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 BiotSavart law [27]:

(14.102) 
Let us assume that

(14.103) 
It follows that we can evaluate the integral on the righthand side of Equation (14.102) at without loss of generality. Now,

(14.104) 
so we get

(14.105) 
where

(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

(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

(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 Fmatrix 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 Lmatrix, , 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 Fmatrix outlined in this section is only approximate. The exact calculation is specified in
References [11] and [22].