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Theory and experiment testing flux-line-cutting physics
Abstract
We discuss predictions of five proposed theories for the critical state of
type-II superconductors accounting for both flux cutting and flux transport
(depinning). The theories predict different behaviours for the ratio $E_y/E_z$
of the transverse and parallel components of the in-plane electric field
produced just above the critical current of a type-II superconducting slab as a
function of the angle of an in-plane applied magnetic field. We present
experimental results measured using an epitaxially grown YBCO thin film
favoring one of the five theories: the extended elliptic critical-state model.
We conclude that when the current density $\bm J$ is neither parallel nor
perpendicular to the local magnetic flux density $\bm B$, both flux cutting and
flux transport occur simultaneously when $J$ exceeds the critical current
density $J_c$, indicating an intimate relationship between flux cutting and
depinning. We also conclude that the dynamical properties of the superconductor
when $J$ exceeds $J_c$ depend in detail upon two nonlinear effective
resistivities for flux cutting ($\rho_c$) and flux flow ($\rho_f$) and their
ratio $r= \rho_c/\rho_f$.
... In addition, the critical current density for a given |B| often increases when B becomes parallel (or close to parallel) to J, causing anisotropic resistivity. Although this effect is of physical interest [11,25,46], it does not have an impact on most power applications. In addition, it is usually assumed that the first critical magnetic field, H c1 , is negligible, as well as the reversible magnetization curve of the superconductor. ...
... An interesting fact is that these variational methods are able to take any vectorial E(J) relation into account (including the double CSM or the anisotropic CSM), where E does not need to be parallel to J [10,11,46,48,69]. This is especially relevant for force-free configurations [25,46]. ...
... Another interesting issue is force-free effects. These appear when there is a component of the magnetic field, B, that is parallel to the current density, causing no macroscopic contribution to the driving force density on superconducting vortices, f = B × J [25]. As a consequence, there is an increase in the critical current density compared to configurations where the magnetic field is perpendicular to the current density, as assumed in usual axisymmetric and infinitely long configurations [25]. ...
... The investigation of general critical states has deserved attention in numerous experiments in the last decades, but it has not been until recently that a clear evidence of the kind of mechanism controlling ∥ J c and ⊥ J c has been available. The relevance of parallel current flow in type-II superconductors was been tested in a dedicated experiment [20] that, provides direct information on the threshold (critical) current density when J and B are at an arbitrary angle. . Basically, the experimental data agree with a dependence of the kind ...
... This formula is a central result of our work, and will be discussed in detail below. Firstly, dimensionless units have been introduced for convenience, through the definitions As a consistency check, one may calculate the angle between the electric field (normal to ϕ ˆ ) and the current density (normal to α ˆ), just by starting with equation (10) Noticeably, this expression coincides exactly with the expression that was earlier derived for the quasi-linear model [5], and also with the experimental observation reported in [20]. What is more, the angle between the electric field and the current density happens to be independent of the power law exponent M. ...
... tr 0 the increasing value of the transport current at a given angle θ 0 with the applied field. Figure 7 shows the main features of our results for Γ = 1 3 c , the actual value reported in [20]. (i) To the left, we plot the sheet current profiles that appear for the extreme condition θ = 0. Notice the increase of the transport current density allowed towards the center of the sample. ...
Based on recent experimental results, and in the light of fundamental physical properties of the magnetic flux in type-II superconductors, we introduce a practical expression for the material law to be applied in numerical modelling of superconducting applications. Focusing on the computational side, in this paper, previous theory is worked out, so as to take the celebrated form of a power-law-like dependence for the current voltage characteristic. However, contrary to the common approach in numerical studies, this proposal suits the general situation of current density flow with components either parallel or perpendicular to the local magnetic field, and different constraints applying on each component. Mathematically, the theory is generated from an elliptic locus defined in terms of the current density vector components. From the physical side, this contour establishes the boundary for the onset of entropy production related to overcritical current flow in different conditions. The electric field is obtained by partial differentiation and points perpendicular to the ellipse. Some numerical examples, inspired by the geometry of a two-layer helical counter-wound cable are provided. Corrections to the widespread use of the implicit isotropic assumption (physical properties only depend on the modulus of the current density vector) are discussed, and essentially indicate that the current carrying capacity of practical systems may be underestimated by using such simplification.
... An additional feature of the general 3D problem is to determine the actual E(J) relation of the superconductor (where E is the electrical field and J is the current density) for J with a component in the magnetic field direction (flux cutting situation), which causes non-parallel E and J. Although there have been interesting theoretical [21,22] and experimental [23] works on this issue, the actual E(J) relation with non-parallel E and J remains mostly unknown. The present work does not investigate this problem, allowing any E(J) relation as input. ...
... where δ jk is 1 when j = k and 0 otherwise. Therefore, we have obtained that the Euler-Lagrange equations from the functional density (19), equations (23) and (24), correspond to (17) and (16) with a global Laplacian operator. Actually, for a general 3D body the part within the Laplacian of (23) and (24) also vanishes, obtaining equations (17) and (16). ...
... For the idealization of infinitely long wires or cables transporting a certain net current, the wire or cable actually contains a returning conductor that closes the circuit (see section 2.4 for details). As a consequence, all fields actually vanish at infinity and equations (23) and (24) imply (17) and (16), respectively. Then, we have found that the ∆A and φ that minimize the functional (18) are the solutions of the magnetostatic problem. ...
Many large-scale applications require electromagnetic modelling with
extensive numerical computations, such as magnets or 3-dimensional (3D) objects
like transposed conductors or motors and generators. Therefore, it is necessary
to develop time-efficient numerical methods. This article develops a general
variational formalism for any ${\bf E}({\bf J})$ relation and apply it to model
coated-conductor coils containing up to thousands of turns, taking
magnetization currents fully into account. The variational principle, valid for
any 3D situation, restricts the computations to the sample volume, reducing the
computation time. Regarding the coils modelling, we use a power law $E(J)$
relation with magnetic-field dependent critical current density, $J_c$, and
power law exponent, $n$. We validated the numerical model by comparing to
analytical formulas for thin strips and experiments for stacks of pancake
coils, finding a very good agreement. Afterwards, we model a magnet-size coil
of 4000 turns (stack of 20 pancake coils). We found that the AC loss is mainly
due to magnetization currents. We also found that for an $n$ exponent of 20,
the magnetization currents are greatly suppressed after 1 hour relaxation. In
addition, in coated conductor coils magnetization currents have an important
impact on the generated magnetic field; which should be taken into account for
magnet design. In conclusion, the presented numerical method fulfils the
requirements for electromagnetic design of coated conductor windings.
... However, the EECSM connects the components of the current density J and the electric field E at J > J c (B, φ) with the use of an additional parameter, namely the ratio r = ρ /ρ ⊥ between the nonlinear effective resistivities for flux-cutting (ρ ) and flux-flow (ρ ⊥ ). In a series of experiments reported on a thin YBCO film subject to in-plane currents and magnetic fields [33], the predictions of the EECSM and the other four models (GDCSM, BMM, OECSM, and BLRM) have been critically verified. A good agreement of the OECSM and the EECSM was found with the measured dependence of J c on the angle φ of the in-plane applied field. ...
... The results obtained in the experimental work [33] have motivated us to investigate how the parameter r, used within the EECSM, influences the magnetic response of hard superconductors subjected to rotating and crossed fields. In a previous work [34], the magnetic response of an irreversible type-II superconductor slab subjected to an in-plane rotating magnetic field was investigated with Bean's model and the OECSM, EECSM, and GDCSM. ...
... The solution of the system of differential equations for E, B, and J requires an appropriate constitutive equation relating these fields. As was mentioned above, we will apply the extended elliptic flux-line cutting critical-state model, recently proposed by Clem [32,33]. The new model introduces the following equations: ...
The magnetic behavior of an irreversible type-II superconducting slab
under the action of in-plane crossed fields is investigated within both
the original elliptic critical-state model and the extended one, which
was recently proposed by Clem. In particular, we study the suppression
of the remanent magnetization of a PbBi specimen by a sweeping external
transverse magnetic field. It is found that both elliptic critical-state
models reproduce the main features of available experimental
magnetization curves. We also show that the average magnetizations,
corresponding to diamagnetic and paramagnetic initial states at a static
bias field Hz, are asymmetrically reduced by the action of an
oscillating transverse field Hy. If the amplitude of the
oscillations of Hy is as large as the first penetration field
HP, the resulting state becomes paramagnetic after various
cycles of Hy. Such a kind of paramagnetism is attributed to
the anisotropy, induced by flux-line cutting effects, in the critical
current density. In PbBi samples, paramagnetism is expected to be
manifest in a wide range of values of the static bias field
Hz.
... This can be done by using a dependence of J c on the angle between the electric field, E, and the current density, J, [41], [62], [63]. Experimental evidence reveals an elliptical dependence [64]. The implementation of 3-D models can be done as follows. ...
... This model requires numerical methods to solve the current distribution [41], [183], [184]. Experimental evidence suggests an elliptical dependence between the electric field and the critical current density [64]. 9) Pulsed applied field: This section first summarizes the main modeling issues for bulk applications (slabs and cylinders ) and, afterwards, for coils in pulse mode. ...
... Even though power-law works in some simulations of simple geometries in three dimensions [261], [262], [263], [32], [264], experimental verification of these studies are missing. However, currently there is ongoing interest in solving this problem [64], [265], [183]. Second, in three dimensions it is not possible to obtain the same discretization level, and thus accuracy, as in two dimensions. ...
Numerical modeling of superconductors is widely recognized as a powerful tool
for interpreting experimental results, understanding physical mechanisms and
predicting the performance of high-temperature superconductor (HTS) tapes,
wires and devices. This is especially true for ac loss calculation, since a
sufficiently low ac loss value is imperative to make these materials attractive
for commercialization. In recent years, a large variety of numerical models,
based on different techniques and implementations, have been proposed by
researchers around the world, with the purpose of being able to estimate ac
losses in HTSs quickly and accurately. This article presents a literature
review of the methods for computing ac losses in HTS tapes, wires and devices.
Technical superconductors have a relatively complex geometry (filaments, which
might be twisted or transposed, or layers) and consist of different materials.
As a result, different loss contributions exist. In this paper, we describe the
ways of computing such loss contributions, which include hysteresis losses,
eddy current losses, coupling losses, and losses in ferromagnetic materials. We
also provide an estimation of the losses occurring in a variety of power
applications.
... One thing to be emphasized is that, in our model, the current and magnetic field are always perpendicular to each other. The problem about how to solve a 3D problem with an angle between the current and magnetic field is still open and readers can refer to [19,23] for information. ...
... Complicated geometry can be found in the helically wound superconducting cables [19] or magnets [20]. To circumvent the geometric modeling challenge, most of the previous cable models simplified the real geometry by neglecting the twisted configuration of the tapes so that they can be considered as straight parallel conductors [21]. ...
... Considering a coaxial HTS cable with concentric phases design, if the tapes are tightly wrapped with no gaps between them, the self-field generated inside the cable has azimuthal and longitudinal components [19]. In this model, the influence of the longitudinal field to the ac loss is studied. ...
A three-dimensional (3D) numerical model is proposed to solve the electromagnetic problems involving transport current and background field of a high-Tc superconducting (HTS) system. The model is characterized by the E–J power law and H-formulation, and is successfully implemented using finite element software. We first discuss the model in detail, including the mesh methods, boundary conditions and computing time. To validate the 3D model, we calculate the ac loss and trapped field solution for a bulk material and compare the results with the previously verified 2D solutions and an analytical solution. We then apply our model to test some typical problems such as superconducting bulk array and twisted conductors, which cannot be tackled by the 2D models. The new 3D model could be a powerful tool for researchers and engineers to investigate problems with a greater level of complicity.
... 11 Despite the great success of the elliptic model, 19 it turns out that there exist phenomena, associated with flux cutting, which are not completely described within such a model. So, in a very recent work, 28 the elliptic critical-state model and other four theoretical approaches for describing the critical state of type-II superconductors (GDCSM, extended GDCSM, 29,30 extended elliptic critical-state model, 28,31 and an elliptic critical-state model based on the variational principle 32 ) were tested. There, the angular dependencies of the critical current density J c and the electric field E (for J just above J c ) were measured, using an epitaxially grown YBCO thin film, and compared with the predictions of the five theories. ...
... 11 Despite the great success of the elliptic model, 19 it turns out that there exist phenomena, associated with flux cutting, which are not completely described within such a model. So, in a very recent work, 28 the elliptic critical-state model and other four theoretical approaches for describing the critical state of type-II superconductors (GDCSM, extended GDCSM, 29,30 extended elliptic critical-state model, 28,31 and an elliptic critical-state model based on the variational principle 32 ) were tested. There, the angular dependencies of the critical current density J c and the electric field E (for J just above J c ) were measured, using an epitaxially grown YBCO thin film, and compared with the predictions of the five theories. ...
... Besides, the smooth angular dependence of the ratio of the transverse to the longitudinal components of the electric field E y /E z for J just above J c , predicted by the three elliptic models, was verified in the experiment. 28 However, the original critical-state model 19 leads to small values of the ratio E y /E z in comparison with the experimental data and the results obtained from the other two elliptic models. On the basis of this detailed comparison between experiment and the five theories, it was concluded in Ref. 28 that the experiment favors only one of the models, namely the extended elliptic critical-state model. ...
The magnetic response of irreversible type-II superconductor slabs subjected
to in-plane rotating magnetic field is investigated by applying the circular,
elliptic, extended-elliptic, and rectangular flux-line-cutting critical-state
models. Specifically, the models have been applied to explain experiments on a
PbBi rotating disk in a fixed magnetic field ${\bm H}_a$, parallel to the flat
surfaces. Here, we have exploited the equivalency of the experimental situation
with that of a fixed disk under the action of a parallel magnetic field,
rotating in the opposite sense. The effect of both the magnitude $H_a$ of the
applied magnetic field and its angle of rotation $\alpha_s $ upon the
magnetization of the superconductor sample is analyzed. When $H_a$ is smaller
than the penetration field $H_P$, the magnetization components, parallel and
perpendicular to ${\bm H_a}$, oscillate with increasing the rotation angle. On
the other hand, if the magnitude of the applied field, $H_a$, is larger than
$H_P$, both magnetization components become constant functions of $\alpha_s$ at
large rotation angles. The evolution of the magnetic induction profiles inside
the superconductor is also studied.
... This is the so-called force-free configuration [35], where J c should be different than the typical J c (B, θ) measurements with B always perpendicular to the transport direction. Since measurements show that J c in force-free configuration is often higher [56,57], this could explain the overestimated cross-field demagnetization in the model. Measurements of solid-angle dependence, J c (B, θ, φ) with φ being the angle of B with the current density, are scarce for any type of sample [56] and missing for this particular tape. ...
... Since measurements show that J c in force-free configuration is often higher [56,57], this could explain the overestimated cross-field demagnetization in the model. Measurements of solid-angle dependence, J c (B, θ, φ) with φ being the angle of B with the current density, are scarce for any type of sample [56] and missing for this particular tape. The cause is the complexity of the measurements, requiring a double goniometer [58][59][60][61]. ...
Stacks of superconducting (SC) tapes can trap much higher magnetic fields than conventional magnets. This makes them very promising for motors and generators. However, ripple magnetic fields in these machines present a cross-field component that demagnetizes the stacks. At present, there is no quantitative agreement between measurements and modeling of cross-field demagnetization, mainly due to the need for a 3D model that takes the end effects and real micron-thick SC layer into account. This article presents 3D modeling and measurements of cross-field demagnetization in stacks of up to 5 tapes and initial magnetization modeling of stacks of up to 15 tapes. 3D modeling of the cross-field demagnetization explicitly shows that the critical current density, J c , in the direction perpendicular to the tape surface does not play a role in cross-field demagnetization. When taking the measured anisotropic magnetic field dependence of J c into account, 3D calculations agree with measurements with less than a 4% deviation, while the error of 2D modeling is much higher. Then, our 3D numerical methods can realistically predict cross-field demagnetization. Due to the force-free configuration of part of the current density, J, in the stack, better agreement with experiments will probably require measuring the J c anisotropy for the whole solid angle range, including J parallel to the magnetic field.
... This is the so-called force-free configuration [35], where J c should be different than the typical J c (B, θ) measurements with B always perpendicular to the transport direction. Since measurements show that J c in force-free configuration is often higher [56,57], this could explain the overestimated cross-field demagnetization in the model. Measurements of solid-angle dependence, J c (B, θ, φ) with φ being the angle of B with the current density, are scarce for any type of sample [56] and missing for this particular tape. ...
... Since measurements show that J c in force-free configuration is often higher [56,57], this could explain the overestimated cross-field demagnetization in the model. Measurements of solid-angle dependence, J c (B, θ, φ) with φ being the angle of B with the current density, are scarce for any type of sample [56] and missing for this particular tape. The cause is the complexity of the measurements, requiring a double goniometer [58][59][60][61]. ...
Stacks of superconducting tapes can trap much higher magnetic fields than conventional magnets. This makes them very promising for motors and generators. However, ripple magnetic fields in these machines present a cross-field component that demagnetizes the stacks. At present, there is no quantitative agreement between measurements and modeling, mainly due to the need of a 3D model that takes the real micron-thick superconducting layer into account. This article presents 3D modeling and measurements of cross-field demagnetization in stacks of up to 5 tapes and initial magnetization modeling of stacks of up to 15 tapes. 3D modeling of the cross-field demagnetization shows that the critical current density, $J_c$, in the $c$-axis does not play a role in cross-field demagnetization. When taking the measured anisotropic magnetic field dependence of $J_c$ into account, calculations agree with measurements with less than 4 \% deviation. Then, our 3D numerical methods can realistically predict cross-field demagnetization. Due to the force-free configuration of part of the current density, $J$, in the stack, better agreement with experiments will require measurement of the $J_c$ anisotropy for the whole solid angle range, including $J$ parallel to the magnetic field.
... In all the above treatments of the critical-state problem, current density and local magnetic field are perpendicular to each other. The behavior of type-II SCs in cases where J has components both perpendicular and parallel to the local field is still in a state of development [126]. Different approaches have been proposed to focus on these kinds of problems. ...
... The generalized double CSM [127]- [131] assumes two different critical parameters J c and J c⊥ acting as thresholds for the components of the current density parallel and perpendicular to the local field, respectively. The idea is that E ⊥ (E ), the electric field component perpendicular (parallel) to the local magnetic field, can occur only when the magnitude of J ⊥ (J ) is above J c⊥ (J c ). J c⊥ is the critical current density for flux depinning, and J c is the critical current density for flux cutting [126]. ...
... Another practically important situation is when the magnetic field and the current are parallel, for example in force-free superconduct-nomena, [13,[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] including flux cutting [19,30,33,34], helical normal/superconducting domains, [35,36] and helical vortex flow. [13,31] However, the vortex behavior responsible for the observed dissipation in parallel magnetic field is still under debate [14,28,30,31]. ...
... However, previous experiments on superconductors in parallel magnetic fields are still controversial. Namely, an electric field, and, correspondingly, dissipation appears along the field/current direction [4][5][6][7][8][9][10][11][12][13][14][15][16]. Moreover, the superconducting critical current appears to increase with the magnetic field [4][5][6][7][8]15]. ...
Significance
Absolute zero resistance of superconducting materials is difficult to achieve in practice due to the motion of microscopic Abrikosov vortices, especially when external currents are applied. Even a partial resistance reduction via vortex immobilization by microscopic material imperfections is the holy grail of superconductivity research. It is commonly believed that the dissipation increases with applied magnetic field since the number of vortices increases as well. Through the example of molybdenum–germanium superconducting nanostrips, we show that resistive losses due to vortex motion can actually be decreased by applying an increasing applied magnetic field parallel to the current. This surprising recovery of superconductivity is achieved through “vortex crowding”: The increased number of vortices impedes their mutual motion, resulting in straight, untwisted vortices.
... . K. [5] . He also developed the theory of reversible mag- 88 netization in the vortex liquid state of HTS materials, which 89 enabled the first determination of their upper critical fields from 90 magnetization measurements [6]operation the coils of such fault current limiters need to be 143 Fig. 1 . ...
... John R. Clem: Contributions to the Phenomenology of Superconductivity This theme culminated in 2011, just two years before he passed 85 away, with his theory of flux cutting [4], beautifully confirmed 86 in experiments conducted at Cambridge University in the 87 [5] . He also developed the theory of reversible mag- 88 netization in the vortex liquid state of HTS materials, which 89 enabled the first determination of their upper critical fields from operation the coils of such fault current limiters need to be 143 Fig. 1 . ...
John Clem, Distinguished Professor of Physics at Iowa State University and Ames Laboratory, who passed away in August 2013, was for many decades one of the world's leading contributors to the phenomenology of superconductivity. Hisfundamental contributions included elucidating domain structures in Type I superconductors, modeling reversiblemagnetization and upper critical field of high temperature superconductors (HTS), introducing the concept of“pancake” vortices in highly anisotropic HTS materials, and developing the theory of flux cutting. He alsotook a great interest in applied superconductivity and consulted for organizations such as IBM, Pirelli Cables and AMSC,deriving the microwave properties of HTS films, elucidating the mechanism of current flow in first generation HTS wires,establishing the transport measurement of ac loss in tapes, and calculating ac loss of fault current limiter coils andHTS power cables. As founder and chief editor of High-Tc Update, he also provided much needed perspective on the torrentof developments during the early years of HTS discoveries.
... We have previously shown that combinations of Equations (3a), (3b) and (3c) can fit data for J c () [10]. In Figure 8 we show data for a YBCO sample measured at 85 K, 1 T [30]. The data is fitted with a combination of Equations (3a) and (3c); the peak has the angular-Lorentzian shape. ...
... We discuss what this fitting implies for the vortex configurations for in-plane fields in Section 4. Figure 8. J c () at 85 K, 1 T for a YBCO thin film of thickness 500 nm. Data from [30]; experiment (■), full fit (▬), fit components (▬,▬). The fitting parameters for these data are summarized in Table 4. Table 4. Parameters of fit components in Figure 8. ...
Maximum entropy inference can be used to find equations for the critical currents (J(c)) in a type II superconductor as a function of temperature, applied magnetic field, and angle of the applied field, theta or phi. This approach provides an understanding of how the macroscopic critical currents arise from averaging over different sources of vortex pinning. The dependence of critical currents on temperature and magnetic field can be derived with logarithmic constraints and accord with expressions which have been widely used with empirical justification since the first development of technical superconductors. In this paper we provide a physical interpretation of the constraints leading to the distributions for J(c)(T) and J(c)(B), and discuss the implications for experimental data analysis. We expand the maximum entropy analysis of angular J(c) data to encompass samples which have correlated defects at arbitrary angles to the crystal axes giving both symmetric and asymmetric peaks and samples which show vortex channeling behavior. The distributions for angular data are derived using combinations of first, second or fourth order constraints on cot theta or cot phi. We discuss why these distributions apply whether or not correlated defects are aligned with the crystal axes and thereby provide a unified description of critical currents in superconductors. For J//B we discuss what the maximum entropy equations imply about the vortex geometry.
... In all the above treatments of the critical-state problem, current density and local magnetic field are perpendicular to each other. The behavior of type-II SCs in cases where J has components both perpendicular and parallel to the local field is still in a state of development [126]. Different approaches have been proposed to focus on these kinds of problems. ...
... The generalized double CSM [127]- [131] assumes two different critical parameters J c and J c⊥ acting as thresholds for the components of the current density parallel and perpendicular to the local field, respectively. The idea is that E ⊥ (E ), the electric field component perpendicular (parallel) to the local magnetic field, can occur only when the magnitude of J ⊥ (J ) is above J c⊥ (J c ). J c⊥ is the critical current density for flux depinning, and J c is the critical current density for flux cutting [126]. ...
We review the modeling progress of the magnetic response of hard type-II superconductors (SCs) using the critical-state model (CSM). We focus on magnetization loops when a uniform field is applied to the SC and on the levitation forces that appear when the applied field is nonuniform. Different analytical and numerical solutions of the CSM are discussed, and the main characteristics and parameters of the magnetization loops and levitation forces are reviewed. Although this paper does not pretend to be an exhaustive review of the modeling of type-II SCs or of the superconducting levitation studies, a general overview of some important theoretical models used to understand the hard type-II SC macroscopic behavior is presented.
... In the limits α = 0° (straight wires) and α = 90° (closed rings), the Garber pattern vanishes and in (1) the initial value I c0 is recovered. For larger currents (I > I t ), the perpendicularly oriented flux fronts from both sides interfere (flux cutting threshold [18] [19]) and the simple CSMbased description used so far loses its validity. ...
Conductors made of high-temperature (HTS) wires helically wound in one or more layers on round tubes (CORT) are compact, flexible, and can carry a large amount of current. Although these conductors were initially developed for DC applications, e.g. in magnets, it is worth considering their use for AC, e.g. in underground cables for medium voltage grids and with currents in the kA-range. In these cases, the major challenge is reducing AC losses. In contrast to a straight superconducting wire, in a helical arrangement, due to superconducting shielding, the current does not follow the direction of the wires, but takes a non-trivial zig-zag path within the individual HTS wires (Garber pattern). This includes current components across the thickness of the superconducting layers, so that the often used thin-shell approximation does not hold. In this contribution, we studied a one-layer three-wire CORT by means of fully three-dimensional simulations, based on the H-formulation of Maxwell's equations implemented in the commercial software package COMSOL Multiphysics. As a result of our simulations, the peculiar current profiles were confirmed. In addition, the influence of current, pitch angle, and frequency on the AC losses was studied. We found an optimum for the pitch angle and that the current profiles strongly depend on frequency.
... Clem introduced this concept into the generalized double-critical-state model (GDCSM) in the critical state and accounted for the mechanism of flux transport and flux cutting [24,25]. The experimental magnetic response of YBCO thin film [26] and the high-temperature superconducting slab [27] was successfully predicted by GDCSM in the case of J inclined to B. GDCSM was extended by Brandt and Mikitik [16], who integrated the coupling effect of flux cutting and depinning into the critical-state model. The authors introduce anisotropy into the critical state [28], transforming a 2D critical current density ellipse into a 3D ellipsoid. ...
This work is an extension of a previous critical-state model proposed by the same author [Li and Gao, SuST (2015)]. We want to show the evolution of critical states for a rather long time, and the physics behind the evolution is also desired. It is another purpose to make clear the role of mechanical deformation played in the evolution process of critical state. We thus develop the model with two new features: a) mechanical loads are applied to both ends of the slab, instead of no external force applied in the previous model; b) the critical current density is now a function of position, rather than a constant over the slab volume as in the previous model. The new results by the model consist of: a) the evolution with time for a given strain value is generally the same as the strain-free case; b) stretching accelerates the development of the critical state, and bending has no significant effect on the critical state; c) the detailed critical-state profiles may provide how to regulate the electromagnetic field by strain in a superconducting device with slab geometry for practical purpose.
... However, some settings in the simulation may generate a quantitative difference. First, B AC, ϕ , which is neglected in the simulation, has an influence on the critical current of the HTS tape [34,54,55]. Second, the field dependence of the critical current, i.e., the I c (B, θ) characteristics, set in the simulation was measured in 2021 for the HTS tapes manufactured in that year [48]; by contrast, the experimental coil was wound using the same type of HTS tape but was manufactured in 2022. ...
High-temperature superconducting (HTS) coils serving as DC magnets can be operated under non-negligible AC fields, like in synchronous machines of maglev trains and wind turbines. In these conditions, dynamic resistance generates in HTS tapes, causing redistribution/bypassing of the transport current inside the no-insulation (NI) coil and its unique operational features. This issue was studied by experiments on an NI coil with DC current supply put into external AC fields. Due to the current redistribution induced by dynamic resistance, the central magnetic field and voltage of the NI magnet initially undergo various transient processes, and eventually exhibit a stable central magnetic field reduction and a DC voltage. These time evolutions have implications for a time-varying torque and loss of an HTS machine. These time evolutions are strongly affected by the contact resistivity distribution, and whether it is the first time of the NI magnet being exposed to the AC field, showing several qualitatively different waveforms (e.g., some are even non-monotonic with time). The magnitudes of the stable central field reductions, and their observed linear correlation with the DC voltages are found to be decided by the local contact resistivity of the innermost and outermost several turns. It is also noted that the non-insulated turn-to-turn contact help lessening the loss induced by the dynamic resistance. A numerical model is established to analyse/explain those experimental results by observing the microscopic current distribution. Two risks of quench are noticed: (i) azimuthal current of the middle part turns increases as the AC field is applied; (ii) a concentration of radial current is observed near the terminals of the NI coil.
... It is the observation at 77 K [48], and has been assumed to be true under all conditions, that the angular variation in I c obtained for fields applied within the plane of the superconductor follows a simple geometric behaviour that sees it pass from some maximum value (limited by flux-line cutting or other effects) in the Lorentz force-free configuration of field and current parallel to a minimum value (under which most characterisation measurements are typically conducted) when the applied field and transport current are perpendicular, generating the maximum Lorentz force on the vortices which causes them to overcome their pinning. Fig. 1.12 Angle dependence of the critical current of an inclined substrate deposited 2G HTS wire sample from THEVA over a range of temperatures (lowest temperature corresponding to highest I c ) at an applied field of 2 T. The in-plane peak in I c is shifted by 30 • from 90 • to 120 • , reflecting the offset in the crystallographic orientation of the substrate. ...
As applications of high-temperature superconductors (HTS) begin to move beyond the demonstrator stage to prototypes of production devices, the need to develop robust design methodologies tailored to efficiency and reliability emerges. Questions of economics also begin to play a role with multiple established manufacturers of HTS wire offering products of widely varying performance at different price-points, although simple availability of material in the lengths required for real-world devices presently remains a factor. With the wide range of proven applications of HTS spanning the full operating parameter space of temperature and magnetic field range, a targeted approach to the specification, selection and characterisation of the source material to be used in these large engineering projects becomes required, tailored to the particular application of interest. Here, we outline a number of aspects of contemporary HTS wire performance of relevance to device design that emerge upon detailed wire characterisation under targeted application conditions, and provide pointers towards those characteristics of wire performance that feed in to the informed selection of the most appropriate wire for a given application.
... Another type of anisotropy is the "force-free" anisotropy, which appears when the current density presents a substantial parallel component with the local magnetic field. The parallel J component does not contribute to the macroscopic driving force (or Lorentz force) on the vortices, F = J × B, being the microscopic vortex dynamics for B J a complex process that includes flux cutting and crossing [26][27][28]. Many power applications with rotating applied fields are influenced with force-free effects. ...
When the magnetic field has a parallel component to the current density ${\bf J}$ there appear force-free effects due to flux cutting and crossing. This results in an anisotropic ${\bf E}({\bf J})$ relation, being ${\bf E}$ the electric field. Understanding force-free effects is interesting not only for the design of superconducting power and magnet applications but also for material characterization. This work develops and applies a fast and accurate computer modeling method based on a variational approach that can handle force-free anisotropic ${\bf E}({\bf J})$ relations and perform fully three dimensional (3D) calculations. We present a systematic study of force-free effects in rectangular thin films and prisms with several finite thicknesses under applied magnetic fields with arbitrary angle $\theta$ with the surface. The results are compared with the same situation with isotropic ${\bf E}({\bf J})$ relation. The thin film situation shows gradual critical current density penetration and a general increase of the magnitude of the magnetization with the angle $\theta$ but a minimum at the remnant state of the magnetization loop. The prism model presents current paths with 3D bending for all angles $\theta$. The average current density over the thickness agrees very well with the thin film model except for the highest angles. The prism hysteresis loops reveal a peak after the remnant state, which is due to the parallel component of the self-magnetic-field and is implicitly neglected for thin films. The presented numerical method shows the capability to take force-free situations into account for general 3D situations with a high number of degrees of freedom. The results reveal new features of force-free effects in thin films and prisms.
[Author version of the article available at https://arxiv.org/abs/1703.05529 ]
... [18,22,23]. It is proven experimentally that there is an elliptic relation with the electric field and current density [24] At this part of the work, losses at superconducting coils when a magnetic field is applied, have been studied. Since understanding the physical mechanisms of heat losses are very essential for superconducting electrical machinery design, a new method is employed to calculate losses at superconducting coils. ...
We calculate magnetic hysteresis loss of HTS coil under rotating magnetic field. No transport current through the coil is assumed other than induced current. In order to find the frequency and applied field direction dependence of the loss, we present the loss versus applied field amplitude curves, for the frequencies f=1, 10, 100 and 1000 Hz, for seven different orientations of magnetic field. The hysteretic loss through the SC component increases with increasing frequency. This behavior is a consequence of higher field penetration to the tape at higher frequencies.
... In general, however, the anisotropy is more complex and influenced by both angles θ and φ. When φ = 0° and θ = 90° for example, Figure 16, the superconductor is in the "forcefree" configuration and consequently the material can transport higher critical currents because the flux lines do not experience a force, if perfectly parallel to the external field [51][52][53]. ...
High temperature superconductors (HTS) and solid oxide fuel cells (SOFCs) both offer the possibility for dramatic improvements in efficiency in power applications such as generation, transmission and use of electrical energy. However, production costs and energy losses prohibit widespread adoption of these technologies. This thesis investigates low-cost methods to tailor the structures of HTS wires and SOFCs to reduce these energy losses. Section I focusses on methods to produce filamentary HTS coated conductors that show reduced AC losses. This includes spark-discharge striation to pattern existing HTS tapes and inkjet printing of different filamentary architectures. The printed structures are directly deposited filaments and ‘inverse’ printed tracks where an initially deposited barrier material separates superconducting regions. Furthermore, the concept and first stages of a more complex ‘Rutherford’ cable architecture are presented. Additionally, Section I investigates how waste material produced during the manufacture of an alternative low-AC loss cable design, the Roebel cable, can be used to make trapped field magnets that produce a uniform magnetic field profile over a large area. This trapped field magnet work is extended to study self-supporting soldered stacks of HTS tape that demonstrate unprecedented magnetic field uniformity. Section II looks at how nanostructuring porous SOFC electrodes via solution infiltration of precursors can improve long-term stability and low temperature performance. Inkjet printing is utilised as a scalable, low-cost technique to infiltrate lab-scale and commercial samples. Anode infiltration via inkjet printing is demonstrated and methods to increase nanoparticle loading beyond ~1 wt% are presented. Symmetric cells with infiltrated cathodes are shown to have improved performance and stability during high temperature aging. Additionally, the sequence of solution infiltration is found to be important for samples dual-infiltrated with two different nanoparticle precursors.
... First of all, we have compared our experimental data with the elliptic critical state model (ECSM) [40]. The main feature of this model is that the critical current (I C ) is dependent on both perpendicular and parallel components of the current (I ⊥ , I ), according to the formula ...
Superconducting 2H−NbSe2 single crystals show intrinsic low pinning values. Therefore, they are ideal materials with which to explore fundamental properties of vortices. (V, I) characteristics are the experimental data we have used to investigate the dissipation mechanisms in a rectangular-shaped 2H−NbSe2 single crystal. Particularly, we have studied dissipation behavior with magnetic fields applied in the plane of the crystal and parallel to the injected currents, i.e., in the force-free state where the vortex helical instability governs the vortex dynamics. In this regime, the data follow the elliptic critical state model and the voltage dissipation shows an exponential dependence, V∝eα(I−IC∥), IC∥ being the critical current in the force-free configuration and α a linear temperature-dependent parameter. Moreover, this exponential dependence can be observed for in-plane applied magnetic fields up to 40° off the current direction, which implies that the vortex helical instability plays a role in dissipation even out of the force-free configuration.
... Flux cutting is believed to be responsible for this enhancement [25,26] . A general critical state model(GCSM) that could describe this phenomenon was firstly proposed by Clem [27,28] , which takes into account both vortex motions induced by transverse current and vortex twist caused by longitudinal currents. This model is purely phenomenological. ...
Nowadays, superconducting high field magnets are used in numerous applications due to their superior properties. High temperature superconductors (HTS) are usually used for production of circular pancake or racetrack coils. However different geometries of HTS coils might be required for some specific applications. In this study, the HTS coil wound on a rectangular frame was fully characterized in homogeneous DC background field. The study contains measurements of critical current angular dependencies. The critical current of the entire coil and two selected strands under different magnitudes and orientations of external magnetic fields are measured. The critical regions of the coil in different angular regimes are determined. This study brings better understanding of the in- field performance of HTS coils wound on frames with right-angles.
... The numerical computations of this article have provided an insight of current penetration. The 3D bending of the current lines indicates that there could be force-free and flux cutting effects [39,40] in superconducting rectangular prisms. The component of the critical current density in the applied field direction may play a role, being the magnetization currents in a bulk and a stack of tapes not identical. ...
[Article available in arXiv at: https://arxiv.org/abs/1703.05529 ]
Bulk superconductors are used in both many applications and material characterization experiments, being the bulk shape of rectangular prism very frequent. However the magnetization currents are still mostly unknown for this kind of three dimensional (3D) shape, specially below the saturation magnetic field. Knowledge of the magnetization currents in this kind of samples is needed to interpret the measurements and the development of bulk materials for applications. This article presents a systematic analysis of the magnetization currents in prisms of square base and several thicknesses. We make this study by numerical modeling using a variational principle that enables high number of degrees of freedom. We also compute the magnetization loops and the saturation magnetic field, using a definition that is more relevant for thin prisms than previous ones. The article presents a practical analytical fit for any aspect ratio. For applied fields below the saturation field, the current paths are not rectangular, presenting 3D bending. The thickness-average results are consistent with previous modeling and measurements for thin films. The 3D bending of the current lines indicates that there could be flux cutting effects in rectangular prisms. The component of the critical current density in the applied field direction may play a role, being the magnetization currents in a bulk and a stack of tapes not identical.
... First of all, we have tested our experimental data with the elliptic critical state model (ECSM) (37). The main feature of this model is that critical current (I C ) is dependent on both perpendicular and parallel components of the current (I ⊥ , I ). ...
2H-NbSe2 single crystals present intrinsic low pinning values. Therefore, they are ideal materials to explore fundamental properties of vortex lattice behavior. (V, I) characteristics are the experimental probes we have used to capture the dissipation mechanisms in a rectangular shape 2H-NbSe2 single crystal. Particularly, we have studied dissipation behavior with magnetic fields applied in the plane of the crystal and parallel to the injected currents, i.e. in the forcefree state where the vortex helical instability governs the vortex dynamics. In this regime, we have found that voltage dissipation follows an exponential dependence. Moreover, this exponential dependence can be observed for in-plane magnetic fields applied up to 40 off the current direction, which implies that the vortex helical instability plays a role promoting dissipation even out of the force-free configuration.
... The situation is more intricate when the field is applied transversally to the remnant magnetization . This effect, called either "abnormal transverse magnetic field effect", "collapse of the static magnetization" or "crossed-field demagnetisation" was observed on low-T c wires [21,28,43], bulk (RE)BCO materials [22,25,28,33,34,40], MgB 2 [44] and recently on stacks of 2G tapes [45]. Up to now, however, the exact processes of the magnetization decay in the case of a crossed field configuration are not completely identified. ...
Due to their ability to trap large magnetic inductions, superconducting bulk materials can be used as powerful permanent magnets. The permanent magnetization of such materials, however, can be significantly affected by the application of several cycles of a transverse variable magnetic field. In this work, we study, at T = 77 K, the long term influence of transverse ac magnetic fields of small amplitudes (i.e. much smaller than the full penetration field) on the axial magnetization of a bulk single grain superconducting GdBCO pellet over a wide range of low frequencies (1 mHz – 20 Hz). Thermocouples are placed against the pellet surface to probe possible self-heating of the material during the experiments. A high sensitivity cryogenic Hall probe is placed close to the surface to record the local magnetic induction normal to the surface.
... Although the behaviour of type-II superconductors is well understood for the case where current density, J, is perpendicular to magnetic induction field, B, the theories for the case where J has components both perpendicular and parallel to B is still in a state of development. Several theoretical studies have been made in the literature aiming to describe the magnetic behaviour of irreversible type-II superconductors subjected to crossed magnetic fields including generalized double criticalstate model, Brandt and Mikitik model, original elliptic criticalstate model, extended elliptic critical-state model and Badia, Lopez, and Ruiz model [12], [13]. ...
This paper investigates the practical effectiveness of employing superconducting stacked tapes to superconducting electric machinery. The use of superconducting bulks in various practical applications has been addressed extensively in the literature. However, in practice, dramatic decrease in magnetization would occur on superconducting bulks due to the crossed field effect. In our study, we employed the superconducting stacked tapes in a synchronous superconducting motor, which was designed and fabricated in our laboratory, aiming to lessen demagnetization due to crossed field effect in comparison with superconducting bulks. Applying the transverse AC field, the effects of frequency, amplitude, and number of cycles of the transverse magnetic field are discussed. Furthermore, a stack of 16 layers of superconducting tapes is modelled and the consequences of applying the crossed magnetic field on the sample are evaluated. The confrontation between experiments and simulation allows us to thoroughly understand the crossed field effects on stacked tapes. At the end, a preventive treatment, based on the shielding characteristic of superconductor and materials with high permeability, i.e. $mu$-metal and metalic glass, is suggested. On the other hand, the shielding feature of aforementioned materials will hinder the penetration of magnetic field and, consequently, reduction of the demagnetization will be attained.
... In the PFM simulation, the bulk was immerged into liquid nitrogen as shown in figure 3, which maintains the temperature around 77 K. The current was assumed to flow parallel to the electric field in the present simulation, which is considered both to be a suitable macroscopic assumption for bulk superconductors and keeps open the problem of flux cutting [55,56]. ...
Large single-grain bulk high-temperature superconducting materials can trap high magnetic fields in comparison with conventional permanent magnets, making them ideal candidates to develop more compact and efficient devices, such as actuators, magnetic levitation systems, flywheel energy storage systems and electric machines. However, macro-segregation of Y-211 inclusions in melt processed Y–Ba–Cu–O (YBCO) limits the macroscopic critical current density Jc of such bulk superconductors, and hence, the potential trapped field. Recently, a new
fabrication technique with graded precursor powders has been developed, which results in a more uniform distribution of Y-211 particles, in order to further improve the superconducting properties of such materials. In order to develop this graded fabrication technique further, a 3D finite-element numerical simulation based on the H-formulation is performed in this paper. The trapped field characteristics of a graded YBCO sample magnetized by the field cooling method are simulated to validate the model, and the simulation results are consistent with the experimental measurements. In addition, the influence of the graded technique and various graded Jc distributions for pulsed field magnetization, recognized widely as a practical route for magnetizing samples in bulk superconductor applications, is also investigated, with respect to the trapped field and temperature profiles of graded samples. This modelling framework provides a new technique for assessing the performance of various sizes and geometries of graded bulk superconductors, and by adjusting the Y-211, and hence Jc, distribution, samples can be fabricated based on this concept to provide application-specific trapped field profiles, such as the generation of either a high magnetic field gradient or a high level of uniformity for the
traditionally conical, trapped field profile.
... We see immediately for this case that H is almost normal to the xy plane, such that the inclined angle 0 . 45 Equations (23), (24), (26), (29) and (31) ...
We introduce a critical-state model incorporating the anisotropy of flux-line pinning to analyze the critical states developing in an anisotropic biaxial superconducting slab exposed to a uniform perpendicular magnetic field and to two crossed in-plane magnetic fields which are applied successively. The theory is an extension of the anisotropic collective pinning theory developed by Mikitik and Brandt. The anisotropic flux-line pinning enters into the critical states by generating the angular dependence of the critical current density and by deviating the direction of the electric field from the current in the plane perpendicular to the vortex line. We find that an enhanced in-plane anisotropy moderates the gradients of the magnitudes of the magnetic field and the electric field along the slab thickness, however increases the gradients of their rotations.
... However, in the FF δV image, as shown in figures 7(d) and (e), we can clearly see that the contrast variation of the δV response (marked with arrows) correlates well with an a-axis grain parallel to the macroscopic current flow direction. The vortex behavior and critical current in the FF configuration has long been an interesting while difficult problem [27][28][29][30]. The electric field detected in the FF configuration is mainly induced by flux cutting. ...
Optimization of vortex pinning in REBCO coated conductors has been very successful in recent years, but here we report that strong current-limiting effects can still be present in even highly optimized samples. We studied a state-of-the-art MOCVD IBAD-MgO coated conductor, finding it to have a global pinning force Fpmax(77 K, H ∥ c axis) that reached 11 GN m−3. Using low temperature laser scanning microscopy (LTLSM), we found that the local electric field in the flux-flow state was very inhomogeneous and dominated by a high density of a-axis grains, which obstruct current flow on dimensions of several µm. By carefully cutting narrow tracks without such grains in the path, Fpmax rose to 17 GN m−3, a value exceeding all but a very few, carefully made research films. That today’s coated conductors can develop exceptional vortex pinning properties, while still losing a significant fraction of the current density to blocking by growth defects, emphasizes that coated conductor development requires simultaneous attention both to enhancement of vortex pinning and to minimization of current-blocking defects.
The no-insulation (NI) winding technique is promising to be applied in the persistent-current mode (PCM) operation of high-temperature superconducting (HTS) coils to provide those advantages. To apply the NI PCM coil, its demagnetization behaviour (i.e., decay of persistent DC current) by external AC field, which occurs in maglev trains, electric machines, and other dynamic magnet systems, is essential to be understood. For this purpose, a 3D FEM model, capturing the full electromagnetic properties of NI HTS coils, is established. This work has studied three kinds of AC fields, observing the impact of turn-to-turn contact resistivity on demagnetization rates, which is attributed to current distribution modulations. Under transverse AC field, the lower contact resistivity attracts more transport current to flow in the radial pathway to bypass the ‘dynamic resistance’ generated in the superconductor, leading to slower demagnetization. Under axial AC field, the demagnetization rate exhibits a non-monotonic relation with the contact resistivity: (1) the initial decrease of contact resistivity leads to concentration of induced AC current on the outer turns, which accelerates the demagnetization; (2) the further decrease of contact resistivity makes the current smartly redistribute to avoid to flow through the loss-concentrated outer turns, thus slowing down the demagnetization. Under the rotating DC field, a hybrid of the transverse and axial fields, the impact of contact resistivity on the demagnetization rate exhibits combined characteristics of the transverse and axial components. Besides, quantitative prediction on the demagnetization rate of NI PCM coil under external AC field is instructive for practical designs and operations, which is tried by this 3D FEM model, and a comparison with experimental result is conducted.
A comprehensive overview of the basic principles of vortex dynamics in superfluids, this book addresses the problems of vortex dynamics in all three superfluids available in laboratories (4He, 3He, and BEC of cold atoms) alongside discussions of the elasticity of vortices, forces on vortices, and vortex mass. Beginning with a summary of classical hydrodynamics, the book guides the reader through examinations of vortex dynamics from large scales to the microscopic scale. Topics such as vortex arrays in rotating superfluids, bound states in vortex cores and interaction of vortices with quasiparticles are discussed. The final chapter of the book considers implications of vortex dynamics to superfluid turbulence using simple scaling and symmetry arguments. Written from a unified point of view that avoids complicated mathematical approaches, this text is ideal for students and researchers working with vortex dynamics in superfluids, superconductors, magnetically ordered materials, neutron stars and cosmological models.
The motion of Abrikosov vortices is the dominant origin of dissipation in type II superconductors subjected to a magnetic field, which leads to a finite electrical resistance. It is generally believed that the increase in the magnetic field results in the aggravation of energy dissipation through the increase in vortex density. Here, we show a distinctive re-entrance of the dissipationless state in quasi-one-dimensional superconducting Ta2PdS5 nanostrips. Utilizing magnetotransport measurements, we unveil a prominent magnetoresistance drop with the increase in the magnetic field below the superconducting transition temperature, manifesting itself as a giant re-entrance to the superconducting phase. Time-dependent Ginzburg-Landau calculations show that this is originated from the suppression of the vortex motion by the increased energy barrier on the edges. Interestingly, both our experiments and simulations demonstrate that this giant re-entrance of superconductivity occurs only in certain geometrical regimes because of the finite size of the vortex.
Ginzburg-Landau theory is one of the most powerful phenomenological theories in physics, with particular predictive value in superconductivity. The formalism solves coupled nonlinear differential equations for both the electronic and magnetic responsiveness of a given superconductor to external electromagnetic excitations. With order parameter varying on the short scale of the coherence length, and the magnetic field being long-range, the numerical handling of 3D simulations becomes extremely challenging and time-consuming for realistic samples. Here we show precisely how one can employ graphics-processing units (GPUs) for this type of calculations, and obtain physics answers of interest in a reasonable time-frame - with speedup of over 100x compared to best available CPU implementations of the theory on a 2563 grid.
In this paper, we present the results of the measurements of HTS tape stack magnetization and magnetic levitation forces between the permanent-magnet NdFeB and the stacks composed of $n = 5$-100 tapes of ReBCO (with the size of 12 mm $\times$ 12 mm) in the field cooling and zero-field cooling modes in a crossed magnetic field. The alternating magnetic field parallel to the surface of the tapes was produced by a copper coil with the amplitude up to 80-Oe frequency was varied in the range from 50 to 400 Hz. Measurements were carried out in liquid nitrogen. The dependencies of the magnetization of the stacks and the levitation force on the magnitude and frequency of the alternating magnetic field were obtained. The suppression of the levitation force and magnetization in a crossed field for the stacks of various thicknesses was shown. The results were compared with the characteristics of the bulk superconductors.
The high interest concerning the investigation of themacroscopicmagnetic properties of type-II superconductors in the mixed state is markedly associated with its relevance to technological and industrial applications achieving elevated transport currents with no discernible energy dissipation. It relates to a wide list of physical phenomena concerning the physics of vortices, which may be analyzed in terms of interactions between the flux lines themselves (lattice elasticity and line cutting), and interactions with the underlying crystal structure averaged by the so-called flux pinning mechanism. In this chapter, a mesoscopic description of real type-II superconductors is described according to the so-called critical state theory, taking account a mean-field approach for a volume containing a big enough number of vortices and, making use of an appropriate material law incorporating the intrinsic magnetic anisotropy of the superconducting material. In a first part, this chapter reviews the classical statements of the critical state theory and derived approaches. Then, the most recent advances on the theoretical development of a generalized critical state theory focusing on novel variational statements and, the establishment of a general material law for 3D critical states with an associated magnetic anisotropy is presented. The so called smooth double
critical state theory is highlighted as the only unified theory for dealing with arbitrary models for the material law, and describing the underlying physics for the mechanism of flux depinning and cutting. Finally, pertinent mathematical and computational details about how to deal with the large-scale nonlinear minimization problems arisen from the critical state theory in 3D systems are revealed
We performed a magneto-optical study of flux distributions in yttrium barium copper oxide (YBCO) crystal under various applied crossed field orientations to elucidate the complex nature of magnetic flux cutting in superconductors. Our work reveals unusual vortex patterns induced by the interplay between flux cutting and vortex pinning. We observe strong flux penetration anisotropy of the normal flux B-perpendicular to in the presence of an in-plane field H-parallel to and associate the modified flux dynamics with the staircase structure of tilted vortices in YBCO and the flux-cutting process. We demonstrate that flux cutting can effectively delay vortex entry in the direction transverse to H-parallel to. Finally, we elucidate details of the vortex cutting and reconnection process using time-dependent Ginzburg-Landau simulations.
Bulk type-II irreversible superconductors can act as excellent passive magnetic shields, with a strong attenuation of low frequency magnetic fields. Up to now, the performances of superconducting magnetic shields have mainly been studied in a homogenous magnetic field, considering only immunity problems, i.e. when the field is applied outside the tube and the inner field should ideally be zero. In this paper, we aim to investigate experimentally and numerically the magnetic response of a high-Tc bulk superconducting hollow cylinder at 77 K in an emission problem, i.e. when subjected to the non-uniform magnetic field generated by a source coil placed inside the tube. A bespoke 3D mapping system coupled with a three-axis Hall probe is used to measure the magnetic flux density distribution outside the superconducting magnetic shield. A finite element model is developed to understand how the magnetic field penetrates into the superconductor and how the induced superconducting shielding currents flow inside the shield in the case where the emitting coil is placed coaxially inside the tube. The finite element modelling is found to be in excellent agreement with the experimental data. Results show that a concentration of the magnetic flux lines occurs between the emitting coil and the superconducting screen. This effect is observed both with the modelling and the experiment. In the case of a long tube, we show that the main features of the field penetration in the superconducting walls can be reproduced with a simple analytical 1D model. This model is used to estimate the maximum flux density of the emitting coil that can be shielded by the superconductor.
A method of solving the critical state in superconductors using the vector potential and commercial software (FlexPDE) is described. It avoids both time dependence and power law resistivity. It uses a material parameter which describes how far flux lines move before most become unpinned. This allows small oscillations and minor hysteresis loops to be modelled. The theory is applied to the problem of demagnetisation in bulks due to crossed fields. It may explain why experimental results do not agree with theory. The theory can be extended to coils, and two and three dimensions. This requires the introduction of a scalar potential Vo. This is not the usual scalar potential, which is due to electrostatic charges as the field is run up, but the integral of this value at the final field after charges have dissipated.
The crossed-magnetic-field effect on the demagnetization factor of stacked second generation (2G) high temperature superconducting tapes is presented. The superconducting sample was initially magnetized along the c-axis by the field cooling magnetization method and after achieving the magnetic relaxation of the sample, an extensive set of experimental measurements for different amplitudes of an applied ac magnetic field parallel to the ab-plane was performed. On the one hand, a striking reduction of the demagnetization factor compared with the reported values for superconducting bulks is reported. On the other hand, the demagnetization factor increases linearly with the amplitude of the ac transverse magnetic field confirming the universal linear behavior for the magnetic susceptibility predicted by Brandt [Phys. Rev. B 54, 4246 (1996)]. The study has been also pursued at different frequencies of the ac transverse magnetic field in order to determine the influence of this parameter on the demagnetization factor measurements. We report an even lower demagnetization factor as long as the frequency of the transverse magnetic field increases. Thus, the significant reduction on the demagnetization factor that we have found by using stacked 2G-superconducting tapes, with higher mechanical strength compared with the one of superconducting bulks, makes to this configuration a highly attractive candidate for the future development of more efficient high-power density rotating machines and strong magnet applications.
As in all research in engineering sciences, the final target in the research of superconductor technology is the construction of new and innovative applications—hopefully with commercial potential. Mesh method based modeling of AC losses in superconductors is an important field of engineering science when considering the commercialization of superconductivity, as AC losses can often be a restricting factor for the feasibility of superconducting AC applications, and thus, analyzing them beforehand is necessary. Hence, it is important that new results and ideas in this field are widely recognized and acknowledged both in scientific and industrial circles. To achieve a wider impact, a clear classification of the research is beneficial. In this topical review, we discuss the possibilities in this field by presenting a classification of research into seven different research directions. We emphasize the roles of these different types of research as connections between steps on the path from theory to applications. Furthermore, we present a review of recent research in these research directions.
This paper deals with the magnetic properties of bulk high temperature superconducting cylinders used as magnetic shields. We investigate, both numerically and experimentally, the magnetic properties of a hollow cylinder with two axial slits which cut the cylinder in equal halves. Finite element method modelling has been used with a three-dimensional geometry to help us in understanding how the superconducting currents flow in such a cut cylinder and therefore how the magnetic shielding properties are affected, depending on the magnetic field orientation. Modelling results show that the slits block the shielding current flow and act as an 'entrance channel' for the magnetic flux lines. The contribution of the slits to the total flux density that enters the cylinder is studied through the angle formed between the applied field and the internal field. The modelled data agree nicely with magnetic shielding properties measured on a bulk Bi-2212 hollow cylinder at 77 K. The results demonstrate that the magnetic flux penetration in such a geometry can be modelled successfully using only two parameters of the superconductor (constant Jc and n value), which were determined from magnetic measurements on the plain cylinder.
Vortex loops are generated by the inhomogeneous stray field of a magnetic dipole on top of a current-carrying mesoscopic superconductor. Cutting and recombination processes unfold under the applied drive, resulting in periodic voltage oscillations across the sample. We show that a direct and detectable consequence of the cutting and recombination of these vortex loops in the present setup is the onset of vortices at surfaces where they were absent prior to the application of the external current. The nonlinear dynamics of vortex loops is studied within the time-dependent Ginzburg-Landau theory to describe the profound three-dimensional features of their time evolution.
We present a statistical mechanical model for critical currents which is successful in describing both the in-plane and out-of-plane magnetic field angle dependence of J
c
. This model is constructed using the principle of maximum entropy, that is, by maximizing the information entropy of a distribution, subject to constraints. We show the same approach gives commonly assumed forms for J
c
(B) and J
c
(T). An expression for two or more variables, e.g. J
c
(B,T), therefore, follows the laws of probability for a joint distribution. This gives a useful way to generate predictions for J
c
(B,T) for the DC critical current in wires, cables or coils.
High temperature superconductor Roebel cables are well known for their large current capacity and low AC losses. For this reason they have become attractive candidates for many power applications. The continuous transposition of their strands reduces the coupling losses while ensuring better current sharing among them. However, since Roebel cables have a true 3D structure and are made of several high aspect ratio coated conductors, modelling and simulation of their electromagnetic properties is very challenging. Therefore, a realistic model taking into account the actual layout of the cable is unavoidably a large scale computational problem. In this work, we present a full 3D model of a Roebel cable with 14 strands. The model is based on the H-formulation, widely used for 2D problems. In order to keep the 3D features of the cable (in particular the magnetization currents near the transpositions), no simplifications are made other than the reduction of the modelled length according to the periodicity of the cable structure. The 3D model is used to study the dependence of AC losses on the amplitude of the AC applied magnetic field or transport current. Beyond the importance of simulating the Roebel cable layout, this work represents a further step into achieving 3D simulation of superconducting devices for real applications.
This paper describes experiments and theories of flux cutting in superconductors. The use of the flux line picture in free space is discussed. In superconductors cutting can either be by means of flux at an angle to other layers of flux, as in longitudinal current experiments, or due to shearing of the vortex lattice as in grain boundaries in YBCO. Experiments on longitudinal currents can be interpreted in terms of flux rings penetrating axial lines. More physical models of flux cutting are discussed but all predict much larger flux cutting forces than are observed. Also, cutting is occurring at angles between vortices of about one millidegree which is hard to explain. The double critical state model and its developments are discussed in relation to experiments on crossed and rotating fields. A new experiment suggested by Clem gives more direct information. It shows that an elliptical yield surface of the critical state works well, but none of the theoretical proposals for determining the direction of E are universally applicable. It appears that, as soon as any flux flow takes place, cutting also occurs. The conclusion is that new theories are required.
A theory for the electromagnetic response of type-II superconductors close
beyond the critical state is presented. Our formulation relies on general
physical principles applied to the superconductor as a thermodynamic system.
Equilibrium critical states, externally driven steady solutions, and transient
relaxation are altogether described in terms of free energy and entropy
production. This approach allows a consistent macroscopic statement that
incorporates the intricate vortex dynamic effects, revealed in non-idealized
experimental configurations. Magnetically anisotropic critical currents and
flux stirring resistivities are straightforwardly included in three dimensional
scenarios.
Starting from a variational form of our postulate, a numerical implementation
for practical configurations is shown. In particular, several results are
provided for the infinite strip geometry: voltage generation in multicomponent
experiments, and magnetic relaxation towards the critical state under applied
field and transport current. Explicitly, we show that for a given set of
external conditions, the well established critical states may be utterly
obtained as diffusive final profiles.
Contents Preface I Electromagnetism of type II superconductors 1 General
Statements Of The Critical State 1.1 The CS In The Maxwell Equations Formalism
1.2 The CS Regime And The MQS Limit 2 Variational Theory for CS Problems 2.1
General Principles Of The Variational Method 2.2 The Material Law: SCs with
magnetic anisotropy 2.2.1 Onto the 1D Critical States 2.2.2 Towards The 3D
Critical States 3 Computational Method Conclusions I References I II Critical
State Problems:Effects & Applications 4 Type-II SCs With Intrinsic Magnetic
Anisotropy 4.1 3D variational statement in slab geometry 4.2 Isotropic
predictions in -3D- configurations 4.3 T-states in -3D- configurations 4.4
CT-states in -3D- configurations 4.5 Smooth critical states in -3D-
configurations Appendix I Critical angle gradient in -3D- configurations 5 The
Longitudinal Transport Problem 5.1 Simplified analytical models and beyond
5.1.1 The simplest analytical model 5.1.2 The SDCST statement and the BM's
approach 5.2 Magnetic anisotropy and the uncommon effects 5.2.1 Extremal case:
The T-states model 5.2.2 Material laws with magnetic anisotropy: CT\chi -
models 6 Electromagnetism For Superconducting Wires 6.1 Theoretical framework
and general considerations 6.2 SC wires subjected to isolated external sources
6.2.1 Wires with an injected AC transport current 6.2.2 Wires under an external
AC magnetic flux 6.2.3 Ultimate considerations on the AC losses 6.3 SC wires
under simultaneous AC excitations (B_{0},I_{tr}) 6.3.1 Synchronous excitations
6.3.2 Asynchronous excitations Conclusions II References II Supplementary
Material II III Microscopical aspects also analyzed 7 E-Ph Theory And The Nodal
Kink Effect In HTSC 8 Is it necessary to go beyond the E-Ph mode? Conclusions
III References III Supplementary Material III IV Addenda
Relying on our theoretical approach for the superconducting critical state
problem in 3D magnetic field configurations, we present an exhaustive analysis
of the electrodynamic response for the so-called longitudinal transport problem
in the slab geometry. A wide set of experimental conditions have been
considered, including modulation of the applied magnetic field either
perpendicular or parallel (longitudinal) to the transport current density. The
main objective of our work was to characterize the role of the macroscopic
material law that should properly account for the underlying mechanisms of flux
cutting and depinning. The intriguing occurrence of negative current patterns
and the enhancement of the transport current flow along the center of the
superconducting sample are reproduced as a straightforward consequence of the
magnetically induced internal anisotropy. Moreover, we show that related to a
maximal projection of the current density vector onto the local magnetic field,
a maximal transport current density occurs somewhere within the sample. The
elusive measurement of the flux cutting threshold (critical value of such
parallel component $J_{c ||}$) is suggested on the basis of local measurements
of the transport current density. Finally, we show that a high correlation
exists between the evolution of the transport current density and the
appearance of paramagnetic peak structures in terms of the applied longitudinal
magnetic field.
We report on the in-plane magnetic field (H) dependence of the critical current density (J{sub c}) in meandered and planar single grain boundaries (GBs) isolated in YBa{sub 2}Cu{sub 3}O{sub 7-{delta}} (YBCO) coated conductors. The meandered boundaries resulted from metal-organic deposition, and the planar boundaries resulted from physical vapor deposition. The J{sub c}(H) properties of the planar GB are consistent with those previously seen in single GBs of YBCO films grown on SrTiO{sub 3} bicrystals. In the straight boundary, a characteristic flux channeling regime when H is oriented near the GB plane, associated with a reduced J{sub c}, is seen. The meandered GB does not show vortex channeling since it is not possible for a sufficient length of vortex line to lie within it.
When a current is applied perpendicular to the vortex lattice (VL), Lorentz force may cause the VL to drift and flux-flow dissipation is observed. When the current is parallel to the applied magnetic field in a force-free (FF) configuration, a dissipation is also observed but at higher values of applied current. It has been suggested that pinning as well as free surfaces play an important role in the stabilization of the VL in the FF configuration. In thin YBa<sub>2</sub>Cu<sub>3</sub>O<sub>7</sub> films, FF configurations can be obtained when Hparab with the current flowing parallel to the ab-planes. In this work we study the influence of thickness, growth method and pinning centers on the dissipation mechanism at variable Lorentz force and FF configurations. Comparisons of experiments done at maximum and variable Lorentz force show that there are two pinning regimes when the field is rotated in these configurations; one consistent with only a decrease in the applied force, indicated by the overlap of the power law exponent of the current-voltage curves as the field is rotated toward the ab-planes, and another very close to the ab-planes, where the dissipation characteristics change.
The longitudinal transport problem (the current is applied parallel to some
bias magnetic field) in type-II superconductors is analyzed theoretically.
Based on analytical results for simplified configurations, and relying on
numerical studies for general scenarios, it is shown that a remarkable
inversion of the current flow in a surface layer may be predicted under a wide
set of experimental conditions. Strongly inhomogeneous current density
profiles, characterized by enhanced transport toward the center and reduced, or
even negative, values at the periphery of the conductor, are expected when the
physical mechanisms of flux depinning and consumption (via line cutting) are
recalled. A number of striking collateral effects, such as local and global
paramagnetic behavior, are predicted. Our geometrical description of the
macroscopic material laws allows a pictorial interpretation of the physical
phenomena underlying the transport backflow.
The magnetic flux dynamics of type-II superconductors within the critical
state regime is posed in a generalized framework, by using a variational theory
supported by well established physical principles. The equivalence between the
variational statement and more conventional treatments, based on the solution
of the differential Maxwell equations together with appropriate conductivity
laws is shown. Advantages of the variational method are emphasized, focusing on
its numerical performance, that allows to explore new physical scenarios. In
particular, we present the extension of the so-called double critical state
model to three dimensional configurations in which only flux transport
(T-states), cutting (C-states) or both mechanisms (CT-states) occur. The theory
is applied to several problems. First, we show the features of the transition
from T to CT states. Second, we give a generalized expression for the flux
cutting threshold in 3-D and show its relevance in the slab geometry. In
addition, several models that allow to treat flux depinning and cutting
mechanisms are compared. Finally, the longitudinal transport problem (current
is applied parallel to the external magnetic field) is analyzed both under T
and CT conditions. The complex interaction between shielding and transport is
solved.
The critical currents of MOD/RABiTS and PLD/IBAD coated conductors have been
measured as a function of magnetic field orientation and compared to films
grown on single crystal substrates. By varying the orientation of magnetic
field applied in the plane of the film, we are able to determine the extent to
which current flow in each type of conductor is percolative. Standard
MOD/RABiTS conductors have also been compared to samples whose grain boundaries
have been doped by diffusing Ca from an overlayer. We find that undoped
MOD/RABiTS tapes have a less anisotropic in-plane field dependence than
PLD/IBAD tapes and that the uniformity of critical current as a function of
in-plane field angle is greater for MOD/RABiTS samples doped with Ca.
We give a theoretical description of the general critical states in which the critical currents in type-II superconductors are not perpendicular to the local magnetic induction. Such states frequently occur in real situations, e.g., when the sample shape is not sufficiently symmetric or the direction of the external magnetic field changes in some complex way. Our study is restricted to the states in which flux-line cutting does not occur. The properties of such general critical states can essentially differ from the well-known properties of the usual Bean critical states. To illustrate our approach, we analyze several examples. In particular, we consider the critical states in a slab placed in a uniform perpendicular magnetic field and to which two components of the in-plane magnetic field are then applied successively. We also analyze the critical states in a long thin strip placed in a perpendicular magnetic field which then is tilted towards the axis of the strip. Comment: 15 pages including 11 figures
The critical current properties of YBa2Cu3O7-δ low-angle grain boundaries where the applied field is rotated in the plane of the film were measured. It was observed that the current is strongly suppressed only when the applied magnetic field is within an angle φk of the grain boundary.
Experimental results and phenomenological theory based on the sponge
model for hysteretic high-field superconductors are given. After developing the
exposition for static magnetization, the response of the superconductors to
alternating fields superimposed upon steady fields is studied. (T.F.H.)
It is shown that a flux vortex, in the presence of a sufficiently large current density applied parallel to its axis, is unstable against the growth of helical perturbations. This instability, which has an analog in magnetohydrodynamics, may play a critical role in current-carrying type-II superconductors subjected to longitudinal magnetic fields.
The resistive state of the current-carrying type 2 superconducting rod in a longitudinal magnetic field was studied for Pb-Tl alloys. The herical structure composed of two kinds of domains was found from the measurement of the potential distribution on the sample surface. The larger domain was confirmed directly to be in a flux flow state and the other domain was estimated to be in a normal state by some theoretical considerations. The direction of the boundary between these domains was dependent only on the direction of magnetic field at the surface.
Current–voltage characteristics and their distribution on the sample surface were measured for a superconducting Pb–In slab at 4.2 K under a longitudinal magnetic field to investigate the mechanism of resistivity under this field geometry. A negative electric field was observed at a certain region of the sample surface at magnetic field higher than 48 mT. It is considered from the estimated direction of the Poynting vector that flux lines are expelled from approximately half of the whole surface area of the sample. The negative electric field was observed only inside this region, and was considered to come from the term, B→×v→ with B→ and v→ denoting the magnetic flux density and the velocity of flux lines, respectively. Although the differential resistivity in the resistive state under the longitudinal magnetic field was complicated because of this term, it increased monotonically with increasing magnetic field in a similar manner as under the transverse magnetic field. This agrees with the Bardeen–Stephen model, which predicts that the resistivity induced by the flux motion is proportional to the magnetic field. It is concluded from these results that the resistivity originates from the motion of flux lines.
A macroscopic theory for bulk superconductors is developed in the framework of the theory for other magnetic materials, where "magnetization" current is separated from "free" current on the basis of scale. This contrasts with the usual separation into equilibrium and nonequilibrium currents. In the present approach magnetization, on a large macroscopic scale, results from the vortex current, while the Meissner current and other surface currents are surface contributions to the Maxwell j⃗. The results are important for the development of thermodynamics in type-II superconductors. The advantage of the description developed here is that magnetization becomes a local concept and its associated magnetic field can be given physical meaning.
A general critical-state theory, including the effects of both flux-line cutting and flux pinning, is proposed for calculations of hysteresis in type-II superconductors in parallel applied magnetic fields that vary in both magnitude and direction. In this theory, if the magnitude of the electrical-current-density component perpendicular to the magnetic induction B⃗ exceeds the corresponding critical value Jc⊥, depinning occurs, and an electric field component E⊥ perpendicular to B⃗ appears; if the magnitude of the current-density component parallel to B⃗ exceeds the corresponding critical value Jc∥, flux-line cutting occurs, and an electric field component E∥ parallel to B⃗ appears. Model calculations are performed to solve for the electrodynamic response of a slab subjected to a parallel, constant magnetic field whose direction undergoes either continuous rotation or periodic oscillation. The relation of the theory to the pioneering experiments of LeBlanc and co-workers is discussed.
Energy dissipation associated with flux-line cutting (intersection and cross-joining of adjacent nonparallel vortices) is considered theoretically. The flux-line-cutting contribution to the dissipation per unit volume, arising from mutual annihilation of transverse magnetic flux, is identified as J⃗∥·E⃗∥. where J⃗∥ and E⃗∥ are the components of the current density and the electric field parallel to the magnetic induction. The dynamical behavior of the magnetic structure at the flux-line-cutting threshold is shown to be governed by a special critical-state model similar to that proposed by previous authors. The resulting flux-line-cutting critical-state model, characterized in planar geometry by a parallel critical current density Jc∥ or a critical angle gradient kc, is used to calculate predicted hysteretic ac flux-line-cutting losses in type-II superconductors in which the flux pinning is weak. The relation of the theory to previous experiments is discussed.
The suppression of the static magnetic moment M of a superconducting plate in the critical state by a sweeping magnetic field h⃗(t), applied perpendicularly to a dc magnetic field H→, has been studied experimentally and theoretically. For every quarter-period of a sweeping field h(t) of changing polarity with an amplitude h0, a noticeable decrease of M can be observed both for the paramagnetic and diamagnetic initial states even for small h0 compared to H. Numerical simulations within the framework of two existing theoretical approaches have been performed in order to study the evolution of the distribution of the magnetic induction and the suppression of the magnetic moment. It turns out that the Clem–Pérez-González double critical-state model describes this process qualitatively well in the first quarter-period for relatively high values of h0. A significant disagreement with the experimental data is observed for small values of the transverse magnetic field. On the other hand, the two-velocity hydrodynamic model provides an adequate explanation of the main features of the suppression of M for both paramagnetic and diamagnetic states and any values of h0.
An anisotropic model for describing the critical state of hard superconductors subjected to crossed and rotating magnetic fields was discussed. The magnetic induction inside a superconducting slab subjected to a dc-bias magnetic field was calculated. The suppression of the magnetization for paramagnetic and diamagnetic initial states occurs was studied. It was found that the anisotropic model reproduced the collapse of the magnetic moment for both small and large amplitudes of the sweeping transverse field.
The static electric potential on the surface of a reversible type II superconducting cylinder carrying a current in an axial field has been investigated as a continuous function of position using a fine sliding contact. Three distinct regions of behaviour are identified, at low, medium and high applied axial fields. The remarkable potential structures displaying negative field regions, which were inferred by Irie, Ezaki and Yamafuji (1974) from point contact measurements, have been traced directly with a resolution of 50 μm. The voltage structure is shown to be very stable and reproducible with a complex fine structure. It is more irregular than supposed by Irie et al. (1974); the negative field regions are narrower and the traces do not appear to be strictly periodic, although there is some evidence that certain features of the traces repeat at regular intervals.
The magnetic response of anisotropic irreversible type-II superconductors is investigated theoretically. Using an elliptic vertical law for the electric field E as a function of the current density J, we have reproduced available experimental magnetization curves of YBCO samples with the c axis lying in the sample plane. Specifically, we could reproduce quantitatively and interpret correctly the appearance of additional extrema and segments with relatively small slopes of the virgin magnetization curves when the direction of the applied magnetic field differs from the principal axes. The notable deformation of magnetization curves in a tilted magnetic field is connected to the strong coupling between the components of the magnetic induction.
While a considerable amount of work has been done in an effort to understand ac losses in power transmission cables made of first generation high temperature superconductor (HTS) wires, use of second generation (2G) HTS wires brings in some new considerations. The high critical current density of the HTS layer in 2G wires reduces the surface superconductor hysteretic losses, for which a new formula is derived. Instead, gap and polygonal losses, flux transfer losses in imbalanced two-layer cables and ferromagnetic losses for wires with NiW substrates constitute the principal contributions. A formula for the flux transfer losses is also derived with a paramagnetic approximation for the substrate. Current imbalance and losses associated with the magnetic substrate can be minimized by orienting the substrates of the inner winding inward and the outer winding outward.
The macroscopic theory of force free magnetic fields in the mixed state of type II superconductors is presented. Solutions for cylindrical geometry are compared with experimental data from a Pb-40 at% Tl alloy. In the experiments the sample is subjected to combinations of field and current in a longitudinal configuration and both axial magnetization and axial resistance are simultaneously monitored. The observed paramagnetic moment is in good agreement with the predictions. An unexpected feature of the experimental results is the appearance of flux flow resistance at high currents. Coupled magnetic and resistive oscillations and discontinuities are observed in the flux flow regime. The system is magnetically irreversible in the resistance less state and reversible during flux flow. The results are discussed in the context of force free field theory.
The behavior of hard superconductors subjected to crossed magnetic fields is theoretically investigated by employing an elliptic critical-state model. Here the anisotropy is induced by flux-line cutting. The model reproduces successfully the collapse of the magnetic moment under the action of a sweeping magnetic field, applied perpendicularly to a dc field, for diamagnetic and paramagnetic initial states. Besides, it explains the transition from the diamagnetic state to the paramagnetic one when the magnitudes of the crossed magnetic fields are of the same order.
Starting from our general critical‐state theory, which includes the effects of both flux‐line cutting and flux pinning, we develop a numerical method for finding the magnetic response of type‐II superconductors subjected to parallel magnetic fields that change in both magnitude and orientation. We describe model calculations of the time‐evolving magnetic profiles when the direction of an applied magnetic field of fixed magnitude oscillates with large amplitude, and we compare our results with related experiments by Cave and LeBlanc. We also report model calculations for the ac behavior when the sample is subjected to a dc bias field and a large‐amplitude ac field at various angles (0°, 45°, and 90°) relative to this field.
We have designed a probe to rotate a sample around two independent axes in a magnetic field of up to 8 T and at temperatures varying from 3 K to room temperature. The sample is in very good thermal contact with a thermometer and a regulated heater, enabling precise temperature control. Pogo pins enable quick electrical connections to the sample. Two computer‐controlled stepper motors rotate the sample around the two independent axes. This probe was primarily designed to investigate the anisotropy of the critical currents of superconducting thin films in a magnetic field.
Experiments have been performed on thin films with the magnetic field applied parallel to the sample current (the force‐free state). Plots of the flux‐flow critical current versus magnetic field show a pronounced temperature independent structure which is sensitive to the film thickness. This structure is interpreted in terms of flux vortex arrangements that are matched to the sample geometry. A flux‐cutting model for the dynamic behavior of vortices in this system is proposed and shown to be in qualitative agreement with the results.
By analyzing trajectories of solid hydrogen tracers in superfluid $^4$He, we
identify tens of thousands of individual reconnection events between quantized
vortices. We characterize the dynamics by the minimum separation distance
$\delta(t)$ between the two reconnecting vortices both before and after the
events. Applying dimensional arguments, this separation has been predicted to
behave asymptotically as $\delta(t) \approx A(\kappa |t-t_0|)^{1/2}$, where
$\kappa=h/m$ is the quantum of circulation. The major finding of the
experiments and their analysis is strong support for this asymptotic form with
$\kappa$ as the dominant controlling feature, although there are significant
event to event fluctuations. At the three-parameter level the dynamics may be
about equally well-fit by two modified expressions: (a) an arbitrary power-law
expression of the form $\delta(t)=B|t-t_0|^{\alpha}$ and (b) a
correction-factor expression $\delta(t)=A(\kappa|t-t_0|)^{1/2}(1+c|t-t_0|)$. In
light of possible physical interpretations we regard the correction-factor
expression (b), which attributes the observed deviations from the predicted
asymptotic form to fluctuations in the local environment and in boundary
conditions, as best describing our experimental data. The observed dynamics
appear statistically time-reversible, which suggests that an effective
equilibrium has been established in quantum turbulence on the time scales
investigated. We discuss the impact of reconnection on velocity statistics in
quantum turbulence and, as regards classical turbulence, we argue that forms
analogous to (b) could well provide an alternative interpretation of the
observed deviations from Kolmogorov scaling exponents of the longitudinal
structure functions.
The suppression of the static magnetic moment M of a superconducting plate by an alternating magnetic field applied perpendicularly to a dc magnetic field has been studied both experimentally and theoretically. Complete disappearance of the hysteresis of the magnetization curve in YBCO melt-textured samples was observed. Nonmonotonous behavior of M with the increase of the ac magnetic field amplitude was found. The dynamics of the collapse of the magnetic moment was studied in the pulse regime of the transverse field. The experimental results indicate homogeneity of the static magnetic induction in the sample regions where the ac field penetrates. The data obtained are interpreted within the framework of a two-liquid hydrodynamic model.
A general critical-state model including the effects of both flux-line cutting and flux pinning is used for calculating B-->-, J-->-, and E-->-field distributions during the approach to the quasisteady state of a type-II superconducting slab subjected to a parallel rotating magnetic field. Three initial magnetic configurations (diamagnetic, paramagnetic, and nonmagnetic) are considered. It is shown that, depending upon sample parameters and magnetic history, multiple-zone structures develop inside the superconductor. It is also shown that, regardless of the specimen's magnetic history, flux-line-cutting B consumption leads to a final state in which B has a diamagnetic profile near the surface of the specimen. The relation of these calculations to the pioneering experiments of LeBlanc and co-workers is discussed.
A general critical-state model including the effects of both flux-line cutting and flux pinning is used to predict the internal-magnetic-field distribution at the critical current of type-II superconductors subjected to parallel magnetic fields. The anticipated behavior is considered for specimens in the form of both slabs or strips and thin-walled cylindrical shells. Striking dependences upon the applied field angle are predicted for the direction of the electric field in slab geometry and for the magnitude and direction of the induced azimuthal current in cylindrical geometry.
Plots of the critical current against the square root of the magnetic field obtained from new experiments on thin type-II superconducting films under conditions of parallel magnetic field and current show regularly spaced steps. These novel results are interpreted as being due to the successive entry of planes of parallel vortices into an oscillating flux distribution associated with coherent flux cutting. The results allow the first estimate of the critical cutting force between vortices, a quantity of general significance for the dynamic response of vortices.
Transport critical current measurements have been performed on 5 degrees [001]-tilt thin film YBa(2)Cu(3)O(7-delta) single grain boundaries with the magnetic field rotated in the plane of the film, phi. The variation of the critical current has been determined as a function of the angle between the magnetic field and the grain boundary plane. In applied fields above 1 T the critical current j(c) is found to be strongly suppressed only when the magnetic field is within an angle phi(k) of the grain boundary. Outside this angular range the behavior of the artificial grain boundary is dominated by the critical current of the grains. We show that the phi dependence of j(c) in the suppressed region is well described by a flux cutting model.
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