Page 276 - Fundamentals of Magnetic Thermonuclear Reactor Design

Plasma Control System Chapter | 8 255

where index i runs over values from 1 to the number of states in the system.
From Eq. (8.19) it follows that

T
xSx
m = i i − 1. (8.20) m =xiTSxixiTL xi−1.
1
T
xL x i 1 1
i
1
−
1
If any matrix (including L*) is multiplied by a positive definite matrix (L L1−1
1
or R), there are no changes to the number of positive or negative eigenvalues, so
there is only one positive m value for one positive unstable mode τ . We denote
γ
i
it as stability margin m . If m < 0, then the plasma is either stable on the passive
s
s
structures’ time constant scale or unstable on an Alfvén timescale.
From Eq. (8.20) it can be shown that the m is only dependent on the con-
s
ductors’ geometry. The x eigenvector, corresponding to stability margin m ,
s
s
determines the in-conductor distribution of currents due to the plasma’s instan-
taneous displacement in the ‘instability’ direction. In physical terms, the m ,
s
if normalised, reflects the difference between a stabilising force acting on the
plasma by currents induced in ideally conducting passive structures at plasma
instantaneous vertical displacements, and a destabilising force due to the plasma
‘elongating’ by the poloidal field. Therefore, the value m determines how close
s
a passive structure has to be to stabilise plasma: the larger the m the closer the
s
structure to the plasma. In the design of a control system, m ≈ 0.5 is currently
s
acceptable. We notice that even at large m , the τ parameter (more important
γ
s
for the plasma stabilisation) may be small due to a passive structure’s poor con-
ductivity. The m and τ must be sufficiently high to enable an acceptable level
γ
s
of plasma stabilisation. Simply put, a well-conducting passive structure should
be as close to the plasma as the design permits.
The linear model (Eq. (8.14)) is used as a basis for synthesising the control-
lers of closed-loop control systems. Thus, it allows obtaining transfer matrix
K(s) that relates sensor signals identifying the plasma position, current and shape
to control coil voltages U(s) = K(s)g(s), with s denoting the Laplace variable.
At the design of the plasma position, current and shape control system, it is
also important to determine the plasma’s possible disturbances, which should
be suppressed by the control system. These include disturbances of the plasma’s
internal inductance l and the β parameter, caused by rearrangement of tem-
i
p
perature and current profiles at ‘minor disruptions.’ We introduce such perturba-
tions in our linear model.
Let δξ be a disturbance vector. Then, electro-technical Eq. (8.7) becomes
d d ∂Ψ d
L 3 I δ ( )+ M p ( δ I p) + δξ ( ) + RI δ = δ U. (8.21) L ddtδI+MpddtδIp+∂Ψ∂ξddtδξ+RδI=δU.
dt dt ξ ∂ dt 3
Eq. (8.9) reflecting plasma current variations becomes
∂Ψ I δ + ∂Ψ I δ + ∂Ψ δξ = 0, (8.22)
I ∂ I ∂ p p ξ ∂ ∂Ψ¯∂IδI+∂Ψ¯∂IpδIp+∂Ψ¯∂ξδξ=0,