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Mathematical modelling of
permanent-magnet brushless
DC motor drives
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c Khalid Salih Mohammad Al-Hadithi
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MATHEMATICAL MODELLING OF
PERMANENT-MAGNET BR USHLESS DC
MOTOR DRIVES
L-.
VY
KHALID SALIH MOHAMMAD AL-HADITHI
A Doctoral Thesis
Submittedin partialfulfilment of the requirements
for the award of the degreeof
Doctor ofPhilosophy
Of
LoughboroughUniversity of Technology
1992
Supervisor
Mr. J. G. Kettleborough
Departmentof ElectronicandElectricalEngineering
0 by Khalid Salih MohammadAl-Hadithi
DedicatedTo
My MotherandFather
ACKNOWLEDGEMENT
I
wish
1.
R.
to
express
Smith
Invaluable
I
would
for
Mr.
and
guidance
and the
research
like
their
to
deepest
my
and
G.
J.
preparation
of
Mr.
Keith
thank
assistance
and
the
this
the
their
course
of
the
thesis.
Gregory
many
Professor
for
Kettleborough
throughout
advice
both
to
gratitude
V.
Dr.
and
V.
discussions
useful
Vadher
that
we
work
and
had.
I
to
am thankful
My sincere
the
of
My
thanks
great
for
technicians
go
to
their
for
my parents
have
they
support
assistance.
hard
their
towards
contributed
the
building
my career.
deserve
brothers
encouragement
I
the
want
to
Scientific
this
Finally,
particularly
thank
the
constant
Ministry
in
of
for
IRAQ
Higher
Education
sponsoring
me
and
while
I
Centre
at
research.
my thanks
Loughborough
their
and support.
Research
undertook
for
thanks
special
to
University
Mr.
David
the
staff
for
in
their
Thomas and Mr.
i
the
help
Geoff
Computer
and
co-operation,
Harris.
ABSTRACT
dc
Brushless
following
popular,
materials
the
control
They
position.
flight
as
drives
the
low
is
in
developed
model
occurrence
in
objective
of
extremely
this
by
is
on
the
numerical
for
formulated
in
to
used
of
the
phase
for
pattern
The
describes
an
current
devices,
phase
changes
current
in
the
design,
is
circuit
to
waveforms
firing
torque
minimize
angle
are
of
ii
and
the
ripple,
inverter
in
changes
switches.
and
testing
MOSFET
using
3-phase
established.
the
are
from
supply
described
being
methods
topology
kW
model
motor
Tensor
PWM inverter,
1.3
a
their
The
the
construction
source
drive
to
A practical
profiling
the
the
with
differential
the
arising
inverter
the
their
effect
for
varying
motor
investigate
system,
frame.
the
the
voltage
experimental
switching
motor.
both
of
mathematical
reduced.
those
with
motor
The
of
solution
of
conduction
be
many
especially
which
for
little
ripples,
drive
may
reference
operation
thesis
they
which
system,
account
discontinuous
the
the
dc
factors
ways
In
to
used
was
establishing
and
equations
speed
and
life,
ratio.
brushless
a typical
magnitude
based
long
undesirable.
thesis
such
actuators,
robot
and
rotor
applications
reliability,
torque
used
the
sense
in
used
torque-to-weight
of
rare-earth
devices
to
and
and
high
presence
speed,
power
systems
a high
and
in
semiconductor
frequently
now
require
maintenance
at
input
control
which
drives
are
the
and
stator
increasingly
become
developments
recent
permanent-magnet
to
have
drives
motor
brushless
implements
which
and
dc
the
The
switches
optimum
of
effect
on
the
torque
ripple
Throughout
verified
by
are
also
the
comparison
examined.
theoretical
thesis,
with
experimental
iii
predictions
results.
are
LIST OF PRINCIPAL SYMBOLS
B
flux-density
Bi
intrinsic
Br
magnetic
C
branch
Ct
transpose
D
viscous
Eb
branch
ek
induced
eo
generated
H
magnetic
Hc
normal
Hcj
intrinsic
h
integration
Ib
branch
IM
mesh
ik
Ik
flux-density
(T)
(T)
remanence
/
transformation
mesh
voltage
in
voltage
phase
KT
torque
KB
back
KG
tachometer
L,
branch
(s)
vector
vector
k
winding
motor
moment
Lm
mesh
I
L'm
inverse
of
LO
average
value
(kg.
M2)
M2)
(V/rps)
constant
inductance
mutual-inductance
inertia
(V/rps)
voltage
Lkj
(kg.
of
(Nm/A)
constant
self-inductance
moment
inertia
of
Lkk
(A)
load
and
constant
emf
(V)
(A/m)
length
step
current
motor
voltage
(A/m)
coercivity
current
(V)
(A/m)
coercivity
JM
k
winding
intensity
field
combined
vector
open-circuit
in
(Nm/krpm)
coefficient
induced
current
matrix
C
of
damping
j
b
(T)
matrix
of
winding
between
inductance
k
(H)
windings
matrix
mesh
of
inductance
Lkk (H)
iv
matrix
k and
J,
k *j
(H)
L2
Lkk
second
average
M2
coefficient
value
the
of
second
p
differential
Rb
branch
Rk
resistance
Rln
mesh resistance
matrix
Te
electromagnetic
torque
TF
frictional
TL
load
Vb
impressed
Vk
voltage
VS
dc supply
voltage
VM
impressed
mesh voltage
Oe
angular
axis
of
Tb
branch
Wk
total
TPb
branch
rotor
pole
harmonic
component
of
rotor
and
the
rotor
and
the
to
the
to
the
pairs
d/dt
operator,
resistance
of
matrix
k
winding
(Q)
(Nm)
(Nm)
torque
(Nm)
torque
branch
across
voltage
vector
k
winding
of
phase
a
vector
in
between
phase
flux
total
flux
a
linkage
flux
in
between
linkage
with
flux
with
permanent-magnet
mesh total
flux
vector
winding
linkage
rotor
rotor
linkage
V
the
degrees
mechanical
permanent-magnet
linkage
the
degrees
electrical
displacement
of
(V)
(V)
displacement
angular
axis
Tin
of
(H)
number
VPrk
component
Lkj (H)
of
P
0r
harmonic
(H)
MO
Lk j
the
of
coefficient
k
(Wb)
due
vector
winding
(Wb)
vector
k
due
IPP,
f lux
mesh
linkage
permanent-magnet
Coe
angular
per
(Or
to
the
rotor
velocity
of
the
rotor
in
electrical
radians
of
the
rotor
in
mechanical
radians
second
angular
per
due
vector
velocity
second
Ar
relative
90
permeability
permeability
of
free
space
vi
(H/m)
CONTENTS
Paap
Acknowledgements
No
i
---------------------------------------
Abstract
-----------------------------------------------
ii
List
principal
iv
of
Contents
symbols
------------------------------
vii
-----------------------------------------------
CHAPTER
I:
INTRODUCTION -----------------------------
CHAPTER
2:
PERMANENT-MAGNET
MATERIALS ---------------
8
2.1
Permanent-magnet
characteristics
8
2.1.1
Magnetic
2.1.2
Normal
2.1.3
Intrinsic
2.1.4
Recoil
lines
2.1.5
Energy
product
energy
product
Curie
Demagnetization
of
CHAPTER 3
magnets
and
line
and
maximum
---------------------
10
---------------------
10
11
------------------
types
11
--------------------
stabilization
12
--------------------------------
PERMANENT-MAGNET
BRUSHLESS
20
DC MOTOR DRIVES -------------------------3.1
Evolution
of
the
3.2
Brushless
dc
motor
3.2.1
brushless
dc
vii
dc
motor
configurations
-------
---------
20
21
half-wave
Single-phase,
brushless
9
10
-----------------------
temperature
2.3
9
H, j -----------
coercivity,
2.1.7
9
Hc---------------
coercivity,
Operating
Permanent-magnet
Br --------------
remanence,
2.1.6
2.2
---------
motor
-----------------
21
3.2.2
Single-phase,
brushless
3.2.3
dc motor ----------------
dc motor ----------------
dc motor ----------------
dc motor ----------------
Three-phase,
3.3.1
dc motor ----------------
23
---------------------
23
description
Machine
Structure
and stator
CHAPTER
the
of
3.3.2
Rotor-positon
3.3.3
Theory
23
full-wave
brushless
3.3
22
half-wave
Four-phase,
brushless
3.2.6
22
half-wave
Three-phase,
brushless
-3.2.5
22
full-wave
Two-phase,
brushless
3.2.4
full-wave
rotor
23
-----------------------
24
sensor --------------
of
operation
25
---------------
3.4
Advantages
4:
INVERTER
4.1
Diode bridge -----------------------------
46
4.2
Power switching
47
4.3
brushless
of
AND CONTROL CIRCUIT
selection
4.2.1
Conventional
4.2.2
Gate-turn-off
4.2.3
Bipolar
4.2.4
Power
Inverter
----------------
4.3.2
Free-wheeling
recovery
48
---------------------------------
48
49
51
51
and reverse
diodes -------------------transient
viii
46
-----------
----------------
circuit
27
------------
MOSFET -----------------------
MOSFETdrive
---
47
thyristor
power circuit
Voltage
drive
DESIGN ------
thyristor
transistor
4.3.1
4.3.3
dc motor
snubber ----------
52
53
4.4
4.5
CHAPTER
Control
circuit
power
supplies
---------------------------
54
Logic
controller
operation
---------------
54
4.5.1
Motor
control
----------------
55
4.5.2
Over
4.5.3
Motor
speed
current
Discussion
5
MATHEMATICAL
of
5.1
Branch
5.2
Mesh
5.3
Branch-Mesh
5.4
System
5.5
Inductance
5.6
Computer
5.7
System
protection
commutation
4.6
BRUSHLESS
CHAPTER
MOSFET drive
and
results
------------
------------------
55
56
56
---------------------
MODEL FOR A 3-PHASE
DC MOTOR DRIVE
----------------
85
frame -------------------
reference
89
frame ---------------------
reference
transformation
mechanical
discontinuities
Turn-on
5.7.2
Turn-off
---------------
96
97
--------------------
implementation
5.7.1
---------------
94
equation
variations
84
------------------
101
-------------------
discontinuity
--------------
discontinuity
98
-------------
102
102
5.8
Master
5.9
Discussion
6:
DETERMINATION OF MOTOR PARAMETERS--------
135
6.1
Stator
phase
resistance
135
6.2
Stator
phase
self-inductance
6.3
Stator
phase
mutual-inductance
6.4
No-load
matrix
---------------------------of
stator
results
winding
Discussion
of results
ix
105
--------------------
------------------
flux
due to the permanent-magnet
6.5
103
---------------------
135
138
linkage
rotor --------
--------------------
139
140
CHAPTER
7:
RIPPLE
7.1
IN
Factors
BRUSHLESS
Effect
of
commutation
7.1.2
Effect
of
phase
7.1.3
Effect
of
incorrect
7.1.4
Effect
of
rotor
7.1.5
Effect
of
stator
7.1.6
Effect
of
ripple
7.2.2
Increasing
the
7.2.3
Adjustment
of
7.2.4
Reducing
rotor
the
inductance
field
152
methods -------
magnet
number
pole
of
arcs ---
phases ----
Improvement
to
stator
the
of
the
slots
--------
profiling
155
155
phase
current
------------------------
implementation
153
current
--------------------------
profiling
152
154
-------------------------
Adjustment
152
angle-153
-------------------------
7.2.6
waveforms
151
-------------
winding
the
current
151
stator
of
Practical
---
teeth
commutation
Skewing
7.4
150
the
7.2.5
Current
-----
----------------------
Adjusting
7.3
commutation
inhomogeneous
7.2.1
7.2.7
of
minimization
controller
150
-----
adjustment
stator
and
149
------
waveforms
on
149
----------
events
emf
sensors
magnetization
CHAPTER
ripple
7.1.1
Torque
148
DC MOTORS------------
torque
effecting
position
7.2
OF TORQUE
AND MINIMIZATION
ANALYSIS
156
156
of
-----------------------
158
8:
CONCLUSIONS
AND RECOMMENDATIONS ----------
191
8.1
Conclusions
-----------------------------
191
8.2
Recommendations
-------------------------
x
193
195
REFERENCES -------------------------------------------
APPENDIX
Al
: A1.1
A1.2
Physical
details
brushless
dc
Main
motor
APPENDIX
A2 : Fourth-Order
motor
------------------------
integration
A3 : Shaft
encoder
specification
APPENDIX
A4 : Paper
entitled
"Current
Pulsation
dc
Brushless
J.
G.
K.
Al-Hadithi,
International
27-30
Profiling
1.
V.
in
September
R.
Vadher,
Conference
Florence,
on
Smith,
The
llth
Electric
Italy,
1992 --------------------
xi
for
Drives",
Kettleborough,
V.
207
-------------
Minimisation
Motor
206
-------------------
APPENDIX
Vehicle,
202
Runge-Kutta
numerical
Torque
202
----------------
tachometer
and
parameters
the
of
209
CHAPTER
I
INTRODUCTION
The
has
increased
due
mainly
drive
of
number
to
a number
of
the
materials.
available
large
permanent-magnet
has
neodymium-boron-iron,
size
the
of
machine
Prior
permanent
could
synchronous
1969,
Buschow,
material
by
National
its
which
this
available,
motor
was
were
research
clearly
and
led
to
competitors
Aeronautics
spacecraft
such
developed
Space
program,
for
unsuitable
1
have
replace
this
become
machines
traditional
early
in
and
brushless
the
March
samarium-cobalt
materials
Administration
to
in
or
a rare-earth
for
more
in
induction
160kjM-3,
to
these
using
reported
confirmed
has
Ferrite
However,
approaching
several
reduce
and
with
drives.
was
then
the
machines
Westendrop[21
figure
both
to
seriously
density
rare-earth
and
alloy
and
servo
which
serious
brushless
and
energy
Since
becoming
The
an
1969
Das(31.
readily
Luiten
with
September
dc
or
switched
efficiencyll).
Alnico
compete
be
are
samarium-cobalt
possible
available,
not
motors
it
its
only
were
-magnets
made
power
which
of
particular
improve
and
1970,
to
materials
in
in
to
development
materials,
brushless
devices
currents
the
and
in
permanent-magnet
semiconductor
enable
electronically,
and
is
This
years.
improvements
by
devices
high-power
20
drives
motor
advances
about
switching
dc
last
the
substantial
brought
semiconductor
brushless
during
considerably
technology,
now
for
applications
machines.
sixties
(NASA)
brushed
application[').
by
the
during
machines
In
recent
brushless
years
robotic,
and
aerospace
because
mainly
develop
brushgear
reduces
machines
to
torque
acceleration
Moreover,
required.
the
operate
at
motors
have
is
which
the
requirements
higher
can
capabilities
of
elimination
much
11,4,5,61,
applications
deceleration
maintenance
for
attractive
producing
and
the
become
vehicle
electric
large
their
high
the
have
drives
often
commutator
and
and
the
allows
brushed
than
speeds
machines.
Brushless
multi-phase
stator
position
and
stator
the
inverter
an
the
signals
between
is
field
the
controls
switches
and
By
is
techniques.
efficient
devices
ratings.
Their
switching
losses,
and
on
has
switching
and they
generated
and
so that
improves
stator
surface(7).
machine
provides
the
motor
windings
and
them.
The
to
the
off
appropriate
should
2
must
over
requires
the
and
voltage
be
fast
be robust
control
advantage
design
adequate
control
extra
important
inverter
speed
for
need
an
the
with
(PWM) techniques
the
have
are
the
permanent-magnet
delivered
turns
reliable
to
the
rotor
to
motor,
the
to
eliminating
available
switching
brushless
the
circuit
to
or
current
construction
the
close
which
PWM inverter
An
This
the
sensor,
and the
excitation
the
devices,
signals
pulse-width-modulation
power.
other
being
power
logic
commutation
of
drive
of
supplying
sequence
direct
speeds.
windings
inverter
feeds
field
stator
all
determine
rotor-position
characteristics
heat-generating
required
the
of
the
at
ensured
thermal
The
inverter
single
rotor,
to
a sensor
which
The
output
synchronism
the
windings,
windings.
from
a permanent-magnet
to
and
current
minimize
easy
to
the
drive
to
reduce
of
the
complexity
various
different
being
MOSFETs
power
the
the
course
An accurate
and
is
flexible
turn-on
overall
includes
and
particular,
the
behaviour
and
built
to
during
be
can
model
to
used
It
as
such
In
switches.
torque
the
predict
implementation
the
drive.
the
inverter
of
analytical
convenient
of
set
discontinuities
the
of
a
of
performance
The
optimised.
consists
a
design
the
enable
be
to
circuit
turn-off
and
drive
providing
the
studying
to
required
thesis
this
equations,
for
is
model
in,
developed
tool
led
available
PWM inverter
the
a brushless
of
differential
are
which
for
Consideration
circuit.
research.
system
performance
model
devices
chosen
this
of
drive
the
of
torque
of
ripple
minimization.
in
Interest
several
low
Most
previous
dqO
aPO
torque
be
windings
be
to
assumed
to
thereby
placed
ripple
prediction[121.
assumptions
dc
motor
non-sinusoidal
be
sinusoidal.
balanced,
on
are
has
19,10,111,
the
unnecessary
a
the
inductance
and
The
phases
of
phase
the
3
induced
the
are
also
are
to
torque
frame
these
analysis
reference
since
emf
generated
variations
machine
restrictions
113,14,151 and,
trapezoidal
the
assume
machine
at
idealized
highly
the
which
rotor
problem.
particular
distributed
application
In
a
considerable
and
the
inappropriate
is
either
used
sinusoidally
voltage
be
aerospace
models[81
this
can
have
for
early
although
frames
reference
some
began
drives
motor
potential
ripple
models
to
In
constant,
when
speeds
their
when
realised.
assumed
was
or
ago,
was
applications
speed
simulating
years
dc
brushAss
with
the
brushless
waveform
rotor
position,
and
it
a much better
provides
This
approach
frames.
associated
Moreover,
simulated
and
currents
are
all
its
essentially
windings,
frames,
it
At
one
has
matrix
obtain
An
the
of
the
All
the
for
of
flux
machine
phase
inductance
of
a
methods(16,171
for
numerical
with
of
the
brushless
the
permanent
linkage.
was measured
provides
system
varying
pattern
of
for
the
a clear
drive.
the
4
topologies
which
drive.
the
conduction
to
used
the
solve
with
possible
model,
are
a brushless
to
changing
behaviour
model
winding
the
significance
efficiently
the
in
the
step
required
switches.
representation
stator
waveform
of
accurate
accurate
to
considered
understanding
time
copes
due
semiconductor
tensor
the
it
since
reference
of
time-varying
equations
reduces
actual
windings.
little
of
the
dqO and aPO
each
is
computers.
Kron's
differential
are
is
reference
frame
to
fictitious
a
at
digital
any
relates
and
simulation.
in
disadvantage
that
this
voltages
the
be
to
reference
the
of
inverted
be
topology
system
saturation
was
the
circuit
both
model
thesis,
approach
it
important
this
equations,
the
to
since
an
time,
high-speed
In
The
to
from
phase
enables
results
machine
the
in
whereas
However,
solution.
modern
correct,
frame
reference
in
reference
frame
may be included
saturation
relates
actual
the
of
aPO
reference
available
inclusion
machine
phase
the
readily
more
dqO and
experimental
since
magnetic
frame,
the
studies.
analytical
complication
the
corresponding
compared,
any
the
with
of
use
directly
Although
for
eliminates
also
transformations
basis
The
at
motor
-magnet
open-circuit
requires
the
component
of
voltage
800 rpm and this
was
determine
to
used
linkage
flux
no-load
an
mathematical
model
significant
loss
permanent
high
coercivity
The
degree
and the
inductance,
position
and
the
and
the
torque
switching
torque
output
stator
generated
optimal
ripple
is
the
the
the
model
a brushless
high
a
developed
emfs[181.
To
generated
for
due
stator
current
is
less
in
as
winding
inductance
the
PWM current
impact
current
thesis,
this
to
an
robotic
stator
in
waveshape
than
The
the
studied
to
winding
result
in
respect
5
the
such
The effect
with
practice,
operation.
deviations
waveforms
generated
for
constant
together
smooth
the
to
be
must
In
applications
requiring
constant
and the
flat-top.
are
which
a
any
phase
waveforms
position
emfs
at
stator
obtain
current
a 120*e
motor
the
instantaneously,
rise
of
is
obtained
mathematical
of
emf
stator
frequency
to
thesis
correlation
product
with
current
the
giving
rotor
determined.
the
the
due
results
phase
of
of
good
of
the
ripple
control
advancing
reducing
to
fringing,
torque
undesirable
on the
torque
the
to
by
described.
generated
and the
due
the
the
technique
do not
currents
a
without
produced
in
developed
both
be trapezoidal
must
The
rotor.
constant
remain
experimental
in
functions
rectangular
120*e
and
the
torque,
output
flux
the
to
with
proportional
and
winding
considerably,
if
model
electromagnetic
currents
stator
samarium-cobalt.
confidence
is
speed
of
of
modelling
The
and
assumed
the
permanent-magnet
simplified
is
simulated
of
the
accuracy,
experimentallyr
between
the
to
of
mathematical
verified
emfs
due
may be
-magnet
for
expression
minimize
the
on the
torque
the
generated
emf,
and
increasing
the
controller
ripple
are
also
examined
are
and
emf
provide
torque
ripple
is
an EPROM to
the
rotor
the
position
The main
(1)
achieved
to
provide
characteristics
(2)
(3)
to
develop
the
motor.
an
design
to
the
validating
Chapter
2 summarises
and the
materials
3 presents
drive
and
and
its
a detailed
explains
dc
specifically
MOSFET PWM inverter,
controller
Chapter
permanent
and
5
drive
are
also
with
the
of
the
power
permanent-magnets
dc motors.
theory
of
3-phase
a
briefly.
of
the
for
voltage
design
details
the
6
drive,
Chapter
dc
motor
operation
types
of
4
Chapter
a
mathematical
brushless
its
Other
design
supplies
:
a brushless
of
with
for
suitable
as follow
examined
together
for
model.
brushless
discusses
the
model
characteristics.
develops
-magnet
is
description
and
motors
brushless
for
to
relation
drive.
mathematical
thesis
available
in
the
and
of
dc motor
characteristics
torque-speed
brushless
deals
the
of
are
mathematical
the
of
position
understanding
a
proposed
The organisation
thesis
accurate
build
and
the
controller
waveform
brushless
the
of
of
instants.
better
a
the
compensation
current
this
of
both
necessary
rotor
a PWM current
and commutation
objectives
the
practice,
stator
is
encoder
on
using
to
ripple
fluctuations
and
information
In
torque
synchronized
deviation
stator.
adjust
is
an incremental
accurate
the
the
minimize
ripple
the
waveforms,
to
which
the
and
events
generated
relative
methods
Since
presented.
commutation
to
other
brushless
model
and
explains
source
of
the
drive.
for
a
its
implementation
methods
for
Chapter
and
torque
the
accurate
measurement
7
presents
a detailed
production
5.
Chapter
the
of
Particular
drive
factors
which
implementation
of
current
profiling
chapter.
for
further
The
conclusions
work
are
of
in
presented
7
the
is
effect
this
parameters
motor
investigation
attention
and
the
of
using
ripple
torque
6 describes
Chapter
computer.
on a mainframe
is
model
to
given
into
the
presented
in
the
this
ripple.
also
presented
thesis
Chapter
and
8.
output
Practical
in
this
recommendations
CHAPTER
2
MATERIALS
PERMANENT-MAGNET
Permanent-magnet
is
materials
maintained
ly-app
external
in
even
lied
field.
special-purpose
rotating
electromechanical
devices.
is
material
in
loop,
the
which
are
opposed.
in
the
B-value
the
includes
flux-density
applied
against
H,
the
resultant
that
intrinsic
to
magnetic
field
normal
shown
flux-density
of
Bi
loop
in
and
material
machine.
B and
is
of
itself
use
the
magnetizing
8
plotted
intrinsic
the
intrinsic
for
in
gOH
is
Bi
only
The
intrinsic
The
flux-density
flux-density
the
of
the
intrinsic
the
represents
2.2.
Figure
it
If
the
flux-density
field
loop
is
which
a
in
shown
loop[211,
both
the
H.
lies
which
loop
hysteresis
and
B
demagnetization
B-H
from
the
hysteresis
loop
typical
field
magnetic
H,
types
flux-density
the
the
normal
proportion
the
permanent-magnet
the
material
the
represents
in
material
its
hysteresis
called
the
as
from
loop
used
devices
of
H and
the
is
resulting
hysteresis
quadrant
contributions
the
other
permanent-magnet
field
of
.
the
of
most
In
known
and
large
a
a permanent-magnet
second
quadrant
[19,203
2.1,
In
portion
second
for
magnetic
That
characteristic
Figure
loop
in
operates
and
which
CHARACTERISTICS
2.1.
Figure
of
frequently
are
machines
hysteresis
in
shown
presence
They
2.1 PERMANENT-MAGNET
A typical
the
field
a magnetic
produce
particular
design
of
Bi,
intensity
a
the
H
are
by the
related
equation
B-g,
Several
terms
in
used
H+
Bi
permanent-magnet
studies
defined
are
below.
MAGNETIC REMANENCE, Br
2.1.1
If
magnet
and
flux-density
displacement
curves,
then
removed,
or
remanence
The remanence
since
this
at
is
an
unmagnetized
there
remains
a
Br due
to
non-elastic
the
between
walls
same for
the
H is
point
to
applied
boundary
the
of
domains(22).
is
force
a magnetizing
the
both
magnetic
the
B and Bi
shown in
as
zero,
residual
Figure
2.2.
NORMAL COERCIVITY, HC
2.1.2
Increasing
eventually
value
2.1.3
the
the
magnetic
coercivity
WRINSIC
The
intensity
zero.
It
material
to
changes
in
several
times
intensity
at
a
withstand
its
for
force
or coercive
Hcj
coercivity
is
flux-density
normal
field
intensity
field
magnetic
B to
this
H
The
zero.
is
situation
Hc.
COERCIVITY, Hci
intrinsic
field
negative
the
reduces
of
termed
the
which
measure
the
of
demagnetizing
than
the
value
intrinsic
the
that
of
9
the
of
magnetic
flux-density
ability
forces
The
magnetization.
greater
is
magnitude
Hc(21,23).
Bi
the
of
without
of
is
magnetic
permanent
Hcj
may
be
2.1.4
RECOIL LINES
If
demagnetization
the
(see
2.3)
Figure
does
demagnetizing
line,
is
is
loop
This
very
termed
the
specified
the
the
the
2.4
the
a
a
the
loop.
minor
slope
a straight
line
this
of
the
of
Along
119,211
of
and
the
strength
the
B
of
and
point
the
energy
maximum
the
of
designated
recoil
line
H
at
the
of
value
energy
product
to
by
Bd
is
energy
corresponding
the
with
corresponding
being
stored
permanent-magnet.
and
force
the
along
material,
coercive
point
product
measure
demagnetization
the
energy
the
a
any
flux-density
the
of
at
permanent-magnet
and
respectively.
is
typical
shows
for
maximum
points
this
Hd
and
obtained
on
the
.
OPERATING LINE
The
the
of
permeability
product
material,
indicates
flux-density
2.1.6
half
by
replaced
permanent
the
It
curve.
product
curves
line
original
the
of
The
intensity
field
magnetic
permanent-magnet
Figure
from
or
is
product
energy
demagnetization
in
recoil
B/H
ENERGY PRODUCT AND MAXIMUM ENERGY PRODUCT
The
and
line[20,211.
the
as
may be
Hb
re-applied
half
recoil
the
lower
Hb is
upper
and
zero,
the
the
along
narrow
value
along
When
the
the
120,221
.
material
2.1.5
rather
follows
to
return
loop.
characteristic
to
reduced
returned
not
but
curve
hysteresis
minor
then
and
characteristic
is
field
point
magnet.
Q in
The
Figure
line
2.5
(a)
intersecting
10
is
the
the
operating
recoil
point
line
at
of
this
is
point
the
point
P
the
magnet.
of
any
additional
is
their
and
demagnetization
curve,
The
in
the
that
is
the
2.4
energy
depends
point
field.
reaction
have
materials
points
in
of
maximum
of
sources
shown
permeance
Figure
operating
operating
as
is
here
permanent-magnet
characteristics
the
on
are
linear
normal
2.5(b).
Figure
CURIE TEMPERATURE
2.1.7
The
Curie
that
a
is
temperature
ferromagnetic
properties
magnetic
main
materials
features
not
it
and
is
which
important
clearly
to
subjected
all
temperature
a
TYPES
the
small
three
types
of
permanent-magnet
machines
rotating
briefly
are
below.
Ceramin-ft-rrfte
magnets
have
These
characteristic,
product
is
of
in
used
summarized
high
lost
at
value.
2.2 PERMANENT-MAGNET
The
temperature
that
are
circuit
this
approaching
(a)
slope
from
presence
Rare-earth
whose
it
since
point,
available
the
The
circuit.
operating
is
line,
operating
magnetic
ideal
on
the
a
a
of
low
force
coercive
around
electrical
resistivity
relatively
cheap
remanence
of
up
to
of
are
used
and
widely
11
OAT,
about
1010 gfIm.
about
demagnetization
250kAm-1,
30kjM-3(20,221
of
and
linear
relatively
in
a
an
moderately
maximum
energy
extremely
Ceramic
small
a
dc
magnets
motors.
high
are
(b) Alnico
magnet
is
This
It
has
high
a more
electrical
resistivity
Samarium-rnhalt
rare-g-nrt-h
This
has
material
force
coercive
than
greater
advantage
on
is
material
used
its
above
is
materials
2.3 DEMAGNETIZATION
When
the
a
flux
armature
reaction,
machine[22,231.
demagnetized
use.
The
it
it
and
and
its
and
the
in
used
decrease
ensure
due
is
necessary
process
12
for
magnets
of
curves
of
the
it
OF MAGNETS
a
machine,
rotating
factors
such
dismantling
the
for
and
falls.
to
or
that
first
availability
cost
AND STABILIZATION
To
the
of
2.6.
Figure
is
great
consequently
demagnetization
in
a
disadvantage
rare-earth
as
product
is
which
was
used,
product
energy
Samarium-cobalt
over-load,
stabilization
maximum
high
a very
energy
equipment
the
may
),
military
permanent-magnet
operating
a
The
widespread
given
0.9T
to
materials,
increasingly
of
demagnetization
a maximum
basis.
disks.
comparison
and
has
other
cost,
becomes
(up
)
It
aerospace
becoming
material
A
of
high
memory
now
750kAm-1
volumetric
for
only
computer
the
a
linear
almost
(20,221.
that
60kjM-3and
around
0.5p0m.
about
remanence
to
400kjM-3
around
are
(up
below
of
magnet
an
a high
of
a very
force
coercive
product
of
ceramic-ferrite.
characteristic,
a low
maximum energy
characteristics,
of
1.2T,
up to
of
l20kAm-1[20,221 Ia
(c)
than
material
demagnetization
a non-linear
remanence
a low
expensive
magnet
to
consists
be
is
of
not
stablized
of
subjecting
as
the
easily
before
the
to
magnet
during
encountered
momentary
the
to
continues
materials
short
operate
with
they
a
a
along
linear
pe rmane nt -magnet
not
characteristic
then
imperative.
almost
from
rotor
magnet
the
the
and
and
13
as
some
this
either
process
it
that
ensures
Permanent-magnet
a
recoil
Other
have
Alnico,
of
line
curves,
stabilizing.
form
that
stalling
demagnetization
need
such
or
have
their
therefore
stator
line.
recoil
by
achieved
Repeating
period[23).
with
materials,
non-linear
may be
This
than
greater
characteristics
coincides
do
slightly
the
stablizes
almost
which
the
of
for
times
effect
operation.
removal
machine
several
and
demagnetizing
a
stabilizing
a
highly
is
+B
+H
-H
-B
Figure 2.1 Hysteresisloop for a typical permanent-magnetmaterial
14
+H
-H
B,
-B,
Figure 2.2 Normal and intrinsic hysteresisloops for a typical
material
permanent-magnet
15
4.)
bi)
cis
+
0
La
AD
ko
16
C.)
I
I
Ivi
ce
E
10
PI,
E4
le
2>
.-
2
C)
U
:5
17
+B
0
I
Recoil
m
Demagn
-H
Magnetic field intensity H
(a) A non-linearmagnet
+B
On
-H
Magnetic field intensity H
(b) A linear magnet
Figure 2.5 Demagnetizationcurve and oPeratingline for
materials
permanent-magnet
18
q Ai!suop-xnlj
te
8
C?
8
C?
8
IT
8
V
0
ýI.
ni
4)
*m
V
9b
8
17
19
Im
2
l=
4)
CHAPTER
3
PERMANENT-MAGNET
BRUSHLESS DC MOTOR DRIVES
brushless
The
dc
characteristic
dc
of
inverter
firing
signals
these
signals,
stator
phase
in
the
controlling
improvements
of
range
aerospace
the
applications
segments
consequence
of
this,
space,
its
axis
with
position.
corresponding
to
response
through
position
the
and
sensing
led
has
This
brushless
to
increased
greatly
dc
and
significant
motors
in
both
drives.
high-performance
dc motor,
through
winding
commutator
the
OF THE BRUSHLESS DC MOTOR
a conventional
armature
made
materials
have
power
and other
is
sequence[241.
rotor
for
solid-state
controls
In
magnetic
performance.
applications
3.1 EVOLUTION
In
input
consists
operation
current
required
both
for
stator
in
directs
the
a
iriverter.
rare-earth
devices
solid-state
it
sensor
solid-state
in
windings
conventional
stator,
rotor-position
inverter
the
Developments
the
the
a
Motor
sensor.
when the
for
of
a wound
rotor,
and a rotor-position
self-synchronous
torque-speed
Constructionally,
motor.
a pe rmane nt -magnet
linear
a
that
to
similar
separately-excited
has
motor
to
the
armature
a direction
When a commutator
armature
the
is
field
is
20
is
the
on
As a
coils.
stationary
in
by the
brush
determined
connected
to
sliding
armature
segment passes
coil
supplied
brushes
stationary
connected
in
current
under
to
a brush
the
either
the
positive
this
Since
negative
switches
Attempts
to
3-phase
the
have
armature
the
are
located
on
on
evolution
from
the
brushless
been
rotating
poles
dc
motor,
by
replaced
the
the
electronic
the
to
that
the
the
of
machine
illustrates
3.1
to
machine
and
the
at
convenient
so
stator
switches
six-switch
coil
more
commutator
led
through
Figure
rotor.
the
a
switches,
conventional
where
by
far
of
have
switches
armature
clearly
cost
prohibitively
achieved
than
field
of
each
rather
coils
the
is
It
switches.
the
be
supplied
of
commutator
segments
clearly
being
sensor.
stationary
many
machine
position
each
reproduce
semiconductor
number
switching
rotor
rotor-position
has
the
reduce
with
of
would
brushless
inverter(251,
correct
a pair
dc motor
a conventional
high.
and
by
to
and
requires
commutator
replaced
semiconductor
terminal,
supply
electronic
be
to
segment
to
an
with
the
the
or
3-phase
the
and
brushgear
have
the
rotor-position
sensor.
3.2 BRUSHLESS DC MOTOR CONFIGURATIONS
is
It
configurations
by
3.2.1
the
and the
As
shown
single-phase
single
Positive
in
winding
semiconductor
torque
way that
over
only
is
current
of
selection
delivered
the
to the
3.2
BRUSHLESS DC MOTOR
(a)
,
this
once
energized
switch
by
different
of
switches.
HALF-WAVE
Figure
number
dc motors
semiconductor
SINGLE-PHASE,
a
envisage
brushless
of
number of phases
windings
to
possible
Sl.
180*e
21
per
Although
of
the
has
arrangement
360*e
the
rotation,
through
a
the
generates
motor
as
shown
in
3.2 (b) ,
Figure
continue
the
the
over
The
3.3(a)
Sl
switches
Figure
3.3(b)
torque
falls
3.2.3
the
3.4
in
is
3.2.4
by
are
indicates
at
any time.
used
half-wave
brushless
displaced
by
point
and
the
that
current
the
winding
use
Provide
at
any
dc motor
120'e.
at
which
as
the
excitation
in
this
sýheme
phase
with
only
time,
and
switch
is
phase
only
its
Its
one
third
the
lack
path
turns
windings
sequence
the
for
3.6
made via
available
at
any
of
the
the
are
the
The
neutral
illustrates
The
main
and the
low
cost
are
the
diodes
current
motor
low
windings
available
free-wheeling
inductive
mutually
time.
disadvantages
of
3-phase
the
3.7
Figure
simplicity
off(7).
22
of
Figure
path
of
package.
50% of
windings
in
shown
are
phase
switching
only
pattern
one
two
BRUSHLESS DC MOTOR
A return
an alternative
semiconductor
and
continuous
motor
that
stator
electronic
usage,
not
with
The
HALF-WAVE
flows
of
100%
consequently
BRUSHLESS DC MOTOR
90'e.
star-connected
advantages
figure
semiconductor
positions
rotor
arrangement
3.5
THREE-PHASE,
The
the
shows
Figure
windings
in
there
of
zero.
displaced
given
is
torque
resulting
two
is
winding
stator
TWO-PHASE, FULL-WAVE
mutually
to
rotation
dc motor
by the
360'e
per
The
shows
to
Figure
of
twice
and S2.
but
utilised,
brushless
full-wave
energized
angular
BRUSHLESS DC MOTOR
FULL-WAVE
single-phase,
is
cause
180*e[26).
remaining
SINGLE-PHASE,
3.2.2
inertia
rotor
has
to
when a
a very
limited
for
potential
inductive
the
where
characteristic
3.2. S FOUR-PHASE,
four
only
of
3.10.
brushless
on
The
are
or
torque
in
given
motor
3-phase
The
3.11
has
which
requires
the
stator
for
the
previous
present
research
in
detail
is
3-phase
PWM inverter
circuits
are
3.3 MACHINE
power
67% and
this
stator
diodes
the
and gives
full-wave
chapter
parallel
is
torque
machine
with
of
of
than
more
used
is
and
commutation
in
later
by
The utilization
The design
the
Figure
energized
in
are
in
shown
winding
output
The
section.
discussed
supply
in
supplies[7).
recovery.
configuration
and
shown
dc motor
configurations(71.
has
time,
as
90*e,
BRUSHLESS DC MOTOR
energy
following
the
two
Feedback
winding
by
2-phase,
the
of
star-connected
inductive
for
switch
that
3.9
Figure
of
displaced
one power
only
brushless
full-wave
motor
any
at
FULL-WAVE
inverter.
6-switch
each
needs
to
a 3-phase
dc
mutually
stator
similar
DC MOTOR
BRUSHLESS
energized
The motor
THREE-PHASE,
3.2.6
the
is
which
performance
3.3.1
large.
sequence
brushless
half-wave
windings
one
Figure
a
logic
required
HALF-WAVE
4-phase
The
a
the
and
is
energy
winding
systems
3.8.
Figure
has
high-performance
either
in
the
in
described
the
associated
control
logic
4.
DESCRIPTION
STRUCTURE OF THE ROTOR AND STATOR
The
research
3-phase
has
6-pole
a
rare-earth
1.3kW
machine
samarium-cobalt
23
used
in
the
permanent
present
-magnet
A non-magnetic
rotor.
to
rotor
magnets
large
centrifugal
stator
star-connected
rotor
profiled
are
(27).
trapezoidal
velocity
operation.
3-phase,
the
concentrated
the
that
the
of
and
the
emf
is
generated
tachometer
a
provides
signal.
Many
methods
position
of
SENSOR
are
transducer,
machine,
diodes.
is
Hall-effect
with
around
the
stator
phases.
to
airgap
sense the
The output
of
logic
high
for
a north
magnetic
pole,
or
vice
versa
when the
The
reverses.
amplified,
to
element
incorporated
respectively
output
a signal
with
its
into
a single
the
output-voltage/flux-density
block
position
to
logic
rotation
of
pole
of
associated
chip.
diagram
the
of
usable
for
characteristic.
24
to the
switches
low for
the
element
magnitude,
3.12(a)
from
a south
rotor
Hall
the
most
60*e apart
relative
flux
is
and each
circuit
is
and (b)
show
electronic
Figure
the
experimental
sensor
each
voltage
give
the
20*m or
spaced
rotor
optical
sensor,
to
rotor
capacitance
resolver,
fitted
elements
the
absolute
A Hall-effect
devices,
these
eg:
motor,
brushless
sensor,
photo
common of
dc
detecting
for
available
brushless
a
Hall-effect
encoder,
Hall
is
dc
presence
house
to
winding
such
the
the
surrounds
high-speed
slots
brushless
A
ROTOR-POSITION
3.3.2
18
The
in
during
has
core
sleeve
place
arising
winding.
poles
in
them
retain
forces
laminated
The
steel
stainless
chip
and
its
3.3.3
THEORY OF OPERATION
In
inverter
output
that
occurs
windings
thus
3.14,
switching
rotor
trapezoidal
Consequently
the
flat-top
phase
in
may be
electronic
commutation
the
supply(l).
corresponding
Only
phase
two
phases
this
MMF moves
travelling
motion
separates
waveforms
in
are
the
them.
and
rectangular.
the
with
direction
The
by
reverse
of
sequence
of
reversing
sequence
stator
This
steps
The
being
60*e
of
rotor
MMF,
any
given
the
and
are
rotation
to
continues
25
time,
de-energized
current
rectangular
discrete
the
at
utilized
120'e
wave[29).
of
are
are
3.16.
for
and
for
phase
phases
than
commutation
of
given
the
rather
of
particular
motor
phase-shifting
180*e,
The
Figure
conducting
Due to
by
current
in
given
by
reversed
sequence
a
3.15127,28).
Figure
The
trapezoidal,
also
the
rotation
for
the
to
coils
This
waveforms
is
and
clockwise
in
current
instant
position.
3.1.
output
torque
motor
shown
power
torque
give
such
given
rotor
continuous
to
switches
commutation
table
stator
circuit
stator
of
the
generated
the
and
rotation
in
causes
any
of
60*e
in
energized
emfs
at
the
rotor-position
inverter
pair
of
maximum
The
on
one
tabulated
and
current.
both
are
from
sequence
logic
the
control
intervals
at
are
figure
60*e
which
switching
The
a commutation
devices
two
sequential
the
by
to
current
3.13.
Figure
the
signals
sensor
switches
in
signals,
only
next
3.11
decoded
are
rotor-position
indicated
as
signals
the
Figure
of
windings,
six
to
response
rather
MMF attempts
the
the
as a
each
60*e.
stator
smooth
follow
to
the
minimize
until
for
waveform,
than
with
angle
next
the
which
switching
V-4
+
cn
Cd
J=
.
1=4
1-4
+
C r.
w-4
+
.0
-4 "-4
b.
0
0
10
cn
.0
,tt
ýZ
c;
Gn
.0
C
C
21
col
-a
c
rA
104
&,
00
C14
Z=
Cf)
'0
1
0.
6
C-4
26
00
1
1
1
1
1
c)
step,
at
process
the
which
of
rotor
dc
to
stator
the
and
The
the
is
speed
the
PWM techniques,
whereby
and
times
By
several
the
changing
average
ratio
voltage
to
at
the
a linear
3.17.
is
The
stalled,
loads
all
due to
produce
to
the
the
varying
is
devices
on-state
to
applied
motor
This
switching
cycle
of
the
winding.
the
than
produces
figure
by
controlled
per
higher
to
permanent-magnet.
rare-earth
stator
torque
in
a
speed
proportional
constant
remains
of
to
when
the
torque
shown
even
as
with
motor
as
flux
coercivity
applied
off
a start/stall
the
and
characteristics
machine,
voltage,
linear
magnetic
motor
voltage
same
60*e
by
advanced
The brushless
is
the
high
the
characteristic
characteristic
the
has
supply
current
torque-speed
since
continued.
motor
torque(7,24).
running
is
permanent-magnet
proportional
the
MMF is
motion
A brushless
conventional
stator
achieved
are
using
on
switched
a chopped
voltage.
the
time,
off-state
stator
average
be
can
winding
controlled(301.
3.4 ADVANTAGES OF BRUSHLESS DC MOTOR DRIVE
A brushless
from
the
brushes
dc
stationary
and
Long
commutator
, of
(b)
Elimination
With
the
and
of
produced
the
may be
by
speed
advantages
the
of
use
arising
elimination
high
summarised
maintenance
reduced
a commutator
(RFI)
(C)
and
several
windings,
and
These
life
has
armature
permanent-magnets.
(a)
drive
motor
of
the
energy-product
as:
due
to
the
lack
brushgear.
the
radio
commutation
frequency
arcing.
imposed
limitations
27
interference
by
the
brushes
and
commutator
of
(d)
is
vacuum
A high
(f)
A
input
excitation
is
(i)
Improved
brushless
applications
electric
size
to
a
light
for
weight
and
high
efficiency,
by the
rotor
magnet.
power
provided
heat
wide
and
a
a
given
as
the
characteristic.
dissipation,
the
since
power
windings
stator.
its
dc
a
environment
explosive
and
torque-speed
on the
over
4
Reduced
A linear
and
ratio.
output.
(h)
speed
capable
possible.
frame
small
Due
an
torque-to-inertia
power
are
in
both
Operation
(e)
high
a
is
motor
131,32).
range
hard
(g)
at
operation
speed
the
eliminated,
excellent
has
motor
such
vehicles,
performance
as
flight
medical
many
replaced
control
systems
28
the
characteristics,,
brushed
actuators,,
and machine
tools.
motors
robotics,
in
+V
-vdc motor
(a) permanent-magnet
+V
-v
dc motor
(b) inverted permanent-magnet
VS
(c) replacementof mechanicalcommutatorby elecrtronic switches
Figure 3.1 Evolution of brushlessdc motor from commutatordc motor
29
Stator
Rotor
VS
Dr
lion
;or
Figure 3.2 (a) Single-phase,half-wave brushlessdc motor drive
4
E2
0
360
180
540
Rotor angle, 'e
Figure 3.2 (b) Torque output for single-phase,half-wave brushlessdc motor
30
Stator
Vs
Rotor
I
N
VS2
c)tor
nition
nsor
Figure 3.3 (a) Single-phase,full-wave brushlessdc motor drive
Rotor angle,'e
Figure 3.3 (b) Torque output for single-phase,full-wave brushlessdc motor
31
z
1ý
0
46
C4
1..
4
Cd
.A.
Pw
6
IA
bo
aD
tz
C14
Ln
C4
cn
II
I
32
Ia
I
si
S2
S3
S4
0
90
270
180
Rotor angle,*e
"1"=S ON
'tO"=S
OFF
,
Figure 3.5 Switching sequenceand correspondingphasecurrent
for
fuU-wave
2-phase,
brushlessdc motor
waveforms
33
360
'0
e
2
CA
gn
P,
m
34
+V
Step I
Step 2
Step 3
+
Figure 3.7 Switching sequencefor 3-phase,half-wave
brushlessdc motor
35
0
1-
E-4
0
JD
1 \
0-4
/
_ _
_
I
sl
c
1
S2 0
S3
0
60
120
"I"=S
ON
180
240
"O"=S
OFF
300
480
360
420
Rotor angle, *e
,
Figure 3.8 Switching sequenceandidealized torque profile for
3-phase,half-wave brushlessdc motor
36
r.
10
rA
iz
37
S
tep I
+V
Step 2
+V
Step 3
+V
Step 4
Figure 3.10 Switching sequencefor 4-phase,half-wave
brushlessdc motor
38
z
C
6.
In
*Z
ý
22
zo
10
Gn
.0
4=.
Cd
0
4
T-4
en
I
39
vcc
V0
utput
Figure 3.12 (a) Block diagramof Hall-effect integrated-circuit
0
0
0
Flux-density
Figure 3.12(b) Characteristicsof Hall integrated-circuit
40
Sensor I
Sensor 2
Sensor3
ia
0
I
Ic
0
60
120
180
240
300
360
420
480
Rotor angle, *e
Figure 3.13 Hall-effect sensorsignalsand correspondingphase
current waveformsfor 3-phase,full-wave brushless
dc motor
41
step I
step
step 4
step
a
step
step
Figure 3.14 Switching sequencefor 3-phase,full-wave brushlessdc
motor
42
Ch
DF
DL
I
si
0
1
S3 0[
1s5 0
1
S4 OL
1-
S6 0
1
S2 Ot
0
60
"1"=S
120
ON,
180
"0"=S
OFF
240
300
420
480
360
Rotor position, *e
Figure 3.15 Switching sequenceand idealizedtorque profile for
3-phase,full-wave brushlessdc motor
43
SensorI
Sensor 2
Sensor3
Code 001
0
Conducting SI, S2
switches
11
S3, S2
S3, S4
S5, S4
1
c
c
000
SI, S6
a
/b
a
bc
bc
bc
00
S5, S6
a
a
a
Cuffent
sequence
10
l
1 Il
ia
Ib(
Ic(
0
60
180
120
240
360
300
Rotor angle, "c
Figure 3.16 Switching sequenceand corresponding phasecurrent waveforms
for reverseoperationof 3-phasebrushlessdc motor.
44
Speed
Figure 3.17 Typical torque-speedcharacteristicfor
brushlessdc motor
45
CHAPTER
4
AND CONTROL CIRCUIT
DESIGN
INVERTER
The
block
overall
circuit
in
shown
supply,
sequentially
to
to
drive
provide
power
Each
circuits.
detail
for
supplies
of
a
control
parts;
an
smooth
dc
dc
the
supply
circuit
a control
inverter
the
these
main
apply
windings,
for
and
provides
to
stator
signals
isolated
low-voltage
control
motor
four
which
inverter
the
inverter
the
comprises
rectifier
3-phase
a
4.1
Figure
bridge
uncontrolled
for
diagram
and
switches
the
drive
and
described
in
switch
is
circuits
below.
4.1 DIODE BRIDGE
dc
The
whose
output
bridge
and
Figure
4.2.
dc
is
is
smoothed
A 10
removed
and
the
as
connected
resistor
capacitor
smoothing
dissipates
diodes
power
both
withstanding
case
of
the
is
the
of
Figure
peak
line
regeneration,
For
current.
voltage
discharge
diode
uncontrolled
capacitor,
a parallel
using
kQ
full-wave
a
auto-transformer
when
energy
in
shown
across
the
the
supply
during
released
braking.
regenerative
in
by
rectified
a 3-phase
from
obtained
discharges
supply
The
is
supply
a
supply
and
voltage
therefore[301
46
4.2
should
voltage,
the
of
with
be
capable
a safety
of
margin
maximum
expected
load
222V
the
line
rms,
peak
VLlw
F2
=,
(pk)
314V
During
rise
VLine(pk)*
above
should
be
current
of
On this
basis,
voltage
capability
41A,
braking,
regenerative
was
If
capable
25A,
of
is
the
in
the
is
470V
and
800R,
M41
800V
of
rise
50%,
diode
the
of
carrying
motor
forward
a
current.
a peak
with
a peak
and
can
voltage
allowable
maximum
diode
power
supply
permitted
withstanding
which
the
used
the
dc
the
reverse
current
of
bridge.
rectifier
4.2 POWER SWITCH SELECTION
four
The
tYpes
main
inverters
power
electronics
4.2.1
CONVENTIONAL
The
It
has
The
three
that
to
are
the
main
to
current
the
which
determines
short
positive
fall
to
a
low
timer
inner
is
An
but
the
the
cathode.
the
value
no
very
increased
maximum
high,
turns
and
gate
acceptable
47
anode
the
current
gate
the
anode
and
gate
a
current
D0,331
.A
to
resistance
into
device
the
the
applied
is
thyristor
improves
power
and
cathode
control
this
gate.
provides
resistance
causes
current
gate
is
the
carries
gate
thereby
and
and
the
forward
thyristor
on
The
while
and
cathode
P-layer
outer
device.
switching
anode,
the
P-region
the
With
resistance
the
connections,
pulse
on-state.
off-state.
to
N-layer
outer
P-N-P-N
terminals;
electrode
the
contact
four-layer,
a
external
contact
below.
summarised
are
in
used
THYRISTOR
is
thyristor
device
switching
power
of
the
dissipation
the
forward
in
the
turn-on
must
be
not
is
thyristors
However,
auxiliary
3-terminal
at
the
application
of
inverter
gating
4.2.3
but,
turn
on-state
voltage
and
can
thyristor,
a
switching
speed
bulky
and
with
by
turned-on
the
To
gate.
in
hence
Like
period[24,33).
a
pass
forward
it
is
is
faster
the
continuous
a
recommend
a high
block
and
Its
to
GTO
the
the
associated
conduction
entire
negative
allows
current
manufacturers
the
unlike
voltage.
high
forward
voltage
when
block
to
unable
than
that
of
thyristor[24,30).
BIPOLAR
The
with
some
turned-on
when
turned-off
the
thyristor,
conventional
the
in
of
A GTO is
of
the
versatile
the
components
pulse
P-N-P-N
to
facility
circuitry(24).
during
current
reverse
to
layer
more
without
circuits
a positive
losses,
current
required
pulse
This
commutating
reduction
conduction
is
construction
brief
a
terminal.
thyristor
conventional
a
transistors.
GTO is
the
by
turned-off
forced
ensure
of
a four
in
similar
However,
gate
of
expensive
to
be
can
construction
a
device,
thyristor.
conventional
is
(GTO)
thyristor
switching
current
available
THYRISTOR
gate-turn-off
it
that
circuit
commutation
GATE-TURN-OFF
because
than
generally
device[34,351.
the
The
of
rating
higher
considerably
an
turn-off
4.2.2
The overload
exceeded.
bipolar
three
the
take
TRANSISTOR
terminals;
transistor
the
is
transistor
device
the
on,
into
a 3-layer
emitter,
a continuous
saturation
48
base
semiconductor
and
device
To
collector.
base current
is
required
to
it
off,
and,
turn
the
base
current
power
loss
is
the
in
the
function
a
to
as
which
base
base
current
current
current
be
must
inexpensive
and
thyristor[331,
can
with
possible.
4.2.4
POWER MOSFET
extremely
short
circuit
is
has
circuit
4.3.
relatively
The
on-state
resistance,
has
a
protective
self
distributed
a positive
the
second
drain
increases
which
over
the
the
by
breakdown
49
shown
current
die,
so
phenomenon
with
drive
in
on
to
that
of
be
it
the
the
and
2001C134).
which
the
Figure
temperature
to
a
device
and
depends
coefficient,
forcing
silicon
25*C
to
is
gate,
current
with
range
temperature
manner
through
a given
2gs
The
gain.
is
1
it
and
MOSFET
the
The
since
drain
than
device,
times.
high
the
relatively
than
switching
very
source,
for
doubles
MOSFET
a
N-channel
an
loss
power
approximately
display
for
symbol
with
the
is
less
turn-off
at
collector
faster
simple,
device
terminals;
f ast
and
the
a
rate
turn-off
which
of
as
has
the
transistor
very
current
reduced
limits
rate
times
of
To
transistor
considerably
a
turn-on
controlled
three
is
MOSFET
power
voltage-
switch
switching
being
a
be
following
and
at
base
should
which
bipolar
The
value
voltage.
The
reduced,
reduced
follow.
can
The
be
can
time
134,351,
transition
off
initial
high
turn-on.
phenomenon
or
collector-emitter
turn-on
after
on
instantaneous
the
the
the
possible
each
of
the
and
reduce
breakdown
second
product
instantaneous
The
zero.
during
dissipation,
power
necessary
rapidly
the
current
the
to
reduced
transistor
of
collector
reduce
be
must
acts
The
in
uniformly
does
bipolar
not
transistor.
their
positive
among
the
Due
high
temperature
to
these
drain
from
electrically
the
and
same
a
current
contains
rating
to
very
high
drain
1361
and
threshold
The
the
as
that,
channel
assumes
The
maximum
expected
normal
a
a
is
these
conditions,
produces
an
pinch-off
the
increase
in
is
voltage
constant
an
value
and
in
Figure
shown
devices
must
be
dc
voltage
and
the
in
the
operating
as
system,
As
current.
50
well
in
the
than
the
4.5
show
drain/source
the
until
drain
current
4.6.
worst
explained
is
Figure
the
capable
as
impedance
current,
reached
voltage
floWS(34,36).
in
drain
switching
supply
zero
current
increase
is
MOSFET
greater
in
the
with
flows
current
given
waveforms
oxide,
The
with
drain
a
isolated
diode
reverse
voltage
source
silicon
drain/source
gate
applied,
of
an
the
the
is
gate
layer
leakage
of
with
and
itself.
the
N-channel
4.4,
that,
the
structure
material
such
positive
the
Figure
transistor
for
chosen
available,
to
internal
small
switching
under
voltage
only
voltage
typical
an
are
The
the
of
research.
The
by
gate/source,
When
.
in
device,
the
sharing
availability
this
of
P-type
of
source
voltage-controlled
applied
MOSFET was
material(301.
the
structure
the
work.
shown
being
ready
restricted
present
N-type
of
current
the
MOSFETs
is
is
body
course
N-channel
the
device
semiconductor
the
follows
for
used
N-channel
and
P and
which
devices
devices,
in
built
description
because
parallel,
forces
to
and
current
both
Although
to
easy
coefficient
advantages,
high
voltage,
are
devices[34,36).
paralleled
PWM inverter
the
MOSFETs
Moreover,
the
in
of
handling
case
peak
rms
value
section
the
current
of
the
4.1,
the
maximum dc supply
and
maximum
450 power
currents
drain
and,
25A
and
the
since
motor
for
IRF
an
current
13A,
of
52A and a drain-to-source
of
rated
respectively,
drain
a continuous
500V was selected
of
voltage
has
current
470V
5.5A
are
MOSFET which
a pulsed
is
voltage
breakdown
inverter.
the
4.3 INVERTER POWER CIRCUIT
4.7
shows
Each
inverter
Figure
circuit.
snubber
and
diode
recovery
in
anti-parallel
diode
used
to
shown
in
4.3.1
MOSFET DRIVE CIRCUIT
the
produce
a
be
about
current
shortest
circuit
circuit
to
has
short,
to
and
drain
recovery
free-wheeling
its
MOSFET and
the
MOSFET
inverter
board
a
the
drive
a series
body
drain
is
diagram
the
rate
2nF)
can
at
be
The
charged
be
this
capacitance
very
the
turn-on
that
the
capable
During
low
sink
switching
51
of
turn-off
losses
sufficient
in
turn-on
the
and
These
are
are
the
drive
gate
rapidly,
times
the
and
supplying
impedance.
is
capacitance
discharged,
turn-off
to
speed
input
during
capacitance
and
small[36).
and
a
terminals
switching
the
which
must
the
device,
gate/source
current.
discharge
have
that
and
voltage-controlled
to
time.
possible
therefore
ensure
charge
a
applied
of
to
proportional
drive
a MOSFET
prevent
complete
is
MOSFET
flow
(typically
the
4.8.
Figure
must
gate
The
inverter
of
the
with
from
voltage
fast
a
diode
Since
contains
with
conducting.
diagram
schematic
arm
circuits,
connected
fast
a
it
must
conditions
both
consequently
very
In
the
pulse
transformer
very
low
is
i. e.
and
negative
and
R4
a
the
bias,
switches
holds
high
and,
since
IC1,
the
output
of
IC3
of
the
buffer
the
C3/R6
to
low,
the
explained
4.3.2
each
fed
back
IC4
and
output
IC1
the
TR2
device
inverter
two
conducting
is
IC2
input
of
the
goes
high
the
at
is
up the
outputs
the
to
drop
is
applied
to
turn-p
of
'A
transistor
MOSFET
goes
transformer
below
IC6
to
+15V
to
the
by
latched
IC4
applying
of
signal
at
of
output
the
control
and
the
condition
from
the
IC1
turns-on,
of
withdrawn
of
The
The output
push/pull
appearing
of
7.5V
this
speed
goes
the
signal
the
of
When
input
the
to
is
charge
spike
FREE-WHEELING
step
TR3
device.
the
This
acts
which
previously.
turn
IC6.
to
the
to
a low
produces
signal
R5 to
For
R3
half
at
added
via
low.
held
Transistor
Transistor
The
every
is
negative
The
to
this
causes
threshold.
gate
to
combination,
secondary
high.
IM
which
divider
control
low.
of
is
positive
IM
spike,
output
by
turn-off
PWM
the
of
alternate
to
the
of
version
voltage
voltage
with
output
The resistance
When the
high,
turned-on
gate,
as
is
TR2.
is
pair
input
IM
formed
transistor
the
positive
the
the
produces
output.
(7.5V).
resulting
windings
a differentiated
the
isolating
the
coupled
and
input
at
goes
output
tightly
square-wave
voltage
high,
has
consequently
usually
supply
of
reactances,
spikes
4 9,
.
Figure
circuit
T1
leakage
transformer
input
drive
gate
are
the
IC2
now
MOSFET
on(37).
AND REVERSE RECOVERY DIODES
switched
MOSFETs
conduct,
period,
one
52
at
intervals
each
device
for
of
60*e
120%
turns-of
At
f
and
and
during
the
end
a
of
second
turns-on.
inductive
The
to
connected
device
the
damaging
potentially
free-wheeling
operations.
4.10,
diode
the
turned-off
and
decreased
to
to
in
-As
reverse
diode
the
the
recovery
S3 begins
time
the
reverse
current
flow
further
external
diode
shown
4.3.3
TRANSIENT
diode
clamping
instantaneous
and
voltage
comprising
a
connected
across
constants,
a
longer
one
typical
voltage
without
a
so
in
slow
that
an
with
used
current
compared
with
fast
external
device.
internal
To
diode,
is
4.11
Figure
internal
transistor.
each
the
through
a
inserted
in
in
snubber
a diode
very
through
and
and
device.
the
one
short
the
through
53
the
not
is
has
Figure
MOSFET
cause
turn-off,
turns-off,
an
4.12
Figure
of
diode
at
recovery
may
circuit
the
provide
forward
a capacitor,
and
waveforms
As
circuit.
its
to
circuit
This
resistor,
current
may
inductance
circuit
The
spike.
4.11
Figure
due
action
stray
resistor,
snubber
SNUBBER
shown
clamping
characteristic,
increased
has
I,
MOSFET.
VOLTAGE
The
an
the
with
when S6
an
as the
is
diode
is
prevent
series
Figure
týe
allow
rating
MOSFET,
diode
anti-parallel
the
by
MOSFET contains
the
of
the
of
during
current
diodes
same current
time
this,
avoid
conduction
when the
a
'
4.3,
recovery
switching
with
way.
the
with
to
illustrated
case
winding
produces
paths
the
The free-wheeling
Figure
the
off
and,
current
conduction
a controlled
shown
However,
ceases
zero.
in
decay
parallel
switched
spike
provide
For
in
is
voltage
in
stored
which
diodes
switching
is
energy
therefore
two
and
time
a
4.13
with
much
shows
and
load
is
current
VDS is
when the
flow
a low
a finite
time
drain
current
into
the
rail
forward
biased
snubber
capacitor
decreases.
by
value
to
the
snubber
and
collapsed.
snubber
capacitor
VDS is
still
Load
current
which
low
quite
continues
VDs reaches
until
load
the
and conducts
voltage
capacitor,
diode
free-wheeling
the
when
diode
the
via
drain/source
The
charge,
has
level,
supply
the
current
at
held
require
to
drain
the
while
into
diverted
the
becomes
current[30).
4.4 CONTROL AND MOSFET DRIVE CIRCUIT POWER SUPPLIES
Since
4.8
Figure
supplies
are
the
are
are
lower
not
multiple
power
12V
square-wave
off,
energizing
individual
circuit
in
Tl
transformer
T2
by
rectified
15V
which
a
single
for
supplies
the
circuits
but
supply.
4.14
has
is
The
diode
60kHz
a
on
and
to
connected
five
to
drives
The
TRl
switches
which
the
The control
power
T6.
to
power
supplies,
supply.
Figure
IC1,
oscillator
transformers
are
power
floating
shown
pulse
pulse
outputs
supply
isolated
MOSFET drive
a common power
further
a
upper
floating
share
MOSFETs shown in
six
same potential,
The three
circuits
requires
the
at
the
of
individual
with
circuit
other
all
required.
provided
three
terminals
source
secondary
the
provide
and
control
[371
.
4. S LOGIC CONTROLLER OPERATION
In
response
IC
controller
which
control
the
the
LS7261
the
control
of
to
motor
Figure
of
inverter.
motor
current
54
position-sensor
4.15
The
using
signals,
generates
controller
an
six
provides
external
the
signals
PWM
control
voltage
and
a
sensor
providing
MOSFET
drive
oscillator,
saw-tooth
with
for
protection
the
,
motor
4.16
Figure
circuits.
an
over-current
windings
the
shows
the
and
controller
circuit.
MOTOR SPEED CONTROL
4.5.1
A saw-tooth
control
voltage
timing
for
connected
to
level
than
that
control
of
the
of
the
control
signal
varying
the
level
can
level
the
be
adjusted,
speed
saw-tooth
the
are
the
output
to
the
output
By
turn-off.
the
on and
switch
4.17,
stator
is
signal
signal,
Figure
motor
the
switches
switches
in
as
When
negative
control
signal,
control
to
the
the
of
14
RC
the
4.16.
more
pin
causing
and
of
is
The
motor.
and
the
and
winding
controlled.
OVER CURRENT PROTECTION
In
from
the
dc
The
motor
voltage
over
to
order
switches
the
low
applied
its
consequently
high,
the
the
Figure
at
variable
comprise
13
pin
signal
of
becomes
voltage
to
when the
that
in
shown
as
for
generator
signal
becomes
durations
average
14,
pin
external
the
provides
control
saw-tooth
saw-tooth
than
positive
4.5.2
the
Conversely,
13
pin
the
with
conjunction
speed
control
signals
turn-on.
off
to
applied
components
network
more
in
used
PWM and consequently
voltage
the
signal
rail
overload,
the
and
current
which
current
the
protect
is
a
0.50
flowing
through
to
applied
(pin
55
of
is
the
this
When
inserted
lower
three
ptl
the
MOSFET
between
MOSFETs.
creates
resistor
potentiometer
12).
the
and
windings
resistor
source
common
sensor
motor
connected
voltage
at
a
to
the
over
current
control
from
sensor
input
brake
the
3
turn-off
turn-on,
which
limit
current
the
of
(pin
Vss,
the
six
is
motor
disconnected
may be
value
output
by
changed
ptl.
is
motor
9).
the
and
the
and
potentiometer
braking
Dynamic
and
the
of
adjustment
to
The
supply.
half
than
more
disabled
are
signals
the
is
The output
the
shorts
effectively
control
+15V
1,2
signals
4p
signals
control
output
by applying
provided
6
5 and
together.
windings
4. S. 3 MOTOR COMMUTATION
The
to
used
based
signals,
and
300*e
this
by
direction
the
of
by
is
to
The motor
to
connected
of
Table
4.1.
the
rotor
direction
Forward
(19)
used
mutually
determines
pin
120*,
and
reverse
OV.
by applying
obtained
60*,
of
20 are
sequence.
+15V
applying
position
sequence
IC
commutator
output
motor
sensors,
1 and
pin
switching
the
control
sensors.
position
both
correct
by changing
selected
three
output
separations
position
so that
the
(19)
direction
is
has
60*e,
achieve
the
motor
1 and 20 ) are
the
of
separation
for
the
motor
the
of
(pins
of
sequence
available
between
research
separated
Pin
correct
are
inputs
the
of
changes
commutation
select
on electrical
Options
sensors.
OV, to
the
select
to
responds
provide
The commutation
windings.
in
to
sensors,
position
240'
IC LS7261
controller
4.6 DISCUSSION OF RESULTS
A selection
section.
signal
Figure
on the
of
results
experimental
4.18
shows
mark-to-space
56
the
effect
ratio
of
are
of
the
presented
varying
the
PWM signal.
in
this
control
It
is
Ul
EP LJ
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+
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1:3
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cc
+
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1:3
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13
C3
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+
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U-
Ln
Ln
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El
[s)
IS
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that
clear
the
vary
output
4.19
Figure
duty
the
frequency
voltage
on the
PWM controller
capable
the
reduces
high
time
period
on-time
for
10gs
a
switching
devices
Since
MOSFET has
the
60ns),
it
is
at
the
from
to
a very
to
signals
speeds
given
are
frequency
of
in
between
synchronism
is
ensured
waveforms
the
of
the
at
drain/source
and
MOSFET S3 turns-off.
free-wheel
the
path
and
a function
of
all
voltage
for
of
the
the
can
the
are
generation.
different
at
be seen
the
that
Thus
speed.
rotor
Figure
15kHz
permanent-magnet
4.21
free-wheeling
shows
diode
the
D6 and
MOSFET S3 when MOSFET SI
turns-on
These
that
phase
anti-parallel
58
it
and
device
the
noise
and the
speeds.
outgoing
of
field
stator
current
the
of
4.20,
than
switching
below
sensors
rotor-position
Figure
(less
in
loss
acoustic
the
amplitude
frequencies
possible
the
is
these
of
of
a 10gs
times.
a high
ripple
power
switching
7.
that
time
such
current
Generally,
because
The output
current
the
excessivi.
be avoided
the
reduce
keep
at
operate
in
turn-off
and
switching
short
the
switching
requires
turn-on
to
chapter
results
This
fast
very
to
order
same time
being
field
have
in
20gs
a
cycle,
off-time.
possible
in
frequency
a 50% duty
(up
consequently
in
results
is
considerably
be shown
as will
frequency
which,
frequency
and
The
research
frequency
switching
ripple,
signal
PWM signal.
this
of
will
windings.
sawtooth
the
course
turn
stator
the
amplitude
ripple
switching
motor
of
switching
torque
and
the
in
which
varying
high
at
current
high-frequency
This
in
increased
An
the
frequency
switching
operating
5OkHz).
of
effect
designed
of
varied,
to
applied
the
shows
may be
cycle
waveforms
falls
confirm
to
zero
diode
in
through
each
the
the
inverter
leg
and that
drain/source
the
Before
voltage
instantr
this
waveforms
for
torque,
PWM control.
these
in
change
in
shown
is
4.24.
Figure
voltage
waveforms
a steady-state
speed
70% duty
at
the
chopping
is
evident.
_Figure
when a load
inertia
from
rest
apparent.
of
the
running
current
start-up
period
analytical
drive
to
and run-up
that
output
study
having
is
for
is
current
the
relatively
these
is
59
smooth
waveforms
given
later
waveforms
voltage
mentioned
and
and
motor
of
of
effect
current
0.7Nm
to
set
reason
speed
a moment
is
started
222rpm.
is
It
times
the
response
during
the
and
linear.
A detailed
and
their
about
speed
motor
torque,
the
on
line
and
The
speed
for
4.25.
Figure
on these
of
as
waveform
PWM control
and the
applied
sudden
1Nm load
phase
a steady-state
that
performance
the
on
commutation
voltage
same
a torque
starting
and
the
voltage,
the
in
seen
a
line
spikes
current
shown
has
and without
to
of
with
be
to
shows
kg. M2 is
0.003
due
frequency.
can
due
are
4.27
voltage
emf
and
supply
are
PWM controller
spikes
it
phase
voltage,
and
phase
that
420rpm
and
current
switching
the
of
these
and
previously.
of
action
Voltage
waveforms
waveforms
20kHz
no-load
of
is
above
300rpm
of
snubber
are
a 70V supply
with
the
the
supply
motor
phase
into
instants
the
at
The
the
shows
cycle
these
described
same conditions
4.26
there
and
level.
evident
speed
moment
rail
a 50V
with
that
current
is
shows
drive
the
clear
phase
Figure
and it
the
supply
generated
4.23
waveforms,
the
the
a steady-state
It
voltage
flows
Figure
waveforms
1.6Nm load
current
speeds
shape.
voltage
the
shows
different
at
trapezoidal
the
4.22
from
conducts
reaches
the
Figure
capacitor.
diode
free-wheeling
the
in
three
effect
chapters
on the
5 and 7.
R
tu
iD
Ei
2
Gn
>b
cn
iz
COS.:
60
S.)
Dl
CPýc0
u
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, 00
C)
cnu
9)
CP
ci in
tn
in
CP
, T] ý-
:-Z.?
:
0
>%
4-
0)
1
9
ei
LL
0
4c
C)
17
ci
>
(0
Ln
C)
jM
U.
C)
.C
61
-0-i
D
CL
C
dro.
gate
G(+)o
inD(+)
3ý
Ed
1-4
source
1_
S
Figure 4.3 Symbol for an N-channelpower MOSFET
Source
Gote
(S " 02)
Dro
Ox i de
n
Figure 4.4 Basic N-channelMOSFET cross-sectional
structure
62
U)
0
bj)
Ln
In
IV
(6)
63
4)
2
0
IR
iuo.un:) u!uja
64
x
ce
;5
65
9
,
t
,
W. 01 s
OML 0,
'Wweis
om. 02
2
I-.
U
U
I...
L74
6
WNSIS
vW-0
al-ols
IDIU-0)
-4IV Not s
, OKLNoý
, 'k. 0 Is
, Diu -0 3
66
LLJ
U. LL.
Ln 0: Ln
w
U.
Ln
Ln
CK)LO
0
f14
10
C\IX-4
MX--4
W-0
(0-Ln
ý-NLO
c
L()CD
LL
mc
r-3
C,j
U-i
cy-
ý
aj-
-Ic
C\j
(i
LO
M
u
CL
LD
U-)
lz
>
Lr)
C'J
c
Cr
U-)
CO
>
>
N
m
C\j
LO
C\jLD
0ý<
m
>
0
Ln
c
C;,
+
67
LS
Rs
s
S5
IF
VS
L-b
J.
S6
S2
1
(a) SI and S6 ON
Ls
p
VS
(b) SI, S2 and D3 ON
LS
F
VS
(c) SI and S2 ON
Figure 4.10 Free-wheelingdiode operation
68
a
BYV29
MOSFET I
BYV29
BYV29
MOSFET 2J
BYV29
Figure 4.11 MOSFETswith externalfree wheeling-diodes
Figure 4.12 MOSFETswith snubbercircuits
69
(a)
-11-
(b)
Figure 4.13 MOSFET turn-off wavefonns (a) with snubber circuit
(b) without snubbercircuit
70
Ln
4
Ln
4>
+
+
41
in4
Ln
4
+
+
to
40
>
4n
-dm f4Pe41IL
i
fly,
ty
ty
co
ti
2KBE
c,
4
Ze
tn
>16
10
v
Ow
IV
a
LO
4
CY
-4
71
i
i
i
i i I-
I
10
Jw
8
iz
". 4
%0
LL
a
iz
c
ii-
a
-
--I
72
I0
-#j
0
in
m
0
g30
C
.-
N
V
V
C
I-1
IV
40
Lo
ci
-IJ
ci
ci
ci
-a
a
M
73
Sawtooth
Vo Ltage
tage
i
0
L11
Sw; tch
ControL
Si gna L
OFF
OFF
i-taff
-1t
an I
I.-TS-4
(a)
i
0
(b)
Figure 4.17 Modulation behaviour(a) with a high control voltage
(b) with a low control voltage
74
+
........................................
control
sawtooth
-------
.............
--------
vertical scale
- --------
-------
------------
-----
-- --
ý- -Wid-W-horizontal scale
2OWdiv
-------------
-
-- -- --------
PWM output
------------------------
--------
-------
(a) control voltage = 11.5V
K
------------------
----
.........................
...............
sawtooth
st
i'
%"Q
control
-a*-
verticalscale
0
5V/div
7- ------------------------------
0.
------
.............
r-:
----
-------------.......
2OWdiv
------ ---- --- ... .... ........
.
- -
...
...
......
LJ
ý-J--:
-1
------
horizontalscale
....
...
...
....
........
I
I'j
------
...
......
--------....
---------------A.......
...
.
PWM
......
output
..........
(b) control voltage = 7.8V
------------------ -------------- ... ..........
.........
--
......
--------
------
--------
-----
-------
..................................
... ..........
.
............
..........
----
................
------
...........
----
....
---------------
sawtooth
------
-
vertical scale
5V/div
horizontal scale
2%LVdiv
...............
..........
......
PWM output
.
..........................
(c) control voltage = 6V
Figure 4.18 Modulation behaviourwith different
control voltage levels
75
control
I
-----------------
...............
- ------------ ----------
UWtDOth
omtrol
......... ..
vertical scale
. 5V/div
0
horizontalSCAIC
------
---
-----
---------
--------
-----
...........
-----
-
2OWdiv
16
---------------------------------
PWM output
0
...............
-------------------
.
-------
...............
........
I -----------------
(a) sawtoothfrequency= 26kHz
...
...........................
.
....................
.
..............
N IN
IN Nt
v
----------------
0
........
.1... ...
.... ..... .... ..... .
...
0
....
....
- -------
...............
............
.... ...
...........
...............
....
sawtooth
control
vertical scale
5V/div
horizontal scale
2OWdiv
... ...
LJ
..........
.....
PWM output
----------------
(b) sawtoothfrequency= 36kHz
control
sawtooth
.... ......t
... ... .... ... ........ .....
----------------------
vertical scale
5V/div
h(xizontal scale
2(ý.Ls/div
--
PWM output
--------------------
-----
-----------
---
.
........
.........
....
(c) sawtoothfirquency = 46kHz
Figure 4.19 Modulation behaviourwith different
sawtoothfrequencies
76
--------------------------------
vertical scale
2OV/div
horizontal scale
20ms/div
........................
.........
....................
.......................
0
sensorl
.........
......
......
0
sensor
----
-------------
---
--------
--------
L sensor
..................
0
------------------------
----------------
...............
------- - -------
-----------------
-----------------
(a) speed= 200 rpm
verticalscale
2OV/div
I
------I................
------------------------
I ------- -----------
........
L
. ...................................
ýorizontalscale
20ms/div
--------
--------------6....
0
0
0
----------------
------- 4--------
sensorl
----------------
-------
sensor
........
................
.......
........
........
sensor
LFý
--------------..................
...............
------------.........................................
........
..............
........
......
(b) speed= 425 rpm
Figure 4.20 Rotor-positionsensorsignalsat different speeds
77
...............
......................................
----------------------
.......
.
............
......
.
.....
.
....
vertical scale
0.2A/ div
horizontal scale
200gs/ div
.............
.......
---------
...........
--------
----------------
f ...................................
......
.................
. ........
-
........
...............
------------
....
...
.. Ir
....
t4r0
............
.
........
....
Figure 411
Free-wheelingdiode (D6) current and power
MOSFET (S3) drain-sourcevoltage waveforins
78
vt,rtical scale
2OV/div
hoxizontalscale
200gs/div
...............
----------
................
----
----------------
........
----------
-------------------
................
-------
------------
--
---------------------- -T-----
...............
------- - -------- r
-----
verticalscale
IOV/div
horizontal scale
lOms/div
............
......
... --------------- ................
--- ------.........I
......
-- - - --- - -------
... .............
J.............
...............
...
A
1,
................
.................
(a) speed= 265 rpm
-----------------------
................
...............
...........
.
...............
..........
4
-------
....... . .......
.............. -------
------
-
-- ------
--
...............
------------------
-------------.............
.
verticalscale
IOV/div
horizontalscale
IOms/div
.......
....... --------------------------
................
---------------
--------
......
-----...........
---------------
-
-------------------------
---------------
. .......
j .......
...
...........
-
------------------------
.
.......
..
L
(b) speed= 500 rpm
Figure 4.22
Generatedphaseemf waveformsfor different speeds
(No-load condition)
79
------
---------
-
----------------
--
-------
---
----------
------
-
------
-----
---
----
.........................
-
-------
-----
---------
vertical scale
IOV/div
horizontal scale
20ms/div
- --------
-- --
... .
-----------T
.....
.
.............
.... ....
.... . ............
- ------------ ----------------------------......................
...I-------------------(a) phasevoltage
........
........
...............
-------- - --------
.....
-----
...
.
---------------
.
....
.
-
...
-------
.
.... ..... . .... .
........
....
.
......
------
L
...............
............
... ...
...
------
.......
..........
-------
- --------
----........
--------
.................
.......
......
........
L.......
..........
6 ................
(b) line voltage
Figure 4.23
Phaseandline voltage waveformswithout pWM
control (Load condition: I-6Nm at 420Tpm)
80
verdcal scale
20V/div
horizontal scale
20ms/div
................
.......................................
------ -- ---------------- -- --- ----
------
.....
--------
.......
. ......
.
-
....
vertical scale
0.5A/div
horizontal scale
20ms/div
...
----- --------
............
-4
...... . .......
- --
--------
- ------------
........
.
....
.........
....
... .......
.............
...............
.......
. ........
.
.............
.....
L
Figure 4.24 Phasecurrent waveform without PWM control
(Load condition: 1.6Nm at 420rpm)
verticalscale
0.5A/div
ýorizontalscale
lOms/div
Figure 4.2S Supply currentwaveform without PWM control
(Load condition: 1.6Nm at 420rpm)
81
0.5A/div
i3rizontalscale
10m.s/div
(a) phasecurrent
L 1. i
.
vertical scale
2OV/div
horizontal scale
lOms/div
10
..................
................
......
.................
...............
.......
........................
(b) phasevoltage
...............
.......
.......
.......
..........
......................
....
I-
..................
0
T
vertical scale
20V/div
horizontal scale
lOms/div
............... ....
Hill
iIII''
---------------........
........
------------
.................
-------
---------------
...........
.... . ..........
- ------
.....
(c) line voltage
Figure 4.26 Phasecurrent,phasevoltage and line
voltage waveforms
with PWM control (Load condition: I Nm at 30orpm)
82
...............
.......
.........
..
.............
.......
-
-- ----------
-
----------
.......
........
......
......
....
Ul
----
-----
.. . ...........
-------- --------
vertical scale
O.SA/div
horizontal scale
50ms/div
.................
11
---------------
..........
.........
.......
...
............
. ......
.
------------------ -- --------........
--------
I..
..
.......
.
.. ........
.....
-------
......
.......
I....
. ...............
...................
...
A ------
(a) phasecurrent
...............
T..............
...............
----------------
........
.....
--------------- ................
r................
---------------........................
............ ---------
0
..................
...................................................................................
...............
.
.......
.......
.
.......
A...........
(b) motor speed
Figure 4.27 Experimentaltransientperformance
( Load condition : 0.7Nm at 222 rpm
83
verticalscale
40rpnVdiv
horizontalscale
50ms/div
CHAPTER 5
MATHEMATICAL MODEL FOR A
3-PBASE BRUSHLESS DC MOTOR DRIVE
This
brushless
dc
The
unit.
frame
d and
two
provision
Under
normal
high
and
eddy
the
similar
to
slot
in
currents
of
be
as
the
the
is
the
the
windings,
may be useful
functions
of
both
by
as
course,
linkages
flux
are
described
be
in
84
analysis
all
seen
in
used
if
by
the
6.
only
and
neglected.
the
on
the
of
The
and
required
design
the
phases.
stator
model,
The
machine
considerations
chapter
calculated
angle
hysteresis
rotor
permanent-magnet
stator
rotor
the
simplify
are
the
the
saturation,
circuits
parameters
measured
expressing
a
Magnetic
airgap.
by
To
having
magnets
included
the
the
with
an
determined
in
operates
of
magnetic
Actual
machine
that
and
of
of
phases
Although
effects
rare-earth
effects
rate-of-change
could,
b,
a,
stator
characteristic,
138,39).
currents
follows,
performance
were
three
circuit
magnetic
of
linkages
machine
influence
damping
phase
coils
where
damper
demagnetization
however
can
flux
which
its
effect
the
on the
windings.
explicit
of
the
working
of
linearising
winding
no
inclusion
the
to
damper
rotor
based
5.1,
Figure
respectively
reluctance
saturation
power-conditioning
is
motor
the
research.
region
very
in
shown
has
the
subsequent
linear
model
motor
for
the
for
model
mathematical
accompanying
of
equivalent
experimental
in
its
and
q correspond
the
and
motor
a
representation
reference
c,
describes
chapter
data
these
data
was
drop
ideal,
be
to
assumed
the
non-conducting,
flexible
as the
turn-on
and turn-off
A tensor
define
as
pattern
system
in
described
reference
frame.
voltages
use
and-mesh
frames
two
as
and
to
study,,
to
as
the
also
integration,
obtain
in
branch
further
tensor
the
definition
a
transformation
and
well
therefore
currents
corresponding
frames,
present
numerical
and
necessitates
reference
by
solved
by
as
apply.
A2[42,431,
obtained
tensors
of
the
changes
linkages
The
sufficiently
topologyJ40,411
which
are
flux
then
are
inverter
Appendix
the
of
in
circuit
equations
equations
is
when
devices.
the
of
used
new
the
of
circuit
variation
The
the
automatically
different
The
is
voltage
-representations,
times
approach[16,171
conduction
the
alternative
are
impedance
developed
software
incorporate
to
forward
or
infinite
an
and
devices
switching
impedance
no
with
conducting
when
inverter
the
Although
available.
the
time
the
mesh
currents
and
transformations.
of
both
branch
between
the
below.
explained
5.1 BRANCH REFERENCE FRAME
branch
The
reference
disconnected
branches
corresponding
matrix
be
in
may
written
frame
is
system
shown
the
of
equation,
the
in
given
concerned
in
full
with
Figure
in
5.2.
Vb = Rb Ib + PTb
(5.2)
where
Vb
is
lb
the
branch
current
Rb
the
branch
resistance
the
impressed
85
branch
voltage
vector,
matrix,
The
5.1,
equation
form
abbreviated
the
vector,
1
0
zr
cm 0
CD
CD
0
wä
CD
CD
0
CD
CD
0
CD
pý
(D
0
0
0
0
0
0
II
18
R1
86
0
%Fb
The
total
functions
rotor
the
position
a-phase
the
mechanical
The
rotor
spatial
at
operator
any
instant
axis
angle
the
flux
variation
is
P the
expressed
0. -
of
number
due
rotorJ391.
The
to
reference
PO.,
pole
Vqare
and
with
specified
described
as
and
position,
rotor
angle
linkages
measured,
and
permanent-magnet
electrical
and
Vat Nfbo Vc*Vd
the
currents,
the
vector,
d/dt.
linkages
to
linkage
flux
total
flux
due
by
various
were
the
winding
linkages
the
branch
winding
of
flux
the
the
0. is
where
pairs.
to
the
in
chapter
permanent-magnet
6,
their
and
as
,Vpja = NfO( cos Oe+ 0.0763cos 30e+ 0.0114cos 50,,)
VF.,b = V, (cos (0, + 120* + 0.0763 cos 3(0, + 120* + 0.0114 cos 5(0. + 120*
Vft,
*.
(cos
(0, - 120' )+0.0763
=
5(0.
120'
))
)+0.0114
3(0,
120*
cos
cos
-
4fm = 4rod( cos 0* + 0.0763 cos 0' + 0.0114 cos 0* )
(5.3)
and NfN =0
where
wo
is
the
the
with
the
of
component
permanent-magnet
Vod
the
the
with
the
VPrk
of
the
damper
the
permanent
be a,
b,
c,
87
-magnet
d or
winding
rotor,
with
q.
linkage
flux
no-load
linkage
fundamental
the
of
permanent-magnet
f lux
the
to
rotor,
d-axis
the
linkage
due
windings
coefficient
component
flux
no-load
stator
fundamental
the
of
coefficient
to
due
to
k
can
and
winding
rotor,
due
k
where
The total
flux
and
Equation
for
linkages
the
various
windings
Nfa
I'aa.is. + Lab ib + Loc ic + Lad id + Laq iq + VPfa
Vb
Lba'a + Lbb ib + Lbc ic + I-bd id + Lbq iq + YPth
vc
I-ta'a + 4b ib + I-tc ic + 1-td id + I-tq iq +4fPtc
Nfd
1-dais + I-Idbb + I-Idcic + Ildd id + VPrd
Nfq
I-Iqais + Lqb ib + lqc ic + I-Iqqiq
Nfs
Lss is
5.4
are
(5.4)
in
may be re-stated
Ilb
the
form
abbreviated
IPPb
Lb
lb
+
=
(5.5)
where
Lb
is
the
branch
4IPb
is
a
vector
f lux
inductance
linkage
the
of
inductance
functions
of
induced
stator
rate-of-change
of
of
winding
voltage
the
flux
total
due
windings
Eb
vector
linkage
time,
of
vector
due
5.6
may be
ELI
b
where
radians
m. is
per
the
angular
to
(5.6)
dT Pb
dLb
bP b+
and the
as
re-written
ý%
-
Uý
velocity
second.
88
the
are
41b is
Eb = Lb Plb + pLb Ib + PTPb
Equation
to
(5.4)
equation
and consequently
position
branch
rotor.
coefficients
rotor
and
the
containing
permanent-magnet
The
matrix,
I+b
of
(5.7)
C%
dO
e
the
rotor
in
electrical
the
first
The
the
represents
term
as
the
a
term
of
term
the
permanent-magnet[441.
in
in
induced
the
speed,
voltages
stator
are
phase
5.5,
equation
the
5.8,
equation
branch
the
which
functions
of
and
current
The
due
terms
it
for
magnetic
is
into
5.9
equation
in
full
in
equation
5.2
given
and
From
properties.
is
vector
(5.9)
5.10.
equation
Substituting
yields
V1, = Rb L-bl (% - Tpb )+ pTb
is
which
in
given
in
full
that
position
Ib = Lb I (Tb - TPb
which
is
5.7
can be seen
rotor
to
are
expression
Equation
from
currents
two
later.
developed
variation
voltages
final
in
second
time[201.
with
rotational
appear
the
and
inductance
the
The
torque
full
no-load
they
since
electromagnetic
given
the
represents
to
variation
angular
final
significant,
due
5.7
equation
of
components
voltage
voltages
rotor
side
right-hand
transformer
rotational
result
the
on
(5.11)
5.12.
equation
5.2 MESH REFERENCE FRAME
The
formed
the
when
each
conducting
switching
period
of
Figure
patterns
differential
5.3,
which
with
being
only
two
in
switches
in
states
Table
may be written
89
stator
normal
the
inverter
5.1.
as
the
is
The
six
of
phases
six
for
sequence,
pattern
for
meshes
operation,
a defined
of
the
with
the
conduction
switching
given
equations
During
occur
typical
the
concerned
connect
supply.
patterns
A
conducting.
switches
dc
the
is
frame
reference
to
motor
discrete
are
mesh
bridge
shown
in
different
system
mesh
09
tn
%wo
e
x
f
00
.4
-
,
"a
.3
'i
AD
-
-
f
fx
.4
-
m
.
-
tr
ja
1
'Ir
W
,o
-
0
00
4r
. 00
AA
car
um
00u
43
90
0
9
J5
to
u
*a
>ý >:ý >ý >? >T
>0
-4
M
CD
cr
CD
.0
JD
0
iz
10
JD
91
04
04
CD
0
U
0
@0
JD
t)
«o
91
cr
OQ
I
In
>ý WN 30? »7 j
)Ir
4r
o
lr
1
.0
0: 0 0 0 0 0
> O >im
92
CDs
tn
I
.0
E9
0
00
0
0
0
u
93
+ p'P,
where
T,,, = L. 1,, + Tp.
5.14,
From equation
the
mesh current
is
vector
Im = L-1 (T. - T, )
PM
Mm
and
into
this
substituting
5.13
equation
V-Rý
MMM
gives
L-1 ("m - TPrn )+ P%
(5.16)
where
VM
is
IM
the
mesh
current
TM
the
mesh
total
flux
TPM
the
mesh
flux
linkage
the
permanent-magnet
RmandLm
impressed
the
are
5.16
pT
and
integrated
Appendix
A2,
A step-by-step
obtain
the
for
rotor,
and
resistance
state-variable
form
the
(5.17)
described
technique
linkage
flux
mesh fluxlinkages
the
to
mesh
Tpm)
M
a new machine
solution
due
matrices.
L-1 (T
using
vector,
vector
the
in
=V-R
M
MMM
numerically,
to
linkage
respectively
may be re-arranged
vector,
vector,
and inductance
Equation
mesh voltage
in
vector
Tm.
be
may thus
obtained.
5.3 BRANCH-MESH
TRANSFORMATION
The mathematical
relevant
mesh
pattern
changes,
transformation
has
model
equations
as
between
the
the
branch
94
generate
switch
is
this
and
to
and
achieved
automatically
diode
by
and mesh reference
the
conduction
defining
frames.
a
The branch/mesh
in
currents
the
of
current
terms
mesh currents
5.4
Figure
in
shown
5.3
Figure
may be obtained
defines
branch
and the
branch
between
The relationship
mesh currents.
of
C
transformation
currents
as
MESH
0
0
00
z
where
the
reversed
0
0
+1
0
that
is
there
An
is
in
invariance
expressed
"*2
between
form
equation
the
in
5.19
If
equation
then,
5.21
by matrix
equation
(5.19)
tensor
5.20
transformation
matrix.
transformations
is
frames(45,461,
which
for
IM
that
may
of
be
any
transposition
95
(5.20)
Substituting
a transpose.
and re-arranging
tmmtm
b -PT b) -(V
holds
This
C 'M
denotes
equation
[C(V
denotes
a0
as
t
superscript
same or
as
( V, - PTI )' I, = (Vm - Oýn
where
the
and
branch.
that
reference
mathematically
has
current,
current
for
(5.18)
current
mesh
in
branch/mesh
constraint
0
branch
abbreviated
the
important
power
the
the
as
lb
C
0
+1
no mesh current
may be written
where
+1
whether
sense
lm3
0
00
±1
denotes
a
lm2
+1
ld
L
0
-PT
) 31
arbitrary
yields
0
'ý
'
(5.21)
current
vector
Im
xy
b-P
b)
=Cv
Ct
is
where
the
transpose
transformation
voltage
and 5.22,
C, and is
of
=CtR
5.2,5.19
equations
Im
bC
(5.23)
5.13
equation
with
branch/mesh
the
gives
V. -PT
and by comparison
termed
Combining
matrix.
and re-arranging,
(5.22)
RM =CtRbC
From
5.22,
equation
branch/mesh
the
(5.24)
flux
linkages
are
related
by
qýn ý-- Ct Tb
may be extended
The
analysis
above
for
the
inductances
mesh
and
their
(5.25)
to
show that
the
rates-of-change
matrices
are
given
by
respectively
Lm
(5.26)
=CtLbc
and
ýL
.a=
d6ý
5.4 SYSTEM MECHANICAL
The
equation
Tý
J
dq
C
(5.27)
dOe
EQUATION
the
relating
Ct
dLb
various
system
torques
(5.28)
+ Dýý + TF + TL
dt
where
1
Te
is
i
the
combined motor
dq/dt
the
rotor
D
the viscous
the
electromagnetic
angular
torque,
and load
acceleration,
damping coefficient,
96
is
inertia,
TF
the
frictional
TL
the
load
The total
power
in
torque,
torque.
the
±:
first
the
where
term
second
the
rate
final
two
terms
of
electromagnetic
energy
change
the
power,
instantaneous
wr - dOr/dt
P((I
C
5.28
power
Or - Oe/ P,
and
t
may be
variation
of
the
5.5 INDUCTANCE
The
functions
of
the
rotor
and
The
of
the
of
the
axis
measurements
the
mechanical
speed
is
(5.30)
di
as
to
(5.31)
the
give
step-by-step
4.
VARIATIONS
variation
inductances
dividing
7ý - D(ý -TF -TL
1
speed
the
with
dTpm
t
numericallyl
rotor
the
result
dL
may be rewritten
integrated
the
by
The
instantaneous
by
M* dOe .1 M
dcý
77
which
associated
obtained
the
and
instantaneous
The
.
(5.29)
energy.
the
power
[201
is
losses
are
the
electromagnetic
T=
Equation
is
copper
inductive
stored
process
torque
electromagnetic
dt
equation
which
conversion
with
of
+It
MM
T!
dqe)
pm.
(L _
dOe dt
),
the
represents
of
dO,
n.
M I' (
dt
M dOe
I' L+
is
frame
mesh reference
dI
PRI+
MMMMmm
and
the
unsaturated
experimental
angle
electrical
using
provide
the
methods
the
97
and
were
measured
machine
0.
between
described
minimum
mutual
self
and
the
axis
a-phase
in
as
chapter
6.
of
the
maximum
inductances
self
and
the
second
harmonic
L2 and
established
that
by
inductance
A
be
stator
the
of
waveforms
and
mutual
influenced
inverter
by
and
harmonic
have
are
waveshape
high-order
of
Investigations
current
the
of
coefficient
self
calculated.
voltage
by
the
value
in
components
the
variationsI12).
model
induced
which
voltages
is
functions,
inverter
therefore
inductances
of
variations
reasonably
the
while
cosinusoidal
as
On
accurate.
experimental
trapezoidal
instants,
switching
inductance
the
the
accurately
represents
and
approximating
the
the
than
switching,
M2 may
average
MO and
and
component
induced
the
the
which
L.
inductances
mutual
inductances
more
from
variations,
machine
basis
this
are[12,13):
L. - 1.2cos 20.
Lbb = I... - L-2cos 2(0, + 2r J3)
L,
= L. - L2 cos 2(0, - 2n/3)
(5.32)
and
L,,b
2(0,
21r/3)
Mo
M2
cos
-
Lac
-mo-
Lbý,
K,
M2
20.
cos
-
M2 cos 2(0. + 2x/3)
(5.33)
5.6 COMPUTER IMPLEMENTATION
Figure
written
numerically
5.5
to
predict
the
reference
frame.
following
algorithm
is
a
flowchart
for
the
motor
performance,
differential
The
equations
solution
process
98
the
computer
based
expressed
is
program
on
in
described
solving
the
by
mesh
the
a.
branch
The
is
assembled
mesh
resistance
matrix
is
matrix
resistance
once
only,
formed.
beginning
the
at
fixed
This
the
of
simulation.
b.
The
is
matrix
is
matrix
conduction
C.
The
dynamic
and
changes
the
with
switch
pattern.
time
using
at
elements
5.32
equations
vary
step,
the
with
Lb is
matrix
5.33.
and
integration
every
which
inductance
branch
varying
determined
formed
from
Rb C
RM=Ct
This
determined
This
it
since
angular
is
matrix
contains
position
the
of
rotor.
d.
The
5.26,
equation
e.
The
inductance
mesh
impressed
its
and
mesh
branch
The
voltage
flux
permanent-magnet
g.
The
mesh
rotor
is
flux
is
linkage
i.
is
5.17
technique
described
vector
The mesh current
using
obtained.
is
determined
from
using
obtained
due
vector
due
vector
to
equation
the
to
the
5.3.
permanent-magnet
from
obtained
Equation
linkage
determined
Vb
linkage
rotor
is
vector
Tpm
h.
is
inverse
VM =Ct
f.
Lm
matrix
=CtY
Pb
integrated
in
numerically,
Appendix
2,
to
using
obtain
Tmvector
4
is
= L" (T
MMM
99
obtained
-
PM
from
the
the
flux
J.
branch
The
current
lb
vector
is
determined
from
lb 2-- C ýM
k.
branch
The
from
obtained
1.
with
respect
The
mesh
derivatives
the
inductance
of
rate-of-change
0..
to
inductance
of
rate-of-change
dLb
dL
m= Ct
dý
dOe
The
is
matrix
from
obtained
m.
inductances
branch
the
of
is
matrix
mesh
derivative
current
C
is
plm
vector
determined
f rom
dL
PI
n.
branch
The
M
0ý Im - PTpn,)
0
L-'
(Vm -RI-T
=
MMMM
derivative
current
Plb is
vector
determined
f rom
P lb ýC
o.
branch
The
voltage
is
vector
Plm
determined
from
dLb
Vb=RbIb
p.
The
new
initial
the
occurred
solution
in
are
the
until
5.30
by
advances
conditions
step,
torque
equations
solution
repeated
the
using
up-dated,
end of
system
is
the
bridge
proceeds
tested
+P T
I
bP b+ -00b
dGý
electromagnetic
obtained
The
+L
the
and
and
topology.
as follows:
100
angular
5.31
and
any
respectively.
At
the
step,
(c)
procedures
changes
If
are
velocity
integration
one
simulation.
for
Pb
the
end of
which
a change
is
(p)
to
each
may have
detected,
1.
Determine
point
2.
Re-integrate
to
its
Form
the
6.
new
mesh
and
the
new
R.,
matrices
equations
of
the
start
current
transformation
new circuit
topology,
and
according
to
L.
and Lm
integrate
and
from
these
discontinuity.
for
system
(6)
If
.
(c)
any
occur,
the
not,
(p)
to
in
changes
If
pattern.
operations
the
to
mesh
the
condition
the
topology.
of
to
step
Ct.
transpose
point
(1)
the
of
from
branch/mesh
the
the
Test
start
discontinuity.
of
new circuit
Form
the
equations
C corresponding
form
the
mesh
point
Re-assemble
the
S.
the
the
matrix
4.
between
time
discontinuity.
of
step
3.
the
the
over
the
repeat
solution
operations
proceeds
integration
next
switch
with
step.
5.7 SYSTEM DISCONTINUITIES
When
a
conduction
also
switch
pattern
system
the
it
operation.,
discontinuity
Turn-on
b.
Turn-off
and
both
of
purpose
is
necessary
ceases
the
of
discontinuity
are
discontinuity.
discontinuity.
are
explained
below.
101
the
conduction,
transformation
accurately
to
occurs.
Two kinds
a.
or
and consequently
For
change.
commences
possible:
matrix
simulating
determine
when
the
the
S. 7.1 TURN-ON
A
discontinuity
turn-on
has
switch
To
a gate
pattern
are
if
Thus,
due
valid,
5.7.2
TURN-OFF
A
turn
period
when
one
switch
as
are
one
the
maintain
commutation
the
commutation,
of
mesh
integration
is
current
unpermitted
determined
the
the
from
mesh
mesh
current
1.2
tested.
When
the
point
interpolation
at
the
beginning
102
it
at
as
and
Im,
builds
At
the
of
the
the
meshes.
decreases.
it
the
outgoing
negative
which
to
commutation?
current
shown
S4,
during
four
becomes
on
meshes
and
c
of
polarity
The
3 turns
D2
to
incoming
4 and
four
phase
three
shown
shown.
condition
the
the
as
and
through
is
It
on.
as
off
commencement
changed
step,
linear
currents
this
through
at
situation),
by
mesh
are
outgoing
each
is
switches
meshes
Im2 passes
flow
Thus
period.
the
system
commutation
turns
shows
5 turns
For
which
current
During
end
If
of
equations
while
the
topologies,
three
switch
5.6(b).
system
up
step.
step,
and
the
another
5.6(a)
produce
when
Figure
required,
a
approximation
during
circuit
Figure
these
occurs
in
shown
of
step-length.
and
off
separate
5.6.
and
period
This
occurs
turns
three
Figure
conducting,
conduction
integration
step
integration
small
discontinuity
in
the
the
end
an
of
conduction.
DISCONTINUITY
by
typically
second
very
in
the
at
instant.
this
at
off
characterised
5
the
end
non-conducting
commences
during
on
the
at
a
changes
only
switched
changed
to
it
and
occur
conduction
are
when
complexity,
to
is
a device
equations
applied
program
assumed
commences
occurs
signal
the
minimize
it
DISCONTINUITY
occurs
by
Figure
end
of
the
(an
is
5.7.
step
are
11
respectively
discontinuity
12,,
and
then
the
time
to
the
point
of
is
II
(5.34)
x
h
where
is
3
switches
integration
the
by
represented
4
and
the
length.
step
are
conducting,
three
and
defined
meshes
After
in
commutation
the
system
Figure
5.6(c).
normal
operating
is
5.8 MASTER MATRIX
Six
distinct
conditions,
These
Figure
with
are
The
meshes
all
current
a
defined
are
further
in
to
six
Table
transformation
during
possible
account
for
commutation.
5.2.
C for
matrix
5.6(a)is
Mesh
1
2
3
a
-1
0
0
b
0
0
0
c
+1
0
0
d
0
+1
0
q
0
0
+1
s
+1
0
0
103
the
case
shown
in
rq
u2
m
u2
CD
9--4
(>
m
cn
lqt
u2
m
u2
u2
kA
Co
et
u2
-4
O-N
v2
cn
c02
\o
v2
9-4
u2
le
in
10
N
u2
"_q
c4
m
j
4
CV2
-u2
m
v2
I-
rq
Gn
CM)
0
v2
104
for
and
shown in
that
is
5.6(b)
Figure
Mesh
2
3
4
0
a
-1
.1
0
b
+1
0
0
0
c
0
+1
0
0
d
0
0
+1
0
q
0
0
0
+1
s
+1
0
0
0
in
C are
on
The
by
method
as
the
of
a master
switch
possible
and
corresponding
be
can
matrix
is
in
branches
switch
Figure
C
into
diode
or
switch-pair
a commutation
given
by
turn-on
and
use
makes
changes
system
formed
loaded
being
column
the
automatically
produced
pattern
conduction
relating
that
The
conduction.
until
diode
matrix(45,461
meshes
relevant
changes
which
to
diode
and
5.8,
all
the
with
the
whenever
C
in
retained
occurs.
5.9 DISCUSSION OF RESULTS
The
computer
model
investigate
the
steady-state
drive
the
motor
using
described
and
transient
given
parameters
105
above
used
was
performance
in
Appendix
of
1.
to
the
Figure
5.9
the
shows
waveforms
from
inertia
of
rest
the
and
motor
by
produced
currents,
at
rotor,
the
rotating
torques.
The
waveforms
motor
is
speed
increases
from
started
currents
linearly
voltage
rise-time
of
less
that
of
inertia
the
motor
rotor.
shown
in
when
Figure
speed
is
this
is
no-load
The
steady-state
load.
with
on
system
the
speed
(260rpm).
new
the
generated
the
moment
of
motor
with
the
removed
condition
as
the
new
the
106
the
winding
emf
and
the
The
at
of
the
moment
inertia
motor
torque
accelerates
condition
the
of
is
change
initially
of
is
value
total
load
running
its
which
expected,
the
torque
steady-state
The
no-load.
a load
rotor
5.10.
period
a typical
is
the
when
no-load
because
speed
start-up
As
is
to
running
and
level
torque.
response
222rpm
on
steady-state
simply
The
When
decreases
a
a
is
and
Figure
emf
to
This
5.11.
suddenly
satisfy
its
to
speed
in
equal
to
the
voltage
initial
friction
the
than
the
decreases
balances
value
since
almost
torque
average
than
zero,
winding
friction
and
torque
shown
during
are
electromagnetic
are
emf
torque
higher
supply
generated
to
almost
phase
rest,
increasing
is
a 24V
with
to
torque,
current,
phase
no-load,
continues
electromagnetic
torque
starting
predicted
on
the
the
As
the
load
the
started
222rpm.
decreases
by
with
generated
electromagnetic
balanced
expected,
the
and
torque.
the
is
torque
field
magnetic
is
of
increasing
speed
torque
motor
speed
the
and
load
the
when
electromagnetic
until
it
As
M2,
steady-state
consequently
which
pulsating
the
the
a 0.7Nm
voltage,
a
torque
current,
0.003kg.
to
up
rises
and
level
supply
of
run
speed
accelerate
and
a 24V
with
a moment
phase
predicted
at
0.7Nm,
decreases
to
its
has
a
,
and
to
new
been
decelerates.
phase
load
torque
is
Figure
5.12
shows
the
reached,
emf
with
speed
current.
In
and
phase
current
rest
and
are
for
5.13
the
the
5.13,
that
the
is
period
result.
The predicted
110ms.
good
between
agreement
5.14p
the
with
the
since
one
can
the
Figure
and
without
The
waveform.
periods
are
current
current
the
the
from
phase
the
be
has
seen
a dip
of
is
in
at
the
inductance
instantaneously
107
time
is
also
in
Close
current
in
does
if
its
value
the
conducting
commutation,
prevents
on
measured
The
waveforms.
the
In
waveform
120'e
during
starting
devices.
with
of
The
the
current
which
not
importance,
switching
instant
Figure
greater
of
and
both
is
great
compared
its
to
start
devices.
harmful
very
the
experimental
simulation
value
non-conducting
winding
changing
and
during
slightly
steady-state
PWM control
60*e
is
from
clear
evident
switching
semiconductor
predicted
clearly
waveform
to
of
the
nature
can
which
the
previously
measured
being
period
the
predict
rating
5.15
in
from
(222rpm).
is
period
drops
started
(218rpm)
and
since
speed
response
result
predicted
transient
waveform
exceed
both
the
of
current
load
speed
values,
motor
from
time
currents
phase
the
with
steady-state
simulated
voltage
simulation
response
start-up
the
response
measured
the
measured
for
account
speed
the
during
waveforms
agreement
measured
on
is
It
speed
good
the
with
motor
described
120ms and the
The predicted
agreement
than
is
value
the
is
motor
waveforms.
predicted
in
start-up
the
motor
in
predicted
conditions
measured
this
of
the
when
load
same
with
steady-state
5.14
the
and
increase
the
effect
and
waveforms
compared
Figure
the
and
Figures
re-applied
the
phase
due
phase
commutation
period
the
and
generated
decay,
voltage
build-up,
and
which
between
predicted
steady
there
not
with
state
a
speed
a 70% duty
cycle
frequency
is
Predicted
and measured
in
shown
during
in
change
waveforms,
5.21
the
in
results
phase
in
evident
voltage
in
both
waveforms
chopping
action
evident.
With
waveform
induced
voltages
windings.
the
during
caused
generated
emf,
The
sudden
5.18.
and
The voltage
spikes
The
108
5.20
changing
to
of
the
waveforms
is
in
occur
and
line
and
effect
on these
due
voltage
phase
measured
non-conducting
rapidly
line
Figures
results.
not
instants
commutation
oscillations
by
is
the
PWM controller
the
PWM
without
to
phase
and
voltage
phase
PWM control.
PWM control,
voltage
of
predicted
with
of
sets
show
the
Figure
the
line
and
are
when
the
at
on
spikes
respectively
voltage
other
current
5.16.
5.19
and
expected,
equal
of
the
of
on no-load
5.18
phase
at
The high
Figure
of
a
set
PWM switching
As
is
torque,
PWM control
waveforms
period
current
frequency.
measured
waveforms
phase
the
Figures
voltage
both
as seen
show
60*e
the
the
conducting,
and
Figure
1Nm load
the
PWM control.
without
phase
current
phase
Good agreement
waveforms
and
predicted
control
is
phase
5.17,
Figure
waveforms
both
the
measured
with
to
current
in
dip
a
switching
due
in
evident
respectively
as
a 20kHz
ripple
inverter
and
the
evident.
voltage,
300rpm
current
is
70V
supply
the
opposes
waveforms
and
with
current
is
The
phases.
assists
phase
predicted
of
motor
commutation.
undergoing
and measured
both
shows
waveforms
consequently
is
current
5.16
incoming
of
the
phase
outgoing
the
that
whereas
the
of
in
emfs
generated
the
phase
period,
due
to
currents
in
the
the
sudden
change
in
the
phase
in
evident
shapes
current
both
the
of
sets
waveforms
reverse
recovery
the
and
5.23
Figure
In
experimental
of
the
capability
supply
close
results
be utilized
will
which
effect
torque
no-load
in
waveforms
between
them
presented
transient
and
are
to
ripple
be minimized.
109
between
predict
chapter
7 to
and the
not
included
phase
voltage
Figure
are
the
5.22.
in
shown
predicted
the
ability
accurately
the
drive.
This
the
of
the
of
evident.
demonstrates
performance
in
is
the
and phase
effects
current
agreement
model
the
shown
also
between
current
diodes
as
agreement
the
that
measured
and
mathematical
steady-state
might
measured
conclusion,
fact
the
trapezoidal
and good
and
factors
both
phase
free-wheeling
the
Predicted
are
Predicted
to
are
The difference
results.
due
instants
commutation
and measured
is
in
model.
waveforms
of
predicted
voltage
in
the
at
investigate
ways by which
the
this
I
0
.0
4)
cn t
cu
bl)
ce
w
0-4
>!4
110
U
rt
.
U
I
>ý
I_r
>r
0
I
P.
.0
cr
Cd
tr
0-4
at
a
04
>19
10
Cd
j
10
0-.4
w
ill
LW
a
iz
v
cn
Ici
In
iz
gn
0.4
112
*a
1-%
JM
iz
gn
2r-
. cn
IRT
iz
C43
04
W
0-4
>r
113
Start
Read motor parameters
Setup:
Initial motor currents, motor speed,
starting time
Choosestep-length
4
Define the circuit transformation matrix from
switching conduction pattern
I Determine trasformed resistancematrix I
3
I Determine winding inductances I
I
Determine rate-of-change of winding inductances
I
Determine flux-linkages due to permanent magnet
Determine generatedemfs due to permanent magnet
Determine transformed inductance matrix
Figure 5.5 Flow chart for motor model
114
continued
...
Deterniine awsformed rate-of-change of inductance matrix
Detem-iinetransformedflux-linkage, I
emf andimpressedvoltagevectors
I Assembleandintegratethe state-variableequationto obtain the I
new meshflux linkagevector
Determine the mesh current vector
Detemiine the branch current vector
Determinethe meshcurrentderivativevector
Determinethe branchcurrentderivativevector
Determine the branch voltage vector
Determine the electromagnetic torque and speed
2
Figure 5.5 Flow chart for motor model
115
continued
...
Yes
Elnd
d
>
I
of simulation?
INo
I Print result and stor
datafor graphical
output
"
Any
%N'.
o,
>.
disconfinuities?
No
Updatevariables
and storedata
for graphical
Stop
Yes
Re-integrateequation
from start of stepto point
of discontinuity
Figure 5.5 Flow chart for motor model
116
LS
R
v
d
q
emr-4
s
(a) State I: S5 and S4 ON
LS
IR
dM
VS
q
rýmy%l
If9l
(b) During commutation S3, S4 and D2 ON
Ls
p
v
m
s
q
I1" I
(c) State2: S3 and S4 ON
Figure 5.6 Typical systemmeshes
117
1
Figure 5.7 Turn-off discontinuity
118
CD
0
9
10
vs
e
00
"-4
Q
-0
C:)
0
Cd
00
COJ
CA
CD
c02
CD
CD
CD
gn
iz
-:2s .
CA
-s
1
tu
1
iz
1
01
Q1
CTI
CA
Ei
JS
.0
HDNVHII
CD
P-4".1q
119
N
0.0
al
0.2
0.3
0.4
0.5
0.6
0.8
0.7
0.9
10
ti
t2
time, S
(a) phase cuffent
3.0
zo
F:
10
0.0 1i1! i1s12112a121a=aM1aa1=IM1aa1aM211!
0.0
0.1
0.7
02
0.3
0.6
0.4
0.5
i-
2aa12saa1i1ia
0.8
0.9
to
ti
time,s
(b) motor torque
*lo2
9
0.0
0.1
02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
to
ti
U
time, s
(c) motorspeed
Figure 5.9 Simulatedwaveforms(Loadcondition: 0.7Nmat 222rpm)
120
13
0.0
0.1
02
0-3
0.4
0.5
0.6
0.7
0.8
0.9
10
11
t2
time, s
(a)' phase current
0
Z
F:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
to
ti
t2
U
time, s
(b) motor torque
$102
E
0.0
0.1
0.2
0.3
0.4
0.5
M6
0.7
0.8
0.9
11
to
time, s
(c) motor speed
Figure 5.10 Simulatedwaveforms (No-load condition)
121
13
0.0
0.1
0.2
0.3
0.4
0.6
0.5
0.7
0.8
0.9
to
ti
t2
time, s
(a) phasecuffent
E
Z
F:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10
1.1
time, s
(b) motor torque
*io2
s
El.
C;
0.0
CLI
02
0.3
0.4
Ob
0.6
0.7
0.8
0.9
to
(c) motorspeed
Figure S.11 Simulatedwaveformsduringloadchange
122
ti
time,s
UU
U
siol
to
0.5
0.0
-0.5
-to
0.0
0.1
02
CO
0.4
0.5
M6
M7
0.9
0.8
*'W time, s
0.7
0.8
*10'
(a) generated phase emf
Ow
s
E6
C;
0.0
0.1
01
13.3
0.4
0.5
0A
(b) motor speed
0.0
0.1
M2
0.3
0.4
CL5
CLS
0.8
CL7
*W
(c) phasecurrent
Figure 5.12 Current inrush phenomenonduring starting
123
0.9
time, S
0.9
time, s
F
CLO
CLO5
al
CL15
0.3
0.25
0.2
0.35
M4
------
-----------
-------------
0.5
time,s
(a) simulatedwaveform
------------------------.......
0.45
----------------------------
----------------------------------------
---------- ---------------
vertical scale
40zpm/div
.................................
............
horizontalscale
50rns/div
------------------------
................
---
-
0
------------------
-------
-------
- -------
................
........
..........................
................
-----------------
----
------
(b) measuredwaveform
Figure 5.13 Motor speedwaveformsduring starting
( Load condition : 0.7Nm at 222 rpm )
124
U
-I
0.0
0.05
0.1
CL15
0.2
0.25
035
0.3
-- --------
0.45
0.5
time,s
(a) simulatedwaveform
---------------
CL4
-----------------------------------------------
- ----------
.......
-- --------
0
verticalscale
0.5A/div
17
horizontalscale
50ms/div
.........
- -------
------------------
-------
---------------
..................
-------------------------
-------
---------
II
.....................
...............
-------
......
j .......
.
.......
.......
..............................................
...............
L8
(b) measuredwaveform
Figure 5.14 Phasecurrent wavcfornis during starting
(Load condition: 0.7Nm at 222 rpm)
125
0.75
0.5
0.25
«
N-4
0.0
--0.
0.95
0.975
1.0
1025
1.05
1125
1075
U75
time,s
(a) simulated waveform
.......
115
----------------------.....
.......
....
.......
...
------------------
.......
---
---
-
------------
----
.........
.......
.......
---------------
---
4-------
----------------
.....
---------
.....
--------------------6--------------------- --------
--------
verdcal scale
0.5A/div
horizontal scale
20ms/div
--
------------------------ ------ -----------------------
--- ---
--- - ---
...........
--
-------- ----
-t-,
--
--------
--------------------------------
-----------------------------------------------
..................
.
...............
------
----
....
-------
.......
.......
----
------------------------
----------------
----------------
L................
(b) measuredwaveform
Figure 5.15 Phase current waveforms without PWM control
(Load condition: 0.7Nm at 222 rpm)
126
0-4
0
-o
-1
-1
time,s
(a) simulatedwaveform.
-----------------------
-
---------------------
-
.......
....
. .....
----- --
-----------------------
......
........
------------------------------
------
-------
--------
--------
--------------
----
.
------.....
---------------------- ------
verticalscale
0.5A/div
----------
horizontalscale
10nis/div
-------
------- - -------
------- - -- - ---
r ........
- ----------
.
....
..... - -- ----- -----------------------------
...........
.
....................
... ---------.
----------------------------------------
------------------------------------------------
.......
-------- - ----------------
(b) measured waveform
Figure 5.16 Phasecurrent waveforms.with PWM control
(Load condition :I Nm at 300 rpm )
127
0.250.200.15
--0.100.05.
0.00--0.05-0.10
-0.150.20
-0.25-0.25
U. 40
0.35
time,s
(a) simulatedwaveform
..........
------- - ---------
vertical scale
50mA/div
--- ---
-------
horizontalscale
25ms/div
-------- .------.........
..........
........ - -----------------------------------
--------- - ------- ---------------------------
... . ..........
-- - --------
......... .
.................
(b) measuredwaveform
Figu re 5.17 Phasecurrent waveforms( No-load condition )
128
OV
1.5
1.0
>lr
0.95
1.0
0.975
V25
ti
tO75
105
1025
1.15
IT75
time,s
(a) simulatedwaveform
---------------
---------------
-
------------------------------------
----------------------
---------------
----------------------------------
----- --
------
--------------
--------
-------
verticalscale
5V/div
--------
horizontalscale
20ms/div
---------
-- --- ------- - -- --
-------
r --------------- ----- - ------------------
I-
-----------------------------
I-------
----- - -------
----
----
-------
... ...
-
-- ------------------------------T
(b) measuredwaveform
Figure 5.18 Phasevoltagewaveformswithout PWM control
(Load condition: 0.7Nrn at 222rpm)
129
OV
zo
ao
0.95
CLS75
10
1.125
11
1075
tO5
1025
1.15
IT75
time, s
(a) simulatedwaveform
...
.......
-------
-
.
..........
.
.....
.......
4----- -
----
.
...
.
.......
--------------------------------------
.......
.
.......
. ...............
. ...
--
--------
-
--
- -----
---
vertical scale
IOV/div
-
----
---
horizontal scale
20ms/div
----------
.. ------
-. -----.....
----------
.............
- ------
L-
-------
-- -------------
...............
----
------------------------
----------
(b) measuredwaveforin
Figure S.19 Line voltage waveformswithout PWM control
(Load condition: 0.7Nm at 222rpm)
130
siol
-540
CL2
CL21
022
CL23
CL24
0.25
CL29
CL27
026
CO
CL31
time, s
(a) simulatedwaveform
........
...............
...............
.......
I
........
.
-------- - -------------------------
------- ....... .......
.........
.............
.........
------- I.......
----------------
j J,
0
IL
-
----------
................
.....................
-
-----
-- - -------------
L
-------
--
.......
(b) measuredwaveform
Figure 5.20 Phasevoltage waveforinswith PWM control
( Load condition :I Nm at 300rpm)
131
vertical scale
20V/div
horizontalscale
lOms/div
-------
......
, -------
*101
7.5
5.0
2.5
-Z5
CL2
0.21
0.22
C123
0.24
0.25
0.26
0.27
0.28
CL29
0.31
time, s
(a) simulatedwaveform
................................
0.3
......................................
- --------
--
vertical scale
20V/div
--------------------------
---------------
-------------------.......................
. .........................
..........
(b) measuredwaveform
Figure 5.21 Line voltage waveformswith PWM control
(Load condition: I Nm at 300rpm )
132
horizontalscale
lOms/div
*101
to
":
I)
0.0
0.85
0.875
0.9
0.925
0.95
0.975
t025
10
t05
tO75
1.15
..
I............
................
.....
........
U25
time,s
(a) simulatedwaveform
........................................
il
..............
..
vertical scale
5V/div
..........
.......
. ... . .....
. ..
horizontalscale
25ms/div
U
.....
........
.
-------
..........
--
..........
..........
...
.........
.......
. .......
...
.......
......
I
.......
.
........
.
.
......
.................
I
.
...............
..
..............
-.
(b) measuredwaveform
Figure 5.22 Generatedphaseemf waveforms(No-load condition )
133
CL9
<
02
0.7
0.6
0.5
0.4
0.2
CLI
iiiiiii11m111!
1m
1.0
0.975
0.95
1.025
--T-
I-II-
1.05
1125
ti
1.075
M
IT75
time,s
(a) simulatedwaveform
-------------
---------------
- -------
...... . ....................... . .......
--
----------------
vertical scale
0.5A/div
horizontalscale
20ms/div
---------------
................
-------
----------------
.... .....................
------- ----------- ---------L7
------------------------
-------
---------
. ...........
.
--------------------------
L
---------------------------------
----------------
(b) measuredwaveform
Figure 5.23 Supply current waveforms (Load condition : 0.7Nm
at 222rpm )
134
CHAPTER
6
DETERMINATION
The
motor
machine
parameters
model
the
the
no-load
This
flux
phase
dc
and
resistance
between
stator
linkages
due
phases
the
to
rotor.
describes
chapter
experimental
mutual-inductance
brushless
the
phase
stator
stator
permanent-magnet
for
required
the
are
self-inductance,
and
OF MOTOR PARAMETERS
machine
how
were
the
1.3kW
the
of
parameters
obtained.
6.1 STATOR PHASE RESISTANCE
The
resistances
an
ambient
at
double-bridge
the
of
three
temperature
(47).
The
phases
stator
25*C
of
using
three
the
of
average
measured
were
Kelvin
a
results
was
1.2780.
6.2 STATOR PHASE SELF-INDUCTANCE
The
at
unsaturated
different
6.1.
R2,
rotor
R4F
the
while
inductance
is
to
adjustment
of
the
across
through
then
the
the
test
are
fourth
is
ratio
test
and
the
winding
Ia.
R41
the
The
current
135
Ia
was
until
also
by
balanced
the
voltage
current
connections
switch
change-over
whose
phase
steady-state
supply
Figure
resistances
stator
bridge
R3 and
zero
was
the
measured
of
circuit
non-inductive
The
arms
was
winding
using
the
bridge
measured.
fluxmeter
reversed
consequence
be
bridge
the
using
the
of
was
self-inductance
phase
angles
legs
Three
R3,
stator
reverses.
and
were
as
At
a
the
end
the
of
again
current
achieved,
there
The
emf
induced
this
period
La.
inductance
La during
by
voltage
through
the
voltage
across
upper
branches
the
phase
winding
is
is
is
is
v
The
voltage
voltage
voltages
bridge
v9
is
across
the
and
R+
R2
R2 is
resistor
across
voltage
dý
R3/R4=R2/R
being
fluxmeter,
the
and
V-J.
R+R2
R3. is
given
by
integral
of
this
the
fluxmeter
voltage[48,491,
R2
V is
so that
136
'ý, dt
R+R2
instantaneous
and
the
the
difference
of
the
1481
di
L
V
9R+
of
in
dt
R+R2
R2_
The deflection
current
di.
balanced
R2
across
for
therefore
VR2_
the
is
current
equation
R2
Since
the
instantaneous
the
and
in
which
the
period,
v/(R3+R4)
the
instantaneous
the
this
to
flux
a voltage
v=(R+R2)ia+La(dia/dt)
instantaneous
the
due
changing
produces
is
condition
period
the
[R3/(R3+R4)]V-
ivL
aa
while
by
during
branches
R3 is
the
transient
a
If
bridge
the
lower
steady-state
fluxmeter.
the
across
is
but
inductance
integrated
the
reversal,
a
a dt
proportional
to
the
time
00
fv
dt
0
R2L
f
di
R2
M-ala
R+R2
and the
stator
phase
is
self-inductance
R2
R+
L
-NL
Ma
a
R2
R3 + R4
_XL
21&
The
with
3A
variation
rotor
is
in
with
dc
rotor
Oe=O
is
angle
remains
If
values
of
L2 of
and
the
Lmin
phase
equation
in
given
a fixed
the
stator
Laa,
stator
Lbb,
6.3.
for
The
a
machine
of
these
of
phase
fixed
clearly
current
and
the
self
-inductances,
maximum
the
and
minimum
terms
are:
(6.1)
02
and
"ýmn
L2
(6.2)
2
and
it
follows
from
Figure
the
unchanged.
Lmax + Lmin
L
Lcc
current
variation
rated
respectively
winding
5.32
the
its
remain
are
and
Figure
to
up
consequently
Lmax and
Oe for
in
current
unsaturated
inductances
-inductances
6.2,
Figure
inductances
self
phase
displacement
angle
given
the
of
R3
6.2
that
137
, approximately,
LO
L. = 9.85 mH
and
L2 = 0.45 niH
6.3 STATOR PHASE MUTUAL-INDUCTANCE
6.4
Figure
mutua 1- inductance
fluxmeter.
the
A reversal
a reversal
the
from
The
I.
by
required
applied
to
to
During
-I..
in
induced
voltage
the
measure
phases(48,50).
voltage
current
instantaneous
the
reversal,
stator
resistance
source
of
the
of
two
any
to
used
circuit
between
Rv provides
resistance
causes
the
shows
phase
the
a
phase
this
b is
di
Vb ý-- Lb
the
and
corresponding
ik
a dt
voltage-time
-1.
f
Vb dt = Lb.
0
If
the
21ar
the
fluxmeter
is
integral
di.
+k
is
deflection
V for
between
mutual-inductance
a current
phases
of
reversal
a and b is
Ilba
21
Figures
variation
6.5
the
of
stator
current
current
for
If
minimum
MO and
Mmax and
M2
3A,
show
their
angle
are
respectively
5.33
Lab,
Lbc,
angular
for
Lca
stator
with
variations
a
(0, =O).
the
maximum
mutual-inductance,
phase-to-phase
equation
the
respectively
inductances
and
rotor
Mmin
of
of
mutual-
of
a fixed
values
6.6
and
are:
138
and
the
the
terms
Ným
xn
(6.3)
2
and
m
Imx
'v'2
and
it
follows
from
-M.
nun
2
6.5'
Figure
(6.4)
that,,
approximately,
M. = 1.95 mH
and
M2 ý
6.4
NO-LOAD
STATOR
0.45 mH
PERMANENT-MAGNET
A
Hewlett
determine
open-circuit
generated
the
a
is
phase
the
voltage
electrical
With
axis.
analyser
of
sin 30
the
terms
are
phase
a is
between
instantaneous
generated
800
If
rpm.
the
considered,
sin 50
C+0.0574
angle
of
speed
the
to
used
was
the
of
rotor
harmonic
three
open-circuit
0.
a
at
45.537
(sinO, +0.229
e
=
oa
where
harmonic
spectrum
voltage
first
the
only
3582A
frequency
phase
DUE TO THE
LINKAGE
ROTOR
Packard
the
FLUX
WINDING
(6.5)
C)
axis
rotor
voltage
given
and
by
dyFýa
eoa
dO.
dXVAa/d0,
where
the
permanent
angular
velocity
is
the
-magnet
in
of
rate-of-change
with
electrical
position
rotor
radians
per
flux
due
linkage
and
second.
W. the
It
rotor
follows
that
ýýPa
dO
30,
0.181
0.
0.229
(
+ 0.0574 sin 50,
+
sin
sin
=
139
to
(6.6)
Integrating
equation
linkage
the
of
Similar
calculations
spatial
variations
is
It
phase
two
from
turns.
the
the
they
are
self
vary
with
6.2
and
the
rotor
in
the
the
due
to
the
the
stator
5.3.
frequency
that
between
is
being
also
the
mutua 1- inductance
that
and
the
times
the
winding
when
a minimum
that
the
stator
phase
between
phases
variation
reluctance
frequency
rotor
a maximum
evident
of
position
the
variation,
and
the
as
path
and
This
length
airgap
flux
any
0,.
position
aligned
It
six
6.5
position,
are
axes
360*e,
every
the
rotor
the
and
twice
a
with
rotor
-inductances
repeat
equation
reluctance
quadrature.
variations
linkages
in
of
the
of
and phase
in
as
and
phases
mutual-inductance
of
reluctance
also
rotor
the
variation
As a result
when
Figures
and
the
inductances
other
flux
given
functions
are
consequently
has
from
self-inductances
arises
a, Vp,,,
phase
OF RESULTS
evident
phases
the
various
are
rotor
for
made
the
of
DISCUSSION
with
rotor
flux
the
30.
0+0.0763
0.0114
)
50,
0.181
(
+
cos
cos
cos
=
e
were
permanent-magnet
0. gives
to
respect
with
permanent-magnet
NfP,
a
6.5
6.6
of
the
rotor
rotation.
Figures
reluctance
negligible
6.3
the
of
figure
demagnetizing
6.6
and
of
5.5A,
effect
that,
show
saturation
the
the
due
probably
permanent
rare-earth
magnetic
self-inductance
rated
and
-magnet,,
evident
Even
mutual-inductance.
stator
on the
current
has
permanent-magnet.
140
in
to
the
there
high
is
both
the
above
the
an insignificant
The values
of
the
stator
6.3
are
phase
self
larger
than
those
of
being
mutually
being
measured
The
used
simulate
in
the
those
L, b and
Lbc*
This
at
the
is
due
by
parameters
mathematical
model
of
to
120*
same rotor
performance
Laa
of
Lc shown
.a
displaced
experimental
the
-inductances
than
mutual-inductances
Lbb and
the
141
and
in
the
and
L,,
the
6.6
these
(0. =O).
obtained
in
developed
brushless
this
in
drive.
of
are
phase
stator
angle
Figure
values
Figure
all
in
shown
the
smaller
windings
inductances
chapter
chapter
were
5 to
stator
phasewinding
La
iR
R2
Figure 6.1 Circuit for measurementof phaseself-inductance
142
11
1
10
gn
C6
3
2
I
0.
0
----.
3D
6D
I.
I.
I.
I.
I.
I.
I.
I.
I.
I
90
120
150
180
210
240
270
300
330
360
Rotor angle, *e
Figure
6.2
Angular variationof statorphaseself-inductanccs
143
12
1
11
ta
10
9
8
7
6
10
r.
-4
4
3
ON
0234567
II
--
I
-.
-.
--I-
--.
-I
Statorcurrent, A
Figure 6.3 Variation of statorself-inductancesL. Lbb, Lc, with
,
statorcurrent for rotor displacement 0 =00
e,
144
--
E
m
9
6
im2
l»i
44
10
04
Lo
a
iz
145
Rotor angle, *e
0
30
60
90
120 150
180
210 240
270
300
330
360
0.0
-03
-1.0
-2.0
-23
-3.0
Figure 6.5 Angular variation of mutual-inductancebetweenstator
phases
146
01
Statorcurrent
,A
123567
-r-r-
aAa
-- x-
-0.5
"1
1.5
.
.
.2
-2.5
-3
Figure
6.6
betweenstator
Variationof mutual-inductance
phaseswith stator current at rotor displacement
0e=00
147
I
CHAPTER
7
ANALYSIS AND MINIMIZATION
OF TORQUE
RIPPLE IN BRUSHLESS DC MOTORS
A major
low
speed,
can
give
precise
drawback
is
the
rise
to
rare-earth
inertia
a faster
and
. The
emf
a flat-top
stator
current[511,
to
waveforms
from
the
in
as
the
at
of
a reduced
expense
of
result
of
predicted
dependent
of
the
field
the
firing
masked
The
the
of
by
affect
computer
simulate
the
phase
current
drive
the
the
ripple.
drive,
'
At
inertia,
performance
program
the
of
described
and
to
148
study
as
the
low
the
drive.
chapter
factors
the
switching
stator
speed,
at
the
and
magnetic
the
affect
also
but
which
a ripple
inverter,
in
current
stator
the
high
however,
position
such
the
waveform
practice
rotor
factors,
and
emf
current
exhibits
the
of
electromagnetic
and
controller,
angle
torque
In
torque
on
Other
events.
magnitude
motor
emf
trapezoidal
this
of
torque.
and the
the
product
generated
ideal,
the
and
and
the
is
motor
since
and,
constant
both
of
periodic
severely
120*e
a rectangular
frequency
to
is
this
in
such
resulted
a brushless
to
produce
commutation
are
in
proportional
needed
differ
is
as shown
spanning
is
the
ripple
generated
torque
is
but
which
introduction
recent
has
rotors
ripples
applications
The
response,
speed
in
problems
robotics.
speed
and
at
especially
7.1.
Figure
with
serious
or
torque
of
permanent-magnet
increased
an
presence
positioning
dc motors,
brushless
with
these
speed
effects
they
may
5 was
written
which
affect
torque
In
ripple
the
particular,
produce
the
and
following
The
torque
waveforms,
and
reduced.
A torque
in
the
study
current
an
present
ripple,,
discuss
be
might
reduced.
waveforms
which
determined.
were
sections
low-speed
this
which
phase
optimum
torque
a smooth
by
ways
by
methods
various
study
it
which
be
might
may be defined
factor17,181
of
current
phase
rectangular
with
ripple
analytical
for
use
as
AT
7; = 7AV
where
AT
is
Tav
7.1
the
the
torque
average
EFFECT OF COMMUTATION
ideal
The
rectangular
7.2(a),
required
achieve
in
from
one
phase
Figure
waveshape,
current
commutation
current
waveforms
waveforms
ripple
than
rather
current
a
torque
and
shown
is
six
The
with
ideal
in
Figure
times
149
rounded
phase
torque
trapezoidal
7.3.
of
these
generated
The
the
in
shown
every
for
profile
the
rectangular
accompanies
that
the
edges
theoretical
the
the
Consequently,
the
ripple
[51,521 -
is
has
waveform
during
prevents
instantaneously.
to
regularly
particular,
inductance
winding
changing
and
In
difficult
commutates
current
another.
the
7.2(b),,
the
is
torque
Figure
of
waveforms
a constant
since
period
EVENTS
phase-current
produce
to
phase
from
current
motor
to
practice,
commutation
torque
and
ripple,
FACTORS AFFECTING TORQUE RIPPLE
7.1.1
emf
torque
peak-to-peak
frequency
stator
stator
phase
phase
of
the
supply
frequency
the
its
and
evident
that
current
value.
7.1.2
between
the
torque
value
for
the
the
7.4
Figure
torque
predicted
higher
a
with
is
It
machine.
larger
the
and
amplitude
experimental
is
the
shows
ripple
ripple
to
proportional
EFFECT OF PHASE EMF WAVEFORMS
Ideally,
brushless
emf
the
motor
waveform
motor,
the
shown
in
rectangular,
less
direct
the
120 *e,
effect
the
airgap
phase
as
on both
current
fluctuations,
which
in
torque.
the
phase
output
emf
waveforms
7.1.3
and
The
ripple[38,54).
shown
in
leads
turn
emf
to
The torque
profile
ideal
with
of
shown
in
This
for
rectangular
spans
has
a
magnitude
an increased
torque
contains
also
on the
not
Figure
emf
generated
waveform
effect
and
because
up and the
a direct
have
in
build
of
rate
as
is
7.5 (b) .
Figure
generated
waveforms
is
the
the
of
in
shown.
as
rounded
flat-top
the
120*ej
spans
distribution
are
corners
phase
brushless
however,
practice
flux-density
Consequently,
than
In
3-phase
a
ideally
waveform
7.5(a).
and
For
generated.
this
of
Figure
the
7.6153).
be
the
of
and a trapezoidal
be rectangular
should
flat-top
distribution
flux-density
airgap
should
fringing,
of
level.
current
stator
approximately
current
operating
relationship
is
magnitude
fluctuations
these
generated
phase
current
7.7.
Figure
EFFECT OF INCORRECT ADJUSTMENT OF POSITION SENSORS
ON COMMUTATION
Inaccurate
further
factor
timing
of
which
can
the
phase
significantly
150
switching
effect
devices
the
is
magnitude
a
of
the
torque
arranged
angle
to
accurately
can
be obtained
Also
they
should
that
the
transition
to
windings
7.1.4
The
ripple.
position
that
ensure
one
in
assembly,
state
the
at
7.2.3.
section
careful
energizing
occur
will
commutation
optimum
by
spaced,
equally
from
an
be
should
sensors
be explained
as will
be
another
three
the
of
so
stator
time.
correct
EFFECT OF ROTOR AND STATOR TEETH
In
the
a permanent-magnet
teeth
stator
produce
the
and
magnetic
torque
a reluctance
the
interaction
saliency
of
machine,
is
which
a
between
the
function
rotor
can
the
of
rotor
position[55).
7.1.5
EFFECT OF STATOR FIELD
The magnetic
interacts
the
with
The
torque
waveform
magnitude
the
material,
not
reaction
effect
flux
on
one
reduction
by
in
side
saturation
The
of
samarium-cobalt,
magnetic
field
the
other
the
of
use
a
a
has
as
there
the
in
the
to
the
maximum
a
net
armature
in
increase
The
always
whether
be
will
machine.
the
and
currents,
is
in
permanent-magnet
similar
pole
stator
of
does,
dc
less
flux
than
is
the
limited
path(56).
magnetic
permanent
a direct
distortion
it
process
side,
strong
type
winding
If
brushed
a
of
the
on
by
distortion
consequent
the
of
to
permanent-magnet,
distribution
the
on
occurs.
flux,
of
the
depends
saturation
the
of
currents
winding
stator
flux-density
of
magnitude
reduction
field
the
of
airgap.
by the
created
main
distortion
cause
or
field
effect
caused
151
-magnet
material,
on minimizing
by
the
stator
such
the
as
main
magnetic
f ield.
Samarium-cobalt
density
and
stator
MMF
a linear
7.1.6
in
most
other
less
the
magnetic
be another
field
for
sometimes
rotor
when the
occurs
surface
and
magnetization
rotor
be
would
the
can
permanent-magnets
in
torque
the
magnetic
This
ripple.
been
have
poles
subsequently
fixed
to
the
a special
using
magnetized
the
METHODS
discusses
section
speed
the
torque
and
ripples
for
available
methods
in
occur
which
a
drive.
practical
ADJUSTING ROTOR MAGNET POLE ARCS
It
the
is
rotor
2-pole
possible
magnet
brushless
inverter,
to
minimize
pole
arcs.
dc
a 120*e
torque
dips
magnetic
pole
arcs
should
either
of
60'e
the
with
conducting
no
torque
by adjusting
ripple
When a 3-phase
by
fed
current
rectangular
the
rotate
the
is
motor
minimize
to
the
yoke[271.
minimizing
7.2.1
than
irregularity
7.2 TORQUE RIPPLE MINIMIZATION
This
the
of
increased
and
to
MAGNETIZATION
additional
distribution
A given
materials[54).
magnetization
reason
energy
disturbance
airgap
EFFECT OF INHOMOGENEOUS
Inhomogeneous
high
coercivity,
characteristic.
causes
distribution
with
a high
demagnetization
therefore
flux-density
case
has
the
at
be
in
phases(531,
152
120*
voltage
is
waveform
commutation
180*e.
change
a
star-connected
the
set
events,
The magnet
source
is
the
thus
flux-density
as shown
in
Figure
To
up.
rotor
able
under
7.8.
7.2.2
INCREASING
Since
THE NUMBER OF PHASES
attention
be paid
must
to
emf required
phase
current
this
over
to
two
phases
shown in
the
for
7.9.
emf waveform
having
a 120*e flat-top,
profile
of
Figure
7.10.
not
constant
is
7.1.2,
section
and the
Figure
7.7.
profile
may be reduced
The effect
as shown in
Figures
for
profiles
6-phase
electronic
of
120'e,
and 7.12,
and 9-phase
increases
of
are required.
7.2.3
ADJUSTMENT OF THE COMMUTATION
The
rate-of-change
depends
the
factor
the
on
of
is
the
the
generated
phase
angle
of
emf
angle,
is
153
torque
This
the
positional
sensors
during
the
commutation
inductance
winding
in
the
ANGLE
defined
defined
torque
of
Another
waveform.
and
current
zero
the
as
on the
complexity
current
winding
such
commutation
injected
commutation
factors,
many
shape
of
as shown in
respectively.
the
which
in
explained
represent
motors
number
generated
number of phases(571,
which
both
and the
circuitry
the
the
more
a waveform
such
torque
constant
as
is
profile
the
a trapezoidal
with
however
by increasing
7.11
however
solution
torque
and
current
phase
has the
full
the
the
60'e
for
In practice
on the
emf waveform
The same motor,
for
constant
of
a rectangular
Figure
emf
for
profile
simultaneously
profile
generated
depends
a sinusoidal
and with
torque
the
torque
constant
and
considerable
of
The emf waveform
range.
energized
corresponding
120*e,
wave-shape
produce
permanent-magnet(27,511,
is
the
distribution
winding
for
torque
each phase produces
as
the
angle
generated
Figure
7.13,
and
important
between
emf.
with
A
the
phase
current
through
generated
is
is
current
defined
30*
as
f igures,
it
between
the
is
supply
winding
the
changing
Figures
7.15,7.16
current,
torque
and
30'e
and
instantaneous
the
the
consequently
steady
7.2.4
the
in
The
effect
waveform.
model
the
waveforms
for
160rpm
For
achieved
torque
with
rises
current
emf
phase
torque
angles
commutation
advance,
E
quicker
of
the
conditionsf
between
5.
chapter
load
a lNm
a 30*e
of
was
motor
predicted
difference
winding
in
these
that
ripple
described
with
since
the
supply
is
larger
to
its
and
desired
state.
REDUCING THE STATOR WINDING
The
does
and this
current
commutated
winding
emf E
winding
the
the
two
difference
of
potential
V and
voltage
these
profile
show
of
is
ripple
From
commutated
on
the
situation
angle,
respectively.
torque
smallest
7.17
speed
the
commutation
computer
speed
7.14,
the
angle
and
a steady-state
0*, 15*
actual
not
edges
stator
of
the
using
the
where
point
instantaneous
the
torque
commutation
investigated
and
the
the
on
the
Figure
V and the
on
hence
and
that
voltage
effect
to
passes
applying
advance[38,58).
by changing
a direct
in
shown
evident
switches
30*e,
commutation
may be controlled
has
as
zero
emf
generated
the
of
by
advanced
is
emf
the
after
When closure
zero.
phase
30*e
applied
and
have
a
winding
commutation
designing
the
phase
the
dip
current
ideal
at
every
the
machine
commutation
and
the
torque
with
a low
154
shown
shape,
rectangular
inductance
on
waveform
INDUCTANCE
stator
but
instant
generated
profile
in
may
emf.
be
inductance,
Figure
7.2(b)
has
rounded
due
to
The
the
effect
minimized
such
by
that
the
time
electrical
its
attains
f inal
inductance
for
value,
160rpm.
is
It
to
proportional
the
torque
magnetic
circuit
to
the
length
a
in
the
stator
the
motor
to
modifications
7.2.5
less
is
turns
area,
a
directly
phase
per
inversely
and
inductance
of
with
Thus
circuit[381.
magnetic
winding
winding
necessitates
design.
SKEWING OF THE STATOR SLOTS
is
It
by
1551
ef f ect
7.2.6
known
well
magnets
one
simulated
torque
or
rotor
torque
reluctance
frequency
frequency
torque
produced
by current
speed
ripple,
rotor
inertia(55).
ripple
which
of
than
magnitudes
research
the
is
155
and
5kHz.
are
much
less
produce
be
filtered
PWM controller
capable
of
7.20.
operating
It
is
high
The
than
only
in
25kHz
at
ripple
at
to
on the
as shown
that
and they
expected
However,
this
torque
commutation
is
controller
7.19
controller
depends
which
Figures
high-frequency
less
current
ripple,
in
given
is
the
PWM current
the
of
the
switching
by
produced
results
that
course
slots
stator
TO THE CURRENT CONTROLLER
ripple
frequency
evident
the
reduces
pitch
a high-frequency
switching
skewing
.
current
causes
the
that
slot
IMPROVEMENT
The
the
the
of
predicted
a speed
is
of
number
rapidly
stator
inductance
cross-sectional
proportional
reduction
the
the
ripple
phase
of
square
the
1Nm and
of
torque
The
value.
current
shows
and
torque
the
that
the
7.18
ripple
load
a
evident
inductance
smaller
and
the
and the
small
Figure
value.
between
relationship
is
constant
those
a small
by
the
designed
in
out
at
5OkHzj
this
and
current
ripple
torque
measured
phase
frequency,
the
ADJUSTMENT
necessitate
not
practicable
the
phase
current
in
the
to
and
switching
novel
design,
the
and
to
the
minimize
adjustment
torque
this
of
this
waveforms
Both
alternative.
ripple
when
current
phase
implementation
detail
torque
minimizing
motor
of
waveforms
practical
explained
switching
increased
an
for
above
adjustment
attractive
the
25kHz
and
ripple.
current
modifications
an
and
predicted
OF THE PHASE CURRENT WAVEFORMS
provides
of
that
high-frequency
show the
5 and
the
reduces
the
at
waveforms
described
methods
7.22
and
confirm
reduces
The
is
a current
which
frequency
7.21
Figures
considerably
consequently
and
magnitude
ripple.
7.2.7
high-frequency
relatively
ripple
are
concept
below.
7.3 CURRENT PROFILING
instantaneous
Since
instantaneous
should
the
current
ripple.
in
product
of
be possible
to
dc motor
proportional
to
the
emf and phase
currentr
it
a smooth
a way which
The instantaneous
a brushless
generated
produce
in
profile
is
torque
the
eliminates
torque
electromagnetic
may be expressed
Te=
by adjusting
torque
torque
developed
as
(e, i, +eb'b+ecic
(7.2)
where
eat eb, e.
are
generated
'at iby ic
are
in
respectively
156
I. -.
the
respectively
phases
the
a,
instantaneous
b and
emfs
c,
instantaneous
currents
in
is
COr
At
any
and
to
occurs
next
are
supply
current.
period
becomes
'
two
the
of
three
one pair
of
7.23
= I.,
are
windings
The
typical
a
phase
60*e
of
the
dc
is
I.
where
instantaneous
total
stator
of
for
ic
phases
position.
rotor
ib '0 0 and
-Isr
The
velocity.
60*e
Figure
and
angular
from
every
in
ia
rotation
only
commutation
defined
currents
rotor
time,
given
conducting,
the
the
b and c,
a,
phases
during
torque
this
(7.3)
(0
With
if
term
is
this
is
(ec-ea)
possible
flat-top.
in
less
than
does
not
60*e
of
rotor
these
Figure
the
ideal
due
factors
7.24.
The resultant
the
phase
deviation
and
fluctuations
minimize
waveforms
1/ (ec-ea)
commutation
the
torque
the
thus
effect
have
over
instant,
ripple.
the
the
the
the
of
60*e
to
This
phase
to
achieve
achieved
157
combination
in
shown
for
any
emf waveforms
and
compensate
The phase
proportion
optimum
in
every
by
considered
the
of
may be minimized
commutation.
in
the
current
a dip
ripple
generated
period
is
the
ripple
be adjusted
to
and
torque
waveforms
in
a flat-top
The
torque
current
has
commutation.
120*e
not
emf
and has
in
result
adjusting
to
to
and
a
is
this
shape
rectangular
position
two
phase
constant
rotation,
having
emf
fluctuations
contains
of
previously,
The generated
120*e,
have
trapezoidal
a
is
torque
60*e
the
over
as explained
practice.
the
current,
constant
with
However,
case
of
dc supply
a constant
the
r
the
to
current
the
term
at
the
and
minimization
program
of
(and
in
introducing
by
practice)
inverter
control
of
the
current
to
follow
shows
the
the
waveforms
speed
of
constant
is
It
this
and
shows
Figure
7.26
waveform
required
to
in
torque
and
7.25.
It
speed
is
speed
ripples
torque
and
the
from
are
considerably
speed
ripples
at
ripples,
the
current
current
boost
The
slight
7.27
the
shown
position
the
shows
to
circuit
following
block
in
phase
the
of
this
torque
implement
and
current
and
still
7.26
and
these
these
minimize
A
modified.
introduced
profiling,
the
on
waveforms.
has
commutation
speed
profiling
Figure
are
current
at
current
and
torque
therefore
the
current
of
further
with
speed
A
ripples.
is
described
section.
IMPLEMENTATION
diagram
Figure
is
effect
in
eliminated
7.4 PRACTICAL
A
together
not
There
To
was
is
the
of
the
Figure
be
to
average
corresponding
that
instants.
commutation
increase
practical
in
and
deviation
the
and
reduced.
evident
loop,
control
virtually
in
during
an
in
the
7.26
has
such
same conditions
Figure
waveform
any
profile
for
the
commutation
7.25
torque
motor
adjusted
emf
for
evident
occur
Figure
the
generated
Figure
and
a fluctuation
compensate
waveforms
clearly
and
in
phase
torque
0.6Nm
the
PWM
with
motor
without
of
that
evident
resulted
the
corresponding
torque
has
waveform.
fluctuations
the
load
force
to
current
with
a
loop
control
proportionality.
phase
for
102rpm.
the
above
together
speed
current
switches
simulated
modification,
into
a
for
the
7.28.
necessary,
OF CURRENT PROFILING
Accurate
to
enable
158
profiling
current
information
the
correct
is
circuit
on
profile
the
rotor
of
the
current
both
synchronized
The
position.
pulses
per
so that
be
to
to
0 to
reset
pin
the
applying
the
the
output
is
pulse
that
of
voltage.
The
Ptl
the
average
compared
signal,
device
switching
to
the
stator
winding.
to
according
the
at
mark-to-space
adjusted
instants
in
relation
and the
subsequently
The
motor
the
to
the
required
current
159
an appropriate
be
subtracted
in
shape
a
the
and
PWM
state
applied
thereby
shaped
changing
EPROM,
PWM signal
position
voltage
on/off
by
by
varied
voltage
is
generated
rotor
applied
generate-
the
current
addresses
of
EPROM is
the
EPROM and,
the
appropriate
ratio
to
controls
and
in
data
the
current
being
waveform
ratio
reference
The resultant
voltage.
sawtooth
mark-to-space
the
values
a
ensuring
required
may
before
7.29,
mark
an EPROM which
to
amplitude
that
zero
shaft
the
data
by
such
360*m,
the
from
the
voltage
counter
encoder
of
converts
the
obtained
addresses
output
counts
is
same
are
which
to
every
the
counter
speed-demand
whose
of
the
Figure
of
with
counter
from
signal
which
potentiometer
the
can
pulses
an AND gate,
The
representation
converter
analogue
60*e.
reset
7.29,
applied
to
60*e,
instants
pulse
output
starts
The digital
a D/A
reset
every
always
a numerical
profile.
is
to
The output
contains
The
binary
high
used
counting
from
60*e.
is
a pulse
1080
per
encoder
Figure
by
reset
counter
also
position.
to
is
in
shown
The
rotor
3 produces
commutation
Ve.
of
accuracy
the
60 pulses
to
is
ripple
and
Appendix
the
and
counter
every
in
detailed
torque
events
corresponding
an
the
since
commutation
position
59 and
gate
the
encoder
rotor
defined
supplied
to
obtained,
revolution,
the
from
be
to
waveform
the
the
can
commutation
may be achieved.
be
In
to
order
store
S2 are
EPROM, switches
Sl
of
comprising
memory,
current
each
motor
7.30.
boost
the
waveform
in
the
the
output
torque.
forms
7.30(c).
to
similarity
can
be
phase
to
ripple
slight
and
speed
the
current
but
tachometer,
accurate
misalignment
readings
due to
results
of
which
the
were,
magnitude.
160
the
mechanical
motor
in
speed
and
it
unfortunately
the
ripples
measure
ripple
output
was difficult
arising
problems
corrupting
any
Figure
7.26,7.27
made to
circuit
converter
voltage
and
gauge
a strain
on
predicted
speed
and
were
in
the
Figures
torque
and
performance,
shown
as
Many attempts
using
to
of
measurement
7.30 (b)
fluctuations
overall
of
effect
consequently
and
between
seen
waveforms
achieve.
frequency
from
best
the
the
deviation
the
in
current
Figure
and
Close
easy
obtain
deviation
such
To achieve
Accurate
of
of
and
given
with
and
waveform
emf
generated
are
minimize
for
compensate
are
7.30.
voltage
to
torque,
output
waveforms
waveform
instants
to
effect
the
shows
necessary,
and
using
waveforms
compensation
measured
torque
phase
current
of
and
not
on
the
minimize
both
output
the
required
to
the
individual
store'the
converter
commutation
commutation
fluctuations
is
to
same
blocks
several
provide
D/A
7.30(a)
Figure
during
the
the
60 bytes,
experimental
corresponding
Figure
of
to
used
the
waveforms.
Several
the
and
in
waveforms
output
several
case,
the
torque
of
small
*102
5
0.0
0.05
0.1
0.15
0.2
0.35
0.3
0.25
0.4
0.45
0.5
time, s
(a) momentof inertia = 0.0005kg.m2
*102
c-
Ll.
C;
0.0
0.05
0.1
0.15
0.2
0.3
0.25
0.35
0.4
time, s
(b) moment of inertia = 0.002kg.m2
*102
0.5
0.45
2.0
ts
to
0.5
0.0 11fi aia1i1aA11iaa1ii!
0.0
0.05
0.1
fi
211A2a1&1
0.15
0.2
0.25
!
ff
0.3
1a&01AihiiIiii!!
0.35
0M "t-t-
0.4
(c) momentof inertia = 0.008kg.mý
Figure 7.1 Simulatedmotor speedwaveformsfor various
momentsof inertia
161
0.45
time, s
0.5
+
Ia 0
+
0
ic
0
0
60
120
180
240
360
300
Rotor angle, *e
(a) Idealized phasecurrents
la
+
0
ic
0
0
60
120
180
240
360
300
Rotor angle, *e
(b) Actual phasecurrents
Figure 7.2 Comparisonbetweenidealized and actual phase
current waveforms
162
ic
ib
ia
+
!2
N,
.........
............
eb
e,
+
ec
/N
rad
L
120T
L
80
ýo
ý
300
360
ý
20
180
Rotor angle, *e
Figure 73 Effectof phasecurrentwaveformson torqueproffle
163
1.0
S
Z
u
Ici
0.8
0.6
0.4
0.2
0.0
0.0
1.0
2.0
3.0
4.0
Statorcurrent, A
Figure 7.4 Variation of torqueripple with statorcurrent
164
5.0
or angle,
(a) Idealizedphaseemf
or angle,*e
(b) Actual phaseemf
Figure 7.5 Comparisonbetweenidealized and
actualphaseemf waveforms
165
2:.
m
30
210 240 270 300 330 360
ýRotor
J--JL--L--L
.IN
- -ý
60 90 120 150 18
angle,ee
Figure 7.6 Typical practical airgap flux-density distribution
166
ic
0
902
ce
ea
eb
e,
w
49
0
E2
60
120
I RO
240
300
360
4 20
80
Rotor angle, *e
Figure 7.7 Effect of phaseemf wavefornison torqueprofile
167
B
+A
..........
s+
IN
+C
+B
Figure 7.8 Brushless dc motor with 120*e rectangular phase
current excitation and rotor magnet with 180*e arc
168
0
60
120
180
240
300
360
420
480
Rotorangle,*e
(a) Sinusoidalgeneratedemf and rectangular
current waveforms
V
F-
0
60
120
180
240
300
360
480
420
Rotor angle, *e
(b) Torque profile
Figure 7.9 Torque profile resulting from sinusoidalgenerated
emf and rectangularcurrent waveforms
169
-'
0
60
120
180
240
300
360
480
420
Rotor angle, *e
(a) Ideal trapeziodalgeneratedemf andrectangular
current waveforms
0
60
120
180
240
300
360
480
420
Rotor angle, *c
(b) Torque profile
Figure 7.10 Torque profile resulting from ideal trapeziodalgenerated
emf and rectangularcurrent waveforms
170
Er
9
Rotor angle: e
Figure 7.11 Torqueprofile for a 6-phasebrushlessdc motor
171
LIM
9
Rotor angle,*e
Figure 7.12 Torqueprofile for a 9-phasebrushlessdc
motor
172
Ici
*Z
Lu
173
t)
i3
Ici
cqs
M
9ý
&0
gL
iz
174
Old
0-4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time, s
(a) phasecurrent
E
Z
F:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time, s
(b) motor torque
*102
E
C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time,
s
(c) motor speed
Figure 7.15 Simulatedwaveformswith O*ecommutationadvance
(Load condition: I Nm at 160rpm)
175
U.U-
-
-to-
-2.0
-3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time, s
(a) phasecurrent
0
Z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time,s
(b) motor torque
s102
IJ
to
s
Lill
S;
0.5
0.0-fL0.0
0.1
0.2
0.3
0.4
0.5
(c) motorspeed
0.6
0.7
0.8
time, s
Figure 7.16 Simulatedwaveformswith 15*ecommutationadvance
Nrn
160rpm)
OL,
:I
at
condition
oad
176
0.0-to- -2.0
-3.0
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time, s
(a) phasecurrent
0
Z
F:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time, s
(b) motor torque
*102
9
0.0
0.1
0.2
0.3
0.4
0.5
(c) motorspeed
0.6
0.7
0.8
time, s
Figure 7.17 Simulatedwaveformswith 30*ecommutationadvance
(Load
condition: I Nm at 160rpm)
177
50
0-1
40
30
20
20
40
60
80
100
Nominal parameters(%)
Figure 7.18 Variation of torqueripple with statorwinding
inductance
178
0.0
0.05
0.1
0.15
0.2
0.25
0.3
035
0.4
0.45
time,s
(a) phasecun-ent
S
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
time, s
(b) motortorque
*102
0.0
0.05
0.1
0.15
0.2
0.25
(c) motor speed
0.3
0.35
0.4
0.45
time, s
Figure 7.19 Simulatedwaveforms(Load condition: 0.7Nrn at 200rpm,
frequency
5kHz
=
switching
179
.o
0-4
0.0
0.05
0.1
0.15
02
025
0.3
0.35
0.4
OA5
time, s
(a) phasecurrent
Ei
Z
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
time, s
(b) motor torque
*102
2-Ot
1.5
to
0.5
0.0 11111111111111211111111111111111111111
0.0
0.05
0.2
0.1
0.15
0.25
0.3
(c) motorspeed
F-4 111111!
0.4
0.35
1
0.45
time,s
Figure 7.20 Simulatedwaveforms(Loadcondition: 0.7Nmat 200rpm,
switchingfrequency= 25kHz
180
-0.25
-0.
-to
0.28
0.29
0.3
0.31
0.33
0.32
0.34
0.35
0.36
0.37
0.38
time, s
(a) simulatedwaveform
--------
--------
------
........
.............
......
-----------
------
......
........
......
-------
.......
verticalscale
0.2A/div
-------
-------
---
.................
-------
horizontalscale
IOms/div
0
-----------------
.......
.......
..............
...................
4-------------------------
------
----
..........
6
.......
----
...........
............
........
(b) measuredwaveform
Figure 711 Phasecurrent waveforms(Load condition: 0.7Nm
at 2OOrpm,switching frequency= 5kHz)
181
.0
6-4
0.28
0.29
0.3
0.33
0.32
0.31
0.34
0.35
0.37
0.38
time,s
(a) simulatedwaveform
........ ...... . ...............
0.36
--------------- - --------
.......
...............................
---
--------
--
----
-
-----
---------------
---------------
--------
r
verticalscale
0.2A/div
......................
horizontalscale
IOms/div
0
.......
................
-
L
-------
.......
........................
.......
............
...............................
-----
--
................
........ .............
.......
........
d
(b) measuredwaveform
Figure 7.22 Phasecurrent waveforms(Load condition : 0.7Nm
at 200rpm, switching frequency= 25kHz)
tj
0-4
to
Ici
L)
en
oft
raq
C4
>!6
183
'b
ic
eb
e,
+
IC
+
w
4g
Cd
V
r
0
60
120
180
240
300
360
420
480
Rotor angle, 'e
Figure 7.24 Effect of phasecurrent and emf waveformson
torqueprofile
184
2
6-4
-2+
0.5
0.6
0.7
0.8
0.9
time,s
(a) phase a current
1
0.8
rz
z
0.6
E-:
0.4
TTTTýMYI
0.2
Oll
0.5
IIIIII..................
0.6
0.7
ýl
0.9
0.8
0.9
0.8
time, s
(b) motor torque
110
E
100
90
-jý0.5
0.6
0.7
(c) motor speed
time, s
Figure 7.25 Simulatedwaveformswithout compensation
(Load condition: 0.6Nrn at 102rpm)
185
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
VON
0.5
.....
0.6
........ .....
0.7
0.8
0.9
time,s
(a) phasea cuiTent
1
0. a
Ei
Z
0.6
ý:
0.4
0.2
0
0.5
0.6
0.8
0.7
0.9
time,s
(b) motor torque
110
100
go-t0.5
0.6
0.7
0.8
(c) motor speed
0.9
time, s
Figure 7.26 Simulatedwaveformswith generatedenafcompensation
(Load condition: 0.6Nm at 102rpm)
186
0.8
0.6
0.4
0.2
0
d
6-4 -0.2
-0.4
-0.6
-0.8
0.5
0.8
0.7
0.6
0.9
time, s
(a) phasea current
0.8
z
F:
0.6
0.4
0.2
oil
0.5
IIIIIIIII..................
0.6
0.5
0.7
U. V
time, s
(b) motor torque
110
100
90+
0.5
0.6
0.7
(c) motor speed
Figure 7.27
0.8
1
0.9
time, s
Simulatedwaveformswith combinedgeneratedemf and
commutationcompensation(Load condition: 0.6Nm at
102rpm)
187
as
a
03
A
zz
188
-3
U) win
E
:
"M
a
Ln
>1
189
vertical scale
0.5A/div
horizontal scale
25ms/div
in
Lj
".ijI%.
I
Lj
(a) phasecuffent waveform with commutation
compensation
-----------------
...............
vertical scale
0.5A/div
horizontal scale
25ms/div
f
-------.......
.......
......
4
I
..... ...... ------................ -- ----- ........
...............
...............I
OP
........I.................
"ITT
........
. .......................
. .......
--------
(b) phasecurrent waveform with generatedemf
compensation
vertical scale
0.5A/div
horizontal scale
25nis/div
(c) phasecurrent waveform with combinedgenerated
ernf and commutationcompensation
Figure 7.30 Measuredphasecurrent and current reference
waveformsfor various current shapingprofiles
(Load condition: 0.6Nm at 102rpm)
190
CHAPTER
8
CONCLUSIONS
This
chapter
presents
for
recommendations
further
the
together
thesis,
with
work.
CONCLUSIONS
A comprehensive
mathematical
investigate
to
used
performance
of
operating
of
the
from
arising
conclusions
in
described
research
8.1
AND RECOMMENDATIONS
the
conditions
The
operation.
the
machine
the
phase
for
is
model
linkage
flux
based
the
Tensor
methods
varying
circuit
topology
arising
in
the
the
of
permanent-magnet
included
and
An
all
in
the
to
this
stator
this
was
was
designed
dc
motor
during
the
course
experimental
motor
parameters,
model,
measured
stages
predicted
and
PWM modes
formulated
in
mechanical
the
account
for
the
in
is
flux
of
at
topologies
model
solution
the
inverter
are
included
the
accurate
linkage
determined
due
to
the
experimentally
model.
drive
were
various
torques
changes
winding
and
to
used
from
experimental
brushless
various
practical
Central
model.
representation
rotor
and
are
under
the
with
and
transient
numerical
equations
defining
shaft.
pattern,
the
on
together
machine
switching
drive
square-wave
differential
equation
developed
and
motor
both
frame,
reference
differential
dc
and
been
steady-state
brushless
a
has
model
results
as
of
in
substantiated
191
the
by
required
described
were
and
built
for
research
the
by
1.3kW
and
the
mathematical
6.
chapter
a
At
experimental
all
investigation,
shape
and
the
of
mathematical
5 and
effect
the
in
and
speed
investigation
an
they
might
that
the
be
inductance
the
of
Also,
wave-shape.
less
of
periodic
the
and
novel
commutation
and
in
proposed
in
relation
in
a way
of
the
the
and
chapter
7.
to
rotor
the
generated
from
introducing
a current
to
emf
drive
for
during
rounded
on
a
phase
192
The
observations,
torque
with
instants,
The
the
the
the
was
waveforms
in
shape
effect
and
in
achieved
a
ripple
current
torque
was
ripple
position
rotor
minimizes
eliminates
commutation.
region
torque
deficiencies
and
loop
every
rectangular
commutation
and
control
due
edges
flat-top
a
these
the
This
which
at
ideal
the
the
output.
by
established
in
both
any
torque
fluctuations.
eliminate
waveforms
a current
boost
has
position
virtually
the
emf
on
Profiling
compensates
which
commutation,
ripples
by
method
ways
commutation
the
Based
events.
effective
to
result
dependent
and
has
then
was
the
the
and
contains
factors
two
and
it
modelt
than
trapezoidal
which
these
of
is
the
rather
120'e,
than
combination
which
position,
rotor
due
the
and
currents
investigations
dip
a
and
predict
emf
affecting
waveform
current
phase
the
drive
the
Practical
minimized.
actual
winding
by
produced
ripples
in
ripple.
factors
the
of
to
stator
in
confidence
established
accuracy
presented
generated
torque
undesirable
both
modelling
ability
of
the
on
the
results
model's
waveform
commutation
consequentially
Having
the
and
and
the
particular,
trapezoidal
of
used
In
validity
developed
7 demonstrated
the
accurately
the
confirmed
in
obtained
correlation
model
described.
chapters
60*e
close
has
magnitude
technique
to
the
and
of
speed
practice
PWM controller
results
presented
in
7.3
section
the
stator
phase
improvements
was
effect
the
of
It
examined.
reduced
by
machine
under
achieved
with
rotor
drive
the
instant
the
is
torque
indicates
This
in
important
be
for
and,
torque
smallest
ripple
could
magnitude
commutation
of
advance.
sensing
the
the
consideration,
position
significant
on the
angle
that
shown
was
a 30*e
modifications
performance.
commutation
advancing
these
provided
waveforms
obtained
results
that
in
instants
commutation
Practical
verified
rig
increase
modest
the
ripple.
current
in
relatively
9% during
of
experimental
the
The
the
a 30% torque
the
to
show
current
eliminated
from
that
the
ripple
that
was
accurate
smooth
producing
shaft
torque.
The
operating
model
the
in
ripple
the
in
the
causes
magnitude
produced
frequency,
of
by
on
the
these
they
commutation
only
The
a small
on
results
produced
torque
ripple,
frequency.
switching
is
the
and
ripple
high-frequency
a
pulsations
produce
current
of
controller
torque.
the
controller
current
the
of
capable
50kHz,
to
up
effect
output
7 show that
controller
of
the
machine
chapter
depends
which
investigate
to
was used
is
controller
frequency
a switching
at
presented
by
PWM current
experimental
The
much
smaller
than
those
and,
due
their
high
to
ripple.
speed
8.2 RECOMMENDATIONS
There
are
to
significant
lead
several
areas
benefits
in
and
below.
193
which
some
further
of
these
research
are
could
discussed
A valuable
development
(2)
would
and
Since
pulsations
are
control
to
eliminate
of
position
The
prediction
(a)
of
excessive
Since
the
developed
technique
may be
in
used
spikes
voltage
the
of
studies
the
on
phase
by considering:
snubber
in
effects
circuits
during
spikes
the
which
prevent
turn-off
of
the
switches.
generated
phase
torque
investigate
flat-top
the
in
undesirable
reverse-recovery
voltage
the
effect
of
diodes.
inclusion
inverter
extremely
may be improved
inclusion
The
the
of
free-wheeling
(b)
optimization
schemes.
waveforms
The
the
the
minimize
motors.
pulsations
control
to
the
profile
would
currents
applications,
torque
be
would
which
enable
controllers
voltage
(4)
phase
off-the-shelf
torque
described
work
system,
motor
This
ripple.
the
smart
the
position
(3)
a
of
automatically
torque
to
extension
emf waveforms
it
ripple,
the
region
of
different
of
effect
would
this
emf
capable
of
on
seem to
be
strongly
widths
the
to
useful
of
the
torque
output
profile.
(5)
Since
the
rotor
damping,
performance
torque
it
of
behaviour,
as a damper
is
model
cage,
be
would
the
including
useful
in
drive,
particular
damping
when additional
are
added
194
to
to
the
rotor.
the
effect
investigate
the
circuits,
of
the
output
such
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T.
electric
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performance
33.
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converters,
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Nehl,
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CR
NASA
No.
Undeland,
and
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27-43.
istor-cont
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32.
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commutated
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R.
earth
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Miller,
A.,
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electronics:
31.
1983,
rare
Magnets
CA,
Diego,
using
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on
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Mexico,
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of
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inductances
PAS-104,
dc
Technical
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41.
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IEEE
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-magnet
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Vol.
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1971,
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1984,
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1985,
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1989.
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October
1987,
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October
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Germany,
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G.,
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England,
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57.
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for
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IEEE-IAS
servo
1, Atlanta,
GA,
Vol.
Record,
Conference
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April
G.,
of
Annual
Brentani,
M.
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56.
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and
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Pt.
"Computation
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1986,
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B,
No.
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commutated
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Meeting
201
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131,
Vol.
of
Record,
902-908.
APPENDIX A]
Al. 1 PHYSICAL DETA111,
S OF THE BRUSHLESS DC MOTOR
A connection
are
shown
the
shows
in
Figures
motor
dc generator
diagram
A1.1
connected
which
acts
and
A1.2
and
to
outline
the
for
inverter
and
coupled
PARAMETERS
Value
Rk
1.278
C1 at
Lo
9.85
mH
L2
0.45
mH
Mo
1.95
mH
M2
0.45
mH
KT
0.966
Nm/A
KB
1.68*10-3
V/rps
KG
3.3*10--l
V/rps
J.
0.002
kg.
TF
0.15
Nm
D
0.03
Nm/krpm
202
the
motor
Figure
respectively.
as a load.
A1.2 MAIN MOTOR AND TACHOMETER
Parameters
drawing
20'C
M2
A1.3
to
the
I
2
3
4
5
6
7
8
9
10
11
12
Blue lead
Brake
Not used
Readlead
Brake
Greanlead
Therrnostat
Greanlead
Ibennostat
Black / White lead
Yellow lead
Purplelead
sensor1
sensor 2
Brown lead
sensor 3
Orangelead
Blue lead
sensor I
sensor 2
Gray lead
sensor3
Redlead
+ 15 V dc:supply
(Position sensor)
Tac en-6F
13
15 V
14
Black lead
15
16,
Ground (position sensor)
White lead
0
0-
0
0-
Phase
,
0
0-
C
Phase
ý
+ 15 V dc supply
(Tacho)
Ground (Tacho)
Phase A
Figure
ALI
Motor connectiondiagram
203
1 3!
Vv
LA, A
%A A<<
L,) U)
out
cr cr
LU
01010
m
UU
" -"1
3: X
LAA
z
cr cx
In M
cc
0
Z;
z CL
_j
-C
z
z
0
x
0x
C)
owD
Lu
cr
_j (L
Lu si
0
Ix
WZ ;:
L) Umx
U,
cx
,j
ix
Z: )
0
m WJ
0
Z
Li
Uuj -t
I
cm
to
17
i0
L'i
u
.4
CL
V)
W
0
x
.8
6 (it
I
-41
'Olmý
.
Adl,
-
t
V
31,
LlILF
ýVr
76:
205
APPENDIX
A2
FOURTH'-ORDER RUNGE-KU7TA
NUMERICAL INTEGRATION
A differential
equation
form
the
of
Mf
Tt
is
readily
Order
solved
on a
method[ 42,431
Runge-Kutta
from
obtained
the
basis
step-by-step
nth
previous
The
value
(n+l)
'n
the
using
th
by
value
4-th
In+1 is
the
applying
formula
IýI
n+l
(KI +2Y%2+2K3 +K4)
+-!
n6
where
K, = h. f On ln )I
I
K2
f
(t
h.
+lh,
=
n 2n
l
I +. K,
2
K3
f(
h.
tn +lh,
=
2n
I +IK2
2
K4 = h. f(tý+
h is
and
The
the
current
the
current
integration
the
quantities
at
at
h, ý+K
the
the
end
of
of
length.
step
tn and
start
3
)l
the
In are
an
integration
step.
206
the
respectively
step,
time
and
and
'n+1 is
APPENDIX
A3
SHAFT ENCODER SPECIFICATION
LEINE AND LINDE LTD
STANDARD PULSE RATES
The following rates (pulses per revolution) are available.
These rates can be doubled by counting both the positive
and negative flank of the pulse. it this is performed on both
channels a quadrupling of the nominal pulse rate will be
achieved.
This doubling or quadrupling can be made in an electronic
counter, in computer soft ware or in a signal converter supplied
by us.
MODEL 63
INCREMENTAL
ENCODER WITH
MARKER PULSE*)
*) Also available
without
5000
4096
4000
3600
2540
2500
2160
marker pulse (Model 53).
2048
2000
1800
1600
1440
/ 1270
1250
1080
1024
1000
900
800
750
720
635
625
600
512
508
500
400
360
300
256
250
254
250
240
200
180
150
128
127
125
100
60
50
SPECIAL MODELS
For special orders, our manufacturing technique enables
us to offer fast delivery of non-standard pulse rates up to
5000.
Rugged photo-electric encoder for industri aI applications
0 Light emitting diodes for long life. Push-pull photo
diode circuit has high noise-immunity and high tolerance
to temperature variations and component ageing.
0 Marker pulse channel gives one pulse per revolution
for use as zero reference.
0 Single supply, 5-24VDC. No separate lamp supply
required.
* Complementary output signals permit use of differential lines to give high immunity to interference on
long lines.
0 Housing is splash proof. Optional shaft seal makes
entire encoder dust and spray proof according to IP65
classification.
Four shaft/bearing options:
60 6mm shaft
06 6mm shaft with screw flat
63 6,35mm (114")shaft
10 10mm shaft with screw flat.
0 Rugged housing with 2 mm walls. Shock and
vibration proof electronics, plastic disc and grating
0 Cable or connector to the rear or to the side.
Line driver standard by 5V.
207
ELECTRICAL SPECIFICATIONS
Light emitting diodes and
Opto components
photo diodes
Line
driver
5=5ý0
Supply current V
12-18
18-30
Current drain at no load
50(70)
130(180) 50(70)
typ. imax) mA
50
50
40
Sink current min mA
35
35
Source
40
Max pulse frequency kHz
200
100
100
better than
(5 V 400 kHz optional)
1.5
1.3
Low output level max V 0.5
Load dependent. see data
High output level
sheet of resp IC-circuit
Complementary (1, i. ý, 2,0,6)
Output signals
Cable length
1.5 rn (8 core screened.
2x0.34 MM2 +6x0.22 mm2)
extra length optional
Cable colour code
A Signal 1 green G Signal 0 brown
B Signal while
H Signal 6 violet
C Signal black
E +E volt red
D Signal 2 yellow F0 volt
blue
Temperature range
-40 C to + 60'C
OUTPUT CIRCUIT OF ENCODER
Supply current V5
12-18
IC-circuit (active pull up) SN75114 see circuit
diagram
Resistance
270Q
5volt
FVolt
0.1p.1-
18-30
see circuit
diagram
2709
i2-30VOlt EV011
OUTPUT SIGNALS
Bidirectional encoders: Two 90* phased (quadrature) square
waves permit direct ion-sensing. Signal 2 comes before signal 1
8t clock-wise rotation (seen from shaft end). One 0-pulse.
The 0-pulse Is synchronized with channel 1 and 2 and Is high
once per revolution at the same time 83 channels 1 and 2
are high.
SIGNAL SEQUENCE
Clock-wiserotation, seen from shalt end.
LUG
ACCURACY
Electricaldegreesare usedas the unit of measurementto
specify the accuracyof the encoder.360electricaldegrees
are defined as the mechanicalshaft anglethat corresponds
to one signal cycle. (360mechanicaldegreesdividedby the
pulse rate of the encoder.)
The dividing error specifiesthe largestdeviationfrom the
nominaldistance betweentwo pulseedgesfrom the sameor
from different signal channelsand consists of mechanical
angleand electronic reproductionerrors.Thedividing error
Increase
Is
does
the
turnedthrougnmore
not
when
encoder
L i ..............................................................
than one completeturn, that Is, the error Is non-cumulative.
Whenmeasuredat arbitraryedgesof different channels:
Max dividing error = =50electricaldegrees.
11
.................. --L
Channelseparationspecifles thedistancebetweenadjacent
edgesof different channels:
Channelseparationa 90 &25electricaldegrees.
............... .........
O
-L j-
T
2--
L
L
rr------
MECHANICAL SPECIFICATIONS
Case
Anodizedaluminium
Shaft
Stainlesssteel
Permanentlylubricatedstainlesssteel
-arings
bearings
Codedisc
Plastic disc with photographically
generatedpattern
Sealing
Dust and moistureproof, exceptshaft
bearing.Optionalshaft seal classifies
encoderasIP65
06,60,63
10
. naft type
diametermm
6
10
Maxshaft load
10
20
radial max N
10
10
axial max N
Torquemax Nrn
2.10-3
4.10-1
Momentof Inertia
14-10-1
18-10-1
maxkgml
Rpmmax
12000
12000
Bearinglife h
50000
50000
Weightkg '
0.4
0.4
Pulserate max
5000
5000
Vibration
better than 10g
Shock
better than 75g
.jLJL
Channel separation
360* el. degrees (1 pulse)
ORDER NUMBERING SYSTEM
63
Model No.
ShaftConnection
I Cable to the rear
0 Connector to the
2 Connector to the
3 Cable to the side
Supply voltage
Pulse rate Shaft seal IP65
Optional -
--T
Specifications subject to change without notice.
208
APPENDIX
A4
The Ilth International Conferenceon Electric Vehicle, Florence, Italy,
27-30 September1992
CURRENT PROFILING FOR TORQUE PULSATION MINIMISATION
BRUSHLESS DC MOTOR DRIVES
IN
JG Kettleborough, IR Smith, K Al-Hadithl &VV Vadher
Loughborough University of Technology, UIC
ABSTRACT
Most mathematical models for rotating electrical machinesassumethat the machine windings are
balanced and sinusoidally distributed, and that the generatedemf waveforms are sinusoidal. The
Trapezoidal Type Brushless DC motor, which is becoming popular for battery powered electric
vehicle applications, has windings which are far from sinusoidally distributed and a generatedemf
whose waveform is almost trapezoidal. Consequently,it is not possible to investigate reliably the
harmonic torque and speed ripple occurring in the machine, particularly at low speed, using one
of these models. To overcome this problem a phasereference frame may be used, since it does
not rely on the above assumptions. This paper outlines the use of the phasereference frame for
the analysis of a brushless DC motor and the resulting model is used in a study of the factors
affecting torque and speed pulsations experiencedby the drive. Particular attention is paid to
profiling the motor phase current, to produce a smooth motor torque. Experimental results from
a practical drive system are shown to compare well with simulation results.
INTRODUCTION
An important item of plant in an electric vehicle is the drive motor and lately, the permanent
Trapezoidal Type Brushless DC motor (BDCM) has become popular for this application,
magnet
0
due to its higher power/weight ratio, smaller inertia and lesser maintenancerequirements. A
typical systemis shown in Figure 1, where the three coils a, b, and c representthe machine stator
d and q represent rotor damper circuits, which may or may not be present,
windings,
and
0
dependingon the machine design. The active rotor is usually a rare-earthpermanent magnet. The
0
MOSFET switches (1 to 6)
typically
stator is supplied from a 3-phaseinverter bridge,
comprising
Z
and inverse parallel diodes (DI to D6) to commutate the current between phases as different
combinationsof switches turn on and off. The conduction pattern of the switches is controlled by
a rotor position sensor, which turns pairs of devices on in a defined pattern according to the rotor
position.
With this type of motor, the stator windings are far-from sinusoidally distributed and the emfs
generatedin the phasewindings are almost trapezoidal. Consequently, the popular dqO and ciýO
0
referenceframes are unsuitable for modelling the system,due to their simplifying assumptions,and
a phasereference frame was chosen to investigate the machine's electrical behaviour [1].
Under normal operating conditions, the magnetic circuit for this type of machine operates in the
linear region of its demagnetisationcurve, since the very high reluctanceof the rare-earth magnets
have a linearising effect similar to an airgap and, consequently, saturation was neglected in the
model. It could however be included, by expressing the stator winding flux linkages as functions
of both the rotor ancrieand the machinecurrents [2]. To simplify the analysis further, slot effects
and hysteresisand eddy currents in the magnetic circuit were neglected.
Although the inverter switching devices are assumedto be ideal, with no impedance or forward
voltage drop when conducting and an infinite impedance when non-conducting, the model
0
developedis sufficiently flexible to incorporate alternative device representations,as well as the
turri-on and tum-off times for the devices.
209
A tensor approach is used [3], to define automatically the circuit topology [4) as the conduction
pattern of the inverter switches changes and consequently the different circuit equations which
apply. The system equationsare solved by numerical integration, to obtain the time variation of
flux linkages and currents in the mesh reference frame and the correspondingbranch voltages and
currents are obtained by further transformations.
A description of the model now follows.
DRIVE SYSTEM MODEL
The use of tensors necessitatesthe definition of branch and mesh reference frames and the
transformation between the two frames.
Branch reference frame.
The branch reference frame is concerned with the voltages across and currents through the
individual windings or branches of the circuit shown in Figure 1. The matrix voltage equation
relating to this reference frame is
V.
Ib
Rb
Vb
*b
1,
R,
V,
p
Rd
V4
(1)
Id
*d
Re
V1.
*4
R:
*$
d
where a, b, c, d, and q relate to the machine windings, s is the supply impedence and pdt
Equation I may be written in the abbreviated form
(2)
V6 - Rb lb + p4r,
where V. is the branch voltage vector, 1, the branch current vect( r, R. the branch resistance
by
matrix and *, the branch total flux linkage
vector
given
0
1.
1ý1
Lab
Lac
Ld
Lq
0.
aa
qTb
41d
Lbb
Lbc
Lb
Lbq
0
Lee
Led
Lcq
0
Ldd
00
L
*q
1C
ld
0
qq
Ib
*Pd
*P9
*PS.
210
inductanceof winding x (x-a, b,c,dq), Ly the mutual inductancebetween
is
L,,
the
self
wherc
(x, y=a, b, c, d, q) and *,, the flux linkages with winding x due to the
y
x
and
windings
permanent magnet
CP
is
form
(3)
equation
i[ri abbreviated
Lb 16 + *pb
inductancematrix and *, the vector of branch flux linkages due to the
is
branch
Lj,
the
where
b
perinanent magnet.
Re-arranging equation(4) and combining with equation (2)
Vb - Rj, L; '(*,, - *J,) + p*,,
Mesh reference frame
formed
the
is
meshes
when conductingswitchesconnect the motor phasesto
with
concemed
-rhis
During
discrete
operation,
DC
normal
six
switching statesoccur in a defined sequence,
supply.
the
instant.
A
is
typical
two
conducting
at
any
switches
conduction
pattern
only
shown in Figure
with
forming
S2
SI
one mesh, togetherwith the permanentmeshesformed by the
and
1, with switches
d
damper circuits and.q. The application of Kirchhoff's voltage law to these meshesresults in
be
differential
the
differential
equations
which
may
expressed
as
mesh
matrix
mesh
equation
three
(5)
p*. - V. - R. L-1(4r. - 4r;,,.)
impressedmeshvoltage vector, 1. the meshcurrent vector,
is
VN,
the
+
where
linkages
flux
linkage
flux
*,,.
due
the
total
vector,
to the permanentmagnet
mesh
vector
the mesh
inductancematrices.
L,
the
mesh
R.
and
respectively
resistance
are
and
and
Equation 5 may be integratednumerically to give a new flux linkage vector *,
be
obtained.
rnay
solution
and a step-by-.step
]3ranch/mesh transformation
has
to generateautomatically the relevant mesh equationsas the switch
model
The mathematical
is achievedby defining a transformation between
this
dic0e
changes
and
pattern
conduction
and
frames.
The
between
branch
the mesh currents shown in
mesh
reference
relationship
and
the
lFigure I and the branch currents may be obtained by inspectionas
+1 00
000
-1
00
1
. 2
1.3.
00
+1
211
+1
00j
(6)
where *I denotes whether the mesh current has the sameor the reversed senseas the branch
current and a0 that there is no mesh current in that branch. Equation 6 may be written in the
abbreviatedform lb = Cl where C is the branch/meshcurrent transformationmatrix. Assuming
ft
power invariance betweenreferenceframes
(Vb - P*. )"b - (Y.
it can be shown that [5]
R. - CRbC and L. - C' LbC
C'4r, and
d6
d6
and that
VM - C' Vb
Where the superscript I denotestranspose.
In the mathematical model, C is changed automatically whenever there is a change in the
conduction pattern. The last two columns of C representthe permanentmeshesof the damper
circuits d and q and theseremain unaltered. The first column (if two devicesare conducting) or
the first two columns (if three devices are conducting, such as occurs during commutation) are
dynamicand changewith the conductionpattern. The elementsof thesecolumnsare obtained from
the mastermatrix shown in Figure 2, which holds the meshesrelating to all practical combinations
of switchesand/or diodes. For the caseshown in Figure 1, mesh (1) would be extractedfrom the
master matrix and inserted in column I of C, as shown in Equation 6.
System mechanical equation
The equationrelating to the various systemtorques is
dw
T. = J=--- +D ca + TL
dt
(7)
where T, is the electromagnetictorque, J the combined motor and load inertia, -ý-u the rotor
C)
dt
D
friction
T,
load
angular
the
velocity,
the
torque.
constant
and
Cý
The electromagnetictorque is given by
0
T. =
(1-, LL
-I-)
d8
(r- d*ý'-)
'
dO
where P is the number of pairs of poles.
Equation7 may be re-arrangedin the form
dca
J-I(T, - Dca - TL)
=
dt
which may be integratednumerically to give a step-by-stepsolution for the rotor speed ca.
Computer implementation
A computerprogram was written to predict the motor performance,using the techniquesdescribed
0
in the previous sections. The solution processis describedby the following algorithm.
The branch resistancematrix is formed. This fixed matrix is assembledonce only, at the
a)
212
b)
C)
d)
C)
f)
g)
h)
j)
k)
beginning of the simulation.
The mesh resistance matrix is determined from R. - C' Rb C. This matrix is dynamic
and,changeswith the switch conduction pattern.
The time varying branch inductance matrix Lb is determined. This matrix is formed at
every integration step, since its elements vary with the angular position of the rotor.
The meshinductancematrix is determined from L. - C' Lb C and its inverse is obtained.
The impressed mesh voltage vector is obtained from V. - C' V6.
The branch flux linkage vector *, due to the permanentmagnetrotor is obtained using
P,,
the relationships given in the Appendix.
The mesh flux linkage vector due to the permanent magnet rotor is obtained from
*P'X W Ct 4rpb
Equation (5) is integrated numerically using the 4th-Order-Runge-Kuttamethod, to obtain
a new value for the flux linkage vector *..
The mesh current vector is obtained from 1. = L-1 (*.
in
The branch current vector is obtained from lb Z Cl'.
The rate-of-changeof branch inductancematrix
dLb
- *PM)
is obtained from the derivative of the
1)
dE)
branch inductance matrix with respect to rotor position.
The mesh rate-of-changeof inductance matrix is obtained from
dLb
dL,.
C,
C
dO
dO
M)
The mesh current derivative vector pl.
n)
The branch current derivative vector pl, is determined from
is determinedfrom
dL_
I
Im P*,
L-. (V. - R. Im PM)
dO
CPI.
=
PIb
0)
The branch voltage vector is obtained from
dLb
Vb
P)
= Rb Ib
* L.
pr,
+
dO
b+
P*,
pb
The new electromagnetictorque is obtained from equation(8) and the new angulaxvelocity
is obtained by integrating numerically equation (9).
The solution advancesby one integration step, the initial conditions are updated and procedures
(c) to (p) are repeateduntil the end of the simulation. At the end of eachstep, the system is tested
for discontinuities (turn-on or turn-off of diodes or MOSFETs) which have occurred in the bridge
conduction pattern. If a change occurs, the following procedure is implemented.
1)
The instant to the time of the discontinuity is obtained using linear interpolation.
2)
Equation (5) is re-integrated from the start of the integration step up to the point of
discontinuity.
3)
A new current transformation matrix C is assembled.
213
4)
5)
6)
New mesh matricesare formed according to the new circuit topology.
Equation (5) is integratedfrom the point of discontinuity to the end of the integration step.
The system is testedfor further changesin the bridge conduction pattern. If any occur,
operation (1) to (6) are repeated. If not, the solution proceedswith operations(c) to (p)
over the next integration step.
PREDICTED AND EXPERIMENTAL PERFORMANCE
A program was written in Fortran 77 and run on a mainframecomputer. The parametersfor the
BDCM model obtainedby test from the actual machine are given in the Appendix.
Figures 3 and 4 show respectivelythe predicted and experimentalsteady-stateperformancefor the
drive, with a supply voltage V, - 24V, load torque T. - 0.7Nm and steady-statespeed
n- 222rpm. The closeagreementbetweenthe two setsof results gives confidencein the models
ability to predict the drive performance. Note in both setsof line-voltage waveforms(Figures 3(c)
and 4(c)) the suddendips in the voltage to zero. Theseoccur at intervals during which one switch
is tuming-on and anotheris turning-off. There is then a period during which both devicesare on
and this causesa short circuit betweentwo lines, or a commutationnotch. In the next section, it
is shown that these notches have an undesirable affect on the motor torque and speedand the
computer program is used to show how the motor phasecurrent may be altered to eliminate this
affect.
Figures 5 and 6 respectively show the predicted and experimentaltransient performanceas the
motor is started from rest, for the same conditions described above. Again, there is good
correspondencebetweenthe two sets of results.
TORQUE RIPPLE NfININTISATION
The emf generatedin a BDCM is ideally trapezoidal, with a flat-top spanning 120*electrical and,
sincethe electromagnetictorque T, is proportional to the product of this emf and the stator Current
[61, a rectangularstatorcurrent waveform is neededto producea constantto*rque.In practice, both
the generatedemf and the stator current waveforms differ from the ideal, and the torque exhibits
a ripple which is periodic and dependenton the rotor position and the commutationevents. At
high speeds,these effects are maskedby the drive inertia, but at low speedthey can
seriously
degradethe drive performance. The computerprogram was usedto study the factors effecting this
torque ripple and the ways by which this might be reducedand in particular, the optimum phase
current waveform to producea smooth torque was determined.
Factors afrecting torque ripple
The major factors effecting the torque ripple are the
non-idealshapeof the phaseemf waveforms
and the commutationevents.
The idealisedand actual phasevoltage waveforms are shownin Figures 7 (a)
and (b). The actual
waveform results from the fact that the airgap flux-density distribution is not rectangular,but has
roundedcomers due to fringing, and consequentlythe emf waveform has fluctuationsas shown.
The actual phaseemf waveform is accurately representedin the
model, since the flux linkages
variation with the rotor position due to the permanentmagnetwere obtained experimentallyand
representedin the computerprogram as a harmonic series.
214
During commutation, the winding inductance prevents the phase current from changing
0
instantaneouslyand consequentlythe waveform has rounded edgesas shown in Figure 8 (a) rather
than the theoretical rectangular waveshape. 7bis phenomenonis inherently predicted by the
computer program.
Current proriling simulations
Figure 8 shows the phasecurrent, motor torque and speedwaveforms for a load torque of 0.6Nm
an average speedof 102rpm. For this condition there is no compensation,and the torque and
speedpulsations are clearly evident. During the period that switches I and 2 both conduct, as
shown in Figure 1, the instantaneousmotor torque is
T.
(e. - e)
where I.,, is the current in mesh 1.
It was shownin Figure 7 (b) that thephasevoltagesare not flat topped. Consequently,to maintainT,
constant, the mesh current has to be adjustedin proportion to the term 11(e. - e) during the
period that switches I and 2 bothconduct. This is achievedin the program (and in practice) by
introducing a current control loop with pulse-width-modulationcontrol of the inverter switchesto
force the motor phasecurrent to follow the aboveproportionality. Figure 9 shows the waveforms
for the same conditions as Figure 8, but it is now evident that the torque and speed ripples are
considerablyreduced. Note the new phasecurrent waveshape,that exhibits the above mentioned
adjustment.
There are still torque and speed pulsations evident in Figure 9 and these clearly occur at
commutation instants. A current boost during commutation was therefore introduced into the
control loop, together with the current profiling, and Figure 10 shows the effect of this on the
waveforms. The slight increasein phasecurrent at commutationhasvirtually eliminated the torque
and speed ripple. The high frequency ripple evident in the current and torque waveforms of
Figures 9 and 10, but absentfrom Figure 8, is due to the PWM switching of the inverter.
0
CONCLUSIONS
The paper haspresenteda mathematicalmodel for a BDCM drive and hasdemonstratedthat good
correspondenceis obtained between theoretical predictions and practical results. Having
establishedconfidence in the model, it was then used to investigate the minimisation of torque
ripple using current profiling. The resultsobtainedshow that the methodcan effectively eliminate
torque and speedpulsationsin the drive.
REFERENCES
[11
[2]
[3]
Smith I R, Snider L A: "Prediction of transient performanceof an isolated synchronous
Vol 119, No 9, Sept 1972, pp 1309-1318.
generator", IEE Proceedings,
CP
Nehl T W, Demerdash N A, Fovad F A: "Impact of winding inductances and other
parameterson the designand performanceof brushlessdc motors", IEEE Trans on Power
Apparatusand Systems,PAS-104, No 8, Aug 19885,pp 2206-2213.
0
Kron G: "Tensors for circuits", Dover Publications, SecondEdition, 1959.
215
[4]
[5]
[6]
KettleboroughJ G, Smith I R, FanthomeB A: "Simulation of a dedicatedaircraft generator
supplying a heavy rectified load", IEE Proceedings,Vol 130, Part B,
No 6, Nov 1983, pp 431-435.
Happ H: 'Diakoptics and networks*, Academic Press, First Edition, 1971.
Jahns T M: "Torque production in permanent-magnetsynchronous motor drives with
rectangualcurrent excitation", IEEE Trans on Industry Applications, Vol IA-20,
No 4, July/August 1984, pp 803-813.
APPENDEK
7be experimentalBDCM had the following nameplaterating:
3 phase, 6 pole, 1.3kW, 2400 rpm
The phaseparametersare as follows:
L,
L4q, L,,., Lq, L.,, Lq
m,
-0
(Assuming
no dampers)
where E)is the angle in electrical radians betweenthe axes of the a-phaseand the rotor pole as
defined in Figure 1.
R., Rb, R,
Rd, Rf
R,
1.280
0 (Assuming no dampers)
10MO
L,. = (9.85 - 045 cos e) mH
LM - (9.85 - 0.45 cos 2(e + 27c/3)) mH
L,, = (9.85 - 0.45 cos 2(0 - 27z/3)) mH
LW Lqc =0 (Assuming no dampers)
L.. - lOmH
L. - (-1.95 - cos2(0 - 27r/3))
b
L,,, = (-1.95 - cos 2(0 + 27r/3))
Lb, = (-1.95 - cos 26)
The elements of the branch flux linkage vector *,, are
0
0.18 (cose + 0.08 cos38 + 0.01 cos5e)
4rpb = 0.18 (cosO + 27r/3) + 0.08 cos3(e + 27c/3) + 0.01 cosS(E) + 2n/3))
*pýý - 0.18 (cos@ - 27r13) + 0.08cos3 (6 - 21r/3) + 0.01 cos5(O - 27r13»
216
3
eDS
3ý
q
D3
c
Cý
VS
Ty
4ý
; 'D4
m2
6
DO
D
D2
ml
Figure
1
Conductin 1
1 Brushless
3
3
5
5
1
1
1
DC Motor
3
3
5
2
2
4
4
6
6I D3 D5 DII D5 DI
Mesh No.
1
2
3
4
5
6
8
Branch
..a
C
d
s
Figure
2 Master
217
5
4
4
6
6
2
2
D3 D6 D21 D4 D2 D4 D6
1
91 101 11 12 13 14 15 16 17 181
Device
71
Drive
Matrix
4
0
H .1
Time ,s
(a)
Phase
Ti? ne, 20ms/Div
Current
(a)
Phase
Current
.v
ass
W=
LM
%a
%=
wn
to
Timets
(b)
Phase
Time,
(b)
Voltage
.. -I
Or
Voltage
Phase
"
- ---
..........
...............
---- -- .......
..... .............. ....
.........
..... ......
........
ýo
%in
%so
up%
to
tm
ts
Time,
(c)
Figure
Line
......
.......
.......
............
rfr
... ....
.........
.......
Nu.
1"
..........................
. ............... . ....... ................
%as
.....................................................................
s
Time,
(c)
Voltage
3 Predicted
Steady-state
Performance
. .......
. ..... ......... .....
>
jj'j
20ms/Div
Figure
218
Line
20ms/Div
Voltage
Steady-state
4 Experimental
Performance
>
... .........
-4
H
a
........ . .....
Ln
;1
0.1
MO
03
UW0.4
0.5
Time,
(a)
HI
..........
s
Time,
(a)
r=
...............
......................... ...
Current
Phase
fl
Phase.
Current
*V
-- ------------
...........
ý4
9)
..........
t5
0.0
111
02
*a
0.4
CLS
Time,
(b)
Figure
s
Rotor
.....
..............
------
C
....
s
Figure
5 Predicted
Transient
Performance
..... .......................
.......................
................................
Time, 50ms/Div
(b)
Speed
... . ... .......
Rotor
Speed
6 Experimental
Transient
Performance
E
U
E
U
or angle, r
ýorangle. e
I
(a)
Idealised
Figure
7 Generated
Phase
219
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Voltage
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