Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton ε µ 1 = n ww 0 a l τ int 2π ∫ 0 B ρ ( ω )sin( ω) dω = n ww 0 µ 0 I p µ τ int 2 1 10.11 Using the expression for B ρ from Equation 7.3 we get µ 1 = ∆ g a l − a l 2R l ⎛ ln a ⎞ ⎜ l ⎝ a p ⎠ ⎟ + ⎛ Λ + 1 ⎞ ⎝ 2 ⎠ 1− a 2 ⎡ ⎛ ⎜ p ⎞ ⎤ ⎟ ⎢ 2 ⎣ ⎝ a l ⎠ ⎥ ⎦ 10.12 The output from the integrated 'modified Rogowski coil each turn of area A, with n 0 cos(ω) turns per unit length, is. 2 π ∫ ε λ 1 = n 0 Aa l B τ ω ( ω)cos( ω ) dω = n 0 Aµ 0 I p int τ int 2 λ 10.13 1 0 Using the expression for B ω from Equation 7.2 we obtain λ 1 = − ∆ g a l = − a l R l − a l 2R l ⎡ ⎛ Λ + ln a ⎞ ⎤ ⎜ l ⎢ ⎟ ⎣ ⎝ a p ⎠ ⎥ − µ 1 ⎦ ⎡ ⎛ ln a ⎞ ⎜ l ⎝ a p ⎠ ⎟ + ⎛ Λ + 1 ⎞ ⎝ 2 ⎠ 1 + a 2 ⎛ p ⎜ ⎞ ⎤ ⎟ ⎢ −1 2 ⎣ ⎝ a l ⎠ ⎥ ⎦ 10.14 Before we can substitute these expressions (Equations 10.13 and 10.14) into Equation 10.10, we must recognize that our equilibrium fields were evaluated in a left handed coordinate system, while this section we have worked in a right handed system. Sorting this out we find λ 1 ⇒ -λ 1 , and µ 1 ⇒ µ 1 , so that ∆ R − ∆ g = a 2 p ⎛ Λ + 1 ⎞ 2R ⎝ l 2⎠ 10.15 This is the difference between the geometric center ∆ g and the current center ∆ R of a circular plasma, under the present approximations. We also note that, after sorting out the coordinates, subtracting the outputs from our coils gives λ 1 − µ 1 = a l R l that is, we can measure ⎛ ⎛ Λ + ln a ⎞ ⎞ ⎜ l ⎜ ⎟ ⎟ 10.16 ⎝ ⎝ a p ⎠ ⎠ 90
Magnetic fields and tokamak plasmas Alan Wootton β I + l i 2 = Λ +1 = 1+ R l a l λ 1 − µ 1 ⎛ ( ) − ln⎜ a l ⎝ a p ⎞ ⎟ 10.17 ⎠ Combining Equations 10.15 and 10.16, we see that we can measure the current center, the geometric center, and Λ, using our modified Rogowski and saddle coils. 91
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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