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 /> | ||
Revision as of 07:49, 10 August 2012
The self-inductance of a current loop is defined as the ratio of the magnetic flux Φ traversing the loop and its current I:
The flux is found by integrating the field over the loop area:
On the other hand, the energy contained in the magnetic field produced by the loop is
The internal inductance is defined as the part of the inductance obtained by integrating over the plasma volume P [3]:
Its complement is the external inductance (L = Li + Le).
In a tokamak, the field produced by the plasma current is the poloidal magnetic field Bθ, so only this field component enters the definition. In this context, it is common to use the normalized internal inductance[4]
(for circular cross section plasmas with minor radius a).
Alternatively, sometimes the internal inductance per unit length is used, defined as[3]
where R0 is the major radius, and similar for the external inductance. This differs from the preceding definition according to .
The value of the normalized internal inductance depends on the current density profile in the toroidal plasma.
References
- ↑ P.M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press (2006) ISBN 0521821169
- ↑ Wikipedia:Inductance
- ↑ 3.0 3.1 J.P. Freidberg, Plasma physics and fusion energy, Cambridge University Press (2007) ISBN 0521851076
- ↑ K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, Springer-Verlag (2005) ISBN 3540242171