# 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

$W = \frac12 L I^2$

## Internal inductance of a plasma

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

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

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

$l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_0^a{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}$

(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 

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

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

The ITER design uses the following approximate definition:

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

which is equal to $l_i$ assuming the plasma has a perfect toroidal shape, $V = \pi a^2 \cdot 2 \pi R_0$.

## 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.