Power Inverters. The distortion factor for an individual harmonic is. µ = = (14.2) nv n (14.3)

Transcription

1 4 Pwr Elctric 4 Pwr Ivrtr Ivri i th cvri f dc pwr t ac pwr at a dird utput vltag r currt ad frqucy. A tatic ivrtr circuit prfrm thi trafrmati. Th trm vltag-fd ad currt-fd ar ud i ccti with ivrtr circuit. A vltag-fd ivrtr i i which th dc iput vltag i tially ctat ad idpdt f th lad currt draw. Th ivrtr pcifi th lad vltag whil th draw currt hap i dictatd by th lad. A currt-fd ivrtr (r currt urc ivrtr) i i which th urc, hc th lad currt i prdtrmid ad th lad impdac dtrmi th utput vltag. Th upply currt cat chag quickly. Thi i achivd by ri dc upply iductac which prvt udd chag i currt. Th lad currt magitud i ctrlld by varyig th iput dc vltag t th larg iductac, hc ivrtr rp t lad chag i lw. Big a currt urc, th ivrtr ca urviv a utput hrt circuit thrby ffrig fault rid-thrugh prprti. Vltag ctrl may b rquird t maitai a fixd utput vltag wh th dc iput vltag rgulati i pr, r t ctrl pwr t a lad. Th ivrtr ad it utput ca b igl-pha, thr-pha r multi-pha. Variabl utput frqucy may b rquird fr ac mtr pd ctrl whr, i cjucti with vltag r currt ctrl, ctat mtr flux ca b maitaid. Ivrtr utput wavfrm ar uually rctiliar i atur ad a uch ctai harmic which may lad t rducd lad fficicy ad prfrmac. ad harmic rducti ca b achivd by ithr filtrig, lctd harmicrducti chppig r pul-width mdulati. Th quality f a ivrtr utput i rmally valuatd i trm f it harmic factr, ρ, ditrti factr, µ, ad ttal harmic ditrti, thd. I cti.6. th firt tw factr wr dfid i trm f th upply currt. Fr ivrtr th factr ar rdfid i trm f th utput vltag harmic a fllw V ρ = = µ > (4.) V Th ditrti factr fr a idividual harmic i V ρ µ = = (4.) V V ρ thd = / V µ = = (4.) Th factr V / i ud ic th harmic currt prducd i a iductiv lad attuat with frqucy. Th harmic currt prduc uwatd hatig ad trqu cillati i ac mtr, althugh uch harmic currt ar t a drawback t th pwr dlivrd t a ritiv hatig lad. 4. dc-t-ac vltag-fd ivrtr bridg tplgi 4.. Sigl-pha vltag-fd ivrtr bridg Figur 4.la hw a H-bridg ivrtr fr prducig a ac vltag ad mplyig witch which may b traitr (MOSFET r IGBT), r at high pwr, thyritr (GTO r CGT). Dvic cducti pattr ar al hw i figur 4.lb ad c. With iductiv lad (t purly ritiv), trd rgy at tur-ff i fd thrugh th bridg ractiv fdback r frwhl did D t D 4. Th fur did clamp th lad vltag t withi th dc upply rail ( t V ). 4..i - Squar-wav utput Figur 4.lb hw wavfrm fr a quar-wav utput (t =t ) whr ach dvic i turd a apprpriat fr 8, (that i ) f th utput vltag cycl (tat quc,,,..). Th lad currt i grw xptially thrugh T ad T (tat ) accrdig t di V = i dt (V) (4.4) Wh T ad T ar turd ff, T ad T 4 ar turd (tat ), thrby rvrig th lad vltag. Bcau f th iductiv atur f th lad, th lad currt cat rvr ad lad ractiv rgy flw back it th upply via did D ad D 4 (which ar i paralll with T ad T 4 rpctivly) accrdig t di V = i dt (V) (4.5) Th lad currt fall xptially ad at zr, T ad T 4 bcm frward-biad ad cduct lad currt, thrby fdig pwr t th lad. Th utput vltag i a quar wav f magitud ± V ad ha a rm valu f V. Fr a impl - lad, with tim ctat τ = /, durig th firt cycl with iitial lad currt, lvig quati (4.4) yild a lad currt t τ i () t = (A) (4.6) Udr tady-tat lad cditi, th iitial currt i I a hw i figur 4.lb, ad quati (4.4) yild

2 Pwr Ivrtr Pwr Elctric fr I I Figur 4.. GCT thyritr igl-pha bridg ivrtr: (a) circuit diagram; (b) quar-wav utput vltag; ad (c) quai-quar-wav utput vltag. t τ i () t = I (A) t t () v = V (V) I (A) a (4.7) Durig th cd half-cycl (t t t ) wh th upply i ffctivly rvrd acr th lad, quati (4.5) yild Vab I b I t τ i () t = I (A) (4.8) t t t () fr v = V (V) I (A) A w tim axi ha b ud i quati (4.8) tartig at t = t i figur 4.lb. Sic i tady-tat I = -I, th iitial tady-tat currt I ca b fud frm quati (4.7) wh, at t = t, i = I yildig t τ t I = -I = = tah (A) t τ τ (4.9) Th zr currt cr-vr pit t x, hw figur 4.b, ca b fud by lvig quati (4.7) fr t wh i =, which yild I t = τ x V (4.) I = τ () V Th avrag thyritr currt, I TH, avrag did currt, I D, ad ma urc currt, I ca b fud by itgrati f th lad currt vr th apprpriatd bud. t IT = i () t dt t tx (4.) t t V V τ τ = ( t t ) τ I t whr i i giv by quati (4.7) ad tx I D = i () t dt t (4.) tx V V τ = t τ I x t whr i i giv by quati (4.8) Ipcti f th urc currt wavfrm i figur 4.b hw that th avrag urc currt i rlatd t th avrag micductr dvic currt by I = ( ITH I D ) t V V (4.) τ = t τ I t Th tady-tat ma pwr dlivrd by th dc upply ad abrbd by th ritiv lad cmpt i giv by

3 Pwr Ivrtr Pwr Elctric t P = V i () t dt V I (W) t = (4.4) whr i (t) i giv by quati (4.7). Th ma lad pwr ca b ud t dtrmi th rm lad currt: P VI I = (A) rm = (4.5) Th rm utput vltag i V ad th utput fudamtal frqucy f i f = t = t. Th itatau utput vltag xprd a a Furir ri i giv by 4 V = V i ω t (V) (4.6) dd whr ω = f = / t ad fr = th magitud f th fudamt frqucy f i 4 V which i a utput rm fudamtal vltag v f v = V =.9 V (V) (4.7) Th lad currt ca b xprd i trm f th Furir vltag wavfrm ri, that i 4 i ( ωt) = V i ( ω t φ ) =,, 5 Z (4.8) = I i ( ω t φ ) I 4 V =,, 5 whr = Z whc = rm I I Z = ( ω ) ω φ = ta Th lad pwr i giv by th um f ach harmic i pwr cmpt, that i I,,5 rm = =,,5 (4.9) P = = I 4..ii - Quai-quar-wav utput Th rm utput vltag ca b varid by prducig a quai-quar utput vltag (t =t, t <t ) a hw i figur 4.c. Aftr T ad T hav b turd (tat ), at th agl -α dvic i turd ff. If T i turd ff, th lad currt lwly frwhl thrugh T ad D 4 (tat ) i a zr vltag lp accrdig t di = i (V) (4.) dt Wh T i turd ff ad T 4 ad T turd (tat ), th rmaiig lad currt rapidly rduc t zr thrugh did D ad D 4. Wh th lad currt rach zr, T ad T 4 bcm frward biad ad th utput currt rvr thrugh T ad T 4. Th utput vltag hw i figur 4.lc cit f a quc f -zr vltag ±V, altratd with zr utput vltag prid. Durig th zr utput prid a did ad witch cduct, firtly T ad D i th firt prid, ad T ad D i th cd zr utput prid. I ach ca, a zr vltag lp i frmd by a witch, did, ad th lad. Th xt tw zr utput quc wuld b T ad D 4 th T 4 ad D, frmig altratig zr vltag lp (quc,,,,,..) rathr tha rpatig a ctiuu T ad D th T ad D quc f zr vltag lp (quc,,,,,.. r quc,,,,,..). By altratig th zr vltag lp (btw tat ad ), l ar uifrmly ditributd btw th micductr, dvic witchig frqucy i half that xpricd by th lad, ad a fir utput vltag rluti i achivabl. With rfrc t figur 4.lc, th lad currt i fr a applid quai quar-wav vltag i dfid a fllw. (i) v > t τ i () t = I t t I (4.) fr I (A) (ii) v = t τ i () t = I t t t (4.) II fr I (A) (iii) v < i () t = I = i () t t t t τ I fr I (A) Th currt I, I, ad I ar giv by t t t τ τ (4.) I = (A) (4.4) t τ t τ I = (A) (4.5) t τ I = I (A) (4.6) Th zr currt cr-vr itat, t x, hw i figur 4.c, i fud by lvig quati (4.) fr t wh i qual zr currt. I t = τ x (4.7) Th avrag thyritr currt, I TH, avrag did currt, I D, ad ma urc currt, I ca b fud by itgrati f th lad currt vr th apprpriatd bud (aumig altratig zr vlt lp).

4 Pwr Ivrtr Pwr Elctric I = i t dt t x t i t dt T whr i i giv by quati (4.) ad (4.) fr th rpctiv itgral, ad I tx = i () t dt I t t i () t dt t t D II t t t () () (4.8) I II t (4.9) whr i i giv by quati (4.) ad (4.) fr th rpctiv itgral. Ipcti f th urc currt wavfrm i figur 4.b hw that th avrag urc currt i rlatd t th avrag micductr dvic currt by I = t i () t dt = ( I I ) (4.) t TH D I Th tady-tat ma lad ad urc pwr ar P = t V i () t dt = V I (W) (4.) t whr i (t) i giv by quati (4.). Th ma lad pwr ca b ud t dtrmi th rm lad currt: I = P VI (A) (4.) rm Th utput fudamtal frqucy f i f = t = t. Th variabl rm utput vltag i vrm = t V dt t (4.) = α ad th utput fudamtal frqucy f i f = t. Thi quati fr rm utput vltag hw that ly th th harmic ca b limiatd wh c½α =, that i fr α = /. I limiatig th th harmic, frm quati (4.4), th magitud f th fudamtal i rducd t 4 V c. Th utput vltag V i it Furir cfficit ri frm i giv by V 4 c ½α = i ω t dd (V) (4.4) ad fr =, th rm fudamtal f th utput vltag v i giv by v = V c½α =.9 V c½ α (V) (4.5) Th charactritic f th lad vltag harmic ar hw i figur 4.. Th lad currt ca b xprd i trm f th Furir vltag wavfrm ri, that i V 4 c½α i ( ωt) = = V i ( ω t φ ) Z =,,.. Z (4.6) = I i ω t φ =,, 5 ( ) Output Vltag pr uit whr = I Z whc I = rm Vrm V V V5 4 V I Z = ( ω ) φ = ta ω Th lad pwr i giv by th um f ach harmic i pwr cmpt, that i I,,5 rm = =,,5 (4.7) P = = I.9 c½α V7 V ½ dlay agl α α Vrm Figur 4.. Full bridg ivrtr utput vltag harmic rmalid with rpct t quar wav rm utput vltag, Vrm=. A variati f th baic fur-witch dc t ac igl-pha H-bridg i th halfbridg vri whr tw ri witch ( pl r lg) ad did ar rplacd by a plit tw-capacitr urc, a hw i figur 4.. Thi rduc th umbr f micductr ad gat circuit rquirmt, but at th xp f halvig th maximum utput vltag. Exampl 4. illutrat th half-bridg ad it tial fatur. Bhaviur charactritic ar a fr th full-bridg, quar-wav, iglpha ivrtr but V i rplacd by ½V i th apprpriat quati. Oly a quar-wav utput vltag ca b btaid. Sic zr vlt lp cat b cratd, rm vltag ctrl i pibl. Th rm utput vltag i ½V. V V V5

5 Pwr Ivrtr 4 4 Pwr Elctric Exampl 4.a: Sigl-pha H-bridg with a - lad A igl-pha H-bridg ivrtr, a hw i figur 4.la, uppli a hm ritac with iductac 5 mh frm a 4 V dc urc. If th bridg i pratig at 5 Hz, dtrmi th lad rm vltag ad currt ad tady-tat currt wavfrm with i. a quar-wav utput ii. a quai-quar-wav utput with a 5 pr ct -tim. Sluti Th tim ctat f th lad, τ =.5mH/Ω = 5 m ad t = m. i. Th utput vltag rm valu i 4 V ac. Equati (4.9) giv th lad currt at th tim wh th upply plarity i rvrd acr th lad, a hw i figur 4.lb, that i t τ I = I = t (A) τ whr t = m. Thrfr 4V I = I = (A) Ω = 5.9A Wh v = 4 V, frm quati (4.7) th lad currt i giv by -t -t i = 4 - (4 5.9) = t m Frm quati (4.) th zr currt cr-vr tim, t x, ccur at 5m ( 5.9A Ω/4V) =.8 m aftr lad vltag rvral. Wh v = -4 V, frm quati (4.8) th lad currt i giv by -t -t i = -4 (4 5.9) = t m Th ma pwr dlivrd t th lad i giv by quati (4.4), that i m -t P = 4V { } m dt = 755 W Frm P = i, th lad rm currt i P i = = 755W = 6.6A rm Ω ii. Th quai-quar utput vltag ha a 5 m -tim, t, ad a 5 m prid f zr vlt. Frm quati (4.) th rm utput vltag i V 5m/m = = 4V rm. Th currt durig th diffrt itrval i pcifid by quati (4.) t (4.6). Altratly, th tady-tat lad currt quati ca b pcifid by dtrmiig th lad currt quati fr th firt fw cycl at tart-up util tady-tat cditi ar attaid. Firt 5 m -prid wh v = 4 V ad iitially i = A -t i = 4-4 ad at 5m, i =.5A Firt 5 m zr-prid wh v = V -t i =.5 ad at 5m, i =7.9A Scd 5 m -prid wh v = -4 V -t i = -4 (47.9) with i = at m ad dig with i = -8.6 A Scd 5 m zr-prid wh v = V -t i = -8.6 dig with i = -6.8A Third 5 m -prid wh v = 4 V -t i = 4 - (46.8) with i = at.9 m ad dig with i = 9. A Third 5 m zr-prid wh v = V -t i = 9. dig with i = 7.A Furth 5 m -prid wh v = -4 V -t i = -4 (47.) with i = at.9 m ad dig with i = -8.9 A Furth 5 m zr-prid wh v = V -t i = -8.9 dig with i = -7.A Stady-tat lad currt cditi hav b rachd ad th lad currt wavfrm i a hw i figur 4.c. Cvrgc f a itrativ luti i mr rapid if th prid cidrd ar much lgr tha th lad tim ctat. Th ma lad pwr fr th quai-quar wav i giv by P = m 4V {4-4 } dt 5m -t = 78 W Th lad rm currt i irm = P = 78W =.74A Ω

6 Pwr Ivrtr 4 4 Pwr Elctric Exampl 4.b: H-bridg ivrtr ac utput factr I ach wavfrm ca f xampl 4.a calculat i. th avrag ad pak currt i th witch ii. th avrag ad pak currt i th did iii. th pak blckig vltag f ach micductr typ iv. th avrag urc currt v. th harmic factr ad ditrti factr f th lwt rdr harmic vi. th ttal harmic ditrti Sluti Squar-wav i. Th pak currt i th witch i I = 5.9 A ad th currt zr cr-vr ccur at t x =.8m. Th avrag witch currt, frm quati (4.) i m t I T = ( ) m dt.8m = 5.7 A ii. Th pak did currt i 5.9 A. Th avrag did currt frm quati (4.) i.8m t I D = ( ) m dt =.66 A iii. Th maximum blckig vltag f ach dvic i 4 V dc. iv. Th avrag upply currt i I = ( ITH I D ) = ( 5.7A -.66A) = 8.A Thi rult i th upply dlivry pwr f 4Vdc 8.A = 754W v. Frm quati (4.6), with th third a th lwt harmic, th ditrti factr ar V hf = ρ = =, that i, pr ct V df µ vi. Frm quati (4.6) V = = V =, that i,. pr ct 9 V thd = / V = ( ) ( ) ( 5 7)... = 46. pr ct Quai-quar-wav, α = ½ (5 m) ad frm quati (4.7) t x =.9m i. Th pak witch currt i 8.9 A. Frm quati (4.8) th avrag witch currt, uig altratig zr vlt lp, i I 5m 5m -t = (4-4 ) 9 m dt.9m 4m dt -t T =.8.5 =.68 A ii. Th pak did currt (ad pak witch currt) i 8.9 A. Th avrag did currt, frm quati (4.9), wh uig altratig zr vlt lp, i giv by I.9m 5m t = ( 4 4 ) 9 m dt 4m dt -t D =.6.5 =.66 A iii. Th maximum blckig vltag f ach dvic typ i 4 V. iv. Th avrag upply currt i I = ( ITH I D ) = (.68A -.66A) = 4.4A Thi rult i th upply dlivry pwr f 4Vdc 4.4A = 74W v. Th harmic ar giv by quati (4.) t (4.) vi. hf = ρ = V V = V ρ df = µ = / =, that i, pr ct 9 = V =, that i,. pr ct thd V = / V = = 46. pr ct

7 Pwr Ivrtr Pwr Elctric Exampl 4.: Harmic aalyi f H-bridg with a - lad Fr ach dlay ca (α = ad α = 9 ) i xampl 4., uig Furir vltag aalyi, dtrmi (igr harmic abv th th ): i. th magitud f th fudamtal ad firt fur harmic ii. th lad rm vltag ad currt iii. lad pwr iv. lad pwr factr Sluti Th apprpriat harmic aalyi i utli i th fllwig tabl, fr α = ad α = 9. Z V I V (α=9 ) I harmic ( 5).9V V.9V V Z c( ½α ) Z Ω V A V A V 6.59A 5.4V.7A Exampl 4.: P i rm 75W pf = = = =.488 fr α = S i v 6.59A 4V rm rm P i rm 76W pf = = = =.486 fr α = 9 S i v.79a 4.4V rm rm Sigl-pha half-bridg with a - lad A igl-pha half-bridg ivrtr a hw i th figur 4., uppli a hm ritac with iductac 5 mh frm a 4 V dc urc. If th bridg i pratig at 5 Hz, dtrmi fr th quar-wav utput v. tady-tat currt wavfrm vi. th lad rm vltag vii. th pak lad currt ad it tim dmai luti, i (t) viii. th avrag ad pak currt i th witch ix. th avrag ad pak currt i th did x. th pak blckig vltag f ach micductr typ xi. th pwr dlivrd t th lad ad rm lad currt Cl Cl Cl i Th magitud f th fudamtal vltag i 6V fr th quar wav ad i rducd t 6V wh a pha dlay agl f 9 i itrducd. Th tabl hw that th harmic magitud rduc a th harmic rdr icra. ½V Cuppr 7V Cu ½ Cu ii Th rm lad currt ad vltag ca b drivd by th quar rt f th um f th quar f th fudamtal ad harmic cmpt, that i, fr th currt irm = I I I... 5 Th lad rm currt, frm th tabl, ar 6.59A ad.7a, which agr with th valu btaid i xampl 4.a. Ntic that th prdictd rm vltag f V ad 5V diffr igificatly frm th valu i xampl 4.a, giv by α, amly 4V ad 4.4V rpctivly. Thi i bcau th magitud f th harmic highr i rdr tha ar t iigificat. Th rrr itrducd it th rm currt valu by igrig th highr rdr vltag i iigificat bcau th impdac icra apprximatly prprtially with harmic umbr, hc th rultat currt i bcm mallr th rdr icra. iii Th lad pwr i th lad i l, that i P = i = 6.59 Ω = 75W fr α = rm P = i =.7 Ω = 76W fr α = 9 rm iv Th lad pwr factr i th rati f ral pwr diipatd t appart pwr, that i ½ -7V I t t ½V.95A tx Clwr.8m I (a) (b) Figur 4.. GCT thyritr igl-pha half-bridg ivrtr: (a) circuit diagram ad (b) quar-wav utput vltag. -.95A Sluti Frm xampl 4. ad 4., τ = 5m. i. Figur 4. hw th utput vltag ad currt wavfrm, with variu circuit cmpt currt wavfrm uprimpd. Nt that zr vltag lp ca b cratd with th half-bridg. Oly lad vltag ±½V, that i ±7V, ar pibl. ii. Th utput vltag wig i ±½V, ±7V, thu th rm utput vltag i ½V, 7V. Thi i, half that f th full-bridg ivrtr uig th am magitud urc vltag V, 4V.

8 Pwr Ivrtr Pwr Elctric iii. Th pak lad currt i half that giv by quati (4.9), that i t τ ½ ½ t = = tah t τ τ I ½ 4V m = tah.95a Ω = 5m Th lad currt wavfrm i dfid by quati (4.7) ad (4.8), pcifically t ½ ½V τ i () t = I I t ½ 4V ½ 4V.95A 5m = Ω Ω ad t 5m fr t m = t ½ ½V τ i () t = I II ½ 4V ½ 4V =.95 Ω Ω t 5m fr t m = By halvig th ffctiv upply vltag, th currt wig i al halvd. t 5m V - T T ON P T T ON m = 7V m = 68.5 W T T ON M V T T ON -t ( ) T T ON T T4 ON T T4 ON - Figur 4.4. Biplar pul width mdulati: (a) carrir ad mdulati wavfrm ad (b) rultat lad pwm wavfrm. dt Vrf T T4 ON T T4 ON T T4 ON (a) (b) iv. Th pak witch currt i I =. 95A. Th avrag witch currt i giv by m t 5m I T = ( ) m dt.8m =.86 A v. Th pak did currt i I =.95A. Th avrag did currt i giv by.8m t 5m I D = dt m =.8 A vi. Wh a witch r did f a paralll pair cduct, th cmplmtary pair f dvic xpric a vltag V, 4V. Thu althugh th lad xpric half th upply vltag, th micductr xpric twic that vltag, th am a with th full bridg ivrtr. vii. Th lad pwr i fud by avragig th itatau lad pwr, that i 4..i - PWM-wav utput Th utput vltag ad frqucy f a igl-pha vltag-fd ivrtr bridg ca b ctrl uig f tw frm f pul-width mdulati, trmd: biplar multi-lvl, uually calld uiplar Bth pwm tchiqu hav b aalyd xtivly fr dc vltag utput wh applid t th tw quadrat ad fur quadrat dc chppr cidrd i Chaptr, cti.5 ad.6. It will b that th am triagular mdulati pricipl ca b applid ad xtdd, wh prducig lw-harmic iglpha ac utput vltag ad currt. Th mai vltag utput diffrc btw th tw mthd i th harmic ctt ar th carrir frqucy ad it harmic. Thr-pha pwm i a aturally xti t th igl-pha ca, xcpt igl-pha pwm ffr mr dgr f flxibility tha it applicati t thr pha ivrtr, althugh thr-pha pwm d hav th attribut f tripl harmic cacllati. Biplar pul width mdulati Biplar mdulati i th implt pwm mthd ad ivlv cmparig a fixd frqucy triagular carrir with th ac wavfrm dird, calld th mdulati wavfrm. Th mdulati i uually a iuid f magitud (mdulati idx) M uch that M.

9 Pwr Ivrtr Pwr Elctric Th wavfrm i figur 4.4 hw that th lad vltag V wig btw th tw vltag lvl, V ad -V, (hc th trm biplar utput vltag), accrdig t T ad T ar wh v rf > v (T ad T 4 ar ff ) uch that V = V T ad T 4 ar wh v rf < v (T ad T ar ff ) uch that V = -V Multi-lvl pul width mdulati Tw multilvl utput vltag tchiqu ca b u with th igl-pha vltag fd ac bridg. Tw triagular carri diplacd by 8 ar ud. i. Th wavfrm i figur 4.5 hw that th lad vltag V wig btw th tw vltag lvl, V ad -V, with itrpacd zr prid (hc th trm multilvl, pcifically thr-lvl i thi ca, V ad ±V ), accrdig t T i wh v rf > v uch that V a = V T 4 i wh v rf < v uch that V a = V T i wh v rf < -v uch that V b = V T i wh v rf > -v uch that V b = V M Th multilvl lad utput vltag i th diffrc btw th tw lg vltag wavfrm ad ca b dfi a fllw: T ad T ar uch that V a = V, V b = V, V ab = V T ad T ar uch that V a = V, V b = V, V ab = -V T ad T ar uch that V a = V, V b = V, V ab = V T ad T 4 ar uch that V a = V, V b = V, V ab = V Th tw zr utput tat ar itrlavd t balac witchig l btw all fur bridg witch. Dvic witchig i at th carrir frqucy, but th bridg lad vltag xpric twic th lg witchig frqucy ic th tw carrir ar diplacd by 8. ii. A cd multilvl utput vltag apprach i hw i figur 4., whr th triagular carrir ar t ly diplacd by 8 i tim, but ar vrtically diplacd, a fr multilvl ivrtr pwm grati, which i t b cidrd i cti 4.4. Th uppr triagl mdulat rfrc valu gratr tha zr, whil th lwr triagl mdulat wh th rfrc i l tha zr. M - fc fc- f fc f fc-4f fc4f fc- f fc f fc- f fc f with igl-pha biplar pwm fh = fr v f - Va V -V Vrf (a) M - f f c f c f c 4 f c (a) fc- f fc f fc- f fc f with igl-pha multilvl pwm fh = fr all (upprd carrir) f Vb (b) f f c f c f c 4 f c Vab V Vab=Va-Vb (b) Figur 4.6. Typical utput frqucy pctrum fr: (a) biplar pwm ad (b) multilvl pwm. - Figur 4.5. Multilvl ( lvl) pul width mdulati: (a) carrir ad mdulati wavfrm ad (b) rultat lad pwm wavfrm. Spctral cmpari btw biplar ad multilvl pwm wavfrm Th ky fatur f th H-bridg utput vltag with biplar pwm ar (fig 4.6a): triagular carrir ha ly dd Furir cmpt, th utput pctrum ly ha carrir cmpt at dd harmic f th carrir frqucy

10 Pwr Ivrtr 4 4 Pwr Elctric th firt carrir cmpt ccur at th carrir frqucy, f c id-bad cmpt ccur pac by f frm thr cmpt, arud ach multipl f th carrir frqucy Frm figur 4.6b, th ky fatur f th H-bridg utput vltag with multilvl pwm ar: th utput witchig frqucy i dubl f c ach lg witchig frqucy f c, ic th witchig f ach lg i tim hiftd, hc th firt carrir rlatd cmpt i th utput ccur at f c ad th at multipl f f c triagular carrir Furir cmpt xit i th utput vltag ic th tw carrir ar 8 apart, ffctivly cacllig athr i pctrum trm id-bad cmpt ccur pac by f frm thr cmpt, arud ach multipl f th carrir frqucy f c 4.. Thr-pha vltag-fd ivrtr bridg Th baic dc t thr-pha vltag-fd ivrtr bridg i hw i figur 4.7. It cmpri ix pwr witch tgthr with ix aciatd ractiv fdback did. Each f th thr ivrtr lg prat at a rlativ tim diplacmt (pha) f ⅔,. Figur 4.7. Thr-pha ivrtr circuit: (a) GCT thyritr bridg ivrtr; (b) tar-typ lad; ad (c) dlta-typ lad. 4..i - 8 () cducti Figur 4.8 hw ivrtr bridg quai-quar utput vltag wavfrm fr a 8 witch cducti pattr. Each witch cduct fr 8, uch that tw micductr witch acr th vltag rail cduct imultauly. Six pattr xit fr utput cycl ad th rat f qucig th pattr pcifi th bridg utput frqucy. Th cductig witch durig th ix ditict itrval ar hw ad ca b ummarid a i Tabl 4.. Tabl 4.. Quai-quar-wav ix cducti tat - 8 cducti. Itrval Thr cductig witch lg tat vltag vctr T T T T v 5 T T T 4 v T 5 v T 4 4 T 4 T 5 5 T 5 T 6 v T v 6 T 6 6 T 6 T T v 4 Th thr utput vltag wavfrm ca b drivd by aalyig a ritiv tar lad ad cidrig ach f th ix ccti pattr, a hw i figur 4.9. Effctivly th ritr rprtig th thr-pha lad ar qutially cycld aticlckwi at a tim, big altratly cctd t ach upply rail. Th utput vltag i idpdt f th lad, a it i fr all vltag urc ivrtr. Altrativly, th grati f th thr-pha vltag ca b aalyd aalytically by uig th rtatig vltag pac vctr tchiqu. With thi apprach, th utput vltag tat frm ach f th thr ivrtr lg (r pl) i cdd a ummarid i th tabl 4., whr a igifi th uppr witch i th lg i, whil a ma th lwr witch i i that lg. Th rultat biary umbr ( bit fr ach f th thr ivrtr lg), rprt th utput vltag vctr umbr (wh cvrtd t dcimal). Th ix vltag vctr ar hw i figur 4. frmig xtat budari, whr th quai-quar utput wavfrm i figur 4.8b i gratd by tppig itatauly frm vctr piti t athr i a aticlckwi dircti. Nt that th rtatial tppig quc i arragd uch that wh rtatig i ithr dircti, ly lg chag tat, that i, dvic tur ff ad th th cmplmtary witch f that lg tur, at ach tp. Thi miimi th ivrtr witchig l. Th dwll tim f th cratd rtatig vctr at ach f th ix vctr piti, i ⅓ ( T) f th cycl prid (T). Nt that th li-t-li zr vltag tat ad ar t ud. Th rprt th cditi wh ithr all th uppr witch (T, T, T 5 ) ar r all th lwr witch (T, T 4, T 5 ) ar witchd. Pha rvral ca b btaid by itrchagig tw pha utput, r a i th prfrrd mthd, th dircti f th rtatig vctr quc i rvrd. vrig i thrfr ffctivly achivd by back-trackig alg ach utput wavfrm.

11 Pwr Ivrtr 4 4 Pwr Elctric T T5 T T / T4 V T4 T T6 V T5 / T T / T6 VB Y T T5 T VY VN = / VBN = - / VYN = / VN = / VBN = - / VYN = - / VB T T6 T6 B Y B VBY B B T T T VN = / VN = - / VY VBN = / VBN = / VYN = - / VYN = - / VN T4 T (i) Y T Y VBN (ib) B T Y T5 Y T5 VN = -/ VN = - / VYN VBN = / VYN = / VBN = - / VYN = / (iy) T6 T4 v 6 v 4 v 5 v v v (b) T4 B Figur 4.8. A thr-pha bridg ivrtr mplyig 8 witch cducti with a ritiv lad: (a) th bridg circuit hwig T, T5, ad T6 cductig (lg tat v6 : ) ad (b) circuit vltag ad currt wavfrm with ach f ix qutial utput vltag vctr idtifid. Figur 4.9. Dtrmiati f th li-t-utral vltag wavfrm fr a balacd ritiv lad ad 8 cducti a illutratd i figur 4.8.

12 . Pwr Ivrtr Pwr Elctric With rfrc t figur 4.8b, th li-t-lad utral vltag Furir cfficit ar giv by c c V = V (4.8) N Th li-t-lad utral vltag i thrfr iωt V = V r,,,.. = (4.9) = 6r± r= that i v = V [ iωt 5i 5ωt 7i 7ωt i ωt...] (V) (4.4) N imilarly fr v YN ad v BN, whr ωt i ubtitutd by ωt⅔ ad ωt-⅔ rpctivly. Th li-t-li vltag, frm quati (4.4) with α = ⅓, giv Furir cfficit dfid by c 4 6 V = V (4.4) Th li-t-li vltag i thu iωt V = V c r,,,... = (4.4) = 6r± 6 r= (th ymbl prvid th ig), that i v = V [ iωt - 5i5ωt - 7i7ωt i ωt...] (V) (4.4) B ad imilarly fr v BY ad v Y. Figur 4.8b hw that v B i hiftd with rpct t v N, hc t btai th thr li vltag whil maitaiig a v N rfrc, ωt huld b ubtitutd with ωt, ωt- ½ ad ωt-, rpctivly. Sic th itrpha vltag cit f tw quar wav diplacd by ⅔, tripl harmic (, 6, 9,...) xit. Th utput cmpri harmic giv by th ri = 6r ± whr r ad i a itgr. Th th harmic ha a magitud f / rlativ t th fudamtal. Itrval # 5 T T5 T6 lg tat v6 = j v v6 Itrval # 4 T4 T5 T6 lg tat v = j Itrval # 6 T T T6 lg tat v4 = j v v4 Itrval # T T4 T5 lg tat v = j Itrval # T T T lg tat v5 = - j Itrval # T T T4 lg tat v = j Figur 4.. Grati ad arragmt f th ix quai-quar ivrtr utput vltag tat. By xamiati f th itrpha utput vltag i figur 4.8 it ca b tablihd that th ma half-cycl vltag i ⅔V ad th rm valu i ⅔ V, amly.86 V. Frm quati (4.4) th rm valu f th fudamtal i 6 V /, amly.78 V, that i / tim th ttal rm vltag valu. Th thr-pha ivrtr utput vltag prprti ar ummarid i Tabl 4.. v v5

13 Tabl 4.. Quai-quarwav vltag prprti Pwr Ivrtr Pwr Elctric Cducti Fudamtal vltag Charactritic prid pak rm Ttal rm Ditrti Factr THD V V rm µ thd V 8 (V) (V) (V) Pha Vltag V - N i Vltag V - =.67V =. V =.45 6 =.78 V =.47 =.86V =.955 = =. 9 =. (V) (V) (V) Pha Vltag V - N i Vltag V - =.55V =.955V 6 =.9 =.67V 6 =.48 =.77V =.955 = =. 9 =. 4..ii - (⅔) cducti Th baic thr-pha ivrtr bridg i figur 4.7 ca b ctrlld with ach witch cductig fr. A a rult, at ay itat ly tw witch ( uppr ad -cmplmtary lwr) cduct ad th rultat quai-quar utput vltag wavfrm ar hw i figur 4.. A 6 (⅓), dad tim xit btw tw ri witch cductig, thrby prvidig a afty margi agait imultau cducti f th tw ri dvic (fr xampl T ad T 4 ) acr th dc upply rail. Thi afty margi i btaid at th xp f a lwr micductr dvic utiliati ad rm utput vltag tha with 8 dvic cducti. Th dvic cducti pattr i ummarid a i Tabl 4.. Figur 4.8b fr 8 cducti ad 4.b fr cducti hw that th li t utral vltag f cducti pattr i prprtial t th li-t-li vltag f th thr. That i frm quati (4.4) with α = ⅓ Figur 4.. A thr-pha bridg ivrtr mplyig witch cducti with a ritiv tar lad: (a) th bridg circuit hwig T ad T cductig ad (b) circuit vltag ad currt wavfrm.

14 v N ad v Y ( ) = ½vY ( ) Pwr Ivrtr 48 (4.44) = V [ iωt - 5i 5ωt - 7i 7ωt i ωt...] (V) ( ) = v ( ) N = [ ωt 5 ωt 7 ωt ωt ] i i5 i7 i... (V) (4.45) Al v Y = v N ad th pha rlatihip btw th li ad pha vltag, f, ha t b rtaid. That i, with rpct t figur 4.b, ubtitut ωt with ωt i quati (4.44) ad ωt ⅓ i quati (4.45). Th utput vltag prprti fr bth ad 8 witch cducti ar ummarid i th Tabl 4.. Tabl 4.. Quai-quarwav cducti tat - cducti i - Variabl vltag dc lik Pwr Elctric Th rm vltag f a quar-wav ca b chagd ad ctrlld by varyig th dc lik urc vltag. A variabl dc lik vltag ca b achivd with a dc chppr a cidrd i chaptr r a ac pha-ctrlld thyritr bridg a cidrd i cti. ad.5. A dc lik -C mthig filtr may b cary. 4..ii - Sigl-pul width mdulati Simpl pul-width ctrl ca b mplyd a cidrd i cti 4..b, whr a igl-pha bridg i ud t prduc a quai-quar-wav utput vltag a hw i figur 4.c. A altrativ mthd f prducig a quai-quar wav f ctrllabl pul width i t trafrmr-add th quar-wav utput frm tw puh-pull bridg ivrtr a hw i figur 4.a. By pha-hiftig th utput by α, a quaiquar um rult a hw i figur 4.b. Itrval Tw cductig dvic T T T T T T 4 4 T 4 T 5 5 T 5 T 6 6 T 6 T 4.. Ivrtr utput vltag ad frqucy ctrl tchiqu It i a cmm rquirmt that th utput vltag ad/r frqucy f a ivrtr b varid i rdr t ctrl th lad pwr r, i th ca f a iducti mtr, t ctrl th haft pd ad trqu. Th ix mdulati ctrl tchiqu t b cidrd ar: Variabl vltag dc lik Sigl-pul width mdulati Multi-pul width mdulati Multi-pul, lctd tchig mdulati Siuidal pul width mdulati Tripl ijcti Tripl ijctd it th mdulati wavfrm Vltag pac vctr mdulati Figur 4.. Vltag ctrl by cmbiig pha-hiftd puh-pull ivrtr: (a) tw ivrtr with tw trafrmr fr ummig ad (b) circuit vltag wavfrm fr a pha diplacmt f α.

15 Pwr Ivrtr Pwr Elctric Th utput vltag ca b dcribd by V = v i ωt (V) (4.46) a dd whr va ½ 4 = V c αdα V c(½ α) (V) = ½ (4.47) Th rm utput vltag i V = V r - α (V) (4.48) ad th rm valu f th fudamtal i V = V c½ α (V) (4.49) A α icra, th magitud f th harmic, particularly th third, bcm igificat cmpard with th fudamtal magitud. Thi typ f ctrl may b ud i high pwr applicati. 4..iii - Multi-pul width mdulati A xti f th igl-pul mdulati tchiqu i multipl-tchig a hw i figur 4.. Th bridg witch ar ctrlld a t vary th t ff tim f ach tch, δ, thrby varyig th utput rm vltag which i giv by V = δ V. Altrativly, th umbr f tch ca b varid. rm Th harmic ctt at lwr utput vltag i igificatly lwr tha that btaid with igl-pul mdulati. Th icrad witchig frqucy d icra th magitud f highr harmic ad th witchig l. 4..iv - Multi-pul, lctd tchig mdulati Slctd limiati f lwr-rdr harmic ca b achivd by prducig a utput wavfrm a hw i figur 4.4. Th xact witchig pit ar calculatd ff-li a t limiat th rquird harmic. Fr witchig pr half cycl, lctd harmic ca b limiatd. I figur 4.4 tw tch pr half cycl ar itrducd; hc ay tw lctd harmic ca b limiatd. Th mr tch, th lwr i th utput fudamtal. Fr xampl, with tw tch, th third ad fifth harmic ar limiatd. Frm ½ 4 b = f ( θ) i θ dθ fr,,,... = (4.5) b = 4 V ( cα cβ) = ad b = 4 V 5 ( c5α c5β) = 5 δ - - δ Figur 4.. Ivrtr ctrl givig variabl duty cycl f fiv tch pr half cycl: (a) lw duty cycl, δ, hc lw fudamtal magitud ad (b) highr duty cycl, δ, fr a high fudamtal vltag utput. Figur 4.4. Output vltag harmic rducti fr a igl-pha bridg uig lctd tchig. Slvig yild α =.6 ad β =.. Th fudamtal rm cmpt f th utput vltag wavfrm i.84 f a quar wav, which i ( /)V. T witchig itrval xit cmpard with tw pr cycl fr a quar wav, hc witchig l ad ctrl circuit cmplxity ar icrad. I th ca f a thr-pha ivrtr bridg, th third harmic d t xit, hc th fifth ad vth (b 5 ad b 7 ) ca b limiatd with α = 6. ad β =.. Th 5th, 7th, th, ad th ca b limiatd with th agl.55, 6.9,.9, ad.87 rpctivly. Bcau th wavfrm hav quartr wav ymmtry, ly agl fr 9 d b trd.

16 Pwr Ivrtr Pwr Elctric Th utput rm vltag magitud ca b varid by ctrllig th dc lik vltag r by trafrmr-addig tw pha-diplacd bridg utput a dmtratd i figur 4.. Th rm magitud ca b chagd by itrducig a xtra ctrait t b atifid, alg with th harmic limiatig ctrait. Th multi pul lctd tchig mdulati tchiqu ca b xtdd t th ptimal pul-width mdulati mthd, whr harmic may t b limiatd, but miimid accrdig t a pcific critri. I thi mthd, th quartr wav utput i cidrd t hav a umbr f witchig agl. Th agl ar lctd a, fr xampl, t limiat crtai harmic, miimi th rm f th rippl currt, r ay thr dird prfrmac idx. Th rultat -liar quati ar lvd uig umrical mthd ff-li. Th cmputd agl ar th trd i a OM lk-up tabl fr u. A t f agl mut b cmputd ad trd fr ach dird lvl f th vltag fudamtal ad utput frqucy. Th ptimal pwm apprach i particularly uful fr high-pwr, high-vltag GCT thyritr ivrtr, which td t b limitd i witchig frqucy by dvic witchig l. 4..v - Siuidal pul-width mdulati (pwm) - Natural amplig lwr triagular carrir wav fc uppr triagular carrir wav fc rfrc mdulati iwav f (a) Sychru carrir Th utput vltag wavfrm ad mthd f grati fr ychru carrir, atural amplig iuidal pwm, uitabl fr th igl-pha bridg f figur 4., ar illutratd i figur 4.. Th witchig pit ar dtrmid by th itrcti f th triagular carrir wav f c ad th rfrc mdulati i wav, f. Th utput frqucy i at th i-wav frqucy f ad th utput vltag i prprtial t th magitud f th i wav. Th amplitud M ( M ) i calld th mdulati idx. Fr xampl, figur 4.a hw maximum vltag utput (M = ), whil i figur 4.b whr th i-wav magitud i halvd (M =.5), th utput vltag i halvd. If th frqucy f th mdulati iwav, f, i a itgr multipl f th triagular wav carrir-frqucy, f c that i, f c = f whr i itgr, th th mdulati i ychru, a hw i figur 4.. If i dd th th pitiv ad gativ utput half cycl ar ymmtrical ad th utput vltag ctai v harmic. I a thr-pha ytm if i a multipl f (ad dd), th carrir i a tripl f th mdulatig frqucy ad th pctrum d t ctai th carrir r it harmic. f = (6 q ) f = f (4.5) c fr q =,,. Siuidal pwm rquir a carrir f much highr frqucy tha th mdulati frqucy. Th gratd rctiliar utput vltag pul ar mdulatd uch that thir durati i prprtial t th itatau valu f th iuidal wavfrm at th ctr f th pul; that i, th pul ara i prprtial t th crrpdig valu f th mdulatig i wav. Figur 4.. Drivati f triggr igal fr multi-lvl aturally ampld pulwidth mdulati wavfrm: (a) fr a high fudamtal utput vltag (M = ) ad (b) fr a lwr utput vltag (M =.5), with cductig dvic hw. If th carrir frqucy i vry high, a avragig ffct ccur, rultig i a iuidal fudamtal utput with high-frqucy harmic, but miimal lwfrqucy harmic. athr tha uig a triagular carrir, which ha a altratig fft a hw i figur 4., a triagular carrir withut a fft ca b ud. Nw th utput ly apprximat th idal. Figur 4. hw thi pwm grati tchiqu

17 Pwr Ivrtr 444 ad vltag utput wavfrm applid t th thr-pha ivrtr i figur 4.7. Th fft carrir i t applicabl t thr-pha pwm grati ic cmplmtary witch acti i rquird. That i, witch i th ivrtr lg mut alway b. It will b ticd that, ulik th utput i figur 4., zr vltag utput prid xit. Thi ha th ffct that, i th ca f GCT thyritr bridg, a larg umbr f cmmutati cycl i rquird. Wh zr utput prid xit, a i figur 4., GCT thyritr i cmmutatd ad th cmplmtary dvic i that lg i t turd. Th prviuly cmmutatd dvic ca b turd back withut th d t cmmutat th cmplmtary dvic, a wuld b rquird with th pwm tchiqu illutratd i figur 4.. Cmmutati l ar rducd, ctrl circuitry implifid ad th liklihd f imultau cducti f tw ri dvic i rducd. Th altratig zr vltag lp ccpt ca b ud, whr i figur 4.b, rathr tha T big ctiuuly durig th firt half f th utput cycl, T i turd ff lavig T, th wh ithr T r T mut b turd ff, T i turd ff lavig T. (b) Aychru carrir Wh th carrir i t a itgr multipl f th mdulati wavfrm, aychru mdulati rult. Bcau th utput frqucy, f, i uually variabl vr a wid rag, it i difficult t ur f c = f. T achiv ychrim, th carrir frqucy mut vary with frqucy f. Simplr gratig ytm rult if a fixd carrir frqucy i ud, rultig i aychrim btw f ad f c at mt utput frquci. ft vr, icmplt carrir cycl crat lwly varyig utput vltag, calld ubharmic, which may b trublm with lw carrir frquci, a fud i high-pwr driv. Natural amplig, aychru iuidal pwm i uually rtrictd t aalgu r ASIC implmtati. Th harmic cquc f aychru-carrir aturalamplig ar imilar t aychru-carrir rgular-amplig i t fllw. 445 Pwr Elctric - gular amplig Aychru carrir Wh a fixd carrir frqucy i ud, uually attmpt i mad t ychri th mdulati frqucy. Th utput wavfrm d t hav quartr-wav ymmtry which prduc ubharmic. Th ubharmic ar iigificat if f c >> f, uually, f c > f. Th implmtati f iuidal pwm with micrprcr r digital igal prcr i cmm bcau f flxibility ad th limiati f aalgu circuitry aciatd prblm. Th digital pwm grati prc ivlv calig, by multiplicati, f th pr uit i-wav ampl trd i OM. Th multiplicati prc i tim-cumig, hc atural amplig i t pibl. I rdr t miimi th multiplicati rat, th iuidal i-wav rfrc i rplacd by a quatid tppd rprtati f th i-wav. Figur 4.7 hw tw mthd ud. Samplig i ychrid t th carrir frqucy ad th multiplicati prc i prfrmd at thr tim th amplig rat fr thrpha pwm grati (c fr ach pha). Figur 4.. Naturally ampld pul-width mdulati wavfrm uitabl fr a thr-pha bridg ivrtr: (a) rfrc igal; (b) cductig dvic ad fudamtal i wav; ad (c) utput li-t-li vltag wavfrm.

18 Pwr Ivrtr Pwr Elctric Symmtrical mdulati Figur 4.7a illutrat th prc f ymmtrical mdulati, whr amplig i at th carrir frqucy. Th quatid i-wav i tppd ad hld at ach ampl pit. Th triagular carrir i th cmpard with th tp i-wav ampl. Th mdulati prc i trmd ymmtrical mdulati bcau th itrcti f adjact id f th triagular carrir with th tppd i-wav, abut th -ampld carrir pak, ar quiditat abut th carrir pak. Th pul width, idpdt f th mdulati idx M, i ymmtrical abut th triagular carrir pak t aciatd with amplig, a illutratd by th uppr pul i figur 4.8. Th pul width i giv by t = ( - M p i f t ) (4.5) fc whr t i th tim f amplig. tp tp M M t Triagular carrir fc frc f frc f t M M i f ymmtry tpa tpa Figur 4.7. gular amplig, aychru, iuidal pul-widthmdulati: (a) ymmtrical mdulati ad (b) aymmtrical mdulati. Figur 4.8. gular amplig, aychru, iuidal pul-widthmdulati, hwig dubl dg: (uppr) aymmtrical mdulati ad (lwr) ymmtrical mdulati.

19 Pwr Ivrtr Pwr Elctric Aymmtrical mdulati Aymmtrical mdulati i prducd wh th carrir i cmpard with a tppd i wav prducd by amplig ad hldig at twic th carrir frqucy, a hw i figur 4.7b. Each id f th triagular carrir abut a amplig pit itrct th tppd wavfrm at diffrt tp lvl. Th rultat pul width i aymmtrical abut th amplig pit, a illutratd by th lwr pul i figur 4.8 fr tw mdulati wavfrm magitud. Th pul width i giv by t = ( -½ M( i ft pa i ft ) ) (4.5) fc whr t ad t ar th tim at amplig uch that t = t /f c. Figur 4.8 hw that a chag i th mdulati idx M vari th pul width ach dg, trmd dubl dg mdulati. A triagular carrir prduc dubl dg mdulati, whil a awtth carrir prduc igl dg mdulati, idpdt f th amplig tchiqu. - Frqucy pctra f pwm wavfrm Th mt cmm frm f iuidal mdulati fr thr-pha ivrtr i rgular amplig, aychru, fixd frqucy carrir, pwm. If f c > f, lw frqucy ubharmic ca b igrd. Th utput pctra cit f th mdulati frqucy f with magitud M. Al prt ar th pctra cmpt aciatd with th triagular carrir, f c. Fr ay amplig, th ar f c ad th dd harmic f f c. (Th triagular carrir f c ctai ly dd harmic). Th dcra i magitud with icraig frqucy. Abut th frqucy f c ar cmpt f f pacd at ± f, which grally dcra i magitud wh furthr away frm f c. That i, at f c th harmic prt ar f c, f c ± f, f c ± 4f, whil abut f c, th harmic prt ar f c ± f, f c ± f,..., but f c i t prt. Th typical utput pctrum i hw i figur 4.9. Th rlativ magitud f th idbad vary with mdulati dpth ad th carrir rlatd frquci prt, f h, ar giv by f = ½ f k ½ f h c ± (4.54) whr k =,,,... (idbad) ad =,,,... (carrir) M - f f with igl-pha uiplar pwm fh = fr dd (upprd carrir ad -dd id bad) Althugh th variu pwm tchiqu prduc thr l prdmiat pctra cmpt th mai diffrc i i th magitud f th carrir harmic ad idbad. Th magitud icra a th pwm typ chag frm aturally amplig t rgular amplig, th frm aymmtrical t ymmtrical mdulati, ad fially frm dubl dg t igl dg. With a thr-pha ivrtr, th carrir ad it harmic d t appar i th li-t-li vltag ic th carrir i c-pha t th thr mdulati wavfrm. 4..vi - Pha dad-badig Dad badig i wh pha (lg) i i a fixd tat, ad th rmaiig pha ar apprpriatly mdulatd that th pha currt rmai iuidal. Th dad badig ccur fr 6 prid f ach cycl with th pha with th largt magitud vltag big prmatly turd. Squtially ach witch i clampd t th apprpriat lik rail. Th lg utput i i a high tat if it i aciatd with th largt pitiv pha vltag magitud, whil th pha utput i zr if it i aciatd with th largt gativ pha magitud. Thu th pha utput ar cycld, big altratly clampd high ad lw fr 6 vry 8 a hw i figur 4.. A cquc f dad badig i rducd witchig l ic ach lg i t witchd at th carrir frqucy fr (tw 6 prid 8 apart). A cquc f dad badig i icrad rippl currt. Dad badig i achivd with dictiuu mdulatig rfrc igal. Dad badig fr a ctiuu pr pha lg i al pibl but th witchig l avig ar t uifrmly ditributd amgt th ix ivrtr witch. m = m = ¼ m = ½ m = ¾ m = ω t f f c f c f c 4 f c Figur 4.9. cati f carrir harmic ad mdulati frqucy idbad, hwig all idbad paratd by fm. 4 Figur 4.. Mdulati rfrc wavfrm fr pha dad badig. 5

20 4..vii - Tripl Ijcti mdulati Pwr Ivrtr 45 Th magitud f th fudamtal ca b icrad frm.87pu t.955pu withut itrducig utput vltag ditrti, by th ijcti f tripl cmpt, which ar c-phaal i a thr-pha ytm, ad thrfr d t appar i th li currt. Tw baic apprach ca b ud t affct thi uditrtd utput vltag magitud icra. Tripl ijcti it th mdulati wavfrm r Vltag pac vctr mdulati Tripl ijctd it th mdulati wavfrm A ivrtr rctitut thr-pha vltag with a maximum magitud f.87 ( /) f th fixd thr-pha iput ac upply. A mtr digd fr th fixd mai upply i thrfr udr-fluxd at ratd frqucy ad t fully utilid a ivrtr. A will b hw, by addig third harmic vltag ijcti, th flux lvl ca b icrad t.955 (/) f that prducd th thr-pha ac mai upply. If vrmdulati (M > ) i t allwd, th th mdulati wav M i ωt i rtrictd i magitud t M =, a hw i figur 4.a. If V N = M iωt pu ad V YN = M i(ωt ⅔) pu th V Y = M i(ωt - ) whr M 45 Pwr Elctric I a thr-pha pwm gratr, th fact that harmic at f (ad multipli f f ) vctrally cacl ca b utilid ffctivly t icra M byd, yt till ur mdulati ccur fr vry carrir frqucy cycl. t V N = M iωt iωt) pu ad V YN = M ( i(ωt ⅔) i (ωt ⅔)) pu th V Y = M i(ωt - ) V N ha a maximum itatau valu f pu at ωt = ±⅓, a hw i figur 4.b. Thrfr V ( ωt = ) = M ' = N that i M ' = M =.55M (4.55) Thu th fudamtal f th pha vltag i M i ωt =.55 M i ωt. That i, if th mdulati rfrc i ωt i ωt i ud, th fudamtal utput vltag i 5.5 pr ct largr tha wh i ωt i ud a a rfrc. Th icrad fudamtal i hw i figur 4.b. Figur 4.. Mdulati rfrc wavfrm: (a) iuidal rfrc, i ωt; (b) third harmic ijcti rfrc, i ωt i ωt; ad (c) tripl ijcti rfrc, i ωt (/ ){9/8iωt - 8/8i9ωt...} whr th ar triagular wavfrm b i half th magitud f th hadd ara. Th patial vltag vctr tchiqu ijct th tripl accrdig t VN r ( ) = M ' iωt i ( r ) ωt r= ( r ) ( r ) (4.56) Th Furir tripl ri rprt half th magitud f th hadd ara i figur 4.c (th wavfrm markd b ), which i frmd by th thr-pha vltag wavfrm. Th patial vltag vctr wavfrm i dfid by i( ωt ) ωt 6 i( ωt ) ωt ½ 6 6 (4.57)

21 Pwr Ivrtr Pwr Elctric Th u f thi rfrc icra th durati f th zr vlt lp, thrby dcraig ivrtr utput rippl. Th maximum mdulati idx i.55. Third harmic ijcti, yildig M =.55, i a atifactry apprximati t patial vltag vctr. Vltag pac vctr pwm Wh gratig thr-pha quai-quar utput vltag, th ivrtr witch tp prgrivly t ach f th ix witch utput pibiliti (tat). I figur 4., wh prducig th quai-quar utput, ach f th ix tat i rprtd by a utput vltag pac vctr. Each vctr ha a ⅓ diplacmt frm it tw adjact tat, ad ach ha a lgth V which i th pl utput vltag rlativ t th ivrtr V rail. Effctivly, th quai-quar thr-pha utput i gratd by a rtatig vctr f lgth V, jumpig uccivly frm utput tat t th xt i th quc, ad i dig cratig ix ctr. Th pd f rtati, i particular th tim fr rtati, dtrmi th ivrtr utput frqucy. Th quc f vltag vctr {v, v, v, v 6, v 4, v 5 } i arragd uch that tppig frm tat t th xt ivlv ly f th thr pl chagig tat. Thu th umbr f ivrtr dvic dig t chag tat (witch) at ach traiti, i miimid. [If th ivrtr witch ar rlablld, uppr witch T, T, T - right t lft; ad lwr witch T 4, T 5, T 6 - right t lft: th th rtatig vltag quc bcm {v, v, v, v 4, v 5, v 6 }] athr tha tppig ⅓ radia pr tp, frm vltag pac vctr piti t th xt, thrby prducig a ix-tp quai-quar fixd magitud vltag utput, th rtatig vctr i rtatd i mallr tp bad th piti big updatd at a ctat rat (carrir frqucy). Furthrmr, th vctr lgth ca b varid, t a magitud l tha V. T icrprat a variabl rtatig vctr lgth (mdulati dpth), it i cary t vary th avrag vltag i ach carrir prid. Hc pul width mdulati i ud i th prid btw ach fiit tp f th rtatig vctr. Pul width mdulati rquir th itrducti f zr vltag utput tat, amly all th tp witch (tat, v 7 ) r all th lwr witch (tat, v ). Th tw xtra tat ar hw i figur 4., at th ctr f th hxag. Nw th pl-t-pl utput vltag ca b zr, which allw duty cycl variati t achiv variabl avrag utput vltag fr ach pha, withi ach carrir prid, prprtial t th magitud f th piti vctr. T facilitat vctr piti (agl) that d t li f th ix quai-quar utput vctr, a itrmdiat vctr V /p jθ i rlvd it th vctr um f th tw quai-quar vctr adjact t th rtatig vctr. Thi prc i hw i figur 4. fr a vltag vctr V /p that li i ctr I, btw utput tat v () ad v (). Th vltag vctr ha b rlvd it th tw cmpt V a ad V b a hw. Th tim rprtd by quai-quar vctr v ad v i th carrir prid T c, i ach ca. Thrfr th prti f T c aciatd with v a ad v b i cald prprtially t v ad v, givig t a ad t b. Itrval # 5 T T5 T6 lg tat v6 = j Itrval # 4 T4 T5 T6 lg tat v = j v v6 SECTO III SECTO IV Itrval # 6 T T T6 lg tat v4 = j v7 SECTO II SECTO V v v4 SECTO I SECTO VI Itrval # T T T lg tat v5 = - j Figur 4.. Itatau utput vltag tat fr th thr lg f a ivrtr. v Itrval # T T4 T5 lg tat v = j Itrval # T T T4 lg tat v = j v v5

22 Pwr Ivrtr Pwr Elctric V / i p t V a a = = Tc v V V iθ / p t V b b = = Tc v V whr v = v ( θ ) (4.58) Th tw i trm i quati (4.58) grat tw i wav diplacd by, idtical t that gratd with tadard carrir bad iuidal pwm. Th um f t a ad t b cat b gratr tha th carrir prid T c, thu t t T a b c (4.59) t t t = T a b c whr th lack variabl t ha b icludd t frm a quality. Th quality dictat that vctr v i ud fr a prid t a, v i ud fr a prid t b, ad durig prid t, th ull vctr, v r v 7, at th ctr f th hxag i ud, which d t affct th avrag vltag durig th carrir itrval T c. A furthr ctrait i impd i th tim dmai. Th rtatig vltag vctr i a fixd lgth fr all rtatig agl, fr a giv ivrtr utput vltag. It lgth i rtrictd i bth tim ad pac. Obviuly th rlvd cmpt lgth cat xcd th pl vctr lgth, V. Additially, th tw vctr magitud ar ach a prti f th carrir prid, whr t a ad t b culd b bth qual t T c, that i, thy bth hav a maximum lgth V. Th amaly i that vltag v a ad v b ar addd vctrially but thir durati (tim t a ad t b ) ar addd liarly. Th lgt tim t a t b pibl i wh t i zr, a hw i figur 4.a ad 4.a, by th hxag budary. Th hrtt vctr t th budary i whr bth rlvig vctr hav a lgth ½V, a hw i figur 4.b. Fr uch a cditi, t a = t b =½T c, that i t a t b = T c. Thu fr a ctat ivrtr utput vltag, wh th rtatig vltag vctr ha a ctat lgth, V / p, th lcu f allwabl rtatig rfrc vltag vctr mut b withi th circl cribd by th maximum lgth vctr hw i figur 4.b. A hw, thi vctr ha a lgth v c, pcifically.866v. Thu th full quai-quar vctr v, v, tc., which hav a magitud f V, cat b ud fr gratig a iuidal utput vltag. Th xc lgth f ach quai-quar vltag (which rprt tim) i accutd fr by uig zr tat vltag vctr fr a prid crrpdig t that xtra lgth (- c at maximum utput vltag). Havig calculatd th cary prid fr th ivrtr pl (t a, t b, ad t ), th carrir prid witchig pattr ca b aigd i tw way. Miimid currt rippl Miimid witchig l, uig dad badig Each apprach i hw i figur 4.4, uig igl dgd mdulati. Th wavfrm ar bad th quivalt f ymmtrical mdulati whr th pul ar ymmtrical abut th carrir trugh. By miimiig th currt rippl, v witchig tat ar ud pr carrir cycl, whil fr l miimiati (dad badig) ly fiv witchig tat ccur, but at th xp f icrad rippl currt i th utput currt. Wh dad badig, th zr vltag tat v i ud i v umbrd xtat ad v 7 i ud i dd umbrd xtat. Sidbad ad harmic cmpt magitud ca b dcrad if dubl-dgd mdulati placmt f th tat i ud, which rquir rcalculati f t a, t b, ad t at th carrir crt, a wll a at th trugh. tb V = V iθ b O/ P V = j? j VVO/P θ / p θ SECTO I V = a VO/ P i( θ) ta v = j V = j? ½v = ½ ½v = ½ (c) Figur 4.. Firt ctr f ivrtr pratial ara ivlvig pl utput ad : (a) gral rtatig vltag vctr; (b) maximum allwabl vltag vctr lgth fr uditrtd utput vltag; ad (c) vr mdulati. Ovr-mdulati i wh th magitud f th dmadd rtatig vctr i gratr tha V p / uch that th zr vltag tim rduc t zr, t =, durig a prti f th tim f rtati f th utput vctr. Iitially thi ccur at ( 6 ( N ctr )) wh th utput vctr lgth rachv p /, a hw i figur 4.b. A th dmad vltag magitud icra furthr, th rgi arud th vctr piti whr t ca t ccur, icra a hw i figur 4.c. Wh th utput rtatial vctr magitud icra t V, th maximum pibl, agl α rduc t zr, ad t ca t ccur at ay rtatial agl. V p / c Tc SECTO I (a) (b) ωt V = j? V 6 -α Tc V > / p / p V / p tb ta < Tc rducd t α Tc Tc v = j tb ta > Tc t tb ta < Tc rducd t Tc Tc v = j Tc

23 Pwr Ivrtr Pwr Elctric Th valu f t a, t b, ad t (if gratr tha zr), ar calculatd a uual, but pul tim ar aigd pr rata t fit withi th carrir prid T c. Φ Φ Y v v v v 7 v 7 v v v ¼ t ½ t a ½ t b ¼ t ¼ t ½ t b ¼ t a ¼ t 4. dc-t-ac ctrlld currt-urcd ivrtr I currt-fd ivrtr (r altrativly currt urcd ivrtr, CSI) th dc upply i f high ractac, big iductiv a t maitai th rquird ivrtr utput bidirctial currt idpdt f th ivrtr lad. 4.. Sigl-pha currt fd ivrtr A igl-pha, ctrlld currt-urcd bridg i hw i figur 4.5a ad it ar quar-wav utput currt i hw i figur 4.5b. N frwhl did ar rquird ad th thyritr rquird frcd cmmutati ad hav t withtad rvr vltag. A ivrtr path mut b maitaid at all tim fr th urc ctrlld currt. Φ B T c (a) v v v 7 v 7 v v ½ t a ½ t b ¼ t ¼ t ½ t b ¼ t a Φ Φ Y Φ B Figur 4.4. Aigmt f pl prid ta ad tb bad : (a) miimum currt rippl ad (b) miimum witchig traiti pr carrir cycl, Tc. T c (b) Figur 4.5. Sigl-pha ctrlld-currt urcd bridg ivrtr: (a) bridg circuit with a currt urc iput ad (b) lad currt wavfrm. Cidr thyritr T ad T ad cductig th ctat lad currt. Th capacitr ar chargd with plat X ad Y pitiv a a rult f th prviu cmmutati cycl. Pha I Thyritr T ad T ar cmmutatd by triggrig thyritr T ad T 4. Th capacitr impr gativ vltag acr th rpctiv thyritr t b cmmutatd ff, a hw i figur 4.6a. Th lad currt i diplacd frm T ad T via th path T -C -D, th lad ad D -C -T 4. Th tw capacitr dicharg i ri with th lad, ach capacitr rvr biaig th thyritr t b cmmutatd, T ad T a wll a did D t D 4. Th capacitr dicharg liarly (du t th ctat currt urc). Pha II Wh bth capacitr ar dichargd, th lad currt trafr frm D t D ad frm D t D 4, which cct th capacitr i paralll with th lad via did D t D. Th plat X ad Y w charg gativ, rady fr th xt cmmutati cycl, a hw i figur 4.6b. Thyritr T ad T ar w frward biad ad mut hav attaid frward blckig ability bfr th tart f pha.

24 Pwr Ivrtr Pwr Elctric Th -gig thyritr autmatically cmmutat th utgig thyritr. Thi rpatd cmmutati qucig i a prcd trmd aut-qutial thyritr cmmutati. Th lad vltag i lad dpdt ad uually ha ctrlld vltag pik durig cmmutati. Sic th GTO ad CGT bth ca b cmmutatd frm th gat, th tw cmmutati capacitr C ad C ar t cary. Cmmutati vrlap i till tial. Al, if th thyritr hav rvr blckig capability, th fur did D t D 4 ar t cary. IGBT rquir ri blckig did, which icra -tat l. I practic, th currt urc ivrtr i ly ud i vry highpwr applicati (>MVA), ad th ratig f th lf-cmmutatig thyritr dvic ca b gratly xtdd if th impl xtral capacitiv cmmutati circuit hw i figur 4.5 ar ud t rduc thyritr tur-ff tr. ar ffctivly cctd i ri. Capacitr C prvid diplacmt currt whilt i paralll, C 5 ad C 5 i ri al prvid thyritr T diplacmt currt, thrby rvr biaig T. Pha II Wh th capacitr hav dichargd, T bcm frward biad, a hw i figur 4.8b, ad mut hav rgaid frward blckig capability bfr th applid pitiv dv/dt. Th capacitr vltag rvr a hw i figur 4.8b ad wh fully chargd, did D ca t cduct. Idpdt f thi cmmutati, lwr thyritr T i cmmutatd by turig T 4, 6 latr (a) (b) Figur 4.6. Ctrlld-currt urcd bridg ivrtr hwig cmmutati f T ad T by T ad T4: (a) capacitr C ad C dichargig ad T, T, D, ad D4 rvrd biad ad (b) C, C, ad th lad i paralll with C ad C chargig. 4.. Thr-pha currt fd ivrtr A thr-pha ctrlld currt-urcd ivrtr i hw i figur 4.7a. Oly tw thyritr ca b at ay itat, that i, th thyritr cducti pricipl hw i figur 4. i ud. A quai-quar li currt rult, a illutratd i figur 4.7b. Thr i a 6 pha diplacmt btw cmmutati f a uppr dvic fllwd by cmmutati f a lwr dvic. A uppr dvic (T, T, T 5 ) i turd t cmmutat athr uppr dvic, ad a lwr dvic (T, T 4, T 6 ) cmmutat athr lwr dvic. Th thr uppr capacitr ar all ivlvd with ach uppr dvic cmmutati, whilt th am ctrait appli t th lwr capacitr. Thyritr cmmutati ccur i tw ditict pha. Pha I I figur 4.8a th capacitr C, C 5, C 5 ar chargd with th hw plariti a a rult f th arlir cmmutati f T 5. T i cmmutatd by turig T. Durig cmmutati, th capacitr btw th tw cmmutatig witch i i paralll with th tw rmaiig capacitr which Figur 4.7. Thr-pha ctrlld-currt urcd bridg ivrtr: (a) bridg circuit with a currt urc iput ad (b) lad currt wavfrm fr pha hwig cducti. A with th igl-pha currt urcd ivrtr, aitd capacitr cmmutati ca gratly imprv th capabiliti f lf-cmmutatig thyritr, uch a th GTO thyritr ad GCT. Th utput capacitr tiff th utput ac vltag. A typical applicati fr a thr-pha currt-urcd ivrtr wuld b t fd ad ctrl a thr-pha iducti mtr. Varyig lad rquirmt ar mt by chagig th urc currt lvl vr a umbr f cycl by varyig th lik iductr iput vltag.