IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 28, NO. 4, JUNE 2018

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Toward Optimization of Multi-Pulse, Pulsed Field
Magnetization of Bulk High-Temperature
Superconductors
Mark D. Ainslie, Senior Member, IEEE, Jan Srpcic , Difan Zhou , Hiroyuki Fujishiro
David A. Cardwell, and John H. Durrell

Abstract—Pulsed field magnetization (PFM) is the most practical method for magnetizing bulk superconducting materials as
trapped field magnets, but the record trapped field achieved by
PFM to date is still significantly less than the true trapped field
capability of these materials. In this paper, a flexible numericalmodeling technique based on the finite element method is used toprovide a comprehensive and realistic picture of multi-pulse PFM,
which has been shown to be effective in increasing the trappedfield/flux over a single pulse. First, the maximum trapped field capability of a representative sample is determined using two typesof numerical model simulating field-cooling and zero-field-coolingmagnetization. Next, various sets of magnetic field pulses are applied to the bulk to analyze multi-pulse PFM. An increase in thetrapped field can be achieved after a second pulse and to do so an
increased amplitude of applied field is required to maximize thetrapped field fully. The numerical analysis shows that this occurs
in subsequent pulses because it is more difficult for the magnetic
flux to penetrate the sample and there is a lower temperature rise.
Index Terms—Bulk high-temperature superconductors, finite element method, numerical simulation, pulsed field magnetization,
trapped field magnets.

I. INTRODUCTIONULK superconducting materials can be used as trappedfield magnets (TFMs) and magnetic fields greater than
17 T have been achieved in large, single-grain (RE)BCO (where
RE = rare earth or Y) bulk high-temperature superconductors[1], [2]. However, developing practical magnetizing techniquesis crucial to using them as TFMs in engineering applications,such as rotating machines, magnetic separation and magnetic
drug delivery systems [3]–[5].

B

Manuscript received September 11, 2017; revised December 23, 2017; accepted December 27, 2017. Date of publication January 1, 2018; date of current
version January 24, 2018. The work of M. D. Ainslie was supported in partby the Royal Academy of Engineering Research Fellowship and in part by theEngineering and Physical Sciences Research Council Early Career Fellowship
EP/P020313/1. The work of H. Fujishiro was supported by JSPS KAKENHI
under Grant 15K04646. (Corresponding author: Mark D. Ainslie.)
M. D. Ainslie, J. Srpcic, D. Zhou, D. A. Cardwell, and J. H. Durrell are with
the Bulk Superconductivity Group, Department of Engineering, University ofCambridge, Cambridge CB2 1PZ, U.K. (e-mail: mark.ainslie@eng.cam.ac.uk).
H. Fujishiro and K. Takahashi are with the Department of Physical Scienceand Materials Engineering, Faculty of Science and Engineering, Iwate University, Morioka 020-8551, Japan (e-mail: fujishiro@iwate-u.ac.jp).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TASC.2017.2788924, Keita Takahashi,

Pulsed field magnetization (PFM) is the most practicalmethod for magnetizing bulk samples as it requires shorter magnetization times (on the order of milliseconds), as well as a morecompact and less complicated magnetization fixture. In contrast,
the zero field cooling (ZFC) and field cooling (FC) techniques
are much slower and require bulky and expensive equipment(usually a large superconducting coil).
The trapped field produced by PFM is generally much smaller
than that of ZFC and FC, particularly at lower operating temperatures, because of the large temperature rise, ΔT, associatedwith the rapid dynamic movement of the magnetic flux in theinterior of the superconductor during the pulse [6].
It has been shown that multi-magnetic pulse techniques, suchas iteratively magnetizing pulsed-field with reducing amplitude(IMRA) [7] and multi-pulse with step-wise cooling (MPSC) [8],[9], can increase the trapped field/flux. The record trapped fieldachieved by PFM to date is 5.2 T at 29 K using a modified MPSC(MMPSC) technique [10]; unfortunately, this is still much lessthan the true trapped field capability of these materials. Numerical models [11]–[13] have been developed that qualitatively
reproduce some of the experimentally observed results; here
we extend this work to provide a comprehensive and realistic
picture of multi-pulse PFM.
In this paper, a 2D axisymmetric finite element method basedon the H-formulation, previously utilised by the authors for single pulse investigations [14]–[20], is extended to the case ofmulti-pulse PFM. Firstly, the maximum trapped field capabilityof a representative bulk sample is estimated using two typesof numerical model simulating FC/ZFC magnetization techniques. This establishes a baseline to compare the simulated
PFM results. Next, for constant operating temperatures of 77,
65 and 50 K, various sets of magnetic field pulses of amplitude up to 8 T are applied to the bulk. The numerical modelingtechnique is extremely flexible and can incorporate detailed information on the material properties. Consequently, the resultspresented will allow the optimization of multi-pulse PFM techniques to enhance the trapped field in bulk high-temperature
superconductors.
II. NUMERICAL SIMULATION FRAMEWORK
Fig. 1 shows the geometry of the 2D axisymmetric model
used in this paper to simulate the solenoid coil magnetization of1051-8223 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

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critical state model [25], [26]. The characteristic voltage criterion E0 = 1 × 10−4 V/m is also assumed.
A pulsed magnetic field, Bapp , is generated by a pulsed current flowing in the magnetizing coil (copper coil subdomain).
An integral constraint is applied to the copper coil subdomain,as described in [17], using the following Ipulse function (3):tt
Ipulse (t) = N · I0 exp 1 −(3)
ττFig. 1. 2D axisymmetric models for numerical simulation of multi-pulse PFM
using a solenoid coil.

a bulk superconductor of diameter (D) 30 mm and thickness (H)
15 mm. Here the sample is cooled from the bottom surface ofthe bulk with a cold head. The point from which the simulated
trapped field is taken is indicated by the red dot, corresponding
to the approximate location of a Hall sensor in an actual experimental setup (z = +0.5 mm above the centre of the top surfaceof the bulk).
The electromagnetic properties are simulated using the 2D
axisymmetric H-formulation, implemented in the commercial software package COMSOL Multiphysics 5.2a. This has
been used previously by the authors to simulate bulk hightemperature superconductors under various magnetization conditions [14]–[18], as well as FC magnetization of MgB2 [19]and iron-pnictide [20] bulks.
The governing equations are derived from Maxwell’s equations – namely, Faraday’s (1) and Ampere’s (2) laws:∇×E+

d (μ0 μr H)
dB
=∇×E+
=0
dt
dt

∇×H =J(1)(2)

where H = [Hr , Hz ] represents the magnetic field components,
J = [Jφ ] represents the current density and E = [Eφ ] represents the electric field. μ0 is the permeability of free space
and, for the superconducting and other sub-domains, the relative
permeability, μr , is simply 1.
The measured in-field critical current density, Jc (B, T ), ofa representative bulk high-temperature superconductor (15wt%
Ag-containing GdBa2 Cu3 O7−δ ) as presented in [17], [18] isused as input data for the model, which is input using a twovariable, direct interpolation [17], [21], [22].
The E-J power law (E α J n ) [23], [24] is used to simulate the non-linear electrical resistivity of the superconductor,
where n defines the steepness of the transition between the superconducting state and normal state and n = 20 as a typicalvalue for HTS materials that is a good approximation of Bean’s

where I0 is the peak magnitude of the current flowing in eachturn of the coil, N is the number turns, and τ = 15 ms is therise time of all pulses in this study, a typical value for standard
PFM experiments carried out at Iwate University [17], [27]. In
other works, the rise time varies between a few ms to several
10s of ms, depending on the type of magnetizing coil, its inductance/resistance, whether any ferromagnetic materials are usedand so on [17], [28], [29]. The magnetizing fixture has the same
coil constant, relating Bapp , the field at the centre of the magnetizing fixture with the bulk removed, to the applied current,
as presented in [17], [18].
Since the temperature of the superconductor can change significantly during PFM [30], the electromagnetic model is coupled with a thermal model, which is based on the following
thermal transient equation (4):dT
= ∇ · (k∇T ) + Q(4)dt
where Q is the heat source calculated from the product of the local electric field and current density, Q = Eφ · Jφ . The thermal
conductivity of the bulk is assumed to be κab = 20 W/(m·K)
along the ab-plane and κc = 4 W/(m·K) along the c-axis [27],[31]. The thermal conductivity of the indium sheet between thebottom surface of the bulk and the cold stage (thickness 0.2 mm)
is set to 0.5 W/(m·K) to represent the finite cooling power of therefrigerator and the thermal contact between the bulk and thecold stage, as described in [31].
The temperature rise during the PFM process takes place
adiabatically: a large proportion of the heat generated occurs
instantaneously, and the low thermal conductivity of the bulk
results in thermal diffusion that can be an order of magnitude
slower than the magnetic flux diffusion [31]. The measured,temperature-dependent specific heat data for the bulk and indium sheet, as presented in [17], is used. The densities of thebulk and indium sheet are 5900 and 7310 kg/m3 , respectively.
ρ·C

III. NUMERICAL SIMULATION RESULTS
A. Maximum Trapped Field Capability (FC/ZFC)
The FC/ZFC magnetization techniques allow us to determine
the maximum trapped field capability of a bulk sample. Here two
types of numerical models are used to estimate the maximum
trapped field, which establishes a reliable baseline to compare
the simulated PFM results. Table I shows the simulated trappedfields above the centre of the top surface of the bulk (r = 0 mm)
at a height z = +0.5 mm using these two models.
Model #1 (FC) is a stationary model that uses COMSOL’s
Magnetic Fields (mf) interface in the AC/DC module. The model

AINSLIE et al.: TOWARD OPTIMIZATION OF MULTI-PULSE, PFM OF BULK HIGH-TEMPERATURE SUPERCONDUCTORS

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TABLE I
FC & ZFC MAGNETIZATION TRAPPED FIELDS
MagnetizationFC
ZFC [5 T]FC
ZFC [10 T]FC
ZFC [20 T]

Time
77 K
—
t = 0 mint = 10 min
t = 20 min
65 K—
t = 0 mint = 10 min
t = 20 min
50 K—
t = 0 mint = 10 min
t = 20 min

Trapped Field
1.544 T1.546 T
1.263 T
1.223 T3.826 T3.827 T
3.256 T
3.158 T7.449 T
7.422 T6.577 T
6.405 T

uses the ‘External Current Density’ node to assume a current
density of Jc (B) flows through the 2D axisymmetric crosssection of the bulk with no flux creep. This is essentially a
Bean-like model that takes into account the in-field dependenceof Jc and finds a self-consistent solution by solving Ampere’s
law using the magnetic vector potential A [32].
Model #2 (ZFC) model is similar to that presented in [15],[20], where isothermal conditions are assumed because the magnetization process is slow (uniform applied magnetic field witha ramp rate of 50 mT/s). The finite n value used (20) accounts
for flux creep relaxation, resulting in a logarithmic decay of thetrapped field after the field is removed. There is good consistency between the FC and ZFC (t = 0 min) results, as should
be expected [30], and trapped fields of around 80% of this is
not an uncommon observation in practical experiments after
waiting for flux creep relaxation. The ZFC(t = 20 min) is usedhereafter as a baseline to compare the PFM results. It should be
noted that these models do not take into account any mechanicalstresses in the bulk that exist during magnetization due to thedevelopment of large Lorentz forces, F L = J × B, for magnetic fields of several tesla or above and can lead to mechanicalfracture without appropriate reinforcement [1], [2], [33], [34].
However, it is not difficult to couple such models together andhas been the subject of studies elsewhere [35]–[37].
B. Pulsed Field Magnetization – Single Pulse Results
Fig. 2 shows the simulated trapped fields at the centre of the
top surface of the bulk (r = 0 mm, z = +0.5 mm), Bt , after
PFM by single pulses of different amplitudes of up to 8 T at
operating temperatures of 77, 65 and 50 K using the coupledelectromagnetic-thermal model described in Section II. The results are given at t = 1 s, as well as t = 120 s. The latter results(t = 120 s) give ample time for both flux creep relaxation andfor the temperature of the bulk to return to the operating temperature, and are used as the input data for subsequent pulses in
the multi-pulse models in the following section. The maximum
Bt (t = 120 s) at 77, 65 and 50 K correspond to approximately
85%, 49% and 29% of the ZFC(t = 20 min) value (see Table I),
respectively.

Fig. 2. Numerical simulation results for the trapped magnetic field at thecentre of the top surface of the bulk (r = 0 mm, z = +0.5 mm) after PFM bysingle pulses of amplitude up to 8 T at operating temperatures of 77, 65 and50 K. The results are given at t = 1 s and 120 s, and the pulse has a rise time,τ = 15 ms.

C. Pulsed Field Magnetization – Multi-Pulse Results – 2nd
Pulse
In this section, the influence of subsequent pulses on thetrapped field and temperature rise in the bulk are investigated
for the multi-pulse PFM technique. The single pulse results fromthe previous section at t = 120 s are used as the input data anda 2nd pulse is applied using (3), but time-shifted such that thepulsed current is Ipulse (t–t2n d ). Since the magnetic history has
an influence on the magnetic flux penetration and temperaturerise during subsequent pulses [38], a number of specific cases
are investigated here, namely:
1) PM: partially-magnetized (the bulk has an ‘M-shaped’[38], [39] trapped field profile across the top surface andlittle magnetic field is seen at the centre).
2) UM: under-magnetized (the bulk is almost fully magnetized, but does not exhibit a full conical shape).
3) FM: fully-magnetized (the bulk is fully magnetized =full conical shape, and attains the maximum trapped fieldvalue for all pulses).
4) OM: over-magnetized (the bulk is fully magnetized, but
the trapped field is less due to a larger temperature risefrom the higher amplitude of the applied pulse, higher
than that of the preceding FM case).
Fig. 3 shows the simulated trapped field profiles across the top
surface of the bulk (z = +0.5 mm) after PFM, corresponding
to the 65 K results at t = 120 s shown in Fig. 2 in Section III-B.
These are examples of the specific cases above that will be used
as input scenarios for the 2nd pulse investigations that follow.
Figs. 4–6 show the simulated trapped fields at the centre of
the top surface of the bulk (r = 0 mm, z = +0.5 mm), Bt , after
PFM (t = 1 s) by the 2nd pulse at operating temperatures of 77,
65 and 50 K, respectively. For all operating temperatures, thetrapped field after the 2nd pulse exhibits two particular characteristics: 1) an increased trapped field, Bt , when the bulk isfully magnetized, which attains a maximum value when the 1stpulse results in full magnetization, and 2) an increased activation
field: the applied field, Bapp , required to maximize the trappedfield fully [27]. These observations are more pronounced as the6800207

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Fig. 3. Numerical simulation results of the trapped magnetic field acrossthe top surface of the bulk (z = +0.5 mm) after PFM, corresponding to the
65 K results at t = 120 s shown in Fig. 2. These examples show the partiallymagnetized (PM, B a p p = 3 T), under-magnetized (UM, B a p p = 4 T), fullymagnetized (FM, B a p p = 4.5 T), and over-magnetized (OM, B a p p = 5.5 T)that are used as input scenarios for the 2nd pulse investigations.

Fig. 6. Numerical simulation results for the trapped magnetic field at thecentre of the top surface of the bulk (r = 0 mm, z = +0.5 mm) after PFM by
a 2nd pulse at 50 K (t = +1 s from pulse start). Three specific cases of initial
magnetization (see Fig. 3) are used as the input data for the 2nd pulse (the OM
case is omitted).

Fig. 4. Numerical simulation results for the trapped magnetic field at thecentre of the top surface of the bulk (r = 0 mm, z = +0.5 mm) after PFM by
a 2nd pulse at 77 K (t = +1 s from pulse start). Four specific cases of initial
magnetization (see Fig. 3) are used as the input data for the 2nd pulse.

Fig. 7. Numerical simulation results for the average temperature of the bulk,
T ave , for the four specific cases shown in Fig. 3, corresponding to single pulses
at 65 K.

Fig. 5. Numerical simulation results for the trapped magnetic field at thecentre of the top surface of the bulk (r = 0 mm, z = +0.5 mm) after PFM by
a 2nd pulse at 65 K (t = +1 s from pulse start). Four specific cases of initial
magnetization (see Fig. 3) are used as the input data for the 2nd pulse.

Fig. 8. Numerical simulation results for the maximum temperature of thebulk, T m a x , for the four specific cases shown in Fig. 3, corresponding to single
pulses at 65 K.

operating temperature is lowered, and the 2nd pulse is less effective when the bulk is only partially-magnetized by the 1stpulse.
Examination of the numerical results in detail permits it to be
deduced that this behaviour occurs because it is more difficult
for magnetic flux to penetrate the bulk due to the existing trappedfield from the 1st pulse (hence, an increased Bapp is necessary),
which has a corresponding induced supercurrent that flows in
the opposite direction. This also results in a lower temperaturerise because of the reduced dynamic movement of the magnetic
flux within the bulk (hence, Bt is higher) [13]. Figs. 7 and 8shows the simulated average temperature of the bulk, Tave , andAINSLIE et al.: TOWARD OPTIMIZATION OF MULTI-PULSE, PFM OF BULK HIGH-TEMPERATURE SUPERCONDUCTORS

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Fig. 9. Numerical simulation results for the maximum average temperatureof the bulk, T ave , m a x , for each of the sets of 2nd pulses, for the four specificcases of initial magnetization (see Fig. 3).

Fig. 11. Numerical simulation results of the current density distribution during
the pulse rise time (t = 5, 10 ms) for a single pulse of 4.5 T (left; corresponding
to the FM case in Fig. 3) and for a 2nd pulse of 4.5 T (right) with results fromthe single pulse used as the initial conditions (1st pulse).

Fig. 10. Numerical simulation results for the maximum temperature of thebulk, T m a x , for each of the sets of 2nd pulses, for the four specific cases of
initial magnetization (see Fig. 3).

maximum temperature, Tm ax , respectively, for the four specificcases shown in Fig. 3, corresponding to single pulses at 65 K.
This shows the clear evolution of Tave and Tm ax , corresponding
to heating during the rise and fall of the pulse, then thermal
diffusion and subsequent cooling, which occurs over a much
longer time scale than the magnetic flux diffusion, during therelaxation time to t = 120 s. Fig. 9 shows the simulated maximum average temperature, Tave,m ax , for each of the sets of 2nd
pulses, which is compared with Tave,m ax for the initial pulses
from Fig. 7. In addition, Fig. 10 shows Tm ax for each of thesets of 2nd pulsed, which is also compared with Tm ax for the
initial pulses from Fig. 8. There is a similar trend of increasing
temperature with increasing amplitude of applied field, as per
Figs. 7 and 8, but the temperature rise is lower during the 2nd
pulse in all cases for the same amplitude of applied field as the
1st pulse.
Fig. 11 (left) shows the simulated current density distribution
during the pulse rise time at t = 5 and 10 ms for a single pulse of
4.5 T, corresponding to the FM case in Fig. 3. Also shown (right)
is the current density distribution at the same times during a 2nd
pulse of 4.5 T when the 4.5 T single pulse results are used as theinitial conditions (1st pulse). The induced supercurrent flowing
in the opposite direction from the 1st pulse is clearly shown and
it is interesting to note that the maximum Bt obtained for the
single/1st pulse results in a small region near the centre where
current flows in the opposite direction to the main magnetizing

Fig. 12. Numerical simulation results of the temperature distribution during
the pulse rise time (t = 5, 10 ms) for a single pulse of 4.5 T (left; corresponding
to the FM case in Fig. 3) and for a 2nd pulse of 4.5 T (right) with results fromthe single pulse used as the initial conditions (1st pulse).

current. This was also observed in [38] for the optimum trappedfield from stacks of coated conductors. There is some asymmetry
in the magnetizing current because the bulk is cooled fromthe bottom, resulting in a non-uniform temperature distribution.
Fig. 12 shows the corresponding temperature distribution for thesame cases, which shows clearly the difference in temperature
related to changes in the magnetic flux penetration. Fig. 13
shows the simulated magnetic flux penetration across the centreof the bulk during the rising pulse to its peak at 5 ms increments(τ = 15 ms) for a single pulse of 4.5 T (solid lines) and for a 2nd
pulse of 4.5 T (dashed lines) with results from the 4.5 T single
pulse used as the initial conditions (1st pulse). The magnetic field
profile within the bulk is consistent with the current density in6800207

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 28, NO. 4, JUNE 2018

ACKNOWLEDGMENT
Additional data related to this publication are available atthe University of Cambridge data repository (https://doi.org/10.
17863/CAM.13608).
APPENDIXPULSED FIELD MAGNETIZATION – MULTI-PULSERESULTS – 3RD PULSE

Fig. 13. Numerical simulation results for the magnetic flux penetration acrossthe centre of the bulk during the rising pulse to its peak (5 ms increments;
τ = 15 ms) for a single pulse of 4.5 T (solid lines) and for a 2nd pulse of 4.5 T(dashed lines) with results from the single pulse used as the initial conditions(1st pulse).

Fig. 11 and it is clear that it is more difficult for the magnetic flux
to penetrate the bulk during the 2nd pulse. Hence, an increased
Bapp is required to fully magnetize the bulk for the 2nd pulse.
These observations have important implications for the determination of the optimum set of pulses that results in the
maximum trapped field at a particular operating temperature.
It has been previously shown experimentally [29], [39] that an
initially partially-magnetized bulk (with an ‘M-shaped’ trappedfield profile) can result in high trapped fields; this is not observed
within this numerical simulation framework. This phenomenon
may be strongly related to flux jumps during the pulse rise timethat can assist the PFM process [18], [28], [29], [40], [41],which also are not observed within this framework. This suggests that some techniques, such as the iteratively magnetizing
pulsed-field method with reducing amplitude (IMRA) [7], may
be not be as effective without assistive flux jumps.
In the Appendix, Figs. 14–16 show the results for Bt for a 3rd
pulse for operating temperatures of 77, 65 and 50 K, respectively.
Similar results are found for the 3rd pulse as for the 2nd pulses;
however, the increase in Bt is marginal, suggesting saturation
has been reached.

Fig. 14. Numerical simulation results for the trapped magnetic field at thecentre of the top surface of the bulk (r = 0 mm, z = +0.5 mm) after PFM by a
3rd pulse at 77 K (t = +1 s from pulse start). The initial conditions for the 2nd
and 3rd pulses correspond to the FM case for the preceding pulse (1st = 2.5 T,
2nd = 2.75 T).

Fig. 15. Numerical simulation results for the trapped magnetic field at thecentre of the top surface of the bulk (r = 0 mm, z = +0.5 mm) after PFM by a
3rd pulse at 65 K (t = +1 s from pulse start). The initial conditions for the 2nd
and 3rd pulses correspond to the FM case for the preceding pulse (1st = 4.5 T,
2nd = 5 T).

IV. CONCLUSION
In this paper, a 2D axisymmetric finite element method basedon the H-formulation is extended to investigate the optimizationof multi-pulse PFM techniques. An increase in the trapped field
can be achieved after a 2nd pulse and this exhibits two particular characteristics, which are more pronounced at lower operating temperatures: 1) an increased trapped field, Bt , when thebulk is fully magnetized, which attains a maximum value whenthe 1st pulse results in full magnetization, and 2) an increased
applied field to fully magnetize the sample. The numerical analysis shows that this occurs because it is more difficult for themagnetic flux to penetrate the sample and there is a lower temperature rise during the 2nd pulse. A partially-magnetized bulk
(with an ‘M-shaped’ trapped field profile) may only be usefulfor pre-magnetization in the case that flux jumps occur during(and assist) the PFM process.

Fig. 16. Numerical simulation results for the trapped magnetic field at thecentre of the top surface of the bulk (r = 0 mm, z = +0.5 mm) after PFM by
a 3rd pulse at 50 K (t = +1 s from pulse start). The initial conditions for the
2nd and 3rd pulses correspond to the FM case for the preceding pulse (1st =6 T, 2nd = 7 T).

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