Kim Critical State Model Research Articles

Kim model for flux-pinning-induced stress in a long cylindrical superconductor

In this work, the flux-pinning-induced stress distribution in a circular cylinder of high-temperature superconductors is studied by adopting the Kim critical state model to describe the relationship between the magnetic flux density and induced current. Based on the plane strain approach, the analytic expressions of the radial and hoop stress in the cylinder are derived for the zero-field cooling and field cooling magnetization processes. It is shown that the stress distributions depend on the activation processes and the values of the dimensionless parameter p in the Kim model, and the overall maximums of the stresses appear at or near the center of the cylinder where cracking may be most likely initiated. In addition, the Kim model has wider applicability than the Bean model, and the influence of p on the stress depends on the activation process. Generally speaking, these results may be useful for understanding the magnetoelastic problem in practical application of bulk superconductors.

The relation between local hysteresis losses and remanent magnetic fields in HTSC films

Various critical state models have been developed to understand the hysteresis loss mechanism of high-temperature superconducting (HTSC) films. The analytic relation between the hysteresis loss and the remanent field was obtained based on Bean's critical state model for thin films in the full-penetration case. Furthermore, numerical calculation of local hysteresis loops was carried out by Kim's critical state model. In this paper, we investigated local hysteresis losses for a GdBCO coated conductor by using low-temperature scanning Hall probe microscopy and reproduced the experimental results by applying the critical state model. Because of the demagnetizing effect in thin films, analysis of local hysteresis losses can be useful approach to understand of total hysteresis losses.

Fracture behaviors of a functionally graded thin superconducting film with transport currents based on the strain energy density theory

In this study, the strain energy density theory is used to investigate a central crack problem for a functionally graded superconducting film with the applied transport currents, where the Kim critical state model is adopted and the shear modulus is assumed to vary along the film’s width in a form of hyperbolic function. The flux and current densities, the stress intensity factors (SIFs) and energy density factors (EDFs) are all analytically obtained. Numerical results show the effects of applied transport currents, model parameters, and crack length on the EDFs and/or SIFs. Among others, in the process of descending transport current, increasing the graded parameter of shear modulus can inhibit crack propagation, and in general, the crack will propagate and grow into the field of shear modulus decreasing. Moreover, the fracture angle is independent of the applied transport currents, and the fracture angle generally increases slightly with either the increasing of material graded parameter or the increasing of crack length. This study should be useful for the application of superconducting devices.

Flux-Pinning-Induced Stress Behaviors in a Long Rectangular Superconducting Slab With a Central Rectangular Hole

In this paper, the rectangular hole problem is investigated for a long high-temperature superconducting slab under electromagnetic forces. The effects of rectangular hole on the magnetic flux density distribution in the slab are considered in either the Bean model or the Kim critical state model in the case of the descending field both for the zero-field cooling (ZFC) and the field cooling (FC) magnetization processes. Based on the finite element method, both the corresponding stress concentration factors near the corners and the distributions of nonzero principle stress along the hole edge are numerically calculated and discussed. Numerical results obtained show that the ZFC activation process has more significant influence on the stress concentration factors than the FC activation process. For every activation process, the peak values of stress concentration factors in the Bean model are larger than the corresponding ones in the Kim model. In addition, in the Kim model, the peak values of stress concentration factors generally decrease with the increasing of the introduced dimensionless parameter p. Additionally, the stress concentration factors usually increase with the increasing of the ratio of length and width of the hole. The present study should be helpful to the design and application of high-temperature superconductors.

Fracture problems of a superconducting slab with a central kinked crack

In this work, the central kinked crack problem is investigated for a long rectangular superconducting slab under electromagnetic forces. The distributions of both the current density and the magnetic flux density in the slab are obtained analytically in the Kim critical state model for both the zero-field cooling and the field cooling magnetization processes. And based on the finite element method, the stress intensity factors at the crack tips for decreasing magnetic fields are numerically calculated. Numerical results obtained show that the zero-field cooling activation process generally has more significant influence on the stress intensity factors than the field cooling activation process, and that for every activation process, as the applied field decreases, the superconducting slab is most dangerous when the currents in the crack region are just be influenced. In general, both the maximal mode-I stress intensity factors (SIFs) and mode-II SIFs decrease with the increasing of either the introduced dimensionless parameter p in the Kim model or the crack length. However, the effects of kinked angles on the SIFs are complex. The present study should be helpful to the design and application of high-temperature superconductors.

Fracture properties of a cylindrical superconductor with a central cross crack

In this work, the central cross crack problem is investigated for a long cylindrical high-temperature superconductor under electromagnetic forces. The distributions of both the current density and the magnetic flux density in the cylinder are obtained analytically in either the Bean or the Kim critical state model for both the zero-field cooling and the field cooling magnetization processes. Based on the finite element method, lots of the stress intensity factors at the crack tips for decreasing magnetic fields are numerically calculated. Numerical results obtained show that the field cooling activation process has more significant influence on the stress intensity factors than the zero-field cooling activation process, and that for every activation process, as the applied field decreases, the superconducting cylinder is most dangerous when the currents in the crack region are just be influenced. The safety of the superconducting cylinder with a cross crack depends on not only the magnetic process but also the crack length. Additionally, the introduced dimensionless parameter p in the Kim model has different effects on the stress intensity factors obtained from the most dangerous field and from the trapped field. The present study should be helpful to the design and application of high-temperature superconductors.

Edge-crack problem in a long cylindrical superconductor

In this work, the edge-crack problem is investigated for a long cylindrical high-temperature superconductor (HTS) under electromagnetic force. The distributions of the current density and the magnetic flux density in the cylinder are obtained analytically in the Kim critical state model. And then, based on the finite element method, the stress intensity factors at the crack tip for different magnetic fields and different lengths of the crack are obtained. The results obtained show that the applied field and the length of the edge crack have significant effects on the safety and validity of an HTS.

DC and AC Shielding Properties of Bulk High-Tc Superconducting Tubes

We have studied numerically and experimentally the magnetic flux penetration in high- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Tc</i> superconducting tube subjected to a uniform magnetic field parallel to its long axis. This study is carried in view of designing low-frequency magnetic shields by exploiting the diamagnetic properties of high- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Tc</i> superconducting ceramics. We have measured the field attenuation for applied magnetic fields in the frequency range 5 mHz-0.1 Hz by Hall probe measurements and at audio frequencies using a sensing coil. A simple 1D analysis using the Kim critical state model was found to be able to reproduce the experimental data satisfactorily. We have also determined the phase shift between the internal and the applied field both experimentally and numerically. Finally, we have studied the sweep rate dependence of the magnetic shielding properties, using data recorded either at several constant sweep rates d <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> /d <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> or at several AC fields of various amplitudes and frequencies. Both methods agree with each other and lead to a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -value of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> ~ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> law equal to ~ 40 at 77 K.

3D-Modeling Numerical Solutions of Electromagnetic Behavior of HTSC Bulk above Permanent Magnetic Guideway

This paper presents a 3D-modeling numerical method using finite element method (FEM) to simulate the electromagnetic behavior of high-temperature superconductors (HTSC). The models are formulated by the magnetic field vector method (H-method). The resolving code was written by FROTRAN language. The electromagnetic properties of HTSC are described though Kim critical-state model. The magnetic fields and current distribution in the bulk HTSC in the applied non-uniform external magnetic fields generated by the permanent magnetic guideway (PMG) are obtained using the proposed method. The magnetic levitation forces by the interaction between the bulk HTSC and the PMG are calculated. In order to validate the method, measurement of the vertical force between a bulk YBaCuO(YBCO) and a PMG is obtained. The measurement and simulation results show good matching. This method could be used in the HTSC magnetic levitation transportation system optimization design.

Kim-Type Critical State Models and Critical Currents of Thallium Based Superconductors

The two extended formulae in the power form of the Kim critical state model were used to fit the critical currents versus the dc applied magnetic field. Two samples of thallium based superconductors: the (Tl0.6Pb0.24Bi0.16)(Sr0.9Ba0.1)2Ca2Cu3Oy film on single-crystalline lanthanum aluminate and the bulk (Tl0.5Pb0.5)Sr2(Ca1.8Gd0.2)Cu2Oy were chosen to test the models. The formulae were compared to the percolation model described by the exponential expression. The first model fits the experimental data better for the thallium based film whereas the second approach is better for the thallium based bulk sample.

AC susceptibility and intergranular critical current density study in pure and Ag doped (La 1− xY x) 2Ba 2CaCu 5O z superconductors

The temperature and AC field amplitude variations of AC susceptibility have been measured on pure and 5 wt% Ag doped (La 1− x Y x ) 2Ba 2CaCu 5O z superconductors. The AC susceptibility as a function of field have been analyzed using Kim's critical state model. The temperature dependence of intergranular critical current density and the effective volume fractions of the grains have been estimated. The Ag doped samples show relatively large critical current density due to the improved intergranular coupling. The exponent of temperature variation of critical current density suggests that the weak links form superconductor–normal metal–superconductor (SNS) type of junctions for all the samples.

Effects of Texturing Mechanisms on the Critical Current Density in Sintered High-Tc Bi-2223 Superconductors

Ac susceptibility measurements on ceramic samples with the nominal composition of Bi1.8Pb0.4Sr2Ca2Cu3O10—δ prepared by various pressing pressures on the pellets (355, 533, 711 and 1067 MPa) and different directional cooling temperature rates (20, 10, and 5 °C/h) using a directional solidification approach with a stationary Pt crucible in a strong temperature gradient have been carried out. The intergranular critical current density (Jcm) distribution has been calculated from ac susceptibility measurements by using Bean's and Kim's critical state models. It is observed that the critical current density improved with increasing pressing pressure and decreasing directional cooling temperature rates.

Magnetic flux profiles inBi1.2Pb0.3Sr1.5Ca2Cu3OyandNdBa2Cu3O7−δsuperconductors and a simulation by critical-state models

The features of the flux profiles recorded for the sintered and press-sintered ${\mathrm{Bi}}_{1.2}{\mathrm{Pb}}_{0.3}{\mathrm{Sr}}_{1.5}{\mathrm{Ca}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$ (BSCCO), and sintered and melt-textured ${\mathrm{NdBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ (NdBCO) superconducting samples with widely different microstructures are analyzed. From the slope of the flux profiles (at ${H}_{\mathrm{ac}}\ensuremath{\rightarrow}0$) recorded for various dc fields, the intergranular ${J}_{\mathrm{ci}}$ ${(H}_{\mathrm{dc}})$ for the BSCCO samples is determined, whose field dependence is found to follow Kim's model. Using the fit parameters thus obtained, the flux profiles for different ${H}_{\mathrm{dc}}$ could be simulated very well including features like nonlinearity at low fields, the presence of a peak, and the oscillatory behavior. The flux profile of the sintered NdBCO recorded with zero ${H}_{\mathrm{dc}}$ could also be simulated well using Kim's critical-state model. An analysis of the intragranular flux profile determined from the measured profile at high fields suggests a wide distribution of ${H}_{g}^{*},$ (the full penetration field of the grain) and thereby of the ${J}_{\mathrm{cg}}.$ This result is in agreement with the broad transition observed in the ac susceptibility of this sample, attributed to the presence of ${\mathrm{Nd}}_{1+x}{\mathrm{Ba}}_{2\ensuremath{-}x}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ precipitates that modify the local ${J}_{\mathrm{cg}}$ values drastically. All the sintered specimen show nearly 45% of flux penetration (into the sample) at low enough fields ${(H}_{\mathrm{dc}}&lt;40\mathrm{Oe}),$ while the flux entry into the melt-textured NdBCO is $&lt;3%.$ At high dc fields (4--8 KOe) the profile exhibits two distinct slopes that are associated with a flux entry initially into the microcracks and then into the grains, from the microstructural considerations.

Analysis of critical-state problems in type-II superconductivity

An efficient numerical scheme is proposed for modeling the hysteretic magnetization of type-II superconductors. Numerical examples are presented for the Bean and Kim critical state models. It is shown that Bean's model is a Hele-Shaw type problem.

Ac-susceptibility study of critical current density in metal-substituted ErBa2Cu3O7- delta.

Studies of the ac susceptibility of the pure ${\mathrm{ErBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ (Er-123) and the substituted samples ${\mathrm{ErBa}}_{2}$ (${\mathrm{Cu}}_{0.995}$${\mathit{M}}_{0.005}$${)}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ (M=Fe, Ni, Co, Ga) have been made at different ac field amplitudes. Analysis of the temperature dependence of the ac susceptibility at these fields [X-T plots] has been done employing Kim's critical state model. The dopants are found to lower the temperature-dependent intergranular critical current density, ${\mathit{J}}_{\mathit{c}}$(T), of the Er-123 system at different rates. Fe doping decreases the ${\mathit{J}}_{\mathit{c}}$ at the highest rate whereas Ni substitution results in the least rate of ${\mathit{J}}_{\mathit{c}}$ lowering; the depression rates for Ga and Co doped samples are similar and intermediate. These results fully support the findings of our previous study [Phys. Rev. B 52, 1320 (1995)] of the flux creep effects on the same batch of samples wherein it was observed that the concerned substituents had a similar effect on the flux creep activation energy values (${\mathit{E}}_{\mathit{a}}$) of the system. The lower ${\mathit{J}}_{\mathit{c}}$ of the Fe, Ni, Co, and Ga substituted samples is attributed to the reduction in the zero-temperature pinning potential ${\mathit{I}}_{0}$(0) brought about by the dopants. \textcopyright{} 1996 The American Physical Society.

Magnetization study of superconducting grains in Bi 2.1Sr 1.9CaCu 2O 8+δ

The field dependence of DC magnetization was measured on a parallelepiped sample of superconducting Bi 2.1Sr 1.9CaCu 2O 8+δ at different temperatures below T c with a maximum applied field of 160 kA/m. The experimental intragranular magnetization extracted from the measured magnetization data was analysed in terms of the Kim critical state model together with the surface barrier modified equilibrium magnetization. Even though at low temperature ( T = 5 K) the experimental data could be explained by magnetization derived from the Kim model, the ‘Z’-shaped loops observed at higher temperatures could be explained only by taking into account the surface barrier modified equilibrium magnetization. Material-dependent parameters, such as critical current density as a function of field, fields for first flux entry ( H 1) and last flux exit ( H 2) and lower critical field ( H c1), were estimated. At higher temperatures, the contribution of equilibrium magnetization ( M eq) is found to be dominating compared to the bulk magnetization. The possible reason for the significant contribution of equilibrium magnetization in granular superconductors is discussed.

Microstructural and magnetisation study in melt-grown Y [sbnd]Ba [sbnd]Cu [sbnd]O samples

The effect of Y 2BaCuO 5 (211) and silver on the microstructure of melt-grown YBa 2Cu 3O 7− x (123) has been systematically investigated. An optimal amount (28 mol%) of 211 in the starting composition of 123 with 5 wt.% of silver gave critical current densities >10 4A/cm 2. Magnetisation measurements using a SQUID magnetometer have been utilised to characterise the superconducting properties of the material at various stages of optimisation. Bean's and Kim's critical-state models are used to analyse the magnetisation of the optimised composition (123 + 28 mol% + 5 wt.% Ag). An anticlockwise tilt observed in the magnetisation curve above 70 K in the optimised sample has been explained as being due to the presence of an additional paramagnetic contribution arising from 211 particles present in the 123 matrix.

Field-dependent critical currents in thin Nb superconducting disks

Magnetic hysteresis loops were measured on thin Nb superconducting disks using a SQUID-based technique; field-dependent critical currents have been observed in thin superconducting disks at low magnetic flux trapping densities (⩽ 4 G). Such phenomena were found to exist over a wide range of temperatures below the transition temperature. The observed field-dependent critical current densities are attributed to long-range interactions between trapped vortices in a superconducting film whose thickness is much less than or comparable to the penetration depth. In the framework of Kim's critical state model, we developed a numerical method to calculate the field-dependent critical currents by fitting the measured magnetic hysteresis loops. Our experimental data were adequately explained by Kim's critical state model. Shielding currents and magnetic field patterns in thin superconducting disks were calculated.

Hysteretical temperature dependence of AC susceptibility in granular high- Tc superconductors

The AC susceptibility, χ AC( T), with superimposed DC magnetic field, H DS a, has been measured as a function of temperature in bulk sintered LaSrCuO, for zero-field cooled (ZFC) and field-cooled (FC) samples. Results are analyzed within Kim's critical-state model extended to a granular superconductor, where the role of grains is to determine the temperature-dependent effective permeability, μ eff( T), and the effective intergranular local field, H eff j ( H DC a, T), which depends on the applied field, H DC a and the thermal and magnetic history, accounting for the observed hysteresis. A method to determine field-independent intergranular critical-state parameters (critical current at zero field and temperature-dependent density of pinning forces as well as μ eff( T)) and the proper scaling of H eff j ( H DC a, T) for ZFC and FC conditions is described. The calculated AC susceptibility for the whole temperature range for both magnetic conditions reproduces the main features of the experimental results.

Straightened voltage effect in high-T c superconductors

A new effect called ‘‘ac-current straightening’’ has been observed in ceramic (Bi,Pb)-2223 slabs carrying ac current Idc+Iac cos(ωt). The current-voltage (I-V) characteristics of the ceramic were measured at 77 K at frequencies ranging from 50 to 20 000 Hz. A spectrum analyzer showed a series of high harmonics in the voltage signal as well as a constant voltage drop. The full set of experimental data has been explained theoretically using the Bean–Kim critical state model with a magnetic field dependent critical current jc(H)=jc(0)/(1+H/H0). A low transport ac current gives a voltage linearly proportional to the frequency and quadratically proportional to the ac-current amplitude Iac. It consists of odd harmonics only. If a bias dc current is switched on, then even harmonics and a dc-voltage drop appear. Their amplitudes are proportional to the small parameter Iac/cH0 and depend on the Idc/Iac ratio.