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Department of Electrical and ComputerEngineering

5-1-1987

Steady State Analysis of a Brushless DC Motor withHalf Bridge InverterTodd J. KazmirskiPurdue University

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Kazmirski, Todd J., "Steady State Analysis of a Brushless DC Motor with Half Bridge Inverter" (1987). Department of Electrical andComputer Engineering Technical Reports. Paper 564.https://docs.lib.purdue.edu/ecetr/564

Todd J. Kazmirski

TR-EE 87-17 May 1987

School of Electrical EngineeringPurdue UniversityWest Lafayette, Indiana 47907

Steady State Analysis of a Brushless DC Motor with Half Bridge Inverter

STEADY STATE ANALYSIS OF A BRUSFILESS DC MOTOR

WITH HALF BRIDGE INVERTER

A Thesis

Submitted to the Faculty

of

Purdue University

by

Todd J. Kazmirski

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Electrical Engineering

Mav 1987

This is dedicated to my family

\

Ill

ACKNOWLEDGMENTS

I would like to thank Professor Paul Krause for his support and gui­

dance throughout this work and my education and, especially, for his sincere

interest in me as a person.

IV

TABLE OF CONTENTS

Page

LIST OF TABLES............................................................. ...................................vi

LIST OF FIGURES............ ..... ................................. .................... ................. ... vii

NOMENCLATURE............ ix

ABSTRACT.......................................................................................... .............xi

CHAPTER 1 - INTRODUCTION........................................................... .............!

CHAPTER 2 - DESCRIPTION OF THE BRUSHLESS DC MOTOR SYSTEM............ ...... ..................................... ....... ........................ 3

2.1 INTRODUCTION........................ 32.2 THE BRUSHLESS DC MOTOR SYSTEM............ ...........2.3 DESCRIPTION OF THE HALF BRIDGE INVERTER................... 42.4 DESCRIPTION OF THE SNUBBER NETWORK.......... ................. ...62.5 DESCRIPTION OF THE BRUSHLESS DC MOTOR...... .............. 72.6 CONCLUSION............................................................................... 8

CHAPTER 3 - DEVELOPMENT OF THE STEADY STATE .MODEL FOR THE SYSTEM........................................... ........ .......................u

3.1 INTRODUCTION......................................................................... .u3.2 OPERATING MODES AND EQUIVALENT CIRCUITS.................. ..123.3 STATE EQUATIONS FOR THE A CONDUCTION PERIOD....... 133.4 OPERATING MODES, EQUIVALENT CIRCUITS, AND

STATE EQUATIONS FOR THE B CONDUCTION PERIOD........143.5 OPERATING MODES, EQUIVALENT CIRCUITS, AND

STATE EQUATIONS FOR THE C CONDUCTION PERIOD 153.6 CONCLUSION........................................ ,«

V

Page

CHAPTER 4 - STEADY STATE DIGITAL COMPUTER SIMULATION ...22

4.1 INTRODUCTION................................... .............. ................. -4.2 DESCRIPTION OF THE COMPUTER PROGRAM..........4.3 MACHINE PARAMETERS................ ....... ............................4.4 DIGITAL SIMULATION RESULTS..............4.5 ANALOG SIMULATION RESULTS.......... ............... ...... .....4.6 TORQUE PULSATIONS AND ZERO TORQUE MINIMA4.7 ELIMINATING THE ZERO TORQUE MINIMA.... ....... ...4.8 CONCLUSION................... .................................. ...... ••••........ .

CHAPTER 5 - STEADY STATE SIMULATION WITH 120 ° < Cp < 180 °'............................................................ .....

5.1 INTRODUCTION ........... ...........................................................................365.2 OPERATING MODES AND STATE EQUATIONS .........................375.3 STEADY STATE DIGITAL COMPUTER SIMULATION................385.4 TORQUE AND EFFICIENCY OPTIMIZATION........ ...... ...............395.5 CONCLUSION.......................... ..............................................................40

CHAPTER 6 - CONCLUDING COMMENTS..............................................60

...22

.24'

.24

.25

.25

28

LIST OF REFERENCES.................................................—•••••••••<>2

APPENDIX............................................................................. ..........................64

VI

LIST OF TABLES

Table Page

3.1 Conduction period A operating modes................... .............. ........ .............19

3.2 Conduction period B operating modes............................... .20

3.3 Conduction period C operating modes.................. ..................... ............21

4.1 Machine parameters................... 35

5.1 Transistor gating sequence for 120 ° < Cp < 180 ° ..................................41

5.2 Conduction period A1 operating modes, 120° < Cp < 180°................... 54

5.3 Conduction period A2 operating modes, 120 < Cp < 180 °................. .55

5.4 Conduction period Bl operating modes, 120° ' < Cp < 180 ° ......... ..56

5.5 Conduction period B2 operating modes, 120 < Cp < 180 ° 57

5.6 Conduction period Cl operating modes, 120 ° < Cp < 180 °.......... 58

5.7 Conduction period C2 operating modes, 120° < Cp < 180 °.......... .......59

Vll

LIST OF FIGURES

Figure Page

2.1 Brushless dc motor with half bridge inverter......... ............................. ......9

2.2 Transistor gating sequence for the half bridge inverter.............................. . 10

3.1 Brushless dc motor system equivalent circuit............................—.........17

3.2 Operating mode 4 equivalent circuit................ 18

4.1 Steady state digital simulation, ias, Vc, 120 ° conduction, ■ > = 0 .......29

4.2 Steady state digital simulation, te, 120 ° conduction,_o — 0 ° ..... .30

4,3; Steady state digital simulation, ias, Vc, 120 ° conduction, o = 30 —31

4.4 Steady state digital simulation, tR, 120 ° conduction,.<;> — 30 32

4.5 Steady state analog simulation, 120 ° conduction, <;> = 0 33

4.6 Transistor gating sequence for the half bridge inverter....... ..................34

5.1 Transistor gating sequence for 120 J < Cp < 180 -41

5.2 Steady state performance, ias, Vc, Cp = 150 3 , ■, = () 3 ........... -42

5.3 Steady state performance, te, Cp = 150° , <i> = 0 3 .......... — .....,...,.,..43

5.4 Steady state performance, ias, V(., Cp = 150° , ■ > = 30 ,.,..,..44

5.5 Steady state performance, te, Cp = 150 J , ■/> = 30 J ....... .,.........,,.,...,,.,.45

Figure

5.6 Steady state performance, ias, Vc, Cp = 180 ° , <!> = 0

5.7 Steady state performance, te, Cp = 180 ° , <b = 0

5.8 Steady state performance, ias, Vc, Cp = 180 , <b = 30

5.9 Steady state performance, te, Cp = 180 ° , <i> = 30

5.10 Steady state average torque output, Cp = 120° , 150

5.11 Steady state average torque output, Cp = 120 , 150

5.12 Steady state system efficiency, Cp = 120 ° , 150

5.13 Steady state system efficiency, Cp = 120 , 150 , 180

IX

NOMENCLATURE

ECM :

PWM

MMF

IMSL

u

R

C

P

T,

I),

electronically commutated motor

pulse width modulation or modulated

magnetomotive force

International Mathematical and Statistical Library

inverter firing angle

inverter conduction period

supply voltage in the brushless dc motor system

machine winding resistance

machine winding self inductance

machine winding leakage inductance

machine winding mutual inductance

permanent magnet flux Linkage

snubber network resistance

snubber network capacitance

number of poles in the machine

transistor i, i = A. B, C

diode L i = A. B, C

rotor mechanical angle

rotor electrical angle

X

• -v'r electrical angular frequency

• >is current in phase I of the machine, i=a, b, c

* vis voltage across phase I of the machine, i=a, b, c

• ve ; voltage across the snubber capacitor

• Te electric torque output of the machine

* Vce collector to emitter voltage of an npn bipolar transistor

• cr.or gating signal for transistor I, i = a, b, c

• on state conducting state of a solid state device

• off state non conducting state of a solid state device

* Yti . voltage across transistor I, i=a,b.c

• vdi voltage across diode I, i=a,b,c

• vei voltage induced in phase I by permanent magnet, i=a,b,

XI

ABSTRACT

Kazmirski, Todd J. M.S., Purdue University. May 1987. Steady State Analysis of a Brushless DC Motor with Half Bridge Inverter. Major Profes­sor: Paul C. Krause.

Brushless do motors have received significant attention in the fractional

horsepower motor industry in recent years. The inverter drive used with this

system has traditionally been of the full bridge configuration. However, it is

possible to use a half bridge inverter to drive the machine. In this paper a

method to analyze the brushless dc motor with half bridge inverter is

developed. The method is integrated into a digital computer program to

simulate the steady state response of the system. The program allows both

the inverter conduction angle and the inverter firing angle to be changed.

This provides an efficient means of evaluating the effects of system parame­

ters as well as inverter operating modes on the performance of the system.

The results of the simulation will demonstrate reductions in efficiency and

torque pulsations as well as an increase in average torque as the conduction

angle of the inverter is increased from 120 degrees to 180 degrees.

1

CHAPTER 1\

INTRODUCTION

The brushless dc motor is a permanent magnet synchronous machine

driven by an inverter drive. This machine also referred to as an

electronically commutated motor ■ (E,CM) has been the subject of growing

interest over the last several years. The reasons for this interest are the

advantages introduced by the brushless dc motor. Among the advantages

are:

- greater efficiency

- variable speed control

- smaller size and weight

- no brushes

- wider speed bandwidth

- bidirectional capability

- durability

- quick acceleration

- reduction of fire and toxicity hazards to the surrounding environment

The main disadvantage is the initial cost of the inverter drive and

inverter controller. The full bridge inverter operating in a discontinuous or

continuous mode has traditionally been used to drive the motor. However, it

2

is also possible to drive the motor using a half bridge inverter. This would

clearly have the advantage of reducing the number of inverter transistors

and diodes by one half. It might also reduce the complexity of the controller

used to switch the inverter.

The work contained herein is a systematic approach to analyzing the

brushless dc motor driven by a half bridge inverter. A description of the

motor, drive and snubber network is given. Next, equations for the machine

are written in machine variables. A description of the various operating

modes of the system is developed. From here, state variable equations are

written which describe the system in each of the operating modes. These

equations are used in a digital computer simulation. The simulation is then

used to study the steady state response of the system and determine the

effects of changing the firing angle of the inverter as well as conduction

period variations. The term steady state implies that the rotor speed of the

machine is held constant and that the phase currents in the machine are

periodic. ■

3

CHAPTER 2

DESCRIPTION OF THE BRUSHLESS DC MOTOR SYSTEM

2.1 INTRODUCTION

The initial step in the solution of any problem is a description of the

problem. From this description^ a readily solvable model can be developed to

obtain a solution. This chapter will begin with a description of the brushless

dc motor system. A description of the major subsystems of the system is

made followed by assumptions made for that component or subsystem. The

description of each subsystem is concluded by describing how that system

will be modeled.

2.2 THE BRUSHLESS DC MOTOR SYSTEM

The brushless do motor system consists of a permanent magnet

synchronous machine, a dc voltage source, a half bridge inverter, and a

snubber network. The system is given in figures 2.1.

2.3 DESCRIPTION OF THE HALF BRIDGE INVERTER

The inverter is of the half bridge type and is made up of three

transistors. In general, these can be any solid state switching device whose

conducting and nonconducting state can be independently controlled.

Throughout the analysis, the conducting state of a transistor will be referred

to as the "on state". This state occurs when the gating signal to the device is

on and the device is properly biased for conduction. The nonconducting

state of a transistor will be referred to as the "off state". This state occurs

when the gating signal is off or when the device is not properly biased for

conduction.

The inverter operates in the 120 degree conduction mode at all times

with the .gating sequence diagram as given in figure 2.2. Each transistor

conducts for 120 degrees and remains in the otf state for 240 degrees. As

shown in the diagram, Ta (transistor A) conducts or 120 degrees followed by

5

T0 conducting for 120 degrees and turning off and finally Tc conducting for

120 degrees and turning off. All angular displacements are in terms of the

electrical angular displacement, This sequence continually repeats itself. The

one variable in the sequence is the firing angle, <:>. The firing angle relates

this sequence to the rotor position.

The choice of the zero reference for the firing angle of the inverter is a

matter of convention. In this analysis, the zero firing angle will be defined to

coincide with A equal to zero. From figure 2.1, the conduction period for

phase A is centered around B equal to zero degrees. Thus, for a firing angle

of ' ’ equal to 0 degrees, ga would be on. for ^ ranging from -60 to 60 degrees.

For a firing angle of O equal to 60 degrees, ga would be on for B ranging from

-120 degrees to 0 degrees. For O equal to -30, ga would be on over the range

of " equal to -30 to 00 degrees. Thus the gating signals given in figure 2.2

can be written as a function of B and o., ' V.;; ' r

zM = gi(^r + '■■'’).■ i=a,b,c (2.1)

For the solution of the problem, the transistors are considered to be

ideal switches. These switches have zero resistance when in the on state and

infinite resistance when in the off state. When the gating to the transistors is

off, or when Vce < 0T the transistors will function as open circuits. When the

gating to the transistors is oil, the transistors will function as short circuits

provided VGe > 0. It is also assumed that there is an accurate wav to sense

tlie pp^ition of the rotor of this machine and a suitable con trol 4yai!ab!e

which will allow for the switching of the inverter at the desired rotor

positions,

2.4 DESCRIPTION OF THE SNUBBER NETWORK

The configuration of the motor inverter system creates problems due to

the abrupt switching of the inverter when windings are conducting. During

typical operation of the system, a transistor will turn on, allow conduction of

its respective phase, and turn off at the end of its 120 degree conduction

period. The switching of the inverter takes place regardless of the currents in

the motor. As one might expect, this will cause a very large voltage to be

induced in the windings due to abruptly forcing the current in the windings

to zero. To reduce these large transient voltages, a snubber network is added

to the the system to absorb the commutating current of the windings and

reduce the large voltages which would occur during the commutation period.

The network consists of a free wheeling diode connecting each phase of

the motor to a common resistor capacitor network as given in figure 2.1.

Throughout the analysis, the diodes will be considered to be ideal. That is.

the diodes are perfect conductors when forward biased and open circuits

when not conducting. These two states will be referred to as the "on state"

and the "off state” respectively.

6

7

2.5 DESCRIPTION OF THE BRUSHLESS DC MOTOR

A 2 pole, brushless dc motor is given in figure 2.1. The windings of the

machine are Y connected with the neutral connected to the positive side of

the dc source. Before writing the equations for this machine several

assumptions will be stated. First of all, the rotor will not conduct eddy

currents[4]. Second, the windings of the machine are sinusoidally distributed.

Finally, the rotor is non-salient[4l. The voltage equations of the machine can

be written in machine variables as

^abcs ^s'abcs d* P'Abcs (2.2)

where

(2.3)

r, 0 0r, 0 r, 0 (2.4)

0 0 r,

(2.5)

8

Ls =

Lis Ls Lm j-,mLrUL|S "t" Lg *--m

Lfn L|g + Lg

(2.6)

V = X'„

sin#r

sin(^r - -y)

sM8r ~ ~)

VeV,eb■v.

(2.7)

and p is the derivative operator d/dt. The equation for torque for this

machine is

Te = (~)xm'[('as ~ '^bs —Xaijcos^ + 'W3(ibs - ics)sin0rj (2.8)

2.6 CONCLUSION

A description of the brushless dc motor system has been made. The

gating sequence for switching the inverter has been presented along with the

zero convention for the tiring angle of the inverter. The models to be used in

the analysis have also been presented. Having completed these preliminary

tasks, the method for developing suitable equations whose solution will

describe the operation of the system will be presented in the next chapter.

9

bs-ixis

Figure 2.1 Brushless dc motor with half bridge inverter

10

g.(0r>4O

gb(^r»^)

g.(M)

A A '

B

a

C

I—<j>

I27T — (j>

9r

Figure 2.2 Transistor gating sequence for the half bridge inverter

11

CHAPTER 3

DEVELOPMENT OF THE STEADY

STATE MODEL FOR THE SYSTEM

3.1 INTRODUCTION

The models developed in the previous chapter will now be used in the

solution of the brushless dc motor system. In this chapter, various modes of

system operation will be examined. For each mode, an equivalent circui t will

be developed. A system of differential equations will then be written for each

circuit, and placed in state variable form. These equations will be integrated

into a digital computer program in the following chapter to simulate the

performance of the ECM system.

. 12

3.2 OPERATING MODES AND EQUIVALENT CIRCUITS

The equivalent circuit for the brushless dc motor system incorporating

the models of Chapter 2 is given in figure 3.1. In this circuit, the conducting

state of each transistor is a function of its respective gating signal and the

collector to emitter voltage. The state of each diode is a function of the

voltage across it. For each conduction period, this circuit can be further

split up into eight equivalent circuits dependent upon the states of the

transistors; and diodes. In other words, the equivalent circuit which models

the system is dependent upon whether the transistors and diodes are

conducting or not conducting. A total of 24 equivalent circuits is needed to

represent the system over all three conduction periods. From the models of

Chapter 2, a transistor will be in the on state if its gating signal is on and

VGe>0. Otherwise, the transistor will be in the off state. A diode will be in

the on state if it is forward biased. It will be in the off state if it is reverse

biased. ' : ■

For each of the three conduction periods, there are eight distinct

operating modes. The eight modes of operation will be outlined for the

conduction period When the gating signal for transistor A is on. The: modes

for the other two conduction periods can be easily derived from the A

conduction period modes. The eight conduction modes for the A conduction

period are listed in table 3.1. Let us examine mode 4. In this mode, is

greater than or equal to zero, and both V(lb and Vdc are less than zero.

Under these conditions, phases B and C are effectively eliminated from the

' 13

circuit. Thus the equivalent circuit of figure 3.2 can be used to represent the

3.3

Circuit equations for each mode can be easily written and placed in the

state variable form

x' = Ax -r Bu

For mode 4 of table 3.1 and the equivalent circuit of figure 3.2 the following

two independent equations can be written.

bsi rs*as ^ea

CV,;'

In matrix form, these equations become

Rx' — Sx -H Tu

where

Ls 0 -rs 0 -l 10 C s = 1 T — 1L J 0 0 —

R R

14

las

i

(0 S»__

____

_j

o>

___

1

U = >

Solving for R 1 and multiplying both sides by R 1

r ' 1 1-— 0 . _ --- . —*--'Ls Ls Ls

o 1x +

0 ——RC o

1---

(3.5)

The set of equations are now in the familiar state variable form of equation

3.1. This form can be easily integrated using a variety of numerical methods.

The A and B matrices for the eight operating modes of conduction period A

are given in the Appendix.

3.4 OPERATING MODES, EQUIVALENT CIRCUITS,

AND STATE EQUATIONS FOR THE B CONDUCTION PERIOD

The operating modes, equivalent circuits, and state equations ot

conduction period B can be easily derived from those of conduction period A.

The derivation is accomplished by a simple change of variables. The change

of variables consists of changing As variables for conduction period A to Bs

variables, Bs variables to C:s, and Gs to As. When .this transformation is

applied to the state equations only the state vector and the input vector are

changed as given in equation 3,6.

15

■ ■

4*bs veb^cs Veclas

u =

Vea;vc :y,

(3-6)

The A and B: matrices remain unchanged which is very convenient for the

digital computer simulation. The operating modes for the B conduction

period are given in table 3.2. A similar transformation can be used to

transform the operating modes, equivalent circuits, and state equations to

the C conduction period.

3.5 OPERATING MODES, EQUIVALENT CIRCUITS,

AND STATE EQUATIONS FOR THE C CONDUCTION PERIOD

The following transformation is used to transform the operating modes,

state equations, and equivalent circuits from the A conduction period to the

C conduction period. Change all A^ variables to Cs variables. Bs to As, and

C, to Bs. Again the A and B matrices remain unchanged with only the state

and input vectors changing. The operating modes for the C conduction

peripd are given in table 3.3 and the state and input vectors are givon in

equation 3.7.

16

V.s v„clas Vea

#1bsU —

Veb

vc V,

(3tf)

3.6 CONCLUSION

The eight operating modes for each conduction period have been defined.

For each state in the A conduction period, an equivalent circuit has been

developed. For each circuit, a set of state equations has been derived whose

solutions will describe the operation of the brushless dc motor system. Two

transformations have been introduced which transform the operating modes,

equivalent circuits, and state equations for the A conduction period to the B

and C conduction periods. The operating modes and the state equations can

now be used as the basis for a digital computer program to simulate the

operation of the brushless dc motor system.

17

db -

Figure 3.1 Brushless dc motor system equivalent circuit

18

Figure 3.2 Operating mode 4 equivalent circuit

19

Table 3.1 Conduction period A operating modes

CONDUCTION PERIOD A OPERATING MODES

Vu- Vdb Vdc operating mode

< 0 < 0 < 0 0

<0 < 0 > 0 1

<0 > 0 < 0 2

< 0 > 0 > 0 3

>o <0 <0 4

>0 < 0 > 0 5

> 0 > 0 < 0 6

> 0 >o > 0 7

20

Table 3.2 Conduction period B operating modes

CONDUCTION PERIOD B OPERATING MODES

vtb ' VdcVda j operating mode

< 0 < 0 <0 0

< 0 < 0 > 0 1

< o > 0 < o 2

<0 > 0 > 0 3

>0 <0 < o 4

>0 <0 > 0 5

>0 >0 < o 6

>0 > 0 > 0 7

Table 3.3 Conduction period C operating modes

CONDUCTION PERIOD C OPERATING MODES

V'.: Vtc ■■■■ Vda V operating mode

< 0 < 0 < 0 0

< 0 < 0 > 0 ; i ^ ^

<0 > 0 <0 2

< 0©A

l > 0 . 3

> 0 <0 < 0 4

> 0 < 0 > 0 : 5 a

> 0 > 0 <0 6 ' ■

> 0■

> 0 >0 7 ■

22

CHAPTER 4

STEADY STATE DIGITAL COMPUTER SIMULATION

4.1 INTRODUCTION

The state equations developed in Chapter 3 are now used in a digital

computer program to simulate the steady state operation of the brushless dc

motor system. Two digital simulations are presented. The difference between

the two is the inverter firing angle used. An analog simulation is also

presented for comparison to the digital simulation. The simulation results

show a large torque pulsation with instantaneous minima torque values

nearly equal to zero. These zero minima are inherent to the ECM with a half

bridge inverter operating in the 120 degree conduction mode. The cause of

the zero torque minima will be discussed and a easily implemented solution

will be presented.

23

4.2 DESCRIPTION OF THE COMPUTER PROGRAM

The method used in the digital computer program is straightforward.

The state equations developed in the preceding chapter are integrated using

the IMSL routine DGEAR. The algorithm is based on the Adams Method of

solving a system of differential equations.

The program can be summarized as follows. Throughout the simulation,

the rotor speed is held constant. Before each integration step, the transistor

and diode voltages are calculated to determine the operating mode of the

system. The proper A and B matrices are then supplied to the integration

routine and the integration is performed. These two steps are repeated until

one complete electrical cycle has been simulated. The final state vector is

then compared to the initial state vector. If the relative difference between

the two is less than the user supplied value, the simulation ends. Otherwise,

a new initial state vector is calculated and the the simulation is repeated.

This process continues until the user specified relative difference is satisfied.

Thus, the initial state vector supplied by the user need not be the steady

state state vector. The results of the integration at each step, namely the

state vector, can be used to calculate any system parameter desired by the

user.

24

4.3 MACHINE PARAMETERS

Results of analog and digital computer simulations of the system are

given in the following sections. The machine parameters used in the

simulation are given in table 4.1. All simulations which follow are performed

with a constant rotor speed of 3600 revolutions per minute.

4.4 DIGITAL SIMULATION RESULTS

Figures 4.1 through 4.4 show the results of two steady state digital

computer simulations of the ECM system with o = 0 degrees and' S == 30

degrees. These results show that this system operating in the 120 degree

conduction mode has a very large pulsating torque. For the case of o — 0,

the torque equals zero at discrete points. For the case of = 30, the torque

nearly equals zero at discrete points. In both cases, these points coincide

with the end of commutation for the winding which is most recently switched

off.

25

4.5 ANALOG SIMULATION RESULTS

Figure 4.5 shows the results of a steady state analog computer

simulation of the ECM system with 0=0 degrees. The model used for this

simulation is not the same as that used for the digital simulation'. The

difference is that the diodes and transistors both have a forward voltage drop

when conducting. In an attempt to compensate for this difference, the dc

source voltage was increased to 14.8 volts in the analog simulation. The

results of the analog simulation are very close to those obtained from the

digital simulation. The slight difference is attributed to the difference in the

models used for the simulations. In both the analog and digital simulations,

the torque equals zero or is very near zero at discrete points. This condition

is undesirable and may be unacceptable for certain applications. A method

to eliminate these near zero torque points will be discussed in the next

section.

4.6 TORQUE PULSATIONS AND ZERO TORQUE MINIMA

The zero torque minima associated with the ECM operating in the 120

degree conduction mode are undesirable. In order to eliminate this problem

it is necessary to understand the source of the torque pulsation. Upon close

examination of figure 4.1, one can see that at certain instants of time none of

the phases of the machine conduct current. These points correspond to the

time at which the last phase to be switched off ends commutation. To

examine this problem more closely, assume the currents which exist in the

machine are those associated with the conduction period of the transistors.

Thus, the currents which flow through the diodes after commutation are

neglected. From figure 2.2 which is repeated as figure 4.6, when ST = —o, Tc

is switched off, the current in phase G will begin commutation and at the end

of commutation, none of the windings conduct current. The resultant

magnetomotive force(MMF) is zero. Next, the A phase immediately begins to

conduct current. This results in an MMF which can be represented by a

vector in the as axis of the machine. As the current increases and decreases

in the A phase the magnitude of the MMF vector changes but its position

remains constant. Thus including the commutation period for the A phase,

the MMF vector will remain in the as axis for 120 electrical degrees. The

vector will then go to zero for a brief instant and then jump to the b3 axis

coinciding with the B conduction period. At the end of commutation of the

B phase, the MMF vector will again go to zero and then jump to the cs axis

due to the current conduction in the C phase. This current conduction

pattern gives rise to an MMF vector which is discretely positioned in one of

three axes dependent upon the phase of the machine which is conducting.

The result is a large pulsating torque which has minima which reach zero at

the end of commutation of each phase.

27

4.7 ELIMINATING THE ZERO TORQUE MINIMA

Ii ilder to eliminate the pulsating torque, the currents in the motor

must produce an MMF which is constant and rotates at the speed of the

rotor. These conditions can only be satisfied if the phase currents are

sinusoidal and displaced by 120 degrees. In order to satisfy these conditions,

a complex controller must be used. This type of controller would certainly be

expensive and may make the ECM system with half bridge inverter an

unacceptable alternative to other drive systems. However, the operating

mode of the existing system can be changed to reduce the torque pulsations

with no additional hardware.

The zero minima of the steady state torque curve can be eliminated if at

least one phase of the machine- is always conducting current. This condition

can be satisfied if the conduction period of the half bridge inverter is

increased from 120 degrees. Using this type of inverter, at least one phase of

the machine will always be conducting current. During the commutation

period, two windings will be conducting. This current conduction pattern

will result in an MMF vector which moves in a continuous fashion.

To visualize this, assume at time tt only phase A conducts current. The

resultant MMF vector is coincident with the as axis of the machine. Since

the conduction period is greater than 120 degrees, phase B will he switched

on before phase A is switched oil". At t.>, the B phase is switched on and the

increasing current in phase B results in an MMF which rotates from the as

axis towards the bs axis. At t3, the A phase is switched off. .As the A phase

28

commutates, the MMF vector continues to rotate towards the bg axis. When

the current in the A phase goes to zero the MMF vector will be coincident

with the bs axis of the machine. The MMF vector will make a similar

transition to the cs and as axes as the three phases of the machine are

switched at the desired rotor positions. It is important to note that although

the MMF vector moves in a continuous fashion, the magnitude and the

rotation speed of the vector is not constant. The result is that the zero

minima of the steady state torque curve are eliminated, but the torque

pulsations will remain although reduced in magnitude.

4.8 CONCLUSION

The results of steady state digital and analog computer simulations have

been presented in this chapter. The results of the digital simulation are in

close agreement with those of the analog simulation. Both simulations show

the undesirable near zero value points occurring in the steady state torque

curves. A solution to this problem has been proposed. The solution involves

extending the conduction period of the half bridge inverter to a value greater

than 120 degrees. This solution will be explored in Chapter 5 with the

presentation of simulation results obtained using several different conduction

periods. . . ' i .

Uc UOLTS

las AMPERES

29

20.00

16.00

12.00 -

8.00

0.0060.00•60.00 0.00 120.00 180 00 240.00 300.00

ELECTRICAL ROTOR POSITION DEGREES

27 oo

26 oo

25.50

24.50

24.00•60.00 0.00 60.00 180.00 240 00 300 00

ELEGTRIGAL ROTOR POSITION DEGREES

Figure 4.1 Steady state digital simulation, ias,' Vc, 120 ° conduction, 0 — 0

torq

ue in

n

30

240.00 300 00

ELECTRICAL ROTOR POSITION DEGREES

Figure 4.2 Steady state digital simulation, te, 120 ° conduction, •.>

UOLTS

las AM

PERES

31

20.00

16.00

12.00

8.00

H.OG

0.00-90.00 30.00 90.00 150.00 270.00210.00

ELECTRICAL ROTOR POSITION DEGREES

27.50

27 00

26.50

26.00 -

25.50

270 00150.00 210.0090.00-30.00 30.00-90.00

DEGREESELECTRICAL ROTOR POSITION

Figure 4.3 Steady state digital simulation, ias, Vc, 120 conduction, O — 30

TOR

QU

E IN N-M

32

\

0.00-90.00 -30.00 30.00 90.00 150.00 210.00 270:00

ELECTRICAL ROTOR POSITION DEGREES

Figure 4.4 Steady state digital simulation, te, 120 ° conduction, d> ==

;

volts

'a

s amps33

0.250

0.125

Figure 4.5 Steady state analog simulation, 120 ° conduction, (h = 0 °

34

g.(0r»$

gb(0r>^)

g•(*„*)

A A

B

<1

C

I—

7T3

9*

5tt

3

!— <£ 27T <P

Figure 4.6 Transistor gating sequence for the half bridge inverter

Table 4.1 Machine parameters

35

MACHINE PARAMETERS

PARAMETER VALUE UNITS

Rs 5.5 ohms

C* 1.0 millifarad

Vs- . 14.0 volts

r 0.1 ohms

:■ 4 0.3 millihenries

L.is 0.03 millihenries

-'in. ' -0.135 millihenries

.....10.38 millivolt—second

radian

poles 6 V

36

CHAPTER 5

STEADY STATE SIMULATION WITH 120 ° < Cp <180°

5.1 INTRODUCTION

In Chapter 4, increasing the conduction period of the inverter from 120

degrees was proposed as a method to reduce the torque pulsation of the ECM

system. In this chapter, the results of a digital computer simulation will be

presented which prove that extending the conduction period does in fact

decrease the torque pulsation. The results also show that larger conduction

periods result in larger average torque values for the system at the expense of

system efficiency.

37

5.2 OPERATING MODES AND STATE EQUATIONS

The same method which is used to analyze the ECM system with 120

degree conduction mode is used to analyze the the more general case of

conduction periods ranging from 120 degrees to 180 degrees. Similar to the

120 conduction mode, there are three symmetrical 120 degree periods. These

three periods can be defined by the rotor position where they start. The three

positions are given in figure 5.1 which shows the transistor gating signal as a

function of rotor position and inverter firing angle. In figure 5.1. the 120

degree symmetrical periods have been further split into two periods and are

determined by the gating signals on at the given rotor position. The period

when Ta is switched on until Tc is switched off is defined as A2. The

subscript two is used to indicate that two transistors are switched on. The

period when Tc is switched off until Tb is switched on is defined as A,. In this

case, the subscript one is used to indicate that only one transistor is switched

on. The periods defined by B and C are symmetrical to the A conduction

period. In other words the state equations for the B and C conduction

periods can be derived from those of the A conduction period by using the

change of variables introduced in Chapter 3. The modes of operation for the

A conduction period are given in tables 5.2 and 5.3 The A and B matrices

corresponding to the state equations for the A, B, and C conduction periods

are given in the Appendix. The modes of operation for the B and C

conduction periods are listed in tables 5.4 through 5.7.

38

5.3 STEADY STATE DIGITAL COMPUTER SIMULATION

AM of the simulations which follow were obtained using a digital

computer program and the same general algorithm used in Chapter 4. The

system parameters are the same as those given in Chapter 4 with the

exception of the conduction period, Cp, which ranges from 120 degrees to 180

degrees. k,::.-

Given in figures 5.2 and 5.3 are the results of a simulation with

Cp = 150 , and 0 — 0 . Figures 5.4 and 5.5 give the results of a simulation

with Cp = 150 , and </> = 30 . Given in figures 5.6 and 5.7 are the results

of a simulation with Cp = 180 ° , and <;> = 0 °. Figures 5.8 and 5.9 give the

results of a simulation with Cp = 180 °, and 4> = 30°. Upon close

examination of these results and those of Chapter 4 several conclusions can

be drawn. First, the near zero torque minima of the 120 degree conduction

mode system are eliminated by increasing the conduction mode to 150 or 180

degrees. Second, a significant increase in the average torque output is

obtained by increasing the conduction period from 120 degrees to 180

degrees. Third, an average torque increase also results when O is advanced

from 0 degrees to 30 degrees for each of the three conduction modes studied.

Finally, the peak capacitor voltage and phase current vary widely as

Cp and <i> are changed. The results also suggest that there may be unique

values of o and Cp to optimize the average torque. Likewise, these two

parameters may be adjusted in order to optimize the efficiency of the ECM

system.

5.4 TORQUE AND EFFICIENCY OPTIMIZATION

39

In order to investigate the effects of Cp and <!) on system efficiency and

average torque, several simulations were run. These simulations consist of

measuring the system efficiency and average torque output for several steady

state operating modes of the system. The efficiency is defined as the

mechanical energy output divided by the electrical energy input into the

system, multiplied by 100%. Results are given in figures 5.10 through 5.13.

Given in figures 5.10 and 5.11 are the average torque output results for

conduction periods of 120, 150, and 180 degrees. Figures o.l2 and 5.13 give

the system efficiency results for conduction periods ol 120, 150, and 180

degrees. These results demonstrate the effects oi' Cp and <!> on the average

torque output and system efficiency. The average torque can be increased by

increasing Cp from 120 degrees. This increase however comes at the expense

of operating efficiency since the 120 conduction mode is the most efficient of

the three modes studied. The firing angle of the inverter on the other hand

can be used to optimize both the average torque output and the system

operating efficiency. As an example, examination of the 120 conduction mode

shows that as the firing angle is changed from 0 to 30 degrees the average

torque increases from 0.15 newton-meters to 0.35 newton-meters. T.he system

efficiency moves from 65% to 70%.

40

5.5 CONCLUSION

A digital computer program was developed and used to compare the

brushless dc motor system with three different conduction modes and several

different inverter firing angles. As expected, the results demonstrate that the

near zero minima of the instantaneous torque output can be eliminated by

changing the 120 degree conduction to 150 or 180 degrees. The results also

demonstrate the use of the conduction period to increase the average torque

output and decrease the system operating efficiency or vice versa. Larger

conduction modes provide for larger torque, but only at the expense of

operating efficiency as compared to the 120 conduction mode. The firing

angle however can be adjusted over various ranges to provide for increased

operating efficiency and torque output for any of the three conduction modes

studied;

41

Table 5.1 Transistor gating sequence for 120° < Cp < 180°

Si ON, gioa CENTER, gic OFF, gi0ff

Sa 1 1 |*o.

-.0. cp

-? + -

gb - (j) - + 120 °: ; 2

- 6 + 120 °CD

- <i> + — + 1202 . .

gc - 6 - — - 120 °' ' 2 .

- <b - 120 ° Vo + ~ 12 0 °

— A2 | '.AT

= Cp - 120

gbC^r*^) x2 = 120 °

B2 ; B1 I, i x3 = 120 °

«*—------ x3 ------- »U»,A)

*2 -| C2 ! Cl ! .* '•

■C,

I/

47T3

n

Figure 5.1 Transistor gating sequence for 120 ° < Cp < 180 °

•JO

LTS

42

HO . 00

33.00

2H 00

16.00

0.0075.00 15.00 105.00 165,00 225.00 285.00

ELECTRICAL ROTOR POSITION

uu -15.00 H5.00 105.00 165.00

electrical rotor position DEGREES

Figure 5.2 Steady state performan

TOR

QU

E IN N-

43

i

225.00 285 0075.00 15.00 45.00 165:00105.00

ELECTRICAL ROTOR POSITION DEGREES:

Figure 5.3 Steady state performance, ter.Cp = 150 ° , <> = 0

UO

LTS

las AMPERES

44

HO 00

32.0 0

24.00

16.00

0 .00255.00195.00135.0075.0015.00■45.00105.00

ELECTRICAL ROTOR POSITION DEGREES

38.00

37.00

36.00

35.00

34.00

33,00 255.00135.00 195 ,0075.. 0015.00105.00

ELECTRICAL ROTOR POSITION DEGREES

Figure 5.4 Steady state performance, ias, Vc, Cp = ISO ° , < > = 30 °

ELECTRICAL ROTOR POSITION DEGREES

Figure 5.5 Steady state performance, te, C

MOLTS

las AM

PERES

46

48.00

40.00

24.00

16.00

8.00

-30.00 30.00 210,00 270.0090.00 90.00 150.00

ELECTRICAL ROTOR POSITION

43,00

39.00

210.00 g70?9B150.0030.0030.00

ELECTRICAL ROTOR POSITION

Figure 5.6 Steady state performance, ias, Vc, Cp =» 180 " , o « () /

TORQUE IN

47

0.00-90.00 -30.00 30.00 90.00 150.00

ELECTRICAL ROTOR POSITION DEGREES

Figure 5.7 Steady state performance, te, Cp = 180 ° , <:> = 0 °

UOLTS

las AM

PERES

48

50.00

HO . 00

20.00

10.00

0.00120.00 •60 00 0.00 60.00 120.00 180.00 2H0.00

ELECTRICAL ROTOR POSITION DEGREES

HH.00

38.00

180.00 gHQ.OQ60.00 120.00120.00 ■60.00

ELECTRICAL ROTOR POSITION DEGREES

5.8 Steady state performance

TOR

QU

E IIS IS-

49

180.00120.0060.000 .00-120.00 -60.00

ELECTRICAL ROTOR POSITION DEGREES

Figure 5.9 Steady state performance, tRt Cp = 1.80 , = 30

AUER

AGE

TORQUE! N-

N

AUER

AGE

TORQ

UE

50

= 120

120.0.90.0060.0030.000 .00-30 00

PHI DEGREES

.20

Cn = 150

-1090.0030.00 60.00-30 00 0.00

PHI DEGREES

Figure 5.10 Steady state average torque output, Cp == 120 ° , 150

51

1.20

90

5 .30

Cn = 180

0.00-30.00 0.00 30.00 60 00 90 00

PHI DEGREES

1.20

Iz:.90

ZD C3 C£ O f—idID<rLdZ><1

.60

30

0 .00

CL = 150

* Cn = 120120.0060.00 90.0030.000 . 00

PHI DEGREES

Figure 5.21 Steady state average torque output, Cp = 120 ° , 150 ° , 180 °

52

100.00

80.00

60 00

h HO.00

CO 20.00

0.00120 .0090 0060.0030 000 . 0030.00

DEGREES

100 .00

70.00

40.00

10.00

20.0 030.00 0 .00 30.00 60.00 90.00 120 .00

DEGREES

Figure 5.12 Steady state system efficiency, C

EFFI

CIE

NC

Y

SYST

EM EFF

ICIE

NC

Y

53

100 .0

80.00

60.00

40 00

0.0030 00 0.00 30 00 60 00 90 00

DEGREES100 o

80.00

60.00

40.00

CO 20.00

0 . 000120 0090 0060.0030.0030.00

DEGREES

Figure 5.13 Steady state system efficiency, C

Table 5.2 Conduction period A1 operating modes, 120 < Cp < 180 °

CONDUCTION PERIOD A1 OPERATING MODES

>db vdc operating mode

< 0 ■■ < 0 < 0 0

< 0 < 0 > 0 :' 1 ■:;

< 0 > 0 < 0 2

<0 > 0 > 0

> 0 < 0 < 0 4 ■■

> 0 ' < o V 0 5

> 0 > 0 <0 6 ■'

> 0 > 0 > 0 7

' 55

Table 5.3 Conduction period A2 operating modes, 120 < Cp < 180

CONDUCTION PERIOD A2 OPERATING MODES

■ - ; Vtc Vdb operating mode

< 0 < 0 <0 8

<0 <0 > 0 9

<0 >0 <0 10

<0 >0 > 0 n

>0 <0 <0 ■ 12 . .

> 0 <0 > 0 : 13

>0 > 0 <0 14

> 0 > 0 > 0 15 ’

56-

Table 5.4 Conduction period Bl operating modes, 120 0 < Cp <180 °

CONDUCTION PERIOD Bl OPERATING MODES

;: ;;vtb : vdc Vda operating mode

< 0 CO < 0 0

< 0 <0 > 0 ■ ■ T'■

< ° . 1 0 : <0 2 ■.

< o >0 >0 3 • "

>0 <0 < 0 i.

> 0 ' <0 > 0 5 .

>0 >0 <0 6

■ . > o ■ _ 0 > 0 fy-i

. 57

Table 5.5 Conduction period B2 operating modes, 120 ° < Cp < 180 °

CONDUCTION PERIOD B2 OPERATING MODES

. ; ' : ; Vtb , . : " Vta . Vdc operating mode

<0 < 0 C 0 , s: .

<0 < o > 0 9

< 0 > 0 <0 10

o > 0 > 0 11

> 0 <0 <0 12

> 0 <0 > 0

>0 >0 <0 14

>0 >0 >0 !5

■. 58

Table 5.6 Conduction period Cl operating modes, 120 ° < Cp <180 °

CONDUCTION PERIOD Cl OPERATING MODES

Vda Vdb operating mode

o ' < 0 < 0

< 0 <0 > 0 i

<0 '■ > 0 <0 2

< o > 0 > 0 . 3

> 0 < 0 < 0 4 . . ' < : ■

> 0 < 0 > 0 5

>0 >0 <0 . ''6

> 0 > 0 > 0 4

; 59

Table 5.7 Conduction period C2 operating modes, 120 ° < Cp < 180 °

CONDUCTION PERIOD C2 OPERATING MODES

Vtb Vda operating mode

A, ' . < 0 <0 8

<0 < 0 > 0 • 9,.

<0 > 0 <0 10

<0 > 0 > 0 : 'n. 'V

> 0 <0 <0 • 12 ...

■ 0 < 0 > 0 ■ 13

> 0 ■ > 0 <0 14

> 0 >0 >0 . 15

60

CHAPTER 6

CONCLUDING COMMENTS

The work presented herein provides mathematical equations whose

solution describe the operation of the brushless dc motor system operating in

conduction modes ranging from 120 to 180 degrees. These equations which

are in state variable form have been integrated into a digital computer

program to provide a simulation of the brushless dc motor system.

The results of a set of simulations have been used to study the use of

the conduction mode and the firing angle of the inverter to control the

operation of the system. Among the conclusions which were drawn from the

simulations is that the conduction mode of the inverter can be used to

eliminate the near zero torque minima encountered with 120 degree

conduction mode system. Also, the conduction mode can be used to increase

the average torque output of the system, but only at the expense of a

decrease in system operating efficiency. The variation of the firing angle of

the inverter was shown to increase both the average torque and the system

efficiency over a limited range.

The system studied was not a pulse width modulated system. In order

tQ study the PWM ECM system, a new set of mathematical equations

describing the system must be developed. The method used to develop the

state equations which describe the non PWM system can also be used to

develop equations for the PWM system. These equations could then be

integrated into the digital computer program by changing the A and B

matrices and changing the logic in the program. The same method could

LIST OF REFERENCES

62

LIST OF REFERENCES

[1] P. C. Krause Analysis of Electric Machinery McGraw-Hill, 1986.

[2] N. A. Demerdash and T. W. Nehl, “Dynamic Modeling of Brushless DC Motors for Aerospace Actuation,” IEEE Trans, on Aerospace arid Electronic Systems,Vol. AES-16, pp. 811-821, November 1980.

[3] S. Williams and C. Eng, DC Motor,” Proc. IEE, Vol. 132, pp 53-56, January, 1985.

4j T. W. Nehl, F. A. Fouad, N. A. Demerdash, and E. A, Maslowski, Electronically Operated Synchronous Machines with Parameters Obtained from Finite Element Field Solution,” IEEE Trans on Industry Applications, Vol, IA-18, pp 172-182, March/April 1982.

[5] F. A. Fouad, T. W. Nehl, and N. A. Demerdash, Operated Synchronous Machines Using Finite Elements,” IEEE Trans, on Power Apparatus and Systems, Vol. PAS-100, pp, 4125-4134, September 1981.

[6] R. S. Ramshaw, A. W. J. Griffin, and K. Lloyd, Control, pp 40-44, January 1966

[7] J. R. Woodbury, IEEE Trans, on Industrial Electronics and Control Instrumentation, ” Vol. IECI-21, pp. 52-80, May 1974.

[8] T. W. Nehl, F. A, Fouad, and N. A. Demerdash, Electronically Commutated DC Permanent Magnet Machines,” IEEE Trans, on Magnetics Vol. MAG-17, pp 3284-3286, November 1981.

[9] T. M. Jahns with Rectangular Current Excitation,” IEEE Trans, on Industrial Applications, Vol. IA-20 No. 4, pp 803-813, July/August 1984.

[10] E. Richter, T. J. E. Miller, T. W. Neumann, T. L. Hudson, Economical Assessment,” IEEE Trans, on Industry Applications, Vol LA-21, No. 4, pp. 644-650, May/June 1985.

[11] R. DeCarlo, Class Notes, 1987.

63

APPENDIX

The conduction modes zero through seven for conduction periods greater

than 120 degrees have the same A and B matrices as the corresponding

modes of the 120 degree conduction mode. The A and B matrices are listed

below.

64

Aq —

0 0 0 0 0 0 0 0 0

0 0 0

000

RC

A, =

0 0 0 0

0 0 —

00

oo —

00_1_'l.

RC

Ao —

0 0

0Ls0 0

— 0c

0

0RC

Aak2

0 0 0 00 —rLs rLm Lm - Ls

® rLs Lm — Lgko ko ko.

0 — —C C RC

A4.;—

~u 00

0 0 0 0 0 0

0 0 0-

0

00

1RC

..'As. _1_k2

-rLs 0 rLm 0 0 0

rLm 0 —rLs k2

Lm0

-Lsk2

RC

A6 k2

-rLg rLm 0 rLm -rLg 0

0 0 0 ko

o — 0c

“Lg0k2

”rc

>\

66

—k2r -k3r -k3r -2k3

II

.

-k3r -^k2r ■^k3r —k2 ~ k3-k3r -k3r —k2.r —k2 — k3

0 illC

k]_C

kiRC

Ag — Aq

Ag — Ao

10

0 0 0 0 0 0 0 0

0 0

0 0

Ls

0

0

1RC

An k2

0 0 0 —rLs

0 rLn k2

0 0

rLs Ljjj

0k2RC

Ai2 — A4

67

A-13 — Ag

A14k2

-rLs 0 rLm 0 0 0

rLm 0 “rLs

0 0 0

000k2

RC

Ais kl

k2r k3rk3r —k2r k3r —k3r

•k3r•k3r■kor

0

”k3

-k2-k3

kiRC

Where

k, = L,3 + 2Lm3 - 3Lm2L,

k, = L,- - Lm-

^3 ^nr^s

68

B00 0 0 0 0 0

0 0 0

0 0 0 _1_

Ls0p1

RC

Bt =

0 0 0 0

0 0

00

0 0 0

00

J_Ls

1RC

B2

0

0

0

0

0J_Ls0

0

0

0

01

0

0

Ls01

RC

b3 k2

0 0 0 0 0 —Lc Lm L, — Lms m s • m

0 Lm —Ls Ls — Lmko

0 0 0RC

GO I

69

B40 0 0 0 0 0

0 0 0

J_Ls001

RC

5 k2

—Le 0 Lm L« — Lms m s m

0 0 0 0 Lm 0 —L, L. — Lmin s s m

o ^2RC

b6 k2

-LsW

0

0

Lm

-Ls0

0

0 L3 — L 0 Ls — L 0 00 >-

RC

mm

b7 _2_ki

■k2

'^3'^3

■k3 —k3 k2 + 2k3 •k2 — k3 k2 + 2k3 •k3 —k2 k2 + 2k3

A.RC

0 0 0

70

B8 Bo

Bg — B,

Bio — Bi

Bn — B3

B12 — B4

Bis — B6

Bh — B5

Bis — B7

71

Where

k, = Ls3 + 2Lm3 - 3Lm2Ls

k2 = Ls2 - Lm2

^3 — Lm‘ — LmLs