Dynamic resistance and voltage response of a REBCO bifilar stack under perpendicular DC-biased AC magnetic fields

The dynamic resistance of REBCO (REBa2Cu3O7-d, RE stands for rare earth), coated conductors (CCs) is a key parameter in many high-temperature superconductor applications where CCs carry DC currents exposed to AC and DC magnetic fields, such as field-triggered persistent current switches, flux pumps, and fault current limiters. In this work, dynamic resistance and dynamic voltage have been studied via experiments and finite element method (FEM) simulations in a REBCO bifilar stack at 77 K, under combined AC and DC magnetic fields with different magnitudes, frequencies, and waveforms. Our results show some distinct features of dynamic resistance and voltage from those under pure AC magnetic fields. With an increasing DC magnetic field, the dynamic resistance exhibits an obvious linearity with the applied AC magnetic field, and becomes less dependent on the AC field frequency. The fundamental frequency of the dynamic voltage under a DC magnetic field becomes the same as that of the applied AC field, which completely differs from the pure AC field case where the fundamental frequency doubles. For the first time, instantaneous threshold field (B th) values are obtained from the dynamic voltage, which are substantially different in the field-increasing and field-decreasing processes. These key differences are attributed to the dominant role of DC magnetic fields in determining the critical current of the superconductor, which significantly dwarfs the influence of AC fields. These new discoveries may help researchers better understand the electromagnetism of superconductors and be useful for relevant applications.


Introduction
REBCO coated conductors (CCs) are becoming the preferred wire choice in many high-temperature superconducting (HTS) applications, such as field-triggered persistent current switches (PCSs) [1][2][3][4][5][6][7][8], flux pumps (FPs) [9][10][11][12], fault current limiters (FCLs) [13][14][15], electrodynamic suspension systems [16][17][18] and linear synchronous motors [19,20]. In these HTS applications, a REBCO CC carries DC currents under AC and DC magnetic fields. When the AC magnetic field exceeds the threshold magnetic field, magnetic flux traverses across the DC current region of the CC, leading to a time-dependent periodic voltage (dynamic voltage) along the CC [21][22][23][24][25][26]. The corresponding time-averaged resistance is termed as 'dynamic resistance' [21], which directly determines the operational performance of these HTS applications, where CCs are inevitably exposed to the self-induced DC magnetic fields from the transport DC currents, or DC bias magnetic fields in the operational conditions. To understand and optimize these HTS applications, it is important to study the dynamic resistance and transient voltage response of the CC under DC bias magnetic fields.
There are extensive reports on the dynamic resistance and voltage response when REBCO CCs carry DC currents under AC magnetic fields, regarding the dependence on the amplitude and frequency of AC magnetic fields [25][26][27][28][29][30][31][32][33][34][35][36], operating temperatures [27,30,35,37], DC currents [25-30, 35, 38, 39] and waveforms of AC magnetic fields [39]. Several reports also present research on the influence of DC magnetic fields on the dynamic resistance and voltage of the CCs [22][23][24]. Mikitik and Brandt [22] derive an analytical equation for the threshold magnetic field of a superconducting strip under AC and DC magnetic fields when the DC magnetic fields are infinitely larger than the AC magnetic fields. Duckworth et al [23] measure the dynamic resistance of different YBCO CCs at 77 K where the AC magnetic fields peak to 25 mT at 60 Hz and the bias DC magnetic fields are up to 1 T. Uksusman et al [24] study the voltage response of a DC-current-carrying YBCO CC when AC fields peak to 20 mT at 15 Hz and DC fields are up to 90 mT. These previous studies have not examined the influence of the waveforms and frequencies of AC magnetic fields on the dynamic resistance and voltage response when CCs are exposed to a wide range of AC and DC magnetic fields. The remaining research gaps need to be addressed to underpin practical applications, such as AC-fieldtriggered HTS switches [1][2][3][4][5][6][7][8] and transformer-rectifier FPs [9][10][11][12], where the waveform and frequency of the external magnetic fields influence their maximum output currents and stability.
A number of these HTS switches and FPs have the configuration of placing a HTS wire in a narrow airgap of a C-shaped iron-core magnet [2,7,12,35]. However, the selffield critical current of the single CC within the iron-core air gap is suppressed by approximately 45%, which directly influences the switching performance and efficiency [7,12,40]. On the other hand, a bifilar stack of two antiparallel REBCO CCs was used as the switching element, which is preferable for gapped-iron-core-based HTS PCSs and HTS FPs. Such a 'noninductive' structure of bifilar stacks effectively minimizes the self-induction of the switch and substantially increases the self-field critical current of the CC, which provides more stable load fields and significantly improves the efficiency of HTS switches and FPs [1,7,12,40].
In this work, the dynamic resistance and voltage of a REBCO bifilar stack is measured at 77 K when carrying DC currents under AC and DC magnetic fields. The amplitudes of the AC magnetic fields are up to 350 mT at 10, 20 and 50 Hz with two different waveforms: sinusoidal and triangular. The DC magnetic fields vary from 0 to 300 mT and DC currents vary between 10 A-40 A. The characteristics of dynamic resistance are analyzed under various operating conditions, including the effect of the amplitude, frequency and waveform of the AC magnetic field, and the magnitudes of the DC currents and magnetic fields.

Sample preparation
The sample is designed as a bifilar stack, assembled from 2 mm wide REBCO CCs, which are manufactured by the SuNAM company. The specifications of the REBCO CC are listed in table 1. The measured self-field critical current of a single REBCO CC is 91 A at 77 K, increased to 108 A after being assembled as a bifilar stack. Figure 1 shows a schematic of the bifilar stack. The two REBCO CCs are vertically stacked together with the superconducting sides facing outwards. One end of the two CCs are soldered together, whilst the other end is used as current feedin terminals. Kapton tape was applied between the two CCs to ensure electrical isolation. The effective length between the voltage taps soldered on the bifilar stack is 6.0 cm. The current feed-in terminals are connected to a pair of current leads via 12 mm wide Superpower CCs soldered to the copper plates, as illustrated in figure 2.
Two G-10 sheets are attached on each side of the stack as the sample holder. The stack, together with the sample holder, are fixed within the airgap of an iron-core magnet where a uniform magnetic field envelope (3 mm × 10 mm × 30 mm) is provided, as shown in figure 2. Figure 3 shows the experimental setup. The iron-core magnet was energized by a Hewlett Packard function generator (3312a, 15 MHz), and a Takasago bipolar power amplifier (BWS40-15) which was operated in a controlled current mode to provide an AC current with a DC offset. The magnet therefore generates an AC field with a DC offset. The currents flowing in the magnet windings were monitored via the voltage across a resistor (AP101-R1-J, 20040610), which are proportional to the amplitudes of magnetic fields for the current experimental study. The relationship between the currents and magnetic fields was calibrated using a Hall sensor. A Keysight DC power supply (6682A) was utilized to provide DC currents 120.0 Self-field critical current @ 77 K (A) 91.0  to the stack. In the measurements, the magnet and sample were immersed in liquid nitrogen at 1 atmosphere (77 K). The voltage signal of the stack and resistor were collected using NI 9238 DAQ (data acquisition) card, with a sampling frequency of 2 kHz. The dynamic resistance (R dyn ) is then obtained from the time-averaged voltage by dividing the DC currents (I dc ) that are fed into the stack.

Experimental system and method
Sinusoidal and triangular AC magnetic fields (B ac ) were generated by the function generator, with amplitudes (B m ) up to 350 mT at 10, 20 and 50 Hz. The bias DC magnetic fields (B dc ) range from 0 to 300 mT and DC currents of the stack were varied from 10 A to 40 A.

Numerical method
The dynamic resistance of the bifilar stack is calculated using finite element modeling, based on H-formulation in COMSOL Multiphysics 5.6. The detailed descriptions of the FEM model have been introduced in earlier publications [25,[27][28][29]. Figure 4 shows the schematics of the model, which is built from the cross section of the stack in the x-y cartesian coordinate. Each superconducting tape comprises a 1 µm thick superconducting layer (REBCO layer) and two 20 µm thick silver layers. There is a 0.1 mm thick air gap between the two superconducting conductors. A structured mesh was applied to the REBCO, silver, and the air gap domain, with 200 elements along the width and five elements along the thickness in each domain. A free triangular mesh was applied to the remaining air domain. The resistivity of the air domains and silver was set as 1 Ω·m and 3.2 × 10 −9 Ω·m [41].
The electromagnetic properties of the REBCO layer are given by the E-J power law of equation (1) [25,[27][28][29],  where E c = 1 µVcm −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. E and J are the electrical field and current density. B ⊥ and B ∥ are the perpendicular and parallel components of the applied magnetic fields with respect to the wide face of the REBCO CC. The constant parameters, n, κ,α and B 0 , were fitted from the measured voltage-current curve of the sample, which were provided by the SuperCurrent facility of Robinson Research Institute [42,43]. In this model, n = 25, κ = 0.26, α = 0.42, and B 0 = 65 mT. The fitted and measured results of the fielddependent critical current are plotted in figure 5. The simulation is implemented in two steps. Firstly, the bifilar stack is fed into DC currents by a ramp function. The DC current is applied to each CC by integral current constraints on REBCO and silver domains as described in equation (3): where R = 10 A s −1 is the constant ramping rate, and t 0 is the ramping time of the DC currents. J and J s are the current density in the superconducting and silver domains, respectively. A HTS and A s are the areas of the superconducting and silver domains.
Once the DC current reaches the desired value, the external magnetic fields are imposed on the boundaries of the air domain. The dynamic resistance (R dyn ) caused by the interaction of DC currents and magnetic fields is calculated using the following equations where f is the frequency of the applied AC magnetic field, t 1 = t 0 + 4/f and t 2 = t 0 + 5/f. Figure 6 shows R dyn of the bifilar stack under various DC magnetic fields (B dc ) plotted as a function of the amplitude of the AC magnetic field when DC currents (I dc ) increase from 10 to 40 A. The simulated results show a good agreement with measured R dyn values when B dc ⩽ 100 mT, although they are slightly smaller when B dc > 100 mT. This discrepancy is attributed to a thermal effect from the loss dissipation, which is not considered in the numerical model. In addition, the fielddependent n-value, and the lateral and longitudinal nonuniformity of critical current in the REBCO CC, may also contribute to the discrepancy. As seen from figure 6, the threshold magnetic field, B th , decreases with increasing B dc for any given I dc , where B th is obtained by fitting a linear line to the R dyn curve and finding the intercept on the x-axis. We attribute this to I c -depression caused by the background B dc . In each subfigure, R dyn increases with increasing B m and B dc . For I dc = 10 A and 20 A in figures 6(a) and (b), R dyn shows a nonlinear relationship with B m when B dc ⩽ 200 mT. However, this increases linearly with B m at B dc = 300 mT when the flux flow resistance that occurs at high B m is ignored . The R dyn (B m ) behaviors mentioned above should be due to the field-dependent I c (B) characteristics shown in figure 5. The I c value of the CC in figure 5 decreases sharply to roughly 25% of the self-field I c value at B dc = 300 mT, whilst it almost linearly decreases with increasing perpendicular B dc . This tendency indicates that the I c value of the bifilar stack is very sensitive to magnetic fields that are less than 300 mT.  [26,31]. These differences are attributed to the induced DC magnetic fields from I dc , which influences the I c (B) characteristics and contributes to the linearity of the R dyn (B m ) curves. Thus, the influence of induced DC magnetic fields from high I dc play a role in determining R dyn behaviors. Hz. When f = 50 Hz, an obvious increase of R dyn compared to the other two frequencies is observed with increasing B m . This deviation in R dyn values at high f is attributed to the thermal effect due to the loss dissipation. When the measurement time is constant, more heat will be accumulated in the stack for the high-f due to the larger number of field cycles. Consequently, an obvious temperature rise within the stack occurs at high-f, leading to the I c -degradation of the CCs. This gives rise to higher R dyn values according to the analytical equation of R dyn [21,22].

Frequency dependence of R dyn at various B dc
In each subfigure, the B m value at which the R dyn curve for f = 50 Hz deviates from the other two curves is marked by a solid line. It is interesting to find that the B m value increases with increasing B dc . In other words, the frequency dependence of R dyn becomes weaker with increasing B dc . We attribute this to the reduced region for the shielding currents due to the I c -depression caused by B dc . The magnetization loss is therefore reduced with increasing B dc , the resultant heating and local temperature rise becoming smaller. Therefore, the thermal effect induced by high f is weakened by the presence of B dc .  values when B m ⩽ 150 mT, while they are smaller than the experimental results in the high-B m region (>150 mT). The differences are explained in the earlier section, which is relevant to the thermal effect. As shown in each subfigure, R dyn (B m ) curves from the sinusoidal and triangular B m almost overlap one another when B dc ⩽ 200 mT, which suggests that R dyn values are insensitive to the waveforms of the AC magnetic fields. In figure 8(e), R dyn curves under the sinusoidal and triangular B m show obvious differences at high B m , where the flux-flow phenomenon occurs because the transport I dc is larger than the field-dependent I c (B). This difference is attributed to the different time periods between the sinusoidal and triangular magnetic fields when I dc ⩾ I c (B). Assuming I dc is larger than the field-dependent I c when B ac (t) > |B ′ |, then we can find that the corresponding time period of the sinusoidal waveform is larger than that of the triangular one, i.e. ∆t 1 > ∆t 2 , as indicated in figure 9. The larger the time period of I dc ⩾ I c (B), the more heating is accumulated within the bifilar stack and the flux-flow phenomenon becomes more evident.   figure 10(a), the measured dynamic voltage fluctuates around zero, with the noise signal within 17 µV, implying the applied magnetic field (B m = 68 mT) is smaller than the threshold magnetic field, B th . In figures 10(b)-(e), measured dynamic voltage waveforms generate nonzero values, arising from R dyn . Based on the working principal of dynamic resistance [21,22,26], dynamic voltage is zero (V(t) = 0) when the applied magnetic field decreases from B m to (B m −2B th ), or increases from (-B m ) to (2B th -B m ). Therefore, B th can be obtained from the dynamic voltage, as indicated in figure 10.

R dyn under sinusoidal and triangular AC magnetic fields
B th values obtained from the measured dynamic voltage vary with B m , which is substantially different with the common assumption in previous works [22, 26-31, 34, 38] where B th is constant. The obtained B th values decrease with increasing B m , being 74.5, 72.5, 37.5 and 22.5 mT when B m = 80, 100, 150 and 200 mT, respectively. The B th value (i.e. 74.5 mT) at B m = 80 mT is consistent with the value from the R dyn (B m ) curve in figure 8(a). Furthermore, the influence of B m on B th values is non-linear. This observed field-dependence of B th is attributed to the field dependent I c (B) characteristics of the CC. Higher B m leads to a lower I c (B) value, resulting in a larger region for I dc . The external magnetic fields that can fully penetrate the CC are therefore reduced, i.e. smaller B th values. The observed B th (B m ) characteristics should be considered for accurate prediction of dynamic resistance and loss.
The corresponding simulation results are plotted in figures 10(a ′ )-(e ′ ), which reproduce the experimental tendency well. As observed from subfigures ((b) vs. (b ′ ), (c) vs. (c ′ )), the simulated dynamic voltage is slightly higher than measured results at low-B ac , with slightly reduced B th value. Compared with figures 10(d) and (e), the simulated voltage waveform shrinks with a higher peak value. However, the simulated B th values at high-B ac agree well with the

Influence of B dc on B th and dynamic voltage.
To further explore the influence of B dc on B th and dynamic voltage, measured V(t) waveforms at various B dc are plotted in figure 11(a), when B m = 80 mT. One obvious feature is that the voltage waveform becomes asymmetric in the field increasing (↑) and decreasing processes (↓). This implies that V(t) has the same fundamental frequency with B ac due to the presence of B dc . Correspondingly, the derived B th values are dramatically different in the field increasing and decreasing process, labeled as B th,↑, and B th,↓ respectively. These distinctive features are attributed to different I c (B dc + B ac ) values over one field cycle. Compared with the time period where B dc and B ac are additive, the stack has higher I c values when B dc and B ac are subtractive, which leads to smaller B th values and lower V(t) values. Figure 11(b) plots the derived B th,↑ , B th,↓ , and their average value B th,ave as a function of B dc . As shown in the figure, B th,↑ is larger than B th,↓ at B ⩾ 50 mT. For convenience, B th,ave is used to characterize the influence of B dc on B th . In general, the B th,ave curve is parallel to the measured I c (B dc ) curve, which indicates the influence of B dc on B th is determined by the I c (B dc ) characteristics.   [24,25,37,39]: (a) the voltage waveform is periodic and has a fundamental frequency twice that of the AC magnetic field; (b) during half a field cycle, the voltage waveform has one peak at low B m (e.g. B m = 100 mT), appearing at B ac (t) = ± B m ; (c) the single peak of the voltage waveform starts to split into double peaks with increasing B m , having a minimum around B ac (t) = 0 due to the influence of AC magnetic fields on I c (B) characteristics; (d) the onset of the nonzero voltage varies with B m -the higher the B m , the earlier the occurrence of the nonzero voltage value.

Influence of Bm on B th and dynamic voltage.
However, the V(t) at B dc = 300 mT shows different patterns, as shown in figures 12(c) and (c ′ ). Firstly, the onset of the nonzero voltage for each B m value occurs almost at the same moment and the voltage waveform shows the minimum at around B ac (t) = −B m , and the maximum around B ac (t) = B m . Secondly, the nonzero voltage waveform has a similar tendency with the applied B ac if ignoring the measured V(t) spike at B ac (t) = B m -monotonously increase over time when B ac increases from −B m to B m and monotonously decrease when B ac decreases from B m to −B m . Moreover, |dV/dt| increases with increasing B m . The voltage waveforms under B dc have the same fundamental frequency as B ac (t), and they are mainly determined by dB ac /dt rather than the instantaneous value of B ac (t). This is because the presence of B dc nullifies the influence of B ac (t) on I c (B) characteristics, as explained in previous sections.
Comparing this with the pure AC field case (figures 12(b) and (b ′ )), we found that dynamic voltage under nonzero B dc is larger and the variation of B th values for different B m values becomes smaller. The differences are mainly attributed to the I c -degradation caused by B dc .

Conclusion
For the first time, we have conducted an experimental and numerical study on dynamic resistance R dyn and instantaneous voltage response to triangular-waveform AC magnetic fields (peak B m ) with different DC offsets B dc . We have made the following new discoveries: (1) R dyn under high B dc shows a linear relationship to B m , which is different from the pure B ac case where the relationship is nonlinear. This is due to the fact that high-B dc nullifies the influence of B ac on the I c (B) characteristics. (2) The frequency-dependence of R dyn becomes weaker with increasing B dc compared with the pure AC field case. This is attributed to the reduced magnetization loss at high B dc , which weakens the thermal effect and hence the frequency-dependence of R dyn . (3) Strong field-dependent B th values are obtained from the dynamic voltage, which are different in the fieldincreasing and field-decreasing process due to the different I c (B dc + B ac ) values when B dc and B ac are additive and subtractive for each half cycle. (4) A dynamic voltage waveform under bias B dc has the same fundamental frequency as the applied B ac . This is different from the pure AC field case where the fundamental frequency doubles.
Our results provide references to relevant HTS applications. The dynamic resistance and voltage are expected to be characterized with a detailed study of the thermal effect in future work.

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