2218IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE 2003
HTS Motor Shape Optimization for Its Maximum
Critical Current of the Field Winding
Joonsun Kang, Joon-Ho Lee, Wansoo Nah, Il-Han Park, Jinho Joo, Young-Kil Kwon, Myung-Hwan Sohn, and
Abstract—Superconducting motors have high efficiency as well
as reduced size and weight. In superconducting motors, the field
winding is composed of HTS tapes (Bi-2223) without any iron core
is limited by the maximum magnetic field in the field winding. To
enhance the performance of superconducting motor, we need to
maximize the critical current of field winding as much as possible.
straint of HTS characteristic (
that improves the critical current of the field winding. Finite ele-
ment analysis and discrete sensitivity approach are used for calcu-
lating the magnetic field of coil and shape optimization.
curve), and proposes a shape
Index Terms—Discrete sensitivity approach, finite element anal-
ysis, shape optimization, superconducting motor.
pollution of the environment. Compared to the conventional
motors, superconducting motors have high efficiency, and re-
duced size and weight. The performance of HTS motor is,
of course, very much dependent on the superconductor char-
acteristic ( –
curves) in its winding section. To have high
performance of HTS motor for a given HTS tape, we need
a new design concept, which maximizes the critical current
of the field winding in the machine.
The approach we had tried to get a high performance HTS
motor is to change the shape of field winding, which enhances
critical current of the field winding. Similar works have been
reported in , , and they reduced the radial magnetic field
component by re-configuring the solenoid shape, finally get-
ting enhanced critical current of the solenoid magnet. But they
did not consider the critical surface constraint simultaneously
during optimization process: they checked the critical surface
constraint after they got various coil shapes.
In this paper, we developed a shape optimization algorithm
to maximize the critical current of the field winding, simulta-
ECENTLY, high efficient and environmental-friendly ma-
chines are required due to the limited resources and the
Manuscript received August 5, 2002. This research was supported by a grant
from Center for Applied Superconductivity Technology of the 21st Century
Frontier R&D Program funded by the Ministry of Science and Technology, Re-
public of Korea.
J. Kang, J.-H. Lee, W. Nah, and I.-H. Park are with the School of Informa-
tion and Communication Engineering, Sungkyunkwan University, Kyunggi-do
440-746, Korea (e-mail: firstname.lastname@example.org).
J. Joo is with the School of Metallurgical and Materials Engineering,
Sungkyunkwan University, Kyunggi-do 440-746, Korea.
Y.-K. Kwon, M.-H. Sohn, and S.-W. Kim are with Korea Electrotechnology
Research Institute, Changwon, 641-120, Korea.
Digital Object Identifier 10.1109/TASC.2003.813050
neously satisfying critical surface of superconductor and fixed
volume constraints. The algorithm uses newly introduced op-
timal shape design sensitivity. This sensitivity analysis takes
into account the anisotropic characteristic of a HTS tape: the
critical current, when the magnetic field is applied normal to the
flat face of a tape, is substantially lower than the critical current
of parallel magnetic field to the tape surface. To check the use-
fulness of the suggested algorithm, it was applied to the 100 hp
HTS motor, which has been being developed by KERI (Korea
Electrotechnology Research Institute). We observed up to 20%
increase of the critical current of the field winding and the mag-
netic field in the air gap increased about 30–50% depending on
the initial coil shape. The sensitivity analysis is described in the
next section, and application results to the 100 hp HTS motor
are coming in the following sections.
II. SHAPE DESIGN SENSITIVITY ANALYSIS
In the shape design process, the design variables are taken
from the nodes of the winding coil, which are allowed to move
to get better shape for minimizing the objective function ,
. In this section the sensitivity formula with critical surface
constraint is introduced in detail.
The standard discrete sensitivity formulation is described as
vectorpotential and theshape design variables, respectively,
. In order to maximize critical current of the superconducting
coil, it is assumed that the operating current density equals
the critical current density
itself. When it comes to current
density, we used overall critical current density rather than
superconductor current density, which was turned out to be
convenient to deal with multifilament tapes . The sensitivity
is calculated as
is the objective function,andare the magnetic
is the adjoint variable. It is defined from the following
is defined as
1051-8223/03$17.00 © 2003 IEEE
KANG et al.: HTS MOTOR SHAPE OPTIMIZATION FOR ITS MAXIMUM CRITICAL CURRENT2219
Alsoandin (2) come from
In the 2-dimensional magneto-static system as in the HTS
motor, the terms in (4) can be written as:
is the maximum flux density in the coil area.
is a first degree shape function for one finite element,
are standard coefficients for the first degree shape
is thearea of one finiteelement. Also,the terms
in (6) are expressed as:
sign variable node. The
step for sensitivity calculation can be summarized as
according to (4)
b) Solve the newly introduced adjoint (3)
and from (5) and (6)
d) Compute the sensitivity from (2).
is the directional cosine vector of the moving de-
terms in (4) and (6) are cal-
III. MOTOR MODEL
The algorithm developed in Section II is to be applied to the
100 hp HTS motor to check its effectiveness. Table I summa-
rizes the specifications of the HTS motor, which has been being
developed by KERI. The field windings are composed of sev-
eral flat racetrack double pancake coils of Bi-2223 tapes.
Fig. 1 shows the 1/8 part of HTS motor cross section. The
racetrack double pancake coils in Fig. 1 are symmetrical along
SPECIFICATIONS OF HTS MOTOR
double pancake coils represent the number of turns.
1/8 part of HTS motor cross section. Numbers beside the race track
correspond to ? ? ?? , 85 , 80 , 75 , 70 , 60 , 50 , 40 , 30 , 15 , and 0 ,
? –? curves of the HTS tape at 77 K. Curves from top to bottom
to the tape surface of the winding section: they are already par-
tially optimized by size parameter optimization technique .
Fig. 2 shows the critical currents versus external magnetic
flux density with various angles to the flat surface of the su-
perconductor tape . The critical current data in Fig. 2 was
2220IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE 2003
Initial shapes of HTS motor and control points, (a) half circle and
obtained at 77 K, and was used in the optimal design calcula-
tion instead of
data at 30 K, which was not available at the
IV. OPTIMAL SHAPE DESIGN OF HTS MOTOR
Since the optimal shape depends heavily on the initial coil
These shapes are basically taken from the original double pan-
cake coil shape in Fig. 1. The two initial coils (one is half circle,
and the other one is parallelogram) have the same center of
gravity and the same volume as the original coil shape.
In order to maximize the critical current, we need to reduce
the maximum field density in the field winding, especially the
magnetic field normal to the tape surface. Since the critical cur-
rent decreases the most with normal magnetic field to the tape
surface as in Fig. 2, we assumed that the maximum -compo-
nent magnetic flux density limits the critical current of the coil:
we used only the
–curve of 0 in Fig. 2 for optimizing cal-
culation. Therefore, the objective function is defined as (6) to
minimize the -component of maximum magnetic flux density
volume of the winding.
field in the calculation region, and
We used 12 and 7 control points to change the coil shape
as in Fig. 3 for (a) and (b), respectively. These control points
were used in
-Spline parameterization . For each case, 40
nodes were taken as design variables to denote the boundary of
the winding section. The nodes were allowed to move in radial-
direction for (a) and in the -direction for (b) in Fig. 3. We used
4th order base function for each
Fig. 4 shows the optimized shape of field winding. It shows
that boththe left and right sidesof the windingsections expands
to radial direction. It should be pointed out here that the calcu-
is the maximum value of
is thevolume of the winding coil, and
in the field winding
is the initial
) is the target magnetic
is the constant with the
density as the iteration number increases. (a) Initial shape of half circle, and
(b) initial shape of parallelogram.
Critical current density and ?-component maximum magnetic flux
true answer .
Fig. 5 shows the
process. After 10 iterations,
increased by 18% and
creased by 12% for the initial shape of half circle, compared to
the initial shape. For the initial shape of parallelogram,
creased by 12% and
decreased by 8%, compared to the
from 45 to 90 at a radius of 157 mm. We observe sinusoidally
distributed magnetic flux density, which means that the opti-
mized coil shape did not disturb the sinusoidal magnetic field
distribution in the air gap. For the initial shape of half circle,
the peak-to-peak value of the magnetic field increased by 17%
with 10 iterations of optimizing
KANG et al.: HTS MOTOR SHAPE OPTIMIZATION FOR ITS MAXIMUM CRITICAL CURRENT2221
157 mm. (a) Initial shape of half circle, and (b) initial shape of parallelogram.
Radial flux density along the air gap from 45 to 90 at a radius of
compared to the initial shape, and 47% increase was observed
compared to the original shape of Fig. 1. For the initial shape of
parallelogram, the peak-to-peak value increased by 18% com-
pared to the initial shape, and the increase was 27% to the orig-
inal shape of Fig. 1, respectively.
In this paper, we employed a newly introduced design sen-
sitivity analysis with HTS characteristics to increase the max-
imum critical current of the field winding in HTS motor. This
approach decreased the maximum field density effectively and
In addition, the radial magnetic field at the air gap increased up
to 50% depending on the initial coil shape. Considering that the
original coil shape was already optimized by size parameter op-
air gap is satisfactory. It was also confirmed that the coil shape
change did not disturb the sinusoidal distribution of magnetic
field in the air gap. It can be concluded that the proposed algo-
rithm is very useful to get high performance HTS motor design.
 J. Kang, J.-H. Lee, W. Nah, D.-H. Kim, I.-H. Park, and J. Joo, “Radial
magnetic field reduction to improve critical current of HTS solenoid,”
Physica C, August 2002, to be published.
 J.-H. Lee, J. Kang, W. Nah, I.-H. Park, and J. Joo, “Reduction of radial
magnetic fields in HTS solenoids with different constraint conditions,”
Cryogenics, to be published.
 W. Nah et al., “Load line analysis of Bi-2223 tape-stacked cable for self
field effects,” IEEE Transactions on Applied Superconductivity, vol. 10,
p. 1158, 2000.
 Y.-S. Jo, Y.-K. Kim, J.-P. Hong, J. Lee, Y.-K. Kwon, and K.-S. Ryu,
“Advanced design approach to the high temperature superconducting
magnet,” Cryogenics, vol. 41, pp. 27–33, 2001.
 J. R. T. Lehtonen, J. A. J. Paasi, A. K. Korpela, J. Pitel, and P. Kovac,
“AC losses in magnets wound of HTS tape conductors,” Advances in
Cryogenic Engineering, vol. 46, pp. 839–846, 2000.
 D. F. Rogers and J. A. Adams, Mathematical Elements for Computer
Graphics: McGraw-Hill, 1990, ch. 5.