CALCULUS OF THE ELECTROMAGNETIC PARAMETERS IN A TOKAMAK

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CALCULUS OF THE ELECTROMAGNETIC PARAMETERS IN A TOKAMAK

I. Montanari, F. Negrini

To cite this version:

I. Montanari, F. Negrini. CALCULUS OF THE ELECTROMAGNETIC PARAMETERS IN A TOKAMAK. Journal de Physique Colloques, 1984, 45 (C1), pp.C1-909-C1-914.

�10.1051/jphyscol:19841186�. �jpa-00223662�

JOURNAL DE PHYSIQUE

Colloque C1, supplbment a u n o 1, Tome 45, janvier 1984 _page C1-909

CALCULUS OF THE ELECTROMAGNETIC PARAMETERS I N A TOKAMAK

I. M o n t a n a r i and

F.

N e g r i n i

Istituto di. EZettrotecniea, FacoZtd di Ingegneria, Universitd di BoZogna, Italy

R6sumd

-

Les c o e f f i c i e n t s d ' i n d u c t i o n a u t o e t mutuelle s o n t c a l c u l e s pour un enroulement t o r o ' i d a l dans un tokamak. Le modBle propose permet l a determina- t i o n d e s f o r c e s e t des moments a g i s s a n t s u r l e s modules de l ' a i m a n t t o r o i d a l pendant l e fonctionnement nominal. Les c a l a u l s s o n t e f f e c t u e s en employant une m6thode d e M o n t e - C a r l o s t r a t i f i e e . On d e c r i t de p l u s un syst2me d t 6 q u a t i o n s , d e r i v e de l a t h 6 o r i e d e s r e s e a u x , q u i permet de c a l c u l e r l e s f o r c e s e t l e s moments en c a s d e d e f a u t e l e c t r i q u e . On a c h o i s i c o m e a p p l i c a t i o n l ' a n a l y s e d ' u n tokamak 2 champ e l e v e

( B T > 8 T ) .

Abstract - The coefficients o f self and mutual induction of the toroidal winding in a tokamak are calculated. The method allows also t u determine the forces and the moments acting o n the modules of the toroidal winding during the nominal operation of the facility. A stratified Monte- Carlo numerical method is used for the calculations.

Further a set of equations which allows t o calculate the forces and the moments during a fault condition is described. The set of equations is derived by means o f the net-work theory. The models are applied t o analize the behaviour o f

a

high field tokamak (6, > 8 T).

1 - INTRODUCTION

The toroidal winding of a tokamak is the object o f the present work. The winding is composed by wedge-shaped modules which constitute angular portions of the torus and contain the coils. I n order t o d e sign correctly this system, i t is necessary t o know the forces and the moments acting on each coil i n nomi- nal operation and particularly i n fault conditions.

I n this paper are illustrated t w o mathematical models. The first model calculates the coefficients of mutual induction and the derivatives of them as regards t o particular directions. This coefficients are need- ed for the evalutation of the forces and of the moments in nominal operation. I n the second model the elec- tromagnetic parameters are utilized for the analysis o f the behaviour of the toroidal winding in fault condi- tions.

2 - THEORETICAL ASPECTS

The toroidal winding i n the tokamak is constituted b y N wedge-shaped coils as shown i n Fig. 1. A local coordinates system is connected t o each modules (Fig. 2). The position o f the local system with re- spect t o the main coordinates system is shown i n Fig. 3. A section of a winding on the x,, y, plane is shown i n Fig. 4. The geometrical parameters indicated i n this figureareemployed as input in the theoretical models.

Fig. 1

Fig. 2

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841186

JOURNAL DE PHYSIQUE

Fig.

3

Fig. 4

I n order t o calculate the coefficients of mutual induction and their derivatives,

t h e

following assump- tions are made:

- closed and stiffly bounded system - toroidal field alone

- stiff module

- quasi-stationary system (every part o f the system is sensible t o a perturbation practically at the same time).

- the relative permeability of the materials is equal t o 1 - the energy is stored only i n the magnetic field

- the modules are identical

The following relations are derived for the k coil:

Eq. 1 assumes the coil as a perfect conductor. This is not a binding assumption because the resistive com- ponent does not contributes t o determine the forces.

When a current goes through the coil a system of electromagnetic forces rises. This system

is

equiva- lent t o a force Fk applied t o 0, and t o a moment dk. The components in the generic Q direction o f Fk a n d X k are given respectively by eq.s 2 and 3

where

wa is the rotation around the axis passing through Ck in the direction of

Q. When a current ik goes through the k coil with self-inductance Lk, the energy is given [I] b y W

=

112 Lkii but

Tk (PI . Ik ^(Q) w = i [ q p ) i i ( p ) d p = - -

, I P - Q I d ~ d Q

therefore

d P d Q

I P - Q I

I n general terms

Po

d P d Q (4)

where V, and Vk are the volumes occupied by the conductors. ji: is the spatial distribution of the current

density in the coil and it is defined b y

The force acting along the l? directipn on the k coil, produced b y h coil, is

Po

[Jk (Q)

x

(Jh (P)

x

I P

--

Q I) ] .

U,

F ; ~ = - ^4~ ih ^{I P --}

^{Q 13}

^{d PdQ}

or, in general terms

[ J , : ( P , ~ ) ~ (J;:(Q,~)~ I P - Q I)].G, d P d Q

al? 4 r I P - Q l3

and

{IP-C, I x [J;(P, t ) x ( j ; ( ~ , t) I P - Q I)]).;,

d P d Q

aw, 4 r I P - Q l3 ⁽⁶⁾

Assuming the current density uniformely distributed i n each section of the coil, the spatial distribution of the current density becames

n -

J;: (P)

=

uOk(p) where p = - *Ro

a (Re - R,) (R, + Re) ( C O ~ 8k + P) _R,

+

Re The calculations of the mutual induction and of the forces i n nominal operation, is then possible.

I n order t o apply the study t o the fault conditions, t w o further assumptions are made:

- the current in the not short-circuited coils is equal t o the current i n nominal operation. This assump- t i o n is imposed b y the feed-back of the supply system

- the coil is adiabatic: there is not heat exchange between adjacent coils or between the coils and the structure o f the magnet.

Assuming as "fault" the short-circuit o f one or more coils, the configuration of the system is described i n the following way. The whole o f the coils is divided i n groups: the 0 group contains aH the not short-cir- cuited coils; the groups w i t h ordinal numbers greater then 0 contain the short-circuited coils. Better spec- ifying: a short-circuited coil constitutes a group; t w o or more short-circuited coils, but connected in series (the short-circuit occurs between the ends o f the series) constitute also a group; t w o or more indipendently short-circuited coils constitute t w o or more groups.

The Kirchoff equations, applied t o the electric circuit defined b y the group structure, assume the form

MEg is the mutual inductance between the k group and the g group (MZ is the self-inductance of the group) R$ is the resistence of the g group, i, and i, are the currents i n the k and g groups. The connection between the mutual inductance of the groups and those of the coils is given b y

Nk Ng

M : = 1. 6 ^ML..

1 1

The equation is obtained equating the expressions of the magnetic energy, written for the t w o parameters.

MkrgS is derived by means of eq. 4. Likewise the resistence Rg is given by

N9

This equation is obtained equating the exprenions of the dissipated power b y Joule effect i n the groups and i n the coils.

F& are the components of the force and of the moment acting o n the k coil in the direction up, produced b y a unitary current going through the k coil. This current is induced by a unitary current going through the g group. Therefore the resulting force and moment acting o n the k coil o f the g group

are

_{NG N}

G

F:

=

ig

f r

Fk ir A: = ig f-Xb ^{i, where F:k}

⁼

xr ^{f~~~ &ik=} x r m : r k

1 6 1 1

fir, and mgk are calculated respectively b y means of the eq.s 5 and 6.

The expression of the resistence of a coil is determined by the equation

JOURNAL DE PHYSIQUE

Rk = lk ^'d"

0

where theelectric conductivity odepends o n the temperature T. The temperature results from the thermal balance equation when neglecting the thermal conductivity term.

Hence the final system o f equations is

The Montecarlo method has been applied for the solution of the eq.s 4 t o 6. I n order t o reduce the computing time, an adaptive method for the reduction o f the variance, has been associated t o i t [2. 31.

The solution of the system 7 has been obtained by means of a method based o n the Adams express- ions [4].

3 - A N APPLICATION O F T H E MODEL

The described model has been applied t o t h e toroidal facility of F T (Frascati Torus) type. The geo- metrical parameters of the toroidal winding are shown in Tab. 1. Fig. 5 shows the behaviour of the nominal current i n the toroidal magnet. The fault conditions examined have been chosen

as

shown i n Tab. 2. Each fault case contemplates t w o groups: the 0 group that contains the not short-circuited coils and the 1 group that contains the series of short-circuited coils.

R I = 0.300E

+

⁰⁰

RE =0.417E

+

⁰⁰

RO =0.101E+01 ALFA =0.709E

-

01 XX =.0.177E

+

⁰⁰

YY = 0.385E

-

01 GAMMA = 0.132E 4- 00

NS =43

AA1 =0.677E

-

10 T A = 0.397E 4- 02 AA2 = 0.163E

-

16 TS = 0.988E

+

⁰¹

TO = 0.770E

+

^0.2

Tab. 1

TlME I.,

case number type o f fault

1 nominal operation (no faults) 2 short-circuit of coil 1

3 short-circuit o f coils 1 and 2 i n series 4 short-circuit o f coils 1, 2 and 5 in series

5

short-circuit o f coils 1, 2, 5 and 6 i n series

Tab.

2

4 - A N A L Y S I S O F RESULTS A N D CONCLUSIONS

Fig.

5

The coefficients of mutual induction and their derivatives are summarized i n Tab. 3. Fig.s 6, 7 and 8 show the variation, with respect t o the nominal operation, of forces and moments acting on coils 1 t o 3 in each fault case. The Fig. 9 shows the behaviour of the current i n the 1 group. From the comparative ana- lysis of the obtained values appear that the heavy fault is the short-circuit of only one coil.

The proposed theoretical model allows the complete analysis of the behaviour of the toroidal winding

in the different operating conditions.

COEFFICIENTS OF MUTUAL INDUCTION AND THEIR DERIVATIVES

I J M C F F CFR

1 1 0.159E

-

2 1 2 0.753E

-

3 1 3 0.266E

-

3 1 4 0.145E

-

3

1 5 0.702E

-

4

1 6 0.450E

-

4 1 7 0.266E

-

4 1 8 0.190E

-

4 1 9 0.133E - 4 1 10 0.106E

-

4 1 11 0.952E

-

5 1 12 0.764E

-

5 1 13 0.729E

-

5 2 3 0.564E

-

3 2 4 0.266E

-

3 2 5 0.121E

-

3 2 6 0.702E

-

4 2 7 0.398E

-

4 2 8 0.266E

-

4 2 9 0.176E

-

4 2 10 0.133E - 4 2 11 0.102E

-

4 2 12 0.865E

-

5 2 13 0.754E

-

5 Total inductance 0.986E

-

1 H

Tab.

3

CMZ

TIME 1%) TIME ($1 TIME (11

CASE 1 CASE 2 CASE 3 CASE 4 CASE 5

C1-914 JOURNAL DE PHYSIQUE

CASE 1 CASE 2 CASE 3 CASE 4 CASE 5

Fig. 7 c 1 -.315E

+

^{7 .318E}

+

7 . 1 7 3 ~

+

⁷ . 1 8 0 ~ + 7 . 1 3 5 ~ + 7 N C2

-

.315E

+

⁷

^-

^.315E

+

^{7 .173E}

+

^{7 .165E+ 7 .118E+ 7}

^N

C3

-

.315E

+

⁷

^-

^{.242E f 7}

^-

^.113E

+

⁷

^-

^.805E

+

⁶

-

.646E

+

⁶

^N

CASE 1 CASE 2 CASE 3 CASE 4 CASE 5

Fig. 8 c 1 .522E

+

^{5 -.526E+ 5 .809E}

+

^{5 .142E}

+

⁶ .123E + 6 Nm C2 -.522E

+

^{5 .202E i- 6}

-

^.809E

+

⁵

-

.356E -I 5 .962E -I- 4 Nm C3 .522E

+

^{5 .231E 6 .335E}

+

^{6 .202E+6 .815E+5 Nrn}

21

^{GROUP 1 ...}

^.

CASE 2 CASE 3 -.389E

+

^{5 -.206E}

+

5 A

CASE 4 CASE 5 . .

. '.. 2.: .

- 4 .'...." ^, ,

0 .I 1.0 1.1 2.0 2.5 3.0 3.5 4.0 4.5 5.0

T~ME 1%)

Fig. 9

REFERENCES

[ I ] JACKSON, J.D., "Classical Electrodynamics", JOHN WILEY & SONS Inc., U.S.A., 1962.

121 STROUD. A.H.. "Approximate calculation of multiple integrals", Prentice Hall Inc. Englewood

- Cliffs, ~ e Jersey. k

131 ZOREMBA. S.K.. "The mathematical basis of MonteCarlo and Quasi MonteCarlo Methods", SlAM .--

Rewiew. ~ d l . l 0 , n 0 3, July 1968.

[4] GEAR, C.W., "Numerical initial value Problems i n Ordinary Differential Equations", Prentice Hall

Inc., Englewood Cliffs, New Jersey 1971.