Superconductor Science and Technology
Supercond. Sci. Technol. 29 (2016) 074003 (16pp)

doi:10.1088/0953-2048/29/7/074003

Enhanced trapped ﬁeld performance of bulk
high-temperature superconductors using
split coil, pulsed ﬁeld magnetization with aniron yoke
M D Ainslie1, H Fujishiro2, H Mochizuki2, K Takahashi2, Y-H Shi1,
D K Namburi1, J Zou1, D Zhou1, A R Dennis1 and D 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 14 March 2016, revised 30 April 2016
Accepted for publication 4 May 2016
Published 31 May 2016

Abstract

Investigating and predicting the magnetization of bulk superconducting materials and developing
practical magnetizing techniques is crucial to using them as trapped ﬁeld magnets in engineering
applications. The pulsed ﬁeld magnetization (PFM) technique is considered to be a compact,
mobile and relative inexpensive way to magnetize bulk samples, requiring shorter magnetization
times (on the order of milliseconds) and a smaller and less complicated magnetization ﬁxture;
however, the trapped ﬁeld produced by PFM is generally much smaller than that of slower zeroﬁeld cooling or ﬁeld cooling techniques, particularly at lower operating temperatures. In thispaper, the PFM of two, standard Ag-containing Gd–Ba–Cu–O samples is carried out using two
types of magnetizing coils: (1) a solenoid coil, and (2) a split coil, both of which make use of aniron yoke to enhance the trapped magnetic ﬁeld. It is shown that a signiﬁcantly higher trapped
ﬁeld can be achieved using a split coil with an iron yoke, and in order to explain these how thisarrangement works in detail, numerical simulations using a 2D axisymmetric ﬁnite element
method based on the H-formulation are carried to qualitatively reproduce and analyze themagnetization process from both electromagnetic and thermal points of view. It is observed thatafter the pulse peak signiﬁcantly less ﬂux exits the bulk when the iron core is present, resulting ina higher peak trapped ﬁeld, as well as more overall trapped ﬂux, after the magnetization process
is complete. The results have important implications for practical applications of bulk
superconductors as such a split coil arrangement with an iron yoke could be incorporated into thedesign of a portable, high magnetic ﬁeld source/magnet to enhance the available magnetic ﬁeld
or in an axial gap-type bulk superconducting electric machine, where iron can be incorporated
into the stator windings to (1) improve the trapped ﬁeld from the magnetization process, and (2)increase the effective air-gap magnetic ﬁeld.
Keywords: bulk superconductors, pulsed ﬁeld magnetization, ﬁnite element method (FEM)
modeling, numerical simulation, numerical modeling, trapped ﬁeld magnets, split coil withiron yoke(Some ﬁgures may appear in colour only in the online journal)0953-2048/16/074003+16$33.00

1© 2016 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 29 (2016) 074003

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1. Introduction
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 [1]. Indeed, it has been shown that (RE)BCO (whereRE=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 criticalcurrent densities, Jcs, of 5×104 A cm−2 at 1 T and 77 K,
resulting in typical trapped ﬁelds of up to between 1 and 1.5 T
for Y–Ba–Cu–O (YBCO) and greater than 2 T for (RE)BCO
at this technologically important temperature, with 3 T thehighest trapped ﬁeld at 77 K so far [4]. As a result, there isgreat interest in using these materials as TFMs in a number ofengineering applications, including magnetic bearings, energy
storage ﬂywheels, magnetic resonance imaging, magnetic
separation and rotating machines [5–7]. 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 [8–11].
Investigating and predicting the magnetization of thesetechnologically important materials and developing practicalmagnetizing techniques is crucial to using them as TFMs inapplications of these types. The pulsed ﬁeld magnetization(PFM) technique is considered to be a compact, mobile andrelative inexpensive way to magnetize bulk samples, requiring shorter magnetization times (on the order of milliseconds)
and a smaller and less complicated magnetization ﬁxture;
however, the trapped ﬁeld produced by PFM is generally
much smaller than that of slower zero ﬁeld cooling (ZFC) or
ﬁeld cooling (FC) techniques, particularly at lower operating
temperatures, due to the large temperature rise ΔT associatedwith the rapid dynamic movement of the magnetic ﬂux in theinterior of the superconductor during the PFM process [12].
Accordingly, the record trapped ﬁeld produced so far by PFM
is only 5.2 T at 29 K [13].
There are many considerations and variables for the PFM
technique, including the duration, magnitude and shape of thepulse, the number of pulses, the temperature(s) at which thepulse(s) is applied, and the shape and type of the magnetizing
coil(s) [12]. In this paper, the last of these considerations isinvestigated in detail and the PFM of two, standard Ag-containing Gd–Ba–Cu–O samples—of the same composition thatrecently achieved the new record trapped ﬁeld [3]—is carriedout using two types of magnetizing coils: (1) a solenoid coil,and (2) a split coil, both of which make use of an iron yoke toenhance the trapped magnetic ﬁeld. Bulk superconductors areoften used in applications that incorporate ferromagnetic
materials, such as in motors and generators [14, 15]. Ferromagnets can also increase the force in levitation systems

Figure 1. Field cooling (FC) magnetization results for the two Ag-

containing GdBCO samples, measured at the center of the top
surface of each sample, with an external applied ﬁeld of 7 T.

[16, 17], and can improve the available ﬂux produced by bulk
superconductors [18–22].
The trapped ﬁeld performance and magnetic ﬂux dynamics of the samples for these two coil arrangements are compared experimentally, and these conﬁgurations are thenmodelled using a 2D axisymmetric ﬁnite element method(FEM) based on the H-formulation to qualitatively reproduce
and analyze the magnetization process from both electromagnetic and thermal points of view. It is shown that using a
split coil arrangement with an iron yoke results in an enhanced
trapped ﬁeld performance for pulsed ﬁelds of equivalent
duration, magnitude and shape, chieﬂy because the iron yokeacts to attract ﬂux to remain in the bulk after the pulse peak.

2. Sample processing details
For this study, two cylindrical Ag-containing (15 wt%)
Gd–Ba–Cu–O (GdBCO) bulk samples of 30 mm in diameterand 15 mm in thickness were grown using the top-seeded
melt growth (TSMG) process described elsewhere [23]. We
have previously developed Ag-containing GdBCO samples
that exhibit good ﬁeld-trapping performance at 77 K [23] andhave shown record trapped ﬁeld capabilities, greater than
17 T, below 30 K in a stack of two such samples [3]. No
signiﬁcant deleterious effect on Jc has been observed for an
Ag content of up to 15 wt% AgO2, and they exhibit a morehomogenous Jc distribution than TSMG-processed Y–Ba–
Cu–O (YBCO) samples [24].

3. FC magnetization
The FC magnetization technique gives the best indication of
the trapped ﬁeld capability of a bulk superconductor sample
and is useful for comparing the maximum potential of2Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

Figure 2. Trapped ﬁeld distributions for each sample, obtained by ﬁeld cooling (FC) magnetization at 77 K in an external magnetic ﬁeld of
1.5 T and measured approximately 1 mm above the top surface using a scanning system consisting of a linear array of Hall probes.

Figure 3. Schematic diagram (top) and photograph (bottom) of the sample holder for PFM experiments using the split coil arrangement. To
ensure a good thermal contact between the samples and the sample holder, a thin ring of indium is placed around the periphery of eachsample. A heater is used to regulate the operating temperature, which is measured using a Pt–Co thermometer.

4. Pulsed ﬁeld magnetizationdifferent samples [25, 26]. Figure 1 shows the FC results forthe two samples, measured by a Hall sensor placed at thecenter of the top surface of the sample, as the temperature isincreased slowly from 40 to 100 K after application of a 7 T
magnetic ﬁeld. The two samples achieved trapped ﬁeld valuesof Bt=6 T (#476) and 5.44 T (#477) at 40 K and
Bt=3.11 T (#476) and 2.02 T (#477) at 65 K. The twodimensional (2D) trapped ﬁeld distributions at 77 K obtainedby FC are also shown for each sample in ﬁgure 2, measured
approximately 1 mm above the top surface using a scanningsystem consisting of a linear array of Hall probes [27].

4.1. Split coil experimental setup

Figure 3 shows a schematic diagram (top) and photograph (bottom) of the sample holder for the PFM experiments using the split coil arrangement. To ensure a goodthermal contact between the samples and the copper sample
holder, a thin ring of indium (of approx. 0.2 mm thickness)is placed around the periphery of each sample. A heater isused to regulate the operating temperature, which is measured using a Pt–Co thermometer. The sample holder is3

Supercond. Sci. Technol. 29 (2016) 074003

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Figure 6. Comparison of trapped ﬁelds for sample #476 after

activation by PFM using the split coil (solid lines) and the solenoid
coil (dashed lines) at operating temperatures of 65 and 40 K,
measured at the center of the top bulk surface.

Figure 4. Geometry and experimental setup, including sample

mounting, for pulsed ﬁeld magnetization using a split coil with aniron yoke.

Figure 7. Comparison of trapped ﬁelds for sample #477 after

activation by PFM using the split coil (solid lines) and the solenoid
coil (dashed lines) at operating temperatures of 65 and 40 K,
measured at the center of the top bulk surface.

Figure 5. Typical evolution of the applied pulsed ﬁelds with time,

estimated from the measured current ﬂowing through a shunt resistor
for the split and solenoid coils with an iron yoke.

experimental setup, including sample mounting, is shownin ﬁgure 4. The initial temperature, Top, of the bulk was set
to 65 and 40 K. Magnetic pulses, Bex, up to around 6 T,with a rise time of tr=18 ms and duration of approximately td=200 ms were applied via a pulse current in thecoil. The typical evolution of the applied pulsed ﬁelds fromthis arrangement, estimated from the measured current
ﬂowing through a shunt resistor, is shown in ﬁgure 5. Afterthe removal of the split coil, the two-dimensional trapped
ﬁeld distributions were measured on the outside surface ofthe vacuum chamber, at a distance approximately 2 mmabove the bulk surface by scanning a Hall sensor using an
x–y stage controller.

connected to the cold stage of a Gifford–McMahon (GM)cycle helium refrigerator in the vacuum chamber, which has
a small vacuum space of 1 mm. A Hall sensor was adheredon the bulk surface to measure the magnetic ﬁeld dynamically during the experiments. A split-type coil with an
inner diameter (ID) 72 mm, outer diameter (OD) of
124 mm, and height of 35 mm, which was submerged inliquid nitrogen, was placed outside the vacuum chamber as
the magnetizing ﬁxture, and a pair of soft iron yokes(60 mm in diameter and 65 mm in height) were inserted in
the hole of the split-type coil. This geometry and4Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

Figure 8. 2D trapped ﬁeld proﬁles measured on the top surface (z=2 mm) for sample #476 after PFM using the split coil with an iron yokeat an operating temperature of 65 K. The central panel shows the corresponding central trapped ﬁeld, as shown in ﬁgure 6.

magnetizing solenoid pulse coil, cooled using liquid nitrogen,was placed outside the vacuum chamber. The direction of
external applied ﬁeld is parallel to the c-axis of the bulk
samples. The magnetizing coil can provide pulsed externalﬁelds up to Bapp=6.4 T with a rise time of tr=12 ms andduration of approximately td=120 ms [24]. The typicalevolution of the applied pulsed ﬁelds from this arrangement,
as measured by the current ﬂowing through a shunt resistor, isshown in ﬁgure 5. After PFM of the sample using the solenoid coil, the two-dimensional trapped ﬁeld distributions were

4.2. Solenoid coil experimental setup

An overview of the PFM experimental setup using a solenoid
coil is described in [24]. The bulk samples were mounted
tightly on a sample holder fabricated from 316 stainless steelof ID slightly larger than the diameter of the samples and OD
56 mm to match the dimensions of the available cold stage ofthe pulse system. Stycast™ (2850 GT) was used to mount the
samples in the holder, which was then mounted on the cold
stage of a GM, closed cycle helium refrigerator, and a copper5Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

Figure 9. 2D trapped ﬁeld proﬁles measured on the top surface (z=2 mm) for sample #476 after PFM using the solenoid coil with an ironyoke at an operating temperature of 65 K. The central panel shows the corresponding central trapped ﬁeld, as shown in ﬁgure 6.

the split coil (solid lines) and the solenoid coil (dashed lines)
at operating temperatures of 65 and 40 K, measured at thecenter of the top bulk surface, for samples #476 and #477,
respectively. It is clear that for both samples, the split coilarrangement results in a signiﬁcantly larger measured trapped
ﬁeld, and at 40 K in particular, not only is the trapped ﬁeld
signiﬁcantly larger, but the activation ﬁeld (deﬁned as theﬁeld required to fully magnetize the bulk sample [24]) is
reduced in comparison to the solenoid coil. The lower activation ﬁeld for the #477 sample (ﬁgure 7) and its lower

measured inside the vacuum chamber using an x–y stage
controller and an axial-type Hall sensor positioned 2 mmabove the top surface of the samples. The trapped ﬁeld close
to the center of the top surface of the samples was measured
dynamically during the application of each pulsed ﬁeld using
the same Hall sensor.
4.3. PFM results

Figures 6 and 7 show a comparison of the measured trapped
ﬁelds (at approx. t=300 ms) after activation by PFM using
6Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

Figure 10. 2D trapped ﬁeld proﬁles measured on the top surface (z=2 mm) for sample #476 after PFM using the split coil with an ironyoke at an operating temperature of 40 K. The central panel shows the corresponding central trapped ﬁeld, as shown in ﬁgure 6.

scanning measurement took place after the split coil (and ironyoke) were moved well away from the sample (in the case of
the solenoid coil arrangement, the iron yoke is still present aftermagnetization and for the scanning measurement since it isembedded in the magnetizing ﬁxture). Hence, there is areduction in the observed peak trapped ﬁeld, but the magnitude
is still higher in the case of the split coil: Bt (65 K)≈1.2 T(split), Bt (65 K)≈0.9 T (solenoid); Bt (40 K)≈1.8 T (split),
Bt (40 K)≈1.3 T (solenoid). The surface plots show that thetrapped ﬁeld by FC magnetization suggests that the overall Jc
is lower in comparison to sample #476.
Figures 8 and 9 show the 2D trapped ﬁeld surface plots forsample #476 measured at 65 K when magnetized by the split
coil (ﬁgure 8) and solenoid coil (ﬁgure 9), and ﬁgure 10 (split)and 11 (solenoid) show the same plots for the same sample
measured at 40 K. The 2D surface plots were taken using thescanning system approximately t=20 min after the applicationof the pulse, and in the case of the split coil measurement, the7Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

Figure 11. 2D trapped ﬁeld proﬁles measured on the top surface (z=2 mm) for sample #476 after PFM using the solenoid coil with an ironyoke at an operating temperature of 40 K. The central panel shows the corresponding central trapped ﬁeld, as shown in ﬁgure 6.

magnetic ﬂux penetrates into both samples in a fairly uniform
manner, similar to the results for the GdBCO sample presented
in [24], which suggests a homogeneous Jc distribution aroundthe ab-plane of the samples.
In order to fully understand the mechanism(s) by which the
split coil with an iron yoke acts to enhance the trapped ﬁeld, a
numerical model is developed in the following section to analyse the electromagnetic and thermal properties of the bulk
during each magnetization process, and thoroughly investigate
the positive effect an iron yoke has on the trapped ﬁeld.

5. Numerical modeling5.1. Modeling framework and assumptions

The numerical model used here combines the electromagnetic
and thermal equations governing the behavior of the superconducting material, based on the 2D axisymmetric H-formulation [22, 25, 28–31] implemented using the commercial FEM software package COMSOL Multiphysics 5.1.
The general form, partial differential equation interface of8Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

In the 2D axisymmetric H-formulation, the governing
equations are derived from Maxwell’s equations—namely,
Faraday’s (1) and Ampere’s (2) laws:

COMSOL is employed for the electromagnetic analysis andthe Heat Transfer module is used for the thermal analysis,
which are coupled together as described below and in [24].

´E+

d (m 0 m r H )
dB
=´E+
= 0,
dt
dt
 ´ H = J,

(1 )
(2 )

where H=[Hr, Hz] represents the magnetic ﬁeld components, J=[Jf] represents the current density and E=[Ef]represents the electric ﬁeld. μ0 is the permeability of freespace and for all sub-domains other than the iron yoke (where
present), the relative permeability is simply μr=1. For theiron yoke, the following equation [22, 31] is used to represent
μr:
m r = m r, max for B < m 0 Msat ,⎛M ⎞m r = ⎜1 + sat ⎟ for B  m 0 Msat ,
⎝
H ⎠

(3 )

where μ0Msat and μr,max are 1.6 T and 2000, respectively, as
representative values for iron. The electrical properties of thesuperconductor are modeled using an E–J power law relation[32, 33], E α J n, where n=21.

Figure 12. Experimental data for Jc(B), measured from a small

sample taken from a representative bulk of the same composition as
the two samples (15 wt% Ag-containing Gd–Ba–Cu–O). The
experimental data is ﬁt up to 10 T using the equation presented in[35] for samples exhibiting a ﬁshtail shape in their Jc(B) curves.

Figure 14. Measured, temperature-dependent speciﬁc heat data for

Stycast™ (epoxy data from [39]), SUS, GdBCO with 15 wt% Ag
and indium [40] for the numerical models.

Figure 13. Measured, temperature-dependent thermal conductivity

data for Stycast™ (2850GT) and SUS for the numerical models.

Table 1. Data ﬁtting coefﬁcients for the Jc(B) characteristics of a small sample using equation (4).

Temperature (K)
85
80
77
70
60
50
40
30

BL

Jc1
8

2.35×10
3.88×108
4.83×108
7.13×108
1.10×109
1.70×109
2.50×109
3.70×109

0.18
0.28
0.34
0.48
0.7
0.8
0.8
0.94

9

Jc2
7

4.90×10
1.48×108
2.20×108
4.40×108
9.00×108
1.49×109
2.20×109
3.00×109

Bmax

y

0.48
0.87
1.12
1.75
2.65
3.5
4.2
5.5

2.5
2
1.87
1.75
1.5
1.2
0.8
0.5

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Figure 16. 2D axisymmetric model setup for numerical simulation of
the solenoid coil with an iron yoke. A thin (0.2 mm) indium sheet isplaced between the bulk and cold stage to provide a good thermal
contact. The bulk is embedded in a SUS ring using Stycast™
(2850GT).

Figure 15. 2D axisymmetric model setup for numerical simulation of
the split coil with an iron yoke. A thin (0.2 mm) indium sheet isplaced around the bulk to provide a good thermal contact to thesample holder.

The results of the numerical model strongly depend on
the Jc(B, T) characteristics of the superconductor [12], and theexperimental data for Jc(B), measured between 30 and 85 K
from a specimen taken from a representative bulk of the same
composition as the two samples in this study (15 wt% Agcontaining GdBCO), is shown in ﬁgure 12. The magnetic
moment hysteresis (m–H) loops were measured using a
Quantum Design Magnetic Property Measurement System
XL SQUID magnetometer by applying the ﬁeld parallel to thec-axis of the specimen and the variation of Jc with magnetic
ﬁeld was determined using the extended Bean model [34].
The experimental data (up to 4 T) is ﬁt up to 10 T using
equation (4) presented in [35] for samples exhibiting a ﬁshtail
shape in their magnetization loop. The coefﬁcients related to thedata ﬁtting are shown in table 1. This data is then input into thenumerical model using a two-variable, direct interpolation, as
described in [36, 37], where it is implemented for the Jc(B, ϑ)data to model the complex anisotropy of high temperature
superconducting (HTS) coated conductors in an HTS coil. Thismethod of data input is simple and direct, and can signiﬁcantly
improve the computational speed of the model [36, 37]

this case, since the magnetization process is slow, isothermal
conditions are assumed; hence, no thermal model is included.
A magnetic ﬁeld of magnitude at least twice the expectedtrapped ﬁeld (based on the FC magnetization data in ﬁgure 1)
is applied using uniform boundary conditions at a slow ramp
rate of 8 mT s−1. It is found that the magnitude of Jc(B)should be adjusted to one third of its small specimen value toadequately reproduce the FC magnetization results presented
in ﬁgure 1, resulting in Bt (77 K)=1.12 T, Bt(70 K)=1.94 T, Bt (60 K)=3.67 T, and Bt (40 K)=8.63 T(also shown for reference in ﬁgure 1). Therefore, the overall
pinning characteristics of the sample are maintained, while
reproducing sensible and realistic trapped ﬁelds in the model.
Suitable parameters for the thermal properties must also
be assumed for the materials and input into the model. The
measured, temperature-dependent thermal conductivity data
for the SUS ring and Stycast™ (2850GT) over the temperature range 0–100 K is shown in ﬁgure 13, and the speciﬁcheat data for the SUS ring, Stycast™ (epoxy data from [39]),
GdBCO and indium sheet [40] is shown in ﬁgure 14. In themodels, the operating temperature of the cold head, Top, is set
on the side of the indium sheet not connected to the bulk
sample, as indicated in ﬁgures 15 and 16 for the split andsolenoid coil, respectively. These ﬁgures show the 2D axisymmetric model setups for the numerical simulations.
In previous work, the magnetizing ﬁeld has been applied
using uniform boundary conditions [24], but in this work, theﬁeld is applied via a pulse current applied to the magnetizing
coils (copper coil subdomain). An integral constraint for the

⎡1 ⎛
⎛ B⎞
⎛ B ⎞y ⎞ ⎤B
Jc (B) = Jc1 exp ⎜ - ⎟ + Jc2
exp ⎢ ⎜1 - ⎜⎟ ⎟ ⎥.
⎢⎣ y ⎝⎝ BL ⎠⎝ Bmax ⎠ ⎠ ⎥⎦
Bmax
(4 )

To validate the Jc(B) data, a ZFC process is employed tocheck the trapped ﬁeld capability of a bulk sample [22, 38]. In
10Supercond. Sci. Technol. 29 (2016) 074003

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Table 2. Other assumed model parameters.

Parameter
Tc
Top
ρb
kab
kc
kindium
E0τDescription

Value

Superconducting transition temperature
Operating temperature (cold head)
HTS bulk density
Thermal conductivity of bulk along ab-plane
Thermal conductivity of bulk along c-axis
Artiﬁcial thermal conductivity of indium sheet
Characteristic voltage
Rise time of applied pulsed current/magnetic ﬁeld

92 K
40, 65 K
5.9×103 kg m−3
20 W m−1 K−1
4 W m−1 K−1
0.5 W m−1 K−1
1×10−4 V m−1
15 ms

Figure 17. Magnetic ﬁeld at the center of the magnetizing ﬁxture atthe peak of the pulsed current, calculated using the numericalmodels with the bulk removed, for (a) solenoid coil with andwithout the iron yoke and (b) split coil with and without theiron yoke.

Figure 18. Numerical simulation results for the trapped magnetic

ﬁeld at the center of the top surface of the bulk sample (r=0 mm) ata height of 0.1 mm at t=300 ms for the split and solenoid coils withand without the iron yoke for two temperatures: (a) Top=65 K and(b) Top=40 K.

current is applied to the copper coil subdomain, as describedin [36], using the following Ipulse function:⎛tt⎞Ipulse (t ) = N ⋅ I0 exp ⎜1 - ⎟ ,⎝tt⎠where I0 is the peak magnitude of the current ﬂowing in eachturn of the coil, N is the number turns, and τ is the rise time ofthe pulse (see table 2). Table 2 also summarizes the otherassumed model parameters, including the thermal

(5 )

11

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Figure 20. Evolution of the (a) trapped ﬁeld and (b) averagetemperature, Tave, from t=300 ms (corresponding to the calculated
trapped ﬁelds shown in ﬁgure 18) to t=10 s for the split andsolenoid coil with an iron yoke, including the case where the ironyoke is removed from the split coil at t=4 s. This particular casecorresponds to Top=65 K, Bex=4 T, which resulted in the
maximum trapped ﬁeld in ﬁgure 18(a).

Figure 19. Numerical simulation results for the average temperatureof the bulk, Tave, at t=300 ms for the split and solenoid coils withand without the iron yoke for two temperatures: (a) Top=65 K and(b) Top=40 K.

conductivity of the indium sheet, which is set to
0.5 W m−1 K−1 to represent the ﬁnite cooling power of therefrigerator and the thermal contact between the bulk and thecold stage, as described in [41]. The rise time of the appliedcurrent, and hence resultant magnetic ﬁeld, is the same value(τ=15 ms) for all models in order to adequately compare
pulsed ﬁelds of equivalent duration, magnitude and shape.
The rise time is slightly different experimentally (see ﬁgure 5)
due to a difference in the inductance of the coils.
Finally, ﬁgure 17 shows the magnetic ﬁeld at the centerof the magnetizing ﬁxture, calculated using the numericalmodels with the bulk removed, for (a) solenoid coil with andwithout the iron yoke and (b) split coil with and without theiron yoke. This ﬁgure is then used a reference to apply
particular pulsed ﬁelds with the bulk present, the results ofwhich are analyzed in the next section. An important result
from ﬁgure 17 is that the presence of an iron yoke for bothmagnetizing coils results in a larger magnetic ﬁeld at thecenter of the ﬁxture for the same applied current, reducing therequirements for the magnetizing ﬁxture.
5.2. Numerical results and discussion

Figure 18 shows the trapped magnetic ﬁeld at the center of the
top surface of the bulk sample (r=0 mm) at a height of
0.1 mm at t=300 ms for the split and solenoid coils with andwithout the iron yoke for two temperatures: (a) Top=65 K
and (b) Top=40 K. In comparison to the case where no yoke
is present, the presence of the iron yoke clearly enhances thecalculated trapped ﬁeld, and this enhancement is particularly
pronounced for the split coil, qualitatively reproducing theexperimental results shown in ﬁgures 6 and 7 (also measured
at t≈300 ms). In the case of no yoke present, the split andsolenoid coil arrangements see essentially the same trapped

12Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et alt=300 ms (corresponding to the calculated trapped ﬁelds
shown in ﬁgure 18) to t=10 s, after removing the iron yokefrom the split coil at t=4 s. Here we examine two particularcases, both of which correspond to the maximum trapped ﬁeld
cases seen in ﬁgure 18: (1) Top=65 K, Bex=4 T, and (2)Top=40 K, Bex=6.5 T.
The removal of the iron yoke is simulated by reducing
the μ0Msat value exponentially to zero with a time constant of
20 ms, i.e., an exp(–t/0.02) function, resulting in an equivalent permeability of the iron yoke of μ0 (μr=1) after a periodof several time constants. The dashed line in ﬁgures 20(a) and21(a) show the calculated trapped ﬁeld when the yoke isremoved at t=4 s. For both temperatures, there is a reductionin the observed trapped ﬁeld when the yoke is removed—at
40 K, the difference is almost negligible—but the trapped
ﬁeld is still higher than in the case of the solenoid coil withthe yoke, where the yoke remains embedded in the ﬁxture.
Another important consideration is the average temperature rise during the entire magnetization process and subsequent cooling rate for each magnetizing ﬁxture.
Figures 20(b) and 21(b) show the calculated Tave in the bulk
as the pulse increases t=2, 4, 6, 8, 10 and 15 ms (pulsepeak), followed by t=30, 100, 300 ms, 1, 3, 5 and 10 s.
Similar to ﬁgure 19 (calculated only at t=300 ms), thetemperature rise during the application of the pulse is very
similar for both the split and solenoid coils with the yoke, but
the bulk cools back down to Top at a faster rate in the case of
the split coil arrangement because the bulk is cooled from theperiphery through the ab-plane, for which the thermal conductivity is several times higher. Hence, the ﬁeld decaysslightly faster with time in the case of the solenoid coil. The
same temperature proﬁle is observed for the split coil whether
or not the yoke is present for both Top cases.
The numerical simulation can also provide an easy way of
examining the ﬂux penetration into the bulk and its subsequent
exit from the sample as the pulse reduces to zero, which isextremely difﬁcult to achieve experimentally. Figure 22(a)shows the ﬂux penetration into the sample (t=2, 4, 6, 8, 10
and 15 ms) for Top=65 K, Bex=4 T as the pulse reaches its
peak value (t=15 ms). The values plotted are for Bz across the
middle of the bulk from the edge to the center. The iron yokeacts to improve ﬂux penetration initially, but at the pulse peak, asimilar ﬂux penetration is observed. As the pulse reduces andthe bulk cools back down towards Top, the iron yoke acts to
attract ﬂux to remain in the bulk, as shown in ﬁgure 22(b), fort=30, 100, 300 ms, 1, 10 s. Signiﬁcantly less ﬂux exits thebulk when the iron core is present, resulting in a higher peaktrapped ﬁeld, as well as more overall trapped ﬂux, after themagnetization process is complete. Figure 23 shows the results
for the Top=40 K, Bex=6.5 T case and a similar trend isobserved for both ﬂux entry and exit.
For the solenoid coil, the enhancement of the ﬁeld incomparison to no yoke can be attributed to both the differencein the ﬂux exiting the bulk and the enhancement effectdescribed in [22] from the presence of a ferromagnetic material on the bottom of the bulk, since the iron yoke is

Figure 21. Evolution of the (a) trapped ﬁeld and (b) averagetemperature, Tave, from t=300 ms (corresponding to the calculated
trapped ﬁelds shown in ﬁgure 18) to t=10 s for the split andsolenoid coil with an iron yoke, including the case where the ironyoke is removed from the split coil at t=4 s. This particular casecorresponds to Top=40 K, Bex=6.5 T, which resulted in the
maximum trapped ﬁeld in ﬁgure 18(b).

ﬁeld at both temperatures. The average temperature, Tave, ofthe bulk, is shown in ﬁgure 19 as calculated at t=300 ms for
all cases for (a) Top=65 K and (b) Top=40 K. There is no
signiﬁcant difference in the average temperature of the bulk,
and the presence of the iron yoke slightly reduces the temperature rise in both cases.
The measured trapped ﬁelds shown in ﬁgures 6 and 7
were measured before the split coil and yoke were removed
for the 2D surface plot scanning measurement, and hence, thepresence of the yoke would be expected to enhance thelocally measured magnetic ﬁeld. This results in the observed
reduction in trapped ﬁeld when analyzing the 2D surface
plots, which are measured approximately 20 min after theapplication of the pulse. In order to assess the ‘true’ trapped
ﬁeld, with the split coil and iron yoke removed, ﬁgures 20(a)and 21(a) show the evolution of the trapped ﬁeld from13

Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

Figure 22. (a) Flux penetration into the sample for Top=65 K, Bex=4 T, for the split coil with and without an iron yoke, at t=2 ms (blue),

4 ms (green), 6 ms (purple), 8 ms (red), 10 ms (orange) and 15 ms (black). The pulse reaches its peak value at t=15 ms. (b) Flux exiting thesample as the pulse reduces and the bulk cools back down towards Top at t=30 ms (blue), 100 ms (green), 300 ms (purple), 1 s (red) and10 s (black).

embedded in the magnetizing ﬁxture and cannot be removed
after the magnetization process.
The experimental results, which have been veriﬁed bynumerical simulation, have important implications for practical applications of bulk superconductors. Such a split coilarrangement with an iron yoke could be incorporated into thedesign of a portable, high magnetic ﬁeld source/magnet toenhance the available magnetic ﬁeld. It also has strong
implications for an axial gap-type bulk superconducting
electric machine, where iron can be incorporated into thestator windings to (1) improve the trapped ﬁeld from themagnetization process, and (2) increase the effective air-gap
magnetic ﬁeld.

6. Conclusion
In this paper, the PFM of two, standard Ag-containing Gd–
Ba–Cu–O samples—of the same composition that recently
achieved the new record trapped ﬁeld—is carried out using
two types of magnetizing coils with an iron yoke: a solenoid
coil and a split coil arrangement. A signiﬁcantly higher
trapped ﬁeld can be achieved using a split coil with an ironyoke, and in order to explain these how this arrangement
works in detail, numerical simulations using a 2D axisymmetric ﬁnite element method based on the H-formulation are
carried to qualitatively reproduce and analyze the magnetization process from both electromagnetic and thermal points

14

Supercond. Sci. Technol. 29 (2016) 074003

M D Ainslie et al

Figure 23. (a) Flux penetration into the sample for Top=40 K, Bex=6.5 T, for the split coil with and without an iron yoke, at t=2 ms(blue), 4 ms (green), 6 ms (purple), 8 ms (red), 10 ms (orange) and 15 ms (black). The pulse reaches its peak value at t=15 ms. (b) Flux
exiting the sample as the pulse reduces and the bulk cools back down towards Top at t=30 ms (blue), 100 ms (green), 300 ms (purple), 1 s(red) and 10 s (black).

trapped ﬁeld from the magnetization process, and (2) increase
the effective air-gap magnetic ﬁeld.

of view. During PFM, the iron yoke in the split coil acts toimprove ﬂux penetration initially, but at the pulse peak, asimilar ﬂux penetration is observed. However, as the pulse
reduces and the bulk cools back down towards the operating
temperature, Top, the iron yoke acts to attract ﬂux to remain in
the bulk. Signiﬁcantly less ﬂux exits the bulk when the ironcore is present, resulting in a higher peak trapped ﬁeld, as well
as more overall trapped ﬂux, after the magnetization process
is complete. The results have important implications forpractical applications of bulk superconductors as such a split
coil arrangement with an iron yoke could be incorporated into
the design of a portable, high magnetic ﬁeld source/magnet toenhance the available magnetic ﬁeld or in an axial gap-typebulk superconducting electric machine, where iron can beincorporated into the stator windings to (1) improve the

Acknowledgments
Mark Ainslie would like to acknowledge the support of a
Royal Academy of Engineering Research Fellowship. Hiroyuki Fujishiro would like to acknowledge support in part by
a Grant-in-Aid for Scientiﬁc Research from the Ministry of
Education, Culture, Sports, Science and Technology, Japan.
This research was also supported in part by a Royal SocietyInternational Exchanges Scheme grant, IE131084. Jin Zou
would like to acknowledge the support of Churchill College,
Cambridge, the China Scholarship Council and the Cambridge Commonwealth, European and International Trust.
15

Supercond. Sci. Technol. 29 (2016) 074003

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