Nd-Fe-B permanent magnets for passive and hybrid magneti bearings with extended appliation temperature range Mihael Weikhmann, Matthias Katter, Bernd Shleede VACUUMSCHMELZE GmbH & Co. KG, Hanau, Germany Index Terms VACODYM R,VACOMAX R magneti Bearing, Surfae harge model, amperien urrent density, oulombian representation, FEA I. CATEGORIES Digest Categories: Magneti bearing Abstrat Passive magneti bearings or hybrid passive magneti bearings are ommonly built up with Sm-Co or with Nd-Fe-B magnets. One of the most important item of those magnets is it s homogeneity in flux density distribution and the preise angularity. A miss-math would ause an undesired displaement. It may therefore be very important to measure these homogeneity s and angular deviation very preisely. Some appliation all for a higher temperature, thus a need of higher remanene B r is inevitable and in some ases the temperature oeffiient T k of remanene B r is also of importane. This may be ahieved by applying the Grain Boundary Diffusion GBD. In the following paper we will give an overview of the urrent development to ensure this future demand. A variation of parameter studies will give an overview of the main influening fators whih at on the bearing fores and stiffness. II. INTRODUCTION Turbo moleular pumps for example are equipped with passive magneti bearings on the high vauum side. The passive magneti bearing are to stabilise the turbine shaft radially, while the axial stabilisation is ahieved by an ative (eletri) magneti bearing or a mehanial bearing. Whenever a parametri analysis of magneti bearings is onduted, the analytial approah by simulating the magnet with surfae harge modules is very onvenient, sine its results may be very rapidly obtained. The programs used in this field are ommonly written in Matlab R /Silab R or in Mathematia R. Of interest are the fores F z obtained in a bearing and its stiffness K z versus displaement. In this paper, we will give a brief introdution of these analytial methods and we will ompare those results with Finite Element Analysis FEA. At the same time we will try to show the realisti influenes of the variation of dimensions, remanene and the magneti orientation onto the fores and the stiffness of a magneti bearing. III. ANALYTICAL APPROACH OF AXIALLY ORIENTED MAGNET RINGS FOR MAGNETIC BEARINGS In general let us first try to simulate a magneti ring by using two surfae harged oils whih may substitute a ring magnet refer to eqn.() and eqn.() as used in [] and []. Figure. Two axially oriented magnet rings as an axial bearing. F z = J J µ f z (r i, r j ) = r i r j z4 z 3 F z = J J µ i= j=3 4 ( ) +i+j {f z (r i, r j )} () z π z i,k= j,l=3 with eqn.() inserted into eqn.() ( Z Z)os(Θ)dzdzdΘ (r i + r j r ir j os(θ) + (z z) ) 3 () 4 ( ) +i+j+k+l F i,j,k,l (3) F i,j,k,l = r i r j g(z k z l, r i + r j + (z k z l ), r i r j ) (4) With A and S being: with A = a b π + g(a, b, ) = A + S (5) (a b) (a + a ) (6) 6 a = log[ ( (a b) ) 3 ] (7)
and a = log[ ( (a b) ) 3 ] (8) S = S + S + S 3 (9) a S = b + ((b + )E[Φ, m ] F [Φ, m ]) () In eqn.() Φ = arsin[ b+ b ] In eqn.() m = arsin[ b b+ ] S = In eqn.() m = b b+ In eqn.() ɛ = b In eqn.() µ = a b + ɛ ( E[m ] µk[m ]) () µ b+ S 3 = ɛβ ((b a )K[µ] + (a b + )Π[ b + a, µ]) () In eqn.() µ = b+, β = b+ b and ɛ = b These analytial alulation and formulae of g(a, b, ) in eqn. ( to ) ontain elliptial integrals K[m], F [Φ, m], E[Φ, m], E[m] and Π[m, n], Π[n, Φ, m] as desribed in the following equations (3) to (8): K[m] = π Where m in eqn.(3) is valid for <=. dφ (3) msin(φ) with α = (r i r j ) + (z k z l ) and m 4 = 4rirj α. And m in eqn.(4) to (8) and n eqn.(7) and eqn.(8) are valid for m, n <= respetively. As the magnet ring is substituted by two surfae oils on the inner and outer irumferene, the magneti orientation is fixed to the axial diretion, thus any deviation of orientation may not be alulated. This alls for a different approah, whih may be found in [3]. In this paper the fous is on a dipole method and gives a D representation of permanent magnet bearings. The results differ slightly from the fully analytial approah given by [] as it is an abbreviation. F z = p J J πµ R 3 S S sin(β + β 3Θ) () For the axial fores versus axial displaement F z is alulated and simulated in eqn. (). K z = K r = p J J πµ 6 R 4 S S os(β + β 4Θ) () In eqn. one may observe, that the radial stiffness is half the axial stiffness, i.e. K r = Kz. This is the reason, why in this paper we onentrate on alulating the axial stiffness K z only. ross setion of an outer ring F [Φ, m] = Φ dφ (4) msin(φ) E[Φ, m] = Φ msin(φ) dφ (5) ross setion of an inner ring E[m] = π msin(φ) dφ (6) Π[n, m] = Π[n, π, m] (7) Φ Π[n, Φ, m] = dφ nsin(φ) msin(φ) (8) The exat analytial formulation of K z as in [] an be expressed in the following equations. where K z = J J µ i,k= k,l=3 4 ( ) +i+j+k+l C i,j,k,l (9) C i,j,k,l = αe[m 4 ] r + r j + (z kz l ) α K[m 4 ] () Figure. D Rings The bearing stiffness K z versus the axial displaement is alulated in eqn. (). The advantage of this approah is, that it allows a deviation of orientation, but only to the extent, that the orientation is uniform over the entire ring. This uniform orientation may give a diverging or onverging flux density above or below the rings pole surfae, whih is also know as hot - Side / old - Side effet [4]. As stated in the paper of Bekinal et. al. [3] there will be a slight differene in results ompared to the Yonnet et. al. publiations [5] and
3 [], thus a omparison with D FEA alulations is therefore useful. A single angular deviation in one single diretion super positioned with the normal axial diretion, e.g. X-diretion or Y-diretion is not possible to simulate using these analytial formulae. This may be done by super positioning a diametrial ring. An analytial approah needs to be elaborated but is not the sope of this paper, thus to obtain any results for this kind of onstellation, it would be neessary to alulate those bearings with 3D FEA. IV. PARAMETER STUDY OF INFLUENCING FACTOR ON BEARING FORCES AND STIFFNESS All analytial alulations as done by e.g. [5] [] [6] [7] [8] [9] [] [] [] are a very fast and onvenient approah to render results, but do not reflet proessing varianes of remanene, dimension and orientation deviation. We therefore have undertaken a parameter study to find the most signifiant influening fators whih at on the bearing fores and stiffness. As a referene we may onsider a passive magneti bearing with the following dimension and magneti properties, homogeneity and orientation deviation (by angular deviation ausing a so alled Hot and Cold Side of eah magneti ring. If one would vary the outer and inner ring magnet of an axial bearing by a parameter study, one would need to vary the dimension of the inner ring denoted with suffix D outer, D inner,height, Br and β and the same for the outer ring denoted by suffix D outer, D inner,height, Br and β aordingly and keeping eah of the five parameter on three levels, i.e. minimal, nominal and maximal. This would need 3 = 5949 ombinations. To run a alulation of so many variations would be a rather endless undertaking. We therefore try to redue the variation by ombining the rings volume and remanene to its magneti moment, i.e. M mag = V ol Br for the inner ring and M mag = V ol Br for the outer ring. By doing so we now redue the number of ombinations, i.e. M mag and β for the inner ring and M mag and β for the outer ring, to 3 4 = 8 ombinations. Nominal LSL USL Magneti Moment M mag inner Ring IR [nvsm] 9.44 8.94. Magneti Moment M mag outer Ring OR [nvsm] 6.965 6.3 7.74 Angle of Orientation β of IR [deg] 88.5 9 9.5 Angle of Orientation β of OR [deg] 88.5 9 9.5 Table II DIMENSION AND PROPERTIES OF OUTER RING MAGNET BEARING V. RESULT OF PARAMETER STUDY The analytial result were ompared with D FEA results to see how good these alulation fit. Figure 3. Fore Fz vs. axial displaement D FEA and analytial alulation It an be seen in fig. 3 the analytial results (red line) fit well. Only the maximum and the minimum differ slightly, the D FEA results (blue dots) show a higher maximum and lower minimum respetively. Nominal LSL USL Outer Ring outer Dia. [mm] 64. 63.98 64. Outer Ring inner Dia. [mm] 44. 43.98 44. Inner Ring outer Dia. [mm] 4. 39.98 4. Inner Ring inner Dia. [mm]. 9.98. Height [mm]. 9.99. Remanene Br [T]..97.3 Angle of Orientation [deg] 9 88.5 9.5 Table I DIMENSION AND PROPERTIES OF INNER RING MAGNET BEARING Tab. I gives an overview of the parametri variation of dimension, magneti properties and deviation of orientation, whih then is redued to the magneti moment and the deviation of orientation as shown in Tab. II. This approah simplifies the number of variation signifiantly. http://www.femm.info/wiki/homepage http://www.ansys.om/produts/simulation+tehnology/systems+&+ Multiphysis/Multiphysis+Enabled+Produts/ANSYS+Maxwell Figure 4. Stiffness Kz vs. axial displaement D FEA and analytial alulation As far as the stiffness is onerned, the D FEA results differ slightly from the analytial results, espeially with the maxima s fig. 4 as desribed by Bekinal [3]. First let us look at the influene of varying the magneti moment and keeping the orientation onstant at 9[deg]. The fores are symmetrial over the displaement, whih an be seen in fig. 5, the maximum in magneti moment
4 Now onsidering the same analysis by varying the angle of orientation, thus, the so alled hot - side old - side effet by varying the angle from 9[deg] + / -.5[deg] symmetrial keeping the magneti moment onstant at the nominal value similar plots an be analysed. Figure 5. Fore Fz vs. axial displaement analytial alulation with orientation 9 [deg] for various magneti moments produes the highest fore where the lowest fore is at 98[N] (blak line) and the highest 8[N] (dark blue line), thus produing a differene of [N]. It also shows, that the smallest inner magneti moment ombined with the highest outer magneti moment (green line) renders a higher fore maximum than the highest inner magneti moment ombined with the lowest outer magneti moment (light blue line). All in all the maximum fore depend very strongly on the magneti moment only, so the higher the magneti moment of both rings is, the higher the maximum fore. Figure 6. Stiffness Kz vs. axial displaement analytial alulation with orientation 9 [deg] for various magneti moments Considering the same with the stiffness Plot fig. 6 a disloation of the inner magnet ring from the symmetry either in upward or downward displaement with varying magneti moment renders a differene from 4.[ N N mm ] to 4.7[ mm ] in stiffness. Sine, aording to eqn., the radial stiffness K r = Kz with the variation of the magneti moments by a total of appr. [%] yields a variation of the radial bearing stiffness by appr. 8[%]. The lowest magneti moment in both rings render the lowest stiffness Kz (blak line). The highest magneti moment renders the highest stiffness Kz (dark blue line). The ombination of the lowest inner magneti moment with the highest outer magneti moment (green line) renders a higher stiffness than the ombination of the highest inner magneti moment with the lowest outer magneti moment (light blue line). Again the Stiffness is only orrelated with the variane of magneti moment. It is important to say, that the respetive maxima are symmetrial, thus the fores and stiffness s are the same regardless the displaement in the positive or negative diretion from the line of symmetry, where the inner ring is within the outer ring. Figure 7. Fore Fz vs. axial displaement analytial alulation with varying angle of orientation When the angle of orientation of the inner and the outer ring are at 88.5 deg (both rings have a diverging field density distribution), the fores shown in fig. 7 are not symmetrial, thus moving the inner ring from the negative to the positive displaement in z-diretion on the negative side the fores are less than on the positive side (blak line). When the angles of orientation of both rings are at 9.5 deg, thus onverging field distribution, the fores are also not symmetrial and on the negative side are higher than on the positive side of the displaement in z-diretion (dark blue line). Combining a diverging inner ring (angle 88.5 deg) with a onverging outer ring (angle 9.5 deg) renders a symmetrial fore (green line). Where as ombining a onverging inner ring (angle 9.5 deg) with a diverging outer ring (angle 88.5 deg) also renders a symmetrial fore and both maxima are on the same level (light blue line). The light blue line overs identially the green line. Figure 8. Stiffness Kz vs. axial displaement analytial alulation with varying angle of orientation As expeted from the fores, the stiffness refer to fig. 8 show the same, if both are diverging rings (i.e. angle 88.5 deg) they render on the negative side of the displaement a lower and on the positive side of the displaement a higher stiffness (blak line). If both rings are onverging (angle 9.5 deg) then the stiffness on the negative side is higher than on the positive side (dark blue line). If the inner ring is diverging and the outer ring onverging, then the stiffness is again symmetrial
5 (green line). If the inner inner ring is onverging and the outer ring diverging, then the stiffness is on the same level as later. Figure 9. Fore Fz vs. axial displaement analytial alulation with varying angle of orientation The above desribed influenes an be analysed in more detail by magnifying fig. 9 in the viinity of the origin. In general, one may dedue, that a unsymmetrial foreor stiffness- distribution may be neutralised by ombining a hot side inner ring with a old side outer ring or vie verse. When two diverging or two onverging magneti rings are ombined within a bearing, a so alled parasiti axial fore at z =. is +/ 8[N] whih an be observed by extrapolating to z =.. These parasit fores may be neutralised by shifting one of the rings in axial diretion by +/.[mm] refer 9. So it is important to measure these effet very preisely. This phenomenon was also patented by VACUUMSCHMELZE GmbH & Co. KG (VAC) 3. Figure. Main Effet angle ating on Fz at z =.. Mean taken over all 8 parameter sets VI. OUTLOOK Most magneti baerings are urrently equipped with Sm- Co magnets. But resently even Nd-Fe-B magnets are being onsidered as an alternative solution. With a new family of grades and tehnologies, one may be able to inrease the share of Nd-Fe-B magnets in magneti bearings, when ever a higher environmental temperature, thus oerivity is needed. Figure. Main Effet magneti moment ating on stiffness Kzmin. Mean taken over all 8 parameter sets Analysing the 8 ombinations possible, one may dedue, that the magneti moment is a so alled main effet, whih ats on the maximum and minimum Fore and Stiffness refer to fig.. Where as the symmetrial or unsymmetrial distribution of fores or Stiffness ause a so alled parasiti axial fore and only depends on the angle of orientation fig. or Hot side and Cold Side of the ring magnets and it may be neutralised by a sorted and paired ombination of rings. 3 Patentshrift DE 3 95 B4 8.. Figure. The inrease in magneti properties with diffusion treatment. A proposal to meet these demands is to use Grain Boundary Diffusion (GBD) magnets as in fig.. By implementing this
6 tehnology the oerivity may be inreased by > 4 ka m and keeping the remanene onstant. As shown in fig by the blue line, whih represents a typial demagnetisation urve of transversal pressed VACODYM 3 TP, while the blak lines shows the inrease of oerivity by applying Dy/Tb Grain Boundary Diffusion. By developing new magnet grades using this advaned proessing step as desribed, VACUUM- SCHMELZE (VAC) has not only met and fulfilled the demand from the market to redue the heavy RE ontent of Nd- Fe-B magnets, but also signifiantly improved the magneti properties. Thus, by reduing the heavy RE ontent, the seurity of supply of RE elements may be enhaned. These additional proess steps of applying RE elements and exposing the magnet to 8 [degc] - [degc] do have an influene on the prodution osts. The overall result may not neessarily be ost neutral, depending of ourse on the level of the raw material pries. However, suh a tehnique demonstrates that new grades an be tailored to meet speifi ustomer and appliation demands, either in the form of higher H J or B r values, enabling preise optimisation of the magnet grade to the working onditions of the ustomers appliation. [] V.Lemarqund R.Ravaud, G.Lemarquand. Fore and stiffness of passive magneti bearings using permanent magnets. part : Radial magnetization. IEEETransations on Magnetis, 45,9:3334 334, 9. [] G.Lemarquand V. Lemarquand. Passive permanent magnet bearing for rotating shaft : Analytial alulation. In Magneti Bearings, Theory and Appliation,. [] Anthony Zander W. Robertson, Ben Cazzolato. A simplified fore equation for oaxial ylindrial magnets and thin oils. Shool of Mehanial Engineering, University of Adelaide, SA 55, Australia, pages 5. VII. CONCLUSION In this paper, we tried to find an analytial method, with whih we are able to find the main influening fators whih at on a axial magneti bearing fores and stiffness. By arrying out a parametri variation, we were able to find that the fores and stiffness not only depend on the mutual magneti moments of the inner and outer magneti ring, but also on their mutual so alled Hot Side or Cold Side effet, whih was represented here by the angular orientation. The approahes by Yonnet et. al. [5] [] through [] [] ould not be used, beause the analytial alulation does not take into aount, that a diverging or onverging orientation of a ring is possible. Only with the work of Bekinal et. al. [3] it is possible to onsider this effet. We also were able to show that a paired ombination of diverging and onverging rings may redue the unsymmetrial fore and stiffness distribution of a magneti bearing. REFERENCES [] V.Lemarquand G.Lemarquand. Passive permanent magnet bearings for rotating shaft : Analytial alulation. Inteh,roatia, ISBN 978-953- 37-48-:3,. [] V. Lemarquand R.Ravaud, G.Lemarquand. Fore and stiffness of passive magneti bearings using permanent magnets. part : Axial magnetization. IEEE Transations of Magnetis, 45,7:996, 9. [3] S. Jana M.I. Peerzade S.I. Bekinal, T.R. Anil. Optimization of halbah magnetized permanent magnet axial bearing. IJEIT,,:5,. [4] M. Weikhmann. Nutzung des Nord- / Sd- Effektes bei permanenterregten Mashinen. FEMAG Anwendertreffen, Offenbah,. [5] J.P. Yonnet. Permanent magnet bearing and ouplings. IEEE Transation on Magnetis, 7:69 73, 98. [6] R. Ravaud G.Lemarquand. Halbah struture for permanent magnets bearing. Progress In Eletromagnetis Researh M, 4:63 77,. [7] R. Ravaud G.Lemarquand. Magneti field produ ed by a parallelepipedi magnet of various and uniform polarization. Progress In Eletromagnetis Researh PIER, 98:7 9, 9. [8] G.Lemarquand R.Ravaud. Halbah strutures for permanent magnets bearings. Progress In Eletromagnets Researh M, 4:63 77,. [9] V.Lemarqund R.Ravaud, G.Lemarquand. Fore and stiffness of passive magneti bearings using permanent magnets. part : Radial magnetization. IEEETransations on Magnetis, 45,9:3334 334, 9.