Analytical modeling of passive electrodynamic levitation systems

Transcription

1 Abbas NAJJAR-KODABAKS Depatment of Electical Engineeing, Islamic Azad Univesit, Najafabad Banch, Najafabad, Esfahan, Ian Analtical modeling of passive electodnamic levitation sstems Abstact. In this pape an analtical model fo an electodnamic levitation sstem is pesented. The sstem consists of a moving pemanent magnet piece levitated ove a conducting plate. The width of conducting plate is assumed to be consideabl geate than magnet. The pemanent magnet piece is modeled b two equivalent cuent sheets, inducing edd cuents in the moving plate. The esultant magnetic field due to the pemanent magnet and the edd cuents ae then calculated. The dag and lift foces ae also computed and thei vaiations with sstem specifications ae investigated. Finall a finite element method is emploed to veif the validit of the analtical model, confiming the desiable accuac of the model. The sstem modeling b the poposed analtical method is fast enough to be used in iteative design and optimization pocedues. Abstact. W pac pzedstawiono model sstemu lewitacji elektodnamicznej. Sstem składa się z uchomego magnesu lewitującego nad pzewodzącą płtką o ozmiaach większch niż ozmia magnesu. Magnes jest modelowan w postaci pzewodzącch pąd akusz indukującch pąd wiowe w płtce. Wpadkowe pole magnetczne jest obliczane. Obliczane są też sił pzciągania i odpchania dla óżnch waunków pac. Pz zastosowaniu metod Elementów Skończonch pzepowadzono walidację modelu osiągając założoną dokładność. (Metoda analiz paswnego sstemu lewitacji elektodnamicznej). Kewods:. Electodnamic evitation, Magnetic levitation, Passive, Analtical modeling, Magnetic fields, Finite Element Method Słowa kluczowe: lewitacja elektodnamiczna, pole magnetczne, siła pzciągania magnesu. Intoduction Maglev tanspotation uses magnetic fields to ceate lift, thust and guidance foces without phsical contact. Thee ae two tpes of magnetic levitation sstems: attactive and epulsive []. In attactive levitation o electomagnetic suspension (EMS), a feomagnetic bod is attacted to a souce of magnetic flu, as a piece of steel is attacted to a pemanent magnet. This tpe of levitation is unstable without feedback contol []. In epulsive levitation o electodnamic suspension (EDS), edd cuents ae geneated in a moving conducting bod (guidewa) when it epeiences a magnetic field povided b supeconducting coils [3]. Passive electodnamic suspension (PEDS) in which the magnetic field is povided b pemanent magnet (PM) has ecentl gained inceasing attention due to its simplicit and cost saving meits [4]. It does not need powe supplies, powe convesion, powe tansmission means, supeconducting coils and cooling facilities. An optimal design of PEDS sstems needs pope modeling and simulation of these sstems and accuate pedictive of lift and dag foces. Diffeent modeling methods have been pesented fo electodnamic suspension sstems. A method based on dnamic cicuit theo models a guidewa b esistances and inductances and the motion b a mutual inductance between the souce of magnetic field and the guidewa [5-8]. The mathematical equations ae then deived and solved using aplace method. This method of modeling cannot accuatel model the sstems with a continuous sheet guidewa. Kaaftmakhe has used a simplified fomula deived b Reitz fo calculation of lift and dag foces of PM pieces above a otating disc [9-]. oweve, these fomulas have some appoimation esulting in a eduction in accuac of computations. An analtical method based on Mawell equations has been pesented fo PEDS, assuming a sinusoidal cuent distibution located above the moving conductive bod []. oweve, this is not the case fo usual EDS sstems which use PM pieces o DC coils as the souce of magnetic field. Finite element method () as a poweful numeical method has been widel used to model diffeent tpes of electodnamic suspension sstems. Two dimensional (D) has been used to calculate life and dag foces in the PEDS [4]. A PEDS sstem including albach magnets and a conductive wheel has been successfull analzed b D and 3D []. The paametes affecting the pefomance of this sstem have been studied using a D stead-state [3]. Also a D comple stead-state fo calculating the lift and thust o beaking foces ceated in an electodnamic wheel [4]. A 3D model using a magnetic chage bounda has been poposed fo accuate and fast modeling of the electodnamic wheel [5]. Although the based models of PEDS sstems pesented so fa accuatel pedict the pefomances of the sstems, like most numeical methods, is time consuming and had to be used in iteative pocedues needed fo the sstem design and optimizations. Theefoe, an accuate and fast method fo modeling of PEDS sstems is to be woked out. This pape poposed a novel and accuate analtical method to model magnetic fields, edd cuents and foces in PEDS sstems. A PM piece is modeled b two equivalent cuent sheets at the ends of the PM. A guidewa is divided to seveal segments and the induced edd cuents ae calculated in each segment The magnetic fields due to the PM and edd cuents ae then calculated and used to compute lift and dag foces. This method is fast enough to be used in iteative sstem design and optimization pocedues. The esults obtain b the poposed method is finall compaed with the ones obtained b. The compaison confims validit and accuac of the poposed method. Sstem Stuctue The stuctue of a PEDS sstem consisting of a moving PM Piece of dimensions D K ove a conducting aluminum plate is shown in Fig.. The PM piece moves with a constant speed with espect to the aluminum sheet along the -ais diection as shown in the figue. An ai gap of g, along the -ais, eists between the PM piece and the plate. 354 PRZEGĄD EEKTROTECNICZNY (Electical Review), ISSN 33-97, R. 86 NR /

2 (6) M M Fig.. Stuctue of electodnamic suspension. Sstem Calculations A phsical model of the PEDS sstem with cetain assumptions and accuac is pesented fist. Assuming unifom magnetization of the PM piece, it can be eplaced b two cuent sheets at magnet ends as seen in Fig.. Fo the sake of simplicit, it is assumed that the sheets etend towads infinit in z diections and the calculation is caied out fo a unit length. The sheets ae located at and with thei cuents going inwad and outwad of the - plane espectivel. D Fig.. Model of pemanent magnet. The cuent densit of sheets is given b: () J ˆ sz Ma z whee, M is the magnetization of PM: M B m d P, ) ( () in which, B m and ae magnet esidual flu densit and vacuum pemeabilit espectivel. If the total cuent of a sheet is denoted b I, fo the coesponding magnetic field at a point P(, ) we have: I (3) whee (4) I M and is the distance between the cuent souce and point of P given as: (5) Theefoe, (3) can be e-witten as: Calculation of magnetic fields The -ais component of magnetic field can be calculated as follows: (7) Sin in which (8) Sin( ) Substituting (8) into (7) ields: M (9) It is noted that the total magnetic field at eve point is a esultant of the fields due to the cuent sheets at and. Theefoe, the -ais component of total magnetic field is obtained b integation (9) ove both cuent sheets as: () M d M d D D M D Tan Tan D Tan Tan Similal fo the -ais component of magnetic field we have: () Cos in which: () Cos( ) Substituting () into () ields: M (3) Again the total magnetic field at eve point is a esultant of the fields due to the cuent sheets at and. Theefoe, the -ais component of total magnetic field is obtained b integating (3) ove both cuent sheets as: (4) D M d D M d M og og 4 D D Equations () and (4) give the magnetic field components poduced b the PM piece in tems of PM dimensions and magnetization. The coesponding flu densit distibutions of these fields ae depicted fo a sstem with a PM of dimensions.5 4 cm in Figs. 3 and 4. PRZEGĄD EEKTROTECNICZNY (Electical Review), ISSN 33-97, R. 86 NR / 355

3 The shapes of flu densit distibutions of Figs. 3 and 4 can be eplained b the PM flu lines as depicted in Fig. 5. In this figue, the flu vectos, at diffeent points unde the PM along a staight path aligned to the -ais, ae decomposed to thei and components. Moving fom left to ight along this path, as shown in Fig. 5 the -component of magnetic flu vecto epeience a athe sinusoidal patten of change eaching a maimum at points 3 and a minimum at point 7 while vanishes at fa points at both sides. This is the same patten shown in Fig. 3. The -component of flu vecto can also be eplained the same wa. oweve, it epeiences a diffeent patten of change. Moving fom the left to the ight along the staight path, this component stats fom a negative value at point, vanishes at point 3 and eaches a maimum at point 5, ight unde the middle of the PM piece. Fig. 3. The distibution of -ais component of magnetic flu densit unde the PM. Fig. 4. The distibution of -ais component of magnetic flu densit unde the PM. Calculation of edd cuents The movement of PM above the aluminum plate poduces edd cuents in the plate accoding to Faada's law. As the PM speed inceases, the edd cuents amplitude inceases too. The modeling of edd cuents can be caied out b seveal methods; a ve common method will be ecalled hee b stating fom the ve basic elationship of: (5) E v B As the diection of movement is aligned with -ais we have (6) E v B v On the othe hand, the electomotive foce is given b [6]: J pz (7) Ez Fom (6) and (7) the cuent densit in the conductive plate is given b: (8) J pz V Now the conducting plate is divided to seveal segments and the cuent in each segment is calculated as follows: I J (9) ind pz The magnetic field due to the cuent of eve segment affects the adjacent segment. The value of the magnetic field at an adjacent segment is given b: Iind () Bind whee is the distance between two segments. Theefoe, the magnetic field in each segment is a sum of its oiginal magnetic field and the induced magnetic field as follows: () Btotal B Bind B epeating this pocedue fo all segments the esultant magnetic field is obtained. Calculation of foces The foce between PM and conductive plate is given b [6]: () F JBdV whee, J, B and dv ae the cuent densit vecto in the plate, magnetic flu densit vecto and plate volume espectivel. Assuming d as a plate thickness, the dag and lift foces ae calculated as follows: (3) FD F Jz d d (4) F F Jz d d Foce [N] ift Foce Dag Foce Fig. 5. flu lines aound the PM Speed [m/s] Fig. 6. Analticall calculated foces 356 PRZEGĄD EEKTROTECNICZNY (Electical Review), ISSN 33-97, R. 86 NR /

4 Substitution () and (4) into the ecent equations, the foces ae detemined. These foces ae shown in Fig. 6 fo the sstem unde stud. It is obseved that the dag foce inceases with an inceasing speed fist and eaches its maimum value at a cetain speed, then deceases gaduall as the edd cuents field pevails ove the PM field; wheeas, the lift foce keeps ising monotonicall with an inceasing speed. oweve, the slope of lift foce cuve deceases gaduall and the foce inceases towads a limiting value in high speeds due to the satuation of edd cuents. Evaluation b Finite Element Time stepping finite element analsis is emploed to evaluate the poposed analtical model. The simulation time is divided into man time steps. Setting the initial conditions, the desied paametes ae then obtained at the end of each time step and used as the initial conditions fo the net time step. This pocedue is epeated fo eve time step and gives all paametes duing the simulation time. The agange multiplie method incopoating sliding inteface is used to model the elative motion. Thee ae thee conventional methods fo foce calculation i.e. Mawell stess tenso, vitual wok and oentz method. The oentz method uses Ampee foce low accoding to which the foce applied to a conducto epeiencing a magnetic field is given b: (5) df J B whee J is the conducto cuent densit and B is the flu densit of etenal field. Theefoe, a foce vecto is computed fo each element and with summation of these foce vectos the total foce is achieved. In Mawell stess method, the nomal and tangential component of foce in each element is given b: (6) (7) Bn Bt dfn dl BnBt dft dl The total foce is then achieved b integating these foces on a specified path. The last method i.e. the vitual wok method calculates foces based on a local co-eneg vaiation: (8) F s W W S This latte method is selected in this pape due to its low sensitivit to mesh shape and its efinement. A flowchat of is shown in Fig. 7. Fig. 7. The algoithm of Now the accuac of the poposed analtical model is investigated with espect to the esults obtained fom the. The magnetic flu densit components unde the PM given b the analtical model and b the ae depicted in Figs. 8 and 9. It is seen that thee is a good ageement between the esults obtained fom the two the methods. Nevetheless, the -component of magnetic flu densit shows a small discepanc in the middle of the PM which is due to the modeling of the PM with two cuent sheets. oweve, the eo is limited to a easonable value. Magnetic Flu Densit (B) [T] MODE Displacment in X Diection [m] Fig. 8. Vaiation of Magnetic Flu Densit (B) [T] B along the PM Displacment in X Diection [m] Fig. 9. Vaiation of B along the PM MODE Also the foces calculated b the analtical method ae compaed with those obtained fom as shown in Fig. fo two diffeent ai gap lengths. It is seen that the esults obtained b the analtical method is ve close to those deived fom which demonstates the validit of the poposed model. It is also seen that the poposed model gives an ecellent accuac when the ai gap is not ve high. Theefoe, the above compaison well confims the validit of the poposed analtical method. Conclusion In this pape an analtical method is pesented fo the pefomance pediction and analsis of magnetic fields, edd cuents and developed foces in a passive electodnamic suspension sstem with a pemanent magnet piece. The pemanent magnet is modeled b two equivalent cuent sheets and the magnetic field components ae computed using Mawell equations. Closed fom fomula fo the field components ae given in tems of sstem paametes. The guidewa is divided into seveal segments and the edd cuents ae calculated in each segment. The esultant magnetic field components PRZEGĄD EEKTROTECNICZNY (Electical Review), ISSN 33-97, R. 86 NR / 357

5 due to the pemanent magnet and the edd cuents ae used to compute lift and dag foces. This method is fast enough to be used in iteative design and optimization pocedues. The esults obtain b the poposed method ae compaed with the ones obtained b a finite element method. Ecellent ageement between the two sets of esults confims the validit of the poposed method. Aigap=4.5 mm ift Foce (N) ift Foce (N) Dag Foce (N) Aigap=3 mm Aigap=4.5 mm Fig.. Compaison of foces obtained b the poposed analtical method and. REFERENCES []. W. ee, K. Kim and J. ee, Review of Maglev Tain Technologies, in: IEEE Tan. Magn., Vol. 4, (6), [] S. Ramtin, Analsis and pefomance impovement of the electodnamic suspension sstem using pemanent magnet, M.Sc. thesis, Depatment of Electical and Compute Eng., Univesit of Tehan, Ian, 5. [3].. Kolm and R. D. Thonton, Electomagnetic flight, in: Scientific Ameican, Vol. 9, (973), 7-5. [4] S. Vaez-Zadeh, S. Ramtin, Pefomance analsis of passive electodnamic suspension sstems, in: Poc. DIA, Kobe- (Awaji, Japan 5) [5] K. R. Dave, Electodnamic Maglev coil design and analsis, in: IEEE Tans. Magn., Vol. 33, (997) [6] T. Sakamoto, A.R. Eastham and G.E. Dawson, Induced cuents and foces fo the split-guidewa electodnamic levitation sstem, in: IEEE Tans. Magn., Vol. 7, (99) [7] K. Dave, Analsis of an electodnamic Maglev sstem, in: IEEE Tans. Magn., Vol. 35, (999) , 999. [8] J.. e, D. M. Rote, and. T. Coffe, Application of the dnamic cicuit theo to maglev suspension sstem, in: IEEE Tans. Magn., Vol. 9, (993) [9] J. R. Reitz, Foces on moving magnets due to edd cuents, in: J. Appl. Phs, vol. 4, (97) [] Y. Kaftmakhe, Maglev fo student, in: Euopean Jounal of Phsics, Vol. 9, (8) [] R. J. ill, Teaching electodnamic levitation theo, in: IEEE Tans. Education., Vol. 33, (99) [] J. Iniguez and V Raposo, aboato scale pototpe of a lowspeed electodnamic levitation sstem based on a albach magnet aa, in: Euopean Jounal of Phsics, Vol. 3, (9) , 9. [3] J. Bid and T. A. ipo, Chaacteistics of an electodnamic wheel using a -D stead-state model, in: IEEE Tans. Magn., Vol. 43, (7) [4] J. Bid and T. A. ipo, Calculating the Foces ceated b an electodnamic wheel using a -D stead-state finite-element method, in: IEEE Tans. Magn., Vol. 44, (8) [5] J. Bid and T. A. ipo, A 3-D magnetic chage finite-element model of an electodnamic wheel, in: IEEE Tans. Magn., Vol. 44, (8) [6]. Gholizad, M. Misalim and M. Mizaei, An impoved Equivalent magnetic cicuit netwok method fo consideation of motional edd cuent in a solid conducto, in: Amikabi Jounal, Vol. 7, (Fall/Winte 7) Dag Foce (N) Aigap=3 mm Autho: Abbas Najja-Khodabakhsh, Depatment of Electical Engineeing, Islamic Azad Univesit, Najafabad Banch, Najafabad, Esfahan, Ian. a.najja@ece.ut.ac.i, I was bon in Isfahan, Ian, in 98. I eceived a B.Sc. degee in electical engineeing fom Islamic Azad Univesit of Najaf Abad, Isfahan, Ian in 4 and an M.Sc. degee in electic powe engineeing fom the Univesit of Tehan, Tehan, Ian in 8. I am a PhD student now in electical engineeing at Islamic Azad Univesit Science and Reseach Campus. M eseach inteests include design, modeling and contol of levitation sstems and high pefomance motos PRZEGĄD EEKTROTECNICZNY (Electical Review), ISSN 33-97, R. 86 NR /