A Three-Dimensional Magnetic Force Solution Between Axially-Polarized Permanent-Magnet Cylinders for Different Magnetic Arrangements

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1 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields A Thee-Dimensional agnetic Foce Solution Between Axially-Polaied Pemanent-agnet Cylindes fo Diffeent agnetic Aangements Abdel-Kaim Daud Electical Engineeing Depatment Palestine Polytechnic Univesity (PPU) Hebon, Palestine daud@ppu.edu Abstact A thee-dimensional field solution is pesented fo axially polaied pemanent magnet cylindes. The field components ae expessed in tems of finite sums of elementay functions ae easily pogammable. They can be used to detemine the opeating point of ae-eath magnet cylindes. They ae also useful fo pefoming apid paametic calculations of field stength as a function of mateial popeties dimensions. The field components ae developed fo diffeent magnet aangements by taking into account the back ion. Also the method of images is used. Using the field equations, thee-dimensional analytical expessions ae deived fo computing the magnetic foce between axiallypolaied pemanent-magnet cylindes fo diffeent magnetic aangements. The field calculated esults ae in good ageement with the expeimental data. Keywods Analytical calculation; magnetic field; pemanent magnet; foce calculation I. INTRODUCTION Thee ae numeous devices that use axially polaied pemanent magnets (Fig. 1). Examples of these include steppe motos, axial-field pemanent-magnet motos, axial couplings linea P geneatos [5-9]. Usually, in these devices, one of the magnets is diven by an extenal while the othe is connected to a load. As the diven magnet moves it impacts a foce to the loaded magnet focing it to move once the loading dag inetial foces have been ovecome. To pedict the foce, vaious assumptions can be made that simplify the analysis. In this aticle, theedimensional analytical expessions ae deived fo computing the field the tansmitted foce. The analysis is based on the assumption of an ideal magnet that is chaacteied by fixed unifom polaiation. The solution method entails use of the vecto potential, ultimately leads to the numeical integation of the fee-space Geen s function ove two of its spatial vaiables. Consequently, the esulting field fomulae ae expessed as discete sums of elementay functions. One of the key featues of this wok is that the field fomulae can be eadily pogammed ae ideally suited fo apid paametic studies of field stength at any point outside the magnet [1-3]. The authos have used these fomulae to detemine the opeating point of existing ae-eath magnets, to detemine the optimum dimensions in ode to achieve a specified field pofile ove an extended spatial egion. The field fomulae ae developed fo diffeent magnet aangements (Fig. 2) with back ion by using of the method of images [4]. The theoy is demonstated with some sample calculations that ae veified with expeimental esults. Of inteest hee, fo foce calculation, only the axial field component of the diven magnet is consideed at axial distance equal to fom the loaded magnet (Fig. 1). The deived foce fomulae can be obtained fo diffeent magnet aangements in vetical motion. They ae applicable fo magnets made of ae eath mateials such as NdFeB. II. THEORY Thee ae numeous techniques fo computing the field due to pemanent magnets. We assume that the magnetic mateial is ideal, unifomly polaied thoughout, then model it as a distibution of equivalent cuents. The analysis stats with the magnetostatic field equations fo cuent fee egions: H = (1) B = (2) whee H is the magnetic field stength B is the magnetic flux density. In magnetic mateials, the two fields H B ae elated to the physical magnetiation, B = ( H + ) (3) It is well-known that the two fist-ode field equations educe to the second-ode equation 2 A = - (4) whee the vecto potential A is given by B = A (5) If the magnetiation is confined to a volume V falls abuptly to eo outside, the solution to (4) can be witten in the following integal fom: 5

2 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields whee S denotes the suface of the magnet J j ae equivalent volume suface cuent densities given by (6) J = (volume cuent density) (7) j = n (suface cuent density) (8) eithe paallel o antipaallel fo these sufaces. Thee ae two emaining sufaces to conside: inne suface = ' = R 11 ' 2p ' (1) ' ' (2) ' = R 12 (12) espectively [4]. Fo the poblem at h, it is assumed that the magnetiation is in the axial () diection, (9) whee the ± indicates the altenating polaity of adjacent poles. It follows fom (7) that the volume cuent density is eo J = =, (1), theefoe, (6) educes to (11) oute suface = R 12 R 11 ʹ(2) ' 2p ' (1) ' ' (2) P ʹ N Pʹ (13) agnet 2 ʹ(1) S N h 2 S Fig. 2. Axially polaied cylindical magnet ings The unit nomals fo these sufaces ae as follows: N (inne suface) = (14) (oute suface) S agnet 1, theefoe, the coesponding suface cuent densities ae given by (inne suface) j = (15) (oute suface) Fig. 1. Two axially polaied cylindical magnet ings The B field can be computed fom (11) (5). Fist, it is necessay to detemine the functional fom of j fo the vaious sufaces as shown in Fig. 2. Fom (8) (9) it follows that j = on the top bottom of the magnet as the magnetiation suface nomal ae Taking into account the esults of (12)-(15), (11) can be ewitten as follows (16) 51

3 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields Note that A () has no component. It can be witten in adial component A aimuthal component A : (17) (18) With (19) (2) It follows fom (16) that the components A A can be witten explicity as The integal in ' can be evaluated numeically. Fo this wok, Simpson s method was found to be adequate (Appendix). Application of this method to (32) gives an equation fo the axial field component, ( ( ( ) ) ( ) ) (27) (21) The emaining integations in ' ae evaluated with (5) of the Appendix. Use of this esult yields (22) Equations (21) (22) give the vecto potential of an axially polaied cylindical magnet; the field fo this magnet B() can be obtained using (5). Expessions fo the thee field components: adial component (B ), aimuthal component (B ), axial component (B ) ae deived in [1]. Fo these deivations, it is useful to intoduce the Geen s function notation which in cylindical coodinates (,,) educes to (23) [ ] (24) III. AXIAL FIELD COPONENT (B ) The axial component of the field follows fom (5), specifically, ( ( ) ) (25) Substitution of (21) (22) into (25) yields ( ( ( ) ) ( )) (28) whee I 1 is defined in Appendix [1]. Eq. (28) gives the axial field due to the entie magnet. IV. EFFECT OF BACK IRON Let us conside the magnetic cicuits shown in Fig. 3. The field in the gap egion can be modeled to fist ode by use of the method of images. Note that the field can be consideed to be a function of the spatial coodinates(,, ) the axial position of the magnet, i.e., ( ) (29) whee (1) (2) ae the positions of the bottom top of the magnet on the axis, espectively. Accoding to the magnetic cicuit shown in Fig. 3a, in which the magnet is in the ai gap egion, the positions (1) (2) ae given in the following equation (1) = -, (2) = (3) With consideing the mio laws of magnetostatic in Catesian coodinates unde the influence of one-sided back ion accoding to Fig.3b, eq.(3) can be witten as (1) = -2, (2) = (31) 52

4 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields While unde the influence of double-sided back ion accoding to Fig.3c, the field components in the ai gap can be expessed as a supeposition of the fields fom an infinite sum of image magnets ( ) whee the axial positions ae given by the following ecusive elations: (33) whee L is the ai gap length between magnet back ion. Fo the magnetic cicuits shown in Fig.3a, b c, the adial positions (1) = R 11 (2) = R 12 emain constant. The idea is to epesent one of the magnets as a distibution of equivalent cuents then to conside the field due to the othe magnet as the extenal field (which can be computed fom eq. (28)). With eqs. (7) (8), the foce on a magnetied body in the pesence of an extenal field is (35) With eq. (1), (35) educes to whee 2 is the magnetiation of the magnet 2 as shown in Fig. 1. Similaity to magnet1, thee ae two sufaces to the magnet 2 to conside a) inne suface = = R 21 2p (1) (2) = R 22 (37) b) oute suface = 2p (38) (1) (2) Substitution of (28), (37) (38) into (36) yields c) L [ ( ( ( ) ) ( ) ] Fig. 3. agnet Systems: a) agnet in ai gap egion b) One-sided back ion c) Double-sided back ion V. FORCE CALCULATION Eq. (39) can be simplified with Simpson s method, ( The foce between two magnets can be computed using the basic elation fo the foce on a distibution of cuent in an extenal field in axial diection (B ) (34) ( ( ) ) ( )) 53

5 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields The emaining integations in ae evaluated with (51) of the Appendix. Use of this esult yields ( ( ( ) ) TABLE 1. PARAETERS OF THE AGNETS agnet 1 agnet 2 1 = 93 ka/m 2 = 93 ka/m R11 = 6 mm R12 = 12.5 mm h1 = 16 mm R21 = 6 mm R22 = 12.5 mm h 2 = 8, 16 o 24mm = - 16 mm ( )) whee I 2 is defined in Appendix the positions (1) (2) ae given in the following equation (1) =, (2) = +h 2 (42) By taking into account the effect of back ion accoding to Fig. 4, the foce can be consideed to be a function of the spatial coodinates the axial positions of the two magnet, i.e., ( ) (43) a) b) h 2 h 2 Accoding to the magnetic cicuit shown in Fig. 4a Fig.4b, the positions (1) (2) ae given in (3) (31), espectively. While unde the influence of double-sided back ion accoding to Fig.4c, the foce equation can be expessed as c) ( ) whee the axial positions ae given by the following ecusive elations (45) whee is the ai gap length between the two magnets. VI. RESULTS The theoy was applied to an axially polaied cylinde type NdFeB magnets fo the diffeent magnet systems shown in Figs. 3 4 veified expeimentally. The paametes of the magnets used in the analysis ae illustated in Table 1. Compute pogams ae developed to calculate the magnetic flux density fo the diffeent magnet aangements. Fo the field calculation as function of axial position, the magnet is oiented symmetically with espect to the x-y plane (i.e = coesponds to the middle of the cylinde). h 2 Fig. 4. agnet Systems fo foce calculation a) agnet in ai gap egion b) One-sided back ion c) Double-sided back ion The field values of B wee computed fo diffeent adial values diffeent magnet aangements in Fig. 3. These esults ae shown in figs. 5 a-c. 54

6 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields Axial Field B (T) Axial Field B (T) Axial Field B (T) Axial Position (mm) (a) (b) (c) = 14 mm = 15 mm = 16 mm = 14 mm = 15 mm = 16 mm Axial Position (mm) = 14 mm = 15 mm = 16 mm Axial Position (mm) Fig. 5. Axial field vs. axial position with adial position as paamete = fo the magnet aangements of Fig. 3: a) Fig. 3a; b) Fig. 3b; c) Fig. 3c Axial Field B (T) Axial Field B (T) Axial Field B (T) Radial Position (mm) (a) (b) = 1 mm = 2 mm = 4 mm measued values (c) Fig. 6. Axial field vs. adial position with axial position as paamete = fo the magnet aangements of Fig. 3: a)fig. 3a; b) Fig. 3b; c) Fig. 3c Radial Position (mm) = 1 mm = 2 mm = 4 mm measued values Radial Position (mm) = 1 mm = 2 mm = 4 mm measued values 55

7 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields Fo the field calculation of B as function of adial position, the magnet is oiented with uppe suface in x-y plane ( = ). The field components B wee computed along adial line above the suface of the magnet ( = 1, 2 4mm, = ). The data fo these calculations appea in figs. 6 a-c). The calculated values ae compaed with expeimental esults fo some cuves as shown in fig. 6 a-c) at = 1 4 mm. The magnetic flux density has been measued by using GAUSS ETER Type 3251 (YOKOGAWA) with its pobe. In geneal, the measued values esults show good ageement with the theoetical esults. Note that in diffeent cases, the computed field values ae slightly highe than the coesponding measued data. The eason fo this diffeence is that the assumption of infinite ion back fo some cases does not take into account the finging (leakage) flux. Nevetheless, the pedicted data ae sufficiently accuate fo paametic design optimiation. The foce values wee obtained fo a seies of sepaation distances which wee on the ode of a few millimetes diffeent magnet height h 2. These values wee obtained fo diffeent magnet aangements of Fig. 4 illustated in Figs. 7,8 9. Fig. 9. agnetic foce vs. accoding to Fig. 4c with h 2 as paamete It is impotant to note that when pogamming the multipole summations in field foce equations, a highe degee of accuacy can be achieved with fewe mesh points if the total of a single summation ove one index is evaluated saved sepaately (as an intemediate step), then the totals of the successive summations ae added togethe to obtain the final sum. VII. CONCLUSION Fig. 7. agnetic foce vs. accoding to Fig. 4a with h 2 as paamete A thee dimensional field solution has been deived fo axially polaied magnetic cylinde fo diffeent magnet aangements. The calculation is based on the assumption of ideal magnetiation is applicable to ae eath mateial such as NdFeB. The expessions fo the field components have been expessed in tems of elementay functions that ae eadily pogammed. oeove, they allow apid paametic studies of field stength could be of consideable use in the design optimiation of numeous devices. The computation esults fo the flux density ae in good ageement with the expeimental data fo diffeent magnet aangements. VIII. APPENDIX In this section, the vaiables '(m), S (m) I 1 that appea in (27) (28) the vaiables (m'), S' ( m') I 2 that appea in (4) (41) ae defined. Specifically = 2pm /N (46), = 2p / (47), Fig. 8. agnetic foce vs. accoding to Fig. 4b with h 2 as paamete S (m) = 1/3 (m=) 4/3 (m=1,3,5, ) 2/3 (m=2,4,6, ) 1/3 (m=n ) (48), 56

8 Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields S θ (m ) = 1/3 (m =) 4/3 (m =1,3,5, ) 2/3 (m =2,4,6, ) 1/3 (m =N θ ) (49) [9] H. B. Etan,. Y. Üctug, R. Colye, A. Consoli, oden electical dives, Kluwe Academic Publishes, 2. [1] A.-K. Daud,, R. Hanitsch, A thee-dimensional field solution fo axially-polaied pemanent-magnet cylindes fo diffeent magnetic aangements, Electomotion, Januay-ach 28, Issue 1,Volume 15, pp Lastly ( ) = ( ) ( ) ( ) fo ( ( ) ) = (5) ( ) fo ( ( ) ) ( ) = ( ) ( ) fo ( ( ) ) = (51) ( ) fo ( ( ) ) REFERENCES [1] E. P. Fulani, A thee-dimensional field solution fo axially polaied multipole disks, J. agn. agn. ate., vol. 135, pp , [2] E. P. Fulani, A thee-dimensional field solution fo pemanentmagnet axial-field motos, IEEE Tans. agn., vol. 33, no. 3, pp , ay [3] Y. N. Zhilichev, Calculation of 3D magnetic field of disk-type micomotos by integal tansfomation method, IEEE Tans. agn., vol. 32, no. 1, pp , Januay [4] D. K. Cheng, Field wave electomagnetic, Addison-Wesley Publishing Company, [5] Y. N. Zhilichev, Thee-dimensional analytic model of pemanent magnet axial flux machine, IEEE Tans. agn., vol. 34, no. 6, pp , Novembe [6] D. A. Gonále, Tapia, J. A. Bettancount, A. L., Design consideation to educe cogging toque in axial flux pemanentmachines, IEEE Tans. agn., vol. 43, no. 8, pp , August 27. [7] J. Aoui, G. Baakat, B. Dakyo, Quasi-3-D analytical modeling of the magnetic field of an axial flux pemanent-magnet synchonous machine, IEEE Tans. Enegy Conves., vol. 2, no. 4, pp , Decembe 25. [8] T.J.E. ille, Bushless pemanent-magnet eluctance moto dives, Claendon Pess, Oxfod,