Physica C 468 (2008) 1494–1497

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Physica C
journal homepage: www.elsevier.com/locate/physc

Finite element analysis of pulsed ﬁeld magnetization process in a cylindricalbulk superconductor
K. Kajikawa a,b,*, R. Yokoo b, K. Tomachi b, K. Enpuku a,b, K. Funaki a,b, H. Hayashi c, H. Fujishiro da

Research Institute of Superconductor Science and Systems, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
Department of Electrical and Electronic Systems Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japanc
Research Laboratory, Kyushu Electric Power Co., Inc., 2-1-47 Shiobaru, Minami-ku, Fukuoka 815-8520, JapandDepartment of Materials Science and Engineering, Iwate University, 3-18-8 Ueda, Morioka, Iwate 020-8550, Japanba r t i c l ei n f oArticle history:
Available online 24 May 2008
PACS:
74.25.Fy
74.25.Ha
74.72.Bk
84.71.Ba
Keywords:
Bulk superconductor
Coupled analysis
Finite element method
Trapped magnetic ﬁelda b s t r a c t
The magnetization properties in a cylindrical superconductor with an inﬁnite length exposed to a pulsed
external magnetic ﬁeld are numerically evaluated by means of a coupled ﬁnite element method. In orderto estimate the distributions of magnetic ﬁeld and temperature inside the superconductor, magnetic diffusion and heat balance equations are alternately and iteratively solved at each time step. It is assumed
that the superconductor has a transport property represented by the power-law model, where the criticalcurrent density depends on the local temperature. The adiabatic condition for the thermal analysis is also
used, so that the temperature rise comes from the local energy dissipation due to the magnetic ﬂux
motion. The inﬂuences of the strength of external applied ﬁeld and the initial temperature in the superconductor on the ﬁnal trapped magnetic ﬁeld are investigated toward an optimal design of bulk superconductor magnet.
Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction
There are two major processes for the magnetization of a bulk
superconductor material. One is a ﬁeld cooling method, which requires a relatively large system to supply a stable magnetic ﬁeld.
The other is a pulsed ﬁeld magnetization (PFM) method [1], which
can be realized inexpensively with a copper coil and a capacitor
bank. In the latter technique, however, the magnitude of appliedﬁeld has to be chosen carefully from both viewpoints of a temperature rise and a ﬁnal trapped magnetic ﬁeld in the superconductor[1–3].
In the PFM methods, the multiple applications of external ﬁelds
to the bulk superconductor are usually carried out to improve theﬁnal trapped magnetic ﬁeld. The typical methods with different sequences of external applied ﬁeld and cooling temperature are illustrated in Fig. 1. Fig. 1a represents a successive magnetic pulse
application (SPA) method [4] that requires several times of apply-* Corresponding author. Address: Research Institute of Superconductor Scienceand Systems, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.
Tel.: +81 92 802 3836; fax: +81 92 802 3829.
E-mail address: kajikawa@sc.kyushu-u.ac.jp (K. Kajikawa).
0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physc.2008.05.062

ing the external magnetic ﬁelds with almost same strength under aconstant cooling temperature. In a modiﬁed multi-pulse techniquewith step-wise cooling (MMPSC) method [5,6] shown in Fig. 1b, on
the other hand, two kinds of cooling temperatures are used effectively. First, the pulsed magnetic ﬁelds with relatively small magnitude are applied twice to the bulk superconductor cooledaround a relatively high temperature. After that, the superconductor is cooled down to a lower temperature and the applications of afew pulsed ﬁelds with larger strength are performed intermittently. By using such a MMPSC process, a high trapped magnetic
ﬁeld of 4.47 T has been achieved at the center of a Gd-based bulk
superconductor in the form of a disk [5]. However, the mechanismof magnetization process for the MMPSC method has not been clariﬁed yet.
In the present study, the magnetization properties in a cylindrical superconductor exposed to multiple pulsed magnetic ﬁelds arenumerically evaluated by means of a ﬁnite element method. The
electromagnetic and thermal analyses are coupled with each otherto take into account the effect of heating due to the penetration ofmagnetic ﬂux into the superconductor on the ﬁnal trapped ﬁeld.
The obtained numerical results are compared with the experimental data published already [5], and the effectiveness of the MMPSC
method is conﬁrmed qualitatively.

K. Kajikawa et al. / Physica C 468 (2008) 1494–1497

abBe

Be0

1st 2nd 3rd 4th 5tht

Ts0

1st 2nd

3rd 4th 5tht

Ts0t0tFig. 1. Schematic diagrams of proﬁles of external applied magnetic ﬁeld andcooling temperature for pulsed ﬁeld magnetization techniques with (a) SPA methodand (b) MMPSC method.

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where C is the heat capacity and j the thermal conductivity. The
term of cooling on the right-hand side in Eq. (4) is disregarded inadvance although the surface of bulk superconductor has beencooled with a refrigerator in the experiment [5], but its treatmentin this study is explained in the ﬁrst part of the following section.
The thermal properties as a function of temperature, C(T) andj(T), are taken into account on the basis of the experimental results
for the Gd-based bulk superconductor [8].
The above-mentioned governing equations for the electromagnetic and thermal analyses have an analogous structure, so thatB, q/l0, 1 and oBe/ot in Eq. (1) correspond to T, j, C and (qJ2) inEq. (4), respectively. This means that the programming code for
one analysis is applicable to the other without major modiﬁcation.
The governing equations are discretized by means of the Galerkinmethod and the backward difference method for space and time,respectively. The obtained two kinds of simultaneous equations
are alternately solved at each time step. Since iterative calculations
at ﬁxed time are also needed for the physical parameters depending on the unknown variables, the successive under-relaxation
method is applied with the relaxation factor of 0.1 [9]. Further-

2. Numerical analysis of PFM process
In order to evaluate the magnetic ﬂux motion and the temperature rise in a bulk superconductor exposed to an external magnetic ﬁeld, their numerical simulation is carried out as follows.
In spite of the use of a Gd-based bulk superconductor disk withdiameter of 45 mm and thickness of 15 mm in the experiment[5], let us consider an inﬁnitely long cylinder with the identical
diameter for the sake of simpliﬁcation. Although such assumption
is unsuitable for quantitative evaluation, the advantages in theMMPSC method are reconﬁrmed numerically in this paper. If
the local magnetic ﬁeld is given by the sum of a uniform externalﬁeld Be applied in the axial direction and a ﬁeld B generated by acurrent induced in the azimuthal direction, the governing equation for the electromagnetic ﬁeld can be obtained with Maxwell’sequations [7],
1 1 ooB
oB oBeqr¼
þ;orot
l0 r orot

ð1Þ

where q is the equivalent resistivity of the superconductor, which isrepresented here with the power-law model for the voltage–current
characteristics asqðJ; TÞ ¼

n1
EcjJj;J c ðTÞ J c ðTÞð2Þ

where Ec is the electric-ﬁeld criterion of the critical current density
Jc and n the n-value. As can be seen in Eq. (2), the resistivity ofsuperconductor is generally a function of the local current density
J and the temperature T. The linear relationship in the critical current density versus temperature is assumed here asJ c ðTÞ ¼ J c0

Tc  T;
Tc  T0

ð3Þ

where Tc is the critical temperature and Jc0 the critical current density at the reference temperature T0. In addition, the ﬁeld dependence of the critical current density is not focused on because theeffect of the decrease in the critical current density due to only
the temperature rise on the ﬁnal trapped ﬁeld is revealed. The following ﬁxed parameters are used: Tc = 93 (K), Ec = 1 (lV/cm),
Jc0 = 20 (kA/cm2) at T0 = 40 (K) and n = 15.
The governing equation for thermal analysis in the model underconsideration is represented by the heat balance equation,

CoT 1 o¼
otr orjroTorþ qJ 2 ;ð4Þ

Fig. 2. Time dependence of local magnetic ﬁelds at ﬁxed positions inside superconductor cylinder. (a) Focuses mainly on a period for the pulsed ﬁeld application,
whereas the convergence of ﬁeld distribution is conﬁrmed for a longer time scale in(b). The peak of external ﬁeld and the initial temperature are 13.0 T and 30 K,
respectively.

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K. Kajikawa et al. / Physica C 468 (2008) 1494–1497

more, the waveform of the external magnetic ﬁeld Be generated
with a pulse coil installed in a capacitor discharge circuit is approximated by [8]"  s1s sBe ðtÞ ¼ Bm

s1
s2

21



 s s2s #1 s1
s2

2

1


t
t
e s2  e s1 ;

ð5Þ

where Bm is the peak of the external applied ﬁeld. In Eq. (5), s1and s2 represent the decay time constants, and they are estimatedas s1 = 8 (ms) and s2 = 19 (ms) for the ﬁrst rising time ofabout 12 ms [8]. The boundary conditions for both analyses are
set toBðR; tÞ ¼ Be ðtÞ;ð6Þ

TðR; tÞ ¼ T s ;ð7Þ

where R and Ts are the radius and cooling temperature of superconductor cylinder, respectively.
3. Numerical results and discussion
Fig. 2 shows a typical example of the time dependence of local
magnetic ﬁelds at several ﬁxed positions inside the superconductor

Fig. 3. Numerical results of trapped magnetic ﬁelds in superconductor cylinder for(a) single pulse and (b) SPA method. The cooling temperature for the SPA method is
ﬁxed at 30 K.

cylinder. The peak of external magnetic ﬁeld and the initial temperature are 13.0 T and 30 K, respectively. It can be seen in Fig.
2a that the local magnetic ﬁelds rise up in sequence with increasing the external applied ﬁeld and then a part of the penetrating
magnetic ﬂux gets out of the superconductor for the decrease inthe external ﬁeld. After the pulsed ﬁeld application, the local magnetic ﬁelds asymptotically approach to their own constants, and it
typically takes about 5 s for their convergence, as shown in Fig. 2b.
In the refrigerated superconductor under consideration, the thermal diffusivity j/C is much smaller than the magnetic diffusivity
q/l0, and therefore their diffusion times are quite different fromeach other. Although a few tens of minutes seem to be needed
for uniform cooldown of a bulk superconductor with a cryocooler
after applying a pulsed ﬁeld in the experiment [5], the numericalcalculation under the adiabatic condition is carried out up to 5 s
and the temperature inside the superconductor is initialized touniform distribution with the ﬁeld proﬁle unchanged in order tosave a computation time.
The dependence of magnetic ﬁeld trapped at the center ofsuperconductor cylinder on the peak of external applied ﬁeld isshown in Fig. 3a, where the pulsed ﬁeld is applied only once. It
can be found that there is an optimum magnitude of external ﬁeldFig. 4. Pulse number dependence of (a) trapped magnetic ﬁeld and (b) maximum
temperature rise for SPA and MMPSC methods. The similar properties obtained in
the experiment have already been presented [5].

K. Kajikawa et al. / Physica C 468 (2008) 1494–1497

that the ﬁnal trapped ﬁeld becomes a maximum. Such a property
shifts to a larger range of magnetic ﬁeld with decreasing the initialcooling temperature. Fig. 3b shows the numerical results of thetrapped magnetic ﬁeld in the superconductor at each step ofpulsed ﬁeld application for the SPA method with the cooling temperature of 30 K. It can be seen that the trapped ﬁelds except forﬁrst pulse application scarcely vary, and that a relatively broad
range of the external ﬁeld for the maximum of ﬁnal trapped ﬁeld
is observed as compared with the results for the single pulse given
in Fig. 3a.
Fig. 4 shows the comparison between the numerical results oftrapped magnetic ﬁeld and maximum temperature rise at eachstep for multiple PFMs with the SPA and MMPSC methods. In
the MMPSC method, the cooling temperature and the external applied ﬁeld strength in the ﬁrst two steps are ﬁxed at 45 K and
11.0 T, respectively. After that, the cooling temperature goes
down to 30 K. The magnitudes of external ﬁelds for third, fourthand ﬁfth pulses are also set to 14.5, 13.5 and 13.5 T, respectively.
It is found in Fig. 4a that the largest trapped magnetic ﬁeld is obtained at fourth application of pulsed ﬁeld in the MMPSC method.
It is also seen in Fig. 4b that the maximum temperature rises in
the MMPSC method are almost same as those for the SPA method
under the identical conditions expect for third pulse application.
It has to be pointed out that the properties of trapped magnetic
ﬁeld and maximum temperature rise given in Fig. 4 agree qualitatively with the experimental results [5]. This means that the present numerical simulation is quite valid to understand theadvantages in the MMPSC method compared with the SPA method although the assumption of the superconductor cylinder with
inﬁnite length differs from an actual shape of specimen in theexperiment [5].

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4. Conclusions
The numerical simulation of the pulsed ﬁeld magnetization inthe cylindrical superconductor was carried out with the coupled ﬁnite element method for electromagnetic and thermal analyses. It
was assumed that the superconductor cylinder had an inﬁnite
length and the magnetic ﬁeld trapped at the center ﬁnally for thepulsed ﬁeld application was evaluated as well as the radial distributions of magnetic ﬁeld and temperature inside it. By taking into
account the power-law model for the voltage–current characteristics with the temperature dependence of critical current density,
the similar properties for ﬁnal trapped magnetic ﬁeld and maximum temperature rise as the experiment were obtained numerically. The further investigation for more quantitative evaluationof the pulsed ﬁeld magnetization process will be needed toward
an optimal design of bulk superconductor magnet.
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