High Temperature Superconducting Magnetic Levitation

226 Ë 7 Numerical simulations of HTS Maglev

Bz=

K

2 (Γ(z,y−y

耠,x−g)−Γ(z,y−y耠,x−(L+g))−Γ(z+t

PM,y−y

耠,x−g)

+Γ(z+tPM,y−y耠,x−(L+g)))儨儨儨儨ww^21 (7.33)

The analytical expressions for the designated PMG can be obtained by translating the
PM along they-axis with the direction of the magnetization considered, and then the
final expressions can be established by translating the segment of the PMG along the
x-axis with the total number of the segments taken into account.

7.5 Two-dimensional modelings and simulations

7.5.1Prigozhin’s model [6, 44, 45]

7.5.1.1Mathematical fundamentals
This model states that the electromagnetic properties of the HTSC in the framework
of Bean’s critical state model can be ruled by a complicated system of equations and
inequalities, which leads to a free boundary problem. It has the shape of variation
formulation as follows,

휇 0

T

X

0

X

R^3

휕H

휕t ⋅휑+

T

X

0

X

ΩHTSC

휌(∇×H) ⋅ (∇×휑)= 0 , (7.34)

whereR^3 denotes the whole computational domain including the HTSC and the
coolant domain, whereasΩHTSCrepresents the superconducting domain only. The
function휑is a “test” vector function whose curl vanishes in the domain outside
the HTSC and whose tangential components are continuous across the boundaryΓ
between the HTSC and the coolant.
Now, we introduce a new variable,h=H−He, satisfying

∇×h= 0 , inΩAir,

|∇×h|⩽Jc(|h+He|), inΩHTSC,

[hΓ]= 0 , onΓ.

(7.35)

Let us define the set of functions

K(h)={|∇×휑|⩽Jc(|h×He|),inΩHTSC} (7.36)

which is dependent onh. Moreover, since

∇×h=∇×H, |∇×H|⩽Jc(H), inΩHTSC, (7.37)