High Temperature Superconducting Magnetic Levitation

218 Ë 7 Numerical simulations of HTS Maglev

along with the analytical model to calculate the magnetic field generated by PMG,
will be introduced at both two- and three-dimensional levels, after a brief description
of the macroscopic electromagnetic properties of HTSC and the basic Maxwell’s
equations for the electromagnetic field. Making recourse to these and using the finite
element method (FEM), we have developed numerical codes, on the basis of which
some practically interesting cases of HTS Maglev have been numerically simulated to
provide the scientific criteria for designing and optimizing a HTS Maglev system with
translational symmetry, i.e. for the Maglev transportation application.

7.2 Maxwell’s equations

Maxwell’s equations [26–28] are a set of four independent integral or differential
expressions that mathematically characterize the generation, interaction, and propa-
gation of the electric and magnetic fields in the nature. These equations, originally
proposed by Maxwell 150 years ago, unify the mathematical expression of the time-
steady and time-dependent electromagnetic fields via the introduction of a displace-
ment current in Ampère’s law and constitute the theoretical foundation of the subject
of electromagnetism.

7.2.1Ampère’s law with Maxwell’s addition

This equation describes the relationship between the magnetic field and its sources,
viz., the current and/or the changing electric field. It can be expressed in either the
differential form

∇×H=J+휕D

휕t

, (7.1)

or the integral form

[

C

H⋅dI=X

S

œJ+휕D

휕t

⋅ds, (7.2)

whereHis the magnetic field intensity,Jis the current density, and휕D/휕tis the
displacement current added by Maxwell to state that a changing electric field can
induce a magnetic field, even without a current – the converse of Faraday’s law.
For situations with a low frequency so that the wavelengths are much longer that
the configurations of interest, the term of displacement current can be reasonably
dropped, which leads to a quasi-static approximation that will be used in the following
simulations.