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Internal inductance: Difference between revisions
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In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition. | In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition. | ||
In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref> | In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref> | ||
:<math>l_i = \frac{ | :<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_P{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math> | ||
(for circular cross section plasmas with [[Toroidal coordinates|minor radius]] ''a''), where angular brackets signify taking a mean value. | (for circular cross section plasmas with [[Toroidal coordinates|minor radius]] ''a''), where angular brackets signify taking a mean value. | ||
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where ''R<sub>0</sub>'' is the [[Toroidal coordinates|major radius]], and similar for the external inductance. | where ''R<sub>0</sub>'' is the [[Toroidal coordinates|major radius]], and similar for the external inductance. | ||
Using Ampère's Law (<math>2 \pi a B_\theta(a) = \mu_0 I</math>), one finds <math>l_i = 2 \pi l_i'</math>. | Using Ampère's Law (<math>2 \pi a B_\theta(a) = \mu_0 I</math>), one finds <math>l_i = 2 \pi l_i'</math>. | ||
The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref> | |||
:<math>l_i(3) = \frac{2 V \left \langle B_\theta^2 \right \rangle}{\mu_0^2I^2 R_0}</math> | |||
which is equal to <math>l_i</math> asuming the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]] | |||
The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile). | The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile). | ||