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1 SPACE ECTOR MODULATION FOR THREE-LEG OLTAGE SOURCE INERTERS.1 THREE-LEG OLTAGE SOURCE INERTER The toology of three-leg voltge soure iverter is show i Fig..1. Beuse of the ostrit tht the iut lies must ever e shorted d the outut urret must lwys e otiuous voltge soure iverter ssume oly eight distit toologies. These toologies re show o Fig... Six out of these eight toologies rodue ozero outut voltge d re kow s o-zero swithig sttes d the remiig two toologies rodue zero outut voltge d re kow s zero swithig sttes. g L I C B A Fig..1. Toology of three-leg voltge soure iverter. 4

2 1() () 3() 4() 5() 6() 7() 8() Fig... Eight swithig stte toologies of voltge soure iverter.. OLTAGE SPACE ECTORS Se vetor modultio (SM) for three-leg SI is sed o the reresettio of the three hse qutities s vetors i two-dimesiol (α, β) le. This hs ee disussed i [1] d is illustrted here for the ske of omleteess. Cosiderig toology 1 of Fig.., whih is reeted i Fig..3 () we see tht the lie voltges,, d re give y 5

3 = g = 0.. (.1) = - g This e rereseted i the α, βle s show i Fig..3(), where voltges,, d re three lie voltge vetors disled 10 i se. The effetive voltge vetor geerted y this toology is rereseted s 1() i Fig..3(). Here the ottio refers to the three legs/hses,, eig either oeted to the ositive d ril () or to the egtive d ril (). Thus orresods to hse eig oeted to the ositive d ril d hses d eig oeted to the egtive d ril g Fig..3(). Toology 1-1() of voltge soure iverter. = g = 0 = - g - g - 1() g Fig..3(). Reresettio of toology 1 i the α, βle. 6

4 Proeedig o similr lies the six o-zero voltge vetors (1-6) e show to ssume the ositios show i Fig..4. The tis of these vetors form regulr hexgo (dotted lie i Fig..4). We defie the re elosed y two djet vetors, withi the hexgo, s setor. Thus there re six setors umered 1-6 i Fig Fig..4. No-zero voltge vetors i the α, βle. Cosiderig the lst two toologies of Fig.. whih re reeted i Fig..5() for the ske of oveiee we see tht the outut lie voltges geerted y this toology re give y = 0 = 0 = 0.. (.) These re rereseted s vetors whih hve zero mgitude d hee re referred to s zero-swithig stte vetors or zero voltge vetors. They ssume the ositio t origi i the α, βle s show i Fig..5(). The vetors 1-8 re lled the swithig stte vetors (SSs). 7

5 7() 8() Fig..5(). Zero outut voltge toologies. = 0 = 0 = 0 7, 8 Fig..5(). Reresettio of the zero voltge vetors i the α, βle..3 SPACE ECTOR MODULATION The desired three hse voltges t the outut of the iverter ould e rereseted y equivlet vetor rottig i the outer lok wise diretio s show i Fig..6(). The mgitude of this vetor is relted to the mgitude of the outut voltge (Fig..6()) d the time this vetor tkes to omlete oe revolutio is the sme s the fudmetl time eriod of the outut voltge. 8

6 Fig..6(). Outut voltge vetor i the α, βle Fig..6(). Outut lie voltges i time domi. Let us osider the situtio whe the desired lie-to-lie outut voltge vetor is i setor 1 s show i Fig..7. This vetor ould e sythesized y the ulse-widthmodultio (PWM) of the two djet SS s 1() d (), the duty yle of eh eig d 1 d d, resetively, d the zero vetor ( 7() / 8() ) of duty yle d 0 [1]: d + d = = m e g jè (.3) d + d + d 1.. (.4) 1 0 = where, 0 m , is the modultio idex. This would orresod to mximum lie-to-lie voltge of 1.0 g, whih is 15% more th ovetiol siusoidl PWM s show [1]. 9

7 () 7() 8() d d 1 θ 1 1() Fig..7. Sythesis of the required outut voltge vetor i setor 1. All SM shemes d most of the other PWM lgorithms [1,4], use (.3), (.4) for the outut voltge sythesis. The modultio lgorithms tht use o-djet SS s hve ee show to rodue higher THD d/or swithig losses d re ot lyzed here, lthough some of them, e.g. hysteresis, e very simle to imlemet d rovide fster trsiet resose. The duty yles d 1, d, d d 0, re uiquely determied from Fig..7, d (.3), (.4), the oly differee etwee PWM shemes tht use djet vetors is the hoie of the zero vetor(s) d the sequee i whih the vetors re lied withi the swithig yle. The degrees of freedom we hve i the hoie of give modultio lgorithm re: 1) The hoie of the zero vetor - whether we would like to use 7() or 8() or oth, ) Sequeig of the vetors 3) Slittig of the duty yles of the vetors without itroduig dditiol ommuttios. Four suh SM lgorithms re osidered i the ext setio, mely: 1) The right liged sequee ( SM1) 10

8 ) The symmetri sequee (SM) 3) The ltertig zero vetor sequee ( SM3) 4) The highest urret ot swithed sequee (SM4). The modultio shemes re desried for the se whe the referee vetor is i setor 1: ll other ses re irulrly symmetri. These modultio shemes re lyzed d their reltive erforme with reset to swithig loss, THD d the ek-to-ek urret rile t the outut is ssessed. The lysis is erformed over the etire rge of modultio idex d for lod ower ftor gle vryig from 90 o to 90 o. All se vetor modultio shemes reseted here ssume digitl imlemettio d, hee, regulr smlig, i.e. ll duty yles re relulted t the egiig of the swithig yle, sed o the vlue of the referee voltge vetor t tht istt..4 MODULATION SCHEMES.4.1 RIGHT ALIGNED SEQUENCE (SM1) A simle wy to sythesize the outut voltge vetor is to tur-o ll the ottom (or to) swithes t the egiig of swithig yle d the to tur them off sequetilly so tht the zero vetor is slit etwee 7() d 8() eqully. This swithig sheme is show i Fig..8 for two smlig eriods. The sigls i the figure rereset the gtig sigls to the uer legs of the iverter. The sheme hs three swith tur-o s d three swith tur-off s withi swithig yle. The erforme of the left liged sequee, where the sequee of vetors is extly oosite to the right liged sequee, is exeted to e similr to the right liged sequee. 11

9 d 0 / d 1 d d 0 / T s d 0 / d 1 d d 0 / T s Fig..8. Phse gtig sigls i SM1..4. Symmetri Sequee (SM) This sheme hs ee show i revious works [4] to hve the lowest THD. This is euse of the symmetry i the swithig wveform s e see i Fig..9. The umer of ommuttios i oe smlig eriod is six. Sie this sheme hs the sme umer of swithigs s SM1, with three swith tur-os d three swith tur-offs, their swithig losses re exeted to e similr. d 0 4 d 1 d d 0 T s d d 1 d 0 4 d 0 4 d 1 d d 0 T s d d 1 d 0 4 Fig..9. Phse gtig sigls i SM. 1

10 .4.3 Altertig Zero etor Sequee (SM3) I this sheme, kow s DI sequee i literture [6], the zero vetors 7() d 8() re used ltertively i djet yles so tht the effetive swithig frequey is hlved, s show i Fig..10. d 1 d d 0 T s d d 1 d 0 T s Fig..10. Phse gtig sigls i SM3. However, the smlig eriod is still T s, sme s i the other shemes. The swithig losses for this sheme re exeted to e idelly 50% s omred to those of the revious two shemes d THD sigifitly higher due to the existee of the hrmois t hlf of the smlig frequey..4.4 Highest Curret Not-Swithed Sequee (SM4) This sheme, kow s DD sequee i literture [6], is sed o the ft tht the swithig losses re roximtely roortiol to the mgitude of the urret eig swithed d hee it would e dvtgeous to void swithig the iverter leg rryig the highest istteous urret. This is ossile i most ses, euse ll djet SS s differ i the stte of swithes i oly oe leg. Hee, y usig oly oe zero vetor, 7() or 8() withi give setor oe of the legs does ot hve to e swithed t ll, s show i Fig..11 (). 13

11 T s T s d 1 d d 0 d 1 d d 0 Fig..11(). Phse gtig sigls i SM4. However, sie the hoie of the o-zero SSs is sed o the desired outut voltge vetor d the hse d mgitude of the urret re determied y the lod, it is ot lwys ossile to void swithig the hse rryig the highest urret. I suh se the hse rryig the seod highest urret is ot swithed d the swithig losses re still redued. For exmle, the hoie of zero vetors i setor 1 is determied usig the flowhrt i Fig..11(). Y I > I d 0 7() N d 0 8() Fig..11(). Choie of zero vetor i setor 1..5 ANALYSIS I this setio the THD, swithig losses d ek-to-ek urret rile of ll the shemes re lyzed over the etire rge of modultio idex, d over vryig lod ower ftor gles. 14

12 .5.1 Totl Hrmoi Distortio The totl hrmoi distortio (THD) of eriodi voltge whih e rereseted y the Fourier series = e = 1 jωt is defied s THD = =.. (.5) 1 d its weighted totl hrmoi distortio is defied s WTHD = =.. (.6) 1 The lysis of THD is doe sed o ovel lgorithm where the Fourier oeffiiets of ll the ulses of give lie/hse voltge i oe fudmetl time eriod re summed u. The lysis is vlid for ll itegrl f s /f o, where f s is the swithig frequey d f o is the fudmetl frequey t the outut of the iverter. Fig..1 shows the lot of tyil hse voltge i oe fudmetl time eriod. The THD of this wveform is lulted y first deomosig the wveform of Fig..1 ito series is ulses whih resemle Fig..13. Phse Phse oltge oltge T s T o Time Time Fig..1. Tyil Phse voltge i oe fudmetl time eriod. 15

13 The the Fourier omoets for eh of these ulses re lulted d summed u to get the effetive Fourier omoets of the etire wveform i oe fudmetl time eriod. This roedure is illustrted here for SM1. Cosiderig ulse show i Fig..13 with fudmetl eriod T o = 1/f o (Fig..1) its Fourier oeffiiets t frequey f o re give y m π = (si( ( dm Ts + T π T o m Tm π )) si( T o )).. (.7) m Tm π π = ( os( ) os( ( dm Ts + Tm ))) π T T o o.. (.8) oltge m T m d m Ts T s Time Fig..13. Outut voltge ulse. where d m is the ulse duty yle, T m is the ulse dely time, m is the ulse mgitude, d T s = 1, is the smlig time eriod. f s Deomositio of the hse voltge swithig wveform for SM1 (Fig..13) to oti series of ulses s i Fig..1 is two ste roess. At first the wveform is deomosed to oti steed ulses s show i Fig..14(), the this steed ulse wveform is further deomosed to oti ulses, similr to Fig..13, s show i Fig..14() d Fig..14(). 16

14 Cosider the lod to e wye oeted d led. The the hse voltges, with reset to eutrl oit (N) hose suh tht + + =0 i y setor for y lod e otied from = - 3, = - 3, = - 3 The hse voltge () i setor1 durig differet SSs (8,1,,7) for SM1 is show i Tle.1 Tle.1. Phse voltge i setor 1: SM1. SS s 8() 1() () 7() duty yle d 0 / d 1 d d 0 / 0 g /3 g /3 0 Fig..14 shows the deomositio of tyil hse voltge i the k th swithig itervl from the egiig of the hse voltge eriod, where f s 0 k (.9) f o g /3 g /3 g /3 g / d 0 / d 1 d d 0 / d 0 / d 1 d 0 /+d d 0 /+d 1 d d 0 / (k-1)t s kt s (k-1)t s kt s (k-1)t s kt s () () () first ulse Fig..14. Phse voltge deomositio i oe smlig eriod. Comrig eh of the omoet ulses with Fig..13 oe fid tht for the 17

15 dm = d 1, Tm = k - g m =, 3 d + T s, (.10) d for the seod ulse d m m = d, T g = 3 m d0 = k d1 T, s.. (.11) The duty yles d 1, d d d 0 re lulted usig (.3) d (.4). The Fourier omoets of these ulses re otied from (.7) d (.8). Thus the Fourier oeffiiets of the hse voltge e foud usig: = fs -1 f o k = 0 k + fs -1 f o k = 0 k.. (.1) The Fourier oeffiiets of the lie urret e otied y I =.. (.13) lie _ urret z where z is the imede of give hse. Altertively, WTHD of lie voltge ould rereset THD of lie urret [,3]. Fig..15 shows the THD of the simulted lie urret d lie voltge d the WTHD of lie voltge for ll the shemes over the etire rge of modultio idex, for the iverter rmeters g = 400, f o = 60 Hz, f s = 160, R(lod) = 3, L(lod) = 1mH. Aedix B lists the MATLAB ode, used to geerte the redited results i Fig

16 Modultio idex(m) () THD THD THD THD 0 Modultio idex(m) () Modultio idex(m) () Fig..15. oltge d urret distortio s futio of modultio idex. ) THD of lie urret 19

17 ) WTHD of lie urret ) THD of lie voltge These results gree with wht is foud i literture. It is iterestig to ote tht the THD (lie urret d voltge) for ll the shemes dereses with irese i modultio idex. This is euse of the irese i the fudmetl omoet of the voltge/urret with irese i modultio idex; the other higher order hrmois eig reltively ostt. It lso e see tht SM (symmeti) hs the lest THD. This e ssoited with the symmetry i the swithig wveform. By itroduig symmetry (SM), the umer of hse voltge ulses i give swithig time-eriod is douled s omred to the other shemes. This would mke the overter look s if it were oertig t twie the swithig frequey. Hee this results i redued THD d redued ek-to-ek rile i the lod urret..5.. Swithig Losses The swithig losses re ssumed to e roortiol to the rodut of the voltge ross the swith d the urret through the swith t the istt of swithig. Sie the voltge ross the swith is the us voltge, it is osidered to e ostt. Thus the losses re roortiol to the urret durig swithig. As first roximtio, the swithig rile is egleted d the losses re estimted sed o the umer of ommuttios required for eh swithig sheme d the urret t the istt of swithig. Losses for sheme SM4 re deedet o the lod ower ftor d their loss erforme hs ee otimized usig the flow hrt 0

18 reseted i Fig..11() for the etire rge of lod ower ftor gle. Fig..16, shows the loss erforme hrteristis of ll the shemes. From Fig..16 we see tht SM4 hs 50 % redutio i losses t high lod ower ftors (>0.866) d the svigs i the losses redue to 37% t low lod ower ftors. Reltive Reltive swithig losses swithig loss Lod Lod ower ftor gle Fig..16. Reltive swithig losses s futio of lod ower ftor gle Pek-to-Pek Curret Rile The ek-to-ek vlue of the urret rile, t mximum vlue of lod urret, is imortt i the desig of idutors - to e used s filters. Some simle formule hve ee derived elow for the urte estimtio of the rile for ll the shemes. Mximum outut urret rile ours whe the volt-seod ross the outut idutors is the lrgest. Cosider hse i Fig..1 d ssume tht outut voltge A (Fig..1) vries slowly with reset to the swithig frequey. The for ssive lod, the mximum volt-seod ross the idutor i hse ours whe the referee vetor i Fig..7 is ollier with the swithig vetor 1(). At this istt the duty yle d = 0 d duty yle d 1 = m, d the resultt hse voltge will e s show i Fig

19 Phse voltge m v d 1 = m d 0 = 1- m d 1 T s T s d 0 T s Time Fig..17. Phse voltge ulse t its ek low frequey vlue. Due to the ssumtio tht the outut voltge A (Fig..1) is ostt, the voltge ross the idutor is Sie d L A ( ) v = (.14) = 3 d v 1 g ; ( ) m m =.. (.15) ( ) v I = L d T 0 s.. (.16) Thus the ek-to-ek urret rile is give y I = g 3L ( 1- m) m T s.. (.17) Equtio (17) is true for SM1, SM3, d SM4. For SM the duty yle d 0 is slit i hlf d hee I g = 3L 1 ( -m) m T s.. (.18)

20 Fig..18 shows the lot of the reltive ek-to-ek rile urret, sed o the ove lysis d sed o the simultio results usig SABER, for the etire rge of modultio idex. Reltive ek-to-ek rile 0 Modultio idex (m) Fig..18. Reltive ek-to-ek urret rile..6 Simultio d Exerimetl Results The iruit i Fig..1 ws simulted usig SABER. Aedix A gives the listig of the se vetor modultio temlte develoed i MAST for SM1. The iruit i Fig..1 ws simulted usig idel swithes, RCD suers, d idel diodes, usig the followig rmeters: g = 16, I lod = 6.5 A, f o = 108 Hz, f s = 3888 Hz, m = 0.6. The outut urret wveforms d their setr re show i Fig..19. Fig..0 shows the wveforms of the outut urret d setrum of outut urret otied from exerimetl iverter ruig uder similr oditios. The figures show firly good greemet etwee simultio d exerimetl results. 3

21 SM1 SM SM3 SM4 Fig..19. Simulted lie urrets d setrum of lie urrets ( f s /f o = 36). 4

22 Fig..0. Exerimetl lie urrets d setrum of lie urrets (f s /f o = 36). 5

23 .7 PERFORMANCE SUMMARY Tle. summrizes the erforme of four se vetor modultio shemes. It e lerly see tht sheme with high THD hs low losses d vie vers d these hrteristis re lod deedet. This ould e trslted s trde-off to e mde etwee the size of the het sik d size of the filters. At low swithig frequeies SM ould e used, sie losses re ot very ritil. At high swithig frequeies SM4 is referred, eseilly t high lod ower ftors due to the 50% redutio i swithig losses. SM3 ould e used t low lod ower ftors. The right liged sequee (SM1) (or the left-liged sequee) does ot seem to hve y rtiulr dvtge if the overter is hrd swithed. However, if softswithig is itrodued the this sheme is rtiulrly useful euse here ll the three legs re eig swithed t the sme time. It hs lso ee oserved tht the erforme of SM4 e imroved y itroduig symmetry. This result s i redued THD d i redued ek-to-ek rile i the lod urret; with the swithig losses remiig the sme.. Tle.. Reltive erforme of vrious modultio shemes (three-leg). Modultio Shemes SM1 SM SM3 SM4 No of ommuttios i T s Reltive Losses ** Domit hrmoi f s f s f s / f s THD t low mod idex Lest THD t high mod. idex Lest Highest Reltive ek-to-ek rile t I mx Numer of swithig sttes i T s ** Deedig o the lod ower ftor * Imortt for digitl modultor imlemettio 6

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