Roebel cables from REBCO coated conductors: a one-century-old concept for the superconductivity of the future

The first HTS Roebel cable was demonstrated by the Siemens Corporate Technology group using BSCCO(2223) tapes as strands [12]. This cable consisted of 13 strands and showed a self-field reduced dc transport current of about 400 A at 77 K. The quite long transposition length of typically 3 m in such cables was caused by the limited in-plane bending ability of the material. The cable achieved the expected reduced ac losses in comparison with stacked tapes and served as a motivation for the application in HTS transformers [13]. The idea of a Roebel structure with REBCO CC was proposed by Martin Wilson in 1997 [14] at the CEC/ICMC conference, but a way to manage the bending, the strand shape forming and the assembling issue was still unknown. The solution was found a few years later thanks to the progress in homogeneity and robustness of CCs and to their commercial availability. The meander-like shaping of the CC as shown in the CAD model of figure 3 and presented by Wilfried Goldacker in 2005 [15] was first realized by means of a precision punching technique at Forschungszentrum Karlsruhe (now Karlsruhe Institute of Technology, (KIT)) and published in 2006 [16]. The first cable was made from 16 punched strands of DyBCO CCs (manufactured by the THEVA company) and carried a self-field transport current of 500 A at 77 K, about half of the sum of the critical currents of the individual tapes. Unfortunately the cable had no copper stabilization and burned through due to the lack of quench protection. The second presented Roebel cable from 12 mm-wide Cu-stabilized SuperPower MOCVD CC tape achieved a dc transport current of 1.02 kA at 77 K in self-field. The 0.36 m long sample (two transposition lengths) already showed the reliability of the strand punching process and excellent homogeneity, thanks to the use of CC from SuperPower [17]. Small resistive contributions in the voltage–current (V–I) characteristics above I_c /2 indicated current redistributions between the strands along the cable. The decrease of the measured transport current by 60% compared to the sum of the critical currents of the individual strands was ascribed to the generated self-field, whose pattern was calculated by means of a numerical method based on a Biot–Savart approach.

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A second group at Industrial Research Limited (IRL) adopted the topic of CC Roebel cables as main subject of investigation, immediately focusing on the industrial production of such cables with machines specifically designed to produce long lengths in an automated fabrication process [18].

Roebel cables from CCs offer a large variety of choices to increase the current, to optimize the cable design and to increase the thermal and mechanical stability by modifying the structure of the composing strands. The potential for further increased transport currents was shown by cabling three-fold stacks of strands into a 45-strand cable of 1.1 m length and 18.8 cm transposition length carrying 2.6 kA (77 K, self-field) [19]. Increasing the transposition length to be able to add more strands is another possibility to increase the current performance in a simple way.

The Roebel cable is made from meander-shaped CC tapes. For the Roebel cable design we follow the nomenclature introduced by IRL [20]. The fundamental geometrical parameters are the original tape width W_T , the transposition length L_TRANS (which is a full period of the meander), the strand width W_R , the strand-edge clearance W_C (clearance in the cable centre), the cross-over width W_X , the inter-strand gap L_ISG , the cross-over angle ϕ, the cut-off fillet radius R, and the inner radius R_i . The parameters are shown in the schematic sketch of figure 4.

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The transposition length is a suitable parameter to determine the number of applicable strands and needs for a specific application. The inter-strand gap is strongly correlated to the assembling procedure, being narrower for hand-made cables (KIT) and wider in the case of fully automatic process (IRL). Numerical calculations on the geometry with the best mechanical performance and current capacity [22] indicate an optimal inner radius R_i = 10 mm. At KIT this parameter is used for punching tapes 12 and 10 mm wide: the resulting strands have a crossover angle of 30°, and W_R = W_X = 5.5 and 4.7 mm, respectively. In the case of 4 mm wide tape, with R_i = 10 mm delamination of the tape was observed and R_i = 2 mm used instead. The resulting strands have a crossover angle of 30°, W_R = 1.8 mm and W_X = 2.5 mm. According to the calculations presented in [22], a sharp outer edge of the cross-over section and an increase of the inner radius leads to a reduction of the von Mises stresses under tensile loads. In the Roebel cables from IRL some parameters were chosen differently, in order to be more compatible with the assembling machinery, namely: a larger clearance width W_c between the two sides of the cable and a smaller number of strands per cable length unit [18]. While producing one single strand from a CC tape, the cross-over width W_X and the angle can be adjusted for best current performance. If several strands are cut in parallel from a wider CC material (e.g. 40 mm), W_X is predetermined by geometrical reasons and the cross over section becomes the current-limiting part [18].

Several methods were investigated for the meander-like shaping process of the CC tapes. Laser cutting with conventional equipment is a very flexible technique, but failed due to heating and melting defects. The newer picoseconds-infrared laser technology avoids producing defects, but it is not very economically attractive in terms of production speed. Mechanical punching was identified both by KIT and IRL as the best method and can achieve a precision of <50 μm for the strand width. Figure 5 shows the reel-to-reel punching tool in use at KIT, which allows using the full tape width and varying the transposition length in a wide range. Advantages of this technique are the possibility of adjusting the tool to the tape conditions and its high production speed, typically 50 m of tape per hour. The loss in current carrying capability from punching can be limited to <2–3% for homogenous CC tapes [17]. Figure 6 shows ten punched strands before and after they are assembled into cable.

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The assembling procedure of the Roebel cables is a rather challenging procedure, which requires a complex bending of the tapes around each other, avoiding over-bending or plastic deformation of the material. IRL solved this problem inventing innovative machineries that are able to produce such cables in long lengths [20]—see figure 7. At KIT sample lengths up to 5 m are currently still made by hand. The advantages are the possibility to obtain a dense packing with a reduced central gap and the possibility to change cable designs, e.g. by multi-stacking of strands or cabling with additional stabilization. In that way the potential of the cable technology can be investigated in a wider range.

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A couple of cable design options exist for the different requests coming from the specific applications or operation conditions, as for example temperature and background field.

2.4.1. Multi-stacking

High transport currents at 77 K can be achieved by adjusting several design parameters. Increasing the transposition length allows increasing proportionally the number of strands, resulting in enhanced critical current. The transposition length, however, needs to meet the requirements and constraints of the prospected application (coil size, loss reduction). The second way of achieving high currents is interlacing stacks of strands instead of individual strands. The potential of this method was demonstrated for both a 12 mm wide (three-fold stacking) [19] and a 4 mm wide cable (three- and five-fold stacking, see figure 8) [23]. Stacking multiple strands is however very challenging to do automatically and so far it has been demonstrated only on cables assembled by hand. Unfortunately multi-stack cables are not very suitable for windings, because they have the general problem of stacks: the tapes in the stacks are not fully transposed and therefore the inner CCs need to be shorter to match the meander pattern of the others. For applications where limited bending occurs, for example in bus bars or straight high-current power lines, this method can be easily applied to enhance the current carrying capability of the cable. The consequences of a multi-stacking arrangement for the cable's ac losses are discussed in section 4.

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2.4.2. Strand coupling

2.4.3. Internal and external stabilization

2.4.4. Current and voltage contacts

2.4.5. Striations of the strands

Striations were successfully applied to Roebel strands by laser grooving (five filaments of 1 mm width, see figure 9) and an effect on the ac loss behaviour was observed [29–32], although further systematic investigations are needed for a deeper understanding. The applied striations, however, already showed the importance of sufficient conductor homogeneity and the absence of defects. By means of Hall probe scans single defects significantly limiting the local transport currents could be identified [33]. Those results confirm the considerations in [34] on the homogeneity criteria to select CC for the preparation of Roebel strands.

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For HTS ac cables other alternative concepts were developed and investigated. According to the assembling geometry of HTS power transmission cables, Danko van der Laan presented the concept of the Cable on Round Core (CORC): the CCs are helically arranged around a cylindrical former in one or several layers. See figure 10 (top). The concept provides the twisting of the strands, it is easy in fabrication (assembling method) and is flexible in terms of current capacity (number of tapes used) [35–37]. Until now this assembly has been done by hand. A disadvantage is that the engineering current density is significantly lower than in Roebel cables. Cables with self-field critical currents of 2800 A and 7561 A at 77 K were demonstrated and first tests on smaller cables at 4.2 K and 20 T were performed [38]. Due to the special arrangement of the tapes in CORC cables self-field effects do not play a significant role and the measured critical currents of the cables are comparable to the sum of critical currents of the single tapes from which they are composed [36, 37].

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Another concept motivated by fusion magnet research was presented by MIT [39]: the idea is to solder three stacks of CCs into grooved copper rods, with twist. Alternatively, a single stack may be inserted into a cylinder with a rectangular cavity, also with twist. This cable design is known as the Twisted Stacked Tape Cable and is shown in figure 10 (bottom). So far this concept has suffered from thermal strains in the composite. In addition, the assembling procedure of thick stacks in long lengths has not been performed yet.

In the frame of further activities to upgrade the Large Hadron Collider at CERN, different cable concepts for MgB₂ and HTS CCs are currently under investigation [40]. In particular, for the CC version two concepts were proposed: a paired-CC stack approach and a layered cylindrical arrangement similar to the CORC.

Transport currents in Roebel cables are strongly influenced by the self-field generated by the currents. A first field pattern for the cable cross-section was calculated via a Biot–Savart-approach and showed a complex field distribution reaching a few hundred mT at the outer edges of the cable [19]. This quite simple approach could explain the observed significant critical current reduction caused by the self-field. Each position of the strands in the cable cross-section experiences a different field, both in terms of magnitude and direction. A finite-element method (FEM) modelling of the self-field pattern allowed a more detailed analysis, including the current anisotropy of the CC and the evaluation of the individual critical currents of the strands.

The highest transport current in a Roebel cable measured so far is 2.6 kA at 77 K in self-field, a cable with 15 × 3-fold stacked strands and a transposition length of 188 mm, made from 12 mm wide SuperPower CC. That particular cable showed some degree of current percolation between the strands, resulting in unusual current–voltage characteristics [17, 19], probably to be ascribed to the fact that the copper contact was soldered on a relatively short length (about half of the transposition length). Usually, soldering the contact over one transposition length avoids the problem of those percolation currents and the current–voltage characteristics of the different strands look very similar to each other, as shown in the example of figure 11.

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Depending on the width of the cable (10, 12 mm or 4 mm) we can define two classes of current capacity. At 77 K the high current class can provide 1–2 kA transport current in self-field, whereas the smaller cables can provide currents in the range 0.3–0.8 kA (no strand stacking). Multi-stacking of the strands can increase the current capacity up to a factor of 2–3 with the restrictions in fabrication and application mentioned in section 2.

Roebel cables are being considered not only for applications at 77 K, but also for much lower temperatures. Cryocooler technology has made big progress during the last few years and offers the possibility of operating devices (motors, generators, transformers and magnets) at much lower temperatures than liquid nitrogen. Closed cooling cycles have already been tested with liquid hydrogen (20 K) and liquid neon (28 K). For bus-bars in accelerator applications helium gas cooling is considered with T = 10–15 K, whereas for future fusion reactor scenarios (DEMO) HTS magnets operating at 50 K or below are being considered. CERN recently performed the first successful current capacity test in the FRESCA test facility at 4.2 K in background fields up to close to 10 T [45]. The result was a remarkable increase of the currents with lowered temperature as shown in figure 12.

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Conductors from KIT and General Cable Superconductors (GCS) were successfully tested; the occurring Lorentz forces were kept under control by fixing the cables on the support by means of lateral pre-compressing. A current of about 14 kA was reached for a 12 mm wide Roebel cable from KIT with ten strands of 5.5 mm width at 4.2 K and a background (parallel) field of approximately 0.5 T. The field dependence was measured for both orientations of the field, perpendicular and parallel to the cable (see figure 12). The data show a behaviour very similar to the I_c (B, T) dependence of a single CC tape and the increase of the transport currents from 77 K to 4.2 K is more than a factor 10. The CC material from SuperPower was significantly different for both kind of cables: BZO doped tape was used by KIT and non-doped by GCS, respectively. Although GCS used 15 strands (width 5 mm) in comparison to ten strands (width 5.5 mm) of KIT, the achieved transport current performance was comparable, which indicates the I_c of the starting materials used in the KIT cable was significantly higher on average. Due to the smaller cable cross-section, the engineering current density of the KIT cable was approximately 50% higher, a big advantage for magnet applications.

In another activity pursued at KIT, the split coil magnet facility FBI was equipped with a heating jacket so that the sample's temperature can be increased beyond 4.2 K [46]. The technological problem of the heat chamber in liquid helium has not been completely solved yet, and the achieved data suffer from the fact that the sample's temperature is not precisely known [21]. The measured current values however indicate that scaling up the currents to other field and temperature conditions works quite similarly to single CCs. The measured results were compatible with the data obtained at CERN within the error margin.

CCs show strong dependence of the critical current on the magnetic field, and specifically a strongly anisotropic dependence on the orientation of the field with respect to the tape. This behaviour and also the temperature dependence of the critical current are strongly correlated with the kind and orientation of artificial pinning centres applied by doping the superconductor. The magnetic field distribution and associated local critical current density reduction inside the Roebel cable cross-section cannot be trivial. A first attempt to get an insight into the behaviour at 77 K was performed with a simple model based on the Biot–Savart law [19, 47]. Later, a more sophisticated analysis was carried out by means of a dedicated FEM model [48], which included the anisotropic dependence of the critical current density J_c on the local magnetic flux density—see figure 13: with those calculations it was possible to estimate the critical current of the strands as they move along the cable length.

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Scanning Hall probe investigations were performed to reveal the magnetic field pattern along the length of Roebel cable samples, which reflects the geometrical arrangement of the CC strands in the cable structure [33]. The width of the space between the strands can be observed in the field map. An example is given in figure 14. Under the action of a moving magnet, higher magnetic fields are found in the middle of the tape, although their relative size decreases as the applied field increases. Scan measurements before and after inter-strand coupling did not show significant differences in the field map.

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The Hall probe scanning technique was also used to investigate the magnetic field and, through the solution of an inverse problem, the current density distribution in individual strands used to assemble Roebel cables [49]. Field and current distributions in the straight part agree with the theory of thin films. A different behaviour for currents penetrating from the edges and currents penetrating from the top and bottom surfaces was found. This results in a non-parallel current flow in the cross-over part of a single Roebel strand. This technique allows the investigation of the local properties of the conductor with high resolution and is therefore a very useful tool to find problems, especially associated with the uniformity of the conductor, in the manufacturing process of Roebel cables. Two examples of current patterns in the presence of defects are shown in figure 15.

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Mechanical induced effects from applied strain have been the subject of FEM investigations [22, 26], which identified the inner radius of the cross-over section as the critical point with the peak values of mechanical stresses. It has also been found that a sharp outer edge of the cross-over section, as in the KIT cables, reduces the peak strain [21].

The field dependences of critical currents of a KIT and a GCS cable were measured at 4.2 K in both magnetic field directions by Fleiter et al [45]. In order to withstand the Lorentz forces that arouse during the measurements, the cables were fixed between two stainless steel plates. Both cables withstood transverse stresses of 45 MPa. Since the cables are not flat, peak stress areas could be identified by means of Fujifilm papers. It was found that the effective section that experiences transverse stress is quite limited and can be estimated to be only about 36% and 24% for the GCS and KIT cables, respectively. This means that a nominal loading up to an average stress of 40 MPa corresponds to a local stress of at least 111 MPa, for the GCS cable, and 167 MPa, for the KIT cables. By disassembling the cables and testing the strands, it was confirmed that no degradation of the superconductor occurred.

In another study, Uglietti et al [50] investigated the effect of transverse compressive loads on the critical current of CC tape of 4 mm width, of a punched CC strand 2 mm wide and of CC Roebel cables (consisting of ten punched strands). While the tapes could withstand a compressive stress larger than 100 MPa without significant degradation of the critical current (less than 2%), the degradation of the critical current of some of the individual strands in the Roebel cables exceeded 30% at a transverse compressive stress as low as 10 MPa. Visual inspection of the extracted strands revealed some local damage of the strand surface corresponding to the meander structure of the strand lying above or under the damaged strand. Subsequent measurements of the local critical current confirmed the reduction of critical current at the positions showing a damaged surface.

Bumby et al [51] performed a similar study (supported also by numerical simulations) on tensile stresses of 12 mm CC tape, 5 mm wide Roebel strands and a 15/5 HTS Roebel cable. They found that irreversible degradation of I_c occurred at 700, 146 and 113 MPa, respectively.

The apparent conflict of those results indicates that there are still many open questions on the mechanical properties of Roebel cable, which need further investigation.

Resin impregnation can help improve the resistance of Roebel cables to mechanical stresses. However, the huge difference of the thermal expansion coefficients of YBCO and resins, on top of the peculiar layered structure of CCs and of the meander shape of the strands in a Roebel cable, makes the impregnation of Roebel cables a quite difficult task, compared for example to the impregnation of conventional LTS cables. First attempts to impregnate a Roebel cable with araldite and silica powder mixture are presented in [52], although the difference of the test conditions used for the impregnated and non-impregnated cables do not allow one to have a conclusive word on the efficacy of the impregnation technique presented there.

In view of the application of Roebel cables in high-field magnets with strong mechanical forces (hundreds of MPa), M Bird has recently proposed the idea of a cable-in-conduit configuration, using a hastelloy jacket to provide high strength and stiffness while also providing similar thermal contraction as the YBCO tape [53].

Very few possible applications of HTS cables, as magnets in fusion devices, big water power or gas turbine generators require an operation current capacity of the cable between 20 and 30 kA. In fusion magnets the projected operation temperature for HTS cables is between 4.2 K and 50 K with approximately 13 T peak field at the conductor. Within a single Roebel cable such performance is hard to achieve, but with further improved CC performance this is not impossible in the future. This performance can be reached by using multi-stacks of strands and a long transposition length, in the order of 0.5–1 m. KIT proposed a Rutherford cable design with Roebel strands as a potential solution [54]. Roebel cables serve as strands wound around a central former, which can be moderately flat or round and can provide the channel for the coolant. This structure provides a transposition of the strands (Roebel cables), which are transposed as well. Efforts to realize a model cable of 0.5 m length pointed out the bottlenecks of this design [55]. The bending of the Roebel strand around the edge is a critical issue: in the experiments carried out in [55], three Roebel cables were wound onto a 10 mm thick stainless steel former with a winding angle of 20°. Of the three cables, one was successfully wound with a current degradation as low as 12%; in the other two samples, the current was drastically reduced because of the solder flowing in between the strands during the soldering of the current contacts and by the subsequent bending of the stiffened area. The open questions are still the behaviour of the bent section upon cooling cycles, the need to avoiding delamination from stresses and how the cable assembling works over long lengths. An advantage is the flexibility of the design, a disadvantage the complex composite and the necessary thickness of the former (15–20 mm). In the future this concept has to compete with the potential of an upgraded single Roebel cable, already showing a self-field critical current of 14 kA at 4.2 K in a standard configuration [45]. Several parameters can multiply the current capacity, for example: extended transposition, wider tapes, improved CC performance (being already shown in short samples by several producers) and finally the multi-stack strands.

Table 1 summarizes the properties of six significant Roebel cable samples manufactured by KIT and GCS in recent years. These cables are reputed significant because of the high-reached critical current (KIT-2, GCS-2) and length (KIT-3, GCS-3). Two cables manufactured with narrow 2 mm wide strands (KIT-1, GCS-1) are also listed. In the table the design critical current refers to the value obtained by simply summing the critical currents of the individual composing strands.

The techniques to measure transport and magnetization loss are very different to each other.

The methods to measure the transport ac loss in Roebel cables essentially follow the same principles used for single tapes [60]. Transport ac losses are measured by recording the voltage component in phase with the transport current, usually by means of a lock-in amplifier [61]. Due to the relatively complex geometry of Roebel cables, however, the question of how and where to fix the voltage taps arises. The most accepted solution seems to be to solder voltage taps on the top of a particular strand over an integer multiple of the transposition length [23, 62] (figure 16). This latter action is necessary in order to average the loss signal over all the positions experienced by the strand inside the cable. Jiang et al showed in [62] that the transport loss of a Roebel cable with N strands can be estimated by taking the mean value of the in-phase voltages V_k measured from different voltage loops attached to the different strands as:

$$\begin{eqnarray*}&&{Q_t} = \frac{{{V_{{\rm{mean}}}}{I_{{\rm{cable}}}}}}{{d \cdot f}} = \frac{{\frac{1}{N}\mathop \sum \limits_{k = 1}^N {V_k}{I_{{\rm{cable}}}}}}{{d \cdot f}},\end{eqnarray*}$$

where I_cable is the total current carried by the cable, d is the length of the voltage loop and f is the frequency.

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Placing many voltage loops can easily become very complicated; alternatively, the voltage taps can be positioned on the current leads. The drawback is that the (usually large) resistive signal of the current leads is measured as well and needs to be subtracted from the measured voltage in order to obtain the loss in the cable. This can be quite tricky, especially because the loss signal coming from the current leads is usually larger than the signal coming from the cable. A good agreement with the 'standard' approach with the taps soldered on a strand has been reported in [23], although the authors acknowledge that it could be coincidental and deserves further investigation. Another situation where the placement of voltage taps on a strand could be difficult is in the case of coils, especially if they are tightly wound [43, 63].

Measurement of magnetization ac losses is typically performed by placing a short piece of Roebel cable (of length at least equal to the transposition length) in a uniform magnetic field usually generated by coils. Losses are then determined either by measuring the magnetization of the sample by means of pick-up coils [64, 65] or by means of the calibration-free method [66]. Measuring a short sample of cable, however, presents the problem that the magnetic flux can enter from the ends of the cable; for this reason, experimental results evaluating the uncoupling between strands (or filaments) should be carefully evaluated in this perspective.

Estimating the ac losses in a Roebel cable is not a very easy task, mainly because of the meander-like shape of the strands and their braid arrangement in the cable structure.

The simplest approach to model a Roebel cable (both in the case of using analytical expressions or building a numerical model) is to consider only the transversal (2D) cross-section of it. This simplified approach is based on the fact that the transposition length is much longer than the strands' width, and the currents crossing the cable's whole width are expected to have negligible effects—see the schematic representation in figure 17.

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With this geometrical simplification, analytical expressions can be used for having an approximate estimate of the loss. In particular, for applied fields much larger than the cable self-field, one can use Bean's slab model [68] paying attention to use the proper value for the slab width w used in the model, i.e. the strand or cable width for the uncoupled and coupled cases, respectively. This model also assumes constant critical current density J_c and the sharp E(J) relation of the critical state model [69]. The ac loss per cycle and cable length in this case is $Q=2{{\mu }_{0}}dw_{s}^{2}{{J}_{c,{\rm uncoupled}}}{{H}_{m}}$ and $\begin{eqnarray*}&&Q={{\mu }_{0}}dw_{c}^{2}{{J}_{c,{\rm coupled}}}{{H}_{m}}\end{eqnarray*}$ for the coupled and uncoupled cases, respectively, where w_c and w_s are the cable and strand width, respectively, d is the total cable thickness, J_c,coupled is the engineering J_c of the whole cable and J_c,uncoupled is the engineering J_c of one column of strands in the cross-section (see figure 13). This equation suggests that the Roebel cable transposition reduces the ac loss by a factor around 2, since the strand width is around half of that of the whole cable and the horizontal gap between strands in figure 13 is small. However, the equation above is not applicable for magnetic-field dependent critical current densities, unless the cable is submitted to a dc magnetic field much larger than the amplitude of the ac magnetic field.

The Norris strip and ellipse formulas [70] provide a rough estimation for the transport loss, assuming a constant J_c .

$$\begin{eqnarray}&&{{Q}_{E}}=\frac{{{\mu }_{0}}I_{c}^{2}}{\pi }\left[ \left( 1-i \right)ln\left( 1-i \right)+\left( 2-i \right)\frac{i}{2} \right]\left( {\rm ellipse} \right),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{Q}_{S}} & = & \frac{{{\mu }_{0}}I_{c}^{2}}{\pi }\left[ \left( 1-i \right)ln\left( 1-i \right)+\left( 1+i \right)ln\left( 1+i \right)-{{i}^{2}} \right] \\ {} & {} & \left( {\rm strip} \right), \\ \end{array}\end{eqnarray} \tag{ 2 }$$

where I = I_a /I_c is the normalized amplitude of the applied current I_a .

The analytical models described above provide however a very rough description of the physics and geometry of the cable. More realistic descriptions of Roebel cables can be obtained by means of numerical models. Most models still describe the cable as two stacks of rectangular tapes, but they have the possibility of including the various parts composing a CC (e.g. substrate, stabilizer) and—more importantly—of controlling the current distribution between the strands. Typically, two extreme situations can be considered: one can leave the current free to distribute between the strands (as if they were electrically in parallel) or force an even current distribution between them. This second condition corresponds to what is expected to happen in reality thanks to the transposition of the strands and results in lower losses [23]. When only an external ac magnetic field is applied, these two situations correspond to complete coupling or uncoupling of the strands. Which of these two situations is most beneficial from the point of view of losses depends on the magnitude of the applied field [23], the geometry of the cable (e.g. cable thickness, gap), and the direction of the applied field. For all cases, at high applied fields the uncoupled case presents the lowest loss (see figure 18). The behaviour at low applied fields depends on the orientation of the field. For parallel applied fields, the uncoupled case is still favourable [71], while for perpendicular ones the coupled case presents lower losses [59]. For the latter situation, the geometry plays an important role. The narrower the horizontal gap, the wider the peak in the loss factor for the uncoupled case (and the higher the loss at low applied fields) [72]. By decreasing the cable thickness, this effect becomes more pronounced.

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The simulation of Roebel cables as two stacks of CC was first proposed in [22], and further refined by Pardo and Grilli in [67] and [73]. Those publications confirmed the results discussed above and in particular [73] showed the importance of the parallel component of the magnetic field. More specifically, the ac loss depends on the parallel component of the applied field for two reasons. First, at low applied fields and low angles, the ac loss due to the penetration across the thickness is important. Second, because for YBCO the field anisotropy in J_c is relatively weak. Then, J_c reduces significantly with increasing parallel component of the applied field for the same perpendicular component of the applied field. This is very promising for solenoids, where the ac loss due to the parallel component of the applied field is important. The transposition of the strands has only moderate effects in reducing the transport losses, as confirmed by simulations and experiments—see figure 20 and section 4.3.1, respectively.

Numerical models have been extended to calculate the losses in coil geometries, by means of axis-symmetric models [43, 71]. In [44] the authors added a dedicated three-dimensional (3D) model for estimating the losses in the copper contacts as well. 2D simulations have also been used to calculate the field profiles in cable made from tapes with magnetic substrate [74].

The first 3D model for a Roebel cable was proposed by Nii et al [75]. The model is, for example, able to calculate the current density distribution and the local dissipation in the crossing points (see figure 19). That work has been recently extended in [76], where the authors come to the conclusion that while the current lines on the strand of the Roebel cable and the mode of magnetic flux penetration are influenced by the 3D transposed structure of the Roebel cable, giving rise to regions of high loss concentration, the ac loss of the Roebel cable averaged along its transposition length is almost the same as those of the stacks of CCs, which simulate bundled conductors with uniform current distribution. For the sake of precision, it has to be mentioned that the model presented in [75, 76] considers the superconductors to be infinitely thin, and as such it cannot take into account the magnetic flux penetration from the wide faces of the superconductor. This limitation has been recently overcome by Zermeno et al [77], who have proposed a fully 3D model for a Roebel cable: while the model is able to solve a fully 3D problem, its routine use may suffer from the intense computational effort and issues related to mesh generation and optimization.

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In this section we report the main experimental results on transport and magnetization loss measurements on Roebel cables. In the following, we focus on cables made from CCs with non-magnetic substrate because of the dwindling use of magnetic substrates in applications, especially in view of recent findings which indicate the possibility of utilizing substrates with virtually non-magnetic behaviour for manufacturing CCs with RABiTS technique [78]. At the end of this section, we present a brief summary of the ac loss behaviour of Roebel cables made of tapes with magnetic substrate and we list the relevant references.

4.3.1. Transport loss

Jiang et al measured the transport loss of a Roebel cable composed of five strands with non-uniform current distribution in a range of frequency from 59 to 354 Hz [62]. It was found that the losses are mostly hysteretic and that the average voltage method described above is a valid estimation of the losses of the cable. A mostly hysteretic loss contribution was also reported in [23], where the transport loss of cables composed of individual strands and stacks of strands was measured. Numerical simulations based on an equally distributed current between the strands agree well with experimental values.

The spacing between the strands (W_c parameter in figure 4) influences the magnetic field experienced by the superconductor, and therefore the ac loss of the cable. In particular, a larger spacing increases the critical current and decreases the transport losses, so spacing the strands may be useful for reducing the cable loss in applications where transport loss is dominant, although this reduces the cable's engineering current density [79].

Evaluating the impact of the Roebel geometry on transport loss characteristics is not straightforward, because different objects can be used for comparison. For this purpose, Norris's formulas can be of help.

One can compare the loss of a Roebel cable composed of N strands with the loss of N conductors bundled together, as done in [62]. In that case, the loss of a conductor in a bundle should be N times higher than the loss of an isolated conductor. However, experimental findings in [62] indicate a smaller loss increase. In a later work [80] the authors compare the transport loss of a Roebel cable composed of eight 2 mm wide strands with a stack composed of four 4 mm wide tapes connected in series to impose the same current in each tape: at low currents the loss is similar, but at medium-high currents the Roebel cable has lower losses; in particular, at I_t  = 0.99I_c the loss of the Roebel cable is about 30% lower than that of the stack. However, more than as a positive effect of the transposition, this can be seen as a simple effect of the central gap inside Roebel cable: the strands in the Roebel cable are more loosely arranged than in the stack (where there is no gap), the self-field is lower and the losses are consequently lower as well. This kind of ac loss reduction has been discussed for two tapes in [81] (see figures 4 and 5 of that article).

Alternatively, one can compare the loss of a Roebel cable to those of conductors (with elliptical or rectangular cross-section) with the same critical current of the Roebel cable. In [23] it was found that the measured loss falls between that of an elliptical and thin rectangular conductor. Numerical calculations for two stacks of tapes with uniform current distribution among them agree well with experiments. For the same Roebel cable geometry, in [67] the authors compared the calculated transport loss for the transposed and untransposed case, finding that the transposition reduces the loss by about 20% (figure 20). This is due to the fact that the Roebel cables under study are thin objects, and for such geometry the current is already approximately balanced between the strands even without transposition; therefore, the transposition can reduce the loss only moderately.

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4.3.2. Magnetization loss

The first measurements of magnetization loss were reported in [82, 83]: a weak frequency dependence was observed, which indicates that the most important loss contribution comes from the superconductor losses. An even weaker frequency dependence was observed on the samples with different architecture presented in [23]. In that paper no special resistive material was used for reducing the coupling loss; this means that the contact resistance between strands is sufficiently high for suppressing the coupling loss. In [84] Lakshmi et al ascribe the observed slight frequency dependence of the loss to the intrinsic frequency dependence of the loss of the superconductor, and they use FEM calculations to support this conclusion—see also [85, 86].

In order to increase the current carrying capability, Terzieva et al [23] prepared cables with stacks of three and five strands. This leads to a better (partial) shielding of the applied magnetic field, to the shift of the field of full penetration to higher amplitudes, and to the reduction of the loss at low applied fields.

In [87] the authors compared the magnetization loss of a five strand Roebel cable with insulated strands to that of an individual strand, showing a reduction of the loss at low and intermediate fields and a convergence of the two loss curves at higher fields, when the penetration of the cable is complete. They also show that the angular dependence of loss obeys a simple scaling law, according to which only the perpendicular component of the magnetic field matters [87]. It has to be remarked, however, that the loss results presented there span over six orders of magnitude, so variations in the order of 20–30% are scarcely visible. In fact, calculations carried out by Pardo and Grilli [73] showed a more complex behaviour caused by the complex angular dependence of J_c on the magnetic field. They found out that, while the dependence of loss only on the perpendicular component of the magnetic field is a reasonable approximation for angles higher than 15–30° (perpendicular field corresponding to 90°), this is no longer the case in general, as can be seen from the plot of the loss function Q/H_m ².

In [87] the authors manufactured a coupled cable (which is more stable against defects) by coupling the strands by means of Cu-bridges: the coupled cable showed the dominance of coupling currents at low frequencies and a saturation of loss at high field, a situation where the cable behaves as a monolithic conductor with higher loss than that of an uncoupled cable.

In [87] Lakshmi et al analyse the frequency dependence of the magnetization loss for two types of Roebel cables: an 'uncoupled cable' (where the strands are electrically insulated from each other) and a 'coupled cable' (where the strands are not individually insulated and some degree of coupling is present). A very weak frequency dependence of the magnetization loss is reported for the uncoupled cable: losses decrease with frequency at low fields and increase with frequency at high fields, the cross-over between the two behaviours occurring approximately at the peak of the loss function. This is due to the intrinsic frequency dependence of the superconductor material due to the flux creep, as confirmed by analytical [88] and numerical [86, 89] calculations. The same dependence is also described in [90], where the two frequency regimes and their cross-over is more visible, and where an additional loss contribution is reported. Overall, for the uncoupled cable, the frequency dependence is in the range of 10% in one decade of frequency, and hence negligible for practical applications. A much stronger dependence is observed for the coupled cable, especially at low fields, and ascribed to the coupling between the strands; at higher fields the coupling contribution is less important and the loss curves for different frequencies converge. Roebel cables with striated strands have been assembled. The strand striation further reduces the loss [29, 31, 32, 91] see figure 21; however, the effectiveness of this method for reducing the loss in practical application has still to be proven, because in the experimental set-up the filaments are electrically isolated from each other at the ends, and hence there are no coupling currents, whereas in real situations the filaments will be coupled at the ends. The concept of additional transposition by means of the more complex structure of a Rutherford cable with Roebel cables used as strands was proposed in [54] and tested with applied Roebel strands recently [55]. However, striated strands were not investigated in that work.

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4.3.3. Combination of transport and magnetization loss

4.3.4. Cables with magnetic substrate