INSTITUTE OF PHYSICS PUBLISHING
SUPERCONDUCTOR SCIENCE AND TECHNOLOGY
Supercond. Sci. Technol. 15 (2002) 754–758
PII: S0953-2048(02)34458-0
Pulse field magnetization of a ring-shaped
bulk superconductor
Hiroyuki Ohsaki1, Tatsuya Shimosaki1 and Naoyuki Nozawa2
1
Department of Advanced Energy, Graduate School of Frontier Sciences,
The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
2
Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro,
Tokyo 153-8505, Japan
Received 16 July 2001
Published 18 April 2002
Online at stacks.iop.org/SUST/15/754
Abstract
We have studied the pulse field magnetization of a YBCO bulk
superconducting ring. In the experiments a 6 ms pulse magnetic field was
applied to a ring-shaped YBCO superconductor with a 46 mm outer
diameter, 16 mm inner diameter and 15 mm height, which was cooled by
liquid nitrogen (77 K). Then, trapped magnetic field profiles were measured
with a Hall sensor. For example, a single 1.2 T pulse field generated a
hollow profile of the trapped magnetic field. In addition, the finite element
method was used to solve the coupled problem of thermal and
electromagnetic fields, where the power-n model was adopted for the E–J
(E is the electric field and J is the current density) relation in the
superconductor. A comparison between the results of experiment and
analysis has shown that n = 6 gave good agreement between them. We have
also discussed the current density distribution, temperature distribution, etc.
1. Introduction
Recently high-performance (RE)BaCuO (RE = rare earth)
bulk superconductors have been developed [1–3]. Among
their applications, a trapped field magnet is considered to be
the most promising [3, 4]. For practical application of trapped
field magnets the magnetization should be more closely
investigated.
There are three typical ways to magnetize a bulk superconductor: field magnetization (FC), zero field magnetization
(ZFC) and pulse field magnetization (PFM). Of the three, PFM
is considered to be the most practical method because it needs
only a small and simple coil to apply a magnetic field and, consequently, no superconducting magnet is needed even to apply
a high magnetic field [5]. Therefore PFM enables the design of
a more compact and lighter electromagnetic system. However,
it is important to consider how to integrate the PFM coils in
the system. Numerical analysis is useful to obtain and understand electromagnetic phenomena in the superconductor during PFM and the design of a total system with in situ PFM coils.
We have studied pulse field magnetization of a ringshaped bulk superconductor because a ring-shaped bulk
superconductor can have a lower ratio of weight to trapped
magnetic flux than a cylindrical bulk superconductor. So
0953-2048/02/050754+05$30.00 © 2002 IOP Publishing Ltd
a lighter system could be designed with ring-shaped bulk
superconductors used as trapped field magnets. We have
performed PFM experiments on a ring-shaped bulk superconductor and numerical analysis of coupled problems of
electromagnetic and thermal fields during PFM. In this paper
we include the results of these PFM experiments, numerical
analysis and a comparison between them.
2. Experiment
In the PFM experiment we used a ring-shaped resinimpregnated YBCO bulk superconductor (QMG), shown in
figure 1, which had an outer diameter of 46 mm, an inner
diameter of 16 mm and a height of 15 mm. For an
examination of the fundamental macroscopic properties of
the superconductor, FC of the superconductor was performed,
where the superconductor was cooled in liquid nitrogen (77 K)
in a uniform magnetic field of 1.0 T. Then a trapped field profile
was measured with a Hall probe on a plane 0.5 mm above the
top surface of the superconductor. A measured trapped field
profile is shown in figure 2. The flux density in the centre was
0.60 T. The profile indicates a rather good homogeneity of the
superconductor. Furthermore the critical current density was
estimated to be 1.0 × 108 A m−2.
Printed in the UK
754
Pulse field magnetization of a ring-shaped bulk superconductor
Figure 1. The ring-shaped YBCO bulk superconductor used in the
PFM experiment. The outer diameter is 46 mm, the inner diameter
is 16 mm and the height is 15 mm. It was produced by the QMG
method with resin impregnation.
(This figure is in colour only in the electronic version)
(a)
(b)
Figure 3. (a) Experimental setup for PFM. (b) Coil current
waveform.
Figure 2. The trapped field profile measured after FC at 1 T. It was
measured on a plane 0.5 mm above the top surface of the
superconductor.
The experimental set-up for the PFM is shown in
figure 3(a). The superconductor in liquid nitrogen (77 K)
was placed between two electric coils, which were used to
apply a pulse magnetic field. Figure 3(b) shows the waveform
of the coil current, which was supplied by a pulse power supply
of 5 kV, 2 kA and 4 kJ with a capacitor bank of 320 µF. Two
experiments were performed separately where the pulse width
was 6 ms, and the peak flux density of the applied magnetic
field at the centre (Ba) was 0.6 T and 1.2 T.
Figure 4(a) shows the trapped field profiles measured
on a horizontal line 0.5 mm above the top surface of the
superconductor after PFM of 0.6 T and 1.2 T. A rapid
change in the externally applied magnetic field induced a large
shielding current in the bulk superconductor, which limits the
penetration of magnetic flux into the superconductor, so the
flux density in the centre region remained low. Although a
higher trapped field is required for practical applications, the
experimental results shown in figure 4(a) are sufficient for a
comparison of the results of experiment and analysis using
the finite element method (FEM). Figure 4(b) shows the flux
density profile on a plane for 1.2 T PFM. A slight asymmetry
coincides with the facet lines of the superconductor. The PFM
with a rapid change in the externally applied magnetic field
is sensitive to varying superconducting properties, so the flux
density profile after PFM becomes asymmetric. However,
the asymmetry of the flux density profiles was not taken into
account in the numerical analysis.
3. Numerical analysis method
The physical phenomena during PFM are described
by electromagnetic and thermal fields. The fundamental
equations on the axisymmetric coordinate (r–z) are as follows:
∂A
∂A
∂
∂ ν ∂
ν
= −σ
(rA) +
(1)
∂r r ∂r
∂z
∂z
∂t
∂ 2T
1 ∂T
∂ 2T
∂T
ρC
= κab 2 + κab
+ κc 2 + Q
∂t
∂r
r ∂r
∂z
(2)
where A is the magnetic vector potential, T is temperature,
ν is magnetic reluctivity, σ is electric conductivity, ρ is mass
755
H Ohsaki et al
(a)
Figure 5. Flux density profiles of FEM analysis where Ba = 1.2 T,
and n = 4, 6 and 8 with no heat generation in the superconductor
taken into account. The experimentally obtained profile is also
shown.
Jc0 (T ) = α 1 −
(b)
Figure 4. (a) Trapped field profiles measured on a horizontal line
0.5 mm above the top surface of the superconductor after PFM at
0.6 T and 1.2 T. (b) Flux density profile on a horizontal plane
0.5 mm above the top surface of the superconductor for a 1.2 T PFM.
density, C is specific heat, kab is thermal conductivity in the ab
plane, kc is thermal conductivity in the c-axis direction, and Q
is heat generation. The current density in the superconductor
J is given by the following equation:
∂A
J = σ E = −σ
(3)
∂t
where E is the electric field. The FEM analysis of the coupled
problem described by these equations was performed. σ is
nonlinear and dependent on E and J. The power-n model
is used to describe the nonlinear E–J characteristics of the
superconductor,
n
J
(4)
E = Ec
Jc
(5)
Jc = σinitial Ec
where Jc is the critical current density, σ initial is the initial
electric conductivity, and Ec is the reference electric field.
The dependence of Jc on temperature and flux density
is taken into account in the FEM analysis. The flux density
dependence of Jc was described by the Kim model,
B0
(6)
Jc = Jc0
|B| + B0
where Jc0 is Jc for B = 0 and B0 is constant. The temperature
dependence was given by the following equation:
756
T
Tc0
2 2
(7)
where Tc0 is the critical temperature at B = 0 and α is constant.
The heat generation in the superconductor Q is given by
Q = J · E.
An iterative calculation is necessary to obtain the
convergence of σ at each time step. A modified σ is calculated
from the following equation:
n−1
1
|Em | n
Jc
Jc
=
(8)
σnew = σinitial
|Em | Ec
Jm
where Em is the electric field at the mth iteration step. A new
σ at the (m + 1)th step is calculated by the following equation:
σm+1 = βσnew + (1 − β)σm
(9)
where β is the relaxation parameter.
The numerical analysis model on the axisymmetric
coordinate (r–z) has geometrical parameters consistent with
the experimental system. In the electromagnetic field analysis
a coil current is a half-cycle sinusoidal wave with a pulse
width of 6 ms. In the thermal field analysis two kinds
of boundary conditions were applied on the superconductor
surface: a fixed temperature boundary condition and an
adiabatic boundary condition. The fixed temperature boundary
condition is simple but rather different from the actual
conditions, while the adiabatic boundary condition seems to be
a good approximation for the short pulse used in PFM. In the
analysis, first-order triangular elements were used; the number
of elements was 8840, and the number of nodes was 4554.
The critical current density of the bulk superconductor was
assumed to be 1.0 × 108 A m−2 based on the FC experiment.
The parameters used in the analysis were as follows: σ initial =
1.0 × 1012 S m−1; Ec = 1.0 × 10−4 V m−1; B0 = 0.4 T; Tc0 =
92 K; α = 1.11 × 108 A m−2; ρ = 6.31 × 103 kg m−3 [6];
C = 1.32 × 102 kg m−3 [6]; kab = 14.5 W m−1 K−1 [7]; and kc =
3.0 W m−1 K−1 [7].
4. Numerical analysis results
Figure 5 shows the flux density profiles for Ba = 1.2 T, and
n = 4, 6 and 8 with no heat generation in the superconductor
Pulse field magnetization of a ring-shaped bulk superconductor
Figure 6. Flux density profiles where Ba = 1.2 T, and n = 4, 6 and 8
with heat generation taken into account using the fixed temperature
boundary condition.
(a)
(b)
(a)
(b)
Figure 7. Flux density profiles for n = 6 with heat generation in the
superconductor taken into account using the adiabatic boundary
condition: (a) Ba = 1.2 T; (b) Ba = 0.6 T.
taken into account, while figure 6 shows those with heat
generation using the fixed temperature boundary condition.
The profile obtained by the experiment is also shown in these
figures. The introduction of heat generation enhances the
Figure 8. Time evolution of flux density profiles for the same model
as shown in figure 7(a) while (a) increasing and (b) decreasing the
applied field.
penetration of magnetic flux and increases the flux density in
the whole region. However, the analysis results deviate largely
from the experimental curves. Figure 7 shows the flux density
profiles for n = 6 with heat generation in the superconductor
taken into account using the adiabatic boundary condition: (a)
for Ba = 1.2 T and (b) for Ba = 0.6 T. The n value of 6 almost
coincides with that reported in [8] on current limiting elements
using bulk superconductors that were prepared by the same
method as used in this research. Tokunaga et al [8] reported
that n was dependent on temperature and samples (7.9 at 77 K
and 5.4 at 87 K for one sample, and 5.7 at 77 K and 4.9 at
87 K for another sample). In the experiment presented in this
paper, during PFM the temperature rises in the superconductor
due to ohmic losses. Thus, it cannot simply be concluded that
the n of the superconductor is 6. However, this condition gave
rather good agreement between the numerical analysis and
experimental results.
Figure 8 shows the time evolution of flux density profiles
for the same model as shown in figure 7(a). The flux density
increased up to 1.2 T at the edge of the superconductor but
was still less than 0.4 T in the central region due to the
shielding current in the superconductor. Then the flux density
decreased with the decreasing applied field, and the maximum
flux density at t = 6 ms was about 0.35 T.
757
H Ohsaki et al
to a large part of the cross section, the applied field changes
direction and a positive current is induced from the peripheral
region. At t = 6 ms the current density distribution becomes
that shown in figure 9, and the maximum is still a little larger
than 3 × 108 A m−2.
The temperature of the superconductor increases while
a pulse field is applied. Figure 10 shows the temperature
distribution in the bulk superconductor at t = 6 ms. Joule
heating increased the temperature in the outer region and
the maximum temperature reached about 83 K at the outer
corners, which decreases the critical current density. The
temperature rise in the outer region influences the final
distribution of flux density.
5. Conclusions
Figure 9. Current density distribution in the superconductor at
t = 6 ms when the pulse magnetic field becomes zero.
We have studied the PFM of a YBCO bulk superconducting
ring. In the PFM experiments a single 6 ms pulse generated
a hollow profile of the trapped magnetic field. FEM analysis
was carried out to solve the coupled problem of the thermal and
electromagnetic fields, where the power-n model was adopted
for the E–J relation of the superconductor. A model, in which
n = 6 using the adiabatic boundary condition for thermal
field analysis, gave good agreement between the results of
experiment and analysis.
References
Figure 10. Temperature distribution in the bulk superconductor at
t = 6 ms.
Figure 9 shows the current density distribution in the
superconductor at t = 6 ms when the pulse magnetic field
becomes zero. A positive current is induced in the outer
region, and a negative current is induced in the middle region.
No current flows in the innermost region. When a pulse
magnetic field is applied, negative current is first induced
from the peripheral region, and then a negative current region
extends towards the centre. But the applied field is not large
enough to spread the negative current into the whole area of
the superconductor. When a negative current region extends
758
[1] Gruss S, Fuchs G, Krabbes G, Verges P, Schaetzle P,
Mueller K-H, Fink J and Schultz L 2001 Trapped field
beyond 14 Tesla in bulk YBa2Cu3O7−δ IEEE Trans. Appl.
Supercond. 11 3720–3
[2] Nariki S, Sakai N and Murakami M 2001 Processing of
GdBa2Cu3O7−γ bulk superconductor and its trapped
magnetic field Physica C 357–360 629–34
[3] Ikuta H, Yanagi Y, Yoshikawa M, Itoh Y, Oka T and Mizutani U
2001 Melt processing and the performance as a
superconducting permanent magnet of RE–Ba–Cu–O/Ag
(RE=Sm, Nd) Physica C 357–360 837–42
[4] Fujimoto H and Kamijo H 2001 Trapped magnetic field of a
mini-bulk magnet using YBaCuO at 77 K Physica C
357–360 852–5
[5] Ikuta H, Ishihara H, Hosokawa T, Yanagi Y, Itoh Y,
Yoshikawa M, Oka T and Mizutani U 2000 Pulse field
magnetization of melt-processed Sm–Ba–Cu–O Supercond.
Sci. Technol. 13 846–9
[6] Tsuchimoto M and Morikawa K 1999 Macroscopic numerical
evaluation of heat generation in a bulk high Tc
superconductor during pulsed field magnetization IEEE
Trans. Appl. Supercond. 9 66–70
[7] Fujishiro H, Ikebe M, Naito T, Noto K, Kobayashi S and
Yoshizawa S 1994 Anisotropic thermal diffusivity and
conductivity of YBCO(123) and YBCO(211) mixed
crystals-I Japan. J. Appl. Phys. 33 4965–70
[8] Tokunaga T, Morita M, Miura O and Ito D 1999 DC transport
properties of QMG current limiting elements IEEE Trans.
Appl. Supercond. 9 1343–6