Superconductor Science and Technology
Supercond. Sci. Technol. 29 (2016) 124004 (9pp)

doi:10.1088/0953-2048/29/12/124004

Flux jump-assisted pulsed ﬁeld
magnetisation of high-Jc bulk
high-temperature superconductors
M D Ainslie1, D Zhou1, H Fujishiro2, K Takahashi2, Y-H Shi1 and J H Durrell1
1

Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Trumpington
Street, Cambridge CB2 1PZ, UK
2
Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate
University, 4-3-5 Ueda, Morioka 020-8551, Japan
E-mail: mda36@cam.ac.uk
Received 15 August 2016, revised 12 September 2016
Accepted for publication 28 September 2016
Published 21 October 2016
Abstract

Investigating, predicting and optimising practical magnetisation techniques for charging bulk
superconductors is a crucial prerequisite to their use as high performance ‘psuedo’ permanent
magnets. The leading technique for such magnetisation is the pulsed ﬁeld magnetisation (PFM)
technique, in which a large magnetic ﬁeld is applied via an external magnetic ﬁeld pulse of
duration of the order of milliseconds. Recently ‘giant ﬁeld leaps’ have been observed during
charging by PFM: this effect greatly aids magnetisation as ﬂux jumps occur in the
superconductor leading to magnetic ﬂux suddenly intruding into the centre of thesuperconductor. This results in a large increase in the measured trapped ﬁeld at the centre of the
top surface of the bulk sample and full magnetisation. Due to the complex nature of the magnetic
ﬂux dynamics during the PFM process, simple analytical methods, such as those based on the
Bean critical state model, are not applicable. Consequently, in order to successfully model thisprocess, a multi-physical numerical model is required, including both electromagnetic andthermal considerations over short time scales. In this paper, we show that a standard numericalmodelling technique, based on a 2D axisymmetric ﬁnite-element model implementing the
H-formulation, can model this behaviour. In order to reproduce the observed behaviour in ourmodel all that is required is the insertion of a bulk sample of high critical current density, Jc. We
further explore the consequences of this observation by examining the applicability of the model
to a range of previously reported experimental results. Our key conclusion is that the ‘giant ﬁeld
leaps’ reported by Weinstein et al and others need no new physical explanation in terms of the
behaviour of bulk superconductors: it is clear the ‘giant ﬁeld leap’ or ﬂux jump-assisted
magnetisation of bulk superconductors will be a key enabling technology for practicalapplications.
Keywords: bulk superconductors, trapped ﬁeld magnets, numerical modelling, pulsed ﬁeld
magnetisation, ﬂux jumps, ﬁnite element method (FEM) modelling(Some ﬁgures may appear in colour only in the online journal)[1, 2], an order of magnitude higher than the maximum ﬁeld
produced by conventional permanent magnets. Such bulk
superconductors can exhibit critical current densities, Jcs, of
50 kA cm−2 at 1 T and 77 K, resulting in trapped ﬁelds of up
to 1–1.5 T for standard Y–Ba–Cu–O (YBCO) and greater

1. Introduction
Large, single grain (RE)BCO (where RE=rare earth or Y)
bulk superconductors, acting as trapped ﬁeld magnets(TFMs), are able to trap magnetic ﬁelds greater than 17 T
0953-2048/16/124004+09$33.00

1© 2016 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 29 (2016) 124004

M D Ainslie et althan 2 T for (RE)BCO materials [3], with 3 T the highest
trapped ﬁeld at 77 K so far in a 65 mm diameter Gd–Ba–Cu–
O (GdBCO) sample [4]. Additionally, neutron-irradiated
YBCO samples, fabricated by the so-called U/n method[5, 6], containing improved broken-columnar pinning centres,
have exhibited much higher Jcs (up to several hundreds ofkA cm−2 [5]) and trapped ﬁelds greater than 2 T at 77 K withsmaller samples only 20 mm in diameter [7].
Investigating and predicting the magnetisation of thesematerials and developing practical magnetising techniques is
crucial to using them as TFMs in a number of engineering
applications, such as electrical machines [8–11], magnetic
separation [12], magnetic resonance imaging, nuclear magnetic resonance [13–15], and magnetic drug delivery systems[16, 17]. The current, best-known method for magnetisingbulk superconductors in practical applications is the pulsed
ﬁeld magnetisation (PFM) technique, whereby a large
magnetic ﬁeld is applied via a pulse on the order of milliseconds; however, a signiﬁcant issue with existing PFM
techniques is that the trapped ﬁeld is generally much less thanthat achieved in comparison to slower ﬁeld cooling (FC) andzero ﬁeld cooling (ZFC) magnetisation techniques, whichthemselves need a large magnetising coil/ﬁxture, and hence
impractical for practical applications. The world record using
PFM, using a modiﬁed multi-pulse, stepwise-cooling technique, is only 5.2 T at 29 K [18], which is much less than thetrue capability of these materials as indicated above. It should
be noted that at higher operating temperatures (closer to Tc,
such as 77 K), ﬁelds have been trapped close to that of FC[7, 19–22].
So-called ‘giant ﬁeld leaps’ have been observed by anumber of research groups investigating PFM [7, 18, 23–26],
and more recently in unpublished experiments carried out inour own research group, where ﬂux jumps occur in the
superconductor, and magnetic ﬂux suddenly intrudes into thecentre of the superconductor, resulting in a large increase in
the measured trapped ﬁeld at the centre of the top surface ofthe bulk sample and full magnetisation. Due to the complex
nature of the magnetic ﬂux dynamics during the PFM process,simple analytical methods, such as those based on the Beancritical state model (CSM), for example, are not applicable[7, 27], and a multi-physical numerical model is required toinclude both electromagnetic and thermal considerations overa very short time scale [3]. To date, this ‘giant ﬁeld leap’effect observed in a number of experiments has not yet beenconclusively explained and no numerical models havereproduced it.
In this paper, it is shown that this effect can be qualitatively reproduced using a standard numerical modelling
technique, based on a 2D axisymmetric ﬁnite-element model
implementing the H-formulation. This numerical model isbased on previous models published by the authors, with a
modiﬁed E–J power law characteristic, representing the normal state resistivity of the superconductor when J>Jc. By
simply inserting a bulk sample of high Jc, as found in highquality standard samples and neutron-irradiated ones, such‘giant ﬁeld leaps’ or ﬂux jumps are observed in the model,
without the need for any new physics to explain the physicalmechanism underlying the effect. The magnetic ﬂux dynamics in high Jc samples, compared with normal and low Jc
samples, are examined using the numerical modelling results,
which have important implications for magnetising bulk
superconductors with high trapped ﬁelds in practical technological applications.

2. Numerical modelling
2.1. Modelling framework

The numerical model used in this paper is based on the 2D
axisymmetric H-formulation presented in [26] for a solenoid
coil magnetising ﬁxture without a soft iron yoke, implemented using the commercial FEM software package COMSOL Multiphysics 5.2a [28]. The general form, partial
differential equation interface of COMSOL is used for theelectromagnetic analysis, and the heat transfer module is used
for the thermal analysis, which are coupled together as
described in [26, 29].
In the 2D axisymmetric H-formulation, the governing
equations are derived from Maxwell’s equations—namely,
Faraday’s (1) and Ampere’s (2) laws:´E+d (m 0 m r H )
dB
=´E+
= 0,
dt
dt
 ´ H = J,

(1 )
(2 )

where H=[Hr, Hz] represents the magnetic ﬁeld components, J=[Jf] represents the current density and E=[Ef]represents the electric ﬁeld. μ0 is the permeability of freespace, and the relative permeability, μr, is 1 for the materials
modelled. The electrical properties of the superconductor aremodelled using an E–J power law relation [30, 31], E α Jn,
where n=20 and is a reasonable approximation of Bean’s
critical state model, for which n→∞ [3]. However, one
addition to the modelling framework in this respect is theinclusion of a normal-state resistivity when the current density
in the superconductor is larger than the critical current density, Jc. A modiﬁed E–J power law relationship is used, as
described in [32–34] and in the FC magnetisation model forbulk MgB2 presented in [35]. Hence, the resistivity tends
towards the normal state resistivity (here ρnormal is assumed tobe 3.5×10−6 Ωm) when J>2–2.5Jc with n=20 [34]. This
gives a more reasonable value for the resistivity than thestandard E–J assumption in such situations, where the normalised current density, J/Jc, can be much larger than 1during the PFM process [36]. Figure 1 shows a comparison of
the resistivity, ρ, and electric ﬁeld, E, for the standard andmodiﬁed
E–J
power
law
relations,whereJc=2×109 A m−2 is assumed, based on the Jc value of thebulk at 40 K (see ﬁgure 2). Figure 1(b) also includes thedifferent regimes of the E–J curve, corresponding to thermally assisted ﬂux ﬂow, ﬂux creep and ﬂux ﬂow [37]. It
allows us to discern whether this is a cause of the ﬂux jumps,
but also improves the convergence properties of the numericalmodel.
2

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M D Ainslie et al

Figure 2. Experimental data for Jc(B), measured from a small sample

taken from a representative bulk sample (15 wt% Ag-containing Gd–
Ba–Cu–O) [26]. The experimental data is ﬁt up to 10 T using theequation presented in [40] for samples exhibiting a ﬁshtail shape in
their Jc(B) curves. © IOP Publishing. Reproduced by permission of
IOP Publishing from [26]. All rights reserved.

Figure 3 shows the 2D axisymmetric model setup for thenumerical simulation. The geometry of the bulk sample isassumed to be 20 mm in diameter and 10 mm thickness, andas a representative magnetising ﬁxture, the solenoid coilexperimental setup presented in [26, 29] is used, with the softiron yoke embedded in the magnetising ﬁxture omitted forsimplicity. Top is the operating temperature of the cold head,
and is assumed to be 40 K in the following simulations.
Since the temperature of the superconductor can change
signiﬁcantly during PFM [3], the electromagnetic model iscoupled with a thermal model, and the thermal behaviour ismodelled using the following thermal transient equation:

Figure 1. Comparison of (a) resistivity, ρ, and (b) electric ﬁeld, E, forthe standard E–J power law relation, E α Jn, and the modiﬁed E–J
power law, where the resistivity tends towards the normal state
resistivity when J>2–2.5Jc. It is assumed that n=20 andρnormal=3.5×10−6 Ωm.

r⋅CdT=  ⋅ (k T ) + Q .
dt

(3 )

The heat source, Q, in the thermal model is calculated fromthe product of the electric ﬁeld and current density throughout
the sample, Q=Ef·Jf.

The results of the numerical model strongly depend on
the Jc(B, T) characteristics of the superconductor [3], and theexperimental data for Jc(B), measured between 40–85 K froma specimen taken from a representative bulk (15 wt% Agcontaining GdBCO) is input into the model using a twovariable, direct interpolation, as described in [26, 38, 39]. In[26], this experimental data (up to 4 T) was ﬁt up to 10 T
using the equation presented in [40] for samples exhibiting aﬁshtail shape in their magnetisation loop, and these Jc(B, T)
characteristics are shown in ﬁgure 2. This method of data
input is simple and direct, and can signiﬁcantly improve thecomputational speed of the model [38, 39]. The 2D axisymmetric model assumes a homogeneous Jc distribution aroundthe ab-plane, neglecting any effects from any inhomogeneityof Jc, e.g., [29, 41–43].

The sample is assumed to be mounted in a 316 stainlesssteel (SUS) sample holder with Stycast™ 2850 GT, and a thinsheet of indium, approximately 0.2 mm thick, is placed
between the cold stage and the sample to ensure a goodthermal contact [26]. The thermal properties (thermal conductivity and speciﬁc heat) of each of these materials, basedon measured experimental data over the temperate range
0–100 K, is input into the model using a linear interpolation
function [26, 35]. The indium sheet has a ﬁxed, ﬁnite thermal
conductivity of 0.5 Wm−1 K−1 to represent the ﬁnite coolingpower of the refrigerator and the thermal contact between thecold stage and the bulk, as described in [44].
Pulsed currents of varying magnitude are applied to thesolenoid coil via an integral constraint on the copper coil3Supercond. Sci. Technol. 29 (2016) 124004

M D Ainslie et al

Figure 4. Numerical simulation results for the trapped magnetic ﬁeld,
Bt, at the centre of the top surface of the bulk samples (r=0 mm) ata height of z=0.1 mm at t=300 ms for applied ﬁelds, Bapp, up to
8 T and at an operating temperature, Top, of 40 K.

maintains the overall pinning characteristics of the sample [22].
• Low Jc
• Normal Jc• High Jc• Ultra-high Jc

Figure 4 shows a comparison of the trapped ﬁeld, Bt, at thecentre of the top surface of the bulk samples (r=0 mm) at aheight of z=0.1 mm at t=300 ms for applied ﬁelds, Bapp,up to 8 T and at an operating temperature, Top, of 40 K.
For the high Jc and ultra-high Jc samples, there is a large
increase in the trapped ﬁeld at the centre of the top surface fora relatively small increase in applied ﬁeld (0.25 T) above 6 T,
which is qualitatively consistent with the experimental results
observed in [7, 18, 23–26]. For the high Jc sample, forexample, Bt is close to zero when Bapp=6 T, but increases to
3.4 T when Bapp is increased by 0.25 to 6.25 T. For increasing
Bapp values after the sample is fully magnetised, the trapped
ﬁeld begins to reduce due to an increasing temperature rise
generated by the rapid movement of ﬂux lines in the sample [29].
It was observed in [24] that standard YBCO samples of asimilar size to this simulated bulk, with average, in-ﬁeld Jcs of
10 kA cm−2 and Bt,max=0.4 T at 77 K, produced similar
results to the simulated low and normal Jc cases, with good
agreement with the CSM, as well as previously reported
numerical simulation results by the authors [26, 29, 45].
However, high-Jc samples fabricated by the U/n method,which introduces improved broken-columnar pinning centres
after a sample doped with a constant mass% of U is irradiatedwith thermal neutrons [24], exhibit average, in-ﬁeld Jcs of
50 kA cm−2 and Bt,max=2 T at 77 K, as well as such jumps(‘giant ﬁeld leaps’) in the measured trapped ﬁeld with only asmall increase in the applied ﬁeld [7, 24, 25].

Figure 3. 2D axisymmetric model setup for numerical simulation ofpulsed ﬁeld magnetisation using a solenoid coil. A thin (0.2 mm)
indium sheet is placed between the bulk and cold stage to provide agood thermal contact. The bulk, which is 20 mm in diameter and of
thickness 10 mm, is embedded in a 316 stainless steel (SUS) ringusing Stycast™ (2850GT). Top is the operating temperature of thecold head, and is assumed to be 40 K in the following simulations.

subdomain, as described in [26], such that
Ipulse (t ) = N ⋅ I0⎛tt⎞
exp ⎜1 - ⎟ ,⎝tt⎠

Jc(B, T)/3
Jc(B, T)
2Jc(B, T)
3Jc(B, T)(4 )where I0 is the peak magnitude of the current ﬂowing in eachturn of the solenoid coil, N is the number of turns, and τ is therise time of the pulse, where τ=15 ms is assumed for eachpulse. This setup has the same coil constant, relating Bapp (theﬁeld at the centre of the magnetising ﬁxture with the bulk
removed) to N·I0 in equation (4), as presented in [26].
2.2. Simulation results

Firstly, we examine the effect of the magnitude of Jc on thesimulated trapped ﬁeld at the centre of the top surface of thebulk (z=0.1 mm) at Top=40 K. Four samples of varying Jc
are studied, with the Jc(B, T) characteristics of the representative Ag-containing GdBCO sample divided or multipliedby an integer value, which varies the magnitude of Jc, but
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Figure 5. Numerical simulation results for the trapped magnetic ﬁeld

Figure 6. Numerical simulation results for the trapped magnetic ﬁeld

proﬁle across the top surface of the low Jc sample (z=0.1 mm) for
increasing Bapp between 3 and 7 T for Top=40 K. No ﬂux jumps areobserved.

proﬁle across the top surface of the high Jc sample (z=0.1 mm) for
increasing Bapp between 5 and 8 T for Top=40 K. A large increasein the trapped ﬁeld, due to ﬂux jumps, is observed when increasing
the applied ﬁeld from 6 to 6.25 T.

It is also important to note that the maximum trapped
ﬁeld Bt=3.4 T for Bapp=6.25 T for the high Jc sample islarger than the permissible value based on the simplistic CSM
prediction, which suggests the ﬁeld trapped at the centre is 
½Bapp for ZFC in the case of an inﬁnitely long slab. Applying
the Biot–Savart law to such a geometry, and assuming theCSM, would suggest that Bapp should be 4 times the trapped
ﬁeld at the surface for full magnetisation of the superconductor. For this particular sample geometry, where theratio of thickness/diameter is ½, an applied ﬁeld of 3.276
times the surface trapped ﬁeld would be required to fully
magnetise the sample. This required applied ﬁeld tends
logarithmically to 2 times in the case of a thin sheet, where
the central ﬁeld can be assumed to be equal to the surface
ﬁeld. The same result is found for the ultra-high Jc sample,
where Bt=3.7 T for Bapp=6.25 T and Bt=3.75 T for
Bapp=6.5 T.
Furthermore, ordinarily for samples with increased Jc, theactivation ﬁeld (the applied pulsed ﬁeld required to fully
magnetise the sample [29]) also increases due to the stronger
pinning forces that need to be overcome for the magnetic ﬂux
to fully penetrate the sample. In the case of the high and ultrahigh Jc samples here, the ﬂux jumps that occur act to reduce
the activation ﬁeld from its expected value, while allowing thesample to be fully magnetised with a high trapped ﬁeld.
Figure 5 shows the numerical simulation results for thetrapped magnetic ﬁeld proﬁle across the top surface of the low
Jc sample (z=0.1 mm) for increasing Bapp between 3 and 7 T
for Top=40 K. No ﬂux jumps are observed, and the magnetic ﬁeld penetration (and resultant trapped ﬁeld) steadily
increases with increasing Bapp, until the sample is fully
magnetised when Bapp=5.5 T. Further increases in Bappabove this value results in full magnetisation, but a reduced
maximum trapped ﬁeld, consistent with previous experimental results observed by the authors [26, 29, 45], becauseof a larger temperature rise due to the higher applied ﬁeld.

Figure 6 shows the same simulated trapped ﬁeld proﬁles forthe high Jc sample (z=0.1 mm) for increasing Bapp between
3 and 7 T for Top=40 K. As shown in ﬁgure 4, the trapped
ﬁeld proﬁle suddenly changes between Bapp=6 and 6.25 T,
to a conical (fully magnetised) trapped ﬁeld proﬁle from apeak-valley shape (partially magnetised), again qualitatively
consistent with the experimental results presented in[7, 24, 25]. One obvious difference between those experimental results and these simulated results is the height of thepeak in the partially magnetised state, which can be explainedby the different magnetising ﬁxture used: in [7, 24, 25], a split
coil of smaller radius is used as the magnetising ﬁxture, forwhich the magnetising mechanism is different in that, insteadof ﬂux penetrating from the periphery (or edge) of the bulk,
the ﬂux penetrates from the top/bottom surfaces [46, 47]; andthe different operating temperature and pinning characteristicsof the sample.
In order to study the magnetic ﬂux dynamics in moredetail, ﬁgure 7 shows the trapped ﬁeld calculated at three
discrete locations above the top surface of the bulk (again atz=0.1 mm): centre, +4 mm, and +8 mm (2 mm from theedge of the 20 mm diameter sample). Figures 7(a)–(c) show
the change in time of the ﬁeld calculated at each point as thepulse is applied, then removed, for the low Jc sample for
Bapp=3, 4.5, and 5 T, respectively. Again, these results are
consistent with CSM predictions and the experimentallyobserved results in [7, 24, 25] for lower Jc, standard YBCO
samples. The combination of a large applied ﬁeld near theedge of the sample and the associated temperature rise due tothe rapid movement of magnetic ﬂux into the sample results
in a localised, suppressed Jc, and the ﬁeld at the +8 mm pointfrom the centre almost follows the applied ﬁeld for increasing
Bapp values. Figures 7(d)–(f) shows similar plots for the highJc sample, where ﬂux jumps act to assist the PFM process,resulting in a sudden, large increase in the trapped ﬁeld at the5Supercond. Sci. Technol. 29 (2016) 124004

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Figure 7. Trapped ﬁeld calculated at three discrete locations above the top surface of the bulk (z=0.1 mm): centre, +4 mm, and +8 mm(2 mm from the edge of the 20 mm diameter sample). Panels (a)–(c) show the change in time of the ﬁeld calculated at each point for the low
Jc sample for Bapp=3, 4.5, and 5 T, respectively. (d)–(f) show the change in time of the ﬁeld calculated at each point for the high Jc sample
for Bapp=5, 6, and 6.25 T, respectively.

centre of the sample. When Bapp=5 T, no ﬂux jump isobserved in the high Jc sample, and the magnetic ﬂux
dynamics are similar to those for the low Jc sample inﬁgure 7(a). However, when Bapp=6 T, as shown inﬁgure 7(e), a ﬂux jump is observed at +4 mm betweent=4.5–5 ms, but there is no change at the centre of thesample. Increasing the applied ﬁeld a further 0.25 T results ina second ﬂux jump between t=11.5–12 ms, as shown in6

Supercond. Sci. Technol. 29 (2016) 124004

M D Ainslie et al

Figure 8. (a) Average temperature, Tave, of the low Jc sample during
the pulse, until the applied ﬁeld reaches its peak value, for Bapp=3,
4.5 and 5.5 T. (b) The sample’s average critical current density,
Jc,ave, over the same period of time for the same applied ﬁelds.

Figure 9. (a) Average temperature, Tave, of the high Jc sample during
the pulse, until the applied ﬁeld reaches its peak value, for Bapp=5,
6 and 6.25 T. (b) The sample’s average critical current density, Jc,ave,over the same period of time for the same applied ﬁelds.

ﬁgure 7(f). Consequently, a large and sudden increase inmagnetic ﬁeld is seen at the centre, resulting in full magnetisation of the sample.
Figure 8(a) shows the average temperature, Tave, of the
low Jc sample during the pulse until the applied ﬁeld reaches
its peak value and ﬁgure 8(b) shows the sample’s average
critical current density, Jc,ave. Both Tave and Jc,ave are calculated by integrating T and Jc, respectively, over the crosssectional area of the bulk, and the result is divided by thisarea. There is a smooth increase in temperature and decreasein Jc due to the magnetic ﬂux penetration during the risingpulse. Since the time constant, τ, of each pulse is the same, ahigher Bapp value results in higher dB/dt and larger ﬁeld
penetration, resulting in a larger temperature rise and further
suppression of Jc. Figures 9(a) and (b) show the same plots
for the high Jc sample. For Bapp=5 T for the high Jc sample,
where no ﬂux jump occurs, there is a similar smooth increasein Tave and reduction in Jc,ave. The ﬂux jumps that occur for
higher applied ﬁelds, Bapp=6, 6.25 T, are accompanied by alarge and sudden temperature rise and a simultaneous,mirrored reduction in Jc,ave. The temperature rise during PFM
takes place adiabatically [3], because a large proportion of theheat generation takes place instantaneously and the coolingpower is ﬁnitely limited. The low thermal conductivity of thebulk also seriously affects the thermal and electromagnetic
responses, for which the thermal diffusion is much slower
than the magnetic diffusion. When modelling typical experimental conditions, and for a typical rise time of τ≈10 ms of
the applied pulse, the magnetic ﬂux propagation can be anorder of magnitude or higher faster than the heat propagation[48]. Another example of such behaviour can be observed inﬁgure 21 in [26].
In [24, 25], it was speculated that a large Lorentz force,
given by FL=J×B, acting on the magnetic ﬂux penetrating the sample during the PFM process is a cause of the ﬂux
jumps into the sample. In order to clarify this hypothesis,
ﬁgure 10 shows the calculated maximum Lorentz force density in each of the bulk samples, until the pulse reaches its
peak value, for (a) Bapp=5 T, (b) Bapp=6 T, and (c)Bapp=7 T. Here, the Lorentz force density in the −r7Supercond. Sci. Technol. 29 (2016) 124004

M D Ainslie et al

sample increases with Jc, and the ﬂux jump in the ultra-high
Jc sample is preceded ﬁrstly by a large FL, followed by aslight reduction in FL, which rapidly reduces when the ﬂux
jump occurs. When Bapp is increased to 6 T, as shown inﬁgure 10(b), this behaviour becomes more pronounced.
Finally, when Bapp is increased to 7 T, as shown inﬁgure 10(c), two ﬂux jumps are observed for the high andultra-high Jc samples, with the latter one causing ﬂux to jump
all the way into the centre of the sample. This secondary ﬂux
jump is preceded by the attainment of a large FL value,comparable to the FL value at which the ﬁrst ﬂux jumpoccurred. These results provide good evidence that the ﬂux
leap in high-Jc bulk superconductors is due to the large
Lorentz force acting on the magnetic ﬂux penetrating thesample, which causes ﬂux to jump further into the sample,
resulting in a sudden and large increase in temperature andreduction in Jc (as shown in ﬁgure 9). The maximum value of
J/Jc during the pulse rise time was also examined to discern
whether this is a cause of the ﬂux jumps, and no correlation
between J/Jc and the ﬂux jumps was observed. In fact, in thecase of the normal Jc bulk, J/Jc exceeded 2 during the pulse
rise time for Bapp=6, 7 T, for example, with no ﬂux jumps
observed.
Hence, if the FL value is large enough, a ﬂux jump to thecentre of the sample can be induced that can fully magnetise asample, with a reduction in the required activation ﬁeld,
which is signiﬁcant for practical applications as such ﬂux
jumps could be exploited to provide higher trapped ﬁelds,
whilst reducing the cost and size of the magnetising ﬁxture.
This numerical simulation framework provides a ﬂexible andcost-effective method for analysing and optimising different
magnetising ﬁxtures, as well as examining the effect of different Jc and pinning characteristics of other families ofsuperconducting materials and/or materials made from different processing techniques.

3. Conclusion
So-called ‘giant ﬁeld leaps’ have been observed by a numberof research groups investigating PFM of bulk high-temperature superconductor, where ﬂux jumps occur in the superconductor, and magnetic ﬂux suddenly intrudes into thecentre of the sample. This phenomenon is assistive to the
PFM process and results in a large increase in the measured
trapped ﬁeld at the centre of the top surface and fullmagnetisation of the sample. In this paper, a 2D axisymmetricﬁnite-element model implementing the H-formulation, with a
modiﬁed E–J power law characteristic representing the normal state resistivity of the superconductor when J>Jc, isused to qualitatively reproduce this phenomenon to goodeffect. By simply inserting a bulk sample of high Jc, as foundin high quality standard samples and neutron-irradiated ones,
such ‘giant ﬁeld leaps’ or ﬂux jumps are observed in themodel, due to the large Lorentz force, FL, generated during
the PFM process that drives the magnetic ﬂux into the sample.
The ﬂux jumps are accompanied by a large and sudden
temperature rise, and a simultaneous, mirrored reduction in

Figure 10. Calculated maximum Lorentz force density, J×B, in the−r direction in the bulk samples, until the pulse reaches its peakvalue, for (a) Bapp=5 T, (b) Bapp=6 T, and (c) Bapp=7 T.

direction is calculated, which drives the magnetic ﬂux
towards the centre of the sample, and is given by −Jfμ0Hz.
For Bapp=5 T, no ﬂux jumps occur for the low, normal andhigh Jc samples, but there is a ﬂux jump betweent=6.5–7 ms for the ultra-high Jc sample. As shown inﬁgure 10(a), the maximum Lorentz force density in the8Supercond. Sci. Technol. 29 (2016) 124004

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105021

Jc,ave. The maximum value of J/Jc during the pulse rise time
was also examined to discern whether this is a cause of theﬂux jumps, and no correlation between J/Jc and the ﬂux
jumps was observed. The numerical simulation framework is
extremely ﬂexible and provides a cost-effective method foranalysing and optimising different magnetising ﬁxtures, aswell as examining the effect on samples of different Jc andpinning characteristics, to exploit such ﬂux jumps to enhance
the trapped ﬁeld in practical, bulk superconductor
applications.

Acknowledgments
Dr Mark Ainslie would like to acknowledge the support of a
Royal Academy of Engineering Research Fellowship. The
authors would like to thank Archie Campbell, University ofCambridge, for his useful discussions.

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