Page 273 - Fundamentals of Magnetic Thermonuclear Reactor Design

252 Fundamentals of Magnetic Thermonuclear Reactor Design

Taking into account Eq. (8.3), linearised Eq. (8.2) is written as follows:
 
d ∂ Ψ
ddt∂Ψ(i)∂I(j)I=I δI+RδI=δU,  () i δ  + R δ = δU, (8.4)
I
I
0
dt   ∂I () j II  
= 0
to describe the dynamics of current deviations in circuits δI(t).
The ψ flux is a linear combination of the flux through circuit, ψ , from ex-
ext
ternal currents and the own current, and the flux from plasma, ψ . The ψ value
p
p
Ψpr,z,tΨpr ,z ,0+∂Ψpi∂IjI=I δ is determined by changes in the plasma shape and the plasma’s own current
0 0
0
I+∂Ψp∂IpI=I δIp, (here and throughout, changes in the shape refer to the plasma column move-
0
ment and deformation). In a linear approximation
∂ Ψ p ∂ Ψ p
,,
I
Ψ (rz t ) ≅ Ψ (r z ,0 )+ δ ( )+ δ ( ), (8.5)
p
, 0
I p
p
0
∂I p II ∂I p II
= 0
= 0
∂ Ψ pi ()
∂Ψpi∂IjI=I 0 where I ∂ is the matrix with elements (i,j), which determines flux devia-
j ()
=
II 0
tions in circuit i due to the plasma shape changes at the unit current deviation in
∂ Ψ p
∂Ψp∂IpI=I 0 circuit j at time t = 0, and ∂ is the vector with elements (i), which deter-
I p
=
II 0
mines changes in flux deviation in circuit i at the unit plasma current deviation
at time t = 0.
∂Ψexti∂IjI=I ddtδI+∂Ψp∂II=I 0 From Eq. (8.5) we see that a linear model must include state δI – the plasma
0
p
ddtδI+∂Ψp∂IpI=I ddtδIp+RδI=δU, current deviation from the initial value. Then, Eq. (8.4) becomes
0
∂ Ψ ext i () d I δ ( )+ ∂ Ψ p d I δ ( )+ ∂ Ψ p d δ ( I p) + R I δ = δ U,
I ∂ j () dt I ∂ = dt ∂ I p dt (8.6)
=
=
II 0 II 0 II 0
∂ Ψ ext i ()
∂Ψexti∂IjI=I 0 where I ∂ j () is the matrix with elements (i,j), which determines flux de-
=
II 0
viation in circuit i at the unit current deviation in circuit j at time t = 0 (a static
inductance matrix). We introduce designations
∂ Ψ ext i () ∂ Ψ pi () ∂ Ψ
L 1 = I ∂ L ; 2 = I ∂ ; M p = ∂ p L ; 3 = L 1 + L 2
=
=
=
L =∂Ψexti∂IjI=I ; L =∂Ψpi∂Ij j () II 0 j () II 0 I p II 0
1
2
0
I=I ; Mp=∂Ψp∂IpI=I ; L =L +L
1
0
2
0
3
to bring Eq. (8.6) to form
d d
δ
L ddtδI+MpddtδIp+RδI=δU. L 3 I δ ( )+ M p I δ ( p) + R I = δ U. (8.7)
3
dt dt