High Temperature Superconducting Magnetic Levitation

220 Ë 7 Numerical simulations of HTS Maglev

7.2.4Conservation of magnetic flux density

This equation states that the magnetic flux lines have no beginning or ending, and
any magnetic flux line that enters an object must exit that object. It can be expressed
in either the differential form

∇ ⋅B= 0 , (7.10)

or the integral form

X

S

B⋅ds= 0. (7.11)

The normal component of the magnetic flux densityBis always continuous in passing
through the surface from Region 1 (B 1 ) to Region 2 (B 2 ), i.e.

n 12 ⋅ (B 2 −B 1 )= 0. (7.12)

7.3 Macroscopic electromagnetic properties of HTSC

7.3.1Nonlinear constitutive equation

HTSC is a nonlinear conducting medium, with the resistivity (conductivity) being
dependent on the parameters of the local electromagnetic field, which requires the
constitutive law between the electric fieldEand current densityJin HTSCs to be
represented by a nonlinear formulation, rather than a linear one of the traditional
Ohm’s law.
The typical model of the nonlinearE-Jconstitutive law in HTSCs is Bean’s model
of the critical state [29], which postulates that the current density in HTSCs is limited
by a critical valueJc, and until this threshold is reached, the electric field is zero. This
assumption results in a discontinuous expression and is very difficult to represent in
numerical simulation. The general form of Bean’s model used in simulations is that
proposed in [30]

J=

. (^66)

(^66)
F
JcE
|E|

(|E| /= 0 ),

휕J

휕t (|E|=^0 ).

(7.13)

In modeling the HTS Maglev, this constitutive law is not appropriate for investigating
problems such as force relaxation [31] and drift under vibration [32] due to the