View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Online Publishing @ NISCAIR Indian Journal of Pure & Applied Physics Vol. 55, April 2017, pp. 261-268 Levitation force analysis of ring and disk shaped permanent magnet-high temperature superconductor Sinan Basaran & Selim Sivrioglu* Department of Mechanical Engineering, Gebze Technical University, Gebze, Turkey Received 4 February 2016; revised 2 January 2017; accepted 27 February 2017 In superconducting magnetic levitation systems, interaction models between a high temperature superconductor and a permanent magnet are useful to analyze the dynamics of the levitated system. In this study, stiffness equations of a superconducting levitation system using a disk and a ring permanent are obtained using frozen image concept. The variation of the stiffness has been analyzed for vertical movements of the PMs. For engineering applications, accuracy of such models should be tested experimentally. An experimental PM-HTS setup has been built to verify the obtained models for different cooling height conditions. Levitation forces computed using the frozen image approach for the disk and ring PMs are converged to the experimental results when the cooling heights are smaller values. Keywords: Superconducting magnetic levitation, Frozen image approach, Interaction model 1 Introduction stiffness’s. As is well known, the frozen image model The technological developments concerning with does not include hysteresis in the HTS. For small superconducting magnetic levitation have great movements of the PM, the frozen image model advancement especially in energy storage systems and successfully predicts the force of interaction between maglev transportation applications 1-3. Superconducting the disk PM and the bulk HTS 7,8. magnetic bearing (SMB) has advantages due to The variation of the levitation force for different passively stable characteristics with considerable permanent magnet shapes can be analyzed using stiffness in the radial and axial directions 4. In bearing current modeling approaches. In this study, a disk PM applications in mechanical systems, there are two and a ring PM over a bulk HTS is modeled and types of bearings configurations considering the stiffness equations are derived for the vertical loading direction such as radial and axial bearings. If displacement of PMs. For the proposed disk and ring a rotor-PM is located inside a hollow HTS cylinder or shaped PM-HTS systems, the stiffness’s are computed a ring HTS, then the resulting levitation structure on the basis of magnetic potentials obtained with the creates a radial or journal magnetic bearing 5. The frozen image concept. Some experimental verification levitation of a permanent magnet over a bulk high are presented and compared with analytical results for temperature superconductor creates an axial type decreasing cooling heights of the disk and ring PMs. bearing configuration. In most axial type PM-HTS 2 Interaction Modeling levitation systems, a disk PM is generally used with a rotor element. In some superconducting 2.1 Disk PM-HTS levitation system levitation applications, it is necessary to understand A disk PM over a bulk HTS is a basic the variation of the levitation forces with different superconducting levitation system as depicted in PM shapes. Fig. 1. Here, Fig. 1(a) shows the PM at cooling The interaction models are used to calculate the position and Fig. 1(b) shows the PM at levitation 6,8-10 levitation force and stiffness between PM and HTS. In position. In references , the frozen image engineering systems, such models are useful to technique is explained in an ideal situation where the predict the dynamics of the levitated system. The superconductor is a large surface with no dimension frozen image approach 6 was used to calculate the and a permanent magnet is levitating above the flat vertical and horizontal levitation forces and surface. In practice, the HTS has a certain dimension ——————— with different shapes. Assuming that the bulk HTS *Corresponding author (E-mail: [email protected]) has a large size comparing to the PM, the geometry 262 INDIAN J PURE & APPL PHYS, VOL. 55 APRIL 2017 illustrated in Fig. 1 can be considered within the [t(uu (uv Zu … (2) scope of the frozen image technique. Moreover, a tilt ͏$ = ʠ wʡ ͨ_ ʚͦ ͯͦ5ʛ angle θ that occurs in levitation from the vertical due ͤͥͥ͡ ͥͦ͡ to the in homogeneities in the magnetism of the PM ͏!-* = ͦ ͦ 4 ͦ and the HTS is accepted in the analysis. Note that the ͬ ͦ + ͭ ͧ + ʚ2͙ − ͮʛ ͨ − 6ͩͬͭ − 6ͪͬʚ2͙ − ͮʛ − 6ͫͭʚ2͙ − ͮʛ ʨ ͩƟ ʩ tilt angle is considered in cooling phase (Fig. 1(a)) ʚͬͦ + ͭͦ + ʚ2͙ − ͮʛͦʛ ͦ due to the fact that the frozen image is not rotating as … (3) well as not moving. Where show the trigonometric ͥ, … , ͫ The frozen image approach is applied to the disk relations and are derived as ͦ , ͥ = 1 + ͗ͣͧ ͦ = PM-HTS levitation system. In this approach, a ͦ ͦ , ͦ ͦ , 1 − 3ͧ͢͝ ͗ͣͧ ͧ = 1 − 3ͧ͢͝ ͧ͢͝ ͨ = 1 − magnetic dipole is assumed to be located in the ͦ , ͦ , geometric center of the PM. A combination of frozen and3͗ͣͧ ͩ = ͧ͢͝ ͧ͗ͣͧ͢͝, and. Here, ͪ = θͧ͗ͣͧ͗ͣͧ͢͝ is the angle dipole (flux pinning) and mirror dipole (diamagnetic betweenͫ = rotational ͧ͗ͣͧͧ͢͢͝͝ magnetization symmetry axes of image) appear inside the HTS material. Considering the PM with the vertical direction and φ is the dipole the superconducting levitation system, the total angle at x− y plane (not seen in Fig. 1). The interaction potential Utot of the system is given as: U= U + U … (1) stiffness in x direction due to vertical displacement tot dia fro of the disk PM is obtained by taking partial derivative where U dia and U fro are the diamagnetic and the of the diamagnetic and the frozen parts and frozen image potentials, respectively. Equations substituting the variables x = 0 , y = 0, z= z , of the diamagnetic and frozen image potentials θ= θ and φ= φ as follows: are obtained as: ∂2U ∂2U k =dia + fro xx_ disk ∂x2 ∂ x 2 µ m m 2λ− 5 λ 0 11 12 2 4 = 5 4π ()2e− z … (4) Also, the stiffness in z direction due to vertical displacement of the disk PM is obtained as: 2 2U ∂ U dia ∂ fro kzz_ disk =2 + 2 ∂z ∂ z µm m 48 λ 12 λ 0 11 12 1 4 =5 + 5 4π ()2ez− 2() 2 ez − … (5) Table 1 shows the magnetic moments and the position vector that used in rderivation of the stiffness equations. In Table 1, m is the magnetic moment of 11 r the dipole from the disk PM, and m is the magnetic Fig. 1 – Disk PM-HTS levitation system for the frozen image 12 approach (a) cooling phase and (b) levitation phase moment of the dipole from the diamagnetic image of Table 1 – Magnetic moments of the disk PM-HTS system Images Magnetic moments Position vectors r r Dipole-diamagnetic mm11= 11 (sinθφ cos xˆ + sin θφ sinyz ˆ + cos θ ˆ) r= rxxˆ + ry y ˆ + rz z ˆ r mm12= 12 ()sinθφ cos xˆ + sin θφ sin yzˆ − cos θ ˆ rx=0, r y = 0, rez z =− 2 2 r r Dipole-Frozen mm11= 11 (sinθφ cos xˆ + sin θφ sinyz ˆ + cos θ ˆ) r= rxxˆ + ry y ˆ + rz z ˆ r rx=,r = yr , =− 2 ez mm12= 12 ()sinθφ cos xˆ + sin θφ sin yzˆ + cos θ ˆ x y z BASARAN & SIVRIOGLU: LEVITATION FORCE OF PARMANENT MAGNET-HIGH TEMPERATURE SUPERCONDUCTOR 263 r height when its supporter is removed after cooling of m11 in the HTS. Also, m11 and m12 are the amplitudes the HTS is completed. In the levitation phase as of these vectorial values. shown in Fig. 2(b), while the frozen image is fixed in 2.2 Ring PM-HTS levitation system e , the diamagnetic image moves with the PM and An interaction modeling for a ring shaped PM is has the levitation height δ from the surface of the not reported in literature. For a disk PM, a magnetic ring HTS. dipole is assumed to be located in the geometric In the ring shaped PM case, PM-HTS interaction is center of the disk PM. Depending on the dipole inside two sided due to hollow shape of the PM. The total the PM, a combination of frozen and mirror dipole magnetic potentials for I and II sides (Fig. 2) are appear inside the HTS material. For a ring shaped written as 11 : PM, if the cross section of the ring is considered, dipoles appear in two sides due to having two (I) ( I) ( I ) Utotal= U dia + U fro geometric centers for the ring HTS. Note that … (6) U()()()II= U II + U II magnetization fields of the ring PM is also two sided total dia fro in an axisymmetric analysis. Therefore, accepting where U (I ) is the dipole and its diamagnetic image two dipole moments may be a good approximation d ia (I) in applying frozen image technique to the ring potential for I side. Also, U fro is the dipole and its shaped PM. frozen image potential. Equations of each potentials The ring shaped permanent magnet over a bulk are written as: high temperature superconductor at the cooling phase ()I µ0m 11 m 12λ 1 of a superconducting levitation as shown in Fig. 2(a). U dia = 3 The ring PM dimension is given by outer diameter, 4π ()2e− 2 z 2 2 2 inner diameter and height such as do× d i × h a . The ()I µ0m 11 m 12 λλλ234x++ y()()()2 ez −+ 6 λλ 56 xy − 62 xez −+ 62 λ 7 yez − U fro = 5 2 coordinate system O xz is fixed to the PM.