# 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*:

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

It can be shown that^{[1]}^{[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]}:

Its complement is the external inductance (*L = L _{i} + L_{e}*).

## Normalized internal inductance

In a tokamak, the field produced by the plasma current is the *poloidal* magnetic field *B _{θ}*, 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*), where angular brackets signify taking a mean value.

Using Ampère's Law (), one obtains ^{[3]}

where *R _{0}* is the major radius, and similar for the external inductance.

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

which is equal to assuming the plasma has a perfect toroidal shape, .^{[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 profile): a small value of corresponds to a broad current profile.

## 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 - ↑ 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 - ↑ Effective plasma radius