Levitation force analysis of ring and disk shaped permanent magnet-high temperature superconductor

Transcription

1 Inn Jounal of Pue & Applied Physics Vol. 55, Apil 017, pp Levitation foce analysis of ing and disk shaped pemanent magnet-high tempeatue supeconducto Sinan Basaan & Selim Sivioglu* Depatment of Mechanical Engineeing, Gebze Technical Univesity, Gebze, Tukey Received 4 Febuay 016; evised Januay 017; accepted 7 Febuay 017 In supeconducting magnetic levitation systems, inteaction models between a high tempeatue supeconducto and a pemanent magnet ae useful to analyze the dynamics of the levitated system. In this study, stiffness equations of a supeconducting levitation system using a disk and a ing pemanent ae obtained using fozen image concept. The vaiation of the stiffness has been analyzed fo vetical movements of the PMs. Fo engineeing applications, accuacy of such models should be tested expeimentally. An expeimental PM-HTS setup has been built to veify the obtained models fo diffeent cooling height conditions. Levitation foces computed using the fozen image appoach fo the disk and ing PMs ae conveged to the expeimental esults when the cooling heights ae smalle values. Keywods: Supeconducting magnetic levitation, Fozen image appoach, Inteaction model 1 Intoduction The technological developments concening with supeconducting magnetic levitation have geat advancement especially in enegy stoage systems and maglev tanspotation applications 1-3. Supeconducting magnetic beaing (SMB) has advantages due to passively stable chaacteistics with consideable stiffness in the al and axial diections 4. In beaing applications in mechanical systems, thee ae two types of beaings configuations consideing the loading diection such as al and axial beaings. If a oto-pm is located inside a hollow HTS cylinde o a ing HTS, then the esulting levitation stuctue ceates a al o jounal magnetic beaing 5. The levitation of a pemanent magnet ove a bulk high tempeatue supeconducto ceates an axial type beaing configuation. In most axial type PM-HTS levitation systems, a disk PM is geneally used with a oto element. In some supeconducting levitation applications, it is necessay to undestand the vaiation of the levitation foces with diffeent PM shapes. The inteaction models ae used to calculate the levitation foce and stiffness between PM and HTS. In engineeing systems, such models ae useful to pedict the dynamics of the levitated system. The fozen image appoach 6 was used to calculate the vetical and hoizontal levitation foces and *Coesponding autho ( s.selim@gtu.edu.t) stiffness s. As is well known, the fozen image model does not include hysteesis in the HTS. Fo small movements of the PM, the fozen image model successfully pedicts the foce of inteaction between the disk PM and the bulk HTS 7,8. The vaiation of the levitation foce fo diffeent pemanent magnet shapes can be analyzed using cuent modeling appoaches. In this study, a disk PM and a ing PM ove a bulk HTS is modeled and stiffness equations ae deived fo the vetical displacement of PMs. Fo the poposed disk and ing shaped PM-HTS systems, the stiffness s ae computed on the basis of magnetic potentials obtained with the fozen image concept. Some expeimental veification ae pesented and compaed with analytical esults fo deceasing cooling heights of the disk and ing PMs. Inteaction Modeling.1 Disk PM-HTS levitation system A disk PM ove a bulk HTS is a basic supeconducting levitation system as depicted in Fig. 1. Hee, Fig. 1(a) shows the PM at cooling position and Fig. 1(b) shows the PM at levitation position. In efeences 6,8-10, the fozen image technique is explained in an ideal situation whee the supeconducto is a lage suface with no dimension and a pemanent magnet is levitating above the flat suface. In pactice, the HTS has a cetain dimension with diffeent shapes. Assuming that the bulk HTS has a lage size compaing to the PM, the geomety

2 6 INDIAN J PURE & APPL PHYS, VOL. 55 APRIL 017 illustated in Fig. 1 can be consideed within the scope of the fozen image technique. Moeove, a tilt angle θ that occus in levitation fom the vetical due to the in homogeneities in the magnetism of the PM and the HTS is accepted in the analysis. Note that the tilt angle is consideed in cooling phase (Fig. 1(a)) due to the fact that the fozen image is not otating as well as not moving. The fozen image appoach is applied to the disk PM-HTS levitation system. In this appoach, a magnetic dipole is assumed to be located in the geometic cente of the PM. A combination of fozen dipole (flux pinning) and mio dipole (magnetic image) appea inside the HTS mateial. Consideing the supeconducting levitation system, the total inteaction potential Utot of the system is given as: U = U + U (1) tot fo whee U and U ae the magnetic and the fo fozen image potentials, espectively. Equations of the magnetic and fozen image potentials ae obtained as: Fig. 1 Disk PM-HTS levitation system fo the fozen image appoach (a) cooling phase and (b) levitation phase = () = (3) Whee,, show the tigonometic elations and ae deived as =1+, = 1 3, =1 3, = 1 3, =, = and =, and. Hee, θ is the angle between otational magnetization symmety axes of the PM with the vetical diection and φ is the dipole angle at x y plane (not seen in Fig. 1). The stiffness in x diection due to vetical displacement of the disk PM is obtained by taking patial deivative of the magnetic and the fozen pats and substituting the vaiables x = 0, y = 0, z = z, θ = θ and φ = φ as follows: k U U fo xx _ disk = + x x µ 0m11m 1 λ 5λ 4 = 5 ( e z ) (4) Also, the stiffness in z diection due to vetical displacement of the disk PM is obtained as: k U U fo zz _ disk = + z z µ m m 48λ 1λ = ( e z ) ( e z ) (5) Table 1 shows the magnetic moments and the position vecto that used in deivation of the stiffness equations. In Table 1, m 11 is the magnetic moment of the dipole fom the disk PM, and m 1 is the magnetic moment of the dipole fom the magnetic image of Table 1 Magnetic moments of the disk PM-HTS system Images Magnetic moments Position vectos Dipole-magnetic ( sinθ cosφ xˆ + sinθ sinφ yˆ + cosθ ) = ˆ ˆ xx+ y y + z m1 = m1 ( sinθ cosφ xˆ + sinθ sinφ yˆ cosθ ) x = 0, y = 0, z = e z Dipole-Fozen ( sinθ cosφ xˆ + sinθ sinφ yˆ + cosθ ) = ˆ ˆ xx + y y + z m = m sinθ cosφ xˆ + sinθ sinφ yˆ + cosθ x = x, y = y, z = e z 1 1 ( )

3 BASARAN & SIVRIOGLU: LEVITATION FORCE OF PARMANENT MAGNET-HIGH TEMPERATURE SUPERCONDUCTOR 63 m 11 in the HTS. Also, m 11 and m1 ae the amplitudes of these vectoial values.. Ring PM-HTS levitation system An inteaction modeling fo a ing shaped PM is not epoted in liteatue. Fo a disk PM, a magnetic dipole is assumed to be located in the geometic cente of the disk PM. Depending on the dipole inside the PM, a combination of fozen and mio dipole appea inside the HTS mateial. Fo a ing shaped PM, if the coss section of the ing is consideed, dipoles appea in two sides due to having two geometic centes fo the ing HTS. Note that magnetization fields of the ing PM is also two sided in an axisymmetic analysis. Theefoe, accepting two dipole moments may be a good appoximation in applying fozen image technique to the ing shaped PM. The ing shaped pemanent magnet ove a bulk high tempeatue supeconducto at the cooling phase of a supeconducting levitation as shown in Fig. (a). The ing PM dimension is given by oute mete, inne mete and height such as d o d i h a. The coodinate system O xz is fixed to the PM. When the bulk HTS is field cooled with the ing PM at a cetain cooling height e, two images appea such as the fozen image and the magnetic image as shown in Fig. (a). The distance of both images fom the suface of the HTS is equal to e befoe levitation is completed. The ing PM loses some Fig. Ring PM-HTS levitation system fo the fozen image appoach (a) cooling phase and (b) levitation phase height when its suppote is emoved afte cooling of the HTS is completed. In the levitation phase as shown in Fig. (b), while the fozen image is fixed in e, the magnetic image moves with the PM and has the levitation height δ fom the suface of the ing HTS. In the ing shaped PM case, PM-HTS inteaction is two sided due to hollow shape of the PM. The total magnetic potentials fo I and II sides (Fig. ) ae witten as 11 : ( I) ( I) ( I) total = + fo U U U ( II ) ( II ) ( II ) total = + fo U U U (6) whee ( I ) U is the dipole and its magnetic image d ia potential fo I side. Also, ( I) U is the dipole and its fo fozen image potential. Equations of each potentials ae witten as: ( I) µ 0m11m 1 λ 1 U = 3 ( e z) U ( I) µ 0m11m 1 λx + λ3 y + λ4 ( e z) + 6λ5xy 6λ6x ( e z) + 6λ7 y ( e z) fo = 5 x + y + ( e z) ( II ) ( I) U = U ( ) ( II ) µ 0m11m 1 λx + λ3 y + λ4 ( e z) 6λ5xy 6λ6x ( e z) 6λ7 y ( e z) U fo = 5 ( x + y + ( e z) ) (7) The stiffness equations due to the vetical displacement ae obtained by taking the patial deivative of the magnetic and the fozen pats and substituting the vaiables x = 0, y = 0, z = z, θ = θ and φ = φ espectively. When the ing PM has vetical movement, the stiffness of the ing PM-HTS in x diection is obtained as: K ( I ) ( I ) ( II ) ( II ) U U fo U U fo xx _ ing = x x x x µ m m λ 5λ = 5 π ( e z) (8) Similaly, the stiffness equation in z diection is deived as K ( I) ( I) ( II) ( II) U U fo U U fo zz _ ing = z z z z µ m m 48λ 1λ = + π ( e z) ( e z) (9)

4 64 INDIAN J PURE & APPL PHYS, VOL. 55 APRIL 017 Table Magnetic moments of the ing PM-HTS system Images Pat II Pat I Magnetic moments Magnetic moments Dipole-magnetic x + y + z m = m x + y z 1 1 Position vectos = xˆ + yˆ + = 0, = 0, = e z x y + z m = m x y z 1 1 Position vectos = xˆ + yˆ + = 0, = 0, = e z Dipole-fozen Magnetic moments x + y + z m = m x + y + z 1 1 Position vectos = xˆ + yˆ + = x, = y, = e z Magnetic moments x y + z m = m x y + z 1 1 ( sinθ cos φˆ sinθ sinφ ˆ cosθ Position vectos = xˆ + yˆ + = x, = y, = e z Table shows the magnetic moments and the position vecto that used in deivation of the Eqs (8) and (9). In Table, m is the magnetic moment of the dipole 11 fom the ing PM, and m is the magnetic moment of 1 the dipole fom the magnetic image of m in the 11 HTS. Also, m 11 and m 1 ae the amplitudes of these vectoial values..3 Levitation foces In a supeconducting levitation system, suppoting foces ae geneated by stiffness effects between supeconducto and PM. Levitation foces of the disk PM-HTS and the ing PM-HTS ae computed using obtained stiffness equations fo diffeent vetical displacements of the PMs. The foce equations ae defined fo the disk and ing PM as follows: F = k z ( _ ) ( _ ) ( _ ) ( _ ) zz _ disk zz disk F = K z zz _ ing zz ing F = k z xx _ disk xx disk F = K z xx _ ing xx ing (10) whee z is the vetical displacement. The paamete values of the disk and ing PMs used in levitation foce computations ae given in Section 3. Figue 3 shows vaiation of the levitation foces F and zz _ disk F when the PMs move fom 5 mm to zz _ ing 1 mm in the vetical diection. As seen in this figue, behavio of the vetical levitation foces is nonlinea. Fig. 3 Vaiation of the levitation foces F xx_disk, F xx_ing with the vetical displacement Z The disk PM poduces lage levitation foces than the ing PM due to having highe magnetization value while the suface aea of both PMs is almost equal. Vaiations of the lateal foces F xx_disk, F xx_ing with vetical displacements ae shown in Fig. 4. Small inceases ae obseved in the lateal foces with the vetical displacements. 3 Expeimental Setup The stuctue of the expeimental levitation system is schematically depicted in Fig. 5. The levitation system is basically consisted of a bulk HTS, a oto with attached PM and a non-contact measuement senso fixed to a suppot.

5 BASARAN & SIVRIOGLU: LEVITATION FORCE OF PARMANENT MAGNET-HIGH TEMPERATURE SUPERCONDUCTOR 65 Fig. 4 Vaiation of the lateal foces F xx_disk, F xx_ing with the vetical displacement Z Fig. 5 Schematic view of the expeimental system The expeimental setup is shown in Fig. 6(a,b). The oto is made fom aluminum and its weight 10 g fo the ing magnet oto and 11 g fo the disk magnet oto. The ing masses ae made fom stainless steel and thei weights 1, g fo each. To decide the vetical levitation foce expeimentally, the extenal ing masses added to the oto, espectively. The noncontact eddy cuent type displacement senso is fixed the top of the oto fo the measuements of the displacements. The single domain bulk HTS which is 34 mm mete and 15 mm height is used in expeiments. The HTS is fixed to the inside to the Styofoam insulated containe and cooled by the liquid nitogen. The expeiment was caied out by fist placing the oto-pm ove the HTS at constant cooling height (CH) which can be adjusted by five diffeent Fig. 6 Expeimental setup (a) complete levitation system and (b) otos and PMs suppotes. The suppote pats ae 15 mm, 10 mm, 5 mm, 3 mm and mm height. Afte the cooling pocess by submeged the HTS into liquid nitogen the suppote pats ae sepaated between PM and HTS. Pemanent magnets have the suface aeas of A disk = m and A ing = m, espectively. Suface aeas of the ing and disk pemanent magnets ae selected appoximately equal to see the effect of the magnet shapes. The ing and the disk PMs ae 5 mm thickness both. Magnetization fields of the pemanent magnets ae computed with axisymmetic analysis using FEMM 4. softwae 1. The ing PM is polaized axially with a magnetization of T and the disk PM is polaized axially with a magnetization of T. The computed magnetization field values ae checked with measued values obtained by the gauss mete pobe. 4 Results and Discussion The expeimental esults ae obtained fo fivediffeent cooling height (CH) condition to undestand the dependency of the fozen image based inteaction model to the initial cooling gap. The expeimental foce values ae obtained by using the weights and the measued vetical displacements. In all levitation expeiments, field cooling (FC) of the HTS is ealized. The expeimental and analytical

6 66 INDIAN J PURE & APPL PHYS, VOL. 55 APRIL 017 model esults ae compaed fo evey cooling height condition in figues. The PM-HTS levitation foces ae evaluated in tems of expeimental and computed esults. The expeimental esults ae given in ode of cooling distance fom lage to smalle values. Figue 7 illustates the vaiation of the vetical foce with the displacement using the disk PM in the supeconducting levitation expeiment established with a stating cooling height of 15 mm. Similaly, the ing PM esult is given in Fig. 8 fo the same stating cooling height. Fo the next cooling distance of 10 mm, the esults ae given fo the disk PM and ing PM in Figs 9 and 10, espectively. Eithe in 15 mm o 10 mm cooling heights, the expeimental esults patially match with the computed values. Figue 11 shows the vaiation of the vetical foce fo the disk PM case fom a stating cooling height of 5 mm. Similaly, the ing PM esult is given in Fig. 1 fo the same stating cooling height. The expeimental foce esults diffe fom the computed esults in both figues but the tendency of expeimental esults given in Fig. 11 matches with the computed esults. The expeimental esults fo the ing PM shown in Fig. 1 does not even show any tendency with the fozen image esults like disk PM configuation.this situation is elated to the diffeent magnetic field chaacteistics of the disk PM and ing PM. Fo the cooling height of 3 mm, the levitation expeiments ae epeated fo the disk PM and the ing PM. Using the expeimental and computed esults, Figs 13 and 14 show the vaiation of vetical foces Fig. 7 Vaiation of the levitation foce with displacement of the disk PM (CH: 15 mm) Fig. 9 Vaiation of the levitation foce with displacement of the disk PM (CH: 10 mm) Fig. 8 Vaiation of the levitation foce with displacement of the ing PM (CH: 15 mm) Fig. 10 Vaiation of the levitation foce with displacement of the ing PM (CH: 10 mm)

7 BASARAN & SIVRIOGLU: LEVITATION FORCE OF PARMANENT MAGNET-HIGH TEMPERATURE SUPERCONDUCTOR 67 Fig. 11 Vaiation of the levitation foce with displacement of the disk PM (CH: 5 mm) Fig. 13 Vaiation of the levitation foce with displacement of the disk PM (CH: 3 mm) Fig. 1 Vaiation of the levitation foce with displacement of the ing PM (CH: 5 mm) fo both PMs, espectively. In these esults, expeimental esults match quite closely with the computed values. Figues 15 and 16 ae obtained fo the cooling height of mm. As seen in these figues, the expeimental esults pefectly matched with the computed esults obtained using fozen image appoach. In most engineeing application, the levitation stiffness is consideed as a paamete athe than the levitation foce.it is possible to have the levitation stiffness fo the disk and ing PMs using the same expeimental data and computed values fo each cooling heights. Although the fozen image model does not include hysteesis, the computed levitation foce and stiffness values ae vey close to measued expeimental data at the smalle cooling heights. Also Fig. 14 Vaiation of the levitation foce with displacement of the ing PM (CH: 3 mm) Fig. 15 Vaiation of the levitation foce with displacement of the disk PM (CH: mm)

8 68 INDIAN J PURE & APPL PHYS, VOL. 55 APRIL 017 PMs ae conveged to the expeimental esults. The fozen image model successfully pedicts the foce of inteaction between the PM-HTS. Acknowledgment This eseach study is suppoted by The Scientific and Technological Reseach Council of Tukey unde the suppot pogam of 1001 with the poject numbe 13M581. Fig. 16 Vaiation of the levitation foce with displacement of the ing PM (CH: mm) it is tue that smalle the cooling height lage the levitation stiffness and foce. 5 Conclusions In this eseach wok, the stiffness vaiation of the disk and ing shaped PM-HTS configuations ae pesented on the basis of magnetic potential equations obtain fo each diection. Evaluation of the fozen image appoach is studied using a disk and ing pemanent magnet (PM) ove a bulk high tempeatue supeconducto (HTS) levitation system fo the diffeent cooling height (CH) distances. Fo smalle cooling heights values, levitation foces computed using the fozen image appoach fo the disk and ing Refeences 1 Wefel F, Floegel-Delo U, Rothfeld F, Riedel T, Wippich D, Schimeiste P & Koenig R, HTS Bulk magnetic application in flywheel enegy stoage systems FESS and MAGLEV tanspotation, Poc Int Conf on MAGLEV, (014). Wefel F, Floegel-Delo U, Rothfeld F, Riedel T, Goebel B, Wippich D & Schimeiste P, Supecond Sci Technol, 5 (01) Sotelo G G, Dias D H N, Andade R, Stephan R M, Del-Valle N, Sanchez A, Navau C & Chen D, IEEE Tans Appl Supecond, 1(5) (011) Tang J, Fang J & Tong W, IEEE Tans Appl Supecond, (6) (01) Sivioglu S & Basaan S, IEEE Tans Appl Supecond, 5(4) (015) Hull J R & Cansiz A, J Appl Phys, 86 (1999) Cansiz A, Hull J R & Gundogdu O G, Supecond Sci Technol, 18 (005) Cansiz A, Physica C, 390 (003) Kodyuk A A, J Appl Phys, 83 (1998) Hull J R, Supecond Sci Technol, 13 (000) Sivioglu S & Cina Y, Supecond Sci Technol, 0 (007) Meeke D, Finite Element Method Magnetic,