Dynamic resistance and total loss in small REBCO pancake and racetrack coils carrying DC currents under an AC magnetic field

In many high-temperature superconducting applications, REBCO (Rare-earth barium copper oxide) coils carry DC currents under AC magnetic fields, such as the field winding of rotating machines, linear synchronous motors and the electro-dynamic suspension system of maglev. In such operating conditions, REBCO coils generate AC loss—total loss which includes the magnetization loss due to the shielding currents, and the dynamic loss arising from dynamic resistance caused by the interaction of DC currents and AC magnetic fields. In this work, dynamic resistance and total loss in a small double pancake coil (DPC) and a small double racetrack coil (DRC) are investigated via experiments in the temperature range between 77 K and 65 K. The DC currents are varied from zero to 70% of the self-field critical currents of the REBCO coils, with AC magnetic fields up to 100 mT. The experimental results in the DPC are well supported by the finite element simulation results using 3D T-A formulation. Our results show that the critical current of the DRC is approximately 2%–5% higher than that of the DPC in the temperature range. For given experimental conditions, the magnetization loss in both coils is much greater than the dynamic loss. The dynamic loss and magnetization loss in the DRC are greater than those in the DPC, which we attribute to the large perpendicular magnetic field component in the straight sections of the DRC.


Introduction
In many high-temperature superconducting (HTS) applications, REBCO coils carry DC currents under a time-varying * Authors to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. magnetic field, such as the field winding of rotating machines [1,2], linear synchronous motors [3,4], and electro-dynamic suspension system of maglev [5][6][7]. In such an operating condition, AC loss is generated in REBCO coils, including the magnetization loss due to the shielding currents, and the dynamic loss arising from dynamic resistance when the external magnetic field exceeds the threshold magnetic field [8][9][10][11][12][13][14][15][16][17]. The sum of dynamic loss and magnetization loss is defined as total loss. The parasitic heating due to the total loss adds to the refrigerator load and increases the risk of quench. In other words, AC loss determines the cooling efficiency, cost, and operational stability of these HTS applications. The understanding and prediction of dynamic resistance and loss characteristics are therefore essential to the design and optimization of these practical HTS applications.
To date, there have been only a few reports on the magnetization loss, dynamic resistance and total loss of REBCO coils [18][19][20] at 77 K. The frequency dependence of total loss in REBCO coils was studied numerically [20], which lacks systematic studies on the influence of DC currents and magnetic fields on the total loss and dynamic resistance. The total loss of a small single REBCO racetrack coil was measured and simulated, where the simulation took 2D approximation only for the straight sections of the racetrack coil [18]. The total loss of a REBCO pancake coil was calculated using a 2D axisymmetric H-formulation model in COMSOL, but without any experimental validation [19]. The total loss predictions in HTS applications usually focused on specific operating conditions such as wind turbines and rotating machines [1,[20][21][22][23]]these studies have not fully explored the underlying behaviors of total loss at various operating conditions. Overall, studies are patchy for the dynamic resistance and total loss of REBCO coils.
We have recently reported the measured and simulated results of dynamic resistance and total loss in a single REBCO CC [10] and a three-tape stack [11]. The current work is a further extension of our previous works to characterize dynamic resistance and total loss in REBCO coils when the coils carry DC currents under AC magnetic fields.
In this work, the dynamic resistance and total loss were measured at 77 K and 65 K in a small REBCO double pancake coil (DPC) and a small REBCO double racetrack coil (DRC). The transport DC currents vary from zero to the 70% self-field critical current of the REBCO coils, and the amplitudes of the AC magnetic field are up to 100 mT at 72.73 Hz, 102.84 Hz and 145.51 Hz. The influence of the coil geometry is analyzed by comparing the critical currents, magnetization loss, dynamic resistance and total loss of the two coils. Furthermore, the different behaviors of the dynamic resistance and magnetization loss in the coils are compared with those in the single REBCO CC [10] and the three-tape REBCO stack [11].

Experimental method
A DPC and a DRC were manufactured from the 4 mm wide REBCO coated conductor (CC). The specifications of the REBCO CC are listed in table 1. The self-field critical current of the REBCO CC is around 108.1 A [10,11], which was used for the loss calculation of the REBCO coils. Before winding, Kapton tapes were attached to the surface of the CC as the turn-to-turn insulation of the REBCO coils.
The DPC was wound on an 18.0 mm diameter G10 former. The DPC consists of two ten-turn pancake coils which are insulated by a 0.2 mm-thick G10 sheet, as shown in figure 1. The effective inner and outer diameters of the DPC are 18.1 mm and 21.7 mm. The DRC was wound on a racetrackshaped G10 former. The DRC is composed of two racetrack coils, each with the same turn number as the DPC. The straight 145.0 Self-field critical current @ 77 K (A) 108.0  The total loss of the small REBCO coils were measured by electrical methods [10,11,15,18], i.e. the calibration-free method for magnetization loss, and the four-probe method for dynamic resistance and its corresponding power dissipationdynamic loss. Prior to the total loss measurement, magnetization loss of the REBCO coils was measured as a loss reference when the REBCO coils carry zero DC current under AC magnetic fields. The details of the experimental system and procedures have been described in the earlier works [10,11]. Figure 2 shows the coil arrangement for the magnetization loss measurement. The DPC coil was attached to the flat surface of a sample holder which was inserted in the AC magnet bore. No voltage signal wires and current leads were soldered on   the coil. By rotating the sample holder, the external magnetic field was applied parallel to the top surface of the coil, as illustrated in figures 2 and 3. The definition of the field angle, θ, is schematized in figure 3, where θ = 0 • represents the perpendicular magnetic field. Under the applied perpendicular magnetic field, the perpendicular magnetic field components are not the same along the circumferences of the DPC and the semi-circular sections of the DRC, but are the same in the straight sections of the DRC. Figure 4 shows the sample arrangement for the total loss measurement when the coil carries a DC current under a perpendicular magnetic field. Voltage signal wires were soldered on the ends of the coil and twisted together to reduce the electromagnetic noise [10,11]. The effective distance between the voltage taps is 1.28 m and 2.81 m for the DPC and the DRC, respectively. The current leads are comprised of two partstwo long pieces of 4 mm-wide REBCO CCs and the copper braids as shown in figure 4. The REBCO CCs are arranged to be parallel to the applied magnetic field, which aims to minimize the coupling between the AC magnet and the DC circuit. The DC currents in the coils are normalized by the self-field critical current of the coils at each temperature i = I dc /I c , ranging from 0.05 to 0.7.

Numerical method
The 3D numerical model based on the T-A formulation is used to calculate magnetization loss, dynamic resistance and total loss in the DPC. The details of the T-A formulation can be found in previous studies [24][25][26][27]. Figure 5(a) shows the 3D geometry for the DPC, which is simplified as a vertical stack of two ten-turn pancake coils. The air gap between these two pancake coils is 0.2 mm, and the height of each pancake coil is 4 mm. Only the superconducting layer of the REBCO CC is considered and approximated as a sheet, while the nonsuperconducting layers are assumed to be air [24][25][26][27]. A cylinder air domain was built, whose diameter and height are three and six times the diameter and height of the DPC. Figure 5(b) shows the meshing of the DPC, where a mapped mesh was applied to the superconducting sheet, with 8 elements and 100 elements along the height and circumference of the coil, respectively. To improve the convergence of the simulation, a diagonal conversion was applied to the mapped mesh. The free tetrahedral mesh was applied to the air domain (not shown in the figure).
The electromagnetic characteristics of the REBCO CC are expressed by the E-J power law in equation (1) [10,11,[24][25][26]. The anisotropy of the critical current is considered using the modified Kim model shown in equation (2) [10,11]. The influence of the applied magnetic field on the n-value is also considered in the model, where E c = 1 µV cm −1 and J c0 is the constant critical current density obtained from the self-field critical current by dividing the cross-section of the superconducting layer at each temperature. B ⊥ and B ∥ are the perpendicular and parallel components of the local magnetic field with respect to the wide-face of the CC. E and J are the electrical field and current density. The constant parameters, κ, α and B 0 , were fitted from the measured E-J curves of the REBCO CC, which can be found in our previous work [10,11]. Figure 6 shows the field-dependent n-values of the REBCO CC at 77 and 65 K, which were fitted from the E-J curves under different perpendicular DC magnetic fields [28,29]. The n-values decrease with increasing magnetic field. The fitted  n(B) curves were directly interpolated into the numerical models for 77 K and 65 K, respectively.
For the magnetization loss calculation, the applied magnetic field is imposed on the air domain via the Dirichlet boundary condition. The magnetization loss per cycle in the coil is calculated as [10,11]: where f is the frequency of the applied AC magnetic fields. S is the surface area of the DPC. δ is the thickness of the superconducting layer. For the dynamic resistance and total loss, DC currents are injected into the DPC via a ramping up process using equation (4). Once DC currents reach the desired values, external magnetic fields are applied to the air domain, where T u and T l are the values of the current vector potential, T, on the upper and lower edges of each pancake coil. R = 5 A s −1 is the constant ramping rate of DC currents, and t 0 is the time required for the DC ramp process. The total loss per cycle in the coil is calculated by equation (5), including the dynamic loss component, Q dyn , and magnetization loss component, Q m,i .
Then Q dyn and Q m,i per cycle is obtained by the following equations: where t 1 = t 0 + 4/f, E a and J dc are the average electrical field and DC current density across the entire length of the DPC. The dynamic resistance, R dyn , per cycle per meter is calculated as: where L is the length of the CC in the DPC.    table 3. At any given temperature, the I c value of the DRC is slightly larger than that of the DPC, and the difference in I c values is less than 5%. This is because the magnetic field at semi-circular sections of the DRC is smaller than that of the DPC due to the existence of the straight sections in the DRC.   The different shielding effect between the DPC and the single REBCO CC can be indicated from their effective penetration fields, B e,p . Figure 8(b) compares B e,p of the DPC and the single CC via the normalized magnetization loss, Γ = Q m /(µ 0 H m ) 2 . Γ is a function of applied magnetic field and has the maximum at B e,p [16,32]. B e,p of the DPC (∼125 mT) is obtained from the numerical results, which is almost five times the B e,p of the single CC (∼26.9 mT). The significant difference in B e,p values implies the strong shielding effect in the DPC, and explains the reduced average loss magnitudes in the DPC. Figure 9 shows the measured magnetization loss (Q m ) of the DPC under various field angles at 77 K and 72.73 Hz. The magnetic field angle, θ, varies from 0 • to 60 • . Figure 9(a) plots Q m as a function of the applied magnetic field. Q m decreases with increasing field angles, which is attributed to the smaller perpendicular magnetic field components at higher field angles. Figure 9(b) re-plots Q m as a function of the perpendicular component of the applied magnetic field. Q m values at different field angles collapse into a common curve, which indicates that the perpendicular magnetic field component determines the magnitudes of magnetization loss [33,34]. Figure 10 shows the normalized measured magnetization loss (Γ = Q m /(µ 0 H m ) 2 ) as a function of the perpendicular magnetic fields under various temperatures at 72.73 Hz. At any given temperature, Γ monotonically increases with increasing magnetic field, when ignoring the scattered data at low magnetic fields. This is because the applied magnetic field is less than the effective penetration field (B e,p ) of the DPC. As discussed in the previous figure 8(b), B e,p of the DPC is around 125 mT at 77 K, which should be much greater at other low temperatures because B e,p is proportional to the critical current of the coil [25,35]. In addition, the Γ values decrease with decreasing temperature, which is attributed to the temperature-dependent critical current characteristics. The increased critical current at lower temperatures leads to a stronger shielding effect of the coil, generating lower magnetization loss. Figure 11 plots the measured Q m of the DRC as a function of the perpendicular magnetic fields (µ 0 H m ) at  three frequencies. At low-µ 0 H m , the measured Q m values at 102.84 Hz and 145.51 Hz are close to each other, but larger than that of 72.73 Hz. This observed frequency dependence for low-µ 0 H m may be caused by the eddy current loss in the copper stabilized layers [18,36,37]. At high-µ 0 H m , Q m values at different frequencies collapse into one common curve, which are much smaller than those of the single REBCO CC due to the strong shielding effect in the coil. Figure 12 compares the measured Q m values between the DPC and the DRC at 77 K and 102.84 Hz. The loss values of the DRC are higher than those of the DPC. This is because the straight sections of the DRC experience a larger perpendicular magnetic field component than the semi-circular sections. In the previous section 4.2.3, we observed that Q m of the DPC decreases with increasing I c in the whole magnetic field range.

Comparison of magnetization loss values between the DPC and DRC.
However, both I c and Q m of the DRC are higher than those of the DPC. This indicates that the coil geometry plays a greater role than I c characteristics of the coils for determining magnetization loss of the REBCO coils.   respectively. These values are much smaller than those of the single REBCO CC [10] and the three-tape REBCO stack [11], as shown in figure 14. For i = 0.1 and 0.2, the R dyn values in the DPC are almost one order smaller than those of the single CC and the stack, which is caused by the strong shielding effect of the DPC. The difference in R dyn values decreases with increasing i, and we attribute the reason to weakening shielding effect and greater I c -degradation of the DPC at higher i. It is also observed that B th values in the DPC are larger than those of a single CC and the stack. By applying a resistance criterion of 0.01 µΩ m −1 Hz −1 , the B th values in the DPC are 50.2, 35.9, 27.5, and 17.1 mT for 0.2 ⩽ i ⩽ 0.7.   applied magnetic fields (µ 0 H m ) in a log-log scale. The power law relationship between R dyn and µ 0 H m is observed at all temperatures. The R dyn values of the DPC and the DRC decrease with decreasing T-a significant reduction in R dyn values is observed at 77 K < T < 72.5 K, while a moderate reduction for 70 K < T < 65 K. The R dyn values of the DRC are larger than those of the DPC at each T, which is attributed to the larger perpendicular magnetic field component in straight sections of the DRC. Figure 16 shows B th values of the DPC and DRC at different temperatures for i = 0.5, which are obtained by applying the criterion of 0.01 µΩ m −1 Hz −1 . The B th values of both coils increase with decreasing temperature due to the temperaturedependent critical current. When the temperature decreases from 77 K to 65 K, B th increases from 27.4 mT to 65.8 mT for the DPC, and from 23.2 mT to 54.8 mT for the DRC. For any given temperature, the B th values of the DRC are around 85% of those for the DPC, which is attributed to the larger perpendicular magnetic field component in the straight section of the DRC.

Dynamic resistance in the DPC and DRC at various temperatures.
Based on the analytical equations of B th in a single REBCO CC (see previous equations (8) and (9) of [10]), the ratio of B th and self-field I c is always constant for any given i. However, the ratio of I c /B th in the DPC and the DRC is not constant, which increases with decreasing temperature, as shown from the different gradients of plotted I c (T) and B th (T) curves. This may be attributed to the field-dependent B th of the coils. Although the i value is equal at all temperatures, the CCs of the coils carry larger DC currents at lower temperatures, which leads to much higher DC magnetic fields generated from other coil turns. Consequently, the measured B th of the coil is smaller than the expected value derived from the selffield critical current values. Moreover, the influence of the 'background DC magnetic fields' on B th becomes more profound with decreasing temperatures, causing a larger I c /B th in the DPC or DRC at lower temperatures.
For the three-tape REBCO stack [11], the dynamic resistance and threshold magnetic field are also influenced by the magnetic fields generated from adjacent REBCO CCs. However, the influence of the magnetic field on the threshold magnetic field is not as profound as that observed in the REBCO coils. Compared with the stack, the CCs of the coils are exposed to a much higher magnetic field, which is not only because of larger turn numbers of the REBCO coils than the stack, but also because of the coil geometry. This observed phenomenon implies I c -degradation of REBCO CCs plays an important role on B th and R dyn in the REBCO coils.  shows the measured and simulated total loss and loss components of the DPC at 77 K and 65 K for different DC levels. Firstly, the simulation results reproduce the measured loss tendency of the DPC at 77 K. Throughout the whole magnetic field range, the dynamic loss, Q dyn , is smaller than the magnetization loss Q m,i at all DC levels. The contribution of Q dyn to Q total can be ignored for i ⩽ 0.3, but increases with increasing i.
Experimental results show Q total , Q dyn and Q m,i of the DPC at 65 K are smaller than those of 77 K at any given i. The onset of Q dyn is much slower at 65 K when compared with the data at 77 K. Moreover, the ratio of Q m,i /Q dyn becomes larger at 65 K. These differences are due to the increased critical current of the DPC at 65 K, which causes a stronger shielding effect in the DPC compared with 77 K. The change in Q dyn values between 77 K and 65 K is more apparent than that of Q m,i at all DC levels. This is because Q dyn is more sensitive to the variation of the critical current than Q m,i [10,11,16,17]    of Q dyn /Q total increases with increasing i; (c) Q m,i dominates Q total .
On the other hand, the loss values of the DPC and DRC are different. The following differences are observed when comparing the loss values in the DPC with those of the DRC: (a) Q m,i and Q dyn in the DPC are smaller; (b) the onset of Q dyn in the DPC is much slower; (c) the contribution of Q dyn to Q total in the DPC is smaller. These differences are mainly caused by the larger perpendicular magnetic field components in the straight sections of the DRC.

Conclusion
The critical current, magnetization loss, dynamic resistance, and total loss have been measured in a small REBCO DPC and a small REBCO DRC from 65 K to 77 K. The amplitudes of the applied magnetic fields are up to 100 mT and transport DC currents in total loss measurements are up to the 70% selffield critical current of the coils.
When the temperature decreases from 77 K to 65 K, the self-field critical current of the DRC varies from 64.1 to 152.7 A, which is slightly higher than that of the DPC at each temperature. The difference in I c values between the DPC and DRC are less than 5%. We attribute the reason to the existence of the straight sections in the DPC-leading to a lower selffield in the semi-circular sections of the DRC than the DPC.
The measured magnetization loss and total loss in the DPC show a good agreement with the simulation results at 77 K, which validates the accuracy of the numerical simulation. In the given magnetic field range, the magnetization loss in the DPC does not show obvious frequency dependence, decreases with decreasing temperatures, and is dominated by the perpendicular magnetic field components. The magnetization loss of the DRC is greater than that of the DPC due to the larger perpendicular magnetic field component in the straight sections than the semi-circular sections.
The dynamic resistance in the DPC and DRC shows a power law relationship with the amplitudes of the applied magnetic fields, rather than a linear relationship. This is because the explored applied magnetic fields in this work are smaller than the effective penetration field of the REBCO coils. The dynamic resistance in the DPC and the DRC increases with increasing DC currents, the applied AC magnetic fields and temperatures, while the threshold magnetic field shows an opposite tendency. We found that the magnetic field from other adjacent CCs has a great influence on the threshold magnetic field of the coil, which should be considered for accurate dynamic loss prediction. In addition, the dynamic resistance of the DRC is larger than that of the DPC, while the threshold magnetic field in the DRC is smaller.
Although the contribution of dynamic loss to total loss increases with DC currents, magnetization loss in the DPC and DRC dominates the total loss at 77 and 65 K at each DC current level. The magnitudes of each loss components of 77 K are higher than those of 65 K. The total loss and each loss component of the DRC are also larger than those of the DPC. This is also attributed to the larger perpendicular magnetic field component in the straight sections of the DRC when compared with the semi-circular sections in the DRC and the DPC.
Compared with the single REBCO CC or the stack, the magnetization loss, dynamic resistance and total loss generated in unit length are much lower in the coils, and the threshold magnetic field of the coils is much larger at low DC currents. These differences are due to the strong shielding effect in the coils.
Our results provide references to relevant HTS applications. The dynamic resistance and total loss of the REBCO coils are expected to be characterized at lower operating temperatures and higher magnetic field range in future work.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).