Page 111 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 95

nonlinear magnetic properties, as well as combined systems with current coils,
permanent magnets and ferromagnetic materials.
The described algorithm allows one to find the threshold of attainable qual-
ity criterion and the complexity margin of a synthesised MS design. Where
MS components fail to meet the design criteria, the magnet design must be
revised. The algorithm provides for an effective numerical implementation.
It is performed through a reduction of a variational problem with inequality
constraints to the sequence of unconditional optimisation problems. The best
solution is found by the trial-and-error method, that is, by varying the regulari-
sation parameter. Such a solution is always a trade-off. On the one hand, it must
provide an economically and technically efficient design. On the other hand, the
solution must allow certain accuracy requirements for the synthesised system
performance to be met. The realisation of this trade-off is not a trivial engineer-
ing exercise.

4.4 ANALYSIS OF ELECTROMAGNETIC TRANSIENTS

4.4.1 Calculation and Methodological Basics
EM transients [7,28–31] in tokamaks are among the major design challenges
due to high EM and thermal loads caused by eddy currents in conductive
structures.
The duration of EM transients typically depends upon the operation mode
and physical processes in the plasma, but in any case, an EM field can be de-
scribed using a quasi-stationary approach [11]. We also note that almost no ma-
terials with nonlinear magnetic properties are employed in which eddy currents
can be induced. Notable exceptions include the ferromagnetic inserts, needed to
decrease the toroidal field ripple, the blanket test modules and some structural
elements of ITER diagnostic systems and buildings.
As known [11,28], when dealing with harmonic fields, about 86% of the
induced current flows mainly through a non-ferromagnetic conductor in its skin
0.5
layer with the depth 2∆ = (8ρ /µ w) , where ρ is the conductor resistivity,
0
0
0
0
and w = 2πf is the oscillation angular frequency. For a copper conductor, the
4
skin layer is about 2.4 mm deep for the characteristic frequency f = 10 Hz. For
4
a non-magnetic steel, it is about 8.9 mm if f = 1.5 × 10 Hz and 28.2 mm if
3
f = 1.5 × 10 Hz.
For a pulse field in tokamaks with impulse front ∆t , the characteristic time ∆t
of the EM process is estimated as oscillation period T = 4 ∆t = 1/f. For the ∆t
0
ITER vacuum vessel steel walls with a thickness of 60 mm, ∆t = 5 ms. For ∆t
an EM field that changes exponentially with the time constant τ , the oscillation
0
−1
frequency is replaced by τ . The skin effect increases when conductors are τ0−1
0
cooled down and weakens when they are warmed up.
These estimates are valid for flat EM waves and semi-infinite conductive
media. The skin effect is less pronounced, if conductor finite dimensions are
taken into account.