IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011

2723

Analysis of Temperature and Magnetic Field
Distribution in Superconducting Bulk During
Pulsed Field Magnetization
Hiroyuki Fujishiro, Tomoyuki Naito, and Daiki Furuta

Abstract—We have constructed the framework of theoreticalsimulation for magnetic field and temperature in the superconducting bulk during pulsed field magnetization (PFM) using finite
element method (FEM). Three successive pulsed field applications
with identical strength (SPA) were performed to the cryo-cooled( ()bulk and time and spatial dependence of local field) were calculated. The trapped fieldand temperature ( (increased and the maximum temperature risedecreasedwith increasing pulse number. The
enhancement results fromthe decrease in temperature rise because of the already trapped
flux. The results of the simulation well reproduced the experimental ones qualitatively.
Index Terms—Pulsed field magnetization, simulation, superconducting bulk.
Fig. 1. Model and dimensions around the bulk and solenoid copper coil for
pulsed field magnetization (PFM).

I. INTRODUCTIONPULSED field magnetization (PFM) for REBaCuO bulks(RE: rare earth element or Y) has been intensively investigated for practical applications as a substitute for field-cooled
by PFM is, howmagnetization (FCM). The trapped field
ever, lower than that by FCM because of the large temperaturerise by the dynamical motion of the magnetic flux. To enhance, it has been believed that the reduction in temperature rise
and the lowering of starting temperatureare effective
because of the enhancement of the critical current density ,similarly for FCM. A multi-pulse application such as an iteratively magnetizing pulsed-field method with reducing amplitude(IMRA) [1] and a multi-pulse technique with step-wise cooling. We have proposed a(MPSC) [2] is effective to enhance
new PFM technique named a modified MPSC (MMPSC) andon
successfully realized the highest field trap ofmm GdBaCuO bulk at 30 K [3], which is a record-highthevalue by PFM to date. To understand the PFM phenomena in devalue, it is necessary to use the aid oftail and to enhance thea theoretical simulation because the experimental research has
been limited.
Several theoretical studies for PFM have been reported [4],[5]. We have also constructed the framework of theoretical simulation in the superconducting bulk with homogeneous superconducting properties after a single magnetic pulse application
and reproduced the experimental results [6]. As described in thisManuscript received August 01, 2010; accepted October 13, 2010. Date ofpublication November 29, 2010; date of current version May 27, 2011.
The authors are with the Faculty of Engineering, Iwate University, Morioka020-8551, Japan (e-mail: fujishiro@iwate-u.ac.jp).
Digital Object Identifier 10.1109/TASC.2010.2089419

paper, we applied the theoretical simulation to the successive
pulsed field application with identical strength (SPA) for the superconducting bulk, and time and spatial dependence of local
and temperature
were calculated.
fieldII. MODELING AND THEORETICAL SIMULATION
Based on our experimental setup around the bulk, the framework of the theoretical simulation was constructed. As shownin Fig. 1, a superconducting bulk disk 46 mm in diameter and15 mm in thickness was stacked on the cold stage of the refrigerator in a vacuum chamber and cooled to the starting temper. In the simulation, the bulk was magnetizedatureusing a solenoid-type copper coil. Physical phenomena during
PFM are described by electromagnetic and thermal equations
on the axisymmetric coordinate in the following, which werereferred from ref. [7] using a commercial software PHOTOEDDY-THERMO (PHOTON Ltd., Japan).
(1)(2)
where is the magnetic vector potential, the magnetic reluctivity, the mass density,
the coil current density, thesuperconducting current density or induced current density, andis the heat generation. In (2),is theandare thespecific heat,
thermal conductivity of the bulk in the -plane and along the-axis, respectively [8].

1051-8223/$26.00 © 2010 IEEE

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TheIEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011

value is given by the following equation:(3)

where is the electric field and the electric conductivity. TheFEM analysis of the coupled problem with electromagnetic field
and heat diffusion was performed.
was used to describe the nonThe power- modelcharacteristic in superconducting bulk [7].
linear(4)
where is the critical current density andis the reference electric field. The magnetic field dependence of
was described by the Kim model:
the(5)
is forand
wheretemperature dependence ofequation,is constant [7]. The
is described by the following

(6)
where
is the critical temperatureand is thestrength of.
was used in the analat 40 K. Iteraysis which represents
tive calculation was performed to obtain the convergence ofin the superconductor at each time step. The applied pulsed fieldwith the rise time of
was approximated in thefollowing equation;

Fig. 2. Experimental results of pulse number dependence of the (a) trappedat the center of the Smfield B and (b) maximum temperature rise TBaCuO bulk. (c) Time dependence of the local field B t and the temperature: .
T t at r for the SPA method of B

()

=01
()
= 4 64 T(7)
Three magnetic pulses from No. 1 to No. 3 with the identicalstrength of
to 8 T were sequentially applied.
III. EXPERIMENTAL RESULTS
We magnetized the SmBaCuO bulk by the SPA method at
40 K experimentally [9]. Fig. 2(a) and (b) present the pulseand the maximum
number dependence of the trapped fieldat the center of the bulk surface. Thetemperature risetime interval between each pulse was 20 min., for which the bulk
was re-cooled to . The largest amount of the flux was trapped
wasduring the No. 1 pulse to the virgin state bulk andgradually enhanced with increasing pulse number. In Fig. 2(b),monotonically decreases and approaches a steady value
with increasing pulse number for each. Fig. 2(c) shows theand the temperaturetime dependence of the local fieldat the center of the bulk surface for the SPA method of. The applied pulsed field
is also shown.
for the No. 1 pulse becomes maximum atand then decreases with increasing time and approaches to the.
for the No. 1 pulse takes a
final value ofand then decreases concomitantly withmaximum atincreasing time.
decreases with increasing pulse number.

Fig. 3. (a) Results of simulation for the pulse number dependence of theat the center of the bulk surface for several applied
trapped field B rfields B . (b) Cross sections of the trapped field B r profiles of the SPA.
for B( = 0)
= 4 and 6 T()

takes a maximum,
It is noteworthy that the time, at which
is about three orders of magnitude faster than the time, at whichtakes a maximum.
anddescribed in Fig. 2 arecommon features for the PFM procedure.
IV. RESULTS OF SIMULATION AND DISCUSSION
Fig. 3(a) depicts the results of the simulation for the pulse
number dependence of the trapped fieldat the center. For the No.
of the bulk surface as a function of applied fieldstarts to increase for, becomes
1 pulse,maximum for 6 T and then decreases with further increase in. The
increases for the No. 2 pulse and tends

FUJISHIRO et al.: ANALYSIS OF TEMPERATURE AND MAGNETIC FIELD DISTRIBUTION IN BULK

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Fig. 4. Time evolution and spatial distribution of the (a) local magnetic field B (t; r) and (b) temperature T (t; r ) on the bulk surface after applying No. 1 pulsed
field of B = 6 T.

Fig. 5. Time evolution and spatial distribution of the (a) local magnetic field B (t; r ) and (b) temperature T (t; r ) on the bulk surface after applying No. 2 pulsed
field of B = 6 T.

to saturate for the No. 3 pulse for each. Fig. 3(b) showsof the SPA
the cross section of the trapped field profile. The positions offor
are the center and edge on the bulk surface, respectively. For, The
profile was concave and the trappedfield slightly increased with increasing pulse number. On the, theprofile was convex forother hand, for
the No.1 pulse andvalue increased fairly for the No.
2 pulse.
Fig. 4(a) and (b), respectively, present the time evolution andand thespatial distribution of the local magnetic fieldtemperatureon the bulk surface after applying a No. 1. The left and right parts for each figure
pulsed field ofand descending
are for the ascending
stages, respectively. For the ascending stage, the magnetic flux
intrudes gradually into the bulk from the bulk periphery andmagnetic gradient increases with approach to the bulk center. For the descending stage, the magnetic field deatcreased gradually at the outer region with increasing time. On
the other hand, local field near the bulk center increased to 3 T
and then decreased gradually toatatat the bulk
the steady state. In Fig. 4(b), the temperature
periphery gradually increased with time at the ascending stage,at. However, the temperature forremained as 40 K. At the descending stage, the generated
heat diffuses to the bulk center gradually. After an isothermal,
temperature profile took place along the -direction at
the temperature decreased with increasing time. In this case,temperature gradient existed along the -direction because thebottom of the bulk was fixed at 40 K.
andon the bulk surFig. 5(a) and (b) depict. For the asface after applying No. 2 pulsed field ofcending stage, the magnetic flux intrudes gradually into the bulk
from the bulk periphery. For the descending stage, the magnetic
field decreased gradually at the outer region with increasing
time. The local field near the bulk center increased to 2.7 T at
and then decreased gradually toat.
behavior is similar time and radius deIn Fig. 5(b), thependences to that for the No. 1 pulse, but the temperature rise isat.
fairly reduced;
Let us summarize the results of the simulation.
Fig. 6(a) and (b) show the pulse number dependence of theand the maximum temperature risetrapped fieldon the bulk center.
increased concomitantly withincreasing pulse number and then saturated and increased with. The
value formonotonicallyincreasing

e.g.,

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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011

and the cooling power and is independent
crystaldecreases with increasing pulseof the pulse number.
peak and thenumber. The time delay between thepeak is of about three orders of magnitude, which is nearly theconsistent of the experimental results as presented in Fig. 2(c).
V. CONCLUSION
We have constructed the framework of theoretical simulation for magnetic field and temperature in the superconducting
bulk during pulsed field magnetization (PFM). Three successive pulsed field with identical strength (No. 1 - No. 3) were applied to the cryo-cooled bulk and time and spatial dependenceand temperature
were calculated.
of local field
The trapped field
increased and the maximum temperature
decreased with increasing pulse number. Theenrise
hancement comes from the decrease in temperature rise becauseof the already trapped flux. The results of the simulation well reproduced the experimental results qualitatively.
REFERENCES

Fig. 6. Results of the simulation for the pulse number dependence of the (a)on the bulk center
trapped field B and (b) maximum temperature rise T
for various B values at 40 K. (c) Time dependence of the local field B t
and the temperature T t at the center of the bulk surface for the SPA method.
of B= 6T

()

1

()

decreases and approaches a steady value with increasing pulsenumber. These results reproduced the experimental resultspresented in Fig. 2(a) and (b) qualitatively. Fig. 6(c) shows theand the temperaturetime dependence of the local fieldat the center of the bulk surface for the SPA method of. The applied pulsed fieldwith a rise time ofis also described.
for the No. 1 pulse takes
and then decreases with increasing
a maximum at. On thetime and approaches to the final value ofother hand, the peak value of
decreases but the final
increases slightly. The simulated
for thevalue ofand thenNo. 1 pulse takes a maximum of 67 K at
takes
decreases with increasing time. The time, at whicha maximum, is decided by the thermal conductivity of the bulk

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