IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 29, NO. 5, AUGUST 2019

6801605

Design Optimization of a Hybrid Trapped Field
Magnet Lens (HTFML)
Sora Namba , Hiroyuki Fujishiro , Mark D. Ainslie , Senior Member, IEEE, Keita Takahashi,
Devendra K. Namburi , Difan Zhou , and Tomoyuki Naito

Abstract—The concept of a hybrid trapped field magnet lens
(HTFML) was recently proposed by the authors, which consistsof a trapped field magnet (TFM) cylinder exploiting the “vortex
pinning effect,” combined with a superconducting bulk magnetic
lens exploiting the “diamagnetic shielding effect.” This HTFML
can generate, within its bore, a magnetic field higher than theapplied magnetic field, even after external field decreases to zero. In
this paper, a design optimization of the inner GdBaCuO magnetic
lens within the GdBaCuO TFM cylinder was carried out using
numerical simulations based on the finite-element method, in orderto maximize the concentrated magnetic field. The HTFML with an
optimized shape and size achieved a concentrated magnetic field of
Bc = 5.6 and 12.8 T at the center of the lens for applied magnetic
fields of Ba p p = 3 and 10 T, respectively. A maximum tensilestress of +135 MPa exists in the outer GdBaCuO TFM cylinder
during the magnetizing process for Ba p p = 10 T, which exceeds thefracture strength of the bulk. This result suggests that mechanicalreinforcement is necessary to avoid mechanical fracture under suchhigh magnetic field conditions.
Index Terms—Hybrid trapped field magnet lens (HTFML), mechanical stress, numerical simulation, REBaCuO bulk superconductor, trapped field magnets.

I. INTRODUCTIONEBACUO (where RE is a rare earth element or Y) superconducting bulks can trap a large amount of magnetic fluxin pinning centers using the “vortex pinning effect,” and can beused as strong trapped-field magnets (TFMs) capable of generating over 17 T [1], [2], which is attractive for various practicalapplications. The performance of TFMs by field-cooled magnetization (FCM) is mostly determined by the critical current
density, Jc (B, T), and the diameter of the bulk material. In addition, a magnetic lens has been also investigated using the“diamagnetic shielding effect” as a magnetic field amplification

R

Manuscript received October 26, 2018; accepted February 7, 2019. Date of
publication February 13, 2019; date of current version March 13, 2019. This research was supported by JSPS KAKENHI under Grant 15K04646. The work of
M. D. Ainslie was supported by the Engineering and Physical Sciences ResearchCouncil (EPSRC) Early Career Fellowship EP/P020313/1. (Corresponding
author: Sora Namba.)
S. Namba, H. Fujishiro, K. Takahashi, and T. Naito are with the Departmentof Physical Science and Materials Engineering, Faculty of Science and Engineering, Iwate University, Morioka 020-8551, Japan (e-mail:,1123sky7@gmail.
com; fujishiro@iwate-u.ac.jp; madokureo@yahoo.co.jp; tnaito@iwate-u.ac.jp).
M. D. Ainslie, D. K. Namburi and D. Zhou are with the Bulk Superconductivity Group, Department of Engineering, University of Cambridge, Cambridge
CB2 1PZ, U.K. (e-mail:,mark.ainslie@eng.cam.ac.uk).
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.2019.2899255

technique, in which the magnetic flux generated from an outersuperconducting coil magnet is concentrated in the center of themagnetic lens [3]. A concentrated field of Bc = 12.42 T has
been achieved at 20 K for a background field of 8 T using a
GdBaCuO magnetic lens [4], and Bc = 30.4 T has been realized at the center of such a lens in a higher background field of
28.3 T [5]. Although the magnetic lens is a promising device
to concentrate high magnetic fields, continuous operation of theexternal magnetic field source is required to produce the higher
concentrated magnetic field.
Recently, we proposed a new concept of a hybrid trappedfield magnet lens (HTFML), which consists of a TFM cylinder
exploiting the vortex pinning effect, combined with a bulk magnetic lens exploiting the diamagnetic shielding effect [6]. The
HTFML can reliably generate a concentrated magnetic field in
the center of the magnetic lens higher than the trapped field of
the TFM and the applied magnetic field, even after the external
magnetizing field decreases to zero.
In this paper, the optimization of the shape and size of the inner GdBaCuO magnetic lens in the HTFML is carried out using
numerical simulations, which is inserted into an outer GdBaCuO
TFM cylinder. The mechanical stress during the magnetization
process, resulting from the combination of the current and magnetic field in the HTFML, is also analyzed.
II. NUMERICAL SIMULATION FRAMEWORKIn this paper, the electromagnetic model presented in our
original paper describing the concept of the HTFML [6] is coupled with a mechanical model and is used to further evaluate
the performance of the HTFML during the magnetizing process. Fig. 1 shows the cross-section of the three-dimensional(3D) numerical model of the HTFML. Assuming that the 10 T
superconducting solenoid magnet (170 mm in outer diameter(O.D.), 120 mm in inner diameter (I.D.), and 200 mm in height(H)) located in Iwate University is used as the magnetizing coil,the HTFML is assembled in a room temperature bore 100 mm
in diameter. The GdBaCuO magnetic lens is inserted inside the
GdBaCuO TFM cylinder (60 mm in O.D., 40 mm in I.D., and
60 mm in H). The magnetic lens part has two slits 10° wide
along the y-direction, whose role is to allow the magnetic flux
to intrude into and be concentrated at the center of the lens.
To optimize the shape and the size of the magnetic lens, fivevariable parameters (outer diameter, OD, inner diameter, ID,
inner diameter 2, ID2, outer height, OH, and inner height, IH)

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6801605

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 29, NO. 5, AUGUST 2019

Fig. 1. Cross-section of three-dimensional numerical model of an HTFML ina magnetizing solenoid coil.

TABLE I
FIVE VARIABLE PARAMETERS OF THE GdBaCuO MAGNETICLENS FOR THE DESIGN OPTIMIZATION

procedure of the HTFML comprises the following four stages
in this simulation.
1) The temperature of the outer GdBaCuO TFM cylinder isheld at T = 100 K (> Tc = 92 K), while the GdBaCuO
magnetic lens is cooled to 20 K (step 0).
2) The external magnetic field, Bex , is increased linearly up
to the maximum magnetic field Bapp = 3 or 10 T at
+0.222 T/min (steps 0–5). The magnetic lens is magnetized under zero-field cooling (ZFC) conditions at 20 K,
where the magnetic flux is concentrated in the bore of thelens.
3) The GdBaCuO TFM cylinder is then also cooled to 20 K(<Tc ) under Bapp , which transitions into the superconducting state (steps 5–6).
4) Finally, Bex is ramped down from Bapp to zero at−0.222 T/min (steps 6–11). In this process, the GdBaCuO
TFM cylinder is magnetized by FCM, which then acts likean external magnetizing coil magnet in a conventional
magnetic lens setup, e.g., [3]. As a result, the magnetic
flux generated from the TFM cylinder is concentrated in
the magnetic lens and a magnetic field higher than Bappis still present after the external Bex decreases to zero.
The n-power law (n = 20) is used to describe the nonlinear
E-J characteristic of the bulk [7]. The Jc (B) characteristics at
20 K of the GdBaCuO bulk is defined by the following equation
proposed by Jirsa et al. [8]B
Jc (B) = Jc1 exp −
BL
 

k 
B
B
1
1−
exp
, (1)
+ Jc2
Bm ax
k
Bm ax
where Jc1 = 9.0 × 109 A/m2 , BL = 1.5 T, Jc2 = 5.4 × 109 A/m2 ,Bm ax = 8.0 T and k = 0.5 are used for the fitting parameters at
20 K [6], [9]. Elastic behavior can be expressed by Hooke’s law,in which the stress tensor, σij , is linearly proportional to strain
tensor, εij , as follows,

Fig. 2. Time step sequence of the temperature, T, of the GdBaCuO magnetic
lens and GdBaCuO TFM cylinder (left vertical axis), and the external magnetic
field, Be x (right vertical axis).

are changed independently, which are indicated in Fig. 1. The
detailed values of the size of each parameter (OD ࣘ 36 mm,
ID2 ࣘ 36 mm, OH ࣘ 60 mm) are shown in Table I, in whichan ID = 10 mm is assumed for all optimization procedures toinsert a Hall sensor for experiments planned in future work.
In the case that both magnetic lens and TFM cylinder are
made of the same bulk superconducting material, it is necessary
to control the temperature of each part independently. Fig. 2
shows the schematic view of the time step (TS) dependence ofthe temperature, T, of the magnetic lens and the cylindrical TFM,
and that of the external magnetic field, Bex . The magnetizing

σij = λ · ek k · δij + 2G · eij ,

(2)

λ=

υE
,
(1 − υ) (1 + 2υ)

(3)

G=

E
,
2 (1 + υ)(4)

where λ and G are the Lame’s coefficients, δij is the Kronecker
delta. Young’s modulus, E = 100 GPa, and Poisson ratio, V =0.33, are assumed as the mechanical parameters for the REBaCuO bulk [10]. The REBaCuO bulk is also assumed to be an
isotropic and linear elastic material. The electromagnetic hoop
stress, σ θ , is calculated for each time step of the magnetizing
process.
III. RESULTS AND DISCUSSION
A. Design Optimization of the GdBaCuO Magnetic Lens
Fig. 3 shows the concentrated magnetic field, Bc , at the center of the magnetic lens (x = y = z = 0) for steps 5 and 11 forNAMBA et al.: DESIGN OPTIMIZATION OF A HYBRID TRAPPED FIELD MAGNET LENS

Fig. 3. The concentrated magnetic field, Bc , at the center of the magnetic lensand the magnetic field amplification factor, Bc /Ba p p , as a function of (a) OD,(b) ID2, (c) OH, (d) IH for Ba p p = 3 T.

Bapp = 3 T, as a function of each variable. The magnetic field
amplification factor, Bc /Bapp , is also shown on the right axis. In
Fig. 3(a), the Bc vs OD relationship is shown, in which the ID2
value is defined as OD minus 6 mm (ID2 = OD − 6 mm),because the ID2 changes depending on the OD value. The Bc
value increases linearly with increasing OD, and a Bc /Bapp of6801605

2.4 is achieved for OD = 36 mm at step 5, where only themagnetic lens is working and the magnetic flux produced by
the magnetizing coil is efficiently concentrated in the bore of
the lens. The Bc value at step 11, at which FCM of the outer
TFM is finished, is lower than that at step 5, since the external
magnetic field decreases during the descending stage and because of flux creep. However, a magnetic field higher than Bapp
= 3 T still remains for all cases of OD, which is the advantageof the HTFML.
Fig. 3(b) shows the ID2 dependence of Bc and Bc /Bapp atsteps 5 and 11 for Bapp = 3 T. For ID2 = 10 mm, i.e., acylindrical-shaped lens, which is a desirable shape from theviewpoint of machining the lens, the Bc /Bapp is as small as 1.3.
The Bc value increases with increasing ID2, similar to the Bc vs
OD relationship shown in Fig. 3(a).
Fig. 3(c) shows the OH dependence of Bc and Bc /Bapp atsteps 5 and 11 for Bapp = 3 T. With increasing OH, the Bc
value increases, takes a maximum for OH = 45 mm and thendecreases slightly for OH = 60 mm, which is the same heightas the outer GdBaCuO TFM cylinder. However, there is no
significant difference between OH = 30–60 mm. This resultmeans that, in the present situation, OH = 30 mm is enoughto maximize the Bc and Bc /Bapp values and to provide a costeffective HTFML.
Fig. 3(d) shows the IH dependence of Bc and Bc /Bapp atsteps 5 and 11 for Bapp = 3 T. For IH = 8 mm, the highest
Bc of 5.67 T is achieved at step 11. The inset of Fig. 3(d)shows the cross-section of the magnetic lens for IH = 8 mm,
where the intersection of the lines extending from the lens taper
corresponds to the lens center (x = y = z = 0). This result
suggests that an appropriate taper angle for the magnetic lens
can maximize the Bc /Bapp value for a cylindrical TFM.
From the viewpoints of both the magnetic field amplificationand cost-effectiveness, and based on these results, it was decided
that the optimum shape of the GdBaCuO magnetic lens, whichis inserted in the GdBaCuO TFM cylinder (60 mm in O.D.,
40 mm in I.D. and 60 mm in H), should be designed with
OD = 36 mm, ID2 = 30 mm, ID = 30 mm, OH = 30 mm and
IH = 8 mm.
Following the same procedure, It was also confirmed that the
optimized shape determined by Bapp = 3 T was also effective
for the case of Bapp = 10 T, and for the case of outer MgB2
TFM cylinder. That is, the optimum shape of the inner lensdepends not on the particular bulk materials used and/or thesuperconducting properties of the outer TFM cylinder, but only
the geometry of the magnetic lens.
Figs. 4(a) and 4(b), respectively, show the time step dependence of the magnetic field profile along the x-direction acrossthe center of the lens with the optimized design for Bapp = 3 and
10 T. The contour maps of the magnetic field within the magnetic lens at step 5 are also shown in each figure. The Bc value
is enhanced up to 6.9 and 15.5 T at the central region (−5 < x< 5) at step 5 for Bapp = 3 and 10 T, respectively. The magnetic
flux penetration into the GdBaCuO lens from both the inner andouter peripheries can be observed with increasing applied field,
as shown in the inset contour maps. As a result, the shielding
effect of the GdBaCuO lens weakens in external magnetic fields

6801605

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 29, NO. 5, AUGUST 2019

Fig. 5. Time step dependence of the electromagnetic hoop stress profiles, σ θ ,across the center of the HTFML (z = 0) for (a) Ba p p = 3 T and (b) Ba p p =10 T. The contour map of σ θ at step 11 is also shown in each figure.

the descending stage (steps 6–11), Bc /Bapp gradually decreases
and final values of 1.86 and 1.28 are achieved for Bapp = 3 and
10 T, respectively.
B. Electromagnetic Hoop Stress, σ θ , During Magnetization

Fig. 4. Time step dependence of the magnetic field, B, profiles across thecenter of the HTFML (z = 0) for (a) Ba p p = 3 T and (b) Ba p p = 10 T. Contour
maps of the magnetic field profiles at step 5 are also shown in each figure. Timestep dependence of (c) the concentrated magnetic field, Bc and (d) the magnetic
field amplification factor, Bc /Ba p p for Ba p p = 3 and 10 T.

as high as 10 T. During the descending stage (steps 8 and 11
in the figure), the GdBaCuO TFM cylinder maintains a trappedfield similar to Bapp at the final step (step 11) as it is magnetizedby FCM. The Bc value gradually decreases and settles to final
values of Bc = 5.6 and 12.8 T, respectively, at the center of thelens.
Figs. 4(c) and 4(d), respectively, show the time step dependence of Bc and Bc /Bapp at the center of the GdBaCuO lens for
Bapp = 3 and 10 T. In the ascending stage (steps 0–5), Bc /Bappincreases linearly and reaches 2.3 (Bc = 6.9 T) for Bapp =3 T. For Bapp = 10 T, however, the Bc /Bapp vs TS curve gradually bends and the Bc /Bapp value at step 5 decreases to 1.55
(Bc = 15.5 T) due to the weakened shielding effect of the lens. In

In high-field applications of REBaCuO bulks, mechanicalfracture due to large electromagnetic forces must be avoided.
Using the coupled electromagnetic-mechanical model, the hoop
stress, σ θ , present in the magnetic lens and TFM cylinder inthe HTFML was estimated. Figs. 5(a) and 5(b) show the time
step dependence of the electromagnetic hoop stress profile, σ θ ,across the center of the lens (z = 0) with the optimized design for Bapp = 3 and 10 T, respectively. During the ascending
stage (steps 0–5), a tensile stress exists at the inner periphery of
the lens, corresponding to the flux penetration area. The shielding current due to the lens’s diamagnetic property results in acompressive stress in the body of the lens.
On the other hand, during the descending stage (steps 6–11),
a large tensile stress in the TFM cylinder, resulting from theLorentz force, increases with increasing time step. The contour
map of σ θ at step 11 is shown in each figure, which details
the stress concentration at the inner edge periphery of the TFM
cylinder. The maximum σ θ value reaches +135 MPa at step
11 for Bapp = 10 T, which exceeds the fracture strength of
typical Ag-doped REBaCuO bulk materials (50 ∼ 70 MPa)[11], [12]. In the magnetic lens, a maximum tensile stress of+35 MPa exists at the inner periphery of the magnetic lens and acompressive stress exists in other regions. These results suggest

NAMBA et al.: DESIGN OPTIMIZATION OF A HYBRID TRAPPED FIELD MAGNET LENS

that adequate mechanical reinforcement is necessary, at least for
the GdBaCuO TFM cylinder, but possibly the magnetic lens as
well. The demonstration experiment of an HTFML consisting
reinforced GdBaCuO bulks will be conducted on the basis ofthese simulation results.
IV. CONCLUSION
In this paper, the optimization of the shape and size of theinner GdBaCuO magnetic lens in a hybrid trapped field magnet
lens (HTFML) was carried out using numerical simulations,which is inserted inside an outer GdBaCuO TFM cylinder. The
mechanical stress in the HTFML with optimized lens was also
analyzed during the magnetizing process. The main conclusions
are as follows.
1) The design optimization of the magnetic lens can be carried out from viewpoints of both the magnetic field amplification and cost-effectiveness.
2) The HTFML with an optimized lens can achieve concentrated magnetic fields of Bc = 5.6 and 12.8 T at the centerof the lens for applied magnetic fields of Bapp = 3 and
10 T, respectively.
3) For Bapp = 10 T, a maximum tensile stress of 35 and135 MPa exist in the magnetic lens and the TFM cylinder, respectively, during the magnetizing process. These
results suggest that adequate mechanical reinforcement isneeded in a practical design in order to avoid mechanicalfracture at high magnetic fields.
ACKNOWLEDGMENT
All data are provided in full in the results section of this paper.

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