Modelling of bulk superconductor
magnetisation: a review
Dr Mark AinslieEngineering & Physical Sciences Research Council (EPSRC)
Research Fellow
Bulk Superconductivity Group, Department of Engineering

Presentation Outline• Overview of modelling bulk superconductor magnetisation• Case studies:• Multi-pulse, pulsed field magnetisation (PFM) of bulk hightemperature superconductors• Split coil PFM with an iron yoke
• Field cooling (FC) magnetisation of iron-pnictide (Ba122) bulks

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Bulk Superconductors• Bulk superconducting materials can‘trap’ large magnetic fields > 17 T
• Achieved by pinning penetrated
magnetic field (quantised flux lines) macroscopic electrical currents• Magnetisation increases withsample volume

A large, single grainbulk superconductor

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Bulk Superconductors• Bulk superconducting materials can‘trap’ large magnetic fields > 17 T
• Achieved by pinning penetrated
magnetic field (quantised flux lines) macroscopic electrical currents• Magnetisation increases withsample volume• Trapped field given by
Btrap = k µ0 Jc R

Typical trapped magnetic
field profile of abulk superconductor

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Bulk Superconductors
Btrap = k µ0 Jc R
where
• From Bean model (infinite slab) +Biot-Savart law• Example of an analytical model• Easier to deal with & faster• Rely on specific simplified geometries & simplified, homogeneous
assumptions:• Constant / uniform Jc, no frequency / time dependence, etc.
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Bulk Superconductors
Btrap = k µ0 Jc R
• Candidate materials
must be able to:• Pin magnetic flux
effectively• Carry large current
density, Jc, over large
length scales• Be insensitive toapplication of large
magnetic fields, Jc(B)Example field dependence of criticalcurrent density, Jc(B), for bulk YBCO

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Magnetisation of Bulk Superconductors• Three magnetisation techniques:

ZFCFC• Field Cooling (FC)
• Zero Field Cooling (ZFC)• Pulsed Field Magnetisation(PFM)• To trap Btrap, need at least Btrap orhigher• FC and ZFC require large
magnetising coils, longmagnetizing times• Impractical for
applications/devices  PFM
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Numerical Modelling of Magnetisation• Numerical models can simulate practical & complex situations,playing a number of crucial roles:• Simulate accurate magnetic field, current, temperature, mechanicalstress distributions• Interpret experimental results & physical mechanisms of bulk
superconductor magnetisation• Design & predict performance of magnetising fixtures & techniques• Design & predict performance of practical bulk superconductor-based
devices

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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE

ELECTROMAGNETIC
FORMULATIONTHERMAL
EQUATIONS &PROPERTIES

Jc(B, T)

• 2D axisymmetric generally sufficient for cylindrical bulkswith a homogeneous Jc distribution• 3D required for an inhomogeneous Jc distribution aroundthe ab-plane; for non-symmetric shapes

E-J POWER LAW

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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE

ELECTROMAGNETIC
FORMULATIONTHERMAL
EQUATIONS &PROPERTIES

Jc(B, T)

• Magnetising fixture: uniform boundary conditions orinserting a copper coil sub-domain• Cooling: using a cold head + vacuum chamber orsubmersion in liquid cryogen

E-J POWER LAW

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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE

ELECTROMAGNETIC
FORMULATION

Finite element method is commonly used & welldeveloped (other techniques do exist)
Governing equations:
Maxwell’s equations (H formulation)

THERMAL
EQUATIONS &PROPERTIES

Faraday’s Law
Ampere’s Law

Jc(B, T)

Other formulations also exist (A-V, T-Ω, Campbell’sequation)

E-J POWER LAW

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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE

ELECTROMAGNETIC
FORMULATION

Thermal behaviour needs to be modelled when thebulk experiences a significant change intemperaturee.g., during PFM, modelling complete FC
magnetisation process
Governing equations:

THERMAL
EQUATIONS &PROPERTIES

Jc(B, T)

E-J POWER LAW

ρ = mass density, C = specific heat, κ = thermal
conductivity, Q = heat source
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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE

ELECTROMAGNETIC
FORMULATIONTHERMAL
EQUATIONS &PROPERTIES

Jc(B, T)

Can use measured
experimental data + fitting
function or interpolation
over a specific temperature
range
Can choose constantparameters for C, κ for
T = Top as a reasonableapproximation

E-J POWER LAW

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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE

ELECTROMAGNETIC
FORMULATIONHTS materials
Kim-like model:
Fishtail effect:
MgB2 materials

THERMAL
EQUATIONS &PROPERTIES

Jc(B, T).
E-J POWER LAW

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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE… or direct interpolation

GdBa2Cu3O7-δ (15wt% Ag)
ELECTROMAGNETIC
FORMULATIONTHERMAL
EQUATIONS &PROPERTIES

Jc(B, T)

Ba0.6K0.4Fe2As2 (Ba122)

E-J POWER LAW

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Numerical Modelling of Magnetisation

BULK GEOMETRY &MAGNETISATIONFIXTURE

ELECTROMAGNETIC
FORMULATION

E-J power law• Conventional materials  non-linear permeability,
linear resistivity• Superconductors  linear permeability (µ0),
non-linear resistivity• Non-linearity is extreme: power law with n > 20

THERMAL
EQUATIONS &PROPERTIES

E = ρJJc(B, T)

E-J POWER LAW

I-V curves for different n values

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Case study #1:
Multi-pulse, pulsed field magnetisation(PFM) of bulk high-temperature
superconductors

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Pulsed Field Magnetisation• PFM technique: compact, mobile, relatively inexpensive• Main issue: Btrap [PFM] < Btrap [FC], [ZFC]• Temperature rise ΔT due to rapid movement of magnetic flux
• Record PFM trapped field: 5.2 T @ 29 K
Top surface of 45 mm diameter Gd-Ba-Cu-OFujishiro et al. Physica C 2006
• Record trapped field by FC: 17.6 T @ 26 K
Centre of 2 x 25 mm diameter Gd-Ba-Cu-O
Durrell et al. Supercond. Sci. Technol. 2014

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Pulsed Field Magnetisation• Many considerations for PFM:
• Pulse magnitude, pulse duration, temperature(s), number of pulses,type of magnetising coil(s), use of ferromagnetic materials• Dynamics of magnetic flux during PFM process• Multi-pulse PFM: effective in increasing trapped field/flux

Fujishiro et al. Physica C 2006

Zhou et al. Appl. Phys. Lett. 2017

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Modelling Trapped Field Capability (FC/ZFC)
Model #1
• Stationary• Comsol Multiphysics• 2D axisymmetric• AC/DC module• Magnetic Field (mf) interface• External Current Density node
• No flux creep• Time taken: ~ 2-3 seconds

Model #2
• Time-dependent• Comsol Multiphysics• 2D axisymmetric• AC/DC module• Magnetic Field Formulation (mfh) interface• E-J power law, E α Jn (flux creep)• Apply + remove background field• Time taken: up to 1-2 hours

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Modelling Trapped Field Capability
Results

GdBa2Cu3O7-δ (15wt% Ag)
Magnetisation

TimeTrapped Field77 KFC
ZFC [5 T]--t = 0 min
t = 10 min
t = 20 min

1.544 T1.546 T
1.263 T
1.223 T79%

3.826 T3.827 T
3.256 T
3.158 T

82.5%

7.449 T
7.422 T6.577 T
6.405 T

86%

65 K
FC
ZFC [10 T]--t = 0 min
t = 10 min
t = 20 min
50 KFCZFC [20 T]

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t = 10 min
t = 20 min

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PFM Modelling Framework
Electromagnetic properties modelled as Model #2 (ZFC);
magnetising fixture assumed as solenoid coil:τ = 15 ms
Thermal behaviour:

Heat source,
coupling with EM model + Jc(B, T)ρ = mass density (bulk 5900 kg/m3, indium 7310 kg/m3)
C = specific heat (measured, temperature-dependent)
κ = thermal conductivity:κab = 20 W/(m·K), κc = 4 W/(m·K), κindium = 0.5 W/(m·K)

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PFM Single Pulse Results• t = 120 s  flux creep relaxation &cooling back to operating temp.
• Percentage of ZFC(t = 20 min):• 77 K 85%• 65 K 49%• 50 K 29%• Four specific cases as initialconditions for 2nd pulse:• Partially-magnetised (PM), socalled ‘M-shaped’ profile• Under-magnetised (UM)• Fully-magnetised (FM)• Over-magnetised (OM)

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PFM 2nd Pulse Results
77 K50 K

65 K
For all T, trapped field after 2nd pulse exhibits
two particular characteristics:
1) Increased trapped field, Bt, when the bulk
is fully magnetised; maximum value whenthe 1st pulse results in full magnetisation
2) Increased activation field: applied field,Bapp, required to fully magnetise the sample
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PFM 2nd Pulse: Magnetic Flux Penetration• Why does this occur?
• More difficult for magnetic flux topenetrate the sample due toexisting trapped field• Existing, induced supercurrent
flows in opposite direction

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PFM 2nd Pulse: Thermal Behaviour• Why does this occur?
• Reduced dynamic movement offlux = lower temperature rise forequivalent next pulse• Can examine maximum averagetemperature, Tave,max, during &after PFM:

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Case study #2:
Split coil PFM with an iron yoke

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Split Coil PFM with an Iron Yoke
Two GdBa2Cu3O7-δ (15wt% Ag) samples:
30 mm diameter, 15 mm thickness#476 sample Bt = 6 T(40 K)
3.11 T (65 K)#477 sample Bt = 5.44 T (40 K)
2.02 T (65 K)

Ainslie, Fujishiro et al. Supercond. Sci. Technol. 29 (2016) 074003

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Split Coil PFM with an Iron Yoke

Ainslie, Fujishiro et al. Supercond. Sci. Technol. 29 (2016) 074003

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Split Coil PFM with an Iron Yoke
Split coil

Solenoid coil

Ainslie, Fujishiro et al. Supercond. Sci. Technol. 29 (2016) 074003

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Split Coil PFM with an Iron Yoke

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Split Coil PFM with an Iron Yoke
Trapped field @ t = 300 ms

Trapped field & Tave with time, incl. yoke removed

Ainslie, Fujishiro et al. Supercond. Sci. Technol. 29 (2016) 074003

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Split Coil PFM with an Iron Yoke
Magnetic flux entry & exit during & after pulse – split coil with & without iron yokezr

Ainslie, Fujishiro et al. Supercond. Sci. Technol. 29 (2016) 074003

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Case study #3:
Field cooling (FC) magnetisation of ironpnictide (Ba122) bulks

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Modelling Iron-Pnictide Bulks• Weiss et al. demonstrated > 1 T trappedin a stack of iron-pnictide (Ba122) bulks• Advantages:• Low anisotropy• High upper critical magnetic field (Hc2)• High homogeneity• Scalable, low-cost fabrication

Ainslie, Fujishiro, Yamamoto et al. Supercond. Sci. Technol. 30 (2017) 105009

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Modelling Iron-Pnictide Bulks• Same ZFC model as before approximates FC, applied field >> full penetration field• n = 50
 observed flux creep much weaker than HTS bulksz = ±0.5 mm

Ainslie, Fujishiro, Yamamoto et al. Supercond. Sci. Technol. 30 (2017) 105009

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Modelling Iron-Pnictide Bulks• Same ZFC model as before approximates FC, applied field >> full penetration field• n = 50
 observed flux creep much weaker than HTS bulks

H2H1Ainslie, Fujishiro, Yamamoto et al. Supercond. Sci. Technol. 30 (2017) 105009

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Modelling Iron-Pnictide Bulks
Influence of geometric parameters can be predicted:
Diameter dependence

Thickness dependence

10 mm thickness• With current state-of-the-art properties, > 2 T @ 5 K (> 1 T @ 20 K) for D = 50 mm• Appropriate aspect ratio 1-1.5 (radius : thickness)
Ainslie, Fujishiro, Yamamoto et al. Supercond. Sci. Technol. 30 (2017) 105009

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Summary• Overview of numerical modelling of magnetisation• Case studies:• Multi-pulse, pulsed field magnetisation (PFM) of bulk hightemperature superconductors• Split coil PFM with an iron yoke
• Field cooling (FC) magnetisation of iron-pnictide (Ba122) bulks

Thank you for listening
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