Electromagnetic modelling of superconductors with a smooth current–voltage relation: variational principle and coils from a few turns to large magnets

Abstract

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 computationally time-efficient but still accurate numerical methods. This article develops a general variational formalism for any E(J) relation and applies 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. However, no additional magnetic materials interacting with the superconductor are taken directly into account. Regarding the coil modelling, we use a power law E(J) relation with magnetic field-dependent critical current density, Jc, and power law exponent, n. We test the numerical model by comparing the results 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 of 200 turns each). 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 fulfills the requirements for electromagnetic design of coated conductor windings. * Final version published as E Pardo, JŠouc and L Frolek 2015 Supercond. Sci. Technol. 28 044003, doi:10.1088/0953-2048/28/4/044003. Several minor typos have been corrected in the published version. 1 ReBCO stands for ReBa 2 Cu 3 O 7−x , where Re is a rare earth, typically Y, Gd or Sm. the flux creep exponent and the AC loss. Critial current density and flux-creep exponentIn this article, we use a ReBCO coated conductor tape from SuperPower [49] for all experiments. This tape is 4 mm wide, with a total of 40 µm copper stabilizer layers, a 1 µm thick superconducting layer, and a self-field critical current at 77 K of 128 A.We measured the dependence of the critical current density J c on the magnetic field magnitude |B| ≡ B and its orientation θ (see sketch in figure 3) at 77 K, as detailed in [50]. In order to extract J c from measurements of the tape critical current, I c , we corrected the spurious effects of the self-field, following the method in [50]. The reader can find the I c measurements and extracted J c for the tape used in this article in [30]. For completeness, we include the extracted J c (B, θ) relation, being J c (B, θ, J) = [J c,ab (B, θ, J) m + J c,c (B) m ] 1/m (60) with J c,ab (B, θ, J) = J 0,ab 1 + Bf (θ,J) B 0,ab β ab ,J c,c (B) = J 0,c

AC Loss Modeling in Superconducting Coils and Motors With Parallel Tapes as Conductor

“…We developed a numerical method to compute the AC loss in the stator of superconducting motors, which combines MEMEP [28] with conventional FEM. This method could also be used for superconducting rotors and fully superconducting motors, as long as the interaction between non-linear eddy currents in the rotor with those in the stator is not important.…”

“…For stand-alone coils, we model the superconductor by the Minimum Electro-Magnetic Entropy Production (MEMEP) [28] method. For the motor, we combine MEMEP with conventional static Finite-Element Methods (FEM).…”

Demagnetization of Cubic Gd-Ba-Cu-O Bulk Superconductor by Crossed-Fields: Measurements and Three-Dimensional Modeling

“…These also exist in several formulations in 3D like those for H [10,11], mixed H and magnetic scalar potential ψ [12], and T effective magnetization [13] (the latter also known as Minimum Electro-Magnetic Entropy Production in 3D, MEMEP 3D). Another practical formulation for 2D is the J formulation [14,15]. All FEM formulations and most varitional method formulations require to solve the surrounding air around the sample, in addition to the sample itself.…”

Simulation of High Temperature Superconductors and experimental validation

“…Additionally, the mixed H-ϕ-Ψ formulation [12,13] of the FE method uses cohomology basis functions in the dielectric region and allows to treat the air as an exact zero conductivity region. An alternative approach to the FE method is the variational method, which is valid for any electric field-current density relation and exists for several formulations: the H-formulation [14,15,16], the effective magnetization T -formulation [17] and the J-φ formulation [18,19,20,21]. Other accurate results have been achieved with circuit models, such as those in [22] for calculating AC losses.…”