A contribution to the development of stepping motors
Citation for published version (APA):
Bakhuizen, A. J. C. (1973). A contribution to the development of stepping motors. Technische Hogeschool
Eindhoven. https://doi.org/10.6100/IR29998
DOI:
10.6100/IR29998
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Published: 01/01/1973
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A CONTRIBUTION
TO THE DEVELOPMENT OF
STEPPING MOTORS
A CONTRIBUTION
TO THE DEVELOPMENT OF
STEPPING MOTORS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. IR. G. VOSSERS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
VRIJDAG 5 OKTOBER 1973 TE 16.00 UUR
DOOR
ARIE JOHANNES CORNELIS BAKHUIZEN GEBOREN TE SLIEDRECHT
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN PROF. DR. IR. J.G. NIESTEN
EN
PROF. P.J. LAWRENSON,
een aandenken aan mijn Ouders
voor
Inge,PREFACE Terminology List of Symbols
CONTENTS
PART I: GENERAL CONSIDERATIONS ON STEPPING MOTORS INTRODUCTION
I. SHAPE AND OPERATION OF STEPPING MOTORS Synopsis 11 12 13 17 17 20
I. I The Primitive Electric Motor 20
1.2 The Basic Permanent Magnet Stepping Motor 22 1.3 The Basic Variable Reluctance Stepping Motor 23 1.4 The Basic Hybrid Stepping Motor 24 1.5 Development of Basic Motors into Industrial Types 26
I .5. I The multistator motor 28
1.5.2 The motor with sectional phase division 30
I. 6 Conclus ion 32
2. THE CALCULATION OF THE QUASI-STATIC TORQUE 33 Synopsis
2.1 The Basic Motors as Used for the Calculation 34
2.2 The Basic V.R. Motor 37
2.3 The Basic P.M. Motor 42
2.4 The Basic Hybrid Motor 45
2.5 Cernparing the Torques of the Three Types of Motor 48 2.5.1 Equal numbers of ampere-turns SI
2.5.2 Optimal excitation 56
2.5.3 Available coil space 59
2. 6 Dimensional Analysis 60
2.7 The Influence of the Number of Poles on the
2.8
2.9 2.10
Average Torque
The Influence of Saturation on the Shape of the Torque-to-Displacement Curve
Torque Calculations for Non-Basic Motors Conclusion
62
64 68 69 3. ON THE DYNAMIC BEHAVIOUR OF STEPPING MOTORS 71
Synopsis
3.1 Voltage and Torque Relations 72
3.2 The Dynamic Behaviour of a Motor with'a Linear
Torque-to-Position Curve 74
3.2.1 Steady state phenomena 79
3.2.2 Transient phenomena 83
3.2.3 Frequency scale 86
3.2.4 Additional damping 87
3.3 The Dynamic Behaviour of a Motor with a
Sinusoidal Torque-to~Position Curve 90
3. 3. I Resonance 92
3.3.2 The upper bound of the slew range 97
3.3.3 Transient phenomena 98
3.3.4 Damping 99
3.4 Conclusion 100
4. REVIEW AND OUTLOOK I 02
PART II: A DUAL MODE SERVOMOTOR INTRODUCTION
I. CONSTRUCTION, SUPPLY AND CONTROL OF THE DUAL MODE MOTOR Synopsis
I. I Construction 1. 1.1 The stator core 1.1.2 The rotor core
1. I. 3 The windings and the· fields 1.2 Supply and Control
1.3 Position Transducer
2. ANALYSIS OF THE PROPERTIES IN THE TWO MODES Synopsis 105 105 110 l i l 113 114 114 117 120 123 2.1 General Considerations on the Analysis of the Motor 124
2. 1.1 Data of the prototype motor 127
2.2 The Induction Motor Mode 131
2.2.1 Verification of parameters 132
2.2.2 Diagram of the stator current, IlO-volt mains,
iron reluctance allowed for 132
2.2.3 Considerations on the higher harmonies in the
stator current 133
2.2.4 Torque vs speed curve, IlO-volt mairts 133
2.2.5 Rotor with single cage 134
2.2.6 Experimental torque curve, IlO-volt mains 135 2.2.7 Additional braking torque owing to hysteresis 136 2.2.8 Comparison of calculated with measured torque 136
2. 2. 9 Synchronous torques !36
2.2.10 Behaviour at higher voltages 138
2.2.1 I Consiclering a correction of the calculation of the
stator current 139
2.3 The Stepping Motor Mode 140.
2.3. I Verification of the method 140
2.3.2 The assumption of constant velocity 141 2.3.3 Torque-to-load angle diagram for 55-volt supply,
cage-less motor 143
2.3.4 Maximum torques as depending on voltage 144
2.3.5 Stator current 145
2.3.6 Correction of currents, allowing for iron influence 147
2.3. 7 Estimating Uactual 148
2.3.8 Torque-to-voltage curve, cage-less motor 149 2.3 .. 9 Current-to-voltage curve, cage-less motor .149 2.3. 10 Torque-to-voltage curve, idealised motor 150 2.3. I I Stator current, correction for influence of iron 151 2.3. 12 Torque-to-voltage diagram, complete motor 152 2.3.13 Current-to-voltage diagram, complete motor 154
2.4 Conclusion 155
3. COMMENT ON THE PROPERTIES OF THE DUAL MODE MOTOR 3.1 Temperature Rise
3.2 Design Considerations Regarding Torque Production· 3.3 Noise
156 156 158 160
4. PERFORMANCE OF THE MOTOR IN A POSITIONING DEVICE 4. I Positioning Accuracy 4.2 Position Approach 4.3 Speed of Positioning 4.4 Duty Cycle APPENDICES A B
c
D ·E FApproximative Calculations with Permanent Magnets in Plane Fields
A. I Some common expressions applying to magnetism A.2 Introduetion of a substitute body
A.3 On the lateral magnetisation
A.4 Calculations of the field in stepping motors A.5 Flux linkages
A.6 Magnetic co-energy
The Response to a Step-wise Excitation, Linear Torque-to-Position Curve
Properties of the Phase Plane Diagram for Linear and Sinusoidal Torque-to-Position Curves
On the Calculation of Magnetic Fields in a Cylindrical Structure
D.I Idealised stator iron, rotor and stator aligned D.I.! Rotor only energised, flux linkage of rotor coil D. 1.2 Rotor and stator both energised, flux linkage
of stator coil
D.2 Idealised iron in stator and rotor, arbitrary position D.2.1 The flux linkage of the stator ~oil
D.2.2 Opposition of stator and rotor excitation D.2.3 Flux linkage, rotor not energised
D.2.4 Air-gap fields in the two extreme conditions Components of the Field in the Air Gap of the DMM E.I Stator coils, inductances
E.2 Stator field • Fourier expansion for one phase energised E.3 Stator field, various ways of excitation
E.3.1 DD1
1:rreecctt current in a- aanndd c-phase; ~a
E.3.2 current in a- c-phase; :a
E.3.3 Direct current in b- and d-phase; 7b E.3.4 Direct current in b- and d-phase; 1b E.3.5 Alternat~ng current; ~a = ~ ie~ ib ~ -E.3.6 A~ternat1ng cur~ent; 1a ~ 1c• ~ = 1d E.3.7 D1rect current 1n four phases
i c. - 1 . c 1d. - 1 . d 1 . d
Method of ~king Calculations Concerning the Induction Motor F.I Stator; fields and currents
F.I.I The development of one energised coil into a set of harmonie current sheets
F.I.2 The combined current sheetsfora four-phase set of coils
F.I.3 The air-gap field caused by a current sheet F.l.4 The air-gap field caused by the energised a-phase
9 162 162 163 165 165 167 167 171 172 176 178 178 180 183 200 202 203 203 204 204 204 205 206 208 208 212 215 215 215 216 216 216 216 216 217 219 219 222 224 225
F.I.5 F.I.6
Flux linkages between stator coils
Revolving fields caused by a four-phase stator
225
excitation 226
F.2 Rotor; fields and currents 227
F.2.1 Flux linkage of a mesfi with_ the reveilving
stator field 228
F.2.2 Flux linkage of a rotor mesh owing to the
system of rotor currents 229
F.2.3 The field by rotor currents 230
F. 2. 4 Mutual induction from ro·tor to stator 231 F.2.5 Rotor currents, resistance and leakage inductance 232
F.3 Rotor with K equal cages 234
F.3.1 Rotor currents and their fields 234 F.3.2 Flux linkage of a rotor mesh caused by the
system of rotor currents 235
F.3.3 Mutual induction from rotor to stator 236
F.4 The differential equations 2)6
F.4.1 Currents 238
F.5 Torques 239
F.6 Stator current 240
G Methad of Making Calculations Concerning the DMM
in the Stepping Mode 242
G.l Calculation of the currents 245
G.2 Calculation of the torque 251
H Estimating the Influence of Saturation and Hysteresis
in Stepping Motors 253
H.I Influence of saturation; constant current excitation 253 H.2 Influence of hysteresis; constant current excitation 255 H.3 Hysteresis torque owing to A.C. excitation 258 H.4 The influence of saturation in the DMM in
the stepping mode 260
H.5 Decrease in the torque in stepping motors owing to
saturation of the teeth 264
I Caiculating Leakage Inductances 267
I. I The induction motor mode 267
I. I. I Slot leakage 268
I.I.2 Coil-end leakage 268
I. 1.3 Zig-zag leakage inductance 268
I. L4 The differential leakage 268
I.l.5 The rotor leakage inductances 269
I.2 The stepping motor mode 270
REPERENCES ACKNOWLEDGEMENTS SUMMARY SAMENVATTING 271 274 275 277
PREFACE
With the rising popularity of digital methods in control engineer-ing, the Department of Electrical Engineering of the Eindhoven University of Technology took an interest in the problems concerning the electric motors that could be used as the activators in numeri-cally controlled positioning devices.
For this purpose the stepping motor has, by its very nature, many advantages over other servo-motors. On further examination, however, it failed to meet ·some high-speed requirements.
An investigation, in an attempt to avoid this deadlock, suggested the design of a motor that effered the choice of behaving either as an induction motor or as a stepping motor, using one set of electric and magnetic circuits in both modes [37] •
For the realisation of such a novel dual mode motor it was consid-ered necessary to make a broad study of the characteristics of various types of stepping motors in order to be able to make a reasonable choice in view of the desired performance.
This study, besides being useful for a proper appreciation of their nature, yielded some basic knowledge covering most types of stepping motors. These general considerations that may be useful also as an introduetion to further analysis are presented as Part I of the thesis. A prototype of the dual mode motor was built, its design based mainly on common sense. A circuit for the proper power distribution in two modes also had to be developed.
After some minor modifications the motor appeared to possess the desired properties to a reasonable extent.
A method for calculating currents and torque was subsequently developed.
Part II of the thesis deals with the details of construction, calculation and behaviour of the prototype motor. In its final chapter some recommendations for improving the performance are made.
TERMINOLOGY
Detent torque: The maximum torque that can be applied to the shaft of an un-excited motor without c·ausing continuous rotation.
Holding torque: The maximum steady torque thàt can be externally
applied to the shaft of an excited motor without causing continuous rotation.
Maximum pulZ-in rate (speed): The maximumswitching rate (speed) at which an unloaded motor can start without losing steps. Maximum pulZ-out rate (speed): The maximum switching rate (speed) which the unloaded motor can fellow without losing steps.
Overshoot: The maximum amplitude .of the oscillation around the final holding position of the rotor after cessation of the switching.
Permanent overshoot: The number of steps the rotor moves after
cessation of the switching pulses.
PulZ-in rate (speed): The maximum switching rate (speed) at which a frictionally loaded motor can start without losing steps. PulZ-in torque: The maximum torque that can be applied to a motor-shaft when starting at the pull-in rate.
PUlZ-out rate (speed): The maximum switching rate (speed) which a frictionally loàded motor can fellow without losing steps. PulZ-out torque: The maximum torque that can be applied to a motorshaft when running at the pull-out rate.
Start range: The range of switching rates within which a motor can start without losing steps.
Step angZe: The nominal angle that the motorshaft must turn through between adjacent step positions.
Stepping rate: The number of step positions passed by a fixed ·point on the rotor. per second.
SZew range: The range of switching rates within which a motor can run unidirectionally and follow the switching rate (within a certain maximum acceleration) without losing steps, but cannot start, stop or reverse.
These terms are borrowed frbm the standardisation_proposal COMEL II/CT/GT6/NL 11, Octobre 1971.
a A b B
c
d f g h H i, I j J J 0 k K 1, L L m M n N p p q r, R s List of Symbols factor=
!f/J area factor = S/J slot width magnetic inductionconstant, Fourier amplitude diameter pitch factor coefficient of friction air-gap width height field intensity current current intensity moment of inertia magnetisation cage number
copper distribution factor number of cages length inductance mesh number number of phases number of meshes shaft speed number of turns
number of control pulses number of pole-pairs power
copper cross sectien radius resistance slip 13 -I sec 2 m -2 sec m Nmsec/rad m m m H r.p.m.
w
2 m ms
t, T T u, u V ww
x,y,z zz
fl y 6 e:e
À V p T w,n
radius of st;ator bere spring constant time torque voltage volume tooth- or slotwidth field energy coordinates number of teeth impedance
factor in Carter analysis A /A
V p angle
angular position factor for stray flux factor for recoil range angular coil width
factor for zone of stable eperation factor for margin
step angle rotor position
rank numher of harmonies permeability of vacuum permeability, relative rank number of harmonies factor in Carter analysis A /A
m p radius vector
stepping time time constant magnetic flux
position of unstable equiiibrium polar coordinate magnetic potential circular frequency m Nm/rad .. sec Nm V 3 m m J rad rad rad rad rad Vsec/Am m sec sec Wb rad rad A rad/sec
Subscripts a,b ,c,d A,B,C,D a a a ab ar ba c d e f fic g m M mut p r rr rs ree ref res s sb sr ss st, tr u V ·a stat Superscripts v, À, h phase indicators
sector or pole indicators air
from a-phase to a-phase from b-phase to a-phase from rotor·to a-phase from a-phase to b-phase coil damper electromagnetic friction fictive gap
mean, magnetic, mesh magnet
mutual pole, pol ar rotor
from rotor to rotor from stator to rotor recoil
reference resonance stator
substitute body from rotor to stator from stator to stator stationary transient unit variation, variable leakage tangential rank numbers IS
PART I
GENERAL CONSIDERATIONS ON STEPPING MOTORS
INTRODUCTION
When getting first aquainted with stepping motors one often gains the impression to be introduced to a somewhat retarded offspring of electrical engineering. Still, the interest in these motors is understandable, consiclering that in the field of control-engineering, digital procedures are rapidly gai~ing and that stepping motors are most suitable to translate pulsewise information into exact step-wise rotation or displacement.
For sucn applications the requirements are usually the following:
a) A load, stipulated by a moment of inertia J
1 and a load-torque T1, is to be given a rotational displacement through an angle that is proportional to the number of signal-pulses applied to tne input of the power-unit that supplies the stepping motor.
The rotation corresponding to one signal-pulse is called the step-angle~ Usually the signal-pulses are accompanied by a signal for the required direction.
b) If no pulses are supplied the position is to be held within close tolerances, the deviation erel from the required position to be smaller than a fraction of the step-angle. (8rel < t:~).
c) Any number, or algebraical sum of, say N, control-pulses must give
rise to a rotation through the angle N.~ , accurate within the same tolerances as indicated under b).
d) The control-pulses must be acceptable at an arbitrary pulse-rate, up to certain limits.
When reviewing the range of devices that claim the surname "stepper", it seems appropriate to limit the field somewhat to strictly rotating types,
barring among others the linear motor and the ratchet-pawl devices [32,33]. The remainder can be devided into three groups:
- The active-rotor type; the rotor of this machine has a number of poles by permanent magnetisation. Usually this type is designated as P.l1. motor [35] •
- The pas~ive-rotor type; here the torque stems solely from the variation of the reluctance of the magnetic circuit through rotor and stator~ Usually this tytie is called the V.R. motor [30].
- The Hybrid-rotor type; the rotor derives a torque due to a combination of the variation of reluctance and the presencè of permanent
magnetisation.
It will be indicated as Hybrid motor [31].
The latter name is introduced here to replace the makers trade name. One can make .the objection that the P.M. motor has a campanion with a wound-rotor; as that type of motor is not available, at least not in Western Europe, it is hard to assess its practical importance [36]. It will be understood that the stepping motor is considered in its. basic feature, i.e. where commutation of the coils is independant of the momentary rotor-position; those motors where ti:Je commutation :is
triggered by a positiontransducer on ti:Je shaft are actually ~.C. motors in disguise.
Among the advantages of stepping motors compared with other ~lectric
servomotors are:
Simple and often cheap construction.
- Each contol pulse results in one unit step movement, thus enabling open-loop control.
- The rotor is held in its appropriate position by the so-called holding torque, without special precautions.
lts disadvantages stem from the following properties:
- As the movement of the rotor is meant to be discontinuous, the ratio of propelling torque to the moment of inertia i,s of foremost
importance, it determines the range of acceptable stepping frequen~. cies. Because, as with all electric machinery, the moment .of iner.tia increases faster with size than the torque does, stepping motors are
*One may wish to make a reservation for the variable reluctance type where saturation of the rotor material is essential for optimum properties; though that motor exists for traction purposes; it has not yet been·announced as a pure stepping moto~
[r].
restricted to the instrument class of servomotors, or must he used only with low or medium stepping frequencies.
- The motor is essentially prone to resonance phenomena; these may cause loss of synchronism if not special precautions have been arranged for. It is the aim of this paper to describe some general and charact~ristic properties of stepping motors.
In chapter some observations are made on the shape and the development of the three types mentioned above.
Chapter 2 deals with a basic comparison of the quasi-static torque that may he expected.
In chapter 3, some problems of vibrations or resonance, common to all types of stepping motors are reviewed.
J. SHAPE AND OPERATION OF STEPPING MOTORS
Synopsis
Stepping motors are available in a number of rather diverging constructions, which even on close examinatien seem to have little in common.
In this chapter it will be shown how they may be thought to be derived from one common ancestor. This has not only its merits for educational purposes, but it also simplifies the analysis and comparison of the various types.
In di$cussing stepping motors, one will preferably use the concepts, definitions, and nomenclature of electrical engineering, and also
. I
draw on the common knowledge of the properties of electric machinery. With this intention the stepp~ng motor will be placed among the other species of the family of electric machines.
1.1 The Primitive Electric Motor
Without reference to actual history, one may think of the primitive motor as sketched in figure 1-J-a as an early precursor.
Figure 1-1-a The Primitive Electria Motor
d b
Figure 1-1-b Conneetion Diagram and Stator Currents
If, as a first refinement, its coils are connected to a four-phase symmetrical A.C. supply, see figure 1-1-b, it will behave as a
synchronous machine; a further development would lead to a stator with cylindrical bore, the coils being replaced by a sophisticated set of windings to obtain cosine/sine-shaped current distribution. Along this route one may think of the development of the full-fledged synchronous machine.
If the primitive machine of figure 1-1-a is supplied by
a
souree ofo.c.,
connected to the a- and c-phases, alternatingly with the b-and d-phases as illustrated in figure 1-2-c, it will operate as astepping motor with a step-angle of 90°. The commutation of the currents in the various coils may be·accomplishedby a device as shown in fig 1-2-b. The speed of the device controls the stepping-frequency, which constitutes one of the attractive properties of stepping motors.
Figure 1-2-a
la=-ic lb=-id r---, I I I I [: I
:-1
I I I c__ _ _ _J L - - - _ _; Figure 1-2-bCurrent Distributor Figure 1-2-c Stator Currents
From this primitive model the three basic types of stepping motor may be derived, sametimes by refinements, often by simplification.
1.2 The Basic Permanent Magnet Stepping Motor
I
The excitation of the rotor of the primitive machine may be replaced by permanent magnetisation, see figure 1-2-a; this offers theadvantage that the construction of the rotor becomes less complicated, as coil and sl~·prtngs may be omitted, and the supply of the rotor-current can be dispensed with; apparently an attractive proposition. The' conneetion of the coils to the current-·source may be as shown in figure 1-2-b, the required current-time diagram is represented in figure 1-2-c.
For proper functioning, the direction of the excitation is required to alternate after every secend step. In the figure this is accomplished by a rotary-switch, but for obvious reasens this device will in practice be replaced by electronic switch-gear. However, reversing the current then constitutes a serieus complication, owing to the nature of the electronic components. Though there are bi-polardrives, usually each pole is equipped with two coils, one for each direction of excitation; the worse exploitation of the available coil volume must be accepted. The motor as shown in figure 1-2-a may be called the 'Basic Stepping Motor with Penrument Magnet Rotor'; usually abbreviated to P.M. motor.
1.3. The Basic Variabie Reluctance Stepping ~otor
~rom the theory of synchronous machines it is well known that the motor of figure 1-1-a would produce a torque, even if the rotor excitation were non-existent. Such a torque is usually called the reluctance torque.
The motor would, however, nat be acceptable as a stepping motor because the movement would not be predictabie in every rotor position; e.g. if the rotor were in rest in alignment with the A and C poles, it might move in any direction when the coils b and d became excited; the rotor might even remain in its present position.
The requirement of unequivocal behaviour leads to the addition of atleast one set of poles, to be excited by a third phase. This results in a model as sketched in figure 1-3-a, known as the 'Basic Variabie Reluctance Stepping Motor', V.R. motor for short.
lts rotor appears to be even simpler than the previous one. It should be realised, though, that the shape of the stator and rotor poles is now of the utmost importsnee because the torque depends on the variatien of reluctance; this type, therefore, requires more precision machining. The basic current-supply may be seen in figure 1-3-b. Contrary to the previous model the direction of the currents need nat be reversed, see figure 1-3-c. This simplification may constitute a campensatien for the additional number of phases, required in the V.R. model.
Fi(JUPe 1-3-a
Baaia VariabZe ReZuctanae Stepping Motor
Figure 1-3-e Statop· CurTents
Pigure 1-3-b CurPent DistributoP
I.4 The Basic Hybrid Stepping Motor
The third variety among the. stepping motors may be visualised, as an attempt to combine the advantages of both the. active and.passive rotor in a motor, or, of the co-operation of the permanent magnetisation with the variation of reluctanee. Such a combination is already. available in the primitive motor of figure I-I-a, but, if the rotor exditation were replaced by permanent magnetisàtion, the use of the. two practica-bie magnet materials alloys or ceramiès, would introduce deficiencies of both alternatives owing to their properties. Thc following may clarify this.
(a) Alloys:
Care should be exercised that the magnetisation never becomes perma-nently impaired. This may happen e.g. during assembly of the motor, but also if the rotor is locked in an unfàvourable position. To avoid adverse consequences of such a position, it is essential to limit the excitation. These restrictions render the realisation of a motor of this shape with alloys unattractive.
(b) Ceramic material (ferrites):
This material has sufficient stamina against. demagnetisation. However, its remanence is considerably less than that of the alloys. Still, it is used in the motors derived from the basic model of figure
I-2-a. The relative permeability, however, is so low that the effect of
!
variation of reluctance may be qualified as negligible. Therefore, the rotor of this type of stepping motor is alwayfo cylindrical.
It experiences a sticking-torque due to the variatien of reluctance to the p.m. flux, which, however, can never he used as a driving torque.
The conclusion is that an attempt to combine the effects of permanent magnetisation and variat.ion of reluctance seems to fail. This deadlock has been solved with a construction as illustrated by figure 1-4-a.
I t may he arrived at by separating two halves of the primitive motor of figure 1-1-a somewhat in the axial direction, forming a pertinent front part and a ditto rear part.
Figure 1-4-a Basic
Hybrid
Stepping MotorThe stator parts are magnetically connected by a yoke, the two rotor parts by a piece of magnetised alloy.
It should be realised that the flux from the perrnanet magnet fellows a path largely different from that of the flux due to the stator excitation.
In figure 1-4-a the course of the fluxes is indicated by way of field-lines; the latter have been drawn as if the fields were allowed to exist independent of each other.
In the axial sectien the field of the permanent magnet in the rotor is illustrated, in the front view the field of the a- and c-coil is shown. It will be clear that the combination of bath fields must yield a
streng H-field in the air-gap under the A-pole of the front part, while under the B-, C- and D-poles the components approximately cancel
each ether, The rear-section shows a mirrered picture of the field-intensities, the directions being reversed.
A closer examinatien of the magnetic circuits will show that those of the permanent magnet are largely de-coupled from the stator coils, and vica versa. This has the advantage that it prevents the rotor magnet from being demagnetised even if the rótor is locked in an unfavourable position. The machine shown in figure 1-4-a may he regarded as a model for the 'Basic Hybrid Stepping Motor'.
This name is here introduced as a substitute for the makers' trade name. This motor has an unequivocal behaviour with a two-phas·e supply, figures 1-4-b and 1-4-c, because the direction of rotatien is to that pole where the H-vector of the stator field is supported by that of the permanent magnet. Therefore,· the consequent reversion of ;the direction of excitation is required, and the motor has to carry sets of double coils as with the P.M. motor.
--1
Figure 1-4-b CU:t':t'ent Distributor
Figure 1-4-e Stator CU:t':t'ents
A discussion of the merits of the three basic types of stepping motor may he postponed till after the analysis in chapter 3.
1.5 Development of the Basic Motors into Industrial Types
In the remainder of this chapter some attention will he paid to the shape of industrial stepping motors.
The proper ties-of the three bas ie models, as far as stepwise movement is concerned, may he summarised as fellows:
Type (figure) P.M. motor (1-2) V.R. motor (1-3) Hybrid motor (1-4) Sequence of excitation in coils, and direction of current (+/-) a, b, --a, -b, a etc. a, b, c, a, etc. a, b, -a, -b, etc. Step-angle Direction of rotation 90° Clockwise 60° Clockwise 90° Cl.ockwise
The direction of rotation can, of cours~ he reversed by a change in the sequence of excitation. It will he clear that certain phases may he energised simultaneously, resulting in a change in the position of equilibrium, with the advantage of a higher resulting torque. There are other more complex patterns of excitation that may he useful in particular circumstances. For the following deliberations the simple pattern of excitation as indicated in the table above is presupposed.
The basic models, having two-pole rotors, have large step angles; most applications require motors with smaller step angles; one may even say that the small step angle is one of the additional attractions. There are types known with 48, 96, 100, 200 and even 400 steps per revolution.
Such motors may he developed from the basic types by increasing the number of poles p, which is illustrated in fip,ures J-5-a for ·the P.M. motor and 1-5-b for the V.R. type.
With the V.R. motor the number of phases m is sametimes
increased to m = 4, 5 or 6. This would give the basic motor with p = I, a step angle of 45°, 36° and 30° respectively.
The consequence of increasing the number of poles may be read from figures 1-5-a and 1-5-b.
The practicability of increasing the number of poles meets as far as the rotor is concerned with little problems from the V.R. and the Hybrid motor; with the P.M. motor the number of poles is restricted, yet p = 6 at least is feasible.
(a) P.M. motor, p
=
6 (b) V.R. motor, p 4 Figure 1-5 Multi-pole Stepping MotorsThe separate excitation of e.g. 24, 50 or even 100 stator poles presents quite a problem, every pole requiring its own coil, more so as the space for coils around each pole reduces with the number of poles. It is this problem that compels the designer to depart from thj basic model to such an extent that the original is hardly recognisab+e in its descendant.
The alternative to the multitude of separate coils is a design,in which the poles of one denomination are made to belong to the magnetic circuit of one coil, or of two coils if symmetry requires so.
This leads to two, far different constructions.
1.5.1 The multistator motor
This concept has been used only for P.M. and V.R. motors. It may be derived from the model of figure 1-5-a as follows:
There are two separate stators, one for. each phase. One stator, e.g. the one for the a-phase, consists of the original stator but with all poles of B and D denomination removed; there remain only A and C poles as illustrated in figure 1-6. 7he sequence of the further simplification is shown in figures 1-6 -a, -b and -c. In figure 1-6-a each pole still carries its own coil; in figure 1-6-b thbse coils are replaced by one continuous winding, which after stretching takes the shape of an annular coil, as shown in figure 1-6-c.
The iron poles have become somewhat deformed in the process, but that will have little consequence for the magnetic field in the air gap.
C
.
-31
(a) (b) (a)
Figure 1-6 SimpZifiaation of the CoiZs of a SingZe-phase Stator In this way it is possihle to create sufficient space for the coil. The complete stator of the P.M. motor is shown in figure 1-7. It will he understood·that the original rotor has to he extended in axial direction in accordance with the increased space for the two staters. For easy reversal of the direction of excitation, the coil is usually wound hi-filarly.
Figure 1-7 Two-stator Permanent Magnet Motor, p
=
6, m=
2 A typical V.R. motor, developed from the motor of figure 1-5-h is shown in figure 1-8. Here three phases require three staters, each carryingonly teeth of one denomination. Here, too, the rotor is extended in axial direction. It will also attract attention that the shape of the teeth differs somewhat from those of figure 1-5-b.
The removal of the B and C teeth from the first stator leaves an abundance of interpolar space, which may be used by widening the pole-faces of the remaining A poles, both in stator and rotor. In the cross-section of figure 1-8 it is easily seen that here the lines lie in radial planes; although the path of the field-lines appears shorter, basically little' h~s been altered. The multi-stator construction has the disadvantage that the active rotor material is utilised inefficiently, viz, only during the time that
the corresponding stator is excited.
Figure 1-8 Three-stator VariabZe ReZuctance Motor, z
=
8, m=
31.5.2 The motor with sectional'phase division
This construction is not suitable for P.M. motors. Starting from the model of figure 1~5-b, once again the aim is to find a construction in which the poles of one denominatien all are in the circuit of one common coil. The stator is now divided into sectors, each phase requires two such sectors, one to contain its "north" poles, the other its "south" poles. The two sectors of one phase will be placed opposite
each other for reasans of mechanical symmetry, otherwise they could as well have been neighbours. In all there are 2 m sectors, M ·= number of phases.
Figure 1-9-a shows the sector-wise division of the stator, where al ready in each sector" the alien teeth have been removed. Such a sector can now be energised by one coil, the coils of- the complimentary sectors connected in series to form one phase; e.g. the coils of the A and the A' sector make the a-phase. In figure 1-9-a a pair of field-lines gives an illustration of the field when the a-phase is energised and the rotor is in equilibrium.
(a} AppZied to motor of
figure 1-5-b
(b} AppZied to a motor
with p
=
18Figure 1-9 SectionaZ Phase Division
This construction seems somewhat odd when applied to a motor with p = 4; in tigure 1-9-b,
a
motor of this type with p = 18 is shown where the construction makes more sense. The space that has been vacated by the removed teeth is used for widening the pole-faces, or tooth-surfaces as they are usually called more adequately.The construction has the drawback that not all of the available rotor teeth are put to use, but only those that happen to be under a sector that is momentarily energised. Apparently this disadvantage to a good approximation balances that of the previous construction, in which the rotor had to be exterided axially and was not put to use all the time.
1.6 Conclusion
With this sketch of the development of steppinp,·motors, (it should be emphasised again that it has no ·historical value) one has arrived at the models the designer probably starts from.
He will aim at an optimal motor that satisfies a number of requirements such as*:
I. Prescribed step angle 2. A certain holding torque
3. Prescribed pull-in rate and the pull-in toique 4. Extension of the start-range and of the slew-range
There may further be limitations due to prescription on weight, volume, permissible temperature rise, not to mention particular specifications in relation to the application of the motor. Apart from industrial constraints, patent rights, etc., one of the first questions will be: Which type of motor roeets the specifications best.
A primary consideration then is what torque can be obtained, or more specifically, how do the torque-to-displacement curves of the three types of motor compare, if the step angles are as required and the moments of inertia are equal.
This problem will be examined in the next chapter.
The design is also influenced by the dynamic behaviour that may be expected from the various types; these problems will he entered upon in chapter 3.
*rypical terms and definitions are listed on page 12.
2. THE CALCULATION OF THE QUASI-STATIC TORQUE Synopsis
A calculation is made of the quasi-static torque in relation to rotor position for the three basic types of stepping motors, each of the two-pole type with rotors of equivalent size.
Further the influence of the increase in the number of poles is considered and the relation of torque to size is traced. An evalu-ation of the results concludes this chapter.
For the calculation of the electromagnetic torque one requires information on:
(a) the magnetic field in the air gap, a field that changes with the displacement of the rotor;
(b) the dependenee of the field intensity H on the excitation N i and on the magnetisation of the permanent magnet, if present. For (a) is necessary a field analysis that requires the Laplace equation of the H field to be solved, meeting the boundary con-ditions. The latter are the magnetic potentials on stator and rotor surfaces. These potentials depend on the outcome of part (b) of the problem.
In most cases the boundaries may he considered to he equipotential surfaces, and then the parts (a) and (b) can he calculated sepa-rately because the shape of the field does not alter with potential. The field intensity is then proportional to the difference of toe potentials at the boundaries. Still, the analysis of that field presents a difficult problem because of the rather complicated shape of the stator sur~ace, and, in case of the V.R. and Hybrid motors, further complicated by the rotorpoles.
With the P.M. motor the boundary conditions at the rotor are influ-enced by the permanent magnetisation.
The problem (b) is usually complicated by the determination of the reluctance of táe iron part of the magnetic circuit, more so if the permeability depends on the magnetic induction. This induction in · its turn depends on the air-gap reluctance, ·which ties a link 'with
These field problems, even if simplified, are clearly cases for computer analysis.
The analysis of the field, and the calculation of the torque has been dealt with by Bakhuizen, for the V.R. motor [2], Suurmeyer,for the P.M. motor
[3]
and by Gommers[ 4]
for the Hybrid motor.If the iron of the teeth becomes saturated, the problem is still more complicated because then the boundary does not constitute an equipotential surface anymore. Now the field equation has to be sol-ved in the region of air and iron together, taking into account the non-linear behaviour of the permeability, see also Appendix H, sec-tion 6. Such a computation is a very elaborate and expensive process and one is probably well advised to settle the problem by experiment. The specific studies mentioned above are directed at types of motors that are commercially available, each with its own refinements and limitations and probably each having merits in its own fieldJ
These niceties prevent a clear-cut comparison as to the question which type of motor is basical!y best suited to which set of require-ments. In the next section the three basic ty·pes are compared as to their specific torques, aeteris paribus. This comparison is made on general lines, a number of simplifications must be taken for granted. It will give a first indication of the possibilities as far as the production of torque is concerned. The latter is to be weighed against other properties such as dynamic behaviour and manufacturing co st.
Perhaps sucn comparison serves most. the purpose of appreciating tne skill of the designers to exploit the possibilities and to cope with drawbacks.·
2. I The Basic Motors as Used for the Calculation
An estimate will be made of the torque of electromagnetic origin Te, in its dependenee of positiori
e
when the motor is energised by ideal current-sources.As stated before, tnese calculations are meant only for camparing tlle potentialÜies of tne three basic machines.
A
number of suppositions will be needed to obtain results in clear expressions.Figure 2-1 Model for Caleulationa with V.R. Motor
Figure 2-2 Model for Caleulatiorw with P.M. Motor
Most of these suppositions are accepted in the theory of electric machinery and they will not be discussed in detail.
The calculations will be made for the two-pole basic roodels as given in the figures 1-2, 1-3 and 1-4, excited by a constant current. As discussed in the pre'vious chapter, all final designs make u se of the separation of phases, multi-stator-wise or sector-wise, hereby gaining,the advantage that the pole-surfaces can be broadened. To account for this feature, the basic roodels are adjusted as illustrated in 'figures 2-1, 2-2 and 2-3, poles covering principally n/2 radians. Of the V.R. motor, figure 2-1 gives only the stator with the A-poles. Of the P.M. motor, figure 2-2, the energised A and C poles are shown in solid lines and those of the b-phase in dotted lines, because the latter may yet influence the resulting torque owing to the magnetised rotor. O,ne should observe tnat the B and D poles are situated in a plane remote from that of the active A and C poles, therefore there is no danger of stator-leakage fields from A pole througn B pole to C pole as may be suggested by the drawing.
To account properly for the processes in the Hybrid motor, all four poles 'are to be considered; tney are shown in figure 2-3. Again the apparent chance of leakage-fields is not real and must be disregarded.
(a) GeneraZ view
(b) Front view
Figur>e 2-3 ModeZ for CaZc:uZations with the Hybrid Motor
motor nas been represented; the permanent magnet in the rotor nas its flux directed perpendicularly to the plane of the drawing and is indicated by the usual
®
or® .
The estimate of the torque will be made for the three basic models separately, afterwards a comparison will be
2.2 The Basic V.R. Motor
The torque will be calculated from the change of the magnetic co-energy with a change of the rotor-position. An estimate must be made of the magnetic flux that crosses the air gap in relation to the position of the rotor poles (relative to the stator). This relation is roughly indicated by figure 2-4-b, to which the following camment may be appropriate:
The positions of maximum and minimum flux are evident, the values of the extremes can be calculated as will óe slioWn. The shape of the curve should óe determined óy calculations of the field.in the air gap, possibly also in the iron of the magnetic circuit. The actual curve is replaced by one that can be expressed in terms that open the
possibility of an algebraic analysis; it approximates the original curve to a reasonable extent as will be discussed below.
The magnetic field in the air gap is to be dealt with first:
For this purpose the assumption is made that the surfaces of the poles or teeth on rotor and stator are equipotential surfaces in the magneta-statie sense. This means that the teeth should show no saturation, even that the reluctance of the teeth is negligible. For the following analysis this assumption will be acceptable, but for an accurate calculation a correction must be considered, see Appendix H-6. The field between two regularly slotted surfaces has been treated by Carter
[s]
and his analysis can be applied to the present case. It is supposed(a) that the machine has sufficient extension in the axial direction so that the field problem may be treated as two-dimensional, (b) that the air gap g is small compared with the diameter 2R of the
rotor and
(c) that there is a sufficient number of teeth to allow the surfaces to be developed as shown in figure 2-4-a.
Carter calculated the flux crossing the air gap for two slotted sur-faces as shown in figure 2-5-a.
He gave also two ways of interpretation of the influence of the slatting, the first is that of an imaginary increase of the air gap, the other, which will be used henceforward, is the one pictured
in figure 2-S-b. It replaces the original surfaces by smooth planes at a distance g, but ~llowing only a flux in limited zones. The width of these zones follows from the calculation and depends on toothwidth and air gap.
\
\
\
\
(a) Developed BUPfaces of stator and rotor
"
31f-·
(b) Flux variation with rotor positionFigure 2-4 On the CalcuZation of the Flux be~eèn Slotted SUPfaces
A similar picture can he found for the situatiort where the teeth are placed exactly opposite the slots as shown in figure 2-5-c; here, too, the flux that crosses the gap can he pictured as homogeneaus blocks, figure 2-S-d. The relevant calculations are to he found in [2] • The magnetic induction in the homo~eneous blocks of fields is equal to that which would he found in the smooth air gap with the width g.
The shape of the poles will henceforward he such that their ~idth w is equal to that of the slots. In [6) are given a number of
diagrams showing the reluctance for other sizes of slots and teeth;
the optimum proportion appears to depend on the tooth-pitch and the air gap; it differs slightly from the 50-50 ratio that is adopted here.
(a) Field lines in position 6
=
0 (a) Field lines in position 6=
~n11111111111
lil
IJ
11111 11111(b) S?-'nrplified following Carter (d) Simplified flux pattem
Figure 2-5 Field Pattem and its Representation
The Carter pictures lead to the interpretation that the pole faces offer a variable passage to the flux, to be expressed by an area A. This area depends on the position of the rotor with respect to the stator. A is to be expressed in relation to the actual pole face A p which here is
Ap = I w,
with axial length of pole, w width of pole.
Because of the two-pole motor, and the agreement on the pole width, here w = !nR
The passage A for the position 6 = 0 is to be designated by A(O), and for the position 6 = !n by A(!n).
The values of bath depend on the ratio of w to g, and are calculated accordinp, to Carter [2] and tabulated below (table 1).
On behalf of calculations, passages A(9) for any intermediate position
e
will be expressed as follows:Table 1 A(O) A(!11) A A A ~ ~ m V V ~ Ct ~ ~ ~ -A-
=x-
A
Ct a -g p A p p p A m w w ~ 2 1.5513 1.4412 1.4962 0.0550 0.0368 0.0275 0 .• 0123 3 1.4491 1.2750 1.3620 0.0871 0.0640 0.0291 01.0157 4 1.3814 1.1026 1.2420 0.1394 0.1122 0.0384 oi.o226 5 I. 3331 0.9890 I. 1610 0.1721 0.1482 0.0344 0.0255 6 I. 2967 0.8982 1.0975 0.1993 0.1816 0.0332 0.0276 10 1.2104 0.6662 0.9383 0.2721 0.2900 0.0272 0.0309 IS I. 1574 0.5124 0.8349 0.3225 0.3863 0.0215 0.0308 20 1.1272 0.4208 o. 7740 0.3532 0.4563o:
0177 0.0296 40 1.0747 0.2544 0.6645 0.4101 0.6172 0.0103 0.0233 60 I. 0541 o. 1868 0.6204 0.4336 0.6968 0.0072 ·o.OJ87 80 1.0428 0.1492 0.5960 0.4468 0.7497 0.0056 0.0158 40A(8) = A m + A cos 28 = Am(l + a cos 28) V
with A m HA(O) + A(!TT) ~
(2-2) A HA(O) - A0TT)}
V
a A /A v m
This expression is in agreement with figure 2-4-b.
As mentioned before, there are good reasons to introduce relation (2-2).
(a) because it is perfectly correat in the extreme rotor-positions and
(b) it gives results that are in fair agreement with the numerical calculations by Brands and Veltkamp
[7] ,
provided the ratio w/g is less than 20; beyond this value the agreement may be qualified as just acceptable.The magnetic induction, to be applied in Carter's field model, will now be calculated. A line-integral over a path C in figure 2-1 yields
Ha 2g +
!':}_
iron number of turns in one coil, with N
i current in the coil
Ha field strength in the air gap.
dl = 2 N i (2-3)
For the present approximation the contribution along the iron part in the magnetic circuit will be neglected, consiclering that it usually requires only a small fraction of the total m.m.f., pro-vided the circuit does not become saturated.
The effect of saturation will be considered insection 7, when discussing the shape of the torque curve.
Expression (2-3) now reduces to
N • i (2-4)
The magnetic flux through the stator pole
~(I)
will now be calcu-sslated, neglecting possible stray fluxes. It follows from (2-2) and (2-4) that
~A
(I +a cos 28)The torque may now be calculated by way of the co-energy concept;
the magnetic co-energy
w'
follows from·m I
W'
=!~.
d i (2-6)m
2 N ~(I) 0
with ~ = flux linkage of the stator circuit,
ss
I excitation current.
Re lation (2-6) with (2-5) yields:
W' lJO - A N2I2 (I + (1 cos 26)
m g m (2-7)
The torque of electromagnetic origin becomes:
àW' N2I2
T
as
m = - 2 lJO - - a A sin 26e g m (2-8)
2.3 The Basic P.M. Motor
The calculation of its torque is complicated by the presence of
permanent magnetisation. The lateral magnetisation, figure
2-9,
is taken into account as if it were generated by a coil fed bY a current-source. As proved in Appendix A, the original rotor can
be replaced by a body of tne same shape, consisting of materikl
with perme·abi lity lJolJrec, enclosed by a co i 1 wi th capper dis tribution
j m j COS Ijl,
the coil being excited by a current
j
• 0
1"'~
r J 11o11rec
Here J
0 is tne ~agnetisatiön înside the homogeneously magnetised
rotor, when fitted in a yoke .of zero reluctance.
42
Fi{Jla'e 2-6
Magnetisation Rep~ced by Current Sheet
It is the obvious course to introduce two field axes, viz., one vertical through the A and C poles, the ether horizontal through the B and D poles.
ft can easily be verified that on the isotropie cylindrical body the windings of figure 2-6 may be replaced by a set of two coils, viz., one with a vertical axis and the ether in the horizontal direction, provided the excitation is reduced by a factor cos
a
for the first and sina
for the secend coil, as shown in figure 2-7.(b)
Figu:r>e 2-7
Roto~ (ImaginaPY) Excitation DissoZved into Ve~ticaZ Component (a)
and
Ho~zontaZ Component (b)The magnetic co-energy may be considered to consist of two components, one depending only on the vertical field, the ether depending solely on the horizontal field.
As the field in the vertical axis is the result of the excitation in the a-phase, combined with tha~ of the imaginary vertical rotor coil, the general expression for the co-energy must be evaluated according to
with k = cp = k i k = I k = W' m I =
f
~k
d ik k=l } ' 0index for designating coils the flux linkage of coil k, the current in coil k, the final value of ik
(2-9)
I to n,
This leads for the energised a-phase to the co-energy in the vertical field through the A- and C-poles:
NSI cos 8 +
~rec(NRI)j
(2-10) (2-1 I)
and 6 is used to account·· for s tray fluxes near the pole tips. If the b-phase, too, is energised,.a similar expression can be found for the. co-energy of the B- and D-poles:
m(hor) = P - - - - -0- sin2
e
+ W'~(
J ) 2 ~o~rec (2-12) Figure 2-8I~~ustration of Leakage Pie~
The objection may be raised that various leakage fields, like the one illustrated in.figure.2.,-8, also contribute to the magnetic co-energy of tbe system. It can be verified that such fields in total do not vary with the rotor position, they are therefore not rele-vant for the calculation.of. the torque. Because with P.M. motors always both phases.are energised simultaneous, the above expres-sions for.the co-energy in vertical and horizontal poles should be added. The torque can now be found by evaluation of
T e
aw•
m
ä"8
If both phases are energised with + I, the torque becomes
T e 44 N I --5--sin (8 -
!n)
(2-:13) (2-14)Reversion of the direction of the excitation must he met hy an appropriate change in the argument.
For later use will he introduced the amplitude of the torque
T e
2.4 The Basic Hyhrid Motor
(2-15)
First an estimate has to he made of the reluctance of stator to rotor, more in particular that of the rotor-head to each of the four poles
shown in figure 2-3.
Referring to section 2.2, it is also here the ohvious course to introduce a variahle pas~age for the flux, viz.:
for the rotor to the A-pole: A(8) A m (I + a cos e), to the B-pole: A(8) A (I + a cos (8 - jn)),
m (2-16)
to the C-pole: A(8) A m (I + a cos (8 - n)), to the D-pole: A(e) A m ( 1 + a cos ce - ~n)) •
These relations are hased on the first approximation of the periodicity of the reluctance; they are exactly true in the
positions of maximum and minimum reluctance, though for intermediate rotor positions they seem to he suspiciously neat, particularly when inspecting figure 2-3.
1~1
- - \ - - - -
·
- ·- -·- ·I
Figu:t'e 2-9 Industria~ Type of Hybrid Motor
I
It should, however, be kept in mind that with actual stepping motors of this kind, having a large number of teeth, these relations are much more evident, as may be understood by inspeetion of figure 2-9. The acceptability of the relations (2-16) is supported by the numerical analysis of the field and the calculations of the torque by Gommers
l4J ,
which were verified by experiment. The values of Am and ~ depend on g and w, as may beseen from table 1.For the rear-section of the motor, the complementary rotor pole determines the reluctance, which means that in the relations (2-16)
the arguments have to be.increased by n.
Each pole carries a coil of N turns, the excitation being e.g. as follows a-phase b-phase pole A: I amps pole B: I amps pole C: -I amps pole D: -I amps
Apart from these coils, the permanent magnet also contributes to the final field configuration.
For the purpose of calculating the combined effort, the perm~nent magnet is thought to be replaced by a cylindrical body, area~ and length lM' its material having a permeability ~o~rec· In accórdance with Appendix A, section A.2, an evenly wound coil provides an
m.m.f. J
~= _ _ ~o~rec o _ l
M (2-17)
For the ease of the analysis, the magnetic circuit of the motor as shown in figure 2-3 is replaced by its electrical analogue, see figure 2-10.
Figu:re 2-10
Eleotrio Analogue·Of Magnetio Cirouit
The batteries with voltages u1 to Ua stand for the coils with N I
ampere-turns, the battery E for the permanent magnet with IM ampere-turns. The resistances R1 to Ra represent the magnetic reluctances of the air gaps between rotor heads and the relevant stator poles; according to (2-16) they can be expressed as
I/R1 1/R (I + a cos a) I /R7 I/R2 1/R (I + a cos (9 - j11)) = I /Ra
I /R3 l/R (I + a cos (9 - 11)) = 1/Rs (2-18) I/R4 l/R (I + a cos (9 - Ij11)) = l/R6
A
witn 1/R = 110 g m
The resistance Z represents the reluctance of the body replacing the permanent magnet, such that
1/Z = 11 11
~
o ree
-r;
(2-!9)The currents i
1 to ia stand for the magnetic fluxes in each of the poles, the current I representing the flux through the permanent magnet,the latter being equal to the return flux through the yoke. Because U= u1 = u2 = US = u6 = - u3 u4 = - u7 = - ua
,
af ter some manipulation the currents appear to bei1 -i7 (jE+ U).R + 2ZU - aZU(cose + sine) R + 2Z RI
i2 -ia qE + U).R + 2ZU - aZU(cosa + sine) R + 2Z R2
i3 -is (jE- U).R
-
R + 2Z 2ZU - aZU(cosa + sine)R3 (2-20)
i4 -i6 = (jE- U).R- 2ZU - aZU(cosa + sine)
R + 2Z R4
I 2E + 2aU(cos9 + sine)
R + 2Z
By substitution of the analogous magnetic quantities, the magnetic co-energy and the torque can be calculated as in the previous section which yields
T e with s = N I
ï"""+'2'S
J0• ~·a /2 sin(a - !n) + A m .a 2 sin 2(8 - in) !Jo· g (2-21)With the excitation of the a-phase and b-phase both with +I, the position of equilibrium becomes !n , as was to be expec:ted. For later use are introduced here:
(2-22)
and A
T = 2 ll N2I2 m a2
re o g(l+s) (2-23)
2.5 Gomparing the Torques of the Three Basic Types of Motor For a proper comparison of the torques of the three basic types of motor it is essential that each type should be equipped with a two-pole rotor having a diameter 2R and an active length L and with the necessary number of stator sections so as to make possible an adequate stepping performance.
Tomeet these requirements, a V.R. basic motor 1~ould exist of an assembly of m single~phase sections as shown in fiqure 2-1. "or simplicity, a four-phase motor (m=4) 1áll be considered, the active lenRth of rotor per phase beinP, !L.
The
P."'.
motor 1~ould be built from two stator sections as shown in figure 2-2, representing the section with A and C poles, the B and D poles heing shown by dotted lines. The active len~th per phase here isjL. Tl1e Hybrid motor has its rotor divided into t1m parts, with a permanent magnet in between. As active parts will be considered the two halves, each with a length jL.
An adequate measure is the torque averaged over one normal step, the coils being excited in such a way as to produce the optimal torque. lVith the four-phase V.R. motor this means that always two phases are energised simultaneously. If, for example, the d~ and a-phases are
energised the rotor has its equilibrium in the position - n/8; one normal step is made upon commutation of the current from the d-phase to the b-phase. ~igure 2-11 shows that the new position of rest is at n/8.
T
'
Figure 2-11Torque vs Position Curves V.R Motor, p
=
1, m=
4Figure 2-12
Torque vs Position Curves P.M. Motor, p = 1, m = 2
With the P.~. motor, always both phases are energised and commutation means the reversal of the direction of excitation in either phase. As
indicated in figure 2-12, the position of rest may he -n/4 owing to excitation of the a-phase with +I amps, and the b-phase with -I amps. Upon commutation of the b-phase, carrying afterwards a current of +I amps, the motor makes a step to -n/4.
The Hybrid motor behaves in this respect as the P.M. motor and will need no further explanation.
For simplicity, a torque-amplitude
T
will he introduced, showing the maximum value of the torque of each machine with two phases energised simultaneously. It is easy to verify that with simultaneous excitation of two phases in any case the larger torque is developed.Such a torque for the V.R. motor may he found from (2-8) by adding the torques of two adjacent stator sections, as fellows:
N2I2 A
a-phase only T e -2 }J -~ m sin 2 8
0 g
A
b-phase only : T -2 ).1 N2I2 ~ ~ sin 2(8 - !n)
e 0 g
