Internal inductance

The self-inductance of a current loop is defined as the ratio of the magnetic flux Φ traversing the loop and its current I:

${\displaystyle L=\Phi /I\,}$

The flux is found by integrating the field over the loop area:

${\displaystyle \Phi =\int _{S}{{\vec {B}}\cdot d{\vec {S}}}}$

On the other hand, the energy contained in the magnetic field produced by the loop is

${\displaystyle W=\int {{\frac {B^{2}}{2\mu _{0}}}d{\vec {r}}}}$

It can be shown that[1][2]

${\displaystyle W={\frac {1}{2}}LI^{2}}$

Internal inductance of a plasma

The internal inductance is defined as the part of the inductance obtained by integrating over the plasma volume P [3]:

${\displaystyle {\frac {1}{2}}L_{i}I^{2}=\int _{P}{{\frac {B^{2}}{2\mu _{0}}}d{\vec {r}}}}$

Its complement is the external inductance (L = Li + Le).

Normalized internal inductance

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]

${\displaystyle l_{i}={\frac {\left\langle B_{\theta }^{2}\right\rangle _{P}}{B_{\theta }^{2}(a)}}={\frac {2\pi \int _{0}^{a}{B_{\theta }^{2}(\rho )\rho d\rho }}{\pi a^{2}B_{\theta }^{2}(a)}}}$

(for circular cross section plasmas with minor radius a), where angular brackets signify taking a mean value.

Using Ampère's Law (${\displaystyle 2\pi aB_{\theta }(a)=\mu _{0}I}$), one obtains [3]

${\displaystyle l_{i}={\frac {L_{i}}{2\pi R_{0}}}{\frac {4\pi }{\mu _{0}}}={\frac {2L_{i}}{\mu _{0}R_{0}}}}$

where R0 is the major radius, and similar for the external inductance.

The ITER design uses the following approximate definition:[5]

${\displaystyle l_{i}(3)={\frac {2V\left\langle B_{\theta }^{2}\right\rangle }{\mu _{0}^{2}I^{2}R_{0}}}}$

which is equal to ${\displaystyle l_{i}}$ assuming the plasma has a perfect toroidal shape, ${\displaystyle V=\pi a^{2}\cdot 2\pi R_{0}}$.[6]

Relation to current profile

The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the ${\displaystyle B_{\theta }(\rho )}$ profile): a small value of ${\displaystyle l_{i}}$ corresponds to a broad current profile.

References

1. P.M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press (2006) ISBN 0521821169
2. Wikipedia:Inductance
3. J.P. Freidberg, Plasma physics and fusion energy, Cambridge University Press (2007) ISBN 0521851076
4. K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, Springer-Verlag (2005) ISBN 3540242171
5. 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