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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 23, NO. 3, JUNE 2013

Numerical Simulation of Trapped Field andTemperature Rise in MgB2 Bulks
Magnetized by Pulsed Field
Hiroyuki Fujishiro, Tomoyuki Naito, Mitsuru Oyama, Takahiro Arayashiki, Takuya Tamura,
Tomohisa Sasaki, and Giovanni Giunchi
Abstract—Pulsed field magnetization (PFM) was performed for
the MgB2 bulk 30 mm in diameter and the applied pulsed-fieldC, temperature rise ΔT and
dependence of the trapped field BT
Ctime dependence of the local field BL(t) were measured on theCbulk. BT = 0.71 T was achieved at 14 K after applying thepulsed field of Bex = 1.55 T, where ΔT of about 10 K was
observed. Numerical simulation of PFM was also performed andCC, ΔT and BL(t) were simulated. The results of the simulationBT
reproduced the experimental ones qualitatively. The flux dynamics
and the heat generation in the MgB2 bulk during PFM were inclear contrast with those in REBaCuO bulks because of a small
specific heat, large thermal conductivity, and narrow temperature
margin against the transition temperature Tc .
Index Terms—MgB2 bulk, numerical simulation, pulsed-fieldmagnetization, trapped field.

I. I NTRODUCTIONT

HE superconducting bulk magnet using a REBaCuO (RE:rare earth element or Y) is one of the typical models
for practical applications such as sputtering cathodes, magnetic separation and drug delivery systems (DDSs) [1], which
produces tesla-order quasi-permanent magnets. However, it isdifficult to fabricate a large single-domain bulk over 100 mm indiameter. MgB2 bulk has also a promising potential as a quasipermanent magnet, which has several attractive natures forbulk magnet, such as low-cost, light-weight and homogeneous
trapped field distribution, which are in clear contrast with the
REBaCuO bulk magnet. The problem of weak-links at grainboundaries can be ignored in the MgB2 polycrystalline bulk
due to their long coherence length, ξ [2]. These characteristics
enable us to realize the better and larger polycrystalline MgB2
bulk magnets below the transition temperature Tc = 39 K. Several groups have already reported the trapped field on the MgB2
Manuscript received October 3, 2012; accepted December 27, 2012. Date of
publication January 4, 2013; date of current version April 10, 2013. This work
supported in part by a Grant-in-Aid for Scientific Research (No. 23560002)
from the Ministry of Education, Culture, Sports, Science and Technology,
Japan, Japan Science and Technology Agency under a Research for Promoting
Technological Seeds 2009 (02-051) and an Adaptable and Seamless Technology Transfer Program through target-driven R&D for an Exploratory Researchof FS stage (AS232Z02579B).
H. Fujishiro, T. Naito, M. Oyama, T. Arayashiki, T. Tamura, and T. Sasaki
are with the Faculty of Engineering, Iwate University, Morioka 020-8551, Japan(e-mail: fujishiro@iwate-u.ac.jp).
G. Giunchi is with EDISON S.p.A., R&D Division, 20121 Milano, Italy(e-mail: giovanni.giunchi@edison.it).
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.2012.2237224

bulk by the field-cooled magnetization (FCM), and attained thetrapped field over 1.5 T at low temperatures [3]–[5]. MgB2
magnets are expected to be applicable in magnetically levitatedtrains (MAGLEVs) and wind power generators using liquid H2
or cryocooler cooling.
Pulsed-field magnetization (PFM) has also been investigatedto magnetize the bulk superconductors because of an inexpensive and mobile experimental setup with no need for a superconducting coil magnet. However, for the REBaCuO bulks,
the trapped field Bz achievable by PFM is nonetheless lower
than that achievable by FCM because of the large temperaturerise caused by the dynamical motion of the magnetic flux.
Several approaches have been performed and succeeded inenhancing Bz using the multi-pulse techniques [6], [7]. We haveexperimentally examined the trapped field Bz on the surface of
cryocooled REBaCuO bulks during PFM for various starting
temperatures Ts and applied fields Bex [8]. To enhance Bz ,the reduction in temperature rise and the lowering of Ts are
effective. Considering the obtained experimental results, we
proposed a new PFM technique named a modified multi-pulse
technique with stepwise cooling (MMPSC) and successfully
realized a highest field trap of Bz = 5.20 T on a GdBaCuO
bulk with 45 mm in diameter at 30 K [9], which is a recordhigh value magnetized by PFM to date.
In the previous paper, we have performed the PFM to a large
MgB2 bulk 50 mm in diameter fabricated by a reactive liquid
Mg infiltration (Mg-RLI) method [10], in which the trappedfield of Bz = 0.47 T was achieved at Ts = 23 K. However,
the trapped field profiles were inhomogeneous because of theinhomogeneous Jc distribution in the bulk. It is necessary
to investigate the flux dynamics and heat generation using ahomogeneous MgB2 bulk experimentally and numerically.
In this study, we applied the PFM technique to the homogeneous MgB2 bulk fabricated by a capsule method [11] andanalyzed the PFM process using the numerical simulation. In
the PFM for the MgB2 bulk, flux dynamics and heat generation
are supposed to be a clear contrast with those in the REBaCuO
bulk because of the small specific heat, large thermal conductivity, and narrow temperature margin against Tc . Using thesimulation, a characteristic aspect in MgB2 during PFM wasdiscussed.
II. E XPERIMENTAL P ROCEDURE
MgB2 bulk was fabricated by the in-situ capsule method[11]. Raw powders of Mg (99% in purity) and amorphous B

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FUJISHIRO et al.: TRAPPED FIELD AND TEMPERATURE RISE IN MgB2 BULK MAGNETIZED BY PULSED FIELD

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magnetizing solenoid coil were the same as those of the experimental condition. Physical phenomena during PFM were
described using electromagnetic and thermal fields, which weretaken from [13]. The power-n model (10 ≤ n ≤ 100) was
supposed to describe the nonlinear E − J characteristic in thebulk. The temperature and magnetic field dependences of thecritical current density Jc (T, B) were described as,    32
2
T
B0
B0
Jc (T, B) = Jc0= α 1−(1)|B| + B0
Tc
|B|+B0C at the center of theFig. 1. (a) Experimental results of the trapped field BT
MgB2 bulk surface as a function of applied pulsed field Bex at Ts = 14 K.
Inset shows the experimental setup. (b) Time dependences of the temperature
T (t) on the bulk surface for each applied field Bex(99% in purity) were weighted with 1.1: 2.0 in molar ratio,
ground and was pressed into pellet 30 mm in diameter and
9 mm in thickness under uniaxial pressure of 12 MPa in air. The
precursor pellet sealed in the stainless steel (SUS304) capsuleby Ar gas was sintered at 800 ◦ C for 6 h in a box furnace andthen cooled to room temperature.
The MgB2 bulk was tightly anchored onto the cold stage ofa Gifford-McMahon (GM) cycle helium refrigerator as shownin the inset of Fig. 1(a). The initial temperature Ts of the bulk
was 14 K. The magnetizing solenoid coil (99 mm I.D., 121 mm
O.D., and 50 mm height), which was dipped in liquid nitrogen,was placed outside the vacuum chamber. A magnetic pulse Bex
up to 1.85 T with a rise time of 0.01 s was applied by flowing thepulsed current to the coil. The time evolutions of the local fieldC
at the bulk center
BLC (t) and the subsequent trapped field BT
were monitored by the Hall sensor (BHT 921; F W Bell), which
was adhered at the center of the bulk surface. Two-dimensional
trapped field profiles of Bz (1 mm) were mapped on the bulk
with 1 mm above the bulk surface, stepwise with a pitchof 1 mm by scanning the axial-type Hall sensor (BHA 921;
F W Bell) using an x − y stage controller. During PFM, thetime dependence of temperature T (t) was also measured at thebulk surface of using the thermometers (Cernox; Lakeshore).
III. M ODEL OF N UMERICAL S IMULATION FOR PFM
Based on the experimental setup, the framework of thenumerical simulation was constructed using the axi-symmetriccoordinate. The detailed procedure of the simulation was
described elsewhere [12]. The sizes of the bulk and thewhere Tc (= 39 K) is the critical temperature at B = 0, andB0 (= 1.3 T) is constant. In (1), α shows the strength of Jc0
and was changed from 2.0 × 108 and 1.0 × 109 A/m2 , which
represents Jc0 = 1.6 × 108 and 8.1 × 108 A/m2 at 14 K, respectively. Although the thermal conductivity κ and the specificheat C values of the MgB2 bulk were steeply temperaturedependent [14], [15], constant values of κ(= 80 W m−1 K−1 )and C(= 17.2 J kg−1 K−1 ) were adopted for simplicity. κ(=
3.0 W m−1 K−1 ) and C(= 30 J kg−1 K−1 ) of the SUS304 capsule, and the mass density ρ of MgB2 (1.4 kg m−3 ) and SUS304(7.93 kg m−3 ) were also supposed to be constant. Commercial
software, Photo-Eddy, combined with Photo-Thermo (PhotonLtd.), was used for analyses of the magnetic field and temperature distribution in the MgB2 bulk during PFM.
IV. E XPERIMENTAL R ESULTS
C
Fig. 1(a) shows the trapped field BT
at the center of the bulk
surface, as a function of the strength of the applied pulsed fieldC
C
increases for Bex > 1 T, takes a maximum of BT=
Bex . BT
0.71 T at Bex = 1.54 T and then decreases with increasing Bex .
Cvs Bex curve is a typical one for PFM in REBCO bulks
The BT[8]. Fig. 1(b) shows the time dependence of the temperaturechange on the bulk surface for each applied field Bex . The
temperature increases abruptly just after the pulse applicationand sharply decreases with increasing time. The magnitude of
the maximum temperature rise increases with increasing Bex .
The sharp temperature change results from the small specificheat and high thermal conductivity of the bulk, compared withthose for the REBaCuO bulk at operating temperatures.
Fig. 2 presents the trapped field profiles Bz (1 mm) for
various applied pulsed fields Bex . For Bex = 1.04 T as shownin Fig. 2(a), the magnetic flux was trapped at the right lower
region of the bulk, where the critical current density Jc seems
to be relatively lower. For Bex = 1.23 T as shown in Fig. 2(b),the magnetic flux was also trapped at the left upper region of thebulk. For Bex = 1.42 T and 1.54 T as shown in Fig. 2(c) and(d), the trapped field profiles are nearly the conical one whichlook like nearly homogeneous, compared with those for the
REBaCuO bulks [8]. Because the MgB2 bulk was polycrystal
and was fabricated by the in-situ sintering method. On the other
hand, the REBaCuO bulk was fabricated by the melt-growth,in which the four-fold growth sector boundaries (GSBs) withhigher Jc exist.
Fig. 3 shows an example of the time dependence of the local
field BLC (t) and pulsed field Bex (t) at Bex = 1.54 T. BLC (t)starts to increase with a slight time delay, takes a maximum

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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 23, NO. 3, JUNE 2013

Fig. 2. Trapped field profiles Bz on the MgB2 bulk 1 mm above the bulk
surface at Ts = 14 K at (a) Bex = 1.04 T, (b) 1.23 T, (c) 1.42 T, and (d) 1.54 T.

C for various J
Fig. 4. Results of the simulation of the (a) trapped field BT
c0
C for various n-values, as a function of the applied fieldvalues and (b) BT
Bex The experimental results were reproduced by the simulation using theparameters of Jc0 = 4.1 × 108 A/m2 and n = 50.

C (t) and the applied pulsed field
Fig. 3. Time dependences of the local field BL
Bex (t) for Bex = 1.54 T.

at 0.015 s and then decreases to a final value due to the flux
flow. The maximum BLC (t) increased with increasing Bex andis smaller than that of Bex (t) because of the shielding effect.
FCM was also performed for the same bulk. The highestCvalue was 1.50 T at 16 K and decreased with increasingBT
FCM
temperature. The trapped field BT(4 mm) at Ts = 28 K was
0.32 T and the trapped field profile was conical, which suggeststhat the Jc distribution in the bulk is nearly homogeneous also
from the profile by FCM.
V. R ESULTS OF N UMERICAL S IMULATION
Fig. 4(a) depicts the results of simulation for the trapped fieldC
at the center of the bulk surface, as a function of the applied
BT
C
field Bex for various Jc0 values (n = 50). In all the cases, BT
starts to increase, takes a maximum, and then decreases with a
C, at whichfurther increase in Bex . The critical applied field Bex
the magnetic flux starts to trap at the bulk center, increases withincreasing Jc0 . Fig. 4(b) shows the results of simulation for theC
as a function of the Bex for various n-values (Jc0 = 4.1 ×
BTC− Bex curve shifts to the low Bex side with
108 A/m2 ). The BT
increasing n-value, but then the shift between n = 50 and 100
was relatively small. These results suggest that the simulation
using the conditions (Jc0 = 4.1 × 108 A/m2 and n = 50) wellFig. 5. Results of the simulation of the radial dependence of the trapped fieldBz (r) at Ts = 14 K for various Bex (Jc0 = 4.1 × 108 A/m2 and n = 50).

reproduces the experimental results. Thereafter, we discuss theflux dynamics and heat generation in the bulk during PFM using
these conditions.
Fig. 5 presents the radial dependence of the trapped field
profile Bz (r) on the bulk for various Bex . The positions of
r = 0 and 15 mm are, respectively, the center and the edge ofthe bulk surface. The Bz (r) profile changes from concave for alower Bex to convex for a higher Bex and then the Bz (r = 0)value decreases with a further increase in Bex .
C(t)Fig. 6 shows the time dependences of the local fields BT
at the center of the bulk surface for various Bex . The timedependence of the magnetic pulse Bex (t) at 1.5 T is also shown.
The local field BLC (t) starts to increase rapidly with a time lagand then decreases.

FUJISHIRO et al.: TRAPPED FIELD AND TEMPERATURE RISE IN MgB2 BULK MAGNETIZED BY PULSED FIELD

Fig. 6. Results of the simulation of the time dependence of the local field
C (t) at the center of the bulk surface at T = 14 K for various applied fields
BLs
Bex (Jc0 = 4.1 × 108 A/m2 and n = 50).

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ture rise and time dependence of the local field BLC (t) were
measured and simulated. Important results and conclusions
obtained in this study are summarized as follows.
1) The maximum trapped field of Bz = 0.71 T was realized
at the center of the bulk surface at Ts = 14 K after themagnetic pulse application of Bex = 1.55 T.
2) The experimental results can be qualitatively reproducedby numerical simulation using electromagnetic and thermal fields.
3) The flux dynamics and the heat generation of the MgB2
bulk during PFM were in clear contrast with those for
REBaCuO bulks because of a small specific heat, large
thermal conductivity, and narrow temperature margin
against Tc in the MgB2 bulk.
R EFERENCES

Fig. 7. Results of the simulation of the time dependence of the temperature
T (t) at the center of the bulk surface at Ts = 14 K for various applied fields
Bex (Jc0 = 4.1 × 108 A/m2 and n = 50).

Fig. 7 shows results of the simulation for the time dependences of temperature T (t) at the center of the bulk surface
for various Bex . T (t) starts to rise rapidly at t = 0.01 s, atwhich Bex (t) takes a maximum. T (t) takes a maximum at
t = 0.02–0.05 s, which becomes shorter with increasing Bex
and then decreases. In Fig. 6, a steep decrease in BLC (t) was
seen for Bex = 1.8 T, which results from the approach of thetemperature to Tc = 39 K because of the large temperature rise.
For higher Bex , the temperature at the bulk periphery exceeded
Tc and most of the trapped flux escaped from the bulk.
The results of the simulation for the MgB2 bulk were inclear contrast with those for REBaCuO bulks [8]; the trappedC
was a broad applied field dependence and T (t) tookfield BTa maximum at 7 s. The differences come from the small specific heat, large thermal conductivity, and narrow temperature
margin against the transition temperature Tc in the MgB2 bulk.
The use of numerical simulation can optimize the operating
parameters for PFM to enhance the trapped field [12].
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