Superconductor Science and Technology
Supercond. Sci. Technol. 34 (2021) 114003 (13pp)

https://doi.org/10.1088/1361-6668/ac2621

Modelling higher trapped fields
by pulsed field magnetisation ofcomposite bulk MgB2 superconductingrings
V Cientanni1,∗, M D Ainslie1, H Fujishiro2 and K Takahashi2
1

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
Department of Physical Science and Materials Engineering, Faculty of Science and Engineering,
Iwate University, Morioka 020-8551, Japan2

E-mail: vc329@cam.ac.uk
Received 10 May 2021, revised 11 August 2021
Accepted for publication 12 September 2021
Published 4 October 2021
Abstract

The recent results of Hirano et al (2020 Supercond. Sci. Technol. 33 085002) reported a hightrapped field of 1.61 T in a composite MgB2 ring comprising copper plates and and a soft ironyoke magnetised by pulsed field magnetisation (PFM). Inspired by these results, an investigation
using systematic modelling methods was conducted to investigate the key parameters leading to
the success of Hirano et al. Our results indicate that composite structures of MgB2 rings present
a viable method of trapping high magnetic fields when magnetised with PFM. Leveraging afinite element method modelling framework with a commercial software package (COMSOLMultiphysics), we have successfully modelled the experimental data with excellent agreement.
We have paid careful attention to the assumptions regarding the thermal physics, which enabled
the successful and accurate modelling of the experiment. Exploiting the flexibility ofcomputational modelling, we extend our studies to investigate the influence of the constituent
elements of the composite bulk on the electromagnetic and thermal behaviour, and discuss indetail how each can enhance the trapped field performance of the bulk. Aided by the models,
it is shown how the number of copper layers influences the elongation of the applied pulse,
reducing the field penetration and the maximum temperature rise of the bulk. The addition of
the iron yoke significantly increases the trapped field, by concentrating flux during and after thepulse.
Supplementary material for this article is available online
Keywords: bulk MgB2 , pulsed field magnetisation, trapped field, numerical modelling,
finite element method, composite structures, soft iron yoke
(Some figures may appear in colour only in the online journal)∗

Author to whom any correspondence should be addressed.

Original Content from this work may be used under theterms of the Creative Commons Attribution 4.0 licence. Any
further distribution of this work must maintain attribution to the author(s) andthe title of the work, journal citation and DOI.
1361-6668/21/114003+13$33.00

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1. Introduction

generation of eddy-currents. A longer pulse rise time reduces
the heating rate on the bulk; which is thought to be beneficial to trapping higher fields when magnetising under the PFM
regime [17]. Applying a pulse with a longer rise time directly by adjusting the current flowing in the magnetising coils[21, 22] is energetically less favourable, requiring bulkier andmore complex equipment. The copper layers thus act passively
to prolong an applied pulse. Further, their high thermal conductivity and heat capacity may be favourable in transferring
heat from the bulk generated during the PFM process (however, large eddy currents may conversely heat the bulk). Meanwhile, the iron yoke inserted into the bore of the composite
MgB2 ring acts to concentrate magnetic flux during the pulse,
assisting the bulk magnetisation. Therefore a large number ofvariables in this experiment may be controlled; from the number of copper layers and their thickness, to the presence of aniron yoke insert compared to no insert. This heeds well fornumerical modelling [23, 24], where parameters such as thesecan be adjusted easily compared to doing so experimentally.
Such flexibility was the impetus behind this work; to leverage
the advantage of numerical modelling, with direct inspirationfrom the record breaking achievement of the experimental results reported in [20].

1.1. Background

Bulk superconductors fabricated from (RE)-Ba–Cu–O (RE =rare-earth, such as Gd or Y) or MgB2 (magnesium diboride)materials show great promise for trapping large magnetic
fields, which can be exploited as permanent magnet analogues in engineering applications such as magnetic separation, superconducting rotating machines, and advanced
NMR/MRI [1–4]. In 2014, a new trapped field record of 17.6 T
at 26 K was achieved in a (RE)-Ba–Cu–O bulk [5], illustrating the significant application of these materials. MgB2
bulks meanwhile, present a number of advantages over (RE)Ba–Cu–O bulks. The lightweight, polycrystalline structure is‘rare-earth-free’, can trap highly homogenous fields due tothe long coherence length [6], and may be fabricated using
a number of techniques with relative ease [7]. The trappedfield performance of MgB2 bulks has continued to improvewith the following achievements of trapped fields in excess
of 3 T; 5.6 T at 11 K with a stacked MgB2 /TiB2 composite
hot-pressed bulk [8], 5.4 T at 12 K with a single hot-pressed
and milled Mg/B powder mix bulk [9], 3.75 T at 5 K with a
C-doped infiltration growth bulk [10], and 3.72 T at 5 K witha ball-milled, pressure-free fabrication technique [11]. Indeed,whilst the record high trapped field of 5.6 T is lower than (RE)Ba–Cu–O, the magnetisation potential is still great. Achieving
these records involved using the field-cooled magnetisation(FCM) technique [12], which ensures full magnetisation of thebulk. This requires a large, static magnetising field from bulky
and expensive magnets that are usually superconducting, so is
usually reserved for sample characterisation purposes. Pulsedfield magnetisation (PFM) meanwhile has greater practicalapplication with an inexpensive and portable setup, despite anoften lower trapped field due to high thermal loading of thesamples [13]. Adopting favourable magnetising coil fixtures,
whilst optimising the magnetisation technique with multiple
pulses and step-wise cooling, has yielded promising results for
PFM [14–18]. However, thermal loads remain a constraint inachieving higher trapped fields. This in particular remains asignificant challenge in MgB2 due to a higher susceptibility
to flux jumps, which results in a significant reduction in thetrapped field [19]. The prevalence of this effect in MgB2 is a
result of a combination of its unique properties: short thermaltransit times, highly non-linear Jc (B, T) characteristics, and a
low operating temperature range due to its low critical temperature, Tc , of 39 K. At lower temperatures the heat capacity
reduces resulting in larger temperature fluctuations per unit ofthermal energy, and thus higher likelihood of a temperature
gradient to induce a flux jump.

2. Modelling framework and assumptions
Figure 1 illustrates the geometry of the 2D axisymmetric models implemented in the FEM software package COMSOLMultiphysics version 5.4. Figure 2 meanwhile illustrates thethree sample configurations modelled. The fabrication of thesamples along with experimental details are outlined in Hirano et al [20]. The various sample configurations were all ofouter diameter (OD): 60 mm, and inner diameter (ID): 20 mmand between 19 and 19.5 mm in height. An iron yoke (ofdiameter 20 mm and height 20 mm) can be optionally inserted into the ring bore. The ‘single bulk’ measured 19 mm
in height, and comprised MgB2 only. It was then sliced into
two smaller annuli of 9 mm in height. Three copper annuli of
0.5 mm height were then placed between the MgB2 slices, as
per figure 2(b). The applied pulsed magnetic fields were simulated by implementing a current constraint on the ‘split coil’subdomains in the model, such that the applied current density(Jpulse ), in the ‘ϕ’ direction, took the form of equation (1). The
value of A has units m−1 , and linearly increases with magnetic
field from 0.46 at 1.04 T to 0.52 at 1.8 T. Constants ts and td ,were adjusted until the form of the current pulse matched thatof the pulses applied experimentally in [20].
()( t )t−
Jpulse = Hext A 1 − e− ts
e td

(1)

The split coil PFM setup with iron yokes is described elsewhere, and has been successfully used to enhance the trappedfield for both (RE)-Ba–Cu–O and MgB2 [25–27]. The magnetic field (Hall) and temperature probes were placed tomeasure the central trapped field and the temperature of thesample via the sample holder, respectively. The Hall probe
was located directly above the centre of the top surface of the

1.2. Motivation

Hirano et al [20] recently trapped a record high field of 1.61 Tin bulk MgB2 magnetised by PFM, by embedding bulk MgB2
superconducting rings between layers of copper, and inserting a soft-iron yoke. The addition of the copper inserts should
increase the effective rise time of the applied pulse through the2Supercond. Sci. Technol. 34 (2021) 114003

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Figure 1. Geometry of the 2D axisymmetric model implemented in COMSOL Multiphysics; complete with split coil magnetisation fixture

and iron yokes. The composite bulk MgB2 sample is embedded in the sample holder within a vacuum space, located in the centre of the coil.
The location of the magnetic field and temperature probes are labelled. The composite bulks measured 19.5 mm in height, whilst the ‘single
bulk’ measured 19 mm.

sample (r = 0, z = [sample height]/2 + 0.5 mm). The spacingof 0.5 mm in the model is to account for the active region of
the Hall probe. The temperature probe meanwhile was placed
radially towards the far edge of the sample holder (r = 48 mm,
z = 0).

∇×E = −d(µH)
dt

∇×H = J(2)(3)

where the vector components of the magnetic field strength
H = [Hz , Hr ], the current density J = [Jϕ ], and electric field
E = [Eϕ ]. In equation (2), µ = µo µr , where µo is the permeability of free space. The values of the relative permeability, µr ,were assumed to equal 1 for all media, except the soft ironyokes for which a B−H curve was modelled following themethod outlined by Ainslie et al [25]. The E−J power law,2.1. Electromagnetic considerations

The governing equations of the simulations involve coupling both electromagnetic and thermal physics. From Maxwell’s equations, Faraday’s law (2) and Ampere’s law (3) wereadopted:3

Supercond. Sci. Technol. 34 (2021) 114003

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Figure 2. Configurations of the modelled composite bulk MgB2 samples investigated experimentally in Hirano et al [20]. (a) ‘single bulk’

comprising only MgB2 , (b) ‘composite without yoke’ comprising MgB2 sandwiched by three layers of copper and (c) ‘composite withyoke’ containing an inserted soft-iron yoke. The copper layer thickness is exaggerated for better visualisation. Reproduced by permission of
IOP Publishing from [20].
2.2. Thermal considerations

Thermal transient properties were applied exclusively to thesample and sample holder, which in the experiments were located within a vacuum space. The thermal transient equation(6) below was utilised, which permits the modelling of the
conductive loss of heat via the outer radius of the sample
holder ri (thus, simulating the connection to the cold-stage).
The value Q is the volumetric heating rate, equal to the productof Eϕ · Jϕ , which provides coupling between the electromagnetic and thermal models (in addition to Jc (B, T)).
ρcp

for MgB2 distribution and associated equation fits for eachtemperature using equation (5). Utilising these equation fits,
COMSOL interpolates between them to generate a Jc (B, T) surface.
Reproduced by permission of IOP Publishing from [29].

represented by equation (4), was used to model the resistivity of the superconductor media; where Eo is the characteristic
electric field, Jc the critical current density of the superconductor, and n the flux creep parameter. The value of n was
assumed to be 45, and Eo = 1 × 10−4 Vm−1 . The Jc distribution was assumed both B and T dependent, as shown infigure 3, interpolated from data representative of the sample[28] and fit using equation (5), where α was obtained from therepresentative data [28], and Bo = 0.85 T.

E = Eo(Jc (B, T) = α 1 −

(

T
Tc

J
Jc

)n

)2 )1.5 − ( B 2 )1.5
B 1− Te o ( Tc )(6)

The flow of heat to the cold stage was determined by iteratively adjusting the thermal contact conductance until thecooling rate at the temperature probe matched the data of
figure 7(b) in [20]. The thermal contact conductance is thus
defined as the average heat energy flowing through a boundary, divided by the temperature gradient across it. The thermal
conductivity, κ, for MgB2 was assumed from the data reference in table 1. All other material thermal properties wereassumed isotropic, homogenous and temperature dependent;
cp , k and ρm , as provided in table 1. The temperature dependent electrical resistivity of each material is also given. A crucial further assumption involved controlling the thermal contact conductance between the composite bulk and the sample
holder. The contact conductance between the copper layers and the sample holder was assumed a temperature independent 2.5 × 104 W m−2 K−1 . For the contact conductance
between the MgB2 bulk and the sample holder however, it was
assumed to vary linearly between 300 W m−2 K−1 at 40 K and1000 W m−2 K−1 at 20 K, i.e. lower contact conductanceat higher temperatures. Whilst this may seem counterintuitive; the modelled results showed significantly better quantitative agreement, in both the time-domain and magnitude of thepulses. We justify our use of a decreasing thermal contact conductance with temperature for the interface between these twomaterials, by analysing their linear thermal expansion. MgB2
exhibits a decreasing linear thermal expansion rate (∆L/Lo ) in
the temperature range modelled: 20–40 K, and is in fact evennegative between 32.5 and 40 K as shown clearly by the insert

Figure 3. Assumed representative measured Jc (B, T) characteristics(∂T= κ∇2 (Tr − Tri ) + Q
∂t

(4)

(5)

4

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Table 1. Table of material properties.

Material
MgB2
Copper
Brass
Iron

Thermalconductivity (k(T))

Specific heat (cp (T))

[32]

−3

[33]

2590 kg m
8940 kg m−3
8730 kg m−37874 kg m−3

NIST [34][36]

Material density (ρm )NIST [34]Resistivity (ρ(T))
Jc (B, T) [28]/equation (4)[35]
3.57 × ρCu
2 × 10−8 m S−1

Figure 5. Comparison of simulation and experimental results for

Figure 4. The increasing linear thermal expansion of Cu contrasts

the trapped field of the sample under field-cooled magnetisation(FCM) from 15 K with a magnetising field of 4 T. The assumed
Jc (B, T) characteristics in the model are based on the interpolated
data presented in figure 3.

to the decreasing thermal expansion of MgB2 between 20 and 40 K
(shown with insert also) from [30] and [31]. This suggests that asboth the brass sample holder and the bulk heat, the contact pressure
between them reduces, reducing the effective thermal contact.
Reproduced by permission of APS Publishing from [30, 31].

but for the 2D axisymmetric case. The composite bulk was
field-cooled to 15 K with a magnetising field of 4 T, then after
removing this field the temperature was gradually increased
to 40 K above its Tc , observing the resulting trapped field. The
results in figure 5 illustrate that the assumed Jc (B, T) characteristics appropriately reproduce the expected trapped field fromFCM.
The magnitude and rise time of the simulated pulse at thecentre of the top surface of the sample (B(Hall)) is directly
influenced by the electrical conductivity and magnetic susceptibility of the materials subject to the applied pulse (Bapp ). The
rise time of the simulated pulse was matched to the experimental pulse by choosing an appropriate resistivity for theiron yoke(s) as given in table 1, whilst seeking well referenced
data for the other parameters listed therein. The pulse magnitude meanwhile was calibrated using the factor A until themagnitude of the applied field Bapp matched the peak measured value of the Hall probe B(Hall) at 40 K. Note that thisfollows the same experimental procedure in [20] and corresponds to the superconductor being in its normal state, T > Tc .
The applied field Bapp is also referred to as ‘Bex (shunt)’ in
figure 6, due to the experimentally generated pulse via a shunt
resistor. It was observed that A should be linearly proportional to the applied field, whose values are given in figure 3.

in figure 4. Copper meanwhile is monotonically increasing inthis range, and at a much larger rate. This suggests that due tothe annular geometry of the sample, the brass sample holder(which we assume to behave similar to copper) may expand
away from the MgB2 when heated in this temperature range.
This would cause a reduction in the contact pressure between
the MgB2 and the sample holder, and thus the effective thermal
conductance should reduce in response. If this effect is indeed
non-negligible, it may help explain why our use of changing
the thermal layer conductance helped achieve results with better agreement between the models and experiments. Figure 4
illustrates the linear thermal expansion (∆L/Lo ) of both materials; with data obtained from [30] and [31].
3. Modelling results
3.1. Pulse calibration

The success and accuracy of the models presented are a resultof carefully setting up the models to replicate the experimentas closely as possible. First, to validate the assumed Jc (B, T)
characteristics of figure 3, the trapped field after FCM was simulated following a similar process to that described in [37],5Supercond. Sci. Technol. 34 (2021) 114003

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Figure 6. A comparison of the time dependence of the

Figure 8. Modelled trapped field (B(Hall)) at the centre of the top

experimentally applied magnetic fields (scatter points) to thosesimulated in the model (line graphs). All data were measured(experiments; symbols) and calculated (simulations; solid lines) at
the same ‘Hall probe’ position, 0.5 mm above the centre of the top
surface of the sample at 40 K. Reproduced by permission of IOP
Publishing from [20].

surface of the sample (+0.5 mm) versus time for the single bulk
configuration compared to the experimental data (see figure 2) for
applied fields of 0.82, 0.85 and 0.88 T at 20 K. Reproduced bypermission of IOP Publishing from [20].

carried out at 40 K with the MgB2 sample in the normal state
above its critical temperature Tc , for both the simulation andthe experiment.

3.2. Single pulse replication

Next, following the experimental procedure outlined by Hirano et al [20], we apply single pulses of varying magnitude
to the three sample configurations at 20 K. These are the‘single bulk’ without copper layers or inserted iron yoke, the‘composite’ bulk with the copper layers and the ‘composite with yoke’ bulk, with the copper layers and inserted ironyoke (see figure 2). Figure 8 shows the simulation results forthe ‘single bulk’ without copper layers and the iron yoke, forapplied fields of 0.82, 0.85 and 0.88 T. There is good qualitative agreement between the simulation and experimental results and reasonable quantitative agreement, with the modelslightly over-predicting the trapped field for all applied fields.
Figure 9 meanwhile illustrates the replication of the trappedfields for the ‘composite without yoke’. The fitting here is
likely improved through the addition of the copper layers,which were modelled to have a higher thermal contact conductance between the sample and the sample holder comparedto that of the MgB2 , as outlined in the thermal assumptions
section. This can also be said for figure 10, which illustrates
the modelled trapped fields of the ‘composite with yoke’ configuration. The yoke was modelled with a fixed electrical conductance of 5×107 S m−1 , and no thermal contact between
the sample and the yoke. Whilst estimations were made forthe thermal contact conductance between the MgB2 and thesample holder, reference data [38] was used for the conductance between the copper and the sample holder. This is a possible reason for the improved agreement between simulated

Figure 7. Applied field (Bapp ) versus the measured field at the centreof the top surface of the sample (B(Hall)). The simulation results
show excellent agreement with the experimental calibration of theapplied field as carried out in Hirano et al [20]. This validates themodelling framework with respect to the magnetising fixture and its
interaction with the sample with the MgB2 in its normal state(T > Tc ). Reproduced by permission of IOP Publishing from [20].

Figure 6 illustrates the temporal dependence of the appliedmagnetic field in the simulations, compared to the experiment.
Figure 7 meanwhile illustrates the relationship between theapplied magnetic field Bapp , to the measured magnetic field atthe Hall probe, B(Hall). In both figures, we observe excellent
agreement between the simulation and experimental results. It
should be noted the pulse calibration in figures 6 and 7 was
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Figure 11. Simulated and experimental results for the maximum

Figure 9. Modelled trapped field (B(Hall)) at the centre of the top

temperature recorded at the temperature probe (r = 48 mm, z = 0)
for all sample configurations with varying applied pulse magnitudes.
Reproduced by permission of IOP Publishing from [20].

surface of the sample (+0.5 mm) versus time for the ‘composite
without yoke’ configuration compared to the experimental data, (seefigure 2) for applied fields of 1.1, 1.2 and 1.3 T at 20 K. Reproducedby permission of IOP Publishing from [20].

despite well agreed field distributions. It is clear that despite well agreed temperature probe measurements between thesimulated and experimental data, the simulated temperaturesurface plots within figure 12 suggest large temperature differences within the sample may exist. It is not practical tomeasure the interior of the sample during a pulse, and thesemodelled plots give insight where the experiment cannot.
The model helps illustrate how the results of Hirano et al
were achieved. Through the addition of the copper plates theapplied pulse was elongated temporally as shown in figure 6,
due to the action of Lenz’s law on the copper. The modelled
thermal contact between these copper slides and the sample
holder also permitted the free conduction of heat energy postpulse, which can be readily interpreted from figure 12. Thisbetter cooling may have helped reduce the effect of heating on
the pinning forces, which would normally reduce their magnitude. The modelled results were heavily dominated by theflow of heat energy and the sample temperature, for which
figure 12 provides interesting insight. Whilst the probe data,
which measures the temperature of the sample indirectly via
the sample holder, shows good agreement; the temperature
within the bulk is significantly higher. It is interesting to further see that despite being both applied with 1.2 T pulses, the‘composite with yoke’ arrangement had a significantly higher
temperature than the ‘composite without yoke’. This is likely
due to the larger motion of flux associated with the inserted
magnetic yoke. The copper slides are seen as dominant outlets for the flow of heat into the sample holder, illustrating
their importance in cooling the bulk. Figures 11 and 12 highlight the difficulties of measuring the temperature directly, butalso the utility and insight that numerical can provide to understand the thermal behaviour. The addition of the iron yoke also
provided benefit to trapping fields within the bulk. This is dueto the assistance the yoke provides in concentrating flux in the

Figure 10. Modelled trapped field (B(Hall)) at the centre of the top

surface of the sample (+0.5 mm) versus time for the ‘composite
with yoke’ configuration compared to the experimental data, (seefigure 2) for applied fields of 1.1, 1.2 and 1.3 T at 20 K. Reproducedby permission of IOP Publishing from [20].

and experimental results; as the trapped field was observed to
be sensitive to this parameter.
Figure 11 illustrates the maximum temperature (Tmax )recorded at the temperature probe, located at r = 48 mm andz = 0, embedded within the sample holder. The model represents an axisymmetric construct with an annular sample holder
and heat flow is isotropic along the radius. This is not the sameas in the experiment as heat flow was anisotropic towards thecold-stage, with the probe on the adjacent side. This may be
why the modelled temperature rises were marginally higher
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Figure 12. Snapshot of the temperature distribution at 0.5 s for each sample configuration. The bulks, in the order presented, were were

subjected to applied pulsed fields of 0.85, 1.2 and 1.2 T, respectively. The probe maximum temperature in the sample holder clearly differsfrom that within the bulk itself.

bore of the bulk during the pulse. The magnetisation of the ironyoke is weakly temperature dependent below 100 K, meaning
it can remain readily magnetised throughout the pulsing. The
magnetising currents of MgB2 meanwhile are strongly temperature (and indeed field) dependent, below 40 K. This fact may
be beneficial to future experimentation aiming to trap greater
fields, and as we explore later in this paper in the extension
studies.

pre-magnetised state of 0.6 T at the centre of the top surface of the sample (+0.5 mm) was achieved. The secondary
applied pulses (of magnitude 1.15 T, 1.25 T and 1.5 T) thenproduced the modelled trapped field represented in figure 13.
There is excellent agreement between the experimental data
and the modelled results, which were produced without any
modification to the simulation assumptions and physics fromthe single pulse framework. Despite the slight reduction inshielding observed on the 1.15 T pulse, this gives a high levelof confidence in the utilisation of these models to carry out theextension studies presented in the following section; which inturn explore the physics of this experiment and attempt to illustrate the primary influencing factors of this complex setup.

3.3. Double pulse replication

Hirano et al [20] also applied a multi-pulse (double-pulse)
technique to the bulks, namely to the ‘composite with yoke’configuration. This pulsing sequence involved applying two
distinct pulses with a large relaxation time between the pulses,
and is described in detail in the literature [39]. Applying thismethod to their samples, Hirano et al [20] achieved the record
trapped field of 1.61 T. It has been shown by a number of
researchers that this technique can lead to an increased trappedfield, however there are a larger number of parameters—
including the number of pulses, the temperature(s), and premagnetised state of the bulk at which they are applied—which
need careful optimisation to achieve this [39–43]. Through theapplication of the first pulse (of magnitude 1.3 T), a uniform

4. Extension studies
Above, we have established with carefully-considered numerical models that it is possible to accurately replicate the experimental data of Hirano et al [20]. We have accurately replicated the electromagnetic and thermal behaviour, with all three
configurations, for the single and multi-pulse cases. Here, we
extend this study by investigating the influence of the copper
layers and the yoke in magnetising the bulks more effectively.

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Figure 13. Modelled trapped field (B(Hall)) at the centre of the top
surface of the sample versus time for the ‘composite with yoke’configuration under the double-pulse sequence as outlined in [20],compared to the experimental data, for applied fields of 1.15, 1.25
and 1.5 T at 20 K. Reproduced by permission of IOP Publishingfrom [20].

Figure 14. Modelled applied field (B(Hall)) at the centre of the top

surface of the sample versus time for the ‘composite without yoke’configuration at 40 K. We modify the number of copper layers (N)in the bulk from three up to ten, spacing them equidistantly.

Despite this negligible change in temperature, the increased
shielding and pulse retardation of more layers is not negligible.
This result indicates that larger, more expensive magnetisingequipment would be required to access higher trapped fields
for samples with many copper layers. Further, elongating thepulse reduces the applied field magnitude. Therefore, to apply
a field of similar magnitude, a higher capacitor bank voltage(and hence stored energy) is required. Given the increased
heating at larger field magnitudes, it seems the choice of threelayers by Hirano et al was a considerably appropriate one.
The ambient temperature of 40 K was chosen to eliminate
the influence of the MgB2 supercurrent, and investigate theinteraction of the copper layers only with the applied field.
Now, in figure 15, we have modelled the ‘composite without
yoke’ sample at 20 K to investigate the trapped field potentialof the samples when superconducting. Single pulses of magnitude 1.03 T and 2.00 T were applied to samples with copper
layer numbers between N = 2, and N = 10. The additional
layer of N = 2 was chosen here to capture the peak at the lower
applied field. These two pulse magnitudes were chosen toillustrate how larger fields are required to effectively penetratesamples with higher layer numbers. The magnetic field plotted(Btrap (centre)) was calculated in the simulation at the centre of
the sample bore (r = 0, z = 0), in contrast to B(Hall) measured
in the location of the Hall probe. This was chosen to make abetter comparison to the sample shielding and penetration. We
define the sample penetration as the displacement along the cut
line z = 0 from the outer sample radius to the inner radius, thathas not shielded the applied pulse. For example, 70% penetration would mean the bulk shielded the pulse up to r = 16 mm(14 mm past the outer radius) along the z = 0 line. For the
1.03 T pulses, the peak trapped field occurred for N = 2 layers.
For higher layer numbers, ‘M-shaped’ magnetisation profiles
were observed with sample penetration dropping from 98% of

4.1. Influence of copper layer number

In the experiment presented by Hirano et al [20], three copper layers were inserted into two of the bulk configurations.
These layers were added to elongate the applied pulse, throughthe induction of eddy-currents in the copper. Hence for eachadded copper layer, the rise-time of the applied pulse can beextended; but simultaneously the volume of superconductor is
reduced. To quantify this effect and determine the optimum
number of copper layers, we vary the number of layers, N,from three to ten, spaced equidistantly throughout the sample.
Figure 14 illustrates the modelled field at the centre of the top
surface of these samples at 40 K, with an applied field (Bapp ) of
1.03 T. This pulse has the same magnitude and rise-time usedby Hirano et al in their 40 K calibration tests (figure 5(a) in[20]). In figure 14, the applied pulse (black line) is comparedto the modelled pulses for varying layer number (coloured). It
can be seen that the applied pulse rise time (trs ) is elongated to
70 ms, and the magnitude reduces to approximately 0.78 T for
the three-layer configuration, showing good agreement withthe experimental data (as seen in figure 6). Comparing thecase for layer number N = 3, and N = 10, we can see thatthe applied field magnitude and rise time changes by a further−0.34 T and 75 ms, respectively. The maximum temperaturerecorded for the N = 3 and N = 10 cases was 40.99 K and
40.83 K respectively, compared to that of 40.63 K for the nolayer case. The difference in temperature rises are small, as the
specific heat of copper is over eight times larger at 40 K than at
20 K. Whilst the applied pulse magnitude (Bapp ) remains constant, the field at the Hall probe (B(Hall)) decreases with each
additional layer. Hence, the small decrease in temperature withadditional layers indicates that increasing the number of copper layers does not significantly reduce the sample heating.
9Supercond. Sci. Technol. 34 (2021) 114003

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Figure 15. Modelled trapped field (Btrap (centre)) at the centre of thesample bore (r = 0, z = 0) for the ‘composite without yoke’configuration at 20 K. The number of copper layers (N) in the bulk
is modified from two up to ten, spacing them equidistantly. On theright y-axis is the maximum temperature rise for each layer number
and pulse.

Figure 16. Modelled trapped field (Btrap ) at the centre of the top

surface of the ‘composite without yoke’ with applied field(B(shunt)), for varying copper layer number (N). The maximum
trapped field is observed for the case of five copper layers.
Reproduced by permission of IOP Publishing from [20].

again indicates that the use of three copper layers was a considerably appropriate choice by Hirano et al [20], when considering an increment of 0.4 T in applied field yields an additional
trapped field of less than 0.2 T for the N = 5 case.

the radius for N = 3, to just 30% for N = 10. The ‘single bulk’(N = 0) trapped negligible or zero field for both pulse magnitudes. The peak trapped field for the 2.00 T pulses occurs
for N = 7, trapping 0.58 T. Sample penetration was 100% for
all layer numbers up to N = 10, where it drops to 95%. For
the 2.00 T pulse, with layer numbers N < 7, the sample could
not trap the large applied field and was heated above 27 K. For
layer numbers N ⩾ 7, the added layers increased the rise-timeand reduced the pulse magnitude, resulting in reduced heating.
The maximum temperature profiles illustrate how both pulses
are moderated by the presence of the layers, with diminishing effect towards the highest layer numbers. Knowing fromfigure 14 that the applied pulse for N = 10 is much lower inmagnitude and greater in rise time, helps explain these smaller
temperature increases. It is clear from figure 15 that whilst larger pulses are required to access and effectively trap uniform
fields at the centre of the bulk, the increased heating limits thetrapped field performance of the bulk. This means that whilst
the thermal performance of samples with a higher layer number is improved, it is not enough to abate the heating of thelarge pulses required to trap a field effectively in the MgB2 .
Finally, in figure 16 we investigate how varying the magnitude of the applied field to the composite bulks affects thetrapped field at the centre of the top surface of the bulk, withincreasing layer number. This should elucidate the optimum
number of layers to trap the maximum field in the sample.
There is good agreement between the experimental data andthe model for the case of three layers. Increasing the layer
number at first increases trapped field performance, but at thecost of needing to apply higher fields. For the case of N = 5
layers, a maximum trapped field of 0.65 T was observed at
1.6 T applied field. As layer number increases, the volume
of superconducting material decreases, and the subsequent
trapped field performance reduces above five layers. This

4.2. Influence of iron yoke insertion

The three sample configurations investigated by Hirano et al[20] were progressively augmented, to first include copper
plates and then also an iron yoke. As a result, the iron yoke wasinvestigated only with the addition of copper plates. Below, we
present a study investigating the case of a ‘single bulk with ironyoke’, to investigate the effect of the iron yoke in magnetisingthe ‘single bulk’, with the absence of copper plates.
First, we examine how the iron yoke modulates the applied
pulse in the absence of copper layers. Figure 17 illustrates how
an applied pulse of 1.03 T is enhanced by the soft-iron yoke at
40 K. The dashed black line shows an applied pulse (Bapp ) of
1.03 T, and the dashed red line (B(Hall)) of the simulated pulse
at the centre of the top surface of the yoke. The pulse rise-timeis increased slightly due to the finite conductivity of the ironyoke, whilst the magnitude is enhanced significantly due toits ferromagnetic properties. Despite this large enhancementof the magnetic field, the maximum temperature of the bulk
was 40.8 K when in the normal state, in part due to assuming
no thermal contact between the iron yoke and bulk. The effectof the iron yoke contrasts significantly to the copper layers,which reduce the applied field magnitude and extend the pulse
rise time more significantly. This also reduces the requirementfrom the viewpoint of the magnetising fixture with a composite iron yoke bulk, due to the enhancing effect of the yoke on
the applied field from 1.03 T to 2.2 T. Figure 17 also illustrates
how three different pulses (of magnitude 0.7 T to 0.8 T) affectthe trapped field of the bulk at 20 K. With an applied field10

Supercond. Sci. Technol. 34 (2021) 114003

V Cientanni et al

Figure 19. Modelled trapped field (Btrap ) distribution along the top

Figure 17. Modelled field (B(Hall)) at the centre of the top surface

surface of the bulk +0.5 mm, for the ‘single bulk with yoke’configuration. A shift in the contribution of magnetisation between
the iron yoke and MgB2 can be observed with increasing applied
pulse magnitude (Bapp ).

of the sample versus time for the ‘single bulk with yoke’configuration at both 20 K and 40 K. The pulses at 40 K illustrate
how the iron yoke enhances the applied pulse, and those at 20 K
illustrate the trapped field behaviour.

rise in the maximum temperature is simultaneously observed
around this peak, followed by a reduced rate of temperature
increase at larger applied fields. This behaviour is observed inthe ‘single bulk’ case of figure 11 with a lower temperature risecompared to the ‘yoke only’ case. It appears that the flux concentrating properties of the iron yoke assists in retaining themagnetisation of the entire bulk, despite the significant heating. The heating rate is reduced once the composite bulk (i.e.
‘single bulk’ with iron insert) magnetisation ‘saturates’ past
Bapp = 0.8 T, which is evident from the two distinct gradientsin the slope of Tmax . Compared to the ‘single bulk’ case, theapplied field which yields the peak trapped field is reduced
from 0.85 T to 0.75 T when inserted with the iron yoke. The
stronger performance of the ‘single bulk with yoke’ sample
at higher applied fields compared to the ‘single bulk’ can be
attributed to the addition of the soft-iron yoke.
Figure 19 illustrates how the magnetic field is trapped
within the ‘single bulk with yoke’ composite, in response to
the applied pulses. The field was sampled at ten points along
the top surface of the bulk +0.5 mm, every 2.5 mm in a stylesimilar to a Hall array. The point of 10 mm was not sampled
due to a saturating effect on the B-field distribution caused by
the sharp corner of the yoke. With an applied pulse of 0.6 T,
the bulk has mostly shielded the applied field. With increasing magnitude however, the bulk begins to trap greater fields.
At 0.7 T applied field, the MgB2 magnetisation peaks with
an M-shaped distribution, and some magnetisation of the ironyoke. At applied fields of 0.75 T and greater, the iron yoke
contributes the greatest magnetisation of the sample, whilst at
lower applied fields, the MgB2 itself contributes greater magnetisation. The peak in magnetic field towards the edge of theiron yoke at 10 mm is caused by the sharp geometry/corner ofthe yoke. The field was not sampled at 10 mm in figure 19 toremove this saturating effect on the figure. Despite forming a

Figure 18. Modelled trapped field (Btrap ) at the centre of the top

surface of the sample for the ‘single bulk with yoke’ configurationand maximum temperature (at the temperature probe) withincreasing applied pulse magnitude (Bapp ). A rapid increase intemperature is observed around the peak trapped field.

of 0.7 T, the bulk has mostly shielded the applied field, trapping only 0.17 T. The peak trapped field for this bulk occurs
for the applied field of 0.75 T, trapping 1.27 T due to the flux
concentrating properties of the iron yoke. At higher appliedfields however, the trapped field begins to decrease. This effectcan be better explained with reference to figure 18, discussed
below.
Figure 18 illustrates the trapped field performance (Btrap )and maximum temperature (Tmax ) of the bulk, against appliedfield magnitude (Bapp ). A sharp peak in trapped field isobserved at 0.75 T applied field, tending towards a trappedfield of approximately 1.0 T at larger applied fields. A rapid
11

Supercond. Sci. Technol. 34 (2021) 114003

V Cientanni et alless uniform field than a disc-shaped bulk, this bulk composite
has enhanced the trapped field at the centre of the bulk. The
iron yoke and MgB2 components of the bulk work collaboratively to enhance the magnetisation of the bulk, and retain
greater and more consistent fields when pulsed at higher magnitudes. This could be useful for other composite bulk arrangements that operate over a larger temperature and field range,such as the hybrid trapped field magnet lens developed by
Takahashi, Fujishiro and Ainslie [44–46].

fields that would normally result in significantly lower trappedfields without the presence of the yoke. The enhancing magnetising effect of the soft-iron yoke may allow for more compact and efficient magnetising coil design, using smaller coils
and lower applied fields.
These results will help guide further experimentation
allowing researchers to determine how these complex parameters may influence the optimal trapped field and produce
better bulk superconducting magnets.

5. Conclusions

Data availability statement

We have demonstrated that the reported data of Hirano et al[20] can be replicated with an accurate FEM modelling framework, whereby a trapped field record in an MgB2 composite bulk using PFM was achieved. With careful calibration of
the modelled physics and geometry, we have validated themodelling framework against the measured FCM performance of the bulk. Further, the experimental PFM data is validated for both single- and multi-pulse regimes, for three sample
configurations: the ‘composite with yoke’ with copper layers and inserted iron yoke, the ‘composite’ with copper layers (no yoke), and the ‘single bulk’ ring (without either copper
layers or inserted yoke). Importantly, we successfully modelled the trapped field record of 1.61 T via PFM using a
multi-pulse technique, with excellent quantitative agreement.
Using these models we have gained insight into the physics and mechanisms that produced the experimental results,
and explained how the transit of heat influenced the results
obtained. We found that decreasing the contact conductance
between the MgB2 bulk and the sample holder at higher temperatures improved the accuracy of the models, and speculate
a physical explanation for this with the help of figure 4.
The models are then extended to investigate the effect oftrapped field with a change in the number of copper layers,which indicated this configuration achieved maximum trappedfield at N = 5 layers for Bapp = 1.6 T. However, this finding
indicates an additional burden on the magnetisation fixture,
where a higher applied pulse current is required to effectively
magnetise samples with a higher number of copper layers. It
is shown how the number of copper layers impacts both theelongation of the applied pulsed field and its magnitude, andhow this affects the trapped field in the composite bulk. We
have also illustrated that the copper layers influence the thermodynamic behaviour of the composite bulk. We found that
increasing the number of copper layers reduces the maximum
temperature rise of the bulk, due to reduced magnetic field penetration and the additional high thermal conductance pathways
created by these layers for cooling the bulk during pulsing. We
conclude that a choice of three layers was a suitable compromise between the benefits and drawbacks of the copper layers.
Finally, the influence of the iron yoke inserted into the composite bulk was investigated in detail, which was not reported
in the original experiments. The soft-iron yoke inserted intothe ‘single bulk’ configuration increases the effective magnitude of the applied pulse, and acts to concentrate flux during the pulse and retain it after the pulse, even for high applied

The data that support the findings of this study are openly
available at the following URL/DOI: https://doi.org/10.17863/
CAM.69446.

Acknowledgments
Vito Cientanni would like to acknowledge financial support
from the EPSRC DTP Fund Dr Mark Ainslie would like toacknowledge financial support from an Engineering and Physical Sciences Research Council (EPSRC) Early Career Fellowship EP/P020313/1. This research was also supported by JSPS
KAKENHI Grant No. 19K05240.
ORCID iDs
V Cientanni  https://orcid.org/0000-0001-6621-2733
M D Ainslie  https://orcid.org/0000-0003-0466-3680
H Fujishiro  https://orcid.org/0000-0003-1483-835X
K Takahashi  https://orcid.org/0000-0002-8278-2688
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