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In the frame of the High-Luminosity Large Hadron Collider construction and Future Circular Collider development program, the magnetic field in the accelerator dipole magnets is being enhanced to 11 T, and 15 T to 16 T level, respectively. Advanced Nb3Sn superconductors with a non-copper critical current density exceeding 2500 A mm−2 at 4.2 K and 12 T, are being developed using the Restacked-Rod-Process (RRP) and Powder-In-Tube (PIT) wire technologies. However, since Nb3Sn is extremely brittle, it is a significant challenge to construct the high-field dipole magnets with such very strain-susceptible superconductor. The high-level of stress acting on the wide face of the Rutherford cables in the coils of 120 MPa to 200 MPa generated by the Lorenz’ force, causes initially a reversible reduction and eventually at some stress level followed by permanent degradation of the critical current when strain goes to high. This study sets out to examine the critical current and upper critical field performance of state-of-the-art RRP and PIT Nb3Sn Rutherford cables in terms of transverse pressure. The variation of the critical current and upper critical field due to the thermal- and mechanical load-cycling was investigated as well. For reference, the critical current of witness wires characterized on standard ITER type barrels were also measured. The results indicate that the RRP type of Nb3Sn Rutherford cables, when fully impregnated with epoxy resin, are able to withstand a transverse stress of 170 MPa to 250 MPa without noticeable irreversible critical current reduction. For the transverse pressure limit for present PIT type of Nb3Sn Rutherford cables somewhat lower values are found at the level of 50 MPa to 120 MPa. Therefore, given the present cables, the high-field dipole magnet construction can be realized using the RRP Nb3Sn Rutherford cables, while for PIT type cables more cable development is requested to enhance the onset of irreversible degradation. The reversible critical current reduction in RRP type of cables of 10% at 150 MPa to 250 MPa needs to be taken into account when predicting magnet performance. Finally, extreme care needs to be taken into account for Nb3Sn coil fabrication, since the experimental results show significant critical current reduction due to stress concentrations already at 0.2° misalignment angles between the pressure applying surface and the surface of the impregnated cable.

The focus of this paper is on the mechanical design of a NbTi-based demonstrator magnet for magnetic density separation (MDS) that is being constructed at the University of Twente. MDS is a new recycling technology that allows the separation of non-magnetic particles based on their mass density, using a vertical magnetic field gradient and a ferrofluid. The unique mechanical design challenge for this type of magnet is the desired minimization of the distance between a sim1 m2 planar array of cryogenic racetrack coils and the ambient-temperature ferrofluid bath. The optimization of the magnet geometry results in a distance between the coils and ferrofluid of 50 mm. This is made possible by opting for conduction-cooling, for the inclusion of room-temperature rods that pass through the cold mass to support the cryostat, and for the geometry of the cassette that reacts to the Lorentz force.

For 30 years, training and unpredictable degradation in accelerator type high-field Nb3Sn magnets have seriously hampered Nb3Sn application. Training and deterioration have to be solved or at least better controlled. The global picture shows that most of the R&D and short model magnets start to train at some 40%-70% of the critical current and then creep up to almost the critical current within some 10-50 training steps. A typical class of failures leading to quenches is largely characterized by cracking and debonding at the interfaces between cable and glass-resin insulation, as well as between insulation and coil former. The study of training by means of testing demonstrator coils is rather expensive and time consuming. However, advances in magnet design and fabrication can also be assessed and benchmarked using BOX, the bonding experiment presented here, that produces maximum uniaxial Lorentz forces at some 7.5 T in a controlled experiment performed in 11 T solenoid facility at the University of Twente. BOX samples use only one meter of Nb3Sn cable inserted in a three-wave meandering slot in a flat metallic sample holder, reproducing magnet-relevant interactions between cable, insulation, impregnated materials and coil former. The meander shape exposes seven straight cable sections to a transverse magnetic field, thereby generating a representative level of shear stress at the interfaces. In this way, characteristic training curves of magnets can be mimicked and solutions studied. We aim to demonstrate with various samples failure mechanisms of high-field Nb3Sn magnets without the need to manufacture complete magnets. BOX may thus be expected to allow for quick and affordable testing of novel insulations, impregnation materials, coatings and interfaces for Nb3Sn magnets achieved by investigating various resins, fillers and more.

The main aim of the EU H2020 project EcoSwing was to demonstrate a technical readiness level of 6–7 for high-temperature superconducting (HTS) technology operating in a wind generator. To reach this goal, a full-scale synchronous HTS generator was successfully designed, built and field-tested in a 3.6 MW turbine. The generator has a rotor with 40 superconducting coils of 1.4 m long. The required >20 km of coated conductor was produced within the project’s time schedule. All coils were tested prior to assembly, with >90% of them behaving as expected. The technical readiness level of HTS coils was thus increased to level 7. Simultaneously, the maturing of cryogenic cooling technology over the last decade was illustrated by the several Gifford-McMahon cold-heads that were installed on-board the rotor and connected with the stationary compressors through a rotating coupling. The cryogenic system outperformed design expectations, enabling stable coil temperatures far below the design temperature of 30 K after only 14 d of cool-down. After ground-based testing at the IWES facility in Bremerhaven, Germany, the generator was installed on an existing turbine in Thyborøn, Denmark. Here, the generator reached the target power range and produced power for over 650 h of grid operation.

In the frame of the EuCARD2 collaboration, aimed at developing the technology for 20 T class accelerator magnets, several demonstrator dipole magnets are being built using high critical current density and fully transposed ReBCO tape-based Roebel-type cables. In accelerator magnets the dynamic magnetic field quality is one of the key parameters, which is affected by the effective inter-strand resistances in the cables. For this reason, measurements of the inter-strand resistances on ReBCO Roebel cables were carried out at 4.2 and 77 K. Acquiring these data is also essential for input of cable simulation models. The cable samples are impregnated with epoxy resin to reduce the effect of transverse stress degradation due to Lorentz forces acting on the strands in the Roebel cables. The measured inter-strand resistance is used to estimate the AC coupling loss in different magnetic field orientations. Moreover, the contributions of diverse interface contact resistances to overall inter-strand resistance of Roebel cables were determined using a novel theoretical model. For validation, the AC loss of cables were examined in various orientations of applied field at 4.2 K. With three analytical models the hysteresis loss was calculated and compared to the measured data. The average inter-strand resistance of the cable samples impregnated with the unfilled epoxy CTD-101K range from 3 to 16 μΩ at 77 K and 1.5 to 9 μΩ at 4.2 K. Between the tapes the copper to copper interface resistance dominates the inter-strand resistance of impregnated Roebel cables. The calculated and measured AC loss for the CTD-101K impregnated Roebel cable lead to equivalent conclusions that the coupling loss is lower than the hysteresis loss within the range of the experiment. These observations substantially differ from earlier results extracted from a similar cable but impregnated with the alumina-filled epoxy resin CTD-101G, which showed considerable coupling loss when exposed to magnetic field parallel the wide face of the cable.

Many high-temperature superconductor (HTS) applications require superconducting cables with high currents while operating in an alternating magnetic field. HTS cables should be composed of numerous superconducting tapes to achieve the required current capacity. Alternating current and magnetic fields cause AC losses in such cables and can provoke conductor instability. AC losses and contact resistances were measured of several cable designs based on commercially available REBCO tapes at the University of Twente. The AC loss was measured under identical conditions for eight REBCO conductors manufactured according to three types of cabling methods—CORC® (Conductor on Round Core), Roebel, and stacked tape, including a full-size REBCO CICC (cable in conduit conductor). The measurements were done at T = 4.2 K without transport current in a sinusoidal AC magnetic field of 0.4 T amplitude and frequencies from 5 to 55 mHz. The AC loss was measured simultaneously by calibrated gas flow calorimeter utilizing the helium boil-off method and by the magnetization method using pick-up coils. Also, the AC loss of two CORC® conductors and a Roebel cable was measured at 77 K. Each conductor was measured with and without background field of 1 T. The measured AC coupling loss in the CORC® and Roebel conductors is negligible at 4.2 K for the applied conditions while at 77 K coupling loss was observed for all conductors. The absence of coupling loss at 4.2 K can be explained by shielding of the conductor interior; this is confirmed with measurement and calculation of the penetration field of CORC® and Roebel cables. The inter-tape contact resistance was measured for CORC® and stacked tape samples at 4.2 and 77 K. It was demonstrated that a short heat treatment of CORC® conductor with solder-coated tapes activates tape-to-tape soldering and decreases the contact resistance. The reduction of contact resistance by two orders in magnitude to tens of nΩm is comparable with the interstrand contact resistance in ITER Nb3Sn type conductors.

The effect of disorder on the normal state resistivity and the superconducting properties of Nb3Sn Sn is explored in a combination of ab initio calculations and microscopic theory. The crystal symmetry is calculated to be preferentially tetragonal at a normal state resistivity below 27.0 ±1.4 μcm, and preferentially cubic above this value, which is shown to be consistent with the experimentally observed transition point. The phonon density of states, the Eliashberg spectrum a2 (w)F (w), the electronphonon coupling constant, the characteristic frequency, the critical temperature Tc, and the upper critical magnetic field at 0 K Hc2 (0) are calculated over a large normal state resistivity range and shown to be consistent with experimental observations. The high degree of consistency between the calculation results and experimental observations is a strong indication that the calculation approach utilized here, a combination of ab initio calculations and microscopic theory, is a useful tool for understanding the superconducting and normal state properties of Nb3Sn.

The current transfer length in composite MgB2 wires with different metallic sheaths and filament numbers was analysed. The electric potential gradient along the wire in the vicinity of the current contact was monitored as a function of transport current in various external magnetic fields up to 14 T. In all samples the characteristic current transfer length is of the order of a millimetre, varying with detailed sheath composition in the range 0.35–2 mm. For the wires with a clear reaction layer at the filament/matrix interface, the electrical resistivity of the material constituting such a barrier was estimated.

The reversible axial strain dependence of the critical current of MgB2 conductors is shown to vary with the temperature and magnetic field. The measured critical temperature and irreversibility field are also found to change reversibly with the axial strain. Combining these effects, we show empirically how the strain dependence of the whole critical surface can be scaled with just three parameters: the strain dependences of its three corner points.

Single-core MgB2 composite wires prepared by in situ and ex situ PIT processes were subjected to axial tension and pressure. Their transport critical currents Ic were measured as a function of axial strain e with a U-shaped spring set-up at liquid helium temperature and external magnetic fields ranging from 3.5 to 6 T. For all wires, Ic changes linearly and reversibly with applied strain, up to a sample-dependent reversible strain limit eirr. The value of eirr depends mainly on the degree of pre-compression, on the mechanical properties of the sheath and also on the structure of the MgB2 core. In contrast to ex situ wires, the aspect ratio of in situ wires does not affect the value of eirr. On the other hand, in situ wires are typically more sensitive to axial strain, i.e. they display a stronger reversible Ic variation and a steeper irreversible degradation, which is attributed mainly to the higher porosity of in situ cores.

Addition of tungsten metallic particles has led to an improvement of the current carrying capacity of ex situ MgB2 wires, due to an improved internal stability. The best performance was observed for 5–10 wt% of W addition and therefore this content was also used for multi-core wires made by the rectangular wire-in-tube technique (RWIT). Transport current measurements show how the variable wire shape (rectangular, circular or tape) as well as additional mechanical deformations (twisting, bending and axial stresses) influence the critical current density of four-filament MgB2–W/Fe wires in external magnetic fields ranging up to 11 T. Critical current degradation by twisting, bending at room temperature and axial stresses at 4.2 K are discussed. It was found that the addition of W particles has no effect on the value of the irreversible strain limit.

International audience; A European consortium has progressed towards elementary and assembled conductors viable for AC applications in the project ACropolis. The main goal was to achieve low AC losses and high critical current densities in tapes developed for AC coils, transformers and cables. In order to verify the performance of tapes an extensive characterization programme was carried out. The dependence of critical current and AC losses on the parameters of the tape manufacturing process were scrutinized. Furthermore, SEM-EDX and DTA/TG analyses were performed to study the material microstructure, and the diffusion coefficient of oxygen through different matrix metals was measured. The homogeneity of the tapes was verified with magnetic knife and Hall sensor experiments. Transport current measurements were performed at different mechanical stress conditions and also the distribution of critical current along long tapes was determined with a quasi-continuous four-point measurement. The results confirmed that the low AC loss conductors can be produced on an industrial scale and they are competitive in power engineering applications.

The development of a 25 T superconducting magnet is usually envisioned with the use of an innermost coil wound with high-temperature superconductor (HTS) material. At present, Bi2Sr2CaCu2Ox and Bi2Sr2Ca2Cu3Ox (BSCCO) conductors are the only candidate conductors available with sufficient length and current carrying capability. Unlike NbTi and Nb3Sn conductors, the BSCCO crystal structure and thus properties are strongly anisotropic. In conductor form, various degrees of anisotropy occur. It is often assumed that this anisotropy simply implies that radial field components must be minimized. We show that this assumption can be misleading. Using the average grain colony misalignment angle to parameterize the anisotropy, three regimes are distinguished in the relationship between average grain colony misalignment and critical current limiting field components. The implications of these separate regimes on the design process for 25 T superconducting magnet design are discussed.

The transport critical current Ic of single- and four-core MgB2/Fe wires under tensile and compressive axial strain e was measured with a U-shaped spring set up. In all wires, Ic increases linearly and reversibly with applied tensile strain, up to a sample-dependent reversible strain limit eirr. The same reversible and linear Ic(e) relation is found when applying compressive strain. This shows how the Ic(e) increase with tension can be understood as a gradual release of the thermal pre-compression strain induced in the MgB2 filaments of the composite. The value of the reversible strain limit eirr depends mainly on the degree of pre-compression, but also on the shape of the composite wire. It is highest in a square wire and lowest in the flattest tape. Tensile strain levels above eirr cause an irreversible degradation of Ic. This Ic(e) degradation, due to filament cracking, is significantly steeper in heat-treated wires than in as-deformed ones. The different Ic(e) regimes are analysed and the mechanical behaviour of the MgB2/Fe composite is compared with that of typical Nb3Sn wires and Bi,Pb(2223) tapes.

Bi,Pb(2223)/Ag tapes exhibit a strong c-axis texturing but are sometimes perturbed by highly misoriented grains that penetrate the Ag sheath: the so-called Bi,Pb(2223) outgrowths. This study focuses on these outgrowths and on their formation mechanism. Extensive SEM analysis has been performed on samples prepared using different techniques and observed from different points of view. The origin of outgrowths appears to be misoriented Bi,Pb(2212) grains, which remain next to the hard agglomerate particles of secondary phases after the deformation. The subsequent growth of misaligned Bi,Pb(2223) platelets easily perforates the soft Ag sheath, and in most cases outgrowths are not related to Ag grain boundaries. Furthermore, we have shown that once these Bi,Pb(2223) grains reach the outer surface of tapes, they melt and crystallize into Bi,Pb(2212) during the cool down. Outgrowth formation can be minimized by using fine-grained precursor powder, which avoids the faults in the Bi,Pb(2212) pre-texture that act as outgrowth nucleation sites.

The intrawire resistance and alternating current (AC) loss of two MgB2 wires with filaments surrounded by Nb barriers have been measured and analyzed. Relatively high values of filament-to-matrix contact resistivity are found in the MgB2 wires; the values are two or three orders higher than those commonly found in NbTi or Nb3Sn wires. Considering the high porosity of the MgB2 filaments, cold high-pressure densification has been applied on the two MgB2 wires to investigate its impact on intrawire resistance and AC loss. The intrawire resistance is measured with a direct four-probe voltage-current method at various temperatures. The AC loss is measured by vibrating sample magnetometer measurements at 4.2 K. In addition to the intrawire resistance measurements, the critical current of MgB2 wires before and after densification is measured with a U-shaped bending spring at 4.2 K as function of axial strain. The critical current in densified MgB2 wires is found to be higher than that in the same wire without densification; it is also less sensitive to the applied axial strain.

The current transfer length of multi-filamentary superconducting NbTi and Nb3Sn strands was measured and analyzed. The aim is to understand and quantify the current distribution process between matrix and superconducting filaments occurring at current injection joints or shunting localized interruptions like originated by transverse cracks or high strain, temperature, or magnetic field in filaments. The current transfer length was investigated on two different multi-filamentary Nb3Sn wires and one NbTi wire. Opposite to earlier clarifications, it was found that the current transfer length cannot be simply represented by a single parameter but depends on the ratio of transport current and critical current and the distance from the current injection point or local interruption of the superconducting path in the filamentary zone. With the aid of our numerical 3D multi-filamentary strand model, simulations were performed showing excellent agreement with the experimental data. For broader use, analytical formulae are proposed to determine the current transfer length for multi-filamentary superconductors with complex cross-sectional layout. The increasing current transfer length with the higher injected current and/or along with the distance away from the current injection point is explained by a progressive current penetration, which is caused by the high resistive matrix layers and complex layout. For the experimental results obtained here, the analytical and numerical simulation results show good agreement against the experimentally measured data from the potential-tips along the strand length.

We report on magnetic measurements in the vicinity of the irreversibility line showing a field-induced dimensional crossover in the vortex ensemble in (Bi, Pb)2Sr2Ca2Cu3O10. The temperature dependence of the irreversibility line changes its behaviour at a crossover field, Hcross = 3.0±0.5 kOe, observed in both DC magnetization and AC susceptibility data, independently of the alternating field and frequency. This yields an estimate for the anisotropy factor in (Bi, Pb)2223, = 45±5. Magnetization fluctuation analysis shows that the applicability limit of the 3D-XY scaling model coincides with the change of behaviour in the IL, which provides further evidence for a dimensionality change in the vortex fluctuations. The pinning potential of the system also shows a change in its field dependence at the same crossover field.

Fluorine-doped Tl(1223) superconductors were prepared starting from mixtures of oxides (carbonates) and fluorides (in particular TlF) under moderately high gas pressure (50 bar). The thermodynamics of the phase formation were found to be modified so that the melting point was lowered and the region where a liquid phase coexists enlarged. 90% phase purity was obtained for bulk materials with the composition Tl0.6Pb0.5Sr1.8Ba0.2Ca1.9Cu3O8.5−δF0.5, prepared by a two-step synthesis including TlF among the starting materials. The critical temperature remained approximately the same as for F-free samples (~115 K), but a remarkable increase of the irreversibility line was observed below 75 K. The samples showed superconducting properties even for the highest amount of fluorine introduced (about one atom per formula unit). Ag-sheathed tapes were prepared by the conventional PIT method, using ex situ reacted Tl(1223) powders or producing the reaction in situ. Transport critical current densities of about 10000 A cm-2 were measured. This value is slightly lower than generally observed for F-free samples, but the magnetic-field dependence was slightly improved. The tape anisotropy could be increased to a factor of 2.5 by performing cycles of mechanical pressing and heating above the formation temperature. The results (in particular neutron scattering diffraction) indicate that, with high-pressure synthesis, F remains in the sample and substitutes for O in the TlO layers in the Tl(1223) phase.

The various fabrication and reaction processes leading to multifilamentary Bi,Pb(2223) tapes with high transport are described. In air, the Bi,Pb(2223) phase in pressed powders is found to be formed by nucleation, following the complete decomposition of the Bi,Pb(2212) intermediate phase. In this work, the precise formation conditions of the Pb-free phase Bi(2223) are described for the first time. It is shown that the presence of Pb leads to a lowering of the melting temperature from 879 to and of that of the `transient liquid' . The latter is formed after decomposition of the 2-layer compound and is necessary to the formation of the 3-layer compound. Thus Pb is found to have no influence on the formation mechanism of the 3-layer compound, which always starts by a nucleation process. Nucleation processes are also found to occur in the first phase of reaction inside Ag sheathed tapes, intercalation being possible at later stages. The highest values of the critical current density in Bi,Pb(2223) tapes reported at the present day are compared, taking into account the analysed tape lengths. The variation of in mono- and multifilamentary Bi,Pb(2223) tapes as a function of the applied magnetic field at 77 and at 4.2 K is described. A new process including deformation by a four-roll machine is presented, and the distribution of inside various tape configurations with similar critical current densities is analysed: for monofilamentary tapes, a maximum of is found at the borders, the ratio being 2.4, while for conventional multifilamentary tapes the maximum is found at the tape centre (, ratio 1.6). For the new four-rolled tapes, a ratio of only 1.1 was measured, indicating a more homogeneous current distribution. In order to lower the AC losses due to the highly conductive Ag matrix, the new concept of `oxide barriers' is introduced. A barrier of surrounds each filament, thus leading to an enhanced radial resistivity, which in turn reduces AC losses. Finally, the current transport mechanism inside Bi,Pb(2223) tapes and the various current limiting mechanisms are discussed.

The current - voltage characteristics and the magnetic field dependence of the critical current of a range of mono-core BSCCO 2223 tapes are presented, illustrating the complementary use of transport and magnetization experiments in determining and analysing the current-limiting dissipation processes in these HTS conductors. Below a magnetic cross-over field the samples resemble a Josephson-linked current network, with the dominant dissipation at the weakest grain boundaries. In this regime, increasing field leads to a gradual fragmentation of the network. The network homogeneity and connectivity can be inferred from screening current length-scale measurements and comparison of transport and magnetization measurements. Above the cross-over field , flux motion within the surviving strongly linked backbone dominates the dissipation. The details of the dependences of the critical current and flux creep rate on magnetic field can be used to examine the intragranular pinning potential in the tapes. We show that, despite the wide range of transport critical-current values of the samples examined, the intragranular pinning proves to be remarkably sample independent. We conclude that while at low fields may be increased by improved processing that yields better intergrain connectivity, the high-field can be enhanced only by strengthening the pinning within the BSCCO 2223 crystallites themselves.

Intra-wire resistance and AC loss of various multi-filamentary MgB2 wires with filaments surrounded by Nb barriers have been measured and analyzed. The intra-wire resistance is measured with a direct four-probe voltage–current method at various temperatures. The AC loss is acquired by both vibrating sample magnetometer (VSM) and magnetization measurements at 4.2 K. With the aid of finite element method simulations, the filament-to-matrix contact resistance and effective transverse resistivity are derived from direct intra-wire resistance measurements. The effective transverse resistivity values are in good agreement with those analytically derived from the AC coupling loss measurements. Surprisingly, very high values of filament-to-matrix contact resistivity are found, being 2 or 3 orders higher than commonly found for NbTi or Nb3Sn wires.

The key factors that make the thallium-based 1223 phase conductors attractive candidates for high-field applications at 77 K are a high transition temperature (about 115 K) and an irreversibility line comparable to that of . However, a third important factor is the grain boundary behaviour. We have performed detailed current - voltage (I - V) studies of a silver-sheathed ( (Tl-1223/Ag) tape. Although the tape has a fairly high self-field critical current density at 77 K (about ), the I - V characteristics display the `weak-link' signatures well known for : linear dissipation above a sharply defined critical current, a pronounced decrease of in a magnetic field H and hysteresis with respect to field cycling. Broader implications for the superconductive behaviour of HTS grain boundaries are discussed.

For a proper characterization of multi-filamentary NbTi and Nb3Sn strands and a better understanding of their performance in short sample tests, as well as for increased understanding of inter-strand current redistribution in cabled conductors, a quantitative knowledge of the inter-filament transverse resistance is essential. In particular, in the case of strain or crack distributions among and along filaments in strain-sensitive superconductors such as Nb3Sn cable-in-conduit conductors, a much better understanding of the voltage–current transition is required as a basis for the analysis of full-size cables. Two particular four-probe voltage–current methods are developed to measure the transverse inter-filament resistance distribution directly, both in well-established and in state-of-the-art superconductors that are presently applied in the ITER, JT-60SA and LHC magnets. To extract values of the filament-to-matrix contact resistance from these direct experiments, some further assumptions are needed. These assumptions are based on FEM simulations and on measurement of the longitudinal strand resistance. An overview is given of a wide range of measurements on various NbTi and Nb3Sn strands, performed at temperatures below 10 K and at various applied magnetic fields. We present the results of the experiments and simulations and demonstrate how the extracted characteristic parameters provide a better insight into the current flow patterns within the strands

Two ITER-type Nb3Sn superconductor strands, one prepared with the bronze route and the other with the internal-tin route, were used to investigate the impact of filament cracking on the strand's transport properties. Careful mechanical polishing allowed unambiguous identification of the microscopic fractures of filaments caused by axial straining of the strands. After application of high axial tensile strain, densely and uniformly spaced cracks were observed in the bronze strand, while fewer but more correlated cracks occurred in the internal-tin strand. Crack initiation was observed in the bronze strand after an applied tensile strain of more than 0.8%, while for the internal-tin strand cracks were found already in the unloaded specimen, with further crack growth beyond 0.3% applied strain. With the Pacman strain device, the voltage–current characteristics at zero applied strain were measured after several successive applications of incrementally increasing tensile strain. Distinct dissimilarities in the voltage–current characteristics were observed between the dispersed and the collective crack distributions. We also modelled the influence of cracks on the voltage–current characteristics of the two strands by considering two limiting cases of the crack behaviour.