Physica C 471 (2011) 889–892

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

Simulation of ﬂux dynamics in a superconducting bulk magnetizedby multi-pulse technique
H. Fujishiro ⇑, T. Naito, M. Oyama
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japana r t i c l e

i n f oArticle history:
Available online 12 May 2011
Keywords:
Pulsed ﬁeld magnetization
Simulation
Temperature rise
Flux intrusion
Bulk superconductora b s t r a c t
We have simulated the time and spatial dependence of local ﬁeld Bz(t, r) and temperature T(t, r) on the
superconducting bulk during pulsed ﬁeld magnetization (PFM) using the ﬁnite element method (FEM).
A modiﬁed multi-pulse technique with step-wise cooling (MMPSC) was performed to the cryo-cooledbulk, which was experimentally conﬁrmed to the effective PFM technique to enhance the trapped ﬁeld
Bz higher than 5 T. In the simulation, the Bz value at the center of the bulk surface was enhanced atthe 2nd stage of the MMPSC method, in which the results of the simulation reproduced the experimental
ones. The enhancement of Bz results from the reduction in the temperature because of the alreadytrapped ﬂux in the bulk at the 1st stage of the MMPSC method.
Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction
REBaCuO superconducting bulk (RE: rare earth elements or Y)
can trap a high magnetic ﬁeld Bz using a pulsed ﬁeld magnetization(PFM) as a substitute for ﬁeld-cooled magnetization (FCM), which
can be used for a magnetic separation, drag delivery system and soon. However, Bz by PFM is smaller than that by FCM because of alarge heat generation due to the dynamical motion of magnetic
ﬂux in the bulk. To enhance Bz by PFM, a multi-magnetic pulse
technique such as an iteratively magnetizing pulsed-ﬁeld methodwith reducing amplitude (IMRA) [1] and a multi-pulse techniquewith step-wise cooling (MPSC) [2] is effective around 77 K. We
have succeeded the highest ﬁeld trap of 5.2 T on the GdBaCuO bulk
45 mm in diameter by a modiﬁed MPSC (MMPSC) at 29 K, which isa record-high value by PFM to date [3]. To explore the desired research direction to enhance the Bz value by PFM still more, it isnecessary to analyze both the ﬂux dynamics and the heat propagation theoretically.
Several theoretical studies for PFM have been reported in previous works [4,5]. We have also constructed the framework of theoretical simulation in the superconducting bulk and simulated thetime and spatial dependence of local ﬁeld Bz(t, r) and temperature
T(t, r) in the superconducting bulk after a single magnetic pulse
application, which reproduced the experimental results qualitatively [6]. However, the Bz value obtained by the simulation was
smaller than the experimental ones. Recently, we have simulated
the Bz(t, r) and T(t, r) for the successive pulsed ﬁeld application⇑ Corresponding author. Tel./fax: +81 19 621 6363.
E-mail address: fujishiro@iwate-u.ac.jp (H. Fujishiro).
0921-4534/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.physc.2011.05.081

with identical strength (SPA), and the enhancement of Bz was conﬁrmed compared with a single pulse application [7]. Kajikawa et al.
reported the enhancement of Bz by the MMPSC method using asimulation [8]. However, time evolution and position dependenceof the magnetic ﬁeld and temperature did not reported. As described in this paper, we performed the simulation to the MMPSC
method for the higher Jc bulk, and investigated the ﬂux dynamics
and the temperature propagation during MMPSC.
2. Theoretical model and simulation
Based on our experimental setup around the bulk, the framework of the theoretical simulation was constructed. The detailed
procedure of simulation and the used parameters were described
elsewhere [6]. Commercial software, Photo-Eddy, combined with
Photo-Thermo (Photon Ltd., Japan) was used for analysis of themagnetic ﬁeld and temperature distribution in superconducting
bulk during PFM. As shown in Fig. 1, a superconducting bulk disk
46 mm in diameter and 15 mm in thickness was stacked on thecold stage of the refrigerator in a vacuum chamber and cooled.
The spacing plate with the thermal conductivity, kcont (=0.5
W m1 K1), between the bulk and the cold stage with 1 mm inthickness was set, which imaginarily represented both the coolingpower of the refrigerator and the thermal contact of the bulk to thecold stage.
Physical phenomena during PFM are described by electromagnetic and thermal equations on the axisymmetric coordinate,
which were referred from Ref. [9].
The power-n model (n = 8) was used to describe the nonlinear
E–J characteristic in superconducting bulk as follows:890

H. Fujishiro et al. / Physica C 471 (2011) 889–892

ﬂuxes with concave Bz proﬁle on the bulk surface. At the 2nd stage,
the bulk was cooled down to T2 (<T1) and a higher pulse ﬁeld of B2(>B1) was applied twice (No. 3 and No. 4 pulses).
3. Experimental results

Fig. 1. Structure and dimensions around the superconducting bulk and solenoid
copper coil for pulsed ﬁeld magnetization (PFM).

E ¼ Ec

 n
J
;
Jc

ð1Þ

where Jc is the critical current density and Ec (=1  106 V/m) is thereference electric ﬁeld. The temperature and magnetic ﬁeld dependence of Jc(T, B) was described as( 2 )32T
B0
J c ðT; BÞ ¼ a 1 ;
Tc
jBj þ B0ð2Þ

where Tc (=92 K) is the critical temperature at B = 0, and B0 (=1.3 T)
is constant. The constant value of a = 1.83  109 A/m2 was used inthis study, which showed Jc (40 K, 0 T) = 1.33  109 A/m2 and wasa typical value of the superconducting bulk at present [10]. Iterative
calculation was performed to obtain the convergence of electricalconductivity r in the bulk at each time step. The pulsed ﬁeld Bex(t)with a rise time of s = 0.013 s and with a pulse duration of 0.13 s
was applied using a solenoid pulsed coil.
The sequence of the MMPSC technique was shown in the inset
of Fig. 2a [3]. Four magnetic pulses were applied at different initial
temperatures T1 and T2 to the bulk. In the simulation, at the 1ststage in the MMPSC, a pulse ﬁeld of B1 was applied twice (No. 1
and No. 2 pulses) at T1 in order to trap a small number of magnetic

Fig. 2. Examples of experimental results for the time evolution of the local ﬁeld
Bz(t) as a function of the distance along the r-direction for the No. 1 pulse (T1 = 45 K,
B1 = 4.5 T) of (a) ascending (t 6 0.013 s) and (b) descending (t P 0.013 s) stages in
MMPSC, respectively [3].

First we show an example of the experimental results of the ﬂux
dynamics in the bulk magnetized by the MMPSC method [3].
Fig. 2a and b shows respectively the ascending (t 6 0.013 s) anddescending (t P 0.013 s) stages of the local ﬁeld Bz(r, t) on the surface of the GdBaCuO bulk (45 mm in diameter and 18 mm in thickness) after applying the No. 1 pulse (B1 = 4.5 T at T1 = 45 K). The
Bz(t) values at positions C (r = 0), M (r = 8 mm) and E (r = 15 mm)
were measured by three Hall sensors adhered on the bulk surface.
For the ascending process, the ﬂux intrusion starts to increase fromthe bulk periphery and the small amount of the magnetic ﬂux arrives at the bulk center. For the descending process, the intruded
ﬂux gradually escaped in the region outer than r = 8 mm and the‘‘M-shaped’’ proﬁle can be obtained.
Fig. 3a and b shows the Bz(r, t) proﬁles for the No. 3 pulse(T2 = 29 K, B2 = 6.7 T) at the 2nd stage of the MMPSC. Bz(t) at theposition E increases, but Bz(t) at the position C and Bz(t) at the position M hardly change for the ascending process. For the descending process, however, Bz(t) at the position C sharply increased at
t = 0.013 s, Bz(t) at the position M also slightly increased and theconical trapped ﬁeld distribution was realized. These behaviors
suggest that the already trapped ﬂux at the 1st stage was pushed
into the bulk center and that the additional ﬂux was supplied fromthe peripheral region. As a result, the highest Bz was trapped at thebulk center by the MMPSC method. The temperature rise at the
2nd stage was decreased experimentally, compared with that after
a single pulse application with the same amplitude to the virginstate bulk because of the already trapped ﬂux at the 1st stage [3].
4. Results of simulation and discussion
Fig. 4a shows the results of the simulation of the trapped ﬁeld
proﬁle Bz(r) on the bulk surface at T = 60 K as a function of applied
pulsed ﬁeld. Bz(r) proﬁle changes from the ‘‘M-shaped’’ proﬁle tothe convex-shaped one with increasing applied ﬁeld. We adopted
B = 6.5 T as the applied ﬁeld of B1 for the 1st stage in the MMPSC
method. Because the ‘‘M-shaped’’ proﬁle at the 1st stage is necessary to enhance Bz at the 2nd stage. Fig. 4b shows the trapped ﬁeld

Fig. 3. Examples of experimental results for the time evolution of the local ﬁeld
Bz(t) as a function of the distance along the r-direction for the No. 3 pulse (T2 = 29 K,
B2 = 6.7 T) of (a) ascending (t 6 0.013 s) and (b) descending (t P 0.013 s) stages in
MMPSC, respectively [3].

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891

Fig. 4. (a) The results of the simulation of the trapped ﬁeld proﬁle Bz(r) on the bulk
surface at 60 K as a function of applied pulsed ﬁeld. (b) The trapped ﬁeld Bz(r = 0)
after the termination of MMPSC method as a function of the applied ﬁeld B2 at the
2nd stage at T2 = 40 K.

Bz(r = 0) after the termination of the MMPSC method as a functionof the applied ﬁeld B2 at the 2nd stage at T2 = 40 K. The ﬁnal
trapped ﬁeld increases with increasing B2 and takes a maximumof 6.2 T at B2 = 15 T, which suggests that an optimum B2 exists
for each condition of T1, T2 and B1.
Fig. 5a and b shows the pulse number dependence of thetrapped ﬁeld proﬁle Bz(r) on the bulk surface for B2 = 11 T and
15 T at T2 = 40 K, respectively. For B2 = 11 T, the magnetic ﬂux
did not sufﬁciently reach the bulk center and the Bz(r) proﬁle is still
concave. On the other hand, for B2 = 15 T, the Bz(r) proﬁle changed
to cone-shaped one and the maximum Bz(r = 0) = 6.1 T was obtained for the No. 4 pulse. In the multi-pulse magnetization process, already trapped ﬂux was treated as the shielding current in
the bulk in the simulation and the additional magnetic ﬂux was
superposed.
Fig. 6a and b, respectively, present the time evolution and spatial
distribution of the local magnetic ﬁeld Bz(t, r) and the temperature
T(t, r) on the bulk surface after applying a No. 1 pulse of B1 = 6.5 T
at T1 = 60 K in MMPSC. The left and right parts for each ﬁgure arefor the ascending (t 6 0.013 s) and descending (t P 0.013 s) stages,
respectively. In Fig. 6a, 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 r = 0 remains to zero at t = 0.1 s and then in-

Fig. 5. The pulse number dependence of the trapped ﬁeld proﬁle Bz(r) on the bulk
surface for (a) B2 = 11 T and (b) B2 = 15 T at T2 = 40 K.

Fig. 6. The time evolution and spatial distribution of the (a) local magnetic ﬁeld
Bz(t, r) and (b) temperature T(t, r) on the bulk surface after applying No. 1 pulse of
B1 = 6.5 T at T1 = 60 K in MMPSC.

creases gradually to Bz = 0.5 T at the steady state. As a result, the concave shaped proﬁle was realized. The results of the simulation
reproduce the experimental ones qualitatively as shown in Figs. 2
and 3.
In Fig. 6b, the temperature T(t) at the bulk periphery gradually
increased with time at the ascending stage, e.g., T = 95 K at
t = 0.013 s. However, T(t) at r = 0 remained to 40 K. At the descending stage, the generated heat diffuses to the bulk center gradually.
After an isothermal temperature proﬁle took place along ther-direction around t = 5 s, T(t) decreased with increasing time. In
this case, temperature gradient existed along the z-direction because the bottom of the bulk was ﬁxed at 60 K.
Fig. 7a and b depict Bz(t, r) and T(t, r) on the bulk surface after
applying No. 3 pulse of B2 = 15 T at the 2nd stage in MMPSC(T2 = 40 K). For the ascending stage, the magnetic ﬂux intrudes
gradually into the bulk from the bulk periphery and Bz(r = 0)reached 2.3 T at t = 0.013 s. For the descending stage, Bz(r = 0) increased to 7 T at t = 0.1 s and then decreased gradually to Bz = 6 T
at t = 5 s. In Fig. 7b, the T(t, r) behavior is the similar time and radius dependences to that for the No. 1 pulse, but the temperaturerise is fairly reduced; Tmax = 56 K at t = 7 s.
Fig. 8a and b respectively show the time dependence of the local
ﬁeld Bz(t) and the temperature T(t) at the center of the bulk surface(r = 0) for the MMPSC method, which were re-plotted from Figs. 6
and 7. The normalized applied pulsed ﬁeld Bex(t) is also describedto the right ordinates in Fig. 8a. Bz(t) for the No. 1 pulse increases
slightly to 0.7 T with time and that for the No. 2 pulse hardly

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H. Fujishiro et al. / Physica C 471 (2011) 889–892

Fig. 8. The time dependence of the local ﬁeld Bz(t) and the temperature T(t) at thecenter of the bulk surface (r = 0) for the MMPSC method, which were re-plottedfrom Figs. 6 and 7. The normalized applied pulsed ﬁeld Bex(t) is also described by adotted line in (a).

Fig. 7. The time evolution and spatial distribution of the (a) local magnetic ﬁeld
Bz(t, r) and (b) temperature T(t, r) on the bulk surface after applying No. 3 pulse of
B2 = 15 T at the 2nd stage in MMPSC (T2 = 40 K).

increases. On the other hand, for the No. 3 pulse at the 2nd stage,
Bz(t) starts to rise around t = 0.1 s, takes a maximum of 7 T, and approaches gradually to the ﬁnal value. For the No. 4 pulse, the similar time dependence of the local ﬁeld can be seen and the ﬁnal
value increased slightly. In Fig. 8b, at the 1st stage (T1 = 60 K),
the simulated T(t) for the No. 1 pulse takes a maximum of 85 K
at t = 7 s and then decreases with increasing time. The maximum
temperature Tmax drastically decreases to 65 K for the No. 2 pulse.
The time, at which T(t) takes a maximum, is decided by the thermal
conductivities of the bulk (kab, kc) and the cooling power, and thetime is independent of the pulse number. For the 2nd stage(T2 = 40 K), T(t) for the No. 3 pulse takes a maximum of 57 K at
t = 7 s and then decreases with increasing time. Tmax drastically decreases to 51 K for the No. 4 pulse.
In the simulation above mentioned, the optimum parameters,
T1 = 60 K, B1 = 6.5 T, T2 = 40 K and B2 = 15 T, were decided, whichwere different with those experimentally obtained; T1 = 45 K,
B1 = 4.5 T, T2 = 29 K and B2 = 6.7 T. The essence of the MMPSC
method is that the lower magnetic pulse B1 is applied twice at
higher temperature T1 to realize the ‘‘M-shaped’’ proﬁle at the
1st stage, and the higher and optimum magnetic pulse of B2 isapplied twice at lower temperature T2 at the 2nd stage. Experi-mentally, the optimum parameters depend strongly on the characteristics of the bulk such as the magnitude and distribution of Jc.
Taking these facts into consideration, the results of the simulation
shown in this paper reproduce the experimental ones qualitatively.
In summary, we have simulated the time and spatial dependence of local ﬁeld Bz(t, r) and temperature T(t, r) on the
cryo-cooled superconducting bulk during a modiﬁed multi-pulse
technique with step-wise cooling (MMPSC). In the simulation,
the Bz value was enhanced at the 2nd stage in the MMPSC method,which reproduced the experimental results qualitatively. The
enhancement of Bz at the 2nd stage results from the reduction in
the temperature because of the already trapped ﬂux in the bulk
at the 1st stage in MMPSC.
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