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Plasma Control System Chapter | 8 271
“overlap” error fields give for ITER case (R = 6.2 m) minimum value B error / B
t0
0
−5
≤ 5 × 10 [9].
8.7.2 Field Perturbation Harmonic Analysis
Error fields are determined in two stages on the basis of the Fourier analysis
using magnetic flux co-ordinates. At the first stage, the normal field B is calcu-
lated on the q = 2 magnetic surface described as
=
Ψ (, = q (/ = (8.50) Ψ(x,y,z)=const, q = (m/n) = 2/1.
mn)2 /1.
xy z,) const,
The magnetic flux Ψ (, Ψ(x,y,z)
xy z,) is produced by the poloidal magnet system and
the plasma current and can be evaluated in a plasma discharge scenario using
the Grad–Shafranov equation. The safety factor is expressed as
q = 1 ∫ dl Bl() , (8.51) q=12πdlBt(l)R(l)⋅Bp(l),
t
2 π Rl() ⋅ Bl()
p
where dl is the poloidal increment of a field line length; Bl() and Bl() are the B t ( l )
Bp(l)
t
p
respective poloidal and toroidal fields on the surface Ψ[X(l),Y(l),Z(l)] = const;
R(l) is the distance between the point [X(l),Y(l),Z(l)] and the tokamak axis.
The toroidal field is evaluated as
µ I
Bl () = 2π 0 Rl) . (8.52) Bt(l)=µ It2πR(l).
t
t
0
(
On unperturbed magnetic field lines [X(l),Y(l),Z(l)], the toroidal angle φ and
the poloidal angle θ are related linearly by
=
=
⋅
φ l () φ (0)− q θ l ( )const; (8.53) φ(l)=φ(0)−q⋅θ(l)=const;
φ
⋅
Xl() = Rl() cos( l); (8.54) X(l)=R(l)⋅cosφ(l);
Y l() = R l() sin( l). (8.55) Y(l)=R(l)⋅sinφ(l).
φ
⋅
The poloidal angle θ is defined by the equation
θ l() = 2 π × ∫ l dl ∫ dl (8.56) θ(l)=2π×(∫l ldlR (l)⋅Bp(l))/(dlR 2
2
0
2
2
l 0 Rl () ⋅ Bl () Rl () ⋅ Bl () (l)⋅Bp(l))
p
p
or through the safety factor q:
⋅
1 l dl Bl)
(
θ l () = ∫ t (8.57) θ(l)=1q∫l ldl⋅Bt(l)R(l)⋅Bp(l)
0
⋅
()
q l 0 Rl Bl()
p
The field B normal to the surface Ψ [X(l),Y(l),Z(l)] = const is determined as
⊥
(
B ⊥ = B ∇ ψ ) | ∇ ψ | . (8.58) B=(Bψ)/|ψ|.