1516 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 21, NO. 3, JUNE 2011Fig. 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 constantssured 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 oscillationmeasurement 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 inThe relationships between the magnetic force and the gap in Section III. According to the results, the mass of the levitationboth 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 beand 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 partIt might be explained that some ﬂuxes pinned in the HTS are is measured by the laser displacement sensor with the samplingreleased from the pinning center resulting in being trapped in frequency of 100 Hz. A representative result of attenuation be-other pinning center , . 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 thebe 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 thetion 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 systemwhere 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
SAKAI AND HIGUCHI: PROPERTIES OF MAGNETIC LEVITATION SYSTEM USING HIGH-TEMP SUPERCONDUCTOR 1517Fig. 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 isintroduced to evaluate the dynamic properties as a function ofthe velocity.B. Viscoelastic Measurement Using Repetitive Control To further investigate dynamic properties of the system, anovel 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 inputfor the next cycle based on the tracking error . A voice coilmotor (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 repetitiveno-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 representthe elasticity and viscosity, respectively. It should be noted that the input current is consistently controlled by a PC, based onrepetitive control using the VCM  is, therefore, useful to tracking error. The sinusoidal input is given until 60 cycles, butevaluate the dynamic properties of the levitation system, be- repetitive control is applied every 3 cycles; each cycle has 1000cause the input oscillation frequency can be ﬂexibly changed samplings. The measurement is conducted in not only the SCwith excellent repeatability. A schematic of the experimental state but also the NC state for comparison. The relationshipssetup is shown in Fig. 8 and the detail of experimental proce- between spring and damping constants and the input frequencydure is as follows. In this experiment, the same procedure is for the oscillation are shown in Figs. 9 and 10, respectively. Theused 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 increasethe steel rod is attached, to move the rod instead of manual of input frequency. Meanwhile the damping constant decreasesmovement. A sinusoidal position oscillation with the amplitude as increase of input frequency and varies more notably than theof 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 infrom the bottom of the cryostat. The position of the rod is mea- the obtained current is signiﬁcantly smaller than the sine wavesured by a linear encoder (Mercury2000, MicroE Systems), and component and the accidental error cannot be ignored.
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 twoFig. 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 decreasedmm 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. REFERENCES  J. R. Hull, J. L. Passmore, T. M. Mulcahy, and T. D. Rossing, “Stable levitation of steel rotors using permanent magnets and high-tempera-Fig. 12. Hysteresis loss vs. input frequency in dynamic measurement and hys- ture superconductors,” J. Appl. Phys., vol. 76, no. 1, pp. 577–580, 1994.teresis loss in static measurement.  Y. Tsutsui and T. Higuchi, “Suspension of soft magnetic materials using high-Tc superconductor,” Electrical Engineering in Japan, vol. 116, no. 3, pp. 116–123, 1996.  H. Ohsaki, M. Takabatake, and E. Masada, “Stable levitation of soft fer- V. ANALYSIS OF HYSTERESIS romagnetic materials by ﬂux pinning of bulk superconductors,” IEEE Trans. Magn., vol. 33, no. 5, pp. 3454–3456, 1997. The source of the damping is the energy loss caused by the  J. Sayama, T. Ueno, M. Ghodsi, and T. Higuchi, “Levitation of soft magnetic material by HTS: Relationship between levitation propertymovement of some ﬂuxes pinned in the HTS . In the lev- and pinning ﬂux density,” in The 19th Symposium on Electromagneticsitation system, the oscillation is attenuated by several factors; and Dynamics, 2007, pp. 380–383.magnetic hysteresis, friction, air resistance, etc. In this work,  M. Ghodsi, T. Ueno, and T. Higuchi, “Improvement of magnetic cir-the hysteresis mentioned in Section III is specially focused on cuit in levitation system using HTS and soft magnetic material,” IEEE Trans. Magn., vol. 41, no. 10, pp. 4003–4005, 2005.and compared in both the static and dynamic measurements. In  M. Ghodsi, T. Ueno, and T. Higuchi, “The characteristics of trappedthe static measurement, hysteresis loss is equal to the area sur- magnetic ﬂux inside bulk HTS in the mixed- levitation system,”rounded by the solid line in Fig. 3. So, the loss can be calcu- in 18th International Symposium on Superconductivity, 2006, pp. 343–346.lated by measuring the magnetic force in the same cycle as the  M. Futamura, T. Maeda, and H. Konishi, “Damping characteristics of aexperiment in Section IV. Fig. 11 shows relationship between magnet oscillating above a YBCO superconductor,” Jpn. J. Appl. Phys.,the magnetic force and the gap in this condition. In a damped vol. 37, no. 7, pp. 3961–3964, 1998.  L. Kuehn, M. Mueller, R. Schubert, C. Beyer, O. de Haas, and L.harmonic oscillator model, the hysteresis loss during a cycle is Schultz, “Static and dynamic behavior of a superconducting magneticequal to the work done by the viscous force , and is expressed bearing using YBCO bulk material,” IEEE Trans. Appl. Supercond.,as follow : vol. 17, no. 2, pp. 2079–2082, 2007.  T. Sugiura, M. Tashiro, Y. Uematsu, and M. Yoshizawa, “Mechanical stability of a high-Tc superconducting levitation system,” IEEE Trans. (3) Appl. Supercond., vol. 7, no. 2, pp. 386–389, 1997.  T. Higuchi and T. Yamaguchi, “Cutting tool positioning by periodic learning control method and inverse transfer function compensation,” System and Control, vol. 30, no. 8, pp. 503–511, 1986.Therefore, the hysteresis losses in the static and dynamic mea-  T. Nonomura, W. Rhie, and T. Higuchi, “Load estimation of voice coilsurements can be compared. The hysteresis losses in both the motor using repetitive control,” in Proceedings of the 2009 JSPE Spring Conference, 2009, vol. 1, pp. 957–958.measurements are plotted in Fig. 12. It should be noted that hys-  G. Sandberg and R. Ohayon, Computational Aspects of Structuralteresis loss in the dynamic measurement is close to that in the Acoustics and Vibration. Wien: Springer, 2009.