1. IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011 1515
Dynamic Properties of Magnetic Levitation System
Using High-Temperature Superconductors
Ichinosuke Sakai and Toshiro Higuchi
Abstract—Pinning effect of a bulk high-temperature supercon-
ductor (HTS) has various possible applications, e.g., it enables a
non-contact and stable levitation without active control. Regarding
the magnetic levitation with HTSs, permanent magnets are gener-
ally employed, but our previous studies have demonstrated that the
permanent magnets can be replaced with soft magnetic materials
such as iron. When the magnetic levitation is applied to large trans-
portation system, the use of this technique has a potential to allow
a low-cost levitation system with a simple structure, in which ex-
pensive permanent magnets can be eliminated. In the mechanical
design, a dynamic evaluation as well as a static evaluation must be
needed. Hence, this paper fundamentally discusses dynamic prop-
erties of the magnetic levitation modulated by a damped harmonic
oscillator model. For the dynamic analysis, this paper proposes a
novel measurement method using repetitive control. The analysis
shows that the dynamic properties of our system are related to the
approach manner—process and speed—of the soft magnetic ma-
terial to the HTS. It is expected that these results may shed further Fig. 1. Magnetic levitation process of a soft magnetic material using a high-
light on the design of efficient levitation system with HTSs. temperature superconductor (HTS).
Index Terms—Dynamic properties, maglev, pinning effect, soft
magnetic material.
II. PRINCIPLE
I. INTRODUCTION The principle of the magnetic gradient levitation system and
the procedure are illustrated in Fig. 1. At first, (a) an HTS is
ULK high-temperature superconductors (HTSs) possess cooled down by liquid nitrogen below the transition tempera-
B many possible applications. Above all, a magnetic lev-
itation using permanent magnets is considered as the most
ture within magnetic field. After magnetic fluxes are pinned in
the HTS, (b) a soft magnetic rod which has the smaller area than
promising technique in industrial field. However, it is hard the flux-pinned area is moved closer to the HTS. Some fluxes
to apply permanent magnets to the large-scale transportation vertically penetrate the top side of the rod, thereby causing an
system. This is because large permanent magnets themselves attractive force between the HTS and the rod. As the gap be-
are relatively expensive and the machining is difficult. To solve tween the two components is reduced, (c) the attractive force is
this problem, alternative levitation systems which use soft enhanced. When the gap is, however, decreased much more, (d)
magnetic materials instead of permanent magnets have been the force becomes weaker as some fluxes far from the rod cannot
proposed [1]–[3]. Especially, Tsutsui and Higuchi have pro- penetrate from the top side. This is because they are strongly
posed a magnetic gradient levitation system, which possesses fixed on the pinning center of the HTS and cannot be sufficiently
a practical attractive force [2]. The authors have improved the distorted to penetrate from the top side. In this case, some fluxes
levitation performance focusing on only the static behavior [4], penetrate the side instead, thereby resulting in reduction of the
[5]. Thus, the dynamic analysis of the levitation system has attractive force. Therefore, the force becomes to have positive
potential to find a key factor in designing the system [7]. In this stiffness and it enables a soft magnetic material to be stably lev-
paper, modulating the system by a damped harmonic oscillator itated without active control at the equilibrium point.
model [7]–[9], the spring and damping constants are measured.
In addition, a novel measurement method which uses repetitive III. STATIC MEASUREMENT
control is proposed. The experimental results would lead a
comprehension of the levitation mechanism. A schematic of experimental setup is shown in Fig. 2. In this
measurement, first, an HTS (YBCO bulk) with 30 mm in di-
ameter and 10 mm in thickness is put into a cryostat which
Manuscript received August 01, 2010; accepted October 15, 2010. Date of has a stainless steel plate (0.4 mm in thickness) at the bottom.
publication November 15, 2010; date of current version May 27, 2011. The HTS is cooled down within magnetic field generated by an
The authors are with Department of Precision Engineering, School of
Engineering, University of Tokyo, Tokyo, 113-8656, Japan (e-mail: ichi-
Nd-Fe-B permanent magnet (30 mm in diameter and 10 mm in
nosuke@aml.t.u-tokyo.ac.jp; higuchi@aml.t.u-tokyo.ac.jp). thickness, with residual flux density of 1.3 T and coercive force
Digital Object Identifier 10.1109/TASC.2010.2089414 of 960 kA/m) without any magnetic material. After the magnetic
1051-8223/$26.00 © 2010 IEEE

2. 1516 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011
Fig. 2. Schematic diagram of experimental setup in static measurement.
Fig. 4. Schematic diagram of experimental setup in a damped oscillation anal-
ysis and a damped harmonic oscillator model.
Fig. 3. Magnetic force vs. gap in superconductivity (SC) state and normal con-
ductivity (NC) state. Fig. 5. Damped oscillation of the levitation part suspended by HTS.
fluxes are pinned in the HTS, a 0.45%C steel rod (8 mm in di- where and are the natural an-
ameter and 50 mm in thickness) is quasi-statically approached gular frequency and damping ratio, respectively. Here, indi-
the bottom of cryostat ranging from 3 mm to 0 mm at 0.1 mm/s. cates the displacement from the equilibrium point of the oscil-
During the experiment, the magnetic force and the gap are mea- lation system. Therefore, the stiffness and damping constants
sured by means of a load cell (LC1205-K020, A&D) and a laser can be calculated by modulating measured oscillation by (2).
displacement sensor (LC2440, KEYENCE), respectively. The A schematic of experimental setup in the damped oscillation
measurement is conducted in not only superconductivity (SC) analysis is shown in Fig. 4. As the procedure, first, a stable lev-
state but also normal conductivity (NC) state for comparison. itation of a steel rod is realized by the properties obtained in
The relationships between the magnetic force and the gap in Section III. According to the results, the mass of the levitation
both the states are shown in Fig. 3. It should be noted that unlike part (including the steel rod) is adjusted so as to shift the equi-
in the NC state, the magnetic force decreases below a certain gap librium point between the gravity and the magnetic force to be
and becomes to have positive stiffness in the SC state—the max- within the region of positive stiffness (Fig. 3). In this condition,
imum attractive force 2.57 N at 1.45 mm. The magnetic force an external impulse force is given to generate damped oscilla-
shows a hysteresis in the approaching and retreating processes. tion in z-direction. The position of levitation part
It might be explained that some fluxes pinned in the HTS are is measured by the laser displacement sensor with the sampling
released from the pinning center resulting in being trapped in frequency of 100 Hz. A representative result of attenuation be-
other pinning center [2], [7]. havior is shown in Fig. 5. The result shows that the natural fre-
quency increases with time transition. It is considered that the
IV. DYNAMIC MEASUREMENT stiffness increases with decrease in the gap within the region
of positive stiffness, as shown in Fig. 3. The relationships be-
A. Damped Oscillation Analysis tween the initial position where the attenuation starts and the
Dynamic properties in the levitation system are discussed by spring and damping constants are shown in Figs. 6 and 7, re-
using a damped oscillation analysis. Dynamic properties can spectively. The graphs show that in the measurement range the
be generally calculated by the attenuation of oscillation. In a spring constant decreases with increase in initial position. Also,
damped harmonic oscillator model as shown in Fig. 4, the mo- the value agrees well with the slope of line at 0.7 mm in the
tion equation of the mass is described as: static measurement (Fig. 3). On the contrary, the damping con-
stant increases with increase in initial position. This implies that
(1) increase of hysteresis affects the attenuation of levitation system
where and are the spring and damping constants, respec- (later we will discuss). Here, it can be said that dynamic proper-
tively. The solution of (1) is given as follow: ties of our system are related to the amplitude of the oscillation.
The damped oscillation analysis, however, can reveal only the
(2) properties depending on the natural frequency on the specific

3. SAKAI AND HIGUCHI: PROPERTIES OF MAGNETIC LEVITATION SYSTEM USING HIGH-TEMP SUPERCONDUCTOR 1517
Fig. 6. Spring constant vs. initial position in damped oscillation analysis.
Fig. 8. Schematic diagram of experimental setup in measurement using repet-
itive control.
Fig. 7. Damping constant vs. initial position in damped oscillation analysis.
Fig. 9. Spring constant vs. input frequency in measurement using repetitive
control.
levitation point. In next section, a new measurement method is
introduced to evaluate the dynamic properties as a function of
the velocity.
B. Viscoelastic Measurement Using Repetitive Control
To further investigate dynamic properties of the system, a
novel measurement method using repetitive control is proposed.
In this method, repetitive control, which is effective for a peri-
odic servo system, is implemented because it is useful to ad-
just an output to track a periodic target, modifying the input
for the next cycle based on the tracking error [10]. A voice coil
motor (VCM) was employed as an actuator to generate the os-
cillation. Because of the repeatability and robustness, the differ-
ence between the current through the HTS under the loaded and Fig. 10. Damping constant vs. input frequency in measurement using repetitive
no-loaded conditions represents the net current to move against control.
the load. When giving the sinusoidal motion by repetitive con-
trol, the sine and cosine components of the current represent
the elasticity and viscosity, respectively. It should be noted that the input current is consistently controlled by a PC, based on
repetitive control using the VCM [11] is, therefore, useful to tracking error. The sinusoidal input is given until 60 cycles, but
evaluate the dynamic properties of the levitation system, be- repetitive control is applied every 3 cycles; each cycle has 1000
cause the input oscillation frequency can be flexibly changed samplings. The measurement is conducted in not only the SC
with excellent repeatability. A schematic of the experimental state but also the NC state for comparison. The relationships
setup is shown in Fig. 8 and the detail of experimental proce- between spring and damping constants and the input frequency
dure is as follows. In this experiment, the same procedure is for the oscillation are shown in Figs. 9 and 10, respectively. The
used as that in the above-mentioned experiments; only the dif- spring constant has the positive stiffness in the SC state and neg-
ference point is that a VCM (X-1741, NEOMAX), on which ative in the NC state. The absolute value decreases as increase
the steel rod is attached, to move the rod instead of manual of input frequency. Meanwhile the damping constant decreases
movement. A sinusoidal position oscillation with the amplitude as increase of input frequency and varies more notably than the
of 0.2 mm is input at the reference point—0.5 or 1.0 mm far spring constant. This is because the cosine wave component in
from the bottom of the cryostat. The position of the rod is mea- the obtained current is significantly smaller than the sine wave
sured by a linear encoder (Mercury2000, MicroE Systems), and component and the accidental error cannot be ignored.

4. 1518 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011
static measurement and the value is larger when the reference
point is closer to the bottom of the cryostat in both measure-
ments. Therefore, in our levitation system, the hysteresis loss
significantly affects the attenuation of oscillation.
VI. CONCLUSION
This work discussed the dynamic properties of the system
in which a soft magnetic material can be levitated by an
HTS. To achieve this, the static and dynamic properties were
investigated, modulating the system by a damped harmonic
oscillator model. The measurements were performed in two
Fig. 11. Magnetic force vs. gap in a cycle of the amplitude of 0.2 mm at 0.5 ways. In damped oscillation analysis, spring constant decreased
mm and 1.0 mm. as increase of initial position of the attenuation and agrees
with the results in static measurement. On the contrary, the
damping constant increased as increase of initial position. To
evaluate the dynamic properties as a function of the oscillating
velocity, a novel measurement method using repetitive control
was proposed. The results showed that both of the spring and
damping constants decreased as increase of the input oscilla-
tion frequency. And evaluation of hysteresis losses in both the
measurements implied that hysteresis significantly affects the
attenuation of oscillation.
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