19th IAEA Fusion Energy Conference14 - 19 October 2002, Lyon, France

A Tokamak with Nearly Uniform Coil Stress Based on Virial Theorem

H.Tsutsui, K.Nayakama, T. Ito, H. Ajikawa,S.Nomura, S.Tsuji-Iio, R.Shimada

Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology

Introduction• We studied the tokamak with the Force-Balanced Coils

which are hybrid helical coils of OH and TF coils and reduced the electromagnetic force.

• The virial theorem, which is derived only form the equilibrium, shows that the tension is required to hold the magnetic energy.

• The virial theorem in magnet systems is derived by the replacement of plasma pressure to stress.

• In this work, we extend the FBC by the virial theorem, and obtain the minimal stress condition.

• The new compact tokamak based on the virial theorem is designed and constructed.

Principle of Force-Balanced Coil

FcI

Fca R

R FhFh

Fc

I

Fc

aR

aR Fh

I

Fh

Hoop Force by Toroidal Current

Centering Force by Poloidal Current

Force-Balanced Coil

Centering force is much reduced,

butstress distribution

is not investigated.

Background

　

• Reduction Error Field• Estimation of Stress• Application of Virial

Theorem

TODOROKI-I Parameter Value

Toroidal Field 1T

Plasma Current 10kA

Time of Discharge 4ms

• The error field by FBC made the control of plasma difficult.

• The force of toroidal direction was reduced in FBC. Is it held in stress ?

Virial Theorem

tensorstress

field magnetic :

densitycurrent :

0

0

0

:S

S

B

j

BjB

Bj

=⋅∇=×∇

=⋅∇+×µ

( )

tensorstress Maxwell

2

02

0

:T

IT

ST

−≡

=+⋅∇

BBBµ１

( )

stress principal :

0d2

d

0d Tr

M0

23

1

i

ii UVBV

V

σµ

σ >≡=

=+

∫∫∑

∫

=

ST Ω

Ω

∫≡

≡

VV

UV

d

~M

σσ

σσ

1~3

1

=∑=i

iσ

• Positive stress (tension) is required to hold the field.• Uniform tension is favorable.• Theoretical limit is determined. 3

1~~~321 === σσσ

Equilibrium Eq.

Application to Helical Coil

N

<σ>

σφ

σθ A=2A=5

A=10

~

~

~

0 5 10 15 20-2

-1

0

1

2

energy.in optimal is 21~~ == φθ σσioAspect Rat :

Coil of Pitch:

1~~

28log2

28log

~

28log2

~

222

222

222

22

AIIN

AAAN

NAAA

AAANAN

φ

θ

φθ

φ

θ

σσ

σ

σ

≡

=+

−+

−−=

−+

−=

Virial-Limit Condition

Shape of Coils

<σφ>=0N=9FBC

<σφ>= <σθ>N=6VLC

<σθ>=0N=4

A

N

σθ=0

σφ=0 σθ=σφ

0 5 10 15

10

20

30

Aspect Ratio A =4

Relations of pitch number and aspect ratio of Virial-Limit Coil (VLC) etc.

Comparison of Toroidal Field

σσTF

ˆUVΩ≡

• In the case of low aspect ratio, 1.5 times stronger magnetic field is created compared with traditional TF coil.

σφ

ˆ2

TFC

=BB

A

<σ>

TFC

^max

σθ=σφ

σθ=0

σφ=0

0 5 10 15

1

2

A

B

σφ=0BTFC

σθ=0

σθ=σφφ

0 5 10 15

1

2

Toroidal EffectEquilibrium of Magnetic Pressure and Stress

z

rR

aρ

θφ

o ∫≡r

Rrrpraru 'd)'(')(

2)(

)(

Rru

RrarpT

rRruT

−−

−=

−=

φ

θ

0

22

2µθφ BB

p−

≡

• Distribution of stress in the toroidal shell with circular cross section is derived analytically by use of magnetic pressure.

Distribution of Stress(large aspect ratio)

A=100

0

22

M

T

2ˆ

µθφ BB

UVp

−≡

θ

pσ

^^

TFC

<σφ>=0

<σφ>=<σθ>σφ

σθ

p

0 1 2 3

0.8

0.9

1

-1

0

1

2

θ

pσ

^^

TFC<σφ>=0<σφ>=<σθ>

σφ

σθ

p

0 1 2 3

0.6

0.8

1

1.2

-1

0

1

2

A=10

• When A=100, distribution of stress is flat.• There is no advantage of helical winding.

Distribution of Stress(low aspect ratio)

• When A<10, distribution of stress is important.

• Assumption of large aspect ratio is not held.

• Optimal distribution is achieved to minimize the stressat θ=π.θ

σ

σφ

σθ^^

^

π[rad]

N=2.0κ=1.5

N=3.0

TFC

TFC

A=2

-2

0

2

4

0

Uniaxial Stress Model

Equilibrium of Electromagnetic Force and Stress

t u

T(s)

T(s+ds)

Fu(s)

Fu(s+ds)

s

coil oflength by coordinate

force neticelectromag

radius,curvatur

force sharing tension,:

0d

d

0dd

:

:

:

:

s

f

R

FT

uu

u

fRT

sF

RF

sT

=++

=+

Tension and Bending Moment

T (kN)

θ (rad)ππ/2

N=2N=3N=4

10

20

30

0

VLC

Mu [

k N

m]

θ [rad]

N=2N=3N=4

ππ/2

-2

0

2

4

0

VLC

• The tension of coil with pitch=3 is reduced and its distribution is flat.• In the fat cable, the bending stress (proportional to bending moment)

is important. • The distribution of bending moment in the coil with pitch 3 is flat.

Stress Analysis of VLC

Why FEM analysis ?3D analysis is required because the virial-limit condition is obtained from the 2D shell model.

Conditions in FEM Analysis Current layer coincide with magnetic surface.

Models in Analysis

VLC with N=3 FBC with N=4

HC with N=3 HC with N=4

・3D-Model with Electromagnetic Force・Structure Analysis by FEM（NASTRAN)　→　Stress

Parameters 　　ValueMajor radius 　0.30 mMinor radius 　0.14 mAspect ratio 　2.14Pitch number N 　24 turnsCoil current 　96 kA/1 poleToroidal field 　1.5 TCross section 　380 mm2Young's modulus 1.26×105 N/mm2Poisson's ratio 　0.33

Stress Analysis of VLC

Distributions of von Mises Stress－ＶＬＣ(N=3)－Max：365 N/mm2

－ＦＢＣ(N=4)－Max ：665 N/mm2

－ＨＣ(N=3)－Max ：627 N/mm2

－ＨＣ(N=4)－Max ：892 N/mm2

VLC(N=3) has no stress concentration and minimum stress compared with those of other coils. VLC realize both nearly uniform distribution of

stress and minimum stress in 3D model.

Outline of ＶＬＣ Tokamak

Easy winding of coil → integer winding pitchLarge Plasma Volume → low aspect ratio

　 Parameter 　　 ValueMajor radius(coil) 0.30 mMinor radius(coil) 0.14 mAspect ratio 　 2.14Pitch number 　　 3Pole number 　　 8inductance 　 1.3 mHMajor radius(vessel) 0.30 mMinor radius(vessel) 0.08 mToroidal field 1.55 TMinor radius(plasma) 0.07 mPlasma current 40.0 kA

Parameters of Tokamak

Outline of Design

Coil:　　　　high-tension Kevlar　cross section of copper 12.72 mm2Vacuum Vessel:　　　SUS304Coil Support:　GFRP (thickness 20 mm)

Materials

Coil orbit between their supports is nearly straight.

VLC (N=3)

Coil SupportVacuum Vessel

Todoroki-II

Vertical Field

Controllability

　→　minimization of mutual inductances to VLC.

Positional Instability

　→　n-index: n (stable condition: 0<n<1.5)

How to design Vertical Field Coil (VFC)

n-index Positions of VFC on the cross section

VLC

Supporting BoardVessel Limiter

0.15 0.20 0.25 0.30 0.35 0.40 0.45

Average

r (m)

φ= 7.5°

0.15 0.20 0.25 0.30 0.35 0.40 0.45

r (m)

φ= 3.75°

0.15 0.20 0.25 0.30 0.35 0.40 0.45

r (m)

0.15

-0.15

-0.10

-0.05

0.05

0.10

0.00

φ= 0°

z (m

)

0.15 0.20 0.25 0.30 0.35 0.40 0.45

r (m)

0.00

1.00

0.25

0.50

1.50

0.75

1.25

coilvessel

Stable in almost all region in the vacuum vessel

Power Supply and Its Operation

Operation

Capacitor x 8 　　　Capacitance　:0.5 mF　　　Max Voltage:12.5 kV

Inductance of VLC 1.3 mH

PLASMA

VLC

VFC

Capacitors

FistIgnitron

SecondIgnitron

0

20

40First Second

0

0.25

0.5

Bφ (T)

Plasma current(kA)

Vloop (V)

VLC currentIpole (kA)

0

50

100

-2 0 2 4 6 80

10

20

Time (msec)

Power Supply

• Discharge period: about 10 msec

Todoroki-IIIn the initial stage, there is notoroidal field because VLC is a hybrid coil of CS and TF coil.

First: Toroidal FieldSecond: Loop Voltage

Breakdown ConditionCase 1: Ideal VLCCase 2: Designed VLCCase 3: Designed VLC+VFC

Required Loop Voltage in Breakdown

Normalized Poloidal Field

200 300 400

0

0.01

0.02

0.03

0.04

0.05

R (mm)B

P/B T

Experiment

　Case 1　Case 2

TZRTP BBBBB 22 +=

Case 1: Ideal Coil Orbit

Case 2: Designed Coil Orbit

Next equation is obtained from Townsend Avalanche Model.

PT

pe X

BB

NNA

eARV

= )log(2 0

1

20min π

R0：major radius　 A1, A2：constants of gas speciesXp：limiter radius ne/n0：multiplication factor of electron

Loop voltage in case 1

( )

Additional vertical field is required for breakdown.

Summary

• The relation of toroidal field and stress is obtained by virial theorem, which shows that the optimal stress configuration is uniform tensile stress.

• When A=2 and κ=2, a virial-limit coil (VLC) makes 1.7 times stronger magnetic field than TF coil.

• VLC winding generates small error fields, and makes room for blanket and other parts in conventional tokamak reactors.

• Nearly uniform stress distribution with VLCconfiguration is obtained from both uniaxial model and FEM analysis.

• A small VLC tokamak Todoroki-II was constructed and its experiments started.