Calculation of Inductance Matrix
In principle, we could determine the relationship between
and the
(which is equivalent to determining the relationship between the
and the
) by solving Equations (14.46) and (14.47) subject to suitable spatial boundary conditions at
and
[11,14,22]. However, in this chapter, we shall adopt a more direct approach [12,13,15,16].
According to the Biot-Savart law [27]:
 |
(14.102) |
Let us assume that
 |
(14.103) |
It follows that we can evaluate the integral on the right-hand side of Equation (14.102) at
without loss of generality. Now,
 |
(14.104) |
so we get
 |
(14.105) |
where
![$\displaystyle G(R,Z;R',Z') = \frac{1}{2}\oint \frac{\left(\cos[(n-1)\,\varphi']...
...arphi'}{\left[R^{2}+R'^{\,2} +(Z-Z')^{2} -2\,R\,R'\,\cos\varphi'\right]^{1/2}}.$](img4181.png) |
(14.106) |
Finally, making use of the standard definition of a toroidal
function [23],
 |
(14.107) |
where
denotes a gamma function [1],
we arrive at
where
![$\displaystyle \eta = \tanh^{-1}\left[\frac{2\,R\,R'}{R^{2}+R'^{\,2}+(Z-Z')^{2}}\right].$](img4187.png) |
(14.109) |
According to Equations (14.60) and (14.66),
 |
(14.110) |
Furthermore, Equations (14.65) and (14.78) yield
 |
(14.111) |
Hence, combining the previous two expressions with Equation (14.105), we obtain the expected normalized inductance
relation [see Equation (14.94)],
 |
(14.112) |
for
,
where [12,13,15,16]
 |
(14.113) |
and
with
![$\displaystyle \eta_{kk'} = \tanh^{-1}\left[\frac{2\,R_k\,R_{k'}}{R_k^{\,2}+R_{k'}^{\,2}+(Z_k-Z_{k'})^{2}}\right].$](img4194.png) |
(14.115) |
Here,
and
index the various rational surfaces in the plasma. Moreover, the double integral in Equation (14.113) is taken around the
th rational surface (cylindrical coordinates
, 0,
; flux coordinates
,
, 0, with
constant; resonant poloidal mode number
) and the
th rational surface (cylindrical coordinates
, 0,
; flux coordinates
,
, 0, with
constant; resonant poloidal mode number
).
Note that
 |
(14.116) |
which, from Equation (14.113), implies that the F-matrix is Hermitian [see Equation (14.100)], as must be the case.
Finally, according to Equations (14.95) and (14.113), the unnormalized inductance matrix
takes the form
 |
(14.117) |
The Hermitian L-matrix,
, specifies the self and mutual inductances of the helical current sheets that flow at the various
rational surfaces within the plasma.
Note that the calculation of the F-matrix outlined in this section is only approximate. The exact calculation is specified in
References [11] and [22].