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

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

Internal inductance of a plasma

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

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

Its complement is the external inductance (L = 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

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

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

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

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

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

The ITER design uses the following approximate definition:

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

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.