Page 293 - Fundamentals of Magnetic Thermonuclear Reactor Design

272 Fundamentals of Magnetic Thermonuclear Reactor Design

At the second stage, the component B [R(l),Z(l),φ(l)] on the q = 2 surfaces
⊥
is decomposed into a double Fourier series. The amplitudes B m,n on unperturbed
magnetic field lines are
×
c ()
s ()
Bm,n=Bm,n(c)+i×Bm,n(s), B mn = B mn + iB mn , (8.59)
,
,
,
1
∫
∫
φ
⋅
θ
φ
θ
⋅
),
[(
(
2
()
[
(
Bm,n=12π dφdθ(l)⋅B[R(l),Z(l),φ( B mn = 2π 2  dφ  dl BR lZ l(),( l)] exp{ in l)− m l)]}, (8.60)
⊥
,
l)]⋅exp{i[nφ(l)−mθ(l)]},
) (
= 1,1
m,n=1,1,2,1,3,1. (mn, ) ( ) ( ,2,1 ,3,1 ).
(0)
Bm,n(0) Amplitude B mn, and phase Ψ mn of the resonance harmonic on the q = 2
,
c ()
c ()
Ψm,n
Bm,n(c) rational surface are expressed through the cosine and sine amplitudes B mn , B mn :
,
,
Bm,n(c) 2 2
c ()
2
+
2
Bm,n(0)=Bm,n(c) +Bm,n(s) , B (0) , = (B mn ) (B mn , s () ) , (8.61)
,
mn
s ()
(
Ψm,n=arctg(Bm,n(s))/Bm,n(c). Ψ mn = arctgB mn ) B , c () . (8.62)
,
,
mn
A Fourier analysis of error fields [10] produced by the poloidal and toroidal
magnet systems may be performed using the matrix method. For instance, the
following relations hold for toroidal coil error fields:
ck 1) 
 b (, +  cosφ − sinφ   b (,
ck) 
bm,n(c,k+1)bm,n(s,k+1)=cosφt−si  mn , sk 1)  =  t t   mn , sk)  , (8.63)
nφtsinφtcosφtbm,n(c,k)bm,n(s,k),  b (, +   sinφ t cosφ t   b (, 
mn ,
mn ,
where φ = 20 degrees is the angle between adjacent toroidal coils, and k is the
t
coil number.
The main contributor to error fields is assembly and manufacture errors
Bm,n(s) of the toroidal and poloidal field coils. The resultant contribution of the coils
to the (m,n) error fields is determined by summing up the respective sine
c ()
s ()
Bm,n(c), and cosine amplitudes B mn ,B mn of error fields. For the ITER poloidal magnet
,
,
system, this procedure, accounting for the individual deviations of the coils,
is described as
12
Bm,n(c,s)(pol)=∑i=112bm,n(c,s)⋅ (, (, ′ ′ ′ ′
cs)
φδ φ ⋅
cs)
(
b
′
,
,
φi';δ'(φi')⋅δ(φi')|φi'=0×Ii+∑i=112b B mn ( pol) = ∑ mn ⋅  i ;( i )  δ φ )| φ =0 × I i
i
i
= i 1
m,n(c,s)⋅φi';δ'(φi')⋅δ(φi')|φi'=π/2× 12
′
′
′
′
(,
φδ φ ⋅
Ii+∑i=112bm,n(c,s)⋅φi';w'(φi')⋅w( + ∑ mn , cs) ⋅ ;( )  δ φ )| ′ × I i
(
b
i
 i
i
=
i
φi')|φi'=0×Ii+∑i=112bm,n(c,s)⋅φi'; = i 1 φπ /2 (8.64)
w'(φi')⋅w(φi')|φi'=π/2×Ii, 12
′
+ ∑ mn , cs) ⋅ ; ′ ( )  ω φ )| φ =0 × I i
′
′
φω φ ⋅
(,
(
b
′
i
 i
i
= i 1 i
12
+ ∑ mn , cs) ⋅ ; ′ ( )  ω φ )| φπ /2 × I ,
′
(,
φω φ ⋅
′
′
(
b
 i
i
′
i
i
=
= i 1 i