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411984
PWM motor control: Model and servo analysis
Rick J. Sandor
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APPROVED BY:
MODEL AND SERVO ANALYSIS
BY
RICK
J.
SANDOR
A THESIS SUBMITTED
IN
PARTIAL FULFILLMENT
OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
ELECTRICAL ENGINEERING
PROF.
R. Unnikrishnan
(THESIS ADVISOR)
PROF.
J?mes E. Palmer
PROF.
[Illegible]
PROF ..
[Illegible]
PROF.
Harry E. Rhody
~4
y
(DEPARTMENT HEAD)
DEPARTMENT OF ELECTRICAL ENGINEERING
COLLEGE OF ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER)
~EW
YORK
1.0
Abstract
In
recent_{years,}
the
performance requirements ofhigh
power servo motor systems
_{utilizing}
pulse width modulated(PWM)
switching
amplifiershave
_{steadily}
increased.
These
PWM
motor amplifiers perform animportant
function
in
the
d.c.
servo system_{by}
_{boosting}
the
low
level
commandsignal
to
the
high
voltage and currentlevels
required_{by}
the
motor.Ideally,
this
power gainis
to
be
constantover all
input
frequencies
but,
in
reality,
gainis
frequency
dependent
which affects systemdynamics.
The
amplifier gain and phase versus
_{frequency}
relationshipsan*i amplifier noise and
d.c.
offsets which_{may}
affect system response mustbe
known
to
the
servodesigner
to
properly
design
the motor control system.The
_{switching}
effects ofthe
PWM
amplifier_{may}
resultin
making
the
overall system unstableif
the
systembandwidth
is
kept
high with respectto
the
PWM
_{switching}
frequency.
Since
the
standard servodesign
techniques
utilizelinear
system modeling,
analysis,
andcompensation,
it
wouldbe
very
advantageousto
the
design
engineerto
have
alinear
model which
best
approximatesthe
true
nonlinearPWM
amplifier.
This
work willlook
atthe
output response ofthe
PWM
amplifier with respectto
_{stability}
and outputthese
_{stability}
and ripple effectsin
a position controlservo system and which
is
valid as system bandwidthreaches onethird
the
PWM
_{switching}
frequency.
This
workextends
the
application ofthe
"Principle
ofEquivalent
Areas"
[141
to
the
bipolar
PWM
amplifier.It
is
then
combined with a
detailed
analysis ofthe
PWM
waveform_{by}
Double
Fourier
Transform
to
yieldthe
uniquePWM
_{switching}
effects
in
a position control servo system.Theoretical
results of
the
_{newly}
derived
_{sampling}
plusharmonic
linear
Acknowledgements
The
author wishesto
expresshis
gratitudeto Professor
R.
Unnikrishnan
for
his
guidance and patience_{during}
this
work.The
author also recognizesthat this
educational_{opportunity}
was made possible
through the
permission and resource ofthe
Eastman Kodak
_{Company}
andthe
encouragement ofthe
author'sSecti
on_{Page}
1.0
_{Abstract}
_{ii.iii}
Acknowledgements
iv
Table
ofContents
v,viList
of_{Figures}
_{vi}_{i}
,vi
i
i
,i
v ,xList
ofSymbols
xi.xii.xiii2.0
Introduction
1
3.0
Background
9
4.0
PWM
Servo
Control
System
Analysis
andModel
28
4.1
Position
Control
Servo
Model
30
4.2
Detailed
Analysis
ofBipolar
PWM
and41
Methods
ofGeneration
4.3
Sample
andHold
Model
for
Uniform
48
Sampling
PWM
Systems
_{by}
the
Principle
ofEquivalent
Areas
4.4
PWM
Amplifier
_{Switching}
Harmonic
Model
56
by
Fourier
Series
andTaylor
Series
Expansion
4.5
Uniform
_{Sampling}
PWM
Amplifier
_{Switching}
70
Harmonic
Model
_{by}
Double
Fourier
Transform
Analysis
4.6
Construction
andAnalysis
ofthe
Total
84
Uniform
_{Sampling}
PWM
Amplifier
Model
4.7
Analysis
andModel
Alteration
for
89
Natural
_{Sampl i ng PWM}
5.0
Computer
Simulation
ofUniform
andNatural
92
Sampling
PWM
andLinearized
Models
5.1
Servo
Control
System
With
Typical
95
Components
6.0
7.0
8.0
5.3
_{Step}
Response
Simulation
Utilizing
Nonlinear
PWM
5.4
_{Step}
Response
Simulation
Utilizing
Linearized
PWM
Models
Conclusions
References
Appendix
105
113
123
128
List
ofFigures
Fi
gu re1
HBridge
Transistor
Configuration
of aTypical
PWM
Amplifier
Output
Stage
Page
4
(a)
Output
Waveform
of aUnipolar
PWM
Amplifier
in
Response
to
aSinusoidal
Input
(b)
Output
Waveform
of aBipolar
PWM
Amplifier
in
Response
to
aSinusoidal
Input
Typical
PWM
Amplifier
Output
Cycle
10
Motor
Current
in
Response
to
A
Typical
PWM
Amplifier
Output
12
(a)
Sample
Data
Model
of aNormalized
Unipolar
14
PWM
_{by}
Andeen
(b)
Input
Waveform
andUnipolar
Output
and14
Equivalent
Sample
Data
Output
Waveforms
Block
Diagram
of aPWM
System
_{Utilizing}
aDescribing
Function
As
aModel
17
Pulse
Width
Modulator
Model
Derived
from
Double
Fourier
Transform
Analysis
_{by}
Tsai
and
Ukrainetz
[33]
21
Development
Block
Diagram
ofthe
PWM
Amplifier
Linear
Model
24
Schematic
Diagram
of anAngular
Position
Control
System
26
10
System
Block
Diagram
of anAngular
Position
Control
Servo
System
31
11
PWM
Generation
_{Using}
aDither
Signal
andRelay
Element
Figure
12
Dither
Signals
Used
in
PWM
Modulation
Page
35
13
_{(a)}
Uniform
_{Sampling}
PWM
(b)
Natural
_{Sampling}
PWM
37
37
14
_{(a)}
Typical
PWM
Input
Signal
andNormalized
Response
(b)
Equivalent
Sample
Data
Model
41
41
15
Equivalent
Sample
andHold
Model
Developed
by
the
Principle
ofEquivalent
Areas
43
16
Block
Diagram
ofthe
Linearized
PWM
Output
Using
Taylor
Series
Expansion
51
17
_{(a)}
PWM
Block
Diagram
Derived
from
Fourier
and53
Taylor
Series
Analysis
(b)
Equivalent
Block
Diagram
53
18
Variations
ofthe
Bessel
Function
in
the
Term
f.
60
19
Linear
Block
Diagram
ofthe
Double
Fourier
Series
PWM
Model
64
20
Block
Diagrams
for
the
Total
Uniform
Sampl
_{ing}
PWM
Model
s(a)
Taylor
Series
Expansion
Model
(b)
Double
Fourier
Series
Model
67
67
21
System
PWM
Block
Diagrams
for
aNatural
Sampl
_{ing}
PWM
Model
by
(a)
Taylor
Series
Expansion
(b)
Double
Fourier
Series
69
Figure
22
_{(a)}
PWM
Amplifier
Representation
on_{Easy}
5
Modeling
andSimulation
Analysis
(b)
Equivalent
PWM
Block
Diagra
mPage
72
72
23
_{(a)}
Servo
System
Block
Diagram
With
Typical
Components
(b)
Equivalent
System
Block
Diagram
75
75
24
_{(a)}
Root
Locus
Plot
ofUncompensated
Zero
Tachometer
System
(b)
Resultant
Block
Diagram
After
Feedback
Compensation
77
77
25
Transient
Response
of aServo
System
Utilizing
the
Simple
PWM
Model
80
26
Transient
Response
of aSystem
_{Utilizing}
a
Uniform
_{Sampling}
PWM
Amplifier
andNatural
_{Sampling}
PWM
Amplifier
82
27
_{Steady}
State
Response
of aSystem
Utilizing
aNatural
_{Sampling}
PWM
Amplifier
and a
Uniform
_{Sampling}
PWM
Amplifier
83
28
Uniform
_{Sampling}
PWM
System
Response
for
Switching
Frequencies
of_{50,}
_{40,}
and30
Hertz
85
29
Natural
_{Sampling}
PWM
System
Response
for
Switching
Frequencies
of_{50,}
_{40,}
and30
Hertz
86
30
Unipolar
PWM
Transient
Response
to
aTwo
Radian
_{Step}
Command
(a)
Double
Fourier
Series
Model
With
First
Harmonic
Gain
of tt/288
Fi
gure31
Transient
Response
ofthe
System
Utilizing
a
Double
Fourier
Series
Model
With
First
Harmonic
Gain
of4/tt
Page
91
32
_{Steady}
State
Response
ofthe
System
Utilizing
aDouble
fourier
Series
Model
With
First
Harmonic
Gain
of4/tt
92
33
Linear
Model
Uniform
Sampling
PWM
System
Response
for
_{Switching}
Frequencies
of_{50,}
_{40,}
and30
Hertz
93
34
Linear
Model
Natural
_{Sampling}
PWM
System
Response
for
_{Switching}
Frequencies
of_{50,}
40.
and30
Hertz
94
35
Final
Linearized
PWM
Models
(a)
Uniform
_{Sampling}
PWM
(b)
Natural
_{Sampling}
PWM
List
ofSymbol
sV1n^)
va(t)
a_{(t)}
maxL
R
U>_{(t)}
KB
A
i
I
av st
H(s)
J
Tx
T^tJ.T^s)
PWM
_{switching}
_{frequency}
PWM
_{switching}
periodPWM
amplifierinput
voltageTime
at whichthe
PWM
output changes state within a single cyclePWM
amplifier output voltage appliedto
the
motorPWM
amplifier peak voltageServo
motor armature currentDuty
cycle ofthe
PWM
output waveformMaximum
value ofV^
(t)
before
saturation or1/2
the
input
voltage rangeMotor
inductance
Motor
resistanceMotor
angular_{velocity}
Motor
back
emfMotor
current variationfrom
averagedue
to
PWM
_{switching}
Average
motor currentComplex
variableTime
Laplace
transform of ahold
networkTotal
inertia
of servo systemload
Coulomb
friction
torque
Motor
windageTortional
tensionin
the
_{coupling}
between
the
motor andthe
load
m
e
_{(t)}
G(a)),g(t),G(s)
OJ CO6(t)
T(t),T(S)
KT
ex(t)
e2(t)
e3(t)
kTACH
d(t)
e(t)
Vin*^)
Ei

Angular
_{position}of
the
motor
Load
inertia

Tortional
coupling
elastic_{damping}
between
motor andload

Tortional
coupling
spring
constantbetween
the
motor andload
Mechanical
time
constant pole ofthe
servo system
Control
system plant
transfer
function
(frequency,
time,
and complex_{variable)}

Angular
frequency
of a sinusoidal signal
Angular
frequency
equalto
t'e
PWM
switching
frequency
Tooling
position and output ofthe
servo control system
Motor
torque
as a
function
oftime
and compl ex variable
Motor
torque
constant

Error
_{signal}between
the
_{command} _{and}tooling
position
Position
_{sensor}conversion constant

Output
of position compensation network

Error
_{signal}between
the
_{command} _{and} motor_{velocity}

Tachometer
_{conversion} _{constant}
PWM
_{amplifier}dither
signal
PWM
_{sum} _{of} V.(t)
andthe
dither
_{signal}in a

Sampled
PWM
input
_{signal}
Magnitude
_{of} sinusoidalinput
_{voltage}
Constant
Fourier
coefficient
Fourier
_{coefficient}N(Eif
w_{)}
R(Joj),
r(t),
_{R(s)}
C(jco),
c(t),
C(s)
K(
a>)
kl
ed(s)
Describing
function
Control
systeminput
signal as afunction
of_{frequency,}
_{time,}
and complex variable
Control
_{system} _{output} signal as afunction
of_{frequency,}
time,
and complex variabl e
Control
_{system}feedback
transferfunction

Feedback
_{gain} _{on}the
tachometer signal
Feedback
_{gain} _{on}the
potentiometer signal
Desired
position of
the
position servosystem
This
section willdefine
the
PWM
motor amplilfier andsummarize
its
advantages.The
_{many}
types
ofPWM
willbe
discussed
andthe
bipolar
PWM
amplifier willbe
presentedin
detail
asthe
subject ofthis
work.Linear
amplifiers arethe
simplest motordrives
and areconstructed
_{using}
linear
gain elementsto
drive
a powerstage.
The
mathematical_{relationship}
between
the
output andinput
voltage canbe
described
_{by}
afirst
orderdiff
erential equation with
_{very}
high
d.c.
gains.With
the
proper choice of voltage or current
_{feedback,}
the
amplifier
canbe
madeto
have
a constant gain_{up}
to
_{very}
high
frequency.
The
amplifierhas
ahigh
performance andits
effects on servo system response are
_{easy}
to
predict.In
general,
linear
motor amplifiers are preferredin
widebandwidth,
lowpower
systems.There
_{is,}
_{however,}
asignificant
disadvantage
in
high
power applications.Linear
amplifiers control motor voltage or current_{by}
controlling
the
voltage appliedto
the
motor.The
amplifier
must_{drop}
a voltage equalto
the
difference
between
the
_{supply}
voltage andthe
voltage appliedto
the
motorwhile
_{delivering}
the
proper current.Thus,
linear
amplifiers
have
the
problem ofhigh
powerdissipation
in
the
supply
voltage_{among}
them.
If
the
_{supply}
voltageis
_{100V,}
the
motor voltageis
_{20V,}
andthe
motor currentis
_{10A,}
the
powerdelivered
to
the
motoris
200W.
This
comparesto
800W
dissipation
in
the
amplifier whichis
undesirableand requires special care
in
_{providing}
for
heat
transfer.
Switching
motor amplifiers offer a solutionto
this
problem.
The
_{switching}
amplifier avoidshigh
powerdissi
pation
in
the
powertransistor
output stage_{by}
_{cycling}
the
output
transistors
from
saturationto
an off state.When
the
output stageis
turned
on,
the
saturation voltageacross
it
is
negligible and whenit
is
turned
off,
the
current
is
zero.In
either_{case,}
the
powerdissipation
in
the
transistors
is
small.Amplifiers
_{operating}
in
this
way,
acquirethe
term
_{switching}
amplifiers.Advantages
ofPWM
amplifiers are_{adequately}
discussed
in
the
_{existing}
literature
_{[1,}
2,
3,
4.
6,
11,
26,
28].
One
form
of_{switching}
amplifiers variesboth
its
_{switching}
frequency
and pulse width as afunction
ofthe
input
signal as well as
the
parameters ofthe
system.This
pulse
_{frequency}
modulated amplifieris
not oftenOther
_{switching}
amplifiers utilize a separatefixed
fre
quency
oscillator andthus
afixed
pulse rate._{Only}
the
pulse width varies with
the
input
signal andthese
amplifiers
are called pulsewidth modulated_{(PWM)}
amplifiers.Control
ofthe
pulse widthis
most often performed_{by}
controlling
the
_{sequencing}
of power transistors arrangedin
anHbridge
configuration as shownin
Figure
1.
Discussion
ofthe
_{many}
PWM
variations and their circuitimplementations
canbe
found
in
the
literature
[1,
2,
3,
1
T>
_{a}
'a'1'
V.(t)
T2
ZA
D2
Motor
Li
V
5
T
4
Figure
1
HBridge
Transistor
Configuration
of aTypical
modes
found
in
servo _{application.}For
the
unipolar modeof
PWM
_{operation,}
the
_{switching}
sequencedepends
onthe
sign of
the
amplifierinput
commandV.
.Let
the
switching
frequency
be
f
andthe
_{switching}
periodbe
T.
Also,
assumethat
the
"on" phaseis
_{during}
the
first
partof
the
period.It
lasts
between
t=0
andt=tj,
andis
followed
_{by}
the
"off" phase whichlasts
through the restof
the
_{switching}
period.The
_{switching}
sequencedepends
on
the
sign ofthe
input
r V .When
V.is
positive,
JA
is
in in r '4
turned
on continuously.Transistor
T,
is
turned
on_{during}
the
"on" phase andT
2
is
turned
onduring
the
"off"
phase.
The
_{resulting}
amplifier_{output,}
V
_{(t),}
appliedto
the
amotor
becomes
va(t)
0
<t
<t
tj
<t
<T
(2.1)
where
V
is
the
peak amplifier output voltage.When
_{V^}
pis
negative,T,
is
turned on continuously.Transistor
T
2
is
turned on_{during}
the
"on"
phase and
Tj
is
turned
onduring
the
"off" phase.The
resulting
amplifier outputbecomes
V.
va(t)
0
<t
<t
tj
<t
<T
shown
in
Figure
2(a).
A
second mode ofPWM
_{operation,}
calledthe
bipolarmode,
switches positive and negative
_{supply}
voltageto
the
motorev9.ry
cycle.Transistors
T,
andJ.
are turned on_{during}
the
"on"phase and
transistors
T
_ andT
_ are turned onduring
the
"off" phase.The
_{resulting}
amplifier outputto
the
motorbecomes
vs
0
<t
<tj
va(t)
V.
(2.3)
tj
<t
<T
1.W
input
signalv
(t)
"V
on
(a)
va(t)
V.
J
n
U
U
u
n
u
f]
JU
u
(b)
Figure
2
(a)
Output
Waveform
ofUnipolar
aPWM
Amplifier
in
Response
to
aSinusoidal
Input
The
bipolar
modedescribed
abovehas
the
disadvantage
that
at
time
t1
one_{transistor}
pair mustbe
turned off andthe
other must
be
turned
on.Since
there
is
adischarge
time
in
the
_{transistors,}
care mustbe
taken
that
two
transist
ors
do
not conduct_{simultaneously}
andform
a short circuitacross
the
power supply.To
avoid_{this,}
a_{delay}
interval
must
be
introduced
between
one transistor pair_{turning}
offand
the
other_{turning}
_{on.}The
_{delay}
interval
has
to
be
longer
than
the
discharge
time.
Also,
the
power transistors
have
afinite
turn on risetime
in
which_{they}
arein
their_{linear,}
high powerdissipation
region.Time
in
this
region mustbe
minimized.The
two
disadvantages
described
above tendto
limit
the
_{switching}
frequency.
Amplifier
manufacturers recognizethis
problem andtend
to
ke*p
switching
frequencies
low
enough suchthat
the
time
taken
_{up}
_{by}
the
dead
band
_{delay}
interval
andthe
trans
istor
risetime
is
a small percentage ofthe
total
cycletime.
The
bipolarPWM
motor amplifieris
the
amplifier_{that,}
from
the author's experience,is
most_{commonly}
usedin
industry.
Thus,
the
bipolarPWM
amplifier willbe
the
focus
of the remainder ofthis
paper.To
avoid unnecessary
complexity,this
analysis will assumethat
the
PWM
amplifier
has
perfect switching,meaning
that
the
previously
discussed deadband andtransistor
risetime
areboth zero.
This
assumptionis
consistent withthe
3.0
_{Background}
This
section summarizesthe
workfound
in
the
referencesconcerning
the
desciption
and analysis of pulse widthmodulation and
its
effects on system performance.Although
each_{study}
offers some_{interesting}
insight
into
some aspect of
PWM
_{operation,}
this
section will alsoexplain
_{why}
these
studies weredeficient
in
_{meeting}
the
goals of
this
work.The
references can_{roughly}
be
divided
into
three
categories:(a)
Simplified
d.c.
_{model,}
average currents and currentvariations
(b)
Sampling
models and_{stability}
studies(c)
Fourier
transform
andfirst
harmonic
modelsPrinciples
which willbe
usedlater
in
this
paper are alsodetailed.
In
orderto
_{properly}
design
a machine control servosystem,
one must understandhow
the
bipolar
PWM
amplifierwill affect system performance.
The
first
attemptsto
analyze motor response
to
the_{switching}
PWM
outpututilized circuit analysis techniques
to
_{determine}
motorPWM
waveformis
dependent
on V.(t)
andis
defined
asr in'
(3.1)
maxwhere
V
is
the
maximum ofV,
or1/2
the
maximuminput
max in
voltage range.
It
follows
that
_{l}
a_{<_}
1
.For
the
PWM
output waveform as shown
in
Figure
3
withthe
assumptionthat
the
input
is
_{relatively}
constant overthe
_{switching}
period,
the
average ordc
value ofthe
output voltageis
given
by:
Vav
=W
_{'Wt)}
dt
= _{aV}O"
(3.2)
Thus,
a simplifiedd.c.
gain model ofthe
PWM
systemis
given as:
VaV^/Vin^
=Vs/Vmax
(3.3)
with the stipulation that the model
is
valid as_{long}
asthe
system bandwidth remainssignificantly
below
the
switching
frequency
andthe
input
is
relatively
constant over the switching period.This
model and stipulationwere presented
_{by}
Kuo
[1],
ElectroCraft
Corporation
[2],
V*'PWM
Amplifier
,<*>vin(t)
a = in
max
V,(t)
V
t)
Vs
h
T
"Vc
_{s}V
1 amax
0
<t
<t.
tj
<t
<T
Figure
3
Although
the
modelis
linear
and canbe
usedin
_{many}
low
response
_{applications,}
it
does
not givewarning
of unknowndestabilizing
effects asthe
bandwidth
is
pushedhigher.
Also
_{lacking}
is
the
ripple_{switching}
effects which aresuspected
to
be
present._{Depending}
onthe
reference_{used,}
"significantly
below
the
_{switching}
frequency"is
defined
between
1/3
to
1/10
the
_{switching}
_{frequency}
f
.The
effect of_{switching}
was analyzedin
these
samereferences
_{by}
circuit analysis over each portion ofthe
output voltage given
in
equation(23).
When
appliedto
adc
permanent magnet servo motor,the
differential equationof the motor
is
given asL
dlfl(t)/dt
+R
_{Ia(t)}
+KB
oj(t)
=_{Va(t)}
_{(3}
where
I
_{(t)}
= motor armature current 3u
(t)
= motor velocity
= motor inductance
= motor resistance
The
solution ofthis
differential
equation resultsin
afalling
and_{rising}
exponential motor current as shownin
Figure
4.
Over
a_{switching}
_{cycle,}
the
motor currenthas
some
dc
value with a maximum current variation_{(Ai)}
whosemagnitude
is
a nonlinearfunction
of_{duty}
cycle.This
relationship, which can
_{accurately}
predict current variations
in
_{steady}
_{state,}
is
inappropriate
for
linear
systemsmodeling
and analysis_{necessary}
for
accurate servodesign.
Circuit
analysistechniques
are also used_{by}
Kuo
_{[1];}
Tal
[41;
Damle
and_{Dubey}
_{[5];}
_{Okamoto, Ichida,}
_{Fujinami,}
andV
t)
V
s
1
t
V s
Figure
4
Motor
Current
in
Response
to
A
Typical
Verma
_{[11],}
and_{Verma,}
_{Baird,}
andAatre
[26]
useboth
acircuit analysis approach and a
lattice
function approachto
look
at current ripple and speed variation of aunipolar
PWM
controlleddc
motor.In
the
circuit analysis_{approach,}
the
systemis
consideredas a simple circuit
_{having}
a_{battery,}
aswitch,
and amotor
in
series.The
switchis
assumedto
be
a timemodulated power switch with conduction periods adjusted
according
to
the
drive
requirements.Interrelations
between
variousparameters,
such asthe
pulse_{width,}
pulseperiod, current ripple
_{factor,}
andthe
motortime
constant,
are established under stalled motor conditions.The
speed analysis ofPWM
controlleddc
motors_{by}
the
second approach
involves
the use ofthe
lattice
function
and the discrete
Laplace
transform concepts.These
techniques yielded good results
in
predicting
the
currentripple
factor
and speed variation_{factor;}
however,
_{they}
do
not
lend
themselvesto
linear
systems analysis andcompensat
ion.
Andeen,
rl41.
recognized the nonlinear nature ofPWM
control and
the
lack
of mathematical means of analysis.Mathematical means
have
been
welldeveloped,
however,
for
pulse amplitude modulation or sample
data
control whichis
a
linear
process.Andeen
applied the"principle
of equivalent
areas"
sample
data
process andthen
applied ztransform compensation
techniques
to
counterthe
_{destabilizing}
effects ofPWM
as systembandwidth
risestoward
the
_{switching}
frequency.
This
equivalent model was shownto
be
quitesuccessful
_{by}
experimentin
_{predicting}
the
_{early}
desta
bilizing
effects ofPWM
asbandwidth
is
increased.
This
model
is
_{valid,}
at_{best,}
_{up}
_{to}
abandwidth
1/2
the
switching
frequency
whichis
adequatefor
applicationto
motor control systems.
The
sampledata
_{model,} whichis
shown
in
Figure
_{5,}
_{along}
withinput
and output_{waveforms,}
does
not showthe
output ripple associated withthe
first
harmonic
ofthe
unipolarPWM
switching.This
effectshould
be
even strongerfor
bipolar
PWM.
The
"principle
of equivalent areas"
will
be
analyzedin
greaterdetail
for
the
bipolar
PWM
_{along}
with_{study}
ofthe
first
harmonic
V*!
"Vx
r
v(tL.
Hold
H(S)
VaCt)
t ""(^
1
UNIPOLAR
PWM
1
1
1
1
(a)
va(t)
V
_u
r
i
r
(b)
Fi
gure5
(a)
Sample
Data
Model
of aNormalized
Unipolar
PWM
_{by}
Andeen
(b)
Input
Waveform
andUnipolar
Output
and_{Equivalent}
Parrish
and_{McVey}
_{[10]}
andSerebryakov
_{[12]}
lend
supportto
the
sampledata
model ofthe
PWM.
These
references usesample
data
to
tak*_{into}
_{account}_{the}
sampling
nature ofthyristor
drive
systems and singlephasesiliconcontrolled rectifier systems.
Kravitz
[221
also uses a sample andhold
modelto
represent
the
closed_{loop}
_{switching}
regulator system.Stability
of pulse width controlleddc
voltage stabilizersis
examined_{by}
Kitayev
andStoyanov
_{[23]}
through a sampleand
hold
model.This
paper alsodemonstrated
that
ztransform
_{theory}
couldbe
usedto
analyze the system.One
method of_{linearizing}
a nonlinear systemis
_{by}
the
method of
_{describing}
functions
whichis
found
in
severalof
the
References
_{[18,}
17,
28,
29].
In
this
_{technique,}
the
input
is
sinusoidal.V1n(t)
=Ei
sin U)t(3.5)
If
the
nonlinear systemis
assumedto
be
symmetric,
there
is
noDC
termin
the
output.In
general,
the
output ofthe
nonlinear system_{Vg(t)}
is
also periodic_{having}
the
same fundamental periods as
that
ofthe
input
signal.Fourier
Series
canbe
usedto
expressthe
output ofthe
00 00
vaU)
=_{An/2}
+_{2}
_{A}
cos nut +I
Bnsin
n cot(3.6)
a U
n1 n n=l
n
where
A
=_{2/tt}
_{/}
Va
(t)
cos nut_{d(u>t)}
nQ
a(3.7)
and
B
=_{2/tt}
_{/}
_{V}
(t)
sin nut_{d(wt)}
nQ
a(3.8)
By
definition,
the
_{describing}
_{function,}
_{N,}
is
an equationexpressing
the ratio ofthe
amplitudes andthe
phase anglebetween
the
fundamental
ofthe
output andthe
input
sinusoids.
M
N(EiiU)
[(Aj
+_{Bj)*/E.]}
_{e}j
Tan'^Aj/Bj)
_{(3.9)}
Higher
order harmonics are neglected sincethe
harmonic
terms
are _{generally} small comparedto
the
fundamental
term
and
the
dynamics
ofthe
servo control system_{inherently}
act as a
low
passfilter.
The
gain and phaseterms
ofthe
describing
function_{N(Ei,w)}
canbe
usedin
a standardblock
diagram or transfer function as shownin
Figure
6
to
r(a>)
+C(u>)
=G(h))N(EilM)
r(u)
l+G(u)N(E.,u))K(u))
Figure
6
Block
Diagram
ofA
PWM
System
_{Utilizing}
A
The'
and
Furmage
_{[18]}
use_{describing}
functions
to
predictsubharmonic
oscillations andto
measurethe
relativedegree
of_{stability}
ofbipolar
PWM
control systems.Their
technique
utilizes an _{envelope} _{created}_{by}
_{a} _{set} ofdescribing
functions
for
the
PWM
whichis
then usedto
test
system_{stability}
_{from}
_{a}_{Nyquist}
_{criterion.}_{From}
_{the}
transfer
function
in
Figure
_{6,}
the
characteristic equationon which.
the
Nyquist
plotis
based
is:
1
+_{G(ja>)}
_{N(E.,u)}
_{K(ju)}
(3.10)
The
normallinear
procedureis
to
obtain a polar plot ofG(joo)
N(E.,W)
K(ju0
(3.11)
which makes
the
point 1 +jO
the
critical pointfor
stability
analysis._{By}
_{rearranging}
the
characteristicequation as:
G(ju)
K(ju)
=1
N
_{(Ei}
,u)(3.12)
the
linear
system functionfactors
are separatedfrom
the
describing
function
factor.
Thus,
G(jw)
_{K(ju>)}
may
be
plotted as a polar transfer
function
_{curve.}_{The}
term
1/N(
E
, u)
may
alsobe
plotted on
the
same sheetto
the
amplitude
Ei
and_{frequency}
_{u.}_{Thus,}
_{plotting}
_{}
l/N(Ei
, a>)results
in
a_{family}
of _{curves.}System
stability
canbe
investigated
_{by}
_{inspecting}
the
relativelocation
of pointsof
the
_{G(j}
_{m)}_{K(jw)}
_{curve} with respectto
the
corresponding
_{}
l/N(Ei
,co) curves.The
systemis
stableif
for
all
_{frequencies}
_{the}
_{points} _{on}_{the}

G(
jco)K(
ju>)
curvelie
outside
the
_{corresponding}
_{1/N(}
Ei
, _{co)}
curves.This
system yielded_{very}
good resultsin
_{predicting}
stability
when comparedto
simulations onthe
_{analog}
computer.
Further,
if
a systemis
found
unstable, seriescompensators can
be
analyzedto
determine
which wouldbring
the
systemG(
_{j}
_{co)}K(
j
co)
curve outsidethe
_{1/N(E,}
co)

curves.Although
_{very}
good_{stability}
predictions were_{obtained,}
plotting
allthe
Nyquist
curvesis
time
_{consuming}
and_{they}
cannot
be
usedin
a standardlinear
systems model andanalysis.
Also,
the
first
harmonic
ripple cannotbe
predicted
in
stable systems.Taft,
Banister,
andSlate
[28]
discuss
severaldifferent
driver
configurationsfor
bipolar
PWM
amplifiers controlling
d.c.
motors.The
input/output
relationshipsfor
both
the
d.c.
gainterm
andthe
first
harmonic
term
ofthe
square wave output are presented
for
each_{configuration.}
saturation
for
most_{configurations,}
the
first
harmonicterm
is
shownto
be
a_{very}
nonlinearfunction
ofthe
input.
_{Describing}
function
analysis was usedto
study
the
limit
cyclePWM
_{oscillator.}
A
pulse width modulatedsignal
is
obtained_{by}
_{closing}
afeedback
_{loop}
around aswitching
elementto
causethe
systemto
have
a highfrequency
limit
cycle.This
approach requires no externaldither
source.Dither
_{sources,}
which are usedin
mostmotor amplifiers encountered
_{by}
the
author,
willbe
further
discussed
in
the
analysis section.Limit
cyclePWM
oscillators will notbe
discussed
in
this
paper.Further
workis
requiredto
linearize
the
nonlinearrelationships generated
[28].
Kolk
[291
derives
_{describing}
functions
for
a wide_{variety}
of unipolar
PWM
situations whichinclude
saturated andunsaturated
_{trailing}
edge modulation and saturated andunsaturated
_{leading}
edge modulation.Assymptotic
approximation
for
rapidPWM
_{sampling}
comparedto
the
input
is
also analyzed.
The
resultis
a_{family}
of curves_{showing}
the
variation of_{describing}
function
magnitude and phase.Experimental investigation verified conclusions
drawn
about stability and
limit
cyclebehavior.
Some
interesting
studieshave
been
done
on electrohydraulic
servo systems utilizingbipolar
PWM
valvedrives
[33,
electrohydraul
i
c_{servovalves}
_{are} _{subject} _{a} number ofnonlinearities
at zero velocity_{When}
an el ectrohydraulic
servo mechanism
is
driven
_{by}
aPWM
control,
the
high
frequency
switching
ofthe
PWM
_{output,}
andthe
_{resulting}
dither
in
the
valve_{spool,}
_{greatly}
_{reduces}_{the}
sensitivity
to
fluid
contamination andthe
effects of suchnonlinearities
as_{hysteresis,}
_{threshold,}
stiction,
dead
zone,
andnull shift.
_{Thus,}
the
PWM
_{switching}
effectsimprove
system _{performance.}
Tsai
andUkrainetz
[331
attemptto
model and analyze aclosed
_{loop}
hydraulic
servo system_{using}
PWM
control andpredict
its
output response_{including}
the
valve spooldither.
Thus
an appropriate transferfunction
ofthe
pulse width modulator
had
to
be
developed.
This
transfer
function
wasderived
from
adouble
Fourier
series analysisof
the
modulated waveform.A
double
Fourier
seriesanalysis of pulse width modulation was earlier presented
by
Black
_{[35]}
along
with adescription
of_{leading}
edge andtrailing
edge modulation and uniform_{sampling}
and naturalsampling
PWM.
Uniform
and natural_{sampling}
willbe
further
described
in
Section
4.
A
double
Fourier
analysiswas used
_{by}
Black
[4]
to
determine
the
spectra ofthe
PWM
signal with
its
infinite
number ofharmonics
andsidebands.
The
resultant pulsewidth _{modulator} _{model}by
Tsai
andUkrainetz
is
shownin
Figure
7.
Note
that,
in
the
average value ofthe
_{output,}
a second systeminput
corresponding
to
the
first
harmonic
has
been
added.This
first
harmonic
input
_{may}
providethe
PWM
_{switching}
effectsin
the
outputthat
one suspectsis
present.Since
the
PWM
switching
wasfound
to
have
negligible effect onthe
electrohydraul
i
ctransient
_{response,}
the
extent of outputripple was omitted
from
experimental verification.Thus,
it
is
notknown
whetherthe
constant amplitude givenfor
the
harmonic
input
is
correct.Also,
there
is
no meansin
this
model of_{predicting}
_{instability}
or_{destabilizing}
effects as system
bandwidth
approaches_{switching}
frequency.
Good
agreement was shownin
this
paperbetween
actual and predicted
_{frequency}
responsefor
the
overalle0(t)
PWM
Amplifier
Ia(t)
H
= _{peaktopeak} _{amplitude} _{of}_{the}
_{modulated} _{output} _{wave}_{form,}
a= _{modulation}
index
Figure
7
Pulse
Width
Modulator
Model
Derived
from
Double
This
paper will_{study}
_{very}
_{closely}
the
double
Fourier
series analysis
technique
appliedto
the
bipolar
PWM
motorampl
i
f
ier
.Ghonaimy
and_{Aly}
_{[19]}
discuss
a phase plane techniquefor
PWM
modeling.This
methodhas
the
disadvantage,
however,
of
_{being}
applicable_{only}
for
second order systems.Phase
plane
is
also utilized_{by}
Karetnyi
andSheryaev
[21].
Many
papers onPWM
discuss
_{stability}
throughLiapunov's
second method.
It
is
not_{clear,}
_{however,}
how
this
method4.0
PWM
_{Servo}
_{Control}
_{System}
_{Analysis}
_{and}_{Model}
This
section presentsthe
analysisdone
onthe
PWM
signaland
its
effect within a motor control servo system.A
linear
modelis
developed
which providesthe
same effectswithin a motor control servo system as
the
true
nonlinearPWM
_{amplifier.}
_{Figure}
_{8}
_{traces}
_{the}
_{development}
_{which} willbe
presentedin
this
section.The
electromechanicalposition control servo problem
_{consisting}
of a piece oftooling
driven
_{by}
a permanent magnetd.c.
motoris
introduced.
The
modeldevelopment
uses_{ordinary}
linear
differential
equationsto
describe
the
system componentsDevelopment
ofthe
Position
Control
Servo
Model
_{Highlighting}
the
PWM
Concerns
Detailed Analysis
ofB ipolar
PWM
andMethods
ofPWM
Generation
Recognition
ofthe
_{Sampling}
andFirst
Harmonic
Nature
ofUniform
Sampling
PWMDevelopment
of aSampling
Model
by
Principle
ofEquivalent
Areas
Development
of aHarmonic
Model
_{by}
Fourier
Series
and
Taylor
Series
Expansion
I
Development
of aHarmonic
Model
_{by}
Double
Fourier
Transform
~7^
Combination
to
Form
aSingle
Linear
Model
andAnalysis
ofTotal
Model
Response
Extension
ofthe
Model
to
Natural
Sampling
PWM
Figure
8
Development
Block
Diagram
ofthe
Obviously
missing
in
this
development
is
alinear
_{descrip}
tion
of_{the}
PWM
motor_{amplifier.}
_{One}
_{is}
_{tempted}
_{to}
_{use} _{a}simple
d.c.
gain asthe
transfer
function
which wouldbe
acceptable
in
alow
_{response}_{application;}
_{however,}
the
servo
designer
is
often askedto
extend response andaccuracy
into
regions which makethe
use ofthis
transfer
function
_{uncomfortable.}
_{One}
_{wishes}_{to}
_{have}
_{a} _{mathematical}description
which would show output ripple andthe
onsetof
_{instability,}
if
_{applicable.}
_{To}
accomplish_{this,}
_{the}
stability
andfirst
harmonic
ripple analysis are attackedseparately.
The
_{sampling}
nature ofPWM
is
explored as aclue
toward
_{stability}
andthe
"Principle
of_{Equivalent}
Areas"
is
usedto
_{develop}
a modelfor
uniform_{sampling}
PWM.
In
_{parallel,}
the
_{switching}
ripple effectis
analyzedfirst
_{by}
Fourier
series andTaylor
series expansion andsecond
_{by}
double
Fourier
transform.
Both
techniques
yieldlinearized
models.The
sampling
andfirst
harmonic
modelsare
then
combined andthe
results ofthis
combinationstudied.
Finally,
the
modelis
modified and presented asa
linearized
description of natural_{sampling}
PWM.
4.1
Position
Control
Servo
Model
A
typical control problem encounteredin
machinedesign
for
_{industry}
is
the
position control of a piece oftooling.
The
position couldbe
linear
or angulardis
schematic
diagram
of an angular position control systemis
shown
in
Figure
9.
The
load
is
a piece of_{tooling}
whichperforms some operation such as _{cutting,} punching,
sealing,
etc.The
_{tooling}
is
driven_{by}
a permanent magnetd.c.
motor powered_{by}
aPWM
motor amplifier.Tooling
position
is
measured_{by}
a sensor such as a potentiometeror encoder and
_{velocity}
is
measured_{by}
the
tachometer onthe
motor._{Velocity}
and position compensation are addedV*>
r(tV
y
Position
Sensor
T(t),u(t)
Figure
9
Schematic
Diagram
ofAn
Angular
Position
The
_{standard}_{tools}
_{used}by
the
servo system designerto
achievethe
desired
system performance arelinear
systemsanalysis and
_{compensation.}
_{Although}
_{no} systemis
completely
_{linear,}
mechanical nonli
neariti
es such asfriction,
dead
_{zones,}
and_{cogging,}
and electricalnonlinearities
such as saturation andhysteresis
areminimized or eliminated
in
a good_{design.}
_{Thus,}
linear
differential
equations are usedto
describe
the
remainderof
the
system and aredescribed
asfollows:
a)
Permanent
Maqnet
d.c.
Motor
The
equivalent electrical circuit ofthe
dc
motorconsists of a series
_{resistor,}
_{inductor,}
and voltagesource.
Motor
windings createthe
resistance andinductance.
The
action ofthe
armature windingscutting
the
lines
of magneticflux
generatesthe
back
electromotive
force
(EMF)
proportionalto
motor speedco
(t).
Torque
generated_{by}
the
motor_{x(t)}
is
proportional
to
armature currentIa(t).
The
_{resulting}
differential equationis
given_{by}
_{Eq}
(3.4).
The
motortransfer function
becomes:
T(s)/Va(s) =