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:

$L=\mathrm{\Phi }/I$

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

$\mathrm{\Phi }=\underset{S}{\int }\overrightarrow{B}\cdot d\overrightarrow{S}$

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

$W=\int \frac{B^2}{2{\mu }_0}d\overrightarrow{r}$

It can be shown that

$W=\frac{1}{2}L{I}^2$

Internal inductance of a plasma

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

$\frac{1}{2}{L}_i{I}^2=\underset{P}{\int }\frac{B^2}{2{\mu }_0}d\overrightarrow{r}$

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

$l_i=\frac{{⟨B_{\theta }^2⟩}_P}{B_{\theta }^2(a)}=\frac{2\pi \underset{0}{\overset{a}{\int }}{B}_{\theta }^2(\rho )\rho d\rho }{\pi a^2{B}_{\theta }^2(a)}$

(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

$l_i=\frac{L_i}{2\pi R_0}\frac{4\pi }{{\mu }_0}=\frac{2L_i}{{\mu }_0{R}_0}$

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

The ITER design uses the following approximate definition:

$l_i(3)=\frac{2V⟨B_{\theta }^2⟩}{{\mu }_0^2{I}^2{R}_0}$

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.