Effect of a DC transport current on the AC loss in no-insulation ReBCO racetrack coils exposed to AC parallel magnetic field at 77 K and 4.2 K

ReBCO coils are developed as DC field coils in linear motor systems to increase the force density, in favor of permanent magnets. Such coils have to sustain a relatively large heat load stemming from the AC magnetic field environment in which they operate. The use of no or partial turn-to-turn insulation can make them more stable against the effects of local heating. Conversely, the radial electrical connections in no-insulation (NI) coils allow for large coupling currents, causing additional AC loss on top of the already significant heat load. Here we report on the AC loss in sub-scale NI, 4 mm wide single-tape, ReBCO racetrack coils exposed to parallel-to-the-tape magnetic field in the frequency range of 10−4 to 1 Hz at 77 K and 4.2 K, while carrying a DC transport current. AC loss is measured magnetically and electrically. The main goal of these experiments is to validate our 2D numerical model, which provides more insight into the origin of the AC loss. At low frequencies, inter-turn coupling currents are spread more or less homogeneously throughout the winding pack. Whereas at high frequencies, the skin effect causes shielding of the interior of the coil and large induced currents only occupy the coil’s outer surface.


Introduction
In recent years, high-temperature superconducting ReBCO tapes have been introduced in industrial applications, such as naval propulsion motors [1] and a wind turbine [2]. The * Author 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. main driver behind these developments is that ReBCO coils are able to produce higher magnetic fields than conventional permanent magnets. In actuators, this enables a higher force density, making them interesting for application in e.g. photolithography machines [3], where a reticle has to be accelerated in rapid repetition [4]. In this particular application, ReBCO stator coils experience a time-varying, non-sinusoidal magnetic field from the moving armature, which has an orderof-magnitude amplitude and frequency of 100 mT and 10 Hz, with higher-order harmonics. This produces a potentially high heat load in the form of AC loss in the superconducting stator coils, which increases further by the presence of a DC transport current [5]. Photolithography machines need to operate reliably for many years with minimal downtime, so both the risk and the consequences of coil quenching must be minimized.
One possible way to mitigate the impact of local heating is to use no-insulation (NI) coils. It is well known that NI coils are less susceptible to local defects in ReBCO tapes and less likely to suffer irreparable damage in the case a quench does occur, because they allow current to bypass a local normal zone or hot spot through the turn-to-turn contact resistance R ct , thus preventing a burn-out. A downside of using such coils in an AC magnetic field environment is the potential of additional loss, specifically coupling loss. In this paper, we therefore investigate the AC loss in sub-scale NI ReBCO racetrack coils to assess their applicability as stator coils in a high-dynamic superconducting linear motor (HSLM) system.
Ever since their conception [6], a lot of research has been invested in NI coils. Extensive efforts have for instance been made to control R ct by co-winding various materials [7], by laminating the conductor surface [8] or by impregnating with solder [9].
Less has been published on AC loss in such coils. Kim et al [10] measured the AC loss due to an AC current in NI pancake coils with various co-wound tapes. They found the lowest AC loss in coils with the lowest R ct value because part of the current bypasses the turns radially. AC loss reduction by this same mechanism is also observed by Zhang et al [11].
AC loss in insulated and NI coils exposed to external ripple fields is reported on by Wang et al [12]. They found that NI coils have much higher AC loss, stemming from induced currents generating Joule heating in the turn-to-turn resistance. Numerical calculations show that the skin effect causes most induced current to flow in the outer turns while shielding the interior.
Li et al [13] demonstrated the characteristics of coupling loss in partially coupled stacks of ReBCO tapes. Their 2D numerical model makes a distinction between loss occurring in the soldered contacts and in the superconductor itself. The latter tends to increase significantly at higher frequencies, which has been demonstrated experimentally as well.
The current distribution inside NI ReBCO pancake coils in a changing magnetic field applied parallel to the tapes' surface has been studied by Zhong et al [14]. They report on induced screening currents in the superconductor and remnant magnetic field, using a similar experimental set-up and winding pack homogenization method to this work.
Here we present experimental data on the AC loss in single 4 mm tape-based NI ReBCO racetrack coils carrying a DC transport current in an external AC magnetic field, applied parallel to the wide face of the tape. The experiments cover transport currents from 0 to 600 A and applied magnetic field frequencies from 0.5 mHz to 1.25 Hz. Measurements are performed both at 77 K and 4.2 K. The DC current/AC field scenario is representative of what the DC stator coils in linear motors experience, except for the frequency, which can be much higher in real motor applications. This work expands on the results published earlier in [15,16]. We use a 2D numerical model to explain the findings and demonstrate that this computation method can be used very well as a design tool to compute AC loss in motor drives using ReBCO coils. Section 2 introduces the different loss mechanisms that contribute in the experiment by evaluating the numerical model in three distinct frequency domains. The effect of a DC transport current on the AC loss is also discussed. Section 3 describes the dedicated sample coils that were made for this experiment as well as the experimental set-up and instrumentation. Section 4 presents the results of the AC loss experiments, considering both magnetization-and transport current losses as a function of applied field frequency and compares them to the model predictions.

AC loss components
Consider a racetrack coil placed in an external time-varying magnetic field along the z-axis, as shown schematically in figure 1.
A current will be induced that tends to counteract the flux change produced by the external field, this current is referred to here as the 'magnetization current'. The loss component generated by this current is called 'magnetization loss' Q mag . When the applied magnetic field is uniform, the magnetization current will flow along the winding direction of the coil, mostly following the superconducting path and only crossing in-between turns to close the current loop. The ohmic dissipation caused by the crossing-over of the magnetization current is called the 'coupling loss' Q coupling . When the magnetization current approaches the critical current density of the tape J c , loss is also generated in the superconductor according to the usual E(J) relation, referred to as the 'superconductor loss' Q superconductor . The induced magnetization current density J mag can reach this level at high frequencies, where the skin effect concentrates current in a shell at the outer surface. The magnitude of J mag decays inward over a characteristic length scale, the skin depth δ = √ ρ ⊥ sin 2 (α)/πµ 0 f [15]. Here, ρ ⊥ is the effective transverse resistivity of the coil, f is the frequency of the applied magnetic field and α = d turn /ℓ turn is the winding angle, i.e. the angle between the path of the conductor and the axis of the straight leg of the racetrack. The sum of the coupling loss and superconductor loss forms the magnetization loss.
Running a DC transport current I dc through the superconducting coil while it is exposed to the AC magnetic field can create additional 'transport current loss' Q trans . The magnetization current displaces the transport current inside the superconductor, requiring the power supply to do extra work to maintain I dc at a constant level [5]. Q trans is further enhanced when part of I dc flows through a resistive medium. This occurs when the current starts to bypass turns as it approaches the critical current I c (which is typical for NI coils) and when magnetization currents occupy the full width of the superconducting tape. The total AC loss Q tot in this system is then given by: (2)

Numerical model
We have developed a 2D numerical model to predict AC loss generated in ReBCO racetrack coils. A detailed description of this computational method can be found in [15]. In brief, it calculates the current distribution in a 2D cross-section of a ReBCO racetrack coil by solving an integral form of Maxwell's equations [17,18]. The winding pack with all its components is approximated as a homogeneous material with anisotropic resistivity, with the effective transverse resistivity ρ ⊥ perpendicular to the turns and the power law ρ ∝ (J/J c ) n−1 along the path of the conductor. In the model result of figures 2 and 3, we assume a uniform J c , thus neglecting the I c -variance between turns because the main focus of these results is to demonstrate the effect of the coil's time constant on the loss characteristics. A J c (B)-characteristic can be included in the model, but this is only done for the measurement data at 4.2 K, since we measured the magnetic field dependency of the tape only at this temperature. A similar homogenization method is used in [19]. In the remainder of this section, we briefly review the various loss components predicted by the model. The character of the AC loss depends strongly on the ratio of the frequency of the applied magnetic field f and the characteristic frequency f c of the sample coil, with f c = 1/2πτ = R c /2π L. Here L is the usual self-inductance and R c the characteristic resistance often used to describe charge-or discharge experiments [20].
Consider an NI ReBCO racetrack coil in a sinusoidal, homogeneous magnetic field of amplitude B a along the z-axis. The racetrack coil carries a DC current and produces a DC magnetic field in the positive z-direction. Magnetization currents oppose the flux change due to the external AC magnetic field, so when dB/dt < 0 they run parallel to I dc and increase the parallel net current density J ∥ . As an example, consider a coil with a characteristic frequency of 2.4 mHz, effective transverse resistivity of 0.82 µΩm and a homogeneous and constant I c value of 72 A and an n-value of 30. One tape is 4 mm wide and 125 µm thick, so the effective J c of the homogenized winding pack becomes 144 A mm −2 . The coil dimensions are set equal to that of the sample coils discussed in section 3 (see figure 1).
We define three distinct frequency ranges where different loss mechanics contribute: f ≪ f c , f ≳ f c and f ≫ f c . In the model, the total AC loss density in W/m 3 is calculated as p tot = E · J, which we separate into p ∥ = E ∥ J ∥ and p ⊥ = E ⊥ J ⊥ , the losses caused by currents flowing parallel and perpendicular to the superconductor, respectively. This distinction will shed light on how AC loss is generated in our coils. Figures 2  and 3 show the evolution of these losses during one magnetic field cycle, with the phase θ = 2π corresponding to one full period of the AC magnetic field. The three distinct frequency regimes are shown separately, each time for two DC current levels (0% and 70% of I c , respectively). Since they are qualitatively different, we describe each frequency regime separately. Corresponding typical current distributions are plotted in figure 4.

f ≪ fc (0.1 mHz).
Here, induced currents can decay faster than the magnetic field oscillates, so they distribute homogeneously throughout the winding pack. As a result, the whole coil experiences the same applied magnetic field and AC loss is spread out evenly over the winding pack. When I dc = 0 A (figure 2(b)), the only power dissipated is due to the coupling currents flowing perpendicular to the superconductor. P ∥ is negligible, because magnetization currents lie far below I c . Adding a transport current (figure 3(b)) produces a flux flow resistance according to the usual E(J) power law because of the finite n-value and thus a DC component is added to P ∥ . The contribution of dynamic resistance, see also the measured and modeled Q trans data in figure 8, is still relatively small here but becomes larger and detectable at around f c . Furthermore, rather than the familiar instantaneous loss variation with 2f, characteristic of 'straightforward' ohmic losses, P ∥ oscillates with the same frequency f as the external AC field. Indeed, I dc breaks the symmetry causing the net current density J ∥ to be larger at θ = π , when the transport-and magnetisation currents flow in the same direction, than at θ = 2π , when they oppose each other. In this low-frequency limit, J ∥ is homogeneous throughout the winding pack, see figure 4.

f ≳ fc (10 mHz).
In this regime, the skin depth δ is smaller than the coil dimensions. Currents induced by the applied magnetic field are therefore largest at the outer surface of the coil and decay inward, see figure 4. As a result, the inner part of the coil becomes partially shielded from the magnetic Modeled AC loss during a single magnetic field cycle, separated into losses caused by currents running parallel and perpendicular to the superconductor. The uniform applied magnetic field is shown in (a). The loss due to parallel and perpendicular currents, as well as the sum of the two, are shown in (b)-(d) for field frequencies of 0.1 mHz, 10 mHz and 1 Hz, respectively. Ic = 72 A and I dc = 0 A. field. Because of this shielding effect, dissipation is concentrated at the outer surface of the coil. The loss is somewhat phase-shifted with respect to B z . However, like in the low-f regime, the bulk of the dissipation is still caused by coupling currents, see figure 2(c). Adding a transport current causes a large peak in P ∥ just before θ = 3π/2 (figure 3(c)). Here, magnetization currents align with I dc and push the net current in the superconductor close to I c . When the magnetization current polarity is opposite (around θ = π/2), minimal change in AC loss with respect to the 0 A result is observed because J ∥ is reduced and thus does not contribute significantly to P ∥ at this point in the field cycle.

f ≫ fc (1 Hz).
At sufficiently high frequencies the skin depth δ becomes so small that the magnetization current density approaches J c in the skin even in the absence of a transport current. This causes a strong increase in P ∥ at the outer surface by fully magnetizing the superconductor (figure 2(d)).  Computed current density (normalized with Jc) flowing parallel to the superconductor in the NI winding pack that is subjected to a 12 mT parallel magnetic field of selected frequencies.
The coil carries a transport current of I dc /Ic = 0.7, indicated in blue in this figure. Only the right leg of the racetrack coil from figure 1 is shown. The current density in this figure is a snapshot when the applied field is 12 mT, i.e. θ = π/2. The magnetization current is such that an opposing field (in the negative z-direction) is produced. At f = 1 Hz, the magnetization currents are concentrated at the outer surface, completely reversing the current in the outer nodes.
The inside of the coil is nearly entirely shielded from the external magnetic field, so dissipation is heavily concentrated at the outer surface. The shielding distorts the magnetic field and a relatively large field is observed at the outer surface. Adding a transport current increases P ∥ in the second half of the magnetic field cycle ( figure 3(d)), similar to what happens at 10 mHz. A second interesting effect is observed around θ = π/2, where the magnetization current opposes I dc and forces it to flow perpendicular to the turns. This results in a large peak in P ⊥ around θ = π/2. Figure 4 also clearly shows this: a current opposing I dc runs along the outer surface of the coil.
To validate this model, an experimental campaign was set up on small NI ReBCO racetrack coils exposed to a uniform AC magnetic field. By covering a wide frequency range, all the loss predictions discussed in this section are verified. In the next section, the experimental set-up, the measurement procedure and the fabricated sample coils are presented.

No-insulation racetrack coil
Two identical 18-turn NI ReBCO racetrack coils are fabricated with single 4 mm wide, InSn solder-coated, tape from SuperOx with a self-field I c of 123 A at 77 K (figure 5). In the set-up, two NI racetrack coils are connected in anti-series to minimize the net torque on the background magnet. For details on the experimental set-up, see [16]. After winding, the turns are electrically connected by heating the whole coil to 150 • C on a hot plate, while adding additional InSn solder from above. This essentially shorts the turns and allows the transport of current from one turn to the next with minimal dissipation. The coil shape is maintained by clamping the winding pack from both sides. Such a solder-filled coil was first introduced by Li et al [9]. The racetracks are characterized for critical current, generated magnetic field and time constant. The full results are reported in [15] and summarized in table 1. A single coil time constant τ is determined by fitting an exponential decay to a sudden discharge test. This models the coil as a straightforward LR-circuit, although some variance in the decay rate is observed during discharging. We incorporate this as an uncertainty in the τ -value. The ∼25% difference in R ct value between the two coils is likely due to slight variations in the soldering procedure. The self-inductance L of both coils is calculated to be 72 µH. Table 1. Measured characteristics of the two racetrack coils. The time constant τ is determined with a sudden discharge test and the contact resistance Rct is calculated from it. The coil Ic is determined measuring the voltage over the coil, assuming an electric field criterion of 100 µVm −1 . Note that this critical current value is about 20% higher than the one used in the numerical model discussed in section 2 due to current bypassing through the soldered turns leading to a more gradual transition. The n-value could not be accurately determined for this same reason. For the model we assume a tape n-value of 30.

Experimental conditions
The AC loss measurements are performed with a uniform, sinusoidally time-varying magnetic field applied in the zdirection, i.e. parallel to the wide face of the superconducting tape. This geometry is schematically shown in figure 1. A first set of measurements is performed in liquid nitrogen at 77 K, followed by a second set in liquid helium at 4.2 K. The explored parameter space is given in table 2. Going to lower temperatures enables a higher applied field amplitude from a NbTi dipole magnet and a larger transport current in the racetrack coils. However, measurements at 4.2 K span a more narrow frequency range, due to voltage limitations of the power supply. At 4.2 K, the used InSn solder in-between the turns is superconducting [21]. To prevent this from interfering with the measurement, a DC offset field B dc of 0.5 T is superimposed on the AC magnetic field, driving the solder in the normal state.

Measurement systems
The AC loss measurements are performed in a set-up capable of measuring magnetization loss Q mag with pick-up coils as well as transport current loss Q trans with voltage taps. Further details on this set-up can be found in [16]. Q mag is calculated by integrating the product of the z-component of the magnetization of the racetrack coils m z and the applied magnetic field B z over a complete field cycle: The relation between the net voltage over the pick-up circuit V pu and m z is derived analytically from the relative positions of the racetrack and pick-up coils: where the geometrical factor K is: This expression is a sum over all pick-up coil wires at positions (y i , z i ) for a leg of the racetrack coil at (y s , z s ). The derivation of equation (5) is described in [15].
The net voltage over the pick-up circuit V pu , from which Q mag is calculated, is measured with two devices. At frequencies of 20 mHz and above, it is recorded with a lock-in amplifier. Using as reference the voltage driving the power supply of the AC dipole magnet. The phase delay between this function generator signal and B z is corrected for with a second lockin amplifier, which records the dipole current with respect to the same reference signal. This phase delay can be significant, approximately 10 • at 1 Hz, and must therefore be taken into account when calculating the AC loss from V pu . The settling time of several field cycles required by the lock-in amplifier makes it impractical for frequencies below ∼20 mHz. Therefore, V pu , as well as B z , are recorded with a nanovolt meter at these frequencies.
A pair of voltage taps on the current terminals of a single coil is used to measure V coil . The voltage is recorded by a Keithley 2000 multimeter. Only a single coil is probed in order to limit the background offset stemming from the joint resistance, since the voltage pair also includes the resistance of the copper terminals and the solder joint between the coil and the terminals. The stationary transport current I dc is measured with a zero-flux transducer. Q trans is then calculated as: Integrating over a full field cycle removes any inductive component. Due to the joint resistances, an offset loss value is measured at zero magnetic field, which is subtracted from subsequent measurements.  Figures 6 and 7 show the magnetization loss in J/cycle as a function of frequency f for several DC transport currents at 77 K and 4.2 K, respectively. In both figures, the AC loss data generally follow the model predictions quite satisfactorily. When f ≪ f c , Q mag scales with f, which indicates that the coupling currents occupy the full volume of the winding pack as predicted in section 2. When f ≫ f c , Q mag scales with 1/ √ f because of the skin effect. The transition between the two regimes, indicated by a peak in Q mag , occurs at the characteristic frequency f c = 1/2πτ (2.5 mHz at 77 K and 1.2 mHz at 4.2 K). Note that the corresponding τ -values (64 s and 133 s, respectively) correspond well to the average of the values found in the independent discharge experiments, see table 1.

Magnetization loss
Adding a DC transport current to the racetrack coil exposed to the AC parallel magnetic field increases the AC loss when f ≫ f c , as can be seen in both figures 6 and 7. The magnitude of the increase in AC loss scales with I dc and starts at lower f for larger I dc , since the superconductor is saturated at a progressively larger skin depth. A more detailed explanation of the acting loss mechanisms is given in section 2.
The 60 A data in figure 6 are noticeably shifted to higher frequencies, indicating a decrease in τ . This is likely caused by the presence of flux flow resistance R sc in the superconductor when the transport current is close to I c . This elevated resistance results in a smaller time constant: τ = L/ (R c + R sc ).
Measured data are compared to numerical simulation results, presented as solid lines in figures 6 and 7. A constant I c value of 72 A at 77 K is assumed, but an experimentally determined J c (B) relation is used at 4.2 K. An n-value of 30 is assumed for both. An additional model solution for I c = ∞ is shown as a dashed line to demonstrate the effect the critical current value has on the AC loss. The effective transverse resistivities ρ ⊥ of the modeled coils are 0.82 µΩm and 0.40 µΩm at 77 K and 4.2 K, respectively. These values are determined from the average time constant observed in the discharge experiments, see also table 1. A qualitative agreement between experimental and numerical data is observed: Q mag increases at low f, reaching a peak around f c and decreasing at higher f. The effect of transport current on Q mag is also reproduced well by the model. It increases the AC loss at high frequencies, more so and starting from lower f, if I dc is larger. The change in time constant observed in figure 6 is also reproduced, which means that the approximation of constant I c is reasonable. In both figures, the measured magnetization loss consistently lies approximately 20% below the calculated values. This systematic difference is likely caused by uncertainties in the positioning of the pick-up coils and the corresponding calibration factor (equation (5)).
Model and experiment deviate substantially for f ≳ 0.1 Hz at 77 K. The model predicts the loss to saturate at a constant value, caused by hysteresis losses in the fully penetrated superconductor. In this range, the experiment does not confirm the prediction. Instead, Q mag continues to scale with 1/ √ f. Since the data behave as was predicted for a coil with infinite J c , it was initially thought that the current might bypass the superconductor to flow through the copper stabilizer instead. However, including such a parallel copper path to the superconductor also in the model did not have a noticeable effect on the AC loss. Interestingly, the onset of a plateau is observed experimentally at 4.2 K and at f = 0.2 Hz, across all the levels of transport current, see figure 7. Why this effect occurs at 4.2 K and not at 77 K is unclear at this moment and subject to further investigation. Frequencies higher than 0.2 Hz could not be studied for this field amplitude, because of limitations of the experimental set-up.

Transport current loss
Q trans is supplied by the current source keeping I dc constant under the effect of the alternating magnetic field. This is observed as a net DC offset voltage over the coils, as shown in figure 8. Only data at 77 K are presented here because the background offset voltage due to the joints was too large at 4.2 K. Q trans starts to become significant at a given frequency where the sum of I dc and the magnetization current approaches I c , soon after which it reaches a maximum. Further increasing f causes the loss per cycle to decrease, due to increased shielding of the interior of the winding pack.
The experimental data in figure 8 are measured only in coil 1 to limit the background offset loss from the joints. They are compared to numerical model results obtained for a single coil with input parameters I c = 72 ± 4 A, n = 30 ± 5 and τ = 58 s. The shaded area around the modeled curves show the sensitivity of the model to this variation in superconducting properties, which is particularly large at the lower end of the considered frequency range, before the peak in Q trans . The modeled 50 A data are more susceptible to these variations because the influence of the uncertainty in the n-value becomes more pronounced as one moves further away from I c .
At low frequencies and when I dc ≪ I c , also the measured Q trans data suffer from a relatively large uncertainty due to the importance of the DC background voltage compared to Figure 9. Frequency dependence of the measured transport and magnetization loss in a single NI racetrack coil at 77 K in a 12 mT amplitude AC magnetic field applied parallel to the tapes' surface, with a DC transport current level of 60 A. The discrete symbols connected by dotted lines are measured data and the solid lines are the numerical results. Data (measured and modeled) are separated into magnetization and transport current loss. The total loss is the sum of these components. the low AC loss, which makes it difficult to determine Q trans accurately. Only when I dc is close to I c does the uncertainty on Q trans become insignificant. Around f = 1 Hz, we systematically measure high Q trans values than the model predicts. This may be explained by the model assumption of a constant J c value, rather than a J c (B) relation. The magnetic field dependence of J c was not measured for this conductor at 77 K. Figure 9 compares the relative contributions of the different loss mechanisms in a single coil when testing at 77 K with a transport current of 60 A. Following equation (2), the total AC loss Q tot is given by the sum of Q mag and Q trans . In the experiment, Q mag and Q trans are measured separately. In principle, Q tot may be measured calorimetrically [16], but in this case the loss is too small to detect with our calorimeter at 77 K. Instead, simply the sum of Q mag and Q trans is plotted.

Loss components
At different f, different loss mechanisms dominate. Given that the coil is DC-excited at roughly 80% of I c , Q trans is much larger than Q mag for f < f c . For smaller I dc values, Q trans is typically small compared to Q mag . Assuming that the transport current is homogeneously displaced by the magnetization currents, the total current stays well below I c everywhere. When f > f c , the induced current is more concentrated near the outer surface of the coil and the loss per cycle decreases. Note that Q mag decreases less steeply than Q trans . This can be qualitatively understood as due to the induced currents that shield most of the interior of the winding pack, allowing I dc to flow undisturbed throughout most of the coil.

Effect of magnetic field amplitude
The measurements at 4.2 K are performed at three magnetic field amplitudes: 12, 100 and 200 mT. Coupling loss is known to vary as B 2 a [22], so to compare the data, they are normalized by this factor. The normalized AC loss is plotted in figure 10, along with the corresponding model predictions.
For frequencies up to ∼10 mHz the data measured at the three amplitudes virtually coincide, i.e. the B 2 a scaling holds. Because the coil behaves linearly when E ∥ ≈ 0, i.e. when J ∥ < J c , the time constant remains the same and the three lines all show a peak at the same value of f c . When J ∥ approaches J c , a shift in f c is observed, such as in figure 6. However, deviations become apparent at the higher frequencies. This behavior is also predicted by the model. The deviations are caused by magnetization currents approaching the critical current I c . This occurs at a lower frequency when the applied field is larger since the induced current is larger. Unfortunately, the set-up did not allow to probe even higher magnetic fields or frequencies.

Conclusion
Turn-to-turn coupling loss in no-insulation single-tape ReBCO racetrack coils has been studied both numerically and experimentally in a uniform, time-varying magnetic field, over a range of magnetic field frequencies on either side of the coils' characteristic frequency, at 77 K and 4.2 K, and in combination with a stationary transport current up to close to the critical current. Two dedicated solder-filled coils were fabricated to achieve sufficiently long time constants in the order of minutes.
It is observed that the coil's time constant and associated characteristic frequency f c , determine the main characteristics of the coupling loss in these NI coils. Applying an alternating magnetic field with a frequency below f c induces coupling currents throughout the winding pack and the loss, expressed in J/cycle scales linearly with frequency. Above f c , due to the skin effect, induced currents can no longer penetrate the full depth of the winding pack, but rather flow along the outer edge.
Combining these magnetization currents with a DC transport current increases the AC loss especially for frequencies above f c . As the skin depth is reduced, the induced magnetization current density increases. At a given point the sum of the shielding and DC transport currents approaches the critical current I c , increasing the resistance in the current path and thereby increasing the loss. Higher transport currents cause this effect to occur at lower frequencies and to become more pronounced. Under these conditions also the transport current loss becomes noticeable. It arises at a frequency where the combined magnetization current and transport current exceed the critical current locally and is usually small relative to the magnetization loss, unless the transport current is close to the critical current I c , approximately at 80%.
The experimental results confirm the predictions of our 2D numerical model. Model and experiments typically agree within 20%, which is quite good considering the data are not fitted to the model after measuring. Instead, the model predicts the losses a priori, using as input the coil geometry and its independently measured I c value, n-factor and decay time constant. The model consistently overestimates the magnetization loss somewhat, which is likely due to the uncertainty in the placement of the pick-up coils. The transport current loss is also reproduced reasonably well by the model. However, it is predicted that the AC loss will reach a plateau at f ≫ f c , but this was not observed experimentally. We can use the numerical model to gain insight into current and loss distributions inside such coils when they are used in an AC magnetic field environment. It can also be used as a design tool for linear superconducting motor drives to determine optimal coil parameters, such as geometry and operating current.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.