- Email: [email protected]
The design and analysis of a novel brushless dc linear motor
~_~ ELSEVIER
Journal of Magnetism and Magnetic Materials 133 (1994) 640-643
~ H journalof magnetism ~ a and 414H magnetic materials
,I¢'ri
The design and analysis of a novel brushless dc linear motor A. Basak *, A.F. Flores Filho Wolfson Centrefor Magnetics Technology, ELSYM, PO Box 917, Newport Road, Cardiff CF2 1XH, Wales, UK
Abstract A novel brushless dc linear m o t o r was designed, with two a r m a t u r e cores a n d seven high energy p r o d u c t magnets. It was t h e n built a n d tested. B o t h the design a n d the analysis were carried o u t assuming a linear m a g n e t i c circuit. T h e design p r o c e d u r e a n d the test results are p r e s e n t e d a n d discussed in this paper. T h e new m o t o r can also b e used as a slotless linear stepping motor.
1. Introduction D e p e n d i n g on t h e application, it may be m o r e app r o p r i a t e to use linear m o t o r s directly i n s t e a d of rotary ones with e i t h e r gears or belts or o t h e r systems to t r a n s f o r m rotary m o t i o n into linear m o v e m e n t . C D players a n d Maglev trains are some of the examples. T h e p u r p o s e of this p a p e r is to show the results obt a i n e d with a novel brushless dc linear motor. T h e motor, which can b e classified as a short secondary type with high energy p r o d u c t magnets, will b e used as a slotless linear stepping m o t o r in the n e a r future.
2. Description of the motor Fig. 1 shows the linear motor. It has a r e c t a n g u l a r s h a p e d a r m a t u r e core, which is m a d e up of two long a n d two short bars of mild steel j o i n e d t o g e t h e r . T h e square cross section of each b a r is 576 m m 2. T h e r e are gaps or inclusions of n o n m a g n e t i c m a t e r i a l along it to avoid m a g n e t i c s a t u r a t i o n a n d to achieve a b e t t e r motor time constant. T h e a r m a t u r e coil is w o u n d with e n a m e l l e d c o p p e r wire along the longer a r m a t u r e core pieces, which d e t e r m i n e the travel l e n g t h of the motor. T h e m o t o r can use two different kinds of winding: (a) a continuous winding forming two s e p a r a t e coils C~ a n d
C 2 that can b e fed with c u r r e n t s of the same magnit u d e I, b u t of different polarities, o n e e a c h at a time; a n d (b) several sections of winding with same n u m b e r of turns, e a c h o n e b e i n g fed with switched currents; in this case w h e n the coils o n C 1 are fed, the coils o n C 2 are not, a n d vice versa. A n o t h e r i m p o r t a n t feature in the latter case is the m e c h a n i c a l d i s p l a c e m e n t of the coils of o n e part with respect to the other. If the length of each section is 2 p , the d i s p l a c e m e n t b e t w e e n two consecutive coils, o n e o n C~ a n d the o t h e r on C2, is p, which is the d i s p l a c e m e n t step of the slider as well. This f e a t u r e characterises the b e h a v i o u r of a linear stepping motor. T h e configuration (a) has b e e n a d o p t e d for this first prototype. Its slider is f o r m e d in a rectangular s h a p e a n d is also m a d e of mild steel. T h e seven m a g n e t s employed o n this m o t o r are N d F e B C r u m a x 315 supplied by Crusteel Magnetics Ltd. Auxiliary m a g n e t s are placed at the i n n e r side of the slider as shown. T h e m a i n magnet, which is placed b e t w e e n C~ a n d C2, is mechanically fixed to t h e slider by two pieces of solid n o n m a g n e t i c material. T h e slider is p r o v i d e d with two supports with linear bearings. It is guided by two cylindrical rods w h o s e lengths are at least equal to the travel length of the motor, which was c h o s e n to b e 200 mm.
3. Analysis and design
* Corresponding author.
F r o m L o r e n t z ' s law, the ideal e q u a t i o n of t h e elect r o m a g n e t i c force resulting from the interaction be-
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved
SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 0 2 2 9 - K
A. Basak, A.F. Flores Filho /Journal of Magnetism and Magnetic" Materials 133 (1994) 640-643 tween the armature current and the magnetic flux provided by the magnets can be obtained as follows:
F = 4 ( N w m / t t ) l L e × Bg,
(1)
where F is the electromagnetic force vector, N is the total number of turns per coil, wm is the width of the magnets, It is the travel length, I is the armature current, L c is the active conductor length vector for each side of the coil in the same direction of I, and Bg is the magnetic flux density vector in the airgap. Eq. (1) is ideal because it takes into account neither that the airgap flux density is non-uniform nor that the magnetic fluxes provided by all the magnets are different due to the magnetic circuit topology. However, for the initial calculations it is adequate. The expressions for the slider velocity v and for the electromagnetic power Pc are:
v = (V, - I r a ) / [ 4 ( N Wm/lt)BgLe],
(2)
and Pe = 4 ( N Wrn/lt)BgLcVl,
(3)
where Vt is the terminal voltage supplied to the armature coil, and r a is the ohmic resistance of the coil. Looking at the topology of the motor (Fig. 1), and adopting a linear magnetic circuit model for the motor, it is possible to realise that the slider can 'see' through the airgap two armature reluctances in parallel. One,
641
Ra, is due to the magnetic path on one half of the motor, and the other, Rb, due to the other half. Both depend on the position of the slider. Eq. (4) gives the value of R a in parallel with Rb, namely Rab: Rab = { [ ( k + 2 x ) / I z L 2 + 21o/tzoSo] -1 + [ ( k + 2 l t - Z x ) / I z L 2 c + 21o/IxoSo]-' }
1,
(4) where k is a constant and is equal to 3L c + 21m + 21g, where Im is the length of the side magnets, lg is the airgap length, lz and iz 0 are the magnetic permeability of the core and air respectively, L c is the active conductor length, l 0 is the length of each of the nonmagnetic gaps along the armature core and S o is their cross section, x is the centre of the slider position with respect to one with the extremities of the long bars of the stator. Referring to Eq. (4), the expression for the magnet load line in this case is
B m / n m = - (t~olmCfLf)//( lg +
2~z0SoR.b),
(5)
where Cf and Lf are the fringing flux and the flux leakage factors respectively, for the magnetic flux provided by the magnets, and the former ones depend on the geometry of the motor. They can be estimated according to the literature or obtained through a finite element analysis. By the analysis of Eqs. (4) and (5), it
Fig. l. Perspective view of the linear motor.
642
A. Basak, A.F. Flores Filho /Journal of Magnetism and Magnetic Materials 133 (1994) 640-643
can be proved that Rab presents a parabolic behaviour with respect to x. Consequently, the magnetic flux provided by the magnets will be minimum at half travel length and maximum at the extremes of the armature core, and so will be the static thrust. This nonlinear force feature can be minimised either using a material of larger ~ or increasing Rab. The latter option will make use of larger magnets to compensate the decrease of flux. The design involved the equations above, conditions such as maximum core saturation, travel length and maximum desired static thrust, and the use of an interactive calculation process. So, it was possible to obtain the dimensions of the magnets and of the magnetic circuit, airgap, and number of turns per coil. As a result, each coil has 1140 turns distributed in four layers, the air gap is 4 mm long, the nominal armature current is 1.5 A, each auxiliary magnet is 5.2 mm thick and the main magnet is 10.4 m m thick.
the slider, thus a relative m o v e m e n t between the armature and the slider takes place. Employing a linearly variable differential transformer ( L V D T ) to measure the position of the armature with respect to the slider, the curves of static thrust versus slider position have been plotted for different values of armature current, By means of the L V D T attached to the slider, the acceleration curve is obtained as well. The thrust and the velocity curves are shown in Figs. 2(a) and (b), where x = 0 means that the slider is at the middle of the track. The sudden increase of static thrust when the slider is placed closer to the armature extremes is explained by the attractive force between the magnets on the slider and the short mild steel bars. This kind of end effect distorts the thrust curve, which was expected to present at least a quadratic behaviour, The static thrust at x = 0 is 8.9 N, which corresponds to a difference of about 13.8% compared to the design value of 10.32 N.
5. Conclusions
4. Test results Some tests were carried out to assess the behaviour of the prototype. The measured coil resistances are 5.72 and 5.67 ohms for coils C 1 and C 2, respectively. The measured electrical time constant is 1.4 ms for each coil section u n d e r n e a t h the slider. A system was conceived to measure the static thrust of the motor along its travel length. To avoid problems of changing the position of the force transducer, i.e. a load cell, with respect to the slider, the latter was kept attached to the load cell which was then fixed to a rigid and standstill support. The armature was placed on a moveable table and the latter moved with respect to
STATIC THRUST (N}
The first prototype of the novel brushless dc linear motor was designed and tested by innovative means. Although the motor works properly, the results indicate that it is necessary to obtain a flatter thrust characteristic without the end effects and obtaining a constant flux density independent of the slider position. It will corroborate to make the control of the motor easy, once the thrust curve is smooth and the acceleration curve is linear. In order to achieve these, the next step of the design will be the modification of the geometry of the magnetic circuit. A 3D finite element analysis package will employed to aid the improvement of the design.
SLIDER VELOCITY [xlO -2 m/s) 45
,
;
~
,
t
,
,
,
~
,
,
[
;
,
,
,
I
~
I
I
,
IL
(b)
10 8 4
eee
2 0 -2
~,.v.v-~r~"v
-4
(a)
35
~---~ I= O.5A G-----El I=IA o . . . . . . - e I= 1.5A
-8
. . . . . . .
-10-
-
-4-2'(~
2
4
6
SLIDER POSITION (xlO-2 m}
L"t_
i ' 10
25
15 -10
L
I
I
I
I
I
I
I
I
-5
,
0
I
,
I
I
I
5
POSITION (xlO-2 m)
Fig. 2. (a) Static thrust versus slider position curves; (b) velocity curve for I = 1.5 A.
L
10
A. Basak, A.F. Flores Filho/Journal of Magnetism and Magnetic Materials 133 (1994) 640-643 Acknowledgements. Mr. Flores Fiiho wishes to thank his sponsor, the Coordenacao de Aperfeicoam e n t o de Pessoal de Nivel S u p e r i o r - C A P E S of the Brazilian Ministry of Education, and the D e p a r t m e n t of Electrical Engineering of the Federal University of Rio G r a n d e do Sul, where he is a lecturer, for the support to his research work.
643
References [1] [2] [3] [4]
A. Basak and F.J. Anayi, in: Proc. UPEC (1989) p. 365. D. Casadei and others, in: Proc. ICEM (1992) p. 132. J. Draeger, in: Proc. ACTUATOR (1988) p. 203. L. Honds and K.H. Meyer, Philips Tech. Rev. 40 (1982) 329. [5] S.A. Nasar and I. Boldea, Linear Electric Motors: Theory, Design, and Practical Applications (Prentice Hall, London, 1987).
Journal of Magnetism and Magnetic Materials 133 (1994) 640-643
~ H journalof magnetism ~ a and 414H magnetic materials
,I¢'ri
The design and analysis of a novel brushless dc linear motor A. Basak *, A.F. Flores Filho Wolfson Centrefor Magnetics Technology, ELSYM, PO Box 917, Newport Road, Cardiff CF2 1XH, Wales, UK
Abstract A novel brushless dc linear m o t o r was designed, with two a r m a t u r e cores a n d seven high energy p r o d u c t magnets. It was t h e n built a n d tested. B o t h the design a n d the analysis were carried o u t assuming a linear m a g n e t i c circuit. T h e design p r o c e d u r e a n d the test results are p r e s e n t e d a n d discussed in this paper. T h e new m o t o r can also b e used as a slotless linear stepping motor.
1. Introduction D e p e n d i n g on t h e application, it may be m o r e app r o p r i a t e to use linear m o t o r s directly i n s t e a d of rotary ones with e i t h e r gears or belts or o t h e r systems to t r a n s f o r m rotary m o t i o n into linear m o v e m e n t . C D players a n d Maglev trains are some of the examples. T h e p u r p o s e of this p a p e r is to show the results obt a i n e d with a novel brushless dc linear motor. T h e motor, which can b e classified as a short secondary type with high energy p r o d u c t magnets, will b e used as a slotless linear stepping m o t o r in the n e a r future.
2. Description of the motor Fig. 1 shows the linear motor. It has a r e c t a n g u l a r s h a p e d a r m a t u r e core, which is m a d e up of two long a n d two short bars of mild steel j o i n e d t o g e t h e r . T h e square cross section of each b a r is 576 m m 2. T h e r e are gaps or inclusions of n o n m a g n e t i c m a t e r i a l along it to avoid m a g n e t i c s a t u r a t i o n a n d to achieve a b e t t e r motor time constant. T h e a r m a t u r e coil is w o u n d with e n a m e l l e d c o p p e r wire along the longer a r m a t u r e core pieces, which d e t e r m i n e the travel l e n g t h of the motor. T h e m o t o r can use two different kinds of winding: (a) a continuous winding forming two s e p a r a t e coils C~ a n d
C 2 that can b e fed with c u r r e n t s of the same magnit u d e I, b u t of different polarities, o n e e a c h at a time; a n d (b) several sections of winding with same n u m b e r of turns, e a c h o n e b e i n g fed with switched currents; in this case w h e n the coils o n C 1 are fed, the coils o n C 2 are not, a n d vice versa. A n o t h e r i m p o r t a n t feature in the latter case is the m e c h a n i c a l d i s p l a c e m e n t of the coils of o n e part with respect to the other. If the length of each section is 2 p , the d i s p l a c e m e n t b e t w e e n two consecutive coils, o n e o n C~ a n d the o t h e r on C2, is p, which is the d i s p l a c e m e n t step of the slider as well. This f e a t u r e characterises the b e h a v i o u r of a linear stepping motor. T h e configuration (a) has b e e n a d o p t e d for this first prototype. Its slider is f o r m e d in a rectangular s h a p e a n d is also m a d e of mild steel. T h e seven m a g n e t s employed o n this m o t o r are N d F e B C r u m a x 315 supplied by Crusteel Magnetics Ltd. Auxiliary m a g n e t s are placed at the i n n e r side of the slider as shown. T h e m a i n magnet, which is placed b e t w e e n C~ a n d C2, is mechanically fixed to t h e slider by two pieces of solid n o n m a g n e t i c material. T h e slider is p r o v i d e d with two supports with linear bearings. It is guided by two cylindrical rods w h o s e lengths are at least equal to the travel length of the motor, which was c h o s e n to b e 200 mm.
3. Analysis and design
* Corresponding author.
F r o m L o r e n t z ' s law, the ideal e q u a t i o n of t h e elect r o m a g n e t i c force resulting from the interaction be-
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved
SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 0 2 2 9 - K
A. Basak, A.F. Flores Filho /Journal of Magnetism and Magnetic" Materials 133 (1994) 640-643 tween the armature current and the magnetic flux provided by the magnets can be obtained as follows:
F = 4 ( N w m / t t ) l L e × Bg,
(1)
where F is the electromagnetic force vector, N is the total number of turns per coil, wm is the width of the magnets, It is the travel length, I is the armature current, L c is the active conductor length vector for each side of the coil in the same direction of I, and Bg is the magnetic flux density vector in the airgap. Eq. (1) is ideal because it takes into account neither that the airgap flux density is non-uniform nor that the magnetic fluxes provided by all the magnets are different due to the magnetic circuit topology. However, for the initial calculations it is adequate. The expressions for the slider velocity v and for the electromagnetic power Pc are:
v = (V, - I r a ) / [ 4 ( N Wm/lt)BgLe],
(2)
and Pe = 4 ( N Wrn/lt)BgLcVl,
(3)
where Vt is the terminal voltage supplied to the armature coil, and r a is the ohmic resistance of the coil. Looking at the topology of the motor (Fig. 1), and adopting a linear magnetic circuit model for the motor, it is possible to realise that the slider can 'see' through the airgap two armature reluctances in parallel. One,
641
Ra, is due to the magnetic path on one half of the motor, and the other, Rb, due to the other half. Both depend on the position of the slider. Eq. (4) gives the value of R a in parallel with Rb, namely Rab: Rab = { [ ( k + 2 x ) / I z L 2 + 21o/tzoSo] -1 + [ ( k + 2 l t - Z x ) / I z L 2 c + 21o/IxoSo]-' }
1,
(4) where k is a constant and is equal to 3L c + 21m + 21g, where Im is the length of the side magnets, lg is the airgap length, lz and iz 0 are the magnetic permeability of the core and air respectively, L c is the active conductor length, l 0 is the length of each of the nonmagnetic gaps along the armature core and S o is their cross section, x is the centre of the slider position with respect to one with the extremities of the long bars of the stator. Referring to Eq. (4), the expression for the magnet load line in this case is
B m / n m = - (t~olmCfLf)//( lg +
2~z0SoR.b),
(5)
where Cf and Lf are the fringing flux and the flux leakage factors respectively, for the magnetic flux provided by the magnets, and the former ones depend on the geometry of the motor. They can be estimated according to the literature or obtained through a finite element analysis. By the analysis of Eqs. (4) and (5), it
Fig. l. Perspective view of the linear motor.
642
A. Basak, A.F. Flores Filho /Journal of Magnetism and Magnetic Materials 133 (1994) 640-643
can be proved that Rab presents a parabolic behaviour with respect to x. Consequently, the magnetic flux provided by the magnets will be minimum at half travel length and maximum at the extremes of the armature core, and so will be the static thrust. This nonlinear force feature can be minimised either using a material of larger ~ or increasing Rab. The latter option will make use of larger magnets to compensate the decrease of flux. The design involved the equations above, conditions such as maximum core saturation, travel length and maximum desired static thrust, and the use of an interactive calculation process. So, it was possible to obtain the dimensions of the magnets and of the magnetic circuit, airgap, and number of turns per coil. As a result, each coil has 1140 turns distributed in four layers, the air gap is 4 mm long, the nominal armature current is 1.5 A, each auxiliary magnet is 5.2 mm thick and the main magnet is 10.4 m m thick.
the slider, thus a relative m o v e m e n t between the armature and the slider takes place. Employing a linearly variable differential transformer ( L V D T ) to measure the position of the armature with respect to the slider, the curves of static thrust versus slider position have been plotted for different values of armature current, By means of the L V D T attached to the slider, the acceleration curve is obtained as well. The thrust and the velocity curves are shown in Figs. 2(a) and (b), where x = 0 means that the slider is at the middle of the track. The sudden increase of static thrust when the slider is placed closer to the armature extremes is explained by the attractive force between the magnets on the slider and the short mild steel bars. This kind of end effect distorts the thrust curve, which was expected to present at least a quadratic behaviour, The static thrust at x = 0 is 8.9 N, which corresponds to a difference of about 13.8% compared to the design value of 10.32 N.
5. Conclusions
4. Test results Some tests were carried out to assess the behaviour of the prototype. The measured coil resistances are 5.72 and 5.67 ohms for coils C 1 and C 2, respectively. The measured electrical time constant is 1.4 ms for each coil section u n d e r n e a t h the slider. A system was conceived to measure the static thrust of the motor along its travel length. To avoid problems of changing the position of the force transducer, i.e. a load cell, with respect to the slider, the latter was kept attached to the load cell which was then fixed to a rigid and standstill support. The armature was placed on a moveable table and the latter moved with respect to
STATIC THRUST (N}
The first prototype of the novel brushless dc linear motor was designed and tested by innovative means. Although the motor works properly, the results indicate that it is necessary to obtain a flatter thrust characteristic without the end effects and obtaining a constant flux density independent of the slider position. It will corroborate to make the control of the motor easy, once the thrust curve is smooth and the acceleration curve is linear. In order to achieve these, the next step of the design will be the modification of the geometry of the magnetic circuit. A 3D finite element analysis package will employed to aid the improvement of the design.
SLIDER VELOCITY [xlO -2 m/s) 45
,
;
~
,
t
,
,
,
~
,
,
[
;
,
,
,
I
~
I
I
,
IL
(b)
10 8 4
eee
2 0 -2
~,.v.v-~r~"v
-4
(a)
35
~---~ I= O.5A G-----El I=IA o . . . . . . - e I= 1.5A
-8
. . . . . . .
-10-
-
-4-2'(~
2
4
6
SLIDER POSITION (xlO-2 m}
L"t_
i ' 10
25
15 -10
L
I
I
I
I
I
I
I
I
-5
,
0
I
,
I
I
I
5
POSITION (xlO-2 m)
Fig. 2. (a) Static thrust versus slider position curves; (b) velocity curve for I = 1.5 A.
L
10
A. Basak, A.F. Flores Filho/Journal of Magnetism and Magnetic Materials 133 (1994) 640-643 Acknowledgements. Mr. Flores Fiiho wishes to thank his sponsor, the Coordenacao de Aperfeicoam e n t o de Pessoal de Nivel S u p e r i o r - C A P E S of the Brazilian Ministry of Education, and the D e p a r t m e n t of Electrical Engineering of the Federal University of Rio G r a n d e do Sul, where he is a lecturer, for the support to his research work.
643
References [1] [2] [3] [4]
A. Basak and F.J. Anayi, in: Proc. UPEC (1989) p. 365. D. Casadei and others, in: Proc. ICEM (1992) p. 132. J. Draeger, in: Proc. ACTUATOR (1988) p. 203. L. Honds and K.H. Meyer, Philips Tech. Rev. 40 (1982) 329. [5] S.A. Nasar and I. Boldea, Linear Electric Motors: Theory, Design, and Practical Applications (Prentice Hall, London, 1987).