Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas
Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas
Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas
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<strong>Magnetic</strong> fields <strong>and</strong> tokamak plasmas<br />
Alan Wootton<br />
ε µ 1<br />
= n ww 0<br />
a l<br />
τ int<br />
2π<br />
∫<br />
0<br />
B ρ<br />
( ω )sin( ω)<br />
dω = n ww 0<br />
µ 0<br />
I p<br />
µ<br />
τ int<br />
2<br />
1<br />
10.11<br />
Using the expression <strong>for</strong> B ρ from Equation 7.3 we get<br />
µ 1<br />
= ∆ g<br />
a l<br />
− a l<br />
2R l<br />
⎛<br />
ln a ⎞<br />
⎜<br />
l<br />
⎝ a p ⎠<br />
⎟ + ⎛<br />
Λ + 1 ⎞<br />
⎝ 2 ⎠ 1− a 2<br />
⎡<br />
⎛<br />
⎜<br />
p ⎞ ⎤<br />
⎟<br />
⎢<br />
2<br />
⎣<br />
⎝ a l<br />
⎠ ⎥<br />
⎦<br />
10.12<br />
The output from the integrated 'modified Rogowski coil each turn of area A, with n 0 cos(ω) turns<br />
per unit length, is.<br />
2 π<br />
∫<br />
ε λ 1<br />
= n 0<br />
Aa l<br />
B<br />
τ<br />
ω<br />
( ω)cos( ω ) dω = n 0<br />
Aµ 0<br />
I p<br />
int<br />
τ int<br />
2 λ 10.13<br />
1<br />
0<br />
Using the expression <strong>for</strong> B ω from Equation 7.2 we obtain<br />
λ 1<br />
= − ∆ g<br />
a l<br />
= − a l<br />
R l<br />
− a l<br />
2R l<br />
⎡ ⎛<br />
Λ + ln a ⎞ ⎤<br />
⎜<br />
l<br />
⎢<br />
⎟<br />
⎣ ⎝ a p ⎠<br />
⎥ − µ 1<br />
⎦<br />
⎡ ⎛<br />
ln a ⎞<br />
⎜<br />
l<br />
⎝ a p ⎠<br />
⎟ + ⎛<br />
Λ + 1 ⎞<br />
⎝ 2 ⎠ 1 + a 2<br />
⎛<br />
p<br />
⎜<br />
⎞ ⎤<br />
⎟<br />
⎢<br />
−1<br />
2<br />
⎣<br />
⎝ a l ⎠<br />
⎥<br />
⎦<br />
10.14<br />
Be<strong>for</strong>e we can substitute these expressions (Equations 10.13 <strong>and</strong> 10.14) into Equation 10.10, we<br />
must recognize that our equilibrium fields were evaluated in a left h<strong>and</strong>ed coordinate system,<br />
while this section we have worked in a right h<strong>and</strong>ed system. Sorting this out we find λ 1 ⇒ -λ 1 ,<br />
<strong>and</strong> µ 1 ⇒ µ 1 , so that<br />
∆ R<br />
− ∆ g<br />
= a 2<br />
p ⎛<br />
Λ + 1 ⎞<br />
2R ⎝<br />
l<br />
2⎠<br />
10.15<br />
This is the difference between the geometric center ∆ g <strong>and</strong> the current center ∆ R of a circular<br />
plasma, under the present approximations. We also note that, after sorting out the coordinates,<br />
subtracting the outputs from our coils gives<br />
λ 1<br />
− µ 1<br />
= a l<br />
R l<br />
that is, we can measure<br />
⎛ ⎛<br />
Λ + ln a ⎞ ⎞<br />
⎜<br />
l<br />
⎜ ⎟ ⎟ 10.16<br />
⎝ ⎝ a p ⎠ ⎠<br />
90
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