AD-RI32 643 CSMP (CONTINUOUS SYSTEM MODSELING PROGRAM) MODELING OF 1/1 IRUSHLESS DC MOTORS(U) NAVAL POSTGRADUATE SCHOOL MONTEREY CA S M THOMAS SEP 84 UNCLASSIFIED F/G 19/2 M
AD-RI32 643 CSMP (CONTINUOUS SYSTEM MODSELING PROGRAM) MODELING OF 1/1IRUSHLESS DC MOTORS(U) NAVAL POSTGRADUATE SCHOOLMONTEREY CA S M THOMAS SEP 84
UNCLASSIFIED F/G 19/2 M
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NAVAL POSTGRADUATE SCHOOLMonterey, California
•S
THETISCSMP MODEL{ING OFBRUSHLESS DC MOTORS
by DTICSteven M. Thomas S R 9Uii!
September 1984
A-J
Thesis Advisor: A. Gerba
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1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER
4I- -1Y4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED*1- Master's Thesis;
CSMP Modelling of Brushless september 1984Septmber1984DC Motors
6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(*) S. CONTRACT OR GRANT NUMBER()
Steven M. Thomas
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK
AREA & WORK UNIT NUMBERS
Naval Postgraduate SchoolMonterey, California 93943
.II CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
Naval Postgraduate School September 1984
13. NUMBER OF PAGESMonterey, California 93943
8888
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Brushless DC Motors
.I .
* 20. ABSTRACT (Continue on reveres side If neceeeary end Identify by block number)
-Recent improvements in rare earth magnets have made it possibleto construct strong, lightweight, high horsepower DC motors.This has occasioned a reassessment of electromechanical actuatorsas alternatives to comparable pneumatic and hydraulic systemsfor use in flight control actuators for tactical missiles. Thisthesis develops a low-order mathematical model for the simulationand analysis of brushless DC motor performance. This model isimplemented in CSMP lanquage. It is used to predict such motor
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;>performance curves as speed, current and power versus torque.Electronic commutation based on Hall effect sensor positionalfeedback is simulated. Steady state motor behavior isstudies under both constant and variable air gap fluxconditions. The variable flux takes two different forms.In the first case, the flux is varied as a simple sinusoid.In the second case, the flux is varied as the sum of asinusoid and one of its harmonics.
D
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Approved for public release; distribution unlimit I.
CSMP Modelling ofPrushless DC Motors
by
Steven M. ThomasLieutemant, United States iay
B.A., Univer sity of Delaware, 1976
Submitted in partial fu'Lfillment of the
requirements fcr the degree of
MASTER OF SCIINCE IN ELECTRICAL ENGINEERING
from the
NAVAI POSTGRADUATE SCHOOL
September 1984
Author: ~2I7z
Approved by: Lt
Inc isA aV --f-
Elect al and Computer Engineering
Lean of Science anEngineering
3
ABS7RACT
Recent improvements in rare earth magnets have made it
possible to construct strong, lightweight, high horsepower
DC motors. This has cccasioned a reassessment of electrome-
chanical actuators as alternatives to comparable pneuratic
and hydraulic systems for use in flight control actuatorsfor tactical missiles. This thesis develops a lcw-crder
mathematical model for the simulation and analysis of brush-
less DC motor performance. The model is implemented in CSN.1P
language. It is used to predict such motor performance
curves as speed, current and power versus torque.
Electronic commutation based on Hall effect senscr Fosi-
tional feedback is sioulated. Steady state motor behavior
is studied under both constant and variable air gap flux
conditions. The variable flux takes two different forms.
In the first case, the flux is varied as a simple sinusoid.
In the second case, the flux is varied as the sum of a sinu-
soid and one of its harmonics.
4
......................................
............................... "........................................
IABLE OF CONTENTS
I INTRODUCTION.................. 9
II. NATURE OF THE PROBLEM ..... .............. 12.
A. PROGRAMMING LANGUAGE .... ............. 12
B. SYSTEM BLCCK DIAGRAM .... ............. 12
C. DEVELOPMENT OF MOTOS SYSTEM EQUATIONS . ... 13
III. CURVE PREDICTICN ....... ................. 19
A. OVERVIEW ........ ................... 19
B. SPEED VS ICRQUE CURVE .... ............ 20
C. CURRENT VS TORQUE CURVE. ... ........... 22
D. OUTPUT POWER VS TORQUE CURVE ......... 22
E. MOTOR REVERSAL ...... ................ 27
IV. ElECTRONIC COEMUTATION ..... .............. 28
A. SWITCHING ACTION ................ 28B. HALL EFFECI SENSOR FEEDBACK ... ......... 30
C. MODEL REVISION o.. . ........ 35
V. VARIABLE FLUX ....... .................. o.36
A. AIR GAP FIUX ....... ................. 36
B. FLUX AS AN AVERAGE VALUE ........... 37
C. SINUSOIDAL FLUX ..................... 38
D. HARMONIC FLUX . ................ 41
Vi. SUMMARY ...... ..................... 47
A. REMARKS ANr CONCLUSIONS ..... ........... 47
B. RECOMMENDATIONS FOR FUTURE STUDY ... ....... 48
APPENrIX A: LISTING CF MODEL PROGRAMS ... ......... 49
A. BASIC PROTCTYPE MODEL .... ............ 49
5
J" .C
B. REVISION CKE ................. 52
C. REVISION T0... . ... . ................. 59
D. REVISION 7HREE . ................ 63
E. REVISION FCUR ...... ................ 69
APPENrIX B: SAMPLE OUTPUT ............... 7
lIST CF FEFERENCES ........ ................... 86
BIBLICGRAPHY .................... ... 87
INITIAL DISTRIBUTION 1IST ...... ................ 88
I6
ii
"
Ir" "
6 °
II
.~ . . .~ ..- * ~t*. -C C C < C .2. .2 * 2. ~ 2t .2 ffi *..
LIST OF TABLES
I. Typical Commercial Motor Parameters ....... 19II. Sensor and Switching Logic .... ............ 3
III. Torque and Speed Ripple Due to Sinusoidal Flux 41
IV. Tcrque and Speed Ripple Due to Harmonic Flux . . 44
.. ..
-h ..' ..i. ; . .i ., ... .; " ; ; : -' ---' " -".G --* ---- .-- i ' i i ' i ' ' -i ' 2 ' ' ' ' ' '' .-.- ' ': -i .' ' ' -' ' :' " -' -" " -.' .' .' i . ..-.- -.
A LIST OF FIGURES
2.1 Eguivalent rC Motor Circuit............14
2.2 Block Diagram of DC Motor ............ 18
3.1 Motor Speed vs Torgue Curve.............23
3.2 Family of S~eed-Torglue Curves.......... .44
3.3 Mctor Current vs Torgue Curve..........2 5
3.4 Output Power vs Torgue Curve ............. 26
4.1 Ccntroller Ccnfiguration. ............ 28
4.2 Switching Logic of a 3-Phase Brushless DC
Motor..........................29
4.3 Equivalent Circuit Pith Electronic
Commutation ........................ 314.4 The Hdali Effect...................33
4.5 Sensor Logic Based on a Hall Effect SeLsor . 314
5.1 Composite Flux Variation for Two Windings . 39
5.2 Back EIIF Waveform Due to Harmonic Flux .. .... 43
8
I. INTRCDUCTION
Direct current electric motors are gaining wider use in
space applications where long life, higa reliability, high
torque and light weight are critical factors. In recent
years, advances in magnet technology and materials, espe-
cially the use of rare earth magnets, have made possitle
significant imprcvements in the torque to inertia ratios of
permanent magnet (PM) motors. Samarium cobalt is cne suchrare earth magnetic material. The great potential of the
rare earth magnets derive from their inherent high flux
density and high coercivity properties. Higher flux densi-
ties mean greater developed motor torque uhile nign coer-
civity means greater innate resistance to demagnetization
which permits thinner magnet sections. The two factors in
combination result in increased mechanical power and energy
product with decreased physical size and weiht.
Brushless dc motors belong to this class of permanent
magnet motors and thus enjoy their improved torque to size
characteristics. In fact, brushless dc motors are signifi-
cantly smaller than either their ac counterparts or conven-
tional brush-type dc motors on an equal horsepower basis.
Compared to its conventional counterpart, a brushless dc
motor'S structure is "inside out." In the brushless config-uration, the permanent magnets are attached to the rotcr and
the ccnducting coils are placed in the stator. As its name
suggests, brushes have been eliminated from this type of
motor. Instead, ccmmutation of current in the stationaryr
armature is accomplished electronically by switching cr the
appropriate windings via transistors specially designed to
provide the high currents needed for motor power. In crder
to sequence the current to the coils, the commutation
circuitry is provided rotor position feedback from sensing
9
.° .. . . . . . ..
T- .7
devices, most commonly from Hall effect sensors. ThesE ar2
simple semiconductor devices that develop a pclarize .
voltage depending upon the magnetic field passing through
them. Small, highly sensitive and reliable, Hall effect
senscrs require little power to operate and thus are widely
used.
Erushless dc motors possess all the advantages tradi-
tionally associated %ith conventional dc motors such as
linear torque-speed characteristics, excellent response
times and high efficiency. In addition, there are a numlerof important advantages that brushless dc motors pcssess
over standard brush-type dc motors. An obvious advantage is
that electronic commutation means the elimination of commu-
tators, the wear and tear of brushes, and the build up of
carbon dust which heretofore significantly limited the life
of the standard dc motor. Since electronic current
switching essentially involves no component wear, therk
results improved reliability and enhanced lifespan. As well
as being subject to wear and tear, the action of brushes
sliding over commutators also generates radio frequency
interference (RFI) which is a severe liability in many
"applications. In fact, it is often in hostile and explosive
environments that the advantages of dc motors are most
needed. Another benefit is in heat transfer characteris-tics. By placing windings in the stator the thermal path to
the environment is made shorter and more direct. "ith the
removal of heat thus enhanced, mechanical and electrical
malfunctions due to heat are reduced. A further important
advantage of the brushless dc motor is in the area of motor
speed variability. Vhereas ac motors usually operate at one
speed fixed by the powerline frequency, any dc motor's speed
can he changed to meet varying power requirements by siml yadjusting the dc input voltage. The electronic commutation
scheme of the brushless dc motor lends itself nicely to just
10
this type of variable speed control. It is a relativz.l
simple matter to attach to the commutation circuit tcar a
small, inexpensive microprocessor whose logic can b e
programmed to run the motor at any of several speeds or in
either direction. A single motor can thus be programmel, in
place if desired, so that its sieed, direction anI tcriue
are matcled to a particular application.
There are at least as many applications for irushless
dc motors as there are for brush-type motor systems aE well
as a hcst of mew applications. Of particulir relevance to
-~ this thesis is the use of brushless dc motors in il. -i
control actuator sytems of Navy tactical missiles, inparticular, the cruise missile. There are three general
types of actuators that are in use in missile systems: pneu-
matic, hydraulic and electromechanical. The maneuvering
requirements of the mcre advanced tactical missiles call for
high contrcl torques which heretofore could only be produce!
by high Fressure pneumatic or hydraulic actuators. Both of
these actuator systems are more susceptible to mecaanical
malfunction than their electrical counterparts. With the
improved torque to inertia ratios resulting from better rare
earth magnet materials and the higher mechanical reliability
due to removal of brushes, as well as other advantages
previously enumerated, the brushless dc motor is in a Eosi-
tion to compete with most pneumatic and hydraulic systems.
This thesis is but one part of a detailed study of the state
of the art in electrcmechanical actuators as applied tc the
cruise missile system (Ref. 1].
11
" --h-__ .." .L'.- .".' '._'.t% '.,-'.'%'-'.7_q'.t'.' .'_........................................................................-'............. -'.........._ .'_, -'.'
II. _VATURE OF THE PROBLEM
A. EBOGEANMING LANGUAGE
The objective of this thesis is to create a simplifiel
low-order mathematical model that will accurately simulate
the response of a biushless dc motor. The IBN Continuous
System Modeling Program (CSMP) was chosen for use since it
is a user-oriented computer language specifically created
for modeling dynamic physical systems and is one of the most
widely used in this area [Ref. 2]. It has the flexibilityof determining the response of a system that is modeled
either in a block diagram format or as a set of ordinary
differential equations. While CS.1P is an offshoot of IBM{'s
Digital Simulation language, it embodies Fortran as its
source language and retains much of Fortran's capability and
flexibility. CSMP is an applications-oriented prcgrao -
intended for use with the IBM mainframe systems. Theutility and ease of using this program language stems from
simplified program control statements that almost exactly
describe the mathematical equations or physical variables of
the system and from the flexibility of its program structure
which contains preprogrammed function blocks that obviate
the need for complicated subroutine programming. In
essence, CSMP was chosen because it permits the user to
concentrate on the details of the physical system rather
than cn the time-consuming complexities of programming.
E. SISTER BLOCK DIAGEAR
he block diagram format was chosen as the basis for
the CS.P program since it is a convenient way of analyzing
the input-output relationships of a physical system and the
12
flow of signals through it without having to refer to the
system equations. 7he basic idea of a block diagram stems
from the application of Laplace transforms to a system'sintegro-differential or differential equations. In effect,
a differential equaticn is rendered into a simpler algetraicexpression which, in turn, becomes a fundamental equaticn of -. -,
a system's functional blocks. Each block indicates the
relationship between its input driving signal and output
response; ie., the transfer function of the input ani
output. The transfer function is defined as the ratio of
the Laplace transform of the output to the transform of the
input with all initial conditions assumed to be zerc. Ablock's transfer function does not include any information
about the internal structure of that part of a system,
merely the transformation of a signal between two pcints in
a system. Furthermore, it should be borne in mind that all
physical systems have certain transfer characteristics that
are actually nonlinear to some extent. A Ic motor system
can be fairly accurately described using the transfer furc-
tion approach to modeling and, when nonlinear transfer char-
acteristics exist, CSMP modeling can usually be obtained
from preprogrammed ncnlinear function routines.
C. DEVEICPMENT OF MCIOR SYSTEM EQUATIONS
In crder to construct the block diagram of a system,
one approach is to identify the various mathematical rela-
tionships between the components of the system and to write
balance equations that will identify each individual block
comprising the system. The system equations for a brushless
dc motor are presented in chapter three where all the
details of the logic control and switching of transistors
for ccmmutation of the motor are developed. 7hat follows is
a simplified input-output characteristics model of the type
13
that has been a standard for brush-type dc moto:s usei in
contrcl system studies [Ref. 3]. The development of thi.-input-output model is the first learring step in the
modeling process for the brushless dc motor and has t Ii P
additional usefulness of being a tool for a concurrent
mechanical engineering study cf load torque roquiremetts for
the motor [Ref. 1].
To write the electrical balance equation describin; a
lasic dc motor, the classical loo. method based on
Kirchoff's voltage law is applied to the equivalent motor
circuit (Figure 2. 1).
R
o L/V
Le,-- +
0 u, - TL
Figure 2.1 Equivalent DC Motor Circuit
1.
E :-..:-::.:. ':...._-..i_,; & .;_i..i).::- :2 .: ::2 :::.i_"i> :::-::-:.-::::::: ::: :::::: .: :2 :: , _
The basic dc motor has the coil windings on the armature and
the magnetic field scurce on the stator. In the case of the
brushless dc motor, the rotor consists of permanent magnets
that produce the magnetic field and the stator contains the
coil windings. The Kirchoff Law for the stator circuit is
es(t) = Ldi/dt + Ri(t) + eb(t) (ecjr 2.1)
where R and L are the stator resistance and inductancerespectively, and eb It) is the back emf. When a conductor
moves in a magnetic field, or there is any relative mction
between the magnetic field source and the conductor, a
voltage is generated across the terminals of the conductor.
In the case of the dc motor, the voltage is proFortional to
the shaft velocity and tends to oppose the current flow.
The relationship between the back emf and the shaft velccity
is
es(t) = (Km*f)*wm (t) = Kb*wm (t) (eqn 2.2)
es (t) = Kb*dm (t)/dt (e-n 2.3)
Substituting and rearranging equations 2.1 and 2.2 into
their differential forms gives the following two equations.
di/dt= (1/L) *es (t) - (F/L) *i (t) - (i/L) *Kb*wm (t) (eqn 2.4)
dem (t)/dt = wm(t) (eqn 2.5)
" .'The basic balance equation that describes the motor as
* . a mechanical system derives from application of Newton's
laws of motion to the system components. The dynamic equa-
tion for a motor coupled to its load is
tm(t) = J*dwmt)/dt + B*wm(t) + tl(t) (egn 2.6)
15
•. ... ...... -.. .. • . .. .. .: - ... .. . .. . .. .- . . . . .- . . .. . .- . . .. .
where tl(t) is the load torque. Rearranging the equation it
tecomes
dwm (t) /dt= (1/J) tm (t)- (1/J) *tl (t) - (3/J) *wm (t) (eqn 2.7)
F is the total viscous friction coefficient of the mctor and
is comprised of the sum of the viscous friction due to the
motor, Bin, and the load, Bi, as seen by the motor shaft.
Likewise, J is the total system inertia comprisea of the su1
of the motor inertia, Jm, and of the load inertia, J1,
reflected through the coupling device to the motor shaft.
Eoth Bi and 31 are functions of the motor's speed rezucin3
mechanism where N is the speed reduction ratio (N>1). Ihe
following equations summarize the foregoing.
B = Bm + B1 (egn 2.8)
J = m J 31 (ein 2.9)
B1 = Blp/N (eqn 2.10)
*i Jl = Jlp/N (en 2. 11)
Jlp and Elp are the inertia and viscous friction seen at the
load side of the system. Since a dc motor is essentially a
torque transducer that converts electrical energy into
mechanical energy, the following equation is necessary to
show the relationshir amoung the developed torque, tm(t),
the air gap flux, #, and the stator current, i(t).
tm(t) = (Km* ) *i(t) (eqn 2. 12)
Because the magnetic field in the motor is assumed uniform
and constant, this can be simplified to
16
. . . . . . . . .. .. .. . . . . . ...... . ..... .
' -. + + ,. .,- . . , -.- . - s- .* --, v o -+ . - _. - - . . .
tm (t) = Kt*i(t) (egn 2. 13)
where Kt is the torque constant of the motor.
Equations 2.1 through 2.13 represent the cause and
effect eguations of the system. The application of es (t)
produces a current flow which causes the torgue and the hack ._
emf tc be generated. The torque produced then causes theangular displacement em(t). Given the differential eiua-
tions of the system, the Laplace transform can he applied
and the block diagram of the system -enerated. As an
example, the block for equation 2.1 is generated as follows
es (t) - eb(t) = L*di/dt + R*i(t) (eqn 2.14)
Zfes(t) - eb(t)) =X[L*di/dt + R*i(t)) (eqn 2.15)
En(s) Es(s) - Eb (s) = (Ls + R)*I (s) (eqn 2.16)
I(s)/En(s) = Is/(Es- Eb) = 1/(Ls + R) (eqn 2.17)
Performing the same operation to the remaining differential
equations, the block diagram in Figure 2.2 results.
It is important to note that while a dc motor is hasi-
cally an open-loop system, the back emf of this type of dc
motor acts as a natural feedback loop that tends to improve
the stability of the motor.
Cnce the basic block diagram is developed, writing a
CSMP program is simple and fairly straightforward. Care
must be taken, however, to put the block transfer functions
into the proper format for CS[+P's functional blocks. A copy
of the CSMP program for a basic dc motor is provided in
Appendix A.
17
LOAD 1OPQUE (IlANGULAR
D1SPLACEM4EN1
1+ 0 sL +P t
BACK EMFMOR
A SPEED
Figure 2. 2 Block Diagram of DC Motor
S.
III. CURVE PREDICTION
A. OVERVIEW
Cne of the purrcses of this model is to produce a set
of performance curves that accurately simulate a givenmotor's behavior when operated under varying load ccndi-
tions. As such the model can be a useful tool for studying
the changing performance due to the configuration changes of
a motor under develoEment or for observin~g the behavior of a
motor under varying conditions of application or environ-
ment. Thus, having developed an input-output computer model
for the DC motor, the next step was to assign values tc the
model parameters and run the program. Program inputs,
constants and parameters were assigned values that are
typical of brushless DC motors commercially available.
Table I provides the parameter values for a given motor that
will he used as a generic motor for the following analysis
and discussion.
TABLE I
Typical Ccmmercial Motor Parameters
Stator Resistance, F 2.74 ohmsStator Inductance L 0.0016 henriesTorque constant, At 15.9 oz-in/amFBacR EMF constant, Kb 0.112 volt/rad/sRotor Inertia, Jm 0.001 oz-in/s 2Viscous Friction
Coefficient, Em unknown oz-in/rad/s
19
19
-. --- .ii-' ....*- .. .. a.. .a...~..+.? .i . .....a...~i.... ''2t..-.' ".i' "a...a.a'...'-.-a-. il.''."i. '-.i2L 2L -'-ILI. . ~ '.ii.i
B. SPEED VS TORQUE CURVE
Specifically, three curves were produced: motor speed
vs. generate! torque, current vs. torque and output power
vs. tortiue. Since permanent magnet DC motors have linear
characteristics, a straight-line curve resulte. under all
load conditions when the parameters of Table I were inserted
into the model and a voltage of 30 VDC was applied. In
order to fully understand the individual contributioLs Of
the various motor parameters on the speed-torque curve, it
was necessary to determine the mathematical relaticnshipsbetween the parameters. Feferring to e'uations 2.1 and 2.5,
the electrical and motor eauations when considered under
steady state conditions become
es(t) = F*i(t) + Kb*wm(t) (egn 3.1)
tm(t) = Kt*i(t) (eqn 3.2)
Solving for i(t) and combining the eluations results in
es (t) = (R/Kt) *tm (t) + Kb*wm (t) (eqn 3.3)
Vhen there is no torque applied to the motor, the current is
small encugh to neglect and the no-load velocity becomes
wnl = es/Kb (egn 3.4)
Given that the applied voltage is a constant, it is apparent
that knowledge of tle value of the back eif constant, Kb,
will prcvide the various no-load speeds for any value of
applied voltage. On the other hand, in the absence of any
known motor specifications, the no-load speed of a given
motor can be empirically measured and then the back emf
constant, Kb, can be mathematically derived.
20
Si
L * * - ~ * * ~ ~ * ; ; . . . - ;. ,"...
Looking next at the slope of the speed-torque curve, it
can be seen that whereas one endpoint of the curve termi-nates at no-load speed, the other ends at stall which is
that value of torque which is large enough to literally stop
the motor. From equations 3. 1 and 3.2, with wm set to zero
the generated torque, tm (t), at stall is ts given by
ts = (Kt/R)*es (eqn 3.5)
Combining equations 3.4 and 3.5, the equation of a straight
line for the speed-torque curve results in
wm(t) = wnl - (Rm*tm(t)) (eqn 3.6)
where
Rm R/(Kb*Kt) (eqn 3.7)
and Bm is the slope factor for the curve, also called the
speed regulation constant. obviously, changing any or all
of the parameters will alter the sloje factor, Rm. However,
the back ezf constant, Kb, and the torque constant, Kt, are
strictly related by the common factor, aiz gap flux, 4. Infact, when both constants are put in the MKS system of
units, Kb in volt/rad/s and Kt in Nm/amp, then they are
equal as equation 3.8 shows.
Kb(volt/rad/s) = Kt (Nm/amp) (eqn 3.8)
Thus, kncwledge of either constant provides knowledge of the
other. Referring to equation 3.7, it can be seen that the
final unknowns are the resistance, R, and the slope factor,
Rm. If the resistance, R, is known from manufacturer speci-
fication sheets, for example, then the slope, Rm, can be
21
easily sclved. On the other hand, if siven an empirically
measured speed-torque curve, it is now obvious that the
curve not only provides Kb which in turn provides Kt, but it
also can provide a derived value for motor resistance, R.
Once acquired, the three basic iarametei.a, Kb, Ft and R,were Flugged into the CSIP model and the speed-torque curve
for a 30 VDC input was generated (see Figure 3.1).
This same procedure can be used to produce a family of
speed-torque curves if it is desired to study a motor's
performance under different input voltages. Figure 3.2
shows the family of curves produced when the values given in
Table I were entered into the model.
C. CURRENT VS TORQUE CURVE
With the speed-tcrque curve understood, the next task
was to repeat the same general procedre for the motor
current vs. torgue. As a starting point, the equation for
the steady-state crrent is given by
iss(t) = (es(t) - Kb*wm(t))/E (egn 3.9)
Since Kb and R were fixed by the speed-torque curve, gener-
ating a current-torque curve was straightforward procedure
for any applied voltage (see Figure 3.3).
D. OUTPUT POWER VS TCRQUE CURVE
The final performance curve to be studied was the
output pcwer vs. torgue curve. The basic equation for power
is given by
P(t) = tm(t)*wm(t) (eqn 3.10)
P(t) = Kt*i(t)*wm(t) (eln 3. 11)
22
-I
I 1- _ 1171"DT
Y.
- I
0 32 ( 1 96 128 IGO"
TO'OUQUE (OZ-IN)
Figure 3.1 Motor Speed vs Torque Curve
where pcwer is measured in watts. Since this eiuat io.n
involves no new constants, 1rolucinj tLe powcr curve .as
only a matter of adding a line to the existing modEl wi.ichalso included a conversion factor for chingin g the tcr,.:lie
units frcm oz-in/amp to Nm/am F in order for the powec to be
23
Sr
N 110
0 32 64 96 128 1G0 192TORQUE (OZ-IN)
Figure 3.2 Family of Speed-Torgue Curves
dimensionally correct in watts. Ajain running the p-cjr-m
with the Farameters given in Table I, the model 1,roluced thE
Fower curve given in Figure 3.a-
. ~ ~~~ . . . .. . . .. . . . .
S(ITEQ,< l: >:-Ix> i
Figure 3.3 Motor Current vs Torque Curve
As Expected both the speed-torjue and cirrent-tor ue
curves were linear. The power-tcr~ue curve is obviously not
linear, especially at hijh icals where the power -Ejir.s to
roll cff fairly sharply. From the molellin; pcint of view
this is due solely tc the fact that the motor sicws .cwn
* . faster than the load torque increases (3e e,- uatior. 3.9)
25
0 . . . - - .. . -.-. •.-.. .. - '. .-.. .'." . . . ." '. -% ...... . .".' . -'-', ... .... ... • .. '.' .. -.-.. '
OlT I,, T II L)VEr V1 N [()rQ( ., (11 t -N',
N
/
~/
C. /
0 32 64 96 128 160
TORQUE (OZ-IN)
Figure 3.4 Output Power vs Torque Curve
Another factor that further accentuates this power rolloff
in an actual motor is the effect due to armature reaction.
Whenever a current flows through the moto armature in a
permanent magnet dc mctor, the arnature heco.es an electro-
magnet which produces a flux that tends to opposE the f.lux
Froduced by the Fermanent magnets. This has the effect of
26
• .. '-.,,. .. ,.,..........-...,.,,--..-........-......,......... ................. ....... ... ..-. ..-. .,
partially demagnetizing the permanent magnets. This
demagnetization is a reversible effect, meaning that as
current returns to zero the permanent magnets returr. t:
their full strength. However, as the current increases and
armature reaction sets in, it has the effect of decreasing
the torque constant, Kt (as well as Kb). As equation 3..9
indicates, this also contributes to the output power.rollcff
at high loads. Though current --ermanent magnet ZC motors
made of rare earth magnets have high coercivity that rezists
armature reaction, at high levels of rated current a small
amount of armature reaction still occurs. For the parroses
of this thesis, the model is considered to accurately siau-
late motor behavior fcr loads up to the peak power load tor
the given applied vcltage. From that load upward, it is
assumed that armature reaction is likely to occur and result
in nonlinear behavior which is not included in this model.
E. MCTOB REVERSAL
The final step in the modeling procedure was to reversethe motor's directior and ensure that mirror images of the.
three curves resulted. To do this, it was first necessary
to replace the positive input voltage with a negative ccint-
erpart. Next, in order to maintain model consistercy whErE
the load torque, tl(t), is treated as a tor.ue that opose-
the motion of the motor, it was necessary to modify the
model so that the load torque was a pcsitive value. This
consisted of creating a CSIP procedure block that identifie2
the input voltage as either positive or negative and then
treated the load torgue accordingly. Once program lugs were
removed, the model accurately simulated motor reversal with
proper speed, current and power values for the aipro~riate
load conditions.
27
0'
i [[ , - . . . . . ' .- . . ,. " -' . - . ° .' ... . ° , . °,*.° .,. . ° . . ° ° J ° " . . "o o . ° "°: , . , . - °. i ,
IV. ELECTRONIC COMMUTATION
A. SWITCHING ACTION
The next major step in this thesis was to convert the
1hasic model of a standard brush-type dc motor into a three-
phase, four-pole brushless dc motor system. Figure 4.1
shows this system with the three-phase stator windings
configured in a standard 'star' connection with each winding
oriented 120 electrical degrees from the others.
F. V.1, do 2 ,, d3
-! 0S
45 4i 5
Figure 4. 1 Controller Configuration
The six transistors are connected to the ends of each stator
leg and through logic-controlled switching action Frcvide
. three-phase full-wave motor ccntrol. Figure 4.2 indicates
the switching logic sequence for counterclockwise rotation.
~ To produce clockwise rotation, the seluence is reversed.
In either case, a pair of transistors will be switched at
the start of each 30 degree (mechanical) interval which
causes current to simultaneously build up in one leg, flow
28
-. ..................................................
_________-____.____.____________,-___.. . . ._ -- -. - -.
0 6O 120 Nmo 240 J00 IbO 60
0 1
0 Z
03
04 ,
as
06o1 a0 OF_ F
____ ___ ___ ___ ___ ___ ___ - Q J
Figure 1.2 Switching Logic of a 3-Phase Brushless CC Motor
steadily in a second, and decay to zero -.ri a third. Given
proper sequencing, the net developed toriie id'.ally zEacte3
a steady state value that causes iotor rotItion at a
constant speed in the desired direction.
For windings configured in a star arran;',ent, it canbe seen that conduction is ccntinuois in orn leg wLile
commutation occurs in the other two les. For exawir, As
Qi and Q5 are energized between 30 and bO ]ej:ecs (ncEcian-
ical), current flows down leg A ani into leg B. At th tpsA;time, the current through leg C due to the energy stz-rj.,
behavior of an inductor decays to zero thro)ijh dl(;e, ,
(see Figure 4.1). In the next sequence, 60 to 90 /zes,C6 is energized and C5 switched off. Current has rac ] -
steady state flow thicugh leg A, but now it flows dcun :eJ
and the current in leg B decays to zero t-rough D2. In the
next sequence, 60 to 120 degrees, Q2 is switched on ai.d ,I
switched off. Current has reached steady state flow through
leg C but now current builds up and flows from leg B wiile
the current in leg A decays through leg C. This sane action
29
0
repeats in subsequent commutation states. Looking at roint
C in Figure 4.1, it can be seen that, since current is
always continuous in one leg whether flowing into or from
node D, Kirchoff's Current Law reguires that the net current
flowing in the other two legs must etlual the f:ow in the
continuous leg. This permits the following important
simplification which was needed later in the modelin;
process. Namely, the three-legged stator can be approxi-
mated by juist two windings serially configured. One leg
will always accurately reflect steady state flow conditions
while the other will be treated as if it were in stead"
state because the build up in one leg is balanced hy decay
in the other. It is recognized that this model is a first
approximation and that future modeles will repuire addi-
tional refinements in order to adeguately represent the
effect of switching transients on the transistor ani diode
elements.
E. HILL EFFECT SENSCE FEEDBACK
As was mentioned in the introduction, the switchin. - ,
action of the commutation circuitry is based upon rotor
position feedback from Hall effect sensors. i: order to
represent electronic switching and postional feedback, as
well as the three phase winding configuration, it was neces-
sdac tc modify the block diagram of Figure 2.1 to that in
Figure 4.3
Comparing Figures 2.1 and 4.3, it is seen that the
chief difference is in the switching logic block and the
addition of two more transfer function blocks to account for
the additional windirgs. To better understand the logic
relationship between the sensors and the current switches,
the truth table for both counterclockwise and clockwise
rotation is presented in Table II.
30
- .
°' .
RPI
0 ________
Figure L~ ~~PP3 Eguaen Cici Wit Eetoiaomuain
frE, th E oet foc giT by Ii xr B (wer h
L 31
.. .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. .................. .. .. .. .
TABLE II
Sensor and Switching Logic
BPS A RPS B RPS C PH A PH B PH C
1 1 0 high low oj'en1 0 0 high open low1 0 1 open high cw0 0 1 low high open0 1 1 low open higho 1 0 open low h i g
COUNTERCIOCKW ISE
0 0 1 low high oer.1 0 1 open high lcw1 0 0 high open low1 1 0 high low open0 1 0 open low h i0 I 1 low open hiah
CLOCKWISE
current, 1 is the length of the conductor and 1 is the
magnetic field) causes the current to bend and electrons to
pile up on cne side and positive charges, or holes, to lo
the same on the other side. Thus a transverse Hall pcten-
tial difference, Vh, develops across the conductor (see -
Figure 4.i4).
Figure 4.5 shows that i;. the presence of a sinuscical
magnetic fiell produced by a rotating magnet, such as cne of
the pole-pairs of a multiple-pole motor, the induced Hall
voltage takes on an ideal square wave shape.
The logic circuitry of the sensor is configured in such
a way as to output a logic level 1 for positive Hall volt-
ages and a 0 for negative voltages. Briefly reviewing the
action of the motor: as the shaft turns, the rotor position
sensors send this information to the switching logic block
which then decides which phases are to be made high, lcw or
open in accordance with Table iI. Ejuivalently, the
voltage, En, is then applied to the two appropriate windings
32
"- " "
-- t _-S A... ... . . . . . . . ..- - - - -° . ..
'. T,
E LECTRO FLO -VTH NO %IAG', TmC F E LD
I ,A G4E TICF -f LDC T OF4. PZGE
ATTEMPTED ELECTRON F LOW IN A M. GNE TIC FIELD
* V
HALL + TI = Ii; I+ MAGj£ TIC
.l OIT OF* PAGE
ACTUAL CONOI TIOS OF CURRENT FLOW IN A MAGN TIC FIEL"
Figure 4.4 The Hall Effect
and zero voltage is applied to the third winling in agrEe-
ment with the assumFtion that current is steady in one leg
while the commutation in the other two legs is etuivalent to
a continuous current in )ne leg. As before, each voltagethen Froduces a current. Because this is assumed to bc a
33
E ~HAL..-4-
V"
Iii Fi -___ ___ __
Figure 4.5 Sensor Logic Based on a Hall Effect Senscr
linear system, the principle of superposition is applicatle
and, thus, the torques developed by current flow through
separate windings car be considered independently and then
summed to give a total motor torque. Beyond this pcint, the
trushless motor compcnents are ilentical to its brush-type
counterpart and behave similarly.
34
IZ
................................ . . .
C. MCDEI REVISION
CSIP modeling of this version of the rushless .I: xotoc
called for two modifications to the brush-type model.
Essentially, a CSME procedure function was constructed to
act as a subroutine that embodies the logic of the switching
action. The logic recognizes that if the mofgr shaft is
within scme degree rarge, for example 30 t. 60 degrees, then
it sets the appropriate rotor position sensors and eiergi2es
the switches for the appropriate two windings and dtener-
gizes the switch for the third. In effect, only the forcing
voltages are passed to the main program, whereas switch an2!
sensor pcsitions are available for output (see ApFendix A).
3i
35
- -,-~~~~~~~~~... ..... ,...,.......,.........°.,......-..........-..-.. ... ....... ....... ,.., _-"._. .. i . ..
V. VARIABLE FLUX
A. AIR GAP FLUX
[ The steady state behavior of both a brush-type and ahrushless dc motor has been studied under the fundamental
assumpticn that the hack emf generated is a product of tueshaft velocity times a constant, Kb (see Equation 2.2).likewise, the motor torque is equated to the product of the
current times a constant, Kt (see Equation 2.12). But as
Equation 2.11 reveals in the case of the torque constant,
Kt, the torque is in fact the product of some other constant
cf propcrticnality, Ktl, times the air ga? flux, 4. lhesame is true for the tack emf Constant, Kb; that is, Kb is
the product of some proportionality constant, Kbl, times the
air gap flux, *. In both cases, the flux, , has beenassumed tc be constant which is in agreement with the
general practice when modeling permanent magnet dc motors.
In point of fact, the flux is not constant but rather isdistributed sinusoidally, generally being the sum of a sinu-soid and one or more of its harmonics. How the motor is - -
physically constructed determines this flux distrituticn.
Two of the chief factors are the structure of the magneticpole faces and, most importantly, how the windings are
distributed in the armature. The latter ircludes such
factors as the number of slots, the number of coils Fer slotand the spacing of the slots (Ref. 5]. The careful desijnerdistrihutes the windings so as to minimize the effects o'
harmonics. Regardless of their constituents, these
composite sinusoids have an average value, of course, anJ so
in practice it is this average value that hecccs the
constant flux term, t.
36
- -. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
B. FLUX AS AN AVERAGE VALUE
A logical next step in the design of a more realistic
model was the deteruination of an average value fz:r the
varitle flux, Oavg, and the correspondi.ng proortionality
constant, KK1 such that the ratio of avg and K'X1 ejuals cneand the relationships of equations 2.2 an2 2.5 arepreserved. The equations for back emf, eb (t), and develo:-ed
torgue, tm(t), now become
eb(t) = Kbp * wm(t) (ein 5.1)
tm(t) = Ktp * i(t) (egn 5.2)
where the back emf ccnstant, Kbp, and the tor4ue constant,
Ktp, are now expressed as
Kbp = Kb * ( avg/KF1) (egn 5.3)
Ktp = Kt * ($avg/KK1) (egn 5.4)
Assuming as a first approximation that the air gap flux
varies as a simple sinusoid of unit magnitude, the first
step toward finding the total system flux, lavg, was to
determine the relaticnship of the individual flux patterns
in each of the three phase windings. Analysis of a -four-pole magnet rotating under three windings separated by 120degrees (electrical) revealed the following relationships
a= sin(2e + T/6) (eqn 5.5)
Ob = sin(29 + 9fr/6) (eqn 5.6)
c= sin (29 5IT/6) (eqn 5.7)
37
where a, OD and tc are the flux coltrilutions off t,.
respective phases. Peturninj to an earlier issumpticn t:.at
the circuit can be aiproximated as two windin3 3i se :s,
ignoring thie third, then the average )lix Car alsc be
approximated from the sum of the flux lev.!opel itr. two wl.-
ings at a time. This approximation is furtner sU-C ZtEd by
the fact that the flux in the decay le has, in fact, an
average value of zero during every comiautation interval.
From thE switching sequence of Figure . 1, the two
conducting legs were identified for each inteval and, froo
Equations 5.5 through 5.7, their respective flux values
calculated and summed. For every interval, the resulting
flux waveshape was identical and had an average value of
4avg = 1.631 (see Figure 5.1).Since KK1 merely normalizes $avg, its value was easily
determined to be KK1 = 0.613. The appropriate substitutions
were then made in the CSMP model (see Appendix A for
Revision Two). Since this amounted to nothing more than a
rearrangement of corstants, the moiel's output behavior
remained unchanged. At this stage, the model simulates tne
behavior of a brushless dc motor being electronically ccmzu-
tated and exhibiting constant, steady state developed tcr:ue
and motor speed due to a fixed, average value air gaE f! ,.
C. SINUSOIDAL FLUX
The final stage of this thesis was to examine the
steady state behavioL of the trushless Ic motor unler vari-
able air gap flux conditions. As a first apecoxindticn, the
flux was modeled as a simple sinusoid -n' the flux relaticn-
ships are those given in euatious 5.5 through 5.7 As
before, only the flux relatinj to tne two conlucting legz
was considered. This composite variable -lux, tvl, is sIc4"r-
in Figure 5.1 Equations 5.3 and 5.4 had to be modified to
account for the different value oz flax.
380>i i
FLUX VARIATION0
--
-r - I I T i T 1 -0 30 80 90 120 150 160 210 240 270 300 330 360
ANGULAR ROTATION (DEGREES)
Figure 5.1 Composite Flux Variation for Two Windings
Kbpp = Kb * ( vl/KK2) (eqn 5.3)
KtFp = Kt * ( vl/KK2) (egn 5.9)
Recall that the objective was to keep the values of KbpF and
Ktpp as near as possible to those of Kb and Kt, respec-tively. In order to roughly normalize 4vl, the constant
value was determined as KK2 = 1.655. Since vl was now avariable quantity, it should be recognized that Kbpf ani
Ktpp were no longer strictly constants and were better
expressed as the follcwing functions.
- KbpF = Kb(ovl) (eqn 5. 10)
039
,,..i:-.:- :. ::. :,>:.._-:- -;....................................................................................."....."...... "" "" '
Ktpp = Kt( vl) (egn 5.11)
The equations for back emf, eb (t), and developed tcrgue,
tm (t), now became
eb(t, vl) = Kbpp * wm(t) (egn 5. 12)
tm(t, vl) = Ktpp * i(t) (eqn 5. 13)
Because the total flux was no longer a constant,
average value, it was expected that variation in developed
torque (and, consequently, in motor speed) would result.
Furthermcre, as the system block diagram of Figure 2.1 indi-
cates, torque is ccnverted by the motor transter function
into a rotational speed. This is a process that is eguiva-
lent to inertia filtering of the tor, ue ripple that results
from variable flux. For the purposes of this thesis, rip-leis defined as the ratio of the amount of variation from the
maximum to the maximum itself. Having made the apFropriate
substitutions in the model (see Appendix A for Revision
Three), the program was run under no-load conditions. Itwas ctserved that tbere was, in fact, ripple in both the
developed torque and motor speed and that some inertia
filtering did occur. The next logical step was to lcad the
motor and study the effect on torque and speed rippe.
Table III indicates that under no-load conditions where
speed was highest, the percentage of ripple in the output
speed was at a minimum but that torque ripple was at a
maximum.
It can be seen that as the load increased the amount of
ripple in the developed torque decreased from about 847 to
8!. A closer exazination of the torque behavior undervarying loads revealed that, even as the motor torlue neces-
sarily grew with increasing loads, in every case there was
40
TABLE III
Torque and Speed Ripple Due to Sinusoidal Flux
MOTCR MOTOR RIPPLElOAD (oz-in) SPEED (F.R.1) TORQUE SPEED
0.0 2557 33.9 1.132.0 2087 21.1 1.6E4.0 1617 12.2 2.696.0 1147 7.8 4.6
approximately a constant 8-9 oz-in variation. This varia-
tion can be attributEd primarily to the flux variaticn. in
Ktpp and to a lesser degree to the variation in i(t) caused
ty the variation in Lack emf. Thus the magnitude of to-ue
variation remained about the same for all loads though the
percentage ripple a,rears to indicate otherwise. In the
case cf motor speed, the amount of ripple appeared to
increase slightly with increasing load and thus decreasing
speed. In this instance, the magnitude of variation did
increase a small amount with decreasing speed - from 2P rprn
at no-load to 52 rpm at 96 oz-ins - and suggests that speed
ripple, at least, is somewhat dependent on load. This is
due in part to the fact that as the motor slows, there is
less rotor momentum and thus less inertia filtering. Inaddition, at slower speeds there are less frictional effects
which at higher speeds also tend to filter out ripple. in
general, it makes intuitive sense that at higher speeds
fre.uency variations are harder to discern than at slower
speeds.
D. HARBCNIC FLUX
Having observed the effects of simple sinusoidal flux
variation, the next step was to replace the single sinusoid
with a flux composed of the sum of a sinusoid and its
harmonics which is mere representative of the air gap flux
41
* . * ~ * - * .- ;.- .. * *.-- * * *.
patterns in an actual motor. The back ea- of a typicaicommercial brushless dc motor is given in Figure 5.2 wherethe voltage between two terminals was measured with the
motor rotating at 1200 rpm.
A harmonics analysis of this waveshape shows that the
principle harmonic was the fifth and that the waveshapE was
closely described by the expression
3.0 sin(e) + 0.59 sin(5e) (eqn 5.14)
when substituted into eguations 5.5 thru 5.7, and the
different angular speed taken into account, the fcllcwir.
relationships for flux result
4a 3.0*sin(2e8+r/6) + 0.59*sin(108E+51r/6) (ein 5. 15)
b= 3.0*sin(28+9Tr/6) + 0.59*sin(10e+9 r/6) (egn 5.16)
Oc = 3.0*sin(2*+51/) + 0.59*sin(108+7r'6) (egn 5. 17)
The same procedures used in the simple sinusoid model
were then applied to the harmonics case where the composite
flux changes to 4v2, a flux factor that varied to a slightly
greater legree than did vl. Equations 5.8 and 5.9 had to
te mcdified as follows
Kbppp Kb * ( v2/.K3) (egn 5. 13)
Ktppp = Kt * ( V2/KF3) (eqn 5. 19)
-where KK3 approximately normalizes the flux and had the
value KK3 = 5.38. As before, Kbppp and Ktppp varied with
the flux and so the new equations for back emf and developed
" .torque became
42
.. .
BACK EMF k9VEFoRII
7-.
LO0 30.0 Ic .* £50.0 Xfl0 20.0 3XO.0 3i.0 W.0
Figure 5.2 Back EMF Waveform Due to Harmonic Flux
eb (t, 'v2) =Kb (Ov 2) *wm (t) (eqn 5. 20)
eb(t,ov2) =Kbppp * m(t) (egn 5. 21)
.............................................
tm (t, v2) = Kt (4v2) * i (t) (egn 5. 22)
tm(t, v2) = Ktppp * i(t) (eqn 5. 23)
The appropriate substitutions were once more made in the
model (see Appendix A for Revision Four) and the mo~e. was
run under various loads. As anticipated, the variable flux,
v2, produced ripple in the developed torque and rotor
speed. Table IV shcws the resultant values for various
loads.
TABLE IV
Torque and Speed Ripple Due to Harmonic Flux
MOTCR MOTOR % RIPPLElOAD (oz-in) SPEED (rpm) TORQUE SPED
0.0 2557 94.6 2.432.0 2087 38.5 3.764.0 1617 22.5 5.696.0 1147 14.7 9.8
Comparing Tables III and IV, it can be seen that the values
for torque ani speed ripple are of similar magnitudes. The
somewhat higher values for torque are due to the fact that
in the Ov2 case there was a 16-21 Oz-in variation as oFjosed
to an average 8-9 cz-in variation in the'ovl case. 1he
speed ripple behavior was much like the 01 case with
proportionate increases resulting from the increased torque
ripple. Thus, for both torque and speed ripple due to 4v2,
- the model performed as expected; that is, slightly higher
[ values of ripple occurred which were clearly attributable to
the increased variability of the flux pattern of ccmpcsite
sinusoidal harmonics.
An important observation made during runs of the vl
model came as a consequence of the motor speed variability,
44
particularly when the motor was unloaded. Since no-load
speed was a function of Kb as given in eluation 3.!, the
variable Kbpp had the effect of increasing the upper-ed
speed beyond the 2557 rpm baseline given in Figure 3.1 This
increased magnitude ccupled with the swing of speed values
created an undesirable effect in the unloaded conditior. In
effect, at certain points in time the additional speel
produced back emf values that exceeded the input voltage
which in turn resulted in negative current and developed
torque values. Though these were transient negative valuesand had small effect on the steady state oieration of the
motor, still their presence was obviously unrealistic. !he
soluticn to returning the motor to all positive values of
current and torque was to increase the friction filtering
effect by adjustment of the viscous friction coefficient,
Bin. It will be recalled from Chapter Three that a value for
the Ba of the commercial motor being modelled was not avail-
able and that for the basic model a value for Bm was derived
from the curve fitting process. Thus, it seemed reasorable
and permissable to repeat the procedure here.
Since it makes sense that increasinj the friction within
a system will slow the system down, in this case increasing
the value of Bm did just that. By increasing Em from
0.00015 to 0. 045 oz-in/rad/s, the upper-end of the motor
speed was reduced to a value that eliminated the undesirable
negative values. Shis caused the no-load speed to he
lowered slightly belcw the desired value of 2557 rpms.
Recalling that no-lcad speed is fixed by the back emf
constant, Kb, this parameter was adjusted from 0.1120 to
0.1089 vclt/rad/s to bring the back up to speed. Since this
adjustment was less than 3% and the manufacturer specifica-
tions allowed a 10 +/- measurement error margin, this
adjustment to Kb was peroissable. Of course, a propor-
tionate adjustment to the torque constant, Kt, was also made
45
....................................................... *.°2.
since Kt and Kb must maintain a constant relationship as
explained in Chapter Three. The effect on motor sp~e-
ripple was neglible. Reviewing the Ov2 model at this Eoint
revealed that similar behavior in the no-load state occurred
but on a larger scale due to the greater variability of v2.
A siiilar set of adjustments was necessary to eliminate the
negative current and torque values in the unloaded condi-
tion. It required an adjustment of Bm from 0.00015 to 0.045
oz-in/rad/s in order to ensure that all values were posi-
tive. In addition, the back eaf constant, Kb, was adjustedto 0.1192 volt/rad/s and a proportionate adjustment to Kt
was made. Again, all adjustments were within tolerances.
Of course, it was earlier established in Equation 3.7
that changes to Kb and Kt %ould change the slope of the
speed-torque curve. Thus, in both cases the slope was
returned to its original value by adjusting the motor resis-
tance, P. In the case of sinusoidal variable flux, F was
reduced from 2.74 to 2.70 ohms. For harmonic variable flux,
R was increased from 2.74 to 3.0 ohms. Again, both adjust-
ments were within the 10% +/- allowable deviation. It was
also cbserved that since both Kb and Kt no longer remaiLed
constant but rather varied within small ranges, the slope
was not constant and varied accordingly in both the vl and
4v2 models with the greatest change occurring when Kt and .5t
simultaneously reached the minimum values of their respec-
tive ranges. Bearing in mind that these deviations in slope
are fairly transient, their average values are most impcr-
tant and are the aFproximate values from which the final
adjustments to R were based.
46
. .* .. . .- . ..
'A VI. SUMMARY
A. REMARKS AND CONCLUSIONS
CSMP language is a convenient tool for modeling a
trushless DC mQtor. Positional sensor feedback and
switching logic being functions of time, the action cf eiec-tronic commutation can be nicely simulated in CS!P. The
model can be fairly easily modified in order to study cther
motor configurations. For example, the three-phase star
configuration used in this thesis could easily be changed to
a grounded neutral, point D in figure 4.1, ani the same
types of analysis performed. In future studies where the
complexity of the mcdel is increased in order to include
more design detail, the advantage of CSMP will beccme even
more apparent.
In the course of simulating the effects of different
air gap flux variaticns, it appeared necessary to make minor
parameter adjustments in order to offset the feedbac
effects cf torque and speed ripple. Within the scope ot
this thesis, adjusting the steady state performance was
reasonable and justifiable. In a larger context, however,
various control systems, particularly tor ue generation
controllers, are specifically designed to reduce tcrzue
ripple and its effects. So, in at least one sense, thes.
adjustments were somewhat artificial. To the extent that
parameter adjustment is necessary and desirable, it must be
understood that this is not a simple procedure since adjust-ment of one parameter almost always requires aijustment of
one cr more others. These adjustments in turn necessitate
readjustment of the criginal parameter. The process is thusan iterative one. This model permits this kind of parareter
47
.'.'. . . -. .. .. .. . .-... ... . . .. . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . .
adjustment, but it is a long and tedi3us process that would
he better performed hy some sort of optimization algorit:,x.
B. RECCMIENDATIONS FCR FUTURE STUDY
The simulation cf ripple effects is particulary rele-
vant to the control engineer who must design the control
circuitry and power ccnditioner to meet specific performance
requirements. This points to several areas for futurea
study. One area is the control of torque generation and the
several principle configurations used in brushless DC
motors. Two commonly used methods are the sinusoidal torque
generation principle and the trapezoidal torque scheme.
[Ref. 5]. In addition to torque generation control, asecond area for investigation is power control of brushless
DC motors. In this thesis, output power was unregulated an!
was studied simply as a function of a constant inut
voltage. In practice, however, power is produced anr
controlled by varying the supply voltage. 7arious scheies
exist for power control; amoung these are linear transistor
control and pulse-width or pulse-frequency contrcl.
[Ref. 5].
A very important area that this thesis did not address
was that of the switching transients that occur as a conse-
quence of commutation. During each switcning interval, each
winding is an inductor that stores energy which causes a
significant amount of current to flow at the instant of
switching. This can have serious effects, especially if
breakdown conditions exist amoung the commutation transis-
tors and other contrcl circuit elements. In this same area,
switching transients also occur within the control circuitry
itself. Thus, a thorough investigation and understanding of
the behavior of power transistors, diodes and other control
elements used with electronic commutation needs to be done.
4~8
~~~~~~~~~~~~~~~~~~~~. ..... ......-...-... ". .....- -.... ".......................... ............. .................. ............ .... ,- -..
APPENDIX A
LISIING OF MODEL PROGRAMS
A. BASIC PROTOTYPE MODEL
The program in this section is a basic program that
simulates the action of a standard brush-type DC motcr. :t
is the prototype from which the brushless DC --ator model an
its several revisions are derived.
49
:;492
.................................................................................
C3Q
LI-Iv-* o2
11 LA 0 or- --c
if I-- (% 0 I - l- <V)I A I uJ -.dtJ Li w.. wI LuJ I -
-. 1 ~ ~ L .1.l - kn z ii- -J9 9-4 LU CD z -
L- L u-c CLi Li UL.)XI Lj .4 -e- -~) W
U.)- -. ca ca I Q u I -j,C)) X.O V4- ) I- U.n c0 wJ-w
Q. cC/)LU o-.- Z~UL. WI. .. w>
V). CL-4 *c. 11*J W" F a-
0- LnU n- 0
Au -j
LU -
00
crfz Lux
V) LUWu
LLJ 0
04
V) LU
4 0.0 z
LU
-. wujI OX
* - LU -q -4 00z I- Li9Ui
0.o W - .- i L. 0732
zA x- r- 0 n~M I--Z xc- Li W-JI n
LC' x -,% W0 W10 -1, I- I-LQC) 0 LI, Li"C 0 LI- U
B. REVISION ONE
lhis is the first revision of the basic prototy;e. 7his
model simulates the switching logic of a brushless ZC Tofo7
which is based on feedback from Hall effect sensors. [n
addition, the windings are treated independently and their
contributions to developed torque are superposed. Since the
superpcsition results in twice as much current flow as a
single lumped coil, the total current is halved to retain
the same overall motcr behavior as in the prototype.
52. I
.- A -3-uc
u4/I-. .1- *.
Q v) -- u Z La -" L
1-- 3 iU I-. LJ.L ._j u CL
-X 13 - I-0QILL.~J Z . - :D~ Q*CL .JZc W:D4(j "A - LiO
>. Zu N I-- ID-x
V% r, - N LU LUQ0 cc
ZOY- -4:0 Q ~ cr mx c . LA.* j(jCU(36 z 11 C00ZW W00C
I/ LAvtuILU.Luw L/1. -j c-4 IL..I 1.11 Lu P-I.SILJZUL Lill- Ac V) C) X.JL)>- t..j -e
JLL(= . LU -- ) n -zLU U .LL L z 1'.- 1
- AZZLUO1Z LL < O'- U i- I-XJLJ.0 0-4--JI-- 7./ 0 LLL -) V *jLU'Qt.-)
u -) < LU )- 11 HNU It LLJ X I -it
V) LIj -. 1
* ~uJ
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LUL/
LU
-
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LUx LU U. F-
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0..I -1 1 L
-~~~L Uj F0- L -4m LUl Or) %AQ.J
CZ- 0x4 c.. - 0 a 0. IO)4 LU7
I- 14-w O'- *-XM3jJ LU .--- 1.- LU 99 LU 0LI) -J, 1~--C) LL MQ"- c u.(j cr JLJ I"9 .-* 0 LL U. LL *-j .O*P-ujcL 1.i -qLj Q6~j LLL9 0
0 uCJ--.4 z. (. "U. 1* LU0 z *LAL W .U. 409 * .IILLULUW9-Z4. 99 w . LL . L .4 ~ -I (x.9 CJ -W9 to 11A iz-l LU 99to I-.0 It -4LJ a J " w. P4..
99 rn) 99X if L% 0.z Z - t- wU C . .jI = ci Lt t-U 99 9999 9 It .A Ll:
W" o- .4 CN4
W(~U)4)(2 4c
H-") Q"4-4 QJLZILUU) z~ z
W-4JI-4 - -
MLn-. IfCL4 A -1
OCLU -9-,
.-001.--4 XX
l'(Ffi.)I '4~ A) 0.J(n LMLA IU4WL tJ0~x
U.1 LA.U
LL = F- JOOO)--; LJ "41:3LOLO3
LU Z OWl-* S*I*U)U*JUVI Z 0N 00000
2(J-.JQWW AC (n4(N~CaL
~~'4 t'd .
'0 1 0 0
o-. In A.) -40 Cii4( M It Ln 1 1 tH it -4
56
ifIu
-Ta. -4
0- -0
QUW X L)
-ju Ln
A. 0 .6. LU
(.j if UJrCjo.Li -4LUVI
Z. W,-41--LJ~ 0 L.MLM
m--so),rno < C,:.- -- j LUUJ r -C-C C - ALJ OU Z-JwzxLULL (a co
20. ci
C. REVISION TWO
In the following program, the motoL's air yap flux is
treated as the average value of the sam of the flux acting
in two windings as explained in Chapter Five. For ccr.ven-
ience of study, both windings are again lumped into a sirile
winding as in the prototype. This facilitates a closer
focussing on the effects of variable flux on motor Lehavicr.
58
. . . .- .-
. . .. . .. . ." °,'.'°, o '- o° p'o.' .'" .'-.. . . . . . . . . . . . . . . . . . .. . . . .-.. . . . . . . . . . . . . . . . . .,° °" i
01
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*U I. L) C:) (a
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I. -4 i LJLI )(4%CZ W I-. 1- -s% :3WZL.JWX z 0 00 .Jcic - ZZ Z!D -rn :)U.x L
I.- - WU. -10 1 Q
I.-iI W Z3-
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L3)
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LU 4%
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- ~ CICV)Z U.
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1-4 - . -CO w > _I. 0W l(.4 MNI . Ul- .L 0J im. 0."-
U.g % X9 * s -, c go 4u-0 nLL. A -416 C\J -o 4 LL CL-4 ac Zk U.1 0I
- 41 ZUCOU * X (. a u/.1.. -. ac'3r~o- -4;1 Q-L 4 .EO A~ -U -9-p I.-4- .A 1 LI)
*of 9 1.- .1, .J- 0 >I-W -- I-OI.-o 04' % O- 0 j 0 qL 0.J =%.Jr W
-j U.. -J . L X: N z - (. U0 0U-( ZA~ gg.n LU U.1 =OaU. .""
LL~ LLJI.
I- LU 16Q
XU-I.J zLn" = LL 0WUi)WJZ X i
LL 0 wm >IL
1,-C.) Cf .r-: ) 'U3L),c
Liur-O1 ) JJ(OJvi L LI
UJ-4ZccZ -9
V) OL 000000
L. JiL'LU LU -.T Uqwv(NtLn()
-LJLL - ] 4-4
I-weeC xUJQ co< (A
C4-4t- ->ZZzzzz
LLI'JISc(nM- XJJ- =a;
CUW. V) 4
0(ze uuV)Z= Wil' 5J 0 5
cr ac I.. I ./ 0~ Go * 0VI (U * 0
*j- - L LS V)J 9 W f.UUI.CILUW Lr J...
--- LLLU /)£CLL L1./lW U.1 0 * 0 9 9000 3 *:)C otnw'J(A
U
0.m
0-
a.
1.-~~~ ~ 1- --)--u -ZI-.U L In
ona.
Ln LC% lo U .)LI 0.1
620
D. REVISION THREE
The purpose of this version of the program is treat thie
air Gap flux as varying in simple sinusoidal fashion. 7he
total system flux is treated as the algebraic sum of the
flux developed in the two active windings.
I.
A6
U. U.~~Ulf
ULJLAJ )
C 4 -j -4 0 )06 Z U6 '..LL z(3Z)(" C X.- - c000 ii o ' Q CLU
-i ~ or 0 x A A LQ) >611.L -4 N 0LU UJ J U)U
L *.-- 0 0i LU L.).
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Ul V'4 )J. (4"N CJ 1UU-); c L. ( C0
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ke -0 , -)U.
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4 - X 9 a -ai InJL- -
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68
- .. *.~* * . . ... .- %A
E. IREVISICE FOUR
This version of the model treats the flux as varying
according to the sum cf a sinusoid of fundamental freguency
and its fifth harmonic as explained in Chapter Five. Ihe
total flux is again ajproximated as the algebraic sue cf the
flux deveJlojed in twc windings at a time.
69
0
U.LLJ j_
0. WL
AV U, - '4
V) U.)L -.
.2DZZ 0 *U 1 W0-1-3J Z0 Q-cj:Ln
lip . -J
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L-j
2: z
(36 C6 6 C4 3z w
444 LU LU
LAULUWJ LU p-1
LL.W
X-
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Q.~4-4 UlN gVV~I9 () u-~'JI .
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OL mm-c qfq ~~ -4 -4 CD*0 N-4 N% 9--0 0 ;l ~-I- ZZ U..1
0 0< WNZ LULUWI 0 C3 - (n.4JL L)
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LU UJLAzz wULLU w r L ~
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-ire APPENDIX B
SAMPLE OUTPUT
The following is a sample of the type of infcrmation
the model can provide. In this particular case, the fourth
revision of the basic prototype, the harmonic flux case, was
used. For completeness, the motor was run at loads from 0.0up through 96.0 oz-in, which was the range of lcads with
which all analysis was performed. In addition to such motor
variables as speed, current and power, a sampling, of the
feedback from Hall effect sensors as well as the switching
action of the power transistors is included.
0
75
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A lIST OF BEFERENCES
1. Fiscal Year 1S83 NAVAIR Strike Wartfare Technci ogvPlan ,Advanced M1issi'e Control Deies bE y R.Dett g7-S7Tmb;9f-7F2 -----
2. Spekhart, F. H. and Green, A. 1., A Guide to Usin~CSMP - The Continuous S stem oe'i-iP o a"015,
3. Kuo, B. C., Automatic Control Systems,' 4th ed.,Prentice-Hall I -rm.
4i. Fitzgerald, A. F. and Kingsley, Jr., Charles ElectriciNachinery, McGraw-Hill Bock Company, Inc., 15"
5. D.C. Motors, Speed Contra is, Servo Systems, AnELQ neerin HanBd-k, TER- e'ff7 -7TfocraTE
86
j .BIBLICGRAPHY
Demer dash, N.A., Miller, P.H., Nenl, T.W. "ComparisonBetween Features and Performance Characteristics of FifteenHP Samarium Cobalt and Ferrite Based brushless DC Ictcrs j.Operated by the Same Power Conditioner," IEEE Transactionson Pcwer stenM _paratus and Systeis, v. P -Ii-- -ry
Demerdash, N.A. and Nehl.T.W., Dynamic Modeling cf BrushlessDC Motor-Power Conditioner U if-- fo -- Iec tro me-EniiXit u1 ?XfZ--Inc p--1cat i- paper Tesef*d ----- 77--3ef'EI -fV6-Eic9-=p~fa s Conference, San riego, Califcrria, 9June 1979.
Miller, 7.3., "Rare Earth Magnets Contribute to Small Size,High Torque of New Motor Designs," Control Enaineerin_, May,19 3.
Murugesan, S., "An Overview of Electric Motors for SpaceAplications " IEEE Transactions on Industrial Electrcnicsan Ccntrol fnst -ien!a---f 7v.TET-2u.- Vr Br7-'9T.-
National Aeronautics and Space Administration REpcrt -1cr-16034, Numerical Simulation of Dynamics of Brushless DCMotors for e-ros ac -anU- -r-A_-f o, -
Naticnal Aeronautics and Space Administration Reorttm-8045, Analytical Modelin of the Dynamics of ErushlessDC Motors T~ rX- osace A1iE i - C.-T
T~~ainDX N7~.AU emer das ... aIn1TE-.-18 August, 1976.
Cppenheimer, M "In IC Form, Hall-Effect Devices Can TakeSany New Ap icaticns, lectron__s, August 2, 1971. .*.
Society of Automotive Engineers, Inc. Technica.l PaperSeries, #780581 Electromechanical Actuator TechnclcgyPRogqra., by J.T. fdge.-7 -IT9-----_
87
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INI71AL DISTRIBUTION LIST
No. Ccpies
1. Defense Technical information Center 2Cameron StationAlexandria, Virginia 22314
2. Litrary, Code 0142 2Naval Postgraduate SchoolMonterey, C liforria 93943
3. Departmnent Chairman, Code 62 1Departmfent of Electrical EngineeringNaval Postgraduate SchoolMonterey, C aliforria 93943
4. Professor Alex Gerla, Jr. Code 62Gz 2Department of Electrical ng2.neeringNaval Postgraduate SchoolMonterey, Califorria 93943
5. Professor George J. Thaler, Code 62Tr 1Department of Electrical Enineerin-gN1onterey, Califorria 93943
6. Naval Weapons Center, China Lake 2Weapcns Power Systems BranchCode 3275
China Lake, California 93555
7. LTI Steven M Thomas 1998 Leahy RoadMonterey, CA 93940
88
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