# 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*:

- $ L = \Phi/I\, $

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

- $ \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

- $ W = \int{\frac{B^2}{2\mu_0} d\vec r} $

It can be shown that^{[1]}^{[2]}

- $ W = \frac12 L I^2 $

## Contents

## 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]}:

- $ \frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r} $

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]}

- $ 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 ($ 2 \pi a B_\theta(a) = \mu_0 I $), one obtains ^{[3]}

- $ l_i = \frac{L_i}{2\pi R_0}\frac{4\pi}{\mu_0} = \frac{2L_i}{\mu_0R_0} $

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

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

- $ l_i(3) = \frac{2 V \left \langle B_\theta^2 \right \rangle}{\mu_0^2I^2 R_0} $

which is equal to $ l_i $ assuming the plasma has a perfect toroidal shape, $ 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 $ B_\theta(\rho) $ profile): a small value of $ l_i $ 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