Page 274 - Fundamentals of Magnetic Thermonuclear Reactor Design

Plasma Control System Chapter | 8 253

The linear system (8.7) is one equation short of the number of states. It
should be complemented with an equation describing the change of the plasma
current δI . To this end, we assume, consistently with the MHD theory, that the
p
poloidal flow is ‘frozen’ into the plasma during the discussed time periods, so
that
1
Ψ = ∫ jds = const,
Ψ
I p s (8.8) Ψ¯=1Ip∫sjΨds=const,
where j is the plasma current distribution, and S is the plasma column cross-
section. We linearise Eq. (8.8) with respect to the δI and δI states, and obtain
p
∂ Ψ I δ + ∂ Ψ I δ = 0,
I ∂ I ∂ p p (8.9) ∂Ψ¯∂IδI+∂Ψ¯∂IpδIp=0,
or

∂ Ψ ∂ Ψ
I δ p =− I ∂ I δ ∂ I p . (8.10) δIp=−∂Ψ¯∂IδI/∂Ψ¯∂Ip.

Through the substitution of Eq. (8.10) into Eq. (8.7) the latter becomes
d
δ
L ∗ I δ ( )+ R I = δ U,
dt (8.11) L∗ddtδI+RδI=δU,
∗
∂
∂
where L = L − M [( Ψ ∂ I) /( Ψ ∂ I )]. This linear system reflects the cur- L∗=L −Mp(∂Ψ¯/∂I)/(∂Ψ¯/∂Ip)
/
/
3
p
p
3
rent behaviour in active and passive circuits, and Eq. (8.9) describes the plasma
current dynamics.
To describe a plasma shape evolution, let us introduce vector g(t) charac-
terising the parameters of plasma shape and position. This vector may include
such parameters as the plasma elongation and triangularity, X-point position,
distance between the plasma boundary and given points on the first wall, and
so on. Then
∂ g ∂ g
gt () = I δ + δ I ,
I ∂ I ∂ p p gt=∂g∂IδI+∂g∂IpδIp,
which leads, through Eq. (8.10), to
 ∂ g ∂ g ∂Ψ ∂Ψ 
gt () =  −  δ I. (8.12) gt=∂g∂I−∂g∂Ip∂Ψ¯∂I/∂Ψ¯∂IpδI.
 I ∂ I ∂ p I ∂ I ∂ p 
∂ g ∂ g Ψ∂ ∂ Ψ
We denote C ≡ − and thus obtain C≡(∂g∂I−∂g∂Ip∂Ψ¯∂I/∂Ψ¯∂Ip)
I ∂ I ∂ p I ∂ ∂ I p
δ
gt () = CI. (8.13) gt=CδI.
We therefore obtain the required MLE describing the dynamics of circuit
currents and plasma shape evolution: