Stability limits for dynamic operation of variablereluctance stepping motor systems

P.J. Clarkson, BA, PhDP.P. Acarnley, BSc, MA, PhD, CEng, MIEE

Indexing terms: Stepping motors, Power electronics, Optimisation

Abstract: Using the simplified approach to thedynamic modelling of a variable reluctance step-ping motor system developed in a companionpaper [1], this paper develops dimensionless pullout torque curves and stability limits. Thesecurves can be used to optimise system per-formance and allow the effects of variations insystem parameters to be appreciated.

List of symbols

B = viscous damping coefficientD = differential operatorh = sampling interval70 = DC component of phase current7X = fundamental component of phase currentJ = sum of rotor and load inertiaskc = torque correction factorLo = average inductanceLx = peak inductance variation with positionn = number of motor phasesp = number of rotor polesr = phase resistances = Laplace operatorT = torque7} = frictional torqueTt = load torqueT'u = torque derivative with respect to uT'd = torque derivative with respect to 3T'z = torque derivative with respect to <5u = fundamental phase voltage frequencyVo = DC component of phase voltageV^ = fundamental component of phase voltageZ = impedancea = damping coefficientP = oscillation frequencyy = exponential constant3 = switching-angleAt = step time6 = position<$> = phase angleO = step length

Paper 6264B (PI) first received 22nd January and in revised form 19thMay 1988Dr Clarkson was with the University Engineering Department, Uni-versity of Cambridge, Cambridge, United Kingdom, and is now withPA Technology, Cambridge Laboratory, Melbourn, Royston, Hert-fordshire SG8 6DP, United KingdomDr Acarnley is with the Department of Electrical and ElectronicEngineering, University of Newcastle upon Tyne, Newcastle upon TyneNE1 7RU, United Kingdom

308

(o = angular speedkb = damping coefficientkt = inductance coefficientkt = torque coefficientkv = voltage coefficient

1 Introduction

The performance of a stepping motor system depends onmany parameters, both within the system, for examplethe drive and motor electrical characteristics, and exter-nal to it, for example the load torque and damping. Inthis paper the pull out torque, stability limits and effi-ciency of the motor system are discussed with respect totheir variation with some of the system parameters. Theemphasis is on the design of a high speed motor system.

The pull out torque describes the limit of torque pro-duction under all forms of motor control. However, acompanion paper [1] has shown that with open loopcontrol, this limit is unstable and the motor cannotoperate at maximum torque. However, with closed loopcontrol, the pull out torque represents the maximumtorque that may be produced. The pull out torque alsoprovides a measure of the motor's acceleration capabilityand determines its maximum speed for a given load.

The open loop performance of the motor, is limited bythe so called mid band instability [2, 3] which describes aregion on the switching angle against speed plane, wherestable motor operation is not possible. The purpose ofthis paper is to develop simple analytical design ruleswhich may be used to optimise the pull out torque, mini-mise the region of midband instability and maximise thesystem efficiency. These rules are derived for the variablereluctance stepping motor with a unipolar switchedvoltage drive. The analysis uses a number of dimension-less coefficients to represent, for example, the motortorque and system damping, enabling the production ofdimensionless design curves which describe the pull outtorque and stability limit for all motor/drive systems ofthis type.

Such an approach is an important alternative torecent published work [4, 5], where the emphasis hasbeen on the characterisation of the oscillations whichresult from the instability. The stability limit presentedhere is an outer bound within which instability mayoccur for particular load conditions. This is a more usefulcriteria for the system designer who wishes to avoidinstability under all conditions.

2 Normalised design curves

2.1 Normalised torque equationsThe operating limits of interest are the pull out torquecurve and the open loop stability limit. These limits have

1EE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988

been discussed in Reference [1] and are derived in theAppendix of that paper for a variable reluctance motorwith a unipolar switched voltage drive.

The pull out torque may be derived directly from thetorque equation (eqn. 67, Reference 1).

T =npkeLxV0

2ryJ{r2+u2L2}

x Fx cos (<j) — 3) —VI (1)

The torque may be maximised at any speed by makingcos (0 — d) equal to unity. The pull out torque is then

(2)_J#cMo_L uLi vo "I

The open-loop stability limit has been derived in Section5.1 of Reference 1 and stability is ensured if

B + pTd, > 0 (3)

npkc

4r(r2T\ cos2((f> - S)

2V%-+ u'

(4)

The stability limit is defined as the contour of pT'd whichis equal in magnitude to the system damping B but ofopposite sign. Therefore, the stability of a motor, oper-ating at a given torque, may be determined by finding theintersection of the torque contour with the pT6 contourcorresponding to the system damping (Fig. 1).

torque contours

contour 2 contour 1

stability limit

Fig. 1 Graphical approach to open loop stability limit

Operation on torque contour 1 of Fig. 1 is possible,except between points A and B, where the torque contourpasses within pT\ contour, and the sum of B and T's isnegative. However, operation on contour 2 is always pos-sible since there is no intersection of the two curves. Thelimit for stable operation may therefore be calculated by

rinding the intersection of the torque contours {T(co, Sj),and the torque derivative contours (T'd(co, 5)). The limit-ing torque for stability may also be derived by finding thetorque for which there is only one point of tangentialintersection. The regions of instability are described bythe range of motor speeds for which operation is not pos-sible.

The analysis may be simplified by using a number ofdimensionless terms:

(a) inductance coefficient

{b) voltage coefficient

(c) torque coefficient

(d) damping

i 0

K =2rubT

npkc V2

(5)

(6)

(7)

npkc

where ub = r/L0.Subscripts p and s will be used to differentiate between

the values of kt relating to the pull out torque and stabil-ity limit, respectively. The torque and torque derivativeexpressions (1 and 4) may now be rewritten using thedimensionless coefficients, and substituting back intoeqns. 7 and 8 gives:

J_ [cos (tan"1 u'-8) 1 u' "|kt — . * . { ( • , . , 2 > ~~ i ' 1 . i 2 \

where u' = u/ub

(9)

1 If~2k2 i + u'2 L

cos2(tan l u' — 5)

u' cos (tan * u' — 6)

+ u'2}+ 2

1 - u'2~\

1 + u'2\(10)

2.2 Normalised pull out torqueThe normalised pull out torque may be derived from eqn.9 by equating cos (tan~ l u' — S) to unity.

k 1

tp ** nn

The normalised pull out torque may be calculated andplotted for a given excitation mode and range of u'. Thecurve shown in Fig. 2 is for the system described in the

1.0

0.8

0.6

0.4

0.2

0.00.1 1 10

Fig. 2 Normalised pull out torque characteristic

Appendix of Reference 1 with one-phase-on excitation.The curve is plotted over a range of values of u' fromub/l0 to l0ub.

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988 309

The normalised pull out torque curve is of the formexpected with ktp decreasing with increasing speed. Thevalue of ktp for zero u' may be deduced from eqn. 11. It isalso possible to determine whether ktp tends to zero athigh speeds or becomes negative.

For zero u'tp

For large values of u'

k

(12)

(13)

Therefore, ktp is positive at low speeds and tends to zeroas u' increases. If /cx > kv, ktp is positive at high speeds. If,however, kx < kv, ktp is negative at high speeds, changingsign at a speed given by

1u = (14)

These results confirm those previously published [6] andshow the importance of the inductance {kY) and voltage(kv) ratios in defining the upper speed limit of the motorsystem.

2.3 Normalised open loop stability limitThe normalised stability limit, defined by the intersectionof the T(o>, S) and T'6(co, 5) curves, may be found byeliminating the switching angle 5 from the equations forkt and kb. This will define the motor speed at these inter-sections, points A and B in Fig. 1, in terms of kt and kb.

Substituting

cos (tan ~ * u' — 8)

from eqn. 9 into eqn. 10 yields

k\{\ + u'2)2 k2 + u'(l + u'2)kt

+ 2(1 - u')2 - (1 + u'2)2 k\ kb) = 0 (15)

This expression is fourth-order in u', but only second-order in kt. Hence, if kb is known, then kt may be calcu-lated for different values of «'. Rearrangement of eqn. 15yields

a1-*1)-(l+u'2)2k2kb} }j - l l (16)

The normalized stability limit (kts) may be calculated forvalues of normalised speed (u') which yield real rootsfrom eqn. 16. At speeds at which eqn. 16 yields complexroots, the motor operation is stable. The range of «'which results in real values of kts defines the speeds forwhich instability may occur under particular load condi-tions.

The corresponding switching-angles (<5) may bederived by substituting kts into eqn. 10:

cos (tan"1 u' — S) u'

-(\+u'2)2k2kb} (17)

There will always be two solutions for <5, but only thelower value is relevant for open loop control since oper-

310

ation at the higher switching angle is unstable, as dis-cussed in Section 2 of Reference 1.

The normalised stability limit may be calculated andplotted for a given system damping, excitation mode ofu'. The excitation mode defines kv (eqn. 6) and kc (Table 1in Section 4.3). The damping coefficient (kb) may bederived from eqns. 3 and 8 for the limit when

B +

then

't = 0

2r3B

np2kcV2L2

(18)

(19)

is produced by the motor. This will be the case if thenormalised pull out torque curve intersects the normal-ised stability limit.

The normalised stability may be approximated by arectangular region within which operation may beunstable (Fig. 3). The extent of this approximate limit is

1.0

0.8

0.6

0.4

0.2

0.0

Mr

/ unstable

0.1 1.0

Fig. 3 Approximate normalised open loop stability limitstability limit

given by the minimum and maximum speeds u'min andiimax, and the maximum torque ktmax for instability.These values of u' and kts may be derived from eqn. 16.

The conditions for stable operation are then:

u' < u'min stable for kt > 0 (20)

« C < "' < timax stable for kt > ktmax (21)

"' > "L* stable for kt > 0 (22)

Inspection of eqn. 16 shows that as u' increases, kts willgo negative before the roots become complex. Themaximum unstable speed-range is therefore defined whenkts equals zero. Hence

i /2

- 1 = 0

and

- « ' 2 ) - ( l +u'2)2k2kb =

(23)

(24)

The maximum range of instability is then defined by thereal roots for u'.

u = —(2k2kb+l)±(Sk2kb

2k\kb(25)

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988

If k\ kh is small, then

'2

min— Jk2 \i

Z K 1 Kb

1.0

0.8

0.6

0.4

0.2

0.00.1 10

Fig. 4 Stable operating regions for open and closed loop controlpull out torquestability limit

and

1~x~ k\kb

and for zero damping

u'in « 1 and u' co

(26)

(27)

For a system with no additional damping, kb is small andthe lower root is near unity and the upper root large. Asdamping is added to the system, the two roots movecloser, until there is only a single root at u' = y/3, whenthe total system damping is defined by

Ski kb+1=0 (28)

This result may also be derived by expressing kb in termsof u'. From eqn. 24:

/2\2u'2)(29)

This defines the damping required to achieve a givenmaximum speed limit for stability. Maximum kb occursfor u' = yj3, when the system damping is defined by eqn.28.

A minimum torque may also be defined for stableoperation at all speeds, and this torque corresponds tothe maximum of the normalised stability limit. The deri-vation of this maximum is difficult, but a good approx-imation may be made by calculating kts for u' = y/3 (Fig.3). Hence

k — 1 +16k2

t(30)

These results correspond directly to those recentlypublished by other authors [4, 5]. Their papers derivemodels for the hybrid and permanent magnet steppingmotors which may easily be transcribed for the variable-reluctance motor. Both develop more complex motormodels than have been presented here, and they are thenreduced to derive simple results. It is the simplified formof their results which correspond to eqns. 24, 28 and 29.The paper by Pickup and Russell [5] is of most interestsince their model predicts the amplitude of stable

midband oscillations for different switching angles androtor inertias. The simplified model is then derived forinfinite rotor inertia and this defines an outer boundwithin which instability may occur. This is the same limitwhich is obtained by the simplified modelling describedhere.

2.4 Summary of the design curvesThe basic form of the design curves has been presented,and Fig. 4 shows the area of stable operation for openand closed loop control. Open loop operation is stable inregion 1 only, and closed loop operation is stable inregions 1 and 2.

3 Variation of external system parameters

The effects of load torque on the system performancehave already been discussed and may be summarised bystating that the region of unstable operation decreases forincreasing load torque. In this Section, the effect ofadding viscous damping to the system will be investi-gated. It has already been shown that the region of insta-bility decreases in size as damping is added and that thesystem is always stable if

Sk2kb + 1 < 0 (31)

This defines a minimum damping to ensure stabilitywhich depends on the excitation mode:

kb = 1/8*?

Hence

= np2kcVlL2/16r

(32)

(33)

When viscous damping (proportional to co) is added tothe system, it reduces the amount of useful or net torqueproduced by the motor. The total or gross torque pro-duced is unchanged, and the difference between the grossand net torque is the damping torque. It is useful toinvestigate the effect of damping on the useful torqueproduced by the system and this may be calculated as afunction of u' for a given excitation mode.

1 1 u' 1

+ «'2} fci"i + «'2_+ kbu' (34)

The normalised net pull out torque and stability limitsfor a drive using one-phase-on excitation with values of

1.0

0 .8

0.6

0.4

0.2

0 . 0

increasing kb

0.1 10

Fig. 5 Variation of normalised net pull-out torque and stability limitswith mechanical damping

pull out torquestability limit

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988 311

kb from —0.05 to zero are shown in Fig. 5. The curvesshow clearly the two fold effect of adding damping to thesystem. First, the normalised net pull out torque isreduced, particularly at high speeds, and this greatlyreduces the speed range of the system. Secondly, theregion of instability is reduced in area with the additionof damping. Therefore, the system designer has to com-promise the speed range of the motor system if stability isto be ensured.

It is also useful to summarise the effect of damping onthe limiting speeds for stability. These limits aredescribed by the maximum and minimum speeds forwhich the motor system will be stable for all loads, andare defined by eqn. 25. Fig. 6 shows the variation of these

0.15

0. 125

0.10

0.05

0.0010

Fig. 6 Effect of damping on limiting speeds for stability for all loadsstability limit

kb1 =Kbklkl

limits with damping, and is applicable for all modes ofexcitation. The maximum of the curve is for u' = ^/3, andthis describes the minimum damping required for a stablesystem as described by eqns. 32 and 34.

4 Variation of internal motor parameters

In this Section the effects of internal motor parametersare considered. Variation of the motor resistance, induc-tance and excitation mode are investigated.

4.1 Phase resistanceThe motor phase resistance is determined by the type ofphase winding and its configuration, but it may beincreased by adding an external series resistance (forcingresistance) such that the mean current in the motorremains constant. Therefore, if the phase resistance isincreased by 50%, the drive voltage must also beincreased by 50%. The addition of resistance reduces thewinding time constant, therefore allowing faster build upof the phase currents and more torque to be produced athigh speeds.

The addition of phase resistance does not affect thenormalised pull out torque curve of ktp against u', sincealthough ub changes with resistance R, ktp does notchange if the ratio of peak drive voltage V to resistanceremains constant (eqn. 11). Note that the peak drivevoltage is proportional to the mean drive voltage Vo.This may be interpreted as an increase of pull out torqueat all speeds. For example, if the phase resistance anddrive voltages were increased by 50%, then the pull outtorque originally produced at 100 steps/sec would then

be produced at 150 steps/sec. Fig. 7 shows the normalisedpull out torque and stability curves for values of phaseresistance ranging from 22.5 to 90 Q.

1.0r

0.8

0.6

0.2

0.0

increasing R

0.1 1

u'

10

Fig. 7 Variation of normalised pull out torque and stability limits withphase resistance

pull out torquestability limit

The normalised stability limit is also affected bychanges in the phase resistance since kb varies with resist-ance (eqn. 16). However, the changes are small and thenormalised stability limit for the lower speeds is almostunchanged. As with the normalised pull out torque, thespeeds represented by u' increase in proportion to thephase resistance. Hence, a 50% increase in phase resist-ance would defer the onset of instability from, forexample, 300 to 450 steps/sec. Similarly, the amount ofadditional damping required for stable operation is givenby eqn. 33 and a 50% increase in phase resistancerequires 50% less damping to stabilise the system.

In conclusion, the addition of a forcing resistance tothe motor drive improves the system's high-speed per-formance. The pull out torque increases and the midbandinstability occurs at a higher speed. However, the effi-ciency of the system is reduced since power is dissipatedin the forcing resistance.

4.2 Phase inductanceThe phase inductance is determined by the type of phasewinding and the mechanical construction of the motor.The inductance is assumed to vary sinusoidally withrotor position and is described by the mean inductance(Lo) and the peak inductance (LJ. The effects of varyingLo and the inductance coefficient (/cj will be investigated.

Firstly, the variation of Lo with constant kv Figure 8shows the normalised pull out torque curves and stabilitylimits for values of Lo ranging 50% above and below thevalue given in appendix B of Reference 1. Note that forconstant ky an increase in Lo requires a proportionateincrease in Lv

The normalised pull out torque curve of ktp against u'does not change (eqn. 11), but both ktp and u' changewith Lo for a given motor torque and speed. Therefore, ifLo increases by 50%, then ktp decreases by 50% and u'increases by 50%. For example, if the motor initially pro-duces 100 mNm at 100 steps/sec and Lo is then increasedby 50%, the operating point on the normalised pull outtorque curve will then represent production of 150 mNmtorque at 50 steps/sec.

The normalised stability limit kts is not affected byvariation of Lo in the absence of any damping (eqn. 16).

312 IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988

However, for a given system damping B, the dampingcoefficient kb varies with Lo, and the normalised stabilitylimit also changes (Fig. 8). The region of instability

i.o

0.8

0.6

0.4

0.2

0.0

increasing l_0

0.1 10

Fig. 8 Variation of normalised pull out torque and stability limits withmean winding inductance

pull out torquestability limit

expands with increasing Lo and the maximum range ofinstability as defined by eqn. 25 increases since kbdecreases in magnitude while kY is constant. Note alsothat ub decreases with increasing Lo. Hence, the onset ofinstability will occur at lower motor speeds as Lo isincreased.

Secondly, the variation of ku the ratio of inductancecoefficients, for constant Lo. Fig. 9 shows the normalised

1.0

0.8

0.6

0.4

0.2

0.0

increasing kl

0.1 10

Fig. 9 Variation of normalised pull out torque and stability limits withratio of inductance coefficients

pull out torquestability limit

pull out torque curves and stability limits for values of kxranging from 1.0 to 2.5. In practice, kx is never as low as1.0, but usually lies between 1.5 and 2.0.

The normalised pull out torque curve varies little withkt and since ktp and u' are independent of kt the pull outtorque is largely independent of kt (eqn. 11). However,the normalised stability limit does vary with ku theregion of instability reducing if kx increases (eqn. 16).Therefore, the minimum torque required for stable oper-ation is also reduced by increasing kv Fig. 10 shows thisminimum torque for a range of kb from —0.05 to zero,and for kx from 1.0 to 2.5. Increasing kx reduces theamount of damping or torque required for stable oper-ation, but note the region of instability is not eliminated

even with the maximum practical value for kx. Increasingthe inductance ratio kx has the effect of stabilising themotor system without reducing the pull out torque.

0.30

0.25

0.20

0.15

0.10

0.05

0.00

increasing kl

M.O

0.00 -0.02 -0.04 -0.06 -0.08kb

-0.10

Fig. 10 Variation of minimum torque for stable operation withdamping and inductance coefficients

stability limit

To conclude, the effects of varying Lo and Lx on thepull out torque and stability limit are complex. Thevariation of pull out torque is well documented in Refer-ence 7, where it is shown that, for high speed operation ofa motor, there are optimum values for the ratio of meaninductance variation kt which depend on the excitationmode. The low speed torque may also be increased byincreasing Lt. However, the variation of the stabilitylimit has not been previously investigated. It has beenshown that the onset of instability occurs at lower speedsif Lo is increased at constant kx but at higher speeds if kxalone is increased. Hence the stability of a motor systemis improved by reducing the mean inductance Lo or thepeak inductance variation Lv

4.3 Excitation modeFinally, the effects of changing the excitation mode arediscussed. Both kv and kc are directly affected by the exci-tation mode and their values are given in Table 1. The

Table 1: Excitation modes

Excitation schemeOne-phase-onHalf steppingTwo-phase-on

kv

0.600.791.20

K1.501.000.75

0.52*,0.79*,1.05*,

normalised pull out torque curve is dependent on kv andhence varies with excitation. It is now more convenient tonormalise the pull out torque with respect to the halfstepping low speed torque, so that a direct comparison ofthe torque produced with each excitation mode may bemade. This is achieved by multiplying ktp by a factor k'cso that it equals one for half stepping at low speed. Thenormalised pull out torque curves and stability limits forthe different excitation modes are shown in Fig. 11.

Let

k'tp = k ' c - k t p a n d Ks = K K s (35)

where

k'c = nkJ4kc

The normalised pull out torque curves show clearlythe difference between single, half and double stepping

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988 313

operation. At low speeds, half stepping produces 15%more torque than the one or two phase-on modes whichproduce the same torque. However, at high speeds the

1.0r

0.8

0.6

0.4

0.2

0.00.1 10

Fig. 11 Variation of normalised pull out torque with excitation mode

pull out torquestability limit

one-phase-on and half stepping modes produce twice asmuch torque as the two-phase-on mode.

The normalised stability limits also show differencesbetween the operating modes. The region of instability issmallest for one-phase-on where operation is theoreti-cally possible at all speeds. The maximum operatingspeeds for half-stepping or two-phase-on are, however,little in excess of u'. Damping may be added to ensurestability for each mode, and the amount of dampingrequired is given by eqn. 28. Note, twice as muchdamping is required to stabilise the two-phase-on modeas for the one-phase-on mode. The damping reduces thenet motor torque (Fig. 12). The curves show that one-

1.0

0.8

0.6

0.4

0.2

0.00.1 10

Fig. 12 Normalised gross and net pull out torques for stable operation

pull out torque

phase-on excitation produces twice the speed-range oftwo-phase-on and twice the torque at speed «'. The lowspeed operation is little changed for all modes.

The efficiency of the motor system also varies withexcitation mode. Mechanical losses are independent ofthe excitation mode. However, electrical losses, of whichthe resistive power loss is dominant, vary with excitationmode. The resistive loss depends on the mean power loss(time average of i2r) for each motor phase and since themean current increases with the number of phasesexcited, the power loss also increases. Hence the efficiency

314

for one-phase-on is higher than for half-stepping which isin turn higher than for two-phase-on [7].

The excitation modes may be summarised as follows:at low speeds two-phases-on has the highest inherentdamping and provides the best single-step and low speedresponse. At moderate speeds, where stability is not aproblem, half-stepping provides more torque and betterresolution. Finally, at high speeds one-phase-on is themost stable mode and provides the most torque with thehighest efficiency.

Note that the design procedure in its present formrelates only to variable-reluctance type motors withsimple viscous damping. Although viscous-inertiadampers can be used to improve stability [9] the pre-sence of such a device increases the order of the charac-teristic equation (eqn. 27 of Reference 1) from second tothird, as well as introducing two additional damperparameters. Therefore a detailed study of viscous-inertiadamper effects is beyond the scope of the present work.

5 System design

5.1 Normalised design curvesFor a given motor system, where the phase inductanceshave been defined by the motor design, the systemdesigner is restricted to varying the mechanical damping,phase resistance and excitation mode. The design curvesof Fig. 13 may then be used to choose the best values forthese parameters. Note that the normalised pull-outtorque is dependent on the excitation mode, but that thenormalised stability limit is independent of the excitation.Therefore, the designer requires a set of curves for eachexcitation mode. Since the curves do not vary signifi-cantly with the phase resistance it is sufficient to use onlycurves which show the effect of varying the mechanicaldamping. To aid comparison of the excitation modes it isproposed that the curves are normalised with respect tothe half-stepping low-speed torque as in Section 4.3.Curves for each excitation mode are shown in Fig. 13 forvalues of kb from —0.05 to zero. Some examples of designcalculations are given in the next Section.

The machine designer would also need to use curveswhich show the effects of variable inductance on thesystem performance. Again, different curves would berequired for each excitation mode. However, rather thanreproduce several sets of design curves for all variationsof the design parameters, it should be noted that thecurves may be easily derived from the equations for k'tpand k'ts for a given motor specification (eqns. 36 and 37).

\ (36)

nuu'2)

1 - - «")

\ l /2 "I

-{\+u>2)2k\kb}\ - l j (37)

where

^~2npk2V2i b npkcV2'

kc = 3/2, 1 or 3/4, u' = u/ub and ub = r/L0

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988

5.2 Applications of system design procedureThis section contains examples illustrating the use of themotor system design curves. For both the examples it is

1.0

0.8

0.6

0.4

0.2

0.00.1

1.0

0.8

0.6

0.4

0.2

0.00.1

1.0

0.8

0.6

0.4

0.2

0.0

increasing kb

•v XTO,OT

10

increasing kb

10

\\

• -O.O2- 0 03- 0 . 0 4 -

• . i i i i .

/ '

\ ' / '

increasing

V N

r ' l f—r—

1 III/ '

kb

x v-^0.00\ ^ ^

N-0.01v r \ . i n

0.1 1 10

Fig. 13 Design curvesa One-phase-onb Half-steppingc Two-phases-on

pull out torquestability limit

assumed that the designer is initially using the motorsystem described in the Appendix of Reference 1. Nor-malised load lines for this motor are plotted in Fig. 14 aswell as various points which are referred to in the exam-ples.

The dimensionless coefficients for this system, with noexternal load in addition to the motor's frictional and

damping torque, are

(a) correction coefficient

kc = 1.5 for single stepping

(b) inductance coefficient

fcx = L0/Lx = 1.58

(c) voltage coefficient

K = VJVt = 0.61

(d) torque coefficient

kt = 2rub T/npkc V20 = 0.056

(e) damping coefficient

kb = 2ru\ rjnpke V\ = -0.0098

(38)

(39)

(40)

(41)

(42)

52.1 Design example 1 — open-loop control;Problem: can the basic motor system be stabilised foropen loop operation up to 400 steps/sec?

Solution: the maximum stable speed, for no externalload, is defined by the intersection of the motor's normal-ised load-line and stability limit corresponding to themotor damping. This intersection is shown as point a inFig. 14a. This defines a lower limit for stability, and theaddition of an external load will increase this limit.

At point a

u' = 1.1 (43)

Hence,

u = 1.1M6 = 463 rad/sec and co = 584 rad/sec (44)

The maximum stepping rate is therefore 221 steps/sec.The stepping rate may be increased to 400 steps/seceither by adding damping or by increasing the forcingresistance.

Damping: the required maximum stepping rate corre-sponds to u' = 2.0 and since this exceeds ,/3 theminimum damping for stable operation defined by eqn.24 must be added.

(45)

This corresponds to a total rotor damping of 0.10 mNmsec. Therefore, additional damping of 0.08 mNm/sec isrequired to stabilise the motor system. The net loadtorque will be reduced by the addition of damping andthe motor's acceleration will be degraded.

Forcing resistance: the normalised pull out torquecurve and stability limit remain unchanged, but ub

increases. An 81% increase in maximum speed isrequired, hence ub, i.e. the phase resistance, must beincreased by 81%. A forcing resistance of 32 Q may beadded in series with each motor phase and the supplyvoltage increased by 81%. The motor torque will beincreased at all speeds, but the system efficiency will bereduced and the low speed response may be impaired.

In practice, a combination of the two stabilisationmethods may be the best solution to the problem. Notethat changing the excitation mode would not greatlyaffect the stability limit and so is not discussed here.

Implementation: the measured stability limit for thebasic motor system was 230 steps/sec. The addition ofdamping increased this limit to 500 steps/sec while theaddition of the forcing resistance increased the limit to

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988 315

390 steps/sec. The experimental results correspond wellwith the model.

5.2.2 Design example 2 — closed-loop control;Problem: can the basic motor system be modified, for

1.0

0.8

0.6

0.2

0.0

increasing kb

0.1 10

increasing kb

1.0

0.8

0.6

0.4

0.2

0.0

increasing kb

0.1 1 10

Fig. 14 Normalised load lines for commercial motora One phase-onb Half-steppingc Two-phases-on

pull out torquestability limitload line

closed loop operation, to produce gross torques of 100mNm at 100 steps/sec and 20 mNm at 750 steps/sec?

Solution: the stability limits are of no importance forclosed loop control and it is the pull torque which limitsthe motor performance. From example 1, ub = 421

rad/sec which is equivalent to 201 steps/sec. The torquecoefficient, for all excitation modes, may be calculatedfrom

Hence

k't = 1.1T

(46)

(47)

The modified system requires k't = 0.77 at u' = 0.50 and/cj = 0.16 at u' = 3.7, marked as points b and c on Figs.14a, b and c. Since the maximum value of k't with one-phase-on excitation is 0.87 at low speed, there are twopossible solutions to the problem. Either single steppingmay be used with a very large forcing resistance, toincrease the torque at all speeds, or half-stepping with amoderate forcing resistance. It is evident from Figs. 14aand 14b that the torque will be easier to achieve withhalf-stepping, which produce more low speed torque.Also, since the addition of a forcing resistance reduces thesystem efficiency, a one-phase-on system would be lessefficient.Half-stepping: from Fig. 14b it is clear that half stepping,in the absence of a forcing resistance, cannot provide suf-ficient torque for either of the required speeds.

Forcing resistance: the normalised pull out torquecurve and stability limit remain unchanged, but ubincreases. Resistance must be added to allow operation atboth 100 and 750 steps/sec. From Fig. 146, the requiredtorque is produced at points b' and c' at u' = 0.4 and 3.1,respectively. Since ub is proportional to the phase resist-ance, if the resistance is increased, points b and c move inthe direction of decreasing u' at constants k't. Sufficientresistance must be added to move points b and c topoints b' and c'. If r0 is the original resistance, then thenew resistance is the greater of (0.50/0.40). r0 and(3.7/3. l).r0. Hence, the resistance must be increased by83% with the addition of 10 Q per motor phase. Theoperating points are now k't = 0.77 at u' = 0.40 and k't =0.17 at u' = 1.30. This corresponds to torques of 100mNm at 100 steps/sec and 22 mNm at 750 steps/sec.

Single-stepping: using a similar analysis to that forhalf-stepping, it can be shown that an additional 34 Qforcing resistance would produce torques of 100 mNm at100 steps/sec and 34 mNm at 750 steps/sec.

Implementation: the measured torques for the basicsingle stepping motor system were 88 mNm at 100 steps/sec and 16 mNm at 750 steps/sec compared to theoreticalvalues of 90 mNm and 17 mNm, respectively. The use ofhalf stepping with an additional 10 Q forcing resistance inseries with each motor phase increased these torques to96 mNm at 100 steps/sec and 18 mNm at 750 steps/sec.The experimental results correspond well with the model.

The alternative solution using single stepping with anadditional forcing resistance of 34 Q produces 95 mNmat 100 steps/sec and 30 mNm at 750 steps/sec. Again theresults are as expected.

The results show that both solutions are feasible. Thepreferred solution may depend on other criteria, such asefficiency.

6 Conclusions

The performance of a high speed stepping motor systemhas been summarised by developing dimensionless pullout torque curves and stability limits. These curves varywith the system parameters and may be used by thesystem designer to optimise the system performance for a

316 IEE PROCEEDINGS, Vol. 135, Pt. B, No. 6, NOVEMBER 1988

given application. The effects of varying the mechanicaltorque and damping, electrical phase resistance, induc-tance and excitation mode have been investigated andthe results presented with dimensionless plots. Theseplots may then be used to predict the performance of anyvariable reluctance motor with unipolar switched-voltagedrive.

7 Acknowledgment

The authors would like to thank the UK Science andEngineering Research Council for the financial supportfor their work on stepping motor systems. They wouldalso like to thank Mrs Pauline Gormley for her assist-ance in the preparation of the typescript.

8 References

1 CLARKSON, P.J., and ACARNLEY, P.P.: 'Simplified approach tothe dynamic modelling of variable-reluctance stepping motors', to bepublished in IEE Proc. B.

2 WARD, P.A., and LAWRENSON, P.J.: 'Backlash, resonance andinstability in stepping motors'. Proceedings of the Sixth annual sym-posium on incremental motion control systems and devices, Uni-versity of Illinois, 1977

3 RUSSELL, A.P., and PICKUP, I.E.D.: 'High-frequency instabilitiesin variable reluctance stepping motors', IEE Proc, 1978, 125, (9), pp.841-847

4 VERGHESE, G.C., LANG, J.H., and CASEY, L.F.: 'Analysis ofinstability in electrical machines', Trans. IEEE, 1986, IA-22, (5), pp.853-864

5 PICKUP, I.E.D., and RUSSELL, A.P.: 'Dynamic instability inpermanent-magnet synchronous/stepping motors', IEE Proc. B, 1987,134, (2), pp. 91-100

6 ACARNLEY, P.P.: 'Stepping motors: a guide to modern theory andpractice'. IEE Control Engineering Series 19 (Peter Peregrinus, 1982)

7 ACARNLEY, P.P., and HUGHES, A.: 'Machine/drive circuit inter-actions in small variable-reluctance stepping and brushless DC motorsystems', Trans. IEEE, IA, 1987

8 CLARKSON, P.J.: 'Modelling and control of stepping motorsystems'. PhD Thesis, University of Cambridge, 1987

9 KUO, B.C., RAJ, K., and MOSKOWITZ, D.: 'Analytical study ofeffects of viscous-inertia dampers on the performance of step motors'.Proceedings of the twelth annual symposium on incremental motioncontrol systems and devices, University of Illinois, 1983

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