Page 279 - Fundamentals of Magnetic Thermonuclear Reactor Design

258 Fundamentals of Magnetic Thermonuclear Reactor Design

associated with the current variations in the jth circuit ∆I . A better precision
j
is achieved when several ∆I current variations, both positive and nega-
j
tive, are taken and then averaged. Current variations must not lead to any
qualitative change in plasma configuration. For instance, when a free-
boundary problem is solved, an X-point plasma configuration must not
become a limiter configuration, and vice versa. As L = L + L , it is in
3
2
1
∂ Ψ
L =∂Ψp∂II=I 0 principle sufficient to determine the L = I ∂ p matrix coefficients. The
2
2
=
∂ Ψ II 0
L =∂Ψext∂II=I 0 L = I ∂ ext matrix is expressed through static inductances that only
1
1
=
II 0
depend upon the conductors’ geometry. Therefore, this matrix can be cal-
culated in advance.
∂ Ψ
Mp=∂Ψp∂IpI=I 0 The coefficients of the M = I ∂ p matrix in Eq. (8.7) are calculat-
p
p
=
II 0
ed in a similar way with the difference that the plasma current is varied:
 ∂ Ψ  ∆ Ψ


Mpj≡∂Ψp,j∂IpI=I ∆Ψp,j∆Ip M pj {} ≡  pj ,  ≅ pj , . Coefficients determined in a similar way
0
 I ∂ p II 0  I ∆ p


=

 ∂ Ψ  ∆ Ψ  ∂ Ψ  ∆ Ψ  ∂ g  ∆ g





∆
Ψ
∂
∂Ψ ∂ ∂ g I i p I I j = I = I 0 0 0  ∆ g ¯ i ∆ ∆ I I j are   ≅ ,   ≅ (see Eq. 8.9) and  i  ≅ i ,
I

¯
∂Ψ¯∂IjI=I ∆Ψ¯∆Ijp


 I ∂ j II 0  I ∆ j  I ∂ p II 0  I ∆ p  I ∂ j II 0  I ∆ j



=

=
=
 ∂ g  ∆ g


∂gi∂IpI=I ∆gi∆Ip  i  ≅ i (see Eq. 8.12).
0

 I ∂ p II 0  I ∆ p

=
The coefficients of the matrices for disturbances factored in the linear model
 ∂ Ψ  ∆ Ψ i  ∂ Ψ  ∆ g




∆
∆
ξ
Ψ
∂
ξ
∂
I
∂Ψi∂ξjI=I ∆Ψi∆ξjj are  i  ≅ ,  i  ≅ i , where the ξ disturbance param-
=
i
j
I
g
i

j
00
 ξ ∂ j II 0  ∆ ξ j  ξ ∂ j II 0  ∆ ξ j




=
=
eter is varied. The ∆Ip and ∆ξ variations too must not lead to any qualitative
change in plasma configuration.
c. After the completion of step b), derive the desired linear model in the form
of Eq. (8.25) or Eq. (8.26). Simultaneously, analyse the linear model char-
acteristics using the passive stabilisation or active stabilisation criteria (by
determining the increment, the stability margin, etc.).
8.4.2 Non-linear Modelling
Non-linear modelling allows one to assess how well certain parameters can be
controlled with poloidal field coils using a given power supply system. In ad-
dition, nonlinear modelling may be employed to optimise discharge scenarios
when planning a plasma experiment. A relevant code should be based on mod-
ern transport concepts in the plasma and be validated through experiments on
the operative machines.