J Supercond Nov Magn
DOI 10.1007/s10948-012-1880-y
ORIGINAL PAPER
The Influence of Excitation Frequency on Magnetic Levitation
Systems with a High-T
c
Superconductor
K. Ben Alia · L. Alloui · F. Bouillault · S.M. Mimoune
Received: 7 November 2012 / Accepted: 30 November 2012
© Springer Science+Business Media New York 2012
Abstract In this paper we present a numerical analysis of
dynamic features of the levitation system generated by an
interaction between a levitated permanent magnet (PM) and
a high-T
c
superconductor (HTSC) excited by an oscillatory
external source. The obtained results show that the value of
the frequency (f
free
) of the PM displacement in the case of
the levitation system generated by an interaction between a
levitated PM and a fixed HTSC is equal at the resonance
frequency (f
re
) of the levitation system generated by an in-
teraction between a levitated PM and HTS excited by an os-
cillatory external source and the resonance frequency (f
re
)
is mainly dependent upon the cooling position (Z
0
) and the
mass of the PM. The numerical problem in this paper is
solved by using the control volume method (CVM).
Keywords High-T
c
superconductor · Control volume
method · Levitation system · Excitation frequency ·
Resonance frequency
K. Ben Alia · L. Alloui ( ) · S.M. Mimoune
Laboratoire de Modélisation des Systèmes Énergétiques LMSE,
Université de Biskra, 07000 Biskra, Algérie
e-mail: lotfi.alloui@lgep.supelec.fr
S.M. Mimoune
e-mail: s.m.mimoune@univ-biskra.dz
L. Alloui · F. Bouillault
Laboratoire de Génie Electrique de Paris, LGEP, CNRS UMR
8507, Supélec, Université Pierre et Marie Curie-Paris 6, Paris,
France
L. Alloui · F. Bouillault
Université Paris Sud-Paris 11, Plateau de Moulon, 11 rue Joliot
Curie, 91192 Gif-Sur-Yvette Cedex, France
F. Bouillault
e-mail: bouillault@lgep.supelec.fr
1 Introduction
High-T
c
superconducting magnetic levitation systems are
able to maintain stable levitation without control. Therefore,
their application to mechanical systems with very little en-
ergy loss is expected. For example, flywheel bearings for
energy storage and maglev vehicles for mass transportation
have been studied and developed so far [1–3]. In design of
these systems, it is important, from the viewpoint of me-
chanical stability, to predict exactly their dynamics in the
presence of external disturbance. However, mechanical res-
onance characteristics of a high-T
c
superconducting mag-
netic levitation system in the mechanical resonant state are
not clearly understood. In this context, we propose in this
paper a 3D numerical method to study the dynamics of a
levitation system in which a permanent magnet is freely levi-
tated above a high-T
c
superconductor excited vertically. The
aims are determining the value of the resonance frequency
and study the influence of some parameters on the resonant
frequency.
Several works have shown that the influence of the flux
creep on dynamic behavior of magnetic levitation systems
with a high-T
c
superconductor is considerable and signif-
icant [4–6], in particular, the numerical works presented
in [7, 8]. In the literature, some macroscopic mathemati-
cal models were proposed to reflect the flux creep and the
flux flow phenomena of superconductivity such as the model
proposed by Anderson–Kim [9] or the model of power law
proposed by Rhyner [10]. These models lead to good agree-
ment with the experimental data [6, 11]. In our paper, the
effect of flux creep in the superconductor is simulated by
using the power law. In this case, to avoid convergence
problems and numerical oscillations involved in the strongly
non-linear character of the problem to be treated, in partic-
ular if the model of power law is used to describe the rela-
tion between the electric field and the current density [12],