Page 94 - Fundamentals of Magnetic Thermonuclear Reactor Design

78 Fundamentals of Magnetic Thermonuclear Reactor Design

A similar technique with the scalar potential formulation is used to model
the fields of magnet systems with retentive materials (permanent magnets). In
many cases, for their description, one can use the ‘idealised ferromagnetic’
model [11] for representing the magnetisation vector M as

M=M +kH, M = M + k H, (4.5)
0
0
where the constant magnetisation M is a preset function of spatial coordi-
0
nates only, and the k coefficient is H-independent. In practical calculations,
such model corresponds to a linear dependence M(H). This is permissible for
present-day retentive materials, such as the NdFeB permanent magnets, whose
demagnetisation curve is actually a straight line within the operating field range.
Based on Eq. (4.3),
H + k)
B=µµ H=µ (H+M +kH)=µ [H(1+ B = µµ 0 H = µ ( H + M + k H = µ [( 1 + M ].
)
0
0
0
0
0
0
0
0
k)+M ].
0
Then, from ∇·B = 0 it follows that
∇⋅B=∇⋅µ (H(1+k)+M )=0 and ∇⋅µ (1+k ∇⋅ B =∇ ⋅ µ H((1 + k) + M ) = 0 and ∇⋅ µ (1 + k) H = −∇ ⋅ µ 0 M .
0
0
0
0
0
0
0
)H=−∇⋅µ M .
0
0
In the absence of the conduction currents ∇ × H = 0, and a scalar potential
 can be introduced, so as H = –∇ . The equation ∇·µ (1 + k) ∇ = ∇·µ M
M
M
0
0
M
0
is used to determine  . The potential describes the field Н generated by mag-
M
netised structures in the absence of conduction currents. A nonlinear relation
between the M and H vectors of the M = M + k (H) H type (for retentive ma-
0
terials) can be factored in with no additional difficulties, as a similar problem
has been solved earlier.
The P and M can be implemented together within the same algorithm. To
0
this end, we assume µ PM = 1 + k for retentive materials, in much the same way
as µ = µ = 1 + χ for ferromagnetics, where χ is the susceptibility. Denoting
FE
µ = µ and µ = µ PM for both materials, we obtain
FE
∇⋅µ µ∇=∇⋅µ µP+∇⋅µ M . ∇⋅ µµ ϕ∇= ∇⋅ µ µ P +∇ ⋅ µ 0 M . (4.6)
0
0
0
0
0
0
0
If we combine the µP and M vectors into one ‘vortex’ vector, we obtain
0
an expression similar to Eq. (4.4). However, the P and M vector definition
0
domains may intersect. This combination can be done informally, based on the
physical nature of vectors P and M .
0
To sum it up, the analysis of combined systems containing current coils and
permanent magnets that takes into account material nonlinear properties, can be
performed using a single methodological and software framework.
The EM forces, induction vector, integral values of loads, energy, induc-
tance, and so on, can be calculated using expressions that link those parameters
to the values of , B and H in the mesh nodes.
In an FE model, element facets in the general case appear to be the surfaces
of ‘strong discontinuity’ of the EM field due to discrete field-quantities as a re-