Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
−n ˜ ζ (s) + k s˜ i s
(s) + k ˜
f
i f
(s) = 0
µ s
s ˜ ζ (s) + (s +1)˜ i s
(s) + m sf
s˜ i f
(s) = 0
µ f
s ˜
⎛
ζ (s) + sm fs˜
i s
(s) + s + 1 ⎞
⎜ ⎟ ˜ v
i f
(s) − ˜ f
(s)
= 0
⎝ τ f ⎠ τ f
v ˜ f
(s) = g ˜
k ζ (s) − ( 1− τ s ) ˜ ζ (s) + x s˜ i (s) + x i ˜ (s)
0 Σ s f f
f
[ ( )]
21.32
where τ Σ = τ a + τ d . Solutions for the displacement ζ, vessel current i s , feedback current i f and
feedback voltage v f can be obtained; for the displacement ζ is given by:
where
˜ ζ (s)
˜ ζ 0
(s) =
A + Bs
D + Es + Fs 2 21.33
A = gk f
B = −g( k s
m sf
− k f )= 0
D = k f
( n + g)+ngx f
E = −gx s
m sf ( n + k s
µ s ) + k f [ n + k s
µ s
+ ( n + k f
µ f )τ f ]− gk f
τ Σ
+gx f
n +k s
m s
−nt S
F = m sf [ gx s
τ Σ
+τ f
k s ( 1− m sf
m fs )]n [ + k s
µ s ]− t S
x f
g( n +k s
m s )
( )
21.34
The term appearing due to the coupling between sensors and feedback windings are in bold. The
remaining terms are identical to those previously derived, except that the values of x s to be
considered are different (negative values are now permitted).
For stability, the Routh criteria can be applied, which requires 1) D > 0, 2) E > 0, and 3) F > 0.
These three criteria define the available operating space in (g,τ Σ ) space for a given instability (i.e.
for a given decay index n).
Application of stability criteria.
Values relevant to TEXT Upgrade are t s = 7ms, t f = 137ms, τ f = 20, m sf = m fs = 0.67, k s = 2.26,
k f = 1.5, µ s = 1.59, µ f = 1.06, and the pairs of values for x s and x f given in table 5. Experimental
observations of open loop vertical motion show τ instability = 1/γ ~ 100 ms, so that a suitable
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