AC loss measurement and simulation in a REBCO coil assembly utilising low-loss magnetic flux diverters

The measurement set-up of the REBCO coil assembly and MPP MFDs are shown in figure 3. The coil assembly used in the measurement is made up of four DPCs from our earlier hybrid coil assembly [39]. The DPCs are wound with 4 mm wide SuperPower GdBCO SCS4050 M3 AP wires. The total length of wire is 33.5 m. Each DPC has $20 \times 2$ turns insulated by co-wound Nomex paper. The critical currents of the GdBCO tape and the coil assembly at 77 K are 89 and 51.6 A, respectively. Tables 1 and 2 show the specifications of the GdBCO tape and the coil assembly. The four REBCO double pancake coils were connected by soldering four 4 mm-wide Bi-2223 strips in parallel in order to minimise the conduct resistance. The resistance in the coil assembly is approximately 0.6 µΩ which was obtained by fitting the linear part of the V–I curve for the coil assembly.

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Table 1. Specifications of the GdBCO tapes.

Parameters	Value
Manufacturer	SuperPower
Tape width (mm)	4
Tape thickness (m${\text{m}}$)	0.1
Ic at 77 K, s.f. (A)	89
Substrate	Hastelloy
Insulation layer	Nomex

Table 2. Specifications of the DPCs assembly.

Parameters	Value
Height (mm)	38.3
Inner diameter (mm)	60
Outer diameter (mm)	73.5
Turns of every DPC	$20 \times 2$
Numbers of DPCs	4
Ic of DPCs assembly (A)	51.6
Inductance of coil assembly at 77 K (mH)	1.53
Total length of used tapes (m)	33.5

The Molypermalloy-power-125 (MPP-125) MFDs used in the measurement were fabricated from toroidal core supplied by Magnetics Inc. The MPP-125 is a compressed powder ferromagnetic material with nominal relative permeability μ_r of 125. The measured electric resistivity of the flux diverter at liquid nitrogen temperature (77 K) is approximately $5 \times {10^4}\ {\Omega } \cdot {\text{m}}$. The eddy current loss in such a high resistivity material at f < 100 Hz is negligible. The inner diameter and outer diameter of them are 48.8 and 78.4 mm, respectively. The top and bottom surfaces of the MFDs were machined flat to remove the rounded corners of the original toroidal cores. They have a cross-section of $14.8{\text{ mm}} \times 13{\text{ mm}}$. The specifications of the MFDs are shown in table 3.

An electrical method is used for AC loss measurement of the coil assembly with and without MFDs [30]. The measurement set-up is shown in figure 4 and the schematic drawing of the measurement circuit is shown in figure 5.

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Two lock-in amplifiers are used for the measurement: one for measuring the current amplitude and phase; the other for measuring the in-phase (with regard to the coil current phase) voltage of the coil assembly. A Rogowski coil (see figure 5) was used as the phase reference. The internal oscillator of one of the lock-in amplifiers supplies a sinusoidal signal to a CROWN K1 audio amplifier. The amplified sinusoidal voltage signal from CROWN K1 powers the primary winding of a toroidal transformer. The output of the secondary winding is used for energising the REBCO DPCs coil assembly immersed in liquid nitrogen at 1 atm. A capacitor bank was connected to the primary winding of the toroidal transformer for impedance matching.

The transport AC loss per cycle in a REBCO coil assembly with and without MFDs can be given as [30]:

$$\begin{equation}{Q_t} = \frac{{{I_{coil,{\text{ }}rms}} \cdot {V_{rms}}}}{f}{\text{ }}\end{equation} \tag{ 1 }$$

where ${I_{coil,{\text{ }}rms}}$ is the rms current in the REBCO coil assembly, ${V_{rms}}$ is the rms value of the in-phase voltage from the voltage taps attached in the ends of the coil assembly, and $f$ is the frequency of the coil current. It is worth noting that the measured data include AC loss both in the coil assembly and the MFDs, and the measurement cannot separate the loss contributions of the HTS coil assembly and the MFDs.

An axisymmetric 2D FEM model based on H-formulation was built to simulate the magnetic field distribution and transport AC loss in the REBCO coil assembly. The MFDs are positioned near the top and bottom parts of the coil assembly and the distance between the assembly and the MFDs, d, can be varied. In order to reduce the computing time, the homogenization method [40] and structured mesh [41] are used in the FEM models. The upper half cross-section in figure 6 shows the homogenized sub-blocks of the REBCO DPCs and the structured meshes. Each DPC was divided into 40 axial elements and 6 radial sub-blocks. Each MPP MFD is divided into 20 axial elements and 30 radial elements.

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The E-J power law [42] in equation (2) is used in the FEM model to simulate the electromagnetic behaviours of the REBCO assembly.

$$\begin{equation}{\boldsymbol{E}} = \frac{{{E_c} \cdot {\boldsymbol{J}}}}{{{J_c}(B)}}{\left( {\frac{J}{{{J_c}(B)}}} \right)^{\left( {n - 1} \right)}}{\rm{ }}\end{equation} \tag{ 2 }$$

where E_c = $1{\ \mu {\rm V}}\ {{\rm{cm}}^{ - 1}}$, ε and J are the electrical field and current density of the REBCO tapes, respectively. n is the power-law exponent of the E-J curve which is set as 25 in this model. The modified Kim model [36] in equation (3) is used to represent the J_c (B) dependence of the REBCO tapes.

$$\begin{equation}{J_c}\left( {\boldsymbol{B}} \right) = {J_c} \cdot {\left( {1 + \frac{{{{(k \cdot {B_\parallel })}^2} + {B_ \bot }^2}}{{{B_0}^2}}} \right)^{ - \alpha }}{\rm{ }}\end{equation} \tag{ 3 }$$

where k, α, and B₀ are the constants which are determined by critical current measurement under magnetic field. In this case, k = 0.43, α = 0.23 and β₀ = 37 mT. B_∥ is the magnetic field component parallel to the tape surface while B_⊥ is the magnetic field component perpendicular to the tape surface. In this numerical model, B_⊥ and B_∥ are B_r and B_z , respectively.

In this numerical model, the relative permeability ${\mu _r} = 1$ is used in the REBCO assembly and air domain. The ${\mu _r}$ of the flux diverters used in this model is from the B-H curve of 'Alloy Powder Core MPP 125' in COMSOL Multiphysics which is shown in figure 7. We experimentally confirmed the ${\mu _r}$ of MPP 125 at 77 K has the same value as at room temperature.

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The AC loss ${Q_{sc}}$ in the superconducting coil winding was calculated from equation (4).

$$\begin{equation}{Q_{sc}} = \mathop \int\!\!\!\int \limits_{{v_{SC}}}^{\rm{ }} {\boldsymbol{E}} \cdot {\boldsymbol{J}}{\rm{ }}dvdt{\rm{ }}\end{equation} \tag{ 4 }$$

where, ${v_{SC}}$ is the volume of the superconducting domain. The eddy current in stabiliser layers is not considered in this model.

The AC loss in MFDs consists of eddy current loss and hysteresis loss. The loss density of the MFDs was measured at 720 and 780 Hz using a conventional toroidal set-up method. A N4L PPA5530 precision power analyser was used for measurement. An MFD with the same dimension as the MFD used in the coil experiments was used. Figure 8 shows the measured Q-B relationship. The frequency-independent loss measurement results confirm hysteretic nature of the AC loss in the MPP MFDs. Eddy current loss can be ignored due to the very high resistivity in the material. AC loss in the MFDs, Q_MFD at different coil currents was calculated using the calculated distribution of peak flux density in the MFDs by interpolating the measured Q-B relationship shown in figure 8. It should be noted that the field configuration and eddy current geometry in the toroidal winding configuration described is not the same as that in the experimental setup with the HTS coils.

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The summation of Q_SC and Q_MFD gives the total loss in the REBCO coil assembly with MFDs.

The measured E-I curves of the REBCO coil assembly without MFDs and with MFDs at different d values are plotted together in figure 9. I_c values of the coil assembly before placing MFDs and after placing MFDs with d = 8 mm, 5 mm, and 2 mm at 77 K are compared. The measured coil I_c values are 51.6 A, 52.4 A, 53.4 A and 55.4 A, respectively. 1 µV cm⁻¹ was taken as the voltage criterion for the coil I_c values. The I_c value is the smallest without MFDs and increases with decreasing d values. This can be explained by the numerical results shown in the numerical results section.

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In figure 10, the measured AC loss values per cycle at three frequencies in the REBCO coil assembly without MFDs, and with MFDs with different d values are plotted as a function of the coil current amplitude. The measurement results in the coil assembly with and without the MFDs at different frequencies agree well from one another. The result indicates the hysteresis nature of the loss characteristic. It should be noted that the measured values are total AC loss in the REBCO coil assembly and the MFDs. There will be difference in the measured AC loss values for different frequencies if there is any loss contribution due to eddy current in the MPP MFDs. There is some scatter in the measured loss values for d = 2 mm. We attribute the result to a lower S/N (signal to noise ratio) than other gaps due to the smaller loss voltage levels.

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Figure 11 compares the induced voltage from a single-turn pick-up coil attached on the surface of the upper half of the top DPC in the coil assembly without MFDs and with MFDs (d = 2 mm) at I_{t , rms} = 10, 15, and 20 A. The rectangular pick-up coil has a dimension of $50{\text{ mm}} \times 10{\text{ mm}}$. The voltage in the pick-up coil is induced due to the time-varying radial magnetic field component and the value of the voltage is proportional to the amplitude of the radial component. The induced voltage for the coil assembly after adding MFDs is significantly reduced compared to that without MFD. At 20 A (rms), the voltage from the assembly with MFDs is approximately 1/3 of that without MFDs. In REBCO coil assemblies, most of AC loss is caused by the radial field (which is perpendicular to the REBCO tape surface) in the end windings [31, 32]. The results in figure 11 directly show the effectiveness of MFDs for supressing the radial field component and hence reducing the AC loss in the coil assembly.

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In figure 12, the measured AC loss results per cycle in the coil assembly without MFDs, with MFDs at three d values are plotted as a function of the coil current amplitude at three frequencies. It is apparent that AC loss in the coil assembly is reduced by the MFDs and it decreases with decreasing d values at all three frequencies. At I_{t , peak} = 30 A (around 60% of the coil I_c ) and f = 13.96 Hz, the AC loss value in the REBCO coil assembly without MFDs is 0.074 J/cycle. After placing MFDs with d = 2 mm, the total AC loss in the coil assembly and MFDs was reduced to 0.015 J cycle⁻¹, i.e. the total AC loss was reduced by 80%. Such a significant AC loss reduction in REBCO coil windings using MFDs has not been reported in previous experimental works. The measurements at the other two frequencies show almost the same results. This unprecedented AC loss reduction may be attributed to the utilisation of low-loss MPP-125 MFDs. Other types of MFDs ferromagnetic materials such as Ni alloy could generate substantial loss in the material themselves although they could reduce AC loss in the REBCO coil windings. However, MPP-125 MFDs have negligible AC loss (as discussed in the following section) compared with the AC loss in the REBCO coil winding. One drawback of the MPP-125 MFD is that it has relatively low saturation field (0.8 T). However, MFDs do not necessarily form part of the main magnetic circuit in actual applications and hence could avoid large magnetic fields. E.g. in HTS transformers, the main circuit is formed through the iron core which couples with HTS coil windings but the MFDs are arranged near the top and bottom of the windings and hence experience a weaker magnetic field [33]. Therefore, a saturation field of 0.8 T might be sufficient for most situations.

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In figure 13, the AC loss reduction rate of the coil assembly with MFDs with three d values at different current amplitudes are compared. Compared with the AC loss in the REBCO coil assembly without MFDs, the average AC loss reduction with MFDs at d = 8, 5, and 2 mm are 36%, 51%, and 83%, respectively. The AC loss reduction rate increases with decreasing d value. The result suggests that MFDs should be placed as close as possible to the HTS winding in order to achieve larger loss reduction ratio in the HTS coil winding.

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Figures 14(a) and (b) show magnetic field lines and the radial magnetic flux density distribution in the upper half cross-section of the coil winding with and without MFDs and at f= 26.62 Hz, I_{t , peak} = 30 A, and d= 2 mm (for the case with MFDs). The flux lines in the top DPC in the coil winding with MFDs become more parallel to the REBCO tape surface (figure 14(b)) compared with the result without MFDs (figure 14(a)). The area filled with large radial field component and the amplitude of the radial field component in the REBCO DPCs for the case without MFDs are much larger than the result with MFDs. The maximum radial field component in the top DPC without MFDs is 0.1 T while the maximum radial field in the top DPC is reduced to 0.05 T with MFDs. Apparently, the difference observed in the above comes from the presence of the flux diverters. On the other hand, the MFDs have large area with high magnetic field concentration as shown in figure 14(b). The results shown in figure 14 clearly explain why the I_c values of the coil assemblies with MFDs shown in figure 9 are larger than that without MFDs.

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In figure 15, the current density distribution in the REBCO DPCs with and without MFDs are compared. Shielding currents induced by the radial magnetic field components are observed in most of the pancake coils except the innermost pancake coil. However, with MFDs, the high current density area (|J/J_c | > 1) in the EWs is reduced remarkably compared with the case without MFDs. In figure 15(b), with MFDs, as most AC loss is generated in the area of |J/J_c | ≥ 1, the reduced area of high |J/J_c | indicates that AC loss reduction is achieved in the EWs. These results mentioned above demonstrate why AC loss in the REBCO coil winding is reduced by the use of MFDs.

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Figure 16 compares the experimentally and numerically obtained total AC loss results in the REBCO coil winding without MFDs and with MFDs with d = 2 mm at 26.62 Hz. A good agreement was obtained between the numerical and experimental results.

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Figure 17 compares the simulated loss components in the REBCO coil winding with MFDs (d = 2 and 8 mm) at 26.62 Hz. It is apparent that AC loss in the MFDs is more than three orders of magnitude smaller than the AC loss in the HTS winding at I_{t , peak} = 45 A and the AC loss in the flux diverters can be ignored. Therefore, total AC loss in the coil assembly with the MFDs is dominated by the AC loss in the HTS windings. Nonetheless the increase of AC loss in the MFDs with decreasing d can be observed due to the increase of the magnetic field inside the MFDs. At both d values, AC loss in the EWs is much greater than that in the CW, and hence AC loss in the EWs dominates AC loss in the HTS winding. Interestingly, with d value varying from 8 to 2 mm, not only AC loss in the EWs becomes smaller but so does that in the CW. However, the change in the AC loss in EWs is much greater than that in the CW. The result at 2 mm indicates that there remains capacity to reduce total AC loss by making the d value smaller, bringing the loss-level of the EWs closer to the loss-level of the CW. However, the d cannot be reduced to 0 mm considering the insulation separation between the MFDs and HTS coil windings required for practical applications.

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As evident from the results in figure 15(b), there is still room for further reducing the AC loss in the double pancake coils next to the end windings. With the shape of the flux diverters described in this work, there is an obvious limit for further reducing the distance between the DPCs and the flux diverters and hence further reduce AC loss for double pancake coils next to the end windings. The limit becomes more obvious for more complicated coil windings comprising stacks of double pancake coils, because influence of the flux diverters can hardly reach DP2 and DP3 (see figure 18(a)) which can also be source of non-negligible AC loss. For coil windings stacked with large number of double pancake coils, combination of flux diverters for the end windings and some of double pancake coils nearby the end windings might be effective for minimising AC loss as shown in figure 18(b). Same strategy of AC loss reduction utilising flux diverters might be applied to hybrid coil windings comprising REBCO double pancake coils as end windings and Bi-2223 pancake coils as centre windings [6]. For 3-phase transformers, AC loss is more concentrated in the very end part of the high voltage and low voltage windings than stand-alone coil windings [32]. Therefore, flux diverters placed near the end windings might suffice for transformer applications.

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