Page 275 - Fundamentals of Magnetic Thermonuclear Reactor Design

254 Fundamentals of Magnetic Thermonuclear Reactor Design

d
 ∗ I δ ( )+ R I = δ U

δ
L
L∗ddtδI+RδI=δUgt=CδI.  dt (8.14)
  gt () = CI.
δ
Eq. (8.14) allows one to analyse the passive stabilisation efficiency of the
structures surrounding the plasma. The characteristic routinely used in this con-
text for vertically elongated plasmas is the plasma instability growth rate (incre-
−L*−1R ment), γ. It is defined as a singular positive eigenvalue of the matrix ()− L * − 1 R,
related to the unstable vertical mode. This value largely determines the param-
eters of power supplies for control by the active coils. Generally, the higher
this value, the greater the power supply frequency characteristics and required
power for control. Another characteristic routinely employed in the passive sta-
bilisation analysis is the stability margin, m . Let us take a close look at it.
s
For unstable plasma configurations, τ , the inverse of the increment, is de-
γ
fined as a singular positive eigenvalue of MLE (8.14), that is
∗
−L∗xu≡S−L xu=τγRxu, −Lx u ≡ ( −SL x = τ R x , (8.15)
γ
1
u
) u
1
*
∂
∂
Ψ
/
¯/
S≡−[L −Mp(∂Ψ¯/∂I)/( L*≡L −S)] where matrix L ≡ L 1 − S; matrix ≡−S L [ 2 − M ( ∂ Ψ ∂I) ∂ ( Ψ ∂I )] ; matrix
/
p
I
2
p
p
1
∂ Ψ
L 1 = ext i () ; R is the resistance matrix, and x is the distribution of currents
I ∂ j () u
=
II 0
induced by the unstable mode, that is, the eigenvector corresponding to the in-
L =∂Ψexti∂IjI=I 0 crement. From Eq. (8.15) follows
1
xL x  xSx 
T
T
τγ=xuTL xuxuTRxuxuTSxuxuTL xu−1. τ = u 1 u  u u − 1.  (8.16)
γ
1
1
T
T
xRx u  xL x u1 
u
u
We now introduce a ‘resistive’ stability margin
T
xSx
µ=xuTSxuxuTL xu−1, µ = u u − 1, (8.17)
1
T
xL x u
1
u
and the passive structure’s senior time constant
T
xL x
τS=xuTL xuxuTRxu−1. τ = u 1 u − 1. (8.18)
S
1
T
xRx u
u
Then, τ = τ µ. From Eq. (8.18) follows that the passive structure’s senior
γ
S
time constant is an averaged value that can be unambiguously determined by the
x . We note that µ depends on the conductors’ resistance through the x .
u
u
To pass to the ‘inductive’ version of the stability margin, we substitute matrix
∂ Ψ ext i ()
L =∂Ψexti∂IjI=I 0 R in Eq. (8.15) with matrix L 1 = I ∂ and find the eigenvalues m and the
i
1
j ()
=
II 0
eigenvectors x in the problem:
i
SL )
*
≡
−L*xi≡S−L xi=miL xi, − Lx ( − 1 x = m Lx , (8.19)
i
i
i 1
i
1
1