Page 113 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 97 BF−d
2
,t=−32 ∂bF0/∂h⋅kt⋅ρ t/µ d 32⋅d-
0
0
2
/π⋅exp−µ d /4ρ t
For a linearly increasing current i = kt, 0 0
B (− d t) = − 32 b () kt ρ ( ⋅ 0 t / µ ) 3 2 ⋅ d / π ⋅ exp(− µ 0 d / 4ρ )
2
∂
0 / h∂⋅
2
,
t
d
0
F
0
F
Field derivative b∂ F (0)/ ∂ h is calculated on surfaces x = 0 for a unit ∂bF(0)/∂h
current. In this case, the characteristic decay time is governed by 1/τ = 4ρ /
0
0
2
(µ d ).
0
Let us discuss an external field penetrating through an infinitely long layer.
We assume that the layer thickness is much smaller than characteristic linear
dimension R , within which the field undergoes substantial changes. This as-
0
sumption allows us to reduce consideration to a single current density compo-
nent j parallel to the layer surface. The normal field components are continuous
F
at the boundaries, B = B , and the tangential field components are related to
x1
x2
each other [28] as n × B ( 2 − B ) = µ 0 ∫ 0 d j dx = µ 0 J . n ×B −B =µ ∫0djFdx=µ JF
2
F
F
1
1
2
2
0
0
When an instant appearance of an external field occurs in region 1, we gen-
erally have a three-stage transient process. The first stage can be described as
2
t « τ = µ d /ρ . At this time, the current is concentrated in layer ∆ < d, while
0
1
0
0
the field is much lower in region 2 behind the layer than in front of it. The field
in region 1 can be calculated assuming ideal conductivity of the layer. For the
field in region 2, the equations for B are valid.
F
The second stage occurs when the current density gets uniform through-
out the layer thickness and begins at t ≥ τ . It lasts until eddy currents have
1
decayed away in the layer. The second-stage duration can be expressed as
τ = µ R d/ρ . For this period, the fields in regions 1 and 2 reach equal
2
0
0
0
levels.
The third stage corresponds to the time longer than τ . At this stage, the eddy
2
current field is small compared with the external field.
A model of a thin conducting shell carrying a uniformly distributed current
is generally employed to describe the second and the third stages of an EM tran-
sient. A general solution for a flat layer at d « R can be found in the literature.
0
For additional information on this topic, please go to Chapter 8.
4.4.2 Sources of Transient Fields
The main sources of transient fields in tokamaks are as follows [4,7–10]:
l toroidal plasma current
l toroidal magnetic flux linked with the plasma
l currents in TFC
l currents in central solenoid and PFCs
l halo current
They provide necessary conditions for reference scenarios of the plasma
discharge and predetermine events, which give rise to plasma instabilities
resulting in the plasma current disruption, fast energy discharge of super-