Computation of Losses in HTS Under the Action of Varying Magnetic Fields and Currents

Francesco Grilli; Enric Pardo; Antti Stenvall; Doan N. Nguyen; Weijia Yuan; Fedor Gömöry

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Computation of Losses in HTS Under the Action of Varying Magnetic Fields and Currents

Francesco Grilli, Enric Pardo, Antti Stenvall, Doan N. Nguyen, Weijia Yuan, Fedor Gömöry

arXiv:1306.6251v1Jun 26, 2013cond-mat.supr-con

TL;DR

Accurate ac-loss computation matters for designing HTS devices and cooling systems, but existing approaches must accommodate complex geometries, materials, and multiple loss mechanisms. This paper reviews analytical, numerical, and application-specific methods for calculating those losses. It concludes that 2-D methods can model realistic applications effectively, while fully 3-D extension remains a major challenge.

Problem
HTS ac-loss calculation must address complex conductor geometries, multiple materials, and hysteresis, eddy-current, coupling, and ferromagnetic loss contributions.
Method
The paper reviews methods that solve electromagnetic state variables and calculate losses for tapes, wires, devices, and varied excitations and applications.
Results
Two-dimensional methods can quickly compute HTS losses with good accuracy for geometries of arbitrary complexity and can handle magnetic materials, ramps, pulses, and combined fields and currents.
Takeaways & Limitations
Ac-loss calculations support cooling-system design and the evaluation of methods for reducing heat loads in HTS power applications.

Abstract

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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.

I. INTRODUCTION

The review motivates numerical ac-loss modeling for complex HTS conductors and introduces the principal loss mechanisms and superconducting material models. It emphasizes that critical-state descriptions provide useful first approximations while numerical methods address more complex geometries and conditions.

I. INTRODUCTION: Numerical ac-loss calculations support the design and commercialization of HTS devices by estimating losses and guiding loss-reduction solutions.High ac loss can make applications unattractive, while loss calculations also inform cooling and conductor design.
I. INTRODUCTION: Analytical loss models are fast and useful for basic understanding, but their geometry and operating-condition limitations restrict accurate estimates for real HTS devices.Numerical models handle increasing complexity at the cost of more complex implementation and longer computation times.
I. INTRODUCTION: Technical HTS conductors exhibit four loss categories: hysteresis, eddy-current, coupling, and ferromagnetic losses.These arise respectively in superconducting material, normal-metal parts, inter-filament paths, and magnetic materials.
I. INTRODUCTION: For a round wire carrying a sinusoidal current, current-density distributions differ at equal current values because vortex arrangements retain the signal’s history.The illustrated case uses a current of 0.6Ic within a sinusoidal signal of amplitude 0.9Ic.
I. INTRODUCTION: The critical-state model represents a macroscopic approximation that predicts important current-density and magnetic-field features without resolving individual-vortex details.Its formulation is based on flux penetration and magnetic history, including zero current in unpenetrated regions.

B. How to solve the electromagnetic quantities

The reviewed computational approaches first solve an electromagnetic state variable and then calculate losses from the resulting time-varying fields or currents. They span analytical, variational, and finite-element formulations for two-dimensional and more complex geometries.

B. How to solve the electromagnetic quantities: Ac-loss simulations solve a state variable such as J, H, T−Ω, or A, with φ sometimes included as an additional unknown.The variables depend on the selected formulation and numerical method.
B. How to solve the electromagnetic quantities: After solving the state variable, losses are calculated for cyclic or noncyclic transport currents, applied fields, or combinations of both.The applied field may be nonuniform and vary in orientation over time.
B. How to solve the electromagnetic quantities: General methods commonly solve two-dimensional cross-sections of infinitely long conductors or bodies with cylindrical symmetry, while surfaces and three-dimensional bodies require additional treatments.The reviewed framework distinguishes mathematically 2-D problems from bi-dimensional surfaces and 3-D bodies.
B. How to solve the electromagnetic quantities: Brandt-type and finite-element methods solve smooth E(J) problems through variables including J, φ, H, T, Ω, A, and φ.FEM models can simplify implementation and analysis through commercial software.
B. How to solve the electromagnetic quantities: Critical-state calculations are generally faster than smooth E(J) simulations but cannot describe relaxation effects or over-current situations.The speed advantage may justify reduced accuracy from the critical-state approximation.
B. How to solve the electromagnetic quantities: The reviewed methods can incorporate magnetic-field dependence and, in principle, spatial dependence of the critical current density.This supports modeling nonuniform conductor properties such as a width-dependent Jc profile.

2) On the A−φ and T−Ωformulations:

The A−φ and T−Ω formulations represent electromagnetic quantities in ways suited to transport currents, isolated filaments, thin surfaces, and three-dimensional bodies. Losses are obtained after solving the electromagnetic state and are based on local power dissipation.

2) On the A−φ and T−Ωformulations:: In the A−φ formulation, φ is the electrostatic potential and is needed when transport-current boundary conditions require compensation of ∂A/∂t.It also enters cases involving isolated filaments or geometries where the electric field must remain tangent to the superconductor surface.
2) On the A−φ and T−Ωformulations:: The T potential generates current density through J = ∇×T, while Ω relates T to H through T−H = ∇Ω.T is gauge-invariant up to a gradient transformation, so the meaning of Ω depends on the gauge of T.
2) On the A−φ and T−Ωformulations:: Thin films can use a scalar current potential g, with J(x,y) = ∇×ẑg(x,y), and can be solved using T-, H-, or variational formulations.The scalar field can also be interpreted as an effective magnetic-dipole density.
2) On the A−φ and T−Ωformulations:: Fully three-dimensional modeling must account for flux cutting when magnetic field and current density are not perpendicular.One approach uses a critical-current dependence on the angle between E and J; experiments indicate an elliptical dependence.
2) On the A−φ and T−Ωformulations:: Three-dimensional implementations include FEM formulations based on T−Ω, A−φ, or H, although the cited calculations assume J is parallel to E with a smooth E(J) relation.Variational 3-D critical-state descriptions are possible in principle but had not been brought into practice in the reviewed account.
2) On the A−φ and T−Ωformulations:: Once electromagnetic quantities are known, local dissipation p = J·E provides the basis for computing ac losses in superconductors and normal conductors.The review derives this relation through vortex motion and notes its use for eddy and coupling currents.

2) Application to the critical state model:

The critical-state treatment computes losses from discontinuous current reversals and simplifies the resulting expressions for specific geometries and excitations. It connects current-front motion, vector potential, and cycle-integrated loss formulas.

2) Application to the critical state model:: Under monotonic penetration, constant Jc, and suitable long-conductor or cylindrical-symmetry assumptions, the loss formulas simplify further.The treatment allows transport current and uniform applied field to be combined proportionally through u(t).
2) Application to the critical state model:: In the critical-state model, each position’s current changes instantaneously between Jc and −Jc once per half-cycle.The resulting time derivative is represented using a Dirac delta at the switching time.
2) Application to the critical state model:: The loss calculation uses the vector potential in Coulomb’s gauge and evaluates the current-front reversals along reverse and returning curves.The current-density sign shifts at the current front, while the initial front encloses a flux-free region with uniform A.
2) Application to the critical state model:: For pure transport current, the general loss expression reduces to the original Norris formula.For a uniform applied field, the general expression reduces to the formula given by Rhyner.
2) Application to the critical state model:: For applied-field excitation, magnetization losses can be derived from the local power-loss density integrated over the superconductor volume.The derivation assumes an applied field varying in time and considers the superconductor’s loss contribution.

3) Magnetization loss:

Magnetization loss is computed from the superconductor’s current and field distributions over a cycle, with formulations that accommodate uniform, nonuniform, and rotating applied fields.

The instantaneous power dissipation is obtained by integrating J_s · E over a volume containing the superconductor.
For periodic fields, the loss per cycle can be expressed through the superconductor field and the time derivative of the applied field.
The applied field need not be uniform and may rotate during the cycle, while the corresponding volume integral is not instantaneous power loss.
Magnetic-material losses can be calculated from the free-energy variation associated with changes in magnetization and applied field.
For infinitely long wires and tapes, volume integrations are replaced by cross-sectional integrations, with magnetic moments replaced by moments per unit length where appropriate.

4) AC losses from the point of view of the power source:

From the power-source perspective, ac loss separates into transport and magnetization contributions under suitable source conditions, but their individual attribution is not universally valid.

A superconductor may be driven by a transport current, an externally generated magnetic field, or both, corresponding to one or two current sources.
For a single conductor or complete coil under pure transport current, the ac loss can be obtained from the voltage delivered by its sole power source.
The first source term is called transport loss, while the second is called magnetization loss.
When current and field are in phase, the two loss terms correspond to losses covered by the transport and magnetization sources, respectively.
For arbitrary phase differences, the total-loss equation remains valid, but its two terms cannot always be assigned separately to the individual sources.

2) AC transport current:

Transport-current losses depend on conductor geometry, current amplitude, and material uniformity; analytical formulas cover simple cases, while other geometries require numerical methods.

Norris’s conformal-mapping results provide analytical transport-loss expressions for thin strips and elliptical wires in the critical-state model.
C-shaped measurement loops must close over at least the strip width to measure thin-strip ac loss properly.
The slab formula estimates only top and bottom losses from current penetration through a film’s wide surfaces.
Numerical methods are usually necessary for other geometries or for an E(J) relation beyond analytically tractable cases.
Degraded superconductor at conductor edges increases ac loss, whereas the reverse material distribution reduces it.
Transport-loss studies also include multiple parallel wires and tapes, where current can distribute freely among conductors.

3) Simultaneous alternating transport current and applied field:

Simultaneous alternating transport current and applied field has limited analytical coverage, while simulations and experiments show that phase strongly affects ac loss.

Simultaneous alternating transport current and applied field: Analytical solutions for in-phase current and field exist only for slabs and strips in the critical-state model.
Simultaneous alternating transport current and applied field: For arbitrary phase shifts, analytical solutions are limited to slabs and strips in the critical-state model and low applied fields.
Simultaneous alternating transport current and applied field: At large applied fields, simulations predict maximum ac loss at intermediate phase shifts for both power-law and critical-state models, confirmed by experiments.
Power-transmission cables: Cylindrical-shell approximations are commonly used for power-transmission cables because they have analytical critical-state solutions.
Power-transmission cables: Straight-tape approximations have negligible error for single-layer cables, while ferromagnetic substrates increase ac losses.
Power-transmission cables: Bending tapes into a slitted tube yields lower ac losses than straight tapes forming a polygon.
Power-transmission cables: Real spiral cable geometries require 3-D models for Bi-2223 tapes, whereas newer coated-conductor models reduce the problem to 1-D or 2-D.

5) Roebel cables:

Roebel-cable loss modeling balances the cable’s complicated transposed geometry against lower-dimensional approximations. Two- and three-dimensional studies identify where losses concentrate and how modeling assumptions affect accuracy.

Roebel cables: A 2-D tape-matrix approximation is appropriate when the transposition length greatly exceeds cable width, because crossed-strand losses are much smaller than straight-part losses.The approximation distinguishes uncoupled and fully coupled cases for perpendicular or arbitrarily angled applied fields.
Roebel cables: Cross-sectional models differ from total-loss calculations by no more than 10% in the reported comparison, although magnetization-loss agreement with experiments is less exact.The mismatch is attributed partly to imperfect experimental strand uncoupling and neglect of field dependence in Jc.
Roebel cables: Full 3-D models reveal high loss density near crossing strands under perpendicular applied fields, while transport-current losses mainly localize along strand edges.Periodic boundary conditions can keep a simulated Roebel periodic cell computationally manageable.

9) Pulsed applied field:

Pulsed-field loss calculations often relate slab responses to periodic ac behavior, while realistic finite cylinders and magnets require numerical or approximation-based modeling. The appropriate approach depends on geometry, field background, and whether detailed current distributions are needed.

Pulsed applied field: A slab’s response to a pulsed applied field is essentially similar to its response to a periodic ac field during corresponding increasing and decreasing segments.For a critical-state model the response is exact, whereas a smooth E(J) relation retains dependence on pulse shape.
Pulsed applied field: Finite cylinders provide more realistic bulk-sample geometries but require numerical computations for current and flux penetration, including models with self-heating.Both critical-state and power-law E(J) formulations have been applied.
Pulsed applied field: DC-magnet pulse-mode losses can be estimated from measured cable losses, slab approximations, or detailed current-distribution calculations for the whole magnet.The slab approximation is reported as good for large magnetic fields.
Pulsed applied field: With an ac field superimposed on a dc component in the same direction, flux penetration follows the pure-ac case after the initial increase, but the critical current density is lower.A dc transport current can increase ac-oscillation losses, whereas a transverse ac field on dc magnetization can strongly reduce magnetization.

B. Computation Methods for Eddy Currents

Eddy-current loss computation uses several electromagnetic formulations and loss-evaluation methods, with model choice depending on geometry, desired accuracy, and available material or contact-resistance data. The review also shows how frequency, current, temperature, and conductor construction affect the importance and estimation of these losses.

Loss evaluation: Eddy-current losses can be computed from solved current-density or magnetic-field distributions using Joule heating, domain integrals, or Poynting-vector surface integration.These methods parallel approaches used for superconducting hysteresis-loss calculations with nonlinear resistivity models.
Numerical formulations: Finite element, integral-equation, and hybrid methods are the main numerical approaches for eddy-current simulations.Hybrid integral-equation methods use either volume or boundary integration for boundary terms, while dense matrices can be accelerated by fast multipole methods.
Application example: A 1 cm YBCO tape example gives 380 mW/m hysteresis loss and approximately 5 mW/m eddy-current loss at 60 Hz, with critical peak current at 77 K.The stated tape has a 0.9 µm YBCO layer, 50 µm copper stabilization, and 260 A self-field critical current.
Loss significance: At power frequencies below 200 Hz, eddy-current losses are often reported as low, but realistic subcritical currents can make them comparable to hysteresis losses.In Bi-based Ag-sheathed tapes, their contribution ranges from 30% to 1% as operation current rises from 0.1Ic to 0.8Ic at 60 Hz and 77 K.
Coupling-loss models: Equivalent-circuit models provide an alternative for coupling-loss simulations but require complicated networks and measurements of contact resistances between many points.They were developed mainly for CICC cables and remain applicable to Roebel cables; accurate FEM modeling likewise requires interface-contact characterization.

B. Influence of Filamentary Coupling on Hysteresis Losses

Filament coupling strongly changes current distributions and hysteresis losses, while intermediate coupling requires models that capture matrix and contact-resistance effects. Magnetic substrates additionally alter field profiles and contribute their own losses.

Modeling implications: Intermediate coupling effects require three-dimensional simulations that include matrix resistivity and filament-to-matrix contact resistance.Common two-dimensional programs typically represent only fully coupled or completely uncoupled limits.
Contact resistance: Below 5000 Hz, coupling loss increases as filament-to-filament resistance decreases, while roughly tenfold higher contact resistance reduces coupling loss by about one order of magnitude.At 50 Hz, coupling loss remains much lower than hysteresis loss across the studied contact resistances.
Coupling cases: Fully coupled filaments interact like a larger filament, whereas uncoupled filaments behave approximately as separate filaments under an applied field.The coupled case shows visible filament interaction; the uncoupled distributions resemble one filament alone.
Coupling cases: Coupling increases superconducting magnetization and therefore produces higher losses than the uncoupled case.The simulations used two 0.5 mm-radius filaments separated by 1 mm under a 50 sin(2πft) mT field.
Magnetic substrates: Magnetic materials modify the field inside the superconductor and add hysteretic losses, so conductor models must account for both effects.RABiTS YBCO conductors typically include a ferromagnetic substrate.

B. Numerical Models

Numerical models for superconductors with magnetic parts primarily use FEM formulations that incorporate field-dependent magnetic properties and estimate losses from simulated field distributions. Their applicability depends on the available material data and validation.

FEM approaches: Two widely used numerical approaches for superconducting devices with magnetic parts employ the finite-element method.They are intended mainly for soft magnetic materials, with extensions possible for hard magnetic materials using approximate permeability data.
H-formulation FEM model: The H-formulation FEM model incorporates field-dependent permeability and ferromagnetic loss by modifying Faraday’s equation when permeability varies with magnetic field.The modified equation accounts for the time dependence introduced through µr(H).
Loss computation: Magnetic hysteresis loss is computed from the area of the local B−H loop as a function of the maximum induction reached during an ac cycle, then integrated over the magnetic material.The Qfe(Bm) relation is fitted from experimental data.
Applicability: The H-formulation model can handle complicated HTS geometries when the magnetic material’s field dependences of permeability and loss are known.For RABiTS conductors, experimentally determined Ni-W substrate functions were fitted and used in calculations compared with measured transport losses.
Alternative FEM model: The alternative FEM technique uses nonlinear permeability treatment with Newton–Raphson iteration, but it has not been validated experimentally.Ferromagnetic loss prediction was not performed in the cited implementation, although a similar approach could be used.
Critical-state implementation: A critical-state implementation assumes a neutral zone with zero current density and monotonic surface flux penetration during monotonic increases of current or applied field.These assumptions enable analytical updates of current density from the calibrated vector potential at the previous time step.

3) Other FEM-based Models:

Other FEM-based models and application studies extend loss analysis to magnetic shields, substrates, and power devices. The review links calculated losses to cooling requirements and identifies design strategies with component-specific trade-offs.

Other FEM-based Models: Magnetic-shield effectiveness depends on both shield shape and permeability when the shield is no larger than the superconductor.Other models use electrostatic–magnetostatic equivalence or adaptive resistivity extended to magnetic materials.
Power-application relevance: Ac-loss calculations are needed to size cooling systems because losses contribute to the thermal load of HTS devices.They also guide exploration of loss-reduction methods.
Loss components: The reviewed loss components are hysteretic, eddy-current, coupling-current, and ferromagnetic losses, each associated with distinct reduction strategies.Strategies include smaller filaments, higher stabilizer or matrix resistivity, and avoiding ferromagnetic materials.
Loss-reduction techniques: Reducing field amplitude or the perpendicular field component, and using flux diverters or shielding layers, can lower ac losses.Coated conductors produce significantly higher losses in perpendicular fields.
Loss-reduction techniques: Filamentization reduces coated-conductor hysteresis losses but introduces coupling losses through conductive barriers, remnant bridges, and interfilament magnetic-flux coupling.The large conductor aspect ratio contributes to high hysteresis loss in external fields.
Transmission cables: For transmission cables, the monoblock model treats helically wound tapes as a superconducting tube for engineering loss estimates.It neglects phase interaction and tape geometry but is frequently used for large-scale applications.

2) Power Transmission Cables:

Power-transmission cable losses depend on conductor geometry, spacing, winding arrangement, current distribution, and magnetic materials. The review also describes application-specific loss burdens and modeling challenges.

Power Transmission Cables: Cable ac losses depend on conductor number, tape width, gaps, former radius, and current distribution.Narrower tapes and smaller gaps are reported as effective loss-reduction strategies; mechanically constrained mono-layer cables may favor 2.5–4 mm tape widths.
Power Transmission Cables: In two-layer counter-wound cables, gap, polygonal, flux-transfer, and ferromagnetic losses are major contributions.Orienting inner substrates inward and outer substrates outward can minimize current imbalance and ferromagnetic losses.
Power Transmission Cables: Multi-layer cable losses are reported to be weakly affected by gaps between adjacent conductors and lateral critical-current distribution.
Power Transmission Cables: Twisting, transposition, filamentarization, and specialized HTS cable architectures reduce coupling or ac losses, although flat-tape conductors complicate transposition.Roebel, Rutherford, and twisted-stack designs are discussed for high-current applications.
Power Applications: 60% of the total thermal load in a commercial 12 kV resistive SFCL is attributed to ac losses.Other reviewed applications include transformers and SMES, where field or current changes can create substantial heat loads.
Conclusion and Outlook: Two-dimensional ac-loss calculations now handle complex geometries, magnetic materials, and non-sinusoidal excitations, while three-dimensional extension remains challenging.The review links application loss calculations to cooling-system design and heat-load reduction.

APPENDIX A ELECTRIC FIELD CREATED BY MOVING VORTICES

The appendix derives the electric field of moving vortices from vector and scalar potentials, then averages it over a volume containing part of a vortex. Symmetry and boundary behavior eliminate the surface contribution under the stated conditions.

Electric Field Created by Moving Vortices: The appendix derives the macroscopic electric field of moving vortices by averaging the vortex-generated field over an integration volume.
Electric Field Created by Moving Vortices: A moving vortex produces E_v = B_v × v + ∇(v · A_v − φ_v), with time dependence arising from vortex motion.A_v and φ_v denote the vector and scalar potentials associated with the vortex.
Electric Field Created by Moving Vortices: The surface-integral term vanishes because the potentials decay far from the vortex and near-core contributions cancel by symmetry.The same reasoning applies when the integration volume is a vortex-lattice cell.
Electric Field Created by Moving Vortices: The averaged field approximates the local field when the integration-volume base is larger than the vortex separation.The analysis is also stated to remain valid for Pearl vortices in thin films under perpendicular fields.
Coulomb Gauge: The derivation uses the Coulomb gauge, defined by Ψ = 0, for the vector and scalar potentials.In this gauge, the vector potential is divergence-free and depends only on current density.