Means to reduce harmonic torque in electromagnetic machines

An electromagnetic machine of the type having a moving member and a stationary member, with a set of structures on the moving member to magnetically interact with a set of structures on the stationary member, has an improved construction comprising displacing by an angle of displacement, .alpha..sub.m, from their normal positions, a first portion of one set of structures whereby the displacement causes a harmonic of the fundamental torque/angle curve to be attenuated. The angle of displacement is determined by the following relationship: EQU .alpha..sub.m =.alpha..sub.e /p, where PA1 .alpha..sub.e =displacement in electrical degrees, PA1 p=number of moving member magnetic pole pairs, PA1 further where .alpha..sub.e =180/h, PA1 where h=an integer equal to the number of a harmonic of the fundamental torque/angle curve to be attenuated.

BACKGROUND OF THE INVENTION 
1. Technical Field 
This invention relates to electromagnetic machines, and more particularly 
to an improved design which reduces a given harmonic of the fundamental 
torque/displacement-angle relationship. While the invention is described, 
for convenience, as applied to rotary motors, it will be understood that 
it may be applied to linear motors, as well as to other electromagnetic 
machines having stationary and moving members, such as signal generating 
devices and electric power generators and other electric power producing 
devices. It has been found to be particularly useful when applied to 
rotary brushless DC, stepping and synchronous inductor motors. 
2. Background Art 
Rotary brushless DC, stepping, and synchronous inductor motors are well 
known in the art. Each type includes a rotor (moving member) and a stator 
(stationary member), with the stator having a plurality of salient poles 
energized by the passage of electric current through coils wound upon the 
poles. The coils are so arranged as to provide at least two electrical 
phases. The rotor includes at least one pair of N-S magnetic poles which 
are flux-linked with the stator poles, so that successive energizations of 
the phases provide rotary motion of the rotor. 
The torque/displacement-angle of the rotor relationship, "torque/angle 
curve", between a rotor pole and each of the stator poles, may be 
expressed in general by the well known Fourier expansion: 
EQU T=k[1+A.sub.1 cos .theta..sub.e +A.sub.2 cos (2.theta..sub.e) . . . A.sub.n 
cos (n.theta..sub.e)+B.sub.1 sin .theta..sub.e +B.sub.2 sin 
(2.theta..sub.e) . . . +B.sub.n sin (n.theta..sub.e)], 
where 
T=torque, 
k=a constant, 
A.sub.1, A.sub.2 . . . A.sub.n=Fourier Coefficients (constants) of the 
cosine terms, 
B.sub.1, B.sub.2 . . . B.sub.n =Fourier Coefficients (constants) of the 
sine terms, and 
.theta..sub.e =the displacement of the rotor in electrical degrees. 
In the above equation, A.sub.1 cos .theta..sub.e +B.sub.1 sin .theta..sub.e 
represents the fundamental (first) harmonic produced as the rotor poles 
pass the stator poles; A.sub.2 cos 2.theta..sub.e +B.sub.2 sin 
2.theta..sub.e is the second harmonic of the fundamental; and so forth. 
In the special case in which .theta..sub.e =o is defined as the rotor 
position for which the centerline of the rotor pole coincides with the 
centerline of the stator pole for which the Fourier expansion is being 
written, the expansion is greatly simplified to 
EQU T=K[B.sub.1 sin .theta..sub.e +B.sub.2 sin (2.theta..sub.e)+ . . . +B.sub.n 
sin (n.theta..sub.e)] 
It is well known that the presence of torque/angle harmonics is especially 
detrimental to the performance of synchronous inductor motors, stepping 
motors, and brushless D.C. motors. In particular, a harmonic of the order 
corresponding to twice the number of phases (e.g. 4th harmonic for a 
2-phase machine, 6th harmonic for a 3-phase machine, etc.) is particularly 
detrimental because of its dominance in the distribution of harmonic 
content. This particular harmonic is responsible for "detent torque", an 
objectionable resistance to rotation of the rotor of a de-energized motor. 
Step accuracy of a step motor, velocity modulation of synchronous inductor 
motors, stepping motors, and brushless D.C. motors, and microstepping 
ability of stepping motors and brushless D.C. motors are all adversely 
affected by torque/angle harmonics, and particularly by the one 
responsible for detent torque as described above. 
It would be advantageous in such motors to be able to minimize the dominant 
harmonic which adversely affects motor performance as described above. 
In copending Application Ser. No. 06/782,932, assigned to the assignee of 
the present application, there is disclosed means for minimizing a given 
harmonic the torque/angle curve of motors by providing a stator with two 
sets of poles, the sets of poles being nonuniformly spaced according to a 
specified relationship. While that arrangement satisfactorily reduces the 
undesired harmonic, the nonuniform stator slot widths cause certain 
complexities, the size of the stator coils is restricted practically by 
the narrowest slot, and the windings must be specially oriented with 
respect to the stator. 
SUMMARY OF THE INVENTION 
The present invention provides for minimization of a given harmonic of the 
torque/angle curve by providing a motor having first and second 
rotor/stator combinations, the two combinations being so arranged that, 
although each produces the harmonic to be minimized, the harmonic produced 
by one cancels the harmonic produced by the other. This result is achieved 
by displacing a first set of stator pole teeth (or poles) from a second 
set of stator pole teeth (or poles) or by displacing a first set of rotor 
teeth from a second set of rotor teeth, such displacement being a given 
increment greater or smaller than any displacement that may exist in 
conventional construction. 
Thus, with the appropriate harmonic substantially reduced, the adverse 
effect on performance in brushless DC, stepping, and synchronous inductor 
motors due to the presence of that harmonic can be reduced, and 
performance in brushless DC and stepping motors can be improved.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
FIG. 1(a) shows diagrammatically in cross section a portion of a 
conventionally constructed motor, which may assumed to be a two-phase, 
eight-pole-stator synchronous inductor motor. Shown are a toothed rotor 10 
and a toothed stator 11, the stator having poles 12 through 19 upon which 
energizing coils (not shown) can be wound. FIG. 1(b) shows a portion of 
the rotor 10 which includes an axially-magnetized permanent magnet 20 and 
pole pieces 21 and 22 mounted with the magnet as shown, with pole piece 21 
having a N magnetization and pole piece 22 having a S magnetization. The 
periphery of each pole piece is formed to have periphery teeth 23 for the 
pole piece 21 and periphery teeth 24 for the pole piece 22. As shown on 
FIG. 1(a), each periphery tooth is identical and the teeth are equally 
spaced; but as shown in FIG. 1(b), the pole piece 21 is placed so that its 
teeth 23 are radially displaced one-half a tooth pitch from the teeth 24 
on the pole piece 22. 
Rotation of the motor shown on FIG. 1(a) is caused by a unidirectional 
magnetic field being attracted to and following a rotating magnetic field. 
The unidirectional field in the motor illustrated is produced by the 
permanent magnet forming a part of the rotor 10. The stator 11 has a two 
phase winding with alternate poles being energized by the same phase. When 
the windings are connected to a source of alternating current, during 
one-half of the cycle of alternating current, each of the poles will have 
a magnetic polarity and on the other half of the cycle they will have the 
opposite polarity. Thus in the poles energized by one phase of the 
winding, one pole will for example be north, the next one south, the next 
north, during one-half of the cycle and in subsequent half cycles they 
become south, north, south, etc. The poles energized by the other phase of 
the winding will have a similar magnetic polarity except that it is about 
90.degree. out of phase with the first set of poles. Accordingly, rotation 
is produced by the unidirectional field being first attracted and then 
repelled by the change in the polarity in the poles. The rotating speed of 
the motor using a constant cycle alternating source is varied by 
mechanical construction features such as the number of poles of the stator 
and the configuration of the rotor and stator poles. 
The fundamental torque/angle curve of each half of the rotor 10 of FIG. 
1(b) is given by 
EQU T=K[B.sub.1 sin .theta..sub.e +B.sub.2 sin 2.theta..sub.e + . . . B.sub.n 
sin (n.theta..sub.e)], 
where the symbols are the same as described above, which relationship is 
shown graphically on FIG. 2. While the fundamental curve is correctly 
shown as an undistorted sine wave, it will be understood that a curve 
representing total torque would be distorted, it being the total of the 
fundamental and harmonics of the fundamental, and it is this distortion 
that contributes to velocity modulations in brushless DC, stepping, and 
synchronous inductor motors and to reduced step and microstep accuracy in 
brushless DC and stepping motors. It has been found that, for the 
two-phase, eight-pole stator of FIG. 1(a), the harmonic that contributes 
most to the accuracy and velocity modulation problems discussed above is 
the fourth harmonic which is also shown on FIG. 2. The present invention 
eliminates or substantially attenuates the fourth harmonic for this case, 
as described below. 
The motor described in connection with FIGS. 1(a) and 1(b) may be modified 
in accordance with the teaching of this invention by displacing one set of 
rotor teeth 23 from the other set of rotor teeth 24 by an angle 
.alpha..sub.e from their normal displacement of 180 degrees electrical to 
eliminate one of the harmonics in the torque vs. angle waveform. Each half 
of the motor rotor 10, may be thought of as an independent contributor to 
the total motor torque, with the torque contributed by the halves 
represented by T.sub.A and T.sub.B, with the total torque represented by 
T.sub.T =T.sub.A +T.sub.B. 
If the halves of the rotor are shifted radially from their normal positions 
by .alpha..sub.e degrees electrical, the torque equations become: 
EQU T.sub.A =K.sub.A [B.sub.1 sin .theta..sub.e +B.sub.2 sin (2.theta..sub.e)+ 
- - - +B.sub.n sin (n.theta..sub.e)] 
and 
EQU T.sub.B =K.sub.B [B.sub.1 sin (.theta..sub.e +.alpha..sub.e)+B.sub.2 sin 
(2.theta..sub.e +2.alpha..sub.e)+ - - - +B.sub.n sin (n.theta..sub.e 
+n.alpha..sub.e)] 
If only the fundamental and the fourth harmonic are present, the above 
equations reduce to (if K.sub.A =K.sub.B =1.0): 
EQU T.sub.A =K.sub.A [B.sub.1 sin .theta..sub.e +B.sub.4 (4.theta..sub.e)] 
EQU T.sub.B =K.sub.B [B.sub.1 sin (.theta..sub.e +.alpha..sub.e)+B.sub.4 sin 
(4.theta..sub.e +4.alpha..sub.e)] 
and 
EQU T.sub.T =T.sub.A +T.sub.B =2B.sub.1 cos (-.alpha..sub.e /2) sin 
(.theta..sub.e +.alpha..sub.e /2)+2B.sub.4 cos (-2.alpha..sub.e) sin 
(4.theta..sub.e +2.alpha..sub.e). 
When .alpha..sub.e =45 degrees electrical, cos (-2.alpha..sub.e)=0, and the 
fourth harmonic term of T.sub.T is reduced to zero, thereby eliminating 
the effects of the fourth harmonic torque component. This is illustrated 
graphically on FIG. 3. T.sub.B leads T.sub.A by 45 degrees electrical and 
the fourth harmonics of T.sub.A and T.sub.B cancel, being displaced 180 
degrees electrical. 
In the same manner, any other torque harmonic can be eliminated by the 
proper choice of displacement angle, .alpha..sub.e. 
The necessary angle of displacement is determined by the relationship 
.alpha..sub.e =180.degree./h, where h is the harmonic to be minimized, 
and, therefore, where the harmonic to be attenuated is the fourth, the 
displacement angle is 45 degrees electrical. Electrical degrees and 
mechanical degrees are related by the expression 
EQU .alpha..sub.m =.alpha..sub.e /p 
where 
.alpha..sub.e =displacement in electrical degrees, 
.alpha..sub.m =displacement in mechanical degrees, and 
p=number of rotor pole pairs. 
FIG. 4 shows the rotor 10 of FIG. 1(b) constructed according to the present 
invention, with the pole pieces 21 and 22 displaced from their normal 
positions by an angle, .alpha..sub.m. It is unimportant to the practising 
of the present invention whether the displacement angle, .alpha..sub.m, is 
in the direction shown or in the opposite direction. 
FIG. 5 shows a two-magnet rotor constructed according to the present 
invention, and generally indicated by the reference numeral 30. Magnets 31 
and 32 have pole pieces 33 and 34, and 35 and 36, respectively, all 
assembled along a common axis as shown, with the teeth of pole pieces 34 
and 35 normally offset 1/2 tooth pitch. Pole piece 33 is offset 1/2 tooth 
pitch from pole piece 34 and pole piece 35 is offset 1/2 tooth pitch from 
pole piece 36. In this embodiment, pole pieces 33 through 36 may be 
magnetized S, N, S, N or N, S, N, S, respectively. The displacement angle, 
.alpha..sub.m, in this embodiment is achieved by radially offsetting the 
two halves of the rotor 30 as shown. 
FIG. 6 shows a two-magnet rotor, constructed according to the present 
invention, and generally indicated by the reference numeral 40. Magnets 41 
and 42 have pole pieces 43 and 44, and 45 and 46, respectively, all 
assembled along a common axis as shown with the teeth of pole pieces 44 
and 45 normally in alignment. Pole piece 44 is offset 1/2 tooth pitch from 
pole piece 44 and pole piece 45 is offset 1/2 tooth pitch from pole piece 
46. Pole pieces 43 and 46 are shown as having a S magnetization and pole 
pieces 44 and 45 are ashown as having a N magnetization, although they 
could have opposite magnetizations, respectively. The displacement angle, 
.alpha..sub.m, in this embodiment is achieved by radially offsetting the 
two halves of the rotor 40 as shown. 
FIG. 7 shows an untoothed magnetic rotor, indicated generally by the 
reference numeral 50, which is radially magnetized, rather than axially 
magnetized as was the case on FIGS. 4, 5, and 6. Here the rotor has N and 
S non-salient poles magnetized on its periphery and uniformly spaced, as 
at 51. The rotor actually comprises two magnets 52 and 53 which are 
displaced by the angle .alpha..sub.m according to the present invention. 
The present invention is not limited to providing minimization of the 
fourth harmonic. For example, in a three-phase motor, it is found that the 
sixth harmonic causes the accuracy and velocity modulation problems 
discussed above. Through the relationship .alpha..sub.e =180.degree./h, it 
is seen that a displacement of 30 degrees electrical is required and such 
may be obtained as hereinbefore discussed. Also, the present invention is 
not limited to the stator/rotor configuration chosen for illustration and 
it will be apparent to one skilled in the art that it can be applied, as 
well, to other stator/rotor configurations. Furthermore, it will be 
understood that the invention disclosed is not limited to those cases 
where it is desired to minimize the problems with velocity modulation and 
accuracy discussed above, but also to any case where it is desired to 
minimize a given harmonic of the fundamental torque/angle curve. 
The foregoing invention is applicable to any number of stator poles, rotor 
teeth, and number of phases. It can also be applied to multiple stacks of 
magnets and rotor pole pieces. 
While the present invention has been described as applied to a motor having 
toothed stator and rotor structures, with halves of the rotor being 
normally displaced, it may be applied as well to other motor 
configurations. As examples: Some motors with toothed stators and rotors 
have the rotor teeth all aligned and halves of the stator teeth normally 
displaced by one-half tooth pitch. In that case the two halves of the 
stator would be offset additionally by the displacement angle determined 
as above. Also, the same principles of the invention apply to motors 
having shifted stator teeth and untoothed rotors, and to motors having 
shifted rotor teeth and untoothed stator poles. Further, it is not 
required for the practising of the invention that the rotor be of the 
permanent magnet type. Moreover, the displacement need not be taken 
entirely in either the stator or the rotor, but both could take part of 
the displacement, as long as the total displacement produces the desired 
result. 
The invention is applicable as well to other types of electromagnetic 
machines having stationary and moving members, such as linear motors, 
signal generating devices, and electric power generators and other 
electric power producing devices. 
It will be understood that what has been disclosed is a novel method for 
minimizing a given harmonic of the fundamental torque/angle curve. 
Since certain changes may be made in carrying out the above invention 
without departing from the scope thereof, it is intended that all matter 
contained in the above description or shown in the accompanying Drawing 
shall be interpreted as illustrative and not in a limiting sense. 
It is also intended that the following claims are intended to cover all of 
the generic and specific features of the invention herein described, and 
all statements of the scope of the invention which, as a matter of 
language, might be said to fall therebetween.