AD-RI32 643 CSMP (CONTINUOUS SYSTEM MODSELING PROGRAM) MODELING OF 1/1IRUSHLESS DC MOTORS(U) NAVAL POSTGRADUATE SCHOOLMONTEREY CA S M THOMAS SEP 84

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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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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

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Naval Postgraduate SchoolMonterey, California 93943

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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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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

Ix

LUL/

LU

-

Id)X

LUx LU U. F-

-1

LliLV)< U

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

OQ L -4 .

*U I. L) C:) (a

-Z u. $!4 z"OZZ -4~ 4d < 0z

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-

Ao -rzLLJ

L3)

I- 'U"J

LU 4%

I-LO

> Lur

LU L

C9J

LWq

iU

=Q9

LU LDL

- ~ CICV)Z U.

-4- -- 40 0 Z-- Za - 1 "" .. A)

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.).

Lu)q 0, 0 w L - I ..-I 7- Q 0.m 1C~ *n '0M %-.

-Jt C0 0 N O z t~ w ) L OLLU.4L. .--J 4 O tw ~ 00 0)

II" " I0 9 -im ..-4 -1

U) C L -4 i.J 9 Z-uw u - .fW QQ MZ

Ul V'4 )J. (4"N CJ 1UU-); c L. ( C0

* ~ ~ L< I. f ofL-. I- UOzxn ~~U 0%

ke -0 , -)U.

LU X

z~ wz

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LU

LL0

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L.J

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4 - X 9 a -ai InJL- -

-o< MN4 -W4 0 C A"u 0

..-4OU4 Q tc~ X4 Li 4i -j LU cc -9

".w x 4 -a 0 04-1-.0L'0 4- -C LUr- uj ZZ .Lj"."..m&"L"j (x C3 C0 0 -U LJ

* .1 01I--- 11 * j W..-C .cp L.-I -

0 0 cN. W WO-I4 0DQ- 0Z 0. "U- W 0- -71L$. -Uto SA x Y. *J *i Z- i "IA z ~ -I

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t.-4J UL LA. /I - U - 00 J0(.V) :)f O uef m ULn.. U.;

cxIf .4 1 Lu L.JL _*I~z .4 %M10 L UUIMJ. W Ljl CZ -J .1W LUt-i * -- .J L It 1 it to 11 L) 11 *1 *1 if to 11 11 '0 11 L) 11 11 1I- 1) LU MLd-J

Lu"

-Jj -6

z 9- Do- M

9- -4 F-0.LU 0 Q LU9

SU. C4U

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-C, L LJ X -

LLI Z

i. -J -JuJu4 L/)

0 0 0 00 00 9c) 00 0 0 0 4LUL ju-Q'

I- l- - L 'J 0 - l '

-4V4("4N~rtQ Zk LU Q .J

LC%~ C30 0. LL. <

LL zw -C

%:~u

0.- t- P- A GL

U- -U Q 00 0.

X.I99 0 rn-L C)-JJ-..~ CL

xZ Z Az.z Za.Co 41 QU(.LJWU U

Li-LU (.j %. em %L cu )JLi * ; 81 *. OIA U.. U -L. U- U..I)L-

!,-D ZZZZZZW wi wU wU w wU- .- 4 ~ ~ a1,cg4 Ur

_j- 0090 90

.ovi -qCOCC a. CU Q11 -4-4-4U. U. U. U. U. t. kJ 1ICNUJr- R . a- 3c X a. a. i LLjuj

a j - U UJJa LU Lu CU LU LL, %J-" ILe a9 0 0 0 0 else* 91-(flLLJow

11 uItC 1o cIfc1 o1 coDc LflZ-.4WetD M

U.LLL.. ULLLL .LL. n.~ - c ~ -M. '00 WU I-

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

l- UJ4;

L-j

2: z

(36 C6 6 C4 3z w

444 LU LU

LAULUWJ LU p-1

LL.W

X-

~JqqJ-) I-ZILU

LA&AA U.

O-W) LU P -b.j J

Q.~4-4 UlN gVV~I9 () u-~'JI .

-4%L - -, 9-Z .- 4JJ LI 0 4 a L U -

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)

- U. * . I%

Ic

I.- I.-

zz

WLL~u-

-3LLWLL(411 7CLJW.."UJ(IX Qi

A9;

ZOCO V) ***

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0 ~ ~ ~ . -1i~ -1 -.1Qi(~

(4i Li-9 9-1 w Li w W

0. JU)4~ (4)i 0 000000x

00 (Az(AZ m -4 .-Cs')nzec *u 0-~)~ ~M .UI-CJ CA/ I 0 * 0

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U, 0

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I-~0 0 0 .0 L LL L 0Z.J CD ZZa~coa A xxx ~ LU

LU UJLAzz wULLU w r L ~

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U m.S J. LU ZZ LALLZLAuL LU LUw LU -U #-

I-- o I .1 -O 11 CO 11 OJ 1 (.3 11 (0. 119/)C'iDUS .--. *0000*z-r - I-N 3-W "La

ca. x.C. xV'.i( a.

U II.0 U.S .i.-IU. U. U . . U

......000 0 04 N 0 0r LUL 0 ~~~ U0CC LnlJ .Jre r- P- LU JI zL.

~Q O LII~r U.---U -J0 a. a. a a. . CA ~ w~s-74

-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

*. . .- °............

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

FILMED

4-85

DTIC