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The design and characteristics of the brushed type of permanent magnet DC motor were reviewed in Section 1. It was seen that the action of the brushes and commutator ensures that the current flow through the rotor winding is always in the same direction relative to that of the permanent magnetic field. It was also found that the current in each turn of the winding has a rectangular AC waveform, alternating in direction as the winding rotates. One of the roles of the brushes and commutator is therefore to act as an inverter, converting DC from the power supply to rectangular AC for use around the turns of the winding. This is the key to the understanding of the brushless motor, where the brushes and commutator are replaced by an electronic inverter. The inverter separates the remaining electromechanical device from the DC supply and provides it with the required AC. The layout of the normal brushed DC motor is fixed by the very use of brushes to effect reversals in the direction of the motor current. It is the rotor of the brushed motor which must carry the load current, and only the stator is available for the magnets. This construction has some disadvantages, not only in the commutation of heavy load currents but also in the generation of losses in the part of the machine most difficult to cool. In the brushless servomotor the load conductors can be placed on the stator and the permanent magnets, which need no external connections, on the rotor.
The i2R losses are more easily removed from this stator than is possible from the rotor of the brushed machine.
Electronic inverters and the method of supplying the current to the brushless motor are described in Section 3. This Section is concerned with the brushless machine itself. The operation of the motor is dealt with from first principles using the basic electromagnetic fundamentals introduced in Section 1. The important features of the magnetic circuit of the motor are covered later on in the Section.
2. Structure and operation
FIG. 3 shows a complete brushless servomotor and FIG. 1 shows its main components. The load conductors are wound on a stator core which is separated from the finned, aluminum case by an electrically insulating sleeve. FIG. 1 shows two rotors of different designs.
The stator core is made of silicon steel, with slots for the current-carrying conductors. The core is laminated in order to minimize eddy currents. The laminations, one of which is shown in FIG. 1, are punched from 0.3-0.5 mm steel sheet. The slots are seen to take up a relatively large part of the total area of the lamination, and have the effect of disrupting the uniformity of the path of the flux. The effect is reduced if the slots are skewed relative to the stator axis, as can be seen in the figure. The skewing presents an overlapping slot pattern and a less irregular path to the flux which enters and leaves the core radially. The windings can sometimes produce audible noise as they vibrate under the operating forces of the motor. An alternative construction is possible where the stator is rigidly bound by a resin mold, reducing its freedom to vibrate.
The permanent magnet rotor
The rotor hub carries the permanent magnets, and is pressed into position on the motor shaft. The hub can be machined from solid, low carbon steel or assembled from laminations punched from the center of the steel sheet used for the stator laminations. The rotors shown in FIG. 1 are of four-pole design. One has magnets of a cylindrical shape, and the other has magnets with non-circular surfaces. For a two-pole rotor with cylindrical magnets, the ideal flux density around the pole circumference would vary as a single-cycle rectangular wave, as shown in FIG. 2. In practice some irregularity remains in the magnetic circuit even when the slots are skewed, and the dotted line along the top of the flux wave shows the effect of the irregularity on the waveform of the flux density in the air gap.
Magnets with a high flux density are used to maximize the torque/rotor volume ratio of the brushless servomotor. These must be very firmly fixed to the hub, and this has been one of the main problems of manufacture. As well as the tensile radial force on the magnets at high rotor speed, there are also shear forces which must be resisted during abrupt acceleration and deceleration. The magnets are bonded to the hub using adhesives with mechanical and thermal expansion coefficients.
FIG. 3 shows one of the most common, where the magnets are bound by fiberglass tape. FIG. 1 shows the rotors before the tape and end rings have been fitted. The extra manufacturing cost involved in bonding and binding the magnets to the hub is a small drawback to the permanent magnet rotor.
The typical stator lamination shown in FIG. 1 has deep slots for the stator conductors, and the conductors are therefore not in the air gap between the rotor and stator surfaces. In Section 1, a simple explanation of the operation of the brushed motor was given without reference to the slotted nature of the rotor. When the torque mechanism was described, the rotor conductors were assumed to cut the air gap flux around the pole faces. Before using the same method with the brushless motor, we should first look at whether or not the assumption is valid.
Flux cutting and flux linkage
In Section 1, expressions for torque and emf were found using the flux-cutting approach. With this method, the starting point in the derivation of emf is the expression
e = Blv
Flux of density B is assumed to be cut by a conductor of length l moving at velocity v, or to pass across one that is stationary.
FIG. 4 shows the conductors of a brushless motor winding lying in a stator slot and almost surrounded by the stator iron. The objection to the flux-cutting method which is sometimes raised is that most of the flux does not cross the conductor but instead passes around the slot, through the iron.
The flux-cutting method overcomes this problem by assuming that all the conductors are effectively on the inner surface of the stator, in the air gap flux at a radius of action equal to half the diameter of the stator bore. With these assumptions, it is found that the method gives the correct basic results for motor torque and back emf. The conceptual difficulty with the flux-cutting approach is avoided in the flux-linkage method. Faraday's law is used to express the emf induced across one meter of a conductor as
E=d lambda / dt
where lambda is the quantity of flux which links the conductor completely at time t. Both methods lead to identical results for the basic analysis of motor torque and back emf, and so we will continue to use the flux-cutting approach.
FIG. 5 shows four positions of a rotor, relative to the stator of a simple two-pole brushless motor. The associated directions of the stator current are shown by using the circled cross and dot convention for current flowing into and out of the paper.
A stator with a single winding has been chosen for the purpose of explaining the operating principle. The stator of a brushless servomotor normally has three windings, but the principle is the same.
In FIG. 5, the current direction in the winding is reversed (using external means) at positions 2 and 4, when the pole centers pass from one side of the winding to the other.
Consequently, at rotor angle 0 the average of the forces BII on the current-carrying conductors is always in the same direction, except at 90 degree and 270 degree when the overall force is zero.
The resulting torque cannot cause rotation of the stator and so the equal and opposite reaction at the rotor provides output torque to the load. The torque is produced in the same way as in the brushed machine and we can express the average output as:
The peak torque at positions 1 and 3 in Figures 5 and 6 is equivalent to the steady output torque of the two-pole brushed motor already described in Figure 1.3. The average torque constant of this particular brushless machine is therefore only half that of the brushed motor. The ripple in the output torque can be described as 100% and would be unacceptable in most servomotor applications. The problem arises when the poles are halfway between one side of the winding and the other, where the net force is zero. We will see shortly how the use of three windings eliminates such nulls to produce a theoretically constant output.
Rotating flux sweeps across the stator winding to produce a back emf with an average half-cycle value of
where KE is half the value of the voltage constant of the brushed machine of Figure 1.3. The back emf alternates in direction as the poles of the magnet change position, as shown in FIG. 6. It is important to note that the back emf at the input terminals of the brushless motor alternates in direction, as does the direct current input. The motor has the same construction as an AC synchronous motor, which normally has a sinusoidal rather than rectangular current waveform. Although the single-phase brushless machine works correctly as a motor, its output torque is 'lumpy' and would be unsuitable for most industrial servomotor applications. The main uses occur at the low power end of the scale where the brushless motor is manufactured with a single winding in very large numbers, for example as fan motors for the cooling of electronic equipment. These are normally exterior-rotor motors, where the fan is mounted on a hollow, cylindrical permanent magnet which rotates around a laminated, cylindrical stator with slots for the winding.
3. The three-winding brushless motor
Most industrial brushless servomotors have three windings, which are normally referred to as phase windings. There are two main types. One is known as the squarewave motor, the name being derived from the (theoretically) rectangular waveform of the current supplied to its windings. The other is supplied with sinusoidal AC and is known as the sinewave motor. Both types are physically very similar to the three- phase AC synchronous motor.
The squarewave motor
The windings of the ideal squarewave motor would be supplied with currents in the form of perfectly rectangular pulses, and the flux density in the air gap would be constant around the pole faces. The squarewave version of the small four-pole motor in FIG. 3 would have the cylindrical magnet rotor shown in FIG. 1. FIG. 7 shows a simple layout for a two-pole machine where each of the three windings, a, b, c, is divided into two coils connected in series; for example, coils b1 and b2 are connected in series to form winding b. The start and finish of, for example, coil b1 are marked b1 and b1'. The two coils of each winding have an equal number of turns and are mechanically spaced apart by 30 degree around the stator.
The effect of distributing each winding into more than one slot is to extend the arc over which the winding is influenced by each pole as the rotor turns. This means that the number of slots should be specified in relation to the number of poles as well as to the number of windings. The stator in FIG. 7 is symmetrical, with three windings, 12 slots and six coils each with an equal number of turns. As a result, each phase will provide the same magnitude of torque and back emf.
Torque production per phase
FIG. 8 shows how the a-phase torque is produced in the squarewave motor when the current has the ideal rectangular waveform shown. The method of supplying the current and its commutation between the phases is described in Section 3. The flux density waveform around the pole faces has not been shown in a rectangular form. Changes in flux direction are less abrupt due to the skewing of the stator slots, and the flux waveform is shown in the diagram with ramp leading and trailing edges as a first approximation. In practice the corners are rounded due to fringing effects near the edges of the poles. Rotation is anticlockwise and the coincidence of the pole divisions with the first coil has been chosen as the starting point. The flux direction is drawn for the N-pole, and the current direction for the upper coil sides in the diagram.
As the rotor moves from the 0 = 0 degree position, N-pole flux starts to cross the upper side of the first coil, and when 0 = 30 degree the second coil comes under the same influence. The lower sides a' of the coils are similarly affected by the S-pole flux. As the rotor turns through 180 degree , the flat topped section of the flux wave moves across the full winding over a window of 120 degree This is the period when the current must be fed in from an electronically controlled supply. Positive torque is produced as the current flows through the winding. The cycle is completed as 0 changes from 180 degree to 360 degree again producing positive torque.
Magnitudes of back emf and torque per phase
The 'ac' nature of the back emf is evident in FIG. 8. When the fiat topped part of the flux wave sweeps across the coils of the 'a' phase, the voltage generated across one side of one turn of either coil is:
where the speed of rotation is w rad s -1, and l is the length of the coil side (into the paper). The voltage generated around a complete turn is
If the winding has Nph turns distributed between the two coils, the total back emf generated around the two series-connected coils is
ea -- 2NphBlrw
The torque produced by one side of one turn of either coil is
t' = Bliar
and the total a-phase torque is
ta = 2NphBlia r
The three windings are symmetrically distributed around the stator, as are the magnetic poles around the rotor, and so
ea -- eb -- ec
ta -- lb -- lc
Before combining these quantities to give the output torque and the back emf at the input terminals, we should first look at how the motor windings are connected together.
Wye (Y) and delta (A) connections
FIG. 9(a) shows the Y or star connection, where the windings are joined to form a star point. The figure also shows the motor currents which flow from an electronically controlled source. Each winding of the star is in series with its supply line, and the same current flows in the line and the winding. One full cycle of each phase current must occur for every 360 degree of rotor movement and so ib and ic are displaced from ia by 0 - 120 degree and 240 degree Note that the sum of the three currents at the star point is zero for all values of 0. Note also that the emf across a pair of motor terminals is the difference (for the chosen reference directions) of the emfs across the respective phase windings.
FIG. 9(b) shows the A connection, where the emf across the windings appears across the motor terminals. The line currents are the same as before but differ here from the phase currents.
The difference between any two phase currents equals the line current flowing to the common point of the two windings.
The line-to-line voltages are no longer trapezoidal, and the phase emfs do not sum to zero. Circulating currents are likely around the closed delta path, with the possibility of motor overheating due to the extra i2R losses. The A connected stator is therefore less useful and most squarewave motors are made with the Y connection.
Three-phase torque and back emf
FIG. 10 shows the patterns of ideal torque and emf for each of the three windings of a Y-connected squarewave motor with the winding and pole layout described in FIG. 8. The squarewave motor is often referred to as the 'trapezoidal' motor in view of the trapezoidal shape of the back emf. The emf across the a-b input terminals in FIG. 9(a) is
eab : ea -- eb
and so the peak emf in FIG. 10 is
eab -- 2ea
e,b = 4NphBlroJ
The back emf across a pair of machine terminals is
eab -- ebc = eta = E = KEW
where KE = 4NphBlr, the voltage constant of the motor.
Looking now at the patterns of torque produced by the motor, we see that each phase works for 240 degree and rests for the remaining 120 degree of each turn of the rotor. However, the combined effort of the three phases does produce the extremely important feature of a theoretically smooth output torque. Only two phases produce torque at any one time and so the motor torque is or
T = 2ta
T = 4NphBllr
where I is the line current input to the motor. The torque can be written in the familiar form
where Kx = 4NphBlr, the torque constant of the motor.
Comparison between the emf and torque expressions confirms that the voltage and torque constants are equal for the squarewave motor. As in the case of the brushed motor, the numerical equivalence exists only when the constants are expressed in SI units.
Practical emf and torque waveforms
The smoothness of the rotor output torque is affected by fringing effects which leave less than 120 degree of the flux wave in a flat-topped form. This is in addition to the effect of the ripple in the flat top caused by stator slotting. Further irregularities in the output torque result from stator current waveforms which are never perfectly rectangular in practice.
The sinewave motor
The ideal squarewave motor has rectangular waveforms of flux density and input current, and has windings concentrated in coils in the stator slots. The ideal sinewave motor has sinusoidal flux and current waveforms and a sinusoidal distribution of its windings.
Sinusoidal AC input current
In common with squarewave motors, most sinewave machines are made with three phases. FIG. 11 (a) shows the three line currents which are supplied to the motor from an electronic inverter.
Sinusoidal flux density in the air gap
There are a number of ways in which the magnetic circuit can be designed to produce a near sinusoidal flux density waveform. A good sinewave can be achieved by tapering the magnets towards the edges as shown in FIG. 1 l(b). The taper of the profile is exaggerated in the diagram. Formation of the waveform is assisted by fringing effects which are encouraged by the use of a relatively small pole arc. FIG. 1 shows a four-pole, sinusoidal rotor where the tapered magnets are mounted on a square section hub.
Sinusoidal winding distribution
The ideal, fully distributed layout of stator conductors for one phase of a two-pole, sinusoidal motor is represented in FIG. 11(c). In practice an irregularity must be present in the distribution due to the bunching of conductors in slots.
The three-phase sinewave motor closely resembles the three- phase AC synchronous motor and its characteristics can be found through the phasor diagram method. However, its ideal torque and back emf can still be found by the method we have used for the squarewave machine. FIG. 12 shows one phase of a two-pole sinewave motor with ideal flux, current and winding distributions. There are Ns conductors on each side of the winding of Ns turns. The reference has been chosen at the moment when the N-S pole axis of the rotor lies horizontally in the diagram, when 0 = 0 and the input current is zero. We assume that the current is controlled (externally) in such a way that it varies sinusoidally with rotor angle 0. Note that the current magnitude varies with 0 and not with stator angle 4 degree. The diagram is drawn for a moment in time when 0 = 90 degree and the conductor current is therefore at its maximum of Iu.
When 0 = 90 degree the emf across a conductor of length I at stator angle 4 degree is
el : Blv
el - BM sin b lrw
The combined emf across the conductors within dO is Ns.
ed~ - -~- sm ~ d~b BM sin ~b lrw
Ns edO -- -~- BM lrw sin 2 ~b d~b
Integrating this expression over 4~ = 0 to 7r gives the total back emf across Ns conductors as
7r EM -- -~ NsBM lrw
EM is the peak back emf, generated at the moment when flux cutting is at a maximum. The peak emf for the full winding, which has 2Ns conductors, is 2EM. The conductors are swept by a sinusoidally distributed flux and the variation of back emf with time must also be sinusoidal. The rms value of the emf across the full winding is therefore
2EM 7r Eph -- 2 2x/~ sBMlr"~
For a Y-connected sinewave motor, the back emfs across the three individual windings form a balanced set of three-phase voltages. The rms emf which appears across the motor input terminals and supply lines will therefore be x/3Eph, or
E- ~NsBMlrw = grw
When 0- 90 degree the force on a conductor at stator angle degree b is f-- BIIM
f= BM sin dpllM
The combined force on the conductors within d~b is
Ns N~ BMIM l sin 2 ~b d~b fd~ - -~-sin ~b d~b BM sin ~b llM -- -~-
Integrating over ~b = 0 to 27r, the total force on the full winding of 2Ns conductors at the moment when 0 = 90 ~ is found to be 71" FM = NSSM/M/
FIG. 12 is drawn for the moment when the force on the winding is at a maximum. The rotor experiences an equal and opposite reaction to give a peak output torque of
71" TM -- ~ NsBM IM lr
TM is the torque when the rotor pole axis lies at an angle of 90 degree to the chosen reference position. When the rotor axis lies at 0 to the reference, the stator current is 1M sin/9 and the flux density effective over the full winding is BM sin0. The torque becomes
To - TM sin20
To is always positive and varies position, as shown in FIG. 13.
sinusoidally with rotor
Three-phase torque and emf
The three-phase sinewave motor has three of the single-phase windings shown in FIG. 12. The axis of each winding is 120 degree from the other two, around the stator. The waveforms of torque against 0 produced by the three phases will therefore be separated by the same angle. The contribution of each phase to the overall torque is shown in FIG. 14. The torque produced by each phase varies sinusoidally around its average value. When the three waveforms are added together, the sinusoidal components cancel out and we are left with the sum of the averages as the constant value of the torque on the rotor. The three-phase torque for the two-pole motor is therefore given by
37f 2x/~Ns BMlrIrms
In the above analysis, the torque of the ideal two-pole, three- phase sinewave motor has been found to have the same value for any particular position of the rotor poles relative to the stator windings. It has also been assumed that the waveform of winding current against time is synchronized with the sine of the rotor angle. Reference  gives a rigorous treatment for rotating poles by multiplying the total rotating ampere- conductor distribution of the stator by the rotating flux distribution of a rotor with p number of pole pairs. The torque is found to be dependent on the cosine of the angle by which the rotor lags behind the rotating field produced by the stator, but independent in the ideal case of the number of pole pairs. The effects of the non-ideal features of the practical windings are also covered in detail.
KT and KE
The last expression above gives the torque constant for the ideal sinewave motor as
37r KT-- 2 v/~ NsBM Ir
The voltage constant has already been seen to be
KE = NsBM lr 2x/~
Comparing the expressions for KT and KE shows that the constants are not numerically equal for the sinewave motor.
The relationship between the numerical values of the constants is given by
KT = x,/3KE
This form of the relationship is valid when KE is expressed as Vrms/rad s -1 and KT is the total three-phase torque constant in Nm/Anns. Other forms are possible when the units are given in terms of peak, or per-phase values .
Torques and ratings
The three Y-connected windings are manufactured in squarewave or sinewave form. Both forms of the winding have the same resistance between the motor terminals. This means that the resistance per phase is 0.5RLL in both cases, where RLL is the value quoted by the manufacturer for the resistance between any two terminals as seen from the supply lines. Both forms of the motor have the same physical size, thermal resistance and torque ratings. At any time, the squarewave motor carries current in only two of its phase windings. The maximum iZR loss is therefore
RLL = Is2qRLL 212q 2
where Isq is the maximum, continuous rms current which may be carried by the two conducting windings without overheating the motor. For equal iZR losses in the sinewave and squarewave versions of the motor we may therefore write
3IsZn RLL _ i2 qRLL 2
or Isn _. ~.2 Isq
where Isn is the maximum, continuous rms current which may be carried by each of the three conducting windings of the sinewave motor. The motors are designed to have the same torque ratings and so
KT(sn)Isn = KT(sq)~sq
Combining the last two equations above gives
v3 KT(sn) -- "~ KT(sq)
where the torque constants refer to the total output torque produced by each form of the motor. Care is needed when the torque constant of the sinewave type is defined. The constant is usually given for the total three-phase torque, but is sometimes given per phase. Table 2.1 shows some figures for the two forms of the small brushless motor shown in FIG. 3. The last row shows the maximum, continuous current which can be supplied without overheating any part of the motor, when the rotor is locked in a stationary position. The continuous stall torque is similarly defined.
The effective resistance
In the trapezoidal form of the motor, the current flows through only two phases at any one time and so the line-to-line resistance is the resistance which is effective in generating the i2R loss. In the sinusoidal case, the loss is generated in the three phases at all times, to give a total of
3 3 x 12 RLL __ 12 X RLL 2
The thermally effective resistance of the sinewave motor is therefore 1.5RLL.
4. Permanent magnets and fields
Up to this point our attention has been on understanding how squarewave and sinewave motors operate, and the 'permanent' field of the magnets has been taken for granted. In the first part of this section we will look at permanent magnets in general, and later at those used in brushless servomotors.
The idea that a magnetic field consists of flux
9 of density B was used in Section 1, when the production of torque and back emf was explained for the brushed DC motor. There are three other concepts which are used in the description of the magnetic fields of both brushed and brushless motors.
Magnetomotive force mmf
The force which pushes magnetic flux along its path is called the magneto-motive force, or mmf. The air-cored and iron-cored coils in FIG. 15 have N turns and carry current I. The force driving the flux is expressed as:
mmf = N I (A-turn)
For example, if each coil has 10 turns and carries 100 A, the mmf will be 1000 A-turn in both cases.
Magnetic field intensity H
The shorter the path of the flux in whatever medium, the greater the amount of flux which can be established in that medium by a given mmf. The mmf per meter of the flux path is called the magnetic field intensity, expressed as A/m.
As its name suggests, permeability tells how easy it is for the mmf to establish flux in a particular medium. The permeability of the medium is
# = #o#r
where #o is the permeability of a vacuum expressed in the unit of henry per meter and #r is the permeability of the medium relative to that of a vacuum.
The density of the flux in a particular case can be seen to be dependent on two factors. One is the intensity of the mmf around the flux path, and the other the permeability of the medium. The flux density is given by
In FIG. 15, H will have about the same value in the two cases. However, ~r for iron is of the order of several thousand and so the flux density will be much greater here than in the air-cored coil, where #r is close to unity.
The hysteresis B-H loops
The 'normal' characteristic which describes the properties of a permanent magnet is the B-H loop shown in FIG. 16. The dotted line is the so-called 'intrinsic' loop. The normal curve shows the full cycle of magnetic states which can be induced in the magnet, starting at the origin with a previously unmagnetized sample. FIG. 17(a) shows such a sample clamped between the ends of an iron core, so that the external magnetizing force NI A-turn can be applied. Note that the x-axis in FIG. 16 is not scaled as H but as #oH, which is the flux density which would exist in the air between the ends of the iron core without the magnet in place. The y-axis gives the density which actually appears in the magnet when in place. Let us now go round the normal B-H loop of FIG. 16, starting at the origin. Assume that the iron core has a very high permeability. This allows the full mmf to appear across the ends of the specimen, without loss along the iron path.
0-A Mmf NI and field intensity H (between the ends of the iron core) are increased from zero until the flux density B in the magnet reaches a maximum at A.
A-Br Current I is switched off at A upon which the flux falls to a residual level of density Br, and not to zero.
Br-C The externally-applied mmf is again increased from zero, but this time in the negative direction. The flux density within the magnet falls to zero at C.
C-D The negative current in the coil is further increased and flux density rises from zero towards its negative peak.
D-A The external mmf is returned to positive values. The D-A return path is usually a mirror image of the A-D route.
Suppose now that the coil is removed, so that B falls to Br, and then an air gap introduced into the magnetic circuit as shown in FIG. 17(b). The value of B around the magnetic circuit will fall below Br due to the low permeability of the gap. The circuit of a brushless motor consists of the magnets in series with paths through the rotor and stator iron, and the gaps between the pole faces and the stator. The working density must therefore be less than Br.
The operating quadrant of the motor is shown in FIG. 18.
Note that the straight line from Br passes through C, and that the knee of the curve is below the horizontal axis. The straight section is the most important feature of the so-called 'hard' materials used in the fabrication of magnets for high performance, brushless servomotors. The term refers to a magnetic rather than a physical toughness. The permanent field of the hard magnet will not be damaged by stray fields provided the flux density in the magnet is not forced to a point on the knee of the curve. The intrinsic curve is used to interpret this behavior.
The intrinsic curve
The vertical axis of the intrinsic curve of FIG. 18 gives the flux density which is potentially still available in the magnet after the externally applied field of intensity H is removed.
Certain points can be defined
Remanence Br The maximum intrinsic flux density when the permeability of the external flux path is infinite.
The maximum external field intensity which does not cause demagnetization which is intrinsically irrecoverable. The value for #oHc is given at the lower end of the straight-line section.
Intrinsic coercivity nci
The applied field intensity which completely demagnetizes the magnet.
For the magnets of the brushless motor, the most important feature of FIG. 18 is that the intrinsic flux density remains at Br following the application and subsequent removal of reverse fields with intensities up to He, but falls below Br when the reverse field has an intensity above Hc.
Permeability of the hard magnet
The vertical axis of the B-H loop of FIG. 18 shows the flux density in tesla which is set up in the magnet by the application of an mmf of magnetic intensity H across its ends. The horizontal axis is also scaled in tesla, and gives the flux density #oH which would exist in the air space occupied by the magnet. For hard magnetic materials the slope of the straight-line working section is B/#oH ~ 1, and so for the magnet itself we have
B ~ #oH
The surprising conclusion is that hard magnets have a low permeability, close to that of air. They are said to have a low recoil permeability. This term is not particularly good as it suggests that the magnet has low rather than high intrinsic ability to recover its flux levels.
In Section 1 we reviewed the permanent-magnet brushed DC motor, where the current-carrying conductors are wound on the rotor. The rotor of the brushed motor is often called the armature of the machine. The flow of current around the armature winding sets up a magnetic field which combines with the field produced by the stator magnets or field windings of the brushed motor. The resulting distortion of the field in the air gap is described as being due to armature reaction. The same term is applied to the brushless motor, even though the source of the effect is the stator and not the rotor.
The mmf developed by the stator winding of the brushless motor may be quite high, especially at full load current. Flux circulates around the conductors, some staying in the stator and some crossing the air gap to pass through the magnet.
However, we have seen that the magnet has much the same permeability as the air gap. The result is that the relatively small amount of stator-induced flux which enters the magnet has little effect on the average operating level of the flux density in the air gap, although it does cause some distortion in the flux distribution.
We shall see in the next Section how the current is supplied to the brushless motor. The supply units use power electronics to control the flow of current, normally with a very high reliability. However, faults can never be entirely ruled out.
The worst effect for the permanent magnet machine is likely to be from the magnetic fields set up by the flow of high overload current through the stator conductors. These fields are obviously stronger than those due to normal armature reaction and the main concern must be to avoid the risk of any permanent effects on the strength of the magnets.
Fortunately the hard magnet has some protection of its own through low permeability and high coercivity. These features are exploited when the cost of the magnet is minimized by reducing the radial length (i.e. in the magnetized direction) to the minimum necessary.
Permanent magnet materials
Two materials have become well established in the manufacture of hard permanent magnets for industrial brushless servomotors.
Samarium cobalt (Sm-Co)
Samarium and cobalt are both rare-earth elements. There are only a few sources of samarium in the world which can supply the quantities needed and so the cost is invariably high. Even so, the Sm-Co magnet is widely used in brushless servomotors. This is mainly because it has superior technical characteristics when compared to ferrite, and is particularly good when compared to metal alloys such as Alnico.
Neodymium iron boron (Nd-Fe-B)
The 'Nib' magnet uses materials which are less expensive than samarium and cobalt. The lower cost of the magnet is, however, its only advantage, as it has no better technical characteristics for motors than the Sm-Co type. One of the most troublesome problems of the Nib magnet is its susceptibility to corrosion and although it has a better second quadrant B-H characteristic than Sm-Co, the advantage is lost at the high end of the operating temperature range.
The Sm-Co magnet has a better high temperature coercivity and a better temperature coefficient of remanence than the Nib type. In FIG. 18, the knee of the B-H characteristic lies below the H-axis. As temperature rises, the knee moves up the curve with the danger of intrusion into the operating quadrant. FIG. 19 shows the relative effects of the knee movements for the two materials in question. The knees below the H-axis move in a way which brings both characteristics closer to the origin, but the movement of the Sin-Co line is small in comparison to that of Ne-Fe-B. The knee of the Sm-Co characteristic does not move up enough to affect the linearity in the operating quadrant, but the movement of Ne-Fe-B is greater and linearity is not maintained. This means that the effects, at typical motor temperatures, of demagnetizing fields with intensities up to the 'hot' value of coercivity will not be permanent for the Sm-Co magnet but may be so for Ne-Fe-B. It should be remembered, however, that even the Sm-Co magnet can still be demagnetized if the maximum allowable motor temperature is exceeded.
The structure of the brushless machine gives it significant advantages in performance over the brushed type.
Performance is enhanced because:
1. There is no brush and commutator transmission of current to the motor and therefore no mechanical commutation limit to the speed at which any particular torque can be supplied.
2. The i2R loss arises in the stator rather than the rotor, allowing the surplus heat to pass more freely into the air surrounding the motor case. Any overheating which does occur is also easier to detect as the effects occur in the accessible component.
There are several ways of comparing the thermal characteristics of brushed and brushless motors. Perhaps the fairest is to compare motors of about the same physical size, the same maximum torque and the same speed range when operated from the same voltage. FIG. 20 shows such a comparison, the shaded area depicting the higher continuous torques available from the brushless motor. Apart from a small region close to maximum speed, the levels of continuous torque available from the brushless motor are of the order of 40% greater than from the brushed type. The intermittent torques available are higher for the brushless motor over most of the diagram, and are generated without the brush and commutator deterioration suffered by the brushed motor when working near to the commutation limit.
The physical size of the most commonly used motors varies quite widely, with motor weights from as little as 1 kg to a substantial 50 kg. The maximum continuous power outputs vary from 50 W to 10 kW, and up to four times these figures intermittently. Three motors from the medium to small end of the range are shown in FIG. 21. The smallest has a length of approximately 12 centimeters. The largest of the three can supply a continuous power demand in the region of 2.5 kW at a nominal speed of 3000 rpm, and its specification includes the details shown in Table 2.2. The motor is a four- pole machine with Sm-Co magnets and is manufactured in either squarewave or sinewave form. Here we should remember that such names refer to the ideal current waveforms. Motor specifications normally classify the motors according to the shape of the back emfs, and the table shows a choice between the trapezoidal and sinusoidal types.
For the trapezoidal form of the motor in Table 2
Ts - KTIs -- 0.84 x 11.7 -- 9.8 Nm
The data given on the specification sheet does not normally include a value for the voltage constant in SI units. However, Kx and KE should have the same numerical value for the trapezoidal motor, and this can be checked from the table.
The back emf is
E = KE~
or 88 = KE • 1000 x 27r/60
KE -- 0.84V/rad s- 1
The continuous stall torque is
Ts = KTIs-- 1.02 x 9.6 = 9.8 Nm
We know that KT and KE do not have the same numerical value for the sinusoidal motor. For the motor in question, the rms line-to-line emf per 1000 rpm is 88/v/2 = K E x 1000 27r/60 giving
-1 KE = 0.59 Vrms/rad s
For the sinusoidal form of the motor, we have the result that KT/KE= 1.02/0.59 = 1.73
This agrees with the theoretical relationship; KT = X/'3KE.
Table 2 includes a figure for cogging torque. A constant output of torque depends on the waveform of flux density in the air gap being either perfectly fiat-topped, or perfectly sinusoidal. Stator slotting produces a non-uniform magnetic flux path which in turn affects the flux patterns and gives rise to a tippling or cogging of the torque. There are two main ways of dealing with the problem: Fractional slotting.
The motor in FIG. 7 has 12 slots and two poles, making an integral slot/pole ratio. For any such integral ratio, all pole edges would line up with slots at the same time as the rotor turns. A reduction in the cogging effect can be made by forming a non-integral slot/pole ratio to produce the so-called fractional-slot winding. It is, however, difficult to make a trapezoidal motor in such a way, and still maintain the necessary length of the fiat-topped section of the rotor flux wave. The technique is easier in the sinewave motor, where fractional slotting is combined with an uneven spacing of the magnets in order to reduce the cogging effect still further.
Skewing The stator slots can be skewed relative to the rotor axis in order to reduce the irregularity of the magnetic circuit, as mentioned in Section 2 and shown in FIG. 1.
FIG. 22 shows the approximate effect of skewing on the cogging torque. Maximum benefit occurs at a skew of one slot pitch, when the end of one slot and the opposite end of the next are aligned along the rotor axis. The amount of skew is in any case limited by the need to maintain the length of the fiat-topped section of the flux waveform in the air gap.
Servomotors are often fitted with a brake, usually in the front end of the motor between the rotor and the front bearing as shown in FIG. 23. When the brake operates, a friction disc is pressed against the fixed steel disc visible in the figure.
The steel disc is large enough to absorb the heat generated during an emergency stop, without deforming. Two types of brake are used. In one, a spring is restrained by a solenoid during normal operation of the motor and released when operation of the brake mechanism is required. In the other type the duty of the spring is performed by a permanent magnet, the effect of the permanent magnetic field being restrained by the effect of a field from a DC winding. For some applications there must be no backlash in the brake mechanism. A brake with a small amount of backlash may be manufactured at a lower cost than one without backlash, whether operated by a spring or by a permanent magnet. However, the method of manufacture makes the permanent magnet more suitable for the zero backlash type.
Both forms of the brake are 'fail-safe', preventing any movement of the motor shaft and connected load following a failure of the power supply to the motor. The brake is also used to hold the motor shaft in position during a shutdown of the work process. In circumstances where the shaft must start exactly from the position reached at the end of the previous operation, the motor must be fitted with the zero backlash form of brake. The brake is not normally used to control the motor during the actual operation. One exception, however, is in a three-dimensional process known as 'two-and-a-half axis' machining, where the product is moved into different positions along a vertical axis by a servomotor, and then machined over two horizontal axes. Here, the motor brake can be used to lock the product into position during the machining process.