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The development of a program system for the design of superconducting poloidal field coils
Nuclear Engineering and D e s i g n / F u s i o n 4 (1986) 11-19 North-Holland, Amsterdam
11
THE DEVELOPMENT OF A PROGRAM SYSTEM FOR THE DESIGN OF SUPERCONDUCTING POLOIDAL FIELD COILS S. T A D O , T. N A R I K A W A , H. T O M I T A , S. Y A M A G U C H I , T. T S U K A M O T O , K. U E D A , T. Y A M A D A , T. F U K U N A G A , T. S A T O W , T. I C H I H A R A , O. A S A I , Y. HATTOR.I, S. S A K A B E a n d B. I K E D A Mitsubishi Electric Corporation, 2-2-3, Marunouchi, Chiyoda-ku, Tokyo, Japan Received August 1985
A program system was developed for the purpose of the computer aided design of superconducting poloidal field coils in tokamak fusion devices. The program system consists of a main program, twelve subroutines and three data files. The main program effects the initialization of the program system and subsequent control of the calculation process. The system can be operated both in iterative mode and batch one by modifying it. The subroutines perform the calculation of electromagnetic variables, the structural design of the coils and the estimation of their design solutions. The data files play the role of a data base. The program system greatly reduces the large amount of labor and time required to design superconducting poloidal field coils. Utilizing the data files, users can readily redesign any part of the coils.
!. Introduction
Tokamak fusion devices consist of coils, a plasma vacuum chamber, a blanket, etc.. It is necessary to arrange these elements in a confined space as close as possible in order to improve the performance of the device. Then, superconducting poloidal field coils of tokamak fusion devices must be designed under strict spatial restraint. In the design of superconducting coils, it is necessary to find out the optimum design solutions of the coils concerning their construction to ensure heavy loads caused by electromagnetic forces and their superconductivity under high magnetic fields. However, it is troublesome to find out these solutions under the spatially limited conditions. The optimum design solutions of superconducting coils of tokamak fusion devices are generally searched by the repetition of the design process based on the design flow as shown in fig. 1. In the design there is need to pick out special values, such as maxima, in the large amount of calculations obtained from the analysis of the magnetic fields and the electromagnetic forces. Moreover, repeating calculations may be required as that design variables cannot be uniquely determined owing to their correlation. These calculation procedures are very troublesome and require
a large amount of labor and time. Therefore, a computer aided design system is requested in order to carry out the design calculation efficiently. Although there are many programs to analyze magnetic fields, electromagnetic forces and coil inductances,
0 1 6 7 - 8 9 9 x / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h : H o l l a n d Physics Publishing Division)
START
T
Basic Design SpecificationJ Shape and Dimension of Coil I d -I Magnetic Field, Electromagnetic Force I d -I
[Coil StructureI -I iConductor Structure D
I
[Electrical Properties i
f
JExciting System I
i
ICoil Design SpecificationJ
l
END Fig. l. Design flow ofsuperconductingcoils.
12
S. Tado et al. / Design of superconducting poloidal field coils
a program system to make the design of superconducting poloidal field coils could not be found. Therefore, the demand for such program system is increasing together with the development of research in tokamak fusion devices. In the present paper we describe the outline and the functions of the computer program system developed for designing superconducting poloidai field coils in tokamak fusion devices.
2. Features and functions of the program system The program system is written in FORTRAN. Fig. 2 shows the constitution of the program system. It consists of a main program, twelve subroutines and three data files. The main program effects the initialization of the program system and subsequent control of the calculation process. The user is able to accomplish his design in sequence in interactive mode. Also this system can be operated in batch mode by rewriting the main program according to a calculation process in the required design flow. Consequently, it is ready to make a design in which iterative calculations must be carried out. The subroutines perform the calculation of electromagnetic variables, the structural design of the coils and the estimation of their design solutions. In every subroutine, the data used in the calculation of the
design variables are directly read from the data files, and, after calculation and output, the results are written into the data files automatically. Every subroutine has a closed form. Consequently, it is ready to revise a n d / o r extend its function. The default values of the design variables are set as many as possible, which the user may change. Then, even a user who does not completely know the design of superconducting poloidal field coils may accomplish the design of the coils at a desirable technical level, which is very effective in case of studies of poloidal coils relating to the total design study of tokamak fusion devices. All the calculation results are output in figures and tables so that the user can quickly and easily examine the results provided by the program system.
3. Description of the subroutines The finer details of the subroutines included in the program system are explained below.
3.1. Plot This subroutine plots the position and shape of the poloidal coils from the input data stored in the data file. With the aid of this routine, the position and shape of the coils may be readily checked.
3.2. Magnetic fieM This subroutine calculates the magnetic flux density, B, and its change rate with time, B, in the cross sections of the coils. These values are requested in selection of superconducting material, in decision of the grading structure and in calculation of heat generated by eddy current, hysteresis and coupling losses of the coils. Additionally, this routine picks out the maximum values of B and /~, and integrates B over the conductor domain. When the i-th coil is graded into k i parts and the l-th part is excited by Ampere turns li.t(t,,,) at time t,,, the magnetic flux density Bj(t,,), and its change rate with time, ~.(t,,,) at the j-th calculating point can be determined from the following formulae: k,
Bj(t,,,)= ~ i=l
~-~.'b,.t.jli.t(t,, ),
(1)
/=]
1
Bj( t,,,)
t,,+l - t,, k,
Fig. 2. Constitution of program system.
x ~ Eb,.,j{l,.,(t.,+,)-l,.,(t.,)}. i = l I=1
(2)
S. Tado et aL / Design of superconducting poloidal field coils kl
li(t,, ) = ~ lid(t,, ),
(3)
I~1
where hi,t, j is the normalized magnetic flux density at the j-th calculating point by the I-th graded part of the i-th coil. The values of B and B are readily calculated for various operating conditions of the coils by using the calculating method mentioned above.
3.3. Electromagnetic force This subroutine calculates the electromagnetic force acting on the coils. These values are requested in the structural design of the coils. In this routine the electromagnetic force caused by the ripple field of the toroidal coils is also calculated. The cross section of the l-th graded part of the i-th coil is divided into m × nt rectangular elements. Then, the electromagnetic force acting on the conductor per unit length along the current path of the i-th coil, Fi(t), is given by the following equation: kI
ttlXn I
r,(t)= t--I E jE ~l
mX
1 nti,,,(t ) [il'j>(Bi'l'j(t)]'
(4)
where it. j is unit current path vector. When the ripple field generated by the toroidal coils is negligibly small, the electromagnetic force is unform along the current path. But, in general, it is not uniform and the bending moment is caused by this ripple field. Then, in this routine, such force is calculated by applying eq. (4) to each segment which is divided along the current path.
3.4. Inductance This subroutine calculates the self inductance, the mutual inductance and coupling coefficient between the poloidal coils, including plasma. These values are requested in evaluating the load of the power source, the stored energy and the induced voltage of the coils. The inductance is calculated as the inductance of a one-turn coil and is output in the form of an inductance matrix. When two or more coils are connected in series in a poioidal coil system, the calculation is performed by regarding the coil group as one coil.
3.5. Stored energy and induced voltage This subroutine calculates the stored energy and the induced voltage of each coil. These values are requested
13
for the estimation of the operating current and the dielectric strength. The stored energy of the i-th coil-group is given by the following equation:
e i ( t ) = ½ Y'~ k~l
Mkilk(t)lj(t
),
(5)
j=l
where Mk / is the coil inductance between the k-th coil and the j-th coil, and k is the number of coils belonging to the i-th coil-group. The voltage induced in one turn of the k-th coil is given by the following equation:
Vk(t., ) = ~. Mkj J=l
lj(t,,,+l)-lj(tm)
(6)
l m + l - - tm
3.6. Operating current and number of turns This subroutine interactively determines the operating current and the number of turns of every coil. The design method to determine the optimum values of these variables has not been fully established. In this routine, the user decides on. the operating current of each coil according to the procedure mentioned below. In view of coil protection on quench, the operating current giving the maximum current density in the conductor lop is given by lop = aE h, where E is the stored energy, and a and b constants [1]. On the other hand lop is given on the basis of past records of the operating current. Then, comparing these two values, the user interactively chooses and inputs one of them. The number of turns and induced voltage are calculated from it automatically. The user can revise the values of lop after computation.
3. 7. Reinforcement This subroutine firstly calculates the sectional areas of the reinforcement required to keep the hoop stress of the conductor below the allowable values under the maximum electromagnetic force. If the result of the calculation is larger than the total sectional area of the coil, then this means that the solution of the structural design does not exist. Therefore, redesign is required for some parts of the former design stage. Second, the dimension of each part of the reinforcement is to be designed using the results of the required sectional area of the structure. In this routine four typical types of structures are prepared for the design of the reinforcement. For each type the dimen-
14
S. Tado et aL / Design of superconducting poloidal field coils
sions of the structure will automatically be decided from minimum data input. Finally, the user could optionally modify some dimensions obtained by processes mentioned above in manual mode. 3.8. Superconducting cable
This subroutine determines the number of the strands and the dimensions of the superconducting filament, CuNi sheath and stabilizing Cu matrix which compose the superconducting cable as shown in fig. 3. The procedure in this routine is as follows: The sectional area of the superconducting filaments of a final cable required for given operating current, S 1,
I
L3
i
Final Cable
is calculated. The number and diameter of the strand are determined on the basis of the dimensions of the cross section occupied by the final cable. Next, another sectional area of the filaments of the final cable, $2, is calculated from the dimensions of the filament, sheath and Cu matrix input by the user. Comparing the two values, the user modifies the dimensions of the filament, sheath and Cu matrix until the condition of S 2 > S 1 is satisfied. Finally, the cooling perimeter and the sectional area of the stabilizing copper matrix are calculated. 3.9. Cross section, volume and weight
This subroutine calculates the sectional area, volume and weight of the components material, such as can, insulator, final cable, reinforcement and helium. These values are necessary to calculate the heat loss and cooling capacity of the coil. The average current density is also provided in this routine. 3.10. Recovery current
2nd Subcable
1st Subcable
This subroutine calculates the recovery current, 1,, in bath cooling by giving the maximum magnetic field, Bmax, the operating current, lop, the sectional area of the stabilizing copper matrix, A c, the cooling perimeter, p, and the cooling efficiency, r/, which are provided by the previous subroutines. The recovery current is given by following equation: (7)
I r = ~/pAcqe~/&,
where pc is the resistivity of the stabilizing copper, which varies with magnetic flux density. The heat flux on normalization, q¢, is obtained on Maddock's "equal area criterion" [2]. Strand
3.11. Strength
This subroutine analyzes the structure of the coil to be designed against the electromagnetic force. Hoop stress, bending stress and allowable buckling load are obtained from the following approximate formulae [3]. Hoop stress and strain:
(8)
s o = ~oE,
Filament '°=
F"m"x {2,~(s:j,c + &G)}
(9) '
Bending stress and strain: Fig. 3. Structure of superconducting cable.
,,, = ~ : ,
(10)
S. Tado et aL / Design of superconducting poloidal field coils Fz .... L2hz % = ( 48~r( Eclcp¢ + E,.j.~.~) } "
(11)
Allowable buckling load: "
Pa=O'8072~rrinEh2r {( 1 - 1 ~u 2
3
. 1/4
hr/2/.2}m ,
(12)
where E is Young's modulus, S the total sectional area in the coil section, r m the mean radius of the coil, I the moment of inertia of cross section, L the span between the supporting points, tin the inner radius of the coil, h r the radial thickness of the coil, Fr and Fz the radial and vertical electromagnetic force, /.% the effectiveness factor, u Poisson's ratio, and the subscripts, c and ss, mean copper and stainless steel respectively.
3.12. Heat loss This subroutine calculates the heat loss caused by the alternating magnetic field. The values calculated in this routine are requested in estimation of heat load of helium cooling system. The following equations are utilized for the heat loss calculations, i.e. Hysteresis loss [4]:
Coupling loss [4]:
P = I---/~2//12A~..
(14)
4. Illustrative Design of P / F
zA
I
Eddy current loss [5]:
,
]
"2
2
Po = ~-T~pB a A,
c -M
N
/"
(15)
where p = pc(or &s), a = ast(Or ass ), A = A~(or Ass), /~ is the change rate of magnetic field, I t the critical current of superconductor, At the sectional area of the superconducting filaments in the final cable, Ac, the sectional area of stabilizer and barrier in the final cable, A c the sectional area of the stabilizing copper matrix in the final cable, Ass the sectional area of the stainless steel reinforcement in the final cable, Jc the critical current density of superconductor, p the resistivity, / the twist pitch of superconducting filament, and ar the diameter of superconducting filament. The losses given by. eqs. (13), (14) and (15) give losser per unit length of the final cable. The losses generated in the coil are obtained by integrating them over the total volume of the coil.
coil
IBM 3081 general purpose computer was used in the operation of the program system and SONY-Tektronix 4014-1 was used as a graphic display unit in interactive mode. As an illustrative design of a superconducting poloidal coil system in a tokamak fusion device, we examined the coil system consisted of twelve coils as shown in fig. 4. These coils are excited in the sequential current patterns as shown in fig. 5. In the subroutine 'Magnetic Field', after computation, the maximum values of magnetic flux .density and its change rate in each coild are displayed in the format as shown in table 1. The user can readily make a suitable choice of superconducting materials a n d / o r grading of the coils. In this illustration, the maximum value of magnetic flux density is less than 6.5 T, so that NbTi may be adopted as the superconductor used in all coils. In the subroutine 'Electromagnetic Force', after computation, the maximum values of radial and vertical electromagnetic force are displayed in format as shown in table 2. These values provide the user with data requested in design of coil structure and coil support. In this routine, moreover, instantaneous electromagnetic force and variations of electromagnetic force are displayed as shown in fig. 6 and fig. 7. After computation of the subroutine 'Reinforcement' and 'Superconducting Cable', the dimension of
\ 2~" ]
p.
15
-',,,,
,
| . I~_.....~
'
I
I ! !
~.
,...," /
",
,
- " ,"
-..Z----L--. "
C7[~X'I V'q
SCALE [m] 0i 0;5 ii Fig. 4. Arrangement of P / F coils.
6~ '/ ~ / - ' ~
..
;:.. y,,;~,,
.. 0 /
1
\
/
\
7,, 2
"'-\\
/
~.
,
v
/3
4
5
6
7
8
u/1"30.,
96
97~'98~99
i00
-3
-5 Fig. 5. Operating sequence of P / F coils.
z~
~
3 [74
NO. ] 2 3 4 5 6 7 8 9 i0 Ii 12
5
~
6 /
R
÷ i!
~ ~
89
7 f
8 SCALE
TIME
=
5.000
ELECTROMAGNETIC
COIL Cl C2 C3 C4 C5 C6 C7 C8 C9 Cl0 CII C12
i
L 2-20xi02
TON
SEC FORCE
AT
THETA
=
0[0
DEG
Fig. 6. Displayofinstantaneouselectromagneticforce. Table 2 Display of electromagnetic force
Table 1 Display of magnetic field [ CALCULATION NO. C O I L
THETA =
]
MAX.ABS
B
(T) 1 2 3 4 5 6 7 8 9 i0 ii 12
Cl C2 C3 C4 C5 C6 C7 C8 C9 CI0 CII C12
6.318E+00 5.959E+00 5.154E+00 2.216E+00 2.474E+00 3.329E+00 3.329E+00 2.474E+00 2.216E+00 5.154E+00 5.959E+00 6.318E+00
[ TOTAL ELECTROMAGNETIC
0.0 DEG
MAX.ABS DB/DT (T/SEC) 8.075E+00 6.960E+00 4.343E+00 1.879E+00 1.840E+00 2.885E+00 2.885E+00 1.840E+00 1.879E+00 4.343E+00 6.960E+00 8.075E+00
NO. COIL 1 2 3 4 5 6 7 8 9 10 ii 12
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 CII C12
MAX. +FR (TON) 6.350E+03 5.641E+03 4.383E+03 6.866E+02 1.041E+03 6.595E+03 6.595E+03 1.041E+03 6.866E+02 4.383E+03 5.641E+03 6.350E+03
FORCE
]
MAX. - F R (TON) -1.572E+00 -3.217E+01 -4.933E+02 -4.367E+00 -2.395E+03 0.0 0.0 -2.395E+03 -4.367E+00 -4.933E+02 -3.217E+01 -1.527E+00
M A X . A B S FZ (TON) 5.640E+02 2.432E+03 1.834E+03 2.823E+02 1.322E+03 2.240E+03 2.240E+03 1.322E+03 2.823E+02 1.834E+03 2.432E+03 5.640E+02
17
S. Tado et al. / Design of superconducting poloidal field coils
/
0 X
4 o 3
2 1 !
0
!
I
410.
50.
60.
710,
80. TIME
-i
910.
1~0. ~ /x l 0 ° [SEC]
l
-2.
COIL
O &
FR FZ
FORCE
AT
C2
ELECTROMAGNETIC
THETA
=
0.0
DEG
Fig, 7. Display of variation of electromagnetic force.
Table 3 List of dimensions of each part in coil and superconducting cable COIL
C12
(TYPE
CASE DIMENSION
B)
OF CASE
COIL DIMENSION OF CONDUCTOR CABLE HEIGHT LENGTH MATERIAL FOR REINFORCEMENT INSULATOR DIMENSION TO EARTH DIMENSION OF TURN TO TURN
COIL
C12
(TYPE
315.00(MM) 0 . 0 (Mbl)
HZ SHZ
= =
821.O0(MM) O. 0 (MM)
LR
315.00(MM)
LZ
=
8 2 1 . 0 0 (MM)
HI L1 T1
L2
=
1 2 8 . 5 0 (MM)
=
1 8 . 2 0 (MM) I10.30(MM) 2.00(MM)
DR SDR
= =
0.0 (MM) 3.00(MM)
DZ SDZ
= =
HR SHR
= =
B)
FILAMENT D I M E N S I O N O F SC F I L A M E N T DIMENSION OF HEXAGON DIMENSION OF BARRIER STRAND DIMENSION OF STRAND DIMENSION OF STABILIZER (CU) DIMENSION OF BARRIER (CUNI) STRUCTURE IST SUB CABLE DIMENSION OF CABLE STRUCTURE 2ND SUB CABLE DIMENSION OF CABLE DIMENSION OF STABILIZER DIMENSION OF BARRIER NO- O F S U B C A B L E
SD . SA DBA
0.01000(MM) 0.00981(MM) 0.00115(MM)
DST DDSTCU DDSTCN
= =
1.29066(MM) 0.09000(MM) 0.04000(MM)
DSUBI
=
2 . 7 8 1 0 0 (MM)
DSUB2 DCU DDCUNI N2ND
= = = =
8 . 3 4 2 9 9 (MM) 2.70099(MM) 0.04000(MM) 27
O. 0 (MM) i0. O0 (MM)
18
S. Tado et al. / Design of superconducting poloidal fieM coils
~ ~ soR
D,
DD
(3O ©©
i
L1 t g
.U
D SHRI
:-DR
L2
03 LR
(
HR
Fig. 8. Structure of cross sections of coil and conductor.
the coil section and superconducting cable are printed out in the format as shown in table 3. Each value in table 3 corresponds to the dimension shown in fig. 3 and fig. 8. For C6 (and C7) coil, variations of the induced voltage between terminals, the resultant stress, the ratio IJlop, the ratio It~lop and the total AC loss in one
operating period are described in table 4 when parameters, such as the number of turns, type of cable and dimension of conductor, are varied. Recovery current I r, critical current I~, and total AC loss vary in a considerably wide range. Then, in order to obtain the optimum design solutions, many case studies may be needed.
Table 4 Main parameters of C6 (C7) coil C o i l N a m e / C6, conductor number
C7
type
1
A
of t u r n s
induced voltage r a t i o of sectional area
case
(kV)
case
2
A
case
3
A
case
204
126
180
180
25.17
15.54
22.21
22.21 0.0386
NbTi
0.0278
0.0421
0.0391
Cu
0.1728
0.1560
0.1549
0.1530
CuNi
0.0840
0.0743
0.0820
0.0825
S.S.
0.1936
0.1936
0.1936
0.1936
space(He)
0.5218
0.5340
0.5304
0.5323
tensile
stress
(kg/mm 2)
27.32
27.32
27.32
27.32
bending
stress
( k g / m m 2)
6.24
6.06
6.19
6.04
4.44
7.70
4.97
4.77
33
17
29
31
diameter subcable
of 2nd (mm)
n u m b e r o f 2nd s u b c a b l e s in c o n d u c t o r
Ir/Iop
1.608
1.150
1.434
1.445
Ic/Iop
2.667
4.042
3.759
3.708
19.48
21.06
16.44
16.26
total AC loss per one cycle
4
B
(kJ)
S. Tado et al. / Design of superconducting poloidal field coils 5. Conclusion
li.z
A computer program system was developed for automatically designing the superconducting poloidal field coils of tokamak fusion devices. This program system features: (1) Labor and time for coil design are greatly reduced since the program system systematically deals with the design process such as the selection of required parameters for coil design, which up to now were manually derived. (2) Quick response to any partial design modification since all final design data are stored in the data files. (3) Very few engineers are able to accomplish a desirable design, which has required many engineers in various technical fields such as magnetic field analaysis, superconductor design and structure analysis. These features are very effective in case studies of poloidal coils relating the total design study of tokamak fusion devices.
I~ lop Ir J¢ / L Mu p P~ P~ P~ Ph q~ tin rm t V
19
exciting Ampere turns in the /-th part of the i-th coil (A), critical current density of superconductor ( A / m 2), operating current (A), recovery current (A), critical current density of superconductor ( A / m 2), twist pitch of filament (m), span between the supporting positions (m), self of mutual inductance (H), cooling perimeter (m), allowable buckling load (N), coupling loss per final cable (W/m), eddy current loss per final cable (W/m), hysteresis !oss per final cable (W/m), heat flux on normalization (W/m2), inner radius of coil (m), mean radius of coil (m), time (s), induced voltage as one turn coil (V).
Greek syrnbols Nomenclature
ar a~, ast A¢
diameter of superconducting filament (m), width of stainless steel reinforcement (m), diameter of strand (m), sectional area of stabilizing copper per conductor (m2), At sectional area of superconducting filament per final cable (m2), A~, sectional area of stabilizer and barrier per final cable (m2 ), A~ sectional area of stainless steel reinforcement per final cable (m2), bi.t. j .normalized magnetic flux density generated at t h e j-th calculating point by the /-th graded part of the i-th coil (T/A), magnetic flux density at the j-th calculating point (T), Bj change rate of magnetic flux density at the j-th calculating point (T/s), ei stored energy of the i-th coil (J), E Young's modulus (Pa), F/ electromagnetic force (N/m), hr radial dimension of winding region in coil section (m), hz axial dimension of winding region in coil section (m),
r/ p,~ u & p~., p, o
strain, cooling efficiency, effectiveness factor, Poisson's ratio, resistivity of copper ($2m), resistivity of stainless steel (I2m), equivalent resistivity between filaments (I2m), stress (Pa).
References
[1] T. Satow, Y. Hattori and M. lwamoto, Optimum current of completely stabilized superconducting coil, in: Proceedings National Meeting of IEEJ 632(1977)789, in Japanese. [2] B.J. Maddock. G.B. James and W.T. Norris, Superconductive composites: Heat transfer and steady state stabilization, Cryogenic 9 (1969) 261-273. [31 Column Research Committee of Japan, Summary of Elastic Stability (Corona Press, Tokyo, 1960) in Japanese. [4] H. Brechna, Superconducting Magnet Systems (SpringerVerlag, Berlin, 1973) pp. 246-260. [5] Japan Atomic Energy Research Institute, Design Study of Superconducting Magnets for Tokamak Experimental Fusion Reactor (II), JAERI-M8666 (1979) in Japanese.
11
THE DEVELOPMENT OF A PROGRAM SYSTEM FOR THE DESIGN OF SUPERCONDUCTING POLOIDAL FIELD COILS S. T A D O , T. N A R I K A W A , H. T O M I T A , S. Y A M A G U C H I , T. T S U K A M O T O , K. U E D A , T. Y A M A D A , T. F U K U N A G A , T. S A T O W , T. I C H I H A R A , O. A S A I , Y. HATTOR.I, S. S A K A B E a n d B. I K E D A Mitsubishi Electric Corporation, 2-2-3, Marunouchi, Chiyoda-ku, Tokyo, Japan Received August 1985
A program system was developed for the purpose of the computer aided design of superconducting poloidal field coils in tokamak fusion devices. The program system consists of a main program, twelve subroutines and three data files. The main program effects the initialization of the program system and subsequent control of the calculation process. The system can be operated both in iterative mode and batch one by modifying it. The subroutines perform the calculation of electromagnetic variables, the structural design of the coils and the estimation of their design solutions. The data files play the role of a data base. The program system greatly reduces the large amount of labor and time required to design superconducting poloidal field coils. Utilizing the data files, users can readily redesign any part of the coils.
!. Introduction
Tokamak fusion devices consist of coils, a plasma vacuum chamber, a blanket, etc.. It is necessary to arrange these elements in a confined space as close as possible in order to improve the performance of the device. Then, superconducting poloidal field coils of tokamak fusion devices must be designed under strict spatial restraint. In the design of superconducting coils, it is necessary to find out the optimum design solutions of the coils concerning their construction to ensure heavy loads caused by electromagnetic forces and their superconductivity under high magnetic fields. However, it is troublesome to find out these solutions under the spatially limited conditions. The optimum design solutions of superconducting coils of tokamak fusion devices are generally searched by the repetition of the design process based on the design flow as shown in fig. 1. In the design there is need to pick out special values, such as maxima, in the large amount of calculations obtained from the analysis of the magnetic fields and the electromagnetic forces. Moreover, repeating calculations may be required as that design variables cannot be uniquely determined owing to their correlation. These calculation procedures are very troublesome and require
a large amount of labor and time. Therefore, a computer aided design system is requested in order to carry out the design calculation efficiently. Although there are many programs to analyze magnetic fields, electromagnetic forces and coil inductances,
0 1 6 7 - 8 9 9 x / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h : H o l l a n d Physics Publishing Division)
START
T
Basic Design SpecificationJ Shape and Dimension of Coil I d -I Magnetic Field, Electromagnetic Force I d -I
[Coil StructureI -I iConductor Structure D
I
[Electrical Properties i
f
JExciting System I
i
ICoil Design SpecificationJ
l
END Fig. l. Design flow ofsuperconductingcoils.
12
S. Tado et al. / Design of superconducting poloidal field coils
a program system to make the design of superconducting poloidal field coils could not be found. Therefore, the demand for such program system is increasing together with the development of research in tokamak fusion devices. In the present paper we describe the outline and the functions of the computer program system developed for designing superconducting poloidai field coils in tokamak fusion devices.
2. Features and functions of the program system The program system is written in FORTRAN. Fig. 2 shows the constitution of the program system. It consists of a main program, twelve subroutines and three data files. The main program effects the initialization of the program system and subsequent control of the calculation process. The user is able to accomplish his design in sequence in interactive mode. Also this system can be operated in batch mode by rewriting the main program according to a calculation process in the required design flow. Consequently, it is ready to make a design in which iterative calculations must be carried out. The subroutines perform the calculation of electromagnetic variables, the structural design of the coils and the estimation of their design solutions. In every subroutine, the data used in the calculation of the
design variables are directly read from the data files, and, after calculation and output, the results are written into the data files automatically. Every subroutine has a closed form. Consequently, it is ready to revise a n d / o r extend its function. The default values of the design variables are set as many as possible, which the user may change. Then, even a user who does not completely know the design of superconducting poloidal field coils may accomplish the design of the coils at a desirable technical level, which is very effective in case of studies of poloidal coils relating to the total design study of tokamak fusion devices. All the calculation results are output in figures and tables so that the user can quickly and easily examine the results provided by the program system.
3. Description of the subroutines The finer details of the subroutines included in the program system are explained below.
3.1. Plot This subroutine plots the position and shape of the poloidal coils from the input data stored in the data file. With the aid of this routine, the position and shape of the coils may be readily checked.
3.2. Magnetic fieM This subroutine calculates the magnetic flux density, B, and its change rate with time, B, in the cross sections of the coils. These values are requested in selection of superconducting material, in decision of the grading structure and in calculation of heat generated by eddy current, hysteresis and coupling losses of the coils. Additionally, this routine picks out the maximum values of B and /~, and integrates B over the conductor domain. When the i-th coil is graded into k i parts and the l-th part is excited by Ampere turns li.t(t,,,) at time t,,, the magnetic flux density Bj(t,,), and its change rate with time, ~.(t,,,) at the j-th calculating point can be determined from the following formulae: k,
Bj(t,,,)= ~ i=l
~-~.'b,.t.jli.t(t,, ),
(1)
/=]
1
Bj( t,,,)
t,,+l - t,, k,
Fig. 2. Constitution of program system.
x ~ Eb,.,j{l,.,(t.,+,)-l,.,(t.,)}. i = l I=1
(2)
S. Tado et aL / Design of superconducting poloidal field coils kl
li(t,, ) = ~ lid(t,, ),
(3)
I~1
where hi,t, j is the normalized magnetic flux density at the j-th calculating point by the I-th graded part of the i-th coil. The values of B and B are readily calculated for various operating conditions of the coils by using the calculating method mentioned above.
3.3. Electromagnetic force This subroutine calculates the electromagnetic force acting on the coils. These values are requested in the structural design of the coils. In this routine the electromagnetic force caused by the ripple field of the toroidal coils is also calculated. The cross section of the l-th graded part of the i-th coil is divided into m × nt rectangular elements. Then, the electromagnetic force acting on the conductor per unit length along the current path of the i-th coil, Fi(t), is given by the following equation: kI
ttlXn I
r,(t)= t--I E jE ~l
mX
1 nti,,,(t ) [il'j>(Bi'l'j(t)]'
(4)
where it. j is unit current path vector. When the ripple field generated by the toroidal coils is negligibly small, the electromagnetic force is unform along the current path. But, in general, it is not uniform and the bending moment is caused by this ripple field. Then, in this routine, such force is calculated by applying eq. (4) to each segment which is divided along the current path.
3.4. Inductance This subroutine calculates the self inductance, the mutual inductance and coupling coefficient between the poloidal coils, including plasma. These values are requested in evaluating the load of the power source, the stored energy and the induced voltage of the coils. The inductance is calculated as the inductance of a one-turn coil and is output in the form of an inductance matrix. When two or more coils are connected in series in a poioidal coil system, the calculation is performed by regarding the coil group as one coil.
3.5. Stored energy and induced voltage This subroutine calculates the stored energy and the induced voltage of each coil. These values are requested
13
for the estimation of the operating current and the dielectric strength. The stored energy of the i-th coil-group is given by the following equation:
e i ( t ) = ½ Y'~ k~l
Mkilk(t)lj(t
),
(5)
j=l
where Mk / is the coil inductance between the k-th coil and the j-th coil, and k is the number of coils belonging to the i-th coil-group. The voltage induced in one turn of the k-th coil is given by the following equation:
Vk(t., ) = ~. Mkj J=l
lj(t,,,+l)-lj(tm)
(6)
l m + l - - tm
3.6. Operating current and number of turns This subroutine interactively determines the operating current and the number of turns of every coil. The design method to determine the optimum values of these variables has not been fully established. In this routine, the user decides on. the operating current of each coil according to the procedure mentioned below. In view of coil protection on quench, the operating current giving the maximum current density in the conductor lop is given by lop = aE h, where E is the stored energy, and a and b constants [1]. On the other hand lop is given on the basis of past records of the operating current. Then, comparing these two values, the user interactively chooses and inputs one of them. The number of turns and induced voltage are calculated from it automatically. The user can revise the values of lop after computation.
3. 7. Reinforcement This subroutine firstly calculates the sectional areas of the reinforcement required to keep the hoop stress of the conductor below the allowable values under the maximum electromagnetic force. If the result of the calculation is larger than the total sectional area of the coil, then this means that the solution of the structural design does not exist. Therefore, redesign is required for some parts of the former design stage. Second, the dimension of each part of the reinforcement is to be designed using the results of the required sectional area of the structure. In this routine four typical types of structures are prepared for the design of the reinforcement. For each type the dimen-
14
S. Tado et aL / Design of superconducting poloidal field coils
sions of the structure will automatically be decided from minimum data input. Finally, the user could optionally modify some dimensions obtained by processes mentioned above in manual mode. 3.8. Superconducting cable
This subroutine determines the number of the strands and the dimensions of the superconducting filament, CuNi sheath and stabilizing Cu matrix which compose the superconducting cable as shown in fig. 3. The procedure in this routine is as follows: The sectional area of the superconducting filaments of a final cable required for given operating current, S 1,
I
L3
i
Final Cable
is calculated. The number and diameter of the strand are determined on the basis of the dimensions of the cross section occupied by the final cable. Next, another sectional area of the filaments of the final cable, $2, is calculated from the dimensions of the filament, sheath and Cu matrix input by the user. Comparing the two values, the user modifies the dimensions of the filament, sheath and Cu matrix until the condition of S 2 > S 1 is satisfied. Finally, the cooling perimeter and the sectional area of the stabilizing copper matrix are calculated. 3.9. Cross section, volume and weight
This subroutine calculates the sectional area, volume and weight of the components material, such as can, insulator, final cable, reinforcement and helium. These values are necessary to calculate the heat loss and cooling capacity of the coil. The average current density is also provided in this routine. 3.10. Recovery current
2nd Subcable
1st Subcable
This subroutine calculates the recovery current, 1,, in bath cooling by giving the maximum magnetic field, Bmax, the operating current, lop, the sectional area of the stabilizing copper matrix, A c, the cooling perimeter, p, and the cooling efficiency, r/, which are provided by the previous subroutines. The recovery current is given by following equation: (7)
I r = ~/pAcqe~/&,
where pc is the resistivity of the stabilizing copper, which varies with magnetic flux density. The heat flux on normalization, q¢, is obtained on Maddock's "equal area criterion" [2]. Strand
3.11. Strength
This subroutine analyzes the structure of the coil to be designed against the electromagnetic force. Hoop stress, bending stress and allowable buckling load are obtained from the following approximate formulae [3]. Hoop stress and strain:
(8)
s o = ~oE,
Filament '°=
F"m"x {2,~(s:j,c + &G)}
(9) '
Bending stress and strain: Fig. 3. Structure of superconducting cable.
,,, = ~ : ,
(10)
S. Tado et aL / Design of superconducting poloidal field coils Fz .... L2hz % = ( 48~r( Eclcp¢ + E,.j.~.~) } "
(11)
Allowable buckling load: "
Pa=O'8072~rrinEh2r {( 1 - 1 ~u 2
3
. 1/4
hr/2/.2}m ,
(12)
where E is Young's modulus, S the total sectional area in the coil section, r m the mean radius of the coil, I the moment of inertia of cross section, L the span between the supporting points, tin the inner radius of the coil, h r the radial thickness of the coil, Fr and Fz the radial and vertical electromagnetic force, /.% the effectiveness factor, u Poisson's ratio, and the subscripts, c and ss, mean copper and stainless steel respectively.
3.12. Heat loss This subroutine calculates the heat loss caused by the alternating magnetic field. The values calculated in this routine are requested in estimation of heat load of helium cooling system. The following equations are utilized for the heat loss calculations, i.e. Hysteresis loss [4]:
Coupling loss [4]:
P = I---/~2//12A~..
(14)
4. Illustrative Design of P / F
zA
I
Eddy current loss [5]:
,
]
"2
2
Po = ~-T~pB a A,
c -M
N
/"
(15)
where p = pc(or &s), a = ast(Or ass ), A = A~(or Ass), /~ is the change rate of magnetic field, I t the critical current of superconductor, At the sectional area of the superconducting filaments in the final cable, Ac, the sectional area of stabilizer and barrier in the final cable, A c the sectional area of the stabilizing copper matrix in the final cable, Ass the sectional area of the stainless steel reinforcement in the final cable, Jc the critical current density of superconductor, p the resistivity, / the twist pitch of superconducting filament, and ar the diameter of superconducting filament. The losses given by. eqs. (13), (14) and (15) give losser per unit length of the final cable. The losses generated in the coil are obtained by integrating them over the total volume of the coil.
coil
IBM 3081 general purpose computer was used in the operation of the program system and SONY-Tektronix 4014-1 was used as a graphic display unit in interactive mode. As an illustrative design of a superconducting poloidal coil system in a tokamak fusion device, we examined the coil system consisted of twelve coils as shown in fig. 4. These coils are excited in the sequential current patterns as shown in fig. 5. In the subroutine 'Magnetic Field', after computation, the maximum values of magnetic flux .density and its change rate in each coild are displayed in the format as shown in table 1. The user can readily make a suitable choice of superconducting materials a n d / o r grading of the coils. In this illustration, the maximum value of magnetic flux density is less than 6.5 T, so that NbTi may be adopted as the superconductor used in all coils. In the subroutine 'Electromagnetic Force', after computation, the maximum values of radial and vertical electromagnetic force are displayed in format as shown in table 2. These values provide the user with data requested in design of coil structure and coil support. In this routine, moreover, instantaneous electromagnetic force and variations of electromagnetic force are displayed as shown in fig. 6 and fig. 7. After computation of the subroutine 'Reinforcement' and 'Superconducting Cable', the dimension of
\ 2~" ]
p.
15
-',,,,
,
| . I~_.....~
'
I
I ! !
~.
,...," /
",
,
- " ,"
-..Z----L--. "
C7[~X'I V'q
SCALE [m] 0i 0;5 ii Fig. 4. Arrangement of P / F coils.
6~ '/ ~ / - ' ~
..
;:.. y,,;~,,
.. 0 /
1
\
/
\
7,, 2
"'-\\
/
~.
,
v
/3
4
5
6
7
8
u/1"30.,
96
97~'98~99
i00
-3
-5 Fig. 5. Operating sequence of P / F coils.
z~
~
3 [74
NO. ] 2 3 4 5 6 7 8 9 i0 Ii 12
5
~
6 /
R
÷ i!
~ ~
89
7 f
8 SCALE
TIME
=
5.000
ELECTROMAGNETIC
COIL Cl C2 C3 C4 C5 C6 C7 C8 C9 Cl0 CII C12
i
L 2-20xi02
TON
SEC FORCE
AT
THETA
=
0[0
DEG
Fig. 6. Displayofinstantaneouselectromagneticforce. Table 2 Display of electromagnetic force
Table 1 Display of magnetic field [ CALCULATION NO. C O I L
THETA =
]
MAX.ABS
B
(T) 1 2 3 4 5 6 7 8 9 i0 ii 12
Cl C2 C3 C4 C5 C6 C7 C8 C9 CI0 CII C12
6.318E+00 5.959E+00 5.154E+00 2.216E+00 2.474E+00 3.329E+00 3.329E+00 2.474E+00 2.216E+00 5.154E+00 5.959E+00 6.318E+00
[ TOTAL ELECTROMAGNETIC
0.0 DEG
MAX.ABS DB/DT (T/SEC) 8.075E+00 6.960E+00 4.343E+00 1.879E+00 1.840E+00 2.885E+00 2.885E+00 1.840E+00 1.879E+00 4.343E+00 6.960E+00 8.075E+00
NO. COIL 1 2 3 4 5 6 7 8 9 10 ii 12
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 CII C12
MAX. +FR (TON) 6.350E+03 5.641E+03 4.383E+03 6.866E+02 1.041E+03 6.595E+03 6.595E+03 1.041E+03 6.866E+02 4.383E+03 5.641E+03 6.350E+03
FORCE
]
MAX. - F R (TON) -1.572E+00 -3.217E+01 -4.933E+02 -4.367E+00 -2.395E+03 0.0 0.0 -2.395E+03 -4.367E+00 -4.933E+02 -3.217E+01 -1.527E+00
M A X . A B S FZ (TON) 5.640E+02 2.432E+03 1.834E+03 2.823E+02 1.322E+03 2.240E+03 2.240E+03 1.322E+03 2.823E+02 1.834E+03 2.432E+03 5.640E+02
17
S. Tado et al. / Design of superconducting poloidal field coils
/
0 X
4 o 3
2 1 !
0
!
I
410.
50.
60.
710,
80. TIME
-i
910.
1~0. ~ /x l 0 ° [SEC]
l
-2.
COIL
O &
FR FZ
FORCE
AT
C2
ELECTROMAGNETIC
THETA
=
0.0
DEG
Fig, 7. Display of variation of electromagnetic force.
Table 3 List of dimensions of each part in coil and superconducting cable COIL
C12
(TYPE
CASE DIMENSION
B)
OF CASE
COIL DIMENSION OF CONDUCTOR CABLE HEIGHT LENGTH MATERIAL FOR REINFORCEMENT INSULATOR DIMENSION TO EARTH DIMENSION OF TURN TO TURN
COIL
C12
(TYPE
315.00(MM) 0 . 0 (Mbl)
HZ SHZ
= =
821.O0(MM) O. 0 (MM)
LR
315.00(MM)
LZ
=
8 2 1 . 0 0 (MM)
HI L1 T1
L2
=
1 2 8 . 5 0 (MM)
=
1 8 . 2 0 (MM) I10.30(MM) 2.00(MM)
DR SDR
= =
0.0 (MM) 3.00(MM)
DZ SDZ
= =
HR SHR
= =
B)
FILAMENT D I M E N S I O N O F SC F I L A M E N T DIMENSION OF HEXAGON DIMENSION OF BARRIER STRAND DIMENSION OF STRAND DIMENSION OF STABILIZER (CU) DIMENSION OF BARRIER (CUNI) STRUCTURE IST SUB CABLE DIMENSION OF CABLE STRUCTURE 2ND SUB CABLE DIMENSION OF CABLE DIMENSION OF STABILIZER DIMENSION OF BARRIER NO- O F S U B C A B L E
SD . SA DBA
0.01000(MM) 0.00981(MM) 0.00115(MM)
DST DDSTCU DDSTCN
= =
1.29066(MM) 0.09000(MM) 0.04000(MM)
DSUBI
=
2 . 7 8 1 0 0 (MM)
DSUB2 DCU DDCUNI N2ND
= = = =
8 . 3 4 2 9 9 (MM) 2.70099(MM) 0.04000(MM) 27
O. 0 (MM) i0. O0 (MM)
18
S. Tado et al. / Design of superconducting poloidal fieM coils
~ ~ soR
D,
DD
(3O ©©
i
L1 t g
.U
D SHRI
:-DR
L2
03 LR
(
HR
Fig. 8. Structure of cross sections of coil and conductor.
the coil section and superconducting cable are printed out in the format as shown in table 3. Each value in table 3 corresponds to the dimension shown in fig. 3 and fig. 8. For C6 (and C7) coil, variations of the induced voltage between terminals, the resultant stress, the ratio IJlop, the ratio It~lop and the total AC loss in one
operating period are described in table 4 when parameters, such as the number of turns, type of cable and dimension of conductor, are varied. Recovery current I r, critical current I~, and total AC loss vary in a considerably wide range. Then, in order to obtain the optimum design solutions, many case studies may be needed.
Table 4 Main parameters of C6 (C7) coil C o i l N a m e / C6, conductor number
C7
type
1
A
of t u r n s
induced voltage r a t i o of sectional area
case
(kV)
case
2
A
case
3
A
case
204
126
180
180
25.17
15.54
22.21
22.21 0.0386
NbTi
0.0278
0.0421
0.0391
Cu
0.1728
0.1560
0.1549
0.1530
CuNi
0.0840
0.0743
0.0820
0.0825
S.S.
0.1936
0.1936
0.1936
0.1936
space(He)
0.5218
0.5340
0.5304
0.5323
tensile
stress
(kg/mm 2)
27.32
27.32
27.32
27.32
bending
stress
( k g / m m 2)
6.24
6.06
6.19
6.04
4.44
7.70
4.97
4.77
33
17
29
31
diameter subcable
of 2nd (mm)
n u m b e r o f 2nd s u b c a b l e s in c o n d u c t o r
Ir/Iop
1.608
1.150
1.434
1.445
Ic/Iop
2.667
4.042
3.759
3.708
19.48
21.06
16.44
16.26
total AC loss per one cycle
4
B
(kJ)
S. Tado et al. / Design of superconducting poloidal field coils 5. Conclusion
li.z
A computer program system was developed for automatically designing the superconducting poloidal field coils of tokamak fusion devices. This program system features: (1) Labor and time for coil design are greatly reduced since the program system systematically deals with the design process such as the selection of required parameters for coil design, which up to now were manually derived. (2) Quick response to any partial design modification since all final design data are stored in the data files. (3) Very few engineers are able to accomplish a desirable design, which has required many engineers in various technical fields such as magnetic field analaysis, superconductor design and structure analysis. These features are very effective in case studies of poloidal coils relating the total design study of tokamak fusion devices.
I~ lop Ir J¢ / L Mu p P~ P~ P~ Ph q~ tin rm t V
19
exciting Ampere turns in the /-th part of the i-th coil (A), critical current density of superconductor ( A / m 2), operating current (A), recovery current (A), critical current density of superconductor ( A / m 2), twist pitch of filament (m), span between the supporting positions (m), self of mutual inductance (H), cooling perimeter (m), allowable buckling load (N), coupling loss per final cable (W/m), eddy current loss per final cable (W/m), hysteresis !oss per final cable (W/m), heat flux on normalization (W/m2), inner radius of coil (m), mean radius of coil (m), time (s), induced voltage as one turn coil (V).
Greek syrnbols Nomenclature
ar a~, ast A¢
diameter of superconducting filament (m), width of stainless steel reinforcement (m), diameter of strand (m), sectional area of stabilizing copper per conductor (m2), At sectional area of superconducting filament per final cable (m2), A~, sectional area of stabilizer and barrier per final cable (m2 ), A~ sectional area of stainless steel reinforcement per final cable (m2), bi.t. j .normalized magnetic flux density generated at t h e j-th calculating point by the /-th graded part of the i-th coil (T/A), magnetic flux density at the j-th calculating point (T), Bj change rate of magnetic flux density at the j-th calculating point (T/s), ei stored energy of the i-th coil (J), E Young's modulus (Pa), F/ electromagnetic force (N/m), hr radial dimension of winding region in coil section (m), hz axial dimension of winding region in coil section (m),
r/ p,~ u & p~., p, o
strain, cooling efficiency, effectiveness factor, Poisson's ratio, resistivity of copper ($2m), resistivity of stainless steel (I2m), equivalent resistivity between filaments (I2m), stress (Pa).
References
[1] T. Satow, Y. Hattori and M. lwamoto, Optimum current of completely stabilized superconducting coil, in: Proceedings National Meeting of IEEJ 632(1977)789, in Japanese. [2] B.J. Maddock. G.B. James and W.T. Norris, Superconductive composites: Heat transfer and steady state stabilization, Cryogenic 9 (1969) 261-273. [31 Column Research Committee of Japan, Summary of Elastic Stability (Corona Press, Tokyo, 1960) in Japanese. [4] H. Brechna, Superconducting Magnet Systems (SpringerVerlag, Berlin, 1973) pp. 246-260. [5] Japan Atomic Energy Research Institute, Design Study of Superconducting Magnets for Tokamak Experimental Fusion Reactor (II), JAERI-M8666 (1979) in Japanese.