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Description  |
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DESCRIPTION
This invention relates to pole-amplitude modulation (P.A.M.),
pole-changing, alternating current electric motors and generators.
Three-phase alternating current, pole-amplitude modulation, pole-changing,
squirrel-cage electric motors and generators are well-known.
The theory of pole-amplitude modulation has been explained in various
patent specifications and technical papers on the assumption that all the
space-distributions of the windings and the imposed modulation waves are
required to be purely sinusoidal and, hence, in practical machines are
required to be as nearly sinusoidal as is possible.
It was found that sinusoidal modulation of pole-amplitude produced certain
m.m.f. harmonics, particularly those harmonics which have been described
as "conjugate" harmonics, which harmonics are additional to those found in
conventional single-speed windings.
Part of the developing technique of pole-amplitude modulation, as it has
been described in the later patent specifications and technical papers,
has been concerned with the reduction of the undesired harmonics by
variation of the modulation sequence or by choice of coil-grouping and
chording. Those harmonics were considered undesirable, in particular,
which resulted in substantial dips, or cusps, or saddles in the
acceleration curve of a motor or which at least caused an overall
reduction of the motor torque at successive points on the acceleration
curve.
This criterion was in practice quite appropriate for small and medium sized
motors because, when the performance of such motors is unsatisfactory, it
is usually by reason of an unsatisfactory torque characteristic.
It has now been found that there may be present in pole-amplitude
modulation machines other undesirable harmonics, the nature and importance
of which has not been previously recognised. These harmonics do not
greatly affect the torque characteristic of a machine, but their presence
is evident because they can react with the main rotating field of the
machine and set up low frequency vibration of the frame of the machine.
These harmonics arise in the process of applying sinusoidal modulation.
This is partly because of the irregular coil-grouping which is inherently
required, and partly because of the divergence, in practice, of the m.m.f.
waveform and of the wave of modulation from their ideal sinusoidal forms.
Such harmonics are additional to the "conjugate" harmonics already referred
to. They are herein described as the "adjacent" harmonics of negative
rotation; that is, they are of Order as near as possible to the Order of
the corresponding main field and rotate in the opposite sense. Usually,
these adjacent harmonics are of Order differing by "2" from the Order of
the main field. In rare cases, they are of Order differing by Unity from
the Order of the main field.
The effect of such adjacent harmonics of negative rotation in causing low
frequency vibration of the frame of a machine is most likely to be
noticed, and is least acceptable, in large motors of high pole-numbers,
although, less noticeably, the effect will still be present in small
motors.
The object of the present invention is to minimise such adjacent harmonics
in pole-amplitude modulation machines.
Accordingly, the invention provides a three-phase alternating current
electric motor or generator for alternative pole-number operation having a
pole-amplitude modulation three-phase stator winding switched to provide
the alternative pole-numbers in accordance with a phase-by-phase
modulation pattern or an overall modulation pattern, said pole-number
switching being effected by including all coils of each phase-winding in
either one or the other of two phase-winding parts, said phase-winding
parts being connected together alternatively in series and in parallel in
order to provide said alternative pole-numbers, and the waveform of said
modulation pattern as determined by phase-winding coil grouping,
approximating to a form representative of sinusoidal form with added
third-harmonic content, whereby the adjacent harmonic content, as herein
defined, of the m.m.f. waveform of the three-phase winding is less for
both pole-numbers than for a modulation waveform of purely sinusoidal
form.
Another form of the present invention provides a method of designing the
stator winding for a three-phase alternating current electric motor or
generator, said stator winding being adapted for switching to provide
alternative pole-numbers by pole-amplitude modulation comprising selecting
a stator slot-number divisible by two factors, the first factor being "6"
or "18", the second factor determining the number of slots for each half
of the initial winding element for said stator winding, examining all
possible phase/slot-number distributions, totalling the number of said
second factor, selecting that said distribution appropriate to said
winding element providing the said alternative pole-numbers with the
minimum content of adjacent harmonics, as herein defined, and symmetrizing
said winding element once at least to provide said stator winding in a
stator of said selected slot-number.
The two halves of the initial winding element must be the same, apart from
signs, in order to permit the parallel/series switching, which is the
characteristic manner of switching for pole-changing with P.A.M. windings.
The principle underlying the present invention is thus the addition, to the
known and basically sinusoidal modulation of pole-amplitude, of an
harmonic modulation specifically to reduce adjacent harmonics in the
resultant m.m.f. waveform. The addition of third-harmonic is found to be
specific for this purpose.
Two methods are possible in practice in applying this principle. According
to the first method, a modulation waveform with added third-harmonic
content is first postulated and the practically possible coil-group
distributions are examined in order to approximate as closely as possible
to the chosen modulation waveform.
The second method is less direct and consists of examining the m.m.f.
adjacent harmonic content for initial half-winding elements having one of
the few possible phase/slot-number distributions and selecting one with a
low adjacent harmonic content. It will then be found that the resultant
modulation waveform has the added third-harmonic content, as though
provided by the first method.
The choice of method for introducing third-harmonic will be seen to be
determined by the stator winding pole-numbers and slot-number. In the
examples which follow herein, alternative pole-numbers which are large
pole-numbers and are of the form P poles/(P=2) poles are chosen, because
such machines both show the effect of adjacent harmonics most clearly and
are a commercially important class of machines.
However, before the specific examples of the invention are described in
detail, it will facilitate understanding of the invention to discuss the
theory which those examples will reflect.
Considering, first, those radial force-waves present in all induction
motors, the rotating magnetic field in all induction motors causes a
rotating radial force-wave, which tends to bend the core and the frame.
The force (per unit area) at any point is proportional to the square of
the magnetic flux-density B.sub.m sin (m.theta.-wt), at that point;
that is, to:
##EQU1##
This gives a double-frequency rotating force-wave, with a pole-number
equal to twice the main pole-number, in addition to the steady radial
force 1/2 [B.sub.m.sup.2 ], where m is the number of main pole-pairs and
.theta. is a mechanical angle.
Considering next, the rotating force-waves in P.A.M. induction motors,
suppose there is superimposed a second rotating magnetic field, B.sub.n
sin (n.theta..-+.wt), of a different pole-number, the second field being
weak in comparison with the main field B.sub.m ; less (say) than 20%.
Their combined effect is to give two additional radial force-waves,
besides that due to the main field. One of these force-waves is
stationary, and one rotates at a speed corresponding to twice the line
frequency. In principle, these additional force-waves, due to harmonics,
occur in all standard induction motors, but they are very small.
The following equations will make matters clear:
Let B.sub.m sin (m.theta.-wt) and B.sub.n sin (n.theta..-+.wt) represent
two rotating magnetic fields in an induction motor, where B.sub.m is very
much greater than B.sub.n. Where the alternative sign is negative, both
fields rotate in the same direction. Where it is positive, the fields
rotate in opposite directions. The total resultant rotating force wave is
now proportional to:
[B.sub.m sin (m.theta.-wt)+B.sub.n sin (n.theta..-+.wt)].sup.2 ( 2)
If B.sub.m is several times as big as B.sub.n, as will always be so for any
P.A.M. winding, the rotating force-wave due to B.sub.n.sup.2 acting alone
can be neglected. Taking the difference between the total resultant
rotating force-wave of Eqn. (2), and the force-wave of Eqn. (1) which is
due to the main field B.sub.m acting alone, and ignoring the force-wave
due to B.sub.n.sup.2, it follows that the extra rotating force-wave in a
P.A.M. induction motor is proportional to:
##EQU2##
Each residual magnetic field can, in principle, cause two resultants,
according to this last equation. (Only resultant magnetic fields are of
importance. Any harmonic m.m.f. will be much larger than the magnetic
field which it produces, and even the harmonic m.m.f.'s in a P.A.M.
winding will be small compared with the main m.m.f.)
The first term in this last expression (3) represents a fixed force-wave of
(m.-+.n) pole-pairs, and the second term represents a rotating force-wave
of (m.+-.n) pole-pairs, with a speed of revolution corresponding to twice
line-frequency. In principle, the fixed force-wave will give a steady
resultant distortion of the core and frame, but no resultant vibration.
The second term represents a rotating force-wave, which may cause
vibration.
The amount of vibration will depend on the magnitude of the force-wave, the
stiffness of the frame and also on the number of poles in the force-wave.
The frame may be considered as a continuous beam; simply-supported at a
number of points, spaced by one pole-pitch. The deflection of a
simply-supported beam of a given section, for a given force, is
proportional to the cube of the distance between the supporting points;
and thus the greater the number of poles the less the vibration, and vice
versa.
The maximum vibration for a given strength of the second magnetic field
B.sub.n will occur with the negative sign for n in the second term of the
expression for the extra force-wave, giving only (m-n) pole-pairs in the
rotating force-wave. This corresponds to a rotation of the second field
B.sub.n in the opposite direction to the main magnetic field B.sub.m. At
the same time, the sign in the first term will be positive, and the fixed
force-wave will have (m+n) pole-pairs, and the resultant fixed distortion
will be trivial.
A second rotating magnetic field will thus be most likely to cause
vibration, if the two fields are nearly of the same Order, and rotate in
opposite senses. In principle, m and n can differ by Unity; but they will
not differ by less than 2 for normal P.A.M. windings. An m.m.f. harmonic
of Order differing only by Unity, 2 or other low integer from the Order of
the fundamental m.m.f. has been defined herein as an adjacent harmonic.
The Orders may differ by Unity for combinations of odd numbers of
pole-pairs, when all Orders of harmonic, Odd and Even, may be present for
both pole-numbers. The important case is 6 poles/10 poles; other cases are
rare in practice.
In earlier P.A.M. windings, special attention was paid to reducing the
magnitude of any higher-Order m.m.f. harmonics (of Orders several times
that of the main pole-number) because those would have a considerable
effect on the speed-torque characteristics.
The principal low-Order harmonics imposed on a P.A.M. winding by the
pre-modulation of the winding are of Orders (p.sub.1 .+-.2), (p.sub.1
.+-.4) etc., where p.sub.1 is the original number of pole-pairs. The
magnitude of these harmonics diminishes very rapidly with increase of
Order. The largest adjacent harmonic before modulation is therefore of
Order (p.sub.1 -2), and is solely or predominantly of negative rotation.
Although the existence of adjacent harmonics has been known for some years,
it is only with the use of P.A.M. pole-changing in large machines that the
effects of vibration have to be considered as unacceptable in practice and
it is only with the present invention that a general remedy has been
found.
In smaller machines, the radial forces exist, but they are there less
likely to cause undue vibration, because frames for small motors are
usually stronger, relatively, than the frames of big motors. Even so,
large adjacent harmonics may cause an appreciable increase in audible
noise. For this reason it may be advantageous to use the remedy of the
present invention in small motors, particularly as the remedy is simple to
apply and quite small in cost.
Considering, now, the Orders of adjacent harmonics, and the direction of
their rotation, these can readily be deduced, as is shown below.
The principal low-Order harmonics imposed on a P.A.M. winding by the
pre-modulation of the winding are of Orders (p.sub.1 .+-.2), (p.sub.1
.+-.4) etc., where p.sub.1 is the original number of pole-pairs. The
magnitude of these harmonics diminishes very rapidly with increase of
Order. The largest adjacent harmonic before modulation is therefore of
Order (p.sub.1 -2), and is solely or predominantly of negative rotation.
The principal low-Order harmonics resulting from modulation arise either
from the third-harmonic component of the phase m.m.f.'s, or from a
third-harmonic component of the modulating wave. The (desired) resultant
of modulation for close-ratio windings is of (p.sub.1 +1) pole-pairs; and
the principal (undesired) harmonic resultant is thus of (p.sub.1 +3)
pole-pairs, where p.sub.1 is the original number of pole-pairs. Expressed
in terms of the main pole-pair number (p.sub.2) for a close-ratio P.A.M.
winding after modulation, the principal undesired adjacent harmonic is
then of Order (p.sub.2 +2), and this is also of negative rotation.
Considering, next, the effect of the squirrel-cage rotor upon the m.m.f.
harmonics present in a P.A.M. induction motor, the rotating field B.sub.m
sin (m.theta.-wt) set up by the main m.m.f. is the fundamental rotating
magnetic field. All the remaining m.m.f. components should in principle be
completely neutralized in normal operation by currents in the squirrel
cage, so that ideally there should be no magnetic flux in the machine
other than the fundamental field, when it is operating normally. In
practice, the larger harmonic m.m.f.'s are not completely neutralized, and
it is the fluxes (B.sub.n), set up by these residual m.m.f.'s, which are
denoted (typically) by B.sub.n sin (n.theta..-+.wt). They will always be
small, because they are proportional only to the un-neutralized part of
the harmonic m.m.f. It is the reactions between B.sub.m and one (or more)
Order(s) of B.sub.n which can set up additional rotating magnetic
force-waves.
The negatively-rotating adjacent harmonic after modulation, of (p.sub.2
+2)=(p.sub.1 +3) pole-pairs, is less likely to be fully neutralized by the
squirrel cage than is the negatively-rotating adjacent harmonic of
(p.sub.1 -2) pole-pairs, before modulation, for a given strength of the
harmonic m.m.f. It will be observed that the Order of the adjacent
harmonic for the higher speed, in close-ratio P.A.M. windings, is always
greater by 5 than the Order of the adjacent harmonic for the lower speed.
The ratio between these two Orders is nearer to Unity for larger
pole-numbers and vice versa. For example, for 8 poles/10 poles the ratio
is 2/7, whereas for 14 poles/16 poles the ratio is 1/2. In consequence,
adjacent harmonics are more nearly of equal significance for both speeds,
for the larger pole-numbers.
SHORT DESCRIPTION OF THE DRAWINGS
In order that the invention may be clearly understood and readily carried
into practice, a number of embodiments will now be described in detail, by
way of example, with reference to the accompanying drawings, in which:-
FIG. 1 is a winding layout and total modulation diagram for an 8 pole/10
pole symmetrical P.A.M. winding in 120 slots, with two phantom coil-groups
per phase;
FIG. 2 shows four graphs, for 8 pole/10 pole windings, relating the number
of coils per coil-group and nominal angular position on the modulating
wave;
FIG. 3 is a winding diagram of the preferred 8 pole/10 pole winding in 120
slots;
FIG. 4 together with FIG. 5 show all six possible alternative 12-slot
elements suitable for a 10 pole/12 pole P.A.M. winding in 108 slots;
FIG. 6 shows the coil-groupings and coil-signs for 10 pole and 12 pole
working for the six alternative 12-slot elements of FIGS. 4 and 5;
FIG. 7 is a winding diagram of one 10 pole/12 pole P.A.M. winding in 108
slots;
FIG. 8 is a winding diagram of another 10 pole/12 pole P.A.M. winding in
108 slots;
FIG. 9 shows 12-pole vector diagrams for a 40-slot winding element (a)
after symmetrization and (b) after the transfer of one coil from phase C
to phase B;
FIG. 10 is a winding diagram for the final preferred 10 pole/12 pole P.A.M.
winding in 120 slots;
FIG. 11 is a winding diagram for a 12 pole/14 pole winding in 108 slots;
FIG. 12 is a winding diagram for a 14 pole/16 pole winding in 126 slots;
FIG. 13 is a clock diagram showing the coil-grouping for a 14 pole/8 pole
winding in 108 slots; and
FIG. 14 is the winding diagram for the final preferred 14 pole/8 pole
P.A.M. winding in 108 slots.
SHORT DESCRIPTION OF THE TABLES
Included, for convenience, with the accompanying drawings are the following
tables of which:
Table 1 tabulates corresponding values for the original (known) winding and
the winding embodying the present invention for four typical slot-numbers
for a symmetrical winding in 8/10 poles;
Table 2 shows typical numbers of slots, and the numbers of corresponding
coils, for nine coil-group combinations showing the possible division,
between the three phases, of coils per half-element of typical windings;
Table 3 shows the harmonic content produced by the six alternative 12-slot
elemental 10 pole/12 pole windings, permitting a choice therefrom for a 10
pole/12 pole P.A.M. winding in 108 slots;
Table 4 shows the harmonic content produced by the four alternative 10-slot
elemental 10 pole/12 pole windings, permitting a choice therefrom for a 10
pole/12 pole P.A.M. winding in 120 slots;
Table 5 shows the harmonic content produced by the six alternative 12-slot
elemental 12 pole/14 pole windings permitting a choice therefrom for a 12
pole/14 pole P.A.M. winding in 108 slots; and
Table 6 shows the harmonic content produced by the seven alternative
14-slot elemental 14 pole/16 pole windings permitting a choice therefrom
for a 14 pole/16 pole P.A.M. winding in 126 slots.
DESCRIPTION OF THE EXAMPLES
General Theory of Harmonic Modulation
The present invention provides a general method of designing a further
class of P.A.M. pole-changing windings, so as to minimise adjacent
harmonics of negative-rotation. This method necessitates an intentional
departure from the pure sinusoidal modulation, which has been the
theoretical basis upon which the known P.A.M. pole-changing induction
motors have been designed.
The theory of harmonic modulation can be applied to P.A.M. windings for any
pole-combination. However, the examples given herein will be for (p.sub.1
/p.sub.2) pole-pairs, where p.sub.1 <p.sub.2 and p.sub.2 =(p.sub.1 +1).
In practice, pole-combinations such as 4/6; 6/8; 8/10; 10/12; etc., where
the pole-numbers differ by 2, have proved to be much the most important,
industrially.
Modulation for such pole-combinations is usually effected, as has been
established by earlier publications, by overall modulation of the stator
winding, by a modulation wave equal to the sum of the alternative
pole-numbers.
The general principle of harmonic modulation is that an undesirable m.m.f.
harmonic can be much reduced by adding an additional component of
modulation to the normal modulating wave. The particular harmonic
modulation which reduces the adjacent m.m.f. harmonics is the third
harmonic of the fundamental phase-modulation wave. The incidence of
double-frequency vibration in large P.A.M. motors can be reduced to a very
low level by the addition of a selected amount of third-harmonic
modulation.
The particular principle of third-harmonic modulation is analysed for the
two modes of overall modulation, as follows:
(A) Overall modulation by the sum of the pole-pairs
(1) From p.sub.1 pole-pairs to p.sub.2 pole-pairs: p.sub.2 =(p.sub.1 +1)
______________________________________
Products of Modulation
Modulation applied
Wanted Inherent
______________________________________
Main modulation
p.sub.1 .+-. (p.sub.1 + p.sub.2)
- p.sub.2 + (2p.sub.1 + p.sub.2)
Third harmonic p.sub.1 .+-. 3
+ (p.sub.1 + 3)
+ (p.sub.1 - 3)
modulation = + (p.sub.2 + 2)
Reduced low-Order
Increased low-
Adjacent harmonic.
Order sub-
harmonic.
Desired. Acceptable.
______________________________________
Conversely, taking modulation in the reverse sense:
(2) From p.sub.2 pole-pairs to p.sub.1 pole-pairs: p.sub.1 =(p.sub.2 -1)
______________________________________
Products of Modulation
Modulation applied
Wanted Inherent
______________________________________
Main modulation
p.sub.2 .+-. (p.sub.1 + p.sub.2)
- p.sub.1 + (2p.sub.2 + p.sub.1)
Third-harmonic p.sub.2 .+-. 3
+ (p.sub.2 - 3)
+ (p.sub.2 + 3)
modulation
= + (p.sub.1 - 2)
Reduced low-Order
Increased high-
Adjacent harmonic.
Order
harmonic.
Desired. Acceptable.
______________________________________
The directions of rotation of all the harmonics are opposite to the
direction of rotation of the main field, in both the cases above. In
particular, the adjacent harmonics are of negative rotation.
Except in symmetrical P.A.M. windings, m.m.f. harmonics are generally
unbalanced. Adjacent harmonics will have both positive and negative
sequence components, but the negative-sequence component is much the
larger. The positive-sequence component reflects second-order differences
between the coil-groupings of the individual phases.
Phase-Modulation
This logic has been established on the basis of overall modulation of the
stator winding, but it is equally applicable to phase modulation. The
phase-modulation wave, in close-ratio modulation, is of 2 poles and the
third-harmonic wave is thus of 6 poles. The spacing between the three
phase-modulation waves in a close-ratio P.A.M. winding is 2.pi./3, and
their third-harmonic components are thus spaced by 2.pi.: that is, they
are electrically coincident. If the third-harmonic modulating components
are impressed on the phase modulating waves, a third-harmonic modulation
will therefore have been impressed on the winding as a whole.
Returning to consideration of overall modulation
(B) Overall modulation by the difference of the pole-pairs
The same logic applies equally to overall modulation by the difference of
the pole-numbers. An example is given later herein for a winding for 8
poles/14 poles. This is a symmetrical P.A.M. winding for which overall
modulation by 6 poles, which is the difference of the alternative
pole-numbers, is used. The harmonic modulation which is needed to correct
the adjacent harmonic is the third-harmonic of 6 poles; that is, 18-pole
harmonic modulation. The only change arising in the numerical
relationships is that the adjacent harmonic for 14 poles is of 10 poles;
that is 4 poles less than the main pole-number instead of 4 poles greater.
Alternative P.A.M. Windings with Added Third-Harmonic Modulation
As has been explained above, when double-frequency vibration is present, in
large motors, it is due to the adjacent harmonic content in the m.m.f.
waveform. The addition of a third-harmonic component in the modulation
waveform reduces and can eliminate the adjacent m.m.f. harmonics of Orders
(p.sub.1 -2) and (p.sub.2 =2), in close-ratio P.A.M. windings for p.sub.1
/p.sub.2) poles.
Before describing the practical winding examples, it is necessary to
consider the two classes of P.A.M. pole-combinations as follows:
Symmetrical P.A.M. pole-combinations, for which neither alternative
pole-number is a multiple of "3"; and
Asymmetrical P.A.M. pole-combinations, for which one of the alternative
pole-numbers is a multiple of "3" and the other is not.
Symmetrical P.A.M. windings
In the first class, the coil-groupings for all the phases are always
identical. It is therefore possible to re-group the coils of all the
phases in the same way, and the windings will remain balanced for both
speeds, whatever coil-grouping is chosen. Normally, the coil-grouping has
to be restricted to low integral values because, in practice, all coils
must be identical, and the number of coils/pole/phase must be low.
A third-harmonic component can be added to the modulating wave by reducing
the number(s) of coils in the centre group(s) of each half phase-winding,
and increasing the number(s) of coils in the outer group(s).
In Table 1 there are set out eight alternative 8 pole/10 pole symmetrical
P.A.M. windings comprising two alternative windings for each of four
stators with different slot-numbers. Of the two alternative windings, the
first is a known 8 pole/10 pole winding, of design based upon sinusoidal
modulation and the second winding represents the one with added
third-harmonic modulation. The characteristics of all eight windings are
tabulated and it will be seen that the adjacent harmonic m.m.f. content is
much reduced for the second alternative winding in each corresponding
stator, that winding with added third-harmonic modulation.
It is to be noted that the coil-pitch is the same for both windings of each
pair.
FIG. 1 shows the winding layout and overall modulation diagram for the 8
pole/10 pole P.A.M. winding in a 120 slot stator having the same
coil-grouping:
7 - 5 - 8 - 0 - - Repeat
for each phase, that is providing two phantom coil groups "0" for each
phase. This is the fourth winding shown in Table 1 and there indicated by
"*".
As shown in FIG. 1, the overall modulation wave, indicated by an arrow, is
8+10=18-pole. The winding shown is 8-pole and is modulated to 10-pole by
reversal in current-carrying sense of those coil-groups which lie inside
the modulation wave which embraces the phase-bands in the outer ring of
the "clock" diagram. This 18-pole wave is 8 - 5 - 7 - 8 -5 -7 - - Repeat -
- Repeat coils per pole.
The middle ring denotes the number of coils in each coil-group. The inner
ring denotes the slot numbers.
The coil pitch is 10 slots, e.g., slot 1 to slot 11, throughout.
The winding connections are:
8-poles--Parallel-Star
10-poles--Series-Delta
For 8 pole/10-pole windings the adjacent harmonics are 4-poles and 14-poles
respectively.
Table 1 shows the adjacent harmonic m.m.f. content, the value of R=(x+z)/2y
where: x - y - z - 0 defines the half-phase coil-grouping, the two winding
factors and the air-gap flux-density ratio for the winding of FIG. 1 and
for the three other "third-harmonic" windings and the four known windings
corresponding.
It will be noted that the adjacent harmonic of 14-poles for the 10-pole
connection is reduced to about one-fifth in the third-harmonic winding
compared with the corresponding known winding and the adjacent harmonic of
4-poles for the 8-pole connection is reduced to about one-half. Of these,
the 14-pole adjacent harmonic is the more likely to produce
double-frequency vibration, which will be in the 10-pole connection of the
winding.
FIG. 2 shows four coil-grouping diagrams relating to 8-pole/10-pole
windings in 144 slots, two of these windings, referenced (b) and (d),
being similarly referenced in Table 1.
As shown in FIG. 2, the coil-grouping of known 8-pole/10-pole windings,
corresponding to "sinusoidal" modulation, would be:
(a) 6 - 12 - 6 - 0 - - Repeat or
(b) 7 - 10 - 7 - 0 - - Repeat
for each phase.
The alternative coil-grouping, representing added third-harmonic
modulation, is:
(c) 9 - 7 - 8 - 0 - - Repeat or
(d) 9 - 6 - 9 - 0 - - Repeat
for each phase.
As previously explained, the addition of third-harmonic modulation
corresponds to increasing the number of coils in two outer coil groups and
decreasing the number of coils in an intermediate coil group. The shape of
the curves (c) and (d) of FIG. 2 exhibits recognizable distortion of the
more sinusoidal (half-wave) shapes of the curves (a) and (b).
The m.m.f. harmonic content corresponding to these coil-grouping examples
is readily evaluated by computer. For the 8-pole/10-pole winding in 144
slots, the coil-grouping examples (b) and (d) would be chosen for the
"sinusoidal" and "third-harmonic" modulation examples and these two are
included in Table 1.
The other examples of Table 1 represent the same logic and have been
selected by the same m.m.f. analysis.
Finally, the winding diagram, for any required 8-pole/10-pole winding, for
any permissible slot number from 72 slots upwards can be readily prepared
using the logic described.
FIG. 3 shows, as a particular example, the third-harmonic modulation
winding form for an 8-pole/10-pole winding in 120 slots using the
coil-grouping: 7 - 5 - 8 - 0 - - Repeat, inserted in the clock diagram of
FIG. 1, and indicated by the "*" in Table 1. Table 1 also lists the
reduced adjacent harmonic content.
If each half phase-winding of any winding designed is represented: x - y -
z - 0, then the average ratio (R) given by (x+z)/2y is, for the 120 slot
windings, 0.75 for the "sinusoidal" winding and 1.5 for the "third
harmonic" winding.
Table 1 sets out four examples of "third harmonic" windings for
8-pole/10-pole machines. Exactly the same design method can be applied for
other symmetrical P.A.M. pole-combinations such as 14-pole/16-pole and
8-pole/14-pole machines.
With stators of the slot-numbers normally used, the numbers of coils per
coil group are fewer for large pole-numbers than for small pole-numbers.
With the necessary limitation, in the change of coil-grouping to provide
third-harmonic modulation, of moving integral numbers of coils from a
central coil group to outer coil groups, the optimum curve shape, see FIG.
2, is not always obtainable. It may sometimes be preferred to use, for
symmetrical P.A.M. pole-combinations, the design method later described
herein as applicable to asymmetrical P.A.M. pole-combinations, thus
departing from simple lumped coil-grouping.
The examples described above with reference to Table 1 are symmetrical
P.A.M. windings using sum overall modulation, that is the (fundamental)
modulation wave applied overall to the three-phase winding has a number of
poles equal to the sum of the alternative pole-numbers.
There is a second sub-class of symmetrical P.A.M. windings using different
overall modulation, that is the (fundamental) modulation wave applied
overall has a number of poles equal to the difference of the alternative
pole-numbers.
A similar logic is applied.
Taking as example an 8-pole/14-pole winding, difference overall modulation
is to be preferred because (14-8) is 6, which is a multiple of "3",
whereas (14+8) is 22, which is not a multiple of "3".
Numerically, the Orders of the adjacent harmonics is changed by the choice,
but the principle of added third-harmonic modulation to reduce the
adjacent harmonic content is unchanged.
Continuing with the 8-pole/14-pole example, the winding provides 8 poles
when unmodulated and (8.+-.6) poles when modulated, that is 14-poles, the
required pole-number and 2-poles as an acceptable subharmonic.
The adjacent harmonic for 14 poles is of 10 poles. The difference of Orders
is still 2, but in this case of lower Order than the working pole-number
of 14 poles.
Third-harmonic modulation provides (8.+-.3.times.6) poles=(8.+-.18)
poles=10 poles, equal to the adjacent harmonic and 26 poles, a high-Order
harmonic which can be reduced by chording.
By choice of the amount of third-harmonic modulation added, the resultant
adjacent harmonic in the 14-pole m.m.f. waveform can be reduced to a low
value.
For the 8-pole connection, the adjacent harmonic is of 4-poles, as it would
be for a machine using sum overall modulation. The 8-pole adjacent
harmonic of 4-poles is similarly reduced by the third-harmonic modulation.
Asymmetrical P.A.M. Windings
Although the same principle of added third-harmonic modulation to reduce
adjacent m.m.f. harmonics is applied also to asymmetrical P.A.M. windings,
the principle is more difficult to apply in practice. Although in
practical machines the fundamental m.m.f.'s will be balanced between
phases, the m.m.f. harmonics are usually unbalanced, for any Order of
harmonic. Further, the coil-grouping of one of the three phases differs
from that of the other two.
The design process for an asymmetrical P.A.M. winding, in a stator of
n-slots, usually 72 or more slots, starts with the choice of an initial
winding element. This winding element will occupy only a few slots, say 2
to 10 slots, per half-element. There follows one or two stages of
symmetrization by which, in the first stage, the initial winding element
is triplicated, with a relative displacement between elements. Similarly,
for the second stage, the first stage resultant winding is triplicated
with a relative displacement between the component windings.
Hence, this design process starts with a winding element of n/9 slots or
n/18 slots per half winding-element. As for all P.A.M. windings, each
phase-winding must have two equal half phase-windings, so that
series/parallel switching with 6 terminals is possible.
So far, the discussion is a summary of published P.A.M. design theory.
Table 2 lists all divisions of total coils between the phases per
half-element, for an initial three-phase winding element, likely to be
used in practice. All are n/18 slots per half winding-element. The
alternatives are tabulated under element/stator slot numbers varying from
2/36 to 10/180, respectively. The number of options, for each stator
slot-number, is shown in brackets at the foot of the respective column.
It may be mentioned here that the invention as applied to asymmetrical
P.A.M. windings is not limited to stators of slot-number divisible by
"18". In the examples which follow, one example, that of FIG. 11, uses a
stator of 120 slots. Nevertheless, it is generally preferred, for
asymmetrical windings, that the slot-number is divisible by "18".
In considering Table 2, it is to be noted that earlier P.A.M. theory
assumed that the coils of any half winding-element should be divided
between the three phases either:
(1) equally between the phases, or
(2) as nearly equally as possible, or
(3) equally between two phases, the third phase having zero coils.
This assumption is untrue and, for all but the lowest slot-number, Table 1
includes coil divisions which are unequal between phases, but which
provide satisfactory windings after successive symmetrization.
Every m.m.f. harmonic in an initial winding element is reduced at each
stage of symmetrization by a factor which is invariant for each particular
harmonic. The fundamental is virtually unchanged. Consequently, the final
m.m.f. harmonic content depends upon the harmonic content of the initial
winding-element and the number of stages of symmetrization.
For any particular harmonic, that half winding-element which has the lowest
content of that harmonic initially, will similarly have the lowest content
finally, for the same number of symmetrization stages. The fir | | |
