Page 288 - Fundamentals of Magnetic Thermonuclear Reactor Design

Plasma Control System Chapter | 8 267

The transient processes in tokamaks are described by a system of ordinary
differential equations, which can be written in a matrix form as
(
dLI)
dt + RI = U, (8.44) dLIdt+RI=U,
where L is the inductance matrix, I is the vector of circuit currents, R is the
diagonal matrix of the circuits’ resistances, and U is the vector of power sup-
ply voltages. The L inductance matrix is determined entirely by the conduc-
tive elements’ geometry. In the context of an axisymmetric description, the
conductive elements are modelled as a system of axisymmetric loops. This
applies to the PFCs and the vacuum vessel and other conductive passive struc-
tural components. The conductive shell structural components are all mod-
elled using the finite-element method. The elements of the R diagonal matrix
are determined by the electrical resistance of the respective structural materi-
als. The model should also include the mathematical description of the PFCs’
power supply.
The goal of the problem is to find such PFCs’ voltage- time dependen-
cies, which would ensure realisation of optimal breakdown conditions at the
breakdown time, that is the simultaneous formation of the electrical field
with the required voltage and compensated stray magnetic fields. The lat-
ter, generally, should be within 2 mT in the breakdown region. The need to
provide the plasma equilibrium imposes additional constraints on the poloi-
dal magnetic field. Starting from breakdown, the magnetic field horizontal
component must be kept close to zero to avoid major plasma vertical dis-
placements. At the same time, its vertical component must remain consistent
with the magnetic equilibrium field corresponding to the given plasma major
radius and current (Shafranov field). These requirements may be written as
follows:
z (
BR Z) ≈ 0
,
µ I  8  R l 3 (8.45) BzR,Z0BzR,Z+µ Ip4πrln8Ra+βp+li2−320.
0
z (
BR Z) + 0 p  ln + β + i −  ≈ 0.
,

p
4π r   a 2 2
To sum it up, the magnetic field in the breakdown region must provide the
plasma in stable equilibrium in the vertical and horizontal directions. This is
especially important during the plasma initiation stage, when the plasma current
is relatively small. The plasma column's circular cross-section, preferable at this
stage, can be achieved, if the decay index of the magnetic field in the breakdown
×
region n<>=− R (/ < B > )( d < B > / dR) is close to 0.5. <n>=−(R/<Bz>)×(d<Bz>/dR)
z
z
In addition, the model should account for the constraints on the values of
PFCs’ currents and voltages, which depend on the choice of a power supply
system and adopted design solutions. Other constraints are possible depend-
ing on specific operational requirements, design features and characteristics of
auxiliary equipment.