leading to higher trapped fields compared to individual samples. For example, a bulk composite consisting of thin disks trapped a field exceeding 17 T at 22.5 K [11], and a two-bulk stack trapped a field of 17.6 T between the two samples at 26 K [7]. Additionally, the stacking assembly eliminates the need for the costly process of producing large, indi- vidual monoliths [154]. (d) Different configurations of TFSs have been summarised, including hybrid stacks, sectioned stacks, angled and pat- terned stacks, self-supporting stacks, and ring-shape tape stack magnets. These concepts aim to achieve higher trapped field strength, enhanced trapped flux, increased uniformity of the trapped field, and improved mechanical strength. Another important concept is the hybrid SC/FM configuration, which can enhance the magnetisation res- ults for both TFBs and TFSs, as well as mitigate the CF demagnetisation of TFMs. (e) Magnetisation by a strong magnetic field is required for TFMs to capture persistent currents, thereby demon- strating high magnetic fields. The highest trapped field achieved to date is 17.89 T, accomplished by a TFS made of 200 EuBCO CCs using the FC method. In addition to the common magnetisation approaches e.g., FC, ZFC, and PFM, the flux pumping method, although exhibiting a relatively low system efficiency, has been summarised as an alternative approach for a compact magnetisation, with three different variants: (a) thermal method, known as TAM method; (b) electrical approach, involving transformer-rectifier flux pumps based on AC field-triggered flux flow; (c) mechanical method, which utilises rotating PMs for mechanical flux pumping. (f) Among the various approaches, PFM is the most prevalent magnetisation method. Despite its advantages e.g., being compact, inexpensive, and time-saving, it cannot fully exploit the field-trapping capabilities due to the heat gen- erated byQp andQv. Various methods have been proposed to improve the magnetisation efficiency and effectiveness for PFM, and those methods are: (a) employing multiple pulses; (b) modifying coil configuration; (c) incorporat- ing an SC/FM structure; (d) mitigating and utilising flux jumps; (e) utilising active waveform control. (g) In various practical applications of TFMs, SC motors emerge as a crucial area, particularly for large-scale propulsion systems requiring high-power density. TFMs offer promising solutions for SCmotors due to their excep- tional magnetic field trapping capabilities. Several prac- tical prototypes have already been developed, incorporat- ing TFMs as field poles. Nevertheless, despite the advant- ages offered by TFMs, such as eliminating brushes and slip rings in rotating rotor parts to prevent friction and heat losses, SC motors with conventional wound-field topolo- gies, e.g. the ASCEND project [374], are under develop- ment. Using this conventional magnet excitation method, stable power output can be maintained by minimising the effects of airgap harmonics. (h) TFMs encounter several challenges. One notable issue is the thermally activated decay of the trapped field, referred to as flux creep. A practical solution to this prob- lem is to lower the Ts of TFMs in applications, which can also increase the Jc since flux creep is temperature- dependent. In addition, rotating and transverse magnetic fields, originating from high-order harmonics in the air- gaps of electrical machines, can also contribute to weak- ening the strength of the trapped magnetic field. Among these factors, CF demagnetisation stands out as particu- larly severe, with the potential to cause the reduction of the magnetic moment of the trapped field in extreme cases. To address this issue, various strategies for mitigating demagnetisation have been proposed, including (a) shield- ing with HTS tapes/bulks; (b) soldering tapes at ends for TFSs; (c) remagnetisation; (d) incorporating the SC/FM materials structure; (e) doping and impregnation. (i) TFBs face challenges such as weak mechanical strength and thermal instability. Consequently, measures to rein- force mechanical strength and enhance the thermal con- ductivity of HTS bulks have been outlined. To improve mechanical strength, methods include material doping with Ag or resin impregnation with a drilled structure, and the use of high-strength metal rings like stainless-steel tubes. For enhanced thermal stability, approaches include impregnation with special materials like Bi–Sn–Cd alloy, augmentation of thermal conductivity with highly con- ductive copper materials for TFBs, lamination of CCs in TFSs with copper foils, and mitigation of flux jumps by diminishing the ramp rate of the magnetisation field. After thoroughly examining the contemporary methodolo- gies employed for HTS TFMs, it is evident that significant pro- gress has been made within this field. This study is expected to enhance the understanding of the TFM technology and serve as a valuable resource for future advancements in the field. It offers insights into key research points and highlights the chal- lenges that need to be addressed for the long-term application of TFMs, thus inspiring further relevant research work in the broader energy conversion community. 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Introduction 1.1. Comparison between TFSs and TFBs 1.1.1. Geometrical flexibility. 1.1.2. Homogenous SC properties. 1.1.3. Enhanced mechanical and thermal properties. 1.1.4. Superior performance under cross-field (CF) demagnetisation. 1.2. Comparison between REBCO bulks and MgB2 bulks 2. Modelling methods 2.1. E–J power law and Jc dependence 2.2. H and T–A formulations 2.2.1. H-formulation. 2.2.2. T–A formulation. 2.3. Multi-layered and homogenisation models 3. Fabrication and macroscopic structures 3.1. HTS TFBs 3.1.1. HTS bulk fabrication. 3.1.2. Seeds for melt growth approaches. 3.1.3. Large-dimension bulks and bulk composites. 3.1.4. Hybrid bulk/FM configuration. 3.2. HTS TFSs 3.2.1. Hybrid magnets. 3.2.2. Sectioned stacks. 3.2.3. Angled stacks and patterned stacks. 3.2.4. Self-supporting stacks. 3.2.5. Ring-shape tape stack magnets. 4. Magnetisation 4.1. Characteristics of trapped field 4.1.1. Trapped field profile. 4.1.2. Influence factors for trapped field strength. 4.1.2.1. Jc and volume of SC materials. 4.1.2.2. CCs' multilayer structure. 4.1.2.3. Tilt angle during magnetisation. 4.1.2.4. TFSs' tape numbers and activation field. 4.1.3. Highest trapped fields of TFMs. 4.1.3.1. TFSs. 4.1.3.2. TFBs. 4.2. Magnetisation methods 4.2.1. FC and ZFC. 4.2.2. PFM. 4.2.2.1. PFM optimisation through multiple pulses. 4.2.2.2. Magnetisation coil configurations. 4.2.2.3. PFM optimisation through FM structure and copper plates. 4.2.2.4. Flux jump mitigation and utilisation. 4.2.2.5. Active waveform control. 4.2.3. Flux pumping. 5. Applications 5.1. Electrical machines 5.1.1. TFB machines. 5.1.2. TFS machines. 5.2. SC levitation 5.3. Halbach arrays 5.4. MRI/NMR 5.5. Other applications 5.5.1. SC undulators (SCUs). 5.5.2. Electromagnetic launchers as linear actuators. 5.5.3. Magnetic separation systems. 6. Challenges and solutions 6.1. Demagnetisation 6.1.1. Flux creep. 6.1.2. Rotating magnetic field demagnetisation. 6.1.3. CF demagnetisation. 6.1.3.1. Demagnetisation of bulk superconductors. 6.1.3.2. Demagnetisation of TFSs. 6.1.4. Methods to suppress demagnetisation. 6.1.4.1. Shielding with HTS tapes/bulks. 6.1.4.2. Soldering tapes at ends for TFSs. 6.1.4.3. Remagnetisation. 6.1.4.4. Incorporating the SC/FM material structure. 6.1.4.5. Doping and impregnation. 6.2. Mechanical strength 6.3. Thermal stability 7. Conclusions References