output voltage and are known as nonzero switching states and the remaining two


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1 SPACE ECTOR MODULATION FOR THREELEG OLTAGE SOURCE INERTERS.1 THREELEG OLTAGE SOURCE INERTER The toology of threeleg 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 ozero 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 threeleg 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 threeleg SI is sed o the reresettio of the three hse qutities s vetors i twodimesiol (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 11() 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 ozero voltge vetors (16) 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 16 i Fig Fig..4. Nozero 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 zeroswithig stte vetors or zero voltge vetors. They ssume the ositio t origi i the α, βle s show i Fig..5(). The vetors 18 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 lietolie outut voltge vetor is i setor 1 s show i Fig..7. This vetor ould e sythesized y the ulsewidthmodultio (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 lietolie 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 odjet 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 ektoek 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 turo 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 turo s d three swith turoff 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 turos d three swith turoffs, 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 NotSwithed 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 ozero 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 ektoek 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 / (k1)t s kt s (k1)t s kt s (k1)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 timeeriod 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 ektoek 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 PektoPek Curret Rile The ektoek 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 voltseod 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 voltseod 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 ektoek 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 ektoek rile urret, sed o the ove lysis d sed o the simultio results usig SABER, for the etire rge of modultio idex. Reltive ektoek rile 0 Modultio idex (m) Fig..18. Reltive ektoek 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 trdeoff 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 leftliged 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 ektoek rile i the lod urret; with the swithig losses remiig the sme.. Tle.. Reltive erforme of vrious modultio shemes (threeleg). 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 ektoek 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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