Physica C: Superconductivity and its applications 536 (2017) 1–10

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Physica C: Superconductivity and its applications
journal homepage: www.elsevier.com/locate/physc

Trapped ﬁeld properties of a Y–Ba–Cu–O bulk by pulsed ﬁeld
magnetization using a split coil inserted by iron yokes with various
geometries and electromagnetic properties
K. Takahashi a,∗, M.D. Ainslie b, H. Fujishiro a, T. Naito a, Y-H. Shi b, D.A. Cardwell bab

Department of Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan
Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UKa r t i c l e

i n f oArticle history:
Received 16 February 2017
Revised 5 March 2017
Accepted 22 March 2017
Available online 29 March 2017
Keywords:
Bulk superconductor
Pulsed ﬁeld magnetization
Numerical simulation
Trapped ﬁeld
Iron based yoke
Electromagnetic propertya b s t r a c t
We have investigated, both experimentally and numerically, the trapped ﬁeld characteristics of a standard
Y–Ba–Cu–O bulk of 30 mm in diameter and 14 mm in thickness magnetized by pulsed ﬁeld magnetization (PFM) using a split coil, in which three kinds of iron yoke are inserted in the bore of the coil: softiron with a ﬂat surface, soft iron with a taper, and permendur (50Fe + 50Co alloy) with a ﬂat surface.
The highest trapped ﬁeld, BTmax , of 2.93 T was achieved at 40 K in the case of the permendur yoke, which
was slightly higher than that obtained for the ﬂat soft iron or the tapered soft iron yokes, and was muchhigher than 2.20 T in the case without the yoke. The insertion effect of the yoke on the trapped ﬁeld
characteristics was also investigated using numerical simulations. The results suggest that the saturation
magnetic ﬂux density, Bsat , of the yoke acts to reduce the ﬂux ﬂow due to its hysteretic magnetization
curve and the higher electrical conductivity, σ , of the yoke material also acts to suppress the ﬂux increase
rate. A ﬂux jump (or ﬂux leap) can be reproduced in the ascending stage of PFM using numerical simulation, using an assumption of relatively high Jc (T, B) characteristics. The insertion effect of the yoke withhigh Bsat and σ is discussed to enhance the ﬁnal trapped ﬁeld from both electromagnetic and thermal
points of view.
© 2017 Elsevier B.V. All rights reserved.

Contents
1.
2.
3.
4.
5.

6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results of numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Magnetic ﬂux behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Yoke insertion effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction
RE-Ba-Cu-O (RE: rare earth element or Y) bulk superconductors have promising potential to be used as a strong trapped ﬁeld
magnet (TFM), which can trap a magnetic ﬁeld of several Tesla be-∗

Corresponding author.
E-mail address: t2216017@iwate-u.ac.jp (K. Takahashi).

http://dx.doi.org/10.1016/j.physc.2017.03.005
0921-4534/© 2017 Elsevier B.V. All rights reserved.

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cause of their high critical current density, Jc [1,2]. This property
can be exploited in practical applications, such as magnetic separation, magnetic drug delivery systems and sputtering cathodes forthin ﬁlm growth [3–5]. The ﬁeld cooling magnetization (FCM) technique, which needs a large and expensive experimental setup using
a superconducting coil magnet, is commonly used for fundamental
measurements on bulk superconductors to give the best indicationof the capability of the sample as a TFM. A record-high trapped

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K. Takahashi et al. / Physica C: Superconductivity and its applications 536 (2017) 1–10

Fig. 1. (a) Experimental setup around the disk bulk for PFM using a split coil. (b) The shape of the yoke pair with a ﬂat surface for the soft iron and the permendur (left)
and with a taper for the soft iron (right). The dimensions of the magnetizing coil and two types of yoke are also shown.

ﬁeld of 17.6 T has been achieved in a stack of two Gd–Ba–Cu–O
bulks at 26 K by FCM [6]. Pulsed ﬁeld magnetization (PFM) has
been also applied as a practical magnetizing method for high TFMs
because of its inexpensive and mobile setup, unlike FCM. However, a signiﬁcant challenge to enhance the trapped ﬁeld by PFM
is the reduction of large temperature rise caused by the dynamical magnetic ﬂux motion, especially at low temperatures, in which
the magnetic ﬂux quickly moves in the bulk on the order of milliseconds [7–10]. Using a multi-pulse technique, named a modiﬁed
multi-pulse technique with stepwise cooling (MMPSC), we successfully achieved a highest trapped ﬁeld of 5.20 T on a 45 mm Gd-BaCu-O disk bulk at 30 K, which is a record-high trapped ﬁeld by PFM
to date [11].
In recent years, a ferromagnetic yoke has been inserted in
the magnetizing ﬁxture to increase the trapped magnetic ﬁeld
[12,13], in which permendur (50Fe +50 Co alloy) and soft iron wereused experimentally as a yoke material. We have also achieved atrapped ﬁeld of over 3 T at 40 K on Gd-Ba-Cu-O disk bulk by PFM,
employing a split coil with a pair of soft iron yokes [14], in which
a symmetric trapped ﬁeld proﬁle can be also realized [15]. In addition, for the MgB2 bulk, we achieved over 1 T at 13 K by PFM without any ﬂux jumps on a high-Jc MgB2 disk bulk using the split coilwith soft iron yoke, which is a record-high trapped ﬁeld by PFM
for the MgB2 bulk to date [16]. Numerical simulation is a powerfultool to consider the obtained experimental results during PFM, andto understand the ﬂux dynamics using electromagnetic and thermal equations. Several studies using the numerical simulation havebeen reported for PFM [17–23]. We have previously conﬁrmed theadvantage of the insertion of a soft iron yoke with a ﬂat surface inthe split coil numerically, and compared this to the case without
the yoke [14,15].
However, there is no systematic investigation yet on the effects of varying the electromagnetic properties and the shape ofthe yokes on the trapped ﬁeld characteristics, both experimentallyand numerically, to enhance the trapped ﬁeld by the PFM technique. In this paper, we investigate the trapped ﬁeld characteristics of a Y–Ba–Cu–O bulk magnetized by PFM using a split coil, in
which three kinds of iron yokes were inserted in the bore of thecoil: a conventional soft iron with a ﬂat surface, soft iron with ataper and a permendur with a ﬂat surface. The effects of the material and the shape of yokes are discussed from electromagnetic
and thermal points of view.
2. Experimental setup
A typical Y–Ba–Cu–O bulk superconductor of 30 mm in diameter and 14 mm in thickness, which was fabricated by the
top-seeded, melt-growth (TSMG) method in Cambridge [24], wasmounted in a copper holder and was connected to the cold stageof a Gifford McMahon (GM) cycle helium refrigerator. A schematicFig. 2. Initial magnetization curves of the soft iron and the permendur at 300 K
under an applied magnetic ﬁeld, Bapp .

view of apparatus is shown in Fig. 1(a). A split coil, which wascooled using liquid nitrogen, was placed outside the vacuum chamber, and the bulk disk was magnetized using the condenser bank
(60 mF) between the two coils of the split coil. Three kinds of ironyokes, as shown in Fig. 1(b), were inserted in the bores of the split
coil: soft iron with a ﬂat surface, soft iron with a taper and permendur with a ﬂat surface. The each yoke were placed at a distanceof 5 mm from the bulk top or bottom surface. Fig. 2 presents theinitial magnetization curve of the soft iron and permendur at 300 K
measured by a SQUID magnetometer under an applied ﬁeld, Bapp ,of up to 6 T. The saturation magnetic ﬂux density, Bsat , was determined to be 1.90 T for the soft iron and 2.13 T for the permendur
at Bex > 1.0 T, respectively. Pulsed applied ﬁelds, Bex , of amplitudes
ranging from 2.24 to 5.67 T were applied individually to the bulk,
cooled to an initial temperature of either Ts = 65 or 40 K. The PFM
experiments were performed for four cases of the yoke: (i) without any yoke, (ii) with soft iron, (iii) with tapered soft iron and (iv)with permendur. The time evolution of the local ﬁeld, By (t), at thecenter of the bulk surface was measured during PFM by using theHall sensor adhered on the center of the bulk and a digital oscilloscope. By (t) at t = 300 ms was referred to as a ﬁnal trapped ﬁeld,
BT , after allowing suﬃcient time to relax the ﬂux ﬂow. The time
evolution of the temperature, T(t), was measured by a CERNOXۛ
thermometer on the side surface of the sample holder. After themagnetic pulse, the split coil was removed and the trapped ﬁeld
proﬁle was mapped on the outer surface of the vacuum chamber,which is 2 mm above the bulk surface, by scanning an axial-type
Hall sensor, with a pitch of 1 mm, using an x-y stage controller.

K. Takahashi et al. / Physica C: Superconductivity and its applications 536 (2017) 1–10

3

Fig. 3. Time evolution of the local ﬁeld, B(t), normalized by those maxima, Bex ,measured at the center of the bulk surface at 100 K for the split coil with andwithout the several types of yokes. The ramp and decay rates of the magnetic pulse
were delayed by the yoke insertion.

The relationship between Bex at the center of the bulk surface andthe coil current ﬂowing through a shunt resistor was calibrated foreach yoke at 100 K above the transition temperature of the bulk.
Fig. 3 shows the time evolution of the local ﬁeld, B(t), normalized
by those maxima, Bex , measured at the center of the bulk surface
at 100 K for the split coil with and without the several types ofyokes. The rise time, τ , at which the magnetic pulse takes a maximum value, was 18 ms for the case without any yoke, and 23 ms
for three cases with a yoke. The presence of the yoke delayed thepulsed ﬁeld during the ascending stage due to the induced eddy
current and the descending stage due to its hysteretic magnetization. Note that the eddy current heating, which comes from thecopper holder, was approximately 1.5 K when applying the pulsed
ﬁeld of 4 T at 100 K whether the yoke was present or not.

Fig. 4. Experimental results of the ﬁnal trapped ﬁeld, BT , at the center of the bulk
surface, as a function of the applied ﬁeld, Bex , using the split coil without any yoke
and with the three types of yokes at (a) 65 K and (b) 40 K.

3. Experimental results
In this section, we present the typical behavior of the magnetic
ﬂux and the inﬂuence of the magnetic properties and geometriesof the yoke on the trapped ﬁeld characteristics. Fig. 4 presents theexperimental results for the ﬁnal trapped ﬁeld, BT , at the centerof the bulk surface, as a function of the applied ﬁeld, Bex , using
the split coil without a yoke and with the three types of yokesat 65 K and 40 K. A critical applied pulsed ﬁeld is necessary forthe full penetration of the magnetic ﬂux into the bulk center. At
65 K, as shown in Fig. 4(a), BT started to increase at Bex = 2.5–3.0 T
and abruptly increased with increasing Bex in all cases. A characteristic behavior, which is so-called ﬂux jumps (or leaps), has beenreported for a high-Jc bulk sample during PFM [25,26], where themagnetic ﬂux suddenly intruded towards the center of the bulk.
We abbreviate the optimum applied ﬁeld as Bex ∗ , at which thebulk achieved the maximum trapped ﬁeld, BTmax . Higher BTmax andhigher Bex ∗ were obtained for the cases with a yoke, compared towithout any yoke. Obvious differences appeared more obviously at
40 K as shown in Fig. 4(b) due to the difference in the magnetic
ﬂux motion and temperature rise between these four cases. The
highest trapped ﬁeld of BTmax = 2.93 T was achieved for the permendur yoke at 40 K, which was higher than that without any yoke
and any other cases (ﬂat or tapered soft iron). Note that the maximum trapped ﬁeld decreased to 2.73 T after removing the split coilwith the yoke for the measurement of the trapped ﬁeld proﬁle.
There were no signiﬁcant differences in the trapped ﬁeld basedon the magnetic properties of the yoke material between the softiron and permendur. The BTmax value for the tapered soft iron yoke

was smaller than that for the soft iron yoke with a ﬂat surface. Although the tapered soft iron yoke was expected to concentrate themagnetic ﬂux towards the center of the sample, there is no significant effect to enhance the trapped ﬁeld.
Fig. 5 shows the trapped ﬁeld proﬁles, BT (2 mm), measured at
2 mm above the bulk surface at 40 K before full magnetization under applying the pulsed ﬁeld of Bex < 4.0 T, and after full magnetization under applying the pulsed ﬁeld of Bex > 4.2 T using the
split coil (a) without any yoke and (b) with soft iron of ﬂat surface, (c) with tapered soft iron and (d) with permendur of ﬂat surface. Before the full magnetization for all cases of (a1)–(d1), thetrapped ﬁeld proﬁles were of a convex shape, showing that themagnetic ﬂux was mainly trapped at the grain sector boundaries(GSBs) due to the inhomogeneous Jc distribution in the bulk, typically observed with TSMG-grown, bulk Y–Ba–Cu–O samples [7].
On the other hand, for the cases of (a2)–(d2) at Bex > 4.2 T, thetrapped ﬁeld proﬁles exhibited a conical shape. The difference inthe trapped ﬁeld between the cases without any yoke (a2) andwith a yoke (b2–d2) remained signiﬁcant, even at 2 mm above thebulk surface, although the magnitude of magnetic ﬁeld is severely
decreased and broadened with increasing distance from the bulk
surface. The magnetic ﬂux density in the case of the soft iron (b2)and permendur (d2) was twice as high as that obtained in the casewithout any yoke (a2). It is clear that a higher trapped ﬁeld witha conical proﬁle can be obtained in the cases with a ﬂat yoke (b2and d2), compared to the tapered soft iron (c2).
Fig. 6 shows the time dependence of the local ﬁeld, By (t), at
the center of the bulk surface after applying the pulsed ﬁeld of

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Fig. 5. Trapped ﬁeld proﬁles, BT (2 mm) measured at 2 mm above the bulk surface at Ts = 40 K below full magnetization after applying Bex < 4.0 T (upper panels), using the
split coil without any yoke (a1) and with three types of yoke: soft iron with a ﬂat surface (b1), soft iron with a taper (c1) and permendur with a ﬂat surface (d1). The
results above full magnetization, after applying Bex > 4.2 T using the split coil without any yoke (a2) and with three types of yoke: soft iron (b2), tapered soft iron (c2) andpermendur (d2), are also shown in the lower panels.

Fig. 6. Time dependence of the ﬁeld, By (t), at the center of the bulk surface after applying the pulsed ﬁeld of Bex ∼= 4.0 T at 65 K (upper panels), and Bex ∼= 5.4 T at 40 K (lower
panels), using the split coil (a) without any yoke and with three types of yoke: (b) ﬂat soft iron, (c) tapered soft iron and (d) ﬂat permendur.

K. Takahashi et al. / Physica C: Superconductivity and its applications 536 (2017) 1–10

Fig. 7. Time dependence of the ﬂux increase rate, RF , at the center of the bulk surface after applying Bex ∼= 4.0 T at 65 K for all cases: with and without the three typesof yoke. RF was calculated from dBy (t)/dt in Figs. 6(a1) – (d1).

Bex ∼= 4.0 T at 65 K (upper panels), and Bex ∼= 5.4 T at 40 K (bottom
panels), using the split coil (a) without any yoke and with three
types of yoke: (b) soft iron with a ﬂat surface, (c) tapered softiron and (d) permendur with a ﬂat surface. The applied pulsed
ﬁeld, Bex (t), is also shown in each ﬁgure. For the case without
any yoke at 65 K, as shown in (a1), the magnetic ﬂux did not
penetrate early in the ascending stage of PFM due to the shielding effect of the bulk, and then abruptly starts to intrude into
the bulk, where a ﬂux jump occurred at t = 10 ms. The magnetic
ﬂux behaved in the same way, regardless of whether the yoke waspresent or not. The ﬂux jump was observed for all the cases, at
65 K and 40 K in the ascending stage of the pulse. Another effectof the yoke insertion can be also seen in the descending stage. The
downward arrow shown in Fig. 6 presents the ﬂux ﬂow causedby the dynamic movement of the magnetic ﬂux associated with
the Lorentz force and the temperature rise. For the case without
any yoke at 65 K (a1), the magnetic ﬁeld decreased signiﬁcantlywith increasing time. However, for the cases with a yoke ((b1)–(d1)) at 65 K, the magnetic ﬁeld still remained and was trappedwith minimal ﬂux ﬂow, even after the whole process of magnetization (t > 300 ms) [15]. The contribution of the yoke insertion during PFM exists mainly in the descending stage. Moreover, for thecases with a yoke ((b2)–(d2)) at 40 K, the maximum By (t) value
was fairly smaller than the maximum Bex (t), which is contrast to
the cases at 65 K. These results suggest that there is still someroom left for further enhancement of the trapped ﬁeld.
Considering the ﬂux increase rate is effective to compare themagnetic ﬂux behavior [27]. Fig. 7 shows the time dependence ofthe ﬂux increase rate, RF , after applying Bex ∼= 4.0 T at 65 K, which
is deﬁned by dBy (t)/dt in Figs. 6(a1) – (d1). The RF value abruptly
increased and took a maximum value at the same time as the occurrence of the ﬂux jump during the ascending stage. Generally,
the maximum RF value increases with the increase in Bex becausea larger amount of magnetic ﬂux must intrude into the bulk within
a ﬁnite time with a constant ramp rate of the pulse. The maximum
RF values for any cases with a yoke were reduced, and these maxima were also delayed, compared to those of the case without any
yoke, showing that the yoke acted to suppress the dynamical motion of the magnetic ﬂux. The RFmax value for the case without any
yoke is larger than that with a yoke because there is no disturbance of ﬂux motion from the yoke.
The heat generation during PFM is caused by the pinning loss
and viscous ﬂow loss due to the dynamical magnetic ﬂux motion[28]. Fig. 8 shows the maximum temperature rise, ࢞T, measured
on the side of the sample holder, as a function of the applied ﬁeld,
Bex , for the cases with and without a yoke at 65 K and 40 K. ࢞T in-

5

Fig. 8. Maximum temperature rise, ࢞T, measured on the side of the sample holder,
as a function of the applied ﬁeld, Bex , using the split coil without any yoke and withthe three types of yoke at 65 K and 40 K.

creased monotonically with increasing Bex , which is in contrast to
the ﬁnal trapped ﬁeld shown in Fig. 4. The ࢞T behavior is relatedwith the RF behavior shown in Fig. 7; the ࢞T and RF values in thecase without any yoke are larger than those in the cases with ayoke. In the cases of the yoke insertion, the ࢞T values for the tapered soft iron yoke are smaller than those for the permendur at
higher Bex . This reduction of the heat generation in the case of thesoft iron yoke also contributes to the enhancement of the trapped
ﬁeld.
4. Numerical modelingIn this section, the yoke insertion effect was considered in detail using numerical simulations, for which the typical behavior of
the magnetic ﬂux dynamics, such as a ﬂux jump in the ascending
stage, is reproduced, and a ferromagnetic yoke is conﬁrmed to havea positive effect on the enhancement of the trapped ﬁeld. Based on
our experimental setup for PFM, as shown in Fig. 1, we constructed
two-dimensional models using the ﬁnite element method (FEM)with a 1.0 mm mesh in size in the bulk. A schematic view of themodeling geometry is shown in Fig. 9 for (a) without any yoke, (b)soft iron and permendur with a ﬂat surface and (c) soft iron with ataper. The physical phenomena during PFM are described by electromagnetic and thermal equations, which were referred from refs.
[29,30]. Commercial software, Photo-Eddy, combined with PhotoThermo (Photon Ltd, Japan), was adopted for the analysis, which
was carried out using a standard personal computer. The simulation procedure and the parameters used were described elsewherein detail [31]. A modiﬁed power-n law (n = 20), described in refs.
[32,33], was used to represent the non-linear E-J characteristic of
the bulk. The temperature and magnetic ﬁeld dependence of critical current density, Jc (T, B), was described asJc (T , B ) = α 1 − T 2  32
Tc

B0

|B| + B0,

(1)

where Tc (=92 K) is the critical temperature of the superconducting bulk. The bulk sample with the same dimensions as those usedin the experiment was set to the sample holder and was cooled
from the periphery to either Ts = 65 K or 40 K before applying thepulsed ﬁeld. The Jc (T, B) characteristics used in the simulation are
shown in Fig. 10. In this simulation, the magnetic ﬁeld dependenceof Jc was described using the Kim model with B0 = 0.04, and aconstant α value of 1.5 × 1010 A/m2 , which is higher than that ofan ordinary Y–Ba–Cu–O bulk, was supposed in order to reproduce

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K. Takahashi et al. / Physica C: Superconductivity and its applications 536 (2017) 1–10

Fig. 9. Schematic view of the modeling framework for PFM using a split coil, (a) without any yoke, (b) with soft iron and permendur with a ﬂat surface and (c) with softiron with a taper. The dimensions of the magnetizing ﬁxture and the bulk are the same as the experimental setup shown in Fig. 1.

Fig. 10. The magnetic ﬁeld dependence of critical current density, Jc , at 65 K and
40 K, represented by a Kim model.

the ﬂux jump in the ascending stage of the PFM process. Notethat we have previously reproduced the ﬂux jumps during PFM
using numerical simulations, in which the Jc (T, B) was input directly using the experimentally measured Jc (T, B) characteristics ofa small sample and the interpolation method [34]. In the thermal
model, the temperature-independent anisotropic thermal conductivities кab = 20 W m−1 K−1 in the ab-plane and кc = 4 W m−1 K−1along the c-axis were assumed for the bulk.
The electromagnetic properties of the yokes were represented
by a saturation magnetic ﬂux density, Bsat , and an electrical conductivity, σ . Bsat for the two yoke materials (soft iron and permendur) was input using the magnetization curves shown in Fig. 2. A
constant σ value of 2.5 × 108 S/m was assumed to match the ramp
rate of the experimentally applied magnetic pulse. The time dependence of the applied ﬁeld, Bex (t), was approximated as

Bex (t ) = Bex

t

τ



exp 1 −


t

τ

,

(2)

where the pulse rise time, τ , is 18 ms. Fig. 11 shows the resultsof the simulation of the time evolution of the normalized appliedﬁelds at the center of the bulk surface at T = 100 K for the split coilFig. 11. Time evolution of the local ﬁeld, B(t), normalized by those maxima, Bex ,at the center of the bulk surface for the split coil with and without the different
yokes.

with and without the different types of yokes. In the ﬁgure, thedashed line shows the case with a yoke by a paramagnetic metalsuch as aluminum or copper, instead of soft iron (or permendur),in which a small delay in Bex (t) can be seen, compared to the casewithout any yoke due to the induced eddy currents. Hence, σ acts
to delay the rise time of the pulse because the yoke suppresses themagnetic ﬁeld by induced eddy currents within it. For the cases ofthe soft iron yokes and permendur yoke, Bsat largely acted to delay
the sweep of the descending stage of the pulse due to its hysteresis
magnetization. These results for the magnetic pulse obtained bysimulation show a good agreement with the experimental resultsin Fig. 3.
5. Results of numerical simulation
Similar results to those presented in the experimental results
are shown in this section. Fig. 12 shows the results of the numerical simulation of the ﬁnal trapped ﬁeld, BT , at the center of thebulk surface, as a function of the applied ﬁeld, Bex , using the split
coil without and with the three types of yoke at 65 K and 40 K.

K. Takahashi et al. / Physica C: Superconductivity and its applications 536 (2017) 1–10

Fig. 12. Numerical simulation results of the ﬁnal trapped ﬁeld, BT , at the center ofthe bulk surface, as a function of the applied ﬁeld, Bex , using the split coil without
any yoke and with the three types of yoke at Ts = 65 K and 40 K.

The maximum trapped ﬁeld, BTmax , for the cases with yoke was
improved by 0.6 T at 65 K and 0.8 T at 40 K, respectively, comparedto the case without any yoke. However, there were no signiﬁcantdifferences in the BTmax and Bex ∗ for the different yoke materials(soft iron or permendur) and different geometries (ﬂat or taper). At
65 K, BT started to increase after applying Bex = 2.5–3.0 T, and drastically increased with only a slight increase in Bex , relating to theﬂux jumps observed in the case of without yoke. The results of thesimulation show a good agreement with the experimental results,
particularly at 65 K (shown in Fig. 4). At Ts = 40 K, the BT vs. Bex
curves are independent of the three types of yoke, but are larger
than that for the case without any yoke. These results roughly reproduce the experimental ones in Fig. 4, in which the BT vs. Bex
curves slightly change depending on the three types yokes fromthe experiments in Fig. 2. The difference may result from that theestimated Jc (B) characteristics are different with those of realistic
ones, as well as ignoring any inhomogeneity in Jc around the abplane, which necessitates a 3D model [7].
5.1. Magnetic ﬂux behavior
The results of the numerical analysis enable more detailed considerations about the magnetic ﬂux behavior inside the bulk. First,
the magnetic ﬂux dynamics during PFM are presented for the casewithout any yoke. Fig. 13 shows the time dependence of the lo-

7

cal ﬁeld, By (t), on the top surface of the bulk at positions r = 0 mm
(center), 7 mm (middle) and 14 mm (edge) from the bulk center,
after applying Bex = 4–5 T using the split coil without any yokeat 40 K. The local ﬁeld, By (t), at r = 14 mm near the bulk periphery essentially follows the applied ﬁeld, Bex (t), indicating that themagnetic ﬂux intrudes into the bulk from its outside. Before fullmagnetization for Bex of (a) 4.00 T and (b) 4.30 T, both of whichwere smaller than Bex ∗ = 5 T, a ﬂux jump occurred from the edge
to the middle position, but the amplitude of Bex and the corresponding ﬂux jump was not enough for full penetration. The ﬂux
jump assisted the magnetic ﬂux intrusion towards the center ofthe bulk with slightly increasing Bex after full penetration, e.g.,
Bex = 4.50 T, as shown in (c). This typical behavior of ﬂux jumps
or leaps, agrees well with our previous work using high Jc (T, B)
characteristics based on experimental results [34].
Fig. 14 shows typical examples of the cross-sections of thetrapped ﬁeld proﬁle in the bulk when a ﬂux jump was conﬁrmed
at each point, after applying Bex = 5.0 T at 40 K using the split coilwithout any yoke, as shown in Fig. 14(d). The enlarged ﬁgure of
Fig. 14(d) is shown again, in which the magnetic ﬂux jumped twiceat the middle and the center of the bulk. The bulk superconductor
hinders the intrusion of the magnetic ﬂux due to the shielding current, ﬂowing within its capability of Jc below Tc . Hence, the magnetic ﬂux suddenly intruded into the bulk and a ﬁrst ﬂux jumpoccurred between t = 8 ms and 10 ms. At t = 8 ms, the gradient ofmagnetic ﬂux became steep around r = 14 mm and then ﬂux was
intrudes into the bulk towards the middle of the bulk, overcoming
the strong pinning force. The second small ﬂux jump occurred between 13 ms and 14 ms near the center, at which the similar ﬂux
concentration can be seen around r = 7 mm and then was relaxed.
In this way, the behavior of the magnetic ﬂux related to the ﬂux
jump can be also conﬁrmed in the cross section of the bulk using
the numerical simulation. We are continuing to research the ﬂux
jump phenomenon during PFM, which requires some careful consideration due to the instabilities in the numerical model during
such jumps. The jumps are related to the speciﬁc heat, and rate ofchange of Jc with respect to temperature (which can rise rapidly
as the pulse is increasing) and/or magnetic ﬁeld (depending onwhether or not a Bean, Kim or other model is used). Qualitatively
the model reproduces the ﬂux jumps we observe experimentally,
but improvements can and will be made quantitatively in the future.
5.2. Yoke insertion effectFig. 15 shows the time dependence of the local ﬁeld on the top
surface of the bulk at each position of r = 0 mm (center), 7 mm

Fig. 13. Time dependence of the local ﬁelds, By (t), on the top surface of the bulk at positions of r = 0 mm (center), 7 mm (middle) and 14 mm (edge) from the bulk center,
after applying Bex of (a) 4.00 T, (b) 4.30 T, (c) 4.50 T and (d) 5.00 T using the split coil without any yoke at 40 K. Flux jumps assist the magnetic ﬂux intrusion towards thecenter of the bulk with increasing Bex .

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K. Takahashi et al. / Physica C: Superconductivity and its applications 536 (2017) 1–10

Fig. 14. Cross-section of the trapped ﬁeld proﬁle in the bulk, when ﬂux jumps occurred at a time of t = 8, 10, 13 and 14 ms, after applying Bex = 5.0 T at 40 K using the split
coil without any yoke (Right panels). The enlarged ﬁgure of Fig. 13(d) is shown in the left panel again.

Fig. 15. Time dependence of the local ﬁeld on the top surface of the bulk at each position r = 0 mm (center), 7 mm (middle) and 14 mm (edge) from the bulk center, after
applying Bex = 3.50 T at 65 K (upper panels) using the split coil (a1) without any yoke and with the three types of yoke, (b1) soft iron, (c1) tapered soft iron and (d1)permendur, and after applying Bex = 5.50 T at 40 K (lower panels), using the split coil (a2) without any yoke and with the three types of yoke, (b2) soft iron, (c2) tapered
soft iron and (d2) permendur.

(middle) and 14 mm (edge) from the bulk center, after applying
the same applied ﬁeld, Bex = 3.50 T at 65 K (upper panels) andBex = 5.50 T at 40 K (lower panels), using the split coil (a) without
any yoke and with the three types of yoke: (b) soft iron, (c) softiron with a taper and (d) permendur. The magnetic ﬂux suddenly
intruded into the bulk, where a ﬂux jump occurs in the bulk in theascending stage at around 10 ms in all cases (a1)–(d1) at 65 K and(a2)–(d2) at 40 K. For the cases without any yoke shown in (a1)and (a2), the magnetic ﬂux decreased with increasing time in thedescending stage. However, for the cases with a yoke (b1)–(d1) or(b2)–(d2), the magnetic ﬂux still remained trapped with minimal
ﬂux ﬂow, even after the whole process of magnetization, in which
the ferromagnetic yoke acts to reduce the ﬂux ﬂow. This behavior of the magnetic ﬂux qualitatively agrees with the experimental
results shown in Fig. 6.

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9

descending stage, and σ acts to suppress RF and reduces the heatgenerated in the bulk, because of eddy currents ﬂowing in the yokethat opposes the magnetization. Hence, a yoke with higher Bsat andσ is desirable to improve the ﬁnal trapped ﬁeld by PFM from bothelectromagnetic and thermal points of view.
6. Summary
We have investigated both experimentally and numerically thetrapped ﬁeld characteristics of a standard Y–Ba–Cu–O bulk magnetized by PFM using a split coil with three kinds of iron yokes
inserted in the bores of coil. The important results and conclusions
obtained from this study are summarized as follows.

Fig. 16. Time dependence of the ﬂux increase rate, RF , at the center of the bulk
surface after applying Bex = 5.5 T at 40 K, for all cases with and without the three
types of yoke. RF was calculated from dBy (t)/dt in Figs. 15(a2) – (d2).

Fig. 17. Maximum temperature rise, ࢞T, extracted from the edge of the bulk, as afunction of the applied ﬁeld, Bex , for the split coil at 65 K and 40 K.

Fig. 16 shows the time dependence of the ﬂux increase rate,
RF , after applying the pulsed ﬁeld, Bex = 5.5 T at 40 K. These ﬁgures, which were calculated from Figs. 15(a2) – (d2), show similarplots to those shown in Fig. 7. The maximum RF value for any casewith a yoke was fairly smaller than without any yoke because themagnetic ﬂux gradually intruded in several stages with small ﬂux
jumps, showing that the yoke acted to suppress the dynamical motion of the magnetic ﬂux. This positive effect of the yoke resulted
from its electrical conductivity, σ , which acted to delay the ﬂux
movement in the bulk by induced eddy currents, ﬂowing within
the yoke.
Fig. 17 shows the maximum temperature rise, ࢞T, estimated on
the edge of the bulk, as a function of the applied ﬁeld, Bex , for the
split coil at 65 K and 40 K. The magnitude of ࢞T was decreasedby the insertion of iron yoke, which was the same manner of themagnitude of RF shown in Fig. 17. The reduction of ࢞T was relatively small, compared to the experimental results in Fig. 8, because the small B0 value of 0.04 in a Kim model, which seriously
suppresses the Jc value under the magnetic ﬁeld, reduces the temperature rise, even though the α value is higher than the presentbulk in Eq. (1).
The effect of the yoke insertion on the PFM results can be related to the electrical properties of the bulk superconductor, andthe saturation magnetic ﬂux density, Bsat , and electrical conductivity, σ , of the yoke material. Bsat acts to reduce the ﬂux ﬂow in the(1) A higher saturation magnetic ﬂux density, Bsat , of the yoke material was effective to reduce ﬂux ﬂow in the descending stageof the pulsed ﬁeld. A maximum trapped ﬁeld of 2.93 T at 40 K
was achieved on the center of the bulk surface using the split
coil with the permendur yoke with a ﬂat surface. The difference in the trapped ﬁeld between the soft iron and permenduris reasonably small because the Bsat value is not signiﬁcantly
different.
(2) A higher and conical trapped ﬁeld can be achieved in the caseof soft iron with a ﬂat surface. In the case of soft iron with ataper, the trapped ﬁeld was expected to be enhanced becauseof the concentration of the magnetic ﬂux. However, the maximum trapped ﬁeld was lower than that for a ﬂat surface. The
magnetic ﬂux cannot concentrate at the center of the coil because the applied ﬁeld is higher than Bsat and thus the tapered
shape is ineffective in this respect.
(3) The results of the numerical simulation qualitatively reproduced the experimental results, such as the applied ﬁeld dependence of the trapped ﬁeld. The typical behavior of the magnetic ﬂux, such as a ﬂux jump, can be also reproduced by anassumption of Jc (B) characteristics based on a Kim model anda higher Jc value under zero magnetic ﬁeld, compared to themeasured one.
(4) The effect of the yoke insertion was also simulated using a realistic saturation magnetic ﬂux density, Bsat , and electrical conductivity, σ , of the yoke materials. The contribution of the yokeon the PFM results is more inﬂuential during the descending
stage of the pulse. Bsat of the yoke material acts to reduce ﬂux
ﬂow, such that the magnetic ﬂux still remains trapped withminimal ﬂux ﬂow, even after the whole process of magnetization. A large σ value of the yoke material also acts to reduce
the ﬂux increase rate, RF , of the ﬂux intruding the bulk becauseof eddy currents that ﬂow in the yoke that oppose the magnetization. This reduces the temperature rise, ࢞T, in the bulk,
which improves the ﬁnal trapped ﬁeld.
Acknowledgments
This research is partially supported by JSPS KAKENHI grant
number 15K04646. Dr. Mark Ainslie would like to acknowledge thesupport of a Royal Academy of Engineering Research Fellowship.
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