Page 97 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 81

We pick out field H = H − H . The vector H describes the field due
ext
M
M
to magnetised structures and satisfies equation ∇ × H = ∇×(H − H ). We
M
ext
further introduce scalar potential  : H = –∇ . From the equations for the
M
M
M
divergence of B and µ H it follows that ∇·(B − µ H ) = 0, or that
ext
0
ext
0
=
∇⋅[ µµ( H M + H ext ) − µ 0 H ext ] =∇ ⋅[ µ µ H M − µ µ −1) H ext ]0. ∇⋅[µ µ(HM+Hext)−µ Hext]=∇⋅[µ 0
(
0
0
0
0
0
µHM−µ (µ−1)Hext]=0.
Finally, to determine  the following equation is used: 0
M
∇
(
∇⋅( µµ ϕ ) = ∇⋅ µ µ −1) H ext . (4.7) ∇⋅(µ µ∇M)=∇⋅µ (µ−1)Hext.
0
0
M
0
0
The algorithm for solving the problem is similar to that used for the earlier
electrical vector potential P. The multiplier (µ − 1) in Eq. (4.7) limits the com-
putation of H to the subdomain with µ ≠ 1 (i.e. within the ferromagnetics).
ext
Note, for this problem the field H is known at least in quadratures and spends
ext
a lot of computation cost compared to the calculation of P.
With respect to a magnet system (MS), all other current sources may be
divided conventionally into ‘external’ j and ‘internal’ j . This is driven by
int
ext
the search for the most effective approach to finding a solution. One can in-
troduce two vector fields, H and P, simultaneously. For these fields, equa-
ext
tions
∇× H ext = j , ∇ × P = j , ∇⋅ µ 0 H ext = 0, H int = −∇ ϕ + P,
int
ext
H = H ext + H int , ∇⋅ B = 0, B = µµ H = µµ( H ext + H int ) ∇×Hext=jext, ∇×P=jint, ∇⋅µ H
0
0
0
ext=0, Hint=−∇+P, H=Hext+
are valid.
From the last two equations, it follows that Hint, ∇⋅B=0, B=µ µH=µ µ(He
0
0
xt+Hint)
∇
)
∇⋅( µµ ϕ = ∇⋅ µ µ P+ ∇⋅ µ µ H . (4.8) ∇⋅(µ µ∇)=∇⋅µ µP+∇⋅µ µHext.
ext
0
0
0
0
0
0
Eq. (4.8) must account for the magnetic properties of materials and the
boundary conditions. As earlier, to avoid many computations, Eq. (4.8) can be
transformed into
∇
∇⋅( µµ ϕ = ∇⋅ µ µ P+ ∇⋅ µ µ −1) H ext . (4.9) ∇⋅(µ µ∇)=∇⋅µ µP+∇⋅µ (µ−1)Hext.
)
(
0
0
0
0
0
0
Among the possible external current sources, first, we consider current coils.
For a wide range of applications, the coils can be modelled as a combination
of ring, rectangular, or arc-shaped conductors with rectangular or polygonal
cross-sections, where current densities are uniformly distributed. The coils are
complemented with ring-shaped current filaments to describe a plasma cur-
rent and with uniformly magnetised volume elements bounded by flat facets to
simulate ferromagnetic structures such as the tokamak building reinforced with
steel rebar and inserts. For a given magnetisation, a field due to such elements
can be calculated analytically or using known quadratures that enables control
of the computation accuracy.