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Actuators with permanent magnets having variable in space orientation of magnetization
Sensors and Actuators A 101 (2002) 168±174
Actuators with permanent magnets having variable in space orientation of magnetization Arvi Kruusing* Tallinn Technical University, Department of Mechatronics, Ehitajate tee 5, EE-19086 Tallinn, Estonia Received 8 January 2002; received in revised form 24 April 2002; accepted 29 April 2002
Abstract The force between a permanent magnet and a coil can be increased if the magnetization orientation of the magnet is suitably varying in space. The force is greatest if the magnet is magnetized along coil ®eld lines (`optimal' magnet). In this article, is studied how the optimisation effect depends on the magnet proportions and coil-magnet distance in case of cylindrical magnets and coaxial planar coils. The dependence of the force on the magnet height and coil-magnet distance for differently magnetized magnets (axial, radial, optimal) is presented. Magnetization of the magnet in the direction of the coil's ®eld can provide a more than two-fold force compared to axial magnetization, especially at thin magnets and small coil-magnet distances. The measured force values differed from the calculated values maximum 2% for a milli-size radial±axial composite magnet. # 2002 Elsevier Science B.V. All rights reserved. PACS: 41.20.Gz; 85.70.Ay Keywords: Magnetic actuator; Permanent magnet; Varying magnetization orientation; Force calculation
1. Introduction In this article, electromagnetic linear actuators, comprising only a permanent magnet and a coil (without soft magnetic parts) are considered. Such actuators have been preferably used in miniature and microsystems [1±11]; see Appendix A. In all cases referred to in the table axisymmetric planar coils in conjunction with co-axial permanent magnets (PM) were used. The coils used were either square of circular and magnets either cylindrical or square prismatic. In some cases [3,4,8] two-layer coils were used to increase the number of turns and to solve simultaneously the problem of inner current feed. Because the magnetic ®eld is divergent in such actuators, the generated force is much smaller than in actuators having closed magnetic circuit. Therefore a careful design is needed. In [8] an optimisation of the dimensions of a planar coilÐaxially magnetized PM actuator was undertaken. In my previous article [12], it was shown that the actuator force could be further increased if magnet's magnetization orientation varies properly in space. *
Present address: University of Oulu, Microelectronics and Materials Physics Laboratories, P.O. Box 4500, FIN-90014 Oulu, Finland. E-mail address: [email protected] (A. Kruusing).
The purpose of this work was to study how much the actuator force can be increased using varying orientation of magnetization and how the force increase effect (further referred as `optimisation effect') depends on the magnet height and on the coil-magnet distance. 2. Calculation of the force between a cylindrical permanent magnet and a coaxial circular flat coil The calculation of the most signi®cant output parameter of the coil-PM actuators, i.e. the force, has been a problem up till the present time. A straight forward summary of the forces between the length elements of the conductor (or volume elements of the coil in assumption of continuous current distribution) and the volume elements of the magnet requires in general the calculation of up to six-fold integrals (three-fold in the case of a cylindrically symmetric system with a zero-thickness coil), in notations of [8]: Z Fz Br rz Hz ~ rd3 r; (1) VM
where
Z 1 ~ r 0 3 0 J ~ r 0 r ~ r d r ; Hz ~ 4p VC j~ r ~ r 0 j3
0924-4247/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 2 ) 0 0 1 5 4 - 1
z
(2)
A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
where Fz is the axial force between the magnet and the coil, Br the remanent magnetic induction ®eld of the magnet, VM the volume of the magnet, Hz the axial component of the magnetic ®eld strength of the coil, VC the volume of the coil, and J(r) the current density in the coil as function of the vector coordinate. The advantage of this approach is that it can be applied also for magnets and coils of low symmetry and for nonaxial coil-magnet systems. With the ®eld of the coil Hz given, the force can be calculated as Z @Hz dV; (3) Fz Mz @z where Mz is the magnetization of the magnet [4,5]. To simplify the calculations, in paper [11] only the ®eld on the symmetry axis was considered and the force was calculated as Z @Hz 0; z dz; (4) Fz Mz @z The formulae for cylindrical coils ®eld may be found for example in [13,14]. With the ®eld of the magnet B given, the force was calculated as [1,2], Fz NSi
dB ; dz
or N Fz 2p i S
(5)
Fig. 1. A cylindrical magnet in the field of a planar coaxial circular coil. Orientation of the magnetization in the direction of the coil's field provides the greatest force (`optimal' magnet). The arrows show the magnetization orientation in the magnet's cross-section.
Brad i; a; r; z
rBr dS;
(6)
where N is the number of the turns, i the current, S the crosssectional area of the coil, and Br the radial component of the magnetic induction in an element of the coil. In all of the aforementioned cases the magnets were magnetized in the symmetry axis direction. Force between a coil and a permanent magnet has also been calculated by the ®nite element method (FEM), using ANSYS and ANSOFT software [10,11]. This method is the best choice when the coil and/or the magnet have low symmetry. For highly symmetrical systems, analytical solutions usually outperform FEM. In the previous article [12], explicit formulae for calculation of force between an arbitrary magnetized cylindrical permanent magnet and a coaxial cylindrical coil were developed. The essentials of the theory and the results for axial, radial and `optimal' magnets are as follows. The system under consideration is shown in Fig. 1. The coil's ®eld is assumed to be a superposition of ring current ®elds m i 1 Baks i; a; r; z 0 2 2p a r z2 1=2 " # a2 r 2 z 2 E k K k ; (7) a r2 z2
m0 i z 1 2 2p r a r z2 1=2 " a2 r 2 z2 E k a r2 z2
# K k ;
(8)
where Baks and Brad are the axial and radial components of the magnetic induction, E(k) and K(k) are elliptic integrals of the argument k
Z
169
s 4ar a r2 z2
(9)
and i is the current in the ring [15,16]. Considering a current distributed evenly in the coil's cross-section, the total force between the coil and the magnet can be calculated by integrating B(a, r, z)M(r, z) (where M is the magnetization; M Br /m0) over the top and bottom surfaces of the magnet and the coil cross-section [12]. Speci®cally, for a zero-thickness circular coil and coaxial cylindrical PM, in three special cases of magnetization (Figs. 1 and 2) simple expressions for the force were found: (a) axially magnetized magnet: (Z Z a r 2p F Br Baks j; x; r; drdxdr m0 0 0 ) Z aZ r Baks j; x; r; d hrdxdr ; 0
0
(10)
Fig. 2. Axially and radially magnetized magnets. Sectional views.
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A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
(b) radially magnetized magnet: (Z Z a r 2p F Br Brad j; x; r; drdxdr m0 0 0 ) Z aZ r Brad j; x; r; d hrdxdr ; 0
0
(11)
(c) magnetization orientation in the coil's field direction (`optimal' magnetization): (Z Z a r 2p F Br B2rad j; x; r; d m0 0 0 B2aks j; x; r; d1=2 rdxdr Z aZ r B2rad j; x; r; d h 0
0
B2aks j; x; r; d
h
1=2
)
rdxdr ;
(12)
Fig. 3. Dependence of the force between a circular planar (zero-thickness) coil and a coaxial cylindrical magnet on the coil-magnet distance d for differently magnetized magnets (Figs. 1 and 2). For notations see Fig. 1. Optimal magnetization means the magnetization in the coil's field direction. Note that for the `optimal' curve, the magnetization of the magnet varies in dependence on the coil-magnet distance d.
where j is the density of the current of the coil and Br is the remanent induction of the magnet. The latter case (case c) will be referred as `optimal' magnetization, because it provides the greatest force between coil and magnet [12]. The formulae (10-12) express the integrals of forces between elementary current rings of the coil and elementary magnet pole rings on the top and bottom of the magnet. If the magnetization on the surface of the magnet is at angle to the symmetry axis, the force can be calculated applying the formulae (10) and (11) both to the axial and the radial components of the magnetization and summing the results. The Eqs. (10)±(12) represent mathematically exact expressions for the force in frames of the assumptions taken. The integration multiplicity of Eqs. (10)±(12) is smaller than of Eqs. (1) and (2), because instead of integrating over the volume of the magnets, in Eqs. (10)±(12) only over the top and bottom surfaces of the magnet is integrated. Of course, the Eqs. (10)±(12) apply only for a rotationally symmetric system with planar coil and ¯at top/¯at bottom magnet. Eqs. (10)±(12) can be easily generalized for the case of a ®nite height coil.
Fig. 4. Dependence of the force between a circular planar coil and a cylindrical magnet on the magnet height h for differently magnetized magnets.
The dependence of the force between the coil and the magnet on the coil-magnet distance for differently magnetized magnets is presented in the Fig. 3. It should be remembered that, in the case of optimal magnetization, each
3. Comparison of actuation force of differently magnetized magnets For comparison of the forces, a coil-magnet system of same dimensions as in [8] was assumed. The force was calculated by Eqs. (10)±(12) using Microsoft Excel 97 (see Appendix B). The results were compared with calculations by Mathematica 4.1 software; the difference was less than the plot resolution of Figs. 3±5. In all calculations in the present section a zero-thickness coil of diameter of 3.2 mm and a homogeneoustly distributed current of density of 750 A/m in the coil was assumed.
Fig. 5. Magnetization optimisation effect as function of the relative magnet height h/r and the relative coil-magnet distance d/r. Coil thickness was assumed to be zero.
A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
value of d corresponds to a differently magnetized magnet, which means that the force on a moving coil does not follow the dependence presented in the diagram. The calculated in this work F(d)Ðcurve for a é 2 mm 4 mm axially magnetized magnet ®tted with the calculations and experiments of Feustel et al. [8] within a 1 % for the d range from 0.1 to 1.1 mm if Br 1:1194 T was assumed. (No Br value was given in [8]). The availability of explicit expressions for coil-magnet force facilitates the study of the dependence of the force on the dimensions of the components of the system and the comparison of the effects of differently magnetized magnets. Fig. 4 presents the actuator's force dependence on the magnet heigth h and Fig. 5 the magnetization optimization effect (ratio of Foptimal/Faxial) in dependence of the relative magnet height h/r and the relative coil-magnet distance d/r. The optimization effect is greatest for thin magnets and small coil-magnet distances, which is particularly of importance in micromechanics. In the case of a magnet magnetized in the direction of the coil's ®eld, the actuator's force may be more than two-fold compared to an axially magnetized magnet. For d 0:1 mm, an optimal 0.25 mm thick magnet generates an equally large force (1 mN) as an axial 1 mm thick magnet. 4. Experiment For proving the concept of force enhancement due to variable in space orientation of magnetization, a composite magnet was composed, Fig. 6.
171
Fig. 7. Schematic of the arrangement for measuring of the force between the magnet and the coil.
The coil had following parameters: i.d. 2 mm, o.d. 6.8 mm, thickness 1 mm, number of turns 100. The dimensions of the magnet and the coil were chosen following the considerations in [12], for to get a maximal force. According to [12], such radial±axial magnet presents an approximation of the optimal magnet. (For optimal magnet the orientation of magnetization varies continuously in space.) For the dimensions used, the magnet in Fig. 6 provides a 26% greater force than an axial magnet, but 12% lower force than the optimal magnet (at d 0:1 mm). The force was measured using an arrangement shown in Fig. 7. Because the micrometer screw was ferromagnetic, there was some attraction force between the magnet and the screw even when there was no current in the coil. This
Fig. 6. Composite magnet used in the experiments. Materials: coreÐNeodymium 35, Br 1:21 T, H c 892 kA/m; shellÐNeodymium 27, Br 1:1 T, H c 740 kA/m (Eneflux Armtek Magnetics, Inc.). (a) 3D view, (b) cross-sectional view, (c) core, (d) piece of the shell, (e) compound magnet in PTFE holder before gluing and final slipping.
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A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
The fabrication methods of magnets with continuously varying orientation of magnetization were reviewed in [12]. 5. Conclusions
Fig. 8. Comparison of measured and calculated forces. d: Distance between the magnet and the coil. Coil's current 49.9 mA. For better distinction the calculated values are given only at intervals of 0.4 mm.
background force was subtracted from or added to the force measured with excited coil. The force was calculated using the Eqs. (10) and (11). Because the coil used in the experiments could not be regarded to have zero-thickness, the force was integrated also over the coil thickness. The magnet was assumed to have cylindrical symmetry with core diameter of 4 mm and o.d. of 6 mm. The Br values provided by the magnets supplier were used. As Fig. 8 demonstrates, the accordance between the calculations and the experiment is pretty good. The measured force was within ( 1.8%, 0.3%) of the computed (by Mathematica) force in coil-magnet distance range d 0:1± 4.9 mm. In contrast to the previous section, this is an absolute comparison. This result proves also, that the square magnetization loop approximation of sintered rare earth permanent magnet materials is valid with good accuracy. Also the calculations of Feustel et al. [8] and Harrison [15] demonstrate, that magnetic actuators with rare-earth permanent magnets can be simulated with great accuracy.
It has been shown, that the analytical expressions Eqs. (10)±(12) can be used for calculating of the force between a cylindrical coil and a coaxial cylindrical rareearth permanent magnet with good accuracy. In contrast to the previous work of other authors, the integration multiplicity has been reduced and no restrictive assumptions of the magnetic ®eld distribution has been used. The presented here computation procedure (based on linear interpolation of the elliptic integral values see Appendix B) is advantageous (a) in case when the dependence of the actuation force on the elements dimensions is of interest, (b) in case of relatively small coil-magnet distance, (c) if the magnet is magnetized in the coil's ®eld direction, (d) if no special mathematical/®nite element programs are available, and (e) if an accuracy of 1±2% is suf®cient. The calculations have shown that the magnetization of the magnet in the direction of the coil's ®eld can provide a more than two-fold force compared to axial magnetization, especially at thin magnets and small coil-magnet distances. Considerable gain in coil-PM actuator force can be achieved also using compound radial±axial magnets. The results are applicable especially for microsystems, where the dimensions of the structures are as rule much greater in the radial direction than in the axial direction. Acknowledgements This work was supported by the Estonian Ministry of Education, Grant no. 0140215s98, and by the Estonian Science Foundation, Grant no. 5150.
Appendix A Parameters of described in literature miniature co-axial permanent coil-magnet actuators (in chronological order) Magnet (all axial)
Coil
Dimensions (mm)
Material
Dimensions (mm)
é 0.5 0.5 é 4 2(?) 552 3 3 1.3 111 é24 é24 é24 552
NdFeB NdFeB SmCo NdFeB NdFeB NdFeB
é 3 0.01 3 é 1.2 100 é 3.3 0.0008 16 é 1.6/é 0.8b 20 &5/&0.7b 30 é 1.6 0.017 12 é 1.2 0.017 9 é 1 0.017 6.5 é 1.6 0.017 12
Number of turns
d (mm)
Current (mA)
4.6 200 46
0.365 0.4 0.4
3 0.9 1.4 3
0.5 0.1±1 0.1±1 0.1±1 0.1±1
4 50 100 1000 200 100 100 100 100
Resistance (O)
Fmax (mN) 7.5 280a 600 230 1600 1100 700 400
Reference
[1] [2,7] [3] [4] [5] [8]
A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
173
Appendix A. (Continued ) Magnet (all axial)
Coil
Dimensions (mm)
Material
Dimensions (mm)
Number of turns
é 2 0.09
Polymer bonded Sr-ferritec
&12.4/&4b
31
NdFeB
é 10d 0.075
é 1.5 0.09 é 1 0.09 552 552 14.15 mm3 56.55 mm3
d (mm)
30 20 20 20
Resistance (O)
3.5 2.5 3.5 3.5
Current (mA)
0.2±4
100
0.2±4 0.2±4 1.5
100 100 1000
Fmax (mN) 12 4.5 2 15000a
Reference
[10]
[11]
d: Coil-magnet distance; Fmax: maximal coil-magnet force. a Calculated. b Inner size. c Screen printed. d Annular circular and square coils; dimensions not given; diameter estimated using the data about the supporting structure.
In this work the effect of differently magnetized magnets and the dimensions of the magnets on the actuator force was studied using Microsoft Excel. This way was found to be most convenient, most fast, less expensive, and suf®ciently accurate. The values of complete elliptic integrals E(k), K(k) can easily be calculated by Microsoft Excel using linear interpolation of their table values. Corresponding Visual Basic macros are given in Appendix C. A quadratic interpolation did not give considerably better result. The E(ind) and KK(ind) values in the macros are standard four-decimal table values of E(k) and K(k), k sino, including 91 values of E and K at integer values of o from
0 to 908. Because K approaches in®nity at 908 (Fig. 9), the value K(89.98) was used in state of K(908). The error of such piecewise linear approximation of E(k) is <0.025%. For of K(k) the error is <2% in range of o 0±898 and <1% in range of o 0±888. If o ! 0, the interpolation errors vanish. The integrals in Eqs. (10)±(12) were calculated by Simpson's formula at 10 10 intervals of a and r. The calculation time for one force value was a fraction of a second on a 200 MHz Pentium PC. Care should be taken close to the coil edge of the magnet and close to the magnet edge of the coil, because the function K takes its greatest value there (Fig. 10). The parameter o should not exceed 888. The Excel worksheet for calculation the force between the magnet and the coil may be obtained from author without charge.
Fig. 9. Elliptic integrals K and E.
Fig. 10. Dependence of the argument of the elliptic integrals o in the Eqs. (7)±(9) on the relative coil radius a/r and the relative coil-magnet distance d/r. Calculated using Eq. (9).
Appendix B. Calculation of the actuator force using linear interpolation of the table values of elliptic integrals
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A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
Appendix C. Visual Basic macros for linear interpolating of elliptic integrals Public function Ellie(k) `for elliptic integral E(k) xx Atn(k/Sqr( k k 1)) oo xx 57:295779513 ind Int(oo) del oo ind Ellie E ind E ind 1 E ind del End function Public function Ellik(k) `for elliptic integral K(k) xx Atn(k/Sqr( k k 1)) oo xx 57:295779513 ind Int(oo) del oo ind Ellik KK ind KK ind 1 KK ind del End function The table of values of elliptic integrals E(ind), KK(ind) should be entered at integers of o; k sino. References [1] W. Affane, T.S. Birch, A microminiature electromagnetic middle-ear implant hearing device, Sens. Actuators A 46/47 (1995) 584±587. [2] J. Behrens, A. Meckes, M. Gebhard, W. Benecke, Electromagnetic actuation for micropums and valves, in: Proceedings of the ACTUATOR 96, Fifth International Conference on New Actuators, Bremen, Germany, 26±28 June 1996, pp. 124±127. [3] A. Feustel, O. Krusemark, U. Lehmann, J. MuÈller, T. Sperling, Electromagnetic membrane actuator with a compliant silicone suspension, in: Proceedings of the ACTUATOR 96, Fifth International Conference on New Actuators, Bremen, Germany, 26±28 June 1996, pp. 76±79. [4] P. Losantos, J.A. Plaza, J. Esteve, C. CaneÂ, Towards an electromagnetically driven micropump system, in: Proceedings of the EUROSENSORS XI, Eleventh European Conference on Solid State Transducers, Warsaw, Poland, 21±24 September 1997, pp. 1591±1594. [5] Y. Shinozawa, T. Abe, T. Kondo, A biproportional microvalve using a bi-stable actuator, in: Proceedings of the Tenth Annual International Workshop on Micro Electro Mechanical Systems, MEMS'97, Nagoya, Japan, 26±30 January 1997, pp. 233±237.
[6] M. Tabib-Azar, Microactuators. Electrical, Magnetic, Thermal, Optical, Mechanical, Chemical and Smart Structures, Kluwer Academic Publishers, Boston, 1998, pp. 96, 115±118. [7] A. Meckes, J. Behrens, W. Benecke, A microvalve with electromagnetic actuator, in: Proceedings of the ACTUATOR 98, Sixth International Conference on New Actuators, Bremen, Germany, 17± 19 June 1998, pp. 152±155. [8] A. Feustel, O. Krusemark, J. MuÈller, Numerical simulation and optimisation of planar electromagnetic actuators, Sens. Actuators A 70 (1998) 276±282. Â [9] D. De BhailõÂs, C. Murray, M. Duffy, J. Alderman, G. Kelly, S.C.O MathuÂna, Modeling and analysis of a magnetic microactuator for use as a micropump, in: Proceedings of the Micromechanics Europe'98, Ninth Workshop on Micromachining, Micromechanics and Microsystems (MME'98), Ulvik in Hardanger, Norway, 4±5 June 1998, pp. 256±259. [10] L.K. Lagorce, O. Brand, M.G. Allen, Magnetic microactuators based on polymer magnets, IEEE J. Microelectromech. Syst. 8 (1999) 2±8. Â. [11] D. De BhailõÂs, C. Murray, M. Duffy, J. Alderman, G. Kelly, S.C.O MathuÂna, Modeling and analysis of a magnetic microactuator, Sens. Actuators A 81 (2000) 285±289. [12] A. Kruusing, Optimizing magnetization orientation of permanent magnets for maximal gradient force, J. Magn. Magn. Mater. 234 (2001) 545±555. [13] A. Weigand, Das magnetische Feld einer von Gleichstrom durchflossenen Zylinderspule, Elektrotechnik 6 (1952) 605±612. [14] R. Sikora, W. Lipinski, Das magnetische Feld einer zylindrischen Spule, Scientia Electrica 21 (1975) 68±74. [15] A.J. Harrison, Composite permanent magnet materials, Ph.D. thesis, University of Nottingham, 1976, p. 35. [16] D. Craik, Magnetism: Principles and Application, Wiley, Chishester, 1995, p. 343.
Biography Arvi Kruusing received the Ph.D degree in electronics from All-Union Institute of Electrotechnics in Moscow, Russia, in 1988 and D.Eng. from Tallinn Technical University, Estonia, in 1998. During 1973±1998 he was employed by the Institute of Energy Research of the Estonian Academy of Sciences, by Special Design Office of the Estonian Academy of Sciences, and by Tallinn Technical University, investigating power semiconductor devices, power converters and developing analytical laser instrumentation. Since 1998, he is employed at Microelectronics Laboratory, University of Oulu, Finland, acting simultaneously as project leader at Tallinn Technical University. His main research interests are laser machining and microsystems design.
Actuators with permanent magnets having variable in space orientation of magnetization Arvi Kruusing* Tallinn Technical University, Department of Mechatronics, Ehitajate tee 5, EE-19086 Tallinn, Estonia Received 8 January 2002; received in revised form 24 April 2002; accepted 29 April 2002
Abstract The force between a permanent magnet and a coil can be increased if the magnetization orientation of the magnet is suitably varying in space. The force is greatest if the magnet is magnetized along coil ®eld lines (`optimal' magnet). In this article, is studied how the optimisation effect depends on the magnet proportions and coil-magnet distance in case of cylindrical magnets and coaxial planar coils. The dependence of the force on the magnet height and coil-magnet distance for differently magnetized magnets (axial, radial, optimal) is presented. Magnetization of the magnet in the direction of the coil's ®eld can provide a more than two-fold force compared to axial magnetization, especially at thin magnets and small coil-magnet distances. The measured force values differed from the calculated values maximum 2% for a milli-size radial±axial composite magnet. # 2002 Elsevier Science B.V. All rights reserved. PACS: 41.20.Gz; 85.70.Ay Keywords: Magnetic actuator; Permanent magnet; Varying magnetization orientation; Force calculation
1. Introduction In this article, electromagnetic linear actuators, comprising only a permanent magnet and a coil (without soft magnetic parts) are considered. Such actuators have been preferably used in miniature and microsystems [1±11]; see Appendix A. In all cases referred to in the table axisymmetric planar coils in conjunction with co-axial permanent magnets (PM) were used. The coils used were either square of circular and magnets either cylindrical or square prismatic. In some cases [3,4,8] two-layer coils were used to increase the number of turns and to solve simultaneously the problem of inner current feed. Because the magnetic ®eld is divergent in such actuators, the generated force is much smaller than in actuators having closed magnetic circuit. Therefore a careful design is needed. In [8] an optimisation of the dimensions of a planar coilÐaxially magnetized PM actuator was undertaken. In my previous article [12], it was shown that the actuator force could be further increased if magnet's magnetization orientation varies properly in space. *
Present address: University of Oulu, Microelectronics and Materials Physics Laboratories, P.O. Box 4500, FIN-90014 Oulu, Finland. E-mail address: [email protected] (A. Kruusing).
The purpose of this work was to study how much the actuator force can be increased using varying orientation of magnetization and how the force increase effect (further referred as `optimisation effect') depends on the magnet height and on the coil-magnet distance. 2. Calculation of the force between a cylindrical permanent magnet and a coaxial circular flat coil The calculation of the most signi®cant output parameter of the coil-PM actuators, i.e. the force, has been a problem up till the present time. A straight forward summary of the forces between the length elements of the conductor (or volume elements of the coil in assumption of continuous current distribution) and the volume elements of the magnet requires in general the calculation of up to six-fold integrals (three-fold in the case of a cylindrically symmetric system with a zero-thickness coil), in notations of [8]: Z Fz Br rz Hz ~ rd3 r; (1) VM
where
Z 1 ~ r 0 3 0 J ~ r 0 r ~ r d r ; Hz ~ 4p VC j~ r ~ r 0 j3
0924-4247/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 2 ) 0 0 1 5 4 - 1
z
(2)
A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
where Fz is the axial force between the magnet and the coil, Br the remanent magnetic induction ®eld of the magnet, VM the volume of the magnet, Hz the axial component of the magnetic ®eld strength of the coil, VC the volume of the coil, and J(r) the current density in the coil as function of the vector coordinate. The advantage of this approach is that it can be applied also for magnets and coils of low symmetry and for nonaxial coil-magnet systems. With the ®eld of the coil Hz given, the force can be calculated as Z @Hz dV; (3) Fz Mz @z where Mz is the magnetization of the magnet [4,5]. To simplify the calculations, in paper [11] only the ®eld on the symmetry axis was considered and the force was calculated as Z @Hz 0; z dz; (4) Fz Mz @z The formulae for cylindrical coils ®eld may be found for example in [13,14]. With the ®eld of the magnet B given, the force was calculated as [1,2], Fz NSi
dB ; dz
or N Fz 2p i S
(5)
Fig. 1. A cylindrical magnet in the field of a planar coaxial circular coil. Orientation of the magnetization in the direction of the coil's field provides the greatest force (`optimal' magnet). The arrows show the magnetization orientation in the magnet's cross-section.
Brad i; a; r; z
rBr dS;
(6)
where N is the number of the turns, i the current, S the crosssectional area of the coil, and Br the radial component of the magnetic induction in an element of the coil. In all of the aforementioned cases the magnets were magnetized in the symmetry axis direction. Force between a coil and a permanent magnet has also been calculated by the ®nite element method (FEM), using ANSYS and ANSOFT software [10,11]. This method is the best choice when the coil and/or the magnet have low symmetry. For highly symmetrical systems, analytical solutions usually outperform FEM. In the previous article [12], explicit formulae for calculation of force between an arbitrary magnetized cylindrical permanent magnet and a coaxial cylindrical coil were developed. The essentials of the theory and the results for axial, radial and `optimal' magnets are as follows. The system under consideration is shown in Fig. 1. The coil's ®eld is assumed to be a superposition of ring current ®elds m i 1 Baks i; a; r; z 0 2 2p a r z2 1=2 " # a2 r 2 z 2 E k K k ; (7) a r2 z2
m0 i z 1 2 2p r a r z2 1=2 " a2 r 2 z2 E k a r2 z2
# K k ;
(8)
where Baks and Brad are the axial and radial components of the magnetic induction, E(k) and K(k) are elliptic integrals of the argument k
Z
169
s 4ar a r2 z2
(9)
and i is the current in the ring [15,16]. Considering a current distributed evenly in the coil's cross-section, the total force between the coil and the magnet can be calculated by integrating B(a, r, z)M(r, z) (where M is the magnetization; M Br /m0) over the top and bottom surfaces of the magnet and the coil cross-section [12]. Speci®cally, for a zero-thickness circular coil and coaxial cylindrical PM, in three special cases of magnetization (Figs. 1 and 2) simple expressions for the force were found: (a) axially magnetized magnet: (Z Z a r 2p F Br Baks j; x; r; drdxdr m0 0 0 ) Z aZ r Baks j; x; r; d hrdxdr ; 0
0
(10)
Fig. 2. Axially and radially magnetized magnets. Sectional views.
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A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
(b) radially magnetized magnet: (Z Z a r 2p F Br Brad j; x; r; drdxdr m0 0 0 ) Z aZ r Brad j; x; r; d hrdxdr ; 0
0
(11)
(c) magnetization orientation in the coil's field direction (`optimal' magnetization): (Z Z a r 2p F Br B2rad j; x; r; d m0 0 0 B2aks j; x; r; d1=2 rdxdr Z aZ r B2rad j; x; r; d h 0
0
B2aks j; x; r; d
h
1=2
)
rdxdr ;
(12)
Fig. 3. Dependence of the force between a circular planar (zero-thickness) coil and a coaxial cylindrical magnet on the coil-magnet distance d for differently magnetized magnets (Figs. 1 and 2). For notations see Fig. 1. Optimal magnetization means the magnetization in the coil's field direction. Note that for the `optimal' curve, the magnetization of the magnet varies in dependence on the coil-magnet distance d.
where j is the density of the current of the coil and Br is the remanent induction of the magnet. The latter case (case c) will be referred as `optimal' magnetization, because it provides the greatest force between coil and magnet [12]. The formulae (10-12) express the integrals of forces between elementary current rings of the coil and elementary magnet pole rings on the top and bottom of the magnet. If the magnetization on the surface of the magnet is at angle to the symmetry axis, the force can be calculated applying the formulae (10) and (11) both to the axial and the radial components of the magnetization and summing the results. The Eqs. (10)±(12) represent mathematically exact expressions for the force in frames of the assumptions taken. The integration multiplicity of Eqs. (10)±(12) is smaller than of Eqs. (1) and (2), because instead of integrating over the volume of the magnets, in Eqs. (10)±(12) only over the top and bottom surfaces of the magnet is integrated. Of course, the Eqs. (10)±(12) apply only for a rotationally symmetric system with planar coil and ¯at top/¯at bottom magnet. Eqs. (10)±(12) can be easily generalized for the case of a ®nite height coil.
Fig. 4. Dependence of the force between a circular planar coil and a cylindrical magnet on the magnet height h for differently magnetized magnets.
The dependence of the force between the coil and the magnet on the coil-magnet distance for differently magnetized magnets is presented in the Fig. 3. It should be remembered that, in the case of optimal magnetization, each
3. Comparison of actuation force of differently magnetized magnets For comparison of the forces, a coil-magnet system of same dimensions as in [8] was assumed. The force was calculated by Eqs. (10)±(12) using Microsoft Excel 97 (see Appendix B). The results were compared with calculations by Mathematica 4.1 software; the difference was less than the plot resolution of Figs. 3±5. In all calculations in the present section a zero-thickness coil of diameter of 3.2 mm and a homogeneoustly distributed current of density of 750 A/m in the coil was assumed.
Fig. 5. Magnetization optimisation effect as function of the relative magnet height h/r and the relative coil-magnet distance d/r. Coil thickness was assumed to be zero.
A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
value of d corresponds to a differently magnetized magnet, which means that the force on a moving coil does not follow the dependence presented in the diagram. The calculated in this work F(d)Ðcurve for a é 2 mm 4 mm axially magnetized magnet ®tted with the calculations and experiments of Feustel et al. [8] within a 1 % for the d range from 0.1 to 1.1 mm if Br 1:1194 T was assumed. (No Br value was given in [8]). The availability of explicit expressions for coil-magnet force facilitates the study of the dependence of the force on the dimensions of the components of the system and the comparison of the effects of differently magnetized magnets. Fig. 4 presents the actuator's force dependence on the magnet heigth h and Fig. 5 the magnetization optimization effect (ratio of Foptimal/Faxial) in dependence of the relative magnet height h/r and the relative coil-magnet distance d/r. The optimization effect is greatest for thin magnets and small coil-magnet distances, which is particularly of importance in micromechanics. In the case of a magnet magnetized in the direction of the coil's ®eld, the actuator's force may be more than two-fold compared to an axially magnetized magnet. For d 0:1 mm, an optimal 0.25 mm thick magnet generates an equally large force (1 mN) as an axial 1 mm thick magnet. 4. Experiment For proving the concept of force enhancement due to variable in space orientation of magnetization, a composite magnet was composed, Fig. 6.
171
Fig. 7. Schematic of the arrangement for measuring of the force between the magnet and the coil.
The coil had following parameters: i.d. 2 mm, o.d. 6.8 mm, thickness 1 mm, number of turns 100. The dimensions of the magnet and the coil were chosen following the considerations in [12], for to get a maximal force. According to [12], such radial±axial magnet presents an approximation of the optimal magnet. (For optimal magnet the orientation of magnetization varies continuously in space.) For the dimensions used, the magnet in Fig. 6 provides a 26% greater force than an axial magnet, but 12% lower force than the optimal magnet (at d 0:1 mm). The force was measured using an arrangement shown in Fig. 7. Because the micrometer screw was ferromagnetic, there was some attraction force between the magnet and the screw even when there was no current in the coil. This
Fig. 6. Composite magnet used in the experiments. Materials: coreÐNeodymium 35, Br 1:21 T, H c 892 kA/m; shellÐNeodymium 27, Br 1:1 T, H c 740 kA/m (Eneflux Armtek Magnetics, Inc.). (a) 3D view, (b) cross-sectional view, (c) core, (d) piece of the shell, (e) compound magnet in PTFE holder before gluing and final slipping.
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The fabrication methods of magnets with continuously varying orientation of magnetization were reviewed in [12]. 5. Conclusions
Fig. 8. Comparison of measured and calculated forces. d: Distance between the magnet and the coil. Coil's current 49.9 mA. For better distinction the calculated values are given only at intervals of 0.4 mm.
background force was subtracted from or added to the force measured with excited coil. The force was calculated using the Eqs. (10) and (11). Because the coil used in the experiments could not be regarded to have zero-thickness, the force was integrated also over the coil thickness. The magnet was assumed to have cylindrical symmetry with core diameter of 4 mm and o.d. of 6 mm. The Br values provided by the magnets supplier were used. As Fig. 8 demonstrates, the accordance between the calculations and the experiment is pretty good. The measured force was within ( 1.8%, 0.3%) of the computed (by Mathematica) force in coil-magnet distance range d 0:1± 4.9 mm. In contrast to the previous section, this is an absolute comparison. This result proves also, that the square magnetization loop approximation of sintered rare earth permanent magnet materials is valid with good accuracy. Also the calculations of Feustel et al. [8] and Harrison [15] demonstrate, that magnetic actuators with rare-earth permanent magnets can be simulated with great accuracy.
It has been shown, that the analytical expressions Eqs. (10)±(12) can be used for calculating of the force between a cylindrical coil and a coaxial cylindrical rareearth permanent magnet with good accuracy. In contrast to the previous work of other authors, the integration multiplicity has been reduced and no restrictive assumptions of the magnetic ®eld distribution has been used. The presented here computation procedure (based on linear interpolation of the elliptic integral values see Appendix B) is advantageous (a) in case when the dependence of the actuation force on the elements dimensions is of interest, (b) in case of relatively small coil-magnet distance, (c) if the magnet is magnetized in the coil's ®eld direction, (d) if no special mathematical/®nite element programs are available, and (e) if an accuracy of 1±2% is suf®cient. The calculations have shown that the magnetization of the magnet in the direction of the coil's ®eld can provide a more than two-fold force compared to axial magnetization, especially at thin magnets and small coil-magnet distances. Considerable gain in coil-PM actuator force can be achieved also using compound radial±axial magnets. The results are applicable especially for microsystems, where the dimensions of the structures are as rule much greater in the radial direction than in the axial direction. Acknowledgements This work was supported by the Estonian Ministry of Education, Grant no. 0140215s98, and by the Estonian Science Foundation, Grant no. 5150.
Appendix A Parameters of described in literature miniature co-axial permanent coil-magnet actuators (in chronological order) Magnet (all axial)
Coil
Dimensions (mm)
Material
Dimensions (mm)
é 0.5 0.5 é 4 2(?) 552 3 3 1.3 111 é24 é24 é24 552
NdFeB NdFeB SmCo NdFeB NdFeB NdFeB
é 3 0.01 3 é 1.2 100 é 3.3 0.0008 16 é 1.6/é 0.8b 20 &5/&0.7b 30 é 1.6 0.017 12 é 1.2 0.017 9 é 1 0.017 6.5 é 1.6 0.017 12
Number of turns
d (mm)
Current (mA)
4.6 200 46
0.365 0.4 0.4
3 0.9 1.4 3
0.5 0.1±1 0.1±1 0.1±1 0.1±1
4 50 100 1000 200 100 100 100 100
Resistance (O)
Fmax (mN) 7.5 280a 600 230 1600 1100 700 400
Reference
[1] [2,7] [3] [4] [5] [8]
A. Kruusing / Sensors and Actuators A 101 (2002) 168±174
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Appendix A. (Continued ) Magnet (all axial)
Coil
Dimensions (mm)
Material
Dimensions (mm)
Number of turns
é 2 0.09
Polymer bonded Sr-ferritec
&12.4/&4b
31
NdFeB
é 10d 0.075
é 1.5 0.09 é 1 0.09 552 552 14.15 mm3 56.55 mm3
d (mm)
30 20 20 20
Resistance (O)
3.5 2.5 3.5 3.5
Current (mA)
0.2±4
100
0.2±4 0.2±4 1.5
100 100 1000
Fmax (mN) 12 4.5 2 15000a
Reference
[10]
[11]
d: Coil-magnet distance; Fmax: maximal coil-magnet force. a Calculated. b Inner size. c Screen printed. d Annular circular and square coils; dimensions not given; diameter estimated using the data about the supporting structure.
In this work the effect of differently magnetized magnets and the dimensions of the magnets on the actuator force was studied using Microsoft Excel. This way was found to be most convenient, most fast, less expensive, and suf®ciently accurate. The values of complete elliptic integrals E(k), K(k) can easily be calculated by Microsoft Excel using linear interpolation of their table values. Corresponding Visual Basic macros are given in Appendix C. A quadratic interpolation did not give considerably better result. The E(ind) and KK(ind) values in the macros are standard four-decimal table values of E(k) and K(k), k sino, including 91 values of E and K at integer values of o from
0 to 908. Because K approaches in®nity at 908 (Fig. 9), the value K(89.98) was used in state of K(908). The error of such piecewise linear approximation of E(k) is <0.025%. For of K(k) the error is <2% in range of o 0±898 and <1% in range of o 0±888. If o ! 0, the interpolation errors vanish. The integrals in Eqs. (10)±(12) were calculated by Simpson's formula at 10 10 intervals of a and r. The calculation time for one force value was a fraction of a second on a 200 MHz Pentium PC. Care should be taken close to the coil edge of the magnet and close to the magnet edge of the coil, because the function K takes its greatest value there (Fig. 10). The parameter o should not exceed 888. The Excel worksheet for calculation the force between the magnet and the coil may be obtained from author without charge.
Fig. 9. Elliptic integrals K and E.
Fig. 10. Dependence of the argument of the elliptic integrals o in the Eqs. (7)±(9) on the relative coil radius a/r and the relative coil-magnet distance d/r. Calculated using Eq. (9).
Appendix B. Calculation of the actuator force using linear interpolation of the table values of elliptic integrals
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Appendix C. Visual Basic macros for linear interpolating of elliptic integrals Public function Ellie(k) `for elliptic integral E(k) xx Atn(k/Sqr( k k 1)) oo xx 57:295779513 ind Int(oo) del oo ind Ellie E ind E ind 1 E ind del End function Public function Ellik(k) `for elliptic integral K(k) xx Atn(k/Sqr( k k 1)) oo xx 57:295779513 ind Int(oo) del oo ind Ellik KK ind KK ind 1 KK ind del End function The table of values of elliptic integrals E(ind), KK(ind) should be entered at integers of o; k sino. References [1] W. Affane, T.S. Birch, A microminiature electromagnetic middle-ear implant hearing device, Sens. Actuators A 46/47 (1995) 584±587. [2] J. Behrens, A. Meckes, M. Gebhard, W. Benecke, Electromagnetic actuation for micropums and valves, in: Proceedings of the ACTUATOR 96, Fifth International Conference on New Actuators, Bremen, Germany, 26±28 June 1996, pp. 124±127. [3] A. Feustel, O. Krusemark, U. Lehmann, J. MuÈller, T. Sperling, Electromagnetic membrane actuator with a compliant silicone suspension, in: Proceedings of the ACTUATOR 96, Fifth International Conference on New Actuators, Bremen, Germany, 26±28 June 1996, pp. 76±79. [4] P. Losantos, J.A. Plaza, J. Esteve, C. CaneÂ, Towards an electromagnetically driven micropump system, in: Proceedings of the EUROSENSORS XI, Eleventh European Conference on Solid State Transducers, Warsaw, Poland, 21±24 September 1997, pp. 1591±1594. [5] Y. Shinozawa, T. Abe, T. Kondo, A biproportional microvalve using a bi-stable actuator, in: Proceedings of the Tenth Annual International Workshop on Micro Electro Mechanical Systems, MEMS'97, Nagoya, Japan, 26±30 January 1997, pp. 233±237.
[6] M. Tabib-Azar, Microactuators. Electrical, Magnetic, Thermal, Optical, Mechanical, Chemical and Smart Structures, Kluwer Academic Publishers, Boston, 1998, pp. 96, 115±118. [7] A. Meckes, J. Behrens, W. Benecke, A microvalve with electromagnetic actuator, in: Proceedings of the ACTUATOR 98, Sixth International Conference on New Actuators, Bremen, Germany, 17± 19 June 1998, pp. 152±155. [8] A. Feustel, O. Krusemark, J. MuÈller, Numerical simulation and optimisation of planar electromagnetic actuators, Sens. Actuators A 70 (1998) 276±282. Â [9] D. De BhailõÂs, C. Murray, M. Duffy, J. Alderman, G. Kelly, S.C.O MathuÂna, Modeling and analysis of a magnetic microactuator for use as a micropump, in: Proceedings of the Micromechanics Europe'98, Ninth Workshop on Micromachining, Micromechanics and Microsystems (MME'98), Ulvik in Hardanger, Norway, 4±5 June 1998, pp. 256±259. [10] L.K. Lagorce, O. Brand, M.G. Allen, Magnetic microactuators based on polymer magnets, IEEE J. Microelectromech. Syst. 8 (1999) 2±8. Â. [11] D. De BhailõÂs, C. Murray, M. Duffy, J. Alderman, G. Kelly, S.C.O MathuÂna, Modeling and analysis of a magnetic microactuator, Sens. Actuators A 81 (2000) 285±289. [12] A. Kruusing, Optimizing magnetization orientation of permanent magnets for maximal gradient force, J. Magn. Magn. Mater. 234 (2001) 545±555. [13] A. Weigand, Das magnetische Feld einer von Gleichstrom durchflossenen Zylinderspule, Elektrotechnik 6 (1952) 605±612. [14] R. Sikora, W. Lipinski, Das magnetische Feld einer zylindrischen Spule, Scientia Electrica 21 (1975) 68±74. [15] A.J. Harrison, Composite permanent magnet materials, Ph.D. thesis, University of Nottingham, 1976, p. 35. [16] D. Craik, Magnetism: Principles and Application, Wiley, Chishester, 1995, p. 343.
Biography Arvi Kruusing received the Ph.D degree in electronics from All-Union Institute of Electrotechnics in Moscow, Russia, in 1988 and D.Eng. from Tallinn Technical University, Estonia, in 1998. During 1973±1998 he was employed by the Institute of Energy Research of the Estonian Academy of Sciences, by Special Design Office of the Estonian Academy of Sciences, and by Tallinn Technical University, investigating power semiconductor devices, power converters and developing analytical laser instrumentation. Since 1998, he is employed at Microelectronics Laboratory, University of Oulu, Finland, acting simultaneously as project leader at Tallinn Technical University. His main research interests are laser machining and microsystems design.