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