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Proceedings of the IEEE, Vol. 82, No. 8, Aug. 1994, pp. 1194 -
1214
Lerb
Pulsewidth Modulation for Electronic Power ConversionJ. Holtz,
Fellow, IEEE
Wuppertal University Germany
Abstract The efficient and fast control of electric powerforms
part of the key technologies of modern automatedproduction. It is
performed using electronic power con-verters. The converters
transfer energy from a source to acontrolled process in a quantized
fashion, using semicon-ductor switches which are turned on and off
at fast repeti-tion rates. The algorithms which generate the
switchingfunctions pulsewidth modulation techniques are man-ifold.
They range from simple averaging schemes to in-volved methods of
real-time optimization. This paper givesan overview.
1. INTRODUCTIONMany three-phase loads require a supply of
variable volt-
age at variable frequency, including fast and high-efficien-cy
control by electronic means. Predominant applicationsare in
variable speed ac drives, where the rotor speed iscontrolled
through the supply frequency, and the machineflux through the
supply voltage.
The power requirements for these applications range
fromfractions of kilowatts to several megawatts. It is preferredin
general to take the power from a dc source and convert itto
three-phase ac using power electronic dc-to-ac convert-ers. The
input dc voltage, mostly of constant magnitude, isobtained from a
public utility through rectification, or froma storage battery in
the case of an electric vehicle drive.
The conversion of dc power to three-phase ac power isexclusively
performed in the switched mode. Power semi-conductor switches
effectuate temporary connections at highrepetition rates between
the two dc terminals and the threephases of the ac drive motor. The
actual power flow in eachmotor phase is controlled by the on/off
ratio, or duty-cycle,of the respective switches. The desired
sinusoidal wave-form of the currents is achieved by varying the
duty-cyclessinusoidally with time, employing techniques of
pulsewidthmodulation (PWM).
The basic principle of pulsewidth modulation is charac-terized
by the waveforms in Fig. 1. The voltage waveform
a) b)
isaisc
j
Imj
Re
isb
i = i exp(jj)sscurrent densitiy
distribution Imj
Re
j
As
sci
sai
sbi
Fig. 2: Definition of a current space vector; (a) cross
sectionof an induction motor, (b) stator windings and stator
currentspace vector in the complex plane
Fig. 1: Recorded three-phase PWM waveforms (suboscilla-tion
method); (a) voltage at one inverter terminal, (b) phasevoltage us
, (c) load current is
at one inverter terminal, Fig. 1(a), exhibits the
varyingduty-cycles of the power switches. The waveform is
alsoinfluenced by the switching in other phases, which createsfive
distinct voltage levels, Fig. 1(b). Further explanation isgiven in
Section 2.3. The resulting current waveform Fig.1(c) exhibits the
fundamental content more clearly, whichis owed to the low-pass
characteristics of the machine.
The operation in the switched mode ensures that theefficiency of
power conversion is high. The losses in theswitch are zero in the
off-state, and relatively low duringthe on-state. There are
switching losses in addition whichoccur during the transitions
between the two states. Theswitching losses increase with switching
frequency.
As seen from the pulsewidth modulation process, theswitching
frequency should be preferably high, so as toattenuate the
undesired side-effects of discontinuous powerflow at switching. The
limitation of switching frequencythat exists due to the switching
losses creates a conflictingsituation. The tradeoff which must be
found here is stronglyinfluenced by the respective pulsewidth
modulation tech-nique.
Three-phase electronic power converters controlled bypulsewidth
modulation have a wide range of applicationsfor dc-to-ac power
supplies and ac machine drives. Impor-tant quantities to be
considered with machine loads are thetwo-dimensional distributions
of current densities and fluxlinkages in ac machine windings. These
can be best ana-lyzed using the space vector approach, to which a
shortintroduction will be given first. Performance criteria will
bethen introduced to enable the evaluation and comparison
ofdifferent PWM techniques. The following sections are or-ganized
to treat open-loop and closed-loop PWM schemes.Both categories are
subdivided into nonoptimal and optimalstrategies.
2. AN INTRODUCTION TO SPACE VECTORS2.1 Definitions
Consider a symmetrical three-phase winding of an elec-tric
machine, Fig. 2(a), reduced to a two-pole arrangementfor
simplicity. The three phase axes are defined by the unityvectors,
1, a, and a2, where a = exp(2pi/3). Neglecting spaceharmonics, a
sinusoidal current density distribution is es-
0 20 mst
10
0
15 A0
ud/2
-ud/2
0ud/2
-ud/2
uL1
is
us
-15 A
a)
c)
b)
-
- 2 -
tablished around the air-gap by the phase currents isa, isb,and
isc as shown in Fig. 2(b). The wave rotates at theangular frequency
of the phase currents. Like any sinusoi-dal distribution in time
and space, it can be represented by acomplex phasor As as shown in
Fig. 2(a). It is preferred,however, to describe the mmf wave by the
equivalent cur-rent phasor is, because this quantity is directly
linked to thethree stator currents isa, isb, isc that can be
directly measuredat the machine terminals:
i a as sa sb sc= + +( )23 2i i i (1)The subscript s refers to
the stator of the machine.
The complex phasor in (1), more frequently referred to inthe
literature as a current space vector [1], has the samedirection in
space as the magnetic flux density wave pro-duced by the mmf
distribution As.
A sinusoidal flux density wave can be also described by aspace
vector. It is preferred, however, to choose the corre-sponding
distribution of the flux linkage with a particularthree-phase
winding as the characterizing quantity. For ex-ample, we write the
flux linkage space vector of the statorwinding in Fig. 2 as
ys s s= l i (2)In the general case, when the machine develops
nonzerotorque, both space vectors is of the stator current, and ir
ofthe rotor current are nonzero, yielding the stator flux link-age
vector as
ys s s h r= +l li i (3)where ls is the equivalent stator winding
inductance and lhthe composite mutual inductance between the stator
androtor windings. Furthermore,
i a ar ra rb rc= + +( )23 2i i i (4)is the rotor current space
vector, ira, irb and irc are the threerotor currents. Note that
flux linkage vectors like ys alsorepresent sinusoidal distributions
in space, which can beseen from an inspection of (2) or (3).
The rotating stator flux linkage wave ys generates in-duced
voltages in the stator windings which are describedby
us
s=
ddty
, (5)where
u a as sa sb sc= + +( )23 2u u u (6)is the space vector of the
stator voltages, and usa, usb, usc arethe stator phase
voltages.
The individual phase quantities associated to any spacevector
are obtained as the projections of the space vector onthe
respective phase axis. Given the space vector us, forexample, we
obtain the phase voltages as
u
u
u
sa s
sb s
sc s
= { }= { }= { }
Re
Re
Re
u
a u
a u
2.
.
(7)
Considering the case of three-phase dc-to-ac power sup-plies, an
LC-filter and the connected load replace the motorat the inverter
output terminals. Although not distributed inspace, such load
circuit behaves exactly the same way as amotor load. It is
permitted and common practice therefore
to extend the space vector approach to the analysis of
equiv-alent lumped parameter circuits.
2.2 NormalizationNormalized quantities are used throughout this
paper.
Space vectors are normalized with reference to the nominalvalues
of the connected ac machine. The respective basequantities are the
rated peak phase voltage 2 Uph R, the rated peak phase current 2
Iph R, and (8) the rated stator frequency sR.Using the definition
of the maximum modulation index insection 4.1.1, the normalized dc
bus voltage of a dc linkinverter becomes ud = pi/2.
2.3 Switching state vectorsThe space vector resulting from a
symmetrical sinusoidal
voltage system usa, usb, usc of frequency s is
us s sexp j= ( )u t. w , (9)which can be shown by inserting the
phase voltages (7) into(6).
A three-phase machine being fed from a switched powerconverter
Fig. 3 receives the symmetrical rectangular three-phase voltages
shown in Fig. 4. The three phase potentialsFig. 4(a) are constant
over every sixth of the fundamentalperiod, assuming one of the two
voltage levels, +Ud/2 or Ud/2, at a given time. The neutral point
potential unp, Fig. 3,of the load is either positive, when more
than one upperhalf-bridge switch is closed, Fig. 4(b); it is
negative withmore than one lower half-bridge switch closed. The
respec-tive voltage levels shown in Fig. 4(b) hold for
symmetricalload impedances.
The waveform of the phase voltage ua = uL1 unp isdisplayed in
the upper trace of Fig. 4(c). It forms a symmet-rical,
nonsinusoidal three-phase voltage system along withthe other phase
voltages ub and uc. Since the waveform unphas three times the
frequency of uLi , i = 1, 2, 3, while itsamplitude equals exactly
one third of the amplitudes of uLi,this waveform contains exactly
all triplen of the harmoniccomponents of uLi . Because of ua = uL1
unp there are notriplen harmonics left in the phase voltages. This
is alsotrue for the general case of three-phase
symmetricalpulsewidth modulated waveforms. As all triplen
harmonicsform zero-sequence systems, they produce no currents inthe
machine windings, provided there is no electrical con-nection to
the star-point of the load, i. e. unp in Fig. 3 mustnot be
shorted.
The example Fig. 4 demonstrates also that a change of
Fig. 3: Three-phase power converter; the switch pairs S1 S4 (and
S2 S5, and S3 S6) form half-bridges; one, and
only one switch in a half bridge is closed at a time.
au
U d12
cubuL1u
npu
U d12
L3
S6
S3
L2
S5
S2
L1
S4
S1
-
- 3 -
any half-bridge potential invariably influences upon theother
two-phase voltages. It is therefore expedient for thedesign of PWM
strategies and for the analysis of PWMwaveforms to analyse the
three-phase voltages as a whole,instead of looking at the
individual phase voltages separate-ly. The space vector approach
complies exactly with thisrequirement.
Inserting the phase voltages Fig. 4(c) into (6) yields
thetypical set of six active switching state vectors u1 ... u6shown
in Fig. 5. The switching state vectors describe theinverter output
voltages.
At operation with pulsewidth modulated waveforms, thetwo zero
vectors u0 and u7 are added to the pattern in Fig. 5.The zero
vectors are associated to those inverter states withall upper
half-bridge switches closed, or all lower, respec-tively. The three
machine terminals are then short-circuited,and the voltage vector
assumes zero magnitude.
Using (7), the three phase voltages of Fig. 4(c) can
bereconstructed from the switching state pattern Fig. 5.
3. PERFORMANCE CRITERIAConsidering an ac machine drive, it is
the leakage induct-
ances of the machine and the inertia of the mechanicalsystem
which account for low pass filtering of the harmoniccomponents
contained in the switched voltage waveforms.Remaining distortions
of the current waveforms, harmoniclosses in the power converter and
the load, and oscillationsin the electromagnetic machine torque are
due to the opera-tion in the switched mode. They can be valued by
perform-ance criteria [2] ... [7]. These provide the means of
compar-ing the qualities of different PWM methods and support
theselection of a pulsewidth modulator for a particular
appli-cation.
3.1 Current harmonicsThe harmonic currents primarily determine
the copper
losses of the machine, which account for a major portion ofthe
machine losses. The rms harmonic current
I T i t i t dth rms T ( ) ( )= [ ]1 1 2 (10)does not only depend
on the performance of the pulsewidthmodulator, but also on the
internal impedance of the ma-chine. This influence is eliminated
when using the distor-tion factor
dI
I=
h rmsh rms six step
(11)
as a figure of merit. In this definition, the distortion
currentIhrms (10) of a given switching sequence is referred to
thedistortion current Ih rms six-step of same ac load operated in
thesix-step mode, i. e. with the unpulsed rectangular
voltagewaveforms Fig. 4(c). The definition (11) values the
ac-sidecurrent distortion of a PWM method independently from
theproperties of the load. We have d = 1 at six-step operationby
definition. Note that the distortion factor d of a pulsedwaveform
can be much higher than that of a rectangularwave, e. g. Fig.
19.
The harmonic content of a current space vector trajectoryis
computed as
I T t t t t dth rms T ( ) ( ) ( ) ( )= ( ) ( )1 1 1i i i i *
(12)from which d can be determined by (11). The asterisc in
(12)marks the complex conjugate.
The harmonic copper losses in the load circuit are pro-portional
to the square of the harmonic current: PLc d2,where d2 is the loss
factor.
3.2 Harmonic spectrumThe contributions of individual frequency
components to
a nonsinusiodal current wave are expressed in a harmoniccurrent
spectrum, which is a more detailed description thanthe global
distortion factor d. We obtain discrete currentspectra hi (k . f1)
in the case of synchronized PWM, wherethe switching frequency fs =
N . f1 is an integral multiple ofthe fundamental frequency f1. N is
the pulse number, orgear ratio, and k is the order of the harmonic
component.Note that all harmonic spectra in this paper are
normalizedas per the definition (11):
h k f I k fIih rms
h rms six-step( ) ( ).
.
11
= . (13)
They describe the properties of a pulse modulation
schemeindependently from the parameters of the connected load.
Nonsynchronized pulse sequences produce harmonic am-
Fig. 4: Switched three-phase waveforms; (a) voltage poten-tials
at the load terminals, (b) neutral point potential, (c)phase
voltages
2pi
ua
0
uc
ub
0
0
1 2 3 4 5 6
uL1
02pi
0
0
uL2
uL3
unp 2pi
a)
c)
b)
tw
ud21
ud21
ud61
ud32
ud32
jIm
Re
( + ) ( + )+
( + )+
( )+ ( + )+
1u
2u3u
4u
5u 6u
( + )( + )ud3
2
7u ( + )+ +0u ( )
Fig. 5: Switching state vectors in the complex plane;
inbrackets: switching polarities of the three half-bridges
-
- 4 -
plitude density spectra hd(f) of the currents, which are
con-tinuous functions of frequency. They generally contain
pe-riodic as well as nonperiodic components and hence mustbe
displayed with reference to two different scale factors onthe
ordinate axis, e. g. Fig. 35. While the normalized dis-crete
spectra do not have a physical dimension, the ampli-tude density
sprectra are measured in Hz-1/2. The normal-ized harmonic current
(11) is computed from the discretespectrum (13) as
d h k f=
ik
( )2 11
. , (14)
and from the amplitude density spectrum as
d h f dff f
=
d ( )20 1,
. (15)
Another figure of merit for a given PWM scheme is theproduct of
the distortion factor and the switching frequencyof the inverter.
This value can be used to compare differentPWM schemes operated at
different switching frequenciesprovided that the pulse number N
> 15. The relation be-comes nonlinear at lower values of N.
3.3 Maximum modulation indexThe modulation index is the
normalized fundamental volt-
age, defined as
mu
u=
11 six step
(16)
where u1 is the fundamental voltage of the modulated switch-ing
sequence and u1 six-step = 2/pi .ud the fundamental voltageat
six-step operation. We have 0 < m < 1, and hence
unitymodulation index, by definition, can be attained only in
thesix-step mode.
The maximum value mmax of the modulation index maydiffer in a
range of about 25% depending on the respectivepulsewidth modulation
method. As the maximum power ofa PWM converter is proportional to
the maximum voltageat the ac side, the maximum modulation index
mmax consti-tutes an important utilization factor of the
equipment.
3.4 Torque harmonicsThe torque ripple produced by a given
switching se-
quence in a connected ac machine can be expressed as
T T T T= ( )max av R , (17)where
Tmax = maximum air-gap torque,Tav = average air-gap torque,TR =
rated machine torque.Although torque harmonics are produced by the
harmon-
ic currents, there is no stringent relationship between bothof
them. Lower torque ripple can go along with highercurrent
harmonics, and vice versa.
3.5 Switching frequency and switching lossesThe losses of power
semiconductors subdivide into two
major portions: The on-state lossesP g u ion on L= ( )1 , ,
(18a)
and the dynamic lossesP f U igdyn s 0 L= ( )2 , . (18b)
It is apparent from (18a) and (18b) that, once the powerlevel
has been fixed by the dc supply voltage U0 and themaximum load
current iL max, the switching frequency fs is
an important design parameter. The harmonic distortion ofthe
ac-side currents reduces almost linearly with this fre-quency. Yet
the switching frequency cannot be deliberatelyincreased for the
following reasons: The switching losses of semiconductor devices
increase
proportional to the switching frequency. Semiconductor switches
for higher power generally pro-
duce higher switching losses, and the switching frequen-cy must
be reduced accordingly. Megawatt switched powerconverters using
GTOs are switched at only a few 100hertz.
The regulations regarding electromagnetic compatibility(EMC) are
stricter for power conversion equipment oper-ating at switching
frequencies higher than 9 kHz [8].Another important aspect related
to switching frequency
is the radiation of acoustic noise. The switched currentsproduce
fast changing electromagnetic fields which exertmechanical Lorentz
forces on current carrying conductors,and also produce
magnetostrictive mechanical deformationsin ferromagnetic materials.
It is especially the magneticcircuits of the ac loads that are
subject to mechanical exci-tation in the audible frequency range.
Resonant amplifica-tion may take place in the active stator iron,
being a hollowcylindrical elastic structure, or in the cooling fins
on theouter case of an electrical machine.
The dominating frequency components of acoustic radia-tion are
strongly related to the spectral distribution of theharmonic
currents and to the switching frequency of thefeeding power
converter. The psophometric weighting ofthe human ear makes
switching frequencies below 500 Hzand above 10 kHz less critical,
while the maximum sensi-tivity is around 1 - 2 kHz.
3.6 Dynamic performanceUsually a current control loop is
designed around a
switched mode power converter, the response time of
whichessentially determines the dynamic performance of the over-all
system. The dynamics are influenced by the switchingfrequency
and/or the PWM method used. Some schemesrequire feedback signals
that are free from current harmon-ics. Filtering of feedback
signals increases the responsetime of the loop [10].
PWM methods for the most commonly used voltage-source inverters
impress either the voltages, or the currentsinto the ac load
circuit. The respective approach determinesthe dynamic performance
and, in addition, influences uponthe structure of the superimposed
control system: The meth-ods of the first category operate in an
open-loop fashion,Fig. 6(a). Closed-loop PWM schemes, in contrast,
inject thecurrents into the load and require different structures
of thecontrol system, Fig. 6(b).
4. OPEN-LOOP SCHEMESOpen-loop schemes refer to a reference space
vector u*(t)
as an input signal, from which the switched three-phasevoltage
waveforms are generated such that the time averageof the associated
normalized fundamental space vector us1(t)equals the time average
of the reference vector. The generalopen-loop structure is
represented in Fig. 6(a).4.1 Carrier based PWM
The most widely used methods of pulsewidth modulationare carrier
based. They have as a common characteristicsubcycles of constant
time duration, a subcycle being de-fined as the time duration T0 =
1/2 fs during which any ofthe inverter half-bridges, as formed for
instance by S1 andS2 in Fig. 3, assumes two consecutive switching
states of
-
- 5 -
opposite voltage polarity. Operation at subcycles of con-stant
time duration is reflected in the harmonic spectrum bytwo salient
sidebands, centered around the carrier frequen-cy fs, and
additional frequency bands around integral multi-ples of the
carrier. An example is shown in Fig. 18.
There are various ways to implement carrier based PWM;these
which will be discussed next.
4.1.1 Suboscillation methodThis method employs individual
carrier modulators in
each of the three phases [10]. A signal flow diagram isshown in
Fig. 7. The reference signals ua*, ub*, uc* of thephase voltages
are sinusoidal in the steady-state, forming asymmetrical
three-phase system, Fig. 8.
They are obtained
from the reference vectoru*, which is split into itsthree phase
componentsua*, ub*, uc* on the basis of(7). Three comparators anda
triangular carrier signalucr, which is common to allthree phase
signals, gener-ate the logic signals u'a, u'b,and u'c that control
the half-bridges of the power con-verter.
Fig. 9 shows the modu-lation process in detail, ex-panded over a
time intervalof two subcycles. T0 is thesubcycle duration. Note
thatthe three phase potentialsua', ub', uc' are of equal mag-nitude
at the beginning and
at the end of each subcycle. The three line-to-line voltagesare
then zero, and hence us results as the zero vector.
A closer inspection of Fig. 8 reveals that the suboscilla-tion
method does not fully utilize the available dc busvoltage. The
maximum value of the modulation index mmax 1= pi/4 = 0.785 is
reached at a point where the amplitudes ofthe reference signal and
the carrier become equal, Fig. 8(b).Computing the maximum
line-to-line voltage amplitude inthis operating point yields
ua*(t1) ub*(t1) = 3 . ud/2 =0.866 ud. This is less than what is
obviously possible whenthe two half-bridges that correspond to
phases a and b areswitched to ua= ud/2 and ub= ud/2, respectively.
In thiscase, the maximum line-to-line voltage amplitude wouldequal
ud.
Measured waveforms obtained with the suboscillationmethod are
displayed in Fig. 1. This oscillogram was takenat 1 kHz switching
frequency and m 0.75.
4.1.2 Modified suboscillation methodThe deficiency of a limited
modulation index, inherent to
the suboscillation method, is cured when distorted refer-ence
waveforms are used. Such waveforms must not con-tain other
components than zero-sequence systems in addi-tion to the
fundamental. The reference waveforms shown inFig. 10 exhibit this
quality. They have a higher fundamentalcontent than sinewaves of
the same peak value. As ex-plained in Section 2.3, such distortions
are not transferred
PWM
3~M
du
=
~
nonlinearcontroller
3~M
=
~
a)b)
ku ku
isus
*is*us
du
0
0
0
0
T0 T0
au*
*bu
cu*
ua'
bu '
u c'
u*u cr
,
cru
tFig. 9: Determination of theswitching instants. T0: subcy-cle
duration
Fig. 6: Basic PWM structures; (a) open-loop scheme, (b)feedback
scheme; uk: switching state vector
ud
su
3~M
=
~32
ucr
s*u
a*u
b*u
c*u
au'
bu'cu'
Fig. 7: Suboscillation method; signal flow diagram
Fig. 8: Reference signals and carrier signal; modulationindex
(a) m = 0.5 mmax, (b) m = mmax
tw
u
tw
u12 d
u12 d
2pi
u12 d
u12 d
2pi
a)
b)
au* bu* cu* cru
au* bu* cu* cru
u ,*ucr
u ,*
cr
0
0
m = m max
mmaxm = 0.5
Fig. 10: Reference waveforms with added zero-sequencesystems;
(a) with added third harmonic, (b), (c), (d) withadded rectangular
signals of triple fundamental frequency
2p3
2p3
a) b)
u*
0
u2d
u2d
d)
u*0
u2d
u2d
u*0
u2d
u2d
c)
u*
0
u2d
u2d
tw tw
tw tw
2p3
2p3
-
- 6 -
to the load currents.There is an infinity of possible additions
to the funda-
mental waveform that constitute zero-sequence systems.The
waveform in Fig. 10(a) has a third harmonic content of25% of the
fundamental; the maximum modulation index isincreased here to mmax
= 0.882 [11]. The addition of rectan-gular waveforms of triple
fundamental frequency leads toreference signals as shown in Figs.
10(b) through 10(d);mmax 2 = 3 pi/6 = 0.907 is reached in these
cases. This isthe maximum value of modulation index that can be
ob-tained with the technique of adding zero sequence compo-nents to
the reference signal [12], [13].
4.1.3 Sampling techniquesThe suboscillation method is simple to
implement in hard-
ware, using analogue integra-tors and comparators for
thegeneration of the triangular car-rier and the switching
instants.Analogue electronic compo-nents are very fast, and
inverterswitching frequencies up to sev-eral tens of kilohertz are
easilyobtained.
When digital signal process-ing methods based on
micro-processors are preferred, the in-tegrators are replaced by
digital timers, and the digitizedreference signals are compared
with the actual timer countsat high repetition rates to obtain the
required time resolu-tion. Fig. 11 illustrates this process, which
is referred to asnatural sampling [14].
To releave the microprocessor from the time consumingtask of
comparing two time variable signals at a high repeti-tion rate, the
corresponding signal processing functions havebeen implemented in
on-chip hardware. Modern microcon-trollers comprise of
capture/compare units which generatedigital control signals for
three-phase PWM when loadedfrom the CPU with the corresponding
timing data [15].
If the capture/compare function is not available in hard-ware,
other samplingPWM methods can beemployed [16]. In thecase of
symmetricalregular sampling, Fig.12(a), the referencewaveforms are
sampledat the very low repeti-tion rate fs which is giv-en by the
switching fre-quency. The samplinginterval 1/ fs = 2T0 ex-tends
over two subcy-cles. tsn are the sam-pling instants. The
tri-angular carrier shownas a dotted line in Fig.12(a) is not
really ex-istent as a signal. Thetime intervals T1 and T2,which
define theswitching instants, aresimply computed in realtime from
the respectivesampled value u*(ts)using the
geometricalrelationships
T T u t1 s( )= +( )12 10 . * (19a)
T T T u t1 s( )= + ( )0 012 1. * (19b)which can be established
with reference to the dotted trian-gular line.
Another method, referred to as asymmetric regular sam-pling
[18], operates at double sampling frequency 2fs. Fig.12(b) shows
that samples are taken once in every subcycle.This improves the
dynamic response and produces some-what less harmonic distortion of
the load currents.
4.1.4 Space vector modulationThe space vector modulation
technique differs from the
aforementioned methods in that there are not separate
mod-ulators used for each of the three phases. Instead, the
com-plex reference voltage vector is processed as a whole
[18],[19]. Fig. 13(a) shows the principle. The reference vectoru*
is sampled at the fixed clock frequency 2 fs. The sampledvalue
u*(ts) is then used to solve the equations
2 f t t ts a a b b s( ). *u u u+( ) = (20a)t f t ts0 a b=
12 (20b)
where ua and ub are the two switching state vectors adjacentin
space to the reference vector u*, Fig. 13(b). The solutionsof (20)
are the respective on-durations ta, tb, and t0 of theswitching
state vectors ua, ub, u0:
t f u t1 s s( )= 12 3 13. * cos sinpi (21a)
t f u t2 s s( )=1
22 3
. * sinpi
(21b)
t f t t0 s 1 2= 1
2 (21c)
The angle in these equations is the phase angle of thereference
vector.
This technique in effect averages the three switchingstate
vectors over a subcycle interval T0 = 1/2fs to equal thereference
vector u*(ts) as sampled at the beginning of thesubcycle. It is
assumed in Fig. 13(b) that the referencevector is located in the
first 60-sector of the complexplane. The adjacent switching state
vectors are then ua = u1and ub = u2, Fig. 5. As the reference
vector enters the nextsector, ua = u2 and ub = u3, and so on. When
programming amicroprocessor, the reference vector is first rotated
back byn . 60 until it resides in the first sector, and then (21)
is
t0
u*
digitizedreference
timer counttn
u*
Fig. 11: Natural sampling
Fig. 12: Sampling techniques; (a)symmetrical regular sampling,
(b)asymmetric regular sampling
0
0
Tn Tn+1 Tn+2
T0
tsn
ts(n+1) ts(n+2)ts(n+3)
Tn+3
0
0
T1nT2n T2(n+1)
2T0
tsnts(n+1)
t
t
a)
b)
T1(n+1)
u*a
ua'
u*a
ua'
0tta bt
Eqn. 21 ud2 sf
u
3~M 0 Re
jIm
a) b)
s(t )*u
ku
*u
0u
au
bu
select =~
*u
Fig. 13: Space vector modulation; (a) signal flow diagram,(b)
switching state vectors of the first 60-sector
-
- 7 -
evaluated. Finally, the switching states to replace the
provi-sional vectors ua and ub are identified by rotating ua and
ubforward by n . 60 [20].
Having computed the on-durations of the three switchingstate
vectors that form one subcycle, an adequate sequencein time of
these vectors must be determined next. Associat-ed to each
switching state vector in Fig. 5 are the switchingpolarities of the
three half-bridges, given in brackets. Thezero vector is redundant.
It can b either formed as u0 (- - -), or u7 (+ + +). u0 is
preferred when the previous switchingstate vector is u1, u3, or u5;
u7 will be chosen following u2,u4, or u6. This ensures that only
one half-bridge in Fig. 3needs to commutate at a transition between
an active switch-ing state vector and the zero vector. Hence the
minimumnumber of commutations is obtained by the switching
se-quence
u u u u0 0 1 1 2 2 7 0t t t t2 2.. .. .. (22a)in any first, or
generally in all odd subcycles, and
u u u u7 0 2 2 1 1 0 0t t t t2 2.. .. .. (22b)for the next, or
all even subcycles. The notation in (22)associates to each
switching state vector its on-duration inbrackets.
4.1.5 Modified space vector modulationThe modified space vector
modulation [21, 22, 23] uses
the switching sequences
u u u0 0 1 1 2 2t t t3 2 3 3.. .. , (23a)u u u2 2 1 1 0 0t t t3
2 3 3.. .. , (23b)
or a combination of (22) and (23). Note that a subcycle ofthe
sequences (23) consists of two switching states, sincethe last
state in (23(a)) is the same as the first state in(23(b)).
Similarly, a subcycle of the sequences (22) com-prises three
switching states. The on-durations of the switch-ing state vectors
in (23) are consequently reduced to 2/3 ofthose in (22) in order to
maintain the switching frequency fsat a given value.
The choice between the two switching sequences (22)and (23)
should depend on the value of the reference vector.The decision is
based on the analysis of the resulting har-monic current.
Considering the equivalent circuit Fig. 14,the differential
equation
ddt li
u us s i= ( )1
(24)
can be used to compute the trajectory in space of the
currentspace vector is. us is the actual switching state vector. If
thetrajectories dis(us)/dt are approximated as linear, the
closedpatterns of Fig. 15 will result. The patterns are shown for
theswitching state sequences (22) and (23), and two
differentmagnitude values, u1* and u2*, of the reference vector
areconsidered. The harmonic content of the trajectories is
de-termined using (12). The result can be confirmed just by avisual
inspection of the patterns in Fig. 15: the harmoniccontent is lower
at high modulation index with the modifiedswitching sequence (23);
it is lower at low modulation indexwhen the sequence (22) is
applied.Fig. 17 shows the corresponding characteristics of the
lossfactor d2: curve svm corresponds to the sequence (22), andcurve
(c) to sequence (23). The maximum modulation indexextends in either
case up to mmax2 = 0.907.
4.1.6 Synchronized carrier modulationThe aforementioned methods
operate at constant carrier
frequency, while the fundamental frequency is permitted tovary.
The switching sequence is then nonperiodic in princi-ple, and the
corresponding Fourier spectra are continuous.They contain also
frequencies lower than the lowest carriersideband, Fig. 18. These
subharmonic components are un-
desired as they produce low-fre-quency torque harmonics. A
syn-chronization between the carrierfrequency and the controling
fun-damental avoids these drawbackswhich are especially prominentif
the frequency ratio, or pulsenumber
N ff=s
1(25)
is low. In synchronized PWM,the pulse number N assumes
onlyintegral values [24].
When sampling techniques areemployed for synchronized car-rier
modulation, an advantage canbe drawn from the fact that thesampling
instants tsn = n /(f1 . N),n = 1 ... N in a fundamental peri-
Fig. 14: Induction motor, equivalent circuit
l
us uiis
0
0
2T0 u*
t
T1(n+1)Tn2Tn1 T2(n+1)
u* tsn( )u* ts(n+1)( )
ua'
Fig. 16: Synchronized regular sampling
Fig. 15: Linearized trajectories of the harmonic current for two
voltage references u1*and u2*: and (a) suboscillation method, (b)
space vector modulation, (c) modifiedspace vector modulation
a) c)b)
disdt ( )1u
2u
2u2u1u
1u1u
0u
0u 0u
1u
6u
disdt ( 2u )2u3u
4u
5u
0u*1u
*2u
1u6u
0u
1u6u
0u
1u6u
0u
-
- 8 -
od are a priori known. The reference signal is u*(t) =
m/mmax
.sin 2pi f1t, and the sampled values u*(ts) in Fig. 16 forma
discretized sine function that can be stored in the proces-sor
memory. Based on these values, the switching instantsare computed
on-line using (19).4.1.7 Performance of carrier based PWM
The loss factor d2 of suboscillation PWM depends on
thezero-sequence components added to the reference signal.
Acomparison is made in Fig. 17 at 2 kHz switching frequen-cy.
Letters (a) through (d) refer to the respective reference
waveforms in Fig. 10.The space vector modulation exhibits a
better loss factor
characteristic at m > 0.4 as the suboscillation method
withsinusoidal reference waveforms. The reason becomes obvi-ous
when comparing the harmonic trajectories in Fig. 15.The zero vector
appears twice during two subsequent sub-cycles, and there is a
shorter and a subsequent larger por-tion of it in a complete
harmonic pattern of the suboscilla-tion method. Fig. 9 shows how
the two different on-dura-tions of the zero vector are generated.
Against that, the on-durations of two subsequent zero vectors Fig.
15(b) arebasically equal in the case of space vector modulation.
Thecontours of the harmonic pattern come closer to the originin
this case, which reduces the harmonic content.
The modified space vector modulation, curve (d) in Fig.17,
performs better at higher modulation index, and worseat m <
0.62.
A typical harmonic spectrum produced by the space vec-tor
modulation is shown in Fig. 18.
The loss factor curves of synchronized carrier PWM areshown in
Fig. 19 for the suboscillation technique and thespace vector
modulation. The latter appears superior at lowpulse numbers, the
difference becoming less significant asN increases. The curves
exhibit no differences at lowermodulation index. Operating in this
range is of little practi-cal use for constant v/f1 loads where
higher values of N arepermitted and, above all, d2 decreases if m
is reduced (Fig.17).
The performance of a pulsewidth modulator based onsampling
techniques is slightly inferior than that of thesuboscillation
method, but only at low pulse numbers.
Because of the synchronism between f1 and fs, the pulsenumber
must necessarily change as the modulation indexvaries over a
broader range. Such changes introduce dis-continuities to the
modulation process. They generally orig-inate current transients,
especially when the pulse numberis low [25]. This effect is
discussed in Section 5.2.3.
4.2 Carrierless PWMThe typical harmonic spectrum of carrier
based pulsewidth
modulation exhibits prominent harmonic amplitudes aroundthe
carrier frequency and its harmonics, Fig. 18. Increasedacoustic
noise is generated by the machine at these frequen-cies through the
effects of magnetostriction. The vibrationscan be amplified by
mechanical resonances. To reduce themechanical excitation at
particular frequencies it may bepreferable to have the harmonic
energy distributed over alarger frequency range instead of being
concentrated aroundthe carrier frequency.
This concept is realized by varying the carrier frequencyin a
randomly manner. Applying this to the suboscillationtechnique, the
slopes of the triangular carrier signal must bemaintained linear in
order to conserve the linear input-output relationship of the
modulator. Fig. 20 shows how arandom frequency carrier signal can
be generated. Whenev-er the carrier signal reaches one of its peak
values, its slopeis reversed by a hysteresis element, and a sample
is taken
m
d2
0
0.02
0.2 0.4 0.6 0.8 10
0.05
0.03
0.01
sub
a)
d)c)b)
svm
osm
Fig. 17: Performance of carrier modulation at fs = 2 kHz; for(a)
through (d) refer to Fig. 9; sub: suboscillation method,svm: space
vector modulation, osm: optimal subcycle meth-od
randomgenerator
ucr
sample & hold
Fig. 19: Synchronized carrier modulation, loss factor d2versus
modulation index; (a) suboscillation method, (b)space vector
modulation
.2 .4 .6 .8 10m
8
4
2
d2
0
6
.2 .4 .6 .8 10ma) b)
mmax1mmax2
9
18
12
9
18
12
N = 6 N = 6
15 15
Fig. 20: Random frequency carrier signal generator
.05
.1
02 40
f6 10kHz
hi
8
Fig. 18: Space vector modulation, harmonic spectrum
-
- 9 -
from a random signal generator which imposes an addition-al
small variation on the slope. This varies the durations ofthe
subcycles randomly [26]. The average switching fre-quency is
maintained constant such that the power devicesare not exposed to
changes in temperature.
The optimal subcycle method (Section 6.4.3) classifiesalso as
carrierless. Another approach to carrierless PWM isexplained in
Fig. 21; it is based on the space vector modula-tion principle.
Instead of operating at constant samplingfrequency 2fs as in Fig.
13(a), samples of the referencevector are taken whenever the
duration tact of the switchingstate vector uact terminates. tact is
determined from the solu-tion of
t t f t t f tact act 1 1 s act 1 2 s ( )u u u u+ + =12 12 . * ,
(26)
where u*(t) is the reference vector. This quantity is
differentfrom its time discretized value u*(ts) used in 12(a). As
u*(t)is a continuously time-variable signal, the on-durations
t1,t2, and t0 are different from the values (20), which introduc-es
the desired variations of subcycle lengths. Note that t1 isanother
solution of (26), which is disregarded. The switch-ing state
vectors of a subcycle are shown in Fig. 21(b). Oncethe on-time tact
of uact has elapsed, ua is chosen as uact for thenext switching
interval, ub becomes ua, and the cyclic proc-
ess starts again [27].Fig. 21(c) gives an example of measured
subcycle dura-
tions in a fundamental period. The comparison of the har-monic
spectra Fig. 21(d) and Fig. 18 demonstrates the ab-sence of
pronounced spectral components in the harmoniccurrent.Carrierless
PWM equalizes the spectral distribution of theharmonic energy. The
energy level is not reduced. To lowerthe audible excitation of
mechanical resonances is a promis-ing aspect. It remains difficult
to decide, though, wheather aclear, single tone is better tolerable
in its annoying effectthan the radiation of white noise.
4.3 OvermodulationIt is apparent from the averaging approach of
the space
vector modulation technique that the on-duration t0 of thezero
vector u0 (or u7) decreases as the modulation index mincreases. t0
= 0 is first reached at m = mmax 2, which meansthat the circular
path of the reference vector u* touches theouter hexagon that is
opened up by the switching statevectors Fig. 22(a). The
controllable range of linear modula-tion methods terminates at this
point.
An additional singular operating point exists in the six-step
mode. It is characterized by the switching sequence u1-
u2 - u3 - ... - u6 and the highest possible fundamental
outputvoltage corresponding to m = 1.
Control in the intermediate range mmax 2 < m < 1 can
beachieved by overmodulation [28]. It is expedient to consid-er a
sequence of output voltage vectors uk, averaged over asubcycle to
become a single quantity uav, as the characteris-tic variable.
Overmodulation techniques subdivide into twodifferent modes. In
mode I, the trajectory of the averagevoltage vector uav follows a
circle of radius m > mmax 2 aslong as the circle arc is located
within the hexagon; uav
six-step modem = 1
Re
jIm
overmodulationrange
m m max2PWM
1u
2u
5u 6u
0u
3u
4u*u
a)
0
w1
Eqn. 26
3~M
select
tact ud
Re
jIm
b)a)0u
bu
ku
*u
u
*u
actu
=
~
a0
w1
Eqn. 26
3~M
select
tact ud
Re
jIm
b)a)0u
bu
ku
*u
u
*u
actu
=
~
a
TTs
1.0
1.2
0.80 2pp
c)w t
4
.1
02
.05
6 kHz8 10f
ih
0d)
Fig. 21: Carrierless pulsewidth modulation; (a) signal
flowdiagram, (b) switching state vectors of the first 60-sector,(c)
measured subcycle durations, (d) harmonic spectrum
Fig. 22: Overmodulation; (a) definition of the overmodu-lation
range, (b) trajectory of uav in overmodulation range I
jIm
Re0
-trajectory
1u
2u
5u 6u
3u
4u
*u
p*u
*u
b)
-
- 10 -
tracks the hexagon sides in the remaining portions (Fig.22(b)).
Equations (21) are used to derive the switchingdurations while uav
is on the arc. On the hexagon sides, thedurations are t0 = 0
and
t Ta =
+0
33
cos sincos sin
, (27a)
t T tb a= 0 . (27b)Overmodulation mode II is reached at m >
mmax 3 = 0.952
when the length of the arcs reduces to zero and the trajecto-ry
of uav becomes purely hexagonal. In this mode, thevelocity of the
average voltage vector is controlled along itslinear trajectory by
varying the duty cycle of the two switch-ing state vectors adjacent
to uav. As m increases, the veloci-ty becomes gradually higher in
the center portion of thehexagon side, and lower near the corners.
Overmodulationmode II converges smoothly into six-step operation
whenthe velocity on the edges becomes infinite, the velocity atthe
corners zero.
In mode II a sub-cycle is made up byonly two switchingstate
vectors. Theseare the two vectorsthat define the hexa-gon side on
which uavis traveling. Since theswitching frequencyis normally
main-tained at constantvalue, the subcycleduration T0 must re-duce
due to the re-duced number ofswitching state vec-tors. This
explaineswhy the distortionfactor reduces at the beginning of the
overmodulation range(Fig. 23).
The current waveforms Fig. 24 demonstrate that the mod-ulation
index is increased beyond the limit existing at linearmodulation by
the addition of harmonic components to theaverage voltage uav. The
added harmonics do not formzero-sequence components as those
discussed in Section4.1.2. Hence they are fully reflected in the
current wave-forms, which classifies overmodulation as a nonlinear
tech-nique.
4.4 Optimized Open-loop PWMPWM inverters of higher power rating
are operated at
very low switching frequency to reduce the switching loss-es.
Values of a few 100 Hertz are customary in the mega-watt range. If
the choice is a open-loop technique, onlysynchronized pulse schemes
should be employed here inorder to avoid the generation of
excessive subharmoniccomponents. The same applies for drive systems
operatingat high fundamental frequency while the switching
frequen-cy is in the lower kilohertz range. The pulse number (25)
islow in both cases. There are only a few switching instants tkper
fundamental period, and small variations of the respec-tive
switching angles k =2pi f1 . tk have considerable influ-ence on the
harmonic distortion of the machine currents.
It is advantageous in this situation to determine the
finitenumber of switching angles per fundamental period by
op-timization procedures. Necessarily the fundamental frequen-cy
must be considered constant for the purpose of definingthe
optimization problem. A solution can be then obtained
Fig. 24: Current waveforms at overmodulation; (a) spacevector
modulation at mmax 2, (b) transition between range Iand range II,
(c) overmodulation range II, (d) operationclose to the six-step
mode
Fig. 23: Loss factor d2 at over-modulation (different d2
scales)
0.8 0.9 1
.2
d2
m
0
1
.8
mmax2mmax3
.1
80 A
20 ms0 10t
is
0
0
0
a)
b)
d)
c)
0
off-line. The precalculated optimal switching patterns arestored
in the drive control system to be retrieved duringoperation in
real-time [29].
The application of this method is restricted to quasi
steady-state operating conditions. Operation in the transient
modeproduces waveform distortions worse than with nonoptimalmethods
(Section 5.2.3).
The best optimization results are achieved with switch-ing
sequences having odd pulse numbers and quarter-wavesymmetry.
Off-line schemes can be classified with respectto the optimization
objective [30].4.4.1 Harmonic elimination
This technique aims at the elimination of a well definednumber
n1 = (N 1)/2 of lower order harmonics from thediscrete Fourier
spectrum. It eliminates all torque harmon-ics having 6 times the
fundamental frequency at N = 5, or 6and 12 times the fundamental
frequency at N = 7, and so on[31]. The method can be applied when
specific harmonicfrequencies in the machine torque must be avoided
in orderto prevent resonant excitation of the driven
mechanicalsystem (motor shaft, couplings, gears, load). The
approachis suboptimal as regards other performance criteria.
4.4.2 Objective functionsAn accepted approach is the
minimization of the loss
factor d2 [32], where d is defined by (11) and (14).
Alterna-tively, the highest peak value of the phase current can
beconsidered a quantity to be minimized at very low pulsenumbers
[33]. The maximum efficiency of the inverter/machine system is
another optimization objective [34].
The objective function that defines a particular optimiza-tion
problem tends to exhibit a very large number of localminimums. This
makes the numerical solution extremelytime consuming, even on
todays modern computers. A setof switching angles which minimize
the harmonic current(d min) is shown in Fig. 25.
Fig. 26 compares the performance of a d min schemeat 300 Hz
switching frequency with the suboscillation meth-od and the space
vector modulation method.
-
- 11 -
Fig. 27: Optimal subcycle PWM; (a) signal flow diagram,(b)
subcycle duration versus fundamental phase angle
select
3~M
=
~
Eqn. 21ud
0tta bt
s(t )
T(k)st +T(k)
ku
s*u
*u
s(t T(k))+ 12*u*u
Fig. 25: Optimal switching angles; N: pulse number
k
k
0 0.5
0
30
60
90
m1
0
30
60
0 0.5m
1
N = 5 N = 7
N = 9 N = 11
90
4.4.3 Optimal Subcycle MethodThis method considers the durations
of switching subcy-
cles as optimization variables, a subcycle being the
timesequence of three consecutive switching state vectors.
Thesequence is arranged such that the instantaneous
distortioncurrent equals zero at the beginning and at the end of
thesubcycle. This enables the composition of the switchedwaveforms
from a precalculated set of optimal subcycles inany desired
sequence without causing undesired currenttransients under dynamic
operating conditions. The approacheliminates a basic deficiency of
the optimal pulsewidthmodulation techniques that are based on
precalculatedswitching angles [35].
A signal flow diagram of an optimal subcycle modulator
is shown in Fig. 27(a). Samples of the reference vector u*are
taken at t = ts , whenever the previous subcycle termi-nates. The
time duration Ts(u*) of the next subcycle is thenread from a table
which contains off-line optimized data asdisplayed in Fig. 27(b).
The curves show that the subcyclesenlarge as the reference vector
comes closer to one of theactive switching state vectors, both in
magnitude as in phaseangle. This implies that the optimization is
only worthwhilein the upper modulation range.
The modulation process itself is based on the space vec-tor
approach, taking into account that the subcycle length isvariable.
Hence Ts replaces T0 = 1/2 fs in (21). A predictedvalue u*(ts + 1/2
Ts(u*(ts)) is used to determine the on-times.The prediction assumes
that the fundamental frequency does
d2
0
0.5
1.0
.2 .4 .6 .8 10m
2.0
fs = 300 Hz
c)
a)
b)
180 Hz
214 Hz
1.5233 Hz
Fig. 26: Loss factor d2 of synchronous optimal PWM, curve(a);
for comparison at fs = 300 Hz: (b) space vector modula-tion, (c)
suboscillation method
fs = 1 kHz
0.40.60.811.21.41.61.8
0 pi/6 pi/2pi/3arg( u*)
TsT0
m = 0.8660.8
0.70.6
0.50.4
Fig. 28: Current trajectories; (a) space vector modulation,(b)
optimal subcycle modulation
i(t)t
5
i(t)t
51010
a) b)
15 A 15 A
not change during a subcycle. It eliminates the perturba-tions
of the fundamental phase angle that would result fromsampling at
variable time intervals.
The performance of the optimal subcycle method is com-pared with
the space vector modulation technique in Fig.28. The Fourier
spectrum lacks dominant carrier frequen-cies, which reduces the
radiation of acoustic noise fromconnected loads.
4.5 Switching conditionsIt was assumed until now that the
inverter switches be-
have ideally. This is not true for almost all types of
semi-conductor switches. The devices react delayed to their
con-trol signals at turn-on and turn-off. The delay times dependon
the type of semiconductor, on its current and voltagerating, on the
controling waveforms at the gate electrode,on the device
temperature, and on the actual current to beswitched.
4.5.1 Minimum duration of switching statesIn order to avoid
unnecessary switching losses of the
devices, allowance must be made by the control logic forminimum
time durations in the on-state and the off-state,respectively. An
additional time margin must be included
-
- 12 -
so as to allow the snubber circuits to energize or deener-gize.
The resulting minimum on-duration of a switchingstate vector is of
the order 1 - 100 s. If the commandedvalue in an open-loop
modulator is less than the requiredminimum, the respective
switching state must be eitherextended in time or skipped (pulse
dropping [36]). Thiscauses additional current waveform distortions,
and alsoconstitutes a limitation of the maximum modulation
index.The overmodulation techniques described in Section 4.3avoid
such limitations.
4.5.2 Dead-time effectMinority-carrier devices in particular
have their turn-off
delayed owing to the storage effect. The storage time Tstvaries
with the current and the device temperature. To avoidshort-circuits
of the inverter half-bridges, a lock-out timeTd must be introduced
by the inverter control. The lock-outtime counts from the time
instant at which one semiconduc-tor switch in a half-bridge turns
off and terminates when theopposite switch is turned on. The
lock-out time Td is deter-mined as the maximum value of storage
time Tst plus anadditional safety time interval.
We have now two different situations, displayed in Fig.29(a) for
positive load current in a bridge leg. When themodulator output
signal k goes high, the base drive signalk1 of T1 gets delayed by
Td, and so does the reversal of thephase voltage uph. If the
modulator output signal k goeslow, the base drive signal k1 is
immediately made zero, butthe actual turn-off of T1 is delayed by
the device storagetime Tst < Td. Consequently, the on-time of
the upper bridgearm does not last as long as commanded by the
controlingsignal k. It is decreased by the time difference Td
Tst,[37].
A similar effect occurs at negative current polarity. Fig.29(b)
shows that the on-time of the upper bridge arm is nowincreased by
Td Tst. Hence, the actual duty cycle of thehalf-bridge is always
different from that of the controllingsignal k. It is either
increased or decreased, depending onthe load current polarity. The
effect is described by
u u u u sig iav d sts
s;= =
* T TT , (28)
where uav is the inverter output voltage vector averaged overa
subcycle, and u is a normalized error vector attributed tothe
switching delay of the inverter. The error magnitude uis
proportional to the actual safety time margin Td Tst; its
direction changes in discrete steps, depending on the
re-spective polarities of the three phase currents. This is
ex-pressed in (28) by a polarity vector of constant magnitude
sig i i a i a is sa sb2
sc( ) + ( ) + ( )= [ ]23 sign sign sign. . , (29)where a =
exp(j2pi/3) and is is the current vector. The
notation sig(is) was chosen to indicate that this
complexnonlinear function exhibits properties of a sign
function.The graph sig(is) is shown in Fig. 30(a) for all
possiblevalues of the current vector is. The three phase currents
aredenoted as ias, ibs, and ics.
The dead-time effect described by (28) and (29) producesa
nonlinear distortion of the average voltage vector trajec-tory uav.
Fig. 30(b) shows an example. The distortion doesnot depend on the
magnitude u1 of the fundamental voltageand hence its relative
influence is very strong in the lowerspeed range where u1 is small.
Since the fundamental fre-quency is low in this range, the
smoothing action of theload circuit inductance has little effect on
the current wave-forms, and the sudden voltage changes become
clearly visi-ble, Fig. 31(a). The machine torque is influenced as
well,exhibiting dips in magnitude at six times the
fundamentalfrequency in the steady-state. Electromechanical
stabilityproblems may result if this frequency is sufficiently
low.Such case is illustrated in Fig. 32, showing one phase cur-rent
and the speed signal in permanent instability.
4.5.3 Dead-time compensationIf the pulsewidth modulator and the
inverter form part of
a superimposed high-bandwidth current control loop, thecurrent
waveform distortions caused by the dead-time ef-fect are
compensated to a certain extent. This may elimi-
Ud
T1
T2
D1
D2
k
0 0
Ud
T1
T2
D1
D2
a) b)
phuphu
Td
T st
k1k2
k
k1k2
Td
T st
Td Td
T 1off
T 1on
T 1off
T 1on
Fig. 29: Inverter switching delay; (a) positive load cur-rent,
(b) negative load current
i(t)
t
i(t)
t
a) b)
40 A 40 A
Fig. 31: Dead-time effect; (a) measured current trajectorywith
sixth harmonic and reduced fundamental, (b) as in (a),with
dead-time compensation
Fig. 30: Dead-time effect; (a) location of the polarity vec-tor
sig(i), (b) trajectory of the distorted average voltage uav
0
0>ia
Re
jIm
avu
icib
00