Pulsewidth Modulation for Electronic Power Conversion

Transcription

1 Lerb Plsewidh odlaion for Elecronic Power Conversion J. Holz, Fellow, IEEE Wpperal Universiy Germany Absrac The efficien and fas conrol of elecric power forms par of he key echnologies of modern aomaed prodcion. I is performed sing elecronic power converers. The converers ransfer energy from a sorce o a conrolled process in a qanized fashion, sing semicondcor swiches which are rned on and off a fas repeiion raes. The algorihms which generae he swiching fncions plsewidh modlaion echniqes are manifold. They range from simple averaging schemes o involved mehods of real-ime opimizaion. This paper gives an overview. A s b crren densiiy disribion a j Re c c exp(jj) b a j Re. INTRODUCTION any hree-phase loads reqire a spply of variable volage a variable freqency, inclding fas and high-efficiency conrol by elecronic means. Predominan applicaions are in variable speed ac drives, where he roor speed is conrolled hrogh he spply freqency, and he machine flx hrogh he spply volage. The power reqiremens for hese applicaions range from fracions of kilowas o several megawas. I is preferred in general o ake he power from a dc sorce and conver i o hree-phase ac sing power elecronic dc-o-ac converers. The inp dc volage, mosly of consan magnide, is obained from a pblic iliy hrogh recificaion, or from a sorage baery in he case of an elecric vehicle drive. The conversion of dc power o hree-phase ac power is exclsively performed in he swiched mode. Power semicondcor swiches effecae emporary connecions a high repeiion raes beween he wo dc erminals and he hree phases of he ac drive moor. The acal power flow in each moor phase is conrolled by he on/off raio, or dy-cycle, of he respecive swiches. The desired sinsoidal waveform of he crrens is achieved by varying he dy-cycles sinsoidally wih ime, employing echniqes of plsewidh modlaion (PW). The basic principle of plsewidh modlaion is characerized by he waveforms in Fig.. The volage waveform L α d / - d / 5 A -5 A ms Fig. : Recorded hree-phase PW waveforms (sboscillaion mehod); ( volage a one inverer erminal, ( phase volage sα, (c) load crren α d / - d / c) sα Fig. : Definiion of a crren space vecor; ( cross secion of an indcion moor, ( saor windings and saor crren space vecor in he complex plane a one inverer erminal, Fig. (, exhibis he varying dy-cycles of he power swiches. The waveform is also inflenced by he swiching in oher phases, which creaes five disinc volage levels, Fig. (. Frher explanaion is given in Secion.. The resling crren waveform Fig. (c) exhibis he fndamenal conen more clearly, which is owed o he low-pass characerisics of he machine. The operaion in he swiched mode ensres ha he efficiency of power conversion is high. The losses in he swich are zero in he off-sae, and relaively low dring he on-sae. There are swiching losses in addiion which occr dring he ransiions beween he wo saes. The swiching losses increase wih swiching freqency. As seen from he plsewidh modlaion process, he swiching freqency shold be preferably high, so as o aenae he ndesired side-effecs of disconinos power flow a swiching. The limiaion of swiching freqency ha exiss de o he swiching losses creaes a conflicing siaion. The radeoff which ms be fond here is srongly inflenced by he respecive plsewidh modlaion echniqe. Three-phase elecronic power converers conrolled by plsewidh modlaion have a wide range of applicaions for dc-o-ac power spplies and ac machine drives. Imporan qaniies o be considered wih machine loads are he wo-dimensional disribions of crren densiies and flx linkages in ac machine windings. These can be bes analyzed sing he space vecor approach, o which a shor inrodcion will be given firs. Performance crieria will be hen inrodced o enable he evalaion and comparison of differen PW echniqes. The following secions are organized o rea open-loop and closed-loop PW schemes. Boh caegories are sbdivided ino nonopimal and opimal sraegies.. AN INTRODUCTION TO SPACE VECTORS. Definiions Consider a symmerical hree-phase winding of an elecric machine, Fig. (, redced o a wo-pole arrangemen for simpliciy. The hree phase axes are defined by he niy vecors,, a, and a, where a exp(π/). Neglecing space harmonics, a sinsoidal crren densiy disribion is es- Proceedings of he IEEE, Vol. 8, No. 8, Ag. 994, pp. 94-4

2 - - ablished arond he air-gap by he phase crrens a, b, and c as shown in Fig. (. The wave roaes a he anglar freqency of he phase crrens. Like any sinsoidal disribion in ime and space, i can be represened by a complex phasor A s as shown in Fig. (. I is preferred, however, o describe he mmf wave by he eqivalen crren phasor, becase his qaniy is direcly linked o he hree saor crrens a, b, c ha can be direcly measred a he machine erminals: ( ) is isa + aisb + a i sc () The sbscrip s refers o he saor of he machine. The complex phasor in (), more freqenly referred o in he lierare as a crren space vecor [], has he same direcion in space as he magneic flx densiy wave prodced by he mmf disribion A s. A sinsoidal flx densiy wave can be also described by a space vecor. I is preferred, however, o choose he corresponding disribion of he flx linkage wih a pariclar hree-phase winding as he characerizing qaniy. For example, we wrie he flx linkage space vecor of he saor winding in Fig. as y s l s () In he general case, when he machine develops nonzero orqe, boh space vecors of he saor crren, and i r of he roor crren are nonzero, yielding he saor flx linkage vecor as y s l s s + l h r i i () where l s is he eqivalen saor winding indcance and l h he composie mal indcance beween he saor and roor windings. Frhermore, ( ) ir ira + airb + a i rc (4) is he roor crren space vecor, i ra, i rb and i rc are he hree roor crrens. Noe ha flx linkage vecors like y s also represen sinsoidal disribions in space, which can be seen from an inspecion of () or (). The roaing saor flx linkage wave y s generaes indced volages in he saor windings which are described by s s d y, (5) d where s ( sa + asb + a sc) (6) is he space vecor of he saor volages, and sa, sb, sc are he saor phase volages. The individal phase qaniies associaed o any space vecor are obained as he projecions of he space vecor on he respecive phase axis. Given he space vecor s, for example, we obain he phase volages as sa Re { s} sb Re { a. s} (7) sc Re { a. s } Considering he case of hree-phase dc-o-ac power spplies, an LC-filer and he conneced load replace he moor a he inverer op erminals. Alhogh no disribed in space, sch load circi behaves exacly he same way as a moor load. I is permied and common pracice herefore o exend he space vecor approach o he analysis of eqivalen lmped parameer circis.. Normalizaion Normalized qaniies are sed hrogho his paper. Space vecors are normalized wih reference o he nominal vales of he conneced ac machine. The respecive base qaniies are he raed peak phase volage U ph R, he raed peak phase crren I ph R, and (8) he raed saor freqency ω sr. Using he definiion of he maximm modlaion index in secion 4.., he normalized dc bs volage of a dc link inverer becomes d π/.. Swiching sae vecors The space vecor resling from a symmerical sinsoidal volage sysem sa, sb, sc of freqency ω s is s s. exp( jw s ), (9) which can be shown by insering he phase volages (7) ino (6). A hree-phase machine being fed from a swiched power converer Fig. receives he symmerical recanglar hreephase volages shown in Fig. 4. The hree phase poenials Fig. 4( are consan over every sixh of he fndamenal period, assming one of he wo volage levels, +U d / or U d /, a a given ime. The neral poin poenial np, Fig., of he load is eiher posiive, when more han one pper half-bridge swich is closed, Fig. 4(; i is negaive wih more han one lower half-bridge swich closed. The respecive volage levels shown in Fig. 4( hold for symmerical load impedances. U d U d L np S S4 L Fig. : Three-phase power converer; he swich pairs S S4 (and S S5, and S S6) form half-bridges; one, and only one swich in a half bridge is closed a a ime. The waveform of he phase volage a L np is displayed in he pper race of Fig. 4(c). I forms a symmerical, nonsinsoidal hree-phase volage sysem along wih he oher phase volages b and c. Since he waveform np has hree imes he freqency of Li, i,,, while is amplide eqals exacly one hird of he amplides of Li, his waveform conains exacly all riplen of he harmonic componens of Li. Becase of a L np here are no riplen harmonics lef in he phase volages. This is also re for he general case of hree-phase symmerical plsewidh modlaed waveforms. As all riplen harmonics form zero-seqence sysems, hey prodce no crrens in he machine windings, provided here is no elecrical connecion o he sar-poin of he load, i. e. np in Fig. ms no be shored. The example Fig. 4 demonsraes also ha a change of a S S5 L b S S6 L c

3 - - d L d L L np d a b c 6 d d Fig. 4: Swiched hree-phase waveforms; ( volage poenials a he load erminals, ( neral poin poenial, (c) phase volages any half-bridge poenial invariably inflences pon he oher wo-phase volages. I is herefore expedien for he design of PW sraegies and for he analysis of PW waveforms o analyse he hree-phase volages as a whole, insead of looking a he individal phase volages separaely. The space vecor approach complies exacly wih his reqiremen. Insering he phase volages Fig. 4(c) ino (6) yields he ypical se of six acive swiching sae vecors... 6 shown in Fig. 5. The swiching sae vecors describe he inverer op volages. A operaion wih plsewidh modlaed waveforms, he wo zero vecors and 7 are added o he paern in Fig. 5. The zero vecors are associaed o hose inverer saes wih all pper half-bridge swiches closed, or all lower, respecively. The hree machine erminals are hen shor-circied, and he volage vecor assmes zero magnide. Using (7), he hree phase volages of Fig. 4(c) can be reconsrced from he swiching sae paern Fig. 5.. PERFORANCE CRITERIA Considering an ac machine drive, i is he leakage indcances of he machine and he ineria of he mechanical sysem which accon for low pass filering of he harmonic componens conained in he swiched volage waveforms. Remaining disorions of he crren waveforms, harmonic losses in he power converer and he load, and oscillaions in he elecromagneic machine orqe are de o he operaion in he swiched mode. They can be valed by performance crieria []... [7]. These provide he means of comparing he qaliies of differen PW mehods and sppor he selecion of a plsewidh modlaor for a pariclar applicaion.. Crren harmonics The harmonic crrens primarily deermine he copper w π π π c) losses of he machine, which accon for a major porion of he machine losses. The rms harmonic crren Ih rms i () i() d T [ T ] () does no only depend on he performance of he plsewidh modlaor, b also on he inernal impedance of he machine. This inflence is eliminaed when sing he disorion facor d Ih rms Ih rms six sep () as a figre of meri. In his definiion, he disorion crren I hrms () of a given swiching seqence is referred o he disorion crren I h rms six-sep of same ac load operaed in he six-sep mode, i. e. wih he nplsed recanglar volage waveforms Fig. 4(c). The definiion () vales he ac-side crren disorion of a PW mehod independenly from he properies of he load. We have d a six-sep operaion by definiion. Noe ha he disorion facor d of a plsed waveform can be mch higher han ha of a recanglar wave, e. g. Fig. 9. The harmonic conen of a crren space vecor rajecory is comped as Ih rms () () () () d T ( i T i ) ( i i ) * () from which d can be deermined by (). The aserisc in () marks he complex conjgae. The harmonic copper losses in he load circi are proporional o he sqare of he harmonic crren: P Lc d, where d is he loss facor. ( + + ) ( + ) ( + + ). Harmonic specrm The conribions of individal freqency componens o a nonsinsiodal crren wave are expressed in a harmonic crren specrm, which is a more deailed descripion han he global disorion facor d. We obain discree crren specra h i (k. f ) in he case of synchronized PW, where he swiching freqency f s N. f is an inegral mliple of he fndamenal freqency f. N is he plse nmber, or gear raio, and k is he order of he harmonic componen. Noe ha all harmonic specra in his paper are normalized as per he definiion (): Ih rms( k. f ) hi( k. f). () Ih rms six-sep They describe he properies of a plse modlaion scheme independenly from he parameers of he conneced load. Nonsynchronized plse seqences prodce harmonic am- d ( + ) 4 Re 7 ( ) 5 ( ) 6 ( + ) ( + + ) Fig. 5: Swiching sae vecors in he complex plane; in brackes: swiching polariies of he hree half-bridges

4 - 4 - plide densiy specra h d (f) of he crrens, which are coninos fncions of freqency. They generally conain periodic as well as nonperiodic componens and hence ms be displayed wih reference o wo differen scale facors on he ordinae axis, e. g. Fig. 5. While he normalized discree specra do no have a physical dimension, he amplide densiy sprecra are measred in Hz -/. The normalized harmonic crren () is comped from he discree specrm () as d hi ( k. f), (4) k and from he amplide densiy specrm as d h d ( f ) df. (5), f f Anoher figre of meri for a given PW scheme is he prodc of he disorion facor and he swiching freqency of he inverer. This vale can be sed o compare differen PW schemes operaed a differen swiching freqencies provided ha he plse nmber N > 5. The relaion becomes nonlinear a lower vales of N.. aximm modlaion index The modlaion index is he normalized fndamenal volage, defined as m (6) six sep where is he fndamenal volage of he modlaed swiching seqence and six-sep /π. d he fndamenal volage a six-sep operaion. We have < m <, and hence niy modlaion index, by definiion, can be aained only in he six-sep mode. The maximm vale m max of he modlaion index may differ in a range of abo 5% depending on he respecive plsewidh modlaion mehod. As he maximm power of a PW converer is proporional o he maximm volage a he ac side, he maximm modlaion index m max consies an imporan ilizaion facor of he eqipmen..4 Torqe harmonics The orqe ripple prodced by a given swiching seqence in a conneced ac machine can be expressed as where T max T av T R T ( Tmax Tav) TR, (7) maximm air-gap orqe, average air-gap orqe, raed machine orqe. Alhogh orqe harmonics are prodced by he harmonic crrens, here is no sringen relaionship beween boh of hem. Lower orqe ripple can go along wih higher crren harmonics, and vice versa..5 Swiching freqency and swiching losses The losses of power semicondcors sbdivide ino wo major porions: The on-sae losses Pon g ( on, il ), (8 and he dynamic losses Pdyn fs g ( U, il ). (8 I is apparen from (8 and (8 ha, once he power level has been fixed by he dc spply volage U and he maximm load crren i L max, he swiching freqency f s is an imporan design parameer. The harmonic disorion of he ac-side crrens redces almos linearly wih his freqency. Ye he swiching freqency canno be deliberaely increased for he following reasons: The swiching losses of semicondcor devices increase proporional o he swiching freqency. Semicondcor swiches for higher power generally prodce higher swiching losses, and he swiching freqency ms be redced accordingly. egawa swiched power converers sing GTO s are swiched a only a few herz. The reglaions regarding elecromagneic compaibiliy (EC) are sricer for power conversion eqipmen operaing a swiching freqencies higher han 9 khz [8]. Anoher imporan aspec relaed o swiching freqency is he radiaion of acosic noise. The swiched crrens prodce fas changing elecromagneic fields which exer mechanical Lorenz forces on crren carrying condcors, and also prodce magneosricive mechanical deformaions in ferromagneic maerials. I is especially he magneic circis of he ac loads ha are sbjec o mechanical exciaion in he adible freqency range. Resonan amplificaion may ake place in he acive saor iron, being a hollow cylindrical elasic srcre, or in he cooling fins on he oer case of an elecrical machine. The dominaing freqency componens of acosic radiaion are srongly relaed o he specral disribion of he harmonic crrens and o he swiching freqency of he feeding power converer. The psophomeric weighing of he hman ear makes swiching freqencies below 5 Hz and above khz less criical, while he maximm sensiiviy is arond - khz..6 Dynamic performance Usally a crren conrol loop is designed arond a swiched mode power converer, he response ime of which essenially deermines he dynamic performance of he overall sysem. The dynamics are inflenced by he swiching freqency and/or he PW mehod sed. Some schemes reqire feedback signals ha are free from crren harmonics. Filering of feedback signals increases he response ime of he loop []. PW mehods for he mos commonly sed volagesorce inverers impress eiher he volages, or he crrens ino he ac load circi. The respecive approach deermines he dynamic performance and, in addiion, inflences pon he srcre of he sperimposed conrol sysem: The mehods of he firs caegory operae in an open-loop fashion, Fig. 6(. Closed-loop PW schemes, in conras, injec he crrens ino he load and reqire differen srcres of he conrol sysem, Fig. 6(. 4. OPEN-LOOP SCHEES Open-loop schemes refer o a reference space vecor () as an inp signal, from which he swiched hree-phase volage waveforms are generaed sch ha he ime average of he associaed normalized fndamenal space vecor s () eqals he ime average of he reference vecor. The general open-loop srcre is represened in Fig. 6(. 4. Carrier based PW The mos widely sed mehods of plsewidh modlaion are carrier based. They have as a common characerisic sbcycles of consan ime draion, a sbcycle being defined as he ime draion T / f s dring which any of he inverer half-bridges, as formed for insance by S and S in Fig., assmes wo consecive swiching saes of

5 - 5 - * s PW k d nonlinear conroller s * Fig. 6: Basic PW srcres; ( open-loop scheme, ( feedback scheme; k : swiching sae vecor opposie volage polariy. Operaion a sbcycles of consan ime draion is refleced in he harmonic specrm by wo salien sidebands, cenered arond he carrier freqency f s, and addiional freqency bands arond inegral mliples of he carrier. An example is shown in Fig. 8. There are varios ways o implemen carrier based PW; hese which will be discssed nex. s * a * b * c * cr 4.. Sboscillaion mehod This mehod employs individal carrier modlaors in each of he hree phases []. A signal flow diagram is shown in Fig. 7. The reference signals a *, b *, c * of he phase volages are sinsoidal in he seady-sae, forming a symmerical hree-phase sysem, Fig. 8. They are obained ' b ' a ' c k Fig. 7: Sboscillaion mehod; signal flow diagram,* cr,* cr d d d d a b c cr m.5 m max Fig. 8: Reference signals and carrier signal; modlaion index ( m.5 m max, ( m m max w a b c cr m m max w d s π π d, cr a ' ' b a b cr c ' c T T Fig. 9: Deerminaion of he swiching insans. T : sbcycle draion from he reference vecor, which is spli ino is hree phase componens a *, b *, c * on he basis of (7). Three comparaors and a rianglar carrier signal cr, which is common o all hree phase signals, generae he logic signals ' a, ' b, and ' c ha conrol he halfbridges of he power converer. Fig. 9 shows he modlaion process in deail, expanded over a ime inerval of wo sbcycles. T is he sbcycle draion. Noe ha he hree phase poenials a ', b ', c ' are of eqal magnide a he beginning and a he end of each sbcycle. The hree line-o-line volages are hen zero, and hence s resls as he zero vecor. A closer inspecion of Fig. 8 reveals ha he sboscillaion mehod does no flly ilize he available dc bs volage. The maximm vale of he modlaion index m max π/4.785 is reached a a poin where he amplides of he reference signal and he carrier become eqal, Fig. 8(. Comping he maximm line-o-line volage amplide in his operaing poin yields a *( ) b *( ). d /.866 d. This is less han wha is obviosly possible when he wo half-bridges ha correspond o phases a and b are swiched o a d / and b d /, respecively. In his case, he maximm line-o-line volage amplide wold eqal d. easred waveforms obained wih he sboscillaion mehod are displayed in Fig.. This oscillogram was aken a khz swiching freqency and m odified sboscillaion mehod The deficiency of a limied modlaion index, inheren o he sboscillaion mehod, is cred when disored reference waveforms are sed. Sch waveforms ms no conain oher componens han zero-seqence sysems in addiion o he fndamenal. The reference waveforms shown in Fig. exhibi his qaliy. They have a higher fndamenal conen han sinewaves of he same peak vale. As explained in Secion., sch disorions are no ransferred d d d d w w p p Fig. : Reference waveforms wih added zero-seqence sysems; ( wih added hird harmonic, (, (c), (d) wih added recanglar signals of riple fndamenal freqency d p d w c) d d w p d)

6 - 6 - o he load crrens. There is an infiniy of possible addiions o he fndamenal waveform ha consie zero-seqence sysems. The waveform in Fig. ( has a hird harmonic conen of 5% of he fndamenal; he maximm modlaion index is increased here o m max.88 []. The addiion of recanglar waveforms of riple fndamenal freqency leads o reference signals as shown in Figs. ( hrogh (d); m max π/6.97 is reached in hese cases. This is he maximm vale of modlaion index ha can be obained wih he echniqe of adding zero seqence componens o he reference signal [], []. a a ' a sn sn T T s(n+) T n T n+ T n+ T (n+) T n T n T(n+) s(n+) s(n+) a ' Fig. : Sampling echniqes; ( symmerical reglar sampling, ( asymmeric reglar sampling s(n+) T n+ digiized reference n imer con Fig. : Naral sampling 4.. Sampling echniqes The sboscillaion mehod is simple o implemen in hardware, sing analoge inegraors and comparaors for he generaion of he rianglar carrier and he swiching insans. Analoge elecronic componens are very fas, and inverer swiching freqencies p o several ens of kiloherz are easily obained. When digial signal processing mehods based on microprocessors are preferred, he inegraors are replaced by digial imers, and he digiized reference signals are compared wih he acal imer cons a high repeiion raes o obain he reqired ime resolion. Fig. illsraes his process, which is referred o as naral sampling [4]. To releave he microprocessor from he ime consming ask of comparing wo ime variable signals a a high repeiion rae, he corresponding signal processing fncions have been implemened in on-chip hardware. odern microconrollers comprise of capre/compare nis which generae digial conrol signals for hree-phase PW when loaded from he CPU wih he corresponding iming daa [5]. If he capre/compare fncion is no available in hardware, oher sampling PW mehods can be employed [6]. In he case of symmerical reglar sampling, Fig. (, he reference waveforms are sampled a he very low repeiion rae f s which is given by he swiching freqency. The sampling inerval / f s T exends over wo sbcycles. sn are he sampling insans. The rianglar carrier shown as a doed line in Fig. ( is no really exisen as a signal. The ime inervals T and T, which define he swiching insans, are simply comped in real ime from he respecive sampled vale ( s ) sing he geomerical relaionships T T ( + ( s) ). * (9 T T + T. ( ( s) ) (9 which can be esablished wih reference o he doed rianglar line. Anoher mehod, referred o as asymmeric reglar sampling [8], operaes a doble sampling freqency f s. Fig. ( shows ha samples are aken once in every sbcycle. This improves he dynamic response and prodces somewha less harmonic disorion of he load crrens Space vecor modlaion The space vecor modlaion echniqe differs from he aforemenioned mehods in ha here are no separae modlaors sed for each of he hree phases. Insead, he complex reference volage vecor is processed as a whole [8], [9]. Fig. ( shows he principle. The reference vecor is sampled a he fixed clock freqency f s. The sampled vale ( s ) is hen sed o solve he eqaions f.( + ) ( ) ( s a a b b s f a b s ( where a and b are he wo swiching sae vecors adjacen in space o he reference vecor, Fig. (. The solions of () are he respecive on-draions a, b, and of he swiching sae vecors a, b, : s f ( s). ( f s) * s cosα sinα π ( f s. ( s) sinα π f a s Eqn. b k ( (c) The angle α in hese eqaions is he phase angle of he reference vecor. This echniqe in effec averages he hree swiching sae vecors over a sbcycle inerval T /f s o eqal he reference vecor ( s ) as sampled a he beginning of he sbcycle. I is assmed in Fig. ( ha he reference vecor is locaed in he firs 6 -secor of he complex plane. The adjacen swiching sae vecors are hen a and b, Fig. 5. As he reference vecor eners he nex secor, a and b, and so on. When programming a microprocessor, he reference vecor is firs roaed back by n. 6 nil i resides in he firs secor, and hen () is d b selec a Re Fig. : Space vecor modlaion; ( signal flow diagram, ( swiching sae vecors of he firs 6 -secor

7 - 7 - l σ ( sn ) ( s(n+) ) T s i Fig. 4: Indcion moor, eqivalen circi evalaed. Finally, he swiching saes o replace he provisional vecors a and b are idenified by roaing a and b forward by n. 6 []. Having comped he on-draions of he hree swiching sae vecors ha form one sbcycle, an adeqae seqence in ime of hese vecors ms be deermined nex. Associaed o each swiching sae vecor in Fig. 5 are he swiching polariies of he hree half-bridges, given in brackes. The zero vecor is redndan. I can b eiher formed as (- - - ), or 7 (+ + +). is preferred when he previos swiching sae vecor is,, or 5 ; 7 will be chosen following, 4, or 6. This ensres ha only one half-bridge in Fig. needs o commae a a ransiion beween an acive swiching sae vecor and he zero vecor. Hence he minimm nmber of commaions is obained by he swiching seqence ( 7 in any firs, or generally in all odd sbcycles, and ( 7 for he nex, or all even sbcycles. The noaion in () associaes o each swiching sae vecor is on-draion in brackes odified space vecor modlaion The modified space vecor modlaion [,, ] ses he swiching seqences...., (...., ( or a combinaion of () and (). Noe ha a sbcycle of he seqences () consiss of wo swiching saes, since he las sae in (() is he same as he firs sae in ((). Similarly, a sbcycle of he seqences () comprises hree swiching saes. The on-draions of he swiching sae vecors in () are conseqenly redced o / of hose in () in order o mainain he swiching freqency f s a a given vale. 4 5 * 6 dis d ( ) d d ( ) 6 Fig. 5: Linearized rajecories of he harmonic crren for wo volage references * and *: and ( sboscillaion mehod, ( space vecor modlaion, (c) modified space vecor modlaion T n T n T (n+) T (n+) Fig. 6: Synchronized reglar sampling The choice beween he wo swiching seqences () and () shold depend on he vale of he reference vecor. The decision is based on he analysis of he resling harmonic crren. Considering he eqivalen circi Fig. 4, he differenial eqaion dis ( d l s i) (4) σ can be sed o compe he rajecory in space of he crren space vecor. s is he acal swiching sae vecor. If he rajecories d ( s )/d are approximaed as linear, he closed paerns of Fig. 5 will resl. The paerns are shown for he swiching sae seqences () and (), and wo differen magnide vales, * and *, of he reference vecor are considered. The harmonic conen of he rajecories is deermined sing (). The resl can be confirmed js by a visal inspecion of he paerns in Fig. 5: he harmonic conen is lower a high modlaion index wih he modified swiching seqence (); i is lower a low modlaion index when he seqence () is applied. Fig. 7 shows he corresponding characerisics of he loss facor d : crve svm corresponds o he seqence (), and crve (c) o seqence (). The maximm modlaion index exends in eiher case p o m max Synchronized carrier modlaion The aforemenioned mehods operae a consan carrier freqency, while he fndamenal freqency is permied o vary. The swiching seqence is hen nonperiodic in principle, and he corresponding Forier specra are coninos. They conain also freqencies lower han he lowes carrier sideband, Fig. 8. These sbharmonic componens are ndesired as hey prodce low-freqency orqe harmonics. A synchronizaion beween he carrier freqency and he conroling fndamenal avoids hese drawbacks which are especially prominen if he freqency raio, or plse nmber f N f s (5) c) 6 6 a ' is low. In synchronized PW, he plse nmber N assmes only inegral vales [4]. When sampling echniqes are employed for synchronized carrier modlaion, an advanage can be drawn from he fac ha he sampling insans sn n/(f. N), n... N in a fndamenal peri-

8 d... sb svm d) osm m Fig. 7: Performance of carrier modlaion a f s khz; for ( hrogh (d) refer o Fig. 9; sb: sboscillaion mehod, svm: space vecor modlaion, osm: opimal sbcycle mehod od are a priori known. The reference signal is () m/ m max. sin π f, and he sampled vales ( s ) in Fig. 6 form a discreized sine fncion ha can be sored in he processor memory. Based on hese vales, he swiching insans are comped on-line sing (9) Performance of carrier based PW The loss facor d of sboscillaion PW depends on he zero-seqence componens added o he reference signal. A comparison is made in Fig. 7 a khz swiching freqency. Leers ( hrogh (d) refer o he respecive reference h i khz f Fig. 8: Space vecor modlaion, harmonic specrm waveforms in Fig.. The space vecor modlaion exhibis a beer loss facor characerisic a m >.4 as he sboscillaion mehod wih sinsoidal reference waveforms. The reason becomes obvios when comparing he harmonic rajecories in Fig. 5. The zero vecor appears wice dring wo sbseqen sbcycles, and here is a shorer and a sbseqen larger porion of i in a complee harmonic paern of he sboscillaion mehod. Fig. 9 shows how he wo differen on-draions of he zero vecor are generaed. Agains ha, he ondraions of wo sbseqen zero vecors Fig. 5( are basically eqal in he case of space vecor modlaion. The conors of he harmonic paern come closer o he origin in his case, which redces he harmonic conen. The modified space vecor modlaion, crve (d) in Fig. 7, performs beer a higher modlaion index, and worse a m <.6. c) A ypical harmonic specrm prodced by he space vecor modlaion is shown in Fig. 8. The loss facor crves of synchronized carrier PW are shown in Fig. 9 for he sboscillaion echniqe and he space vecor modlaion. The laer appears sperior a low plse nmbers, he difference becoming less significan as N increases. The crves exhibi no differences a lower modlaion index. Operaing in his range is of lile pracical se for consan v/f loads where higher vales of N are permied and, above all, d decreases if m is redced (Fig. 7). The performance of a plsewidh modlaor based on sampling echniqes is slighly inferior han ha of he sboscillaion mehod, b only a low plse nmbers. Becase of he synchronism beween f and f s, he plse nmber ms necessarily change as he modlaion index varies over a broader range. Sch changes inrodce disconiniies o he modlaion process. They generally originae crren ransiens, especially when he plse nmber is low [5]. This effec is discssed in Secion N 6 N 6 d 6 m max m max m m Fig. 9: Synchronized carrier modlaion, loss facor d verss modlaion index; ( sboscillaion mehod, ( space vecor modlaion 4. Carrierless PW The ypical harmonic specrm of carrier based plsewidh modlaion exhibis prominen harmonic amplides arond he carrier freqency and is harmonics, Fig. 8. Increased acosic noise is generaed by he machine a hese freqencies hrogh he effecs of magneosricion. The vibraions can be amplified by mechanical resonances. To redce he mechanical exciaion a pariclar freqencies i may be preferable o have he harmonic energy disribed over a larger freqency range insead of being concenraed arond he carrier freqency. This concep is realized by varying he carrier freqency in a randomly manner. Applying his o he sboscillaion echniqe, he slopes of he rianglar carrier signal ms be mainained linear in order o conserve he linear inpop relaionship of he modlaor. Fig. shows how a random freqency carrier signal can be generaed. Whenever he carrier signal reaches one of is peak vales, is slope is reversed by a hyseresis elemen, and a sample is aken random generaor sample & hold cr Fig. : Random freqency carrier signal generaor

9 - 9 - Eqn. 6 selec ac k d w a b Re ac from a random signal generaor which imposes an addiional small variaion on he slope. This varies he draions of he sbcycles randomly [6]. The average swiching freqency is mainained consan sch ha he power devices are no exposed o changes in emperare. The opimal sbcycle mehod (Secion 6.4.) classifies also as carrierless. Anoher approach o carrierless PW is explained in Fig. ; i is based on he space vecor modlaion principle. Insead of operaing a consan sampling freqency f s as in Fig. (, samples of he reference vecor are aken whenever he draion ac of he swiching sae vecor ac erminaes. ac is deermined from he solion of acac + + f ac. () s fs, (6) where () is he reference vecor. This qaniy is differen from is ime discreized vale ( s ) sed in (. As () is a coninosly ime-variable signal, he on-draions,, and are differen from he vales (), which inrodces he desired variaions of sbcycle lenghs. Noe ha is anoher solion of (6), which is disregarded. The swiching sae vecors of a sbcycle are shown in Fig. (. Once he on-ime ac of ac has elapsed, a is chosen as ac for he nex swiching inerval, b becomes a, and he cyclic process sars again [7]. Fig. (c) gives an example of measred sbcycle draions in a fndamenal period. The comparison of he harmonic specra Fig. (d) and Fig. 8 demonsraes he absence of prononced specral componens in he harmonic crren. Carrierless PW eqalizes he specral disribion of he harmonic energy. The energy level is no redced. To lower he adible exciaion of mechanical resonances is a promising aspec. I remains difficl o decide, hogh, wheaher a clear, single one is beer olerable in is annoying effec han he radiaion of whie noise. 4. Overmodlaion I is apparen from he averaging approach of he space vecor modlaion echniqe ha he on-draion of he zero vecor (or 7 ) decreases as he modlaion index m increases. is firs reached a m m max, which means ha he circlar pah of he reference vecor oches he oer hexagon ha is opened p by he swiching sae vecors Fig. (. The conrollable range of linear modlaion mehods erminaes a his poin. An addiional singlar operaing poin exiss in he sixsep mode. I is characerized by he swiching seqence and he highes possible fndamenal op volage corresponding o m. Conrol in he inermediae range m max < m < can be achieved by overmodlaion [8]. I is expedien o consider a seqence of op volage vecors k, averaged over a sbcycle o become a single qaniy av, as he characerisic variable. Overmodlaion echniqes sbdivide ino wo differen modes. In mode I, he rajecory of he average volage vecor av follows a circle of radis m > m max as long as he circle arc is locaed wihin he hexagon; av six-sep mode m. T s T. 4 α PW m m max Re.8 c) p p w. 5 6 overmodlaion range h i.5 4 Re 4 d) 6 8 khz f Fig. : Carrierless plsewidh modlaion; ( signal flow diagram, ( swiching sae vecors of he firs 6 -secor, (c) measred sbcycle draions, (d) harmonic specrm 5 6 p -rajecory Fig. : Overmodlaion; ( definiion of he overmodlaion range, ( rajecory of av in overmodlaion range I

10 - - racks he hexagon sides in he remaining porions (Fig. (). Eqaions () are sed o derive he swiching draions while av is on he arc. On he hexagon sides, he draions are and cos α sin α a T cos α+ sin, (7 α b T a. (7 Overmodlaion mode II is reached a m > m max.95 when he lengh of he arcs redces o zero and he rajecory of av becomes prely hexagonal. In his mode, he velociy of he average volage vecor is conrolled along is linear rajecory by varying he dy cycle of he wo swiching sae vecors adjacen o av. As m increases, he velociy becomes gradally higher in he cener porion of he hexagon side, and lower near he corners. Overmodlaion mode II converges smoohly ino six-sep operaion when he velociy on he edges becomes infinie, he velociy a he corners zero. In mode II a sbcycle is made p by only wo swiching sae vecors. These are he wo vecors ha define he hexagon side on which av is raveling. Since he swiching freqency is normally mainained a consan vale, he sbcycle draion T ms redce de o he redced nmber of swiching sae vecors. This explaines why he disorion d.8.. m max m max.8.9 m Fig. : Loss facor d a overmodlaion (differen d scales) facor redces a he beginning of he overmodlaion range (Fig. ). The crren waveforms Fig. 4 demonsrae ha he modlaion index is increased beyond he limi exising a linear modlaion by he addiion of harmonic componens o he average volage av. The added harmonics do no form zero-seqence componens as hose discssed in Secion 4... Hence hey are flly refleced in he crren waveforms, which classifies overmodlaion as a nonlinear echniqe. 4.4 Opimized Open-loop PW PW inverers of higher power raing are operaed a very low swiching freqency o redce he swiching losses. Vales of a few Herz are csomary in he megawa range. If he choice is a open-loop echniqe, only synchronized plse schemes shold be employed here in order o avoid he generaion of excessive sbharmonic componens. The same applies for drive sysems operaing a high fndamenal freqency while he swiching freqency is in he lower kiloherz range. The plse nmber (5) is low in boh cases. There are only a few swiching insans k per fndamenal period, and small variaions of he respecive swiching angles α k π f. k have considerable inflence on he harmonic disorion of he machine crrens. I is advanageos in his siaion o deermine he finie nmber of swiching angles per fndamenal period by opimizaion procedres. Necessarily he fndamenal freqency ms be considered consan for he prpose of defining he opimizaion problem. A solion can be hen obained 8 A ms Fig. 4: Crren waveforms a overmodlaion; ( space vecor modlaion a m max, ( ransiion beween range I and range II, (c) overmodlaion range II, (d) operaion close o he six-sep mode off-line. The precalclaed opimal swiching paerns are sored in he drive conrol sysem o be rerieved dring operaion in real-ime [9]. The applicaion of his mehod is resriced o qaseadysae operaing condiions. Operaion in he ransien mode prodces waveform disorions worse han wih nonopimal mehods (Secion 5..). The bes opimizaion resls are achieved wih swiching seqences having odd plse nmbers and qarer-wave symmery. Off-line schemes can be classified wih respec o he opimizaion objecive [] Harmonic eliminaion This echniqe aims a he eliminaion of a well defined nmber n (N )/ of lower order harmonics from he discree Forier specrm. I eliminaes all orqe harmonics having 6 imes he fndamenal freqency a N 5, or 6 and imes he fndamenal freqency a N 7, and so on []. The mehod can be applied when specific harmonic freqencies in he machine orqe ms be avoided in order o preven resonan exciaion of he driven mechanical sysem (moor shaf, coplings, gears, load). The approach is sbopimal as regards oher performance crieria Objecive fncions An acceped approach is he minimizaion of he loss facor d [], where d is defined by () and (4). Alernaively, he highes peak vale of he phase crren can be considered a qaniy o be minimized a very low plse nmbers []. The maximm efficiency of he inverer/ machine sysem is anoher opimizaion objecive [4]. The objecive fncion ha defines a pariclar opimizaion problem ends o exhibi a very large nmber of local minimms. This makes he nmerical solion exremely ime consming, even on oday s modern compers. A se of swiching angles which minimize he harmonic crren (d min) is shown in Fig. 5. Fig. 6 compares he performance of a d min scheme a Hz swiching freqency wih he sboscillaion mehod and he space vecor modlaion mehod. c) d)

11 - - α k α k N 5 N 7 N 9 N.5 m Fig. 5: Opimal swiching angles; N: plse nmber.5 m 4.4. Opimal Sbcycle ehod This mehod considers he draions of swiching sbcycles as opimizaion variables, a sbcycle being he ime seqence of hree consecive swiching sae vecors. The seqence is arranged sch ha he insananeos disorion crren eqals zero a he beginning and a he end of he sbcycle. This enables he composiion of he swiched waveforms from a precalclaed se of opimal sbcycles in any desired seqence wiho casing ndesired crren ransiens nder dynamic operaing condiions. The approach eliminaes a basic deficiency of he opimal plsewidh modlaion echniqes ha are based on precalclaed swiching angles [5]. A signal flow diagram of an opimal sbcycle modlaor d f s Hz Hz m is shown in Fig. 7(. Samples of he reference vecor are aken a s, whenever he previos sbcycle erminaes. The ime draion T s () of he nex sbcycle is hen read from a able which conains off-line opimized daa as displayed in Fig. 7(. The crves show ha he sbcycles enlarge as he reference vecor comes closer o one of he acive swiching sae vecors, boh in magnide as in phase angle. This implies ha he opimizaion is only worhwhile in he pper modlaion range. The modlaion process iself is based on he space vecor approach, aking ino accon ha he sbcycle lengh is variable. Hence T s replaces T / f s in (). A prediced vale ( s +/ T s (( s )) is sed o deermine he on-imes. The predicion assmes ha he fndamenal freqency does c) 4 Hz 8 Hz Fig. 6: Loss facor d of synchronos opimal PW, crve (; for comparison a f s Hz: ( space vecor modlaion, (c) sboscillaion mehod T s T s+t(k) ( s) T(k) f s khz Eqn. selec * s Fig. 7: Opimal sbcycle PW; ( signal flow diagram, ( sbcycle draion verss fndamenal phase angle i() a b ( s + T(k)) k m π/6 π/ π/ arg( ) no change dring a sbcycle. I eliminaes he perrbaions of he fndamenal phase angle ha wold resl from sampling a variable ime inervals. The performance of he opimal sbcycle mehod is compared wih he space vecor modlaion echniqe in Fig. 8. The Forier specrm lacks dominan carrier freqencies, which redces he radiaion of acosic noise from conneced loads. 5 A i() 5 A 5 5 Fig. 8: Crren rajecories; ( space vecor modlaion, ( opimal sbcycle modlaion d 4.5 Swiching condiions I was assmed nil now ha he inverer swiches behave ideally. This is no re for almos all ypes of semicondcor swiches. The devices reac delayed o heir conrol signals a rn-on and rn-off. The delay imes depend on he ype of semicondcor, on is crren and volage raing, on he conroling waveforms a he gae elecrode, on he device emperare, and on he acal crren o be swiched inimm draion of swiching saes In order o avoid nnecessary swiching losses of he devices, allowance ms be made by he conrol logic for minimm ime draions in he on-sae and he off-sae, respecively. An addiional ime margin ms be inclded

12 - - so as o allow he snbber circis o energize or deenergize. The resling minimm on-draion of a swiching sae vecor is of he order - µs. If he commanded vale in an open-loop modlaor is less han he reqired minimm, he respecive swiching sae ms be eiher exended in ime or skipped (plse dropping [6]). This cases addiional crren waveform disorions, and also consies a limiaion of he maximm modlaion index. The overmodlaion echniqes described in Secion 4. avoid sch limiaions Dead-ime effec inoriy-carrier devices in pariclar have heir rn-off delayed owing o he sorage effec. The sorage ime T s varies wih he crren and he device emperare. To avoid shor-circis of he inverer half-bridges, a lock-o ime T d ms be inrodced by he inverer conrol. The lock-o ime cons from he ime insan a which one semicondcor swich in a half-bridge rns off and erminaes when he opposie swich is rned on. The lock-o ime T d is deermined as he maximm vale of sorage ime T s pls an addiional safey ime inerval. U d k k k T d T T T d T on T s D D T off ph ph We have now wo differen siaions, displayed in Fig. 9( for posiive load crren in a bridge leg. When he modlaor op signal k goes high, he base drive signal k of T ges delayed by T d, and so does he reversal of he phase volage ph. If he modlaor op signal k goes low, he base drive signal k is immediaely made zero, b he acal rn-off of T is delayed by he device sorage ime T s < T d. Conseqenly, he on-ime of he pper bridge arm does no las as long as commanded by he conroling signal k. I is decreased by he ime difference T d T s, [7]. A similar effec occrs a negaive crren polariy. Fig. 9( shows ha he on-ime of he pper bridge arm is now increased by T d T s. Hence, he acal dy cycle of he half-bridge is always differen from ha of he conrolling signal k. I is eiher increased or decreased, depending on he load crren polariy. The effec is described by T d Ts av * ; sig is, (8) Ts where av is he inverer op volage vecor averaged over a sbcycle, and is a normalized error vecor aribed o he swiching delay of he inverer. The error magnide is proporional o he acal safey ime margin T d T s ; is U d T T D D k k k T d T s T d T on T off Fig. 9: Inverer swiching delay; ( posiive load crren, ( negaive load crren direcion changes in discree seps, depending on he respecive polariies of he hree phase crrens. This is expressed in (8) by a polariy vecor of consan magnide sig is sign( is+ a. sign( is+ a. sign( isc), (9) where a exp(jπ/) and is he crren vecor. The noaion sig( ) was chosen o indicae ha his complex nonlinear fncion exhibis properies of a sign fncion. The graph sig( ) is shown in Fig. ( for all possible vales of he crren vecor. The hree phase crrens are denoed as i as, i bs, and i cs. > ib < [ ] The dead-ime effec described by (8) and (9) prodces a nonlinear disorion of he average volage vecor rajecory av. Fig. ( shows an example. The disorion does no depend on he magnide of he fndamenal volage and hence is relaive inflence is very srong in he lower speed range where is small. Since he fndamenal freqency is low in his range, he smoohing acion of he load circi indcance has lile effec on he crren waveforms, and he sdden volage changes become clearly visible, Fig. (. The machine orqe is inflenced as well, exhibiing dips in magnide a six imes he fndamenal freqency in he seady-sae. Elecromechanical sabiliy problems may resl if his freqency is sfficienly low. Sch case is illsraed in Fig., showing one phase crren and he speed signal in permanen insabiliy. i() < i a sig( ) > > i c < Fig. : Dead-ime effec; ( locaion of he polariy vecor sig(i), ( rajecory of he disored average volage av 4 A i() 4 A av Fig. : Dead-ime effec; ( measred crren rajecory wih sixh harmonic and redced fndamenal, ( as in (, wih dead-ime compensaion 4.5. Dead-ime compensaion If he plsewidh modlaor and he inverer form par of a sperimposed high-bandwidh crren conrol loop, he crren waveform disorions cased by he dead-ime effec are compensaed o a cerain exen. This may elimi- sig(is) Re

13 - - Fig. : Elecromechanical insabiliy de o he dead-ime effec nae he need for a separae dead-ime compensaor. A compensaor is reqired when fas crren conrol is no available, or when he machine orqe ms be very smooh. Dead-ime compensaors can be implemened in hardware or in sofware. k 5 A a & & s down SB p clock k' a w T k k sign( ph ) min Fig. : Dead-ime compensaion; ( compensaion and delay circi per phase, ( signal waveforms; T: inerlock circi The hardware compensaor Fig. ( operaes by closedloop conrol [8]. Idenical circis are provided for each bridge leg. Each compensaor forces a consan ime delay beween he logic op signal k of he plse modlaor and he acal swiching insan. To achieve his, he insan a which he phase volage changes is measred a he inverer op. A logic signal sign( ph ) is obained ha is fed back o conrol an p-down coner, which in rn conrols he bridge: A posiive con conrols a negaive phase volage, and vice versa. Fig. ( shows he signals a posiive load crren. The half-bridge op is negaive a he beginning, and sign( ph ). The coner holds he measred sorage ime T s of he previos commaion. I sars downconing a fixed clock rae when he modlaor op k rns high. The inverer conrol logic receives he on-signal k afer T s, and hen insers he locko ime T d before k rns he bridge on. The oal ime delay of he rn-on process amons o T d + T s, and an idenical delay is provided for he rn-off process. The swiching seqence ges delayed in ime, b is dy-cycle is conserved. When T s changes following a change of he crren polariy, he iniial con of T s is wrongly se, and he nex commaion ges displaced. Thereafer, he dy-cycle is k con k' k sign( ph ) T s T d T d w T s again mainained as he coner sars wih a revised vale of T s. Sofware compensaors are mosly designed in he feedforward mode. This eliminaes he need for poenial-free measremen of he inverer op volages. Depending on he sign of he respecive phase crren, a fixed delay ime T s is eiher added or no o he conrol signal of he halfbridge. As he acal sorage delay T s is no known, a complee compensaion of he dead-ime effec may no be achieved. The changes of he error volage vecor ac as sdden disrbances on he crren conrol loop. They are compensaed only a he nex swiching of he phase leg. The remaining ransien error is mosly olerable in indcion moor drive sysems; synchronos machines having sinsoidal back-emf behave more sensiively o hese effecs as hey end o operae parly in he disconinos crren mode a ligh loads. The reason for his adverse effec is he absence of a magneizing crren componen in he saor crrens. Sch machines reqire more elaborae swiching delay compensaion schemes when applied o high-performance moion conrol sysems. Alernaively, a d-axis crren componen can be injeced ino he machine o shoren he disconinos crren ime inervals a ligh loads [9]. 5. CLOSED-LOOP PW CONTROL Closed-loop PW schemes generae he swiching seqences inherenly in a closed conrol loop, Fig. 6(. The feedback loop is esablished eiher for he saor crren vecor or for he saor flx vecor. The conrol is generally fas enogh o compensae he nonlinear effecs of plse dropping and variable swiching delay. 5. Nonopimal mehods 5.. Hyseresis crren conrol The signal flow diagram in Fig. 4( shows hree hyseresis conrollers, one for each phase. Each conroller deermines he swiching sae of one inverer leg sch ha he error of he corresponding phase crren is mainained wihin he hyseresis bandwidh + i. The conrol mehod is simple o implemen, and is dynamic performance is excellen. There are some inheren drawbacks, hogh [4]: There is no inercommnicaion beween he individal hyseresis conrollers of he hree phases and hence no sraegy o generae zero volage vecors. This increases he swiching freqency a lower modlaion index. * i i limi cycle doble error i* Fig. 4: Hyseresis crren conrol; ( signal flow diagram, ( basic crren waveform- d s