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Title:
FLUX IMPULSE MOTOR
Document Type and Number:
WIPO Patent Application WO/2002/101907
Kind Code:
A1
Abstract:
A motor (1) has a stator (2) and a rotor (10) mounted for rotation in the stator. A commutating salient pole (11) of the stator has a winding (7), and a field-producing salient pole (12) has a winding (8). The rotor has at least two salient poles (10 a, b). An outer back-iron (13) magnetically joins bases of said stator poles. An inner back-iron (14) extends from the field pole around the rotor to such an extent that a substantial amount of the magnetic field in the rotor between its poles is short-circuited by the inner back-iron, at least for a part of its rotation. An electrical circuit powers the windings to drivingly rotate the rotor.

Inventors:
LEFLEY PAUL (GB)
Application Number:
PCT/GB2002/002442
Publication Date:
December 19, 2002
Filing Date:
June 11, 2002
Export Citation:
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Assignee:
HOBBY ROGER BARRY (GB)
LEFLEY PAUL (GB)
International Classes:
H02K19/10; H02K21/16; (IPC1-7): H02K21/38
Foreign References:
US5717316A1998-02-10
US5117144A1992-05-26
EP0455578A21991-11-06
Other References:
See also references of EP 1396068A1
Attorney, Agent or Firm:
Harrison, Goddard Foote (Leopold Street, Sheffield S1 2QD, GB)
Download PDF:
Claims:
CLAIMS
1. A motor comprising: a stator ; a rotor mounted for rotation in the stator ; a commutating pole of the stator; a commutating winding; a fieldconnecting pole of the stator; a pair of salient poles of the rotor; an outer backiron magnetically joining said stator poles ; an inner backiron, extending from said field connecting pole around the rotor to such an extent that a substantial amount of the magnetic field in the rotor between said poles of the rotor is short circuited by said inner backiron for at least a part of the rotation of the rotor; and an electrical circuit to power the commutating winding to drivingly rotate the rotor.
2. A motor as claimed in Claim 1, in which the angular extent of said poles of the rotor is about the same as the total angular extent of the inner back iron, about (270/n) °, where n is the number of rotor pole pairs.
3. A motor as claimed in Claim 1 or 2, in which the angular extent of one of said poles of the rotor is about the same as the angular extent of the commutating pole, preferably about (90/n)', where n is \ the number of rotor pole pairs.
4. A motor as claimed in Claim 1, 2 or 3, in which said poles of the rotor have a varying radius across their angular extent such that said poles are short circuited over an angle a equal to about (45/n) °, where n is the number of rotor pole pairs.
5. A motor as claimed in any preceding claim, in which there are an equal plurality each of commutating, and fieldconnecting poles alternately disposed around the stator, and twice as many poles of the rotor as there are fieldconnecting poles.
6. A motor as claimed in Claim 5, in which there are two fieldconnecting poles, two commutating poles and a cross shaped rotor having two pole pairs.
7. A motor as claimed in Claims 4 and 6, in which the angle a is about 22°.
8. A motor as claimed in any preceding claim, in which the rotor is magnetised so that said rotor poles are oppositely magnetised, said electrical circuit being controlled to produce an alternating magnetic field in the commutating pole of the stator to attract each pole of the rotor as it approaches the commutating pole and to repel each pole of the rotor as it moves away from the commutating pole, said field alternating as many times per revolution of the rotor as there are poles of the rotor.
9. A motor as claimed in any of Claims 1 to 7, in which the rotor is not magnetised, and instead a permanent magnetisation is provided in said inner backiron, said electrical circuit being controlled to produce a switched magnetic field in said commutating pole of the stator and which field is energised to attract one of said poles of the rotor until a minimum reluctance is achieved, whereupon the field is de energised until the other of said poles approaches the commutating pole of the stator.
10. A motor as claimed in any preceding claim, in which said poles of the stator are salient.
11. A motor as claimed in any preceding claim, in which said commutating winding is around the commutating pole of the stator.
12. A motor as claimed in any preceding claim, further comprising a field winding around the fieldconnecting pole.
13. A motor as claimed in Claim 11 and 12, in which said commutating and field windings are in series.
14. A motor as claimed in Claim 13, in which the field winding develops a larger magnetic field than the commutating winding.
15. A reciprocating flux impulse motor, substantially as hereinbefore described with reference to Figures 2, 3,6b and 7 of the accompanying drawings.
16. A unidirectional flux impulse motor, substantially as hereinbefore described with reference to Figures 1, 4,5,6a and 8 of the accompanying drawings.
Description:
FLUX IMPULSE MOTOR The present invention relates to electric motors of the brushless type.

Such motors comprise a stator and a rotor, the rotor rotating in the stator. The rotor is often, but not necessarily, magnetised. The stator usually comprises salient poles mounting windings forming part of an electrical circuit. By appropriate switching of the circuit the rotor can be driven to rotate at a speed determined by the circuit, usually in a synchronous manner.

There are numerous designs of motor and EP-A-0455578 discloses one arrangement. In this, a hybrid single phase variable reluctance motor is disclosed. Here, two commutating stator poles have two other poles between them, one carrying a permanent magnet to park the rotor in a position in which the motor will start when the commutating poles are energised, and the other carrying a position-sensing device. The arrangement is very simple and effective, with the circuit arrangements being easy to manufacture resulting in a low-cost motor suitable for applications such as driving fans.

However, in this simple mode, the torque produced is very pulsatile, leading to uneven operation.

It is an object of the present invention to provide a simple motor which not only parks the rotor in a startable position but which also provides a smooth operation.

In accordance with the present invention, there is

provided a motor comprising: a stator; a rotor mounted for rotation in the stator ; a commutating pole of the stator ; a commuta. ting winding; a field-connecting pole of the stator ; a pair of salient poles of the rotor ; an outer back-iron magnetically joining said stator poles; an inner back-iron, extending from said field- connecting pole around the rotor to such an extent that a substantial amount of the magnetic field in the rotor between said poles of the rotor is short-circuited by said inner back-iron for at least a part of the rotation of the rotor ; and an electrical circuit to power the commutating winding to drivingly rotate the rotor.

The short-circuiting of the magnetic field through the rotor by the inner back-iron provides a low reluctance path which has both a positive and negative impact on the torque applied to the rotor. The effect is positive when reducing reluctance is experienced, as the short-circuiting commences, and torque is applied to the rotor, but is negative when the short-circuiting ceases and an equivalent negative torque is applied. This has a smoothing effect on the overall torque curve since the negative effect can be arranged to coincide with the main driving"pulse"of the motor. This reduces the size of the driving pulse, which is instead seen"translated" into the positive effect of the, short-circuited field.

The poles of the stator are preferably salient. The commutating winding may be around the commutating pole of 'the stator. A field winding might be provided around the

field-connecting pole. Moreover, the commutating and field windings may be in series. Indeed, the field winding could develop a larger magnetic field than the commutating winding.

Preferably, the angular extent of said poles of the rotor is about the same as the total angular extent of the inner back-iron, about (270/n), where n is the number of rotor pole pairs. By"angular extent"is meant the angle of the segment (s) of a circle which includes both said poles of the rotor, or, in the case of the inner back-iron, the inner back-iron.

Preferably, the angular extent of one of said poles of the rotor is about the same as the angular extent of the commutating pole, which is preferably about (90/n) Preferably, said poles of the rotor have a varying radius across their angular extent such that said poles are short-circuited over an angle a equal to about (45/n) °.

When there is just one of said poles then said angle a is about 45°.

Preferably, there are an equal plurality each of commutating and field-connecting poles, alternately disposed around the stator, and twice as many poles of the rotor as there are field-connecting poles.

Preferably there are two field poles, two commutating poles and a cross-shaped rotor. In this event, the angle a is about 22°.

Preferably, the rotor is magnetised, said rotor poles being oppositely magnetised and said electrical circuit being controlled to produce an alternating magnetic field in the commutating pole of the stator to attract each pole of the rotor as it approaches the

commutating pole and repel each pole.. of the rotor as it moves away from the commutating pole, said field alternating as many times per revolution of the rotor as there are poles of the rotor.

When the rotor is cross-shaped with two pole pairs, the poles are alternately magnetised. Such an arrangement is preferable from a torque perspective but requires an electrical circuit having at least two switches in order to change the direction of magnetisation of the commutating pole (as many times per revolution as there are poles of the rotor).

Alternatively, the rotor may not be magnetised, and instead a permanent magnet is disposed in said inner back-iron, said electrical circuit being controlled to produce a switched magnetic field in said commutating pole of the stator and which field is energised to attract one of said poles of the rotor until a minimum reluctance is achieved, whereupon the field is de- energised until the other of said poles approaches the commutating pole of the stator, This arrangement has the advantage that only a single switch is required since the direction of magnetisation of the commutating pole can be the same, regardless of the orientation of the rotor. In this event, the field through the rotor switched as the rotor rotates. Thus, this arrangement provides the simplest electrical circuit, although it cannot produce the same torque per ampere of current in the windings.

The invention is further described hereinafter, by way of example, with reference to the accompanying drawings in which:

Figures A to. D are schematic diagrams of a known motor, not forming part of the present invention; Figures la to d are schematic diagrams of a motor in accordance with the invention having a two-pole rotor, in each drawing the rotor being in a different angular position ; Figures 2a to d are the same as Figures la to d, except here the rotor is permanently magnetised ; Figures 3a to c are similar views of a further embodiment of the present invention in which a four-pole rotor is employed having alternate permanent magnetisation of its poles ; Figure 4 is a similar view to Figure 3c of a further embodiment., except that here, like Figures la to d, the rotor is not magnetised, permanent magnetisation being incorporated in the stator ; Figure 5 is a view similar to Figure 4 of a simpler embodiment of the present invention.

Figures 6a and b are torque curves for the motors of Figures 5 and 3 respectively.

Figures 7a and b are different circuit arrangements for powering the motors of Figures 2 and 3; Figure 8 is a circuit arrangement for powering the motors of Figures 1, 4 and 5; Figures 9 a to c show the flux distribution of the motor of Figure 5 when a south pole is formed at the commutating poles; Figure 10 shows the flux distribution of the motor of Figure 5 when a north pole is formed at the commutating poles.

\ Figure A shows a basic arrangement of a known two pole flux impulse motor a, comprising a two pole rotor b, a stator consisting of two commutating poles c and d, and two field poles e and f : The field poles may

either be permanently magnetised with one possible arrangement (as shown) or there is a field winding (around the field poles) with a dc current flowing to produce the magnetisation shown. Coils (not shown) are wound around the commutating poles to form a winding called the commutating winding. The poles g of the rotor may not have a constant radius arc at the pole extremities. Instead, a tapering curved leading edge h of the poles is provided, such that the radial air gap i created between the poles of the rotor and any of the stator poles varies during rotation. There may also be other features of the rotor poles to enhance the asymmetry of the two rotor poles. For example, instead of the (preferred) infinitely varying gap i shown in Figures A to D, a step or shoulder may be formed at the pole. tip.

In the absence of a current in the commutating winding the rotor is at a position of rest as shown in Figure A. If a dc current is applied to the commutating winding a field is created on the commutating poles with a north pole on pole c and a south pole on pole d (see Figure B). As a result of the field, the rotor b rotates from a position of rest to another position of equilibrium shown in Figure B. If the magnitude of the current in the commutating winding is increased, then the rotor rotates further to another position-of equilibrium, for example as shown in Figure C. If the current in the commutating winding is now switched off, (and in the absence of any rotational inertia of the rotor. in a clockwise sense), the rotor may rotate anticlockwise and settle in the initial position of equilibrium shown in figure A.

This demonstrates a possible starting problem with this type of motor. In order to ensure that the motor is able to start up from rest and accelerate, the magnitude of the magnetic field produced at the commutating poles has to be much greater than the permanent field produced by the field poles. To ensure a continuous rotation, the commutating poles must pull the rotor to the position shown in Figure D. In this position, and upon releasing the current in the commutating winding, the field poles will pull the rotor clockwise to the position shown in Figure A (completing a rotation of 180 degrees).

Clearly a large pulse of current is required to start the motor, or the magnitude of the permanent field needs to be weak, at least in order to avoid excessive currents. As soon as the rotor is rotating at some speed, large impulses of flux produced at the commutating poles pull the rotor from the field alignment flux. The inertia of the rotor (and some torque produced by the field alignment flux) carries the rotor to the next commutation position. Thus the current in the commutating winding is not continuous and has a duty of typically around 50%.

To ease the starting and running problems it may be advantageous to switch off the field at certain times.

However, this means that permanent magnets may not be used to produce the field flux, and that there must be appropriately timed commutation of the field winding current, in addition to the commutating winding current.

This adds extra complexity to the design of the motor and the power electronics.

Turning to the present invention therefore, in Figure 1 to the drawings, a motor l comprises a stator 2

and a rotor 10 mounted for rotation within the stator 2.

The stator 2 has a commutating pole 11 and a field pole 12. The stems or bases of the commutating pole 11 and field pole 12 are joined by an outer back-iron 13a, b. The commutating pole 11 has a commutating winding 7 which is in series (or parallel) with a field winding 8 on the field pole 12. However, there may be some advantage in keeping the commutating pole coils 7 separate from the field connecting pole coils 8.

The angular extent x of the commutating pole 11 is about 90°. The rotor 10 has two salient poles 10a, b, the angular extent of which is likewise about 90°. Each pole is provided with a shoulder so as to provide an enlarged air gap 9a on the leading edge of the rotor, and a thin air gap 9b on the trailing edge of the rotor. This enlarged air gap 9a ensures rotation of the rotor 10 in the direction of the arrow A. It means that flux connection between the rotor occurs over only about half the angular extent of the rotor, ie about 45°. Although shown as a step, it is preferred that an arrangement similar to that shown in Figures A to D is employed which provides a smoother transition between flux connection and isolation. For convenience, however, a stepped shoulder is shown here, although and nothing at all is shown in Figures 3 to 5, and 9 and 10, below.

The field pole 12 is provided with two limbs 12a, b which, between them, define an inner back-iron 14. The angular extent of the inner back-iron is about 270°. As mentioned above, the poles 10a, b of the rotor (ie, those parts presenting the minimum air gap 9b with the stator poles) subtend an angle of about 45° each. Accordingly, as can be seen in Figures lc and d, the angle a of

rotation of the rotor over which both poles 10a, b of the rotor 10 lie adjacent the back-iron 14 (ie are short- circuited by it) is about 45°.

In operation, the rotor 10 is rotating in the direction of the arrow A in Figure la. That Figure shows the rotor 10 in the zero angle position. In this position there is is excitation of the commutating coils 7 and a north pole is presented at the commutating pole 11. With reference also to the torque curve in Figure 6a, it can be seen that, in the zero position there is a small positive torque. This is because the minimum reluctance position has not yet arrived but occurs at about 15°,.. at which point, the power to the windings 7,8 is turned off. The rotor is then drawn with high torque being applied by the magnetisation of the limb 12b of the back-iron 14. This torque is applied until the minimum reluctance position of the rotor 10 within the confines of the back-iron 14 is achieved at about 100°, as shown in Figure lc.

However, at about 90°, ie before the minimum reluctance position is reached, the windings 7,8 are re- energised and a positive torque is applied as the pole 10b is attracted by the strong field produced by the commutating and field coils 7,8. Torque therefore continues to be applied to the rotor until beyond the 180° position, which corresponds with the zero position of Figure la. The cycle thereafter repeats each 180 mechanical degrees.

However, between about 135° and 180°, the developing flux of the magnetic field (see dashed arrows in Figure ld) caused by the excitation of the commutating and field-connecting coils 7,8 interacts with the closed-

circuit flux loop (solid arrows) caused by the magnetisation of the inner back iron. See the opposite directions of the flux lines in the limb 12c of the back- iron 14. Thus, while the closed loop exists, it reduces the torque developed by the commutated magnetic flux.

Hence the drop in torque seen at about 160° in Figure 6a.

Nevertheless, there is a dual effect taking place. On the one hand, the pole 10b progressively"pinches-off" the air gap across which the closed loop flux crosses, so that the negative effect of the closed loop reduces. On the other hand, the pole 10b progressively connects with the commutating pole 11, so that the commutated flux leaving pole 10a and opposing the closed loop flux in limb 12c, increases.

In any event, the effect of the enlarged field pole 12 producing an inner back-iron 14 is that, not only does the rotor park (in either of the Figure lc or Id positions) in a position at which it will start when power is first applied, but also it encourages combining of the fields produced by each source. The effect of this seems to be that the current impulse to force the rotor to deflect from its low reluctance position (ie Figure ld) need not be as large as required in the prior art arrangements shown in Figures A to D or exemplified by EP-A-455578 where the field produced are orthogonal.

The power delivery to the rotor is also smoothed, reducing the need for inertial or other smoothing,.

Turning to Figures 2a to d, a variation on the Figure 1 embodiment is shown in that the rotor 10'is magnetised, whereas the field pole limbs 12a, b are not.

Otherwise this embodiment is identical with Figure 1, although the driving causes are different.

For example, about 10 after the zero position shown in Figure 2a, the commutating windings 7 (and field- connecting windings 8) are energised to create a north pole at commutating pole 11. This repels the rotor 10' in the direction of rotation (Arrow A). At about 45°, a reluctance effect in the developing magnetic short- circuit through the inner back-iron 14 produces further driving torque (beyond what it would have been without it as shown in phantom lines in Figure 6). This diminishes at about 90°, however, when the short-circuit is complete.

On the other hand, at this point, the south pole 10b of the rotor 10'is now attracted by the north commutating pole, which attraction increases between 135° and 190°.

Such increase would normally also increase the torque felt by the rotor (dashed line in Figure 6) except that, at this phase, the low reluctance effect of the magnetic short-circuit (arrows, Figure 2d) is being broken. At about 190°, the current through the field windings 7,8 is reversed to present a south pole at the commutating pole 11. Thereafter, the cycle is repeated in reverse but leading to the same shape of torque curve.

As can be seen from the upper curve in Figure 6, there is some smoothing of the torque applied to the rotor.

Although the present invention works, in principle, with a two-pole rotor, the arrangements shown in Figures 3 and 4 are preferred because, over-180° of mechanical rotation twice as many peaks and troughs in the torque \ curve are experienced. For the same inertial mass of rotor, this will lead to smoother rotation, despite the fact that the electrical circuit required is the same.

In Figure 3a, a four-pole rotor 10"is in the form

of a cross and is magnetised to present, alternating north and south poles 10a, b, c and d around the cross. Each pole is stepped or curved (not shown), as in the embodiments of Figures 1 and 2, to present a variable air gap for rotation direction control.

When the coils are energised in the Figure 3a position, each commutating pole 11"presents a north magnetic pole to the north poles of the rotor 10". This repels the two north poles 10a, c of the rotor and so the rotor moves clockwise. The inner iron ring 14 now starts to provide a low reluctance path between the north and south pole pairs 10a, b and 10c, d of the rotor 10".

The rotor rotates to a position of minimum reluctance, a few degrees clockwise beyond the position shown in Figure 3b. This will naturally occur with or without energisation from the stator windings. When the rotor is at the position shown in Figure 3b the stator coils are energised (if not already-that is to say, if turned off for a period between the Figure 3a and b positions) so that a north pole is present at the airgap surface of commutating poles, 11. Now the south poles of the rotor are attracted to the commutating poles. The method of torque production at this step is by a combination of electromagnetic alignment torque and reluctance torque.

With the rotor in the position as shown in Figure 3c, the current in the stator windings is reversed as quickly as possible. This impulse in the magnetic field kicks'the south poles of the rotor by applying an opposing field and thus pushes the rotor again with a clockwise rotation. The process is repeated for every 90 degrees of rotation, but each time the directions of

current are reversed. As with the previously discussed embodiments, one electrical cycle may be seen to occur every 180° of mechanical rotation.. The reversal of the field in the commutating poles may describe this machine as a reciprocating flux impulse motor.

A simpler version of the motor is shown in Figure 4, where the field in the commutating poles and back iron is unidirectional (ie, it does not reverse), and the inner iron ring 14"possesses a magnetisation as shown. This arrangement corresponds with the Figure la to d embodiment of the present invention. Here the inner iron has a four-pole magnetisation to attract the four-poles of the rotor when at rest.

Also, for a four-pole rotor, a two-pole magnetisation may also be applicable as shown in Figure 5 However, with this arrangement, the permanent field in the inner iron 14"'will be shared with the back iron 13" When current is applied to the coils 8,9, the commutating poles will attract the poles of the rotor.

The direction of current is important as this will enhance the permanent field in the inner iron or try to oppose it and may affect the torque production mechanism at commutation. However, if sufficient ampere-turns is applied, the commutating poles will attract the nearest rotor poles and (mostly) reluctance torque will be produced. As there is a variable air gap between the commutating poles 11"'and the rotor poles 10"', the commutating poles will pull the rotor to a position similar to that shown in Figure 3a.

Upon releasing the current, the rotor will continue

to rotate clockwise due to the permanent magnetisation of the inner iron. As there is no permanent magnetisation of the rotor 10"', the commuting current may be unidirectional. This applies also to the Figure 4 arrangement.

Typical electrical drive circuits for the reciprocating flux impulse motor (ie the embodiments of Figures 2 and 3) are shown in Figure 7. Figure 7a shows an arrangement using two transistor switches 16, allowing a bi-directional field to be produced using a bifilar winding 15a, b for the commutating and field coils 8,9 respectively. Alternatively, a bi-directional field may be produced using an H-bridge arrangement shown in Figure 7b.

For the reciprocating flux impulse motor to function satisfactorily the rapid reversal of the flux is important. This can be achieved by the circuit in Figure 7a which employs a snubber arrangement 17 to controllably allow the turn-off voltage across the switch 16 to rise to a maximum voltage. Thus this large voltage opposes the inductive current in the winding and so rapidly forcing it to zero. Simultaneously turning on the other switch allows the field to build up in the other 1 direction. The stored energy absorbed during the turn- off of the first switch may be used to forcibly and rapidly increase (or kick-start') the current in the other winding, rather than simply allowing the current to steadily build up with a normal supply voltage. This is because the stored energy can be arranged to be at a much higher voltage than the voltage of the power supply.

Alternatively, the stored energy in the snubber 17 can just be returned to the supply, rather than being

dissipated in a resistance. In either case the snubber circuit is said to be regenerative, where the recovered energy is not lost, and is, therefore, more efficient than a conventional RCD (resistor-capacitor-diode) snubber.

A typical circuit for the simple (unidirectional) flux impulse motor of Figures 1,4 and 5, is shown in Figure 8 using only a single winding, 15, a single switch, 16, and a snubber circuit that may be regenerative.

In both types of motor the firing of the drive circuit ist synchronised to the rotor position relative to the stator by a suitable encoder. On the other hand, it is envisaged that sensorless techniques may be used to achieve the same objective.

Turning finally to Figures 9 and 10, the unidirectional flux impulse motor of the present invention has parking magnets (N) in the inner iron of the stator, as shown in Figure 9a. When commutation is about to take place, the direction of the magnetic field in the commutating poles 11 affects the torque producing mechanism of the rotor. If the parking magnets presents north'poles into the airgap of the motor and the direction of the current in the coils allow the commutating poles to be south'poles at the airgap then the magnetic flux path in the motor is shown in Figure 9a. It can be seen that the field due to the magnets and the field due to the commutating poles are coincident and the net attractive force on two of the rotor poles is great. The rotor is forced to move to a position as shown in Figure 9b. At this point the other two neighbouring poles of the rotor are approaching the

vicinity of the parking magnets. Instead of the rotor stopping at the position shown in Figure 9b, the magnetic field finds a new (lower reluctance) path via the neighbouring poles, see Figure 9c. Motoring torque is maintained in the situation of Figure 9c, where the neighbouring poles of the rotor are being pulled into alignment with the combined field passing through the magnets.

If, however, the current in the coils is in the reverse direction, such that the commutating poles presented north'magnetic poles to the airgap (like the magnets) then the magnetic fields due to the magnets and the stator excitation will not be coincident. Instead, they will seek independent paths throughout the iron of the motor, see Figure 10. This situation does not effectively and efficiently commutate the motor and may result in stalling the machine. This arrangement should therefore be avoided.

The flux impulse motor of the present invention may typically operate as a variable speed drive utilising one or two power transistors to commutate the current in the commutating winding. Other arrangements are possible using more than two transistors including the possibility to commutate the field winding current, but this adds complexity and cost to the drive. The speed of the machine is controlled by either varying the magnitude or duration (or both) of the current in the commuting winding (and possibly the field winding). The effect is to control the magnitude of thé torque producing impulse of flux at the commutating poles. The magnitude of the current is varied by chopping the current (usually at some high frequency). This may cause starting problems though, so phase angle control of the current is an

alternative (if not a preferred) option. Phase angle control operates by introducing a variable delay at turn- on of the current in the commutating winding. This delay is typically zero at start-up and is increased to achieve the desired operating speed. This may be achieved by utilising some form of closed loop feedback control system.




 
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