Critical State Model Research Articles

Magnetic Moment of a Flat Type-II Superconducting Tape Carrying a Current

I have used the critical state model for investigating the magnetic moment of a flat type-II superconducting strip which carries a current such that the magnetic flux density B generated by this current depends solely on the coordinate along the width of the tape. To do so, I have also calculated the magnetic flux distribution in the film. The calculations clearly show that the magnetic moment of the material exposed to this current is increased as the thickness of the sample is decreased; a behavior similar to the thickness dependence of the critical current density.

Experimentally Determined Magnetization ac Losses of Mono and Multifilamentary MgB2 Wires

Magnetization AC loss measurements have been performed by the calibration free method on mono, 19 filaments un-twisted and 19 filaments twisted MgB2 superconducting wires with magnetic fields perpendicular to the wire axis at various frequencies and temperatures between 24 and 40 K. AC losses were also calculated based on critical state model for mono-filamentary Nb sheathed MgB2 wire with conditions corresponding to experiments. Good agreement is observed between model and experimental data. AC losses go through a peak when the wire was cooled below Tc. In terms of AC magnetization losses, the twisted wire has the minimum losses in order of 10−5 J/m at 30 K in AC magnetic field with 20 mT amplitude.

Enhancement of Flux Pinning in YBa2Cu3O7−δ + Ag Nanocomposites

YBa2Cu3O7−δ/Ag thin film is prepared through pulsed laser deposition. The presence of Ag brings about significant inter- and intragranular modification in the mi- crostructure of the composites. The resistive broadening in the tail part is suggested to be associated with the link be- tween the grains and to be extremely sensitive to magnetic fields. It is assumed that the zero resistance state at the Tc0 region of the above high temperature superconductor is gov- erned by the excitations in the weak link network. Invoking Beans critical state model, enhancement in critical current density Jc is observed.

Magnetization of a thin type-II superconducting circular washer with and without a radial transport current

In this paper, we have studied the magnetization of a thin type-II superconducting circular washer subjected to a normal applied magnetic field in the absence and also in the presence of a transport radial current using the Bean critical state model in which the critical current density is assumed to be independent of the local magnetic field. The results of our calculations on the magnetizations in the absence and in the presence of a radial transport current are compared with each other and also with the other existing theoretical works carried out using different methods.

Dependence of levitation force on frequency of an oscillating magnetic levitation field in a bulk YBCO superconductor

The dependence of the magnetic field strength required for levitation of a melt textured, single domain YBCO superconductor disk on the frequency of the current generating the levitating magnetic field has been investigated. The magnetic field strength is found to be independent of frequency between 10 and 300Hz. This required field strength is found to be in good experimental and theoretical agreement with the field strength required to levitate the same superconductor with a non-oscillating magnetic field. Hysteretic losses within the superconductor predicted by Bean’s critical-state model were also calculated. The measured data rules out any significant Bean’s model effects on the required levitation field strength within the measured frequency range.

Effect of Ce substitution on structural and superconducting properties of Bi-2212 system

In this study, Bi2Sr2Ca1−xCexCu2Oy, where x = 0, 0.01, 0.05, 0.1 and 0.25, superconducting samples were prepared by solid state method and subsequently used as feed in a laser induced directional solidification (LFZ) process. The physical properties of the samples were investigated by powder X-ray diffraction, scanning electron microscopy, dc-electrical resistance, magnetization, and magnetic-hysteresis loops measurements. It has been found that no significant difference has been observed in the critical transition temperatures of samples except for the sample with the highest Ce additions, which shows the lowest T C compared with the other doped samples. Magnetic hysteresis measurements have shown that the hysteresis loop is greater than the doped samples. In addition, critical current density values obtained from the hysteresis loops measurements by using Bean’s critical state model show a decrease with Ce-addition. All the results indicate that Cerium substitution for Ca produces the deterioration on the superconducting properties, compared with the undoped sample.

Flux motion and isotropic effects in MgB2 near the critical temperature

The anisotropic behaviour of polycrystalline MgB2 was studied in details by means of angle dependent magnetovoltage measurements (V-θ curves) at the temperatures near the critical temperature Tc, where θ is the angle between the external magnetic field H and the transport current I. The measurement of V-θ curves was carried out at fixed temperatures below Tc for different values of H and I in a range of θ = −20° −200°. Magnetovoltage measurements (V-H curves) were performed for perpendicular and parallel orientations of H with respect to I, i.e., the V⊥-H curves for\( \vec H \bot \vec I \) and V∥-H curves for \( \vec H\parallel \vec I \). The analysis of V-θ curves does not give any agreement with V(θ) ~ sin2(θ) or any modified angular relationship, so that the measured dissipation is nearly independent of the angle between H and I. The isotropic effects observed in MgB2 were discussed in the frame of the continuum approximation, the discrete case and the flux line cutting associated with the double critical state model derived in the literature. Finally, it is concluded that structural disorder induced isotropic effects can explain the isotropic angular dissipation observed in V-θ, I-V and V-H curves.

Stress and magnetostriction in an infinite hollow superconducting cylinder with a filling in its central hole

In this paper, the stress and magnetostriction induced by flux pinning in an infinite hollow cylinder of type-II superconductor with a non-superconductive filling in its central hole are analytically investigated. Based on the exponent model of critical state of superconductor, the magnetic and current distributions in the hollow cylinder superconductor are obtained firstly. The stress and magnetostriction of the composite superconductive cylinder are then formulated and the magnetoelastic behaviors are characterized analytically. The results show that the hoop stress concentration near the central hole is dominant due to the tension Lorentz force and it is certainly suppressed by filling a non-superconductive material in the hole. Without change of the magnetization characteristic of the superconductor, the filling material provides effective way to remedy the stress state at the verge of the hollow in the superconductive cylinder by adjusting its Young modulus. The magnetostriction of the composite cylinder under the cycled magnetic field is further presented. Effect of the applied maximum field, complete penetration field parameter and filling material parameter on the profile of the cycled magnetostriction for the composite superconducting cylinder is discussed in detail.

Remagnetization effects due to lateral displacement above a PMG on bulk HTS magnet

For a high-Tc superconducting (HTS) maglev system with large force requirements, the use of magnetized bulk high-Tc superconductor magnets (MBSCMs) is a good candidate because of its strong flux pinning ability and corresponding high trapped flux. Different from the rare-earth permanent magnet (PM), the trapped flux of a MBSCM is sustained by the supercurrent produced by a magnetizing process, so the trapped flux is sensitive to variations of the supercurrent. The lateral displacement of a MBSCM above a PM guideway (PMG) will provide disturbance of the applied field and then alter the supercurrent as a process of remagnetization. Different magnetization histories will bring different remagnetization characteristics and consequently diverse levitation performances for a MBSCM during the lateral displacements. When the MBSCMs are applied into the HTS maglev system, the influence of lateral displacements on levitation performance should be taken into consideration. This article investigates the remagnetization characteristics of a MBSCM when it is subject to the lateral displacements above a PMG with different trapped magnetic flux and opposite magnetization polarities. Relevant analyses about the internal supercurrent configuration based on the critical state model are also included to better understand the remagnetization characteristic of a MBSCM.

A numerical study to investigate magnetization, current density and trapped field properties of doped‐Sm123 bulk superconductor under different magnetic fields

AbstractIn this study, an FEM code based on a critical state model is used to further understand the electromagnetic behaviour of Yb211‐doped Sm123 bulk superconductor by calculating the current and magnetic field distributions in doped samples both in uniform and non‐uniform external field conditions. The results of magnetization loops from the numerical modelling have been compared with experimental results and they show a good match. In the numerical calculation result, the highest trapped field value, Bz, is 0.49 T for x = 0.25. In addition, the value of Bz trapped field in a non‐uniform applied field is calculated as 18% larger than in uniform applied field and the current density of samples as a function of radius increase particularly at the sample edge with a non‐uniform applied field for the x = 0.25 sample. It can be said that this numerical method will be a useful candidate to get an easy solution to many electromagnetic problems of doped‐superconductor with complex geometries and in different field conditions as the performed numerical study is successful in clarifying magnetization, trapped field and pinning properties of Yb211‐doped Sm123 superconductor.

Modeling non-local electrodynamics in superconducting films: the case of a right angle corner

We consider magnetic flux penetration in a superconducting film with a concave corner. Unlike convex corners, where the current flow pattern is easily constructed from Bean’s critical state model, the current flow pattern at a concave corner is highly nontrivial. To address the problem, we do a numerical flux creep simulation, where particular attention is paid to efficient handling of the non-local electrodynamics, characteristic of superconducting films in the transverse geometry. We find that the current stream lines at the concave corner are close to circular, but the small deviation from exact circles ensures that the electric field is continuous inside the sample. Yet, the electric field is, as expected, very high at the concave corner. At low fields, the critical state penetration is deeper from the concave corner than from the straight edges, which is a consequence of the electrodynamic non-locality. A magneto-optical experiment on Y Ba2Cu3Ox displays an almost perfect match with the magnetic flux distribution from the simulation, hence verifying the necessity of including electrodynamic non-locality in the modeling of superconducting thin films.

Electric field formulation for thin film magnetization problems

We derive a variational formulation for thin film magnetization problems for type-II superconductors written in terms of two variables: the electric field and the magnetization function. A numerical method, based on this formulation, makes it possible to accurately compute all variables of interest, including the electric field, for any value of the power in the power law current–voltage relation characterizing the superconducting material. For high power values we obtain a good approximation to the critical state model solution. Numerical simulation results are presented for simply and multiply connected films, and also for an inhomogeneous film.

2D electromagnetic modelling of superconductors

Some issues concerning the numerical analysis of superconductors are discussed and a novel approach to 2D modelling is proposed. Both axial and translational symmetric as well as current driven and voltage driven systems are examined in detail. The E–J power law is chosen instead of the critical state model as a constitutive relation of the material and the need to modify this relation in order to account for the normal state transition at high currents is discussed. A linear space reconstruction of the current density by means of nodal shape functions is used in order to build the finite dimensional model. A method to relax the tangential continuity of the current density, which is inherent to the discretization method used, is discussed. The performance of the proposed approach, both in terms of current distribution and AC loss, is evaluated with reference to some cases of practical interest involving composite materials. The role of the electric field as a natural state variable for superconducting problems is also pointed out. The use of the method as an alternative to the circuit approach or edge elements for modelling the superconductors is finally discussed.

Low Field ac Susceptibility and High Harmonics Studies in PbMo6S8 Polycrystalline Superconductor

The ac susceptibility and high harmonic response of polycrystalline PbMo6S8 superconductor are measured in the presence of small ac excitation field and dc magnetic field applied on it. Our experimental results could be explained by the classical Bean critical state model, when the current does not depend on the field and even harmonics are not observed.

A critical state model for saturated and unsaturated interfaces

Based on the concept of independent stress state variables to consider the impact of unsaturated conditions, an elasto-plastic critical state constitutive model for saturated and unsaturated interfaces is introduced in this paper. The proposed model is capable of predicting many characteristics of unsaturated interface behavior, such as the dependence of initial tangent modulus, peak shear stress, dilatancy, and ultimate strength on matric suction, net normal stress, and the interface state measured with respect to the critical state line. To this aim, two distinct yield mechanisms are employed in the model. While change in stress ratio generates plastic deformation in the first mechanism, plastic deformations are due to an increase in net normal stress, decrease in matric suction, or both, in the second mechanism. The presence of appropriate state dependent ingredients enables the model to provide realistic predictions over a wide range of variations of density, net normal stress, and matric suction. By direct comparison of the model predictions with experimental data, the predictive capacity of the proposed model is evaluated.

Secondary Peak on Asymmetric Magnetization Loop of Type-II Superconductors

Asymmetric magnetization loops with a second peak effect were parameterized by the extended critical state model. The magnetic field distribution in a sample is considered. Expression is suggested for a peak of the critical current density and corresponding depression on field dependence of the depth of surface layer with equilibrium magnetization. These functions determine the width and the asymmetry of a magnetization loop. Asymmetry of the secondary peak height on magnetization branches for increasing and decreasing field is reproduced on the computed magnetization curves.

Time-dependent magnetization of a type-II superconductor numerically calculated by using the flux-creep equation

The macroscopic magnetic behaviors of a type-II superconductor, such as the field- or the temperature-dependent magnetization, have been described by using critical state models. However, because the models are time-independent, the magnetic relaxation in a type-II superconductor cannot be described by them, and the time dependence of the magnetization can affect the fieldor the temperature-dependent magnetization curve described by the models. In order to avoid the time independence of critical state models, we try the numerical calculation used by Qin et al., who mainly calculated the temperature dependence of the ac susceptibility χ(T). Their calculation showed that the frequency-dependent χ(T) could be obtained by using the flux-creep equation. We calculated the field-dependent magnetization and magnetic relaxation by using a numerical method. The calculated field-dependent magnetization M(H) curves shows the shapes of a typical type-II superconductor. The calculated magnetic relaxation do not show a logarithmic decay of the magnetization, but the addition of a surface barrier to the relaxation calculation caused a clear logarithmic decay of the magnetization, producing a crossover at a mid-time. This means that the logarithmic magnetic relaxation is caused by not only flux creep but also a combination of flux creep and a surface barrier.

Integral Equations for Computing AC Losses of Radially and Polygonally Arranged HTS Thin Tapes

In this paper, we derive integral equations for radially and polygonally arranged high-temperature superconductor thin tapes, and we solve them by finite-element method. The superconductor is modeled with a nonlinear power law, which allows the possibility of considering the dependence of the parameters on the magnetic field or the position. The ac losses are computed for a variety of geometrical configurations and for various values of the transport current. Differences with respect to existing analytical models, which are developed in the framework of the critical state model and only for certain values of the transport current, are pointed out.

DC and AC susceptibility study of sol–gel synthesized Bi2Sr2CaCu2O8+δ superconductor

We study DC susceptibility along with amplitude and frequency dependence of AC susceptibility of sol–gel synthesized polycrystalline samples of Bi2Sr2CaCu2O8+δ (Bi-2212) sintered at different temperatures. In particular, it is demonstrated that susceptibility techniques are an effective tool to characterize granular characteristics of high temperature superconductors. The results are discussed in the framework of Bean's critical state model whence the field and temperature dependence of critical current density is determined. Flux pinning force density is calculated and possibility of the pinning mechanisms prevalent in type II superconductors are investigated. Flux creep activation energy is determined in the light of vortex dynamics exhibited by frequency dependence of AC susceptibility. Since polycrystalline samples are granular in nature, we calculated grain volume fraction and separated the contribution of grain and matrix susceptibility from total measured AC susceptibility. We establish that increase in the sintering temperature not only changes the grain morphology but affects the superconducting properties significantly, validating the impact of grain boundaries on superconducting performance of studied Bi-2212 superconductor.

The effect of a constraining metal tube on flux pinning induced stress in a long cylindrical superconductor

The use of an alloy tube to impose pressure on a superconducting cylinder during magnetizing reduces pinning-induced tensile stress in high temperature superconductors has been well established. In this paper the natural contact state between the superconducting cylinder and the metal tube is modeled. An exact solution is obtained for the isotropic magnetoelastic problem with the superconductor behaving magnetically, and an expression for the contact pressure exerted on the superconductor by the metal tube is obtained. This expression explicitly gives the contribution of the ratio of Young’s modulus of the superconductor to that of the metal and the ratio of the internal to external radii of the metal tube. The stress profile in the superconductor, subjected to the restriction of metal tube, with both field cooled activation and pulse field activation is analyzed in terms of the Bean critical-state model. The results show that the metal tube can prevent radial expansion of the superconductor and can reduce the maximum tension for field-cooled and pulsed-field activations. These results are important for the selection of materials as well as the optimization of sizes of the alloy tube.