peripherals (printers, paper tape readers, plotters, etc.) .... To do this, we first turn out attention to the energy stored in the magnetic field. To simplify the analysis ...
Comp~t. & Elect. Eagag Vol. 7, pp. 243-266 © PergamonPress Ltd., I~0. Printed in Great Britain
0045-7906/80/1201-0243/$02.00/0
STEPPING MOTORS---A REVIEW SPIROSG. PAPAIOANNOU Department of Manufacturing Engineering, Boston University, Boston, Mass., U.S.A. (Received for publication 2 July 1980) Abstract--Rarely have two independently conceived devices turned out to have so perfectly matching characteristics as the digital computer and the stepping motor. The nature of this basic compatibility, however, must be thoroughly understood if pitfalls are to be avoided in the selection and use of stepping motors in digital control systems. This paper attempts to consolidate in one place enough information to allow intelligent decisions to be made in this regard. Topics covered are the principle of operation of variable reluctance and permanent magnet motors, electrohydraulic stepping motors and the design of low and high performance stepping motor drives.
1. INTRODUCTION Stepping motors have been known for several decades. Recently, their popularity has grown as a result of the proliferation of digital systems [I]. A stepping motor is essentially an electromechanical transducer which converts a train of digital pulses into mechanical motion. The pulse train is normally generated by a computer or other digital source. In response to each pulse, the motor shaft rotates through a small fixed angle. Consequently, the angular position of the shaft is proportional to the number of incoming pulses. This important property implies that by properly timing the driving pulses, the computer through its software exercises complete control over the stepping motion. Although in most applications comparable results can be obtained with conventional d.c. and a.c. motors, stepping motors have certain unique advantages: 1. A stepping motor is inherently a pulse driven device. It can, therefore, be more easily interfaced with pulse generating (digital) systems. 2. Its positioning error is generally small (typically 5% of a tooth pitch). This error is also non-cumulative. 3. The angular position of the motor shaft is solely determined by the input pulse train. If the pulses are generated by a general purpose digital computer, the position can be programmed. 4. Since under normal conditions the shaft position is independent of load dynamics, reliable positioning can be achieved with open-loop control. Currently, stepping motors are widely used for generating incremental motion in computer peripherals (printers, paper tape readers, plotters, etc.), wire-wrapping machines, drafting machines, welding machines, fabric cutting machines, scientific instruments and a wide variety of numerically controlled machine tools. The list of applications is a large one and still growing. To satisfy these diverse needs, stepping motors are commercially available in a large variety of sizes and resolutions. The resolution is represented by the step angle, namely, the angle through which the motor shaft rotates in response to an input pulse. Typical step angles are listed in Table I. Table 1. Typicalstep angles Step Angle(in°)
Pulses per Revolution
0.72 1.8 2.0 2.5 5.0 8.0 15.0
500 200 180 144 72 45 24 243
244
S.G. PAP^IOAlqlqOU
Despite their increasing popularity, however, stepping motors do have their limitations. At the present state of the art, they can be used to position only small to medium size loads. Their torque is limited by about 200 lb-in., enough to drive a 2 hp milling machine. Higher loads demand a more powerful type of motor. These requirements are met by coupling a stepping motor to a hydraulic torque amplifier. The resulting electrohydraulic stepping motor can generate torques of the order of 2000 lb-in. Such motors are primarily used for driving NC machines [2, 3]. Another limitation of stepping motors is their speed of response. SLO-SYN series motors [4], for example, made by Superior Electric can drive inertial loads at speeds up to 4000 steps per sec (sps) or 1200 rpm. Although this speed is fairly high, it may not be high enough for certain applications. For these reasons, in the field of numerical control stepping motors are by no means a universal choice. The prevailing design philosophy seems to confine the use of stepping motors to mostly open-loop NC actuators. Despite the high reliability of such systems, however, many designers still fret (understandably so) at the idea of missed steps which in an open-loop system will go undetected. Consequently, in systems in which the demand for accuracy is high, incremental motion is generated with servo motors rather than stepping motors and the required accuracy is obtained through the use of feedback control loops [5]. These systems will not be covered here. A recent survey conducted by American Machinist[6] disclosed that out of 106 numerical control systems currently on the market, 37 or 34.9% employ stepping motors. The same survey also revealed that 41 systems already use microcomputers, a trend which is likely to continue as the current microprocessor explosion reduces the cost of these devices and increases their power and versatility. I. THE PRINCIPLE OF MINIMUM RELUCTANCE
The most common type of stepping motor operates on the principle of minimum reluctance. This principle is simply the recognition of the fact that in a magnetic circuit with at least one moving member, a position of minimum reluctance is a position of stable equilibrium. Consider the magnetic circuit of Fig. 1, in which the armature AB is pivoted at the bottom so that it can freely rotate. Assume that the coil has N turns and carries a current i. A basic relationship in a magnetic circuit is the so-called magnetic law of Ohm
F=R~
(1)
where F = Ni is the magnetomotive force (MMF), • is the induced magnetic flux and R is the total reluctance of the circuit. The reluctance is made up of the individual reluctances of the circuit components, i.e.
R=ERi (2)
where /~i, li and A~ are the permeability, length and cross-sectional area of component i, respectively. Normally, there are two types of components, those made of ferromagnetic material, collectively known as the ferromagnetic core or simply iron core and one or more air gaps. Ferromagnetic materials range from cast iron and steel to various alloys of iron with other metals (nickel, aluminum, cobalt, etc). known as permalloys which exhibit very high permeability. The permeability of a given substance is usually expressed as a product = ~," ~0
(3)
where/zo is the permeability of free space or vacuum (4~r x 10-7 Henry/meter) and/z, is the relative permeability of the substance, a dimensionless number. The relative permeability of
Stepping motors--a review
245
sheet steel, for example, is 4500 while the relative permeability of air is approximately 1. The large difference in permeability values between iron core members and air is crucial. Since (2), the permeability of a circuit component is inversely proportional to its reluctance, it follows that the reluctance of the air gap is much higher than that of the core. In fact, as a first approximation the core reluctance may be neglected and the entire reluctance may be considered as being concentrated in the gap. In the circuit of Fig. 1, for example, suppose that the core is made of sheet steel and that the length of the gap is 0.1 ~m. Then, the ratio of core reluctance to gap reluctance is Rc = lzg x lc Rg Izc x lg
50 4500 x 0.1 =0.11. The fact that most of the reluctance is concentrated in the air gap makes a number of circuit characteristics strongly dependent on gap length. By changing the length of the gap, for example, the magnetic flux through'the circuit can be made to vary over a wide range. The attractive force between the armature and the rest of the core, being proportional to the flux, is also strongly dependent on the length of the gap. Our next task will be to investigate the relationship between the electromagnetic force and gap length, since it is from this relationship that the principle of minimum reluctance is derived. To do this, we first turn out attention to the energy stored in the magnetic field. To simplify the analysis, thermal losses in the coil due to resistance, eddie currents, etc. willl be ignored. As long as the length of the gap and the current i are constant, the circuit remains in a steady state. The voltage drop across the coil is zero and the electric source does not provide any power to the circuit. The energy accumulated in the magnetic field must, therefore, remain constant. The field energy E~ is computed by considering a rise in current from zero to its present value i. From eqn (1). this rise in current is accompanied by a rise in magnetic flux which according to Faraday's law induces a voltage d v = d t (¢N)
(4)
across the coil winding. This equation yields d/Ni
\
= d(Li)
(5)
where N 2
L =~
(6)
A
i
T 3_
N turns
1
50 cm
Fig. I. A single-loop magnetic circuit with rotating armature. CAEEVol.7, No. 4--B
/
246
S, G. PAPAIOANNOU
is the inductance of the coil. In order to sustain this voltage, the electric source supplies energy
Ef = fot vi dt= fot i d (Li)dt =
fo'
i d(Li)
(7)
to the coil. Assuming that the air gap and hence the inductance are held constant, all energy provided by the electric source is stored in the field. The accumulated field energy then is
Ef=L
idi
1 2 = ~Li.
(8)
The magnetic field manifests itself in various ways. One of them is the development of a force of attraction between the two faces bounding the gap. The face from which magnetic lines emanate is, in general, taken as the north pole, the opposite face as the south pole. The exertion of an attractive force between these two opposite magnetic poles is consistent with our everyday experience of permanent magnet behavior. The magnitude of the attractive force can be estimated from the principle of virtual work. Assume that the armature is allowed to rotate through a small angle dO corresponding to a change in gap length from 1 to 1 +dl. The attractive force can be derived from the energy balance equation associated with this hypothetical or "virtual" displacement. Let the rotation occur in time dt and assume that during the rotation the current i remains constant. The change in gap length induces a voltage across the winding
v = d ( L i ) = 'z-~-. dL
(9)
To sustain this voltage, the electric source supplies energy
dEs = vi d t = i2 dL.
(10)
Part of this energy is stored in the magnetic field. The rest is converted into mechanical work. From (8), the energy stored in the field is dEt = d(½Li 2) = ½i 2 dL.
(11)
Let d W be the resulting amount of work. The energy balance equation requires that
dEs = dE/+ d W
(12)
d W = des - d E f = ½i 2 dL.
(13)
from which
Thus, under ideal (no loss) conditions, half of the energy supplied by the source ends up in the magnetic field, while the remaining half is converted into mechanical work through the rotating armature. On the mechanical side, the work d W associated with the displacement of the armature can be expressed in two ways, namely, as a product of force times the linear displacement in the gap Fdl or as a product of torque times the angular displacement of the armature T d0. In the first instance, we obtain
F dl = ½i2 dL I .2
dL
(14)
Stepping motors--a review
247
This relationship applies to all electromechanical transducers[7]. It states that the force developed by a linear electromechanical transducer is proportional to the rate of change of inductance and acts in the direction in which the inductance increases. For a rotational transducer, eqn (13) yields T dO = ½i2 d L
T =½ i ~dL d--O
(15)
so that the torque developed by a rotational transducer is also proportional to the rate of change of the transducer's inductance and acts in the direction in which the inductance increases. A similar conclusion can be reached concerning the relationship between mechanical force or torque and reluctance. Substituting (6) into (14) and (15), we obtain F=-I
(Ni~2dR
_l-2dR
_ I (Ni]2dR (~2dR T - - ~ k-~] - ~ = - ½ d--O
(16) (17)
from which it is seen that both force and torque act in the direction in which reluctance decreases. It follows that a position of minimum reluctance is a position of stable equilibrium. Any displacement away from this position generates a force or torque which tends to move the transducer back to the position where the reluctance assumes its minimum value. III. VARIABLE RELUCTANCE STEPPING MOTORS
The principle of minimum reluctance suggests that a motor could be made to rotate by establishing several positions of stable equilibrium along the periphery of the stator and then inducing the rotor to move from one such position to the next, in continuous succession. In order to create positions of minimum reluctance, teeth are machined on the rotor and also on each stator pole. Figure 2 is a longitudinal view of a 3-phase variable reluctance stepping motor. Each section carries one electric phase. Figure 3 shows a cutaway view of one section. Notice that both stator and rotor have the same tooth pitch, so that a perfect alignment of the teeth is possible. The position of perfect alignment is a position of minimum reluctance. When the phase is excited, the magnetomotive force F = Ni induces four magnetic loops, each carrying a portion of the total flux. The torques developed by these loops act in the same direction, since they tend to rotate the rotor towards the position of minimum reluctance. Continuous rotation is obtained by indexing the position of minimum reluctance among the phases. Thus, while the teeth on all rotor sections are aligned, the teeth on each stator pole are displaced with respect to the teeth on the previous pole by 1/3 of a tooth pitch (Fig. 4). Consequently, if the three phases are excited once in the sequence ABC, the motor shaft, Phase A
Phase B
Phase C
Fig. 2. Axial view of a 3-phase variable reluctance stepping motor.
248
S. G. PAPAIOANNOU
///'/
,_
\\\
Fig. 3. Cross sectionof phase A.
passing through successive positions of equilibrium, will rotate counterclockwise by a full tooth pitch. Reversal of the excitation sequence causes the motor to rotate clockwise. Addition of more phases is possible. For a motor with N phases, the stator teeth are displaced by I / N of a tooth pitch from one section to the next. Under these circumstances, the step angle is o =
360°
,~n--~
(18)
where n,. is the number of teeth on each rotor section. In practice, however, there is an upper as well as a lower limit on the number of phases. The lower limit is imposed by the fact that for N < 3, directional control is lost. For N -- l, this is evident. For N = 2, successive positions of equilibrium in both the clockwise and counterclockwise direction differ by ½of a tooth pitch (Fig. 5). Consequently, whenever the next phase is excited, the motor is equally likely to rotate in one direction as in the other, so that the actual direction of rotation becomes a matter of chance. The upper limit on the number of phases is dictated by the practical necessity to keep the unsupported length of the motor shaft from becoming excessive. Four and 5-phase motors are common, while motors with up to eight phases are commercially available. Another version of the variable reluctance motor has a single section (Fig. 6). Since positions of equilibrium must be indexed within the section, the number of teeth on the stator and on the rotor are necessarily different. To obtain proper indexing, the number of stator teeth ns should be ns = n, -+ np
Fig. 4. Teetharrangementin the motorof Fig. 2.
(19)
Stepping motors--areview
Stator"A"
249
I ~
Fig. 5. Teetharrangementin a 2-phasevariablereluctancemotor,illustratingthe loss of directioncontrol.
where n~ is number of teeth on the rotor and np is the number of stator teeth per phase. In Fig. 6, n~ = 5, np= 1 and ns = 4. The angular position shown is the position of equilibrium when phase Na is excited. If the current is next shifted to phase Nb, the rotor will rotate .18° clockwise in an attempt to align tooth II with that phase. Notice that this angle is the difference between the stator tooth pitch (90°) and the rotor tooth pitch (72°). In general, the step angle is given by
O : tp 360 -~ 360) ~,, = tp360n
(20)
where n=
1
(21)
1----1
ns
nr
and tp is the number of stator teeth per pole. Driving a stepping motor requires a power source which is capable of exciting the motor phases in the proper sequence and with the right amount of voltage and current. Occasionally, as for example, in the case of a 3-phase variable reluctance motor (Fig. 2), these requirements can be met with an ordinary 3-phase a.c. power supply. This is known as the a.c. mode of operation and generally results in low speeds unless the frequency of the a.c. system can be raised above the standard 50 or 60 Hz. During each voltage cycle, all phases are excited once and the motor shaft rotates through a full tooth pitch. The obtained motor speed is
s
= vvj
nr
rpm
(22)
where f is the frequency of the 3-phase system in Hz and n, is the number of teeth on each rotor section. For f = 60 Hz and nr = 50, s = 72 rpm.
~
7
2
• Ax,o,
~
A ~
x
i
s
,or
ofstator Stator
Fig.6.Crosssectionofasinglephasevariablereluctancemotor.
250
S.G. PAPAIOANNOU
Higher speeds are normally obtained by driving the motor from a digital source. The speed can then be computed from the formula 60(sps) rpm v = (spr)
(23)
where sps is the pulse or step rate expressed in pulses or steps per sec and spr is the number of steps per revolution. An alternative expression can be obtained by observing that spr = 360°/0, so that v=~
rpm
(24)
For a motor with N sections, spr = n,N and 0 = 360/nrN. By changing the pulse rate, speed can be made to vary over a wide range. If the source transmits pulses serially over a single line, an N-bit circular shift register, also known as a Johnson or ring counter, loaded with a single circulating 1 can distribute the pulses to successive phases (Fig. 7). If the pulses are generated by a digital computer, the phases may be connected in parallel to one of the computer's I/O ports and the pulses distributed under software control.
IV. THE PERMANENT MAGNET MOTOR
Unlike a variable reluctance motor in which the rotor is unexcited, in a permanent magnet stepping motor, the rotor consists of two identical disks mounted on a cylindrical permanent magnet (Fig. 8). To establish a proper magnetic relationship, the teeth on one disk are displaced angularly with respect to the teeth on the other by ½of a tooth pitch. Figure 9 is a cross sectional view of a permanent magnet motor having five teeth on each rotor section. The stator has four salient poles carrying a 2-phase winding. This example is rather artificial and serves only to illustrate the principle of operation. In commercially available motors the number of teeth is generally higher, while the number of poles is usually eight. Since the rotor is a permanent magnet, the operation of this type of motor can best be understood by considering the attractive forces between poles of opposite polarity. Assume; for example, that the stator poles are magnetized as shown in Fig. 10(a). Notice that the north pole of phase 2 is aligned with one tooth of the south pole section of the rotor, while the south pole of phase 2 is aligned with a tooth on the rear section of the rotor which is a north magnetic pole. The attractive forces between these two opposite pole pairs lock the rotor in a state of stable equilibrium. Next, consider what happens if suddenly the magnetizing current in phase 2 reverses direction, thereby switching the polarities of the two poles in that phase (Fig. 10b). The change in polarities converts the rotor position from one of stable equilibrium, to one of unstable equilibrium. As a result, the rotor advances to a new stable equilibrium position in which the north and south pole of phase 1 are aligned with a south and north polarized rotor tooth, respectively. Subsequent moves of the rotor can be explained in a similar fashion.
F-]
QC - -
J
IJ
L
)CLX
r
I 1
QD--
QB
~
m
QA
_J
! I I
Fig. 7. Ring counter waveforms.
I L I I__ I I
Stepping motors--a review
~
251
Statorlamirmtion Stotor winding .~
~Rotor
--,.~
.LJ L '
i ~ I Fig. 8. Axial view of a permanent magnet motor. The magnet is magnetized axially, making one of the disks a north pole and the other a south pole. Teeth are machined along the periphery of the disks and on each stator pole.
Pho~~~Pho0Phase 2 se
Fig. 9. Cross section of an 180 permanent magnet step motor.
~'118°/"~-i t
Phase 2 1
Phose,~~~/~Phose I (a) SWI=+
SW2 = - " ~ 1 o 2 L , / Phase 2
r-%
(c) SWI =%SW2=+
~i~~
(b) S W I = + , S W 2 = + ' ~ . I / y 2 I'qv~54o
(d) SWl =-, SW2 =-
[~"~72
(e) SWl =+, SW2---
Fig. 10. Successive rotor positions generated through changes in stator polarities.
o
252
S. G. PAPAIOANNOU
I Fig. ! 1. Cross section of a commercial permanent magnet motor.
The angle of rotation in each step is again the difference between the stator and rotor tooth pitches so that eqns (20) and (21) are valid here, too. In general, to obtain proper indexing, the following relationship must be maintained.
nr = m n , + ~
(25)
where n~ = nr = p = m =
number of teeth on the stator number of teeth on the rotor number of poles per phase a positive integer.
Figure I1 shows a SLO-SYN motor made by Superior Electric. This motor has nr = 50, n~ = 48, p = 4, m = 1 and tp = 6. For the step angle, eqn (20) yields o
=
. {360 o
360]
1"8°"
=
Substituting this value into (24), we obtain v = 0.3 (sps)rpm J_
SWl o / .~+ I
?-
(a)
L t ~ Phase 2
7Step
(b)
CW Rotation SWI SW2
CCW Rotation SWI SW2
I
+
4"
+
4"
2
-
+
+
-
4
+
-
-
+
Fig. 12. (a) Basic circuit for driving a permanent magnet motor from a dual-voltage supply and (b) Four-step switching sequence for CW and CCW rotation.
Stepping motors--a review
253
Figure 12 shows a circuit for driving the two phases of a permanent magnet motor and the required switching sequence for clockwise rotation. If counterclockwise rotation is desired, the switching sequence is simply applied in reverse order. The switch positions have also been marked in Fig. 10 in order to relate them to the various angular positions of the rotor. Notice that this circuit reverses the direction of current flow by exciting each phase through a separate power supply. In order to eliminate the need for a second power supply, commercial motors are usually equipped with bifilar windings (Fig. 13). A bifilar winding consists of two identical halfs wound around the same stator poles but in opposite directions. Only one half of each winding is excited at any given time. The magnetic polarity is changed by shifting the current to the other half. To keep our explanations simple, the actual profile of the current rise or fall has up to this point been deliberately glossed over. The question is important since torque is a function of current. Ideally, when a winding is activated, current should rise fast in order to produce enough torque to drive the shaft towards the next equilibrium position in a short period of time. When this position is passed, the winding must be deactivated in order to diminish the torque which now acts in a direction opposite to that of rotation and allow the torque developed by another winding to take over. However, because of the energy storage properties of inductors, a step increase or decrease in voltage does not produce an instantaneous step increase or decrease in current. Rather, the current rises or falls exponentially with a time constant L/R. As a result, a finite amount of time is required until the torque develops its full value or until from this value it returns to zero. Figure 14 illustrates typical torque-speed characteristics of a stepping motor. In general, the motor generates its highest torque at standstill. As the step rate increases, the delaying action of the phase inductances prevents the currents from attaining their steady-state values and the output torque drops. For each motor, the manufacturer normally provides a starting and a running torque-speed characteristic. The starting characteristic specifies the speed at which the motor can start a given load without missing steps. Once the motor gets started, the digital source moves the operating point towards the running torque-speed characteristic, by gradually increasing the step rate. The torque characteristic of a given motor can be substantially improved by reducing the current rise and fall times. Ways for doing so are examined in Section VI. V. ELECTROHYDRAULIC STEPPING MOTORS
Because of their limited torque, stepping motors are often unable to drive heavy loads directly. This limitation motivated the development of electrohydraulic motors, in which a small REO
O 'IIp-GREEN/WHITE
STEP I 2 3 4 I
SWl ON ON OFF OFF ON
SW2 OFF OFF ON ON OFF
SW3 ON OFF OFF ON ON
SW4 OFF ON ON OFF OFF
CW
CCW
Fig. 13. Circuit schematic and switching sequence for driving bifllaxwindings.
254
S.G. PAPAIOANNOU
250
(la.o) ...,_.-
200 Z
.u
~¢
N 13.
o ~ IzJ O a:
(14.4)
- - R U N N I N G START-STOP W I T H O U T ERROR
--.. ~ q,Q,,.,
-x,,
150
(10.8] 100 17.2)
SO 13.61
0
I..-
1
0
75
150
225
300
375 450
525
600
675
750
SPEED (STEPS PER SECOND)
Fig. 14. Typical stepping motor torque-speed characteristics.
stepping motor is used to control a powerful hydraulic motor. In this manner, the advantages of direct digital control can be extended to a much wider horsepower range. Consider, for example, the torque-speed characteristic of one of the highest rated stepping motors (model MI72-FD301) made by Superior Electric[4] (Fig. 15a). This model has a step angle 0 = 1.8° corresponding to 200 spr. Horsepower is the product of torque and angular velocity, i.e. ToJ P = 5-3-6hp
(26)
where 550 is the conversion factor from hp to ft-lb/sec. From Fig. 15(a) the maximum power this motor can deliver is 1 800 2zrx2000 Pi = 550 16x 12 200 = 0.48 hp.
(o)
1250
lgO.Ol 1000 (72.0} i
!
bJ
~
730 t54.oi
~
----...
5oo 136.0}
0 0 I-
250
1la.Ol
!
250
0
500
750
1000 1230
1500 1750
2000 2230
2500
SPEED (STEPS PER SECOND) MI72-FD3OI
MOTOR
(b) 2000 PSI
I000 =; i. m .J
I00
0
400
1300 PSi ~ . .
600 1000 PSI
0 F-
5 0 0 PSI
200
0
200
400
SPEED,
600
RPM
Fig. 15. Torque-speed characteristics of (a) a purely electric stepping motor and (b) an electro-hydraulic motor (Superior Electric).
Stepping motors--a review
255
This power is obtained at 2000 sps. The same company makes an electrohydraulic motor whose torque-speed characteristics at various pressures are shown in Fig. 15(b). This motor produces its maximum power l 9502~r × 600 P2 = 550 12 ~ = 9.04 hp at 2000 psi, 600 rpm. Thus, the use of hydraulic power can increase the drive capabilities of these motors by a factor of 20 or more. An electrohydraulic stepping motor (EHSM) contains three major components: an electric stepping motor, a servovalve, and a hydraulic motor. These elements, when integrated in a particular fashion, cause the hydraulic motor to accurately follow the rotary motion of the stepping motor but with a much higher torque output. A hydraulic motor is a flow-operated device. In an EHSM, the presence of the servovalve is necessary since the stepping motor cannot control the flow into the hydraulic motor directly. The valve acts as the interface between the two. Figure 16 illustrates the operation of a four-way sliding spool valve connected to a power cylinder. The spool may be shifted left or right. If shifted to the left, port A is connected to the pressure port P, while port B is connected to the drain. As a result, the piston moves to the right. Conversely, if the spool is shifted to the right, the pressure causes the piston to move to the left. An axial piston hydraulic motor uses a number of power cylinders (typically (9)) to convert hydraulic power into mechanical motion (Fig. 17). The cylinders are oriented axially along the periphery of a cylinder block. Whenever the controlling valve is off its neutral position, half of the cylinders are connected to the pressure port while the rest are connected to the drain. As the pistons move in and out of their bores, they transmit axial forces to a tilted shafted or bearing, forcing it to rotate. Since this shaft is coupled to the pistons, its rotation causes the pistons and the cylinder block to rotate at the same speed. Figure 18 illustrates how the tilted plate converts the pressure forces to torque. On the high pressure side of the cylinder block, each piston transmits an axial force Ap where A is the piston area and p the hydraulic pressure. This force may be resolved into a component normal to the plate which is balanced by a bearing reaction and a torque generating component Apsina. The resulting toque is
(27)
T = ~ Apsina R sin 0i.
Figure 19 shows an axial view of an EHSM [8]. Notice that the spool of the control valve is coupled to the stepping motor through a two-gear train, while at the other end, it is threaded into a nut rigidly attached to the hydraulic motor shaft. The main characteristic of this configuration is that rotary motion of either motor causes the spool to move longitudinally. Od under pressure
Drain I Drain
t I1
t [
_
...e.-,=.
Fig. 16. Four-way sliding spool valve controlling a power cylinder.
256
S.G. PAPAIOANNOU
Out~ In |
(a)
shaft
(b) Fig. 17. Axial piston hydraulicmotor. (a) Schematicand (b) Typicaldesign of commerciallyavailable motors. Assume that the hydraulic motor is at rest. To understand the operating principle of the EHSM, consider the effects of an angular step performed by the stepping motor. As the motor shaft rotates, it rotates the spool through the gear coupling. Since the spool rotates inside a stationary nut, it moves longitudinally away from its neutral position and opens up the high pressure port letting pressure into the hydraulic motor. As the hydraulic motor rotates, so does the nut attached to its shaft. Now, however, the geared end of the spool is held stationary by the stepping motor and the spool is prevented from rotating. As a result, the spool moves axially in the opposite direction and closes the supply port forcing the hydraulic motor to come to a stop. In this manner, the hydraulic motor matches the angular motion of the stepping motor step for step. The torque required for driving the spool can be provided with a relatively small stepping motor. This torque is to a large extent independent from the load imposed on the hydraulic motor. Thus, the stepping motor remains unaffected by load fluctuations and the possibility of missed steps is practically nonexistent: The EHSM is also largely free from overshoot because of the damping influence of the hydraulic fluid on each angular step. VI. STEPPING MOTOR DRIVES A stepping motor drive has several functions to fulfil (Fig. 20). It must generate the necessary switching sequence and have provision for direction control. It must amplify the
I
Fig. 18.Torquegenerationmechanism.
257
Stepping motors--a review Outlet
~Powersupply ~
~Drive$crewL~ Output shLJaftJ
Fig. 19. Electrohydraulic stepping motor (EHSM).
weak TTL type pulses transmitted by the digital source both in voltage and current. It must suppress inductive surges. Finally, it must enable the motor to meet the requirements of the load both in terms of torque and speed. If the digital source is a general purpose computer, the sequencing logic can be easily implemented in software. In that case, the pulse amplifier is directly connected to one of the computer's I/O ports. Otherwise, the drive must include a sequence generator. This last approach is favored by stepping motor manufacturers, who normally provide off-the-shelf drives for their motors, since it makes more drive alternatives possible. If the motor happens to be of the variable reluctance type, distributing the input pulses to the phases on a one-at-a-time basis requires no more than a shift register wired as a ring counter (Fig. 7). Any standard bidirectional shift register such as the 7499 or the 74194 will do. Figure 21 shows the 74194 wired as a 4-bit ring counter. The input pulses are applied to the clock input, while the states of the two MODE inputs determine what the outputs will do at the next low-to-high transition ( t ) of the clock. The register is first initialized by loading the internal flip-flops with the states of the parallel inputs A B C D (1000). The input pulses are then distributed to the motor phases by enabling the right or left shift mode depending on the desired direction rotation. For a permanent magnet motor, a special counter can be easily designed from MSI components. Figure 22 shows one possible configuration. Notice the correspondence between the truth table and the switch positions in Fig. 13. Special semiconductor chips are also available. An example is the SAA 1027 driver chip made by North American Philips Controls. Apart from generating the right switching sequence, this device has a limited pulse amplification capability (up to 12 V, 350 mA) which allows it to drive small stepping motors directly (Fig. 23). Techniques for pulse amplification and inductive surge suppression are more or less standardized. To begin with, often the driving capability of totem-pole type digital outputs is so limited that, in the high state, they cannot even source the few milliamperes of current required to drive a transistor. This deficiency can be easily corrected by digital buffering. Suitable buffers are the 7406 inverting or the 7407 noninverting buffer drivers. Both devices feature open-collector outputs which in the high state become disconnected allowing the power supply to provide up to 30 mA of current to each power transistor base through the pull-up resistor. A convenient starting point is the ciruit shown in Fig. 24. This circuit embodies the fundamental principles used in the control of inductive loads such as relays, solenoids, etc. In
INPUTPULSES~IsEQUENCE HPULSE OIRECT,ON
-[OENERATOR]
[AMPLIFIER
t
POWER SUPPLY
INDUCTIVEH STEPPING I PROTECTIOINIMOTOR
H I
Fig. 20. Block diagram of a stepping motor drive.
S. G. PAPAIOANNOU
258 *$v
T F
_ MODE CLEAR
Q
A
QC
S0 S1
QB QA
CLK | C
D GN
E
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H
H
J'
PARALLEL LOAD
L
I,I
~'
SHIFT RIGHT
H
t
~
SHIFT LEFT
1111 !
(a)
(b)
Fig. 21. (a) A 74194shiftregisterwiredas a 4-bit ringcounterand (b) Truthtable.
this case, L and R represent the inductance and resistance of a single stepping motor phase. Note that the phase receives its power from the power supply when the power transistor is turned on. The diode and resistor Ra provide an outlet for the phase current when the transistor is turned off. This path dissipates the accumulated magnetic energy and, at the same time, it protects the power transistor from overvoltage. A detailed analysis of this circuit will serve to illustrate the basic principles involved in the design of stepping motor drives. Suppose that a pulse of duration T is transmitted to the phase by the sequence generator. The rising edge of the pulse saturates the power transistor bringing its collector (point C) to ground. As a result, current starts building up and the phase is progressively energized. The rising phase current produces voltage drops vL and ve across the phase inductance and resistance, respectively. The polarities of these voltages are such that the diode remains reverse biased. The operation of the circuit at this stage is described by the equation: (28)
v = ve + VL = R i + d ( L i ) . Ul
direction ~ 4 - S N 7 4 0 0 N
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A2
BI
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I
I
O
I
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2
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O
I
3
O
I
O
I
4
O
I
I
O
I
I
O
I
O
CW
IT CCW
tb) Fig. 22. Sequencegeneratorfor permanentmagnetmotors(a) logiccircuitand (b) Truthtable.
Stepping motors--a review
259
SAA1027 T , ~ TRIGGER~._..
STAGE
-ts° I-
s
STAGE
3fDIRECTIONAL 4 POSITION SYNCHRONOUS COUNTER
STAGE
,
m
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.
D
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.
m
1
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STEPPER MOTOR
(c) Fig. 23. (a) SAA 1027 stepping motor driver chip (North American Philips Controls), (b) SAA 1027 block diagram and (c) SAA 1027-based stepping motor drive.
The derivative term must be developed with caution since, as the motor rotates, both L and i vary with time. L depends on gap geometry and, hence, on the angular position 0 of the rotor. The relationship between L and 0 can be approximated as shown in Fig. 25 by the equation: Lmax + Lmin Lmax- Lmin 2 + 2 cos
L -
(29)
nrO
where nr is the number of rotor teeth and the angle 0 is measured from the angular position in which the phase and rotor teeth are fully aligned. In order to introduce this relationship, eqn (28) is differentiated as follows
.dLdO v=Ri+L ~t +Zd0 dt.
~
Rpu
BUFFER
"1
L
(30)
R
I +V
/
I
"POWER
._[__ TRANSISTOR
°
Fig. 24. Basiccircuit for driving a steppingmotorphasefrom a digital source.
260
S.G. PAPAIOANNOU L tO) L,~o,
L,.i,.
I
|
I J
f
Fig. 25. Approximate variation of inductance with angular position.
The last term is known as the back electromotive force (EMF). Its presence makes the solution of this equation a difficult proposition even when the function (dL/dO) is precisely known, since the angular velocity (dO/dt) depends on the dynamics of the entire system. However, the back EMF is not the dominant term and as a first approximation it may be dropped to obtain di v = Ri + L ~ .
(31)
The solution of the resulting first order differential equation is (32)
i(t) = If + (Io - If) e -'/"
where I0 and Ir are the initial and steady-state value of the current and z is the time constant. In this case Io = 0, If = (V[R) and r = (L/R). Therefore,
i(t) =
V
L (1 - e-t/'), ~"= ~.
(33)
Figure 26 is a plot of i(t). Several points can be made regarding this curve. First notice that since the energizing stage lasts as long as the input pulse, only the initial portion of the curve between time 0 and T is valid. The form of the curve is also important because of the strong dependence of torque on current. A large amount of torque can generally be extracted from a given motor by maintaining a high level of current flow. It follows that the ideal current profile is a rectangular pulse whose height equals the maximum amount of current the phase can withstand without overheating. This ideal situation, however, can only be reached in the limit as the time constant approaches 0. The significance of the time constant is now evident. For a given steady-state current, it defines the current profile and, consequently, the amount of torque that can be obtained from the motor. Thus, a given time constant sets a limit on torque and indirectly on motor speed. Typically, the time constant of phase windings is of the order of 10msec. To generate substantial torque, the current must be given a chance to reach a sufficiently high level. In general, the current reaches 63.2% of its final valve in one time constant, 86.5% in two time constants and 95.0% in three time constants. A judicious if somewhat arbitrary limit can be established by requiring the width T of the excitation pulse to be at least 3 time constants or l(t)
Ideol profile
/••ck
emf neglected
With back emf
t 0
T
Fig. 26. Ideal and actual phase current profiles.
Stepping motors--a review
261
30 msec. Since in permanent magnet motors each phase is excited at ½the step rate (Fig. 12), a winding receives 95% of full current at a step rate 2/(30 x 10-3) = 66 sps. Beyond this rate the torque deteriorates rapidly and the motor becomes unable to meet its load. Notice that for a 1.8° motor, this figure represents only 0.33 revolutions per second. This is a severe limitation which could normally disqualify the motor for most applications. As we shall see, however, performance can be improved to the point that the same motor can deliver 15,000 sps if properly driven. As a final remark on Fig. 26, notice the detrimental effect of the back EMF on the current profile. The actual current profile was determined by using simulation techniques [9]. The instant t -- T at which the input pulse expires marks the beginning of the deenergizing stage. At this point, the driving transistor is cut off blocking the flow of current to ground. At the same time, the diode opens up and the current is diverted into the dissipation loop. Thus, by providing an outlet for the phase current, the diode prevents the occurrence of an inductive voltage surge vL = L(d//dy) which could damage the transistor. Notice that the dissipation loop does not include the power supply. Therefore, due to resistive losses, both the current and the magnetic energy which sustains it start decreasing until they simultaneously reach zero. In fact, the change in current slope from positive to negative provides the mechanism through which the current forces its way through the dissipation loop, since a sign change in di/dt reverses the polarity of vL thereby enabling the diode to open. The current profile is a decaying exponential
i(t) =
V
e -'t'd,
L
7d = R + Rd"
(34)
This function is obtained from eqn (32) by substituting Ir = 0 and I0 = V/R. The effect of the dissipation time constant 7a on driving torque remains to be examined. Recall that when a phase is energized, the motor shaft rotates towards a position of stable equilibrium with respect to that phase. When the phase is cut-off, the shaft has normally passed this position. From this point on, the phase current generates a torque which opposes shaft rotation since it tends to move the shaft back to the equilibrium point which has already been passed. Consequently, in order to minimize the adverse effect of this torque, the phase current must be dissipated as fast as possible. This, in turn requires a small time constant. A high dissipation rate is obtained by including a large resistance Rd in the bypass loop. Large values of Rd, however, may cause excessively high voltages to appear at the transistor collector. Let Vcmax by the maximum collector voltage the transistor can withstand without breaking down. The maximum voltage imposed on the collector occurs at the time the transistor is cut-off. This voltage must not exceed Vcmax or V + V R d
