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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=12πdlBt(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=(Bψ)/|ψ|.
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