248 Ë 7 Numerical simulations of HTS Maglev
Eq. (7.65) is reduced to the following form when the Coulomb gauge and boundary
condition Eq. (7.67) are considered
C(P)T(P)=^1
4 휋
X
V(∇耠×T(P耠))×∇耠^1
R(P,P耠)
dV耠−^1
4 휋
X
S(n耠⋅T(P耠))∇耠^1
R(P,P耠)dS耠. (7.68)TheB-Hconstitutive law of the HTSC can be assumed to be linear as that in vacuum
to a good approximation because its applicable conditions [34] can be easily satisfied
in a levitation system using bulk Y-Ba-Cu-O due to its small lower critical fieldBc1and
large applied field as well as its geometry. Thus,
B=휇 0 H. (7.69)
The induced fieldBsgenerated by the current in the HTSC can be expressed as a
function of the vectorTvia combining Eq. (7.68) with Biot-Savart’s law,
Bs=휇 0 C(P)T(P)+휇 0
4 휋X
S(n⋅T(P耠))∇耠1
R(P,P耠)dS耠. (7.70)
When an equivalent conductivity휎s, which is nonlinear and dependent on the local
electrical field, is introduced, the traditional Ohm’s law in the HTSC has the following
form:
J=휎s(|E|)E. (7.71)By substituting Eqs. (7.64) and (7.71) into Eq. (7.65) and consideringB=Be+Bs, where
Eeis the applied field, the following equation is found:
∇×^1
휎s(∇×T)+휕(Be+Bs)
휕t= 0. (7.72)
The governing equation of the HTSC based on the variableTis finally derived from
Eqs. (7.70) and (7.72) as follows,
∇×^1
휎s(∇×T)+휇 0 C(P)휕T
휕t+휇^0
4 휋
X
S휕(n耠⋅T耠)
휕t∇耠^1
R(P,P耠)
dS耠+휕Be
휕t