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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

Seog-Whan Kim

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

becauseofmagneticsaturation,andthecurrentinthefieldwinding

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.

Thispaperintroducestheshapeoptimizationmethodwiththecon-

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.

I. INTRODUCTION

R

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 [1], [2], 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: wsnah@yurim.skku.ac.kr).

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 [1],

[2]. In this section the sensitivity formula with critical surface

constraint is introduced in detail.

The standard discrete sensitivity formulation is described as

(1)

where

vectorpotential and theshape design variables, respectively[1],

[2]. 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 [3]. The sensitivity

is calculated as

is the objective function,andare the magnetic

(2)

where

adjoint equation:

is the adjoint variable. It is defined from the following

(3)

where

is defined as

(4)

1051-8223/03$17.00 © 2003 IEEE

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KANG et al.: HTS MOTOR SHAPE OPTIMIZATION FOR ITS MAXIMUM CRITICAL CURRENT2219

Alsoandin (2) come from

(5)

(6)

where

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.

(7)

(8)

(9)

where

is a first degree shape function for one finite element,

are standard coefficients for the first degree shape

function, and

is thearea of one finiteelement. Also,the terms

in (6) are expressed as:

and

(10)

(11)

(12)

(13)

(14)

(15)

(16)

where

sign variable node. The

culatedfrom

step for sensitivity calculation can be summarized as

a) Assemble

according to (4)

b) Solve the newly introduced adjoint (3)

c) Calculate

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-

–dataofHTStape asinFig.2.Thecomputing

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

the -axis,andarearrangedtoreducethemagneticfieldsnormal

TABLE I

SPECIFICATIONS OF HTS MOTOR

Fig. 1.

double pancake coils represent the number of turns.

1/8 part of HTS motor cross section. Numbers beside the race track

Fig. 2.

correspond to ? ? ?? , 85 , 80 , 75 , 70 , 60 , 50 , 40 , 30 , 15 , and 0 ,

respectively.

? –? 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 [4].

Fig. 2 shows the critical currents versus external magnetic

flux density with various angles to the flat surface of the su-

perconductor tape [5]. The critical current data in Fig. 2 was

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2220IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE 2003

(a)(b)

Fig. 3.

(b) parallelogram.

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

moment.

IV. OPTIMAL SHAPE DESIGN OF HTS MOTOR

Since the optimal shape depends heavily on the initial coil

shape,twobasicinitialshapesweretakenas(a)and(b)inFig.3.

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

Subject to(6)

In (6),

section,

volume of the winding.

field in the calculation region, and

range of

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 [2]. 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-

latedcoilshapeddoesnotrepresenttheultimateoptimumshape,

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

(

.

-spline curves.

(a) (b)

Fig.4.

of parallelogram.

Optimizedshapesfor(a)initialshapeofhalfcircle,and(b)initialshape

(a)

(b)

Fig. 5.

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

whichgloballyminimizesthe -componentmaximummagnetic

fieldsinthecoilsection:itisnotusuallypossibletogiveasingle

true answer [6].

Fig. 5 shows the

and

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

initial shape.

Fig.6showstheradialmagneticfluxdensityalongtheairgap

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

de-

in-

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KANG et al.: HTS MOTOR SHAPE OPTIMIZATION FOR ITS MAXIMUM CRITICAL CURRENT2221

(a)

(b)

Fig. 6.

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.

V. CONCLUSION

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

increasedthecriticalcurrentdensityofthefieldwindingindeed.

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-

timizationtechnique[4],theincreaseofthemagneticfieldatthe

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.

REFERENCES

[1] 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.

[2] 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.

[3] 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.

[4] 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.

[5] 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.

[6] D. F. Rogers and J. A. Adams, Mathematical Elements for Computer

Graphics: McGraw-Hill, 1990, ch. 5.