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Review of the AC Loss Computation for HTS using the H-formulation
Boyang Shen, Francesco Grilli, Tim Coombs
TL;DR
AC losses constrain HTS applications because refrigeration at cryogenic temperatures makes dissipation costly. This review examines FEM models based on the H formulation across equations, geometries, scaling strategies, and applications. The reviewed studies generally agree with experiments within 10% to 50%, and the formulation remains widely used despite implementation and numerical considerations.
Problem
AC losses are an important limitation for HTS commercialization because refrigeration costs make cryogenic power dissipation burdensome, while analytical models are limited to simple geometries and critical-state laws.
Method
The review analyzes FEM H formulation models that solve Faraday’s equation using magnetic-field components as state variables across HTS tapes, cables, windings, geometries, and applications.
Results
Calculated AC losses generally agree with experiments, with reported accuracy varying between 10% and 50%.
Takeaways & Limitations
The H formulation is a widespread, de facto standard numerical tool whose flexibility supports AC-loss studies across varied HTS scenarios.
Takeaways & Limitations
The review considers AC losses at constant temperature and does not consider coupling with thermal models.
Abstract
from arXiv · showhide
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.
1 Introduction
HTS offer strong application potential, but AC losses impose costly refrigeration burdens that must be estimated accurately. This review examines the H formulation as a flexible numerical approach for realistic HTS geometries and applications, while excluding thermal coupling.
- Motivation: HTS are promising for superconducting applications because of their high current and power density, in-field behavior, and mechanical strength.Their performance has improved while prices have decreased, supporting proposed uses in magnets, machines, transformers, and other power applications.
- Motivation: AC losses limit HTS commercialization because refrigeration at very low temperatures makes power dissipation especially burdensome.At 77 K, the Carnot specific power is about 2.9, while realistic cooling-system efficiency is only 10-20%.
- Modeling gap: Analytical AC-loss models are restricted by simple geometries and critical-state constitutive laws, whereas FEM H formulation models address realistic applications.The review contrasts analytical expressions for tapes and infinite stacks with numerical modeling of electrodynamic responses in practical geometries.
- Modeling gap: Finite-turn coil models using the H formulation provide AC-loss estimates closer to measured data than single-turn or infinite-stack analytical limits.Analytical models supply lower and upper limits for a real double pancake coil, while the numerical model represents its finite geometry.
- Review scope: The H formulation uses magnetic-field components as state variables and became a de facto standard after implementations in home-made codes and COMSOL Multiphysics.The approach was first proposed in 2003 and later implemented independently in commercial FEM software.
- Review scope: The review covers mathematical implementations, practical HTS applications, computational efficiency, implementation advantages and drawbacks, and conclusions.It addresses multiple geometries and modeling strategies, while restricting the review to constant-temperature AC-loss calculations without thermal coupling.
2 H formulation
The H formulation computes AC losses by solving for magnetic-field components in finite elements, with material laws, current constraints, and geometry-specific adaptations. The review also describes homogenized and multi-scale strategies for making large HTS systems tractable.
- 2.1 Basic equations: The model solves Faraday’s equation with magnetic-field components as finite-element state variables, while enforcing a divergence-free magnetic flux density through the initial condition.The divergence-free condition persists in time, but its numerical preservation can depend on solver robustness and element choice.
- J · dAk, (6): The formulation can impose transport currents through integral constraints over surfaces intersecting individual conductors, including current redistribution among tapes in non-transposed cables.Fully transposed cables require additional constraints if every tape must carry identical current.
- J · dAk, (6): The superconductor is represented by a nonlinear power-law resistivity whose parameters include critical current density Jc and flux-creep index n, with field and spatial dependencies optionally included.The model can also represent resistive terminations by adding a resistance-current contribution for each superconducting tape.
- J · dAk, (6): AC losses are obtained by integrating local dissipation J · E over the superconducting domain and averaging over a cycle after excluding the initial transient.The same expression can be adapted to arbitrary time-dependent excitations.
- 2.2 Equations in 2D for longitudinal and axisymmetric problems: For long conductors and cylindrically symmetric windings, a 2D cross-sectional model uses two magnetic-field components and assumes material properties are constant along the conductor length.Cylindrical cases use the corresponding equations in cylindrical coordinates.
- 2.4 Homogenized and multi-scale models: The homogenization method was extended to 3D racetrack coils, while the multi-scale method simulates one conductor at a time using environmental-field boundary conditions and can parallelize the resulting calculations.The multi-scale approach can build an AC-loss map by moving the simulated tape among selected device positions.
3 Applications
This section reviews studies that use different forms of the H formulation to calculate AC losses across practical applications and scenarios.
- The reviewed studies apply different forms of the H formulation to AC-loss calculations.
- The application review covers practical cases and operating scenarios.
- The section connects formulation variants with AC-loss analysis across the reviewed applications.
3.1 Tapes and coils
The H formulation is applied to tapes, coils, and related structures, with modeling choices addressing thin geometries, material losses, magnetic substrates, and three-dimensional effects. These studies report validation against experiments or other models while identifying computational and physical limitations.
- Modeling strategies: Artificially expanding the superconducting layer thickness while reducing Jc, together with structured elongated elements, substantially accelerates coated-conductor simulations.A discretization of 50–100 elements along tape width is generally sufficient, and one element through thickness is often used.
- Tapes and coils: The H formulation has been used to model eddy-current losses, bifilar coils, low-inductance coils, and coils under varying waveform gradients.Copper eddy-current power loss is proportional to the square of frequency and contributes substantially above 1 kHz.
- Tapes and coils: H-formulation models calculate AC losses in tapes and coils with magnetic substrates, whose effects can increase superconducting losses and alter turn-level loss profiles.At low current, ferromagnetic substrate losses can dominate; as current increases, substrate losses saturate and superconducting-layer losses become dominant.
- Tapes and coils: Flux diverters can significantly reduce HTS-coil transport losses, with high saturation field and low remnant field identified as desirable material properties.Simulations also examine how diverter geometry affects transport AC losses.
- Three-dimensional modeling: Three-dimensional models capture twist-angle effects, end effects, coupling losses, and cases where planar two-dimensional models diverge at medium and high current.For a twisted HTS cable, transport losses increase with twist angle because twisting generates an additional shielding current.
- Limitations and validation: Comparisons report reasonable H-formulation computation speed and boundary-condition settings, but low air resistivity can cause current leakage and underestimate losses.For DC currents with AC ripples, slowly relaxing current profiles make evaluated cyclic losses depend on where the loss curve is sampled.
3.2 High-current cables
The H formulation has been applied to multiple HTS cable architectures, using two-dimensional approximations, homogenized or simplified three-dimensional models, and full three-dimensional simulations. The reviewed studies address coupling, frequency, magnetic substrates, shielding, end effects, and experimental agreement.
- Roebel cables: Roebel cables are often approximated by two stacks of tapes because their transverse cross section is sufficient for many AC-loss calculations.Two strand-coupling scenarios provide limits for the actual cable losses, and results have been compared with the MMEV model.
- Roebel cables: Roebel-cable studies combine transport current and magnetic-field excitation, frequency sweeps, and experiments across cables and pancake coils.One study varied frequency from 50 Hz to 10 kHz, while dedicated three-dimensional simulations enabled agreement with experimental coil data.
- Roebel cables: A full 3D model of a 14-strand Roebel cable found losses similar to 2D results but revealed localized high dissipation near strand corners.The corner dissipation contributes little to whole-cable cyclic losses but may represent a stability issue.
- CORC cables: CORC-cable models account for strand coupling, shielding, coverage ratio, and end effects, with uncoupled strands providing a better match to experimental results in one study.Short-cable magnetization-loss studies require correction for end effects when experimental samples are used to estimate longer-cable behavior.
- Cable design effects: Magnetic substrates increase transport loss in HTS stacks, especially centrally, whereas cross-conductor cables require operation at a low fraction of critical current to maintain acceptable losses.Quasi-isotropic strands and former-material choices are additional design approaches discussed for cable-loss control.
- Stacked-tape cables: For twisted stacked-tape cables in transverse AC fields, magnetization losses are governed by the field component perpendicular to the tape and can be scaled from a straight 2D model by 2/π.Termination resistances can be represented using separate non-connected domains.
- Multifilamentary wires: Three-dimensional models are necessary for some twisted multifilamentary and MgB2 wires because coupling losses can contribute significantly, especially at high applied fields.Geometric transformations can reduce computation time, while nonlinear matrix permeability can add substantial losses.
3.3 High field magnets and power applications
The H formulation has been applied to diverse HTS magnets, machines, fault current limiters, transformers, and SMES devices, often with homogenization to make large systems tractable. These studies report validation against experiments, substantial computational savings, and application-specific loss distributions and operating limits.
- High-field magnets: Homogenized and multi-scale H-formulation models enabled AC-loss analysis of large coils with no significant accuracy loss, while homogenization was 50-60 times faster than the full-tape reference.The reference case modeled five 100-turn pancakes after exploiting symmetry.
- Electrical machines: A 2D H-formulation model incorporating anisotropic Jc dependence successfully matched measured AC losses in HTS racetrack armature windings.Measurements used both electrical and calorimetric methods.
- Electrical machines: A single-tape, unidirectionally coupled H-formulation estimate placed steady-state AC loss below 60 W for a 10 MW-class Bi-2223 wind-turbine generator.The tape model used magnetic fields computed from the machine model.
- Fault current limiters: Fault-current-limiter studies used H-formulation calculations alongside experiments and found good agreement above 50% of critical current, while outer HTS layers can experience higher losses than central layers.The outer-layer concentration indicates where quench is more likely in the saturated-core device.
- Fault current limiters: Losses in a three-phase 35 kV/90 MVA saturated iron-core fault current limiter reached the kW level and increased with DC-bias-current amplitude.The study identifies DC bias as a design quantity requiring particular attention.
- Transformers and SMES: Homogenized H-formulation calculations reproduced transformer AC-loss features with disagreement below 20% relative to experiments and MMEV predictions without mesh optimization.The transformer model represented approximately one thousand HV turns and Roebel-cable LV windings.
- Transformers and SMES: For a roughly 7000-turn hybrid HTS SMES magnet, AC loss concentrated at the top and bottom ends and increased with steady-state current, dynamic current, and ramp rate.The study reported a steady-state current below 100 A as an appropriate choice for reducing loss in the investigated SMES.
- Transformers and SMES: In a 1 MW/5 s SMES comparison, losses were dominated by the applied field, and the quantitative loss indicator was higher for toroidal than solenoidal geometry.The indicator combined loss per unit length with conductor length exposed to each perpendicular-field level.
4 Discussion
The H formulation is popular because it is broadly capable, relatively easy to implement and exchange, and generally agrees with experiments. Its drawbacks include commercial-software dependence, difficult open-source implementation, inefficient air-domain discretization, and limited parallelization.
- Advantages and popularity: The H formulation has been used across many HTS topologies and operating scenarios, with COMSOL Multiphysics accounting for the overwhelming majority of published implementations.Other implementations include FlexPDE, Matlab, GetDP, and home-made FEM codes.
- Advantages and popularity: COMSOL implementation is attractive because its relative ease helps new users become productive quickly and supports rapid knowledge transfer within research groups.
- Advantages and popularity: Publicly available models for numerous topologies and scenarios make H-formulation models easy to exchange, including advanced cable and winding models.
- Accuracy: Calculated AC losses generally agree with experiments, with reported accuracy varying between 10% and 50%, comparable to similar numerical models.
- Accuracy: Accuracy is constrained by uncertain tape-property characterization, including the angular dependence of Jc on magnetic field.
- Accuracy: Assuming uniform tape properties along the length overlooks common variations of approximately 5-10%.
- Accuracy: Cable and coil models can be affected by tape misalignment relative to theoretical positions.
- Drawbacks: COMSOL's closed code and license cost can limit accessibility, while open-source implementations are nontrivial and time-consuming for first-time users.
5 Conclusion
The review concludes that the H formulation is a widespread, flexible FEM tool for HTS AC-loss calculations, extended from individual conductors to complex and large systems. Despite competition from the T−A formulation for thin superconductors, its flexibility is expected to preserve its leading role.
- The H formulation has become the de facto standard numerical tool for calculating AC losses in HTS.Its popularity is attributed partly to its ease of implementation in commercial software, alongside successful home-made implementations.
- The model has been extended from individual conductors to nonlinear magnetic materials, cables, medium-size coils, and large superconducting magnet systems.
- Multi-scale and homogenization methods avoid simulating every individual tape in large superconducting magnet systems.
- The H formulation faces serious competition from the T−A formulation for thin superconductors such as HTS coated conductors.
- Its flexibility across different scenarios, including cases that T−A cannot handle, is likely to preserve the H formulation’s leading role in HTS AC-loss simulation.