Internal inductance

From FusionWiki
Jump to navigation Jump to search

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=Φ/I

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

Φ=SBdS

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

W=B22μ0dr

It can be shown that[1][2]

W=12LI2

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]:

12LiI2=PB22μ0dr

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[4]

li=Bθ2PBθ2(a)=2π0aBθ2(ρ)ρdρπa2Bθ2(a)

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

Using Ampère's Law (2πaBθ(a)=μ0I), one obtains [3]

li=Li2πR04πμ0=2Liμ0R0

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

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

li(3)=2VBθ2μ02I2R0

which is equal to li assuming the plasma has a perfect toroidal shape, V=πa22πR0.[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 Bθ(ρ) profile): a small value of li corresponds to a broad current profile.

References

  1. P.M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press (2006) ISBN 0521821169
  2. Wikipedia:Inductance
  3. 3.0 3.1 J.P. Freidberg, Plasma physics and fusion energy, Cambridge University Press (2007) ISBN 0521851076
  4. K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, Springer-Verlag (2005) ISBN 3540242171
  5. 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
  6. Effective plasma radius