Superconductor Science and Technology
Supercond. Sci. Technol. 27 (2014) 065008 (9pp)

doi:10.1088/0953-2048/27/6/065008

Modelling and comparison of trapped fieldsin (RE)BCO bulk superconductors for
activation using pulsed field magnetization
M D Ainslie1 , H Fujishiro2 , T Ujiie2 , J Zou1 , A R Dennis1 , Y-H Shi1 andD A Cardwell1
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 17 January 2014, revised 7 March 2014
Accepted for publication 11 March 2014
Published 8 April 2014

Abstract

The ability to generate a permanent, stable magnetic field unsupported by an electromotive
force is fundamental to a variety of engineering applications. Bulk high temperature
superconducting (HTS) materials can trap magnetic fields of magnitude over ten times higher
than the maximum field produced by conventional magnets, which is limited practically torather less than 2 T. In this paper, two large c-axis oriented, single-grain YBCO and GdBCO
bulk superconductors are magnetized by the pulsed field magnetization (PFM) technique attemperatures of 40 and 65 K and the characteristics of the resulting trapped field profile areinvestigated with a view of magnetizing such samples as trapped field magnets (TFMs) in situ
inside a trapped flux-type superconducting electric machine. A comparison is made between
the temperatures at which the pulsed magnetic field is applied and the results have strong
implications for the optimum operating temperature for TFMs in trapped flux-type
superconducting electric machines. The effects of inhomogeneities, which occur during the
growth process of single-grain bulk superconductors, on the trapped field and maximum
temperature rise in the sample are modelled numerically using a 3D finite-element model
based on the H-formulation and implemented in Comsol Multiphysics 4.3a. The results agree
qualitatively with the observed experimental results, in that inhomogeneities act to distort thetrapped field profile and reduce the magnitude of the trapped field due to localized heating
within the sample and preferential movement and pinning of flux lines around the growth
section regions (GSRs) and growth sector boundaries (GSBs), respectively. The modelling
framework will allow further investigation of various inhomogeneities that arise during theprocessing of (RE)BCO bulk superconductors, including inhomogeneous Jc distributions andthe presence of current-limiting grain boundaries and cracks, and it can be used to assist
optimization of processing and PFM techniques for practical bulk superconductor applications.
Keywords: bulk (RE)BCO superconductors, pulsed field magnetization, trapped field magnets,
numerical modelling, magnetic flux dynamics(Some figures may appear in colour only in the online journal)0953-2048/14/065008+09$33.00

1

c 2014 IOP Publishing Ltd

Printed in the UK

Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et al

1. Introduction

The ability to generate a permanent, stable magnetic field
unsupported by an electromotive force is fundamental to avariety of engineering applications. The magnetization of
conventional permanent magnets, such as NdFeB and SmCo,
which is a useful measure of the ability to generate magnetic
fields, is independent of sample volume and is therefore
limited by the material properties of the permanent magnet(i.e., it cannot be increased by subtle changes to processing).
Bulk high temperature superconducting (HTS) materials, on
the other hand, trap magnetic flux via the generation ofmacroscopic electrical currents, which leads directly to anincrease in magnetization with sample volume. This, in turn,potentially overcomes this fundamental limit in the size of field
generated by conventional permanent magnets.
The peak trapped magnetic flux density, Btrap , at the centre
of a (RE)BCO (where RE = a rare earth element or yttrium)
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
Btrap = Aµ0 Jc dFigure 1. YBCO (left) and GdBCO (right) bulk superconductor

samples. The growth sector boundaries (GSBs) are shown in eachsample.

comparatively lower magnitude is applied in the FC process
to a superconductor 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 [5,6]. This invariably requires large magnetizing coils, which isimpractical for most applications of these materials. The PFM
technique is similar to ZFC, except that the large magnetic field
is applied via a pulse on the order of milliseconds, rather thanramped up and down slowly over a period of many minutes oreven hours. Achieving reliable, in situ magnetization is crucial
to producing a competitive and compact machine design intrapped flux-type rotating machines, and the PFM techniqueshows great promise as a compact, mobile and relatively
inexpensive magnetization technique for the magnetization ofsuch devices. However, one issue with this technique is that thetrapped field produced using PFM is much smaller than thatof ZFC or FC, due to the large temperature rise 1T associatedwith the rapid dynamic movement of the magnetic flux in theinterior of the superconductor during the PFM process [7].
Compared with the record trapped field (17 T at 29 K), which
was generated by a field-cooled technique, the record trappedfield produced by PFM is only 5.2 T at 29 K [8]. Therefore, it isimportant to understand the flux dynamics when magnetizing
a bulk superconductor using the PFM technique in order toachieve an optimum trapped field profile and to investigate howthese results translate into a practical magnetizing techniquefor a trapped flux-type superconducting electric machine.
In this paper, we investigate the magnetization of two,large c-axis oriented single-grain Y–Ba–Cu–O (YBCO) and
Gd–Ba–Cu–O (GdBCO) bulk superconductors by the PFM
technique at temperatures of 40 and 65 K. We report the characteristics of the trapped field produced by this technique witha view to developing a practical in situ magnetizing process for
TFMs in trapped flux-type superconducting electric machines.

(1)

where A is a geometrical constant, µ0 is the permeability offree space, Jc is the critical current density of the superconducting material and d is the sample diameter. This implies
two main approaches for improving the field-trapping abilityof a bulk superconductor: (1) by enhancing the critical current
density through improved flux pinning and increased sample
homogeneity, and (2) by increasing the sample size.
Bulk superconductors can trap magnetic fields of magnitude over ten times higher than the maximum field producedby conventional magnets, which is limited practically to rather
less than 2 T. Indeed, the world record field generated byan arrangement of two bulk superconductors currently stands
at 17 T at 29 K [1]. Bulk superconductors can exhibit Jc sof 5 × 104 A cm−2 at 1 T and 77 K (the boiling point ofliquid nitrogen), resulting typically in trapped fields of up to
between 1 and 1.5 T for YBCO and greater than 2 T for(RE)BCO at this technologically important temperature, withthe highest trapped field at 77 K currently standing at 3 T [2].
As a result, there is great interest in using these materials
as trapped field magnets (TFMs) in a number of engineering
applications, including magnetic levitation, magnetic bearings,
energy storage flywheels, magnetic resonance imaging, magnetic separation and rotating machines [3]. Significantly, thehigher magnetic loading in rotating machines would provide
an increased torque/power density, resulting potentially in amachine that is smaller and lighter in weight than conventional
devices of the same rating [4]. However, developing a practicalmagnetizing technique is crucial to using bulk superconductors
as TFMs in applications of these types.
There are three magnetization techniques for magnetizing
a bulk superconductor that are in common use: zero fieldcooling (ZFC), field cooling (FC) and pulse field magnetization (PFM). In ZFC, the superconductor is cooled below
its critical temperature, Tc , prior to the application of a large
magnetic field, typically of several Teslas. A magnetic field of

2. Bulk superconductor sample details

The pulsed field magnetization of the YBCO and GdBCO
single-grain superconducting samples shown in figure 1 has
been investigated as part of this study. The YBCO sample has adiameter of 32 mm and thickness 15 mm. The GdBCO sample,
which contains 10 wt% Ag2 O, has a diameter of 41 mm andthickness 16 mm. Both samples were fabricated by the topseeded melt-growth (TSMG) method, similar to that described2Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et al

Figure 2. Trapped field distribution at 77 K obtained by field cooling (FC) for each bulk superconductor: YBCO (left), GdBCO (right). The
peak trapped field at the centre of the top surface of each sample was 0.692 T and 1.19 T, respectively.

in previous research [9]. Precursor powders for fabricationof the GdBCO sample with a composition 75 wt% Gd123 + 25 wt% Gd-211 + 10 wt% Ag2 O + 1.0 wt% BaO2 +0.1 wt% Pt were mixed and ground thoroughly using anelectrical mortar and pestle. 190 g of this mixed powder
was pressed uniaxially into pellets 50 mm in diameter andplaced on ZrO2 rods inside a box furnace. A generic seedof composition 1 wt% MgO–NdBCO [9] was placed on the
top surface of the pressed pellet to control both nucleationand grain orientation. The sample was heated to 1045 ◦ C,
held for 1 h, then cooled appropriately through its peritectic
growth window, until finally being furnace cooled to room
temperature. The as-grown single grain was then oxygenatedin a tube furnace at temperatures between 360 and 440 ◦ C
for 300 h. The YBCO sample was fabricated using a similar
process, but with the key processing temperatures varied
accordingly for this system. The samples were mounted ona sample holder fabricated from 316 stainless steel of innerdiameter 46 mm and outer diameter 56 mm to match thedimensions of the available cold stage of the pulse system.
Stycast 2850 FT, Catalyst 23 LV was used to mount the
samples in the holder, with the epoxy set under vacuum toensure void-free embedment.
The trapped field distributions at 77 K obtained by fieldcooling for each sample are shown in figure 2. This resulted
in peak trapped fields at the centre of the top surface of eachsample of 0.692 T (YBCO) and 1.19 T (GdBCO).
Figure 3 shows the magnitude of trapped field obtainedby FC for each bulk sample as a function of temperature, inwhich the vertical axis shows the normalized trapped field
divided by the sample diameter d, measured by a Hall sensor
located at the centre of the top surface of the bulk samples.
This normalization accounts for the difference in sample sizes
and can be considered an average critical current density foreach sample, averaged over the sample volume and including
the field-dependent nature of the critical current density [10].
After initially magnetizing the sample at around 40 K by
FC with an applied field of 7 T, the temperature was increased
slowly to 100 K at a rate of 0.3 K min−1 . The peak trappedfield at the centre of the bulk at 46 K was 5.5 T for GdBCO
bulk and 4.8 T for YBCO. After correcting the data for the

Figure 3. Trapped field normalized to units of kA cm−2 obtained by

FC for the GdBCO and YBCO bulk samples investigated in thisstudy as the temperature is increased from 40 to 100 K for an appliedfield of 7 T. The value of the vertical axis represents the pinningstrength as the average critical current density of each sample.

difference in size of the bulk samples (the GdBCO is 41 mm
and the YBCO is 32 mm), it is clear from this figure thatpinning in the GdBCO sample is significantly stronger thanthat in the YBCO sample investigated in this study.
3. Pulsed field magnetization experimental results

An overview of the pulsed field magnetization experimental
arrangement is shown in figure 4. The bulk samples were
mounted tightly on the cold stage of a Gifford–McMahon(GM), closed cycle helium refrigerator, and a copper magnetizing solenoid pulse coil, cooled using liquid nitrogen, wasplaced outside the vacuum chamber. This magnetizing coilgenerates pulsed fields up to Bapp = 6.4 T with a rise time oftr = 12 ms and duration of approximately td = 120 ms. The
typical characteristics of the applied pulsed fields are shownin figure 5. The time dependence of the externally applied
pulsed field Bext (t) was monitored by the current i(t) flowing
in a shunt resistor connected in series with the pulse coil. The
central axis of the applied field coincides with that of the bulk,
such that the direction of the applied field is perpendicular to
the top surface of the samples.
3

Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et al

Figure 6. Typical evolution of the trapped field with time measured

close to the centre of the sample using a Hall sensor located on the
top surface of the samples. These results are from measurements of
the YBCO sample at 65 K for the applied pulsed fields shown infigure 5.

Figure 4. Schematic illustration of the pulsed field magnetization
experimental arrangement.

Figure 5. Typical applied pulsed field characteristics. These fields
were applied to the YBCO sample at 65 K.

Different amplitudes of pulsed fields of up to 6.3 T
were applied to each sample and the two-dimensional trappedfield distributions were measured inside the vacuum chamberusing an x–y stage controller and an axial-type Hall sensor
positioned 1 mm above the top surface of each sample. The
applied field and the trapped field close to the centre of thesample were measured dynamically during the application ofeach pulse using the same Hall sensor (located on the top
surface of the samples).
Figure 6 shows the typical evolution of the trapped fieldwith time, measured close to the centre of the sample using a
Hall sensor located on the top surface of the samples. These
results are derived from measurements of the YBCO sample
at 65 K for the applied pulsed fields shown in figure 5. The
value at t = 300 ms, after allowing for adequate relaxation of
the magnetic flux, is referred to hereafter as Btrap , the trappedfield at the centre of the top surface of the sample.
Two-dimensional trapped field distributions measured at
1 mm above the top surface of the YBCO sample at 65 K and
40 K are shown in figures 7 and 8, respectively. Figures 9 and 10
show the trapped field distributions for the GdBCO sample at
65 K and 40 K, respectively. The central panels of figures 7–10
show the maximum trapped field for fully magnetized samples
and the trapped field close to the centre of the sample for cases
where the sample was not fully magnetized.

Figure 7. Trapped field distributions measured at 1 mm above the

top surface of the YBCO sample at 65 K. The central panel shows
the maximum trapped field for fully magnetized samples and thetrapped field close to the centre of the sample prior to fullmagnetization.

Figures 7 and 8 show that the YBCO sample becomes
magnetized initially along the growth sector boundaries(GSBs), which are highlighted for reference in figure 1. Of thefour GSBs, two in particular exhibit preferential pinning along
their length. Once the magnitude of the applied pulsed field is
sufficiently high that it magnetizes the sample fully, the trappedfield distribution changes to a characteristic, approximately
conically-shaped trapped field. The field required to fully
magnetize the bulk sample will be hereafter referred to as the‘activation field’ (for example, the activation field in figure 74Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et al

Figure 9. Trapped field distributions measured at 1 mm above the

Figure 8. Trapped field distributions measured at 1 mm above the

top surface of the GdBCO sample at 65 K. The central panel shows
the maximum trapped field for fully magnetized samples and thetrapped field close to the centre of the sample prior to fullmagnetization.

top surface of the YBCO sample at 40 K. The central panel shows
the maximum trapped field for fully magnetized samples and thetrapped field close to the centre of the sample prior to fullmagnetization.

cryogenics without compromising significantly on material
performance. An additional consideration, and an argument
for higher operating temperature, is that stronger pinning at
lower temperatures (or between different samples, as described
below) results in a higher activation field that would require alarger and more complex magnetizing assembly.
The trapped field distributions for the GdBCO sample
are shown in figures 9 and 10. The results indicate a much
more homogeneous sample in comparison to YBCO, whichis particularly obvious for fields below the activation field.
The bulk, single-grain sample should be as homogeneous as
possible to enhance the trapped field produced by the PFM
technique. If large inhomogeneities, such as those present
in the YBCO sample, exist within the bulk between the
GSBs and growth sector regions (GSRs), the trapped field
profile becomes inhomogeneous, particularly for fields below
the activation field. In addition, the magnetization profile ofthe trapped field is not necessarily enhanced, even though
the trapped field generated by ZFC/FC techniques may be
maximized in the same samples. It is also clear from theseresults that the flux pinning strength is greater for the GdBCO
sample, which results in a higher trapped field, but also in alarger magnetizing device for an electric machine design due
to the higher activation field, as described above.

is around 3.4 T). There is no significant increase in trappedfield for larger applied fields than the activation field, and, infact, the trapped field begins to reduce for a sufficiently large
field due to the temperature rise in the bulk generated by the
rapid movement of flux lines in the sample interior [11].
Cooling the sample to a lower temperature (e.g., from65 to 40 K) results in stronger flux pinning and a higher Jc
within the sample, which results in much higher localized fielddensities (the behaviour seen in the trapped field distribution
is qualitatively the same). However, the maximum trappedfield is not increased significantly at 40 K, which differsfrom the slower ZFC and FC techniques (see figure 3).
With respect to the magnetization of bulk superconductors
in a trapped flux-type superconducting machine, the choiceof temperature range is crucial, and for PFM, it is clear
from these results that it is not necessarily true that a lower
temperature results in a significantly higher trapped field
as observed for the ZFC/FC techniques. The lower specificheat and stronger pinning forces at lower temperatures resultin a larger amount of heat generated in the sample, which
suppresses Jc and hence trapped field. As a result, thesedata show that sample performance in terms of trapped field
is not improved significantly at 40 K compared to that at
65 K. The activation field for the YBCO sample is higher
at 40 K (4.6 T at 40 K compared with 3.6 T at 65 K),
although the resulting trapped field is only marginally bigger.
This would justify, for example, the selection of a machine
operating temperature range of 65–77 K, simplifying the

4. Modelling inhomogeneous behaviour

In this section, the influence of inhomogeneities on trappedfield is investigated qualitatively using a three-dimensional
5Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et al

Figure 11. Spatial distribution of the critical current density Jc (θ ) in

the ab-plane (x y-plane) of the bulk superconductor for the two cases
being analysed (homogeneous and inhomogeneous).

where τ = 12 ms. This waveform approximates the typicalexperimentally applied pulsed fields shown previously infigure 5, with a rise time of 12 ms and a pulse width ofapproximately 120 ms for;
(1) A homogeneous bulk sample with an average Jc of 1 ×
108 A m−2 .
(2) An inhomogeneous bulk sample with a Jc of 1 × 108 A m−2
with x–y spatial variation ±0.1 × 108 A m−2 .

Figure 10. Trapped field distributions measured at 1 mm above the

top surface of the GdBCO sample at 40 K. The central panel shows
the maximum trapped field for fully magnetized samples and thetrapped field close to the centre of the sample prior to fullmagnetization.

In order to simulate the thermal effects of an external fieldon trapped field, a thermally-isolated model of a bulk superconductor is used. The simulation is carried out using the COMSOL Heat Transfer module, coupled with the H-formulation,
which uses the general partial differential equation (PDE)
module. The HTS bulk sample is assumed to be cooled byliquid nitrogen, with the sample environment described by thethermal model parameters listed in table 1, which are typicalfor these (RE)BCO bulk superconductors [24].
The following thermal transient equation is used in thismodel:(3D) finite-element model. The finite-element model is basedon the H-formulation, which has been applied variously
to analysing high temperature superconductors for over a
decade [12–22], and is implemented using COMSOL Multiphysics 4.3a [23]. To model the electromagnetic and thermal
properties of a bulk superconductor in 3D, we have extended
previous models of HTS coated conductors [17–19, 21] anddrawn additional inspiration from references [20, 24, 25].
Inhomogeneities occur during the growth process of
c-axis seeded, single-grain (RE)BCO bulk superconductors,
resulting in the formation of GSBs and GSRs described above
and shown in figure 1, with a higher critical current density
for the GSBs in comparison with the GSRs [26–29]. Here thetrapped field distribution and maximum temperature rise foran applied pulsed magnetic field is compared for a completely
homogeneous bulk superconducting sample with an ab-plane
Jc of 1 × 108 A m−2 and an inhomogeneous sample with thesame average Jc , but with Jc varying as a cosine function in
the ab-plane [20], as shown in figure 11. The thick, dashedlines represent the GSBs. Other than this spatial variation of
Jc , a constant Jc approximation is considered for the in-field
behaviour for the purposes of this analysis. The superconductor
is modelled using an E–J power law relation [30, 31], Eα J n ,where n = 21. An external pulsed magnetic field is applied tothe bulk along the z-axis, perpendicular to the top surface ofthe sample, by setting appropriate boundary conditions suchthattt
Bext (t) = Bapp e(1− τ )
(2)
τ

ρ ·C

dT
= ∇ · (k∇T ) + Q
dt(3)

Jc0 (T ) is the temperature-dependent critical current densityof the superconductor, given by:"T
Jc0 (T ) = α 1 −Tc

2 #1.5(4)where α is the critical current density extrapolated to T = 0 K.
The heat source Q in the thermal model is derived from theproduct of the electric field and current density throughout
the bulk superconductor,
defined as Q =√E norm · Jnorm , where√
E norm = (E x2 + E y2 + E z2 ) and Jnorm = (Jx2 + Jy2 + Jz2 ).
The maximum trapped field at the centre of the top surface
for the homogeneous and inhomogeneous Jc distribution
models are compared in figure 12. Pulsed fields of between
0 and 5 T are applied to the bulk sample at 77 K. Similar
6Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et al

Figure 12. Maximum trapped field at the centre of the top surface, as well as the maximum temperature rise in the superconductor, for the
homogeneous and inhomogeneous Jc distribution models. Pulsed fields of between 0 and 5 T are applied to the bulk samples at 77 K.

Table 1. Thermal model parameters.

Parameter

Description

Value

Tc
ρb
ρn
Cb
Cn
kab
kc
kn
E0
α
τ
Bapp

Transition temperature
HTS bulk density
Nitrogen density
Heat capacity of bulk
Heat capacity of liquid nitrogen at 77 K
Thermal conductivity of bulk along ab-plane
Thermal conductivity of bulk along c-axis
Thermal conductivity of liquid nitrogen
Characteristic voltage
Critical current density extrapolated to T = 0 K
Rise time of applied magnetic field
Peak value of applied magnetic field

92 K
5.9 × 103 kg m−3
808.4 kg m−3
1.32 × 102 J kg−1 K−11040 J K−1 kg−1
20 W m−1 K−1
4 W m−1 K−1
0.026 W m−1 K−1
1 × 10−4 V m−1
6.1 × 108 A m−2
12 ms
0–5 T

to the experimental results shown in the central panels of
figures 7–10, there is an optimal activation field, above which
the trapped field reduces with increasing field magnitude due
the temperature rise associated with the movement of flux
lines. Figure 13 shows the two-dimensional trapped field
distributions at the top surface of the bulk for a range ofapplied fields (Bapp = 1, 3 and 5 T) at time t = 300 ms,
which gives the flux some time to relax after the pulse isapplied. The inhomogeneous distribution of pinning sites
results in a distorted trapped field profile, where flux istrapped preferentially in regions of stronger pinning (higher
Jc ). Although flux enters the superconductor more easily via
the GSRs due to weaker pinning (lower Jc ), it also leaves thisregion more easily and this results in localized heating acting
to reduce Jc /pinning strength further.
Figure 12 also shows the maximum temperature risein the superconductor for each model for the same appliedfield pulse. The inhomogeneous model produces a larger
maximum temperature rise, which results in a lower trappedfield. The inhomogeneous distribution of pinning centres acts
to reduce the maximum trapped field potential of the sample as
described above. Finally, figure 14 compares the temperature
distributions generated by the homogeneous (right) and inhomogeneous (left) models at the pulse peak (t = 12 ms) andpulse end (t = 120 ms) for an intermediate applied pulsed
magnetic field Bapp = 3 T. There are contributions to theheat generated in the bulk superconductor from magnetic flux
moving into (t ≤ 12 ms) and out of (t > 12 ms) the sample.
This modelling framework allows the presence of thevarious inhomogeneities that arise during the processing of(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
[32, 33], for example) and the presence of current-limiting
grain boundaries and cracks. It can be used to assist optimization of processing techniques, such as samples with novel seedarrangements [34–37], as well as PFM techniques and pulse
coil design, for practical bulk superconductor applications.
7Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et al

Figure 13. Trapped field distributions for the homogeneous and inhomogeneous Jc distribution models for pulsed fields Bapp = 1, 3 and 5 T

at time t = 300 ms, allowing flux to relax after the pulse is applied.

trapped field magnets (TFMs) in situ inside a trapped flux-type
superconducting electric machine.
Inhomogeneities within the bulk affect the dynamics ofthe flux entering the sample, causing a distorted trapped field
profile with flux trapped preferentially in regions of stronger
pinning (higher Jc ). It was found by comparing the YBCO andGdBCO samples that the latter has stronger flux pinning, inaddition to a more homogeneous distribution of pinning sites,
resulting in an enhanced trapped field capability.
It is apparent by comparing the temperature at which
the pulsed magnetic field is applied that a lower magnetizing
temperature does not necessarily result in much higher trappedfield for PFM, and this has implications for the optimum operating temperature for TFMs in trapped flux-type superconducting electric machines. The magnitude of trapped field increases
with reducing temperature for slower zero field cooling (ZFC)
and field cooling (FC) magnetizing techniques, which requirelarge magnetizing fixtures and timescales. However, there is nosuch simple relationship for PFM due to the reduced specificheat of the material at lower temperature and the somewhat
complex dynamics of the flux movement, which generates
heat.
The effects of inhomogeneities, which occur during the
growth process of single-grain bulk superconductors, on thetrapped field and maximum temperature rise in the sample
were modelled numerically using a 3D finite-element model
based on the H-formulation and implemented in Comsol
Multiphysics 4.3a. The results agree qualitatively with theobserved experimental results, in that inhomogeneities act todistort the trapped field profile and reduce the magnitude of thetrapped field due to localized heating within the sample andpreferential movement and pinning of flux lines around thegrowth section regions (GSRs) and growth sector boundaries(GSBs), respectively. The modelling framework will allow

Figure 14. Comparison of the temperature distributions generated

by the homogeneous (right) and inhomogeneous (left) models at thepulse peak (t = 12 ms) and pulse end (t = 120 ms) for an applied
pulsed magnetic field Bapp = 3 T. There are contributions to theheat generated in the bulk superconductor from magnetic flux
moving into (t ≤ 12 ms) and out of (t > 12 ms) the sample.

5. Conclusion

In this paper, two large c-axis oriented, single-grain YBCO andGdBCO bulk superconductors were magnetized by the pulsed
field magnetization (PFM) technique at temperatures of 40
and 65 K and the characteristics of the resulting trapped field
were investigated with a view of magnetizing such samples as8Supercond. Sci. Technol. 27 (2014) 065008

M D Ainslie et alfurther investigation of various inhomogeneities that arise
during the processing of (RE)BCO bulk superconductors,
including inhomogeneous Jc distributions and the presenceof current-limiting grain boundaries and cracks, and it can beused to assist optimization of processing and PFM techniques
for practical bulk superconductor applications.

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Acknowledgments

Dr Mark Ainslie would like to acknowledge the support ofa Royal Academy of Engineering Research Fellowship. This
research was also supported in part by a Grant-in-Aid for
Scientific Research (No. 23560002) from the Ministry of
Education, Culture, Sports, Science and Technology, Japan,
and an IET Travel Award from the Institution of Engineeringand Technology, UK.
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