IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 29, NO. 3, APRIL 2019

8000705

Influence of Jc(B, T) Characteristics on the Pulsed
Field Magnetization of REBaCuO Disk Bulks
Tatsuya Hirano , Keita Takahashi , Fumiya Shimoyashiki, Hiroyuki Fujishiro, Tomoyuki Naito ,and Mark D. Ainslie , Senior Member, IEEE

Abstract—The trapped field properties during pulsed-fieldmagnetization (PFM) have been investigated numerically using
three different assumptions relating to the Jc (B, T) characteristics(Jirsa, Kim, and Bean models) and compared with experimental
results. The trapped field properties using the Jirsa model withthe so-called “peak effect,” in which a realistic Jc (B, T) is assumed,
rather than the Kim model, result in a more realistic numericalsimulation. The trapped field properties using a Kim model witha monotonically decreasing Jc (B) also show similar results to
those using the Jirsa model. The trapped field properties using a
Bean model, for which Jc is independent of magnetic field, are notnecessarily enhanced because of a larger temperature rise. The numerical results suggest it is necessary to fabricate REBaCuO bulkswith Jc (B, T) characteristics with moderate magnetic field andtemperature dependences to enhance the trapped field by PFM.
Index Terms—Jc (B, T) characteristics (Jirsa model, Kim model,
Bean model), numerical simulation, pulsed-field magnetization(PFM), REBaCuO bulk.

I. INTRODUCTION
EBaCuO (RE: rare earth element or Y) superconducting
bulks have a significant potential as strong trapped fieldmagnets (TFMs), which can be used in a variety of engineering
applications. Pulsed-field magnetization (PFM) is a practicalmagnetizing technique to realize TFMs without the need for a
superconducting magnet, in contrast to field-cooled magnetization (FCM), because of its relatively compact, inexpensive,and mobile experimental setup. However, the trapped field by
PFM is much lower than that by FCM because of a large temperature rise due to the rapid and dynamic motion of magnetic
flux. Several improvements have been made experimentally andR

Manuscript received January 14, 2018; revised August 25, 2018; accepted
September 16, 2018. Date of publication October 1, 2018; date of current
version January 8, 2019. This work was supported in part by JSPS KAKENHI
under Grant 15K04646. The work of M. D. Ainslie was support by Engineeringand Physical Sciences Research Council (EPSRC) Early Career Fellowship
EP/P020313/1. This paper was recommended by Associate Editor M. Izumi.
(Corresponding author: Tatsuya Hirano.)
T. Hirano, K. Takahashi, F. Shimoyashiki, H. Fujishiro, and T. Naito arewith the Department of Physical Science and Materials Engineering, Faculty of
Science and Engineering, Iwate University, Morioka 020-8551, Japan (e-mail:,g0318132@iwate-u.ac.jp; fujishiro@iwate-u.ac.jp).
M. D. Ainslie is with the Bulk Superconductivity Group, Department ofEngineering, University of Cambridge, Cambridge CB2 1PZ, U.K. (e-mail:,mark.ainslie@eng.cam.ac.uk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TASC.2018.2872819

numerically for the PFM technique to enhance the trapped field;
the insertion of soft iron yoke from a magnetic point of view[1], and the multipulse application from a thermal point of view[2], [3]. The trapped field of TFMs by PFM is determined by acomplex relationship between Jc (B, T), the thermal properties(specific heat and thermal conductivity) of the bulk material,cooling condition (the thermal contact and magnetizing temperature), the rise time of the magnetic pulse, etc. In general, theresults from numerical simulation can provide insights that are
difficult to realize experimentally, and can also verify experimental results. Therefore, it is a powerful tool to analyze such
behaviors and predict the performance of bulk superconductors
as TFMs. [4].
Several numerical analyses have been performed to understand the magnetizing mechanism and to enhance the trappedfield by PFM [5]. There are several Jc (B) characteristic models
used in the literature. In the Jirsa model, experimental Jc (B)characteristics that exhibit a peak effect are fitted at each temperature [6]. In the classical Bean model, the Jc value is independent of magnetic field at each temperature [7]. In the Kim
model, the Jc (B) characteristics monotonically decrease withincreasing magnetic field at each temperature [8]. Until now,
numerical simulations during PFM using three different Jc (B,T) characteristics have not been investigated using an identicalnumerical model. Furthermore, such simulations have not been
compared with the experimental results.
In this paper, to understand the complex trapped field mechanism and to clarify the desirable Jc (B, T) characteristics of the
REBaCuO bulk, we performed numerical simulations of PFM
for a REBaCuO disk bulk using three different assumptions of
the Jc (B, T) characteristics: the Jirsa model [6], the Bean model[7], and the Kim model [8]. These numerical results are compared with experimental results. The most desirable approachto enhance the trapped field, as well as the most appropriate
assumptions for the simulation closely to reproduce the experimentally observed results are discussed from the viewpoints ofthe magnetic and thermal behavior during PFM.
II. EXPERIMENTAL SETUP AND NUMERICALSIMULATION FRAMEWORK
A. Experimental Setup
A GdBaCuO disk bulk superconductor (Nippon Steel & Sumitomo Metal) of 64 mm in outer diameter (O.D.) and 20 mm in

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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 29, NO. 3, APRIL 2019

Fig. 1. Schematic view and the dimensions of the experimental setup for the
PFM experiments using a magnetizing solenoid coil.

height (H), mounted in a stainless steel (SUS316L) ring 5 mm
in width, was attached to the cold stage of a Gifford-McMahon(GM) cycle helium refrigerator. A copper solenoid magnetizing
coil (inner diameter (I.D.) = 100 mm, O.D. = 120 mm, H =50 mm), which was cooled using liquid nitrogen, was placedoutside the vacuum chamber, as shown in Fig. 1. The detailed
experimental setup is described elsewhere [9]. After the bulk
was cooled to Ts = 65 K, a single magnetic pulse, Bex , ranging
from 3 to 6 T and with a rise time of 13 ms, was applied tothe bulk. During PFM, the time dependence of the local fieldBz (t) and the final trapped field, Bt , were measured using a
Hall sensor located at the center of the top surface of the bulk.
The time dependence of the temperature, T(t), was measured
on the side surface of the SUS316L ring using a CERNOX
thermometer.

Fig. 2. Magnetic field and temperature dependences of the critical current
density, Jc (B, T), between 65 and 80 K used in the simulation for (a) the Jirsa
and the Bean models and (b) the Jirsa and the Kim models.
TABLE INUMERICAL PARAMETERS FOR THE Jc (B, T) CHARACTERISTICS USING JIRSAMODEL AT 65, 70, 75, AND 80 K IN (1)B. Numerical Simulation Framework
Based on our experimental setup shown in Fig. 1, a twodimensional (2-D) numerical model was constructed and numerical simulations were performed using the finite element method(FEM). The physical phenomena during PFM are described by
the fundamental electromagnetic and thermal equation in the
2-D axisymmetric coordinate system [10], [11]. Commercial
software package, Photo-Eddy, combined with Photo-Thermo(Photon Ltd, Japan) was adopted for the analysis. The simulation procedure and the parameters used are described elsewherein detail [12].
Fig. 2 shows the Jc (B, T) profiles used in the simulation. The
Jirsa model with the peak effect is represented by the following
equation [6]:BB
Jc (B, T ) = Jc1 (T ) exp −+ Jc2 (T )Bl (T )
Bm ax (T )α (T ) 
B
1
× exp
1−
.
α (T )
Bm ax (T )(1)

The experimental Jc (B, T) data [13] were fit up to 10 T between
65 K and 80 K using (1) and the determined parameters (Jc1 , Bl ,Jc2 , Bmax , and α) at each temperature are shown in Table I. The
Jc (B, T) profiles at intermediate magnetic field and temperature
are interpolated using each parameter.
In the Bean model, the temperature dependence of Jc3 (T) is
assumed to be the following equation, which is the same as
Jc1 (T) in the Jirsa model
Jc3 (T ) = Jc1 (T ) = aT 4 + bT 3 + cT 2 + dT + e(2)

where a, b, c, d, and e are constant values shown in Table II.
In the Kim model, temperature and magnetic field dependenceof Jc (B, T) is expressed in the following equation: 2 32T
B0(3)
Jc (B, T ) = Jc4 1 −Tc|B| + B0

HIRANO et al.: INFLUENCE OF Jc (B, T) CHARACTERISTICS ON THE PFM OF REBaCuO DISK BULKS

8000705

TABLE IINUMERICAL PARAMETERS FOR THE Jc (T) CHARACTERISTICS USING
BEAN MODEL IN (2)Fig. 4. Time evolution of the local field, Bz , for (a) Jirsa model and (b) Kim
model at the center of the bulk surface for various applied pulsed fields, Be x , inwhich the temperature changes due to the coupling of the thermal model.

Fig. 3. Applied field dependence of (a) trapped field, Bt , at the center of thebulk surface and (b) maximum temperature rise, ΔTmax , from 65 K in
the numerical simulations using the three kinds of Jc (B, T) characteristics.
The experimental results are also shown.

where B0 = 1.3 T is constant and Jc4 = 3.45 × 109 A/m2 is theextrapolated Jc value at T = 0 K and B = 0 T, which corresponds
to Jc (0 T, 65 K) = 1.2 × 109 A/m2 and is the same value as thatin the Jirsa model. The magnitude of Jc (B, T) values shown in(1)–(3) was adjusted to one third of its small specimen value toadequately reproduce the experimental results [1].
The anisotropic thermal conductivities κab = 20 Wm−1 K−1
in the ab-plane and κc = 4 Wm−1 K−1 along the c-axis of the
REBaCuO bulk were assumed to be independent of temperaturefor simplicity [1]. The temperature dependent thermal conductivity, κSUS , and specific heat, CSUS , of the SUS316L ring wereused [1]. The bulk was cooled to Ts = 65 K and the pulsed
field, Bex (t) with a rise time of 10 ms was applied. Using theframework, we investigated the trapped field characteristics numerically for these three Jc assumptions.
III. RESULTS AND DISCUSSION
Fig. 3(a) shows the numerical and experimental results ofthe trapped field, Bt, at 65 K at the center of the bulk surface,

as a function of the applied pulsed field, Bex . The experimental
results of the Bt versus Bex profile were qualitatively reproducedby the numerical simulation using the Jirsa model, better thanwhen using Bean and Kim models, which suggests that the Jirsa
model should be used in the simulation. When the parameters in
the Jirsa model are optimized more accurately, the discrepancy
may be minimized. Fig. 3(b) shows the maximum temperaturerise, ΔTmax from 65 K during PFM, as a function of Bex , which
was estimated at the same position as that in the experimentshown in Fig. 1. The temperature rise in the simulation increased
with the increase in the applied pulsed field, Bex . The ΔTmax
value of the experiment was larger than that of the simulation.
When using the Bean model, the trapped field by FCM is likely
to increase, because Jc is not reduced by the presence of themagnetic field. On the other hand, during the PFM process of a
Bean model, a larger applied field is necessary for the magnetic
flux intrusion into the bulk because of the independence of B in
Jc . As a result, a larger temperature rise happens after the flux
intrusion and then the trapped field is reduced, which is in clear
contrast to the FCM process.
It is interesting to consider what kind of Jc (B, T) profile isdesired to enhance the trapped field by PFM. For the results of
the Kim model, as shown in Fig. 3, the activation field, Bex ∗ ,which was defined as the magnetic field required to fully magnetize the bulk [9], becomes lower, compared to the Jirsa model.
The trapped field of the Kim model is slightly larger because of
the moderate Jc degradation with increasing temperature and/or
magnetic field. In practical applications, lowering the Bex ∗ value
is preferable because the size of the capacitor bank, and ultimately the magnetization fixture, can be reduced. To enhance
the trapped field by PFM, a weak temperature dependence of
Jc (T) is preferable, rather than the existence of the peak effectin Jc (B).
Fig. 4(a) and (b) shows the time evolution of the local field,
Bz (t), at the center of the bulk surface for the Jirsa model andthe Kim model, respectively, for various applied fields, Bex , inwhich the temperature variation was permitted (with thermal
model). For the Kim model, the magnetic flux is easy to penetrate the bulk center even for lower Bex , compared to the Jirsa

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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 29, NO. 3, APRIL 2019

Fig. 5. Time evolution of the local field, Bz , for (a) Jirsa model and (b) Kim
model at the center of the bulk surface for various applied pulsed fields, Be x ,in which the temperature is fixed at 65 K assuming isothermal conditions (nothermal model is included).

Fig. 6. Trapped field, Bt , for the Jirsa and Kim models combined with (w/)
and without (w/o) thermal model, as a function of applied pulsed field, Be x .

model because of the absence of the peak effect which enhances
the pinning strength at intermediate and higher applied field. For
higher Bex , the time dependence of Bz (t) and the final trappedfield, Bt , are nearly the same for both models, which may result from the complex relationship between Jc (B, T) and heatgeneration during PFM.
Fig. 5(a) and (b) shows the time evolution of the local field,
Bz (t), at the center of the bulk surface for the Jirsa model and theKim model, respectively, for various applied fields, Bex , in casethat the temperature is fixed at 65 K, i.e., isothermal conditions
are assumed. The magnetic flux is difficult to penetrate into thebulk center for the lower applied pulsed field because of theabsence of temperature rise. As a result, for higher Bex than
7.5 T, the final trapped field for the Jirsa model is higher thanthat for the Kim model due to the peak effect.
Fig. 6 shows the numerical results for the trapped field, Bt ,at 65 K, as a function of applied pulsed field, Bex , for the Jirsa
model and Kim model, in which the results with (w/) and without(w/o) thermal model are shown. When the temperature is fixed

Fig. 7. Experimental results of the time evolution of the local field, Bz (t), at
the center of the bulk surface for various applied pulsed fields, Be x .

at 65 K (w/o) for each model, the activation field, Bex ∗ , shifts tohigh magnetic field and the Bt value increases with increasing
the applied pulsed field, Bex . The Bt value for the Jirsa modelis larger than that for the Kim model due to the peak effectin Jc (B). The Bt value without thermal model is larger thanthat with thermal model. These results suggest that the presenceof the peak in Jc (B) enhances the final trapped field, if thetemperature rise is reduced.
Fig. 7 shows the experimental results of the time evolution of
the local field, Bz (t), at the center of the bulk surface for various
applied pulsed fields, Bex . Because of the large temperature riseduring the actual PFM, the local field, Bz (t), changes according
to the thermal model, as shown in Fig. 4. The Bt value increases
with increasing Bex . The flux flow at t > 20 ms increases withincreasing Bex due to the temperature rise. Compared to Fig. 4,
the experimental results showed similar trends to the numericalresults for higher magnetic fields (Bex ࣛ 5 T), while the experimental results for lower magnetic fields (Bex < 5 T) differedfrom the numerical results. However, these results suggest thatthe relationship between the Bt and Bex can be reproduced bynumerical simulation.
IV. CONCLUSION
The trapped field properties during PFM of a REBaCuO
disk bulk were investigated numerically using three different
Jc (B, T) characteristics (Jirsa, Kim, and Bean models) and compared with the experimental results. The bulk with the fieldindependent Jc characteristics like the Bean model, does notachieve higher trapped field by PFM because of a larger temperature rise. In the Jirsa model, the peak effect in Jc (B) is effective
to enhance the trapped field, but a steep reduction in Jc withtemperature rise can result in a decrease in the trapped field.
∗, is lower than thatIn the Kim model, the activation field, Bex
for the Jirsa model due to the absence of peak effect and thetrapped field is slightly larger than or similar to that for the
Jirsa model because of the weak temperature dependence of
Jc . Therefore, to enhance the trapped field during PFM, Jc (B,T) characteristics with the peak effect and a weak temperature

HIRANO et al.: INFLUENCE OF Jc (B, T) CHARACTERISTICS ON THE PFM OF REBaCuO DISK BULKS

dependence are required. If a REBaCuO bulk with moderatemagnetic field and temperature dependences in Jc (B, T) can be
fabricated, the trapped field should be enhanced by PFM basedon these analyses.
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Authors’ biographies not available at the time of publication.