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Mechanism of magnetic flux trapping on superconducting bulk magnetized by pulsed fieldusing a vortex-type coilThis article has been downloaded from IOPscience. Please scroll down to see the full text article.
2011 Supercond. Sci. Technol. 24 075015(http://iopscience.iop.org/0953-2048/24/7/075015)
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IOP PUBLISHING

SUPERCONDUCTOR SCIENCE AND TECHNOLOGYSupercond. Sci. Technol. 24 (2011) 075015 (7pp)

doi:10.1088/0953-2048/24/7/075015

Mechanism of magnetic flux trapping onsuperconducting bulk magnetized bypulsed field using a vortex-type coil
Hiroyuki Fujishiro, Tomoyuki Naito and Mitsuru Oyama
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan
E-mail: fujishiro@iwate-u.ac.jp

Received 25 January 2011, in final form 28 April 2011
Published 1 June 2011
Online at stacks.iop.org/SUST/24/075015
Abstract
Numerical simulations of trapped field Bz and temperature T have been performed for acryo-cooled superconducting bulk disc after applying a magnetic pulse using a vortex-type coil.
Results of the simulation qualitatively reproduced experimental results reported by Ida et al(2004 Physica C 412–414 638). Using the vortex-type coil, the magnetic flux does not intrude
into the bulk from the periphery but intrudes from both surfaces of the bulk disc. The
temperature rise was less than that obtained when using a conventional solenoid coil. The effectof the trapped field enhancement using a long pulse application is small, which suggests that
down-sizing of the condenser bank used for the pulsed field magnetization is possible and thatthe use of the vortex-type coil is desirable for practical applications.

which were used as pole field magnets in the rotor platein a synchronous motor [8]. They demonstrated that, using
the vortex-type coil, the magnetic flux starts to trap on thebulk centre, even for the lower pulsed field Bex , and that the
maximum trapped field was better enhanced than that obtainedusing the solenoid coil. Recently, a higher magnetic field has
been trapped in the superconducting bulk, extending it by asmuch as 140 mm in diameter using a vortex-type coil [9].
However, the mechanism of the field trap has not been clarified.
Several studies of theoretical simulations for PFM havereported using a solenoid coil [10–12]. Experimental research
has remained limited and has shown little progress in exploring
the enhancement of the trapped field using PFM. Therefore, wehave also constructed the framework of theoretical simulationin the superconducting bulk during PFM using the solenoid coiland have simulated the time and spatial dependence of the local
field Bz (t, r ) and temperature T (t, r ) in the superconducting
bulk for a single magnetic pulse application [13], for successive
pulsed field application with identical strength (SPA) [14], andfor MMPSC [15], all of which reproduced the experimental
results qualitatively.
As described in this paper, we simulated the trapped field
and temperature rise in the superconducting bulk magnetizedby a pulsed field using the vortex-type coil. We examined
the mechanism of the field trap using the vortex-type coil

1. Introduction
As a technique for magnetizing bulk REBaCuO (RE: rareearth element or Y), pulsed field magnetization (PFM) has
been investigated intensively for practical applications as
a substitute for field-cooled magnetization (FCM) because
PFM is an inexpensive and mobile experimental set-up that
obviates the use of a superconducting solenoid coil. Usinga solenoid copper coil, the trapped field Bz achievable by
PFM is nonetheless lower than that achievable by FCM
because of the large temperature rise caused by the dynamical
motion of the magnetic flux. Several approaches have been
undertaken to enhance Bz using a multi-pulse technique,including an iteratively magnetizing pulsed field method with
reducing amplitude (IMRA) [1] and a multi-pulse techniquewith stepwise cooling (MPSC) [2]. Considering our obtained
experimental results for the time and spatial dependences of
the temperature T (t, r ) and local field Bz (t, r ), and the trappedfield Bz on the surface of cryo-cooled REBaCuO bulk during
PFM [3–5], we proposed a new PFM technique designated as
modified MPSC (MMPSC) [6] and produced the highest field
trap of Bz = 5.20 T on a 45 mm GdBaCuO bulk at 30 K [7],which is a record high value by PFM to date.
Ida et al proposed a new PFM technique using a vortextype coil. The coil consists of a couple of copper coils andwas developed for the magnetization of the bulks by PFM
0953-2048/11/075015+07$33.00

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Supercond. Sci. Technol. 24 (2011) 075015

H Fujishiro et al

superconducting bulk as
E = Ec

J
Jc

n
,

(1)

where Jc is the critical current density in the ab -plane andE c (=1 × 10−6 V m−1 ) is the reference electric field. The
temperature and magnetic field (along the c-axis) dependenceof the critical current density Jc (T, B) were described asJc = α 1 −



T
Tc

2  32

B0,|B| + B0(2)

where Tc is the critical temperature (=92 K) at B = 0
and B0 (=1.3 T) is constant. The constant value of α =4.6 × 108 A m−2 was used for this study, which showed
Jc (40 K, 0 T) = 3.3 × 108 A m−2 . Iterative calculation was
performed to obtain the convergence of electrical conductivity
σ in the bulk at each time step.
The time dependence of the pulsed field Bex (t) along thec-axis with a rise time of τ was approximated as


t
t
.
Bex (t) = Bex exp 1 −(3)ττFigure 1. Geometric configuration for the superconducting bulk andthe pulsed coil. (a) Vortex S-coil, (b) vortex L-coil, and (c) solenoidcoil.

The simulation was performed mainly for τ = 0.01 s. It
was also performed for τ = 0.001 and 1 s for comparison insection 3.3. The strength of the pulsed field Bex along the caxis was defined at the position of the bulk centre in the absenceof the bulk. Detailed procedures of the simulation and otherparameters, including the anisotropic thermal conductivitiesκab = 20 W m−1 K−1 in the ab-plane and κc = 4 W m−1 K−1along the c-axis, are described elsewhere [13].

and compared the results with those obtained using theconventional solenoid coil.

2. Theoretical model and simulation
Figure 1 portrays the geometric configuration for thesuperconducting bulk and the pulsed coil. In the simulation,a set of vortex-type coils was replaced by that of split coilswith a small inner dimension. For the vortex-type coil, thebulk (46 mm diameter and 15 mm thickness) was cooled
to Ts = 40 K from the side surface (along the ab -plane
direction) and was placed between a set of vortex-type coils(ID = 4 mm, H = 35 mm). In this study, vortex-type coilsof two types were examined for the simulation: the vortex Scoil (OD = 30 mm) and the vortex L-coil (OD = 62 mm),as portrayed in figures 1(a) and (b). The distance between thebulk surface and the coil edge was 10 mm. For comparison, thesimulation was also performed for the bulk using the solenoid
coil (ID = 82 mm, OD = 116 mm, H = 50 mm), where thebulk was cooled to Ts = 40 K along the c-axis, as presented
in figure 1(c). For all cases, the spacing plate of 1 mm inthickness with thermal conductivity, κcont (=0.5 W m−1 K−1 ),was set between the bulk and the supporting brass block,which imaginarily represented both the cooling power of therefrigerator and the thermal contact of the bulk to the brass
block.
Physical phenomena during PFM are described using
electromagnetic and thermal equations on the axisymmetricco-ordinate, which were obtained from [12]. Based on the setup around the bulk, the framework of the theoretical simulation
was constructed.
The power-n model (n = 8) was used to describe
the nonlinear E – J characteristic in the ab -plane of the

3. Results of simulation and discussion
3.1. Trapped field profile and temperature rise
Figures 2(a) and (b) respectively portray the trapped field Bz(r = 0) and maximum temperature Tmax (r = 0) at the centreof the bulk surface after applying the pulsed field Bex using
the vortex S-coil, vortex L-coil, and the solenoid coil. For thesolenoid coil, Bz (r = 0) starts to increase at Bex = 4 T,
becomes a maximum at Bex = 6 T and then decreases witha further increase in Bex . The temperature Tmax (r = 0)
increased monotonically with increasing Bex . The results of thesimulation for the solenoid coil reproduced the experimental
ones qualitatively [4]. However, in the case using the vortex Scoil, the magnetic flux starts to be trapped for Bex as low as 1 T
and Bz (r = 0) increased concomitantly with increasing Bex .
The maximum Bz (r = 0) was 2.6 T at Bex = 9 T, which
was higher than that attained using the solenoid coil. The
temperature rise for the vortex S-coil increases concomitantlywith increasing Bex , but it was about one-third or one-quarteras large as that obtained using the solenoid coil. Figure 2(c)presents experimental results obtained by Ida et al for thetrapped field Bz on the centre of the GdBaCuO bulk surface
at 77 K as a function of applied pulsed field Bex using both thevortex-type coil and the solenoid coil, as re-written from [8].
2Supercond. Sci. Technol. 24 (2011) 075015

H Fujishiro et al

Figure 2. (a) The trapped field Bz (r = 0) and (b) the maximum temperature Tmax (r = 0) at the centre of the bulk surface as a function of theapplied pulsed field Bex using the vortex S-coil, vortex L-coil, and the solenoid coil. (c) Reported experimental results of remnant
magnetization as a function of applied pulsed field using a vortex-type coil and solenoid coil by Ida et al [8].

Figure 3. Trapped field Bz (r ) profiles on the bulk surface for various applied fields Bex using (a) the vortex S-coil and (b) the solenoid coil.

Comparing with figure 2(a), the results of the simulation
qualitatively reproduced the experimental results reported by
Ida et al.
For the vortex L-coil, Bz (r = 0) starts to increase forBex  4 T, reaches a maximum at 7 T and then decreases
concomitantly with increasing Bex . The Bz (r = 0) and Tmax
(r = 0) for the vortex L-coil show intermediate behaviours
between those of the cases using the vortex S-coil and thesolenoid coil. To enhance the Bz value, the OD value should
be decreased and then the Bz value using the vortex-type coil islarger than that using the solenoid coil. Hereafter, we focus on
the vortex S-coil and compare the characteristics of the trappedfield and temperature with those using the solenoid coil.
Figures 3(a) and (b) respectively present the trapped fieldBz (r ) profiles on the bulk surface for various applied fields Bex

using the vortex S-coil and the solenoid coil. The positionsof r = 0 and 23 mm are, respectively, the centre and edge
on the bulk. For the vortex S-coil, the convex Bz (r ) profile
was achieved even for the lower Bex and Bz (r ) increases
concomitantly with increasing Bex . The convex shape of thetrapped field profile at even lower Bex is advantageous for therealization of the superconducting bulk magnet. However, forthe solenoid coil as portrayed in figure 3(b), the Bz (r ) profilechanges from concave for lower Bex to convex for higher Bex ;then the Bz (r = 0) value decreases with further increase in
Bex . These behaviours using both coils are consistent withexperimental results as shown in figure 2(c).
Figure 4 depicts the trapped magnetic flux distribution
in the cross section of the bulk after applying the magnetic
pulse of Bex = 4 and 6 T using the vortex S-coil (left)
3

Supercond. Sci. Technol. 24 (2011) 075015

H Fujishiro et al

Figure 4. Trapped magnetic flux distribution in the cross section of the bulk after applying the magnetic pulse of Bex = 4 and 6 T using thevortex S-coil (left) and the solenoid coil (right).

and the solenoid coil (right). In each panel, arrows indicate
the magnitude and the direction of the trapped magnetic fieldat each position, but the absolute value of the magnitude isarbitrary. For the vortex S-coil, at Bex = 4 T, the trappedmagnetic flux was oriented towards the r -direction in the innerposition and was oriented towards the z -direction on the bulk
surface. No flux trap exists in the core of the bulk becauseof the low Bex value. The magnetic flux starts to intrude into
the core of the bulk at Bex = 6 T. In addition, the direction of
the trapped magnetic field thoroughly aligned to the z -direction
with a further increase in Bex . As a result, the trapped field Bz
along the z -direction at the centre of the bulk surface increased
concomitantly with the increase in Bex .
However, for the solenoid coil, at Bex = 4 T, the magnetic
flux was trapped mostly around the bulk periphery. No trapped
flux exists at the bulk centre. The direction of the trapped fieldin the bulk was along the z -direction because the magnetic
flux intrudes into the bulk from the periphery. The magnetic
flux can be trapped at the bulk centre at Bex = 6 T andthe shape of the trapped field along the z -direction is convex.
The magnitude of the magnetic field decreased with a further
increase in Bex because of the large temperature rise. In thisway, the trapped field profile in the bulk and the magnetizing
mechanism differ greatly among magnetizing coils.

3.2. Time dependence of the magnetic flux movement andtemperature rise
In section 3.1, the trapped field profile in the bulk differs
greatly in the case using each magnetizing coil because of
the different mechanism of flux intrusion. In this section,
we investigate the time evolution of the flux intrusion andthe temperature rise in the bulk using each magnetizing coil.
Figures 5(a) and (b) show the time evolution of the local fieldBz (t) at r = 0, 10, and 20 mm on the bulk surface after
applying the magnetic field of Bex = 6 T using the vortex
S-coil and the solenoid coil, respectively. In the figures, themagnetic pulsed field Bex (t) defined by equation (3) is also
shown. Using the vortex S-coil, the local fields Bz (t) at all
positions take a maximum around t = 0.01 s and then decreaseconcomitantly with increasing time. The local field Bz (r =0 mm) is the maximum on the bulk surface. However, using
the solenoid coil, Bz (t) reaches a maximum at r = 20, 10,
0 mm in this order with time and then decreases concomitantlywith increasing time.
Figures 6(a) and (b) present the time evolution of thelocal temperature T (t) at r = 0, 10, and 20 mm on the bulk
surface after applying the magnetic field of 6 T using the vortex
S-coil and the solenoid coil, respectively. Using the vortex
S-coil, T (t) at r = 20 and 10 mm reach their respective
maxima of 50 and 52 K at around t = 0.01 s; then they4Supercond. Sci. Technol. 24 (2011) 075015

H Fujishiro et al

Figure 5. Time evolution of the local field Bz (t) (r = 0, 10, and 20 mm) on the bulk surface after applying the magnetic pulse field of
Bex = 6 T using (a) the vortex S-coil and (b) the solenoid coil. The magnetic pulsed field Bex (t) defined by equation (3) is also shown.

Figure 6. Time evolution of the local temperature T (t) (r = 0, 10, and 20 mm) on the bulk surface after applying the magnetic pulse field of
6 T using (a) the vortex S-coil and (b) the solenoid coil.

decrease in Bz (t) at each position because of the difference ofthe cooling condition. Figures 7(b) and (c) show respectively
the trapped field Bz (r = 0) and the maximum temperature
Tmax (r = 0) for both cooling conditions as a function of theapplied field Bex . Bz (r = 0) for the bulk cooled from the bulk
periphery is slightly larger and Tmax (r = 0) is slightly lower,
compared with those for the bulk cooled from the bottom.
Because, using the solenoid coil, the heat generation takes
place at the bulk periphery mainly and the generated heat was
exhausted effectively from the side surface in the bulk cooled
from the periphery. As a result, the cooling condition was
hardly influenced for the field trap and the differences in thefield trap between the vortex-type coil and the solenoid coil, as
shown in figure 2(a), result from the coil configuration.

decrease concomitantly with increasing time. The temperature
T (t) at the centre of the bulk surface reaches a maximum of
46.5 K at t = 0.5 s and then decreases concomitantly withincreasing time. However, using the solenoid coil, T (t) rises
and reaches a maximum at r = 20, 10, 0 mm in this orderwith time and then decreases concomitantly with increasingtime because of the flux intrusion from the bulk periphery. In
those figures, the scales of the ordinates are different. At thecentre of the bulk surface (r = 0 mm), using the vortex S-coil,
the temperature rise is small (Tmax = 46.5 K) and the time
tm at which the temperature rise reaches a maximum is short(t = 0.5 s). Using the solenoid coil, the temperature rise islarge (Tmax = 66 K) and tm is 5 s. These results qualitatively
reproduce the experimental ones obtained using the solenoid
coil and the split coil, which resembles the vortex-type coil [4].
In figures 5 and 6, not only the coil configuration but also thecooling condition differ between the vortex-type coil and thesolenoid coil in the results of the simulation. To separate them,an additional simulation was performed for the bulk cooled
from the bulk periphery using the solenoid coil.
Figure 7(a) shows the time evolution of the local fieldBz (t) at r = 0, 10, and 20 mm on the bulk cooled from thebulk periphery after applying the magnetic field of Bex = 6 T
using the solenoid coil. The coil configuration is illustrated
in the inset. Let us compare the results in figure 7(a) andthose in figure 5(b). It should be noted that the results infigure 7(a) closely resemble those in figure 5(b) with a slight

3.3. Effect of the rise time of the magnetic pulse using avortex-type coil
In a previous paper, we presented a long magnetic pulse
application during PFM as a possible solution to enhance thetrapped field using the solenoid coil [13]. In this section, we
discuss the effect of the rise time elongation of the magnetic
pulse on the trapped field characteristics using the vortex Scoil. Figures 8(a) and (b) depict the trapped field Bz (r = 0)
and maximum temperature Tmax (r = 0) at the centre of thebulk surface as a function of the applied field Bex for various
rise times τ of the magnetic pulse. The results for τ = 0.01 s
5Supercond. Sci. Technol. 24 (2011) 075015

H Fujishiro et al

Figure 7. (a) Time evolution of the local field Bz (t) (r = 0, 10, and 20 mm) on the bulk cooled from the bulk periphery after applying amagnetic pulse field of 6 T using the solenoid coil. (b) The trapped field Bz (r = 0) and (c) the maximum temperature Tmax (r = 0) for bothcooling conditions as a function of the applied field Bex (see text).

Figure 8. (a) The trapped field Bz (r = 0) and (b) the maximum temperature Tmax (r = 0) at the centre of the bulk surface as a function of theapplied field Bex for various rise times τ of the magnetic pulse.

are also shown. For Bex  7 T, Bz (r = 0) increases slightly
with increasing τ , and for Bex  8 T, Bz (r = 0) decreases
slightly with increasing τ . However, it should be noted that nolarge difference exists for each τ value, which differs greatly
from that obtained when using the solenoid coil [13]. These
results show that a similar trapped field is obtainable even forsmall τ values and suggest that the condenser bank capacity
used for PFM can be down-sized. In PFM, after a large amountof electricity is charged in the capacitance in the condenser
bank, it is discharged through the magnetizing coil. The risetime τ becomes short with the decrease in the inductance Lof the coil and capacitance C of the condenser bank. As
a result of the short τ value, the volume of the condenserbank can be down-sized, although it is necessary to enhance

the charged voltage. The suggestion is quite favourable forpractical applications.

4. Summary
Numerical simulations of magnetic flux propagation andthe temperature rise were performed for a cryo-cooled
superconducting bulk disc using a vortex-type coil after
applying a magnetic pulse, compared with those using a
conventional solenoid coil. A summary of important resultsof the simulation and conclusions obtained from this study ispresented below.
(1) The applied pulsed field dependence of the trapped fieldBz using the vortex-type coil qualitatively reproduced the6Supercond. Sci. Technol. 24 (2011) 075015

H Fujishiro et al

experimental results reported by Ida et al. The magnetic
field can be trapped even for a lower applied pulsed field
and the maximum trapped field can be enhanced more thanwhen using a solenoid coil. To enhance the Bz value, theouter diameter (OD) of the vortex-type coil must be lessthan that of the superconducting bulk disc.
(2) The time and spatial dependence of the local field andtemperature in the bulk were simulated, revealing a clear
difference in the magnetizing mechanism. For the vortextype coil, the magnetic flux intrudes mainly into the bulk
from the surfaces. However, for the solenoid coil, allmagnetic fluxes intrude into the bulk from the periphery.
(3) The temperature rise during the flux intrusion for thevortex S-coil is one-third or one-quarter as large as that forthe solenoid coil. Reduction in the maximum temperaturerise also enhances the trapped field.
(4) The effect of rise time elongation of the pulsed field on
the trapped field enhancement has been simulated: the risetime τ was changed from 0.001 to 1 s. No large differenceswere found in the trapped field for each τ value. In otherwords, a similar trapped field is obtainable even for small
τ values. These results suggest that the capacity of thecondenser bank used in PFM can be down-sized, which
is favourable for practical applications. The results of thesimulation for the vortex-type coil must be reproduced andverified through further detailed experimentation.

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