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Open and closed loop control of step motors through the IBM parallel port

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Title:
Open and closed loop control of step motors through the IBM parallel port theory and practice
Creator:
Harrer, James Alfred
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Language:
English
Physical Description:
vii, 222 leaves : illustrations (2 folded) ; 29 cm

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Subjects / Keywords:
Stepping motors -- Design and construction ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references.
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering.
Statement of Responsibility:
by James Alfred Harrer.

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Source Institution:
University of Colorado Denver
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
26922710 ( OCLC )
ocm26922710
Classification:
LD1190.E54 1992m .H37 ( lcc )

Full Text
OPEN AND CLOSED LOOP CONTROL OF STEP MOTORS THROUGH
THE IBM PC PARALLEL PORT; THEORY AND PRACTICE
by
James Alfred Harrer
B.S., University of Toledo, 1969
M.S., University of Toledo, 1972
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
1992
"'I
*
J


1992 by James Alfred Harrer
All rights reserved.


This thesis for the Master of Science
degree by
James Alfred Harrer
has been approved for the
Department of
Electrical Engineering
by
Edward Wall
Marvin Anderson
Michael Radenkovic
2-1/fZ-
Dai


Harrer, James Alfred (M.S., Electrical Engineering)
Open and Closed Loop Control of Step Motors Through the IBM PC Parallel
Port: Theory and Practice
Thesis directed by Professor Edward T. Wall
Abstract
This thesis reviews the theory, construction and application of the DC
brush motor and step motor, with computer simulations of the step responses
for each motor type. Four different step motor drive techniques are compared
in theory and three drive methods are demonstrated in hardware.
Experimental results of the single step response for the Voltage drive, Bilevel
drive, and Constant Current or Chopper drive are presented for the step motor.
The results for open and closed loop step motor control are given. It is
demonstrated that closed loop PID control of the step motor through the
parallel port of the IBM PC compatible computer is not only possible but
provides a dramatic improvement to the step motor velocity profile. Several
software program listings in BASIC and C source code are included.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed ___________
Edward T. Wall
IV


Acknowledgements
Science and engineering are never done in isolation but always with the
support of colleagues, teachers and friends, even in the darkest hours late at
night near thesis deadline with a negative result. I wish to acknowledge the
following; my teachers for their knowledge and support, particularly Dr. Wall
for suggesting this topic in the first place and for his patience and continued
help and encouragement; Hannah Kelminson, at the UCD Writing Center, for
editing this text; my friends and colleagues at Martin Marietta, particularly
Dave Crosson, for his invaluable critiques of my circuit designs and help in
troubleshooting the motor drives; Bob Rice, for his excellent comments and
suggestions on experimental procedures; Tim Anderson, for patiently
answering each of my sometimes desparate questions on the C programming
language; and Charles Overy, for showing me the intricacies of scanning
images into the text; my employer, for making resources available to me after
hours to do this work; and my wife Judy, for everything. It was this support
that allowed me to persevere and turn the sometimes negative results into a
positive outcome. I have learned a great deal by having the opportunity to do
this thesis, and I hope that by its completion I am returning something to the
University for the education of other students as well as myself.
v


Contents
Part I. Comparing the DC Brush and Step Motors: Theoretical
Considerations.
1. Introduction ............................................................1
2. Permanent Magnet DC Brush Motors.........................................8
2.1 Construction and Operation........................................8
2.2 Mathematical Model for the DC Brush Motor.......................10
2.3 Example of a DC Brush Motor Transfer Function...................18
3. Step Motors ...........................................................22
3.1 The Variable Reluctance Step Motor..............................22
3.2 The Permanent Magnet Step Motor.................................24
3.3 The Hybrid Step Motor............................................28
3.4 Step Motor Dynamic Characteristics...............................35
3.4.1 Step Response.............................................36
3.4.2 Mechanical Resonance......................................37
3.4.3 Speed-Torque Curve........................................42
3.4.4 Start/Stop Rates and Ramped Open
Loop Velocity Profiles..........................................44
3.5 Step Motor Driver Classifications................................47
3.5.1 The Hybrid Step Motor Stepping Sequences.................47
3.5.2 Bipolar Drive Circuits....................................52
3.5.3 Unipolar Drive Circuits...................................53
3.5.4 Drive Methods to Control Motor Current..................56
3.5.4.1 Voltage Drive......................................59
v


3.5.4.2 L/R Drive
61
3.5.4.3 Bilevel Drive....................................63
3.5.4.4 Constant Current Drive...........................65
3.6 Mathematical Model for a Hybrid Step Motor...................71
4.0 Computer Simulation of the Mathematical Models.....................81
4.1 DC Brush Motor Step Response.................................82
4.2 Step Motor Single-step Response with L/R Drive...............86
4.3 Step Motor Single-step Response with
Constant Current Drive........................................89
5. Closed Loop Step Motor Control.......................................91
6. Summary and Conclusions for Part 1..................................95
Part 13. Hardware Design and Experimental Results.
7. Introduction ........................................................97
8. Checking Out Second Hand Motors......................................97
9. Motor Parameter Measurements........................................105
9.1 DC Brush Motor Transfer Function Measurement.................105
9.2 Step Motor Parameter Measurements............................107
9.3 Step Motor Step Moptor Single Step Response..................112
9.4 Step Motor Multiple Step Response............................127
9.5 Speed-Torque Curves for the Step Motor.......................135
10. Interfacing Motors to the IBM PC Compatible Computer...............137
vi


10.1 Interfacing the IBM Parallel Port for Motor control...........138
10.2 Software Trade Study...........................................143
11. Step Motor Interface and Drive Electronics...........................145
11.1 The Bilevel Motor Drive Electronics............................146
11.2 The Constant Current Motor Drive Electronics..................148
12. DC Brush Motor Interface and Drive Electronics.......................152
13. Software Programming for Motor Control...............................152
13.1 Description of the Programs....................................153
13.2 Rate Profile Results for the Open and Closed Loop
Systems........................................................163
14. Recommendations for Future Study......................................169
Appendices
A. Computer Simulation Software Code and Results.........................172
B. Motor and Electronics Sources.........................................181
C. Electronic Schematics and Data Sheets.................................183
D. Step Motor Motion Control Programs in Microsoft C....................191
References ...............................................................217
vu


Part I. COMPARING THE DC BRUSH AND STEP MOTORS:
THEORETICAL CONSIDERATIONS.


1. Introduction
This thesis describes the theory and control of the step motor as a motion
control device in open loop and closed loop operation. As a basis of
comparison, the more familiar DC brush motor is also described. Part I
discusses the theory and operation of these motors, and Part II presents the
hardware and software design and integration together with the experimental
results obtained by controlling a step motor via the parallel port of an IBM PC
compatible computer.
Step motors position loads by operating in discrete increments, or steps,
unlike DC brush motors which are controlled to operate at constant speeds [4].
Another major difference between a DC brush motor and a step motor is that in
the brush motor, commutation is done mechanically through the brushes. In the
step motor, there are no brushes to wear out and the commutation is done
electronically by switching motor phases on or off. This is an advantage in
applications with explosive or vacuum environments where brush motors
cannot be used [39]. Motor brushes may produce dangerous sparking in
operating room or welding environments. In vacuum environments, absence of
oxygen and moisture prevent standard motor brushes from forming a good
electrical contact with the commutator [17]. Step motors or brushless DC
motors are therefore usually selected for space-based applications.
1


Step motors (also called stepper motors, stepping motors, or steppers) are
described by Kuo [39] as "electromagnetic incremental-motion actuators which
convert digital pulse inputs to analog output motion." When properly controlled
by switching the current or voltage input to the motor in a programmed manner,
a step motor indexes in angular or linear increments which are equal in number
to the number of input command pulses. Each input pulse advances the angular
(or linear [23]) increment by one step, as in a clock-work mechanism, and the
motor magnetically latches precisely into its new step position until the next
input pulse is given. Any position error, usually less than 5 percent of the
step length, is non-cumulative, as each step length is fixed by motor
construction [39]. The step motor moves as slow or as fast (up to the speed
limit of the motor itself) as the stream of input pulses. By counting the number
of input pulses, very accurate position control can be obtained with
microprocessor or digital electronic control and without the need for expensive
feedback sensors required for DC brush motor servo systems. In many step
motor designs, such as the hybrid design described here, a residual or "detent
torque" holds the motor in its last step position, even when motor power is
removed [15].
A conventional DC brush motor servo system (Figure 1.1) consists of: 1) a
command input; 2) a system compensator such as a PID controller; 3) the driver
amplifier, 4) the DC brush motor itself; 5) the load; and 6) a required feedback
device such as an encoder or tachometer [15, 17].
2


Input
Figure 1.1 Brush Motor Servo System.
Similarly, a step motor is always part of a motion control system (Figure
1.2) which consists of: 1) a pulse generator; 2) a translator for electronic
commutation of the motor phases; 3) the drive amplifier, 4) the step motor itself;
5) the load; and 6) an optional position sensor such as an encoder [17]. An
ASCII keyboard and microprocessor can be used to provide a pulse stream
input to permit control of motor direction, stepping rate, number of steps, and
velocity profiles. The optional encoder may be used simply in a step-
confirmation scheme to ensure that the number of motor steps taken is equal to
the number of steps commanded. A more sophisticated form of encoder
feedback in a step motor system can provide additional compensation to form a
brushless DC servo for rate or position control [1,10]
3


Figure 1.2 Step Motor System.
Step motors have been available for many years and have made important
contributions to position control, primarily because of the precision of motion
they can provide with simple, open loop control. A list of patents from the
United States and Great Britain on devices with step motor action dates back to
1919 [40], but commercial exploitation only began in the 1960's when step
motors were made more torque efficient and improved transistors capable of
switching large DC currents (greater than one amp) in the motor winding were
made readily available [3]. Since that time, step motors have found wide
application (Table 1.1 and Appendix B), and the market for these motors has
increased fourfold in the last decade (Acamley, p viii). Kuo [39] identifies six
categories of step motors (Table 1.2), but only a few types enjoy commercial
success.
4


Table 1.1 Categories of Step Motors (Kuo, [39])
1. Solenoid-ratchet
2. Variable-reluctance
3. Permanent magnet
4. Hybrid
5. Electro-mechanical harmonic drive
6. Electrohydraulic
Because of the simplified control requirements and the freedom from
expensive feedback sensors, step motors have become viable alternatives to
pneumatic, hydraulic and servo motor systems [4,13,61]. Step motor
manufacturers recommend their motors for applications requiring low speed,
moderate torque, low inertia and high positional accuracy, as in
instrumentation, machine tools, laser or optical pointing, industrial controls and
robotics. DC brush motor servo systems should then be chosen for
applications with speeds above 2000 RPM and torques above 2000 oz-in.
Some motor control manufacturers [63, 64] dispute this guideline and claim that
the step motor will eventually be completely replaced by modem, competitively
priced DC brush or brushless motor servos with tachometer or encoder
feedback. Prior to the mid 1980's, step motors were less expensive than the
more recently designed brush and brushless DC industrial servo systems, for
which the motor manufacturers had not recovered their tooling and engineering
development costs. These costs were passed on to the consumer. Now,
however, prices for the DC servo systems, including encoder or tachometer
5


feedback, are becoming more competitive and the DC servo manufacturers are
replacing step motors in several of the applications listed in Table 1 [63]. For
example, the list price as of January 1992 for a 100 oz-in step motor, 2.25"
diameter by 4" long, such as used in the experimental section of this thesis, is
$190 (Superior Electric P/N MO63-LS09). List price for a DC brush motor of
similar size and torque is $176 (Reliance Motion Control/Electro-Craft P/N
0540-04-32). A 200 line quadrature encoder may be purchased for an
additional $40 (US Digital P/N SP-200-B). To remain competitive, step motor
manufacturers are also offering feedback control step motor systems for step
and distance confirmation (Compumotor Catalog). The unique characteristics
of the step motor (such as open loop control, brushless commutation, and
tolerance of a one to ten times change in rotor to load inertia ratios) will
probably meet many motion control requirements for years to come.
Table 1.2 Typical present-day step motor applications [39, 61].
Aircraft control systems
Automatic typesetting machines
Clocks and watches
Coil-winding machines
Cut-to-length of metal, plastic, fabric
Digitally controlled milling and painting machines
Fiberglass mass production
Medical equipment (motor-controlled microscope stages)
Nuclear reactor control rods
Packaging machines
Photocopying machines
Photographic and laser/optical positioning equipment
Pick-and-place automatic loading and unloading
Read/write head control on floppy disk drives
Robotics
Serial and line printers
6


Sheet metal machines
Silicon crystal growing
Sun track control for solar panels
Textile machines
Welding
Wire-wrapping machines
X-Y microfiche access machines
X-Y plotters
The step motor provides a good subject for study in a master's thesis
project. Several step motors are available for student use in the university
controls laboratory, and they may be purchased at surplus for under fifteen
dollars (Appendix A). The unique control requirements of the step motor draw
on knowledge from electrical engineering courses in circuit analysis, analog and
digital electronics, controls and dynamics. Step motors are studied in several
undergraduate controls engineering laboratories, both in in this country and
Great Britain [2, 38, 62]. The hardware and software provided with this
project may serve as basis for other controls lab experiments or senior projects
at the University of Colorado. Finally, study of the theory and operation of the
step motor serves as a useful background for understanding the operation of
other motors, such as the brushless DC motor.
7


2. Permanent Magnet DC Brush Motors
The Permanent magnet DC brush motor construction, operation and
mathematical model will be discussed briefly here to serve as a basis of
comparison to the step motor.
2.1 Construction and Operation
The conventional DC brush motor uses either permanent magnets or
separately excited field coils placed concentrically around an iron core motor
shaft to produce a radial magnetic field (Figure 2.1).
Figure 2.1 Conventional DC Brush Motor and Motor Drive. (Courtesy
ElectroCraft Corp. [16])
8


The availability of powerful ceramic and rare earth magnets allows
permanent magnet motors to be built with sufficient torque for most
applications, and the separately excited field coil DC motors are becoming less
common [9]. The rotor windings, or armature, of a DC brush motor consists
of slotted steel laminations wound with wire forming one or more armature
phase windings [15]. Current is applied to the armature windings, which
interact with the magnetic field of the stator magnets to produce torque. As the
motor rotates, a commutator mechanically switches current flow through the
windings to maintain rotation. Brushes make electrical contact with the
commutator and deliver current to the armature windings as the rotor turns.
The DC motor is well-suited for applications which require controlled
speed, acceleration, deceleration, or torque [15,17]. An outstanding feature of
the motor is its nearly linear speed to torque relationship (Figure 2.2).
A
Torque
Speed
Figure 2.2 Conventional DC brush motor speed-torque curve [16].
9


2.2 Mathematical Model for the DC Brush Motor
Two basic equations for DC motor operation are based on the electrical
and dynamic behavior of the motor [16]. The electrical equation of the motor is
derived from considering the motor equivalent circuit in Figure 2.3 [15]. The
dynamic equation of the motor is derived from a mechanical model of the
motor, load and shaft shown in Figure 2.6.
Figure 2.3 Electrical model of a DC brush motor.
Resistance R represents the series resistance of the brushes, commutator,
and motor windings. La represents the inductance of the armature winding.
Any parallel impedances due to electrical or magnetic losses are usually ten
times greater than R or La and may be ignored [16]. Eg represents a generated
voltage which results when the rotor coils move through the magnetic field flux
established by the permanent magnets.
+
o
10


Using Kirchoff s law, the voltage equation around the loop in Figure 2.4
is:
(2.1)
Nonlinearities in this relation, such as changes in field flux as the motor
turns, are usually insignificant [16]. Since Eg is proportional to motor velocity,
we may write
where w is the motor speed. Ke, the back EMF term for the motor, is
measured by driving the motor in the generator mode at constant speed with a
second motor, and measuring the voltage generated at the motor terminals.
Combining equations 2.1 and 2.2 determines the electrical equation of the
motor.
The dynamic model for the motor must account for the mechanical linkage
between the motor and the load, whether belt, gear or direct drive. In this
thesis, only direct drive is used between motor and encoder or motor and load.
However, even with direct drive, flexible shaft couplers (Figure 2.5) are often
used to eliminate problems caused by shaft misalignment, such as excess
bearing wear and increased position error [22].
Eg = KEG)
(2.2)
Vi = Ria + La^- + KEG)
(2.3)
11


Figure 2.4 Flexible shaft coupling joining two shafts with exaggerated
skew.
Under acceleration or start-stop conditions, the flexible shaft may present
some spring wind-up, torsional resonance or hysteresis problems. Theoretical
considerations for a flexible shaft coupler will therefore briefly be considered
here. The relation between generated torque, Tg, and the opposing torques due
to motor and load inertias, damping, spring rate of the shaft, and friction is
often written down without derivation in many standard motor textbooks. The
following derivation using Lagrange's method lends some insight into the
dynamics of the motor and load, and applies to both brush motors and step
motors [56].
Lagrange's general equation for systems with damping and excitation is
[56]
dS(KE) S(KE) + S(DE) =
dt 5qi 8qi 8qi
where 1 = 1,2, 3,...n degrees of freedom in the system
Qi =ith external excitation applied to the system
qi = ith generalized coordinate of the system
qi = ith first derivative with respect to time
12


The mechanical system in Figure 2.6 models the motor and load inertias,
the shaft spring between rotor and load, viscous damping and bearing friction.
T
g
f
Figure 2.5 Mechanical model of the motor, shaft and load.
From equation 2.6 and Figure 2.5, we may write
dS(KE)
dt 50i
dS(KE)
dt 802
8(KE) 8(PE) 5(DE) + + Tpi = Tg (2.5)
50i 50i 50i
8(KE) 8(PE) + ^ + T 0 (2.6)
802 802 502
where Tg = generated motor torque
Ji+ J2 = motor and load inertias
Bi + B2 = motor, load and bearing damping constants
Tpi + Tk, = running torque friction
Tsi + Ts2 = starting torque friction
Ks = spring constant between motor and load
01, 02 =motor and load shaft angular positions
0i, 02 =motor and load shaft angular rates
. 2 *2
KE = Kinetic Energy = J101 + ^2 2
. 2 *2
DE = Damping Energy = yBi0i + JB2Q2
PE = Potential Energy = ^Ks(02 0i)2
Solving for the terms in equations 2.5 and 2.6,
13


=Jl0i; 5(KE)=0; 5(PE) = .Ks(02.e1);^E)=B1e1(2.7)
dt
80i
501
60i
60i
=Jle2; Mi)=0; 5(PH) = ^3(02-00; ^1)=B202 (2.
dt
502
80i
502
8)
602
Substituting these terms back into equations 2.7 and 2.8 gives us the two
equations of motion for the mechanical system.
Jii + Bi0i + Ks(0i -02) + Tpi + Tsi = Tg
J202 + B202 + Ks(02 -0l) + Tf2 + Ts2 = 0
(2.9)
(2.10)
These last two equations are more complex than what is usually required,
and additional assumptions may be made to simplify them. The friction terms
TS1 and Ts2 are the stiction torques which occur only at start-up and are often
modeled as finite pulse functions [62]. Since they are often small in value
compared to the running friction, TS1 and TS2 may be neglected [16]. For the
flexible shaft couplers used in this experiment (Helical P/N AR 075-8-8),
torsional stiffness is rated at 0.68 degrees/lb-in. When the 100 oz-in step motor
and shaft coupler used here are run at 40% of rated torque, a displacement error
of 1.7 between motor shaft and load is developed, which is almost a full step
length. However, for loads of small inertia requiring small torque as in
constant velocity applications, 01 and 02 can be considered equal. With these
assumptions we may simplify and combine equations 2.9 and 2.10 to write the
dynamic equation as
14


(2.11)
J0 + B0 +Tp = Tg
where J = Ji+ J2
B = Bi + B2
Tf = Tfi +Tp2
Tsi =Ts2 = 0
One other important equation relates the electrical and dynamic properties
of the DC motor. If we assume the magnetic field is constant and the armature
windings are not saturated with current, then the motor torque generated, Tg, is
proportional to the motor current.
Tg Kjia
(2.12)
Applying the Laplace transform to equations 2.3, 2.11 and 2.12, we
obtain the system equations
Vi(s) = (sLa + R)Ia(s) + Keco(s) (2.13)
Tg(s) = (sJ + B)co(s) = (sJ + B)s0(s) (2.14)
Tg(s) = KTIa(s) (2.15)
The equations may be combined to form the transfer functions for velocity,
Gw(s), and position, Gq(s)
co(s) _____________Kj_____________
Gco(s)
Ge(s) =
Vi(s) (sLa + R)(sJ + B) +KeKt
0(s) ________________Kj______________
Vi(s) s[(sLa + R)(sJ + B) +KeKt]
(2.16)
(2.17)
15


The block diagram of the system equations 2.16 and 2.17 is given in
Figure 2.7.
V.
Figure 2.6 DC brush motor block diagram.
The transfer function in equation 2.16 has two negative and real poles for
all practical cases, and may be written as
Gm(s)=#t
_L
JLa (S pi)(s p2)
(2.18)
In most cases the damping factor, B, is negligible [16], and equation 2.16
may be written as
_ Q)(s) _____________Kj___________
" Vi(s) s2LaJ + sRJ + KeKt
The roots are
Pi,2
-RJ V(RJ)2 4LaJKEKT
2LaJ
(2.19)
(2.20)
16


A further simplification of the motor transfer function is possible for small
La [16], where
(2.21)
For this case, the radical expression in equation 2.19 may be approximated
by
V(RJ)2 4LaJKEKT RJ(1 2LaKEKT (2.22)
R2J
The transfer function reduces to
Gco(s)
Kj/LJ
(s^X-£)
(2.23)
and the mechanical and electrical poles are
_pi =Ke&l = J_
P RJ Tm
-P2 = f
(2.24)
where ^ and te are defined as the mechanical and electrical time constants
of the motor (ElectroCraft). Electrocraft also recommends Equation 2.23 only
when p2 > 10(pj); otherwise equation 2.19 is more accurate.
17


2.3 Example of a DC Brush Motor Transfer Function
The manufacturer's specifications for a brush motor used in the
experimental part of this thesis (RAE Part No. 231096.0) are given in Table
2.1 below.
Table 2.1 Manufacturer's specifications for the RAE 231096.0 Brush
Motor.
Ke = 3.83 V/KRPM 10% = 0.03658 V/rad/sec
KT = 5.18 in-oz/Amp 10% = 36.579E-3 Nm/A
Ra = 1.3989 £2 7.5% (armature resistance)
RT = 1.7486 £2 12.5% (terminal resistance)
FT = 2.50 in-oz = 17.65E-3 Nm (frictional torque)
FTmax = 5-00 in-oz = 35.3IE-3 Nm (start-up torque)
JA = 0.0025 oz-in-sec2 = 17.65E-6 kg-m2
La = 4.387 mH
te = 2.509 msec
tm = 22.905 msec
Bandwidth = 6.9 Hz
Since no viscous damping is provided in this list and the manufacturer
claims it is negligible, equations 2.19 or 2.23 can be used to calculate the open
loop transfer function of the motor. The manufacturer has already determined te
and tm, and so from equation 2.24 we may calculate
pi =-43.66 rad/sec n
p2 = -398.6 rad/sec K
Since p2 and pj are not an order of magnitude apart, a better pole
approximation may be given by equation 2.19, which yields
18


pi = -49.61 rad/sec
P2 = -349.6 rad/sec
(2.26)
and the transfer function is
Gco(s)
472.305E3
(s + 349.6)(s + 49.61)
472.305E3
s2 + 399.21s + 17.344E3
t
(2.27)
which is a classic, overdamped second order system where con = 131.7 rad/sec
and £ = 1.52. The PC MATLAB calculations [46] for the Bode plot and step
response of the open loop transfer function given in equation 2.27 is shown in
Figures 2.8 and 2.9.
19


Degrees dB (Rate/V)
Frequency, rad/sec
Figure 2.7 Magnitude and phase frequency response for a DC brush
motor.
20


30
Step response
Figure 2.8 Step response for a DC brush motor.
The steady state velocity calculated from the transfer function in equation
2.27 and as shown in figure 2.8 is 27.23 rad/sec.
21


3. Step Motors
The basic concept of the step motor has existed over 50 years [39], but of
all the various electromagnetic ratcheting systems devised, only three types of
step motor have achieved commercial success by producing significant torque
from a reasonable volume [3]. These three types are the variable reluctance
(VR), the permanent magnet (PM), and the hybrid step motors. The motors are
readily found in surplus supply stores, and one of each type is available in the
lab. The design of these motors is derived from the ac synchronous inductor
motor [17] and their construction will be briefly described in this section.
Special focus is given the hybrid step motor, which combines elements of the
first two and is used in the experimental part of this thesis.
3.1 The Variable Reluctance Step Motor
The VR step motor uses a magnetically soft iron rotor having teeth and
slots. Opposing pairs of stator teeth are wound by a wire winding which
produces a magnetic field when excited by a DC current (Fig. 3.1; Leenhouts
[45]).
22


Figure 3.1 Cross section through variable reluctance step motor
showing the flux paths through the motor with phase A energized (Courtesy
Leenhouts [45]).
The magnetic force thus established forces the rotor teeth to align with the
stator teeth through the major flux path. Rotating the magnetic field by
switching the DC current to an adjacent winding (from phase A to Phase B)
causes the rotor to rotate to the next stator tooth. VR step motors require a
minimum of three phases, or switched cycles of DC current, to produce
bidirectional motion. These phase windings may be arranged in a single-stack
or in multiple stacks (Figure 3.2) to provide more torque.
23


stack A stack B stack C outer case
Figure 3.2 Multiple stack VR step motor cut lengthwise to show three
stacks of phase windings and rotor sections (adapted from Acamley [3]).
Since VR step motors do not have magnets, they have no detent torque like
the PM and hybrid step motors, and the motor shaft can be turned by hand
smoothly. They generally operate with step angles from 2 to 15 [3].
3.2 The Permanent Magnet Step Motor
The PM step motor is probably the most popular step motor in commercial
use today [32]. It is also sometimes called the claw motor or the tin can type
24


step motor, as the stator structure consists of toothed or claw-like pieces of
sheet metal welded together, and the motor case and stator cups can be
economically stamped out of sheet metal ([3,28]; Figures 3.3 and 3.4). More
expensive, instrument grade PM motors are also manufactured, however, and
the parts are machined from stainless steel (Rapidsyn [53]; Figure 3.5).
Figure 3.3 Cutaway view of the PM "tin can" step motor (Courtesy
AIRPAX Corp. [4]).
The PM step motor has a construction similar to the VR type motor, but the
rotor has a smooth, permanently magnetized surface made of a permanent
magnet material such as ferrite, Alnico, samarium cobalt or neodynium [32].
The rotor is radially magnetized with alternating North-South poles (Figure
3.4). The number of pole-pairs on the PM motor varies from one to over
twelve but is limited by the difficulty in manufacturing multiple poles in the
same magnetic material. The practical resolution to which a permanent magnet
may be radially magnetized is twelve pole-pairs [32].
COIL 8
COIL A
25


Figure 3.4 PM "claw motor" rotor showing one of the claw-shaped
stator poles (Leenhouts [45]).
The number of stator phases and the number of rotor poles determines the
step length of the PM step motor.
Steps/rev. = phases x states x Pole-pairs (3.1)
For a twelve pole-pair motor with two phases and two states each as shown in
Figures 3.3 and 3.4, there are 48 steps per revolution [32].
In the industrial grade PM motor example shown in Figure 3.4, the rotor
has one pole pair and two stator phases that can be energized with plus or minus
current This creates a step length of 90. Unlike the VR motor, only two motor
phases are required for bidirectional motion. One wire wrappped around two
opposed stator poles forms phase A+ and A-. Similarly, the same wire carries
the current around stator poles B+ and B-. For positive current in the winding
of phase A, the rotor position aligns as shown in Figure 3.4. Turning off phase
A and switching on positive current in phase B, as indicated by the arrow,
26


produces one 90 clockwise step. Negative current in phase B would produce a
counterclockwise step.
Figure 3.5 Diagram of a two-phase PM step motor (Courtesy of
Acamley [3]).
Commercially available PM step motors have resolutions or step lengths
from 7.5 to 90, or 4 to 48 steps per revolution. These step lengths are longer
than in the hybrid step motors (discussed in section 3.3 below) and are useful in
moving a load a great distance in a few steps when high resolution and high
torque are not required. The torque per unit volume of most PM motors is poor
relative to similarly sized brush motors and hybrid step motors [3]. Newer
developments in magnet fabrication using injection molding to form isotropic
neodynium magnets [9] may increase the possible number of rotor poles,
27


increase the resolution and torque, and decrease the total motor weight in this
popular step motor design.
3.3 The Hybrid Step Motor
For better step resolution and higher torque capability, the hybrid step
motor combines concepts of the VR and PM step motors. A hybrid motor
(Figures 3.5 and 3.6) has a permanent magnet mounted on the rotor, like the
PM motor, and also has salient poles, like the VR motor. The permanent
magnet of the hybrid motor is not radially magnetized as in the PM motor, but
has its north-south poles oriented axially. The hybrid motor usually has two
phases, A and B, but motors with up to five phases are constructed [45].
Hybrid construction provides greater torque for a given motor volume than for
the VR or the PM type step motors [3]. Bifilar windings, described in section
3.4.3, are generally used in the hybrid step motor, so a single source power
supply can be used. The motor can be driven one phase on at a time, or two
phases on at a time for more torque. Even with the windings of the hybrid
motor not powered, the magnetic flux produces a small "detent torque" which
holds the motor in a step position. Hybrid motors have a step length range of
0.9 to 5, but 1.8 is most common. This step length has become a standard in
the industry for hybrid step motors, which are often attached to lead screws for
x-y positioning of loads in machining operations. A standard five pitch lead
screw (five threads per inch) gives a good mechanical advantage to the hybrid
28


motor and advances the load by exactly 0.001" for a 1.8 step which is very
useful for machining and operations, and disc drive track heads [3].
The hybrid motor usually has eight stator poles, with each pole having
between two and 6 teeth. The stator windings are similar to the PM motor
diagram shown in Figure 3.4, except in the hybrid there are more stator poles.
In the hybrid step motors studied in the experimental section of this thesis, there
are eight stator poles, each with five teeth (Figure.3.4; [45]). The rotor also has
50 salient poles, or teeth, machined into the axial North-South magnetic
material, to guide the magnetic flux to preferred locations in the air gap between
stator and rotor. The 50 rotor teeth interact with the 40 stator teeth to provide
200 stable equilibrium or rest points in the motor, which causes the detent
torque felt when the motor shaft is turned slowly by hand. There may be one to
three stacks of North-South rotor cups for additional torque. In Figure 3.5 is a
simple sketch illustrating some of these features of the rotor for the hybrid
motor.
29


South Pole
Tooth Pitch
Figure 3.6 Sketch of rotor for the hybrid step motor showing one stack
of rotor cups.
Figure 3.6 shows cross-sections through the North and South rotor cups
of a single-stack, two phase hybrid step motor [45].
30


1
s

A
1
S
2
A. North pole rotor cup B. South pole rotor cup
Figure 3.7 Cross sections through North and South pole rotor cups with
phase A energized. (Courtesy of Leenhouts, [45].)
In Figure 3.7, a continuous wire wrapped around stator poles 1, 3, 5 and 7
forms phase A, and phase B is formed by the wire wrapped around stator poles
2,4,6 and 8. Phase A is energized to produce South magnetization at the teeth
of stator poles 1 and 5, but North magnetization at stator poles 3 and 7. Figure
3.7 shows the preferred alignment or stable equilibrium position of the rotor for
the flux paths thus established. The figure shows how the teeth on the rotor
over the North pole cup are offset 1/2 tooth pitch from the teeth on the rotor
over the South pole cup to provide proper alignment of the rotor teeth with the
stator teeth. The rotor and stator teeth are aligned at poles 1 and 5, but are
offset at poles 3 and 7 for the North rotor cup (Figure 3.7A). In the South rotor
cup (Figure 3.7B), the poles 3 and 7 are aligned with the rotor. When phase A
is de-energized and phase B is energized, stator poles 2 and 6 become South
poles, while stator poles 4 and 8 become North poles. The new flux paths
31


move the rotor 1/4 tooth pitch, or 1/4 the distance between rotor teeth, from the
alignment in Figure 3.7 to the alignment in Figure 3.8. In this way, the rotor
advances one step, or 1/4 rotor tooth pitch for each phase change. For 50 rotor
teeth, 4 X 50 or 200 phase changes or steps are required to produces one
complete revolution.
A. North pole rotor cup B. South pole rotor cup
Figure 3.8 Cross sections through North and South pole rotor cups
with phase B energized. (Courtesy of Leenhouts [45]).
Figures 3.7 and 3.8 show the rotor positions for the first two motor steps;
phase A is energized for step one in Figure 3.7, then phase B is energized for
step two in Figure 3.8. For the third step, phase A is energized again but with
the current in the stator winding reversed, or with the field direction in the
opposite sense. Stator poles 1 and 5 are now made North, and stator poles 3
and 7 are South, just the opposite of the case in Figure 3.7. Magnetic attraction
advances the rotor another 1/4 rotor tooth pitch. The fourth step is produced by
energizing phase B again, but with the field direction in the opposite sense.
32


If the motor is forced more than half a rotor tooth pitch away from its stable
equilibrium position, it will skip over one or more rotor teeth to find a new rest
point. Thus a step motor, when the load torque becomes too great, will not
"lose or "skip" just a single step, but will skip at least one rotor tooth length or
four steps at a time [45].
The stepping sequence may be understood by referring to a sinusoidal
approximation of the torque curves for the hybrid step motor shown in Figure
3.9 A and B.
33


A+ & B+ A- & B-
B. Two-phase-on excitation
Figure 3.9 Static torque/rotor characteristics for the hybrid step motor with
one-phase-on excitation (A) and two-phase-on excitation (B). (Adapted from
Acamley [3]).
The set of curves in plot A is obtained by energizing one winding at a time and
measuring the torque vs. rotor position for each winding, and then
34


superimposing the four resultant curves on the same graph [34]. The plot in B
is obtained by energizing two windings at a time and again plotting the torque
as a function of rotor position. The curves are mutually displaced by one step
length, or p/2Nr =1.8 mechanical degrees in the hybrid motor, where Nr is the
number of rotor teeth. The points marked on the X axis correspond to the
equilibrium points or preferred rest positions for the rotor with one phase on. If
the rotor is at rest in the phase A+ equilibrium position as shown in Figure 3.7,
or the 0 mark in Figure 3.9A above, then energizing phase B+ applies positive
torque and the rotor moves CW one step to the phase B+ equilibrium, p/2Nr.
Conversely, if negative torque is applied by energizing phase B- when the rotor
is at 0 phase A+ equilibrium, the rotor moves CCW one step to the B- rest
point, -p/2Nr. Energizing the phases sequentially as A+, B+, A-, and B-
applies continuous positive torque and advances the rotor CW. Reversing the
sequence steps the motor CCW.
Figure 3.9B gives the static torque curves for two motor phases turned on
at once. Two-phase on operation provides 1.4 times more torque and requires a
different stepping sequence, as will be described in section 3.4.1.
3.4 Step Motor Dynamic Characteristics
All of the step motor types discussed above have similar dynamic
characteristics which are useful in describing and understanding motor
performance. Four important dynamic characteristics include the step response,
35


the speed-torque curve, mechanical resonance, and starting/stopping rates
[3,33,45].
3.4.1 Step Response
A plot of the step response describes what happens to the rotor position
each time a step command is given. The single-step response for a step motor
can be modeled as an underdamped, second order system (Figure 3.10, below;
[7, 33, 34]), in contrast to the DC brush motor, which has an overdamped step
response (Figure 2.8)
Figure 3.10 Typical single-step response for a step motor.
36


The overshoot, settling time and natural frequency of oscillation of the step
motor speeds up the step response, but these properties can cause a problem in
some applications. For example, if a step motor is used to drive the carriage of
a dot matrix printer, the printer must pause after each line feed until the carriage
comes to rest before printing another character [3]. The longer the oscillation,
the slower the overall operating speed of the printer. Step motor oscillations are
often damped out by adding more friction to the load, or by electronically
braking the moror by momentarily reversing motor torque [39].
3.4.2 Mechanical Resonance
Another consequence of the highly oscillatory single-step response is the
phenomenon of mechanical resonance, which occurs at stepping rates that are
multiples of the natural frequency of the rotor about the equilibrium position
[3]. Figure 3.11 shows multiple step responses of a step motor at two different
step rates.
37



/ loss of
/ synchronism
| stepping rate at natural frequency
^ stepping rate at 0.6 x natural frequency
Figure 3.11 Multiple step response (adapted from Acamley [3]).
38


The solid line traces a normal multiple step response of a step motor. After each
step, the motor oscillation quickly dampens out to a uniform position before the
next step command is given, and the motor never overshoots more than 75%.
In the other response shown by the dotted line, the stepping rate is about equal
to the natural frequency. For this case, at the end of its first step where the
rotor still has significant positive velocity, the motor is given another step
command. The response to the second command is more oscillatory, and the
rotor swings further from its equilibrium step position. At each successive
step, the oscillation increases until the rotor lags or leads the demanded step
position by more than half a rotor tooth (more than 3.6 or two step lengths).
Once this oscillation amplitude is exceeded, the rotor tries to realign with
another magnetic field by moving ahead or behind a full rotor tooth, which is
7.2 or 4 step lengths, from the expected position. There no longer is a one-for-
one correspondence between steps commanded and steps taken, which can be a
problem in open loop control of step motors. This problem may be resolved by
avoiding speeds at or near the natural frequency of the system, which changes
as load inertia changes, or by adding friction, such as a fluid-filled viscous
damper, to the load [39,41]. Unfortunately, too much friction may cause loss
of synchronism at low speeds [45].
The resonant frequency can be derived from the dynamic equation for the
motor (Equation 2.11) and the sinusoidal torque curves in Figure 3.9 [45].
Referring to Figure 3.9A, with phase A+ on, the rotor is held at the stable
39


equilibrium position of 0, where the torque stiffness, dT/dq, is at a maximum.
In equation form, this is expressed as
r = dX = _d_(TpksinNre)
d0 d0 (3.2)
= NrTpkcosNr0 = NrTpk, at 0
Now if the rotor is displaced a small amount, q, by applying an opposing
torque, -Tq, we may rewrite equation 2.11 as
J0 + B0 + TF = Tg T0 (3.3)
For a motor with light damping and minimal friction, we may simplify
equation 3.3 and take the Laplace transform,
(Js2 + T)0(s) = Tg(s) (3.4)
The resonant frequency for this simple harmonic system is
(3.5)
Equation 3.5 shows that as the peak torque is increased or the system
inertia is decreased, the frequency of mechanical resonance will increase by the
square root of the torque stiffness. For one of the step motors used here
(Superior Electric MQ63-FD), the frequency of mechanical resonance for the
motor without load is determined as follows:
Tpk = 100 oz-in; Nr = 50; J = 4.56E-3 oz-in (rotor inertia)
40


Then Torque stiffness = T = 5000 oz-in
and wn = 1,047 rad/sec or fn = 167 Hz
This mechanical resonance effect can also be seen in the speed-torque curve
for a step motor, as shown in Figure 3.12.
Figure 3.12 Speed-torque curve for a step motor (Adapted from
Acamley [3]).
41


3.4.3 Speed-Torque Curve
The speed-torque curve for the step motor is measured at different step rates in
steps per second (SPS) by measuring with a dynamometer the maximum torque
the motor will produce at each speed. This measurement is called the "pull-out
torque," because if the load torque exceeds this value, the rotor is pulled out of
synchrony with the step commands and the motor stalls [3]. The speed-torque
curve for the step motor is highly nonlinear, compared to the DC brush motor in
Figure 2.2; however, three regions can be identified. At low step rates (below
100 SPS), when a phase is turned on, the current climbs rapidly to its steady
state value and remains at its peak level for a substantial part of the time for
which the phase is excited. Higher overall current levels in the windings results
in higher torque at the lower speeds [3]. As motor speed increases, motor back
EMF increases linearly and the torque decreases through the linear region
between 100 and 300 SPS.
Two sharp dips in the curve occur at 20 and 40 SPS, caused by the
mechanical resonance effect in the motor and load. Resonance is likely to occur
at frequencies which command a step when the rotor is in the positive overshoot
portion of the step-response curve in Figure 3.11 [3, 33, 45]. These
frequencies are at the natural frequency and the associated sub-harmonics of the
motor and load, or
42


f
Resonant frequencies (SPS) = , k = 1, 2, 3, ...
(3.6)
where fn equals the frequency measured direcdy from the step-response in
Figure 3.10 or calculated by equation 3.4. Thus a motor with a natural
frequency of 40 Hz may have notches in the speed-torque curve at 40,20, 10,
5,... steps per second. At speeds other than these frequencies or above the
natural frequency of the system, the motor is not receiving a step command
when the rotor has a positive oscillation velocity, and resonance is not a
problem.
For high speeds (greater than 300 SPS in Figure 3.12), motor inductance
becomes the dominant circuit element. The time constant for the current rise
and decay occupies a significant portion of the total phase excitation time. This
results in lower overall current levels in the windings, and lower torque levels at
higher speeds, as shown by the hyperbolic portion at the right of the speed-
torque curve.
A more realistic representation of the speed-torque curve is shown in
Figure 3.13. Here, the loss of torque at or near the resonant frequency and
associated subharmonics of the motor are apparent on the left of the curve. In
the middle of the curve is another region of torque drop-off, termed "midrange
instability." This phenomenon can occur as the motor makes a transition from
the nearly linear portion of the speed torque curve at lower speeds where motor
impedance and back EMF circuit elements predominate in the motor torque
43


equation, to the region of high speed where motor inductance predominates.
This problem can be alleviated by half-stepping the motor, as will be shown in
section 3.5.2, or by using an external damper or a current drive circuit [45].
Figure 3.13 A more realistic plot of speed vs. motor pull-out torque for
a step motor (Courtesy Leenhouts [45]).
3.4.4 Start/Stop Rates and Ramped Open Loop Velocity Profiles
The simplest form of open-loop motor control of a step motor is a constant
stepping rate which is applied to the motor until the load reaches its target
position [3]. The maximum stepping rate at which the motor can start from rest
and respond without loss of steps is the "starting rate." The "stopping rate" is
the maximum stepping command rate which can be immediately turned off
without allowing the motor to overshoot the target position. Theoretically,
these rates are the same and can be derived from the dynamic equation of the
motor as follows [3,45]. From equation 2.11, for light damping and minimal
friction, the maximum acceleration the motor can experience is given by
44


(3.7)
Integrating twice with all initial conditions at zero, the step length and step time
are related by
For the unloaded motor referenced above in section 3.4.2 (Superior
Electric M063FD), the torque to inertia ratio is 21,930 rad/sec2 and the step
length is 1.8 or 0.0314 rads. Using equation 3.9, the maximum theoretical
start/stop rate for this motor is 836 steps per second or 250 RPM. In practical
applications, to allow for motor wear and slight changes in load torque, the rate
for a step motor operated at a fixed period is usually set at 10% below the
maximum start/stop step rate.
Improved motor performance can be obtained by using a variable instead of
a constant step command timing sequence [48, 62]. The motor can be run at a
higher speed than the theoretical start/stop speed if it is first started at a low rate,
then rapidly accelerated through the regions of motor resonance to a much
higher slew rate, and finally decelerated to below the stop rate before turning off
the step command sequence. Changing the period of the step command is
(3.8)
and the start/stop rate is given by
(3.9)
45


referred to as ramping strategies, which can be linear or exponential and are
designed to bring the motor, under open loop control, to a speed above the
start/stop rate and back to zero velocity without failure. A comparison of these
control strategies is given in Figure 3.14.
Figure 3.14 Comparison of constant rate control with linear and
exponential ramping velocity profiles.
The total move time in moving the load from the start position to the target
position is significantly improved by using a linear ramp velocity profile, and
better still for the exponential velocity profile, since the average speed is greater.
46


3.5 Step Motor Driver Classifications
To produce rotor motion, the phases A+ and A-, and B+ and B- in the two
phase hybrid step motor are excited in a precise order, called the stepping
sequence, by changing the direction of the field in phase windings A and B
[61]. This is accomplished by using either a bipolar or a unipolar drive circuit
and the appropriate logic circuits to control the phase switching [45]. The
stepping sequence for the hybrid motor will be given in section 3.5.1 below for
a bipolar drive circuit. Bipolar drive circuits (section 3.5.2) are designed to
provide bidirectional current flow in the same winding. Unipolar drive circuits
(section 3.5.3) utilize motors wound with two windings per stator pole. Both
types will be discussed briefly, and then a math model for the unipolar drive
and bifilar motor, which is more common and is used in the experimental part
of this thesis, will be developed. Motor drive methods which control current to
the motor include voltage drive, L/R drive, and "chopper" or constant current
drive, and are discussed in section 3.5.4. The L/R and constant current drive
types are used in the experimental part of this thesis.
3.5.1 The Hybrid Step Motor Stepping Sequences
The switches for a bipolar drive circuit are shown schematically for a two
phase hybrid step motor in Figure 3.15. In this figure, both the transistor
switches and their associated bypass diodes, which must be used whenever
switching inductive loads, are simply represented by S1 through S4. Three
47


step processes to move the hybrid step motor CW or CCW are shown in Table
3.1 A, B and C, and are described as the "one-phase-on full step", the "two-
phases-on full step", and the "half step" stepping sequences [61].
48


Figure 3.15 Bipolar drive circuit with two supplies.
Table 3.1 A, B, and C One-phase-on, two-phase-on and half step stepping
sequences for the hybrid step motor [61].
CCW
CW
Switch Position
(* is off)
Step Phase SI S3 S2 S4
1 A+ ON * *
2 B+ * ON * *
3 A- * ON *
4 B- * * ON
1 A+ ON * * *
Switch Position
(* is off)
Step Phases On SI S3 S2 S4
1 A+ & B+ ON ON * *
2 B+ & A- * ON ON *
3 A- &B- ON ON
4 B- & A+ ON * ON
1 A+ & B+ QN QN * *
A. One-phase-on stepping sequence.
B. Two-phase-on stepping sequence.
CCW
CW
Switch Position
Half-Steps Phases On SI S3 S2 S4
1 A+ ON * * *
2 A+&B+ ON ON *
3 B+ * ON * *
4 B+& A- * ON ON *
5 A- * * ON *
6 A-&B- * * ON ON
7 B- * * * ON
8 B-&A+ ON * * ON
1 A ON * * *
C. Half-step stepping sequence.
49


The simplest of the three stepping sequences above is the "one-phase-on"
sequence. Referring to the static torque curves for one-phase-on excitation in
Figure 3.9A, the motor advances when positive torque is applied by energizing
sequentially the phases A+, B+, A-, B-, which corresponds to switches SI,
S3, S2, and S4, respectfully, in Figure 3.15. Phases A and B are energized
sequentially by connecting the positive power supply to the coils through switch
SI for step one and S3 for step two. In steps three and four, the negative
power supply is connected through S2 and S4, as shown in Table 3.1 A.
Clockwise (CW) motion is produced by sequentially energizing the phase
windings from steps one to four. Counterclockwise (CCW) motion is
produced by moving backwards through the sequence from four to one. In this
step sequence, only one phase is turned on at a time, there are four steps in the
sequence, and at each phase change, the motor moves a full step of 1.8. Note
that by connecting a four bit microprocessor register to the appropriate transistor
switches in Figure 3.15, the motor motion, direction and speed can be
completely controlled. Rotating a "one" left or right through the register would
drive the motor CW or CCW.
The second stepping method is given in Table 3. IB, labeled the four step
"two-phases-on" sequence. Referring to the static torque curves for two-
phases-on excitation in Figure 3.9B, the motor advances CW when positive
torque is applied by energizing sequentially two phases at a time: A+ and B+;
B+ and A-; A- and B-; and B- and B+. This corresponds to the sequential
switch closures listed in Table 3.IB. The motor direction is reversed by
50


reversing this sequence. In this step sequence, two phases are turned on at a
time, there are four steps in the sequence, and at each phase change, the motor
moves a full step of 1.8.
The third step method is given in Table 3.1C, the "half step" sequence, and
is an eight step sequence which combines the other two sequences. With this
method, the motor alternates between "one-phase-on" and "two-phases-on"
drive, thus forcing the rotor to take half steps between the equilibrium positions
of one-phase-on and two-phases-on. Referring again to the torque curves in
Figure 3.9A, at the end of step one, the rotor is at the A+ equilibrium position,
0. At the end of step two, the rotor moves to the A+ and B+ equilibrium
position, p/4Nr shown in Figure 3.9B. At the end of step three, the rotor
moves another half step to the B+ equilibrium position, p/2Nr, and so on for
the entire eight steps of the CW sequence. Rotor direction is reversed by
reversing this sequence.
Both the "one-phase-on full step" and the "two-phase-on full step" have
four steps in the phase sequence and take a full 1.8 per step; however, because
the two phase on step sequence energizes two phases per step, the motor has
about 40% more torque (see section 3.6, equations 3.42-3.45). In the "half
step" input sequence there are eight steps in the phase sequence and the motor
takes 0.9 per step or 400 steps per revolution. In the half step mode, the motor
has less torque but smoother operation than in the full step mode.
51


A fourth method for control of step motors is called microstepping or
ministepping [60]. This technique provides finer positioning resolution and
smoother operation than can be obtained with simple on/off control of the phase
currents. To microstep a step motor, two digitally reconstructed sine waves 90
apart are used to fractionally turn on transistor switches SI through S4 in
Figure 3.15. Thus as the current in phase B is fractionally increased, the
current in phase A is fractionally decreased, and the motor advances a
"microstep." Commercial microstepping drives are available which subdivide a
standard 1.8 step by from to over 128 times [60].
3.5.2 Bipolar Drive Circuits
All bipolar drives work by allowing bidirectional current flow in the motor
coils [45]. One bipolar drive method requiring two power supplies and two
transistor switches per motor phase was given above in Figure 3.15. Another
bipolar drive method provides bidirectional current to the phase windings by
using one supply and an "H" bridge switching network. This method is shown
for one phase winding in Figure 3.16. Figure 3.16A illustrates a high power
method in which the switches S1-S4 might be power MOSFET transistor
switches. For positive current flow in the direction indicated, transistor
switches SI and S4 are closed. For negative current flow, switches S3 and S2
are closed. Figure 3.16B is a low power modified H bridge design in which
52


current is limited by the pull-up resistors for the two transistor switches S1 and
S2.
S1
S3
S4
S1
S2
A. High power H Bridge design. B. Modified H bridge
Figure 3.16 Bipolar drive method for Phase A of a step motor using a single
supply and an "H" bridge design.
3.5.3 Unipolar Drive Circuits
The bipolar drive method changes the field direction in the stator poles of
phase A and phase B as required to advance the motor, but requires two
supplies and two switches per phase (Figure 3.15), or one supply with four
switches per phase (Figure 3.16A). This results in greater expense and weight.
53


A simpler and less costly drive method is the unipolar drive using "bifilar"
wound hybrid motors, which are offered by most step motor manufacturers [3].
A conventional winding and a bifilar winding are compared in Figure 3.17.
Direction of field
i L
+ current
- current

A. Conventional Winding
Direction of field
+ phase
winding
- phase
winding

B. Bifilar Winding
Figure 3.17 Comparison of conventional and bifilar windings around a
single stator pole (Adapted from Acamley [3]).
In the conventional or bipolar winding, a bidirectional current flow in the
same winding produces a bidirectional field in the stator poles. In the bifilar or
unipolar winding, there are actually two wires wrapped in opposite senses
around the same pole. Positive current flowing into the dot of winding 1
54


produces a field direction opposite to the field direction produced by current
flow into the dot of winding 2. For a slightly more complicated motor
construction, a simpler drive scheme than the bipolar method can be used, as
shown in Figure 3.18. This drive sscheme is used in Part II of this thesis.
S1
Figure 3.18 Unipolar drive for bifilar wound step motor.
In the unipolar drive, one supply and four transistor switches are sufficient
to switch the field directions back and forth in phases A and B, as required to
advance the motor. This bifilar wound motor is still referred to as a two pole
step motor, even though it now has four separate coils, because only two stator
poles are involved; in the actual motor, stator pole windings A and B are
alternately distributed around the motor, as shown in Figures 3.7 and 3.8. The
55


bifilar motor shown in Figure 3.18 can also be connected to the bipolar drive
shown in Figure 3.15 simply by leaving the center tap wires to phases A and B
disconnected. As with the bipolar drive, three different step modes are
available: "one-phase-on full step", "two phases on full step", and "half step."
They are identical to the switching sequences given in Table 3.1 A, B and C.
3.5.4 Drive Methods to Control Motor Current
To construct a motor drive for the hybrid step motor, the switches
diagrammed in Figure 3.18 cannot be simply replaced by power transistors and
electronically switched according to Table 3.1. We also need to consider the
charge and discharge paths for motor current, and the time constants for the
motor phases. The motor drive must be able to inject current quickly into a coil
to turn a motor phase on, and discharge the current from that coil to turn the
phase off. For example, consider the switch and coil shown in Figure 3.19
below, which could be for a solenoid drive [8].
56


+20V
Q
A
1 ohm
1 mH
\
B
SI
V
Figure 3.19 Charge and discharge of a coil.
After the switch is closed, the current will rise to a steady state level in a time
determined by the L/R time constant of the circuit When the switch is opened,
voltage across the coil from A to B will rise rapidly to try and keep the current
flowing according to the equation
Vab = L& (3.10)
If switch SI is a power MOSFET transistor (International Rectifier Part
Number IRF130, as used in the experimental part of this thesis), the switch will
change from an on resistance of less than 0.2 W to an off resistance of over 10
MW in 50 nsec or less. Thus for a coil with 1 mH inductance and 1 W
resistance, the voltage across the coil will try to rise toward 400 KV!
Vab = L = (lE-3)2^-2 0 = 400 KV (3.11)
^ 50E-9
57


However, the voltage will never rise to this level, as the IRF130 breakdown
voltage is 300 volts, and the FET begins conducting again when the voltage
rises to this level, discharging the coil. Although this is no longer a destructive
action for the IRF130, as it would be for many other lesser transistor switches,
it is not good practice to let circulating currents rise to such high voltage levels
[18, 19]. The "inductive kick" caused by opening the switch in Figure 3.19 can
be discharged through a bypass diode connected in parallel with the coil as
shown in Figure 3.20.
Figure 3.20 Bypass diode used to solve the "inductive kick" problem in
a switched coil.
In this circuit, when switch S1 is closed, current rises to a steady state level and
diode D1 is reverse biased. When SI is opened, the coil voltage rises rapidly
until D1 is forward biased and the energy is returned to the supply. This
method is often employed in solenoids circuits [19]. In the bifilar wound step
motor, the "inductive kick" is returned to the supply by the transformer
+20V
p
1 ohm
1 mH
V
58


coupling between the bifilar coils. The return path for the coil voltage in the
bifilar wound step motor will be illustrated below.
Step motor torque and speed are directly related to the amount of current in
the active phases. The current in the phase, in turn, depends on the L/R time
constant. Several drive methods have been developed to inject current into the
motor coil as quickly as possible, without exceeding the power rating of the
coil, which would overheat and possibly damage the motor. Four of these
methods are illustrated in Figures 3.21 through 3.24 for a single phase of a step
motor. The other phases of the motor not shown have identical drive
electronics. The four circuits are all unipolar drives for the bifilar wound step
motor, and are identified as the Voltage drive, L/R drive, Bilevel drive and the
Chopper or Current drive. Current paths in each circuit when the phase
winding is turned on and off are traced for each case with dashed lines. In each
of the figures, phase winding A= is active, and the discharge current path is
through phase winding A-. The last three current control methods, L/R. Bilevel
and Chopper drive, are demonstrated in the experimental part of this thesis.
3.5.4.1 Voltage Drive
The first of the four drive types discussed here is the Voltage drive (Figure
3.21), which is the least expensive drive type but gives the poorest motor
performance [45, 62].
59


Figure 3.21 Voltage drive for one phase of a bifilar wound step motor.
The dashed lines in the figure trace the current paths through the circuit. Switch
S2 is open at all times when exciting phase A+. When switch SI closes,
current passes left through phase A+ and the coil resistance, then through SI to
ground. Current entering the side of the coil with the dot causes CW motion in
the rotor When S1 opens, energy stored in the left hand coil is transferred to the
right hand coil, A-, via the inductive coupling of the bifilar windings. The
current path is from ground, through bypass diode D2, the right-hand side
resistance and inductance of winding A-, and back to the supply. Current
enters the dotted side of phase A- and thus continues to apply motor torque in
the CW direction as it returns to the supply. Current is controlled by limiting
the motor supply +VSS, which provides just the rated voltage and current for the
60


motor coil, for example five volts at one amp. Steady state current is limited by
the winding resistance R to the rated current, IR = 1 amp. Writing the loop
equation in Figure 3.21 for the left-hand side of the circuit (neglecting the back
EMF term of the motor) yields
Vss = Rii+^1 (3.12)
dt
Solving equation 3.12 gives the time response for the current in the coil after
transistor switch S1 is closed.
it
(3.13)
The time constant for the winding is L/R.
3.5.4.2 L/R Drive
The circuit for the L/R drive in Figure 3.22 is identical to the Voltage drive
circuit, except for the addition of series resistor Rs [45].
61


+K*VSS
Figure 3.22 L/R drive for one phase of a bifilar wound step motor.
The current paths for exciting phase A+ are traced with dashed lines in the
above figure. Switch S2 is open at all times and SI is closed to excite phase
A+. The charge path when SI is closed is from the supply through resistor Rs,
through phase winding A+ and down switch SI to ground. The discharge path
when S1 is opened is from ground through bypass diode D2, phase winding A-
and resistor Rs back to the supply. Series resistor Rs now limits current to the
coil. As Rs is increased in value, the voltage supply, +VSS, must be increased
to K*VSS to provide the same amount of rated steady state current, IR. The
62


value K is termed the overdrive voltage ratio [62] and is typically between five
and ten. Neglecting the back EMF, the time response of the current for the L/R
drive is
^KYgLc.e-^1) (3.14)
The time constant of the winding is now L/(R + Rs). This equation shows that
adding series resistance decreases the rise time of the current compared to the
Voltage drive, but the steady state current remains the same. Also, the effect of
back EMF produced by the motor, which subtracts from the effective voltage
supply (discussed in section 3.6), is reduced by increasing the supply voltage.
The reduced back EMF effect and decreased current rise time for the L/R drive
result in allowing higher operating speeds [62]. However, more power is
consumed as heat by the series resistor, and this inefficiency precludes the use
of L/R drives to low power applications [45].
3.5.4.3 Bilevel Drive
This inefficiency problem is solved in the Bilevel drive, but it requires two
supply voltages (Figure 3.23; [45]).
63


Figure 3.23 Bilevel Motor drive for one phase of a bifilar wound step
motor.
The high power supply, K*VSS, is five to ten times or more the rated
voltage of the coil and is applied to the coil whenever the phase current is first
turned on by closing switches S1 and S2. The current paths for i3 are shown
by the dashed lines. When S1 and S3 are closed, diode D4 is reverse biased
and the current in phase A+ rapidly rises toward the peak value until, after a
timed interval, the high voltage supply is switched off (S2 is opened). Diode
64


D4 now becomes forward biased, and the winding A+ current is maintained by
the low voltage supply, +VSS. The low voltage supply is again chosen so that
Vss/R = rated current. When the next step command is given, the high voltage
supply is switched on to phase B (not pictured in Figure 3.23) and the cycle is
repeated. Neglecting the back EMF effect, the time response for the bilevel
motor drive is given by
i3=^ss VR R (3.15)
t>ti
For optimal performance, the time, tj, at which the voltage is switched from
high to low power must be adjusted according to the time constant of the
winding and the step time of the motor (Leenhouts). Other Bilevel drives do
not use a timed interval to switch the supplies, but use a current sensing resistor
in the grounded leg of the motor coil with a comparator to turn off S2 as soon
as the desired current is obtained. This latter method is used in the commercial
step motor drive demonstrated in the experimental part of this thesis (Anaheim
Part Number MBL 511).
3.5.4.4 Constant Current Drive
The "Chopper" or Current drive overcomes the drawbacks of the other
methods by providing much tighter current control with one high power voltage
65


supply. One such Chopper driver is shown in Figure 3.24, which represents
the circuit design built in the experimental part of this thesis.
Figure 3.24 Chopper or Current drive for one phase of a bifilar wound
step motor.
The Chopper drive gets its name from the fact that switch S1 chops the
voltage supply to the coil on and off rapidly (10 KHz or more) to maintain a
constant current in the coil. Rf is a current sensing resistor used to measure the
phase winding current, which triggers the comparator above a certain peak
level. The comparator output is logically "ANDed" with the switch control
command, effectively chopping S1 on and off to maintain a constant phase
66


current as long as the step input command signal is high. The current paths are
traced by the dashed lines. Whenever S1 turns off, excess energy in coil A+ is
transferred by transformer coupling to coil A- and back to the supply, as
shown. The phase winding R and L, together with the comparator and switch,
form an oscillator which can be adjusted with hysteresis in the comparator
circuit for a chopping frequency above 10 KHz [21]. For a single supply at
several times the rated voltage of the motor phase, the phase current will rise
quickly to its rated current. Neglecting back EMF, the time response for the
Current drive is given by
U = - e i<4Ir (3.16a)
14 = Ir = Maximum Rated Current, i4>lR (3.16b)
The value Rf is typically smaller than the stator winding resistance R, so very
little power is wasted in heating the added series resistance. Equation 3.16a
shows that the current rises rapidly to the peak rated current, IR because of the
overdrive voltage ratio K. Current is then limited to the IR value (Equation
3.16b) by the chopping action of the circuit.
The effect of motor drive on current rise time in the motor winding may be
graphically illustrated by solving equations 3.13 to 3.16 for an actual motor.
For one of the hybrid motors used in the experimental part of this thesis
67


(Superior Electric Part Number M061-FD02), the motor and circuit parameters
for the four drive types are given in Table 3.2.
Table 3.2 Motor and circuit parameters for the Superior Electric M061-
FD02 step motor.
Vss = 5.0 Volts
IR = 1.0 Amps
R = 5.0 Ohms
L= lOmH
K = 6.0
Rs = 25 Ohms
Rf = 1 Ohm
tj = .4 msec
Substituting the values from Table 3.2 into equations 3.13 to 3.16 yields
11 = 1_e-500t (3.17)
12 = 1 -
i3 = 6(1 e500t), t < 0.4 msec
Vcc
13 = Ir = 77^-, t > 0.4 msec
Rm
14 = 6(1 e'600t). U ^ 1 Amp
14 = 1 Amp, i > 1 Amp
The results are plotted in Figure 3.25. The Current and Bilevel motor drives are
nearly tied for the fastest current rise time, and thus the highest motor speed and
torque performance.
(3.18)
(3.19)
(3.20)
68


1.2
0.0 | | i | - -r | i
0 1 2 3 4 5
Time, msec
Voltage
LVR
Bilevel
Current
Figure 3.25 Comparison of current rise time in a step motor winding
using four different motor drives.
Figure 3.26 A, B and C gives manufacturer's data for the same hybrid
motor run on an L/R drive, Bilevel drive and Chopper drive (Rapidsyn Motor
P/N 23E-6108 [53].)
69


40 40
in) GO O in) CO o

N O. 20
o 10 I'O h-
h-
n 0
250 500 750 Speed (SPS) A. L7R Drive 1000 500 2000 3000 Speed (SPS) B. Bilevel Drive 40C
1000 2000 3000 4000
Speed (SPS)
C. Chopper Drive
Figure 3.26 Comparison of the speed-torque curves in a 1.8 hybrid step
motor with different drive types. A) L/R drive; B) Bilevel drive; C) Chopper
drive (Courtesy Rapidsyn [53]).
The vertical axis'of each plot shows the maximum torque the motor will
produce before stalling (pull-out torque). The horizontal axis is the speed of the
motor in steps per second (SPS). For the 1.8 hybrid step motor shown here,
one thousand SPS is 300 RPM. The plots show that for the same motor, the
70


L/R drive is speed limited to less than 1000 SPS, but the Chopper drive
produces usable torque up to to 4000 SPS. Note that the manufacturer's data
does not reflect the mechanical resonance or midrange instability described for
the Voltage drive in Figure 3.13.
71


3.6 Mathematical Model for a Hybrid Step Motor
Several models with various degrees of complexity have been developed
for step motors. Since these models become nonlinear and time variant as the
phases are switched on and off, they are usually solved by numerical integration
techniques. The model developed here is for the two phase hybrid bifilar
wound step motor, which is used in the experimental part of this thesis.
However, the model can also be modified to describe the VR, PM and hybrid
bipolar motors as well. The model described here is based on models by Kuo-
Singh [41, 58] Hair [20] and Taft et al. [62].
The electrical and dynamic equations of the step motor are similar to the
equations for the DC brush motor as derived in section 2.2, but with some
added complexity to describe the nonlinear, switching nature of the step motor.
To describe the electrical characteristics of the two phase bifilar wound step
motor, the motor can be considered to have two pairs of mutually coupled coils.
The electrical analysis is based on a model for the unipolar voltage drive as
shown in Figure 3.27 Steady state current in the motor coils is limited simply
by setting supply voltage Vs equal to ImaxR The motor phases are numbered
in the order in which they are energized for clockwise motion, steps one
through four. Circuit dynamics must be included in any step motor model, as
the type of current limiter used in the motor drive strongly influences the motor
performance [20], as was demonstrated in Figure 3.26. The model can be
72


modified for the other types of current limiting circuits, such as the L/R drive
and the Current drive, as required.
+V
Figure 3.27 Electrical model for the two phase bifilar wound step motor
with unipolar voltage drive, numbered in order for one-phase-on CW
excitation.
The variables for the unipolar drive, bifilar wound hybrid step motor
model given in Figure 3.27 are defined in Table 3.3.
73


Table 3.3 Variables for the unipolar drive step motor in Figure 3.27
V1,2 3 4 = Voltages applied across phases A+, B+, A-, B-.
il 2 3 4 = Current in phases A+, B+, A-, B-.
Ri 2 3 4 = Resistance of phase windings A+, B+, A-, B-.
Li 2 3 4 = Self-inductance of motor coils in phases A+, B+, A-, B-.
M = Mutual inductance betwqeen the bifilar windings of phases A+ and A-
, or B+ and B-.
Egi 2 3 4 = Back EMF induced voltage generated by the moving rotor
magnet in phases A+, B+, A-, B-.
Si 2 3 4 = Transistor switches and bypass diodes for phases A+, B+,
A-, B-.
9 = mechanical angle, or rotor angular displacement in rads.
co = rotor angular velocity in rads/sec.
Nr = number of rotor teeth = 50.
0e = Nrq = electrical angle of motor in rads.
The voltage equations for the phase windings in the circuit of Figure 3.27
may be written as
V, =ilRl+L,4!l~M^-E (3.21)
V2 = i2R2 + L2^2- -E82 dt dt 8 (3.22)
V3 = i3R3 + L3^--M^-Eg3 (3.23)
V4 = i4R4 + U?Â¥- -Eg4 dt dt 8 (3.24)
M represents the mutual inductance between the motor coils, which is
significant only for the bifilar windings, between phases one and three, and
between phases two and four. Some small amount of mutual inductance does
exist between the magnetically isolated coils and could be included [73], but is
74


considered negligible in this model. Mutual inductance and self inductance for
two coils are related by [15]
M = kVTIt^
(3.25)
where k is the coefficient of flux coupling between the two coils. For the bifilar
windings which are closely wrapped around the same stator pole, nearly all the
flux is coupled and the coefficient is close to 1.0. Any flux not coupled in this
way becomes "leakage flux" and may influence the other coils, but is
considered negligible in this model. Assuming equal self and mutual
inductances, we may write
Also, because of the mutual inductance between the coupled coils, we may
write
M = Lj = L/2 = L3 = L4 = L
(3.26a)
(3.26b)
andL^i = -L^
dt dt
Substituting these equalities into equations 3.21 to 3.24, we obtain
(3.27)
V2 =i2R2 + 2L^. _Eg2
V3 = i3R3+2L^- Eg3
(3.28)
(3.29)
75


(3.30)
V4 = LjR4 + 2L^_ Eg4
The terms Egl through Eg4 represent the back EMF induced voltages in
coils one to four generated by the moving rotor magnet. These terms may be
directly measured by driving the step motor at constant speed with another
motor and measuring the induced voltage across each of the open coils of the
step motor. The waveforms measured are sinusoidally shaped and are mutually
offset by p/Nr rads or 1.8 mechanical degrees, as are the static torques curves
shown in Figure 3.8A. As is the case for the DC brush motor (Equation 2.2),
the back EMF term subtracts from the power supply voltage and is proportional
to velocity, (0, but it is also sinusoidal and current dependent at high speeds
[35]. Neglecting the current dependence, we may approximate the induced
voltage terms as [37]
Egi =-KBesin(Nr0) (3.31)
Eg2 = -KB9sin(Nr0 £) = KB0 cos(Nr0) (3.32)
Eg3 = -KB9sin(Nr0 jc) = KB0 sin(Nr0) (3.33)
Eg4 = -KB0sin(Nr0 ^) = -KB0 cos(Nr0) (3.34)
where 0 is the mechanical or rotor angle position in radians and Nr9 represents
the electrical angle in radians. There are Nr or 50 electrical degrees per
mechanical degree, or as many electrical degrees per mechanical degree as there
are rotor teeth. Since there are four steps for each electrical angle, the four
76


induced voltage waveforms are 90 electrical degrees apart, as described in
equations 3.31 through 3.34.
The dynamic equation for the step motor is similar to the DC brush motor
(equation 2.11)
J0 + B0 + TD = Tg (3.35)
where TD is the detent torque of the step motor. This detent torque can be
observed by turning the shaft of a hybrid motor or PM motor by hand and
feeling the cogging or ripple effect. The magnitude KD of the detent torque is
measured by a torque wrench with the motor unenergized, and can be modeled
as a sinusoidal function
Td = KDsin(4Nr0) (3.36)
where 4Nr is the number of steps per revolution.
Tg is the total torque generated by the motor and is the sum of all torques
generated in each phase.
Tg = Ti + T2 + T3 + T4 (3.37)
The individual torques in each phase are proportional to current, as for the
brush motor (equation 2.15), but are also sinusoidal like the induced voltage
terms in equations 3.31 through 3.34. Figure 3.28 represents a typical static
torque curve in a 1.8 hybrid motor with one phase energized. The torque is
77


plotted against rotor displacement in mechanical degrees, 0, and in electrical
degrees, 0e. For the 1.8 hybrid step motor, one electrical cycle of the torque
curve represents 7.2 mechanical degrees of the motor shaft, since 0e = Nr0.
+ Peak 0 180 Nr 0 =360 electrical
Figure 3.28 Static torque curve for one phase of a 1.8 hybrid step
motor.
The actual torque curve may not be perfectly sinusoidal (Figure 3.28), but
equations 3.38 to 3.40 provide a good approximation to the actual torque curves
[35, 42]. Each torque curve is displaced by 90 electrical degrees, or p/(2Nr) =
1.8 mechanical degrees in the hybrid step motor. From the static torque curves
shown in Figure 3.9A we may write
Ta+ = Ti = -KTiisin(Nr0) = -TPKsin(Nr0) (3.38)
Tb+ = T2 = -KTi2sin(Nr0 -1) = KTi2cos(Nr0) = TPKcos(Nr0) (3.39)
78


Ta_ = T3 = -KTi3sin(Nr0 71) = KTi3sin(Nr0) = TPKsin(Nr0) (3.40)
Tb_ = T4 = -KTi4sin(Nr0 ^-) = -KTi4COs(Nr0) = -TPKcos(Nr0) (3.41)
The magnitude KT can be measured with a torque wrench while applying
current to the winding and plotting peak torque, TpK, vs. current.
Note that for two-phases-on excitation shown in Figure 3.9B, equations
3.18 to 3.21 may be combined to form
TA+ + TB+=1.4TPKsin(Nr- (3.42)
TB+ + TA.= 1.4TPKsin(Nr-2ZL) (3.43)
TB+ + TA.= 1.4TPKsin(Nr-5jL) (3.44)
TB+ + TA. = 1.4TPKsin(Nr-2t) (3.45)
These equations show that exciting a pair of phases together increases the peak
static torque by a factor of 1.4 over one-phase-on excitation (Figure 3.9A).
This set of equations cannot be reduced to a single closed form transfer
function as was done for the DC brush motor (Equation 2.16). Numerical
integration techniques may be used to solve equations 3.27 to 3.30 for the
currents, and equations 3.37 to 3.41 for the torques for some small increment in
time, Dt. Finally, the dynamic equation 3.35 can be solved and numerically
integrated to give motor rate and position as a function of time for single and
multiple step responses of the motor. Examples of this procedure for a single-
79


step, single phase step motor with the L/R and current drives is given in the
next section.
80


4. Computer Simulation of the Mathematical Models
Given the electrical and dynamic equations for the DC brush and step
motors developed in the mathematical models of sections 2.2 and 3.6, a
numerical integration routine can be used to solve the equations by computer in
a step by step process to find the rate and postion of the rotor over all time.
Several types of integration routines are available and can be written in Fortran,
BASIC, or C code, or even in a programmable calculator [62]. Two
straightforward integration routines are rectangular and trapezoidal integration.
In this section, the method of rectangular integration are outlined for the
electrical and dynamic equations of the DC brush and step motors. The results
of rectangular integration are compared to results of trapezoidal integration
obtained for the DC brush motor step response. The simulation results for the
one-phase-on step motor single step response are also presented. The
rectangular and trapezoidal integration routines for the DC brush motor and step
motor simulations are written in the BASIC programming language and are
listed with typical outputs in Appendix A.
81


4.1 DC Brush Motor Step Response
To illustrate the method of rectangular integration in a Basic program [14,
30], consider the dynamic equation for the DC brush motor, section 2.2
equation 2.11, which is
J0 + B0 + Tf = Tg (4.1)
From the definition of the derivative, we may write
0 = dQ- or d0 = 0dt
. dt (4.2)
and 0 =4^- or d0 = 0dt
dt
We may approximate the infinitesimally small time period of dt by a finite
interval Dt, which we define as a constant in the computer program. Thus,
starting at time equal to zero and with the initial conditions of rate and position,
we may write
0(Ax) = 0(0) + 0(O)Ax (4.3)
The velocity for all time, ^(nAx)^ can be found at each time interval by
continuing the routine:
0(2Ax) = 0(Ax) + 0(Ax)Ax
0(3At) = 0(2Ax) + 0(2Ax)Ax
: (4.4)
0(nAx) = 0[(n-l)Ax] + 0[(n-l)Ax]Ax
82


This routine is called rectangular integration [62] and develops 0(nAx) in a step
by step process. The position for all time, 0(nAx), may be developed similarly.
To apply the rectangular integration method and simulate the DC brush
motor step response in a BASIC program, we begin with the electrical and
dynamic equations developed in section
2.2:
Vi = Ria + La^f + KE0 (4.5)
J0 + B0 + TF = KTia (4.6)
Assigning Basic variables for the notation in the equations,
Let V = Vj = 1.0 volt, 10 = ia, L = La> II = di^dt
also let PI = 0, P2 = 0, and T = Dt
Rewriting equations 4.5 and 4.6 in the BASIC notation and solving for the
highest order derivative term in each equation yields:
I1=V/L-(R/L)*I0-(KE/L)*P1 (4.7)
P2=(KT/J)*I0-(B/J)*P1 (4.8)
The motor parameters are the same as those used in section 2.3 for the RAE
brush motor:
KE = KE = .03658V/rad/sec; KT = KT = 5.18 oz-in/Amp;
83


J = 0.0025 oz-in-sec2; Tp = 0; B = 0
The BASIC statements which implement the rectangular integration are fairly
simple, and are listed in Table 4.1 and Appendix A.
Table 4.1 BASIC program listing simulating the DC brush motor step
response.
10 REM: BRUSH MOTOR STEP RESPONSE FOR RAE 231096.0
20 LET 10=0:11=0: P2=0: P1=0
30 LET V=l: R=1.7486: L=.004387; KE=.03658: KT=5.18: J=.0025
40 LET T=.0001: TS=2000
50 FOR X=0 TO TS*T STEP T
60 I1=V/L-(R/L)*I0-(KE/L)*P1
70 I0=T*I1+I0
80 P2=(KT/J)*I0-(B/J)*P1
90 P1=T*P2+P1
100 PRINT X,T1,I1,I0,P1
110 NEXT X
120 END
Referring to the above table, the initial conditions are declared in line 20,
and the motor parameters are declared in line 30. The electric and dynamic
equations 4.6 and 4.7 are in lines 60 and 80, respectively. Successive
integrations are performed in the "FOR...NEXT" loop at lines 70 and 90. The
results for time, X; di/dt, II; current, 10; and velocity in rads/sec, PI, are
tabulated by line 100. Output is to the CRT if the "PRINT" statement is used,
or to the printer if "LPRINT" is used. Line 40 defines the sampling interval, T,
and the total time steps to be taken, TS. For T = .001 and TS = 200, the
routine will print out values at every millisecond over a total of 200
84


milliseconds. Not every value may be required to print out for plotting
purposes, so every mth value can be printed out to avoid too many lines of data.
Critical to the accuracy of this method is the value chosen for the time
increment, T. If T is made too small (less than 1 msec), the simulation takes
much longer to run and may require double-precision arithmetic. T must not be
larger than half the L/R time constant of the motor, or 1.25 msec in this
example, to satisfy the Nyquist criteria for sampling and to avoid causing the
solution to go unstable. The velocity results of this integration routine for the
DC brush motor are given in Figure 4.1 by the solid line. Plotted on the same
graph (dotted line) are the results for a trapezoidal integration routine using a
sampling interval of 0.1 msec instead of 1 msec; this routine, also listed in
Appendix A, yields slightly more accurate results.
Rectangular
Integration,
T = 1 msec.
Trapezoidal
Integration,
T = 0.1
msec
Figure 4.1 Computer simulation results for the step response of the DC
brush motor.
85


Note that the plots essentially reproduce the PC MATLAB step response shown
in Figure 2.8 of section 2.3. In addition, the data output for the simple program
in Table 4.1 can be used to provide plots of other system parameters vs. time,
such as di/dt and rotor acceleration, if desired.
4.2 Step Motor Single-Step Response with L/R Drive
For single-phase on operation of the step motor using an L/R drive, the
electrical equations 3.27 to 3.30 in section 3.6 simplify to (with only phase A+
active)
Vi = K*VSS = iiRT +2L^- KB0sin(Nr0) dt (4.9)
V2 = off = 0= i2R2 + 2L^- Eg2 (4.10)
V3 = V4 = off (4.11)
where RT = R + Rs as shown in Figure 3.22 for the L/R drive. For single-
phase on operation, we consider only the single step response for switches S1
closed and switches S2 to S4 open. The rotor is considered at rest at the phase
B- equilibrium point, -1.8, prior to closing SI. Phase windings B+ and B- are
inactive. Phase winding A+ is energized and producing the step response we
are simulating here. Although phase A- is off, current transients in phase A- are
2L^ii-
coupled by transformer action to phase A+. This is accounted for by the dt
term in equation 4.9.
86


The generated motor torque for the single step response with only phase
A+ on is found from equations 3.37 and 3.38
Tg = J0 +B9 + KDsin(4Nr0) = -KTiisin(Nr0) (4.12)
The electrical and dynamic equations for the single-step, phase A+ on step
motor with L/R drive can be rearranged as follows for integration to find the
variables ij, 0, 0, and 0 for all time:

0 = ^iisin(NR0) ^0(Nr0) ^sin(4Nr0)
(4.13)
(4.14)
Here the running torque friction, Tp, is considered negligible; however, this
term could be added in other simulations. A trapezoidal integration algorithm
written in Basic is given in Appendix A. The results obtained for the step
response with an experimentally estimated damping factor of 0.5 are plotted in
Figure 4.2 for two different L/R drives and the same step motor, Superior
Electric M063-FD-326, which is used in the experimental part of this thesis.
The motor parameters are listed in Table 4.2. The first L/R drive, with results
plotted in Figure 4.2A, has K*VSS = 30 volts, RT = 30 W, and Imax = 1.0
amp, and is based on a theoretical design with an overvoltage ratio of 13. The
second L/R drive, with results plotted in Figure 4.2B, has K*VSS = 5 volts, RT
= 11 W, and Imax = 0.45 amp, and is based on a design used in the
experimental part of this thesis.
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Table 4.2 Motor parameters for the Superior Electric M063-FD-326 step
motor [61].
tpk Minimum Holding Torque 100 oz-in
Kt Torque Constant 21.74 oz-in/A
kb Back EMF Constant 0.230 V/rad/sec
kd Minimum Residual Torque 2.5 oz-in
J Motor Inertia 0.00456 oz-in-s:
vss Nominal volts 2.25 V
Imax Rated Amps 4.6A
R Resistance per phase in
L Inductance per phase 1.27 mH
B Damping Constant 0.5 oz-in-s
(Estimated)
Figure 4.2 Computer simulation results of the step response for the Superior
Electric M063 step motor. A) L/R drive, Vss = 30V, RT = 30 Q, Imax =1.0
A, Damping = 0.5; B) L/R drive, Vss = 5V, RT = 11 Q, Imax = 0.45A,
Damping = 0.5.
These plots are typical of the step response for a step motor, as shown
previously in Figure 3.10. The response begins at -1.8, the stable equilibrium
point of the rotor before phase A+ is energized, rises quickly after the phase is
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energized at time zero, and eventually settles about the new stable equilibrium
point at 0. The results demonstrate that driving the motor at a higher current
produces a faster rise time (4 msec vs. 6 msec), but creates more overshoot
(70% at 1 amp vs. 64% at 0.45 amp).
4.3 Step Motor Single-Step Response with Constant Current
Drive
Single-step, single-phase on operation of the step motor using a constant
current drive as shown in Figure 3.24 reduces the general electrical equations
3.27 to 3.30 to the following (with only phase A+ active)
Vi=icontRT (4.15)
V2 = off = 0= i2R2 Eg2 (4.16)
V3 = V4 = off (4.17)
where RT = R + Rp. Current in the coil is controlled at a constant level, icont,
by current feedback, and the di/dt terms may be ignored.
The generated motor torque for the single-step response with only phase
A+ active is found from equations 3.37 and 3.38
Tg = J9 +B0 + KDsin(4Nr9) = -KTiisin(Nr0) (4.18)
The dynamic equation given in equation 3.35 remains unchanged as
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(4.19)
0 = ^LilSin(NR0) £O(NR0) -^sin(4Nr0)
where the running torque friction, TF, is again ignored.
A trapezoidal integration algorithm written in BASIC to solve equation
4.18 for position over time is given in Appendix A. The results obtained for the
step response using the motor parameters listed in Table 4.2 are plotted in
Figure 4.3 for the constant current drive. The current is set at 1.0 A in Figure
4.3A and 0.45 A in Figure 4.3B to compare to the two results for the L/R drive
run at the same current levels in Figure 4.2.
Tims, ms sc. Tims, mssc.
A. Current Drive, I = 1.0A B. Current Drive, I = 0.45A
Figure 4.3 Computer simulation of the step response for the Superior
Electric M063 step motor. A) Current drive, I = 1.0 A, Damping = 0.5; B)
Curent drive, I = 0.45A, Damping = 0.5.
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5. Closed Loop Step Motor Control
Leenhouts [45] makes a valid point that the step motor can already be
considered a closed loop plant without the addition of external sensing devices.
The detent torque provides position feedback to the motor torque, which holds a
step position at standstill and helps dampen overshoot when the motor is
advanced one step. The back EMF is a velocity dependent term which also
provides feedback damping. In many applications, therefore, open loop control
of the step motor is satisfactory. Start/stop rates or ramped velocity profiles can
be applied to the motor to avoid mechanical resonance and provide reasonably
accurate position control. However, when speed or positional accuracies are
critical, a closed loop scheme can be utilized to prevent loss of motor
synchronism and ensure that motor steps commanded and motor steps actually
taken are equal. Several such methods exist [1,3, 36,45,47, 66] and five will
be described here. Three of the methods are investigated in the experimental part
of this thesis. Each method requires feedback from a shaft encoder,
tachometer, or from current in the phase winding.
1) Step Confirmation
2) Control by Switching Angle
3) Control by Torque Angle
4) Velocity Feedback for Damping
5) Control as a Two Phase Brushless Motor
In step confirmation, an encoder is mounted on the output shaft and
increments a counter each time a step is taken. The counter tally is compared to
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