Page 272 - Fundamentals of Magnetic Thermonuclear Reactor Design

Plasma Control System Chapter | 8 251

8.4 MATHEMATICAL MODELLING OF ELECTROMAGNETIC
PROCESSES

8.4.1 Derivation of Linear Models
We assume that plasma column displacements are small and axisymmetric and
ignore plasma electrical resistance and viscosity. In other words, we confine
ourselves to considering ideal, non-inertial plasma in frame of the classical
magnetohydrodynamic(MHD) theory. We also assume that plasma is stable on
an Alfvén timescale and that our linear model describes only the processes on
the timescale comparable with the time constants of the conducting structures.
Our aim is to obtain matrixes of linear equations (MLE) describing the behav-
iour of plasma and currents in surrounding conducting structures under equi-
librium conditions. In a general case, the evolution of interest is described by a
system of nonlinear partial differential equations, namely, the Grad–Shafranov
equation for equilibrium and electro-technical equations for plasma and circuit
currents. To solve this, we use numerical methods. The model of the response
of the free boundary equilibrium plasma on the disturbance of the currents in
the surrounding conductors is used for the linearisation. Here, we follow the
approaches described in Refs [1, 2].
To linearise any equations, we have to define a set of states, or variables
that determine the linear model dynamics. Plasma position and shape evolution
depends on currents in poloidal field coils and passive circuits.
It is therefore natural to use deviations of those currents relative to initial
(base) values as linear model states:
δ It () = It ()− I , (8.1) δIt=It−I ,
0
0
where I(t) denotes currents at time t, and I refers to currents for the initial (base)
0
configuration (t = 0).
The dynamics of circuit currents (states) is expressed by electro-technical
equations of the form
dΨ + RI = U, (8.2) dΨdt+RI=U,
dt
where ψ is the poloidal flux vector averaged over the circuit cross-section, R is
the matrix of circuit resistances, and U is the vector of external voltages applied
to the circuits. In a linear case, vector ψ can be expressed as
 
Ψ (rz t,, ) ≅ Ψ (r z, 0 ,0 )+  ∂Ψ i ()  δ I (8.3) Ψr,z,tΨr ,z ,0+∂Ψ(i)∂I(j)I=I δI,
0
0 0
0
  ∂I j () II  
 ∂Ψ  = 0
where   i ()   is the matrix with elements (I, j), which determines the flux ∂Ψ(i)∂I(j)I=I 0
 I ∂ j () II 0 
=
deviation in circuit i at the unit current deviation in circuit j at time t = 0.