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Simulation of temperature and magnetic field distribution in superconducting bulk during
pulsed field magnetization

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2010 Supercond. Sci. Technol. 23 105021(http://iopscience.iop.org/0953-2048/23/10/105021)
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IOP PUBLISHING

SUPERCONDUCTOR SCIENCE AND TECHNOLOGYSupercond. Sci. Technol. 23 (2010) 105021 (8pp)

doi:10.1088/0953-2048/23/10/105021

Simulation of temperature and magnetic
field distribution in superconducting bulk
during pulsed field magnetization
Hiroyuki Fujishiro and Tomoyuki Naito
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan
E-mail: fujishiro@iwate-u.ac.jp

Received 4 July 2010, in final form 3 September 2010
Published 17 September 2010
Online at stacks.iop.org/SUST/23/105021
Abstract
A theoretical simulation of electromagnetic and thermal fields was performed for a cryocooled
superconducting bulk disc after applying a magnetic pulse. The results of the simulation
qualitatively reproduced the experimental ones for the time and applied field dependences of thetrapped field Bz and the temperature T on the bulk surface. For magnetic pulse application witha rise time of τ = 0.01 s, the magnetic flux propagation was about two orders of magnitude
faster than the heat propagation because of the low thermal conductivity of the bulk. The results
show that the intruding magnetic flux escaped because of the delayed temperature rise. For alonger magnetic pulse application with τ = 1 or 10 s, the flux propagation speed becomes slow
and approaches the heat propagation speed. In this case, the magnetic flux escape attributable tothe flux creep becomes small and a higher trapped field can be achieved. The method of
exploring the enhancement of the trapped field using pulsed field magnetization is discussed.

new PFM technique named modified MPSC (MMPSC) [6] andsuccessfully realized a highest field trap of Bz = 5.20 T on a
45 mm GdBaCuO bulk at 30 K [7], which is a record-highvalue by PFM to date. However, experimental research has
been limited. Results have shown little progress to explore theenhancement of the trapped field using PFM. It is necessary
to use a theoretical simulation to elucidate the multi-physics inthe bulk during PFM.
Several studies of theoretical simulations for PFM havebeen reported. Many of the studies were performed underconditions in which the bulk was immersed in liquid nitrogen.
Simulations have been performed also for cryocooled bulks;
Kajikawa et al reported theoretical results of the MMPSC
method, in which the experimental results were reproduced
qualitatively through the simulation [8]. Komi et al reported
magnetic and thermal fields during PFM in a cryocooled
bulk with inhomogeneous superconducting properties [9]. We
investigated heat propagation in drilled holes in the bulk bothexperimentally [10] and theoretically [11]. The results suggestthat the heat generation took place adiabatically because ofits greater heat generation than the cooling power of therefrigerator.
As described in this paper, we constructed the frameworkof the theoretical simulation in the superconducting bulk with

1. Introduction
Recently, pulsed field magnetization (PFM) for REBaCuO
bulk (RE: rare earth element or Y) has been investigatedintensively for practical applications as a substitute forfield-cooled magnetization (FCM) because PFM is an
inexpensive and mobile experimental set-up with no need fora superconducting magnet. 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
performed and have succeeded in enhancing Bz including
an iteratively magnetizing pulsed field method with reducing
amplitude (IMRA) [1] and a multi-pulse technique with stepwise cooling (MPSC) [2]. We have experimentally examined
the time and spatial dependences of the temperature T (t, x),local field Bz (t, x), and the trapped field Bz on the surface of
cryocooled REBaCuO bulks during PFM for various starting
temperatures Ts and applied fields Bex [3–5]. To enhance Bz ,it is believed that the reduction in temperature rise T andthe lowering of Ts are effective because of the enhancementof the critical current density Jc , similar to the case for FCM.
Considering the obtained experimental results, we proposed a
0953-2048/10/105021+08$30.00

1© 2010 IOP Publishing Ltd Printed in the UK & the USA

Supercond. Sci. Technol. 23 (2010) 105021

H Fujishiro and T Naito

Their fundamental equations on the axisymmetriccoordinate are as follows.




∂A
∂ ν ∂
∂
(r A) +
ν
= J0 + J(1)∂r r ∂r
∂z
∂z




∂
1 ∂
∂T
∂T
∂T
=
r κab
+
κc
+ Q. (2)
ρC
∂t
r ∂r
∂r
∂z
∂z
Therein, A represents the magnetic vector potential, J0 is theforced current density or coil current density, J denotes thesuperconducting current density or induced current density,
and Q signifies the heat generation. In equation (2), T
denotes the temperature, C is the specific heat, and κab andκc respectively denote the thermal conductivity in the ab-plane
and along the c-axis.
The J value is given as

J = σ E = −σFigure 1. Structure and dimensions around the bulk and solenoid
copper coil for pulsed field magnetization (PFM). The 1 mm thickspacing plate with thermal conductivity of κcont was placed between
the bulk and the cold stage of the refrigerator, which imaginarily
represents the cooling power of the refrigerator and the thermal
contact of the bulk on the cold stage.

∂A
,
∂t

(3)

where E is the electric field and σ is the electricalconductivity. Then FEM analysis of the coupled problem withelectromagnetic field and heat diffusion was performed.
The power-n model was used to describe the nonlinear E –J characteristic in superconducting bulk, as nJE = Ec,(4)Jchomogeneous superconducting properties after a magnetic
pulse application using the finite element method (FEM). We
reproduced the experimental results using the simulation. We
also investigated the effect of a magnetic pulse application witha longer rise time of τ = 1 or 10 s on the Bz (t) and T (t) values
than those using a magnetic pulse with τ = 0.01 s, which
was used experimentally because the long pulse application
enhanced the trapped field experimentally [12]. The direction
to explore the enhancement of the trapped field by PFM is
discussed herein.

where Jc is the critical current density and E c is the referenceelectric field. The magnetic field dependence of the Jc0 was
described using the Kim model [13] as

Jc = Jc0

B0,|B| + B0

(5)

where Jc0 is Jc for B = 0 and B0 is constant. The temperaturedependence of Jc0 is described by the following equation,



2. Modelling and theoretical simulation

Jc0 = α 1 −Based on our experimental set-up around the bulk [5], theframework of the theoretical simulation was constructed. A
schematic view is presented in figure 1. A superconducting
bulk disc of 46 mm diameter and 15 mm thickness was stacked
on the cold stage of a refrigerator in a vacuum chamber andcooled to the starting temperature Ts = 40 K. We set the
spacing plate with thermal conductivity κcont between the bulk
and the cold stage to 1 mm thickness, which imaginarily
represents both the cooling power of the refrigerator and thethermal contact. Then κcont was set as 0.5 W mK−1 basedon the experimental results of T (t) on the bulk surface [6].
In the simulation, the bulk was magnetized using a solenoidtype copper coil (82 mm I.D., 116 mm O.D., and 50 mm
height). Physical phenomena during PFM are describedusing electromagnetic and thermal fields. The equations andparameters were taken from [9]. Commercial software, PhotoEddy, combined with Photo-Thermo (Photon Ltd, Japan),
was used for analyses of the magnetic field and temperature
distribution in superconducting bulk during PFM.



T
Tc

2  32,(6)

where Tc is the critical temperature (=92 K) at B = 0 and
α is constant. Heat generation Q in the conductor is given as
Q = J E . Iterative calculations were performed to obtain theconvergence of σ in the superconductor at each time step.
The applied pulsed field Bex (t) with the rise time of τ =0.01 s was approximated in the following equation:


t
t
.
Bex (t) = Bex exp 1 −(7)ττThe numerical values for the parameters used in the simulation
are presented in table 1. The κab , κc and C values were
presumed to be temperature-independent for simplicity.

3. Example of experimental results
Let us show an example of the experimental results of thetrapped field on the bulk magnetized by PFM. Figure 2(a)2Supercond. Sci. Technol. 23 (2010) 105021

H Fujishiro and T Naito

Figure 2. Example of experimental results for the trapped field on the GdBaCuO bulk magnetized by PFM [4]. (a) Trapped field Bz (r = 0) at
the centre of the bulk at various starting temperatures Ts after applying a pulsed field Bex with a rise time of τ = 0.01 s. (b) Cross-section of
the Bz profiles for various applied pulsed fields Bex at Ts = 70 K. In the inset, an example of the two-dimensional Bz profile is shown for
Bex = 2.66 T.

the inset, an example of the two-dimensional Bz profile isshown for Bex = 2.66 T. The trapped field profile changes
from a concave shape for Bex  3.84 T to a convex shapefor Bex  4.52 T. For lower Ts = 40 K, similar changes of
the Bz profile were observed as a function of Bex (not shown).
However, Bz (r = 0) at the bulk centre was small, even if apulsed field with identical strength was applied as that at 70 K.
The maximum Bz (r = 0) increases to 1.8 T for Bex = 6.29 T,
but the convex profile cannot be realized because of the strong
pinning force at low temperatures.

Table 1. Numerical values for parameters used in the simulation.
SymbolParameter

Value

Tc

Transition temperature inequation (6)Mass density of bulk inequation (2)Specific heat of bulk inequation (2)Thermal conductivity ofbulk along the ab-plane inequation (2)Thermal conductivity ofbulk along the c-axis inequation (2)n -value in equation (4)Constant in equation (4)Constant in equation (5)Constant in equation (6)Electrical conductivity( T > Tc ) in equation (3)Rise time of the pulse inequation (7)Thermal conductivity ofspacing plate

92 K

ρ
C
κab
κc
n
Ec
B0
α
σ
τ
κcont

5.9 × 103 kg m−3
1.32 × 102 J kg−1 K−1
20 W mK−1
4 W mK−1

4. Results of simulation and discussion
4.1. Applied field dependence of trapped field

8
1 × 10−6 V m−1
1.3 T
0.23–1.83 × 109 A m−2
1 × 103 S cm−1

Figure 3(a) depicts the results of the simulation of the trappedfield Bz (r = 0) at the centre of the bulk surface as a function of
the applied field Bex for various α values in equation (6). The
Jc0 values at 40 K for α = 1.83 × 109 , 9.2 × 108 , 4.6 × 108
and 2.3 × 108 are, respectively, Jc0 = 1.33 × 109 , 6.6 × 108 ,3.3 × 108 and 1.6 × 108 A m−2 . The Jc0 values for a typicalsuperconducting bulk are, respectively, 2–3 × 109 A m−2 at
40 K and 1–2 × 108 A m−2 at 77 K [14]. Therefore, α =1.83 × 109 is a typical value of the bulk at 40 K. In allcases, the Bz (r = 0) starts to increase concomitantly withincreasing applied pulsed field Bex , becomes maximum andthen decreases with a further increase in Bex . The criticalC, at which the magnetic flux starts to trap at
applied field Bex
the bulk centre, increases concomitantly with increasing α .
The results presented in figure 3(a) can also be interpreted as
the Ts dependence of Bz (r = 0) for various Bex for the bulk
with constant α , which is consistent with the experimental
results shown in figure 2(a). Although the trapped field BzFC
achieved by FCM increases concomitantly with increasing
Jc , the trapped field Bz achieved by PFM is not necessarily
enhanced for higher Jc because of the large temperature rise,as described in section 4.2.

0.01, 1, 10 s
0.5 W mK−1

shows the trapped field Bz (r = 0) at the centre of
the GdBaCuO bulk (45 mm in diameter and 18 mm inthickness) after applying a pulsed field Bex at various starting
temperatures Ts [4]. The rise time τ of the pulsed field was
0.01 s. Bz (r = 0) at Ts = 70 K starts to increase for Bex 
2.7 T, takes a maximum at Bex = 4.4 T and then decreases
with increasing Bex . For lower Ts , the applied field, at which
the magnetic flux starts to be trapped at the bulk centre, andthe peak of the Bz (r = 0) value increase with decreasing Ts
because of the enhancement of the critical current density Jc .
Figure 2(b) presents a vertical cross-section of the Bz (r )profiles after applying a pulsed field at Ts = 70 K [4]. In
3

Supercond. Sci. Technol. 23 (2010) 105021

H Fujishiro and T Naito

Figure 3. (a) Results of simulation of applied field dependence of the trapped field Bz at the bulk centre (r = 0) for various α values
C( Ts = 40 K). The inset shows the time dependence of the normalized magnetic pulse. Arrows indicate the critical applied field Bex
for each α
8value (see the text). (b) Cross-section of the trapped field profiles for α = 4.6 × 10 at Ts = 40 K as a function of Bex (simulation).

t = 0.1 s and then decreases gradually to Bz = 1.7 T at thesteady state.
In figure 4(b), the temperature T (t) at the bulk peripheryincreases gradually with time at the ascending stage, e.g. T =78 K at t = 0.01 s. However, the temperature for r = 0
remains as 40 K. At the descending stage, the generated heatdiffuses to the bulk centre gradually. In line with the isothermal
temperature profile along the r -direction at t = 5 s, thetemperature decreases concomitantly with increasing time. In
this case, a temperature gradient exists along the z -direction
because the bottom of the spacing plate was fixed at 40 K.
Figures 5(a) and (b) show the time dependence of local
fields B(r, t) and the temperatures T (r, t) at r = 0, 10 and
20 mm on the bulk surface, respectively, which were re-plottedfrom figure 4. The time dependence of the magnetic pulse
Bex (t) is also shown. In figure 5(a), the local field Bz (t) near
the bulk periphery starts to increase rapidly; it then decreases.
On the other hand, Bz (t) at the inner position rises with a timelag, decreases gradually, and approaches the final value fort > 10 s.
In figure 5(b), the temperature T (t) near the bulk
periphery starts to rise rapidly. It then decreases. The timeat which the temperature starts to rise becomes delayed at
the inner measuring position. At the bulk centre (r = 0),
T (t) takes a maximum at t = 7 s and is independent of themeasuring position. It is noteworthy that the time at whichBz (t) becomes maximum is about two orders of magnitude
faster than the time at which T (t) reaches its maximum value.
These results are attributable to the lower thermal conductivityof the bulk and smaller heat transfer to the cold stage. After
Bz (t) reaches its maximum value, T (t) becomes maximum.
As a result, the local field decreases because of the flux creep.
For t > 7 s, Bz (t) becomes constant because the temperature
decreases.
Figures 6(a) and (b) respectively show a simulation of thetime dependence of the local field Bz (t) and temperature T (t)at the centre of the bulk surface (α = 4.6 × 108 ) after applying
pulsed fields of Bex = 5, 6 and 8 T. The peak height of Bz (t)In the simulation, the maximum temperature Tmax (r = 0)
at the centre of the bulk surface increased concomitantly withincreasing Bex , which was independent of the α value. These
results suggest that the generated heat was determined by thestrength of Bex and that the temperature rises adiabaticallybecause of the smaller heat transfer to the cold stage. The
result reproduces the experimental results, in which Tmax for
the YBaCuO bulk with a lower pinning force is nearly the sameas that for the GdBaCuO bulk with a higher pinning force [5].
Figure 3(b) presents a cross-section of the trapped fieldBz (r ) profile for α = 4.6 × 108 for various applied fields Bex .
The positions of r = 0 and 23 mm are, respectively, the centre
and edge on the bulk surface. The Bz (r ) profile changes fromconcave for lower Bex to convex for higher Bex and then theBz (r = 0) value decreases with further increases in Bex . Thesebehaviours are apparent for each α value and are consistent
with the experimental ones as shown in figure 2(b). In thisway, the results of the numerical simulation reproduced theexperimental ones qualitatively.
4.2. Time dependence of local field and temperature
Figures 4(a) and (b) respectively show the time evolution andspatial distribution of the local magnetic field Bz (t, r ) andthe temperature T (t, r ) on the bulk surface after applying apulsed field of Bex = 6 T for α = 4.6 × 108 ( Jc0 =3.3 × 108 A m−2 ). The left and right parts of each figure
respectively show ascending (t  0.01 s) and descending(t  0.01 s) stages. For the ascending stage, the magnetic
flux intrudes gradually into the bulk from the bulk peripheryand the magnetic gradient increases with approach to the bulk
centre at t = 0.01 s. It should be noted that the magnetic
flux concentration took place outside the bulk because of the
diamagnetism of superconducting bulk, and that a magnetic
field of 6.8 T was applied along the z -axis at r = 23 mm. For
the descending stage (t  0.01 s), the magnetic field decreased
gradually at the outer region with increasing time. On the other
hand, the local field near the bulk centre increases to 3 T at
4

Supercond. Sci. Technol. 23 (2010) 105021

H Fujishiro and T Naito

Figure 4. Time and distance dependence of (a) the local magnetic field B(t, r ) and (b) the temperature T (t, r ) on the bulk surface after
applying a pulsed field of Bex = 6 T at Ts = 40 K for α = 4.6 × 108 . The left and right parts of each figure respectively show the ascending(t  0.01 s) and descending (t  0.01 s) stages.

Figure 5. Time dependence of (a) the local field Bz (t) and (b) the temperature T (t) for r = 0, 10 and 20 mm on the bulk surface after
applying a pulsed field of Bex = 6 T (α = 4.6 × 108 ).

increases concomitantly with increasing Bex . However, thetime at which Bz (t) takes a maximum value is about 0.05 s; it
is independent of Bex . In fact, Bz (t) approaches the final value
for t > 10 s. The temperature T (t) increases and reaches its
peak at about 7 s, which is independent of Bex . The peak heightof T (t) increases with increasing Bex .
The results of the simulation shown in figures 5 and 6
well reproduce the experimental results reported in previous
papers [4, 5]. It is noteworthy that the simulation model used
for this study can qualitatively demonstrate Bz (t) and T (t) inthe bulk. However, some differences from the experimental
results do exist. In the experiments at Ts = 40 K, the magnetic
flux started to intrude and trap from 3 T at the centre of thebulk with 45 mm diameter and 15 mm thickness, which was as
high-performance as α = 1.83 × 109 [4]. In existing bulks,

an inhomogeneous Jc distribution exists, which arises fromthe crystal growth mechanism. The magnetic flux intrudes
through the lower Jc paths and traps them inhomogeneously.
The numerical simulation must be performed for the bulk withan inhomogeneous Jc distribution.
4.3. Application of long magnetic pulse
An extremely useful PFM technique is zero-field cooling(ZFC), of which the ascending and descending periods areof the order of 100–1000 s. Considering the Bean model,a magnetic field resembling that by FCM can be trapped by
ZFC after a magnetic field twice as high as that for FCM
is applied. We have reported that a trapped field Bz can beenhanced using a longer ascending magnetic pulse [12]. In
5

Supercond. Sci. Technol. 23 (2010) 105021

H Fujishiro and T Naito

Figure 6. Time dependence of (a) the local field Bz (t) and (b) the temperature T (t) at the centre of the bulk surface after applying a pulsed
field of Bex = 5, 6 and 8 T (α = 4.6 × 108 ).

Figure 7. (a) Applied field dependence of the trapped field Bz (r = 0) at the bulk centre for τ = 0.01, 1 and 10 s. (b) Cross-section of thetrapped field profiles for τ = 10 s as a function of applied pulsed field Bex (α = 1.83 × 109 ).

section 4.2, we presented a simulation for the bulk with lower
Jc (α = 4.6 × 108 ; Jc0 = 3.3 × 108 A m−2 at 40 K) using apulsed field with a rise time of τ = 0.01 s. In this subsection,
we present a simulation of Bz on the bulk with higher Jc(α = 1.83 × 109 ; Jc0 = 1.33 × 109 A m−2 at 40 K) using
a longer magnetic pulse field. We discuss the origin of the
Bz enhancement using a longer pulsed field and discuss the
direction for the Bz enhancement using PFM.
Figure 7(a) presents the calculated trapped field Bz (r = 0)on the bulk surface (α = 1.83 × 109 ) as a function of an appliedfield Bex with a rise time of τ = 1 and 10 s. The results forτ = 0.01 s are also shown. With increasing rise time τ , themaximum trapped field Bz (r = 0) increases concomitantlyCwith increasing τ . In addition, the applied field Bex, at which
the magnetic flux starts to be trapped at the bulk centre,
decreases. For example, for τ = 10 s, Bz (r = 0) reaches
C
4.3 T and Bex
decreases to 5 T. Figure 7(b) shows a typicalcross-section of the trapped field profiles for τ = 10 s. The
Bz (r ) profile changes from concave to convex with increasing
Bex and maintains a higher Bz value. It is noteworthy that theBz (r = 0) value is enhanced for the longer τ values.

Figures 8(a) and (b) show the time evolution of local fieldBz (t) and temperature T (t) at the centre of the bulk surface(α = 1.83 × 109 ) after applying a pulsed field of Bex = 8 Twith τ = 1 and 10 s, respectively. In both figures, the time tTP atwhich T (t)(r = 0) peaks is about 10 s, which is independentof the τ value because the thermal conductivity (κab , κc ) of
the bulk and that of the spacing plate (κcont ) are fixed. On theother hand, the time t BP , at which Bz (t)(r = 0) peaks, increases
concomitantly with increasing τ . In fact, t BP were, respectively
4 s and 15 s for τ = 1 and 10 s. For a longer pulse application,t BP increases and approaches tTP = 10 s. For τ = 1 s presented
in figure 8(a), Bz (t) decreases and saturates for t > 10 s,
at which T (t) decreases concomitantly with increasing time.
The results show that the flux creep was suppressed and thata large amount of the magnetic flux remained near the bulk
centre. For τ = 10 s presented in figure 8(b), t BP is nearly
equal to tTP ; their similar behaviour is visible. This scenario
is the main reason why the higher field was trapped for thelong magnetic pulse application. In this regard, the longpulse application is effective for trapping a higher magnetic
field. To realize a longer magnetic pulse experimentally, it is6Supercond. Sci. Technol. 23 (2010) 105021

H Fujishiro and T Naito

Figure 8. Time dependence of the local field Bz (t)(r = 0) and temperature T (t)(r = 0) for a pulsed magnetic field of Bex = 8 T with(a) τ = 1 s and (b) τ =10 s at Ts = 40 K. Time dependences of the applied magnetic pulse Bex (t) are also shown as a dotted line.

(a)Figure 9. Time dependence of (a) the local field Bz (t)(r = 0) and (b) the temperature T (t)(r = 0) for a pulsed magnetic field of Bex = 9 Twith τ = 0.01 s for various thermal conductivities (κab , κc , κcont ) (α = 1.83 × 109 ). (c) Trapped field profiles for respective conditions.

necessary to use a condenser bank to store a large amount ofelectrical energy. However, using such a device is unrealistic
for practical applications at present because such condenser
banks are prohibitively expensive.

4.4. Effect of thermal conductivity enhancement on thetrapped field
As explained in section 4.3, a long pulsed field application
is a possible solution to realize a large Bz value. However,7Supercond. Sci. Technol. 23 (2010) 105021

H Fujishiro and T Naito

intruding magnetic flux escapes by the flux creep becauseof the delayed temperature rise.
(3) A magnetic pulse application as long as τ = 1 or 10 s,
which enhanced the trapped field experimentally, reduces
the speed of the magnetic flux propagation. Then the
difference in the propagation speed between magnetic flux
and temperature becomes small. The results show that the
influence on the flux creep becomes slight and the trappedfield maintains a higher value. A long magnetic pulse
application during PFM is a possible solution to enhance
the trapped field.
(4) Based on analyses of long pulse application, we canenhance the trapped field for short pulse application
through enhancement of bulk thermal conductivity andcooling power, which are two or three orders of magnitude
higher than those of the real system. However, thatsimulation is unrealistic. Using numerical simulation, wemust explore another direction to enhance the trappedfield such as a multi-pulse technique for a bulk with an
inhomogeneous Jc distribution.

it is difficult to prepare a large condenser bank for a longer
magnetic pulse. Using a simulation in which we can investigate
an unrealistic case, exploration was performed to enhance Bz
using a conventional condenser bank with τ = 0.01 s. To
enhance the Bz value, the time difference between t BP and tTP
should be minimized. In this subsection, the effects of thethermal conductivity enhancement of the bulk and the spacing
plate on the trapped field Bz are discussed.
Figures 9(a) and (b) respectively show the time evolutionof a local field Bz (t) and temperature T (t) for the bulk withvarious thermal conductivities of the bulk (κab , κc ) and thespacing plate (κcont ) after applying a pulsed field of Bex =9 T at 40 K. Figure 9(c) presents the trapped field profiles
for each condition. For a real condition with κab = 20,
κc = 4 and κcont = 0.5 W mK−1 , the trapped field profileis shown as concave. When κcont increases to 1000 times(κcont = 500 W mK−1 ), the trapped field does not change,
although the temperature increase was reduced for t > 10 s.
When the thermal conductivity increases to 100 times (κab =
2000, κc = 400 W mK−1 ), the generated heat diffuses overthe bulk. However, the trapped field is convex but remains
small because of the smaller heat transfer to the cold stage.
When the thermal conductivities of both the bulk and thespacing plate increase, trapped field enhancement is predictedbecause the time delay between Bz (t) and T (t) becomes small.
These results suggest that similar phenomena took place for
applying a pulsed field with τ = 0.01 s to those for the longpulse application (τ = 10 s). However, the enhancementof the thermal conductivity for the Ag-doped bulk is as low
as 10–20% [15] and enhancement of the contact thermal
conductivity, i.e. enhancement of κcont cannot be realized in
actual experimental set-ups.

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[4] Fujishiro H, Hiyama T, Miura T, Naito T, Nariki S, Sakai N and
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[5] Fujishiro H, Hiyama T, Tateiwa T, Yanagi Y and Oka T 2007
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5. Summary
A theoretical simulation of electromagnetic and thermal fields
has been performed for a cryocooled superconducting bulk discwith homogeneous superconducting properties after applying amagnetic pulse. Important simulation results and conclusions
obtained from this study are summarized as follows.
(1) Simulation results for bulks with various critical current
densities Jc0 qualitatively reproduced experimental results
such as time and applied field dependences of the trappedfield Bz (t, r ) and the temperature rise T (t, r ) on the bulk
surface.
(2) For a rise time τ = 0.01 s of the magnetic pulse, which
is an ordinary experimental condition, the magnetic flux
propagation is about two orders of magnitude faster thanthe heat propagation at the bulk centre because of the
low thermal conductivity in the bulk. Consequently, the8