On the coupling current losses in a multifilamentary superconducting composite

The coupling current losses in a superconducting multifilamentary composite exposed to trapezoidally varying external magnetic field and carrying a small transport current are investigated for the volume pinning force density described with the dependence: F_v = α B^1?γ. Such a model allows estimation of the deviations from the solution based on Bean's (γ = 0) critical state model. Results indicate that there exists a region of small magnetic field amplitudes for which discrepancies are largest. They are also very sensitive to the rate of magnetic field change.     

Losses in a twisted multifilamentary superconducting composite submitted to any space and time variations of the electromagnetic surrounding

Using a circuit model, loss calculations have been carried out for coils and short samples, made of twisted multifilamentary composites or flat braided cables, which undergo any space and time-variations of the magnetic induction, of the transport current and of the temperature. This model has been tested by measuring losses in short samples submitted to a rotating magnetic field and to a longitudinal magnetic field and in solenoids and ‘special coils’ which look like small Tokamaks, where the composite which carries a time dependent current, is submitted to simultaneous transverse and longitudinal magnetic induction changes. Experimental and theoretical values are in good agreement within about 20% for hysteretic losses and for eddy current losses.     

A general treatment of losses in multifilamentary superconductors

Losses are calculated for a range of shapes of superconducting wire and cable. It is shown that the coupling losses can nearly always be expressed to a first approximation in terms of two parameters. One is the shape of the coil, the other (which contains most of the material parameters) is the time constant.The time constant is calculated for several cable types, including rectangular conductors, in which an exact solution for the field can be found at low frequencies. Maximum values for the loss are calculated and general conclusions for the design of cables are drawn.     

Coupling losses of finite superconducting cables

The calculations of coupling losses in superconducting cables are generalized for ac fields with time variations comparable with, or even larger than, the time cosntant τ of the coupling currents. The losses are calculated as function of the frequency ω and the cable length. At small frequencies, the losses are a monotonically increasing function of the cabling length, whereas for larger frequencies the losses reach a maximum at lengths about l₀/(1 + ωτ) (l₀ is the cabling length) and then they decrease with further increase of the length. Moreover, for systems with larger values of V = $\text{2 π}{}^2\text{(R + D)}{}^2\text{l>}{}_0{}^2$ (R is radius of the strand, D is the distance between them) the oscillations dependent on the cable length are more pronounced. For a given cable length, three characteristic forms of the frequency dependence of losses are found.The results can be important when comparing the losses from measurements on finite samples with the expected losses in larger systems (eg magnets for fusion reactors, generators, etc.).     

Coupling losses in cables in spatially changing ac fields

The coupling losses between strands of multifilamentary superconductors can be important in alternating and pulsing magnetic fields. Up to now, mainly spatially homogeneous applied fields were considered.In this paper, the coupling losses in cables of the EURATOM-LCT-conductor type are calculated for transverse ac fields periodically changing along the cable. Sinusoidal, triangular and trapezoidal field distributions are taken into account as special cases. The losses are enhanced by some orders of magnitude for the sinusoidal field, when the ‘wave length’ of the field I_m comes close to the cabling length I_o. This enhancement is expressive also for triangular and trapezoidal fields, if the wave length of the Fourier components is close to I_o.Some consequences of this effect are pointed out, important for designing larger magnets (eg for the fusion, for generators, etc.)     

High-field superconducting properties of multifilamentary nbtihf superconductors

A Nb-61.7at%Ti-3at%Hf alloy was fabricated into multifilamentary wires and studied in terms of high field superconducting properties. The effects of precipitation heat treatment and cold working after the heat treatment on the superconducting properties were examined. To obtain high J_c in high fields, a process emerges where the heat treatment producing a maximum upper critical field, B_c2, is followed by severe cold working. The superconducting properties at 1.8 K were then investigated which indicated that the temperature scaling law may hold for this NbTiHf superconductor.     

Effect of the respective positions of filament bundles and stabilizing copper on coupling losses in superconducting composites

Superconductor stabilization is ensured by embedding filaments in a copper matrix. Calculations show that a central copper region always leads to an increase in loss in comparison with the case where the filaments are located in a central region surrounded by an outer copper sheath. Practical cases with a copper matrix and also mixed matrix conductors are discussed. Attention is drawn to the comparative roles played by the effective transverse resistivity of the matrix and the copper resistivity.     

Losses in a multifilamentary superconducting wire caused by a simultaneous sweep of current and magnetic field

A theoretical discussion is presented on the energy loss in a multi-filamentary superconducting wire when an applied transport current and an external transverse magnetic field are varied simultaneously with a repeating pulsive wave form. In the present calculation, the effects of the ‘uniforming time constant’ which has been introduced by the authors as a characteristic time constant for the change in the transport-current distribution inside the wire is taken into account, together with the field dependence of the critical current density of superconducting filaments.Thus the present analytic expression for the energy loss of multi-filamentary wire is available to the whole range of the external magnetic field. It is shown that the contribution of the dynamic resistance loss to the total loss is strongly dependent on the position of the wire inside a coil.     

Coupling-current loss in a multifilamentary superconducting wire with a normal-metal core

Coupling-current losses in a multifilamentary superconducting wire with a normal metal core were calculated for the case of transverse magnetic fields. The results showed that the dependence of the coupling-current loss of such a wire on the rate of field variation is markedly different from that of a wire without the core, the former being characterized by two time constants while the latter by one. The theoretical result was confirmed experimentally. It was pointed out that in the case when a wire with Cu/CuNi matrices is to be used in a rapidly changing magnet field, Cu for a stabilization should be arranged at the centre of the wire rather than at the surface of the wire in order to reduce the loss.     

Multifilamentary superconducting materials for large scale applications

The February 1971 edition of this journal carried a ten page article by the author and his colleagues reviewing the then ‘state of the art’ in multifilamentary superconducting composites.¹ At the time we attempted to cover in a fairly comprehensive manner the first decade of development of these materials since the construction of the first high field superconducting magnet.² Now more than another decade of development has passed and obviously the field can no longer be reviewed comprehensively in the same few pages. This article will attempt to give a brief overview of these materials from the point of view of one engaged in their development and supply for large applications. Hopefully, the references to books,^3,4 review articles,^5,6 and the proceedings of the various conferences^7–11 will serve as a source of further information for those who require more details.     

The influence of filament bundle location on coupling losses in superconducting composites. part i: mixed matrix conductor

The ac losses in mixed matrix multifilamentary superconducting composites with different superconducting filament bundle positions have been measured using the magnetization method to establish the relation between filament bundle position and coupling losses. Experimental results indicate that coupling losses between filaments in the composite with the filament bundle located in the central region is smaller than the composite with the filament bundle located in the outer region. It is also pointed out that a copper stabilizer, divided by the CuNi barrier into small regions, like a honeycomb, shows anomalous increase in its resistivity due to Ni duffusion during heat treatment.     

AC losses in multilayer superconducting tapes

ac losses have been investigated experimentally as well as theoretically in tapes having on the surface several normal conducting and/or superconducting layers. The superconducting layers under investigation have been Nb₃Sn on a niobium substrate and Nb₃Ge on a stainless steel substrate. It has been proved that the layered structure of the tapes is well reflected by the stepwise character of the ac losses dependence on the amplitude of the surface magnetic field. The magnetic flux passing through a surperconducting layer or surface barrier into the inside of the tape enhances the losses in the passed barrier or layer.     

Theoretical and experimental study of the saturation of a superconducting composite under fast changing magnetic field

The behaviour of a superconducting composite subjected to a transverse time-varying magnetic field has been simulated with an electrical network. The problem of the current distribution inside the composite has been solved with a computer code.The theoretical results clearly show the existence of different zones (saturated, intermediate, and saturation-free zones) as well as their evolutions when the field variation rate ? increases. The loss power has been also calculated.We found that the penetration of the saturated zones in the middle plane of the composite is relatively easy, and that the centre is reached for a finite value of ? However, the total saturation of the whole composite is achieved asymptotically only when ? becomes infinite. Experiments using high values of ? have confirmed the theoretical predictions about the evolution of the saturated zones inside the composite.     

Alternating field losses in superconducting wires carrying dc transport currents. Part 2: multifilamentary composite conductors

Investigations of transient field losses in multifilamentary composite conductors carrying dc transport currents, I, are presented. As shown previously, the total loss is best characterized by a normalized transport current i = l/l_c and a dimensionless field-change rate β = τB_e/B_p, where τ is the relaxation time of the coupling current and B_p is the full-penetration field of a solid conductor equivalent to the multifilamentary composite. For β ? 1, the composite conductor behaves like a solid conductor and a saturation effect occurs in both the magnetization and the ac losses. The characteristic feature of the composite appears for β ? 1, and the total losses for transport currents below i < 1 ? β are almost independent of i. Beyond the limit given, losses rise sharply. Experimental results over a wide range of the field change rate and the transport-current level agree sufficiently with the calculations.     

Transient field losses in multifilamentary composite conductors carrying dc transport currents

Theoretical studies are presented for transient field losses in multifilamentary composite conductors carrying a dc transport current. The analysis of the total loss is split into separate calculations of the magnetization loss and the loss due to the dynamic resistance. Analytical expressions are derived for a field change of trapezoidal shape. The parameter which characterizes the total loss is a dimensionless sweep rate of the external field $\text{β = τ}\text{B}\text{?}{}_{\mathrm{<mtext>e</mtext>}}\text{/B}{}_{\mathrm{<mtext>p</mtext>}}$, where τ is the relaxation time of the eddy current and B_p is the full-penetration field of the composite conductor. Graphs for a wide range of the field change rate are given and they reveal the strong influence of the transport current.     

Influence of adiabatic conditions on the losses in composite superconductors

The losses in composite multifilamentary superconductors during a pulse of a transverse magnetic field under adiabatic conditions have been studied. A considerable difference in value and character of the loss dependences under adiabatic conditions, as compared to those under isothermal conditions, was found. The influence of various factors and conductor parameters on the losses was investigated.     

The influence of solder-filling on the ac losses of pulsed superconducting cables

AC losses of various kinds of pulsed conductors were measured. First we found that the ac losses of braid filled with Pb-Sn solder were much greater than those of the braid without filler. Since the braid consists of ‘three component strands’ with the mixed matrix of Cu and CuNi, the intra-strand coupling loss should be quite small. Therefore, we concluded that the large increase in ac losses due to solder-filling was caused by the inter-strand coupling through the outer copper sheath. To verify this, we measured the ac losses of two kinds of braids composed of Cu- and CuNi-sheathed strands, respectively. Both samples had the same size and characteristics except that one had copper sheath, while the other had cupro-nickel one. By replacing Cu sheath with CuNi, the ac losses were decreased to the level of those of the braid with Cu- sheathed strands.We also measured the ac losses of the compacted cables, one, filled with Ag-Sn solder and two, without solder. Again, solder filling caused an increase in the ac losses.When cables are not solder-filled, contact resistance among strands cuts the inter-strand coupling effectively.     

Effect of field independent surface barriers on hysteresis losses in type II superconductors

The behaviour of hysteresis losses, W(H_o, ΔH), per cycle and unit volume, as a function of the amplitude h_o and the normalized height of field independent surface barriers, ΔH/h_o, has been examined computationally. These model calculations reveal that, in many instances, the losses when a barrier is acting are higher than when no barrier is present. We present a catalogue of twelve sets of families of curves of W(h_o, ΔH)/W(h_o, O) versus ΔH_o for various fixed amplitudes $\text{h}{}_{\mathrm{o}}\text{? H}{}_1$, where H₁, is the field required for full penetration of the flux front. These twelve sets arise from considering, (i) infinite plane and cylinder geometry, (ii) half-wave and full-wave cycles where the applied magnetic field varies between 0 and 2h_o and between h_o and -h_o, (iii) symmetric barriers which oppose entry and exit of flux equally and asymmetric barriers which oppose entry only, and (iv) bulk pinning causing the critical current density j_c to be independent of the magnetic induction B and inversely proportional to B. This matrix generates sixteen combinations, but only twelve different cases.     

Hysteresis losses in a superconducting plate

On the basis of theoretical analysis expressions have been derived suitable for calculating the hysteresis losses in the presence of variable and constant components of both external magnetic field and transport current in a superconducting layer. The calculation scheme can be applied to any arbitrary shape of the field variation cycle, provided the variable components of the field and current are proportional. It has been assumed that the critical current density is dependent only on the external field. For the more common forms of this relationship the results have been expressed in an analytical form. Application of the results is illustrated by calculating the losses in a superconducting solenoid under variable current in the winding and the external field.