Calculation of alternating current losses in stacks and coils made of second generation high temperature superconducting tapes for large scale applications
TL;DR: In this article, a homogenization method to model a stack of second generation High Temperature Superconducting tapes under AC applied transport current or magnetic field has been obtained, where the idea is to find an anisotropic bulk equivalent for the stack such that the geometrical layout of the internal alternating structures of insulating, metallic, superconducting, and substrate layers is washed out while keeping the overall electromagnetic behavior of the original stack.
Abstract: A homogenization method to model a stack of second generation High Temperature Superconducting tapes under AC applied transport current or magnetic field has been obtained. The idea is to find an anisotropic bulk equivalent for the stack such that the geometrical layout of the internal alternating structures of insulating, metallic, superconducting, and substrate layers is “washed” out while keeping the overall electromagnetic behavior of the original stack. We disregard assumptions upon the shape of the critical region and use a power law E–J relationship allowing for overcritical current densities to be considered. The method presented here allows for a computational speedup factor of up to 2 orders of magnitude when compared to full 2-D simulations taking into account the actual dimensions of the stacks without compromising accuracy.
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TL;DR: The basics of the finite-element method (FEM) based on theinline-formula, its evolution from 2D to 3D, its application for calculating critical currents and AC losses as well as magnetization of HTS bulks and tape stacks, and its application to study the dynamic resistance of superconductors and flux pumps are reviewed.
Abstract: This paper reviews the modeling of high-temperature superconductors (HTS) using the finiteelement method (FEM) based on the H-formulation of Maxwell's equations. This formulation has become the most popular numerical modeling method for simulating the electromagnetic behavior of HTS, especially thanks to the easiness of implementation in the commercial finite-element program COMSOL Multiphysics. Numerous studies prove that the H-formulation is able to simulate a wide scope of HTS topologies, from simple geometries such as HTS tapes and coils, to more complex HTS devices, up to large superconducting magnets. In this paper, we review the basics of the H-formulation, its evolution from 2D to 3D, its application for calculating critical currents and AC losses as well as magnetization of HTS bulks and tape stacks. We also review the use of the H-formulation for large-scale HTS applications, its use to solve multi-physics problems involving electromagnetic-thermal and electromagnetic-mechanical couplings, and its application to study the dynamic resistance of superconductors and flux pumps.
142 citations
Cites methods from "Calculation of alternating current ..."
...The challenge of simulating superconductors with very large width-to-thickness ratio (such as HTS coated conductors) was first tackled by artificially expanding the superconducting layer’s thickness [36] and then, for systems characterized by a large number of tapes, with the homogenization method [37]....
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TL;DR: In this article, a general review of the status of numerical modeling applied to the design of high temperature superconductor devices is presented, and the main limitations of existing numerical models are reported.
Abstract: In this paper, we present a general review of the status of numerical modelling applied to the design of high temperature superconductor devices. The importance of this tool is emphasized at the beginning of the paper, followed by formal definitions of the notions of models, numerical methods and numerical models. The state-of-the-art models are listed, and the main limitations of existing numerical models are reported. Those limitations are shown to concern two aspects: on the one hand, the numerical performance (i.e. speed) of the methods themselves is not good enough yet; on the other hand, the availability of model file templates, material data and benchmark problems is clearly insufficient. Paths for improving those elements are indicated in the paper. Besides the technical aspects of the research to be further pursued, for instance in adaptive numerical methods, most recommendations command for an increased collective effort for sharing files, data, codes and their documentation.
134 citations
TL;DR: In this article, the authors present a numerical model based on the critical state with angular field dependence of Jc to extract the Jc(B,theta) relation from experimental data.
Abstract: Numerical models for computing the effective critical current of devices made of HTS tapes require the knowledge of the Jc(B,theta) dependence, i.e. of the way the critical current density Jc depends on the magnetic flux density B and its orientation theta with respect to the tape. In this paper we present a numerical model based on the critical state with angular field dependence of Jc to extract the Jc(B,theta) relation from experimental data. The model takes into account the self-field created by the tape, which gives an important contribution when the field applied in the experiments is low. The same model can also be used to compute the effective critical current of devices composed of electromagnetically interacting tapes. Three examples are considered here: two differently current rated Roebel cables composed of REBCO coated conductors and a power cable prototype composed of Bi-2223 tapes. The critical currents computed with the numerical model show good agreement with the measured ones. The simulations reveal also that several parameter sets in the Jc(B,theta) give an equally good representation of the experimental characterization of the tapes and that the measured Ic values of cables are subjected to the influence of experimental conditions, such as Ic degradation due to the manufacturing and assembling process and non-uniformity of the tape properties. These two aspects make the determination of a very precise Jc(B,theta) expression probably unnecessary, as long as that expression is able to reproduce the main features of the angular dependence. The easiness of use of this model, which can be straightforwardly implemented in finite-element programs able to solve static electromagnetic problems, is very attractive both for researchers and devices manufactures who want to characterize superconducting tapes and calculate the effective critical current of superconducting devices.
121 citations
TL;DR: In this article, a review of the finite element method (FEM) model based on the $H$ formulation of Maxwell's equations used to calculate AC losses in high temperature superconductor (HTS) tapes, cables and windings for different applications is presented.
Abstract: This article presents a review of the finite element method (FEM) model based on the $H$ formulation of Maxwell's equations used to calculate AC losses in high temperature superconductor (HTS) tapes, cables and windings for different applications. This model, which uses the components of the magnetic field as state variables, has been gaining a great popularity and has been in use in tens of research groups around the world. This contribution first reviews the equations on which the model is based and their implementation in finite element method programs for different cases, such 2D longitudinal and axis-symmetric geometries, 3D geometries. Modeling strategies to tackle large number of HTS tapes, such as multi-scale and homogenization methods, are also introduced. Then, the second part of the article reviews the applications for which the $H$ formulations has been used to calculate AC losses, ranging from individual tapes, to complex cables and large magnet windings. Afterwards, a section is dedicated to the discussion of the $H$ formulation in terms of computational efficiency and easiness of implementation. Its pros and cons are listed. Finally, the last section draws the main conclusions.
119 citations
TL;DR: In this article, an efficient two-dimensional T-A formulation based approach is proposed to calculate the electromagnetic characteristics of tape stacks and coils made of second generation high temperature superconductors.
Abstract: An efficient two dimensional T-A formulation based approach is proposed to calculate the electromagnetic characteristics of tape stacks and coils made of second generation high temperature superconductors. In the approach, a thin strip approximation of the superconductor is used in which the superconducting layer is modeled as a 1-dimensional domain. The formulation is mainly based on the calculation of the current vector potential T in the superconductor layer and the calculation of the magnetic vector potential A in the whole space, which are coupled together in the model. Compared with previous T-based models, the proposed model is innovative in terms of magnetic vector potential A solving, which is achieved by using the differential method, instead of the integral method. To validate the T-A formulation model, it is used to simulate racetrack coils made of second generation high temperature superconducting (2G HTS) tape, and the results are compared with the experimentally obtained data on the AC loss. The results show that the T-A formulation is accurate and efficient in calculating 2G HTS coils, including magnetic field distribution, current density distribution, and AC loss. Finally, the proposed model is used for simulating a 2000 turn coil to demonstrate its effectiveness and efficiency in simulating large-scale 2G HTS coils.
117 citations
References
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Book•
01 Mar 1993
TL;DR: The Finite Element Method in Electromagnetics, Third Edition as discussed by the authors is a leading textbook on the finite element method, incorporating major advancements and further applications in the field of electromagnetic engineering.
Abstract: A new edition of the leading textbook on the finite element method, incorporating major advancements and further applications in the field of electromagneticsThe finite element method (FEM) is a powerful simulation technique used to solve boundary-value problems in a variety of engineering circumstances. It has been widely used for analysis of electromagnetic fields in antennas, radar scattering, RF and microwave engineering, high-speed/high-frequency circuits, wireless communication, electromagnetic compatibility, photonics, remote sensing, biomedical engineering, and space exploration.The Finite Element Method in Electromagnetics, Third Edition explains the methods processes and techniques in careful, meticulous prose and covers not only essential finite element method theory, but also its latest developments and applicationsgiving engineers a methodical way to quickly master this very powerful numerical technique for solving practical, often complicated, electromagnetic problems.Featuring over thirty percent new material, the third edition of this essential and comprehensive text now includes:A wider range of applications, including antennas, phased arrays, electric machines, high-frequency circuits, and crystal photonicsThe finite element analysis of wave propagation, scattering, and radiation in periodic structuresThe time-domain finite element method for analysis of wideband antennas and transient electromagnetic phenomenaNovel domain decomposition techniques for parallel computation and efficient simulation of large-scale problems, such as phased-array antennas and photonic crystalsAlong with a great many examples, The Finite Element Method in Electromagnetics is an ideal book for engineering students as well as for professionals in the field.
3,705 citations
"Calculation of alternating current ..." refers background in this paper
...Ï.7;H75, (23) where HÜÝ is the length of the edge joining the vertices E and F as presented in FIG. 12. Then, the magnetic field t within the element is given by: t L ÍzÜ*Ü 7 Ü@5 , (24) 17 where *Ü#TAB#is the tangential component of the magnetic field on the Eth edge. Just like in the previous section, it is easy to see that: ∙ L0 (25) and Ï Ht L 1 Δ Í*ÜHÜ 7 Ü@5 G à. (26) Ag...
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TL;DR: In this paper, a critical current density relation alpha /J = B/sub 0/ + B is deduced for Nb/sub 3/Sn and 3Nb-Zr.
Abstract: The magnetization in high-fteld superconductors has been investigated using tubular samples. When the sample assumes a critical state, wherein every region of the sample carries critical current density J(B) determined only by the local magnetic field B, the magnetization can be predicted quantitatively from the critical current density J(B). Using observed magnetization data, a critical current density relation alpha /J = B/sub 0/ + B is deduced for Nb/sub 3/Sn and 3Nb-Zr. Alpha is a direct measure of the current carrying capacity of a sample, and B/sub 0/ coincides approximately with the thermodynamic critical field of the material. Since this relation implies JB = alpha = const for B >> B/sub 0/, the Lorertz force plays an important role in determining the critical current density. (auth)
616 citations
"Calculation of alternating current ..." refers background in this paper
...etize the magnetic field L :*ë,ì;. FIG. 12 Triangular edge element For this purpose, the area coordinates :.5,6,7; are considered as follows: .Ý:,U; L 1 2∆ k=Ý E>ÝT E?ÝU o, (19) where ∆ 5 6 :T5:U6 FU7; ET6:U7 FU5; ET7:U5 FU6;; is the surface area of the triangular element and the constant coefficients =Ý, >Ý and ?Ý are given by: =5 LT6...
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TL;DR: In this paper, a numerical method is proposed to analyse the electromagnetic behavior of systems including high-temperature superconductors (HTSCs) in time-varying external fields and superconducting cables carrying AC transport current.
Abstract: A numerical method is proposed to analyse the electromagnetic behaviour of systems including high-temperature superconductors (HTSCs) in time-varying external fields and superconducting cables carrying AC transport current. The E–J constitutive law together with an H-formulation is used to calculate the current distribution and electromagnetic fields in HTSCs, and the magnetization of HTSCs; then the forces in the interaction between the electromagnet and the superconductor and the AC loss of the superconducting cable can be obtained. This numerical method is based on solving the partial differential equations time dependently and is adapted to the commercial finite element software Comsol Multiphysics 3.2. The advantage of this method is to make the modelling of the superconductivity simple, flexible and extendable.
428 citations
TL;DR: In this article, a new numerical model for computing the current density, field distributions and AC losses in superconductors is presented, based on the direct magnetic field H formulation without the use of vector and scalar potentials.
Abstract: This paper presents a new numerical model for computing the current density, field distributions and AC losses in superconductors. The model, based on the direct magnetic field H formulation without the use of vector and scalar potentials (which are used in conventional formulations), relies on first-order edge finite elements. These elements are by construction curl conforming and therefore suitable to satisfy the continuity of the tangential component of magnetic field across adjacent elements, with no need for explicitly imposing the condition . This allows the overcoming of one of the major problems of standard nodal elements with potential formulation: in the case of strong discontinuities or nonlinearities of the physical properties of the materials and/or in presence of sharp corners in the conductors' geometry, the discontinuities of the potentials' derivatives are unnatural and without smoothing artifices the convergence of the algorithm is put at risk. In this work we present in detail the model for two-dimensional geometries and we test it by comparing the numerical results with the predictions of analytical solutions for simple geometries. We use it successively for investigating cases of practical interest involving more complex configurations, where the interaction between adjacent tapes is important. In particular we discuss the results of AC losses in superconducting windings.
421 citations
"Calculation of alternating current ..." refers background in this paper
...n from Thakur et al.20. Once multiplied by the volume fraction, the equivalent , ,ä:n; dependence is given by: 8 ,,ä:n; L ,Ö , BÁÍÌ É Ç1 §G6+$ ∥+ 6 E|$D|6 $4 Ì Ê , (14) where $4 L42.65#TAB#mT, Ö , L28#TAB#GA/m6, G L0.29515, Ù L0.7, and $ ∥ and $D are respectively, the parallel and perpendicular components of the magnetic flux density with respect to the tape’s surfa...
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Book•
17 Dec 1997
TL;DR: In this article, the Scalar Magnetic Potential (SGP) approach is used to solve the problem of approximating the SGP with respect to the Wasserstein equation. But this approach is not suitable for all applications.
Abstract: Maxwell Equations: Overview. Magnetostatics: "Scalar Potential" Approach. Solving for the Scalar Magnetic Potential. The Approximate Magnetic Potential: Properties and Shortcomings. New Tools: Whitney Elements, Symmetry. Complementary in Magnetostatistics. Magnetostatistics in Infinite Domains. Eddy-Current Problems. Mathematical Background. Appendices. Subject Index.
408 citations
"Calculation of alternating current ..." refers methods in this paper
...he element is given by: t L ÍzÜ*Ü 7 Ü@5 , (24) 17 where *Ü#TAB#is the tangential component of the magnetic field on the Eth edge. Just like in the previous section, it is easy to see that: ∙ L0 (25) and Ï Ht L 1 Δ Í*ÜHÜ 7 Ü@5 G à. (26) Again, recalling that the tangential components *Ü,E∈<1,2,3, are space constants and using Ampere’s law Ï Ht Lv, it is easy to see that the cur...
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