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Calculation of the local current density in a
High Temperature Superconducting coil using
a Volume Integral Formulation
Blandine ROZIER1, Brahim RAMDANE1, Arnaud BADEL1,2, Gérard MEUNIER1
Grenoble Alpes, CNRS, Grenoble INP, G2ELab, 38000 Grenoble – France
2High Field Laboratory for Superconducting Materials, IMR, Tohoku University, Sendai 980-8577, Japan
1Univ.

Introduction
High Temperature Superconducting (HTS) materials represent a very attractive option for high magnetic field generation as they exhibit high current carrying capabilities without dissipation
even under high magnetic field. However, some key points have still to be investigated in order to reach their theoretical promising performances, especially regarding protection issues. In this
work, we developed a transient 2D axisymmetric electromagnetic model using a volume integral formulation based on a generalization of the Partial Element Equivalent Circuit method to
compute the local current density distribution inside a superconducting magnet, taking into account the effects of local performance variations. The model is then applied to the computation of
a HTS coil’s critical current, helping to define appropriated operating margins

J-Formulation

Nonlinear E(J) characteristic

Q (integration point)

𝒞

•

θ

2D axisymmetric condition (VIM) :

𝐀𝟐𝐃 𝐚𝐱𝐢 𝐉 = μ0

K
Ωc 4πR

2−k

2

Power law model [1]

z-axis

𝝆𝐬𝐜

r

∙ J1 k − 2J2 k

h

K

∙ 𝐉 dΩc + 𝐀𝟎

R

With :
• J1 : complete elliptic integral of first order
• J2 : complete elliptic integral of second order
• k = 4rR/D²

•

𝐰𝐟𝐣 Ij

•

Electrical circuit (dual mesh) :
𝐌

𝐌 ∆𝐕 = [𝟎

•

𝐭

• Power Law : Jc = 3.108
A/m² and n = 20
• Mapped mesh (800
elements for the bulk

1
𝐰𝐟𝐤 ∙ 𝐧 dS = ±
Sk
Sk

j

Validation
Benchmark #4 [2] : HTS
bulk cylinder magnetization

n

Interpolation (Whitney facet elements) :

|𝐉|
𝐉𝐜 (𝐓, 𝐁

P (computation
point)
2D domain

G2Daxi : 2D axisymmetric
Green Kernel

𝐉=

𝐄𝐜
=
𝐉𝐜 (𝐓, 𝐁

𝐧−𝟏

𝐈𝐋 = [𝐈𝐁

z

Bz [T]

Btrap
2 mm
1

10 mm

Arbitrarily shaped superconducting domain (2D)
Magneto-quasistatic approximation assumption

r

Time [s]
0

10

5

15

12.5 mm

region)

Weak formulation (no external branches) :

𝐌

R ij =

𝛛
[𝐑 +
[𝐋
𝛛𝐭

𝐌

𝐭

𝐈𝐋

μ0
Lij =
4π

wi ρsc J wj 2πri dΩc
Ωc

𝛛
= [𝐀 𝟎
𝛛𝐭

wi
Ωc

A0i

𝜕
=−
𝜕t

wi A0 2πri dΩc
Ωc

External source vector

wj G2D axi dΩ′c 2πri dΩc
Ωc

Inductance matrix
(Full)

Resistivity matrix
(Sparse)

Application to HTS coils
Problem description

Magnetostatic study with Flux (FEM)

Objective : estimation of the critical current of a HTS
coil to locate the most vulnerable zone (quench
initiation) and define appropriated operating margins

(homogeneous current distribution assumption)
z

z

r

r

Geometry :

Rout

Rin

Rin

z
θ

Rin

r

…

-672 mT

672 mT

Icturn =

Rout

Bz

Br

25 turns coil’s
cross-section

-1.36 T

Ic 4.2 K, B, θ, ri , z dz

Transient study (VIM)
(constant current increase)
Mesh :
•
•
•

Comparison & Conclusion

Thin regions approximation (1 turn: 2 μm thick –
12 mm wide)
Piecewise constant approximation
Each turn is divided into 100 elements

Iccoil = min(Icturn
turn

Magnetostatic

Transient

1269 A
(innermost turn)

1452 A
(6th turn)

 Homogeneous current distribution assumption too strong, leading
to a false Icturn distribution
 Gradual penetration of the current density from edges to the
centre depending on the turn position (due to Jc anisotropy) to be
taken into account
J/Jc
1.0

0.5

Power Law parameters :
•
•

Jc(4.2 K, B, θ) is measured on short sample REBCO
tapes [3]
n = 25

Icturn such as: E Icturn = 1 μV/cm

Conclusion & Perspectives
 Implementation of the 2D axisymmetric J-Formulation (internal platform) validated by comparison to FEM
on a superconducting problem
 J-Formulation particularly adapted to HTS coils modelling  light mesh (active regions being actually only
the superconducting layers which are 1-3 μm thick) and current conservation
 Improvements are considered to speed up the computation time in order to model bigger size problems

0.0
I = 830 A

I = 1286 A

I = 1450 A

References
[1] J. Rhyner, Phys. C Supercond., vol. 212, no 3-4, p. 292-300,1993
[2] « HTS MODELING WORKGROUP: Benchmarks ». [Online]. Available on:
http://www.htsmodelling.com/?page_id=2
[3] T. Benkel et al., IEEE Trans. Appl. Supercond., vol. 26, no 3, p. 1-5, 2016.

Contact : blandine.rozier@g2elab.grenoble-inp.fr



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