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Chapter 10. Stepping Motors and TheirPower Electronic Drives

Topics to cover:1) Types and Principles2) Drive Circuits3) Characteristics

4) Low Speed Operation5) High Speed Operation

Introduction

Stepping motors have been developed in response to the demand for a device capable of producing a definite angular displacement in a driven shaft and holding its positionagainst a torque applied to the driven shaft. For example, in machine tool applications,such motors could, through gearing, position both the cutting tool and the workpiecevery accurately, and hold them in those positions during the subsequent cuttingoperation. Fig.1 illustrates the principle of a numerically controlled milling machineusing three stepping motors to control the three axis movement of the workpiece, wherethe motor governing the Z axis is installed below the table. Typical applications arecomputer peripherals, such as printers (Fig.2), plotters (Fig.3), and disk drives (Fig.4),watches, and other devices which require accurate position control.

Moreover, since the stepping motor isoperated by current pulses, it may be

controlled by command signals storeddigitally. It is therefore suitable for computer controlled operations.

Types and Principles of Stepping Motors

While there are many configurations of stepping motors, they may be grouped intothree basic types: variable reluctance,

permanent magnet, and hybrid steppingmotors.

Variable Reluctance Stepping Motors

(1) Single Stack VR Motors

Fig.5 shows the cross section and winding arrangement of a three phase single stack variable reluctance (VR) stepping motor. Current to each phase is controlled in theON/OFF mode by their respective switches. If a current is applied to the coils of Ph.1, or in other words if Ph.1 is excited, the magnetic flux will occur as shown in Fig.6. The

rotor will then be positioned so that the stator teeth I and I' and any two of the rotor teeth

Fig.1 Principle of numerically controlled millingmachine using three stepping motors [1]


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Page 10-2

are aligned. Thus when the rotor teeth and stator teeth are in alignment, the magneticreluctance is minimized, and this state provides a rest or equilibrium position.

Fig.2 External view, internal mechanism, and fundamental construction of character-impact type serial printer [1]


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48541 EMS Chapter 10. Stepping Motors and Drives

Page 10-3

Fig.3 A recent XY-plotter and the pen drive system [1]

(a)

(b)Fig.4 Floppy disc drive (a) Floppy disc structure and mounting mechanism;

(b) driving mechanism of the head [1].


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Page 10-4

Fig.7 illustrates how a step motion proceeds when excitation is switched from Ph.1 toPh.2. The motor reluctance seen from the DC power supply will be suddenly increased

just after the switching takes place. The rotor will, then, move through a step angle of 30o counter-clockwise so as to minimize the reluctance. This motion through a stepangle at each switching of excitation is called a step. After completing a rotor-tooth-

pitch rotation in three steps, the rotor will apparently return to its original position. Thisis illustrated in Fig.8.

Fig.5 Cross sectional model of a three phase VR stepping Fig.6 Equilibrium position

motor and winding arrangement [1] with phase 1 excited [1]

Fig.7 How a step motion proceeds when excitation is switched from Ph1 to Ph2 [1]

Fig.8 Step motions as switching sequence proceeds in a three phase VR motor [1]

The relationship between step angle s, number of phases m, rotor teeth N r , and stepnumber S is given by

S = 360/ s = mN r (1)

Fig.9 is the picture of a four-phase 7.5 o motor which has 16 stator teeth and 12 rotor teeth. In order to reduce the step angle, the number of rotor teeth must be increased. Anexample of a typical stepping motor is shown in cross section in Fig.10. This machinehas four winding phases and 50 rotor teeth, and the step angle is 1.8 o.

The step angle can be further reduced by the so called 'half-step excitation mode', whichis a combination of the single phase and two phase excitation. The switching sequencefor the motor in Fig.5 is illustrated in Table 1.


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Page 10-5

Fig.9 Stator and rotor of a four phase VR motor of 7.5 o step angle [1]

Fig.10 Cross sectional view of a four phase VR stepping motor; number of rotor teeth being 50, step number 200, and step angle 1.8 o [1]

Table 1 Switch sequence of half step excitation modeStep 1 2 3 4 5 6S1 1 1 0 0 0 1S2 0 1 1 1 0 0S3 0 0 0 1 1 1

Switch state: 1 stands for ON, and 0 for OFF.

Obviously, the step angle is halved, and the step angle s and step number S can becalculated by

S = 360/ s = 2mN r (2)

(2) Multi-Stack VR Motors

Fig.11 is the cut away view of a three stack variable reluctance stepping motor. In eachstack, as illustrated in Fig.12, the rotor and stator have equal numbers of teeth. Adjacentstator teeth have equal and opposite magnetomotive forces produced by windings

carrying the phase current. On the same shaft, normally, there are m similar stacks. Eachof these has either its rotor or its stator displaced sequentially from its neighbour by a


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Page 10-6

tooth pitch divided by m, and each is excited by a phase current. The step angle can still be calculated by (1).

Fig.11 Cutaway view of a three stack VR stepping Fig.12 One phase of a multistack VR motor (After Ref.[2]) stepping motor (After Ref.[3])

Permanent Magnet Stepping Motors

An example of a basic two phase permanentmagnet (PM) stepping motor is shown inFig.13. A change in excitation between the twowindings produces a step of 90 o. The current

polarity is important in the permanent magnetmotor; the rotor position illustrated is for

positive current in winding A, a switch to positive current in winding B would produce aclockwise step, whereas negative excitation of B would give anticlockwise rotation. Anadvantage of this type of machine is that therotor tends to remain in its last position when

phase current is removed. It is difficult tomanufacture a small permanent magnet motor with a great number of poles and consequentlystepping motors of this type are restricted tostep lengths in the range 30 o-90o.

Hybrid Stepping Motors

Another type of stepping motor having a permanent magnet in its rotor is the hybridmotor. The term 'hybrid' derives from the fact that the motor is operated under thecombined principles of the permanent magnet and variable reluctance motors. Thecutaway view and the cross sectional/axial diagram of the hybrid motor in wide usetoday are shown in Fig.14. The stator core structure is the same as, or very close to, that

of the VR motor shown in Fig.10, but the windings and coil connections are differentfrom those in the VR motor. In the VR motor only one of the two coils of one phase is

Fig.13 Permanent magnet stepping motor

(After Ref.[2])


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Page 10-7

wound on one pole, while in the four phase hybrid motor, coils of two different phasesare not wound on the same pole as shown in Fig.14(b). Therefore one pole does not

belong to only one phase. The two coils at a pole are wound in the so called bifilar scheme that will be discussed later; they produce different magnetic polarities onexcitation.

Another important feature of the hybrid motor is its rotor structure. A cylindrical shapedmagnet lies in the core of the rotor as shown in Fig.15, and it is magnetized lengthwiseto produce a unipolar field as shown in Fig.16(a). Each pole of the magnet is coveredwith uniformly toothed soft steel. The teeth on the two sections are misaligned withrespect to each other by half tooth pitch. In some motors, the rotor teeth are aligned witheach other, but the stator core has a misalignment as shown in Fig.17. The toothedsections are normally made of laminated silicon steel, but sintered steel or solid siliconsteel is employed in some cases.

The magnetic field generated by the stator coils is heteropolar field as shown inFig.16(b).

(a)

(b)

Fig.14 A typical hybrid motor; (a) cutaway view; (b) construction (After Ref.[1]).


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Page 10-8

Fig.15 Rotor structure of a hybrid motor (After Ref.[1])

Fig.16 Magnetic paths in a hybrid motor;(a) the flux due to the rotor's magnet producing a unipolar field, while(b) shows the heteropolar distributed flux due to the stator currents(After Ref.[1]).

Fig.17 Misalignment in stator teeth (After Ref.[1])

Fig.18 Split and unrolled model of a four phase hybrid motor;the upper being for the south pole cross section,and the lower the north pole cross section (After Ref.[1]).


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In this type of motor, torque is created by the interaction of these two types of magneticfields in the toothed structure in the airgaps. For an explanation let us have a look at thesplit and unrolled model of Fig.18. In this schematic diagram, the stator tooth pitch isthe same as the rotor tooth pitch. In some motors, however, the stator tooth pitch is alittle larger than the rotor's to reduce detent torque and increase positioning accuracy.

The upper half of this figure is the cross sectional view of the south pole side of themagnet while the lower half represents the north pole side. We are concerned with themagnetic fields under the teeth of pole I and III in this model. Pole I is now excited to

produce the north pole, and pole III the south pole and they build field distributions asdrawn by the solid curves. The dotted curves represent the flux due to the magnet.

Firstly it should be noted that no effective torque is generated by the magnetic field dueto the coil alone as in the VR motor, because the rotor teeth in the north pole side andthose in the south pole side are misaligned with respect to each other by half a tooth

pitch. The permanent magnet produces some detent torque, but this is not a veryimportant factor in the hybrid motor. Let us see what happens when the magnetic fieldsdue to the coils and the permanent magnet are superimposed. The results are suggestedin the same figure. A driving factor toward the left will appear in the upper half section

because both fields reinforce each other in the toothed structure under pole I soincreasing the left oriented force, while both components neutralize each other and soweaken the right oriented force under pole III. The same force is produced as well in thelower half section, as the stator field and rotor field are in the same direction under poleIII, while they are in opposing direction under pole I. Hence, the resultant force will betoward the left. After the rotor has moved a quarter tooth pitch in this direction, thedriving force is reduced to zero, and an equilibrium position is reached.

If the old excitation is turned off, and new poles are excited to produce a south pole anda north pole, respectively, the rotor will make another step. As seen above, the

permanent magnet plays an important role in creating the driving force. But it should benoted that the toothed structures in both stator and rotor are designed to realize a smallstep angle in the hybrid stepping motor.

The most popular hybrid motor is the four phase 200 step motor, the step angle being1.8o. In order to raise the torque, multi-stack hybrid motors such as shown in Fig.19 areemployed.

Fig.19 Three-stack hybrid motor designed to increase torque (After Ref.[1])


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Drive Circuits

Unipolar Drive Circuit

A simple unipolar drive circuit suitable for use with a three phase variable reluctance

stepping motor is shown in Fig.20. Each phase winding is excited by a separate drivecircuit, which is controlled by a low power 'phase control signal'. This control signalmay require several stages of switched amplification before is attains the power levelrequired at the base of the phase transistor.

Fig.20 Three phase unipolar drive circuit (After Ref.[2])

In general the phase winding has a considerable inductance, so its natural electrical time

constant (inductance/resistance) is long. The build-up of phase current to its rated valuewould be too slow for satisfactory operation of the motor at high speeds. By adding theforcing resistance, with a proportional increase in supply voltage, the phase electricaltime constant can be reduced, enabling operation over a wider speed range.

Another consequence of the finite phase winding inductance is that the phase currentcannot be switched instantaneously. If the base drive of the switching transistor wassuddenly removed a large induced voltage would appear between the transistor collector and emitter, causing permanent damage to the drive circuit. This possibility is avoided

by providing an alternative current path - known as the freewheeling circuit - for the phase current. When the switching transistor is turned off the phase current can continueto flow through the path provided by the freewheeling diode and freewheeling resistor.

Bipolar Drive Circuit

One phase of a transistor bridge bipolar drive circuit, suitable for use with a hybrid or permanent magnet stepping motor, is shown in Fig.21. The transistors are switched in pairs according to the current polarity required. For positive excitation of the phasewinding transistors T1 and T4 are turned on, so that the current path is from the supply,through transistor T1 to the phase winding and forcing resistance, then throughtransistor T4 back to the supply. In the opposite case the transistors T2 and T3 are

turned on so that the current direction in the phase winding is reversed.


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Fig.21 One phase of a transistor bridge bipolar drive circuit (After Ref.[2])

The four switching transistors in the bridge require separate base drive to amplify thetwo (positive and negative) phase control signals. In the case of the 'upper' transistors(T1 and T2) the base drive must be referred to the positive supply rail, which may be at

a variable potential. For this reason the phase control signals to these upper base drivesare often transmitted via a stage of optical isolation.

A bridge of four diodes, connected in reverse parallel with the switching transistors, provides the path for freewheeling currents. In the illustration of Fig.21 the freewheelingcurrent path, via diodes D2 and D3, corresponding to the situation immediately after turn-off of transistors T1 and T4. The freewheeling path includes the d.c. supply andtherefore some of the energy stored in the phase winding inductance at turn-off isreturned to the supply. The consequent improvement in overall system efficiencyrepresents a significant advantage of the bipolar bridge drive over the unipolar drive andfor this reason most large stepping motors, including variable reluctance types, areoperated from bipolar drives.

Freewheeling currents in the bipolar drive decay more rapidly than in the unipolar drive, because they are opposed by the d.c. supply voltage. Therefore it is not necessary toinclude additional freewheeling resistance in the bipolar bridge drive.

Bifilar Windings

The transistor bridge bipolar drive circuit requires four transistor/diode pairs per phase,whereas the simple unipolar drive requires only one pair per phase, so drive costs for a

hybrid stepping motor are potentially higher than for the variable reluctance type; a two- phase hybrid motor drive has eight transistors and diodes, but a three-phase variablereluctance motor drive has only three transistors and diodes. The bridge configurationhas the additional complication of base drive isolation for the pair of switchingtransistors connected to the positive supply rail. From the view point of drive costs theconventional hybrid motor has a severe disadvantage and therefore many manufacturershave introduced 'bifilar-wound' hybrid motors, which can be operated with a unipolar drive.

A bidirectional current flowing in the hybrid motor windings produces a bidirectionalfield in the stator poles. With a bifilar winding the same results is achieved by two polewindings in opposite senses, as illustrated for one pole in Fig.22. Depending on the fielddirection, one of the windings is excited by a unidirectional current; in Fig.22 the field


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Page 10-12

produced by a positive current in the conventional arrangement is available by excitingthe bifilar +winding with positive current. The effect of negative current in theconventional winding is then achieved by positive excitation of the bifilar -winding.

Each of the bifilar pole windings must have as many turns as the original winding and

the same rated current, so a bifilar winding has twice the volume of a conventionalwinding. This additional volume does, of course, increase the manufacturing costs butfor small size of hybrid motor this is outweighed by resultant reduction in drive costs.

Fig.22 Comparison of conventional and bifilar windings (After Ref.[2])

The two bifilar windings of one phase may be excited by separate unipolar drive circuitsof the type discussed in Section 3.1, but one alternative is to 'share' the forcing resistance

between the two bifilar windings, as shown in Fig.23. There are now only twotransistor/diode pairs per phase, so the two-phase hybrid motor with bifilar windingsrequires four transistors and diodes in its complete drive circuit and has comparabledrive costs to a three-phase variable-reluctance motor. The freewheeling path of the

bifilar drive does not return energy to stored to in the inductance at turn-off to the d.c.supply, so the drive has a lower efficiency than the bipolar bridge drive. This reductionin efficiency, coupled with the extra winding costs, is very significant for larger sizes of stepping motor, which are therefore rarely bifilar-wound.

Fig.23 Unipolar drive circuit for one phase of a bifilar-wound motor (After Ref.[2])


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Bilevel Drive Circuit

In the bilevel drive there are two supply voltages. A high voltage is used when the phasecurrent is to be turned on or off, while a lower voltage maintains the current at its ratedvalue during continuous excitation.

The circuit diagram for one phase of a unipolar bilevel drive is shown in Fig.24. Whenthe winding is to be excited both transistors (T1 and T2) are switched on, so the voltageapplied to the phase winding is equal to the sum of the two supply voltage (V L+V H), thediode D2 being reverse-biased by V H. There is no series resistance to limit the current,which therefore starts to rise towards a value which is many times the rated windingcurrent. After a short time, however, transistor T2 is switched off and the windingcurrent flows from the supply voltage V L via diode D2 and transistor T1. The ratedwinding current is maintained by the voltage V L, which is chosen so that V L/R=ratedcurrent. At the end of the phase excitation interval transistor T1 is also switched off andthe winding current is left to flow around the path through diodes D1 and D2. Rapid

decay of the current is assured, because the high supply voltage V H is included in thisfreewheeling path.

A typical current waveform for one excitation interval is illustrated in Fig.25.

Chopper Drive Circuit

This drive circuit - illustrated in its unipolar form in Fig.26 - has a high supply voltagewhich is applied to the phase winding whenever the current falls below its rated value. If the phase excitation signal is present, the base drive for transistor T2 is controlled by thevoltage V c dropped across the small resistance R c by the winding current. At the

beginning of the excitation interval the transistor T1 is switched on and the base drive toT2 is enabled. As the phase current is initially zero there is no voltage across V c and thetransistor T2 is switched on. The full supply voltage is therefore applied to the phasewinding, as shown in the timing diagram, Fig.27.

Fig.24 The bilevel drive and the effective circuits during the excitation interval(a) at turn-on; (b) continuous excitation; and (c) at turn-off. (After Ref.[2])


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Fig.25 Phase current waveform for a bilevel drive (After Ref.[2])

Fig.26 Chopper drive current waveform and transistor switching times (After Ref.[2])

Fig.27 Copper drive current waveform and transistor switching times (After Ref.[2])


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The phase current rises rapidly until it slightly exceeds its rated value (I). Consequentlythe control voltage is R cI+e and this is sufficient to switch off transistor T2. There isnow no voltage applied to the phase winding and the current decays around a path whichincludes T1, R c and diode D1. This current path has a small resistance and no opposingvoltage, so the decay of current is relatively slow. As the resistance Rc is still included

in the circuit the winding current can be monitored and when the control voltage hasfallen to R cI-e the transistor T2 is switched on again. The full supply voltage is appliedto the winding and the current is rapidly boosted to slightly above rated. This cycle isrepeated throughout the excitation time, with the winding current maintained near itsrated value by an 'on-off' closed-loop control.

At the end of the excitation interval both transistors are switched off and the windingcurrent freewheels via diodes D1 and D2. The current is now opposed by the supplyvoltage and is rapidly forced to zero. A high proportion of the energy stored in thewinding inductance at turn-off is returned to the supply and therefore the system has ahigh efficiency.

The chopper drive incorporates more sophisticated control circuitry, e.g. the T2 basedrive requires a Schmitt triggering of the control voltage V c to produce the transitionlevels. If these levels are not well-separated the transistor T2 switches on and off at avery high frequency, causing interference with adjacent equipment and additional ironlosses in the motor. However the chopper drive does have the advantage that theavailable supply voltage is fully utilised, enabling operation over the widest possiblespeed range, and the power losses in forcing resistors are eliminated, giving a goodsystem efficiency.

Specification of Stepping Motor Characteristics

In this section, technical terms used for specifying the characteristics of a steppingmotor are studied.

Static Characteristics

The characteristics relating to stationary motors are called static characteristics.

(1) T/ characteristics

The stepping motor is first kept stationary at a rest (equilibrium) position by supplying acurrent in a specified mode of excitation, say, single-phase or two phase excitation. If anexternal torque is applied to the shaft, an angular displacement will occur. The relation

between the external torque and the displacement may be plotted as in Fig.28. Thiscurve is conventionally called the T/ characteristic curve, and the maximum of statictorque is termed the 'holding torque', which occurs at =M in Fig.28. At displacementslarger than M, the static torque does not act in a direction towards the originalequilibrium position, but in the opposing direction towards the next equilibrium

position. The holding torque is rigorously defined as 'the maximum static torque that can be applied to the shaft of an excited motor without causing continuous motion'. The


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Page 10-17

relation between the frictional load torque and the maximum pulse frequency withwhich the motor can synchronize is called the pull-out characteristic (see Fig.30). The

pull-out curve is greatly affected by the driver circuit, coupling, measuring instruments,and other conditions.

Fig.30 Dynamic characteristics (After Ref.[1])

(3) The maximum starting frequency

This is defined as the maximum control frequency at which the unloaded motor can startand stop without losing steps.

(4) Maximum pull-out rate

This is defined as the maximum frequency (stepping rate) at which the unloaded motor can run without losing steps, and is alternatively called the 'maximum slewing

frequency'.

(5) Maximum starting torque

This is alternatively called 'maximum pull-in torque' and is defined as the maximumfrictional load torque with which the motor can start and synchronize with the pulsetrain of a frequency as low as 10Hz.

Low Speed Operation

Step Response

At low speeds, each individual step is discernable and the behavior is a series of stepinput transients. The motor must be modeled by a set of differential equations, which arein general nonlinear. Approximations may be made to linearize the equations for analytical solutions. Otherwise the solution must be marched out in time using acomputer numerical method, as explained in the Principles of ElectromechanicalEnergy Conversion. For a VR stepping motor with one phase winding excited, the stateequation model is comprised of the following equations

( )didt L

R dLd

i L

vr = + +1 1 (3)


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d dt J

T J

T r load = 1 1 (4)

andd dt r

= (5)

The motor response for a given supply can be obtained by solving the above state

equations together with the initial conditions: i it = =0 0 , r t r = =0 0 , and t = =0 0 .

Fig.31 illustrates the experimental step response of a VR motor. It is shown that for thismotor the electrical time constant is much smaller than the mechanical time constant.

0

50

100

150

200

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

Ti me (s)

Voltage (V) & Position (degree) Curr ent (V)

v(t)

i(t)

(t)

Fig.31 Experimental step response of a VR motor

Static Torque/position Characteristics and Static Position Error

Fig.32 illustrates the static torque/rotor position characteristics against the phase current.It is shown that the higher the phase current, the higher the holding torque, whereas the

profile of the characteristics are the same. When a stepping motor is employed to drive aload, the equilibrium position will be away from the ideal step position since the torque

produced by the motor must balance the load torque.

The difference between the actual equilibrium position and the ideal step position isdefined as the static posi tion er ror . Fig.33 illustrates the static position error of astepping motor of eight rotor teeth and a holding torque of 1.2 Nm. When carrying aload of 0.75 Nm, the static position error of this is motor is 8 o. When a sinusoidalapproximation is used for the static torque/rotor position characteristic, the static

position error can be estimated approximately by

( )

error L pk

r

T T

N

sin 1(6)

where T L is the load torque and T pk is the holding torque.


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Fig.32 Static torque/rotor position characteristics against phase current

Fig.33 Static position error for a stepping motor of eight rotor teeth anda holding torque of 1.2 Nm

Pull-out Torque/Speed Characteristics

Pul l-out torque is the maxi mum torque a steppin g motor can develop at a speed.

Fig.34 illustrates the pull-out torque/speed characteristic of a typical stepping motor.Due to the low speed resonance (chaotic behavior), there could be some dips in thecharacteristic.

To understand the pull-out torque, let us consider first the step by step operation of astepping motor at no load, as shown in Fig.35. The static torque/position characteristicsof phases A, B, and C are approximated by sine waves. Since there is no mechanicalload, the rotor would stop at the step position of phase A when phase A winding isenergized. For next step, phase B is energized. From the diagram it can be seen that thetorque produced by the motor is greater that zero when phase B is energized, and hence

the rotor will rotate for another step and stops at the step position of phase B. Thesituation for further steps will be similar.


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Page 10-20

Fig.34 Pull-out torque/speed characteristic

If the motor carries a load, the equilibrium position where the rotor stops when phase Ais energized will be different from the ideal step position since the motor torque should

balance the load torque. When phase B is energized for the next step, the rotor willrotate for a step if the torque produced by the motor is higher than the load torque. For amotor of small inertia, the maximum load torque it can carry or the pull-out torque can

be determined by the intersection of the static torque/position characteristics of twoadjacent phases, as illustrated in Fig.36. For a motor of large inertia, the pull-out toqueis higher since the kinetic energy stored by the inertia helps the rotor to rotate forward.In this case, the pull-out torque can be determined by averaging the static torque/rotor

position characteristics of adjacent phases. As illustrated in Fig.37, the pull-out torquefor a three phase stepping motor is 50% of the holding torque if the motor inertia is low,and 83% of the holding torque if the motor inertia is high. For a four phase motor, thesetwo figures are closer.

Fig.35 Rotor position at phase switching times for no load


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Page 10-21

Fig.36 Rotor position at phase switching times for pull-out load

Fig.37 Low and high inertia pull-out torques of (a) three and (b) four phase stepping motors

Low Frequency Resonance

In the pull-out torque/speed characteristic in Fig.34, there exist some torque dips. Thischaotic behavior is caused by low frequency resonance. Fig.38 shows a typical stepresponse. When the rotor oscillation frequency equals the natural frequency of themotor, a low frequency resonance occurs and the pull-out torque drops dramatically or the motor simply stalls. To overcome this problem, a viscously-coupled inertia damper (VCID) as shown in Fig.39 can be employed. Fig.40 shows the effects of VCID.


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Page 10-22

Fig.38 A typical step response

Fig.39 Viscously-coupled inertia damper (a) parallel and (b) perpendicular to the shaft

Fig.40 Effect of VCID


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High Speed Operation

At high speeds, the steps merge into one another, and alternative methods may be usedto set up and/or solve the differential equations:

(1) Step ripple present

The equations must be marched-out in time as before, or the equations can be linearized for small oscillations.

(2) Speed ripple assumed zeroIt may be possible to use time-domain methods such as Fourier analysisand/or phasor analysis.

(3) Speed ripple assumed zero and current assumed sinusoidal A.C. phasor analysis can be used, (Same model as that for synchronousmotors).

Steady State Phasor Analysis

Fig.41 illustrates the equivalent circuit in time domain of a phase winding. The circuitequation for one phase excitation can be written as

( )v Ri L

didt

dLd

id dt

Ri Ldidt

r m= + + +

= + +

(7)

where L is the stator winding inductance, m the stator winding flux linkage due to the

permanent magnet, and

( )

edL

d i

d dt r

m= +

(8)

v(t)

R L

e

i(t)

Fig.41 Per phase equivalent circuit in time domain

(1) Phasor Expression of VR Stepping Motor

For a VR stepping motor, we have

m =0 (9)and


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Page 10-24

( ) L L L N o r = + 1 sin (10)

Since unipolar drive is employed, we may express the fundamental components of thevoltage and current in the stator phase winding as

( )v t V V t o= + 1 cos (11)and ( ) ( )i t I I t o= + 1 cos (12)

Substituting (9) - (12) into (7) and neglecting the high frequency terms, we obtain thevoltage and current relations as

V RI o o= (13)and

( ) ( )

( )V t RI t L I t

L I t o

o

1 1 1

1

cos cos sin

cos

= + (14)

In phasor expression, the above voltage-current relationship becomes

V RI j L I E o= + + (15)where E L I o= 1 (16)

(2) Phasor Expression of PM and Hybrid Stepping Motors

For PM and hybrid motors, L can be considered as independent of the rotor position.The fundamental component of the voltage and current can be expressed as

( )v t V t = cos (17)and ( ) ( )i t I t = cos (18)

Assuming the flux linkage of the stator winding due to the permanent magnet is

( ) m m t = $

sin (19)we obtain

( ) ( )

( )V t RI t LI t

t m

cos cos sin

cos

= + (20)

In phasor expression, the above voltage-current relationship becomes

V RI j LI E = + + (21)where

E m

= $

(22)


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Page 10-25

(3) Equivalent Circuit in Frequency Domain

A common phasor expression for all stepping motors is

V RI j LI E = + + (23)where E L I o= 1 (24)

for a VR stepping motor, and

E m= $

(25)

for a PM or hybrid stepping motor.

Fig.42 illustrates the corresponding equivalent circuit in frequency domain, and Fig.43shows the corresponding phasor diagram.

V

R L

E

I

Fig.42 Equivalent circuit in frequency domain

Fig.43 Phasor diagram of stepping motors

(4) Pull-out Torque Expression

From the phasor diagram, it can be derived that the electromagnetic torque of a steppingmotor can be expressed as

( )

( )T

pmEI

R L

pmE R

R L=

+

+

cos

2 2 2

2

2 2 2(26)


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Page 10-26

where m is the number of phases, and p = N r /2 the pole pairs of the motor.

The pull-out torque is the maximum torque for a certain speed, and can be determined by letting = . Therefore,

( )T pmEI

R L

pmE R R Lmax

=+

+ 2 2 22

2 2 2 (27)

Fig.44 plots the predicted pull-out torque against the rotor speed by (27), where

K LV L V

= 1 00 1

(28)

for a VR stepping motor, and

K VL R

m=$

(29)

for a PM or hybrid stepping motor.

Fig.44 Predicted pull-out torque against rotor speed

(5) High Frequency Instability

The second term in the torque equation reaches maximum when

= =crit R L (30)

For an excitation frequency higher than this critical frequency, the stepping motor woulddevelop low frequency oscillations, and may drop out, as shown in Fig.34 by the highspeed torque dip due to instability. This oscillation can be prevented by damping (rotor cage), or by rotor position feedback (brushless dc motor). Fig.45 illustrates the effects of

damping.


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Page 10-27

Fig.45 Prevention of high speed instability by damping

REFERENCES

[1] T. Kenjo, "Stepping motors and their microprocessor controls", Oxford University Press, 1984[2] P.P. Acarnley, "Stepping motors: a guide to modern theory and practice", Peter Peregrinus Ltd.,

1982[3] G.R. Slemon, "Electric machines and drives", Addison-Wesley, 1992

Exercises:

1. List and describe the constructions and principles of the main types of steppingmotors.

2. What type of drive circuit would be suitable for each stepping motor listed in problem (1)?

3. Summarise the features of bilevel and chopper drive circuits.

4. A single stack, four phase VR stepping motor is required to produce an 18 o stepmotion. Determine the number of rotor teeth and the sequence of excitation of thestator phases. Draw a cross sectional view of the stepping motor.

5. A three stack, four pole VR stepping motor has eight teeth on the rotor as well as onthe stator. Determine the step angle as excitation is changed from one stack to thenext.

6. A permanent magnet stepping motor has the following parameters:8 rotor poles, 3 stator phases, stator winding resistance is 3 per phase, windinginductance 5 mH per phase, and induced emf equals 12 V at 1000 rev/min.

Calculate (assume sinusoidal emf and current):(a) Supply frequency in Hz and rad/s at 1000 rev/min,

(b) Maximum flux linkage,(c) Maximum torque per ampere,


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(d) Stepping rate (steps/s) at 1000 rev/min,(e) Step angle,(f) The supply voltage magnitude and phase angle for a current of 0.2 A in phase

with the back emf at 1000 rev/min,(g) The phasor diagram (sketch) for (f) above,(h) The output power and torque for (f) above, and(i) The input power for (f) above.

7. A stepping motor has a torque/angle curve which may be represented by a sinewave as shown below

3015 Mech.

Torque

0

(Nm)0.1

Deg.

Calculate:(a) The holding torque,(b) The maximum torque the motor could step at low speeds with a low inertia

load, or at very low speeds,(c) The maximum torque the motor could step at low speeds with a high inertia

load, or at slightly higher speeds,(d) The torque available for acceleration, and the acceleration for an inertia of 0.004 kgm 2, if the load torque is 0.02 Nm, for operation in region (c) above.

8. For the motor of Problem 6, calculate:(a) The pull-out torque at 1000 rev/min with a supply voltage of 16 V (AC rms

value),

(b) The torque/angle curve for a DC supply of 16

4 32

1

2

V, and

(c) sketch the pull-out torque/speed curve.

kenjo book.pdf

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