High Temperature Superconducting Magnetic Levitation

232 Ë 7 Numerical simulations of HTS Maglev

with the magnetic vector potential along thex-directionAxbeing decomposed into
Asc,xandAex,x, whereAsc,xrepresents the vector potential induced by the supercur-
rent, andAex,xserves as the magnetic excitation. The gradient of the electric scalar
potential∇Vis invariant across they-zplane and only depends on timet, as bothE
andAare nontrivial merely along thex-axis and independent of thex-axis [46]. If we
denote the value of (∇V)xat an arbitrary timetnasC(tn), Eq. (7.45) can be rewritten as

Ex= −

휕áAsc,x+

tn

X

0

C(t)dt+Aex,xé

휕t. (7.46)

Making use of Ampère’s law [Eq. (7.1)] within the quasi-static approximation and
exploiting Eq. (7.46), the electromagnetic properties in the HTSC are governed by

−

1

휇 0 ¬

휕^2 Asc,x

휕y^2 +

휕^2 Asc,x

휕z^2 ­+휎

휕(Asc,x+

tn

X

0

C(t)dt)

휕t +휎

휕Aex,x

휕t =^0 , (7.47)

where we have assumed that magnetic field intensity and magnetic flux density in the
HTSC are related linearly with the vacuum permeability휇 0.

Since

tn

X

0

C(t)dtis independent ofyandz, it is expedient to adopt a generalized

vector potentialA耠sc, which is defined as

A耠sc,x=Asc,x+

tn

X

0

C(t)dt, (7.48)

to make the integral term in Eq. (7.46) implicit. Eq. (7.47) is thus reduced to

−^1

휇 0

¬

휕^2 A耠sc,x

휕y^2

+

휕^2 A耠sc,x

휕z^2

­+휎

휕A耠sc,x

휕t

+휎

휕Aex,x

휕t

= 0 , (7.49)

whereA耠sc,xis the unknown to solve. It is worth noting that the contribution of the
electric scalar potential will dominate in the computational region far from the HTSC,
where the vector potentialAsc,xdue to the supercurrent is negligible, i.e.A耠sc,x≈
tn
X
0

C(t)dtholds (if the HTSC is subjected to a symmetric magnetic field,C(tn)≡0 stands,

and it is not needed to introduce the generalized vector potential).