Electromagnetic modelling using T-A formulation for high-temperature superconductor (RE)Ba 2 Cu 3 O x high field magnets

: Second generation (2G) high-temperature superconductor (HTS) (RE)Ba 2 Cu 3 O x (REBCO) shows a great potential in building high field magnets beyond 23.5 T. The electromagnetic modelling is vital for the design of HTS magnet, however, this always suffers the challenge of huge computation for high field magnets with large number of turns. This study presents a novel electromagnetic modelling based on T-A formulation for REBCO magnets with thousands of turns. An equivalent turn method is proposed to reduce the number of turns in calculation, so that the computation cost can be reduced significantly, and meanwhile the key electromagnetic behaviour of HTS magnet can be simulated with enough accuracy. The ramping operation of a fully HTS magnet with 12,000 turns are analysed using both the original T-A model with actual turns and improved T-A model with equivalent turns. The two models show a good agreement on the key electromagnetic behaviours of the magnet: distribution of current density, magnetic fields, screen current induced field and magnetisation loss, so that this improved T-A model using equivalent turns is validated. The T-A modelling of REBCO magnet is a powerful tool for the electromagnetic analysis of industry-scale high field magnets.


Introduction
Superconducting high field magnet plays an important role in scientific instruments, accelerator magnets, nuclear magnetic resonances and energy storage, which always prefer higher magnetic fields [1][2][3][4]. Current superconducting magnets are based on low-temperature superconductors (LTS) like NbTi and Nb3Sn, and their fields are limited to 23.5 T, which is the critical fields of Nb3Sn. The second generation (2G) high-temperature superconductor (HTS), (RE)Ba 2 Cu 3 O x (REBCO), shows significant advantages on high critical fields and high current density, which has been improved in recent two decades [5,6]. It has a great potential to generate extra high magnetic fields up to 100 T at low temperature far exceeding the possible field range of LTS can reach [7]. Therefore, REBCO conductor is also called high field superconductor [8,9]. Significant process has been achieved on HTS high field magnets in recent 5 years: 32 T field was achieved in 2017 by a LTS/HTS hybrid magnet with a 17 T REBCO insert and 15 T LTS outset in National High Magnetic Field Laboratory (NHMFL), USA [9][10][11]. A 24 T LTS/HTS hybrid magnet with 9 T insert HTS magnet was developed by the Chinese Academy of Sciences in 2015 [12][13][14][15][16]. Then a 26 T full REBCO magnet was developed by MIT and SuNAM in 2016 [17]. More ambitious HTS magnet projects beyond 30 T have been announced at several institutes in recent 3 years [18,19].
An efficient numerical modelling tool is vital to study and predict electromagnetic behaviours of HTS magnets, especially the screen current and AC loss. However, numerical modelling of HTS magnet has always suffered the challenges of highly non-linear E-J power relationship, high aspect ratio of coated conductor and hard convergence [20][21][22][23][24][25]. REBCO high field magnet often consists of thousands of turns, which makes the modelling much more difficult because of the huge computation. Some analytical methods have been developed to calculate the screen current in superconductor, which is simple and fast [26][27][28][29]. However, these methods cannot do a comprehensive electromagnetic analysis on the HTS, and are also not suitable for complex electromagnetic environments and operations. Finite-element method (FEM) based on H-formulation has been widely applied on the electromagnetic modelling of HTS applications [30][31][32][33], and homogenous Hformulation model has been developed to reduce the computation cost [34,35]. Its governing equation uses magnetic intensity H as solution variable, and the whole model can be built and solved in commercial FEM software Comsol Multi-physics. T-w model has also been developed for the electromagnetic modelling of HTS, which couples T-formulation (Faraday's law) and integration equation (Biot-Savart Law) [36][37][38][39].
A new T-A formulation model has been developed for REBCO conductors in 2016, which is based on the previous T-w model [40]. The key feature of T-A formulation model is to neglect the thickness of REBCO conductors and treat it as thin shell, meanwhile keep the dimension of all the other domains, then apply a T-formulation model on the superconducting shell with lower dimension, and apply an A-formulation model on all the domains. This can significantly reduce the computation cost because of dimension and variables reduction, which makes it possible and easy to complicate electromagnetic modelling of HTS applications [41][42][43][44][45]. However, for high field magnets using REBCO tapes, T-A model still suffers the problem of huge computation, especially for magnets with thousands of turns. Too huge computation can significantly reduce the practicability and convergence of the HTS electromagnetic modelling.
This paper presents an improved T-A model for REBCO high field magnets with large number of turns, which can significantly reduce computation cost, and therefore make the electromagnetic simulation of high field magnets fast, easy and practical. This model is partly inspired by the conventional homogenous Hformulation model, and some techniques and fundamentals of homogenous H-formulation are introduced. The actual turns on the inner part of the pancake coils are replaced with equivalent turns whose number is much less than that of these actual turns. The equivalent turns can significantly reduce the number of meshes as well as computation cost, and meanwhile keep all the key electromagnetic characteristics of the REBCO coil, such as distribution of current density, magnetic field induced, ramping loss and screen current induced field (SCIF). Then, to validate this model, case comparisons are conducted on a 15 T REBCO magnet with more than 10,000 turns, and key electromagnetic characteristics of these high field magnets are analysed.

T-A formulation
The REBCO coated conductor has a multilayer structure, and the thickness of the superconducting layer is <1% of the total thickness, as shown in Fig. 1a. Since the ramping of high field magnets is relatively slow, current only flows in superconducting layer below critical current. Therefore, the metallic layers are neglected in the electromagnetic modelling below critical current. Meanwhile, it is reasonable to treat the REBCO conductor as thin shell considering the high aspect ratio of superconducting layers, as shown in Fig. 2 [37,40]. The key feature of T-A formulation is to apply the T-formulation on this REBCO thin shell only to calculate the current distribution, and meanwhile apply the A-formulation on all the domains to calculate magnetic fields induced, as shown in Fig. 1b [40,41]. T is the current vector potential. Current flowing in the thin shell does not have component normal to the shell surface, thus current vector potential only has the normal component on the shell. Therefore, the current in thin shell J can be expressed as where A is magnetic potential, B is magnetic flux density, E is electric field. In the T-formulation, the transport current is imposed by adding a Dirichlet boundary condition on the edges of the REBCO shell [40] where I op is the transport current, d is the thickness of the REBCO tape (1 μm in this study). T 1 and T 2 are the current vector potential on the two edges of the REBCO shell, as shown in Fig. 1b. The current distribution obtained from the T-formulation is input into A-formulation as external current. Therefore, a Neumann boundary condition is added on the REBCO shell in the A-formulation model where μ is the permeability, J is current density obtained from Tformulation, B 1 and B 2 are the magnetic fields on the two sides of the REBCO shell. The magnetic fields obtained from Aformulation will be fed back to the T-formulation equations. The T-A model is solved in commercial FEM software COMSOL Multiphysics.
The T-formulation model is solved in a PDE module, and the A-formulation model is solved in magnetic field module. E-J power law is applied on the REBCO shell to represent the critical state of HTS where n = 28, E 0 = 1 μV/cm, J c is the critical current density. Its field dependence is expressed by following formulas in this study, which is suitable for high fields [46] where B is magnetic field norm, θ is the field angle related to caxis of the tape. More details about these formulas have been presented in reference.

Modelling of REBCO pancake coils
The electromagnetic fields of REBCO pancake coils show a 2D axisymmetric distribution, thus it can be simulated by a 2D axisymmetric FEM model [47,48]. As shown in Fig. 2a  In REBCO pancake coils, the distributions of magnetic fields and current density show sharply variations on turns near the inner side and outer side of the coil, but have very flat variation on the middle turns of the coil. It is reasonable to apply coarse meshes on the turns with same computational accuracy, which is the fundamental mechanism of previous homogenous H-formulation [34]. In this study, we improve the T-A model by replaying the actual turns on middle zones with equivalent turns, as shown in Fig. 2b 1 . The number of equivalent turns is much less than that of physical turns, thus the number of meshes is significantly reduced, as shown in Fig. 2b 2 . Equivalent critical current I c, eq and transport current I op, eq is applied on these equivalent turns I c, eq = n t I c I op, eq = n t I op (7) where I c and I op are the critical current and transport current of physical turns replaced. n t is number of physical turns replaced, which is 2-10 in this study. It can increase with the number of physical turns of pancake coils. Therefore, this technique can prevent the increase of computation cost with number of turns of HTS magnets, and therefore, make the electromagnetic modelling of large REBCO magnet system more practical and easier.

REBCO magnet for analysis
A 15 T test coil using insulated REBCO tapes has been designed as insert magnet in a high magnetic magnet. This REBCO coil consists of 40 single pancake coils (SPC), and each coil has 300 turns, thus it has 12,000 turns in total. The above model proposed will be validated by simulating the electromagnetic behaviour of this magnet during ramping operations. The mechanical stress analysis on this coil is not considered in this paper. Parameters of the REBCO tape used are based on tape of Superpower. Its width and thickness are 4.1 and 0.1 mm, respectively. Thickness of its superconductor layer is 1 μm. This magnet is designed to generate 15 T magnetic fields at axial centre, with transport current 200 A which is well below the critical current of the Superpowers's tape of the art-of-state. More specifications of this magnet are shown in Table 1. As shown in Fig. 3, only upper half of the magnet (20 SPCs) is calculated in the simulation, considering the symmetric characteristics of electromagnetic fields. All the pancake coils are named SPC1-SPC40 from top coil to the bottom coil. The actual number of turns in original T-A model is 6000, and it is reduced to 1560 turns by applying the equivalent turn method, as shown in Fig. 3. Table 2  In the simulation, the operating current of this HTS magnet is 200 A, and it is ramped up and down with ramping rate 1 A/s, as shown in Fig. 4. Note that only the HTS coil is ramped in this study, the background field is not considered. The solution time is significantly reduced by applying the equivalent turns, as shown in Table 2.

Distribution of current and fields
Same ramping operation is simulated by both original T-A model with actual turns and improved T-A model with equivalent turns.
The H-formulation model, which is most popular electromagnetic modelling method of HTS, is also applied on the magnet here to validate the T-A model. There is a slight difference on the transferring zones between actual turns and equivalent turns, as shown in Fig. 6. This is led by position difference between actual turns and equivalent turns, which induces a slight difference on the distribution of magnetic fields. The normalised current from both T-A models shows a same distribution among turns with that from H-formulation, which validates the correctness of both T-A models.
The results in Fig. 6 also show that coils near the top and bottom of the magnet have much more penetration than the coils near the middle zones of the magnet. This is because the coils on top and bottom have much higher radial background fields, which are perpendicular to the tape surface. This generates two effects: first, due to the anisotropy of critical current of REBCO tapes, the tapes of top coils have lower critical current, and radial magnetic field has more penetration on the superconductors. Second, higher radial fields induce more screen current on superconducting layers because of more flux coupling. The main component of magnetic fields on middle coils is axial field, which is parallel to the tapes surface. This parallel field is hard to induce screen current in superconducting layers of REBCO tape because of its thin thickness (1 μm). Both effects lead to much more penetration on top coils. Figs. 5 and 6 only show the top six pancakes of the magnet, where the distribution of fields and current shows a more significant variation among pancakes. In the other pancakes, the current and fields of original T-A model are also accurately reproduced by the improved T-A model. Fig. 7 shows the distribution of current density magnitude on the middle turns of each pancake coil, which matches the red dashed line in Fig. 3. Note that the real current density in the equivalent turns of improved T-A model is n t (n t = 5 in this study) times of that in the original T-A model, since the transport current and critical current have been enlarged n t times in these equivalent turns, as shown in (7). In Fig. 7, the data from improved T-A model has been shrank to 1/n t of its calculated value for a better comparison. Results from T-A improved model show a good agreement with that from original model. Generally, the current density magnitude increases from the middle coils to the top coils, due to the increases of radial fields, which leads to relatively low current density on middle coils (like SPC 30). However, the top coil (SPC 1) also has a lower current density magnitude than others because it has lower critical current. Small oscillations occur on the virgin zone of the tape, where should have no current. This is induced by numerical errors in solution, and same situations have also occurred on H-formulations [32].

Screen current induced field
Screen currents are induced by the radial fields, part of them are opposite to the transport current, as shown in Figs. 5 and 6. SCIF can lead to significant reduction and non-uniformity on magnetic fields, which is a great challenge for many high field magnet applications like MRI/NMR and accelerator [49][50][51]. Fig. 8 shows the distribution of magnetic fields at t = 400 s, when the magnet is ramped down to 0 A for the first time. There is a considerable field residue (up to 4.6 T) on the REBCO coils when the transport current is ramped down to zero. Fig. 9 shows the normalised current density at t = 400 s. Although the total transport current of each turn is zero at t = 400 s, there are several screen currents with opposite direction in each tape, which generate the magnetic fields in Fig. 8. Higher radial fields at top coils induce much more screen currents, which generate higher SCIF on these coils. This screen current also generates a residual field 0.5 T at the centre of the magnet though the transport current is 0 A at the same time.
Here we also calculated the magnetic field induced by the magnet without screen current, which has a uniform current distribution in all the tapes. The field difference ΔB at magnet centre is used to represent the influence of SCIF where B SC is the magnetic field at magnet centre considering the screening current. B uniform is the magnetic field at magnet centre with uniform current density distribution, which is accurately proportional to the transport current. Fig. 10 shows the dependence of this field difference ΔB on the central magnetic field induced by uniform current density B uniform , which is used to represent the SCIF. During the ramping cycle, the variation of transport current is as shown in Fig. 4. Significant magnetic hysteresis is observed during the ramping up and down operations of HTS magnet, which is induced by SICF. The field reduction induced by SCIF can be ∼0.9 T under transport current 200 A. Measures have to be developed to eliminate or reduce the SCIF, which will be discussed in future publications.
The results in Figs. [8][9][10] show that, for the study of SCIF in HTS magnet, the improved T-A model with equivalent turns can reproduce accurately the data of original T-A model with actual turns, like distribution of current density, magnetic fields and magnetic hysteresis induced by SCIF.

Ramping loss
Magnetisation loss is generated in superconducting layers during the ramping operation of HTS magnet, due to flux creep and jump [52]. This is the main part of ramping loss of HTS magnet below critical current, which may lead to a potential quench and ramp failure [33,53]. In T-A models, this magnetisation loss can be calculated by where W m and Q m are the magnetisation loss power and energy, respectively. Fig. 11a shows total magnetisation loss power W m of the 20 pancake coils during the ramping operation in Fig. 4. Fig. 11b shows the distribution of total magnetisation loss energy Q m generated in first ramping up to 200 A among pancake coils. The magnetisation loss shows a significant non-uniform distribution among pancakes. The top coils generate much more loss energy than the middle coils because of much more screen current and flux penetration on top coils. During the first ramping cycle t = 0-200 s, the magnetisation loss power W m increases fast with the transport current and reach to peak value when the transport current stops increasing at t = 200 s. Then the magnet starts to ramp down and the magnetisation loss power drops fast to nearly zero with the decrease of transport current. However, the valley point does not occur at t = 400 s when the transport current ramp down to 0 A. Due to the magnetic hysteresis induced by screen current, magnetisation loss power drops to valley point at t = 230 s and then increases gradually with the decreases of transport current. It increases continuously in the next ramping up operation to − 200 A, and reaches to a higher peak value than the peak of first ramping up at t = 560 s. The accumulation of screen current induces more penetration on the REBCO coils, as shown in Fig. 12. There is a time gap between peak points of magnetisation loss power (t = 560 s) and transport current (t = 600 s), which is also induced by the screen current.
Figs. 11 and 12 show that, for the magnetisation loss generated in the ramping of HTS magnet, the results from improved T-A model with equivalent turns have a good agreement with that from the original T-A model with actual turns.

Conclusion
This paper presents an electromagnetic modelling based on T-A formulation for high field magnets using REBCO tapes. To solve the problem of huge computation, an equivalent turn method is proposed for REBCO magnet with large number of turns. This method can considerably reduce the number of turns calculated in solution, prevent the approximately linear increase of meshes with the number of actual turns, and therefore significantly reduce the computation cost, make the electromagnetic modelling of industryscale HTS high field magnets practical and easy.
This improved T-A model with equivalent turns is validated by comparing its results from original T-A model with actual turns. A REBCO test coil of 15 T consisting of 40 SPCs and 12,000 turns is analysed, and its key characteristics during ramping operation are calculated and compared: distribution of current density and magnetic fields, SCIF and magnetisation loss during ramping. For all these issues, the results from improved T-A model with equivalent turns have a very good agreement with that from original T-A model with actual turns, which validate the feasibility of the improved T-A model with equivalent turns.
The high field magnets consisting of multiple REBCO pancakes always suffer from a serious problem of SCIF, which is induced by radial fields. The SICF can lead to a considerable field reduction on the centre of this magnet, which is 0.9 T for the 15 T magnet studied. It can also induce a high residual field on the magnet when the magnet is fully discharged, 0 A transport current. Highest residual fields often occur on the top and bottom coils of the magnet, which reaches to 4.6 T for this 15 T magnet. This is a potential risk for the mechanical stability and safety of high field magnet, which needs a special attention in the magnet design. The top and bottom coils have higher radial fields than other coils, which induce higher screen current and magnetisation loss on these coils. During the ramping up and down cycle, more magnetisation loss is generated in superconductors after the first ramping up operation, which is induced by screen currents.