Superconductor Science and Technology
Supercond. Sci. Technol. 28 (2015) 053002 (20pp)

doi:10.1088/0953-2048/28/5/053002

Topical Review

Modelling of bulk superconductor
magnetization
M D Ainslie1 and H Fujishiro2
1

Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Trumpington
Street, Cambridge CB2 1PZ, UK
2
Department of Materials Science and Engineering, Faculty of Engineering, Iwate University, 4-3-5 Ueda,
Morioka 020-8551, Japan
E-mail: mda36@cam.ac.uk
Received 6 December 2014, revised 14 January 2015
Accepted for publication 2 February 2015
Published 30 March 2015
Abstract

This paper presents a topical review of the current state of the art in modelling the magnetizationof bulk superconductors, including both (RE)BCO (where RE = rare earth or Y) and MgB2
materials. Such modelling is a powerful tool to understand the physical mechanisms of theirmagnetization, to assist in interpretation of experimental results, and to predict the performanceof practical bulk superconductor-based devices, which is particularly important as manysuperconducting applications head towards the commercialization stage of their development in
the coming years. In addition to the analytical and numerical techniques currently used byresearchers for modelling such materials, the commonly used practical techniques to magnetize
bulk superconductors are summarized with a particular focus on pulsed ﬁeld magnetization(PFM), which is promising as a compact, mobile and relatively inexpensive magnetizing
technique. A number of numerical models developed to analyse the issues related to PFM and
optimise the technique are described in detail, including understanding the dynamics of themagnetic ﬂux penetration and the inﬂuence of material inhomogeneities, thermal properties,
pulse duration, magnitude and shape, and the shape of the magnetization coil(s). The effect ofexternally applied magnetic ﬁelds in different conﬁgurations on the attenuation of the trapped
ﬁeld is also discussed. A number of novel and hybrid bulk superconductor structures aredescribed, including improved thermal conductivity structures and ferromagnet–superconductor
structures, which have been designed to overcome some of the issues related to bulk
superconductors and their magnetization and enhance the intrinsic properties of bulk
superconductors acting as trapped ﬁeld magnets. Finally, the use of hollow bulk cylinders/tubes
for shielding is analysed.
Keywords: numerical modelling, bulk superconductors, (RE)BCO, MgB2, bulk superconductor
magnetization, pulsed ﬁeld magnetization, trapped ﬁeld magnets(Some ﬁgures may appear in colour only in the online journal)
1. Introduction

conventional permanent magnets, such as NdFeB and SmCo, is
independent of sample volume and is therefore limited by thematerial properties of the permanent magnet. Bulk superconducting materials, on the other hand, trap magnetic ﬂux via
the generation of macroscopic electrical currents, which leads
directly to an increase in magnetization with sample volume.

The ability to generate a permanent, stable magnetic ﬁeld
unsupported by an electromotive force is fundamental to avariety of engineering applications. The magnetization—a
useful measure of the ability to generate a magnetic ﬁeld—of
0953-2048/15/053002+20$33.00

1© 2015 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

magnetic bearings, energy storage ﬂywheels, magnetic resonance imaging, magnetic separation and rotating machines[24–26]. Signiﬁcantly, the higher magnetic loading in rotatingmachines would provide an increased torque/power density,
resulting potentially in a machine that is smaller and lighter inweight than conventional devices of the same rating [27, 28].
However, investigating and predicting the magnetization ofthese technologically important materials and developing
practical magnetizing techniques is crucial to using them as
TFMs in applications of these types.
A number of analytical and numerical models have been
proposed for simulating the properties of bulk superconducting materials and modelling is a powerful tool tounderstand the physical mechanisms of their magnetization,
to reproduce and assist in the interpretation of experimental
results, and predict the performance of practical bulk superconductor-based devices. In addition, in the case of bulk
superconductors, such numerical modelling can be used to
predict and propose new magnetizing techniques, which ismore difﬁcult to achieve experimentally.
Analytical models can be easier, faster and more accuratecompared to numerical methods, such as those based on theﬁnite element method (FEM), but are limited to speciﬁc
geometries and simpliﬁed assumptions [29]. Numericalmodels can overcome some of these limitations and simulate
more complex geometries and situations, but this comes at theexpense of increased computational requirements, more
complex software implementation and longer computationaltimes [30]. An example is the multi-physical numerical model
required for the pulsed ﬁeld magnetization (PFM) of a bulk
superconductor, which requires both electromagnetic andthermal considerations over a very short time scale.
Numerical modelling is a particularly important and costeffective method to guide both superconducting material
processing and practical device design as many superconducting applications head towards the commercialization
stage of their development in the coming years. Therefore,
this paper presents a topical review of the current state of theart in modelling the magnetization of bulk superconductors,
including both (RE)BCO and MgB2 materials. In addition to
the analytical and numerical techniques currently used byresearchers for modelling, the paper also reviews some of theexperimental results that support the modelling results, andthe implications of these for designing practical trapped ﬂuxtype devices, such as bearings and electric machines.
In section 2, a simple analytical method to calculate thetrapped ﬁeld of a bulk superconductor based on the Bean critical state model is presented, as well as an overview of analytical and numerical techniques used to model
superconducting materials in general. In section 3, the commonly used practical magnetization techniques are summarizedwith a particular focus on the PFM technique, including
understanding the role of inhomogeneities, multi-pulse techniques and pulse coil conﬁgurations, on the ﬂux dynamics ofthe magnetization process. In section 4, the effect of externallyapplied magnetic ﬁelds in different conﬁgurations on theattenuation of the trapped ﬁeld is discussed, and in section 5, a
number of novel and hybrid bulk superconductor structures are

This, in turn, potentially overcomes this fundamental limit in
the size of ﬁeld generated by conventional permanent magnets.
Bulk superconductors, acting as trapped ﬁeld magnets(TFMs), can trap magnetic ﬁelds of magnitude over ten times
higher than the maximum ﬁeld produced by conventional
magnets, which is limited practically to rather less than 2 T, lessthan the saturation magnetization of iron [1]. Indeed, it has beenshown that (RE)BCO (where RE = rare earth or Y) bulk
superconductors can trap ﬁelds greater than 17 T—the longstanding world record ﬁeld generated by an arrangement of two
bulk superconductors of 17.24 T at 29 K [2] was recently
exceeded by 17.6 T at 26 K in [3]. Additionally, bulk superconductors can exhibit Jcs of 5 × 104 A cm−2 at 1 T and 77 K
(the boiling point of liquid nitrogen), resulting typically intrapped ﬁelds of up to between 1 and 1.5 T for YBCO andgreater than 2 T for (RE)BCO at this technologically important
temperature, with 3 T the highest trapped ﬁeld at 77 K so far[4]. It should be noted that the values quoted are with respect tostandard, non-irradiated YBCO and (RE)BCO samples, but
neutron-irradiated, radioactive YBCO samples have exhibited
higher Jc values [5, 6] and trapped ﬁelds over 2 T at 77 K [5–8].
Another superconductor that shows promise in bulk form
as TFMs is MgB2. MgB2 was ﬁrst reported in [9] and although
the transition temperature for this material is low (39 K), which
requires a lower operating temperature (15–20 K) and more
complex cryogenic system than that for (RE)BCO bulks, thematerial is cheaper, lighter weight and has a more homogeneous Jc distribution. Furthermore, the relative ease of fabrication of MgB2 materials, as well as their long coherence
length [10], lower anisotropy and strongly linked supercurrent
ﬂow in untextured polycrystalline samples [11, 12], hasenabled a number of different processing techniques to be
developed. Signiﬁcant improvements continue to be made interms of in-ﬁeld Jc and trapped ﬁeld capability:• 5.4 T at 12 K was achieved with a single 20 mm diameter
MgB2 bulk fabricated by hot-pressing ball-milled Mg
and B powders [13];• 4 T at 11 K and 3 T at 20 K with a pair of 30 mm
diameter MgB2 bulks fabricated by conventional sintering [14];• 3.6 T at 13.2 K and 2.8 T at 20 K with a single 38 mm
diameter Ti-doped MgB2 bulk fabricated by the hot
isostatic pressing method [15];• 3.14 T at 17.5 K with a pair of 25 mm diameter MgB2
bulks fabricated by uniaxial hot pressing [16];• 1.5 T at 16.4 K with a 30 mm diameter MgB2 bulk
fabricated by a capsule method [17]; and• 1.3 T at 15 K with a 55 mm diameter sample (with a
6 mm central hole) fabricated by the reactive Mg liquid
inﬁltration technique [18].
There has also been some recent work on the possibilityof utilizing stacks of (RE)BCO tapes as pseudo-bulk TFMs
[19–23]: up to 7.34 T has been trapped at 4.2 K using a stackof 120 layers of 12 mm square high-temperature superconducting (HTS) coated conductor [19].
As a result, there is great interest in using these materials
as TFMs in a number of engineering applications, including2Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

presented in [17, 33–35]:

Btrap (z) =⎡⎛μ 0 Jc ⎢a+(z + t ) ln ⎜⎜
2 ⎢⎣
⎝
⎛
a+
− z ⋅ ln ⎜⎜⎝described, including improving the thermal properties of bulk
superconductors, ferromagnet–superconductor structures, andhollow bulk cylinders/tubes for shielding. Finally, the paper isconcluded in section 6 with a view towards the future and theremaining challenges in this active ﬁeld of research.
2. Calculating trapped ﬁelds
2.1. Analytical techniques

Based on the critical state model presented by Bean [31, 32],
the peak trapped magnetic ﬂux density, Btrap, at the centre of
the top surface of a single-grain bulk superconductor, oriented
with its thickness parallel to the c-axis, due to an induced,
persistent supercurrent is given in its simplest form by(1)

where μ0 is the permeability of free space, Jc is the criticalcurrent density of the superconducting material, and a is the
sample radius. k is a correction factor to the simple Bean(slab) approximation due to the ﬁnite thickness, t, of a discshaped bulk superconductor sample, based on application of
the Biot–Savart law, given by [16, 17]:
k=

⎛
t
a
ln ⎜⎜ +
2a ⎝ t⎞⎛ a ⎞2
1 + ⎜ ⎟ ⎟⎟ .
⎝t⎠
⎠

⎞⎤
a2 + (z + t )2 ⎟ ⎥
⎟ ⎥.
z+t
⎠⎦

(3)

Equation (1) implies two main approaches for improving
the ﬁeld trapping ability of a bulk superconductor: (1) byenhancing the critical current density through improved ﬂux
pinning and increased sample homogeneity, and (2) byincreasing the sample size. In addition to these factors, theperformance of a bulk superconductor can be limited bystrong Lorentz forces that result in large mechanical stresseson the material that can induce cracks in the sample andseverely degrade the trapped ﬁeld [3], and a heat exchange
rate with the cryogenic ﬂuid or cooling apparatus that is too
low can result in signiﬁcant temperature rises in the material [36].
An overview of analytical methods and formulae formodelling HTS materials is presented in [37], but this work
focuses chieﬂy on thin strips and stacks thereof, and istherefore most applicable to HTS wires, cables and related
devices. Analytical expressions have been derived for thecurrent and ﬁeld distributions using the Bean assumption(constant Jc) for an inﬁnitely thin disc for the static non-linear
case [38], as well as for perpendicular time-varying periodic
external magnetic ﬁelds [39]. Numerical solutions also exist
for superconducting discs of ﬁnite thickness [40–43], and in[44], Brandt calculated the electromagnetic response of a discof ﬁnite thickness in a perpendicular time-varying magnetic
ﬁeld, extending his work on a thin strip in [45]. Whilst analytical methods are easier and faster to implement comparedto numerical methods, they are limited to speciﬁc geometries
and simpliﬁed, homogeneous assumptions regarding thesuperconductor’s electromagnetic and thermal behaviours.
Such methods are usually based on the critical state model(constant Jc and no frequency dependence) and can only deal
with simple, uniform ﬁelds. Numerical methods, which are
introduced in the following section, can be used model more
complex, practical situations, for arbitrary geometries andapplied ﬁelds, and where the superconductor’s behaviour canvary depending on a number of different variables.

Figure 1. Zero ﬁeld cooling (ZFC) and ﬁeld cooling (FC) bulk
superconductor magnetization techniques using the Bean model[31, 32] approximation. Reproduced with permission from [84]. For
ZFC (left), a magnetic ﬁeld equal to the full penetration ﬁeld, Hp, isapplied to a pre-cooled sample (top panel), which is increased to 2Hp(middle panel), and then the ﬁeld is removed, trapping a ﬁeld
H0 = Hp (bottom panel). For FC (right), a magnetic ﬁeld H0 isapplied to a sample above its critical temperature Tc (top panel),
which is then cooled below Tc and the ﬁeld is slowly removed(middle panel; where the applied ﬁeld is reduced to 0.5H0), trapping
a ﬁeld equal to the applied ﬁeld H0 (bottom panel).

Btrap = kμ 0 Jc a,⎞
a2 + (z + t )2 ⎟⎟z+t⎠

2.2. Numerical techniques

There exist a number of different numerical techniques thathave been applied variously to the modelling of superconducting materials [30, 46], including the ﬁnite difference
method [47], the combined sand-pile/Biot–Savart method[48–51], the Fourier Transform method [29], minimization orvariational techniques [52–58] and FEM. In general, FEM
techniques are the most commonly used and developed
methods, and these can be applied to superconducting material problems using a variety of formulations [30, 59, 60]:
namely the A–V [61–63], T–Ω [64–66], H [67–78] and E(2)

A more general formula for the ﬁeld at any height (z)above a superconducting disc of radius a and thickness t was3Supercond. Sci. Technol. 28 (2015) 053002

Topical Review[59, 79–81] formulations, and Campbell’s equation [63, 82].
Maxwell’s equations can be written in each of these formulations and these formulations are equivalent in principle,
but the solutions of the corresponding partial differential
equations can be very different [83]. It should be noted thatmany of these examples are applied to geometries other thansuperconducting discs in the above references, but in theremainder of this paper, recent applications of such numerical
techniques related speciﬁcally to bulk superconductor magnetization are highlighted in more detail.

experimentally conﬁrmed that the trapped ﬁeld by PFM
cannot exceed the trapped ﬁeld by FC. Accordingly, therecord trapped ﬁeld produced by PFM is only 5.2 T at 29 K
[89]. Therefore, it is important to understand the ﬂux
dynamics when magnetizing a bulk superconductor using the
PFM technique in order to achieve an optimum trapped ﬁeld
proﬁle and to investigate how these results translate into apractical magnetizing technique for applications such as
trapped ﬂux-type superconducting electric machines [90].
There are many experimental results published in theliterature and the following issues have been conﬁrmed for
the PFM technique:

3. Investigating practical magnetization techniques• The temperature rise increases with increasing magnitudeof the applied pulsed magnetic ﬁeld and with decreasing
operating temperature for an identically applied ﬁeld,
which comes from the stronger magnetic forces and thedecreased heat capacity of the bulk at lower temperatures,
respectively [91];• The temperature rise during PFM increases for increased
Jc values of the bulk, which is strongly related to thepinning force, Fp [92];• The temperature rise, which arises due to the pinning loss
and the viscous loss [92, 93] takes place adiabaticallyduring PFM, because a large amount of the heatgeneration takes place instantaneously and the coolingpower is ﬁnitely limited. The low thermal conductivity ofthe bulk also seriously affects the thermal and electromagnetic responses; and• The experimental setup, including the thermal pathways
between the cooling system (either cryogenic ﬂuid or
cryocooler) and the sample, and the type and form of theepoxy impregnation of the bulk (if present), can heavily
inﬂuence the resultant magnetization [91].

There are three magnetization techniques for magnetizing abulk superconductor that are in common use: zero ﬁeld
cooling (ZFC), ﬁeld cooling (FC) and PFM. The basic principles behind the ZFC and FC techniques are shown inﬁgure 1 using the Bean model [31, 32] approximation, whichassumes a constant Jc. The ﬁeld, Hp, in ﬁgure 1 is called thefull penetration ﬁeld and is directly related to equation (1) fora disc of ﬁnite thickness (for an inﬁnitely long slab, Hp = Jca,
where a is the half-width of the slab).
In ZFC, the superconductor is cooled below its criticaltemperature, Tc, prior to the application of a large magnetic
ﬁeld, typically of several tesla. A magnetic ﬁeld of comparatively lower magnitude is applied in the FC process to asuperconductor at a temperature above Tc, which is thencooled below Tc. In either case, to trap the maximum possible
ﬁeld corresponding to the sample’s ﬂux-trapping ability, themagnitude of the applied ﬁeld needs to be at least Btrap (atleast Btrap in FC; 2Btrap in ZFC), assuming Bean’s
model [31, 32].
The ZFC and FC magnetization techniques invariably
require large magnetizing coils, which is impractical for mostapplications of these materials. The PFM technique is similar
to ZFC, except that the large magnetic ﬁeld is applied via apulse on the order of milliseconds, rather than ramped up anddown slowly over a period of many minutes or even hours.
Achieving reliable, in situ magnetization is crucial to producing a competitive and compact machine design in trapped
ﬂux-type rotating machines, and the PFM technique shows
great promise as a compact, mobile and relatively inexpensive
magnetization technique for the magnetization of these andother TFM-based devices.

All of these results could be clariﬁed by measuring the
maximum temperature, Tmax, of the bulk surface after theapplication of the pulsed ﬁeld. Because of signiﬁcant temperature rises and the inverse relationship between Jc and thetemperature, T, the bulk cannot be magnetized to its highest
potential, but also the bulk is not irreversibly damaged by
PFM, as far as has been observed thus far.
3.2. Understanding flux dynamics

The most important characteristic of a superconductor for any
practical superconducting device is the maximum electricaltransport current density it can maintain without resistance, or
its critical current density, Jc [94]. Therefore, one of the most
crucial aspects of superconducting bulk material and device
modelling is providing accurate data for the in-ﬁeld performance, including anisotropy, of Jc at different temperatures,i.e., Jc(B, T). If this behaviour is known in detail, it is possible
to accurately predict the magnitude and shape of the trapped
ﬁeld proﬁle and the ﬂux dynamics of the magnetization
process.
The results of the numerical simulation strongly dependon Jc(B, T) and it is necessary to adjust this relationshipprecisely to the experimentally obtained one. Kim [95] and3.1. PFM technique

One signiﬁcant issue with the PFM technique is that thetrapped ﬁeld produced is generally smaller than that of ZFC
or FC, particularly at lower operating temperatures, due to thelarge temperature rise ΔT associated with the rapid dynamic
movement of the magnetic ﬂux in the interior of the superconductor during the PFM process [85]. However, at higher
operating temperatures (closer to Tc, such as 77 K) ﬁelds havebeen trapped close to that of FC [8, 35, 86–88].
Compared with the record trapped ﬁeld (17.6 T at 26 K),
which was generated by the slower FC technique, it has been4Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

Anderson [96] showed that the critical current in ‘hard,’ ortype II, superconductors (for example, HTS materials) exhibit
a strong dependence on temperature, as well as local magnetic
ﬁeld, which is expressed in the following, generalized
empirical relationship:
Jc (B , T ) =Jc0 (T )
β
⎛
B⎞
⎜1 +
⎟
B0 ⎠
⎝

,

introduce into a model once the dependence on magnetic
ﬁeld, temperature and position is known [107].
The role of inhomogeneities in the PFM process has been
analysed numerically in [90, 98, 99, 113]. In [90], a model ispresented based on the three-dimensional (3D) FEM Hformulation [75, 77, 90, 114], but in principle, as described in
section 2.2, other formulations are interchangable. There also
exist two-dimensional (2D) inﬁnitely long [69, 70, 72–74, 115]
and 2D axisymmetric [78, 116] versions of the H-formulation
for the reader’s reference.
In the 3D H-formulation, the governing equations arederived from Maxwell’s equations—namely, Faraday’s (5)and Ampere’s (6) laws:(4)

where Jc is the local critical current density, B is the magnitude of the local ﬁeld, B0 and β are material-dependent constants that can vary with temperature and Jc0 is the magnitudeof critical current density when the local ﬁeld is zero. Jc0 can
be considered a constant value or as a temperature-dependentfunction Jc0(T) if thermal considerations are necessary.
Equation (4) has been applied variously to bulk superconductors in [97–102]. The material-dependent constants
can be estimated via the trapped ﬁeld proﬁle of the bulk (forexample, [97]) or by measurements of the magnetic momenthysteresis (m-H) loops of sub-specimens taken from the bulk
(for example, [16, 103, 49]) at different temperatures.
Equation (4) does require some modiﬁcation to include
the so-called ‘ﬁsh-tail’ or ‘peak’ effect in some (RE)BCO
superconductors, where there is a ‘second peak anomaly’
found in some bulk samples [26, 104, 105]. An exponential
ﬁtting function has also been proposed to represent such
behaviour [106]. It also requires some modiﬁcation in thecase of HTS wire where there is a dependence of Jc on boththe magnitude and orientation of the magnetic ﬁeld seen by
the superconductor [107]. Furthermore, as described in
section 3.2.2, a different equation is needed to represent the
behaviour of MgB2 materials because there is no peak effectand there is a drastic suppression of Jc in the presence of amagnetic ﬁeld.
During PFM, since there are localized (and sometimes
large) changes in temperature and magnetic ﬁeld, it isimportant to introduce a precise relationship for Jc(B, T). Thisis largely due to the role of inhomogeneities in the superconductor and this is the subject of the next section.

×E+

(

)

d μ 0μr H
dB
=×E+
= 0,
dt
dt
 × H = J,

(5)
(6)

where H = [Hx, Hy, Hz] represents the magnetic ﬁeld
components, J = [Jx, Jy, Jz] represents the current density
and E = [Ex, Ey, Ez] represents the electric ﬁeld components.
μ0 is the permeability of free space and for the superconducting and air sub-domains, the relative permeability issimply μr = 1. However, this formulation also allows theintroduction of magnetic subdomains into the model byintroducing a relative permeability μr ≠ 1, which may be notconstant, but varies with magnetic ﬁeld, i.e. μr(H), where H is
the magnitude of the magnetic ﬁeld [73, 74, 76].
It is assumed that the electric ﬁeld E is parallel to thecurrent density J [69, 72, 73, 75] such that J = σE or E = ρJ,
where σ and ρ are the conductivity and resistivity,
respectively, and both highly non-linear. The electricalbehaviour of the superconducting material is modelled by
the E-J power law [117, 118], where E is proportional to Jn,
as shown in (7).
⎛ J ⎞nE = E0 ⎜ ⎟ ,⎝ Jc ⎠(7)

where E0 = 1 μV cm−1 is the characteristic electric ﬁeld and n
is an appropriate value for the superconductor. For HTS
materials, n usually varies between 5 (strong ﬂux creep) and50 (limiting value between HTS and LTS materials)[60, 61, 73], and for n > 20, it becomes a good approximationof Bean’s critical state model, for which n → ∞. In the case of[90], n = 21.
In order to simulate the thermal effects of an externalﬁeld on the trapped ﬁeld, a thermally-isolated model of a bulk
superconductor can be used to simulate liquid nitrogencooling [90] or a cold head and vacuum [78, 103, 113] can be
included in the model with or without a spacing plate between
the cold head and sample.
The following thermal transient equation can be used inthis model to model the thermal properties of the system:

3.2.1. Inhomogeneities in (RE)BCO bulk superconductors.

Inhomogeneities occur during the growth process of c-axis
seeded, single grain (RE)BCO bulk superconductors,
resulting in the formation of growth section regions (GSRs)
and growth sector boundaries (GSBs), as shown in ﬁgure 2,
with a higher Jc for the GSBs in comparison with the GSRs
[98, 108–110]. These play a signiﬁcant role in themagnetization of these materials, particularly for the PFM
technique, where they can affect the magnitude and shape ofthe trapped ﬁeld and the maximum temperature rise, anddifferent results are observed in comparison to FC and ZFC,
particularly for lower applied ﬁelds [111]. Jc can also vary as
a function of position in a sample: generally a lower Jc is
found at the edge with respect to the centre and at the bottom
with respect to the top [26, 33, 112]. In general, and for moststandard cases, the equation for Jc is relatively easy toρ⋅CdT=  ⋅ (κΔT ) + Q ,dt(8)where ρ, C and κ are the mass density, speciﬁc heat andthermal conductivity of the bulk sample, respectively.
5Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

Figure 2. YBCO (left) and GdBCO (right) bulk superconductor samples. The growth sector boundaries (GSBs) are shown for each sample.

© IOP Publishing. Reproduced by permission of IOP Publishing from [90]. All rights reserved.

In the simulation, suitable assumptions must be maderegarding the speciﬁc heat and thermal conductivity for both
MgB2 and (RE)BCO bulks. Figure 3 shows a comparison of
the typical measured speciﬁc heat [119, 120] and thermal
conductivity [121, 122] for MgB2 and (RE)BCO bulk
superconductors. At the typical operating temperature foreach bulk (Ts = 15 K for MgB2 and Ts = 40 K for (RE)BCO),
the speciﬁc heat per unit volume of MgB2 is three orders ofmagnitude smaller and the thermal conductivity is about threetimes larger, compared with those of the (RE)BCO bulk.
These results suggest that a larger temperature rise is expectedin the MgB2 bulk under identical heat generation, but thegenerated heat diffuses faster.
The heat source Q in the thermal model is derived fromthe product of the electric ﬁeld and current density throughout
the bulk superconductor, deﬁned as Q = E · J, where
E = Ex2 + Ey2 + Ez2 and J = Jx2 + Jy2 + Jz2 , describing
the adiabatic condition where the temperature rise comes fromthe local dissipation due to magnetic ﬂux motion.
Jc0(T) is the temperature dependent critical current
density of the superconductor, given by:
2 ⎤m
⎡⎛
T⎞
Jc0 (T ) = α ⎢ ⎜ 1 − ⎟ ⎥ ,⎢⎣ ⎝Tc ⎠ ⎥⎦(9)

where α is the critical current density extrapolated to T = 0 K,
Tc ≈ 90–92 K in the case of a (RE)BCO bulk superconductor,
and m = 1.5 [78, 90, 103, 123] or 2 [124, 125]. There are also
examples where a linear Jc(T) relationship is assumed([99, 101] and see sections 3.3.1 and 4). It is common
practice for modellers to use an equation, such as equation (4)(or equation (19) for MgB2 in section 3.2.2), to ﬁt themeasured experimental data at a particular operating temperature, Top, and then use equation (9) to represent thethermal behaviour of the superconductor for variations intemperature around Top, which assumes that the materialdependent parameters in equation (4) do not vary signiﬁcantly
for such temperature variations.

Figure 3. Comparison for MgB2 and (RE)BCO bulk superconductorsof (a) speciﬁc heat and (b) thermal conductivity.

Although there can be some temperature rise during theslower FC and ZFC magnetization techniques [126], theeffects are not as pronounced as for PFM and generally, for aslow enough applied ﬁeld, an electromagnetic model isusually sufﬁcient.
6Supercond. Sci. Technol. 28 (2015) 053002

Topical Review(2) An inhomogeneous bulk sample with Jc of
1 × 108 A m−2
withx–y
spatialvariation± 0.1 × 108 A m−2
Consistent with the experimental results presented in[90], the inhomogeneous Jc distribution results in a distorted
trapped ﬁeld proﬁle, where ﬂux is trapped preferentially inregions of stronger pinning (higher Jc). Flux does enter theregions of weak pinning GSRs more easily, but ﬂux can also
leave these regions more easily, and localized heating acts to
reduce Jc and pinning strength further. The homogeneousmodel traps a larger overall magnetic ﬁeld and a lower
temperature rise is observed in comparison to the inhomogeneous one [90].
This kind of modelling framework allows the presence ofthe various inhomogeneities that arise during the processingof (RE)BCO bulk superconductors to be considered, including inhomogeneous Jc distributions (between top and bottom
regions or between the seed and outer edge of the bulk
[33, 128], for example) and the presence of current limiting
grain boundaries and cracks. It can also be used to assist
optimization of processing techniques, such as samples withnovel seed arrangements [129–133], as well as PFM
techniques and coil design, for practical bulk superconductor
applications.

Figure 4. Spatial distribution of the critical current density Jc(θ) in
the ab-plane (xy-plane) of the bulk superconductor for the two cases
being analysed (homogeneous and inhomogeneous). © IOP
Publishing. Reproduced by permission of IOP Publishing from [90].
All rights reserved.

An external magnetic ﬁeld is usually applied to the bulk
along the z-axis, perpendicular to the top surface of thesample, by setting appropriate boundary conditions in themodel. It is also possible to introduce the magnetizing coilitself as a separate entity in the model to investigate theinﬂuence of the coil type and shape on the trapped ﬁeld, e.g.,solenoid [78, 98], vortex [127] and split [100] coils, which is
described in more detail in section 3.3.2.
The applied ﬁeld can take a variety of forms—sinusoidal,
trapezoidal, triangular and so on—but the following equation
is often used in the case of PFM to represent the practicalexperimental waveform observed when discharging capacitors through a magnetizing coil [90, 103]:tt
Bext = Bapp e( 1 − τ ),τ

Homogeneous MgB2 bulk superconductors. In
contrast to the Jc distribution of (RE)BCO bulk
superconductors, bulk MgB2 has a more homogeneous Jc
distribution, making studies on such systems somewhat easier
because simpliﬁed assumptions regarding the geometry and Jc
distribution can be made. Such studies can also assist inunderstanding more deeply the physics of (RE)BCO bulk
superconductor magnetization.
In [126], a 2D axisymmetric model is implemented using
the A–V formulation in combination with measured criticalcurrent density Jc(B, T) proﬁles and the experimental resultsfor MgB2 bulk samples magnetized by the FC technique werereproduced well. The value of the peak trapped magnetic ﬁeld
Btrap and the trapped ﬁeld proﬁle can be estimated precisely
for bulks of any shape, for various applied ﬁelds andoperating temperatures.
In the A-V formulation, the magnetic vector potential A isdeﬁned by
3.2.2.

(10)

where τ is an appropriate value for the rise time of the pulse.
In [90], the trapped ﬁeld distribution and maximum
temperature rise for an applied pulsed magnetic ﬁeld iscompared for a completely homogeneous bulk superconducting sample with an ab-plane Jc of 1 × 108 A m−2 and an
inhomogeneous sample with the same average Jc, but with Jc
varying as a cosine function in the ab-plane [75], as shown inﬁgure 4. The thick, dashed lines represent the GSBs. Other
than this spatial variation of Jc, a constant Jc approximation isconsidered for the in-ﬁeld behaviour for the purposes ofcomparing the role a varying Jc between GSBs and GSRs
plays on the magnetization of the sample, but in theory, aﬁeld- or position-dependent Jc may be introduced into themodel. In this case, there are two models: × A = B,(11)

 ⋅ A = 0.

(12)

Since

×

dA
dB
=
= − × E.
dt
dt

(13)

dA
− V ,
dt(14)

we haveE =−where V is an arbitrary scalar and can be considered thepotential due to electrostatic charges from (12). This
formulation has two main advantages: the contour of A

(1) A homogeneous bulk sample with an average Jc of
1 × 108 A m−2
7

Supercond. Sci. Technol. 28 (2015) 053002

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Figure 6. Estimated Jc(B) characteristics for both MgB2 bulks in[126] at 10 K and 20 K. The solid lines show the ﬁtting curves usedin the simulation. © IOP Publishing. Reproduced by permission of
IOP Publishing. All rights reserved.

Figure 5. Schematic view of the experimental setup and the basis forthe numerical model of bulk MgB2 superconducting discs
magnetized by FC presented in [126]. © IOP Publishing.
Reproduced by permission of IOP Publishing. All rights reserved.

following relation [134]:

directly gives the magnetic lines of force in 2D Cartesiangeometry and there are a large number of problems in which
A is scalar, requiring only one variable, instead of three, todescribe the vector ﬁeld B [46].
Using Ampere’s law (6), we obtain⎛1⎞dA−  ⋅ V.
 × ⎜  × A⎟ = −σ⎝μ⎠dt

⎡ ⎛ B ⎞γ ⎤Jc (B) = Jc0 exp ⎢ −⎜ ⎟ ⎥ ,⎢⎣ ⎝ B0 ⎠ ⎥⎦where B0 and γ are the ﬁtting parameters for each temperature.
In addition, the E-J power law (7) with n = 100 and the Jc0(T)relationship given by (9) are used with Tc = 39 K.
The modelling results reproduce the experimental results
well, as shown in ﬁgure 7, and the slight difference between
the two may be explained by slight inhomogeneities in the Jc
and/or temperature distributions in the bulk. When the appliedmagnetic ﬁeld is removed, a bell-shaped trapped ﬁeld proﬁlematerializes (see ﬁgure 7), which may originate from thesteep magnetic ﬁeld dependence of Jc. These results are incontrast with those using Bean’s critical state model describedin sections 2 and 3, highlighting the limitations of such amodel, which results in a conical proﬁle. It also highlights thenecessity of accurate Jc(B, T) data for modelling bulk
superconducting magnetization. It was also found that
increasing the diameter of the bulk is not an effective routeto improving the peak trapped ﬁeld, but efforts should focus
on improving the Jc(B) properties.

(15)

Since  ⋅ J = 0, we also obtain⎡ ⎛ dA⎞⎤ ⋅ ⎢σ ⎜+ V ⎟ ⎥ = 0 .
⎝
⎠⎦
⎣
dt

(16)

Based on this formulation and the relationships describedthus far, the following electromagnetic and thermal equations
were derived in [126] for a 2D axisymmetric systemrepresenting a cylindrical bulk superconducting disc.
⎞d ⎛V dd ⎛ dA ⎞⎜⎜V⎟ = J0 + J ,(rA) ⎟ +⎝⎠dr r drdz ⎝ dz ⎠(17)

dT
1 d ⎛ dT ⎞d ⎛ dT ⎞
⎜ rκ
⎟ +
⎜κ
⎟ + Q,=
dt
r dr ⎝ dt ⎠ dz ⎝ dz ⎠

(18)

ρC

(19)

This model also includes the magnetizing coil, instead of
uniform boundary conditions representing the applied ﬁeld, as
shown in ﬁgure 5. In (17), J0 is the ‘forced’ coil current
density and J is the induced superconducting current density,and (18) is the 2D axisymmetric form of (8). The heatgenerated in the bulk, Q, is again calculated as Q = E · J.
As described in section 3.2, a suitable approximation for
the Jc(B, T) characteristics of the bulk is required and thehomogeneity of the MgB2 bulk means a relatively smaller
number of sub-specimens are required for data ﬁtting,especially with respect to varying position, in comparison to(RE)BCO bulks. In [126], Jc(B, T) for each MgB2 bulk at
10 K and 20 K was ﬁtted, as shown in ﬁgure 6, using the

3.3. Other considerations for magnetization
3.3.1. Multi-pulse techniques. There are many considerationsand variables for the PFM technique, including the pulse
duration, magnitude and shape, the temperature(s) at which
the pulse is applied, the number of pulses and the shape andtype of the magnetizing coil(s), and some novel PFM
techniques have been proposed to improve the trapped ﬁeld
because the trapped ﬁeld from a single pulse is often afraction of the sample’s trapped ﬁeld capability as indicatedby FC magnetization [90, 125]. For example, multi-magnetic
pulse techniques, such as an iteratively magnetizing pulsedﬁeld method with reducing amplitude [135] and a multi-pulse
8Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

Figure 8. Schematic diagrams of the proﬁles of external appliedmagnetic ﬁeld and cooling temperature for PFM with (a) the SPA
method and (b) the MMPSC method. Reprinted from [139],
copyright 2008, with permission from Elsevier.

conditions for the thermal analysis. For SPA, several pulsed
ﬁelds of similar magnitude are applied to a sample, which isheld at a constant cooling temperature. For MMPSC, pulsed
ﬁelds of relatively smaller magnitude are applied initially to asample that is cooled to a relatively higher temperature, andthen the sample is cooled further and subject to pulses oflarger magnitude. A schematic diagram of these twoprocesses is shown in ﬁgure 8.
The numerical results for a single pulse agree qualitatively with experimental PFM results whereby increasing themagnitude of the applied pulse after full magnetization results
in an increasing reduction in trapped ﬁeld [90, 92, 103, 110].
The SPA multi-pulse technique results in a trapped ﬁeld thatsaturates with successive pulses, with no signiﬁcant additional
ﬂux trapped for further pulses. The MMPSC technique, on theother hand, results in a higher trapped ﬁeld after a few pulses,
but a similar temperature rise in the sample in comparison to
SPA. The numerical simulation conﬁrmed the advantage ofthe MMPSC technique and the experimental results in [140].
In [141], the authors applied the numerical technique in[103] to evaluate the MMPSC technique. Extending theresults above [139], the real cylindrical geometry of thesample was modelled and the time evolution and position
dependence of the magnetic ﬁeld and temperature werereported, in order to further understand the ﬂux dynamics andtemperature propagation when using MMPSC. It was found,in particular, that the increase in trapped ﬁeld after the second
pulse could be explained by a reduction in temperature risefrom the pulse due to the ﬂux already trapped from the initial
pulse. Such numerical analyses can assist in developing andoptimizing novel magnetizing techniques in terms of thenumber of pulses, temperature proﬁles and so on, in order tomaximize the trapped ﬁeld.

Figure 7. (a) Simulation results for the trapped ﬁeld Bz (z = 0 mm) at
10 and 20 K for each MgB2 bulk. The solid lines show theexperimental results. (b) Cross-sections of the trapped ﬁeld proﬁles
at 20 K for each bulk at z = 0 (bulk surface) and 3 mm [126]. © IOP
Publishing. Reproduced by permission of IOP Publishing. All rights
reserved.

technique with step-wise cooling (MPSC) [136, 137]. The
record achievement of 5.2 T at 29 K using PFM was carriedout using a modiﬁed MPSC (MMPSC) technique [89, 93].
In [138], the authors use a 2D axisymmetric model, basedon the critical state model and a Jc(B) relation given by theKim model, to reproduce the experimentally observed trapped
ﬁeld proﬁle for a bulk that is magnetized further by a second
pulsed ﬁeld again after being partially magnetized by aninitial pulsed ﬁeld. Re-pulsing the bulk sample withtemperature control is found to be effective for increasing
the magnetization for a partially magnetized sample.
In [139], the H-formulation presented in [68] is extended
to the numerical analysis of the MMPSC technique tocompare magnetization from a successive magnetic pulse
application (SPA) method and evaluate the underlying
physical mechanisms governing the process. For thisqualitative analysis, the model considers an inﬁnitely long
cylinder of width the same as the diameter of the sample, an
E-J power law (7), a linear relationship for Jc(T) and adiabatic

3.3.2. Different magnetization coil arrangements. Anotherimportant practical consideration for the magnetization of
bulk superconductors is the shape and type of the magnetizing
coil(s). Solenoid coils are most commonly used for9Supercond. Sci. Technol. 28 (2015) 053002

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diameter of the coil is smaller than the bulk diameter, and an
L-coil, where the coil is larger than the bulk diameter. The
simulation results support many of the observed experimental
results and suggest that the maximum trapped ﬁeld can belarger for lower applied ﬁelds in comparison to a solenoid
coil. Figure 10 shows the trapped ﬁeld and maximum
temperature rise at the centre of the bulk surface for thethree coils and the results are compared with the experimental
results of [154]. Both [100, 127] show that with split/vortex
coils, the magnetizing mechanism is different in that, insteadof ﬂux penetrating from the periphery (or edge) of the bulk,
the ﬂux penetrates from the top/bottom surfaces. The process
is most effective with coils smaller in diameter than the bulk.
Furthermore, it is shown that there is signiﬁcantly less heatgenerated when pulsing with vortex coils, and a longer pulse
application time, which is recognized to be important in thecase of solenoid-type coils [103], is not necessary for suchcoils. These results are promising for reducing the size of thearray of capacitors required for PFM systems.
Such numerical analyses can assist in optimizing theshape and type of the magnetizing coil(s) to take into accountpractical considerations for TFM-based devices that require acompact and efﬁcient magnetization technique that also
maximizes the trapped ﬁeld.

Figure 9. Geometric conﬁguration for modelling vortex-type coilmagnetization of bulk superconductors (a) vortex S-coil, (b) vortex
L-coil and (c) solenoid coil. © IOP Publishing. Reproduced bypermission of IOP Publishing from [127]. All rights reserved.

fundamental measurements on such materials, but anumber of other coil structures have been investigated to
improve the performance of the magnetization process andwith consideration to magnetizing bulk superconductors
practically in TFM-based applications.
Split coil magnetization [100, 125, 142, 143], where apair of magnetizing coils are placed above and below thebulk, instead of around the periphery of the bulk (solenoidcoil), has a number of merits: heat generated in the sample
may be removed along the sample’s ab-plane, which has athermal conductivity an order of magnitude higher than along
the c-axis [103]; coils smaller than the diameter of the sample
itself can be used, which allows for a more compact
magnetization structure; and it has been reported that suchcoils can increase the total trapped ﬂux signiﬁcantly [144]. A
special type of split coil arrangement, known as vortex-typecoils [145–149], has also been developed and implemented in
an axial gap-type HTS synchronous machine [146, 150] tomagnetize a rotor utilizing bulks as TFMs.
In [151, 152], ﬂat racetrack-shaped magnetizing coils
were developed in order to use PFM to magnetize an array ofbulk superconductors as ﬁeld poles in a radial ﬂux-type HTS
synchronous motor. It may also be possible to use superconducting materials of different Tcs and a dual coolingsystem to develop an in situ FC magnetization process for
YBCO bulk plates using the superconducting stator coils of
an electric machine, as described in [153].
Numerical simulations of the PFM process using splitand vortex-type magnetization coils were presented in [100]and [127], respectively. Figure 9 shows the geometricconﬁguration for a reference solenoid coil and two types ofvortex coils analysed in [127]: an S-coil, where the outer

4. AC losses and demagnetization effects
It is well known that, at a ﬁnite temperature, thermal energy
can allow ﬂux lines inside a superconductor to move from one
pinning point to another due to the Lorentz force arising fromthe interaction between the current and ﬂux density [155].
This is known as ﬂux creep [156–158] and leads to a slow,
logarithmic decay (log t) of trapped ﬁeld in a bulk superconductor over time [33]. Hence, a consequence of the ﬁnite nvalue in the power law relationship, given by (7), is therelaxation of the magnetization due to ﬂux creep, which ismore suitable to describe the behaviour of HTS materials thantrue critical state models [115]. Because of the logarithmic
time dependence of the trapped ﬁeld, the decay in trapped
ﬁeld is large immediately after magnetization, so it isimportant experimentally to know the time at which thetrapped ﬁeld measurement was taken, which can be a cause of
mismatch between the experimental results and numericalsimulation.
When a magnetized bulk superconductor, magnetizedparallel to the c-axis, is subjected to an external magnetic ﬁeldthat may be parallel or perpendicular (the so-called ‘crossed
ﬁeld effect’ [22, 115, 159, 160]) to its pre-magnetization, it
can suffer a partial or full decay of the trapped ﬁeld [115],
which is also known as the ‘collapse of the magnetic moment’[159]. This can signiﬁcantly reduce the bulk’s potential as a
TFM and can result in the failure of such bulk magnets inpractical applications [116]. In a trapped ﬂux-type superconducting synchronous machine [150, 161–163], for example, where the HTS bulks act as TFMs, bulk (RE)BCO
samples can be employed in the rotor and magnetized parallelto the c-axis before the machine rotates. In these rotating10Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

Figure 10. (a) Trapped ﬁeld Bz (r = 0) and (b) maximum temperature Tmax (r = 0) at the centre of the bulk surface as a function of applied ﬁeld
Bex using the three coils (vortex S-coil, vortex L-coil and solenoid coil) shown in ﬁgure 9. (c) Reported experimental results by Ida et al in[154]. © IOP Publishing. Reproduced by permission of IOP Publishing from [127]. All rights reserved.

machines, the position of bulk sample is expected to follow
continuously the rotating ﬁeld generated by the three phasestator/armature winding. In practice, however, any ripples in
the applied torque on the shaft may cause the trapped ﬁeld tobecome misaligned with the direction of stator over a shorttime period [164]. Due to vibration, lateral movement or an
inhomogeneous stator ﬁeld, an HTS bulk in a practical device
may experience periodic variation of the external applied ﬁeld
[165, 166]. Although the most severe inﬂuence of external acﬁeld arises when the ﬁeld direction is perpendicular to theinitial trapped ﬁeld of HTS bulk [115, 167], the trapped ﬁeld
can decay or even be erased by the inﬂuence of an external
time-varying ﬁeld parallel to the direction of trapped ﬁeld
even when the amplitude is much smaller than the trappedmagnetic ﬁeld [167–169].
In [115], the properties of bulk (RE)BCO samples wereanalysed numerically in the crossed ﬁeld conﬁguration, i.e.,
the externally applied ﬁeld is orthogonal to the magnetizationof the superconductor, using the 2D inﬁnitely long H-formulation. The numerical model qualitatively reproduced theobserved experimental results, and indicated that the suppression of magnetization comes predominately from theredistribution of the supercurrent in the bulk, leading to arotation of the plane of symmetry of the current distribution
such that it opposes the variation in magnetic ﬂux imposed by
the external ﬁeld. If a ﬁeld-dependent Jc(B) is included in themodel, then the effect is increased whereby the magnitude of
Jc reduces. In [159], a similar model is used to analyse theremagnetization process of a (RE)BCO sample partiallydemagnetized by an orthogonal ﬁeld by applying a small
magnetic ﬁeld between transverse cycles that is parallel to theoriginal magnetization. Such models allow analysis of the
dynamics of the complex supercurrent ﬂowing in the bulk, asshown in ﬁgure 11, for various combinations of appliedexternal ﬁelds [159].
In addition to the attenuation of trapped ﬁeld in bulk
superconductors caused by ﬂux creep and shielding current
redistribution within the HTS bulks, careful considerationmust also be given to changes in the thermal properties related
to heat generated in the bulk. Similar to the considerations forPFM (and to a lesser degree for FC and ZFC) describedearlier in this section, there is heat generated in the sample
due to the penetration of an external time-varying ﬁeld into
the bulk, resulting in a temperature rise [170]. The
11

Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

Figure 11. Modelled data of the complex current density distributions Jx(y, z) within the cross-section of the bulk sample at selected
times during each of three cycles of an applied magnetic ﬁeld of magnitude Hmax = 0.5Hp. The magnetic moment normalized to the initialvalue, mz/m0, is also given below each current distribution. More details on the conﬁguration of the applied ﬁelds can be found in [159].
© IOP Publishing. Reproduced by permission of IOP Publishing. All rights reserved.

experimentally observed temperature rise due to an AC ﬁeld
has been described successfully qualitatively using a linear
temperature dependence Jc(T) with Bean’s model [169] or aﬁeld-dependent Jc(B) and n(B) [101] and assuming a constantconvective heat transfer between the sample and coolant [171].
In [172], analytical expressions were derived for the steadystate (equilibrium) temperature of the bulk sample subject to an
ac ﬁeld based on Bean’s model for an inﬁnite slab and an
inﬁnite cylinder with a constant heat transfer coefﬁcient. By
deﬁning two speciﬁc threshold magnetic ﬁeld Htr1 and Htr2, themodel can predict a ‘low’ or ‘high’ steady-state temperature
due to the alternating ﬁeld, where thermal runaway occurs
above Htr2. In [173], the authors use a semi-analytical approach
for a superconducting disc of ﬁnite thickness based on Brandt’s
algorithm [174, 175], which uses the Biot–Savart law todetermine the current density J(r, z, t) inside the superconductor [54, 174–176]. This algorithm is used to compute,for one period of ac ﬁeld, the power dissipated locally, P(r, z),
for a given Jc(r, z) distribution for a constant temperature, andis coupled to a heat diffusion algorithm, which is used tocompute the time evolution of the local temperature, T(r, z), fora given time interval ΔtH, caused by a given P(r, z) distribution
and speciﬁc thermal boundary conditions. The superconducting
disc is discretized with steps Δr and Δz (Nr x Nz rings of widthΔr and thickness Δz), as shown in ﬁgure 12. The time evolutionof the ring temperature is calculated from the distribution of thedissipated power P(r, z) using (20):4ρ ⋅ ΔV ⋅ cp

ΔT
= ∑Q i + P (r , z) ⋅ ΔV ,
Δt
i=1

(20)

where ΔT is the ﬁnite variation of temperature during the ﬁnite
time interval Δt, corresponding to a number of periods ofapplied ﬁeld (e.g., 10–20 cycles ≈ 0.2 ms). cp, ρ and ΔV are thespeciﬁc heat, density and ﬁnite volume of each ring element,
respectively, and the sum of Qi elements are the four heat
ﬂuxes shown in ﬁgure 12: Qup, Qdown, Qin and Qex. Using sucha method, the authors were able to reproduce the experimental
result of heat diffusing from the corner of the sample towards
the centre and to investigate the space- and time-dependent
temperature distribution during the application of an ac ﬁeld.
12Supercond. Sci. Technol. 28 (2015) 053002

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Some research continues to be carried out on reducing
the effect of such alternating ﬁelds for practical applications
using shielding [177–179], and predicting the effect of different shielding materials, such as conventional conductors,
superconducting wire (which was presented in [177] using theA–V formulation and a constant Jc approximation based on[180]) and ferromagnetic materials, is an interesting area of
research for numerical modelling that should receive someattention in the near future.

5. Novel and hybrid superconductor structures
In this section, some novel and hybrid bulk superconductor
structures designed to overcome a few of the aforementioned
issues and enhance the intrinsic properties of bulk superconductors acting as TFMs are described, including improved
thermal conductivity structures and ferromagnet–superconductor structures. Finally, the analysis and use of hollow
bulk cylinders/tubes for shielding is discussed.
5.1. Improved thermal conductivity composite structures

As described in detail in section 3.1, heat generation is a signiﬁcant problem when carrying out PFM of bulk superconductors, having a signiﬁcant detrimental effect on the
maximum trapped ﬁeld from this magnetization process. In[78], Patel and Glowacki analyse the effect of various thermal
conductivities in the ab-plane (kab) and along the c-axis (kc) on
the trapped ﬁeld following PFM. The 2D axisymmetric Hformulation to simulate PFM by a solenoid coil for two types ofcooling environment—a thermally isolated bulk (representing
conduction cooling by helium gas) and a bulk cooled via a coldhead in a vacuum. For an isolated bulk, a higher kab increases
the overall trapped ﬂux, but decreases the peak trapped ﬁeld at
the centre of the top surface; kc has a negligible effect on both.
For the cold head system, increasing kc can signiﬁcantly
increase the trapped ﬁeld and it is unnecessary for a high kab.
Since it is difﬁcult to enhance the thermal conductivitiesof the superconducting material to the levels analysed in [78],the authors introduce a composite structure for the bulk using
metal plates or discs embedded in the bulk, as shown inﬁgure 13. It was found that structures A and C (see ﬁgure 13),in particular, could improve the trapped ﬁeld and overall
trapped ﬂux by around 20–30% in comparison to the bulk
alone. In fact, this is considered one advantage of using
stacked tape pseudo-bulks [19–23], in addition to theimproved mechanical strength from the presence of a metalalloy substrate and a more homogeneous and uniform Jc in
the radial and axial directions for the tape [20]. Stacked tape
pseudo-bulks are a relatively new area of research andnumerical modelling can play an important role in theunderstanding and optimization of such structures, and more
concretely identify their advantages and disadvantages. An
initial attempt has been made in [181] to model a composite
stacked tape bulk using an effective thermal conductivity
based on values for Hastelloy (the tape’s substrate material),
YBCO and silver contributions, but improving the

Figure 12. (a) Cross-section of the superconducting disc modelled in[173] showing the thermal boundary conditions. The lateral (r = a)and top (z = b) surfaces exchange heat with the cryogenic ﬂuid byconvection. The central plane (z = 0) and the axis (r = 0) are
adiabatic. (b) Cross-section of a ring element of width Δr andthickness Δz. The heat ﬂux directions, Qup, Qdown, Qin and Qex, areindicated by the arrows. © IOP Publishing. Reproduced bypermission of IOP Publishing. All rights reserved.

FEM techniques can also be applied to such problems andin [116] the inﬂuence of a time-varying ﬁeld, applied parallel tothe initial magnetization, on trapped ﬁeld attenuation wasinvestigated using the 2D axisymmetric H-formulation. Electromagnetic and combined electromagnetic-thermal models
were used to investigate the inﬂuence on the trapped ﬁeld
proﬁle of ﬂux creep, shielding current redistribution and theheat generated by the penetration of the time-varying ﬁeld into
the bulk. Without the inclusion of thermal effects, it was foundthat the larger penetration depth for lower frequencies causes
greater attenuation of the trapped ﬁeld. However, when
including the thermal model, the higher temperature rise forhigher frequencies decreases the trapped ﬁeld signiﬁcantly. As
found experimentally [167, 168], external ﬁelds of large
enough magnitude can attenuate the trapped ﬁeld severely,
including cases where the trapped ﬁeld in the bulk material isalmost erased. Such models can help better explain the maincauses of trapped ﬁeld attenuation, with regard to penetration
depth, redistribution of supercurrent, and thermal effects due toheat generated from ac losses.
13Supercond. Sci. Technol. 28 (2015) 053002

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Figure 14. YBCO bulks with drilled holes before and after ﬁllingwith soft ferromagnetic powder (a) Sample A before impregnation.
(b) Sample A after impregnation. (c) Sample B before impregnation.
(d) Sample B after impregnation [187]. © IOP Publishing.
Reproduced by permission of IOP Publishing. All rights reserved.

In [189], the 3D combined electromagnetic-thermal
model based on the H-formulation (see section 3.2.1) is used
to analyse the local heating effects for holes with embedded
copper and it was found that heat removal is improved forsuch a structure. In [187], the A–V formulation presented in[36] was used to simulate the effect of ﬁlling the holes with
soft ferromagnetic powder, as shown in ﬁgure 14, by introducing a relative permeability μr for the ferromagnetic subdomains. The concentration of ﬂux lines inside the ﬁlled holes
increases the average magnetic ﬂux density in the sample andthere is an increase in trapped ﬂux. This technique canovercome the inherent reduction in trapped ﬁeld/ﬂux resulting
from drilling the holes, which was a key ﬁnding in [36],where this model was originally used to simulate the effect ofdifferent arrays of holes on the trapped ﬁeld proﬁle. It was
found that a polar triangular lattice of drilled holes, in comparison the other arrays shown in ﬁgure 15, resulted in thelargest trapped ﬁeld, though the trapped ﬁeld is considerably
less than an undrilled bulk.

Figure 13. Composite bulk superconductor structures with enhanced

thermal conductivity modelled in [78] for a 46 mm diameter bulk
cooled by a cold head. The grey-coloured domains represent the bulk
superconductor and green represents the added metal plates or discs
embedded in the bulk and the cold head. © IOP Publishing.
Reproduced by permission of IOP Publishing. All rights reserved.

assumptions of the model and modelling realistic tape layers
in terms of their electromagnetic and thermal properties is anarea of interest for the future.
Although MgB2 bulk materials can be made with ahigher thermal conductivity in comparison to (RE)BCO bulks(see ﬁgure 3(b) and [78, 182]), the thermal and magnetic
stability of such bulks with higher Jc(B) characteristics can be
unstable, resulting in ﬂux jumps during magnetization thatcan dramatically affect the trapped ﬁeld [183]. Indeed, suchcomposite structures may also have a positive effect on themagnetization of MgB2 bulks.
The drilling of holes in the sintered powder before thecrystal growth [184–186] has been proposed as a method toimprove the processing of single grain bulk HTS materials,
which offers a larger surface area for oxygen diffusion,
enabling a higher oxygen content and a reduction in thenumber of macrocracks formed during oxygen annealing[187]. The mechanical resistance of the TFM can be increased
by impregnating the holes with a resin [2, 186] and research
has also been carried out investigating the role of other ﬁller
materials, such as solder [188], copper [189] and soft ferromagnetic powder [187], for which the experimental results
suggest that all of these techniques can contribute to anincreased trapped ﬁeld and/or ﬂux.

5.2. Hybrid ferromagnet–superconductor structures

It is well known that using a combination of ferromagnetic
and superconducting materials can enhance the performanceof the superconductor and macroscopic hybrid ferromagnet–
superconductor structures have been considered in various
applications. Magnetic covers, or ﬂux diverters, have beenused in a number of conﬁgurations with the wire form ofsuperconductors: to totally enclose the ﬁlaments [190] or ﬁll
the slits between ﬁlaments [191] in a striated YBCO conductor, as a sheath around a BSCCO-2223 multi-ﬁlamentary14

Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

Figure 15. (a) The four lattices of drilled holes analysed numerically in [36]. (b) Trapped ﬁeld proﬁles along the bulk diameter calculated by
the model, with a sweep rate of 10 mT s−1 and n = 25, including a reference bulk superconductor without holes. © IOP Publishing.
Reproduced by permission of IOP Publishing. All rights reserved.

such as cylinders and rings, as shown in ﬁgure 16, in order tomodify and improve the trapped ﬁeld proﬁle.
Numerical modelling using the Brandt method [174, 175]
was used initially to determine the current distribution as afunction of applied ﬁeld based on equation (4) and Campbell’s equation [63, 82] was then used to understand andcompare the dynamics of the ﬂux penetration for the hybrid
structures. The soft ferromagnetic property of the FeNi alloy
was incorporated by modifying the left hand side of Campbell’s equation from  × ( × A) to  ×  × A/μr , whereμr is represented by the function:(μr = μr, max

for Bapp = 0,

)

(21a)

⎛ μ
⎞⎤
sgn Bapp μ 0 Msat ⎡⎢r , max Bapp ⎟ ⎥⎜1 − exp −,μr = 1 +⎢⎜Bapp
μ 0 Msat ⎟⎠ ⎥
⎝
⎣
⎦

(

Figure 16. Cross-sections and dimensions of the hybrid ferromag-

netic (shaded)–superconductor (white) conﬁgurations (F/S hybrids)
investigated experimentally and numerically in [97]. Reprinted from[97], copyright 2014, with permission from Elsevier.

)for Bapp ≠ 0,

(21b)

where μr,max = 1.7 × 103 and μ0Msat = 1.4 T based on experimental measurements of the material’s B–H curve. Similarly,
a more precise ﬁtting function can be used to represent theﬁeld dependence of the ferromagnetic material, μr(H), as
described for the H-formulation section 3.2.1 and in[73, 74, 76, 202]. The numerical modelling in [97] provides avaluable insight into the difference in the contour lines of A
when the ferromagnet is added to the bulk, as shown inﬁgure 17, and how this affects its magnetization. The ferromagnet acts as a shield by concentrating and deﬂecting themagnetic ﬂux lines from the TFM, but the overall trapped ﬂux
is enhanced in such structures.

tape [192, 193] or in a C-shape to cover only the edges[194, 195] and as a horse-shoe cover for a YBCO conductor[196]. Most of these conﬁgurations resulted in a reduction inloss but [192, 193] reported a loss increase, indicating that theshape and location play an important role in loss reduction.
An ac loss reduction can also be achieved using ferromagnetic
covers on the edges of multiple superconducting tapes in astack [197], and diverters have also been investigated for
power transmission cables [198] and synchronous generators[199]. It is also possible to reduce the ac loss in superconducting coils using such techniques [76, 200].
As described in section 5.1, soft ferromagnetic powder
can be used to ﬁll drilled holes in a bulk superconductor [187]and it was shown in [1, 201] that the insertion of an iron yokein the central hole of a YBCO ring can modify the trappedmagnetic ﬁeld proﬁle and enhance the trapped ﬂux. In [97],the authors investigate the combination of a YBCO bulk witha soft FeNi ferromagnetic alloy machined into various shapes,

5.3. Hollow bulk cylinders and tubes for shielding

In addition to acting as TFMs with high magnetic ﬁelds,
hollow, cylindrical or tube-shaped bulk superconductors arealso able to shield magnetic ﬁelds in a more efﬁcient way thanan equivalent ferromagnetic cylinder [203–205]. This
15

Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

Figure 17. (a) Contour lines of the vector potential A in the fully
magnetized remanent state for the superconductor without any
ferromagnetic alloy addition in [97]. The results are obtained bysolving Campbell’s equation [63, 82]. (b) The same contour lines forthe hybrid ferromagnet–superconductor structure C1 shown inﬁgure 16. Reprinted from [97], copyright 2014, with permission
from Elsevier.

shielding property has also been exploited in bulk superconductor reluctance machines [206, 207], where the magnetic ﬁeld along the direct axis is enhanced by reinforcing the
low reluctance path provided by the ferromagnetic material in
the rotor.
In [204], the authors present an analysis of the magnetic
shielding properties of a polycrystalline Bi-2223 superconducting tube subject to ac and dc axial magnetic ﬁelds. A
numerical model based on [174, 175] is again used to simulate the experimental results. The tube can effectively shield
an axial ﬁeld up to a threshold axial ﬁeld, Blim, that isdetermined by the length of the tube and its mean radius, butalso the exact Jc(B) dependence of the superconductor. For
decreasing tube length, Blim can be severely reduced due todemagnetization effects. Two mechanisms for the ﬁeld
penetration in the superconductor are described: one from theexternal surface and one from the opening ends of the tube;
the latter being supressed for long tubes. Finally, the shielding
factor is increased for increasing frequency, which is
explained in terms of the power law, shown in equation (7).
However, the effect of heat generation due to the motion ofvortices was not examined, which may have a signiﬁcant
inﬂuence as described in [116].
In [205], this study is extended to analyse the effect of acap on the ends of the tube, effectively closing the tube, andfor tubes of shorter length, this has a noticeable, positiveeffect on the maximum shieldable ﬁeld and the uniformity ofthe shielding. In addition, the effect of a non-superconducting
joint and of inhomogeneities in the superconducting properties is investigated, which can both act to reduce the magnitude and location of maximum ﬁeld attenuation.
Finally, in [208], the authors investigate the shielding
properties of hybrid ferromagnetic-superconductor shielding
tubes, which had previously been studied mostly experimentally [209–211]. The addition of a ferromagnetic cylinder
either inside or outside the superconducting cylinder, as
shown in ﬁgure 18, can substantially improve the shielding
limit, which is maximized using the Ferro Out conﬁguration,i.e., the ferromagnetic tube is placed outside the HTS tube.

Figure 18. (a) Conﬁgurations of the ferromagnet–superconductor

tubes considered in [208]. The ferromagnetic tube (black) is placed
either (2) inside (Ferro In) or (3) outside (Ferro Out) the HTS tube.
(b) The shielding factor (SF) as a function of applied magnetic ﬂux
density for the three conﬁgurations. © 2010 IEEE. Reprinted, withpermission, from [208].

With the assistance of numerical simulations, which include
the electrical properties of the superconductor and saturationof the ferromagnet, the shielding effect for the ferro out
conﬁguration is decomposed into two processes: attenuationfrom the ferromagnetic layer (independent of the HTS layer)
and a decrease in magnetic ﬂux density in the wall of the HTS
tube, according to its own shielding factor. The mechanism
for the Ferro In conﬁguration is slightly different in that thepresence of the ferromagnetic tube improves the magnetic
ﬁeld decay in the HTS tube. The height of the ferromagnetic
tube also plays an important role, which can be optimized
using numerical methods.

6. Conclusion and view towards the future
In summary, this paper provides a comprehensive review of
the current state of the art in modelling the magnetization of
bulk superconductors, including both (RE)BCO and MgB2
materials. Numerical modelling is a powerful tool to understand the physical mechanisms of their magnetization, toassist in interpretation of experimental results, and to predictthe performance of practical bulk superconductor-based
devices. It is also a particularly important and cost-effective
method to guide both superconducting material processing
and practical device design as many superconducting
16Supercond. Sci. Technol. 28 (2015) 053002

Topical Review

applications head towards the commercialization stage of
their development in the coming years.
A simple analytical method to calculate the trapped ﬁeldof a bulk superconductor based on the Bean critical statemodel is described, in addition to an overview of the analytical and numerical techniques used to model superconducting
materials in general. The commonly used practical techniques
to magnetize bulk superconductors are summarized with aparticular focus on PFM, which is promising as a compact,
mobile and relatively inexpensive magnetizing technique. A
number of numerical models developed to analyse the issues
related to PFM and optimize the technique are described indetail, including understanding the dynamics of the magnetic
ﬂux penetration and the inﬂuence of material inhomogeneities, thermal properties, pulse duration, magnitude and shape,
and the shape of the magnetization coil(s). The effect ofexternally applied magnetic ﬁelds in different conﬁgurations
on the attenuation of the trapped ﬁeld is also discussed. A
number of novel and hybrid bulk superconductor structures
are described, including improved thermal conductivity
structures and ferromagnet–superconductor structures, whichhave been designed to overcome some of the issues related tobulk superconductors and their magnetization and enhance
the intrinsic properties of bulk superconductors acting as
TFMs. Finally, the use of hollow bulk cylinders/tubes forshielding is analysed.
With a view towards the future, many of the numerical
techniques developed so far provided detailed qualitative
results based on valid assumptions, but more quantitative
results can be made with more accurate assumptions,
including the input data used for the models—for example,
improving the Jc(B, T) characterization of (RE)BCO bulk
superconductors using a number of sub-specimens from thesample. This task, however, is more challenging than merely
taking more specimens from a sample because often theestimated macro-Jc of the whole sample is smaller than the
micro-Jc measured for small samples. As with all numericalmodelling, trade-offs need to be made based on valid
assumptions, where possible using data that have been veriﬁed experimentally.
The complexity of numerical models continues toincrease, allowing researchers to simulate increasingly complex geometries and situations with various realistic parameters and conditions. Again there are trade-offs for thenumerical technique itself, including, for example, the number and type of mesh elements, error tolerances and the use ofsymmetry, and analytical models are a helpful tool to checkthe sensibility of the numerical model in its basic form. The
use of symmetry in the case of a cylindrical, disc-shaped bulk
greatly alleviates the computational requirements, but there
are several cases where a 3D model is needed. A 2D axisymmetric model may take up to an hour or two, but 3Dmodels of even a single 3D bulk can require hundreds ofthousands of elements to ensure sufﬁcient accuracy and cantake the good part of a day or longer to solve for one particular set of parameters. For models to extend to more complex shapes and arrays in 3D, which is required forsuperconducting device modelling, as well as for modellingstacked tape pseudo-bulks, there is a strong need for
improving the computational speed and memory requirements
by the reﬁnement of existing techniques and the developmentof new ones.

Acknowledgments
Dr Mark Ainslie would like to acknowledge the support of a
Royal Academy of Engineering Research Fellowship. Professor Hiroyuki Fujishiro would also like to acknowledge
support in part by a Grant-in-Aid for Scientiﬁc Research from
the Ministry of Education, Culture, Sports, Science andTechnology, Japan. This research was also supported in partby a Royal Society International Exchanges Scheme grant,
IE131084.

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