High Temperature Superconducting Magnetic Levitation

7.3 Macroscopic electromagnetic properties of HTSC Ë 221

assumption that the current flowing in the HTSCs is constant with time and the lack
of material-related parameters.
The second model of the nonlinearE-Jconstitutive law in HTSCs is the so-called
power law [33, 34], which is expressed as

E=Ecœ

|J|

Jc

n J

|J|, (7.14)

with 1⩽n⩽∞. This power law corresponds to a logarithmic current dependence of
the activation energyU(J)=Ucln(Jc/J), which inserts into an Arrhenius law yielding
E(J)=E,exp(−U/휅훩)=Ec(J/Ic)n. Indexnis defined asn=U 0 /휅훩, whereU 0 is the
pinning potential of the HTSC at an absolute temperature훩and휅is the Boltzmann
constant. At the extremes, the power law reduces to the Ohm’s law withn=1 and to
the Bean’s model withn→∞.
The third model of the nonlinearE-Jconstitutive law in HTSCs is the so-called flux
flow and creep model, which was derived from the Anderson’s theory [35] and has the
form [31],

E=

. (^66)
6

(^66)
6
F
2 휌cJcsinhœU^0 |J|
휅훩Jc
expœ−U^0
휅훩

J

|J|

, 0 ⩽|J|⩽Jc,

Ec+휌fJcœ|J|

Jc

 J

|J|

, |J|>Jc,

(7.15)

where휌cand휌fare the creep and flow resistivity, respectively, andEc=휌cJc[ 1 −
exp(− 2 U 0 /휅훩)]≈휌cJc.
Recently, a smoothed Bean’s model of the critical state in the hyperbolic tangent
approximation [12, 36],

J=Jctanhœ|E|

E 0

E

|E|

(7.16)

was proposed to describe the nonlinearJ-Econstitutive law in HTSCs, whereE 0 is
a characteristic electric field that determines the electrical conductivity휎 0 at zero
electrical field by휎 0 =Jc/E 0. The continuity of the current density around zero electric
field makes the associated numerical simulation stable and it is not necessary to
introduce a residual resistivity for numerical stability.
Additionally, the field-dependent property of the critical current density is gene-
rally characterized by the Kim’s model [37],

J=Jc0œ

B 0

|B|+B 0

, (7.17)