242 Ë 7 Numerical simulations of HTS Maglev
to describe this characteristic in simulations, we assume that the resistance of the
currents flowing along thec-axis are three times larger than that within thea-b
plane, viz.,
. 6
> 6
F
Jsc,x⩽Jc,
Jsc,y⩽Jc/ 3 ,
Jsc,z⩽Jc.(7.55)
Eq. (7.51) can be rewritten as
Esc=( 0
8Esc,x
Esc,y
Esc,z) 1
9
=
1
휎sc. 6
> 6
F
( 0
8
Jsc,x
Jsc,y
Jsc,z) 1
9
+( 0
8
0
2 Jsc,y
0) 1
9
/ 7
? 7
G
=
1
휎sc(Jsc+Q), (7.56)whereQ=( 0 , 2 Jsc,y, 0 )T. By this method, the HTSC is mathematically composed of
two parts, one is a homogeneous HTSC as bulkAand the other has only thec-axis
conductivity which is not equal to zero, as bulkB. By substituting Eq. (7.56) and
B=휇Hinto Faraday’s law [Eq. (7.4)], we have
∇×Esc=∇×^1
휎scJsc+∇×^1
휎scQ= −휕
휕t(휇H). (7.57)
If we further substitute Ampère’s law [Eq. (7.1)] into Eq. (7.57), we find the PDE that
governs the electromagnetic properties in the HTSC,
휕
휕t(휇H)+∇×^1
휎sc∇×H+∇×^1
휎scQ= 0. (7.58)
Taking all computational regions shown in Fig. 7.16 into account and considering that
Jy=휕Hx/휕z−휕Hz/휕x, Eq. (7.58) can be reshaped as follows by defining an auxiliary
parameter휆to identify the medium involved,
휕
휕t(휇H)+∇×1
휎∇×H+휆∇×
1
휎Q=^0 , (7.59)
where휎=휎air,휆=0 for regionΩ 1 and휎=휎sc,휆=1 for regionΩ 2 (see Fig. 7.16), andQ
is defined as
Q= 0 , 2 휕Hx
휕z−휕Hz
휕x, 0
T. (7.60)
A stable resolution of PDE needs some suitable boundary conditions. For Eq. (7.59),
Dirichlet and Neumann boundary conditions are used to describe the electromagnetic