Page 121 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 105

the reduced magnetic scalar potential . The potentials are quantified over
3
the entire metric domain R . However, the vector potential P is nonzero by
its physical interpretation only in simply connected conductive structures.
From ∇× P = j, it follows that P is determined accurate to an arbitrary sca- ∇×P=j
lar gradient. To resolve the ambiguity, TORNADO applies an additional
(gauge) condition P × u = 0, where u = u(r) is an arbitrary stationary vector
field without closed field lines, with u(r) ≠ 0 over those subdomains, where
P is nonzero.
The total field H is a superposition H = H + H of field H associ-
tot
tot
ext
ated with eddy currents and field H due to given external sources. The
ext
electric and magnetic potentials are found by solving the following system
of equations:
ρ ∇×
t
∇× (( P + µ ∂ µ( P −∇ ϕ) / ∂= − µ ∂ µ H ext / ∂t;
))
0
0
µ
))
µ ∇⋅(( P −∇ ϕ =− µ ∇⋅(( µ −1) H ext );
0
0
H = P − ∇ ϕ;
Pu ⋅= 0. ∇×(ρ(∇×P))+µ ∂µ(P−∇)/∂t=−µ ∂
0
0
µHext/∂t;µ ∇⋅(µ(P−∇))=−µ ∇⋅((µ
0
0
In the absence of ferromagnetic materials, µ = 1, the equations take the µ=1 −1)Hext);H=P−∇;P⋅u=0.
form
∇× ρ ( ( ∇× P ) + µ ∂( P −∇ ϕ) / ∂= − µ ∂ H ext / ∂t;
)
t
0
0
)
∇⋅( P −∇ ϕ = 0;
H = P − ∇ ϕ;
Pu⋅= 0. ∇×ρ(∇×P)+µ ∂(P−∇)/∂t=−µ ∂He
0
0
xt/∂t;∇⋅(P−∇)=0;H=P−∇;P⋅u=0.
While the first equation is valid for conductive structures with resistivity ρ,
3
the second one holds for the entire domain R . The eddy current density vector
must fulfil the condition ∇ × P = j.
To determine the eddy currents, we need to know the spatial–temporal distri-
butions of the field H over the 3D conductive structures. This may be obtained
ext
from KLONDIKE simulations.
In the TORNADO simulation of EM transients, we can consider the giv-
en current sources of an external field. These current sources provide the
given time dependence of the total eddy current flowing through dedicated
cross-sections of conductors. This model is used to determine the halo cur-
rent. In such simulation, the total electric vector potential P is represented
tot
by the superposition P = P + P . Here, P is an unknown electric vector
tot
src
potential related to the eddy current density as ∇× P = ; P is the vector ∇×P=j
j
src
electric potential with the curl equal to the predetermined current density
vector j = ∇ × P . The latter expression can be transformed into the in-
src
srt
tegral equation