High Temperature Superconducting Magnetic Levitation

7.5 Two-dimensional modelings and simulations Ë 233

For a superconducting unit containingmconstituents, the magnetic excitation
to an arbitrary constituentiis generated by the external source as well as by all the
others, which requires modification of the electromagnetic master equation for the
constituentito be,

−

1

휇 0 ¬

휕^2 A耠sc,x,i

휕y^2 +

휕^2 A耠sc,x,i

휕z^2 ­+휎à

휕

휕t

m

H

j= 1

A耠sc,x,j−

휕

휕t

m

H

j= 1 ,j/=i

A耠sc,x,j,Γè

+휎

휕Asc,x

휕t

= 0 , (7.50)

HereA耠sc,x,j,Γrefers to the boundary unknown of the constituentj(j/=i), whereA耠sc,x,j,Γ≈
(∇V)x,jstands. In this case,mdifferent unknowns,A耠sc,x, 1 ,A耠sc,x, 2 ,⋅ ⋅ ⋅,A耠sc,x,mshould
be defined at each finite element node, and they are coupled through the second term
of the left-hand side of Eq. (7.50).
The coolant to provide a cryogenic environment for the HTSC is supposed to be
dielectric, and the second term of the left-hand side of Eqs. (7.49) and (7.50) thus
vanishes in the coolant. The electromagnetic master equation in the coolant is actually
represented by the Laplace’s equation. The continuity ofA耠sc,xas well as its normal
derivative applies to the interior bounds between the superconductor and the coolant,
and the value ofA耠sc,xon the outer bounds is not constrained.

7.5.2.2Example of applications: mutual effect among the superconducting
constituents in a levitation system with translational symmetry [14]
The PDEs governing the electromagnetic properties of HTSCs as well as the surroun-
ding coolant, illustrated above, were numerically discretized by resorting to the FEM
technique and FDM scheme, in the spatial and temporal domain. Considering the two
different configurations of PMG shown in Fig. 7.8, both of which were derived from
the Halbach array, we had applied this model to study the mutual effect among the
superconducting constituents in a HTS unit composed of three HTSCs.
In these simulations, the dimension of each HTS constituent was 30 mm in width
and 15 mm in thickness, and the gap between the constituents was 2 mm. We hade
usedJc0= 1. 5 × 108 A/m^2 ,E 0 = × 10 −^6 V/m, andB 0 =0.25 T, to roughly represent the ma-
terial properties of bulk Y–Ba–Cu–O at 77 K. According to the geometrical and material
characteristics of the PMGs in Fig. 7.8, the starting position of vertical movement was
120 mm to approach the ZFC condition over both PMGs, where the strength of applied
magnetic field was negligible. The minimum vertical distance between the HTSC and
the PMG was 5 mm, and the speed of the HTSC was 1 mm/s to create a quasi-static state
for all cases estimated in this section.
Two cases will be estimated and compared. One is the actual case with all con-
stituents present, and the other is an envisaged case of considering one constituent