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Internal inductance: Difference between revisions
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It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) ISBN 0521821169</ref><ref>[[:Wikipedia:Inductance]]</ref> | It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) ISBN 0521821169</ref><ref>[[:Wikipedia:Inductance]]</ref> | ||
:<math>W = \frac12 L I^2</math> | :<math>W = \frac12 L I^2</math> | ||
The internal inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref>J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>: | The internal inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>: | ||
:<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math> | :<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math> | ||
Its complement is the external inductance (''L = L<sub>i</sub> + L<sub>e</sub>''). | Its complement is the external inductance (''L = L<sub>i</sub> + L<sub>e</sub>''). | ||
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 per unit length, defined as | 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{L_i}{2\pi R_0}\frac{4\pi}{\mu_0} = \frac{2L_i}{\mu_0R_0}</math> | :<math>l_i = \frac{\int_P{B_\theta^2 d\vec r}}{\pi a^2 B_\theta^2(a)} </math> | ||
and similar for the external inductance. | (for circular cross section plasmas with [[Toroidal coordinates|minor radius]] ''a''). | ||
Alternatively, sometimes the internal inductance per unit length is used, defined as<ref name="Freidberg"/> | |||
:<math>l_i' = \frac{L_i}{2\pi R_0}\frac{4\pi}{\mu_0} = \frac{2L_i}{\mu_0R_0}</math> | |||
where ''R<sub>0</sub>'' is the [[Toroidal coordinates|major radius]], and similar for the external inductance. | |||
This differs from the preceding definition according to <math>l_i' = l_i/(2\pi R_0)</math>. | |||
The value of the normalized internal inductance depends on the current density profile in the toroidal plasma. | The value of the normalized internal inductance depends on the current density profile in the toroidal plasma. | ||
== References == | == References == | ||
<references /> | <references /> | ||